JHEP09(2019)056 Springer July 12, 2019 May 20, 2019 : : August 19, 2019 : September 9, 2019 : Revised Received Accepted Published e , Published for SISSA by https://doi.org/10.1007/JHEP09(2019)056 and Justin Vines d [email protected] , Astronomy, University of Waterloo, & . 3 Alexander Ochirov 1812.06895 a,b,c The Authors. c Scattering Amplitudes, Black Holes

We provide evidence that the classical scattering of two spinning black holes , [email protected] [email protected] ETH Z¨urich,Institut f¨urTheoretische Physik, Wolfgang-Pauli-Str. 27, 8093 Z¨urich,Switzerland Max Planck Institute forAm 1, Gravitational M¨uhlenberg Physics Potsdam (Albert 14476, Einstein Germany Institute), E-mail: Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada Department of Physics Waterloo, ON N2L 3G1, Canada CECs Valdivia & Departamento deCasilla Universidad F´ısica, 160-C, de Concepci´on,Chile Concepci´on, b c e d a Open Access Article funded by SCOAP These connections link thedeeper computation orders in of the higher-multipole soft interactions expansion. toKeywords: the study of ArXiv ePrint: tion. A four-point gravitational Comptontheorem, amplitude equivalent is to obtained gluing froman two an exponential exponential extrapolated operator. three-point soft amplitudes, Thetree-level construction and scattering uses becomes angle these at itself amplitudes all to: ordersinteraction, in 1) 3) spin, recover match 2) the recover a known theat previous known quadratic one-loop conjectural order linear-in-spin expression in spin, for 4) the propose one-loop new scattering one-loop angle results through quartic order in spin. ing previously known resultsof to the higher classical orders limiting in is massive spin spinor-helicity accomplished variables. at via The one-loopparticles the three-point order. amplitude on-shell minimally for leading-singularity coupled arbitrary-spin The method massive infinite-spin to extraction and limit gravity us- it is matches expressed the effective in stress-energy an tensor of exponential the form, linearized and Kerr solu- in the Abstract: is controlled by thetion soft of expansion Cachazo-Strominger of soft exchanged factors,used gravitons. acting to on We find show massive spin higher-spin how contributions amplitudes, an to can exponentia- the be aligned-spin scattering angle, conjecturally extend- soft factors Alfredo Guevara, Scattering of spinning black holes from exponentiated JHEP09(2019)056 (1.1) , ) 2 k acting on them ( , i.e. O . The sum is over µν i ν µν + J ε J n µ 6 ε M = ]. Indeed, it was already  13 2 6 , µν ) 5 ε 8 µν i k · J 19 corresponds to the external soft ν i ε p 17 k µ k 11 ( 28 1 2 ], the subleading behaviour of gravity , and the operators 4 µ i − , p 3 17 ) µν i J ν – 1 – ε k 22 15 µ · k i 20 p )( 31 30 ], whose universality is associated to the equivalence 7 ε 2 ] showed that the soft limit of tree-level gravity ampli- · 1 i p ( i 6 + 2 ) k ε · · i i p p (  1 n =1 i 26 X = +1 n 2.1.1 Spinor-helicity2.1.2 recap Spin-1 amplitude in spinor-helicity variables M 3.4 Second post-Minkowskian order 2.4 Factorization and soft theorems 3.1 Linearized stress-energy3.2 tensor of Kerr Kinematics solution and3.3 scattering angle for First aligned post-Minkowskian spins order 2.1 Massive spin-1 matter 2.2 Exponential form2.3 of three-particle amplitude Exponential form of gravitational Compton amplitude the remaining external particlesinclude with momenta both orbital andthe spin standard parts Weinberg soft ofprinciple. factor the [ angular Following momentum. theamplitudes QED The was results first first of term studied Low is long [ simply ago by Gross and Jackiw [ up to sub-subleading order.graviton, and Here we the have soft constructed momentum its polarization tensor as 1 Introduction In 2014 Cachazo and Stromingertudes [ is controlled by the action of the angular momentum operator B Spin tensor for spin-1 matter C Angular-momentum operator 4 Discussion A Three-point amplitude with spin-1 matter 3 Scattering angle as leading singularity Contents 1 Introduction 2 Multipole expansion of three- and four-point amplitudes JHEP09(2019)056 ] 1 ], p 8 , 19 ]. − 7 – 2 p 21 k 15 = , k 8 | , k 1 [ i k ) | ], as illustrated HCL k , = 27 k ]. Spin effects require = (0 ]. On a different front, (b) 30 k , ] by employing the soft ]. Indeed, rotating black 13 , 28 , 22 26 of which can be reproduced ], although it is known that 12 – 27 s ) to higher orders in the soft 11 23 – NR Classical Limit 9 1.1 Soft Expansion 0 k – 2 – b b , m , m b b , s , s 4 3 ] to derive the spectrum of the radiated power in black- p p 22 ]. 29 minimally coupled particles exchanging gravitons [ – k (a) ... s ] has led to impressive and wide-reaching developments [ 27 14 ] that the subleading soft theorem follows from gauge invariance (see [ 6 , 5 . The matching between these amplitudes with spin and a non-relativistic a a (a) Four-point amplitude involving the exchange of soft gravitons, which leads to classical 1a , m , m a a ] for a recent review. Following such correspondence, an infinite tower of Ward , s , s 2 1 20 Here we present a complementary picture to the one of [ Recently, a classical version of the soft theorem up to sub-subleading order has been p p (PN) framework [ theorem in the conservative sectorholes (i.e. and no at external theexpansion. gravitons), same focusing time on This extending rotating is the black soft achieved factor in in the ( following way: It was shown by one of the authors computed from the scattering ofholes massive can be point-like treated sources via [ aby spin-multipole expansion, scattering the order spin- 2 in figure potential for black-hole scattering has been performed explicitly in the post-Newtonian identities has indeed been proposed to follow from all ordersused in by the Laddha soft expansionhole and [ scattering Sen with [ externalthat soft conservative and graviton non-conservative insertions. long-range effects This of relies interacting black on holes the can remarkable be fact subleading order. Starting atmatter sub-subleading content and order EFT the operators softgauge present invariance expansion still in provides can the partial depend information theory atthe [ on all realization the orders that [ soft theoremsat correspond to null Ward identities infinity for [ asymptoticsee symmetries [ leading order in HCL through the soft expansion. observed in [ for a modern perspective), and because of this, it also adopts a universal form up to Figure 1. observables. The external massive statesbetween are the interpreted HCL as and two the black-holesubleading sources. non-relativistic orders limit (b) in in Comparison the the COM nonrelativistic frame (NR) [ classical