The QCD $\beta$-function On The String Worldsheet

TL;DR

This paper explores a novel connection between the Yang--Mills $eta$-function and string worldsheet anomalies, revealing a deep link between gauge theory and string theory in confining strings.

Contribution

It establishes a direct relation between the Yang--Mills $eta$-function and the Polchinski--Strominger amplitude on the string worldsheet, extending to theories with adjoint quarks.

Findings

Relation between $eta$-function and worldsheet amplitude

Connection between scale and Weyl anomalies

Extension to theories with adjoint quarks

Abstract

We consider confining strings in pure gluodynamics and its extensions with adjoint (s)quarks. We argue that there is a direct map between the set of bulk fields and the worldsheet degrees of freedom. This suggests a close link between the worldsheet $S$-matrix and parton scattering amplitudes. We report an amusing relation between the Polchinski--Strominger amplitude responsible for the breakdown of integrability on the string worldsheet and the Yang--Mills $\beta$-function \[ b_0={D_{cr}-D_{ph}\over 6}\;. \] Here $b_0=11/3$ is the one-loop $\beta$-function coefficient in the pure Yang--Mills theory, $D_{cr}=26$ is the critical dimension of bosonic strings and $D_{ph}=4$ is the dimensionality of the physical space-time we live in. A natural extension of this relation continues to hold in the presence of adjoint (s)quarks, connecting two of the most celebrated anomalies---the scale anomaly in quantum chromodynamics (QCD) and the Weyl anomaly in string theory.