Advances in High Energy Physics

Volume 2018, Article ID 2312498, 21 pages

https://doi.org/10.1155/2018/2312498

## Hamiltonian Approach to QCD in Coulomb Gauge: A Survey of Recent Results

Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany

Correspondence should be addressed to H. Reinhardt; ed.negnibeut-inu@tdrahnier.h

Received 6 June 2017; Revised 4 September 2017; Accepted 2 October 2017; Published 19 February 2018

Academic Editor: Ralf Hofmann

Copyright © 2018 H. Reinhardt et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We report on recent results obtained within the Hamiltonian approach to QCD in Coulomb gauge. Furthermore this approach is compared to recent lattice data, which were obtained by an alternative gauge-fixing method and which show an improved agreement with the continuum results. By relating the Gribov confinement scenario to the center vortex picture of confinement, it is shown that the Coulomb string tension is tied to the spatial string tension. For the quark sector, a vacuum wave functional is used which explicitly contains the coupling of the quarks to the transverse gluons and which results in variational equations which are free of ultraviolet divergences. The variational approach is extended to finite temperatures by compactifying a spatial dimension. The effective potential of the Polyakov loop is evaluated from the zero-temperature variational solution. For pure Yang–Mills theory, the deconfinement phase transition is found to be second order for and first order for , in agreement with the lattice results. The corresponding critical temperatures are found to be and , respectively. When quarks are included, the deconfinement transition turns into a crossover. From the dual and chiral quark condensate, one finds pseudocritical temperatures of and , respectively, for the deconfinement and chiral transition.

#### 1. Introduction

One of the most challenging problems in particle physics is the understanding of the phase diagram of strongly interacting matter. By means of ultrarelativistic heavy ion collisions the properties of hadronic matter at high temperature and/or density can be explored. From the theoretical point of view we have access to the finite-temperature behavior of QCD by means of lattice Monte-Carlo calculations. Due to the sign problem, this method fails, however, to describe baryonic matter at high density or, more technically, QCD at large chemical baryon potential [1]. Therefore, alternative, nonperturbative approaches to QCD which do not rely on the lattice formulation and hence do not suffer from the notorious sign problem are desirable. In recent years, much effort has been devoted to develop nonperturbative continuum approaches. These are based on either Dyson–Schwinger equations [2–7] or functional renormalization group flow equations [8, 9], or they exploit the variational principle in either the Hamiltonian [10, 11] or covariant [12, 13] formulation of gauge theory. There are also semiphenomenological approaches assuming a massive gluon propagator [14] or the Gribov–Zwanziger action [15]; see [16].

In this talk, I will review some recent results obtained within the Hamiltonian approach to QCD in Coulomb gauge both at zero and at finite temperatures; for earlier reviews, see [17, 18]. After a short introduction to the basic features of this approach, I will summarize the essential zero-temperature results for pure Yang–Mills theory and compare them to recent lattice data which were obtained by an alternative gauge-fixing method, which is expected to yield results closer to the continuum theory. After that, I will show by means of lattice calculations that the so-called Coulomb string tension is linked not to the temporal but to the spatial string tension. In this context, I will demonstrate that the Gribov–Zwanziger confinement scenario is related to the center vortex picture of confinement. I will then report on new variational calculations carried out for the quark sector of QCD. After that I will extend the Hamiltonian approach to QCD in Coulomb gauge to finite temperatures by compactifying a spatial dimension [19]. Numerical results will be given for the Polyakov loop and the chiral and dual quark condensates. Finally, I will give some outlook on future research within the Hamiltonian approach.