limit, but can be fully determined at the JHEP09(2019)056 , , ]. ). ν  ε . ) 33 µ 3 # 1b (1.3) (1.4) (1.5) (1.2) ε ) k ) ( 3 2 k O k ( ( O + πδ 2 ]. Now, the + ) k 2 32 ·  a ) by 2 ( µν corresponds to the ε k S ( µν · , ν 2 , η ρ ) ε ) p ) parts, the condition µν p 1 2 ε µ ~ h G · k ( ( µν + p  S O O σ 1 2 amplitude can be extracted a +  ρ − s ρ λµνρ k is identified with the angular µν  p ε 2 ρ S µν S · ) ν ε 1 S ν m µνρσ ε p · ) ν 2 µ i ε p k ik µ = i − k ∗ i λ − is the rescaled spin vector of the black a µν a  ], which drops − η  1 31 /m ν exp exp( " µ p 2 2 µ µ ) ) S ( – 3 – ε ε ε p ⇒ ) · · ) = ε k p p · · µ )( )( between the massive sources is null. On the support p p a k k ( )( k σ · · k a p p πδ · ρ ( ( p δ δ p ]: ) ) ( , and 2 2 δ σ 32 ) ) = 2 k k -independent) piece of the spin- k µνρσ 2 ( ( k ~  ρ δ δ k is exactly the radius of its ring singularity. Here we have per- a ( 2 2 − ) ) ( = δ a 2 π π νρσ ) µν µν µ π T  S ). Here it naturally appeared as a rewrite of the exponential structure = (2 carries the intrinsic angular momentum of the black hole (see figure = = 0, after which it becomes ) = (2 1.1 S k ν ν of the matter particles. ) = (2 p − µ k i ( ) J µν − k ( µν S ∗ . After we take the graviton to be on-shell and replace T µν ) a , where T µν k n ) ( ] that the classical ( T S k Even though the fact that classical gravitational quantities can be reproduced from To see the soft expansion more explicitly, consider the energy-momentum tensor of a ( µν n 29 µν = 0 reduces the amplitude to a purely classical expansion in spin multipoles of the form h k h µν 2 h way beyond what iscoupled’ guaranteed by nature universality of and thecontribution it Kerr of is solution. the a energy-momentum consequence Note tensor of of that the the the ‘minimally linearized prefactor SchwarzschildQFT solution ( computations [ has been known for a long time, the precise conceptual foundations of The terms inside thepansion parentheses in look eq. precisely ( likeof an the exponential linearized completion Kerr of energy-momentum the tensor. ex- We will see that this structure extends satisfying where we have used thea support simple of form by the introducing delta the functions. spin This tensor expression can be written in formed a Fourier transforminteraction vertex of between the a graviton worldline− and formulas a (18) massive source and correspondsthe (32a) to vertex the of becomes contraction [ where ( hole. The magnitude This precisely matches thethe soft graviton expansion momentum oncemomentum and the the momentum transfer classical is spin recognized vector as single linearized Kerr black hole,form which by one has of recently the been authors written [ down in an exponential from a covariant Holomorphicsuch Classical that Limit the (HCL), momentum whichof transfer sets the the leading-singularity external kinematics (LS)k construction [ ∼ in [ JHEP09(2019)056 ] , ]. s E 2 ]. 35 i 28 2 (1.6) ] can 34 , ] and | ], i.e. . Both 41 25 40 26 , or – E 2 s p 2 i 36 , 2 ]. Moreover, − 1 ] in terms of | 1 , as appearing 33 p 30 , , 30 µν , J = 32 28 , , 28 , ] was simply to show 26 27 27 29 , p s 2 i 1 | . However, the explicit match  G µν ε J · ν ε p µ k i  0 limit. It is only after computing the ef- → exp s – 4 – ~ 2 | and one graviton is given by 2 h we show that the three-point scattering amplitude s 2 ) s ε 2.2 2 · p m The goal of one of the authors in [ 2( 1 ×  In section 2 κ ). This operator acts naturally on the product states −  1.1 ], can be computed with spinning particles directly from the LS. The ) = 40 , − , k 38 2 , , p ]. For these reasons a more direct conversion from the LS into a gravitational 1 36 , p 30 ( ) 32 s ] the new massive spinor-helicity variables of Arkani-Hamed, Huang and Huang [ ( 3 Here we will show that the natural extension of the scattering angle, for aligned spins The computation of the classical piece of the amplitude was made direct, through the Very recent progress on relating classical observables to quantum amplitudes has been made in [ 29 1 M where the exponential operator isin the generated soft by theorem the ( angular momentum exponentiated form, which fits naturallysecond into the post-Minkowskian Fourier (1PM transform and leading 2PM) to scattering the angles firstSummary and in of a results. resummed form. between two massive particles of spin as in [ building blocks needed for this computation areamplitude the for three-point amplitude massive and spinning the Compton soft particles expansion interacting with with soft respect gravitons. to We the will internal use gravitons the to write the building blocks in an observable is evidentlyamplitudes needed. setup to Very evaluatethe recently, the deflection a angle scattering direct of angle twodemonstrated approach of massive that was particles for classical in scalar proposed general the particlesbe in relativity the large-impact-parameter obtained scattering regime. [ the via angle It a computed was simple by 2D Westphal [ Fourier transform of the classical limit of the amplitude. Such potential is notand gauge-invariant, non-canonical i.e. transformations not thatpart an become of observable, the cumbersome and phase when can space.subtraction spin undergo of Moreover, is tree-level canonical at pieces considered one andspin-1 loop as suffers the [ from Born some approximation (apparent) inconsistencies itself already requires at the the post-Minkowskian (PM) expansion (seemany e.g. recent more discussion references in [ therein),to i.e. the standard at QFT a(which amplitude fixed corresponds was power only to of performed thePN up effective standard potential to through PN spin-1 the and expansion). Born leading approximation suffers order Moreover, some in complications the [ computation of the and the standard QFTfective amplitude potential in from the thisgeneral amplitude relativity. that one matches the post-Newtonianleading potential singularity, for of arbitrary spinthe and tree-level all and orders one-loop in the versions center-of-mass of energy this computation correspond to a single order in the agreement of the LSin method [ with the previous computationswere of implemented [ to constructthen operators matched, carrying trough spin multipoles. apolarization change These vectors of and operators basis, Dirac were to spinors, those enabling a constructed systematic in translation [ between the LS the matching are still lacking. JHEP09(2019)056 4 ). k , ) (the 3 1.5 (1.7) (1.8) (1.9) b (1.10) G ( (1.12a) , O  + µν 4  J as a function ) ε ν ]. The Fourier a 4 . We compute ) (1.11) · χ , b ε 3 29 2 . p s , a µ 2 G s b 4 k/ ( a k k/ − ( i ...µ p O − 1 f and p  − ) a ,µ + ). s = = a 1 ( n 2 1 i m s ε ) p p exp s b 1.8

