JHEP09(2020)129 Springer July 7, 2020 : a August 24, 2020 : September 21, 2020 , : Received Accepted Published and A

JHEP09(2020)129 Springer July 7, 2020 : a August 24, 2020 : September 21, 2020 , : Received Accepted Published and A. Strominger , a Published for SISSA by https://doi.org/10.1007/JHEP09(2020)129 N. Paul [email protected] a,b sruthi , M. Pate, 1 [email protected] -matrix deﬁned with an infrared (IR) cutoﬀ factorizes into , a, S . 3 -matrix in gravity 2005.13433 -matrix. The Authors. S S S.A. Narayanan, c Scattering Amplitudes, Space-Time Symmetries, Gauge Symmetry

a The gravitational , [email protected] Corresponding author. 1 Cambridge, MA 02138, U.S.A. E-mail: [email protected] [email protected] Center for the FundamentalCambridge, Laws of MA Nature, 02138, Harvard U.S.A. University, Society of Fellows, Harvard University, b a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: translations. The second is itsbroken symplectic supertranslations. partner, the Goldstone The currenting current for the algebra spontaneously gravitational has cusp anfurther anomalous off-diagonal shown dimension level that and structure the the involv- within gravitational logarithm the memory of soft effect the is IR contained cutoff. as It an is IR safe observable Abstract: hard and soft factors.divergences. Here The we soft show,the factor in soft is a factor universal momentum and is spacesphere. basis, contains fully The that all reproduced first the of the by these intricate IR two is expression the boundary and for supertranslation collinear currents, current, which which generates live spacetime on super- the celestial E. Himwich, The soft JHEP09(2020)129 ) C CS , (1.1) ]. In U(1) gauge 6 , 5 3 ]. In particular, SL(2 2 , 7 1 . CS i n 6 -matrix S ···O 1 9 – 1 – i → hO 4 in 5 |S| 5 out h 8 ]. ]: 9 – 4 7 , 3 3 1 8 ]. -matrix for massive particles 12 – S 10 Soft factorization is a closely related universal IR feature of scattering amplitudes. It These symmetries are realized as current algebras on the celestial sphere. This struc- 6.1 Gravity 6.2 Gravitational memory 4.1 Gravitational Wilson4.2 lines Summary of current algebra ture both governs thelatter form are of needed andtudes in explains [ order the to origin set of to infrared zero (IR)posits all divergences: that conservation-law-violating the scattering exclusive amplitudes ampli- (with an IR cutoff) can be factored into hard and soft The asymptotic symmetries additionallyservation include laws supertranslations, are whose associated equivalenttheory, to con- the Weinberg’s soft soft photon gravitonlarge theorem gauge theorem symmetries is [ [ the conservation laws following from the asymptotic at null infinity. This observationcorrelation lends functions scattering of amplitudes a aon holographically natural the reinterpretation dual celestial as “celestial sphere conformal [ field theory” (CCFT) 1 Introduction The scattering amplitudes ofform four-dimensional covariantly asymptotically under asymptotic flat symmetries, quantum which gravitymetry include trans- the group infinite-dimensional sym- of aLorentz transformations two-dimensional act conformal as field the theory global conformal [ symmetry on the celestial sphere 7 Soft 5 Infrared divergences from virtual6 gravitons Current algebra correlators and the soft 3 Supertranslations and Weinberg’s soft4 graviton theorem Currents on the celestial sphere Contents 1 Introduction 2 Preliminaries JHEP09(2020)129 we intro- ]. 4.1 18 ]. The gravitational 15 , 14 , ] to gravitational theories. 