JHEP02(2019)086 Springer February 10, 2019 February 14, 2019 December 10, 2018 : : : Accepted Received Published Published for SISSA by https://doi.org/10.1007/JHEP02(2019)086 [email protected] , . 3 1808.03288 The Authors. c Gauge Symmetry, Scattering Amplitudes

We explore the logarithmic terms in the soft theorem in four dimensions by ana- , [email protected] Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhusi, Allahabad 211019, India E-mail: Open Access Article funded by SCOAP we expect the results to be valid more generally. Keywords: ArXiv ePrint: Abstract: lyzing classical scattering with generic incomingscattering and amplitudes. outgoing states The and one classicalAlthough loop most and quantum of quantum our results analysisa are in theory consistent quantum of theory with (charged) is scalars each interacting carried other. via out gravitational for and one electromagnetic loop interactions, amplitudes in Biswajit Sahoo and Ashoke Sen the soft theorem in four dimensions Classical and quantum results on logarithmic terms in JHEP02(2019)086 ]. Indeed, 44 , 18 27 clear how to interpret a ] produced a more direct 17 88 28 a priori ]. Ref. [ 34 , 31 , 13 16 3 20 15 – 1 – ] where more care may be needed [ 41 4 ]. However these arguments break down in four space-time 39 ] and also by direct analysis of amplitudes in field theory and 41 6 84 36 , – 83 13 , 12 81 10 38 ] have been analyzed recently from different perspectives, both using ]. There are general arguments establishing their validity in any space- 1 12 87 – – 1 37 Although soft theorem is a relation between quantum scattering amplitudes — ampli- 7.2 Soft photon theorem 7.1 Soft graviton theorem 3.2 Effect of3.3 gravitational interactions Effect of electromagnetic and gravitational interactions 2.1 Summary of2.2 the results Discussion of2.3 results Special cases 3.1 Effect of electromagnetic interactions tudes with soft photon or gravitonalso to amplitudes relate without soft soft photon theorem orthis to graviton can — classical one be scattering can done amplitudes. viarelation between asymptotic In soft symmetries four theorem [ space-time anddirectly dimensions classical taking scattering the in classical generic limit space-time of dimensions a by quantum scattering amplitude. This relates various — general coordinate invariancesoft for photon soft theorem graviton [ theoremdimensions and due to U(1) infrared divergences gauge [ invariancesince for the S-matrix itselfrelation is whose both infrared sides divergent, are it divergent. is not Soft theorems [ asymptotic symmetries [ string theory [ time dimensions in any theory as long as one maintains the relevant gauge symmetries 1 Introduction 6 Soft graviton theorem in7 gravitational scattering Generalizations 4 How to treat momentum5 conservation and infrared Soft divergences photon theorem in scalar QED 3 Classical analysis Contents 1 Introduction 2 Summary and analysis of the results JHEP02(2019)086 τ ω as and τ ) ln ) and ]. Soft in the . The ω µ τ p 94 ω ( τ ν – µ b ln x ] the scat- 91 approaches − µ 89 b ν µ p x µ b , where µ p ν is independent of x ] by considering several µν − j 89 ν p 1 µ x . Therefore computed this way to evaluate the soft µν approaches a constant and → ∞ j µ τ p replacement rule. We also analyze directly the as – 2 – 1 α − . This has been tested in [ ω 1 ]. The subleading terms in the soft theorem contain a and ln − ω µ 90 c → , of the individual particles involved in the scattering, with τ 89 in the soft expansion. This analysis yields results consistent µν j 1 ], is that we do not insist on a power series expansion in − ω 41 , making the subleading soft factor divergent. µν j with constant τ . This gives a logarithmically divergent term of the form ( µ µ x α p + µ c and we can use the asymptotic value of in the naive expression by ln τ ) label the asymptotic coordinates and momenta of the particle as a function of the The purpose of this paper is two fold. In all the examples considered in [ Since we do not expect the radiative component of the metric or gauge fields to diverge Since classical scattering is well defined even in four space-time dimensions, one can The existence of various logarithmic terms in classical scattering has been known earlier [ τ 1 ( → ∞ µ calculating the coefficients of the poweranalytic series terms expansion. of Instead order we ln allow for possible non- theorem provides asubleading systematic soft procedure factor without for detailed knowledge computing of the the forces coefficient responsible of for the the scattering. logarithmic term in the this assumption and consider a generalscattering scattering have process masses where of all the particlesclassical same involved soft in order, the factor and using then the determinequantum ln the subleading logarithmic soft terms factor in byin the considering the one presence loop ofprevious scattering gravitational analysis, of and e.g. charged electromagnetic in scalar [ interaction. fields The difference with the examples of classical scattering in four space-time dimensions. tering process considered had oneitational heavy field, center producing and the other long particlesthe range carrying influence Coulomb smaller of or the grav- masses long were range taken to fields produced be by moving the under heavy center. In this paper we relax in classical scattering inthe four subleading space-time soft dimensions, factor thisof is suggests the a that soft breakdown the particle.natural of divergence guess Therefore in the is the that power soft the series softof factor factor ln expansion must at in contain the non-analytic subleading the order terms energy is in given by replacing the factors factor. However in four space-timemagnetic dimensions forces the acting long on range the particles gravitationalexpression produce and/or for an electro- additional term ofin the the form expression for the orbital contribution to thep angular momentum given by proper time. In dimensionsthe larger form than four, τ sions. Since inpart higher of dimensions the the electromagnetic soft andmenta theorem gravitational of expresses fields incoming in the and terms lowsame outgoing of formula frequency will momenta finite radiative continue and energy to angulardure hold particles, mo- we in the encounter four an naive dimensions. obstacle guess Howeverfactor [ in of will carrying angular be out momentum this that proce- the and gravitational fields in classicalthe scattering logic, in one generic can space-timesoft use dimensions. factors. the Reversing classical scattering data to give anhope alternative to definition use the of classical the definition of soft factors to understand soft theorem in four dimen- terms in soft theorem to appropriate terms in the radiative part of the electromagnetic JHEP02(2019)086 ] we 84 , ]. To ]. In 7 83 , 84 98 , , 81 97 83 , 81 ], and also that the ] by taking the soft 97 98 contains a summary and a 2 describes one loop quantum computation 5 – 3 – describes some general strategy for dealing with the 4 ]. ] can be shown to be in perfect agreement with our results for 96 98 , describes a similar computation in the soft graviton theorem in = 1 units. 6 95 ] gave a general derivation of soft theorem including loop corrections as πG ]. We have been informed by G. Veneziano that for massless particle 84 The papers quoted above have now appeared in the arXiv [ , = 8 96 83 c , , = 95 81 describes the analysis of the logarithmic terms in the soft expansion for gen- ~ 3 Classical gravitational radiation during a high energy scattering process has been an- The rest of the paper is organized as follows. Section Since [ 2 Summary and analysis of theIn this results section we shallresults. first Finally summarize we our shall results considershall and some use then special discuss limits various and aspects compare of with the known results. We Note added. particular the results ofscattering [ of massless particles. alyzed in [ scattering, results related tologarithmic the ones terms described in here thelimit were classical of found scattering the in results have [ of been [ derived in [ consider charged scalars interacting via bothand gravitational determine and the electromagnetic interaction, onetromagnetic loop interaction contribution and to one the loopto quantum gravitational contribution soft interaction. to graviton the factor quantum due soft to photon elec- factor due infrared divergent part ofing the use S-matrix of and momentumof extracting conservation. the the logarithmic Section quantum terms soft(scaler in factor QED). by the Section mak- softa photon theory theorem of in charge neutral scalar scalars quantum interacting electrodynamics with the gravitational field. In section usual soft theorem, and need to be computed explicitly. discussion of our results whereSection we also discuss variouseral special classical cases scattering. of our Section classical result. The first is the regionenergy of of integration in the which external the softto loop particle. be momentum valid, is In and large this we comparedapplying region to find the we the that expect usual the the soft contributionsource arguments operator from is of this on the [ region region the canexternal of amplitude indeed soft integration without be in momentum. the obtained which The soft by the contribution graviton. loop from momentum The this is other small region compared cannot to be the derived using the we interpret as the result of back reaction of thelong as radiation 1PI on vertices the do motionextent not of we generate the could soft particles. derive factor thethis in end results the we of note denominator, that the one there current could are paper two ask distinct using to sources the of what logarithmic result terms of in the [ soft theorem. with the classical results, although the quantum results contain additional real part which JHEP02(2019)086 .

2 2 b can (2.4) (2.1) (2.2) (2.3) terms m , is φ 1 2 a R − m ω 3 − 2 ) and momenta a for gauge fields } .p and ln a b µ q and an amplitude 1 p , { A − ε µ 2( 2 ω ) ε  } as the infrared regulator. / , ) ρ a 3 = 4 ρ ρ a R p } = ρ p ε, k 2 b µ b h D ( µ b p m p . S ε.φ 2 a µν | − − η ω µ a ~x 1 from the scattering center, the best m | µ a 2 p 2 1 p ρ b − R ρ b = 2 p 2 − / p ) { ( ) for a 2) 2 b 4. Our interest will be in analyzing µν 2 − .p / and polarization h m b 3 D ε, k ,R 2 a ( ( p | } ) is the ratio of an amplitude with an k = ( 2 b  ~x m S | { ] that in the classical limit the quantum D > m a µν b ε, k 2 a q 88 ~x/ ( q ω π a iωR πiR a µ a 4 S m e 2 .p q .k ≡ p b a −  µ p ˆ n – 4 – p 2 ε i =1 a πR ) , e b ), containing terms of order 6= 4 a η 2 b a X iωR , a / .p η X ) e − -th particle is ingoing (outgoing). Then the classical ) b ε, k ˆ a b n particles carrying electric charges . For the quantum part, the natural infrared cut-off is provided ρ p ( , = µν ( k R arises as an infrared cut-off. For the classical result the ln η { n . Therefore it is again natural to take .k ) = µ em k.p (1 a ε R 1 S =1 − ω a p /R a b − ~x,t 6= η q b − b ( = X a µν b X η η g 1) if the a = ρ ε.φ X ) − k k 1 1 ) the radiative part of the metric or electromagnetic field at a .k = ( µ − − a iωt ε p a ω R µν ~x,t q ( h dt e ln . In our convention the momenta/charges carry extra minus sign if . For both electromagnetism and gravity we define classical soft factor φ i n a = 4. +ln Z µν X − for a scattering event around the origin. For electromagnetic field, 1 e 1 a ··· D − t − q , ω ω µ a space-time dimensions via the relation: .k p ln for gravity, and a (ln = 1 µ to be takes value +1 ( p ε D π π i i φ a a µν is the polarization tensor of the soft particle so that 4 8 at time η e a ε X + + ~x ) in µν for ε We consider the scattering of = In this and subsequent expressions } 2 ε, k a ( em p S arise due to long rangeing gravitational force center on to the the soft detector photonby over or the a graviton resolution during distance of its the journeyenergy detector. from resolution the possible For scatter- a is detector of order placed 1 at a distance result for the soft photon factor the situation in { they are outgoing. The particlesinteractions are besides taken to other interact via short electromagneticsymbol range and gravitational interactions whose nature we need not know. The On the other handoutgoing the soft quantum photon soft orwithout graviton factor such with a momentum softsoft particle. factor It reduces was to shown the in classical [ soft factor for where and and take S In order to give awe uniform treatment shall of denote the classical by point soft photon and soft gravitonbe theorem, directly identified with the gauge field. For the gravitational field we define 2.1 Summary of the results JHEP02(2019)086 . ξ (2.5) ), we   2.4   2 b 2 b p p term in the 2 a 2 a . This makes 2 b p p that are cubic 0 limit follows µ . m 1 − − − → ! 2 2 ) and momentum ) ) ω 2 for any constant b b b ) 2 b µ 2.5 ˆ k m .p .p . a a b m p p ξ k p 0 limit. The coefficient ) should not be directly ( ( ( is the leading soft factor + q q

→ µ 2.5 ε b (0) em + − is a vector constructed out of S m b b → ) ln ’s to zero. On the other hand if .p .p U µ .k c ) is small compared to ( a a  ε b q p p p aµ ( 2.5   3 where ∂ ∂p ) and b ln for any mass parameter ν a (0) ), we can replace the ln em X p 2 2

2.4 µ 2 b µ a − /µ p p 2 b 2 a .k µ k.U S p a aν ε m p ∂ 1 a − q − – 5 – ∂p 2 b ) by ln ln( 2 p µ a R ) 2 a b p b a that are linear in p p 2.5 ) to compute the radiative component of the classical  X .p , the quantum correction ( b 1 a − ) ν − p 2.2 1 .k 2 k P ω ) a − µ 2.2 1 b p has additional terms: ε − R .p ) a + 2 ( a 2 q em b , the quantum results are ambiguous and are defined up to p π ) to compute the classical electromagnetic field produced by ( S .p a + ln 4 a X of the form ln q 1 2.2 p − 1 b = (8 q − em ω a U S ω q in the last line of ( (ln 2 ln ) into (  2 1 2 1 − π   π 8 2.4 1 a R ) are invariant under gauge transformation 6= 16 − b X . We shall discuss later the conditions under which we expect the quantum results to be ) and substituted into ( ~ 2.5 = 2.4 cannot be changed this way, but in this case the finiteness of ), ( ’s to zero. em 1 c S q − ’s. By choosing 2.4 a ∆ ω If we want to consider the situation where we ignore the effect of gravity, then we need As discussed in section As will be discussed in section The quantum result for p Note however that when we express the results in terms of the frequency/wavelength of the soft pho- 3 in the ton/graviton and momenta ofany the power finite of energy particles,small neither compared the to classical the classical nor results. the quantum result has to set the terms proportionalwe to want ln to consider thebetween situation the where particles during we scattering ignore (but the stillsoft effect use photon electromagnetic emission of interaction process), electromagnetic to we compute interaction have to set the terms proportional to ln the coefficient of ln manifest the fact that theof ln coefficient is not divergentas in a consequence the of cancellationconservation. between the second and third line of ( can substitute ( a scattering event. addition of a termgiven to by the first term on the right hand side of ( by working with one loopthat amplitudes ( in scalar QED coupled to gravity. It isadded easy to to ( check electromagnetic field. Rather, when the contribution ( The classical results areparticles. universal, We independent expect that of the the quantum results theory are and also universal, the but nature we have of derived them external the flux to thisthe order. electromagnetic field However do the receive flux subleadingfor for contribution. gravity. circular An identical polarization situation and/or prevails the wave-form of Since for real polarization the subleading contribution is purely imaginary, it does not affect JHEP02(2019)086 | . . ) ’s µ τ

c | ξ 2 b q 2.4 (2.6) (2.7) ) are m 2 a   2.7 m to arrive   3 2 b 2 b 1 ), ( p p − − independent 2 a 2 a 2 ω p p ) 2.6 c a q − − .p b 2 2 ) ) p by ln b b 2 | 2( } / .p .p ρ a  3 , τ a a | ) p } p p 2 ρ a 2 b µ b ( ( ) p that are quadratic in p term in the coefficient of 2 b ˆ k m ], the soft factor involves µ b . q q 1 for any constant vector 2 a − p b 2 b m − 89 p µ a µ + − ). The second lines of ( m − m ( ω p k b b µ a ρ b − ν p by exploiting the ambiguity in p ln 2.6 ξ .p .p 2 ρ b  ) { a a 2 p a + p p 2 b .k ( aµ µ ), ( b ν .p ∂ 2 m   p b k / . One can check that ( 2 a ∂p p 3 µ ), the ln 2.4 ( ln ν a b ] of replacing ln ξ 4 } m { ) are the result of direct application of p 2 b X 2.5

