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+{"text":"\\section{Introduction}\nVery few symmetries in nature are manifestly realized. Why do I \nthink then that the breaking of CP invariance is very special -- \nmore subtle, more fundamental and more profound than parity \nviolation? \n\\begin{itemize}\n\\item \nParity violation tells us that nature makes a difference between \n\"left\" and \"right\" -- but not which is which! For the \nstatement that neutrinos emerging from pion decays are \nleft- rather than right-handed implies the use of \npositive instead of negative pions. \"Left\" and \n\"right\" is thus defined in terms of \"positive\"  \nand \"negative\", respectively. This is like saying that \nyour left thumb is on your right hand -- certainly \ncorrect, yet circular and thus not overly useful. \n\nOn the other hand CP violation manifesting itself through \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{{\\rm BR}(K_L \\rightarrow l^+ \\nu \\pi ^-)}\n{{\\rm BR}(K_L \\rightarrow l^- \\bar \\nu \\pi ^+)} \\simeq 1.006J\\neq 1 \n\\eeq \nallows us to define \"positive\" and \"negative\" in \nterms of \n{\\em observation} rather than {\\em convention}, and \nsubsequently likewise for \"left\" and \"right\". \n\\item \nThe limitation on CP invariance in the $K^0 - \\bar K^0$ \nmass matrix \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\rm Im} M_{12} \\simeq 1.1 \\cdot 10^{-8} \\; \\; {\\rm eV} \\; \\; \n\\hat = \\; \\; \\frac{{\\rm Im} M_{12}}{m_K} \\simeq 2.2 \\cdot  \n10^{-17} \n\\eeq\nrepresents the most subtle symmetry violation \nactually observed to date. \n\n\\item \nCP violation constitutes one of the three essential ingredients in any \nattempt to understand the observed baryon number of the universe \nas a dynamically generated quantity rather than an initial \ncondition \\cite{DOLGOV2}. \n\n\\item \nDue to CPT invariance -- on which I will not cast any doubt during   \nthese lectures -- CP breaking implies a violation of \ntime reversal invariance \n\\footnote{{\\em Operationally} one defines time reversal as the \nreversal \nof {\\em motion}: $\\vec p \\rightarrow - \\vec p$, $\\vec j \\rightarrow - \\vec j$ for \nmomenta $\\vec p$ and angular momenta $\\vec j$.}. That nature \nmakes an \nintrinsic distinction between past and future on the \n{\\em microscopic} level that cannot be explained by statistical \nconsiderations is an utterly amazing observation.   \n\n\\item \nThe fact that time reversal represents a very peculiar operation can \nbe expressed also in a less emotional way, namely through \n{\\em Kramers' Degeneracy} \n\\cite{KRAMERS}. The time reversal operator $\\bf T$ \nhas to be {\\em anti}-unitary; ${\\bf T}^2$ then has eigenvalues \n$\\pm 1$. Consider the sector of the Hilbert space with \n${\\bf T}^2 = -1$ and assume the dynamics to conserve ${\\bf T}$; \ni.e., the Hamilton operator $\\bf H$ and ${\\bf T}$ commute. It is \neasily shown that if $|E\\rangle$ is an eigenvector of $\\bf H$, so is \n${\\bf T}|E\\rangle$ -- with the {\\em same} eigenvalue. Yet \n$|E\\rangle$ and ${\\bf T}|E\\rangle$ are -- that is the main \nsubstance of this theorem -- orthogonal to each other! Each \nenergy eigenstate in the Hilbert sector with ${\\bf T}^2 = -1$ \nis therefore at least doubly degenerate. This degeneracy is realized \nin nature through {\\em fermionic spin} degrees. Yet it is \nquite remarkable that the time reversal operator $\\bf T$ already \nanticipates this option -- and the qualitative difference \nbetween fermions and bosons -- through ${\\bf T}^2 = \\pm 1$ -- \n{\\em without} any explicit reference to spin! \n\n\\end{itemize}\n\n\\subsection{General Description of Particle-Antiparticle Oscillations}\nA symmetry $\\bf S$ can be manifestly realized in two different \nways:    \n\\begin{itemize}\n\\item \nThere exists a pair of degenerate states that transform into each \nother under $\\bf S$. \n\\item \nWhen there is an {\\em un}paired state it has to be an eigenstate of \n$\\bf S$. \n\\end{itemize}\nThe observation of $K_L$ decaying into a $2\\pi$ state -- which \nis CP even -- and a CP odd $3\\pi$ combination therefore \nestablishes CP violation only because $K_L$ and $K_S$ are \n{\\em not} mass degenerate. \n\nIn general, decay rates can exhibit CP violation in three different \nmanners,  \nnamely through \n\\begin{itemize}\n\\item \nthe {\\em existence} of a reaction, like \n$K_L \\rightarrow \\pi \\pi$, \n\\item \na {\\em difference} in CP conjugate rates, like \n$K_L \\rightarrow l^- \\bar \\nu \\pi ^+$ vs. \n$K_L \\rightarrow l^+  \\nu \\pi ^-$, \n\\item \na decay rate evolution that is {\\em not a purely \nexponential} function of the proper time of decay; i.e., \nif one finds for a CP {\\em eigenstate} $f$ \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{d}{dt} e^{\\Gamma t}{\\rm rate} \n(K_{neutral}(t) \\rightarrow f) \\neq 0 \n\\eeq  \nfor all (real) values of $\\Gamma$, then CP symmetry must be \nbroken. This is easily proven: \nif CP invariance holds, the decaying state \nmust be a CP eigenstate like the final state $f$; yet in that case \nthe decay rate evolution must be purely exponential -- unless \nCP is violated. Q.E.D. \n\n\\end{itemize} \nThe whole formalism of particle-antiparticle oscillations is \nactually a straightforward application of basic quantum mechanics. \nI will describe it in terms of strange mesons; the generalization to \nany other flavour or quantum number (like beauty or charm) is \nobvious. In the absence of weak forces one has two mass degenerate \nand stable mesons $K^0$ and $\\bar K^0$ carrying definite \nstrangeness $+1$ and $-1$, respectively, since the strong and \nelectromagnetic  \nforces conserve this quantum number. The addition of the weak \nforces changes the picture qualitatively: strangeness is no longer \nconserved, kaons become unstable and the new mass eigenstates \n-- being linear superpositions of $K^0$ and $\\bar K^0$ -- no longer \ncarry definite strangeness. The violation of the quantum number \nstrangeness has lifted the degeneracy: we have two physical \nstates $K_L$ and $K_S$ with different masses and lifetimes: \n$\\Delta m_K = m_L - m_K \\neq 0 \\neq \\Delta \\tau = \n\\tau _L - \\tau _S$. \n\nIf CP is conserved in the $\\Delta S=2$ transitions the mass \neigenstates $K_1$ and $K_2$ have to be CP eigenstates as pointed out \nabove: $|K_1\\rangle = |K_+\\rangle$, $|K_2\\rangle = |K_-\\rangle$,  \nwhere ${\\bf CP}|K_{\\pm}\\rangle \\equiv \\pm |K_{\\pm}\\rangle$. \nUsing the phase convention \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n|\\bar K^0 \\rangle \\equiv - {\\bf C} |K^0\\rangle \n\\eeq \nthe time evolution of a state that starts out as a $K^0$ is given by \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n|K^0(t)\\rangle = \n\\frac{1}{\\sqrt{2}} e^{im_1t} e^{-\\frac{\\Gamma _1}{2}t} \n\\left( |K_+\\rangle + e^{i\\Delta m_Kt} e^{-\\frac{\\Delta \\Gamma}{2}t}\n|K_-\\rangle \\right) \n\\eeq \nThe intensity of an initially pure $K^0$ beam traveling in vacuum  will then exhibit the \nfollowing time profile: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nI_{K^0}(t) = |\\langle K^0|K^0(t)\\rangle |^2 = \n\\frac{1}{4} e^{-\\Gamma _1 t } \n\\left( 1 + e^{\\Delta \\Gamma _Kt} + 2e^{\\frac{\\Delta \\Gamma _K}{2} t} \n{\\rm cos}\\Delta m_K t\\right)  \n\\eeq \nThe orthogonal state $|\\bar K^0 (t)\\rangle$ that was absent initially \nin this beam gets regenerated {\\em spontaneously}: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nI_{\\bar K^0}(t) = |\\langle \\bar K^0|K^0(t)\\rangle |^2 = \n\\frac{1}{4} e^{-\\Gamma _1 t } \n\\left( 1 + e^{\\Delta \\Gamma _K t} - 2e^{\\frac{\\Delta \\Gamma _K}{2} t} \n{\\rm cos}\\Delta m_K t\\right)  \n\\eeq \nThe oscillation rate expressed through $\\Delta m_K$ and \n$\\Delta \\Gamma _K$ is naturally calibrated by the average \ndecay rate $\\bar \\Gamma _K \\equiv \n\\frac{1}{2}( \\Gamma _1 + \\Gamma _2)$: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nx_K \\equiv \\frac{\\Delta m_K}{\\bar \\Gamma _K} \\simeq 0.95 \n\\; \\; \\; , \\; \\; \\; \ny_K \\equiv \\frac{\\Delta \\Gamma _K}{2\\bar \\Gamma _K} \\simeq 1  \n\\eeq \nTwo comments are in order at this point: \n\\begin{itemize}\n\\item \nIn any such binary quantum system there will be two lifetimes. \nThe fact that they differ so spectacularly for neutral kaons  \n-- $\\tau (K_L) \\sim 600 \\cdot \\tau (K_S)$ -- is \ndue to a kinematical accident: the only available nonleptonic \nchannel for the CP odd kaon is the 3 pion channel, for which \nit has barely enough mass.  \n\\item \n$\\Delta m_K \\simeq 3.7 \\cdot 10^{-6}$ eV is often related \nto the kaon mass: \n\\begin{equation}}   \\def\\eeq{\\end{equation}  \n\\frac{\\Delta m_K}{m_K} \\simeq 7 \\cdot 10^{-15} \n\\label{STRIKING} \n\\eeq \nwhich is obviously a very striking number. Yet \nEq.(\\ref{STRIKING}) somewhat overstates the point. \nThe kaon mass has nothing really to do with the \n$K_L-K_S$ mass difference \n\\footnote{It would not be much more absurd to relate \n$\\Delta m_K$ to the mass of an elephant!} and actually is  \nmeasured relative to $\\Gamma _K$. There is however \none exotic application where it makes sense to state \nthe ratio $\\Delta m_K\/m_K$, and that is in the context \nof antigravity where one assumes matter and antimatter \nto couple to gravity with the opposite sign. The gravitational \npotential $\\Phi$  \nwould then produce a {\\em relative} phase between $K^0$ and \n$\\bar K^0$ of  2 $m_K\\Phi t$. In the earth's potential this would \nlead to a gravitational oscillation time of \n$10^{-15}$ sec, which is much shorter than the lifetimes or \nthe weak oscillation time; $K^0 - \\bar K^0$ oscillations could \nthen not be observed \\cite{GOOD}. \nThere are some loopholes in this argument -- \nyet I consider it still intriguing or at least entertaining. \n\n\n\\end{itemize}\n\n  \n\\section{CP Phenomenology in $K_L$ Decays}\n\\subsection{General Formalism}\n\nOscillations become more complex once CP symmetry is broken in \n$\\Delta S=2$ transitions, as seen from solving the (free) \nSchr\\\" odinger equation \n\\begin{equation}}   \\def\\eeq{\\end{equation} \ni\\frac{d}{dt} \\left( \n\\begin{array}{ll}\nK^0 \\\\\n\\bar K^0\n\\end{array}  \n\\right)  = \\left( \n\\begin{array}{ll}\nM_{11} - \\frac{i}{2} \\Gamma _{11} & \nM_{12} - \\frac{i}{2} \\Gamma _{12} \\\\ \nM^*_{12} - \\frac{i}{2} \\Gamma ^*_{12} & \nM_{22} - \\frac{i}{2} \\Gamma _{22} \n\\end{array}\n\\right) \n\\left( \n\\begin{array}{ll}\nK^0 \\\\\n\\bar K^0\n\\end{array}  \n\\right) \n\\label{SCHROED} \n\\eeq\nCPT invariance imposes \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nM_{11}= M_{22} \\; \\; , \\; \\; \\Gamma _{11} = \\Gamma _{22} \\; . \n\\label{CPTMASS}\n\\eeq  \n\\begin{center} \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $ \\\\ \n{\\em Homework Problem \\#1}: \n\\end{center}\nWhich physical situation is \ndescribed by an equation analogous to Eq.(\\ref{SCHROED}) \nwhere however the two diagonal matrix elements differ \n{\\em without} violating CPT? \n\\begin{center} \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $\n\\end{center} \nThe mass eigenstates obtained through diagonalising this matrix \nare given by (for details see \\cite{LEE,BOOK}) \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n|K_S\\rangle \\equiv |K_1\\rangle = \n\\frac{1}{\\sqrt{|p|^2 + |q|^2}} \\left( p |K^0 \\rangle + \nq |\\bar K^0\\rangle \\right) \n\\eeq \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n|K_L\\rangle \\equiv |K_2\\rangle = \n\\frac{1}{\\sqrt{|p|^2 + |q|^2}} \\left( p |K^0 \\rangle -  \nq |\\bar K^0\\rangle \\right) \n\\eeq \nwith \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{q}{p} = \\sqrt{\\frac{M_{12}^* - \\frac{i}{2} \\Gamma _{12}^*}\n{M_{12} - \\frac{i}{2} \\Gamma _{12}}}\n\\eeq \nand eigenvalues \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nM_S - \\frac{i}{2}\\Gamma _S = M_{11} - \\frac{i}{2} \\Gamma _{11} \n- \\sqrt{\\left( M_{12} - \\frac{i}{2} \\Gamma _{12}\\right) \n\\left( M^*_{12} - \\frac{i}{2} \\Gamma ^*_{12}\\right)} \n\\eeq \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nM_L - \\frac{i}{2}\\Gamma _L = M_{11} - \\frac{i}{2} \\Gamma _{11} \n+ \\sqrt{\\left( M_{12} - \\frac{i}{2} \\Gamma _{12}\\right) \n\\left( M^*_{12} - \\frac{i}{2} \\Gamma ^*_{12}\\right)} \n\\eeq \nThese states are conveniently expressed in terms of the \nCP eigenstates $|K_{\\pm}\\rangle$: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n|K_S \\rangle = \\frac{1+q\/p}{\\sqrt{2(1+ |q\/p|^2)}} \n\\left( |K_+\\rangle + \\bar \\epsilon |K_-\\rangle\\right) \n\\eeq \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n|K_L \\rangle = \\frac{1+q\/p}{\\sqrt{2(1+ |q\/p|^2)}} \n\\left( |K_-\\rangle + \\bar \\epsilon |K_+\\rangle\\right) \n\\eeq \nwhere the parameter \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\bar \\epsilon \\equiv \\frac{1-q\/p}{1+q\/p} \n\\eeq  \nreflects the CP impurity in the state vector. \n\nA few comments -- some technical, some not -- \nmight elucidate the situation: \n\\begin{itemize}\n\\item \nIf there is no relative phase between $M_{12}$ and \n$\\Gamma _{12}$ \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\rm arg} \\frac{M_{12}}{\\Gamma _{12}} =0 \n\\eeq \nthen $q\/p =1$ and the state vectors conserve CP: \n$\\bar \\epsilon =0$. \n\\item \nYet for our later \ndiscussion one should take note that \n$q\/p$ -- and therefore also $\\bar \\epsilon$ --  \n{\\em by itself cannot} be an observable. For a change in the \nphase {\\em convention} adopted for defining $\\bar K^0$ does \nnot leave it invariant: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n|\\bar K^0 \\rangle \\rightarrow e^{i\\xi}|\\bar K^0 \\rangle \\; \n\\Longrightarrow \\; \n(M_{12}, \\Gamma _{12}) \\rightarrow e^{i\\xi} \n(M_{12}, \\Gamma _{12}) \\; \n\\Longrightarrow \\; \n\\frac{q}{p} \\rightarrow e^{-i\\xi} \\frac{q}{p} \\; !\n\\eeq\nOn the other hand $|q\/p|$ is independant of the phase \nconvention and its deviation from unity is one measure \nof CP violation. \n\\item \nOn very general grounds -- without recourse to any model -- \none can infer that CP violation in the neutral kaon system \nhas to be small. CP invariance implies the two mass eigenstates \n$K_L$ and $K_S$  to be orthogonal -- as can be read off \nexplicitely from the general expression \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\langle K_L |K_S \\rangle = \\frac{1 -|q\/p|^2} {1 +|q\/p|^2}\n\\eeq\nThe Bell-Steinberger relation allows to place a bound on \nthis scalar product from inclusive decay rates \n\\cite{LEE,BOOK}: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\langle K_L |K_S \\rangle \\leq \\sqrt{2} \n\\sum _f \\sqrt{\\frac{\\Gamma _L^f\\Gamma _S^f}{\\Gamma _S^2} } \n\\leq \\sqrt{2} \\sqrt{\\frac{\\Gamma _L}{\\Gamma _S}} \n\\simeq 0.06 \n\\label{BELL} \n\\eeq \nThere is no input from any CP measurement. What is essential, \nthough, is the huge lifetime ratio. \n\\item \nThere are actually two processes underlying the transition \n$K_L \\rightarrow 2\\pi$: $\\Delta S=2$ forces generate the mass eigenstates \n$K_L$ and $K_S$ whereas $\\Delta S=1$ dynamics drive the decays \n$K \\rightarrow 2\\pi$. Thus CP violation can enter in two a priori independant \nways, namely through the $\\Delta S=2$ and the $\\Delta S=1$ \nsector. This distinction can be made explicit in terms of the \ntransition amplitudes: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\eta _{+-} \\equiv \n\\frac{A(K_L \\rightarrow \\pi ^+ \\pi ^-)}{A(K_S \\rightarrow \\pi ^+ \\pi ^-)} \\equiv \n\\epsilon _K+ \\epsilon ^{\\prime} \\; , \\; \n\\eta _{00} \\equiv \n\\frac{A(K_L \\rightarrow \\pi ^0 \\pi ^0)}{A(K_S \\rightarrow \\pi ^0 \\pi ^0)} \\equiv \n\\epsilon _K- 2\\epsilon ^{\\prime}  \n\\eeq\nThe quantity $\\epsilon _K$ describes the CP violation common to the \n$K_L$ decays; it thus characterizes the decaying {\\em state} and is \nreferred to as {\\em CP violation in the mass matrix} or \n{\\em superweak CP violation}; $\\epsilon ^{\\prime}$ on the other \nhand differentiates between different channels and thus \ncharacterizes {\\em decay} dynamics; it is called {\\em direct \nCP violation}.   \n\\item  \n{\\em Maximal} parity and\/or charge conjugation violation \ncan be defined by saying there is no right-handed neutrino \nand\/or left-handed antineutrino, respectively. Yet \n{\\em maximal} CP violation {\\em cannot} be defined in an analogous \nway: for the existence of the right-handed antineutrino which is the \nCP conjugate to the left-handed neutrino is already required by \nCPT invariance. \n\n\\end{itemize} \n\n\\subsection{Data}\nThe data on CP violation in neutral kaon decays are as follows: \n\\begin{enumerate} \n\\item \n{\\em Existence} of $K_L \\rightarrow \\pi \\pi$:\n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\begin{array}{l}\n{\\rm BR}(K_L \\rightarrow \\pi ^+ \\pi ^-) = (2.067 \\pm 0.035) \\cdot 10^{-3} \\\\ \n{\\rm BR}(K_L \\rightarrow \\pi ^0 \\pi ^0) = (0.936 \\pm 0.020) \\cdot 10^{-3} \\\\ \n\\end{array}\n\\label{ETADATA}\n\\eeq \n\\item \nSearch for {\\em direct} CP violation: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{\\epsilon ^{\\prime}}{\\epsilon _K} \\simeq \n{\\rm Re} \\frac{\\epsilon ^{\\prime}}{\\epsilon _K} = \n\\left\\{   \n\\begin{array}{ll}\n(2.3 \\pm 0.65) \\cdot 10^{-3} & NA\\; 31 \\\\\n(1.5 \\pm 0.8) \\cdot 10^{-3} & PDG \\; '96\\;  average \\\\\n(0.74 \\pm 0.52 \\pm 0.29) \\cdot 10^{-3} & E\\; 731 \\\\\n\\end{array} \n\\right. \n\\label{DIRECTCPDATA} \n\\eeq \n\\item \nRate {\\em difference} in semileptonic decays: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\delta _l\\equiv \n\\frac{\\Gamma (K_L \\rightarrow l^+ \\nu \\pi ^-) - \n\\Gamma (K_L \\rightarrow l^- \\bar \\nu \\pi ^+)}\n{\\Gamma (K_L \\rightarrow l^+ \\nu \\pi ^-) + \n\\Gamma (K_L \\rightarrow l^- \\bar \\nu \\pi ^+)}  = \n(3.27 \\pm 0.12)\\cdot 10^{-3} \\; , \n\\label{SLDIFFDATA} \n\\eeq\nwhere an average over electrons and muons has been taken. \n\\item \n{\\em T violation}: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{\\Gamma (K^0 \\Rightarrow \\bar K^0) - \n\\Gamma (\\bar K^0 \\Rightarrow K^0)} \n{\\Gamma (K^0 \\Rightarrow \\bar K^0) + \n\\Gamma (\\bar K^0 \\Rightarrow K^0)} = \n(6.3 \\pm 2.1 \\pm 1.8) \\cdot 10^{-3} \\; \\; \\; \nCPLEAR \n\\label{CPLEAR}\n\\eeq \nfrom a third of their data set \\cite{CPLEARBLOCH}. \nIt would be premature to claim this asymmetry has been \nestablished; yet it represents an intriguingly direct test of time \nreversal violation and is sometimes referred to as the Kabir \ntest . It requires tracking the flavour identity of \nthe {\\em decaying} meson as a $K^0$ or $\\bar K^0$ through its \nsemileptonic decays -- $\\bar K^0 \\rightarrow l^- \\bar \\nu \\pi ^+$ vs. \n$K^0 \\rightarrow l^+  \\nu \\pi ^-$ -- and also of the {\\em initially \nproduced} kaon. The latter is achieved through correlations \nimposed by associated production. The CPLEAR collaboration \nstudied low energy proton-antiproton annihilation \n\\begin{equation}}   \\def\\eeq{\\end{equation} \np \\bar p \\rightarrow K^+ \\bar K^0 \\pi ^- \\; \\; vs. \\; \\; \np \\bar p \\rightarrow K^-  K^0 \\pi ^+ \\; ; \n\\eeq \nthe charged kaon reveals whether a $K^0$ or a $\\bar K^0$ was \nproduced in association with it. In the future the CLOE collaboration \nwill study T violation in $K^0 \\bar K^0$ production at DA$\\Phi$NE: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \ne^+ e^- \\rightarrow \\phi (1020) \\rightarrow K^0 \\bar K^0 \n\\eeq  \n\\end{enumerate}\n\n\\subsection{Phenomenological Interpretation}\n\\subsubsection{Semileptonic Transitions}\nCPT symmetry imposes constraints well beyond the equality of \nlifetimes for particles and antiparticles: certain {\\em sub}classes of \ndecay rates have to be equal as well. For example one finds \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\Gamma (\\bar K^0 \\rightarrow l^- \\bar \\nu \\pi ^+) = \n\\Gamma (K^0 \\rightarrow l^+ \\nu \\pi ^-)  \n\\eeq \nThe rate asymmetry in semileptonic decays listed in \nEq.(\\ref{SLDIFFDATA}) thus reflects pure superweak CP violation: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\delta _l = \\frac{ 1 - |q\/p|^2}{1+|q\/p|^2} \n\\eeq\nFrom the measured value of $\\delta _l$ one then obtains \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\left| \\frac{q}{p}\\right| = 1 + (3.27 \\pm 0.12) \\cdot 10^{-3} \n\\eeq  \nSince one has for the $K^0 - \\bar K^0$ system specifically \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\left| \\frac{q}{p}\\right| \\simeq \n1 + \\frac{1}{2} {\\rm arg}\\frac{M_{12}}{\\Gamma _{12}} \n\\eeq \none can express this kind of CP violation through a phase: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\Phi (\\Delta S=2) \\equiv {\\rm arg}\\frac{M_{12}}{\\Gamma _{12}} = \n(6.54 \\pm 0.24)\\cdot 10^{-3} \n\\label{PHI2}\n\\eeq\nThe result of the Kabir test, Eq.(\\ref{CPLEAR}), yields: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\Phi (\\Delta S=2) = (6.3 \\pm 2.1 \\pm 1.8)\\cdot 10^{-3}\\; ,  \n\\eeq  \nwhich is of course consistent with Eq.(\\ref{PHI2}). \n\nUsing the measured value of $\\Delta m_K\/\\Delta \\Gamma _K$ one \ninfers \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{M_{12}}{\\Gamma _{12}} = - (0.4773 \\pm 0.0023)\n\\left[ 1 - i (6.54 \\pm 0.24)\\cdot 10^{-3})\\right] \n\\eeq  \n\\subsubsection{Nonleptonic Transitions}\nFrom Eq.(\\ref{ETADATA}) one deduces \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\begin{array}{l} \n|\\eta _{+-}| = (2.275 \\pm 0.019)\\cdot 10^{-3} \\\\ \n|\\eta _{00}| = (2.285 \\pm 0.019)\\cdot 10^{-3} \\\\ \n\\end{array}\n\\eeq \nAs mentioned before the ratios $\\eta _{+-,00}$ are sensitive also \nto direct CP violation generated by a phase between the \ndecay amplitudes $A_{0,2}$ for $K_L\\rightarrow (\\pi \\pi )_I$, where the \nsubscript $I$ denotes the isospin of the $2\\pi$ system: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\Phi (\\Delta S=1) \\equiv {\\rm arg}\\frac{A_2}{A_0} \n\\eeq\nOne finds \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\eta _{+-} \\simeq  \\frac{i \\tilde x}{2\\tilde x+i}\n\\left[ \\Phi (\\Delta S=2)  + 2 \\omega \\Phi (\\Delta S=1) \\right] \\; , \n\\eeq  \nwith  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\tilde x \\equiv  \\frac{\\Delta m_K}{\\Delta \\Gamma _K} = \n\\frac{\\Delta m_K}{\\Gamma (K_S)} \n=\\frac{1}{2} x_K \\simeq 0.477 \\; \\; , \\; \\; \n\\omega \\equiv \\left| \\frac{A_2}{ A_0}\\right| \\simeq 0.05\n\\eeq\nwhere the second quantity represents the observed\nenhancement of $A_0$ for which a name -- \"$\\Delta I=1\/2$ rule\" -- \nyet no quantitative dynamical explanation has been found. \nEquivalently one can write   \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{\\epsilon ^{\\prime}}{\\epsilon _K} \\simeq 2 \\omega \n\\frac{\\Phi (\\Delta S=1)}{\\Phi (\\Delta S=2)}\n\\eeq \nThe data on $K_L \\rightarrow \\pi \\pi$ can thus be expressed as \nfollows \\cite{WINSTEIN} \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\begin{array}{l} \n\\Phi (\\Delta S=2) = (6.58 \\pm 0.26) \\cdot 10^{-3} \\\\ \n\\Phi (\\Delta S=1) = (0.99 \\pm 0.53) \\cdot 10^{-3}  \n\\end{array}\n\\eeq  \n\\subsubsection{Resume}\nThe experimental results can be summarized as follows: \n\\begin{itemize}\n\\item \nThe decays of neutral kaons exhibit unequivocally CP violation \nof the superweak variety, which is expressed through the \nangle $\\Phi (\\Delta S=2)$. The findings from \nsemileptonic and nonleptonic transitions concur to an impressive \ndegree. \n\\item \nDirect CP violation still has not been established. \n\\item \nA theorist might be forgiven for mentioning that the evolution of the \nmeasurements over the last twenty odd \nyears has not followed the straight line this brief summary \nmight suggest to the uninitiated reader. \n\\end{itemize} \n\n\\section{Theoretical Implementation of CP Violation}\n\n\\subsection{Some Historical Remarks}\nTheorists can be forgiven if they felt quite pleased with the state \nof their craft in 1964:\n\\begin{itemize}\n\\item \nThe concept of (quark) families had emerged, at least in a \nrudimentary form. \n\\item \nMaximal parity and charge conjugation violations had been found in \nweak charged current interactions, yet CP invariance apparently \nheld. Theoretical pronouncements were made ex cathedra why this \nhad to be so!\n\\item \n{\\em Pre}dictions of the existence of two kinds of neutral \nkaons with different lifetimes and masses had been \nconfirmed by experiment \\cite{PAIS}. \n\\end{itemize}\nThat same year the reaction $K_L \\rightarrow \\pi ^+ \\pi ^-$ was \ndiscovered \\cite{FITCH}! Two things should be noted here. \nThe Fitch-Cronin experiment had predecessors: rather than \nbeing an isolated effort it was the culmination of a whole \nresearch program. Secondly there was at least one theoretical \nvoice, namely that of Okun \\cite{OKUN}, who in 1962\/63 had listed a \ndedicated search for $K_L \\rightarrow \\pi \\pi$ as one of the most important \nunfinished tasks. Nevertheless for the vast majority of the community the \nFitch-Cronin observation came as a shock and caused considerable \nconsternation among theorists. Yet -- to their credit -- these data \nand their consequence, namely that CP invariance was broken, \nwere soon accepted as facts. This was phrased -- though \n{\\em not explained} -- in terms of the Superweak Model \n\\cite{WOLFSW} later that same year. \n\nIn 1970 the renormalizability of the $SU(2)_L\\times U(1)$ \nelectroweak gauge theory was proven. I find it quite amazing \nthat it was still not realized that the physics known at that time \ncould not produce CP violation. As long as one had to struggle \nwith infinities in the theoretical description one could be forgiven \nfor not worrying unduly about a tiny quantity like \nBR$(K_L \\rightarrow \\pi ^+ \\pi ^-) \\simeq 2.3 \\cdot 10^{-3}$. Yet no such \nexcuse existed any longer once a renormalizable theory had been \ndeveloped! The existence of the Superweak Model somewhat \nmuddled the situation in this respect: for it provides merely \na classification of the dynamics underlying CP violation rather \nthan a dynamical description itself. \n\nThe paper by Kobayashi and Maskawa \\cite{KM}, \nwritten in 1972 and published in 1973, was the first \n\\begin{itemize}\n\\item \nto state clearly that the \n$SU(2)_L\\times U(1)$ gauge theory even with two complete \nfamilies \\footnote{Remember this was still before the $J\/\\psi$ \ndiscovery!} is necessarily CP-invariant and \n\\item \nto list the possible extensions that could generate CP \nviolation; among them -- as one option -- was the three (or more) \nfamily scenario now commonly referred to as the KM ansatz. They \nalso discussed the impact of right-handed currents and of a \nnon-minimal Higgs sector. \n\\end{itemize}\n\n\\subsection{The Minimal Model: The KM Ansatz}\nOnce a theory reaches a certain degree of complexity, many potential \nsources of CP violation emerge. Popular examples of such a scenario \nare provided by models implementing supersymmetry or its \nlocal version, supergravity; hereafter both are referred to as \nSUSY. In my lectures I will however focus on the minimal \ntheory that can support CP violation, namely the Standard Model \nwith three families. All of its dynamical elements have been \nobserved -- except for the Higgs boson, of course. \n\n\\subsubsection{Weak Phases like the Scarlet Pimpernel}\n\nWeak interactions at low energies are described by four-fermion \ninteractions. The most general expression for spin-one \ncouplings are \n$$ \n{\\cal L}_{V\/A} = \\left( \\bar \\psi _1 \\gamma _{\\mu}\n(a + b \\gamma _5)\\psi _2\\right) \n\\left( \\bar \\psi _3 \\gamma _{\\mu}\n(c + d \\gamma _5)\\psi _4\\right) + \n$$\n\\begin{equation}}   \\def\\eeq{\\end{equation} \n+ \\left( \\bar \\psi _2 \\gamma _{\\mu}\n(a^* + b^* \\gamma _5)\\psi _1\\right) \n\\left( \\bar \\psi _4 \\gamma _{\\mu}\n(c^* + d^* \\gamma _5)\\psi _3\\right)\n\\eeq \nUnder CP these terms transform as follows: \n$$  \n{\\cal L}_{V\/A} \\stackrel{CP}{\\Longrightarrow} \nCP {\\cal L}_{V\/A} (CP)^{\\dagger} = \n\\left( \\bar \\psi _2 \\gamma _{\\mu}\n(a + b \\gamma _5)\\psi _1\\right) \n\\left( \\bar \\psi _4 \\gamma _{\\mu}\n(c + d \\gamma _5)\\psi _3\\right) + \n$$ \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n+ \\left( \\bar \\psi _1 \\gamma _{\\mu}\n(a^* + b^* \\gamma _5)\\psi _2\\right) \n\\left( \\bar \\psi _3 \\gamma _{\\mu}\n(c^* + d^* \\gamma _5)\\psi _4\\right)\n\\eeq \nIf $a,b,c,d$ are real numbers, one obviously has \n${\\cal L}_{V\/A}= CP {\\cal L}_{V\/A} (CP)^{\\dagger} $ and CP \nis conserved. Yet CP is {\\em not necessarily} broken if these \nparameters are complex, as we will explain specifically \nfor the Standard Model. \n\n{\\em Weak Universality} arises naturally whenever the weak \ncharged current interactions are described through a \n{\\em single} non-abelian gauge group -- $SU(2)_L$ in the case \nunder study. For the single {\\em self}-coupling of the gauge bosons \ndetermines also their couplings to the fermions; \none finds for the quark couplings to the charged $W$ bosons: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\cal L}_{CC} = \ng \\bar U_L^{(0)}\\gamma _{\\mu}D_L^{(0)} W^{\\mu} + \n\\bar U_R^{(0)} {\\bf M}_U U_L^{(0)} + \n\\bar D_R^{(0)} {\\bf M}_D D_L^{(0)} + h.c. \n\\eeq   \nwhere $U$ and $D$ denote the up- and down-type quarks, \nrespectively: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nU= (u,c,t) \\; \\; \\; , \\; \\; \\; \\; D=(d,s,b) \n\\eeq \nand ${\\bf M_U}$ and ${\\bf M_D}$ \ntheir 3$\\times$3 mass matrices. In general \nthose will not be diagonal; to find the physical states, one has to \ndiagonalize these matrices: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\bf M}^{diag}_U = {\\bf K}^U_R{\\bf M}_U ({\\bf K}^U_L)^{\\dagger} \\; \\; , \\; \\; \n{\\bf M}^{diag}_D = {\\bf K}^D_R{\\bf M}_D ({\\bf K}^D_L)^{\\dagger}\n\\eeq \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nU_{L,R} = {\\bf K}^U_{L,R} U^{(0)}_{L,R} \\; \\; , \\; \\; \nD_{L,R} = {\\bf K}^D_{L,R} D^{(0)}_{L,R}\n\\eeq  \nwith ${\\bf K}_{L,R}^{U,D}$ representing four unitary 3$\\times$3 matrices. \nThe coupling of these physical fermions to $W$ bosons is then given \nby \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\cal L}_{CC} = \ng \\bar U_L({\\bf K}_L^U)^{\\dagger}{\\bf K}_L^D\\gamma _{\\mu}D  W^{\\mu} + \n\\bar U_R {\\bf M}^{diag}_U U_L + \n\\bar D_R {\\bf M}^{diag}_D D_L + h.c. \n\\eeq   \nand the combination $({\\bf K}_L^U)^{\\dagger}{\\bf K}_L^D \n\\equiv {\\bf V}_{CKM}$ \nrepresents the KM matrix, which obviously has to be unitary \nlike $K^U$ and $K^D$. Unless the \nup- and down-type mass matrices are aligned in flavour \nspace (in which case they would be diagonalized by the \nsame operators ${\\bf K}_{L,R}$) one has ${\\bf V}_{CKM} \\neq 1$.  \n\nIn the neutral current sector one has \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\cal L}_{NC} = g^{\\prime} \n\\bar U_L^{(0)} \\gamma _{\\mu}U_L^{(0)}Z_{\\mu} = \ng^{\\prime} \n\\bar U_L \\gamma _{\\mu}U_LZ_{\\mu}\n\\eeq \nand likewise for $U_R$ and $D_{L,R}$; i.e. {\\em no} \nflavour changing neutral currents are generated, let alone \nnew phases. CP violation thus has to be embedded into the \ncharged current sector. \n\nIf ${\\bf V}_{CKM}$ is real (and thus orthogonal), CP symmetry is \nconserved in the weak interactions. Yet the occurrance of \ncomplex matrix elements does not {\\em automatically} signal \nCP violation. This can be seen through a straightforward \n(in hindsight at least) algebraic argument. A unitary \n$N\\times N$ matrix contains $N^2$ independant real \nparameters; $2N-1$ of those can be eliminated through \nre-phasing of the $N$ up-type and $N$ down-type fermion \nfields (changing all fermions by the {\\em same} phase obviously \ndoes not affect  ${\\bf V}_{CKM}$). Hence there are $(N-1)^2$ \nreal physical parameters in such an $N \\times N$ matrix. \nFor $N=2$, i.e. two families, one recovers a familiar result, \nnamely there is just one mixing angle, the Cabibbo \nangle. For $N=3$ there are four real physical parameters, \nnamely three (Euler) angles -- and one phase. It is the latter \nthat provides a gateway for CP violation. For $N=4$ Pandora's \nbox opens up: there would be 6 angles and 3 phases. \n\nPDG suggests a \"canonical\" parametrization for the $3\\times 3$ CKM \nmatrix: \n$$ \n{\\bf V}_{CKM} = \n\\left(  \n\\begin{array}{ccc} \nV(ud) & V(us) & V(ub) \\\\\nV(cd) & V(cs) & V(cb) \\\\\nV(td) & V(ts) & V(tb) \n\\end{array} \n\\right) \n$$\n\\begin{equation}}   \\def\\eeq{\\end{equation} \n= \\left( \n\\begin{array}{ccc} \nc_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i \\delta _{13}}  \\\\\n- s_{12}c_{23} - c_{12}s_{23}s_{13}e^{i \\delta _{13}} &\nc_{12}c_{23} - s_{12}s_{23}s_{13}e^{i \\delta _{13}} & \nc_{13}s_{23} \\\\\ns_{12}s_{23} - c_{12}c_{23}s_{13}e^{i \\delta _{13}} &\n- c_{12}s_{23} - s_{12}c_{23}s_{13}e^{i \\delta _{13}} &\nc_{13}c_{23} \n\\end{array}\n\\right) \n\\label{PDGKM} \n\\eeq\nwhere \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nc_{ij} \\equiv {\\rm cos} \\theta _{ij} \\; \\; , \\; \\;  \ns_{ij} \\equiv {\\rm sin} \\theta _{ij}\n\\eeq \nwith $i,j = 1,2,3$ being generation labels. \n\nThis is a completely general, yet not unique parametrisation: a \ndifferent set of \nEuler angles could be chosen; the phases can be shifted around \namong the matrix elements \nby using a different phase convention. In that sense one can refer to \nthe KM phase as the Scarlet Pimpernel: \"Sometimes here, sometimes \nthere, sometimes everywhere!\"  \n\nUsing just the observed hierarchy \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n|V(ub)| \\ll |V(cb)| \\ll |V(us)| , |V(cd)| \\ll 1\n\\label{HIER}\n\\eeq \none can, as first realized by Wolfenstein, expand \n${\\bf V}_{CKM}$ in powers of the Cabibbo angle $\\theta _C$: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\bf V}_{CKM} = \n\\left( \n\\begin{array}{ccc} \n1 - \\frac{1}{2} \\lambda ^2 & \\lambda & \nA \\lambda ^3 (\\rho - i \\eta + \\frac{i}{2} \\eta \\lambda ^2) \\\\\n- \\lambda & 1 - \\frac{1}{2} \\lambda ^2 - i \\eta A^2 \\lambda ^4 & \nA\\lambda ^2 (1 + i\\eta \\lambda ^2 ) \\\\ \nA \\lambda ^3 (1 - \\rho - i \\eta ) \\\\\n& - A\\lambda ^2 & 1 \n\\end{array}\n\\right) \n+ {\\cal O}(\\lambda ^6) \n\\label{WOLFKM}\n\\eeq \nwhere \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\lambda \\equiv {\\rm sin} \\theta _C\n\\eeq\nFor such an expansion in powers of $\\lambda$ to be self-consistent, \none has to require that $|A|$, $|\\rho |$ and $|\\eta |$ are of order \nunity. Numerically we obtain  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\lambda = 0.221 \\pm 0.002  \n\\label{KMNUm1} \n\\eeq \nfrom $|V(us)|$,  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nA = 0.81 \\pm 0.06 \n\\label{KMNUM2}\n\\eeq \nfrom $|V(cb)| \\simeq 0.040 \\pm 0.002|_{exp} \\pm 0.002|_{theor}$ \nand \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\sqrt{\\rho ^2 + \\eta ^2} \\sim 0.38 \\pm 0.11 \n\\label{KMNUM3} \n\\eeq\nfrom $|V(ub)| \\sim (3.2 \\pm 0.8)\\cdot 10^{-3}$. \n\nWe see that the CKM matrix is a very special unitary matrix: \nit is almost diagonal, it is almost symmetric and the matrix \nelements get smaller the more one moves away from the \ndiagonal. \nNature most certainly has encoded a profound message in \nthis peculiar pattern. Alas -- we have not succeeded yet in \ndeciphering it! I will return to this point at the end of my \nlectures.  \n\n\\subsubsection{Unitarity Triangles}\n\nThe qualitative difference between a two and a three family scenario \ncan be seen also in a less abstract way.  \nConsider $\\bar K^0 \\rightarrow \\pi ^+ \\pi ^-$; it can proceed through \na tree-level process \n$[s\\bar d] \\rightarrow [d \\bar u][u\\bar d]$ , in which case its \nweak couplings are \ngiven by $V(us)V^*(ud)$. Or it can oscillate first to $K^0$ \nbefore decaying; i.e., on the quark level it is the \ntransition $[s\\bar d] \\rightarrow [d\\bar s] \\rightarrow [d \\bar u][u\\bar d]$ \ncontrolled by \n$\\left( V(cs)\\right) ^2 \\left( V^*(cd)\\right) ^2 V^*(us)V(ud)$. \nAt first sight it would seem that those two combinations of \nweak parameters are not only different, but should also exhibit a \nrelative phase. Yet the latter is not so -- if there are two families \nonly! In that case the four quantities $V(ud)$, $V(us)$, $V(cd)$ \nand $V(cs)$ have to form a unitary $2\\times 2$ which leads to the \nconstraint \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nV(ud)V^*(us) + V(cd)V^*(cs) = 0 \n\\label{UNIT2FAM}\n\\eeq \nUsing Eq.(\\ref{UNIT2FAM}) twice one gets  \n$$  \n\\left( V(cs) V^*(cd)\\right) ^2 V^*(us)V(ud) = \n- |V(cd)V(cs)|^2 V(cs)V^*(cd) = \n$$ \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n= |V(cd)V(cs)|^2 V^*(ud) V(us) \\; ; \n\\eeq  \n i.e., the two combinations $V^*(ud)V(us)$ and \n$\\left( V(cs)\\right) ^2 \\left( V^*(cd)\\right) ^2 V^*(us)V(ud)$ are \nactually parallel to each other with {\\em no} \nrelative phase. A penguin \noperator with a charm quark as the internal fermion \nline generates another contribution to \n$K_L \\rightarrow \\pi ^-\\pi ^+$, this one controlled by $V(cs)V^*(cd)$. Yet \nthe unitarity condition Eq.(\\ref{UNIT2FAM}) forces this contribution \nto be antiparallel to $V^*(ud)V(us)$; i.e., again no relative phase. \n\nThe situation changes fundamentally for three families: the weak \nparameters $V(ij)$ now form a $3\\times 3$ matrix and the condition \nstated in  Eq.(\\ref{UNIT2FAM}) gets extended: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nV(ud)V^*(us) + V(cd)V^*(cs) + V(td)V^*(ts) = 0 \n\\label{UNIT3FAM}\n\\eeq \n{\\em This is a triangle relation in the complex plane.} \nThere emerge now \nrelative phases between the weak parameters and the \nloop diagrams with internal charm and top quarks can generate \nCP asymmetries. \n\nUnitarity imposes altogether nine algebraic conditions on the \nmatrix elements of ${\\bf V}_{CKM}$, of which six are triangle relations \nanalogous to Eq.(\\ref{UNIT3FAM}). \nThere are several nice features about this representation in terms of \ntriangles; I list four now and others later: \n\\begin{enumerate}\n\\item \nThe {\\em shape} of each triangle is independant of the phase \nconvention adopted for the quark fields. Consider for example \nEq.(\\ref{UNIT3FAM}): changing the phase of any of the \nup-type quarks will not affect the triangle at all. Under \n$|s\\rangle \\rightarrow |s\\rangle e^{i \\phi _s}$ the whole triangle will \nrotate around the left end of its base line by an angle \n$\\phi _s$ -- yet the shape of the triangle -- in contrast \nto its orientation in the complex plane -- remains the same! \nThe angles inside the triangles are thus observables;  \nchoosing an orientation for the triangles is then a matter \nof convenience. \n\\item \nIt is easily shown that all six KM triangles possess \nthe same area. \nMultiplying Eq.(\\ref{UNIT3FAM}) by the phase \nfactor $V^*(ud)V(us)\/|V(ud)V(us)|$, which does not change the \narea, yields \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n|V(ud)V(us)| + \\frac{V^*(ud)V(us)V(cd)V^*(cs)}{|V(ud)V(us)|} + \n\\frac{V^*(ud)V(us)V(td)V^*(ts)}{|V(ud)V(us)|} = 0 \n\\eeq \n$$  \n{\\rm area (triangle \\; of \\; Eq.(\\ref{UNIT3FAM})})  = \n\\frac{1}{2} |{\\rm Im}V(ud)V(cs)V^*(us)V^*(cd)| = \n$$ \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n= \\frac{1}{2} |{\\rm Im}V(ud)V(ts)V^*(us)V^*(td)|\n\\eeq\nMultiplying Eq.(\\ref{UNIT3FAM}) instead by the phase \nfactors $V^*(cd)V(cs)\/|V(cd)V(cs)|$ or $V^*(td)V(ts)\/|V(td)V(ts)|$ \none sees that the area of this triangle can be expressed in other ways \nstill. Among them is \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\rm area (triangle \\; of \\; Eq.(\\ref{UNIT3FAM})})  = \n\\frac{1}{2} |{\\rm Im}V(cd)V(ts)V^*(cs)V^*(td)|  \n\\eeq \nDue to the unitarity relation \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nV^*(cd)V(td) + V^*(cb)V(tb) = - V^*(cs)V(ts) \n\\label{UNIT3FAM2}\n\\eeq\none has  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\rm area ( triangle \\; of \\; Eq.(\\ref{UNIT3FAM})})  = \n\\frac{1}{2} |{\\rm Im}V(cd)V(tb)V^*(cb)V^*(td)|  \n\\eeq \n-- yet this is exactly the area of the triangle defined by \nEq.(\\ref{UNIT3FAM2})! \nThis is the re-incarnation of the original \nobservation that there is a {\\em single irreducible}  \nweak phase for three families. \n\\item \nIn general one has for the area of these triangles \n$$  \nA_{CPV}(\\rm every \\; triangle) = \\frac{1}{2} J \n$$ \n\\begin{equation}}   \\def\\eeq{\\end{equation}  \nJ = {\\rm Im}V^*(km) V(lm)V(kn)V^*(ln) = \n{\\rm Im}V^*(mk) V(ml)V(nk)V^*(nl) \n\\label{KMAREA}\n\\eeq\nirrespective of the indices $k,l,m,n$; $J$ is obviously re-phasing \ninvariant. \n\\item  \nIf there is a representation of $V_{CKM}$ where \nall phases were confined to a $2\\times 2$ \nsub-matrix exactly rather than approximately, then one can  \nrotate all these phases away; i.e., CP is conserved in such a \nscenario! Consider again the triangle described by \nEq.(\\ref{UNIT3FAM}): it can always be rotated such that its \nbaseline -- $V(ud)V^*(us)$  -- \nis real. Then Im$V(td)V^*(ts)$ = - Im$V(cd)V^*(cs)$ holds. \nIf, for example, there were no phases in the third row and column, \none would have Im$(V(td)V^*(ts)) =0$ and therefore \nIm$V(cd)V^*(cs) =0$ as well; i.e., $V(ud)V^*(us)$ and \n$V(cd)V^*(cs)$ were real relative to each other; \ntherefore $J=0$, i.e. all six triangles had zero area meaning \nthere are no relative weak phases! \n\\end{enumerate}  \n\n\\subsection{Evaluating $\\epsilon _K$ and $\\epsilon ^{\\prime}$} \nIn calculating observables in a given theory -- in the case under \nstudy $\\epsilon _K$ and $\\epsilon ^{\\prime}$ \nwithin the KM Ansatz -- one is faced with the `Dichotomy of the Two \nWorlds', namely \n\\begin{itemize}\n\\item \none world of {\\em short}-distance physics where even the strong \ninteractions can be treated {\\em perturbatively} in terms of \nquarks and gluons and in which theorists like to work, and \n\\item \nthe other world of {\\em long}-distance physics where one has \nto deal with hadrons the behaviour of which is controlled \nby {\\em non}-perturbative dynamics and where, by the way, \neveryone, including theorists, lives. \n\\end{itemize}\nAccordingly the calculational task is divided into two \nparts, namely first determing the relevant \ntransition operators in the short-distance world and then \nevaluating their matrix elements in the hadronic world. \n\n\\subsubsection{$\\Delta S=2$ Transitions}\nSince the {\\em elementary} interactions in the \nStandard Model can change strangeness at most by one unit, \nthe $\\Delta S=2$ amplitude driving $K^0 - \\bar K^0$ \noscillations is obtained by iterating the \nbasic $\\Delta S=1$ coupling: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\cal L}_{eff} (\\Delta S=2) = {\\cal L}(\\Delta S=1) \\otimes \n{\\cal L}(\\Delta S=1) \n\\eeq  \nThere are actually two ways in which the $\\Delta S=1$ transition \ncan be iterated: \n\n{\\bf (A)} \nThe resulting $\\Delta S=2$ transition is described by \na {\\em local} operator. The celebrated box diagram makes \nthis connection quite transparent. The contributions that do \n{\\em not} depend on the mass of the internal quarks cancel against \neach other due to the GIM mechanism. Integrating over the internal \nfields, namely the $W$ bosons and the top and charm quarks \n\\footnote{The up quarks act merely as a subtraction term here.} \nthen yields a convergent result: \n$$  \n{\\cal L}_{eff}^{box}(\\Delta S=2, \\mu ) = \n\\left( \\frac{G_F}{4\\pi }\\right) ^2 \\cdot \n$$ \n\\begin{equation}}   \\def\\eeq{\\end{equation}  \n\\cdot \\left[  \\xi _c^2 E(x_c) \\eta _{cc} + \n\\xi _t^2 E(x_t) \\eta _{tt} + \n2\\xi _c \\xi _t E(x_c, x_t) \\eta _{ct}\n \\right] \\cdot  [\\alpha _S(\\mu ^2)]^{-6\/27} \n\\left( \\bar s \\gamma _{\\mu}(1- \\gamma _5) d\\right) ^2 \n\\label{LAGDELTAS2}\n\\eeq  \nwith $\\xi _i$ denoting combinations of KM parameters \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\xi _i = V(is)V^*(id) \\; , \\; \\; i=c,t \\; ; \n\\eeq \n$E(x_i)$ and $E(x_c,x_t)$ reflect the box loops with equal and \ndifferent internal quarks, respectively \\cite{INAMI}:  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nE(x_i) = x_i \n\\left(   \n\\frac{1}{4} + \\frac{9}{4(1- x_i)} - \\frac{3}{2(1- x_i)^2} \n\\right) \n- \\frac{3}{2} \\left( \\frac{x_i}{1-x_i}\\right) ^3 \n{\\rm log} x_i \n\\eeq \n$$  \nE(x_c,x_t) = x_c x_t \n\\left[ \\left( \n\\frac{1}{4} + \\frac{3}{2(1- x_t)} - \\frac{3}{4(1- x_t)^2} \\right) \n\\frac{{\\rm log} x_t}{x_t - x_c} + (x_c \\leftrightarrow x_t) - \n\\right. \n$$ \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\left. - \\frac{3}{4} \\frac{1}{(1-x_c)(1- x_t)} \\right]  \n\\eeq\n\\begin{equation}}   \\def\\eeq{\\end{equation} \nx_i = \\frac{m_i^2}{M_W^2} \n\\eeq  \nand $\\eta _{ij}$ containing the QCD radiative corrections from \nevolving the effective Lagrangian from $M_W$ down to \nthe internal quark mass. The factor $[\\alpha _S(\\mu ^2)]^{-6\/27}$ \nreflects the fact that a scale \n$\\mu$ must be introduced at which the four-quark operator \n$\\left( \\bar s \\gamma _{\\mu}(1- \\gamma _5) d\\right) ^2 $ is \ndefined. This dependance on the auxiliary variable \n$\\mu$ drops out when one takes the matrix element of this \noperator (at least when one does it correctly).  \nIncluding next-to-leading log \ncorrections one finds (for $m_t \\simeq 180$ GeV) \\cite{BURAS}: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\eta _{cc} \\simeq 1.38 \\pm 0.20 \\; , \\; \\; \n\\eta _{tt} \\simeq 0.57 \\pm 0.01 \\; , \\; \\; \n\\eta _{cc} \\simeq 0.47 \\pm 0.04 \n\\eeq                       \n\n{\\bf (B)} \nHowever there is also a {\\em non}-local $\\Delta S=2$ \noperator generated from the iteration of ${\\cal L}(\\Delta S=1)$. \nIt presumably provides a major contribution to $\\Delta m_K$. \nYet for $\\epsilon _K$ it is not sizeable within the KM ansatz \n\\footnote{This can be inferred from the observation that \n$|\\epsilon ^{\\prime}\/\\epsilon _K|\\ll 0.05$} and will be \nignored here. \n\nEven for a local four-fermion operator it is non-trivial to \nevaluate an on-shell matrix element \nbetween hadron states since that is \nclearly controlled by non-perturbative dynamics. Usually one \nparametrizes this matrix element as follows: \n$$  \n\\matel{\\bar K^0}{(\\bar s \\gamma _{\\mu}(1-\\gamma _5)d) \n(\\bar s \\gamma _{\\mu}(1-\\gamma _5)d)}{K^0} = \n$$ \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n= \\frac{4}{3} B_K \n\\matel{\\bar K^0}{(\\bar s \\gamma _{\\mu}(1-\\gamma _5)d)}{0} \n\\matel{0}{(\\bar s \\gamma _{\\mu}(1-\\gamma _5)d)}{K^0} = \n\\frac{4}{3} B_K f_K^2m_K\n\\label{BAGFACT} \n\\eeq \nThe factor $B_K$ is -- for historical reasons of no consequence now -- \noften called the bag factor; $B_K = 1$ is referred to as \n{\\em vacuum saturation} or {\\em factorization ansatz} since it \ncorresponds to a situation where inserting the vacuum intermediate \nstate into Eq.(\\ref{BAGFACT}) reproduces the full result \nafter all colour contractions of the quark lines have been included. \nSeveral theoretical techniques have been employed to estimate the \nsize of $B_K$; their findings are listed in \nTable \\ref{TABLEBAG}.  \n\\begin{table}\n\\begin{tabular} {|l|l|}\n\\hline   \nMethod & $B_K$ \\\\ \n\\hline \n\\hline \nLarge $N_C$ Expansion & $\\frac{3}{4}$\\\\\n\\hline \nLarge $N_C$ Chiral Pert. with loop correction & $0.66 \\pm 0.1$\\\\\n\\hline \nLattice QCD & $0.84 \\pm 0.2$ \\\\\n\\hline \n\\end{tabular}\n\\centering\n\\caption{Values of $B_K$ from various theoretical techniques} \n\\label{TABLEBAG} \n\\end{table} \nThese results, which are all consistent with each other and with \nseveral phenomenological studies as well, can be summarized as \nfollows:  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nB_K \\simeq 0.8 \\pm 0.2 \n\\label{BAG} \n\\eeq \nSince the size of this matrix element is determined \nby the strong interactions, one indeed expects $B_K \\sim 1$. \n\nWe have assembled all the ingredients now for calculating \n$\\epsilon _K$. The starting point is given by \n\\footnote{The exact expression is \n$|\\epsilon _K|  = \\frac{1}{\\sqrt{2}}\n\\left| \\frac{{\\rm Im}M_{12}}{\\Delta m_K} - \\xi _0\\right| $ \nwhere $\\xi _0$ denotes the phase of the \n$K^0 \\rightarrow (\\pi \\pi )_{I =0}$ isospin zero amplitude; its \ncontribution is \nnumerically irrelevant.}: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n|\\epsilon _K|  \n\\simeq \n\\frac{1}{\\sqrt{2}}\n\\left| \\frac{{\\rm Im}M_{12}}{\\Delta m_K} \\right| \n\\eeq \nThe CP-odd part Im$M_{12}$ is obtained from \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\rm Im} M_{12} = \n{\\rm Im}\\matel{K^0}{{\\cal L}_{eff}(\\Delta S=2)}{\\bar K^0} \n\\eeq \nwhereas for $\\Delta m_K$ one inserts the experimental value,   \nsince the long-distance contributions to $\\Delta m_K$ are not under \ntheoretical control. One then finds \n$$  \n|\\epsilon _K|_{KM} \\simeq |\\epsilon _K|_{KM}^{box} \\simeq  \n$$ \n$$ \n\\simeq \\frac{G_F^2}{6 \\sqrt{2} \\pi ^2} \n\\frac{M_W^2m_K f_K^2 B_K}{\\Delta m_K} \n\\left[ {\\rm Im}\\xi _c^2 E(x_c) \\eta _{cc} + \n {\\rm Im}\\xi _t^2 E(x_t) \\eta _{tt} + \n2{\\rm Im}(\\xi _c\\xi _t) E(x_c,x_t) \\eta _{ct} \\right] \n$$ \n$$  \n\\simeq 1.9 \\cdot 10^4 B_K \\left[ {\\rm Im}\\xi _c^2 E(x_c) \\eta _{cc} + \n {\\rm Im}\\xi _t^2 E(x_t) \\eta _{tt} + \n2{\\rm Im}(\\xi _c\\xi _t) E(x_c,x_t) \\eta _{ct} \\right] \n\\simeq \n$$ \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\simeq \n7.8 \\cdot 10^{-3} \\eta B_K (1.3 - \\rho ) \n\\label{EPSKM} \n\\eeq \nwhere I have used the numerical values for the KM parameters \nlisted above and \n$x_t \\simeq 5$ corresponding to $m_t = 180$ GeV. \n\nTo reproduce the observed value of $|\\epsilon _K|$ one needs \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\eta \\simeq \\frac{0.3}{B_K} \\frac{1}{1.3 - \\rho } \n\\label{EPSREP} \n\\eeq\nFor a given $B_K$ one thus obtains another $\\rho - \\eta $ \nconstraint. Since $B_K$ is not precisely known \n\\footnote{Some might argue that this is an understatement.} \none has a fairly broad band in the $\\rho - \\eta $ plane \nrather than a line. Yet I find it quite remarkable and very \nnon-trivial that Eq.(\\ref{EPSREP}) {\\em can} be \nsatisfied since \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{0.3}{B_K} \\sim 0.3 \\div 0.5 \n\\eeq \nwithout stretching any of the parameters or bounds, in particular \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\sqrt{\\rho ^2 + \\eta ^2} \\sim 0.38 \\pm 0.11 \\; .   \n\\eeq  \n While this does of course not amount to a {\\em pre}diction, \none should keep in mind for proper perspective \nthat in the 1970's and early \n1980's values like $|V(cb)| \\sim 0.04$ and \n$|V(ub)|  \\sim 0.004$ would have seemed quite unnatural; claiming \nthat the top quark mass had to be 180 GeV would have been \noutright preposterous even in the 1980's! Consider a \nscenario with $|V(cb)| \\simeq 0.04$ and $|V(ub)| \\simeq 0.003$, \nyet $m_t \\simeq 40$ GeV; in the mid 80's this would have appeared \nto be quite natural (and there had even been claims that top quarks \nwith a mass of $40\\pm 10$ GeV had been discovered). In that \ncase one would need \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\eta \\sim \\frac{0.75}{B_K} \n\\label{EPS40} \n\\eeq \nto reproduce $|\\epsilon _K|$. Such a large value for $\\eta$ would hardly \nbe compatible with what we know about $|V(ub)|$ \n\\footnote{For some time it was thought that $B_K \\simeq 0.3 \\div \n0.5$ was the best estimate. This would make satisfying \nEq.(\\ref{EPS40}) completely out of the question!}. \n\\begin{center} \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $ \\\\ \n{\\em  Homework Problem \\# 2}: \n\\end{center}\nEq.(\\ref{EPSKM}) suggests that \na non-vanishing value for $\\epsilon _K$ is  generated from the \nbox diagram with internal charm quarks only -- \nIm$\\xi _c^2\\; E(x_c) = - \\eta A^2 \\lambda ^6 E(x_c) \\neq 0$ -- \n{\\em without} top quarks. How does this match up with the \nstatement that the intervention of three families \nis needed for a CP asymmetry to arise? \n\\begin{center} \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $\n\\end{center} \n\n\\subsubsection{$\\Delta S=1$ Decays}\nAt first one might think that no {\\em direct} CP asymmetry \ncan arise in $K \\rightarrow \\pi \\pi$ decays since it requires the interplay \nof three quark families. Yet upon further reflection one realizes \nthat a one-loop diagram produces the so-called Penguin \noperator which changes isopin by half a unit only, \nis {\\em local} and contains a CP {\\em odd} component since it involves virtual charm \nand top quarks. With direct CP violation thus being  \nof order $\\hbar$, i.e. a pure quantum \neffect, one suspects already at this point that it will be reduced \nin strength.  \n\nThe quantity $\\epsilon ^{\\prime}$ is suppressed relative to \n$\\epsilon _K$  due to two other reasons: \n\\begin{itemize}\n\\item \nThe GIM factors are actually quite different for $\\epsilon _K$ and \n$\\epsilon ^{\\prime}$: in the former case they are of the type \n$(m_t^2 - m_c^2)\/M_W^2$, in the latter log$(m_t^2\/m_c^2)$. Both \nof these expressions vanish for $m_t=m_c$, yet for the realistic \ncase $m_t \\gg m_c$ they behave very differently: $\\epsilon _K$ \nis much more enhanced by the large top mass than \n$\\epsilon ^{\\prime}$. This means of course that \n$|\\epsilon ^{\\prime}\/\\epsilon _K|$ is a rather steeply decreasing \nfunction of $m_t$. \n\\item \nThere are actually two classes of Penguin operators contributing \nto $\\epsilon ^{\\prime}$, namely strong as well as electroweak \nPenguins. The latter become relevant  \nsince they are more enhanced than the former for very heavy top \nmasses due to the coupling of the longitudinal virtual $Z$ boson \n(the re-incarnation of one of the original Higgs fields) to \nthe internal top line. Yet electroweak and strong Penguins contribute \nwith the opposite sign! \n\\end{itemize}\nCPT invariance together with the measured $\\pi \\pi$ phase shifts \ntells us that the two complex quantities $\\epsilon ^{\\prime}$ and \n$\\epsilon _K$ are almost completely real to each other; i.e., \ntheir ratio is practically real: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{\\epsilon ^{\\prime}}{\\epsilon _K} \\simeq 2 \\omega \n\\frac{\\Phi (\\Delta S=1)}{\\Phi (\\Delta S=2)}\n\\label{EPSPOVEREPSTH} \n\\eeq \nwhere, as defined before,  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\omega \\equiv \\frac{|A_2|}{|A_0|} \\simeq 0.05 \\; \\; , \\; \\; \n\\Phi (\\Delta S=2) \\equiv {\\rm arg}\\frac{M_{12}}{\\Gamma _{12}} \n\\; \\; , \\Phi (\\Delta S=1) \\equiv {\\rm arg} \\frac{A_2}{A_0}\n\\eeq\nEq.(\\ref{EPSPOVEREPSTH}) makes two points obvious:\n\\begin{itemize}\n\\item \nDirect CP violation -- $\\epsilon ^{\\prime} \\neq 0$ -- \nrequires a relative phase between the isospin 0 and 2 amplitudes; \ni.e., $K \\rightarrow (\\pi \\pi )_0$ and $K \\rightarrow (\\pi \\pi )_2$ have to exhibit \ndifferent CP properties. \n\\item \nThe observable ratio $\\epsilon ^{\\prime} \/ \\epsilon _K$ is \n{\\em artifically reduced} by the \nenhancement of the $\\Delta I =1\/2$ amplitude, as \nexpressed through $\\omega$. \n\\end{itemize} \nSeveral $\\Delta S=1$ transition operators contribute to \n$\\epsilon ^{\\prime}$ and their renormalization has to be treated \nquite carefully. Two recent detailed analyses yield \n\\cite{BURASPRIME,CIUCHINIPRIME}\n\\begin{equation}}   \\def\\eeq{\\end{equation} \n-2.1 \\cdot 10^{-4} \\leq \\frac{\\epsilon ^{\\prime}}{\\epsilon _K} \\leq \n13.3 \\cdot 10^{-4} \n\\label{BURAS1} \n\\eeq \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{\\epsilon ^{\\prime}}{\\epsilon _K} =  \n(4.6 \\pm 3.0 \\pm 0.4)\\cdot 10^{-4} \n\\label{CIUCHINI} \n\\eeq \nThese results are quite consistent with each other and show  \n\\begin{itemize}\n\\item  \nthat the KM ansatz leads to a prediction typically in the range \n{\\em below} $10^{-3}$, \n\\item \nthat the value could happen to be zero or \neven slightly negative and \n\\item \nthat large theoretical uncertainties persist due to cancellations among \nvarious contributions. \n\\end{itemize} \nThis last (unfortunate) point can be illustrated also by comparing \nthese predictions with older ones made before top quarks were \ndiscovered and their mass measured; those old predictions \n\\cite{FRANZINI} are \nvery similar to Eqs.