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

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.

6 pages, 3 figures

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• From QCD Strings to WZW
• 3D Yang-Mills Glueballs vs Closed Effective Strings