# Soft theorems Soft theorems are statements about scattering amplitudes in the limit where certain momenta or energies become large. They play an important role in [[0010 Celestial holography|celestial holography]]. Soft theorems are also essential for controlling [[0295 Infrared divergences in scattering amplitude|IR divergences]] and getting IR finite inclusive cross-sections. ## General refs - reviews - [[2017#Strominger (Lectures)]] - [[2021#Miller (Review)]] - conformal soft theorems - [[2018#Donnay, Puhm, Strominger]] - $U(1)$ KM symmetry of gauge theory and BMS supertranslation symmetry of gravity are generated by spin-1 and spin-2 primaries with $\Delta=1$ - [[2019#Pate, Raclariu, Strominger]] - YM conformal soft theorem - [[2019#Puhm]] - [[2019#Guevara]] notes - [[2019#Fan, Fotopoulos, Taylor]] - [[0328 w(1+infinity)|w(1+infinity)]] - [[2021#Guevara, Himwich, Pate, Strominger]] - [[0453 Jacobi identity or associativity of celestial OPE|Jacobi identity]] - [[CheungDelaFuenteSundrum2016]] - [[2020#Himwich, Narayanan, Pate, Paul, Strominger]] ## Understanding soft theorems - [[2020#DeLisle, Wilson-Gerow, Stamp]] - [[2019#Adamo, Mason, Sharma]] - in terms of features on the [[0022 Celestial sphere]] - [[Kalyanapuram202105]][](https://arxiv.org/abs/2105.04314) - [[Kalyanapuram202107]][](https://arxiv.org/pdf/2107.06660.pdf) - relating gravity subsubleading to gauge subleading - [[BenekeHagerSzafron202110]][](https://arxiv.org/pdf/2110.02969.pdf) - [[0509 Soft-collinear EFT|soft-collinear EFT]] - [[BenekeHagerSzafron202112]][](https://arxiv.org/pdf/2112.04983.pdf): an effective theory of soft and collinear gravity - action principle - [[2023#Kim, Kraus, Monten, Myers]] - [[0331 BRST quantisation|BRST]] formulation - [[2024#Baulieu, Wetzstein]] ## Examples (tree-level) ### General - infinite soft theorems - [[LiLinZhang2018]] - [[2018#Hamada, Shiu]] - [[2021#Guevara, Himwich, Pate, Strominger]] currents from poles - [[2021#Strominger]] $\rm w_{1+\infty}$ - geometric soft theorems - [[CheungHelsetParra-Martinez2021]][](https://arxiv.org/pdf/2111.03045.pdf) - [[2022#Cheung, Helset, Parra-Martinez]] - [[2024#Derda, Helset, Parra-Martinez]] ### Gravitons - leading soft graviton theorem <-> BMS supertranslation - soft theorem itself [[Weinberg1965]] - [[HeLysovMitraStrominger2014]] - $\Delta=1$ - $\mathcal{M}_{n+1}\left(k_{1}, k_{2}, \ldots k_{n}, q\right)=S^{(0)} \mathcal{M}_{n}\left(k_{1}, k_{2}, \ldots k_{n}\right)+\mathcal{O}\left(q^{0}\right)$ with $S^{(0)} \equiv \sum_{a=1}^{n} \frac{E_{\mu \nu} k_{a}^{\mu} k_{a}^{\nu}}{q \cdot k_{a}}$ - subleading soft graviton <-> [[0032 Virasoro algebra]] symmetry - $\Delta=0$ - $\mathcal{M}_{n+1}\left(k_{1}, k_{2}, \ldots k_{n}, q\right)$=S^{(0)} \mathcal{M}_{n}\left(k_{1}, k_{2}, \ldots k_{n}\right)+S^{(1)} \mathcal{M}_{n}\left(k_{1}, k_{2}, \ldots k_{n}\right)+\mathcal{O}\left(q^{1}\right)$ - with $S^{(1)} \equiv-i \sum_{a=1}^{n} \frac{E_{\mu \nu} k_{a}^{\mu}\left(q_{\rho} J_{a}^{\rho \nu}\right)}{q \cdot k_{a}}$ - first discussed in [[White2011]] using eikonal methods - proof [[2014#Cachazo, Strominger]] - $\overline{SL(2,\mathbb{C})})$ ([[2020#Banerjee, Ghosh, Paul]]) - [[2019#Adamo, Mason, Sharma]] - [[KapecLysovPasterskiStrominger2014]] and [[CampigliaLaddha2014]] - explained in [[2020#Donnay, Pasterski, Puhm]] - subsubleading soft graviton - $\Delta=-1$ - $\mathcal{M}_{n+1}\left(k_{1}, k_{2}, \ldots k_{n}, q\right)=\left(S^{(0)}+S^{(1)}+S^{(2)}\right) \mathcal{M}_{n}\left(k_{1}, k_{2}, \ldots k_{n}\right)+\mathcal{O}\left(q^{2}\right)$ with $S^{(2)} \equiv-\frac{1}{2} \sum_{a=1}^{n} \frac{E_{\mu \nu}\left(q_{\rho} J_{a}^{\rho \mu}\right)\left(q_{\sigma} J_{a}^{\sigma \nu}\right)}{q \cdot k_{a}}$ - [[2019#Guevara]] also mentions $\Delta=1-\mathbb{Z}_+$ - [[2014#Cachazo, Strominger]] - [[CampigliaLaddha2016]] - [[CondeMao2016]] - [[2018#Hamada, Shiu]] - via Einstein equations: [[2021#Freidel, Pranzetti, Raclariu (Nov)]] and [[2021#Freidel, Pranzetti, Raclariu (Dec)]] - identifies the [[0060 Asymptotic symmetry]] responsible for this ### Photons - leading soft photon theorem <-> angle-dependent U(1) gauge transformation [[2014#He, Mitra, Porfyriadis, Strominger]] - QED subleading soft photon theorem - the soft theorem itself [[1958#Low]] - [[2014#Lysov, Pasterski, Strominger]] - current algebra [[2019#Himwich, Strominger]] - other refs include [[CampigliaLaddha2016]] and [[CondeMao2016]] - more subleading [[Sahoo2020]] [arXiv](https://arxiv.org/pdf/2008.04376.pdf) - in general even dimensions - [[KapecLysovStrominger2014]] - relation of large gauge transformation and little group - [[HamadaSeoShiu2017]] - in the flat limit of AdS - [[BanerjeeFernandesMitra2021]][](https://arxiv.org/pdf/2102.06165.pdf) ### Gluons - [[0107 Soft gluon symmetry]] - leading $\Delta=1$ - [[0069 Kac-Moody algebra|KM algebra]], or large gauge transformations - turns out holomorphic - see [[2015#He, Mitra, Strominger]] - subleading $\Delta=0$ - proof [[2014#Casali]] - does not correspond to a pure gauge (see [[2019#Adamo, Mason, Sharma]]) - negative $\Delta$ - generated by the two above (see [[2021#Guevara, Himwich, Pate, Strominger]]) ### Other theories - scalars in the sigma model - [[2022#Kapec, Law, Narayanan]][](https://arxiv.org/abs/2205.10935) - EFT containing photons - [[2017#Laddha, Mitra]] - with Goldstone bosons - [[KampfNovotnyShifmanTrnka2019]][](https://arxiv.org/abs/1910.04766) - [[0479 BFSS matrix model|BFSS]] - [[2022#Miller, Strominger, Tropper, Wang]] - scalar-YM - [[2022#Melton, Narayanan, Strominger]] - phonons - [[2023#Cheung, Derda, Helset, Parra-Martinez]] - dilaton - [[Marotta2022]][](https://arxiv.org/pdf/2203.07957.pdf) - scalars - [[2022#Biswas, Semenoff]] - scalar fields with broken Lorentz boosts - [[2024#Du, Stefanyszyn]] ## Loop corrections - Maxwell - [[2019#Campiglia, Laddha]] - [[0071 Yang-Mills|YM]] - leading - [[1998#Bern, Del Duca, Schmidt]] - both IR-finite and infinite parts obtained - [[2008#Dixon, Magnea, Sterman]] - [[DixonGardiMagnea2009]][](https://arxiv.org/abs/0910.3653) - reviewed in [[2014#Bern, Davies, Nohle]] - subleading - IR-finite [[2014#He, Huang, Wen]] - gravity - none at leading order: Weinberg - subleading: - IR finite - [[2014#He, Huang, Wen]] - no change at 1-loop - IR infinite - [[2014#Bern, Davies, Nohle]] - only to 1-loop - [[DonnayNguyenRuzziconi2022]][](https://arxiv.org/abs/2205.11477) - subsubleading - IR-divergent - [[2014#Bern, Davies, Nohle]] - only to 2-loop - IR-finite - [[BroedelDeLeeuwPlefkaRosso2014]] - [[2014#He, Huang, Wen]] - modified at 1-loop - more refs - IR-finite [[2014#He, Huang, Wen]] - [[BernChalmers1995]][](https://arxiv.org/abs/hep-ph/9503236) - celestial context - [[2019#Campiglia, Laddha]] - [[2017#He, Kapec, Raclariu, Strominger]] - general study - [[2023#Krishna, Sahoo]] - [[0060 Asymptotic symmetry|asymptotic symmetries]] for log soft theorems - [[2024#Choi, Laddha, Puhm (Mar)]] - [[2024#Choi, Laddha, Puhm (Dec)]] - soft currents - [[2024#Bhardwaj, Srikant]] ## Higher derivative corrections - $F^3$, $R^3$ and $R^2\phi$ - [[2014#Bianchi, He, Huang, Wen]]: with $F^3$ and $R^3$ interactions, the leading soft theorems are not modified because they are suppressed in the soft limit - in bosonic string (graviton and dilaton) - [[DiVecchiaMarottaMojaza2016]] - general EFT with massless particles - [[2016#Elvang, Jones, Naculich]]: subleading soft photon theorem and subsubleading soft graviton theorems are corrected by a finite number of higher-dimensional operators (which are nonminimally coupled to gravity) - [[2021#Jiang (Aug)]]: reproduces [[2016#Elvang, Jones, Naculich]]'s results about soft theorem corrections by EFT operators; provides an argument that there are no EFT corrections to chiral symmetry algebra - higher derivative corrections to soft photon theorem in EFT - [[2017#Laddha, Mitra]] - including negative helicity soft currents - [[2021#Mago, Ren, Srikant, Volovich]]: finds constraints from Jacobi identity ## Susy - D-dimensional N-extended supergravities - [[2024#Kallosh]] - general - [[2024#Tropper]] ## Higher dimensions - subleading soft graviton - [[2014#Schwab, Volovich]] - [[2014#Afkhami-Jeddi]] - higher spin - [[2020#Campoleoni, Francia, Heissenberg]] - renormalisation issues - [[Lionetti2022]][](https://arxiv.org/pdf/2209.10889.pdf): extra divergences in higher dimensions need to be renormalised ## Higher spin - [[2022#Tran (Dec)]] - [[2020#Campoleoni, Francia, Heissenberg]] ## Infinite tower - [[2018#Hamada, Shiu]] - [[LiLinZhang2018]] - this leads to [[0328 w(1+infinity)]] ## Extensions/applications - classical soft theorems - [[0294 Classical soft graviton theorem]] - SSB of Lorentz boosts (in inflationary geometry) - [[GreenHuangShen2022]][](https://arxiv.org/pdf/2208.14544.pdf) - [[0170 TTbar|TTbar]] deformation - [[HeMaoMao2022]][](https://arxiv.org/abs/2209.01953) - via helicity constraints - [[RabearinoroRasoanaivoRaboanary2022]][](https://arxiv.org/abs/2212.00722) - correction of soft graviton theorem for small cosmological constant - [[BanerjeeBhattacharjeeMitra2020]] [](https://arxiv.org/pdf/2008.02828.pdf) - multiple soft photons - [[LiuMao2021]][](https://arxiv.org/pdf/2107.03240.pdf) - light-cone gauge - [[2022#Ananth, Pandey, Pant]] - de Sitter - [[2022#Bhatkar, Jain]] - [[2023#Mao, Zhang]] - relation to [[0152 Colour-kinematics duality|BCJ relations]] - [[2023#Wei, Zhou]] - finite temperature - [[2023#Solanki, Bhattacharjee]] - AdS - [[2024#Chowdhury, Lipstein, Mei, Mo]] - [[2025#Mei, Mo]] - fermions in dS and AdS - [[2024#Chowdhury, Chowdhury, Moga, Singh]] - [[0419 Carrollian CFT|Carrollian]] - [[2024#Bagchi, Dhivakar, Dutta]] ## Boundary results - [[2023#Herderschee, Maldacena (Dec, b)]]: [[0479 BFSS matrix model|BFSS]] ## Two types of scaling 1. symmetric - $k_{n}^{\alpha \dot{\alpha}} \rightarrow \delta k_{n}^{\alpha \dot{\alpha}}, \quad \lambda_{n}^{\alpha} \rightarrow \sqrt{\delta} \lambda_{n}^{\alpha}, \quad \tilde{\lambda}_{n}^{\dot{\alpha}} \rightarrow \sqrt{\delta} \tilde{\lambda}_{n}^{\dot{\alpha}}$ - then $M_{n}^{\text {tree }} \rightarrow\left(\frac{1}{\delta} S_{n}^{(0)}+S_{n}^{(1)}+\delta S_{n}^{(2)}\right) M_{n-1}^{\text {tree }}+\mathcal{O}\left(\delta^{2}\right)$ 2. holomorphic - $k_{n}^{\mu} \rightarrow \delta k_{n}^{\mu}, \quad \lambda_{n}^{\alpha} \rightarrow \delta \lambda_{n}^{\alpha}, \quad \tilde{\lambda}_{n}^{\dot{\alpha}} \rightarrow \tilde{\lambda}_{n}^{\dot{\alpha}}$ - then $M_{n}^{\text {tree }} \rightarrow\left(\frac{1}{\delta^{3}} S_{n}^{(0)}+\frac{1}{\delta^{2}} S_{n}^{(1)}+\frac{1}{\delta} S_{n}^{(2)}\right) M_{n-1}^{\text {tree }}+\mathcal{O}\left(\delta^{0}\right)$ - related by little group scaling ## Two prescriptions for momentum 1. impose momentum conservation for n and n-1 - [[2014#Cachazo, Strominger]] 2. use same momenta - [[2014#Bern, Davies, Nohle]] - shift contribution between different orders \[*Acknowledgement: I thank Temple He for correspondence and Atul Sharma, Lecheng Ren, and Anders Schreiber for many discussions on this topic.*\]