+ 89 b ν a m 2 b ν a − q π 2.6 p 2 a p p a µν 4 a µ a 2 a µ a q .k ε .k aν m p p p a .p a ∂ b p p − =1 µν ∂p → a p − µν – 6 – 2 2 b b ε ε 6= µ a ) η 2 p b X ) and ( a a ) p µν 2 a η b a a ε p  .p X ρ X b .p 2.4 ν k a p − b ) k ( ν a p 1 .k 2 { ρ a p .k ) a − p k.p to zero. b a p =1 a µν R 1 p b 1 .p µρ 6= ε η − + 2 ( ) can be replaced by ln b a ε X − b a = b p η b X ω a ( η 2.7 ρ .p a + ln X a k X ) q 1 1 ν a p 1 .k ). Again the classical results are valid universally. The quantum − − 1 b p a − q ˆ n − ω ω p a , µν R ω q ln ε (ln 2 i ln in four dimensions due to the long range gravitational/electromagnetic  a 2 +ln − = (1 2 τ X 1 π 1   ν a π 1 8 1 − a p − µ a 6= ω 16 .k − b k/ω X ω p a p − = (ln ln µν ε π π = i i gr 8 4 in the last line of ( S a ˆ k X + + 1 ∆ If we want to consider the situation where we ignore the effect of electromagnetic The classical result for soft graviton factor takes the form − = R gr S orbital angular momenta ofthe initial elapsed and time final particlesforce and on these the diverge logarithmically incomingin in and the outgoing trajectory. particles We thatat follow generates the the a first prescription term of and proportional [ third to lines ln of the classical results ( terms in the coefficient of ln 2.2 Discussion ofFirst results we shall brieflylater outline sections. how these The resultsclassical classical are results soft derived; ( theorem more to details subleading can order. be found As in described in [ interactions, then we need to setto zero. the terms On proportional the to othergravitational ln hand interaction if between we the want to particles consider duringinteraction the scattering to situation (but compute where still we soft use ignore graviton gravitational the emission effect of process), we have to set the results are obtained fromexpect one them loop to calculation beln in universal. scalar As QED in coupledthe the to definition case gravity, of of but the we ( invariant soft under factor gauge discussed transformations in section where The quantum result has additional terms JHEP02(2019)086 , 1 − 2 ω ) have dif- in the first ) represents b 1 2.7 q − a q . This also has R ) for soft photon ), ( R 2.5 2.6 in these expressions. + ln 1 R , and express this as the − ω ω ) and ( and larger than the infrared 2.4 ) and in ( ω ]. As mentioned in footnote 2.5 94 , but small compared to the energies ), ( 93 ω , 2.4 ]. 91 41 in ( 1 ). The term proportional to − ω ) proportional to (ln – 7 – 2.6 2.6 terms in the scattering amplitude of multiple finite 1 introduced in [ − 0 n ω σ ) for the soft graviton. Even though the S-matrix elements 2.7 ) represents the effect of late time gravitational radiation due to and ln 1 2.6 − term, for the soft graviton to travel to a distance ) and ( ω 1 − 2.6 . This term has appeared e.g. in [ and is responsible for the terms proportional to ln 1 R − R R ) arise from additional phases that are not directly determined by soft theorem. by the ln the effect of inducingterm. backscattering In of the the quantum soft computationintegration graviton, these where terms represented the arise loop by from momentum thecut-off region is ln of smaller loop than momentum reliable, this needs tothese be particles. taken to be large compared to thethe Schwarzschild radii effect of of gravitationalparticles drag in on the final the state. soft graviton This due has to the the effect of other causing finite a energy time delay, represented computation both these termswhere arise the loop from momentum the is region largeof compared of the to loop other momentum particles.largest integration In length this scale case involvedtypical the in energy scale the carried of by quantum these the scattering logarithms finite which energy is is external again the states. set inverse by For of one the loop the result to be expect the scale ofthe these classical logarithms scattering to process, be e.g.the set the by particles typical the involved distance largest in oforder length closest the of scale approach the scattering. involved between in Schwarzschild radiiwave-lengths This of of is the the particles taken particles and to involved much be larger in larger than the the than scattering. Compton or In of the the quantum one loop line represents the effectacceleration of of the late particles time viathe long last gravitational range line radiation electromagnetic of due interaction.the ( late The to time term acceleration the in of late the particles time via long range gravitational interaction. We 2.6 The term in the second line of ( We begin with the classical result ( The different terms proportional to ln Quantum results are the result of direct one loop computation in a field theory of 2. 1. length cut-off This is related to the quantity ferent origin. We shall explainof them soft in photon the factor context of is the very soft similar. graviton factor, but the case energy scalars and an outgoingproduct soft photon of or the graviton amplitude ofthat without energy we the call the soft soft photon factor.and or the The graviton latter sum and is of given awith by ( and multiplicative the without sum factor the of soft ( particlethe are ratio infrared divergent, of much the of this two. cancels when The we remaining take infrared divergent part is regulated by the infra-red They represent the effectgraviton of which long causes range the gravitational soft force particle on to the slow outgoing downmultiple soft and charged photon also scalars, or backscatter. coupledevaluate to the electromagnetic order and gravitational fields. We simply and ( JHEP02(2019)086 ) 2.4 if the ) and 1 − 2.6 R and apply the since we cannot ω 1 − ) and the first two ] by direct classical and large compared R 89 2.6 ω ] requires that the total cannot be recovered this 88 ], based on general coordi- ω and not by direct calculation 83 , 1 ) in the quantum theory is that − 81 ω 2.7 ). Therefore we should expect that the but small compared to the energies of the ) that arise in the quantum computation. 2.6 ω 2.7 in ( terms are imaginary for real polarizations. The – 8 – 1 a from the scattering center. − ω R . 1 ). The rest of the contribution that arises from integration − R 2.7 ), can be generated using a simple algorithm. As discussed earlier, the , namely the terms in the first and third line of ( ω 2.7 ), but we have not done this. 2.6 to the detector for the classical scattering and the resolution of the detector for the generate inverse power of softlong momentum, as remain the valid in loop four momentum dimensions is as large well compared as to the external soft momentum. factor, we recover preciselythe the first results two given lines in ofregion the where ( first the and loop thirdway. momentum line This is of indicates small ( that comparednate the invariance to of general 1PI argument effective action of and [ power counting assuming that loops do not lines of ( amplitude without the softLet graviton us has call an this the infrared IRwe divergent factor. restrict factor the If multiplying in loop the it. momentumusual integration integration over subleading to loop soft momenta be differential of large operator this compared IR that to factor arises in higher dimensions to this IR other particles, while the termintegration in where the the third loop lineto momentum arise the is from infrared region small cut-off of compared loop to momentum pared to First note that bothresult where these the terms coefficients are ofterms ln real in for the real firstthe two polarizations loop lines unlike momentum come is the from large classical compared regions to of loop momentum integration where theorem the logarithmically divergent termsof by electromagnetic ln and gravitationalcases radiation during the classical equivalence scattering. ofcomputation. In these special two In procedures principleand was similar ( tested tests in can [ be done for the general formulae ( R quantum scattering. The lattermeasure in the turn energy has of a the lowerdetector limit outgoing is set particle placed by with at an a accuracy distance better than the scale of these logarithms is set by the effective infrared cut-off, e.g. the distance In the quantum computation the terms that arise from loop momenta large com- We now turn to the additional terms ( We emphasize that the classical results are obtained by replacing in the classical soft Since the real infrared divergent part of the amplitude reflects the effect of real graviton 5. 4. 3. energy carried by the soft radiation is small. end note that theenergy carried validity by of soft the radiationenergy should classical objects remain limit taking small part described compared inwhich to the in the scattering. are [ energies not Here of ‘soft included the radiation’extra in finite represents terms those the particles arising sum in over the quantum theory should be small in the limit when the total emission, our interpretation of thethey extra reflect contributions the ( effect of backreaction of soft radiation on the classical trajectories. To this JHEP02(2019)086 . ) 0 M 2.7 →  to have ) similar ) changes ε ) becomes ), of order 2.7 2.7 . This again 2.7 2.6 M ) to modify the 2.7 ] relied on the fact ) are small compared 88 . This shows that it is a ), due to the argument ) as a test of when the 2.7 p to be the heavy particle a b 2.7 , q a p → − ( a p 0 limit, a careful analysis shows come in approximately equal and } → − a → ) p  a { , q a p as the initial state heavy particle and the final ], then we need to first consider multiple soft ) represent the incoming and the corresponding and also under – 9 – b b 88 p a, b ). Therefore we should not use ( and then carefully evaluating the result in the ], where the radiated energy remains small compared ). However in the quantum correction (  → − 88 2.6 + b 2.6 a p p . In this case the dominant contribution to ( − ) and also under ( b M = to be one of the light particles and , q b b ). For this we shall work in a frame in which the heavy particle a p , i.e. the pairs ( p term, and we need to explicitly evaluate this and show that it ( a 2.6 p a ). Instead we should use the smallness of ( → − = ) is valid. An identical discussion holds for electromagnetism. ' − ) b b and the other particles are lighter carrying energy small compared to b 2.6 2.6 p , , q a b M q p − ), then setting = 2.7 b q We must emphasize however that the quantum analysis is carried out for single soft Another situation discussed in [ In order to test this hypothesis we need to consider a scattering where the energy , comes from choosing breaks down when thethe total energies of energy the carried hardcomparable by particles to the — the soft precisely classical when particlesclassical result the becomes result ( quantum ( comparable correction ( to classical result ( graviton emission. Ifof we the want classical to gravitationalgraviton relate field emission the as and quantum then in resultthat take [ the to the classical the soft limit. factors radiativeeach associated The component other, analysis with i.e. of different the [ probability binsnot of in depend emitting the on certain phase number how space of many soft are soft particles independent particles in of are one emitted bin does in the other bin. This independence in the second andcontribution cancels third between line the choice ofstate of heavy ( particle, andshows we that do quantum not corrections are get small any compared contribution to proportional the to classical result in this limit. to the classical resultis ( initially at rest,only spatial and components. using gaugebut After it invariance the is choose scattering small theM the compared polarization to heavy tensor particle acquires a momentum that the result vanishes. This confirms that quantum correctionsto are the small energies in of thisa the limit. large hard mass particles,We is shall the now verify probe that limit in this in case which too one the of quantum corrections the ( particles has outgoing particle. In thisnot case include there is the novanishes. other term This that can cancels beline this of checked since ( explicitly the by sumlimit. first does evaluating Even the though derivatives individual in terms the diverge second in the the last term changes signsmall under for small deflection scattering.sign The under first ( term onof the right the hand log side ofapproximately getting ( cancel inverted making under the eachwhen result of small. these There operations. is one This exception to shows this that that the arises terms carried away by softobjects. radiation One remains way smallthat to each compared incoming achieve particle to this gets deflected the isthis by a process to energies small remains amount small. consider of and the In scattering finiteopposite energy this at radiated case pairs energy during the large — momenta impact the incoming parameter and so the corresponding outgoing particle. Now in eq. ( JHEP02(2019)086 1 at − for + 1 . If ω = 0. ξ (2.9) (2.8) M ), the , then M N ) 2 a 2 a bj = 1 for ~ β +1 p = 2  m | / N − a 3 b 3 p a +1 ~ ) β =1 e | 2 a − N b (1 / 2 a by m 2 a P e = 0 we see that 0 , − m b , spatial momenta )(2 M 2 a q is negative, we can 2 a ) with and a set of neutral e = for any vector = 0 e a ( b m = 2 a ~ β µ − ], this produces a late .p e bj , M k ( a a p ν q 90 p (1 a ξ ' aj a and ] where a neutral massive a ~ β +1 =1 β 6= 0 e b b 1 + X N ˆ 90 n. ai − . In this case if we take the M ν m β 2 2 ai k − p ij ) with = , µ (or both) represents a massless m 1 ε ξ ij a b ε a + 2.6 p ··· + e 2 or ~p +1 =1 + µν =1 N a X N a a ε X 2 p a 1 ~ β and are therefore smaller in the limit of 0 as − 3 = → 1 0 ω M 3 2 − − 1 ]. As discussed in [ | M e µν ln ω a − ε ~ 90 β ω π | i + 1 ), either – 10 – 4 . Examining ( are both outgoing then 3 and ln a 0 ~ b β + 2.6 comes from the terms where we choose 2 and momentum π i 2 p M ~p 8 1 aj a − p aj in ( M + + ~ β ω β ai 2 1 and |  b ) vanishes. This would be the situation during binary a p ˆ n. ai aj  m ~ a β ij β +1 β p 2.8 − ε a ˆ n. ij p i N ai ) could be significant even though their contribution will be and 1 ε p m β | − +1 finite energy states. Using the relation =1 a = a ij 2.8 a X e 1 N ε 1 ) proportional to ln 1 N e a and =1 N − e a X 2.6 ω 0 0 =1 N ln M a X at rest decays into a heavy object of mass M 0 π i 1 terms in ( 4 − M M ' − ω 1 , the expression ( ··· − ln +1 . Our goal will be to write down the classical soft graviton factor for this case. ω contain terms without a factor of and energies N 0 N π N i . This agrees with the results of [ p ln ~p 8 0 ≤ ··· − labels any of the π ··· i = M , 4 a a Another special case we can consider is when a charge neutral object of mass Note that when all the final state light particles are massless, so that ≤ and = 1 ~p term in the classical soft graviton factor continues torest vanish. breaks apart into two charge neutral objectspolarization of tensor masses of the soft graviton to have components only along the spatial direction, where in the lastTherefore we step see that we even have without making used any approximation, conservation the of coefficient of spatial the ln momentum suppressed by the ratioparent of system. the total We shall energythis now carried case evaluate away in the by the resultparticle. radiation sum without to Recalling over making the that any mass when express approximation. of the the In terms in ( 1 black hole merger when theprocesses final the state particles radiation aresystem, carries only the away gravitons. an However since appreciable in fraction such of the mass of the parent where large time tail in the gravitational wave-form that falls off as inverse power of time. and net contribution takes the form: the spatial direction, sinceinvariance under the the result gauge forwe transformation denote the the other momentum componentswe carried have may by be final found statedominant heavy by term object proportional using of to mass ln As a special caseobject we of can mass consider thelight situation objects described carrying in mass [ a We shall take the polarization tensor of the soft graviton to have components only along 2.3 Special cases JHEP02(2019)086 ) 2.6 (2.14) (2.13) (2.10) (2.11) (2.12) denotes , ν a p a . µ a massive p  . P k.p 2 µν ν − ε − P in the summand in µ

