In this work, a covariant formulation of the gluon self-energy in presence of ellipsoidal anisotropy is considered. It is shown that the general structure of the gluon self-energy can be written in terms of six linearly independent projection tensors. Similar to the spheroidal anisotropy, mass scales can be introduced for each of the collective modes considering the static limits. With a simplified ellipsoidal generalization of the Romatschke-Strickland form, the angular dependencies of the mass scales are studied. It is observed that, compared to the spheroidal case, additional unstable mode may appear in presence of ellipsoidal anisotropy depending upon the choice of the parameters.
We calculate the quark self-energy in a quark-gluon plasma that possesses an ellipsoidal momentum-space anisotropy in the local rest frame. By introducing additional transverse momentum anisotropy parameters into the parton distribution functions, we generalize previous results which were obtained for the case of a spheroidal anisotropy. Our results demonstrate that the presence of anisotropies in the transverse directions affects the real and imaginary parts of quark self-energy and, consequently, the self-energy depends on both the polar and azimuthal angles in the local rest frame of the ma…
It has been shown recently that there exist analytical solutions of the Yang-Mills equations for non-Abelian Weizsäcker-Williams fields which describe the distribution of gluons in large nuclei at small x. These solutions however depend on the color charge distribution at large rapidities. We here construct a model of the color charge distribution of partons in the fragmentation region and use it to compute the boost covariant momentum distributions of wee gluons. The phenomenological applications of our results are discussed.
The quark propagator in presence of an arbitrary gluon field is calculated gauge and Lorentz covariantly order by order in terms of powers of gluon field and its derivatives. The result is independent of path connecting ends of propagator and leading order result coincides with the exact propagator in the trivial case of vanishing gluon field.
We introduce an efficient general method for calculating the self-energies, collective modes, and dispersion relations of quarks and gluons in a momentum-anisotropic high-temperature quark-gluon plasma. The method introduced is applicable to the most general classes of deformed anisotropic momentum distributions and the resulting self-energies are expressed in terms of a series of hypergeometric basis functions which are valid in the entire complex phase-velocity plane. Comparing to direct numerical integration of the self-energies, the proposed method is orders of magnitude faster and provide…
We compute the gluon and ghost propagators of Yang-Mills theory in linear covariant gauges from the coupled system of Dyson-Schwinger equations. For small values of the gauge fixing parameter $ξ\sim 0.1$, the deviations to the Landau gauge already become clearly visible. For the ghost dressing function, this is reflected in a logarithmic infrared suppression. Also, the gluon dressing function changes - but only quantitatively - and the gluon propagator remains finite at zero momentum. From the results, a running coupling is extracted.
We analyze the mixing of the scalar glueball with the scalar-isoscalar quarkonia states above 1 GeV in a non-local covariant constituent approach. Similarities and differences to the point-particle Klein-Gordon limit and to the quantum mechanical case are elaborated. Model predictions for the two-photon decay rates in the covariant mixing scheme are indicated.
We compute the vector-meson, nucleon and delta/omega-baryon masses and their evolution with the current-quark mass using a covariant generalized Bethe-Salpeter equation approach. The interaction kernel is truncated to a dressed gluon exchange. We study the model dependence of our results with the quark-gluon dressing to assess the validity of the truncation.
We discuss QCD condensate contributions to the gluon propagator both in the fixed-point gauge and in covariant gauges for the external QCD vacuum gluon fields with the conclusion that a covariant gauge is essential to obtain a gauge invariant QCD vacuum energy density difference and to retain the unitarity of the quark scattering amplitude. The gauge-invariant QCD condensate contributions to the effective one-gluon exchange potential are evaluated by using the effective gluon propagator which produces a gauge-independent quark scattering amplitude.
We report on recent results of a calculation of the nucleon and delta masses in a covariant bound-state approach, where to the simple rainbow-ladder gluon-exchange interaction kernel we add a pion-exchange contribution to account for pion cloud effects. We observe good agreement with lattice data at large pion masses. At the physical point our masses are too large by about five percent, signaling the need for more structure in the gluon part of the interaction.