12 13 ]. This effect is encoded in the , by modifying the Wilson line , we recast the statements from ] for QED to be celestially realized 17 7 4 13 , we relate the OPEs determined pre- -matrix for an arbitrary number of soft presents notation and conventions. In S to include external massive particles. An 2 ], we find that the celestial algebra includes 6.2 5 6 , 5 – 2 – , we review the soft graviton mode, which realizes the 4.2 ]. , we show that the soft 16 , we turn to IR divergences in scattering amplitudes arising from in- 6.1 5 , we review the relation between supertranslation symmetry and IR behavior as current algebra identities on the celestial sphere. In subsection 3 3 -matrix. Inequivalent vacua are related by supertranslations and the memory ef- -matrix was reproduced from a 2D current algebra on the celestial sphere. S S The paper is organized as follows. Section Generic scattering processes induce transitions between inequivalent vacua, which can The purpose of this paper is to extend the analysis of [ -matrix and demonstrate that it is reproduced by correlation functions of the Goldstone gravitons and hard particlescurrents is and reproduced Goldstone by correlators bosons.viously involving to both In the supertranslation subsection gravitationaloperators, we memory generalize effect. the analysisextension in In of section this section analysis to a conformal primary basis will be included in [ Goldstone mode. In subsection supertranslation current on thealgebra. celestial In sphere, section andternal soft derive graviton the exchanges. OPEs WeS review of the the Feynman diagrammatic associated formulamode. for the In soft subsection section of gravitational scattering matrixsection elements. In section duce, for hard massless external particles, Wilson line-like operators constructed from the soft fect measures theory transformation is of an the IRIR Goldstone safe finite observable boson operator and underGoldstone we product the boson. show expansion it symmetry. (OPE) is reproduced between Mem- in the the supertranslation current current algebra and by the cusp anomalous dimension, which appears into the soft level factorization of the in Goldstone [ current, was related be measured through the gravitational memory effect [ to the gravitational cuspsoft anomalous part dimension of andcurrent the gravitational algebra logarithm correlation scattering functions. of amplitudes In thesoft direct is IR piece analogy shown cutoff. with is the a to gaugeGoldstone The correlation theory be boson function analysis, the and of entirely Wilson supertranslation reproducedtheir line-inspired currents. relation by to operators gravitational Gravitational constructed soft Wilson dressings from were line the discussed operators in [ and We determine, in a momentum spacegoverns basis, the the leading current IR algebra behaviorpreviously-studied on supertranslation of the current gravitational celestial [ scattering sphere that amplitudes.a In second addition “Goldstone to current” the spontaneously broken that supertranslation is symmetry. It the has a gradient non-vanishing of level proportional the Goldstone mode associated to of the full amplitude. Thisas factorization the was familiar shown in factorization [ of 2Dsoft” CFT current correlations algebra functions into factor asoft and universal “conformally a theory-dependent “conformally hard” factor. The full pieces, where the soft piece is given by a universal formula that contains all IR divergences JHEP09(2020)129 . . n 2 I | ij (3.2) (2.5) (2.6) (3.1) (2.2) (2.3) (2.4) z | 2 . i . in + ≡ − ν k 2 ε | |S| j + z µ k ε out − ) and polarization h ¯ . i z = k k ) z z | , k ω + z, crosses null infinity 2 ¯ z ( µν k k − , , ε µ η k ) − z k ˆ q ¯ p z z ) ω ( n =1 ¯ z µ k X . ) = ˆ q = j ¯ z. 2 ω, z, ¯ k z = ¯ µ ( z − , i q ∂ ], defined as j ω, z, dzd ( z in 2 out , 2 6 ( † − 1 in , r ) = µ √ † − 5 )) ¯ z q |S| [ ¯ ) (2.1) ωa z )ˆ + i = k ωa