M i ,s ) a ) + (0) 4 ...µ 2). In the operator form, s )–( s b 1 ( 3 ( 4 2 µ , M , a ε 1.7  a s > hM hM = a we write this as = ( µν ) , ) ε s f s J s ν ( 4 b →∞ · ( 3 →∞ ν ε s b ˆ ε ...ν p ˆ µ lim m ) lim ,s 1 M µ M ε ]):  a k ,ν ) k s , and the proper impact parameter · 1 2 i 44 v b ∂ ε ∂b · k p corresponds to the four-point amplitude of s  s ( ( k i ) δ E e b ...ν ...µ ) 2 πδ 1 1 exp 2 ,s , 2 – 5 – a s ,ν ) k ,µ k s s 2 ( 1 πG ( 4 2 π (0) 3 i = 2 δ ε d 1 2 s ...µ (2 − | ) M 1 M ) µν s  µ n π (appearing only for ...µ ( 4 and spin quantum numbers h Z , which simply amounts to a change of basis. The soft b 1 = ˆ 3 (2 ˆ a µ 2 J b 2 M ) ε M ε 1 2 ) ∂ ∂b s s s ( 3 m − v ) are the two building blocks needed to compute the scat- 2 2 | ˆ ) a ...µ 2 − M 1 a and ) = 1.8 h γv ,µ k s 3 and b (1 − 2 2 k 1 , the rescaled spins (ring radii, intrinsic angular momenta per ε − b a E b m ( m a m = m + µν m ) and ( ]. Denoting the operator by i b T ) ) = ) a s (2 2 35 ( n − k 4 1.7 ) and ( − + v , k a ) and its plus-helicity version. a µν hM = ], , the relative velocity at infinity + 3 a for definitions). Here b h m θ a 1.7 , k (1 + + corresponds to the amplitude for a massive scalar emitting a graviton. In ) in this case is extrapolated in an exponential form, and corresponds to the 43 2 , b 3.2 on the l.h.s. is the linearized stress-energy tensor of the Kerr black hole ( we extend this result to the distinct-helicity Compton amplitude, showing that  , p 1.1 (0) 3 . We first show that, with and 1 42 , with masses , s µν 2 p a can be replaced by ( v 2.3 M T a ) GE 26 1a 4 s ( 4 ε We find the following expression for the aligned-spin scattering angle The formulas ( = M θ impact parameter separating the zeroth-order/asymptotichole’s worldlines defined Tulczyjew spin by each supplementary black condition [ this amplitude at both treetransform and can one-loop be levels performed using using the the LS proposed exponential forms in [ ( of the masses mass) as in [ (see section figure where We then construct the aligned-spin scattering angle, for two-spinning-black-hole scattering, generalized expectation value (GEV) Here we focus on integer-spinfor particles spin for simplicity, therefore we use polarization tensors theorem ( simple statement of factorizationgiven of by the eq. Compton ( amplitudes into three-point amplitudes tering angle. In order to recover the classical observables we introduce and compute the up to corrections ofand fifth order in Huang and Huang [ where section which are constructed from the new spinor-helicity variables introduced by Arkani-Hamed, JHEP09(2019)056 ]. ], = = 40 38 κ k (2.1) (2.2) (2.3) )[ 2 ) from 2 (1.12c) (1.12b) G , 2.1 ) 2 p ], the spin-1 − 1 , 30 . , ) p 5 (  ] σ 1 2 ν 28 (
, 2 = O . (with momentum ε 27 ) ) + 1 ν ε ε vσ ! · µ 2 ε + k / b 3 as being (proportional to) the (  = ] , p 2 + ( ) ν 2 va i ) is valid up to quartic order in 1 , and ε ] µν 4 5 ε 2 ν 2 µ va ε ) ) [ 1 / . ε 2 a 1 ε − µ (2 τ 1 ε [ 1 2 − 2 ε · ε 1.12b )  τ − in front of tree-level amplitudes, we use ν 2 − k 1 σ ε 2 v p ( 2 ε ) µ κ − σ µν,σ k − κ n µ 2 = [ 2 + ˆ Σ ε p 2)  + σ κ 2 − ( 2 , which also motivates that the term involving  κ/ – 6 – 1 ε ) ( ( = (1 m  2 A − ε γ = κ − · v ] 1 4 ν 2 µν ε ε S a, ). To rewrite this contribution in terms of multipoles, )( µ with + 1 [ ε ε b + · γ 2 2.1 b − p σ  via a two-particle expectation value/matrix element, which ( m

