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+{"text":"\\section{Introduction}\nWhy do electrically charged particles exist but magneti-cally charged particles, i.e., magnetic monopoles, appa-rently do not~\\cite{Rajantie:2012xh,Rajantie:2016paj}? If they did, Maxwell's equations would have a perfect duality symmetry between electricity and magnetism.\nIn 1931, Dirac showed that (static) magnetic monopoles are compatible with quantum mechanics~\\cite{Dirac:1931kp}, provided that their magnetic charge $g$ and the electric charges $e$ of all electrically charged particle satisfy the Dirac quantisation condition,\n\\begin{equation}\n\\frac{eg}{2\\pi}\\in{\\mathbb Z}.\n\\label{equ:Diraccondition}\n\\end{equation}\nThis implies that both electric and magnetic charges have to be quantised, and if the elementary electric charge is the charge of the positron, then the elementary magnetic charge is the Dirac charge,\n\\begin{equation}\ng_D=\\frac{2\\pi}{e}\\approx 20.7.\n\\end{equation}\n\n\n\n\n\n\nBecause $g_D\\gg 1$, quantum field theory of magnetic monopoles cannot be studied using standard perturbation theory techniques.\nOne consequence of this failure of perturbation theory is that it is not possible to compute the production cross section of magnetic monopoles in particle collisions. Collider searches of magnetic monopoles can therefore only report upper bounds on the cross section, rather than actual constraints on the theory parameters such as the monopole mass. In practice, it is customary for the experiments to quote mass bounds obtained by assuming the tree-level Drell-Yan cross section, but these bounds cannot be taken literally and serve mainly as a way of comparing the performance of different experiments.\n\nFurthermore, for solitonic monopoles such as the 't~Hooft-Polyakov monopole~\\cite{tHooft:1974kcl,Polyakov:1974ek}, there are semiclassical arguments that their production cross section in elementary particle collisions would be exponentially suppressed~\\cite{Witten:1979kh,Drukier:1981fq}. The argument does not apply, at least in the same form, to elementary magnetic monopoles, but it still raises the question of whether their production cross section may also be very different from the Drell-Yan estimates.\n\nIn this paper, I will discuss a different monopole production process, namely Schwinger pair production in strong magnetic fields~\\cite{Affleck:1981ag,Affleck:1981bma,Gould:2017zwi}. Its rate can be computed using semiclassical instanton techniques, without having to assume a weak coupling. Therefore it does not suffer from the same strong-coupling issue as the perturbative calculations. It is also largely independent of the microscopic details of the monopoles, and within limits, the same results will therefore apply to both elementary and solitonic monopoles. This means that it is possible to obtain actual, largely model-indepenent lower bounds on the magnetic monopole mass.\n\nThe strongest known magnetic field in the Universe are in certain neutron stars called magnetars and in heavy ion collision experiments. I discuss the Schwinger process in both cases, and review the monopole mass bounds obtained from them.\n\n\\section{Elementary and Solitonic Monopoles}\nIn principle, magnetic monopoles can appear in quantum field theory as either elementary or composite particles. In the former case, there is a separate quantum field associated with the monopole, whereas in the latter case, they are states made up of other fields. At weak coupling, this state is usually a semiclassical soliton solution, and therefore I will refer to these as solitonic monopoles. \n\nThe best known example of a solitonic monopole is the 't~Hooft-Polyakov monopole solution~\\cite{tHooft:1974kcl,Polyakov:1974ek} in the Georgi-Glashow model~\\cite{Georgi:1972cj}, an ${\\rm SU}(2)$ gauge theory with an adjoint scalar field $\\Phi$. When the scalar field has a non-zero vacuum expectation value, it breaks the gauge symmetry to ${\\rm U}(1)$, and therefore the low energy effective theory corresponds to electrodynamics.\n\nIn the classical theory, the 't~Hooft-Polyakov monopole is a smooth solution of the field equations, of the form\n\\begin{equation}\n\\Phi^a(\\vec{x})\\propto x^a,\\quad A_i^a \\propto \\epsilon_{iaj}x_j,\n\\end{equation}\nand it has the magnetic charge $g=4\\pi\/e=2g_D$, where $e$ is the ${\\rm SU}(2)$ gauge coupling and also corresponds to the elementary electric charge. The monopole mass is given by the energy of the solution,\nand is approximately\n\\begin{equation}\nM\\approx \\frac{4\\pi m}{e^2}\\approx 137m,\n\\label{equ:tPmass}\n\\end{equation}\nwhere $m$ is the mass of the massive gauge bosons. The monopole also has a finite size,\n\\begin{equation}\nR\\approx \\frac{1}{m} \\approx \\frac{4\\pi}{e^2 M}\\approx 137M^{-1}.\n\\label{equ:tPradius}\n\\end{equation}\n\nThe same monopole exists also in the quantum theory as a non-perturbative state, and at weak coupling it is well approximated by the classical solution. Quantum corrections to it can be calculated perturbatively~\\cite{Kiselev:1988gf}, because the coupling constant of the theory is $e\\ll 1$, not $g=4\\pi\/e\\gg 1$. One can also go beyond perturbation theory by using lattice Monte Carlo methods~\\cite{Rajantie:2005hi,Rajantie:2011nq}.\n\n\nThe 't~Hooft-Polyakov solution can be found in any Grand Unified Theory (GUT)~\\cite{Preskill:1979zi}, and therefore the existence of magnetic monopoles is an unavoidable consequence of grand unification.\nThe mass of these GUT monopoles (\\ref{equ:tPmass}) would typically be around\n$10^{16}~{\\rm GeV}$, which is well above the energies of any foreseeable particle collider experiments. However, there are theories that have lighter solitonic monopole solutions~\\cite{Cho:2013vba,Ellis:2016glu}, possibly even within the reach of the Large Hadron Collider.\n\nAlthough most theoretical work has focused on solitonic monopoles, it is also important to consider the case of elementary monopoles.\nIn practice, theoretical calculations with them are difficult, not only because of their strong charge, which makes perturbation theory invalid, but also because the existing quantum theory formulations are cumbersome~\\cite{Schwinger:1966nj,Zwanziger:1970hk,Milton:2006cp}. However, that is not an argument against their existence.\nIf the magnetic monopole is an elementary particle, its mass is a free parameter. It is therefore perfectly possible that it is at the TeV scale or even lower, as long as it is compatible with the current experimental bounds.\n\n\nBecause of quantum effects, even elementary monopoles would have to have a non-zero effective size~\\cite{goebel1970spatial,goldhaber1982monopoles},\n\\begin{equation}\nR\\gtrsim R_{\\rm QM}=\\frac{g^2}{8\\pi M},\n\\label{equ:quantumradius}\n\\end{equation}\nwhere $M$ is the monopole mass. This is significantly larger than their Compton radius, and therefore even elementary monopoles would not actually appear fully pointlike. However, because of the calculational difficulties due to the strong magnetic charge, there is no detailed understanding of how this finite size arises. It is, nevertheless, interesting to note that it agrees with the size (\\ref{equ:tPradius}) of the 't~Hooft-Polyakov monopole. Indeed, the distinction between elementary and solitonic monopoles is not necessarily clear-cut, and there are examples of theories which have two dual descriptions, in one of which the monopoles are solitonic and in the other one elementary~\\cite{Montonen:1977sn}.\n\n\n\n\n\n\n\n\n\\section{Production Cross Section}\n\nThere have been several searches for magnetic monopoles in particle colliders~\\cite{Patrizii:2015uea}, including LEP, Tevatron and the LHC. The more recent results are from the ATLAS and MoEDAL experiments at the LHC~\\cite{Aad:2019pfm,Acharya:2019vtb}. \nBecause there has been no positive discovery, these experiments place upper bounds on the monopole production cross section.\nIn order to constrain actual theory parameters such as the monopole mass, one would need a reliable theoretical prediction for the production cross section. Sadly such a prediction does not exist for monopole production in collisions of elementary particles.\n\nThe main obstacle to the calculation of the production cross section of magnetic monopoles is their strong magnetic charge $g=ng_D$, $n\\in\\mathbb{Z}$, which the Dirac quantisation condition (\\ref{equ:Diraccondition}) requires to be much greater than one. This means that the calculation cannot be carried out using perturbation theory. \nThere are also strong arguments that the production cross section of solitonic monopoles, such as GUT monopoles, is suppressed by an exponential factor $\\exp(-4\/\\alpha)\\sim 10^{-238}$~\\cite{Witten:1979kh,Drukier:1981fq}. This is because they are highly ordered coherent lumps of field consisting of $O(1\/\\alpha)$ quanta. Even if there is enough energy available to produce the monopoles, it is much more likely that the same energy gets distributed to a large number of particles in a less ordered fashion. It is not known if the production cross section of elementary monopoles is suppressed by a similar factor, because we currently do not have the tools to carry out the calculation.\n\nBecause of this theoretical uncertainty, experiments tend to quote nominal mass bounds based on the tree-level Drell Yan cross section.\nFor monopoles with a single Dirac charge, $g=g_D$, this gives~\\cite{Aad:2019pfm} $M\\gtrsim 1850~{\\rm GeV}$ or $M\\gtrsim 2370~{\\rm GeV}$, depending on whether they have spin $0$ or $1\/2$, respectively.\nHowever, these nominal bounds are mainly useful for comparing different experiments and should not be interpreted as actual lower bounds on monopole masses. Much lighter monopoles can exist if their production cross sections is low. In order to obtain an actual mass bound, one therefore has to consider other processes than monopole pair production from elementary particle collisions.\n\n\n\\section{Schwinger Pair Production}\nIn addition to elementary particle collisions, magnetic monopoles can also be produced in a strong external magnetic field by the Schwinger process. \n\nIt was shown by Sauter~\\cite{Sauter:1931zz} and Schwinger~\\cite{Schwinger:1951nm} that electrically charged particles are\npair produced in a sufficiently strong electric field. This can be understood as tunneling through the Coulomb potential barrier. The rate of this process can be calculated using semiclassical instanton techniques even at strong coupling~\\cite{Affleck:1981ag,Affleck:1981bma}.\n\nIf state $|\\Omega\\rangle$ is unstable, then its decay probability is given by\n\\begin{eqnarray}\n\tP&=&1-\\left|\\langle\\Omega|\\hat{S}|\\Omega\\rangle\\right|^2\n\t=1-\\exp\\left[2\\,{\\rm Im}\\left(i\\log \\langle\\Omega|\\hat{S}|\\Omega\\rangle\\right)\\right]\n\t\\nonumber\\\\\n\t\t&=&1-\\exp\\left[2\\,{\\rm Im}\\left(\\log \\langle\\Omega|\\hat{S}_E|\\Omega\\rangle\\right)\\right],\n\\end{eqnarray}\nwhere the S-matrix $\\hat{S}$ is the time evolution operator from past infinity to future infinity in Minkowski space and $\\hat{S}_E$ the corresponding operator in Euclidean space. In the semiclassical approximation, the exponent is given by\n\\begin{equation}\n{\\rm Im}\\left(\\log \\langle\\Omega|\\hat{S}_E|\\Omega\\rangle\\right)\\sim e^{-S_{\\rm inst}},\n\\end{equation}\nwhere $S_{\\rm inst}$ is the action of the instanton solution, which is a classical solution of the Euclidean equations of motion with one negative mode, i.e. a saddle point solution. One negative mode is needed so that the solution gives an imaginary contribution to the path integral.\n\nFor an electrically charged point particle with mass $m$ and electric charge $e$ in a background gauge field $A^{\\rm ext}_\\mu$, the Euclidean action is\n\\begin{eqnarray}\nS_E[x^\\mu]&=&m\\int_0^1  d\\tau\\, \\left(\\dot{x}_\\mu \\dot{x}^\\mu\\right)^{1\/2}-ie\\oint dx^\\mu A^{\\rm ext}_\\mu\n\\nonumber\\\\\n&&\n-\\frac{e^2}{8\\pi^2}\\int d\\tau d\\tau'\n \\frac{\\dot{x}^\\mu(\\tau) \\dot{x}_\\mu(\\tau')}{|x(\\tau)-x(\\tau')|^2}\n,\n\\end{eqnarray}\nwhere $\\tau\\in [0,1)$ is a parameter along the worldline and $A^{\\rm ext}_\\mu$ is the background gauge field. The last term corresponds to the self-interactions of the particle.\n\nIn a constant background electric field $\\vec{E}$, rotation symmetry implies that the solution is a circle in the plane defined by the field $\\vec{E}$ and the time direction. Denoting the radius of the circle by $r$, its action is\n\\begin{equation}\nS_E(r)=2\\pi m r - e|\\vec{E}| \\pi r^2 - \\frac{e^2}{4},\n\\end{equation}\nwhere the unstable direction corresponds to change of $r$. \nTherefore the saddle point corresponds to the radius\n\\begin{equation}\nr=r_{\\rm inst}=\\frac{m}{e|\\vec{E}|},\n\\label{equ:instantonradius}\n\\end{equation}\nwhich maximises the action, and gives\n\\begin{equation}\nS_{\\rm inst}=S_E\\left(r_{\\rm inst}\\right)=\\frac{\\pi m^2}{e|\\vec{E}|} - \\frac{e^2}{4}.\n\\end{equation}\nThe solution has zero modes corresponding to translations in the four Euclidean directions, which contribute a spacetime volume factor, which means that there is a non-zero, finite rate per unit spacetime volume~\\cite{Affleck:1981bma}\n\\begin{equation}\n\\Gamma= D e^{-S_E}=\n\\frac{e^2|\\vec{E}|^2}{8\\pi^3}\n\\exp\\left( - \\frac{\\pi m^2}{e|\\vec{E}|} + \\frac{e^2}{4} \\right),\n\\label{equ:GammaE}\n\\end{equation}\nwhere the prefactor $D$ is given by a functional determinant of the second derivatives of the action.\n\nThe result (\\ref{equ:GammaE}) means that if the field is sufficiently strong,\n\\begin{equation}\n|\\vec{E}|\\gtrsim \\frac{\\pi m_{\\rm e}^2}{e}\\approx 10^{18}~{\\rm V\/m},\n\\end{equation}\nwhere $m_{\\rm e}$ is the electron mass,\nelectron-positron pairs are produced at an unsuppressed rate. This field is a few orders of magnitude stronger than what can be  currently reached with the most powerful lasers, and therefore Schwinger pair production has not yet been confirmed directly in experiments~\\cite{Turcu:2016dxm,Gould:2018efv}.\n\nBy the electromagnetic duality, if magnetic monopoles exist, they would be pair produced by the same mechanism in sufficiently strong external magnetic field. The rate can be obtained from Eq.~(\\ref{equ:GammaE}) by replacing $e\\rightarrow g$ and $\\vec{E}\\rightarrow \\vec{B}$~\n\\cite{Affleck:1981ag,Affleck:1981bma},\n\\begin{equation}\n\\Gamma=\n\\frac{g^2|\\vec{B}|^2}{8\\pi^3}\n\\exp\\left( - \\frac{\\pi M^2}{g|\\vec{B}|} + \\frac{g^2}{4} \\right).\n\\label{equ:GammaB}\n\\end{equation}\nBecause $g\\gg 1$, the second term in the exponent is important, and therefore the field strength needed to produce monopoles of mass $M$ is\n\\begin{equation}\n|\\vec{B}|\\gtrsim \\frac{4\\pi M^2}{g^3}.\n\\end{equation}\nCorrespondingly, if monopole production is not observed in field $\\vec{B}$, it implies a lower mass bound\n\\begin{equation}\nM\\gtrsim \\sqrt{\\frac{g^3|\\vec{B}|}{4\\pi}}.\n\\label{equ:vacuumbound}\n\\end{equation}\n\nThis calculation assumes that the monopoles are pointlike.\nThe radius of the instanton is given by the electromagnetic dual of Eq.~(\\ref{equ:instantonradius}),\n\\begin{equation}\nr_{\\rm inst}=\\frac{M}{g|\\vec{B}|},\n\\label{equ:instantonradius2}\n\\end{equation}\nso this assumption is justified if the monopole size (\\ref{equ:quantumradius}) is less than this,\n$R_{\\rm QM}\\ll r_{\\rm inst}$. It is easy to check that this is true if the monopole mass satisfies Eq.~(\\ref{equ:vacuumbound}).\n\nAs a simple application, one can consider Schwinger pair production of monopoles by the LHC magnets,\nwhich have field strength $|\\vec{B}|\\approx 8.3~{\\rm T}\\approx 1.6\\times  10^{-15}~{\\rm GeV}^2$. Even before any particle collisions were carried out, the fact that this field did not produce magnetic monopoles when the magnets were first switched on, implies the lower mass bound\n\\begin{equation}\nM\\gtrsim 1.5\\left(\\frac{g}{g_D}\\right)^{3\/2}~{\\rm keV}\n\\label{equ:LHCbound0}\n\\end{equation}\nfor the mass of monopoles.\n\n\\section{Neutron Stars}\nTo improve the bound (\\ref{equ:LHCbound0}), one needs to find stronger magnetic fields. The strongest known magnetic fields currently existing in the Universe and in neutron stars known as magnetars, 23 of which have been found~\\cite{Olausen:2013bpa}. \nThe one with the strongest field is SGR 1806-20, with $|\\vec{B}|\\approx 2\\times 10^{11}~{\\rm T}\\approx 4\\times 10^{-5}~{\\rm GeV}^2$.\nIts temperature is low in comparison, $T\\approx 0.55~{\\rm keV}$, and therefore the Schwinger pair production rate is given by\nthe zero-temperature expression~(\\ref{equ:GammaB}).\n\nAt the surface of SGR 1806-20, the ratio of the gravitational and magnetic forces acting on a monopole\nis\n\\begin{equation}\n\\frac{F_G}{F_B}=\\frac{G_N M_{\\rm NS} M}{g|\\vec{B}|R_{\\rm NS}^2}\n\\approx\n\\left(\n\\frac{g}{g_D}\\right)^{-1}\n\\frac{M}{1.8\\times 10^{17}~{\\rm GeV}},\n\\end{equation}\nwhere $R_{\\rm NS}\\sim 10~{\\rm km} \\approx  5\\times 10^{19}~{\\rm GeV}^{-1}$ is the radius of the magnetar and $M_{\\rm NS}\\sim 1.5M_{\\odot}\\approx 1.6\\times 10^{57}~{\\rm GeV} $ is its mass. \nIf a pair of magnetic monopoles with mass $M\\ll 1.8\\times 10^{17}~{\\rm GeV}$ are produced near the surface of a magnetar, the magnetic field would therefore pull one of them to the surface of the star and expel the  other one into space. This would reduce the strength of the magnetic field, in contradiction with observations~\\cite{Gould:2017zwi}. Using Eq.~(\\ref{equ:vacuumbound}), one therefore obtains a bound\n\\begin{equation}\nM\\gtrsim 0.17\\left(\\frac{g}{g_D}\\right)^{3\/2}~{\\rm GeV}.\n\\label{equ:NSbound}\n\\end{equation}\nA more detailed calculation, which takes into account the long time scale over which the field has to survive and grow, gives a somewhat stronger bound $M\\gtrsim 0.31~{\\rm GeV}$ for $g=g_D$~\\cite{Gould:2017zwi}.\n\n\nThe bound (\\ref{equ:NSbound}) was obtained by considering the magnetic field outside the magnetar, but the magnetic fields in the interior are even stronger.\nThere are also many other open questions about magnetars,\nand with a better understanding of them, one may well be able to improve the bound further.