Beta function without UV divergences

TL;DR

This paper develops a regulator-free, functorial formulation of two-dimensional quantum field theory to define and compute the beta function through conformal variations of the partition function. By treating local observables as derived from the theory space and studying marginal perturbations via multiple deformations, the authors show that a conformal anomaly emerges as observables acquire logarithmic scaling dimensions. The main result is the second-order beta function, which reproduces the known one-loop form in two dimensions and is expressed in terms of OPE structure constants, highlighting the geometric meaning of renormalization-group data within the functorial framework. The work provides a conceptually new, algebraically flavored view of RG flow, with potential extensions to higher orders and connections to A∞-algebraic structures and contact-term phenomena.

Abstract

In this paper, we construct the beta function in the functorial formulation of two-dimensional quantum field theories (FQFT). A key feature of this approach is the absence of ultraviolet divergences. We show that, nevertheless, in the FQFT perturbation theory, the local observables of deformed theories acquire logarithmic dimension, leading to a conformal anomaly. The beta function arises in the functorial approach as an infinitesimal transformation of the partition function under the variation of the metric's conformal factor, without ultraviolet divergences, UV cutoff, or the traditional renormalization procedure.

Figures (1)

• Figure 1: The yellow disk with holes $X$ shown in (a) is cut along the dotted line $\Gamma$, resulting in a union of the blue disk $X_{1}$ with a new hole and the red disk $X_{2}$, shown in (b).