(\\ref{BURAS1},\\ref{CIUCHINI}), once the now \nknown value of $m_t$ has been inserted. \n\nTwo new experiments running now -- NA 48 at CERN and KTEV at \nFNAL -- and one expected to start up soon -- CLOE at DA$\\Phi $NE -- \nexpect to measure $\\epsilon ^{\\prime}\/\\epsilon _K$ with a \nsensitivity of $\\simeq \\pm 2\\cdot 10^{-4}$. Concerning their future \nresults one can distinguish four scenarios: \n\\begin{enumerate}\n\\item \nThe `best' scenario: \n$\\epsilon ^{\\prime}\/\\epsilon _K \\geq 2 \\cdot 10^{-3}$. One would \nthen \nhave established unequivocally direct CP violation of a strength that \nvery probably reflects the intervention of new physics beyond the \nKM ansatz. \n\\item \nThe `tantalizing' scenario: \n$1\\cdot 10^{-3}\\leq \\epsilon ^{\\prime}\/\\epsilon _K \n\\leq 2 \\cdot 10^{-3}$. It would be tempting to interprete this \ndiscovery of direct CP violation as a sign for new physics -- yet \none could not be sure! \n\\item \nThe `conservative' scenario: \n$\\epsilon ^{\\prime}\/\\epsilon _K \\simeq \n{\\rm few}\\cdot10^{-4} > 0$. This strength of direct CP violation could \neasily be accommodated \nwithin the KM ansatz -- yet no further constraint would \nmaterialize. \n\\item \nThe `frustrating' scenario: \n$\\epsilon ^{\\prime}\/\\epsilon _K \\simeq 0$ within errors! \nNo substantial conclusion could be drawn then concerning the \npresence or absence of direct CP violation, and the allowed KM \nparameter space would hardly shrink. \n\\end{enumerate}\n\n\n\\section{`Exotica'}\nIn this section I will discuss important possible manifestations \nof CP and\/or T violation that are exotic only in the sense that \nthey are unobservably small with the KM ansatz.  \n\n\\subsection{$K_{3\\mu}$ Decays}\nIn the reaction  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nK^+ \\rightarrow \\mu ^+ \\nu \\pi ^0\n\\eeq \none can search for a transverse polarisation of the emerging \nmuons: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nP_{\\perp}^{K^+}(\\mu ) \\equiv \n\\langle \\vec s (\\mu) \\cdot \n(\\vec p (\\mu ) \\times \\vec p (\\pi ^0))\\rangle  \n\\eeq\nwhere $\\vec s$ and $\\vec p$ denote spin and momentum, \nrespectively. \nThe quantity $P_{\\perp}(\\mu )$ constitutes a {\\em T-odd}  \ncorrelation: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\left. \n\\begin{array}{l}\n\\vec p \\Rightarrow - \\vec p \\\\ \n\\vec s \\stackrel{T}{\\Rightarrow} - \\vec s\n\\end{array} \n\\right\\} \\leadsto \nP_{\\perp}(\\mu ) \\stackrel{T}{\\Rightarrow} - P_{\\perp}(\\mu ) \n\\eeq   \nOnce a {\\em non-}vanishing value has been observed for a \nparity-odd correlation one has unequivocally found a manifestation \nof parity violation. From $P_{\\perp}^{K^+}(\\mu ) \\neq 0$ one can \ndeduce that T is violated -- yet the argument is more subtle as can \nbe learnt from the following homework problem. \n\\begin{center} \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $ \\\\ \n{\\em Homework Problem \\#3}: \n\\end{center}\nConsider \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nK_L \\rightarrow \\mu ^+ \\nu \\pi ^- \n\\eeq \nDoes $P_{\\perp}^{K_L}(\\mu ) \\equiv \n\\langle \\vec s(\\mu ) \\cdot (\\vec p(\\mu ) \\times \\vec p(\\pi ^-)) \n\\rangle \\neq 0$ necessarily imply that T invariance does not hold in \nthis reaction?\n\\begin{center}  \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $ \\\\ \n\\end{center}\nData on $P_{\\perp}^{K^+}(\\mu )$ are still consistent with zero \n\\cite{SCHMIDT}: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nP_{\\perp}^{K^+}(\\mu ) = ( -1.85 \\pm 3.60) \\cdot 10^{-3} \\; ; \n\\label{YALE}\n\\eeq \nyet being published in 1981 they are ancient by the standards of our \ndisciplin. \n\nOn general grounds one infers that \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nP_{\\perp}^{K^+}(\\mu ) \\propto {\\rm Im} \\frac{f_-^*}{f_+} \n\\label{POLTH} \n\\eeq \nholds where $f_-\\, [f_+]$ denotes the chirality changing \n[conserving] decay amplitude. Since $f_-$ practically vanishes \nwithin the Standard Model, one obtains a fortiori \n$P_{\\perp}^{K^+}(\\mu )|_{KM} \\simeq 0$. \n\nYet in the presence of charged Higgs fields one has \n$f_- \\neq 0$. CPT implies that \n$P_{\\perp}^{K^+}(\\mu ) \\neq 0$ represents CP violation as \nwell, and actually one of the {\\em direct} variety. A rather \nmodel independant guestimate on how large such an effect \ncould be is obtained from the present bound on \n$\\epsilon ^{\\prime}\/\\epsilon _K$: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nP_{\\perp}^{K^+}(\\mu ) \\leq 20 \\cdot \n(\\epsilon ^{\\prime}\/\\epsilon _K) \\cdot \\epsilon _K \\leq \n10^{-4} \n\\eeq \nwhere the factor 20 allows for the `accidental' reduction of \n$\\epsilon ^{\\prime}\/\\epsilon _K$ by the $\\Delta I=1\/2$ rule: \n$\\omega \\simeq 1\/20$. This bound is a factor of 100 larger \nthan what one could obtain within KM.  It could actually be bigger \nstill since there is a loophole in this generic \nargument: Higgs couplings to leptons could be strongly \nenhanced through a large ratio of vacuum expectation values $v_1$ \nrelative to $v_3$, where $v_1$ controls the couplings to \nup-type quarks and $v_3$ to leptons. \nThen \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nP_{\\perp}^{K^+}(\\mu )|_{Higgs} \\leq {\\cal O}(10^{-3}) \n\\eeq \nbecomes conceivable with the Higgs fields as heavy as \n80 - 200 GeV \n\\cite{GARISTO}. Such Higgs exchanges would be quite insignificant \nfor $K_L \\rightarrow \\pi \\pi $! \n\nSince $K_{\\mu 3}$ studies provide such a unique \nwindow onto Higgs dynamics, I find it mandatory to probe for \n$P_{\\perp}(\\mu ) \\neq 0$ in a most determined way. \nIt is gratifying to note that an on-going KEK experiment will be \nsensitive to $P_{\\perp}(\\mu )$ down to the $10^{-3}$ level -- \nyet I strongly feel one should not stop there, but push \nfurther down to the $10^{-4}$ level. \n\\subsection{Electric Dipole Moments}\nConsider a system -- such as an elementary particle or  \nan atom -- in a weak external electric field $\\vec E$. The \nenergy shift of this system due to the electric field \ncan then be expressed through an expansion in powers \nof $\\vec E$ \\cite{BERN}: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\Delta E = \\vec d \\cdot \\vec E + d_{ij}E_i E_j + \n{\\cal O}(|\\vec E|^3) \n\\label{ESHIFTDIP}\n\\eeq \nwhere summation over the indices $i,j$ is understood. The \ncoefficient $\\vec d$ of the term linear in $\\vec E$ is called \nelectric dipole moment or sometimes permanent \nelectric dipole moment (hereafter referred to as EDM) whereas that \nof the quadratic \nterm is often named an {\\em induced} dipole moment. \n\nFor an elementary object one has \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\vec d = d \\vec j \n\\eeq \nwhere $\\vec j$ denotes its total angular momentum since that \nis the only available vector. Under time reversal one finds  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\begin{array}{l}\n\\vec j \\; \\stackrel{T}{\\Rightarrow} \\; - \\vec j \\\\ \n\\vec E \\; \\stackrel{T}{\\Rightarrow} \\vec E \\; . \n\\end{array} \n\\eeq \nTherefore \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\rm T \\; invariance}  \\leadsto d = 0 \\; ; \n\\eeq \ni.e., such an electric dipole moment has to vanish, unless T is \nviolated (and likewise for parity). \n\nThe EDM is at times confused with an induced electric dipole \nmoment objects can possess due to their internal structure. To \nillustrate that consider an atom with two {\\em nearly degenerate} \nstates of opposite parity: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\bf P}|\\pm \\rangle = \\pm |\\pm \\rangle \\; , \\; \n{\\bf H}|\\pm \\rangle = E_{\\pm} |\\pm \\rangle \\; , \\; E_+ < E_- \\; , \\; \n\\frac{E_- - E_+}{E_+} \\ll 1 \n\\eeq \nPlaced in a constant external electric field $\\vec E$ the states \n$|\\pm \\rangle $ will mix to produce new energy eigenstates; \nthose can be found by diagonalising the matrix of the \nHamilton operator: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nH = \n\\left( \n\\begin{array}{ll}\nE_+ & \\Delta \\\\ \n\\Delta & E_- \n\\end{array} \n\\right) \n\\eeq \nwhere $\\Delta = \\vec d_{ind} \\cdot \\vec E$ with  \n$\\vec d_{ind}$ being the transition matrix element between \nthe $|+\\rangle$ and $|-\\rangle$ states induced by the electric \nfield. The two new energy eigenvalues are \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nE_{1,2} = \\frac{1}{2} (E_+ + E_-) \\pm \n\\sqrt{\\frac{1}{4} (E_+ - E_-)^2 + \\Delta ^2} \n\\eeq \nFor $E_+ \\simeq E_-$ one has \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nE_{1,2} \\simeq \\frac{1}{2} (E_+ + E_-) \\pm |\\Delta | \\; ; \n\\eeq \ni.e., the energy shift appears to be linear in $\\vec E$: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\Delta E = E_2 - E_1 = 2 |\\vec d_{ind} \\cdot \\vec E| \n\\label{FAKELINEAR} \n\\eeq   \nYet with $\\vec E$ being sufficiently small one arrives at \n$4(\\vec d_{ind}\\cdot \\vec E)^2 \\ll (E_+ - E_-)^2$ and therefore \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nE_1 \\simeq E_- + \\frac{(\\vec d_{ind}\\cdot \\vec E)^2}{E_- - E_+} \\; , \\; \nE_2 \\simeq E_+ - \\frac{(\\vec d_{ind}\\cdot \\vec E)^2}{E_- - E_+} \\; ; \n\\eeq\ni.e., the induced energy shift is {\\em quadratic} in $\\vec E$ \nrather than  \nlinear and therefore does {\\em not} imply T violation! The distinction \nbetween an EDM and an induced electric dipole moment is somewhat \nsubtle -- yet it can be established in an unequivocal way by \nprobing for a linear Stark effect with weak electric fields. A more \ncareful look at Eq.(\\ref{FAKELINEAR}) already indicates that. \nFor the energy shift stated there does not change under \n$\\vec E \\Rightarrow - \\vec E$ as it should for an EDM which also \nviolates parity! \n\nThe data for neutrons read: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nd_n = \\left\\{   \n\\begin{array}{l} \n(-3 \\pm 5)\\cdot 10^{-26} \\; ecm \\; \\; \\; \\; \\; {\\rm ILL} \\\\ \n(2.6 \\pm 4 \\pm 1.6)\\cdot 10^{-26} \\; ecm \\; \\; \\; \\; \\; {\\rm LNPI}  \n\\end{array}\n\\right. \n\\label{EDMNEUT}\n\\eeq \nThese numbers and the experiments leading to them are very \nimpressive: \n\\begin{itemize} \n\\item \nOne uses neutrons emanating from a reactor and \nsubsequently cooled down \nto a temperature of order $10^{-7}$ eV. This is comparable to the \nkinetic energy a neutron gains when dropping 1 m in the \nearth's gravitational field. \n\\item \nExtrapolating the ratio between the neutron's radius \n-- $r_N \\sim 10^{-13}$ cm -- with its EDM of no more than \n$10^{-25}$ ecm to the earth's case, one would say that it \ncorresponds to a situation where one has searched for a displacement \nin the earth's mass distribution of order $10^{-12} \\cdot r_{earth} \n\\sim 10^{-3} {\\rm cm} = 10$ microns!  \n\\end{itemize} \n\nA truly dramatic increase in sensitivity for the \n{\\em electron's} EDM has \nbeen achieved over the last few years: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nd_e = (-0.3 \\pm 0.8) \\cdot 10^{-26} \\; \\; e \\, cm \n\\label{EDMEL}\n\\eeq \nThis quantity is searched for through measuring electric dipole \nmoments of {\\em atoms}. At first this would seem to be a \nlosing proposition theoretically: for according to Schiff's \ntheorem an atom when placed inside an external electric \nfield gets deformed in such a way that the electron's EDM is \ncompletely shielded; i.e., $d_{atom} = 0$. This theorem \nholds true in the nonrelativistic limit, yet is vitiated by relativistic \neffects. Not surprisingly the latter are particularly large for \nheavy atoms; one would then expect the electron's \nEDM to be only partially shielded: $d_{atom} = S\\cdot d_e$ with \n$S < 1$. Yet amazingly -- and highly welcome of \ncourse -- the electron's EDM can actually get magnified by two to three \norders of magnitude in the atom's electric dipole moment; for \nCaesium one has \\cite{BERN} \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nd_{Cs} \\simeq 100 \\cdot d_e \n\\eeq\nThis enhancement factor is the theoretical reason behind the \ngreatly improved sensitivity for $d_e$ as expressed through \nEq.(\\ref{EDMEL}); the other one is experimental, namely the great \nstrides made by laser technology applied to atomic physics. \n\nThe quality of the number in Eq.(\\ref{EDMEL}) can be illustrated \nthrough a comparison with the electron's magnetic moment. \nThe electromagnetic form factor $\\Gamma _{\\mu}(q)$ \nof a particle like the electron evaluated at momentum \ntransfer $q$ contains two tensor terms: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nd_{atom} =  \n\\frac{1}{2m_e}\\sigma _{\\mu \\nu} q^{\\nu}\\left[ i F_2(q^2) + \nF_3(q^2) \\gamma _5 \\right]  + ... \n\\eeq  \nIn the nonrelativistic limit one finds for the EDM: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nd_e = - \\frac{1}{2m_e} F_3(0) \n\\eeq \nOn the other hand one has \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{1}{2}(g-2) = \\frac{1}{e} F_2(0) \n\\eeq \nThe {\\em precision} with which $g-2$ is known for the \nelectron -- $\\delta [(g-2)\/2] \\simeq 10^{-11}$ -- \n(and which represents one of the great success stories of \nfield theory) corresponds to an {\\em uncertainty} in the electron's \n{\\em magnetic} moment \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\delta \\left[ \\frac{1}{2m_e} F_2(0)\\right] \\simeq \n2\\cdot 10^{-22} \\; \\; e\\, cm\n\\eeq \nthat is several orders of magnitude larger than the bound on its \nEDM! \n\nSince the EDM is, as already indicated above, described by a \ndimension-five operator in the Lagrangian \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\cal L}_{EDM} = - \\frac{i}{2} d \n\\bar \\psi \\sigma _{\\mu \\nu}\\gamma _5 \\psi F^{\\mu \\nu} \n\\eeq\nwith $F^{\\mu \\nu}$ denoting the electromagnetic field strength \ntensor, one can calculate $d$ within a given theory of CP violation \nas a finite quantity.  Within the KM ansatz one finds that the \nneutron's EDM is zero for all practical purposes \n\\footnote{I ignore here the Strong CP Problem, which is \ndiscussed in the next section.}:  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\left. d_N\\right| _{KM} < 10^{-30} \\; \\; e\\, cm \n\\eeq\nand likewise for $d_e$. Yet again that is due to very specific features \nof the KM mechanism and the chirality structure of the Standard \nModel. In alternative models -- where CP violation enters \nthrough {\\em right}-handed currents or a non-minimal \nHiggs sector (with or without involving SUSY) -- one finds \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\left. d_N\\right| _{New\\; Physics} \\sim \n10^{-27} - 10^{-28} \\; \\; e\\, cm \n\\eeq  \nas reasonable benchmark figures. \n\n\\section{The Strong CP Problem}\n\\subsection{The Problem}\n\nIt is often listed among the attractive features of QCD that it `naturally' conserves \nbaryon number, flavour, parity and CP. Actually the last two points \nare not quite true,   \nwhich had been overlooked for \nsome time \nalthough it can be seen in different ways \n\\cite{PECCEI}. Consider \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\cal L}_{eff} = \\sum _q \\bar q \\left( i \\not{D} - m_q\\right) q \n- \\frac{1}{4} G \\cdot G + \n\\frac{\\theta g_S^2}{32 \\pi ^2} \nG\\cdot \\tilde G \n\\label{QCDCP} \n\\eeq \nwhere $D_{\\mu}$, \n$G$ and \n$\\tilde G$ denote the covariant derivative, the \ngluon field strength tensor and its dual, respectively:\n\\begin{equation}}   \\def\\eeq{\\end{equation} \nD_{\\mu} = \\partial _{\\mu} + i g_S A^i_{\\mu}t^i \n\\eeq  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nG_{\\mu \\nu} \\equiv G_{\\mu \\nu}^i t^i \\; \\; , \\; \\; \nG_{\\mu \\nu} ^i = \\partial _{\\mu} A^i_{\\nu} - \n\\partial _{\\nu} A^i_{\\mu} + g_S if_{ijk}A^j_{\\mu}A^k_{\\nu} \\; \\; \n, \\; \\; \\tilde G_{\\mu \\nu} \\equiv \\frac{i}{2} \n\\epsilon _{\\mu \\nu \\alpha \\beta} G_{\\alpha \\beta} \n\\eeq  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nG\\cdot G \\equiv G_{\\mu \\nu} G_{\\mu \\nu} \\; \\; , \n\\; \\; \nG\\cdot \\tilde G \\equiv G_{\\mu \\nu} \\tilde G_{\\mu \\nu} \n\\eeq\nIn adding the operator $G \\cdot \\tilde G$ \nto the usual QCD Lagrangian we have followed a general \ntenet of quantum field theory:  any Lorentz scalar gauge invariant \noperator of dimension four has to be included in the Lagrangian \nunless there is a specific reason -- in particular a symmetry \nrequirement -- that enforces its absence. For otherwise \nradiative corrections will resurrect such an operator with \na (logarithmically) divergent coefficient! \n\nSuch an operator exists also in an \nabelian gauge theory like QED where the field strength tensor \ntakes on a simpler form: $F_{\\mu \\nu} = \n\\partial _{\\mu} A_{\\nu} - \\partial _{\\nu} A_{\\mu}$.  One \nthen finds \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nF_{\\mu \\nu} \\tilde F_{\\mu \\nu} = \n\\partial _{\\mu} K^{QED}_{\\mu} \\; , \n\\; \\; K^{QED}_{\\mu} = 2 \\epsilon _{\\mu \\alpha \\beta \\gamma } \nA_{\\alpha}  \\partial _{\\beta} A_{\\gamma} \\; ; \n\\eeq \ni.e., this extra term can be reduced to a total \nderivative which is usually dropped without further ado \nas physically irrelevant. \n\nFor nonabelian gauge theories one obtains \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nG_{\\mu \\nu}\\tilde G_{\\mu \\nu} = \\partial _{\\mu} K_{\\mu} \\; , \n\\; \\; K_{\\mu} = 2 \\epsilon _{\\mu \\alpha \\beta \\gamma } \n\\left( A_{\\alpha}  \\partial _{\\beta} A_{\\gamma} + \n\\frac{2}{3} i g_S A_{\\alpha}A_{\\beta} A_{\\gamma} \n\\right) \\; . \n\\label{KTOPO} \n\\eeq \nThe extra term is still a total derivative and our first reaction \nwould be to just drop it for that very reason. \nAlas this time we would be wrong in doing so!  \nLet us recapitulate the usual argument. If a term in the \nLagrangian can be expressed as a total divergence \nlike $\\partial _{\\mu}K_{\\mu}$ than its contribution to the \n{\\em action} which determines the dynamics can be \nexpressed as the integral of the current $K$ over a surface \nat infinity. Yet with physical observables having to \nvanish rapidly at infinity to yield finite values for energy etc., \nsuch integrals are expected to yield zero. The field strength indeed \ngoes to zero at infinity -- but not necessarily the gauge potentials \n$A_{\\mu}$! The field configuration at large \nspace-time distances has to approach that of a \nground state for which $G_{\\mu \\nu} = 0$ holds. Yet the \nlatter property \ndoes not suffice to define the ground state {\\em uniquely}: \nit \nstill allows ground states to differ by pure gauge configurations \nwhich obviously satisfy $G_{\\mu \\nu} = 0$. \nThis is also true for abelian gauge \ntheories, yet remains without dynamical significance. The structure  \nof nonabelian gauge theories on the other hand is much more \ncomplex and they possess an infinity of {\\em inequivalent}  \nstates defined by $G_{\\mu \\nu}=0$ \n\\cite{REBBI}. Their differences can be expressed \nthrough topological characteristics of their gauge field \nconfigurations.  To be more precise: these states can be characterised \nby an integer, the so-called {\\em winding number}; accordingly they \nare denoted by $|n \\rangle$. They are {\\em not} gauge \ninvariant. Not surprisingly \nthen transitions between states \n$|n_1\\rangle$ and $|n_2\\rangle $ with $n_1 \\neq n_2$ \ncan take place. The net change $\\Delta n$ in winding number between \n$t=-\\infty$ and $t=\\infty$ is described by their $K$ charge, \nthe space integral of the zeroth component of the current \n$K_{\\mu}$ defined in Eq.(\\ref{KTOPO}).  \nA gauge invariant state is constructed as a \nlinear superposition of the states $|n \\rangle$ labeled by a real \nparameter $\\theta$  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n|\\theta \\rangle = \\sum _n e^{-i n\\theta }|n\\rangle \\; . \n\\eeq \nOne easily shows that for a \ngauge invariant operator $O_{g.inv.}$ \n$\\matel{\\theta}{O_{g.inv.}}{\\theta ^{\\prime}}=0$ \nholds if $\\theta \\neq \\theta ^{\\prime}$. \nWe thus see that the state space of QCD consists of \n{\\em disjoint} sectors built up from ground states $|\\theta \\rangle $.  \n\nFor vacuum-to-vacuum transitions one then finds \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\langle \\theta _+|\\theta _-\\rangle = \n\\sum _{n,m} e^{i\\theta (m - n)}\\langle m_+|n_-\\rangle = \n\\sum _{\\Delta n}e^{i \\Delta n \\theta}\n\\sum _n \\langle (n+\\Delta n)_+| n_- \\rangle \\;  , \n\\; \\; \\Delta n = m - n \\; , \n\\eeq \nwhich can be reformulated in the path integral formalism \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\langle \\theta _+|\\theta _-\\rangle = \n\\sum _{\\Delta n} \\sum _{\\rm fields} \ne^{i \\int d^4x {\\cal L}_{eff}} \n\\delta (\\Delta n - \\frac{g_S^2}{16 \\pi ^2} \\int d^4x G \\cdot \n\\tilde G) \\; ;  \n\\eeq \ni.e., the $G \\cdot \\tilde G$ term in Eq.(\\ref{QCDCP}) acts as a \nLagrangian multiplier implementing the change in winding \nnumber $\\Delta n$. \n\nThis is easily generalized to any transition amplitude, and \nthe situation can be summarized as follows: \n\\begin{itemize}\n\\item \nThere is an infinity set of {\\em inequivalent} groundstates in QCD \nlabeled by a real parameter $\\theta$. \n\\item \nThe dependance of observables on $\\theta$ can be determined by \nemploying the {\\em effective} Lagrangian of \nEq.(\\ref{QCDCP}). \n\n\\end{itemize} \n\nThe problem with this additional term in the Lagrangian is \nthat $G\\cdot \\tilde G$ -- in contrast to $G \\cdot G$ -- \nviolates both parity and time reversal invariance!   \nThis is best seen by expressing $G_{\\mu \\nu}$ and its dual \nthrough the colour electric and colour magnetic fields $\\vec E$ \nand $\\vec B$, respectively: \n\\begin{eqnarray} \nG\\cdot G \\propto |\\vec E|^2 +|\\vec B|^2 \n&\\stackrel{{\\bf P}, {\\bf T}}{\\Longrightarrow}& \n|\\vec E|^2 +|\\vec B|^2 \\\\  \nG \\cdot \\tilde G \\propto 2\\vec E \\cdot \\vec B \n&\\stackrel{{\\bf P}, {\\bf T}}{\\Longrightarrow}& \n- 2\\vec E \\cdot \\vec B\n\\end{eqnarray}   \nsince  \n\\begin{eqnarray} \n\\vec E \\stackrel{{\\bf P}}{\\Longrightarrow} - \\vec E \n\\; \\; \\; &,& \\; \\; \\;      \n\\vec B \\stackrel{{\\bf P}}{\\Longrightarrow}  \\vec B \\\\    \n\\vec E \\stackrel{{\\bf T}}{\\Longrightarrow}  \\vec E  \n\\; \\; \\; &,& \\; \\; \\;      \n\\vec B \\stackrel{{\\bf T}}{\\Longrightarrow} - \\vec B \\; ; \n\\end{eqnarray}  \ni.e., for $\\theta \\neq 0$ neither parity nor time reversal invariance \nare fully conserved by QCD. This is the {\\em Strong CP Problem}. \n\nThe problem which resides in gluodynamics spreads into \nthe quark sector through the `chiral'  anomaly \n\\cite{ABJANOM}: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\partial _{\\mu} J_{\\mu}^5 = \n\\partial _{\\mu} \\sum _q \\bar q_L \\gamma _{\\mu}q_L = \n\\frac{g_S^2}{32\\pi ^2} G \\cdot \\tilde G \\neq 0 \\; ; \n\\label{TRIANOM} \n\\eeq   \ni.e., the axial current of massless quarks, which is conserved \n{\\em classically}, ceases to be so on the quantum level \n\\footnote{This is why it is called an anomaly.}. This \nchiral anomaly is also called the `triangle' anomaly because it \nis produced by a diagram with a triangular fermion loop.  \n\nThere are two further aspects to the anomaly expressed in \nEq.(\\ref{TRIANOM}): \n\\begin{itemize}\n\\item \nThe anomaly actually solves one long standing puzzle of \n{\\em strong} dynamics, the `U(1) Problem': \nIn the limit of massless \n$u$ and $d$ quarks QCD would appear to have a {\\em global} \n$U(2)_L \\times U(2)_R$ invariance. While the vectorial part \n$U(2)_{L+R}$ is a manifest symmetry, the axial part \n$U(2)_{L-R}$ is spontaneously realized leading to the emergence \nof four Goldstone bosons. In the presence of quark masses those \nbosons acquire a mass as well. The pions readily play the part, \nbut the $\\eta$ meson does not \n\\footnote{An analogous discussion can be given with $s$ quarks included. The spontaneous breaking of the global \n$U(3)_{L-R}$ symmetry leads to the existence of nine \nGoldstone bosons, yet the $\\eta ^{\\prime}$ meson is far \ntoo heavy for this role.}! Yet from the anomaly one infers that \ndue to quantum corrections the axial $U(1)_{L-R}$ was never there \nin the first place even for massless quarks: therefore only \nthree Goldstone bosons are predicted, the pions! \n\\item \nOn the other hand the anomaly aggravates the \nStrong CP Problem \nwhen electroweak dynamics are included. For the quarks \nacquire their masses from the Higgs mechanism driving \nthe phase transition \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nSU(2)_L \\times U(1) \\leadsto U(1)_{QED} \n\\eeq \nThe resulting quark mass ${\\cal M}_{quark}$ matrix cannot be expected to be diagonal and Hermitian ab initio; \nit will have to be diagonalized through  \nchiral rotations of the quark fields: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\cal M}_{quark}^{diag} = \nU_R^{\\dagger} {\\cal M}_{quark} U_L\n\\eeq  \nExactly because of the axial anomaly this induces an additional \nterm in the Lagrangian of the Standard Model: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\cal L}_{SM,eff} = {\\cal L}_{QCD} + {\\cal L}_{SU(2)_L \\times U(1)} \n+ \\frac{\\bar \\theta g_S^2}{32 \\pi ^2} G \\cdot \\tilde G \n\\eeq \nwhere \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\bar \\theta = \\theta _{QCD} + \\Delta \\theta _{EW}\\; , \n\\; \\;  \\Delta \\theta _{EW} = {\\rm arg \\; det} U_R^{\\dagger} U_L \n\\eeq \nSince the electroweak sector has to contain sources of \nCP violation other than $G \\cdot \\tilde G$, the second term \nin $\\bar \\theta$ has no a priori reason to vanish. \n\\end{itemize}\n\n\n\\subsection{The Neutron Electric Dipole Moment}\nSince the gluonic operator $G \\cdot \\tilde G$ does not change \nflavour one suspects right away that its most noticeable impact \nwould be to generate an electric dipole moment (EDM) \nfor neutrons. \nThis is indeed the case, yet making this connection more concrete \nrequires a more sophisticated argument. In the context of the Strong CP Problem one views the neutron \nEDM -- $d_N$ -- as due to the photon coupling to a virtual \nproton or pion in a fluctuation of the neutron: \n$ \nn \\; \\Longrightarrow \\; p^* \\pi ^* \\; \\Longrightarrow  \\; n\n$.  \nOf the two effective pion nucleon couplings in this one-loop process \none is produced by ordinary strong forces and conserves \nP, T and CP; \nthe other one is induced by $G\\cdot \\tilde G$. The first \nestimate was obtained by Baluni \n\\cite{BALUNI} in a nice paper using bag model \ncomputations of the transition amplitudes between \nthe neutron and its excitations: \n$d_N \\simeq 2.7 \\cdot 10^{-16} \\; \\bar \\theta \\; e cm$. \nIn \\cite{CREWTHER} \nchiral perturbation theory instead was employed: \n$d_N \\simeq 5.2 \\cdot 10^{-16} \\; \\bar \\theta \\; e cm$. \nMore recent estimates yield values in roughly the same range: \n$d_N \\simeq (4\\cdot 10^{-17} \\div 2\\cdot 10^{-15}) \\bar \\theta \n\\; e\\; cm$ \\cite{PECCEI}. Hence \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nd_N \\sim {\\cal O}(10^{-16} \\bar \\theta ) \\; \\; e \\; cm \n\\eeq \nand \none infers  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nd_N \\leq 1.1 \\cdot 10^{-25} \\; \\; e \\; cm \\; \\;  \\; 95\\% \\; C.L. \n\\; \\; \\; \\leadsto \\; \\; \\; \n\\bar \\theta < 10^{-9 \\pm 1} \n\\label{THETABOUND}\n\\eeq \nAlthough $\\theta _{QCD}$ is a QCD parameter it might not necessarily \nbe of order unity; nevertheless its truly tiny size begs for \nan explanation. The only kind of explanation that is usually \naccepted as `natural' in our community is one based on \nsymmetry. Such an explanation has been put forward based \non a so-called Peccei-Quinn symmetry. Yet before we start speculating too wildly, we want \nto see whether there are no more mundane explanations.   \n\n\\subsection{Are There Escape Hatches?}\nOne could argue that the Strong CP Problem is fictitious \nusing one of two lines of reasoning: \n\\begin{itemize}\n\\item \nBeing the coefficient of a dimension-four operator \n$\\bar \\theta$  \ncan in general \\footnote{Exceptions will be mentioned below.} \nbe renormalized to any value, including zero. \nThis is technically correct; however $\\theta \\leq {\\cal O}(10^{-9})$ \nis viewed as highly `unnatural':  \n\\begin{itemize} \n\\item \nA priori there is no reason why $\\theta _{QCD}$ and \n$\\Delta \\theta _{EW}$ should practically vanish. \n\\item \nEven if $\\theta _{QCD} = 0 = \\Delta \\theta _{EW}$ were set \n{\\em by fiat} quantum corrections to $\\Delta \\theta _{EW}$ are \ntypically much larger than $10^{-9}$ and ultimately actually \ninfinite. \n\\item \nTo expect that $\\theta _{QCD}$ and $\\Delta \\theta _{EW}$ \ncancel as to render $\\bar \\theta$ sufficiently tiny would \nrequire fine tuning of a kind which would have to strike even \na skeptic as unnatural. For $\\theta _{QCD}$ reflects dynamics \nof the strong sector and $\\Delta \\theta _{EW}$ that of the \nelectroweak sector. \n\\item \nIn models where CP symmetry is realized in a spontaneous \nfashion one has $\\theta _{tree} = 0$ and \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\bar \\theta = \\delta \\theta _{ren} \n\\eeq \nturns out to be a finite and calculable quantity that has no \napparent reason to be smaller than, say, $10^{-4}$. \n\\end{itemize}\n\\item \nA more respectable way out is provided by the \nfollowing observation: if one of the quark masses \nvanishes   \nthe resulting chiral invariance would remove any \n$\\bar \\theta$ dependance of observables by rotating it \n-- through the anomaly -- into the quark mass matrix with its zero \neigenvalue. However most authors argue quite forcefully that \nneither the up quark nor a fortiori the down quark mass can \nvanish \\cite{PECCEI}: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nm_d (1\\; \\,\\mbox{GeV}) > m_u (1\\; {\\,\\mbox{GeV}}) \\simeq 5\\; \\,\\mbox{MeV} \n\\eeq \nwhere the notation shows that one has to use the running mass \nevaluated at a scale of 1 GeV. \n\\end{itemize} \n\n\n\\subsection{Peccei-Quinn Symmetry}\nAs just argued $\\bar \\theta \\leq {\\cal O}(10^{-9})$ \ncould hardly come about accidentally; an organizing principle had \nto arrange various contributions and corrections in such a way as to \nrender the required cancellations.  There is the general \nphilosophy that such a principle has to come from an underlying \nsymmetry. We have already sketched such an approach: a global chiral invariance allows to rotate the \ndependance on $\\bar \\theta$ away; we failed however in our \nendeavour because this symmetry is broken by $m_q \\neq 0$. Is \nit possible to invoke some other variant of chiral symmetry for \nthis purpose even if it is spontaneously broken?  \nOne \nparticularly intriguing ansatz is to re-interprete a physical \nquantity that is conventionally taken to be a constant as a \ndynamical degree of freedom that relaxes itself to a certain \n(desired) value in response to forces acting upon it. One early \nexample is provided by the original Kaluza-Klein theory \n\\cite{KALUZA} \ninvoking a six-dimensional `space'-time manifold: two compactify \ndynamically and thus lead to the quantization of electric and \nmagnetic charge. \n\nSomething similar has been suggested by \nPeccei and Quinn \\cite{PQ}. They augmented the Standard Model \nby a global $U(1)_{PQ}$ symmetry -- now referred to as \nthe Peccei-Quinn symmetry -- that is axial. The spontaneous breaking of this symmetry gives rise to a Goldstone boson -- named the axion -- \nwith zero mass on the Lagrangian level. Goldstone couplings \nto other fields usually have to be derivative, i.e. involve \n$\\partial _{\\mu}a(x)$, but not $a(x)$ directly. Since \n$U(1)_{PQ}$ is axial it exhibits a triangle anomaly again; \nthis is implemented in the effective Lagrangian by having a \nterm linear in the axion field coupled to $G\\cdot \\tilde G$: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\cal L}_{eff} = {\\cal L}_{SM} + \\frac{g_S^2\\bar \\theta}{32\\pi ^2} \nG \\cdot \\tilde G + \\frac{g_S^2}{32\\pi ^2}\\frac{\\xi}{\\Lambda _{PQ}} \naG \\cdot \\tilde G - \\frac{1}{2}\\partial _{\\mu} a \\partial _{\\mu} a \n+ {\\cal L}_{int}(\\partial _{\\mu} a,\\psi ) \n\\label{LAGAXION}\n\\eeq\nThe size of the parameters $\\Lambda _{PQ}$ and \n$\\xi$ and the form of \n${\\cal L}_{int}(\\partial _{\\mu} a,\\psi )$ describing the \n(purely derivative) coupling of the axion field to other fields \n$\\psi$ \ndepend on how the Peccei-Quinn symmetry is specifically \nrealized. \n\nThe term $a G \\cdot \\tilde G$ represents an {\\em explicit} \nbreaking of the $U(1)_{PQ}$ symmetry. This gives \nrise to an axion mass. Yet the primary role of \n$a G \\cdot \\tilde G$ is to make the $ G \\cdot \\tilde G$ term \ndisappear from the effective Lagrangian. $U(1)_{PQ}$ \ninvariance being realized {\\em spontaneously} means \nthat $a(x)$ acquires a vacuum expectation value (=VEV) \n$\\langle a\\rangle$; the physical axion excitations  \nare then described by the shifted field \n$ \na_{phys}(x) = a(x) - \\langle a \\rangle \n$  \nand one rewrites Eq.(\\ref{LAGAXION}) as follows: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\cal L}_{eff} = {\\cal L}_{SM} + \\frac{g_S^2}{32\\pi ^2} \n\\bar \\theta  G \\cdot \\tilde G - \n\\frac{1}{2}\\partial _{\\mu} a_{phys} \\partial _{\\mu} a_{phys}  \n+\\frac{g_S^2}{32\\pi ^2}\\frac{\\xi}{\\Lambda _{PQ}} \na_{phys} G \\cdot \\tilde G \n+ {\\cal L}_{int}(\\partial _{\\mu} a_{phys},\\psi ) \n\\eeq \nwhere now \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\bar \\theta  = \\theta _{QCD} + \n{\\rm arg\\, det} U_R^{\\dagger} U_L - \n\\frac{\\langle a\\rangle }{f_a} \\; , \n\\; \\; f_a = \\frac{\\Lambda _{PQ}}{\\xi}\\; ; \n\\eeq \ni.e., the size of the \ncoefficient of the $G \\cdot \\tilde G$ operator is \ndetermined by the VEV of the axion field. \n\nThe term $a G \\cdot \\tilde G$ generates an effective potential \nfor the axion field; its minimum defines the ground state: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\langle \\frac{\\partial V_{eff}}{\\partial a}\\rangle \\equiv \n- \\frac{\\xi }{\\Lambda _{PQ}} \\frac{g_S^2}{32 \\pi ^2} \n\\langle G  \\cdot \\tilde G \\rangle = 0 \n\\eeq \nThat means that the term $a G \\cdot \\tilde G$ -- the reincarnation \nof the anomaly -- singles out one of the previously degenerate \n$\\bar \\theta$ states as the true ground state. This is not \nsurprising since $a G \\cdot \\tilde G$ is {\\em not} invariant \nunder $U(1)_{PQ}$. It is a pleasant surprise, though, that \nfor this lowest energy state $G\\cdot \\tilde G$ settles into a \n{\\em vanishing} expectation value thus banning \nthe Strong CP Problem {\\em dynamically}. \n\n\\subsection{The Dawn of Axions -- and Their Dusk?}\nRather than ending here the story contains another twist or \ntwo. The breaking of $U(1)_{PQ}$ gives rise to a Nambu-Goldstone \nboson. Actually the axion is, as already mentioned, a \npseudo-Nambu-Goldstone \nboson; for it acquires a mass due to the anomaly: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nm_a^2 \\sim \\frac{G \\cdot \\tilde G}{\\Lambda _{PQ}^2} \n\\sim {\\cal O}\\left(  \\frac{\\Lambda _{QCD}^4}{\\Lambda _{PQ}^2} \n\\right) \n\\eeq \nSince one expects on general grounds \n$\\Lambda _{PQ} >> \\Lambda _{QCD}$ one is dealing with a \nvery light boson. The question is how light would the \naxion be. \n\nThe electroweak scale $v_{EW} = \n\\left( \\sqrt{2}G_F\\right) ^{-\\frac{1}{2}} \\simeq 250$ GeV \nprovides the discriminator for two scenarios: \n\\begin{itemize} \n\\item \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\Lambda _{PQ} \\sim v_{EW} \n\\eeq \nIn that case axions can or even should be seen in accelerator based \nexperiment. Such scenarios are referred to as {\\em visible} axions. \n\\item \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\Lambda _{PQ} \\gg v_{EW} \n\\eeq \nSuch axions could not be \nfound in accelerator based experiments; therefore they are \ncalled {\\em invisible} scenarios. Yet that does not mean that \nthey necessarily escape detection! For they could be of \ngreat significance for the formation of stars, whole galaxies \nand even the universe. \n\n\\end{itemize} \n\n\\subsubsection{Visible Axions}\nThe simplest scenario involves two $SU(2)_L$ doublet Higgs fields \nthat possess opposite hypercharge \n\\footnote{In the Standard Model the Higgs doublet and its \ncharge conjugate fill this role.}. \nThey also carry a $U(1)$ \ncharge {\\em in addition} to the hypercharge; this second (and \nglobal) $U(1)$ is identified with the PQ symmetry, and the axion is \nits pseudo-Nambu-Goldstone boson. \nThe anomaly  induces a mass for the axion: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nm_a \\simeq \\frac{m_{\\pi}F_{\\pi}}{v}N_{fam} \n\\left(  x + \\frac{1}{x} \\right)  \n\\frac{\\sqrt{m_um_d}}{(m_u + m_d)} \\simeq \n25 \\; N_{fam} \\left(  x + \\frac{1}{x} \\right) \\; \\; {\\rm KeV} \n\\eeq \nwhere $N_{fam}$ denotes the number of families. \nSuch an axion is almost certainly much lighter than the \npion. Depending on the axion's mass two cases have to be \ndecided:\n\\begin{itemize}\n\\item \nIf $m_a > 2 m_e$, the axion decays very rapidly into electrons and \npositrons: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\tau (a \\rightarrow e^+ e^-) \\simeq 4 \\cdot 10^{-9}\n\\left( \\frac{1\\; {\\rm MeV}}{m_a} \\right) \n\\frac{x^2 \\; {\\rm or} \\; 1\/x^2}{\\sqrt{1- \\frac{4m_e^2}{m_a^2}}} \\; \\; {\\rm sec} \n\\eeq \n\\item \nIf on the other hand $m_a < 2 m_e$ then the axion decays \nfairly slowly into two photons:  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\tau (a \\rightarrow \\gamma \\gamma ) \\sim \n{\\cal O}\\left( \\frac{100 \\; {\\rm KeV}}{m_a}  \n\\right) \\; \\; {\\rm sec} \n\\eeq \n\\end{itemize} \nI have presented here a very rough sketch of scenarios \nwith visible axions since we can confidently declare that they \nhave been  \nruled out experimentally.  They have been looked for in beam \ndump experiments -- without success. Yet the more telling \nblows have come from searches in rare decays:\n\\begin{itemize}\n\\item \nFor long-lived axions -- $m_a < 2 m_e$ -- one expects a \ndominating contribution to $K^+ \\rightarrow \\pi ^+ + \\; nothing$ from \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nK^+ \\rightarrow \\pi ^+ + a \n\\label{KTOAXION}\n\\eeq \nwith the axion decayig well outside the detector. For the two-body \nkinematics of Eq.(\\ref{KTOAXION}) one has very tight bounds from \npublished data \\cite{ADLER}: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nBR(K^+ \\rightarrow \\pi ^+ X^0) < 5.2 \\cdot 10^{-10} \\; \\; \\; 90\\%\\; C.L.\n\\eeq\nfor $X^0$ being a practically massless and noninteracting particle. \nTheoretically one would expect:\n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\left. BR(K^+ \\rightarrow \\pi ^+ a) \\right|_{theor} \\sim \n3\\cdot 10^{-5} \\cdot \\left( x + 1\/x \\right) ^{-2} \n\\label{KTOAXTH}\n\\eeq\nAlthough Eq.(\\ref{KTOAXTH}) does not represent a precise prediction \nthe discrepancy between expectation and observation is \nconclusive. \n\\item \nOne arrives at the same conclusion that \n{\\em long-lived visible} axions \ndo not exist from the absence of {\\em quarkonia} decay \ninto them: neither $J\/\\psi \\rightarrow a\\gamma$ nor \n$\\Upsilon \\rightarrow a \\gamma$ has been seen. \n\\item \nThe analysis is a bit more involved for \n{\\em short-lived} axions -- $m_a > 2 m_e$. Yet again \ntheir absence has been established through a combination \nof experiments. Unsuccessful searches for \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\pi ^+ \\rightarrow a e^+ \\nu \n\\eeq  \nfigure prominently in this endeavour. Likewise the \n{\\em absence} of axion driven \n{\\em nuclear de-excitation} has been \nestablished on a level that appears to be conclusive \n\\cite{NUCLEAR}. \n\\subsubsection{Invisible Axions}\nInvisible axion scenarios involve a complex \nscalar field $\\sigma$ that \n(i) is an $SU(2)_L$ {\\em singlet}, \n(ii) \ncarries a PQ charge and \n(iii)  \npossesses a huge VEV $\\sim \\Lambda _{PQ} >> v_{ew}$. \n\\end{itemize}\nThe reasons underlying those requirements should be obvious; \nthey can be realized in two distinct (sub-)scenarios: \n\\begin{enumerate}\n\\item \nOnly presumably very heavy new quarks carry a PQ charge. This \nsituation is referred to as {\\em KSVZ} axion \\cite{KSVZAX}. The minimal \nversion can do with a single $SU(2)_L$ Higgs doublet. \n\\item \nAlso the known quarks and leptons carry a PQ charge. Two \n$SU(2)_L$ Higgs doublets are then required in addition to \n$\\sigma$. The fermions do not couple directly to \n$\\sigma$, yet become sensitive to PQ breaking through \nthe Higgs potential. This is referred to as {\\em DFSZ} axion \n\\cite{DFSZAX}. \n\\end{enumerate} \nFrom current algebra one infers for the axion mass in either \ncase \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nm_a \\simeq 0.6 \\; {\\rm eV} \\cdot \\frac{10^7 \\; {\\rm GeV}}{f_a} \n\\eeq \nThe most relevant coupling of such axions is to two photons \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\cal L}( a \\rightarrow \\gamma \\gamma ) = \n- \\tilde g_{a\\gamma \\gamma} \\frac{\\alpha}{\\pi} \n\\frac{a(x)}{f_a} \\vec E \\cdot \\vec B \\; ,  \n\\eeq \nwhere $\\tilde g_{a\\gamma \\gamma}$ is a model dependant \ncoefficient of order unity. \n\nAxions with such tiny masses have lifetimes easily in excess \nof the age of the universe. Also their couplings to other \nfields are so minute that they would not betray their \npresence -- hence their name {\\em invisible} axions -- \nunder ordinary circumstances! \nYet in astrophysics and cosmology more favourable \nextra-ordinary conditions can arise. \n\nThrough their couplings to electrons axions would provide \na cooling mechanism to {\\em stellar evolution}. Not \nsurprisingly their greatest impact occurs for the lifetimes of \nred giants and the supernovae like SN 1987a. The actual \nbounds depend on the model -- whether it is a \nKSVZ or DFSZ axion -- but relatively mildly only. \nAltogether astrophysics tells us that {\\em if} \naxions exist one has \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nm_a < 3 \\cdot 10^{-3} \\; \\; {\\rm eV} \n\\eeq \n\n{\\em Cosmology} on the other hand provides us with  a \n{\\em lower} bound through a very intriguing line of \nreasoning. At temperatures $T$ above $\\Lambda _{QCD}$ \nthe axion is massless and all values of \n$\\langle a(x)\\rangle$ are equally likely. For \n$T \\sim 1$ GeV the anomaly induced potential \nturns on driving $\\langle a(x)\\rangle$ \nto a value as to yield $\\bar \\theta =0$ at the new potential \nminimum. The energy stored previously as {\\em latent heat} \nis then released into axions oscillating around its new VEV. \nPrecisely because of the invisible axion's couplings are so \nimmensely suppressed the energy cannot be dissipated into \nother degrees of freedom. We are then dealing with a fluid of \naxions. Their typical momenta is the inverse of their \ncorrelation length which in turn cannot exceed their horizon; \none finds \n\\begin{equation}}   \\def\\eeq{\\end{equation} \np_a \\sim \\left( 10^{-6}\\; {\\rm sec}\\right) ^{-1} \n\\sim 10^{-9}\\; {\\rm eV} \n\\eeq \nat $T \\simeq 1$ GeV; i.e., the axions despite their minute \nmass form a very {\\em cold} fluid and actually represent \na candidate \nfor cold dark matter. Their contribution to the density of the \nuniverse relative to its critical value is \\cite{SIKIVIE}\n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\Omega _a = \\left(  \\frac{0.6 \\cdot 10^{-5}\\; {\\rm eV}}\n{m_a}\\right) ^{\\frac{7}{6}}\\cdot \n\\left(  \\frac{200\\; {\\rm MeV}}\n{\\Lambda _{QCD}}\\right) ^{\\frac{3}{4}}\\cdot \n\\left(  \\frac{75 \\; {\\rm km\/sec\\, \\cdot Mpc}}\n{H_0}\\right) ^2 \\; ; \n\\eeq \n$H_0$ is the present Hubble expansion rate. For axions not to overclose the universe one thus has to require: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nm_a \\geq 10^{-6} \\; {\\rm eV} \n\\eeq\nor \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\Lambda _{PQ} \\leq 10^{12} \\; {\\rm GeV} \n\\eeq  \nThis means also that we might be existing in a bath \nof cold axions still making up a significant fraction of the \nmatter of the universe. \n\nIngenious suggestions have been made to search for such \ncosmic background axions. The main handle one has on them \nis their coupling to two photons. They can be detected \nby stimulating the conversion \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\rm axion} \\; \\;  \\stackrel{\\vec B}\n{\\longrightarrow} \\; \\;  {\\rm photon} \n\\eeq \nin a strong magnetic field $\\vec B$: \nthe second photon \nwhich is virtual in this process effects the interaction \nwith the inhomogeneous magnetic field in the cavity.  The \navailable microwave technology allows an impressive \nexperimental sensitivity. No signal has been found yet, but the search continues and soon should reach a level where one has a \ngood chance to see a signal \\cite{LLNL}. \n\n\\subsection{The Pundits' Judgement}\nThe story of the Strong CP Problem is a particularly intriguing one. \nWe -- like most though not all of our community -- find the \ntheoretical arguments persuasive that there is a problem that \nhas to be resolved. The inquiry has been based on an impressive \narsenal of theoretical reasoning and has inspired fascinating \nexperimental undertakings. \n\nLike many modern novels the problem -- if its is indeed \none -- has not found any resolution. On he other hand it has \nthe potential to lead the charge towards a new paradigm \nin high energy physics. \n\n\n\\section{Summary on the CP Phenomenology with Light Degrees of \nFreedom}\nTo summarize our discussion up to this point: \n\\begin{itemize} \n\\item \nThe following data represent the most sensitive probes:  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\rm BR}(K_L \\rightarrow \\pi ^+ \\pi ^-) = 2.3 \\cdot 10^{-3} \\neq 0 \n\\eeq  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{{\\rm BR}(K_L \\rightarrow l^+ \\nu \\pi ^-)}\n{{\\rm BR}(K_L \\rightarrow l^- \\nu \\pi ^+)} \\simeq 1.006 \\neq 1 \n\\eeq \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\rm Re} \\frac{\\epsilon ^{\\prime}}{\\epsilon _K} = \n\\left\\{ \n\\begin{array}{l} \n(2.3 \\pm 0.7) \\cdot 10^{-3} \\; \\; NA\\, 31 \\\\ \n(0.6 \\pm 0.58 \\pm 0.32 \\pm 0.18) \\cdot 10^{-3} \\; \\; \nE\\, 731 \n\\end{array}  \n\\right. \n\\eeq\n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\rm Pol}_{\\perp}^{K^+}(\\mu ) = (-1.85\\pm 3.60) \\cdot 10^{-3} \n\\eeq\n\\begin{equation}}   \\def\\eeq{\\end{equation} \nd_N < 12 \\cdot 10^{-26} \\; \\; e\\, cm \n\\eeq   \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nd_{Tl} = (1.6 \\pm 5.0) \\cdot 10^{-24} \\; \\; e\\, cm \n\\stackrel{theor.}{\\leadsto} \nd_e = (-2.7 \\pm 8.3) \\cdot 10^{-27} \\; \\; e\\, cm\n\\eeq \n\\item \nAn impressive amount of experimental ingenuity, acumen and \ncommitment \nwent into producing this list. We know that CP violation \nunequivocally exists in nature; it can be \ncharacterized by a {\\em single} non-vanishing quantity: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\rm Im} M_{12} \\simeq 1.1 \\cdot 10^{-8} \\; eV \\neq 0 \n\\eeq  \n\\item \nThe `Superweak Model' states that there just happens to \nexist a $\\Delta S=2$ interaction that is fundamental or effective -- \nwhatever the case may be -- generating Im$M_{12} =$ \nIm$M_{12}|_{exp}$ while $\\epsilon ^{\\prime} =0$. It \nprovides merely  \na {\\em classification} for possible dynamical implementations \nrather than such a dynamical implementation itself.  \n\n\\item \nThe KM ansatz allows us to incorporate CP violation into the \nStandard Model. Yet it does not regale us with an understanding. \nInstead it relates the origins of CP violation to central \nmysteries of the Standard Model: Why are there families? \nWhy are there three of those?  What is underlying \nthe observed pattern in the fermion masses? \n\\item \nStill the KM ansatz succeeds in {\\em accommodating} the data in an \nunforced way: $\\epsilon _K$ emerges to be naturally \nsmall, $\\epsilon ^{\\prime}$ naturally tiny (once the huge \ntop mass is built in), the EDM's for neutrons [electrons] \nnaturally (tiny)$^2$ [(tiny)$^3$] etc. \n\n\\end {itemize}  \n\n\n\n\\section{CP Violation in Beauty Decays -- The KM Perspective}\nThe KM predictions for strange decays and electric dipole \nmoments given above will be subjected to sensitive tests in the \nforeseeable future. Yet there is one question that most naturally \nwill come up in this context: \"Where else to look?\" \nI will show below that on very general grounds one has to \nconclude that the decays of beauty hadrons provide by far the \noptimal lab. Yet first I want to make some historical remarks. \n\n\\subsection{The Emerging Beauty of $B$ Hadrons}\n\\subsubsection{Lederman's Paradise Lost -- and Regained!}\nIn 1970 Lederman's group studying the Drell-Yan process \n\\begin{equation}}   \\def\\eeq{\\end{equation} \np p \\rightarrow \\mu ^+ \\mu ^- X\n\\eeq \nat Brookhaven observed a shoulder in the di-muon mass \ndistribution around 3 GeV. 1974 saw the `Octobre Revolution' when \nTing et al. and Richter et al. found a narrow resonance -- the $J\/\\psi$ \n-- with a mass of 3.1 GeV at Brookhaven and SLAC, respectively, and \nannounced it. In 1976\/77 Lederman's group working at \nFermilab saw \na structure -- later referred to as the Oops-Leon -- around 6 GeV, \nwhich then disappeared. In 1977 Lederman et al. discovered \nthree resonances in the mass range of 9.5 - 10.3 GeV, the \n$\\Upsilon$, $\\Upsilon ^{\\prime}$ and $\\Upsilon ^{\\prime \\prime}$! \nThat shows that persistence can pay off -- at least sometimes and for  \nsome people. \n\n\\subsubsection{Longevity of Beauty}\nThe lifetime of weakly decaying beauty quarks can be related \nto the muon lifetime \n\\begin{equation}}   \\def\\eeq{\\end{equation}  \n\\tau (b) \\sim \\tau (\\mu ) \\left( \\frac{m_{\\mu}}{m_b} \\right) ^5 \n\\frac{1}{9} \\frac{1}{|V(cb)|^2} \\sim \n3 \\cdot 10^{-14} \\left|  \\frac{{\\rm sin}\\theta _C}{V(cb)}\\right| ^2  \n\\; {\\rm sec} \n\\eeq  \nfor a $b$ quark mass of around  5 GeV; the factor 1\/9 reflects \nthe fact that the virtual $W ^-$ boson in $b$ quark decays can \nmaterialize as a $d \\bar u$ or $s \\bar c$ in three colours each and \nas three lepton pairs. I have ignored phase space corrections here. \nSince the $b$ quark has to decay outside its own family one would \nexpect $|V(cb)| \\sim {\\cal O}({\\rm sin}\\theta _C) = \n|V(us)|$. Yet starting in 1982 data showed a considerably longer \nlifetime \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\tau ({\\rm beauty}) \\sim 10^{-12} \\; {\\rm sec} \n\\eeq \nimplying \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n|V(cb)| \\sim {\\cal O}({\\rm sin}^2\\theta _C) \\sim 0.05 \n\\eeq \nThe technology to resolve decay vertices for objects of such \nlifetimes happened to have just been developed -- for charm \nstudies! \n\n\\subsubsection{The Changing Identity of Neutral $B$ Mesons}\nSpeedy $B_d - \\bar B_d$ oscillations were discovered by ARGUS in \n1986: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nx_d \\equiv \\frac{\\Delta m(B_d)}{\\Gamma (B_d)} \\simeq \n{\\cal O}(1)\n\\eeq \nThese oscillations can then be tracked like the decays. This \nobservation was also the first evidence that top quarks had to be \nheavier than orginally thought, namely $m_t \\geq M_W$. \n\n\\subsubsection{Beauty Goes to Charm (almost always)}\nIt was soon found that $b$ quarks exhibit a strong preference \nto decay into charm rather than up quarks  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\left| \\frac{V(ub)}{V(cb)} \\right| ^2 \\ll 1 \n\\eeq \nestablishing thus the hierarchy \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n|V(ub)|^2 \\ll |V(cb)|^2 \\ll |V(us)|^2 \\ll 1\n\\eeq \n\n\\subsubsection{Resume}\nWe will soon see how all these observations form crucial inputs \nto the general message that big CP asymmetries should emerge in \n$B$ decays and that they (together with interesting rare decays)  \nare within reach of experiments. It is for this reason that I strongly \nfeel that the only appropriate name for this quantum number is \n{\\em beauty}! A name like bottom would not do it justice. \n\n\\subsection{The KM Paradigm of Huge CP Asymmetries}\n\\subsubsection{Large Weak Phases!}\nThe Wolfenstein representation expresses the CKM matrix as an \nexpansion: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nV_{CKM} = \n\\left( \n\\begin{array}{ccc} \n1 & {\\cal O}(\\lambda ) & {\\cal O}(\\lambda ^3) \\\\ \n{\\cal O}(\\lambda ) & 1 & {\\cal O}(\\lambda ^2) \\\\ \n{\\cal O}(\\lambda ^3) & {\\cal O}(\\lambda ^2) & 1 \n\\end {array} \n\\right) \n\\; \\; \\; , \\; \\; \\; \\lambda = {\\rm sin}\\theta _C \n\\eeq \nThe crucial element in making this expansion meaningful is the \n`long' lifetime of beauty hadrons of around 1 psec. That number \nhad to change by an order of magnitude -- which is out of the \nquestion -- to invalidate the conclusions given below for the \nsize of the weak phases. \n\nThe unitarity condition yields 6 triangle relations: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\begin{array}{ccc} \nV^*(ud)V(us) + &V^*(cd)V(cs) + &V^*(td) V(ts) = \n\\delta _{ds}= 0 \\\\\n{\\cal O}(\\lambda ) & {\\cal O}(\\lambda ) & {\\cal O}(\\lambda ^5) \n\\end{array} \n\\label{TRI1} \n\\eeq \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\begin{array}{ccc} \nV^*(ud)V(cd) + &V^*(us)V(cs) + &V^*(ub) V(cb) = \n\\delta _{uc}= 0 \\\\\n{\\cal O}(\\lambda ) & {\\cal O}(\\lambda ) & {\\cal O}(\\lambda ^5) \n\\end{array} \n\\label{TRI2} \n\\eeq \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\begin{array}{ccc} \nV^*(us)V(ub) + &V^*(cs)V(cb) + &V^*(ts) V(tb) = \n\\delta _{sb}= 0 \\\\\n{\\cal O}(\\lambda ^4) & {\\cal O}(\\lambda ^2) & {\\cal O}(\\lambda ^2) \n\\end{array} \n\\label{TRI3} \n\\eeq \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\begin{array}{ccc} \nV^*(td)V(cd) + &V^*(ts)V(cs) + &V^*(tb) V(cb) = \n\\delta _{ct}=0 \\\\\n{\\cal O}(\\lambda ^4) & {\\cal O}(\\lambda ^2) & {\\cal O}(\\lambda ^2) \n\\end{array} \n\\label{TRI4} \n\\eeq \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\begin{array}{ccc} \nV^*(ud)V(ub) + &V^*(cd)V(cb) + &V^*(td) V(tb) = \n\\delta _{db}=0 \\\\\n{\\cal O}(\\lambda ^3) & {\\cal O}(\\lambda ^3) & {\\cal O}(\\lambda ^3) \n\\end{array} \n\\label{TRI5} \n\\eeq \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\begin{array}{ccc} \nV^*(td)V(ud) + &V^*(ts)V(us) + &V^*(tb) V(ub) = \n\\delta _{ut}=0 \\\\\n{\\cal O}(\\lambda ^3) & {\\cal O}(\\lambda ^3) & {\\cal O}(\\lambda ^3) \n\\end{array}\n\\label{TRI6}  \n\\eeq \nwhere below each product of matrix elements I have noted \ntheir size in powers of $\\lambda $. \n\nWe see that the six triangles fall into three categories: \n\\begin{enumerate}\n\\item \nThe first two triangles are extremely `squashed': two sides are \nof order $\\lambda $, the third one of order $\\lambda ^5$ and their \nratio of order $\\lambda ^4 \\simeq 2.3 \\cdot 10^{-3}$; \nEq.(\\ref{TRI1}) and Eq.(\\ref{TRI2}) control the situation in \nstrange and charm decays; the relevant weak phases there \nare obviously tiny. \n\\item \nThe third and fourth triangles are still rather squashed, yet less so: \ntwo sides are of order $\\lambda ^2$ and the third one of order \n$\\lambda ^4$. \n\\item \nThe last two triangles have sides that are all of the same \norder, namely $\\lambda ^3$. All their angles are therefore \nnaturally large, i.e. $\\sim$ several $\\times$ $10$ degrees! Since to \nleading order in $\\lambda$ one has \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nV(ud) \\simeq V(tb) \\; , \\; V(cd) \\simeq - V(us) \\; , \\; \nV(ts) \\simeq - V(cb) \n\\eeq \nwe see that the triangles of Eqs.(\\ref{TRI5}, \\ref{TRI6}) \nactually coincide to that order. \n\\end{enumerate} \nThe sides of this triangle having naturally large angles are \ngiven by $\\lambda \\cdot V(cb)$, $V(ub)$ and \n$V^*(td)$; these are all quantities that control important \naspects of $B$ decays, namely CKM favoured and disfavoured \n$B$ decays and $B_d - \\bar B_d$ oscillations! \n\n\\subsubsection{Different, Yet Coherent Amplitudes!