\n\n\n\n\\section{SPS Heavy Ion Collisions}\nEven stronger magnetic fields than those around magnetars are present in relativistic heavy ion collisions. The heavy ions are nuclei of heavy elements such as Au or Pb, and therefore they have a high electric charge $Q=Ze\\sim 100e$.  In these experiments, these nuclei are collided at relativistic speeds. When the collision is not head-on, one therefore has two very high electric currents moving in opposite directions past each other at the time of the collision. This induces a very strong magnetic field for a short period of time.\n\nThe most recent published monopole search in heavy ion collisions was carried out at CERN Super Proton Synchrotron (SPS) in 1997~\\cite{He:1997pj}.\nIt was a fixed-target Pb collision with beam energy $160A~{\\rm GeV}$, corresponding to a centre-of-mass energy\n$\\sqrt{s_{NN}}\\approx 17~{\\rm GeV}$ per nucleon. This produces a magnetic field \n$|\\vec{B}|\\approx 0.01~{\\rm GeV}^2$~\\cite{Skokov:2009qp} and a fireball with a high temperature~$T\\approx 0.185~{\\rm GeV}$~\\cite{Schlagheck:1999aq}. \n\nThe Schwinger process at non-zero temperature was studied in Ref.~\\cite{Gould:2017fve,Gould:2018ovk}. \nTo calculate the rate in the semiclassical approximation, one needs to find the instanton in Euclidean space with a compact imaginary time direction of length $\\beta=1\/T$. This breaks the Euclidean rotation symmetry and therefore the instanton is no longer a circle. Because of the strong coupling, the solutions have to be found numerically, and this was done in Ref.~\\cite{Gould:2017fve}. The result for the SPS case is, however, simple. Because the temperature is sufficiently high, the relevant instanton is a time-dependent sphaleron solution, which corresponds to a static monopole-antimonopole pair. \n\nThe energy of a static monopole-antimonopole pair separated by the distance $\\vec{r}$ in a constant magnetic field $\\vec{B}$ is\n\\begin{equation}\nE(\\vec{r})=2M-\\frac{g^2}{4\\pi |\\vec{r}|}-g\\vec{B}\\cdot\\vec{r}.\n\\end{equation}\nThe sphaleron configuration corresponds to the distance\n\\begin{equation}\n|\\vec{r}|=r_{\\rm sph}=\\sqrt{\\frac{g}{4\\pi |\\vec{B}|}},\n\\end{equation}\nwhich maximises the energy, which gives the sphaleron energy\n\\begin{equation}\nE_{\\rm sph}\n=E(r_{\\rm sph})=2M-2\\left(\\frac{g^3|\\vec{B}|}{4\\pi}\\right)^{1\/2}.\n\\end{equation}\nIncluding the prefactor,\nthe pair production rate is~\\cite{Gould:2018ovk}\n\\begin{equation}\n\\Gamma\\approx\n\\left(\\frac{M^5 T^9}{64\\pi^7 g B^3}\\right)^{1\/2}\\exp\\left[-\\frac{2M}{T}\\left(1-\\sqrt{\\frac{g^3|\\vec{B}|}{4\\pi M^2}}\\right)\\right],\n\\end{equation}\nand the predicted monopole pair production cross section is\n\\begin{equation}\n\\sigma_{M\\overline{M}}\\approx \\sigma_{\\rm tot}{\\cal V}\\Gamma,\n\\end{equation}\nwhere $\\sigma_{\\rm tot}\\approx 6.3~{\\rm b}$ is the total inelastic cross section, and ${\\cal V}$ is the spacetime volume of the collision.\nThe failure of SPS heavy ion collisions to produce magnetic monopoles implies an upper bound \n$\\sigma_{M\\overline{M}}\\lesssim 1.9~{\\rm nb}$ on the monopole pair production cross section~\\cite{He:1997pj}.\nThis translates to a bound on the monopole mass~\\cite{Gould:2017zwi},\n\\begin{equation}\nM\\gtrsim\n\\left(2.0+2.6\\left(\\frac{g}{g_D}\\right)^{3\/2}\\right)~{\\rm GeV}.\n\\label{equ:SPSbound}\n\\end{equation}\n\n\n\\section{LHC Heavy Ion Collisions}\nThe magnetic field produced by a heavy ion collision increases with the collision energy, and therefore the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) should be able to provide much stronger bounds on monopole pair production and, therefore, on the monopole mass. There is no published data on monopole searches at RHIC, so I will focus on the LHC, which carried out a month-long heavy ion run in November 2018, with collision energy per nucleon\n$\\sqrt{s_{NN}}=5.02~{\\rm TeV}$.\n\nThe evolution of the electromagnetic fields during the collision was studied in Ref.~\\cite{Deng:2012pc}. The field strength grows linearly with collision energy, and is highest for collisions with impact parameter $b\\approx 13~{\\rm fm}$, which corresponds to the diameter of the nucleus. Therefore we are mainly interested in peripheral collisions, and can ignore thermal effects.\n\nFor the LHC energy, the peak magnetic field strength is\n$B\\approx 7.3~{\\rm GeV}^2$, reached at the time of collision. The field is highly time-dependent, and can be approximately described by the analytic fit~\\cite{Gould:2019myj}\n\\begin{eqnarray}\n\\vec{B}&=&\\frac{B\\hat{y}}{2}\\left[\n\\left(1+\\omega^2\\left(t-\\frac{z}{v}\\right)^2\\right)^{-3\/2}\n\\right.\n\\nonumber\\\\&&\n\\left.\n+\n\\left(1+\\omega^2\\left(t+\\frac{z}{v}\\right)^2\\right)^{-3\/2}\n\\right],\n\\label{equ:timedepB}\n\\end{eqnarray}\nwhere\n$\\hat{y}$ is the direction of the impact parameter, and\n$\\omega\\approx 73~{\\rm GeV}$ parameterises the rate of the field evolution in time.\nThe time-dependence is important if the\nthe ratio of this to the inverse radius (\\ref{equ:instantonradius2})) of the constant-field instanton,\n\\begin{equation}\n\\xi=\\omega r_{\\rm inst}=\\frac{\\omega M}{gB}\n\\approx \\left(\\frac{g}{g_D}\\right)^{-1}\\frac{M}{2~{\\rm GeV}},\n\\end{equation}\nis large. Therefore\nwe see that time dependence cannot be ignored for monopoles heavier than a few ${\\rm GeV}$.\n\nIn Ref.~\\cite{Gould:2019myj}, the pair production rate was computed\nfor the time-dependent field (\\ref{equ:timedepB}), by analytically continuing it to Euclidean time $\\tau$, \n\\begin{eqnarray}\n\\vec{B}\\longrightarrow \\vec{B}_E&=&\\frac{B\\hat{y}}{2i}\\left[\n\\left(1+\\omega^2\\left(i\\tau-\\frac{z}{v}\\right)^2\\right)^{-3\/2}\n\\right.\\nonumber\\\\&&\\left.\n+\n\\left(1+\\omega^2\\left(i\\tau+\\frac{z}{v}\\right)^2\\right)^{-3\/2}\n\\right],\n\\label{equ:timedepBE}\n\\end{eqnarray}\nand then finding the instanton solution in this Euclidean background field.\n\nIn the zeroth-order approximation, in which the monopole worldline self-interactions are ignored completely,\nthe instanton solution can be found analytically~\\cite{Dunne:2005sx,Dunne:2006st,Gould:2019myj} to be an ellipse with semi-major axis $a_\\tau=M\/gB\\sqrt{1+\\xi^2}$ in the Euclidean time direction and semi-minor axis\n$a_y=M\/gB(1+\\xi^2)$ in the $y$ direction.\nIts action is~\\cite{Gould:2019myj}\n\\begin{equation}\nS_E^{(0)}=\\frac{4M^2}{gB\\xi^2}\n\\left[\n{\\bf E}(-\\xi^2)\n-{\\bf K}(-\\xi^2)\n\\right]\\sim \\frac{4M^2}{gB\\xi}\n,\n\\label{equ:SEzero}\n\\end{equation}\nwhere ${\\bf E}$ and ${\\bf K}$ are elliptic integrals, and the last expression is valid at $\\xi\\gg 1$, which corresponds to $M\\gg 1~{\\rm GeV}$. At the zeroth order, the prefactor in the rate $\\Gamma$ can also be computed analytically~\\cite{Dunne:2005sx,Dunne:2006st}. \nImportantly, the result (\\ref{equ:SEzero}) shows that the rate decreases when $\\xi$ increases.\n\nOne can also go beyond the zeroth-order approximation analytically, and compute the next-to-leading order self-interaction correction to \nthe instanton action~\\cite{Gould:2019myj},\n\\begin{equation}\nS_E^{(1)}=-\\frac{g^2}{8}\\left(\n\\sqrt{1+\\xi^2}\n+\\frac{1}{\\sqrt{1+\\xi^2}}\n\\right)\n\\sim -\\frac{g^2}{8}\\xi\n.\n\\label{equ:SEone}\n\\end{equation}\nComparing Eq.~(\\ref{equ:SEzero}) and (\\ref{equ:SEone}), we see that\nthe total next-to-leading order action $S_E^{(0)}+S_E^{(1)}$ is negative if\n$\\xi^2>32M^2\/g^3B$, which is always satisfied for LHC collisions, irrespective of the monopole mass $M$.\nAt the face value this would imply that the pair production is unsuppressed, but in practice it is more likely to be a signal that the approximations have broken down.\n\nIt is possible to find the instanton to all orders in self-interactions by solving the full non-linear equations numerically~\\cite{Gould:2019myj}. Because the equations are non-local, this is a computationally heavy task, and therefore it has currently only been done for relatively small $\\xi$, where the NLO approximation should be valid, and indeed, the results agree with it.\n\nAgain, this calculation is based on the assumption of a pointlike monopole. \nFor it to be valid, the monopole size (\\ref{equ:quantumradius}) needs to be smaller than the semi-minor axis $a_y$ of the instanton, which gives the condition $\\xi^2\\lesssim 8\\pi M^2\/g^3 B$. This coincides approximately with the value where the NLO action in the pointlike approximation becomes negative, and is not satisfied for any monopole masses in LHC heavy ion collisions. For an accurate and reliable estimate of the production rate, it will therefore be necessary to include the non-zero monopole size.\nFor 't~Hooft-Polyakov monopoles this can be done, at least in principle, by finding the instanton solution in the full field theory.\n\nAs a rough estimate of the production cross section, one can use the locally constant field approximation, in which the constant-field rate (\\ref{equ:GammaB}) is integrated over the spacetime volume were the fields are strongest. \nWithin this approximation,\nif the LHC searches do not find monopoles,  Eq.~(\\ref{equ:vacuumbound}) would give a very rough bound\n\\begin{equation}\nM\\gtrsim 70~{\\rm GeV}.\n\\label{equ:LHCbound}\n\\end{equation}\nBecause the effect of the time-dependence appears to enhance the production, this can be expected to be a conservative estimate. A complete calculation may therefore make the bound significantly stronger. \n\n\n\n\n\n\n\\section{Conclusion}\nThe electromagnetic dual of\nSchwinger pair production provides a new way of searching for magnetic monopoles.\nIf monopoles exist, they would be produced in a sufficiently strong magnetic field. Conversely, if this does not happen, monopoles either do not exist or their mass is too high. By considering physical instances of strong magnetic fields, we can therefore derive lower bounds on magnetic monopole masses. This requires a theoretical calculation of the pair production rate, which can be carried out using the semiclassical instanton method, which does not require perturbation theory or the assumption of a weak coupling. Therefore it can be applied to magnetic monopoles, whose coupling to the electromagnetic field is necessarily strong. \n\nIn a constant field at zero temperature, the rate can be computed analytically and the result is independent of the microscopic details of the monopoles and whether they are elementary and solitonic. The resulting mass bounds are therefore universal. In a time-dependent field, the full result requires a numerical calculation, and the finite monopole size needs to be taken into account. Therefore the precise result will also depend on the microscopic nature of the monopoles. For solitonic 't~Hooft-Polyakov monopoles, the numerical calculation is straightforward in principle, although computationally very demanding.\n\nUsing this approach, the magnetic fields of magnetars imply a bound~(\\ref{equ:NSbound}) of slightly below one ${\\rm GeV}$, and heavy ion collisions at SPS an order of magnitude stronger~(\\ref{equ:SPSbound}). The LHC should be able to improve significantly on these. The results of the one-month heavy-ion run in November 2018 have not yet been published, but the estimate (\\ref{equ:LHCbound}) suggests that if monopoles with mass $M\\lesssim 70~{\\rm GeV}$ exist, they would have been produced then. If data shows no monopoles, it will therefore imply a lower mass bound of the same order. However, obtaining the precise mass bound will need further theoretical work in order to take the time dependence of the collision and the finite monopole size into account.\n\nAt the face value, this appears to be $20-30$ times lower than the current bounds from ATLAS and MoEDAL. However, it is important to remember that all existing mass bounds from proton-proton collisions are based on perturbation theory, which is not actually valid for magnetic monopoles because of their strong magnetic charge. Therefore one cannot currently rule out the existence of monopoles with masses of even a few ${\\rm GeV}$, and the only way to do that is to carry out these calculations and experiments.\n\n \n\\acknowledgments\nThe author would like to thank O.~Gould, D.L.-J.~Ho, S.~Mangles, S.~Rose, D.J.~Weir and C.~Xie and the whole MoEDAL collaboration for useful discussions and collaboration on this topic. This work was supported by the UK Science and Technology Facilities Council grant ST\/P000762\/1.\n\n\n\\input RSTA_Rajantie_arxiv.bbl\n\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{}\n\n\n\n\\section{\\(CP\\)-violating weak phase \\(\\phis\\) and \\(\\DGs\\) from flavor-tagged\n         time-dependent angular analysis of \\(\\Bst\\) by ATLAS}\n\\label{delgamma}\n  New phenomena beyond the predictions of the Standard Model (SM) may\n  alter \\(CP\\) violation in \\(B\\)-decays. A channel that is expected to\n  be sensitive to new physics contributions is the decay\n  \\(\\overline{\\ensuremath{B_{s}^{0}}\\xspace}\\ensuremath{\\rightarrow}\\xspace\\ensuremath{J\/\\psi}\\phi\\) (quark content\n  \\(\\aBs\\equiv\\ensuremath{b}\\xspace\\bar{\\ensuremath{s}\\xspace}\\)~\\cite{Beringer:1900zz}).\n  \\(CP\\) violation in the \\(\\overline{\\ensuremath{B_{s}^{0}}\\xspace}\\ensuremath{\\rightarrow}\\xspace\\ensuremath{J\/\\psi}\\phi\\) decay\n  occurs due to interference between direct decays and decays occurring\n  through \\BsaBs mixing.  The oscillation frequency of \\ensuremath{B_{s}^{0}}\\xspace\\ meson mixing\n  is characterized by the mass difference \\dms\\ of the heavy (\\BH) and\n  light (\\BL) mass eigenstates. The \\(CP\\)-violating phase \\phis\\ is\n  defined as the weak phase difference between the \\( \\BsaBs\\) mixing\n  amplitude and the \\(b\\ensuremath{\\rightarrow}\\xspace{c}\\ensuremath{\\overline c}\\xspace{s}\\) decay amplitude. In the SM the\n  phase \\phis\\ is small and can be related to CKM quark mixing matrix\n  elements; a value of\n  \\(\\phis\\simeq-2\\beta_{s}=-0.0368\\pm0.0018\\)~rad\n  is predicted in the SM~\\cite{Bona:2006sa}.\n  Many models describing a new physics predict\n  larger \\(\\phis\\) values whilst satisfying all existing constraints,\n  including the precisely measured value of\n  \\dms\\ \\cite{Abulencia:2006ze, Aaij:2011qx}.  Another physical quantity\n  involved in \\(\\BsaBs\\) mixing is the width difference\n  \\(\\DGs=\\GL-\\GH\\) of \\(\\BL\\) and \\(\\BH\\).  Physics beyond the SM is not\n  expected to affect \\DGs\\ as significantly as \\phis~\\cite{Lenz:2011ti}.\n  The decay of the pseudoscalar \\(\\aBs\\) to the vector-vector\n  final-state \\(J\/\\psi\\phi\\) results in an admixture of \\(CP\\)-odd and\n  \\(CP\\)-even states, with orbital angular momentum \\(L = 0\\), \\(1\\) or\n   \\(2\\). The \\(CP\\) states are separated statistically through the\n  time-dependence of the decay and angular correlations among the\n  final-state particles.\n\\par\n  The analysis is based on a data sample of an integrated luminosity\n  \\(4.9\\,\\ifb \\) collected in 2011 by the ATLAS detector in \\(\\ensuremath{pp}\\xspace\\)\n  collisions at \\(\\sqrt{s} = 7\\TeV\\) with di-muon triggers selecting\n  \\(\\ensuremath{J\/\\psi}\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\) candidates~\\cite{atlas-conf-phis}.  The triggers\n  select di-muon events requiring both muons to have\n  \\(\\mbox{$p_{\\rm T}$}\\xspace(\\ensuremath{\\mu^\\pm}\\xspace)>4\\gevc\\), or with asymmetric requirements of\n  \\(\\mbox{$p_{\\rm T}$}\\xspace(\\mu_{1})>6\\gevc\\), and \\(\\mbox{$p_{\\rm T}$}\\xspace(\\mu_{1})>4\\gevc\\) with a rapidity\n  range of \\(\\left|\\eta(\\ensuremath{\\mu^\\pm}\\xspace)\\right|<2.4\\) for both cases.  The\n  candidates for \\(\\phi\\ensuremath{\\rightarrow}\\xspace\\ensuremath{K^-}\\xspace\\ensuremath{K^+}\\xspace \\) are reconstructed from all pairs of\n  oppositely charged particles with \\(\\mbox{$p_{\\rm T}$}\\xspace(h)>1\\gevc\\) and\n  \\(\\left|\\eta(h)\\right|<2.5\\), that are not identified as muons.\n  The \\(\\aBs\\) candidates are reconstructed using measurements provided\n  by the inner tracking detectors and the muon\n  spectrometers~\\cite{Aad:2008zzm}.\n  Candidates for\n  \\(\\aBs\\ensuremath{\\rightarrow}\\xspace\\ensuremath{J\/\\psi}(\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\mathrm{\\mu^+ \\mu^-}})\\phi(\\ensuremath{\\rightarrow}\\xspace\\ensuremath{K^-}\\xspace\\ensuremath{K^+}\\xspace)\\)~\\footnote{Unless otherwise\n    stated all references to the specific charge combination imply the\n    charge conjugate combination as well.}\n  are sought by fitting the tracks for each combination of\n  \\(\\ensuremath{J\/\\psi}\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\mathrm{\\mu^+ \\mu^-}} \\) and \\(\\phi\\ensuremath{\\rightarrow}\\xspace\\ensuremath{K^-}\\xspace\\ensuremath{K^+}\\xspace\\) to a common vertex with\n  \\(m(\\ensuremath{\\mathrm{\\mu^+ \\mu^-}})\\) constrained to the world average \n  \\(m(\\ensuremath{J\/\\psi})\\)~\\cite{Beringer:1900zz}.\n  In total \\(N(\\ensuremath{B_{s}^{0}}\\xspace)=131,000\\) candidates are collected within a mass\n  range of \\(m(\\ensuremath{B_{s}^{0}}\\xspace)\\in(5.15,\\,5.65)\\gevcc\\) to be used in the likelihood\n  fit.\n  To infer the initial flavor of the \\(\\ensuremath{B}\\xspace\\)-meson (\\(\\aBs\\) or \\(\\ensuremath{B_{s}^{0}}\\xspace\\))\n  the flavor of the \\(\\ensuremath{B}\\xspace\\)-hadron originating from the other \\(b\\)-quark\n  (OS tagging) is determined.  The flavor tagging probabilities are\n  evaluated using the calibration mode \\(\\ensuremath{B^-}\\xspace\\ensuremath{\\rightarrow}\\xspace{J}\/{\\psi{K}^{-}}\\)\n  reconstructed in the same data sample.\n  A maximum likelihood fit is performed over an unbinned set of the\n  reconstructed mass \\(m\\), the measured proper decay time, \n  \\(t\\equiv\\mbox{$L_{xy}$}\\xspace\\cdot{m}(\\ensuremath{B_{s}^{0}}\\xspace)\/({c}\\cdot\\mbox{$p_{\\rm T}$}\\xspace(\\ensuremath{B_{s}^{0}}\\xspace))\\), the\n  measured mass and proper decay time uncertainties \\(\\sigma_{m}\\) and\n  \\(\\sigma_{t}\\), and the transversity angles \\(\\Omega\\) of each \n  \\(\\overline{\\ensuremath{B_{s}^{0}}\\xspace}\\ensuremath{\\rightarrow}\\xspace\\ensuremath{J\/\\psi}\\phi\\) decay\n  candidate satisfying the selection criteria~\\cite{atlas-conf-phis}.