Summing planar diagrams by an integrable bootstrap

Abstract

Correlation functions of matrix-valued fields are not generally known for massive renormalized field theories. We find the large-N limit of form factors of the (1 + 1)-dimensional sigma model with SU(N) × SU(N) symmetry. These form factors give a correction to the free-field approximation for the N = ∞ Wightman function. The method is a combination of the 1/N-expansion of the S-matrix and Smirnov's form-factor axioms. We expand the renormalized field in terms of a free massive Bosonic field as N → ∞.

Nontrivial thermodynamics in ’t Hooft’s large-Nlimit

“…It has been shown that the form factors of operators are not trivial at large N [3][4] [5]. As we will see, the computation of thermal expectation values can be done by summing over form factors.…”

Correlation functions of theSU(∞)principal chiral model

“…In Section III, we build on earlier results [4], [5], [6] to find all the form factors of the current operator. We use these to write down an expression for the vacuum expectation value of two currents in Section IV.…”

“…Some of the conventional wisdom concerning the 't Hooft limit can be tested. For example, the operators defining the Zamolodchikov-Faddeev algebra have been identified with a free Gaussian master field [4], from which the scaling field and other operators can be constructed.…”

“…This paper is not an introduction to the form-factor program at large N , but should be accessible to readers who understand the main points in References [4], [5] and [6].…”

“…Recently, the integrable bootstrap method was used to calculate all the form factors of the renormalized-field operator Φ(x) at large N [4], [5]. This was used to find an exact expression for the Wightman function, i.e., the non-time-ordered vacuum expectation value of two renormalized field operators…”