2 h a + ε B 2 ) i · ε a m 1 vb 1 · 2 ] and through linear order in spin at one loop (at order ε p = + 2 − 36  2( 2 b , ) = = − m 32 ) resums those contributions in a compact form, including higher orders σ, a µν + ( )[ ) = S 2 a f G 1.12 m , k 2 q , p 1 is the average momentum of the spin-1 particle before and after the graviton = ) is split into two massless polarization vectors. The derivation of eq. ( p 2 ( p p E 3 . Also note that we work in the mostly-minus metric signature. can be thought of as an angular-momentum contribution to the scattering. In other ] − However, we now face our first challenge: as explained in [ M 2 ν πG We omit the constant-coupling prefactors 1 ε 2 p 32 µ 1 [ we call the generalized expectation value (GEV) √ we can use a redefined spin tensor It is introduced in appendix words, we are temptedclassical to spin interpret tensor. the combination amplitude contains up to quadrupolelinear interactions, piece i.e. is quadratic apparent in in spin, eq. whereas only ( the where emission and the polarization− tensor of thethe Proca graviton action isε detailed in appendix 2.1 Massive spin-1We matter start our discussion ofsion the by multipole two expansion massive by vector dissecting fields. the case The of corresponding graviton three-particle emis- amplitude reads one of the spinshigher-spin (but amplitudes. to all orders in the other2 spin) according to the Multipole minimally expansion coupled of three- and four-point amplitudes This agrees withlinear previous order classical in computationsas to well all as ordersMoreover, with in eq. the ( spin conjectural atin one-loop tree spin. quadratic-in-spin level expression (at We presented have in indicated [ that the expression ( with where JHEP09(2019)056 ] ) ) 2 ]. ]. = of 2
C ε 50 49 , i 45 · . In µν , 0 (2.7) (2.4) (2.5) (2.6) i µν k S 2 S 44 F
)( µν 1 µν ε . Indeed, F · ˆ Σ S k µν ∝ , , F   and using a helicity ) ) h 2 2 ) ε k ) ε ε 2 · 2 6= · · ε ε p 2 · k · k i 1 1 )( )( ε ε 1 µν ( 1 ( ε 2 ε ˆ 2 Σ · · m m µν k k ) satisfies the above SSC by ( ( F Following the non-relativistic h 3 + 1.4 + . ( which is well suited to describe µν µν µν µν S S ε 4 S ν · ], S ν ε µν ε p µ F 35 µ k k . ε 2 i · m √ p mx − i = 0 2 1 − ) as the term linear in both  ] to be the quadrupole interaction µν √ 2 – 7 – 1 S x 30  1.5 = µ 2 ), we rewrite the above amplitude as ) – p 2 x m ε 27 2.5 · − 1 ε = ( 2 ] ,τ x 1 2 46 ε σ 1 m ε στ 3 σ − 2 M ε ], as the classical spin tensor σ . This is because the statement is true at the levels of spin operators, 2 ) = ε 32 µν , k 2 S = i , p µν 3 1 F p ( 3 hM first introduced in [ M is constructed as an angular-momentum operator shifted in such a way that its ]. Indeed, in a non-covariant form, this was already related to the soft expansion long ago [ x µν , analogously to the magnetic dipole moment 48 ] , ˆ Σ ν Inserting this spin tensor in eq. ( 47 ε We thank Yu-tin Huang for emphasizing to us the analogyThe to spinor-helicity the conventions used electromagnetic in Zeeman the coupling, present see paper are detailed in the latest arXiv version of [ µ 3 4 [ k scattering amplitudes for massivethis particles with formalism spin. allows Much to like its construct massless all counterpart, ofe.g. the [ scattering kinematics from basic SL(2 spinor-helicity variables. 2.1.1 Spinor-helicity recap This subsection can beformalism skipped of if Arkani-Hamed, the Huang reader and is familiar Huang with [ the massive spinor-helicity limit, the third term was identifiedfor in spin-1. [ It may seemas a the priori square puzzling of that we wishbut to not regard at the interaction the ( order level to of expose (generalized) the expectation exponentialspin values, structure i.e. operators described in at the any introduction and order, construct we such are going to recast the multipole expansion in terms of we recognize the dipoleparticles coupling of with eq. spin ( 2 couple naturally to the field-strength tensor of the graviton (at higher points it becomesin gauge-dependent the but GEV can of still the be amplitude, used as a shorthand). Now, where for further conveniencevariable we also expressed the scalar products definition. The purposethe of spin this tensor of SSCreference an is point object for to the to constrain intrinsicof vanish spin the mass. in of mass-dipole its a rest components spatially frame. extended object In at a its classical rest-frame setting center it puts the GEV satisfies the Fokker-Tulczyjew covariant spin supplementary condition (SSC) [ In this paper wehole find computation this of condition [ to be crucial for the matching to the rotating-black- Here JHEP09(2019)056 , (2.8) (2.9) + , ), we 1 (2.11) (2.10) i x ) (2.12c) (2.12a) (2.12b) 2 − b 2.11 2 = k ). Choosing h , ] i   k , i i | 2.1 ) 1 r k p [ b 2 | ( b r = 1 2 2 m [ k k 12 p − , ih ih ε · ˙ k , k βa = ) i ) ˜ λ 2 12 ) 2 − 2 a a p a b α ε 1 1 2 h λ h k ν external wavefunctions. For 2 = p 1 . = 2 1 ih 2). This is in contrast to the ˙ mx mx µ ˙ S β m 1 β , , x 2 p b | ˜ 2 (not to be confused with the λ 22 p ( ] i a , ε α 2 k · p − | i − [ λ = 1 k r k ) p i − α 22 | h 2 1 i ih b , µν p = 1 = r b a on the three-point on-shell kinematics. ( 2 η ε h k m ˙ ) ) β p 2 r | | 2 2 ) = 0 = = ˙ a a 2 = k β, . . . [ a 1 1 = h α 1 ab p + 11 p ˙ α, i ih h β  ε ε x ) represent the physical spin-projection num- | i · k pνab k  a, b, . . . · b | 1 i 1 ε p b ab p 11 a k [ ( ( – 8 – p = 1 ab pµ 2 α 1 to construct spin- ε a ε i ˙ 1 ⇒ h ( β 2 and ˙ a a b ˙ 2 µ 1 α ( β          , p h x 1 | σ ˜ λ 2 h 2 µ i 1 ] ab 2 k k  m m = 1 1 i | 2 m 2 r k ⇒ µ = and = ¯ σ √ 2[ − − ˙ | ˙ β a β r α α √ µ α ] [ = = ) λ k σ b ) = 2 2 − µ p b b 2 | α, β, . . . p 1 1 b ) can be written in terms of massless spinors as m b 2 b µ = 1 are convenient to built massless polarization vectors ( 2 b 2 2 ε σ = a ˙ ε | · µ − β 1 2.6 √ ˙ a · 2 β a ˜ λ ( a α  k p 1 p h a 1 )( µν ε 2 S and ) indices = a , ε 1 − ν ] C i a 1 α ε k , ab pµ ε | µ λ ε · r k µ k h ) σ k 1 with respect to a spin quantization axis, as chosen by the massive spinor basis. 2 | 2 ( r ε − √ h is independent of the reference momentum , · 0 1 x , ε = ( Let us also point out here that the massless polarization vectors and hence the associ-  µ + ε 2.1.2 Spin-1 amplitudeWe in can spinor-helicity now variables obtain concretethe spinor-helicity polarization expressions of for the the graviton amplitude to ( be negative, we have where Note that the vectormust indices, always as be well contracted as and their do dotted not and represent undotted aated spinorial physical helicity quantum counterparts, variable number. ( where the symmetrized little-group indices ( bers 1 Now just as can use the massiveinstance, spinors massive polarization vectors are explicitly massless case, where thecomplex little nature of group massless is two-spinors U(1), so its index is naturally hidden inside the The massive little group is SU(2), so the Pauli-matrix map from two-spinors to momenta involves a contraction ofspinorial the SL(2 SU(2) indices spinors that transform covariantly with respect to the little group of the associated particle. JHEP09(2019)056 , ) 2 2 β , as α ] via . i i ) B 35 (2.16) (2.17) (2.15) (2.13) (2.14) b 2 1 b for the 1 2 α . 2 β ( k  using the

