Effective theories of finite volume QCD

Finite volume QCD close to the chiral limit cannot be described by chiral Perturbation Theory using the usual p-expansion when the correlation length of pions becomes larger than the size of the box. An alternative approach to this problem was proposed by Gasser and Leutwyler in 1987, it is referred to as $\epsilon$-expansion. In 1993 Shuryak and Verbaarschot conjectured that the spectral properties of the leading order of this alternative expansion were shared with a simpler theory called chiral Random Matrix Theory. In the following years this equivalence was widely used. In the first part of this work we prove this equivalence for any value of masses and for both zero and non-zero chemical potential. In particular the equivalence of all the low energy spectral properties imply the equivalence of the individual eigenvalue distributions, which are particularly useful to determine low energy constants from Lattice QCD with chiral fermions. In the second part, working in $\epsilon$-expansion with an accuracy up to the next to the leading order, we determine the volume and mass dependence of scalar and pseudoscalar two-point functions in $N_f$-flavour QCD, in the presence of an isospin chemical potential. Thanks to the non-vanishing chemical potential these correlation functions show a dependence on both chiral condensate and pion decay constant already at leading order.