z, ¯ z z ¯ z z, ( out , ( , − P µ h i ) + µ k k µ ˆ q z ¯ ) + z # ˆ z q dudr ( µ ¯ z ), in which the standard Cartesian coordi- ( ν k r µ ¯ − z µ p ˆ q is conformally mapped to a plane on which q ˆ q µ k + · k ω, z, = p k , n µ ( ω, z, ω , – 3 – p ν ( CS k + µν ) 1) , ε ¯ ε un out + in + η z u, r, z, dx ) ( − z k µ , = 1 2 n ωa ωa =1 ¯ z 0 − k X , , µ k dx -matrix. k 1 0 = ¯ ¯ p z z ω z , , S µν ∂ ∂ " ( ) µ η 0 0 ¯ z ¯ µ z x ˆ q ∂ → → = k 0 − lim lim = (1 ω ω z 2 z → ∂ µ − − lim ( ω 2 i n ds 1 − = = − √ − + = z, z z = 1 for outgoing and incoming particles respectively, and i + ¯ + κP κP in | µ k , Weinberg’s soft graviton theorem for a scattering process involving ε − = z ¯ z, z P k πG z η 32 0, √ S −S ≥ + z = k P | κ ) = (1 + ω ) labels the point where a massless particle with momentum ¯ z k Introducing the soft graviton mode operators We employ flat Bondi coordinates ( ¯ z becomes out z, , h ( k µ µν z ˆ q where hard massless particles and one softε graviton of momentum 3 Supertranslations and Weinberg’s softIn this graviton section, theorem wesupertranslation review symmetry the of equivalence the between Weinberg’s soft graviton theorem and In these coordinates, the( celestial sphere where The Minkowski line element becomes The polarization tensor for a positive helicity outgoing gravitonnates is are where Polarization tensors for spinningvectors massless particles are constructed from the polarization We consider four-dimensionalMinkowski background. perturbative We parametrize gravity null momenta coupled as to massless matter on a 2 Preliminaries JHEP09(2020)129 , S act Q (3.6) (3.7) (3.8) (3.3) (3.4) (3.5) S + Q i H in ]. Q ]: O ] and refer- 6 |S| 4 = Q, [ i out h Q = ) -matrix is z O S δ − , is k + − z Q and then integrating the I ( | . The soft charges z (2) i zz i -matrix elements presented δ . k . C in i k S p ¯ | z − | z ω ∂ in ) − S | k P k η Q ¯ z − − H , is canonically paired with the soft with charge Q + + k n =1 1 I z k X | ( , C ) δφ f ¯ zz S − S z . C. k + S + C 2 z = 0 S − S z, ω C ¯ z ∂ φ ( Q k ∂ f. + H | − ) with respect to ¯ − iη → zf Q Q = = | 2 φ = out 3.2 = d C zz – 4 – i + − ): out f −h k I Z N ¯ z δ | p ¯ z S − S | zz = ≡ h z, + 1, respectively. For the full charge H ( i C Q f du ∂ iQ in | = − ∞ = z k −∞ P i η Z k p | = f δ + z S − S ) is identified as + z implement the action of the supertranslation symmetry on matter. GP P 3.5 | 4 H is the radiative component of the gravitational field (see [ Q out zz h ¯ z C ∂ u ) ∂ and related to a boundary component of the asymptotic metric [ ¯ z as the action of a current algebra on the celestial sphere. = acts on states with and boundary components of the asymptotic metric + z, z ( 3 P + H zz f I z Q N π 2 2 d Supertranslation symmetry is spontaneously broken in the standard Minkowski vac- The statement of supertranslation invariance of the gravitational on The operator equation for a classical symmetry 1 Z + z 4 Currents on the celestialIn sphere this section, wein recast section the supertranslation symmetry of The Goldstone boson transformsneous under shift: an infinitesimal supertranslation by an inhomoge- uum, giving rise tograviton a Goldstone boson, denoted where the left-hand side ofby ( adding soft gravitons, performing a non-trivial vacuum transformation. The hard charges A single hard massless asymptotic state transforms as which follows from taking theresult derivative of against ( an arbitrary function ences therein for definitions). Similar expressions hold for P where A Fourier mode decomposition of the graviton field in turn relates the soft graviton mode JHEP09(2020)129 CS (4.8) (4.5) (4.6) (4.7) (4.1) (4.2) (4.3) (4.4) . 