A i ih i 2 k b k i ⊗ | = a 2 1 | . 1 ih b 2 1 (  h k . On the other β β 2 2 | 1 | 2 1 ) it has the form

1 ), the quadrupole k α α and i mx ) 2 ih | 2.5 i h 1.5 a k B k and | h 1 2 i − | h 12

) b h ] is always possible [

. 2 2 a i  a A a p 1 k | 1 a 1 ) mx 1 h ˙ h p β , ∂ − 2 ˜ λ 2 | ⊗ h x ∂ ] = 1 1 2

b α a a i 1 unless stated otherwise. From ( ˙ ( 2 m p a 1 − 12 ˜ h λ ) as h [1 +  . i αβ 2  α + intrinsic 2 i 1 k 5 a m µ + 2.12a ) into the amplitude, we obtain p | ih ˙ ∂ − β ∂p k ˙ m α , 1 ν  = i 2.12 = a k ip 2 and the antichiral ones ) ˙ β b ] p β

– 9 – i h i 2 a , whereas in our amplitude ( − i ∂ a h p 2 2 | p 1 12 ∂λ | ) ν | ˙ β h −  α a α ] to the chiral spinor basis of ∂ | ( µν x ∂p b p p k 2 2 S µ − | mx λ ν ih  ε ip k i and − µ ˙ ] = α k = 2 | ] and k . One then could wonder if in some sense the latter is the I a