}\n$B^0 - \\bar B^0$ oscillations provide us with two different \namplitudes that by their very nature have to be coherent: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nB^0 \\Rightarrow \\bar B^0 \\rightarrow f \\leftarrow B^0 \n\\eeq \nOn general grounds one expects oscillations to be speedy for \n$B^0 - \\bar B^0$ (like for $K^0 - \\bar K^0$), yet slow for \n$D^0 - \\bar D^0$ \n\\footnote{$T^0 - \\bar T^0$ oscillations cannot \noccur since top quarks decay before they hadronize \n\\cite{RAPALLO}.}. \nExperimentally one indeed finds \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{\\Delta m(B_d)}{\\Gamma (B_d)} = 0.71 \\pm 0.06 \n\\label{OSCBD}\n\\eeq  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{\\Delta m(B_s)}{\\Gamma (B_s)} \\geq 10  \n\\label{OSCBS}\n\\eeq\nWhile Eq.(\\ref{OSCBD}) describes an almost optimal \nsituation the overly rapid pace of $B_s - \\bar B_s$ \noscillations will presumably cause experimental \nproblems. \n\nThe conditions are quite favourable also for {\\em direct} \nCP violation to surface. Consider a transition amplitude \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nT(B \\rightarrow f) = {\\cal M}_1 + {\\cal M}_2 = \ne^{i\\phi _1}e^{i\\alpha _1}|{\\cal M}_1| + \ne^{i\\phi _2}e^{i\\alpha _2}|{\\cal M}_2| \\; . \n\\eeq    \nThe two partial amplitudes ${\\cal M}_1$ and ${\\cal M}_2$ are \ndistinguished by, say, their isospin -- as it was the case for \n$K \\rightarrow (\\pi \\pi )_{I=0,2}$ discussed before; $\\phi _1$, $\\phi _2$ \ndenote the phases in the {\\em weak} couplings and \n$\\alpha _1$, $\\alpha _2$ the phase shifts due to \n{\\em strong} final state interactions. For the CP conjugate reaction \none obtains \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nT(\\bar B \\rightarrow \\bar f) =  \ne^{-i\\phi _1}e^{i\\alpha _1}|{\\cal M}_1| + \ne^{-i\\phi _2}e^{i\\alpha _2}|{\\cal M}_2| \\; . \n\\eeq \nsince under CP the weak parameters change into their \ncomplex conjugate values whereas the phase shifts remain \nthe same; for the strong forces driving final state \ninteractions conserve CP. The rate difference is then given by \n$$ \n\\Gamma (B \\rightarrow f) - \\Gamma (\\bar B \\rightarrow \\bar f) \\propto \n|T(B \\rightarrow f)|^2 - |T( \\bar B \\rightarrow \\bar f)|^2 = \n$$\n\\begin{equation}}   \\def\\eeq{\\end{equation} \n= - 4 {\\rm sin}(\\phi _1 - \\phi _2) \\cdot \n{\\rm sin}(\\alpha _1 - \\alpha _2) \\cdot \n{\\cal M}_1 \\otimes {\\cal M}_2  \n\\eeq \nFor an asymmetry to arise in this way two conditions need to be \nsatisfied simultaneously, namely \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\begin{array}{l} \n\\phi _1 \\neq \\phi _2 \\\\ \n\\alpha _1 \\neq \\alpha _2\n\\end{array}\n\\eeq \nI.e., the two amplitudes ${\\cal M}_1$ and ${\\cal M}_2$ have to \ndiffer both in their weak and strong forces! The first condition \nimplies (within the Standard Model) that the reaction has to \nbe KM suppressed, whereas the second one require the intervention \nof nontrivial final state interactions. \n\nThere is a large number of KM suppressed channels in $B$ \ndecays that are suitable in this context: they receive significant \ncontributions from weak couplings with large phases -- \nlike $V(ub)$ in the Wolfenstein representation -- and there \nis no reason why the phase shifts should be small in general \n(although that could happen in some cases). \n\n\\subsubsection{Resume}\nLet me summarize the discussion just given and anticipate the \nresults to be presented below. \n\\begin{itemize}\n\\item \n{\\em Large} CP asymmetries are {\\em pre}dicted {\\em with} \nconfidence to occur in $B$ decays. If they are not found, there is \nno plausible deniability for the KM ansatz. \n\\item \nSome of these predictions can be made with high \n{\\em parametric} reliability. \n\\item \nNew theoretical technologies have emerged that will allow us to \ntranslate this {\\em parametric} reliability into \n{\\em numerical} precision. \n\\item \nSome of the observables exhibit a high and unambiguous \nsensitivity to the presence of New Physics since we are \ndealing with coherent processes with observables depending  \n{\\em linearly} on New Physics amplitudes and where the \nCKM `background' is (or can be brought)  \nunder theoretical control. \n\\end{itemize}\n\n\n\\subsection{General Phenomenology}\nDecay rates for CP conjugate channels can be expressed as follows: \n\\begin{equation}}   \\def\\eeq{\\end{equation}  \n\\begin{array} {l} \n{\\rm rate} (B(t) \\rightarrow f) = e^{-\\Gamma _Bt}G_f(t) \\\\  \n{\\rm rate} (\\bar B(t) \\rightarrow \\bar f) = \ne^{-\\Gamma _Bt}\\bar G_{\\bar f}(t)  \n\\end{array} \n\\label{DECGEN}\n\\eeq \nwhere CPT invariance has been invoked to assign the same lifetime \n$\\Gamma _B^{-1}$ to $B$ and $\\bar B$ hadrons. Obviously if \n\\begin{equation}}   \\def\\eeq{\\end{equation}\n\\frac{G_f(t)}{\\bar G_{\\bar f}(t)} \\neq 1 \n\\eeq \nis observed, CP violation has been found. Yet one should \nkeep in mind that this can manifest itself in two (or three) \nqualitatively different ways: \n\\begin{enumerate} \n\\item \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{G_f(t)}{\\bar G_{\\bar f}(t)} \\neq 1 \n\\; \\; {\\rm with} \\; \\; \n\\frac{d}{dt}\\frac{G_f(t)}{\\bar G_{\\bar f}(t)} =0 \\; ; \n\\label{DIRECTCP1}\n\\eeq   \ni.e., the {\\em asymmetry} is the same for all times of decay. This \nis true for {\\em direct} CP violation; yet, as explained later, it also \nholds for CP violation {\\em in the oscillations}.  \n\\item \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{G_f(t)}{\\bar G_{\\bar f}(t)} \\neq 1 \n\\; \\; {\\rm with} \\; \\; \n\\frac{d}{dt}\\frac{G_f(t)}{\\bar G_{\\bar f}(t)} \\neq 0 \\; ; \n\\label{DIRECTCP2}\n\\eeq   \nhere the asymmetry varies as a function of the time of decay. \nThis can be referred to as CP violation {\\em involving \noscillations} \\footnote{This nomenclature falls well short of \nShakespearean standards.}. \n\\end{enumerate} \n\nQuantum mechanics with its linear superposition principle makes \nvery specific statements about the possible time dependance of \n$G_f(t)$ and $\\bar G_{\\bar f}(t)$; yet before going into that \nI want to pose another homework problem: \n\\begin{center} \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $ \\\\ \n{\\em Homework Problem \\# 4}: \n\\end{center}\nConsider the reaction \n\\begin{equation}}   \\def\\eeq{\\end{equation} \ne^+ e^- \\rightarrow \\phi \\rightarrow \n(\\pi ^+\\pi ^-)_K  (\\pi ^+\\pi ^-)_K \n\\eeq \nIts occurrance requires CP violation. For the {\\em initial} state -- \n$\\phi $ -- carries {\\em even} CP parity whereas the \n{\\em final} state with the two $(\\pi ^+\\pi ^-)$ combinations \nforming a P wave must be CP {\\em odd}: \n$(+1)^2 (-1)^l = -1$! Yet Bose statistics requiring identical \nstates to be in a symmetric configuration would appear to \nveto this reaction; for it places the two $(\\pi ^+\\pi ^-)$ states \ninto a P wave which is antisymmetric. What is the flaw in this \nreasoning? The same puzzle can be formulated in terms of \n\\begin{equation}}   \\def\\eeq{\\end{equation} \ne^+ e^- \\rightarrow \\Upsilon (4S) \\rightarrow B_d \\bar B_d \\rightarrow \n(\\psi K_S)_B (\\psi K_S)_B \\; . \n\\eeq \n\\begin{center} \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $\n\\end{center} \nA straightforward application of quantum mechanics yields  \nthe general expressions \n\\cite{CARTER,BS,CECILIABOOK}:\n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\begin{array}{l}\nG_f(t) = |T_f|^2 \n\\left[ \n\\left( 1 + \\left| \\frac{q}{p}\\right| ^2|\\bar \\rho _f|^2 \\right) + \n\\left( 1 - \\left| \\frac{q}{p}\\right| ^2|\\bar \\rho _f|^2 \\right) \n{\\rm cos}\\Delta m_Bt \n+ 2 ({\\rm sin}\\Delta m_Bt) {\\rm Im}\\frac{q}{p} \\bar \\rho _f \n\\right]  \\\\ \n\\bar G_{\\bar f}(t) = |\\bar T_{\\bar f}|^2 \n\\left[ \n\\left( 1 + \\left| \\frac{p}{q}\\right| ^2|\\rho _{\\bar f}|^2 \\right) + \n\\left( 1 - \\left| \\frac{p}{q}\\right| ^2|\\rho _{\\bar f}|^2 \\right) \n{\\rm cos}\\Delta m_Bt \n+ 2 ({\\rm sin}\\Delta m_Bt) {\\rm Im}\\frac{p}{q} \\rho _{\\bar f}  \n\\right]  \n\\end{array}\n\\eeq \nThe amplitudes for the instantaneous $\\Delta B=1$ \ntransition into a \nfinal state $f$ are denoted by \n$T_f = T(B \\rightarrow f)$ and $\\bar T_f = T(\\bar B \\rightarrow f)$ and  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\bar \\rho _f = \\frac{\\bar T_f}{T_f} \\; \\; , \n\\rho _{\\bar f} = \\frac{T_{\\bar f}}{\\bar T_{\\bar f}} \\; \\; , \n\\frac{q}{p} = \\sqrt{\\frac{M_{12}^* - \\frac{i}{2} \\Gamma _{12}^*}\n{M_{12} - \\frac{i}{2} \\Gamma _{12}}}\n\\eeq \nStaring at the general expression is not always very illuminating; \nlet us therefore consider three very simplified limiting cases: \n\\begin{itemize}\n\\item\n$\\Delta m_B = 0$, i.e. {\\em no} $B^0- \\bar B^0$ oscillations: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nG_f(t) = 2|T_f|^2 \\; \\; , \\; \\; \n\\bar G_{\\bar f}(t) = 2|\\bar T_{\\bar f}|^2 \n\\leadsto \\frac{\\bar G_{\\bar f}(t)}{G_{ f}(t)} = \n\\left|\n\\frac{\\bar T_{\\bar f}}{T_{ f}}\n\\right|^2 \\; \\; , \\frac{d}{dt}G_f (t) \\equiv 0 \\equiv \n\\frac{d}{dt}\\bar G_{\\bar f} (t) \n\\eeq \nThis is explicitely what was referred to above as {\\em direct} \nCP violation. \n\\item \n$\\Delta m_B \\neq  0$   \nand $f$ a flavour-{\\em specific} final state with {\\em no} \ndirect CP violation; i.e., \n$T_{f} = 0 = \\bar T_{\\bar f}$ and $\\bar T_f = T_{\\bar f}$   \n\\footnote{For a flavour-specific mode one has in general \n$T_f \\cdot T_{\\bar f} =0$; the more intriguing case arises  \nwhen one considers a transition that requires oscillations \nto take place.}: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\begin{array} {c} \nG_f (t) = \\left| \\frac{q}{p}\\right| ^2 |\\bar T_f|^2 \n(1 - {\\rm cos}\\Delta m_Bt )\\; \\; , \\; \\; \n\\bar G_{\\bar f} (t) = \\left| \\frac{p}{q}\\right| ^2 |T_{\\bar f}|^2 \n(1 - {\\rm cos}\\Delta m_Bt) \\\\ \n\\leadsto \n\\frac{\\bar G_{\\bar f}(t)}{G_{ f}(t)} = \\left| \\frac{q}{p}\\right| ^4 \n\\; \\; , \\; \\; \\frac{d}{dt} \\frac{\\bar G_{\\bar f}(t)}{G_{ f}(t)} \\equiv 0  \n\\; \\; , \\; \\; \\frac{d}{dt} \\bar G_{\\bar f}(t) \\neq 0 \\neq \n\\frac{d}{dt} G_ f(t)\n\\end{array} \n\\eeq \nThis constitutes CP violation {\\em in the \noscillations}. For the CP conserving decay into the flavour-specific \nfinal state is used merely to track the flavour identity of the \ndecaying meson. This situation can therefore be denoted also \nin the following way: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{{\\rm Prob}(B^0 \\Rightarrow  \\bar B^0; t) - \n{\\rm Prob}(\\bar B^0 \\Rightarrow  B^0; t)}\n{{\\rm Prob}(B^0 \\Rightarrow \\bar B^0; t) + \n{\\rm Prob}(\\bar B^0 \\Rightarrow  B^0; t)} = \n\\frac{|q\/p|^2 - |p\/q|^2}{|q\/p|^2 + |p\/q|^2} = \n\\frac{1- |p\/q|^4}{1+ |p\/q|^4} \n\\eeq \n\n\\item \n$\\Delta m_B \\neq  0$ with $f$ now being a  \nflavour-{\\em non}specific final state -- a final state {\\em common} \nto $B^0$ and $\\bar B^0$ decays -- of a special nature, namely \na CP eigenstate -- $|\\bar f\\rangle = {\\bf CP}|f\\rangle = \n\\pm |f\\rangle $ -- {\\em without} direct CP violation --  \n$|\\bar \\rho _f| = 1 = |\\rho _{\\bar f}| $: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\begin{array} {c} \nG_f(t) = 2 |T_f|^2 \n\\left[ 1 + ({\\rm sin}\\Delta m_Bt) \\cdot \n{\\rm Im} \\frac{q}{p} \\bar \\rho _f \n\\right] \\\\  \n\\bar G_f(t) = 2 |T_f|^2 \n\\left[ 1 - ({\\rm sin}\\Delta m_Bt )\\cdot \n{\\rm Im} \\frac{q}{p} \\bar \\rho _f \n\\right] \\\\ \n\\leadsto \n\\frac{d}{dt} \\frac{\\bar G_f(t) }{G_f(t)} \\neq 0\n\\end{array} \n\\eeq \nis the concrete realization of what was called CP violation \n{\\em involving oscillations}. \n\\end{itemize} \n\n\\subsubsection{CP Violation in Oscillations}\nUsing the convention blessed by the PDG \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nB = [\\bar b q] \\; \\; , \\; \\; \\bar B = [\\bar q b] \n\\eeq\nwe have \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\begin{array} {c} \nT(B \\rightarrow l^- X) = 0 = T(\\bar B \\rightarrow l^+ X) \\\\ \nT_{SL} \\equiv  T(B \\rightarrow l^+ X) = T(\\bar B \\rightarrow l^- X) \n\\end{array} \n\\eeq  \nwith the last equality enforced by CPT invariance. The \nso-called Kabir test can then be realized as follows: \n$$  \n\\frac\n{{\\rm Prob}(B^0 \\Rightarrow \\bar B^0; t) - \n{\\rm Prob}(\\bar B^0 \\Rightarrow B^0; t)} \n{{\\rm Prob}(B^0 \\Rightarrow \\bar B^0; t) + \n{\\rm Prob}(\\bar B^0 \\Rightarrow B^0; t)} = \n$$ \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n= \\frac\n{{\\rm Prob}(B^0 \\Rightarrow \\bar B^0 \\rightarrow l^-X; t) - \n{\\rm Prob}(\\bar B^0 \\Rightarrow B^0 \\rightarrow l^+X; t)} \n{{\\rm Prob}(B^0 \\Rightarrow \\bar B^0 \\rightarrow l^-X; t) + \n{\\rm Prob}(\\bar B^0 \\Rightarrow B^0 \\rightarrow l^+X; t)} = \n\\frac{1 - |q\/p|^4}{1+|q\/p|^4} \n\\eeq \nWithout going into details I merely state the results here \n\\cite{CECILIABOOK}: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n1 - \\left|  \\frac{q}{p} \\right| \\simeq \n\\frac{1}{2} {\\rm Im}\\left( \\frac{\\Gamma _{12}}\n{M_{12}} \\right) \\sim \n\\left\\{ \n\\begin{array}{ccc} \n10^{-3} & \\; {\\rm for} \\; & B_d=(\\bar bd) \\\\ \n10^{-4} & \\; {\\rm for} \\; & B_s =(\\bar bs) \\\\ \n\\end{array} \n\\right. \n\\eeq\ni.e.,  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \na_{SL} (B^0) \\equiv \\frac{\\Gamma (\\bar B^0(t) \\rightarrow l^+ \\nu X) - \n\\Gamma ( B^0(t) \\rightarrow l^- \\bar \\nu X)}\n{\\Gamma (\\bar B^0(t) \\rightarrow l^+ \\nu X) +  \n\\Gamma ( B^0(t) \\rightarrow l^- \\bar \\nu X)} \\simeq \n\\left\\{ \n\\begin{array}{ccc} \n{\\cal O}(10^{-3}) & \\; {\\rm for} \\; & B_d \\\\ \n{\\cal O}(10^{-4}) & \\; {\\rm for} \\; & B_s \\\\ \n\\end{array} \n\\right. \n\\eeq  \nThe smallness of the quantity $1-|q\/p|$ is primarily due to \n$|\\Gamma _{12}| \\ll |M_{12}|$ or $\\Delta \\Gamma _B \\ll \n\\Delta m_B$. Within the Standard Model this hierarchy \nis understood (semi-quantitatively at leaast) as due to the \nhierarchy in the GIM factors of the box diagram \nexpressions for $\\Gamma _{12}$ and $M_{12}$, namely \n$m_c^2\/M_W^2 \\ll m_t^2\/M_W^2$. \n\nFor $B_s$ mesons the phase between $\\Gamma _{12}$ and \n$M_{12}$ is further (Cabibbo) suppressed for reasons that \nare peculiar to the KM ansatz: for to leading order in the \nKM parameters quarks of the second and third family only \ncontribute and therefore \narg$(\\Gamma _{12}\/M_{12}) = 0$ to that order. If New \nPhysics intervenes in $B^0 - \\bar B^0$ oscillations, it would \nquite naturally generate a new phase between \n$\\Gamma _{12}$ and $M_{12}$; it could also reduce \n$M_{12}$. Altogether this CP asymmetry could get \nenhanced very considerably: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \na_{SL}^{New \\; Physics} (B^0) \\sim 1 \\% \n\\eeq \nTherefore one would be ill-advised to accept the somewhat \npessimistic KM predictions as gospel. \n\nSince this CP asymmetry does not vary with the time of decay, \na signal is not diluted by integrating over all times. It is, \nhowever, essential to `flavour tag' the decaying meson; i.e., \ndetermine whether it was {\\em produced} as  a \n$B^0$ or $\\bar B^0$. This can be achieved in several ways \nas discussed later.  \n \n\n\n\\subsubsection{Direct CP Violation}\n\nSizeable direct CP asymmetries arise rather naturally in \n$B$ decays. Consider \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nb \\rightarrow s \\bar u u \n\\eeq \nThree different processes contribute to it, namely \n\\begin{itemize}\n\\item \nthe tree process \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nb \\rightarrow u W^* \\rightarrow u (\\bar u s)_W  \\; , \n\\eeq \n\\item \nthe penguin process with an internal top quark which is \npurely local (since $m_t > m_b$) \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nb \\rightarrow s g^* \\rightarrow s u \\bar u \\; , \n\\eeq \n\\item \nthe penguin reaction with an internal charm quark. Since \n$m_b > 2m_c + m_s$, this last operator is {\\em not} \nlocal: it contains an absorptive part that amounts to a \nfinal state interaction including a phase shift. \n\\end{itemize}\nOne then arrives at a guestimate \\cite{SONI,CECILIABOOK} \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{\\Gamma (b \\rightarrow s u \\bar u) - \n\\Gamma (\\bar b \\rightarrow \\bar s u \\bar u)}\n{\\Gamma (b \\rightarrow s u \\bar u) + \n\\Gamma (\\bar b \\rightarrow \\bar s u \\bar u)} \n\\sim {\\cal O}(\\% ) \n\\eeq  \nInvoking quark-hadron duality one can expect  \n(or at least hope) that this quark level analysis -- rather \nthan being washed out by hadronisation -- yields \nsome average asymmetry or describes the \nasymmetry for some inclusive subclass of nonleptonic \nchannels. I would like to draw the following lessons \nfrom these considerations: \n\\begin{itemize}\n\\item \nAccording to the KM ansatz the natural scale for direct \nCP asymmetries in the decays of beauty hadrons \n(neutral or charged mesons or baryons) is the \n$10^{-2}$ level -- not $10^{-6} \\div 10^{-5}$ as in \nstrange decays! \n\\item \nThe size of the asymmetry in {\\em individual} channels -- \nlike $B\\rightarrow K \\pi$ -- is shaped by the strong final state \ninteractions operating there. Those are likely to differ \nconsiderably from channel to channel, and at present we \nare unable to predict them since they reflect long-distance \ndynamics. \n\\item \nObservation of such an asymmetry (or lack thereof) will not provide \nus with reliable numerical information on the parameters of the \nmicroscopic theory, like the KM ansatz. \n\\item \nNevertheless comprehensive and detailed studies are an \nabsolute must!\n\\end{itemize} \nLater I will describe examples where the relevant long-distance \nparameters -- phase shifts etc. -- can be {\\em measured} \nindependantly. \n\n\\subsubsection{CP Violation Involving Oscillations}\nThe essential feature that a final state in this category has to \nsatisfy is that it can be fed both by $B^0$ and $\\bar B^0$ decays \n\\footnote{Obviously no such common channels can exist for \ncharged mesons or for baryons.}. However for convenience reasons \nI will concentrate on a special subclass of such modes, namely \nwhen the final state is a CP eigenstate. A more comprehensive \ndiscussion can be found in \\cite{CECILIABOOK,BOOK}. \n\nThree qualitative observations have to be made here: \n\\begin{itemize}\n\\item \nSince the final state is shared by $B^0$ and $\\bar B^0$ decays \none cannot even define a CP asymmetry unless one acquires \n{\\em independant} information on the decaying meson: was it \na $B^0$ or $\\bar B^0$ or -- more to the point -- was it originally \nproduced as a $B^0$ or $\\bar B^0$? There are several \nscenarios for achieving such {\\em flavour tagging}: \n\\begin{itemize} \n\\item \nNature could do the trick for us by providing us with \n$B^0$ - $\\bar B^0$ production asymmetries through, say, \nassociated production in hadronic collisions or the use of \npolarized beams in $e^+ e^-$ annihilation. Those production \nasymmetries could be tracked through decays that are \nnecessarily CP conserving -- like $\\bar B_d \\rightarrow \\psi K^- \\pi ^+$ vs.  \n$B_d \\rightarrow \\psi K^+ \\pi ^-$. It seems unlikely, though, that such \na scenario could ever be realized with sufficient statistics. \n\n\\item \n{\\em Same Side Tagging}: One undertakes to repeat the success \nof the $D^*$ tag for charm mesons -- $D^{+*} \\rightarrow D^0 \\pi ^+$ \nvs. $D^{-*} \\rightarrow \\bar D^0 \\pi ^-$ -- through finding a conveniently \nplaced nearby resonance -- $B^{-**} \\rightarrow \\bar B_d \\pi ^-$  vs. \n$B^{+**} \\rightarrow B_d \\pi ^+$ -- or through employing correlations \nbetween the beauty mesons and a `nearby' pion (or kaon \nfor $B_s$) as pioneered by the CDF collaboration. This method can be calibrated by \nanalysing how well $B^0 - \\bar B^0$ oscillations are reproduced.  \n \n\n\\item \n{\\em Opposite Side Tagging}: With electromagnetic and strong \nforces conserving the beauty quantum number, one can employ \ncharge correlations between the decay products (leptons and kaons) \nof the two beauty hadrons originally produced together.  \n\n\\item \nIf the lifetimes of the two mass eigenstates of the neutral $B$ \nmeson differ sufficiently from each other, then one can wait \nfor the short-lived  component to fade away relative to the \nlong-lived one and proceed in qualitative analogy to the $K_L$ \ncase. Conceivably this could become feasible -- or even \nessential -- for overly fast oscillating $B_s$ mesons \n\\cite{DUNIETZ}. \n\n\\end{itemize} \nThe degree to which this flavour tagging can be achieved is a crucial \nchallenge each experiment has to face. \n\n\\item \nThe CP asymmetry is largest when the two interfering amplitudes \nare comparable in magnitude. With oscillations having to provide \nthe second amplitude that is absent initially at time of production, \nthe CP asymmetry starts out at zero for decays that occur right after \nproduction and builds up for later decays. The (first) maximum \nof the asymmetry \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\left|  1 - \\frac{1 - {\\rm Im}\\frac{q}{p}\\bar \\rho _f \n{\\rm sin}\\Delta m_Bt}\n{1 + {\\rm Im}\\frac{q}{p}\\bar \\rho _f \n{\\rm sin}\\Delta m_Bt}\n\\right| \n\\eeq \nis reached for \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{t}{\\tau _B} = \\frac{\\pi }{2} \\frac{\\Gamma _B}{\\Delta m_B} \n\\simeq 2 \n\\eeq \nin the case of $B_d$ mesons. \n\n\\item \nThe other side of the coin is that very rapid oscillations -- \n$\\Delta m_B \\gg \\Gamma _B$ as is the case for $B_s$ mesons -- \nwill tend to wash out the asymmetry or at least will severely \ntax the experimental resolution.  \n\n\\end{itemize} \n\n\\subsubsection{Resume}\nThree classes of quantities each describe the three types of CP \nviolation: \n\\begin{enumerate} \n\\item \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\left| \\frac{q}{p} \\right| \\neq 1 \n\\eeq \n\\item \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\left| \n\\frac{T(\\bar B \\rightarrow \\bar f)}{T( B \\rightarrow f)} \n\\right| \\neq 1\n\\eeq \n\n\\item \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\rm Im} \\frac{q}{p} \\frac{T(\\bar B \\rightarrow \\bar f)}{T( B \\rightarrow f)} \n\\neq 0\n\\eeq \n\\end{enumerate}\nThese quantities obviously satisfy one necessary condition \nfor being observables: they are insensitive to the phase convention \nadopted for the anti-state. \n\n\\subsection{Parametric KM Predictions}\nThe triangle defined by \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\lambda V(cb) - V(ub) + V^*(td) = 0 \n\\eeq\nto leading order controls basic features of $B$ transitions. As \ndiscussed before, it has naturally large angles; it usually is called \n{\\em the} KM triangle. Its angles are given by KM matrix elements \nwhich are most concisely expressed in the Wolfenstein \nrepresentation: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \ne^{i\\phi _1} = - \\frac{V(td)}{|V(td)|} \\; \\; ,  \\; \\; \ne^{i\\phi _2} =  \\frac{V^*(td)}{|V(td)|} \\frac{|V(ub)|}{V(ub)}\\; \\; , \\; \\; \ne^{i\\phi _3} =  \\frac{V(ub)}{|V(ub)|} \n\\eeq \nThe various CP asymmetries in beauty decays are expressed in \nterms of these three angles. I will describe `typical' examples now. \n\n\\subsubsection{Angle $\\phi _1$}\nConsider \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\bar B_d \\rightarrow \\psi K_S \\leftarrow B_d\n\\eeq \nwhere the final state is an almost pure odd CP eigenstate. On the \nquark level one has two different reactions, namely one \ndescribing the direct decay process \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\bar B_d = [b \\bar d] \\rightarrow [c\\bar c] [s \\bar d] \n\\label{BDTREE}\n\\eeq \nand the other one involving a $B_d - \\bar B_d$ oscillation: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\bar B_d = [b \\bar d] \\Rightarrow B_d = [\\bar b d] \n\\rightarrow [c \\bar c] [ \\bar sd] \n\\label{BDOSC} \n\\eeq \n\\begin{center} \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $ \\\\ \n{\\em Homework Problem \\# 5}: \n\\end{center}\nHow can the $[s \\bar d]$ combination in \nEq.(\\ref{BDTREE}) interfere with \n$[\\bar sd]$ in Eq.(\\ref{BDOSC})? \n\\begin{center} \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $\n\\end{center} \nSince the final state in $B\/\\bar B \\rightarrow \\psi K_S$ can carry \nisospin 1\/2 only, we have for the {\\em direct} \ntransition amplitudes: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\begin{array}{c}\nT(\\bar B_d \\rightarrow \\psi K_S) = V(cb)V^*(cs) \ne^{i\\alpha _{1\/2}} |{\\cal M}_{1\/2}| \\\\ \nT(B_d \\rightarrow \\psi K_S) = V^*(cb)V(cs) \ne^{i\\alpha _{1\/2}} |{\\cal M}_{1\/2}| \n\\end{array}\n\\eeq \nand thus \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\bar \\rho _{\\psi K_S} = \\frac{V(cb)V^*(cs)}{V^*(cb)V(cs)} \n\\eeq \nfrom which the hadronic quantities, namely the \nphase shift $\\alpha _{1\/2}$ and the hadronic matrix \nelement $|{\\cal M}_{1\/2}|$ -- both of which {\\em cannot} \nbe calculated in a reliable manner -- have dropped out. \nTherefore \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\left| \\bar \\rho _{\\psi K_S} \\right| = \n\\left|  \n\\frac{T(\\bar B_d \\rightarrow \\psi K_S}{T(B_d \\rightarrow \\psi K_S}\n\\right|  = 1 \\; ; \n\\eeq \ni.e., there can be {\\em no direct} CP violation in this channel. \n\nSince $|\\Gamma _{12}| \\ll |M_{12}|$ one has \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{q}{p} \\simeq \\sqrt{\\frac{M^*_{12}}{M_{12}}} = \n\\frac{M^*_{12}}{|M_{12}|} \\simeq \n\\frac{V^*(tb)V(td)}{V(tb)V^*(td)}\n\\eeq \nwhich is a pure phase. Altogether one obtains \n\\footnote{The next-to-last (approximate) equality \nin Eq.(\\ref{SIN2PHI1}) holds in the Wolfenstein \nrepresentation, although the overall result is general.}\n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\rm Im} \\frac{q}{p} \\bar \\rho _{\\psi K_S} = \n{\\rm Im} \\left( \n\\frac{V^*(tb)V(td)} {V(tb)V^*(td)} \n\\frac{V(cb)V^*(cs)} {V^*(cb)V(cs)}  \n \\right) \n\\simeq {\\rm Im} \\frac{V^2(td)}{|V(td)|^2} = \n-{\\rm sin}2\\phi _1 \n\\label{SIN2PHI1} \n\\eeq \nThat means \nthat to a very good approximation the observable \nIm $\\frac{q}{p} \\bar \\rho _{\\psi K_S}$, which is the amplitude \nof the oscillating CP asymmetry, is in general given by \n{\\em microscopic} parameters of the theory; within \nthe KM ansatz they combine to yield the angle $\\phi _1$ \n\\cite{BS}. \n\nSeveral other channels are predicted to exhibit a CP asymmetry \nexpressed by sin$2\\phi _1$, like $B_d \\rightarrow \\psi K_L$ \n\\footnote{Keep in mind that \nIm$\\frac{q}{p}\\bar \\rho _{\\psi K_L} =- \n{\\rm Im}\\frac{q}{p}\\bar \\rho _{\\psi K_S}$ holds because \n$K_L$ is mainly CP odd and $K_S$ mainly CP even.}, \n$B_d \\rightarrow D \\bar D$ etc. \n\n\\subsubsection{Angle $\\phi _2$}\nThe situation is not quite as clean for the angle $\\phi _2$. \nThe asymmetry in $\\bar B_d \\rightarrow \\pi ^+ \\pi ^-$ vs. \n$B_d \\rightarrow \\pi ^+ \\pi ^-$ is certainly sensitive to $\\phi _2$, \nyet there are two complications:  \n\\begin{itemize}\n\\item \nThe final state is described by a superposition of {\\em two} \ndifferent isospin states, namely $I = 0$ and $2$. The \nspectator process contributes to both of them. \n\\item \nThe Cabibbo suppressed Penguin operator \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nb \\rightarrow d g^* \\rightarrow d u \\bar u \n\\eeq \nwill also contribute, albeit only to the $I=0$ amplitude.  \n\\end{itemize} \n\nThe direct transition amplitudes are then expressed as follows: \n$$  \nT(\\bar B_d \\rightarrow \\pi ^+ \\pi ^-) = \nV(ub)V^*(ud)e^{i \\alpha _2}|{\\cal M}_2^{spect}| + \n$$ \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n+ e^{i \\alpha _0}\\left( V(ub)V^*(ud) |{\\cal M}_0^{spect}| + \nV(tb)V^*(td) |{\\cal M}_0^{Peng}| \n\\right)  \n\\eeq   \n$$  \nT(B_d \\rightarrow \\pi ^+ \\pi ^-) = \nV^*(ub)V(ud)e^{i \\alpha _2}|{\\cal M}_2^{spect}| + \n$$ \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n+ e^{i \\alpha _0}\\left( V^*(ub)V(ud) |{\\cal M}_0^{spect}| + \nV^*(tb)V(td) |{\\cal M}_0^{Peng}| \n\\right)  \n\\eeq  \nwhere the phase shifts for the $I=0,2$ states have been factored \noff. \n\n{\\em If} there were no Penguin contributions, we would have \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\rm Im} \\frac{q}{p} \\bar \\rho _{\\pi \\pi} = \n{\\rm Im} \\frac{V(td)V^*(tb)V(ub)V^*(ud)}\n{V^*(td)V(tb)V^*(ub)V(ud)} = \n- {\\rm sin}2\\phi _2 \n\\eeq \nwithout direct CP violation -- $|\\bar \\rho _{\\pi \\pi }| =1$ --  \nsince the two isospin amplitudes still contain the same weak \nparameters. The Penguin contribution changes the picture in \ntwo basic ways: \n\\begin{enumerate}\n\\item \nThe CP asymmetry no longer depends on $\\phi _2$ alone: \n$$  \n{\\rm Im} \\frac{q}{p} \\bar \\rho _{\\pi \\pi } \\simeq  \n- {\\rm sin} 2\\phi _2  + \\left|  \\frac{V(td)}{V(ub)} \\right|  \n\\left[  {\\rm Im}\\left( e^{-i\\phi _2}\n\\frac{{\\cal M}^{Peng}}{{\\cal M}^{spect}}\\right)  \n- {\\rm Im}\\left( e^{-3i\\phi _2}\n\\frac{{\\cal M}^{Peng}}{{\\cal M}^{spect}}\\right) \n\\right] + \n$$ \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n+ {\\cal O}(|{\\cal M}^{Peng}|^2\/|{\\cal M}^{spect}|^2) \n\\label{PENGPOLL} \n\\eeq \nwhere \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\cal M}^{spect} = e^{i \\alpha _0}|{\\cal M}^{spect}_0|  + \ne^{i \\alpha _2}|{\\cal M}^{spect}_2| \\; \\; , \\;  \\; \n{\\cal M}^{Peng} = e^{i \\alpha _0}|{\\cal M}^{Peng}_0| \n\\eeq \n\\item \nA direct CP asymmetry emerges:\n\\begin{equation}}   \\def\\eeq{\\end{equation} \n|\\bar \\rho _{\\pi \\pi}| \\neq 1 \n\\eeq \n\\end{enumerate}\nSince we are dealing with a Cabibbo suppressed Penguin operator, \nwe expect that its contribution is reduced relative to the \nspectator term: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\left| \\frac{{\\cal M}^{Peng}}{{\\cal M}^{spect}}  \n\\right| \n< 1 \\; , \n\\eeq \nwhich was already used in Eq.(\\ref{PENGPOLL}). \nUnfortunately this reduction might \nnot be very large. This concern is based on the observation \nthat the branching ratio for $\\bar B_d \\rightarrow K^- \\pi ^+$ appears to \nbe somewhat larger than for $\\bar B_d \\rightarrow \\pi ^+ \\pi ^-$ \nimplying that the Cabibbo favoured Penguin amplitude  \nis at least not smaller than the spectator amplitude. \n\nVarious strategies have been suggested to unfold the \nPenguin contribution through a combination of additional \nor other  \nmeasurements (of other $B \\rightarrow \\pi \\pi $ channels \nor of $B\\rightarrow \\pi \\rho$, $B \\rightarrow K\\pi$ etc.) and supplemented by \ntheoretical considerations like $SU(3)_{Fl}$ symmetry \n\\cite{PENGTRAP}.  \nI am actually hopeful that the multitude of exclusive \nnonleptonic decays (which is the other side of the \ncoin of small branching ratios!) can be harnessed to \nextract a wealth of information on the strong dynamics that \nin turn will enable us to extract \nsin$2\\phi _2$ with decent accuracy. \n\n\\subsubsection{The $\\phi _3$ Saga}\nOf course it is important to determine $\\phi _3$ as accurately  \nas possible. This will not be easy, and one better keep \na proper perspective. I am going to tell this saga now in \ntwo installments. \n\n{\\bf (I)} {\\em CP asymmetries involving $B_s - \\bar B_s$ \nOscillations}: In principle one can extract $\\phi _3$ from \nKM suppressed $B_s$ decays like one does $\\phi _2$ from \n$B_d$ decays, namely by measuring and analyzing the difference \nbetween the rates for, say, $\\bar B_s(t) \\rightarrow K_S \\rho ^0$ and \n$B_s(t) \\rightarrow K_S \\rho ^0$: \nIm$\\frac{q}{p}\\bar \\rho _{K_S\\rho ^0} \\sim {\\rm sin}2\\phi _3$. \nOne has to face the same complication, namely that in \naddition to the spectator term a \n(Cabibbo suppressed) Penguin amplitude contributes to $\\bar \\rho \n_{K_S\\rho ^0}$ with different weak parameters. Yet the situation \nis much more challenging due to the rapid \npace of the $B_s - \\bar B_s$ oscillations. \n\n\\noindent A more promising way might be to compare the rates for \n$\\bar B_s (t) \\rightarrow D_s^+ K^-$ with $B_s (t) \\rightarrow D_s^- K^+$ as a function \nof the time of decay $t$ since there is no Penguin contribution. \nThe asymmetry depends on sin$\\phi _3$ rather than \nsin$2\\phi _3$ \n\\footnote{Both $D_s^+K^-$ and $D_s^- K^+$ are final states common to \n$B_s$ and $\\bar B_s$ decays although they are not CP \neigenstates.}.  \n\n{\\bf (II)} {\\em Direct CP Asymmetries}: The largish direct CP \nasymmetries sketched above for $B \\rightarrow K \\pi$ depend on \nsin$\\phi _3$ -- and on the phase shifts which in general are neither  \nknown nor calculable. Yet in some cases they can be determined \nexperimentally -- as first described for \n$B^{\\pm} \\rightarrow D_{neutral} K^{\\pm}$ \n\\cite{WYLER}. There are \n{\\em four independant} rates that can be measured, namely \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\Gamma (B^- \\rightarrow D^0 K^-) \\; , \\; \n\\Gamma (B^- \\rightarrow \\bar D^0 K^-) \\;  , \\;  \n\\Gamma (B^- \\rightarrow D_{\\pm} K^-) \\; , \\; \n\\Gamma (B^+ \\rightarrow D_{\\pm} K^+) \n\\eeq  \nThe {\\em flavour eigenstates} $D^0$ and $\\bar D^0$ are defined \nthrough flavour specific modes, namely \n$D^0 \\rightarrow l^+ X$ and $\\bar D^0 \\rightarrow l^- X$, respectively; \n$D_{\\pm}$ denote the even\/odd CP eigenstates \n$D_{\\pm} = (D^0 \\pm \\bar D^0)\/\\sqrt{2}$ defined by \n$D_+ \\rightarrow K^+K^-, \\, \\pi ^+ \\pi ^-,$ etc., \n$D_- \\rightarrow K_S\\pi ^0, \\, K_S \\eta ,$ etc. \\cite{PAISSB}. \n\nFrom these four observables one can (up to a binary ambiguity) \nextract the four basic quantities, namely the moduli of the two \nindependant amplitudes ($|T(B^- \\rightarrow D^0 K^-)|$, \n$|T(B^- \\rightarrow \\bar D^0 K^-)|$), their strong phaseshift -- and \nsin$\\phi _3$, the goal of the enterprise! \n\n\\subsubsection{A Zero-Background Search for New \nPhysics: $B_s \\rightarrow \\psi \\phi , \\, D_s^+ D_s^-$}\nThe two angles $\\phi _1$ and $\\phi _2$ will be measured in \nthe next several years with decent or even good accuracy. I \nfind it unlikely that any of the direct measurements \nof $\\phi _3$ sketched above will yield a more precise \nvalue than inferred from simple trigonometry: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\phi _3 = 180^o - \\phi _1 - \\phi _2 \n\\label{180}\n\\eeq \nEq.(\\ref{180}) holds \nwithin the KM ansatz; of course the real goal is to uncover \nthe intervention of New Physics in $B_s$ transitions. It then \nmakes eminent sense to search for it in a reaction where \nKnown Physics predicts a practically zero result. \n$B_s \\rightarrow \\psi \\phi , \\, \\psi \\eta , \\, D_s \\bar D_s$ fit this bill \n\\cite{BS}: \nto leading order in the KM parameters the CP asymmetry has \nto vanish since on that level quarks of the second and third \nfamily only participate in $B_s - \\bar B_s$ oscillations -- \n$[s \\bar b] \\Rightarrow t^* \\bar t^* \\Rightarrow [b \\bar s]$ -- and \nin these direct decays -- $[b \\bar s] \\rightarrow c \\bar c s \\bar s$. Any \nCP asymmetry is therefore Cabibbo suppressed, i.e. $\\leq 4$\\% . \nMore specifically \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\left. {\\rm Im} \\frac{q}{p}\\bar \\rho _{B_s \\rightarrow \\psi \\eta , \n\\psi \\phi , D_s \\bar D_s} \\right| _{KM} \\sim 2\\% \n\\eeq  \nYet New Physics has a good chance to contribute to \n$B_s - \\bar B_s$ oscillations; if so, there is no reason for \nit to conserve CP and asymmetries can emerge that are easily \nwell in excess of 2\\% . New Physics scenarios with non-minimal \nSUSY or flavour-changing neutral currents could actually \nyield asymmetries of $\\sim 10 \\div 30 \\%$ \n\\cite{GABB} -- completely \nbeyond the KM reach!  \n\n\n\\subsubsection{The HERA-B Menu}\nQuite often people in the US tend to believe that a restaurant that \npresents them with a long menu must be a very good one. The \nreal experts -- like the French and Italians -- of course know \nbetter: it is the hallmark of a top cuisine to concentrate on a \nfew very special dishes and prepare them in a spectacular fashion \nrather than spread one's capabilities too thinly. That is exactly the advice \nI would like to give the HERA-B collaboration, namely to focus \non a first class menu consisting of three main dishes and one side \ndish, namely \n\\begin{enumerate} \n\\item \nmeasure $\\Delta m(B_s)$ which within the Standard Model \nallows to extract $|V(td)|$ through \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{\\Delta m(B_d)}{\\Delta m(B_s)} \\simeq \n\\frac{Bf_{B_d}^2} {Bf_{B_s}^2} \\left|  \\frac{V(td)}{V(ts)} \n\\right| ^2 \\; ; \n\\eeq \n\n\\item \ndetermine the rates for $\\bar B_d \\rightarrow \\psi K_S$ and \n$B_d \\rightarrow \\psi K_S$ to obtain the value of \nsin$2\\phi _1$; \n\n\\item \ncompare $\\bar B_s \\rightarrow \\psi \\phi , \\, D_s \\bar D_s$ with \n$B_s \\rightarrow \\psi \\phi , \\, D_s \\bar D_s$ \nas a clean search for New Physics and \n\n\\item \nas a side dish: measure the $B_s$ lifetime separately in \n$B_s \\rightarrow l \\nu D_s^{(*)}$ and $B_s \\rightarrow \\psi \\phi , \\, D_s \\bar D_s$ where \nthe former yields the algebraic average of the $B_{s, short}$ \nand $B_{s,long}$ lifetimes and the latter the $B_{s, short}$ \nlifetime. One predicts for them \\cite{URALTSEV}:\n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{\\tau (B_s \\rightarrow l \\nu D_s^{(*)}) - \\tau (B_s \\rightarrow \\psi \\phi \n,\\, D_s \\bar D_s)}\n{\\tau (B_s \\rightarrow l \\nu D_s^{(*)})} \\simeq 0.1 \\cdot \n\\left(  \\frac{f_{B_s}}{200\\; {\\rm MeV}} \\right) ^2\n\\eeq \n\n\\end{enumerate} \nIf the HERA-B chefs succeed in preparing one of these  \nmain dishes, then they have achieved three star status!\n\n\n\\subsection{Theoretical Technologies in Heavy Flavour Decays}\nOne other intriguing and gratifying aspect of heavy flavour \ndecays has become understood just over the last several \nyears, namely that the decays in particular of beauty hadrons \ncan be treated with a reliability and accuracy that before would \nhave seemed to be unattainable. These new theoretical \ntechnologies can be referred to as {\\em Heavy Quark Theory} \nwhich combines two basic elements, namely an \nasymptotic symmetry principle on one hand and a dynamical \ntreatment on the other, which tells us how the asymptotic limit \nis approached. The symmetry principle is Heavy Quark \nSymmetry stating that all sufficiently heavy quarks behave \nidentically under the strong interactions. The dynamical treatment \nis provided by $1\/m_Q$ expansions allowing us to express \nobservable transition rates through a series in inverse powers \nof the heavy quark mass. This situation is qualitatively similar \nto chiral considerations which start from the limit of chiral \ninvariance and describe the deviations from it through \nchiral perturbation theory. In both cases one has succeeded in \ndescribing nonperturbative dynamics in special cases. \n\nThe lessons we have learnt can be summarized as follows \n\\cite{HQEREV}: we have \n\\begin{itemize}\n\\item \nidentified the sources of the non-perturbative corrections;  \n\\item \nfound them to be smaller than they could have been; \n\\item \nsucceeded in relating the basic quantities of the Heavy Quark \nTheory -- KM paramters, masses and kinetic energy of heavy quarks, \netc. -- to various a priori independant observables with a fair \namount of  redundancy; \n\\item \ndeveloped a better understanding of incorporating perturbative \nand nonperturbative corrections without double-counting.  \n\\end{itemize}  \nIt has been shown that the heavy quark expansion has to be \nformulated in terms of short distance masses rather than \npole masses. One finds \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\begin{array} {c} \nm_b - m_c = 3.50 \\pm 0.04 \\; {\\rm GeV} \\\\ \nm_b(1 \\; {\\rm GeV}) = 4.64 \\pm 0.05 \\; {\\rm GeV} \n\\end{array} \n\\eeq  \nThis information is then used to extract $|V(cb)|$ from the \nobserved semileptonic $B$ width with the result \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n|V(cb)|_{incl} = 0.0412 \\cdot \n\\sqrt{\\frac{{\\rm BR}(B\\rightarrow l X)}{0.105}} \\cdot \n\\sqrt{\\frac{1.6 \\; {\\rm psec}}{\\tau _B}}\\cdot \n\\left( 1 \\pm 0.05 |_{theor} \\right) \n\\label{VCBIN}\n\\eeq \nAlternatively one can analyze the exclusive mode \n$B\\rightarrow l \\nu D^*$ and extrapolate to the kinematical \npoint of zero recoil to obtain \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n|F_{D^*}(0) V(cb)| = 0.0339 \\pm 0.0014 \n\\eeq \nFrom Heavy Quark Theory one infers \\cite{HQEREV} \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nF_{D^*}(0) = 0.91  \\pm 0.06  \n\\eeq \nto arrive at \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n|V(cb)|_{excl} = 0.0377 \\pm 0.0016|_{exp} \\pm 0.002 |_{theor}\n\\label{VCBEX}\n\\eeq \nThe two determinations in Eqs.(\\ref{VCBIN}) and (\\ref{VCBEX}) \nare systematically very different both in their experimental and \ntheoretical aspects. Nevertheless they are quite consistent with \neach other with the experimental and theoretical uncertainties \nbeing very similar. A few years ago it would have seemed \nquite preposterous to claim such small theoretical uncertainties! \nI am actually confident that those can be reduced from the \npresent 5\\% level down to the 2\\% level in the foreseeable future.  \n\n$|V(ub)|$ (or $|V(ub)\/V(cb)|$) is not known with an even remotely \nsimilar accuracy, and so far one has relied on models rather \nthan QCD proper to extract it from data. Yet we can be confident \nthat over the next ten years $|V(ub)|$  will be determined with a \ntheoretical uncertainty below 10\\% . It will be important to obtain \nit from systematically different semileptonic distributions and \nprocesses; Heavy Quark Theory provides us with the \nindispensable tools for combining the various analyses in a \ncoherent fashion. \n\nThis theoretical progress can embolden us to hope that in the end \neven $|V(td)|$ can be determined with good accuracy -- say \n$\\sim 10 \\div 15\\%$ -- from \n$\\Gamma (K^+ \\rightarrow \\pi ^+ \\nu \\bar \\nu )$, \n$\\Delta m(B_s)$ vs. $\\Delta m(B_d)$ or \n$\\Gamma (B \\rightarrow \\gamma \\rho \/\\omega )$ vs. \n$\\Gamma (B \\rightarrow \\gamma K^* )$ etc. \n\n\\subsection{KM Trigonometry}\nOne side of the triangle is exactly known since the base line can be \nnormalized to unity without affecting the angles: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n1 - \\frac{V(ub)}{\\lambda V(cb)} + \n\\frac{V^*(td)}{\\lambda V(cb)} = 0 \n\\eeq \nThe second side is known to some degree from semileptonic $B$ \ndecays: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\left| \n\\frac{V(ub)}{V(cb)} \n\\right| \n\\simeq 0.08 \\pm 0.03\n\\eeq\nwhere the quoted uncertainty is mainly theoretical and amounts \nto little more than a guestimate. In the Wolfenstein representation \nthis reads as \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\sqrt{\\rho ^2 + \\eta ^2} \\simeq 0.38 \\pm 0.11\n\\eeq  \nThe area cannot vanish since $\\epsilon _K \\neq 0$. Yet at present \nnot much more can be said for certain. \n\nIn principle one would have enough observables -- namely \n$\\epsilon _K$ and $\\Delta m(B_d)$ in addition to \n$|V(ub)\/V(cb)|$ -- to determine the two KM parameters \n$\\rho$ and $\\eta$ in a {\\em redundant} way. In practise, though, \nthere are two further unknowns, namely the size of the \n$\\Delta S=2$ and $\\Delta B=2$ matrix elements, as expressed through \n$B_K$ and $B_B f_B^2$. For $m_t$ sufficiently large $\\epsilon _K$ \nis dominated by the top contribution:   \n$d \\bar s \\Rightarrow  t^* \\bar t^* \\Rightarrow s \\bar d$. The same holds \nalways for $\\Delta m(B_d)$. In that case things are simpler: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{|\\epsilon _K|}{\\Delta m(B_d)} \\propto \n{\\rm sin}2\\phi _1 \\simeq \n0.42 \\cdot UNC \n\\label{SIN2BETA} \n\\eeq \nwith the factor $UNC$ parametrising the uncertainties \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nUNC \\simeq \\left( \\frac{0.04}{|V(cb)|}\\right) \n\\left( \\frac{0.72}{x_d}\\right) \\cdot \n\\left( \\frac{\\eta _{QCD}^{(B)}}{0.55}\\right) \\cdot \n\\left( \\frac{0.62}{\\eta _{QCD}^{(K)}}\\right) \\cdot \n\\left( \\frac{2B_B}{3B_K}\\right) \\cdot \n\\left( \\frac{f_B}{160\\, {\\rm MeV}}\\right) ^2  \n\\eeq\nwhere   \n$x_d \\equiv \\Delta m(B_d)\/\\Gamma (B_d)$; \n$\\eta _{QCD}^{(B)}$ and $\\eta _{QCD}^{(K)}$ denote the \nQCD radiative corrections for ${\\cal H}(\\Delta B=2)$ and \n${\\cal H}(\\Delta S=2)$, respectively; $B_B$ and $B_K$ \nexpress the expectation value of ${\\cal H}(\\Delta B=2)$ or \n${\\cal H}(\\Delta S=2)$ in units of the `vacuum saturation' \nresult which is given in terms of the decay constants \n$f_B$ and $f_K$ (where the latter is known). The main \nuncertainty is obviously of a theoretical nature related to \nthe hadronic parameters $B_B$, $B_K$ and $f_B$; as discussed \nbefore, state-of-the-art theoretical technologies yield \n$B_B \\simeq  1$, $B_K \\simeq 0.8 \\pm 0.2$ and \n$f_B \\simeq 180 \\pm 30 \\, {\\rm MeV}$ where the latter range \nmight turn out to be anything but conservative! \nEq.(\\ref{SIN2BETA}) represents an explicit illustration that some CP \nasymmetries in $B^0$ decays are huge. \n\nFor $m_t \\simeq 180$ GeV the $c \\bar c$ and $c\\bar t + t \\bar c$ \ncontributions to $\\epsilon _K$ are still sizeable; nevertheless \nEq.(\\ref{SIN2BETA}) provides a good approximation. Furthermore \nsin$2\\phi _1$ can still be expressed reliably as a function of the \nhadronic matrix elements: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\rm sin}2\\phi _1 = f(B_Bf_B^2\/B_K) \n\\eeq  \nIt will become obvious why this is relevant. \n\nThe general idea is, of course, to construct the triangle as accurately \nas possible and then probe it; i.e. search for inconsistencies that \nwould signal the intervention of New Physics. A few remarks on that \nwill have to suffice here. \n\nAs indicated before we can expect the value of \n$|V(ub)\/V(cb)|$ to be known to better than 10\\% and hope for \n$|V(td)|$ to be determined with decent accuracy as well. \nThe triangle \nwill then be well determined or even overdetermined. Once the \nfirst asymmetry in $B$ decays that can be interpreted reliably -- \nsay in $B_d \\rightarrow \\psi K_S$ -- has been measured and $\\phi _1$ \nbeen determined, the triangle is fully constructed from \n$B$ decays alone. Furthermore one has arrived at the first \nsensitive consistency check of the triangle: one  \ncompares the measured value of sin$2\\phi _1$ with \nEq.(\\ref{SIN2BETA}) to infer which value of \n$B_Bf_B^2$ is thus required; this value is inserted into the \nStandard Model expression for $\\Delta m(B_d)$ together \nwith $m_t$ to see whether the experimental result is \nreproduced. \n\nA host of other tests can be performed that are highly sensitive to \n\\begin{itemize}\n\\item \nthe presence of New Physics and \n\\item \nto some of their salient dynamical features. \n\\end{itemize}\nDetails can be found in the ample literature on that subject.  \n\n\n\n\n\\section{Oscillations and CP Violation in Charm Decays -- \nThe Underdog's Chance for Fame}\nIt is certainly true that \n\\begin{itemize}\n\\item \n$D^0-\\bar D^0$ oscillations proceed very slowly in the \nStandard Model and \n\\item \nCP asymmetries in $D$ decays are small or even tiny within \nthe KM ansatz. \n\\end{itemize}\nYet the relevant question quantitatively is: how slow and how small? \n\n\\subsection{$D^0-\\bar D^0$ Oscillations}\nBounds on $D^0 - \\bar D^0$ oscillations are most cleanly   \nexpressed through `wrong-sign' semileptonic decays: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nr_D = \\frac{\\Gamma (D^0 \\rightarrow l^-X)}{\\Gamma (D^0 \\rightarrow l^+X)} \n\\simeq \\frac{1}{2} \\left( x_D^2 + y_D^2\\right)  \n\\eeq \nwith $x_D = \\Delta m_D\/\\Gamma _D$, \n$y_D = \\Delta \\Gamma _D\/2\\Gamma _D$. It is often stated that  \nthe Standard Model predicts \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nr_D \\leq 10^{-7} \\; \\; \\hat = \\; \\; \nx_D, \\, y_D \\leq 3 \\cdot 10^{-4} \n\\eeq \nI myself am somewhat flabbergasted by the boldness of such \npredictions. For one should keep the following in mind for \nproper perspective: there are quite a few channels that can drive \n$D^0 - \\bar D^0$ \noscillations -- like $D^0 \\Rightarrow K\\bar K , \\; \\pi \\pi \\Rightarrow  \n\\bar D^0$ or $D^0 \\Rightarrow K^- \\pi ^+ \\Rightarrow \n\\bar D^0$ -- and \nthey branching ratios on the $({\\rm few})\\times 10^{-3}$ \nlevel \n\\footnote{For the $K^- \\pi ^+$ mode this represents the average of \nits Cabibbo allowed and doubly Cabibbo suppressed incarnations.}. \nIn the limit of $SU(3)_{Fl}$ symmetry all these contributions have \nto cancel of course. Yet there are sizeable violations of $SU(3)_{Fl}$ \ninvariance in $D$ decays, and one should have little confidence in an \nimperfect symmetry to ensure that a host of channels with branching \nratios of order few$\\times 10^{-3}$ will cancel as to render \n$x_D, \\, y_D \\leq 3\\cdot 10^{-4}$. To say it differently: \nThe relevant question in this context is {\\em not} whether \n$r_D \\sim 10^{-7} \\div 10^{-6}$ is a possible or even reasonable \nStandard Model estimate, but whether \n$10^{-6} \\leq r_D \\leq 10^{-4}$ can {\\em reliably be ruled out}! I \ncannot see how anyone could make such a claim with the \nrequired confidence. \n\nThe present experimental bound is \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nr_D |_{exp} \\leq 3.4 \\cdot 10^{-3} \\; \\; \\hat = \\; \\; \nx_D, \\, y_D \\leq 0.1 \n\\eeq \nto be compared with a {\\em conservative} Standard Model bound \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nr_D |_{SM} < 10^{-4} \\; \\; \\hat = \\; \\; y_D, \\, x_D|_{SM}\n\\leq 10^{-2} \\; \n\\eeq \nNew Physics on the other hand can enhance $\\Delta m_D$ \n(though not \n$\\Delta \\Gamma _D$) very considerably up to \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nx_D |_{NP} \\sim 0.1 \\; , \n\\eeq \ni.e. the present experimental bound. \n\n\\subsection{CP Violation involving $D^0 - \\bar D^0$ Oscillations}\nOne can discuss this topic in close qualitative analogy to \n$B$ decays. First one considers final states that are CP \neigenstates like $K^+K^-$ or $\\pi ^+ \\pi ^-$ \n\\cite{BSDDBAR}: \n$$  \n{\\rm rate}(D^0(t) \\rightarrow K^+ K^-) \\propto e^{-\\Gamma _D t} \n\\left(  \n1+ ({\\rm sin}\\Delta m_Dt) \\cdot {\\rm Im}\\frac{q}{p}\n\\bar \\rho _{K^+K^-}   \n\\right) \n\\simeq \n$$\n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\simeq e^{-\\Gamma _D t} \n\\left(  \n1+ \\frac{\\Delta m_Dt}{\\Gamma _D}\\cdot \n\\frac{t}{\\tau _D} \n\\cdot {\\rm Im}\\frac{q}{p} \\bar \\rho _{K^+K^-}   \n\\right) \n\\eeq \nWith $x_D|_{SM} \\leq 10^{-2}$ and \nIm$\\frac{q}{p} \\bar \\rho _{K^+K^-}|_{KM} \\sim {\\cal O}(10^{-3})$ one \narrives at an asymmetry of around $10^{-5}$, i.e. for all practical \npurposes zero, since it presents the product of two \nvery small numbers. \nYet with New Physics one conceivably has $x_D|_{NP} \\leq 0.1$, \nIm$\\frac{q}{p} \\bar \\rho _{K^+K^-}|_{NP} \\sim {\\cal O}(10^{-1})$ \nleading to an asymmetry that could be as large as of order 1\\%. \nLikewise one should compare the doubly Cabibbo suppressed transitions \\cite{BIGIBERK,NIR}\n$$ \n{\\rm rate}(D^0(t) \\rightarrow K^+ \\pi ^-) \\propto \ne^{-\\Gamma _{D^0} t} {\\rm tg}^4\\theta _C|\\hat \\rho _{K\\pi }|^2 \\cdot \n$$ \n$$ \n\\cdot \\left[ 1 - \\frac{1}{2}\\Delta \\Gamma _D t + \n\\frac{(\\Delta m_Dt)^2}{4{\\rm tg}^4\\theta _C|\\hat \\rho _{K\\pi }|^2} \n+ \\frac{\\Delta \\Gamma _Dt}\n{2{\\rm tg}^2\\theta _C|\\hat \\rho _{K\\pi }|}\n{\\rm Re}\\left( \n\\frac{p}{q}\\frac{\\hat \\rho _{K\\pi }}{|\\hat \\rho _{K\\pi }|}   \n\\right)  \n- \\right. \n$$ \n\\begin{equation}}   \\def\\eeq{\\end{equation}\n\\left. - \\frac{\\Delta m_Dt}{{\\rm tg}^2\\theta _C|\\hat \\rho _{K\\pi }|} \n{\\rm Im}\\left( \n\\frac{p}{q}\\frac{\\hat \\rho _{K\\pi }}{|\\hat \\rho _{K\\pi }|}   \n\\right)  \n\\right] \n\\eeq \n$$ \n{\\rm rate}(\\bar D^0(t) \\rightarrow K^- \\pi ^+) \\propto \ne^{-\\Gamma _{D^0} t} {\\rm tg}^4\\theta _C|\\hat{\\bar \\rho }_{K\\pi }|^2 \\cdot \n$$ \n$$ \n\\cdot \\left[ 1 - \\frac{1}{2}\\Delta \\Gamma _D t + \n\\frac{(\\Delta m_Dt)^2}{4{\\rm tg}^4\\theta _C\n|\\hat{\\bar \\rho}_{K\\pi }|^2} \n+ \\frac{\\Delta \\Gamma _Dt}\n{2{\\rm tg}^2\\theta _C|\\hat{\\bar \\rho }_{K\\pi }|}\n{\\rm Re}\\left( \n\\frac{p}{q}\\frac{\\hat{\\bar \\rho }_{K\\pi }}\n{|\\hat{\\bar \\rho }_{K\\pi }|}   \n\\right)  \n+\\right. \n$$ \n\\begin{equation}}   \\def\\eeq{\\end{equation}\n\\left. + \\frac{\\Delta m_Dt}{{\\rm tg}^2\\theta _C\n|\\hat{\\bar \\rho }_{K\\pi }|} \n{\\rm Im}\\left( \n\\frac{p}{q}\\frac{\\hat{\\bar \\rho }_{K\\pi }}{|\\hat{\\bar \\rho }_{K\\pi }|}   \n\\right)  \n\\right] \n\\eeq\nwhere \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\rm tg}^2\\theta _C \\cdot \\hat \\rho _{K\\pi } \\equiv \n\\frac{T(D^0 \\rightarrow K^+ \\pi ^-)}{T(D^0 \\rightarrow K^- \\pi ^+)}\\; \\; , \\; \\; \n{\\rm tg}^2\\theta _C \\cdot \\hat{\\bar \\rho }_{K\\pi } \\equiv \n\\frac{T(\\bar D^0 \\rightarrow K^- \\pi ^+)}{T(\\bar D^0 \\rightarrow K^+ \\pi ^-)}\\; ; \n\\eeq  \nin such New Physics scenarios \none would expect a considerably enhanced asymmetry \nof order $1\\%\/{\\rm tg}^2 \\theta _C \\sim 20\\%$ -- at the cost \nof  smaller statistics. \n\nEffects of that size would unequivocally signal the intervention \nof New Physics! \n\n\n\\subsection{Direct CP Violation}\nAs explained before a direct CP asymmetry requires the presence \nof two coherent amplitudes with different weak and different \nstrong phases. Within the Standard Model (and the \nKM ansatz) such effects can occur in Cabibbo suppressed \n\\footnote{The effect could well reach the $10^{-3}$ \nand exceptionally the $10^{-2}$ level.}, yet not \nin Cabibbo allowed or doubly Cabibbo suppressed modes. There \nis a subtlety involved in this statement. Consider for example  \n$D^+ \\rightarrow K_S \\pi ^+$. At first sight it appears to be \na Cabibbo allowed mode described by a single amplitude without \nthe possibility of an asymmetry. However \\cite{YAMAMOTO}  \n\\begin{itemize} \n\\item \ndue to $K^0 - \\bar K^0$ mixing the final state \ncan be reached also through a doubly Cabibbo suppressed \nreaction, and the two amplitudes necessarily interfere; \n\\item \nbecause of the CP violation in the $K^0 - \\bar K^0$ complex \nthere is an asymmetry that can be predicted on general grounds \n$$  \n\\frac{\\Gamma (D^+ \\rightarrow K_S \\pi ^+) - \\Gamma (D^- \\rightarrow K_S \\pi ^+)} \n{\\Gamma (D^+ \\rightarrow K_S \\pi ^+) + \\Gamma (D^- \\rightarrow K_S \\pi ^+)} \n\\simeq - 2 {\\rm Re}\\, \\epsilon _K  \\simeq - 3.3 \\cdot 10^{-3} \n\\simeq \n$$  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\simeq \n\\frac{\\Gamma (D^+ \\rightarrow K_L \\pi ^+) - \\Gamma (D^- \\rightarrow K_L \\pi ^+)} \n{\\Gamma (D^+ \\rightarrow K_L \\pi ^+) + \\Gamma (D^- \\rightarrow K_L \\pi ^+)} \n \\; ; \n\\label{DIRECTCPEPS}\n\\eeq  \n\\item\n If New Physics contributes to the doubly Cabibbo suppressed \namplitude $D^+ \\rightarrow K^0 \\pi ^+$ (or $D^- \\rightarrow \\bar K^0 \\pi ^-$) then \nan asymmetry could occur quite conceivably on the few percent \nscale; \n\\item \nsuch a manifestation of New Physics would be unequivocal; against \nthe impact of $\\epsilon _K$, Eq.(\\ref{DIRECTCPEPS}) it \ncould be distinguished not only through the size of the asymmetry, \nbut also how it surfaces in $D^+ \\rightarrow K_L \\pi ^+$ vs. \n$D^- \\rightarrow K_L \\pi ^-$: if it is New Physics one has \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\frac{\\Gamma (D^+ \\rightarrow K_S \\pi ^+) - \\Gamma (D^- \\rightarrow K_S \\pi ^+)} \n{\\Gamma (D^+ \\rightarrow K_S \\pi ^+) + \\Gamma (D^- \\rightarrow K_S \\pi ^+)} \n= - \n\\frac{\\Gamma (D^+ \\rightarrow K_L \\pi ^+) - \\Gamma (D^- \\rightarrow K_L \\pi ^+)} \n{\\Gamma (D^+ \\rightarrow K_L \\pi ^+) + \\Gamma (D^- \\rightarrow K_L \\pi ^+)} \n\\eeq \ni.e.,  the CP asymmetries in $D \\rightarrow K_S \\pi$ and $D \\rightarrow K_L \\pi$ \ndiffer in sign -- in contrast to Eq.