nLab soft graviton theorem

Idea

In perturbative quantum gravity, the soft graviton theorem (Weinberg 1965) re-formulates scattering amplitudes of a set of finite energy external particles with one or more low energy external gravitons, in terms of the amplitude without the low energy gravitons.

In the classical limit, there is a different manifestation of the same theorem (Saha, Sahoo & Sen 2019): here it determines the low frequency component of the gravitational wave-form produced during a scattering process in terms of the momenta and the spin of the incoming and outgoing objects, without any reference to the interactions responsible for the scattering.

References

Original reference:

• Steven Weinberg, Infrared photons and gravitons, Phys. Rev. 140, B516 (1965) (doi:10.1103/PhysRev.140.B516)

Review with focus on photons in electromagnetism:

• Noah Miller, From Noether’s Theorem to Bremsstrahlung: a pedagogical introduction to large gauge transformations and classical soft theorems (arXiv:2112.05289)

Exposition:

• Ashoke Sen: Gravitational Wave Tails from Soft Theorem: A Short Review [arXiv:2408.08851]

Derivation of sub-leading soft graviton theorem (the Sahoo-Sen soft theorem) from the analysis of gravitational loop-amplitudes:

• Biswajit Sahoo, Ashoke Sen: Classical and Quantum Results on Logarithmic Terms in the Soft Theorem in Four Dimensions, J. High Energ. Phys. 2019 86 (2019) [doi:10.1007/JHEP02(2019)086, arXiv:1808.03288]

• Hare Krishna, Biswajit Sahoo: Universality of loop corrected soft theorems in 4d, J. High Energ. Phys. 2023 233 (2023) [doi:10.1007/JHEP11(2023)233, arXiv:2308.16807]

• Martin Beneke, Patrick Hager, Robert Szafron, Soft-Collinear Gravity and Soft Theorems, in section Effective Quantum Gravity of Handbook of Quantum Gravity, Springer (2023) [arXiv:2210.09336]

Discussion of application to gravitational waves and gravitational tail memory in the classical limit:

• Arnab Priya Saha, Biswajit Sahoo, Ashoke Sen: Proof of the Classical Soft Graviton Theorem in $D=4$, J. High Energ. Phys. 2020, 153 (2020) [doi:10.1007/JHEP06(2020)153, arXiv:1912.06413]

• Biswajit Sahoo: Classical Sub-subleading Soft Photon and Soft Graviton Theorems in Four Spacetime Dimensions [arXiv:2008.04376&rbrack

• Debodirna Ghosh, Biswajit Sahoo: Spin-dependent gravitational tail memory in $D=4$, J. High Energ. Phys. 2020 70 (2020) [doi:10.1007/JHEP12(2020)070, arXiv:2106.10741]

Discussion in the BFSS matrix model:

• Noah Miller, Andrew Strominger, Adam Tropper, Tianli Wang, Soft Gravitons in the BFSS Matrix Model [arXiv:2208.14547]

• Adam Tropper, Tianli Wang, Lorentz Symmetry and IR Structure of The BFSS Matrix Model [arXiv:2303.14200]

• Aidan Herderschee, Juan Maldacena, Soft Theorems in Matrix Theory [arXiv:2312.15111]

Analogous discussion for gluons:

• Peng Cheng, Pujian Mao, Soft gluon theorems in curved spacetime [arXiv:2211.00031]

Analogous discussion for phonons:

• Clifford Cheung, Maria Derda, Andreas Helset, Julio Parra-Martinez: Soft Phonon Theorems, J. High Energ. Phys. 2023 103 (2023) [arXiv:2301.11363, doi:10.1007/JHEP08(2023)103]

See also:

• Avi Wadhwa: Soft Theorems in Chern-Simons Matter Theories [arXiv:2509.02542]

In the context of 2-group global symmetry:

• Jonah Berean-Dutcher, Maria Derda, Julio Parra-Martinez: Soft Theorems from Higher Symmetries [arXiv:2505.03566]