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+{"text":"\\section{Introduction}\n\nDiffusion was invented by Fourier to describe the dynamics of heat \\cite{fourier1822theorie}. Heat or energy transport is ubiquitous and of relevance to essentially any physical system at nonzero temperature. In modern parlance, diffusion is understood more generally as the universal late-time hydrodynamic description of systems governed by a single conservation law, barring spontaneously broken symmetries. The diffusion constant may itself depend on the background value of the quantity that is being transported. Classically, this results in the diffusion equation being augmented to a non-linear differential equation and can lead to rich phenomena (see e.g.~\\cite{crank1979mathematics}), in analogy with turbulence in the Navier-Stokes equation. At the quantum or statistical physics level, the consequence of these non-linearities is that the effective theory of hydrodynamic fluctuations is interacting.\n\nThe traditional approach to address this class of problems is to couple the degrees of freedom of interest to stochastic noise fields and solve perturbatively a non-linear Langevin equation \\cite{PhysRevA.8.423,POMEAU197563}. This approach is also familiar in the context of the KPZ equation \\cite{PhysRevLett.56.889}. These methods have revealed striking effects in hydrodynamics such as long-time tails \\cite{DESCHEPPER19741,POMEAU197563,Kovtun:2003vj,PhysRevB.73.035113} and potentially large renormalizations of transport parameters \\cite{PhysRevA.16.732,Kovtun:2011np,Kovtun:2014nsa}. However, where standard classical hydrodynamic stood on firm symmetry principles \\cite{chaikin1995principles}, the physical principles governing stochastic hydrodynamics -- in particular how `noise' fields interact with conserved densities -- were less transparent. This was remedied recently by a decade-long effort culminating in a first principles construction of the general effective field theory of hydrodynamic fluctuations about a thermal equilibrium state \\cite{Dubovsky:2011sj,Grozdanov:2013dba,Haehl:2015pja,Crossley:2015evo,Jensen:2017kzi}, see \\cite{Glorioso:2018wxw} for a recent review.\n\nAnother motivation for a systematic study of hydrodynamic fluctuations is thermalization. The local thermalization (or equilibration) time $\\tau_{\\rm th}$ is loosely defined as the time it takes a system to reach local thermodynamic equilibrium. At times $t>\\tau_{\\rm th}$, hydrodynamics governs the slower relaxation to global thermodynamic equilibrium.\nIt is tempting to identify the thermalization time with the exponential decay of non-hydrodynamic correlators $\\langle \\mathcal{O}(t) \\mathcal{O}\\rangle\\sim e^{-t\/\\tau_{\\rm th}}$. Such correlators are however sensitive in general to hydrodynamic long-time tails and therefore strictly do not decay exponentially \\cite{POMEAU197563,Kovtun:2003vj}. A better understanding of long-time tails may therefore help provide a sharp definition of $\\tau_{\\rm th}$. See e.g.~\\cite{Han:2018hlj,Lucas:2018wsc} for recent alternative approaches to $\\tau_{\\rm th}$.\n\n\n\nIn the following we use the general formalism of Ref.~\\cite{Crossley:2015evo} to uncover the universal structure of late-time  response functions for interacting systems with a single continuous symmetry, focusing on time translation invariance (and therefore heat transport) for concreteness. We find that the thermal dc conductivity and diffusion constant both receive independent non-vanishing radiative corrections, even in the case of a single conserved density, and that the correction is not sign definite. Both of these statements are different to the results obtained from a traditional approach \\cite{Kovtun:2014nsa}, for reasons we shall explain. Moreover, we compute the one-loop retarded Green's function $G^R_{\\varepsilon \\varepsilon}(\\omega,k)$ at finite frequency and wavevector, revealing its analytic structure. We conclude by discussing experimental signatures of hydrodynamic fluctuations with applications to insulators, bad metals and cold atoms.\n\n\n\n\\section{Formalism}\n\nOur objective is to understand the structure of energy density correlation functions in non-integrable quantum systems at nonzero temperature\n\\begin{equation}\\label{eq_corr}\n\\langle \\varepsilon (t, x) \\varepsilon (t', x') \\cdots\\rangle_\\beta\n\t\\equiv \\Tr \\Bigl(\\rho_{\\beta}\\,  \\varepsilon (t,x) \\varepsilon(t',x')\\cdots \\Bigr)\\, ,\n\\end{equation}\nwhere the thermal density matrix $\\rho_\\beta = e^{-\\beta H}\/\\Tr e^{-\\beta H}$. Here we will be interested in the case where energy is the only conserved quantity. The systematic study of a single diffusive charge was initiated in Ref.~\\cite{Crossley:2015evo}. In that formalism, furthermore, the contribution of ghosts (or lack thereof) has been well understood \\cite{Gao:2018bxz}. A self-contained review of this special case of the formalism is given in appendix \\ref{app_Hong}. The output of this method is an effective field theory that provides a perturbative expansion for computing the correlators \\eqref{eq_corr}:\n\\begin{equation}\nZ = \\int D \\varepsilon D \\varphi_a \\, e^{i \\int  \\mathcal L[\\varepsilon,\\, \\varphi_a]} \\,.\n\\end{equation}\nHere $\\varepsilon$ is the energy density and $\\varphi_a$ is an auxiliary field (the $a$ subscript is not an index). The most general Lagrangian to cubic order in fields was constructed in Ref.~\\cite{Crossley:2015evo}. In appendix \\ref{app_Hong} we extend their construction to quartic interactions, which will play a role below. The resulting Lagrangian to leading order in derivatives that is at most quartic in fields is given by\n\\begin{equation}\\label{eq_action_main}\n\\mathcal L\n\t= iT^2\\kappa (\\nabla \\varphi_a)^2 - \\varphi_a \\left(\\dot \\varepsilon -\n\t D \\nabla^2 \\varepsilon\\right) \n\t+ \\nabla^2\\varphi_a\\left[\\frac12 \\lambda\\varepsilon^2 + \\frac13\\lambda'\\varepsilon^3\\right] +\n\ticT^2(\\nabla\\varphi_a)^2 \\left[\\widetilde\\lambda\\varepsilon + \\widetilde\\lambda'\\varepsilon^2\\right]\n\t+ \\cdots \\, ,\n\\end{equation}\nwhere $T$ is the temperature, $D$ the diffusivity, $c$ the specific heat and $\\kappa = c D$ the thermal conductivity. These are all `bare' values that will be renormalized by the interactions in (\\ref{eq_action_main}). The couplings $\\lambda,\\lambda',\\widetilde \\lambda, \\widetilde \\lambda'$ themselves can be written as linear combinations of the following derivatives of the transport parameters\n\\begin{equation}\\label{eq_couplings_raw}\nT\\partial_T \\kappa\\, , \\qquad\\quad\nT^2\\partial_T^2 \\kappa\\, , \\qquad\\quad\nT\\partial_T D\\, , \\qquad\\quad\nT^2\\partial_T^2 D\\, .\n\\end{equation}\nTheir explicit expressions are given in \\eqref{eq_couplings}.\n\nThe traditional stochastic approach to hydrodynamic fluctuations with Gaussian noise \\cite{PhysRevA.8.423,POMEAU197563,PhysRevLett.56.889,Kovtun:2003vj} can be recovered from the general effective action \\eqref{eq_action_main} when the interactions that are quadratic in auxiliary fields (i.e. the $\\widetilde\\lambda$ and $\\widetilde \\lambda'$ terms) are absent, by performing a Legendre transform and introducing the noise field $\\xi = \\partial \\mathcal L \/\\partial \\varphi_a$ \\cite{Crossley:2015evo}. However, when $\\widetilde \\lambda$ or $\\widetilde \\lambda'$ is non-vanishing, the resulting theory will contain  interactions of the form $\\xi^2 \\varepsilon$ or $\\xi^2 \\varepsilon^2$. The noise correlations will therefore not be strictly Gaussian because they now depend on energy fluctuations.%\n\t\\footnote{That these interactions should arise is already clear from the stochatistic approach, where the fluctuation dissipation theorem imposes $\\langle \\xi(x)\\xi\\rangle = -2T^2\\kappa(\\varepsilon) \\nabla^2 \\delta(x)$. These interactions only vanish if $\\kappa(\\varepsilon) = \\rm const$, which is also apparent in \\eqref{eq_couplings}.} \n\nIn the remainder we will show precisely how\nthe interactions in (\\ref{eq_action_main}) lead to non-analyticities in response functions and renormalize the transport parameters themselves \\cite{DESCHEPPER19741,PhysRevB.73.035113,PhysRevA.16.732,Kovtun:2011np,Kovtun:2012rj}. Concretely, we are interested in the one-loop correction to the retarded Green's function, which is simply diffusive in the absence of interactions\n\\begin{equation}\\label{eq:bare}\nG_{\\varepsilon \\varepsilon}^{R,0}(\\omega,k)\n\t= \\frac{i\\kappa T k^2}{\\omega + iDk^2}\\, .\n\\end{equation}\nThe diagrams contributing at one loop are shown in Fig. \\ref{fig_loops}, and computed in appendix \\ref{app_loops}. These loops are all UV divergent and should be truncated at the hydrodynamic cutoff $k_{\\rm max} = 2\\pi\/\\ell_{\\rm th}$, which defines the thermalization length $\\ell_{\\rm th}$. Perturbation theory is controlled because all couplings in the Lagrangian (\\ref{eq_action_main}) are power counting irrelevant, and therefore have small effects at low, hydrodynamic energy scales. Indeed, the appropriate dimensional analysis is set by the diffusive pole in (\\ref{eq:bare}), so that $[\\omega] = 2 [k]$ and $[D]=0$. It follows that $[\\varphi_a] = [\\varepsilon]= \\frac{d}{2}[k]$ and hence the cubic couplings $[\\lambda] = [\\widetilde\\lambda] = -\\frac{d}{2}[k]$ are irrelevant. These dimensions suggest that the one loop corrections to tree-level diffusion, which are quadratic in coupling, will be of the schematic form $Dk^2(1+\\lambda^2 k^d)$, as we verify below.\n\n\n\n\\begin{figure}[h]\n\\centerline{\n\\subfigure{\\label{sfig_label1}\n\\includegraphics[width=0.22\\linewidth, angle=0]{fig\/d1.png}}\n\\subfigure{\\label{sfig_label2}\n\\includegraphics[width=0.22\\linewidth, angle=0]{fig\/d4.png}}\n\\subfigure{\\label{sfig_label2}\n\\includegraphics[width=0.22\\linewidth, angle=0]{fig\/tadpole2.png}}\n}\n\\centerline{\n\\subfigure{\\label{sfig_label2}\n\\includegraphics[width=0.22\\linewidth, angle=0]{fig\/d2.png}}\n\\subfigure{\\label{sfig_label2}\n\\includegraphics[width=0.22\\linewidth, angle=0]{fig\/d3.png}}\n\\subfigure{\\label{sfig_label2}\n\\includegraphics[width=0.22\\linewidth, angle=0]{fig\/tadpole1.png}}\n}\\caption{\\label{fig_loops}The one-loop diagrams contributing to $G_{\\varepsilon \\varepsilon}$. Solid lines denote the energy density field $\\varepsilon$ and squiggly lines denote the auxiliary field $\\varphi_a$.}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\\section{Results}\n\nThe diagrams shown in Fig. \\ref{fig_loops} sum up to give the one loop retarded Green's function\n\\begin{equation}\\label{eq_GR}\nG^R_{\\varepsilon \\varepsilon}(\\omega,k)\n\t= \\frac{i \\left(\\kappa + \\delta\\kappa(\\omega,k)\\right) Tk^2}{\\omega+ i D k^2 + \\Sigma(\\omega,k)}\\, ,\n\\end{equation}\nwhere both $\\delta\\kappa(\\omega,k)$ and $\\Sigma(\\omega,k)$ receive analytic and non-analytic contributions. Separating these contributions as\n\\begin{equation}\\label{eq_ana_nonana}\n\\delta \\kappa(\\omega,k)\n\t= \\delta \\kappa + \\kappa_\\star(\\omega,k) \\, , \\qquad\\qquad\n\\Sigma(\\omega,k)\n\t= i\\delta D k^2 + \\Sigma_\\star(\\omega,k) \\, ,\n\\end{equation}\none finds that the analytic pieces have the form\n\\begin{equation}\\label{eq_res_ana}\n\\frac{\\delta \\kappa}{\\kappa} = \\frac{f_d}{c\\, \\ell_{\\rm th}^d} \\lambda_\\kappa\\, , \\qquad\\qquad\n\\frac{\\delta D}{D} = \\frac{f_d}{c\\, \\ell_{\\rm th}^d} \\lambda_D\\, ,\n\\end{equation}\nwith $f_d = {\\rm Vol} (B_{d})  =  2,\\, {\\pi},\\, \\frac{4\\pi}{3}$ for spatial dimensions $d=1,2,3$, and where $\\lambda_\\kappa$, $\\lambda_D$ are dimensionless effective couplings. Their explicit form will be given below. The non-analytic parts of \\eqref{eq_ana_nonana} have the form\n\\begin{equation}\\label{eq_res_nonana}\n\\kappa_\\star(\\omega,k)\n\t= f_\\kappa(\\omega,k) \\alpha_d (\\omega,k) \\, , \\qquad\\qquad\n\\Sigma_\\star(\\omega,k)\n\t= k^2 f_\\Sigma(\\omega,k) \\alpha_d (\\omega,k) \\, , \n\\end{equation}\nwhere $f_\\kappa$, $f_\\Sigma$ are analytic functions, shown below, that do not depend on dimension, and the non-analyticity is \n\\begin{subequations}\\label{eq_alpha}\n\\begin{align}\n&&\\alpha_1(\\omega,k) &= \\frac{1}{4} \\left[k^2 - \\frac{2i\\omega}{D}\\right]^{-1\/2}\\, , \n& (d=1)\\\\\n&&\\alpha_2(\\omega,k) &= -\\frac{1}{16\\pi}\\log \\left[k^2 - \\frac{2i\\omega}{D}\\right]\\, , \n& (d=2)\\\\\n&&\\alpha_3(\\omega,k) &= -\\frac{1}{32\\pi} \\left[k^2 - \\frac{2i\\omega}{D}\\right]^{1\/2}\\, .\n& (d=3)\n\\end{align}\n\\end{subequations}\nThe effect of these non-analyticities is suppressed by powers of momenta and frequency appearing in $f_\\kappa$, $f_\\Sigma$, as we will see below.\n\nThe retarded Green's function is analytic in the upper-half frequency plane, as required by causality. \nThe interactions have induced a branch point at $\\omega = -\\frac{i}{2} D k^2$. Moreover, the diffusive pole is split into two poles with small real parts $\\omega = -i(D+\\delta D)k^2 \\pm O(k^2 |k|^{d}).$\\footnote{Additional poles in \\eqref{eq_GR} are outside the validity of the resummation. The non-analytic corrections to diffusion are seen to be more important than the $O(k^4)$ higher derivative corrections to the diffusion equation in $d=1$ and also in $d=2$, where the dispersion receives an additional imaginary part $O(k^4 \\log k)$. \n}  The location of the branch point can be understood from simple kinematics, by putting both internal legs on-shell (in either the retarded or advanced Green's functions) as in Fig.~\\ref{fig_non_ana}. The frequencies $\\omega$ for which the on-shell condition is satisfied form a half-line in the complex plane parametrized by the loop momentum $k'$, where the Green's function has a branch cut. The branch point is located at the smallest frequency $\\omega$ (in magnitude) that can satisfy the on-shell conditions:\n\\begin{equation}\n\\omega_\\star\n\t= -i D \\min_{k'} \\left[ k^2 + 2k\\cdot k' + 2 k'^2\\right]\n\t= -\\frac{i}{2}Dk^2\\, .\n\\end{equation}\n\n\n\\begin{figure}\n\\centerline{\n\\subfigure{\\label{sfig_label1}\n\\begin{overpic}[width=0.40\\textwidth,tics=10]{fig\/cut}\n\t \\put (55,47) {$\\omega'=iD k'^2$}\n\t \\put (55,16) {$\\omega+\\omega'=-iD (k+k')^2$}\n\t \\put (13,36) {$\\omega,k$} \n\t \\put (29,49) {$\\omega',k'$} \n\\end{overpic} \n}\n\\hspace{50pt}\n\\subfigure{\\label{sfig_label2}\n\\begin{overpic}[width=0.35\\textwidth,tics=10]{fig\/non_ana_new}\n\t \\put (5,66) {$\\omega$}\n\t \\put (18,43) {$ - \\frac{i}{2}D k^2$} \n\t \\put (-4,26) {$ - i(D+\\delta D)k^2$} \n\\end{overpic}\n}\n}\n\\caption{\\label{fig_non_ana} On-shell condition for the two internal legs (left), and analytic structure of the retarded Green's function $G^R_{\\varepsilon\\varepsilon}(\\omega,k)$ at one loop (right). In imposing the on-shell condition, it is important to consider two poles in opposite halves of the complex $\\omega'$ plane, otherwise the loop contribution vanishes. The pole in the upper half plane arises from an advanced Green's function in the loop: $G^A = \\left(G^R\\right)^*$.}\n\\end{figure}\n\nIn previous treatments, similar physics to what we have just described was found in the coupled diffusion of two modes \\cite{PhysRevB.73.035113,Kovtun:2014nsa}. Due to the absence of a systematic formalism for hydrodynamic fluctuations at that time, those works did not account --- among other things --- for interactions that are quadratic in the auxiliary field (in particular $\\widetilde\\lambda$), nor the quartic terms $\\lambda'$ and $\\widetilde\\lambda'$. While the systematic approach modifies the results for two coupled modes, see appendix \\ref{app_2n}, the most qualitative difference is seen for diffusion of a single conserved density. We have found that renormalization of the diffusion constant and conductivity occurs even in this case, \ncontrolled by the effective couplings in \\eqref{eq_res_ana}\n\\begin{equation}\\label{eq:ll}\n\\lambda_\\kappa = \\frac{c^2 T^2 }{D}\\tilde\\lambda'\\, , \\qquad\\qquad\n\t\\lambda_D = - \\frac{c^2 T^2}{2D^2} \\left[\\lambda(\\lambda+\\tilde \\lambda) + 2\\lambda' D\\right]\\,,\n\\end{equation}\nwhich are not sign-definite in general.\n\nFurthermore, we have also found non-analytic corrections to the Green's function even with a single diffusing mode. These are not as strong as those arising with two modes, as we now explain.\nThe functions $f_\\kappa,\\,f_\\Sigma$ appearing in the non-analytic contributions \\eqref{eq_res_nonana} are\n\\begin{equation}\nf_\\kappa(\\omega,k)\n\t= \\frac{cT^2}{D^2} k^2 \\lambda\\tilde \\lambda\\, , \\qquad\\qquad\nf_\\Sigma(\\omega,k)\n\t=\\frac{cT^2}{D^2}\\left[\\omega \\lambda(\\lambda+\\tilde \\lambda) + iDk^2 \\lambda\\tilde \\lambda\\right]\\, .\n\\end{equation}\nWhile $f_\\Sigma$ has both the $O(\\omega)$ and $O(k^2)$ terms expected at this order in the derivative expansion, $f_{\\kappa}$ only has an $O(k^2)$ term.\nThe subsequent suppression of $f_{\\kappa}$ as $k \\to 0$ implies that the optical conductivity does not receive non-analytic corrections, and is instead constant in the hydrodynamic regime\n\\begin{equation}\nT\\kappa(\\omega)\n\t\\equiv \\lim_{k\\to 0} \\frac{\\omega}{k^2}\\Im  G^R_{\\varepsilon \\varepsilon}(\\omega,k)\n\t= T(\\kappa + \\delta \\kappa)\\, .\n\\end{equation}\nThis result can be contrasted with the case of two interacting diffusive densities, wherein the optical conductivity receives a non-analytic fluctuation correction \\cite{DESCHEPPER19741,PhysRevB.73.035113}. We revisit this case in appendix \\ref{app_2n}, where the analytic structure is discussed in the light of a systematic inclusion of fluctuation effects. We find a non-analytic correction to the optical conductivity of the form (we use $\\sigma$ to denote a generic conductivity)\n\\begin{equation}\n\\delta\\sigma(\\omega,k)\n\t\\sim \\omega\\, \\alpha_d(\\omega,k) + \\cdots\\, ,\n\\end{equation}\nwhere $\\alpha_d$ is as in (\\ref{eq_alpha}) and $\\cdots$ denote terms that are further $k^2$ suppressed. In particular, the correction in $d=2$ is $\\delta\\sigma(\\omega) \\sim \\omega\\log\\omega$.\n\n\n\\section{Discussion and Applications}\n\nStrong renormalization of the transport parameters due to hydrodynamic fluctuations occurs if the ratios in (\\ref{eq_res_ana}) are large. When the dimensionless couplings are order unity, $\\lambda_\\kappa\\sim \\lambda_D \\sim 1$,\\footnote{Using equations (\\ref{eq:ll}) and (\\ref{eq_couplings}), this holds in any scaling regime where $\\kappa \\sim T^a$ and $c \\sim T^b$.} the strength of fluctuations is controlled by the specific heat per `thermal volume' $c\\ell_\\text{th}^d$. This quantity can be thought of as the number of degrees of freedom in the smallest volume that can reach local thermodynamic equilibrium. When there are many degrees of freedom in a thermal volume, fluctuation effects are small. One might further expect that a sufficient number of degrees of freedom are necessary in order for a region to locally thermalize, and hence $c\\ell_\\text{th}^d \\gtrsim 1$. Indeed, such a bound has been established in the presence of operators with microscopic positivity properties, using the eigenstate thermalization hypothesis \\cite{Delacretaz:2018cfk,ETHnew}. Thus hydrodynamic fluctuation corrections to thermal transport parameters are expected to be at most comparable to the bare values.\n\nIf microscopic interactions are weak then $\\ell_\\text{th} \\sim \\ell_\\text{mfp}$, the (inelastic) quasiparticle mean free path, will be large. Fluctuation corrections to transport are therefore small in weakly interacting systems. In contrast, in strongly correlated systems $\\ell_\\text{th}$ can become very short and fluctuations may be important. For example, at high temperatures in a lattice model, with order one degrees of freedom per unit cell,\nthe bound mentioned above is only saturated when \n$\\ell_\\text{th} \\sim a$, the lattice spacing. This roughly coincides with the `minimal' mean free path for thermal transport by well-defined phonons in insulators \\cite{SLACK19791, PhysRevB.49.9073}. In strongly correlated regimes, however, the notion of a mean free path is likely not a useful concept. Recent measurements of thermal diffusivity in cuprates \\cite{Zhang:2016ofh,2018arXiv180807564Z} and perovskites \\cite{doi:10.1063\/1.3371815,PhysRevLett.120.125901,AhBh} suggest (assuming that the microscopic sound speed is the relevant velocity) that a transport lengthscale reaches and possibly surpasses the lattice spacing at high temperatures. The specific heat in these materials is roughly $c a^2 \\sim 40$ and $c a^3 \\sim 15$, respectively. It may be interesting to look for signatures of diffusive fluctuations in the thermal transport of these systems.\n\nFluctuation effects can also become important for transport close to a thermal phase transition. The thermalization length diverges as $\\ell_\\text{th} \\sim \\tau^{-\\nu}$ as the reduced temperature $\\tau \\to 0$, while the specific heat scales as $c \\sim \\tau^{2-\\alpha}$. It follows that $\\delta D\/D \\sim \\tau^{\\alpha + d\\nu-2} \\sim 1$ if hyperscaling is obeyed, so fluctuations are important in that case. Above the upper critical dimension hyperscaling is violated and fluctuations are small.  A more sophisticated discussion must include fluctuations of the order parameter in the analysis \\cite{RevModPhys.49.435}.\n\nTransport lengthscales approaching or exceeding the lattice spacing are also seen in `bad metals' \\cite{PhysRevLett.74.3253,RevModPhys.75.1085,hussey}. All of our expressions above are easily adapted to describe the diffusion of a single conserved $U(1)$ charge, instead of heat.\\footnote{If thermoelectric effects are strong, one should instead work with coupled heat and charge diffusion, as in appendix \\ref{app_2n}.} Particle-hole symmetry should be broken, typically by a background charge density, otherwise many terms we have considered are forced to be zero.\nThe correction to the dc electrical conductivity $\\sigma$, for example, is found to be\n\\begin{equation}\\label{eq:sigma}\n\\frac{\\delta \\sigma}{\\sigma}\n\t= \\frac{f_d}{\\ell_{\\rm th}^d} \\frac{T}{\\chi \\mu^2} \\lambda_\\sigma\\, .\n\\end{equation}\nHere $\\chi$ is the charge susceptibility and $\\mu$ the chemical potential.\nIn the definition of the couplings in (\\ref{eq_action_main}) in terms of the thermodynamic derivatives (\\ref{eq_couplings_raw}), one replaces $T \\to \\mu$.\n\nCondensed matter systems --- including most bad metals --- are typically at degenerate temperatures $T < E_F$, below the Fermi energy. At these temperatures $\\chi \\sim k_F^d\/E_F$ and $\\mu \\sim E_F$. Here $k_F$ is the Fermi momentum. The contribution (\\ref{eq:sigma}) of fluctuations to the conductivity is therefore small, even when the thermalization length becomes of order $\\ell_\\text{th} \\sim a \\sim 1\/k_F$. This is the shortest length consistent with local thermalization \\cite{ETHnew}. In contrast, at high temperatures where fermions are non-degenerate, $\\chi \\sim 1\/(T a^d)$. If the total charge is held fixed, then $\\mu \\sim T$. It follows that as the thermalization length becomes short, of order $\\ell_\\text{th} \\sim a$, fluctuation corrections to the conductivity are order one. Diffusive transport by strongly correlated but non-degenerate fermions has recently been probed in an ultracold atom realization of the Hubbard model \\cite{2018arXiv180209456B}, and earlier in e.g. \\cite{Schneider2012}, as well as in numerics \\cite{2018arXiv180308054M, 2018arXiv180608346H}. Indeed, $\\ell_\\text{th}$ is found to saturate around the lattice scale at high temperatures, and so fluctuation effects may be important.\n\n\nFinally, diffusion with a short thermalization length has also been seen in spin transport in strongly interacting ultracold atoms in a trap \\cite{2018arXiv180505354E}. The formulae we have developed can be applied directly to longitudinal spin diffusion in a magnetic field (to break spin reversal symmetry) or to transverse spin diffusion without a magnetic field or spontaneous magnetization (so that isotropy prevents mixing of the two transverse modes).\nAt temperatures $T \\lesssim E_F$, with electrons on the verge of becoming non-degenerate, the thermalization length is found to be $\\ell_\\text{th} \\sim 1\/k_F$ (there is no lattice scale in these experiments). Diffusive fluctuations may therefore again be important for transport.\n\nIn summary, long wavelength fluctuations about diffusive dynamics may be relevant in condensed matter and cold atom systems of widespread interest. We have seen that a systematic derivation of these effects leads to different results than previous, more phenomenological, approaches. For this reason, it will be important to revisit the computation of fluctuations in relativistic hydrodynamics \\cite{Kovtun:2011np, Stephanov:2017ghc}, which includes a sound mode in addition to transverse momentum diffusion. Fluctuations in relativistic hydrodynamics may have direct consequences for the quark-gluon plasma.\n\n\n\n\n\\section*{Acknowledgments} \n \nWe would like to thank Erez Berg, Debanjan Chowdhury, Paolo Glorioso and Andrew Lucas for illuminating discussions. SAH and LVD are partially supported by the US Department of Energy Office of Science under Award Number DE-SC0018134. XCL is supported by the Knut and Alice Wallenberg Foundation.\n\n\\pagebreak\n\n\\bibliographystyle{ourbst}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nThe Unites States (U.S.) is one of the countries with the highest firearms--related death rate, exceptionally high compared to other industrialized countries \\citep{pallin19}. Furthermore, it is the country with the highest firearms possession rate by citizens, thanks to loose legislation in several of its member states \\citep{hemenway17}. The relation between firearms--related death rate and firearms possession rate has been largely investigated and a causal relation going from the latter to the former seems very plausible, e.g. \\citet{krug98, bangalore13} and \\citet{siegel13}. If this fact has prompted part of the public opinion to ask for stricter guns laws, the constitutional right to posses and bear firearms has been strenuously defended by the opposite faction, rendering this one of the most popular and controversial topic in the country.\\par\nAlmost every aspect of the relation between firearms legislation and gun violence has been extensively researched. Besides the mentioned studies that have tried to establish a causal link between firearms--related death rate and firearms possession rate \\citep{krug98, bangalore13, siegel13}, others have investigated the effect of firearms legislation\\textbackslash possession rate on suicide rates \\citep{kellermann92, anestis18}, on pediatric firearms--related mortality \\cite{goyal19}, on homicides rate \\citep{duggan01, kovandzic13, siegel13, siegel14}, and on the death rate of police officers on duty \\citep{lester87, mustard01, swedler15}.\\par\nWith approximately 1000 deaths per year, United States (U.S.) holds another inglorious record among industrialized countries: the highest rate of homicides committed by police forces \\citep{hemenway19}. This raises the question of whether the high rate of police fatal shootings results from relaxed firearms legislation\\textbackslash high possession rate. From a speculative point of view, one can argue that the diffusion of firearms increases the probability of police officers to face armed people while on duty, thus increasing the probability of being involved in potentially dangerous situations that require the use of guns from their part. Furthermore, the increased probability to face dangerous situations is a factor of stress that may lead law enforcement officers to overreact or, more generally, to commit mistakes. A positive relation between firearms diffusion and deadly assaults to police officers \\citep{swedler15} corroborates the theory of an increase danger to officers for higher levels of firearms ownership.\\par\nThanks to the availability of independent datasets that remedy the underreporting of police fatal use of force in official statistics \\citep{conner19}, this topic has recently been investigated. \\citet{hemenway19} find a positive association between firearms prevalence and fatal police shooting rates. \\citet{kivisto17} report that U.S. states with stricter firearms legislation have lower incidence rates of fatal police shootings. Both studies are, however, cross-sectional and thus, despite the use of several controls common in the dedicated literature, may suffer from the omitted variable problem. \\citet{kivisto17}, for example, mention, among the limitations of their paper, exactly this problem, stating that ``it is possible that states with stricter gun legislation also have better training for police officers and more stringent hiring practices, or that states that are already safe are more likely to implement stricter gun laws''. \\par \nThe novelty of the present study is to use a panel dataset to investigate the relation between firearms legislation\\textbackslash possession rate and fatal police shootings. This allows to control for unobserved fixed characteristics at state level that may have biased previous analysis, providing, therefore, more robust results.\n\\section{Literature Review}\nAs discussed before, firearms legislation and ownership is a strongly investigated topic, particularly in the United States. A number of studies document negative impacts of firearms legislation and prevalence, such as increased suicide and homicide rate, although evidence is not unambiguous. From an extensive literature review, \\citet{kleck15} concludes that guns diffusion is a positive determinant of crime rate, but this relation looses statistical significance in the most methodologically rigorous papers. \\citet{branas09} find that possessing a gun increases the probability of being shot during an assault, thus dismantling the opinion of weapons having a protective role. On the other side, \\citet{kleck93}, by comparing 170 U.S. cities, find scarce evidence that guns restrictions have some positive role in reducing the rate of violence. Similarly, \\citet{altheimer08} evidences that gun availability has no effect in determining the number of total individual assaults and robbery, but only increases the number of the ones committed with a fire--weapon.\\par \nRegarding one of the most serious violent crimes, homicide, the results are mixed. \\citet{duggan01}, \\citet{siegel13} and \\citet{siegel14} find a significant and positive relation between guns diffusion and the number of homicides. However, this view is strongly opposed by \\citet{kovandzic13}, which, on the contrary, document a negative relation. They cite a potential deterrent effect of guns as an explanation for their results. A similar debate has surrounded the permission of carrying concealed weapons, with \\citet{lott97} showing a positive role of such law in reducing violent crimes whereas \\citet{dezhbakhsh98} rejecting this finding and claiming the opposite. Findings regarding firearms diffusion and legislation are therefore very discordant even when considering aspects such as crime and homicide rate that are among the most studied.\\par \nShifting the attention on police forces, there is a paucity of research regarding the association between firearms diffusion and legislation and the number of killings of, and by, police officers. When considering the former, the killing of law enforcement officers, we have again contrasting findings. \\citet{lester87} and \\citet{swedler15} find that increasing levels of households gun ownership are a clear factor of risk for police officers, whereas \\citet{mustard01}, limiting the attention to the possibility of carrying concealed weapons, puts in evidence a potential protective role of this law. \\par\n\\citet{kivisto17} and \\citet{hemenway19} investigate the relation between firearms and killings committed by police officers. The two studies both rely on independent and open source databases to retrieve the number of fatal police shootings: \\citet{kivisto17} on \\href{https:\/\/www.theguardian.com\/us-news\/series\/counted-us-police-killings}{The Counted}, maintained by The Guardian, and \\citet{hemenway19} on \\href{https:\/\/www.washingtonpost.com\/graphics\/investigations\/police-shootings-database\/}{Fatal Force}, created by The Washington Post. Furthermore, they both rely on a cross sectional analysis using the fraction of suicides with a fire--weapon on the total number of suicides as a proxy for guns diffusion, as previously done in other papers, e.g. \\citet{kleck04} and \\citet{azrael04}. Compared to \\citet{hemenway19}, whose focus is exclusively on guns diffusion as a cause of fatal police shootings, \\citet{kivisto17} further consider firearms legislation, using the \\href{https:\/\/www.bradyunited.org\/}{Brady Campaign} scorecards as an indicator of law strength. In both papers, firearms ownership is found to positively affect the number of fatal police shootings. Even after controlling for firearms prevalence, firearms regulations on gun trafficking and on child and consumer safety significantly reduces fatal police shootings \\citep{kivisto17}.\\par \nGiven the contrasting evidence emerged in other topics related to firearms diffusion and legislation and since both the last mentioned papers rely on a cross sectional analysis that may be plagued by the omitted variable bias, the extension to a panel data setting seems a necessary further step. This may help to strengthen the findings reached so far or to contest their validity as the result of a biased analysis.   \n\\section{Data and Methods}\nFollowing \\citet{kivisto17} and \\citet{hemenway19}, our units of observation are the 50 U.S. states, with District of Columbia having being excluded for lack of data in several covariates. The covered time period is from January 1, 2012, to December 31, 2018, and all variables are expressed as yearly values, forming a dataset with seven time periods. Different databases have been consulted and merged in order to have all the variables of interest and the necessary controls: \\hyperlink{https:\/\/fatalencounters.org\/}{Fatal Encounters}, \\href{https:\/\/giffords.org\/lawcenter\/resources\/scorecard\/}{Giffords scorecards}, the \\href{https:\/\/www.census.gov\/}{U.S. Census Bureau} data portal, the \\href{https:\/\/crime-data-explorer.fr.cloud.gov\/explorer\/national\/united-states\/crime}{Federal Bureau of Investigation's (FBI's) Crime Data Explorer} and the Centers for Disease Control\nand Prevention's (CDC's) \\href{https:\/\/www.cdc.gov\/injury\/wisqars\/index.html}{Web-based Injury Statistics Query and Reporting System} (WISQARS). Following is a description of all variables.\n\\subsection{Description of Variables}\nOur dependent variable, the number of deaths caused by police shooting per million inhabitants (\\textit{Pol\\_Shoot}), is retrieved from the \\hyperlink{https:\/\/fatalencounters.org\/}{Fatal Encounters} database. We choose an independent database to alleviate the likely problem of underreporting of such episodes in the FBI's official statistics \\citep{williams19}. Compared to other open source repositories, e.g. \\href{https:\/\/www.theguardian.com\/us-news\/series\/counted-us-police-killings}{The Counted}, \\href{https:\/\/www.washingtonpost.com\/graphics\/investigations\/police-shootings-database\/}{Fatal Force}, \\href{https:\/\/mappingpoliceviolence.org\/}{Mapping Police Violence} and \\href{https:\/\/www.gunviolencearchive.org\/}{Gun Violence Archive}, the Fatal Encounters database covers the longest time span -- from 2000 to present. Specific cases of police shooting can be retrieved by selecting the category ``Deadly use of force'' and the subcategory ``Gunshot''.\\par\nOur independent variable of interest is the strength of firearms regulations at state level (\\textit{Giff\\_Score}). In order to obtain a synthetic measure of the strength of a state legislation, we rely on the \\href{https:\/\/giffords.org\/lawcenter\/resources\/scorecard\/}{Giffords scorecards}, available for the period 2010-2018, with the exclusion of the year 2011, hence the need to drop the year 2010. The overall score is an aggregation of seven component scores, namely: background checks and access to firearms (\\textit{BCAF}), other regulations of sales and transfers (\\textit{ORST}), classes of weapons and magazines\\textbackslash ammunitions (\\textit{CWAM}), consumers and child safety (\\textit{CCS}), gun owner accountability (\\textit{GOA}), firearms in public places (\\textit{FPP}) and a residual class (\\textit{OTH}). Disaggregation of the overall score allows us to test the role of each component in explaining fatal police shootings, following \\citet{kivisto17}. This is helpful in identifying the areas where intervention should be prioritized to reduce police shooting episodes. Since the scoring system has been slightly modified several times during the study period, we have implemented a harmonization procedure, retaining only the sub-indicators that remained unaltered over time. \\citet{kivisto17}, using data from the same source, eliminate the weighting system in favor of a ``1 law = 1 point'' scale. They argue that a weighting system necessarily entails a degree of arbitrariness. However, we think that the equal weighting implied by the ``1 law = 1 point'' scale is analogously arbitrary. We, therefore, prefer to rely on the weights assigned by professional lawyers, thus leaving the Giffords scorecard scale unaltered.\\par\nStricter legislation on firearms may reduce the quantity of fire--weapons owned by citizens, but may also promote safer use, e.g. by denying dangerous subjects access to guns or by increasing the safety of circulating weapons. Besides examining if laws to promote safe gun use are effective in reducing fatal police shootings, we can test if the effect of firearms legislations operate via the former channel by looking at the relation between fatal police shootings and gun diffusion. Lacking state--level data on gun ownership for our study period, we rely on a commonly adopted proxy (\\textit{Suicide}) -- the percentage of suicides committed with a fire--weapon over the total of suicides \\citep{kleck04, azrael04, kivisto17, hemenway19}. These data are retrieved from the Web-based Injury Statistics Query and Reporting System (WISQARS).\\par \n\\begin{table}[H]\n\t\\centering\n\t\\caption{Descriptive Statistics}\\label{tab1}\n\t{\\renewcommand{\\arraystretch}{1.3}\n\t\t\\begin{tabular}{lccccc}\n\t\t\t\\hline\\hline\n\t\t\t\\textbf{Variable} & \\textbf{N. Obs.} & \\textbf{Mean} & \\textbf{Std. Dev.} & \\textbf{Min.} & \\textbf{Max.} \\\\\n\t\t\t\\hline\\hline\n\t\t\t\\textit{Pol\\_Shoot} & 350 & 3.51 & 2.11 & 0 & 10.85 \\\\\n\t\t\t\\textit{Giff\\_Score} & 350 & 31.98 & 24.42 & 4 & 105.50 \\\\\n\t\t\t\\textit{Crime} & 350 & 3627.32 & 1388.63 & 1026.24 & 8849.56 \\\\\n\t\t\t\\textit{Suicide} & 350 & 51.54 & 12.35 & 13.20 & 74.30 \\\\\n\t\t\t\\textit{PC\\_Income} & 350 & 29712.58 & 4828.93 & 20119 & 52500.00 \\\\\n\t\t\t\\textit{Urban} & 350 & 73.59 & 14.44 & 38.70 & 95 \\\\\n\t\t\t\\textit{Poverty} & 350 & 9.99 & 2.80 & 4 & 19.20 \\\\\n\t\t\t\\textit{White} & 350 & 76.95 & 12.67 & 24.30 & 95.10 \\\\\n\t\t\t\\textit{Low\\_Edu} & 350 & 11.19 & 2.97 & 6.10 & 18.60 \\\\\n\t\t\t\\textit{Unemp.} & 350 & 3.96 & 1.18 & 1.80 & 7.90 \\\\\n\t\t\t\\textit{Young} & 350 & 23.15 & 1.31 & 19.80 & 27.50 \\\\ \\hline\n\t\t\t\\multicolumn{6}{c}{\\textit{\\textbf{Giff\\_Score disentangled}}} \\\\\n\t\t\t\\textit{BCAF} & 350 & 7.19 & 5.51 & 0 & 22 \\\\\n\t\t\t\\textit{ORST} & 350 & 4.12 & 5.89 & 0 & 24 \\\\\n\t\t\t\\textit{CWAM} & 350 & 1.82 & 3.86 & 0 & 14 \\\\\n\t\t\t\\textit{CCS} & 350 & 2.25 & 2.02 & 0 & 9 \\\\\n\t\t\t\\textit{GOA} & 350 & 2.76 & 5.00 & 0 & 17.50 \\\\\n\t\t\t\\textit{FPP} & 350 & 9.16 & 4.36 & 0 & 19\\\\ \n\t\t\t\\textit{OTH} & 350 & 4.67 & 2.18 & 0 & 10\\\\ \\hline\\hline\n\t\\end{tabular}}\n\\end{table}\n\\indent \nRegarding control variables, we retrieved data on the number of violent crimes (per million inhabitants, \\textit{Crime}) from the FBI's Crime Data Explorer, where a crime is defined as any of the four offenses -- murder and non-negligent manslaughter, rape, robbery, and aggravated assault. All the other controls are retrieved from the U.S. Census Bureau. Note that all values are projections on the 2010 census data. These controls include per--capita income in 2010 inflation--adjusted dollars (\\textit{PC\\_Income}) and the percentage of people living in urban areas (\\textit{Urban}). It must be noted that, for this last variable, only figures for the year 2010 were available, thus it is treated as a time--invariant covariate. Other socio--economic characteristics, such as poverty rate (\\textit{Poverty}),  unemployment rate (\\textit{Unemp.}), and the percentage of adults with an education lower than high school diploma (\\textit{Low\\_Edu}), are also included. The percentage of young population, aged 18--34, (\\textit{Young}) and the percentage of white Caucasians (\\textit{White}) over the whole population are controlled for in the analysis. The last variable is added since several studies find a racial bias in police shootings \\citep{ross15, nix17, mesic18}. Table \\ref{tab1} reports some key statistics of all the mentioned variables.     \n\\subsection{Statistical Analysis}\nThe statistical analysis is divided into two main parts. In the first part, we focus on the role of the legislative strength as a whole, thus considering the overall score provided by Giffords for each U.S. state. In the second part, we analyze the component Giffords scores separately. This analysis should provide more specific policy indications with regard to the legislative field where intervention may be more productive in reducing fatal police shootings.\\par \nThe effects of any changer in legislation may take time to be observed. In all our analysis, therefore, both the overall and the component Giffords scores enter in their first lags. The inclusion of lags more distant in time (two or three years) is precluded by the limited number of time periods at our disposal. We have actually run regressions with the contemporaneous level of the Giffords scores, but none of them has resulted in significant coefficients (results are available from the authors upon request). The possibility to include both the lag and the current level is prevented by their high correlation ($\\rho = 0.99$) that most likely causes a problem of collinearity. \\par \nFigure \\ref{fig1} shows the Spearman's rank correlation coefficients of all variables, except the component Giffords score. Note \\textit{Giff\\_Score\\_L}  denotes the lagged Giffords score. From Figure \\ref{fig1} it is possible to observe that several covariates have a relatively low correlation coefficient. The exceptions are the violent crime rate ($\\rho = 0.49$) and the per--capita income ($\\rho = -0.29$). \\par \n\\begin{figure}[htbp!]\n\t\\centering\n\t\\caption{Correlation Matrix}\\label{fig1}\n\t\\includegraphics[scale=0.75]{Fig\/Corr_fig.eps}\n\\end{figure} \n\\indent  \nDespite the low correlation of these covariates with the dependent variable, our full specifications (specifications (1) and (2) in Table \\ref{tab2}) include all of them. Their coefficients are not significant in these models. Failing to reject the null hypothesis that the joint significance of the socio--economic covariates (\\textit{Poverty}, \\textit{White}, \\textit{Low\\_Edu}, \\textit{Unemp.} and \\textit{Young}) is equal to zero (p--value = 0.6098 for model (1) and p--value = 0.8595 for model (2)), we omit these variables in any subsequent analysis. Despite its lack of significance, \\textit{Urban} is kept in all models because of its high correlation with the lagged Giffords score ($\\rho = 0.62$) and because it is a significant control in previous studies \\citep{hemenway19}. Models (3) and (4) in Table  \\ref{tab2} report the results with the parsimonious set of explanatory variables. From Table \\ref{tabs1} in the Appendix, it is possible to observe non significant p--values for the RESET and for the Mundlak test. Compared to models (1) and (2), the coefficients of models (3) and (4) are negligibly different.