(\\ref{DIRECTCPEPS}). \n\\end{itemize} \n\n\\section{Baryogenesis  in the Universe}\n\\subsection{The Challenge}\n\nOne of the most intriguing aspects of big bang \ncosmology is to `understand' nucleosynthesis, i.e. to reproduce the \nabundances observed for the nuclei in the universe as \n{\\em dynamically} generated rather than merely dialed as \ninput values. This challenge has been met successfully, in \nparticular for the light nuclei, and actually so much so that \nit is used to obtain information on dark matter in the universe, \nthe number of neutrinos etc. It is natural to ask whether such \na success could be repeated for an even more basic quantity, \nnamely the baryon number density of the universe which is defined \nas the difference in the abundances of baryons and antibaryons: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\Delta n_{Bar} \\equiv n_{Bar} - n_{\\overline{Bar}} \n\\eeq \nQualitatively one can summarize the observations through \ntwo statements: \n\\begin{itemize}\n\\item \nThe universe is not empty. \n\\item \nThe universe is almost empty. \n\\end{itemize}\nMore quantitatively one finds\n\\begin{equation}}   \\def\\eeq{\\end{equation} \nr_{Bar} \\equiv \\frac{\\Delta n_{Bar}}{n_{\\gamma}} \\sim {\\rm few} \n\\times 10^{-10} \n\\label{BNUMUNI}\n\\eeq \nwhere $n_{\\gamma}$ denotes the number density of photons in the \ncosmic background radiation. Actually we know more, namely \nthat at least in our corner of the universe there are \npractically no primary antibaryons: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nn_{\\overline{Bar}} \\ll n_{Bar} \\ll n_{\\gamma}\n\\label{TWOHIER}\n\\eeq\nIt is conceivable that in other \nneighbourhoods antimatter dominates and that the universe \nis formed by a patchwork quilt of matter and antimatter \ndominated regions with the whole being \nmatter-antimatter symmetric. Yet it is widely held \nto be quite unlikely -- primarily because no mechanism has been \nfound by which a matter-antimatter symmetric universe following \na big bang evolution can develop sufficiently large regions \nwith non-vanishing baryon number. While there will be \nstatistical fluctuations, they can be nowhere near large \nenough. Likewise for dynamical effects: baryon-antibaryon \nannihilation is by far not sufficiently effective to create pockets \nwith the observed baryon number, Eq.(\\ref{BNUMUNI}). For the \nnumber density of {\\em surviving} baryons can be estimated as \n\\cite{DOLGOV1} \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nn_{Bar} \\sim \\frac{n_{\\gamma}}{\\sigma _{annih}m_N M_{PL}} \n\\simeq 10^{-19}n_{\\gamma} \n\\eeq\nwhere $\\sigma _{annih}$ denotes the cross section of nucleon \nannihilation, $m_N$ and $M_{Pl}$ the nucleon and Planck mass, \nrespectively. Hence we conclude for the universe as a whole \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n0 \\neq \\frac{n_{Bar}}{n_{\\gamma}} \\simeq \n\\frac{\\Delta n_{Bar}}{n_{\\gamma}} \\sim {\\cal O}(10^{-10}) \n\\eeq  \nwhich makes more explicit the meaning of the statement \nquoted above that the universe has been observed to be \nalmost empty, but not quite. Understanding this double \nobservation is the challenge we are going to address now.  \n\\subsection{The Ingredients}\nThe question is: under which condition can one have a situation \nwhere the baryon number of the universe that vanishes at the initial time -- \nwhich for all practical purposes is the Planck time --  develops \na non-zero  value later on: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\Delta n_{Bar}(t = t_{Pl}\\simeq 0) = 0 \n\\; \\; \\stackrel{?}{\\Longrightarrow} \\; \\; \n\\Delta n_{Bar}(t = `today') \n\\neq 0 \n\\eeq \nOne can and should actually go one step further in the task one is \nsetting for oneself: explaining the observed baryon number as \ndynamically generated \n{\\em no matter what its initial value was!} \n\nIn a seminal paper that appeared in 1967 Sakharov \nlisted the three ingredients that are essential for the feasibility \nof such a program \\cite{SAKH,DOLGOV2} : \n\\begin{enumerate}\n\\item \nSince the final and initial baryon number differ, there have to be baryon number violating transitions: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\cal L}(\\Delta n_{Bar} \\neq 0) \\neq 0 \n\\eeq  \n\\item \nCP invariance has to be broken. Otherwise for every baryon number \nchanging transition $N \\rightarrow f$ there is its CP conjugate one \n$\\bar N \\rightarrow \\bar f$ and no net baryon number can be generated. \nI.e., \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\Gamma (N \\; \\; \\stackrel{{\\cal L}(\\Delta n_{Bar} \\neq 0)}\n{\\longrightarrow} \\; \\;  f ) \n\\; \\; \\; \\neq \\; \\; \\; \n\\Gamma (\\bar N \\; \\; \\stackrel{{\\cal L}(\\Delta n_{Bar} \\neq 0)}\n{\\longrightarrow} \\; \\; \\bar f ) \n\\eeq \nis needed. \n\\item \nUnless one is willing to entertain thoughts of CPT violations, the \nbaryon number and CP violating transitions have to proceed \nout of thermal equilibrium. For in thermal equilibrium time \nbecomes irrelevant globally and CPT invariance reduces to \nCP symmetry which has to be avoided, see above: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n{\\rm CPT \\; invariance} \\; \\; \\; \n\\stackrel{thermal \\; equilibrium}\n{\\Longrightarrow} \\; \\; \\;  {\\rm CP \\; invariance} \n\\eeq \n\\end{enumerate}  \nIt is important to keep in mind that these three conditions have \nto be satisfied {\\em simultaneously}. The other side of the coin is, \nhowever, the following: once a baryon number has been \ngenerated through the concurrance of these three effects, \nit can be washed out again by these same effects.\n\n\\subsection{GUT Baryogenesis}\nSakharov's paper was not noticed (except for \\cite{KUZMIN}) \nfor several years until the concept of Grand Unified Theories \n(=GUTs) emerged starting in 1974 \\cite{PATI}; \nfor those naturally provide all three necessary ingredients: \n\n\\begin{enumerate} \n\\item \nBaryon number changing reactions have to exist in GUTs. For placing \nquarks and leptons into common representations of the underlying \ngauge groups -- the hallmark of GUTs -- means that gauge interactions exist changing baryon and lepton numbers.  \nThose gauge bosons are generically \nreferred to as $X$ bosons and have two couplings to fermions \nthat violate baryon and\/or lepton number: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nX \\leftrightarrow qq \\;  , \\; \\;  q \\bar l  \n\\label{XDEC} \n\\eeq  \n\n\\item  \nThose models are sufficiently complex to allow for several \npotential sources of CP violation. Since $X$ bosons have (at least) \ntwo decay channels open CP asymmetries can arise \n\\begin{eqnarray}  \n\\Gamma (X \\rightarrow qq) = (1+\\Delta _q) \\Gamma _q \n&,& \\; \\; \n\\Gamma (X \\rightarrow q\\bar l) = (1-\\Delta _l) \\Gamma _l \\\\   \n\\Gamma (\\bar X \\rightarrow \\bar q \\bar q) = \n(1-\\Delta _q) \\Gamma _q \n&,& \\; \\; \n\\Gamma (\\bar X \\rightarrow \\bar q l) = (1+\\Delta _l) \\Gamma _l \n\\end{eqnarray}   \nwhere  \n\\begin{eqnarray}  \n{\\bf \\rm CPT} \\; \\; &\\Longrightarrow & \\; \\; \n\\Delta _q \\Gamma _q = \\Delta _l \\Gamma _l \\\\  \n{\\bf \\rm CP} \\; \\; &\\Longrightarrow & \\; \\; \\Delta _q = 0 = \\Delta _l \\\\  \n{\\bf \\rm C} \\; \\; &\\Longrightarrow & \\; \\; \\Delta _q = 0 = \\Delta _l \n\\end{eqnarray}   \n\n\\item  \nGrand Unification means that a phase transition takes \nplace around an energy scale $M_{GUT}$. For temperatures \n$T$ well above the transition point -- $T \\gg M_{GUT}$ -- \nall quanta are relativistic with a number density \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nn(T) \\propto T^3\n\\label{NREL} \n\\eeq \nFor temperatures around the phase transition -- \n$T \\sim M_{GUT}$ -- some of the quanta, in particular those \ngauge bosons generically referred to as $X$ bosons aquire \na mass $M_X \\sim {\\cal O}(M_{GUT})$ and their \nequilibrium number \ndensity becomes Boltzmann suppressed: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nn_X(T) \\propto (M_XT)^{\\frac{3}{2}} {\\rm exp}\n\\left( - \\frac{M_X}{T}\\right) \n\\label{BOLTZ} \n\\eeq \nMore $X$ bosons will decay according to Eq.(\\ref{XDEC}) \nthan be regenerated from $qq$ and $q \\bar l$ collisions \nultimately bringing the number of $X$ bosons down to the \nlevel described by Eq.(\\ref{BOLTZ}). Yet that will take some time; \nthe expansion in the big bang cosmology leads to a cooling rate \nthat is so rapid that thermal equilibrium cannot be maintained \nthrough the phase transition. That means that $X$ bosons \ndecay -- and in general interact -- out of thermal \nequilibrium \\cite{DOLGOV2}. \n\\end{enumerate} \nTo the degree that the back production of $X$ \nbosons in $qq$ and $q\\bar l$ collisions can be ignored one finds \nas an order-of-magnitude estimate  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nr_{Bar} \\sim \\frac{\\frac{4}{3}\\Delta _q \\Gamma _q - \n\\frac{2}{3}\\Delta _l \\Gamma _l}{\\Gamma _{tot}} \\frac{n_X}{n_0} \n= \\frac{\\frac{2}{3}\\Delta _q \\Gamma _q }{\\Gamma _{tot} }\\frac{n_X}{n_0}\n\\label{GUTRESULT} \n\\eeq \nwith $n_X$ denoting the initial number density of $X$ \nbosons and $n_0$ the number density of the light decay \nproducts \n\\footnote{Due to thermalization effects one can have \n$n_0 \\gg 2n_X$.}. \nThe three essential conditions for baryogenesis are thus \nnaturally realized around the GUT scale in big bang cosmologies, as \ncan be read off from Eq.(\\ref{GUTRESULT}): \n\\begin{itemize} \n\\item \n$\\Gamma _q \\neq 0$ representing baryon number violation; \n\\item \n$\\Delta _q \\neq 0$ reflecting CP violation and \n\\item \nthe absence of the back reaction due to an absence of thermal \nequilibrium. \n\\end{itemize}   \n\nThe fact that this problem can be formulated in GUT models \nand answers obtained that are very roughly in the right \nballpark is a highly attractive feature of GUTs, in particular \nsince this was {\\em not} among the original motivations \nfor constructing such theories. \n\nOn the other hand it would be highly misleading to claim \nthat baryogenesis has been understood. There are  \nserious problems in any attempt to have baryogenesis \noccur at a GUT scale: \n\\begin{itemize}\n\\item \nA baryon number generated at such high temperatures is \nin grave danger to be washed out or diluted in the subsequent \nevolution of the universe. \n\\item \nVery little is known about the dynamical actors operating at \nGUT scales and their characteristics -- and that is putting it mildly. Actually \neven in the future we can only hope to obtain some \nslices of indirect information on them. \n\\end{itemize} \nOf course it would be premature to write-off baryogenesis at \nGUT scales, yet it might turn out that it is best \ncharacterised as a proof of principle -- namely that the \nbaryon number of the universe can be understood as dynamically \ngenerated -- rather than as a semi-quantitative realization. \n\\subsection{Electroweak Baryogenesis}\nBaryogenesis at the electroweak scale \n\\cite{RUBAKOV} \nis the most actively analyzed scenario at present. For it possesses several highly attractive features: \n\\begin{itemize}\n\\item \nWe know that dynamical landscape fairly well.\n\\begin{itemize}\n\\item \nIn particular CP violation has been found to exist there. \n\\item \nA well-studied phase transition, namely the spontaneous breaking \\begin{equation}}   \\def\\eeq{\\end{equation} \nSU(2)_L\\times U(1) \\; \\; \\Longrightarrow  U(1)_{QED}\n\\eeq \ntakes place. \n\\end{itemize}\n\\item \nFuture experiments will certainly probe that dynamical regime \nwith ever increasing sensitivity, both by searching for the \non-shell production of new quanta -- like SUSY and\/or Higgs \nstates -- and the indirect impact through quantum corrections on \nrare decays and CP violation.  \n\\end{itemize}\nHowever at this point the reader might wonder: \"What \nabout the third required ingredient, baryon number violation? At the \nelectroweak scale?\" It is often not appreciated that \nthe electroweak forces of the Standard Model by themselves \nviolate baryon number, though in a very subtle way.  \nWe find here what is called an anomaly: the baryon \nnumber current is conserved on the classical, \nyet {\\em not} on the quantum level:  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\partial _{\\mu} J_{\\mu}^{Bar} = \n\\partial _{\\mu}\\sum _q (\\bar q_L \\gamma _{\\mu}q_L) = \n\\frac{g^2}{16 \\pi ^2} G_{\\mu \\nu} \\tilde G_{\\mu \\nu} \\neq 0 \n\\label{ANOM} \n\\eeq  \nwhere $G_{\\mu \\nu}$ denotes the electroweak field strength \ntensor \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nG_{\\mu \\nu} = \\partial _{\\mu} A_{\\nu} - \\partial _{\\nu} A_{\\mu} \n+ g[A_{\\mu}, A_{\\nu}] \n\\eeq \nand $\\tilde G_{\\mu \\nu} $ its dual: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\tilde G_{\\mu \\nu}  = \\epsilon _{\\mu \\nu \\alpha \\beta }\nG_{\\alpha \\beta} \n\\eeq \nThe right hand side of Eq.(\\ref{ANOM}) can be written as \nthe divergence of a current \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nG_{\\mu \\nu} \\tilde G_{\\mu \\nu} = \\partial _{\\mu} K_{\\mu} \\; , \n\\; \\; K_{\\mu} = 2 \\epsilon _{\\mu \\nu \\alpha \\beta } \n\\left(  A_{\\nu} \\partial _{\\alpha} A_{\\beta} + \n\\frac{2}{3} ig A_{\\nu}A_{\\alpha}A_{\\beta} \n\\right) \n\\label{KCURRENT}  \n\\eeq \nA total derivative is usually unobservable since partial \nintegration allows to express its contribution \nthrough a surface integral at infinity. The field strength \ntensor $G_{\\mu \\nu}$ indeed vanishes at infinity -- but \nnot necessarily the gauge potential $A_{\\mu}$. \nTo be more specific: The field configuration at infinity is \nthat of a ground state for which $G_{\\mu \\nu} = 0$ \nholds. Yet that property does not define it uniquely:  \nground states get differentiated by the value of their  \n$K$ charge, i.e. the space integral  \nof $K_0$, the zeroth component of the current $K_{\\mu}$ \nconstructed from their gauge field configuration. \nThis integral reflects differences in the gauge topology \nof the ground states and therefore is called the \n{\\em topological charge}. \nWhile this charge is irrelevant for abelian gauge theories \nwhere the \nlast term in Eq.(\\ref{KCURRENT}) necessarily vanishes,  \nit becomes relevant for non-abelian theories.  \nWe have encountered \nthis phenonemon already in our discussion of the \nStrong CP Problem that is driven by the axial quark current \nnot being conserved in the strong interactions of QCD. It is often \nreferred to as `Chiral' Anomaly since it breaks chiral invariance, \nor `Triangle' Anomaly since it is produced by a triangular fermion \nloop diagram or `Adler-Bell-Jackiw' Anomaly named after the authors who discovered it. \n\nThe concrete impact of the triangle anomaly on the physics \ndepends on the specifics of the theory: here \nbecause of the chiral \nnature of the weak interactions it induces baryon number \nviolation. \nEq.(\\ref{ANOM}) and Eq.(\\ref{KCURRENT}) show that the difference \n$J_{\\mu}^{Bar} - K_{\\mu}$ is conserved. The transition from one \nground state to another which represents a tunneling \nphenomenon is thus accompanied by a change in \nbaryon number. Elementary quantum mechanics tells \nus that this baryon number violation is described as a \nbarrier penetration and exponentially suppressed at low \ntemperatures or energies \\cite{THOOFTBAR}: \n$Prob(\\Delta n_{Bar}\\neq 0) \\propto \n{\\rm exp}(-16 \\pi ^2\/g^2) \\sim {\\cal O}(10^{-160})$ --   \na suppression that reflects the tiny size of the \nweak coupling.  \n\nThere is a corresponding anomaly for the lepton number \ncurrent implying that lepton number is violated as \nwell with the selection rule \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\Delta n_{Bar} - \\Delta n_{lept} = 0 \\; . \n\\label{B-L}\n\\eeq\nThis is usually referred to by saying that $B-L$, the difference between baryon and lepton number, is still conserved. \n\nAt sufficiently high energies this huge suppression of baryon \nnumber changing transition rates will evaporate since \nthe transition between different ground states can be \nachieved classically through a motion {\\em over} \nthe barrier. The question then is at which energy scale this \nwill happen and how quickly baryon number violation will \nbecome operative. Some semi-quantitative observations can \nbe offered and answers given \n\\cite{RUBAKOV2,DOLGOV2}. \n\nThere are special field configurations -- called sphalerons -- \nthat carry the topological $K$ charge. In the Standard Model they \ninduce effective multistate interactions among left-handed \nfermions that change baryon and lepton number by three \nunits each: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\Delta n_{Bar} = \\Delta n_{lept} = 3 \n\\eeq \nAt high energies where the weak bosons $W$ and $Z$ \nare massless, the height of the transition barrier between \ndifferent groundstates vanishes likewise and the change \nof baryon number can proceed in an unimpeded way and \npresumably faster than the universe expands. Thermal \nequilibrium is then maintained and any baryon \nasymmetry existing before this era is actually washed out \n\\footnote{To be more precise, only $B+L$ is erased \nwithin the Standard Model whereas $B-L$ remains \nunchanged.}! \nRather than generate a baryon number sphalerons act to \ndrive the universe back to matter-antimatter \nsymmetry at this point in its evolution. \n\nAt energies below the phase transition, i.e. in the broken phase \nof $SU(2)_L \\times U(1)$ baryon number is conserved for \nall practical purposes as pointed out above. \n\nThe value of \n$\\Delta n_{Bar}$ as observed today can thus be generated \nonly in the transition from the unbroken high energy to the \nbroken low energy phase. With $\\Delta n_{Bar}\\neq 0$ \nprocesses operating there the issue now turns to the \nstrength of the phase transition: is it relatively smooth \nlike a second order phase transition or violent like a \nfirst order one? Only the latter scenario can support \nbaryogenesis. \n\nA large amount of interesting theoretical work has been  \non the thermodynamics of the Standard Model in an \nexpanding universe. Employing perturbation theory \nand lattice studies one has arrived at the following result: \nfor light Higgs masses up to around 70 GeV, the phase \ntransition is first order, for larger masses it is second \norder \n\\cite{SHAP}. Since no such light Higgs states have been observed \nat LEP, one infers that the phase transition is second order \nthus apparently foreclosing baryogenesis occurring at the \nelectroweak scale. \n\nWe have concentrated here on the questions of thermal \nequilibrium and baryon number while \ntaking CP violation for \ngranted since it is known to operate at the electroweak scale. \nYet most authors -- with the exception of some notable \nheretics -- agree that the KM ansatz is not at all \nup to {\\em this} task: it fails by several orders of \nmagnitude. On the other hand New Physics \nscenarios of CP violation -- in particular \nof the Higgs variety -- can reasonably be called upon to perform \nthe task. \n\n\\subsection{Leptogenesis Driving Baryogenesis}\nIf the electroweak phase transition is indeed a \nsecond order one, \nsphaleron mediated reactions cannot drive baryogenesis \nas just discussed and they will wipe out any \npre-existing $B+L$ number. Yet if at some high \nenergy scales a lepton number is generated the very \nefficiency of these sphaleron processes can \n{\\em communicate} this asymmetry to the baryon \nsector through them maintaining conservation of \n$B-L$. \n\nThere are various ways in which such scenarios can \nbe realized. The simplest one is to just add \n{\\em heavy right-handed Majorana} neutrinos \nto the Standard Model. This is highly attractive \nin any case since it enables us \nto implement the see-saw mechanism for explaining \nwhy the observed neutrinos are (practically) massless; \nit is also easily embedded into $SO(10)$ GUTs. \n\nThe basic idea is the following \\cite{FUKUGITA}: \n\\begin{itemize} \n\\item  \nA primordial lepton asymmetry is generated at high \nenergies well above the electroweak phase transition: \n\\begin{itemize} \n\\item \nSince a Majorana neutrino $N$ is its own CPT mirror image, \nits dynamics necessarily violate lepton number. It will \npossess at least the following classes of decay channels: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \nN \\rightarrow l \\bar H \\; , \\; \\; \\bar l H\n\\eeq \nwith $l$ and $\\bar l$ denoting a light charged or neutral lepton \nor anti-lepton and $H$ and $\\bar H$ a Higgs or anti-Higgs field, \nrespectively. \n\\item \nA CP asymmetry will in general arise  \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\Gamma (N \\rightarrow l \\bar H) \\neq \n\\Gamma (\\bar N \\rightarrow \\bar l H)\n\\eeq \nthrough a KM analogue in the neutrino mass \nmatrix (which can be quite different from the \nmass matrix for charged leptons).  \n\\item \nThese neutrino decays are sufficiently slow as to occur \nout of thermal equilibrium around the energy scale \nwhere the Majorana masses emerge. \n\\end{itemize} \n\\item \nThe resulting lepton asymmetry is transferred into a \nbaryon number through sphaleron mediated processes \nin the unbroken high energy phase of \n$SU(2)_L \\times U(1)$: \n\\begin{equation}}   \\def\\eeq{\\end{equation} \n\\langle \\Delta n_{lept}\\rangle = \n\\frac{1}{2}\\langle \\Delta n_{lept} + \\Delta n_{Bar}\\rangle + \n\\frac{1}{2}\\langle \\Delta n_{lept} - \\Delta n_{Bar}\\rangle \n\\Longrightarrow \n\\frac{1}{2}\\langle \\Delta n_{lept} - \\Delta n_{Bar}\\rangle \n\\eeq \n\\item \nThe baryon number thus generated survives through the \nsubsequent evolution of the universe. \n\\end{itemize}  \n\n\n\n\\subsection{Wisdom -- Conventional and Otherwise}\nWe understand how nuclei were formed in the universe \ngiven protons and neutrons. Obviously it would be \neven more fascinating if we could understand how \nthese baryons were generated in the first place. \nWe do not possess a specific and \nquantitative theory successfully describing baryogenesis. \nHowever leaving it at that statement would -- we believe -- \nmiss the main point. We have learnt which kinds of \ndynamical ingredients are neccessary for baryogenesis to \noccur in the universe. We have seen that these ingredients \ncan be realized naturally: \n\\begin{itemize} \n\\item \nGUT scenarios for baryogenesis \nprovide us with a proof of principle that such a program \ncan be realized. In practical terms however they suffer from \nvarious shortcomings: \n\\begin{itemize} \n\\item \nSince the baryon number is generated \nat the GUT scales, very little is and not much more might \never be known about that dynamics. \n\\item \nIt \nappears quite likely that a baryon number produced at such \nhigh scales is  subsequently washed out. \n\\end{itemize} \n\\item \nThe highly fascinating proposal of baryogenesis at the electroweak \nphase transition has attracted a large degree of attention -- \nand deservedly so: \n\\begin{itemize} \n\\item \nA baryon number emerging from \nthis phase transition would be in no danger of being \ndiluted substantially. \n\\item \nThe dynamics involved here \nis known to a considerable degree and will be probed \neven more with ever increasing sensitivity over the \ncoming years. \n\\end{itemize} \nHowever it seems that the electroweak phase transition is \nof second order and thus not \nsufficiently violent. \n\\item \nA very intriguing variant turns some of the vices of sphaleron \ndynamics into virtues by attempting to understand \nthe baryon number of the universe as a reflection of a \n{\\em primary} lepton asymmetry. \nThe required new dynamical entities  \n-- Majorana neutrinos and their decays -- obviously would \nimpact on the universe in other ways as well.  \n\\end{itemize} \nThe challenge to understand baryogenesis has already inspired \nour imagination, \nprompted the development of some very intriguing \nscenarios and thus has initiated many \nfruitful studies -- and in the end we might even be \nsuccessful in meeting it! \n\n\n\\section{The Cathedral Builders' Paradigm}\n\\subsection{The Paradigm}\n\nThe dynamical ingredients for numerous and multi-layered \nmanifestations of CP and T violations do exist or are likely to exist. Accordingly one searches \nfor them in many phenomena, namely in  \n\\begin{itemize}\n\\item \nthe neutron electric dipole moment probed with ultracold \nneutrons at ILL in Grenoble, France; \n\\item \nthe electric dipole moment of electrons studied through the \ndipole moment of atoms at Seattle, Berkeley and Amherst in the US; \n\\item \nthe transverse polarization of muons in \n$K^- \\rightarrow \\mu ^- \\bar \\nu \\pi ^0$ at KEK in Japan; \n\\item \n$\\epsilon ^{\\prime}\/\\epsilon _K$ as obtained from $K_L$ \ndecays at FNAL and CERN and soon at DA$\\Phi$NE in Italy; \n\\item \nin decay distributions of hyperons at FNAL; \n\\item \nlikewise for $\\tau$ leptons at CERN, the beauty factories and BES \nin Beijing; \n\\item \nCP violation in the decays of charm hadrons produced \nat FNAL and the beauty factories; \n\\item \nCP asymmetries in beauty decays at DESY, at the beauty \nfactories at Cornell, SLAC and KEK, at the FNAL collider and \nultimately at the LHC. \n\n\\end{itemize} \nA quick glance at this list already makes it clear \nthat frontline research on this topic \nis pursued at all high energy labs in the world -- and then some; \ntechniques from several different branches of physics -- \natomic, nuclear and high energy physics -- are harnessed in \nthis endeavour together with a wide range of set-ups. \nLastly, experiments are performed at the lowest temperatures \nthat can be realized on earth -- ultracold neutrons -- and at the \nhighest -- in collisions produced at the LHC. And all of that dedicated \nto one profound goal. \nAt this point I can explain what I mean by the term \n\"Cathedral Builders' Paradigm\". \nThe building of cathedrals required interregional collaborations, \nfront line technology (for the period) from many different fields \nand commitment; it had to be based on solid foundations -- and \nit took time. The analogy to the ways and needs of high energy \nphysics are obvious -- but it goes deeper than that. \nAt first sight a cathedral looks \nlike a very complicated and confusing structure with something \nhere and something there. Yet further scrutiny reveals that \na cathedral is more appropriately \ncharacterized as a complex rather than a complicated \nstructure, one that is multi-faceted and multi-layered -- \nwith a coherent theme! One cannot (at least for \nfirst rate cathedrals) remove any of its elements \nwithout diluting (or even destroying) its technical soundness and \nintellectual message. Neither can one in our efforts to come to grips \nwith CP violation!  \n\n\\subsection{Summary}\n\\begin{itemize} \n\\item \nWe know that CP symmetry is not exact in nature since \n$K_L \\rightarrow \\pi \\pi $ proceeds and presumably because we  \nexist, i.e. because the baryon number of the universe does \n{\\em not} vanish. \n\\item \nIf the KM mechanism is a significant actor in $K_L \\rightarrow \\pi \\pi$ \ntransitions then there must be large CP asymmetries in the decays \nof beauty hadrons. In $B^0$ decays they \nare naturally measured in units of 10 \\%!  \n\\item \nSome of these asymmetries are predicted with high parametric \nreliability. \n\\item \nNew theoretical technologies will allow us to translate such \nparametric reliability into quantitative accuracy. \n\\item \nAny significant difference between certain KM predictions for the \nasymmetries and the data reveals the intervention of New Physics. \nThere will be no `plausible deniability'.  \n\\item \nWe can expect 10 years hence the theoretical uncertainties in \nsome of the \npredictions to be reduced below 10 \\% . \n\\item \nI find it likely that deviations from the KM predictions will show up \non that level. \n\\item \nYet to exploit this discovery potential to the fullest one will have to \nharness the statistical muscle provided by beauty production \nat hadronic colliders. \n\\end{itemize}\n\n\n\\subsection{Outlook}\nI want to start with a statement about the past: \n{\\em The comprehensive study of kaon and hyperon physics \nhas been instrumental in guiding us to the Standard Model.}  \n\\begin{itemize}\n\\item \nThe $\\tau -\\theta $ puzzle led to the realization that parity is not \nconserved in nature. \n\\item \nThe observation that the production rate exceeded the decay rate \nby many orders of magnitude -- this was the origin of the \nname `strange particles' -- was explained through postulating \na new quantum number -- `strangeness' -- conserved by the strong, \nthough not the weak forces. This was the beginning of the second \nquark family. \n\\item \nThe absence of flavour-changing neutral currents was incorporated \nthrough the introduction of the quantum number `charm', which \ncompleted the second quark family. \n\\item \nCP violation finally led to postulating yet another, the third \nfamily. \n\\end{itemize}\nAll of these elements which are now essential pillars of the Standard \nModel were New Physics at {\\em that} time! \n\nI take this historical \nprecedent as clue that a detailed, comprehensive and thus \nneccessarily long-term program on beauty physics will lead to a \nnew paradigm, a {\\em new} Standard Model! \n\nCP violation is a fundamental as well as mysterious phenomenon \nthat we have not understood yet. This is not surprising: after all \naccording to the KM mechanism CP violation enters through the \nquark mass matrices; it thus relates it to three central \nmysteries of the Standard Model: \n\\begin{itemize}\n\\item \nHow are fermion masses generated? \n\\footnote{Or more generally: how are masses produced in \ngeneral? For in alternative models CP violation enters through \nthe mass matrices for gauge bosons and\/or Higgs bosons.}  \n\\item \nWhy is there a family structure?\n\\item \nWhy are there three families rather than one?\n\\end{itemize} \nIn my judgement it would be unrealistic to expect that these \nquestions can be answered through pure thinking. I strongly \nbelieve we have to appeal to nature through experimental efforts to \nprovide us with more pieces that are surely missing in the \npuzzle. CP studies are essential in obtaining the full dynamical \ninformation contained in the mass matrices or -- in the language \nof v. Eichendorff's poem quoted in the beginning, \"to find the \nmagic word\" that will decode nature's message for us. \n\nConsiderable progress has been made in theoretical engineering \nand developing a comprehensive CP phenomenology from \nwhich I conclude: \n\\begin{itemize}\n\\item \n$B$ decays constitute an almost ideal, certainly optimal and unique \nlab. Personally I believe that even if no deviation \nfrom the KM predictions were uncovered, we would find \nthat the KM parameters, in particular the angles of the \nKM triangle, carry special values that would give us \nclues about New Physics. Some very interesting \ntheoretical work is being done about how GUT dynamics in \nparticular of the SUSY (or Supergravity) variety operating \nat very high scales would shape the observable \nKM parameters. \n\\item \nA comprehensive analysis of charm decays with special emphasis on  \n$D^0 - \\bar D^0$ oscillations and CP violation is a moral \nimperative! Likewise for $\\tau$ leptons. \n\\item \nA vigorous research program must be pursued for light \nfermion systems, namely in the decays of kaons and hyperons \nand in electric dipole moments. After all it is conceivable \nof course that no CP asymmetries are found \nin $B$ decays on a measurable level. \nThen we would know that the KM ansatz is {\\em not} a \nsignificant actor in $K_L \\rightarrow \\pi \\pi$, that New Physics drives it -- \nbut what kind of New Physics would it be? Furthermore even if \nlarge CP asymmetries were found in $B$ decays, it could \nhappen that the signals of New Physics are obscured by \nthe large `KM background'. This would not be the case \nif electric dipole moments were found or a transverse \npolarization of muons in $K_{\\mu 3}$ decays. \n\\item Close feedback between experiment and theory will be \nessential. \n\n\\end{itemize}\n As the final summary: insights about Nature's \nGrand Design that can be obtained from a \ncomprehensive and detailed program of CP studies \n\\begin{itemize}\n\\item \nare of fundamental importance, \n\\item \ncannot be obtained any other way and \n\\item \ncannot become obsolete! \n\n\n\\end{itemize} \n  \n\n\n\\bigskip \n\n{\\bf Acknowledgments:} \\hspace{.4em} \nThis work was supported by the National Science Foundation under\nthe grant numbers PHY 92-13313 and PHY 96-05080.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWe consider the dynamics of the Heisenberg-Ising spin-1\/2 chain   with  anisotropy parameter $\\Delta$, also known as the XXZ spin-1\/2 chain,\non the one-dimensional lattice $\\mathbb{Z}$ with domain wall initial conditions.  We start with an initial state of $N$ up-spins at the\nsites $\\{1,2,\\ldots, N\\}$ in a sea of down-spins; and by utilizing ideas from coordinate Bethe Ansatz \\cite{Bethe, Gaudin, Suth, YY} to solve the\nSchr\\\"odinger equation, we find the quantum state $\\Psi_N(t)$ at time $t$ is\n\\[ \\Psi_N(t)=\\sum_X \\psi_N(X,t) e_X, \\]\nwhere the sum is over all $X=\\{x_1<x_2<\\cdots<x_N\\}$ and $e_X$ denotes the state with up-spins at $X$. Alternatively we can view a spin up at site $x_j$ as a particle and a spin down as an empty lattice site or hole. The ``Bethe-coordinates'' $\\psi_N(X,t)$ are given below in Theorem 1.\\footnote{Since the Hamiltonian $H_{\\Delta}$ \nof the Heisenberg-Ising model is a \n(non-unitary) similarity transformation of the Markov generator of the ASEP \\cite{Gwa-Spohn}, the results in \\cite{TW1} give immediately\nthe Bethe coordinates of Theorem 1 once\nan identification of parameters is made (see Section 3.3).} They have the\ninterpretation that $\\left\\vert\\psi_N(X;t)\\right\\vert^2$ is the probability the system is in state $X$ at time $t$. Observe that the $\\psi_N(X,t)$ have the standard Bethe Ansatz structure\nas a sum over the permutation group $\\mathcal{S}_N$; where now, each term in the summand is an $N$-dimensional contour integral.\n\\par\n\\subsection{One-Point Functions}\nIf $X_1(t)$ denotes the position of the left-most particle at time $t$, then\n\\[ \\mathbb{P}_N(X_1(t) = x)=\\sum_{X,x_1=x}\\left\\vert\\psi_N(X,t)\\right\\vert^2 \\]\nwhere the sum is over all $X=\\{x_1=x<x_2<\\cdots<x_N\\}$.  In ASEP the analogous quantity involves a \\textit{single sum} over $\\mathcal{S}_N$ where as now\nwe have a \\textit{double sum} over $\\mathcal{S}_N$.\nIn \\cite{TW1} an identity involving the sum over the permutation group\\footnote{See equation (1.6) in \\cite{TW1}.} was used to reduce the\nsum to a single  $N$-dimensional integral.  Cantini, Colomo, and Pronko \\cite{CCP} have  generalized the single sum permutation\nidentity to a double sum permutation identity, which also generalize to the (spin) Hall-Littlewood functions \\cite{P, WZJ}. Employing this new identity reduces the expression for $\\mathbb{P}_N(X_1(t)=x)$ to\na single $2N$-dimensional integral whose integrand involves the famous Izergin-Korepin determinant \\cite{Iz,Ko}.  The resulting expression\nis given in Theorem 2. This part of the paper overlaps the recent work of J.~M.~St\\'ephan \\cite{St1, St2}.\n\\par For the special case $\\Delta=0$,  the analysis simplifies considerably.  Using Toeplitz operators and their determinants,  we show the $N\\rightarrow\\infty$ limit\ncan be taken resulting in the representation\n\\[ \\lim_{N\\rightarrow\\infty}\\mathbb{P}_N(X_1(t)\\ge x )=\\det(I-L)\\]\nwhere $L$ is an integral operator whose kernel is the  \\textit{discrete Bessel kernel} \\cite{Bo,BOO, Jo}. (See also Chapter 8 in\n\\cite{BDS}.) This makes connections to the distribution\nof the length of the longest\nincreasing subsequence in a random permutation \\cite{BDJ, BDS}.  See Theorem 3 below.  From this identification it follows that\n\\[ \\lim_{t\\rightarrow\\iy}\\lim_{N\\rightarrow\\iy}\\mathbb{P}_N\\left(\\frac{X_1(t)+2t}{t^{1\/3}}\\ge -y\\right)=F_2(y)\\]\nwhere $F_2$ is the $\\textrm{TW}_2$ distribution \\cite{TW0a, TW0b}. This last result appears to be well-known in the physics literature since\nthe case $\\Delta=0$ is\nreducible to a ``free fermion'' model \\cite{ER, SZ, St1, VSDH}.\n\n\\subsection{Contour Deformations and a Conjecture}\n\nTaking the contour integral functions for the one-point function to the infinite time statistics is another major challenge. In the case of the ASEP, this was achieved by Tracy-Widom \\cite{TW1a} by deforming the contours to obtain a Fredholm determinant. Then, in a later work by the same authors \\cite{TW2}, the Fredholm determinant was further analyzed by deforming the kernels to obtain the Tracy-Widom distribution. We also deform the contour integrals for our one-point function, in Section \\ref{s:contour_deformations}, to obtain a type of series expansion.\n\n\\textsc{Theorem 4.} Let $X_1(t)$ be the location of the left-most particle in the Heisenber-Ising spin-1\/2 chain with $N$ particles, initial conditions $Y = (y_1< y_2 < \\cdots < y_N )$, and $\\D \\in \\mathbb{R}$ so that $\\D \\neq 0$. Then, $\\mathbb{P}(X_1(t) \\geq x)$ is equal to\n\\begin{equation}\n\\sum_{n=0}^N \\sum_{\\tau \\in \\mathcal{T}_n}\\oint_{\\mathcal{C}_R}\\cdots \\oint_{\\mathcal{C}_R} \\oint_{\\mathcal{C}_{R'}} \\cdots \\oint_{\\mathcal{C}_{R'}} I_N(\\xi, \\zeta; \\tau) f(\\xi, \\zeta; \\tau)\\, \\left(\\prod_{j \\in J} d\\zeta_j \\right)\\, d^N \\xi\\hspace{5ex}\n\\end{equation}\nwhere the integrand is given by (\\ref{e:integrands}), the summation is take over the set of maps $\\mathcal{T}_n$ given by (\\ref{e:maps_cont}), and the contours $\\mathcal{C}_R$ and $\\mathcal{C}_{R'}$ are circles centered at zero with radii $R, R' > 0$ that satisfy the following inequalities $\\max\\{2|\\D|^{-1} , 2(1+2|\\D|)\\} < R < \\max\\{4|\\D|^{-1}, 4(1 + 2|\\D|)\\} < R'\/2$.\n\nWe expect this series expansion to to give rise to a series expansion of a Fredholm determinant in the infinite time limit. In fact, we may deform the contours in the previous formula to the steepest descent to contours in an effort to obtain the infinite time limit by a saddle point analysis. The result is given by our \\textsc{Conjecture 8.1}. Aside from technical details of certain bounds and approximations, there are some terms that we still can't control after the saddle point analysis. Recent results \\cite{BK,CdLV, St1, St1a}, based on numerical, hydrodynamic and analytical arguments are inconclusive in the appropriate scaling, i.e.~$t^{1\/2}$ versus $t^{1\/3}$, for the fluctuations of the one-point function in the infinite time limit. Based on our conjecture, we expect the location of the left-most particle to be at $-2t$ with fluctuations on the order of $t^{1\/3}$ but the limiting distribution is still unclear.\n\n\n\n\n\n\\section{XXZ Quantum Spin-$\\frac{1}{2}$ Hamiltonian}\\label{s:XXZ}\nThe definition of the quantum spin chain Hamiltonian on the infinite lattice $\\mathbb{Z}$ requires some  explanation since there is the problem of making sense of  infinite tensor products in the construction of a Hilbert space of states. The general construction uses the Gelfand-Naimark-Segal (GNS) construction; but in the case considered here, there is an elementary treatment \n\\cite{NSS} which we now describe.\n\\par\n Let $\\mathcal{H}_{0}=\\mathbb{C}$.  For each positive integer $N$ we define\n\\[ \\mathcal{X}_N:=\\left\\{X=\\{x_1,\\ld,x_N\\}\\in\\mathbb{Z}^N: x_1<\\cdots<x_N\\right\\} \\]\nand \n\\[ \\mathcal{H}_N:=\\ell^2(\\mathcal{X}_N).\\]\nThe Hilbert space of states is \n\\[ \\mathcal{H}:=\\bigoplus_{N=0}^\\iy \\mathcal{H}_N.\\]\nThe normalized state $\\Omega=1\\in\\mathcal{H}_0$ is the ground state of all spins down.  In physicists' notation\n\\[ \\Omega= \\vert\\cd\\downarrow\\cd\\downarrow\\cd\\downarrow\\cd\\rangle.\\]\nGiven $N\\in\\mathbb{Z}^{+}$ and $X=\\{x_1,\\ld,x_N\\}\\in\\mathcal{X}_N$, define $e_X\\in\\mathcal{H}_N$ by\n\\[ e_X(Y)=\\dl_{X,Y}.\\]\nThe set $\\{e_X\\}_{X\\in\\mathcal{X}_N}$ defines a natural orthonormal basis of $\\mathcal{H}_N$. The physical interpretation of\n$e_X$ is the state with up spins at $x_1<\\cd<x_N$ in a sea of down spins:\n\\[ e_X=\\vert \\cd \\underset{x_1}{\\uparrow}\\cd\\underset{x_2}{\\uparrow}\\cd\\underset{x_N}{\\uparrow}\\cd\\rangle. \\]\nThis is a model\nof a quantum lattice gas (see, for example, \\S6.1.6 of \\cite{Suth}).  We will frequently use this particle interpretation.\\par\nWe  introduce the \\textit{Pauli operators} $\\s_j^\\a$, $j\\in\\mathbb{Z}$, $\\a=3,\\pm$.\n\\begin{eqnarray}\n \\s_j^3 e_X&=&\\left\\{\\begin{array}{ll}\\hspace{1.5ex} \\> e_X&\\textrm{if}\\>\\>\\>j\\in X=\\{x_1,\\ld,x_N\\},\\\\\n\t\t\t\t\t\t\t\t-\\> e_X & \\textrm{otherwise}.\\end{array}\\right.\\\\ \n\\s_j^{+} e_X&=&\\left\\{\\begin{array}{ll} 0 &\\textrm{if}\\>\\>\\> j\\in \\{x_1,\\ld,x_N\\},\\\\\ne_{X^{+}} & \\textrm{where}\\>\\>\\> X^+=\\{x_1,\\ld,x_k, j, x_{k+1},\\ld, x_N\\}, \\> x_k<j<x_{k+1}\\end{array}\\right.\\\\\n \\s_j^{-} e_X&=&\\left\\{\\begin{array}{ll} 0 &\\textrm{if}\\>\\>\\> j\\notin X=\\{x_1,\\ld,x_N\\}\\\\\ne_{X^{-}}& \\textrm{where}\\>\\>\\>X^{-}=\\{x_1,\\ld,x_{k-1},x_{k+1},\\ld,x_N\\},  \\>\\>\\> j=x_k\\end{array}\\right.\\end{eqnarray}\n\nIn words, $\\s_j^{+}:\\mathcal{H}_N\\rightarrow\\mathcal{H}_{N+1}$  acts as the identity except at the site $j$ where\nit takes $\\downarrow \\leadsto \\uparrow$ and annihilates a $\\uparrow$ state.  Similarly, $\\s^{-}_j:\\mathcal{H}_{N}\\rightarrow\\mathcal{H}_{N-1}$ acts as the identity except at the site $j$\nwhere it takes $\\uparrow\\leadsto \\downarrow$ and annihilates a $\\downarrow$ state.  By definition \n$\\s_j^3\\Omega=-\\Omega$, $\\s_j^{-}\\Omega=0$ and $\\s_j^{+}\\Omega=e_{\\{j\\}}$.    We also recall the \nPauli  operators  $\\s_j^1=\\s_j^{+}+\\s_j^{-}$ and $\\s_j^2=-i\\s_j^{+}+i\\s_j^{-}$.\nDefine\n \\[ h_{j,j+1}=\\frac{1}{2}\\left(\\s_j^1\\s_{j+1}^1\n+\\s_{j}^2\\s_{j+1}^2+\\D(\\s_j^3\\s_j^3-1)\\right) =\\s_j^{+}\\s_{j+1}^{-}+\\s_j^{-}\\s_{j+1}^{+}+\\frac{\\D}{2}(\\s_j^3\\s_{j+1}^3-1)\\]\nand\n\\begin{equation} H_{XXZ}=\\sum_{j\\in\\mathbb{Z}} h_{j,j+1} \\label{XXZ}.\\end{equation}\nThe operator $H_{XXZ}$ is the \\textit{Heisenberg-Ising spin-$\\frac{1}{2}$ chain Hamiltonian}; or more briefly, the  $XXZ$ spin Hamiltonian.\nIt's clear from the above definitions that $H_{XXZ}:\\mathcal{H}_N\\rightarrow\\mathcal{H}_N$.  Since the number of particles is conserved under the dynamics of $H_{XXZ}$, we can work in a sector $\\mathcal{H}_N$.\n\\par\nA state $\\Psi_N=\\Psi_N(t)\\in\\mathcal{H}_N$ can be represented by\n\\begin{equation}  \\Psi_N(t)=\\sum_{X\\in\\mathcal{X}_N} \\psi_N(X,t) e_X.\\label{BigPsi}\\end{equation}\nThe initial condition is $\\Psi(0)=e_Y$, $Y=\\{y_1,\\ld, y_N\\}\\in\\mathcal{X}_N$,  so that $\\psi_N(X;0)=\\dl_{X,Y}$.  The dynamics is determined by the Schr\\\"odinger equation\n\\begin{equation} \\textrm{i}\\, \\frac{\\pl\\Psi_N}{\\pl t}=H_{XXZ}\\Psi_N.\\label{SchroEqn}\\end{equation}\nThe Hamiltonian $H_{XXZ}$ is self-adjoint and so by Stone's theorem there\nexists a unitary operator \\newline $U=\\exp(-\\textrm{i} t H_{XXZ})$ such that $\\Psi_N(t)=U(t)\\Psi_N(0)$.  We have\n\\[ \\langle \\Psi_N(t),\\Psi_N(t)\\rangle =\\sum_{X\\in \\mathcal{X}_N} \\left\\vert \\psi_N(X;t)\\right\\vert^2=1.\\]\n\n \n\n\nThe goal is to describe the dynamics $\\Psi_{DW}(t)$ starting from the \\textit{domain wall} (DW) initial state \n\\[ e_{\\mathbb{N}}=\\vert\\cd \\downarrow\\da\\underset{0}{\\downarrow}\\underset{1}{\\uparrow}\\uparrow\\ua\\cd\\rangle.\\]\nOne immediately sees the difficulty in that $e_{\\mathbb{N}}$ is not an element of $\\mathcal{H}_N$ for any \n$N$.\\footnote{Presumably,\none could construct a domain wall Hilbert space $\\mathcal{H}_{DW}$ by replacing the state $\\Omega$ by\n$e_{\\mathbb{N}}$.  Unfortunately, we do not know how to proceed with a Bethe Ansatz solution in this space.}  If\n$X_m(t)$ denotes the position of the $m$th particle on the left, we define\n\\[ \\mathbb{P}_{\\mathbb{N}}(X_m(t)=x)=\\lim_{N\\rightarrow\\iy} \\mathbb{P}_{\\{1,\\ldots,N\\}}(X_m(t)=x).\\]\n\n\\section{Bethe Ansatz Solution $\\Psi_N(t)$}\\label{s:bethe_ansatz}\nThis section closely follows \\cite{TW1, YY}.  We first note that \n\\begin{eqnarray}\nh_{j,j+1}\\lvert \\cd \\underset{j}{\\uparrow}\\underset{j+1}{\\uparrow}\\cd\\rangle&=&0,\\label{h1}\\\\\nh_{j,j+1}\\lvert \\cd \\underset{j}{\\downarrow}\\underset{j+1}{\\downarrow}\\cd\\rangle&=&0,\\label{h2}\\\\\nh_{j,j+1}\\lvert \\cd \\underset{j}{\\uparrow}\\underset{j+1}{\\downarrow}\\cd\\rangle&=&\n-\\D \\lvert \\cd \\underset{j}{\\uparrow}\\underset{j+1}{\\downarrow}\\cd\\rangle +\n\\lvert \\cd\\underset{j}{\\downarrow}\\underset{j+1}{\\uparrow}\\cd\\rangle,\\label{h3}\\\\\n h_{j,j+1}\\lvert \\cd \\underset{j}{\\downarrow}\\underset{j+1}{\\uparrow}\\cd\\rangle&=&\n-\\D \\lvert \\cd \\underset{j}{\\downarrow}\\underset{j+1}{\\uparrow}\\cd\\rangle +\n\\lvert \\cd\\underset{j}{\\uparrow}\\underset{j+1}{\\downarrow}\\cd\\rangle.\\label{h4}\n\\end{eqnarray}\n\n\\subsection{$N=1$}\nLet $\\Psi_1(t)=\\sum_{x_1}\\psi_1(x_1;t) e_{\\{x_1\\}}$, then\n\\begin{eqnarray*} H_{XXZ}\\Psi_1(t)&=&\\sum_{j} h_{j,j+1}\\sum_{x_1}\\psi_1(x_1;t) e_{\\{x_1\\}}\\\\\n&=&  \\sum_j\\psi_1(j;t) h_{j,j+1} e_{\\{j\\}}+\\psi_1(j+1;t)h_{j,j+1}e_{\\{j+1\\}}\\\\\n&=& \\sum_j \\psi_1(j;t)\\left[ -\\D e_{\\{j\\}}+e_{\\{j+1\\}}\\right]+\\psi_1(j+1;t)\n\\left[-\\D e_{\\{j+1\\}}+e_{\\{j\\}}\\right]\\\\\n&=&\\sum_j \\left[ -2\\D\\psi_1(j;t)+\\psi_1(j-1;t)+\\psi_1(j+1;t)\\right] e_{\\{j\\}}\n\\end{eqnarray*}\nThus the coordinates $\\psi_1(x;t)$ must satisfy\n\\[ i\\frac{\\pl\\psi_1(x;t)}{\\pl t}= \\psi_1(x-1;t)+\\psi_1(x+1;t)-2\\D \\psi_1(x;t),\\>\\>x\\in\\mathbb{Z}, t\\ge 0,\\]\nwith initial condition\n\\[ \\psi_1(x;0)=\\dl_{x,y}.\\]\nThe solution is\n\\begin{equation} \\psi_1(x;t)=\\frac{1}{2\\pi i}\\,\\int_{\\mathcal{C}_r}\\xi^{x-y-1} e^{-i t\\ve(\\xi)}d\\xi\\, =e^{2i\\D t} (-i)^{x-y} \\, J_{x-y}(2t)\\label{psi1}\\end{equation}\nwhere\n\\[ \\ve(\\xi)=\\xi+\\frac{1}{\\xi}-2\\D\\]\nand $J_\\nu(z)$ is the Bessel function of order $\\nu$.\n\\par\nIt is obvious probabilistically, and is easily verified analytically, that\n\\[ \\sum_{x\\in\\mathbb{Z}} \\vert \\psi_1(x;t)\\vert^2 =1 \\]\nfor all $y\\in\\mathbb{Z}$.  We also have (setting $y=0$)\n\\[ \\sum_{x\\in \\mathbb{Z}} x \\vert\\psi_1(x;t)\\vert^2=0,\\>\\> \\sum_{x\\in\\mathbb{Z}} x^2\\vert\\psi_1(x;t)\\vert^2=t,\n\\>\\>\\textrm{and}\\>\\>\\sum_{x\\in\\mathbb{Z}} x^4\\vert\\psi_1(x;t)\\vert^2=t^2+3 t^4.\n\\]\n\\subsection{$N=2$}\nFor $N=2$ we set\n\\[ \\Psi_2(t)=\\sum_{x_1<x_2}\\psi_2(x_1,x_2;t) e_{\\{x_1,x_2\\}}\\]\nFor $x_2>x_1+1$ the action of $H_{XXZ}$ on $\\Psi_2$ is the same as above in each coordinate $x_i$:\n\\begin{equation} i \\frac{\\pl \\psi_2(x_1,x_2;t)}{\\pl t}=\\psi_2(x_1-1,x_2;t)+\\psi_2(x_1+1;x_2;t)+\\psi_2(x_1,x_2-1;t)+\\psi_2(x_1,x_2+1)-4\\D\\psi_2(x_1,x_2;t).\n\\label{n2a}\\end{equation}\nIf $x_2=x_1+1$ then due to (\\ref{h1}) and (\\ref{h2}) there are terms missing with the result that\n\\begin{equation} i \\frac{\\pl \\psi_2(x_1,x_2;t)}{\\pl t}=\\psi_2(x_1-1,x_2;t)+\\psi_2(x_1,x_2+1)-2\\D\\psi_2(x_1,x_2;t).\\label{n2b}\\end{equation}\nWe now require (\\ref{n2a}) to hold for all $(x_1,x_2)\\in\\mathbb{Z}^2$ and to satisfy the boundary condition\n\\[ \\psi_2(x_1,x_1;t)+\\psi_2(x_1+1,x_1+1;t)-2\\D\\psi_2(x_1,x_1+1;t)=0,\\>\\> x_1\\in\\mathbb{Z}.\\]\nIf this last boundary condition is satisfied then in region $\\mathcal{X}_2$ equation (\\ref{n2b}) is satisfied.\n\n\nDefine \\cite{YY} (the Yang-Yang $S$-matrix)\n\\begin{equation} S_{21}(\\xi_2,\\xi_1)=-\\frac{1+\\xi_1\\xi_2-2\\D\\xi_2}{1+\\xi_1\\xi_2-2\\D\\xi_1},\\quad  \\xi_1,\\xi_2\\in\\mathbb{C}.\\label{YY}\\end{equation}\nWith this choice of $S$,\n\\[ \\left\\{ \\xi_1^{x_1-y_1-1}\\xi_2^{x_2-y_2-1}+S_{21}(\\xi_2,\\xi_1) \\xi_2^{x_1-y_2-1}\n\\xi_1^{x_2-y_1-1}\\right\\} e^{-it(\\ve(\\xi_1)+\\ve(\\xi_2))}\\]\nsatisfies (\\ref{n2a}) and the boundary condition (\\ref{n2b}); however it does not satisfy the initial condition.\nWe take\\footnote{ Here and later all differentials $d\\xi$ and $d\\zeta$ incorporate the factor $(2\\pi \\textrm{i})^{-1}$.} ``linear combinations''\n\\begin{equation} \\psi_2(x_1,x_2;t)=\\int_{\\mathcal{C}_r}\\int_{\\mathcal{C}_r}\\left\\{ \\xi_1^{x_1-y_1-1}\\xi_2^{x_2-y_2-1}+S_{21}(\\xi_2,\\xi_1) \\xi_2^{x_1-y_2-1}\n\\xi_1^{x_2-y_1-1}\\right\\} e^{-it(\\ve(\\xi_1)+\\ve(\\xi_2))}\\, d\\xi_1 d\\xi_2.\\label{psi2}\\end{equation}\nA residue calculation shows that in the physical region\\footnote{$x_1<x_2$ and $y_1<y_2$} the initial condition $\\psi(x_1,x_2;0)=\\delta_{x_1,y_1}\\delta_{x_2,y_2}$ will be  satisfied if the radius of the circle $\\mathcal{C}_r$,\ncentered at the origin, is sufficiently small so that the only singularities of the integrand\nlying inside $\\mathcal{C}_r$ are those at 0.\n\\subsection{General $N$}\n\nThe generator of the finite $N$ asymmetric simple exclusion process (ASEP) is a similarity transformation \n(\\textit{not} a unitary transformation!) of the Heisenberg-Ising Hamiltonian.  Because of this the Schr\\\"odinger equation (\\ref{SchroEqn})\nfor the quantum spin chain is essentially identical to the master equation (Kolmogorov forward equation) for the Markov process ASEP assuming\nthe identification of parameters\n\\[ \\xi_i= \\xi_i^\\prime \/\\sqrt{\\t} ,\\>\\>\\t=\\frac{p}{q},\\>\\>2\\D=\\frac{1}{\\sqrt{pq}},\\>\\>\nS^{XXZ}_{\\b\\a}(\\xi_{\\b},\\xi_{\\a})=S_{\\b\\a}^{ASEP}(\\xi_{\\b}',\\xi_{\\a}'),\\>\\>\n\\ve^{XXZ}(\\xi)=\\frac{1}{\\sqrt{pq}}\\ve^{ASEP}(\\xi').\\]\n\\par\nThus given the  ASEP result \\cite{TW1, TW3} and the above identifications, we have\n\\par\n\\textsc{Theorem 1.}  \nFor $\\s\\in\\mathcal{S}_N$, define\n\\begin{equation} A_\\s(\\xi)=\\prod\\left\\{S_{\\b\\a}(\\xi_\\b,\\xi_\\a): \\{\\b,\\a\\}\\>\\> \\textrm{is an inversion in}\\>\\> \\s\\right\\}, \\label{e:A_coeff}\\end{equation}\nthen the solution to (\\ref{SchroEqn}) satisfying the initial condition $\\psi_N(X;0)=\\delta_{X,Y}$ is\n\\begin{equation} \\psi_N(X;t)=\\sum_{\\s\\in\\mathcal{S}_N}\\int_{\\mathcal{C}_r}\\cd\\int_{\\mathcal{C}_r} A_\\s(\\xi) \\prod_i\\xi_{\\s(i)}^{x_i} \\prod_i \\left(\\xi_i^{-y_{i}-1}\\,\n\\textrm{e}^{-\\textrm{i} t \\ve(\\xi_i)}\\right)\n d\\xi_1\\cd d\\xi_N\\label{psi_N}\\end{equation}\nwhere $\\mathcal{C}_r$ is a circle centered at zero with radius $r$ so small that all the poles of $A_\\s$ lie outside of\n$\\mathcal{C}_r$.\n\nAdditionally, we have a contour integral formula with large contours instead of small contours as above in \\textsc{Theorem 1}. Below, we will use a combination of the small and large contour formulas.\n\n\\textsc{Theorem 1a.}  \nFor $\\s\\in\\mathcal{S}_N$, define\n\\[ A_\\s(\\xi)=\\prod\\left\\{S_{\\b\\a}(\\xi_\\b,\\xi_\\a): \\{\\b,\\a\\}\\>\\> \\textrm{is an inversion in}\\>\\> \\s\\right\\},\\]\nthen the solution to (\\ref{SchroEqn}) satisfying the initial condition $\\psi_N(X;0)=\\delta_{X,Y}$ is\n\\begin{equation} \\psi_N(X;t)=\\sum_{\\s\\in\\mathcal{S}_N}\\int_{\\mathcal{C}_R}\\cd\\int_{\\mathcal{C}_R} A_\\s(\\xi) \\prod_i\\xi_{\\s(i)}^{x_i} \\prod_i \\left(\\xi_i^{-y_{i}-1}\\,\n\\textrm{e}^{-\\textrm{i} t \\ve(\\xi_i)}\\right)\n d\\xi_1\\cd d\\xi_N\\label{psi_N_large}\\end{equation}\nwhere $\\mathcal{C}_R$ is a circle centered at zero with radius $R$ so large that all the poles of $A_\\s$ lie inside of $\\mathcal{C}_R$.\n\nThe proof of this statement is an adaptation of the arguments in \\cite{TW1} that give the proof of \\textsc{Theorem 1}. For completeness, we give the proof of \\textsc{Theorem 1a} in \\Cref{s:appendix_a}.\n\n\n\n \\section{Probability $\\mathcal{P}_Y(x,m;t)$}\\label{s:one_point}\nIf the initial state is $e_Y\\in\\mathcal{H}_N$, $Y\\in\\mathcal{X}_N$, then\nat time $t$ the system is in state $\\Psi_N(t)=\\sum_{X\\in\\mathcal{X}_N} \\psi_N(X;t) e_X$ where\n$\\psi_N(X;t)$ is given by (\\ref{psi_N}) or (\\ref{psi_N_large}).  The\nquantity\n\\[ \\left\\vert\\langle e_X,\\Psi_N(t)\\rangle\\right\\vert^2=\\left\\vert\\psi_N  (X;t)\\right\\vert^2 ,\\>\\> X\\in\\mathcal{X}_N,\\]\nis the probability that the system is in state $e_X$ at time $t$. \n\\par\nDenote by $\\mathcal{P}_Y(x,m;t)$ the probability that at time $t$ the state has the $m$th particle from the left at position $x$\ngiven initially the state is $Y$. \nLet $X=\\{x_1,x_2,\\ldots, x_N\\}\\in\\mathcal{X}_N$, $1\\le m\\le N$, and define the projection operator\n\\begin{equation} P_{x,m}e_X=\\left\\{\\begin{array}{cc}e_X&\\textrm{if}\\>\\> x_m=x,\\\\\n0& \\textrm{otherwise}.\\end{array}\\right.\\end{equation} \nThen the outcome of the measurement yielding  ``the $m$th spin from the left is at position $x$ at\ntime $t$'' is that the system is now in state\n\\[\\Psi_N(x,m;t):= P_{x,m}\\Psi_N(t)=\\sum_{\\substack{X\\in\\mathcal{X}_N \\\\xi_m=x}}  \\psi_N(X;t) e_X.\\]\nThus the probability of this outcome is\n\\begin{equation} \\mathcal{P}_Y(x,m;t):=\\langle\\Psi_N(x,m;t),\\Psi_N(x,m;t)\\rangle=\n\\sum_{\\substack{X\\in\\mathcal{X}_N \\\\xi_m=x}}\\left\\vert\\psi_N(X;t)\\right\\vert^2. \\label{1ptProb}\\end{equation}\n\n\\subsection{Distribution of left-most particle}\nWe now restrict to the case $m=1$, i.e.\\ $\\mathcal{P}_Y(x,1;t)$.\nLet\n\\[ x_1=x,\\> x_2=x+v_1,\\>\\ldots, x_N=x+v_1+v_2+\\cdots+v_{N-1},\\> v_i\\ge 1,\\]\nand  note that  $\\overline{\\Psi_N(x;t)}=\\Psi_N(x;-t)$. Then, using (\\ref{psi_N}) for $\\Psi(x;t)$ and (\\ref{psi_N_large}) for $\\Psi(x;-t)$ with $R \\, r < 1$, followed by performing the\ngeometric sums (since $R\\,r <1$, the summations may be brought inside) \n\\begin{eqnarray*}\n\\mathcal{P}_Y(x,1;t)&=&\\sum_{\\substack{X\\in\\mathcal{X}_N \\\\xi_1=x}} \\psi_N(X;t)\\psi_N(X;-t)\\\\\n&=&\\sum_{\\s,\\mu\\in \\mathcal{S}_N}\\int_{\\mathcal{C}_R}\\cdots\\int_{\\mathcal{C}_r} \\sum_{v_i\\ge 1} A_\\s(\\xi) A_\\mu(\\zeta)\\,\n(\\xi_{\\s(2)}\\zeta_{\\mu(2)})^{v_1} (\\xi_{\\s(3)}\\zeta_{\\mu(3)})^{v_1+v_2}\\cdots \n(\\xi_{\\s(N)}\\zeta_{\\mu(N)})^{v_1+\\cdots+v_{N-1}}\\\\\n&&\\times\n\\prod_j (\\xi_j\\zeta_j)^{x-y_j-1} \\textrm{e}^{-\\textrm{i} t(\\ve(\\xi_j)-\\ve(\\zeta_j))}\\,d\\zeta_1\\cdots d\\zeta_N d\\xi_1\\cdots d\\xi_N\\\\\n&\\hspace{-15ex} =&\\hspace{-10ex}\\sum_{\\s,\\mu\\in \\mathcal{S}_N}\\int_{\\mathcal{C}_R}\\cdots\\int_{\\mathcal{C}_r}  A_\\s(\\xi) A_\\mu(\\zeta)\n\\frac{\\xi_{\\s(2)}\\zeta_{\\mu(2)} \\xi_{\\s(3)}^2\\zeta_{\\mu(3)}^2\\cdots\n \\xi_{\\s(N)}^{N-1} \\zeta_{\\mu(N)}^{N-1}}{(1-\\xi_{\\s(2)}\\zeta_{\\mu(2)}\\cdots \\xi_{\\s(N)}\\zeta_{\\mu(N)})\n(1-\\xi_{\\s(3)}\\zeta_{\\mu(3)}\\cdots \\xi_{\\s(N)}\\zeta_{\\mu(N)})\\cdots (1-\\xi_{\\s(N)}\\zeta_{\\mu(N)})}\\\\\n&&\\times\\prod_j (\\xi_j\\zeta_j)^{x-y_j-1} \\textrm{e}^{-\\textrm{i} t(\\ve(\\xi_j)-\\ve(\\zeta_j))}\\,d\\zeta_1\\cdots d\\zeta_N d\\xi_1\\cdots d\\xi_N \n\\end{eqnarray*}\nIn the formulas above, we have $2N$ contour integrals with the contour $\\mathcal{C}_r$ for the first $N$ contours and the contours $\\mathcal{C}_R$ for the following $N$ contours.\nNow, at the analogous step in ASEP, an identity\\footnote{See (1.6) in \\cite{TW1}.}  was\nderived that simplified the sum over $\\mathcal{S}_N$\n resulting in a single multidimensional integral.\\footnote{See Theorem 3.1 in \\cite{TW1}.}\n Now we have a \\textit{double sum} over $\\mathcal{S}_N$ and we need a new identity.  Fortunately\n such an identity has been discovered by Cantini, Colomo, and Pronko \\cite{CCP}.\n Let\n \\begin{equation} d(x,y):=\\frac{1}{(1-x \\,y)(x+y -2\\Delta\\,  x\\, y)}\\>\\>\\textrm{and}\\>\\> \nD_N(\\xi,\\zeta)=\\det\\left(d(\\xi_i,\\zeta_j)\\vert_{1\\le i,j\\le N}\\right),\\label{IK}\\end{equation}\nthen\n\\begin{eqnarray}\n\\sum_{\\s,\\mu\\in\\mathcal{S}_N} A_\\s(\\xi) A_{\\mu}(\\zeta) \\frac{\\xi_{\\s(2)}\\zeta_{\\mu(2)} \\xi_{\\s(3)}^2\\zeta_{\\mu(3)}^2\\cdots\n \\xi_{\\s(N)}^{N-1} \\zeta_{\\mu(N)}^{N-1}}{(1-\\xi_{\\s(2)}\\zeta_{\\mu(2)}\\cdots \\xi_{\\s(N)}\\zeta_{\\mu(N)})\n(1-\\xi_{\\s(3)}\\zeta_{\\mu(3)}\\cdots \\xi_{\\s(N)}\\zeta_{\\mu(N)})\\cdots (1-\\xi_{\\s(N)}\\zeta_{\\mu(N)})}\\nonumber\\\\\n=\\frac{(1-\\prod_j \\xi_j\\zeta_j) \\, \n\\prod_{i,j=1}^N(\\xi_i+\\zeta_j-2\\Delta \\xi_i\\zeta_j)}{\n\\prod_{i<j} (1+\\xi_i\\xi_j-2 \\Delta \\xi_i) (1+\\zeta_i\\zeta_j-2 \\Delta \\zeta_i)}\\>D_N(\\xi,\\zeta)&&\\label{1ptIden}\n\\end{eqnarray}\nRemarks:\\vspace{-4ex}\n\\begin{itemize}\n\\item  The identity (\\ref{1ptIden}) is Proposition 6 of \\cite{CCP} (with a change of notation). The identity (\\ref{1ptIden}) also appears in a more general setting of (spin) Hall-Littlewood functions in \\cite{P, WZJ}, which specializes to the ASEP case as shown in Corollary 7.1 in \\cite{P}.