We construct the multiplicatively renormalizable effective potential for the mass dimension two local composite operator A^2 in linear covariant gauges. We show that the formation of <A^2> is energetically favoured and that the gluons acquire a dynamical mass due to this gluon condensate. We also discuss the gauge parameter independence of the resultant vacuum energy.
The gluon polarization functional is evaluated for a generic linear covariant gauge and for any space dimension in pure Yang-Mills SU(N) theory up to second order in a generalized perturbation theory, where the zeroth order action is freely chosen and can be determined by some variational method. Some numerical data are given for the gluon propagator in Landau gauge and compared with Feynmann gauge. A comparison is given for several variational methods that can be set up by the knowledge of the second-order polarization.
In this paper, a study of two-photon and two-gluon decays in the context of p-wave heavy quarkonia is presented. Within the covariant light-front framework, the annihilation rates of scalar and tensor quarkonium states are derived. In the absence of free parameters in this case, the results for the charmonium decay widths are consistent with the experimental data. However, in comparison to other theoretical calculations, there are large discrepancies in our results regarding bottomonia.
Explicit analytical expressions are derived for the gluon propagator in a generic linear covariant $R_ξ$ gauge, by a screened massive expansion for the exact Faddeev-Popov Lagrangian of pure Yang-Mills theory. At one-loop, if the gauge invariance of the pole structure is enforced, the gluon dressing function is entirely and uniquely determined, without any free parameter or external input. The gluon propagator is found finite in the IR for any $ξ$, with a slight decrease of its limit value when going from the Landau gauge ($ξ=0$) towards the Feynman gauge ($ξ=1$). An excellent agreement is fou…
We study the non-perturbative production of gluon pairs from a constant SU(3) chromo-electric background field via the Schwinger mechanism. We fix the covariant background gauge with an arbitrary gauge parameter α. We determine the transverse momentum distribution of the gluons, as well as the total probability of creating pairs per unit space time volume. We find that the result is independent of the covariant gauge parameter αused to define arbitrary covariant background gauges. We find that our non-perturbative result is both gauge invariant and gauge parameter αindependent.
Gluon mass generation is investigated for 4-dimensional $SU(N)$ Yang-Mills in conventional covariant and in background field gauges within an effective description that, through a parameterization, can be regarded as a massive gluon model, or as a Nambu-Jona-Lasinio-like expansion around a massive leading order while preserving the Yang-Mills Lagrangian. We employ a renormalization scheme that introduces the ratio of the gluon mass parameter $m$ to the saturation value of the gluon propagator. This, along with the mass $m(μ)$ and the strong coupling $α_s(μ)$, provided the fit parameters for co…
A calculation of the masses and electromagnetic properties of the Delta and Omega baryon together with their evolution with the current quark mass is presented. Hereby a generalized Bethe-Salpeter approach with the interaction truncated to a dressed one-gluon exchange is employed. The model dependence is explored by investigating two forms for the dressed gluon exchange.
