The low-energy effective theory of QCD at small quark masses in a finite volume

At low energies the theory of quantum chromodynamics (QCD) can be described effectively in terms of the lightest particles of the theory, the pions. This approximation is valid for temperatures well below the mass difference of the pions to the next heavier particles. We study the low-energy effective theory at very small quark masses in a finite volume V. The corresponding perturbative expansion in V^{-1/2} is called epsilon expansion. At each order of this expansion a finite number of low-energy constants completely determine the effective theory. These low-energy constants are of great phenomenological importance. In the leading order of the epsilon expansion, called epsilon regime, the theory becomes zero-dimensional and is therefore described by random matrix theory (RMT). The dimensionless quantities of RMT are mapped to dimensionful quantities of the low-energy effective theory using the leading-order low-energy constants Sigma and F. In this way Sigma and F can be obtained from lattice QCD simulations in the epsilon regime by a fit to RMT predictions. For typical volumes of state-of-the-art lattice QCD simulations, finite-volume corrections to the RMT prediction cannot be neglected. These corrections can be calculated in higher orders of the epsilon expansion. We calculate the finite-volume corrections to Sigma and F at next-to-next-to-leading order in the epsilon expansion. We also discuss non-universal modifications of the theory due to the finite volume. These results are then applied to lattice QCD simulations, and we extract Sigma and F from eigenvalue correlation functions of the Dirac operator. As a side result, we provide a proof of equivalence between the parametrization of the partially quenched low-energy effective theory without singlet particle and that of the super-Riemannian manifold used earlier in the literature. Furthermore, we calculate a special version of the massless sunset diagram at finite volume without constant mode which was not known before. Apart from the universal regime of QCD, random matrix models can be used as schematic models that describe certain features of QCD such as the chiral phase transition. These schematic models are defined at fixed topological charge instead of fixed vacuum angle. Therefore special care has to be taken when different topological sectors are combined. We classify different schematic random matrix models in terms of the topological domain of Dirac eigenvalues, i.e., the part of eigenvalues that is affected by topology. If the topological domain extends beyond the microscopic eigenvalues, additional normalization factors need to be included to allow for finite topological fluctuations. This is important since the mass of the pseudoscalar singlet particle eta' is related to topological fluctuations, and the normalization factors thus solve the corresponding U(1)_A problem