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Soft pion theorem, asymptotic symmetry and new memory effect

Yuta Hamada, Sotaro Sugishita

TL;DR

The paper extends the infrared triangle to a theory with a massless Nambu-Goldstone pion arising from spontaneous axial U(1) breaking, showing that the soft pion theorem can be recast as a Ward-Takahashi identity for an infinite set of asymptotic charges. By analyzing massless and massive field asymptotics at null and timelike infinity, the authors define soft and hard charges whose conservation encodes a pion memory effect in the 1/r^2 component of the pion field. They provide both a quantum (Ward-Takahashi) and classical (Green’s function) derivation of the memory relation, establishing a triangular relation between soft theorems, asymptotic symmetries, and memory for NG bosons. The work suggests a broader applicability of asymptotic symmetries beyond gauge theories and sets the stage for future investigations into sub-subleading soft theorems, loop effects, and potential connections to quantum gravity. Overall, it demonstrates that memory-like observables can arise from spontaneously broken global symmetries in a well-behaved, infrared-finite setting.

Abstract

It is known that soft photon and graviton theorems can be regarded as the Ward-Takahashi identities of asymptotic symmetries. In this paper, we consider soft theorem for pions, i.e., Nambu-Goldstone bosons associated with a spontaneously broken axial symmetry. The soft pion theorem is written as the Ward-Takahashi identities of the $S$-matrix under asymptotic transformations. We investigate the asymptotic dynamics, and find that the conservation of charges generating the asymptotic transformations can be interpreted as a pion memory effect.

Soft pion theorem, asymptotic symmetry and new memory effect

TL;DR

The paper extends the infrared triangle to a theory with a massless Nambu-Goldstone pion arising from spontaneous axial U(1) breaking, showing that the soft pion theorem can be recast as a Ward-Takahashi identity for an infinite set of asymptotic charges. By analyzing massless and massive field asymptotics at null and timelike infinity, the authors define soft and hard charges whose conservation encodes a pion memory effect in the 1/r^2 component of the pion field. They provide both a quantum (Ward-Takahashi) and classical (Green’s function) derivation of the memory relation, establishing a triangular relation between soft theorems, asymptotic symmetries, and memory for NG bosons. The work suggests a broader applicability of asymptotic symmetries beyond gauge theories and sets the stage for future investigations into sub-subleading soft theorems, loop effects, and potential connections to quantum gravity. Overall, it demonstrates that memory-like observables can arise from spontaneously broken global symmetries in a well-behaved, infrared-finite setting.

Abstract

It is known that soft photon and graviton theorems can be regarded as the Ward-Takahashi identities of asymptotic symmetries. In this paper, we consider soft theorem for pions, i.e., Nambu-Goldstone bosons associated with a spontaneously broken axial symmetry. The soft pion theorem is written as the Ward-Takahashi identities of the -matrix under asymptotic transformations. We investigate the asymptotic dynamics, and find that the conservation of charges generating the asymptotic transformations can be interpreted as a pion memory effect.

Paper Structure

This paper contains 15 sections, 94 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Soft theorem in $U(1)_A$ model
  3. Model
  4. Soft pion theorem
  5. Soft pion theorem as the Ward-Takahashi identity
  6. Ward-Takahashi identity for spontaneously broken symmetry
  7. Asymptotic symmetry
  8. Asymptotic behaviors of massless fields
  9. Asymptotic behaviors of massive fields
  10. Charge conservation as memory effect
  11. Expression of charge in terms of the asymptotic fields
  12. Classical derivation of the memory effect
  13. Summary and discussions
  14. Convention and formulas
  15. Coordinate systems

Figures (2)

  • Figure 1: Triangular relation among soft theorem, asymptotic symmetry and memory effect.
  • Figure 2: A part of a diagram relevant to the soft limit. The circle represents the other part of the diagram. The momentum $p^\mu-\omega q^\mu$ approaches to on-shell in the soft limit $\omega \to 0$.