=1 . 1 a 2 2 a P ν b X − η p m or both zero. Therefore ω ) and µν ) 2 1 µ a ε b p 1 m m 3 − 2.11 µν in the classical formula ( ε ω − massive or 1 π 2 i ln P − a ) 4 , ω 2 − a π m i ~p  + p 2 P ( 1 + .k + − n.~p 2 k. b = e p 1 1 ν a ) vanishes and we need to only compute X a 1 + ˆ p − η ν a e 2 µ a .k p ω 2.6 e p − a 2( µ a .k p ln p  a µν + = = 1, we can express the term proportional to p ε 2 π µν i a / b ε 4 – 11 – 3 p η =1 ˆ } n.~p a π b − i 2 2 = 1, and either η η 1 4 =1 − = b a = m X a η a,b η 1 2 1 − X ν a ). Defining e 2 p a,η m ≡ a .k ~p µ a 6=  b 2.6 b p − P ) p + = 1, k.p 2 2 vanishes. Also the coefficient of ln µν to be a final state massless state in the term in the second ν a 2 e ) a ε e p b 2 η b k.P 1 µ a q ~p + .k e 1 a p =1 a a 1 q a − p + X e η µν ( ω is negative for 2 ε b take the form j e b 1 ln p π i 1 i e 4 .p k.p ( p − a π i { ij ω p 2 1  ε − 1 = − × b X − massless η ) for massless particles as b ω 1 denotes total outgoing momentum as defined in ( ). The net contribution from such terms is given by ln 2.6 − ω P π i 2.6 8 in ( − ln ) with the restriction 1 π i Next we consider terms that involve a pair of final state momenta at least one of which More generally one can show that for a general scattering process involving both mas- Next special case we shall analyze is that of scattering of massless particles, again − 2.6 4 ω to be an initial state and of ( the term proportional to the last two lines simplifies to the total outgoing momentum carriedpend by the explicitly on massive particles. the momenta Thereforeconservation. of this the does outgoing not massless de- states except through momentum is massless. This term receives contribution from all three lines on the right hand side line of ( where sive and massless particles, theis terms not proportional sensitive to tomomentum ln the conservation. details To of seestate this the massless let final particle us momenta state and first masslessa the consider particles initial terms state except that momenta. through could These involve overall come a from final choosing Note that this involves onlythe the momenta momenta of of the the finalanalysis initial state we state particles. are particles This considering and asymmetry soft is is particle insensitive related to only to in the the fact final that state in and our not in the initial state. and the fact that ln focussing on the classical result ( the contribution from aproportional pair to of ln final states. This can be easily evaluated and the terms then the contribution from the initial state to ( JHEP02(2019)086 . a
J may (3.3) (3.1) (3.2) (2.15) (2.16) and or both ν em massive also the a S b P b p , or a or q a ). a µ massive P 2.2 − , but in the second ν a P µ = P b -th particle, counted with . , a , µν ε . ρµ a ρµ a 1 . This left over contribution ν b J J ··· − a .k p ρ ρ a .k ω µ a k k + p a ν a p = µ | p ln ] as described in ( p ε σ b µν | π i 88 a µν ε 4 q ε ln = 1 a µ a a c − X X    term. Therefore the first two terms almost = i i ν b b + η p a + + = µ a σ X a a p = µ a η ν a q ; – 12 – 1 p p b − µ a µ a a a,b .k includes the term where = .k p 1 p a b either a or b massless a m µ b η p p ε µν = 1 a X a ε − η η ; a ω and a X a and b massive a,b X ln a ) = = − σ π = ν b i ( 4 p µ a em gr µ a r ) to four dimensional theories, the complication arises from the S p S 1 . The net contribution from the terms where either 3.2 a − = b runs over all the incoming and outgoing particles, and = from the orbital angular momentum. They are computed from the η ), ( b a = X a µν a 3.1 η ; represents the classical angular momenta carried by the external particles. J is a differential operator involving derivatives with respect to the external a a,b a J    J can now be added to the last term to include in the sum over µν ν a ε p 1 µ a − p ω In applying ( In dimensions larger than 4, the soft factors for photons and gravitons are given re- µν ln ε π i π i 4 4 contribution to form of the asymptotic trajectories: tum theory, momenta. However inmacroscopic, the classical limitIn in this limit which the soft the factorsand describe external gravitational the fields radiative finite part during of energy a the states low classical frequency scattering are electromagnetic [ Here the sum over denote the charge, momentumpositive and angular sign momentum for of analso the ingoing contain particle a and non-universal term negative at sign the for subleading an order. outgoing For particle. S-matrix elements in quan- spectively by and 3 Classical analysis The goal of this sectionspace-time will dimensions be by to examining calculate the them logarithmic in terms the in classical the limit. soft factors in four This also does notexcept depend for on the the total details momentum carried of by the these momenta particles. of massless final state particles This can be rewritten as and the third term thecancel, sum leaving excludes the behind a contribution wherecontribution we where set represent massless state is then In the first term the sum over JHEP02(2019)086 are ) at (3.6) (3.7) (3.4) (3.5) x µ a is the ( c ) a b ( µ , m b A µ b p m ]. This phase b ) and replace represents the η | . 3.2 94 σ ' , | . , ) ··· ) , reflecting the fact ) σ 93 ρ a , µ a ( ρ a + , but when a variable p p p | σ dσ bµ µ a 91 µ a σ ) and ( c | c dr − 3.1 − ≡ .k µ a ) ln µ a denotes the asymptotic four a .k p µ a ) p p a ρ a p b ρ a σ c p independent asymptotic value. c ν a ( V ( ( c approaches zero asymptotically. µ ρ b ρ σ ) into ( k k µ a − ). ν a µ p σ p ν a ε 3.4 ( p ,V 0 b a µν µ a ) q ε c 2 and )) a > r a is large and negative for incoming particles σ X X ( 0 σ 1 /dσ b 1 x r µ a − − ω ω – 13 – − dr a ln x + spin = ( ln η ( i a i µ a − p m ( + ) + . We have + a σ b ν a δ ( q is the frequency of the outgoing soft radiation — we can p ν ) a ) = µ a ’s denote the asymptotic values of µ a r .k 0 σ σ .k p µ a p ( a ( k a µ p − p µ a p ε bµ µν comes from orbital angular momentum and is universal. p ν a = ε V p 1 b a ) ω may contain a non-universal term at the subleading order, the − q a X σ ω ) the X ( b , | µ a = em r σ = S 3.4 | due to particle ] indicates that if we substitute ( em dσ η ' gr S S x ), ( 89 Z µν a — where ) it denotes the value of the variable at proper time J π 3.3 σ 1 1 2 -th particle and the proper time − due to electromagnetic and gravitational interactions. We know from explicit a ω µ a denotes the usual Dirac delta function with the understanding that we have to is positive for incoming particles and negative for outgoing particles, c ) = a x + ( η δ ) by ln b ( µ | Irrespective of what forces are operative during the scattering, the coefficient Analysis of [ A σ | space-time point where choose the zero of the argument for which will also be determined below. 3.1 Effect ofWe electromagnetic shall interactions first study theelectromagnetic effect interaction. of logarithmic For correction this to the we trajectory need due to to compute long the range gauge potential compute comparison with known resultstions that there in are the no case additionalrange of phases interaction scattering in there the via is soft electromagnetic ansoft factor, interac- additional photon but phase or in reflecting the soft the case graviton of effect in gravitational of the long backscattering background of gravitational the field [ Note that although term proportional to ln determined only by the long rangeThese forces will acting on be the taken incoming to and the be outgoing electromagnetic particles. and/or gravitational interaction. We shall now and that the difference between ln recover the logarithmic termsoverall in phases: the soft factors up to overall phases. This gives, up to Here and in thean following argument we ( shall useis the convention written that without whenTherefore a an variable in argument, is ( followed we by take it to be its mass of the and large and positive for outgoingeffect particles. of long The range term electromagnetic proportionalThis to and/or gives, ln gravitational for interaction between large the particles. where JHEP02(2019)086 , ) 2 -th ) at a , x (3.8) (3.9) σ (3.12) (3.14) (3.15) (3.11) (3.13) (3.16) (3.10) µ b ( 2 + 0 b p / 2 3 2 b ) . } 2 b m 2 .x > x 1 , 2 a / b m 3 2 bµ − 0 V 2 a } m / } ( V x ρ a 3 1 = m . p } + p + 2 b µ b 2 − 2 a µ a − p / + 2 m p V 3 aµ 2 ) bµ 2 a b ) . } .x − V a V 2 a b b 2 m .p µ a ν } x V .V / a .p p .V x b µ a 3 b . p − ρ b 1, a + p } + V p p 2 b 2 − 2 2 b ( V ν b ( σ − 2 / { ) ( { p bµ { ) 3 m a m 2 b δ bν V } = b ) we get 2 a , .x | − b V .p 1 q π m b 2 b q aν π µ σ a b m ν a -th particle if either both 2 a 2 4 . V a p 4 V q aµ x − b p V ( ( q σ c 2 3.13 + m − µ b b { 2 { − x a ) p η 1 2 a b b .x a ) { =1 a a η ), ( q b q bν m b + π π a 2 b η 6= bµ η a .V b V 4 4 b V 2 X a | b .p q − V η ) m η b b 2 =1 V a aµ 2 a 3.12 b q p ( .x = 6= V η ( − =1 { b b b a X m ' a b { η η 6= V ' − η 2 2 ) b = aµ ( X ) we shall ignore the logarithmic corrections a b ) -th particle we get, using r η b q 2 π 2 x 2 1 a q p dσ a 4 σ / ··· π σ ( ρ 3.7 d q µ 3 ) 4 b k b – 14 – π + a } 1 ( µ a .k V µ ' b 4 2 1 η a ε η =1 m a ) we get A ) σ b . This gives, using p + a ν 6= a − η σ ' of the b σ q X + ∂ η a ( µ = 2 3.7 a ) η b ν a ) a V ) x V − ). Comparing ( ' η a a b ( V | − σ X ) .x σ ) ( .V σ b b ' )) .V x 1 | )) ( µ 3.3 b = ( a V a ) σ dσ − aµ σ V ) A ( V ( b µ a V ( σ ω 4 ( ν a ( a p { dp r − b +2 r ν a A ( ln r ( b 2 c ) µ ) q b π = x i b ∂ ( a ( µν 4 µν − q − σ − F F ( ν a a b a p + V η ) = q µ a δ =1 a x b c -th particle takes the form ( 6= ) now give µ a = η b =1 ) a X a .k a p b b a η a 6= ( η µ ) = µν b r 3.6 -th particle. In evaluating ( X p a ε 2 b F a η q )) a a σ 1 X = ( m -th particle will feel the field produced by the b ) a -th particle are outgoing or if both particles are ingoing. Therefore the equation r = ) and ( − b σ − ( = x em 3.5 dσ ( aµ S µ a c Note that even if we assume that the logarithmic corrections to the trajectories are generated predom- − dp 4 ( + inantly by electromagnetic interaction,the the gravitational radiation resulting during acceleration the can scattering. generate logarithmic corrections to and Eqs. ( where in the last step we used ( On the other hand we have Now the and the of motion for the At the location the solution. Substituting this into ( From this we calculate to the trajectory and take δ where the sign in front of the square root has been chosen to ensure that velocity of the JHEP02(2019)086 . µν  e

a a (3.21) (3.23) (3.18) (3.24) (3.17) (3.19) (3.20) .V .V b b . . , V V 3 

. 3 2 b  } − 2
} − 3 / -th particle is m bµ a ρ a ) 3 b 2 a 3 a p } .V ) 2 b b µ b m a .xV .V p 3 V b b m . .V 3 V 2 a V } b − ) − 2 V 2 + − bρ m 2( µ a ) . This gives . µ )) a 3 p  2( 2 ρ ) − ρ b x σ ρ  ) .p a .xV p ( { due to the 2 ρ b /σ h b a b α { ) b p .V α a r  V V x a 2 b b V c µν µ 2( -th particle b V + ρτ .p 1 2 − η − m b  a ρ η V 2 a 2 1 2( x p ) x . 1 2 + ( ( − ρ a { m { 2 − and its trace reversed version µ α p α + − a a b x  ( µ b b V V V bτ p µν q + π bν µν ρ 1 2 µτ + b 1 2 a V δ h 4 V h − η q V 2 ) − bρ ( µ a ) 1 2 + = σ bµ  ) V p 2 ( .x V α =1 ρ b a / + a  2 b b 3 b p µν bν 6= η / V 3.9 b ( V bτ } X 3 a V ( m 1 2 η 1 V 2 } }− ) / 1 ) (3.22) ρ − p − σ 3 bµ bτ ( 2 k } σ V  1 − – 15 – ) ( 2 b ν a π , e .k 1 bµ a τ 2 p 4 1 a 2 a 2 .xV V ) m  / b p / .V V a 2 a b 3 µν ) b ' V a ) } ε .V V αµ m ) σ 1 µν + b .p ( η ( x b η − τ { a V ρ ( 2 − p a 2 ( dσ m X x ) b / ) − b { V 2 { 3 1 1 ( µν a ) m } b Z e − a ))  2 a µν .p ) the left hand side is given by ω b m σ π g x .V 1 µρ ( m ( p 2 b η ( a ln + 3.3 V 1 =1 r { a 2 1 2 ≡ ( we get the analog of ( i =1 b a ( ) 6= b η { b 6= α η X ) for the gravitational field produced at ) = + a b ) − X =1 .x b a µν a η b x b b η ( ρτ bρ ··· 6= h ( η ν a b m V 3.7 X ) 2 a V Γ p 2 ( b η + ( µν µ a { .k 1 bµ e 1 =1 p =1 πσ a a a σ πσ π b b V b 1 4 6= π p 6= η b η 8 8 b b µν X X  a 4 a m a V ε η η η = = + − ρ a − π a 1 p ) = 4 X µ a ) = ' − = σ a c ( x = η ) b ( − r σ α gr µ a 2 ) = ( b p S α a ( ρτ α a ρ a r dσ c Γ c 2 d and On the other hand using ( From this we can write down the equation of motion of the The associated Christoffel symbol is given by, in the weak field approximation, Using tational interaction. We introduce thevia graviton the field equations Then the analog of ( 3.2 Effect ofLet gravitational us interactions now suppose that the logarithmic correction to the trajectories arise due to gravi- and JHEP02(2019)086 . ) ) ] ρ a ρ a | p t p | (3.28) (3.29) (3.30) (3.31) (3.25) (3.26) (3.27) µ b µ b p p ) ln − − 0 µ a µ a p p . ρ b ρ b ω m p α b p ( ( p + ) is a null vector 2 1 by comparing the 2 / ˆ n ) as / − 3 , b 3 µ = } } b X is a four vector to be k.~m 2 b ~ η m 2 b ( 3.27 i µ m m π = (1 1 2 a 2 a 4 m a n a m m − . .p .p b ] exp[ − b − = p p 2 2 ) α ) ˆ n.~x b a a , V .p iω , | .p b b b + finite τ ρ p | p | + m ( t ( | dτ 1 { dx { ln − b iωt = ν µ b X =1 a ) ln ), we can express ( =1 − a η b ) the four momentum of the soft particle, dτ m 0 b 6= η dx 6= η b ˆ n X b a up to overall phases: X , π a m , η 1 + η . i 4 µ νρ 3.27 5

τ ρ (1 n ρ Γ 2 b

k = ω µ k 2 b – 16 – = exp[ − ν a − m .k .k µ n 3 − p i i 2 a a m a ε ) = 2 a p p

a m µν | m using ( .n q t ε µ m 3 ) = b | 2 ) = 0 ) we get, 3 ) into the equation of motion x τ V + ( ( 2 x a ( − a dτ , k t µ d X − X ) ln 0 2 α 3.6 i b ≡ x ) 0 , we get the following expression for 1 2 k 1 3.27 n a V ) t − − νρ µ a 3 .p ω = | ω b = ( ˆ n m .p b b i b p ln ln k x p − ) and ( m 2( n.V π ) of Γ i 2( π |  i ~m 8 3.5 8  ( 1 × − + × + 3.21 b − = a b in terms of X ν a η q ˆ nt p τ µ a µ a π .k 1 .k − p p 4 a a µ sign reflecting the fact that it is an outgoing particle, the wave-function of p ~x p ε µν −  ε − = a k. ~ a i X is the affine parameter associated with the trajectory, α X − = m τ h = terms on the two sides of the equations of motion: In this case we expect the wave-form of the gauge field/metric to also have an additional em 2 Even if the logarithmic correction to the trajectory is generated by gravitational interaction, the parti- gr 5 exp S S /τ cles can emit electromagnetic waves.and This a happens neutral for particle. example if we have a scattering of a charged particle Therefore if we denote by the overall the particle will be proportional to Now eliminating and using the form ( 1 where along the asymptotic direction ofdetermined. motion Now of substituting the ( soft particle and phase factor reflecting theother effect particles. of For this the let gravitational us drag characterize on the asymptotic the trajectory soft of particle the soft due particle to as the and Substituting this into ( JHEP02(2019)086 , . .