\n  The fit finds the following values for the physics parameters of the interest:\n\\begin{align*}\n   \\phis                & = 0.12\\pm0.25\\,\\rm{(stat.)}\\pm0.11\\,\\rm{(syst.)}\\,rad\\\\\n   \\DGs                 & = 0.053\\pm0.021\\,\\rm{(stat.)}\\pm0.009\\,\\rm{(syst.)}\\,{ps}^{-1}\\\\ \n   \\ensuremath{ \\Gamma_{s}     }                  & = 0.677\\pm0.007\\,\\rm{(stat.)}\\pm0.003\\,\\rm{(syst.)}\\,{ps}^{-1}\\\\\n   |A_{0}(0)|^2         & = 0.529\\pm0.006\\,\\rm{(stat.)}\\pm0.011\\,\\rm{(syst.)}\\\\\n   |A_{\\parallel}(0)|^2 & = 0.220\\pm0.008\\,\\rm{(stat.)}\\pm0.009\\,\\rm{(syst.)}\\\\\n   {\\delta_{\\perp}}     & = 3.89\\pm0.46\\,\\rm{(stat.)}\\pm0.13\\,\\rm{(syst.)}\\,{rad}\n\\end{align*}\n  The resulting contours for the several confidence intervals are\n  produced using a profile likelihood method and are shown in\n  Fig.~\\ref{fig:Contour}.\n  \\begin{figure}[hbt]\n  \\begin{center}\n    \\includegraphics[width=0.34\\textwidth]{.\/figures\/atl-dgamma.png}\n    \\caption{Likelihood contours in the \\(\\phis{-}\\DGs\\) plane.  Three\n             contours show the \\(68\\%,\\,90\\%\\) and \\(95\\%\\) confidence intervals\n             (statistical errors only). }\n  \\end{center}\n  \\label{fig:Contour}\n  \\end{figure}\n  The values are consistent with those obtained in the previous untagged\n  analysis~\\cite{Aad:2012kba}, and as expected improving significantly\n  on the overall uncertainty on \\(\\phis\\). These results are also\n  consistent with theoretical expectations, in particular \\(\\phis\\) and\n  \\DGs\\ are in good agreement with the\n  values predicted in the Standard Model.\n\\section{Angular analysis of the decay \\(\\ensuremath{B^0}\\xspace\\ensuremath{\\rightarrow}\\xspace{K^{*0}}\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\)}\n\\label{kstarmumu}\n   Another productive area for indirect searches of new phenomena, in\n   flavor physics, is the study of flavor-changing neutral current\n   decays of \\(b \\) hadrons such as the semileptonic decay mode\n   \\(\\ensuremath{\\Bbar^0}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Kbar^{*0}}\\xspace}\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\).\n   This decay is forbidden at tree level in the SM, resulting in small\n   SM rates. From the theoretical side, robust calculations are now\n   possible for much of the phase space of this decay and the\n   calculations also indicate that new physics could give rise to\n   readily observable effects. Finally, this decay mode is relatively\n   easy to select and reconstruct at hadron colliders.  Two important\n   observables\n   in the \\(\\ensuremath{\\Bbar^0}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Kbar^{*0}}\\xspace{(892)}}(\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{K^-}\\xspace\\ensuremath{\\pi^+}\\xspace})\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\) \n   decay are the forward-backward asymmetry of the muons, \\({A_{FB}}\\),\n   and the longitudinal polarization fraction of the\n   \\(\\ensuremath{\\Kbar^{*0}}\\xspace{(892)}\\), \\({F_{L}}\\).\n   The relevant angular variables are the angle \\(\\theta_{l}\\) defined\n   as the angle between the positive (negative) muon momentum and the\n   direction opposite to the \\(\\ensuremath{\\Bbar^0}\\xspace\\,(\\ensuremath{B^0}\\xspace)\\) in the dimuon reference\n   frame and the angle \\(\\theta_{K}\\) defined as the angle between the\n   kaon momentum and the direction opposite to the \\(\\ensuremath{\\Bbar^0}\\xspace\\,(\\ensuremath{B^0}\\xspace)\\) in\n   the \\(\\ensuremath{\\Kbar^{*0}}\\xspace\\,(\\ensuremath{K^{*0}}\\xspace)\\) rest frame.  The decay rate distribution\n   of \\(\\ensuremath{\\Bbar^0}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Kbar^{*0}}\\xspace}\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\) is described as a function of\n   \\(\\theta_{l}\\) and \\(\\theta_{K}\\) and is measured in several\n   \\({q^{2}}\\) (\\(\\equiv{m^{2}(\\ensuremath{\\mathrm{\\mu^+ \\mu^-}})}\\)) bins. The main results of the\n   analysis, \\({F_{L}}\\) and \\({A_{FB}}\\) are extracted from unbinned\n   extended maximum likelihood fits to three variables: \\(m(\\ensuremath{\\Bbar^0}\\xspace)\\)\n   and the two angular variables.  The results yielded by fits for every\n   \\({q^{2}}\\) bin are compared to SM\n   predictions~\\cite{Kruger:1999xa}. Deviations from the SM predictions\n   may indicate new phenomena.\n\\par  \n   The CMS analysis~\\cite{cms-conf-afb} uses the data sample of\n   \\({\\ensuremath{\\int\\!{\\cal{L}}\\,dt}}\\xspace\\approx5.2\\,\\ifb \\) collected by several flavors of CMS dimuon\n   trigger.  The CMS trigger acceptance criteria are\n   \\(\\left|\\eta(\\ensuremath{\\mu^\\pm}\\xspace)\\right|<2.2\\), \\(\\mbox{$p_{\\rm T}$}\\xspace(\\ensuremath{\\mu^\\pm}\\xspace)>3,\\,4,\\,4.5,\\,5\\gevc\\)\n   (depending on trigger flavor) and \\(\\mbox{$p_{\\rm T}$}\\xspace(\\ensuremath{\\mathrm{\\mu^+ \\mu^-}})>6.9\\gevc\\). The CMS\n   trigger fits \\(\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\) pairs to a common point required to be\n   displaced from a vertex of the origin of \\(\\ensuremath{pp}\\xspace\\) interaction. The\n   \\(\\ensuremath{\\Kbar^{*0}}\\xspace\\) candidates are reconstructed through their decay mode\n   \\(\\ensuremath{\\Kbar^{*0}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{K^-}\\xspace\\ensuremath{\\pi^+}\\xspace\\)\n   and the \\(\\ensuremath{\\Bbar^0}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Kbar^{*0}}\\xspace{(892)}}(\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{K^-}\\xspace\\ensuremath{\\pi^+}\\xspace})\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\)\n   is reconstructed by fitting the identified \\(\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\) pair and the two\n   hadron tracks each with \\(\\mbox{$p_{\\rm T}$}\\xspace(h)>0.75\\gevc\\) to a common vertex.\n\\par  \n   The ATLAS analysis~\\cite{atlas-conf-afb} is based on\n   \\({\\ensuremath{\\int\\!{\\cal{L}}\\,dt}}\\xspace\\approx4.9\\,\\ifb\\) of data collected by ATLAS single muon and\n   dimuon triggers.  The main ATLAS triggers select di-muon events\n   requiring both muons to have \\(\\mbox{$p_{\\rm T}$}\\xspace(\\ensuremath{\\mu^\\pm}\\xspace)>4\\gevc\\) or alternatively\n   \\(\\mbox{$p_{\\rm T}$}\\xspace(\\mu_{1})>6\\gevc\\) and \\(\\mbox{$p_{\\rm T}$}\\xspace(\\mu_{2})>4\\gevc\\) with a rapidity\n   range of \\(\\left|\\eta(\\ensuremath{\\mu^\\pm}\\xspace)\\right|<\\)2.4. In most ATLAS triggers no constraint\n   on the di-muon invariant mass is applied. The \\ensuremath{\\Kbar^{*0}}\\xspace candidates are\n   reconstructed from hadron tracks with \\(\\left|\\eta(h)\\right|<\\)2.5 and\n   \\(\\mbox{$p_{\\rm T}$}\\xspace(h)>0.5\\gevc\\).\n\\par  \n   The results are shown in Fig.~\\ref{fig:fl-afb}.  No deviations from\n   the SM predictions~\\cite{Kruger:1999xa} are found by both\n   experiments. There is a slight tension between ATLAS data and SM\n   expectations for low \\({q^{2}}\\) bins.\n  \\begin{figure}[hbt]\n  \\begin{center}\n       \\subfigure{%\n           \n         \\includegraphics[width=0.25\\textwidth]{.\/figures\/cms-fl-data.png}\n       }%\n       \\subfigure{%\n           \n         \\includegraphics[width=0.25\\textwidth]{.\/figures\/atl-fl-data.png}\n       }\\\\%\n       \\subfigure{%\n           \n         \\includegraphics[width=0.25\\textwidth]{.\/figures\/cms-afb-data.png}\n        }%\n       \\subfigure{%\n           \n          \\includegraphics[width=0.25\\textwidth]{.\/figures\/atl-afb-data.png}\n       }%\n  \\end{center}\n  \\caption{ Results of the measurement of \\({F_{L}}\\) (upper plots) and\n    \\({A_{FB}}\\) (bottom plots) versus the dimuon \\({q^{2}}\\).  The\n    \\({q^{2}}\\) bins corresponding to \\(\\ensuremath{J\/\\psi} \\) and \\(\\ensuremath{\\psi{(2S)}}\\xspace\\)\n    resonances are not filled.  Good agreement with SM predictions~\\cite{Kruger:1999xa}\n    is found. }\n  \\label{fig:fl-afb}\n  \\end{figure}\n\\section{\\(\\Y1S,\\,\\Y2S,\\,\\Y3S\\) cross section measurements by CMS}\n\\label{y123s}\n  In this section we present the production of the lowest\n  \\(\\Upsilon(nS)\\) states in \\(\\ensuremath{pp}\\xspace \\) collisions studied with the CMS\n  detector. These bottomonium (\\(\\ensuremath{b\\overline b}\\xspace\\)) states are produced promptly,\n  contrary to charmonium (\\(\\ensuremath{c\\overline c}\\xspace\\)) states originating partially from\n  weak \\ensuremath{b}\\xspace-decays.  Their dominant production mechanism is through the\n  fragmentation of partons, \\(\\exegrat\\), gluons, \\(\\ensuremath{g}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{b\\overline b}\\xspace\\) though some\n  fraction of \\(S\\)-wave states result from the strong or radiative\n  decays of higher, \\(\\exegrat\\), \\({P}\\)-wave \\(\\ensuremath{b\\overline b}\\xspace\\) states.  There are\n  several bottomonium production models which predict different shapes\n  of \\(\\mbox{$p_{\\rm T}$}\\xspace\\) production spectra at their high range in \\(\\ensuremath{pp}\\xspace \\)\n  collisions~\\cite{Cho:1995vh}.  The experimental measurements from LHC\n  provide access to high-\\mbox{$p_{\\rm T}$}\\xspace range of \\(\\Upsilon(nS)\\) production\n  spectra to make a useful comparison with the predictions.\n\\par  \n  Earlier, the ATLAS Collaboration has published the production\n  cross section measurements of all three \\(\\Y1S,\\,\\Y2S,\\,\\Y3S\\)\n  states~\\cite{Aad:2012yna}. The analysis was based on\n  \\({\\ensuremath{\\int\\!{\\cal{L}}\\,dt}}\\xspace\\approx1.8\\,\\ifb\\) collected by ATLAS dimuon triggers. The total\n  production cross-section over \\(p_{\\rm T}^{\\Upsilon}<70\\gevc \\) and\n  \\(\\left|y^{\\Upsilon}\\right|<2.25\\) and the \\(\\mbox{$p_{\\rm T}$}\\xspace \\) spectra in central\n  rapidity \\(\\left|y^{\\Upsilon}\\right|<1.2\\) and forward\n  \\(1.2<|y^{\\Upsilon}|<2.25\\) rapidity intervals have been\n  measured~\\cite{Aad:2012yna}.\n\\par  \n  The latest analysis by the CMS Collaboration presented here is based on an\n  unprescaled dimuon trigger involving the tracker and muon\n  systems~\\cite{cms-conf-yns}. The larger data sample of\n  \\({\\ensuremath{\\int\\!{\\cal{L}}\\,dt}}\\xspace\\approx4.9\\,\\ifb\\) contains \\(\\Upsilon(nS)\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\) candidates\n  with vertex fitted and produced in a central area,\n  \\(\\left|y^{\\Upsilon}\\right|<0.6\\) with a broad \\(\\mbox{$p_{\\rm T}$}\\xspace\\) range of\n  \\(10-100\\gevc\\). The experimental spectra are found from fits of the\n  invariant mass \\(m(\\ensuremath{\\mathrm{\\mu^+ \\mu^-}})\\) distributions of the candidates\n  reconstructed in several \\mbox{$p_{\\rm T}$}\\xspace bins of \\(\\left|y^{\\Upsilon}\\right|<0.6\\)\n  rapidity range.  The raw spectra are corrected by the trigger\n  efficiency, by the acceptance of analysis criteria and normalized to\n  the luminosity and \\({\\ensuremath{\\cal B}\\xspace}(\\Upsilon(nS)\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}) \\).\n  The results are shown in Fig.~\\ref{fig:prod-yns} for \\(\\Y1S\\) and\n  \\(\\Y2S\\) states~\\cite{cms-conf-yns}.  The \\mbox{$p_{\\rm T}$}\\xspace-spectra reveal at\n  \\(\\mbox{$p_{\\rm T}$}\\xspace\\gsim{20}\\gevc\\) the change of an exponential shape to a\n  power-law behavior.\n  To emphasize the transition area the production\n  ratio\n  \\({\\frac{d\\sigma}{d{p_{T}}}}\\cdot{\\ensuremath{\\cal B}\\xspace}(\\Y2S)\/{\\frac{d\\sigma}{d{p_{T}}}}\\cdot{\\ensuremath{\\cal B}\\xspace}(\\Y1S)\\)\n  is presented at the bottom right plots both for CMS and\n  ATLAS. Interestingly enough, the recently published ATLAS\n  spectra~\\cite{Aad:2012yna} do show the similar break-down of the shape\n  around \\(\\mbox{$p_{\\rm T}$}\\xspace\\sim20\\gevc \\) as the right bottom plot of\n  Fig.~\\ref{fig:prod-yns} demonstrates. In a summary, new measurements\n  of\n  \\({\\frac{d\\sigma}{d{p_{T}}}}\\Big|_{|y|<0.6}\\times{{\\ensuremath{\\cal B}\\xspace}\\big(\\Upsilon(nS)\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\big)}\\)\n  in a wide range of \\(\\mbox{$p_{\\rm T}$}\\xspace\\in(10-100)\\gevc\\) are made with CMS detector.\n  The transition from nearly exponential cross section decrease with\n  \\(\\mbox{$p_{\\rm T}$}\\xspace \\) to power-law behavior for all three \\(\\Upsilon(nS)\\) is\n  observed presenting a challenge to theoretical models.\n\\begin{figure}[hbt]\n  \\begin{center}\n       \\subfigure{%\n           \n         \\includegraphics[width=0.25\\textwidth]{.\/figures\/cms-xs_fit_1S.pdf}\n       }%\n       \\subfigure{%\n           \n         \\includegraphics[width=0.25\\textwidth]{.\/figures\/cms-xs_fit_2S.pdf}\n       }\\\\%\n       \\subfigure{%\n           \n         \\includegraphics[width=0.25\\textwidth]{.\/figures\/cms-Ratio2S-1S.pdf}\n        }%\n       \\subfigure{%\n           \n          \\includegraphics[width=0.25\\textwidth]{.\/figures\/atlas-ratio2S-1S.png}\n       }%\n  \\end{center}\n  \\caption{ CMS results: \\(\\Y1S \\) production cross section spectrum\n            (upper left plot), \\(\\Y2S \\) production spectrum cross section\n            (upper right plot) and the ratio \n            \\({\\frac{d\\sigma}{d{p_{T}}}}\\cdot{\\ensuremath{\\cal B}\\xspace}(\\Y2S)\/{\\frac{d\\sigma}{d{p_{T}}}}\\cdot{\\ensuremath{\\cal B}\\xspace}(\\Y1S)\\)\n            of production spectra (bottom left plot).\n            ATLAS result to compare: ATLAS\n            \\((\\Y2S)\/(\\Y1S)\\) \n            production ratio (bottom right plot). The same change in\n            shape of the production ratio at high \\(\\mbox{$p_{\\rm T}$}\\xspace \\) is observed.}\n  \\label{fig:prod-yns}\n\\end{figure}\n\\section{\\(\\ensuremath{B^+}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{J\/\\psi}\\ensuremath{K^+}\\xspace\\) cross section measurements by ATLAS}\n\\label{bpxsec}\n  Measurements of the \\ensuremath{b}\\xspace-hadron production cross section in \\(\\ensuremath{pp}\\xspace\\)\n  collisions at LHC provide further tests of QCD calculations for\n  heavy-quark production at higher center-of-mass energies and in wider\n  transverse momentum (\\mbox{$p_{\\rm T}$}\\xspace) and rapidity (\\(y\\)) ranges, thanks to the\n  extended coverage and excellent performance of the LHC\n  detectors. ATLAS and CMS are sensitive in the central rapidity region,\n  so their measurements are complementary to open beauty measurements\n  with LHCb detector.\n\\par  \n  In this section we present the production cross section measurement\n  for \\(\\ensuremath{B^+}\\xspace \\) reconstructed in its fully exclusive decay mode to\n  \\(\\ensuremath{J\/\\psi}(\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\mathrm{\\mu^+ \\mu^-}})\\ensuremath{K^+}\\xspace \\) using the ATLAS detector. The data for this\n  analysis correspond to an integrated luminosity\n  \\({\\ensuremath{\\int\\!{\\cal{L}}\\,dt}}\\xspace\\approx2.4\\,\\ifb\\) collected at \\(\\mbox{$\\sqrt{s}$}\\xspace=7\\tev \\) by a dimuon\n  trigger, which requires the presence of at least two muon candidates\n  of \\(\\mbox{$p_{\\rm T}$}\\xspace(\\ensuremath{\\mu^\\pm}\\xspace)>4.0\\gevc\\) and \\(\\left|\\eta(\\ensuremath{\\mu^\\pm}\\xspace)\\right|<2.4\\) each.  Offline,\n  the events are required to contain at least one pair of reconstructed\n  muons, and each pair is fitted using a vertexing algorithm. The\n  corresponding \\(\\ensuremath{J\/\\psi}\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\mathrm{\\mu^+ \\mu^-}} \\) candidate with fitted common vertex\n  is selected with the invariant mass, \\(m(\\ensuremath{\\mathrm{\\mu^+ \\mu^-}})\\in(2.7,\\,3.5)\\gevcc \\).\n  The muon tracks of the selected \\(\\ensuremath{J\/\\psi}\\) are again fitted to a common\n  vertex with an additional hadron track of \\(\\mbox{$p_{\\rm T}$}\\xspace(h)>1\\gevc\\) and with\n  the \\(\\ensuremath{K^\\pm}\\xspace\\) mass assigned.  The three-track vertex fit is performed\n  by constraining the muon tracks to the \\(\\ensuremath{J\/\\psi} \\) world average\n  mass~\\cite{Beringer:1900zz}.  The \\(\\ensuremath{B^+}\\xspace\\) candidates with\n  \\(\\mbox{$p_{\\rm T}$}\\xspace(\\ensuremath{B^+}\\xspace)>9\\gevc\\) and \\(\\left|{y(\\ensuremath{B^+}\\xspace)}\\right|< 2.3\\) in the mass range\n  \\(m(\\ensuremath{B^+}\\xspace)\\in(5.040,\\,5.800)\\gevcc\\) are kept for further analysis.\n  The number of reconstructed \\(\\ensuremath{B^+}\\xspace\\) mesons is obtained using a binned\n  maximum likelihood fit to the invariant mass of the selected\n  candidates per every \\((\\Delta{\\mbox{$p_{\\rm T}$}\\xspace},\\,\\Delta{y})\\) bin.\n  The cross section \\(d^{2}\\sigma(\\ensuremath{pp}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{B^+}\\xspace\\,+X)\/d{\\mbox{$p_{\\rm T}$}\\xspace}d{y}\\) for\n  four \\(\\Delta{y} \\) and eight \\(\\Delta{\\mbox{$p_{\\rm T}$}\\xspace} \\) intervals, covering the\n  range of \\(\\left|{y}\\right|< 2.25\\) and \\(\\mbox{$p_{\\rm T}$}\\xspace\\in(9,\\,120)\\gevc \\) are\n  presented in Fig.~\\ref{fig:prod-bp}~\\cite{ATLAS:2013cia}.\n  The measured differential cross section is compared with the\n  {\\sc QCD NLO} calculations.  The predictions are obtained using\n  {\\sc Powheg+Pythia} and {\\sc MC$@$NLO+Herwig} and are quoted with an\n  uncertainty from renormalization and factorization scales and \\ensuremath{b}\\xspace-quark\n  mass of the order of \\((20-40)\\% \\). Within these uncertainties,\n  {\\sc Powheg+Pythia} predictions are in agreement both in absolute scale\n  and in the shape with the measured \\(\\mbox{$p_{\\rm T}$}\\xspace\\) and \\(y\\) double\n  differential distributions. At low \\(\\left|{y}\\right|\\), \n  {\\sc MC$@$NLO+Herwig} predicts lower production cross section and a\n  softer \\(\\mbox{$p_{\\rm T}$}\\xspace \\) spectrum than the one observed in data, which becomes\n  harder for \\(\\left|{y}\\right|>1.0\\).  An {\\sc FONLL} calculation\n  with \\({f_b}\\rightarrow\\ensuremath{B^+}\\xspace=(40.1\\pm1.3)\\%\\) to fix the overall scale,\n \n \n  is in a good agreement with the measured spectra, in particular at \n  \\(\\mbox{$p_{\\rm T}$}\\xspace<30\\gevc\\) range.\n\\begin{figure}[hbt]\n  \\begin{center}\n       \\subfigure{%\n           \n            \\includegraphics[width=0.25\\textwidth]\n            {.\/figures\/atl-bp-xsec-pt.pdf}\n       }%\n       \\subfigure{%\n          \n           \\includegraphics[width=0.25\\textwidth]\n           {.