Classical Yang-Mills Equations Research Articles

The onset of hydrodynamical flow in high energy heavy ion collisions

Abstract In this proceedings, we present recent improvements of the Color Glass Condensate effective theory that amounts to taking fully into account the first α s correction to the leading order solution, as well as a subset of higher order corrections. All these corrections are coming from the initial condition, and one can therefore apply the classical-statistical approximation (CSA) that amounts to solve the classical Yang–Mills equation of motion with fluctuating initial condition. On the numerical side, this leads to a fast isotropization of the pressure tensor at already small couplings. In addition, the system seems to experience an anomalously small shear viscosity over entropy ratio.

High-energy QCD evolution from BRST symmetry

We show that the (gauge-fixed) classical action of the Color Glass Condensate is invariant under a suitable Becchi–Rouet–Stora–Tyutin symmetry, which holds after the gluon modes are split into their fast, semi-fast, and soft components, according to the longitudinal momenta they carry. This entails the existence of a corresponding Slavnov–Taylor identity which in turn strongly constrains the effective field theory arising when integrating out the semi-fast modes. Specifically, we prove that this identity guarantees the gauge invariance of the resulting effective theory. In addition, we use it to demonstrate that the integration over the semi-fast modes does not deform the classical Yang–Mills equations of motion, thus validating a key assumption in the usual procedure adopted when deriving the renormalization group equation governing the evolution with energy of the effective theory. As far as the latter are concerned, we finally prove that its functional form is common, and it is determined by symmetry arguments alone. The formal properties of these equations valid in different regimes and/or approximations (e.g., the JIMWLK equation and its BFKL limit) can be therefore derived in a unified setting within this algebraic approach.

Turbulent thermalization process in heavy-ion collisions at ultrarelativistic energies

The non-equilibrium evolution of heavy-ion collisions is studied in the limit of weak coupling at very high energy employing lattice simulations of the classical Yang-Mills equations. Performing the largest classical-statistical simulations to date, we find that the dynamics of the longitudinally expanding plasma becomes independent of the details of the initial conditions. After a transient regime dominated by plasma instabilities and free streaming, the subsequent space-time evolution is governed by a nonthermal fixed point, where the system exhibits the self-similar dynamics characteristic of wave turbulence. This allows us to distinguish between different kinetic scenarios in the classical regime. Within the accuracy of our simulations, the scaling behavior found is consistent with the ``bottom-up" thermalization scenario.

Isotropization of the quark gluon plasma

We report here recent analytical and numerical work on the theoretical treatment of the early stages of heavy ion collisions, that amounts to solving the classical Yang-Mills equations with fluctuating initial conditions. Our numerical simulations suggest a fast isotropization of the pressure tensor of the system. This trend appears already for small values of the coupling constant $\alpha_s$ . In addition, the system exhibits an anomalously small shear viscosity.

Yang-Mills lattice on CUDA

Abstract The Yang-Mills fields have an important role in the non- Abelian gauge field theory which describes the properties of the quarkgluon plasma. The real time evolution of the classical fields is given by the equations of motion which are derived from the Hamiltonians to contain the term of the SU(2) gauge field tensor. The dynamics of the classical lattice Yang-Mills equations are studied on a 3 dimensional regular lattice. During the solution of this system we keep the total energy on constant values and it satisfies the Gauss law. The physical quantities are desired to be calculated in the thermodynamic limit. The broadly available computers can handle only a small amount of values, while the GPUs provide enough performance to reach out for higher volumes of lattice vertices which approximates the aforementioned limit.

Integral form of Yang-Mills equations and its gauge invariant conserved charges

Despite the fact that the integral form of the equations of classical electrodynamics is well known, the same is not true for non-Abelian gauge theories. The aim of the present paper is threefold. First, we present the integral form of the classical Yang-Mills equations in the presence of sources and then use it to solve the long-standing problem of constructing conserved charges, for any field configuration, which are invariant under general gauge transformations and not only under transformations that go to a constant at spatial infinity. The construction is based on concepts in loop spaces and on a generalization of the non-Abelian Stokes theorem for two-form connections. The third goal of the paper is to present the integral form of the self-dual Yang-Mills equations and calculate the conserved charges associated with them. The charges are explicitly evaluated for the cases of monopoles, dyons, instantons and merons, and we show that in many cases those charges must be quantized. Our results are important in the understanding of global properties of non-Abelian gauge theories.

