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

Abstract

We compute the free energy in the presence of a chemical potential coupled to a conserved charge in the effective SU(N ) × SU(N ) scalar field theory to third order for asymmetric volumes in general d-dimensions, using dimensional regularization (DR). We also compute the mass gap in a finite box with periodic boundary conditions.

On the rotator Hamiltonian for the SU$(N)\times\,$SU$(N)$ sigma-model in the delta-regime

“…where = L t /L with no power-like corrections! This is in agreement with the O(n) rotator result (2.3) of [3] at n = 4 and with the χPT for SU(N ) × SU(N ) at N = 2 [2].…”

“…which is in agreement with eq. (4.48) of [2] obtained by χPT to NNL order for general N . 5 Here we assume N ≥ 3…”

“…The coordinates in the two descriptions are: U ∈ SU (2), where 2) , (σ k the Pauli matrices) or equivalently, in the O(4) picture s = (s 0 , s 1 , s 2 , s 3 ) ∈ S 3 , (s 2 = 1) . The wave functions have the form ψ(U ) or ψ(s) .…”

“…The set of four wave functions ψ(U ) ∈ U 11 , U 12 , U 21 , U 22 belong to the representation with j L = j R = 1/2. In general, the Hilbert space of the system splits into irreps of SU(2)×SU (2) for which j L = j R = j 1 .…”

“…In a previous paper [2] we pointed out that by comparing the already obtained NNLO results for the isospin susceptibility from χPT at large ≡ L t /L with that computed from the standard rotator, one can establish, under reasonable assumptions, that at 3-loops there is a correction to the rotator Hamiltonian proportional to the square of the Casimir operator, with a proportionality constant determined by the NNLO low energy constants (LEC's) of χPT.…”