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In the Color Glass Condensate, the inclusive spectrum of produced quarks in a heavy ion collision is obtained as the Fourier transform of a $2$-fermion correlation function. Due to its non-locality, the two points of this function must be linked by a Wilson line in order to have a gauge invariant result, but when the quark spectrum is evaluated in a background that has a non-zero chromo-magnetic field, this procedure suffers from an ambiguity related to the choice of the contour defining the Wilson line. In this paper, we use an analytically tractable toy model of the background field in order to study this contour dependence. We show that for a straight contour, unphysical contributions to the spectrum in $p_\perp^{-2}$ and $p_\perp^{-3}$ cancel, leading to a spectrum with a tail in $p_\perp^{-4}$. If the contour defining the Wilson line deviates from a straight line, the path dependence is at most of order $p_\perp^{-5}$ if its curvature is bounded, and of order $p_\perp^{-4}$ otherwise. When the contour is forced to go through a fixed point, the path dependence is even larger, of order $p_\perp^{-2}$., Comment: 28 pages, 9 figures

We use an expansion in angular mode functions in order to solve the Boltzmann equation for a gluon plasma undergoing longitudinal expansion. By comparing with the exact solution obtained numerically by other means we show that the expansion in mode functions converges rapidly for all cases of practical interest, and represents a substantial gain in numerical effort as compared to more standard methods. We contrast the cases of a non expanding plasma and of longitudinally expanding plasmas, and follow in both cases the evolutions towards thermalization. In the latter case, we observe that, although the spherical mode function appears to be well reproduced after some time by a local equilibrium distribution function depending on slowly varying temperature and chemical potential, thereby suggesting thermalization of the system, the longitudinal and transverse pressures take more time to equilibrate. This is because the expansion hinders the relaxation of the first angular mode function. This feature was also observed in a simpler context where the Boltzmann equation is solved in terms of special moments within the relaxation time approximation, and attributed there to the particular coupling between the first two moments of the distribution function. The present analysis confirms this observation in a more realistic setting., 23 pages, 16 figures, version published in Nuclear Physics A

We present recent results on in-medium spectral functions of vector and axial-vector mesons, the electromagnetic (EM) spectral function and dilepton rates using the Functional Renormalization Group (FRG) approach. Our method is based on an analytic continuation procedure that allows us to calculate real-time quantities like spectral functions at finite temperature and chemical potential. As an effective low-energy model for Quantum Chromodynamics (QCD) we use an extended linear-sigma model including quarks where (axial-)vector mesons as well as the photon are introduced as gauge bosons. In particular, it is shown how the $\rho$ and the $a_1$ spectral function become degenerate at high temperatures or chemical potentials due to the restoration of chiral symmetry. Preliminary results for the EM spectral function and the dilepton production rate are discussed with a focus on the possibility to identify signatures of the chiral crossover and the critical endpoint (CEP) in the QCD phase diagram., Comment: Quark Matter 2018 proceedings

We present results of numerical lattice simulations of anomalous and topological effects in Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD) in far from equilibrium situations. Based on the classical-statistical approximation to the Schwinger-Keldysh path integral formalism, we perform extensive numerical studies including dynamical Wilson and overlap fermions. Using advanced algorithmic techniques, we study the real-time dynamics of the axial anomaly relevant for strong field laser physics beyond the Schwinger limit and we observe novel dynamical refringence effects caused by the anomaly. Furthermore, motivated by recent interest in the physics of the Chiral Magnetic Effect in ultra-relativistic heavy ion collisions, we study the real time dynamics of fermions during and after a sphaleron transition and anomalous transport in the presence of strong magnetic fields.