\\par\nA conditional fixed effect (FE) and a random effect (RE) Poisson regressions (models (1) and (2) in Table \\ref{tab2}) are our main models of interest. The functional form specification is checked through a RESET test, by adding the squared residuals of the Poisson regressions (FE and RE) and checking their significance \\citep{ramsey74}. Results are reported in Table \\ref{tabs1} in the  \\hyperref[app]{Appendix} together with the coefficients of the year dummies (year 2013 as base) included in all models. The p--value of the squared residuals is far above the 10\\% significance level, thus dismissing possible concerns about misspecification. The choice to report both the FE and RE estimates is due to the fact that the Mundlak test -- also reported in Table \\ref{tabs1} -- has a significance level very close to the 5\\% level \\citep{mundlak78}. Although this test, chosen compared to the more common Hausman test given the presence of year dummies and a time invariant covariate (\\textit{Urban}), suggests the use of the random effect model, the closeness of the p--value to the threshold level and the concerns about the distributional assumptions of the Poisson RE model lead us to report the fixed effect estimates as well. However, it can be observed that the estimated coefficients are very similar in both specifications.\\par\n\\begin{table}[H]\n\t\\centering\n\t\\caption{Regressions Results: Total Giffords Score (Lagged)}\\label{tab2}\n\t{\\renewcommand{\\arraystretch}{1.3}\n\t\t\\begin{tabular}{lcccc}\n\t\t\t\\hline\\hline\n\t\t\t\\multirow{3}{*}{} & \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}(1) \\\\ Poisson FE\\end{tabular}}} & \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}(2) \\\\ Poisson RE\\end{tabular}}} & \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}(3) \\\\ Poisson FE\\end{tabular}}} & \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}(4) \\\\ Poisson RE\\end{tabular}}} \\\\\n\t\t\t&  &  &  &  \\\\\\hline\\hline\n\t\t\t& \\multicolumn{1}{l}{\\textit{Pol\\_Shoot}} & \\multicolumn{1}{l}{\\textit{Pol\\_Shoot}} & \\multicolumn{1}{l}{\\textit{Pol\\_Shoot}} & \\multicolumn{1}{l}{\\textit{Pol\\_Shoot}} \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Giff\\_Score\\_L}} & 0.990* & 0.992+ & 0.990* & 0.993+ \\\\\n\t\t\t& (-2.05) & (-1.65) & (-2.10) & (-1.82) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Crime}} & 1.000 & 1.000*** & 1.000 & 1.000*** \\\\\n\t\t\t& (1.61) & (4.04) & (1.46) & (4.69) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Suicide}} & 0.993 & 1.007 & 0.993 & 1.008 \\\\\n\t\t\t& (-0.77) & (1.03) & (-0.76) & (1.21) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{PC\\_Income}} & 1.000* & 1.000* & 1.000 & 1.000+ \\\\\n\t\t\t& (-2.15) & (-2.05) & (-1.49) & (-1.94) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Urban}} &  & 1.004 &  & 1.005 \\\\\n\t\t\t&  & (0.90) &  & (1.32)\\\\\n\t\t\t\\multirow{2}{*}{\\textit{Poverty}} & 0.958 & 0.965 &  &  \\\\\n\t\t\t& (-1.51) & (-1.04) &  &  \\\\\n\t\t\t\\multirow{2}{*}{\\textit{White}} & 0.958 & 0.996 &  &  \\\\\n\t\t\t& (-1.21) & (-0.66) &  &  \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Low\\_Edu}} & 1.025 & 1.012 &  &  \\\\\n\t\t\t& (0.90) & (0.38) &  &  \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Unemp.}} & 1.064 & 1.032 &  &  \\\\\n\t\t\t& (1.20) & (0.80) &  &  \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Young}} & 0.946 & 1.008 &  &  \\\\\n\t\t\t& (-1.01) & (0.23) &  &  \\\\ \\hline\\hline\n\t\t\t\\multicolumn{5}{l}{\\footnotesize{t-statistics in parentheses. P-value: + p \\textless{}0.1, * p \\textless{}0.05, ** p \\textless{}0.01, *** p \\textless{}0.001.}}\n\t\\end{tabular}}\n\\end{table}\n\\indent\nAs a robustness check, we also report the results of linear models, both FE and RE, after that the dependent variable has been transformed with a Yeo--Johnson power transform in order to render its distribution more normal--like \\citep{yeo00}. The results of the linear models are shown in Table \\ref{tabs2}. Although the transformation of the dependent variable prevents the computation of meaningful marginal effects, the sign and significance of the coefficients serve to confirm the results of, or to signal a possible problem in, the Poisson regressions. The distribution of the dependent variable before and after the Yeo--Johnson transformation is shown in Figure \\ref{figs1} in the \\hyperref[app]{Appendix}.\\par \nFor our second purpose, we regress the rate of fatal police shootings on the component Giffords scores rather than the overall score. Here we present only the parsimonious models. Four models have been run, two Poisson -- FE and RE -- and two analogous linear regressions. Results are reported in Table \\ref{tab3}, models (5), (6), (7) and (8), and auxiliary information can be found in Table \\ref{tabs1} in the \\hyperref[app]{Appendix}. From this last table, it is possible to see that the p--values of the RESET and of the Mundlak test are all above conventional significance levels. As for the previous models, the difference in the significance of the coefficients of the Poisson and of the linear regressions are very modest, so as the difference in the coefficients between the FE and RE models. \\par \n\\begin{table}[H]\n\t\\centering\n\t\\caption{Regressions Results: Linear Models}\\label{tabs2}\n\t\\resizebox{!}{9cm}{\n\t\t\\begin{tabular}{lcccc}\n\t\t\t\\hline\\hline\n\t\t\t\\multirow{2}{*}{} & \\textbf{\\begin{tabular}[c]{@{}c@{}}(1b) \\\\ Linear FE\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}(2b) \\\\ Linear RE\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}(3b) \\\\ Linear FE\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}(4b) \\\\ Linear RE\\end{tabular}} \\\\\n\t\t\t\\hline\\hline\n\t\t\t& \\textit{Pol\\_Shoot} & \\textit{Pol\\_Shoot} & \\textit{Pol\\_Shoot} & \\textit{Pol\\_Shoot} \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Giff\\_Score\\_L}} & -0.00879* & -0.00924** & -0.00925* & -0.00803* \\\\\n\t\t\t& (-2.32) & (-2.86) & (-2.34) & (-2.52) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Crime}} & 1.47-E004+ & 1.36-E004*** & 1.60-E004+ & 149-E004*** \\\\\n\t\t\t& (1.76) & (3.75) & (1.95) & (4.43) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Suicide}} & -0.00515 & 0.00193 & -0.00562 & 0.00237 \\\\\n\t\t\t& (-0.55) & (0.36) & (-0.59) & (0.45) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{PC\\_Income}} & -1.83-E005* & -1.95-E005* & -1.44-E005+ & -155-E005* \\\\\n\t\t\t& (-2.07) & (-2.22) & (-1.75) & (-2.12) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Urban}} &  & 0.00255 &  & 0.00357 \\\\\n\t\t\t&  & (0.66) &  & (0.92) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Poverty}} & -0.0248 & -0.0259 &  &  \\\\\n\t\t\t& (-0.80) & (-0.87) &  &  \\\\\n\t\t\t\\multirow{2}{*}{\\textit{White}} & -0.0159 & -0.00411 &  &  \\\\\n\t\t\t& (-0.41) & (-1.03) &  &  \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Low\\_Edu}} & 0.0142 & 0.0124 &  &  \\\\\n\t\t\t& (0.51) & (0.52) &  &  \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Unemp.}} & 0.0393 & 0.0327 &  &  \\\\\n\t\t\t& (0.64) & (0.69) &  &  \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Young}} & -0.0476 & -0.0138 &  &  \\\\\n\t\t\t& (-0.92) & (-0.47) &  &  \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Const.}} & 4.304 & 2.195* & 1.938** & 1.291* \\\\\n\t\t\t& (1.34) & (2.06) & (2.83) & (2.42) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{2014}} & 0.0538 & 0.0639 & 0.0313 & 0.0418 \\\\\n\t\t\t& (0.76) & (0.92) & (0.46) & (0.60) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{2015}} & 0.0661 & 0.0829 & 0.0359 & 0.0522 \\\\\n\t\t\t& (0.84) & (1.15) & (0.52) & (0.76) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{2016}} & 0.137 & 0.151+ & 0.113* & 0.120* \\\\\n\t\t\t& (1.55) & (1.95) & (2.24) & (2.48) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{2017}} & 0.163 & 0.189+ & 0.129+ & 0.146* \\\\\n\t\t\t& (1.42) & (1.75) & (1.99) & (2.31) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{2018}} & 0.255* & 0.282* & 0.220** & 0.236*** \\\\\n\t\t\t& (2.11) & (2.40) & (3.10) & (3.36) \\\\ \\hline\\hline\n\t\t\t\\multicolumn{5}{l}{\\footnotesize{t-statistics in parentheses. P-value: + p \\textless{}0.1, * p \\textless{}0.05, ** p \\textless{}0.01, *** p \\textless{}0.001.}}\n\t\\end{tabular}}\n\\end{table}\n\\indent\n\n\\begin{table}[H]\n\t\\centering\n\t\\caption{Regressions Results: Giffords Score Disentangled (Lagged)}\\label{tab3}\n\t{\\renewcommand{\\arraystretch}{1.3}\n\t\t\\begin{tabular}{lcccc}\n\t\t\t\\hline\\hline\n\t\t\t& \\textbf{\\begin{tabular}[c]{@{}c@{}}(5) \\\\ Poisson FE\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}(6) \\\\ Poisson RE\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}(7) \\\\ Linear FE\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}(8) \\\\ Linear RE\\end{tabular}} \\\\ \\hline\\hline\n\t\t\t& \\textit{Pol\\_Shoot} & \\textit{Pol\\_Shoot} & \\textit{Pol\\_Shoot} & \\textit{Pol\\_Shoot} \\\\\n\t\t\t\\multirow{2}{*}{\\textit{BCAF\\_L}} & 1.001 & 1.000 & 0.00131 & -6.63-E004 \\\\\n\t\t\t& (0.15) & (0.03) & (0.22) & (-0.12) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{ORST\\_L}} & 1.032 & 1.008 & 0.0231 & 0.0110 \\\\\n\t\t\t& (1.44) & (0.47) & (1.17) & (0.93) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{CWAM\\_L}} & 0.981 & 1.014 & -0.0168 & 0.00372 \\\\\n\t\t\t& (-0.93) & (0.44) & (-1.01) & (0.22) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{CCS\\_L}} & 0.926 & 0.958 & -0.0732 & -0.0409 \\\\\n\t\t\t& (-1.09) & (-1.24) & (-1.19) & (-1.47) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{GOA\\_L}} & 0.963* & 0.962* & -0.0351* & -0.0352** \\\\\n\t\t\t& (-2.57) & (-2.32) & (-2.58) & (-2.93) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{FPP\\_L}} & 0.984 & 0.983 & -0.0191 & -0.0193 \\\\\n\t\t\t& (-1.25) & (-1.38) & (-1.54) & (-1.63) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Crime}} & 1.000 & 1.000*** & 1.58-E004+ & 1.57-E004*** \\\\\n\t\t\t& (1.45) & (4.96) & (1.90) & (4.64) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Suicide}} & 0.992 & 1.002 & -0.00597 & -0.00118 \\\\\n\t\t\t& (-0.83) & (0.37) & (-0.63) & (-0.21) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{PC\\_Income}} & 1.000 & 1.000* & -1.39-E005+ & -1.62E-005* \\\\\n\t\t\t& (-1.48) & (-2.18) & (-1.70) & (-2.14) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Urban}} &  & 1.004 &  & 0.00351 \\\\\n\t\t\t&  & (1.19) &  & (0.94) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Const.}} &  &  & 1.955** & 1.503** \\\\\n\t\t\t&  &  & (2.72) & (2.66) \\\\ \\hline\\hline\n\t\t\t\\multicolumn{5}{l}{\\footnotesize{t-statistics in parentheses. P-value: + p \\textless{}0.1, * p \\textless{}0.05, ** p \\textless{}0.01, *** p \\textless{}0.001.}}\n\t\\end{tabular}}\n\\end{table}   \n\\section{Results and Discussion}    \nIn the presentation of results, we will focus solely on the Poisson regressions, given the difficult interpretation of the coefficients of the linear models discussed earlier. Note that all the reported coefficients relative to the Poisson models are incidence rate ratios (IRR). From Table \\ref{tab2}, it is possible to observe that the Giffords score (lagged) is always statistically significant, although only at 10\\% level in the RE models, while at 5\\% in the FE models. An increase of one point in the overall Giffords score causes an approximately 1\\% reduction in the number of fatal police shootings per million of inhabitants if considering the FE model. The percentage reduction is slightly lower for the RE model: 0.8\\% when including all controls -- model (3) -- and 0.7\\% when excluding the subset of jointly non-significant covariates -- model (4).\\par \nAn important point to notice is the lack of significance, in all models of Table \\ref{tab2}, of the proxy for firearms diffusion: \\textit{Suicide}. The strong correlation with the lag of the Giffords score ($\\rho = -0.83$) may suggest that the effect of this last masks the one of the former. However, if running the same regressions without the inclusion of the Giffords score, the p--value of \\textit{Suicide} remains above 0.1 in all models: 0.152, 0.106, 0.140 and 0.118 for, respectively, models (1), (2), (3) and (4) without \\textit{Giff\\_Score\\_L} (full results available from the authors upon request). \\par \nWhen considering Table \\ref{tab3} and the disentangled categories composing the Giffords score, only the coefficient related to the lag of one category, gun owner accountability, is significant (at 5\\% level). This happens to be true both in the FE and in the RE Poisson regressions, so as in the linear models. In particular, an increase of one point in the strength of the gun owner accountability category is associated with a decrease in per--million inhabitants fatal police shootings of 3.7\\% (FE model) or of 3.8\\% (RE model).\\par \nWe can further notice that in all models, both in Table \\ref{tab2} and \\ref{tab3}, the sign of the coefficients of the main control variables is as expected. In particular, the number of violent crimes positively impacts the number of fatal police shooting episodes whereas per--capita income has the opposite effect. A last word is dedicated to the significance of the violent crime rate that is very high (0.1\\%) in all RE Poisson models but absent in the FE models. This is possibly due to the persistent nature in time of this phenomenon that, in the FE model, gets captured by the fixed effect component.           \n\\subsection{Discussion}\nThe present study shows that increasing levels of firearms regulation are significantly associated with a lower number of fatal police shooting cases. In particular, a point increase in the overall Giffords score leads to a decrease of in fatal police shootings of 0.7\\%--1\\%, depending from the model. When considering separately the various categories composing the Giffords score, one point increase in the strength of gun owner accountability leads to a decrease of approximately 3.7\\%--3.8\\% in the number of people killed by police officers. The diffusion of fire--weapons, instead, has no statistically significant role in determining the considered outcome. This finding has been achieved through the use of a panel dataset, thus controlling for unobserved heterogeneity through the use of FE Poisson models.\\par\nIt is interesting to compare our results with previous findings. Regarding the effect of firearms diffusion, our results clearly contradicts the previous findings of \\citet{kivisto17} and of \\citet{hemenway19}. In fact, we do not find a statistically significant effect of guns diffusion in determining the number of fatal police shootings. \\par \nConsidering the strength of firearms regulations, our analysis basically confirms the findings of \\citet{kivisto17}. However, this is true only for the overall score. When evaluating each category separately, significant differences emerge. First of all, it must be noted that the results are not easily comparable, given the different scoring system used in \\citet{kivisto17}, namely the Brady scorecards, and in the present analysis. However, a comparison is not impossible. In \\citet{kivisto17}, two categories remained significant after all controls were added, namely promoting safe storage via child and consumer safety laws and curbing gun trafficking. The former corresponds to the category consumers and child safety (\\textit{CSS}) in the Giffords scorecards and the latter is included in other regulation of sales and transfers (\\textit{ORST}), both not significant in our models. The gun owner accountability, the category found significant in the present analysis, is instead composed by three elements: licensing of gun owners and purchasers, having the highest weights, followed by registration of firearms and reporting lost or stolen firearms. This is an important difference, with potentially relevant implications for policy--makers. Furthermore, it is reasonable to think that the first sub-category, namely the need of gun owners to have a license, has a great discriminant power in determining the final identity of gun owners. This further suggests that the qualitative side of gun diffusion (who owns a gun) is more important in limiting the number of police shootings than the quantitative side (how many guns are owned).     \n\\section*{Conclusions}\nThe present analysis has shown that police shooting episodes are significantly reduced by stricter levels of firearms law. While this finding partially confirms what emerged in previous studies, we also find that the diffusion of fire--weapons is inconsequential in determining the number of police shootings, thus contradicting the precedent evidence.\\par \nThe policy recommendations that can be derived from the present paper are quite straightforward. Improving the strength of firearms regulations seems an effective way for reducing the number of people killed by law enforcement officers. Actually, the policy prescriptions can be even more specific. In fact, from the analysis it emerges that the most effective intervention for reducing fatal police shooting episodes would be to strengthen the rules of gun owner accountability, namely licensing of gun owners and purchasers, registration of firearms and reporting lost or stolen firearms. These policy prescriptions are different from the ones provided by previous studies.\\par\nAnother important lesson, and a departure from previous findings, is the lack of statistical significance of the diffusion of guns in causing fatal police shootings. This suggests that the cause may be more qualitative (who owns the guns) rather than quantitative (how many guns). The fact that the only significant legislative category emerged from this study is the gun owner accountability further strengthens this hypothesis.\\par  \nThere are several possible ways in which the analysis could be expanded in order to have more precise and specific prescriptions. One possibility would be to further disaggregate each category of the Giffords score into its subcategories. We have not pursued this road due to the limited number of observations at our disposal. Another interesting extension would be to consider the episodes of police shootings directed towards unarmed citizens \\citep{hemenway19}. The lack of this information in the Fatal Encounters database has prevented us to conduct this analysis.    \n\\newpage \n\\bibliographystyle{apalike}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\bf NOTATION AND DEFINITIONS}\\label{nota} \n\n\\noindent $\\bullet$ Throughout this article units are used in which \n$\\hbar = c = 1$.  \n\n\\noindent $\\bullet$ Everywhere the {\\it repeated} indexes imply the\nsummation.\n\n\\noindent $\\bullet$ $\\delta^{ik} = \\delta_{ik} = \\delta^i_k = \\delta^k_i$\n$(i,k = 1, \\ldots , n)$ is Kronecker symbol,\n\\[\n\\delta^{ik} = 0 \\quad \n i \\ne k, \\quad \\delta^{11} = \\delta^{22} = \\ldots \\delta^{nn} = 1.\n\\]\n\\noindent Metric tensor in Minkowski space $g^{\\mu \\nu} = g_{\\mu \\nu}$ \n$(\\mu, \\nu = 0, 1, \\ldots , n)$ equals:\n\\[\ng^{\\mu \\nu} = 0 \\quad  \\mu \\ne \\nu, \\quad \ng^{00} = 1, \\quad g^{11} = g^{22} = \\ldots = g^{nn} = -1.\n\\]\n\\noindent The tensor $g^{\\mu \\nu}$ is used for raising and lowering of the \nLorentz subscripts and superscripts. \n\n\\noindent $\\bullet$ The scalar products of any two $p$ and $q$ vectors (both \nin Euclidean and in Minkowski spaces) is denoted as follows: \n\\[\n p \\cdot q \\quad {\\rm or} \\quad (pq), \\quad {\\rm i.e.} \\quad p \\cdot q = \n(pq).\n\\]\nThe scalar products of any two $p$ and $q$ Euclidean vectors would be also\ndenoted as:\n\\[\n\\vec p \\vec q, \\quad {\\rm i.e.} \\quad \\vec p \\vec q = p \\cdot q = (pq).\n\\]\n\n\\noindent $\\bullet$ 4--vector $p^{\\mu}$ in Minkowski space is given by\n\\[\np^{\\mu} \\equiv (E, \\, \\vec p) \\; = \\; (p_0, \\, p_1, \\, p_2, \\, p_3), \\; \\;\np_{\\mu} = g_{\\mu \\nu} p^{\\nu} \\, = \\, (E, \\, -\\vec p).\n\\]\nThe scalar product of any two vectors $p$ and $q$ in Minkowski space is given\nby\n\\[\np^{\\mu} g_{\\mu \\nu} q^{\\nu} \\, = \\, p^{\\mu} q_{\\mu} \\, = \\, p_{\\nu} q^{\\nu} \\,\n = \\, p_0 q_0 - p_1 q_1 - p_2 q_2 - p_3 q_3. \n\\]\nThe products of the 4--vector $p^{\\mu}$ with Dirac $\\gamma^{\\mu}$ matrix \ndenotes as usual\n\\[\n \\hat p \\; \\equiv \\; p^{\\mu} g_{\\mu \\nu}  \\gamma^{\\nu} \\, = \\,\n p^{\\mu} \\gamma_{\\mu} \\, = \\, p_{\\nu} \\gamma^{\\nu}. \n\\]\n\n\\noindent $\\bullet$ {\\bf Totally antisymmetric tensor\n $\\varepsilon^{A B \\ldots N}$}.\n\n\\noindent $\\bullet$ {\\it $\\varepsilon$-symbol in two dimensions}:\n$\\varepsilon^{AB} \\quad (A,B=1, 2)$:\n\\begin{eqnarray*} \n && \\varepsilon_{12}=\\varepsilon^{12} = 1; \\; \\varepsilon_{21}=\n \\varepsilon^{21} = -1; \\; \n \\varepsilon_{AB}\\ =\\ \n \\left( \\begin{array}{c c} 0\\ &\\ 1 \\\\ -1 \\ & \\ 0 \\end{array} \\right); \\\\\n && \\varepsilon^{AB}=\\varepsilon_{AB}, \\; \\varepsilon_{BA}=-\\varepsilon_{AB}, \n \\;   \\varepsilon_{AB} \\varepsilon^{AB}\\ =\\ 2; \\;  \n\\varepsilon_{AB} \\varepsilon^{BC}\\ =\\ - \\delta_A^C; \\\\\n&& \\varepsilon_{AB} \\varepsilon^{CD}\\ =\\ \n\\delta_A^C \\delta_B^D \\ -\\ \\delta_A^D \\delta_B^C; \\\\ \n &&\\varepsilon_{AB} \\varepsilon_{CD} + \\varepsilon_{AC} \\varepsilon_{DB} +\n\\varepsilon_{AD} \\varepsilon_{BC} =\\ 0.\n\\end{eqnarray*}\n $\\varepsilon^{AB}$--symbol is used for rising and lowering of the spinor \nindexes (see Subsection~\\ref{sigpaul}). \n\n\\noindent $\\bullet$ {\\it $\\varepsilon$-symbol in three dimensions}: \n$\\varepsilon^{ijk} \\quad (i,j,k = 1,2,3)$: \n\\begin{displaymath}\n \\varepsilon^{123}=\\varepsilon_{123}=1, \\quad \n \\varepsilon^{ijk}=\\varepsilon_{ijk}, \\quad \n\\varepsilon_{ijk}\\varepsilon^{lmn}= \n\\left| \\begin{array}{ccc}\n\\delta_i^l & \\delta_j^l & \\delta_k^l \\\\\n\\delta_i^m & \\delta_j^m & \\delta_k^m \\\\\n\\delta_i^n & \\delta_j^n & \\delta_k^n \n\\end{array} \\right|\n\\end{displaymath}\n\n\\[\\varepsilon_{ijk}\\varepsilon^{lmk}= \n\\delta_i^l\\delta_j^m-\\delta_i^m\\delta_j^l, \\ \\ \\ \n\\varepsilon_{ijk}\\varepsilon^{ljk}=2\\>\\delta^{l}_{i},\\ \\ \\\n\\varepsilon_{ijk}\\varepsilon^{ijk}=6. \\]\n{\\it Schouten identity}. For any 3--vector $p^i$ one has:\n\\[ p_{i_1} \\varepsilon_{i_2 i_3 i_4}\n- p_{i_2} \\varepsilon_{i_1 i_3 i_4}\n+ p_{i_3} \\varepsilon_{i_1 i_2 i_4}\n- p_{i_4} \\varepsilon_{i_1 i_2 i_3}\\ =\\ 0. \\]\n\n\\noindent $\\bullet$ {\\it $\\varepsilon$-symbol in four--dimensional Minkowski\n space}: \n$\\varepsilon^{\\alpha \\beta \\mu \\nu}$ $(\\alpha, \\ldots \\nu = 0,1,2,3)$:\n\\[\\varepsilon^{0123}\\ =\\ -\\ \\varepsilon_{0123}\\ =\\ 1. \\]\n\\begin{displaymath}\n\\varepsilon_{\\mu\\nu\\alpha\\beta}\\varepsilon^{\\lambda\\rho\\sigma\\tau} =\n-\\left| \\begin{array}{cccc}\n\\delta_\\mu^\\lambda & \\delta_\\nu^\\lambda & \\delta_\\alpha^\\lambda &\n\\delta_\\beta^\\lambda \\\\\n\\delta_\\mu^\\rho & \\delta_\\nu^\\rho & \\delta_\\alpha^\\rho & \\delta_\\beta^\\rho \\\\\n\\delta_\\mu^\\sigma & \\delta_\\nu^\\sigma & \\delta_\\alpha^\\sigma &\n\\delta_\\beta^\\sigma \\\\\n\\delta_\\mu^\\tau & \\delta_\\nu^\\tau & \\delta_\\alpha^\\tau & \\delta_\\beta^\\tau\n\\end{array} \\right|, \\; \\; \n\\varepsilon_{\\mu\\nu\\alpha\\beta}\\varepsilon^{\\lambda\\rho\\sigma\\beta} = -\n\\left| \\begin{array}{ccc}\n\\delta_\\mu^\\lambda & \\delta_\\nu^\\lambda & \\delta_\\alpha^\\lambda \\\\\n\\delta_\\mu^\\rho & \\delta_\\nu^\\rho & \\delta_\\alpha^\\rho \\\\\n\\delta_\\mu^\\sigma & \\delta_\\nu^\\sigma & \\delta_\\alpha^\\sigma \n\\end{array} \\right|,\n\\end{displaymath}\n\\[\\varepsilon_{\\mu\\nu\\alpha\\beta}\\varepsilon^{\\lambda\\rho\\alpha\\beta}\n=\\ -2 (\\delta^\\lambda_\\mu \\delta^\\rho_\\nu -\n\\delta^\\rho_\\mu \\delta^\\lambda_\\nu),\\ \\ \\ \\\n\\varepsilon_{\\mu\\nu\\alpha\\beta}\\varepsilon^{\\lambda\\nu\\alpha\\beta}\n =\\ -6 \\delta_\\mu^\\lambda, \\ \\ \\ \\ \n\\varepsilon_{\\mu\\nu\\alpha\\beta}\\varepsilon^{\\mu\\nu\\alpha\\beta} =\\ -24. \\]\n\\noindent {\\it Schouten identity}. For any 4--vector $p_{\\mu}$ one has:\n\\[p_{\\mu_1} \\varepsilon_{\\mu_2 \\mu_3 \\mu_4 \\mu_5}\n+ p_{\\mu_2} \\varepsilon_{\\mu_3 \\mu_4 \\mu_5 \\mu_1}\n+ p_{\\mu_3} \\varepsilon_{\\mu_4 \\mu_5 \\mu_1 \\mu_2}\n+ p_{\\mu_4} \\varepsilon_{\\mu_5 \\mu_1 \\mu_2 \\mu_3} \n+ p_{\\mu_5} \\varepsilon_{\\mu_1 \\mu_2 \\mu_3 \\mu_4} = 0. \\]\n\n\\noindent $\\bullet$ {\\it Generalized Kronecker deltas} \\\\\nSometimes one can make no difference between a vector and\nindex. For example, one can write:\n\\[ \n\\varepsilon^{p_1 p_2 p_3 p_4} \\; {\\rm or} \\; \n\\varepsilon (p_1, p_2, p_3, p_4) \\quad {\\rm instead} \\; {\\rm of} \\; \n\\varepsilon_{\\mu\\nu\\rho\\sigma} p_1^\\mu p_2^\\nu p_3^\\rho p_4^\\sigma.\n\\] \nThese notation can be used in operations with generalized Kronecker deltas:\n\\begin{displaymath}\n\\delta_{i_1 ... i_n}^{j_1 ... j_n}\\equiv\n\\left| \\begin{array}{ccc}\n\\delta_{i_1}^{j_1} & ... & \\delta_{i_n}^{j_1} \\\\\n... & ... & ... \\\\\n\\delta_{i_n}^{j_1} & ... & \\delta_{i_n}^{j_n} \n\\end{array} \\right| , \\ \\mbox{or} \\ \n\\delta_{p_1 ... p_n}^{q_1 ... q_n}\\equiv\n\\left| \\begin{array}{ccc}\np_1\\cdot q_1 & ... & p_n\\cdot q_1 \\\\\n... & ... & ... \\\\\np_1\\cdot q_n & ... & p_n\\cdot q_n \n\\end{array} \\right|.\n\\end{displaymath}\nIn $n$-dimensional Euclidean space one has:\n\\[\n\\delta_{p_1 ... p_m}^{q_1 ... q_m}=\\frac{1}{(n-m)!}\n\\varepsilon^{q_1 ... q_m \\alpha_{m+1} ... \\alpha_n}\n\\varepsilon_{p_1 ... p_m \\alpha_{m+1} ... \\alpha_n}.\n\\]\nIn Minkowski space the minus sign appears:\n\\[ \\delta_{p_1 p_2 p_3}^{q_1 q_2 q_3}=\\ -\n\\varepsilon_{p_1 p_2 p_3 \\mu} \\varepsilon^{q_1 q_2 q_3 \\mu}, \\ \\ \n\\delta_{p_1 p_2}^{q_1 q_2} =\\ -\\frac{1}{2}\n\\varepsilon_{p_1 p_2 \\mu \\nu}\\varepsilon^{q_1 q_2 \\mu \\nu}. \\]\n\n\\noindent $\\bullet$ {\\it Matrices}\n\n\\noindent For any matrix $A = (a_{ik}) \\quad (i,k = 1, \\ldots n)$ we use the \nfollowing notation: \\\\\n\\noindent $I$ is the {\\it unit} matrix, i.e. $I = \\delta_{ik}$ (sometimes, the\nunit matrix will be denote just $1$); \\\\\n\\noindent $A^{-1}$ is the {\\it inverse} matrix, i.e. \n$A^{-1} A = A A^{-1} = I$;  \\\\\n\\noindent $A^{\\top}$ is the {\\it transposed} matrix, i.e. \n$a^{\\top}_{ik} = a_{ki}$; \\\\\n\\noindent $A^{\\ast}$ is the {\\it complex conjugated} matrix, i.e. \n$(a^{\\ast})_{ik} = (a_{ik})^{\\ast}$; \\\\\n\\noindent $A^{\\dagger}$ is the {\\it Hermitian conjugated} matrix, i.e. \n$a^{\\dagger}_{ik} = a^{\\ast}_{ki}$; \\\\\n$H$ -- {\\it Hermitian} and  $U$ -- {\\it unitary} matrices\nshould satisfy the following conditions:\n\\begin{eqnarray*}\n &&  H^{\\dagger} = H, \\\\\n &&U = (U^{\\dagger})^{-1}, \\quad {\\rm hence} \\quad \n U^{\\dagger} = U^{-1}, \\quad  U U^{\\dagger} = U^{\\dagger} U = I.\n\\end{eqnarray*}\n\\noindent $\\det A $ is the {\\it determinant} of matrix $A$\n\\begin{eqnarray*}\n\\det A &=& \\varepsilon^{i_1 i_2 \\ldots i_n} a_{i_1 1} a_{i_2 2} \\cdots \na_{i_n n} \\\\\n &=& \\frac{1}{n !} \\varepsilon^{i_1 i_2 \\ldots i_n} \n                   \\varepsilon^{k_1 k_2 \\ldots k_n} \n  a_{i_1 k_1} a_{i_2 k_2} \\cdots a_{i_n k_n}.\n\\end{eqnarray*}\n\n\n\\noindent ${\\rm Tr} A $ is the {\\it trace} of matrix $A$ : \n ${\\rm Tr} A = a_{ii} \\, (= \\sum_{i=1}^n a_{ii})$. The chief properties of the \ntrace are  as follows (below $\\lambda$ and $\\mu$ are parameters):\n\\begin{eqnarray*}\n&& {\\rm Tr} (\\lambda A + \\mu B) = \\lambda {\\rm Tr} A + \\mu {\\rm Tr} B, \\\\\n&& {\\rm Tr}A^{\\top} = {\\rm Tr}A, \\quad {\\rm Tr} A^{\\ast} = \n {\\rm Tr} A^{\\dagger} = ({\\rm Tr}A)^{\\ast},\n\\quad {\\rm Tr} I = n, \\\\\n&&{\\rm Tr} (AB) = {\\rm Tr} (BA), \\qquad \\det(e^A) = e^{{\\rm Tr}A}. \n\\end{eqnarray*}\n\n\\noindent For any two matrices $A$ and $B$ the {\\it commutator} \n$[A,B]$ and {\\it anticommutator}  $\\lbrace A,B \\rbrace$ are denoted as usual:\n\\[\n [A,B] \\equiv AB - BA, \\quad \\lbrace A,B \\rbrace \\equiv AB + BA.\n\\]\n\n\n\n\n\\section{\\bf PAULI MATRICES}\\label{pauli} \n\n\\subsection{\\it Main Properties}\nThe Pauli matrices $\\sigma_i$ $(i=1,2,3)$ are generators of the group\n$SU(2)$. The $\\sigma_i$ are equal \\cite{bogol,land,itzu,okun}:\n\\begin{displaymath}\n \\sigma_1 =\\sigma^1 = \\left( \\begin{array}{ccc} 0 & 1 \\\\ 1 & 0 \\end{array} \\right),\n\\quad\n \\sigma_2 = \\sigma^2 = \\left( \\begin{array}{ccc} 0 & -i \\\\ i & 0 \\end{array} \\right),\n\\quad  \n \\sigma_3 =  \\sigma^3 =  \\left( \\begin{array}{ccc} 1 & 0 \\\\ 0 & -1 \\end{array} \\right). \n \\end{displaymath}\nThe main properties of $\\sigma_i$ are as follows:\n\\begin{eqnarray}\n  \\sigma^{\\dagger}_i \\, = \\, \\sigma_i, \\quad {\\rm Tr}\\sigma_i \\,=\\,0,\\quad \n \\det \\sigma_i \\, = \\, -1, \\quad \n\\sigma_i \\sigma_k \\, = \\, i\\varepsilon_{ikj} \\sigma_j \\, + \\, \\delta_{ik}. \n\\label{p1} \n\\end{eqnarray}\nUsing relation (\\ref{p1}), one gets:\n\\begin{eqnarray*}\n&& \\sigma^{2}_i \\, = \\, I, \\quad [\\sigma_i, \\sigma_k] \\, = \\, \n 2 i\\varepsilon_{ikj} \\sigma_j, \\quad \n \\lbrace \\sigma_i, \\sigma_k \\rbrace  \\, = \\, 2\\delta_{ik}, \\\\\n&& \\sigma_i \\sigma_k \\sigma_l \\, = \\, i\\varepsilon_{ikl}I + \\delta_{ik} \n\\sigma_l  - \\delta_{il} \\sigma_k + \\delta_{kl} \\sigma_i, \\\\\n&& {\\rm Tr} (\\sigma_i \\sigma_k) \\, = \\, 2\\delta_{ik}, \\quad \n {\\rm Tr} (\\sigma_i \\sigma_k \\sigma_l) \\, = \\, 2 i\\varepsilon_{ikl}, \\\\\n&&{\\rm Tr}(\\sigma_i \\sigma_k \\sigma_l \\sigma_m) \\, \n = 2(\\delta_{ik} \\delta_{lm} \\, \n+  \\, \\delta_{im} \\delta_{kl} \\, - \\, \\delta_{il} \\delta_{km}).\n\\end{eqnarray*}\n\n\\subsection{\\it Fiertz Identities}\nThe Fiertz identities for the Pauli matrices have the form: \n\\begin{eqnarray} \n \\sigma^i_{AB} \\sigma^i_{CD} \\, &=& \\, 2 \\delta_{AD}\\delta_{CB} \n - \\delta_{AB} \\delta_{CD}, \\label{p2} \\\\\n \\sigma^i_{AB} \\sigma^i_{CD} \\, &=& \\, \\frac{3}{2} \\delta_{AD}\\delta_{CB} \\,\n - \\, \\frac{1}{2} \\sigma^i_{AD} \\sigma^i_{CB}. \\label{p3}\n\\end{eqnarray}\n\nUsing (\\ref{p2}), one can obtain the following relations:\n\\begin{eqnarray*} \n \\delta_{AB} \\sigma^i_{CD} \\, &=& \\, \\frac{1}{2}[\\delta_{AD}\\sigma^i_{CB} +\n\\sigma^i_{AD} \\delta_{CB} + i\\varepsilon^{ikl} \\sigma^k_{AD} \\sigma^l_{CB}], \\\\\n \\sigma^i_{AB} \\delta_{CD} \\, &=& \\, \\frac{1}{2}[\\delta_{AD}\\sigma^i_{CB} +\n\\sigma^i_{AD} \\delta_{CB} - i\\varepsilon^{ikl} \\sigma^k_{AD} \\sigma^l_{CB}], \\\\\n \\delta_{AB} \\sigma^i_{CD} &+& \\sigma^i_{AB} \\delta_{CD} \\, = \\, \n \\sigma^i_{AD} \\delta_{CB} + \\delta_{AD} \\sigma^i_{CB}, \\\\\n  \\sigma^i_{AB} \\sigma^k_{CD} \\, &=& \\, \\frac{1}{2} \n [\\sigma^i_{AD} \\sigma^k_{CB} + \\delta^{ik} \\delta_{AD} \\delta_{CB}\n - \\delta^{ik} \\sigma^l_{AD} \\sigma^l_{CB} + \\\\\n &&  + i\\varepsilon^{ikl} \\sigma^l_{AD} \\delta_{CB}\n  - i\\varepsilon^{ikl} \\delta_{AD} \\sigma^l_{CB}].\n\\end{eqnarray*}\n\n\\subsection{$\\sigma_+$ {\\it and} $\\sigma_-$ {\\it Matrices}}\nThe $\\sigma_+$ and $\\sigma_-$ matrices are defined as follows:\n\\begin{displaymath}\n \\sigma_+ \\equiv \\frac{1}{2}(\\sigma_1 + i \\sigma_2)  \n = \\left( \\begin{array}{ccc} 0 & 1 \\\\ 0 & 0 \\end{array} \\right), \n\\quad\n \\sigma_- \\equiv  \\frac{1}{2}(\\sigma_1 - i \\sigma_2)  \n  = \\left( \\begin{array}{ccc} 0 & 0 \\\\ 1 & 0 \\end{array} \\right).\n \\end{displaymath}\nThe relations for these matrices are given by \n\\begin{eqnarray*}\n && (\\sigma_{\\pm})^{\\dagger} = \\sigma_{\\mp}, \\quad {\\rm Tr}\\sigma_{\\pm}=0,\\quad\n \\det \\sigma_{\\pm} = 0, \\\\\n && [\\sigma_{\\pm}, \\sigma_1] = \\pm \\sigma_3, \\quad\n [\\sigma_{\\pm}, \\sigma_2] = i \\sigma_3, \\quad\n [\\sigma_{\\pm}, \\sigma_3] = \\mp 2 \\sigma_{\\pm}, \\quad\n  [\\sigma_+, \\sigma_-] = \\sigma_3, \\\\\n && \\lbrace \\sigma_{\\pm}, \\sigma_1 \\rbrace  = I, \\quad\n \\lbrace \\sigma_{\\pm}, \\sigma_2 \\rbrace = \\pm iI, \\quad\n \\lbrace \\sigma_{\\pm}, \\sigma_3 \\rbrace  = 0, \\quad\n \\lbrace \\sigma_+, \\sigma_- \\rbrace = I, \\\\\n &&  \\sigma_+^2 = \\sigma_-^2 = 0, \\quad \n \\sigma_+ \\sigma_3 = - \\sigma_+, \\quad\n \\sigma_3 \\sigma_+ =  \\sigma_+, \\\\ \n && \\sigma_+ \\sigma_- = \\frac{1}{2} ( I + \\sigma_3), \\quad\n  \\sigma_- \\sigma_+ = \\frac{1}{2} ( I - \\sigma_3), \\\\\n && (\\sigma_+ \\sigma_-)^n = \\sigma_+ \\sigma_-, \\quad \n  (\\sigma_- \\sigma_+)^n = \\sigma_- \\sigma_+.\n\\end{eqnarray*}\nFor any parameter $\\xi$ one gets:\n\\[\n exp(\\xi \\frac{\\sigma_3}{2}) \\sigma_{\\pm} exp(-\\xi \\frac{\\sigma_3}{2}) =\n  \\sigma_{\\pm} exp(\\pm \\xi).\n\\] \nIf $f(\\sigma_+ \\sigma_-)$ (or $f(\\sigma_- \\sigma_+))$ is an arbitrary function\nof $\\sigma_+ \\sigma_-$ (or of $\\sigma_- \\sigma_+)$, and this function can \nbe expanded into power series with respect to $\\sigma_+ \\sigma_-$ (or with\nrespect to $\\sigma_- \\sigma_+)$,\nthen\n\\begin{eqnarray*}\n f(\\sigma_+ \\sigma_-) = f(0) + [f(1) - f(0)] \\sigma_+ \\sigma_-, \\\\\n f(\\sigma_- \\sigma_+) = f(0) + [f(1) - f(0)] \\sigma_- \\sigma_+.\n\\end{eqnarray*}\n\n\\subsection{\\it Various Relations} \nAny $2 \\times 2$ matrix $A$ can be expanded over the set $\\{ I, \\sigma_i \\}$:\n\\[\n A = a_0 I + a_i \\sigma_i,\n\\]\nwhere $a_0 = \\frac{1}{2}{\\rm Tr} A$, and $a_i = \\frac{1}{2}\n {\\rm Tr}(\\sigma_i A)$.\n\n\\noindent Let $\\alpha_i$ be the 3--vector. Then\n\\begin{eqnarray}\ne^{\\alpha_i \\sigma_i} = \\cosh \\sqrt{\\vec \\alpha^2} + \n \\frac{\\sinh \\sqrt{\\vec \\alpha^2}}{\\sqrt{\\vec \\alpha^2}} (\\alpha_i\\sigma_i)\n = p_0 + p_i \\sigma_i.  \\label{p4}\n\\end{eqnarray}\nThe components of the 4--vector $p^{\\mu}$ equal:\n\\begin{eqnarray}\n p_0 = \\cosh \\sqrt{\\vec \\alpha^2}, \\quad \n p_i = \\frac{\\sinh \\sqrt{\\vec \\alpha^2}}{\\sqrt{\\vec \\alpha^2}} \\alpha_i, \\quad\n p_0^2 - \\vec p^{\\,2} = p^2 = 1, \\label{p5}\n\\end{eqnarray}\nand we have \n\\begin{eqnarray}\n \\alpha_i = \\frac{p_i}{\\sqrt{\\vec p^{\\, 2}}} \\ln (p_0 + \\sqrt{\\vec p^{\\, 2}}). \n\\label{p6}\n\\end{eqnarray}\nLet $p$ and $q$ be two 4--vectors, and $p^2 = q^2 = 1$, then \n\\begin{eqnarray}\n (p_0 + p_i \\sigma_i) (q_0 + q_k \\sigma_k) = a_0 + a_l \\sigma_l = \n e^{\\beta_i \\sigma_i}, \\label{p7}\n\\end{eqnarray}\nwhere $a_0 = p_0q_0 + (\\vec p \\vec q), \\quad a_j = p_0 q_j + p_j q_0 +\ni\\varepsilon^{jkl} p_k q_l$, and the 3--vector $\\beta_i$ in the relation \n(\\ref{p7}) is expressed through $a_0$ and $\\vec a$ as in (\\ref{p6}). \n\n\\subsection{\\it 4--dimensional $\\sigma^{\\mu}$ Matrices} \\label{sigpaul}\nHere we present the various properties of $2 \\times 2$ matrices $\\sigma^{\\mu}$ \nand  $\\bar \\sigma^{\\mu}$ ($\\mu \\: = \\: 0,1,2,3$): \n\\begin{eqnarray}\n \\sigma^{\\mu}_{A \\dot B} \\equiv (I, -\\sigma_i); \\quad\n \\bar \\sigma^{\\mu \\dot A B} \\equiv (I, \\sigma_i), \\quad \n\\mu = 0, \\, 1, \\, 2, \\, 3, \\label{p10}\n\\end{eqnarray}\nwhere $\\sigma_i$ are Pauli matrices. \\\\\nWith the help of $\\sigma^{\\mu}$--matrices any tensor in Minkowski space can be\nunambiguously rewritten in spinorial form. In order to deal only with \nLorentz--covariant expressions one should clearly distinguish between dot and\nundot, lower and upper Weyl indices. The $\\varepsilon$--symbol (see\nSection~\\ref{nota}) used here for rising and lowering indices. \n\n\\noindent The main properties of the $\\sigma^{\\mu}$ matrices are as follows: \n\\begin{eqnarray*}\n &&  \\bar \\sigma^{\\mu \\dot A A} = \\varepsilon^{\\dot A \\dot B} \\varepsilon^{AB} \n \\sigma^{\\mu}_{B \\dot B}, \\quad\n \\sigma^{\\mu}_{\\dot A A} = \\varepsilon_{AB} \\varepsilon_{\\dot A \\dot B} \n \\bar \\sigma^{\\mu \\dot B B}, \\\\ \n &&  (\\sigma^{\\mu})^{\\dagger} = \\sigma^{\\mu}, \\quad \n  (\\bar \\sigma^{\\mu})^{\\dagger} = \\bar \\sigma^{\\mu}, \\quad \n \\det \\sigma^{\\mu} = \\det \\bar \\sigma^{\\mu} = 1(-1), \\: {\\rm for}\n\\: \\mu = 0(1,2,3).\n\\end{eqnarray*}\nFor any 4--vector $p^{\\mu}$ one has: \n\\[ \n\\det p_{\\mu} \\sigma^{\\mu} = \\det p_{\\mu} \\bar \\sigma^{\\mu} = p^2.\n\\]\n\n\n\\noindent Various products of $\\sigma^{\\mu}$ matrices have the form: \n\\begin{eqnarray*}\n && \\sigma^{\\mu}_{A \\dot C} \\bar \\sigma^{\\nu \\dot C B}\n+ \\sigma^{\\nu}_{A \\dot C} \\bar \\sigma^{\\mu \\dot C B} = \n 2 g^{\\mu \\nu} \\delta_A{}^B, \\quad \n \\bar \\sigma^{\\mu \\dot A C} \\sigma^{\\nu}_{C \\dot B}\n + \\bar \\sigma^{\\nu \\dot A C} \\sigma^{\\mu}_{C \\dot B} =\n 2 g^{\\mu \\nu} \\delta^{\\dot A}{}_{\\dot B}, \\\\\n && \\sigma^{\\mu}_{A \\dot C} \\bar \\sigma_{\\mu}^{\\dot C B}\n =  4 \\delta_A{}^B, \\quad \\bar \\sigma^{\\mu \\dot A C} \\sigma_{\\mu C \\dot B}\n = 4 \\delta^{\\dot A}{}_{\\dot B}, \\\\\n && \\sigma^{\\mu}_{A \\dot A} \\sigma_{\\mu B \\dot B} \\varepsilon^{\\dot A \\dot B} \n= 4 \\varepsilon_{AB},  \\quad \n \\sigma^{\\mu}_{A \\dot A} \\sigma_{\\mu B \\dot B}  \\varepsilon^{A B} \n= 4 \\varepsilon_{\\dot A \\dot B}, \\\\\n && \\bar \\sigma^{\\mu \\dot A A} \\bar \\sigma_{\\mu}^{\\dot B B}\n  \\varepsilon_{\\dot A \\dot B} = 4 \\varepsilon^{AB},  \\quad \n \\bar \\sigma^{\\mu \\dot A A} \\bar \\sigma_{\\mu}^{\\dot B B}  \\varepsilon_{A B} \n= 4 \\varepsilon^{\\dot A \\dot B}, \\\\\n&& \\sigma^{\\mu}_{A \\dot A} \\sigma^{\\nu}_{B \\dot B} \\varepsilon^{A B} \n \\varepsilon^{\\dot A \\dot B} \\, = \\, \n \\bar \\sigma^{\\mu \\dot A A} \\bar \\sigma^{\\nu \\dot B B}  \\varepsilon_{A B} \n \\varepsilon_{\\dot A \\dot B} = \\, 2 g^{\\mu \\nu}, \\\\\n&&\\sigma^{\\mu} \\bar \\sigma^{\\lambda} \\sigma^{\\nu} = g^{\\mu \\lambda}\\sigma^{\\nu}\n + g^{\\nu \\lambda}\\sigma^{\\mu} - g^{\\mu \\nu}\\sigma^{\\lambda}\n   -i\\varepsilon^{\\mu \\lambda \\nu \\rho}\\sigma^{\\rho}, \\\\\n&& \\bar \\sigma^{\\mu} \\sigma^{\\lambda} \\bar \\sigma^{\\nu} = \ng^{\\mu \\lambda}\\sigma^{\\nu}+g^{\\nu\\lambda}\\sigma^{\\mu}\n -g^{\\mu \\nu}\\sigma^{\\lambda}\n   +i\\varepsilon^{\\mu \\lambda \\nu \\rho}\\sigma^{\\rho}, \\\\\n&& \\varepsilon^{\\mu\\nu\\rho\\lambda}=i\\sigma^{\\mu\\dot A A}\n\\sigma^{\\nu\\dot B B}\\sigma^{\\rho\\dot C C}\\sigma^{\\lambda \\dot D D} \n(\\varepsilon_{AC}\\varepsilon_{BD}\\varepsilon_{\\dot A\\dot D}\n \\varepsilon_{\\dot B\\dot C} -\n\\varepsilon_{AD}\\varepsilon_{BC}\\varepsilon_{\\dot A\\dot C}\n\\varepsilon_{\\dot B\\dot D}). \n\\end{eqnarray*}\nThe commutators of $\\sigma^{\\mu}$ and $\\bar \\sigma^{\\mu}$ matrices have the\nspecial notation:\n\\[ \\sigma^{\\mu\\nu B}_A \\equiv \\frac{1}{4}\n  (\\sigma^{\\mu}_{A \\dot C} \\bar \\sigma^{\\nu \\dot C B} -\n  \\sigma^{\\nu}_{A \\dot C} \\bar \\sigma^{\\mu \\dot C B}), \\; \\;  \n \\bar \\sigma^{\\mu\\nu \\dot A}{}_{\\dot B} \\equiv \\frac{1}{4}\n  (\\bar \\sigma^{\\mu \\dot A C} \\sigma^{\\nu}_{C \\dot B} -\n  \\bar \\sigma^{\\nu \\dot A C} \\sigma^{\\mu}_{C \\dot B}).\n\\]\nThe main properties of $\\sigma^{\\mu \\nu}$ are as follows:\n\\begin{eqnarray*}\n &&  \\sigma^{0i}=\\frac{1}{2}\\sigma^i, \\, \\, \n \\sigma^{ik}=-\\frac{i}{2} \\varepsilon^{ikl} \\sigma^l, \\quad\n \\bar \\sigma^{0i}=-\\frac{1}{2}\\sigma^i, \\, \\, \n \\bar \\sigma^{ik} = \\sigma^{ik}, \\\\\n&& \\sigma^{\\mu \\nu} = - \\sigma^{\\nu \\mu}, \\quad\n \\bar \\sigma^{\\mu \\nu} = - \\bar \\sigma^{\\nu \\mu}, \\\\ \n&& (\\sigma^{\\mu} \\bar \\sigma^{\\nu})_A{}^B =\n  g^{\\mu \\nu} \\delta_A{}^B + 2 \\sigma^{\\mu\\nu B}_A, \\quad\n (\\bar \\sigma^{\\mu} \\sigma^{\\nu})^{\\dot A}{}_{\\dot B} = \ng^{\\mu\\nu} \\delta^{\\dot A}{}_{\\dot B}+2 \\bar \\sigma^{\\mu\\nu \\dot A}{}_{\\dot B}, \n\\\\\n&& \\sigma^{\\mu\\nu K}_A \\varepsilon_{KB} = \n   \\sigma^{\\mu\\nu K}_B \\varepsilon_{KA}, \\quad\n \\bar \\sigma^{\\mu\\nu \\dot A}{}_{\\dot K} \\varepsilon^{\\dot K \\dot B} = \n  \\bar \\sigma^{\\mu\\nu \\dot B}{}_{\\dot K} \\varepsilon^{\\dot K \\dot A}, \\\\\n&& \\varepsilon^{\\mu \\nu \\lambda \\rho} \\sigma_{\\lambda \\rho} = \n     -2i \\sigma^{\\mu \\nu}, \\quad \n \\varepsilon^{\\mu \\nu \\lambda \\rho} \\bar \\sigma_{\\lambda \\rho} =\n     2i \\bar \\sigma^{\\mu \\nu}. \n\\end{eqnarray*}\n\n\\subsection{\\it Traces of $\\sigma^{\\mu}$ Matrices} \n\\begin{eqnarray*}\n&&{\\rm Tr}\\sigma^{\\mu}={\\rm Tr}\\bar \\sigma^{\\mu}\\,=\\,2(0)\\quad {\\rm for} \\quad \n \\mu = 0 \\, (1,2,3), \\\\\n&&{\\rm Tr}\\sigma^{\\mu \\nu}={\\rm Tr}\\bar \\sigma^{\\mu \\nu} = 0, \\quad\n{\\rm Tr}(\\sigma^{\\mu} \\bar \\sigma^{\\nu})\n  = {\\rm Tr}(\\bar \\sigma^{\\mu} \\sigma^{\\nu}) = 2 g^{\\mu \\nu}, \\\\ \n&&{\\rm Tr}(\\sigma^{\\mu} \\bar \\sigma^{\\nu} \\sigma^{\\lambda} \\bar \\sigma^{\\rho})\n =  2(g^{\\mu \\nu}g^{\\lambda \\rho} + g^{\\mu \\rho}g^{\\nu \\lambda} \n - g^{\\mu \\lambda}g^{\\nu \\rho} - i \\varepsilon^{\\mu \\nu \\lambda \\rho}), \\\\\n&&{\\rm Tr}(\\sigma^{\\mu \\nu} \\sigma^{\\lambda \\rho}) =\n {\\rm Tr}(\\bar \\sigma^{\\mu \\nu}\\bar \\sigma^{\\lambda \\rho}) = \n \\frac{1}{2} (g^{\\mu \\rho}g^{\\nu \\lambda} - g^{\\mu \\lambda}g^{\\nu \\rho}\n   - i \\varepsilon^{\\mu \\nu \\lambda \\rho}).\n\\end{eqnarray*}\n\n\\subsection{\\it Fiertz Identities for $\\sigma^{\\mu}$ Matrices} \n\nThe Fiertz identities for $\\sigma^{\\mu}$ equal:\n\\begin{eqnarray}\n\\sigma^{\\mu}_{A \\dot A} \\bar \\sigma_{\\mu}^{\\dot B B} = \n 2 \\delta_A{}^B \\delta_{\\dot A}{}^{\\dot B}, \\; \\;\n\\sigma^{\\mu}_{A \\dot A} \\sigma_{\\mu B \\dot B} = \n 2 \\varepsilon_{AB} \\varepsilon_{\\dot A \\dot B}, \\; \\;\n \\bar \\sigma^{\\mu \\dot A A} \\bar \\sigma_{\\mu}^{\\dot B B} = \n 2 \\varepsilon^{\\dot A \\dot B} \\varepsilon_{AB}.\n\\label{p11} \n\\end{eqnarray}\nFrom the relations (\\ref{p11}) one gets:\n\\begin{eqnarray*}\n\\sigma^{\\mu}_{A \\dot A} \\bar \\sigma^{\\nu \\dot B B} &=& \n \\frac{1}{2} g^{\\mu \\nu} \\delta_A{}^B \\delta_{\\dot A}{}^{\\dot B} \n - \\delta_A{}^B \\bar \\sigma^{\\mu \\nu \\dot B}{}_{\\dot A}  \n + \\sigma^{\\mu \\nu B}_A \\delta_{\\dot A}{}^{\\dot B} \n + 2 \\sigma^{\\nu \\lambda B}_A \\bar \\sigma^{\\mu \\lambda \\dot B}{}_{\\dot A}, \\\\\n \\sigma^{\\mu}_{A \\dot A} \\sigma^{\\nu}_{B \\dot B} &=& \n \\frac{1}{2} g^{\\mu \\nu} \\varepsilon_{AB} \\varepsilon_{\\dot A \\dot B} \n + \\varepsilon_{AB} (\\varepsilon_{\\dot A \\dot C} \n     \\bar \\sigma^{\\mu \\nu \\dot C}{}_{\\dot B}) + \n  (\\sigma^{\\mu \\nu C}_A \\varepsilon_{CB}) \\varepsilon_{\\dot A \\dot B} \\\\\n && - 2 (\\sigma^{\\mu \\lambda C}_A \\varepsilon_{CB}) \n  (\\varepsilon_{\\dot A \\dot C} \\bar \\sigma^{\\nu \\lambda \\dot C}{}_{\\dot B}), \\\\\n \\bar \\sigma^{\\mu \\dot A A} \\bar \\sigma^{\\nu \\dot B B} &=& \n \\frac{1}{2} g^{\\mu \\nu} \\varepsilon^{AB} \\varepsilon^{\\dot A \\dot B} \n + (\\varepsilon^{AC} \\sigma^{\\mu \\nu B}_C) \\varepsilon^{\\dot A \\dot B} \n +  \\varepsilon^{AB}\n  (\\bar \\sigma^{\\mu \\nu \\dot A}{}_{\\dot C} \\varepsilon^{\\dot C \\dot B}) \\\\  \n && - 2 (\\varepsilon^{AC} \\sigma^{\\mu \\lambda C}_B) \n  (\\bar \\sigma^{\\nu \\lambda \\dot A}{}_{\\dot C}\\varepsilon^{\\dot C \\dot B}), \n\\end{eqnarray*}\n\\[ \n (\\varepsilon^{AC} \\sigma^{\\mu \\lambda C}_B)  \n  (\\bar \\sigma^{\\nu \\lambda \\dot A}{}_{\\dot C}\\varepsilon^{\\dot C \\dot B}) = \n (\\varepsilon^{AC} \\sigma^{\\nu \\lambda C}_B) \n  (\\bar \\sigma^{\\mu \\lambda \\dot A}{}_{\\dot C}\\varepsilon^{\\dot C \\dot B}).\n\\] \n\n\n\n\\section{\\bf DIRAC MATRICES}\\label{dirac}\n\n\\subsection{\\it Main Properties}\n\n\\noindent The main properties of the Dirac $\\gamma$-matrices are as follows \n\\cite{bogol,land,itzu,okun,velt}:\n\\begin{eqnarray}\n&& \\gamma^{\\mu} \\gamma^{\\nu} + \\gamma^{\\nu} \\gamma^{\\mu} = 2 g^{\\mu \\nu}, \n  \\label{d1} \\\\\n && (\\gamma^0)^2 = I, \\quad (\\gamma^i)^2 = -I, \\quad, (\\gamma^0)^{\\dagger} = \n \\gamma^0, \\quad (\\gamma^i)^{\\dagger} = - \\gamma^i. \\label{d2} \\\\\n &&  \\sigma^{\\mu \\nu} \\; \\equiv \\; \\frac{1}{2} \n(\\gamma^{\\mu} \\gamma^{\\nu} - \\gamma^{\\nu} \\gamma^{\\mu}), \\quad\n\\sigma^{\\mu \\nu} \\; = \\; - \\sigma^{\\nu \\mu} \\label{d13}\n\\end{eqnarray}\nThe definition of the $\\gamma^5$ matrix and its properties are as follows: \n\\begin{eqnarray}\n&& \\gamma^5 \\equiv i\\gamma^0 \\gamma^1 \\gamma^2 \\gamma^3 \\, = \n\\, -\\frac{i}{4!