\/figures\/atl-cms-bp-xsec-pt.pdf}\n       }%\n  \\end{center}\n  \\caption{ Double-differential cross section of \\ensuremath{B^+}\\xspace production as a\n            function of \\mbox{$p_{\\rm T}$}\\xspace for different rapidity ranges: (left plot) The data\n            points are compared to {\\sc NLO} predictions from {\\sc Powheg} and\n            {\\sc MC$@$NLO}; \n            (right plot) the data points are compared with \n            {\\sc  FONLL} predictions (see also inset) and \n            the CMS results as well.\n  }\n  \\label{fig:prod-bp}\n\\end{figure}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and main results}\n\nWe begin by introducing some notation. By $H$ we denote the Hilbert space\nwith inner product $\\langle \\cdot,\\cdot \\rangle$ and norm $\\|\\cdot\\|$,\nand by $A$ we denote an unbounded operator from its domain $D(A)\n\\subset H$ to $H$.\n\nIf $A$ is self-adjoint and has a compact resolvent operator, then it has an orthonormal basis of eigenvectors. Unfortunately, even a slight perturbation of $A$ can destroy the self-adjointness of $A$ and so also the orthonormal basis property of the eigenvectors. However, in general the (normalized) eigenvectors, $\\{\\phi_n\\}_{n \\in {\\mathbb N}}$  will still form a Riesz basis, i.e., their span is dense in $H$ and there exist (positive) constants $m$ and $M$ such that\n\\begin{equation}\n  \\label{eq:1.1}\n    m \\sum_{n=1}^N |\\alpha_n|^2 \\leq \\left\\|\\sum_{n=1}^N \\alpha_n \\phi_n \\right\\|^2 \\leq  m \\sum_{n=1}^N |\\alpha_n|^2\n\\end{equation}\nfor every sequence $\\{\\alpha_n\\}_{n=1}^N$.\nIf $A$ processes a Riesz basis of eigenvectors, then many system theoretic properties like stability, controllability, etc.\\ are easily checkable, see e.g.\\ \\cite{CuZw95}. \n\nSince this Riesz-basis property is so important there is an extensive\nliterature on this problem. We refer to the book of Dunford and\nSchwartz \\cite{DuSc71}, where this problem is treated for discrete\noperators, i.e., the inverse of $A$ is compact. They apply these\nresults to differential operators. Riesz spectral properties for\ndifferential operators is also the subject of Mennicken and Moller\n\\cite{MeMo03} and Tretter \\cite{Tret00}. In these references no use is\nmade of the fact that for many differential operators the abstract\ndifferential equation\n\\begin{equation}\n  \\label{eq:1.1a}\n  \\dot{x}(t) = A x(t), \\qquad x(0)=x_0\n\\end{equation}\non the Hilbert space $H$ has a unique solution for every initial\ncondition $x_0$, i.e., $A$ is the infinitesimal generator of a\n$C_0$-semigroup. In \\cite{XuYu05} this property is used. The property\nwill also be essential in our paper. In many applications the differential\noperator $A$ arises from a partial differential equation, for which it is\nknown that (\\ref{eq:1.1a}) has a solution. Hence the assumption that $A$ generates a\n$C_0$-semigroup is not strong. However, for our result the semigroup\nproperty does not suffice, we need that $A$ generates a group, i.e.,\n(\\ref{eq:1.1a}) possesses a unique solution forward and backward in\ntime. Since we also assume that the eigenvalues lie in a strip\nparallel to the imaginary axis, the group condition is not very\nrestrictive. For more information on groups and semigroups, we refer\nthe reader to \\cite{CuZw95,EnNa00}.\n\nThe approach which we take to prove our result is different to the one\ntaken in \\cite{DuSc71, MeMo03, Tret00}, and \\cite{XuYu05}. We use\nthe fact that every generator of a group has a bounded ${\\mathcal\n  H}_{\\infty}$-calculus on a strip. This means that to every complex\nvalued function $f$ bounded and analytic in a strip parallel to the\nimaginary axis, there exists a bounded operator $f(A)$. For more detail\nwe refer to \\cite{Haas06}. Note that the ${\\mathcal\n  H}_{\\infty}$-calculus extends the functional calculus of Von~Neumann\n\\cite{Neum96} for self-adjoint operators.\n\nWe formulate our main results.\n\\begin{theorem}\n  \\label{T1.1}\n  Let $A$ be the infinitesimal generator of the $C_0$-group $(T(t))_{t\n    \\in {\\mathbb R}}$ on the Hilbert space $H$. We denote the\n  eigenvalues of $A$ by $\\lambda_n$ (counting with multiplicity), and\n  the corresponding (normalized) eigenvectors by $\\{\\phi_n\\}$. If the\n  following two conditions hold,\n  \\begin{enumerate}\n  \\item The span of the eigenvectors form a dense set in $H$;\n  \\item The point spectrum has a {\\em uniform gap}, i.e.,\n    \\begin{equation}\n    \\label{eq:1.2}\n    \\inf_{n \\neq m} |\\lambda_n - \\lambda_m| > 0.\n    \\end{equation}\n  \\end{enumerate}\n  Then the eigenvectors form a Riesz basis on $H$.\n\\end{theorem} \n\n\\mbox{}From this result, we have some easy consequences.\n\\begin{remark}%\n\\label{R1.2}\n  \\begin{itemize}\n  \\item Since we are counting the eigenvalues with multiplicity, we see\n    from (\\ref{eq:1.2}) that all eigenvalues are simple.\n  \\item If the operator possesses a Riesz basis of eigenvectors, then\n    the spectrum equals the closure of the point spectrum. Using\n    (\\ref{eq:1.2}), we see that the spectrum of $A$, $\\sigma(A)$,\n    is pure point spectrum, and $\\sigma(A) =\\{\\lambda_n\\}_{n\\in\n      {\\mathbb N}}$.\n  \\item It is easy to see that for $\\lambda \\in \\rho(A)$, the\n    finite-range operator $\\sum_{n=1}^N \\frac{1}{\\lambda-\\lambda_n}\n    \\phi_n$ converges uniformly to $(\\lambda I - A)^{-1}$. Thus under\n    the conditions of Theorem \\ref{T1.1}, the resolvent operator of\n    $A$ is compact, i.e., $A$ is discrete.\n  \\end{itemize}\n\\end{remark}\n\nIf the eigenvalues do not satisfy (\\ref{eq:1.2}), then Theorem \\ref{T1.1} need not to hold. A simple counter example is given next.\n\\begin{example}\n  \\label{E1.3}  Let $H$ be a Hilbert space with orthonormal basis $\\{e_n\\}_{n \\in {\\mathbb N}}$. Define $A$ as\n  \\begin{equation}\n    \\label{eq:6}\n    A \\sum_{n \\in {\\mathbb N}} \\alpha_n e_n = \\sum_{k \\in {\\mathbb N}} (ki \\alpha_{2k-1} + \\alpha_{2k} ) e_{2k-1} + (k+\\frac{1}{k})i \\alpha_{2k} e_{2k}\n  \\end{equation}\n  with domain\n  \\begin{equation}\n    \\label{eq:5}\n    D(A) =\\left\\{ x= \\sum_{n=1}^{\\infty} \\alpha_n e_n \\in H \\mid \\sum_{n=1}^{\\infty} |n\\alpha_n|^2 < \\infty \\right\\}.\n  \\end{equation}\n  Hence the operator $A$ is block diagonal, i.e.,\n  \\begin{equation}\n    \\label{eq:7}\n    A = \\mathrm{diag}\\, \\left[ \\begin{array}{cc} ki & 1 \\\\ 0 & (k +\\frac{1}{k})i \\end{array} \\right].\n  \\end{equation}\n  Using this structure, it is easy to see that $A$ generates a strongly continuous group on $H$, and that its eigenvalues are given by $\\{ ki, (k+\\frac{1}{k})i; k \\in {\\mathbb N}\\}$, with the normalized eigenvectors $\\phi_{2k-1} = e_{2k-1}$ and $\\phi_{2k} = \\frac{1}{\\sqrt{k^2+1}} \\left( ki e_{2k-1} + e_{2k} \\right)$, $k \\in {\\mathbb N}$. \n\n  Since\n  \\[\n     \\inf_k \\| i \\phi_{2k-1} - \\phi_{2k} \\| =0\n  \\]\n  we have that the eigenvectors do not from a Riesz basis.\n\\end{example}\n  \n  It is trivial to see that the eigenvalues in the above example can be written as the union of two sets with every subset satisfying (\\ref{eq:1.2}). Furthermore, if we would group the eigenvectors as $\\Phi_n = \\{ \\phi_{2n-1},\\phi_{2n} \\}$, then the (spectral) projections, $P_n$, on the span of $\\Phi_n$ satisfy\n  \\begin{equation}\n    \\label{eq:8}\n    \\sum_{n} \\|P_n x \\|^2 = \\|x\\|^2.\n  \\end{equation}\nThis is equivalent to the fact that the projections are orthonormal. The concept of a Riesz basis is an extension of the concept of an orthonormal basis. Similarly, we can extend the concept of orthonormal projections.\n\\begin{definition}\n  \\label{D1.4}\n  The family of projections $\\{P_n, n \\in {\\mathbb N}\\}$ is a Riesz family if there exists constants $m_1$ and $M_1$ such that\n  \\begin{equation}\n    \\label{eq:9}\n    m_1 \\|x\\|^2 \\leq \\sum_{n} \\|P_n x\\|^2 \\leq M_1 \\|x\\|^2\n  \\end{equation}\n  for all $x \\in H$.\n\\end{definition}\n\nPlease note that in \\cite[section I.1.4]{AvIv95} the range of the projections is called a (Riesz) basis. We found it confusing with the standard definition of a Riesz basis, and therefor we use Riesz family. This concept is equivalent to Riesz basis in parenthesis, see \\cite{Shka86}.\nWermer \\cite{Werm54} proved the following characterization.\n\\begin{lemma}\n\\label{L1.5}\n  A family of projections is a Riesz family if and only if there exists a $M_2$ such that for every subset ${\\mathbb J}$ of ${\\mathbb N}$ there holds\n  \\begin{equation}\n    \\label{eq:10}\n    \\|\\sum_{n \\in {\\mathbb J}} P_n \\| \\leq M_2.\n  \\end{equation}\n\\end{lemma}\n\nIn the example we saw that we had a Riesz family of spectral projections. The following theorem states that this always hold when the eigenvalues can be decomposed in  a finite number of sets with every set satisfying (\\ref{eq:1.2}).\n\\begin{theorem}\n\\label{T1.6}\n  Let $A$ be the infinitesimal generator of the $C_0$-group $(T(t))_{t\n    \\in {\\mathbb R}}$ on the Hilbert space $H$. We denote the\n  eigenvalues of $A$ by $\\lambda_n$ (counting with multiplicity). If the\n  following two conditions hold,\n  \\begin{enumerate}\n  \\item The span of the (generalized) eigenvectors form a dense set in $H$;\n  \\item The eigenvalues $\\{\\lambda_n\\}$ can be decomposed into $K$ sets, with every set having a uniform gap. \n  \\end{enumerate}\n  Then there are spectral projections $P_n$, $n \\in {\\mathbb N}$ such that\n  \\begin{enumerate}  \n  \\item $\\sum_n P_n = I$, i.e., for every $x\\in H$ there holds $\\lim_{N\\rightarrow \\infty} \\sum_{n=1}^N P_n x = x$;\n  \\item The dimension of the range of $P_n$ is at most $K$;\n  \\item The family of projections $\\{P_n, n \\in {\\mathbb N}\\}$ is a Riesz family.\n  \\end{enumerate}\n\\end{theorem}\n\nSo the difference with Theorem \\ref{T1.1} is that we allow for non-simple eigenvalues, and that the eigenvalues may cluster. Apart from these differences, the other remarks of Remark \\ref{R1.2} still hold. An operator satisfying the conditions of Theorem \\ref{T1.6} has pure point spectrum and has compact resolvent.\n\n\\section{Functional calculus for groups.}\n\\label{sec:2}\n\nWe begin by introducing some notation. \nSince $\\left(T(t)\\right)_{t \\in {\\mathbb R}}$ is a group, there exists\na $\\omega_0$ and $M_0$ such that $\\|T(t)\\| \\leq M_0 e ^{\\omega_0 |t|}$.\n\nFor $\\alpha>0$ we define a strip parallel to the imaginary axis by $S_{\\alpha}:=\\{ s\n\\in {\\mathbb C} \\mid - \\alpha < \\mathrm{Re}(s) < \\alpha \\}$.\n\nBy ${\\mathcal H}^{\\infty}(S_{\\alpha})$ we denote the linear space of\nall functions from $S_{\\alpha}$ to ${\\mathbb C}$ which are analytic\nand (uniformly) bounded on $S_{\\alpha}$. The norm of a function in\n${\\mathcal H}^{\\infty}(S_{\\alpha})$ is given by\n\\begin{equation}\n  \\label{eq:2.1}\n  \\|f\\|_{\\infty}  = \\sup_{s \\in S_{\\alpha}} |f(s)|.\n\\end{equation}\nIn Haase \\cite{Haas04,Haas06} it is shown that the generator of the\ngroup, $A$, has a ${\\mathcal H}^{\\infty}(S_{\\alpha})$-calculus for\n$\\alpha > \\omega_0$. This we explain in a little bit more detail.\n\nChoose $\\omega_1$ and $\\omega$ such that $\\omega_0 < \\omega_1 <\\alpha < \\omega$. Furthermore, let $\\Gamma = \\gamma_1 \\oplus \\gamma_2$ with $\\gamma_1 = -\\omega_1- ir$, $\\gamma_2=\\omega_1 + i r$, $r \\in {\\mathbb R}$. For $f \\in {\\mathcal H}^{\\infty}(S_{\\alpha})$ and $x \\in D(A^2)$ we define\n\\begin{equation}\n\\label{eq:2.2}\n  f(A)x = \\frac{1}{2\\pi i} \\int_{\\Gamma} \\frac{f(z)}{z^2- \\omega^2} (z I - A)^{-1} d z \\cdot (A^2 - \\omega^2)x .\n\\end{equation}\nFor $\\lambda \\not\\in \\overline{S_{\\alpha}}$ we have that \n\\[\n  \\left( \\frac{1}{\\lambda - \\cdot} \\right)(A)x = (\\lambda I - A)^{-1} x\n\\]\nFurthermore, the operator defined in (\\ref{eq:2.2}) extends to a bounded operator on $H$, and\n\\begin{equation}\n  \\label{eq:2.3}\n  \\|f(A)\\| \\leq c \\|f\\|_{\\infty},\n\\end{equation}\nwith $c$ independent of $f$.\n\nIn the following lemma we show that this functional calculus behaves\nlike one would expect from the functional calculus of von~Neumann and\nDunford.\n\\begin{lemma}\n  \\label{L2.1}\n  Let $A$ be the infinitesimal generator of a group and let $\\phi_n$\n  be an eigenvector for the eigenvalue $\\lambda_n$. Then for every $f\n  \\in {\\mathcal H}^{\\infty}(S_{\\alpha})$ there holds that\n  \\begin{equation}\n    \\label{eq:2.4}\n      f(A) \\phi_n = f(\\lambda_n) \\phi_n, \\qquad n \\in {\\mathbb N}.\n  \\end{equation}\n  Furthermore, if $\\phi_{n,j}$ is the $j$-th generalized eigenvector for the eigenvalue $\\lambda_n$, i.e., $(A-\\lambda_n) \\phi_{n,j} =\\phi_{n,j-1}$, $j\\geq 1$, with $\\phi_{n,0}=\\phi_n$, then\n  \\begin{equation}\n    \\label{eq:1}\n     f(A) \\phi_{n,j} = \\sum_{m=0}^j \\frac{f^{(j-m)}(\\lambda_n)}{(j-m)!} \\phi_{n,m}, \\qquad n \\in {\\mathbb N},\n \\end{equation}\n  where $f^{(\\ell)}$ denotes the $\\ell$-th derivative of $f$.\n\\end{lemma}\n  {\\bf Proof}:\\\/ Since $\\phi_n$ is an eigenvector, it is an element of\n  $D(A^2)$ and so we may use equation (\\ref{eq:2.2}).\n  Hence\n  \\begin{eqnarray*}\n    f(A)\\phi_n &=&  \\frac{1}{2\\pi i} \\int_{\\Gamma} \\frac{f(z)}{z^2- \\omega^2} (z I - A)^{-1} d z \\cdot (A^2 - \\omega^2) \\phi_n\\\\\n    &=& \\frac{1}{2\\pi i} \\int_{\\Gamma} \\frac{f(z)}{z^2- \\omega^2} (z I - A)^{-1} d z\\, (\\lambda_n^2-\\omega^2)\\phi_n\\\\\n    &=& (\\lambda_n^2-\\omega^2) \\frac{1}{2\\pi i} \\int_{\\Gamma} \\frac{f(z)}{z^2- \\omega^2} (z I - \\lambda_n)^{-1}\\phi_n d z.\n  \\end{eqnarray*}\n  Since $f$ is bounded on $S_{\\alpha}$, we see that the integrand\n  converges quickly to zero for $|z|$ large. Hence we may apply Cauchy\n  residue theorem. The only pole within the contour is $\\lambda_n$,\n  and so we obtain\n  \\[\n    f(A) \\phi_n = (\\lambda_n^2-\\omega^2) \\frac{f(\\lambda_n)}{\\lambda_n^2- \\omega^2} \\phi_n = f(\\lambda_n) \\phi_n.\n  \\]\n  This shows equation (\\ref{eq:2.4}). We now prove the assertion for the generalized eigenvectors. \n\n  By induction it is easy to show that\n  \\begin{equation}\n    \\label{eq:2}\n    (zI-A)^{-1} \\phi_{n,j} = \\sum_{m=0}^j \\frac{1}{(z-\\lambda_n)^{j+1-m}} \\phi_{n,m}.\n  \\end{equation}\n  Using (\\ref{eq:2.2}) we have that\n  \\begin{align*}\n    f(A) \\phi_{n,j}  &=  \\frac{1}{2\\pi i} \\int_{\\Gamma} \\frac{f(z)}{z^2- \\omega^2} (z I - A)^{-1} d z \\cdot (A^2 - \\omega^2) \\phi_{n,j}\\\\\n      &= \\frac{1}{2\\pi i} \\int_{\\Gamma} \\frac{f(z)}{z^2- \\omega^2} (z I - A)^{-1} d z ((\\lambda_n^2 - \\omega^2)\\phi_{n,j} + 2\\lambda_n \\phi_{n,j-1} + \\phi_{n,j-2})\\\\\n      &=  (\\lambda_n^2-\\omega^2) \\frac{1}{2\\pi i} \\int_{\\Gamma} \\frac{f(z)}{z^2- \\omega^2} \\sum_{m=0}^j (z I - \\lambda_n)^{-(j-m+1)}\\phi_{n,m} d z + \\\\\n   &\\qquad 2\\lambda_n \\frac{1}{2\\pi i} \\int_{\\Gamma} \\frac{f(z)}{z^2- \\omega^2} \\sum_{m=0}^{j-1} (z I - \\lambda_n)^{-(j-m)}\\phi_{n,m} d z + \\\\\n   &\\qquad\n  \\frac{1}{2\\pi i} \\int_{\\Gamma} \\frac{f(z)}{z^2- \\omega^2} \\sum_{m=0}^{j-2} (z I - \\lambda_n)^{-(j-m-1)}\\phi_{n,m} d z \\\\\n   &= (\\lambda_n^2-\\omega^2) \\sum_{m=0}^{j} \\frac{g^{(j-m)}(\\lambda_n)}{(j-m)!} \\phi_{n,m} + \\\\\n   &\\qquad 2\\lambda_n \\sum_{m=0}^{j-1} \\frac{g^{(j-1-m)}(\\lambda_n)}{(j-m-1)!} \\phi_{n,m} +\n   \\sum_{m=0}^{j-2} \\frac{g^{(j-2-m)}(\\lambda_n)}{(j-m-2)!} \\phi_{n,m},\n  \\end{align*}\n  where we have introduced $g(z)=f(z)\/(z^2-\\omega^2)$. Using the fact that $f(z) = (z^2-\\omega^2) g(z)$, we see that for $\\ell \\geq 2$\n  \\[\n     f^{(\\ell)}(z) = (z^2-\\omega^2) g^{(\\ell)}(z) + 2 z\\ell g^{(\\ell-1)}(z) + \\ell(\\ell-1) g^{(\\ell-2)}(z) \n  \\]\n  and\n  \\[\n     f^{(1)}(z) = (z^2-\\omega^2) g^{(1)}(z) + 2 z g(z).\n  \\]\n  This implies that\n  \\begin{eqnarray*}\n   f(A) \\phi_{n,j} &=& \\sum_{m=0}^{j-2} \\left[ (\\lambda_n^2-\\omega^2) \\frac{g^{(j-m)}(\\lambda_n)}{(j-m)!} +\\right. \\\\\n   && \\hphantom{\\sum_{m=0}^{j-2} \\left[ \\right]}\n   \\left. 2\\lambda_n \\frac{g^{(j-1-m)}(\\lambda_n)}{(j-m-1)!} +\n   \\frac{g^{(j-2-m)}(\\lambda_n)}{(j-m-2)!} \\right] \\phi_{n,m} + \\\\\n   &&\n  \\left[ (\\lambda_n^2 - \\omega^2) g^{(1)}(\\lambda_n) + 2 \\lambda_n g(\\lambda_n) \\right] \\phi_{n,j-1}+ (\\lambda_n^2 - \\omega^2) g(\\lambda_n) \\phi_{n,j}\\\\\n  &=&  \\sum_{m=0}^{j-2} \\frac{1}{(j-m)!}f^{(j-m)}(\\lambda_{n}) \\phi_{n,m} + f^{(1)}(\\lambda_n)  \\phi_{n,j-1}  + f(\\lambda_n) \\phi_{n,j}.\n  \\end{eqnarray*}\n  Hence we have proved the assertion. \\hfill$\\Box$\n\\medskip\n\n\\section{Interpolation sequences}\n\nFor the proof of Theorem \\ref{T1.1} we need the following\ninterpolation result, see \\cite[Theorem VII.1.1]{Garn07}.\n\\begin{theorem}\n  \\label{T2.2}\n  Consider the sequence ${\\mu_n}$ which satisfies $\\beta_1 >\n  \\mathrm{Re}(\\mu_n) > \\beta_2 >0$ and $\\inf_{n\\neq m} |\\mu_n - \\mu_m|\n  >0$. Then for every bounded sequence $\\{\\alpha_n\\}_{n\\in {\\mathbb\n      N}}$ of complex numbers there exists a function $g$ holomorphic\n  and bounded in the right-half plane ${\\mathbb C}_+:=\\{ s \\in\n  {\\mathbb C} \\mid \\mathrm{Re}(s) >0\\}$ such that\n  \\begin{equation}\n    \\label{eq:2.5}\n      g(\\mu_n) = \\alpha_n.\n    \\end{equation}\n  Furthermore, there exists an $M$ independent of $g$ and $\\{\\alpha_n\\}_{n \\in {\\mathbb N}}$ such that\n  \\begin{equation}\n    \\label{eq:2.6}\n      \\sup_{\\mathrm{Re}(s) >0} |g(s)| \\leq M \\sup_{n \\in {\\mathbb N}} |\\alpha_n|.\n  \\end{equation}\n\\end{theorem}\n\nA sequence $\\{\\mu_n\\}_{n \\in {\\mathbb N}}$ satisfying the conditions of the above theorem is called an {\\em interpolation sequence}. It is well known that for any interpolation sequence, we can find a Blaschke product with exactly this sequence as its zero set. Using Schwartz lemma, and Lemma VII.5.3 of \\cite{Garn07} the following is easy to show.\n  \\begin{lemma}\n  \\label{L3.2}\n    Let $\\{\\mu_n\\}_{n \\in {\\mathbb N}}$ be an interpolating sequence, and let $0< \\delta := \\inf_{n\\neq m} |\\mu_n - \\mu_m|$. Furthermore, let $B(a, r)$ denote the ball in the complex plane with center $a$ and radius $r$. \n\n  Let $f$ be a function defined on $B(\\mu_n,\\delta\/2)$ and which is analytic and bounded on this set. Furthermore, $f$ is zero at $\\mu_n$, i.e., $f(\\mu_n)=0$. Then there exists a constant $m_1$ independent of $n$ and $f$ such that\n  \\begin{equation}\n    \\label{eq:11}\n      \\sup_{s \\in B(\\mu_n,\\delta\/2)} \\left| \\frac{f(s)}{Bl(s)} \\right| \\leq m_1 \\sup_{s \\in B(\\mu_n,\\delta\/2)} |f(s)|,\n  \\end{equation}\n  where $Bl(s)$ is Blaschke product with zeros $\\{\\mu_n\\}_{n \\in {\\mathbb N}}$.