1 is invariant − k ˜ O then gives = [mass] C zz g 1 r , ) , , we introduce operators of i k k ¯ z . n ] = . , k ) O C k k O z ), implying that p alone create physical scattering ( ( ···O k , k k ) 1 4.1 ˜ . k O W . have the OPE O ¯ z ) ) , k hO w k k k . k i ¯ ¯ z k z z . W − ( k z − , , z − W ) in ( k nor k C ω k k P k k z ˜ z k z z k − O p k ω ( ( ω η k ( − z ), which transform under supertranslations k k − ∼ f f k and W k η k k iη + W ) z z p O ), z e n ω ω =1 ¯ ( P w – 5 – k X P k k = k ∼ 3.8 k ≡ iη iη O k w, = ) = ( z k O i W p C P n z ( ) = ) = z k k k P P in terms of the Goldstone operator ¯ z p ) becomes ( W , k ···O k k z 1 3.2 W O ( f k O δ z ) implies that follows from ( W P h f ) for k 4.3 δ W 4.2 ): 3.6 ), the soft theorem ( 2 1.1 . z The decomposition ( Massless particles in momentum eigenstates are associated to unique points on Note that the argument of the exponent is dimensionless because [ P 2 Using the expression ( which is immediately recognized as theby Ward identity of a Kac-Moody symmetry generated Using ( states. Physical states are dual to the4.2 composite operator Summary ofOn current the algebra celestial sphere, Weinberg’sa soft current graviton theorem can be recast as the insertion of The transformation of and accounts for theunder full transformation supertranslations. of Note that neither the form and decompose and are represented byaccording local to operators ( To isolate the supertranslation transformation properties of Weinberg’s soft graviton theorem, astotic reviewed states in of the hard previous massless particles section, transformwe implies non-trivially recast under that supertranslations. asymp- the Here, actionWilson of line-inspired supertranslations operators on constructed celestialtherefore from correlators also gravitational as include boundary in the modes, the transformation which supertranslation of we current algebra. 4.1 Gravitational Wilson lines JHEP09(2020)129 . i ) k k 1 C ω − in O 5.8 (5.1) (4.9) |S| (4.10) (4.11) (4.12) ]. The \ 21 out h ], implying 20 ) rather than i IR λ in |S| \ out is a spin-one operator of h z   P j ], p 2 · , so we expect an OPE of the form 19 i µ p 1 2 2 , . − ) ) 2 , ) conformal symmetry, under which ) ¯ an arbitrary mass scale and 1 2 w w C ln w follows from that fact that the argument . , µ − − j 0 1 C, − p z C z z · ∼ (¯ ( z i∂ ], virtual graviton exchange contributes a uni- i ( k w p with = 20 – 6 – P ∼ ∼ z 3 z =1 n ˜ w P w P X ). As shown in [ i,j ˜ ˜ P P 1 2 z z π , ˜ G P 2 P 1 2 , the exponent becomes proportional to log( has dimension 1 z IR   ˜ P λ ). The weight of 1 2 , and its action raises the conformal dimension of operators − ) must have net zero conformal weight and that energies = exp hard massless particles factorizes as , -matrix. This section reproduces the virtual soft exchange factor 1 2 1 2 i S n , 4.2 in − 3 2 |S| implies that C out h dimensional regularization ), we immediately find the OPE 2 4.8 current algebra. − . is a constant that we will determine in the following section (see equation ( 1 2 k CS , = 4 1 2 We begin by recalling the general formula for IR divergences derived in e.g. [ To deduce the OPE of two supertranslation currents, note that soft gravitons have When the IR regulator is a cutoff d 3 -matrix for scattering in IR finite. in the S In addition to singularities arising fromare the captured emission by and absorption the of soft softries graviton gravitons, of theorem, which gravity scattering contain amplitudes IR in divergencesexternal arising four-dimensional legs. from theo- virtual As soft Weinberg first gravitonsversal explained exchanged soft [ between factor to the where therein). 5 Infrared divergences from virtual gravitons The dimension of and from ( vanishing energy and do not couple atthat leading order in a low-energy expansion [ In analogy with the supertranslation current, we define a Goldstone current has conformal weight ( of the exponentialtransform in with ( conformal weightconformal ( dimension by This OPE is of the general form allowed by SL(2 JHEP09(2020)129 k k W W and (5.6) (5.7) (5.8) (5.2) (5.3) (5.4) (5.5) z P , 2 | ij z . | i . ln -point correlation in j ), involving a sum   n i . p |S| · ) ) of operators accord- i j 5.4 ¯ \ p z = 0 , out j 4.3 j h n 6= z , ! i X (   . 2 | iµ i 2 C = | p n ij ) ij i z ˜ | O ¯ z z n | =1 2 , i | X i ln -matrix ··· z ij ln