b 2 b 2 b m m (3.33) (3.34) (3.35) (3.32) k.p 2 a 2 a 1 m m − 3 b 3 = b X − − η 2 2 ) ) a a (0) gr .p .p S b b p
p 1 ). This has been re- 2( 2( −   ) ) R . ρ a ρ a R ω p p    µ b µ b b ) as p p + ln 1 − − k.p ν a − a µ a µ a 3.31 p 1 q p p ) to ln( ω µ a − .k ρ b ρ b b ]. p µ a a = p p b .k X ln p p ( ( η a µν 90 µ 3.32 2 2 p ε ε / / π R i 3 3 4 a a } } in ( ln X 2 b 2 b X b b π R m m i 4 2 a 2 a a a k.p k.p − m m .p 1 .p 1 ] that the effect of gravitational backscattering b    b − − − − and b p p b = 2 2 = – 17 – b X ) ) b X 94 η , η a a
.p .p
1 b b , 93 1 , b p p − ] = exp − is the distance of the soft particle from the scattering ( ( { { 91 R R R k.p R ) we get the net soft factors to be =1 =1 a a 1 ln b b 6= 6= − η η b +ln b b +ln X X a a = 1 η η 1 b X 3.26 η − ρ − ρ ik.m k where k ω ω ν a .k µ .k (0) em p a ln ε a ln R S p p a exp[ µν q ) and (
ε π π i i 1 ), we can express the second factor in ( | ∼ 4 4 t a − | a X + 3.25 + X R 1 ) in the expansion, and multiplying this by the leading soft factor, we get 3.29 a ν a 1 − q p − µ a ω µ a .k ωR ω + ln p .k ]. It is natural to absorb this multiplicative factor in the wave-form into the def- a p ln a p µ 1 ln µν p ln( 89 ε − π ε i π i ω 8 ω 8 a a X + ln X + It follows from the analysis of [ = = π i gr 4 em S S 3.3 Effect of electromagneticWe and now gravitational combine interactions the resultsfor the of soft last factor two when subsectionsresponsible both to gravitational for interaction write the and down electromagnetic logarithmic the interactions corrections are general expression to the trajectory. The logarithmic terms get and Adding these to ( viewed in [ inition of the soft factors.of Expanding order the exponential in aadditional power contributions series, to picking up the the soft term photon and soft graviton factor at the subleading order Since this is aeffect pure on phase the it electromagnetic/gravitational does wave-form not [ affect the flux.of However the it soft does photon/graviton produce actually observable converts ln soft factor. Using center, and eq. ( The second factor can be regarded as an additional infrared divergent contribution to the JHEP02(2019)086 . . , n n

p b 2 and 2 b (4.1) ) m m (3.37) (3.36) ··· n ’s and a 2 2 a with ( , a 1 p m m p 1) 3 3 n, − ( − and Γ 2 2 ) ) 1) a a n, .p .p ( b b p p = 0, the amplitude 2( 2( ,   k ) ) ) a ρ n ρ a p + p p b µ µ b a p p p ··· , − ν a a − a 1 p a µ µ a q p 2 µ a p P p 2 .k } ( / µ a } b ρ p / ρ b ) a ρ a 3 .k ρ a 3 p p n p p ) depend only on the charges and p } a and one soft particle of momentum ( ( p } µ µν ( 2 b µ b p 2 b ε ε µ b 2 n 2 p / p / m )Γ p m 3 3 3.37 2 a } a − a 2 a − } } a X X µ a 2 b 2 b µ a m ··· p m p b b p . There is however a potential problem. , { ρ b m − m ρ b 1 − ; 2 a p gr 2 a 2 p p 2 a k.p a k.p ) { S ) { m a m 1 1 ) and ( 2 b .p a .p 2 b ε, k b − − b − ( .p − b b m or = 0. Therefore we cannot keep the .p p = = p b m 2 – 18 – 2 b b b 2 a S X X ) 2 a p ) η η 3.36 p a ( a a ( m em p ' m {

{ .p .p S a 1 1 ) b b b b q − − p π p q π ( P a , k ( 4 ). Usually this problem is overcome by including the a R R 4 { q { finite energy external states carrying momenta n q p =1 a n =1 4.1 a b =1 b 6= a ) respectively. η =1 6= a b +ln +ln b η X b b a 6= ··· η X a 6= 1 1 b η η X , b a η has momentum conservation X a 2.6 − − η 1 ρ η ρ p ω k ω k ρ ( 1) ρ ν a .k µ .k k k ln ln n, p a 1) ε a ν a ( .k µ .k p p with just a n, p a ε a µν ( q ) p π π p ) and ( ε a i i Γ n µν 4 4 q ( a ε a 2.4 ) is the soft factor X + + a X } 1 a a ν a X 1 a q − X p 1 p − µ a ω 1 µ a .k { − ω .k p p a − ; ω a ln µ p ω ln p µν ε π ln i ε ε, k π ln 8 i i ( a 8 i a S X + − X + − has momentum conservation = Note that the soft factors given in ( = ) as independent variables in ( n em to an amplitude Γ gr ( S S where While the amplitude Γ Γ k momentum conserving delta-functions in the definition of the amplitudes Γ finite energy external states carrying momenta k This relation takes the form there is no derivativeshall with carry respect out to some the explicit external quantum computations momenta. to In examine to the4 what next extent few this sections holds. How we to treatIn momentum quantum conservation theory, and single infrared soft divergences theorem is expected to relate an amplitude Γ These reproduce ( momenta carried by theplicative external soft states. factors in Therefore the these full can quantum be theory — reinterpreted since as there multi- is no angular momentum and added up, yielding the results: JHEP02(2019)086 , ) ) b } R n ( reg a p and ln { (4.3) (4.4) (4.2) ) ; multi- , since ] since n U ( 1) ε, k − ( n, k.Q of the soft ( reg S k.Q = ) for any ) appearing k a } Q p a and hence of p a ’s) since by the { a ) we see that the 1) ; P p ( n, ( reg δ k.Q ε, k 4. However since in } . ( µ b we can add a term of ) S = 0. This has the effect n . p a . Choosing K 1) p D > ∂/∂p R n, ··· a ( reg { , besides the ones that appear terms that we are after since µ ) as (1 + 1 factors, the term proportional P ]Γ k p 1 1) ( 1 . 6 + in the expression for Γ − K ) n, k.Q − ) ( n ω k ( reg } ω R a vanishes. However addition of such p independent. However this can affect ]. Since the logarithmic term in )Γ { a to the extreme left. } 41 ; ω p = exp[ a gives S p a is 1) { 1) ε, k ] the infrared divergent factor of Γ ; P n, ( n, ( Q ( , ) by a multiplicative factor of exp[ K of the finite energy particles. This makes Γ Γ ) (0) ε, k n ’s amounts to having an additive contribution will be reflected in the infrared divergent con- a ( ( ) in power series of the momentum 4.2 a p still contains some residual infrared divergences – 19 – S , k p 1) ) n 1) n, ' + in ( ( reg ]. Expanding exp( ( reg n, ) a ) is now replaced by (which could be a function of the ( reg R k.U S ]Γ p term and 1) have infrared divergences which can be represented as , k a k.Q ) n, 1 Q 4.1 ln K n ( reg n − ( p P − ω ( δ ··· , 1 = exp[ and Γ p ) ) by exp[ ( n } 1) ). ( 1) a } n, Γ p n, ( a we can include terms proportional to ln ( reg { p ; Γ { Q for any vector ; constructed from the a ε, k U p ( ε, k ( a S S ) as a relation between distributions. The soft factor P ). This is due to the fact that in the definition of } 4.1 Q. a p of the form . In any case we shall denote by exp[ { ) is treated as a differential operator that also acts on the delta function and generates ) ; n (1) There is however a potential ambiguity in the definition of Γ In four space-time dimensions there are additional issues due to infrared divergence. ( The situation here is somewhat different from the one in [ 4.1 S is in general a function of the momenta = 4 we only analyze subleading terms containing ln 6 ε, k ( to that we are after is beingdivergent represented factor as on a the multiplicative factor right instead hand of side a differential can operator, be the moved infrared past plying the leading softthe factor. leading soft It factor does hasthe no not genuine ln affect infrared the divergent terms ln in proportional the to ln definition of for some vector the form momentum conserving delta functiona in term Γ changes the definitionthe momentum of conserving Γ delta functionof in multiplying Γ additional contribution appears at the subleading order, and has the form of The residual infrared divergencestributions in to Γ S K free from infrared divergences,for but gravitational Γ interaction. Eq. ( in Γ define regulated amplitudes via the relation: to derivative of the delta function will not appearBoth in the our amplitudes analysis. Γ overall multiplicative factors multiplyinginteractions infrared these finite factors amplitudes. aregravity common For there and electromagnetic is can a be residual factored infrared out divergent of factor the in amplitudes Γ but for in ( the Taylor series expansion of particle. The subleading termis in included this in expansion, theD given full by subleading soft theorem in dimensions treating ( JHEP02(2019)086 . , ` n b p p (5.3) (5.1) (5.2) ` ··· , -th leg, + 1 b represent b

p p of masses a γ → φ . n a ∗ ` ) φ ` φ (c) ··· ab a 2 µν ( − ··· m a K ` , γ p 1 − − φ ) + k a γ µν -th leg to the a φ + p η a = 0. We further assume µ a p k a = A . q -scalars. Then the relevant a ) o scalars 3 ) n iq ab -th particle is outgoing it will =1 µν ( , carrying momenta ab n a n a µν ( n − φ G a P b φ p + ,G and µ ··· ` ) ) , ∂ µ 1 i + ab µν )( ( φ b A a ∗ . p − . K flowing from the φ 1) 2 → n # µ ` ` ) n, ∗ n ` A ( i ··· (b) , satisfying φ + a n + i − iq q .` b ··· b 2 – 20 – p , with p + ∗ 1 ` ` and Γ ··· a involving internal photon line connecting two different a ∗ (2 , − − the sum over tree and one loop contribution to this ) . 1 )(2 φ λ φ ) n a 6= k 1) q has been shown in figure ( µ ` p = i 1) b n, ∂ γ γ + + ) ( ( n, − n a + ( n i (  p k ` a 2 φ for p µν ` − n =1 b η 2 a (2 X ··· − ` 1 i − .` b a − p p λ φ µν ` F (2 + + ν b µν ` to the leg p µ F ` a → 1 4 ` . We denote by Γ will denote the amplitude without the external soft photon to one loop . Our goal will be to analyze this amplitude at one loop order, involving = (a) ··· − γ 0 a ) 2 k ) p n " ( ab + µν ( x a 4 ` p and carrying U(1) charges and K d − 1 n . One loop contribution to Γ external states corresponding to the fields a Z k m p n γ + ] a We shall use Feynman gauge and decompose the photon propagator of momentum In our analysis we shall ignore graphs with self energy insertions on external legs and ··· p k 99 , and 1 as [ where, assume that we follow on-shellthe renormalization tree so level that propagatorsthe the are external mass the scalars parameters physical cancel appearing masses. between in Γ The wave-function renormalizationconnecting of the leg an internal photon connectingin two figures matter lines. Theamplitude. relevant diagrams Γ have beenorder. shown One loop contribution to Γ We consider in this theoryk an amplitude with oneAll external momenta outgoing are photon counted of as momentum positivehave if negative ingoing so that if the part of the action takes the form 5 Soft photon theoremConsider in a scalar theory QED m containing a U(1)that gauge there field is a non-derivative contact interaction between the Figure 1 legs. The thick linesphotons. represent There scalar are particles other diagrams and related the to thin this lines by carrying permutations the of the symbol external scalar particles. JHEP02(2019)086 (5.5) (5.4) denotes , ` , and not b  2 c m (a)). i and − + 1 a , 2 c p ··· ··· ` − = 0 2 c γ . Sum over all insertions ε.` m a γ 2 c 4 p a + γ p k i q i 2 ) k , it is pure gauge. This allows 2 − ` ν ` − µ + b ` )] c p p k ` ( +  we do not carry out any decomposition. + c c b b q p p − = (2 → a ε. = ` c ··· j ··· 2 c ··· γ i q – 21 – m ` γ . For involving internal photon line connecting two different − b + ` γ a ) γ a 1) i 2 p k γ . There are other diagrams related to this by permutations ) p − n, ) − ` ( k ` n a + k ( a p + to leg ` p c a p ( + 2 c ) p µ ` (2 + ε. c cµ refer to the external momenta flowing into the legs p i q [ b (2 c ··· ··· p q c γ ` i q γ and µ a a ` γ γ a p p . 2 c p 2 k ` k ) . One loop contribution to Γ m . One loop contribution to Γ i .k − + a p c 2 ( Since the K-photon polarization is proportional to p / Figure 2 points on the samescalar leg. particles. There are Inwith other the diagrams mass related last renormalization toto term that this 1 the has by permutations + to of on be the the adjusted external scalar to line cancel represents the a net counterterm contribution associated proportional and whose diagrammatic representations have been shown in figure the momentum flowing from leg us to sum over K-photon insertions using Ward identities of the external scalar particles. Note that necessarily the momenta of thehave lines additional to contribution which from the external photon soft propagator momentum, attaches e.g. (which in may figures Figure 3 JHEP02(2019)086 . k and , (5.8) (5.9) (5.6) (5.7) ) can takes a (5.10) = 0 ) o n 5.8 ( 1) /k.p Γ n, for which ( self a em 2 . γ γ ⇑ S ε.p # + Γ ] a ) = q 1) 99 i n, 1) =1 ( G , − n a n, ( ) 2 , ` P ) i + Γ γ . ` γ (1) em + − 1) o + 2 ) n, .` b ( tree ` b n ) ( G p iλ S p ` Γ + + + n (2 + + . ] )(2 .` ) + Γ b a a b ` i p em ) p n − γ ε.p f f ( tree (2 k.p + K i λ . γ . k a )(2 a Γ ) 2 ` p q ). The arrow on the photon line represents ` ` = i n (2 ) − − n =1 5.5 + − em n a X a ( tree 2 .` = exp [ S p ` a iλ , p 1) (2 = − = Γ 7 n, ) and ( (2 = f ( ) 1) .` is the leading soft factor – 22 – n Γ a i n, ( G 1) 5.4 produces an exponential factor [ ( self p denotes the sum of diagrams in figure n, vanishes when we replace the internal photon by 1 − (0) γγ em ( self 1) (2 2 S + Γ 3 1) n, + Γ ` ) on the left hand side has been expanded in a power series in ( i , ) k n, + Γ 4 ( self 1) n o 1 − ( tree ) ` + ) n, 1) 4 π Γ ( G 2 a n or Γ with the polarization of the incoming photon stripped off. ( G d ` p n, where ( G (2 ) c is equal to the first term on the right hand side up to terms a 4 q n − + Γ ) ( ` ) are multiplied by the appropriate delta functions. We also implicitly P 1) + Γ − (1) 4 Z em ( π + Γ 1) δ = 0, and we have: n, d b ) 5.8 S ( tree (2 q n ) n, 1) ( tree ( tree n a + ( G n, Γ Γ q ( tree Z are computed by replacing the internal photons by the G-photons in n b Γ (0) em ] q 1) a ` ~ ` ~ ). This gives ) ` ~ | d ` .` .k ~ | k 2 − | and | 5.16 | ( 3 b c ` π (2 d i µ b 0 d p p a are positive, i.e. both lines represent incoming states. ε ~v − ` 4 (2 | c − ( ’s from the denominators that never vanish. ) ` b reg reg , q 4 ` i ~ /E π 0 i Z . Z ) E a π d reg b c | ~p (2 2 + i Z d` ~v b b 1 defined in ( X ~v ) run over the entire real axis since the regions outside the allowed range .` E = b . To this end, note that once we have imposed the range restriction on i given in ( reg a and a ∞ = | − = ( 1 − em a E −∞ p Z ` ~ ∞ i E | a a 1) 0 ~v a Z 5.22 b K ` (1) em , -axis with q ~v n, to E E 3 | i + ( G 0 b z S ) a ` = ( will run over the range where p q 3 1 π 2 = − ` 0 d ) a −∞ ` 2 = (2 π 6= ab ` a,b ) and b b X I (2 i i 4 E reg b i 2 ) we now see that the net contribution to the logarithmic terms in ) integration runs over a limited range, one might wonder why we are choosing the ` integral in ( , Z E 4 π 0 a 0 µ b a | − d ≡ ` p ` ` ~ is the factor (2 E | E 5.11 1 a = reg em reg = E em a = ( reg K E Z − K 0 0 ab ` I Let us first analyze the second term. Since the result should be Lorentz invariant, it So essentially we need to evaluate From ( ≡ = Since the 8 ab -axis, we can express the second term in ( I integration range from we can let the do not generate any logarithmic contribution. should not depend onare the chosen along frame. the positive Forz simplicity choose a frame in which Note that we have removed the So now if we close thefrom contour in the lower half plane we have to take the pole contributions where we have assumed that The integrand has simple poles at, the loop momentum to the momenta of the finite energy external states: obtained by dividing Γ where JHEP02(2019)086 . ),    ). and   a 5.24 b 2 b 2 (5.28) (5.27) (5.26) ~v p p 5.26 to some a 2 a 2 p p ω  − − ). Finally if if necessary, θ are negative 2 2 ) ) . µ b b b , `   . Therefore the cos E 5.26 .p .p are parallel. 2 b − a 2 b 2 b µ ’s in the last two a a a ` v b p p p p v m i 2 a 2 a ~p ( ( to 2 a p p and − µ q q m ` a 1 − − → −  − + and − 2 2 E µ b b ) ) ) ) − | | ` 2 a 1 b b b b .p .p ) ~p θ b ~p ~p a a .p .p | p p a a .p p p a + − | cos   ( ( b p b b b ( v v ln q q E E , since | q π − i )( )( b + − | | 2 1 1 b b a a .~p l ~ − . ~p ~p a −  .p .p | b ~p 1 ω ) are reversed. But this can be brought a a , | a ~v b p p − | + v 1 η ln = a a a   1 η − | − | 5.22 l E E δ ~ b π b | ln ) in the integral, and using the fact that the 1 ( ( ~p 4 v    2 b  l || ~ /x . p ) . We get b 2 a | a 2 a – 26 – ln p ~p = a (1 will continue to be described by ( ~v | p a 2 b | ~v 1 p | P b i cosθ E − ~v | ( | − − b > 2 a l ~ − d 2 | ) 1 1 ) + ~p | − ) .p ` ~ b b b . x a 1 | a 1 b ~v l ( .p ~ p | 1 ~v | − .p 1 − | ~v | a a 2 Z a p ) by making a change of variables p 1 ( = iπδ ( 3 E 1 − | ) 0 l are outgoing instead of ingoing, then ~ − b q ` 3 q ω π ~p b 5.22 ω d ) = | 1 1 (2 ln − i ln 1 − ω b − . We will again evaluate this integral in the frame in which ω -axis with and in the last two terms in ( − reg b ω z E ab ln Z ln x E a a i 00 ( a 1 ln b 2 / I π i 1 E i 4 π E π E 2 2 . However the imaginary part of the last term makes the integral non- i 8 b piece of the last term the integral vanishes since the integrand is an odd a 4 i θ π π q E 8 8 − ) come with the same sign. By changing variables from a i q ' for which our approximation of the integrand is valid ranges from = = = = ) we can now write down the expression for the logarithmic term in the sub- a | 6= a,b 5.22 b ` ~ 0 ab X ab | 00 I i 2 5.20 I = Combining these results we get We now turn to the contribution from the first term on the right hand side of ( If both the legs are parallel to the reg em b K Using ( It is easy tolegs check are that outgoing the or form one of leg the is incoming contribution and remains the unchanged other even leg when is both outgoing. we can ensure that bothlower the half poles plane. are In in this the case upper there half will plane be andwhich no close we analog the will of contour call to the the contribution given in ( and the signs of the back to the form givennet in result ( for the residueone at of the momenta isterms outgoing of and ( the other is ingoing, then both the where in the intermediate stage we used function of cos vanishing. Using 1 value of finite energy, we get, Without the ~v JHEP02(2019)086  b (5.30) (5.31) (5.32) (5.33) (5.29) k.p ). This defined a a .p k.p reg em 5.20 + K b . . ) .p  n ( tree − aµ Γ ∂  given by the second ∂p 2 (1) em / ν a b S 3 (1) as given in ( em p } reduces to S + 2 b − (1) p em ) 2 a . However if we make an ad em S n p 2 b ` aν ( tree , ∂ K p Γ − ) 2 a ∂p 2 } n p ) ( µ a b . p Γ em . .p }   K a b 2 ν p  em / (1) em ( .k ρ a (1) k 3 em b a { K S  k.p p µ b S 2 b p ε µ b a a {   + p (1) em a m 2 b 2 b b + q 2 a S .p − p p k.p vanishes at one loop order, we get (0) em 2 a 2 a ρ b ) m a ) p p + p = S n ( tree n b X µ a { − ( G − − – 27 – Γ p 2 2 2 (1) em  ) = = .p ) ) b 2 b S (0) b b em − 1) .p S m (1) .p .p em a  2 a n, a a b S p ( = 2 b p p ) holds if instead of scalars we have interacting fermions. ( m ( ( Γ ) to replace the left hand side, and throwing away terms p  2 a 1) q q p ρ n, 5.29 b ( G .k k , + − 5.10 − the integration over loop momentum runs over all range and a denotes that we are using the differential operator form that b b .p µ a 2 p .k  ) a .p .p a b + Γ p em a a b ε.p (1) p of the external soft photon, then .p p p q K 1) S a a 2 a   q n, p ω q ( self ( , using ( ln ) and the fact that Γ b a (1) b em q X =1 i λ a q which vanishes, we get 2 + Γ b a S 5.7 6= η 2 a q b X = a = ) q 1) a η n a 6= ) ( tree n, a,b b ( tree 9 (0) X 6= em n a,b Γ n b =1 ( tree X ) and we recover the correct logarithmic terms in ). The rest of the contribution is extra. Γ S a X 1 1 (1) 1 em − − b − S ω 5.21 3.16 ω ω ln ln ln than the energy in ( suggests an ad hoc rule for computing the logarithmic terms in the soft expansion In the definition of we have an infrared divergencehoc from the restriction region that of the small loop momentum integral will run in the range much larger Then using ( Using Γ like arises in the quantum theory: theorem: where the ‘hat’ on 2 2 Suppose we assume the validity of the naive version of the subleading soft photon We have also checked that ( We end this section by making some observation on the results derived above: 1 1 π π π Since the presence of the logarithmic term makes the finite part ambiguous, we consider only the i 1. 9 8 4 4 + − − logarithmic terms in the subleading factor. The term in theterm first of line agrees ( with the classical expression for This confirms that the logarithmicof the correction particle. to the soft factor is independent of the spin leading soft factor JHEP02(2019)086 . # (6.1) (6.2) ∗ n φ ··· ∗ 1 ]. λ φ 99 + n φ ··· 1 = 0 can then be avoided λ φ ], that assumes existence of 2 ` + 83 ) the Feynman propagator for ,