We calculate gluon and ghost propagators in Yang-Mills theory in linear covariant gauges. To that end, we utilize Nielsen identities with Landau gauge propagators and vertices as the starting point. We present and discuss numerical results for the gluon and ghost propagators for values of the gauge parameter $0<ξ\le 5$. Extrapolating the propagators to $ξ\to \infty $ we find the expected qualitative behavior. We provide arguments that our results are quantitatively reliable at least for values $ξ\lesssim 1/2$ of the gauge fixing parameter. It is shown that the correlation functions, and in par…
Gribov quantization is a method to improve the infrared dynamics of Yang-Mills theory. We study the thermodynamics and transport properties of a plasma consisting of gluons whose propagator is improved by the Gribov prescription. We first construct thermodynamics of Gribov plasma using the gauge invariant Gribov dispersion relation for interacting gluons. When the Gribov parameter in the dispersion relation is temperature dependent, one expects a mean field correction to the Boltzmann equation. We formulate covariant kinetic theory for the Gribov plasma and determine the mean-field contributio…
We decompose the quark propagator in the presence of an arbitrary gluon field with respect to a set of Dirac matrices. The four-dimensional integrals which arise in first order perturbation theory are rewritten as line-integrals along certain field lines, together with a weighted integration over the various field lines. It is then easy to transform the propagator into a form involving path ordered exponentials. The resulting expression is non-perturbative and has the correct behavior under Lorentz transformations, gauge transformations and charge conjugation. Furthermore it coincides with the…
We construct a quark target model (QTM) to incorporate intrinsic glue into effective low-energy models of QCD, which often contain only quark degrees of freedom. This method guarantees the gauge invariance of observables order-by-order in the strong coupling. The quark and gluon PDFs for the dressed quarks are obtained in the QTM at leading order. We demonstrate gauge invariance of the results by comparing both covariant and light cone gauges, with the former including an explicit Wilson line contribution. A key finding is that in covariant gauges the Wilson line can carry a significant amount…
The Gribov copies and their consequences on the infrared behavior of the gluon propagator are investigated in Euclidean Yang-Mills theories quantized in linear covariant gauges. Considering small values of the gauge parameter, it turns out that the transverse component of the gluon propagator is suppressed, while its longitudinal part is left unchanged. A Green function, G_{tr}, which displays infrared enhancement and which reduces to the ghost propagator in the Landau gauge is identified. The inclusion of the dimension two gluon condensate <A^2> is also considered. In this case, the transvers…
Using the effective potential approach for composite operators, we have formulated a general method of calculation of the truly nonperturbative Yang-Mills vacuum energy density in the covariant gauge QCD ground state quantum models. It is defined as an integration of the truly nonperturbative part of the full gluon propagator over the deep infrared region (soft momentum region). A nontrivial minimization procedure makes it possible to determine the value of the soft cutoff in terms of the corresponding nonperturbative scale parameter, which is inevitably present in any nonperturbative model fo…
We examine the two solutions (massive and scaling) for the covariant Yang-Mills Dyson-Schwinger equations within stochastic quantization, and find that the scaling solution does not survive outside Landau gauge. We also see that the (rainbow) massive solution has less Faddeev-Popov effective action. Finally, we argue that gluon confinement has only been marginally established in experiment and suggest further empirical work.
We study the collective modes of gluons in an anisotropic thermal medium in the presence of a constant background magnetic field using the hard-thermal loop (HTL) perturbation theory. The momentum space anisotropy of the medium has been incorporated through the generalized $`$Romatschke-Strickland' form of the distribution function, whereas, the magnetic modification arising from the quark loop contribution has been taken into account in the lowest Landau level approximation. We consider two special cases: (i) a spheroidal anisotropy with the anisotropy vector orthogonal to the external magnet…
A covariant description of the interactions of axion-like particles and hadrons
University of California Santa Cruz（加州大学圣克鲁兹分校）；Santa Cruz Institute for Particle Physics（圣克鲁兹粒子物理研究所）；Technion – Israel Institute of Technology（以色列理工学院）；Massachusetts Institute of Technology（麻省理工学院）
A double perturbation idea is presented in framework of the quark-gluon plasma kinetic theory. A solvable set of equations from the 'double perturbation' is derived and the equations are showed to be gauge-independent. The formalism of Landau damping rate for the plasmon at zero momentum is given and discussed.
We analyze the uncertainties induced by different definitions of the momentum fraction $z$ in the application of gluon to heavy quarkonium fragmentation function. We numerically calculate the initial $g \to J / ψ$ fragmentation functions by using the non-covariant definitions of $z$ with finite gluon momentum and find that these fragmentation functions have strong dependence on the gluon momentum $\vec{k}$. As $| \vec{k} | \to \infty$, these fragmentation functions approach to the fragmentation function in the light-cone definition. Our numerical results show that large uncertainties remains w…
A general expression of the hard thermal loop gluon propagator is given for a variety of abelian gauge fixing conditions (Coulomb gauge, axial gauge, covariant gauge, temporal gauge ...) The current --- current interaction energy, in particular the Coulomb energy, is shown to be independent of the gauge fixing condition (not only of the parameter).
