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Soft Theorems from Higher Symmetries

Jonah Berean-Dutcher, Maria Derda, Julio Parra-Martinez

Abstract

We describe the connection between spontaneously-broken higher symmetries and soft theorems for scattering amplitudes of their associated Nambu-Goldstone bosons. Our main result is a new sub-leading double soft pion theorem in theories with a spontaneously-broken continuous 2-group global symmetry, which intertwines amplitudes with different numbers of pions and photons. We also provide a novel derivation of the leading soft photon theorem from the Ward identity of an emergent 1-form global symmetry in effective field theories where antiparticles are integrated out. Our derivations of these soft theorems use the algebra of spacetime currents and do not rely on asymptotic symmetries or diagrammatic arguments.

Soft Theorems from Higher Symmetries

Abstract

We describe the connection between spontaneously-broken higher symmetries and soft theorems for scattering amplitudes of their associated Nambu-Goldstone bosons. Our main result is a new sub-leading double soft pion theorem in theories with a spontaneously-broken continuous 2-group global symmetry, which intertwines amplitudes with different numbers of pions and photons. We also provide a novel derivation of the leading soft photon theorem from the Ward identity of an emergent 1-form global symmetry in effective field theories where antiparticles are integrated out. Our derivations of these soft theorems use the algebra of spacetime currents and do not rely on asymptotic symmetries or diagrammatic arguments.
Paper Structure (16 sections, 109 equations, 2 figures)

This paper contains 16 sections, 109 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Review: Soft pions from ordinary symmetry
  3. Pion Adler zero
  4. Double soft pion theorem
  5. Soft photons from higher-form symmetry
  6. Photons as Nambu-Goldstone bosons
  7. Soft photon theorems
  8. Emergent higher-form symmetry in heavy-particle EFT
  9. New soft theorem from higher-group symmetry
  10. 2-group double soft pion theorem
  11. Proof
  12. Examples
  13. Conclusions
  14. Select form factor soft limits
  15. Soft vector current
  16. ...and 1 more sections

Figures (2)

  • Figure 1: Illustration of the decomposition of a momentum $\ell^\mu=p^\mu+k^\mu$, as a large component on the mass shell, $p^\mu$, and a small component $k^\mu \ll p^\mu$. The field $\varphi_v^+$ describes small fluctuations around the mass shell in the shaded region, whereas all modes outside this region are integrated out in the effective theory.
  • Figure 2: Graphical representation of the singular contributions comprising the 2-group soft factor $S_\kappa^{(1)}$ in \ref{['eq:double-soft-theorem-2-group-schematic']} and \ref{['eq:double-soft-next-to-leading-order-2-group']}.