Holographic and exact RG beta function computations of the Sine-Gordon model[HBNI Th168]

Abstract

The exact renormalization group is used to study the RG flow of quantities in field theories. The basic idea is to write an evolution operator for the RG flow and evaluate it in perturbation theory. This is easier than directly solving the differential equation. This is illustrated by reproducing known results in the four dimensional φ 4 field theory and the two dimensional Sine-Gordon theory. It is shown that the calculation of beta function is somewhat simplified. The technique is also used to calculate the c-function in two dimensional Sine-Gordon theory. This agrees with other prescriptions for calculating c-functions in the literature. If one extrapolates the connection between central charge of a CFT and entanglement entropy in two dimensions, to the c-function of the perturbed CFT, then one gets a value for the en- tanglement entropy in Sine-Gordon theory that is in exact agreement with earlier calculations. Next, the Sine Gordon theory is generalized to include many scalar fields and several cosine terms. This is similar to the world sheet description of a string propagating in a tachyon background. This model is studied as a (boundary) 2d euclidean field theory and also using an AdS 3 holographic bulk dual. The beta functions for the cosine vertex of this modified theory are first computed in the boundary using techniques based on the exact RG. The beta functions are also computed holographically using position space and momentum space techniques. The results are in agreement with each other and with earlier computations. The cosine perturbation is of the form cos bX. Due to wave function renormalisation the parameter b, and thus the dimension of the cosine, get renormalised. The beta function for this parameter is thus directly related to the anomalous dimension of the X field. We compute this beta function in position space. They match with the earlier results in [22].