Matching effective chiral Lagrangians with dimensional and lattice regularizations

Abstract

Abstract:We compute the free energy in the presence of a chemical potential coupled to a conserved charge in effective O(n) scalar field theory (without explicit symmetry breaking terms) to third order for asymmetric volumes in general d-dimensions, using dimensional (DR) and lattice regularizations. This yields relations between the 4-derivative couplings appearing in the effective actions for the two regularizations, which in turn allows us to translate results, e.g. the mass gap in a finite periodic box in d = 3 + 1 dimensions, from one regularization to the other. Consistency is found with a new direct computation of the mass gap using DR. For the case n = 4, d = 4 the model is the low-energy effective theory of QCD with N f = 2 massless quarks. The results can thus be used to obtain estimates of low energy constants in the effective chiral Lagrangian from measurements of the low energy observables, including the low lying spectrum of N f = 2 QCD in the δ-regime using lattice simulations, as proposed by Peter Hasenfratz, or from the susceptibility corresponding to the chemical potential used.

Finite volume mass gap and free energy of the $\mathrm{SU}(N)\times\mathrm{SU}(N)$ chiral sigma model

“…In particular we could convert the computation of the mass gap in a periodic box, by Niedermayer and Weiermann [7] using lattice regularization to a result involving parameters of the dimensionally regularized effective theory, and we verified this result by a direct computation [6] (which disagrees slightly with the previous computation [5]).…”

“…In the latter references the computation was done using lattice regularization. Here we will employ dimensional regularization as we did in [6]. The dynamical fields U (x) are now defined in a volume…”

“…where R(0) is dimensionally regularized [6]. So for the energy shift, first computed by Leutwyler [4],…”

“…Some relations between different conventions for the O(4) couplings to connect with those used in ref. [6] l i = l r i +…”

Casimir squared correction to the standard rotator Hamiltonian for the O(n) sigma-model in the delta-regime

“…I nm;D are 1-loop dimensionally regularized sums over momenta (cf eq. (3.44) of [7]) 2 . Finally W in (2.7) is an integral associated with a two-loop vacuum massless sunset diagram, which is discussed in Appendix B.…”