Infrared Divergence and Low Energy Theorem in Non-Abelian Gauge Theory

TL;DR

This paper reviews infrared divergence cancellation in QED, proves factorization in QCD at all orders using Ward-Takahashi identities, and establishes the low energy theorem for soft gluon emissions, highlighting the role of gauge invariance.

Contribution

It provides a comprehensive proof of infrared divergence cancellation and factorization in QCD at all orders, utilizing Ward-Takahashi identities and axial gauge, and extends Low's theorem to soft gluon emissions.

Findings

Infrared divergences cancel among gauge-invariant graphs in QCD.

Factorization of infrared divergences in QCD is proven at all orders.

Low energy theorem for soft gluon emission is established at all orders.

Abstract

In the thesis, first, the cancellation of infrared divergences in QED is reviewed. Next, two examples in QCD, the quark-quark scattering and the quark-gluon scattering are examined at one loop, from which the importance of Ward-Takahashi identities becomes manifest for the cancellation to occur. The factorization of infrared divergences in QCD is proved at all orders, by full usage of the Ward-Takahashi identities. In these proofs, the axial gauge condition is used. The cancellation of infrared divergences in QCD occurs among the gauge invariant set of graphs. In this way, the low energy theorem of F. E. Low, for emission of one or two soft gluons, are proved at all orders. From this, the differential equation controlling the infrared divergences in QCD is derived. In QCD, the coupling constant renormalization introduces the other infrared divergence, which is governed by the beta-function in the pure Yang-Mills theory. In the Epilogue 2022, a motivation for recently translating the thesis in English is briefly stated.