\n\n  Similarly,  let $f$ be a function defined on $B(\\mu_n,\\delta\/2)$ and which is analytic and bounded on this set. Furthermore, $f$ has $\\ell$ zeros at $\\mu_n$, i.e., $f(\\mu_n)=\\cdots = f^{(\\ell)}(\\mu_n)=0$. Then there exists a constant $m_\\ell$ independent of $n$ and $f$ such that\n  \\begin{equation}\n    \\label{eq:11a}\n      \\sup_{s \\in B(\\mu_n,\\delta\/2)} \\left| \\frac{f(s)}{Bl(s)^\\ell} \\right| \\leq m_\\ell \\sup_{s \\in B(\\mu_n,\\delta\/2)} |f(s)|.\n  \\end{equation}\n  \\end{lemma}\n\\begin{theorem}\n  \\label{T3.3} \n  Let $\\{\\mu_n\\}_{n \\in {\\mathbb N}}$ be a sequence of numbers in the right-half plane, and let $\\{\\zeta_n\\}_{n \\in {\\mathbb N}}$ be an interpolation sequence. Furthermore, let $K$ be a positive natural number. Suppose that we can find positive real numbers $r_n$ such that \n  \\begin{itemize}\n  \\item $0< \\inf_n r_n \\leq \\sup_n r_n < \\infty$;\n  \\item $\\{\\mu_n\\}_{n \\in {\\mathbb N}} \\subset \\cup_{n\\in {\\mathbb N}} B(\\zeta_n,r_n)$;\n  \\item The number of $\\mu_n$'s in $B(\\zeta_k,r_k)$ is bounded by $K$, and \n  \\item The distance between every two balls $B(\\zeta_n,r_n)$ and $B(\\zeta_m,r_m)$, $n \\neq m$, is bounded away from zero.\n  \\end{itemize}\n  Let $\\nu$ be an element of ${\\mathbb N} \\cup \\{0\\}$. Under these assumptions we have that for every bounded sequence $\\{\\alpha_n\\}_{n \\in {\\mathbb N}}$ there exists a function $g$ holomorphic and bounded in the right-half plane ${\\mathbb C}_+$ such that\n  \\begin{equation}\n    \\label{eq:4}\n     g(\\mu_k) = \\alpha_n \\quad \\mbox{if}\\quad  \\mu_k \\in B(\\zeta_n,r_n)\n  \\end{equation}\n  and \n  \\begin{equation}\n    \\label{eq:4a}\n     g^{(q)}(\\mu_k) = 0 \\quad \\mbox{for}\\quad  1\\leq q \\leq \\nu,\\mbox{ and } \\mu_k \\in B(\\zeta_n,r_n)\n  \\end{equation}\n  Furthermore, there exists an $M$ independent of $g$ and $\\{\\alpha_n\\}_{n \\in {\\mathbb N}}$ such that\n  \\begin{equation}\n    \\label{eq:16}\n    \\sup_{\\mathrm{Re}(s) >0} |g(s)| \\leq M \\sup_{n \\in {\\mathbb N}} |\\alpha_n|.\n  \\end{equation}\n  Hence the function $g$ interpolates the points $\\mu_k$, but points close to each other are given the same value.\n\\end{theorem}\n  {\\bf Proof}:\\\/ We present the proof for the case that $\\kappa =2$ and $\\nu=1$. For higher values of $\\kappa$ and $\\nu$ the proof goes similarly. We denote the points $\\mu_n \\in B(\\zeta_n,r_n)$ by $\\xi_n$ and $\\gamma_n$, and we assume that $\\gamma_n \\neq \\xi_n$.  By the assumptions, we know that these are interpolation sequences.\n\nLet $Bl_1(s)$ and $BL_2(s)$ be the Blaschke products with zeros $\\{\\xi_n\\}_{n \\in {\\mathbb N}}$ and $\\{\\gamma_n\\}_{n \\in {\\mathbb N}}$, respectively.\n\nWe write $g$ in the following form\n\\begin{equation}\n  \\label{eq:12}\n  g(s) = g_1(s) + Bl_1(s)g_2(s) + Bl_1(s)^2 g_3(s) + Bl_1(s)^2Bl_2(s) g_4(s), \n\\end{equation}\nand we construct bounded analytic functions $g_j$ such that (\\ref{eq:4})--(\\ref{eq:16}) are satisfied. For our situation, the interpolation conditions become $g(\\xi_n)=g(\\gamma_n)=\\alpha_n$ and $g^{(1)}(\\xi_n)=g^{(1)}(\\gamma_n)=0$. Using the form (\\ref{eq:12}), these conditions are equivalent to\n\\begin{align}\n  \\label{eq:3}\n   g_1(\\xi_n) = & \\, \\alpha_n\\\\ \n   \\label{eq:13}\n   g_2(\\xi_n) = & -\\frac{g_1^{(1)}(\\xi_n)}{Bl_1^{(1)}(\\xi_n)}\\\\\n   \\label{eq:14}\n   g_3(\\gamma_n) =& \\,\\frac{\\alpha_n - g_1(\\gamma_n)- Bl_1(\\gamma_n) g_2(\\gamma_n)}{Bl_1(\\gamma_n)^2} \\\\\n   \\label{eq:15}\n   g_4(\\gamma_n) = &  \\left[ g_1^{(1)}(\\gamma_n) + Bl_1(\\gamma_n)g_1^{(1)}(\\gamma_n) + Bl_1^{(1)}(\\gamma_n) g_2(\\gamma_n) +\\right.\\\\\n\\nonumber\n    &\\left. 2 Bl_1(\\gamma_n) Bl_1^{(1)}(\\gamma_n)g_3(\\gamma_n) + Bl_1(\\gamma_n)^2g_3^{(1)}(\\gamma_n)\\right]\\left[Bl_1(\\gamma_n)^2Bl_2^{(1)}(\\gamma_n)\\right]^{-1}.\n\\end{align}\nSince $\\{\\xi_n\\}$ is an interpolation sequences, and since $\\{\\alpha_n\\}$ is a bounded sequence, we have by Theorem \\ref{T2.2} the existence of a bounded $g_1$ for which (\\ref{eq:2.6}) holds.\n\n\\mbox{} Define the function $f(s)=-g_1(s)+\\alpha_n$. This is clearly bounded and analytic on $B(\\zeta_n,r_n)$ and zero at $s=\\xi_n$. By Lemma \\ref{L3.2} the function $f(s)\/Bl_1(s)$ is bounded (uniformly in $n$) in this ball. l'Hopital gives that the value at $s=\\xi_n$ equals the right-hand side of (\\ref{eq:13}), and so the right hand-side is a bounded sequence. Furthermore, since $\\{\\xi_n\\}_{n \\in {\\mathbb N}}$ is an interpolation sequence we can apply Theorem \\ref{T2.2} and find a bounded analytic function $g_2$ satisfying (\\ref{eq:13}).\n\nConsider the function\n\\begin{equation}\n  \\label{eq:17}\n  q_1(s) = \\frac{\\alpha_n - g_1(s)- Bl_1(s) g_2(s)}{Bl_1(s)^2} \n\\end{equation}\nBy (\\ref{eq:3}) and (\\ref{eq:13}), we have that the denominator has two zero's in $\\xi_n$. Furthermore, it is analytic on $B(\\zeta_n,r_n)$ and bounded independently of $n$. Lemma \\ref{L3.2} implies that $q_1(s)$ is uniformly bounded. In particular, the sequence $q_1(\\gamma_n)$ is uniformly bounded. By Theorem \\ref{T2.2} we construct a bounded $g_3$ satisfying (\\ref{eq:14}).\n\nIt remains to show that the right hand-side of (\\ref{eq:15}) is a uniformly bounded sequence. The sequence $-\\frac{g_3^{(1)}(\\gamma_n)}{Bl_2^{(1)}(\\gamma_n)}$ is uniformly bounded for the same reason why the sequence in (\\ref{eq:13}) was bounded. So we may disregard that term in (\\ref{eq:15}). Using the value of $g_3(\\gamma_n)$ as found in (\\ref{eq:14}), we have that the denominator of (\\ref{eq:15}) becomes\n\\begin{align}\n  \\label{eq:18}\n    g_1^{(1)}(\\gamma_n) + Bl_1(\\gamma_n)g_1^{(1)}(\\gamma_n) +& Bl_1^{(1)}(\\gamma_n) g_2(\\gamma_n) + \\\\\n\\nonumber\n   &2 Bl^{(1)}_1(\\gamma_n) \\frac{\\alpha_n - g_1(\\gamma_n)- Bl_1(\\gamma_n) g_2(\\gamma_n)}{Bl_1(\\gamma_n)}.\n\\end{align}\nBased on this and using equation (\\ref{eq:17}) we define in $B(\\zeta_n,r_n)$ the bounded analytic function\n\\begin{equation}\n  \\label{eq:20}\n  f(s)= g_1^{(1)}(s) + Bl_1(s)g_1^{(1)}(s) + Bl_1^{(1)}(s) g_2(s) + 2 Bl_1(s)Bl^{(1)}_1(s) q_1(s).\n\\end{equation}\nUsing (\\ref{eq:13}), we have that $f(\\xi_n)=0$, and using that \n\\[\n   q_1(\\xi_n) = \\frac{g_1^{(2)}(\\xi_n) - Bl_1^{(2)}(\\xi_n) g_2(\\xi_n) - 2 Bl_1^{(1)}(\\xi_n) g_2^{(1)}(\\xi_n)}{2 \\left( Bl_1^{(1)}(\\xi_n)\\right)^2}\n\\]\nwe find that $f^{(1)}(\\xi_n)$ is zero as well. By Lemma \\ref{L3.2}, we conclude that \n  \\begin{align*}\n    \\left[ g_1^{(1)}(\\gamma_n) + Bl_1(\\gamma_n) \\right.  & g_1^{(1)}(\\gamma_n) + Bl_1^{(1)}(\\gamma_n) g_2(\\gamma_n) +\\\\\n     & \\left. 2 Bl_1(\\gamma_n) Bl_1^{(1)}(\\gamma_n)g_3(\\gamma_n) + Bl_1(\\gamma_n)^2g_3^{(1)}(\\gamma_n)\\right]\\left[Bl_1(\\gamma_n)^2\\right]^{-1}\n  \\end{align*}\n  is uniformly bounded. Since $Bl_2(s)$ is a Blaschke product with zeros $\\{\\gamma_n\\}$ we have that $Bl_2^{(1)}(\\gamma_n)$ is bounded away from zero. Hence the right hand-side of (\\ref{eq:15}) is a bounded sequence, and so by Theorem \\ref{T2.2} we can find the interpolating $g_4$. This concludes the construction.\n \\hfill$\\Box$\n\\medskip\n\nNow we have all the ingredients for the proof of Theorem \\ref{T1.1} and \\ref{T1.6}.\n\n\n\\section{Proof of Theorem \\ref{T1.1} and \\ref{T1.6} }\n\nAs may be clear from the formulation of the Theorems \\ref{T1.1} and \\ref{T1.6}, Theorem \\ref{T1.1} is a special case of Theorem \\ref{T1.1}. However, since the Riesz basis property is for applications more important than Riesz family, we we decided to formulate them separately. The proof of Theorem \\ref{T1.1} is more simple than that of the general theorem, but the underlying ideas are the same. \n\\medskip\n\n\\noindent{\\bf Proof of Theorem \\ref{T1.1}}: \nLet $\\alpha$ be the positive number defined at the beginning of Section \\ref{sec:2}. We define the complex numbers $\\mu_n$ as\n\\begin{equation}\n  \\label{eq:2.7}\n  \\mu_n = \\lambda_n + \\alpha \\qquad n \\in {\\mathbb N}.\n\\end{equation}\nBy the conditions on $\\alpha$ and $\\lambda_n$, we see that $\\{\\mu_n\\}_{n \\in {\\mathbb N}}$ satisfies the conditions of Theorem \\ref{T2.2}.\n\nLet ${\\mathbb J}$ be a subset of ${\\mathbb N}$. Since the eigenvalues\nsatisfy (\\ref{eq:1.2}), we conclude by Theorem \\ref{T2.2} there exists\na function $g_{{\\mathbb J}}$ bounded and analytic in ${\\mathbb C}_+$\nsuch that\n\\begin{equation}\n  \\label{eq:2.8}\n  g_{\\mathbb J}(\\mu_n)= \n  \\begin{cases} 1, & \\quad \\mbox{if } n \\in {\\mathbb J}\\\\\n                0, & \\quad \\mbox{if } n \\not\\in {\\mathbb J}. \n  \\end{cases} \n\\end{equation}\nFurthermore, see (\\ref{eq:2.6})\n\\begin{equation}\n  \\label{eq:2.10}\n  \\sup_{s \\in {\\mathbb C}_+} |g_{{\\mathbb J}}(s)| \\leq M.\n\\end{equation}\n\nGiven this $g_{\\mathbb J}$ we define $f_{\\mathbb J}$ as\n\\begin{equation}\n  \\label{eq:2.11}\n  f_{\\mathbb J}(s) = g_{\\mathbb J}(s+ \\alpha), \\qquad s \\in S_{\\alpha}.\n\\end{equation}\nThen using the properties of $g_{\\mathbb J}$ we have that $f_{\\mathbb J} \\in {\\mathcal H}_{\\infty}(S_{\\alpha})$, \n\\begin{equation}\n  \\label{eq:2.12}\n  f_{\\mathbb J}(\\lambda_n)= \n  \\begin{cases}  1, & \\quad \\mbox{if } n \\in {\\mathbb J}\\\\\n                 0, & \\quad \\mbox{if } n \\not\\in {\\mathbb J},\n  \\end{cases} \n\\end{equation}\nand there exists a $M>0$ independent of $f_{\\mathbb J}$ such that \n\\begin{equation}\n  \\label{eq:2.13}\n  \\|f_{{\\mathbb J}}\\|_{\\infty} \\leq M.\n\\end{equation}\n\nNext we identify the operator $f_{\\mathbb J}(A)$. Combining\n(\\ref{eq:2.4}) with (\\ref{eq:2.12}) gives\n\\[\n  f_{\\mathbb J}(A)\\phi_n = \n  \\begin{cases}  \\phi_n, & \\quad \\mbox{if } n \\in {\\mathbb J}\\\\\n                 0, & \\quad \\mbox{if } n \\not\\in {\\mathbb J}.\n  \\end{cases}\n\\]\nSince $f_{\\mathbb J}(A)$ is a linear operator, we obtain that\n\\begin{equation}\n  \\label{eq:2.14}\n  f_{\\mathbb J}(A) \\left(\\sum_{n=1}^N \\alpha_n \\phi_n\\right) = \\sum_{n\\in {\\mathbb J} \\cap \\{1,\\cdots,N\\}} \\alpha_n \\phi_n.\n\\end{equation}\nBy assumption the span of $\\{\\phi_n\\}_{n \\in {\\mathbb N}}$ is dense in\n$H$. Furthermore, $f_{\\mathbb J}(A)$ is a bounded operator. So we\nconclude that $f_{\\mathbb J}$ is the spectral projection associated to\nthe spectral set $\\{ \\lambda_n \\mid n \\in {\\mathbb J}\\}$.\n\nCombining (\\ref{eq:2.3}) with (\\ref{eq:2.13}) we have that these\nspectral projections are uniformly bounded. Since the eigenvalues are\nsimple, this implies that the (normalized) eigenvectors form a Riesz\nbasis, see Lemma \\ref{L1.5}. \\hfill$\\Box$\n\\medskip\n\n\\noindent{\\bf Proof of Theorem \\ref{T1.6}}:\nAs in the previous proof we can shift the eigenvalues by $\\alpha$ such that they all lies in the right half plane, and they are bounded away from the imaginary axis. We denote these shifted eigenvalues by $\\mu$. \nSince the $\\lambda_n$'s can be decomposed into $K$ interpolation sequences, the same holds for $\\mu_n$. Hence we can group the $\\mu_n$'s as in Theorem \\ref{T3.3}. Note that we are counting the eigenvalues $\\lambda_n$'s, and thus $\\mu_n$, with their multiplicity, and so in every ball there can at most be $K$ different values, and the multiplicity of every value is also bounded by $K$. Let us renumber the $\\mu_n$'s such that the values in the $n$'th ball are given by $\\mu_{n,k}$, $k=1,\\cdots,n_k$.  The eigenvectors corresponding to $\\lambda_{n,k}=\\mu_{n,k}-\\alpha$ are denoted by $\\phi_{n,k,0}$, and the generalized eigenvectors by $\\phi_{n,k,j}$, $j =1,\\cdots, j_{n,k}$. By construction we have that \n\\begin{equation}\n  \\label{eq:21}\n  \\sum_{k=1}^{n_k} \\left[ j_{n,k}+1\\right] \\leq K.\n\\end{equation}\n\nLet ${\\mathbb J}$ be a subset of ${\\mathbb N}$. By the above, we conclude from Theorem \\ref{T3.3} that there exists a function $g_{\\mathbb J}$ bounded and analytic in ${\\mathbb C}_+$ such that\n\\begin{equation}\n  \\label{eq:2.15}\n  g_{\\mathbb J}(\\mu_{n,k})= \n  \\begin{cases} 1, & \\quad \\mbox{if } n \\in {\\mathbb J}\\\\\n                0, & \\quad \\mbox{if } n \\not\\in {\\mathbb J}. \n  \\end{cases} \n\\end{equation}\nand\n\\begin{equation}\n  \\label{eq:2.16}\n   g_{\\mathbb J}^{(j)}(\\mu_{n,k})=0, \\qquad j=1,\\cdots, K.\n\\end{equation}\nFurthermore, see (\\ref{eq:16})\n\\begin{equation}\n  \\label{eq:2.17}\n  \\sup_{s \\in {\\mathbb C}_+} |g_{{\\mathbb J}}(s)| \\leq M.\n\\end{equation}\n\nGiven this $g_{\\mathbb J}$ we define $f_{\\mathbb J}$ as\n\\begin{equation}\n  \\label{eq:2.18}\n  f_{\\mathbb J}(s) = g_{\\mathbb J}(s+ \\alpha), \\qquad s \\in S_{\\alpha}.\n\\end{equation}\nThen using the properties of $g_{\\mathbb J}$ we have that $f_{\\mathbb J} \\in {\\mathcal H}_{\\infty}(S_{\\alpha})$, \n\\begin{equation}\n  \\label{eq:2.19}\n  f_{\\mathbb J}(\\lambda_{n,k})= \n  \\begin{cases}  1, & \\quad \\mbox{if } n \\in {\\mathbb J}\\\\\n                 0, & \\quad \\mbox{if } n \\not\\in {\\mathbb J},\n  \\end{cases} \n\\end{equation}\n\\begin{equation}\n  \\label{eq:2.20}\n   f_{\\mathbb J}^{(j)}(\\lambda_{n,k})=0, \\qquad j=1,\\cdots, K.\n\\end{equation}\nand there exists a $M>0$ independent of ${\\mathbb J}$ such that \n\\begin{equation}\n  \\label{eq:2.21}\n  \\|f_{{\\mathbb J}}\\|_{\\infty} \\leq M.\n\\end{equation}\n\nNext we identify the operator $f_{\\mathbb J}(A)$. Combining\n(\\ref{eq:1}) with (\\ref{eq:2.19}) and (\\ref{eq:2.20}), gives\n\\[\n  f_{\\mathbb J}(A)\\phi_{n,k,j} = \n  \\begin{cases}  \\phi_{n,k,j}, & \\quad \\mbox{if } n \\in {\\mathbb J}\\\\\n                 0, & \\quad \\mbox{if } n \\not\\in {\\mathbb J}.\n  \\end{cases}\n\\]\nSince $f_{\\mathbb J}(A)$ is a linear operator, we obtain that\n\\begin{equation}\n  \\label{eq:2.22}\n  f_{\\mathbb J}(A) \\left(\\sum_{n=1}^N \\sum_{k=1}^{n_k} \\sum_{j=0}^{j_{n,k}} \\alpha_{n,k,j} \\phi_{n,k,j}\\right) = \\sum_{n\\in {\\mathbb J} \\cap \\{1,\\cdots,N\\}}\\sum_{k=1}^{n_k} \\sum_{j=0}^{j_{n,k}}  \\alpha_{n,k,j} \\phi_{n,k,j}.\n\\end{equation}\nBy assumption the span of $\\{\\phi_{n,k,j}\\}_{n \\in {\\mathbb N}, k=1,\\cdots,n_k,,j=0,\\cdots,j_{n,k}}$ is dense in\n$H$. Furthermore, $f_{\\mathbb J}(A)$ is a bounded operator. So we\nconclude that $f_{\\mathbb J}$ is the spectral projection associated to\nthe spectral set $\\{ \\lambda_{n,k} \\mid n \\in {\\mathbb J}\\}$.\n\nCombining (\\ref{eq:2.21}) with (\\ref{eq:2.22}) we have that these\nspectral projections are uniformly bounded. By Lemma \\ref{L1.5} we conclude that the spectral projections $\\{P_n\\}_{n \\in {\\mathbb N}}$, where $P_n$ is the spectral projection associated to the eigenvalues in the $n$-th ball, are a spectral family. From (\\ref{eq:21}) we see that the dimension of the range of $P_n$ is bounded by $K$. \\hfill$\\Box$\n\\medskip\n \n\n\n\n\n\n\n\n\\section{Closing remarks}\n\\label{sec:3}\n\nA natural question is the following: {\\em If $A$ is the infinitesimal generator of a group and $A$ has only point spectrum with multiplicity one, is the span over all eigenvectors dense in $H$}? \n\nIn general the answer to this question is negative. On page 665 of Hille and Phillips \\cite{HiPh57} one may find an example of a generator of a group without any spectrum. \n\nHowever, there are some interesting cases for which the answer is positive. If $A$ generates a bounded group, i.e., $\\sup_{t \\in {\\mathbb R}}\\|T(t)\\| < \\infty$, then $A$ is similar to a skew-adjoint operator, and so there is a complete spectral measure, see \\cite{Cast83,Zwar01}. Another interesting situation is the following. Since $A$ generates a group, it can be written as $A=A_0+Q$, where $A_0$ generates a bounded group, and $Q$ is a bounded linear operator, see \\cite{Haas04}. If $A_0$ has only point spectrum which satisfies (\\ref{eq:1.2}), then by Theorem XIX.5.7 of \\cite{DuSc71} we know that condition 1.\\ holds for $A$.\n\nWhen calculating the eigenvalues of a differential operator, one normally finds that these eigenvalues are the zeros of an entire function. If this function has its zeros in a strip parallel to the imaginary axis, and on the boundary of this strip the function is bounded and bounded away from zero, then its zeros can be decomposed into finitely many interpolation sequences, see Proposition II.1.28 of \\cite{AvIv95}. They name this class of entire function {\\em sine type functions}, but in Levin \\cite{Levi96} this name is restricted to a smaller class of functions.\n\n\n\\subsection*{Acknowledgment}\n\nThe author wants to thank Markus Haase and Jonathan Partington for their help .\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nEarthquake catalogs are a key seismological tool for problems involving location, precursors, and earthquake source parameters \\citep{Gutenberg56,Scholz68,AkiRichards,Meier2017}. Catalogs are fundamental to determining where, when, and how a fault has ruptured \\citep{AkiRichards,Scholz2002} and they are central to studies of earthquake precursors and seismic hazard based on both field observations \\citep{Marsan2014,Bouchon2008,Bouchon2013,Huang2017,Meng2018} and theory \\citep{AkiRichards,Ferdowsi2013,Kazemian}. Laboratory studies of earthquake precursors also rely on event catalogs \\citep{Lei2014,JohnsonPrecursor,riviere2018}. Such studies include the temporal evolution of the Gutenberg-Richter b-value, which is a fundamental parameter relating laboratory and field studies of earthquake physics \\citep{Smith1980,Wang2016,Scholz2015,Gulia,riviere2018,Kwiatek2014}. The seismic b-value has emerged as a key parameter for estimating seismic hazard and for connecting laboratory and field studies of seismicity \\citep{Scholz2002}.