} \\varepsilon_{\\alpha\\beta\\mu\\nu} \\gamma^{\\alpha}\n\\gamma^{\\beta}\\gamma^{\\mu}\\gamma^{\\nu}. \\label{d3} \\\\\n&&  (\\gamma^5)^2 \\, = \\, I, \\quad  (\\gamma^5)^{\\dagger} \\, = \\, \\gamma^5, \\quad\n\\gamma^5 \\gamma^{\\mu} + \\gamma^{\\mu} \\gamma^5 = \\{\\gamma^{\\mu}, \\, \\gamma^5 \\}\n= 0 \\nonumber\n\\end{eqnarray}\nNote, that\n\\begin{eqnarray*}\n  \\gamma^{0} = \\gamma_{0}, \\; \\gamma^{i} = -\\gamma_{i}, \\;\n  (\\gamma^{\\mu})^{\\dagger} =  \\gamma^0 \\gamma^{\\mu} \\gamma^{0} = \\gamma_{\\mu} \n\\end{eqnarray*}\n  \n\\noindent The {\\bf \\it Dirac conjugation} of any $4\\times4$--matrix $A$ is\ndefined as follows: \n\\begin{eqnarray}\n \\bar A \\equiv \\gamma^0 A^{\\dagger} \\gamma^0. \\label{d4}\n\\end{eqnarray}\nFrom (\\ref{d4}) one gets:\n\\begin{eqnarray*} \n && \\overline{\\gamma^{\\mu}} = \\gamma^{\\mu}, \\quad \n\\overline{ \\gamma^5} = -\\gamma^5, \\quad\n \\overline{\\gamma^{\\alpha} \\gamma^{\\beta} \\cdots \\gamma^{\\lambda}} =\n \\gamma^{\\lambda} \\cdots \\gamma^{\\beta} \\gamma^{\\alpha}, \\\\\n && \\overline{\\gamma^{\\alpha} \\gamma^{\\beta} \\cdots \\gamma^{\\sf m } \\gamma^5\n  \\cdots \\gamma^{\\lambda}} =\n \\gamma^{\\lambda}\\cdots (-\\gamma^5) \\gamma^{\\sf m}\\cdots \\gamma^{\\beta}\n \\gamma^{\\alpha} = \\gamma^{\\lambda} \\cdots \\gamma^{\\sf m} \\gamma^5 \\cdots\n \\gamma^{\\beta} \\gamma^{\\alpha}.\n\\end{eqnarray*}\nIn this Section for the string of the $\\gamma$--matrices we shall use the\nspecial notation:\n\\begin{eqnarray}\n S = S^n \\equiv \\gamma^{\\alpha_1} \\gamma^{\\alpha_2} \\cdots \n       \\gamma^{\\alpha_n}, \\quad \n S_R = S^n_R \\equiv \\gamma^{\\alpha_n} \\cdots \\gamma^{\\alpha_2}  \n       \\gamma^{\\alpha_1}.  \\label{dd1}\n\\end{eqnarray}\nOdd-- and even--numbered string of $\\gamma$--matrices will be denoted as \nfollows:\n\\begin{eqnarray}\n S^{odd} \\equiv \\gamma^{\\alpha_1} \\gamma^{\\alpha_2} \\cdots \n       \\gamma^{\\alpha_{2k+1}}, \\quad \n S^{even}  \\equiv \\gamma^{\\alpha_1}  \\gamma^{\\alpha_2}  \\cdots\n       \\gamma^{\\alpha_{2k}}.  \\label{dd2}\n\\end{eqnarray}\n\n\\subsection{\\it Representations of the Dirac Matrices} \\label{ref_gamma}\n\nThe non--singular transformation $\\gamma \\to U \\gamma U^{\\dagger}$ connects the\ndifferent representations of the $\\gamma$--matrices (Pauli lemma). Here we  \npresent three representations of the Dirac matrices.\n\n\\noindent $\\bullet$ {\\bf Dirac (standard)} representation\n\\begin{displaymath}\n \\gamma^0_D = \\left( \\begin{array}{ccc} 1 & 0 \\\\ 0 & -1 \\end{array} \\right) ,\n\\quad\n \\gamma^i_D = \\left( \\begin{array}{ccc} 0 & \\sigma^i \\\\ -\\sigma^i & 0 \n\\end{array} \\right) , \\quad\n \\gamma^5_D = \\left( \\begin{array}{ccc} 0 & 1 \\\\ 1 & 0 \\end{array} \\right).\n\\end{displaymath}\n\n\\noindent $\\bullet$ {\\bf Chiral (spinorial)} representation \n\\begin{eqnarray*}\n  &&  \\gamma^{\\mu}_C \\, = \\,  U_C \\, \\gamma^{\\mu}_D U_C^{\\dagger}, \\qquad \n U_C \\, = \\, \\frac{1}{\\sqrt{2}}(1 - \\gamma^5_D \\gamma^0_D) \\, = \\,\n  \\frac{1}{\\sqrt{2}}\n \\left( \\begin{array}{ccc} 1 & 1 \\\\ -1 & 1 \\end{array} \\right), \\\\\n &&  \\gamma^0 = \\left( \\begin{array}{ccc} 0 & -1 \\\\ -1 & 0 \\end{array} \\right) ,\n\\quad\n \\gamma^i = \\left( \\begin{array}{ccc} 0 & \\sigma^i \\\\ -\\sigma^i & 0 \n\\end{array} \\right) , \\quad\n \\gamma^5 = \\left( \\begin{array}{ccc} 1 & 0 \\\\ 0 & -1 \\end{array} \\right), \\\\\n&& \\gamma_{R}=\\frac{1}{2} \\left(1 \\, + \\, \\gamma^5 \\right)  = \\, \n\\left( \\begin{array}{ccc} 1 & 0 \\\\ 0 & 0 \\end{array} \\right), \\quad \n\\gamma_{L}=\\frac{1}{2} \\left( 1 \\, - \\, \\gamma^5 \\right) \\, = \\, \n\\left( \\begin{array}{ccc} 0 & 0 \\\\ 0 & 1 \\end{array} \\right).\n\\end{eqnarray*}\n\n\n\\noindent $\\bullet$ {\\bf Majorana} representation \n\\begin{eqnarray*}\n&& \\gamma^{\\mu}_M \\, = \\,  U_M \\, \\gamma^{\\mu}_D U_M^{\\dagger}, \\qquad \n U_M \\, = \\frac{1}{\\sqrt{2}}\n\\left( \\begin{array}{ccc} 1 & \\sigma_2 \\\\ \\sigma_2 & -1 \\end{array} \\right), \n \\\\\n&& \\gamma^0 = \\left( \\begin{array}{ccc} 0 & \\sigma_2 \\\\ \\sigma_2 & 0 \n\\end{array} \\right), \\quad\n \\gamma^1 = \\left( \\begin{array}{ccc} i\\sigma_3 & 0 \\\\ 0 & i\\sigma_3 \n\\end{array} \\right), \\quad\n \\gamma^2 = \\left( \\begin{array}{ccc} 0 & -\\sigma_2 \\\\ \\sigma_2 & 0 \n\\end{array} \\right), \\quad \\\\\n&& \\gamma^3 = \\left( \\begin{array}{ccc} -i\\sigma_1 & 0 \\\\ 0 & -i\\sigma_1 \n\\end{array} \\right), \\qquad\n\\gamma^5 = \\left( \\begin{array}{ccc} \\sigma_2 & 0 \\\\ 0 & -\\sigma_2 \n \\end{array} \\right).\n\\end{eqnarray*}\n\n\\subsection{\\it Expansion of $4 \\times 4$ Matrices}\n\nThe following 16 matrices $\\Gamma_A \\quad (A=1, \\ldots , 16)$\n\\begin{eqnarray}\nI, \\quad \\gamma^5, \\quad \\gamma^{\\mu}, \\quad \\gamma^5 \\gamma^{\\mu}, \\quad\n\\sigma^{\\mu \\nu}  \\label{d5} \n\\end{eqnarray}\nare the full set of $4 \\times 4$--matrices. \\\\\nThe main properties of $\\Gamma_A$ are as follows:\n\\begin{eqnarray}\n {\\rm Tr} I = 4, \\quad {\\rm Tr} \\gamma^5 = {\\rm Tr} \\gamma^{\\mu} = \n {\\rm Tr} \\gamma^5 \\gamma^{\\mu} = {\\rm Tr} \\sigma^{\\mu \\nu} = 0 \\label{d7}\n\\end{eqnarray}\nAny $4 \\times4$--matrix $A$ can be expanded over set of the\n$\\Gamma_A$-matrices:\n\\begin{eqnarray}\n A = a_0 I + a_5 \\gamma^5 + v_{\\mu} \\gamma^{\\mu}+ a_{\\mu} \\gamma^5 \\gamma^{\\mu}\n  + T_{\\mu \\nu} \\sigma^{\\mu \\nu}, \\label{d8}\n\\end{eqnarray}\nwhere the coefficients could be found from the following relations:\n\\begin{eqnarray*}\n a_0 = \\frac{1}{4} {\\rm Tr} A, \\quad \n a_5 = \\frac{1}{4} {\\rm Tr} (\\gamma^5 A), \\quad \n v^{\\mu} = \\frac{1}{4} {\\rm Tr}(\\gamma^{\\mu} A), \\\\\n a^{\\mu} = -\\frac{1}{4} {\\rm Tr}( \\gamma^5 \\gamma^{\\mu} A), \\quad\n T^{\\mu \\nu} = -T^{\\nu \\mu} = -\\frac{1}{8} {\\rm Tr}(\\sigma^{\\mu \\nu} A).\n\\end{eqnarray*}\nFor the expansion of a matrix $A$ one can use another set of $\\Gamma'_A$ \n($\\Gamma'_A = X, \\; Y, \\; U^{\\mu}, \\; V^{\\mu}, \\; \\sigma^{\\mu \\nu}$):\n\\begin{eqnarray*}\n && X = I + \\gamma^5, \\quad Y = I - \\gamma^5, \\quad \n U^{\\mu} = (I + \\gamma^5)\\gamma^{\\mu},  \\quad \n V^{\\mu} = (I - \\gamma^5)\\gamma^{\\mu}, \\\\\n &&X^2 = 2X, \\quad Y^2 = 2Y.\n\\end{eqnarray*}\nThese matrices have the following properties: \n\\begin{eqnarray*} \n && U^2 = V^2 = XY = YX = X U^{\\mu} =  Y V^{\\mu} = 0, \\\\\n && {\\rm Tr} X = {\\rm Tr} Y = 4, \\; {\\rm Tr} U^{\\mu} = {\\rm Tr} U^{\\mu} =\n {\\rm Tr} \\sigma^{\\mu \\nu} = 0.\n\\end{eqnarray*}\nThe expansion of any $4 \\times4$--matrix $A$ over set of $\\Gamma'$-matrices has\nthe form:\n\\begin{eqnarray*}\n A = a_x X + a_y Y + b_{\\mu} U^{\\mu} + c_{\\mu} V^{\\mu}\n  + T_{\\mu \\nu} \\sigma^{\\mu \\nu}, \n\\end{eqnarray*}\nwhere \n\\begin{eqnarray*}\n && a_x = \\frac{1}{8} {\\rm Tr} (XA), \\quad \n a_y = \\frac{1}{8} {\\rm Tr} (YA), \\quad \n b^{\\mu} = \\frac{1}{8} {\\rm Tr}(V^{\\mu} A), \\quad \n c^{\\mu} = \\frac{1}{8} {\\rm Tr}(U^{\\mu} A), \\\\\\\n && T^{\\mu \\nu} = -\\frac{1}{8} {\\rm Tr}(\\sigma^{\\mu \\nu} A).\n\\end{eqnarray*}\n\n\\subsection{\\it Products of the Dirac Matrices}\n\\begin{eqnarray*}\n  \\gamma^{\\mu} \\gamma^{\\nu} &=& g^{\\mu \\nu} + \\sigma^{\\mu \\nu}, \\quad\n \\sigma^{\\mu \\nu} = \\gamma^{\\mu} \\gamma^{\\nu} - g^{\\mu \\nu} = \n - \\gamma^{\\nu} \\gamma^{\\mu} + g^{\\mu \\nu}, \\\\\n  \\gamma^5 \\gamma^{\\mu} \\gamma^{\\nu} &=& g^{\\mu \\nu} \\gamma^5 \n + \\frac{i}{2} \\varepsilon^{\\mu \\nu \\alpha \\beta} \\sigma_{\\alpha \\beta}, \\quad \n  \\gamma^5 \\sigma^{\\mu \\nu} = \n  + \\frac{i}{2} \\varepsilon^{\\mu \\nu \\alpha \\beta} \\sigma_{\\alpha \\beta}, \\\\ \n \\gamma^{\\lambda} \\sigma^{\\mu \\nu} &=& (g^{\\mu \\lambda} \\gamma^{\\nu} \n  -g^{\\nu \\lambda} \\gamma^{\\mu}) - i\\varepsilon^{\\lambda \\mu \\nu \\alpha}\n   \\gamma^5 \\gamma_{\\alpha}, \\\\\n \\sigma^{\\mu \\nu} \\gamma^{\\lambda} &=& -(g^{\\mu \\lambda} \\gamma^{\\nu} \n  -g^{\\nu \\lambda} \\gamma^{\\mu}) - i\\varepsilon^{\\lambda \\mu \\nu \\alpha}\n   \\gamma^5 \\gamma_{\\alpha}, \\\\\n \\gamma^5 \\gamma^{\\lambda} \\sigma^{\\mu \\nu} &=& \n  (g^{\\mu \\lambda} \\gamma^5 \\gamma^{\\nu} \n  -g^{\\nu \\lambda} \\gamma^5 \\gamma^{\\mu}) \n   - i\\varepsilon^{\\lambda \\mu \\nu \\alpha} \\gamma_{\\alpha}, \\\\\n  \\sigma^{\\mu \\nu} \\gamma^5 \\gamma^{\\lambda} &=&\n  -(g^{\\mu \\lambda} \\gamma^5 \\gamma^{\\nu} \n  -g^{\\nu \\lambda} \\gamma^5 \\gamma^{\\mu}) \n   - i\\varepsilon^{\\lambda \\mu \\nu \\alpha} \\gamma_{\\alpha}, \\\\\n  \\sigma^{\\alpha \\beta} \\sigma^{\\mu \\nu} &=& \n  g^{\\alpha \\nu} g^{\\beta \\mu} - g^{\\alpha \\mu} g^{\\beta \\nu} \n  - i\\varepsilon^{\\alpha \\beta \\mu \\nu} \\gamma^5 \\\\\n &&+ ( g^{\\alpha \\nu} g^{\\beta \\lambda} g^{\\mu \\sigma} \n   - g^{\\alpha \\mu} g^{\\beta \\lambda} g^{\\nu \\sigma}\n   - g^{\\beta \\nu} g^{\\alpha \\lambda} g^{\\mu \\sigma}\n + g^{\\beta \\mu} g^{\\alpha \\lambda} g^{\\nu \\sigma}) \\sigma_{\\lambda \\sigma}, \\\\\n \\sigma^{\\alpha \\beta} \\sigma^{\\mu \\nu} & + &\\sigma^{\\mu \\nu} \n\\sigma^{\\alpha \\beta}= \n 2(g^{\\alpha \\nu} g^{\\beta \\mu} - g^{\\alpha \\mu} g^{\\beta \\nu} \n  - i\\varepsilon^{\\alpha \\beta \\mu \\nu} \\gamma^5).\n\\end{eqnarray*}\n\nThe totally antisymmetric tensor $\\gamma^{\\mu \\nu \\lambda}$ is defined as \nfollows:\n\\begin{eqnarray*}\n&&  \\gamma^{\\mu \\nu \\lambda} \\equiv \\frac{1}{6} \n (\\gamma^{\\mu} \\gamma^{\\nu} \\gamma^{\\lambda} \n +\\gamma^{\\nu} \\gamma^{\\lambda} \\gamma^{\\mu} \n +\\gamma^{\\lambda} \\gamma^{\\mu} \\gamma^{\\nu} \n -\\gamma^{\\nu} \\gamma^{\\mu} \\gamma^{\\lambda} \n -\\gamma^{\\lambda} \\gamma^{\\nu} \\gamma^{\\mu} \n -\\gamma^{\\mu} \\gamma^{\\lambda} \\gamma^{\\nu}), \\\\\n&& \\gamma^{\\mu} \\gamma^{\\nu} \\gamma^{\\lambda} = \\gamma^{\\mu \\nu \\lambda}\n + g^{\\mu \\nu} \\gamma^{\\lambda} - g^{\\mu \\lambda} \\gamma^{\\nu} \n + g^{\\nu \\lambda} \\gamma^{\\mu}, \\\\\n&& \\gamma^{\\mu \\nu \\lambda} = -i \\varepsilon^{\\mu \\nu \\lambda \\alpha} \n  \\gamma^5 \\gamma_{\\alpha}, \\quad \n \\gamma^5 \\gamma^{\\alpha}=\\frac{i}{6} \\varepsilon^{\\alpha \\mu \\nu \\lambda} \n \\gamma_{\\mu \\nu \\lambda}.\n\\end{eqnarray*} \n\nThe products of the type $\\sum_{A=1}^{16} \\Gamma^A \\Gamma^B \\Gamma^A$ are\npresented in the Table~\\ref{dirac}.1. \n\n\\noindent \n\\underline{ {\\bf Table~\\ref{dirac}.1.}}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\\hline\n $\\Gamma^B$ & $\\gamma^5 \\Gamma^B \\gamma^5$ &\n $\\gamma^{\\nu} \\Gamma^B \\gamma^{\\nu}$ & \n $\\gamma^5 \\gamma^{\\nu} \\Gamma^B \\gamma^5 \\gamma^{\\nu}$ & \n $\\sigma^{\\mu \\nu} \\Gamma^B \\sigma^{\\mu \\nu}$ \\\\ \\hline\n $I$ & $I$ & $4$ & $-4$ & $-1$ \\\\ \\hline\n $\\gamma^5$ & $\\gamma^5$ & $-4\\gamma^5$ & \n $-4\\gamma^5$ & $-12\\gamma^5$ \\\\ \\hline\n $\\gamma^{\\alpha}$ & $-\\gamma^{\\alpha}$ & \n  $-2\\gamma^{\\alpha}$ & $-2\\gamma^{\\alpha}$ & $0$ \\\\ \\hline\n $\\gamma^5 \\gamma^{\\alpha}$ &\n  $-\\gamma^5 \\gamma^{\\alpha}$ & $2\\gamma^5 \\gamma^{\\alpha}$ &\n  $2\\gamma^5 \\gamma^{\\alpha}$ & $0$ \\\\ \\hline\n $\\sigma^{\\alpha \\beta}$ & $\\sigma^{\\alpha \\beta}$ &\n $0$ & $0$& $4\\sigma^{\\alpha \\beta}$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\n\\noindent So--called Chisholm identities are given by \\cite{velt}:\n\\begin{eqnarray}\n &&\\gamma^{\\mu} S^{odd} \\gamma_{\\mu} = -2 S^{odd}_R,  \\label{dd3} \\\\\n &&\\gamma^{\\mu} S^{even} \\gamma_{\\mu} = \n \\gamma^{\\mu} \\gamma^{\\lambda} S'^{odd} \\gamma_{\\mu} = \n 2 \\gamma^{\\lambda} S'^{odd}_R + 2 S'^{odd}\\gamma^{\\lambda}, \\label{dd4}\n\\end{eqnarray}\nwhere in the last relation $ S^{even} = \\gamma^{\\lambda} S'^{odd}$. \\\\\nUsing the relations (\\ref{dd3}) and (\\ref{dd4}), one gets:\n\\begin{eqnarray*}\n \\gamma^{\\mu} S^{even} \\gamma_{\\mu} &=& \n {\\rm Tr} (S^{even}) I - {\\rm Tr} (\\gamma^5S^{even}) \\gamma^5,  \\\\\n \\hat p S^{even} \\hat p &=& - p^2 S^{even}_R + \\frac{1}{2}\n {\\rm Tr}(\\hat p \\gamma^{\\alpha} S^{even}_R) \\gamma_{\\alpha} \\hat p, \\\\\n \\hat p S^{even} \\hat p &=& - \\hat p S^{even}_R \\hat p, \n \\quad {\\rm for} \\quad p^2=0, \\\\ \n \\gamma^{\\mu} S^{odd} \\gamma_{\\mu} &=& -\\frac{1}{2}\n {\\rm Tr} (\\gamma^{\\alpha}S^{odd}) \\gamma_{\\alpha} + \\frac{1}{2}\n  {\\rm Tr} (\\gamma^5\\gamma^{\\alpha}S^{odd}) \\gamma_{\\alpha} \\gamma^5,  \\\\\n \\hat p S^{odd} \\hat p &=& - p^2 S^{odd}_R + \\frac{1}{2}\n {\\rm Tr}(\\hat p S^{odd}_R) \\hat p + \\frac{1}{2} \n  {\\rm Tr}(\\gamma^5 \\hat p S^{odd}_R) \\hat p \\gamma^5, \\\\\n S^{odd} &=& \\frac{1}{4} {\\rm Tr}(\\gamma^{\\alpha} S^{odd}) \\gamma_{\\alpha}\n + \\frac{1}{4} {\\rm Tr}(\\gamma^5 \\gamma^{\\alpha} S^{odd}) \n    \\gamma_{\\alpha}\\gamma^5,\\\\\n S^{odd} &+& S^{odd}_R = \\frac{1}{2} {\\rm Tr}(\\gamma^{\\alpha} S^{odd})\n    \\gamma_{\\alpha}.\n\\end{eqnarray*}\nUsing (\\ref{dd3}) and (\\ref{dd4}), one can write also the well known relations \nfor $S^1$, $S^2$, $S^3$, $S^4$:\n\\begin{eqnarray*}\n\\gamma^{\\mu} \\gamma^{\\alpha} \\gamma_{\\mu} &=& -2 \\gamma^{\\alpha}, \\quad\n\\gamma^{\\mu} \\gamma^{\\alpha} \\gamma^{\\beta} \\gamma^{\\delta} \\gamma_{\\mu} =\n -2 \\gamma^{\\delta} \\gamma^{\\beta} \\gamma^{\\alpha}, \\\\\n \\gamma^{\\mu} \\gamma^{\\alpha} \\gamma^{\\beta} \\gamma_{\\mu} \n   &=& 4 g^{\\alpha \\beta}, \\\\\n \\gamma^{\\mu} \\gamma^{\\alpha_1} \\gamma^{\\alpha_2} \\gamma^{\\alpha_3} \n  \\gamma^{\\alpha_4} \\gamma_{\\mu} &=& \n 2( \\gamma^{\\alpha_4} \\gamma^{\\alpha_1} \\gamma^{\\alpha_2} \\gamma^{\\alpha_3}\n +\\gamma^{\\alpha_3} \\gamma^{\\alpha_2} \\gamma^{\\alpha_1} \\gamma^{\\alpha_4})  \\\\\n &=& 2( \\gamma^{\\alpha_1} \\gamma^{\\alpha_4} \\gamma^{\\alpha_3} \\gamma^{\\alpha_2}\n +\\gamma^{\\alpha_2} \\gamma^{\\alpha_3} \\gamma^{\\alpha_4} \\gamma^{\\alpha_1}), \\\\\n \\gamma^{\\mu} \\sigma^{\\alpha \\beta} \\gamma_{\\mu} &=& 0, \\quad\n \\gamma^{\\mu} \\sigma^{\\alpha \\beta} \\gamma^{\\delta} \\gamma_{\\mu} = \n 2 \\gamma^{\\delta} \\sigma^{\\alpha \\beta}, \\quad\n \\gamma^{\\mu} \\gamma^{\\delta} \\sigma^{\\alpha \\beta} \\gamma_{\\mu} = \n 2 \\sigma^{\\alpha \\beta} \\gamma^{\\delta}.\n\\end{eqnarray*}\n\n\\subsection{\\it Fiertz Identities}\n\nFiertz identities for $\\gamma$--matrices could be obtained from\nthe basic formula: \n\\begin{eqnarray}\n\\delta_{ij} \\delta_{kl} = \n\\frac{1}{4} [ \\delta_{il} \\delta_{kj}\n + \\gamma^5_{il} \\gamma^5_{kj} + \\gamma^{\\mu}_{il} \\gamma_{\\mu\\;kj}  \n - (\\gamma^5 \\gamma^{\\mu})_{il} (\\gamma^5 \\gamma_{\\mu})_{kj}  \n - \\frac{1}{2} \\sigma^{\\mu \\nu}_{il} (\\sigma_{\\mu \\nu})_{kj}]. \\label{d11}\n\\end{eqnarray}\nUsing (\\ref{d11}) one can obtain the well known relations:\n\\begin{eqnarray}\n \\Gamma^M_{ij} \\Gamma^M_{kl} = \n \\sum^{16}_{N=1} C_{MN} \\Gamma^N_{il} \\Gamma^N_{kj}, \\label{d12}\n\\end{eqnarray}\nThe coefficients $C_{MN}$ are presented in Table~\\ref{dirac}.2, where we  use \nthe traditional notations:\n\\[\nS=I, \\quad P=\\gamma^5, \\quad V = \\gamma^{\\mu}, \\quad \nA = \\gamma^5 \\gamma^{\\mu}, \\quad T = \\sigma^{\\mu \\nu}.\n\\]\n\\vspace{0.5cm}\n\\noindent \n\\underline{ {\\bf Table~\\ref{dirac}.2.}}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\\hline\n   & $N = S$ & $V$ & $T$ & $A$ & $P$ \\\\ \\hline\n $M=I$ & $\\frac{1}{4}$ & $\\frac{1}{4}$ & $-\\frac{1}{8}$ & $-\\frac{1}{4}$ & \n $\\frac{1}{4}$ \\\\ \\hline\n $V$ &  $1$ & $-\\frac{1}{2}$ & $0$ & $-\\frac{1}{2}$ & $-1$ \\\\ \\hline\n $T$ & $-3$ & $0$  & $-\\frac{1}{2}$ & $0$ & $-3$ \\\\ \\hline\n $A$ & $-1$ & $-\\frac{1}{2}$ & $0$ & $-\\frac{1}{2}$ & $1$ \\\\ \\hline\n $P$ & $\\frac{1}{4}$ & $-\\frac{1}{4}$ & $-\\frac{1}{8}$ & $\\frac{1}{4}$ & \n $\\frac{1}{4}$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\nUsing relation (\\ref{d11}) one gets:\n\\begin{eqnarray*}\n&& (1 \\pm \\gamma^5)_{ij} \\delta_{kl} = \\\\\n&&\\frac{1}{8} [ 2(1 \\pm \\gamma^5)_{il} (1 \\pm \\gamma^5)_{kj}\n + 2((1 \\pm \\gamma^5)\\gamma^{\\mu})_{il} ((1\\mp \\gamma^5)\\gamma_{\\mu})_{kj} \n  - ((1 \\pm \\gamma^5)\\sigma_{\\mu \\nu})_{il} \\sigma^{\\mu \\nu}_{kj}], \\\\\n&& \\delta_{ij} (1 \\pm \\gamma^5)_{kl} = \\\\\n&& \\frac{1}{8} [ 2(1 \\pm \\gamma^5)_{il} (1 \\pm \\gamma^5)_{kj}\n + 2((1\\mp \\gamma^5)\\gamma^{\\mu})_{il} ((1 \\pm \\gamma^5)\\gamma_{\\mu})_{kj} \n  - ((1 \\pm \\gamma^5)\\sigma_{\\mu \\nu})_{il} \\sigma^{\\mu \\nu}_{kj}], \\\\\n&& (1 \\pm \\gamma^5)_{ij} (1 \\pm \\gamma^5)_{kl} = \n \\frac{1}{4} [ 2(1\\pm \\gamma^5)_{il} (1\\pm \\gamma^5)_{kj}\n  - ((1\\pm \\gamma^5)\\sigma_{\\mu \\nu})_{il} \\sigma^{\\mu \\nu}_{kj}], \\\\\n&& (1\\pm \\gamma^5)_{ij} (1\\mp \\gamma^5)_{kl} =  \\frac{1}{2} \n [(1\\pm \\gamma^5)\\gamma^{\\mu}]_{il}   [(1\\mp \\gamma^5)\\gamma_{\\mu}]_{kj}, \\\\\n&& [(1\\pm \\gamma^5)\\gamma^{\\mu}]_{ij} [(1\\pm \\gamma^5)\\gamma_{\\mu}]_{kl} = \n -[(1\\pm \\gamma^5)\\gamma^{\\mu}]_{il} [(1\\pm \\gamma^5)\\gamma_{\\mu}]_{kj}, \\\\ \n&& [(1\\pm \\gamma^5)\\gamma^{\\mu}]_{ij} [(1\\mp \\gamma^5)\\gamma_{\\mu}]_{kl} = \n 2 (1\\pm \\gamma^5)_{il} (1\\mp \\gamma^5)_{kj}, \\\\ \n && (\\gamma^{\\mu})_{ij} (\\gamma_{\\mu})_{kl} + \n  (\\gamma^5 \\gamma^{\\mu})_{ij} (\\gamma^5 \\gamma_{\\mu})_{kl} = \n - [(\\gamma^{\\mu})_{il} (\\gamma_{\\mu})_{kj} + \n  (\\gamma^5 \\gamma^{\\mu})_{il} (\\gamma^5 \\gamma_{\\mu})_{kj}].\n\\end{eqnarray*}\n \n\\subsection{\\it Traces of the $\\gamma$-matrices} \n\nThe trace of any odd--numbered string of $\\gamma$--matrices\n(including any number of $\\gamma^5$ matrices) and trace of the \n$\\gamma^5 \\gamma^{\\mu} \\gamma^{\\nu}$ product are equal to zero:\n\\[\n {\\rm Tr} S^{odd} = {\\rm Tr} \\bigl( S^{odd}(\\cdots \\gamma^5 \\cdots) \\bigr) = \n {\\rm Tr} (\\gamma^5 \\gamma^{\\mu} \\gamma^{\\nu})  = 0.\n\\]\nIn this Subsection we use the following notation:\n\\[\nT^{\\mu_1 \\mu_2 ... \\mu_n} \\equiv \\frac{1}{4} {\\rm Tr}(\\gamma^{\\mu_1} \n\\gamma^{\\mu_2} \\, ... \\, \\gamma^{\\mu_n}). \n\\]\nThen\n\\begin{eqnarray*}\n&&T^{\\mu \\nu} = g^{\\mu \\nu}, \\qquad\nT^{\\alpha \\beta \\delta \\sigma} = g^{\\alpha \\beta} g^{\\delta \\sigma} + \ng^{\\alpha \\sigma} g^{\\beta \\delta} - g^{\\alpha \\delta} g^{\\beta \\sigma}, \\\\\n &&T^{\\alpha \\beta \\delta \\lambda \\rho \\sigma} = \n  g^{\\alpha \\beta} T^{\\delta \\lambda \\rho \\sigma} \n - g^{\\alpha \\delta} T^{\\beta \\lambda \\rho \\sigma} \n + g^{\\alpha \\lambda} T^{\\beta \\delta \\rho \\sigma}\n - g^{\\alpha \\rho} T^{\\beta \\delta \\lambda \\sigma} \n + g^{\\alpha \\sigma} T^{\\beta \\delta \\lambda \\rho}, \\\\\n&&{\\rm Tr}(\\gamma^5) = 0, \\quad \n  {\\rm Tr}(\\gamma^5\\gamma^{\\mu}\\gamma^{\\nu}) = 0, \\quad\n{\\rm Tr}(\\gamma^5\\gamma^{\\alpha}\\gamma^{\\beta}\\gamma^{\\delta}\\gamma^{\\lambda})\n= -4i\\varepsilon^{\\alpha\\beta\\delta\\lambda}, \\\\\n&&{\\rm Tr}(\\gamma^5 \\gamma^{\\alpha_1} \\gamma^{\\alpha_2} \\gamma^{\\alpha_3}\n \\gamma^{\\alpha_4} \\gamma^{\\alpha_5} \\gamma^{\\alpha_6}) = \n4i(g^{\\alpha_1 \\alpha_2}\\varepsilon^{\\alpha_3\\alpha_4\\alpha_5\\alpha_6} \n - g^{\\alpha_1 \\alpha_3}\\varepsilon^{\\alpha_2\\alpha_4\\alpha_5\\alpha_6} \\\\\n && + g^{\\alpha_2 \\alpha_3}\\varepsilon^{\\alpha_1\\alpha_4\\alpha_5\\alpha_6} \n  + g^{\\alpha_4 \\alpha_5}\\varepsilon^{\\alpha_1\\alpha_2\\alpha_3\\alpha_6} \n - g^{\\alpha_4 \\alpha_6}\\varepsilon^{\\alpha_1\\alpha_2\\alpha_3\\alpha_5} \n + g^{\\alpha_5 \\alpha_6}\\varepsilon^{\\alpha_1\\alpha_2\\alpha_3\\alpha_4}), \\\\\n&&{\\rm Tr} \\sigma^{\\alpha \\beta} \\sigma^{\\mu \\nu} = \n 4 (g^{\\alpha \\nu} g^{\\beta \\mu} - g^{\\alpha \\mu} g^{\\beta \\nu}).\n\\end{eqnarray*}\nUsing the relation (\\ref{d11}), one can rewrite the trace of the \nproduct of two $4 \\times 4$ matrices $A$ and $B$ as follows:\n\\begin{eqnarray*}\n 4 {\\rm Tr}(AB) &=& {\\rm Tr}(A) {\\rm Tr}(B) + {\\rm Tr}(\\gamma^5 A) \n {\\rm Tr}(\\gamma^5 B) + {\\rm Tr}( \\gamma^{\\mu} A) {\\rm Tr}(\\gamma_{\\mu} B) \\\\ \n &-&{\\rm Tr}( \\gamma^5 \\gamma^{\\mu} A) {\\rm Tr}(\\gamma^5 \\gamma_{\\mu} B)\n - \\frac{1}{2}{\\rm Tr}(\\sigma^{\\mu \\nu} A){\\rm Tr}(\\sigma_{\\mu \\nu} B).\n\\end{eqnarray*}\nThe additional equation can be obtained using the Chisholm identities \n(\\ref{dd3}) and (\\ref{dd4}):\n\\[\n {\\rm Tr}(A \\gamma^{\\mu} B) {\\rm Tr}(\\gamma_{\\mu} S^{odd}) = \n2\\Bigl [{\\rm Tr}(A S^{odd}B) + {\\rm Tr}(A S^{odd}_R B) \\Bigr ]. \n\\]\n\n\\subsection{\\it Dirac Matrices Algebra in $n$--dimensions}\n\nIn the framework of dimensional regularization one gets:\n\\begin{eqnarray*}\n && {\\rm Tr} \\; I \\; = f(n), \\;\\; f(4) = 4, \\quad g^{\\mu\\nu}g_{\\mu\\nu} = n, \\\\\n && \\gamma^{\\mu} \\gamma^{\\nu} \\, + \\,  \\gamma^{\\nu} \\gamma^{\\mu} \\, = \\,\n 2g^{\\mu \\nu}, \\\\\n&& \\gamma_{\\mu} \\gamma^{\\alpha}\\gamma^{\\mu} \\, = \\, (2-n)\\gamma^{\\alpha}, \\\\\n&& \\gamma_{\\mu} \\gamma^{\\alpha}\\gamma^{\\beta} \\gamma^{\\mu} \\, = \\, \n 4g^{\\alpha\\beta} + (n-4)\\gamma^{\\alpha}\\gamma^{\\beta}, \\\\\n&& \\gamma_{\\mu} \\gamma^{\\alpha} \\gamma^{\\beta}\\gamma^{\\delta}\\gamma^{\\mu} \\, =\n \\, -2\\gamma^{\\delta}\\gamma^{\\beta}\\gamma^{\\alpha} + \n (4-n) \\gamma^{\\alpha} \\gamma^{\\beta}\\gamma^{\\delta}.\n\\end{eqnarray*}\n\n\n\\section{\\bf GELL--MANN MATRICES}\\label{gellmann}\n\n\\subsection{\\it The main properties}\n\nThe Gell-Mann $3 \\times 3$ matrices $\\lambda_i (i=1, \\ldots ,8)$ are\ngenerators\nof the group $SU(3)$. Their properties were presented elsewhere \n \\cite{itzu,okun,gw,ditn,cvit}. \\\\\nUsually in QCD instead of $\\lambda_i$ one deals with matrices $t_i$:\n\\[\nt_i \\equiv \\frac{1}{2} \\lambda_i.\n\\]\nEight $\\lambda_i$ matrices equal: \n\\begin{displaymath}\n \\lambda_1 = \\left( \\begin{array}{ccc} 0 & 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 0\n\\end{array} \\right) ,\n\\quad\n \\lambda_2 = \\left( \\begin{array}{ccc} 0 & -i & 0 \\\\ i & 0 & 0 \\\\ 0 & 0 & 0\n\\end{array} \\right) ,\n\\quad\n \\lambda_3 = \\left( \\begin{array}{ccc} 1 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & 0\n\\end{array} \\right) ,\n\\end{displaymath}\n\\begin{displaymath}\n \\lambda_4 = \\left( \\begin{array}{ccc} 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ 1 & 0 & 0\n\\end{array} \\right) ,\n\\quad\n \\lambda_5 = \\left( \\begin{array}{ccc} 0 & 0 & -i \\\\ 0 & 0 & 0 \\\\ i & 0 & 0\n\\end{array} \\right) ,\n\\quad\n \\lambda_6 = \\left( \\begin{array}{ccc} 0 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 1 & 0\n\\end{array} \\right) ,\n\\end{displaymath}\n\\begin{displaymath}\n \\lambda_7 = \\left( \\begin{array}{ccc} 0 & 0 & 0 \\\\ 0 & 0 & -i \\\\ 0 & i & 0\n\\end{array} \\right) ,\n\\quad\n \\lambda_8 = \\frac{1}{\\sqrt{3}}\n\\left(\\begin{array}{ccc} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & -2\n\\end{array} \\right).\n\\end{displaymath}\nThe main properties of $t_i$ (or $\\lambda_i$) are as follows:\n\\begin{eqnarray}\n &&t^{\\dagger}_i = t_i, \\quad  \\det t_i = 0, \\quad (i=1, \\ldots ,7), \\quad \n \\det t_8 = -\\frac{2}{\\sqrt{3}}, \\nonumber \\\\\n&& [ t^a , t^b] = i f^{abc} t^c, \\qquad \\{ t^a , t^b \\} = \n\\frac{1}{3}\\delta^{ab}+d^{abc}t^c, \\label{t1}\n\\end{eqnarray}\nwhere $d^{abc} (f^{abc})$ is totally symmetric (anti-symmetric) tensor. The\nnon-zero elements of $f^{abc}$ and $d^{abc}$ are equal to: \n\\begin{eqnarray*}\n&& f_{123} = 1, \\, f_{147}=-f_{156}=f_{246}=f_{257}=f_{345}=-f_{367}=\n\\frac{1}{2}, \\, f_{458}=f_{678}=\\frac{\\sqrt{3}}{2}, \\\\\n&& d_{146} = d_{157}=-d_{247}=d_{256}=d_{344}=d_{355}=-d_{366}=\n-d_{377} = \\frac{1}{2}, \\\\\n&& d_{118} = d_{228}=d_{338}=-d_{888} = \\frac{1}{\\sqrt{3}}, \\, \nd_{448} =d_{558}=d_{668}=d_{778}=-\\frac{1}{2\\sqrt{3}}.\n\\end{eqnarray*}\nThroughout this Section we use two additional notations: \n\\begin{eqnarray*}\n && h^{abc} = d^{abc} + i f^{abc}, \\quad \n h^{abc} = h^{bca} = h^{cab}, \\quad h^{aab} = 0, \\\\\n && S(a_1 a_2 \\ldots a_n) \\equiv t^{a_1} t^{a_2} \\ldots t^{a_n}, \\; \n S_R(a_1 a_2 \\ldots a_n) \\equiv t^{a_n} \\ldots t^{a_2} t^{a_1}.\n\\end{eqnarray*}\nThus, from (\\ref{t1}) one has:\n\\begin{eqnarray}\n t^a t^b = \\frac{1}{6}\\delta^{ab} + \\frac{1}{2}(d^{abk}+if^{abk})t^k \\; \n = \\frac{1}{6}\\delta^{ab} +  \\displaystyle {\\frac{1}{2}} h^{abk} t^k. \\label{t2}\n\\end{eqnarray}\n\n\\subsection{\\it Traces of the $t^a$--matrices} \n\nTrace of any string of $t^a$ matrices can be evaluated recursively using the \nrelation (\\ref{t2}):\n\\begin{eqnarray}\n{\\rm Tr} S(a_1 a_2 \\ldots a_n) = \\frac{1}{6} \\delta^{a_{n-1} a_n} \n {\\rm Tr}S(a_1 \\ldots a_{n-2}) +  \\displaystyle \\frac{1}{2}  h^{a_{n-1} a_n k} \\, \n {\\rm Tr} S(a_1 \\ldots a_{n-2} k) \\label{t3} \n\\end{eqnarray}\nUsing (\\ref{t1}) and (\\ref{t3}) one gets: \n\\begin{eqnarray*}\n&&{\\rm Tr}(t^a) = 0, \\quad {\\rm Tr}(t^at^b) = \\frac{1}{2}\\delta^{ab}, \\quad \n{\\rm Tr}(t^at^bt^c) = \\frac{1}{4}(d^{abc}+if^{abc}) = \\frac{1}{4}h^{abc}, \\\\\n&&{\\rm Tr}(t^at^bt^ct^d) = \\frac{1}{12}\\delta^{ab}\\delta^{cd} + \n\\frac{1}{8} h^{abn} h^{ncd}, \\\\ \n&&{\\rm Tr}(t^at^bt^ct^d t^e) = \\frac{1}{24} h^{abc}\\delta^{de} + \n\\frac{1}{24} \\delta^{ab} h^{cde} + \\frac{1}{16} h^{abn} h^{nck} h^{kde}.\n\\end{eqnarray*}\n\n\\subsection{\\it Fiertz Identity}\n\nThe Fiertz identity for $t^a$ has the form:\n\\begin{eqnarray}\n t^a_{ik} t^a_{jl} = \\frac{1}{2}(\\delta_{il}\\delta_{kj}  \n - \\frac{1}{3} \\delta_{ik}\\delta_{jl}). \\label{t5}\n\\end{eqnarray}\nAny $3 \\times 3$ matrix $A$ can be expanded over set $\\{I, t^a\\}$:\n\\[\nA=a_0I + a^i t^i, \\quad {\\rm where} \\quad a_0 = \\frac{1}{3} {\\rm Tr} A, \\quad\n a^i = 2 \\, {\\rm Tr} (t^i A).\n\\]\nDecomposition of the two $u_i$ and $\\bar u_i$ color spinors products into\ncolor--singlet and color--octet parts has the form:\n\\[u_i \\bar u_j = \\frac{\\delta_{ij}}{\\sqrt{3}} + \\sqrt{2}\\varepsilon^kt^k_{ij},\n\\quad \\varepsilon^k \\varepsilon^l = \\delta^{kl}.\n\\]\n\n\\subsection{\\it Products of the $t^a$--matrices}\n\nThe product of $n$ matrices $t^a$ could be written in the form\n$a_0 + a_i t^i$\nusing the following relations (see ({\\ref{t2})):\n\\begin{eqnarray}\nS(a_1 a_2 \\ldots a_n) = \n \\frac{1}{6} \\delta^{a_{n-1} a_n} S(a_1 a_2 \\ldots a_{n-2})\n+  \\displaystyle \\frac{1}{2} h^{a_{n-1} a_n k} S(a_1 a_2 \\ldots a_{n-2} k) \\label{t4}\n\\end{eqnarray}\nThus, the products of two, three, and four matrices equal: \n\\begin{eqnarray*}\n t^a t^b &=& \\frac{1}{6}\\delta^{ab} + \\frac{1}{2}(d^{abk}+if^{abk})t^k \\; \n = \\frac{1}{6}\\delta^{ab} + \\frac{1}{2} h^{abk} t^k, \\\\ \nt^at^bt^c &=& \\frac{1}{6}\\delta^{ab}t^c + \\frac{1}{12} h^{abc} \n +  \\frac{1}{4} h^{abk} h^{kcn} t^n, \\\\\nt^at^bt^ct^d &=& \\frac{1}{36}\\delta^{ab}\\delta^{cd} \n  + \\frac{1}{24} h^{abk} h^{kcd} + \\frac{1}{12} [ h^{abk} \\delta^{cd} \n+ \\delta^{ab} h^{cdk}] t^k\n \n  +  \\frac{1}{8} h^{abn} h^{cdk} h^{nkp} t^p.\n\\end{eqnarray*}\nThe products of the type $t^k S t^k$ have the form:\n\\begin{eqnarray*}\n && {\\sf t^k} S {\\sf t^k} = \\frac{1}{2} {\\rm Tr}(S) - \\frac{1}{6}S, \\\\\n&& t^k t^k = \\frac{4}{3}I, \\quad {\\sf t^k}t^a{\\sf t^k} = -\\frac{1}{6}t^a, \\quad \n{\\sf t^k}t^a t^b {\\sf t^k} = \\frac{1}{4}\\delta^{ab} -\n \\frac{1}{6}t^at^b, \\quad  \n{\\sf t^k}t^at^bt^c{\\sf t^k}  = \\frac{1}{8} h^{abc} -\\frac{1}{6}t^at^bt^c, \\\\\n&&{\\sf t^k}t^at^bt^ct^d{\\sf t^k} = -\\frac{1}{6}t^at^bt^ct^d +\n \\frac{1}{24}\\delta^{ab}\\delta^{cd} + \n \\frac{1}{16} h^{abn} h^{ncd}.\n\\end{eqnarray*}\nThe products of the type $S S^{\\Pi}$ (here $ S^{\\Pi}$ is denoted any\npermutation of the $t^{a_i}$--matrices) are given by\n\\begin{eqnarray*}\n&&t^{a_1}t^{a_2} \\ldots t^{a_n} \\; \\; t^{a_n} \\ldots t^{a_2}t^{a_1} = \n \\Bigl ( \\frac{4}{3} \\Bigr )^n, \\\\\n && t^a t^b t^a t^b = -\\frac{2}{9}I, \\; \\; t^a t^b t^b t^a = \\frac{16}{9}I.\n\\end{eqnarray*}\nThe products of the $S(abc) S^{\\Pi}(abc)$ and $S(abcd) S^{\\Pi}(abcd)$ are\npresented on the following tables (in these tables symbol $(abc)$ stands for\n$t^a t^b t^c$, etc). \n\n\\noindent \n\\underline{ {\\bf Table~\\ref{gellmann}.1.}} \n  The products of the $(abc)$ on the \n  $(abc)^{\\Pi}$. All products are contain the common factor\n $ \\displaystyle \\frac{1}{27}I$.\n\\begin{center} \n\\begin{tabular}{|c|c|c|c|c|c|}\\hline\n   $(abc)$ & $10$ & $(bac)$ &  $1$ & $(cab)$ & $-8$ \\\\ \\hline\n   $(acb)$ & $1$  & $(bca)$ & $-8$ & $(cba)$ & $64$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\n\\vspace{0.5cm}\n\\noindent \n\\underline{ {\\bf Table~\\ref{gellmann}.2.}} \n The products of the $(abcd)$ on the $(abcd)^{\\Pi}$. All products are contain \nthe common factor $\\displaystyle \\frac{1}{81}I$.\n\\begin{center}\n\\begin{tabular}{|c|c||c|c|c|c|c|c|}\\hline\n$(abcd)$ & $-14$ & $(bacd)$ & $+31$ & $(cabd)$ & $-5$ & $(dabc)$ & $+40$ \n  \\\\ \\hline\n$(abdc)$ & $+31$ & $(badc)$ & $+\\frac{71}{2}$ & $(cadb)$ & $-\\frac{1}{2}$ & \n   $(dacb)$ & $+4$    \\\\ \\hline\n$(acbd)$ & $+31$ & $(bcad)$ & $-5$ & $(cbad)$ & $-\\frac{1}{2}$ & \n   $(dbac)$ & $+4$   \\\\ \\hline\n$(acdb)$ & $-5$ & $(bcda)$ & $+40$ & $(cbda)$ & $+4$ & \n   $(dbca)$ & $-32$   \\\\ \\hline\n$(adbc)$ & $-5$ & $(bdac)$ & $-\\frac{1}{2}$ & $(cdab)$ & $+4$ & \n   $(dcab)$ & $-32$   \\\\ \\hline\n$(adcb)$ & $-\\frac{1}{2}$ & $(bdca)$ & $+4$ & $(cdba)$ & $-32$ & \n   $(dcba)$ & $+256$   \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\n\\subsection{\\it Convolutions of $d^{abc}$ and $f^{abc}$ with $t^a$}\nThe convolutions of the coefficients $d^{abc}$ and $f^{abc}$ with the \n$t^a$--matrices equal:\n\\begin{eqnarray*}\n&&d^{abc} t^c = t^a t^b + t^b t^a -\\frac{1}{3}\\delta^{ab}, \\quad \n f^{abc} t^c = i(t^b t^a - t^a t^b), \\\\\n &&h^{abc} t^c = 2 t^a t^b - \\frac{1}{2} \\delta^{ab}\n \\end{eqnarray*}\n \\begin{eqnarray*}\n&&d^{abk} d^{kcl} t^l = \n(t^a t^b t^c + t^b t^a t^c + t^c t^a t^b + t^c t^b t^a) -\\frac{1}{3} d^{abc}I \n - \\frac{2}{3} \\delta^{ab} t^c, \\\\\n&&d^{abk} f^{kcl} t^l = \n i(-t^a t^b t^c - t^b t^a t^c + t^c t^a t^b + t^c t^b t^a), \\\\\n&&f^{abk} d^{kcl} t^l = \n i(-t^a t^b t^c + t^b t^a t^c - t^c t^a t^b + t^c t^b t^a) \n      -\\frac{1}{3} f^{abc}I, \\\\ \n&&f^{abk} f^{kcl} t^l = \n (-t^a t^b t^c + t^b t^a t^c + t^c t^a t^b - t^c t^b t^a), \\\\\n&&d^{abc}t^at^bt^c = \\frac{10}{9}I, \\; \nf^{abc}t^at^bt^c = 2iI, \\; h^{abc}t^at^bt^c = -\\frac{8}{9}I, \\\\ \n && d^{abc}t^at^b = \\frac{5}{6}t^c, \\; \nf^{abc}t^at^b = \\frac{3}{2} i t^c, \\; \nh^{abc}t^at^b = -\\frac{2}{3} t^c.\n\\end{eqnarray*}\nThe Jacobi identities for the coefficients $f^{abc}$ and $d^{abc}$ equal: \n\\begin{eqnarray*}\nf_{abk}f_{kcl} \\, &+& \\,f_{bck}f_{kal}\\, +\\,f_{cak}f_{kbl}\\, = 0, \\\\\nd_{abk}f_{kcl} \\, &+& \\,d_{bck}f_{kal}\\, +\\,d_{cak}f_{kbl}\\, = 0.\n\\end{eqnarray*}\nThe various relations of a such type were presented in \\cite{ditn}:\n\\begin{eqnarray*}\n&&d_{abk}d_{kcl} \\, + \\,d_{bck}d_{kal}\\, +\\,d_{cak}d_{kbl}\\, = \\,\\frac{1}{3} \n(\\delta_{ab}\\delta_{cl}\\,+\\,\\delta_{ac}\\delta_{bl}\\,+\\,\n\\delta_{al}\\delta_{bc}), \\\\\n&& f_{abk}f_{kcl} \\, = \\,\\frac{2}{3}(\\delta_{ac}\\delta_{bl}\\,-\\,\n\\delta_{al}\\delta_{bc})\\,+\\,d_{ack}d_{blk}\\, -\\,d_{alk}d_{bck}, \\\\ \n&&3d_{abk}d_{kcl} \\, = \\,\\delta_{ac}\\delta_{bl}\\,+\\,\\delta_{al}\n\\delta_{bc}\\,-\\,\n\\delta_{ab}\\delta_{cl}\\,+\\,f_{ack}f_{blk}\\, +\\,f_{alk}f_{bck}, \\\\ \n&& d_{aac} \\; = \\; f_{aac} \\; = \\; d_{abc} f_{abm} \\; = \\; 0.\n\\end{eqnarray*}\n\\begin{eqnarray*}\nf_{akl}f_{bkl} &=& 3 \\delta_{ab}, \\qquad d_{akl}d_{bkl} = \\frac{5}{3}\n\\delta_{ab}, \\\\\nf_{pak}f_{kbl}f_{lcp} &=& -\\frac{3}{2}f_{abc}, \\quad \nd_{pak}f_{kbl}f_{lcp} = -\\frac{3}{2}d_{abc}, \\\\\nd_{pak}d_{kbl}f_{lcp} &=& \\frac{5}{6}f_{abc}, \\quad \nd_{pak}d_{kbl}d_{lcp} = -\\frac{1}{2}d_{abc}, \\\\  \nd_{piq}d_{qjm}d_{mkt}d_{tlp} &=& \\frac{1}{36}\n(13 \\delta_{ij} \\delta_{kl} - 7 \\delta_{ik} \\delta_{jl} \n + 13 \\delta_{il} \\delta_{jk} - d_{ikm} d_{mjl}), \\\\ \nd_{piq}d_{qjm}d_{mkt}f_{tlp} &=& \\frac{1}{12}\n(-7 d_{ijm} f_{mkl} + d_{ikm} f_{mjl} + 9 d_{ilm} f_{mjk}), \\\\\nd_{piq}d_{qjm}d_{mkt}d_{tlp} &=& \\frac{1}{36}\n(-21 \\delta_{ij} \\delta_{kl} + 19 \\delta_{ik} \\delta_{jl} \n - \\delta_{il} \\delta_{jk} ) \\\\\n& +& \\frac{1}{6} (d_{ikm} d_{mjl}  - 4 d_{ilm} d_{mjk}), \\\\\nd_{piq}f_{qjm}f_{mkt}d_{tlp} &=& \\frac{3}{4}\n(d_{ikm} f_{mil} + d_{ilm} f_{mkj}), \\\\ \nf_{piq}f_{qjm}f_{mkt}f_{tlp} &=& \\frac{1}{4}\n(5 \\delta_{ij} \\delta_{kl} + \\delta_{ik} \\delta_{jl} \n + 5 \\delta_{il} \\delta_{jk}  - 6 d_{ikm} d_{mjl}).\n\\end{eqnarray*}\n\n\\begin{eqnarray*}\n  \\begin{array}{lcrlcr}   \nd^{abc} d^{abc} & = & 40\/3,  &  f^{abc} f^{abc} &=& 24, \\\\ \nh^{abc} h^{abc} & = & -32\/3, & h^{abc} h^{bac}  &=& 112\/3, \\\\ \n \\displaystyle d^{abk}d^{klc}d^{cbn}d^{nla} &  = & - 20\/3, &\nd^{abk}d^{klc}d^{cbn}f^{nla} &=& 0, \\\\\nd^{abk}d^{klc}f^{cbn}f^{nla} & = & 20, &\nd^{abk}f^{klc}d^{cbn}f^{nla} &=& -20, \\\\\nd^{abk}f^{klc}f^{cbn}f^{nla} & =&  0, & \nf^{abk}f^{klc}f^{cbn}f^{nla}  &=& 36, \\\\\n\\displaystyle h^{abk}h^{klc}h^{cbn}h^{nla} & = & - 32\/3 & {}\n\\end{array} \n \\end{eqnarray*} \n\n\\subsection{\\it Gell-Mann Matrices  in $n$--dimensions}\nThe generators for an arbitrary $SU(n)$ group are:\n\\begin{eqnarray}\n  t^a, \\quad a = 1,2,..,N, \\quad N=n^2-1  \\label{sun-1}\n\\end{eqnarray}\nwith the properties as follows:\n\\begin{eqnarray}\n &&  (t^a)^{\\dagger} = t^a, \\;\\; Tr t^a = 0,  \\label{sun-2} \\\\\n  && [t^a, t^b] = i \\, f^{abc} \\, t^c   \\label{sun-3} \\\\\n  && \\{t^a, t^b\\} =  d^{abc} \\, t^c \\, + \\frac{1}{n} \\delta^{ab}  \\label{sun-4} \\\\\n  &&   [t^a, t^b] \\equiv t^a t^b - t^b t^a, \\quad\n  \\{t^a, t^b\\} \\equiv t^a t^b + t^b t^a, \\nonumber \n\\end{eqnarray}\nThe constants $f^{abc} (d^{abc})$ is totally antisymmetric (symmetric) tensor.\n\\\\\n$SU(n)$ group has the invariants as follows:\n\\begin{eqnarray}\n   f^{a \\, kl} f^{b \\, kl} = C_A \\delta^{ab}, \\;\\;\n  d^{\\, a \\, kl} d^{\\, b \\, kl} = C_D \\delta^{ab}, \\;\\;\n  t^a t^a = C_F I, \\;\\; Tr \\, t^at^b = T_F \\delta^{ab}\n  \\label{app-gr-5}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n  C_A = n, \\;\\; C_F = \\frac{n^2-1}{2n}, \\;\\; C_D = \\frac{n^2-4}{n}, \\;\\;\n  T_F = \\frac{1}{2} \\label{app-gr-6}\n\\end{eqnarray}\nThe Jacobi identities have the form: \n\\begin{eqnarray*}\n && f^{ab \\, K} f^{K\\, cl} + f^{bc \\, K} f^{K\\, al} + f^{ca \\, K} f^{K\\, bl}  = 0 \\\\ \n && d^{ab \\, K} f^{K\\, cl} + d^{bc \\, K} f^{K\\, al} + d^{ca \\, K} f^{K\\, bl}  = 0  \n\\end{eqnarray*}\n\n\n\\section{\\bf VECTOR ALGEBRA}\\label{vecal}\n\nLet $\\{ p_1, \\ldots ,p_n\\}$ be some basis and scalar products $p_i \\cdot p_j$ \ndefine a matrix $M$: $M_{ij}=p_i\\cdot p_j$. The dual basis is the set of \nvectors $\\{\\xi_1,...,\\xi_n\\}$, which satisfy the conditions:\n\\[ \\xi_i\\cdot p_j = \\delta_{ij},\\ \\ \\ \n\\xi_i \\cdot \\xi_j=(M^{-1})_{ij}. \\]\nThen\n\\[\\xi_i^\\alpha=\\delta^{p_1,...,p_{i-1},\\alpha,p_{i+1},...,p_n}\n_{p_1, .\\ .\\ .\\ .\\ .\\ .\\ .\\ .\\ .\\ .\\ ,p_n}\/\\Delta_n, \\]\nwhere $\\Delta_n=\\delta_{p_1,...,p_n}^{p_1,...,p_n}$.\nSometimes one needs to represent some vector $Q$ in the form \\cite{vermaseren}:\n\\[ Q_\\alpha = \\cal{P}_\\alpha+V_\\alpha, \\]\nwhere $\\p_\\alpha$ is a linear combination of $p_1,...,p_m\\ \\ (m<n)$,\nand $V\\cdot p_i=0$ for $i=1,...,m$.\n\\begin{eqnarray*}\nm=1 &  {\\cal P}_{\\alpha}=\\frac{p_1 \\cdot Q}{p_1 \\cdot p_1} p_{1\\alpha}, \\; \nV_\\alpha=\\frac{1}{p_1\\cdot p_1}\\delta^{p_1\\alpha}_{p_1 Q} & \\\\\nm=2 &  {\\cal P}_\\alpha= \\frac{1}{\\Delta_2} \n\\delta^{Q\\mu}_{p_1 p_2}\\delta^{\\mu \\alpha}_{p_1 p_2}=\\frac{1}{\\Delta_2}\n\\left(\\delta^{Q p_2}_{p_1 p_2}p_{1\\alpha}\\right.\n\\left.+\\delta^{p_1 Q }_{p_1 p_2}p_{2\\alpha}\\right) & \\\\\n & V_\\alpha=\\frac{1}{\\Delta_2}\\delta^{p_1 p_2 \\alpha}_{p_1 p_2 Q} \\\\\nm=3 & {\\cal P}_\\alpha= \\frac{1}{2\\Delta_3} \n\\delta^{Q\\mu\\nu}_{p_1 p_2 p_3}\\delta^{\\alpha \\mu\\nu}_{p_1 p_2 p_3}\n=\\frac{1}{\\Delta_3} \\left(\\delta^{Q p_2 p_3}_{p_1 p_2 p_3}p_{1\\alpha}\\right.\n+\\delta^{p_1 Q p_3}_{p_1 p_2 p_3}p_{2\\alpha}\n\\left.+\\delta^{p_1 p_2 Q }_{p_1 p_2 p_3}p_{3\\alpha}\\right) & \\\\\n & V_\\alpha=\\frac{1}{\\Delta_3}\n\\delta^{p_1 p_2 p_3 \\alpha}_{p_1 p_2 p_3 Q}. &\n\\end{eqnarray*}\n\n\\subsection{\\it Representation of 3--dimensional Vectors, Reflections and \nRotations Using the Pauli Matrices}\n\\noindent $\\bullet$ {\\bf Vectors} \\\\\nAny vector $\\vec x$ in 3--dimensional Euclidean space can be represented in \nthe matrix form:\n\\begin{eqnarray*}\n\\hat x\\ &\\equiv &  \\vec x \\cdot \\vec \\sigma\\ = \n x^i\\sigma^i \\ =\\ \\left( \\begin{array}{cc}\nx^3 & x^1-ix^2 \\\\\nx^1+ix^2 & -x^3 \n\\end{array} \\right), \\\\\n\\det\\> \\hat x\\ &=&\\ -\\vec x\\cdot \\vec x \\ =\\ -(x_1^2+x_2^2+x_3^2),\n\\end{eqnarray*}\nwhere $\\sigma^i$ is the Pauli matrices (see Section~\\ref{pauli}). \\\\\nThe fundamental property of this representation is\n\\begin{equation}\n(\\hat x)^2= \\vec x\\cdot \\vec x\\> I,\\ \\ \\mbox{hence}\\ \\ \\\n\\hat x\\hat y+\\hat y\\hat x =2\\> (\\vec x\\cdot \\vec y)\\> I.\n\\end{equation}\nOne should also note that \n\\[\\hat x\\hat y-\\hat y\\hat x\\ =\\ 2i\\> \\widehat{\\vec x\\times \\vec y}.\\]\nIf components of $\\vec x$ are real, then $\\hat x^\\dagger =\\hat x$.\nHowever, in some practically important cases $(\\hat x)^2=0$ and,\nhence, $\\vec x\\cdot \\vec x=0$, components of $\\vec x$ are complex, say,\n$x^2=ix^0$. Then the matrix\n\\begin{displaymath}\n\\hat x=\\left( \\begin{array}{cc}\nx^3 & x^1+x^0 \\\\\nx^1-x^0 & -x^3\n\\end{array}\\right)\n\\end{displaymath}\nrepresents a vector from 3--dimensional space--time, in that\ncase $\\hat x^\\dagger \\neq \\hat x$.\n\n\\noindent $\\bullet$ {\\bf Reflections} \\\\ \nLet $\\vec x$ be an arbitrary vector and $S$ be the plane\northogonal to some unit vector $\\vec s$. Then, vector $\\vec x'$\nwhich results from $\\vec x$ after the reflection in the plane $S$\nis equal to: \n\\[\\vec x' = \\vec x - 2 (\\vec x \\vec s) \\vec s, \\]\nor, in the considered matrix representation\n\\[ \\hat x'  = - \\hat s\\hat x\\hat s. \\]\n\n\\noindent $\\bullet$ {\\bf Rotations} \\\\ \nLet $\\vec p$ and $\\vec q$ be the two unit vectors with the\nangle $\\theta\/2$ between them:\n\\[ \\vec p^{\\; 2} = \\vec q^{\\; 2} = 1, \\ \\ (\\vec p\\vec q) = \\cos(\\theta\/2).\\]\nSince any spatial rotation is a composition of two reflections,\nthe rotation by the angle $\\theta$ in the direction from $\\vec p$\nto $\\vec q$ is given by the matrix \n\\[ M\\ =\\ \\hat q\\hat p, \\]\ni.e. an arbitrary vector $\\vec x$ transforms as follows:\n\\[ \\hat x' = M\\hat x M^{-1} = \\hat q\\hat p\\hat x\\hat p\\hat q. \\]\nThe matrix $M$ can be rewritten in widely used form:\n\\begin{eqnarray} \nM = \\hat q\\hat p = \\vec q \\cdot \\vec p\\> I + i\\varepsilon_{qpr}\n\\hat r = \\cos(\\theta\/2) I\\>-\\>i\\>\\vec n\\vec \\sigma \\> \\sin(\\theta\/2), \n\\label{ve2}\n\\end{eqnarray} \nwhere $\\vec n\\ \\sin(\\theta\/2)\\ =\\ \\vec p \\times \\vec q,\\ \\ \n-\\pi<\\theta<\\pi$, {\\bf positive values of $\\theta$ correspond to \ncounterclockwise rotations if one sees from the head of\nvector $\\vec n$}.\n\n\\noindent So, we get the two forms of representation of a spatial rotation:\n\\begin{itemize}\n\\item{} The rotation by angle $\\theta$ about a unit vector $\\vec n$\nis given by \n\\[\nM = \\cos(\\theta\/2)I \\>-\\>i\\>\\vec n\\vec \\sigma \\> \\sin(\\theta\/2).\n\\] \n\\item{}The rotation in the plane of unit vectors $\\vec p$ and\n$\\vec q$ which transforms $\\vec p$ into $\\vec q$ is represented by\n\\[M\\ =\\ \\frac{I\\ +\\ \\hat q\\hat p}{\\sqrt{2(1+\\vec q\\cdot \\vec p)}}\\]\n\\end{itemize}\n\n\\subsection{\\it Representation of 4--dimensional Vectors, Reflections and \nRotations Using the Dirac Matrices}\n\n\\noindent $\\bullet$ {\\bf Vectors} \\\\\n$4\\times 4$ matrix $\\hat x$, which represents the 4--vector $x^{\\mu}$ in\nMinkowski space looks as follows:\n\\begin{displaymath}\n\\hat x \\equiv x^\\mu\\gamma_\\mu=\\left(\\begin{array}{cccc}\n0 & 0 & -x^0-x^3 & -x^1+ix^2 \\\\\n0 & 0 & -x^1-ix^2 & -x^0+x^3 \\\\\n-x^0+x^3 & x^1-ix^2 & 0 & 0 \\\\\nx^1+ix^2 & -x^0-x^3 & 0 & 0 \n\\end{array} \\right).\n\\end{displaymath}\nThis matrix satisfies the fundamental property\n\\begin{eqnarray} \n(\\hat x)^2 =x\\cdot x I \\ \\ \\Rightarrow \\ \\ \n\\hat x\\hat y +\\hat y\\hat x=2\\> x\\cdot y. \\label{ve3}\n\\end{eqnarray} \n\n\\noindent $\\bullet$ {\\bf Reflections} \\\\ \nUsing the relation (\\ref{ve3}) one can easily derive formulas for the\nreflections in 3--hyperplanes. Let $x^{\\mu}$ be an arbitrary vector and \n$S$ be the 3-hyperplane orthogonal to some unit vector $s$. Then, vector $x'$\nwhich results from $x$ after the reflection in the hyperplane $S$\nis equal to \n\\[ x' =  x - 2 \\ x\\cdot s \\ s, \\]\nor, in the considered matrix representation\n\\[ \\hat x'  = - \\hat s\\hat x\\hat s. \\]\n\n\\noindent $\\bullet$ {\\bf Lorentz transformations} \\\\ \nLet $p$ and $q$ be the two unit time-like vectors:\n\\[ p\\cdot p = q\\cdot q = 1.\\]\nLorentz transformation, which is a composition of the reflections\nin 3-hyper\\-pla\\-nes determined by the vectors $p$ and $q$ is given by\nthe matrix:\n\\[ M\\ =\\ \\hat q\\hat p, \\]\ni.e. an arbitrary vector $x$ transforms as follows:\n\\[ \\hat x' = M\\hat x M^{-1} = \\hat q\\hat p\\hat x\\hat p\\hat q. \\]\nThe Lorentz transformation in the 2-plane (defined by the vectors $p$ and $q$), \nwhich transforms $p$ into $q$, is represented by\n\\[M\\ =\\ \\frac{I\\ +\\ \\hat q\\hat p}{\\sqrt{2(1+ q\\cdot p)}}. \\]\nFor space-like unit vectors $p$ and $q$ we arrive at \n\\[M\\ =\\ \\frac{- I\\ +\\ \\hat q\\hat p}{\\sqrt{2(1 - q\\cdot p)}}. \\]\n\nLet $\\Lambda^{\\mu}_{\\nu}$ be the matrix of Lorentz boost\nthat moves the vector\\\\\n$m\\mathbf{e_0}^\\nu=(m,0,0,0)\\ $ to $\\ p^\\mu=(p^0, p^1, p^2, p^3)$, \n$$\np^\\mu = \\Lambda^{\\mu}_{\\nu} \\ m\\mathbf{e_0}^\\nu\\ ,\n$$\nand leaves all vectors orthogonal to $p$ and $\\mathbf{e_0}$\ninvariant. Then \n$$\n\\hat p = S(\\Lambda)\\; m \\gamma^0\\; S^{-1}(\\Lambda)\\;,\n$$\nwhere\n$$\nS(\\Lambda) = {\\displaystyle {\\hat p \\gamma^0 + m \\over \\sqrt{2m(m+p_0)}}  }\\;.\n$$\nThis being so,\n\\begin{eqnarray*}\n\\Lambda^{\\mu}_{\\nu} = {1\\over 4} \\;T\\!r\\; \\gamma^\\mu S(\\Lambda) \n\\gamma_\\nu S^{-1}(\\Lambda)\\;=\\;\\left(\n\\begin{array}{cc}\n\\displaystyle{{p^0\\over m}} & \\displaystyle{{p^i\\over m}}\\\\[4mm]\n\\displaystyle{{p^i\\over m}} & \\displaystyle{\\delta^{ij}+{p^ip^j\\over m(p^0+m)}}\n\\end{array}\\right)\\ .\n\\end{eqnarray*}\n\n\n\n\\noindent \nThe reference frame where\nthe momentum of the reference body is equal to $p$\nis usually referred to as the laboratory frame {\\it Lab}-frame;\nthe frame where the it is equal to $m\\mathbf{e_0}$\nis the rest frame {\\it R}-frame;\n\nLet $k^{\\mu}$ be some 4-vector defined in the {\\it Lab}-frame  and\n$k^{* \\mu}$ be the same 4-vector in the {\\it R}-frame;\n$k^{\\mu}=\\Lambda^{\\mu}_{\\nu} k^{* \\nu}$ and\n$k^{* \\mu}=\\big(\\Lambda^{-1}\\big)^{\\mu}_{\\nu} k^{\\nu}\\;.$\n\nThe Lorentz transormations form {\\it R}(est)-frame to {\\it Lab}-frame and vice versa \ncan also be represented in the form\n\\begin{eqnarray}\n \\begin{array}{l} { k^{*} \\to k} \\\\ {R \\to Lab} \\end{array}: \\; { \\left \\{ \\begin{array}{ccl}\n k_0 & = & \\displaystyle \\frac{k_{0}^{*}p_{0} + (\\vec{k}^*\\vec{p})} {m} \\\\[2mm]\n \\vec{k} & =  & \\vec{k}^{*} + \\alpha \\vec{p} \n   \\end{array} \\right. } \\quad\n \\begin{array}{r} {k \\to k^{*}} \\\\ {Lab \\to R} \\end{array}: \\;\n              { \\left \\{ \\begin{array}{ccl}\n k_0^{*} & = & \\;\\; \\displaystyle \\frac{(pk)} {m} \\\\[2mm]\n \\vec{k}^* & =  & \\vec{k} - \\alpha \\vec{p} \n   \\end{array} \\right. } \\quad\n \\label{lorentz-10}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray*}\n \\alpha = \\frac {k^{*}_0 + k_0} {p_0 + m}\n\\end{eqnarray*}\n\n\n\n\n\\section{\\bf DIRAC SPINORS}\\label{spin2}\n\n\\subsection{\\it General Properties}\n\nDirac spinors $u(p, s)$ and $v(p, s)$ describe the solutions of the Dirac \nequation with positive and negative energy:\n\\begin{eqnarray}\n  && (\\hat p -m)\\> u(p,s)=0,\\ \\  \\ (\\hat p+m)\\> v(p,s)=0 \\\\\n  && \\bar u(p,n)(\\hat p-m)=0, \\ \\ \\ \\bar v(p,n)(\\hat p+m)=0 \n\\end{eqnarray}\nwhere  The {\\it conjugated} spinors are defined as \nfollows:\n\\[ \n\\bar u=u^\\dagger \\gamma^0,\\ \\ \\ \\bar v=v^\\dagger \\gamma^0,\n\\] \nThese spinors are the functions of 4-momentum $p^{\\mu}$ on the mass shell \n$p^0=\\sqrt{\\vec p^{\\; 2}+m^2}$. The normalization conditions are\nas follows:\n\\begin{eqnarray*}\n\\bar u(p, s) u(p,s) &=&+2m, \\\\\n\\bar v(p, s) v(p,s) &=& -2m.