2 ( ), we find z | 1 S 2 | µ j | ˜ j ij C O p h ij z 2.1 ) ω | j z 2 i ih j | ω n µ ω j ω G i π . j j 2 ω ω η i η , we find G i j π 1 ω , we fix the level of the Goldstone current 2 η η C j i z ln( 4 ···W 1 η = ˜ η P i 1 − i j – 7 – η n =1 ) = n 6= j X j j i X hW ¯ k z n 6= ln i ) of , X 1 2 . Thus expressing the correlation function of j j = ) = z C − p π j G i ( ·   n i ω 4.10 1 , note that the decomposition ( C j p ) η − i CS ¯ z =1 ) becomes   n , ···O ln( X i i,j = exp 1 j z 5.1 ( p i = · n hO C i h = exp p j i , we now show how these results can be used to calculate the soft p 2 · =1 in i C n ), we immediately deduce ···W µ p X i,j 1 2 correlators are the soft (IR divergent) and hard (IR finite) factors, |S| 5.4 ) to be k hW ˜ out ln O h j , can be reproduced by a correlation function of exponential operators 4.12 p -matrix in gravity, including the emission of soft gravitons and the gravitational · i i, j S in ( p and k k =1 n X i,j W It follows that the general form of the IR divergent piece in ( To recast this result on the Working with the momentum parametrization ( = 0, and invoking total momentum conservation to obtain identities such as 2 i 6 Current algebra correlatorsHaving and fixed the the soft Goldstone leading operators singularities inpart the of the OPEs ofmemory the effect. soft photon currents While the level of thethat it Goldstone does current is not affect IR IR divergent, we safe will observables such show as in gravitational the memory. next section Comparing with ( from which, using thealgebra definition ( function of theoperators exponentiated in field terms of correlation functions of The respectively. over pairs for which the two-point function is the only non-vanishing connected ing to their supertranslationcorrelation symmetry functions: transformation properties implies factorization of With these simplifications, ( where the last equalityp follows from expanding the logarithm into a sum of terms, using JHEP09(2020)129 . k W (6.7) (6.4) (6.5) (6.6) (6.1) (6.2) (6.3) ): 4.3 .   2 | k` z | . , i ln , i . and operators n 2 i i n i | ˜ z ) to obtain the last n O n n k` P z , and also only involves 5.5 | ··· ` ···O w 1 ˜ ω ···O ···W 1 ···W P ˜ and noting that it can be k O z 1 1 1 to compute the soft part of O ˜ ω P z ih z ` 5 . hO n P η P hW hW i i h k   η   i in in i j and ` | , not = ) in z j n 6= − S| w 4 j ···W z i k X ω ˜ ¯ z + P 1 n − P z i j |S| , hard massless particles is represented z η n i π j G P W | |S z P z n 1 ( m out z h n − C ···W =1 out out P j z X ) imply a further factorization: h ···W   1 P 1 −h h m =1 W = ··· 4.9 j i Y z 1 exp = i ω hW   z – 8 – P j n i i   h P n j = h iη in j z = i ] ω = n ) and ( S| n − j 5 i =1 ), we find ···O i j ) X i η + z ···O 1 z j n   4.7 ¯ z 1 5.7 P ) for a single insertion of O | , soft gravitons and n z =1 j = ···W j X hO z P 6.2 1 i ( h out m ···O n h m =1 1 2 1 C W i Y z ) and ( m   O is holomorphic away from other operator insertions, subleading P z m h = P z z j 5.6 ···W i P 1 ω P n j ··· W 1 iη ··· z z 1 P P z n ···W =1 h h j X P 1 h W m z P ··· 1 : z -matrix given by the correlation function of soft photon currents + We next use crossing symmetry [ We begin by focusing on ( Finally, combining ( If we assume that z P S h P to relate this ratio ofof correlation functions to a scattering amplitude with a single insertion where we use theequality. factorization property of correlation functions ( Rearranging, we find In this subsection, we recomputeThe the key gravitational point memory is effect that from gravitationalratios memory the of involves current algebra. amplitudes. It is therefore an IRwritten safe as observable. 