Chromomagnetic Field Research Articles

QCD with 2+1 flavors at the physical point in external chromomagnetic fields

We investigate full quantum chromodynamics (QCD) with (2+1)-flavor of highly improved staggered quarks fermions at the physical point in the presence of uniform Abelian chromomagnetic background fields. Our focus is to conduct an exploratory study on the renormalized light and strange chiral condensates around the pseudocritical temperature. We find that in the confined region the gauge system is subjected to the chromomagnetic catalysis that turns into the inverse catalysis in the high-temperature regime. We further observe that the chiral condensates are subjected to the so-called thermal hysteresis. Our estimate of the deconfinement temperature indicates that the critical temperature begins to decrease in the small field region, soon after it seems to saturate and finally increase with the strength of the chromomagnetic field. Published by the American Physical Society 2024

Quarkonium polarization in medium from open quantum systems and chromomagnetic correlators

We study the spin-dependent in-medium dynamics of quarkonia by using the potential nonrelativistic QCD (pNRQCD) and the open quantum system framework. We consider the pNRQCD Lagrangian valid up to the order rM0=r and r0M=1M in the double power counting. By considering the Markovian condition and applying the Wigner transformation upon the diagonal spin components of the quarkonium density matrix with the semiclassical expansion, we systematically derive the Boltzmann transport equation for quarkonia with polarization dependence in the quantum optical limit. Unlike the spin-independent collision terms governed by certain chromoelectric field correlators, new gauge invariant correlators of chromomagnetic fields determine the recombination and dissociation terms with polarization dependence at the order we are working. We also derive a Lindblad equation describing the in-medium transitions between spin-singlet and spin-triplet heavy quark-antiquark pairs in the quantum Brownian motion limit. The Lindblad equation is governed by new transport coefficients defined in terms of the chromomagnetic field correlators. Our formalism is generic and valid for both weakly coupled and strongly coupled quark gluon plasmas. It can be further applied to study spin alignment of vector quarkonia in heavy ion collisions. Published by the American Physical Society 2024

Chromoelectric flux tubes within non-Abelian Proca theory

Flux tube solutions within non-Abelian SU(3) Proca theory with external sources are obtained. It is shown that such tubes have a longitudinal chromoelectric field possessing two components (nonlinear and gradient), as well as a transverse chromomagnetic field whose force lines create concentric circles with the center on the axis of the tube. The scenario of a possible relationship between non-Abelian Proca theory and quantum chromodynamics is considered. In such scenario: (a) the components of color fields have different behavior: those which are almost classical, and those which are purely quantum; (b) the second components create a gluon condensate that is a source of the field for the almost classical components of the Proca field; (c) Proca masses may appear as a result of an approximate description of the gluon condensate; (d) the question of gauge invariance is considered. It is shown that the results obtained are in good agreement with the results of lattice calculations. We make an assumption that an approximate description of a flux tube in quantum chromodynamics can be carried out using classical Proca equations but with a mandatory account of a gluon condensate.

Off-lightcone Wilson-line operators in gradient flow

Off-lightcone Wilson-line operators are constructed using local operators connected by time-like or space-like Wilson lines, which ensure gauge invariance. Off-lightcone Wilson-line operators have broad applications in various contexts. For instance, space-like Wilson-line operators play a crucial role in determining quasi-distribution functions (quasi-PDFs), while time-like Wilson-line operators are essential for understanding quarkonium decay and production within the potential non-relativistic QCD (pNRQCD) framework. In this work, we establish a systematic approach for calculating the matching from the gradient-flow scheme to the MS¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\ extrm{MS}} $$\\end{document} scheme in the limit of small flow time for off-lightcone Wilson-line operators. By employing the one-dimensional auxiliary-field formalism, we simplify the matching procedure, reducing it to the matching of local current operators. We provide one-loop level matching coefficients for these local current operators. For the case of hadronic matrix element related to the quark quasi-PDFs, we show at one-loop level that the finite flow time effect is very small as long as the flow radius is smaller than the physical distance z, which is usually satisfied in lattice gradient flow computations. Applications include lattice gradient flow computations of quark/gluon quasi-PDFs, gluonic correlators related to quarkonium decay and production in pNRQCD, and spin-dependent potentials in terms of chromoelectric and chromomagnetic field insertions into a Wilson loop.