a , 81 ) φ 5.22 ∗ a n ( φ Γ 2 a by comparing the two sides up to one o m (1) gr + (1) gr S a S φ + , we shall postulate a relation of the form ν ∂ 5 (0) gr ∗ a – 28 – S φ µ n ∂ = µν g 1)  n, ( Γ n =1 a X ). If however we replace in ( − R 5.29 πG 1 16 ), since the contribution from the pole at ), the quantum result found here has an extra term given in the second " is known to be valid to all orders in perturbation theory [ g 5.29 5.29 em det K − p x seems to bequantum to result replace by the retarded propagator. Feynman propagator of the photon in the loop in the and third line of ( the photon by theline retarded of propagator, we ( getby only appropriate the choice contribution of fromquantum contour. electrodynamics, the Therefore the first at rule least for for relating the the soft quantum and photon the theorem classical in result only at one loopsince order, the result may be valid to all orders inresults. perturbation As theory, already noted,line compared of to ( the classical result that agrees with the first down for the contributionsoft where momenta. the loop Onthe momentum the region is other of smaller integration handmomenta than where we or the the larger. do external loop not momentum expectThis is argument any of also suggests large the that although contribution order we from of have carried the out the external explicit calculation the action of the differentialmomentum operator integration on the to amplitude, lie restrictingthan in the the momenta region a of of the range loop finitebe larger energy justified particles. than by noting With the that hindsight, the this soft1PI general prescription effective momenta arguments can action of but with [ no smaller powers of soft momenta coming from the vertices, breaks in quantum theory — begin with the usual soft expansion and explicitly evaluate 4 The second observation concerns the relation between the classical and the quantum d 2. Z and gravitational interaction. As in section and try to determineloop the order. logarithmic terms in Even though in this caseplex we in could order take to the extend scalar the fields analysis to to be the real, case we where have the kept scalars them have both com- electromagnetic 6 Soft graviton theoremWe in now turn gravitational to the scattering analysis ofinteracting the via soft gravity, graviton to theorem in one the loop scattering of order. scalar The particles, action is taken to be We shall now writeof down observation 1 the works. resultsobservation 2. for We shall the also other explore cases if and the test results if satisfy the the generalization generalization of JHEP02(2019)086 ) is and 3 (6.6) (6.7) (6.3) (6.4) (6.5) k ν ↔ µ β scalar fields is . 2
p n and the Lorentz
α νβ 1   2 η αβ p ) has a complicated k
and Lorentz indices η αβ αγ µν η 2 βγ αγ − µν
η 6.6 αγ η η k η η η 1 µν , , γα µσ k η − νσ . µσ 1 σγ η η µβ  η 1 2 η ν −
k 1 ) η η 2 β 2 2 σβ σ .k µα 1 p ρσ − 1 η = 3 .k ) attaches to η m k µ η k µ 3 1 3 k ν 3 3 ν νβ . 3 k p − µν 3 k k µν .k η k + i η k ν and the graviton carrying ingoing 1 2 + 1 αβ k µα 2 + σγ .p k − + η γµ µα η p 1 η αγ and η η + p ,  + βγ η − νρ γβ ( νβ 1 2 ). Even though ( η βγ βσ η η 2 2 η αγ p k µσ η η µν βα η νσ .p η m , µσ η η σα . 6.6 ) respectively, then the 3-graviton vertex µα 1 η νσ 1 αν η η µσ η p νβ η α η k + νγ − ν
η µν 3 3 η
σγ 3 η + 1 in ( } µ 3 k 1 µν ), is given by k σ .k νβ 2 ν µ .k η 2 γ µ 2 1 .k .k η p νσ 2 3 2 k – 29 – 2 3 1 p k µν ν k η k iκλ η k µ 1 k k αβ β ↔ 3 .k 2 p + η µρ + 2 k + + p + + ) and ( σ η 1 2 k 3 + 3 σγ + αβ + βγ ν η αβ νβ + .k + 1 2 η .k η 2 ν 2 η 2 2 and αγ 2 p k αβ µν k βγ αν i η p µ µν .p ), ( η β η η 1 η β 1 η + i γ 1 + p µβ − µ p is given by: µν 2 2  2 νσ p µα η 2 ↔ σγ η 2 k ` η k { ν ` 2 η .k σ 1 .k βν σ ν 1 1 i κ 1 1 1 k .k η µα 10 , αµ k k 1 k k .k − σ η η − α 2 3 2 k 1 2 αµ 4 k k 2 .k and Lorentz index ( η ↔ 1 + 2 − + + 2 + − + − 2 k − µ p = 1 in our convention. The vertex involving two scalars carrying ingoing  h ) it is understood that the vertices need to be symmetrized under the − 2 , and two gravitons carrying ingoing momenta 1 2 6.6 i κ πG p i κ p ) is given by ) and 8 2 , − 1 √ µν 6.5 p = in ( ) and ( κ β The vertex where a graviton carrying Lorentz index ( We shall carry out our computation in the de Donder gauge in which the propagator 6.5 ) and ( In writing this and other vertices we already include the symmetry factor related to exchange of identical ↔ 10 αβ given by: particles. Therefore if weno were further to symmetry use factor this is vertex to necessary. compute tree level two graviton, two scalar amplitude, In ( exchange of the pairα of Lorentz indices carriedform, by we each shall external need graviton,small the e.g. compared form to of the the others. vertex In when this one limit of it the simplifies. external momenta (say If we label theindices ingoing carried graviton by momenta them by takes by the ( form: ( momentum where momenta of a graviton of momentum For our analysis we also needvertex, the with vertices involving the the scalars graviton. carrying The ingoing scalar-scalar-graviton momenta JHEP02(2019)086 . 8 (6.8) ggg scalar receives n ) n ( . Therefore we 1 scalar amplitude with the internal with all photons − 3 n ) that are needed for 2 ω ) attach to 6.1 ρσ ······ and . However there are also 1 . . 10 ) ) n ) and ( ( denote gravitons and the thicker νρ η g µν and . At one loop order Γ b µσ 9 p 1) η n, ` ( − ggggg − b νσ g p η ` µρ η . We can also have a diagram where both ends of ) that are analogous to figure ← ··· n ··· − g ( 7 g – 30 – ρσ η -scalar vertex, but this vanishes in dimensional regular- ` n include the analogs figures µν + η a ( 1) . Diagram contributing to Γ a p n, λ p ( 2 i κ − Figure 7 . However even without knowing the form of this vertex one can see 10 -scalar and one soft graviton amplitude Γ n . This diagram shows various vertices induced from the action ( . Another diagram contributing to Γ and The relevant diagrams for Γ We can use these vertices to compute one loop contribution to the ) n ( our computation. Herelines the denote thinner scalars. lines carrying the symbol Figure 6 contribution from diagrams shown in figure photon replaced by a graviton. There are also somereplaced additional diagrams by shown gravitons. in This figure some have extra been diagrams shown that in we figures shall list below: We also need the vertexdiagram containing of two scalars figure and threethat gravitons this for diagram evaluating the doeshave fifth not not written generate down contributions the proportional expression for to this ln vertex. Γ The vertex where twofields gravitons is carrying given Lorentz by: index ( Figure 8 the internal graviton are attachedization to and the so we have not displayed them. JHEP02(2019)086 b p ` − b g -scalar p n ` ··· ` (c) ··· ← -point vertex + n a a g ` p p + k . . k . The first diagram g a 12 11 + ··· ` p 13 a g p k , but there will be some a g 5 p k b p where two ends of the internal ··· jiihj ` g 14 − b p a g p ` k ` ··· (b) ← ··· gg ` ` g – 31 – a g + p involving internal graviton line connecting two different involving internal graviton line connecting two different a k k 1) p 1) + n, ( n, a ( p k ). These have been shown in figure ). These have been shown in figures ··· ` 6.8 jijih 6.7 -scalar vertex. In dimensional regularization these diagrams g ). Examples of these are shown in figure n b a g p p ) or ( 6.6 ` k ··· − 6.7 ` b p g ` a g p (a) ··· ← k k g + a ` p . One loop contribution to Γ = 0. The second diagram has no logarithmic terms. Therefore we shall ignore + ··· . One loop contribution to Γ g a k ρ ρ p graviton attach to the vanish. Therefore we shall ignore these diagrams in our analysis. via the coupling ( can be made toε vanish by taking thethese external diagrams graviton in polarization subsequent discussions. to be traceless: the cubic coupling ( vertex via the coupling ( + There are diagrams of the type shown in figure There are diagrams where one end of the internal gravitonThere attaches are to diagrams the where the external graviton attaches to the scalar There are diagrams where the external graviton couples to the internal graviton via g Our analysis of these diagrams will proceed as in section a a g p k p 4. 2. 3. 1. k ` important differences that we shall point out below. For an internal graviton of momentum Figure 10 points on the same leg. Figure 9 legs. The thicker lines represent scalar particles and the thinner lines represent gravitons. JHEP02(2019)086 g does not g ··· k ··· g g g -point vertex. n b p -point vertex. n ` − k ··· g k -point vertex. The first diagram ··· g g n g since the soft momentum ··· ··· ` g ` g g 1 + − a g ω p a p g g ··· g g ` b – 32 – p − k + ` b − k ··· k g g . Diagrams involving 3-graviton vertex. g ··· ··· g g ` g g ` p + ··· g a Figure 11 p a p . Diagrams where the internal graviton attaches to the g g . Diagrams where both ends of the internal graviton attach to the ··· . Diagrams where the external graviton attaches to the g ··· g Figure 12 g g Figure 14 In dimensional regularizationdiagrams these cannot diagrams have any vanish. contributionflow proportional through to Even any ln loop. if we use momentum cut-off, these Figure 13 vanishes if we takelogarithmic the terms. external graviton polarization to be traceless. The second diagram has no JHEP02(2019)086 b , ) } ` (6.9)  i ρ (6.15) (6.14) (6.10) (6.11) (6.12) (6.13) + ` − σ k ) ) ` + b ν p ) p ( − ` 2 . b  p. + i p 2 − ) k  , k ` µν .` } ( ) η + b + ( 2 . , a `. p p )+ # σ ( m ` . αβ ρσ ` )( β η `, p  ρ b η } }{ + , p ( flowing from the leg ) 1 2 )+ , p ) ) is given by .p `  i i ` αβ b ` a µ αµ η − (2 ) o ` p − + η − − ) , p a ) ν µν,ρσ ` ( p b k. 2 ) ) a µν a ) ( p a a ` η  + K − ( p p with p. 1 2 k β `, p `, p + { − µν ) + 2( ) ( b ( , is pure gauge and allows us to   + ` flowing from the vertex towards a − η `
+ 2 .` + 2 µ p, k, `, ξ, ζ µν p p ) ) ` a ( ` + − ` ( ` η ) ` νβ ρσ ν p -scalar vertex and the other end is ( ( ab a ν k A η µν,ρσ ) µν,ρσ and ( η ( p. + n + ( k `. ` − `. )( + ν b µ K K a 1) µν ν µα a ) βν p k + ` ` η η .p p ( η − }{ − µ ( a }{ a ( , with . α  + ) i p ) + −
p a ` p αµ i +2 p 2 a p η ( ν 2( − ρσ 12 νρ + , − m µν " k 2 µ + ( η ) η η `, p ) p ) − a ` 8 + ν ( ` µ µν p µν 2 ) ` + µσ β ` ( – 33 – b ζ η − µ ) ν η  + − ) ) ) ` ) b + .p ` a  ) α ` − a p a + µ µν,ρσ ( + a ( p k p p . + µν ( + k G ( β νρ b . νσ ζ k 2 ξ a , the analog of Grammer-Yennie decomposition of the η a p n `, p + ` ) η , with the quantity p p ( + ` 2(2 + p 9 ( µσ e 1 2 { , ( C µρ p  −  β }{ η + ( ν 15 η k − 2 b 7 ) p k i i µν µ b α p ( + p ) = 2 a ξ i − − αµ ( a , p p νσ i η iξ. ) a 2 αβ ) = ` η ` ) = η 2 a − iξ.pζ `, p a 2 ( µρ `, p + +2 + − +2 ) − ( η `, p b a ( `, p C p ) ( .p ) = 2 ( a e . a µν,ρσ C ( ) a p a ) = ) = ’th scalar leg as in figures , as in figures p K µν,ρσ ( b b a a 2( { G h , p , p , we do not carry out any Grammer-Yennie decomposition. a a ) = p,k,`,ξ,ζ b 11 `, p `, p ( ( ( , p A a ) ) Due to this additional term, theterms sum that over will K-gravitons will be leave discussed behind below. some residual sum over K-graviton insertionshave using been Ward shown identities. in The figure relevant Ward identities ab ab ’th leg, we express the propagator as: µν,ρσ ( µν,ρσ ( `, p The K-graviton polarization, being proportional to The decomposition into G and K-gravitons is not arbitrary but has been chosen to a ( G K 1. C , whose two ends are attached to two scalar lines For internal gravitons whose oneas end in is attached figure to a 3-graviton vertex instead of aensure scalar, two properties: where and attached to the the where If one end of an internal graviton is attached to the towards the leg graviton propagator will be taken to be ` JHEP02(2019)086 . c is ξ.p gr 2 K . The will be ). − µ k ) gr ν there is a ξ 6.15 K + 1) n, ν , where ( ) k n µ ( ξ , k, `, ξ, ζ c p ( without external soft to Γ . Therefore, even after A ) , 1) gr n for gravity can be com- ( n, ). The left hand figure is ( f 7 k = . In the first diagram the circle iλK + ρσ ζ a ]. , we are left with an additional are not the same, one carries a ) p gr ( n ( can be carried out similarly, lead- e 16 may be evaluated similarly, with K ξ. 8 + 12 gr is taken to be equal to g g ⇑ K ]. There are however some left-over terms − and k gr is taken to be has no infrared divergence and we shall not e 9 exp[ K k , the circled vertices are momentum dependent. gr + – 34 – e iλ K 15 gr . K ) while the circle on the right denotes a vertex gr . The relevant part of the expression for k e K + em ] factor multiplying Γ exp[ f iλ g c K p g em vanish by construction. Therefore the net contribution to ( (0) gr e K that comes from lack of complete cancellation among terms ξ. 8 for gravity. The arrow on the graviton line represents that the , in the sum over K-graviton insertions in Γ 2 + from sum over K-gravitons that must be accounted for. 4 A = − iλ S gr 15 1) and n, K + + 7 ( while the right-hand figure is relevant for Γ , leading to a contribution of the form ) while the other carries a factor of gg g g ⇑ n 5 ( ↑ k a → c ξ.p p g i i g ↓ ↑ ) appearing on the right hand side of the second diagram is given in eq. ( ` k . Analog of figure → c p Therefore the two circled verticesfactor shown of in figure relevant for Γ factoring out exp[ contribution to Γ residual contribution where a K-graviton is insertedinto to the the scalar-scalar-graviton two vertex. sides of a scalar-scalar-graviton vertex and graviton, the result vanishes if we replace the internal graviton by G-graviton. − to one loop order may be written as As shown in figure As explained in the caption of figure In any one loop diagram contributing to the amplitude Γ , k, `, ξ, ζ The K-graviton contributions to figures With this convention the K-graviton contribution to figure c ) 1. 2. 2. p n ( ( Γ the factorized term giving arising as follows: puted as in section the gravitational counterpart of described later. The K-gravitoning contribution to to figure an expressionwrite of down the its expression form explicitly althoughcontributions it to is figures straightforward to do so. The G-graviton Figure 15 polarization of thepolarization graviton of the carrying graviton momentum carryingon momentum the left denotesA a vertex JHEP02(2019)086 ) . n (  11 i and (6.17) (6.16) − 9 2 l . The G-  i 1) + and the other n, 1 .l ( residual ω b p . The result takes , but only the first  ω i 11 . − and the net contribution .l (1) gr a ) takes the form: 1) p  n, ( G iλ S . In principle we should also 4 5.11 ) l = 4 π d receives logarithmic contribution g (2 k 9 , but we ignore them since they do 1) graviton 1) g reg n, − n, ( residual Z ( 3 14 a a . We shall analyze the contribution from p + Γ ω 2 a p .ε.p a p  . This receives contribution from figure – 35 – 2 b graviton 1) will be denoted by Γ p and figure − n, 2 a ( 3 1) . Individually these diagrams suffer from collinear p 12 , and the other due to the momentum dependence 1 13 n, − ( residual − h h + Γ 2 g . Finally the contribution to the diagrams in figure ω ······ 15 . It turns out that the residual part of the K-graviton ) and b 1) 1) 9 .p n, arises from the five diagrams in figure n, 15 ( G ( self a p does not have any logarithmic term. On the other hand the 2( a + Γ  p a 1) 12 graviton 1) n =1 6= n, b b − X ( self n, ( 3 Γ n =1 a X i 2 ) will be called Γ iλ ( 10 − . Figure illustrating the difference in the factorized K-graviton contribution to Γ . . As explained above, there are two kinds of terms: one due to the right hand . = +1) 5 n 12 ( 1) Contribution to Γ We shall now briefly describe how we evaluate these contributions and then give the n, ( residual Γ such graphs after usingthese momentum diagrams. conservation. The Therefore net contribution wefrom from always two these work regions diagrams receive with — logarithmic one sumwhere contribution the where of loop the momentum loop isthe momentum small compared is region to large of compared small to loop momentum later. Contribution from the region where the loop This contribution maysection be evaluated following a procedure similartwo give to terms the proportionaldivergence one to from region ln used of integration in parallel where the to momenta that of the of internal the gravitons become external graviton, but these divergences cancel in the sum over the form: side of the secondof figure the of circled vertices figure contribution in from figure figure residual part of the K-gravitononly contribution from from figure the region where the loop momentum is large compared to final result. Firstfigure let us consider Γ graviton contributions to figures from figure involving 3-graviton coupling willinclude be the denoted contributions by from Γ not figure generate logarithmic terms. In this case the analog of ( Figure 16 and Γ We shall denote the sum of these two types of residual contributions as Γ JHEP02(2019)086 1 − . ω #  may , the (6.20) (6.18) (6.19) ) i 12 , we can 15 ), we can − .ε.` satisfying a `.ε.` 2 .k 11 ξ p ` a and ( p

