QCD β -function on the string worldsheet

Sergei Dubovsky

doi:10.1103/physrevd.98.114025

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 $\ensuremath{\beta}$-function ${b}_{0}=\frac{{D}_{cr}\ensuremath{-}{D}_{ph}}{6}.$ Here ${b}_{0}=11/3$ is the one-loop $\ensuremath{\beta}$-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.

Highlights

11=3 is the one-loop β-function coefficient in the pure Yang–Mills theory, Dcr 1⁄4 26 is the critical dimension of bosonic strings and Dph 1⁄4 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
Our description of strong interactions is embarrassingly incomplete without understanding of strings responsible for quark confinement

Summary

Sergei Dubovsky

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. The axion is a pseudoscalar both with respect to the Oð2Þ group of rotations in the transverse plane, and with respect to the two-dimensional Poincaresymmetry ISOð1; 1Þ along the worldsheet Both at D 1⁄4 3 and D 1⁄4 4 this is a matter content of an integrable theory enjoying the nonlinearly realized target space Poincaresymmetry ISOð1; D − 1Þ. The corresponding integrable phase shift coincides with the Dray–’t Hooft [18] gravitational shock wave phase shift e2iδðsÞ 1⁄4 eil2s s=4: ð2Þ This phase shift describes integrable scattering on the worldsheet of critical (super)strings [19].

SERGEI DUBOVSKY

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