Lattice simulations along with studies in continuum QCD indicate that non-perturbative quantum fluctuations lead to an infrared regularisation of the gluon propagator in covariant gauges in the form of an effective mass-like behaviour. In the present work we propose an analytic understanding of this phenomenon in terms of gluon condensation through a dynamical version of the Higgs mechanism, leading to the emergence of color condensates. Within the functional renormalisation group approach we compute the effective potential of covariantly constant field strengths, whose non-trivial minimum is …
We study gluon production in high energy proton-nucleus collisions in the semi-classical framework of the Color Glass Condensate. We develop a general formalism to compute gluon fields in covariant gauge to lowest order in the classical field of the proton and to all orders in the classical field of the nucleus. The use of the covariant gauge makes the diagrammatic interpretation of the solution more transparnt. k_t-factorization holds to this order for gluon production -- Our results for the gluon distribution are equivalent to the prior diagrammatic analysis of Kovchegov and Mueller. We also…
The gauge dependence of the hot gluon self energy is examined in the context of Pisarski's method for resumming hard thermal loops. Braaten and Pisarski have used the Ward identities satisfied by the hard corrections to the n-point functions to argue the gauge fixing independence of the leading order resummed QCD plasma damping rate in covariant and strict Coulomb gauges. We extend their analysis to include all linear gauges that preserve rotational invariance and display explicitly the conditions required for gauge fixing independence. It is shown that in covariant gauges the resummed damping…
The two-loop three-gluon vertex is calculated in an arbitrary covariant gauge, in the limit when two of the gluons are on the mass shell. The corresponding two-loop results for the ghost-gluon vertex are also obtained. It is shown that the results are consistent with the Ward-Slavnov-Taylor identities.
It is argued by the author that the canonical form of the quark energy-momentum tensor with a partial derivative instead of the covariant derivative is the correct definition for the quark momentum and angular momentum fraction of the nucleon in covariant quantization. Although it is not manifestly gauge invariant, its matrix elements in the nucleon will be non-vanishing and are gauge invariant. We test this idea in the path-integral quantization by calculating correlation functions on the lattice with a gauge-invariant nucleon interpolation field and replacing the gauge link in the quark latt…
We study, to one loop order, the behavior of the gluon self-energy in the non covariant Coulomb gauge at finite temperature. The cancellation of the peculiar energy divergences, which arise in such a gauge, is explicitly verified in the complete two point function of the Yang-Mills theory. At high temperatures, the leading T^2 term is determined to be transverse and nonlocal, in agreement with the results obtained in covariant gauges. The coefficient of the sub-leading ln(T) contribution, is non transverse but local and coincides (up to a multiplicative constant) with that of the ultraviolet p…
We present first results for the quark mass function in Minkowski space in both the spacelike and timelike regions calculated from the same quark-antiquark interaction kernel used in the latest meson calculations using the Gross equation. This kernel consists of a Lorentz vector effective one-gluon-exchange-type interaction, a vector constant, and a mixed scalar-pseudoscalar covariant linear confining interaction that does not contribute to the mass function. We analyze the gauge dependence of our results, prove the gauge independence of the constituent quark mass and mass gap equation, and id…
In this work we use two different but complementary approaches in order to study the ghost propagator of a pure SU(3) Yang-Mills theory quantized in the linear covariant gauges, focusing on its dependence on the gauge-fixing parameter $ξ$ in the deep infrared. In particular, we first solve the Schwinger-Dyson equation that governs the dynamics of the ghost propagator, using a set of simplifying approximations, and under the crucial assumption that the gluon propagators for $ξ>0$ are infrared finite, as is the case in the Landau gauge $(ξ=0)$. Then we appeal to the Nielsen identities, and expre…