On New Solutions of Classical Yang-Mills Equations with Cylindrical Sources

Strong fields generated by big electric currents are examined within the framework of the Yang-Mills nonlinear generalization of the Maxwell electrodynamics proposed in our earlier papers. First we consider the case of stationary currents and find a new exact solution to the Yang-Mills equations. Then we study a Yang-Mills field inside a thin circular cylinder with nonstationary plasma and find expressions for field strengths in it. Obtained results are applied to interpret several puzzling natural phenomena.

First correction to JIMWLK evolution from the classical equations of motion

We calculate some corrections to the JIMWLK kernel in the framework of the light-cone wavefunction approach to the high-energy limit of QCD. The contributions that we consider originate from higher order corrections in the strong coupling and in the density of the projectile to the solution of the classical Yang–Mills equations of motion that determine the Weizsäcker–Williams fields of the projectile. We study the structure of these corrections in the dipole limit, showing that they are subleading in the limit of large number of colours N and that they cannot be fully recast in the form of dipole degrees of freedom.

Infrared degrees of freedom of Yang-Mills theory in the Schrödinger representation

We set up a new calculational framework for the Yang-Mills vacuum transition amplitude in the Schroedinger representation. After integrating out hard-mode contributions perturbatively, we perform a gauge invariant gradient expansion of the ensuing soft mode action which renders a subsequent saddle point expansion for the vacuum overlap manageable. The standard "squeezed" approximation for the vacuum wave functional then allows for an essentially analytical treatment of physical amplitudes. Moreover, it leads to the identification of dominant and gauge invariant classes of gauge field orbits which play the role of gluonic infrared (IR) degrees of freedom. Those emerge as a rich variety of (mostly solitonic) solutions to the saddle point equations which are characterized by a common relative gauge orientation of the underlying gluon fields. We discuss their scale stability, guaranteed by a virial theorem, and other general properties including topological quantum numbers and action bounds. We then find important saddle point solutions explicitly and examine their physical impact. Some of them are related to tunneling solutions of the classical Yang-Mills equation, i.e. to instantons and merons, while others appear to play unprecedented roles. A remarkable new class of IR degrees of freedom comprises vortex and knot solutions of Faddeev-Niemi type, potentially related to glueballs.

Gluon propagation inside a high-energy nucleus

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 corrections.

Searching for KvBLL calorons in SU(3) lattice gauge field ensembles

We discuss Kraan - van Baal - Lee - Lu (KvBLL) solutions of the classical Yang-Mills equations with temperature in the context of SU(3) lattice gauge theory. We present discretized lattice versions of KvBLL solutions and other dyonic structures, obtained by cooling in order to understand their variety and signature. An analysis of the zero modes of the lattice Dirac operator for different fermionic boundary conditions gives clear evidence for a KvBLL-like background of finite T lattice subensembles with Q = +/-1. Using APE-smearing we are able to study the topological charge density q(x) of the configurations and to corroborate this interpretation.

Calorons, instantons, and constituent monopoles in SU(3) lattice gauge theory

We analyze the zero-modes of the Dirac operator in quenched SU(3) gauge configurations at non-zero temperature and compare periodic and anti-periodic temporal boundary conditions for the fermions. It is demonstrated that for the different boundary conditions often the modes are localized at different space-time points and have different sizes. Our observations are consistent with patterns expected for Kraan - van Baal solutions of the classical Yang-Mills equations. These solutions consist of constituent monopoles and the zero-modes are localized on different constituents for different boundary conditions. Our findings indicate that the excitations of the QCD vacuum are more structured than simple instanton-like lumps.