\n\\item\n  In Appendix B of \\cite{CCP}, the authors show that\n (\\ref{1ptIden}) reduces to (1.6) of \\cite{TW1} \nin the limit $\\xi_j\\rightarrow \\sqrt{\\frac{q}{p}}\\,\\xi_j$ and $\\zeta_j\\rightarrow \\sqrt{\\frac{p}{q}}$.\n\\item The determinant $D_N(\\xi,\\zeta)$ ``is nothing but the well-known \\textit{Izergin-Korepin\ndeterminant} \\cite{Iz, Ko} in disguise'' \\cite{W}. \n\\end{itemize}\n\\par\nWe thus have\n\\vspace{-2ex}\n\\begin{equation} \\hspace{-5ex}\\mathcal{P}_Y(x,1;t)=\\int_{\\mathcal{C}_R}\\cdots\\int_{\\mathcal{C}_r}\\frac{(1-\\prod_j \\xi_j\\zeta_j) \\, \n\\prod_{i,j=1}^N(\\xi_i+\\zeta_j-2\\Delta \\xi_i\\zeta_j)}{\n\\prod_{i<j} (1+\\xi_i\\xi_j-2 \\Delta \\xi_i) (1+\\zeta_i\\zeta_j-2 \\Delta \\zeta_i)}\\>D_N(\\xi,\\zeta)\n\\prod_j (\\xi_j\\zeta_j)^{x-y_j-1} \\textrm{e}^{-\\textrm{i} t(\\ve(\\xi_j)-\\ve(\\zeta_j))}\\,d^N\\zeta d^N\\xi\n\\label{detRep}\\end{equation}\nThe factor $(1-\\prod_j \\xi_j\\zeta_j)$ is eliminated if we consider \n\\begin{equation} \\mathcal{F}_N(x,t):=\\mathbb{P}_Y(X_1(t)\\ge x)=\\sum_{n=x}^\\iy\\mathcal{P}_Y(n,1;t)\\label{cFn}\\end{equation}\n\\par\n\nFrom \\cite{CCP}\n\\begin{equation}  \\prod_{1\\le j,k\\le N} (\\xi_j+\\zeta_k-2 \\D \\xi_j \\zeta_k)\\,\\cdot\\, D_N(\\xi,\\zeta)\n=\\frac{\\D_N(\\xi)\\D_N(\\zeta)}{\\prod_{j,k} (1-\\xi_j\\zeta_k)} Q_{N}(\\xi,\\zeta) \\label{Wfn} \\end{equation}\nwhere $Q_N$ is a ``polynomial of degree $N-1$ in each variable, separately symmetric under permutations\nof the variables within each set'' \\cite{CCP, W}.\\footnote{For example\n\\begin{eqnarray*} Q_1(\\xi,\\zeta)&=&1,\\\\\nQ_2(\\xi,\\zeta)&=&4 \\Delta ^2 \\zeta _1 \\zeta _2 \\xi _1\n   \\xi _2-2 \\Delta  \\zeta _1 \\zeta\n   _2 \\xi _1-2 \\Delta  \\zeta _1\n   \\zeta _2 \\xi _2-2 \\Delta  \\zeta\n   _1 \\xi _1 \\xi _2-2 \\Delta  \\zeta\n   _2 \\xi _1 \\xi _2+\\zeta _1 \\zeta\n   _2 \\xi _1 \\xi _2+\\zeta _1 \\zeta\n   _2+\\xi _1 \\xi _2+1,\n   \\end{eqnarray*}\n   $Q_3$ in expanded form has 459 terms, and $Q_4$ has 60,820 terms.} Here $\\D_N(\\xi)$ is the Vandermonde product $\\prod_{1\\le j<k\\le N}(\\xi_k-\\xi_j)$ (not to\nbe confused with the constant $\\D$).\n   It's useful to define\n\\[ U(\\xi,\\xi'):=\\frac{1+\\xi\\x'-2\\D\\xi}{\\xi'-\\xi} .\\]\n \n\n\n   The identity (\\ref{1ptIden}) can be rewritten as\n   \\begin{eqnarray} \\sum_{\\s,\\mu}\\prod_{i<j} U(\\xi_{\\s(i)},\\xi_{\\s(j)}) U(\\zeta_{\\mu(i)},\\zeta_{\\mu(i)})\\,\n   \\frac{\\xi_{\\s(2)}\\zeta_{\\mu(2)} \\xi_{\\s(3)}^2\\zeta_{\\mu(3)}^2\\cdots\n \\xi_{\\s(N)}^{N-1} \\zeta_{\\mu(N)}^{N-1}}{(1-\\xi_{\\s(2)}\\zeta_{\\mu(2)}\\cdots \\xi_{\\s(N)}\\zeta_{\\mu(N)})\n(1-\\xi_{\\s(3)}\\zeta_{\\mu(3)}\\cdots \\xi_{\\s(N)}\\zeta_{\\mu(N)})\\cdots (1-\\xi_{\\s(N)}\\zeta_{\\mu(N)})}\\nonumber \\\\\n&\\hspace{-160ex}=&\\hspace{-80ex} \\frac{1-\\prod_j\\xi_j \\zeta_j}{\\prod_{j,k}(1-\\xi_j\\zeta_k)}\\, Q_N(\\xi,\\zeta)\n\\label{idenU}\n\\end{eqnarray}\nThe close relationship of (\\ref{idenU}) to (1.6) of \\cite{TW1} (see also Identity $1_L$ in \\cite{TW4}) is now clearer.\nWe have proved\n\\par\n\\textsc{Theorem 2.}\n$\\mathcal{F}_N(x,t)=\\mathbb{P}_Y(X_1(t)\\ge x)$ equals\n\\begin{equation}\n\\int_{\\mathcal{C}_R}\\cdots\\int_{\\mathcal{C}_r} \\frac{\\prod_{j,k}(\\xi_j+\\zeta_k-2\\D\\xi_j\\zeta_k)}{\\prod_{j<k}(1+\\xi_j\\xi_k-2\\D\\xi_j)(1+\\zeta_j\\zeta_k-2\\D\\zeta_j)}\\,\nD_N(\\xi,\\zeta)\n\\prod_j (\\xi_j \\zeta_j)^{x-y_j-1} \\textrm{e}^{-\\textrm{i} t(\\ve(\\xi_j)-\\ve(\\zeta_j))}\\, d^N\\zeta \\,d^N\\xi\\hspace{5ex}\\label{Fprob1}\\end{equation}\n\\begin{equation} = \\int_{\\mathcal{C}_R}\\cdots\\int_{\\mathcal{C}_r} \\frac{\\D_N(\\xi)\\D_N(\\zeta)}{\\prod_{j<k}(1+\\xi_j\\xi_k-2\\D\\xi_j)\n(1+\\zeta_j\\zeta_k-2\\D\\zeta_j)}\\,\n\\frac{Q_N(\\xi,\\zeta)}{\\prod_{j,k}(1-\\xi_j\\zeta_k)}\n\\prod_j (\\xi_j \\zeta_j)^{x-y_j-1} \\textrm{e}^{-\\textrm{i} t(\\ve(\\xi_j)-\\ve(\\zeta_j))}\\, d^N\\zeta \\,d^N\\xi\\hspace{5ex}\\label{FprobQ}\n\\end{equation}\nwhere $\\mathcal{C}_r$ (resp.~$\\mathcal{C}_R$) is a circle centered at zero with radius $r$ (resp.~$R$) so small (resp.~large) that all the poles of the integrand except for the the poles at the origin (resp.~infinity) lie outside $\\mathcal{C}_r$ (resp.~inside $\\mathcal{C}_R$) and $R\\,r <1$.\n\\section{Special case $\\D=0$}\\label{s:delta_zero}\nWhen $\\D=0$, (\\ref{FprobQ}) reduces to\n\\begin{eqnarray}\n\\mathcal{F}_N(x,t)\\big\\vert_{\\D=0}&=&\n    \\int_{\\mathcal{C}_R}\\cdots\\int_{\\mathcal{C}_r} \\frac{\\D_N(\\xi)\\D_N(\\zeta)}{\\prod_{j,k}(1-\\xi_j\\zeta_k)}\n\\prod_j (\\xi_j \\zeta_j)^{x-y_j-1} \\textrm{e}^{-\\textrm{i} t(\\ve(\\xi_j)-\\ve(\\zeta_1))}\\, d^N\\zeta \\,d^N\\xi\\hspace{5ex}\n\\label{FprobQ2}\\\\\n&=& \n\\int_{\\mathcal{C}_R}\\cdots\\int_{\\mathcal{C}_r} \\det\\left(\\frac{1}{1-\\xi_j\\zeta_k}\\right)\n\\prod_j (\\xi_j \\zeta_j)^{x-y_j-1} \\textrm{e}^{-\\textrm{i} t(\\ve(\\xi_j)-\\ve(\\zeta_1))}\\, d^N\\zeta \\,d^N\\xi\\hspace{5ex}\n\\label{FprobQ3}\n\\end{eqnarray}\nsince\n\\[ \\lim_{\\D\\rightarrow 0}\\frac{Q_N(\\xi,\\zeta)}{\\prod_{j<k}(1+\\xi_j\\xi_k-2\\D\\xi_j)(1+\\zeta_j\\zeta_k-2\\D\\zeta_j)}=1.\\]\nand\n\\[ \\det\\left(\\frac{1}{1-\\xi_j\\zeta_k}\\right)=\\frac{\\D_N(\\xi)\\D_N(\\zeta)}{\\prod_{j,k} (1-\\xi_j\\zeta_k)}.\\]\n More directly since $A_\\s\\big\\vert_{\\Delta=0}=\\textrm{sgn}(\\s)$,  we use the identity\n    \\[\n    \\sum_{\\s,\\mu}\\textrm{sgn}(\\s)\\textrm{sgn}(\\mu) \\frac{\\xi_{\\s(2)}\\zeta_{\\mu(2)} \\xi_{\\s(3)}^2\\zeta_{\\mu(3)}^2\\cdots\n \\xi_{\\s(N)}^{N-1} \\zeta_{\\mu(N)}^{N-1}}{(1-\\xi_{\\s(2)}\\zeta_{\\mu(2)}\\cdots \\xi_{\\s(N)}\\zeta_{\\mu(N)})\n(1-\\xi_{\\s(3)}\\zeta_{\\mu(3)}\\cdots \\xi_{\\s(N)}\\zeta_{\\mu(N)})\\cdots (1-\\xi_{\\s(N)}\\zeta_{\\mu(N)})}\\]\n   \\begin{equation} =\\det\\left(\\frac{1}{1-\\xi_j\\zeta_k}\\right)\n    \\end{equation}\n    \\par\n    \\subsection{Fredholm determinant representation}\nDefine\n\\[ \\phi_j(\\xi)=\\xi^{x-y_j-1}\\textrm{e}^{-\\textrm{i} t\\ve(\\xi)},\\>\\> \n\\psi_j(\\zeta)=\\zeta^{x-y_j-1}\\textrm{e}^{\\textrm{i} t\\ve(\\zeta)}\\]\nand\n\\begin{equation} K(j,k)=\\frac{\\phi_j(\\xi_j) \\psi_k(\\zeta_k)}{1-\\xi_j\\zeta_k}\\label{K}\\end{equation}\nThus\n\\begin{equation} \n\\mathcal{F}_N(x,t)\\big\\vert_{\\D=0}=\n    \\int_{\\mathcal{C}_R}\\cdots\\int_{\\mathcal{C}_r} \\det(K)\\, d^N\\zeta \\,d^N\\xi= \\int_{\\mathcal{C}_r}\\cdots\\int_{\\mathcal{C}_r} \\det(K)\\, d^N\\zeta \\,d^N\\xi\\hspace{5ex}\\label{Kdet}\\end{equation}\n    \\par\n    For the second identity, we deformed the contours from $\\mathcal{C}_R$ to $\\mathcal{C}_r$ for all the $\\zeta$-variables. When we deform the contours, we don't cross any poles since the poles, given by $1- \\xi_j \\zeta_k=0$, are located outside of the contour $\\mathcal{C}_R$ since we have taken $R\\, r <1$. Additionally, note that the variable $\\xi_j$ appears only in row $j$ and $\\zeta_k$ appears only in column $k$. It follows that the multiple integral\n    is gotten by integrating each $K(j,k)$ with respect to $\\xi_j$, $\\zeta_k$.\n    Therefore the multiple integral (\\ref{Kdet}) equals the determinant with $j,k$ entry\n    \\[ K_N(j,k)=\\int_{\\mathcal{C}_r}\\int_{\\mathcal{C}_r} \\frac{\\phi_j(\\xi)\\psi_k(\\zeta)}{1-\\xi\\zeta}\\,d\\zeta d\\xi \\]\n    \\par\n    We consider step initial condition, so that $y_j=j$. In preparation\n    for taking the limit as $N\\rightarrow\\iy$, we make the replacements\n    $j\\rightarrow j+1$, $k\\rightarrow k+1$, so that the indicies run for $0$ to $N-1$  rather than\n    $1$ to $N$. Then, in preparation for eventual steepest descent, we make\n    the substitutions $\\xi\\rightarrow\\textrm{i}\\,\\xi$, $\\zeta\\rightarrow\\zeta\/\\textrm{i}$.  Aside from the factor $\\textrm{e}^{\\textrm{i}\\pi(j-k)\/2}$, which will not\n    affect the determinant, the kernel becomes\n        \\[ L_N(j,k)=\\int_{\\mathcal{C}_r}\\int_{\\mathcal{C}_r}\\frac{\\xi^{x-j-2}\\,\\zeta^{x-k-2}}{1-\\xi\\zeta} \n        \\textrm{e}^{t (\\theta(\\xi)+\\theta(\\zeta))}\\, d\\zeta d\\xi,\\]\n    where we have set $\\theta(\\xi)=\\xi-1\/\\xi$.  We write the above as\n    \\[ \\sum_{\\ell=0}^\\iy\\int_{\\mathcal{C}_r}\\int_{\\mathcal{C}_r} \\xi^{x-j+\\ell-2}\\zeta^{x-k+\\ell-2}\\,\\textrm{e}^{t(\\theta(\\xi)+\\theta(\\zeta))}\\,d\\zeta d\\xi.\\]\n    \\par\n    We may take all integrations over the unit circle $\\mathcal{C}_1$ and in the $\\zeta$-integral make the substitution\n    $\\zeta\\ra1\/\\zeta$.  We obtain\n    \\[ L_N(j,k)=\\sum_{\\ell=0}^\\iy \\int_{\\mathcal{C}_1}\\int_{\\mathcal{C}_1} \\xi^{x-j+\\ell-2}\\zeta^{-x+k-\\ell}\\,\\textrm{e}^{t(\\theta(\\xi)-\\theta(\\zeta))}\\, d\\zeta d\\xi.\\]\n    In Toeplitz terms this is the operator\n    \\[ P_N T(a) T(a^{-1}) P_N,\\]\n    where $P_N$ is the projection from $\\ell^2(\\mathbb{Z}^+)$\\footnote{$\\mathbb{Z}^+$ denotes\n    the set of nonnegative integers.} to $\\ell^2([0,\\ldots,N-1])$ and where $a$ is the symbol\n    \\[ a(\\xi)=\\xi^{x-1} \\textrm{e}^{t\\,\\theta(\\xi)}.\\]\n    It it known (see, e.g.\\ \\S5.1 in \\cite{BS})  that $T(a) T(a^{-1})$ is of the form $I$+trace class and so $\\det(K_N)$ has\n    the limit $\\det(T(a) T(a^{-1}))$ on $\\ell^2(\\mathbb{Z}^+)$.\\footnote{One can show that for $x>1$ the determinant\n    of the product is zero.}\n    \\par\n    By a well-known identity, $T(a) T(a^{-1})=I-H(a) H(\\tilde{a}^{-1})$, where $H(a)$ denotes\n    the Hankel operator and $\\tilde{a}(\\xi)=a(\\xi^{-1})$.  In this case $\\tilde{a}=a^{-1}$ and the square\n    of $H(a)$ has kernel\\footnote{Recall that the $i,j$-entry of $H(f)$ is $f_{i+j+1}=\\int \\xi^{-i-j-2} f(\\xi)\\,d\\xi$.}\n    \\[ L(j,k)=\\sum_{\\ell=0}^\\iy \\int_{\\mathcal{C}_1}\\int_{\\mathcal{C}_1} \\xi^{x-j-\\ell-3} \\zeta^{x-k-\\ell-3} \\,\\textrm{e}^{t(\\theta(\\xi)+\\theta(\\zeta))}\n    \\, d\\zeta d\\xi,\\]\n    and we are interested in $\\det(I-L)$.   The substitutions $\\xi\\ra1\/\\xi$, $\\zeta\\rightarrow 1\/\\zeta$ give\n    \\begin{equation} L(j,k)=\\sum_{\\ell=0}^\\iy \\int_{\\mathcal{C}_1}\\int_{\\mathcal{C}_1}\\xi^{-x+j+\\ell+1} \\zeta^{-x+k+\\ell+1}\\,\\textrm{e}^{-t(\\theta(\\xi)+\\theta(\\zeta))}\\, d\\zeta d\\xi.\\label{Lexpand}\\end{equation}\n    If we take our integrals over $\\mathcal{C}_r$ and sum we obtain\n    \\begin{equation} L(j,k)=\\int_{\\mathcal{C}_r}\\int_{\\mathcal{C}_r}\\frac{\\xi^{-x+j+1}\\zeta^{-x+k+1}\\,\\textrm{e}^{-t(\\theta(\\xi)+\\theta(\\zeta))}}{1-\\xi\\zeta}\\ d\\zeta d\\xi\n    \\label{L}\\end{equation}\n    The kernel $L(j,k)$ is known as the \\textit{discrete Bessel kernel} \\cite{Bo} (see also Chapter 8 in \\cite{BDS}) due to the following representation.\n    Using the Bessel generating function\n    \\[\\exp(t\\theta(\\xi))=\\sum_{n=-\\iy}^\\iy \\xi^n J_n(2t) \\] \n    in (\\ref{Lexpand}) and the identity, $\\nu\\neq \\mu$,\n    \\begin{equation} \\sum_{n=0}^\\iy J_{\\nu+n}(t) J_{\\mu+n}(t)= \\frac{t}{2(\\nu-\\mu)}\\left[J_{\\nu-1}(t) J_{\\mu}(t)-J_{\\nu}(t) \n    J_{\\mu-1}( t)\\right]\\label{besselIden}\\end{equation}\nwe find\n\\[ L(j,k) =t\\, \\frac{J_{j-x+1}(2t) J_{k-x+2}(2t)-J_{j-x+2}(2t) J_{k-x+1}(2 t)}{j-k}\\]\nFor $j=k$ one lets $\\mu\\rightarrow\\nu$ in (\\ref{besselIden}) to find\n\\[L(j,j)=\\sum_{n=0}^\\iy J_{\\nu+n}(2t)^2 =t\\,\\left[ J_\\nu(2t) \\frac{\\pl J_\\mu}{\\pl\\mu}\\big\\vert_{\\mu=\\nu-1} -J_{\\nu-1}(2t) \n\\frac{\\pl J_\\mu}{\\pl \\mu}\\big\\vert_{\\mu=\\nu-1}\\right],\\>\\nu=-x+j+1.\\] \nFor $x\\le 1$ and domain wall initial condition $Y=\\mathbb{N}$,  we have the Toeplitz representation \n\\begin{equation} \\mathbb{P}_{\\mathbb{N}}(X_1(t)\\ge x)\\big\\vert_{\\D=0}=\\det(I-L)_{\\ell^2(\\{1-x, 2-x, \\dots\\})}=\\textrm{e}^{-t^2} \\det\\left(I_{j-k}(2 t)\\right)\n \\Big\\vert_{j,k=0,\\ldots,-x}\n\\label{BOiden}\\end{equation}\nwhere the last equality\\footnote{$I_\\nu(z)$ is the modified Bessel function of order $\\nu$.} was proved in \\cite{BOk}.\n  \\par\n  If $\\mathcal{L}(t)$ denotes the length of the longest increasing subsequence of a random permutation of size $\\mathcal{N}$ where\n  $\\mathcal{N}$ is a Poisson random variable with parameter $t^2$, then \\cite{BDJ, BDS, G}\n  \\[ \\mathbb{P}(\\mathcal{L}(t)\\le n) =\\textrm{e}^{-t^2} \\det(I_{j-k}(2t))_{j,k=0,\\ldots,n-1}\\]\n  \n\\textsc{Theorem 3.} For $x \\leq 1$ and domain wall initial conditions $Y = \\mathbb{N}$, we have\n\\begin{equation}\n\\mathbb{P}_{\\mathbb{N}}(X_1(t)\\ge x)\\big\\vert_{\\D=0}=\\mathbb{P}(\\mathcal{L}(t)\\le 1-x)\n\\end{equation}\nwhere $\\mathcal{L}(t)$ denotes the length of the longest increasing subsequence of a random permutation of size $\\mathcal{N}$ so that $\\mathcal{N}$ is a Poisson random variable with parameter $t^2$.\n\n\\subsection{Asymptotics}\nFrom the classic work of Baik, Deift, and Johnasson \\cite{BDJ} (see also Chapter 9 in \\cite{BDS}), we know that the limiting distribution of\n$\\mathcal{L}(t)$ is\n\\begin{equation} \\lim_{t\\rightarrow\\iy}\\mathbb{P}\\left(\\frac{\\mathcal{L}(t)-2 t}{t^{1\/3}}\\le x\\right) =F_2(x)\\end{equation}\nwhere $F_2$ is the $\\beta=2$  TW distribution  \\cite{TW0a, TW0b}.\nIn the present problem, $\\D=0$,  we can therefore conclude \nthat the left-most particle for domain wall initial condition $Y=\\mathbb{N}$  has the limiting distribution\n\\begin{equation} \\lim_{t\\rightarrow\\iy} \\mathbb{P}\\left(\\frac{X_1(t)+2t}{t^{1\/3}}\\ge -y\\right)= F_2(y).\\label{TWF2}\\end{equation}\n\n\n\n\\section{Steepest descent curve}\\label{s:steepest_descent}\n\n\\subsection{Spectral functions}\nWe introduce a pair of functions \n\\begin{equation}\\label{e:spectral_function}\n        G(\\xi) = x \\log \\xi - i t (\\xi + \\xi^{-1}), \\quad H(\\zeta) = -x \\log \\zeta - i t (\\zeta + \\zeta^{-1}),\n\\end{equation}\nwhich we call the \\emph{spectral functions}. Note that the spectral functions appear in the integrand of the formula for $\\mathcal{F}_N(x,t)$ given by (\\ref{Fprob2}). In particular, we have\n\\begin{equation}\n    (\\xi_j \\zeta_j)^{x} e^{- i t (\\varepsilon(\\xi_j) - \\varepsilon(\\zeta_j))} = \\exp \\left \\{ G(\\xi_j) - H(\\zeta_j)\\right\\}.\n\\end{equation}\nIn the following, we will deform the contours in the contour integral formula for $\\mathcal{F}_N$ given by (\\ref{Fprob2}) so that the real part of the difference of the spectral function is negative, $\\mathrm{Re}(G- H) <0$. Thus, making $\\mathcal{F}_N$ suitable for asymptotic analysis.\n\n\\subsection{Critical points}\nThe steepest descent contours in the contour integral formula $\\mathcal{F}_N$ given by (\\ref{Fprob1}) are determined by the critical points of the spectral functions. We have\n\\begin{equation}\n        G'(\\xi)  = \\frac{- i t \\xi^2 + x \\xi + i t}{\\xi^2}, \\quad H'(\\zeta) = \\frac{-i t \\zeta^2 - x \\zeta + it }{\\zeta^2}.\n\\end{equation}\nso that the critical points are given by\n\\begin{equation}\n    \\xi = \\frac{x \\pm \\sqrt{x^2 -4 t^2}}{2 i t},\\quad \\zeta = \\frac{-x \\pm \\sqrt{x^2 -4 t^2}}{2 i t}.\n\\end{equation}\nNote that each function, $G$ and $H$, has a double critical point when $x = \\pm 2 t$ and the critical point are\n\\begin{equation}\n    \\xi_0  = \\begin{cases} - i , \\quad x= 2 t \\\\ i , \\quad x = -2t \\end{cases}, \\quad \\zeta_0  = \\begin{cases}  i , \\quad x= 2 t \\\\ -i , \\quad x = -2t \\end{cases},\n\\end{equation}\nrespectively. Physically, we expect the point $x=-2t$ to correspond to the left-edge of the up-spins and the point $x= 2t$ to correspond to the right-edge of the up-spins. Thus, we restrict our attention to the critical point given by $x= -2t$ and take $(\\xi_0, \\zeta_0)= (i, -i)$.\n\n\\subsection{Steepest descent curve: local}\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[scale=0.3]{local_SD_curve.pdf}\n    \\caption{The curves show the directions of large increase of magnitude for the real part of $G$ (i.e.~steepest descent) near the critical point, which is centered to be at the origin. We label the curves so that the sign of $Re\\, G$ is clear.}\n    \\label{f:local_SD}\n\\end{figure}\n\nThe steepest descent curve depends locally on the value of the third derivative for the double critical points. In particular, we have\n\\begin{equation}\n    G^{(3)}(- 2t) = 2 i t, \\quad H^{(3)}(- 2 t) = 2 i t.\n\\end{equation}\nThis means that the steepest descent curves for both spectral functions are locally the same. That is,\n\\begin{equation}\\label{e:Taylor}\n    G(\\xi) - G(\\xi_0) = \\frac{i t}{3} (\\xi - \\xi_0)^3 + \\mathcal{O}((\\xi-\\xi_0)^4), \\quad H(\\zeta) - H(\\zeta_0) = \\frac{i t}{3} (\\zeta - \\zeta_0)^3 + \\mathcal{O}((\\zeta-\\zeta_0)^4) \n\\end{equation}\nA diagram for the local steepest descent curve is give in Figure \\ref{f:local_SD}.\n\n\n\n\\subsection{Steepest descent curve: global}\n\n\\begin{figure}\n     \\centering\n     \\begin{subfigure}[h!]{0.45\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{global_SD_Curve2a.png}\n         \\caption{}\n        \n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[h!]{0.45\\textwidth}\n         \\centering\n        \\includegraphics[width=\\textwidth]{global_SD_Curve2b.png}\n         \\caption{}\n        \n     \\end{subfigure}\n        \\caption{The level curves for the spectral functions, $G$ and $H$, with $\\xi_0 = i$, $\\zeta_0=-i$, and $x = -2 t$. The component corresponding to the steepest descent path is denoted by red arrows with the direction along the steepest descent.}\n        \\label{f:global_SD_2}\n\\end{figure}\n\nGlobally, we obtain the steepest descent curve by an implicit equation through the Cauchy-Riemann equations for holomorphic functions. In short, we have that the direction of greatest change in the real part of a holomorphic function is the same direction of no change in the imaginary part of a the same holomorphic function (away from singular points). Therefore, the steepest descent curve is given by the level curves of the imaginary part of the spectral functions:\n\\begin{equation}\n    \\{\\xi \\in \\mathbb{C} : Im\\, G(\\xi) = Im\\, G(\\xi_0)\\}, \\quad \\{\\zeta \\in \\mathbb{C} : Im\\, H(\\zeta) = Im\\, H(\\zeta_0)\\}\n\\end{equation}\nwith $(\\xi_0 , \\zeta_0) = ( i, -i)$. The level curves will have multiple components, but we are only interested in the components that (i) are closed curves in $\\mathbb{C} \\cup \\{ \\infty\\}$ and (ii) $Re \\, G(\\xi) - G(\\xi_0) \\leq 0$ and $Re \\, H(\\xi) - H(\\xi_0) \\geq 0$.\\par\n\nThe steepest descent curves are components of the curves given by\n\\begin{equation}\\label{e:steepest_descent}\n\\begin{split}\n    &2 \\tan^{-1}\\left(\\frac{y}{x} \\right) + \\frac{x^3 + x y^2 +x}{x^2 +y^2} = \\begin{cases}  \\pi, &\\quad x>0 \\\\ -\\pi, &\\quad x<0 \\end{cases},\\\\\n    & 2 \\tan^{-1}\\left(\\frac{v}{u} \\right) - \\frac{u^3 + u v^2 +u}{u^2 +v^2} =  \\begin{cases}  -\\pi, &\\quad u>0 \\\\ \\pi, &\\quad u<0 \\end{cases}\n\\end{split}\n\\end{equation}\nwith $\\xi = x + i y$ and $\\zeta = u + i v$. We plot these curves using \\emph{Mathematica}, see Figure \\ref{f:global_SD_2}. The components of the level curves corresponding to the steepest descent path is determined by the local behaviour determined in Figure \\ref{f:local_SD}.\n\n\n\\subsection{Steep descent curve}\n\nWe want a friendly version of the steepest descent curves given implicitly by (\\ref{e:steepest_descent}) or, rather, a more explicit version. We introduce the \\emph{steep descent contours} given by three segments on three regions: in the region near the critical points, we take straight lines coming emanating from the critical point at angles $\\pm\\pi\/6$ and $\\pm5\\pi\/6$; in an intermediate region, we take horizontal lines emanating from the end points of the straight lines in region near the critical point; in the region far away from the critical point, we take a segment of the circle $\\mathcal{C}_R$ that connects with the horizontal lines. We use these contours so that we may explicitly determine the location of the poles when we deform to these \\emph{steep descent contours}. Although these contours don't follow the path of steepest descent for the real part of the spectral function, we show below that we still have the main property that $Re\\,\\{ G(\\xi) - G(\\xi_0)\\} \\leq 0$ and $Re \\, \\{H(\\zeta) - H(\\zeta_0)\\} \\geq0$ along these steep descent contours.\n\nWe now give a precise definition for the steep descent contours. We give a piece-wise description based on the proximity to the critical points. Let $\\mathcal{B}(z, r)$ be a ball centered at $z\\in \\mathbb{C}$ of radius $r>0$ and $\\mathcal{B}(z,r)^c$ be its complement. Then, we take the components \n\\begin{equation}\\label{e:contour_parts}\n    \\begin{split}\n    \\Gamma_{\\pm}^{(1)}&=\\{\\pm i + x  e^{ \\pm\\pi i\/6} \\mid 0 \\leq x \\} \\cap \\mathcal{B}(\\pm i, 1),\\\\\n    \\Gamma_{\\pm}^{(2)}&=\\{\\pm i + x e^{ \\pm 5\\pi i\/6} \\mid 0 \\leq x \\} \\cap \\mathcal{B}(\\pm i, 1)\\\\\n    \\Gamma_{\\pm}^{(3)}&=\\{\\pm i +  e^{\\pm \\pi i\/6} +x \\mid 0 \\leq x \\} \\cap \\mathcal{B}(\\pm i, 1)^c \\cap \\mathcal{B}(0, R_{\\pm}),\\\\\n    \\Gamma_{\\pm}^{(4)}&=\\{\\pm i + 1 e^{\\pm 5 \\pi i\/6} -x \\mid 0 \\leq x \\} \\cap \\mathcal{B}(\\pm i, 1)^c \\cap \\mathcal{B}(0, R_{\\pm})\\\\\n    \\Gamma_{\\pm}^{(5)}&= \\mathcal{C}_R \\cap \\{z\\in \\mathbb{C} \\mid Im \\, \\{z\\} \\leq (\\pm1)Im\\, \\{\\pm i + e ^{\\pm \\pi i \/6}\\} \\}\n    \\end{split},\n\\end{equation}\nwith radii $R_{\\pm} > \\sqrt{3}$. The bound on the radii is chosen so that the horizontal segments of the contours are non-trivial. Then, the steep descent contours are given by\n\\begin{equation}\\label{e:steep_contours}\n    \\Gamma_{k} = \\Gamma_k^{(1)} \\cup \\Gamma_k^{(2)} \\cup \\Gamma_k^{(3)} \\cup \\Gamma_k^{(4)} \\cup \\Gamma_k^{(5)}\n\\end{equation}\nfor $k=\\pm$. See Figure \\ref{f:global_SD_3}.\n\n\\begin{figure}\n     \\centering\n     \\begin{subfigure}[h]{0.45\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{SDConttour4a.pdf}\n         \\caption{}\n        \n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[h]{0.45\\textwidth}\n         \\centering\n        \\includegraphics[width=\\textwidth]{SDContour3a.pdf}\n         \\caption{}\n        \n     \\end{subfigure}\n        \\caption{The components of the $\\Gamma_{\\pm}$ contours.}\n        \\label{f:global_SD_3}\n\\end{figure}\n\n\\textsc{Lemma 6.1} Let $x=-2t$ and take the contours $\\Gamma_k$, $k=\\pm$, given by (\\ref{e:contour_parts}) and (\\ref{e:steep_contours}). Additionally, take $t^{-\\alpha} \\leq T \\ll 1$ with $1\/4 < \\alpha < 1\/3$. Then, we have\n\\begin{equation}\\label{e:bound1}\n    Re\\,\\{ G(\\xi) - G(\\xi_0)\\} \\leq 0, \\quad Re \\, \\{H(\\zeta) - H(\\zeta_0)\\} \\geq 0 \n\\end{equation}\nif $\\xi \\in \\Gamma_{+}$ and $\\zeta \\in \\Gamma_{-}$. Moreover, if $\\xi \\in \\Gamma_{+} \\cap \\mathcal{B}(\\textrm{i}, t^{-\\alpha})^c$ and $\\zeta \\in \\Gamma_{-} \\cap \\mathcal{B}(-\\textrm{i}, t^{-\\alpha})^c$, we have\n\\begin{equation}\\label{e:bound2}\n    Re\\,\\{ G(\\xi) - G(\\xi_0)\\} < - c_1(T)\\, t^{1- 3\\alpha}, \\quad Re \\, \\{H(\\zeta) - H(\\zeta_0)\\} > c_2(T)\\, t^{1-3\\alpha}, \n\\end{equation}\nfor some constants $c_1(T), c_2(T)>0$ that depend only on $T$.\n\n\\begin{proof}\nWe prove the bounds by showing that derivative of the real part of the functions are monotone along the different segments of the contours $\\Gamma_{\\pm}$ as parameterized in (\\ref{e:contour_parts}). Since $G(\\xi) - G(\\xi_0) = 0$ for $\\xi = \\xi_0$ and $H(\\zeta) - H(\\zeta_0) = 0$ for $\\zeta = \\zeta_0$, the first bounds (\\ref{e:bound1}) then follow by monotonicity. Moreover, since the real part of the functions are monotone, we establish the bounds (\\ref{e:bound2}) by bounding the real part of the functions on the boundary of the segment $\\Gamma_{\\pm} \\cap \\mathcal{B}(\\pm \\textrm{i}, t^{-\\alpha})$. \n\nThe arguments for both functions are the same, except for some negative signs here and there. So, we focus solely on the case for the $G$ function. Additionally, the arguments are fairly routine and standard. So, we just sketch the main idea needed for the bounds.\n\nTake $\\xi \\in \\Gamma_{+}^{(1)} \\cup \\Gamma_{+}^{(2)}$. In this case, we have $\\xi =\\textrm{i} + x e^{\\pi i \/6}$ or $\\xi  = \\textrm{i} +x  e^{5\\pi i \/6}$, with $0 \\leq x \\leq 1$ since $\\Gamma_{+}^{(1)} \\cup \\Gamma_{+}^{(2)} \\subset \\mathcal{B}(i, 1)$. Then, we may write the real part of the $G$ function explicitly and show that it is monotone by taking its derivative. For instance, we have\n\\begin{equation}\n    \\frac{d}{dx} Re\\,\\{G(i +  x e^{\\pi i \/6} ) - G(i)\\} = \\frac{t}{2} \\left(1 - \\frac{2 +4 x}{1 +x + x^2} + \\frac{1 + 4x +x^2}{(1 + x+x^2)^2} \\right).\n\\end{equation}\nOne may now check that the derivative is zero when $x=0$ and negative if $0 < x < 1 + \\sqrt{3}$. Thus, the bound (\\ref{e:bound1}) follows for this segment.\n\nTake $\\xi \\in \\Gamma_{+}^{(3)} \\cup \\Gamma_{+}^{(4)}$. In this case, we have $\\xi = i +  e^{\\pi i \/6} + x$ or $\\xi  = i+ e^{5\\pi i \/6} -x$, with $x$ non-negative and bounded since $\\Gamma_{+}^{(3)} \\cup \\Gamma_{+}^{(4)} \\subset \\mathcal{B}(0, R_{+})$. Then, we may write the real part of the $G$ function explicitly and show that it is monotone by taking its derivative. For instance, we have\n\\begin{equation}\n    \\frac{d}{dx} Re\\,\\{G(i +   e^{\\pi i \/6} + x) - G(i)\\} = -t \\left(1 - \\frac{3(\\sqrt{3}  + 2 x)}{2(3 + \\sqrt{3}\\, x + x^2)^2} \\right).\n\\end{equation}\nForm this, one may show that the derivative is strictly negative for all $x \\geq 0$. The bound (\\ref{e:bound1}) follows for this segment.\n\nTake $\\xi \\in \\Gamma_{+}^{(5)}$. In this case, we have $\\xi = R_{+}\\, e^{i \\theta}$, with $- \\pi\/2 \\leq \\theta \\leq \\phi_1 < \\pi\/2$ and $\\pi\/2 < \\phi_2 \\leq \\theta \\leq 3\\pi\/2$ for some constants $\\phi_1$ and $\\phi_2$ since $\\Gamma_{+}^{(5)} \\subset \\{z \\in \\mathbb{C} \\mid Im\\, \\{z\\}  \\leq Im\\, \\{ i +  e^{\\pi i \/6}\\} \\}$. In this case, we have\n\\begin{equation}\n    Re\\,\\{G(\\xi) - G(i)\\} = -2 t\\,\\log R_{+} + t(R_{+}+R_{+}^{-1}) \\sin \\theta.\n\\end{equation}\nSince $R_{+}>1$, one may then show that this function is monotone on $\\theta$ for each of the segments $- \\pi\/2 \\leq \\theta \\leq \\phi_1 < \\pi\/2$ and $\\pi\/2 < \\ph_2 \\leq \\theta \\leq 3\\pi\/2$. The bound (\\ref{e:bound1}) follows for this segment.\n\nThe bound (\\ref{e:bound2}), now that we have established that the function is monotone along all the segments of the contours, follows by evaluating the function on the boundary of the segment $\\Gamma_{+} \\cap \\mathcal{B}(i, t^{-\\alpha})$. That is, we evaluate the function at the points $\\xi = \\xi_0 + t^{-\\alpha} \\, e^{ \\pi \\textrm{i}\/6}$ and $\\xi = \\xi_0 + t^{-\\alpha} e^{5 \\pi \\textrm{i}\/6 }$. In particular, we use the Taylor expansion\n\\begin{equation}\n    G(\\xi) - G(\\xi_0) = -\\frac{1}{3}x^3 t^{1- 3 \\alpha} + \\mathcal{O}(t^{1-4\\alpha})\n\\end{equation}\nto approximate the function at the desired points. Since $t^{-\\alpha}< T \\ll 1$, we obtain the bound (\\ref{e:bound2}). \n\\end{proof}\n\n\n\\section{Contour Deformations}\\label{s:contour_deformations}\n\n\\subsection{Small to Large Contour deformations}\n\nWe deform the contours in the probability function for the left-most particle given by (\\ref{Fprob1}). In particular, we deform the contours $\\mathcal{C}_r$, for the $\\zeta$-variables, to some contour $\\mathcal{C}_{R'}$ with a large radius $R'>0$. Let\n\\begin{equation}\\label{e:deformation_contour}\n    \\Omega(\\xi) :=  \\mathcal{C}^{(0)} \\cup -\\mathcal{C}^{(1)} \\cup -\\mathcal{C}^{(2)} \\cup \\cdots \\cup -\\mathcal{C}^{(N)}  \n\\end{equation}\nbe the union of $(N+1)$ circles so that $-\\mathcal{C}^{(j)}$, for $j=1, \\dots, N$, is a negatively oriented circle centered at $\\xi_j^{-1}$ with radius $ r' >0$ and $\\mathcal{C}^{(0)}$ is a positively oriented circle centered at the origin with radius $R'>0$. We give precise conditions on the radii in the statement of \\textsc{Lemma 7.1} below. Then, as we deform the $\\mathcal{C}_r$ contour, we will encounter poles at $\\zeta_i = \\xi_j^{-1}$ for $i, j =1, \\dots, N$. As a result, we obtain the contour $\\Omega(\\xi)$ when we deform the contour $\\mathcal{C}_r$ to $\\mathcal{C}_{R'}$. This result and the proof for the contour deformations, given by \\textsc{Lemma 7.1} below, is similar to the contour deformation in \\cite{CDP}.\n\n\\textsc{Lemma 7.1}\nFor $\\Delta \\neq 0$, $\\mathcal{F}_N(x,t)=\\mathbb{P}_Y(X_1(t)\\ge x)$ equals\n\\begin{equation}\n\\int_{\\mathcal{C}_R}\\cdots\\int_{\\Omega(\\xi)} \\frac{\\prod_{j,k}(\\xi_j+\\zeta_k-2\\D\\xi_j\\zeta_k)}{\\prod_{j<k}(1+\\xi_j\\xi_k-2\\D\\xi_j)(1+\\zeta_j\\zeta_k-2\\D\\zeta_j)}\\,\nD_N(\\xi,\\zeta)\n\\prod_j (\\xi_j \\zeta_j)^{x-y_j-1} \\textrm{e}^{-\\textrm{i} t(\\ve(\\xi_j)-\\ve(\\zeta_j))}\\, d^N\\zeta \\,d^N\\xi\\hspace{5ex}\\label{Fprob2}\\end{equation}\nwhere the contour $\\mathcal{C}_R$ for the $\\xi$-variables is a circle centered at zero with radius $R>0$ and the contour $\\Omega(\\xi)$ for the $\\zeta$-variables is given by (\\ref{e:deformation_contour}) with radii $R' >0$ and $r' = 1\/(2 R)$, so that the radii satisfy the following inequalities $\\max\\{2|\\D|^{-1}, 2(1+2|\\D|)\\}< R< \\max\\{ 4|\\D|^{-1}, 4(1 + 2|\\D|)\\} < R' \/2$.\n\n\\begin{proof}\nWe take formula (\\ref{Fprob1}) with radius $R$ as given in the conditions in the Lemma and radius $r>0$ so that $\\max\\{ 4|\\D|^{-1}, 4(1 + 2|\\D|)\\}<r^{-1} < R' \/2$. Note that the conditions on the contours $\\mathcal{C}_R$ and $\\mathcal{C}_r$ given in \\textsc{Theorem 2} (i.e.~all the poles lie inside\/outside of the contours) are satisfied for our choice of radii. Then, we deform the contour in (\\ref{Fprob1}) for the $\\zeta$-variables to a large radius $R'>0$, with $R'$ satisfying the conditions given in the Lemma. We begin by deforming the contour for $\\zeta_N$, then the contour for $\\zeta_{N-1}$, and continue successively until we deform the contour for $\\zeta_1$. When we deform the contour for the $\\zeta_n$ variable, we encounter three types of poles\n\\begin{equation}\n    (a)\\, 1 - \\xi_i \\zeta_n= 0; \\quad (b)\\, 1 + \\zeta_i \\zeta_n -2 \\Delta \\zeta_i = 0, \\, i < n;\\quad (c)\\, 1 +\\zeta_n \\zeta_j - 2 \\Delta \\zeta_n, \\, n < j\n\\end{equation}\nfor any $i, j =1, \\dots, N$. The contribution for a type $(a)$ pole is given by the contour integral with respect to the variable $\\zeta_n$ with contour $- \\mathcal{C}^{(i)}$, i.e.~a negatively oriented circle centered at $\\xi_i^{-1}$ with radius $r'>0$ as given in the conditions of the Lemma. Note that the only pole, with respect to the variable $\\zeta_n$, inside the contour $-\\mathcal{C}^{(i)}$ is given by $\\zeta_n = \\xi_i^{-1}$ because $r'$ is chosen to be small enough. The result then follows by showing that the type $(b)$ and $(c)$ poles contribute no residue.\n\nAssume we have already deformed the $\\zeta_j$ variables for $j >n$ so that $\\zeta_i \\in \\mathcal{C}_r$ for $i <n$ and $\\zeta_j \\in \\Omega(\\xi) $ for $j >n$. We then deform the contour for the $\\zeta_n$ variable. Below, we consider the residue contribution from the type $(b)$ and $(c)$ poles.\n\n\\noindent \\textbf{Case (b).} We compute the residue at\n\\begin{equation}\n    \\zeta_n = (2 \\Delta \\zeta_{\\ell} - 1)\/\\zeta_{\\ell}\n\\end{equation}\nfor $\\ell <n$. The result is a $(2N-1)$-fold contour integral with the same integrand, say $I_N(\\xi, \\zeta; t)$, except that the term $1 + \\zeta_{\\ell} \\zeta_n - 2 \\Delta \\zeta_{\\ell}$ is replaced by $\\zeta_{\\ell}$ and the variable $\\zeta_n$ is evaluated at $(2 \\Delta \\zeta_{\\ell} - 1)\/\\zeta_{\\ell}$ for the rest of the terms. \n\nWe then compute the integral with respect to the $\\zeta_{\\ell}$ variable for the resulting residue term. The integral is computed by analyzing the poles and residues inside the contour $\\mathcal{C}_{r}$ for $\\zeta_{\\ell}$. The possible poles are given by\n\\begin{equation}\n    \\begin{split}\n    &1 - \\xi_k \\zeta_n = 0, \\quad \\hspace{11mm} k=1, \\dots, N\\\\\n    &1 + \\zeta_{\\ell} \\zeta_j - 2 \\Delta \\zeta_{\\ell} = 0, \\quad \\ell < j,\\, \\zeta_j \\in \\Omega(\\xi)\\\\\n    &1 + \\zeta_i \\zeta_{\\ell} - 2 \\Delta \\zeta_{i}=0, \\quad i < \\ell, \\, \\zeta_i \\in \\mathcal{C}_r\\\\\n    &1 + \\zeta_i \\zeta_n - 2 \\Delta \\zeta_i =0, \\quad i < n, \\, \\zeta_i \\in \\mathcal{C}_r\\\\\n    &1 + \\zeta_n \\zeta_j + 2\\Delta \\zeta_n = 0, \\quad n < j, \\, \\zeta_j \\in \\Omega(\\xi)\\\\\n    &\\zeta_{\\ell}^{x-y_j-1} \\zeta_n^{x- y_n-1} =0, \\quad j \\neq n .\n    \\end{split}\n\\end{equation}\nIn particular, the location of the possible poles is given by the following\n\\begin{equation}\n    \\begin{split}\n    \\zeta_{\\ell} = \\frac{\\xi_k}{2 \\Delta \\xi_k -1}  \\quad &\\Rightarrow \\quad    \\left| \\frac{\\xi_k}{2 \\Delta \\xi_k -1} \\right| > r \\\\\n    \\zeta_{\\ell} = \\frac{1}{2 \\Delta - \\zeta_j} \\quad &\\Rightarrow \\quad \\zeta_n = 2 \\Delta - \\zeta_{\\ell}^{-1} = \\zeta_j\\\\\n    \\zeta_{\\ell} = 2 \\Delta - \\zeta_i^{-1} \\quad &\\Rightarrow\\quad \\left| 2\\Delta - \\zeta_i^{-1}\\right| >r\n    \\\\\n    \\zeta_n = 2\\Delta - \\zeta_i^{-1} \\quad &\\Rightarrow \\quad \\zeta_{\\ell} = \\zeta_i\\\\\n    \\zeta_{\\ell} = \\frac{2\\D - \\zeta_j}{4\\D^4- 2 \\D \\zeta_j -1} \\quad &\\Rightarrow \\quad \\left| \\frac{2\\D - \\zeta_j}{4\\D^2- 2 \\D \\zeta_j -1}\\right|>r\\\\\n    \\zeta_{\\ell}^{y_n - y_{\\ell}} \\quad &\\Rightarrow \\quad y_n- y_{\\ell}\\geq 1.\n    \\end{split}\n\\end{equation}\nWe use the assumptions on the radii $R, R', r'>0$ given in the statement of the Lemma and the condition on the radius $r>0$ fixed at the beginning of the proof to establish the inequalities above. For the first two inequalities, it suffices to have $R , r^{-1} > 1 + 2 |\\D|$. For the third inequality, we have to consider two cases $\\zeta_j \\in \\mathcal{C}^{(0)}$ or $\\zeta_j \\in \\mathcal{C}^{(k)}$ with $k\\neq 0$. In the first case when $\\zeta_j \\in \\mathcal{C}^{(0)}$, we have that $|\\zeta_j| = R'$ and we use the bounds $R'>16 |\\D|$ and $R' > 8 |\\D|^{-1}$ that follow from the condition on the statement of the Lemma. In the second case when $\\zeta_j\\in\\mathcal{C}^{(k)}$ with $k \\neq 0$, we have that $|\\zeta_j| \\leq (3\/2)R^{-1}$ and we use the bound $(3\/2)R^{-1} < |\\D|$ that follows from the statement of the Lemma. Then, in all the cases above except for the second and fourth case, the poles lie outside the contour $\\mathcal{C}_r$, meaning that there is no residue contribution. In the second and fourth cases, the determinant term $D_N(\\xi, \\zeta)$ vanishes because two columns in the matrix of the determinant are equal to each other since two $\\zeta$ variables are equal to each other. In the last case, there is no pole since the exponent is positive. Then, the pole from the denominator and the zero from the determinant cancel out, meaning that these cases don't produce a residue. \n\nTherefore, by computing the integral with respect to the $\\zeta_{\\ell}$ variable, we have that the residues from the type $(b)$ poles vanish.\n\n\\noindent\\textbf{Case (c).} We compute the residue at \n\\begin{equation}\n    \\zeta_n = \\frac{1}{2 \\D - \\zeta_{\\ell}}\n\\end{equation}\nwith $n <\\ell$. The result is a $(2N-1)$-fold contour integral with the same integrand, say $I_N(\\xi, \\zeta; t)$, except that the term $1 + \\zeta_{n} \\zeta_{\\ell} - 2 \\Delta \\zeta_{n}$ is replaced by $2\\D-\\zeta_{\\ell}$ and the variable $\\zeta_n$ is evaluated at $1\/(2\\D - \\zeta_{\\ell})$ for the rest of the terms. \n\nIn this case, we have have $\\zeta_{\\ell} \\in \\Omega(\\xi)$ since $\\ell >n$. Thus, we have two possibilities: (i) $\\zeta_{\\ell} \\in \\mathcal{C}^{(0)}= \\mathcal{C}_{R'}$ , or (ii) $\\zeta_{\\ell} \\in - \\mathcal{C}^{(k)}$ for some $k =1, \\dots, N$ (i.e.~a negatively oriented small circle of radius $r'$ centered at $\\xi_k^{-1}$). In the first case, we will not cross a pole in the contour deformation and there will be no residue to consider. In the second case, the pole will cancel out with a zero from the numerator and, again, there will be no residue to consider. We give more details below. \n\nIn the first case, when $\\zeta_{\\ell} \\in \\mathcal{C}_{R'}$, we have $\\zeta_n = 1\/(2 \\Delta - \\zeta_{\\ell})$. This pole lies inside the contour $\\mathcal{C}_r$ since $R'\\,r > 2 $ and $r < (1 + 2 |\\D|)^{-1} $. Thus, we don't cross this pole when we deform the $\\mathcal{C}_r$ contour to $\\mathcal{C}^{(0)} = \\mathcal{C}_{R'}$.\n\nIn the second case, when $\\zeta_\\ell  \\in - \\mathcal{C}^{(k)}$, we first compute the residue at $\\zeta_{\\ell} = \\xi_k^{-1}$. We obtain an $(2N-1)$-fold contour integral with the same integrand, say $I_N(\\xi, \\zeta; t)$, except that the determinant $D_N(\\xi, \\zeta)$ is replaced the same determinant with the $k^{th}$ row and the ${\\ell}^{th}$ removed and multiplied by the factor $(1 + \\xi_k^2 - 2\\Delta \\xi_k)^{-1}$, and the rest of the terms are the same with the variable $\\zeta_{\\ell}$ evaluated at $\\xi_k^{-1}$.\n\nWe now deform the contour for $\\zeta_{n}$ to the contour $\\mathcal{C}^{(0)}$. After taking the $\\zeta_{\\ell} = \\xi_k^{-1}$ residue, it turns out that the terms giving rise to the pole $\\zeta_n= 1\/(2\\Delta - \\zeta_{\\ell})$ becomes\n\\begin{equation}\n    (1 +  \\zeta_n \\zeta_{\\ell} - 2\\Delta \\zeta_n) = \\xi_k^{-1}(\\xi_k + \\zeta_n - 2\\Delta \\xi_k \\zeta_n).\n\\end{equation}\nNote that this term also appears in the numerator of the integrand, meaning that this term cancels out and there is no residue in this case.\n\nTherefore, when we deform the contour for the $\\zeta_n$ to infinity, we don't cross any type $(c)$ poles. Moreover, this, along with the argument for the type $(b)$ poles, means that we only cross the poles due to the type $(a)$ poles. This establishes the result.\n\n\\end{proof}\n\n\\subsection{Series Expansion}\n\nWe write the contour formula (\\ref{Fprob2}) as a summation by expanding the integrals over the contour $\\Omega(\\xi)$, given by (\\ref{e:deformation_contour}), as a summations of $N+1$ integrals. We introduce some notation to encode the different terms in the summation.\n\nTake the set of all maps from the index set $\\{ 1, \\dots, N\\}$ to the set $\\{0, 1, \\dots, N\\}$ and denote it by\n\\begin{equation}\\label{e:maps}\n    \\mathcal{T} := \\{\\tau : \\{1, \\dots, N\\} \\rightarrow \\{0, 1, \\dots, N\\} \\} = \\mathrm{Hom}\\left(\\{1, \\dots, N\\}, \\{0,1, \\dots, N\\}\\right).\n\\end{equation}\nIn the following, a map $\\tau \\in \\mathcal{T}$ will correspond to a term with contours $\\mathcal{C}^{(\\tau(k))}$, given by (\\ref{e:deformation_contour}), for the $\\zeta_k$ variable and $k=1, \\dots, N$. Moreover, we will show that some contour integrals will vanish for certain $\\tau \\in \\mathcal{T}$. We consider the set of maps that map injectively to the elements $\\{1, \\dots, N\\}$ in the image and the cardinality of the preimage $\\sigma^{-1}(0)$ is fixed; \n\\begin{equation}\\label{e:maps_cont}\n    \\mathcal{T}_n := \\{\\tau \\in \\mathcal{T} \\mid |\\tau^{-1}(0)| =n; |\\tau^{-1}(k)| \\leq 1,\\, k =1, \\dots, N  \\}.\n\\end{equation}\n\n\\textsc{Lemma 7.2}\nFor $\\D \\neq 0$, $\\mathcal{F}_N(x,t)=\\mathbb{P}_Y(X_1(t)\\ge x)$ equals\n\\begin{equation}\n\\sum_{n=0}^N \\sum_{\\tau \\in \\mathcal{T}_n}\\oint_{\\mathcal{C}_R}\\cdots\\oint_{\\mathcal{C}_R}  \\oint_{\\mathcal{C}^{(\\tau(1))}} \\cdots \\oint_{\\mathcal{C}^{(\\tau(N))}}I_N(\\xi, \\zeta; x, t)\\, d^N\\zeta \\,d^N\\xi\\hspace{5ex}\\label{Fprob3}\\end{equation}\nwhere the integrand $I_N(\\xi, \\zeta; x, t)$ is the same integrand as in (\\ref{Fprob2}), the summation is take over the set of maps $\\mathcal{T}_n$ given by (\\ref{e:maps_cont}), the contour $\\mathcal{C}_R$ is a circle centered at zero with radius $R>0$, the contours $\\mathcal{C}^{(\\tau(k))}$ are given by (\\ref{e:deformation_contour}) with radii $r' = R^{-1}\/2, R' >0$ so that the radii satisfy the bounds $\\max\\{2 |\\D|^{-1}, 2(1+2|\\D|) \\} < R < \\max \\{4 |\\D|^{-1}, 4(1+2|\\D|) \\}< R'\/2$ \n\n\\begin{proof}\nWe take the contour formula (\\ref{Fprob2}) from \\textsc{Lemma7.1}. We then expand the integrals over the contours $\\Omega(\\xi)$ as a sum of $N+1$ integrals with contours given by the right side of (\\ref{e:deformation_contour}). The result is a summation over the set of maps $\\mathcal{T}$ given by (\\ref{e:maps}),\n\\begin{equation}\\label{e:sum1}\n    \\mathcal{F}_N(x,t) = \\sum_{\\tau \\in \\mathcal{T}} \\oint_{\\mathcal{C}_R}\\cdots \\oint_{\\mathcal{C}_R}  \\oint_{\\mathcal{C}^{(\\tau(1))}} \\cdots \\oint_{\\mathcal{C}^{(\\tau(N))}}I_N(\\xi, \\zeta; x, t)\\, d^N\\zeta \\,d^N\\xi.\n\\end{equation}\nThe result of this lemma follows by showing that some terms vanish, i.e.~if $\\tau \\notin \\mathcal{T}_n$ the corresponding contour integral will vanish. Below, we show that a term in the summation vanishes if $\\tau(j) = \\tau(k) >0$ with $j \\neq k$.\n\nTake $\\tau \\in \\mathcal{T}$ with $\\tau(j') = \\tau(k') =\\ell >0$ with $n \\neq m$ and $j', k'=1, \\dots, N$. We show that the term in the summation (\\ref{e:sum1}) with this $\\tau \\in \\mathcal{T}$ vanishes by taking the integrals with respect to the variables $\\zeta_{j'}$ and $\\zeta_{k'}$. We take the integral with respect to the $\\zeta_{j'}$ and $\\zeta_{k'}$ variables by taking the residues at the poles given by $\\zeta_{j'} = \\xi_{\\ell}^{-1}$ and $\\zeta_{k'} = \\xi_{\\ell}^{-1}$. Note that the poles given by $\\zeta_{j'} = \\xi_{\\ell}^{-1}$ and $\\zeta_{k'} = \\xi_{\\ell}^{-1}$ correspond to the $(\\ell, j')$-entry and the $(\\ell, k')$-entry of the matrix for the $D_N(\\xi, \\zeta)$ determinant. First, we take the residue at $\\zeta_{j'} = \\xi_\\ell^{-1}$, the determinant transforms as follows\n\\begin{equation}\\label{e:transform1}\n    D_N(\\xi, \\zeta) = \\det \\left( \\frac{1}{(1 - \\xi_j \\zeta_k)(\\xi_j+ \\zeta_k - 2 \\Delta \\xi_j \\zeta_k)} \\right)_{j, k=1}^N \\longrightarrow \\frac{(-1)^{\\tau(j')-j'-1}}{1+\\xi_{\\ell}^2 - 2\\Delta \\xi_{\\ell}} \\det \\left( \\frac{1}{(1 - \\xi_j \\zeta_k)(\\xi_j+ \\zeta_k - 2 \\Delta \\xi_j \\zeta_k)} \\right)_{j \\neq \\ell, k \\neq j'}.\n\\end{equation}\nFor the rest of the factors in the integrand, one evaluates $\\zeta_{j'} = \\xi_\\ell^{-1}$ when we take the residue at $\\zeta_{j'} = \\xi_{\\ell}^{-1}$. One may check that this doesn't introduce any poles with respect to the $\\zeta_{j'}$ variable inside the $\\mathcal{C}^{(\\ell)}$ contour. Then, the residue at $\\zeta_{j'} = \\xi_{\\ell}^{-1}$ doesn't have a pole at $\\zeta_{k'} = \\xi_{\\ell}^{-1}$ since the pole at $\\zeta_{k'} = \\xi_{\\ell}^{-1}$ is removed when we take the residue and no other pole is introduced. Thus, by taking the residue at $\\zeta_{k'} = \\xi_{\\ell}^{-1}$ after taking the residue at $\\zeta_{j'} = \\xi_{\\ell}^{-1}$, we have that the term vanishes. That is,\n\\begin{equation}\\label{e:vanish_term1}\n    \\oint_{\\mathcal{C}_R}\\cdots \\oint_{\\mathcal{C}_R}  \\oint_{\\mathcal{C}^{(\\tau(1))}} \\cdots \\oint_{\\mathcal{C}^{(\\tau(N))}}I_N(\\xi, \\zeta; x, t)\\, d^N\\zeta \\,d^N\\xi = 0\n\\end{equation}\nif $\\tau(n) = \\tau(m) =\\ell >0$ with $n \\neq m$ and $n, m=1, \\dots, N$.\n\n\n\nThe result of the lemma then follows by taking the summation representation given by (\\ref{e:sum1}) and noting that the terms with $\\tau \\notin \\mathcal{T}_n$ for some $n =0, 1, \\dots, N$ vanish due to the identities (\\ref{e:vanish_term1}).\n\\end{proof}\n\n\\subsection{Residue Computations}\n\nWe compute the contour integrals with respect to the $\\zeta_k$ variables with $\\tau(k)\\neq 0$ for each of the terms in the series expansion given by (\\ref{Fprob3}). First, we introduce some notation to represent the resulting integrand after the residue computations.\n\nFix $\\tau \\in \\mathcal{T}_{N-M}$ with $0 \\leq M \\leq N$ and $\\mathcal{T}_{N-M}$ given by (\\ref{e:maps_cont}). Then, define the following sets\n\\begin{equation}\\label{e:index_sets}\n    K_1 := \\tau^{-1}(0), \\quad K_2 := K_1^{c} = \\{k_1 < \\cdots < k_M \\}, \\quad J_2=\\tau \\left( K_2 \\right) = \\{\\tau_1 = \\tau(k_1), \\dots, \\tau_{M}=\\tau(k_M) \\}, \\quad J_1 = J_2^c.\n\\end{equation}\nWe also introduce the following functions\n\\begin{equation}\\label{e:integrands}\n    \\begin{split}\n    I_N(\\xi, \\zeta; \\tau)&=\\frac{\\prod_{j\\in J_1, k \\in K_1}(\\xi_j + \\zeta_k - 2\\D \\xi_j \\zeta_k)D_N(\\xi, \\zeta; \\tau)}{\\prod_{\\substack{j<k\\\\ j, k \\in J_1}}(1 + \\xi_j \\xi_k - 2\\D \\xi_j) \\prod_{\\substack{j<k\\\\j,k \\in K_1}}(1 +\\zeta_j \\zeta_k - 2\\D \\zeta_j)} \\prod_{j \\in J_1}\\xi_j^{x-y_j-1} e^{-\\textrm{i} t \\epsilon(\\xi_j)} \\prod_{k \\in K_1}\\zeta_k^{x-y_k-1} e^{\\textrm{i} t \\epsilon(\\zeta_k)}\\\\\n   \n    f(\\xi, \\zeta; \\tau)&= \\prod_{\\ell=1}^M\\left(\\prod_{\\substack{\\tau_{\\ell} < k \\\\ k \\neq \\tau_{\\ell+1}, \\dots, \\tau_M}}\\left(\\frac{1 + \\xi_{\\tau_{\\ell}} \\xi_k - 2\\D \\xi_k}{1 + \\xi_{\\tau_{\\ell}} \\xi_k - 2\\D \\xi_{\\tau_{\\ell}}}\\right)\\prod_{\\substack{k_{\\ell} < k \\\\ k \\neq k_{\\ell+1}, \\dots, k_M}}\\left(\\frac{\\xi_{\\tau_{\\ell}} +  \\zeta_k - 2\\D \\xi_{\\tau_{\\ell}} \\zeta_k}{\\xi_{\\tau_{\\ell}} +  \\zeta_k - 2\\D }\\right) \\right) \\prod_{\\ell=1}^{M}\\xi_{\\tau_{\\ell}}^{y_{k_{\\ell}}- y_{\\tau_{\\ell}}-1}\\\\\n    D_N(\\xi, \\zeta; \\tau) &= (-1)^{\\sum_{\\ell=1}^M \\tau_{\\ell} - k_{\\ell}} \\det\\left(d(\\xi_j, \\zeta_k)\\right)_{j \\in J_1, k \\in K_1} = (-1)^{\\sum_{\\ell=1}^M \\tau_{\\ell} - k_{\\ell}} \\sum_{\\gamma: K_1 \\rightarrow J_1} \\prod_{k \\in K_1}(-1)^{\\gamma(k)- k} d(\\xi_{\\gamma(k)}, \\zeta_{k})\n    \\end{split},\n\\end{equation}\nwhere the function $d(\\xi, \\zeta)$ is given by (\\ref{IK}), the sum on the last line is taken over all bijections $\\gamma: K_1 \\rightarrow J_1$, and the sets $K_1$, $J_1$ are given by (\\ref{e:index_sets}) and $M = N - |\\tau^{-1}(0)|$. Note that $I_N(\\xi, \\zeta; \\tau)$ is equal to the integrand of contour integrals (\\ref{Fprob1}), (\\ref{Fprob2}) and (\\ref{Fprob3}) if $|\\tau^{-1}(0)| = N$.\n\n\\textsc{Lemma 7.3} Fix $\\tau \\in \\mathcal{T}_{N-M}$, with $0 \\leq M \\leq N$, and take the notation from (\\ref{e:index_sets}). Then, for $\\D \\neq 0$, we have\n\\begin{equation}\n    \\oint_{\\mathcal{C}_R}\\cdots \\oint_{\\mathcal{C}_R} \\oint_{\\mathcal{C}^{\\tau(1)}} \\cdots \\oint_{\\mathcal{C}^{\\tau(N)}} I_N(\\xi, \\zeta) d^N \\zeta d^N \\xi = \\oint_{\\mathcal{C}_R}\\cdots \\oint_{\\mathcal{C}_R} \\oint_{\\mathcal{C}_{R'}} \\cdots \\oint_{\\mathcal{C}_{R'}} I_N(\\xi, \\zeta; \\tau) f(\\xi, \\zeta; \\tau) \\left(\\prod_{k \\in K_1} d\\zeta_j \\right) d^N \\xi\n\\end{equation}\nwhere the integral on the left side is a $2N$-fold contour intergal and the integral on the right side is a $(N+|K_1|)$-fold contour integral, the integrand on the left side equal to the integrand in (\\ref{Fprob2}) and the integrand on the right side is given by (\\ref{e:integrands}), and the contours are the same as in the statement of \\textsc{Lemma 7.2} so that the $\\zeta$ variables are integrated with respect to $\\mathcal{C}_{R'}$ contours.\n\n\\begin{proof}\nWe obtain the identity in this lemma by computing the integrals with respect to the $\\zeta_{k_{\\ell}}$ variables with $k_{\\ell} \\in K_2$. In particular, the contours are given by $-\\mathcal{C}^{(\\tau_{\\ell})}$, which are negatively oriented circles of radius $r'= 1\/(2 R)$ and centered at $\\xi_{\\tau_{\\ell}}^{-1}$, for the integrals with respect to $\\zeta_{k_{\\ell}}$ and $k_{\\ell} \\in K_2$. Then, we compute the integrals by taking the residues at $\\zeta_{k_{\\ell}} = \\xi_{\\tau_{\\ell}}^{-1}$. We start by taking the residue at $\\zeta_{k_{M}} = \\xi_{\\tau_{M}}^{-1}$ and continue successively until we take the residue at $\\zeta_{k_1} = \\xi_{\\tau_1}^{-1}$.\n\nLet's take the residue with respect to $\\zeta_{k_M} = \\xi_{\\tau_M}^{-1}$. Note that the pole corresponding to this residue comes from the $(\\tau_M, k_M)$-entry of the matrix of the $D_{N}(\\xi, \\zeta)$ determinant. Then, when we take the residue, the determinant is replaced by a determinant of the same matrix with the $\\tau_M$-row and $k_M$-column removed and a prefactor $(-1)^{\\tau_M-k_M}(1 + \\xi_{\\tau_M}^2 - 2\\D\\, \\xi_{\\tau_M})^{-1}$. That is,\n\\begin{equation}\n    \\det \\left( \\frac{1}{(1 - \\xi_i \\zeta_j)(\\xi_i+ \\zeta_j - 2 \\Delta \\xi_i \\zeta_j)} \\right)_{i, j=1}^N \\longrightarrow \\frac{(-1)^{\\tau_M-k_M-1}}{1+\\xi_{\\tau_M}^2 - 2\\Delta \\xi_{\\tau_M}} \\det \\left( \\frac{1}{(1 - \\xi_i \\zeta_j)(\\xi_i+ \\zeta_j - 2 \\Delta \\xi_i \\zeta_j)} \\right)_{i \\neq \\tau_M, j \\neq k_M}.\n\\end{equation}\nThe other terms of the integrand, when we compute the residue, transform by evaluating $\\zeta_{k_M} = \\xi_{\\tau_M}^{-1}$. Then, the result after taking the residue is\n\\begin{equation}\n    \\begin{split}\n    &\\frac{\\prod_{j \\neq \\tau_M, k \\neq k_M}(\\xi_j + \\zeta_k - 2\\D \\xi_j \\zeta_k)}{\\prod_{\\substack{j<k\\\\ j, k \\neq \\tau_M}}(1 + \\xi_j \\xi_k - 2\\D \\xi_j) \\prod_{\\substack{j<k\\\\j,k \\neq k_M}}(1 +\\zeta_j \\zeta_k - 2\\D \\zeta_j)} \\prod_{j \\neq \\tau_M}\\xi_j^{x-y_j-1} e^{-\\textrm{i} t \\epsilon(\\xi_j)} \\prod_{k \\neq k_M}\\zeta_k^{x-y_k-1} e^{\\textrm{i} t \\epsilon(\\zeta_k)}\\\\\n    \\times & (-1)^{\\tau_M- k_M}\\det\\left(\\frac{1}{(1- \\xi_j \\zeta_k)(\\xi_j +\\zeta_k -2\\D \\xi_j \\zeta_k)}\\right)_{j \\neq \\tau_M, k \\neq k_M}\\\\\n    \\times&\\xi_{\\tau_{M}}^{y_{k_{M}}- y_{\\tau_{M}}-1} \\prod_{\\tau_{M} < k }\\left(\\frac{1 + \\xi_{\\tau_{\\ell}} \\xi_k - 2\\D \\xi_k}{1 + \\xi_{\\tau_{\\ell}} \\xi_k - 2\\D \\xi_{\\ell}}\\right)\\prod_{k_{M} < k }\\left(\\frac{\\xi +  \\zeta_k - 2\\D \\xi_{\\ell} \\zeta_k}{\\xi_{\\ell} +  \\zeta_k - 2\\D }\\right)\n    \\end{split}.\n\\end{equation}\nThe sign infront of the determinant changed by negative one since we are taking the integral over a negatively oriented circle.\n\nWe continue taking the integrals with respect to the variables $\\zeta_{k_{\\ell}}$, successively with $\\ell$ decreasing, and evaluating the residues at $\\zeta_{k_{\\ell}} = \\xi_{\\tau_{\\ell}}^{-1}$. The computations are similar to the base case $\\zeta_{k_{M}} = \\xi_{\\tau_{M}}^{-1}$. In particular, the pole giving rise to residue comes from the $(\\tau_{\\ell}, k_{\\ell})$-entry of the determinant. Then, when we take the residue, the determinant transforms by removing the $\\tau_{\\ell}$-row and the $k_{\\ell}$-column and adding a prefactor. The other terms in the integrand transform by evaluating $\\zeta_{k_{\\ell}} = \\xi_{\\tau_{\\ell}}^{-1}$. We skip the details here since the computations are very similar to the base case. The result follows by computing all the integrals with respect to the $\\zeta_{k_{\\ell}}$ variable with $k_{\\ell} \\in K$.\n\\end{proof}\n\n\\textsc{Theorem 4.} For $\\D \\neq 0$, $\\mathcal{F}_N(x,t)=\\mathbb{P}_Y(X_1(t)\\ge x)$ equals\n\\begin{equation}\n\\sum_{n=0}^N \\sum_{\\tau \\in \\mathcal{T}_n}\\oint_{\\mathcal{C}_R}\\cdots \\oint_{\\mathcal{C}_R} \\oint_{\\mathcal{C}_{R'}} \\cdots \\oint_{\\mathcal{C}_{R'}} I_N(\\xi, \\zeta; \\tau) f(\\xi, \\zeta; \\tau)\\, \\left(\\prod_{k \\in K_1} d\\zeta_k \\right)\\, d^N \\xi\\hspace{5ex}\\label{Fprob4}\n\\end{equation}\nwhere the integrand is given by (\\ref{e:integrands}), the summation is take over the set of maps $\\mathcal{T}_n$ given by (\\ref{e:maps_cont}), and the contours $\\mathcal{C}_R$ and $\\mathcal{C}_{R'}$ are circles centered at zero with radii $R, R'>0$ so that $\\max\\{2 |\\D|^{-1}, 2(1 + 2|\\D|)\\} < R < \\max\\{4 |\\D|^{-1}, 4(1 + 2|\\D|)\\} < R'\/2$.\n\n\\begin{proof}\nThe result is a direct consequence of \\textsc{Lemma 7.2} and \\textsc{Lemma 7.3}.\n\\end{proof}\n\n\\subsection{Deformation to Steep Descent Contours}\n\nWe take the series expansion formula (\\ref{Fprob4}) and deform the contours to the steep descent contours given by (\\ref{e:steep_contours}).\n\nLet $\\widehat{\\Gamma}$ be a positive oriented rectangle centered at zero, with length equal to $2 L = 2 \\sqrt{R^2-1}$ and height equal to $2$, and two half-circle bumps as indicated on Figure \\ref{f:rec_contour}. The bump centered at $\\textrm{i}$ has radius $\\epsilon_1$ and the bump centered at $\\textrm{i} + 2\\D$ has radius $\\epsilon_2$ so that $0 < \\epsilon_2 \\ll \\epsilon_1 \\ll 1$. \n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[scale=0.35]{RecContour3a.png}\n    \\caption{The contour $\\widehat{\\Gamma}$.}\n    \\label{f:rec_contour}\n\\end{figure}\n\n\n\\textsc{Lemma 7.5} Fix $\\tau \\in \\mathcal{T}_{N-M}$, with $0 \\leq M \\leq N$, take the notation from (\\ref{e:index_sets}). Then, for $\\D \\neq 0$, we have\n\\begin{equation}\n    \\begin{split}\n    &\\oint_{\\mathcal{C}_R}\\cdots \\oint_{\\mathcal{C}_R} \\oint_{\\mathcal{C}_{R'}} \\cdots \\oint_{\\mathcal{C}_{R'}} I_N(\\xi, \\zeta; \\tau) f(\\xi, \\zeta; \\tau)\\, d^J \\zeta\\, d^N \\xi= \\oint_{\\Gamma_{+}} \\cdots \\oint_{\\Gamma_{-}} I_{N}(\\xi, \\zeta; \\tau)\\, \\left(  \\oint_{\\widehat{\\Gamma}} \\cdots \\oint_{\\widehat{\\Gamma}}f(\\xi, \\zeta; \\tau) d^{J_2} \\xi\\right) d^{K_1}\\zeta\\, d^{J_1}\\xi\n    \\end{split}\n\\end{equation}\nwhere the integrand is given by (\\ref{e:integrands}), the differentials $d^S \\xi$ or $d^S \\zeta$ are $|S|$-fold differential over the variables $\\xi_s$ or $\\zeta_s$ with $s\\in S$, the contours $\\Gamma_{\\pm}$ are given by (\\ref{e:steep_contours}) with $R_{+} = R$ and $R_{-} = R'$ so that $\\xi_{j} \\in \\Gamma_{+}$ and $\\zeta_{k} \\in \\Gamma_{-}$ for $j\\in J_1$ and $k \\in K_1$, the contour $\\widehat{\\Gamma}$ is given by Figure \\ref{f:rec_contour}, the contours $\\mathcal{C}_R$ and $\\mathcal{C}_{R'}$ are circles centered at zero with radii $R,R' >0$ so that $\\max\\{2|\\D|^{-1}, 2(1 + 2|\\D|)\\} < R < \\max\\{4|\\D|^{-1}, 4(1+2 |\\D|)\\}< R'\/2$.\n\n\n\\begin{proof}\nWe obtain the result by deforming the contours and showing that we don't cross any poles. We begin by deforming the contour, for the $\\xi_j$ variables with $j \\in J_2$, from $\\mathcal{C}_R$ to $\\widehat{\\Gamma}$. Then, for the $\\xi_j$ variables with $j \\in J_1$, we deform the contours $\\mathcal{C}_R$ to the contours $\\Gamma_{+}$. Finally, for the $\\zeta_k$ variables, we deform the contours $\\mathcal{C}_{R'}$ to the contours $\\Gamma_{-}$.\n\nConsider the integral with respect to $\\xi_{\\ell} \\in \\mathcal{C}_R$ and $\\ell \\in J_2$. We deform the contour $\\mathcal{C}_R$ to the contour $\\widehat{\\Gamma}$. Note that the factor $I_N(\\xi,\\zeta; \\tau)$ is independent of the $\\xi_{\\ell}$ variable. Then, the only possible poles are given by\n\\begin{equation}\\label{e:poles1}\n    1 + \\xi_{\\ell} \\xi_k -2\\D \\xi_{\\ell}= 0 , \\quad \\xi_{\\ell} + \\zeta_k - 2\\D =0.\n\\end{equation}\n\nIn the first case of (\\ref{e:poles1}), the location of the pole is given by $(2\\D - \\xi_k)^{-1}$ with $\\xi_k \\in \\mathcal{C}_R$ or $\\xi_k \\in \\widehat{\\Gamma}$, depending on the index and if the contour for the variable has been deformed. If $\\xi_k \\in \\mathcal{C}_R$, the location of the pole $(2\\D - \\xi_k)^{-1}$ clearly lies inside the unit circle since $R > 1+ 2|\\D|$. In particular, we don't cross this pole when we deform from the contour $\\mathcal{C}_R$ to the contour $\\widehat{\\Gamma}$, since the contour $\\widehat{\\Gamma}$ lies outside the unit circle. If $\\xi_k \\in \\widehat{\\Gamma}$, we note that the location of the pole $(2 \\D - \\xi_k)^{-1}$ also lies inside the unit circle except for the region with the small half-circle bump of radius $\\epsilon_2$. We then consider $\\xi_k$ lying on the small half-circle bump of $\\widehat{\\Gamma}$ and we write $\\xi_k = \\textrm{i} + 2 \\D + \\epsilon_2\\, e^{\\textrm{i} \\phi}$. Then, the location of the pole is given by\n\\begin{equation}\n    (2 \\D - \\xi_k)^{-1} = (- \\textrm{i} - \\epsilon_2\\, e^{\\textrm{i} \\phi})^{-1} = \\textrm{i} - \\epsilon_2\\, e^{\\textrm{i} \\phi} + \\mathcal{O}(\\epsilon_2^2),\n\\end{equation}\nwhere the last equality follows from $0 < \\epsilon_2 \\ll 1$. Moreover, since $\\epsilon_2 \\ll \\epsilon_1$, we have that the location of the pole $(2 \\D - \\xi_k)^{-1}$ lies inside the large bump of the contour $\\widehat{\\Gamma}$, when $\\xi_k$ lies on the small bump. Then, we have that the pole $(2 \\D - \\xi_k)^{-1}$ lies inside the unit circle if $\\xi_k$ doesn't lie on the small bump, and the pole lies inside the large bump if $\\xi_k$ lies on the small bump. In particular, if $\\xi_k \\in \\mathcal{C}_R \\cup \\widehat{\\Gamma}$, the location of the pole lies inside the contour $\\widehat{\\Gamma}$ and we don't cross any poles, given by the first case of (\\ref{e:poles1}), when we deform form the contour $\\mathcal{C}_R$ to the contour $\\widehat{\\Gamma}$.