In this paper, we discuss the gluon propagator in the linear covariant gauges in $D=2,3,4$ Euclidean dimensions. Non-perturbative effects are taken into account via the so-called Refined Gribov-Zwanziger framework. We point out that, as in the Landau and maximal Abelian gauges, for $D=3,4$, the gluon propagator displays a massive (decoupling) behaviour, while for $D=2$, a scaling one emerges. All results are discussed in a setup that respects the Becchi-Rouet-Stora-Tyutin (BRST) symmetry, through a recently introduced non-perturbative BRST transformation. We also propose a minimizing functiona…
We study the binding energy of a heavy quark-antiquark ($q\bar{q}$) pair using the first-order path integral formalism. This makes the Yang-Mills constraint equation explicit, and highlights that it is valid without relying on a semiclassical approximation. A generalized gauge-covariant Coulomb gauge is chosen to allow for a decomposition of the chromoelectric field into a gauge-covariant generalization of transverse and longitudinal parts. This decomposition makes it clear that the $q\bar{q}$ binding energy is determined solely by the solution to the constraint equation. Assuming that the low…
Resummation of the soft gluon radiative corrections for the quark-vector boson vertex is performed within the path-integral (world-line) approach. The leading order expression for the vacuum averaged Wilson integral for an arbitrary gauge field is found in n-dimensional space-time. The cusp anomalous dimension of the color non-singlet Sudakov form factor of on-mass-shell quark is calculated in an arbitrary covariant gauge in the one-loop order, and the leading double-logarithmic asymptotical behavior is obtained from the corresponding evolution equation.
We extend to a general class of covariant gauges an approach which relates the thermal Green functions to forward scattering amplitudes of thermal particles. A brief discussion of the non-transversality of the thermal gluon polarization tensor is given in this context. This method is then applied to the calculation of the ln(T) contributions associated with general configurations of 2 and 3-point gluon functions. The results are Lorentz covariant and have the same structure as the ultraviolet divergent contributions which occur at zero temperature.
We show that, in the light-cone gauge, it is possible to derive in a very simple way the solution of the classical Yang-Mills equations for the collision between a nucleus and a proton. One important step of the calculation is the derivation of a formula that describes the propagation of a gluon in the background color field of the nucleus. This allows us to calculate observables in pA collisions in a more straightforward fashion than already proposed. We discuss also the comparison between light-cone gauge and covariant gauge in view of further investigations involving higher order correction…
The $S$-matrix pinch technique is used to derive an effective gluon self-energy to one-loop order, when the theory is quantized in the Coulomb gauge (CG) and in the temporal axial gauge (TAG). When the pinch contributions are added, the gluon self-energies calculated in CG and TAG turn out to be identical and coincide with the result previously obtained with covariant gauges. The issue of gauge independence of several quantities in hot QCD is discussed from the pinch technique point of view. It is also pointed out that the spurious singularities which appear in TAG calculations cancel out once…
In this work we explore the applicability of a special gluon mass generating mechanism in the context of the linear covariant gauges. In particular, the implementation of the Schwinger mechanism in pure Yang-Mills theories hinges crucially on the inclusion of massless bound-state excitations in the fundamental nonperturbative vertices of the theory. The dynamical formation of such excitations is controlled by a homogeneous linear Bethe-Salpeter equation, whose nontrivial solutions have been studied only in the Landau gauge. Here, the form of this integral equation is derived for general values…
The approach which relates thermal Green functions to forward scattering amplitudes of on-shell thermal particles is applied to the calculation of the gluon self-energy, in a class of temporal gauges. We show to all orders that, unlike the case of covariant gauges, the exact self-energy of the gluon is transverse at finite temperature. The leading T^2 and the sub-leading ln(T) contributions are obtained for temperatures T high compared with the external momentum. The logarithmic contributions have the same structure as the ultraviolet pole terms which occur at zero temperature.
