All order I.R. finite expansion for short distance behavior of massless theories perturbed by a relevant operator

Abstract

We consider here renormalizable theories without relevant couplings and present an I.R. consistent technique to study corrections to short distance behavior (Wilson O.P.E. coefficients) due to a relevant perturbation. Our method is the result of a complete reformulation of recent works on the field, and is characterized by a more orthodox treatment of U.V. divergences that allows for simpler formulae and consequently an explicit all order (regularization invariant) I.R. finitess proof. Underlying hypotheses are discussed in detail and found to be satisfied in conformal theories that constitute a natural field of application of this approach.

Operator product expansion coefficients of the 3D Ising model with a trapping potential

“…where the O i are a complete set of operators in CFT, the < .. > h is the expectation value over the action perturbed with a small relevant parameter h. The Wilson coefficients C k ij (r, h) can be expanded as a Taylor series with respect to the perturbation parameter in a regular way without infrared divergences, so that the coefficients of the expansion are the derivatives calculated at h = 0 [20]. In particular we note that the OPE coefficients can be extracted from the first correction term of the critical correlator.…”

“…The method exploits the short distance expansion of the perturbed twopoint functions, written as a power series of the Wilson coefficients derivatives and the one-point functions. This procedure was already known [20]- [23], but it was never applied to the three dimensional case. In [19] the practical feasibility of the method was shown and in particular it was used to extract the OPE coefficients of the 3D Ising model, setting the magnetic field as perturbation, and the following results were found: C σ σ = 1.07(3) and C = 1.45 (30).…”

Isomorphism of critical and off-critical operator spaces in two-dimensional quantum field theory

“…Perturbative considerations [2,3,4] suggest that, up to symmetry breaking effects, such a structure should survive a perturbation of the fixed point leading to a massive theory. The existence in two dimensions of massive integrable theories provides the chance of testing nonperturbatively this conjecture, but also poses a problem which is intriguing and challenging at the same time.…”

On correlation functions in the perturbed minimal models

“…In what follows, we concentrate on the minimal models of CFT M p,p ′ (with coprime integers 1 < p < p ′ ) perturbed by the field Φ 13 . These massive field theories having an infinite number of conservation laws are integrable [4].…”