How large is the space of covariantly constant gauge fields

The covariantly constant gauge fields are solutions of the sourceless Yang-Mills equation and represent classical vacuum fields. We found that the moduli space of the covariantly constant gauge fields is infinite-dimensional and is therefore much larger than the space of constant chromomagnetic fields. These solutions represent a space lattice of non-perturbative magnetic flux tubes/vertices oriented in the opposite directions, each of which has a Dirac quantum flux. The geometrical structure of the solutions is self-sustaining without presence of any Higgs field support. They are similar to a condensate of the Nielsen-Olesen magnetic flux tubes of opposite orientations. The solutions have a non-vanishing Hopf invariant density.

Covariantly constant Yang Mills vacuum fields and condensation of magnetic fluxes

Covariantly constant gauge fields are solutions of the sourceless Yang-Mills equation and represent classical vacuum fields. We found that the moduli space of covariantly constant gauge fields is infinite-dimensional and is much larger than the space of constant chromomagnetic fields. These solutions represent non-perturbative chromomagnetic flux tubes (vortices) and are the superposition of Nielsen-Olesen chromomagnetic vortices of opposite orientations. It is suggested that the solutions describe the condensate of superposed tubes (vortices) of opposite orientations and represent a dual analog of the Cooper pairs condensate in a superconductor.

The QCD Vacuum as a Disordered Chromomagnetic Condensate

An attempt is made to describe from first principles the large-scale structure of the confining vacuum in quantum chromodynamics. Starting from our previous variational studies of the SU(2) pure gauge theory in an external Abelian chromomagnetic field and extending Feynman’s qualitative analysis in (2+1)-dimensional SU(2) gauge theory, we show that the SU(3) vacuum in three-space and one-time dimensions behaves like a disordered chromomagnetic condensate. Color confinement is assured by the presence of a mass gap together with the absence of color long-range correlations. We offer a clear physical picture for the formation of the flux tube between static quark charges that allows us to determine the color structure and the transverse profile of the flux-tube chromoelectric field. The transverse profile of the flux-tube chromoelectric field turns out to be in reasonable agreement with lattice data. We, also, show that our quantum vacuum allows for both the color and ordinary Meissner effect. We find that for massless quarks, the quantum vacuum can accommodate a finite non-zero density of fermion zero modes leading to the dynamical breaking of the chiral symmetry.

Stability of initial glasma fields

A system of gluon fields produced in the earliest phase of relativistic heavy-ion collisions, which is called , can be described in terms of classical fields. Initially there are chromoelectric and chromomagnetic fields along the collision axis. A linear stability analysis of these fields is performed, assuming that the fields are space-time uniform and using the SU(2) gauge group. We apply Milne coordinates and the gauge condition, which are usually used in studies of glasma. The chromoelectric field is in the Abelian configuration with the corresponding potential linearly depending on coordinates, but the chromomagnetic field is in the non-Abelian configuration generated by the potential of noncommuting components. The chromomagnetic field is found to be unstable, and the growth rate of the unstable mode is derived. Our findings are critically debated and confronted with the numerical simulations by Romatschke and Venugopalan, who found that the evolving glasma is unstable due to the Weibel instability, which is well known in electromagnetic plasma. Published by the American Physical Society 2024

Width of the flux tube in SU(3) pure gauge theory

In this paper, we present results obtained from the flux tubes between a quark and an antiquark in SU(3) pure gauge lattice field theory. We fit the transverse distribution of the energy density and the parallel components of the chromoelectric and chromomagnetic fields with several functions and used the coefficients obtained from the fit for computation of the width of the flux tube. Subsequently, we looked into in the dependence of the width of the flux tube on the distance between the static quark-antiquark sources and temperatures. Finally, we compared our results with those from other similar studies. We observed that the numerical value of the width of the flux tube computed from the pure gauge theory is larger than that from the full QCD theory, and the width of the flux tube widened until Tc, and then decreased above the critical temperature.

Photons as a Signal of Deconfinement in Hadronic Matter under Extreme Conditions

The photon production by conversion of gluons gg→γ via quark loop in the framework of the mean-field approach to the QCD (quantunm chromodynamics) vacuum is studied here. According to the domain model of QCD vacuum, the confinement phase is dominated by Abelian (anti-)self-dual gluon fields, while the deconfinement phase is characterized by a strong chromomagnetic field. In the confinement phase, photon production is impossible due to the random spacial orientation of the statistical ensemble of vacuum fields. However, the conditions of Furry theorem are not satisfied in the deconfinement phase, the conversion of gluons is nonzero and, in addition, photon distribution has a strong angular anisotropy. Thus, the photon production in the discussed process acts as one of the important features of transition in quark-gluon plasma to the deconfinement phase.