In the idealized high-energy limit of heavy-ion collisions, the system right after a collision is described as an over-occupied gluonic plasma expanding in the longitudinal direction. The understanding of the quark dynamics in such a pre-equilibrium state is of prime importance since it has a direct connection to electromagnetic probes. We report on a numerical study of the Boltzmann equation including two-to-two scatterings of gluons and quarks in the expanding geometry. In the gluon sector, we confirm the self-similar scaling behavior of the one-particle momentum distribution that is consistent with the bottom-up thermalization scenario. Also in the quark sector, we find that the distribution function show scaling behavior like those of gluons.

We present a parametric estimate of photon production at early times in heavy-ion collisions based on a consistent weak coupling thermalization scenario. We quantify the contribution of the off-equilibrium Glasma phase relative to that of a thermalized Quark-Gluon Plasma. Taking into account the constraints from charged hadron multiplicity data, the Glasma contribution is found to be significant especially for large values of the saturation scale., 4 pages, 4 figures, proceedings for Quark Matter 2017

We report on a numerical study of the Boltzmann equation including $2\leftrightarrow 2$ scatterings of gluons and quarks in an overoccupied Glasma undergoing longitudinal expansion. We find that when a cascade of gluon number to the infrared occurs, corresponding to an infrared enhancement analogous to a transient Bose-Einstein condensate, gluon distributions qualitatively reproduce the results of classical-statistical simulations for the expanding Glasma. These include key features of the distributions that are not anticipated in the "bottom-up" thermalization scenario. We also find that quark distributions, like those of gluons, satisfy self-similar scaling distributions in the overoccupied Glasma. We discuss the implications of these results for a deeper understanding of the self-similarity and universality of parton distributions in the Glasma., 34 pages, 21 figures; version accepted in PRD

Recent classical-statistical numerical simulations have established the "bottom-up" thermalization scenario of Baier et al. as the correct weak coupling effective theory for thermalization in ultrarelativistic heavy-ion collisions. We perform a parametric study of photon production in the various stages of this bottom-up framework to ascertain the relative contribution of the off-equilibrium "Glasma" relative to that of a thermalized Quark-Gluon Plasma. Taking into account the constraints imposed by the measured charged hadron multiplicities at RHIC and the LHC, we find that Glasma contributions are important especially for large values of the saturation scale at both energies. These non-equilibrium effects should therefore be taken into account in studies where weak coupling methods are employed to compute photon yields., 33 pages, 9 figures; version accepted in PRC, title changed

Pair creation in a uniform classical electromagnetic field (Schwinger mechanism) is studied focusing on the time evolution of the distribution of created particles. The time evolution of the distribution in time-dependent fields is also presented as well as effects of back reaction. Motivated by the Glasma flux tube, which may be formed at the initial stage of heavy-ion collisions, we investigate effects of a magnetic field parallel to an electric field, and find that the magnetic field makes the evolution of a fermion system faster., Comment: 57pages, 44figures. v2: typos corrected, references added, to appear in Ann.Phys

We investigate axial charge production in two-color QCD out of equilibrium. We compute the real-time evolution starting with spatially homogeneous strong gauge fields, while the fermions are in vacuum. The idealized class of initial conditions is motivated by glasma flux tubes in the context of heavy-ion collisions. We focus on axial charge production at early times, where important aspects of the anomalous dynamics can be derived analytically. This is compared to real-time lattice simulations. Quark production at early times leading to anomalous charge generation is investigated using Wilson fermions. Our results indicate that coherent gauge fields can transiently produce significant amounts of axial charge density, while part of the induced charges persist to be present even well beyond characteristic decoherence times. The comparisons to analytic results provide stringent tests of real-time representations of the axial anomaly on the lattice., 12 pages, 10 figures; To appear in Phys. Rev. D

In this article, we review recent theoretical works on the Schwingermechanism of particle production in external electrical fields. Although the non-perturbative Schwinger mechanism is at the center of this discussion, many of the approaches that we discuss can cope with general time and space dependent fields, and therefore also capture the perturbative contributions to particle production., Review for Progress in Particle and Nuclear Physics. 76 pages, 36 figures

