Phases of QCD and QCD-like theories

The phase diagram of quantum chromodynamics (QCD) will be investigated at non-vanishing temperature and chemical potential with functional methods. While Monte-Carlo simulations are plagued by the sign problem at non-zero chemical potential, this region, which is central for upcoming experiments at FAIR and NICA, is directly accessible for functional methods. Also physical values for the quark masses can be used, which is computationally very demanding for lattice methods. The challenge with functional equations lies in devising appropriate truncations of the infinite sets of equations. Based on recent investigations of three- point functions in the vacuum, which shed new light on the convergence properties of functional equations, existing truncation schemes will be improved. To assess the reliability of the developed truncations we will also perform calculations for QCD-like theories, where the gauge group is replaced by SU(2) or G2. For these theories lattice results at non-vanishing chemical potential are available and direct comparisons are thus possible. For functional methods, on the other hand, the sign problem does not manifest so that the truncation effects for a QCD-like theory and QCD itself should be very similar. As another test for the truncation dependence of functional equations at non-vanishing chemical potential we consider the U(1) gauge Higgs model, for which the sign problem was solved. To start with, a newly developed truncation for the vacuum will be adapted to non-zero temperature. This allows detailed comparisons with lattice results also for SU(3) so that we can construct a reliable basis for the introduction of a chemical potential. As usual the crossover/phase transition lines will be determined by calculating the quark and the dual quark condensates. An important aspect of this project is the high level of self-consistency of the developed truncation to reduce the dependence on model input. Since this requires the calculation of many correlation functions, we take advantage of a recently developed framework for functional equations (DoFun, CrasyDSE) that automates several aspects of their handling and their calculation. While in some earlier calculations lattice results from zero chemical potential were used as input, which limits to a certain extent the quantitative reliability of the predictions, we here go beyond this approximation and include all quark back coupling effects at non-vanishing chemical potential.

In this project the behavior of quarks and gluons for various ranges of temperatures and densities was investigated. Quarks and gluons are the elementary particles that interact via the strong interaction which is described by the theory called quantum chromodynamics, or short QCD. The behavior of quarks and gluons depends strongly on the temperature and the density. For example, very high temperatures describe the situation at the beginning of the universe, while a certain range of densities is relevant for neutron stars where quarks are tightly packed. Current experiments, like ALICE at the LHC, and experiments under construction measure the behavior for different conditions. In the past, progress was made in particular for the description at zero or small densities. At higher densities, various problems exist which hinder further progress. The method employed in this project works at all densities, but we would need to solve infinitely many equations for an exact description. Thus, one takes only a few equations and solves them. However, it is very difficult to estimate how good the calculations are. Typically, one compares the results to other methods to get an idea about the reliability, but for large densities such a comparison cannot be done due to the lack of results. This is where QCD-like theories come in. They are very much like QCD but with a decisive difference: A specific method, so-called lattice simulations which fail for QCD at large densities, is applicable for QCD-like theories at large densities. Thus, in this case we can perform both types of calculations and compare them. The goal is to identify in such theories the relevant equations to learn which ones are important for QCD. Though QCD-like theories themselves do not exist in nature, they can be used by theoreticians to learn about nature. We studied a widely used choice out of the infinitely many equations and its applicability to QCD. These equations contain the interaction between quarks and gluons via a model, it is not calculated. We found that this model works very well, but the model parameters need to be fixed separately for different temperatures and densities. This limits its applicability at high densities where too little information is available. Thus, we conclude that one should overcome this shortcoming and calculate the interaction between quarks and gluons. Unfortunately, this necessitates to know several other quantities which did not need to be known before. Within this project, the subtleties of calculating them were worked out. For one specific situation explicit calculations were performed. This paves the way for future studies at various temperatures and densities.