\n\\end{eqnarray*}\nSymbol $s$ stands for the polarization of the fermion. The axial--vector \n$s^\\mu$ of the fermion spin is defined by the relations: \n\\begin{eqnarray*}\n && \\bar u(p,s)\\gamma^\\mu \\gamma^5 u(p,s)\\ =\\ m\\> s^\\mu, \\quad  \n  (s \\, s) = -1,\\ \\ \\ (s \\, p) = 0. \n\\end{eqnarray*}\n\n\\noindent It is helpful to clear up the relation between $n$\nand the Pauli--Lubanski pseudovector\n$$\nW_\\mu=\\varepsilon_{\\mu\\alpha\\beta\\nu}M^{\\alpha\\beta}P^{\\nu}\\;,\n$$\nwhere $M^{\\alpha\\beta}$ and $P^{\\nu}$ are the generators of \nLorentz transformations and translations; it acts on the\nfermion fields as follows:\n\\begin{eqnarray*}\n&& \n  W_\\mu \\psi(x) = {i\\over 2}\\;\\sigma_{\\mu\\nu}\\gamma^5 \\partial^{\\nu}\\psi(x),\n  \\quad \n\\gamma^5 \\hat s = -\\;{2\\over m}\\; s\\cdot W \n\\end{eqnarray*}\nTo make a bridge between the above formulas and nonrelativistic\nconcept of spin it is well to recollect that the operator of spin\nof the fermion at rest is given by\n\\begin{displaymath}\n\\Sigma^k = {1\\over 2} \n\\left( \\begin{array}{ccc} \\sigma^k & 0 \\\\ 0 & \\sigma^k \\end{array} \\right) \n\\ =\\;-\\;{1\\over 2}\\; \\gamma^5 \\gamma^k \\gamma^0 \n\\end{displaymath}\nFor the fermion of momentum $p$, the spin has the form:\n$$\n\\Sigma^\\mu = {1\\over 2} \\gamma^5 \\gamma^\\mu {\\hat p\\over m} \n$$\nIts projection on the direction defined by \nan arbitrary 4-vector $s$ such that  $(s\\, p)=0$ and $(s \\, s)=-1$\nis given by the formula:\n$$\n\\Sigma\\cdot s = {1\\over 2} \\gamma^5 \\hat s\\; {\\hat p\\over m}\\;.\n$$\nThus, the spin is the Pauli--Lubanski pseudovector times the sign of energy:\n$$\n\\Sigma = W\\; {\\hat p\\over m}\\;,\n$$\nThat is why the spin of the fermions $u(p,s)$ and $v(p,-s)$ is \n$\\displaystyle \\Sigma= {s\\over 2}$ and the spin of the fermions\n$u(p,-s)$ and $v(p,s)$ is \n$\\displaystyle \\Sigma = -\\;{s\\over 2}$.\n\nNote that the axial vector $s$ describing  \nthe spin of the fermion has only spatial non-zero components in its rest\nframe  \nand transforms together with the vector $p$ under Lorentz transformations.\n\nThe spinor $u(p,s)$ describes a fermion with momentum $p$ and the vector of \nspin $n$. The spinor $v(p,s)$ describes an antifermion with momentum $p$ and \nthe vector of spin $-s$. (One should note, that axial vector $s$ describing  \nspin of a fermion has only spatial non-zero components in the rest frame of \nthis fermion. However, it transforms together with the vector $p$ under \nLorentz transformations.)\n\n\\noindent Spinors $u(p,s)$ and $v(p,s)$ satisfy the following relations:\n\\begin{eqnarray} \nu(p,s)\\> \\bar u(p,s)\\ &=&\\ \\frac{(\\hat p+m)\\>(1+\\gamma^5\\hat s)}{2}, \\\\\nv(p,s)\\> \\bar v(p,s)\\ &=&\\ \\frac{(\\hat p-m)\\>(1+\\gamma^5\\hat s)}{2}, \\\\\n\\hat s \\gamma^5 u(p,s) = u(p,s),\\ && \\ \\hat s \\gamma^5 v(p,s) = v(p,s),\n\\end{eqnarray} \nas well as the Gordon identities:\n\\begin{eqnarray*}\n \\bar u(p_1,s_1)&\\gamma^\\mu& u(p_2,s_2) \\\\ &=& \n\\frac{1}{2m}\\bar u(p_1,s_1)\n\\left[ (p_1+p_2)^\\mu+\\sigma^{\\mu\\nu}(p_1-p_2)_\\nu\\right] u(p_2,s_2),\\\\\n \\bar u(p_1,s_1)&\\gamma^\\mu &\\gamma^5 u(p_2,s_2) \\\\ &=& \n\\frac{1}{2m}\\bar u(p_1,s_1)\n\\left[ (p_1-p_2)^\\mu\\gamma^5+\\sigma^{\\mu\\nu}(p_1+p_2)_\\nu\\gamma^5 \\right]\nu(p_2,s_2),\n\\end{eqnarray*}\nBoth $(p^{\\mu}+mn^{\\mu})$ and $(p^{\\mu}-mn^{\\mu})$ are light-like vectors.\n\n\\subsection{\\it Bispinors in the Dirac represenetaion of the\n  $\\gamma$ matrices}\nFor the massive fermion with the mass $m$ and 4-momentum $p_{\\mu}$\n\\[\np^{\\mu} = (p^0, p^1, p^2, p^3) = (p^0, \\vec{p}\\,), \\;;\np^0 = \\sqrt{m^2 + \\vec{p}^{\\,\\, 2}}\n  \\]\nWe define two reference frames, namely, {\\it R}-frame (the fermion rest-frame)\n  and {\\it L}-frame (where the fermion momentum $p^{\\mu}$ defined above):\n \\begin{eqnarray*}\n  && \\hbox{{\\it R}-frame}: \\quad p^{\\mu} = (m, 0) \\\\\n   && \\hbox{{\\it L}-frame}: \\quad p^{\\mu} = (p^0, p^1, p^2, p^3)\n \\end{eqnarray*}\nIn the fermion rest-frame we introduce four orts are as follows:\n\\begin{eqnarray}\n&& {\\mathbf e_0}^\\mu = (1,0,0,0), \\;\\; {\\mathbf e_1}^\\mu=(0,1,0,1), \\;\\;\n {\\mathbf e_2}^\\mu = (0,0,1,0), \\;\\; {\\mathbf e_3}^\\mu=(0,0,0,1)\n \\label{orts} \\\\\n &&  {\\mathbf e_0}^2 = 1, \\;\\; {\\mathbf e_i}^2 = -1, \\; i=1,2,3, \\;\\;\n  ({\\mathbf e_i} \\, {\\mathbf e_j}) = 0, i,j=0,1,2,3, \\;  i \\ne j \\nonumber\n\\end{eqnarray} \nIn the {\\it L}-frame the spatial orts have the form: \n\\begin{eqnarray*}\n && e_i^{\\mu} = {\\mathbf e_i}^{\\mu} + \\frac{p^i}{m} \\, V^{\\mu}; \\; \\; i =1,2,3, \\quad \nV^{\\mu} = \\left( 1;  \\frac{\\vec{p}}{m+p_0} \\right); \\;\nV^2 = \\frac{2m}{m+p_0} \\\\  \n&&   (e_i \\,  e_j) = -\\delta_{ij}, \\; (e_i \\, p) = 0, \\;\\; i=1,2,3  \n\\end{eqnarray*}\nThe standard spinors $u$ and $v$\nare most conveniently defined  in the fermion {\\it R}est-frame,\n\\begin{eqnarray*}\n & u_1^{(0)}  \n  =   \\left( \\begin{array}{c} \\sqrt{2m} \\\\ 0 \\\\ 0 \\\\ 0 \\end{array} \\right),\n  \\;\\; \n& \\bar{u}_1^{(0)}  = \n \\left( \\begin{array}{rrrr} \\sqrt{2m}, & 0, & 0, & 0 \\end{array} \\right), \\\\\n& u_2^{(0)} = \n \\left( \\begin{array}{c} 0 \\\\ \\sqrt{2m} \\\\ 0 \\\\ 0 \\end{array} \\right),  \\;\\; \n &\\bar{u}_2^{(0)} = \n \\left( \\begin{array}{rrrr} 0, & \\sqrt{2m}, & 0, & 0 \\end{array} \\right), \\\\\n & v_1^{(0)} = \n \\left( \\begin{array}{c} 0 \\\\ 0 \\\\ \\sqrt{2m} \\\\ 0 \\end{array} \\right),  \\;\\; \n & \\bar{v}_1^{(0)} = \n \\left( \\begin{array}{rrrr} 0, & 0, & -\\;\\sqrt{2m}, & 0 \\end{array} \\right),\n \\\\\n & v_2^{(0)} = \n \\left( \\begin{array}{c} 0 \\\\ 0 \\\\ 0 \\\\ \\sqrt{2m} \\end{array} \\right),  \\;\\; \n & \\bar{v}_2^{(0)} = \n \\left( \\begin{array}{rrrr} 0, & 0, & 0, & -\\; \\sqrt{2m}\n \\end{array} \\right)  \n\\end{eqnarray*}\n These spinors $u_1$ and $v_2$ describe the positive polarization\n  along the $Z$ axis, while $u_2$ and $v_1$ correspond to the\n  negative polarization.\n\nThen, the standard spinors in the {\\it L}-frame of the fermion  \n  can be obtained by making the Lorentz boost $S_1$:\n  \\begin{eqnarray*}\n&&\\hat p = S_1\\ m\\hat{\\mathbf e_0} \\ S_1^{-1} = S_1\\ m\\gamma^0 \\ S_1^{-1} \\\\\n && S_1 = {\\displaystyle {\\hat p \\gamma^0 + m \\over \\sqrt{2m(m+p_0)}}  } \\;=\\;\n {\\displaystyle {1\\over \\sqrt{2m(m+p_0)}}  }\n  \\left( \n  \\begin{array}{cc} (p^0 + m) I & \\sigma^i p^i \\\\ \\sigma^i p^i & (p^0+m) I\n  \\end{array} \\right)\\; ; \\\\ \n  && S_1^{-1}=S_1^\\dagger = {\\displaystyle {\\gamma^0 \\hat p  + m \\over\n      \\sqrt{2m(m+p_0)}}  } \\;=\\;\n {\\displaystyle {1\\over \\sqrt{2m(m+p_0)}}  }\n  \\left( \n  \\begin{array}{cc} (p^0 + m) I & -\\;\\sigma^i p^i \\\\ -\\;\\sigma^i p^i & (p^0+m)\n    I \\end{array} \\right)\\; ; \\\\ \n&&\nu_r(p,z)\\;=\\;S_1 u_r(m {\\mathbf e_0}, {\\mathbf e_3} ), \\qquad\nv_r(p,z)\\;=\\;S_1 v_r(m {\\mathbf e_0}, {\\mathbf e_3} ), \\qquad r=1,2\n\\end{eqnarray*}\nThus, int the {\\it L}-frame one has:\n\\begin{eqnarray*}\n & u_1(p) = {\\displaystyle  \\frac{1}{\\sqrt{m+p^0}} }\n \\left( \\begin{array}{c} m+p^0 \\\\ 0 \\\\ p^3 \\\\ p_+ \\end{array} \\right),  \\;\\; \n& \\bar{u}_1(p) = \\frac{1}{\\sqrt{m+p^0}}  \n \\left( \\begin{array}{rrrr} m+p^0\\;, & 0\\;, & -p^3\\;, & -p_- \\end{array} \\right), \\\\\n& u_2(p) = {\\displaystyle \\frac{1}{\\sqrt{m+p_0}} }\n \\left( \\begin{array}{c} 0 \\\\ m+p^0 \\\\ p_- \\\\ -p^3 \\end{array} \\right),\n \\;\\; \n &\\bar{u}_2(p) = \\frac{1}{\\sqrt{m+p^0}}  \n \\left( \\begin{array}{rrrr} 0\\;, & m+p^0\\;, & -p_+\\;, & p^3 \\end{array} \\right), \\\\\n & v_1(p) = {\\displaystyle \\frac{1}{\\sqrt{m+p^0}} }\n \\left( \\begin{array}{c} p^3 \\\\ p_+ \\\\ m+p^0 \\\\ 0 \\end{array} \\right),  \\;\\; \n & \\bar{v}_2(p) = \\frac{1}{\\sqrt{m+p^0}}  \n \\left( \\begin{array}{rrrr} p^3\\;, & p_-\\;, & -(m+p^0)\\;, & 0  \\end{array} \\right),\n \\\\ \n & v_2(p) = {\\displaystyle  \\frac{1}{\\sqrt{m+p^0}} }\n \\left( \\begin{array}{c} p_- \\\\ -p^3 \\\\ 0 \\\\ m+p^0 \\end{array} \\right),  \\;\\; \n & \\bar{v}_1(p) = \\frac{1}{\\sqrt{m+p^0}}  \n \\left( \\begin{array}{rrrr} p_+\\;, & -p^3\\;, & 0\\;, & -(m+p^0) \\end{array}\n \\right)\n\\end{eqnarray*}\nwhere\n\\[\n p_{\\pm} = p^1 \\pm i p^2 \n\\] \nThese spinors have the properties as follows:\n\\begin{eqnarray*} \n&& (\\hat{p} -m)u_i=0; \\; i=1,2; \\quad \n \\bar{u}_1 u_1 = \\bar{u}_2 u_2 = 2m; \\; \\;\n  \\bar{u}_1 u_2 = \\bar{u}_1 u_2 = 0 \\\\\n&& (\\hat{p} + m)v_i=0; \\; i=1,2; \\quad \n  \\bar{v}_1 v_1 = \\bar{v}_2 v_2 = -2m; \\; \\;\n  \\bar{v}_1 v_2 = \\bar{v}_1 v_2 = 0  \\\\\n&& \nu_1\\bar{u}_1 + u_2 \\bar{u}_2 = \\hat{p} + m; \\;\\; \nv_1\\bar{v}_1 + v_2 \\bar{v}_2 = \\hat{p} - m  \\label{spinor-11}\n\\end{eqnarray*}\n\nIn the both frames there are the following relations:\n\\begin{eqnarray*}\n  && \\hbox{{\\it R}-frame}: \\;\\; \\leftrightarrow \\;\\;\n  {\\mathbf e_i}^\\mu, \\;\\; u^{(0)}, \\; v^{(0)}  \\\\\n   && \\hbox{{\\it L}-frame}: \\;\\; \\leftrightarrow \\;\\;\n    e_i^\\mu, \\;\\; u(p), \\; v(p) \\\\ \n &&U-spinors \\\\\n &&\n{\\begin{array}{llcll}  \n \\gamma^5 \\hat{e}_1 u_1 = u_2 &  \\gamma^5 \\hat{e}_1 u_2 = u_1 & &\n \\bar{u}_1 \\gamma^5 \\hat{e}_1 = \\bar{u}_2 & \\bar{u}_2 \\gamma^5 \\hat{e}_1 =\n \\bar{u}_1 \\\\\n\\gamma^5 \\hat{e}_2 u_1 = i u_2 & \\gamma^5 \\hat{e}_2 u_2 = -i u_1 & &\n\\bar{u}_1 \\gamma^5 \\hat{e}_2 = -i \\bar{u}_2 & \n \\bar{u}_2 \\gamma^5 \\hat{e}_2 = i \\bar{u}_1 \\\\ \n\\gamma^5 \\hat{e}_3 u_1 = u_1 & \\gamma^5 \\hat{e}_3 u_2 = -u_2 & &\n \\bar{u}_1 \\gamma^5 \\hat{e}_3 = \\bar{u}_1 &\n \\bar{u}_2 \\gamma^5 \\hat{e}_3 = -\\bar{u}_2 \n\\end{array} }\n\\\\\n&& V-spinors \\\\\n&& { \\begin{array}{llcll}  \n \\gamma^5 \\hat{e}_1 v_1 = v_2 &  \\gamma^5 \\hat{e}_1 v_2 = v_1 & &\n \\bar{v}_1 \\gamma^5 \\hat{e}_1 = \\bar{v}_2 & \\bar{v}_2 \\gamma^5 \\hat{e}_1 =\n \\bar{v}_1 \\\\\n\\gamma^5 \\hat{e}_2 v_1 = i v_2 & \\gamma^5 \\hat{e}_2 v_2 = -i v_1 & &\n\\bar{v}_1 \\gamma^5 \\hat{e}_2 = -i \\bar{v}_2 & \n \\bar{v}_2 \\gamma^5 \\hat{e}_2 = i \\bar{v}_1 \\\\ \n\\gamma^5 \\hat{e}_3 v_1 = v_1 & \\gamma^5 \\hat{e}_3 v_2 = -v_2 & &\n \\bar{v}_1 \\gamma^5 \\hat{e}_3 = \\bar{v}_1 &\n \\bar{v}_2 \\gamma^5 \\hat{e}_3 = -\\bar{v}_2 \n\\end{array} }\n\\end{eqnarray*}\n\n\\noindent \nThe polarization vector \n$z$ is defined by the relation \n$$\n\\hat{z} = S_1\\; {\\hat{\\mathbf e_3}}\\; S_1^{-1} = S\\; (-\\;\\gamma^3)\\; S_1^{-1}\\;\n$$\nIt has the form\n$$\nz^\\mu={1\\over m(p^0+m)} \\Big( p^3(p^0+m)\\,, \\ \\ p^1p^3, \\ \\ p^2p^3,\n\\ \\ (p^3)^2+m(p^0+m) \\Big)\\ .\n$$\nSince $z^2=-1$ and $z\\cdot p=0$, any polarization vector $n$ \nof the fermion of momentum $p$ can be obtained \nby the Lorentz transformation $S_2$ from the Wigner little group of the\nmomentum $p$,\n$$\n\\hat s = S_2\\hat z S_2^{-1}, \\quad\nS_2={\\hat s \\hat z +1\\over \\sqrt{2(1 + z\\cdot s)}}.\n$$\nThus we arrive at the explicit expressions\nfor the spinors $u(p,s)$ and $v(p,s)$:\n$$\nu_r(p,s)=S_2 u_r(p,z), \\qquad v_r(p,s)=S_2 v_r(p,z)\\;;\n$$\nto put it differently, the spinors $u_1(p,s)$,\n$u_2(p,s)$, $v_1(p,s)$, and $v_2(p,s)$ are the columns \nof the matrix \n$$\n{\\sqrt{2m}} S_2 S_1 = {\\hat s\\hat z +1\\over \\sqrt{2(1 + z\\cdot s)}}\n{\\hat p \\gamma^0 + m \\over \\sqrt{(m+p_0)}}\\; ,\n$$\nwhere\n$$\n\\hat z = -\\; {(\\hat p \\gamma^0 + m)\\gamma^3(\\gamma^0\\hat p  + m)\n  \\over {2m(m+p_0)}}\\; .\n$$\n\n\n\\noindent\nThe case when the spatial component $\\vec s$ of the polarization vector $s^{\\mu}$\nis directed along the 3-momentum $\\vec p$ is of particular interest.\nIn this case, $u_r$ and $v_r$ describe states with definite helicity.\n\nGiven $p=(p^0\\;, p^1\\;, p^2\\;, p^3\\;)=\n(E\\;,\\ |\\vec p|\\cos\\, \\theta\\, \\cos\\varphi\\;, \\ |\\vec p|\\cos\\, \\theta\\,\n\\sin\\varphi\\;,\n\\ |\\vec p|\\sin\\, \\theta\\,)$,\nwe obtain in the both ({\\it R} and {\\it L}) frames\n\\begin{eqnarray} \nR:  s^{(0)\\;\\; \\mu} = \\left( 0, \\, \\frac{\\vec{p}}{|{\\vec{p}}\\,| } \\right), \\;\\;\nLf:  s^{\\mu} = \\frac{1}{ m |\\vec p\\,|} ( |\\vec{p}\\,|^2, \\,\n  p^0 \\vec{p} \\,) \\label{vspin}\n\\end{eqnarray}\nAs a result  we can obtain the so-called ``helicity'' spinors with\nthe properties as follows:\n\\begin{eqnarray} \n  \\left\\{ \\begin{array}{l|l}\n    \\gamma^5 \\hat{s} \\; u_R = u_R  &  \\gamma^5 \\hat{s} \\; u_L = -u_L  \\\\\n    \\gamma^5 \\hat{s} \\; v_R = -v_R &  \\gamma^5 \\hat{s} \\; v_L = v_L \n        \\end{array}\\right.\n   \\label{spir-1}\n\\end{eqnarray}\nFor the massless fermions, these spinors have the well-known projectors\n\\begin{eqnarray} \n  \\left\\{ \\begin{array}{ll|ll}\n     P_R u_R = u_R & P_L u_R = 0 &  P_R u_L = 0 & P_L u_L = u_L  \\\\\n     P_R v_R = v_R & P_L v_R = 0  & P_R v_L = 0 & P_L v_L = v_L\n  \\end{array}\\right.; \\quad\n  P_{R\/L} = \\frac{1}{2} \\left(1 \\pm \\gamma^5 \\right) \\quad \n   \\label{spir-2}\n\\end{eqnarray}\nThe explicit expressions for these spinors (in the Dirac representation of the\n$\\gamma$-matrices) are as follows:\n\\begin{eqnarray*} \\displaystyle\n  &&u_R =  \\left( \\begin{array}{r}\n     \\chi_1 \\\\  \\chi_2 \\end{array}\\right), \\;\\; \n  \\bar{u}_R =  \\left( \\chi^{\\dagger}_1, \\, - \\chi^{\\dagger}_2 \\right) \n\\\\\n  &&u_L =  \\left( \\begin{array}{r}\n     \\chi_3 \\\\   \\chi_4 \\end{array}\\right), \\;\\; \n  \\bar{u}_L = \\left( \\chi^{\\dagger}_3, \\, - \\chi^{\\dagger}_4 \\right) \n  \\\\\n && v_R =  \\left( \\begin{array}{r}\n      \\chi_2 \\\\   \\chi_1 \\end{array}\\right), \\;\\; \n  \\bar{v}_R = \\left( \\chi^{\\dagger}_2, \\, -\\chi^{\\dagger}_1 \\right) \n  \\\\\n && v_L =   \\left( \\begin{array}{r}\n     \\chi_4 \\\\   \\chi_3 \\end{array}\\right), \\;\\; \n   \\bar{v}_L = \\left( \\chi^{\\dagger}_4, \\, -\\chi^{\\dagger}_3 \\right)\n\\end{eqnarray*}\nwhere\n\\begin{eqnarray*}\n  \\chi_1 = \\left( \\begin{array}{c}\n    \\sqrt{E+m}\\;\\cos {\\theta\\over 2} \\\\[2mm] \n    \\sqrt{E+m}\\;\\sin {\\theta\\over 2}\\ e^{i\\varphi}\n\\end{array}     \\right)\n    =   \\sqrt{\\frac{p_0 + m}{2 |\\vec{p}\\, | (|\\vec{p}\\, | + p_3)}}\n    \\left( \\begin{array}{c}\n      |\\vec{p}\\, | + p^3 \\\\[2mm] \n      p_{+}\n    \\end{array} \\right)\n \\\\\n \\chi_2= \\left( \\begin{array}{c}\n    \\sqrt{E-m}\\;\\cos {\\theta\\over 2} \\\\[2mm] \n    \\sqrt{E-m}\\;\\sin {\\theta\\over 2}\\ e^{i\\varphi}\n\\end{array}     \\right)\n  =  \\sqrt{\\frac{p_0 - m}{2 |\\vec{p}\\,| (|\\vec{p}\\,| + p_3)}}\n  \\left( \\begin{array}{c}\n    |\\vec{p}\\,| + p^3 \\\\[2mm] \n      p_{+}   \\end{array} \\right) \\\\\n\n\n  \\chi_3= \\left( \\begin{array}{c}\n    -\\sqrt{E-m}\\;\\sin {\\theta\\over 2} e^{-i\\varphi} \\\\[2mm] \n     \\sqrt{E-m}\\;\\cos {\\theta\\over 2}\n \\end{array}     \\right)\n  =  \\sqrt{\\frac{p_0 - m}{2 |\\vec{p}\\,|  (|\\vec{p}\\,| + p_3)}}\n  \\left( \\begin{array}{c}\n    -p_{-} \\\\[2mm]\n    |\\vec{p}\\,| + p^3\n  \\end{array} \\right)\n  \\\\\n  \\chi_4 = \\left( \\begin{array}{c}\n    \\sqrt{E+m}\\;\\sin {\\theta\\over 2}\\ e^{-i\\varphi}\n    \\\\[2mm] \n    -\\sqrt{E+m}\\;\\cos {\\theta\\over 2}\n\\end{array}     \\right)\n    =   \\sqrt{\\frac{p_0 + m}{2 |\\vec{p}\\,| (|\\vec{p}\\,| + p_3)}}\n \\left( \\begin{array}{c} p_{-} \\\\[2mm] -|\\vec{p}\\,| - p^3  \\end{array} \\right)\n\\end{eqnarray*}\nNote, that\n\\[\n v_R = \\gamma^5 u_R, \\quad v_L =  \\gamma^5 u_L \\nonumber \n\\]\n\n\n\\section{\\bf STANDARD MODEL LAGRANGIAN}\\label{sml}\n\nIn this Section we present the basic Lagrangian of the Standard Model(SM),\ncorresponding to the $SU(3)\\times SU(2)\\times U(1)$ local gauge symmetry\n(see, \nfor example, \\cite{itzu,okun,aoki}). The algebra of the semisimple group \n$SU(3)\\times SU(2)\\times U(1)$ is generated by Gell-Mann matrices \n$t^a = \\frac{1}{2}\\lambda^a$ (a =1,...8) (Section~\\ref{gellmann}), \nPauli matrices $\\tau^i = \\sigma^i\/2$ (Section~\\ref{pauli}) \nand hypercharge $Y$ with the following commutation relations\n\\begin{eqnarray*}\n\\left[ t^a , t^b \\right] &=& i\\;f^{abc} t^c, \\\\ \n\\left[ \\tau^i , \\tau^j \\right] &=& i\\;\\epsilon^{ijk} \\tau^k, \\\\\n\\left[ \\tau^i , Y \\right] &=& \\left[ t^a, \\tau^j \\right] = \n\\left[ t^a, Y \\right] = 0.\n\\end{eqnarray*}\nThe full SM Lagrangian has the form \\cite{itzu,okun}:\n\\begin{equation}\n{\\cal L = L_G + L_F + L_H + L_M + L_{GF} + L_{FP}.} \\label{sm91}\n\\end{equation}\nHere ${\\cal L_G}$ is the Yang-Mills Lagrangian without matter fields\n\\begin{eqnarray}\n{\\cal L_G} = -\\frac{1}{4} F^i_{\\mu\\nu}(W) F^{\\mu\\nu}_i(W) \n -\\frac{1}{4} F^{\\mu\\nu}(W^0) F^{\\mu\\nu}(W^0) \n-\\frac{1}{4} F^a_{\\mu\\nu}(G) F^{\\mu\\nu}_a(G), \\label{sm92}\n\\end{eqnarray} \nwhere $ F^i_{\\mu\\nu}(W), F^a_{\\mu\\nu}(G),  F_{\\mu\\nu}(W^0)$ are given by\n\\begin{eqnarray*}\n F^i_{\\mu\\nu}(W) &=& \\partial_{\\mu}W^i_{\\nu} - \\partial_{\\nu}W^i_{\\mu}\n+ g\\;\\epsilon^{ijk} W^j_{\\mu} W^k_{\\nu}, \\\\\nF_{\\mu\\nu}(W^0) &=& \\partial_{\\mu}W^0_{\\nu} - \\partial_{\\nu}W^0_{\\mu}, \\\\\nF^a_{\\mu\\nu}(G) &=& \\partial_{\\mu}G^a_{\\nu} - \\partial_{\\nu}G^a_{\\mu}\n+ g_s\\;f^{abc} G^b_{\\mu} G^c_{\\nu},\n\\end{eqnarray*}\nwith $W^i_{\\mu}, W^0_{\\mu}$ the $SU(2)\\times U(1)$ original gauge fields \nand $G^a_{\\mu}$ the gluon fields. The infinitesimal gauge transformations \nof these fields are given by\n\\begin{eqnarray*}\n\\delta W^0_{\\mu} &=& \\partial_{\\mu}\\theta(x), \\\\\n\\delta W^i_{\\mu} &=& \\partial_{\\mu}\\theta^i -\ng\\epsilon^{ijk}\\theta^j W^k_{\\mu} = {\\cal D}^{ij}_{\\mu}(W) \\theta^j \\\\\n\\delta G^a_{\\mu} &=& \\partial_{\\mu}\\epsilon^a -\ng_s f^{abc}\\epsilon^b G^c_{\\mu} = {\\cal D}^{ab}_{\\mu}(G) \\epsilon^b\n\\end{eqnarray*}\nHere ${\\cal D}^{ij}_{\\mu}(W)$ and $  {\\cal D}^{ab}_{\\mu}(G)$ stand for the \ncovariant derivatives, $g_s$ and $g$ are the $ SU(3)$ and $SU(2)$ gauge \ncoupling constants, respectively, $\\epsilon$ and $\\theta^{a(i)}$ are an \narbitrary functions depending on the space-time coordinates. It can be easily \nchecked that Lagrangian (\\ref{sm92}) is invariant under \nthese gauge transformations. \n\nLagrangian ${\\cal L_F}$ describes coupling of fermions with gauge fields. \nFor simplicity we shall consider one lepton generation, say $e^-$ and\n$\\nu_e$, and three quark generations. Fermions constitute only doublets and \nsinglets in $SU(2)\\times U(1)$\n\\begin{eqnarray*}\nR &=& e^-_R, \\ \\ \\ L = \n\\left( \\begin{array}{c} \\nu_L \\\\ e^-_L \\end{array} \\right)   \\\\\nR_I &=& \\left( q_I \\right)_R, \\ \\ R_i = \\left( q_i \\right)_R, \\ \\ \nL_I = \\left( \\begin{array}{c} q_I \\\\ \\ \\\\ V^{-1}_{Ii}q_i \\end{array} \\right)\n\\end{eqnarray*}\nwhere $L$ and $R$ denote left- and right-handed components of the spinors, \nrespectively: \n\\begin{eqnarray*}\ne_{R,L} = \\frac{ 1\\pm \\gamma_5}{2} e.\n\\end{eqnarray*}\nThe neutrino is assumed to be left-handed, while right-handed components of \nboth up- and down-quarks enter in the ${\\cal L_F}$. Indices $I$ and $i$ \nnumerate three quark generations: $I,i = 1,2,3$, and $I(i)$ refers to the \nup (down) quarks. A possible mixing of quark generations was taken into \naccount by introduction of Kobayashi-Maskava matrix $V_{iI}$ (see, for\nexample, \n\\cite{okun,pdg} for details).\nThe infinitesimal gauge transformations of fermion fields looks as follows:\n\\begin{eqnarray*}\n\\delta \\psi_{lep} &=& \\left(\nig'\\frac{Y}{2}\\theta(x) + ig\\frac{\\sigma^i}{2}\\theta^i(x) \\right)\n\\psi_{lep},  \\\\\n\\delta \\psi_{quark} &=& \\left(\nig'\\frac{Y}{2}\\theta(x) + ig\\frac{\\sigma^i}{2}\\theta^i(x)\n +ig_s t^a \\theta^a(x) \\right)\n\\psi_{quark},\n\\end{eqnarray*}\nwhere $g'$ is $U(1)$ gauge coupling constant.\nObviously, lepton and quark fields belong to the fundamental \nrepresentation of the $ SU(3)\\times SU(2) \\times U(1)$.\nUnder the requirements of the $ SU(3)\\times SU(2) \\times U(1)$\nlocal gauge symmetry and renormalizability of the theory, the Lagrangian\n${\\cal L_F}$ acquires the following expression\n\\begin{eqnarray}\n{\\cal L_F} = i \\bar {L} \\hat {D}_L L +  i \\bar {R} \\hat {D}_R R \n+ i \\sum_{I}\\left( \\bar {L}_I \\hat {D}_L^q L_I + \n \\bar {R}_I \\hat {D}^q_R R_I \\right)\n+i\\sum_{i} \\bar {R}_i \\hat {D}^q_R R_i, \\label{sm93}\n\\end{eqnarray}\nwhere covariant derivatives are given by\n\\begin{eqnarray*}\nD_{L\\;\\mu} &=& \\partial_{\\mu} - ig'\\frac{Y}{2}W^0_{\\mu} -\n ig\\frac{\\sigma^i}{2}W^i_{\\mu}, \\\\\nD_{R\\;\\mu} &=& \\partial_{\\mu} - ig'\\frac{Y}{2}W^0_{\\mu},  \\\\\nD^q_{L\\;\\mu} &=& \\partial_{\\mu} - ig'\\frac{Y}{2}W^0_{\\mu} -\n ig\\frac{\\sigma^i}{2}W^i_{\\mu} - ig_s t^a G^a_{\\mu}, \\\\\nD^q_{R\\;\\mu} &=& \\partial_{\\mu} - ig'\\frac{Y}{2}W^0_{\\mu} -\n ig_s t^a G^a_{\\mu}.\n\\end{eqnarray*}\nWe remind that the value of hypercharge $Y$ is determined by the following\nrelation $Q~ =~ \\tau_3~ +~ Y\/2$ with $Q$ being the charge operator.\n\nBoth the gauge fields and fermion ones described above have zero mass, while \nin the reality all charged fermions are massive and intermediate bosons are \nknown to be very heavy. To make the weak bosons massive one can use Higgs \nmechanism of spontaneous breakdown of the $SU(2)\\times U(1)$ symmetry to the \n$U(1)$ symmetry. The widely accepted way to do that consists in the \nintroduction of the Higgs $SU(2)$ doublet $\\Phi$ (with $Y=1$). This doublet \nacquires the nonzero vacuum expectation value:\n\\begin{eqnarray*}\n <\\Phi> = \\left( \\begin{array}{c} 0 \\\\ \\ \\\\ \n \\frac{v}{\\sqrt2} \\end{array} \\right). \n\\end{eqnarray*}\nThe potential term $V(\\Phi)$, which can give rise to the symmetry violation,\n reads\n\\begin{eqnarray*}\nV(\\Phi) = -\\mu^2 \\Phi^+ \\Phi + \\lambda \\left(\\Phi^+\\Phi\\right)^2.\n\\end{eqnarray*}\nOne can easily verify that the vacuum expectation value satisfies to the \nconditions:\n\\begin{eqnarray*}\n\\tau^i <\\Phi> &=& \\frac{1}{2}\\sigma^i <\\Phi> \\neq 0, \\\\\nQ<\\Phi> &=& \\frac{1}{2}\\left(\\sigma_3+ Y\\right) = 0.\n\\end{eqnarray*}\nIt means, that only the symmetry generated by $Q$ is not broken on\nthis vacuum. \nLet us choose the Lagrangian for the Higgs field interaction with\ngauge fields in the form:\n\\begin{equation}\n{\\cal L_H} = \\left( D_{L\\;\\mu}\\Phi\\right)^+\\left( D_L^{\\mu}\\Phi\\right)\n-V(\\Phi).\n\\end{equation}\nThen one finds that only gauge boson coupling to $Q$ ( i.e. photon)\nremains massless, while other bosons acquire masses. Diagonalization of the\nmass matrix gives\n\\begin{eqnarray}\nW^{\\pm}_{\\mu} &=& \\frac{1}{\\sqrt2}( W^1_{\\mu} \\mp iW^2_{\\mu} ), \\;\\; \n M_W = \\frac{1}{2} g v, \\\\ \nZ_{\\mu} &=& \\frac{1}{\\sqrt{g^2+g'^2}} (g W^3_{\\mu} - g'W^0_{\\mu} ), \\;\\; \n M_Z = \\frac{1}{2}\\sqrt{g^2 +g'^2}v, \\\\\nA_{\\mu} &=& \\frac{1}{\\sqrt{g^2+g'^2}} (g' W^3_{\\mu} + g W^0_{\\mu} ), \\;\\;\n M_A = 0,\n\\end{eqnarray}\nwhere $W^{\\pm}_{\\mu}, Z_{\\mu}$ are charged and neutral weak bosons, $A_{\\mu}$ \nis the photon. It is suitable to introduce rotation angle $\\vartheta_W$\nbetween $(W^3,W^0)$ and $(Z,A)$, which is called the {\\it Weinberg angle} \n\\begin{equation}\n\\sin\\vartheta_W \\equiv g'\/\\sqrt{g^2+g'^2}.\n\\end{equation}\nThe relation of constants $g,g'$ with electromagnetic coupling constants $e$ \nfollows from (\\ref{sm93}). Since  the photon coupling with charged particles\nis proportional to $g g' \/\\sqrt{g^2 +g'^2}$, we should identify this \nquantity with the electric charge $e$:\n\\begin{equation}\n e = \\frac{g g'}{\\sqrt{g^2+g'^2}}.\n\\end{equation}\nIn order to find mass spectrum in the Higgs sector, let us \nexpress doublet $\\Phi$ in the form\n\\begin{eqnarray*}\n\\Phi = \\left( \\begin{array}{c} i\\omega^+ \\\\\n\\frac{1}{\\sqrt2}\\left( v +H -i z \\right)  \\end{array}\\right).\n\\end{eqnarray*}\nOne can verify that Nambu-Goldstone bosons $\\omega^{\\pm}, z$ have zero masses \nand may be cancelled away by suitable choice of the $SU(2)\\times U(1)$ \nrotation. The only {\\bf physical component of the Higgs doublet}\nis $H$, which acquires mass \n\\[ \nm_H = \\sqrt2 \\mu.\n\\] \n\nThe Lagrangian ${\\cal L_M}$ generates fermion mass terms. Supposing the \nneutrinos to be massless, we write the Yukawa interaction of the \nfermions with Higgs doublet in the form\n\\begin{eqnarray}\n{\\cal L_M} = - f_e \\bar {L}\\Phi R  - \\sum_{i} f_i \\bar {L}_i\\Phi R_i\n-\\sum_{I} f_I \\bar {L}_I \\left(i\\sigma^2\\Phi^*\\right)R_I + h.c.\n\\end{eqnarray}\nHere we introduced doublet $L_i $ related with $L_I$ by\n\\begin{eqnarray*}\nL_i = V_{i\\;I}\\;L_I,\n\\end{eqnarray*}\nand $f_{I,\\;i}$ are the Yukawa coupling constants. Then the masses of \nfermions in the tree approximation are given by\n\\begin{equation}\nm_{I,\\;i} = \\frac{f_{I,\\;i}\\;v}{\\sqrt2}.\n\\end{equation}\n\nIt is well known that quantization of dynamical systems is governed by \nLagrangians having local gauge symmetry requires an additional care. Freedom \nof redefining gauge and matter fields without changing the Lagrangians leads \nto the vanishing of some components of the momenta, canonically conjugate to \nthe gauge fields, say\n\\begin{eqnarray*}\n\\frac{\\delta L}{\\delta \\partial_0 A_{\\mu}} = - F^{0\\;\\mu} = 0 \n \\quad ({\\rm for} \\;\\; \\mu = 0).\n\\end{eqnarray*}\nTo perform the quantization procedure, one should add to the Lagrangian \na gauge fixing terms, breaking explicitly the local symmetry. In the \nfunctional integral formulation it leads, in the case of non-Abelian gauge \nsymmetry, to modification of the path integral measure \\cite{faddeev}. As a \nresult, the measure of the path integral will be multiplied by functional \ndeterminant $\\Delta (W^a_{\\mu})$. In order to apply the well known methods of \nperturbation theory, one may \nexponentiate $\\Delta(W^a_{\\mu})$ and redefine the initial Lagrangian. It can \nbe made by introducing auxiliary fields $c^a$ and $\\bar c^a$ which are scalar \nfields anticommuting with themselves and belonging  to the adjoint \nrepresentation of the Lie algebra.\nThe fields $c^a$ and $\\bar {c}^a$ are called Faddeev-Popov ghosts (FP ghosts).\n\nThe gauge fixing terms are usually chosen in the form \n\\begin{eqnarray*}\nL_{GF} = B^a F^a(W) + \\frac{\\xi}{2}\\left( B^a\\right)^2,\n\\end{eqnarray*}\nwhere $B^a$ are auxiliary fields introduced to linearize this expression,\n$\\xi$ is the gauge parameter, $ F^a = \\partial^{\\mu}W^a_{\\mu}$.\nThen FP ghosts enter in the Lagrangian in the following way\n\\begin{eqnarray}\nL_{FP} = -\\bar {c}^a \\frac{\\partial F^a}{\\partial W^c_{\\mu}}\nD^{cb}_{\\mu}(W) c^b. \\label{sm912}\n\\end{eqnarray}\nAs it was pointed above, these additional terms violate local gauge invariance, \nbut the final Lagrangian becomes invariant under the global transformations  \nmixing the gauge fields and FP ghosts. This symmetry, found by by Becchi, \nRouet, and Stora, was called BRS symmetry. The BRS infinitesimal \ntransformations are defined by the following relations\n\\begin{eqnarray*}\n\\delta^{BRS}\\psi(x) &=& i\\beta g c^a(x) t^a \\psi(x),  \\;\\;  \n\\delta^{BRS}W^a_{\\mu}(x) = \\beta D^{ab}_{\\mu}c^b(x),  \\\\\n\\delta^{BRS}\\bar {c}^a (x) &=& \\beta B^a(x), \\;\\;\n\\delta^{BRS} c^a(x) = -\\frac{\\beta}{2} g f^{abc} c^b(x) c^c(x),  \\\\\n\\delta^{BRS} B^a(x) &=& 0.\n\\end{eqnarray*}\nHere $\\psi$ denotes any matter field, the parameter $\\beta$ does not depend on \n$x$ and anticommutes with $c^a$ and $\\bar {c}^a$, as well as with all fermion \nfields. Using these relations, the formula (\\ref{sm912}) can be written in the \nbrief form:\n\\begin{eqnarray}\nL_{GF} = \\frac{\\delta}{\\delta \\beta}\\left(\n\\bar {c}^a \\delta^{BRS}\\left( \\partial^{\\mu}W^a_{\\mu}\\right)\\right),\n\\end{eqnarray}\nwhere $ \\delta\/\\delta \\beta$ means left differentiation.\n\nIn our case we choose the gauge fixing part of the Lagrangian in the form\n\\begin{eqnarray}\n{\\cal L_{GF}} &=& B^+ (\\partial^{\\mu} W^-_{\\mu} + \\xi_W M_W\n\\omega^- ) + B^- (\\partial^{\\mu} W^+_{\\mu} + \\xi_W M_W\n\\omega^+ ) \\\\\n&+& B^Z (\\partial^{\\mu} Z_{\\mu} + \\xi_Z M_Z z) \n+ B^A (\\partial^{\\mu} A_{\\mu})\n+ B^a (\\partial^{\\mu} G^a_{\\mu})  \\nonumber \\\\\n&+& \\xi_W B^+ B^- + \\frac{\\xi_Z}{2} B^Z B^Z\n + \\frac{\\xi_A}{2} B^A B^A + \\frac{\\xi_G}{2} B^a_G B^a_G,\n\\nonumber \n\\end{eqnarray}\nthen FP--ghost Lagrangian looks as follows:\n\\begin{eqnarray}\n&&{\\cal L_{FP}} =   \\\\\n&&\\frac{\\delta}{\\delta \\beta} \\Bigl \\{\n\\bar {c}^{\\;+} \\delta^{BRS}\\left( \\partial^{\\mu}W^-_{\\mu} + \\xi_W \nM_W \\omega^- \\right) +\n\\bar {c}^{\\;-} \\delta^{BRS}\\left( \\partial^{\\mu}W^+_{\\mu} + \\xi_W \nM_W \\omega^+ \\right)  \\nonumber \\\\\n&&+ \\bar {c}^{\\;Z} \\delta^{BRS}\\left( \\partial^{\\mu}Z_{\\mu} + \\xi_Z\nM_Z \\;z \\right)  \n+ \\bar {c}^{\\;A} \\delta^{BRS}\\left( \\partial^{\\mu}A_{\\mu} \\right)\n+ \\bar {c}^{\\;a} \\delta^{BRS}\\left( \\partial^{\\mu}G^a_{\\mu} \\right)\\Bigr\\},\n \\nonumber\n\\end{eqnarray}\nwhere the fields $c^A,c^Z$ are constructed from original ghosts\n$c^0,c^3$ just like the bosons $Z_{\\mu}, A_{\\mu}$ from initial fields\n$W^0_{\\mu}, W^3_{\\mu}$.\n\n{\\bf The total Lagrangian of the Standard Model} \\\\\nNow, we are ready to present the total Lagrangian of the \n{\\bf Standard Model} rewritten in the terms of physical fields \\cite{aoki}.\n\\begin{eqnarray}\n{\\cal L_G} &=& -\\frac{1}{2}F^+_{\\mu\\nu}F^{-\\;\\mu\\nu}\n -\\frac{1}{4}( F^Z_{\\mu\\nu})^2  -\\frac{1}{4}( F^A_{\\mu\\nu})^2\n -\\frac{1}{4}( G^a_{\\mu\\nu})^2     \\label{sm917} \\\\\n&+& i e \\cot\\vartheta_W \\left(  \ng^{\\alpha\\gamma}g^{\\beta\\delta} - g^{\\alpha\\delta}g^{\\beta\\gamma} \\right)\n\\Bigl( W^-_{\\gamma}Z_{\\delta}\\partial_{\\alpha}W^+_{\\beta}    \n+Z_{\\gamma}W^+_{\\delta}\\partial_{\\alpha}W^-_{\\beta} \\nonumber \\\\\n&+& W^+_{\\gamma}W^-_{\\delta}\\partial_{\\alpha}Z_{\\beta} \\Bigr)        \n + i e  \\left( g^{\\alpha\\gamma}g^{\\beta\\delta} -\ng^{\\alpha\\delta}g^{\\beta\\gamma} \\right)\n\\Bigl(  W^-_{\\gamma}A_{\\delta}\\partial_{\\alpha}W^+_{\\beta}  \\nonumber \\\\\n& + & A_{\\gamma}W^+_{\\delta}\\partial_{\\alpha}W^-_{\\beta} \n+W^+_{\\gamma}W^-_{\\delta}\\partial_{\\alpha}A_{\\beta} \\Bigr)          \n- \\frac{1}{2}\ng_s f^{abc} G^a_{\\mu} G^b_{\\nu} \\partial^{\\mu} G^{c\\;\\nu}  \\nonumber \\\\\n&+& e^2 \\left( g^{\\alpha\\gamma}g^{\\beta\\delta} -\ng^{\\alpha\\beta}g^{\\gamma\\delta} \\right)\nW^+_{\\alpha}W^-_{\\beta} A_{\\gamma}A_{\\delta}   \\nonumber \\\\\n& + & e^2  \\cot^2\\vartheta_W \\left( g^{\\alpha\\gamma}g^{\\beta\\delta} -\ng^{\\alpha\\beta}g^{\\gamma\\delta} \\right)\nW^+_{\\alpha}W^-_{\\beta} Z_{\\gamma}Z_{\\delta})  \\nonumber \\\\\n &+& e^2 \\cot\\vartheta_W \\left(\n g^{\\alpha\\delta}g^{\\beta\\gamma} + g^{\\alpha\\gamma}g^{\\beta\\delta} \n- 2 g^{\\alpha\\beta}g^{\\gamma\\delta} \\right)\nW^+_{\\alpha}W^-_{\\beta}A_{\\gamma}Z_{\\delta} \\nonumber \\\\\n &+& \\frac{e^2}{2\\sin^2\\vartheta_W} \\left(\ng^{\\alpha\\beta}g^{\\gamma\\delta} - g^{\\alpha\\gamma}g^{\\beta\\delta} \\right)\nW^+_{\\alpha}W^+_{\\beta}W^-_{\\gamma}W^-_{\\delta} \\nonumber \\\\\n &-&\\frac{1}{4} g^2_s f^{rab} f^{rcd} G^a_{\\mu} G^b_{\\nu} G^{c\\;\\mu} \nG^{d\\;\\nu},    \\nonumber \n\\end{eqnarray}\nwhere the field sthrenghtes $G^a_{\\mu\\nu}, F^+_{\\mu\\nu}, \\ldots$\nare given by\n\\begin{eqnarray*}\n&&F^+_{\\mu\\nu} = \\partial_{\\mu}W^+_{\\nu} - \\partial_{\\nu}W^+_{\\mu}, \\\\\n&&G^a_{\\mu\\nu} = \\partial_{\\mu}G^a_{\\nu} - \\partial_{\\nu}G^a_{\\mu}, \\\\\n&& \\cdots\n\\end{eqnarray*}\n\n\\begin{eqnarray}\n{\\cal L_F} &=&  i \\bar {e} \\hat {\\partial} e + \ni \\bar{\\nu}_L \\hat {\\partial} \\nu_L  + \ni \\sum_{n} \\bar {q}_n \\hat {\\partial} q_n  \\label{sm918} \\\\\n&+& \\frac{e}{\\sqrt2 \\sin\\vartheta_W} \\left(\n\\bar {\\nu}_L \\hat {W}^+ e_L +\\bar {e}_L \\hat {W}^- \\nu_L \\right)\n+ \\frac{e}{\\sin 2\\vartheta_W} \n\\bar {\\nu}_L \\hat {Z} \\nu_L   \\nonumber \\\\\n&+& \\frac{e}{\\sin 2\\vartheta_W } \\left(\n\\bar {e}\\hat {Z} ( 2\\sin^2\\vartheta_W - \\frac{1 - \\gamma_5}{2} ) e \\right)\n- e  \\bar {e} \\hat {A} e  \\nonumber \\\\\n&+& \\frac{e}{\\sqrt2 \\sin\\vartheta_W} \\sum_{I,i} \\left(\n\\bar {q}_I \\hat {W}^+ q_{i\\;L} (V^+)_{Ii} \n+ \\bar {q}_i \\hat {W}^- q_{I\\;L} V_{iI} \\right)  \\nonumber \\\\\n&+& \\frac{e}{\\sin 2\\vartheta_W}\\sum_{I} \\left(\n\\bar {q}_I \\hat {Z} ( \\frac{1 - \\gamma_5}{2} - 2 Q_I\\sin^2\\vartheta_W ) q_I\n\\right) \\nonumber \\\\\n&+& \\frac{e}{\\sin 2\\vartheta_W}\\sum_{i} \\left(\n\\bar {q}_i \\hat {Z} ( \\frac{-1 + \\gamma_5}{2} - 2 Q_i\\sin^2\\vartheta_W ) q_i\n\\right) \\nonumber \\\\\n&+& e \\sum_{n} Q_n  \\bar {q}_n \\hat {A} q_n\n+ g_s \\sum_{n} \\bar {q}_n G^a_{\\mu} \\gamma^{\\mu} t^a q \\nonumber\n\\end{eqnarray}\n\n\\begin{eqnarray}\n{\\cal L_H} &=& \\frac{1}{2} (\\partial_{\\mu}H)^2 - \\frac{m^2_H}{2}H^2\n+ \\frac{1}{2} (\\partial_{\\mu} z)^2 + \n\\partial_{\\mu}\\omega^+ \\partial^{\\mu}\\omega^-   \\label{sm919}  \\\\\n&+& M^2_W W^+_{\\mu}W^{-\\;\\mu} +  \\frac{1}{2}M_Z^2 Z^2_{\\mu} -M_W \\left(\nW^-_{\\mu}\\partial^{\\mu}\\omega^+ + W^+_{\\mu}\\partial^{\\mu}\\omega^-\n\\right)                                       \\nonumber \\\\\n&-& M_Z Z_{\\mu} \\partial^{\\mu} z                   \n+ \\frac{e M_W}{\\sin\\vartheta_W} H W^+_{\\mu} W^{-\\;\\mu}\n+ \\frac{e M_Z}{\\sin2\\vartheta_W} H\\,  Z^2_{\\mu}  \\nonumber \\\\\n&+& \\frac{e}{2\\sin\\vartheta_W} W^{+\\;\\mu} \\left(\\omega^-  \n\\stackrel{\\leftrightarrow}{\\partial}_{\\mu} ( H - i z)\\right) \n+ \\frac{e}{2\\sin\\vartheta_W} W^{-\\;\\mu} \\left(\\omega^+ \n\\stackrel{\\leftrightarrow}{\\partial}_{\\mu} ( H + i z)\\right) \\nonumber  \\\\\n&+& i e ( A^{\\mu}+ \\cot 2\\vartheta_W Z^{\\mu}) \\left(\\omega^-\n\\stackrel{\\leftrightarrow}{\\partial}_{\\mu}\\omega^+\\right) \n+ \\frac{e}{\\sin 2\\vartheta_W} Z^{\\mu} \\left( z\n\\stackrel{\\leftrightarrow}{\\partial}_{\\mu} H \\right)   \\nonumber \\\\\n&+& i e M_Z \\sin\\vartheta_W Z^{\\mu} ( W^+_{\\mu}\\omega^- \n- W^-_{\\mu}\\omega^+ )  +\ni e M_W A^{\\mu} (  W^-_{\\mu}\\omega^+ -  W^+_{\\mu}\\omega^-) \\nonumber \\\\  \n&+& \\frac{e^2}{4\\sin^2\\vartheta_W} H^2 ( W^+_{\\mu}W^{-\\;\\mu} +\n2\\;Z^2_{\\mu} )\n+ \\frac{i e^2}{2\\cos\\vartheta_W} H Z^{\\mu} ( W^+_{\\mu}\\omega^-\n- W^-_{\\mu}\\omega^+ )                           \\nonumber \\\\\n&+& \\frac{i e^2}{2\\sin\\vartheta_W} H A^{\\mu} ( W^-_{\\mu}\\omega^+\n- W^+_{\\mu}\\omega^- )\n+ \\frac{e^2}{4\\sin^2\\vartheta_W} z^2 ( W^+_{\\mu}W^{-\\;\\mu} +\n 2\\;Z^2_{\\mu })        \\nonumber    \\\\\n&+& \\frac{e^2}{2\\cos\\vartheta_W} z Z^{\\mu} ( W^+_{\\mu}\\omega^-\n+ W^-_{\\mu}\\omega^+ ) \n- \\frac{e^2}{2\\sin\\vartheta_W} z A^{\\mu} ( W^+_{\\mu}\\omega^-\n+ W^-_{\\mu}\\omega^+ )                                \\nonumber \\\\\n&+& \\frac{e^2}{2\\sin^2\\vartheta_W}\\omega^+\\omega^- W^+_{\\mu}W^{-\\;\\mu}\n+ e^2 \\cot^2 2\\vartheta_W \\omega^+\\omega^- Z_{\\mu}^2\n+ e^2 \\omega^+\\omega^- A_{\\mu}^2    \\nonumber \\\\\n&+& 2 e^2 \\cot(2\\vartheta) \\omega^+\\omega^- A^{\\mu}Z_{\\mu}  \n-\\frac{e m^2_H}{4 M_W \\sin\\vartheta_W} H^3\n- \\frac{e m^2_H}{2 M_W \\sin\\vartheta_W} \\omega^+\\omega_- H \\nonumber \\\\\n&-& \\frac{e m^2_H}{4 M_W \\sin\\vartheta_W} z^2 H\n-\\frac{e^2 m^2_H}{32 M^2_W \\sin^2\\vartheta_W} H^4\n-\\frac{e^2 m^2_H}{32 M^2_W \\sin^2\\vartheta_W} z^4   \\nonumber \\\\\n&-& \\frac{e^2 m^2_H}{8 M^2_W \\sin^2\\vartheta_W}\\omega^+ \\omega^- \n( H^2 + z^2)\n-\\frac{e^2 m^2_H}{16 M^2_W \\sin^2\\vartheta_W} z^2 H^2\n\\nonumber \\\\\n&-& \\frac{e^2 m^2_H}{8 M^2_W \\sin^2\\vartheta_W} \n(\\omega^+\\omega^-)^2 \\nonumber\n\\end{eqnarray}\nHere symbol $f\\stackrel{\\leftrightarrow}{\\partial}_{\\mu} g$ is\nused as usual: $f\\stackrel{\\leftrightarrow}{\\partial}_{\\mu} g \\equiv \nf \\partial_{\\mu} g - ({\\partial}_{\\mu} f) g$.\n\n\\begin{eqnarray}\n{\\cal L_M} &=& -\\frac{e m_e}{M_Z \\sin 2\\vartheta_W}\nH \\bar {e} e  - \\frac{e}{M_Z \\sin 2\\vartheta_W}\n\\sum_{n} m_n H \\bar {q}_n q_n        \\label{sm920} \\\\\n&+& \\frac{i e \\sqrt2 m_e}{M_Z \\sin 2\\vartheta_W} \\left(\n\\omega^- \\bar {e} \\nu_L - \\omega^+ \\bar {\\nu}_L e \\right)\n+ \\frac{i e m_e}{M_Z \\sin 2\\vartheta_W} z\n\\bar {e} \\gamma_5 e                    \\nonumber \\\\\n&+& \\frac{i e  }{\\sqrt2 M_Z \\sin 2\\vartheta_W}\n\\omega^+ \\sum_{I,i}( V^+)_{Ii} \\bar {q}_I \\left(\nm_I - m_i - ( m_I + m_i) \\gamma_5 \\right) q_i  \\nonumber \\\\\n&+& \\frac{i e  }{\\sqrt2 M_Z \\sin 2\\vartheta_W}\n\\omega^- \\sum_{I,i}( V)_{iI} \\bar {q}_i \\left(\nm_i - m_I - ( m_I + m_i) \\gamma_5 \\right) q_I  \\nonumber \\\\\n&-& \\frac{i e  }{ M_Z \\sin 2\\vartheta_W} \\sum_{I}\nm_I\\bar {q}_I \\gamma_5 q_I\n + \\frac{i e  }{ M_Z \\sin 2\\vartheta_W} \\sum_{i}\nm_i\\bar {q}_i \\gamma_5 q_i \\nonumber \n\\end{eqnarray}\n\n\\begin{eqnarray}\n{\\cal L_{FP}} &=& -\\bar {c}^{\\;+} (\\partial^2 + \\xi_W M_W^2 )c^-\n-\\bar {c}^{\\;-} (\\partial^2 + \\xi_W M_W^2 )c^+ \n -\\bar {c}^{\\;A} \\partial^2 c^A      \\label{sm921} \\\\\n&-&\\bar {c}^{\\;Z} (\\partial^2 + \\xi_Z\\; M_Z^2 )c^Z           \n-\\bar {c}^{\\;a} \\partial^2 c^a\n+ i e \\cot\\vartheta_W W^{+\\;\\mu}\\left(\n\\partial_{\\mu}\\bar {c}^{\\;-} c^Z - \\partial_{\\mu}\\bar {c}^{\\;Z} c^-\n\\right)                                    \\nonumber \\\\\n&+& i e W^{+\\;\\mu}\\left(\n\\partial_{\\mu}\\bar {c}^{\\;-} c^A - \\partial_{\\mu}\\bar {c}^{\\;A} c^-\n\\right)\n- i e \\cot\\vartheta_W W^{-\\;\\mu}\\left(\n\\partial_{\\mu}\\bar {c}^{\\;+} c^Z - \\partial_{\\mu}\\bar {c}^{\\;Z} c^+\n\\right)                                    \\nonumber \\\\\n&-& i e W^{-\\;\\mu}\\left(\n\\partial_{\\mu}\\bar {c}^{\\;+} c^A - \\partial_{\\mu}\\bar {c}^{\\;A} c^+\n\\right)\n+ i e \\cot\\vartheta_W Z^{\\mu}\\left(\n\\partial_{\\mu}\\bar {c}^{\\;+} c^-  - \\partial_{\\mu}\\bar {c}^{\\;-} c^+\n\\right)                                  \\nonumber \\\\\n&+& i e A^{\\mu}\\left(\n\\partial_{\\mu}\\bar {c}^{\\;+} c^-  - \\partial_{\\mu}\\bar {c}^{\\;-} c^+\n\\right)       \\nonumber \\\\\n&+& i \\omega^+ \\left(\n-\\xi_W e M_W \\cot 2\\vartheta_W \\bar {c}^{\\;-} c^Z\n-\\xi_W e M_W \\bar {c}^{\\;-} c^A\n+\\frac{\\xi_Z e}{2\\sin\\vartheta_W} M_Z \\bar {c}^{\\;Z} c^-\n\\right)                                    \\nonumber \\\\\n&+& i \\omega^- \\left(\n\\xi_W e M_W \\cot 2\\vartheta_W \\bar {c}^{\\;+} c^Z\n+ \\xi_W e M_W \\bar {c}^{\\;+} c^A\n-\\frac{\\xi_Z e}{2\\sin\\vartheta_W} M_Z \\bar {c}^{\\;Z} c^+\n\\right)                                     \\nonumber \\\\\n&+& \\frac{i \\xi_W e}{2} M_Z \\cot\\vartheta_W z \\left(\n\\bar {c}^- c^+ - \\bar {c}^{\\;+} c^- \\right)  \n- \\frac{ \\xi_W e}{2\\sin\\vartheta_W} M_W H \\left(\n\\bar {c}^{\\;-} c^+ + \\bar {c}^{\\;+} c^-\n\\right)                                \\nonumber \\\\\n&-& \\frac{ \\xi_Z e}{\\sin 2\\vartheta_W} M_Z H \n\\bar {c}^{\\;Z} c^Z   \\nonumber\n\\end{eqnarray}\n\n\n\n\\section{\\bf FEYNMAN RULES}\\label{fr}\n\n\\subsection{\\it General Remarks}\nThis Section presents the complete list of Feynman rules\ncorresponding to the Lagrangian of SM (see (\\ref{sm917} -- \\ref{sm921})).\n\nFirst of all we define the propagators by the relation\n\\begin{equation}\n\\Delta_{ij}(k) = i \\int d^4 x\\; e^{-ikx} <0|T(\\phi_i(x)\\phi_j(0)|0>,\n\\end{equation}\nwhere $\\phi_i$ presents any field. Curly, wavy and zigzag lines denote \ngluons, photons and weak bosons respectively, while full, dashed and dot\nlines \nstand for fermions (leptons and quarks), Higgs particles and ghosts fields,\nrespectively. Arrows on the propagator lines show : for the $W^+$ and \n$\\omega^+$ fields the flow of the positive charge, for the fermion that of\nthe \nfermion number, and for the ghost that of the ghost number.\n \nThe vertices are derived using ${\\it L_I}$, instead of usual usage of\n${\\it i\\; L_I}$. All the momenta of the particles are supposed to flow in.\nThe only exception was made for the ghost fields, where\ndirection of momentum  coincides with the direction of ghost number flow.\nThis convention permits to minimize the number of times when the \nimaginary unit $i$ appears. \n\nIt should be noted ones more, that all fields can be ''divided'' into two\nparts:  \n\\begin{verse}\n$\\bullet$ {\\it physical fields}: \n$A$ (photon), $W^{\\pm}$, $Z$, $G$ (gluon), $\\psi$, $H$ (Hiigs). \\\\\n$\\bullet$ {\\it non--physical fields}: \n$\\omega^{\\pm}$, $z$ (pseudogoldstones), $c^{\\pm}$, $c^z$, $c^A$, $c_a$ (\nghosts).\n\\end{verse}\n\n\\noindent Charged fermions have the electric charges (in the positron charge\n$e$ units):\n\\begin{eqnarray*}\n &&Q(e^-) = Q(\\mu^-) = Q(\\tau^-) = -e, \\\\ \n &&Q(e^+) = Q(\\mu^+) = Q(\\tau^+) = +e, \\\\\n &&Q(u) = Q(c) = Q(t) = +\\frac{2}{3}e, \\\\ \n &&Q(d) = Q(s) = Q(b) = -\\frac{1}{3}e. \n\\end{eqnarray*}\nThe electric charge $e$ (or strong coupling constant $g_s$ in QCD) is related\nto the fine structure constant $\\alpha$ (or $\\alpha_s$ in QCD) as follows:\n\\[\n\\alpha_{QED} \\equiv \\alpha = \\frac{e^2}{4 \\pi}, \\quad\n\\alpha_{QCD} \\equiv \\alpha_s = \\frac{g_s^2}{4 \\pi}.\n\\]\nThe electric charge, the $\\sin \\vartheta_W$, and Fermi constant\n$G_F$ are related as follows:\n\\begin{eqnarray}\n \\frac{e}{2 \\sqrt{2} \\sin \\vartheta_W} = M_W \\sqrt{\\frac{G_F}{\\sqrt{2}}}.\n \\label{vr3}\n\\end{eqnarray}\nFinally, every loop integration is performed by the rule\n\\begin{equation}\n\\int \\frac{d^d \\;k}{i \\, (2\\pi)^d},\n\\end{equation}\nand with  every fermion or ghost loop we associate extra factor $(-1)$.  \n\n\\subsection{\\it Propagators}\\label{prop}\n\\vspace{5mm}\n\n\\includegraphics[scale=0.95]{figs\/fig_prop.pdf} \n\n\n\\vspace{5mm}\n\n\\subsection{\\it Some Popular Gauges}\\label{gauges} \n\nHere we discuss the explicit forms of the propagators for some popular\ngauges.\nLet us consider a theory with free  boson Lagrangian:\n\\[\n{\\cal L} = -\\frac{1}{4} F^2_{\\mu \\nu}, \\quad F_{\\mu \\nu} = \n\\partial_{\\mu} A_{\\nu} - \\partial_{\\nu} A_{\\mu}.\n\\]\nOne can fix a gauge in one of three ways \\cite{itzu,lieb}: \n\\begin{itemize}\n\\item{\\bf i} to impose a gauge condition,\n\\item{\\bf ii} to add a {\\it Gauge Fixing Term} (GFT) to the Lagrangian\n\\item{\\bf iii} to fix a form of the Hamiltonian.\n\\end{itemize}\n\nIn a rigorous theory one should impose two gauge conditions. However,\nas it is \nusually accepted, we write only one condition. It should be considered\nrather \nas a symbol which denotes acceptable for a given gauge procedure of \nquantization, described somewhere in literature.\n\nIn practical calculations one needs an explicit form of a propagator with \nsatisfactory prescription for poles (which plays a key role in the loop \ncalculations). For this purposes it is sufficient to fix a gauge as\nmentioned \nin {\\bf ii} and {\\bf iii}. Polarization vectors of physical bosons and\nghosts \nshould be chosen in accordance with a detailed quantization procedure\napplicable for a given gauge. \n\n{\\bf Covariant gauges.}\n\n1. {\\it Generalized Lorentz gauge.}\n  \n\\begin{itemize}\n\\item{Notation}  $\\partial ^\\mu A_\\mu (x) = B(x)$\n\\item{GFT} $L_{GF}=-\\displaystyle \\frac{1}{2\\xi}(\\partial ^\\mu A_\\mu)^2$\n\\item{Propagator}\n\\[\nD^{\\mu\\nu} = \\frac{1}{k^2+i\\varepsilon}[g^{\\mu\\nu} \n- (1-\\xi)\\frac{k^{\\mu}k^{\\nu}}{k^2+i\\varepsilon}].\n\\]\n\\item{Comments} \n\n$\\xi =1$ is Feynman gauge, while $\\xi =0$ is Landau gauge. For the photon \n(gluon) propagator one should write $\\xi_G (\\xi_A)$ (see\nSubsection~\\ref{prop}). \n\\end{itemize}\n\n2. {\\it 't Hooft gauges ($R_\\xi$-gauges).}\n\n\\begin{itemize}\n\\item{Notation}  $\\partial ^\\mu A^a_\\mu (x) -i\\xi (v, \\tau^a \\phi) = B^a(x)$\n\\item{GFT} $L_{GF}= \\displaystyle -\\frac{1}{2\\xi}(\\partial ^\\mu A^a_\\mu)^2$\n\\item{Propagator}\n\\[\nD^{ab}_{\\mu\\nu} = \\frac{\\delta^{ab}}{k^2-M^2+i\\varepsilon}[g_{\\mu\\nu} \n- (1-\\xi)\\frac{k_{\\mu}k_{\\nu}}{k^2-\\xi M^2+i\\varepsilon}].\n\\]\n\\item{Comments} \n\nThe gauge parameter $\\xi = \\xi_W (\\xi_Z)$ for the case of $W(Z)$ boson (see \nSubsection~\\ref{prop}), $v\/\\sqrt{2}$ is the vacuum expectation\nvalue of the gauge \nfield, $\\tau^a$ are generators, $M$ is the vector boson mass.  $\\xi =1$ is \n't Hooft--Feynman gauge, $\\xi = 0$ is Landau gauge, $\\xi\\rightarrow\\infty$\ncorresponds to {\\it unitary} gauge. Non--physical gauge bosons should\nalso be \ntaken into account in loop calculations. They also have gauge--dependent \npropagator, see Subsection~\\ref{prop}. \n\\end{itemize}\n\n{\\bf Non-covariant gauges}\n\n3. {\\it Coulomb gauge.}\n\\begin{itemize}\n\\item{Notation}  $\\vec \\partial \\vec A =0,\\ k=1,2,3.$ \n\\item{GFT} $L_{GF}= \\displaystyle -\\frac{1}{2\\xi}(\\partial_kA_k)^2.$\n\\item{Propagator}\n\\[\nD_{\\mu\\nu} = \\frac{1}{k^2+i\\varepsilon}[g_{\\mu\\nu} \n- \\frac{k_{\\mu}k_{\\nu}-k_0 k_\\mu g_{\\nu 0} - k_0 k_\\nu g_{\\mu 0}}\n{|\\vec k |^2} -\\frac{\\xi k^2k_\\mu q_\\nu}{|\\vec k|^4}].