\n\nLaboratory-based studies of seismic hazard and earthquake forecasting have traditionally relied on earthquake catalogs. However, recent work shows that lab earthquakes can be predicted based on continuous acoustic data \\citep{rouet17ML,BRLFriction,Hulbert2017SlowSlip}. These works show that statistical characteristics of the continuous seismic signal emanating from lab fault zones can predict the timing of future failure events as well as the frictional state of the fault zone. Moreover, recent work has extended this approach to field observation by showing that statistical characteristics of continuous seismicity recorded in Cascadia contain a fingerprint of the megathrust displacement rate that can be used to predict the timing of episodic tremor and slip events \\citep{RouetCascadia18}. The methods are based on machine learning (ML) and rely exclusively on continuous measurements of acoustic emission (AE). Thus, a key question involves whether this approach can be extended to catalog based measurements of seismic activity.\n\nApplications of ML to seismology and geosciences are becoming increasingly common \\citep{lary2016,khawaja18,kenta18,zefeng18,Holtzman18}. Here, we apply the ML Random Forest method \\citep{Breiman2001} to study laboratory earthquake catalogs.  We find that as the catalog degrades, so does our ability to infer fault physics, as quantified the ability to predict shear stress and the time to and since failure.  We begin by describing the biaxial shearing experiment, our methods to construct a catalog from the continuous waveform data, the statistical descriptors we extract from the catalog, and the random forest method used to make inferences from these descriptors. This results in successful models for estimating the fault physics variables using local-in-time catalog data.  Then, from the original catalog we construct a series of progressively more degraded catalogs, each of which contains only the events that exceed a variable magnitude of completeness.  We examine the performance of this method as a function of the magnitude of completeness, and observe that the learned signatures of fault physics degrade dramatically if small enough events are not detected. However, the performance plateaus for low magnitudes of completeness; the smallest 80\\% events in the catalog can be dropped without sacrificing performance. Our results show that ML methods based on event catalogs can successfully model laboratory faults, with the caveat that predictive power requires a sufficiently complete catalog.\n\n\\section{Data Collection}\n\n\\subsection{Biaxial Shear Experiment}\n\nWe used data from laboratory friction experiments conducted with a biaxial shear apparatus \\citep{marone98,JohnsonPrecursor} pictured in Fig.~\\ref{figstress}a. Experiments were conducted in the double direct shear configuration in which two fault zones are sheared between three rigid forcing blocks. Our samples consisted of two 5 mm-thick layers of simulated fault gouge with a nominal contact area of 10 x 10 cm$^2$. Gouge material consisted of soda-lime glass beads with initial particle size between 105-149 {\\textmu}m. Additional details about the apparatus and sample construction can be found in \\citet{Anthony05}, \\citet{riviere2018}, and Text S1.  Prior to shearing, we impose a constant fault normal stress of 2 MPa using a servo-controlled load-feedback mechanism and allow the sample to compact. Once the sample has reached a constant layer thickness, the central block is driven down at constant rate of 10~\\textmu m\/s. In tandem, we collect an AE signal continuously at 4~MHz (Fig.~\\ref{figstress}b, yellow curve) from a piezoceramic sensor embedded in a steel forcing block $\\approx 22$ mm from the gouge layer. \n\nFigure~\\ref{figstress}c shows the shear stress measured on the fault interface for a full experiment.  Experiments begin with a run-in stage where the shear stress increases and macroscopic shearing of the fault zone transitions from stable sliding to stick-slip failure. The repetitive cycles of loading and failure represent laboratory seismic cycles and transition from periodic to aperiodic as a function of load-point displacement in our experiments \\citet{Anthony05,JohnsonPrecursor}. Here, we focus on aperiodic slip cycles and measure the slip history for a set of 23 failure events that occur over 268~seconds~\\ref{figstress}c. We define large failure events as times for which stress drop exceeds 0.05 MPa within 1 ms.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=32pc]{fig1.pdf}\n\n \\caption{{\\bf a)} Diagram of biaxial shear apparatus. The blue arrow indicates the direction of shear, and the red arrow locates the piezoceramic sensor that records AE. {\\bf b)} Example cataloged events (grey circles) derived from continuously recorded AE (yellow) using the statistics of the smoothed signal envelope (black). {\\bf c)} Observed shear stress for an experiment at fixed strain rate. Sharp drops in stress correspond to failure events. Inset: The data segment analyzed in this paper. An event catalog is constructed from AE data sampled at 4 MHz. Our models are constructed using event data from the training segment (blue) and evaluated on data from the test segment (green). \n }\n \\label{figstress}\n \\end{figure}\n\n\n\\subsection{Event Catalog}\n\\label{sec:events}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=32pc]{fig2.pdf}\n \\caption{ {\\bf a)} Normalized count of training events with magnitude exceeding $M$. The dashed line represents ideal Gutenberg-Richter scaling with a $b$-value of 1.66. {\\bf b)} Event rates vs. TTF stacked over all 23 failure cycles. {\\bf c)} Stacked event statistics for median event magnitude vs. time to failure.}\n \\label{figstack}\n \\end{figure}\n\nOur event catalog is built from the acoustic signal (example in Fig.~\\ref{figstress}b, yellow) during the time interval noted in Fig.~\\ref{figstress}c, inset. The catalog is composed of event amplitudes  $A(t_k)$ and times $t_k$, where $k$ is an event index, built following methods described in \\citet{riviere2018} and Text S2. We define laboratory magnitudes $m_k = \\log_{10}{A(t_k)}$. The catalog contains  $N_\\textrm{tot} \\approx 3.8 \\times 10^5$ slip events with laboratory magnitudes that range from about 1.0 to 4.2. Our lab magnitudes are based only on acoustic signal amplitude, and although they do not connect directly to earthquake magnitude, previous works have demonstrated the connection between seismic b-value and lab frequency magnitude statistics \\citep{Scholz2002}.\n\nTo visualize the distribution of magnitudes, we construct the cumulative event distribution\n\\begin{equation}\nN(M) = \\sum_{m_k \\geq M} 1. \\label{eq:n_M}\n\\end{equation}That is, $N(M)$ is the count of all events with magnitude at least $M$. We plot $\\log N(M) \/ N_\\textrm{tot}$ in Fig.~\\ref{figstack}a and observe reasonable Gutenberg-Richter scaling with a $b$-value of 1.66. \n\nFigure~\\ref{figstack}b shows the rate of events as a function of time to failure (TTF), averaged over all slip cycles.  The rate of events increases significantly as failure is approached, similar to what has been observed in prior works \\citep{JohnsonPrecursor,weeks1978,Ponomarev1997,riviere2018,Goebel2013}.  Figure~\\ref{figstack}c shows that the median event magnitude changes steadily but slowly from approximately $M_{\\textrm{median}}=1.4$ to $M_{\\textrm{median}}=1.5$  as a function of time during the lab seismic cycle. The trends of the catalog event count and size reflected in Fig.~\\ref{figstack}b and Fig.~\\ref{figstack}c motivate the creation of machine learning features that describe the evolution event count and size in more detail.\n\n\\section{Machine Learning Methods}\n\n\\subsection{Regression with Random Forests}\n\nAbstractly, a regression model maps input points $\\mathbf{X}_i$ to continuous output labels  $\\hat y_i$. Input points are represented by a feature vectors $\\mathbf X_i = (X_{i,1}, X_{i,2},\\dots X_{i, N_\\textrm{features}})$ that are designed to capture the relevant information about $\\hat y_i$. The training dataset $\\{(\\mathbf X_i, y_i)\\}$ consists of many points $\\mathbf X_i$ associated with true labels $y_i$. From this training set, one aims to learn a model that is successful at predicting labels $\\hat y_i$ for {\\em new} data points that were unseen within the training process. In this work, the features will be statistics extracted from the catalog, the labels will describe the macroscopic state of the fault (such as shear stress); both are indexed by time window $i$. \n\nWe  use the random forest (RF) algorithm \\citep{Breiman2001} as it is implemented in the scikit-learn package \\citep{pedregosa2011scikit}. A single RF model contains many decision trees, each of which is a simple regression model constructed stochastically from the training data. To make a prediction, the decision tree begins at its root node and asks a series of yes\/no questions. Each question has the form: {\\em Is some input feature above or below some threshold value?} Eventually a unique leaf node is reached, from which the decision tree makes its prediction. The RF makes a prediction by averaging over many decision trees, and tends to be more robust than any individual tree. Additional details about our training procedure are available in Text S3. We investigated other regression methods \\citep{Friedman2001,hui2005,tibshirani96,hoerl1970} besides the RF, and although all were successful to some extent, none offered definitive improvement over the RF (see Table~S1). \n\nA regression model is successful if its predicted labels $\\hat y_i$ agree with the true labels $y_i$ for data points in the {\\em testing dataset}, which is separate from the training dataset. We use the dimensionless $R^2$ value to quantify the performance of the model,\n\n\\begin{equation}\nR^2 = 1 - \\frac{\\sum_i (\\hat{y_i}-y_i)^2}{\\sum_i (\\bar{y}-y_i)^2},\n\\end{equation}\nwhere $\\bar{y}$ is the mean output label for the testing dataset. Let us review the significance of the $R^2$ metric, which is maximized at $R^2=1$, and can take on arbitrarily low values. Using the mean label as a constant predictor for all data points, $\\hat y_i = \\bar y$, yields $R^2=0$. Thus, $R^2<0$ indicates that a model performs worse than a constant learned only from the labels, without utilizing the features; if $R^2 \\leq  0$, a model has failed to discover useful correlations between the features and the labels. The condition $R^2=1$ would indicate that a model makes perfect predictions for all labels in the test set.\n\n\\subsection{Feature and Label Creation}\n\nOur task is to build a regression model that uses the event distribution within a {\\em local} time window to understand and make predictions about the stress state and times to and from large failure events. Each time window is labeled, and predictions of the labels will depend only on features from within the window. Thus, each test prediction is independent of prior predictions, and the model cannot exploit the quasi-periodicity of the fault state.\n\nThere is significant interest in predicting the TTF for an upcoming earthquake event. \\citet{rouet17ML} and \\citet{BRLFriction} showed that the continuous AE from a fault is remarkably predictive of the TTF and instantaneous friction. In both previous works, $\\mathbf X_i$ is a collection of statistical features extracted from the {\\em continuous} AE signal for the time window. Here, we build features for each time window using only the event catalog, rather than the continuous acoustic signal. \n\nWe divide our full event catalog (cf. Sec.~\\ref{sec:events}) into sub-catalogs that contain only the events occurring in non-overlapping time windows of length $\\Delta T = 1$ second. The feature vector contains information about events in the sub-catalog for this time window. Our analysis region consists of 270 seconds, which we divide into 1 second windows. We train on the first $60\\%$ (159 windows) and test on the remaining $40\\%$ (111 windows).\n\nWe construct regression models for three labels: (a) the shear stress, (b) the time to the next failure event (TTF), and (c) the time since the last failure event (TSF). The shear stress label is assigned using the mean shear stress over that window. TTF is measured following the end of time window, and TSF is measured preceding the beginning of the time window. For windows including failures, both TTF and TSF are zero.\n\nAbstractly, our the features for each time window describe the cumulative statistics of event counts and amplitudes from the sub-catalog evaluated at magnitude thresholds extracted from the full catalog. More precisely, the feature vector $\\mathbf X_i$ for time window $i$ will have components $X_{i,1}, X_{i,2}, \\dots X_{i,N_\\textrm{features}}$. Each feature component $X_{i,j}$  is associated with a characteristic magnitude $M_j$. For each $j$, we define $M_j$ to be the largest observed magnitude that satisfies\n\\begin{equation}\n\\frac{N(M_j)}{N_\\textrm{tot}} \\geq \\alpha^j,\n\\end{equation}\nwhere the $j$ in right hand side is applied as an exponent; $\\alpha$ is a parameter of range $ 0 < \\alpha <1$ that controls the fineness of the magnitude bins and the total number of bins, $N_\\textrm{features}$. Recall from Eq.~\\eqref{eq:n_M} that $N(M)$ is the cumulative event count of our {\\em entire} training catalog, and that $N_\\textrm{tot} \\approx 3.8\\times 10^5$. In this work, we select $\\alpha=0.7$, and this leads to $N_\\textrm{feature} = 35$ non-empty bins. We emphasize that the characteristic magnitudes $M_j$ are derived from the entire training catalog, independent of window index $i$.\n\nWe now define the feature vector $\\mathbf{X}_i$ for each window $i$ from the sub-catalog of event magnitudes $m^{(i)}_k$, where $k$ indexes the events within the window. We test two simple schemes for $\\mathbf X_i$, sensitive to the counts and amplitudes in the catalog. Mathematically, the $j$th components of these two feature vectors are\n\\begin{align}\nX^{\\textrm{count}}_{i,j} &=\\sum_{m^{(i)}_k \\geq M_j} 1 \\\\\nX^{\\textrm{ampl}}_{i,j} &= \\sum_{m^{(i)}_k \\geq M_j} 10^{m^{(i)}_k}  \\label{eq:ampl_features}\n\\end{align}\nwhere the sums run over all sub-catalog events $k$ with magnitude $m^{(i)}_k$ at least $M_j$. In words,  $X_{i,j}^\\textrm{count}$ is the count of events in the $i$th window with magnitude exceeding $M_j$ (analogous to $N(M_j)$, but restricted to sub-catalog $i$). The features $X_{i,j}^\\textrm{ampl}$ measure the total amplitude of all events in window $i$ above the magnitude $M_j$.\n\n\n\\subsection{Catalog Ablation Test}\n\nTo understand how our catalog-RF model performs with decreasing information, we vary the magnitude of completeness of our catalog by discarding all events with magnitude below a cutoff,  $M_\\textrm{cut}$. In practice, we implement this ablation by removing features  $X_{i,j}$ for indices  $j$ that correspond to magnitudes $M_j$ below the cutoff $M_{\\textrm{cut}}$. We can then quantify the performance as a function of the ablation cutoff $M_\\textrm{cut}$.\n\n\\section{Results}\n\\subsection{Performance on Full Catalogs}\n\nTable \\ref{tabregression} shows the $R^2$ performance of our catalog-RF models for shear stress, TSF, and TTF, trained using the full catalogs.  The TTF models ($R^2 = 0.551, R^2=0.612$) do not capture fine details. The TTF prediction is worse than the TSF and shear stress models ($R^2 > 0.8$)---in other words, it is harder to predict the future than to estimate aspects of the past or present. Table \\ref{tabregression} also indicates that the cumulative counts  $\\mathbf{X}^\\textrm{count}$ are better for predicting the shear stress, and the cumulative amplitudes  $\\mathbf{X}^\\textrm{ampl}$ are slightly better for TTF and TSF. The count-based catalog features predict shear stress with an accuracy of $R^2 = 0.898$, nearly equal to the continuous acoustic approach of \\citet{BRLFriction}, who report $R^2 = 0.922$ using the same data (that is, same train and test segments on the same experiment). We note that although the same data are analyzed, the methods cannot be compared exactly; for example, the time window in \\citet{BRLFriction} is $1.33$ seconds, and each time window has $90\\%$ overlap with the previous window. (Recall that in our work the time window length is $1$ second, and windows do not overlap.)  Figure~\\ref{figregression} shows the predictions compared to the true labels over time for the amplitude-based features $\\mathbf{X}^\\textrm{ampl}$, as well as scatter-plots showing a comparison of the true and predicted labels. Figure~S1 shows that models perform nearly as well for shear stress and TSF when analyzing a larger region of 573 s of data (44 failure cycles), but with reduced performance for TTF. This fact can be accounted for by drift in cycle length over time; TTF performance is restored by randomly assigning cycles to testing data from throughout the entire data analysis region (Fig.~S2).\n\n\\begin{table}[h]\n \\centering\n \\begin{tabular}{l  r  r  r }\n \\toprule\n   & Shear Stress & TSF  & TTF \\\\\n \\midrule\n  $\\mathbf{X}^\\textrm{count}$ & 0.898 & 0.842 & 0.551 \\\\\n  $\\mathbf{X}^\\textrm{ampl}$ & 0.848 & 0.882 & 0.612 \\\\\n  \\bottomrule\n \\end{tabular}\n \\caption{$R^2$  performance for regression of shear stress, TSF, and TTF for catalog-RF models on complete catalogs. The choice of count or amplitude based features ( $\\mathbf{X}^\\textrm{count}$ or  $\\mathbf{X}^\\textrm{ampl}$) affects performance slightly.}\n \\label{tabregression}\n\\end{table}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=32pc]{fig3.pdf}\n\n \\caption{Regression performance for the catalog-ML model using amplitude-based features. The rows show performance for shear stress, TSF, and TTF, respectively. The left panel of each row shows predictions on training and testing data over time. True values are shown as black squares, and predicted values as colored circles. Time windows for features are contiguous and non-overlapping. Each point is plotted at the end of the time window represented.  The right panel of each row plots the prediction vs. true values for each label. Testing data is shown with dark markers, and training data is shown with faint markers.}\n \\label{figregression}\n \\end{figure}\n\n\\subsection{Performance on Ablated Catalogs}\n\n\n \\begin{figure}[h]\n \\centering\n \\includegraphics[width=32pc]{fig4.pdf}\n\n \\caption{The accuracy of the catalog-RF model decreases as training sub-catalogs are artificially limited to varying magnitudes of completeness, $M_\\textrm{cut}$. Left: $R^2$ performance vs. $M_{\\textrm{cut}}$. The drop of $R^2$ for $M_\\textrm{cut}$ between 2.0 and 3.0 indicates that this range of event magnitudes plays a key role in our RF models. The dashed vertical line at $M_{\\textrm{cut}} = 3.67$ denotes the smallest event magnitude associated with a large stress drop. Right: $R^2$ is plotted against $N(M_\\textrm{cut})\/N_\\textrm{tot}$, i.e. the fraction of events remaining in the full catalog after ablation. One observes that most events contribute very little to the catalog-RF performance.