Soft Theorem and its Classical Limit Ashoke Sen Harish-Chandra - PowerPoint PPT Presentation

Soft Theorem and its Classical Limit Ashoke Sen Harish-Chandra Research Institute, Allahabad, India Florence, April 2019 1

A partial preview of the results Alok Laddha, A.S. arXiv:1806.01872 Biswajit Sahoo, A.S. arXiv:1808.03288 + earlier papers 2

Consider an explosion in space D A bound system at rest breaks apart into fragments carrying four momenta p 1 , p 2 , · · · with a ) 2 + � p 2 a + m 2 p 2 a ≡ − ( p 0 p 2 a = 0 , a = 1 , 2 , · · · , a This process emits gravitational waves Detector D placed far away detects h µν ≡ ( g µν − η µν ) / 2 Physical components: h TT (transverse, traceless) ij 3

We shall be interested in the late time tail of the radiation – the radiation at a large time u after the passage of the peak It has the form ij + 1 h TT = A TT uB TT ij + · · · ij p µ a p ν A µν = − 2 G u . v = − u 0 v 0 + � � a u .� p a . n , v R a � B µν = 2 G 2 n . p b � p µ a p ν 2 a R n . p a a , b n ρ p ( ν � p b . p a 2 ( p b . p a ) 2 − 3m 2 p a . n ( p µ ) a a p ρ b − p µ ) b p ρ � a m 2 � � + a ) { ( p b . p a ) 2 − m 2 b a m 2 b } 3 / 2 a , b b � = a n=(1, ˆ ˆ n), n: unit vector towards the detector R: distance to detector, G: Newton’s constant 4

ij + 1 h ij = A TT uB TT ij , for large u p µ a p ν A µν = − 2 G � a p a . n R a � B µν = 2 G 2 n . p b � p µ a p ν 2 a R n . p a a , b � n ρ p ( ν p b . p a 2 ( p b . p a ) 2 − 3m 2 p a . n ( p µ ) b − p µ ) � a a p ρ b p ρ � a m 2 � + a ) { ( p b . p a ) 2 − m 2 b a m 2 b } 3 / 2 a , b b � = a Note: The result is given completely in terms of the momenta of final state particles without knowing what caused the explosion or how the particles moved during the explosion – consequence of soft graviton theorem (a result for quantum S-matrix) 5

ij + 1 h ij = A TT uB TT ij , for large u A ij : memory term – a permanent change in the state of the detector after the passage of gravitational waves Zeldovich, Polnarev; Braginsky, Grishchuk; Braginsky, Thorne; · · · – connected to leading soft theorem Strominger; · · · B ij : tail term – connected to logarithmic terms in the subleading soft theorem Laddha, A.S. 6

A similar result exists for a general scattering process – gives the gravitational wave-form ˜ h ij at low frequency in terms of momenta of incoming and outgoing particles. TT ˜ x , ω ) = C TT ij ( ~ ij ( ω ) h   p µ a p ν 2 G   C µν � a  1 + 2 i G ln ( ω − 1 R − 1 ) � = η a k . p b i R p a . k a b ,η b = − 1  k ρ p ( ν + 2 G 2 R ln ω − 1 � � p a . k ( p µ ) a a p ρ b − p µ ) b p ρ a ) a b � = i η a η b = 1 p b . p a 2 ( p b . p a ) 2 − 3m 2 a m 2 � � × + finite . { ( p b . p a ) 2 − m 2 b a m 2 b } 3 / 2 � 1 ,� x / | � � k = − ω x | , η a : +1 if a is incoming, − 1 if a is outgoing. – Matches explicit results in special cases Peters; Ciafaloni, Colferai, Veneziano; Addazi, Bianchi, Veneziano 7

In the rest of the talk we shall try to explain the origin of this result from soft graviton theorem – combination of logic + guesswork + test Units: � = c = 8 π G = 1 Some earlier references: Weinberg; . . . White; Cachazo, Strominger; Bern, Davies, Di Vecchia, Nohle; Elvang, Jones, Naculich; . . . Klose, McLoughlin, Nandan, Plefka, Travaglini; Saha Bianchi, Guerrieri; Di Vecchia, Marotta, Mojaza; . . . Strominger; He, Lysov, Mitra, Strominger; Strominger, Zhiboedov; Campiglia, Laddha; . . . Bern, Davies, Nohle; Cachazo, Yuan; He, Kapec, Raclariu, Strominger 8

What is soft graviton theorem? Take a general coordinate invariant quantum theory of gravity coupled to matter fields Consider an S-matrix element involving – arbitrary number N of external particles of finite momentum p 1 , · · · p N – M external gravitons carrying small momentum k 1 , · · · k M . Soft graviton theorem: Expansion of this amplitude in power series in k 1 , · · · k M in terms of the amplitude without the soft gravitons. 9

Under some assumptions one can give a completely general derivation of soft graviton theorem A.S.; Laddha, A.S.; Chakrabarti, Kashyap, Sahoo, A.S., Verma – generic theory (including string theory) – generic number of dimensions – arbitrary mass and spin of elementary / composite finite momentum external states e.g. gravitons, photons, electrons, massive string states, nuclei, molecules, planets, stars, black holes Main ingredient: Graviton coupling with zero or one derivative is fixed completely by general coordinate invariance. 10