= 2 b | 2 p i µν | a ˙ [2 β

1 J i m 12 mx ] for the massless case. For a massive particle of momentum 2 h ˙ α,β =  k − α h 2 51 I β J 2 i x henceforth, i.e. it carries helicity a

p I i | , −  k i we construct the differential form of the angular-momentum operator ). We now show that this is precisely the case if the angular momentum ) = ˙ x 2 αβ 1 k p

− a µν | C we find that the differential operator for the total angular momentum is 1 1 S h for , k ∝ h h 1 ν 2 ˙ 2 βa ) ε x − 1 p 2 µ , p x m ε ˜ λ 1 k · a p − ) we can see that going to the chiral spinor basis has both an advantage and a ( pα k ] = 3 λ = k )( a 2 1 M 2.12 = In appendix Now in the multipole expansion of the Kerr stress-energy tensor ( The main advantage of the spinor-helicity variables for what we wish to achieve in ε ε [1 The transition between the chiral spinors · · ˙ 5 β 1 α ε k given by the Dirac equations which involves the standardspin. orbital This piece operator and admitsto a the much the “intrinsic” simpler contribution one realization in dependentp derived terms on in of spinor [ variables, similar square of ( is realized as a differential operator. in momentum space starting from its definition operator is of the( simple form ( Here the operators haveand their the lower notation indices assumesmomentum. symmetrized, that Combining all the i.e. the reader ( terms keeps in in eq. ( mind the spins associated with each this paper is that nowrepresentations we of can the switch massive-particle tosymmetrized states spinor tensor 1 tensors product, and we 2. can rewrite Introducing eq. the ( symbol disadvantage. On the one hand,that the the multipole expansion spin becomes orderhand, transparent of the in exponential the a structure sense term ofterms. the is vector identified basis However, by is this spoiled theanswer is by obtained leading a just from shift power the an by of generalized higher artifact multipole expectation of value is the the chiral same. basis, and we should see that the following identities for the three-point kinematics, We also use eq. ( where we have reduced all [1 JHEP09(2019)056 . , i ). 2 2 i a 1 2.7 p | (2.21) (2.22) (2.18) (2.20) 2 (2.19c) (2.19a) (2.19b)  : i ⊗ | 2 1 i . µν 1 a − a 1 b J | ε p δ 2 and use it | . i − ν · , / 2 ] ε i k 1  i | ν = a µ i p 1 p ¯ 2 σ | 2 a ik = µ [

p  i i i |  2 i| a a a . 2 iσ p b  | p p 3 | 2 ∂ k p i| = ∂λ h µν 1 − ≥ ih b p 1 2 k J ε k h | µν ∂ · − ν  ∂λ ); σ = ε 1 . k µ = 2 p 2 i⊗ | h i k 1 1.1 i k a 2 2 p a  k | ih mx h 1 2 k k p 2 + | 6 i , i ih − , i k 2 k ⊗ 2 a , a p 1 i  i p I h p ∂ 2 i⊗| ∂λ ⊗| µν k p + 1 1 i| − x h  a 1 k 2 J ε I i 2 p | · 1 on the product state i − ν k p ih m a ⊗ ε k 1 a + p | ih ) chiral generator | µ p µν i i| 2 k ) and its spin-1 amplitude representation ( k k 2 | C k p p b , J x a , h 2 ν 2 ih 2  p – 10 – ∂ i mx i 2 2 ∂λ ε 1.5 , j k i 1 | mx m µ | k √ i⊗| h k 1  = = = = 0 1 + − a  2 2 2 i  µν 1 k p i i i b = 2 − µν J | p p p ε ih ˙ , it is direct to check that it acts in the same way as the − | | | J β 2 k p − ν k · j 2 ε h h | − ν ε  1 µν · 2   ˙ ε µ α,β p = = J x p µ α µν µν 2 ik µν J − J ik

˙ − − J J β ε i  − ν ε ε ˙ ) = α · a − ν 2 − ν ε · · |  p ε ε − | p µ 2 β ) p µ p µ h λ , k ik µν ik ik 2 α = J ) of the standard SL(2  λ   i − ν , p 2 µν ε 1 2 . µ p 1 σ k √ k ( 4 C 3 ih ⊗ 2( k I √ = 1 M i ) as the non-zero powers of this dipole operator acting on the state + ih µν I 12 J h ⊗ − ν 2.15 correspond to the scalar, spinthe dipole Kerr energy and momentum quadrupole tensor interactions ( Note in the that expansion the of signthe algebraic flip and in differential theappendix Lorentz dipole generators, as term pointed comes out from in the the beginning sign of difference between match the differential operators of the soft expansion ( ε 2 µ These identities allow us to reinterpret the last two terms in the amplitude for- We can now act with the operator µν mx More explicitly, we have • • k σ 6 ( − with similar manipulations for higher powers. It is now clear that these terms and rewrite the amplitude as differential operator above: mula ( Although it is theis differential easy operator to obtain that on− realizes three-particle the kinematics. soft Indeed, theorem,as if an its we algebraic take algebraic realization a form of tensor-product version For the negative helicity of the graviton, we have Applying the spinor differential operator above, we find JHEP09(2019)056 , x s 2 i 1 | (2.26) (2.27) (2.23) (2.24) (2.25) # j ), which  yields the | annihilates . 2 k | 2 2.25 j i 2 . On the other ih ) 2 h mx | 2 ) collapses into k | 2 µν 7 x − s J s 2 that enables us to 2 − ν 2.15  ε . Concentrating our s m j µ s

