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BMS Supertranslation Symmetry Implies Faddeev-Kulish Amplitudes

Sangmin Choi, Ratindranath Akhoury

TL;DR

This work shows that infrared-finite scattering amplitudes in perturbative gravity emerge when one imposes conservation of the BMS supertranslation charges. By constructing eigenstates of the soft graviton charge and analyzing how graviton clouds dress external particles, the authors prove that any charge-conserving amplitude is equivalent to a Faddeev-Kulish amplitude, effectively moving soft clouds across the scattering operator without changing IR properties. They demonstrate clouds weakly commute with the S-matrix and establish the precise equality M = M_c = M_FK, linking asymptotic symmetries to IR finiteness. The analysis is applied to decoherence studies, arguing that while a naive tracing over soft degrees can yield decoherence, the properly dressed FK framework yields IR-finite density matrices with well-defined diagonal elements. The results suggest a deep connection between asymptotic symmetries and the infrared structure of gravity, with potential extensions to QED and QCD.

Abstract

We show explicitly that, among the scattering amplitudes constructed from eigenstates of the BMS supertranslation charge, the ones that conserve this charge, are equal to those constructed from Faddeev-Kulish states. Thus, Faddeev-Kulish states naturally arise as a consequence of the asymptotic symmetries of perturbative gravity and all charge conserving amplitudes are infrared finite. In the process we show an important feature of the Faddeev-Kulish clouds dressing the external hard particles: these clouds can be moved from the incoming states to the outgoing ones, and vice-versa, without changing the infrared finiteness properties of S matrix elements. We also apply our discussion to the problem of the decoherence of momentum configurations of hard particles due to soft boson effects.

BMS Supertranslation Symmetry Implies Faddeev-Kulish Amplitudes

TL;DR

This work shows that infrared-finite scattering amplitudes in perturbative gravity emerge when one imposes conservation of the BMS supertranslation charges. By constructing eigenstates of the soft graviton charge and analyzing how graviton clouds dress external particles, the authors prove that any charge-conserving amplitude is equivalent to a Faddeev-Kulish amplitude, effectively moving soft clouds across the scattering operator without changing IR properties. They demonstrate clouds weakly commute with the S-matrix and establish the precise equality M = M_c = M_FK, linking asymptotic symmetries to IR finiteness. The analysis is applied to decoherence studies, arguing that while a naive tracing over soft degrees can yield decoherence, the properly dressed FK framework yields IR-finite density matrices with well-defined diagonal elements. The results suggest a deep connection between asymptotic symmetries and the infrared structure of gravity, with potential extensions to QED and QCD.

Abstract

We show explicitly that, among the scattering amplitudes constructed from eigenstates of the BMS supertranslation charge, the ones that conserve this charge, are equal to those constructed from Faddeev-Kulish states. Thus, Faddeev-Kulish states naturally arise as a consequence of the asymptotic symmetries of perturbative gravity and all charge conserving amplitudes are infrared finite. In the process we show an important feature of the Faddeev-Kulish clouds dressing the external hard particles: these clouds can be moved from the incoming states to the outgoing ones, and vice-versa, without changing the infrared finiteness properties of S matrix elements. We also apply our discussion to the problem of the decoherence of momentum configurations of hard particles due to soft boson effects.

Paper Structure

This paper contains 7 sections, 86 equations, 3 figures.

Table of Contents

  1. Introduction
  2. BMS charge and eigenstates
  3. Relation to Faddeev-Kulish amplitudes
  4. Moving the graviton clouds
  5. Equality of the amplitudes
  6. Soft gravitons and decoherence of momentum configurations of hard matter particles
  7. Discussion

Figures (3)

  • Figure 1: Diagrams (a)-(d) represent processes with Faddeev-Kulish asymptotic states. Diagrams (e)-(h) represent the same processes with the incoming cloud moved to the outgoing state. Notice the "wrong" sign $+R_f$ compared to a normal outgoing cloud with $-R_f$.
  • Figure 2: An example of an incoming cloud being moved to the out-state. Each boson connecting this cloud to an external propagator obtains two factors of $(-1)$, one from the soft factor and the other from the "wrong" sign of $R_f$. These two factors cancel, and thus the overall amplitude is unaffected by such a change.
  • Figure 3: Diagrams contributing to an amplitude with external soft boson. The first two diagrams cancel the last two diagrams, and only the diagram in the middle remains, which is of zeroth order in the soft momentum.