#### 2. Variational Hamiltonian Approach to Yang–Mills Theory

For pedagogical reason let me first summarize the basic features of the Hamiltonian approach in Coulomb gauge for pure Yang–Mills theory. The Hamiltonian approach to Yang–Mills theory starts from Weyl gauge and considers the spatial components of the gauge field as coordinates. The momenta are introduced in the standard fashion and turn out to be the color electric field . The classical Yang–Mills Hamiltonian is then obtained aswhereis the non-Abelian color magnetic field with being the coupling constant. The theory is quantized by replacing the classical momentum with the operator . The central issue is then to solve the Schrödinger equation for the vacuum wave functional . Due to the use of Weyl gauge, Gauß’s law (with being the covariant derivative in the adjoint representation) has to be put as a constraint on the wave functional, which ensures the gauge invariance of the latter. Instead of working with explicitly gauge invariant states, it is more convenient to fix the gauge and explicitly resolve Gauß’s law in the chosen gauge. This has the advantage that any (normalizable) wave functional is physically admissible for a variational approach, while the price to pay is a significant complication of the gauge-fixed Hamiltonian. A particular convenient choice of gauge for this method turns out to be Coulomb gauge .

After canonical quantization in Weyl gauge and resolution of Gauß’s law in Coulomb gauge one finds the following gauge-fixed Hamiltonian [20]:withwhereis the Faddeev–Popov determinant andis the so-called Coulomb term with the color charge densityThis expression contains besides the charge density of the matter fields also a purely gluonic part. Due to the implementation of Coulomb gauge, the scalar product in the Hilbert space of wave functionals is defined byHere, the functional integration is over transverse spatial gauge fields and the Faddeev–Popov determinant appears due to Coulomb gauge fixing with the standard Faddeev–Popov method. The Faddeev–Popov determinant (5) in the integration measure represents the Jacobian of the change of variables from “Cartesian” to “curvilinear” variables in Coulomb gauge. With the gauge-fixed Hamiltonian (3) one has to solve the stationary Schrödinger equation for the vacuum wave functional . Once is known, all observables and correlation functions can, in principle, be calculated. This has been attempted by means of the variational principle using Gaussian type ansatz for the vacuum wave functional [21, 22]. However, the first attempts did not properly include the Faddeev–Popov determinant, which turns out to be crucial in order to describe the confinement properties of the theory. Below, I will discuss the variational approach developed in [10, 11], which differs from previous attempts by the ansatz for the vacuum wave functional, the treatment of the Faddeev–Popov determinant and, equally important, by the renormalization; see [23] for further details.

##### 2.1. Variational Solution of the Schrödinger Equation

The ansatz for the vacuum wave functional is inspired by the quantum mechanics of a particle in a spherically symmetric potential for which the ground state wave function is given by , where the radial wave functional satisfies a standard one-dimensional Schrödinger equation and represents (the square root of the radial part of) the Jacobian of the transformation from the Cartesian to spherical coordinates. Our ansatz for the vacuum wave functional is given byThe inclusion of the preexponential factor has the advantage that it eliminates the Faddeev–Popov determinant from the integration measure in the scalar product (8). Furthermore, for the wave function (9), the gluon propagator is given up to a factor of by the inverse of the variational kernel . As the numerical calculation shows [24] in the Yang–Mills sector the Coulomb term (6) can be ignored. The reason is the presence of the Faddeev–Popov determinant in the pure Yang–Mills term of (see (6)), which ensures that in the numerator of the expectation value the gluon energy occurs only in the combination (see [10]), which is infrared vanishing. The Faddeev–Popov determinant , however, drops out from in the quark sector, since commutes with the charge density of the quarks; see (31).

Calculating the expectation value of the remaining parts of the Yang–Mills Hamiltonian (4) with the wave functional (9) up to two loops, the minimization of the energy density with respect to yields the following gap equation in momentum space (Due to translational and rotational invariance, kernels such as can be Fourier transformed as where the new kernel in momentum space depends on only. For simplicity, we will use the same symbol for the kernel in position and momentum space and go back and forth between both representations with impunity.):where is a finite renormalization constant resulting from the tadpole andrepresents the ghost loop ( is the transverse projector). This can be expressed in terms of the ghost propagator:which is evaluated with the vacuum wave functional (9) in the rainbow-ladder approximation, resulting in a Dyson–Schwinger equation for the form factorof the ghost propagator which is diagrammatically illustrated in Figure 1. This equation has to be solved together with the gap equation (10).