\n\\]\n\\item{Comments} \n\nThe proper {\\it Coulomb gauge} corresponds to the case $\\xi = 0$.\n\\end{itemize}\n\n\n4. {\\it The general axial gauge.} \n\n\\begin{itemize}\n\\item{Notation} $n^\\mu A_\\mu(x)=B(x).$\n\\item{GFT} $L_{GF}= \\displaystyle -\\frac{1}{4\\xi}[n^*\\cdot \\partial\\ \nn\\cdot A]^2.$ \n\\item{Propagator}\n\\[\nD_{\\mu\\nu} = \\frac{1}{k^2+i\\varepsilon}[g_{\\mu\\nu} \n- \\frac{(n_{\\mu}k_{\\nu} + k_{\\mu}n_{\\nu})\\ n^*\\cdot k}\n{n\\cdot k\\ n^*\\cdot k+i\\varepsilon} \n- \\frac{(\\xi k^2 - n^2)\\ (n^*\\cdot k)^2}{(n\\cdot k\\ n^*\\cdot k\n +i\\varepsilon)^2} k_{\\mu}k_{\\nu}].\n\\]\n\\item{Comments}\n\nFeynman rules in this gauge usually do not contain ghosts. As it has been shown\nin \\cite{lieb} one has to consider an additional gauge vector $n^{\\ast \\mu}$ in\norder to have a correct prescription for poles. The quantization\nin this gauge was considered, for example, in \\cite{burnel,bassetto}.\n\nThe gauge vector $n^\\mu$ has the form: \n\\[ n^\\mu\\ =\\ (n_0;\\vec n)\\ =\\ (n_0; \\vec n_\\bot,n_3)\\ =\\ (n_0;n_1,n_2,n_3).\\]\nThe explicit form of the component structure of $n^\\mu$ and $n^{\\ast \\mu}$ \nshould be considered separately in the cases $n^2>0,\\ n^2=0$ and $n^2<0$. \nThe following widely used gauges are obtained in the limit $\\xi = 0$:\n\\end{itemize}\n\n4a. {\\it Temporal gauge:} $n^2>0$.\n\n\\[\nD_{\\mu\\nu} = \\frac{1}{k^2+i\\varepsilon}[g_{\\mu\\nu} \n- \\frac{(n_{\\mu}k_{\\nu}+k_{\\mu}n_{\\nu})\\ n^*\\cdot k}\n{n\\cdot k\\ n^*\\cdot k+i\\varepsilon} + \\frac{n^2\\ \n(n^*\\cdot k)^2}{(n\\cdot k\\ n^*\\cdot k+i\\varepsilon)^2} k_{\\mu} k_{\\nu}],\n\\]\n\\[\nn^\\mu\\ =\\ (n_0; \\vec n_\\bot,-i|\\vec n_\\bot|);\\ \\ \nn^{\\ast \\, \\mu}\\ =\\ (n_0; \\vec n_\\bot, i| \\vec n_\\bot|).\n\\]\n\n4b. {\\it Light--cone gauge:} $n^2=0$.\n\\[\nD_{\\mu\\nu} = \\frac{1}{k^2+i\\varepsilon}[g_{\\mu\\nu} \n- \\frac{(n_{\\mu}k_{\\nu}+k_{\\mu}n_{\\nu})\\ n^*\\cdot k}\n{n\\cdot k\\ n^*\\cdot k+i\\varepsilon} ],\n\\]\n\\[\nn^\\mu\\ =\\ (|\\vec n|;\\vec n);\\ \\ n^{* \\, \\mu}\\ =\\ (|\\vec n|;-\\vec n).\n\\]\n\n4c. {\\it Proper axial gauge:} $n^2<0$.\n\\[\nD_{\\mu\\nu} = \\frac{1}{k^2+i\\varepsilon}[g_{\\mu\\nu} \n- \\frac{(n_{\\mu}k_{\\nu}+k_{\\mu}n_{\\nu})\\ n^*\\cdot k}\n{n\\cdot k\\ n^*\\cdot k+i\\varepsilon} + \\frac{ n^2\\ \n(n^*\\cdot k)^2}{(n\\cdot k\\ n^*\\cdot k+i\\varepsilon)^2} k_{\\mu} k_{\\nu}],\n\\]\n\\[\nn^\\mu\\ =\\ (|\\vec n_\\bot|;\\vec n);\\ \\ n^{* \\, \\mu}\\ =\\ (|\\vec n_\\bot|;-\\vec n).\n\\]\n\n5. {\\it Planar gauge.}\n  \n\\begin{itemize}\n\\item{Notation} $n^\\mu A_\\mu(x)=B(x),\\ n^2\\neq 0.$\n\\item{GFT} $L_{GF}=\\displaystyle \\frac{1}{2n^2}[\\partial_\\mu (n\\cdot A)]^2.$ \n\\item{Propagator}\n\\[\nD_{\\mu\\nu} = \\frac{1}{k^2+i\\varepsilon} \\left [ g_{\\mu\\nu} \n- \\frac{(n_{\\mu}k_{\\nu}+k_{\\mu}n_{\\nu})\\ n^*\\cdot k}\n{n\\cdot k\\ n^*\\cdot k+i\\varepsilon} \\right ],\n\\]\n\\[\nn^\\mu\\ =\\ (n_0; \\vec n_\\bot,-i| \\vec n_\\bot|),\\ \\ \nn^{* \\, \\mu}\\ =\\ (n_0; \\vec n_\\bot,i|\\vec n_\\bot|),\\ \\ \\mbox{if}\\ \\ n^2>0;\n\\]\n\\[\nn_\\mu\\ =\\ (| \\vec n_\\bot|;\\vec n),\\ \\ n^*_\\mu\\ =\\ (|\\vec n_\\bot|;-\\vec n)\n\\ \\ \\mbox{if} \\ \\ n^2<0.\n\\]\n\n\\item{Comments}\n\nYang-Mills theory is not multiplicatively renormalizable in this gauge. \nQuantization in this gauge is also poorly understood. This gauge has the same \ndenotation as the axial gauge, that is not suitable. However, that should not \nlead to confusion (see the beginning of this Subsection).\n\n\\end{itemize}\n\n\n\n\\section{\\bf INTEGRATION IN $N$-DIMENSIONS}\\label{dimreg}\n\n\\subsection{\\it Dimensional Regularization }\n\nA powerful method of the evaluation of the loop integrals (which very often\nare\ndivergent) is {\\it Dimensional Regularization } (DR) \\cite{hooft}. The idea\nof\nDR is to consider the loop integral as an {\\it analytic function} of $n$ --\nnumber of dimensions. Then one can calculate this integral in that region of\nthe complex $n$ plane, where this function is convergent.\n\nA typical loop integral looks as follows:\n\\[\n\\int_{}^{} \\frac{d^4 p} {(2 \\pi)^4} \\frac{P(q_i^{\\nu}, m_i, p^{\\nu})}\n{ \\prod_{i=1}^{l} ( m_i^2 - (p - k_i)^2)},\n\\]\nwhere $q_i$ ($m_i$) are 4--momenta (masses) of external particles; \n$P(q_i^{\\nu}, m_i, p^{\\nu})$ is a function of masses $m_i$ and momenta \n$q_i$ and $p$.\n\nTo use the DR method one needs to transform the product of denominators into\nexpression such as : $p^2 + (pk) + M^2$, where $k^{\\nu}$ is the linear\ncombination of $q_i$ momenta and $M$ is a combination of $q_i^2$, $(q_i q_j)$,\nand $m_i^2$. That can be done by using of {\\it Feynman parameterization}:\n\\begin{eqnarray*}\n\\frac{1}{a^{\\alpha}b^{\\beta}} &=& \n\\frac{\\Gamma(\\alpha+\\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)}\n\\int_{0}^{1} dx \\frac{x^{\\alpha-1}(1-x)^{\\beta-1} }\n { [ax+b(1-x)]^{\\alpha+\\beta}}, \\\\ \n\\frac{1}{a^n} - \\frac{1}{b^n} &=& \\int_{0}^{1}  \n \\frac{n(a-b) dx}{ [(a-b)x+b]^{n+1} }, \\\\\n\\frac{1}{b_1^{\\alpha_1}b_2^{\\alpha_2} \\ldots b_m^{\\alpha_m}} &=& \n\\frac{\\Gamma(\\alpha_1+ \\ldots +\\alpha_m)}\n{\\Gamma(\\alpha_1)\\Gamma(\\alpha_2) \\ldots \\Gamma(\\alpha_m)}\n\\int_{0}^{1} dx_1 \\int_{0}^{x_1} dx_2 \\ldots \\int_{0}^{x_{m-2}} dx_{m-1} \\\\\n&& \\frac{\nx_{m-1}^{\\alpha_1-1}(x_{m-2}-x_{m-1})^{\\alpha_2-1} \\ldots \n(1-x_1)^{\\alpha_m-1}}\n{ [b_1x_{m-1}+b_2(x_{m-2}-x_{m-1})+\\ldots+b_m(1-x_1)]^\n{\\alpha_1+\\ldots+\\alpha_m}},\n\\end{eqnarray*}\nwhere $\\Gamma(z)$ is the Euler {\\it Gamma function}.\n\nUsing the Wick rotation $p_0 \\; \\to \\; i p_0$ and replacement $4 \\to n$, one\ncan obtain a typical integral in $n$-dimensional Euclidean space:\n\\[\n J = \\int d^np \\frac{P(q_i^{\\nu}, m_i, p^{\\nu})}\n{(p^2+2(pk)+M^2)^{\\alpha}}, \\qquad {\\rm Re} \\; \\alpha > 0.\n\\]\nThe differential $d^n p$ has the form:\n\\begin{eqnarray}\n d^n p &=& p^{n-1} dp \\; d\\Omega_n, \\quad \n \\int d\\Omega_n = \\Omega_n = 2\\pi^\\frac{n}{2} \/ \\Gamma(\\frac{n}{2}), \n \\label{int1} \\\\\n d\\Omega_n &=& (\\sin^{n-2}\\vartheta_{n-1} d\\vartheta_{n-1}) d\\Omega_{n-1}  \n \\label{int2} \\\\\n & = &  (\\sin^{n-2}\\vartheta_{n-1} d\\vartheta_{n-1})\n(\\sin^{n-3}\\vartheta_{n-2} d\\vartheta_{n-2}) \\ldots  d\\vartheta_1, \\nonumber\n\\end{eqnarray}\nwhere $0 \\leq \\vartheta_i \\leq \\pi, \\quad  0 \\leq \\vartheta_1 \\leq 2\\pi.$\n(The\nlast equality in (\\ref{int2}) obeys for the integer $n$.)\n\n\\subsection{\\it Integrals}\n\nLet us introduce the following notation:\n\\begin{eqnarray}\n \\hat J \\; f(p) \\; &\\equiv& \\; \\int d^np  \\; \\frac{1} \n{(p^2+2(pk)+M^2)^{\\alpha}} \\; f(p), \\label{int3} \\\\\nJ_0 \\; &\\equiv& \\; \\frac{i\\pi^{n\/2}\\;\\; i^n}{(M^2-k^2)^{\\alpha-\\frac{n}{2}}\n\\Gamma(\\alpha)}.   \\label{int4}\n\\end{eqnarray} \nThen :\n\\begin{eqnarray*}\nI_0 &=& \\hat J \\; 1 =  \\frac{i\\pi^{n\/2}}{(M^2-k^2)^{\\alpha-n\/2}} \n\\frac{\\Gamma(\\alpha-\\frac{n}{2})}{\\Gamma(\\alpha)} =\n\\Gamma(\\alpha-\\frac{n}{2}) \n J_0. \\\\\nI^{\\mu} &=& \\hat J \\; p^{\\mu} = (-k^{\\mu}) I_0, \\\\\nI_2  &=& \\hat J \\; p^2 = J_0 \\{ k^2\\Gamma(\\alpha-\\frac{n}{2}) + \n\\frac{n}{2}\\Gamma(\\alpha-1-\\frac{n}{2})(M^2-k^2)\\}, \\\\\nI^{\\mu\\nu} &=& \\hat J \\; p^{\\mu}p^{\\nu} = J_0 \\{k^{\\mu}k^{\\nu} \n\\Gamma(\\alpha-\\frac{n}{2}) + \\frac{1}{2}\\Gamma(\\alpha-1-\\frac{n}{2})\n g^{\\mu\\nu}(M^2-k^2) \\}, \\\\\nI^{\\mu\\nu\\lambda} &=& \\hat J \\; p^{\\mu}p^{\\nu}p^{\\lambda} = \n J_0 \\{-k^{\\mu}k^{\\nu}k^{\\lambda}\\Gamma(\\alpha-\\frac{n}{2}) \\\\\n &&- \\frac{1}{2}\\Gamma(\\alpha-1-\\frac{n}{2})(M^2-k^2)(g^{\\mu\\nu}k^{\\lambda}\n + g^{\\mu\\lambda}k^{\\nu} + g^{\\nu\\lambda}k^{\\mu}) \\}, \\\\\nI_2^{\\mu} &=&  \\hat J \\; p^2p^{\\mu}= \n -J_0 \\, k^{\\mu} \\, \\{k^2\\Gamma(\\alpha-\\frac{n}{2}) +\n \\frac{n+2}{2}\\Gamma(\\alpha-1-\\frac{n}{2})(M^2-k^2) \\}.\n\\end{eqnarray*}\nFor calculation of the basic integral $I_0$ one can use the well--known \nrelation \\cite{be}:\n\\[ \\displaystyle\n\\int_{0}^{\\infty} \\frac{x^{\\beta}}{(x^2+M^2)^{\\alpha}} dx =\n\\frac{ \\Gamma \\left(\\displaystyle \\frac{\\beta+1}{2}\n  \\right) \\Gamma \\left( \\displaystyle \\frac{2\\alpha-\\beta-1}{2} \\right) }\n{2\\Gamma(\\alpha)\\displaystyle (M^2)^{\\alpha - \\frac{\\beta+1}{2}} }.\n\\]\n\n\\subsection{\\it Spence Integral (Dilogarithm)}\n\nAs a rule the final expressions for the loop integrals include so--called\n{\\it Spence integral} or {\\it Euler dilogarithm} \\cite{be,pbm,lewin}:\n\\begin{eqnarray}\n\\li_2(z) = \\li(z) \\equiv - \\int_{0}^{z}\\frac{\\ln(1 - t)}{t}dt =\n  \\int_{0}^{1}\\frac{\\ln t}{t - z^{-1}}dt \\;\\; [arg(1-z) < \\pi].  \\label{int5} \n\\end{eqnarray}\nDilogarithm is a special case of the polylogarithm \\cite{be,pbm,lewin}:\n\\begin{eqnarray}\n\\li_{\\nu}(z) \\equiv \\sum_{k=1}^{\\infty} \\frac{z^k}{k^{\\nu}} \\quad \n[|z| < 1, \\;\\;\\; {\\rm or} \\, |z| = 1, \\, {\\rm Re} \\nu > 1].\n\\end{eqnarray}\nThe main properties of $\\li(z)$ are as follows:\n\\begin{eqnarray*}\n&&\\li_n(z) + \\li_n(-z) = 2^{1-n} \\li_n(z^2), \\\\\n&&\\li_n(iz) + \\li_n(-iz) = 4^{1-n} \\li_n(z^4) - 2^{1-n} \\li_n(z^2), \\\\\n&&\\li_n(iz) - \\li_n(-iz) = 2i \\sum_{k=0}^{\\infty} \\frac{(-1)^k z^{2k+1}}\n{(2k+1)^n}  \\quad  [|z| < 1], \\\\\n&&\\li_n(z) = \\int_0^z \\frac{\\li_{n-1}(t)}{t} dt \\quad (n = 1, 2, \\ldots), \\\\\n&&\\li_0(z) = \\frac{z}{1-z}, \\quad \\li_1(z) = -\\ln(1-z).\n\\end{eqnarray*}\nThe Riemann sheet of the $\\li_2(z)$ has a cut along the real axes when $z > 1$,\nand\n\\[\n{\\rm Im} \\; \\li_2(z \\pm i\\varepsilon) = \\pm \\pi \\Theta(z-1) \\ln (z),\n\\]\nwhere the $\\Theta(x)$ is the step function (see Subsection~\\ref{miscel}). \\\\\nThe equation ${\\rm Re} \\li_2(z) = 0$ has two solutions on the real axes\n\\[\nz_1 = 0, \\;\\;\\; {\\rm and} \\;\\; z_2 \\approx 12.6.\n\\]\n${\\rm Re} \\li_2(z)$ achieves its maximum at $z = 2$:\n\\[\n\\li_2(2) = \\frac{\\pi^2}{4},\n\\]\nand at this point the $\\li_2(z)$ has the expansion as follows \\cite{vermaseren}:\n\\[\n\\li_2 (2 - \\delta) = \\frac{\\pi^2}{4} - \\frac{\\delta^2}{4} \n- \\frac{\\delta^3}{6} - \\frac{5 \\delta^4}{48} - \\frac{\\delta^5}{15} - \\ldots\n\\]\nOne easily gets:\n\\begin{eqnarray*}\n&&\\li_2(0) = 0, \\quad \\li_2(1) = \\frac{\\pi^2}{6}, \\quad \n\\li_2(-1) = -\\frac{\\pi^2}{12}, \\\\\n&&\\li_2(\\frac{1}{2}) = \\frac{\\pi^2}{12} - \\frac{1}{2} \\ln^2 2, \\\\\n&&\\li_2(\\pm i) = -\\frac{\\pi^2}{48} \\pm i {\\bf G}, \\;\\; \n{\\bf G} = \\sum_{k=0}^{\\infty} \\frac{(-1)^k}{(2k+1)^2} = 0.915965594\\ldots \n\\end{eqnarray*}\nThe various relations with $\\li_2$ are as follows \\cite{be,pbm,lewin}: \n\\begin{eqnarray*}\n&&\\li_2(z) = -\\li_2(1-z) + \\frac{\\pi^2}{6} - \\ln z \\, \\ln(1-z) \\;\\; \n[|argz|, \\, |arg(1-z)| < \\pi], \\\\\n&&\\li_2(z) = -\\li_2(\\frac{1}{z}) - \\frac{1}{2} \\ln^2 z + i \\pi \\ln z\n+ \\frac{\\pi^2}{3}  \\;\\;\\; [|arg(-z)| < \\pi], \\\\\n&&\\li_2(z) = \\li_2(\\frac{1}{1-z}) + \\frac{1}{2} \\ln^2 (1-z) - \\ln (-z) \\ln (1-z)\n- \\frac{\\pi^2}{6} \\\\ \n && \\hspace{10mm} [|arg(-z)| < \\pi].\n\\end{eqnarray*}\nThe Hill identity has the form \\cite{vermaseren,pbm}:\n\\begin{eqnarray*}\n\\li_2(\\omega z)& =& \\li_2(\\omega) + \\li_2(z) \n- \\li_2 \\left ( \\frac{\\omega - \\omega z}{1-\\omega z} \\right ) \n- \\li_2 \\left ( \\frac{z - \\omega z}{1-\\omega z} \\right ) \\\\\n &-& \\ln \\left (\\frac{1-\\omega}{1-\\omega z} \\right )\n  \\ln \\left (\\frac{1-z}{1-\\omega z} \\right ) \\\\\n&-& \\eta \\left [ 1- \\omega, \\; \\frac{1}{1- \\omega z} \\right ] \\; \\ln \\omega \\; \n - \\eta \\left [ 1- z, \\; \\frac{1}{1- \\omega z} \\right ] \\; \\ln z,\n\\end{eqnarray*} \nwhere the function $\\eta$ compensates for the cut in the Riemann\nsheet of the logarithm \\cite{vermaseren}:\n\\[\n\\ln x y = \\ln x + \\ln y + \\eta (x,y).\n\\]\nA typical integral, which can be expressed via the dilogarithm, is, for \nexample: \n\\begin{eqnarray*}\n\\int_a^b \\frac{\\ln(p+qt)}{t} dt = \\ln p \\ln \\frac{b}{a} \n- \\li_2(-b \\frac{q}{p}) + \\li_2(-a \\frac{q}{p}). \n\\end{eqnarray*}\nThe Euler {\\it Gamma function} $\\Gamma(z)$ is given by the integral \nrepresentation \\cite{be}:\n\\[ \n \\Gamma(z) \\equiv \\int_{0}^{\\infty} dt \\, t^{z-1} \\, e^{-t}, \n\\qquad {\\rm Re} \\; z > 0.\n\\]\nThe main properties of the $\\Gamma(z)$ are as follows \\cite{be}:\n\\begin{eqnarray*}\n&& \\Gamma(1+z) = z \\Gamma(z), \\quad \\Gamma(n+1) = n !, \\\\\n&& \\Gamma(z) \\Gamma(-z) = - \\frac{\\pi}{z \\; \\sin(\\pi z)}, \\;\\;\\;\n   \\Gamma(z) \\Gamma(1-z) =  \\frac{\\pi}{\\sin(\\pi z)}, \\\\\n&& \\Gamma(\\frac{1}{2}+z) \\Gamma(\\frac{1}{2}-z) = \n     \\frac{\\pi}{ \\cos(\\pi z)}, \\;\\;\\;\n \\Gamma(2z) =  \\frac{2^{(2z-1)}}{\\sqrt{\\pi}} \\Gamma(z) \\Gamma(\\frac{1}{2}+z),\\\\\n&&\\Gamma(1) = \\Gamma(2) = 1, \\;\\;\\; \\Gamma \\left ( \\frac{1}{2} \\right ) = \n    \\sqrt{\\pi}, \\\\\n&&\\Gamma(z)|_{z \\to 0} \\simeq \\frac{1}{z} + \\Gamma'(1);  \\quad \n    \\Gamma'(1) = \\Gamma(1) \\Psi(1) = \\Psi(1) = -\\gamma = -0.57721 \\, 56649\n    \\ldots, \n\\end{eqnarray*}\nwhere $\\gamma$ is Euler constant.\n\n\n\\section{\\bf KINEMATICS}\\label{kinem} \n\nThe nice book by E.~Byckling and K.~Kajantie \\cite{bk} contains a lot of \ninformation about relativistic kinematics. Here we present a brief description \nof\nrelativistic kinematics following the Review of Particle Properties \\cite{pdg}.\n\n\\subsection{\\it Lorentz transformation}\nLet  $p^{\\mu}$ be some four-momentum of the massive particle with the mass M:\n\\begin{eqnarray} \n  p^{\\mu} = (p_0, \\vec{p}); \\;\\; \\vec{p} = (p_x, p_y, p_z),  \\;\n  p_0=\\sqrt{M^2 + \\vec{p}^2} \\label{kin_eq-1}\n\\end{eqnarray} \n\n\\noindent The reference frame where this $p^{\\mu}$-vector is defined is usually\nreferred as the {\\it laboratory} frame ({\\it L}-frame). This momentum in its\nrest frame ({\\it R}-frame) is as follows:\n\\begin{eqnarray} \n  Rf: \\;\\; p^{\\mu} = (M, 0)  \\label{kin_eq-2}\n\\end{eqnarray}\n\n\\noindent \nLet $k^{\\mu}$ be some 4-vector (4-momentum) defined in the {\\it Lab}-frame\nand $k^{* \\mu}$\nbe the same 4-vector in the {\\it R}-frame. \nThe Lorentz transormations form {\\it R}(est)-frame to {\\it Lab}-frame and\nvice versa can be written in the form as follows:\n\\begin{eqnarray}\n  \\begin{array}{l} { k^{*} \\to k} \\\\ {R \\to Lab} \\end{array}: \\;\n     { \\left \\{ \\begin{array}{ccl}\n k_0 & = & \\displaystyle \\frac{k_{0}^{*}p_{0} + (\\vec{k}^*\\vec{p})} {M} \\\\\n \\vec{k} & =  & \\vec{k}^{*} + \\alpha \\vec{p} \n   \\end{array} \\right. } \\quad\n     \\begin{array}{r} {k \\to k^{*}} \\\\ {Lab \\to R} \\end{array}: \\;\n          { \\left \\{ \\begin{array}{ccl}\n k_0^{*} & = & \\;\\; \\displaystyle \\frac{(pk)} {M} \\\\\n \\vec{k}^* & =  & \\vec{k} - \\alpha \\vec{p} \n   \\end{array} \\right. } \\quad  \\quad \\quad  (\\ref{lorentz-10}) \\nonumber\n\\end{eqnarray}\nwhere\n\\begin{eqnarray*}\n \\alpha = \\frac {k^{*}_0 + k_0} {p_0 + M}\n\\end{eqnarray*}\n\n\n\\subsection{\\it Variables} \n\nInitial (final) particles total momentum (energy) squared will be denoted by:\n\\begin{eqnarray}\n s \\equiv \\Big( \\sum_{initial} p_i \\Big)^2 = \\Big( \\sum_{final} p_j \\Big)^2. \n \\label{kin33}\n\\end{eqnarray}\nLet $E$ and $\\vec p$ be energy and momentum of a particle. The energy and \nmomentum of this particle ($E', \\vec p'$) in the frame moving with the velocity\n$\\vec \\beta$ are given by the Lorentz transformation:\n\\begin{eqnarray}\n E' = \\gamma (E + \\beta p _{||}), \\quad  p'_{||} = \\gamma (p_{||} + \\beta E), \n\\quad \\vec p{\\;}'_{\\top} = \\vec p_{\\top}, \\label {kin1}\n\\end{eqnarray}\nwhere $\\gamma = 1\/\\sqrt{1 - \\beta^2}$ and $\\vec p_{\\top} (p_{||})$ are the \ncomponents of $\\vec p$ perpendicular (parallel) to $\\vec \\beta$. \\\\\nThe beam direction choose along the $z$--axes.  4--momentum of a particle \n$p^{\\mu} = (E, \\vec p)$ can be written as:\n\\begin{eqnarray}\n &&E = p_0, \\; \\vec p_{\\top} = (p_x, p_y), \\; p_z, \\nonumber \\\\\n && p_x = |\\vec p| \\cos \\phi \\sin \\vartheta, \\;\n p_y = |\\vec p| \\sin \\phi \\sin \\vartheta, \\;\n p_z = |\\vec p| \\cos \\vartheta, \\label{kin2}\n\\end{eqnarray}\nwhere $\\phi$ is the azimuthal angle $(0\\leq~\\phi~\\leq~2\\pi)$;\n $\\vartheta$ is the polar angle $(0\\leq \\vartheta \\leq \\pi )$.\\\\\nAnother parameterization of $p^{\\mu}$ looks as follows:\n\\begin{eqnarray}\n E = m_{\\top} \\cosh y, \\; p_x, \\; p_y, \\; p_z = m_{\\top} \\sinh y, \\label{kin3}\n\\end{eqnarray}\nwhere $ m_{\\top}^2 = m^2 + p_{\\top}^2$ is the transverse mass (''old''\ndefinition), $y$ is the rapidity. \\\\\n{\\it Rapidity} $y$ is defined by\n\\begin{eqnarray}\n y \\equiv \\frac{1}{2} \\ln \\Big (\\frac{E+p_z}{E-p_z}\\Big) = \\ln \\Big(\n\\frac{E + p_z} {m_{\\top}} \\Big) =\n \\tanh^{-1} \\Big(\\frac{p_z}{E}\\Big). \\label{kin4}\n\\end{eqnarray}\nUnder a boost along $z$--direction to a frame with velocity $\\beta$,\n\\[\ny \\; \\to \\; y + \\tanh^{-1} \\beta.\n\\] \n{\\it Pseudorapidity} $\\eta$ is defined by:\n\\begin{eqnarray}\n &&\\eta \\equiv -\\ln (\\tan(\\vartheta \/2)), \\\\\n&& \\sinh \\eta = \\cot \\vartheta, \\; \\; \\cosh \\eta = \\frac{1}{\\sin \\vartheta}, \n \\; \\;  \\tanh \\eta = \\cos  \\vartheta. \\nonumber\n\\end{eqnarray}\nFor $p \\gg m$ and $\\vartheta \\gg  1 \/\\gamma$ one has : $\\quad \\eta \\approx y$.\n\n\\noindent Feynman's $x_F = x$ variable is given by\n\\begin{eqnarray}\n x = \\frac{p_z}{p_{z\\; {\\hbox to 0pt {max\\hss}}}} \\hspace{5mm}\n\\approx \\frac{(E + p_z)}{\n (E + p_z)_{\\hbox to 0pt {max\\hss}}}, \\quad \n  {\\rm in} \\; {\\rm cms} \\;\\; x = \\frac{2 p_z}{\\sqrt{s}}.\n\\end{eqnarray}\nThe last equation is valid for two particles collisions, and here $s$ is\ntotal energy squared (see (\\ref{kin33})).  \\\\\nIn the collider's experiments the following additional variables are used:\n\\begin{eqnarray*}\n\\begin{array}{lcrl}\nE_{\\bot} &=& E \\sin\\vartheta & \\; {\\rm -} \\; {\\rm transverse} \\; {\\rm energy},\n  \\\\\n \\vec{p}_{\\bot mis} &=& -(\\Sigma \\vec{p}_{\\bot}) & \\; {\\rm - } \\; { \\rm missing } \n \\; {\\rm transverse} \\; {\\rm momentum},\n \\\\\n \\vec{ E}_{\\bot mis} &=& -(\\Sigma \\vec E_{\\bot}) & \\; {\\rm -} \\; {\\rm  missing} \n \\; {\\rm transverse} \\; {\\rm energy}\n\\end{array}\n\\end{eqnarray*}\nwhere sum is performed over all detected particles. \\\\\nThe ''distance'' in $(\\eta, \\phi)$--plane between two particles (clusters)\n$1$ and $2$ is given by\n\\[\n\\Delta R \\equiv \\sqrt{(\\Delta \\phi)^2 + (\\Delta \\eta)^2 }, \\;\n \\Delta \\phi = \\phi_1 - \\phi_2, \\; \\Delta \\eta = \\eta_1 - \\eta_2.\n\\]\nThe \"transverse\" mass of the particle (cluster) $c$ with momentum $\\vec p_c$ \nand the \"missing\" transverse momentum (energy) $\\vec p_{\\bot \\; mis}$\n($\\vec E_{\\bot \\; mis}$) is given by:\n\\[\nM_{\\bot}^2(c, \\vec p_{\\bot \\; mis}) \\equiv \n(\\sqrt{m_c^2 + p^2_{\\bot c}} + p_{\\bot \\; mis})^2 -\n (\\vec p_{\\bot c} + \\vec p_{\\bot \\; mis})^2.\n\\]\n\n\\subsection{\\it Event Shape Variables} \n\nIn this Subsection we describe in brief event shape variables for $n$--particle\nfinal state (for details, see, for example \\cite {sj}). None of the\nvariables presented in this Subsection are Lorentz invariant. \n\n\\noindent $\\bullet$ {\\bf Sphericity} \\\\\nThe {\\it sphericity} tensor is defined as \\cite{sj,bjbr}:\n\\begin{eqnarray}\n S^{ab} \\equiv \\frac{ \\sum_i p^a_i p^b_i}{\\sum_i |\\vec p_i|^2},\n\\end{eqnarray}\nwhere $a,b = 1,2,3$ corresponds to the $x, y$ and $z$ components. By\nstandard diagonalization of $S^{ab}$ one can find three eigenvalues\n\\[ \n \\lambda_1 \\geq \\lambda_2 \\geq \\lambda_3, \\quad {\\rm with} \\quad\n \\lambda_1 + \\lambda_2 + \\lambda_3 = 1.\n\\]\nThen, the {\\it sphericity} is defined as:\n\\begin{eqnarray} \n S  \\equiv \\frac{3}{2} ( \\lambda_2 + \\lambda_3), \\quad 0 \\leq S \\leq 1. \n\\end{eqnarray}\nEigenvectors $\\vec s_i$ can be found that correspond to the three eigenvalues\n$\\lambda_i$. The $\\vec s_1$ eigenvector is called the {\\it sphericity\naxes}, while the {\\it sphericity event plane} is spanned by $\\vec s_1$ and\n $\\vec s_2$. \\\\\nSphericity is essentially a measure of the summed $\\vec p_{\\top}$ with\nrespect to sphericity axes. So, one can use another definition of the\n sphericity:\n\\begin{eqnarray} \n S = \\frac{3}{2} \\min_{\\vec n} \\frac{\\sum_i \\vec p_{\\top i}^{\\; 2}}\n  {\\sum_i |\\vec p_i|^2}, \\label{kin5} \n\\end{eqnarray}\nwhere $\\vec p_{\\top i}$ is a component of $\\vec p_i$ perpendicular to\n$\\vec n$. So, the sphericity axes $\\vec s_i$ given (\\ref{kin5}) by the\n$\\vec n$ vector for which minimum is attained. A 2--jet event corresponds\nto $S \\approx 0$ and isotropic event to $S \\approx 1$. \n\nSphericity is not an infrared safe quality in QCD perturbation theory.\nSometimes one can use the generalization of the sphericity tensor, given by\n\\begin{eqnarray} \n S^{(r)ab} \\equiv \\frac{\\sum_i |\\vec p_i|^{r-2} p^a_i p^b_i} \n  {\\sum_i |\\vec p_i|^r}, \\label{kin6} \n\\end{eqnarray}\n\n\\noindent $\\bullet$ {\\bf Aplanarity} \\\\\nThe {\\it aplanarity} $A$ is define as \\cite{sj,mrkj}:\n\\begin{eqnarray} \n A \\equiv  \\frac{3}{2} \\lambda_2, \\quad 0 \\leq A \\leq \\frac{1}{2}.\n\\end{eqnarray}\nThe aplanarity measures the transverse momentum component out of the event\nplane. A planar event has $A \\approx 0$ and isotropic one $A \\approx \n\\frac{1}{2}$. \\\\\n\n\\noindent $\\bullet$ {\\bf Thrust} \\\\\nThe {\\it thrust} $T$ is given by \\cite{sj,bran} \n\\begin{eqnarray} \n T \\equiv  \\max_{|\\vec n|=1} \\frac{\\sum_i |(\\vec n \\vec p_i)|}\n   {\\sum_i |\\vec p_i|}, \\quad  \\frac{1}{2} \\leq T \\leq 1.\n\\end{eqnarray}\nand the {\\it thrust axes} $\\vec t_i$ is given by the $\\vec n$ vector for\nwhich maximum is attained. 2--jet event corresponds to $T \\approx 1$ and\nisotropic event to $T \\approx \\frac{1}{2}$. \n\n\\noindent $\\bullet$ {\\bf Major and \\bf minor values} \\\\\nIn the plane perpendicular to the thrust axes, a {\\it major axes} $\\vec m_a$ \nand {\\it major value} $M_a$ may be defined in just the same fashion as thrust\n\\cite{sj}, i.e. \n\\begin{eqnarray} \n M_a \\equiv \\max_{|\\vec n|=1, \\; (\\vec n \\vec t_1)=0} \n \\frac{\\sum_i |(\\vec n \\vec p_i)|} {\\sum_i |\\vec p_i|}. \n\\end{eqnarray}\nFinally, a third axes, the {\\it minor axes}, is defined perpendicular to\nthe thrust ($\\vec t_1$) and major ($\\vec m_a$) axes. The {\\it minor value}\n$M_i$ is calculated just as thrust and major values.\n\n\\noindent $\\bullet$ {\\bf Oblatness} \\\\\nThe {\\it oblatness} $O$ is given by \\cite{sj}\n\\[\nO \\equiv M_a - M_i.\n\\]\nIn general, $O \\approx 0$, corresponds to an event symmetrical around the\nthrust axes $\\vec t_1$ and high $O$ to aplanar event.\n\n\\noindent $\\bullet$ {\\bf Fox--Wolfram moments} \\\\\nThe {\\it Fox--Wolfram moments} $H_l$, $l = 0, 1, 2, \\ldots,$ are defined by\n\\cite{sj,fox}:\n\\begin{eqnarray} \n H_l \\equiv \\sum_{i,j = 1} \\frac{|\\vec p_i| |\\vec p_j|}{E^2_{vis}} \n P_l(\\cos \\vartheta _{ij}),\n\\end{eqnarray}\nwhere $\\vartheta _{ij}$ is the opening angle between hadron $i$ and $j$,\nand $E_{vis}$ is the total visible energy of the event. $P_l(z)$ are the\nLegendre polynomials \\cite{be}:\n\\begin{eqnarray*} \n && P_0(z) = 1, \\; P_1(z) = z, \\; P_2(z) = \\frac{1}{2}(3 z^2 - 1), \\ldots \\\\\n && P_k(z) = \\frac{1}{k} \\big [ (2k-1)z P_{k-1}(z) - (k-1) P_{k-2}(z) \\big ].\n\\end{eqnarray*}\nNeglecting the masses of all the particles, one gets $H_0 = 1$. If\nmomentum is balanced, then $H_1=0$. 2--jet events tend to give $H_l \\approx 1$\nfor $l$ even and $H_l \\approx 0$ for $l$ odd. \n\nThe summary of the discussed quantities are presented in Table~\\ref{kinem}.1. \n\n\\vspace{0.5cm}\n\\noindent \n\\underline{ {\\bf Table~\\ref{kinem}.1.}} \\\\\n Summary of event shape variables. \n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\\hline\n & $S$ & $A$ & $T$ & $O$ &\n  $\\begin{array}{c} H_0 \\\\ {\\rm all} \\; m_i=0 \\end{array}$ & $H_l$ \\\\ \\hline\n  isotropic & $1$ & $\\frac{1}{2}$ & $\\frac{1}{2}$ & - & $1$ & - \\\\ \\hline\n  2--jet  &  $0$  & - & $1$ & $0$ & $1$ & \n  $\\begin{array}{c} H_1 = 0 \\\\ H_{2k} \\approx 1, \\; H_{2k+1} \\approx 0 \n   \\end{array} $ \\\\  \\hline\n   planar & - & $0$ & - & $\\gg 0$ & $1$ & - \\\\ \\hline \n\\end{tabular}\n\\end{center}\n\n\\subsection {\\it Two--body Final State}\n\nIn the collision of two particles of mass $m_1$ and $m_2$ and momenta $p_1$\nand $p_2$ \n\\[\n s = (p_1 + p_2)^2 = m_1^2 + m_2^2 + 2 E_{1\\;Lab}m_2,\n\\]\nwhere the last equation is valid in the frame, where one particle (second one)\nis at rest (Lab frame).\n\nThe energies and momenta of the particles $1$ and $2$ in their center--of--mass\nsystem (cms) are equal to:\n\\begin{eqnarray}\n&& E^{\\ast}_1 = \\frac{s + m_1^2 - m_2^2}{2 \\sqrt{s}}, \\quad\n  E^{\\ast}_2 = \\frac{s - m_1^2 + m_2^2}{2 \\sqrt{s}}, \\label{kin7} \\\\\n&& \\vec p_1^{\\; \\ast} = - \\vec p_2^{\\; \\ast}, \\quad\n|\\vec p_1^{\\; \\ast}| = \\frac{\n \\sqrt{[s - (m_1 + m_2)^2][s - (m_1-m_2)^2]}}{2 \\sqrt{s}}, \\label{kin8}\n\\end{eqnarray}\nor\n\\[\n|\\vec p_1^{\\; \\ast}| = \\frac{1}{2\\sqrt{s}} \\lambda^{1\/2}(s, m^2_1, m^2_2),\n\\]\nwhere $\\lambda (x,y,z)$ is the so--called {\\it kinematical function} \\cite{bk}:\n\\begin{eqnarray}\n \\lambda (x,y,z) & \\equiv & (x - y - z)^2 - 4yz \\label{kint1} \\\\\n & = & x^2 + y^2 + z^2 - 2xy - 2yz - 2 zx   \\nonumber \\\\\n & = & \\bigl \\{ x - (\\sqrt{y} + \\sqrt{z})^2 \\bigr \\}\n  \\bigl \\{ x - (\\sqrt{y} - \\sqrt{z})^2 \\bigr \\}.\n\\end{eqnarray}\nLet us now consider the two--body reaction (4--momenta of the particles are \npresented in the parentheses):\n\\begin{eqnarray*}\na(p_a) + b(p_b) &\\to& 1(p_1) + 2(p_2) \\\\\n p_a  + p_b &=& p_1 + p_2\n\\end{eqnarray*}\nThe Lorentz--invariant Mandelstam variables for reaction $2 \\to 2$ are defined \nby:\n\\begin{eqnarray}\ns &=& (p_a  + p_b)^2 = (p_1 + p_2)^2, \\quad  \nt = (p_a  - p_1)^2 = (p_b - p_2)^2, \\label{kin11} \\\\ \nu &=& (p_a  - p_2)^2 = (p_b - p_1)^2, \\nonumber\n\\end{eqnarray}\nand they satisfy\n\\[\ns + t + u = m_a^2 + m_b^2 + m_1^2 + m_2^2.\n\\]\nTwo limits of t (corresponding to $\\vartheta_{cm} = 0$ and $\\pi$) equal:\n\\begin{eqnarray}\n t_{\\pm} &=& m^2_a + m^2_1 - 2E^{\\ast}_a E^{\\ast}_1 \\pm 2|\\vec p^{\\; \\ast}_a|\n|\\vec p^{\\; \\ast}_1| = \\\\\n   &=& m^2_a + m_1^2 - \\frac{1}{2s}(s + m_a^2 - m_b^2)(s+m_1^2-m_2^2) \n\\nonumber \\\\\n &&\\pm \\frac{1}{2s} \\lambda^{1\/2}(s, m^2_a, m^2_b) \n                    \\lambda^{1\/2}(s, m^2_1, m^2_2).  \\nonumber\n\\end{eqnarray}\n\n\\subsection {\\it Three--body Final State}\n\nLet us consider three--body decay of particle $a$ with mass $M$ \n\\[ \n a(P) \\to 1(p_1) + 2(p_2) + 3(p_3).\n\\]\nDefining\n\\begin{equation}\np_{ij} \\equiv p_i + p_j, \\;\\; m_{ij}^2 \\equiv p_{ij}^2, \\label{kin10}\n\\end{equation}\nthen\n\\begin{eqnarray*}\n && m_{12}^2 + m_{23}^2 + m_{13}^2 =  M^2 + m_1^2 + m_2^2 + m_3^2, \\\\\n{\\rm and} \\;\\; &&m_{ij}^2 = (P - p_k)^2 = M^2 + m_k^2 - 2ME_k.\n\\end{eqnarray*}\nThe $1 \\to 3$ decay is described by two variables (for example, $m_{12}$\nand $m_{13}$). If $m_{12}$ is fixed, then limits of $m_{13}^2$ variation are\nequal to:\n\\begin{eqnarray*}\n \\Big(m_{13}^2 \\Big)_{\\pm} &=& m_1^2 + m_3^2 - \n\\frac{1}{2m^2_{12}}(m^2_{12}-M^2+m^2_3)(m^2_{12}+m^2_1- m^2_2) \\\\\n &&\\pm \\frac{1}{2m^2_{12}} \\lambda^{1\/2}(m^2_{12}, M^2, m^2_3) \n                    \\lambda^{1\/2}(m^2_{12}, m^2_1, m^2_2) = \\\\  \n &=& (E^{\\ast}_1 + E^{\\ast}_3 )^2 - (\\sqrt{E^{\\ast\\, 2}_1-m^2_1} \\mp\n   (\\sqrt{E^{\\ast\\, 2}_3-m^2_3})^2,\n\\end{eqnarray*}\nwhere $E^{\\ast}_1 = \\frac{m^2_{12}+m^2_1-m^2_2}{2m_{12}}$ and\n $E^{\\ast}_3 = \\frac{M^2 - m^2_{12} - m^2_3}{2m_{12}}$. \n\n\\noindent $2 \\to 3$ scattering is described by five independent variables.\nFor example, \n\\[\ns = (p_a + p_b)^2, \\; m^2_{12}, \\; m^2_{23}, \\; t_1 = (p_q - p_1)^2, \n\\; t_2 = (p_b - p_3)^2.\n\\]\n\n\\subsection{\\it Lorentz Invariant Phase Space}\n\nLorentz invariant phase space (LIPS) of $n$ particles with 4--momenta $p_j$\n($j = 1, 2, \\ldots n$) and the total momentum \n$ P = \\sum^{n}_{j=1} p_j$ is given by:\n\\begin{eqnarray} \nd R_n(P; \\, p_1, p_2, \\ldots p_n)  \\equiv \\delta^{(4)}(P - \\sum_{j=1}^n p_j) \n \\prod_{j=1}^{n} \\frac{ d^3p_j}{(2\\pi)^3 2E_j}.\n\\end{eqnarray}\nThrough of this Subsection we use the following notation:\n\\[\n s \\equiv P^2.\n\\]\nThis LIPS can be generated recursively as follows \\cite{pdg,bk}:\n\\begin{eqnarray} \nd R_n = d R_2(P; \\, p_n, q) (2\\pi)^3 dq^2 \\, d R_{n-1}(q; \\, p_1, \n\\ldots p_{n-1}), \\label{lips2}\n\\end{eqnarray}\nwhere $q = \\sum^{n-1}_{i=1} p_i$ and $(m_1+m_2+ \\ldots + m_{n-1})^2 \\leq q^2\n\\leq (\\sqrt{P^2} - m_n)^2$, or:\n\\begin{eqnarray} \nd R_n = d R_{n-j+1}(P; \\, q,  p_{j+1}, \\ldots p_n) (2\\pi)^3 dq^2 \\, \nd R_{j}(q; \\, p_1, \\ldots p_{j}), \\label{lips3}\n\\end{eqnarray}\nhere $q = \\sum^j_{l=1} p_l$ and\n\\[\n(m_1 + \\ldots + m_j)^2 \\leq q^2 \\leq (\\sqrt{P^2} - \\sum^n_{l=j+1}m_l)^2.\n\\]\n\n\\noindent The integrated LIPS for $m_1 = m_2= \\ldots = m_n = 0$ equals:\n\\[\nR_n(0) = \\frac{1}{(2\\pi)^{3n}} \\frac{(\\pi \/ 2)^{n-1}}{(n-1)!(n-2)!} \n(P^2)^{n-2}.\n\\]\n\n\\noindent Two--particle LIPS equals:\n\\[\nR_2 = \\frac{1}{(2\\pi)^6} \\frac{p^{\\ast}_1}{4\\sqrt{P^2}} \\int d \\Omega^{\\ast}_1\n =  \\frac{1}{(2\\pi)^6} \\frac{\\pi p^{\\ast}_1}{\\sqrt{P^2}} \n =  \\frac{1}{(2\\pi)^6} \\frac{\\pi p^{\\ast}_1}{\\sqrt{s}} ,\n\\]\nwhere $p^{\\ast}_1$ is momentum of first (second) particle in cms \n(see (\\ref{kin8})). \\\\\n\n\\noindent Different choice of $m_1$ and $m_2$ leads to:\n\\begin{eqnarray*} \n R_2 &=& \\frac{1}{(2\\pi)^6} \n \\frac{\\pi \\sqrt{[s - (m_1+m_2)^2][s - (m_1-m_2)^2]}}{2 s}, \n\\quad (m_1 + m_2) \\leq \\sqrt{s}, \\\\\n R_2 &=& \\frac{1}{(2\\pi)^6} \\frac{\\pi}{2} \\sqrt{1 - \\frac{4 m^2}{s}}, \n    \\quad m_1 = m_2 = m, \\\\\n R_2 &=& \\frac{1}{(2\\pi)^6} \\frac{\\pi}{2} ( 1 - \\frac{m^2_1}{s}), \n    \\quad m_2=0, \\\\\n R_2 &=& \\frac{1}{(2\\pi)^6} \\frac{\\pi}{2}, \\quad m_1=m_2=0. \n\\end{eqnarray*}\n\n\\noindent Three body decay final state LIPS equals:\n\\begin{eqnarray*} \n d R_3 = \\frac{1}{(2\\pi)^9} \n \\frac{\\pi^2}{4s} d m^2_{12} d m^2_{13}  \n  = \\frac{1}{(2\\pi)^9} \\pi^2 d E_1 \\, d E_2,\n\\end{eqnarray*}\nwhere $m_{12}$ and $m_{13}$ are defined in (\\ref{kin10}), $E_1(E_2)$ is the\n energy of\nthe first (second) particle in $P$ rest frame. This is the standard form of the\nDalitz plot.\n\n\\subsection{\\it Width and Cross Section}\n\nThe partial decay rate ({\\it partial width}) of a particle of mass $M$ into \n$n$ bodies in its rest\nframe is given in terms of the Lorentz--invariant matrix element $M_{fi}$ by:\n\\begin{eqnarray}\nd\\Gamma = \\frac{(2\\pi)^4}{2M} |M_{fi}|^2 d R_n(P; \\; p_1, p_2, \\ldots, p_n).\n \\label{kin12}\n\\end{eqnarray}\nThe differential cross section of the reaction $a \\; + b \\; \\to \\; 1 \\; + 2 \\;\n+ \\ldots + \\; n$ ($p_a+p_b \\equiv P)$  is given by:\n\\begin{eqnarray}\n&&d\\sigma = \\frac{(2\\pi)^4}{2 I} |M_{fi}|^2 d R_n(P; \\; p_1, p_2, \\ldots, p_n),\n \\label{kin13}  \\\\ \n&& I^2 = [s-(m_a+m_b)^2][s-(m_a-m_b)^2] = 4[(p_a p_b)^2 - m^2_a m^2_b].\n\\nonumber\n\\end{eqnarray}\n\n\n\\section{\\bf DECAYS}\\label{decays}\n\n\\subsection{\\it Standard Model Higgs Decays Rates} \n\nStandard Model Higgs is expected to have a mass between 45 Gev and 1 TeV, and,\nsince it couples directly to the masses of other particles, to decay into\nheaviest available particles. The SM Higgs decay rates, calculated without \nradiative corrections  are as follows (see \\cite{pl} and references therein):\n\\begin{eqnarray*}\n &H& \\to f \\bar f, \\quad \\Gamma = \\frac{N_c G_F m^2_f}{4 \\sqrt{2}\\pi} m_H \n\\beta^3, \n\\end{eqnarray*}\nwhere $\\beta = \\sqrt{1 - 4m^2_f \/ m^2_H}$ and $N_c = 1(3)$ for $f=$~lepton \n(quark).\n\\begin{eqnarray*}\n &H& \\to W^+ W^- (ZZ), \\quad \\Gamma = \\frac{G_F^2 M^2 m_H}{8 \\sqrt{2}\\pi}\n \\frac{\\sqrt{1-x}}{x} (3x^2-4x+4), \n\\end{eqnarray*}\nwhere $x = 4M^2 \/ m^2_H$, $M$ is $W^{\\pm}(Z)$--boson mass. \nHiggs decay into two photons or two gluons proceeds via loops. Its decay rates\nare equal \\cite{hf}: \n\\begin{eqnarray*}\n &H& \\to \\gamma \\gamma, \\quad \\Gamma = \\frac{\\alpha^2 G_F}{8 \\sqrt{2}\\pi^3}\n m^3_H |I|^2, \n\\end{eqnarray*}\nwhere $I = I_{lepton} + I_{hadron} + I_W + \\ldots$, and $|I| \\approx O(1)$.\n\\begin{eqnarray*}\n &H& \\to g g, \\quad \\Gamma = \\frac{\\alpha^2_s G_F}{4 \\sqrt{2}\\pi^3}\n \\frac{ m^3_H}{9}  |N|^2, \n\\end{eqnarray*}\nwhere $N \\equiv  3 \\sum_j N_j$ is the sum of the quark's contributions \n$j = 1,2,\\ldots,$ given by \\cite{ggmn}:\n\\[ \nN_j  = \\int^1_0 dx \\int^{1-x}_0 dy \n\\frac{1 - 4xy}{1 - xy \\frac{m^2_H}{m^2_j} - i\\varepsilon} = \n  2 \\lambda_j + \\lambda_j (4 \\lambda_j -1) G( \\lambda_j),\n\\] \nwhere $\\lambda_j \\equiv  m^2_j \/ m^2_H$, and\n\\begin{eqnarray*}\n G(z) &=& -2 [\\arcsin(\\frac{1}{2\\sqrt{z}})]^2, \\quad z \\geq \\frac{1}{4}, \\\\\n G(z) &=& \\frac{1}{2} \\ln^2 \\left [ \\frac{1+\\sqrt{1-4z}}{1-\\sqrt{1-4z}} \n \\right ]\n - \\frac{\\pi^2}{2} + i \\pi \\ln \\left [ \\frac{1+\\sqrt{1-4z}}{1-\\sqrt{1-4z}} \n\\right ],  \\quad z \\le \\frac{1}{4}.\n\\end{eqnarray*}\n$N_q$ vanishes for $m_q \\ll m_H$ and $N_q \\to 1\/3$ for $m_q \\gg m_H$.\n\n\\begin{eqnarray*}\n H \\to W^{\\pm} f \\bar f', \\quad &\\Gamma& = \\frac{g^4 m_H}{307 \\pi^3} \n F(\\epsilon), \\quad \\epsilon = \\frac{m_W}{m_H},\n\\end{eqnarray*}\n\\begin{eqnarray*} \n H \\to W^{\\pm} \\sum f \\bar f', \\quad &&({\\rm except} \\; W^+ \\to t \\bar b) \\; \\\\\n &  \\Gamma& = \\frac{3 g^4 m_H}{512 \\pi^3} F(\\epsilon),\n\\quad \\epsilon = \\frac{m_W}{m_H}, \\\\\n H \\to Z \\sum f \\bar f, \\quad &\\Gamma& = \\frac{g^4 m_H}\n {2048 \\pi^3 \\cos^4 \\vartheta_W} \\\\\n &\\times& (7 - \\frac{40}{3} \\sin^2 \\vartheta_W\n  + \\frac{160}{9} \\sin^4 \\vartheta_W) F(\\epsilon'),\n\\quad \\epsilon' = \\frac{m_Z}{m_H}, \\\\\n F(z) &=& \\frac{3(1-8z^2 + 20 z^4)}{\\sqrt{4 z^2 - 1}} \\arccos\n (\\frac{3z^2-1}{2z^3}) \\\\\n & - &(1-z^2)(\\frac{27}{2}z^2-\\frac{13}{2}+\\frac{1}{z^2}) \n - 3(1-6z^2+4z^4) \\ln z. \n\\end{eqnarray*}\n\n\\subsection{\\it $W$ and $Z$ Decays} \n\nThe partial decay widthes for gauge bosons to decay into massless fermions \n$f_1 \\bar f_2$ are equal to \\cite{okun,pdg}: \n\\begin{eqnarray*}\n W^+ \\to e^+ \\nu_e, \\quad \\Gamma &=& \\frac{G_F M_W^3}{6\\sqrt{2}\\pi} \\approx \n 227 \\pm 1 \\; {\\rm MeV}, \\\\ \n W^+ \\to u_i \\bar d_i, \\quad \\Gamma &=& C\\frac{G_F M_W^3}{6\\sqrt{2}\\pi} \n  |V_{ij}|^2 \\approx (707 \\pm 3) |V_{ij}|^2 \\; {\\rm MeV}, \\\\ \n Z \\to \\psi_i \\bar \\psi_i, \\quad \\Gamma &=& C\\frac{G_F M_Z^3}{6\\sqrt{2}\\pi} \n  [g_{iV}^2 + g_{iA}^2] \\approx \\\\\n &=& \\left \\{\n \\begin{array}{cc} 167.1 \\pm 0.3 \\; {\\rm MeV} \\; (\\nu \\bar \\nu), & \n  83.9 \\pm 0.2 \\; {\\rm MeV} \\; (e^+ e^-), \\\\ \n 298.0 \\pm 0.6 \\; {\\rm MeV} \\; (u \\bar u), & \n 384.5 \\pm 0.8 \\; {\\rm MeV} \\; ( d \\bar d), \\\\ \n 375.2 \\pm 0.4 \\; {\\rm MeV} \\; (b \\bar b), & {} \\end{array} \\right.\n\\end{eqnarray*}\nFor lepton $C=1$, while for quarks $C=3(1+ \\frac{\\alpha_s(M_V)}{\\pi} \n + 1.409 \\frac{\\alpha_s^2}{\\pi^2} - 12.77 \\frac{\\alpha_s^3}{\\pi^3})$,\n where $3$ is due to color and the factor in parentheses is a QCD\n correction~\\cite{Chetyrkin:1979bj}.\n\n\\subsection{\\it Muon Decay}\n\nIn the SM the total muon decay width is equal (up to $100 \\%$ accuracy) to the\nwidth of the decay\n\\[\n\\mu^- \\to e^- \\bar \\nu_e \\nu_{\\mu}.\n\\]\nThe matrix element squared for this decay equals \\cite{okun}:\n\\[\n|M|^2 = 128 G_f^2 (p_{\\mu} p_{\\nu_e}) \\, (p_e p_{\\nu_{\\mu}}).\n\\]\nThen the total muon width is given by \\cite{mudec}:\n\\begin{eqnarray}\n \\Gamma^{tot}_{\\mu} =  \\frac{G_F^2 m^5_{\\mu}} {192 \\pi^3} \nF(\\frac{m_e^2}{m^2_{\\mu}}) (1 + \\frac{3}{5}\\frac{m^2_{\\mu}}{M^2_W})\n [ 1 +\\frac{\\alpha(m_{\\mu})}{2 \\pi} (\\frac{25}{4} - \\pi^2)],\n\\end{eqnarray}\nwhere $F(x) = 1 - 8x + 8x^3 - x^4 - 12 x^2 \\ln x$, and\n\\[\n\\alpha(m_{\\mu})^{-1} = \\alpha^{-1} - \\frac{2}{3 \\pi} \\ln (\\frac{m_{\\mu}}{m_e})\n + \\frac{1}{6 \\pi} \\approx 136.\n\\]\nFor pure $V-A$ coupling (and neglecting of the electron mass) in the rest \nframe of the polarized muon ($\\mu^{\\mp}$) the differential decay rate is:\n\\[\n d \\Gamma (\\mu^{\\mp}) = \\frac{G_F^2 m^5_{\\mu}}{192 \\pi ^3} [3-2x \\pm (1-2x) \n  \\cos \\vartheta] x^2 dx d(\\cos \\vartheta),\n\\]\nwhere $\\vartheta$ is the angle between the $e^{\\pm}$  momentum and the $\\mu$ spin,\nand $x = 2 E_{\\mu} \/ m_{\\mu}$. \n\n\\subsection{\\it Charged Meson Decay}\n\n\\noindent The decay constant $f_P$ for pseudoscalar meson $P$ is defined by\n\\cite{pdg}\n\\[\n <0|A_{\\mu}(0)|P(k)> = i f_P k_{\\mu}.\n\\]\nThe state vector is normalized by $<P(k)|P(k')> = (2\\pi)^3 2 E_q \\delta^3\n(\\vec k - \\vec k')$. The annihilation rate of the \n$P(q_1 \\bar q'_2) \\to f \\bar f'$ decay is given by\n\\begin{eqnarray}\n\\Gamma(P \\to f \\bar f') = C \\frac{G_F^2 |V_{q_1 q'_2}|^2}{8 \\pi}\n f^2_P m^2_f M_P (1 - \\frac{m^2_f}{M^2_P}),\n\\end{eqnarray}\nwhere $C = 1$ for $P \\to l \\nu_l$ decay and $C = (3 |V_{q_3 q'_4}|^2)$ for\n$ P \\to q_3 \\bar q'_4$ one, and $m_f$ is the heaviest final fermion mass.\n\n\\subsection{\\it Quark Decay} \n\nIn the region $m_q \\ll M_W$ the total quark width is given by \\cite{okun}:\n\\begin{eqnarray}\n\\Gamma(Q \\to q_2 q_3 \\bar q'_4)  = \\frac{G_F^2 m^5_Q} {64 \\pi^3}\n |V_{Q q_2}|^2 |V_{q_3 q'_4}|^2.\n\\end{eqnarray}\nFor the case of $m_Q \\gg m_W + m_q$ the width of the heavy quark decay \n$Q \\to W + q$ equals \\cite{bigi}: \n\\begin{eqnarray}\n\\Gamma (Q \\rightarrow W + q) &=& \\frac{G_F m_Q^3}{8\\pi \\sqrt{2}} \\,\n |V_{Qq}|^2 \\,  \\frac{2k}{m_Q} f_Q(\\frac{m^2_q}{m^2_Q}, \\frac{M^2_W}{m^2_Q}) \\, \n \\approx  180 \\; ({\\rm MeV}) \\, \\quad |V_{Qq}|^2 \n \\left ( \\frac{m_Q}{m_W} \\right )^3, \\nonumber\n\\end{eqnarray}\nwhere\n\\begin{eqnarray*} \nf_Q(x,y) &=& (1-x)^2 + (1+x)y - 2y^2, \\;\\;\\; \nk = \\frac{1}{2 m_Q}\\sqrt{ [ m^2_Q - (m_W+m_q)^2][ m^2_Q - (m_W-m_q)^2]},\n\\end{eqnarray*}\nhere $k$ is $W$ (or $q$) momentum in the $Q$--quark rest frame. \\\\\nThe width of the heavy $Q$ decay\n\\[\n  Q \\to q + W (\\to l \\nu)\n\\]\nis given by \\cite{bigi}: \n\\[\n\\Gamma (Q \\rightarrow q + W(\\rightarrow l \\nu) = \n \\frac{G_F^2 m_Q^5}{192 \\pi^3} |V_{Qq}|\n F(\\frac{m_Q^2}{m_W^2};\\frac{m_q^2}{m_Q^2};\\frac{\\Gamma_W^2}{m_W^2}),\n\\] \nwhere\n\\begin{eqnarray*}\n && F(a,b,c) = 2\\int_{0}^{(1-\\sqrt{b})^2} d t \n\\frac{ f_Q(b,t) \\sqrt{1+b^2+t^2 - 2(b+bt+t)} }\n{[(1-at)^2+c]}, \\\\\n &&F(a,0,c) = \\\\\n &&\\frac{2}{a^4} {[c-3(1-a)]A+2a(1-a)-a[3(2-a)c-(2+a)(1-a)^2]B}, \\\\\n&& A = \\ln \\frac{c+1}{c+(1-a)^2}, \\quad\n B = \\frac{1}{a\\sqrt{c}} [ \\arctan (\\frac{1}{\\sqrt{c}}) - \n\\arctan (\\frac{1-a}{\\sqrt{c}})].\n\\end{eqnarray*}\n\n\\subsection{\\it Heavy Quarkonia $(Q \\bar Q)$ Decays} \n\nSuppose that the matrix element of the vector state $V$ decay $V \\to l^+ l^-$\nis given by\n\\[ \n M = g_V e^{\\nu}_V \\bar u(l^+) \\gamma^{\\nu} u(-l^-).\n\\] \nThen\n\\begin{eqnarray*}\n \\Gamma(V \\to l^+ l^-) = \\frac{g^2_V}{12\\pi}M_V, \n\\quad  g_V = \\sqrt{ \\frac{12\\pi \\Gamma(V \\to l^+ l^-)}{M_V}}.\n\\end{eqnarray*}\nDenote $R^2_0 \\equiv 4 \\pi |\\psi(0)|^2$, where $\\psi(0)$ is bound state wave\nfunction in the origin.\n\n\\noindent The width of the decay of the quark antiquark vector state $1^{--}$\nequals: \n\\[\n\\Gamma (1^{--} \\rightarrow l^+ l^-) = N_c\\frac{4 }{3} \\frac{\\alpha^2Q^2_q}\n{M^2} R^2_0. \n\\]\nwhere $N_c = 1(3)$ for colorless (color) quarks, $Q_q$ is the effective \ncharge: \n\\begin{displaymath}\n\\begin{array}{ccccccccc}\n\\rho & = & \\frac{1}{\\sqrt{2}} (u \\bar u - d \\bar d) &\\Rightarrow&\nQ^2_q& =& |\\frac{1}{\\sqrt{2}}(\\frac{2}{3} + \\frac{1}{3})|^2 &=& \\frac{1}{2}, \\\\\n\\omega &=& \\frac{1}{\\sqrt{2}} (u \\bar u + d \\bar d) &\\Rightarrow&\n Q^2_q &=& |\\frac{1}{\\sqrt{2}}(\\frac{2}{3} - \\frac{1}{3})|^2& =&\n \\frac{1}{18}, \\\\\n \\phi &=& s \\bar s  &\\Rightarrow& Q^2_q & = & \\frac{1}{9},& & \\\\\n J \/ \\psi &=& c \\bar c  &\\Rightarrow& Q^2_q & = & \\frac{4}{9},& & \\\\\n \\Upsilon  &=& b \\bar b  &\\Rightarrow& Q^2_q & =& \\frac{1}{9}.& &\n\\end{array}\n\\end{displaymath}\nFor positron annihilation (with $Q_e = 1$) one has: \n\\begin{eqnarray*}\n&&\\Gamma (0^- \\rightarrow \\gamma \\gamma ) = \\frac{4\\alpha^2}{M^2} R_0^2, \\\\\n&&\n\\Gamma (1^{--} \\rightarrow \\gamma\\gamma\\gamma ) = \\frac{16}{9\\pi}\n(\\pi^2-9)\\frac{\\alpha^3}{M^2} R_0^2.\n\\end{eqnarray*}\nFor quarkonia annihilation one gets:\n\\[\n\\Gamma (0^- \\rightarrow \\gamma \\gamma ) = \\frac{12\\alpha^2Q_q^4}{M^2} R^2_0.\n\\]\nFor the two (three) gluon annihilation one need to change :\n$\\alpha^2 Q^4_q \\rightarrow 2\\alpha^2_s \/9$ \n($\\alpha^3 \\rightarrow 5\\alpha^3_s \/18$): \n\\begin{eqnarray*}\n&&\\Gamma (0^- \\rightarrow gg) = \\frac{8\\alpha^2_s}{3M^2} R_0^2, \\\\\n&&\\Gamma (1^{--} \\rightarrow ggg) = \\frac{40}{81\\pi}\n(\\pi^2-9)\\frac{\\alpha^3_s}{M^2} R_0^2.\n\\end{eqnarray*}\n\n\n\\section{\\bf CROSS SECTIONS }\\label{cros}\n\n\\subsection {\\it $e^+ e^-$ Annihilation}\\label{annihil}\nFor pointlike spin--$\\frac{1}{2}$ fermions the differential cross section\nin the cms for $e^+ e^- \\to f \\bar f$ via single photon and $Z$--boson (with\nmass $M_Z$ and total width $\\Gamma_Z$) is given by \\cite{pdg}:\n\\begin{eqnarray}\n\\frac{d \\sigma}{d \\Omega} &=& \\frac{\\alpha^2}{4s}\\beta Q^2_f \n\\left \\{1 + \\cos^2\\vartheta + (1-\\beta^2)\\sin^2\\vartheta \\right \\} \n\\label {sig1} \\\\\n &+& \\frac{\\alpha^2}{4s}\\beta \\chi_2 \\Bigl \\{ V^2_f(1+V^2)\n [1 + \\cos^2\\vartheta + (1-\\beta^2)\\sin^2\\vartheta] \\label{sig2} \\\\\n &&+ \\beta^2 a^2_f(1+V^2)[1+\\cos^2\\vartheta] - 8\\beta V V_f a_f \n \\cos \\vartheta \\Bigr \\} \\nonumber  \\\\\n &-& \\frac{\\alpha^2}{4s}\\beta 2Q_f \\chi_1 \\Bigl \\{ V V_f\n[1 + \\cos^2\\vartheta + (1-\\beta^2)\\sin^2\\vartheta] \\label{sig3} \\\\\n && - 2a_f\\beta \\cos \\vartheta \\Bigr \\},  \\nonumber \n\\end{eqnarray}\nwhere $\\beta = \\sqrt{1 - 4m^2_f\/s}$ is the velocity of the final\nstate fermion in the center of mass, $Q_f$ is the charge of the\nfermion in units of the proton charge,\n\\begin{eqnarray*}\n \\chi_1 &=& \\frac{1}{16 \\sin^2\\vartheta_W \\cos^2\\vartheta_W}\n \\frac{s(s-M^2_Z)}{(s-M^2_Z)^2 + \\Gamma^2_Z M^2_Z}, \\\\\n \\chi_2 &=& \\frac{1}{256 \\sin^4\\vartheta_W \\cos^4\\vartheta_W}\n \\frac{s^2}{(s-M^2_Z)^2 + \\Gamma^2_Z M^2_Z}, \\\\\n V &=& -1 + 4 \\sin^2\\vartheta_W, \\quad \n V_f = 2T_{3f} - 4 Q_f \\sin^2\\vartheta_W, \\quad \n a_f = 2 T_{3f}, \n\\end{eqnarray*}\nhere the subscript $f$ refers to the particular fermion and\n\\begin{eqnarray*}\n T_3 &=& + \\frac{1}{2} \\quad {\\rm for} \\quad \\nu, u, c, t, \\\\\n T_3 &=& - \\frac{1}{2} \\quad {\\rm for} \\quad l^-, d, s, b.\n\\end{eqnarray*}\nThe first (\\ref{sig1}), second (\\ref{sig2}), and third\n(\\ref{sig3}) terms correspond to the $e^+ e^- \\to f \\bar f$\nprocess via single photon annihilation, via $Z$--boson exchange,\nand photon~--~$Z$--boson interference, respectively. \\\\\nFor $s \\gg m^2_f$ (i.e. $\\beta \\to 1$) the annihilation via\nsingle photon exchange (\\ref{sig1}) tends to:\n\\begin{eqnarray}\n\\sigma = \\frac{4 \\pi \\alpha^2}{3s} Q^2_f \\approx \\frac{86.3 Q^2_f}\n{s \\; ({\\rm GeV}^2)} \\; {\\rm nb}.\n\\end{eqnarray}\n\n\\subsection {\\it Two--photon Process at $e^+ e^-$ Collisions}\nWhen an $e^+$ and $e^-$ collide with energies $E_1$ and $E_2$,  they emit\n$d n_1$ and $d n_2$ virtual photons with energies $\\omega_1$ and\n$\\omega_2$ and 4--momenta $q_1$ and $q_2$. In the equivalent\nphoton approximation (EPA) \\cite{eqf}, the cross section for the reaction\n\\begin{eqnarray}\n e^+ e^- \\to e^+ e^- X \\label {sig4}\n\\end{eqnarray}\nis related to the cross section for $\\gamma \\gamma \\to X$ by:\n\\begin{eqnarray}\nd \\sigma_{EPA} (s) \\equiv d \\sigma_{e^+ e^- \\to e^+ e^- X}(s) = d n_1 \\, d n_2 \n d \\sigma_{\\gamma \\gamma \\to X} (W^2), \\label{sig5}\n\\end{eqnarray}\nwhere $s = 4 E_1 E_2$, $\\;\\;$ $W^2 = 4\\omega_1 \\omega_2$ and\n\\[\n d n_i = \\frac{\\alpha}{\\pi} \\Bigl [ 1 - \\frac{\\omega_i}{E_i} \n + \\frac{\\omega_i^2}{2E_i^2} \n - \\frac{m^2_e \\omega_i^2}{(-q^2_i)E_i^2} \\Bigr ] \n \\frac{d \\omega_i}{\\omega_i} \\frac{d q^2_i}{q^2_i}.