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Sub-subleading Soft Graviton Theorem in Generic Theories of Quantum Gravity

Alok Laddha, Ashoke Sen

TL;DR

The paper derives a sub-subleading soft graviton theorem for generic quantum gravity theories, showing the amplitude with one soft graviton decomposes into a universal part depending only on the hard amplitude and a non-universal part tied to the theory’s two- and three-point data. The result is established by covariantizing the 1PI effective action in a soft-graviton background, and it holds for tree amplitudes in any dimension and loop amplitudes in $D\ge5$, consistent with known QFT and string-theory results. Consistency checks show the universal structure depends only on on-shell data, and the non-universal piece is fixed by on-shell three-point amplitudes and higher-derivative couplings. The paper includes detailed comparisons with Einstein–Maxwell theory, fermions, higher-derivative four-dimensional theories, and tree-level string theory, addressing IR/collinear divergences and confirming the theorem across contexts.

Abstract

We analyze scattering amplitudes with one soft external graviton and arbitrary number of other finite energy external states carrying arbitrary mass and spin to sub-subleading order in the momentum of the soft graviton. Our result can be expressed as the sum of a universal part that depends only on the amplitude without the soft graviton and not the other details of the theory and a non-universal part that depends on the amplitude without the soft graviton, and the two and three point functions of the theory. For tree amplitudes our results are valid in all space-time dimensions while for loop amplitudes, infrared divergences force us to restrict our analysis to space time dimensions five or more. With this restriction the results are valid to all orders in perturbation theory. Our results agree with known results in quantum field theories and string theory.

Sub-subleading Soft Graviton Theorem in Generic Theories of Quantum Gravity

TL;DR

The paper derives a sub-subleading soft graviton theorem for generic quantum gravity theories, showing the amplitude with one soft graviton decomposes into a universal part depending only on the hard amplitude and a non-universal part tied to the theory’s two- and three-point data. The result is established by covariantizing the 1PI effective action in a soft-graviton background, and it holds for tree amplitudes in any dimension and loop amplitudes in , consistent with known QFT and string-theory results. Consistency checks show the universal structure depends only on on-shell data, and the non-universal piece is fixed by on-shell three-point amplitudes and higher-derivative couplings. The paper includes detailed comparisons with Einstein–Maxwell theory, fermions, higher-derivative four-dimensional theories, and tree-level string theory, addressing IR/collinear divergences and confirming the theorem across contexts.

Abstract

We analyze scattering amplitudes with one soft external graviton and arbitrary number of other finite energy external states carrying arbitrary mass and spin to sub-subleading order in the momentum of the soft graviton. Our result can be expressed as the sum of a universal part that depends only on the amplitude without the soft graviton and not the other details of the theory and a non-universal part that depends on the amplitude without the soft graviton, and the two and three point functions of the theory. For tree amplitudes our results are valid in all space-time dimensions while for loop amplitudes, infrared divergences force us to restrict our analysis to space time dimensions five or more. With this restriction the results are valid to all orders in perturbation theory. Our results agree with known results in quantum field theories and string theory.

Paper Structure

This paper contains 17 sections, 152 equations, 7 figures.

Table of Contents

  1. Introduction and summary
  2. Sub-subleading soft graviton theorem
  3. Evaluation of Fig. \ref{['f2']}
  4. Evaluation of Fig. \ref{['f1']}
  5. Final result
  6. Consistency check
  7. Comparison with tree level results for massless fields
  8. Einstein-Maxwell theory
  9. Fermions with minimal coupling to gravity
  10. Four dimensional quantum field theories with higher derivative corrections
  11. Comparison with results from tree level string theory
  12. Two tachyon two graviton amplitude
  13. Scattering of gravitons and dilatons
  14. Amplitude for two tachyons, one graviton and one massive particle
  15. Infrared divergences
  16. ...and 2 more sections

Figures (7)

  • Figure 1: The leading contribution.
  • Figure 2: The subleading contribution.
  • Figure 3: The amplitude without the soft graviton.
  • Figure 4: Possible sources of correction to the sub-subleading soft graviton theorem for two graviton, two tachyon scattering in string theory. $S$ denotes the soft graviton, $T$ denotes external tachyon, $H$ denotes finite energy external or internal graviton and $\phi$ denotes a finite energy internal dilaton. These represent additional contribution to Fig. \ref{['f1']} besides the one arising from the three point vertex (\ref{['e4.8n']}) representing minimal coupling.
  • Figure 5: Potentially infrared divergent contribution to the amplitude.
  • ...and 2 more figures