New findings for topological excitations in SU(3) lattice gauge theory

We probe the SU(3) vacuum using eigenvectors of the Dirac operator with an arbitrary phase for the temporal boundary condition. We consider configurations with topological charge | Q|=1 near the QCD phase transition and at low temperatures on a torus. For all our ensembles we show that the zero mode of the Dirac operator changes its position as one changes the phase of the boundary condition. For ensembles near the QCD phase transition our results closely resemble the behavior of zero modes for Kraan–van Baal solutions of the classical Yang–Mills equations where the individual lumps are interpreted as monopoles. Our findings near T c and on the torus show that for both cases an excitation with topological charge | Q|=1 is built from several separate lumps.

Forced tunneling and turning state explosion in pure Yang-Mills theory

We consider forced tunneling in QCD, described semiclassically by instanton--anti-instanton field configurations. By separating topologically different minima we obtain details of the effective potential and study the turning states, which are similar to the sphaleron solution in electroweak theory. These states are alternatively derived as minima of the energy under the constraints of fixed size and Chern-Simons number. We study, both analytically and numerically, the subsequent evolution of such states by solving the classical Yang-Mills equations in real time, and find that the gauge field strength is quickly localized into an expanding shell of radiating gluons. The relevance to high-energy collisions of hadrons and nuclei is briefly discussed.

Classical Yang-Mills equations invariant under subgroups of the conformal group without coupling to matter

Classical Yang-Mills field equations on Minkowski space with gaugegroup SU(2) are determined. Solutions invariant up to a gauge transformation under a four-dimensional subgroup of the conformal group are evaluated for the Yang-Mills field in the absence of matter. Some properties of the solutions of the equations such as their asymptotic behaviour are obtained.PACS No.: 03.50-z

Gluon minijet production in nuclear collisions at high energies

We show explicitly that the leading soft gluon $p_T$ distribution, predicted by Kovner, McLerran, and Weigert after solving classical Yang-Mills equations, can be understood in terms of conventional QCD perturbation theory. We also demonstrate that the key logarithm in their result represents the logarithm in DGLAP evolution equations.

Classical Yang-Mills equations invariant under subgroups of the conformal group without coupling to matter

Classical Yang-Mills field equations on Minkowski space with gaugegroup SU(2) are determined. Solutions invariant up to a gauge transformation under a four-dimensional subgroup of the conformal group are evaluated for the Yang-Mills field in the absence of matter. Some properties of the solutions of the equations such as their asymptotic behaviour are obtained.PACS No.: 03.50-z

On the possibility of the dual Meissner effect induced by instantons

The classical Yang-Mills equation of motion is numerically investigated in the Lorentz gauge for a SU(2) gauge group. The color-electric field of two point-like charges is studied in the “empty” vacuum and in a state with an instanton present. The major effect for a fixed orientation of the instanton is that the color-electric field lines are expelled or attracted from the instanton region depending on the orientation of the instanton. If over the orientations of the instanton is averaged, this effect drops out. In this case of a random instanton orientation, we find that the external color-electric field is expelled from the instanton core. The origin of this effect is discussed.

Analogue of Black Strings in Yang-Mills Gauge Theory

The classical Yang–Mills equations areanalyzed within the geometrical framework of aneffective gravity theory. Exact analytical solutions arederived for the cylindrically symmetric configurationsof the coupled gauge and isoscalar fields. Itturns out that there is an infinite family of solutionsparametrized by two real parameters, one of whichdetermines the asymptotic behavior of fields near the symmetry axis and in infinity, while the secondlocates the singularity. These configurations have asimple pole at a finite value of the radial coordinate,and physically they represent “thickstring”-like objects which possess the confinementproperties. It is demonstrated that the particles withgauge charge cannot move classically and quantummechanically out of the interior region. Such objectsare thus direct analogues of the “blackstring” gravitational configurations reportedrecently in the literature.

On the possibility of the dual Meissner effect induced by instantons

The classical Yang-Mills equation of motion is numerically investigated in the Lorentz gauge for a SU(2) gauge group. The color-electric field of two point-like charges is studied in the ''empty'' vacuum and in a state with an instanton present. The major effect for a fixed orientation of the instanton is that the color-electric field lines are expelled or attracted from the instanton region depending on the orientation of the instanton. If over the orientations of the instanton is averaged, this effect drops out. In this case of a random instanton orientation, we find that the external color-electric field is expelled from the instanton core. The origin of this effect is discussed.