6.2 Gravitational memory which is simplysphere. the manifestation of Weinberg’s soft graviton theorem on the celestial the terms in its OPEthe with leading other singularities operators given cannot by ( contribute to correlation functions. Then, A scattering process involving by a celestial correlation function, which factors according to the decomposition ( Now, we will use the current algebra specified in sections 6.1 Gravity JHEP09(2020)129 µν : k 2 h ratio (7.3) (7.1) (7.2) (6.8) (6.9) m (6.10) (6.11) . − µ k ˆ p = k k 2 boundary is m ], where p k η 17 ∞ ) involves a ≡ = ρ )) -matrix for massive 6.5 k . S ¯ , z i . k # n i OPE. Note that unlike ) z ) ¯ n z ν j ¯ z ν j − p C p z, z µ j z, ( (1 µ j ···O p P . ( ˆ q 2 k p ···O ˆ 1 q · hard particles [ ρ + µν · 1 + µν j O ε ¯ z j n ε z p , p 1+ hO P h )) " − dzd n =1 ¯ z ¯ 2 , z j X . ) = ρ ∂ z, k π ( ¯ µν G − z 2 + µ ) ˆ − ¯ q z 2 = κh 2 r 2 k ρ j ρ z dρ − + j ( z r, z, 2 k ω + ( j µν ] for gravitational memory due to the scattering − 2 µ iρ η – 9 – ). Massive momenta are naturally parametrized ) = η τ z µν ¯ ¯ n z z 22 − ( h = , + z, = ρ ) τ ∆ 2 k 2 i µν r, z, z ) µν + ( g j dτ = + ¯ ¯ z rε µν − , k µ j h z x z ( = ∆ ( 2 k →∞ lim 2 r induced by the scattering of C µν + , ρ z ds ) ¯ z P rε k µν h ¯ ∂ z h j k π ω z G →∞ 2 j lim are constant curvature hyperbolic slices and their r iη τ r ) on the hyperboloid, defined by the mass-shell condition (1+ k 2 k ¯ z ρ , k is IR finite and directly related to an IR safe observable: ( , z (1+ k -matrix for massive particles i k ρ ) and S k C z m ρ k 2 P 6.5 η h , i Massive particles in momentum eigenstates are not canonically associated to points = µ k -matrix for massless external particles correctly reproduces the soft p CC Surfaces of constant the celestial sphere, labelled byby points points ( ( in which the line element becomes external particles. on the celestialthree-dimensional sphere, hyperboloid. but We instead thusMinkowski to use space: coordinates points that on give a a hyperbolic resolution slicing of of timelike infinity by a 7 Soft In this section, we showS that the Goldstone two-point function derived above from the soft The gravitational memory formula ish thus determined by the of scattering amplitudes that precisely cancels the IR divergences due to virtual gravitons. we obtain the Braginsky-Thorne formula [ of massive bodies: Using ( the asymptotic metric ∆ is the perturbation of the metric about Minkowski space Altogether, we find In turn, the right-hand side can be interpreted as the expectation value of the change in JHEP09(2020)129 ) ,   i 4.2 ) (7.7) (7.8) (7.9) (7.4) (7.5) (7.6) ¯ z . z,   ( ! C ) i j ¯ ρ ρ w ) determined + w, i 5.7 j ( ρ , ρ C h k` + . ) γ ¯ z ij γ ) z, !# ) on the hyperboloid at ¯ ; z ) k j . cosh ¯ z ¯ ν k z p z, n (ˆ , p ( ) z, k )cosh µ k 2 ( C | j (3) p ≡ − ˆ ) q k , z ρ G ¯ i · z ] z k ) + µν ρ ) becomes ρ ε k ¯ 2 w z, 23 − | p n k ; z 7.5 ρ k k` w, |

, 2ln( p z ; 2 k | 2 ¯ z (ˆ i ` ρ − p ∂ 2 ρ (ˆ 1 2 | (3) ) k k ¯ G z ρ (3) z z ij (1 + G 2 z, + γ − z ( d 2 2 1) ` k z C | ρ ρ π Z k − z wd 2 π – 10 – ρ 2 2 k + 2 n d d j ( + ` m k 2 m k ρ ρ Z +4cosh k i Z 1 j ρ i iη approach the points ( ij ) = m ¯ j γ − m z k 1 2 i terms, which arise from Wick contractions within a single are given generally by [ η 2 " p i 1 z, − − j ) m η ; j n ) of massive momenta, ( ( k η = = i sinh =1 p G n ` ) = i η (ˆ 7.3 X ˆ ) ]. ¯ p i,j z ) = exp ). We will also make use of the following inner products: n · ij k =1 ( ) = exp 20 π n z, γ p G k k X 4 G ( ( i,j ˆ p p ˆ

