The exact mass-gaps of the principal chiral models

Abstract

An exact expression for the mass-gap, the ratio of the physical particle mass to the Λ-parameter, is found for the principal chiral sigma models associated to all the classical Lie algebras. The calculation is based on a comparison of the free-energy in the presence of a source coupling to a conserved charge of the theory computed in two ways: via the thermodynamic Bethe Ansatz from the exact scattering matrix and directly in perturbation theory. The calculation provides a non-trivial test of the form of the exact scattering matrix.

D-theory: field quantization by dimensional reduction of discrete variables

“…Similarly, using the exact mass gap of the 2-d Sp(N) and SO(N) chiral models [32] one obtains corresponding results for the correlation length. For the Sp(N) chiral spin models one gets…”

Exact results for integrable asymptotically-free field theories

“…and, from (20), each of the densities ǫ 1 and ǫ2 contributes an amount to the ground-state energy given by (9). We have therefore succeeded in reducing the problem to the relatively simple case which we already know how to handle.…”

“…First, in order to simplify the analysis of the TBA system we must choose Q so that the new ground-state consists of a restricted number of particle types. In the models considered in [4,5,6,7,8,9] it was possible to find a generator Q such that the new ground-state contained a single particle type. In the present theory, however, we know that the lowest energy states must come in degenerate supersymmetric multiplets, since supersymmetry commutes with the SU(n) invariance.…”

“…To minimize the amount of work involved, it is clearly desirable to try to choose Q so as to produce a classical term in the perturbative expression for the ground-state energy (1) because, as we have explained, the mass-gap can then be found from the TBA analysis in conjunction with a one-loop perturbative calculation. It can be shown [12] that such a choice of Q is possible for any bosonic or supersymmetric sigma model based on a symmetric space G/H, a result which unifies the treatment of various examples considered previously in [4,5,8,9,10,11]. For the purely fermionic Gross-Neveu models, however, it seems that generically the classical contribution vanishes, and so a three-loop calculation is necessary in order to find the mass gap [6,7].…”

Duality-invariant class of two-dimensional field theories

“…There is, however, a particular domain in parameter space, where for a wide 1 Examples include the O(N ) [1], the principal chiral [2] and the Gross-Neveu models [3], for which the complete S-matrix was found through the existence of higher-spin-conserved currents that lead to its factorization property. Building on work in [4], comparison between the S-matrix results and those obtained by perturbative techniques in the ultraviolet (UV) regime was made for the O(N ) σ-model [5], the Principal Chiral models for SU (N ) [6], SO(N ) and Sp(N ) [7] and the O(N ) Gross-Neveu model [8], finding perfect agreement. range of energies in the UV, the description is effectively perturbative with a UV-stable fixed point corresponding to the point where the gauge invariance develops.…”