\n\nIn the second case of (\\ref{e:poles1}), the location of the pole is given by $2 \\D - \\zeta_k$. Additionally, we have that $\\zeta_k \\in \\mathcal{C}_{R'}$. Given the conditions on the radii $R, R'>0$, it follows that $R < R' -2 |\\D|$. Then, the pole given by $2 \\D - \\zeta_k$ lies outside the contour $\\mathcal{C}_R$. In particular, we don't cross the pole when we deform the contour form $\\mathcal{C}_R$ to $\\widehat{\\Gamma}$. Thus, we don't cross any poles, given by the second case of (\\ref{e:poles1}), when we deform the contours from $\\mathcal{C}_R$ to $\\widehat{\\Gamma}$.\n\nConsider now the integral with respect to $\\xi_{\\ell} \\in \\mathcal{C}_{R}$ with $\\ell \\in J_1$. We deform the contour $\\mathcal{C}_{R}$ to the contour $\\Gamma_{+}$ with $\\xi_j \\in \\widehat{\\Gamma}$ for $j \\in J_2$. The location of the possible poles are given by\n\\begin{equation}\\label{e:poles3}\n    (2\\D - \\xi_j)^{-1}, \\quad 2\\D - \\xi_j^{-1}, \\quad \\zeta_k^{-1}.\n\\end{equation}\n\nIn the first case of (\\ref{e:poles3}), the variable $\\xi_j$ may lie on the the contours $\\Gamma_{+}$ or $\\mathcal{C}_R$, depending on the index. In particular, if $\\xi_j \\in \\widehat{\\Gamma}$, then $j = \\tau_{k}$ for some $k$, see (\\ref{e:index_sets}). Moreover, $I_N(\\xi, \\zeta; \\tau)$ is independent of $\\xi_{j} = \\xi_{\\tau_{k}}$ and the pole due to the $f(\\xi, \\zeta; \\tau)$ function is of the form $(2\\D- \\xi_{\\tau_{k}})^{-1}$; see (\\ref{e:integrands}). Thus, for the first case, $\\xi_j$ will never lie on the contour $\\widehat{\\Gamma}$ and only lie on the contours $\\mathcal{C}_{R}$ or $\\Gamma_{+}$. If $\\xi_j \\in \\mathcal{C}_R$, the location of the pole $(2\\D - \\xi_j)^{-1}$ clearly lies inside the unit circle since $R- 2|\\D| >1$. In particular, we don't cross this pole when we deform from the contour $\\mathcal{C}_R$ to the contour $\\Gamma_{+}$, since the contour $\\Gamma_{+}$ lies outside the unit circle. If $\\xi_j \\in \\Gamma_{+}$, the location of the pole $(2\\D - \\xi_j)^{-1}$ will also lie outside the unit circle. This due to the fact the $\\D$ is a real number and $R-2|\\D| >1$. In particular, if $\\xi_j \\in \\mathcal{C}_R \\cup \\Gamma_{+} $, we don't cross a pole, given by the first case of (\\ref{e:poles3}) when we deform from the contour $\\mathcal{C}_R$ to the contour $\\Gamma_{+}$.\n\nIn the second case of (\\ref{e:poles3}), the variable $\\xi_j$ may lie on the the contours $\\Gamma_{+}$, $\\mathcal{C}_R$, or $\\widehat{\\Gamma}$, depending on the index. In all three cases, we have that the $- \\xi_j^{-1}$ point lies inside the unit circle since the contours lie outside the unit circle. Then, the pole $2\\D - \\xi_j^{-1}$ will lie inside $\\Gamma_{+}$ since $\\D$ is a real number and $2(1 + 2|\\D|) < R$. In particular, if $\\xi_j \\in \\mathcal{C}_R \\cup \\Gamma_{+} \\cup \\widehat{\\Gamma}$, we don't cross a pole, given by the second case of (\\ref{e:poles3}), when we deform from the contour $\\mathcal{C}_R$ to the contour $\\Gamma_{+}$.\n\nIn the third case of (\\ref{e:poles3}), we have $\\zeta_k \\in \\mathcal{C}_{R'}$. Then, the location of the pole $\\zeta_k^{-1}$ lies completely inside the unit circle. Then, since $\\Gamma_{+}$ lies outside the unit circle, we don't cross a pole when we deform the contour $\\mathcal{C}_R$ to the contour $\\Gamma_{+}$.\n\n\nLastly, consider the integral with respect to $\\zeta_{\\ell} \\in \\mathcal{C}_{R'}$ with $\\ell \\in K_1$. We deform the contour $\\mathcal{C}_{R'}$ to the contour $\\Gamma_{-}$. The location of the possible poles is given by\n\\begin{equation}\\label{e:poles2}\n    (2\\D - \\zeta_j)^{-1}, \\quad 2\\D - \\zeta_j^{-1}, \\quad 2\\D- \\xi_k, \\quad \\xi_k^{-1}\n\\end{equation}\nwhere the variables may lie on different contours depending on the indexes.\n\nIn the first case of (\\ref{e:poles2}), the variable $\\zeta_j$ may lie on the contour $\\mathcal{C}_{R'}$ or on the contour $\\Gamma_{-}$. In either case, the location of the pole lies completely inside the unit circle. When $\\zeta_j \\in \\mathcal{C}_{R'}$, this follows from the bound $R>2(1 +2 |\\D|)$. When $\\zeta_j \\in \\Gamma_{-}$, in addition the bound $R>2(1 +2 |\\D|)$, we also need the fact that $\\D$ is a real number, which means that $(2\\D - \\zeta_j)$ lies outside the unit circle for $\\zeta_j \\in \\Gamma_{-}$. Then, we have that the location of the pole $(2 \\D - \\zeta_j)^{-1}$ lies completely inside the unit circle and we don't cross any poles when we deform the contour $\\mathcal{C}_{R'}$ to the contour $\\Gamma_{-}$.\n\nIn the second case of (\\ref{e:poles2}), the variable $\\zeta_j$ may lie on the contour $\\mathcal{C}_{R'}$ or on the contour $\\Gamma_{-}$. In either case, we know that $\\zeta_j^{-1}$ lies inside the unit circle since $\\mathcal{C}_{R'}$ and $\\Gamma_{-}$ lie outside the unit circle. Then, since $\\D$ is a real number and $4(1+ 2|\\D|) < R'$ , we have that the location of the pole $2\\D -\\zeta_j^{-1}$ lies completely inside the contour $\\Gamma_{-}$. Thus, we don't cross any poles when we deform the contour $\\mathcal{C}_{R'}$ to the contour $\\Gamma_{-}$.\n\nIn the third case of (\\ref{e:poles2}), the variable $\\xi_k$ may lie on $\\widehat{\\Gamma}$ since this pole is due to the $f(\\xi, \\zeta; \\tau)$ factor in the integrand; see (\\ref{e:integrands}). In this case, the location of the pole $2\\D - \\xi_k$ lies completely inside the contour $\\Gamma_{-}$ due to the bumps of the contour $\\widehat{\\Gamma}$. Since $0 < \\epsilon \\ll 1$, the large bump of the contour $\\widehat{\\Gamma}$ lies completely above the horizontal section of the contour $\\Gamma_{-}$. Since the small bump in the contour $\\widehat{\\Gamma}$ lies inside the rectangle, the small bump will also lie completely above the V-section of the $\\Gamma_{-}$ contour. Additionally, since $R + 2|\\D|< R'$, the rest of the contour $\\widehat{\\Gamma}$ will lie completely inside the contour $\\Gamma_{-}$. Then, we don't cross any poles when we deform the contour $\\mathcal{C}_{R'}$ to the contour $\\Gamma_{-}$.\n\nIn the fourth case of (\\ref{e:poles2}), we have $\\xi_k \\in \\Gamma_{+}$. Then, the location of the pole $\\xi_k^{-1}$ lies completely inside the unit circle, since the contour $\\Gamma_{+}$ lies outside the unit circle. Then, since $\\Gamma_{-}$ lies outside the unit circle, we don't cross a pole when we deform the contour $\\mathcal{C}_{R'}$ to the contour $\\Gamma_{-}$.\n\nWe have now shown that we don't cross any poles in any case when we deform the contours. Thus, the result follows.\n\n\\end{proof}\n\n\\textsc{Proposition 7.6} For $\\D \\neq 0$, $\\mathcal{F}_N(x,t)=\\mathbb{P}_Y(X_1(t)\\ge x)$ equals\n\\begin{equation}\n\\sum_{n=0}^N \\sum_{\\tau \\in \\mathcal{T}_n}\\oint_{\\Gamma_{+}} \\cdots \\oint_{\\Gamma_{-}} I_{N}(\\xi, \\zeta; \\tau)\\,\\left(  \\oint_{\\widehat{\\Gamma}} \\cdots \\oint_{\\widehat{\\Gamma}}f(\\xi, \\zeta; \\tau) d^{J_2} \\xi \\right) d^{K_1}\\zeta\\, d^{J_1}\\xi\\,\\hspace{5ex}\\label{Fprob5}\n\\end{equation}\nwhere the integrand is given by (\\ref{e:integrands}), the sets $J_1, J_2, K_1, K_2$ are given by (\\ref{e:index_sets}), the summation is take over the set of maps $\\mathcal{T}_n$ given by (\\ref{e:maps_cont}), and the contours $\\Gamma_{\\pm}$ and $\\widehat{\\Gamma}$ are given by (\\ref{e:deformation_contour}) and Figure \\ref{f:rec_contour} with $R_{+}=R, R_{-}= R'$ so that $\\max\\{2 |\\D|^{-1}, 2(1 + 2|\\D|) \\} < R < \\max\\{4 |\\D|^{-1}, 4(1 + 2|\\D|) \\} < R'\/2$.\n\n\\begin{proof}\nThe result is a direct consequence of \\textsc{Proposition 7.4} and \\textsc{Lemma 7.5}.\n\\end{proof}\n\n\n\n\\section{Asymptotic Analysis, a Conjecture}\\label{s:conjecture}\n\nWe believe that the formula for the probability of the left-most particle given by (\\ref{Fprob4}) in \\textsc{Theorem 7.6} may be suitable for asymptotic analysis when $t \\ll N \\rightarrow \\infty$. Note that we have decomposed the integrand into two factors, $I_N(\\xi, \\zeta; \\tau)$ and $f(\\xi, \\zeta; \\tau)$. In particular, note that that the factor $f(\\xi, \\zeta; \\tau)$ is independent of time $t$. Additionally, for the variables of the term $I_N(\\xi,\\zeta; \\tau)$, we have deformed the contours to steepest descent paths. Thus, in the asymptotic limit, we expect the main contribution for the $I_N(\\xi,\\zeta;\\tau)$ term to come from the saddle point $(\\xi_0,\\zeta_0) =(i,-i)$. Moreover, we expect the asymptotic limit of $I_N(\\xi, \\zeta ; \\tau)$ to be given by the Airy kernel.  We give some details of the computation below but, unfortunately, we don't give all the technical details here. The arguments below need more careful consideration.\n\nFix $\\tau \\in \\mathcal{T}_n$ and let's consider the contribution of the contour integrals near the saddle point. We use the following notation for the index sets:\n\\begin{equation}\n\\begin{split}\n    K_1 := \\tau^{-1}(0) , \\quad K_2 := (K_1)^c, \\quad \n    J_1 := \\tau(K_2)^c , \\quad J_2 := \\tau(K_2)\n\\end{split}\n\\end{equation}\nThe sets $K_1$ and $K_2$ will be used to index the $\\zeta$-variables and the sets $J_1$ and $J_2$ will be used to index the $\\xi$-variables. In particular, variables with index from the sets $K_1$ and $J_1$ will lie on the contours $\\Gamma_{\\pm}$, respectively, and the variables with index from the set $J_2$ will lie on the contour $\\widehat{\\Gamma}$. There are no variables with index from the set $K_2$ because these variable have been integrated out, but nonetheless, this index set will appear in our formulas. Note $K_1 \\cup K_2 = J_1 \\cup J_2 = \\{1, \\dots, N \\}$.\n\n\nRecall that the spectral function $G$ and $H$, given in (\\ref{e:spectral_function}), have a double critical point at $\\xi = i$ and $\\zeta=-i$, respectively, when $x =-2t$. Let $\\mathcal{B}(z,r)$ be an open ball centered at $z \\in \\mathbb{C}$ of radius $r>0$ and $\\mathcal{B}(z,r)^c$ be its complement. Then, we take the following scaling\n\\begin{equation}\\label{e:scaling}\n    x= -2t - st^{1\/3}, \\quad \\xi = \\textrm{i} + \\textrm{i}\\, \\tilde{\\xi}\\, t^{-1\/3}, \\quad \\zeta = -\\textrm{i} + \\textrm{i}\\, \\tilde{\\zeta}\\, t^{-1\/3},\\quad y_j+1 = v_j \\, t^{1\/3}\n\\end{equation}\nif $\\xi \\in \\mathcal{B}(i, t^{- \\alpha})$ and $\\zeta \\in \\mathcal{B}(-i, t^{-\\alpha})$ with $1\/3 < \\alpha < 1\/4$.\n\nWe also have that the integrand $I_N(\\xi, \\zeta; \\tau)$ is exponentially small if $\\xi_j \\in \\mathcal{B}(i, t^{-\\alpha})^c$, for $j \\in J_1$, or $\\zeta_j \\in \\mathcal{B}(-i, t^{- \\alpha})^c$, for $j \\in K_1$. This follows from \\textsc{Lemma 6.1}. Additionally, we may uniformly bound the factor $f(\\xi, \\zeta; \\tau)$, independently of $t$, on all the $\\xi$ and $\\zeta$ variables. Then, we may restrict the contours $\\Gamma_{\\pm}$ to the a neighborhood around the saddle points and only lose an exponentially small term. That is,\n\\begin{equation}\\label{e:approx1}\n    \\begin{split}\n    &\\oint_{\\Gamma_{+}} \\cdots \\oint_{\\Gamma_{-}} I_{N}(\\xi, \\zeta; \\tau)\\, \\left( \\oint_{\\widehat{\\Gamma}} \\cdots \\oint_{\\widehat{\\Gamma}}f(\\xi, \\zeta; \\tau) d^{J_2} \\xi \\right) d^{K_1}\\zeta\\, d^{J_1}\\xi\\\\\n    &= \\oint_{\\Gamma_{+} \\cap \\mathcal{B}(\\textrm{i}, t^{-\\alpha})} \\cdots \\oint_{\\Gamma_{-} \\cap \\mathcal{B}(-\\textrm{i} , t^{-\\alpha})} I_{N}(\\xi, \\zeta; \\tau)\\, \\left( \\oint_{\\widehat{\\Gamma}} \\cdots \\oint_{\\widehat{\\Gamma}}f(\\xi, \\zeta; \\tau) d^{J_2} \\xi\\right) d^{K_1}\\zeta\\, d^{J_1}\\xi\\, + \\mathcal{O}(e^{- C t^{1-3\\alpha}})\n    \\end{split}\n\\end{equation}\nfor some positive constant $C>0$, based on \\textsc{Lemma 6.1}, and $1\/3 < \\alpha <1\/4$.\n\nLet us now approximate the integrands $I_N(\\xi, \\zeta;\\tau)$ and $f(\\xi, \\zeta;\\tau)$ when $\\xi_j \\in \\Gamma_{+} \\cap \\mathcal{B}(i, t^{- \\alpha})$, for $j \\in J_1$, and $\\zeta_k \\in \\Gamma_{-} \\cap \\mathcal{B}(-i, t^{-\\alpha})$, for $k \\in K_1$. In particular, we take the scaling (\\ref{e:scaling}) for the variables with indexes in the sets $J_1$ and $K_1$, for the $\\xi$-variables and $\\zeta$-variables respectively. \n\nNote that $I_N(\\xi, \\zeta;\\tau)$ only depends on the variables with indexes from the sets $K_1$ and $J_1$. Then, we have\n\\begin{equation}\\label{e:approx2}\n    \\begin{split}\n    I_N(\\xi, \\zeta; \\tau)&= (-1)^{|\\mathcal{X}_1|+\\sum_{j \\in \\mathcal{Z}_2}\\tau(j)-j}\\sum_{\\gamma : \\mathcal{Z}_1 \\rightarrow \\mathcal{X}_1} \\prod_{k \\in \\mathcal{Z}_1}(-1)^{\\gamma(k) - k} (\\textrm{i})^{y_{\\gamma(k)}- y_k} g(\\tilde{\\xi}_{\\gamma(k)}, \\tilde{\\zeta}_k; v_{\\gamma(k)}, v_k)\\, t^{n\/3} + \\mathcal{O}(t^{(n-1)\/3})\\\\\n    g(\\xi, \\zeta; x, z)&=  \\frac{\\exp\\left( \\frac{1}{3}\\xi^3 - \\frac{1}{3}\\zeta^3 - (s + x)\\,\\xi +(s + z)\\,\\zeta\\right)}{(\\xi - \\zeta)},\n    \\end{split}\n\\end{equation}\nwhere the sum is taken over all bijections $\\gamma : K_1\\rightarrow J_1$. This approximation is obtained by expanding the determinant in the term $I_{N}(\\xi, \\zeta;\\tau)$, given by (\\ref{e:integrands}) and taking the scaling (\\ref{e:scaling}). More details regarding this approximation are given in Appendix \\ref{s:I_approx}.\n\nNow, consider the approximation of the term $f(\\xi, \\zeta;\\tau)$ when $\\xi_j \\in \\Gamma_{+} \\cap \\mathcal{B}(i, t^{- \\alpha})$, for $j \\in J_1$, and $\\zeta_k \\in \\Gamma_{-} \\cap \\mathcal{B}(-i, t^{-\\alpha})$, for $k \\in K_1$. We introduce the following function\n\\begin{equation}\\label{e:B_fun}\n    B(\\xi;\\tau) = \\prod_{\\substack{j < k, j,k\\in K_2 \\\\ \\tau(k) < \\tau(j)}} \\left(\\frac{1 + \\xi_{\\tau(k)} \\xi_{\\tau(j)}- 2\\D \\xi_{\\tau(j)}}{1 + \\xi_{\\tau(k)} \\xi_{\\tau(j)}- 2\\D \\xi_{\\tau(k)}} \\right),\n\\end{equation}\nwith the indexes $j,k \\in K_2$ and $\\tau(k), \\tau(j) \\in J_2$. Also, let us denote the number of inversions of the $\\tau$ map as follows,\n\\begin{equation}\\label{e:nu1}\n    \\begin{split}\n    \\nu_1(j; \\tau) &: = \\# \\{j' \\in K_2 \\mid j' < j , \\quad \\tau(j')>\\tau(j)\\ \\}\\\\\n    \\nu_2(j; \\tau) &: = \\# \\{j' \\in K_2 \\mid j < j' , \\quad \\tau(j)>\\tau(j')\\ \\}\\\\\n    \\nu(j;\\tau) &:= j-\\tau(j) +\\nu_2(j; \\tau) - \\nu_1(j; \\tau)\n    \\end{split}.\n\\end{equation}\nNote that, in the case $K_2 = \\{1, 2, \\dots, N\\}$, we have $\\nu(j;\\tau)  = 0$ for $j=1, \\dots, N$. Then, by taking the scaling (\\ref{e:scaling}), we obtain\n\\begin{equation}\\label{e:approx3}\n    f(\\xi, \\zeta; \\tau) = B(\\xi;\\tau) \\prod_{j \\in K_2}\\left( \\frac{\\xi_{\\tau(j)} - (2\\D +\\textrm{i})}{(2\\textrm{i} \\D +1) \\xi_{\\tau(j)}- \\textrm{i}}\\right)^{\\nu(j;\\tau)} \\prod_{j\\in K_2} \\xi_{\\tau(j)}^{y_{j}- y_{\\tau(j)}-1} + \\mathcal{O}(t^{-1\/3}).\n\\end{equation}\nThis approximation is obtained by applying the scaling (\\ref{e:scaling}) and taking the leading term in the $t^{-1\/3}$ expansion of the $f(\\xi,\\zeta; \\tau)$ function. More details regarding this approximation are given in Appendix \\ref{s:f_approx}. \n\nWe now combine the approximations (\\ref{e:approx1}), (\\ref{e:approx2}) and (\\ref{e:approx3}), given above. Note that the leading term of the approximation (\\ref{e:approx3}) is independent of the $\\tilde{\\xi}$ and $\\tilde{\\zeta}$ variables. We then introduce the term\n\\begin{equation}\\label{e:F}\n    F(\\tau) = (\\textrm{i})^{|K_2|}\\oint_{\\widehat{\\Gamma}} \\cdots \\oint_{\\widehat{\\Gamma}} B(\\xi;\\tau) \\prod_{j \\in K_2}\\left( \\frac{\\xi_{\\tau(j)} - (2\\D +\\textrm{i})}{(2\\textrm{i} \\D +1) \\xi_{\\tau(j)}- \\textrm{i}}\\right)^{\\nu(j; \\tau)} \\prod_{j\\in K_2} (\\textrm{i}\\,\\xi_{\\tau(j)})^{y_{j}- y_{\\tau(j)}-1}d^{J_2} \\xi,\n\\end{equation}\nwhere we have taken the leading term of the $f(\\xi,\\zeta;\\tau)$ function and also incorporated the $(\\textrm{i})^{y_{\\gamma(k)} - y_k}$ term from the approximation of $I_N(\\xi, \\zeta;\\tau)$ given by (\\ref{e:approx2}), noting that $\\sum_{k\\in K_1}y_{\\gamma(k)} - y_k +\\sum_{k\\in K_2}y_{\\tau(k)} - y_k =0$. Then, for fixed $\\tau \\in \\mathcal{T}_n$, we obtain the following approximation near the saddle point\n\\begin{equation}\\label{e:approx4}\n    \\begin{split}\n    &\\oint_{\\Gamma_{+}} \\cdots \\oint_{\\Gamma_{-}} I_{N}(\\xi, \\zeta; \\tau)\\,\\left( \\oint_{\\widehat{\\Gamma}} \\cdots \\oint_{\\widehat{\\Gamma}}f(\\xi, \\zeta; \\tau) d^{J_2} \\xi\\right) \\, d^{K_1}\\zeta\\, d^{J_1}\\xi\\\\\n    &=t^{-n\/3}(-1)^{|J_1|+\\sum_{k \\in K_2}\\tau(k)-k} \\sum_{\\gamma : K_1 \\rightarrow J_1}  F(\\tau) \\prod_{k \\in K_1}(-1)^{\\gamma(k) - k} \\mathbf{K}_{Ai}\\left(s + v_{\\gamma(k)}, s + v_k \\right)\\\\\n    &\\quad \\quad + \\mathcal{O}(t^{(1-n)\/3}) + \\mathcal{O}(e^{-Ct^{1- 3 \\alpha}}).\n    \\end{split}\n\\end{equation}\nThe $t^{-n\/3}$ term and the Airy kernel $\\mathbf{K}_{Ai}$ are obtained by taking the change of variables (\\ref{e:scaling}) and the following expression for the Airy kernel\n\\begin{equation}\\label{e:Airy}\n    \\mathbf{K}_{Ai}(x,z) = \\int_{\\infty\\, e^{- 2\\pi \\textrm{i} \/3}}^{\\infty\\, e^{2\\pi \\textrm{i} \/3}} \\int_{\\infty\\, e^{- \\pi \\textrm{i} \/3}}^{\\infty\\,e^{ \\pi \\textrm{i} \/3}} \\frac{\\exp\\left(\\frac{1}{3}\\xi^3  - \\frac{1}{3}\\zeta^3 - x\\, \\xi + z\\, \\zeta\\right)}{\\xi - \\zeta}\\, d\\xi\\, d\\zeta,\n\\end{equation}\nwhere the contours for the $\\xi$ (resp.~$\\zeta$) variable starts at $\\infty\\, e^{-\\pi i \/3}$ (resp.~$\\infty\\, e^{-2\\pi i \/3}$) goes through the origin and ends at $\\infty\\, e^{\\pi i \/3}$ (resp.~$\\infty\\, e^{2\\pi i \/3}$).\n\nLet's now consider the formula (\\ref{Fprob5}) and, in particular, the summation over $\\mathcal{T}_n$ and $n$. We substitute the term in the summation by the right side of the approximation (\\ref{e:approx4}). The result is a summation over $\\mathcal{T}_n$, $n$, and injective maps $\\gamma: K_1\\rightarrow J_1$. More precisely, the summation is over a pair of bijective maps\n\\begin{equation}\n    \\tau: K_2 \\rightarrow J_2, \\quad \\gamma: K_1 \\rightarrow J_1,\n\\end{equation}\nwhere $K_1 \\cup K_2 = J_1 \\cup J_2 = \\{1, 2, \\dots, N\\}$. This means that we may write the summation, over $\\mathcal{T}_n$, $n$ and the injective maps $\\gamma: K_1 \\rightarrow J_1$ and $\\tau: K_2 \\rightarrow J_2$, as the summation over permutations of the set $[N] =\\{1, 2, \\dots, N\\}$. In particular, we may uniquely identify a pair of bijective maps $(\\tau, \\gamma)$ with a  permutation $\\sigma \\in \\mathcal{S}_N$ and a subset $S \\subset [N]$ so that $(\\tau, \\gamma) = (\\sigma|_{S^c}, \\sigma|_{S})$, where the right side are restrictions of the permutation to the indicated sets. Then, under this identification, we rewrite some the notation introduced earlier. For $(\\sigma, S)$ with $\\sigma|_{S^c} = \\tau$, we have\n\\begin{equation}\n    B(\\xi;\\sigma, S) = B(\\xi; \\tau) = \\prod_{\\substack{j, k \\in S^c, j<k \\\\ \\sigma(j) > \\sigma(k)}} \\left(\\frac{1 + \\xi_{\\sigma(k)} \\xi_{\\sigma(j)} - 2\\D \\xi_{\\sigma(j)}}{1 + \\xi_{\\sigma(k)} \\xi_{\\sigma(j)} - 2\\D \\xi_{\\sigma(k)}} \\right).\n\\end{equation}\nAdditionally, for $(\\sigma, S)$ with $\\sigma|_{S^c} = \\tau$, we write the inversion sets as follows,\n\\begin{equation}\n    \\begin{split}\n    \\nu_1(j; \\sigma, S) &= \\nu_1(j; \\tau)  = \\# \\{j' \\in S^c \\mid j' < j , \\quad \\sigma(j')>\\sigma(j)\\ \\}\\\\\n    \\nu_2(j; \\sigma, S) &= \\nu_2(j; \\tau)  = \\# \\{j' \\in S^c \\mid j < j' , \\quad \\sigma(j)>\\sigma(j')\\ \\}\\\\\n    \\nu(j; \\sigma, S) &= \\nu(j;\\tau) = j-\\sigma(j) +\\nu_2(j; \\sigma, S) - \\nu_1(j; \\sigma, S).\n    \\end{split}.\n\\end{equation}\nLastly, for $(\\sigma, S)$ with $\\sigma|_{S^c} = \\tau$, we write\n\\begin{equation}\n    F(\\sigma, S) = F(\\tau) = (\\textrm{i})^{|S^c|} \\oint_{\\widehat{\\Gamma}} \\cdots \\oint_{\\widehat{\\Gamma}} B(\\xi;\\sigma, S) \\prod_{j \\in S^c}\\left( \\frac{\\xi_{\\sigma(j)} - (2\\D +\\textrm{i})}{(2\\textrm{i} \\D +1) \\xi_{\\sigma(j)}- \\textrm{i}}\\right)^{\\nu(\\sigma, S)} \\prod_{j\\in S^c} (\\textrm{i} \\, \\xi_{\\sigma(j)})^{y_{j}- y_{\\tau(j)}-1}d^{\\sigma(S^c)} \\xi.\n\\end{equation}\nThen, under the identification of the pair of injective maps and the permutations, we have\n\\begin{equation}\n    \\begin{split}\n    &\\sum_{n=0}^N \\sum_{\\tau \\in \\mathcal{T}_n}\\oint_{\\Gamma_{+}} \\cdots \\oint_{\\Gamma_{-}} I_{N}(\\xi, \\zeta; \\tau)\\,\\left( \\oint_{\\widehat{\\Gamma}} \\cdots \\oint_{\\widehat{\\Gamma}}f(\\xi, \\zeta; \\tau) d^{J_2} \\xi\\right)\\, d^{K_1}\\zeta\\, d^{J_1}\\xi\\\\\n    &=\\sum_{\\sigma \\in \\mathcal{S}_N} (-1)^{\\sigma} \\sum_{S \\subset [N]} (-1)^{|S|} t^{- |S|\/3} \\left(F(\\sigma, S)\\prod_{k \\in S} \\mathbf{K}_{Ai}\\left( s + \\frac{y_{\\sigma(k)}+1}{t^{1\/3}}, s + \\frac{y_k+1}{t^{1\/3}}\\right) + \\mathcal{O}(t^{-1\/3}) + \\mathcal{O}(e^{-C t^{1- 3\\alpha}})\\right)\n    \\end{split}.\n\\end{equation}\n\nAssuming that the error terms don't contribute in the limit, we have the following conjecture.\n\n\\textsc{Conjecture 8.1} As $t\\ll N \\rightarrow \\infty$, $\\mathcal{F}_N(x,t)=\\mathbb{P}_Y(X_1(t)\\ge x)$, with $x = -2t -s \\, t^{-1\/3}$ and $y_j +1 = v_j\\, t^{1\/3}$, equals to the limit of\n\\begin{equation}\\label{Fprob6}\n    \\sum_{\\sigma \\in \\mathcal{S}_N} (-1)^{\\sigma} \\sum_{S \\subset [N]} (-1)^{|S|} t^{- |S|\/3} F(\\sigma, S)\\prod_{k \\in S} \\mathbf{K}_{Ai}\\left( s + v_{\\sigma(k)}, s +v_k\\right) \n\\end{equation}\nwhere $F$ is given by (\\ref{e:F}) and the Airy kernel $\\mathbf{K}_{Ai}$ is given by (\\ref{e:Airy}).\n\nAt the moment, we are not able to control the limit of (\\ref{Fprob6}) when $t \\ll N \\rightarrow \\infty$. The main obstacle is the term $F(\\sigma, S)$ on (\\ref{Fprob6}). However, under some assumptions, we may simplify (\\ref{Fprob6}) as a determinant of the difference of two kernels. For instance, assume\n\\begin{equation}\\label{e:assump}\n    F(\\sigma,S) = \\prod_{j \\in S^c} \\mathbf{Q}(\\sigma(j), j)\n\\end{equation}\nfor some kernel $\\mathbf{Q}$ on the set $\\{1, \\dots, N\\}$. Then, we have\n\\begin{equation}\n    \\begin{split}\n    &\\sum_{\\sigma \\in \\mathcal{S}_N} (-1)^{\\sigma} \\sum_{S \\subset [N]} (-1)^{|S|} t^{- |S|\/3} F(\\sigma,S)\\prod_{k \\in S} \\mathbf{K}_{Ai}\\left( s + \\frac{y_{\\sigma(k)}+1}{t^{1\/3}}, s + \\frac{y_k+1}{t^{1\/3}}\\right) \\\\\n    &=\\sum_{\\sigma \\in \\mathcal{S}_N} (-1)^{\\sigma} \\sum_{S \\subset [N]} (-1)^{|S|} t^{- |S|\/3} \\prod_{j \\in S^c} \\mathbf{Q}(\\sigma(j), j) \\prod_{k \\in S} \\mathbf{K}_{Ai}\\left( s + \\frac{y_{\\sigma(k)}+1}{t^{1\/3}}, s + \\frac{y_k+1}{t^{1\/3}}\\right)\\\\\n    &= \\sum_{\\sigma \\in \\mathcal{S}_N} (-1)^{\\sigma} \\prod_{k=1}^N\\left(\\mathbf{Q}(\\sigma(k), k) - t^{-1\/3}\\, \\mathbf{K}_{Ai}\\left(s + \\frac{y_{\\sigma(k)}+1}{t^{1\/3}}, s + \\frac{y_k+1}{t^{1\/3}} \\right) \\right)\\\\\n    & = \\det \\left( \\mathbf{Q}(j, k) - t^{-1\/3}\\mathbf{K}_{Ai}\\left(s + \\frac{y_{j}+1}{t^{1\/3}}, s + \\frac{y_k+1}{t^{1\/3}} \\right)\\right)_{j,k=1}^N,\n    \\end{split}\n\\end{equation}\ngiven the assumption (\\ref{e:assump}). In fact, when $\\Delta = 0$, one may check the assumption to be true and we have\n\\begin{equation}\n    F(\\sigma,S) = \\mathds{1}\\left(\\sigma|_{S^c} = \\mathrm{Id}_{S^c} \\right) = \\prod_{j \\in S^c} \\mathds{1}(\\sigma(j) = j),\n\\end{equation}\nwhere the functions with $\\mathds{1}$ are indicator functions. This identity is easy to check since the first two terms in the intergand for $F(\\sigma,S)$, given by (\\ref{e:F}), are identically equal to one when $\\Delta =0$. Then, we have\n\\begin{equation}\n    \\begin{split}\n    &\\det \\left( \\mathbf{Id}(j, k) - t^{-1\/3}\\mathbf{K}_{Ai}\\left(s + \\frac{y_{j}+1}{t^{1\/3}}, s + \\frac{y_k+1}{t^{1\/3}} \\right)\\right)_{j, k=1}^N\\\\\n    &=\\sum_{\\sigma \\in \\mathcal{S}_N} (-1)^{\\sigma} \\sum_{S \\subset [N]} (-1)^{|S|}t^{- |S|\/3} F(\\sigma, S)\\prod_{k \\in S} \\mathbf{K}_{Ai}\\left( s + \\frac{y_{\\sigma(k)}+1}{t^{1\/3}}, s + \\frac{y_k+1}{t^{1\/3}}\\right)\n    \\end{split}\n\\end{equation}\nwhen $\\Delta = 0$. This means that \\textsc{Conjecture 8.1} is true when $\\Delta =0$. Moreover, if $\\Delta=0$ and $y_j=j$, we may take the limit $t \\ll N \\rightarrow \\infty$. The right side becomes a sum of Riemann integrals, corresponding to the series expansion of a Fredholm determinant. Then, we have\n\\begin{equation}\n    \\begin{split}\n    \\lim_{N \\rightarrow \\infty}\\mathbb{P}_Y\\left(\\frac{X_1(t)+2t}{t^{1\/3}}\\ge -s\\right) &= \\lim_{t \\ll N \\rightarrow \\infty}  \\sum_{\\sigma \\in \\mathcal{S}_N} \\sum_{S \\subset [N]} t^{- |S|\/3} \\det \\left( \\mathbf{K}_{Ai}\\left( s + \\frac{j+1}{t^{1\/3}}, s + \\frac{k+1}{t^{1\/3}}\\right) \\right)_{j, k \\in S}\\\\\n    &= \\det \\left(\\mathbf{Id} - \\mathbf{K}_{Ai} \\right)_{L^2(s, \\infty)}\\\\\n    &=F_2(s).\n    \\end{split}\n\\end{equation}\nThis matches the earlier result (\\ref{TWF2}) for $\\Delta =0$.\n\nWe also may compute the terms in (\\ref{Fprob6}) when $S=\\emptyset$ and $S^c = \\{1, \\dots, N\\} =[N]$. In that case, the formula for $F(\\sigma, \\emptyset)$ simplifies as follows\n\\begin{equation}\n    F(\\sigma, \\emptyset) = \\oint_{\\widehat{\\Gamma}} \\cdots \\oint_{\\widehat{\\Gamma}}\\prod_{\\substack{j < k \\\\ \\sigma(k) < \\sigma(j)}} \\left(\\frac{1 + \\xi_{\\tau(k)} \\xi_{\\tau(j)}- 2\\D \\xi_{\\tau(j)}}{1 + \\xi_{\\tau(k)} \\xi_{\\tau(j)}- 2\\D \\xi_{\\tau(k)}} \\right)  \\prod_{j=1}^N \\xi_{\\tau(j)}^{y_{j}- y_{\\tau(j)}-1}d^N \\xi,\n\\end{equation}\nwhere $i,j= 1, 2, \\dots, N$ on the first product of the integrand. Additionally, we may deform the contours $\\widehat{\\Gamma}$ to arbitrarily large circles centered at the origin. Note that $(-1)^{\\sigma}F(\\sigma,\\emptyset)$ is equal to the integral inside the sum of (\\ref{psi_N_large}) with $x_i =y_i$, for $i=1, \\dots, N$, and $t=0$. Then, by \\textsc{Theorem 1a}, we have\n\\begin{equation}\n    \\sum_{\\sigma \\in \\mathcal{S}_N} (-1)^{\\sigma}F(\\sigma, \\emptyset) = 1\n\\end{equation}\nfor any $N>0$.\n\n\n\n\\section*{Acknowledgement}\n\nThe authors thank F.~Colomo, B.~Nachtergaele, and L.~Petrov for their helpful communications. This work was supported by the National Science Foundation under the grant DMS--1809311 (second author). This work was also supported by the National Science Foundation under Grant No. DMS--1928930 while the first author participated in the program \\emph{``Universality and Integrability in Random Matrix Theory and Interacting Particle Systems\"} hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2021 semester. Additionally, the first author was partially supported by the \\emph{Engineering and Physical Sciences Research Council (EPSRC)} through grant EP\/R024456\/1.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\\subsection{Motivations}\nThe recent trend of using a large number of parameters to model large datasets in machine learning and statistics has created a strong demand for optimization algorithms that have low computational cost per iteration and exploit model structure. Empirical risk minimization is a class of important optimization problems in this area. Such problems can be written as\n\\begin{align}\n\\min_{\\mb{x} \\in \\mc{P}} F(\\mb{x}) \\equiv \\frac{1}{n}\\sum_{i = 1}^n f_i(\\mb{x}), \\label{erm_problem}\n\\end{align}\nwhere $\\mc{P}$ is a compact polyhedral set in $\\mr{R}^p$ and each $f_i(\\cdot)$ is a convex function. A popular approach for solving (\\ref{erm_problem}) is the proximal gradient method which solves a projection sub-problem in each iteration. The major drawback of this method is that the projection step can be expensive in many situations. As an alternative, the Frank-Wolfe (FW) algorithm \\cite{FW56}, also known as the conditional gradient method, solves a linear optimization sub-problem in each iteration which is much faster than the standard projection technique when the feasible set is a simple polytope \\cite{N15}. On the one hand, when the number of observations $N$ in problem (\\ref{erm_problem}) is large, calculating the gradient of $F(\\cdot)$ in every FW iteration becomes a computationally intensive task. The question of whether `cheap' stochastic gradient can be used as a surrogate in FW immediately arises. On the other hand, when the dimension of the parameter space $d$ in problem (\\ref{erm_problem}) is large, taking advantage of any underlying structure in the problem can further accelerate the algorithm. In this paper, we present a linearly convergent semi-stochastic Frank-Wolfe algorithm with away-steps for empirical risk minimization and extend it to problems with block-coordinate structure. Specifically, we apply the proposed algorithm to structural support vector machine and graph-guided fused lasso problems.\n\\subsection{Contributions}\nThe contribution of this paper has two aspects. On the theoretical side, it presents the first stochastic conditional gradient algorithm that converges linearly in expectation. In addition, this algorithm uses an adaptive step size rather than one determined by exact line search. On the application side, the algorithm can be applied to various statistical and machine learning problems including support vector machine, multi-task learning, regularized generalized linear models and many others. Furthermore, when the problem has a block-coordinate structure, performance of this algorithm can be enhanced by extending it to a randomized coordinate version which solves several small sub-problems instead of a large problem.\n\\subsection{Related Work}\nThe Frank-Wolfe algorithm was proposed sixty years ago  \\cite{FW56} for minimizing a convex function over a polytope and is known to converge at an $O(1\/k)$ rate. In \\cite{LP66} the same convergence rate was proved for compact convex constraints. When both objective function and the constraint set are strongly convex, \\cite{GH15} proved that the Frank-Wolfe algorithm has an  $O(1\/k^2)$ rate of convergence with a properly chosen step size.  Motivated by removing the influence of ``bad\" visited vertices, the away-steps variant of the Frank-Wolfe algorithm was proposed in \\cite{Wol70}. Later, \\cite{GM86} showed that this variant converges linearly under the assumption that the objective function is strongly convex and the optimum lies in the interior of the constraint  polytope. Recently, \\citep{GH13} and \\cite{LJJ13} extended the linear convergence result by removing the assumption of the location of the optimum and \\cite{BS15} extended it further by relaxing the strongly convex objective function assumption. Stochastic Frank-Wolfe algorithms have been considered by \\cite{LAN13} and \\cite{LWM15} in which an $O(1\/k)$ rate of convergence in expectation is proved. In addition, the Frank-Wolfe algorithm has been applied to solve several different classes of problems, including non-linear SVM \\cite{OG10}, structural SVM \\cite{LJJSP13}, and comprehensive principal component pursuit \\cite{MZWG15} among many others.\n\\subsection{Notation}\nBold letters are used to denote vectors and matrices, normal fonts are used to denote scalers and sets. Subscripts represent elements of a vector,  while superscripts with parentheses represent iterates of the vector, i.e. $\\mathbf{x}^{(k)}$ is a vector at iteration $k$, and $x_i^{(k)}$ is the $i$-th element of $\\mathbf{x}^{(k)}$. The cardinality of the set $I$ is denoted by $\\vert I \\vert$. Given two vectors $\\mathbf{x}, \\mathbf{y} \\in \\mathbb{R}^n$, their inner product is denoted by $\\langle \\mathbf{x}, \\mathbf{y} \\rangle$. Given a matrix $\\mathbf{A}\\in \\mathbf{R}^{m \\times n}$ and vector $\\mathbf{x} \\in \\mathbb{R}^n$, $\\norm{\\mathbf{A}}$ denotes the spectral norm of $\\mathbf{A}$, and $\\norm{\\mathbf{x}}$ denotes the $l_2$ norm of $\\mathbf{x}$. $\\mathbf{A}^\\top$ represent the transpose of $\\mathbf{A}$. We denote the $i$th row of a given matrix $\\mathbf{A}$ by $\\mathbf{A}_i$, and for a given set $I \\subset \\{1, \\ldots, m\\}$, $\\mathbf{A}_I \\in \\mathbf{R}^{\\vert I \\vert \\times n}$is the submatrix of $\\mathbf{A}$ such that $(\\mathbf{A}_I)_j = \\mathbf{A}_{I_j}$ for any $j = 1, \\ldots, \\vert I \\vert$. Given matrices $\\mathbf{A} \\in \\mathbb{R}^{n \\times m}$ and $\\mathbf{B} \\in \\mathbb{R}^{n \\times k}$, the matrix $[\\mathbf{A}, \\mathbf{B}] \\in \\mathbb{R}^{n \\times (m + k)}$ is their horizontal concatenation. Let $\\lceil x \\rceil$ be the ceiling function that rounds $x$ to the smallest integer larger than $x$.\n\n\\section{Stochastic Condtional Gradient Method for Empirical Risk Minimization}\n\\subsection{Problem description, notation and assumptions}\nConsider the minimization problem\n\\begin{align}\n\\min_{\\mb{x} \\in \\mc{P}} \\Big\\{F(\\mb{x}) \\equiv \\frac{1}{n}\\sum_{i = 1}^n f_i(\\mb{a}^\\top_i \\mb{x}) + \\langle \\mb{b}, \\mb{x} \\rangle \\Big\\}, \\label{general_problem} \\tag{P1}\n\\end{align}\nwhere $\\mathcal{P}$ is a non-empty compact polyhedron given by $\\mathcal{P} = \\{x \\in \\mathbb{R}^p : \\mathbf{Cx} \\leq \\mb{d}\\}$ for some $\\mathbf{C} \\in \\mathbb{R}^{m \\times p}$, $\\mathbf{d} \\in \\mathbb{R}^m$. For every $i = 1, \\ldots, n$, $\\mb{a}_i \\in \\mr{R}^p$, and $f_i : \\mr{R} \\rightarrow \\mr{R}$ is a strongly convex function with parameter $\\sigma_i$ and has a Lipschitz continuous gradient with constant $L_i$. $\\mb{b}$ in the linear term of (\\ref{general_problem}) is a vector in $\\mr{R}^p$. Note that the gradient of $F(\\cdot)$ is also Lipschitz continuous with constant $L \\leq (\\sum_{i=1}^nL_i \\norm{a_i}) \/ n$. However, because of the affine transformation in the argument of each $f_i(\\cdot)$, $F(\\cdot)$ may not be a strongly convex function.\n\n\\noindent\\textbf{Remark:}\n\\begin{description}\n\\item[1] Many statistics and machine learning problems can be modeled as problem (\\ref{general_problem}). For example:\n\\begin{description}\n\\item[i] The LASSO problem : $\\min_{\\boldsymbol\\beta \\in \\mathbb{R}^p} \\sum_{i=1}^n (y_i - \\mb{x}_i^\\top\\boldsymbol\\beta)^2 + \\lambda \\vert\\vert \\boldsymbol\\beta \\vert\\vert_1$, where $n$ is the sample size and for $i = 1, 2,\\ldots, n$, $y_i$'s are responses and $\\mb{x}_i$'s are the covariates. $\\boldsymbol\\beta$ is the regression coefficient to be estimated by solving the minimization problem and $\\lambda$ is a regularization parameter for sparsity.\n \n\\item[ii]$l_1$-Regularized Poisson Regression:  $\\min_{\\boldsymbol\\beta \\in \\mathbb{R}^p} \\sum_{i = 1}^n \\exp\\{\\mathbf{x}_i^\\top\\boldsymbol\\beta\\} - y_i\\mathbf{x}_i^\\top\\boldsymbol\\beta +\\log(y_i!) +  \\lambda \\vert\\vert \\boldsymbol\\beta \\vert\\vert_1 $, where $n$ is the sample size and for $i = 1, 2, \\ldots, n$, $\\mb{x}_i$'s are the covariates and $y_i \\in \\mr{N}$ are the responses; that is, $y_i$'s are obtained from event counting. $\\boldsymbol\\beta$ is the regression coefficient to be estimated by solving the minimization problem and $\\lambda$ is a regularization parameter for sparsity.\n\n\\item[iii] $l_1$-Regularized Logistic Regression: $\\min_{\\boldsymbol\\beta \\in \\mathbb{R}^p}\\sum_{i = 1}^n \\log(1 + \\exp\\{-y_i\\mathbf{x}_i^\\top\\boldsymbol\\beta\\})  + \\lambda \\vert\\vert \\boldsymbol\\beta \\vert\\vert_1$, where $n$ is the sample size and for $i = 1, 2, \\ldots, n$, $\\mb{x}_i$'s are the covariates and $y_i \\in \\{0, 1\\}$ are the responses; that is, $y_i$'s are binary labels obtained from classification. $\\boldsymbol\\beta$ is the regression coefficient to be estimated by solving the minimization problem and $\\lambda$ is a regularization parameter for sparsity.\n\n\\item[iv] Dual Problem of $l_1$-loss Support Vector Machine: given training label-instance pairs $(y_i, \\mb{z}_i)$ for $i = 1, \\ldots, l$, the problem is formulated as \n\\begin{align*}\n&\\text{minimize}_{\\boldsymbol\\alpha}\\quad  \\frac{1}{2}\\boldsymbol\\omega^\\top\\boldsymbol\\omega - \\boldsymbol1^\\top\\boldsymbol\\alpha\\\\\n&\\text{subject to}\\quad  \\boldsymbol\\omega = \\mathbf{A}\\boldsymbol\\alpha, \\\\\n&\\quad \\quad \\quad \\quad \\quad \\; 0 \\leq \\alpha_i \\leq C, i = 1, \\ldots, l.\n\\end{align*}\nwhere $\\mb{A} = [y_1\\mb{z}_1, \\ldots, y_l\\mb{z}_l]$, $\\boldsymbol1$ is the vector of ones, and $C$ is a given upper bound. This problem can be transformed to the form of (\\ref{general_problem}) by replacing the $\\boldsymbol\\omega$ is in the objective function by $\\mb{A}\\boldsymbol\\alpha$.\n\\end{description}\n\\item[2] The objective functions in the unconstrained problems (i),(ii) and (iii) all involve a non-smooth regularization term. They however, can be modeled as (\\ref{general_problem}). For example, in the LASSO problem, we can always take $\\boldsymbol\\beta = \\boldsymbol0$ to get an upper bound on the function value and add the constraint $\\norm{\\boldsymbol\\beta}_1 \\leq \\sum_{i=1}^n y_i^2$ to the original problem. Then by introducing new variables $u_i$, $i = 1, \\ldots, p$, we can express $\\norm{\\boldsymbol\\beta}_1$ as $\\sum_{i=1}^p u_i$ after adding the constraints $\\beta_i \\leq u_i$ and $-\\beta_i \\leq u_i$ for all $i = 1, \\ldots, p$ to the problem. We can apply the same method to $l_1$-regularized logistic regression and $l_1$-regularized Poisson regression problems.\n\\end{description}\n\n\\subsection{The Main Result}\nLet $\\mathcal{O}: \\mathbb{R}^p \\rightarrow \\mathcal{P}$ be a linear oracle that given $\\mathbf{c} \\in \\mathbb{R}^p$, returns $\\mathbf{z} = \\mathcal{O}(\\mathbf{c}) \\in \\mathcal{P}$ that $\\mathbf{c}^\\top\\mb{z} \\leq \\mathbf{c}^\\top \\mb{x}$ for every $\\mb{x} \\in \\mc{P}$; i.e. $\\mb{z} = \\arg\\min\\{\\mb{c}^\\top \\mb{x} \\; \\vert \\; \\mb{x} \\in \\mc{P}\\}$. Let $V$ be the set of vertices of polytope $\\mc{P}$.\n\n\\begin{algorithm}\n\\caption{Semi-Stochastic Frank-Wolfe algorithm with Away-Steps}\n\\label{cond_grad_1}\n\\begin{algorithmic}[1]\n\\STATE {\\bfseries Input:}  $\\mathbf{x}^{(1)} \\in V$, $f_i$, $\\mb{a}_i$, $\\mb{b}$ and $L$\n\\STATE Set $\\mu^{(1)}_{\\mathbf{x}^{(1)}} = 1$, $\\mu_\\mathbf{v}^{(1)} = 0$ for any $\\mathbf{v} \\in \\mathcal{V} \/ \\{\\mathbf{x}^{(1)}\\}$ and $U^{(1)} = \\{\\mathbf{x}^{(1)}\\}$.\n\\FOR{$k = 1, 2, \\ldots$}\n\\STATE Uniformly sample $\\mc{J} =\\{j_1, \\ldots, j_{m_k}\\}$ from $\\{1, \\ldots, n\\}$ without replacement, and denote  $\\mathbf{g}^{(k)} = \\frac{1}{m_k}\\sum_{i=1}^{m_k}f'_{j_i}(\\mb{a}_{j_i}^\\top\\mb{x}^{(k)})\\mb{a}_{j_i} + \\mb{b}$.\n\\STATE Compute $\\mathbf{p}^{(k)} = \\mathcal{O}(\\mathbf{g}^{(k)})$.\n\\STATE Compute $\\mathbf{u}^{(k)} \\in {\\arg\\!\\max}_{\\mathbf{v} \\in U^{(k)}}\\langle \\mathbf{g}^{(k)}, \\mathbf{v} \\rangle$.\n\\IF{$\\langle \\mathbf{g}^{(k)}, \\mathbf{p}^{(k)}  + \\mathbf{u}^{(k)}- 2\\mathbf{x}^{(k)} \\rangle \\leq 0$}\n\\STATE Set $\\mathbf{d}^{(k)} = \\mathbf{p}^{(k)} - \\mathbf{x}^{(k)}$ and $\\gamma^{(k)}_{\\text{max}} = 1$. \n\\ELSE\n\\STATE Set $\\mathbf{d}^{(k)} = \\mathbf{x}^{(k)} - \\mathbf{u}^{(k)}$ and $\\gamma^{(k)}_{\\text{max}} = \\frac{\\mu^{(k)}_{\\mathbf{u}^{(k)}}}{1 - \\mu^{(k)}_{\\mathbf{u}^{(k)}}}$.\n\\ENDIF\n\\STATE Set $\\gamma^{(k)} = \\min\\{-\\frac{\\langle \\mb{g}^{(k)}, \\mb{d}^{(k)}\\rangle}{L\\norm{\\mb{d}^{(k)}}^2}, \\gamma^{(k)}_\\text{max}\\}$\n\\STATE Set $\\mathbf{x}^{(k+1)} = \\mathbf{x}^{(k)} + \\gamma^{(k)}\\mathbf{d}^{(k)}$.\n\\STATE Update $U^{(k + 1)}$ and $\\mathbf{\\mu}^{(k+1)}$ by Procedure VRU.\n\\ENDFOR\n\\STATE{\\bfseries Return:} $\\mathbf{x}^{(k+1)}$.\n\\end{algorithmic}\n\\end{algorithm}\n\\noindent The following algorithm updates a vertex representation of the current iterate and is called in Algorithm \\ref{cond_grad_1}.\n\\begin{center}\n\\begin{algorithm}[h]\n\\caption{Procedure Vertex Representation Update (VRU)}\n\\begin{algorithmic}[1]\n\\STATE {\\bfseries Input:} $\\mb{x}^{(k)}$, $(U^{(k)}, \\boldsymbol\\mu^{(k)})$, $\\mb{d}^{(k)}$, $\\gamma^{(k)}$, $\\mb{p}^{(k)}$ and $\\mb{v}^{(k)}$.\n\\IF{$\\mb{d}^{(k)} = \\mb{x}^{(k)} - \\mb{u}^{(k)}$}\n\\STATE Update $\\mu^{(k)}_\\mb{v} = \\mu_\\mb{v}^{(k)}(1 + \\gamma^{(k)})$ for any $\\mb{v} \\in U^{(k)} \/ \\{\\mb{u}^{(k)}\\}$.\n\\STATE Update $\\mu^{(k+1)}_{\\mb{u}^{(k)}} = \\mu^{(k)}_{\\mb{u}^{(k)}}(1 + \\gamma^{(k)}) - \\gamma^{(k)}$.\n\\IF{$\\mu^{(k+1)}_{\\mb{u}^{(k)}} = 0$}\n\\STATE Update $U^{(k+1)} = U^{(k)} \/\\{\\mb{u}^{(k)}\\}$\n\\ELSE\n\\STATE Update $U^{(k+1)} = U^{(k)}$\n\\ENDIF\n\\ENDIF\n\\STATE Update $\\mu_\\mb{v}^{(k+1)} = \\mu^{(k)}_\\mb{v}(1 - \\gamma^{(k)})$ for any $\\mb{v} \\in U^{(k)} \/ \\{\\mb{p}^{(k)}\\}$.\n\\STATE Update $\\mu^{(k+1)}_{\\mb{p}^{(k)}} = \\mu^{(k)}_{\\mb{p}^{(k)}}(1 - \\gamma^{(k)}) + \\gamma^{(k)}$.\n\\IF{$\\mu_{\\mb{p}^{(k)}}^{(k+1)} = 1$}\n\\STATE Update $U^{(k+1)} = \\{\\mb{p}^{(k)}\\}$.\n\\ELSE\n\\STATE Update $U^{(k+1)} = U^{(k)} \\cup \\{\\mb{p}^{(k)}\\}$.\n\\ENDIF\n\\STATE (Optional) Carath\\'eodory's theorem can be applied for the vertex representation of $\\mb{x}^{(k+1)}$ so that $\\vert U^{(k+1)}\\vert = p+1$ and $\\boldsymbol\\mu^{(k+1)} \\in \\mr{R}^{p+1}$.\n\\STATE {\\bfseries Return:} $(U^{(k+1)}, \\boldsymbol\\mu^{(k+1)})$\n\\end{algorithmic}\n\\end{algorithm}\n\\end{center}\n\n\\noindent We need to introduce some definitions before presenting the theorems in this paper. Write $\\mb{A} = [\\mb{a}_1, \\mb{a}_2, \\ldots, \\mb{a}_n]^\\top$ and $\\mb{G}(\\mb{x})= (1\/n)[f_1'(\\mb{a}_1^\\top\\mb{x}), \\ldots, f_n'(\\mb{a}_n^\\top\\mb{x})]^\\top $. Define $D = \\sup\\{\\norm{\\mb{x} - \\mb{y}}\\, \\vert \\, \\mb{x}, \\mb{y} \\in \\mc{P}\\}$ and $G = \\sup\\{\\norm{\\mb{G}(\\mb{x})}\\, \\vert\\, \\mb{x} \\in \\mc{P}\\}$. It follows from compactness of $\\mc{P}$ and continuity of $F(\\cdot)$ that $D < \\infty$ and $G < \\infty$. Write $\\kappa =\\theta^2 \\{D\\norm{\\mb{b}} + 3GD\\norm{\\mb{A}} + \\frac{2n}{\\sigma_F}(G^2+1)\\}$ where $\\sigma_F = \\min\\{\\sigma_1, \\ldots, \\sigma_n\\} > 0$ and $\\theta$ is the Hoffman constant associated with matrix $[\\mb{C}^\\top,\\mb{A}^\\top, \\mb{b}^\\top]$ that is $\\theta = \\max\\{1 \/ \\lambda_{\\min}(\\mb{BB}^\\top) \\;\\vert \\; \\mb{B} \\in B\\}$, $\\lambda_{\\min}(\\mb{BB}^\\top)$ is the smallest eigenvalue of $\\mb{BB}^\\top$, where $B$ is the set of all matrices constructed by taking linearly independent rows from the matrix $[\\mb{C}^\\top,\\mb{A}^\\top, \\mb{b}^\\top]$. We denote by $I(\\mb{x})$, the index set of active constraints at $\\mb{x}$; that is,  $I(\\mb{x}) = \\{i \\in {1, \\ldots, m} \\; \\vert \\; \\mb{C}_i\\mb{x} = d_i\\}$. In a similar way, we define the set of active constraints for a set $U$ by  $I(U) = \\{i \\in \\{1, \\ldots, m\\} \\; \\vert \\; \\mb{C}_i\\mb{v} = d_i, \\forall \\mb{v} \\in U\\} = \\cap_{\\mb{v} \\in U}I(\\mb{v})$. Let $V$ be the set of vertices of $\\mc{P}$, then define $\\Omega_\\mathcal{P} = \\frac{\\zeta}{\\phi}$ where \n\\begin{align*}\n\\zeta &= \\min_{\\mathbf{v} \\in V, i \\in \\{1, \\ldots, m\\}: d_i > \\mathbf{C}_i \\mathbf{v}} (d_i - \\mathbf{C}_i\\mathbf{v}), \\\\\n\\phi &= \\max_{i \\in \\{1, \\ldots, m\\} \/ I(V)}\\norm{\\mathbf{C}_i}.\n\\end{align*}\n\\noindent\\textbf{Remark:} The vertex representation update procedure can also be implemented by using Carath\\'eodory's theorem so that each $\\mb{x}^{(k)}$ can be written as a convex combination of at most $N = p+1$ vertices of the polytope $\\mc{P}$. Then the set of vertices $U^{(k)}$ and their corresponding weights $\\boldsymbol\\mu^{(k)}$ can be updated according to the convex combination.\n\\begin{Theorem}\nLet $\\{\\mathbf{x}^{(k)}\\}_{k \\geq 1}$ be the sequence generated by Algorithm \\ref{cond_grad_1} for solving Problem $(\\ref{general_problem})$, $N$ be the number of vertices used to represent $\\mb{x}^{(k)}$ (if VRU is implemented by using Carath\\'eodory's theorem, $ N = p + 1$, otherwise $N = \\vert V \\vert$) and $F^*$ be the optimal value of the problem. Let $\\rho = \\Omega_{\\mathcal{P}}^2 \/\\{8N^2 \\kappa D\\max (G, LD)\\}$ and set $m_k = \\lceil n \/ (1 + n(1 - \\rho)^{2\\alpha k})\\rceil$ for some $0 < \\alpha < 1$. Let $D^{(k)}$ be the event that the algorithm removes a vertex from the current convex combination at iteration $k$, that is, the algorithm performs a `drop step' at iteration $k$. Assume $\\mr{P}(D^{(k)}) \\leq (1 - \\rho)^{\\beta k}$ for some $ 0 < \\beta < 1$.\nThen for every $k \\geq 1$\n\\begin{align}\\label{rate_of_convergence}\n\\mathbb{E}\\{F(\\mathbf{x}^{(k+1)}) - F^*\\} \\leq C_3(1 - \\rho)^{\\min (\\alpha, \\beta)k}\n\\end{align}\nwhere\n\\begin{align*}\nC_3 = F(\\mb{x}^{(1)}) - F^*  + \\frac{G\\sqrt{C_2}}{L\\{1 - (1 - \\rho)^{1 -\\alpha}\\}} + \\frac{G^2}{2L\\{1 - (1 - \\rho)^{1 - \\beta}\\}}\n\\end{align*}\nand \n\\begin{align*}\nC_2 &= \\sup_{\\mb{x} \\in \\mc{P}}[ \\frac{1}{n}\\sum_{i=1}^n \\{f_i'(\\mb{a}_i^\\top\\mb{x})\\}^2\\mb{a}_i^\\top\\mb{a}_i \\\\\n&\\quad - \\frac{1}{n(n-1)}\\sum_{i \\neq j}f_i'(\\mb{a}_i^\\top\\mb{x})f_j'(\\mb{a}_i^\\top\\mb{x})\\mb{a}^\\top_i\\mb{a}_j ] < \\infty.\n\\end{align*}\n\\end{Theorem}\n\\noindent Proof of the Theorem 1 is presented in the appendix. It is worth noting that by dividing both sides of (\\ref{rate_of_convergence}) by $F^{(1)} - F^*$, this linear convergence result for Algorithm 1 depends on relative function values. Therefore, the linear convergence of the proposed algorithm is invariant to scaling of the function.\n\n\\noindent\\textbf{Remarks}: The constant $\\rho$ depends on the \"vertex-face\" distance of a polytope as discussed in \\cite{BS15}. This also gives some intuition as to why constraint set $\\mc{P}$ has to be a polytope for linear convergence of the Frank-Wolfe algorithm with away-steps. When the boundary of the constraint set is `curved' as is the case for the $l_2$-ball, every point on the boundary is an extreme point. Then, faces can always be constructed so that the vertex-face distance is infinitesimal. Thus, we cannot get linear convergence when we apply the proof to a general convex constraint.\n\n\\section{Block Coordinate Semi-Stochastic Frank-Wolfe Algorithm with Away-Steps}\nIn this section, we assume the domain $\\mc{P}$ takes the form $\\mc{P} = \\mc{P}_{[1]} \\times \\mc{P}_{[2]} \\times \\cdots \\times \\mc{P}_{[q]}$, where each $\\mc{P}_{[i]}$ is a compact polytope that can be expressed as $\\mc{P}_{[i]} = \\{\\mb{x} \\in \\mr{R}^{p_i} \\, \\vert \\, \\mb{C}_{[i]}\\mb{x} \\leq \\mb{d}_{[i]}\\}$ where $ \\mb{C}_{[i]} \\in \\mr{R}^{m_i \\times p_i}$, $\\mb{d}_{[i]} \\in \\mr{R}^{m_i}$, $\\sum_{i = 1}^q m_i = m$ and $\\sum_{i=1}^q p_i = p$. Hence, $\\mc{P}$ still has the polytopic representation $\\mc{P} = \\{\\mb{x} \\in \\mr{R}^p \\: \\vert \\: \\mb{C}\\mb{x} \\leq \\mb{d}\\}$ where\n\\begin{align*}\n\\mb{C}= \\begin{bmatrix}\n\\mb{C}_{[1]} & & &\\\\\n& \\mb{C}_{[2]} & & \\\\\n& &\\ddots & \\\\\n& & & \\mb{C}_{[q]}\n\\end{bmatrix} \\quad \\quad \\text{ and }\\quad  \\quad \\mb{d} = \\begin{bmatrix}\n\\mb{d}_{[1]}\\\\\n\\mb{d}_{[2]}\\\\\n\\vdots\\\\\n\\mb{d}_{[q]}\n\\end{bmatrix}.\n\\end{align*}\nExamples of such Cartesian product constraints include the dual problem of structural SVMs, fitting marginal models in multivariate regression and multi-task learning. Let $V_{[i]}$ be the set of vertices of polytope $\\mc{P}_{[i]}$. We use subscripts $[i]$ to denote vectors, matrices, sets and other quantities correspond to the $i$-th block of constraints. Specifically,\nlet $\\mb{x}_{[i]} \\in \\mr{R}^{p_i}$ and $\\mb{x} = [\\mb{x}_{[1]}^\\top, \\mb{x}_{[2]}^\\top, \\cdots, \\mb{x}_{[q]}^\\top]^\\top \\in \\mr{R}^p$. Define $\\mb{U}_{[i]} \\in \\mr{R}^{p\\times p_i}$ as the sub-matrices of $p \\times p$ identity matrix that corresponds to the $i$-th block of the product domain, that is  $[\\mb{U}_{[1]},\\mb{U}_{[2]}, \\ldots, \\mb{U}_{[q]}] = \\mb{I}_{p \\times p}$. Hence $\\mb{x}_{[i]} = \\mb{U}_{[i]}^\\top\\mb{x}$ and $\\mb{x} = \\sum_{i = 1}^q \\mb{U}_{[i]}\\mb{x}_{[i]}$. Let $\\mc{O}_{[i]}$ be the linear oracle corresponding to the $i$-th block polytope in a lower dimension space. With the above notation, we are ready to present the block-coordinate version of the semi-stochastic Frank-Wolfe algorithm with away-steps and prove its linear convergence.\n\n\\begin{algorithm}[t]\n\\caption{Block Coordinate Semi-Stochastic Frank-Wolfe Algorithm with Away-Steps}\n\\label{block_cond_grad_1}\n\\begin{algorithmic}[1]\n\\STATE {\\bfseries Input:}  $\\mathbf{x}^{(1)} \\in V_{[1]} \\times \\cdots \\times V_{[q]}$, $f_i$, $\\mb{a}_i$, $\\mb{b}$ and $L$\n\\STATE Let $\\mu^{(1)}_{\\mathbf{x}^{(1)}_{[i]}} = 1$, $\\mu_\\mathbf{v}^{(1)} = 0$ for any $\\mathbf{v} \\in V_{[i]} \/ \\{\\mathbf{x}^{(1)}_{[i]}\\}$ and $U^{(1)}_{[i]} = \\{\\mathbf{x}^{(1)}_{[i]}\\}$.\n\\FOR{$k = 1, 2, \\ldots$}\n\\STATE Uniformly sample $\\mc{J} =\\{j_1, \\ldots, j_{m_k}\\}$ from $\\{1, \\ldots, n\\}$ without replacement, and denote  $\\mathbf{g}^{(k)} = \\frac{1}{m_k}\\sum_{i=1}^{m_k}f'_{j_i}(\\mb{a}_{j_i}^\\top\\mb{x}^{(k)})\\mb{a}_{j_i} + \\mb{b}$.\n\\STATE Uniformly sample $\\mc{L} = \\{l_1, \\ldots, l_r \\}$ from $\\{1, \\ldots, q\\}$ without replacement\n\\FOR{$i = 1, 2, \\ldots, r$, in parallel}\n\t\\STATE Compute $\\mathbf{p}^{(k)}_{[l_i]} = \\mathcal{O}_{[l_i]}(\\mb{U}_{[l_i]}^\\top\\mathbf{g}^{(k)})$\n\t\\STATE Compute $\\mathbf{u}^{(k)}_{[l_i]} \\in {\\arg\\!\\max}_{\\mathbf{v} \\in U^{(k)}_{[l_i]}}\\langle\\mb{U}_{[l_i]}^\\top \t\t\\mathbf{g}^{(k)}, \\mathbf{v} \\rangle $.\n\\IF{$\\langle \\mb{U}_{[l_i]}^\\top\\mathbf{g}^{(k)}, \\mathbf{p}^{(k)}_{[l_i]}  + \\mathbf{u}^{(k)}_{[l_i]}- 2\\mathbf{x}^{(k)}_{[l_i]}\\rangle \\leq 0$}\n\\STATE Set $\\mathbf{d}^{(k)}_{[l_i]} = \\mathbf{p}^{(k)}_{[l_i]} - \\mathbf{x}^{(k)}_{[l_i]}$ and $\\bar{\\gamma}^{(k)}_{[l_i]} = 1$. \n\\ELSE\n\\STATE Set $\\mathbf{d}^{(k)}_{[l_i]} = \\mathbf{x}^{(k)}_{[l_i]} - \\mathbf{u}^{(k)}_{[l_i]}$ and $\\bar{\\gamma}^{(k)}_{[l_i]} = \\frac{\\mu^{(k)}_{\\mathbf{u}^{(k)}_{[l_i]}}}{1 - \\mu^{(k)}_{\\mathbf{u}^{(k)}_{[l_i]}}}$.\n\\ENDIF\n\\STATE Set $\\gamma^{(k)}_{[l_i]} = \\min\\{-\\frac{\\langle \\mb{U}_{[l_i]}^\\top \\mb{g}^{(k)}, \\mb{d}^{(k)}_{[l_i]} \\rangle}{L\\norm{\\mb{d}^{(k)}_{[l_i]}}^2}, \\bar{\\gamma}^{(k)}_{[l_i]} \\}$.\n\t\\STATE Update $U^{(k + 1)}_{[i]}$ and $\\mathbf{\\mu}^{(k+1)}_{[i]}$ by Procedure VRU.\n\\ENDFOR\n\\STATE Update $\\mathbf{x}^{(k+1)} = \\mathbf{x}^{(k)} + \\sum_{i= 1}^r \\gamma^{(k)}_{[l_i]}\\mb{U}_{[l_i]}\\mathbf{d}^{(k)}_{[l_i]}$.\n\\ENDFOR\n\\STATE {\\bfseries Return:} $\\mathbf{x}^{(k+1)}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{Theorem}\nLet $\\{\\mathbf{x}^{(k)}\\}_{k \\geq 1}$ be the sequence generated by Algorithm \\ref{block_cond_grad_1} for solving Problem $(\\ref{general_problem})$ with block-coordinate structure constraints ,$N$ be the number of vertices used to represent $\\mb{x}^{(k)}$ (if VRU is implemented by using Carath\\'eodory's theorem, $ N = p + 1$, otherwise $N = \\vert V \\vert$) and $F^*$ be the optimal value of the problem. Let $\\hat{\\rho} = r\\Omega_{\\mathcal{P}}^2 \/\\{8N^2 \\kappa q^2D \\max (G, LD)\\}$ and set $m_k = \\lceil n \/ (1 + n(1 - \\hat{\\rho})^{2\\mu k})\\rceil$ for some $0 < \\mu < 1$. Let $D^{(k)}_{[i]}$ be the event that the algorithm removes a vertex from the current convex combination in block $i$ at iteration $k$, that is, the algorithm performs a `drop step' at iteration $k$. Assume $\\max_{i}\\{\\mr{P}(D^{(k)}_{[i]})\\} \\leq (1 - \\hat{\\rho})^{\\lambda k}$ for some $ 0 < \\lambda < 1$ and all $k \\geq 1$. Then for any $k \\geq 1$\n\\begin{align*}\n\\mathbb{E}\\{F(\\mathbf{x}^{(k+1)}) - F^*\\}\n&\\leq C_4(1 - \\hat\\rho)^{\\min(\\mu, \\lambda)}\n\\end{align*}\nwhere\n\\begin{align*}\nC_4 = \\{F(\\mb{x}^{(1)}) - F^*  + \\frac{rG\\sqrt{C_2}}{qL\\{1 - (1 - \\hat{\\rho})^{1 -\\mu}\\}} + \\frac{rG^2}{2L\\{1 - (1 - \\hat{\\rho})^{1 - \\lambda}\\}}\\}.\n\\end{align*}\n\\end{Theorem}\n\\noindent Proof of the Theorem 2 can be found in the appendix.\n\n\\noindent\\textbf{Remark:} All of the sub-problems in the inner loop of the proposed algorithm can be solved in parallel. This feature can further boost the performance of the algorithm. It seems that the rate of convergence of the algorithm for block-coordinate problems is worse since $\\hat{\\rho} < \\rho$. One reason for this is the algorithm only uses sampled blocks instead of a full data pass in every iteration. Another reason is that the theoretical result is based on the worst case scenario happening at the same time in every sampled block (which won't happen in real implementations) for the ease of proving linear convergence. The worst case scenario is the so called `drop-step' and the detailed analysis can be found in the supplementary material.\n\\section{Numerical Studies}\nIn this section, we apply the semi-stochastic Frank-Wolfe algorithm with away-steps to two popular problems in machine learning. The first one is a graph guided fused LASSO problem for multi-task learning using a simulated data set. The second one is a structural support vector machine (SVM) problem using a real data set in speech recognition.\n\\subsection{Multi-task Learning via Graph Guided Fused LASSO}\n\\begin{figure*}\n\\centering\n\\begin{tabular}{rrl}\n\\includegraphics[scale = 0.28]{true_coeff.eps}&\n\\includegraphics[scale = 0.28]{proxgrad.eps} &\n\\multirow{2}*[2.7cm]{\\includegraphics[scale=0.5]{function_value.eps}}\\\\\n\\includegraphics[scale = 0.28]{bcfwas.eps} &\n\\includegraphics[scale = 0.28]{Correlation_Block.eps}&\n\\end{tabular}\n\\caption{(a) is the true regression coefficient matrix in the simulated data for GFLASSO problem, (b) is the estimated regression coefficient matrix by using proximal gradient method (c) is the estimated regression coefficient matrix by using block-coordinate Frank-Wolfe algorithm with away-steps (BCFWAS), (d) is the truncated correlation matrix of the outputs $\\mb{Y}$ in the simulated data set base on which the graph in the GFLASSO problem is constructed and (e) is the plot of the logarithmic objective function values of both methods. Median of the 10 sample-paths when running both algorithms are plotted in the lines. The shaded areas shows the upper and lower bounds at each iteration in the 10 replications. The detailed plot in the middle shows that the BCFWAS beats the Prox-Grad after the 20th iteration in this experiment.}\n\\label{gflasso}\n\\end{figure*}\n\nConsider the following multivariate linear model:\n\\begin{align*}\n\\mathbf{y}_i = \\mathbf{X}\\boldsymbol\\beta_i + \\boldsymbol\\epsilon_i \\quad\\quad i = 1,\\ldots, N\n\\end{align*}\nwhere $\\mathbf{X} \\in \\mathbb{R}^{N \\times J}$ denotes the matrix of input data for $J$ covariates over $N$ samples, $\\mathbf{Y}=[\\mathbf{y}_1, \\ldots, \\mathbf{y}_K] \\in \\mathbb{R}^{N \\times K}$ denotes the matrix of output data for $K$ tasks, $\\mathbf{B} = [\\boldsymbol\\beta_1, \\ldots, \\boldsymbol\\beta_K] \\in \\mathbb{R}^{J \\times K}$ denotes the matrix of regression coefficients for the $K$ tasks and the $\\boldsymbol\\epsilon_i$'s denote the noise terms that are independent and identically distributed. Let $G = (V, E)$ be a graph where $V$ is the set of vertices and $E$ is the set of edges. In \\citep{CLKCX11}, the graph is constructed by taking each task as a vertex and each non-zero off-diagonal entry of the correlation matrix of $\\mb{Y}$, as an edge. When the correlation between two tasks is small, it will be truncated to $0$ when calculating the correlation matrix and hence there won't be edges between such tasks. The graph guided fused LASSO (GFLASSO) problem for multi-task sparse regression problem is formulated as \n\\begin{align*}\n\\min_{\\mathbf{B}\\in \\mathbb{R}^{J \\times K}} \\frac{1}{2}\\norm{\\mathbf{Y} - \\mathbf{XB}}_F^2 + \\Omega(\\mathbf{B}) + \\lambda\\norm{\\mathbf{B}}_1,\n\\end{align*}\nwhere $$\\Omega(\\mathbf{B}) = \\gamma \\sum_{e = (m, l) \\in E} \\vert r_{ml}\\vert\\sum_{j=1}^J \\vert \\beta_{jm} - \\text{sign}(r_{ml})\\beta_{jl}\\vert, $$ $r_{ml}$ is the $(m, l)$-th entry of \\textbf{cor}($\\mathbf{Y}$), the truncated correlation matrix of ${\\mathbf{Y}}$ at a pre-determined level, $\\norm{\\cdot}_F$ denotes the Frobenius norm, and  $\\gamma$ and $\\lambda$ are the regularization parameters.