Volume 38-3  博士論文

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

菅本 晶夫 (お茶の水女子大学)

素粒子論研究・電子版 Vol. 38 (2022) No. 3

2022年9月30日受理

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概要

This is the English translation of the Doctor thesis submitted to the University of Tokyo in March 1978. Its Japanese version was published in Soryushiron Kenkyu 60-2 (1979-11) pp. 47–117. An appendix is newly added, which is the excerption of a relevant mathematical part of a report submitted to Prof. Mikio Sato in August 1976. In the thesis, first, the cancellation of infrared divergences is reviewed. The cancellation occurs between a virtual process with soft photon corrections and the other process with real soft photon emissions. The factorization of infrared divergences is shown by using the eikonal approximation, or by the (renormalization group like) differential equation controlling the infrared divergences. Next, two examples in QCD, the fermion (quark)-fermion (quark) scattering and the fermion (quark) gauge-boson (gluon) scattering are examined at one loop, from which the importance of Ward-Takahashi identities becomes manifest for the cancellation to occur. (The identities represent the group properties.) After these preparations, the factorization of infrared divergences in QCD is proved at all orders in the perturbation theory, by full usage of the Ward-Takahashi identities. The identites are also proved at all orders. In these proofs, the axial gauge condition is used, which simplifies the derivation of the Ward-Takahashi identities as well as the proof of the unitarity. The cancellation of infrared divergences in QCD occurs among “the gauge invariant set of graphs”, if the quantum numbers of color are averaged and summed over the initial and final states, respectively. In this way, the low energy theorem by F. E. Low, for emission of one or two soft gauge bosons, are proved at all orders in the perturbation expansion. By joining the two emitted soft gauge bosons, the differential equation controlling the infrared divergences in QCD can be derived. In QCD, the coupling constant renormalization introduces the other infrared divergence, which is governed by the beta function β(g), or by the ultraviolet divergences in the pure Yang-Mills theory. In the Appendix, the cancellation of infrared divergences is examined mathematically in terms of the singular spectrum dS.S of Mikio Sato’s microfunction. In the Epilogue (2022), a motivation for recently translating the thesis in English is briefly stated.

キーワード

Infrared divergence、Low energy theorem、Non-Abelian gauge theory