The quark production in classical color fields is investigated with a focus on the induction of an electromagnetic current by produced quarks. We show that the color SU(2) and the SU(3) theories lead significantly different results for the electromagnetic current. In uniform SU(2) color fields, the net electromagnetic current is not generated, while in SU(3) color fields the net current is induced depending on the color direction of background fields. Also the numerical study of the quark production in inhomogeneous color fields is done. Motivated by gauge field configurations provided by the color glass condensate framework, we introduce an ensemble of randomly distributed color electric fluxtubes. The spectrum of photons emitted from the quarks by a classical process is shown., 19 pages, 19 figures; To appear in Phys. Rev. D

We revisit the problem of quark production in high energy heavy ion collisions, at leading order in $\alpha_s$ in the color glass condensate framework. In this first paper, we setup the formalism and express the quark spectrum in terms of a set of mode functions of the Dirac equation. We determine analytically their initial value in the Fock-Schwinger gauge on a proper time surface $Q_s\tau_0\ll 1$, in a basis which makes manifest the boost invariance properties of this problem. We also describe a statistical algorithm to perform the sampling of the mode functions., Comment: 40 pages, 3 figures, alternate derivation in the new appendix B

We study the Boltzmann equation with elastic point-like scalar interactions in two different versions of the the classical approximation. Although solving numerically the Boltzmann equation with the unapproximated collision term poses no problem, this allows one to study the effect of the ultraviolet cutoff in these approximations. This cutoff dependence in the classical approximations of the Boltzmann equation is closely related to the non-renormalizability of the classical statistical approximation of the underlying quantum field theory. The kinetic theory setup that we consider here allows one to study in a much simpler way the dependence on the ultraviolet cutoff, since one has also access to the non-approximated result for comparison., 37 pages, 21 figures

In this paper, we show how classical statistical field theory techniques can be used to efficiently perform the numerical evaluation of the non-perturbative Schwinger mechanism of particle production by quantum tunneling. In some approximation, we also consider the back-reaction of the produced particles on the external field, as well as the self-interactions of the produced particles., 28 pages, 28 pdf figures; title changed, to appear in Phys.Rev.D

The time evolution of a system where a uniform and classical SU(3) color electric field and quantum fields of quarks are dynamically coupled with each other is studied focusing on non-perturbative pair creation and its back reaction. We characterize the color direction of the electric field in a gauge invariant way, and investigate its dependence. Momentum distributions of created quarks show plasma oscillation as well as quantum effects such as the Pauli blocking and interference. Pressure of the system is also calculated, and we show that pair creation moderates degree of anisotropy of pressure. Furthermore, enhancement of pair creation and induction of chiral charge under a color magnetic field which is parallel to the electric field are discussed., 20 pages, 40 figures; v2: minor changes, to appear in Ann. Phys

Pair creation of scalar particles in a boost-invariant electric field which is confined in the forward light cone is studied. We present the proper-time evolution of momentum distributions of created particles, which preserve the boost invariance of the background field. The two-particle correlation of the created particles is also calculated. We find that long-range rapidity correlations may arise from the Schwinger mechanism in the boost-invariant electric field., Comment: 21 pages, 10 figures; v2: minor changes, to appear in Phys. Rev. D

We discuss gluon production by the Schwinger mechanism in collinear color-electric and magnetic fields which may be realized in pre-equilibrium stages of ultra-relativistic heavy-ion collisions. Fluctuations of non-Abelian gauge fields around a purely color-magnetic field contain exponentially growing unstable modes in a longitudinally soft momentum region, which is known as the Nielsen--Olesen instability. With a color-electric field imposed parallelly to the color-magnetic field, we can formulate this instability as the Schwinger mechanism. This is because soft unstable modes are accelerated by the electric fields to escape from the instability condition. Effects of instability remain in the transverse spectrum of particle modes, leading to an anomalously intense Schwinger particle production., Comment: 9 pages, 1 figure, v2: Sec.4 is added. To appear in Phys.Lett.B