\n\\]\nAfter integration (including that over $q^2_i$ in the region \n$m^2_e \\omega^2_i \/E_i (E_i \\omega_i) \\leq -q^2_i \\leq (-q^2)_{max}$), the \ncross section is\n\\begin{eqnarray}\n \\sigma_{EPA}(s) = \\frac{\\alpha^2}{\\pi^2} \n\\int^1_{z_{th}} \\frac{dz}{z}\\Biggl [f(z) \\left ( \\ln \\frac{(-q^2)_{max}}\n{m^2_ez} - 1 \\right )^2 - \\frac{\\ln^3 z}{3}\\Biggr] \n \\sigma_{\\gamma \\gamma \\to X} (zs), \\label{sig6}\n\\end{eqnarray}\nwhere $z = W^2\/s$, and\n\\[\nf(z) = (1+ \\frac{z}{2})^2 \\ln\\frac{1}{z} - \\frac{1}{2} (1-z)(3+z).\n\\]\nThe value $(-q^2)_{max}$ depends on properties of the produced\nsystem $X$. For example, $(-q^2)_{max} \\sim m^2_{\\rho}$ for  hadron \nproduction $(X = h)$, and $(-q^2)_{max} \\sim M^2_{ll}$ for\nthe lepton pair production $(X = l^+ l^-)$. \\\\\nFor the production of a resonance of mass $M_R$ and spin $J \\neq 1$  one has:\n\\begin{eqnarray}\n \\sigma_{EPA}(s) &=& (2J+1) \n \\frac{8\\alpha^2 \\Gamma(R \\to \\gamma \\gamma)}{M^3_R} \\label{sig7} \\\\\n &\\times& \\Biggl [ f(\\frac{M^2_R}{s})(\\ln \\frac{s M^2_0}{m^2_eM^2_R}-1)^2 \n - \\frac{1}{3}(\\ln\\frac{s}{M^2_R})^3 \\Biggr], \\nonumber \n\\end{eqnarray}\nwhere $M_0$ is the mass that enters into the from factor of the\n$\\gamma \\gamma \\to R$ transition: $M_0 \\sim m_{\\rho}$ for $R =\n\\pi^0, \\rho^0, \\omega, \\phi, \\ldots$ and $M_0 \\sim M_R$ for $R\n= c \\bar c$ or $b \\bar b$ resonances.\n\n\\subsection {\\it $l \\; h $ Reactions}\nThe reaction of the lepton hadron deep inelastic scattering (DIS)\n\\begin{eqnarray}\nl(k, m_l) \\quad h(P, M) \\; \\to \\; l'(k', m_{l'}) \\quad X, \\label{sig8}\n\\end{eqnarray}\nis described by the following invariant kinematic variables (the\n4--momenta and masses of the particles are denoted in the parentheses)\n\\cite{pdg}: \n\\begin{description}\n\\item[$q = k - k'$] is four--momentum transferred by exchanged particle\n($\\gamma$, $Z$, or $W^{\\pm}$) to the target, \n\\item[$\\nu = \\frac{q \\cdot P}{M} = E - E'$] is the lepton's energy\nloss in the lab frame, $E$ and $E'$ are the initial and final\nlepton energies in the lab,\n\\item[$Q^2 = -q^2 = 2(E E' - \\vec k \\cdot \\vec k') - m^2_l - m^2_{l'},$] if \n $E E'\\sin^2(\\vartheta \/ 2) \\gg m^2_l, \\; m^2_{l'},$ then $Q^2  \\approx \n 4E E'\\sin^2(\\vartheta \/ 2)$, where $\\vartheta$ is the lepton's\nscattering angle in the lab,\n\\item[$x = \\frac{Q^2}{2 M \\nu} = \\frac{Q^2}{2 q \\cdot P},$] in\nthe parton model, $x$ is the fraction of the target hadron's\nmomentum carried by the struck quark,  \n\\item[$y = \\frac{ q \\cdot P}{ k \\cdot P} = \\frac{\\nu}{E},$] is\nthe fraction of the lepton's energy lost in the lab,\n\\item[$W^2 = (P + q)^2 = M^2 + 2 M \\nu - Q^2,$] is the mass\nsquared of the system recoiling against the lepton,\n\\item[$s = (P + k)^2 = M^2 + \\frac{Q^2}{xy}.$] \n\\end{description}\nThe differential cross section of the reaction (\\ref{sig8}) as a\nfunction of the different variables is given by\n\\[\n\\frac{d^2 \\sigma}{ dx d y} = \\nu(s-M^2) \\frac{d^2 \\sigma}{ d\\nu d Q^2} =\n\\frac{2\\pi M\\nu}{E'}\\frac{d^2 \\sigma}{ d\\Omega_{lab}d E'} = \n x (s-M^2) \\frac{d^2 \\sigma}{ d x d Q^2}.\n\\] \nParity conserving neutral current process, $l^{\\pm} h \\to l^{\\pm} X$, \ncan be written in terms of two structure functions $F^{NC}_1(x, Q^2)$\nand $F^{NC}_2(x, Q^2)$:\n\\begin{eqnarray}\n\\frac{d^2 \\sigma}{d x d y} &=& \\frac{4 \\pi \\alpha^2 (s-M^2)}{Q^4} \n \\label{sig9} \\\\\n &\\times& \\Bigl [ (1-y)F^{NC}_2 +y^2 x F^{NC}_1 \n-\\frac{M^2}{(s-M^2)} x y F^{NC}_2 \\Bigr ]. \\nonumber \n\\end{eqnarray}\nParity violating charged current processes, $l h \\to \\nu X$ and\n$\\nu h \\to l X$, can be written in terms of three structure functions \n$F^{CC}_1(x, Q^2)$, $F^{CC}_2(x, Q^2)$, and $F^{CC}_3(x, Q^2)$:\n\\begin{eqnarray}\n\\frac{d^2 \\sigma}{d x d y} &=& \\frac{G^2_F (s-M^2)}{2 \\pi} \n \\frac{M^4_W}{(Q^2  + M^2_W)^2}   \\label{sig10} \\\\\n &\\times& \\Bigl \\{ [(1-y - \\frac{M^2 xy}{(s-M^2)}]F^{CC}_2 +y^2 x F^{CC}_1 \n \\pm (y - \\frac{y^2}{2})x F^{CC}_3 \\Bigr \\}, \\nonumber\n\\end{eqnarray}\nwhere the last term is positive for $l^-$ and $\\nu$ reactions\nand negative for $l^+$ and $\\bar \\nu$ reaction. \n\n\\subsection {\\it Cross Sections in the Parton Model}\nIn the {\\it parton model} framework the reaction\n\\begin{eqnarray}\n h_1 \\quad h_2 \\; \\; \\to \\; \\; C \\quad X, \\label{sig11}\n\\end{eqnarray}\nwhere $C$ is a particle (or group of the particles) with large mass \n(invariant mass) or with high $p_{\\top}$  can be considered  as a result \nof the hard interaction of the one $i$--parton from  $h_1$ hadron with\n$j$--parton from $h_2$ hadron. Then the cross section of the  reaction \n(\\ref{sig11}) can be written as follows:\n\\begin{eqnarray}\n\\sigma(h_1 h_2 \\to C X) = \\sum_{ij} \\int f^{h_1}_i(x_1, Q^2)\n f^{h_2}_j(x_2, Q^2) \\hat \\sigma(i j \\to C) d x_1 d x_2, \\label{sig12}\n\\end{eqnarray}\nwhere sum is performed over all partons, participating in the  subprocess \n$i j \\to C$; $f^{h}_i(x, Q^2)$ is {\\it parton distribution} in $h$--hadron;\n$Q$ is a typical momentum transfer in partonic process $ij \\to C$ and \n$\\hat \\sigma$ is partonic cross section.\n\n\\newpage\n\\subsection {\\it Vector Boson Polarization Vectors}\\label{vecpol}\n\nLet us consider a vector boson with mass $m$ and 4--momentum $k^{\\mu} \\; \n(k^2=m^2)$. Three polarization vectors of this boson can expressed in terms of\n$k^{\\mu}$,\n\\[ \n k^{\\mu} = (E, k_x, k_y, k_z), \\;  k_{\\top} = \\sqrt{ k^2_x + k^2_y} \n\\]\nas folows \\cite{hz}:\n\\begin{eqnarray} \n\\displaystyle\n\\left. \\begin{array}{l}\n \\varepsilon^{\\mu}(k, \\lambda = 2) = \\frac{1}{k_{\\top}}\n (0, \\, k_y, \\, -k_x, 0), \\\\\n  \\varepsilon^{\\mu}(k, \\lambda=1) = \\frac{1}{|\\vec k| k_{\\top}}\n (0, \\, k_x k_z, \\, k_y k_z, \\, -k^2_{\\top}), \\\\\n \\varepsilon^{\\mu}(k, \\lambda = 3) = \\frac{E}{ m |\\vec k|}\n (\\frac{\\vec k^{\\; 2}}{E}, \\, k_x, \\, k_y, \\, k_z). \n\\end{array} \\right \\} \\label{vp1}\n\\end{eqnarray} \nIt is easy to verify that\n\\begin{eqnarray}\n p^{\\mu} \\varepsilon_{\\mu}(k, \\lambda) = 0, \\quad \n \\varepsilon^{\\mu}(k, \\lambda) \\varepsilon_{\\mu}(k, \\lambda') = \n -\\delta^{\\lambda \\lambda'}. \\label{vp2}\n\\end{eqnarray} \nFor $k_{\\top} = 0$ (i.e. $k^{\\mu} = (E, 0, 0, k)$) these polarization\nvectors can be chosen as follows:\n\\begin{eqnarray} \n\\displaystyle\n\\left. \\begin{array}{l}\n \\varepsilon^{\\mu}(k, \\lambda=1) = {}(0, \\, 0, \\, 1, \\, 0), \\\\ \n \\varepsilon^{\\mu}(k, \\lambda = 2) = {}(0, \\, 1, \\, 0, \\, 0), \\\\\n \\varepsilon^{\\mu}(k, \\lambda = 3) = \\frac{1}{m}\n (k, \\, 0, \\, 0, \\, E). \n\\end{array} \\right \\} \\label{vp3}\n\\end{eqnarray}\n\n\\noindent\nGluon is the massless vector boson. Any massless vector boson has only two\npolarization states, $\\lambda=1$ and $2$, on its mass-shell. \\\\\nThe gluon density matrix (in the axial gauge) has the form\n(see Subsection~\\ref{gauges}):\n\\begin{eqnarray} \n\\displaystyle\n\\rho^{\\mu \\nu} = -g^{\\mu \\nu} + \\frac{k^{\\mu} n^{\\nu} + k^{\\nu} n^{\\mu}}{k\\cdot n } -\n\\frac{n^2 k^{mu} k^{\\nu}} {(k\\cdot n)^2} \\; = \\;\n\\epsilon^{\\mu}_{1} \\epsilon^{\\nu}_{1} + \\epsilon^{\\mu}_{2} \\epsilon^{\\nu}_{2},\n\\end{eqnarray}\nwhere $n$ is axial gauge fixing vector and $\\epsilon^{\\mu}_{i}$ are the gluon\npolarization vectors.\nIn the axial gauge there appears an additional condition (see Subsection~\\ref{gauges}): \n\\[ \n\\epsilon^{\\mu}_i  n^{\\mu} = 0, \\;\\; i=1,2 \n\\]\nFor this case polarization vectors $\\epsilon^{\\mu}_g(p, \\lambda=1,2)$ can\nbe chosen as follows:\n\\begin{equation}\n\\epsilon^{\\mu}_g(p, \\lambda) = \\varepsilon^{\\mu}(p, \\lambda) \n - \\frac{ \\varepsilon(p, \\lambda) \\cdot n}{p \\cdot n} p^{\\mu}, \\label{vp5}\n\\end{equation}\nwhere $\\varepsilon^{\\mu}(p, \\lambda)$ are given in (\\ref{vp1}) or (\\ref{vp3}). \\\\\nFor numerical calculations it is convenient to set\n\\begin{equation}\n  n^{\\mu} = (1, \\vec{0})\n\\end{equation}\nAs a result the first two vectors from (\\ref{vp1}) can be used:\n\\begin{eqnarray} \n\\displaystyle\n\\left. \\begin{array}{l}\n \\epsilon^{\\mu}_1 = \\frac{1}{k_{\\top}}\n (0, \\, k_y, \\, -k_x, 0) \\\\\n \\epsilon^{\\mu}_2 = \\frac{1}{|\\vec k| k_{\\top}}\n (0, \\, k_x k_z, \\, k_y k_z, \\, -k^2_{\\top}) \n\\end{array} \\right \\}, \\; k_{\\top} >0; \\;\\;\\;\n\\left. \\begin{array}{l}\n \\epsilon^{\\mu}_1 = (0, \\, 1, \\, 0, \\, 0) \\\\\n \\epsilon^{\\mu}_2 = (0, \\, 0, \\, 1, \\, 0) \n\\end{array} \\right \\}, k_{\\top} =0\n\\label{vp10}\n\\end{eqnarray}\nFor gluon being {\\bf off}-shell we should introduce third ``polarization'' vector:\n\\begin{eqnarray} \n\\displaystyle\n\\rho^{\\mu \\nu} = \\epsilon^{\\mu}_{1} \\epsilon^{\\nu}_{1} + \\epsilon^{\\mu}_{2} \\epsilon^{\\nu}_{2}\n+ \\epsilon^{\\mu}_{3} \\epsilon^{\\nu}_{3}, \\;\\;\n\\epsilon^{\\mu}_{3} \\cdot \\epsilon^{\\mu}_{3} = - \\frac{k^2}{k^2_0}\n\\end{eqnarray}\nwhere one has:\n\\begin{eqnarray} \n\\displaystyle\n \\begin{array}{ll}\nk^2 > 0: \\epsilon^{\\mu}_3 = \\frac{\\sqrt{k^2}}{k_0 |\\vec{k}|} (0, \\, \\vec{k}),  \\\\\nk^2 < 0: \\epsilon^{\\mu}_3 = i \\frac{\\sqrt{|k^2|}}{|k_0| |\\vec{k}|} (0, \\, \\vec{k})\n\\end{array}\n\\label{vp101}\n\\end{eqnarray}\nNote, that for space-like momentum third vector becomes a complex one.  \n\n\\noindent $\\bullet$ {\\bf Two Photons (Gluons) System}\n \nFor the system of two photons (gluons) with momenta $p_1$ and $p_2$ the \npolarization vectors $\\varepsilon^{\\mu}_{1(2)}$ can be written in the\nexplicitly covariant form: \n\\begin{equation}\n\\varepsilon^{\\mu}_i(\\pm) \\, = \\, \\frac{1}{\\sqrt{2\\Delta_3}}\n  \\bigl [ (p_1 p_2) q^\\mu - (q p_2) p_1^{\\mu} - (q p_1) p_2^{\\mu} \n  \\pm i \\varepsilon^{\\mu \\nu \\alpha \\beta} q^{\\nu} \n  p_{1 \\, \\alpha} p_{2 \\, \\beta}  \\bigr]. \\label{vp6}\n\\end{equation}\nwhere sign $+(-)$ corresponds to positive (negative) helicity, $q$ is any \narbitrary vector, which is independent on $p_1$ and $p_2$ (it may be a \nmomentum of some  particle), and \n\\[ \\Delta_3=\\delta_{qp_1p_2}^{qp_1p_2} = (p_1 p_2) \n(2\\> (q p_1) (q p_2) - q^2 (p_1 p_2)). \\]\nThese vectors were considered also in Subsection~\\ref{sbc65}. \\\\\nProjectors on various combinations of the helicity states look as follows:\n\\begin{eqnarray*}\n&&\\frac{1}{2}\n\\left(\\varepsilon^\\mu_1(+) \\varepsilon^\\nu_2(-)  \n    + \\varepsilon^\\mu_1(-) \\varepsilon^\\nu_2(+) \\right)\n = \\frac{1}{2(p_1 p_2)} (p_1^\\nu \\> p_2^\\mu - (p_1 p_2)\\>g^{\\mu\\nu}), \\\\\n&&\\frac{1}{2}\n\\left(\\varepsilon^\\mu_1(+) \\varepsilon^\\nu_2(-)  \n    - \\varepsilon^\\mu_1(-) \\varepsilon^\\nu_2(+) \\right)\n=-\\frac{i}{2\\> (p_1 p_2)}\\varepsilon^{p_1p_2\\mu\\nu}, \\\\\n&&\\frac{1}{2}\n\\left(\\varepsilon^\\mu_1(+) \\varepsilon^\\nu_2(+)  \n    + \\varepsilon^\\mu_1(-) \\varepsilon^\\nu_2(-) \\right)\n=\\frac{1}{2\\Delta_3}\\{2 [(p_1p_2)(q p_1)(q p_2) g^{\\mu\\nu} \\\\\n&& + q^\\mu q^\\nu (p_1 p_2)^2  \n -(p_1 p_2) ((q p_1) p_2^\\mu q^\\nu + (q p_2) p_1^\\nu q^\\mu)] \\\\\n&&+q^2 (p_1 p_2) (p_1^\\nu p_2^\\mu - (p_1 p_2) g^{\\mu\\nu})\\}, \\\\\n&&\\frac{1}{2}\n\\left(\\varepsilon^\\mu_1(+) \\varepsilon^\\nu_2(+)  \n    - \\varepsilon^\\mu_1(-) \\varepsilon^\\nu u_2(-) \\right) = \\\\\n &&\\frac{i}{2\\Delta_3}\\{ ((p_1 p_2) q^\\mu - (q p _1) p_2^\\mu )\n\\varepsilon^{\\nu  q p_1p_2}  \n+((p_1 p_2) q^\\nu - (q p_2) p_1^\\nu)\\varepsilon^{\\mu q p_1p_2} \\},  \\\\ \n&&= \\frac{i\\ (p_1 p_2)}{2\\Delta_3}\n\\left( q^\\mu \\varepsilon^{\\nu q p_1p_2}+q^\\nu \\varepsilon^{\\mu q p_1p_2}\\right.\n(qp_1) \\varepsilon^{p_2 q\\mu\\nu}+ (qp_2) \\left. \\varepsilon^{p_1q\\mu\\nu}\\right).\n\\end{eqnarray*}\n\n\n\n\\section{\\bf MATRIX ELEMENTS }\\label{matrel}\n\n\\subsection {\\it General Remarks}\nIn this Section we present the matrix elements squared $|M|^2$ for various \nprocesses in the Standard Model. Almost all of these $|M|^2$ were presented \nin the book by R.~Gastmans and Tai~Tsun~Wu \\cite{gw}. The symbol $|M|^2$ is \nused to denote the square of the absolute value of the matrix element $M$\nsummed over the {\\bf initial} and {\\bf final} degrees of freedom (polarization \nand color), but {\\bf without} averaging over the {\\bf initial} state \ndegrees of freedom.\n\nSo, one can use  the  well--known crossing  relations to obtain\n$\\overline{|M|^2}$ for processes differing from each other by repositioning the\nfinal and\/or initial particles. The averaged over the initial state degrees of \nfreedom matrix element squared $\\overline{|M|^2}$ can be obtained from\n$|M|^2$ by trivial procedure:\n\\begin{eqnarray*}\n e^+ e^-, \\; e^{\\pm} \\gamma, \\; \\gamma \\gamma &:& \n\\frac{1}{2} \\cdot \\frac{1}{2} \\;({\\rm spin}) \\hspace{22mm} \\; \\Rightarrow \n   \\overline{|M|^2} = \\frac{1}{4} |M|^2, \\\\\n q \\bar q, \\; q q, \\; \\bar q \\bar q &:& \n\\frac{1}{2} \\cdot \\frac{1}{2} \\;({\\rm spin}) \n \\frac{1}{3} \\cdot \\frac{1}{3} \\;({\\rm color}) \\; \\Rightarrow \n   \\overline{|M|^2} = \\frac{1}{36} |M|^2, \\\\\n g q, \\; g \\bar q &:& \n\\frac{1}{2} \\cdot \\frac{1}{2} \\;({\\rm spin}) \n \\frac{1}{8} \\cdot \\frac{1}{3} \\;({\\rm color}) \\; \\Rightarrow \n   \\overline{|M|^2} = \\frac{1}{96} |M|^2, \\\\\n g g &:& \n\\frac{1}{2} \\cdot \\frac{1}{2} \\;({\\rm spin}) \n \\frac{1}{8} \\cdot \\frac{1}{8} \\;({\\rm color}) \\; \\Rightarrow \n   \\overline{|M|^2} = \\frac{1}{256} |M|^2.\n\\end{eqnarray*}\nFor the $2 \\to 2$ processes the differential cross section is\nrelated to the \n$\\overline{|M|^2}$ as follows: \n\\begin{eqnarray}\n \\frac{d \\sigma(2 \\to 2)}{dt} = \\frac{\\overline{|M|^2}}{16 \\pi I^2}, \\quad\n I^2 \\approx s^2,\n\\end{eqnarray}\nwhere $t$ and $I$ are defined in (\\ref{kin11}) and (\\ref{kin13}).\n\nThe notations, used through of this Section, are the same as in \nSection~\\ref{fr}:\n\\begin{description}\n\\item[$e$] is the electric charge of the positron,\n  $\\alpha_{QED} \\equiv \\alpha \n = \\frac{e^2}{4 \\pi} \\approx \\frac{1}{137}$,\n\\item[$Q_f$] is the charge of the quark in units of the positron charge,\n\\item[$g_s$] is the QCD coupling constant, $\\alpha_{QCD} \\equiv \\alpha_s = \n\\frac{g_s^2}{4 \\pi} $,\n\\item[$G_F$] is the Fermi constant. \n\\end{description}\nAs in Section~\\ref{kinem} for the reaction $2 \\to 2$ \n\\begin{eqnarray*}\na(p_1) + b(p_2) &\\to& 1(q_1) + 2(q_2) \\\\\n p_1  + p_2 &=& q_1 + q_2\n\\end{eqnarray*}\nthe Lorentz--invariant Mandelstam variables for reaction are given by \n\\begin{eqnarray*}\n&& s = (p_1  + p_2)^2 = (q_1 + q_2)^2, \\quad  \nt = (p_1  - q_1)^2 = (p_2 - q_2)^2, \\\\ \n&&u = (p_1  - q_2)^2 = (p_2 - q_1)^2, \\\\\n&&s + t + u = m_a^2 + m_b^2 + m_1^2 + m_2^2.\n\\end{eqnarray*}\n\n\\subsection{\\it Matrix Elements} \n\n\\subsubsection{\\it $e^+ e^- \\to f \\bar f$ (no $Z$--boson exchange)}\n\\noindent $\\bullet$ $e^+ e^- \\to l^+ l^-$ ($l \\ne e$, $l = \\mu, \\tau$). \n\\begin{eqnarray}\n|M_e|^2 &=& 8 e^4 \\frac{1}{s^2} \\bigl [ t^2 + u^2 + (m_e^2 + m_f^2)\n(2s - m_e^2 - m_f^2) \\bigr ], \\label{ms2} \\\\\n &=& 8 e^4 \\frac{t^2 + u^2}{s^2}, \\quad {\\rm for} \\; \\; m_e = m_f=0. \\nonumber\n\\end{eqnarray}\n\\noindent $\\bullet$ $e^+ e^- \\to q \\bar q$ \n\\[ \n|M_q|^2  = 3 Q^2_f |M_e|^2.\n\\]\nThe detailed description of the process\n$e^+ e^- \\to f \\bar f$ with $Z$--boson\nexchange is presented in Subsection~\\ref{annihil}.\n\n\\subsubsection{\\it $e^+ e^- \\to e^+ e^-$ (no $Z$--boson exchange)}\n\\begin{eqnarray}\n|M|^2 &=& 8 e^4 \\Bigl \\{ \\frac{1}{s^2} \\bigl [ t^2 + u^2 + 8m^2(s - m^2) \n \\bigr]   \n + \\frac{2}{st} (u - 2 m^2) (u - 6 m^2) \\Bigr \\},  \\label{ms3}  \\\\\n &=& 8 e^4 \\frac{s^4 + t^4 + u^4}{s^2 t^2}, \\quad {\\rm for} \\; \\; m = 0. \n\\nonumber\n\\end{eqnarray}\n\n\\subsubsection{\\it $e^+ e^- \\to \\gamma \\gamma \\gamma$} \n\\noindent $\\bullet$ $e^+(p_1) + e^-(p_2) \\to \\gamma(k_1) + \\gamma(k_2) \n  + \\gamma(k_3), \\;\\;m_e = 0.$\n\\begin{eqnarray}\n|M|^2 = 8 e^6 \\;\\; \\frac{\\sum\\limits_{i=1}^{3} (p_1 k_i) (p_2 k_i) \n \\bigl [ (p_1 k_i)^2 + (p_2 k_i)^2 \\bigr]}\n {\\prod\\limits_{i=1}^{3} (p_1 k_i) (p_2 k_i) }. \\label{ms4}\n\\nonumber\n\\end{eqnarray}\n\n\\noindent\n$\\bullet$ $e^+ e^- \\to \\gamma \\gamma \\gamma, \\;\\; m_e = m \\ne 0.$ \\\\\nFor the case of $s = (p_{e^+} + p_{e^-})^2 \\to 4m^2$, i.e. in the limit \n$p_{e^+} = p_{e^-} = (m, 0)$, \nthe $|M|^2$ is given by \\cite{land}: \n\\begin{eqnarray}\n|M|^2 = 64 e^6 \n \\left [ \\Bigl (\\frac{m -\\omega_1}{\\omega_2 \\omega_3} \\Bigr )^2 \n  + \\Bigl (\\frac{m -\\omega_2}{\\omega_1 \\omega_3} \\Bigr )^2\n  + \\Bigl (\\frac{m -\\omega_3}{\\omega_1 \\omega_2} \\Bigr )^2 \\right ],\n \\label{ms5}\n\\end{eqnarray}\nwhere $\\omega_i$ is $i$--photon energy in cms.\n\n\\subsubsection{\\it $e^+ e^- \\to l^+ l^- \\gamma$} \n\\[\n e^+(p_1) + e^-(p_2) \\to l^+(q_1) + l^-(q_2) + \\gamma(k),\n  \\;\\; m_e = m_{l} = 0.\n\\]\nInvariants:\n\\begin{eqnarray}\n  s &=& 2(p_1 p_2), \\quad t = -2(p_1 q_1), \\quad u = -2(p_1 q_2),\n  \\label{ms6} \\\\\n s' &=& 2(q_1 q_2), \\quad t' = -2(p_2 q_2), \\quad u' = -2(p_2 q_1). \\nonumber\n\\end{eqnarray}\n\n\\noindent $\\bullet$ $l \\ne e$, for example, $e^+ e^- \\to \\mu^+ \\mu^- \\gamma$\n\\begin{eqnarray}\n|M|^2 = -4 e^6 (v_p - v_q)^2 \\frac{t^2 + t'^2 + u^2 + u'^2}{s s'}. \\label{ms7}\n\\end{eqnarray}\n\n\\noindent $\\bullet$ $l = e$, i.e. $e^+ e^- \\to e^+ e^- \\gamma$\n\\begin{eqnarray}\n|M|^2 = -4 e^6 (v_p - v_q)^2 \n \\frac{s s'(s^2 + s'^2) + t t'(t^2 + t'^2) + u u'(u^2 + u'^2)}{s s' t t'}. \n \\label{ms8}\n\\end{eqnarray}\nwhere in (\\ref{ms7}) and (\\ref{ms8}) we use:\n\\begin{eqnarray} \n v_p^{\\mu} \\equiv \\frac{p_1^{\\mu}}{(p_1 k)} - \\frac{p_2^{\\mu}}{(p_2 k)}, \\quad\n v_q^{\\mu} \\equiv \\frac{q_1^{\\mu}}{(q_1 k)} - \\frac{q_2^{\\mu}}{(q_2 k)}.\n\\label{ms9}\n\\end{eqnarray}\n\n\\subsubsection{\\it $e^+ e^- \\to q \\bar q g$}\nFor this reaction the invariants are the same as in (\\ref{ms6}).\n\\begin{eqnarray}\n|M|^2 = 16 e^4 Q^2_f g_s^2 \\frac{t^2 + t'^2 + u^2 + u'^2}{s (q_1k) (q_2 k)}. \n\\label{ms10}\n\\end{eqnarray}\n\n\\subsubsection{\\it $e^+ e^- \\to q \\bar q \\gamma$}\nFor this reaction the invariants are the same as in (\\ref{ms6}).\n\\begin{eqnarray}\n|M|^2 = - 12 e^6 (v_p + Q_f v_q)^2  \\frac{t^2 + t'^2 + u^2 + u'^2}{s s'},\n \\label{ms11}\n\\end{eqnarray}\nwhere $v_p$ and $v_q$ are defined in (\\ref{ms9}). \n\n\\subsubsection{\\it $g g \\to q \\bar q$, $m_q = m \\ne 0$} \nThe final $q \\bar q$--pair can be in color {\\it singlet} or color {\\it octet} \nfinal states.\n\\begin{eqnarray}\n|M_{singl}|^2 &=& 16 g_s^4 \\chi_0 \\left [ \\frac{1}{3} \\right ], \\quad\n|M_{oct}|^2 = 16 g_s^4 \\chi_0 \\left [ \\frac{7}{3} - 6 \\chi_1 \\right ], \n \\nonumber \\\\\n|M_{tot}|^2 &=& |M_{singl}|^2 + |M_{oct}|^2 = 16 g_s^4 \\chi_0 \n  \\left [ \\frac{8}{3} - 6 \\chi_1 \\right ], \\label{ms12}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n \\chi_0 & = & \\frac{m^2 - t}{m^2 - u} + \\frac{m^2 - u}{m^2 - t} \n + 4 \\left ( \\frac{m^2}{m^2 - t} + \\frac{m^2}{m^2 - u} \\right )\n -4 \\left ( \\frac{m^2}{m^2 - t} + \\frac{m^2}{m^2 - u} \\right )^2  \\label{ms13}  \\\\\n \\chi_1 & = & \\frac{(m^2 - t)(m^2 - u)}{s^2} \\label{ms14}\n\\end{eqnarray}\nFor $m_q = 0$,\n\\[\n\\chi_0 = \\frac{t^2 + u^2}{ut}, \\quad \\chi_1 = \\frac{ut}{s^2}.\n\\]\n\n\\subsubsection{\\it $\\gamma g (\\gamma \\gamma) \\; \\to \\; f \\bar f$}\n\n\\noindent $\\bullet$ $g \\gamma \\to q \\bar q: \\quad$ \n$\\quad  \n|M|^2 = 32 g_s^2 e^2 Q^2_f \\chi_0.\n$\n\n\\noindent $\\bullet$ $\\gamma \\gamma \\to q \\bar q: \\quad$ \n$\\quad  \n|M|^2  = 24 e^4 Q^4_f \\chi_0.\n$\n\n\\noindent $\\bullet$ $\\gamma \\gamma \\to e^+ e^-:  \\quad$ \n$ \\quad  \n|M|^2  = 8 e^4 \\chi_0.\n$\n\n\\subsubsection{\\it $q \\bar q \\; \\to \\; Q \\bar Q$, $m_q = 0$, $m_Q = m \\ne 0$}\n\\begin{eqnarray}\n |M|^2  = 16 g^4_s \\frac{t^2 + u^2 + 2m^2(2s - m^2)}{s^2}. \\label{ms15}\n\\end{eqnarray}\n\n\\subsubsection{\\it $q q \\; \\to \\; q q$, $m_q = 0$} \n\\begin{eqnarray}\n |M|^2  = 16 g^4_s \\left [\\frac{s^4 + t^4 + u^4}{t^2u^2} - \\frac{8}{3}\n\\frac{s^2}{tu} \\right ]. \\label{ms16}\n\\end{eqnarray}\n\n\\subsubsection{\\it $q \\bar q \\; \\to \\; q \\bar q$, $m_q = 0$} \n\\begin{eqnarray}\n |M|^2  = 16 g^4_s \\frac{1}{s^2 t^2} [s^4 + t^4 + u^4 - \\frac{8}{3}stu^2].\n \\label{ms17}\n\\end{eqnarray}\n\n\\subsubsection{\\it $g g \\; \\to \\; g g$}\n\\begin{eqnarray}\n |M|^2  = 288 g^4_s \\frac{(s^4 + t^4 + u^4)(s^2+t^2+u^2)}{s^2t^2u^2}.\n \\label{ms18}\n\\end{eqnarray}\n\n\\subsubsection{\\it $ f_1 \\bar f_2 \\to W^{\\ast} \\to f_3 \\bar f_4$}\n\\[\n f_1(p_1) + \\bar f_2(p_2) \\to f_3(p_3) + \\bar f_4(p_4), \\quad\n m_{1,2,3,4} \\ne 0.\n\\]\n\\begin{eqnarray}\n|M|^2 = C 128 G^2_F M^4_W \\frac{(p_1 p_4) (p_2 p_3)}\n {(s - M^2_W)^2 + \\Gamma^2_W M^2_W}, \\label{ms19}\n\\end{eqnarray}\nwhere $C=1$ for $l^- \\bar \\nu \\to l'^- \\bar \\nu'$, \n$C=3$ for $l^- \\bar \\nu \\to q \\bar q' ( q \\bar q' \\to l^- \\bar \\nu)$, and \n$C=9$ for $q_1 \\bar q_2 \\to q_3 \\bar q_4$, $M_W$ and $\\Gamma_W$ are the mass \nand total width of the $W$--boson.\n\n\\subsubsection{\\it $ l^- \\bar \\nu \\to d \\bar u g $}\n\n\\[\n l^-(p_1) + \\bar \\nu(p_2) \\to d(p_3) + \\bar u(p_4) + g(k), \\quad\n m_{d, u} \\ne 0.\n\\]\n\\begin{eqnarray}\n|M|^2 &=& 256 G^2_F M^4_W g_s^2 \\frac{ A_1 - A_2 - A_3}\n {(s - M^2_W)^2 + \\Gamma^2_W M^2_W}, \\label{ms20} \\\\\n A_1 &=& \\frac{1}{(k p_3) (k p_4)} \\Bigl \\{ s \\bigl [(p_1 p_4)^2 + (p_2 p_3)^2\n \\bigr ] \\nonumber \\\\\n &-& (m^2_u+m^2_d) \\bigl [\\frac{s}{2} ((p_1 p_4)+(p_2p_3)) \n  - (p_1p_3)(p_2p_3) - (p_1p_4)(p_2p_4)\\bigr] \\Bigr \\}, \\nonumber \\\\\n A_2 &=& \\frac{2m_u^2}{(k p_4)^2} (p_2 p_3) \\bigl[(p_1 k) + (p_1 p_4) \\bigr],\n  \\quad \n A_3 = \\frac{2m_d^2}{(k p_3)^2} (p_1 p_4) \\bigl[(p_2 k) + (p_2 p_3) \\bigr].\n  \\nonumber \n\\end{eqnarray}\n\n\n\\section{\\bf MISCELLANEA}\\label{misc} \n\n\\subsection{\\it Miscellanea}\\label{miscel} \n\n\n\\noindent $\\bullet$ Let us consider the recursion $A_n = aA_{n-1}+bA_{n-2}$ \nfor given $A_0$ and $A_1$. Then \n\\begin{eqnarray*}\n&& A_n = \\alpha z_1^n + \\beta z_2^n,\n \n  z_{1,2} = \\frac{a}{2}[1 \\pm \\sqrt{1+4b\/a^2}], \\quad\n\\alpha = \\frac{A_1-z_2A_0}{z_1-z_2}, \\quad\n\\beta = \\frac{z_1A_0-A_1}{z_1-z_2}.\n\\end{eqnarray*}\n\n\\noindent $\\bullet$ Various representations of the {\\it Dirac} \n$\\delta$--function:\n\\begin{eqnarray*}\n&& \\delta(x) \\equiv \\frac{1}{(2 \\pi)} \\int^{\\infty}_{-\\infty}\n e^{ixt} dt, \\\\\n&& \\delta(x, \\alpha) = \\frac{\\alpha}{\\pi(\\alpha^2 x^2 + 1)}, \\; \n \\alpha \\to \\infty; \\quad \n \\delta(x, \\beta) = \\frac{\\beta}{\\pi(x^2 + \\beta^2)}, \\; \n \\beta \\to 0, \\\\ \n&& \\delta(x, \\alpha) = \\frac{\\alpha}{\\sqrt{\\pi}}\n e^{-\\alpha^2 x^2}, \\;\\;\\;  \\alpha \\to \\infty, \\quad \n \\delta(x, \\alpha) = \\frac{\\alpha}{\\pi}\n \\frac{ \\sin(\\alpha x)}{(\\alpha x)}, \\;  \\alpha \\to \\infty, \\\\ \n&& \\frac{1}{x \\pm i\\varepsilon} = \\p \\frac{1}{x} \\mp i \\pi \\delta(x).\n\\end{eqnarray*}\n\n\\noindent $\\bullet$ {\\it Step}--functions $\\Theta (x)$ and $\\varepsilon (x)$\n\\begin{eqnarray*}\n&& \\Theta (x) \\equiv \\frac{1}{(2 \\pi i)} \\int^{\\infty}_{-\\infty}\n \\frac{e^{ixt}}{t - i \\varepsilon} dt \\; = \\; \n\\left \\{ \\begin{array}{rl} 1, & x>0 \\\\ 0, & x<0 \\end{array} \\right. \n \\\\\n&& \\varepsilon (x) \\equiv \\frac{1}{(i \\pi )} \\p \\int^{\\infty}_{-\\infty}\n \\frac{e^{ixt}}{t} dt \\; \\hspace{5mm} = \\;\n\\left \\{ \\begin{array}{rl} 1, & x>0 \\\\ -1, & x<0 \\end{array} \\right. \n\\end{eqnarray*}\n\n\\noindent $\\bullet$ \n\\begin{eqnarray*}\n&& \\frac{1}{(a - i \\varepsilon)^k} = \\frac{i^k}{\\Gamma(k)} \n\\int^{\\infty}_0 e^{i \\alpha (-a + i \\varepsilon)} \\alpha^{k-1} d \\alpha, \n \\; k \\ge 0, \\\\\n&& \\int^{\\infty}_0 (e^{i t a} - e^{i t b}) \\frac{d t}{t} \\; = \\; \n \\ln \\biggl ( \\frac{b + i \\varepsilon}{a + i \\varepsilon} \\biggr ).\n\\end{eqnarray*}\n\n\\subsection{\\it Properties of Operators} \n\nThe various properties of the operators can be found, for example, in\n\\cite{velt,louis,wilcox}. Let $f(A)$ be any function from the operator \n(matrix) $A$, \nwhich can expanded into series with respect to operators (matrices) $A^n$:\n\\[ f(A) = \\sum_{n=0}^\\infty c_n A^n.\n\\]\n\n\\noindent $\\bullet$ Let $\\xi$ be a parameter, then: \n\\begin{eqnarray*}\n&& e^{\\xi A} e^{-\\xi A} \\, =\\, 1, \\quad e^{\\xi A} A e^{-\\xi A}\\,=\\, A,\n  \\quad e^{\\xi A} A^n e^{-\\xi A} \\, =\\, A^n, \\quad\n   e^{\\xi A} f(A) e^{-\\xi A}\\,=\\, f(A).\n\\end{eqnarray*}\n\n\\noindent $\\bullet$ Let $A$ and $B$ be noncommuting operators, $\\xi$ and\n$n$ be parameters ($n$ integer). Then: \n\\begin{eqnarray*}\n&& e^{\\xi A} B^n e^{-\\xi A} = (e^{\\xi A} B e^{-\\xi A})^n, \\\\\n&& e^{\\xi A} F(B) e^{-\\xi A} = F(e^{\\xi A} B e^{-\\xi A}), \\\\\n&& e^{\\xi A} B e^{-\\xi A} = B + \\xi[A,B] + \\frac{\\xi^2}{2!} [A,[A,B]] \n+ \\frac{\\xi^3}{3!} [A,[A,[A,B]]] + \\cdots\n\\end{eqnarray*}\n\n\\noindent $\\bullet$ Let $A$ be an operator and there exists the inverse \noperator $A^{-1}$. Then for any integer $n$ :\n\\[ \n A B^n A^{-1} = (A B A^{-1})^n, \\qquad   A f(B) A^{-1} = f(A B A^{-1}).\n\\]\n\n\\noindent $\\bullet$ Let $A(x)$ be an operator, depending on the scalar\nvariable\n$x$, then \n\\begin{eqnarray*}\n  \\frac{d A^{-1}(x)}{d x} = -A^{-1}(x) \\frac{d A(x)}{d x} A^{-1}(x),\n  \\quad \\frac{d e^{A(x)}}{d x} = \\int_0^1 e^{(1-t)A(t)} \n \\frac{d A(t)}{d t} e^{tA(t)} dt.\n\\end{eqnarray*}\n\n\\subsection {\\it The Baker-Campbell-Hausdorff Formula}\n\n Let $A$ and $B$ be non--commuting operators, then :\n\\begin{equation} \ne^A \\; e^B = e^{ \\sum^{\\infty}_{i = 1} Z_i}, \\label{misc10}\n\\end{equation} \nwhere\n\\begin{eqnarray}\n Z_1 & = & A+B; \\label{misc1} \\\\ \n Z_2 &=& \\frac{1}{2}[A,B] ; \\label{misc2} \\\\ \n Z_3 & = & \\frac{1}{12} \\bigl [A,[A,B]\\bigr] + \\frac{1}{12}\\bigl [[A,B],B\n\\bigr]; \\label{misc3} \\\\ \n Z_4 & = & \\frac{1}{48}\\Bigl[A,\\bigl[[A,B],B\\bigr]\\Bigr] + \n  \\frac{1}{48}\\Bigl[\\bigl[A,[A,B]\\bigr],B\\Bigr]; \\label{misc4} \\\\ \n Z_5 & = & \\frac{1}{120}\\biggl[\\Bigl[A,\\bigl[[A,B],B\\bigr]\\Bigr],B\\biggr] + \n  \\frac{1}{120} \\biggl[A,\\Bigl[\\bigl[A,[A,B]\\bigr],B\\Bigr]\\biggr] \n \\label{misc5} \\\\ \n   & - & \\frac{1}{360} \\biggl[A,\\Bigl[\\bigl[[A,B],B\\bigr],B\\Bigr]\\biggr] - \n \\frac{1}{360} \\biggl[\\Bigl[A,\\bigl[A,[A,B]\\bigr]\\Bigr],B\\biggr] \\nonumber \\\\ \n   & - & \\frac{1}{720}\\biggl[A,\\Bigl[A,\\bigl[A,[A,B]\\bigr]\\Bigr]\\biggr] - \n \\frac{1}{720} \\biggl[\\Bigl[\\bigl[[A,B],B\\bigr],B\\Bigr],B\\biggr], \\ldots \n\\nonumber \n\\end{eqnarray}\nThe other terms can be evaluated from the relation (see \\cite{velt,wilcox}): \n\\begin{eqnarray} \n \\sum_{k=0}^{\\infty} \\frac{1}{(k+1)!} [\\![ Z^k,Z' ]\\!] = A + \n \\sum_{j=0}^{\\infty} \\xi^j \\frac{ [\\![ A^j,B ]\\!] }{j!}, \\label{o1} \n\\end{eqnarray}\nwhere $e^Z = e^{\\xi A} e^{\\xi B}$; $Z = \\sum_{n=1}^{\\infty} \\xi^n Z_n$; \n$Z' = \\sum_{n=1}^{\\infty} n \\xi^{n-1} Z_n$.\nThe repeated commutator bracket is defined as follows\n\\begin{eqnarray*}\n [\\![A^0,B ]\\!] = B, \\quad [\\![ A^{n+1},B ]\\!] = \\biggl [ A, [\\![\nA^n,B ]\\!] \\biggr ].\n\\end{eqnarray*}\nSince relation (\\ref{o1}) must be satisfied identically in $\\xi$, one can \nequate the coefficients of $\\xi^j$ on the two sides of this relation. In\nparticular,  $j=0,1,2,3,4$ gives (\\ref{misc1}, \\ref{misc2}, \\ref{misc3}, \n\\ref{misc4}, \\ref{misc5}), respectively.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}Introduction}\n\nThe propagation of mechanical vibrations in particulate matter is a nontrivial phenomenon that has implications in several fields, both in basic and applied research~\\cite{deGennes1999,Porter2015}. The complexity of the contact network~\\cite{Radjai1996,Lherminier2014} and the nonlinearity of the interaction force between particles~\\cite{Popov2010} determines how the waves propagate in granular media~\\cite{Norris1997}. In dry packings, the repulsive nonlinear interaction between two spheres with radius $R$ and elastic modulus $E$ relies on the Hertz contact force, $F_H \\propto ER^{1\/2}\\delta^{3\/2}$~\\cite{Landau1986}. The latter implies that the contact stiffness increases rapidly with the deformation $\\delta$ and vanishes in absence of mechanical contact. In the dry case, there is thus no tensile force, and the particles can eventually loose contact~\\cite{Tournat2004}. In one-dimensional lattices of particles~\\cite{Nesterenko2001}, the nonlinear elastic interaction given by the Hertzian contact dictates the dynamics; a mechanical impact can propagate weakly to strongly nonlinear waves~\\cite{Daraio2005,Job2005} depending on the amplitude of the pulse. Similarly, in random granular packing, a low-amplitude mechanical excitation generates a linear ballistic waves, both in the longitudinal (namely a {\\em P-wave}) and in the transverse (namely a {\\em S-wave}) directions~\\cite{Norris1997,Makse2004,Johnson2005}. The coherent perturbation, resulting from an ensemble average~\\cite{Page1995}, travels straight from the source to the receiver, as in an effective medium~\\cite{Norris1997,Makse2004}. Owing to the randomness, the coherent pulse is followed by an incoherent and long-lasting {\\em coda wave}, which corresponds to the multiple scattering~\\cite{Snieder2006} of the initial excitation, across the contact network. Interestingly, the ballistic waves possess the reminiscent features of the microscopic scale~\\cite{Makse2004}. Recently, a continuous description bridging the linear and the nonlinear regimes in random packing of particles has been proposed~\\cite{Wildenberg2013}, depending on the confinement pressure applied to the packing and the amplitude of the propagating impulse~\\cite{Santibanez2016}.\\\\\n\nWeakening of granular materials is another nonlinear mechanism that is reported to occur in particulate systems. Its origin is a softening induced by an acoustic fluidization mechanism~\\cite{Jia2011, Espindola2012, Giacco2015, Olson2015}. Such nonlinear process has been demonstrated to be responsible for triggering secondary earthquakes after the occurrence of a main seismic event~\\cite{Johnson2005}. The propagation of the mechanical impulse interacts with the grains in the material and mobilizes the particles with weaker contacts~\\cite{Johnson2008, Johnson2016}. This leads to a modification of the contact network that can trigger major faults and lead to the emission of new events~\\cite{Johnson2005,Johnson2008}. Since the material weakening involves weakly consolidated contacts, it can be modified by the presence of an interstitial fluid, owing to an enhanced cohesion due to capillary effects~\\cite{Dorostkar2018} or to the viscous lubrication between the grains~\\cite{Galaz2018}.\\\\\n\nWet granular media are ubiquitous in nature, as for instance the sediments, the mud, or the sand of a beach. In this paper, we look at understanding how the presence of an interstitial fluid can affect the nonlinear dynamics of particulate matter. The behavior of wet particles has been investigated in recent years due to the great number of industrial applications, from mining to food and pharmaceutical industries. A wet granular medium has a fluid phase that partially occupies the interstitial volume available between grains and their motions are likely determined by interactions mediated by the fluid. In the literature, it has been shown that viscous forces, surface tension, and capillary bridges among others, have a great importance in systems where the energy dissipation occurs due to liquid films trapped either in the asperities~\\cite{Brunet2008} or nearby the contact region~\\cite{Marshall2011}. On one hand, capillarity bridges between grains~\\cite{Semprebon2016} have been studied owing to significant effects on the static cohesion of granular packings~\\cite{Pacheco2012,Saingier2017}. On the other hand, wave propagation in wet granular media relies on the dynamics of the interstitial fluid~\\cite{Herminghaus2005,Moller2007}, which generates a viscous repulsion~\\cite{Donahue2010,Arutkin2017} that generally cannot be neglected at acoustic frequencies; the fluid modifies the acoustic features due to a velocity-dependent viscous contribution~\\cite{Herbold2006,Job2008,Griffiths2010}, in addition to increasing the overall dissipation~\\cite{Brunet2008}. When the fluid is highly confined, in the lubrication regime~\\cite{Davis1986}, and under high-frequency or high-amplitude vibrations, the viscous forces can ultimately induce elastic deformations of the particles, via an elastohydrodynamic interaction~\\cite{Leroy2011,Leroy2012,Villey2013,Wang2015}. Understanding all these behaviors is fundamental for unraveling all the phenomena ranging from the stability of a pile of wet grains~\\cite{Scheel2008} to the dynamics of dense suspensions~\\cite{Waitukaitis2012,Han2016,Peters2016,Buttinoni2017}.\\\\\n\nIn this paper, we aim (i) at ruling out the mechanisms involved in the propagation of mechanical impulses in a wet model granular medium and (ii) at unraveling how the fluid can affect the dynamic weakening of such a material. In Sec.~\\ref{sec:setup_observations}, we present the experimental setup, the protocols of analysis, and a description of the experimental observations. In Sec.~\\ref{sec:model}, we analyze and interpret the features of the high-frequency spectrum of the transmitted pulses, in terms of the propagation velocity. These features allow probing the elasticity of the medium, and how it is affected by the presence of an interstitial fluid. In Sec.~\\ref{sec:weakening}, we analyze the low-frequency spectrum of the mechanical response. Our measurements reveal the elastic weakening of dry granular media at low confinement and high amplitude, which disappears either at large confinement pressure or when the contacts are lubricated by a sufficiently viscous fluid. Finally, Sec.~\\ref{sec:conclusion} summarizes the results and the observations presented in this paper.\\\\\n\n\\section{\\label{sec:setup_observations} Experimental Setup and observations}\n\n\\begin{figure}[b]\n\\begin{center}\n\\includegraphics[width=0.450\\textwidth]{1808_03150_v2_Fig_1.pdf}\n\\caption{\\label{fig:setup} Sketch of the experimental setup depicting a cylindrical granular medium enclosed in a soft elastic sheet, the shock initiator pendulum, and the positions of the dynamic sensors.}\n\\end{center}\n\\end{figure}\n\nThe experimental sample under study (see Fig.~\\ref{fig:setup} and Ref.~\\cite{Santibanez2016}) consists in a packing of approximately $3000$ spherical glass particles (density $\\rho_g=2400$~kg\/m$^3$, radius $R_g=2.5$~mm, Young modulus $E_g=69$~GPa, Poisson's ratio $\\nu_g=0.2$, and surface roughness $Ra\\simeq10$~nm~\\cite{Brunet2008,Buttinoni2017}) confined inside a thin deformable latex sheet. The sheet is hermetically sealed and clamped between two square plates made of a rigid plastic. The two plates are $1$~cm thick and have a $5$~cm in diameter circular aperture. Thin lateral holes on the side of the plates allow the emergence of sensors cables and a vacuum hose. The soft sheet allows maintaining a controlled isotropic stress on the granular medium, by evacuating the interstitial air from the hose with a vacuum pump. While the sample is set under pressure, it can be molded by hand in the form of a cylinder of length $L_s=15$~cm and radius $R_s=2.5$~cm that fits in the holes of the two supporting plastic plates. This finally leads to a solidlike granular medium with compactness approximately equal to the random close packing fraction, $\\phi_s\\simeq0.63$. The pump allows reaching a hydrostatic pressure of as high as $P_0\\approx83$~kPa, which is probed by a static pressure sensor ({\\em Honeywell 19C015PV5K} with its {\\em INA114} low-noise amplifier). At one extremity of the sample, a dynamic force sensor ({\\em PCB Piezotronics 208C01}) is placed in direct contact with the grains through a central and hermetic hole cut in the sheet. At the opposite side of the sample, a miniature accelerometer ({\\em PCB Piezotronics 352A24}) is located on the axis of the cylindrical sample. A single short mechanical impulse is initiated in the sample by impacting the back of the dynamic force sensor with a $21$~cm long pendulum. The impacting head of the pendulum consists in a piece of brass in front of which a spherical glass particles has been glued. The contact duration depends on the mass ($57$~g) of the head~\\cite{Landau1986}, which has been chosen so that the initial excitation is about a fraction of a millisecond~\\cite{Santibanez2016}. A collision thus generates a broadband excitation from dc to few kilohertz. The strength of the impact is controlled by adjusting the initial release angle of the pendulum. At the opposite end, the accelerometer records the outgoing pulse that travels through the sample. The signals of the dynamic force sensor and the accelerometer are routed through a signal conditioner ({\\em PCB Piezotronics model 482C}) and acquired simultaneously, together with the signal of the static pressure, with an analog-to-digital converter ({\\em National Instruments USB-6356}) at $500$~kS\/s sample rate.\\\\\n\n\\begin{figure*}\n\\includegraphics[width=0.900\\textwidth]{1808_03150_v2_Fig_2.pdf}\n\\caption{\\label{fig:signals_low_P} Examples of experimental data at low confinement pressure ($P_0=3.2$~kPa). The left column (0) shows the input force for each experiment: the solid line corresponds to the dry configuration, the dashed line is $\\mu=1$~Pa.s and the dotted line is $\\mu=30$~Pa.s, whose acceleration outputs are shown in the next three columns (1)--(3), respectively. Rows (a)--(c) correspond to different strength of the initial excitation, as shown in the right-side text labels.}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.900\\textwidth]{1808_03150_v2_Fig_3.pdf}\n\\caption{\\label{fig:signals_high_P} Examples of experimental data at high confinement pressure ($P_0=83$~kPa). The left column (0) shows the input force for each experiment: the solid line corresponds to the dry configuration, the dashed line is $\\mu=1$~Pa.s and the dotted line is $\\mu=30$~Pa.s, whose acceleration outputs are shown in the next three columns (1)--(3), respectively. Rows (a)--(c) correspond to different strength of the initial excitation, as shown in the right-side text labels.}\n\\end{figure*}\n\nThe experimental setup allows probing the acoustic response of dry and wet granular media by measuring the speed of the waves that propagate through the samples. The wet samples are prepared by homogeneously filling a fraction of the interparticle space with a viscous fluid. The amount of added liquid corresponds to a third of the available volume; given the estimated packing fraction $\\phi_s$, the liquid thus approximately occupies $12\\%$ of the total volume of a sample. If all the spherical particles were homogeneously lubricated~\\cite{Galaz2018}, the liquid coating would have a thickness $D_{coat}=R_g([1+(1-\\phi_s)\/3\\phi_s]^{1\/3}-1)\\simeq 150$~$\\mu$m. Before pouring the wet particles into the soft elastic sheet, we make sure that all the grains are uniformly coated with fluid by carefully stirring the mixture in a container. After having used a fluid, the grains are washed several times with alcohol and then gently dried in an oven. The fluids are silicone oils from {\\em Sigma Aldrich}. Four different fluids' viscosity were considered in this study, ranging over more than two decades, from $\\mu=0.1$~Pa.s to $\\mu=30$~Pa.s at room temperature. The sound speed and the mass density of these fluids are close to the features of water, $c_f\\simeq1500$~m\/s and $\\rho_f\\simeq1000$~kg\/m$^3$. Their surface tension is $\\gamma_f\\simeq21.5$~mN\/m~\\cite{Langley2017} and their storage modulus is $G\\simeq5$~kPa~\\cite{Oswald2014}, both independently of the viscosity. In particular, these fluids remain Newtonian below a critical flow's shear rate $\\dot{\\gamma}_f=G\/\\mu$~\\cite{Oswald2014,Guyon2001}. This condition is always satisfied in all our experiments, as demonstrated in Sec.~\\ref{sec:model}. Above the critical shear rate, the behavior of these fluids is known to be relatively independent of the viscosity~\\cite{Oswald2014,Barlow1964}.\\\\\n\nA set of typical waveforms propagated through different samples under $P_0=3.2$~kPa and $P_0=83.0$~kPa confinement pressures is shown in Figs.~\\ref{fig:signals_low_P} and~\\ref{fig:signals_high_P}, respectively. In both figures, the rows stand for three different excitation amplitude. The leftmost columns show the initial force perturbation as a function of time, resulting from the impact of the pendulum. The short initial impact is about $0.2$~ms wide: its spectrum (not shown; see Ref.~\\cite{Santibanez2016}) extends from dc to $8$~kHz approximately. The three rightmost columns present the acceleration of the transmitted waves, first in a dry medium and then with two different interstitial viscosities. The long-lasting transmitted {\\em coda wave} corresponds to the multiple scattering of the incident pulse~\\cite{Snieder2006}, along different paths through the random network of particles. The very first transmitted event [see the magnified plots in Figs.~\\ref{fig:wave_speed}(a) and~\\ref{fig:wave_speed}(b)] corresponds to the fastest ballistic contribution~\\cite{Page1995}, that is a longitudinal pressure wave traveling straight from the source to the receiver~\\cite{Norris1997, Makse2004, Johnson2005}. In all the transmitted signals presented in Figs.~\\ref{fig:signals_low_P} and~\\ref{fig:signals_high_P}, one clearly distinguishes two distinct frequency components. On one hand, the ballistic wave carries a high-frequency content, lying in the range of $1$~kHz [see the rise time in Figs.~\\ref{fig:wave_speed}(a) and~\\ref{fig:wave_speed}(b)]. This spectrum matches the bandwidth of the excitation attenuated by the frequency-dependent scattering~\\cite{Page1995} and the viscoelastic dissipation at the contact between grains~\\cite{Job2005, Popov2010}. In the presence of a fluid, the amplitude of the ballistic pulse decreases slightly more: the viscous dissipation enhances the attenuation~\\cite{Herbold2006,Job2008,Brunet2008}. The analysis of the speed of the ballistic wave is presented in more details in Sec.~\\ref{sec:model}. On the other hand, the transmitted waves shown in Figs.~\\ref{fig:signals_low_P} and~\\ref{fig:signals_high_P} also exhibit a long-lasting, superimposed, low-frequency oscillation of the order of $100$~Hz. It has been demonstrated to rely on a longitudinal resonance of the sample~\\cite{Santibanez2016} and to reveal the nonlinear weakening~\\cite{Jia2011, Espindola2012, Olson2015, Giacco2015, Johnson2008, Johnson2016} of the granular matter. The detailed analysis of the low-frequency oscillation is presented in Sec.~\\ref{sec:weakening}.\\\\\n\n\\begin{figure}[b]\n\\includegraphics[width=0.450\\textwidth]{1808_03150_v2_Fig_4}\n\\caption{\\label{fig:wave_speed} Examples of the time of flight (tof) measurement for (a) dry and (b) wet ($\\mu=30$~Pa.s) cases, at low confinement pressure. Wave speed as a function of the magnitude of the initial collision force in dry and wet granular media, for both low (c) and high (d) confinement pressures. Markers correspond to different interstitial fluid viscosity: $\\circ:$ dry, $\\triangle:0.1$~Pa.s, $\\square:1$~Pa.s, $\\triangledown:10$~Pa.s, and $\\diamond:30$~Pa.s. The dashed lines are guides for the eyes; the vertical arrows point toward increasing viscosities.}\n\\end{figure}\n\nThe measurements of the speed of the ballistic pulses are presented in Fig.~\\ref{fig:wave_speed}. The wave speed $c$ is estimated as the distance $L_s$ between the input (force sensor) and the output (accelerometer) divided by the time of flight. The latter is measured as the time difference between the earliest in and out events [see for instance the dashed line in Figs.~\\ref{fig:wave_speed}(a) and~\\ref{fig:wave_speed}(b)], which are detected systematically as the instants at which the amplitude of each waveform emerges above a threshold (three times the average noise level). In Figs.~\\ref{fig:wave_speed}(c) and~\\ref{fig:wave_speed}(d), we show the propagation velocity for both low and high confinement pressures, respectively. In both cases, the wave speed is measured first in a dry medium as a reference, then with the four fluids with different viscosities. The dry measurements at low confinement pressure, $P_0=3.2$~kPa, are shown as circles in Fig.