Structure functions of the 2d non-linear sigma models

Abstract

We investigate structure functions in the 2-dimensional (asymptotically free) nonlinear O(n) σ-models using the non-perturbative S-matrix bootstrap program. In particular the exact small (Bjorken) x behavior is derived. Structure functions in the special case of the n = 3 model are accurately computed over the whole x range for −q 2 /M 2 < 10 5 , and some moments are compared with results from renormalized perturbation theory. Some results concerning the structure functions in the 1/n approximation are also presented.

On form factors in nested Bethe Ansatz systems

“…These equations are restrictive enough, so that supplied with a few additional assumptions (possibly depending on the operator in question); they uniquely determine the form factors. In massive field theories, it is sufficient to obtain explicit expressions for the form factors with a small number of particles, because they typically saturate the spectral series for the vacuum correlations even at small distances [41,42].…”

Bethe-Salpeter Wave Functions in Integrable Models

“…We now outline the information which can be gained from perturbative field theory; for any undefined notation we refer the reader to [20]. We start with the short distance expansion are coefficient functions (which can, in principle, be calculated in perturbation theory).…”

“…The constant occurring in the coefficient α cannot be calculated since we do not know the relative normalization of the operator τ ab (R) with respect to t ab whose normalization is fixed non-perturbatively by (3.28). We also do not know the numerical value of the non-perturbative constant D n for general n. However, we do know [20] D 3 = 4/π and this enables us to write for n = 3 (and, for simplicity, at zero energy):…”

Asymptotic factorization of n-particle SU(N) form factors

“…Remarkably, due to integrability, it is possible to obtain exact form factors of local operators [6][7][8][9]. 1 In the remarkable papers [19][20][21] Balog and Weisz define analogs of the structure functions in two-dimensional integrable quantum field theories. In particular, they consider form factors of the current operator (related to the structure-function) of the O(3) sigma model, which are accurately computed over the whole x range; in addition, the structure functions and some moments are compared with renormalized perturbation theory.…”