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

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 $$b_0=\frac{D_{cr}-D_{ph}}{6}\mathrm{.}$$ Here $b_0=11\mathrm{/}3$ 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.