QCD: Low temperature expansion and finite size effects (original) (raw)

1988, Nuclear Physics B - Proceedings Supplements

Measurements of QCD correlation functions on a lattice are performed at finite volume and at nonzero temperature. It is not a simple matter to extrapolate the numerical results to infinite volume and to zero temperature, because the correlation lengths associated with the lightest bound states (~, K,n) are large. The reason why these correlation lengths are large was identified long agol: the pseudoscalar mesons are the Goldstone bosons generated by the spontaneous breakdown of chiral symmetry. If the quark masses were exactly zero, the pseudoscalar mesons would be massless and their correlation lengths would be infinite. In the real world, three of the quark masses (mu,md,ms) are small -this is why the pseudoscalars w,K,n are the lightest bound states of the theory. In fact, chiral symmetry not only controls the masses of the Goldstone bosons, but it also determines their mutual interactions at low energy. In a recent series of papers 2-5, we have shown that this information suffices to calculate the leading terms in the low temperature expansion of the theory, thus converting the familiar low energy theorems for pion scattering 6 into temperature theorems for the partition function. Furthermore, we have shown that chiral symmetry also controls the leading finite size effects. In this talk, I briefly review this work, including some unpublished results. In the first part, I consider the infinite volume limit and set the quark masses equal to zero. As is well-known, the Hamiltonian of mass-less QCD is invariant under the group SU(N)RXSU(N)L, where N is the number of flavours. I assume that the standard picture is correct: AI) The ground state spontaneously breaks chiral symmetry down to SU(N)R+ L with a nonzero quark condensate <oI~@Io>. A2) There are N2-1 massless Goldstone bosons ("pions") with a nonzero coupling to the axial current <OIA r~> o ip F (1) (F is the pion decay constant of the massless theory; in the real world, F = 93 MeV). I assume that the spectrum does not contain any other massless particles. It is easy to see why, under these circumstances, chiral symmetry governs the behaviour of the partition function at low temperatures. (i) Multipion states dominate the thermal average. States containing massive particles, such as the nucleon or the o-meson are suppressed exponentially. (ii) The typical pion energies are small, of order E ~ T. Chiral symmetry implies that the interaction among low energy pions is weak: the connected part of the scattering amplitude associated with the collision of any number of pions is of order E 2 6. The interaction can therefore be treated as a perturbation.