~\\ref{fig:wave_speed}(c): the log-log representation reveals that the wave speed $c$ approximately increases as a power law of the magnitude of the excitation, as it is predicted from a Hertzian interaction~\\cite{Norris1997, Makse2004, Santibanez2016, Wildenberg2013}, see Sec.~\\ref{sec:model}. In contrast, with interstitial fluids, the wave speed increases more slowly with the strength of the impact and quickly tends to a constant value, independent of the input force, at large viscosities. Remarkably, the propagation velocity noticeably increases with the viscosity of the fluid; this nontrivial feature has been observed yet in one-dimensional granular media~\\cite{Herbold2006,Job2008} and is analyzed in Sec.~\\ref{sec:model}. On the opposite, at the highest confinement pressure, $P_0=83$~kPa, the propagation velocity shown in Fig.~\\ref{fig:wave_speed}(d) appears independent on the amplitude of the perturbation in the dry configuration: the dynamical stress being negligible compared to the static pressure, the response is linear. With fluid, the wave speed does not depend either on the perturbation strength and a weaker dependence of the wave speed with the viscosity of the fluid is observed, compared to the low confinement case.\\\\\n\nThese observations suggest a competition between the elastic deformations of the particles resulting from the confinement or the dynamic pressure, and an effect of the viscous fluid, which likely resides in between the particles and at the periphery of their contacts~\\cite{Marshall2011}. Here, it is possible to rule out the contribution of capillary effects since all our fluids have the same surface tension, $\\gamma_f\\simeq21.5$~mN\/m, independently of their viscosity~\\cite{Langley2017}: it thus cannot explain the viscosity-dependent wave speed observed in the wet media. Moreover, equating the Laplace pressure~\\cite{Guyon2001} to the confinement pressure, $P_0\\simeq \\gamma_f\/R_c$, indicates excessively small capillary bridges curvatures, compared to the size of the particles and the liquid's filling fraction, to produce a significant contribution: $R_c\\simeq P_0\/\\gamma_f\\sim 7$~$\\mu$m at $P_0=3.2$~kPa and $R_c\\sim 0.25$~$\\mu$m at $P_0=83$~kPa. The next section aims at deriving an alternative framework, in order to relate the increase of the wave speed, i.e., the enhancement of the effective elasticity of the medium, as an effect of the viscous flow in the interstices of the particles.\\\\\n\n\\section{\\label{sec:model} Wave speed analysis and interpretation}\n\n\\subsection{\\label{subsec:model_dry} Wave speed in dry granular media}\n\nMeasuring the speed of mechanical waves gives access to the elastic features of the media, at an effective macroscopic scale. Considering that the fastest event corresponds to a ballistic pressure wave in the longitudinal direction~\\cite{Norris1997, Makse2004}, one can extract the effective longitudinal modulus $M$ from the experimental measurements of the wave speed $c$ shown in Fig.~\\ref{fig:wave_speed},\n\\begin{equation}\nM = K+(4\/3)G = \\rho_g\\phi_s c^2.\\label{eq:model_Mexp}\n\\end{equation}\n\nAccording to the effective medium theory (EMT)~\\cite{Norris1997, Makse2004}, the effective bulk and shear moduli, $K$ and $G$, of a random packing of frictional spheres can be related to the interactions at the interparticles scale; these moduli are given by\n\\begin{eqnarray}\nK &\\propto& (Z\\phi_s\/R_{\\ast})\\kappa_n,\\label{eq:model_Kemt}\\\\\nG &\\propto& (Z\\phi_s\/R_{\\ast})[\\kappa_n+(3\/2)\\kappa_t],\\label{eq:model_Gemt}\n\\end{eqnarray}\nwhere $\\kappa_{n,t}=\\partial F_{n,t}\/\\partial \\delta_{n,t}$, $F_{n,t}$, and $\\delta_{n,t}$ are the normal and tangential stiffnesses, forces, and deformations at a single contact between two particles, respectively. $Z$ is the coordination number and $R_\\ast=R_g\/2$ is the reduced radius of curvature at the contacts. The interparticles stiffness depends on the size of the mechanical contact; in the case of dry spheres, the Hertz-Mindlin interaction potential~\\cite{Landau1986, Popov2010} provides\n\\begin{eqnarray}\na_{dry} &=& (R_\\ast\\delta_n)^{1\/2}\\propto R_\\ast(p\/E_\\ast)^{1\/3},\\label{eq:model_Adry}\\\\\n\\kappa_{n,t} &\\propto& F_{n,t}\/\\delta_{n,t} \\propto E_\\ast a_{dry},\\label{eq:model_Kdry}\n\\end{eqnarray}\nwhere $a_{dry}$ is the radius of the flat contact disk between two spheres, $E_{\\ast}=E_g\/2(1-\\nu_g^2)$ is the reduced elastic modulus and $p\\propto Z\\phi_sF_n\/R_{\\ast}^2$~\\cite{Norris1997} is the confining pressure. Note that the scaling given in Eq.~\\ref{eq:model_Kdry} relies on the estimation of the normal force, $F_n\\propto\\pi a_{dry}^2p_{max}\\propto E_\\ast a_{dry}\\delta_n$, from the maximal pressure inside the contact region given by the Hooke's law, $p_{max}\\propto E_\\ast(\\delta_n\/a_{dry})$. Note also that in Eq.~\\ref{eq:model_Gemt}, the tangential contribution $\\kappa_t$ stands for a nonsliding (i.e., sticking) contact resulting from an infinite Coulomb's friction coefficient between particles. In the case of frictionless particles, the tangential stiffness is zero, $\\kappa_t=0$. Hence, it turns out that the effective longitudinal modulus non-linearly depends on the confinement pressure $p$,\n\\begin{equation}\n(M_{dry}\/E_{\\ast}) = \\alpha_{dry}(p\/E_{\\ast})^{1\/3},\\label{eq:model_Mdry}\n\\end{equation}\nwhere $\\alpha_{dry}$ is a numerical prefactor of the order of unity, which only depends on the topology of the granular packing (via $Z$ and $\\phi_s$) and wether the contacts of the particles stick or slip. Quantitatively (see Eqs. (1), (12), (13) and (14) in Ref.~\\cite{Makse2004}), a lower and an upper bounds of the prefactor are $\\alpha_{dry}\\simeq 0.37$ for frictionless particles with $Z=2$ and $\\alpha_{dry}\\simeq 1.70$ for frictional particles with $Z=6$, at $\\phi_s=0.63$.\\\\\n\n\\begin{figure}[b]\n\\includegraphics[width=0.450\\textwidth]{1808_03150_v2_Fig_5.pdf}\n\\caption{\\label{fig:elastic_moduli} (a) Effective longitudinal modulus in the dry case as a function of the pressure $p$, where $p$ stands either for the static pressure $P_0$ at high confinement or for the magnitude of the perturbation in the low confinement limit. (b) Ratio of the wet to dry longitudinal modulus, $M_{wet}\/M_{dry}$, as a function of $\\mu\\omega\/p$. (c) Ratio of the elastohydrodynamic contribution to the dry longitudinal modulus as a function of $\\mu\\omega\/p$, with $M_{ehd}=M_{wet}-M_{dry}$, see Eq.~\\ref{eq:model_Mwet}. (d) Linear plot of the elastohydrodynamic contribution to the elastic modulus as a function of $(\\mu\\omega\/E_{\\ast})^{1\/3}$. In (a)--(d), the markers refer to the definition given in Fig.~\\ref{fig:wave_speed} and the shaded region show the $50\\%$ mean deviation error. In (a) and (c), the straight lines have a slope of $1\/3$, according to Eqs.~\\ref{eq:model_Mdry},~\\ref{eq:model_Mehd} and~\\ref{eq:model_Mwet}.}\n\\end{figure}\n\nThe estimations of the dry longitudinal modulus, $M_{dry}$, obtained from the measurements of the wave speeds $c$ shown in Fig.~\\ref{fig:wave_speed}, are presented in Fig.~\\ref{fig:elastic_moduli}(a) in a nondimensional form. Experimentally, $p$ stands for the static pressure at the highest confinement, $p=P_0+P_m\\simeq P_0\\gg P_m$, where $P_m=F_m\/\\pi R_s^2$ is an estimation of the magnitude of the dynamic stress and $F_m$ is the magnitude of the measured excitation force. At the lowest confinement, $p$ stands for the magnitude of the perturbation, $p\\simeq P_m\\gg P_0$. Matching the data shown in Fig.~\\ref{fig:elastic_moduli}(a) to the Eq.~\\ref{eq:model_Mdry} provides two estimations for the prefactor, both being of the order of unity: $\\alpha_{dry}=0.25\\pm38\\%$ at low confinement and $\\alpha_{dry}=0.88\\pm3\\%$ at high confinement. The order of magnitude of these coefficients are in fair agreement with the EMT prediction given in Eq.~\\ref{eq:model_Mdry}. In particular, the experimental data at low confinement pressure reveal the nonlinear nature of the amplitude-dependent elastic modulus. In this regime, the slightly lower exponent, in comparison to the $1\/3$ expectation, and the slightly low experimental prefactor $\\alpha_{dry}$ are presumably a trace of the nonlinear softening of the material~\\cite{Jia2011, Espindola2012, Giacco2015, Olson2015, Johnson2016} described in Sec.~\\ref{sec:weakening}.\n\n\\subsection{\\label{subsec:model_wet}elastohydrodynamic interactions mediated by the fluid}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.450\\textwidth]{1808_03150_v2_Fig_6.pdf}\n\\caption{\\label{fig:sketch_contact} Sketch of the normal deformation at the contact between two spheres in the dry case (dashed line) and with an interstitial fluid (solid line), depicting the spatial extent of the deformations, $a_{dry}$ and $a_{ehd}$, and a fluid layer with thickness $D_c$.}\n\\end{center}\n\\end{figure}\n\nGetting a deeper insight on the fluid-dependent elastic features shown in Fig.~\\ref{fig:elastic_moduli} can be achieved by analyzing the interaction between two elementary elastic spheres separated by a thin layer of viscous fluid, as sketched in Fig.~\\ref{fig:sketch_contact}. This case has been extensively addressed in the literature untill recently, via the elastohydrodynamic rebound of an elastic sphere on a liquid layer~\\cite{Davis1986} or via the oscillatory excitations of a spherical tip immersed in a fluid near an elastic plane~\\cite{Leroy2011, Leroy2012, Villey2013}. The details presented in these four studies are recalled in the following, in order to derive a useful description of our observations. In the following, $D>0$ denotes the initial separation between two spheres separated by an interstitial layer of viscous fluid in the lubrication limit, $D\\ll R_\\ast$. The relative displacement of the particles and their elastic deformation, mediated by the fluid, are denoted by $d(t)$ and $\\delta_n(r,t)$, respectively. They result from a normal collision, along the direction $z$ in Fig.~\\ref{fig:sketch_contact}, at relative velocity $v_n=\\dot{d}\\propto\\omega d$. Here, $\\omega=2\\pi f$ is the angular frequency relying on the duration of the shock, $\\tau\\propto\\omega^{-1}$. The thickness of the fluid as a function of the radial coordinate $r$ and the time $t$ thus reads $\\Delta=D+\\delta_n(r,t)-d(t)+r^2\/2R_\\ast$.\\\\\n\n{\\em Rigid particles ($\\delta_n=0$)}. The case of rigid particles is considered first. The weak collision ($d\\ll D$) between the two spheres squeezes the fluid out from the interstitial region and induces a radial flow, which is assumed laminar and incompressible~\\cite{Davis1986, Guyon2001}. Within the incompressible assumption, the mean radial velocity of the fluid $\\langle v_r \\rangle=(1\/\\Delta)\\int_0^\\Delta{v_rdz}$ can be estimated from the flow rate conservation, $\\pi r^2v_n=2\\pi r\\Delta\\langle v_r \\rangle$, as $\\langle v_r \\rangle\\propto r\\dot{d}\/(D+r^2\/2R_\\ast)$. The latter expression reveals the radial extent of the hydrodynamic field $a_{h}=(2R_\\ast D)^{1\/2}$~\\cite{Davis1986}: the radial velocity vanishes at $r=0$ and $r\\gg a_h$, and is maximal and of the order of $\\langle v_r \\rangle \\propto a_h\\dot{d}\/D$ at $r\\propto a_h$. The laminar assumption implies that $v_r$ is parabolic along the normal $z$ axis~\\cite{Leroy2011} ($v_r=0$ at the solid\/fluid interfaces due to the non-slip condition and is maximal at $z=0$ for symmetry reason). The local shear rate in the fluid is thus $\\dot{\\gamma}=\\partial v_r\/\\partial z\\propto a_h\\dot{d}\/D^2$ and its mean value over the whole interstitial region is $\\langle\\dot{\\gamma}\\rangle=(1\/\\pi R^2)\\int_0^R{\\dot{\\gamma}2\\pi rdr}\\propto(a_h\/R)^2\\dot{\\gamma}\\propto\\dot{d}\/a_h$. The flow generates a hydrodynamic pressure given by the Stokes equation, $(\\partial p_h\/\\partial r\\propto p_h\/a_h)\\simeq(\\mu\\partial\\dot{\\gamma}\/\\partial z\\propto \\mu a_h\\dot{d}\/D^3)$~\\cite{Davis1986,Leroy2011}, such that $p_h\\propto\\mu a_h^2\\dot{d}\/D^3\\propto\\mu\\omega R_\\ast d\/D^2$. In the case of rigid particles, the interstitial flow consequently induces the well-known Reynolds force~\\cite{Guyon2001}, which counteracts the relative approach of the particles, $F_h\\simeq\\pi a_h^2p_h\\propto \\mu\\omega R_\\ast^2 d\/D$.\\\\ \n\n{\\em Interstitial fluid flow}. The details of the interparticles flow regime can be inferred from the long-wavelength experimental data shown in Figs.~\\ref{fig:signals_low_P},~\\ref{fig:signals_high_P}, and~\\ref{fig:wave_speed}. The largest acceleration is typically $\\Gamma\\propto\\omega V\\sim10$~m\/s$^2$, see Figs.~\\ref{fig:signals_low_P} and~\\ref{fig:signals_high_P}, with $V$ denoting the velocity field. The rise duration revealing a frequency content at around $f\\sim1$~kHz, see Fig.~\\ref{fig:wave_speed}, then $V\\sim1.6$~mm\/s at the most. The normal relative velocity between two particles, $v_n=\\dot{d}$, can be deduced from an estimation of the gradient of the velocity field, $V\/\\lambda$ where $\\lambda=c\/f$ is the wavelength, as $v_n\\propto RV\/\\lambda\\propto R\\Gamma\/c$. The wave speed being at least $c\\sim200$~m\/s with fluids, see Fig.~\\ref{fig:wave_speed}, then $\\dot{d}\\sim0.25$~mm\/s at the most. This allows estimating then the typical fluid's shear rate in the interstitial region, $\\langle\\dot{\\gamma}\\rangle\\propto\\dot{d}\/a_h\\propto (R\/D)^{1\/2}(\\Gamma\/c)$. The latter requires an estimation of the fluid's thickness $D$. As a worst situation, the surface roughness of the particles, $Ra\\sim10$~nm~\\cite{Brunet2008,Buttinoni2017}, can be considered as the minimal achievable separation, $D\\sim Ra$, below which the fluid may be trapped in between asperities~\\cite{Marshall2011,Buttinoni2017}. At worst, the typical shear rate is thus $\\langle\\dot{\\gamma}\\rangle\\sim25$~s$^{-1}$. This value remains an order of magnitude, or smaller, below the critical shear rate, $\\langle\\dot{\\gamma}\\rangle\\ll\\dot{\\gamma}_f=G\/\\mu$, above which the fluid becomes non-Newtonian: with $G\\simeq5$~kPa~\\cite{Oswald2014}, the critical shear rate is $\\dot{\\gamma}_f\\sim166$~s$^{-1}$ at the highest viscosity $\\mu=30$~Pa.s and $\\dot{\\gamma}_f\\sim50.000$~s$^{-1}$ at the lowest viscosity $\\mu=0.1$~Pa.s. The fluid thus remains Newtonian in all our experiments. Finally, the largest radial velocity of the fluid, $\\langle v_r \\rangle \\propto (R\/D)^{1\/2}(\\Gamma R\/c)\\sim6.25$~cm\/s, indicates a Mach number $M_f=\\langle v_r \\rangle\/c_f\\ll1$ and a Reynolds number, $Re=\\rho_f \\langle v_r \\rangle D\/\\mu\\ll1$ well below unity. The interstitial flow thus remains incompressible and laminar~\\cite{Guyon2001}, in agreement with previous assumptions. As a consequence, the increase of the effective elasticity observed in wet samples cannot be attributed either to a non-Newtonian viscoelastic behavior of the fluid, or to an effect of its compressibility.\\\\\n\n\\begin{figure*}\n\\includegraphics[width=0.882\\textwidth]{1808_03150_v2_Fig_7.pdf}\n\\caption{\\label{fig:weakening_spectra} Frequency spectra at $P_0=3.2$~kPa (solid black line) and $P_0=83$~kPa (dashed red line) for increasing amplitudes of excitation. (a) dry contacts, (b) $\\mu=1$~Pa.s, (c) $\\mu=10$~Pa.s, and (d) $\\mu=30$~Pa.s show the weakening behavior of the lowest mode. The vertical arrows are guides for the eye, pointing toward increasing amplitudes of excitation.}\n\\end{figure*}\n\n{\\em Elastic particles ($\\delta_n>0$)}. Nevertheless, the hydrodynamic pressure field $p_h\\propto\\mu\\omega R_\\ast d\/D^2$ can reach sufficiently high values, for instance when the fluid's thickness vanishes, to involve the elasticity of the particles~\\cite{Davis1986,Leroy2011}. Given the Hooke's law, $p_h\\propto E_\\ast(\\delta_n\/a_h)$, the elastic deformation of the particles can be rewritten as a fraction of their relative displacement, $(\\delta_n\/d)\\propto (D_c\/D)^{3\/2}$, where~\\cite{Leroy2011}\n\\begin{equation}\nD_c=8R_\\ast(\\mu\\omega\/E_\\ast)^{2\/3}\n\\label{eq:model_D_cutoff}\n\\end{equation}\nis a cutoff thickness at which the deformation accommodates the displacement, $\\delta_n=d$. This situation was referred to as an {\\em elastic confinement} of the fluid~\\cite{Villey2013}: the fluid being clamped by its viscosity, it does not flow but instead mediates the elastic deformations of the bodies as a rigid layer. The cutoff thickness thus stands as a minimal achievable fluid thickness. Quantitatively, it is approximately $D_c\\simeq70$~nm with the lowest viscosity and $D_c\\simeq3$~$\\mu$m with the highest viscosity, under our experimental conditions. These values are larger than the typical surface roughness, $D_c\\gg Ra\\sim10$~nm, and smaller than the typical thickness of the liquid coating, $D_c\\ll D_{coat}\\sim150$~$\\mu$m, this regime is thus accessible in our experiments. In such an elastohydrodynamic regime, the typical extent of the field and the normal stiffness thus become, respectively\n\\begin{eqnarray}\na_{ehd} &=& (2R_\\ast D_c)^{1\/2}\\propto R_\\ast(\\mu\\omega\/E_\\ast)^{1\/3}, \\label{eq:model_Aehd}\\\\\n\\kappa_n &\\propto& F_{ehd}\/\\delta_n\\propto E_{\\ast}a_{ehd}, \\label{eq:model_Kehd}\n\\end{eqnarray}\nwhere the elastohydrodynamic force $F_{ehd}\\propto\\pi a_{ehd}^2p_{ehd}$ is proportional to the pressure given by the Hooke's law, $p_{ehd}\\propto E_\\ast(\\delta_n\/a_{ehd})$, with $p_{ehd}\\propto \\mu\\omega R_\\ast d\/D_c^2$. In addition, the tangential interaction presumably becomes frictionless in presence of the lubricating layer of fluid, $\\kappa_t=0$. Consequently, the effective longitudinal modulus given by Eqs.~\\ref{eq:model_Kemt},~\\ref{eq:model_Gemt},~\\ref{eq:model_Aehd}, and~\\ref{eq:model_Kehd} becomes\n\\begin{equation}\n(M_{ehd}\/E_{\\ast}) = \\alpha_{ehd} (\\mu\\omega\/E_{\\ast})^{1\/3},\\label{eq:model_Mehd}\n\\end{equation}\nwhere $\\alpha_{ehd}$ is a numerical prefactor depending on $Z$ and $\\phi_s$ only. Interestingly, the comparison of the expressions of the dry and the wet elastic moduli, given by Eqs.~\\ref{eq:model_Mdry} and~\\ref{eq:model_Mehd}, respectively, shows that their ratio is a function of a single nondimensional parameter, $(M_{ehd}\/M_{dry})\\propto(\\mu\\omega\/p)^{1\/3}$. This result substantiates the observation of a nontrivial competition between a viscous contribution of the fluid and the elastic deformation of the particles under the action of the confinement pressure. This feature is confirmed by representing the experimental wet-to-dry elastic moduli as a function of $\\mu\\omega\/p$, see Fig.~\\ref{fig:elastic_moduli}(b). The data set, aggregating two different confinement pressures and four different viscosities, indeed fairly collapses along a master curve when represented in such a nondimensional form. However, the ratio of the elastic moduli asymptotically saturates at one for small values of $\\mu\\omega\/p$ in experiments. This is consistent with the fact that a wet sample under a high pressure and with a small interstitial viscosity should tend to behave as a dry sample, $M_{wet}\\simeq M_{dry}$ at $\\mu\\omega\/p\\ll1$. Moreover, within a constant confinement pressure only, the fluid quasistatically flows out from the contact between particles, leaving a central region in mechanical contact surrounded by a peripheral region filled with fluid~\\cite{Marshall2011}, see Fig.~\\ref{fig:sketch_contact}. A more convenient ansatz would thus correspond to a Hertzian elastic response, coming from the central and flat dry region, acting in parallel to a peripheral elastohydrodynamic response, due to the fluid pinched at the edge between the elastic solids:\n\\begin{equation}\nM_{wet} = M_{dry}+M_{ehd}.\\label{eq:model_Mwet}\n\\end{equation}\n\nThe ansatz given by Eq.~\\ref{eq:model_Mwet} is probed in Fig.~\\ref{fig:elastic_moduli}(c), which represents the ratio $(M_{ehd}\/M_{dry})=(M_{wet}\/M_{dry}-1)$ as a function of $(\\mu\\omega\/p)$. As in Fig.~\\ref{fig:elastic_moduli}(b), the data set still demonstrates fair correlations, but now shows a clear power law with an exponent close to $1\/3$, in agreement with our analysis, see Eqs.~\\ref{eq:model_Mdry} and~\\ref{eq:model_Mehd}. Quantitatively, the fit of the data shown in Fig.~\\ref{fig:elastic_moduli}(c) provides $(\\alpha_{ehd}\/\\alpha_{dry})=1.01\\pm43\\%$ at low confinement pressure and $(\\alpha_{ehd}\/\\alpha_{dry})=0.46\\pm20\\%$ at high confinement pressure; this corresponds to prefactors of the order of unity, $\\alpha_{ehd}\\simeq0.25$ and $\\alpha_{ehd}\\simeq0.40$, respectively. Finally, the Fig.~\\ref{fig:elastic_moduli}(d) shows $(M_{ehd}\/E_{\\ast})$ as a function of $(\\mu\\omega\/E_{\\ast})^{1\/3}$ for the high confinement pressure data set. The plot confirms that $M_{ehd}$ increases monotonically according to Eq.~\\ref{eq:model_Mehd}, i.e., the effective elastic modulus increases with the viscosity of the fluid; matching the curve to Eq.~\\ref{eq:model_Mehd} provides a consistent value of the prefactor, $\\alpha_{ehd}=0.37\\pm37\\%$.\\\\\n\nThe analysis of the experimental wave speed presented in Fig.~\\ref{fig:wave_speed}, based on an effective medium theory, thus demonstrates that the propagation of mechanical waves in wet granular samples induces an elastohydrodynamic mechanism at the interparticle level. The mechanical response of a wet sample results from the competition between (i) an elastic contribution related to the static confinement pressure within the contact region between grains and (ii) an elastohydrodynamic interplay between the particles and the fluid, which resides at the periphery of the contacts. The crossover between these two contributions is fairly described by a unique nondimensional number, $(\\mu\\omega\/p)$.\n\n\n\\section{\\label{sec:weakening} Material weakening}\n\nMaterial weakening has been proposed as a triggering mechanism for the emission of secondary pulses after the passage of a principal mechanical event. Several authors~\\cite{Johnson2005,Jia2011,Johnson2016} have linked the breaking of unconsolidated and weak contacts with the loss of material strength and the dynamical change of the shape of the {\\em coda wave}. Here, the material softening is probed by tracking the frequency shift of the lowest prominent mode in the spectrum of the long-lasting outgoing acceleration. The dry configuration is first considered, see Fig.~\\ref{fig:weakening_spectra}(a), at both low and high confinement pressure. It is observed that for the high confinement pressure, the low-frequency component, at around $100$~Hz, remains unaffected by the strength of the dynamical perturbation. This results is coherent with the fact that the dynamics of the sample is linear at high confinement, owing to a negligible dynamical perturbation: the sample is highly consolidated and no material weakening is observable. On the other hand, at the lowest confinement pressure, the low-frequency component decreases with increasing impulse amplitude: a $20\\%$ variation in respect to the initial frequency is observed, consistently with the observations available in the literature~\\cite{Johnson2005}. The decrease of the prominent frequency reveals the nonlinear nature of the contact dynamics and the weakening of the material as the dynamical strength is progressively increased.\\\\\n\n\\begin{figure}[t]\n\\includegraphics[width=0.450\\textwidth]{1808_03150_v2_Fig_8.pdf}\n\\caption{\\label{fig:weakening_frequency} Frequency of the lowest mode as a function of the input force, for different fluid viscosities at (a) low and (b) high confinement pressures. The markers refer to the definitions given in Fig.~\\ref{fig:wave_speed}.}\n\\end{figure}\n\nNext, the same procedure is repeated in samples containing interstitial fluids with different viscosities, see Figs.~\\ref{fig:weakening_spectra}(b)--\\ref{fig:weakening_spectra}(d). In Fig.~\\ref{fig:weakening_frequency}, the evolution of the lowest mode for every fluid is shown as a function of the impulse amplitude. The weakening of the sample remains observable for the lowest viscosity only, and progressively disappears when the viscosity is increased. This means that an interstitial fluid with a sufficiently high viscosity consolidates the sample and tends to linearize its mechanical response. Indeed, using the nondimensional parameter $(\\mu\\omega\/p)$ derived in Sec.~\\ref{sec:model} as an indicator of the regime, one find a crossover viscosity $\\mu_{\\ast}=p\/\\omega$. Below this value, the dynamics is essentially Hertzian and nonlinear, see Eqs.~\\ref{eq:model_Mdry}. Above the crossover, the dynamics tends to becomes linear owing to a dominant elastohydrodynamic interplay, see Eqs.~\\ref{eq:model_Mehd} and~\\ref{eq:model_Mwet}. Such a crossover is $\\mu_{\\ast}\\simeq3.6$~Pa.s at low confinement pressure, where $f\\sim140$~Hz, in agreement with our observations. Similarly, the crossover is $\\mu_{\\ast}\\simeq82$~Pa.s at high confinement pressure, where $f\\sim160$~Hz. In this case, the behavior of all the wet samples relies on the Hertzian elastic interaction: it is thus reminiscent to the dry configuration, which proved to be consolidated and where no weakening is expected.\\\\\n\n\\section{\\label{sec:conclusion} Conclusions}\n\nThe experimental evidences presented in this paper showed that the presence of an interstitial fluid in a granular media significantly modifies the contact dynamics between particles and, consequently, the features of mechanical waves propagation in the long wavelength approximation. In both low and high confinement cases, it was observed that the fluid induces an elastohydrodynamic mechanism that enhances the rigidity of the contacts, rendering a higher wave speed as compared with the dry configuration. All our results were discussed in terms of an effective mean-field theory, coupled to a description of the elastohydrodynamic interaction between spherical elastic particles mediated by an interstitial Newtonian viscous fluid. Our analysis suggested a nontrivial competition between the elastohydrodynamic interaction and the elastic deformation due to the confining pressure. The interplay was shown to be fairly described by a unique nondimensional parameter $\\mu\\omega\/p$, which allowed defining a threshold viscosity above which the effect of the fluid dominates. It was also observed that the dry granular material weakens when submitted to strong dynamical perturbations, due to the breaking of unconsolidated contacts. This nonlinear softening can be impeded by either increasing the confinement pressure, or by adding an interstitial fluid with a viscosity above the threshold. Our analysis and description qualitatively match the experimental observations; all these results might prove to be useful in practical situations and pave the way to the need of a more quantitative and precise description.\\\\\n\n\\begin{acknowledgments}\nThe authors thank Francisco Melo for many fruitful discussions. R.Z. acknowledges CONICYT National Doctoral Program Grant No. 21161404 and the financial support of {\\em Supm{\\'e}ca} during his stay in France. S.J. acknowledges the {\\em Pontificia Universidad Cat{\\'o}lica de Valpara{\\'i}so} and the Franco-Chilean {\\em Laboratoire International Associ{\\'e} LIA-MSD} for the financial support during his stay in Chile. F.S. acknowledges the financial support from FONDECYT Project No. 11140556.\n\\end{acknowledgments}\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nIn the present paper, we study spectral estimates for the logarithmic Laplacian \n$L_{\\text{\\tiny $\\Delta \\,$}}\\!= \\log (-\\Delta)$, which is a (weakly) singular integral operator with Fourier symbol $2\\log |\\eta|$ and arises as formal derivative $\\partial_s \\Big|_{s=0} (-\\Delta)^s$ of fractional Laplacians at $s= 0$. The study of $L_{\\text{\\tiny $\\Delta \\,$}}\\!$ has been initiated recently in \\cite{HW}, where its relevance for the study of asymptotic spectral properties of the family of fractional Laplacians in the limit $s \\to 0^+$ has been discussed. A further motivation for the study of $L_{\\text{\\tiny $\\Delta \\,$}}\\!$ is given in \\cite{jarohs-saldana-weth}, where it has been shown that this operator allows to characterize the $s$-dependence of solution to fractional Poisson problems for the full range of exponents $s \\in (0,1)$.  The logarithmic Laplacian also arises in the geometric context of the $0$-fractional perimeter, which has been studied recently in \\cite{DNP}. \n\nFor matters of convenience, we state our results for the operator $\\mathcal H= \\frac{1}{2}L_{\\text{\\tiny $\\Delta \\,$}}\\!$ \nwhich corresponds to the quadratic form \n\\begin{equation}\n\\label{log-quadratic}\n\\varphi \\mapsto (\\varphi,\\varphi)_{log} := \\frac{1}{(2\\pi)^{d}} \\, \\int_{\\Bbb R^d} \\log(|\\xi|)\\,  |\\widehat{\\varphi}(\\xi)|^2\\, d\\xi.\n\\end{equation}\nHere and in the following, we let $\\widehat{\\varphi}$ denote the Fourier transform \n$$\n\\xi \\mapsto \\widehat{\\varphi}(\\xi)= \\int_{{\\mathbb R}^d} e^{-ix \\xi} \\varphi(x)\\,dx\n$$\nof a function $\\varphi\\in L^2({\\mathbb R}^d)$. Let $\\Omega\\subset \\Bbb R^d$ be an open set of finite measure, and let \n${\\mathbb H}(\\Omega)$ denote the closure of $C^\\infty_c(\\Omega)$ with respect to the norm\n\\begin{equation}\n  \\label{eq:def-norm--star}\n\\varphi \\mapsto \\|\\varphi\\|_{*}^2:= \\int_{\\Bbb R^d} \\log(e + |\\xi|)\\,  |\\widehat{\\varphi}(\\xi)|^2\\, d\\xi.\n\\end{equation}\nThen $(\\cdot,\\cdot)_{log}$  defines a closed, symmetric and semibounded quadratic form with domain ${\\mathbb H}(\\Omega) \\subset L^2(\\Omega)$, see Section~\\ref{sec:prel-basic-prop} below. Here and in the following, we identify $L^2(\\Omega)$ with the space of functions $u \\in L^2({\\mathbb R}^d)$ with $u \\equiv 0$ on ${\\mathbb R}^d \\setminus \\Omega$. Let \n$$\n\\mathcal H : {\\mathcal D}(\\mathcal H) \\subset L^2(\\Omega) \\to L^2(\\Omega) \n$$\nbe the unique self-adjoint operator associated with the quadratic form. The eigenvalue problem for $\\mathcal H$ then writes as \n\\begin{equation}\\label{D}\n\\left\\{\n  \\begin{aligned}\n\\mathcal H \\varphi &= \\lambda \\varphi, &&\\qquad \\text{in $\\Omega$,}\\\\\n\\varphi &= 0, &&\\qquad \\text{on ${\\mathbb R}^d \\setminus \\Omega$.}\n  \\end{aligned}\n\\right.\n\\end{equation}\nWe understand (\\ref{D}) in weak sense, i.e. \n$$\n\\varphi \\in {\\mathbb H}(\\Omega) \\quad \\text{and}\\quad (\\varphi,\\psi)_{log}= \\lambda \\int_{\\Omega}\\varphi(x)\\psi(x)\\,dx \\quad \\text{for all $\\psi \\in {\\mathbb H}(\\Omega)$.}\n$$\nAs noted in \\cite[Theorem 1.4]{HW}, there exists a sequence of eigenvalues \n$$\n\\lambda_1(\\Omega)< \\lambda_2(\\Omega) \\le \\dots, \\qquad \\lim_{k \\to \\infty} \\lambda_k(\\Omega) = \\infty \n$$\nand a corresponding complete orthonormal system of eigenfunctions. We note that the discreteness of the spectrum is a consequence of the fact that the embedding ${\\mathbb H}(\\Omega) \\hookrightarrow L^2(\\Omega)$ is compact. In the case of bounded open sets, the compactness of this embedding follows easily by Pego's criterion~\\cite{Pego}. In the case of unbounded open sets of finite measure, the compactness can be deduced from \\cite[Theorem 1.2]{jarohs-weth} and estimates for $\\|\\cdot\\|_*$, see Corollary~\\ref{cor-compact-embedding} below. \n\nIn Section \\ref{sec:prel-basic-prop}, using the results from \\cite{HW} and \\cite{FKV}, we discuss properties  of functions from  $\\mathcal D(\\mathcal H)$. In particular, we show that $e^{ix\\xi}\\big|_{x\\in\\Omega} \\in \\mathcal D(\\mathcal H)$, $\\xi\\in\\Bbb R^d$, provided $\\Omega$ is an open bounded sets with Lipschitz boundary.\n\nIn Section \\ref{sec:deriving-an-upper-1} we obtain a sharp upper bound for the Riesz means and for the number of eigenvalues  $N(\\lambda)$ of the operator $\\mathcal H$ below $\\lambda$. Here we use technique developed in papers \\cite{Bz1}, \\cite{Bz2}, \\cite{LY} and \\cite{L}. In \\cite{Lap} it was noticed that such technique could be applied for a class of pseudo-differential operators with Dirichlet boundary conditions in domains of finite measure without any requirements on the smoothness of the boundary. \n\n We discuss  lower bounds for $\\lambda_1(\\Omega)$ in Section \\ref{sec:lower-bound-lambd}. In Theorem \\ref{lower-bound-lambda_1-first} we present an estimate that is valid for arbitrary open sets of finite measure. For sets with Lipschitz boundaries, H.Chen and T.Weth \\cite{HW} have proved a Faber-Krahn inequality for the operator $\\mathcal H$ that reduces the problem to the estimate of \n$\\lambda_1(B)$, where $B$ is a ball satisfying $|B| = |\\Omega|$, see Corollary \\ref{cor-faber-krahn}. In Theorem \\ref{lower-bound-lambda-1-second} we find an estimate for $\\lambda_1(B_d)$, where $B_d$ is the unit ball, that is better in lower dimensions than the one obtained in Theorem \\ref{lower-bound-lambda_1-first}. We also compare our results with bounds resulting from previously known spectral inequalities obtained in  \\cite{BK} and \\cite{B}.\n\nIn Section \\ref{LowB1} we obtain asymptotic lower bounds using the coherent states transformation approach given in \\cite{G}. It allows us to derive, in Section \\ref{Weyl}, asymptotics for the Riesz means of eigenvalues in  Theorem \\ref{3.1} and for $N(\\lambda)$ in Corollary \\ref{3.2}. Here $\\Omega \\subset {\\mathbb R}^N$ is an arbitrary open set of finite measure without any additional restrictions on the boundary. \n\nFinally in Section \\ref{LowB2} we obtain uniform bounds on the Riesz means of the eigenvalues using the fact that for bounded open sets with Lipschitz boundaries we have $e^{ix\\xi}\\big|_{x\\in\\Omega} \\in \\mathcal D(\\mathcal H)$.\n\n\n\n \n\n\n\\section{Preliminaries and basic properties of eigenvalues}\n\\label{sec:prel-basic-prop}\n\nAs before, let $(\\cdot,\\cdot)_{log}$ denote the quadratic form defined in (\\ref{log-quadratic}), and let, for an open set $\\Omega \\subset {\\mathbb R}^d$, \n ${\\mathbb H}(\\Omega)$ denote the closure of $C^\\infty_c(\\Omega)$ with respect to the norm $\\|\\cdot\\|_*$ defined in (\\ref{eq:def-norm--star}). \n\n\\begin{lem}\n\\label{closed-semibounded}\nLet $\\Omega \\subset {\\mathbb R}^d$ be an open set of finite measure. Then $(\\cdot,\\cdot)_{log}$  defines a closed, symmetric and semibounded quadratic form with domain ${\\mathbb H}(\\Omega) \\subset L^2(\\Omega)$.\n\\end{lem}\n\n\\begin{proof}\nObviously, the form $(\\cdot,\\cdot)_{log}$ is  symmetric. For functions $\\varphi \\in C^\\infty_c(\\Omega)$, we have \n\\begin{equation}\n  \\label{eq:basic-fourier-ineq}\n(2\\pi)^{d} \\|\\varphi\\|_2^2 =\\|\\widehat \\varphi\\|_2^2 \\le \\|\\varphi\\|_*^2. \n\\end{equation}\nMoreover, with $c_1:= \\log (e+2)+ \\sup \\limits_{t \\ge 2}\\frac{\\log (e+t)}{\\log t}$\nwe have\n\\begin{align}\n\\frac{\\|\\varphi\\|_{*}^2}{c_1} &\\le \\| \\widehat \\varphi\\|_{2}^2+  \\int_{|\\xi| \\ge 2}\\ln |\\xi| |\\widehat{\\varphi}(\\xi)|^2  \\, d\\xi \\nonumber\\\\\n&\\le (2\\pi)^d \\bigl( \\|\\varphi\\|_{2}^2 +  (\\varphi,\\varphi)_{log}\\bigr) \n- \\int_{|\\xi| \\le 2}\\ln |\\xi| |\\widehat{\\varphi}(\\xi)|^2  \\, d\\xi \\nonumber \\\\\n&\\le (2\\pi)^d \\bigl( \\|\\varphi\\|_{2}^2 +  (\\varphi,\\varphi)_{log}\\bigr) \n+ \\bigl\\| \\ln |\\cdot| \\bigr\\|_{L^1(B_2(0))} \\|\\widehat{\\varphi}\\|_\\infty^2 \\label{closed-semibounded-est-1}\n\\end{align}\nwhile \n\\begin{equation}\n  \\label{eq:closed-semibounded-est-2}\n\\|\\widehat{\\varphi}\\|_\\infty^2 \\le \\|\\varphi\\|_1^2 \\le |\\Omega|\\, \\|\\varphi\\|_2^2.\n\\end{equation}\nConsequently, \n\\begin{align}\n(\\varphi,\\varphi)_{log} &\\ge   \\frac{\\|\\varphi\\|_*^2}{(2\\pi)^d c_1}-\n\\left(1+ \\frac{|\\Omega|\\, \\bigl\\| \\ln |\\cdot| \\bigr\\|_{L^1(B_2(0))}}{(2\\pi)^{d}}\\right)\\|\\varphi\\|_2^2 \\label{intermediate-est}\\\\\n&\\ge  \\left(\\frac{1}{c_1}\\,-\\,1\\,-\\,\\frac{|\\Omega|\\, \\bigl\\| \\ln |\\cdot| \\bigr\\|_{L^1(B_2(0))}}{(2\\pi)^{d}}\\right)\\|\\varphi\\|_2^2. \\nonumber\n\\end{align}\nIn particular, $(\\varphi,\\varphi)_{log}$ is semibounded. Moreover, it follows from (\\ref{intermediate-est}) and the completeness of $({\\mathbb H}(\\Omega),\\|\\cdot\\|_*)$ that the form $(\\varphi,\\varphi)_{log}$ is closed on ${\\mathbb H}(\\Omega)$.  \n\\end{proof}\n\n\\begin{lem}\n\\label{equivalent-norms}\nLet $\\Omega \\subset {\\mathbb R}^d$ be an open set of finite measure. Then \n\\begin{equation}\n  \\label{eq:def-norm-double-star}\n\\varphi \\mapsto \\|\\varphi\\|_{**}^2:= \n\\int \\!\\!\\! \\int_{|x-y|\\le 1} \\frac{(\\varphi(x)-\\varphi(y))^2}{|x-y|^d}\\,dxdy\n\\end{equation}\ndefines an equivalent norm to the norm $\\|\\cdot\\|_*$ defined in (\\ref{eq:def-norm--star}) on $C^\\infty_c(\\Omega)$. \n\\end{lem}\n\n\\begin{proof}\nLet $\\varphi \\in C^\\infty_c(\\Omega)$. By \\cite[Lemma 2.7]{FKV}, we have \n\\begin{equation}\n  \\label{eq:FKV-lemma}\n\\|\\varphi\\|_2 \\le c_2 \\|\\varphi\\|_{**}  \\qquad \\text{with a constant $c_2>0$ independent of $\\varphi$.}  \n\\end{equation}\nIn particular, $\\|\\cdot\\|_{**}$ defines a norm on $C^\\infty_c(\\Omega)$.\nNext we note that, by \\cite[Theorem 1.1(ii) and Eq. (3.1)]{HW},   \n$$\n(\\varphi,\\varphi)_{log} = \\frac{1}{2}\\int_{{\\mathbb R}^d}[L_{\\text{\\tiny $\\Delta \\,$}}\\! \\varphi(x)]\\varphi(x)\\,dx = \\kappa_d \\|\\varphi\\|_{**}^2 - \\int_{\\Bbb R^d} [j * \\varphi] \\varphi \\,dx + \\zeta_d  \\|\\varphi\\|_2^2\n$$\nwith \n\\begin{equation}\n  \\label{eq:def-zeta_d}\n\\kappa_d:= \\frac{\\pi^{- \\frac{d}{2}}  \\Gamma(d\/2)}{4}, \\qquad \\zeta_d:= \n\\log 2 + \\frac{1}{2}\\left(\\psi\\left(d\/2\\right) -\\gamma\\right)\n\\end{equation}\nand \n$$\nj: {\\mathbb R}^d \\setminus \\{0\\} \\to {\\mathbb R}, \\qquad j(z)= 2 \\kappa_d 1_{{\\mathbb R}^d \\setminus B_d}(z)|z|^{-d}.\n$$\nHere $\\psi:= \\frac{\\Gamma'}{\\Gamma}$ is the Digamma function and $\\gamma= -\\Gamma'(1)$ is the Euler-Mascheroni constant. \nConsequently, we have \n\\begin{align}\n\\Bigl|(\\varphi,\\varphi)_{log}- \\kappa_d \\|\\varphi\\|_{**}^2\\Bigr| &\\le \n\\|j\\|_\\infty \\|\\varphi\\|_1^2 + \\zeta_d  \\|\\varphi\\|_2^2 \\nonumber\\\\\n&\\le \\Bigl(\\|j\\|_\\infty |\\Omega| +\\zeta_d\\Bigr)\\|\\varphi\\|_2^2.  \\label{modulus-ineq-quad-form}\n\\end{align}\nAs a consequence of (\\ref{eq:basic-fourier-ineq}) and (\\ref{modulus-ineq-quad-form}), we find that \n\\begin{align*}\n\\|\\varphi\\|_{**}^2 &\\le \\frac{1}{\\kappa_d}\\Bigl[ (\\varphi,\\varphi)_{log} + \n\\bigl(\\|j\\|_\\infty |\\Omega|+\\zeta_d\\bigr) \\|\\varphi\\|_2^2 \\Bigr]\\\\\n&\\le \\frac{1}{(2\\pi)^d \\kappa_d}\\Bigl(1  + \\|j\\|_\\infty |\\Omega|+\\zeta_d\\Bigr)\\|\\varphi\\|_*^2.\n\\end{align*}\nMoreover, by (\\ref{closed-semibounded-est-1}), (\\ref{eq:closed-semibounded-est-2}), (\\ref{eq:FKV-lemma}) and (\\ref{modulus-ineq-quad-form}) we have \n\\begin{align*}\n&\\frac{\\|\\varphi\\|_{*}^2}{c_1}  \n\\le (2\\pi)^d \\bigl(\\|\\varphi\\|_{2}^2 + (\\varphi,\\varphi)_{log}\\bigr) \n+   \\bigl\\| \\ln |\\cdot| \\bigr\\|_{L^1(B_2(0))} |\\Omega| \\|\\varphi\\|_2^2\\\\\n&\\le (2\\pi)^d \\Bigl( \\kappa_d \\|\\varphi\\|_{**}^2\n+ \\bigl(1+ \\|j\\|_\\infty |\\Omega| +\\zeta_d\\bigr)\\|\\varphi\\|_2^2 \\Bigr) \n+   \\bigl\\| \\ln |\\cdot| \\bigr\\|_{L^1(B_2(0))} |\\Omega| \\|\\varphi\\|_2^2\\\\\n&\\le c_3 \\|\\varphi\\|_{**}^2\n\\end{align*}\nwith $c_3 = (2\\pi)^d \\kappa_d + c_2\\bigl[(2\\pi)^d \\bigl(1+ \\|j\\|_\\infty |\\Omega|+\\zeta_d\\bigr) +\\bigl\\| \\ln |\\cdot| \\bigr\\|_{L^1(B_2(0))}|\\Omega| \\bigr]$. Hence the norms $\\|\\cdot\\|_{*}$ and $\\|\\cdot\\|_{**}$ are equivalent on $C^\\infty_c(\\Omega)$.\n\\end{proof}\n\n\\begin{cor}\n\\label{cor-compact-embedding}\nLet $\\Omega \\subset {\\mathbb R}^d$ be an open set of finite measure. Then the embedding ${\\mathbb H}(\\Omega) \\hookrightarrow L^2(\\Omega)$ is compact. \n\\end{cor}\n\n\\begin{proof}\nLet $\\tilde {\\mathbb H}(\\Omega)$ be defined as the space of functions $\\varphi \\in L^2({\\mathbb R}^d)$ with $\\varphi \\equiv 0$ on ${\\mathbb R}^d \\setminus \\Omega$ and \n$$\n\\int \\!\\!\\! \\int_{|x-y|\\le 1} \\frac{(\\varphi(x)-\\varphi(y))^2}{|x-y|^d}\\,dxdy <\\infty.\n$$  \nBy \\cite[Theorem 1.2]{jarohs-weth}, the Hilbert space $(\\tilde {\\mathbb H}(\\Omega),\\|\\cdot\\|_{**})$ is compactly embedded in $L^2(\\Omega)$. \nSince, by Lemma~\\ref{equivalent-norms}, the norms $\\|\\cdot\\|_*$ and $\\|\\cdot\\|_{**}$ are equivalent on $C^\\infty_c(\\Omega)$, the space ${\\mathbb H}(\\Omega)$ is embedded in $\\tilde {\\mathbb H}(\\Omega)$. Hence the claim follows.    \n\\end{proof}\n\n\n\\begin{cor}\n\\label{space-equivalence}\nLet $\\Omega \\subset {\\mathbb R}^d$ be a bounded open set with Lipschitz boundary.\n\\begin{enumerate}\n\\item[(i)] The space ${\\mathbb H}(\\Omega)$ is equivalently given as the set of functions $\\varphi \\in L^2({\\mathbb R}^d)$ with $\\varphi \\equiv 0$ on ${\\mathbb R}^d \\setminus \\Omega$ and \n  \\begin{equation}\n    \\label{eq:kernel-finiteness-cond}\n\\int \\!\\!\\! \\int_{|x-y|\\le 1} \\frac{(\\varphi(x)-\\varphi(y))^2}{|x-y|^d}\\,dxdy <\\infty.\n  \\end{equation}\n\\item[(ii)] ${\\mathbb H}(\\Omega)$ contains the characteristic function $1_\\Omega$ of $\\Omega$ and also the restrictions of exponentials $x \\mapsto 1_\\Omega(x) \\, e^{ix \\xi}$, $\\xi \\in {\\mathbb R}^d$.\n\\end{enumerate}\n\\end{cor}\n\n\\begin{proof}\n(i) Let, as in the proof of Corollary~\\ref{cor-compact-embedding}, $\\tilde {\\mathbb H}(\\Omega)$ be the space of functions $\\varphi \\in L^2({\\mathbb R}^d)$ with $\\varphi \\equiv 0$ on ${\\mathbb R}^d \\setminus \\Omega$ and with (\\ref{eq:kernel-finiteness-cond}), endowed with the norm $\\|\\cdot\\|_{**}$. Since $\\Omega \\subset {\\mathbb R}^d$ be a bounded open set with Lipschitz boundary, it follows from \\cite[Theorem 3.1]{HW} that $C_0^\\infty(\\Omega) \\subset \\tilde {\\mathbb H}(\\Omega)$ is dense. Hence the claim follows from Lemma~\\ref{equivalent-norms}.\n\n(ii) follows from (i) and a straightforward computation.   \n\\end{proof}\n\nNext we note an observation regarding the scaling properties of the eigenvalues $\\lambda_k(\\Omega)$. \n\\begin{lem}\n\\label{lemma-scaling-properties}  \nLet $\\Omega \\subset {\\mathbb R}^d$ be a bounded open set with Lipschitz boundary, and let \n$$\nR\\Omega:= \\{R x\\::\\: x \\in \\Omega\\}. \n$$\nThen we have\n$$\n\\lambda_k(R \\Omega) = \\lambda_k(\\Omega) -  \\log R \\qquad \\text{for all $k \\in {\\mathbb N}$.}\n$$\n\\end{lem}\n\n\\begin{proof}\nSince $C_0^\\infty(\\Omega) \\subset {\\mathbb H}(\\Omega)$ is dense, it suffices to note that\n\\begin{equation}\n  \\label{eq:scaling-test-functions}\n(\\varphi_R,\\varphi_R)_{log} = (\\varphi,\\varphi)_{log} - \\log R \\|\\varphi\\|_{L^2({\\mathbb R}^d)}^2 \\qquad \\text{for $\\varphi \\in C^\\infty_c({\\mathbb R}^d)$}\n\\end{equation}\nwith $\\varphi_R \\in C^\\infty_c({\\mathbb R}^d)$ defined by $\\varphi_R(x)= R^{-\\frac{d}{2}}\\varphi(\\frac{x}{R})$, whereas $\\|\\varphi_R\\|_{L^2({\\mathbb R}^d)}= \\|\\varphi\\|_{L^2({\\mathbb R}^d)}$. Since \n$$\n\\widehat{\\varphi_R}= R^{\\frac{d}{2}} \\widehat{\\varphi}(R \\,\\cdot \\,)\n$$\nwe have \n\\begin{align*}\n&(\\varphi_R,\\varphi_R)_{log}\\\\\n&= \\frac{1}{(2\\pi)^{d}} \\, \\int_{\\Bbb R^d} \\log(|\\xi|)\\,  |\\widehat{\\varphi_R}(\\xi)|^2\\, d\\xi = \\frac{R^{d}}{(2\\pi)^{d}} \\, \\int_{\\Bbb R^d} \\log(|\\xi|)\\,  |\\widehat{\\varphi}(R \\xi)|^2\\, d\\xi\\\\\n&= \\frac{1}{(2\\pi)^{d}} \\, \\int_{\\Bbb R^d} \\bigl(\\log(|\\xi|)-\\log R\\bigr)\\,  |\\widehat{\\varphi}(\\xi)|^2\\, d\\xi =(\\varphi,\\varphi)_{log} - \\log R \\|\\varphi\\|_{L^2({\\mathbb R}^d)}^2,\n\\end{align*}\nas stated in (\\ref{eq:scaling-test-functions}).\\\\ \n\\end{proof}\n \n\\section{An upper trace bound}\n\\label{sec:deriving-an-upper-1}\n\n\nThroughout this section, we let $\\Omega \\subset {\\mathbb R}^d$ denote an open set of finite measure. Let $\\{\\varphi_k\\}$ and $\\{\\lambda_k\\}$ be the orthonormal in $L^2(\\Omega)$ system of eigenfunctions and the eigenvalues of the operator $\\mathcal H$  respectively. In what follows we denote\n$$\n(\\lambda - t)_+ = \n\\begin{cases}\n\\lambda - t, & {\\rm if} \\quad t <\\lambda, \\\\\n0, \\quad & {\\rm if} \\quad  t \\ge \\lambda.