Stability of Yang Mills vacuum state

We examine the phenomena of the chromomagnetic gluon condensation in the Yang-Mills theory and the problem of stability of the chromomagnetic vacuum fields. The apparent instability of the chromomagnetic vacuum fields is a result of quadratic approximation. The stability is restored when the nonlinear interaction of negative/unstable modes is taken into account in the case of chromomagnetic vacuum fields and the interaction of the zero modes in the case of (anti)self-dual covariantly-constant vacuum fields. All these vacuum fields are stable and indicate that the Yang-Mills vacuum is highly degenerate quantum state.

Photons production in heavy-ion collisions as a signal of deconfinement phase

The photon production due to conversion of two gluons into a photon, $$gg\rightarrow \gamma $$ , in the presence of the background gauge fields is studied within the specific mean-field approach to QCD vacuum. In this approach, mean field in the confinement phase is represented by the statistical ensemble of almost everywhere homogeneous abelian (anti-)self-dual gluon configurations, while the deconfined phase can be characterized by the purely chromomagnetic fields. The probability of gluon conversion of two gluons into a photon vanishes in the confinement phase due to the randomness of the background field configurations. The anisotropic strong electromagnetic field, generated in the collision of relativistic heavy ions, serves as a catalyst for deconfinement with the appearance of an anisotropic purely chromomagnetic mean field. Respectively, deconfined phase is characterized by nonzero probability of the conversion of two gluons into a photon with strongly anisotropic angular distribution.

Tachyon condensation in a chromomagnetic background field and the groundstate of QCD

I consider the chromomagnetic vacuum in SU(2). The effective Lagrangian in one loop approximation is known to have a minimum below zero which results in a spontaneously generated magnetic field. However, this minimum is not stable; the effective action has an imaginary part. Over the past decades, there were many attempts to handle this situation which all were at some point unsatisfactory. I propose an idea for a new solution by assuming that the tachyonic mode, at low temperature, acquires a condensate and, as a result, undergoes a phase transition like in the Higgs model. I consider the approximation where all gluon modes are dropped except for the tachyonic one. For this mode, we have a O(2)-model with quartic self-interaction in two dimensions. I apply the CJT (2PI) formalism in Hartree approximation. As a result, at zero and low temperatures, a minimum of the effective action at a certain value of the condensate and of the background fields is observed and there is no imaginary part. Raising the temperature, this minimum becomes shallower and at a critical temperature, the perturbative state becomes that with lower effective potential; the symmetry is restored. The physical interpretation says that the unstable mode creates tachyons until these come into equilibrium with their repulsive self-interaction and form a condensate. The relation to the Mermin-Wagner theorem is discussed.

Spontaneous magnetization of a vacuum in high temperature gluodynamics (two-­loop approximation)

In SU(N) gluodynamics, at high temperature the spontaneous magnetization, b(T) ≠ 0, of a vac­uum happens in the approximation to the effective potential ­- the tree plus the one-­loop, plus daisy diagrams, W(b) = b2/2g2 + W(1)(b) + Wdaisy(b). At the same time, in two­-loop approximation, W(A0) = W(1) (A0) + W(2)(A0), other classical field ­ A0 condensate directly related to the Polyakov loop - ­is also spontaneously generated. To investigate the creation of the condensates together, the two-loop effective po­tential of both fields should be calculated. This program was realized recently for SU(2) in [1]. However, the generation of magnetic field in two­-loop order was not studied in detail. In the present paper, we compute the value of chromomagnetic field b(T) for latter case. Then, considering the spectrum of color charged gluons at the background of both condensates, we conclude that the A0 stabilizes the magnetized vacuum at high temperature. This is in agreement with the lattice simulations carried out already and clarifies the mechanism of the magnetic field stabilization.

Preliminary results on deconfinement phase transition in full QCD

We explore the profiles of the flux tube connecting a quark and an antiquark in hightemperature SU(3) lattice gauge theory in close vicinity to the critical temperature of thephase transition. In this work, we consider the more realistic case of the flux tube withdynamical quarks, extending the previous study to SU(3) gauge group and making use ofthe Gradient flow method in smoothing procedure for noise reduction. The profiles of thechromoelectric and chromomagnetic field strengths in the flux tube have been measuredfrom Polyakov loop-plaquette correlations using the highly improved staggered quark(HISQ) action on a lattice with temporal extent.

