Effective theory for heavy quark QCD at finite temperature and density with stochastic quantization

In this thesis we presented the derivation as well as the numerical and analytical treatment of an effective theory for lattice Quantum Chromodynamics (LQCD). We derived the effective theory directly from LQCD, which allows us to systematically introduce further improvements. The derivation was performed by means of an expansion around the limit of infinite quark masses and infinite gauge coupling. Using this theory we were able to derive results in the region of large densities. This region is, due to the sign problem, inaccessible to standard LQCD approaches. Although LQCD simulations at large densities have been performed recently by applying stochastic quantization, those are still limited to lattice with low numbers of timeslices and therefor can not reach the low temperature region. Furthermore, they can not be crosschecked with Monte-Carlo simulations. Since the equivalence between stochastic quantization and Monte-Carlo is unproven for the case of finite density systems, new approaches to access the cold dense region of the QCD phase diagram are desirable. The effective theory presented in this thesis provides such an approach. We introduced continuum QCD in chapter 2. In chapter 3 we presented how LQCD, i.e. QCD in a discretized space-time, can be formulated and used as a tool to explore the non-perturbative regions of the QCD phase diagram. Special emphasis was placed on simulations at finite baryon densities and the numerical problems that arise in this region. These problems are caused by the complexification of the action and are known as the sign problem. We gave a detailed presentation of the derivation of our effective theory in chapter 4. For this we performed expansions around the limit of strong coupling and static quarks, κ=β=0, introducing corrections order by order in the expansion parameters κ and β. Truncating the theory at different orders allowed us to determine the parameter region where the convergence to full LQCD is good. The gauge corrections are sufficient to reach β∼6, which translates to lattice spacings down to a ∼0.1 fm. Furthermore we determined the convergence in κ by simulating the action truncated at different orders. Due to the three dimensional nature of our theory the convergence depends on the temporal extent N_τ. We concluded that our theory converges well up to values of at least (N_τκ²)/(3) ∼ 0.04. Both results can be improved by deriving further corrections. In chapter 5 we presented the numerical treatment of our theory. While the sign problem is still present, it is mild compared to the case of full LQCD. This allowed us to use both Monte-Carlo with reweighting and stochastic quantization in order to crosscheck results. This confirms the validity of stochastic quantization for our theory, which is our method of choice since, in contrast to reweighting, it is not limited to small lattice volumes. We presented results for two parameter regions, the region of large density and low temperatures, and the region of high temperature and low density. For the cold dense region we calculated several thermodynamical quantities and performed continuum extrapolations. This allows us to make a connection to continuum QCD, although in a parameter region far away from the physical point. The results show the onset from the vacuum to the region of finite density, displaying Silver Blaze behavior. We furthermore demonstrated the existence of a finite binding energy between baryonic states, which in the continuum are responsible for the formation of nuclear matter. Although experiments show the transition from the vacuum to the region of finite density to be of first order for low enough temperatures, the convergence region of our theory is not large enough to reproduce this. Nevertheless, we where able to find signals for a change from a crossover to a true phase transition when we left this region. This demonstrates that our theory is in principle able to reproduce the qualitative features of cold and dense nuclear matter. In the region of high temperatures and low densities, we investigated the chiral condensate and the nature of the critical endpoint of the Roberge-Weiss transition. In both cases LQCD calculations exist for comparison. While the behaviour of the chiral condensate can be modeled quantitatively, the second tricritical Roberge-Weiss endpoint located in [19] is absent in our theory. We discussed possible reasons for this, but can not offer a final conclusion so far. Chapter 6 demonstrated how, in the cold dense region, our effective theory can be solved analytically. We used this to accurately reproduce the numerical results from chapter 5. We furthermore derived analytic expressions for different thermodynamical observables, demonstrating how to leading order the binding energy can be described by a Yukawa potential. We finally showed how the analytic results can be resummed, potentially extending their range of validity beyond the range of the original effective theory. Future research perspectives lie in the possibility to systematically improve the theory. Using the methods described in chapter 4, higher orders can be derived in order to extend the convergence region. This is much simplified in the limit of low temperature and high density presented in chapter 4, which simultaneously is the most interesting parameter region due to the lack of LQCD simulations. Together with the resummation scheme from chapter 6 it should be possible to extend the theory far beyond the parameter range presented here. This will allow for better continuum extrapolations and the use of lighter quarks. Simultaneously the use of lighter quarks will mean that it is no longer possible to use the pure gauge beta function in order to set the scale, so those will have to come from LQCD simulations including fermions.