q k 1 ( 1 8 · k × W − k − → ∞ ˆ p W     τ = exp = exp i i is implicitly defined to be real for any pair (outgoing or incoming) of on-shell n -point correlation function of such operators takes the form n n k` γ Explicitly evaluating this expression using the two-point function ( ···W ···W vanishes as the points are taken to be coincident. This reflects the well-known fact that collinear 1 The To generalize the Wilson line operators to massive particles, we begin by rewriting ( 4 1 Note that in the massless analysis, we do not encounter such terms because the two-point function . 4 C k hW hW of divergences cancel in gravity [ previously, we find where here the sum includes W where the Green’s functions Using the parametrization ( where momenta. as timelike infinity ( Massive particles with momenta JHEP09(2020)129 , i in , (7.10) (7.11) (7.12) ] to be |S| \ 24 , , out , h . Using, in   . 20 , = 0 = 0 ). 1 2   5 ]. jµ jµ 1 2 p p ij , (2014) 152 7.11 γ ij n n 2 =1 =1 j j γ X X 07 SPIRE in ( 2 µ i µ Yang-Mills Theory IN p n ) i iπ ][ i D i 4 ρ m − ρ i JHEP +4cosh η , ij ln( ij γ + 4 cosh γ n n =1 =1 2 i -matrix, shown in [ X i 1 X ij → S 2 γ 2 ) simplifies to 2 ij − − 1 sinh γ Semiclassical Virasoro symmetry of the = = 7.9 j ) ]. j sinh 1 ρ p , j arXiv:1406.3312 · ), ( j i η [ i m p ij η j ) 2.5 γ i SPIRE η j ρ IN iπδ m n =1 i ][ j X ln( − – 11 – Kac-Moody Symmetry of i m ij j i =1 n γ m η (2014) 058 D ρ i ( X i i,j j η η 2 08 m i − n =1 =1 n Evidence for a New Soft Graviton Theorem i X X m i,j = ), which permits any use, distribution and reproduction in j ]. ]. η ij π i = 2 ]. G JHEP γ 4 η , 1 i j =1 n − ρ ρ SPIRE SPIRE arXiv:1503.02663 X = 1, we analytically continue i,j   SPIRE [ )cosh IN IN + j [ j π IN i η j [ G ][ ρ -matrix i 4 ρ i CC-BY 4.0 ρ η S 1 ρ Lectures on the Infrared Structure of Gravity and Gauge Theory On BMS Invariance of Gravitational Scattering This article is distributed under the terms of the Creative Commons = exp j − ln( i   j m n i (2016) 137 m i m j m 10 η i = exp j ···W η i η is the constant null vector defined in ( 1 i η in =1 n n hW arXiv:1312.2229 X i,j JHEP arXiv:1703.05448 [ arXiv:1404.4091 quantum gravity =1 |S| n A. Strominger, A. Strominger, F. Cachazo and A. Strominger, D. Kapec, V. Lysov, S. Pasterski and A. Strominger, T. He, P. Mitra and A. Strominger, 2 X i,j [4] [5] [1] [2] [3] out h Open Access. Attribution License ( any medium, provided the original author(s) and source areReferences credited. We are grateful tocussions. Nima Arkani-Hamed, Aditya This Parikh, work andand was Betty Ana supported Moore Raclariu by Foundation forInitiative. and useful DOE John dis- N.P. grant Templeton was DE-SC/0007870 Foundation supported grantsof and by via a by the the Junior the Purcell Black Fellowship at Gordon fellowship. Hole the M.P. Harvard acknowledges Society the of support Fellows. provided that when Acknowledgments This correlation function reproduces the soft factor of the where particular, Invoking total momentum conservation simplifies the result, as in section JHEP09(2020)129 , 01 , ]. JHEP ]. , ]. SPIRE IN (2011) 060 SPIRE Kac-Moody ][ IN 05 SPIRE ]. [ D ]. 2 IN ][ ]. SPIRE JHEP Adv. Theor. Math. Phys. SPIRE , IN , IN Celestial Amplitudes from ]. , Cambridge University Press ][ ]. ]. ][ SPIRE ]. (1965) B516 ]. ]. IN ][ SPIRE Infrared Divergences in QED, SPIRE 140 arXiv:1910.05882 SPIRE IN [ SPIRE IN SPIRE SPIRE IN ][ New Symmetries of Massless QED IN IN ][ IN ][ ][ arXiv:1708.05717 ][ ][ ]. 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