2 [12] on the helicity variable k  . . µν 2 2 s i J ) = ) as the multipole contri- =0 2 1] 1 ν j X | | − ε . The starting point in this µ  "  s k s , k ) when acting on 2.15 2 2 2 ε µν | µν · + 1) ) is self-adjoint. − 2 J , p J 2 ε h ε ε . 1 − p + ν · − ν 2 · 2 · ( p ε ε / − x ( p p p µ 2 µ ) s 2 ( s k 2 x µν k 2 ( 3 / i i J m ν   µν ε M µ J = k ν s exp exp ε 2 ) = [12] 2 µ 2

| − | k  2 [2 h , k i – 11 – 2 , 2 2 1 1 2 x x k , p − 2 1 ih x − mx p s k s ) = ) = ( 2 2 2 3 ) imply that the amplitude formula ( i h + − m ) should become 12 , k , k M ε + h 2 2 · 2.13 i ). The purpose of the insertion of the differential operators p , p , p ( 21 1 1 / 2.5 ) = h p p µν ( ( +  3 3 J 2 ν 3, the three terms can be obtained from an exponential , k − ε 2 2 s M M µ 2 ≥ x implicitly relies on the fact that the action of ) individually. Furthermore, as the operator ( k ]: , p ε j 1 m · p 2.5 35 ( p ) s ) = ( 3 − M , k 2 ): , p 1 p ( ) 2.23 In the next section we extend this procedure to arbitrary spin. Let us point out that It can be checked explicitly that acting with the operator on the state s The division by ( 3 7 M where we have takenperform advantage of the the binomial symmetrized expansion tensor (we product have suppressedvanishes. the Note also identity that factors in the tensor spin tensor is completely hidden.amplitude In in order the to chiral restore basis it, we need to write the minus-helicity case is the three-pointlittle-group amplitudes sense for [ massive matter minimally coupled to gravity in the As explained in the previous section, in such a compact form all the dependence on the will then be matched to the Kerr black2.2 hole. Exponential formIn of this three-particle section amplitude we generalizeattention the on previous discussion integer to spin arbitrary allows spin us to ignore factors of ( However, let us stress thatexplicit this in form the completely vector hides formis the ( precisely spin to structure extract that the was spin-dependent already pieces from the minimal coupling ( the explicit amplitude can befact, brought into the a three-point compact identities form ( by changing the spinor basis. In same result, i.e. inhand, this choosing the sense other the helicityeq. operator of ( the graviton will yield the parity conjugated version of butions with respect to themultipoles chiral in spinor eq. basis, ( despitethe the spin-1 fact state that for they do not equal the In this way, we interpret the three terms in the amplitude ( JHEP09(2019)056 (2.31) (2.32) (2.28) (2.30) (2.29) s ) matches the 2 , namely ε i ] corresponds s · 1 s . | p Of course, this 2 ) i 29 ( s, s. i s , s, s. / ( 3 2 2 1 s 8 2 2 | 2 k p . ˆ µν j of [ i ) can be regarded as a s ≤ ). From this we can M ih S  ≤ 1 | s | ν k 2 k ε | |  µν 2.28 µ 2 − ih 2.20 k J h = , j , j > ε k µν , j , j > ), as can be seen by squaring | s − ν · i − j 2 J ε j ε 1 p p ] to perform the matching with  µ 2.4 − ν · − m i| i k ε s 29 p p i 2 µ the exponential can be realized = k ⊗ k p ∂  I i ) ∂λ ! mx ih s 1 j , since the arbitrary spin version ( 3  k k

2 | j ih G =0 M → ∞ ∞ ⊗  used in [ j X exp j  s s s k p | S − 2 2 h | | s · k ) to product states of spin- 2 2 2 i h h k ih mx p k | ) is nilpotent of order 2 | = = , 2.19 )! s   2 j i , )! )! 2.28 – 12 – i s 1 µν − j | p − )! J ) that the spin operator s # (2 ε s − j ∂ − ν · ∂λ (2 ε 0 s (2  p ) the spinorial differential operator ( µ k | 2.27 (2 k ) as an exponential 0    k ih i ih 2.29 =     mx k k p | 2.27 s h = 2 i exp j 1 p  | mx