9 . ( in the numerator 2 b . The result is: i `  p 2 ω 2 a ) i − .k f p 2 a a p − .` ( − , a 2 2 p ` .ξ p ) i  a b p 2 1 − .p 1  ) + a 2 a ) for this amplitude based i p ` .k 2( − a + .ε.p  . We find that the G-graviton  p a .` ( } i 2 5 p a f ( i p ). On the other hand the diagrams − − ν a  for an arbitrary vector 2 a p 2 .k − 2 a p ` a µ a µ 4 p 2 2 ) p p ` k ` ( ) 4 π ν   b − µν d ξ }{ i ε (2 " i .k f + term, since the expression for the amplitude ) in the second diagram of figure .ε.p `.k + a may be analyzed following the argument  ` b ν 2 reg ). 1 − 1 p .` k i Z − − 2 b µ .ξ p ) 10 ) p a 1 ξ k − a ` – 36 – 6.18 i p 2   ` − + 2 ( − by .ε.p  2 2 b i k a ` ( p p 2 µν . These diagrams have the same structure as in scalar { ] − may be approximated as ) 2 a ε a p . It turns out that it receives logarithmic contribution i 9 .` = ω a 1) − 1 p n =1 n, .ε.p i  ( self a X a .` -dependent denominator and therefore cannot have a ln a p 4 ) generates infrared finite integral. − i given in figure 2 k p ) ` [ 2 ) 4 π ) 1 1) d ` 6.19 has no logarithmic contribution. Therefore we are left with the 4 iλ (2 ) n, ` ( ( self − 4 π ) + 2 ( = 0. We have also used the fact that in the expression for the 12 − d b k reg (2 ( a Z .p = p a a ). We assume a general form p reg n =1 6= 1) b b Z X ) and hence Γ =1 n, n a )( ( self .k b 5.16 n =1 Γ a n =1 P a X . Due to the presence of the non-vanishing right-hand side in figure p a X for this choice of polarization may be evaluated using the Ward identity given ( .ε.p i 4 i f 2 a 15 10 Similar manipulations will be used in other terms as well. ) p ) iλ iλ 11 8 ( ( ( One loop contribution from the diagrams involving G-gravitons in figures Contribution to Γ = 0. Then the amplitude reduces to 2 − This manipulation can be carried out only for terms containing at least two powers of − −  11 G-graviton contributions to figure QED and can be evaluatedsignificant similarly. contribution As only in from the the case region of where scalar the QED, loop these momentum diagramsso is receive that large each compared of the terms in ( This cancels the term in the last line ofbe ( evaluated following thecontribution procedure to described figure in section result does not vanish. Comparingcompute this with the expectedfrom result region 2 of integration where the loop momentum is large compared to given below ( on Lorentz invariance and replace k.ξ in figure in figure and ignore the contribution from theinvolving ( this term hasterm. no momentum graviton propagator carrying momentum ( use the identity In arriving at this result we have used integration by parts and also conservation of total momentum is large compared to JHEP02(2019)086 .    . We !#   (6.23) (6.25) (6.21) (6.22) (6.24) 2 b 2 b ω 12 . These .l p p .ε.l a 2 a 2 a ω a p p p p . − − from regions ) 2 2 ) ) i b b + 2 .p .p a + from the region of a a p p .` graviton 1) 2 a ( ( b ) already involves an p .ε.p − n, p ( 3 a q q  p 1 )( + − i . 6.24 ,

graviton 1) b b i  # − n, −  .p .p ( 3  i − 2 b a a , can be expressed as 2 .l l p p p a ω .` aµ − 2 a k.l ∂ p a p 2     p l scalars, with the understanding ∂p  ( and Γ i − ln ν a 2 b n   p p 2 i π + i ) 2 a i 2 b 1 − p 1 − .l 1) b , .p + − 2 n, − p a ` aν 1 ( residual 1 , .l p ∂ 2 b reg gr b 4 ) η   ∂p ) p b 2( ` ,Γ a K  4 µ a i η π .p 1) has more terms — ( d p δ terms. Explicit evaluation gives the following   a (2 n, − (1) p  gr − ( self 1    i gr b .l S – 37 – ) − ν 2(