}\n \\label{figablation}\n \\end{figure}\n\nNext we build our catalog-RF models on ablated catalogs that include events only above an imposed magnitude of completeness, $M_\\textrm{cut}$. For this study, we use the amplitude features $\\mathbf{X}^\\textrm{ampl}$ of Eq.~\\eqref{eq:ampl_features}.  Figure~\\ref{figablation}, left side, shows the $R^2$ performance as a function of the imposed magnitude cutoff.  As one might expect, catalog-RF models generally perform worse when provided less information. However, it is interesting to note that $R^2$ decays relatively slowly until $M_\\textrm{cut}$ reaches about 2.0, after which the performance drops precipitously, plateauing near magnitude 3.0. In this regard we say that $M_{\\textrm{cut}} = 2.0$ provides a {\\em sufficient magnitude of completeness} for the catalog-RF model with the given window size.  In Fig.~\\ref{figablation}, right side, we plot $R^2$ as a function of the remaining event fraction $N(M_\\textrm{cut})\/N_\\textrm{tot}$. This highlights that the performance drop at $M_{\\textrm{cut}} \\approx 2.0$ corresponds to ablating $\\approx 80\\%$ of the events from the catalog. Likewise, the sharp decay in performance indicates that events $M \\lesssim 2.5$ are {\\em required} to make an accurate determination of the fault state. The performance of the method for $M_{\\textrm{cut}} \\gtrsim 3 $ plateaus at a non-zero value of $R^2$ because the algorithm is still able to differentiate between windows that contain a large failure and those that do not, but is otherwise unable to discern useful differences between sub-catalogs. \n\nA study of the ablation curve as a function of window size (Fig.~S3) shows that the sharp decay in performance shifts to smaller cutoff magnitudes as the window size is decreased; while regression can be performed with small window sizes, the sufficient magnitude of completeness is lower. This leads us to hypothesize that the sharp decay in performance may be related to the number of events within a window; to keep the number of events in a window fixed, one must lower the magnitude of completeness along with the window size. \n\n\\section{Conclusions}\n\nWe analyze a dense acoustic event catalog obtained from biaxial shearing friction experiment.  The device produces dozens of aperiodic stick-slip events during an experiment. Stacking catalog statistics over many cycles shows that as failure approaches, events are more common and typically larger (Fig.~\\ref{figstack}).  We then use a machine learning workflow to show that physical characteristics of the fault (shear stress, TSF, and TTF) can be obtained from catalog statistics over a short time window using a Random Forest regression model. Previous work employed the continuous AE, and showed remarkable ability to forecast failure \\citep{rouet17ML}, as well as to infer the instantaneous shear stress and friction \\citep{BRLFriction}.  Here, we demonstrate that the continuous acoustic waveform is not needed if a catalog {\\em with sufficient completeness} is available. Our models achieve similar accuracy to the continuous approach.\n\nWe note, however, that our catalogs are extraordinarily well resolved. To create these catalogs, we recorded AE at very high frequency (4 MHz) using a sensor very near the fault (distance $\\approx 22$ mm). One would not expect such fidelity in real earthquake catalogs, except perhaps if the sensors happened to be located very close to the slip patch.  We find that the smallest events in our catalog contribute little to the prediction accuracy; increasing $M_\\textrm{cut}$ from 1.0 to 2.0 only slightly diminishes the catalog-RF model performance (visible in Fig.~\\ref{figablation}).\n\nThe fact that the smallest events are not {\\em required} offers hope that a catalog-based ML approach could eventually contribute to the understanding of natural seismicity. We note a few potential differences between laboratory conditions and natural seismicity that may pose challenges. Laboratory data corresponds to a single isolated fault. Tectonic fault zones are typically comprised of many sub-faults, and predictions may be vastly more difficult. While precursors are frequently observed in laboratory studies, they are not as reliably observed in natural faults. Moreover, large seismic cycles in the earth are far slower \\citep{nelson2006,Ellsworth2013}, and thus fewer cycles are available to train ML models. \n\nWe note that \\citet{RouetCascadia18} have applied the continuous approach developed in the laboratory to slow slip in Cascadia with surprisingly good results.  Thus we are hopeful that the catalog-based approach described here may prove fruitful to the study of real earthquakes, given sufficient catalog quality. It may have particular importance in regions where continuous data are not yet available, where the continuous data is quite noisy, or for examining archival catalogs.\n\n\\acknowledgments\nWork supported by institutional support (LDRD) at the Los Alamos National Laboratory.  We gratefully acknowledge the support of the Center for Nonlinear Studies. CM was supported by NSF-EAR1520760 and DE-EE0006762. We thank Andrew Delorey, Robert A. Guyer, Bertrand Rouet-Leduc, Claudia Hulbert, and James Theiler for productive discussions. The data used here are freely available from the Penn State Rock Mechanics lab at \\url{http:\/\/www3.geosc.psu.edu\/~cjm38\/}.\n\n\n\\section*{Contents}\n\n\\begin{enumerate}\n\\item Text S1: Experimental details\n\\item Text S2: Catalog generation from acoustic emissions\n\\item Text S3: Random forest details\n\\item Table S1: Performance of various regression algorithms\n\\item Figure S1: Regression for long time segment with contiguous train\/test split\n\\item Figure S2: Regression for long time segment with random cycle train\/test split\n\\item Figure S3: Ablation analysis for varying time-window sizes\n\\end{enumerate}\n\n\\section*{Text S1: Experimental details}\n\nWe constructed samples for the double direct shear (DDS) assembly following the procedures detailed in previous works \\citep{Anthony05,riviere2018}. Steel guide plates are mounted on the side blocks of the DDS to prevent gouge loss on the front and back edges. The faces of the forcing blocks in contact with the gouge layers are grooved to ensure that shear occurs within the layer; grooves are 0.8 mm deep and spaced every 1mm perpendicular to the direction of shear. In addition, a small rubber jacket covers the bottom half of the sample to minimize material loss throughout the experiment. Forces on faults are measured with strain-gauge load cells placed in series with the horizontal and vertical pistons. Displacements on the fault are measured using direct current displacement transducers that are coupled to the horizontal and vertical pistons. Stress and displacement data are measured continuously at 1kHz throughout the experiment from a 24-bit recording system. All measurements of shear stress and displacement for the DDS configuration are relative to one fault. In addition, we have instrumented 36 broad-band piezoceramic sensors (~0.02-2 MHz) into steel 10 x 10 cm\\textsuperscript{2} blocks. The sensors are in blind holes 2-mm from the block face and the block is placed adjacent to the fault zone to monitor acoustic emission activity \\citep{riviere2018}. The acoustic emission data analyzed here are recorded continuously at 4 MHz from a single piezoceramic sensor using a 14-bit Verasonics data acquisition system.\n\n\\section*{Text S2: Catalog generation from acoustic emissions}\n\n\nWe use a simple event detection algorithm to construct a catalog from the acoustic emission time series signal.  The algorithm has been designed to account for both the response of the sensor and the background noise level. The first step is to compute the envelope of the signal and use a moving average to smooth it over the time scale $\\tau_\\textrm{smooth}$. From this smoothed envelope we examine peaks and label them as events if they satisfy the following criteria: 1) The peak is above a minimum amplitude threshold, $A_\\textrm{min}$, related to the average background noise. 2) The peak is separated from previous peaks by a minimum inter-event time threshold, $\\tau_\\textrm{min}$. 3) The peak does not occur during in the coda of an event -- this is defined using an exponential ring-down function computed over the ten previous events. Mathematically, event $k$ is not part of a coda if the peak amplitude $A(t_k)$ satisfies\n \\begin{equation}\nA(t_k) > A(t_{k-i})\\exp{\\frac{-{(t_k-t_{k-i})}}{\\tau_\\textrm{ring-down}}}\n\\label{eq:A_t0}\n\\end{equation}\nfor $i = 1...10$.  The parameters of this algorithm are selected by evaluating the catalog quality on random 5~$\\mu s$ snapshots of the acoustic signal using trial parameters and visual evaluation of catalog quality. A high-quality catalog maximizes the number of distinct visual phenomena selected as events, while at the same time minimizes the assignment of multiple events to a single phenomenon. In the end, we selected $A_\\textrm{min} = 15$, $\\tau_\\textrm{smooth} = 7.75$ {\\textmu}s,  $\\tau_\\textrm{min} = 150$ {\\textmu}s, and $\\tau_\\textrm{ring-down} = 156.7$ {\\textmu}s.\n\n\\section*{Text S3: Random forest details}\nWe employed 100 decision trees per random forest, each of which sees a bootstrap resampling of the training set, with a mean-squared-error cost function. Before training a random forest model (i.e., designing each tree's yes\/no questions from the data), one may specify the maximum depth of the decision trees, which controls the allowable complexity of the random forest model. (If the maximum depth is larger, the random forest can fit more complicated datasets, but also becomes more prone to overfitting irrelevant details of the data.) For each regression task, we used 10-fold cross-validation of the training set to select the maximum depth; the possible depths are 1, 2, 4, 6, 8, and 12. The folds for cross-validation are selected by randomly partitioning the training set. Cross-validation uses the $R^2$ score as the criterion for the best model.  After identifying the tree depth which cross-validates most successfully, we retrain a forest with that maximal depth to the entire training set. Other regularization hyperparameters such as the minimal leaf size and random feature sub-selection were not utilized.\n\n\\begin{table}[h]\n \\centering\n\\begin{tabular}{llrrrrr}\n\\toprule\n           &       &    RFR &    GBR &  ElasticNet &  Lasso &  Ridge \\\\\nRegression task & Feature vector &        &        &               &          &          \\\\\n\\midrule\nShear stress & $\\mathbf X^\\textrm{count}$ &  {\\bf 0.901} &  0.893 &         0.715 &    0.715 &    0.800 \\\\\n           & $\\mathbf X^\\textrm{ampl}$ &  {\\bf 0.860} &  0.827 &         0.817 &    0.817 &    0.796 \\\\\nTSF & $\\mathbf X^\\textrm{count}$ &  0.843 &  \\bf{0.886} &         0.630 &    0.634 &    0.743 \\\\\n           & $\\mathbf X^\\textrm{ampl}$ &  0.888 &  \\bf{0.891} &         0.779 &    0.779 &    0.771 \\\\\nTTF & $\\mathbf X^\\textrm{count}$ &  0.547 &  \\bf{0.579} &         0.533 &    0.533 &    0.571 \\\\\n           & $\\mathbf X^\\textrm{ampl}$ &  \\bf{0.618} &  0.578 &         0.588 &    0.588 &    0.471 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{$R^2$ performance for various regression models. We compare random forest regression (RFR) \\citep{Breiman2001}, gradient boosting regression (GBR) \\citep{Friedman2001}, and linear regression with ElasticNet \\citep{hui2005}, lasso \\citep{tibshirani96}, and ridge \\citep{hoerl1970} regularization penalties as implemented in scikit-learn \\citep{pedregosa2011scikit}. The best performing model for each regression task and feature vector type is shown in bold. RFR and GBR produce models of similar quality and both outperform linear regression schemes.}\n \\label{methodchoices}\n\\end{table}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=32pc]{figS1.pdf}\n\n \\caption{Regression performance for the catalog-ML model using amplitude-based features using 573 seconds of data. The rows show performance for shear stress ($R^2 =0.869$) , TSF ($R^2 =0.813$), and TTF ($R^2 =0.284$), respectively. The left panel of each row shows predictions on training (region shaded blue) and testing (region shaded green) data over time. True values are shown as black squares, and predicted values as colored circles. The right panel of each row plots the prediction vs. true values for each label. Testing data is shown with bold markers, and training data is shown with faint markers.}\n \\label{figregression}\n \\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=32pc]{figS2.pdf}\n\n \\caption{Regression performance for the catalog-ML model using amplitude-based features using 573 seconds of data, where each failure cycle is individually assigned to either training or testing data. The rows show performance for shear stress ($R^2 =0.881$), TSF ($R^2 =0.869$), and TTF ($R^2 =0.511$), respectively. The left panel of each row shows predictions on training (region shaded blue) and testing (region shaded green) data over time. True values are shown as black squares, and predicted values as colored circles. The right panel of each row plots the prediction vs. true values for each label. Testing data is shown with bold markers, and training data is shown with faint markers.}\n \\label{figregression}\n \\end{figure}\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=20pc]{figS3.pdf}\n\n \\caption{Ablation analysis for time since failure (TSF) showing $R^2$ performance vs. artificial magnitude of completeness for different window sizes. As the window size decreases, performance decreases. Furthermore, the curve shifts left; the magnitude of completeness necessary to achieve a given $R^2$ performance lowers along with window size. This figure demonstrates that while regression can be performed with small window sizes, the process relies on having information about progressively smaller events.}\n \\label{fig:windowsizeabalation}\n \\end{figure}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nNon-Hermitian physics has stimulated significant interest in recent years~\\cite{PTreview1,PTreview2,PTreview3,Uedareview}, where particular attention has been devoted to its unconventional dynamics, peculiar critical behavior, and exotic band topology. A major driving force behind the booming field is the experimental implementation or simulation of these intriguing phenomena, particularly in open dissipative quantum systems~\\cite{dalibard,daley,openbook}.  Therein, the system undergoes particle or energy loss to its environment, and a non-Hermitian effective Hamiltonian becomes relevant by imposing postselection~\\cite{dalibard,daley}.\nSo far, a wide spectrum of non-Hermitian phenomena, ranging from parity-time (PT)-symmetry breaking and non-Hermitian criticality~\\cite{benderreview,ptcrit,kznc}, to non-Hermitian skin effects and non-Bloch topology~\\cite{WZ1,Budich,alvarez,mcdonald,ThomalePRB,Lee,WZ2,murakami}, have been experimentally implemented and explored in quantum mechanical systems such as the single-photon interferometry network~\\cite{xuept,crit1,photonskin}, cold atoms~\\cite{luole,bryce,yan,NHSOCexp,yanzeno}, nitrogen-vacancy centers~\\cite{crit2,epencircle}, superconducting qubits~\\cite{scpt}, and trapped ions~\\cite{chenion,zhangion}. While most of these experiments investigate the single-particle aspects of the non-Hermitian physics, the interplay of non-Hermiticity and interaction is a fast-growing frontier with many open questions and fresh challenges~\\cite{sf1,sf2,nonHtwobody,Yu,Cui}.\n\nOne of the key issues here is the relevance of non-Hermitian many-body Hamiltonians to open dissipative quantum systems~\\cite{fu,michishita}.\nWhile the latter is naturally characterized by trace-preserving density-matrix dynamics, the former requires a biorthogonal construction to ensure othornormality and recover the bosonic\/fermionic statistics~\\cite{DCB}.\nFurther, the postselection framework in a many-body setting requires an unchanged particle number~\\cite{michishita}, thus exacting a stringent limit on the timescale within which the non-Hermitian description can be applied.\nBy contrast, this is not an issue with a non-interacting system, as is the case with the recent observations of PT transition and exceptional-point encircling in cold atoms~\\cite{luole,NHSOCexp}. Therein,\na non-interacting atomic gas undergoes particle loss to the environment through optical pumping---a single-particle process. The dynamics is driven by a non-Hermitian effective Hamiltonian derived from the Lindblad master equation by dropping the quantum jump term $c\\rho c^\\dag$ (here $c$ is the atomic annihilation operator and $\\rho$ the full density matrix of the non-interacting system).\nThis is equivalent to focusing only on atoms that are not lost to the environment, in the spirit of postselection.\nSpecifically, under the quantum-trajectory description~\\cite{daley}, since dynamics of individual atoms are decoupled, they constitute an ensemble of independent trajectories with no quantum jumps, all driven by the non-Hermitian effective Hamiltonian.\nTherefore, given a large number of atoms, the corresponding non-unitary dynamics can be probed by making measurements on the remaining atoms. Should interactions exist, however, dynamics of atoms in general would not decouple. Applying postselection would then amount to requiring the complete absence of quantum jumps for any single atom within the ensemble, which becomes exponentially unlikely with an increasing atom number.\n\n\n\nNevertheless, we demonstrate in this work that, key non-Hermitian physics can still be probed in the full density-matrix dynamics over fairly long times, provided the open many-body system be dominated by few-body correlations.