We solve analytically the ellipsoidally expanding fireball hydrodynamics with source terms in the momentum and the energy equations, using the non-relativistic approximation. We find that energy transport from high pt jets of gluons to the medium leads to a transient, exponential inflation of the fireballs created in high energy heavy ion collisions. In this transient, inflatory period, the slopes of the single particle spectra increase, while the HBT radius parameters decrease exponentially with time. This effect is shown to be similar to the development of the homogeneity of our Universe due…
The next-to-leading order contribution $\d¶\omn (ø,\vc q )$ to the polarization function of the hot gluon system is analysed at non-zero wave vectors $\vc q \,$. Using Braaten-Pisarski resummation and general covariant gauges, $\d¶\omn$ is found to be gauge-fixing independent and transverse on the longitudinal mass-shell. The real part of the longitudinal component $\d¶_\ell$ is UV and IR stable (for real $q$). At imaginary $q$ it is IR singular, and at the point $ø=0$, $q^2=-3m^2$ it coincides with the result of Rebhan for next-to-leading order Debye screening. When $q$ approaches the lightco…
In covariant gauges, the gluonic mass gap in Yang-Mills theory manifests itself in the basic observation that the massless pole in the perturbative gluon propagator disappears in nonperturbative calculations, but the origin of this behavior is not yet fully understood. We summarize a recent study of the respective dynamics with Dyson-Schwinger equations in Landau-gauge Yang-Mills theory. We identify the parameter that distinguishes the massive Yang-Mills regime from the massless decoupling solutions, whose endpoint is the scaling solution. Similar to the PT-BFM scheme, we find evidence that ma…
In nonabelian gauge theory the three-gluon vertex function contains important structural information, in particular on infrared divergences, and is also an essential ingredient in the Schwinger-Dyson equations. Much effort has gone into analyzing its general structure, and at the one-loop level also a number of explicit computations have been done, using various approaches. Here we use the string-inspired formalism to unify the calculations of the scalar, spinor and gluon loop contributions to the one-loop vertex, leading to an extremely compact representation in all cases. The vertex is compu…
The Wilson contour integral approach is applied to resum the soft gluon radiative correctins to the quark form factors in the Sudakov regime. The one-loop order results for the quark-photon (color singlet form factor) and quark-gluon (color non-singlet form factor) vertices are presented. The explicit expressions for the vacuum averaged contour integrals in $g^2$ accuracy are derived for an arbitrary gauge field. The corresponding one-loop cusp anomalous dimensions are found in the case of perturbative gluon field in arbitrary covariant gauge. It is shown that the gauge dependence drops out fr…
We propose a new reformulation of Yang-Mills theory in which three- and four-gluon self-interactions are eliminated at the price of introducing a sufficient number of auxiliary fields. We discuss the validity of this reformulation in the possible applications such as dynamical gluon mass generation, color confinement and glueball mass calculation. Moreover, we set up a new $1/N_c$ color expansion in the $SU(N_c)$ Yang-Mills theory based on this reformulation. In fact, we give the Feynman rules of the $1/N_c$ expansion in the manifestly Lorentz covariant gauge. The Yang-Mills theory is defined …
We propose a vacuum condensate of mass dimension 2 consisting of gluons and ghosts in the framework of the manifestly covariant gauge fixing of the SU(N) Yang-Mills theory. This quantity is both BRST and anti-BRST invariant for any gauge. It includes the ghost condensation $C^a \bar{C}^a$ proposed first in the modified Maximal Abelian gauge and reduces to the gluon condensates $(\mathscr{A}_μ)^2$ of mass dimension 2 proposed recently in the Landau gauge of the Lorentz gauge fixing. The vacuum condensate of dimensions 2 can provide the effective mass for gluons and ghosts. The possible existenc…
We investigate the relevant gluon-momentum region for confinement in lattice QCD on $16^4$ at $β$=5.7, 5.8 and 6.0, based on the Fourier expansion. We find that the string tension $σ$, i.e., the confining force, is almost unchanged even after removing the high-momentum gluon component above 1.5GeV in the Landau gauge. In fact, the confinement property originates from the low-momentum gluon component below 1.5GeV, which is the upper limit to contribute to $σ$. In the relevant region, smaller gluon momentum component is more important for confinement. Next, we develop a manifestly gauge-covarian…