K a reg a : − ) 2 b ,Γ k  2 b ω 1 p Z p a ρ ]. But we have not verified this by explicit computation. .k p .l 1)  iλ 2 a − p − a 2 a a ( .ε.p p n,  ω reg p p gr ) p ( G 4 a µρ 2 b 2 1  ) l − p K ε p .k 4 ( π 2 2 a b − 4 d ) p p ) 2 b l b 2 a (2 ) 4 2 1 p X π .p b d )( 2 a b Z (2 .p − = p − a ( .p 2  p a ) for gravitational scattering, namely it is the factor that appears ) Z a ( (1) b gr b p q a b  S .p .k n =1 6= 4( reg acting on exp[ em b b a a X .ε.p 1 .ε.p " p p a K a − ( (2) gr p p n =1 ω b S  a X  )( ln a i b 2 i 2 6= a,b b ) X .p ) π 1 a 4 i 2 iλ iλ p a ( is the quantum subleading soft graviton operator 6= 8( ≡ a,b b − X + ( " i (1) 2 gr = reg gr b S = is the analog of K and small compared to the momenta of finite energy particles. The net logarithmic 1) At this stage the only remaining terms are the contributions to Γ The total logarithmic terms in Γ n, It is natural to conjecture that this pattern continues to hold also for subsubleading soft graviton ( G ω reg gr reg gr 12 Γ K of loop momentum integration where the loop momentum istheorem, small i.e. the compared universal part to ofsoft the graviton subsubleading operator contribution is given by the action of the subsubleading expression for the terms involving ln in the exponent ofthat the the soft integration factor overnote in loop however the that momentum the scattering is full of approximation expression restricted that for to the loop the momentumsince region is this small larger compared is than to the the region energies that of generates external ln lines and K where integration where the loop momentum is large compared to contributions from these diagrams is given by to JHEP02(2019)086 . , is R ,  ) ).    # ` a    and ˆ n 2 νρ 2 (7.2) (7.1) 2 φ , ) ) η q ) (6.28) (6.26) (6.27) ν 2 b ˆ k 2 b ˆ k .k . .k . A µσ b b m a b and ln m a η p p = (1 p p ( 1 ( ( iq + − ) and Lorentz ln b − ln ω νσ k/ω 2 a .k η k b .k − φ .ε.p b p ν b µρ p ∂ = p η b  )( and ˆ k b 2( X 2 ∗ a X 1 ν a − φ .k k p ν a ) 1 µ µ a p .k k .k A p µ a a .k a a p p a + p µν p iq ε µν  ε )( σ + b 1 a ∗ a k X a ρ φ . X 2 .ε.p π µ 1 . k # a 2 ∂ 2 p ∗ n ( 1 π + − φ i 8 2( µν ν a σ " g runs in the region where the components 2 σµ p ··· − i k η µ a `  .k ∗ 1 ρ ν we get the terms involving ln p ν a a i 1 1 p p n =1 k . In the first diagram there are two relevant k µν + µ a a X λ φ .k iλ ρ  ε 2 p 2 a ` 11 k − is small, but they are related to each other by p + – 38 – µν a µν  n ε ` X η + R i φ b − a − σν + πG X 1 η ··· k k.p .`  µ b 1 1 a 16 2 µ 1 − p 2 k b = ρ k k.p + λ φ  b X 1 ν η 1 i 1 k + i − µν k . The result may be expressed as b + =    + F b ω  X 2 ) ) and dividing by η a ` 1 + µν ρµ φ − + η π ∗ a F i µν ν R 4 symmetry. In the second diagram the relevant region is when 6.27 φ 1 4 1 η 2 a k.`    k b 2 2 − σ m )  +ln 2 .k 1 ), and the two photons carry ingoing momenta " 1 ↔ k 1 4 ) to ( + − is small and when ) g ` − k respectively, then the graviton-photon-photon vertex is given by: ρσ a 4 + R π ` ω d ν − reg gr 6.22 (2 det ρν  (ln η K is an infrared lower cut-off on momentum integration and and − + ln µ Z π 1 and 2 ρσ ` 1 4 p (1) k gr η n − /R =1 µ b σ h S b + − x X : 1 ω 4 k k = d are small compared to n Adding ( (1) =1 gr i κ  a X (ln S ` (1) Z gr → − − λ iλ S Lorentz index ( indices For this analysis wegraviton-photon-scalar-scalar need vertex. two If new the vertices, graviton carries the an graviton-photon-photon ingoing vertex momentum and the In this section we shall considerand the gravitational case where interaction the via scalars the interact via action: both electromagnetic 7 Generalizations in where 1 with the understanding that the integrationof over regions: when ` small. The net contribution from these regions may be approximated by come from the first two diagrams in figure JHEP02(2019)086 1 ), p and 5.3 (7.4) (7.3) ) holds n ). This ( 4 20 where the we arrive at 19 1) n, following ( and momenta ( γγg . q 17 i − ρ , ) . It turns out however by following the pro- and a photon carrying 2 q p 1 , obtained by replacing, 21 k 18 . Therefore we have not − . The sum over K-photons 1 18 1 − p 19 ( , ω µν (1) gr ] that cancels between Γ η em − i λ S µ K ) and figure 2 = p and figure 17 1) − 18 n, ) and momentum 1 ( γγg = 0 and have not been displayed. We carry p ( µν ρ ρ + Γ νρ ε – 39 – η 1) . Denoting this contribution by Γ + n, ( G ν requires the two scalar, two graviton and one photon 19 ) 2 , all counted ingoing, is given by ). We replace the external graviton polarization by a ) and apply Ward identity to evaluate the sum over the 23 + Γ p 2 µ k k 1) − ν 5.16 n, 1 ξ ( self p represented by the graphs in figure ( Γ , the external photon by a graviton but keeping the internal + ) with an additional contribution to the left hand side given by 1) ν µρ 5 k η n, ( self µ . In this case the Ward identity has an additional contribution as h and gives the factor of exp[ ξ 5.11 5 18 i κ q needs the vertex containing two scalars, two photons and one graviton, − -scalar vertex vanish for 18 and momentum n ). None of the terms have any infrared divergence, and therefore there are . We shall describe below the organization of the various terms and then ρ ω does not generate any logarithmic contribution. , we also have the diagrams of figure 6.16 6 1) n, ( self , a graviton carrying Lorentz indices ( pure gauge form ( graphs in figure shown in the right handthat side its of contribution the to second the diagram amplitudeΓ in does figure not have any logarithmic term. Therefore cedure described below ( 2 . In this case there is no residual contribution since the analog of figure One can analyze Γ p There are two other vertices that are needed for our analysis. For example the sixth 1) 1. n, ( no logarithmic terms from thecompared region to of integration instate which the the final loop result: momentum is small diagrams of the form shownthe in relation figure with the understanding that bothappear sides represent in contributions ( in addition to what already but not for diagrams of thefactorize form as shown in in figure section Γ with the upper photonleads in to the the second analog identity of replaced ( by a graviton (see figure in the diagrams inline section as a photon.external We graviton also connects have to an the additional internalattaches set photon. to of Diagrams the diagrams in shown whichout the in external figure Grammer-Yennie graviton decomposition for the internal photons in figure written down the expressions for these vertices. 7.1 Soft gravitonWe theorem first consider the soft gravitonin theorem. section In this case besides the contributions analyzed soft photon theorem. diagram of figure whereas the sixth diagram of figure vertex. However even withoutdiagrams knowing do the not form of generate these contributions vertices proportional one to can ln see that these and Lorentz index In this theory we shall analyze the extra terms in both the soft graviton theorem and the On the other hand the vertex with a pair of scalars carrying charges JHEP02(2019)086 .    2 also (7.6) (7.5) ) b b 2 ˆ k p . 17 . b m p ω ··· ( ln .k a γ b g denote photons. (c) ··· p p γ k b X a ν ··· ` p a g a µ p .k p a k p µν a g γ ε p is given by: ) to get the total logarithmic k a from the first two diagrams of 1) X ··· j b n, 1 6.28 γ π ( G 1 p − 2 ω g − . a a ν p p and Γ reg a µ em k ` .k p a 1) from the first two diagrams in figure K p ··· (b) n, µν ··· ( γγg 1) ε (1) gr n, ` b ( G S – 40 – a ) X γ a b from the region where the loop momentum is large g iλ p a g γ ( 1 p k k.p − 1 ω k − ··· j b = γ b X η i    b a g ) p p 1 k ` −
factor, we have to add this to ( R ··· reg gr ` denote gravitons and the thin lines carrying the symbol . i λ : K ω g +ln γ 1 + γ (1) gr (a) ··· a − g S p reg em ω , from the region where the loop momentum is large compared to k K receives contribution proportional to ln (ln 19 π . One loop contribution to soft graviton amplitude involving internal photon line con- a . Contribution to soft graviton amplitude due to internal photon whose two ends are con- ··· 1 1) 4 p (1) γ gr g n, b γγg ( S + figure has terms proportional to ln compared to k = Γ Finally, the G-photon contribution Γ The net logarithmic contribution from Γ a g p 2. 3. (1) gr k ` S After removing the contribution to Figure 18 necting two points on the same leg. Figure 17 nected to two different scalarcarrying lines. the Here symbol the thickest lines denote scalars, lines of medium thickness JHEP02(2019)086 ) 1 } ) ` 5.28 + k + g p ··· γ γ (2 ) after using ( ξ. `. , and two additional 2.7 ) + 23 ` , + f and k ) and ( . The circled vertex has been  + = 0 22 2.6 p (2 ⇑ ξ.k . { c − i q 2 in the sum of ( ··· γ 1 − – 41 – − g = γ γγ ω g gg γ . There is also an additional diagram obtained by 25 f + + . ⇑ 15 for graviton in the presence of a photon. The graviton carries a f f 4 = k ` , but also extra diagrams where the internal photon of figures ) and the photon carries a polarization and the other where one end of the internal graviton is attached µ g gg ` 5 γ γγ − ν gg g g ⇑ 24 ξ ↑ k → + c ν p + + ` µ g ξ ··· . The Ward identity for the photon in the presence of a graviton-scalar-scalar vertex. γ γ . Analog of figure . Diagrams containing graviton-photon-photon vertex that contribute to the soft photon ). -scalar vertex as in figure f f n k ` is replaced by an internal graviton, as shown in figures 6.25 2 − Figure 20 considered in section and sets of diagrams:photon one as where in one figure to end the of the internal graviton connects to the external This reproduces terms proportional to ln and ( 7.2 Soft photon theorem Next we shall consider the soft photon theorem. In this case we have all the diagrams Figure 21 polarization ( explained in the caption of figure Figure 19 contribution to the soft graviton theorem. JHEP02(2019)086 b p ··· ··· g g (c) ··· a γ γ γ p k ··· ` g γ a p a γ k p γ ··· k γ g b ··· jjiih g p γ a p k ` gγ ··· (b) ··· ··· γ γ ` g – 42 – g a γ a p p k k ··· jjiih g b a p γ p g k ` ··· γ ··· ` γ g g (a) ··· a γ p k γ ··· γ g . Diagrams involving graviton-photon-photon vertex that need to be included in com- . One loop contribution to soft photon amplitude involving internal graviton line con- . Diagrams in which the external photon and both ends of the internal graviton attach a ··· p g γ k a γ p k ` Figure 23 to the same scalar leg. Figure 24 puting the soft graviton contribution to the soft photon theorem. Figure 22 necting two different legs. JHEP02(2019)086 . 1) n, ( G (7.7) or from ). In this ω 5.16 ] which cancels ). The result of will be denoted gr e K ··· g 23 6.14 + gr )–( , K . 6.9 do not have any left-over ω will be denoted by Γ (1) gr γ , the factorized contribution 20 25 i λ S ) goes as follows: 16 ··· g and = using Grammer-Yennie decompo- is replaced by an external photon. 7.7 and 4 16 25 22 1) n, γ ( residual , does not receive any logarithmic terms and + Γ ··· 25 22 g in figure 1) ) takes the form: k n, – 43 – . However there will be residual part that will ( γγg ) and the external graviton by the external photon, but n γ ( 5.11 + Γ 14 22 will generate the factor of exp[ 1) , and the contributions from figure differ. The only difference in the present case is that n, 1) 1) ( G 1) n, , given by the left-over contribution after summing over n, ( γγg γ n, ( ( + Γ 1) . Another residual contribution arises due to the momentum . 1) ··· n, g ( residual 5 n, ( self 21 and Γ Γ ) n ( ··· g -point vertex. , we shall denote these residual contributions in the sum over K-gravitons γ n 6 can be shown to vanish using the same argument given below ( . The G-graviton contribution to figures . Diagrams with external soft photon and an internal graviton where the internal graviton 1) , will be denoted by Γ . Then the generalization of ( n, ( self 1) 1) extra contributions. K-graviton insertions in figures either from the regionregions of of loop loop momentum momentum integration integration small large compared compared to to case the relevant Ward identities given in figures 24 n, n, ( residual ( self It turns out that Γ Γ Analysis of various terms on the left hand side of ( We shall analyze the diagrams in figures 2. 1. those described in section figure by Γ again with the understanding that both sides represent additional contribution besides of K-gravitons for Γ the external graviton carrying momentum As in section by Γ The contribution from diagrams involving the coupling of graviton to photon, as shown in summing over K-gravitons in Γ a similar factor inbe the left over expression due ofright-hand to Γ non-cancellation side of of the figure dependence sum over of K-graviton the insertions circled reflected vertices; in as the illustrated in figure replacing in the first diagramthis of figure vanishes in dimensional regularization. sition for the internal graviton following the rules described in ( Figure 25 attaches to the JHEP02(2019)086 , . .       11 ! (7.9) (7.8) ! 2 (7.10) ) , from 2 2 b ) ˆ k ) to get receives 2 b p . ˆ k b p . 24 − . b p 1) − ( ω p 5.20 n, ( ( γγg