Assumptions 1. The scattering is described by a general coordinate invariant one particle irreducible (1PI) effective action – tree amplitudes computed from this give the full quantum results 2. The vertices do not contribute powers of soft momentum in the denominator – breaks down in D=4 Work in D > 4 for now – to be rectified at the end 11

Result: Let Γ be the scattering amplitude of any set of finite energy (hard) particles. Scattering amplitude of the same set of states with M additional soft gravitons of polarization { ε r } and momentum { k r } (1 ≤ r ≤ M) takes the form S ( { ε r } , { k r } ) Γ up to subleading order in expansion in powers of soft momentum. S ( { ε r } , { k r } ) : Known, universal operator, involving derivatives with respect to momenta of hard particles and matrices acting on the polarization of the hard particles. 12

Classical limit Laddha, A.S. We take the limit in which 1. Energy of each hard particle becomes large (compared to M pl ) –represented by wave-packets with sharply peaked distribution of position, momentum, spin etc. 2. The total energy carried by the soft particles should be small compared to the energies of the hard particles – a necessary criterion for how small the momenta should be so as to be declared soft. 13

In this limit the multiple soft theorem takes the form � M � S gr = S ( 0 ) + S ( 1 ) + · · · � S gr ( ε r , k r ) Γ , r = 1 N ( p a · k ) − 1 ε µν p µ � S ( 0 ) ( ε, k ) ≡ a p ν a a = 1 N ( p a · k ) − 1 ε µν p µ � a k ρ J ρν S ( 1 ) ( ε, k ) = i a a = 1 J µν a : classical angular momentum of the a-th hard particle All (angular) momenta are counted positive if ingoing. If in the far past / future the object has trajectory x µ a = c µ p µ a + m − 1 a τ a then J µν = ( x µ a p µ a ) + spin = ( c µ a p µ a p ν a − x ν a p ν a − c ν a ) + spin a S gr is large in the classical limit since p a and J µν are large. 14 a

�� M � Amplitude: Γ soft ≡ r = 1 S gr ( ǫ r , k r ) Γ Probability of producing M soft gravitons of • polarisation ε , • energy between ω and ω ( 1 + δ ) • within a small solid angle Ω around a unit vector ˆ n � M � 1 1 1 M ! | Γ soft | 2 × 2 ω ω D − 2 ( ω δ ) Ω = | Γ | 2 A M / M ! , ( 2 π ) D − 1 1 1 2 ω ω D − 2 ( ω δ ) Ω , | S gr ( ε, k ) | 2 k = − ω ( 1 , ˆ A ≡ n ) ( 2 π ) D − 1 Note: A is large in the classical limit 15

| Γ | 2 A M / M ! is maximised at ∂ � | Γ | 2 A M / M ! � = 0 ∂ M ln Assuming that M is large, ∂ ⇒ ∂ M ( M ln A − M ln M + M ) = 0 ⇒ M = A In the classical limit M is large since A is large Probability distribution of M is sharply peaked Note: the value of M does not change if we allow soft radiation in other bins. 16

1 2 D π D − 1 | S gr ( ε, k ) | 2 ω D − 2 Ω δ no. of gravitons = A = Total energy radiated in this bin 1 2 D π D − 1 | S gr ( ε, k ) | 2 ω D − 1 Ω δ A ω = This can be related to the radiative part of the metric field ⇒ gives a prediction for the low frequency radiative part of the metric field during classical scattering (up to overall phase and gauge transformation) N 1 ω � ( D − 2 ) / 2 � TT x , ω )) TT = � ( p a · n ) − 1 � � p µ a − i ω n ρ J ρ ( ν p µ ) ( h µν ( � a p ν a a 2 ω 2 2 π iR a = 1 1 ,� x / | � R = | � � � n = x | , x | – tested in many explicit examples involving classical scattering 17

D=4 The S-matrix suffers from IR divergence, making soft factor ill-defined. However we can still use the radiative part of the gravitational field during classical scattering to define soft factor. Naive guess: Soft factor defined this way is still given by the same formulæ: N S ( 0 ) ≡ ( p a · k ) − 1 ε µν p µ � a p ν a a = 1 N S ( 1 ) = i ( p a · k ) − 1 ε µν p µ � a k ρ J ρν a a = 1 18

Problem: Due to long range force on the initial / final trajectories due to other particles, the trajectory of the a-th particle takes the form: x µ a = c µ p µ a τ + b µ a + m − 1 a ln | τ | a for some constants b µ a . J µν = ( x µ a p µ a ) = ( c µ a p µ a ) + ( b µ a p µ a p ν a − x ν a p ν a − c ν a p ν a − b ν a ) ln | τ | a Due to the ln | τ | term, the soft factors do not have well defined | τ | → ∞ limit 19

Next guess: The soft expansion has a ln ω − 1 term at the subleading order, given by S ( 1 ) with ln | τ | replaced by ln ω − 1 . ω ≡ k 0 N S ( 1 ) = i ( p a · k ) − 1 ε µν p µ a k ρ J ρν � a a = 1 N ( p a · k ) − 1 ε µν p µ a k ρ ( b ρ a p ρ � a p ν a − b ν a ) ln ω − 1 = i i = 1 + finite This has been tested by studying explicit examples of gravitational radiation during scattering in D=4 – perfect agreement with all the cases studied. 20