\n\\end{cases}\n$$\nThen we have\n\\begin {thm}\\label{1.1}\nFor the eigenvalues of the problem \\eqref{D} and any $\\lambda\\in \\Bbb R$ we have\n\\begin{equation}\\label{BU}\n\\sum_{k}(\\lambda - \\lambda_k)_+ \\le \\frac{1}{(2\\pi)^{d}}\\, |\\Omega|\\, e^{d\\lambda} \\, |B_d|\\, d^{-1},\n\\end{equation}\nwhere $|B_d|$ is the measure of the unit ball in $\\Bbb R^d$.\n\\end{thm}\n\n\\begin{proof}\nExtending the eigenfunction $\\varphi_k$ by zero outside $\\Omega$ and using the Fourier transform we find\n\\begin{multline*}\n\\sum_{k}(\\lambda - \\lambda_k)_+ = \\sum_{k}\\left(\\lambda (\\varphi_k, \\varphi_k) - (\\mathcal H\\varphi_k, \\varphi_k) \\right)_+ \\\\\n= \\frac{1}{(2\\pi)^{d}}\\, \\left(\\sum_k \\int_{\\Bbb R^d} \\left(\\lambda - \\log(|\\xi|) \\right) \\, |\\widehat{\\varphi_k}(\\xi)|^2 \\, d\\xi \\right)_+\\\\\n\\le\n \\frac{1}{(2\\pi)^{d}}\\, \\int_{\\Bbb R^d} \\left(\\lambda - \\log(|\\xi|) \\right)_+ \\,  \\sum_k |\\widehat{\\varphi_k}(\\xi)|^2 \\, d\\xi. \n\\end{multline*} \nUsing that $\\{\\varphi_k\\}$ is an orthonormal basis in $L^2(\\Omega)$ and denoting \n$e_\\xi = e^{-i (\\cdot,\\xi)}$we have \n$$\n\\sum_k |\\widehat{\\varphi_k}(\\xi)|^2 = \\sum_k |(e_\\xi, \\varphi_k)|^2 = \\|e_\\xi\\|^2_{L^2(\\Omega)} = |\\Omega|,\n$$\nand finally obtain\n\\begin{align*}\n\\sum_{k}(\\lambda - \\lambda_k)_+ & \\le  \\frac{1}{(2\\pi)^{d}}\\, |\\Omega|\\, \\int_{\\Bbb R^d} \\left(\\lambda - \\log(|\\xi|) \\right)_+ \\\\\n& =  \\frac{1}{(2\\pi)^{d}}\\, |\\Omega|\\, e^{d\\lambda} \\,  \\int_{|\\xi|\\le 1} \\log(|\\xi|^{-1}) \\, d\\xi.\n\\end{align*}\nWe complete the proof by computing the last integral. \n\\end{proof}\n\n\\noindent\nLet $\\eta >\\lambda$ and let us consider the function\n$$\n\\psi_\\lambda(t) = \\frac{1}{\\eta - \\lambda} (\\eta - t)_+.\n$$\nDenote by $\\chi$ the step function \n$$\n\\chi_\\lambda (t) = \n\\begin{cases} \n1,  \\quad & {\\rm if} \\quad t<\\lambda,\\\\\n0,\\quad & {\\rm if} \\quad t \\ge \\lambda,\n\\end{cases}\n$$\nand let  \n$$\nN(\\lambda) = \\# \\{k:\\, \\lambda_k<\\lambda\\},\n$$\nbe the number of the eigenvalues below $\\lambda$ of the operator $\\mathcal H$.\n\nThen by using the previous statement we have \n$$\nN(\\lambda) \\le \\frac{1}{\\eta - \\lambda} \\, \\sum_k (\\eta - \\lambda_k)_+ \\le \n\\frac{1}{\\eta - \\lambda} \\,  \\frac{1}{(2\\pi)^{d}}\\, |\\Omega|\\, e^{d\\eta} \\,  |B_d|\\, d^{-1}.\n$$\nMinimising the right hand side w.r.t. $\\eta$ we find $\\eta = \\lambda + \\frac1d\n$ and thus obtain the following\n\n\\begin{cor}\n\\label{cor-N-lambda}\nFor the number $N(\\lambda)$ of the eigenvalues of the operator $\\mathcal H$ below $\\lambda$ we have\n\\begin{equation}\\label{Numb}\nN(\\lambda) \\le \n e^{\\lambda d +1} \n\\frac{1}{(2\\pi)^{d}}\\, |\\Omega| \\, |B_d|.\n\\end{equation}\n\\end{cor}\n\n\n\n\\section{A lower bound for $\\lambda_1(\\Omega)$}\n\\label{sec:lower-bound-lambd}\n\nIn this section, we focus on lower bounds for the first eigenvalue $\\lambda_1= \\lambda_1(\\Omega)$. From Corollary~\\ref{cor-N-lambda}, we readily deduce the following bound. \n\n\\begin{thm} \n\\label{lower-bound-lambda_1-first}\nLet $\\Omega \\subset {\\mathbb R}^d$ be an open set of finite measure. Then we have \n\\begin{equation}\n  \\label{eq:est-lambda_1-first}\n\\lambda_1(\\Omega) \\ge \\frac{1}{d} \\log \\frac{(2\\pi)^{d}}{e |\\Omega| \\, |B_d|}.\n\\end{equation}\nIn particular, if $|\\Omega| \\le \\frac{(2\\pi)^{d}}{e\\, |B_d|}$, then the operator $\\mathcal H$ does not have negative eigenvalues.\n\\end{thm}\n\n\\begin{proof}\nIf $\\lambda < \\frac{1}{d} \\log \\frac{(2\\pi)^{d}}{e |\\Omega| \\, |B_d|}$, then \n$N(\\lambda)<1$ by (\\ref{Numb}), and therefore $N(\\lambda)=0$. Consequently, $\\mathcal H$ does not have eigenvalues below $\\frac{1}{d} \\log \\frac{(2\\pi)^{d}}{e |\\Omega| \\, |B_d|}$.\n\\end{proof}\n\n\n\\begin{rem}\nNote that the inequalities \\eqref{BU}, \\eqref{Numb} and \\eqref{eq:est-lambda_1-first} hold for any open set $\\Omega$ of finite measure without any additional conditions on its boundary.\n\\end{rem}\n\n\nIn the following, we wish to improve the bound given in Theorem~\\ref{lower-bound-lambda_1-first} in low dimensions $d$ for open boundary sets with Lipschitz boundary. We shall use the following Faber-Krahn type inequality.\n\n\\begin{thm} (\\cite[Corollary 1.6]{HW})\\\\\n\\label{sec:faber-Krahn-main}\nLet $\\rho>0$. Among all bounded open sets $\\Omega$ with Lipschitz boundary and $|\\Omega| = \\rho$, the ball $B=B_r(0)$ with $|B|=\\rho$ minimizes $\\lambda_1(\\Omega)$.\n\\end{thm}\n\n \n\\begin{cor}\n\\label{cor-faber-krahn}\nFor every open bounded sets $\\Omega$ with Lipschitz boundary we have \n\\begin{equation}\n  \\label{eq:sharp-lower-bound}\n\\lambda_1(\\Omega) \\ge \\lambda_1(B_d) + \\frac{1}{d}\\log \\frac{|B_d|}{|\\Omega|},\n\\end{equation}\nand equality holds if $\\Omega$ is a ball.   \n\\end{cor}\n\n\\begin{proof}\nThe result follows by combining Theorem~\\ref{sec:faber-Krahn-main} with the identity \n$$\n\\lambda_1(B_r(0)) = \\lambda_1(B_d) + \\log \\frac{1}{r}\\qquad \\text{for $r>0$,}\n$$\nwhich follows from the scaling property of $\\lambda_1$ noted in Lemma~\\ref{lemma-scaling-properties}.\n\\end{proof}\n\nCorollary~\\ref{cor-faber-krahn} gives a sharp lower bound, but it contains the  \nunknown quantity $\\lambda_1(B_d)$. By Theorem~\\ref{lower-bound-lambda_1-first}, we have \n\\begin{align}\n\\lambda_1(B_d) &\\ge \\frac{1}{d} \\log \\frac{(2\\pi)^{d}}{e |B_d|^2} = \n\\log (2\\pi) -\\frac{1}{d}\\bigl(1+ 2 \\log |B_d|\\bigr) \\nonumber \\\\\n&= \\frac{2}{d} \\log \\Gamma\\left(d\/2\\right)  + \\log 2 + \\frac{2}{d} \\log \\frac{d}{2} -\\frac{1}{d}.\n\\label{lower-bound-lambda-1-first}  \n\\end{align}\nThe following theorem improves this lower bound in low dimensions $d \\ge 2$. \n\n\\begin{thm} \n\\label{lower-bound-lambda-1-second}\nFor $d \\ge 2$, we have \n\\begin{equation}\n  \\label{eq:est-lambda-1-first}\n\\lambda_1(B_d) \\ge \\log \\bigl(2 \\sqrt{d +2}\\bigr) -  \n\\frac{2^{d+1} |B_d|^2 (d +2)^{\\frac{d}2} }{d (2\\pi)^{2d}}.\n\\end{equation}\n\\end{thm}\n\n\\begin{proof}\n Let $u \\in L^2(B_d)$ be radial with $\\|u\\|_{L^2}=1$. Then $\\widehat u$ is also radial, and \n\\begin{align*}\n|\\widehat u(\\xi)|&=|\\widehat u(s)|= s^{1-\\frac{d}{2}}\\left|\\int_{0}^{1} u(r)J_{\\frac{d}{2}-1}(rs)r^{\\frac{d}{2}}dr\\right| \\\\\n  &\\le s^{1-\\frac{d}{2}} \\left( \\int_0^{1}r^{d-1} u^2(r)\\,dr\\right)^{1\/2} \n\\left(\\int_0^{1}rJ_{\\frac{d}{2}-1}^2(sr) \\,dr\\right)^{1\/2}\\\\ \n  &=\\frac{s^{1-\\frac{d}{2}}}{\\sqrt{|S^{d-1}|}} \n\\left(s^{-2} \\int_0^{s} \\tau J_{\\frac{d}{2}-1}^2(\\tau) \\,d\\tau\\right)^{1\/2} \\\\\n&=\\frac{s^{-\\frac{d}{2}}}{\\sqrt{|S^{d-1}|}} \n\\left(\\int_0^{s} \\tau J_{\\frac{d}{2}-1}^2(\\tau) \\,d\\tau\\right)^{1\/2}\\qquad \\text{for $\\xi \\in {\\mathbb R}^d$ with $s = |\\xi|$.}\n\\end{align*}\nConsequently, \n$$\n|S^{d-1}| |\\widehat u(s)|^2 \\le s^{-d} \\int_0^{s} \\tau J_{\\frac{d}{2}-1}^2(\\tau) \\,d\\tau.\n$$\nIn the case where, in addition, $u$ is a radial eigenfunction of (\\ref{D}) corresponding to $\\lambda_1$ in $\\Omega= B_d$, it follows that, for every $\\lambda \\in {\\mathbb R}$,  \n\\begin{align*}\n&(2\\pi)^{d}[\\lambda-\\lambda_1] =  \\int_{{\\mathbb R}^d} (\\lambda -\\ln |\\xi|)|\\widehat u(\\xi)|^2\\,d\\xi \\le  \\int_{{\\mathbb R}^d} (\\lambda -\\ln |\\xi|)_+|\\widehat u(\\xi)|^2\\,d\\xi\\\\\n&=|S^{d-1}| \n\\int_0^\\infty s^{d-1} (\\lambda -\\ln s)_+|\\widehat u(s)|^2\\,ds \\le \\int_0^\\infty \\frac{(\\lambda -\\ln s)_+}{s}  \\int_0^{s} \\!\\!\\!\\tau J_{\\frac{d}{2}-1}^2(\\tau) \\,d\\tau \\,ds\\\\\n&= \\int_0^\\infty  \\tau J_{\\frac{d}{2}-1}^2(\\tau) \\int_{\\tau}^\\infty \\frac{(\\lambda -\\ln s)_+}{s}  \\,ds d\\tau= \\int_0^{e^\\lambda}  \\tau J_{\\frac{d}{2}-1}^2(\\tau) \\int_{\\tau}^{e^\\lambda} \\frac{\\lambda -\\ln s}{s} \\,ds d\\tau\\\\\n&= \\int_0^{e^\\lambda}  \\tau J_{\\frac{d}{2}-1}^2(\\tau) \\int_{\\ln \\tau}^{\\lambda}(\\lambda - s) \\,ds d\\tau= \\int_0^{e^\\lambda}  \\tau J_{\\frac{d}{2}-1}^2(\\tau) \\int_{0}^{\\lambda- \\ln \\tau} s \\,ds d\\tau\\\\\n&= \\frac{1}{2}  \\int_0^{e^\\lambda}  \\tau J_{\\frac{d}{2}-1}^2(\\tau)\n\\bigl(\\lambda- \\ln \\tau \\bigr)^2 \\,d\\tau = \\frac{e^{2\\lambda}}{2}  \\int_0^{1}  \\tau J_{\\frac{d}{2}-1}^2(e^\\lambda \\tau) \\ln^2 \\tau \\,d\\tau. \n\\end{align*}\nWe now use the following estimate for Bessel functions of the first kind:\n\\begin{equation}\n  \\label{eq:bessel-est-proof}\nJ_\\nu(x) \\le \\frac{x^\\nu}{2^\\nu \\Gamma(\\nu+1)} \\quad \\text{for $\\:\\nu > \\sqrt{3}-2$, $\\:0 \\le x < 2 \\sqrt{2(\\nu+2)}$.}\n\\end{equation}\nA proof of this elementary estimate is given in the Appendix. We wish to apply (\\ref{eq:bessel-est-proof}) with $\\nu = \\frac{d}{2}-1$. This gives \n$$\ne^{2\\lambda} J_{\\frac{d}{2}-1}^2(r_0 e^\\lambda \\tau) \\le e^{d\\lambda} \\frac{\\tau^{d-2}}{2^{d-2}\\Gamma^2 (\\frac{d}{2})}=\n \\frac{d^2 |B_d|^2e^{d\\lambda}}{(2\\pi)^{d}} \\tau^{d-2}\n \\qquad \\text{for $\\tau \\in [0,1]$} \n$$ \nif $d \\ge 2$ and $e^\\lambda \\le 2 \\sqrt{d +2}$, i.e., if   \n\\begin{equation}\n\\label{condition-proof}\nd \\ge 2 \\quad \\text{and}\\quad \\lambda \\le \\log \\bigl(2 \\sqrt{d +2}\\bigr).\n\\end{equation}\nHere we used that $|B_d|= \\frac{2}{d} \\frac{\\pi^{\\frac{d}{2}}}{\\Gamma(d\/2)}$. Consequently, if (\\ref{condition-proof}) holds, we find that  \n$$\n(2\\pi)^{d}[\\lambda-\\lambda_1] \\le  \\frac{d^2 |B_d|^2e^{d\\lambda}}{(2\\pi)^{d}}\\int_0^{1} \n \\tau^{d-1} \\ln^2 \\tau \\,d\\tau,\n$$\nwhere \n$$\n\\int_0^{1} \\tau^{d-1} \\ln^2 \\tau d\\tau = - \\frac{2}{d}\n\\int_0^1 \\tau^{d-1} \\ln \\tau d\\tau = \\frac{2}{d^2} \\int_{0}^1 \\tau^{d-1}d\\tau\n= \\frac{2}{d^3}.\n$$\nHence \n$$\n(2\\pi)^{d}[\\lambda -\\lambda_1] \\le \\frac{2|B_d|^2}{d (2\\pi)^{d}}e^{d\\lambda}, \\quad \\text{i.e.,}\\quad \\lambda_1 \\ge \\lambda- \\frac{2|B_d|^2 }{d (2\\pi)^{2d}} e^{d\\lambda}.\n$$\nInserting the value $\\lambda = \\log \\bigl(2 \\sqrt{d +2}\\bigr)$ from (\\ref{condition-proof}), we deduce that \n$$\n\\lambda_1= \\lambda_1(B_d) \\ge  \\log \\bigl(2 \\sqrt{d +2}\\bigr) - \\frac{2^{d+1} |B_d|^2 (d +2)^{\\frac{d}2} }{d(2\\pi)^{2d}},  \n$$\nas claimed. \n\\end{proof}\n\n\n\\begin{rem}{\\rm \n\\label{rem-comparison-of-other-bounds}\nIt seems instructive to compare the lower bounds given in (\\ref{lower-bound-lambda-1-first}) and (\\ref{eq:est-lambda-1-first}) with other bounds obtained from spectral estimates which are already available in the literature. We first mention Beckner's logarithmic estimate of uncertainty \\cite[Theorem 1]{B}, which implies that\\footnote{We note here that a different definition of Fourier transform is used in \\cite{B} and therefore the inequality looks slightly different}\n\\begin{equation*}\n(\\varphi,\\varphi)_{log} \\ge \\int_{{\\mathbb R}^d} \\left[\\psi\\left(d\/4\\right)+ \\log \\frac{2}{|x|}\\right]\\varphi^2(x) dx \\ge \\left[\\psi\\left(d\/4\\right)+ \\log 2\\right]\\|\\varphi\\|_2^2 \n\\end{equation*}\nfor functions $\\varphi \\in C^\\infty_c(B_d)$ and therefore \n\\begin{equation}\n  \\label{eq:beckner-lambda-1-est}\n\\lambda_1(B_d) \\ge \\psi\\left(d\/4\\right)+ \\log 2 .\n\\end{equation}\nHere, as before, $\\psi = \\frac{\\Gamma'}{\\Gamma}$ denotes the Digamma function. Next we state a further lower bound for $(\\varphi,\\varphi)_{log}$ which follows from \\cite[Proposition 3.2 and Lemma 4.11]{HW}. We have\n\\begin{equation} \n\\label{cw-inequality}\n(\\varphi,\\varphi)_{log} \\ge \\zeta_d \\|\\varphi\\|_2^2 \\qquad \\text{for $\\varphi \\in C^\\infty_c(B_d)$,}\n\\end{equation}\nwhere $\\zeta_d$ is given in (\\ref{eq:def-zeta_d}), i.e., \n$$\n\\zeta_d =  \\log 2 + \\frac{1}{2}\\left( \\psi(d\/2)-\\gamma\\right) = \\left\\{\n  \\begin{aligned}\n  &- \\gamma + \\sum_{k=1}^{\\frac{d-1}{2}} \\frac{1}{2k-1},&&\\qquad \\text{$d$ odd,}\\\\\n  &\\log 2 - \\gamma + \\sum_{k=1}^{\\frac{d-2}{2}} \\frac{1}{k},&&\\qquad \\text{$d$ even.}   \n  \\end{aligned}\n\\right.\n$$\nInequality (\\ref{cw-inequality}) implies that\n\\begin{equation} \n\\label{cw-lambda-1-bound}\n\\lambda_1(B_d) \\ge \\zeta_d.\n\\end{equation}\nThe latter inequality can also be derived from a lower bound of Ba$\\rm{\\tilde{n}}$uelos and Kulczycki for the \nfirst Dirichlet eigenvalue $\\lambda_1^\\alpha(B_d)$ of the fractional Laplacian $(-\\Delta)^{\\alpha\/2}$ in $B_d$. In \n\\cite[Corollary 2.2]{BK}, it is proved that \n$$\n\\lambda_1^\\alpha(B_d) \\ge 2^\\alpha \\frac{\\Gamma(1+\\frac{\\alpha}{2}) \\Gamma(\\frac{d+\\alpha}{2})}{\\Gamma(\\frac{d}{2})}\\qquad \\text{for $\\alpha \\in (0,2)$.}\n$$\nCombining this inequality with the characterization of $\\lambda_1(B_d)$ given in \\cite[Theorem 1.5]{HW}, we deduce that   \n$$\n\\lambda_1(B_d)= \\lim_{\\alpha \\to 0^+}\\frac{\\lambda_1^\\alpha(B_d)-1}{\\alpha}\\ge \\frac{d}{d\\alpha}\\Big|_{\\alpha=0}\\, 2^\\alpha \\frac{\\Gamma(1+\\frac{\\alpha}{2}) \\Gamma(\\frac{d+\\alpha}{2})}{\\Gamma(\\frac{d}{2})} = \\zeta_d, \n$$\nas stated in (\\ref{cw-lambda-1-bound}). \n\nWe briefly comment on the quality of the lower bounds obtained here in low and high dimensions. In low dimensions $d \\ge 2$, (\\ref{eq:est-lambda-1-first}) is better than the bounds (\\ref{lower-bound-lambda-1-first}), (\\ref{eq:beckner-lambda-1-est}) and (\\ref{cw-lambda-1-bound}). In dimension $d=1$ where the bound (\\ref{eq:est-lambda-1-first}) is not available, the bound (\\ref{lower-bound-lambda-1-first}) yields the best value. The following table shows numerical values of the bounds $b_1(d)$, $b_2(d)$, $b_3(d)$ resp. $b_4(d)$ given by (\\ref{lower-bound-lambda-1-first}), (\\ref{eq:est-lambda-1-first}), (\\ref{eq:beckner-lambda-1-est}), (\\ref{cw-lambda-1-bound}), respectively. \n\\medskip\n\n\\begin{center}\n{\\tiny\n\\renewcommand{\\arraystretch}{1.6}\n    \\begin{tabular}{ l | l | l | l | l |l | l | l | l | l | l |}\n    $d$ & 1 & 2 & 3 & 4&5&6&7&8&9&10\\\\ \\hline\n    $b_1(d)$ & $-0,55$ & $0,19$ & $0,55$ & $0,79$ &$0,97$&$1,12$ & $1,25$ & $1,36$ & $1,46$ & $1,55$\n\\\\ \\hline\n$b_2(d)$ & $\\quad\/$& $1,28 $& $1,48 $ & $1,59 $& $1,67$&$1,73$ &$1,79$ & $1,84$& $1,89$ & $1,94$\\\\ \\hline\n$b_3(d)$ &$-3.53$ & $-1,27$ & $-0,39$ &$0,12$&$0,47$ & $0,73$&$0,94$& $1,12$ &$1,27$ & $1,40$ \\\\ \\hline\n$b_4(d)$ &$-0,58$& $0,12$ &$0,42$ &$0,62$& $0,76$ &$0,87$ &$0,96$ & $1,03$ & $1,10$ & $1,16$\n\\end{tabular}\n\\renewcommand{\\arraystretch}{1}\n}\n\\end{center}\n\\medskip\n\nTo compare the bounds in high dimensions, we consider the asymptotics as $d \\to \\infty$. \nSince $\\frac{\\log \\Gamma(t)}{t} = \\log t - 1 + o(t)$ as $t \\to \\infty$, the bound (\\ref{lower-bound-lambda-1-first}) yields \n\\begin{equation}\n\\label{lower-bound-lambda-1-first-asymptotics}\n\\lambda_1(B_d) \\ge \\log d - 1 + o(1) \\qquad \\text{as $d \\to \\infty$,} \n\\end{equation}\nwhereas (\\ref{eq:est-lambda-1-first}) obviously gives \n\\begin{equation}\n\\label{lower-bound-est-lambda-1-first-asymptotics}\n\\lambda_1(B_d) \\ge \\log \\sqrt{d+2} + \\log 2  + o(1) \\qquad \\text{as $d \\to \\infty$,} \n\\end{equation}\nMoreover, from (\\ref{eq:beckner-lambda-1-est}) and the fact that \n\\begin{equation}\n  \\label{eq:Digamma-asymptotics}\n\\psi(t) = \\log t + o(1)\\qquad\\text{as $t \\to \\infty$,}   \n\\end{equation}\nwe deduce that \n\\begin{equation}\n  \\label{eq:beckner-lambda-1-est-asymptotics}\n\\lambda_1(B_d) \\ge \\log d - \\log 2 + o(1) \\qquad \\text{as $d \\to \\infty$,} \n\\end{equation}\nFinally, (\\ref{cw-inequality}) and (\\ref{eq:Digamma-asymptotics}) yield\n\\begin{equation} \n\\label{cw-lambda-1-bound-asymptotics}\n\\lambda_1(B_d) \\ge \\log \\sqrt{d} +  \\log 2  -\\frac{\\gamma}{2} + o(1) \\qquad \\text{as $d \\to \\infty$.} \n\\end{equation}\nSo (\\ref{eq:beckner-lambda-1-est-asymptotics}) provides the best asymptotic bound as $d \\to \\infty$.\n\nNumerical computations indicate that the bound (\\ref{eq:est-lambda-1-first}) is better than the other bounds for $2 \\le d \\le 21$, and (\\ref{eq:beckner-lambda-1-est}) is the best among these bounds for $d \\ge 22$.\n}\n\n\\end{rem}\n\n\n\n\n\n\\section{An asymptotic lower trace bound}\\label{LowB1}\n\nThroughout this section, we let $\\Omega \\subset {\\mathbb R}^d$ denote an open set of finite measure. In this section we prove the following asymptotic lower bound. A similar statement was obtained in \\cite{G} for the Dirichlet boundary problem for a fractional Laplacian.\n\n\\begin {thm}\\label{2.1}\nFor the eigenvalues of the problem \\eqref{D} and any $\\lambda\\in \\Bbb R$ we have\n\\begin{equation}\\label{BLow}\n\\liminf_{\\lambda\\to\\infty} e^{-d\\lambda} \\sum_{k}(\\lambda - \\lambda_k)_+ \\ge \\frac{1}{(2\\pi)^{d}}\\, |\\Omega|\\, \\, |B_d|\\, d^{-1}.\n\\end{equation}\n\\end{thm}\n\n\\begin{proof}\nLet us fix $\\delta>0$ and consider \n$$\n\\Omega_\\delta = \\{ x\\in \\Omega: \\, {\\rm dist}(x, \\Bbb R^d \\setminus \\Omega) >\\delta\\}.\n$$\nSince $\\delta$ is arbitrary it  suffices to show the lower bound \\eqref{BLow}, where $\\Omega$ is replaced by $\\Omega_\\delta$.\nLet $g\\in C_0^\\infty(\\Bbb R^d)$ be a real-valued even function, $\\|g\\|_{L^2(\\Bbb R^d)} = 1$ with support in $\\{x\\in \\Bbb R^d: \\, |x| \\le \\delta\/2\\}$. For $\\xi\\in \\Bbb R^d$ and $x\\in \\Omega_\\delta$ we introduce the \\lq\\lq coherent state\" \n$$\ne_{\\xi,y}(x) =  e^{-i\\xi x} g(x-y).\n$$\nNote that $\\|e_{\\xi,y}\\|_{L^2(\\Bbb R^d)} = 1$. Using the properties of coherent states \\cite[Theorem 12.8]{LL} we obtain\n$$\n\\sum_{k}(\\lambda - \\lambda_k)_+ \\ge \n \\frac{1}{(2\\pi)^{d}}\\, \\int_{\\Bbb R^d} \\int_{\\Omega_\\delta} (e_{\\xi,y}, (\\lambda - \\mathcal H)_+ e_{\\xi,y})_{L^2(\\Omega)} \\, dy d\\xi.\n$$\nSince $t \\mapsto (\\lambda-t)_+$ is convex then applying Jensen's inequality to the spectral measure of $\\mathcal H$ we obtain\n\\begin{equation}\\label{jensen}\n\\sum_{k}(\\lambda - \\lambda_k)_+ \\ge   \\frac{1}{(2\\pi)^{d}}\\, \\int_{\\Bbb R^d} \\int_{\\Omega_\\delta}\n\\left(\\lambda - (\\mathcal H e_{\\xi,y}, e_{\\xi,y})_{L^2(\\Omega)} \\right)_+ \\, dy d\\xi.\n\\end{equation}\nNext we consider the quadratic form \n\\begin{multline*}\n\\left(\\mathcal H e_{\\xi,y},  e_{\\xi,y} \\right)_{L^2(\\Omega)} = \\frac{1}{(2\\pi)^d} \\, \\int_{\\Bbb R^d} \\int_{\\Omega} \\int_{\\Omega} e^{i(x-z)(\\eta-\\xi)} g(x-y)g(z-y) \\log(|\\eta|)  \\, dz dx d\\eta \\\\\n=\n\\frac{1}{(2\\pi)^d} \\, \\int_{\\Bbb R^{d}} \\int_{\\Omega} \\int_{\\Omega} \ne^{i(x-z)\\rho} g(x-y)g(z-y) \\log(|\\xi-\\rho|) \\, dz dx d\\rho\\\\\n= \\frac{1}{(2\\pi)^d} \\, \\int_{\\Bbb R^{d}} \\int_{\\Omega} \\int_{\\Omega} \ne^{i(x-z)\\rho} g(x-y)g(z-y) \\left( \\log|\\xi| + \\log \\left(\\left|\\xi -\\rho\\right|\/|\\xi| \\right)\\right)\n\\, dz dx d\\rho\\\\\n=  \\log|\\xi| + R(y,\\xi).\n\\end{multline*}\nSince $g\\in C_0^\\infty(\\Bbb R^d)$ we have for any $M>0$\n\\begin{multline*}\nR(y,\\xi) =  \\\\\n\\frac{1}{(2\\pi)^d} \\, \\int_{\\Bbb R^{d}} \\int_{\\Omega} \\int_{\\Omega} \ne^{i(x-y)\\rho} g(x-y) e^{i(y-z)\\rho} g(z-y)  \\log \\left(\\left|\\xi -\\rho\\right|\/|\\xi| \\right)\n\\, dz dx d\\rho\\\\\n= \\int_{\\Bbb R^{d}} |\\widehat{g}|^2\\, \\log \\left(\\left|\\xi -\\rho\\right|\/|\\xi| \\right) \\, d\\rho\n \\le C_M\\,\n\\int_{\\Bbb R^{d}} (1+ |\\rho|)^{-M}   \\log \\left(\\left|\\xi -\\rho\\right|\/|\\xi| \\right)  \\,  d\\rho\\\\\n\\le C\\, |\\xi|^{-1}.\n\\end{multline*}\nTherefore from \\eqref{jensen} we find \n\\begin{equation}\\label{below1}\n\\sum_{k}(\\lambda - \\lambda_k)_+  \\ge (2\\pi)^{-d}\\, |\\Omega_\\delta| \\, \\int_{\\Bbb R^d}  (\\lambda -  \\log|\\xi| -  C|\\xi|^{-1})_+ \\, d\\xi.\n\\end{equation}\nLet us redefine the spectral parameter $\\lambda =  \\ln \\mu$.\nThen introducing polar coordinates we find\n\\begin{multline}\\label{below22} \n\\int_{\\Bbb R^d}  (\\lambda -  \\log|\\xi| -  C|\\xi|^{-1})_+ \\, d\\xi = \\left|\\Bbb S^{d-1} \\right| \\, \\int_0^\\infty \\left(\\ln \\frac{\\mu}{r} - \\frac{C}{r}\\right)_+ \\, r^{d-1}dr\\\\\n= \n\\mu^d\\, \\left|\\Bbb S^{d-1} \\right| \\, \\int_0^\\infty \\left(\\ln \\frac{1}{r} - \\frac{C}{\\mu r}\\right)_+ \\, r^{d-1}dr\n\\end{multline} \nThe expression in the latter integral is positive if $ - r\\ln r > C\\mu^{-1}$. The function $ -r\\ln r $ is concave. \n\n\\smallskip\n\n\\qquad\\qquad\\qquad \\qquad \\qquad{\\centering \n{\\includegraphics[scale=.4]{concave.png}} }\n \n\\smallskip\n \\noindent\nIts maximum is achieved at $r=1\/e$ at the value $1\/e$. The equation\n$ - r\\ln r = C\\mu^{-1}$ has two solutions $r_1(\\mu)$ and $r_2(\\mu)$ such that $r_1(\\mu) \\to 0$ and $r_2(\\mu)\\to 1$ as $\\mu \\to\\infty$\nTherefore \n\\begin{multline}\\label{below3}\n\\int_0^\\infty \\left(\\ln \\frac{1}{r} - \\frac{C}{\\mu r}\\right)_+ \\, r^{d-1}dr \\ge \n\\int_{r_1(\\mu)}^{r_2(\\mu)}     \\left(\\ln \\frac{1}{r} - \\frac{C}{\\mu r}\\right)  \\, r^{d-1}dr \\\\\n=-\\frac1d\\, r^d \\ln r  \\Big|_{r_1(\\mu)}^{r_2(\\mu)}+ \\frac{C}{\\mu(d+1)} r^{d+1}  \\Big|_{r_1(\\mu)}^{r_2(\\mu)}\n+ \\frac{1}{d^2}r^d   \\Big|_{r_1(\\mu)}^{r_2(\\mu)} \\to \\frac{1}{d^2} \\quad {\\rm as} \\quad \\mu\\to\\infty.\n\\end{multline} \nPutting together \\eqref{below1}, \\eqref{below22} and \\eqref{below3} and using $\\mu = e^\\lambda$ we obtain\n$$\n\\liminf_{\\lambda\\to\\infty} e^{-d\\lambda} \\sum_{k}(\\lambda - \\lambda_k)_+ \\ge \\frac{1}{(2\\pi)^{d}}\\, |\\Omega_\\delta|\\, \\, |B_d|\\, d^{-1}.\n$$\nSince $\\delta>0$ is arbitrary we complete the proof of  Theorem \\ref{2.1}.\n\n\\end{proof}\n\n\\section{Weyl asymptotics}\\label{Weyl}\n\n\\noindent\nThroughout this section, we let $\\Omega \\subset {\\mathbb R}^d$ denote an open set of finite measure. Combining Theorems \\ref{1.1} and \\ref{2.1} we have\n\n\n\\begin {thm}\\label{3.1}\nThe Riesz means of the eigenvalues of the Dirichlet boundary value problem  \\eqref{D} satisfy the following asymptotic formula \n\\begin{equation}\\label{Weyl1}\n \\lim_{\\lambda\\to\\infty} e^{-d\\lambda}\\, \\sum_{k}(\\lambda - \\lambda_k)_+ = \\frac{1}{(2\\pi)^{d}}\\, |\\Omega|\\,  |B_d|\\, d^{-1}.\n\\end{equation}\n\\end{thm}\nAs a corollary we can obtain asymptotics of the number of  the eigenvalues of the operator $\\mathcal H$.\n\n\n\\begin{cor} \\label{3.2}\nThe number of the eigenvalues $N(\\lambda)$ of the Dirichlet boundary value problem  \\eqref{D} below $\\lambda$ satisfies the following asymptotic formula \n\\begin{equation}\\label{Weyl2}\n\\lim_{\\lambda\\to\\infty} e^{-d\\lambda} \\, N(\\lambda) =   \\frac{1}{(2\\pi)^{d}}\\, |\\Omega|\\,  |B_d|.\n\\end{equation}\n\\end{cor} \n\n\\begin{proof}\nIn order to prove \\eqref{Weyl2} we use two simple inequalities. If $h>0$, then\n\\begin{equation}\\label{Nabove}\n\\frac{(\\lambda + h - \\lambda_k)_+   - (\\lambda - \\lambda_k)_+}{h} \\ge 1_{\\text{\\tiny $(-\\infty,\\lambda)$}}(\\lambda_k)\n\\end{equation}\nand\n\\begin{equation}\\label{Nbelow}\n\\frac{(\\lambda - \\lambda_k)_+   - (\\lambda - h- \\lambda_k)_+}{h}\n \\le 1_{\\text{\\tiny $(-\\infty,\\lambda)$}}(\\lambda_k)\n\\end{equation}\n\nThe inequality \\eqref{Nabove} implies, together with Theorems \\ref{1.1} and~\\ref{2.1}, that \n\\begin{align*}\n&\\limsup_{\\lambda \\to \\infty}e^{-d\\lambda}N(\\lambda) \\le \n\\limsup_{\\lambda \\to \\infty}e^{-d\\lambda} \\sum_{k}\\frac{(\\lambda + h - \\lambda_k)_+   - (\\lambda - \\lambda_k)_+}{h}\\\\\n&\\le \\frac{1}{h}\\Bigl[e^{dh} \\limsup_{\\lambda \\to \\infty}e^{-d(\\lambda+h)} \\sum_{k}(\\lambda + h - \\lambda_k)_+   -\\liminf_{\\lambda \\to \\infty}e^{-d\\lambda} \\sum_{k}(\\lambda - \\lambda_k)_+\\Bigr]\\\\\n&\\le \\frac{|\\Omega| |B_d|}{d(2\\pi)^d}\\:\\frac{e^{dh}-1}{h} \\qquad \\text{for every $h>0$}\n\\end{align*}\nand thus \n\\begin{equation}\n  \\label{eq:limsup-N-lambda-ineq}\n\\limsup_{\\lambda\\to\\infty}e^{-d\\lambda}N(\\lambda) \\le \\frac{|\\Omega| |B_d|}{d(2\\pi)^d}\\lim_{h \\to 0^+}\\frac{e^{dh}-1}{h}= \\frac{|\\Omega|\\,  |B_d|}{(2\\pi)^{d}}.\n\\end{equation}\nMoreover, \\eqref{Nabove} implies, together with Theorems \\ref{1.1} and~\\ref{2.1}, that \n\\begin{align*}\n&\\liminf_{\\lambda \\to \\infty}e^{-d\\lambda}N(\\lambda) \\ge \n\\liminf_{\\lambda \\to \\infty}e^{-d\\lambda} \\sum_{k}\\frac{(\\lambda - \\lambda_k)_+   - (\\lambda -h - \\lambda_k)_+}{h}\\\\\n&\\ge \\frac{1}{h}\\Bigl[e^{dh} \\liminf_{\\lambda \\to \\infty}e^{-d \\lambda} \\sum_{k}(\\lambda - \\lambda_k)_+   -e^{-dh}\\limsup_{\\lambda \\to \\infty} e^{-d(\\lambda-h)} \\sum_{k}(\\lambda -h - \\lambda_k)_+\\Bigr]\\\\\n&\\ge \\frac{|\\Omega| |B_d|}{d(2\\pi)^d}\\:\\frac{1-e^{-dh}}{h} \\qquad \\text{for every $h>0$}\n\\end{align*}\nand therefore \n\\begin{equation}\n  \\label{eq:liminf-N-lambda-ineq}\n\\liminf_{\\lambda\\to\\infty}e^{-d\\lambda}N(\\lambda) \\ge \\frac{|\\Omega| |B_d|}{d(2\\pi)^d}\\lim_{h \\to 0^+}\\frac{1-e^{-dh}}{h}= \\frac{|\\Omega|\\,  |B_d|}{(2\\pi)^{d}}.\n\\end{equation}\nThe claim follows by combining (\\ref{eq:limsup-N-lambda-ineq}) and (\\ref{eq:liminf-N-lambda-ineq}).  \n\\end{proof}\n\n\n\\section{An exact lower trace bound}\\label{LowB2}\n\nIn this section we prove the following exact lower bound in the case of bounded open sets with Lipschitz boundary.\n\n\\begin {thm}\\label{2.1-new-lower-bound}\nLet $\\Omega \\subset {\\mathbb R}^d$, $N \\ge 2$ be an open bounded set with Lipschitz boundary, let $\\tau \\in (0,1)$, and let \n\\begin{equation}\n\\label{def-C-Omega}\nC_{\\Omega,\\tau} := \\frac{1}{|\\Omega|(2\\pi)^d} \\int_{\\Bbb R^d}(1+|\\rho|)^\\tau \\log(1+|\\rho|) |\\widehat{1_\\Omega}(\\rho)|^2\\,d\\rho, \n\\end{equation}\nwhere $1_\\Omega$ denotes the indicator function of $\\Omega$. \n\nFor any $\\lambda \\ge 2 C_{\\Omega,\\tau}$,  we have\n\\begin{equation}\\label{BL}\n\\sum_{k}(\\lambda - \\lambda_k)_+\n\\ge \\frac{|\\Omega|\\, |B_d|}{(2\\pi)^{d}\\,d} \\Bigl[e^{d\\lambda} \\,- \\,a_\\tau\\, C_{\\Omega,\\tau}\\,e^{(d-\\tau)\\lambda} \\,- \\,b_\\tau\\, C_{\\Omega,\\tau}^2\\, \ne^{(d-2\\tau)\\lambda} \\,-\\, (d \\lambda + 1)  \\Bigr] \\nonumber\n\\end{equation}\nwith $a_\\tau:= \\frac{d(d-\\tau)-1}{d-\\tau}$ and $b_\\tau := 4d \\tau$.\n\\end{thm}\n\n\\begin{rem}\nIn the definition of $C_{\\Omega,\\tau}$, we need $\\tau<1$, otherwise the integral might not converge. In particular, if $\\Omega=B_d$ is the unit ball in ${\\mathbb R}^d$, we have \n$$\n\\widehat{1_\\Omega}(\\rho)= (2\\pi)^{\\frac{d}{2}} |\\rho|^{-\\frac{d}{2}}J_{\\frac{d}{2}}(|\\rho|)\n$$\nwhere $J_{\\frac{d}{2}}(r)= O(\\frac{1}{\\sqrt{r}})$ as $r \\to \\infty$. Hence the integral defining $C_{\\Omega,\\tau}$ converges if $\\tau <1$. A similar conclusion arises for cubes or rectangles, where \n$$\n\\widehat{1_\\Omega}(\\rho) = f_1(\\rho_1) \\cdot \\dots \\cdot f_d(\\rho_d)\n$$\nand $f_j(s) = O(\\frac{1}{s})$ as $|s| \\to \\infty$, $j=1,\\dots,d$.\n\nOn the other hand, if $\\Omega \\subset {\\mathbb R}^d$ is an open bounded set with Lipschitz boundary, we have \n\\begin{equation}\n  \\label{eq:C-Omega-tau-finiteness}\nC_{\\Omega,\\tau}<\\infty \\qquad \\text{for $\\tau \\in (0,1)$.}  \n\\end{equation}\nIndeed, in this case, $\\Omega$ has finite perimeter, i.e., $1_\\Omega \\in BV({\\mathbb R}^d)$. Therefore, as noted e.g. in \\cite[Theorem 2.14]{Lombardini}, $\\Omega$ also has finite fractional perimeter\n$$\nP_\\tau(\\Omega)= \\int_{\\Omega}\\int_{{\\mathbb R}^d \\setminus \\Omega} |x-y|^{-d-\\tau}\\,dxdy = \\frac{1}{2} \\int \\!\\! \\int_{{\\mathbb R}^{2d}}\\frac{(1_\\Omega(x)-1_\\Omega(y))^2}{|x-y|^{d+\\tau}}\\,dxdy\n$$\nfor every $\\tau \\in (0,1)$. Moreover, $P_\\tau(\\Omega)$ coincides, up to a constant, with the integral\n$$\n\\int_{\\Bbb R^d}|\\rho|^\\tau |\\widehat{1_\\Omega}(\\rho)|^2\\,d\\rho\n$$\nwhich therefore is also finite for every $\\tau \\in (0,1)$. Since moreover $1_\\Omega$ and therefore also $\\widehat{1_\\Omega}$ are functions in $L^2(\\Bbb R^d)$ and for every $\\varepsilon>0$ there exists $C_\\varepsilon>0$ with \n$$\n(1+|\\rho|)^\\tau \\log(1+|\\rho|) \\le C_\\varepsilon \\bigl(1 + |\\rho|^{\\tau+\\varepsilon}\\bigr) \\qquad \\text{for $\\rho \\in {\\mathbb R}^d$,}\n$$\nit follows that (\\ref{eq:C-Omega-tau-finiteness}) holds. \n\\end{rem}\n\nIn the proof of Theorem~\\ref{2.1-new-lower-bound}, we will use the following elementary estimate. \n\n\\begin{lem}\n\\label{elem-lemma}\nFor $r \\ge 0$, $s>0$ and $\\tau \\in (0,1)$, we have \n\\begin{equation}\n  \\label{eq:first-elem-ineq}\n\\log\\left(1 + \\frac{r}{s}\\right) \\le \\frac{1}{s} \\log(1+r) \\qquad \\text{if $s \\in (0,1)$}\n\\end{equation}\nand \n\\begin{equation}\n  \\label{eq:second-elem-ineq}\n\\log\\left(1 + \\frac{r}{s}\\right) \\le \\frac{(1+r)^\\tau}{s^\\tau} \\log(1+r) \\qquad \\text{if $s \\ge 1$.}\n\\end{equation}\nIn particular, \n$$\n\\log\\left(1 + \\frac{r}{s}\\right) \\le \\max \\left\\{\\frac{1}{s}, \\frac{1}{s^\\tau} \\right\\} \n(1+r)^\\tau\\log(1+r) \\qquad \\text{for $r,s>0$.}\n$$\n\\end{lem}\n\n\\begin{rem}\nThe obvious bound $\\log(1 + \\frac{r}{s}) \\le \\frac{r}{s}$ will not be enough for our purposes. We need an upper bound of the form $g(s)h(r)$ where $h$ grows less than linearly in $r$.    \n\\end{rem}\n\n\\begin{proof}[Proof of Lemma~\\ref{elem-lemma}]\nLet first $s \\in (0,1)$. Since \n$$\n\\log\\left(1 + \\frac{r}{s}\\right)\\Big|_{r=0} = 0 = \\frac{1}{s} \\log(1+r)\\Big|_{r=0}\n$$\nand, for every $r>0$, \n$$\n\\frac{d}{dr} \\log\\left(1 + \\frac{r}{s}\\right) = \n\\frac{1}{s+r} \\le \\frac{1}{s+ sr} =  \\frac{d}{dr} \\frac{1}{s} \\log(1+r),\n$$\ninequality (\\ref{eq:first-elem-ineq}) follows. To see (\\ref{eq:second-elem-ineq}), we fix $s>1$, and we note that\n$$\n\\log\\left(1 + \\frac{r}{s}\\right)\\Big|_{r=0} = 0 = \\frac{(1+r)^\\tau}{s^\\tau} \\log(1+r)\\Big|_{r=0}.\n$$\nMoreover, for $0 < r \\le s-1$, we have  \n\\begin{align*}\n&\\frac{d}{dr} \\frac{(1+r)^\\tau}{s^\\tau} \\log(1+r)= \n\\frac{(1+r)^{\\tau-1}}{s^\\tau}(1+ \\tau \\log(1+r))\\\\\n&\\ge \\frac{(1+r)^{\\tau-1}}{s^\\tau} \n\\ge \\frac{1}{s}\\ge \\frac{1}{s+r}= \\frac{d}{dr} \\log\\left(1 + \\frac{r}{s}\\right),\n\\end{align*}\nso the inequality holds for $r \\le s-1$. If, on the other hand, $r \\ge s-1$, we have obviously\n$$\n\\log\\left(1 + \\frac{r}{s}\\right) \\le \\log(1 + r) \\le \\frac{(1+r)^\\tau}{s^\\tau} \\log(1+r).\n$$\n\\end{proof}\n\nWe may now complete the \n\n\\begin{proof}[Proof of Theorem~\\ref{2.1-new-lower-bound}]\nFor $\\xi \\in {\\mathbb R}^d$, we define $f_\\xi \\in L^2(\\Bbb R^d)$ by $f_\\xi(x)= \\frac{1}{\\sqrt{|\\Omega|}}1_{\\Omega} e^{-i x \\xi}$. Note that  $\\|f_{\\xi}\\|_{L^2(\\Bbb R^d)} = 1$ for any $\\xi \\in \\Bbb R^d$. We write\n\\begin{align*}\n\\sum_{k}(\\lambda - \\lambda_k)_+ &= \\sum_{k}(\\lambda - \\lambda_k)_+ \\|\\varphi_k\\|_{L^2(\\Omega)}^2 \n= \\frac{1}{(2\\pi)^{d}} \\sum_{k}(\\lambda - \\lambda_k)_+ \n\\|\\widehat{\\varphi_k}\\|_{L^2(\\Bbb R^{d})}^2 \n\\\\\n&=\\frac{|\\Omega|}{(2\\pi)^{d}}  \\sum_{k}(\\lambda - \\lambda_k)_+ \\int_{\\Bbb R^{d}} |\\langle f_\\xi,\\varphi_k \\rangle|^2\\,d\\xi\n\\\\\n&=\\frac{|\\Omega|}{(2\\pi)^{d}}  \\int_{\\Bbb R^{d}} \\sum_{k}(\\lambda - \\lambda_k)_+  |\\langle f_\\xi,\\varphi_k \\rangle|^2\\,d\\xi.\n\\end{align*}\nSince $\\sum \\limits_{k} |\\langle f_\\xi,\\varphi_k \\rangle|^2 = \\|f_{\\xi}\\|_{L^2(\\Bbb R^d)}^2 = 1$ for $\\xi \\in \\Bbb R^d$, Jensen's inequality gives \n\\begin{align}\n\\sum_{k}(\\lambda - \\lambda_k)_+ &\\ge \n\\frac{|\\Omega|}{(2\\pi)^{d}}  \\int_{\\Bbb R^{d}} \\Bigl( \\lambda \\sum_{k}|\\langle f_\\xi,\\varphi_k \\rangle|^2    - \\sum_k \\lambda_k  |\\langle f_\\xi,\\varphi_k \\rangle|^2\\Bigr)_+\\,d\\xi \\nonumber\\\\\n&=\\frac{|\\Omega|}{(2\\pi)^{d}}  \\int_{\\Bbb R^{d}} \\Bigl( \\lambda   - \\sum_k \\lambda_k  |\\langle f_\\xi,\\varphi_k \\rangle|^2\\Bigr)_+\\,d\\xi \\nonumber\\\\\n&=\\frac{|\\Omega|}{(2\\pi)^{d}}  \\int_{\\Bbb R^{d}} \\Bigl( \\lambda   - ( \\mathcal H  f_\\xi, f_\\xi ) \\Bigr)_+\\,d\\xi. \\label{jensen-new}\n\\end{align}\nHere, since \n$$\n\\sqrt{|\\Omega|}\\, \\widehat {f_\\xi}(\\eta-\\xi)= \n\\int_{\\Omega}e^{-i(\\eta-\\xi) x}e^{-ix \\xi}\\,dx = \n\\int_{\\Omega}e^{-i \\eta x} \\,dx = \\widehat{1_\\Omega}(\\eta)\n$$\nfor $\\eta,  \\xi \\in {\\mathbb R}^d$, we have \n\\begin{align}\n&|\\Omega|(2\\pi)^d \\left(\\mathcal H  f_{\\xi}, f_{\\xi}    \\right) = |\\Omega| \\int_{\\Bbb R^d}\n\\log |\\eta| |\\widehat {f_\\xi}(\\eta)|^2d \\eta = |\\Omega| \\int_{\\Bbb R^d}\n\\log |\\eta-\\xi| |\\widehat {f_\\xi}(\\eta-\\xi)|^2d \\eta \\nonumber\\\\\n&=\\int_{\\Bbb R^d} \\log|\\eta-\\xi| |\\widehat{1_\\Omega}(\\eta)|^2\\,d\\eta \\le \\int_{\\Bbb R^d}\n\\left[ \\log |\\xi| +\\log (1+|\\eta|\/|\\xi|)\\right] |\\widehat{1_\\Omega}(\\eta)|^2\\,d\\eta \\nonumber\\\\\n&\\le \\log |\\xi| \\int_{\\Bbb R^d} |\\widehat{1_\\Omega}(\\eta)|^2\\,d\\eta+ \\max \\left\\{ \\frac{1}{|\\xi|}, \\frac{1}{|\\xi|^\\tau}\\right\\}\n\\int_{\\Bbb R^d}(1+|\\eta|)^{\\tau} (\\log(1+|\\eta|)|\\widehat{1_\\Omega}(\\eta)|^2\\,d\\eta\n\\nonumber\\\\\n&= |\\Omega|(2\\pi)^d \\Bigl(\\log |\\xi| +  \\max \\left\\{ \\frac{1}{|\\xi|}, \\frac{1}{|\\xi|^\\tau}\\right\\} C_{\\Omega,\\tau} \\Bigr)\\qquad \\text{for $\\xi \\in {\\mathbb R}^d$,}\\label{jensen-new-compl} \n\\end{align}\nwhere $C_{\\Omega,\\tau}$ is defined in (\\ref{def-C-Omega}). Here we used Lemma~\\ref{elem-lemma}. Combining (\\ref{jensen-new}) and (\\ref{jensen-new-compl}), we get\n\\begin{equation}\n\\sum_{k}(\\lambda - \\lambda_k)_+ \\ge \n\\frac{|\\Omega|}{(2\\pi)^{d}}  \\int_{\\Bbb R^{d}} \\Bigl(\\lambda - \\log |\\xi| - \n\\max \\left\\{ \\frac{1}{|\\xi|}, \\frac{1}{|\\xi|^\\tau}\\right\\} C_{\\Omega,\\tau}\\Bigr)_+d\\xi. \\label{intermediate-new} \n\\end{equation}\nLet us redefine the spectral parameter $\\lambda =  \\log \\mu$ again.\nThen we find\n\\begin{align}\n&\\int_{\\Bbb R^{d}} \\Bigl(\\lambda - \\log |\\xi| - \n\\max \\left\\{ \\frac{1}{|\\xi|}, \\frac{1}{|\\xi|^\\tau} \\right\\} C_{\\Omega,\\tau} \\Bigr)_+d\\xi \\nonumber\n\\\\\n&= \\left|\\Bbb S^{d-1} \\right| \\, \\int_0^\\infty \\left(\\log \\frac{\\mu}{r} - \n\\max \\left\\{ \\frac{1}{r}, \\frac{1}{r^\\tau}\\right\\} C_{\\Omega,\\tau}\n\\right)_+ \\, r^{d-1}dr  \\nonumber \\\\\n&= \n\\mu^d\\, \\left|\\Bbb S^{d-1} \\right| \\, \\int_0^\\infty \\left(-\\log r  - \n\\max \\left\\{ \\frac{1}{\\mu^{1-\\tau} r}, \\frac{1}{r^\\tau}\\right\\} \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}\n\\right)_+ \\, r^{d-1}dr\n \\nonumber \\\\\n&\\ge \n\\mu^d\\, \\left|\\Bbb S^{d-1} \\right| \\, \\int_{\\frac{1}{\\mu}}^\\infty \\left(-\\log r  - \n\\frac{1}{r^\\tau}\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}\n\\right)_+ \\, r^{d-1}dr.  \\label{below2}\n\\end{align}\nFor the last inequality, we used the fact that $\\frac{1}{\\mu^{1-\\tau} r} \\le \n\\frac{1}{r^\\tau}$ for $r \\ge \\frac{1}{\\mu}$.\n\nNext we note that the function $r \\mapsto f_\\mu(r) = \n-\\log r  - \n\\frac{1}{r^\\tau}\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}$ satisfies \n\\begin{equation}\n  \\label{eq:boundary-conditions}\nf_\\mu(r)<0 \\quad \\text{for $r \\ge 1$}\\qquad \\text{and}\\qquad \n\\lim_{r \\to 0^+}f_\\mu(r)= -\\infty.\n\\end{equation}\nMoreover, this function has two zeros $r_1(\\mu), r_2(\\mu)$ with $0<r_1(\\mu)< \\frac{1}{\\mu} < r_2(\\mu)<1$ and \n$$\nf_\\mu(r)\\ge 0 \\qquad \\text{if and only if}\\quad r_1(\\mu) \\le r \\le r_2(\\mu).\n$$\nTo see this, we write \n$$\nf_\\mu(r)= \\frac{1}{r^\\tau}g(r^\\tau) \\qquad \\text{with}\\qquad g(s)=-\\frac{s}{\\tau} \\log s - \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}\n$$\nand note that $g$ is strictly concave since $s \\mapsto g'(s)= -\\frac{1}{\\tau} - \\log s$ is strictly decreasing. Consequently, $g$ has at most two zeros, and the same is true for $f$. Combining this with (\\ref{eq:boundary-conditions}) and the fact that \n$$\nf(1\/\\mu) = \\log \\mu  - C_{\\Omega,\\tau} > 0 \n$$\nsince $\\lambda \\ge 2 C_{\\Omega,\\tau} >C_{\\Omega,\\tau}$ by assumption, the claim above follows. From (\\ref{below2}), we thus obtain the lower bound\n\\begin{align}\n  \\label{eq:r-2-mu-est}\n\\int_{\\Bbb R^{d}} &\\Bigl(\\lambda - \\log |\\xi| - \n\\max \\bigl\\{ \\frac{1}{|\\xi|}, \\frac{1}{|\\xi|^\\tau} \\bigl\\} C_{\\Omega,\\tau} \\Bigr)_+d\\xi \\\\\n&\\ge \\mu^d\\, \\left|\\Bbb S^{d-1} \\right| \\, \\int_{\\frac{1}{\\mu}}^{r_2(\\mu)} \\left(-\\log r  - \n\\frac{1}{r^\\tau}\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}\n\\right)_+ \\, r^{d-1}dr. \\nonumber  \n\\end{align}\nNext, we claim that\n\\begin{equation}\n\\label{r-2-mu-lower-bound}  \nr_2(\\mu) \\ge r_3(\\mu):= e^{\\frac{1}{2\\tau}\\bigl(\\sqrt{1- \\frac{4\\tau C_{\\Omega,\\tau}}{\\mu^\\tau}}\\;-1\\bigr)}.\n\\end{equation}\nHere we note that $\\frac{4\\tau C_{\\Omega,\\tau}}{\\mu^\\tau}=\\frac{4\\tau C_{\\Omega,\\tau}}{e^{\\tau \\lambda}} <1$ since $\\lambda \\ge 2 C_{\\Omega,\\tau}$ by assumption. \nTo see (\\ref{r-2-mu-lower-bound}), we write \n$$\nr_3(\\mu)= e^{- c_\\mu \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}}\\qquad \\text{with}\\qquad  \nc_\\mu = \\frac{\\mu^\\tau}{2 \\tau  C_{\\Omega,\\tau}}\\Bigl(1 - \\sqrt{1- \\frac{4\\tau C_{\\Omega,\\tau}}{\\mu^\\tau}}\\Bigr),\n$$\nnoting that \n$$\n\\frac{\\tau C_{\\Omega,\\tau}}{\\mu^\\tau} c_\\mu^2 -c_\\mu +1= 0\n$$\nand therefore \n\\begin{align*}\n&f(r_3(\\mu)) = f(e^{-\\frac{c_\\mu  C_{\\Omega,\\tau}}{\\mu^\\tau}})=\\frac{c_\\mu C_{\\Omega,\\tau}}{\\mu^\\tau} - \n\\frac{1}{e^{-\\tau \\frac{c_\\mu C_{\\Omega,\\tau}}{\\mu^\\tau}}}\n\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}\\\\\n&=\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau e^{-\\tau \\frac{c_\\mu C_{\\Omega,\\tau}}{\\mu^\\tau}}} \\Bigl( c_\\mu e^{-\\tau \\frac{c_\\mu C_{\\Omega,\\tau}}{\\mu^\\tau}} - 1\\Bigr) h\\ge \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau e^{-\\tau \\frac{c_\\mu C_{\\Omega,\\tau}}{\\mu^\\tau}}}\\Bigl( c_\\mu \\bigl(1 - \\tau \\frac{c_\\mu C_{\\Omega,\\tau}}{\\mu^\\tau}\\bigr)-1\\Bigr) = 0.\n\\end{align*}\nThis proves (\\ref{r-2-mu-lower-bound}). As a consequence of the inequality $\\sqrt{1-a} \\ge 1-\\frac{a}{2} -\\frac{a^2}{2}$ for $0 \\le a \\le 1$, we also have \n$$\nr_3(\\mu) \\ge e^{- \\bigl(\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}+\\frac{4\\tau C_{\\Omega,\\tau}^2}{\\mu^{2\\tau}}\\bigr)} = : r_4(\\mu).\n$$\nConsequently, \n\\begin{align*}\n\\int_{\\Bbb R^{d}} &\\Bigl(\\lambda - \\log |\\xi| - \n\\max \\bigl\\{ \\frac{1}{|\\xi|}, \\frac{1}{|\\xi|^\\tau} \\bigl\\} C_{\\Omega,\\tau} \\Bigr)_+d\\xi \\\\\n&\\ge \\mu^d\\, \\left|\\Bbb S^{d-1} \\right| \\, \\int_{\\frac{1}{\\mu}}^{r_4(\\mu)} \\left(-\\log r  - \n\\frac{1}{r^\\tau}\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}\n\\right)_+ \\, r^{d-1}dr\\\\  \n&= \n  \\mu^d\\, \\left|\\Bbb S^{d-1} \\right| \\, \\Bigl[-\\frac{r^d}{d} \\log r + \\frac{1}{d^2}r^d - \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau(d-\\tau)}r^{d-\\tau}  \\Bigr]_{\\frac{1}{\\mu}}^{r_4(\\mu)}\\\\\n&= \n  \\mu^d\\, \\left|\\Bbb S^{d-1} \\right| \\,\\Bigl( \\Bigl[-\\frac{r_4(\\mu)^d}{d} \\log r_4(\\mu) + \\frac{1}{d^2}r_4(\\mu)^d - \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau(d-\\tau)}r_4(\\mu)^{d-\\tau}  \\Bigr]\\\\\n&- \\Bigl[\\frac{\\mu^{-d}}{d} \\log \\mu + \\frac{1}{d^2}\\mu^{-d} - \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau(d-\\tau)}\\mu^{\\tau-d}  \\Bigr]\\Bigr),\n\\end{align*}\nwhich implies that \n\\begin{align*}\n\\int_{\\Bbb R^{d}} &\\Bigl(\\lambda - \\log |\\xi| - \n\\max \\bigl\\{ \\frac{1}{|\\xi|}, \\frac{1}{|\\xi|^\\tau} \\bigl\\} C_{\\Omega,\\tau} \\Bigr)_+d\\xi \\\\\n&\\ge\n  \\mu^d\\, \\left|\\Bbb S^{d-1} \\right| \\,\\Bigl(\\frac{1}{d^2}r_4(\\mu)^d - \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau(d-\\tau)}r_4(\\mu)^{d-\\tau}- \\frac{\\mu^{-d}}{d} \\log \\mu - \\frac{1}{d^2}\\mu^{-d}  \\Bigr)\\\\\n&=\n  \\mu^d\\, \\left|\\Bbb S^{d-1} \\right| \\,\\Bigl(\\frac{1}{d^2}e^{- d\\bigl(\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}+\\frac{4\\tau C_{\\Omega,\\tau}^2}{\\mu^{2\\tau}}\\bigr)}\n - \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau(d-\\tau)}e^{- (d-\\tau)\\bigl(\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}+\\frac{4\\tau C_{\\Omega,\\tau}^2}{\\mu^{2\\tau}}\\bigr)}\\\\\n&- \\frac{\\mu^{-d}}{d} \\log \\mu - \\frac{1}{d^2}\\mu^{-d}  \\Bigr).\n\\end{align*}\nSince \n$$\ne^{- d\\bigl(\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}+\\frac{4\\tau C_{\\Omega,\\tau}^2}{\\mu^{2\\tau}}\\bigr)} \\ge 1-  d\\bigl(\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}+\\frac{4\\tau C_{\\Omega,\\tau}^2}{\\mu^{2\\tau}}\\bigr)\n$$\nand \n$$\ne^{- (d-\\tau)\\bigl(\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}+\\frac{4\\tau C_{\\Omega,\\tau}^2}{\\mu^{2\\tau}}\\bigr)} \\le 1,\n$$\nwe conclude that \n\\begin{align*}\n\\int_{\\Bbb R^{d}} &\\Bigl(\\lambda - \\log |\\xi| - \n\\max \\bigl\\{ \\frac{1}{|\\xi|}, \\frac{1}{|\\xi|^\\tau} \\bigl\\} C_{\\Omega,\\tau} \\Bigr)_+d\\xi \\\\\n&\\ge \n  \\mu^d\\, \\frac{\\left|\\Bbb S^{d-1} \\right|}{d^2} \\,\\Bigl(1-  d\\bigl(\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}+\\frac{4\\tau C_{\\Omega,\\tau}^2}{\\mu^{2\\tau}}\\bigr)\n - \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau(d-\\tau)}- \\mu^{-d}(d \\log \\mu + 1) \\Bigr)\\\\\n&= \n\\frac{\\left|B_d \\right|}{d} \\,\\Bigl(\\mu^d \n- C_{\\Omega,\\tau}(d-\\frac{1}{d-\\tau})\\mu^{d-\\tau} - 4d\\tau C_{\\Omega,\\tau}^2 \\mu^{d-2\\tau} - (d \\log \\mu + 1) \\Bigr)\\\\\n&= \n\\frac{\\left|B_d \\right|}{d} \\,\\Bigl(e^{d \\lambda} \n- \\frac{d(d-\\tau)-1}{d-\\tau} C_{\\Omega,\\tau}e^{(d-\\tau)\\lambda} - 4d\\tau C_{\\Omega,\\tau}^2 \ne^{(d-2\\tau)\\lambda} - (d \\lambda + 1) \\Bigr).\n\\end{align*}\nCombining the last estimate with (\\ref{intermediate-new}), we get the asserted lower bound.\n\\end{proof}\n\n\n\\section{Appendix: Note on a bound for Bessel functions}\n\\label{sec:appendix:-note-bound}\n\nThe following elementary bound might be known but seems hard to find in this form. \n\n\\begin{lem}\nFor $\\nu \\ge \\sqrt{3}-2$ and $0 \\le x \\le 2 \\sqrt{2(\\nu+2)}$ we have \n$$\n|J_\\nu(x)| \\le \\frac{x^\\nu}{2^\\nu \\Gamma(\\nu+1)}.\n$$\n\\end{lem} \n\n\\begin{proof}\nWe use the representation \n$$\nJ_\\nu(x)= \\Bigl(\\frac{x}{2}\\Bigr)^{\\nu} \\sum_{m=0}^\\infty \\frac{(-1)^m}{m! \\Gamma(m+\\nu + 1)} \\Bigl(\\frac{x}{2}\\Bigr)^{2m}.\n$$\nFor $0 \\le x \\le 2 \\sqrt{2(\\nu+2)}$ and $m \\ge 1$, we have \n$$\n\\Bigl(\\frac{x}{2}\\Bigr)^{2} \\le (m+1)(m+\\nu+1) = \\frac{(m+1) \\Gamma(m+\\nu + 2)}{\\Gamma(m+\\nu + 1)}\n$$\nand therefore \n\\begin{equation}\n  \\label{eq:bessel-proof-1}\n\\frac{\\Gamma(\\nu+1)}{(m+1)! \\Gamma(m+\\nu + 2)} \\Bigl(\\frac{x}{2}\\Bigr)^{2(m+1)} \n\\le \\frac{\\Gamma(\\nu+1)}{m! \\Gamma(m+\\nu + 1)} \\Bigl(\\frac{x}{2}\\Bigr)^{2m}.\n\\end{equation}\nConsequently, \n\\begin{align*}\nJ_\\nu(x) &= \\frac{x^\\nu}{2^\\nu \\Gamma(\\nu+1)}\\Bigl[1 + \\sum_{m=1}^\\infty \\frac{(-1)^m \\Gamma(\\nu+1)}{m! \\Gamma(m+\\nu + 1)} \\Bigl(\\frac{x}{2}\\Bigr)^{2m}\\Bigr]\\le \\frac{x^\\nu}{2^\\nu \\Gamma(\\nu+1)}.\n\\end{align*}\nFrom (\\ref{eq:bessel-proof-1}) we also deduce that \n\\begin{align*}\nJ_\\nu(x) &\\ge \\frac{x^\\nu}{2^\\nu \\Gamma(\\nu+1)} \\Bigl[1 - \\frac{\\Gamma(\\nu+1)}{\\Gamma(\\nu + 2)} \\Bigl(\\frac{x}{2}\\Bigr)^{2}+ \\frac{\\Gamma(\\nu+1)}{2\\Gamma(\\nu + 3)} \\Bigl(\\frac{x}{2}\\Bigr)^{4}- \\frac{\\Gamma(\\nu+1)}{6\\Gamma(\\nu + 4)} \\Bigl(\\frac{x}{2}\\Bigr)^{6}\\Bigr]\\\\\n&= \\frac{x^\\nu}{2^\\nu \\Gamma(\\nu+1)} \\Bigl[1-\\frac{1}{\\nu + 1}f\\bigl(\\bigl(\\frac{x}{2}\\bigr)^{2} \\bigr)\\Bigr]\n\\end{align*}\nwith $f: {\\mathbb R} \\to {\\mathbb R}$ given by $f(t)= t - \\frac{t^2}{2(\\nu+2)}+ \\frac{t^3}{6(\\nu+2)(\\nu+3)}$. Since \n$$\nf'(t)= 1- \\frac{t}{\\nu+2} + \\frac{t^2}{2(\\nu+2)(\\nu+3)}, \\qquad \\text{and}\\qquad f''(t)= \\frac{1}{\\nu+2}\\bigl(\\frac{t}{\\nu+3}- 1\\bigr)\n$$\nwe have \n$$\nf'(t) \\ge f'(\\nu+3) = 1-  \\frac{\\nu+3 }{\\nu+2} + \\frac{\\nu+3}{2(\\nu+2)}\n= 1 - \\frac{1}{2}\\frac{\\nu+3 }{\\nu+2} \\ge 0 \n\\quad \\text{for $t \\in {\\mathbb R}$ if $\\nu \\ge -1$}\n$$\nand therefore \n$$\nf(t) \\le f(2(\\nu +2))= 2(\\nu+2) - \\frac{[2(\\nu+2)]^2}{2(\\nu+2)}+ \\frac{[2(\\nu+2)]^3}{6(\\nu+2)(\\nu+3)}= \\frac{4(\\nu+2)^2}{3(\\nu+3)} \n$$\nfor $t \\le 2(\\nu+2)$ if $\\nu \\ge -1$. Since $\\frac{4(\\nu+2)^2}{3(\\nu+3)} \\le \\frac{2}{\\nu+1}$ for $\\nu \\ge \\sqrt{3}-2$, we conclude that\n$$\nJ_\\nu(x) \\ge \\frac{x^\\nu}{2^\\nu \\Gamma(\\nu+1)} \\Bigl[1-\\frac{1}{\\nu + 1}f\\bigl(\\bigl(\\frac{x}{2}\\bigr)^{2} \\bigr)\\Bigr]\\ge - \\frac{x^\\nu}{2^\\nu \\Gamma(\\nu+1)}.\n$$\nfor $\\nu \\ge \\sqrt{3}-2$ and $0 \\le x \\le 2 \\sqrt{2(\\nu+2)}$. The claim thus follows. \n\\end{proof}\n\n\\noindent\n{\\it Acknowledgements}.\nAL was supported by the RSF grant  19-71-30002.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}