\nThe GFLASSO was first proposed by \\cite{KSX09} for problems in quantitative trait network in genomic association studies. To solve GFLASSO, \\cite{KSX09} formulated it as a quadratic programming problem and proposed using an active-set method. Later \\cite{CLKCX11} proposed a smoothing proximal gradient method which is more saclable than the active-set method. We propose to use the block-coordinate semi-stochastic Frank-Wolfe algorithm with away-steps to solve this problem. When the underlying graph is a union of several connected components (as it is the case in most genetic studies) instead of being fully connected, the proposed algorithm can be applied by considering each connected component as a block. Such a structure effectively transforms a large original problem into several small subproblems which are easier to solve. When the regression coefficient matrix $\\mb{B}$ is sparse, which is also common in most genetic studies, the sparse update in each step of the Frank-Wolfe type algorithms will also have the advantage of being able to extract information from the data sets efficiently. In our numerical example, we follow the data generation procedure in \\cite{CLKCX11} which simulates genetic association mapping data. We set $N = 200$, $J = 50$, $K = 20$, entries of $\\mb{X}$ and $\\boldsymbol\\epsilon_k$'s are generated as standard normal random variables, $\\boldsymbol\\beta_k$'s are generated such that the correlation matrix of outputs of $\\mb{y}_k$'s has a block structure and all non-zero elements of the $\\boldsymbol\\beta_k$'s are equal to 1. The entries in the correlation matrix of $\\mb{Y}$ are truncated to $0$ when their absolute value is smaller than $0.7$. In the illustration, fitting GFLASSO using the proposed algorithm reveal the structures of the coefficients very quickly and the proposed algorithm converges much faster than the Prox-Grad method that was proposed in \\cite{CLKCX11}.\n\n\\noindent\\textbf{Remark:} Initial values for Algorithm \\ref{block_cond_grad_1} must be vertices of each `block' polytope and even with a relatively bad initialization, the proposed algorithm has a better performance than the Prox-Grad algorithm in this problem. To obtain the same initialization as in the Prox-Grad algorithm, we can write the initial value as a convex combination of the vertices in each block and set $U_{[i]}^{(1)}$ and $\\boldsymbol\\mu_{[i]}^{(1)}$ accordingly. It is worth noting that due to the numerical error, the points returned by the oracles may not be the vertices of the constraint polytopes. As a result, we may obtain a smaller function value than the optimal value due to the tiny in-feasibilities. Such problems do not occur when the vertices are integral, which is the case of our next example.\n\n\\subsection{Structural Support Vector Machines (SVM)} \nThe idea of using the Frank-Wolfe algorithm to solve the dual of structural SVM problems was introduced by \\cite{LJJSP13}. Briefly speaking, structural SVM solves multi-label classification problems through a linear classifier (with parameter $\\mb{w}$) $h_\\mb{w}(\\mb{x}) = \\arg\\!\\max_{\\mb{y} \\in \\mc{Y}(\\mb{x})}\\langle \\mb{w}, \\phi(\\mb{x}, \\mb{y}) \\rangle$, where $\\mb{y}$ is a structured output given an input $\\mb{x}$, $\\mc{Y}(\\mb{x})$ denotes the set of all possible labels for an input $\\mb{x}$, $\\phi: \\mc{X} \\times \\mc{Y} \\rightarrow \\mr{R}^d$ is a given feature or basis map, and $\\mc{X}$ denotes the set of all possible inputs. To learn $\\mb{w}$ from training samples $\\{(\\mb{x}_i, \\mb{y}_i), i = 1, \\ldots, n\\} $, we solve \n\\begin{align*}\n\\min_{\\mb{w}, \\boldsymbol\\xi} \\quad &\\frac{\\lambda}{2}\\norm{\\mb{w}} + \\frac{1}{n}\\sum_{i=1}^n \\xi_i \\\\\n\\text{s.t.}                    \\quad          & \\langle \\mb{w}, \\psi_i(\\mb{y}) \\rangle \\geq L_i(y) -\\xi_i \\\\\n                                             & \\mb{y} \\in \\mc{Y}_i \\\\\n                                             & i = 1, 2, \\ldots, n,\n\\end{align*}\nwhere $\\psi_i(\\mb{y}) = \\phi(\\mb{x}_i, \\mb{y}_i) -\\phi(\\mb{x}_i, \\mb{y})$ is the potential function, $L_i(\\mb{y})$ denotes loss incurred by predicting $\\mb{y}$ instead of the observed label $\\mb{y}_i$, $\\mc{Y}_i = \\mc{Y}(\\mb{x}_i)$,  $\\xi$'s are the surrogate losses for the data points and $\\lambda$ is a regularization parameter. Now consider the Lagrange dual of the above problem. Let $\\alpha_i(\\mb{y}) \\in \\mr{R}$ be the dual variable associated with training sample $i$ and possible output $\\mb{y} \\in \\mc{Y}_i$, $\\boldsymbol\\alpha_i \\in \\mr{R}^{m_i}$ be the concatenation of all $\\alpha_i(\\mb{y})$ over all $ \\mb{y} \\in\\mc{Y}_i$,  and $\\boldsymbol\\alpha \\in R^m$ be the column concatenation of all $\\alpha_i(\\mb{y})$, where $m_i = \\vert \\mc{Y}_i \\vert$ and $m = \\sum_{i=1}^n m_i$. Then the dual problem can be written as\n\\begin{align*}\n\\min_{\\boldsymbol\\alpha} \\quad &F(\\boldsymbol\\alpha) := \\frac{\\lambda}{2}\\norm{\\mb{A}\\boldsymbol\\alpha}^2 -\\mb{b}^\\top \\boldsymbol\\alpha \\\\\n\\text{s. t.}   \\quad                & \\boldsymbol\\alpha \\geq \\mb{0} \\\\\n\t\t\t\t\t\t\t\t\t  & \\sum_{\\mb{y} \\in \\mc{Y}_i}\\alpha_i(\\mb{y}) = 1\\\\\n\t\t\t\t\t\t\t\t\t  & i = 1, \\ldots, n,\n\\end{align*}\nwhere $A \\in \\mr{R}^{d \\times m}$ whose columns are given by $\\psi_i(\\mb{y}) \/(\\lambda n), \\mb{y} \\in \\mc{Y}_i, i = 1, 2, \\ldots, n$ and $b \\in \\mr{R}^m$ whose entries are $L_i(\\mb{y})\/n, \\mb{y} \\in \\mc{Y}_i, i = 1, 2, \\ldots, n$. Given a dual solution $\\hat{\\boldsymbol\\alpha}$, the corresponding primal solution can be retrieved from the relation $\\hat{\\mb{w}} = \\mb{A}\\hat{\\boldsymbol\\alpha}$ which is implied by KKT conditions. Observe that $\\norm{\\mb{A}\\boldsymbol\\alpha}^2 = \\sum_{i=1}^d (\\mb{A}_i \\boldsymbol\\alpha)^2$ and the constraint set for the dual problem is a Cartesian product of polytopes that can be written as $\\mc{M} := \\Delta_{\\vert \\mc{Y}_1 \\vert} \\times \\cdots \\times \\Delta_{\\vert \\mc{Y}_n \\vert}$ where $\\Delta_{\\vert\\mc{Y}_i\\vert}$ is the simplex generated by the elements in $\\mc{Y}_i$. This suggests the block coordinate stochastic Frank-Wolfe algorithm with away-steps can be applied to this problem if we have linear oracles for each block which solves\n\\begin{align*}\n\\min_{\\boldsymbol\\alpha_i \\in \\Delta_{\\vert \\mc{Y}_i \\vert}} \\langle \\nabla_{[i]} F(\\cdot), \\boldsymbol\\alpha_i \\rangle\n\\end{align*}\nwhere $\\nabla_{[i]} F(\\cdot)$ denotes the partial derivative of $F(\\cdot)$ with respect to $i$-th block of variables.\nSince the gradient of $F$, $\\nabla F = \\lambda \\mb{A}^\\top \\mb{A}\\boldsymbol\\alpha - \\mb{b} = \\lambda\\mb{A}^\\top\\mb{w} - \\mb{b}$ whose $(i, \\mb{y})$-th entry is $\\{\\langle \\mb{w}, \\psi_i(\\mb{y}) \\rangle - L_i(\\mb{y})\\} \/ n := -H_i(\\mb{y}; \\mb{w}) \/ n$, the above linear oracle is equivalent to the following maximization oracle\n\\begin{align*}\n\\max_{\\mb{y} \\in \\mc{Y}_i} H_i(\\mb{y}; \\mb{w}).\n\\end{align*}\n\\begin{algorithm}[tb]\n\\caption{Block Coordinate Semi-Stochastic Frank-Wolfe Algorithm with Away-Steps for Structural SVM}\n\\label{block_cond_grad_2}\n\\begin{algorithmic}[1]\n\\STATE Let $\\mb{y}^{(1)} \\in \\mc{Y}_1 \\times \\cdots \\times \\mc{Y}_n$ where $\\mu^{(1)}_{\\mb{y}^{(1)}_{[i]}} = 1$, $\\mu_\\mathbf{v}^{(1)} = 0$ for any $\\mathbf{v} \\in \\mathcal{Y}_i \/ \\{\\mb{y}^{(1)}_{[i]}\\}$ and $U^{(1)}_{[i]} = \\{\\mathbf{y}^{(1)}_{[i]}\\}$.\n\\STATE Set $\\mb{w}^{(1)}_i = \\frac{1}{\\lambda n}\\psi_i(\\mb{y}^{(1)}_{[i]})$, $\\ell^{(1)}_i = \\frac{1}{n}L_i(\\mb{y}^{(1)}_{[i]})$, $\\mb{w}^{(1)} = \\sum_{i=1}^n \\mb{w}^{(1)}_i$, and $\\ell^{(1)}= \\sum_{i=1}^n \\ell^{(1)}_i$.\n\\FOR{$k = 1, 2, \\ldots$}\n\\STATE Uniformly sample $\\mc{J} =\\{j_1, \\ldots, j_{m_k}\\}$ from $\\{1, \\ldots, d\\}$ without replacement.\n\\STATE Randomly pick $i$ from $\\{1, \\ldots, n\\}$\n\\STATE Compute $\\mathbf{p}^{(k)}_{[i]} = \\arg\\!\\max_{\\mb{y} \\in \\mc{Y}_i}H_i(y; I_\\mc{J}\\mb{w}^{(k)})$\n\\STATE Compute $\\mathbf{u}^{(k)}_{[i]} \\in {\\arg\\!\\max}_{\\mathbf{v} \\in U^{(k)}_{[i]}}\\langle I_\\mc{J}\\mb{w}^{(k)}, \\psi_i(\\mathbf{v}) \\rangle - L_i(\\mb{v})$. \n\t\\IF{$\\langle I_\\mc{J}\\mb{w}^{(k)}, \\frac{1}{\\lambda n}\\psi_i(\\mb{p}^{(k)}_{[i]}) - \\mb{w}^{(k)}_i \\rangle + \\ell^{(k)}_i - \\frac{1}{n}L_i(\\mb{p}_{[i]}^{(k)}) \\leq \\langle I_\\mc{J}\\mb{w}^{(k)},  \\mb{w}^{(k)}_i - \\frac{1}{\\lambda n}\\psi_i(\\mb{u}^{(k)}_{[i]}) \\rangle + \\frac{1}{n}L_i(\\mb{u}_{[i]}^{(k)}) - \\ell^{(k)}_i  $}\n\\STATE Set $\\mb{d}^{(k)}_{i} = \\frac{1}{\\lambda n}\\psi_i(\\mb{p}^{(k)}_{[i]}) - \\mb{w}^{(k)}_i$, ${e}^{(k)}_{i} = \\frac{1}{n}L_i(\\mb{p}_{[i]}^{(k)}) - \\ell_i^{(k)}$ and $\\bar{\\gamma}^{(k)}_{i} = 1$.\n\t\\ELSE\n\t\\STATE Set $\\mathbf{d}^{(k)}_{i} = \\mathbf{w}^{(k)}_{i} - \\frac{1}{\\lambda n}\\psi_i(\\mathbf{u}^{(k)}_{[i]})$, ${e}^{(k)}_{i} = \\ell_i^{(k)} - \\frac{1}{n}L_i(\\mb{u}_{[i]}^{(k)}) $, and $\\bar{\\gamma}^{(k)}_{i} = \\frac{\\mu^{(k)}_{\\mathbf{u}^{(k)}_{[i]}}}{1 - \\mu^{(k)}_{\\mathbf{u}^{(k)}_{[i]}}}$.\n\t\\ENDIF\n\t\\STATE Set $\\gamma^{(k)} = \\max\\{0, \\min\\{\\frac{-\\lambda\\langle \\mb{w}^{(k)}, \\mb{d}^{(k)}_{i}\\rangle + e^{(k)}_{i}}{\\lambda\\norm{\\mb{d}^k_{i}}^2}, \\bar{\\gamma}^{(k)}_{i}\\} \\}$\n\t\\STATE Update $\\mb{w}^{(k+1)}_i = \\mb{w}^{(k)} + \\gamma^{(k)} \\mb{d}^{(k)}_{i}$\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\nAccording to \\cite{LJJSP13}, this maximization problem is known as the \\textit{loss-augmented decoding} subproblem which can be solved efficiently. The last thing which may prohibit us from applying the proposed algorithm is the potentially exponential number $\\vert \\mc{Y}_i \\vert$ of dual variables due to its combinatorial nature, which makes maintaining a dense $\\boldsymbol\\alpha$ impossible. This is also the reason why other algorithms, for example QP, become intractable for this problem. However, the simplex structure of the constraint sets enables us to keep the only non-zero coordinate of the solution to the linear oracle which corresponds to the solution of the loss-augmented decoding sub-problem in each iteration. Thus, we only need to keep track of the initial value $\\boldsymbol\\alpha^{(0)}$ and a list of previously seen solutions to the maximum oracle. Another way to keep track of the vertices is to maintain a list of primal variables from the linear transformation $\\mb{w} = \\mb{A}\\boldsymbol\\alpha$ when the dimension of $\\mb{w}$ is moderate. Denote the vertex set of $\\Delta_{\\vert \\mc{Y}_i \\vert}$ as $\\mc{V}_i$ and for an set $\\mc{J} \\subset \\{1, 2, \\ldots, d\\}$, let $I_\\mc{J}$ be a $d \\times d$ matrix whose $j$-th diagonal entry equals $1$ for all $j \\in \\mc{J}$ and all other entries are zero. With above notation, we can apply the stochastic block coordinate Frank-Wolfe algorithm with away-steps the the structural SVM problem.\nWe apply the above algorithm on the OCR dataset ($n = 6251, d = 4028$) from \\cite{TGK03} and compare it with the block-coordinate Frank-Wolfe algorithm. To make a fair comparison, the initial vertex $\\mb{y}^{(1)}_i$ in $i$-th coordinate block is chosen to be the observed tag $\\mb{y}_i$ corresponding to the choice of primal variables $\\mb{w}_i^{(1)} = 0$ as in the implementation of \\cite{LJJSP13}. It is worth noting that in the experiments, the performance of both algorithms was worse when the initializations were changed. Computational results for OCR dataset with regularization parameters $\\lambda = 0.05, 0.01 $ and $0.001$ are presented here. From the figures, we can see that the stochastic Block-Coordinate Frank-Wolfe Algorithm with Away-Steps (BCFWAS) dominates the Block-Coordinate Frank Wolfe algorithm (BCFW) for every $\\lambda$ when an accurate solution is required.\n\\begin{figure*}\n\\vskip 0.2in\n\\begin{center}\n\\subfigure[]{\\includegraphics[scale = 0.55]{lambda_05.eps}}\n\\subfigure[]{\\includegraphics[scale = 0.55]{lambda_01.eps}}\n\\subfigure[]{\\includegraphics[scale = 0.55]{lambda_005.eps}}\n\\subfigure[]{\\includegraphics[scale = 0.55]{lambda_001.eps}}\n\\caption{Each figure is plotted the number of effective passes of data versus the relative duality gap. In the implementation, we used the weighted average technique introduced in \\cite{LJJSP13} in both algorithms  which outputs the series $\\bar{\\mb{w}}^{(k+1)} = k\/(k+2)\\bar{\\mb{w}}^{(k)} + 2\/(k+2)\\mb{w}^{(k+1)}$ and $\\bar{\\mb{w}^{(0)}} = \\mb{w}^{(0)}$. The embedded figures are the corresponding (by frame color) details in the original plots.}\n\\label{pics:ssvm}\n\\end{center}\n\\vskip -0.2in\n\\end{figure*}\n\\section{Conclusion, Discussion and Future Work}\nIn this paper, we established the linear convergence of a semi-stochastic Frank-Wolfe algorithm with away-steps for empirical risk minimization problems and extended it to problems with block-coordinate structure. We applied the algorithms to solve the graph-guided fused LASSO problem and the structural SVM problem. Numerical results indicate the proposed algorithms outperform competing algorithms for these two problems in terms of both iteration cost and number of effective data passes. In addition, the stochastic nature of the proposed algorithms can use an approximate solution to the sub-problems, that is, an inexact oracle which can further reduce the computational cost. Possible extensions of this work include:\n\\begin{description}\n\\item[1] In the algorithms, we assume that the Lipschitz constants of the gradient of $F(\\cdot)$ are known. This assumption might be avoided by performing back-tracking. \n\\item[2] The algorithms are semi-stochastic since we need to use all data to calculate gradients at the final steps. Variance reduced gradient techniques such as the one proposed by \\cite{JZ13} might be applicable in stochastic versions of the Frank-Wofe algorithm with away-steps to make it fully stochastic and linearly convergent in high probability.\n\\item[3] The compact polytope constraint is crucial in the theoretical analysis of the proposed algorithms. Finding ways to relax this assumption is another direction for future work.\n\\end{description}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Abstract}\\label{ch1}\n\nWe present a new efficient method to compute Boltzmann collision integral for all two-particle interactions in relativistic plasma. Plasma is assumed to be homogeneous in coordinate space and isotropic in momentum space. The set of reactions consists of: Moeller and Bhabha scattering, Compton scattering, two-photon pair annihilation, and two-photon pair production, which are described by QED matrix elements. In our method exact energy and particle number conservation laws are fulfilled. Drastic improvement in computation time with respect to existing methods is achieved. Reaction rates are compared, where possible, with the corresponding analytical expressions and convergence of numerical rates is demonstrated.\n\\section{Introduction}\\label{ch2}\n\nRelativistic plasma, for which $kT\\geq mc^2$, where $k$ is Boltzmann constant, $c$ is speed of light, $m$ is electron mass, $T$ is temperature, is relevant in different branches of astrophysics. In the early universe ultrarelativistic electron-positron pairs contribute to the matter contents of the Universe \\cite{weinberg2008cosmology}. X-ray and gamma-ray radiation from numerous astrophysical sources such as gamma-ray bursts \\cite{1999PhR...314..575P,2010PhR...487....1R,2015PhR...561....1K}, active galactic nuclei \\cite{2012AAT...27..557A, blandford2013active}, and X-ray binaries \\cite{2006ARA&A..44..323F} points out to existence of relativistic electron-positron plasma in these objects.\nThe upcoming high-energy laser facilities aiming at generation of femtosecond laser pulses with intensity more than $10^{21}W\/cm^2$ aim generation of relativistic plasma by interacting laser pulses. At present relativistic electron-positron jets are generated by interaction of laser pulses with condensed matter \\cite{2015NatCo...6E6747S,0741-3335-53-1-015009,1742-6596-454-1-012016,PhysRevLett.102.105001}. \n\n\nSolving the Boltzmann equations with collision integral containing a quantum cross-section represents the most general and complete method to describe a behavior of relativistic plasma \\cite{vereshchagin2017relativistic,cercignani2012relativistic,groot1980relativistic}. \nThe one-particle distribution function (DF) is defined on a seven dimensional space, three dimensions for the physical space and three dimensions for the momentum space, and one dimension for the time. Thus one has a multidimensional problem which is a real challenge from the computational point of view. Beside the dimensionality problem, there are other difficulties which are related to kinetic equations in general \\cite{1990JMP....31..245B,1991RvMaP...3..137B}. \nOur main goal in this paper is to tackle the challenge associated with the calculation of the collision integral, dealing with two key issues. First, the computational cost related to the evaluation of the collision operator involving multidimensional integrals which should be solved in each point of the coordinate space. Second, the presence of multiple scales requires the development of adapted numerical schemes capable of solving stiff dynamics. Different deterministic approaches are used to tackle collision integral from a numerical point of view: finite volume, semi-Lagrangian and spectral schemes \\cite{1991RvMaP...3..137B,dimarco_pareschi_2014,2006MaCom..75.1833M,DIMARCO2017,WU201327}. While the deterministic methods could normally reach high order of accuracy, the probabilistic ones, such as Monte-Carlo (MC) method, are often faster.\n\nMC methods are traditionally used to model Coulomb interactions in non-relativistic plasma\\cite{SHERLOCK20082286,HUTHMACHER2016535,TURRELL2015144,BOBYLEV2013123}. As a rule MC techniques are based on the random pairing of particles in close vicinity and the calculation of a scattering angle due to the interaction. Small-angle Coulomb collisions which allow small energy and momentum transfer are often described in diffusion approximation by the Fokker-Planck equation \\cite{1981phki.book.....L}. The principal feature of relativistic plasma is a presence of pair creation\nand pair annihilation processes, which are often included in MC based models \\cite{0741-3335-53-1-015009,1742-6596-454-1-012016}. However Fokker-Planck approximation is no longer valid in relativistic plasma \\cite{2009PhRvD..79d3008A}.\n\n\nClassical Boltzmann equation does not take into account quantum statistics of particles. The generalization of classical Boltzmann equation including quantum corrections is Uehling-Uhlenbeck (U-U) equation, which contains additional Pauli blocking and Bose enhancement multipliers that give rise to equilibrium solution with Bose-Einstein and Fermi-Dirac distributions \\cite{1934PhRv...46..917U,1933PhRv...43..552U}. The main problem of the MC methods in application to U-U equations is that total reaction rate is unknown as distribution function is unknown too. Compensation methods include smoothing of the delta-function distribution of MC-particles over cells in the phase space, but it suffers from a large number of simulation particles and cells needed to reproduce Bose-Einstein steady state distribution. Spectral methods based on the Fourier transformation of the velocity distribution function require very dense computational grid to reach high accuracy \\cite{2017JCoPh.330.1010Y,Hu2015,huying12,PhysRevE.68.056703,2010arXiv1009.3352F}.\nProcess-oriented approach to the U-U collision integral presented in this work allows one to get high accuracy results with low computational cost.\n\nIn this paper we further develop the method first used in the work \\cite{2004ApJ...609..363A}. This method was successfully applied to follow the thermalization of relativistic plasma \\cite{2007PhRvL..99l5003A,2009AIPC.1111..344A,2009PhRvD..79d3008A,2010AIPC.1205...11A} and to investigate thermalization timescales for an electron-positron plasma \\cite{2010PhRvE..81d6401A}. In section \\ref{ch3} we recall Boltzmann and UU equations and present usual scheme of their analytic treatment. Section \\ref{ch4} is devoted to the description of our numerical scheme, while section \\ref{ch5} shows comparison between our code results and known analytic formulae for non-degenerate case. Conclusion follows.\n\n\\section{Formulation}\\label{ch3}\n\nThe Boltzmann equation governs an evolution of one-particle distribution function $f(\\mathbf x,\\mathbf p,t)$. We assume that plasma is homogeneous and isotropic in coordinate space and isotropic in momentum space, thus distribution function depends on absolute value of momentum (energy) and time. DF is normalized on particles concentration, so that $n=\\int f(\\mathbf p,t)d^3p$.\\\\\nConsider an interaction of two initial particles of type I and II which are in states 1 \u0438 2, correspondingly, and creation of two final particles of type III \u0438 IV which are in states 3 \u0438 4, correspondingly. Let us image the process by the following scheme:\n\\begin{equation}\\label{2pdir}\nI_1 + II_2 \\rightarrow III_3 + IV_4.\n\\end{equation}\nThe corresponding inverse process is:\n\\begin{equation}\\label{2pinv}\nIII_3 + IV_4\\rightarrow I_1 + II_2.\n\\end{equation}\nIf every particle has momentum $p_i$, which lies in interval $d^3p_i$, then a number of interactions in unit time and unit space volume is:\n\\begin{equation}\nw(p_1,p_2;p_3,p_4)f_{I} f_{II} d^3p_1d^3p_2d^3p_3d^3p_4,\n\\end{equation}\nfunction $w$ is called a transition rate for a given reaction. \\\\\nAn effective cross-section is defined by the formula:\n\\begin{equation}\nd\\sigma=\\frac{w}{v}d^3p_3 d^3p_4,\n\\end{equation}\nwhere $v=c\\epsilon_1^{-1}\\epsilon_2^{-1}\\sqrt[]{\\left( \\epsilon_1\\epsilon_2-(\\mathbf{p_1}\\mathbf{p_2})c^2 \\right)^2-(m_1 m_2 c^4)^2}$ is a relative velocity of particles. \\\\\nIn quantum field theory an expression for interaction cross-section is:\n\\begin{equation}\\label{dsigma}\nd\\sigma=\\frac{\\hbar^2 c^6}{(2\\pi)^2}\\frac{1}{v}\\frac{|M_{if}|^2}{16\\epsilon_1\\epsilon_2\\epsilon_3\\epsilon_4}\n\\delta(\\epsilon_1+\\epsilon_2-\\epsilon_3-\\epsilon_4)\\delta(\\mathbf p_1+\\mathbf p_2-\\mathbf p_3-\\mathbf p_4)d^3p_3d^3p_4,\n\\end{equation}\nwhere $|M_{if}|$ are a matrix elements calculated with a methods of quantum field theory. \\\\\nComparing two last formulas one can derive the following expression for transition rate:\n\\begin{equation}\\label{transw}\nw(p_3,p_4;p_1,p_2)=\\frac{\\hbar^2 c^6}{(2\\pi)^2}\\frac{|M_{if}|^2}{16\\epsilon_1\\epsilon_2\\epsilon_3\\epsilon_4}\\delta(\\epsilon_1+\\epsilon_2-\\epsilon_3-\\epsilon_4)\n\\delta(\\mathbf p_1+\\mathbf p_2-\\mathbf p_3-\\mathbf p_4),\n\\end{equation}\nNow let us write the Boltzmann equation for DF of particle I for a given process:\n\\begin{multline}\\label{StfI}\n\\dot{f_I}=\\int d^3p_2 d^3p_3 d^3p_4 [w(p_3,p_4;p_1,p_2)f_{III}(\\mathbf p_3,t)f_{IV}(\\mathbf p_4,t)\\\\\n-w(p_1,p_2;p_3,p_4)f_{I}(\\mathbf p_1,t)f_{II}(\\mathbf p_2,t)],\n\\end{multline}\nwhere a dot denotes time derivative. Equations for particle DFs of remaining types can be derived by the corresponding replacement of indices.\\\\\nSpecifically, for a scattering with $I=III$ and $II=IV$ the inverse process is the same as the direct one since pairs of indices $(1,2)$ and $(3,4)$ can be interchanged. The relation $w(p_3,p_4;p_1,p_2)=w(p_1,p_2;p_3,p_4)$ holds for all processes listed in Table \\ref{2pptable}.\nThe right hand side of the Boltzmann equation is a collision integral denoted as St$f_I$. The first term in collision integral describes particle outcome and the second term describes particle income, we will denote it as $\\text{St}^-f$ and $\\text{St}^+f$, respectively.\\\\\n\n\\begin{table}\n\\caption{Two-particle processes in electron-positron-photon plasma\\label{2pptable}}\n\\center\n\\begin{tabular}[c]{|l|l|l|l|l|}\n\\hline\nProcess (q) & I & II & III & IV \\\\\n\\hline\nCompton Scattering (CS) & $e^{\\pm}$ & $\\gamma$ & $e^{\\pm}$ & $\\gamma$\\\\\n\\hline\nBhabha Scattering (BS) & $e^{\\pm}$ & $e^{\\mp}$ & $e^{\\pm}$ & $e^{\\mp}$\\\\\n\\hline\nM{\\o}ller Scattering (MS) & $e^{\\pm}$ & $e^{\\pm}$ & $e^{\\pm}$ & $e^{\\pm}$\\\\\n\\hline\nPair Annihilation (PA) & $e^{-}$ & $e^{+}$ & $\\gamma$ & $\\gamma$\\\\\n\\hline\nPair Creation (PC) & $\\gamma$ & $\\gamma$ & $e^{-}$ & $e^{+}$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nThe generalization of Boltzmann equation for the case of particles obeying quantum statistics is U-U equation. For the particle $I$ in the state $1$ U-U equation has the following form:\n\\begin{multline}  \\label{uustfI}\n\\dot{f_I} =\\int d^{3} \\mathbf{p}_{2} d^{3} \\mathbf{p}_{3} d^{3} \\mathbf{p}_{4}  \\\\\n\\shoveleft{\\ \\times\\biggl[w(p_3,p_4;p_1,p_2)\nf_{III}(\\mathbf{p}_{3},t) f_{IV}(\\mathbf{p}_{4},t)\n\\left( 1+\\eta \\frac{f_{I}(\\mathbf{p}_{1},t)}{2 h^{-3}}\\right)\\left( 1+\\eta \\frac{f_{II}(\\mathbf{p}_{2},t)}{2 h^{-3}}\\right)}\\\\\n\\shoveleft{\\quad -w(p_1,p_2;p_3,p_4)\nf_{I}(\\mathbf{p}_{1},t)f_{II}(\\mathbf{p}_{2},t)}\n\\left(1+\\eta \\frac{f_{III}(\\mathbf{p}_{3},t)}{2 h^{-3}}\\right) \\left(1+\\eta \\frac{f_{IV}(\\mathbf{p}_{4},t)}{2 h^{-3}}\\right) \\biggr],\n\\end{multline}\nwhere $\\eta$ is defined through\n\\begin{equation}\n    \\eta=\n    \\begin{cases}\n        +1, & \\text{for Bose-Einstein statistics,}\\\\\n \t  -1, & \\text{for Fermi-Dirac statistics,}\\\\\n         0, & \\text{for Maxwell-Boltzmann statistics.}\n    \\end{cases}\n\\end{equation}\nWhen incoming or outgoing particles coincide ($I=II$ and\/or $III=IV$) quantum indistinguishability gives the term $\\frac12$ in front of the corresponding\noutcome and income terms, see e.g. \\cite{1973rela.conf....1E}, \\cite{groot1980relativistic}.\n\nFor numerical evaluation phase space is divided into zones, in calculations we approximate continuous DF by its averaging over each zone (see Eq.~\\eqref{Ydef}). For this purpose we add an integral over $\\mathbf{p}_{1}$ in UU equation \\ref{uustfI}, the RHS of resulting equation will have the same form for each particle type differing only by sign and its integration limits:\n\\begin{multline}\\label{uustfIint}\n\\pm \\int d^{3} \\mathbf{p}_{1} d^{3} \\mathbf{p}_{2} d^{3} \\mathbf{p}_{3} d^{3} \\mathbf{p}_{4}  \\\\\n\\shoveleft{\\ \\times\\biggl[w(p_3,p_4;p_1,p_2)\nf_{III}(\\mathbf{p}_{3},t) f_{IV}(\\mathbf{p}_{4},t)\n\\left( 1+\\eta \\frac{f_{I}(\\mathbf{p}_{1},t)}{2 h^{-3}}\\right)\\left( 1+\\eta \\frac{f_{II}(\\mathbf{p}_{2},t)}{2 h^{-3}}\\right)}\\\\\n\\shoveleft{\\quad -w(p_1,p_2;p_3,p_4)\nf_{I}(\\mathbf{p}_{1},t)f_{II}(\\mathbf{p}_{2},t)}\n\\left(1+\\eta \\frac{f_{III}(\\mathbf{p}_{3},t)}{2 h^{-3}}\\right) \\left(1+\\eta \\frac{f_{IV}(\\mathbf{p}_{4},t)}{2 h^{-3}}\\right) \\biggr],\n\\end{multline}\nwhere the upper sign corresponds to particle type I and II and the lower sign corresponds to particle type III and IV.\n\nEvaluating collision integral in the framework of reaction-oriented approach we use one expression \\eqref{uustfIint} and distribute the result to each particle type according to integration limits in \\eqref{uustfIint}.\n\nIn this paper we deal with all two-particle QED processes in relativistic plasma, which are collected in Table \\ref{2pptable}. The exact QED matrix elements for these processes can be found in the standard textbooks, e.g. \\cite{2003spr..book.....G,1982els..book.....B}.\\\\\n\nLet us make a notice connected with a conservation laws for interacting particles. Energy and momentum conservations read\n\\begin{gather}\\label{2pconslaws}\n\\hat\\varepsilon = \\varepsilon_{1} + \\varepsilon_{2}=\\varepsilon_{3} +\n\\varepsilon_{4},\\qquad \\hat{\\mathbf p}=\\mathbf p_{1} + \\mathbf p_{2}=\\mathbf p_{3} + \\mathbf\np_{4}.\n\\end{gather}\nThere are 4 delta-functions in Eq.~\\eqref{transw} representing conservation of energy and momentum \\eqref{2pconslaws}. Three integrations over momentum of particle $III$ can be performed immediately\n\\begin{equation}\n    \\int d\\mathbf p_3 \\delta^3(\\mathbf{p}_{1}+\\mathbf{p}_{2}-\\mathbf{p}_{3}-\\mathbf{p}_{4})\\longrightarrow 1.\n\\end{equation}\nIn the integration over energy $\\varepsilon_{4}$ of particle $IV$ it is necessary to take into account that $\\varepsilon_3$ is now a function of energy and angles of particles $I$ and $II$, as well as angles of particle $IV$, so we have\n\\begin{equation}\n    \\int d\\varepsilon_4 \\delta(\\varepsilon_{1}+\\varepsilon_{2} -\\varepsilon_{3}-\\varepsilon_{4}) \\longrightarrow\n        \\frac{1}{1-(\\beta_{3}\/\\beta_{4})\\mathbf n_{3}\\cdot\\mathbf n_{4}},\n\\end{equation}\nwhere $\\mathbf n=\\mathbf p\/p$ is the unit vector in the direction of particle momentum, $p=|\\mathbf p|=\\sqrt{(\\varepsilon\/c)^2 -m^2c^2}$ is the absolute value of particle momentum, $\\beta=pc\/\\varepsilon$, and a dot denotes scalar product of 3-vectors.\\\\\nWe use spherical coordinates in momentum space: $\\{\\varepsilon, \\mu ,\\phi \\}$, $\\mu=\\cos\\vartheta$, where $\\varepsilon$ is the particle energy, and $\\vartheta$ and $\\phi$ are polar and azimuthal angles, respectively. Then energy and angles of particle $III$ and energy of particle $IV$ follow from energy and momentum conservation \\eqref{2pconslaws} and relativistic energy-momentum relation, namely\n\\begin{gather}\\label{KinematicsStart}\n    \\varepsilon_4=c\\sqrt{p_4^2+m_{IV}^2c^2},\\qquad \\mathbf p_4=p_4 \\mathbf n_4,\\\\\n    \\varepsilon_3=\\hat\\varepsilon-\\varepsilon_4,\\qquad\n    \\mathbf p_3=\\hat{\\mathbf p}-\\mathbf p_4,\\nonumber\\\\\n    \\mathbf n_3=\\frac{\\mathbf p_3}{p_3},\\qquad \\mathbf n_4=\\frac{\\mathbf p_4}{p_4},\\\\\n    \\mathbf{n}_3=\\left(\\sqrt{1-{\\mu_3}^2}\\cos\\phi_3, \\sqrt{1-{\\mu_3}^2}\\sin\\phi_3,\n    \\mu_3\\right),\\\\\n    \\mathbf{n}_4=\\left(\\sqrt{1-\\mu_4^2}\\cos\\phi_4, \\sqrt{1-\\mu_4^2}\\sin\\phi_4,\n    \\mu_4\\right),\\\\\n    p_4=\\frac{AB\\pm\\sqrt{A^2+4m_{IV}^2c^2(B^2-1)}}{2(B^2-1)},\\label{Kinematics}\\\\\n    A=\\frac{c}{\\hat\\varepsilon}[\\hat{p}^2+(m_{III}^2-m_{IV}^2)c^2]\n    -\\frac{\\hat\\varepsilon}{c},\\qquad\n    B=\\frac{c}{\\hat\\varepsilon}\\mathbf n_4\\cdot\\hat{\\mathbf p}.\\nonumber\n\\\\\n    \\mathbf n_{3}\\cdot\\mathbf n_{4}=\\mu_3\\mu_4+\\sqrt{(1-\\mu_3^2)(1-\\mu_4^2)}\\cos(\\phi_3-\\phi_4).\n    \\label{KinematicsEnd}\n\\end{gather}\n\nThen we introduce these relations into collision integral \\eqref{uustfIint}. We also use spherical symmetry in momentum space to fix angles of the particle $I$: $\\mu_1=1,\\phi_1=0$, and to perform the integration over azimuthal angle of particle $II$: $\\int d\\phi_2\\longrightarrow2\\pi$, setting $\\phi_2=0$ in the remaining integrals. Then final expression for collision integral is\n\\begin{multline}\\label{stfI1}\n\\text{St}f_I=\\frac{\\hbar^2}{32\\pi}\\int d\\varepsilon_2 d\\mu_2 \\ d\\mu_4 d\\phi_4\n\\frac{p_2 p_4|M_{fi}|^2}{\\varepsilon_{1}\\varepsilon_{3}[1-(\\beta_{3}\/\\beta_{4})\\mathbf n_{3}\\cdot\\mathbf n_{4}]}\\\\\n\\shoveleft{\\ \\times\\biggl[f_{III}(\\varepsilon_{3},t) f_{IV}(\\varepsilon_{4},t) \\left( 1+\\eta \\frac{f_{I}(\\varepsilon_{1},t)}{2 h^{-3}}\\right)\n\\left( 1+\\eta \\frac{f_{II}(\\varepsilon_{2},t)}{2 h^{-3}}\\right)}\\\\\n\\shoveleft{\\quad -f_{I}(\\varepsilon_{1},t)f_{II}(\\varepsilon_{2},t)}\\left(1+\\eta\\frac{f_{III}(\\varepsilon_{3},t)}{2 h^{-3}}\\right)\n\\left(1+\\eta\\frac{f_{IV}(\\varepsilon_{4},t)}{2 h^{-3}}\\right) \\biggr].\n\\end{multline}\n\nFor numerical integration, however, another expression is proved useful\n\\begin{multline}\\label{stfI2}\n\\int d\\varepsilon_1 \\text{St}f_I =\\frac{\\hbar^2}{32\\pi}\\Biggl[ \\int d\\varepsilon_3\\ d\\varepsilon_4 d\\mu_4 \\ d\\mu_2 d\\phi_2\n\\frac{p_2 p_4|M_{fi}|^2}{\\varepsilon_{1}\\varepsilon_{3}\\,[1-(\\beta_{1}\/\\beta_{2})\\mathbf n_{1}\\cdot\\mathbf n_{2}]}\\\\\n    \\times f_{III}(\\varepsilon_{3},t) f_{IV}(\\varepsilon_{4},t)\\left(1+\\eta\\frac{f_{I}(\\varepsilon_{1},t)}{2 h^{-3}}\\right) \\left(1+\\eta \\frac{f_{II}(\\varepsilon_{2},t)}{2 h^{-3}}\\right)\\\\\n-\\int d\\varepsilon_1 \\ d\\varepsilon_2 d\\mu_2 \\ d\\mu_4 d\\phi_4\n\\frac{p_2 p_4|M_{fi}|^2}{\\varepsilon_1 \\varepsilon_{3}\\,[1-(\\beta_{3}\/\\beta_{4})\\mathbf n_{3}\\cdot\\mathbf n_{4}]}\\\\\n\\shoveleft{\\ \\times f_{I}(\\varepsilon_1,t)f_{II}(\\varepsilon_{2},t)}\\left(1+\\eta\\frac{f_{III}(\\varepsilon_{3},t)}{2 h^{-3}}\\right) \\left(1+\\eta \\frac{f_{IV}(\\varepsilon_{4},t)}{2 h^{-3}}\\right)\\Biggr],\n\\end{multline}\nwhere the first term is expressed in the form ready for replacement by the sum over incoming particles $III$ and $IV$. In this term $\\varepsilon_1,\\mu_1,\\phi_1,\\varepsilon_2$ are given by relations \\eqref{Kinematics} with indices exchange $1\\leftrightarrow3$, $2\\leftrightarrow4$, $I\\leftrightarrow III$, $II\\leftrightarrow IV$.\n\nThis collision integral of any of two-particle processes is a four-dimensional integral in momentum space. In Sec.~\\ref{ch4} we show how such integral is computed numerically on finite grid.\n\nHere we note that in the case of homogeneous and isotropic pair plasma one has to satisfy only two conservation laws, namely of energy and particle number. Momentum conservation should be added for nonisotropic in momentum space DF, see e.g.~\\cite{BENEDETTI2013206}. In our method electric charge is conserved due to conservation of particles because we use between cell interpolation for the same kind of particles described in the next Section. \n\\section{Numerical Scheme}\\label{ch4}\n\n\nThe phase space is divided in zones. The zone $\\Omega^\\alpha_{a,j,k}$ for particle specie $\\alpha$ corresponds to energy $\\varepsilon_a$, cosine of polar angle $\\mu_j$ and azimuthal angle $\\phi_k$, where indices run in the following ranges $1\\leq a\\leq a_{\\mathrm{max}}$, $1\\leq j\\leq j_{\\mathrm{max}}$, and $1\\leq k\\leq k_{\\mathrm{max}}$. The zone boundaries are $\\varepsilon_{a\\mp1\/2}$, $\\mu_{j\\mp1\/2}$, $\\phi_{k\\mp1\/2}$. The length of the $a$-th energy zone $\\Omega^\\alpha_{a}$ is $\\Delta\\varepsilon_{a} \\equiv \\varepsilon_{a+1\/2} - \\varepsilon_{a-1\/2}$. On finite grid $f_\\alpha$ does not depend on $\\mu$ and $\\phi$, and number density of particle $\\alpha$ in zone $a$ is\n\\begin{multline}\\label{Ydef}\nY^\\alpha_{a}(t)=4\\pi\\int_{\\varepsilon_{a-1\/2}}^{\\varepsilon_{a+1\/2}}c^{-3}\\varepsilon\\sqrt{\\varepsilon^2-m_\\alpha^2c^4}\nf_\\alpha(\\varepsilon,t)d\\varepsilon\\\\= 4\\pi c^{-3}\\varepsilon_a\\sqrt{\\varepsilon_a^2-m_\\alpha^2c^4} f_\\alpha(\\varepsilon_a,t)\\Delta\\varepsilon_a.\n\\end{multline}\nIn this variables discretized U-U equation for particle $I$ and energy zone $a$ reads\n\\begin{equation}\\label{discrBoltzmann}\n    \\frac{d Y^\\alpha_a(t)}{dt}=\\sum \\left[\\text{St}^+Y^{I}_a -\\text{St}^-Y^{I}_a \\right],\n\\end{equation}\nwhere the sum is taken over all processes involving particle $I$. Coefficients of particles income and outcome on the grid are obtained by integration of \\eqref{stfI2} for two-particle processes over the zone. The corresponding integrals are replaced by sums on the grid.\nFor instance, coefficient of particle $I$ outcome in two-particle process \\eqref{2pdir} is\n\\begin{multline}\\label{YrateI}\n    \\text{St}^-Y^{I}_a=\\frac{\\hbar^2c^{4}}{8(4\\pi)^2} \\sum_{b,j,s,k}\\Delta\\mu^{II}_{j}\\Delta\\mu^{IV}_{s} \\Delta\\phi^{IV}_{k} |M_{fi}|^2 \\frac{p_4}{\\varepsilon_{3}[1-(\\beta_{3}\/\\beta_{4})\\mathbf n_{3}\\cdot\\mathbf n_{4}]}\\times \\\\\n\\times  \\frac{Y^{I}_{a}(t)}{\\varepsilon^{I}_{a}}\\frac{Y^{II}_{b}(t)}{\\varepsilon^{II}_{b}}\n    \\times\\left[1+\\eta \\frac{Y^{III}_c(t)}{\\bar Y^{III}_c}\\right]\\left[1+\\eta \\frac{Y^{IV}_d(t)}{\\bar Y^{IV}_d}\\right],\n\\end{multline}\nand coefficient of particle $I$ income in process (\\ref{2pinv}) from integration of \\eqref{stfI2} is\n\\begin{multline}\\label{YrateII}\n    \\text{St}^+Y^{I}_a=\\frac{\\hbar^2c^{4}}{8(4\\pi)^2} \\sum_{c,d,j,s,k} C_a(\\varepsilon_1) \\Delta\\mu^{IV}_{j}\\Delta\\mu^{II}_{s} \\Delta\\phi^{II}_{k} |M_{fi}|^2 \\frac{p_2}{\\varepsilon_{1}[1-(\\beta_{1}\/\\beta_{2})\\mathbf n_{1}\\cdot\\mathbf n_{2}]} \\times \\\\\n\\times  \\frac{Y^{III}_{c}(t)}{\\varepsilon^{III}_{c}}\\frac{Y^{IV}_{d}(t)}{\\varepsilon^{IV}_{d}}\n    \\times\\left[1+\\eta \\frac{Y^{I}_a(t)}{\\bar Y^{I}_a}\\right]\\left[1+\\eta \\frac{Y^{II}_b(t)}{\\bar Y^{II}_b}\\right],\n\\end{multline}\nwhere $\\bar Y^\\alpha_{a}=4\\pi\\int_{\\varepsilon_{a-1\/2}}^{\\varepsilon_{a+1\/2}}\n        c^{-3}\\varepsilon\\sqrt{\\varepsilon^2-m_\\alpha^2c^4}\\ (2 h^{-3})d\\varepsilon\n        = 8\\pi(hc)^{-3}\\varepsilon_a\\sqrt{\\varepsilon_a^2-m_\\alpha^2c^4} \\Delta\\varepsilon_a,$\nand\n\\begin{equation}\\label{csol}\n    C_a(\\varepsilon_1)=\n    \\begin{cases}\n        \\dfrac{\\varepsilon_{a}-\\varepsilon_1}{\n            \\varepsilon_{a}-\\varepsilon_{a-1}}, &\n        \\varepsilon_{a-1} < \\varepsilon_1 < \\varepsilon_{a},\\\\[2.5ex]\n        \\dfrac{\\varepsilon_{a+1}-\\varepsilon_1}{\n            \\varepsilon_{a+1}-\\varepsilon_{a}}, &\n        \\varepsilon_{a} < \\varepsilon_1 < \\varepsilon_{a+1},\\\\[2.5ex]\n        0, & \\text{otherwise.}\n    \\end{cases}\n\\end{equation}\nIn integration of (\\ref{stfI2}) over the zone one can integrate out the $\\delta$-function $\\int \\delta(\\varepsilon_1-\\varepsilon)d\\varepsilon_1\n\\longrightarrow 1$. However, when energies of incoming particles are fixed on the grid, the energies of outgoing particles are not on the grid. Hence an interpolation \\eqref{csol} is adopted, which enforces the exact number of particles and energy conservation in each two-particle process due to redistribution of outgoing particle $\\alpha$ with energy $\\varepsilon$ over two energy zones $\\Omega^{\\alpha}_{n}, \\Omega^{\\alpha}_{n+1}$ with $\\varepsilon_{n} < \\varepsilon < \\varepsilon_{n+1}$. Further we denote this technique as particle splitting.\n\nThe redistribution of final particles should also satisfy requirements of quantum statistics. Therefore if a process occurs, when fermionic final particle should be distributed over the quantum states which are fully occupied, such process should be forbidden. Thus we introduce the Bose enhancement\/Pauli blocking coefficients in (\\ref{YrateI}) and (\\ref{YrateII}) as\n\\begin{gather}\n    \\left[1+\\eta \\frac{Y^{\\alpha}_a(t)}{\\bar Y^{\\alpha}_a}\\right]=\n    \\min\\left(1+\\eta \\frac{Y^{\\alpha}_{n}(t)}{\\bar Y^{\\alpha}_{n}},\n    1+\\eta \\frac{Y^{\\alpha}_{n+1}(t)}{\\bar Y^{\\alpha}_{n+1}}\\right).\n\\end{gather}\nThe sum over angles $\\mu_j, \\mu_s, \\phi_k$ can be found once and for all at the\nbeginning of the calculations. We then store in the program for each set of the\nincoming and outgoing particles the corresponding terms and redistribution\ncoefficients given by Eq.~(\\ref{csol}).\n\nRepresentation of discretized collisional integral for particle $I$ and energy zone $a$ in processes \\eqref{2pdir},\n\\eqref{2pinv} is\n\\begin{multline}\\label{BoltzmannFinal}\n      \\dot{Y}^I_a=\n    \\sum P_{abcd}\\times Y^{III}_{c}(t) Y^{IV}_{d}(t)\n    \\times \\left[1+\\eta \\frac{Y^{I}_{a}(t)}{\\bar Y^{I}_{a}}\\right]\n    \\left[1+\\eta \\frac{Y^{II}_{b}(t)}{\\bar Y^{II}_{b}}\\right]\\\\\n-\\sum R_{abcd}\\times Y^{I}_{a}(t) Y^{II}_{b}(t)\n    \\times \\left[1+\\eta \\frac{Y^{III}_{c}(t)}{\\bar Y^{III}_{c}}\\right]\n    \\left[1+\\eta \\frac{Y^{IV}_{d}(t)}{\\bar Y^{IV}_{d}}\\right],\n\\end{multline}\nwhere constant coefficients $P,R$ are obtained from the summation over angles in the sums \\eqref{YrateI}, \\eqref{YrateII}. In the nondegenerate case of Boltzmann equation the indices $b$ in the first sum and $c,d$ in the second sum become dummy, equation \\eqref{BoltzmannFinal} can be partially summed and takes the following form:\n\\begin{gather}\\label{BoltzmannFinal2}\n      \\dot{Y}^I_a=\n    \\sum P_{acd}\\times Y^{III}_{c}(t) Y^{IV}_{d}(t)\n    -\\sum R_{ab}\\times Y^{I}_{a}(t) Y^{II}_{b}(t),\n\\end{gather}\nwhere $P_{acd}=\\sum_{b} P_{abcd},\\ R_{ab}=\\sum_{c,d} R_{abcd}$. The last quantity is essentially \\emph{reaction rate} usually used for description of binary processes and simply connected to the total cross section.\n\nThe full U-U equation \\eqref{BoltzmannFinal} contains similar sums for all processes from Table \\ref{2pptable}. Each individual term in these sums appears in the system of discretized equations four times in emission and absorption coefficients for each particle entering a given process. Then each term can be computed only once and added to all\ncorresponding sums, that is the essence of our \\emph{\"reaction-oriented\" approach} \\cite{ivanphd,2015AIPC.1693g0007S}.\n\nWe point out that unlike classical Boltzmann equation for binary interactions\nsuch as scattering, more general interactions are typically described by four\ncollision integrals for each particle that appears both among incoming and\noutgoing particles. \n\\section{Numerical results}\\label{ch5}\n\nThe results of numerical calculations are presented below. As all known analytical expressions for reaction rates in relativistic plasma concern nondegenerate case, here we compare our results for collision integral to that of nondegenerate plasma. Notice that for Coulomb scattering we have implemented a cutoff scheme based on minimal scattering angle \\cite{2009PhRvD..79d3008A,1988A&A...191..181H}.\n\nWe consider mildly relativistic plasma with\n\\begin{equation}\n0.01\\lesssim e \\lesssim100,\n\\end{equation}\nwhere $e$ is particle kinetic energy divided by electron rest energy, this range contains both relativistic and non-relativistic domains. The upper limit is chosen to avoid thermal production of other particles such as neutrinos and muons, while the lower limit is required to have sufficient pair density.\n\nWe introduce logarithmic energy grid with $a_{max}=40$ nodes for all calculations and different homogeneous grids for angular variables, $\\phi$-grid is 2 time denser then $\\mu$-grid (typically $\\mu$-grid contains $j_{max}=64$ nodes). To compare results with known analytical expressions we use definition of angle-averaged reaction rate per pair of particles \n\\begin{equation}\\label{sv1}\n\\overline{v\\sigma}(e_1,e_2)=\\int_{-1}^{+1} \\frac{d\\mu_{2}}2 \\int_{\\vec{p}_3,\\vec{p}_4} v d\\sigma,\n\\end{equation}\nand angle-averaged emissivity per pair of particles \n\\begin{equation}\\label{sv2}\n\\overline{v\\frac{d\\sigma}{de_3}}(e_1,e_2,e_3)=\\int_{-1}^{+1} \\frac{d\\mu_{2}}2 \\int_{\\vec{p}_3,\\vec{p}_4} v \\frac{d\\sigma}{de_3},\n\\end{equation} introduced by Svensson~\\cite{1982ApJ...258..321S}, where $d\\sigma$ is given by standard definition \\eqref{dsigma} and we have used spherical symmetry as described before Eq.~\\eqref{stfI1}. We use Coppi \\& Blandford \\cite{1990MNRAS.245..453C} analytical expressions (2.3), (3.2), (4.3) for $\\overline{v\\sigma}$, which corresponds to $R_{ab}$. Svensson \\cite{1982ApJ...258..321S} formula (55), Peer \\& Waxman \\cite{2005ApJ...628..857P} formulae (19, 28) are used for quantity $\\overline{v\\frac{d\\sigma}{de}}$, which corresponds to $P_{abc}\/\\Delta e_a$.\n\nTo compare numerical results with analytical ones, we introduce the following quantity for each process\n\\begin{equation}\nQ=\\frac{1}{a_\\text{max}^2}\\sum_{a,b} |R_{ab}\/\\overline{v\\sigma}(e_a,e_b)-1|,\n\\end{equation}\nexpressing average relative deviation of numerical results from analytical ones for all energy grid nodes. Table \\ref{qtab} presents values of $Q$ for selected number of angular grid nodes. It is evident that the relative error decreases with increasing of number of angular grid nodes reaching about 1~\\% with 128 nodes. This demonstrates convergence of numerical results to the corresponding analytical ones.\n\n\\begin{table}[tbp]\n\\caption{Values of $Q$ for selected number of angular grid nodes ($a_{max}=40, k_{max}=2j_{max}$). \\label{qtab}} \\center\n\\begin{tabular}[c]{|l|l|l|l|l|}\n\\hline\nProcess\/$j_{max}$ & 16 & 32 & 64 & 128 \\\\\n\\hline\nCS & 0.0855 & 0.0403 & 0.0207 & 0.0145\\\\\n\\hline\nPA & 0.0231 & 0.00693 & 0.00313 & 0.00138\\\\\n\\hline\nPC & 0.146 & 0.0657 & 0.0303 & 0.0116\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nBelow we present some representative plots for the reaction rates of all reactions together with analytical curves (where they are known). Energy is measured in electron rest energy units. Presented results reproduce both nonrelativistic and relativistic energy cases. All computations were carried on Intel Core i3-7100 CPU @3.90 GHz processor using one processor core. The code is written in C and compiled in Windows 7 environment with Microsoft Visual Studio 2015 in fully optimized x64 mode. Computation time of initial angular integration of collision integrals for each reaction from Table~\\ref{2pptable} is shown in Table~\\ref{timing}. It shows even lower than expected $O(j_{max}^3)$ behaviour due to kinematic cuts on the phase space of reactions.\n\n\\begin{table}[tbp]\n\\caption{CPU time (in seconds) of each reaction initial angular integration for selected number of angular grid nodes ($a_{max}=40, k_{max}=2j_{max}$), and its exponent of computational cost $O(j_{max}^n)$. \\label{timing}} \\center\n\\begin{tabular}[c]{|l|l|l|l|l||l|}\n\\hline\nProcess\/$j_{max}$ & 16 & 32 & 64 & 128 & n \\\\\n\\hline\nCS & 2.215 & 14.48 & 113.2 & 590.1 & 2.7 \\\\\n\\hline\nPA & 2.106 & 14.73 & 100.2 & 543.1 & 2.7 \\\\\n\\hline\nPC & 0.531 & 3.619 & 28.82 & 223.2 & 2.9 \\\\\n\\hline\nMS & 2.418 & 16.87 & 130.5 & 1030 & 2.9 \\\\\n\\hline\nBS & 3.354 & 22.74 & 178.6 & 1113 & 2.8 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nCompton scattering presents well-known challenge for numerical treatment as all the analytical formulas for scattering rate behave badly numerically in different parameter areas, see e.g. \\cite{2005ApJ...628..857P, 2009A&A...506..589B}. We easily bypass this difficulty as we numerically integrate well-behaved differential cross-section, as one can see for non-relativistic regime in Fig.~\\ref{CS_D} and for relativistic regime in Fig.~\\ref{CS_I}. Figure~\\ref{CS_D} presents analytic photon spectrum for the reaction $\\gamma+e^{\\pm}\\rightarrow\\gamma'+e^{\\pm}{}'$ as solid line and our numerical results shown by dots. Overall there is good agreement between numerical and analytical results. Small deviations in high-energy of the spectrum arise from leakage of the particles to kinematically forbidden area at the boundary of energy zones. Due to particle splitting (between cell interpolation) some final paricles would be placed on a grid node, that is kinematically forbidden, and it is indeed the case of Fig.~\\ref{CS_D}. To show this effect we enlarge the plot  range especially on this figure. On the other spectrum figures these points appear to be outside the presented plot range. There the maximum allowed photon energy is $e_{max}=0.291$, but we have particles of energies from 0.275 up to $e_{max}$ that are splitted between energy zones of 0.275 and 0.327 -- the second is kinematically forbidden.\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=70mm]{fig1}\n\\caption{Compton scattered-photon distribution $\\overline{v\\frac{d\\sigma}{de_\\gamma'}}(e_\\gamma,e_{\\pm},e_\\gamma ').$}\\label{CS_D}\n\\end{figure}\nFigure~\\ref{CS_I} shows the total reaction rate of the same process. Again there is good agreement between numerical and analytical results. Small discrepancy arises from truncation of reactions where final particles get out of the grid to higher or lower energies. As a result numerical reaction rates are systematically lower than analytic ones.\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=70mm]{fig2}\n\\caption{Compton scattering rate $\\overline{v\\sigma}(e_\\pm,e_\\gamma)$.}\\label{CS_I}\n\\end{figure}\n\nAnnihilation photon spectrum for reaction $e^{+}+e^{-}\\rightarrow\\gamma+\\gamma'$ is illustrated in Fig.~\\ref{AN_D} and total reaction rate in this process in Fig.~\\ref{AN_I}. Figure~\\ref{AN_D} shows that the method is able to accurately reproduce the spectrum of annihilation photons in the range of more than two orders of magnitude. Reaction truncation errors, hardly seen at Fig.~\\ref{AN_I}, are much lower for annihilation as low-energy photons are rare in this process.\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=70mm]{fig3}\n\\caption{Distribution for pair annihilation $\\overline{v\\frac{d\\sigma}{de_\\gamma}}(e_\\gamma,e_{+},e_{-}).$}\\label{AN_D}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=70mm]{fig4}\n\\caption{Pair annihilation rate $\\overline{v\\sigma}(e_{+},e_{-}).$}\\label{AN_I}\n\\end{figure}\n\nBalance between pair creation and annihilation represent an independent test for the numerical scheme, as it is not automatically satisfied due to different numerical treatment of incoming and outgoing particles in the reactions. Pair creation spectra for reaction $\\gamma_1+\\gamma_2\\rightarrow e^{+}+e^{-}$ are reproduced well, see Fig.~\\ref{PC_D}, as well as total reaction rates, see Fig.~\\ref{PC_I}. Numerical balance can be checked by the form of particle distributions in numerical equilibrium, that was verified to be within 5~\\% of corresponding Boltzmann distributions.\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=70mm]{fig5}\n\\caption{Distribution for pair production $\\overline{v\\frac{d\\sigma}{de_\\pm}}(e_{\\gamma_1},e_{\\gamma_2},e_{\\pm}).$}\\label{PC_D}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=70mm]{fig6}\n\\caption{Pair production rate $\\overline{v\\sigma}(e_{\\gamma_1},e_{\\gamma_2}).$}\\label{PC_I}\n\\end{figure}\n\nFor completeness we present also the results for M{\\o}ller and Bhabha scattering, they show that these processes are indeed dominant for electrons and positrons in relativistic plasma, compare Figs.~\\ref{BS_I}, \\ref{MS_I} with Figs.~\\ref{CS_I}, \\ref{AN_I}, \\ref{PC_I}.\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=70mm]{fig7}\n\\caption{Bhabha scattering rate $\\overline{v\\sigma}(e_+,e_-).$}\\label{BS_I}\n\\end{figure}\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=70mm]{fig8}\n\\caption{M{\\o}ller scattering rate $\\overline{v\\sigma}(e_{\\pm},e_{\\pm}).$\\label{MS_I}}\n\\end{figure}\n\n\n\\begin{figure}[ptb!]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=70mm]{figron1}\n\\end{tabular}\n\\caption{Time evolution of energy density in components of electron-positron-photon plasma: photon energy density (blue), electron\/positron energy density (orange), total energy density (green). Solid lines correspond to Boltzmann case, dashed lines correspond to Uehling-Uhlenbeck case.}\\label{evolro}\n\\end{figure}\n\\begin{figure}[ptb!]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=70mm]{figron2}\n\\end{tabular}\n\\caption{Time evolution of number density in components of electron-positron-photon plasma: photon concentration (blue), electron\/positron concentration (orange), total concentration (green). Solid lines correspond to Boltzmann case, dashed lines correspond to Uehling-Uhlenbeck case. Note the difference in the final pair and photon density due to rest mass of elecron\/positron (in both cases) and difference in statistics (for U-U case).}\\label{evoln}\n\\end{figure}\n\n\\begin{figure}[ptb!]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=70mm]{figsp10} &\\includegraphics[width=70mm]{figsp11}\n\\end{tabular}\n\\caption{Final energy spectra at $t_{final}=10^{-14}$~s. Solid lines are equilibrium Boltzmann and Bose-Einstein\/Fermi-Dirac fits of numerical results: photon Boltzmann energy spectrum (blue), photon Bose-Einstein energy spectrum (cyan), pairs Boltzmann energy spectrum (orange), pairs Fermi-Dirac energy spectrum (red).}\\label{evolge}\n\\end{figure}\n{Finally, we present a time evolution of energy density and concentration to demonstrate the difference between the classical Boltzmann and U-U equations. Both systems \\eqref{BoltzmannFinal} and \\eqref{BoltzmannFinal2} were solved numerically with the same initial conditions under $a_{max}=60, j_{max}=64, k_{max}=2j_{max}$ and using Gear's method for resulting stiff ODE system \\cite{1976oup..book.....H}.\n\nThe energy spectrum $d\\rho\/d\\varepsilon$ is shown instead of the distribution function $f$, that are related by $d\\rho\/d\\varepsilon=4\\pi |\\mathbf{p}|\\varepsilon^2 c^{-2}f$. We chose an initial state without electrons and positrons but with photons only, initial spectrum has a power-law shape $d\\rho\/d\\varepsilon=a (\\varepsilon\/\\varepsilon_0)^b$, with $a=3.63\\times10^{28}\\text{ cm}^{-3}$ and $b=-0.438$, $\\varepsilon_0=1$~erg, between $e=0.157$ and $e=157$. The initial spectrum corresponds to a total energy density $\\rho=4.10\\times 10^{26}\\text{ erg cm}^{-3}$ and a total number density of particles $n=8.15\\times 10^{31}\\text{ cm}^{-3}$. In general, initial spectrum can have an arbitrary shape and thermalization process transforms it to an equilibrium form. Fig.~\\ref{evolge} represents energy spectra at final equilibrium state. They attain corresponding shapes of Boltzmann and Bose-Einstein\/Fermi-Dirac with some deviations in high-energy tails that are attributed to reaction truncation errors described before. We note that total energy and number densities do not change in time due to particle splitting applied, this feature does not depend on a form of a system of equations or a type of numerical ODE solver.} \\\\\n\n\\section{Conclusions}\\label{ch6}\n\nIn this paper, we propose a new numerical method to accurately calculate Uehling\u2013Uhlenbeck collision integral for two-particle interactions in relativistic plasma. Exact energy and particle number conservation laws are achieved by using interpolation scheme \\eqref{csol}. After calculation of collision integral discretized Uehling\u2013Uhlenbeck equations transforms into system of ODEs, which can be treated by various methods suitable to solve stiff ODEs. The method admits parallelization on GPU\/CPU. Improvement in computation time with respect to previous work is achieved. Our reaction-oriented approach can be easily applied to any other types of particles and any other binary interactions, for instanse, weak interactions of neutrinos or electromagnetic ones of protons.\nGeneralization of the proposed method for triple interactions is straightforward.\n\nOur results show that reaction rates in relativistic plasma are well reproduced with moderate number of grid nodes in energy and angles (see Figures and Table \\ref{2pptable}) both for non-relativistic and relativistic particle energies. This allows development of an efficient method of solution for relativistic Uehling\u2013Uhlenbeck equation. \n\n\n\\section{Acknowledgements}\nWe thank anonymous referees for their remarks which improved the presentation of our results.\n\\newpage\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nIn this paper we give very concrete applications of \ncontinuous logic in group theory. \nWe consider classes of (locally compact) metric groups \nwhich can be also viewed as (reducts of) axiomatizable \nclasses of continuous structures. \nThen by some standard logical tricks we obtain \nseveral interesting consequences. \nUsually we concentrate on classes which are \ntypical in geometric group theory.  \\parskip0pt \n\nThe following notion is one of the main objects of the paper. \nA class of groups $\\mathcal{K}$ is called {\\em bountiful} if \nfor any pair of infinite groups $G\\le H$ with $H\\in \\mathcal{K}$ \nthere is $K\\in \\mathcal{K}$ such that $G\\le K\\le H$ and $|G|=|K|$. \nIt was introduced by Ph.Hall and was studied in \npapers \\cite{KM}, \\cite{phillips}, \\cite{sabbagh} and \\cite{thomas}. \nSome easy logical observations from \\cite{KM} show that if \n$\\mathcal{K}$ is a reduct of a class axiomatizable in \n$L_{\\omega_1 \\omega}$ then $\\mathcal{K}$ is bountiful. \n\\parskip0pt \n\nWhen one considers topological groups, the definition of \nbountiful classes should be modified as follows. \n\\begin{definition} \nA class of topological groups $\\mathcal{K}$ is called  bountiful \nif for any pair of infinite groups $G\\le H$ with $H\\in \\mathcal{K}$ \nthere is $K\\in \\mathcal{K}$ such that $G\\le K\\le H$ and \nthe density character of $G$ \n(i.e. the smallest cardinality of a dense subset of the space) \ncoincides with the density character of $K$. \n\\end{definition} \nWe mention paper \\cite{HHM} where similar questions were studied \nin the case of locally compact groups. \nWe will see below that under additional assumptions of metricity \nlogical tools become helpful in this class of groups.  \nWe should only replace first-order logic (or $L_{\\omega_1 \\omega}$) \nby continuous one. \nWe concentrate on negations of properties {\\bf (T)}, {\\bf FH}, \n{\\bf F}$\\mathbb{R}$ (\\cite{BHV}, \\cite{HV}) and on negations \nof boundedness properties classified in \\cite{rosendalN}. \n\nIn the final part of the paper we consider separable locally compact groups \nwhich have separably categorical continuous theory, i.e. the group \nis determined uniquely (up to metric isomorphism) by its continuous theory \nand the the density character. \nIt is interesting that some basic properties of the automorphism groups \nof such structures are strongly connected with some classes examined on \nbountifulness below. \n\n\nIn the rest of this introduction we briefly remind \nthe reader some preliminaries of continuous logic. \nThen we finish this section by some remarks on sofic groups. \n\n\\bigskip \n\n\n\\paragraph{Continuous structures.} \n\nWe fix a countable continuous signature \n$$\nL=\\{ d,R_1 ,...,R_k ,..., F_1 ,..., F_l ,...\\}. \n$$  \nLet us recall that a {\\em metric $L$-structure} \nis a complete metric space $(M,d)$ with $d$ bounded by 1, \nalong with a family of uniformly continuous operations on $M$ \nand a family of predicates $R_i$, i.e. uniformly continuous maps \nfrom appropriate $M^{k_i}$ to $[0,1]$.   \nIt is usually assumed that to a predicate symbol $R_i$ \na continuity modulus $\\gamma_i$ is assigned so that when \n$d(x_j ,x'_j ) <\\gamma_i (\\varepsilon )$ with $1\\le j\\le k_i$ \nthe corresponding predicate of $M$ satisfies \n$$ \n|R_i (x_1 ,...,x_j ,...,x_{k_i}) - R_i (x_1 ,...,x'_j ,...,x_{k_i})| < \\varepsilon . \n$$ \nIt happens very often that $\\gamma_i$ coincides with $id$. \nIn this case we do not mention the appropriate modulus. \nWe also fix continuity moduli for functional symbols. \nNote that each countable structure can be considered \nas a complete metric structure with the discrete $\\{ 0,1\\}$-metric. \n\nBy completeness continuous substructures of a continuous structure are always closed subsets. \n\nAtomic formulas are the expressions of the form $R_i (t_1 ,...,t_r )$, \n$d(t_1 ,t_2 )$, where $t_i$ are terms (built from functional $L$-symbols). \nIn metric structures they can take any value from $[0,1]$.   \n{\\em Statements} concerning metric structures are usually \nformulated in the form \n$$\n\\phi = 0 \n$$ \n(called an $L$-{\\em condition}), where $\\phi$ is a {\\em formula}, \ni.e. an expression built from \n0,1 and atomic formulas by applications of the following functions: \n$$ \nx\/2  \\mbox{ , } x\\dot- y= max (x-y,0) \\mbox{ , } min(x ,y )  \\mbox{ , } max(x ,y )\n\\mbox{ , } |x-y| \\mbox{ , } \n$$ \n$$ \n\\neg (x) =1-x \\mbox{ , } x\\dot+ y= min(x+y, 1) \\mbox{ , } sup_x \\mbox{ and } inf_x . \n$$ \nA {\\em theory} is a set of $L$-conditions without free variables \n(here $sup_x$ and $inf_x$ play the role of quantifiers). \n   \nIt is worth noting that any formula is a $\\gamma$-uniformly continuous \nfunction from the appropriate power of $M$ to $[0,1]$, \nwhere $\\gamma$ is the minimum of continuity moduli of $L$-symbols \nappearing in the formula. \n\nThe condition that the metric is bounded by $1$ is not necessary. \nIt is often assumed that $d$ is bounded by some rational number $d_0$. \nIn this case the (dotted) functions above are appropriately modified.  \nSometimes predicates of continuous structures map $M^n$ to some \n$[q_1 ,q_2 ]$ where $q_1 ,q_2 \\in \\mathbb{Q}$.  \n\nThe following theorem is one of the main tools of this paper. \n\\bigskip \n \n{\\bf L\\\"{o}wenheim-Skolem Theorem.} (\\cite{BYBHU}, Proposition 7.3) \n{\\em Let $\\kappa$ be an infinite cardinal number and assume \n$|L|\\le \\kappa$. \nLet $M$ be an $L$-structure and suppose $A\\subset M$ has \ndensity $\\le\\kappa$. \nThen there exists a substructure $N\\subseteq M$ containing $A$ such that \n$density(N) \\le\\kappa$ and $N$ is an elementary substructure of $M$, i.e. \nfor every $L$-formula $\\phi (x_1 ,...,x_n)$ and $a_1 ,...,a_n \\in N$ \nthe values of $\\phi (a_1 ,...,a_n )$ in $N$ and in $M$ are the same.}  \n\n\\bigskip \n\n\\begin{remark} \\label{Hofmann}\n{\\em \nIt is proved in \\cite{HHM} that for any locally compact group $G$, \nthe entire interval of cardinalities between $\\aleph_0$ and $w(G)$, \nthe weight of the group, is occupied by the weights of closed \nsubgroups of $G$.   \nWe remind the reader that the weight of a topological space $(X,\\tau )$ \nis the smallest cardinality which can be realized as \nthe cardinality of a basis of $(X,\\tau )$.  \nIf the group $G$ is metric, the weight of $G$ coincides \nwith the density character of $G$. \nThis yelds the following version of the L\\\"{o}wenheim-Skolem Theorem. }\n\n Let $G$ be a locally compact group which is a continuous structure. \nThen for any cardinality $\\kappa < density(G)$ there is \na closed subgroup $H<G$ such that $density(H)=\\kappa$ \nand $H$ is an elementary substructure of $G$. \n\\end{remark} \n\n\\begin{remark} \\label{FaHaSh} \n{\\em \nFollowing Section 4.2 of \\cite{fhs} we define a topology \non $L$-formulas relative to a given continuous theory $T$. \nFor $n$-ary formulas $\\phi$ and $\\psi$ of the same sort set \n$$ \n{\\bf d}^T_{\\bar{x}} (\\phi ,\\psi) = sup \\{ |\\phi (\\bar{a}) -\\psi (\\bar{a} )|: \\bar{a} \\in M, M\\models T\\} . \n$$ \nThe function ${\\bf d}^T_{\\bar{x}}$ is a pseudometric. \nThe language $L$ is called {\\em separable} if \nfor every $L$-theory $T$ and any tuple $\\bar{x}$ \nthe density character of ${\\bf d}^T_{\\bar{x}}$ is countable. \nBy Proposition 4.5 of \\cite{fhs} in this case for every \n$L$-model $M$ the set of all interpretations of $L$-formulas \nin $M$ is separable in the uniform topology. \nBy Corollary 4.7 of \\cite{fhs} if in the formulation of \nthe L\\\"{o}wenheim-Skolem theorem we replace the assumption \n$|L|\\le \\kappa$ by the condition that $L$ is separable \nthen the statement also holds for $\\kappa =\\aleph_0$. \nThis will be applied in Section 2.  \n}\n\\end{remark} \n\n\n\n\n\n\n\n\n\nDefinability in continuous structures is introduced as follows. \n\n\\begin{definition} \nLet $A\\subseteq M$. \nA predicate $P:M^n \\rightarrow [0,1]$ is definable in $M$ over $A$\nif there is a sequence $(\\phi_k (x) :k\\ge 1 )$ of $L(A)$-formulas \nsuch that predicates interpreting $\\phi_k (x)$ in $M$ \nconverge to $P(x)$ uniformly in $M^n$. \n\\end{definition} \n\nWe define the automorphism group $Aut(M)$ of $M$ to be the subgroup \nof $Iso (M,d)$ consisting of all isometries preserving \nthe values of atomic formulas. \nIt is easy to see that $Aut(M)$ is a closed subgroup with \nrespect to the pointwise convergence topology on $Iso (M,d)$. \n\nThe following statement is Corollary 9.11 of \\cite{BYBHU}. \n\\begin{quote} \nLet $M$ be an $L$-structure with $A\\subseteq M$ and suppose \n$P:M^n \\rightarrow [0,1]$ is a predicate. \nThen $P$ is definable in $M$ over $A$ if and only if \nwhenever $(N,Q)$ is an elementary extension of $(M,P)$, \nthe predicate $Q$ is invariant under all automorphisms of $N$ \nthat leave $A$ fixed pointwise. \n\\end{quote}  \n\nA tuple $\\bar{a}$ from $M^n$ is {\\em algebraic} in $M$ over \n$A$ if there is a compact subset $C\\subseteq M^n$ such that \n$\\bar{a}\\in C$ and the distance predicate $dist(\\bar{x},C)$ \nis definable in $M$ over $A$. \nLet $acl(A)$ be the set of all elements algebraic over $A$. \nIn continuous logic the concept of algebraicity is \nparallel to that in traditional model theory \n(see Section 10 of \\cite{BYBHU}).  \n\nFor every $c_1 ,...,c_n \\in M$ and $A\\subseteq M$ \nwe define the $n$-type $tp(\\bar{c}\/A)$ of $\\bar{c}$ over $A$ \nas the set of all $\\bar{x}$-conditions with parameters from $A$ \nwhich are satisfied by $\\bar{c}$ in $M$.  \nLet $S_n (T_A )$ be the set of all $n$-types over $A$ \nof the expansion of the theory $T$ by constants from $A$. \nThere are two natural topologies on this set. \nThe {\\em logic topology} is defined by the basis consisting of \nsets of types of the form $[\\phi (\\bar{x})<\\varepsilon ]$, \ni.e. types containing some $\\phi (\\bar{x})\\le \\varepsilon'$ with \n$\\varepsilon '<\\varepsilon$.   \nThe logic topology is compact. \n\nThe $d$-topology is defined by the metric \n$$\nd(p,q)= inf \\{ max_{i\\le n} d(c_i ,b_i )| \\mbox{ there is a model } M \\mbox{ with } M\\models p(\\bar{c})\\wedge q(\\bar{b})\\}. \n$$ \nBy Propositions 8.7 and 8.8 of \\cite{BYBHU} the $d$-topology is finer \nthan the logic topology and $(S_n (T_A ),d)$ is a complete space. \n\n\n\\paragraph{Separable categoricity.} \n\nA theory $T$ is {\\em separably categorical} if any \ntwo separable models of $T$ are isomorphic. \nBy Theorem 12.10 of \\cite{BYBHU} a complete theory $T$ is separably \ncategorical if and only if for each $n>0$, every $n$-type $p$ is principal. \nThe latter means that for every model $M\\models T$, the predicate \n$dist(\\bar{x},p(M))$ is definable over $\\emptyset$. \n\nAnother property equivalent to separable categoricity states that \nfor each $n>0$, the metric space $(S_n (T),d)$ is compact.  \nIn particular for every $n$ and every $\\varepsilon$ there is \na finite family of principal $n$-types $p_1 ,...,p_m$ so that \ntheir $\\varepsilon$-neighbourhoods cover $S_n(T)$. \n\nIn first order logic a countable structure $M$ is \n$\\omega$-categorical if and only if $Aut(M)$ is \nan {\\em oligomorphic} permutation group, i.e. \nfor every $n$, $Aut(M)$ has finitely many orbits \non $M^n$. \nIn continuous logic we have the following modification.  \n\n\\begin{definition} \nAn isometric action of a group $G$ on a metric space $({\\bf X},d)$ \nis said to be approximately oligomorphic if for every $n\\ge 1$ and $\\varepsilon >0$ \nthere is a finite set $F\\subset {\\bf X}^n$ such that \n$$ \nG\\cdot F = \\{ g\\bar{x} : g\\in G \\mbox{ and } \\bar{x}\\in F\\}\n$$\nis $\\varepsilon$-dense in $({\\bf X}^n,d)$. \n\\end{definition} \n\nAssuming that $G$ is the automorphism group of a non-compact \nseparable continuous metric structure $M$, $G$ is approximately \noligomorphic if and only if  the structure $M$ is separably categorical \n(C. Ward Henson, see Theorem 4.25 in \\cite{scho}). \nIt is also known that separably categorical structures are \n{\\em approximately homogeneous} in the following sense: \nif $n$-tuples $\\bar{a}$ and $\\bar{c}$ have the same types \n(i.e. the same values $\\phi (\\bar{a})=\\phi (\\bar{b})$ for all $L$-formulas $\\phi$) \nthen for every $c_{n+1}$ and $\\varepsilon >0$ there is \nan tuple $b_1 ,...,b_n ,b_{n+1}$ of the same type with \n$\\bar{c},c_{n+1}$, so that $d(a_i, b_i )\\le \\varepsilon$ for $i\\le n$.  \nIn fact for any $n$-tuples $\\bar{a}$ and $\\bar{b}$ there is \nan automorphism $\\alpha$ of $M$ such that \n$$\nd(\\alpha (\\bar{c}),\\bar{a})\\le d(tp(\\bar{a}),tp(\\bar{c})) +\\varepsilon . \n$$  \n(i.e $M$ is {\\em strongly $\\omega$-near-homogeneous} in the sense\nof Corollary 12.11 of \\cite{BYBHU}). \n\n\\begin{definition} \nA topological group $G$ is called Roelcke precompact \nif for every open neighborhood of the identity $U$, \nthere exists a finite subset $F\\subset G$ such that $G=UFU$. \n\\end{definition} \n\nThe following theorem is a combination of the remark above, \nTheorem 6.2 of \\cite{rosendal},  \nTheorem 2.4 of \\cite{tsankov} and Proposition 1.20 of \\cite{rosendalN}. \n\n\\begin{theorem} \nLet $G$ be the automorphism group of a non-compact separable structure $M$. \n\nThen \\\\ \n(i) the group $G$ is approximately oligomorphic if and only if $M$ is separably categorical; \\\\  \n(ii) if $G$ is Roelcke precompact and approximately oligomorphic for 1-orbits, \nthen $M$ is separably categorical;\\\\  \n(iii) if the structure $M$ is separably categorical, then $G$ is Roelcke precompact. \n\\end{theorem} \n\n\n\n\\paragraph{Axiomatizability in continuous logic, topological properties and  sofic groups. } \n\nSuppose $\\mathcal{C}$ is a class of metric $L$-structures. \nLet $Th^c (\\mathcal{C})$ be the set of all closed $L$-conditions \nwhich hold in all structures of $\\mathcal{C}$. \nIt is proved in \\cite{BYBHU} (Proposition 5.14 and Remark 5.15) \nthat every model of $Th^c (\\mathcal{C})$ is elementary equivalent \nto some ultraproduct of structures from $\\mathcal{C}$. \nMoreover by Proposition 5.15 of \\cite{BYBHU} we have the following statement. \n\\begin{quote} \nThe class $\\mathcal{C}$ is axiomatizable in continuous logic \nif an only if it is closed under metric isomorphisms and \nultraproducts and its complement is closed under ultrapowers. \n\\end{quote} \nLet $Th^c_{\\sup} (\\mathcal{C})$ be the set of all closed \n$L$-conditions of the form \n$$ \nsup_{x_1} sup_{x_2} ... sup_{x_n} \\varphi =0 \n\\mbox{ ( $\\varphi$ does not contain $inf_{x_i}$, $sup_{x_i}$ ), } \n$$\nwhich hold in all structures of $\\mathcal{C}$.\nSome standard arguments also give the following theorem. \n\n\\begin{theorem} \\label{axiom} \n(1) The class $\\mathcal{C}$ is axiomatizable in continuous logic \nif an only if it is closed under metric isomorphisms, ultraproducts and \ntaking elementary submodels. \\parskip0pt  \n\n(2) The class $\\mathcal{C}$ is axiomatizable in continuous logic \nby $Th^c_{sup} (\\mathcal{C})$ if an only if it is closed under \nmetric isomorphisms, ultraproducts and taking substructures. \n\\end{theorem} \n\nIt is worth noting that when one considers classes axiomatizable \nin continuous logic it is obviously assumed that all operations \nand predicates are uniformly continuous. \nThis shows that some topological properties cannot be described \n(axiomatized) in continuous logic. \n\nSome other obstacles arise from the fact that existentional \nquantifiers cannot be expressed in continuous logic. \nFor example consider the class of all \nmetric groups which are discrete in their metrics (with $id$ as continuity moduli). \nThis class is not closed under metric ultraproducts but \nif we replace all metrics by the $\\{ 0,1\\}$-one \nwe just obtain the (axiomatizable) class of all groups. \n\nIt may also happen that when we extend an axiomatizable \nclass of structures with the $\\{ 0,1\\}$-metric \n\\footnote{in this case axiomatizability in continuous logic is equivalent to axiomatizability in first-order logic} \nby (abstract) structures from this class with all possible \n(not only possible discrete) metrics we lose axiomatizability. \nA nice example of this situation is the class of non-abelian groups \nwith $[0,1]$-metrics. \nFor example there is a sequence of non-abelian groups $G_n \\le Sym (2^n +3 )$ \nwith $G_n \\cong \\mathbb{Z}(2)^n \\times S_3$ so that their metric unltraproduct \n with respect to Hamming metrics is abelian (an easy exercise). \n\nContinuous axiomatizability appears in one of the most active areas in group theory \nas follows. \n\\begin{quote} \nAn abstract group is {\\em sofic} if it is embeddable into a metric \nultraproduct of finite symmetric groups with Hamming metrics. \n\\end{quote} \nLet $\\mathcal{S}$ be the class of complete $id$-continuous \nmetric groups of diameter 1, which are embeddable as closed subgroups  \nvia isometric morphisms into a metric ultraproduct of finite symmetric \ngroups with Hamming metrics.  \nThis class is axiomatizable by Theorem \\ref{axiom}. \nWe call it the class of {\\em metric sofic groups}. \n\n\\begin{corollary} \nThe class of metric sofic groups is $sup$-axiomatizable (i.e. by its theory $Th^c_{sup}$).  \n\\end{corollary} \n\nIt is folklore that any abstract sofic group can be \nembedded into a metric ultraproduct of finite symmetric \ngroups as a discrete subgroup \n(see the proof of Theorem 3.5 of \\cite{pestov}). \nThis means that the set of all abstract sofic groups consists \nof all discrete structures of the class $\\mathcal{S}$.  \n\n\n\\section{Boundedness properties} \n\nIt is worth noting that many classes from geometric \ngroup theory  are just universal. \nFor example if a group has free isometric actions \non real trees (resp. Hilbert spaces) then any its subgroup \nhas the same property. \nSimilarly a closed subgroup of  a locally compact  amenable \ngroup is amenable. \n\\footnote{the class of discrete initially amenable groups (see \\cite{cornulier})  is  universal too} \nThus these classes are bountiful. \n\nOn the other hand if we extend these classes by non-compact \nlocally compact groups without Kazhdan's property ${\\bf (T)}$\nor by groups admitting isometric actions on real trees  \nwithout fixed points then we lose universality.  \nAre these classes still bountiful? \nWe may further extend our classes by so called \nnon-{\\em boundedness properties} introduced in \\cite{rosendalN}. \nFor example consider metric groups which satisfy non-${\\bf OB}$ \n(in terms of \\cite{rosendalN}): they have isometric strongly \ncontinuous actions (i.e. the map $g\\rightarrow g\\cdot x$ defined on \n$G$ is continuous for each $x$) on metric spaces with unbounded orbits. \nThe first part of this section is devoted to some modifications of this property.  \nWe will show how continuous logic can work in these cases. \nIn fact metric groups from these classes can be presented \nas reducts of continuous metric structures  \nwhich induce some special actions.  \n\nIn the second part of the section we consider non-${\\bf (T)}$ and \nnon-${\\bf F\\mathbb{R}}$ (of fixed points for isometric actions on real trees).    \nNote that ${\\bf (T)}$ and property ${\\bf FH}$ \n(that any strongly continuous isometric affine action on a real Hilbert space has a fixed point)  \nare equivalent for $\\sigma$-compact locally compact groups \n(see Chapter 2 in \\cite{BHV}).  \nSince definitions of these properties require Hilbert spaces (or unbounded trees), \nwe will here apply a many-sorted version of continuous logic \n(as in Section 15 of \\cite{BYBHU}). \nWe will also present our groups as a union of an increasing \nchain of subsets of bounded diameters treating each subset as a sort.   \nThis situation is very natural if the group is $\\sigma$-compact \n(i.e. a union of an increasing chain of compact subsets). \n\nIt is worth noting that by Section 1.10 of \\cite{rosendalN} \nin the case of $\\sigma$-locally compact groups \n(=$\\sigma$-compact locally compact) Roelcke precompatness coincides \nwith all boundedness properties studied in \\cite{rosendalN} excluding only ${\\bf FH}$.  \nIn particular it coincides with compactness and property ${\\bf OB}$. \nOn the other hand an elementary submodel of a non-compact (resp. compact) continuous \nstructure is also non-compact (resp. compact, see \\cite{BYBHU}, Section 10). \nThus by the L\\\"{o}wenheim-Skolem theorem (in a 1-sorted language) \nnon-compacness is bountiful in the class of locally compact groups.  \nWhen the property non-${\\bf OB}$ coincides with non-compacness \n(as in the case of locally compact Polish groups) it is also bountiful. \nThis explains why in the first part of the section we do not \nassume that a group is locally compact or Polish. \n\n\nIt is worth noting that our methods do not work \nfor the classes of (locally compact) groups satisfying \nproperties {\\bf (T)}, {\\bf F}$\\mathbb{R}$ and {\\bf FH} \n(see discussion before Proposition \\ref{discr}). \nThe case of amenable Polish groups is open and  looks very intresting. \nA topological group $G$ is called {\\em amenable} \nif every $G$-flow admits an invariant Borel probability measure. \nIn the case of locally compact groups this definition coincides with \nthe classical one. \nIt is noticed in \\cite{kechrisN}, that the group $Sym(\\omega )$ \nof all permutations of $\\omega$ is amenable. \nSince it has closed non-amenable subgroups, the class of amenable \nPolish groups is not universal (with respect to taking closed subgroups).  \n\n\n\\subsection{Negations of strong boundedness and OB} \n\nAn abstract group $G$ is {\\em Cayley bounded} if for every generating subset\n$U\\subset G$ there exists $n\\in \\omega$ such that every element\nof $G$ is a product of $n$ elements of $U\\cup U^{-1}\\cup\\{ 1\\}$.\nIf $G$ is a Polish group then $G$ is {\\em topologically Cayley bounded} \nif for every analytic generating subset $U\\subset G$ \nthere exists $n\\in \\omega$ such that every element\nof $G$ is a product of $n$ elements of $U\\cup U^{-1}\\cup\\{ 1\\}$.\nIt is proved in \\cite{rosendal} that for Polish groups property \n{\\bf OB} is equivalent to topological Cayley boundedness together \nwith {\\em uncountable topological cofinality}: $G$ is not the union \nof a chain of proper open subgroups. \n\n\\paragraph{Discrete groups.} \n\nLet us consider the abstract (discrete) case. \nA group is {\\em strongly bounded} if it is Cayley bounded and\ncannot be presented as the union of a strictly increasing chain\n$\\{ H_n :n\\in \\omega\\}$ of proper subgroups\n(has {\\em cofinality} $>\\omega$).  \nIt is known that strongly bounded groups have property {\\bf FA}, \ni.e. any action on a simplicial tree fixes a point. \n\nThe class of strongly bounded groups is not bountiful. \nIndeed, by \\cite{dC} for any finite perfect group $F$ and \nan infinite $I$ the power $F^I$ is strongly bounded. \nSince $F^I$ is locally finite, any its countable subgroup \nhas cofinality $\\omega$. \nSimilar arguments can be applied to property ${\\bf FA}$.  \n\nIt is shown in \\cite{dC}, that strongly bounded groups\nhave property {\\bf FH}.   \nIt can be also deduced from \\cite{dC} that strongly bounded groups \nhave property ${\\bf F}\\mathbb{R}$ that every isometric action of $G$ \non a real tree has a fixed point (since such a group acting on \na real tree has a bounded orbit, all the elements are elliptic \nand it remains to apply cofinality $>\\omega$). \nIt is now clear that the bountiful class of groups having \nfree isometric actions on real trees (or on real Hilbert spaces) \nis disjoint from strong boundedness. \n \n\n\\begin{proposition} \\label{discr} \nThe following classes of groups are reducts of axiomatizable \nclasses in $L_{\\omega_1 \\omega}$: \\\\ \n(1) The complement of the class of strongly bounded groups; \\\\ \n(2) The class of groups of cofinality $\\le \\omega$; \\\\  \n(3) The class of groups which are not Cayley bounded; \\\\ \n(4) The class of groups presented as non-trivial free products with amalgamation \n(or HNN-extensions); \\\\   \n(5) The class of groups having homomorphisms onto $\\mathbb{Z}$.  \n\nAll these classes are bountiful. \nThe class of groups which do not have property {\\bf FA} is bountiful too. \n\\end{proposition} \n\n{\\em Proof.} \n(1) We use the following characterization of strongly bounded \ngroups from \\cite{dC}.  \n\\begin{quote} \nA group is strongly bounded if and only if for every \npresentation of $G$ as $G=\\bigcup_{n\\in \\omega} X_n$ for \nan increasing sequence $X_n$, $n\\in \\omega$, with \n$\\{ 1\\} \\cup X^{-1}_n \\cup X_n \\cdot X_n \\subset X_{n+1}$ \nthere is a number $n$ such that $X_n =G$. \n\\end{quote} \nLet us consider the class $\\mathcal{K}_{nb}$ of all structures \n$\\langle G, X_n \\rangle_{n\\in\\omega}$ with the axioms \nstating that $G$ is a group, $\\{ X_n \\}$ is a sequence of \nunary predicates on $G$ defining a strictly increasing \nsequence of subsets of $G$ with \n$\\{ 1\\} \\cup X^{-1}_n \\cup X_n \\cdot X_n \\subset X_{n+1}$ \n(these axioms are first-order) and \n$$ \n(\\forall x) (\\bigvee_{n\\in\\omega} x\\in X_n ). \n$$\nBy the L\\\"{o}wenheim-Skolem theorem for countable fragments of \n$L_{\\omega_1 \\omega}$ (\\cite{keisler}, p.69)\nany subset $C$ of such a structure is contained in an elementary \nsubmodel of cardinality $|C|$ (the countable fragment which we consider \nis the minimal fragment containing our axioms). \nThis proves bountifulness in case (1). \\\\ \n(2) The case groups of cofinality $\\le \\omega$ is similar. \\\\ \n(3) The class of groups which are not Cayley bounded is \na class of reducts of all groups expanded by an unary predicate  \n$\\langle G,U\\rangle$ with an $L_{\\omega_1 \\omega}$-axiom stating \nthat $U$ generates $G$ and with a system of first-order axioms \nstating that there exists an element of $G$ which is not a \nproduct of $n$ elements of $U\\cup U^{-1} \\cup \\{ 1 \\}$. \nThe rest is clear.   \\\\ \n(4) The class of groups which can be presented as \nnon-trivial free products with amalgamation is the class \nof reducts of all groups expanded by two unary predicates  \n$\\langle G,U_1 ,U_2 \\rangle$ with first-order axioms that $U_1$ and $U_2$ \nare subgroups and with $L_{\\omega_1 \\omega}$-axioms \nstating tha $U_1 \\cup U_2$ generates $G$ and a word in the \nalphabeth  $U_1 \\cup U_2$ is equal to 1 if and only if this \nword follows from the relators of the free product of $U_1$ \nand $U_2$ amalgamated over $U_1 \\cap U_2$. \nThe rest of (4) is clear.  \\\\ \n(5) Groups having homomorphisms onto $\\mathbb{Z}$ \ncan be considered as reducts of structures in the language \n$\\langle \\cdot ,...U_{-n},...,U_0 ,...,U_{m},...\\rangle$, \nwhere predicates $U_t$ denote preimages of the corresponding integer numbers.\n\nTo see that the class of groups without {\\bf FA} is \nbountiful, take any infinite $G\\models {\\bf notFA}$.  \nIt is well-known (\\cite{serre}, Section 6.1) \nthat such a group belongs to the union of \nthe classes from statements (2),(4) and (5).  \nThus $G$ has an expansion as in one of the cases (2),(4) or (5). \nNow applying the L\\\"{o}wenheim-Skolem theorem, for any  \n$C\\subset G$ we find a subgroup of $G$ of cardinality $|C|$ \nwhich contains $C$ and does not satisfy {\\bf FA}.   \n$\\Box$\n\n\\bigskip \n\n\n\\paragraph{Topological groups.} \n\nAs we already mentioned in Introduction separably categorical structures have \nRoelcke precompact automorphism groups. \nIn the following definition we consider several versions of this property. \n\n\\begin{definition} \nLet $G$ be a topological group. \\\\ \n(1) The group $G$  is called bounded if for any open $V$ containing $1$ there is \na finite set $F\\subseteq G$ and a natural number $k>0$  such that $G=FV^k$. \\\\ \n(2) The group $G$ is Roelcke bounded if for any open $V$ containing $1$ there is \na finite set $F\\subseteq G$ and a natural number $k>0$  such that $G=V^k FV^k$. \\\\ \n(3) The group $G$ is Roelcke precompact if for any open $V$ containing \n$1$ there is a finite set $F\\subseteq G$ such that $G=VFV$. \\\\ \n(4) The group $G$ has property ${\\bf (OB)_k}$ if for any open symmetric \n$V\\not=\\emptyset$ there is a finite set $F\\subseteq G$ such that $G=(FV)^k$. \n\\end{definition} \nIt is known that for Polish groups property ${\\bf OB}$ is equivalent \nto the property that for any open symmetric $V\\not=\\emptyset$ there is \na finite set $F\\subseteq G$ and a natural number $k$ such that $G=(FV)^k$.  \nThus when $G$ is non-{\\bf OB}, there is an non-empty open $V$ such that \nfor any finite $F$ and a natural number $k$, $G\\not=(FV)^k$. \nNote that for such $F$ and $k$ there is a real number $\\varepsilon$ \nsuch that some $g\\in G$ is $\\varepsilon$-distant from $(FV)^k$. \nIndeed, otherwise $(FV)^k V$ would cover $G$.  \n \nThis explains why in order to  define a suitable class which is \ncomplementary to {\\bf OB} we consider the following property. \n\n\\begin{definition} \nA metric group $G$ is called uniformly non-{\\bf OB} if there is an open \nsymmetric $V\\not=\\emptyset$ so that for any natural numbers $m$ and $k$ \nthere is a real number $\\varepsilon$ such that for any $m$-element subset \n$F\\subset G$ there is $g\\in G$ which is $\\varepsilon$-distant from $(FV)^k$. \n\nUniform non-boundedness, uniform non-Roelcke boundedness, uniform non-Roelcke \nprecompactness and uniform non-${\\bf (OB)_k}$ are defined by the same scheme. \n\\end{definition}  \n\nIt is clear that in the case discrete groups if a symmetric \nsubset $V$ has the property that $G\\not= (FV)^k$ for all finite \n$F\\subset G$ and natural numbers $k$, then the corresponding \nuniform version also holds. \n\n\n\\begin{proposition} \\label{uniform} \nThe following classes of metric groups are bountiful: \\\\ \n(1) The class of uniformly non-bounded groups; \\\\ \n(2) The class of uniformly non-Roelcke bounded groups; \\\\  \n(3) The class of uniformly non-Roelcke precompact groups; \\\\ \n(4) The class of uniformly non-${\\bf (OB)_k}$-groups; \\\\   \n(5) The class of uniformly non-${\\bf (OB)}$-groups.  \n\\end{proposition} \n \n{\\em Proof.} \nLet us consider the class of uniformly non-${\\bf (OB)}$-groups. \nLet $\\mathcal{K}_{0}$ be the class of all continuous \nmetric structures $\\langle G, P,Q \\rangle$ with the axioms \nstating that $G$ is a group and $P:G\\rightarrow [0,1]$ and   \n$Q:G\\rightarrow [0,1]$ are  unary predicates on $G$ with $Q(1)=0$ so that  \n$$ \nsup_x min (P(x),Q(x))= sup_x |P(x)-P(x^{-1})|=0 \\mbox{ and } inf_x |P(x)-1\/2| =0,  \n$$\n$$ \nsup_x |Q(x)-Q(x^{-1})|=0 \\mbox{ and } inf_x |Q(x)-1\/2| =0,  \n$$\n$$\n\\mbox{ and for all rational }\\varepsilon \\in [0,1]  \n$$ \n$$\nsup_x  min ( \\varepsilon \\dot{-} Q(x), inf_y (max(d(x,y)\\dot{-} 2\\varepsilon , \\varepsilon \\dot{-}  P(y)))=0.   \n$$ \nNote that the last axiom implies that any neighbourhood of an element from the nullset of $Q$ \ncontains an element with non-zero $P$. \n\nFor any natural $m$ and $k$ and any rational $\\varepsilon$ let us consider the following condition (say $\\theta (m,k,\\varepsilon )$): \n$$ \nsup_{x_1 ...x_m }  inf _{x}   sup_{y_1 ...y_k} min(P(y_1),...,P(y_n), (\\varepsilon \\dot{-} min_{w\\in W_{m,k}}(d(x,  w))))=0, \n$$ \n$$ \n\\mbox{ where } W_{m,k} \\mbox{ consists of all words of the form } x_{i_1} y_1 x_{i_2} y_2 ...x_{i_k}y_k . \n$$ \nIf $G$ is a uniformly non-${\\bf (OB)}$-group, then find an open symmetric $V$ \nsuch that for any natural numbers $m$ and $k$ \nthere is a real number $\\varepsilon$ such that for any $m$-element subset \n$F\\subset G$ there is $g\\in G$ which is $\\varepsilon$-distant from $(FV)^k$. \nWe interpret $Q(x)$ by $d(x,V)$ and $P(x)$ by $d(x, G\\setminus V)$ \n(possibly normalizing them to satisfy the axioms of $\\mathcal{K}_0$). \nThen observe that $\\langle G,P,Q\\rangle\\in \\mathcal{K}_0$ and for any \nnatural numbers $m$ and $k$ there is a rational number $\\varepsilon$ \nso that $\\theta(m,k,\\varepsilon )$ holds in $(G,P,Q)$.  \nBy the L\\\"{o}wenheim-Skolem theorem for continuous logic  \nany infinite subset $C$ of such a structure is contained in an elementary \nsubmodel of the same density character as $C$. \nTo verify uniform {\\bf non-(OB)} in such a submodel take the complement \nof the nullset of $P(x)$ as an open symmetric subset. \nThis proves statement (5).  \n\nAll remaining cases are considered in a similar way. \n$\\Box$ \n\\bigskip \n\n\n\n\\subsection{Unbounded actions} \n\\paragraph{{\\bf Negation of (T).}}\n\nLet a topological group $G$ have a strongly continuous unitary \nrepresentation on a Hilbert space ${\\bf H}$.  \nA closed subset $Q\\subset G$ \nhas an {\\em almost $\\varepsilon$-invariant unit vector} in ${\\bf H}$ if \n$$ \n\\mbox{ there exists }v\\in {\\bf H} \\mbox{ such that } \nsup_{x\\in Q} \\parallel x\\circ v - v\\parallel < \\varepsilon\n\\mbox{ and } \\parallel v\\parallel =1.  \n$$ \nWe call a closed subset $Q$ of the group $G$ a {\\em Kazhdan set} \nif there is $\\varepsilon$ with the following property: \nevery unitary representation of $G$ on a Hilbert space \nwith almost $(Q,\\varepsilon )$-invariant unit vectors also has \na non-zero invariant vector.  \nIf the group $G$ has a compact Kahdan subset then it is said that \n$G$ has property ${\\bf (T)}$ of Kazhdan. \n\nIf we want to consider unitary representations in \ncontinuous logic we should fix continuity moduli \nfor the corresponding binary functions \n$G\\times B_n \\rightarrow B_n$ induced by the action, \nwhere $B_n$ is the $n$-ball of the corresponding Hilbert space. \nIn fact if $G$ is $\\sigma$-locally compact, \nthen we can present $G$ as the union of a chain \nof compact subsets $K_1 \\subseteq K_2 \\subseteq ...$ \nand consider continuity moduli for the corresponding \nfunctions $K_m \\times B_n \\rightarrow B_n$. \nNote that each $B_k$ and $K_l$ will be considered as sorts \nof a continuous structure. \nIn this version of continuous logic we do not assume that \nthe diameter of a sort is bounded by 1. \nIt can become any rational number.  \n\nWe can now slightly modify the definition of a Kazhdan set \nas follows. \n\n\\begin{definition} \nLet $G$ be the union of a chain of closed subsets \n$K_1 \\subseteq K_2 \\subseteq ...$  of bounded diameters.  \nLet $\\mathcal{F} = \\{ F_1 , F_2 , ... \\}$ be a family of continuity \nmoduli for continuous function $K_i \\times B_i \\rightarrow B_i$. \n\nWe call a closed subset $Q$ of the group $G$ \nan $\\mathcal{F} $-Kazhdan set if there is $\\varepsilon$ \nwith the following property: \nevery $\\mathcal{F} $-continuous unitary representation of $G$ \non a Hilbert space with almost $(Q,\\varepsilon )$-invariant unit vectors \nalso has a non-zero invariant vector.  \n\\end{definition} \n\nLet us consider such actions in continuous logic. \nWe treat a Hilbert space over $\\mathbb{R}$ \nexactly as in Section 15 of \\cite{BYBHU}. \nWe identify it with a many-sorted metric structure \n$$\n(\\{ B_n\\}_{n\\in \\omega} ,0,\\{ I_{mn} \\}_{m<n} ,\n\\{ \\lambda_r \\}_{r\\in\\mathbb{R}}, +,-,\\langle \\rangle ),\n$$\nwhere $B_n$ is the ball of elements of norm $\\le n$, \n$I_{mn}: B_m\\rightarrow B_n$ is the inclusion map, \n$\\lambda_{r}: B_m\\rightarrow B_{km}$ is scalar \nmultiplication by $r$, with $k$ the unique integer \nsatisfying $k\\ge 1$ and $k-1 \\le |r|<k$; \nfuthermore, $+,- : B_n \\times B_n \\rightarrow B_{2n}$ \nare vector addition and subtraction and \n$\\langle \\rangle : B_n \\rightarrow [-n^2 ,n^2 ]$ \nis the predicate of the inner product. \nThe metric on each sort is given by \n$d(x,y) =\\sqrt{ \\langle x-y, x-y \\rangle }$.   \nFor every operation the continuity modulus is standard.  \nFor example in the case of $\\lambda_r$ this is $\\frac{z}{|r|}$. \n\nStating existence of infinite approximations of orthonormal bases \n(by a countable family of axioms, see Section 15 of \\cite{BYBHU}) \nwe assume that our Hilbert spaces are infinite dimensional. \nBy \\cite{BYBHU} they form the class of models of a complete \ntheory which is $\\kappa$-categorical for all infinite $\\kappa$, \nand admits elimination of quantifiers. \n\nThis approach can be naturally extended to complex Hilbert spaces, \n$$\n(\\{ B_n\\}_{n\\in \\omega} ,0,\\{ I_{mn} \\}_{m<n} ,\n\\{ \\lambda_c \\}_{c\\in\\mathbb{C}}, +,-,\\langle \\rangle_{Re} , \\langle \\rangle_{Im} ). \n$$\nWe only extend the family \n$\\lambda_{r}: B_m\\rightarrow B_{km}$, $r\\in \\mathbb{R}$, \nto a family $\\lambda_{c}: B_m\\rightarrow B_{km}$, $c\\in \\mathbb{C}$, \nof scalar products by $c\\in\\mathbb{C}$, with $k$ \nthe unique integer satisfying $k\\ge 1$ and $k-1 \\le |c|<k$. \n\nWe also introduce $Re$- and $Im$-parts of the inner product. \n\nIf we remove from the signature of complex Hilbert spaces \nall scalar products by $c\\in \\mathbb{C}\\setminus \\mathbb{Q}[i]$, \nwe obtain a countable subsignature \n$$\n(\\{ B_n\\}_{n\\in \\omega} ,0,\\{ I_{mn} \\}_{m<n} ,\n\\{ \\lambda_c \\}_{c\\in\\mathbb{Q}[i]}, +,-,\\langle \\rangle_{Re} , \\langle \\rangle_{Im} ),\n$$\nwhich is {\\em dense} in the original one: \\\\ \nif we present $c\\in \\mathbb{C}$ by a sequence $\\{ q_i \\}$ \nfrom $\\mathbb{Q}[i]$ converging to $c$, \nthen the choice of the continuity moduli of \nthe restricted signature still guarantees that \nin any sort $B_n$ the functions $\\lambda_{q_i}$ \nform a sequence which converges to  $\\lambda_c$ \nwith respect to the metric \n$$ \nsup_{x\\in B_n} \\{ |f^M (x) - g^M (x)| : M \\mbox{ is an $L$-structure } \\}.  \n$$  \nThis obviously implies that the original language \nof Hilbert spaces is {\\em separable}. \nIn particular we may apply Remark \\ref{FaHaSh}. \n\n\n\n\\bigskip \n\n\n\n\n\nLet us consider a class of continuous metric structures \nwhich are unions of the many-sorted structures \n$$ \n(\\{ K_n \\}_{n\\in\\omega}, \\cdot , ^{-1}, 1 ),  \n$$\ncorresponding to groups $G$ presented as $\\bigcup_{n\\in\\omega} K_n$, \ntogether with metric structures of complex Hilbert spaces  \n$$ \n(\\{ B_n\\}_{n\\in \\omega} ,0,\\{ I_{mn} \\}_{m<n} ,\n\\{ \\lambda_c \\}_{c\\in\\mathbb{C}}, +,-,\\langle \\rangle_{Re} , \\langle \\rangle_{Im} ).\n$$\nThe operation $\\cdot$ (and $^{-1}$) is considered as \na family of maps $K_n \\times K_n \\rightarrow K_{n+1}$. \n(maps $K_n\\rightarrow K_{n+1}$ respectively), $n\\in\\omega$.  \nSuch a structure (say $A(G,{\\bf H})$) also contains \na binary operation $\\circ$ of an action which is defined \nby a family of appropriate maps  \n$K_n \\times B_m \\rightarrow B_{m}$. \nWhen we add the obvious continuous $sup$-axioms that \nthe action is linear and unitary, we obtain an axiomatizable \nclass $\\mathcal{K}_{GH}$. \nWe do not state exactly which continuity moduli would \ncorrespond to these operations. \nIn fact this depends on groups and actions we want \nto have in  $\\mathcal{K}_{GH}$.  \nBy $\\mathcal{K}_{GH}(\\mathcal{F} )$ we denote the \ncorresponding class with continuity moduli $\\mathcal{F} $. \n\nAssuming that continuity moduli $\\mathcal{F} $ are fixed \nlet $\\mathcal{K}_{aiv}(\\mathcal{F} )$ be the subclass of \n$\\mathcal{K}_{GH}$ axiomatizable by the axioms \n$$ \ninf_{v\\in B_{m}} sup_{x\\in K_{n}} \nmax(\\parallel x\\circ v - v\\parallel \\dot- \\mbox{ } {1\\over n}, |1- \\parallel v\\parallel \\mbox{ } |)=0\n\\mbox{ , } m,n\\in\\omega \\setminus \\{ 0\\}, \n$$ \nwhich in fact say that each $K_n$ has an almost \n${1\\over n}$-invariant unit vector in ${\\bf H}$.\n\nBelow we will only consider metric groups which have presentations \n$G=\\bigcup_{i\\in\\omega} K_i$, where $\\{ K_i :i\\in\\omega\\}$ \nis an increasing sequence of closed  subsets of diameters \n$d_1 \\le d_2 \\le ... $ with \n$\\{ 1 \\} \\cup K_n \\cdot K_n \\cup K^{-1}_n \\subseteq K_{n+1}$. \nMoreover we will assume that every function \n$K_n\\times K_n \\rightarrow K_{n+1}$ induced by the \nmultiplication is uniformly continuous with respect to \nsome fixed family of continuity moduli $\\mathcal{F}_0$.  \nWhen $G$ is a $\\sigma$-locally compact group \nthen such $\\mathcal{F}_0$ and such a decomposition obviously exist. \n\n\n\\begin{proposition} \\label{nT} \nLet $G$ be a metric group having a presentation \n$G= \\bigcup_{i\\in\\omega} K_i$ into an icreasing \nsequence of closed subsets of bounded diameters as above, \nso that no  $K_i$ is an $\\mathcal{F} $-Kazhdan set for $G$. \nThen there is a metric structure in $\\mathcal{K}_{aiv}(\\mathcal{F} )$  \nwhich naturally expands \n$(\\{ K_n \\}_{n\\in\\omega}, \\cdot , ^{-1}, 1 )$ \nso that $G$ does not have non-zero fixed vectors. \nIn particular any $\\sigma$-locally compact group $G$ \nwithout property {\\bf (T)} for $\\mathcal{F} $-actions \nhas such an expansion to ${\\bf H}$. \n\\end{proposition} \n\n{\\em Proof.} \nConsider the particular case of the proposition. \nLet $G=\\bigcup_{i\\in\\omega} K_i$, where $\\{ K_i :i\\in\\omega\\}$ \nis an increasing sequence of compact neighborhoods of $1$ \nwith $K_n \\cdot K_n \\cup K^{-1}_n \\subseteq K_{n+1}$. \nWe know that for any natural $n$ and rational $0<q<1$ there is \na unitary $\\mathcal{F}$-continuous representation of $G$ \non a Hilbert space ${\\bf H}$ which has an almost $(K_n ,q)$-invariant \nunit vector but does not have a non-zero invariant vector. \nDecomposing a basis of ${\\bf H}$ into a union of an infinite family \nof pairwise disjoint infinite subsets  (labelled by pairs $(n,q)$) \nand defining representations above on the corresponding subspaces, \nwe find an unitary $\\mathcal{F}$-representation of $G$ on ${\\bf H}$ \nwithout non-zero invariant vectors so that for any natural $n$ and rational \n$q<1$ there is an almost $(K_n ,q)$-invariant unit vector in ${\\bf H}$. \n\nLet us define a required metric structure \n(denoted by $A(G,{\\bf H})$ as above) \nas a union of the many-sorted structure \n$$ \n(\\{ K_n \\}_{n\\in\\omega}, \\cdot , ^{-1}, 1 ),  \n$$\ncorresponding to the group $G$, together with \nthe metric structure of the separable Hilbert space \n$$ \n(\\{ B_n\\}_{n\\in \\omega} ,0,\\{ I_{mn} \\}_{m<n} ,\n\\{ \\lambda_c \\}_{c\\in\\mathbb{C}}, +,-,\\langle \\rangle ).\n$$\nThe operation $\\cdot$ (and $^{-1}$) is considered as \na family of maps $K_n \\times K_n \\rightarrow K_{n+1}$. \n(maps $K_n\\rightarrow K_{n+1}$ respectively), $n\\in\\omega$.  \nThe structure $A(G,{\\bf H})$ also contains a binary \noperation $\\circ$ of an action which is defined by \na family of appropriate maps  \n$K_n \\times B_m \\rightarrow B_m$. \nSince the representation is unitary, any element \nof $G$ preserves each $B_m$.  \nThe axioms of $\\mathcal{K}_{inv}(\\mathcal{F})$ obviously \nhold in $A(G,{\\bf H})$. \n\nThe argument in the general case is the same. \n$\\Box$ \n\n\\bigskip \n\n\\begin{remark} \n{\\em When any compact subset of $G$ is \ncontained in some $K_i$, the group $G$ from \nthe formulation above does not have property {\\bf (T)} \nfor $\\mathcal{F}$-continuous representations. \nFor example this happens when each $K_i$ \nis the closed $d_i$-ball of $1$. }\n\\end{remark}\n\\bigskip \n\nThe following corollary can be considered \nas a kind of bountifulness for groups without {\\bf (T)}. \nIt follows from the proposition and remark above and \nthe L\\\"{o}wenheim-Skolem theorem for continuous logic \n(Remark \\ref{FaHaSh}). \n\n\\begin{corollary} \nLet $G$ be a metric group having a presentation \n$G= \\bigcup_{i\\in\\omega} K_i$ into an icreasing \nsequence of closed balls of $1$ of bounded diameters, \nso that no  $K_i$ is an $\\mathcal{F}$-Kazhdan set for $G$. \nThen for any infinite subset $C\\subset G$ there is a closed \nelementary (in continuous logic) subgroup of $G$ containing $C$, \nwith the same density character as $C$ \nand without property {\\bf (T)} for \n$\\mathcal{F}$-continuous representations. \n\\end{corollary} \n\n\n\\paragraph{{\\bf Non-FH-actions.}} \n\nTo consider non-{\\bf FH} let us fix a binary function \n$\\nu :\\omega\\times\\omega \\rightarrow\\omega$ \nwhich is increasing in each argument. \nWe now define $\\mathcal{K}_{GH}(\\nu )$, a class \nof continuous metric structures which are unions \nof many-sorted $\\mathcal{F}_0$-continuous structures \n$$ \n(\\{ K_n \\}_{n\\in\\omega}, \\cdot , ^{-1}, 1 ),  \n$$\ncorresponding to groups $G$ presented as \n$\\bigcup_{n\\in\\omega}K_n$ (with assumptions as before Proposition \\ref{nT}), \ntogether with metric structures of real Hilbert spaces  \n$$ \n(\\{ B_n\\}_{n\\in \\omega} ,0,\\{ I_{mn} \\}_{m<n} ,\n\\{ \\lambda_c \\}_{r\\in\\mathbb{R}}, +,-,\\langle \\rangle ).\n$$   \nWe also add a binary $\\mathcal{F}$-continuous operation \n$\\circ$ of an action defined by a family of appropriate \nmaps $K_n \\times B_m \\rightarrow B_{\\nu (n,m)}$. \nObvious continuous $sup$-axioms that the action \nis isometric give an axiomatizable \nclass $\\mathcal{K}_{GH}(\\nu ,\\mathcal{F} )$. \n\n\nThe following statement is a straightforward application of the continuous L\\\"{o}wenheim-Skolem theorem. \n\n\\begin{proposition} \nLet $G$ be a metric group having a presentation  \n$G= \\bigcup_{i\\in\\omega} K_i$ into an icreasing \nsequence of closed subsets of bounded diameters, \nso that there is an isometric action of $G$ on \na real Hilbert space inducing a structure from \n$\\mathcal{K}_{GH}(\\nu, \\mathcal{F} )$ without fixed points. \nThen for any infinite subset $C\\subset G$ \nthere is a closed continuously elementary subgroup of $G$ \ncontaining $C$, with the same density character as $C$ \nand without property {\\bf FH} for $\\mathcal{F}$-actions. \n\\end{proposition} \n\nNote that any $\\sigma$-locally compact metric group \n$G$ without property {\\bf FH} for $\\mathcal{F}$-actions \nbelongs to the class $K_{GH} (\\nu ,\\mathcal{F})$ \n(for appropriate $\\nu$ and $\\mathcal{F}_0$). \nWe will now see this in a slightly stronger form. \nFix a function $\\eta$ which assigns to a natural number $k$ \na pair $(l,s)\\in \\omega\\times\\omega$.  \nTo obtain $\\mathcal{K}_{nFH}(\\nu ,\\eta ,\\mathcal{F})$ take the \nsubclass of $\\mathcal{K}_{GH}(\\nu )$ axiomatizable by the axioms \n$$ \nsup_{v\\in B_{k}} inf_{x\\in K_{l}} \n({1\\over s} \\dot{-}  \\parallel x\\circ v - v\\parallel  )=0  \n\\mbox{ , for } \\eta (k) =(l,s)   \n$$ \n(saying that each vector of $B_k$ is moved by some  \nelement of $K_l$ by approximately $ {1\\over s}$). \n\n\\begin{proposition} \nFor any $\\sigma$-locally compact metric group $G$ \nwithout property {\\bf FH} for $\\mathcal{F}$-continuous \nactions there are functions $\\nu$ and $\\eta$ so that \n$G$ belongs to the class $\\mathcal{K}_{nFH}(\\nu ,\\eta ,\\mathcal{F})$. \n\\end{proposition} \n\n{\\em Proof.} \nLet $G=\\bigcup_{i\\in\\omega} K_i$, where $\\{ K_i :i\\in\\omega\\}$ \nis an increasing sequence of compact neighborhoods of $1$ \nwith $K_n \\cdot K_n \\cup K^{-1}_n \\subseteq K_{n+1}$. \nFix appropriate $\\mathcal{F}_0$. \nWe know that there is an affine $\\mathcal{F}$-continuous \nisometric action of $G$ on a real Hilbert space ${\\bf H}$ \nwhich does not have a fixed vector. \nSince for every $v\\in {\\bf H}$ the map \n$$\nG \\rightarrow {\\bf H} \\mbox{ , } g \\rightarrow gv \n$$ \nis continuous there are $k\\in\\omega$ and an open subset \nof $G$ which maps $0$ into $B_k$. \nIn particular for any $n$ we can find such a $k$ so that \nall elements of $K_n$ map $0$ into $B_k$. \nThis means that $K_n \\circ B_m \\subseteq B_{k+m}$. \nThis defines a function \n$\\nu :\\omega\\times\\omega \\rightarrow\\omega$ so that \nthe continuous structure corresponding to the action \nbelongs to $\\mathcal{K}_{GH}(\\nu ,\\mathcal{F} )$. \n\nLet us show that this structure belongs to \n$\\mathcal{K}_{nFH}(\\nu ,\\eta ,\\mathcal{F})$ \nfor appropriate $\\eta$. \nSince $G$ does not fix any point,  each orbit of $G$ \nis unbounded (Proposition 2.2.9 of \\cite{BHV}). \nThus there is $g\\in G$ so that $B_k \\cap g (B_k) =\\emptyset$. \nIn particular there is $s\\in \\mathbb{N}$ such that \n$ {1\\over s} \\le \\parallel g\\circ v - v\\parallel$ \nfor all $v\\in B_k$.  \nWe define $\\eta (k)$ to be $(l,s)$, where $l$ is chosen so \nthat $K_l$ contains $g$ as above. \n$\\Box$ \n\n\\bigskip \n\n\\paragraph{{\\bf Non-F$\\mathbb{R}$-actions.}}   \n\nTo consider non-{\\bf F}$\\mathbb{R}$ we apply similar ideas. \nLet us fix a binary function \n$\\nu :\\omega\\times\\omega \\rightarrow\\omega$ \nwhich is increasing in each argument.  \nWe now define $\\mathcal{K}_{GR}(\\nu ,\\mathcal{F} )$, a class \nof continuous metric structures \nwhich are unions of the many-sorted structures \n$$ \n(\\{ K_n \\}_{n\\in\\omega}, \\cdot , ^{-1}, 1 ),  \n$$\ncorresponding to groups $G$ presented as $\\bigcup_{n\\in\\omega} K_n$, \ntogether with metric structures of pointed real trees  \n$$ \n(\\{ B_n\\}_{n\\in \\omega} ,0,\\{ I_{mn} \\}_{m<n} ), \n$$   \nwhere $B_n$ is the $n$-ball of $0$ in the tree. \nIt is shown in \\cite{carlisle} \nthat the class of pointed real trees is axiomatizable \nin contimuous logic by axioms of $0$-hyperbolicity and \nthe approximate midpoint property. \n \nWe also add a binary operation $\\circ$ of an action \ndefined by a family of appropriate $\\mathcal{F}$-continuous \nmaps $K_n \\times B_m \\rightarrow B_{\\nu (n,m)}$. \nObvious continuous $sup$-axioms that the action \nis isometric give an axiomatizable \nclass $\\mathcal{K}_{GR}(\\nu ,\\mathcal{F})$. \n\nAs in the non-{\\bf FH}-case we have the following straightforward statement. \n\n\\begin{proposition} \\label{211} \nLet $G$ be an $\\mathcal{F}_0$-continuous group with respect \nto a presentation $G= \\bigcup_{i\\in\\omega} K_i$ into \nan icreasing sequence of closed subsets of bounded diameters, \nso that there is an isometric action of $G$ on a real tree \ninducing a structure from $\\mathcal{K}_{GR}(\\nu ,\\mathcal{F})$ \nwithout fixed points. \nThen for any infinite subset $C\\subset G$ there \nis a closed continuously elementary subgroup of $G$ \ncontaining $C$, with the same density character as $C$ \nand without property {\\bf F}$\\mathbb{R}$ for \n$\\mathcal{F}$-continuous actions. \n\\end{proposition} \n\nFix a function $\\eta$ which assigns to a natural \nnumber $k$ a pair $(l,s)\\in \\omega\\times\\omega$.  \nTo obtain $\\mathcal{K}_{nFR}(\\nu ,\\eta ,\\mathcal{F})$ take \nthe subclass of $\\mathcal{K}_{GR}(\\nu )$ axiomatised  \nby the axioms \n$$ \nsup_{v\\in B_{k}} inf_{x\\in K_{l}} \n(  {1\\over s} \\dot{-} d(x\\circ v,v  ))=0  \\mbox{ , for } \\eta (k)=(l,s)   \n$$ \n(saying that each element of $B_k$ is moved by some  \nelement of $K_l$ by approximately $ {1\\over s}$). \n\n\n\\begin{proposition} \nFor any $\\sigma$-locally compact metric \ngroup $G$ without property {\\bf F}$\\mathbb{R}$ \nfor strongly $\\mathcal{F}$-continuous actions there are functions \n$\\nu$ and $\\eta$ so that $G$ belongs to the class \n$\\mathcal{K}_{nFR}(\\nu ,\\eta ,\\mathcal{F})$. \n\\end{proposition} \n\n{\\em Proof.} \nLet $G=\\bigcup_{i\\in\\omega} K_i$, where $\\{ K_i :i\\in\\omega\\}$ \nis an increasing sequence of compact neighborhoods of $1$ \nwith $K_n \\cdot K_n \\cup K^{-1}_n \\subseteq K_{n+1}$. \nFind appropriate moduli $\\mathcal{F}_0$ making $G$ \nan $\\mathcal{F}_0$-continuous structure. \nWe know that there is an $\\mathcal{F}$-continuous \nisometric action of $G$ on a real tree $T$ which \ndoes not have a fixed point. \nSince for every $v\\in T$ the map \n$$\nG \\rightarrow T \\mbox{ , } g \\rightarrow gx \n$$ \nis continuous there are $k\\in\\omega$ and an open subset \nof $G$ which maps $0$ into $B_k$. \nIn particular for any $n$ we can find a $k$ so that \nall elements of $K_n$ map $0$ into $B_k$. \nThis means that $K_n \\circ B_m \\subseteq B_{k+m}$. \nThis defines a function \n$\\nu :\\omega\\times\\omega \\rightarrow\\omega$ so that \nthe continuous structure corresponding to the action \nbelongs to $\\mathcal{K}_{GR}(\\nu ,\\mathcal{F} )$. \n\nLet us show that this structure belongs to \n$\\mathcal{K}_{nFR}(\\nu ,\\eta ,\\mathcal{F})$ for appropriate $\\eta$. \nIf $G$ has a hyperbolic element $g$ of hyperbolic length $r$ \n(i.e. there is a line $L$ so that $g$ $r$-shifts all points of $L$), \nthen $\\eta$ is constant where $l$  is chosen so \nthat $K_l$ contains this hyperbolic element \nand $s$ is chosen so that $ {1\\over s} \\le r$. \n\nConsider the case when $G$ consists of elliptic elements (i.e. fixing points). \nSince $G$ does not fix any point, by a well-known \nargument $G$ fixes an end (\\cite{serre}, Section 6.5, Exercise 2). \nLet $L_0$ be the half-line starting from $0$ which \nrepresents this end and let $v_1 ,....,v_i ,....$  \nbe a cofinal $\\omega$-sequence in $L_0$ with $d(v_i ,v_{i+1})\\ge 1$. \nThen we may assume that $G$ is the union of a strictly \nincreasing chain of stabilizers $G_i$ of $v_i$. \nSince the action is continuous, all $G_i$ are closed. \n\nHaving $k$ find $j$ with $v_{j-1} \\not\\in B_k$ (thus $v_j \\not\\in B_k$). \nSince any arc linking $v_j$ with an element from $B_k$ must \ncontain $v_{j-1}$, we see that if $g\\in G_j$ fixes a point \nof $B_k$ then it fixes $v_{j-1}$. \nIn particular $G_j$ does not fix any element of $B_k$. \nSince $d(v_j ,v_{j-1}) \\ge 1$ any point of $B_k$ can \nbe taken by some element of $G_j$ at a distance greater than 1.   \nThus to define $\\eta (k)=(l,s)$, we choose $l$ so that $K_l$ contains \nan element of $G_j$ not fixing $v_{j-1}$. \nWe define $s=1$. \n$\\Box$\n\n\n\n\n\n\n\\bigskip \n\n\n\n\n\n\\section{Separably categorical locally compact groups} \n\nIf $G$ is a locally compact group, then $G$ admits \na compatible complete left invariant metric $d(x,y)$ \n(\\cite{becker}, 3.C.2). \nWe may assume that $d(x,y)$ satisfies $d(x,y) < 1$ \n(it can be replaced by $\\frac{d(x,y)}{d(x,y)+1}$). \nWe thus may consider locally compact groups in continuous logic \nas a class of continuous metric structures  \n$$ \n(G, d, \\cdot , ^{-1}, 1 ),  \n$$ \ntogether with fixed continuity moduli for functional symbols. \n\n\n\\begin{theorem} \\label{catlc}\nLet $G$ be a separably categorical locally compact \nnon-compact group. \nThen there is a compact clopen subgroup $H<G$ \nwhich is invariant with respect to all metric automorphisms of $G$, \nand the induced action of $Aut(G,d)$ on the coset space $G\/H$ \nis oligomorphic. \n\nIf the connected component of the unity $G^0$ is not trivial, \n$H$ can be taken to be $G^0$. \nIn this case and in the case when $d$ is two-sided-invariant, \nthe subgroup $H$ is normal and $G\/H$ is an $\\omega$-categorical \ndiscrete group. \n\\end{theorem} \n\nWe start with the following preliminaries. \nWe may assume that $G$ is not discrete. \nThere is a rational number $\\rho <1$ such that the $\\rho$-ball \nof the unity $B_{\\rho} (1)=\\{ x\\in G: d(x,1)\\le\\rho\\}$ is compact. \nIn particular $B_{\\rho}(1)$ is a subset of $acl(\\emptyset )$ in $G$ \n(the condition $d(x,1)\\le \\rho$ defines a totally bounded, complete subset in any elementary extension of $G$). \nThus any $B^n_{\\rho}(1)$ also is a subset of $acl(\\emptyset )$. \nLet $G_{\\rho}$ be the subgroup generated by $B_{\\rho}(1)$. \nNote that for any $g\\in G_{\\rho}$ the open ball \n$$\nB_{<\\rho}( g)=\\{ x\\in G: d(x,g)<\\rho\\}= \\{ x\\in G: d(g^{-1} x,1) <\\rho \\}\n$$\nis a subset of $G_{\\rho}$; thus $G_{\\rho}$ is an open (in fact clopen) subgroup. \nIf $G$ has a non-trivial connected component of the unity $G^0$, \nwe may assume that $G_{\\rho}=G^0$. \nNote that when $d$ is a two-sided-invariant metric, $G_{\\rho}$ \nis a normal subgroup of $G$. \n \n\\begin{lemma} \nAssume that $G$ is a separably categorical \nlocally compact group. \nThen under the circumstances above \nthe predicate $P(x)=d(x,G_{\\rho})$ is definable in $G$. \n\\end{lemma} \n\n{\\em Proof.} \nSince the space $(S_n(T),d)$ is compact, \nfor every $\\varepsilon$ there is \na finite set of types which is $\\varepsilon$-dense \nin the set of types of elements of $\\bigcup_{n>0} B^{n}_{\\rho}(1)$. \nThus there is a number $n$ such that \nthe $\\varepsilon$-neighbourhood of $B^n_{\\rho}(1)$ \ncontains the zeroset of  $P(x) = dist (x,G_{\\rho})$.\nIf $(N,Q)$ is an elementary extension of $(G,P)$ then \n$(N,Q)$ satisfies the condition \n$$ \nsup_x inf_{y_1} ...inf_{y_n}max(d(y_1 ,1)\\dot{-}\\rho ,...,  d(y_n ,1)\\dot{-}\\rho, |Q(x) -d(x,y_1 \\cdot ...\\cdot y_n )|\\dot{-}\\varepsilon )=0, \n$$ \ni.e. the $\\varepsilon$-neighbourhood of $B^n_{\\rho}(1)$ \ncontains the zeroset of $Q(x)$.   \nIn particular the zeroset of $Q$ coincides with the closure of $G_{\\rho}$, \ni.e is $G_{\\rho}$ itself and is a subset of $acl(\\emptyset )$. \nSince $(N,Q)$ is an elementary extension of $(G,P)$,\n$Q(x)$ is the distance from the zeroset of $Q$ (see Theorem 9.12 in \\cite{BYBHU}). \nIn particular any automorphism of $N$ preserves $Q$. \nUsing Corollary 9.11 of \\cite{BYBHU} (cited in  Introduction above)   \nwe see that  $P(x)$ is a definable predicate. \n$\\Box$\n\n\\bigskip \n\n \n\\begin{lemma} \nUnder the circumstances above  \nthere is a natural number $n$ so that \n$G_{\\rho}=B^n_{\\rho}(1)$. \nIn particular $G_{\\rho}$ is compact. \n\\end{lemma} \n\n{\\em Proof.} \nIf $G_{\\rho}\\not=B^n_{\\rho}(1)$ for all $n\\in\\omega$, there are \npositive rational numbers $\\varepsilon_1 ,...,\\varepsilon_n ,...$ so that \nthe $\\varepsilon_n$-neighbourhood of $B^n_{\\rho}(1)$ does not cover $G_{\\rho}$.    \nThus all statements \n$$\nsup_{x_1 ...x_n}(min(\\varepsilon_n \\dot{-} d(x,x_1 \\cdot ....\\cdot x_n ), \\rho\\dot{-}d(1,x_1 ),..., \\rho\\dot{-}d(1,x_n )))=0\n$$ \nare finitely consistent together with $P(x)=0$. \nBy compactness of continuous logic we obtain a contradiction. \n$\\Box$  \n\n\\bigskip \n\nSince $G_{\\rho}$ is a characteristic subgroup of $G$ with respect to \nthe automorphism group of the metric structure $G$, \nwe see that $Aut(G,d)$ acts correctly on $G\/G_{\\rho}$ \nby permutations of $G\/G_{\\rho}$. \nNote that $G\/G_{\\rho}$ is a discrete space with respect to \nthe topology induced by the topology of $G$. \n\n\\begin{lemma} \nThe action of $Aut(G,d)$ on $G\/G_{\\rho}$ is oligomorphic. \n\\end{lemma} \n\n{\\em Proof.} \nSince $(G,d)$ is separably categorical, $Aut(G,d)$ \nis approximately oligomorphic on $(G,d)$. \nThus for every $n$ there is a finite set $F$ of \n$n$-tuples from $G$ such that the set of orbits \nmeeting $F$ is $\\rho$-dense in $(G,d)$. \nIn particular for any $g_1 ,...,g_n \\in G$ \nthere is a tuple $(h_1 ,...,h_n )\\in F$ and \nan automorphism $\\alpha \\in Aut(G,d)$ \nsuch that $g^{-1}_i \\alpha (h_i )\\in G_{\\rho}$ \nfor all $i\\le n$. \n$\\Box$ \n\n\\bigskip \n\nTo see that Theorem \\ref{catlc} follows from lemmas above \njust take $H$ to be $G_{\\rho}$. \n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}