)ln ) after using the )ln .k b 2.5 .k p b ( p ( n =1 b X n =1 ) and ( b X µ a p µ a .k 2.4 µ p a ε .k µ each has contribution propor- p a a ε q p a only from the G-graviton contri- ). q 24 1 n =1 have collinear divergence from the − a X n =1 6.25 ω 2 a X 1) 1 π 2 . n, ), we have to add them to ( 8 ( γγg 1 π 8 reg − gr from the small loop momentum region is 7.9 a − in the sum of ( ) and ( K q a 1) 1 q µ a n, from the first two diagrams in figure (1) .k em − p ( γγg µ a – 44 – a b 5.28 µ S ω .k 1 p p ε a ) and ( µ − p ε i λ ω a 7.8 X a b X b k.p given in ( 1 k.p − 1 b reg = gr − b X b η = K b X η π , from region of integration where the loop momentum is larger i . This gives 4 factors from (
π i from the region of integration where the loop momentum is large and    (1) 4 em 22 ) 1 , but the sum of these contributions vanishes. Finally, Γ reg S gr 1    i λ − reg ω − em ) K 1 ω R K − + is given by R . The second and third diagrams of figure ω reg em +ln . 1 receives contributions proportional to ln K 24 ω +ln − 1 1) ω − (1) em n, ( G b ω S the region where the momentum of the internal graviton is smaller than momentum of the externalfigure photon. Thistional cancels to in ln thecompared sum to over allcontributions diagrams proportional in to ln bution to figure than region where the momenta of the internal graviton and photon are parallel to the The individual diagrams contributing to Γ Γ = +(ln (ln 4. 3. (1) em iλ S This reproduces terms proportional toexplicit ln forms of is that the divergent contributionmomentum comes becomes only small, from and theThis not reflects region when the where the fact the internal that internalparticles, while photon graviton photons the momentum feel graviton, becomes the being small. longAfter charge range removing gravitational neutral, force the does due not tothe feel other total any soft long factor range Coulomb force. One difference from the previous diagrams of this type, e.g. the ones shown in figure On the other handgiven by: the contribution to Γ The net logarithmic contribution fromis the larger region of than integration where the loop momentum JHEP02(2019)086 ]. , 20 ]. , Phys. SPIRE 166 , IN ]. B 345 [ SPIRE Nucl. Phys. Phys. Rev. IN , , [ SPIRE IN (2014) 152 [ Phys. Rev. Phys. Rev. Lett. (1975) 2937 , , 07 Nucl. Phys. (1969) 1894 , ]. D 11 184 (1964) B1049 JHEP , SPIRE (1965) B516 ]. 135 IN ][ 140 Phys. Rev. SPIRE Phys. Rev. , IN , ][ ] for informing us of their results on the Phys. Rev. , 98 BMS supertranslations and Weinberg’s soft , Phys. Rev. 97 , ]. – 45 – arXiv:1401.7026 [ ]. Scattering of low-energy photons by particles of spin SPIRE arXiv:1408.2228 IN SPIRE [ [ Asymptotic symmetries and subleading soft graviton theorem Extension of the low soft photon theorem ]. ), which permits any use, distribution and reproduction in IN On the low-burnett-kroll theorem for soft-photon emission ]. [ (2015) 151 Low-Energy Theorem for Graviton Scattering ]. ]. 05 SPIRE SPIRE IN ]. IN [ Soft Dilations and Scale Renormalization in Dual Theories ]. SPIRE (1954) 1433 SPIRE (1969) 62 ]. ][ IN JHEP CC-BY 4.0 IN (2014) 124028 [ , [ 96 On BMS Invariance of Gravitational Scattering SPIRE This article is distributed under the terms of the Creative Commons On the Renormalization of Dual Models High-energy Bremsstrahlung Theorems for Soft Photons SPIRE Photons and Gravitons in s Matrix Theory: Derivation ofInfrared Charge photons Conservation and gravitons IN A 60 SPIRE IN [ D 90 Low-Energy Theorems for Massless Bosons: Photons and Gravitons IN Bremsstrahlung of very low-energy quanta in elementary particle collisions [ (1958) 974 [ Low-energy theorem for Compton scattering ]. (1975) 221 110 Phys. Rev. (1968) 1623 , SPIRE arXiv:1312.2229 IN [ graviton theorem Phys. Rev. B 94 [ (1968) 1287 168 Nuovo Cim. (1990) 369 and Equality of Gravitational and Inertial Mass Rev. (1968) 86 1/2 S. Weinberg, D.J. Gross and R. Jackiw, V. Del Duca, S. Weinberg, S. Saito, T.H. Burnett and N.M. Kroll, J.S. Bell and R. Van Royen, M. Gell-Mann and M.L. Goldberger, F.E. Low, T. He, V. Lysov, P. Mitra and A. Strominger, M. Campiglia and A. Laddha, M. Ademollo et al., J.A. Shapiro, A. Strominger, R. Jackiw, [8] [9] [6] [7] [3] [4] [5] [1] [2] [14] [15] [11] [12] [13] [10] References also by the Infosys Chair Professorship. Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. We would like to thankearly Alok stages Laddha for of manyVeneziano this useful for discussions work. discussions and and collaboration the We authorssoft at of would limit the [ also of scattering likepart amplitudes to by of thank the massless J.C. Massimo states. Bose Bianchi The fellowship and of work the of Gabriele Department A.S. of was Science supported and in Technology, India and Acknowledgments JHEP02(2019)086 ]. , , , 04 ] 05 (2017) 12 SPIRE IN 05 ][ , JHEP JHEP , JHEP , ]. , JHEP ]. , ]. SPIRE ]. ]. Gravity ]. IN SPIRE ]. arXiv:1707.08016 ][ IN +2 SPIRE [ d IN ][ ]. SPIRE SPIRE SPIRE ][ IN ]. IN arXiv:1612.05886 IN ]. SPIRE [ ][ [ ][ IN SPIRE ][ Equations of Motion as Constraints: IN SPIRE SPIRE ][ Loop-Corrected Virasoro Symmetry of Higher-Dimensional Supertranslations IN IN ]. ]. ]. (2017) 261602 ][ ][ (2017) 136 Asymptotic Charges at Null Infinity in Any On higher-spin supertranslations and New Gravitational Memories arXiv:1709.03850 119 [ 03 SPIRE SPIRE SPIRE Color Memory: A Yang-Mills Analog of Gravitational Memory in Higher Dimensions arXiv:1701.00496 [ IN IN IN arXiv:1506.05789 [ ][ ][ ][ ]. ]. ]. – 46 – arXiv:1509.01406 arXiv:1502.07644 arXiv:1605.09094 [ , [ JHEP arXiv:1703.01351 , [ (2018) 132 SPIRE SPIRE SPIRE arXiv:1712.09591 Gravitational Memory, BMS Supertranslations and Soft ]. Burg-Metzner-Sachs symmetry, string theory and soft arXiv:1411.5745 [ (2017) 050 IN IN IN 05 [ arXiv:1505.05346 ][ ][ ][ Sub-subleading soft gravitons: New symmetries of quantum Subleading soft photons and large gauge transformations Sub-subleading soft gravitons and large diffeomorphisms Asymptotic symmetries of QED and Weinberg’s soft photon Asymptotic symmetries of gravity and soft theorems for New symmetries for the Gravitational S-matrix [ Phys. Rev. Lett. 08 ]. (2015) 094 , (2017) 218 -Dimensional Stress Tensor for Mink SPIRE (2016) 026003 d IN (2017) 120 Asymptotic Symmetries and Subleading Soft Photon Theorem in 12 JHEP A ][ (2018) 47 , BMS Supertranslations and Not So Soft Gravitons JHEP SPIRE 05 arXiv:1712.01204 arXiv:1605.09677 arXiv:1608.00685 4 D 93 , (2016) 086 [ [ [ B 764 IN [ (2015) 115 JHEP 01 , 07 Lectures on the Infrared Structure of Gravity and Gauge Theory JHEP , Universe arXiv:1711.04371 arXiv:1502.06120 arXiv:1502.02318 JHEP , [ [ [ (2018) 138 (2016) 012 (2017) 036 Phys. Rev. , Phys. Lett. JHEP ]. , , , 06 11 01 arXiv:1612.08294 Quantum Gravity [ SPIRE D IN and Weinberg’s Soft Graviton Theorem theorem (2015) 076 (2016) 053 (2018) 186 JHEP Dimension Gravitational Wave Memory [ Effective Field Theories superrotations arXiv:1703.05448 060 4 Superselection Rules, Ward Identities JHEP JHEP theorems massive particles gravity? Theorems M. Pate, A.-M. Raclariu and A. Strominger, A. Campoleoni, D. Francia and C. Heissenberg, A. Laddha and P. Mitra, D. Kapec and P. Mitra, A. Campoleoni, D. Francia and C. Heissenberg, A. Strominger, M. Pate, A.-M. Raclariu and A. Strominger, T. He, D. Kapec, A.-M. Raclariu and A. Strominger, M. Asorey, A.P. Balachandran, F. Lizzi and G. Marmo, M. Campiglia and A. Laddha, M. Campiglia and A. Laddha, E. Conde and P. Mao, M. Campiglia and A. Laddha, M. Campiglia and A. Laddha, D. Kapec, V. Lysov, S. Pasterski and A. Strominger, M. Campiglia and A. Laddha, S.G. Avery and B.U.W. Schwab, M. Campiglia and A. Laddha, S. Pasterski, A. Strominger and A. Zhiboedov, A. Strominger and A. Zhiboedov, [34] [35] [32] [33] [29] [30] [31] [27] [28] [24] [25] [26] [22] [23] [19] [20] [21] [17] [18] [16] JHEP02(2019)086 , , ]. Phys. ]. ] (2014) , ] SPIRE (2014) IN 10 [ SPIRE ]. (2014) 062 IN (2011) 060 ][ B 737 ]. JHEP (2014) 077 08 , , 05 SPIRE 08 (2015) 375 IN SPIRE ][ ]. arXiv:1405.3533 JHEP IN arXiv:1406.6987 arXiv:1803.03023 , The method of regions JHEP , ][ [ [ , JHEP Phys. Lett. B 742 , , arXiv:1405.3413 SPIRE , IN ][ ]. arXiv:1404.7749 [ ]. Phys. Lett. SPIRE (2014) 084035 (2018) 106019 arXiv:1406.6574 , IN [ ]. ]. Constraining subleading soft gluon and ][ SPIRE arXiv:1405.1015 Low-Energy Behavior of Gluons and [ IN D 90 D 97 More on Soft Theorems: Trees, Loops and ][ SPIRE SPIRE (2014) 101601 Note on Soft Graviton theorem by KLT Relation IN IN arXiv:1406.5155 – 47 – ][ ][ [ Double soft graviton theorems and 113 (2014) 065024 Phys. Rev. Phys. Rev. Subleading Soft Theorem in Arbitrary Dimensions from , , (2014) 085015 On Loop Corrections to Subleading Soft Behavior of Gluons ]. ]. Loop Corrections to Soft Theorems in Gauge Theories and Evidence for a New Soft Graviton Theorem arXiv:1405.1410 D 90 [ ]. ]. ]. ]. Are Soft Theorems Renormalized? Next to subleading soft-graviton theorem in arbitrary dimensions ]. D 90 arXiv:1405.2346 SPIRE SPIRE (2015) 065022 [ IN IN ][ ][ SPIRE SPIRE SPIRE SPIRE arXiv:1407.5982 arXiv:1408.4179 Phys. Rev. Lett. SPIRE [ [ IN IN IN IN , D 92 IN Soft Graviton Theorem in Arbitrary Dimensions (2014) 115 Phys. Rev. [ ][ ][ ][ ][ , Subleading Soft Factor for String Disk Amplitudes 12 Phys. Rev. Conformal Invariance of the Subleading Soft Theorem in Gauge Theory Sub-sub-leading soft-graviton theorem in arbitrary dimension , Diagrammatic insights into next-to-soft corrections Factorization Properties of Soft Graviton Amplitudes (2014) 087701 Soft sub-leading divergences in Yang-Mills amplitudes (2015) 107 (2014) 090 ]. ]. ]. JHEP Phys. Rev. , 01 11 , D 90 arXiv:1406.7184 arXiv:1407.5936 [ [ SPIRE SPIRE SPIRE arXiv:1410.6406 IN arXiv:1406.4172 IN arXiv:1404.5551 IN arXiv:1103.2981 arXiv:1404.4091 [ Scattering Equations JHEP and next-to-soft corrections in Drell-Yan[ production 216 148 JHEP graviton theorems Gravitons from Gauge Invariance [ [ Strings Rev. [ and Gravitons Gravity Bondi-Metzner-Sachs symmetries [ [ Y.-J. Du, B. Feng, C.-H. Fu and Y. Wang, D. Bonocore, E. Laenen, L. Magnea, L. Vernazza and C.D. White, M. Zlotnikov, C. Kalousios and F. Rojas, Z. Bern, S. Davies, P. Di Vecchia and J. Nohle, C.D. White, B.U.W. Schwab, M. Bianchi, S. He, Y.-t. Huang and C. Wen, J. Broedel, M. de Leeuw, J. Plefka and M. Rosso, F. Cachazo and E.Y. Yuan, N. Afkhami-Jeddi, Z. Bern, S. Davies and J. Nohle, S. He, Y.-t. Huang and C. Wen, A.J. Larkoski, E. Casali, B.U.W. Schwab and A. Volovich, C.D. White, F. Cachazo and A. Strominger, A.H. Anupam, A. Kundu and K. Ray, [53] [54] [51] [52] [49] [50] [46] [47] [48] [44] [45] [41] [42] [43] [39] [40] [37] [38] [36] JHEP02(2019)086 , ]. ]. JHEP (2016) , SPIRE D 92 IN 06 (2016) 173 SPIRE ]. ][ (2015) 140 IN ]. (2015) 143 Nuovo Cim. 04 Fortsch. Phys. , in ][ , , 03 12 JHEP SPIRE , ]. SPIRE IN Phys. Rev. JHEP , IN ][ (2015) 166 , JHEP ][ JHEP , SPIRE , ]. 06 Double-Soft Limits of IN ]. ]. ][ arXiv:1502.05258 SPIRE [ JHEP IN SPIRE , SPIRE arXiv:1507.08882 ]. ][ [ IN IN ][ ][ arXiv:1604.03355 SPIRE ]. [ arXiv:1604.02834 (2015) 137 IN [ ][ ]. 05 -point amplitudes with a single n arXiv:1504.05558 SPIRE ]. ]. [ (2015) 065024 Subsubleading soft theorems of gravitons and Soft theorem for the graviton, dilaton and the Double-soft behavior for scalars and gluons from ]. IN Soft Theorems from String Theory ][ SPIRE (2016) 054 – 48 – The Double Copy Structure of Soft Gravitons JHEP Double Soft Theorems in Gauge and String Theories IN , (2016) 060 arXiv:1512.00803 SPIRE SPIRE D 92 06 ][ [ SPIRE IN IN arXiv:1601.03457 arXiv:1507.00938 07 IN Extensions of Theories from Soft Limits New Double Soft Emission Theorems [ ][ ][ ][ ]. (2015) 135 ][ On the soft limit of closed string amplitudes with massive On the soft limit of tree-level string amplitudes On the soft limit of open string disk amplitudes with massive JHEP 07 ]. ]. ]. 0555 arXiv:1505.05854 , ]. JHEP Scattering Equations, Twistor-string Formulas and Double-soft [ SPIRE , (2016) 188 Phys. Rev. IN Soft theorems from anomalous symmetries , (2015) 150 ][ SPIRE SPIRE SPIRE JHEP SPIRE arXiv:1504.05559 , IN IN IN 12 [ Note on Identities Inspired by New Soft Theorems IN ][ ][ ][ B 905 ][ arXiv:1507.08829 (2015) 164 A Note on Soft Factors for Closed String Scattering [ Soft behavior of string amplitudes with external massive states arXiv:1503.04816 arXiv:1412.3699 arXiv:1511.04921 Soft Theorems from Conformal Field Theory JHEP 09 [ [ [ , (2015) 095 , Rome, Italy, July 12–18, 2015, vol. 4, pp. 4157–4163 (2017) Nucl. Phys. JHEP 07 (2016) 221 , , arXiv:1604.03893 [ (2016) 389 (2015) 070 arXiv:1604.00650 arXiv:1509.07840 arXiv:1504.01364 arXiv:1411.6661 DOI:10.1142/9789813226609 [ dilatons in the bosonic string Limits in Four Dimensions 170 Volumes) [ states Proceedings, 14th Marcel Grossmann MeetingExperimental on General Recent Developments Relativity, Astrophysics in and Theoretical and Relativistic Field Theories (MG14) (In 4 negative-helicity graviton [ 64 string theory C 39 Gluons and Gravitons JHEP states (2015) 065030 [ [ 03 Kalb-Ramond field in the bosonic string S. He, Z. Liu and J.-B. Wu, F. Cachazo, P. Cha and S. Mizera, J. Rao and B. Feng, P. Di Vecchia, R. Marotta and M. Mojaza, M. Bianchi and A.L. Guerrieri, M. Bianchi and A.L. Guerrieri, Y.-t. Huang and C. Wen, P. Di Vecchia, R. Marotta and M. Mojaza, P. Di Vecchia, R. Marotta and M. Mojaza, A.L. Guerrieri, S.D. Alston, D.C. Dunbar and W.B. Perkins, A. Volovich, C. Wen and M. Zlotnikov, M. Bianchi and A.L. Guerrieri, F. Cachazo, S. He and E.Y. Yuan, A.E. Lipstein, T. Klose, T. McLoughlin, D. Nandan, J. Plefka and G. Travaglini, A. Sabio Vera and M.A. Vazquez-Mozo, P. Di Vecchia, R. Marotta and M. Mojaza, B.U.W. Schwab, [72] [73] [70] [71] [68] [69] [66] [67] [63] [64] [65] [61] [62] [58] [59] [60] [56] [57] [55] JHEP02(2019)086 , ]. , 09 ] JHEP , ] D 46 SPIRE Phys. ] JHEP ]. IN , , [ ]. SPIRE ]. IN (2017) 123 SPIRE Phys. Rev. (2016) 020 ][ , IN 11 ]. ][ (1970) 1559 12 SPIRE (2016) 165 arXiv:1702.03934 IN ]. [ arXiv:1705.06175 ][ 09 D 1 [ JHEP arXiv:1801.05528 Subleading Soft Theorem for SPIRE , [ JHEP ]. IN , A Periodic Table of Effective SPIRE ]. ][ IN JHEP ][ , (2017) 113 ]. SPIRE SPIRE (2017) 001 IN arXiv:1611.02172 Phys. Rev. 06 Next-to-soft corrections to high energy IN [ , ][ arXiv:1611.07534 09 ][ SPIRE [ (2018) 201601 ]. IN arXiv:1707.06803 JHEP ][ [ , Soft Photon and Graviton Theorems in Effective ]. 120 JHEP Soft behavior of a closed massless state in Double-soft behavior of the dilaton of , (2017) 052 SPIRE ]. arXiv:1706.00759 – 49 – IN [ [ 01 SPIRE arXiv:1611.03137 Infinite Soft Theorems from Gauge Symmetry [ IN (2017) 231601 (2017) 150 SPIRE ][ IN arXiv:1702.02350 12 arXiv:1805.11079 JHEP [ ][ 118 [ , ]. ]. ]. Hereditary effects in gravitational radiation (2017) 065 Phys. Rev. Lett. arXiv:1802.03148 , Universality of soft theorem from locality of soft vertex operators (2017) 020 10 Infinite Set of Soft Theorems in Gauge-Gravity Theories as [ JHEP Observational Signature of the Logarithmic Terms in the Soft Gravity Waves from Soft Theorem in General Dimensions Sub-subleading Soft Graviton Theorem in Generic Theories of Logarithmic Terms in the Soft Expansion in Four Dimensions SPIRE SPIRE SPIRE , 02 IN IN IN ]. ][ ][ ][ (2018) 400 arXiv:1806.01872 JHEP , (2017) 045002 , JHEP arXiv:1804.09193 SPIRE , Phys. Rev. Lett. [ IN , Relativistic gravitational bremsstrahlung B 936 arXiv:1801.07719 [ (2018) 045004 D 96 Double soft limit of the graviton amplitude from the Cachazo-He-Yuan formalism Double Soft Theorem for Perturbative Gravity [ ]. ]. ]. Soft Theorems in Superstring Theory Subleading Soft Graviton Theorem for Loop Amplitudes D 98 (2018) 056 SPIRE SPIRE SPIRE IN arXiv:1703.00024 IN IN arXiv:1607.02700 arXiv:1610.03481 Graviton Theorem (1992) 4304 Nucl. Phys. (2018) 105 10 [ Rev. Quantum Gravity Multiple Soft Gravitons Ward-Takahashi Identities [ spontaneously broken conformal invariance [ Phys. Rev. [ scattering in QCD and gravity Field Theory Field Theories [ superstring and universality in[ the soft behavior of the dilaton A. Laddha and A. Sen, P.C. Peters, L. Blanchet and T. Damour, A. Laddha and A. Sen, A. Laddha and A. Sen, Z.-Z. Li, H.-H. Lin and S.-Q. Zhang, S. Higuchi and H. Kawai, S. Chakrabarti, S.P. Kashyap, B. Sahoo, A. Sen and M. Verma, Y. Hamada and G. Shiu, P. Di Vecchia, R. Marotta and M. Mojaza, A. Laddha and A. Sen, A.P. Saha, A. Sen, A. Sen, H. Elvang, C.R.T. Jones and S.G. Naculich, C. Cheung, K. Kampf, J. Novotny, C.-H. Shen and J. Trnka, P. Di Vecchia, R. Marotta and M. Mojaza, A. Luna, S. Melville, S.G. Naculich and C.D. White, A.P. Saha, [90] [91] [92] [88] [89] [86] [87] [84] [85] [82] [83] [79] [80] [81] [77] [78] [75] [76] [74] JHEP02(2019)086 , , ] ]. ]. SPIRE IN ]. ][ SPIRE IN [ ]. arXiv:1512.00281 ]. [ SPIRE ]. Unified limiting form of graviton IN SPIRE ][ SPIRE IN IN SPIRE [ [ IN ][ arXiv:1211.6095 [ (2016) 044052 Black hole mass dynamics and renormalization Infrared features of gravitational scattering and arXiv:1901.10986 , (1973) 4332 Soft gravitational radiation from ultra-relativistic D 93 – 50 – arXiv:1409.4555 [ D 8 Improved treatment for the infrared divergence problem in arXiv:1812.08137 (2014) 124033 arXiv:0912.4254 , Gravitational Radiation from Massless Particle Collisions [ Gravitational radiative corrections from effective field theory Phys. Rev. , D 89 Phys. Rev. , (2016) 125012 33 (2010) 124015 Phys. Rev. , D 81 ]. SPIRE IN radiation in the eikonal approach [ collisions at sub- and sub-sub-leading order quantum electrodynamics radiation at extreme energies Phys. Rev. group evolution Class. Quant. Grav. M. Ciafaloni, D. Colferai and G. Veneziano, G. Grammer Jr. and D.R. Yennie, M. Ciafaloni, D. Colferai, F. Coradeschi and G. Veneziano, A. Addazi, M. Bianchi and G. Veneziano, W.D. Goldberger, A. Ross and I.Z. Rothstein, A. Gruzinov and G. Veneziano, W.D. Goldberger and A. Ross, [98] [99] [96] [97] [94] [95] [93]