\nUsing a minimal model of either two or three interacting fermions in a one-dimensional lattice, we first show that, under the non-Hermitian effective Hamiltonian of the corresponding Lindblad master equation, two-body scattering states of the system feature state- and interaction-dependent PT transitions. We then evolve the three-fermion dissipative system using the quantum trajectory scheme, taking into account the quantum jump processes. Remarkably, the decay of two-body correlations in this three-fermion open system follows the imaginary components of the complex eigenenergies of the two-body non-Hermitian scattering states. This enables us to extract the global exceptional point of the underlying two-body non-Hermitian system in the three-atom dissipative dynamics, from which the impact of interaction on the exceptional point is identified.\nOur results suggest that, non-Hermitian many-body physic can in principle be probed in the context of an open dissipative many-body setting,\nat least when both are dominated by few-body correlations.\n\nOur paper is organized as follows. In Sec.~II, we present the model configuration, the corresponding Lindblad master equation, and the non-Hermitian effective Hamiltonian. We solve the two-body eigen problem of the non-Hermitian Hamiltonian using exact diagonalization in Sec.~III, where we reveal the existence of PT transitions and exceptional points in the scattering states. In Sec.~IV, we solve the Lindblad master equation for a three-fermion system using the quantum trajectory approach, and show the relevance between the two-body correlations therein and the complex eigenenergies of the non-Hermitian scattering states. We summarize in Sec.~V.\n\n\\section{Dissipative many-body system and non-Hermitian Hamiltonian}\n\nAs illustrated in Fig.~\\ref{fig:model}, we consider fermionic atoms with two hyperfine states in a one-dimensional optical lattice potential. The corrsponding Hamiltonian is given by\n\\begin{align}\nH= & \\sum_{k, \\sigma} \\epsilon_{k} c_{k, \\sigma}^{\\dagger} c_{k, \\sigma}+J\\sum_{k}\\left(c_{k, \\uparrow}^{\\dagger} c_{k, \\downarrow} + c_{k, \\downarrow}^{\\dagger} c_{k, \\uparrow}\\right)\\nonumber \\\\\n&+\\frac{U_s}{\\mathcal{L}} \\sum_{k, k^{\\prime}, q} c_{k+q, \\uparrow}^{\\dagger}\nc_{k^{\\prime}-q, \\downarrow}^{\\dagger} c_{k^{\\prime}, \\downarrow}\nc_{k, \\uparrow}\\nonumber\\\\\n&+\\frac{U_p}{\\mathcal{L}} \\sum_{k, k^{\\prime}, q, \\sigma} \\sin(\\frac{k-k^\\prime+2q}{2})\\sin(\\frac{k-k^\\prime}{2})\\nonumber\\\\\n&\\quad \\quad\\quad\\quad\\quad \\times c_{k+q, \\sigma}^{\\dagger}\nc_{k^{\\prime}-q, \\sigma}^{\\dagger} c_{k^{\\prime}, \\sigma}\nc_{k, \\sigma}.\n\\label{eq:H}\n\\end{align}\nHere $c_{k, \\sigma} $ ($c^\\dag_{k, \\sigma} $) annihilates (creates) a fermionic atom with quasimomentum $k$ ($k\\in [-\\pi,\\pi)$) in the hyperfine state $|\\sigma\\rangle$ ($\\sigma=\\uparrow,\\downarrow$), $\\epsilon_k=-t [\\cos(ka_0)-1]$ with a hopping rate $t$ under the tight-binding approximation, $J$ is the radio-frequency (r.f.) coupling rate between different hyperfine spins, and $\\mathcal{L}$ is the quantization length. We consider both $s$-wave and $p$-wave interactions, characterized by $U_s$ and $U_p$~\\cite{pinteraction}, respectively, with $U_s,U_p<0$.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=0.4\\textwidth]{figure1}\n\\caption{Schematics of the dissipative lattice gas driven by the master equation (\\ref{eq:lind}). The single-particle loss (with rate $\\Gamma$) is induced by optical pumping, via an electronically excited state, to a third state (not drawn) that is not trapped by the lattice potential. The r.f. coupling rate $J$ and hopping rate $t$ are defined in the main text.\n}\n\\label{fig:model}\n\\end{figure}\n\nWe further consider the case where one of the hyperfine spin states ($\\left|\\downarrow\\right\\rangle$) is subject to optical pumping, via an electronically excited state, out of the lattice potential. Under the Markovian approximation, the dynamics of the system is captured by the Lindblad master equation\n\\begin{align}\n\\frac{d \\rho}{dt}=-i\\Big(H_{\\text{eff}}\\rho-\\rho H^\\dag_{\\text{eff}}\\Big)+\\Gamma\\sum_{k}c_{k,\\downarrow} \\rho c_{k,\\downarrow}^{\\dagger},\n\\label{eq:lind}\n\\end{align}\nwhere $\\rho$ is the density matrix, the non-Hermitian effective Hamiltonian of the Lindblad equation is given by $H_{\\text{eff}}=H-i\\frac{\\Gamma}{2}\\sum_k c^\\dag_{k,\\downarrow}c_{k,\\downarrow}$, and $\\Gamma$ is the single-particle loss rate.\n\nWhile the full dynamics of the open system is governed by Eq.~(\\ref{eq:lind}), under the quantum trajectory framework, the dynamics is understood as a non-unitary time evolution driven by $H_{\\text{eff}}$, which is further interrupted by quantum jumps $\\{c_{k,\\downarrow}\\}$ with relative probabilities $\\{\\Gamma\\delta t|c_{k,\\downarrow}|\\psi(t)\\rangle|^2\\}$. Here $\\delta t$ is the coarse-grained time step, and $|\\psi(t)\\rangle$ is the instantaneous state of the system. It is often argued that, when the effects of quantum jumps are negligible, the open system would evolve under the non-Hermitian Hamiltonian $H_{\\text{eff}}$. Such a postselection argument plays a key role in connecting realistic open quantum systems to the rich and exotic non-Hermitian physics that has attracted much attention of late~\\cite{Uedareview}.\n\nFor a non-interacting atomic gas with $U_s=U_p=0$, imposing postselection is conveniently equivalent to focusing only on the dynamics of atoms that are not lost to the environment.\nSince the full density matrix is just a direct product of single-particle density matrices, dynamics of each individual atom is driven by the same\nmaster equation. Further, as atoms that remain necessarily have not undergone the quantum jump process, the trajectories of remaining atoms are\nnon-unitary evolutions driven by the same non-Hermitian effective Hamiltonian.\nThis is indeed the case with the recent experimental demonstrations of PT symmetry and exceptional-point encircling in cold atoms~\\cite{luole,NHSOCexp}.\n\nHowever, in an interacting system, the full density matrix can no longer be decomposed into single-particle ones.\nPostselection thus corresponds to a complete absence of quantum jumps, i.e., it requires an unchanged total particle number.\nThe non-Hermitian effective Hamiltonian is then applicable at short times,\nwhen the impact of quantum jump terms are negligibly small. This is equivalent to the practice in Ref.~\\cite{Yu}, which projects the time evolution of the Lindblad equation onto the maximum-atom-number subspace. However, the corresponding time scale should become exponentially short with increasing particle number.\nGenerally, consider an $N$-particle system undergoing single-particle loss with the rate $\\Gamma$.\nThe probability that not a single quantum jump occurs scales as $\\sim e^{-N\\Gamma \\tau}$, where $\\tau$ is the evolution time.\nTherefore, the probability of all $N$ particles still remain at the time $1\/\\Gamma$ (for an evolution starting at $t=0$) is of the order $e^{-N}$, and the time scale at which the non-Hermitian effective Hamiltonian dominates should be $t< 1\/N\\Gamma$.\nNevertheless, we show in the following that, two-body physics under the non-Hermitian effective Hamiltonian {\\it can} be probed in the full density-matrix dynamics of the corresponding Lindblad equation, on time scales that exceed $1\/\\Gamma$.\n\n\n\n\n\\section{Non-Hermitian two-body scattering state}\n\nWe first characterize the two-body problem under the non-Hermitian effective  Hamiltonian $H_{\\text{eff}}$. To connect with previous cold-atom experiments on PT symmetry~\\cite{luole,NHSOCexp}, we define the PT symmetric Hamiltonian $H_{\\text{PT}}=H_{\\text{eff}}+i\\Gamma$. While the addition of the pure imaginary energy shift $i\\Gamma$ does not change key physics such as the emergence and location of exceptional points, it renders Hamiltonian $H_{\\text{PT}}$ PT symmetric in the non-interacting limit, with purely real (imaginary) eigenspectrum for $J>\\Gamma\/4$ ($J<\\Gamma\/4$).\n\n\\begin{figure}[tbp]\n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth] {figure2.pdf}\n\t\\caption{(Color online) Complex energy spectra of two fermions under the PT symmetric Hamiltonian $H_{\\text{PT}}$ for (a) $ U_s=0$, $U_p=0$, $J\/t=0.04 $; (b) $ U_s=0$, $U_p\/t=-0.2$, $J\/t=0.04$; and (c) $ U_s\/t=-2$, $U_p\/t=-0.2$, $J\/t=0.04 $. The left (right) panel shows the real (imaginary) components of the eigenspectra, with $K_c$ the center-of-mass momentum of the two-body state. For all numerical calculates, we take a one-dimensional lattice with $N=16$ sites.}\n\\label{fig:fig1}\n\\end{figure}\n\nIn Fig.~\\ref{fig:fig1}, we show the numerically evaluated eigenspectra for two fermions along a lattice of $N=16$ sites, with the parameters $J\/t=0.04$ and $\\Gamma\/t=0.1$. Here $K_c$ is the center of mass of the two-body state, which is a good quantum number of the system. In the non-interacting case [see Fig.~\\ref{fig:fig1}(a)(b)], $H_{\\text{PT}}$ is in the PT-unbroken regime, with purely real eigenspectra. The PT symmetry is broken under a sufficiently large $p$-wave interaction, as the eigenspectra acquire imaginary components under a finite $U_p$ [see Fig.~\\ref{fig:fig1}(c)(d)]. This in contrast to the $s$-wave interaction, which does not affect the imaginary components of the eigenspectra [see Fig.~\\ref{fig:fig1}(e)(f)]. Note that while discrete two-body bound states can be identified, for instance in Fig.~\\ref{fig:fig1}(e), it is the two-body scattering states within the continuum that acquire imaginary components.\nImportantly, we expect the PT transition point (or the exceptional points) of the non-interacting Hamiltonian be shifted by the $p$-wave interaction.\n\n\\begin{figure}[tbp]\n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth] {figure3.pdf}\n\t\\caption{(Color online) Complex energy spectra of $ H_{\\text{PT}} $ as functions of $ J\/t$ for (a) $ K_c=0$ and (b) $ K_c=\\pi $. As $ J\/t$ decreases, the scattering states coalesce in pairs at state-dependent exceptional points, through which their real components merge and imaginary components bifurcate. The full eigenspectra become completely real above a critical $J_c\/t$, which is identified as the global exceptional point (or the global PT transition point).\n}\n\\label{fig:fig2}\n\\end{figure}\n\nThis is confirmed in Figs.~\\ref{fig:fig2} and \\ref{fig:fig3}. Specifically, in Fig.~\\ref{fig:fig2}, we show the splitting of exceptional points in different $K_c$ sectors under a finite $U_p$. With increasing $J$, scattering states sequentially coalesce in pairs at an array of second-order exceptional points. We identify the exceptional point with the largest $J$ as the global PT transition point under the $p$-wave interaction,denoted by $J_c$. The resulting PT phase diagram is shown in Fig.~\\ref{fig:fig3}. Apparently, $p$-wave interactions shift the global exceptional point toward larger $J$, consistent with the results in Fig.~\\ref{fig:fig1}.\n\n\n\\begin{figure}[tbp]\n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth] {figure4.pdf}\n\t\\caption{(Color online) PT phase diagram for $H_{\\text{PT}}$ with $U_s=0$ and $U_p<0$. The global exceptional point $J_c\/t$ increases with larger $p$-wave interaction.\nThe parameter used in Fig.~\\ref{fig:fig2} is indicated by the vertical dashed line.}\n\\label{fig:fig3}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\t\\centering\n\\includegraphics[width=0.45\\textwidth] {figure5.pdf}\n\\caption{(Color online) Comparison of the total particle-number evolution $\\sum_{k,\\sigma}\\text{Tr}\\Big[ \\rho(\\tau)c^\\dag_{k,\\sigma}c_{k,\\sigma}\\Big]$ under the Lindblad master equation for systems initialized with\nthree atoms (blue solid), two atoms (red solid), and a single atom (yellow dashed), respectively, in the states $|\\uparrow,k=-\\pi\\rangle\\otimes |\\uparrow,k=0\\rangle\\otimes |\\uparrow, k=\\pi-2\\pi\/N$ ($N=8$), $|\\uparrow,k=\\pi\\rangle\\otimes |\\uparrow,k=0\\rangle$, and $|\\uparrow,k=0\\rangle$. The interaction parameters are (a) $U_p\/t=-0.5$, (b) $U_p=0$, with $U_s=0$, $J\/t=0.08$, and $\\gamma\/t=0.1$ for both cases. To facilitate comparison, the particle numbers in all cases are normalized to one.\n}\n\\label{fig:N}\n\\end{figure}\n\n\\section{Probing non-Hermitian physics in open system}\n\nWe now show that the global exceptional point $J_c$ of the two-body scattering states in Fig.~\\ref{fig:fig3} can be probed from the density-matrix dynamics under the full Lindblad equation. Under interactions, information of the two-body exceptional point is difficult to extract from the particle-number dynamics of a many-body system. This is illustrated in Fig.~\\ref{fig:N}, where we compare the particle-number evolution under the Lindblad master equation for systems initialized with different particle numbers, either with (Fig.~\\ref{fig:N}(a)) or without (Fig.~\\ref{fig:N}(b)) interactions.\nWhile the dynamics for different initial particle numbers appear to be the same without interactions, they generally differ under a finite $U_p$.\nThis shows that non-Hermitian two-body physics cannot be directly probed using particle-number dynamics in a three-body dissipative system. Note that non-Hermitian two-body physics can be fully captured by a Lindblad equation initialized in the two-body sector, since quantum jump terms in a two-body sector only couple to three-body states~\\cite{Yu}.\n\nInstead, other observables should be adopted. For a minimal demonstration, we solve the Lindblad equation in a three-fermion system, initialized in the state $|\\psi^{(S)}(K_c=-\\pi)\\rangle\\otimes |\\uparrow,k=\\pi-2\\pi\/N\\rangle$. Here $|\\uparrow,k\\rangle$ is a single-particle state with hyperfine spin $|\\uparrow\\rangle$ and momentum $k$, $|\\psi^{(S)}(K_c)\\rangle$ is the two-body scattering state of $H_{\\text{eff}}$ with a center-of-mass momentum $K_c$, and $|\\psi^{(S)}(K_c=\\pi)\\rangle$ is the eigenstate with the largest imaginary eigenenergy component, denoted as $\\text{Im}(E^{(S)})$. We take the system size $N=8$ for numerical calculations.\nFurther, we define a normalized two-body correlation function\n\\begin{align}\nG_{\\alpha\\beta}(k_1,k_2)=\\frac{\\text{Tr} \\Big[\\rho(\\tau) c_{k_1,\\alpha}^{\\dagger} c_{k_2,\\beta}^{\\dagger} c_{k_2,\\beta} c_{k_1,\\alpha}\\Big]}{\\text{Tr} \\Big[\\rho(0) c_{k_1,\\alpha}^{\\dagger} c_{k_2,\\beta}^{\\dagger} c_{k_2,\\beta} c_{k_1,\\alpha}\\Big]}. \\label{eq:G}\n\\end{align}\n\nWe evolve the Lindblad equation using the quantum trajectory approach, and plot the correlation function $ G_{\\uparrow\\uparrow}(k_1=-\\pi,k_2=0) $ in Fig.~\\ref{fig:fig4}.\nDue to the particular choice of initial state and the two-body correlation function, the decay of the correlation function follows the imaginary component of the two-body scattering state with the largest critical $J$.\nThus, by fitting the exponents of decay in $G$, we are able to map out the global exceptional points in the phase diagram Fig.~\\ref{fig:fig3}, and retrieve key properties of a two-body non-Hermitian Hamiltonian from the full dynamics of a interacting three-body open system.\nWe expect that this would hold true for many-body open systems, as long as the dominant correlations remain few-body in nature.\n\nFinally, we note that, should we choose a different initial state and two-body correlation function, we would be able to extract information of other two-body scattering states.\n\n\n\n\n\n\\begin{figure}[tbp]\n    \\centering\n   \\includegraphics[width=0.45\\textwidth] {figure6.pdf}\n   \\caption{(Color online) (a)(b)(c) Time evolution of the normalized two-body correlation function $ G_{\\uparrow\\uparrow}(k_1=-\\pi,k_2=0)$ (blue lines) under the quantum trajectory approach for (a) $ J\/t=0.04 $, (b) $ J\/t=0.08 $, and (c) $ J\/t=0.1 $. We take $ U_p\/t=-0.2$, $U_s=0$, and $\\Gamma\/t=0.1 $ for numerical calculations.\n  \n   The red lines show the time evolution of the norm of the corresponding two-body eigenstate of the non-Hermitian Hamiltonian $ H_{\\text{PT}} $.\n   (d) Comparison between the imaginary component of the two-body eigenenergy under $ H_{\\text{PT}} $ (red line), with the numerically fitted exponent of the correlation-function decay (blue line and symbol). For the exponent, we numerically fit the time-dependent correlation function up to the time $\\tau t=5$. For all our numerical calculations here, we take the lattice size $N=8$. We average over $2000$ trajectories for the quantum-trajectory calculations.}\n      \\label{fig:fig4}\n\\end{figure}\n\n\n\\section{Summary and Discussion}\\label{sec:sum}\n\nAdopting a minimal model of a few dissipative fermions on a one-dimensional lattice, we show that PT transitions exist in the scattering states of the non-Hermitian Hamiltonian, and are shifted by the $p$-wave inter-atomic interactions. The interaction-shifted global exceptional point can be probed by measuring two-body correlations in the trace-preserving density-matrix dynamics driven by the Lindblad master equation. We therefore explicitly demonstrate a minimal scenario where key properties of a non-Hermitian interacting Hamiltonian can be probed in the context of an open system.\n\nIn particular, while one expects the non-Hermitian many-body Hamiltonian to be of relevance on a short time scale of $1\/N\\Gamma$ (see discussions in Sec.~II), it is remarkable that the two-body correlation captures key features of the two-body non-Hermitian scattering states at time scales even longer than $1\/\\Gamma$ (see Fig.~\\ref{fig:fig4}). We attribute such a phenomenon to the dominance of few-body correlations in the open dissipative system. Our result is complementary to previous attempts at connecting non-Hermitian many-body Hamiltonians with open dissipative systems~\\cite{fu,michishita}, and is relevant to cold atomic gases where few-body correlations dominate.\n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\nWe thank Xiaoling Cui for helpful discussions.\nThis work has been supported by the Natural Science Foundation of China (Grant No. 11974331) and the National Key R\\&D Program (Grant Nos. 2016YFA0301700, 2017YFA0304100).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}