Addendum to “Stability of classical chromodynamic fields”

A system of chromodynamic fields, which can be treated as classical, is generated at the earliest stage of relativistic heavy-ion collisions. Numerical simulations show that the system is unstable but the nature of the instability is not well understood. We study the problem systematically. In the first paper, we have performed a linear stability analysis of space-time uniform chromoelectric and chromomagnetic fields. There they have been considered the Abelian configurations of single-color potentials linearly depending on coordinates and non-Abelian ones where the fields are generated by the multicolor noncommuting uniform potentials. Here we extend and supplement the analysis. We discuss the parallel chromoelectric and chromomagnetic fields that occur simultaneously. We also consider a general non-Abelian configuration of the uniform fields. Finally, we discuss the gauge dependence of our results.

Classical Yang Mills equations with sources: Consequences of specific scalar potentials

Some well known gauge scalar potential very often considered or used in the literature are investigated by means of the classical Yang Mills equations for the SU(2) subgroups of Nc=3. By fixing a particular shape for the scalar potential, the resulting vector potentials and the corresponding color-charge distributions are found. By adopting the spherical coordinate system, it is shown that spherically symmetric solutions, only dependent on the radial coordinate, are only possible for the Abelian limit, otherwise, there must have angle-dependent component(s). The following solutions for the scalar potential are investigated: the Coulomb potential and a non-spherically symmetric generalization, a linear potential A0(r→)∼(κr), a Yukawa-type potential A0(r→)∼(Ce−r/r0/r) and finite spatial regions in which the scalar potential assumes constant values. The corresponding chromo-electric and chromo-magnetic fields, as well as the color-charge densities, are found to have strong deviations from the spherical symmetric configurations. We speculate these types of non-spherically symmetric configurations may contribute (or favor) for the (anisotropic) confinement mechanism since they should favor color charge-anti-charge (or three-color-charge) bound states that are intrinsically non spherically symmetric with (asymmetric) confinement of fluxes. Specific conditions and relations between the parameters of the solutions are also presented.

The effective potential of gluodynamics in the background of Polyakov loop and colormagnetic field

In SU(N) gluodynamics, above the de-confinement temperature, the effective potential has minima at non-zero A_0-background fields in the two-loop approximation. Also, it has a minimum at non-zero chromomagnetic background field, known as ’Savvidy’-vacuum, which shows up on the one-loop level. In this paper, we join these two approaches. We formulate, at finite temperature, the effective action, or the free energy, in SU(2) gluodynamics on the two-loop level, with both, A_0 background and magnetic background present at the same time, which was not done so far. We provide the necessary representations for both, effective numerical calculation and high-temperature expansions. The results are represented as a 3D plot of the real part of the effective potential. Also, we reproduce for zero either, the A_0-background or the magnetic background, the known minima and compare them. The imaginary part is, on the two-loop level, still present. We mention that, as is known from literature for the case without A_0-background, the imaginary part is compensated by the ring (’daisy’) diagrams. However, in our two-loop approximation, the results reveal an unnatural, singular behavior of the real part of the effective potential in the region, where the imaginary part sets in. Our conclusion is that one has to go beyond the two-loop approximation and its ring improved version, in order to investigate the minimum of the effective action as a function of A_0 and chromomagnetic field, and its stability, at least in the approximation of super daisy diagrams, i.e., the Hartree approximation in the CJT formalism.

Stability of classical chromodynamic fields

A system of gluon fields generated at the earliest phase of relativistic heavy-ion collisions can be described in terms of classical fields. Numerical simulations show that the system is unstable but a character of the instability is not well understood. With the intention to systematically study the problem, we analyze a stability of classical chromomagnetic and chromoelectric fields which are constant and uniform. We consider the Abelian configurations discussed in the past where the fields are due to the single-color potentials linearly depending on coordinates. However, we mostly focus on the nonAbelian configurations where the fields are generated by the multi-color non-commuting constant uniform potentials. We derive a complete spectrum of small fluctuations around the background fields which obey the linearized Yang-Mills equations. The spectra of Abelian and nonAbelian configurations are similar but different and they both include unstable modes. We briefly discuss the relevance of our results for fields which are uniform only in a limited spatial domain.

Relativistic Topological Insulator Model

A relativistic topological insulator model in three spatial dimensions which is a non trivial extension of the non-abelian Landau problem is proposed. The model is exactly soluble and energy levels have both a discrete and a continuous degeneracy. The chromomagnetic field is strong and the fermions are confined in a plane and the physical effects that appear reflects the ${\bm Z}_2$ symmetry.