String techniques for the calculation of renormalization constants in field theory

TL;DR

This work shows that off-shell continuations of open bosonic string amplitudes yield renormalization constants in Yang–Mills theory that reproduce the background-field method results and Ward identities, by mapping string moduli-space regions to Feynman-diagram classes. It develops three complementary approaches: extracting UV divergences from field-theory limits of M-gluon amplitudes, a regularized Green function method that isolates one-particle-irreducible contributions, and a generating-functional partition function for strings in a non-abelian background field that encodes multiloop effective actions. A single string integral Z(d) governs the one-loop renormalization and extends to higher loops through the background-field partition function, providing a coherent, gauge-invariant framework. The results illustrate how string theory organizes gauge-theory perturbation theory and offer a practical pathway toward multiloop calculations within a consistent off-shell prescription.

Abstract

We describe a set of methods to calculate gauge theory renormalization constants from string theory, all based on a consistent prescription to continue off shell open bosonic string amplitudes. We prove the consistency of our prescription by explicitly evaluating the renormalizations of the two, three and four-gluon amplitudes, and showing that they obey the appropriate Ward identities. The field theory limit thus performed corresponds to the background field method in Feynman gauge. We identify precisely the regions in string moduli space that correspond to different classes of Feynman diagrams, and in particular we show how to isolate contributions to the effective action. Ultraviolet divergent terms are then encoded in a single string integral over the modular parameter $τ$. Finally, we derive a multiloop expression for the effective action by computing the partition function of an open bosonic string interacting with an external non-abelian background gauge field.

Figures (11)

• Figure 1: Representative diagram for the type $I$ region of the three-gluon amplitude.
• Figure 2: Representative diagram for the type $II$ region $\hat{\nu}_{32}=O(\tau^{-1})$ of the three-gluon amplitude.
• Figure 3: Representative diagram for the pinching region $\hat{\nu}_{32}=O(\tau^{-2})$ of the three-gluon amplitude.
• Figure 4: Representative diagram for the type $II$ region $1-\hat{\nu}_{3}=O(\tau^{-1})$ of the three-gluon amplitude.
• Figure 5: Representative diagram for the pinching region $\nu_{2}\to 0$ of the three-gluon amplitude.