9 j s (0) 3 j  2  = representation.  limit) of the Compton amplitude is not yet known. µν M µν s − ] only when the spin tensor satisfies the SSC ( J − s µν ) corresponds to the operator ε J =0 2 ε . Moreover, in the formal limit − − ν · 29 j J X − ν · ε ) = ε ε p − ν µν · " µ 2.30 − p → ∞ µ ε s J k p 2 µ k i | s , k − ν i k 2 2 ε  i h  µ used in [ , p k 1 S p · ( = 1, eq. ( ) . It can be read from eq. ( k s s j ( 3 ], here we recast this into exponential form by inserting the differential angular mo- ˆ M Interestingly, due to its property ( For 29 8 9 ladder operator for a spin- the standard QFT amplitude.quantity We note, however,both that terms. the classical quantity is still present andThis can framework be will mapped be(and to hence particularly classical the useful observables at such order as the scattering angle. order 2 precisely to as a linear operator thatthe does exponential not operator truncate! in However, let us stress that even for finite spins where we note that the exponential expansion, albeit valid to all orders, becomes trivial at Therefore, we can rewrite eq. ( In other words, inalso admits general an the algebraic operator realization,derive which ( the is formal extends the relations formula ( Indeed, it is easy to generalize the formulae ( of [ mentum operator product). Even though this already corresponds to an expansion in the “spin operator” JHEP09(2019)056 , ) ε 3 1.1 ] the (2.34) (2.33) (2.35) 29 . s ]. in [ 2 1] | 29 ) , s . In particular, we ( 3 s 26 2 ˆ i 10 M in the denominator, 1 s | 2 ε |  . · [2  µν s , agreeing with previously p 2 4 J − 4 1 µν ν ε S ε m J − 4 · and the polarization vector · ε ν p ε µ = p k 4 µ ) k k s ( 3 i i   M in the exponential operator may act corresponding to the scalar piece (the exp exp 2 µν . s x 2 J | 2 (0) 4 G we will use the Compton amplitude as an 2 m h 3 , , as was shown directly in [ M 2  and = = G s – 13 – p µν ) = 2 (0) 3 + J − 4 is in a sense reminiscent of Low’s original derivation of the ε 1] | + ν · M , k ε  p 2.4 µ + 3 k µν i , k J + 3 we will use this compact form to compute the scattering 2  ν ε ) make explicit the fact that the higher-spin amplitude is + 3 3 · , p ), with ε 1 p µ exp p 3 ] the classical contribution in the spinless case was identified 2.33 ( 1.5 k ) (0) 3 s i . We will see that this exponential form is extremely suitable for 26 4 ( 2  ]. ˆ M S 3 M ) is our first main result. Note that this holds for the full three-point exp ) and ( ) = s 2 2.32 + | 2.32 [2 , k 2 ]. However, despite the appearance of the factor , p 1 35 p ( ) s ( 3 ˆ The importance of this amplitude (as opposed to the same-helicity case) is that it The formula ( The forms ( Analogously, it can be shown that the transition to the positive helicity amounts to M Historically, the Compton amplitude was the prototype in the discovery of subleading soft theorems [ ]. The construction provided in section 10 6 , known results at order the computation of the latter as a Fourier transform. 5 subleading factor in QED [ controls the classical contribution atclassical order piece was arguedIn to the lead to new the approachby correct of computing 2PN the [ potential scattering after angle.input a for In computing Fourier section the transform. scattering angle with spin up to order either on massive statecan 1 be or associated 2. to Moreover, either the of momentum the two gravitons. Explicitly, we have will show that for the cases of interest the following holds Here the linear and angular momentum angle of two Kerr black holes at linear2.3 order in Exponential formThe of task gravitational of Compton this amplitude sectionCompton is amplitude, to without extend the the support construction of presented three-particle in kinematics. the previous one to the simply arise when gluing these three-point amplitudes. amplitude with no classicalenergy-momentum tensor limit ( whatsoever.Schwarsczhild case). This In formula section matches precisely the Kerr non-local [ the exponential factor isrecognize gauge-invariant due in to the the argument three-particlethe of kinematics. Cachazo-Strominger the soft We exponential theorem. further the Inthe same fact, extended as structure soft will as be factoramplitude the made of one explicit of Cachazo appearing in higher-spin the and in next particles. Strominger section, is The just poles an present instance in of the a three-point extended soft factor ( exchanging angle brackets with square brackets: JHEP09(2019)056 . , ) ) ). i 3 (0) 3 | 1.1 1.1 2.34 M (2.39) (2.37) (2.38) (2.36) . , we see i . 4 , | [43], so the ) 2 2.4 / + 3 i | . 4 4 i . | [3 , k i i i a 2 2 2 | 3 4 34 | ∂ 1 4] yields the last ), we recall from , p h | ∂ | ∂λ 2 is in the kernel of 1 4 ∂λ 4 p h √ 4 ( ih 2.36 (3) 0 ) ) acts always trivially. ), ih s a = = ( 3 does not depend on M 2 43 − 4 42 ) h 3] and 2 ε 2.36 h s M 2.36 | 3] i ( 3 2 | 2 i i 1 | √ 34   √ M h 4 h − µν µν − 2 3 4 4 J J ], in order for the action of the ε ε = = = ν ν 1 · · 4 4 2 3 ε ε 3 2 µν 3 x µν 2 µ µ p k 2 J 4 4 J then vanishes by going to the chosen ν k k ν m 4 4 ) is in the contraction with i i amplitude 2 ε ε ) µ   = s 4 µ 4 4 ε k 2.37 · (0) 3 exp exp 3 2 2 k M ) ) 2. We leave the problem of obtaining the case – 14 – ( 4 4 4 4 , giving the same result. k k ε ε s · · ≤ : · · ∝ 2 ⇒ | i 2 3 , k 2 3 s i 1 shows that the scalar piece p [4 k in the spin- p k | . As explained in [ i a 1 ( ( i i s 4 , | 2 4 ∂ or s i + b ∂λ + 43 ˙ 2 s β ] for h 2 s = i ˙ 2 4 α h 2 12  / ] |  4 | h appears in eq. ( ih b m 12 ) 2 52 k a h µν , 2 [3 h 1 β 3 | 4 i a 3 J ∂ ε 4 is not guaranteed. Here we simply fix 35 h 41 (0) 3 ν | [1 ∂λ · h 4 i 2 4 α ε 1 2 a ) we first propose an all-order extension of the soft expansion ( ). Now to evaluate the third term in eq. ( ε M i µ p √ 3( · √ 4 ) vanishes, as we will show in a moment. The second problem is that i 4 1 k iλ p h − = i 2.34 2.37 ) =  + 3 = + 3 = 2.36 ) either on = 2 ε 4 , k i ε ˙ µν in terms of independent variables. Solving for 2 exp 1 β 4 · | J 2 2 , p p ν ) ˙ | α,β 4 1 ( (0) 3 4 4 1 α k p / ε | ε self-dual C 3 ( · µ 4 · M ) 4 J µν h 1 s 1 3 ( k p J p ν ( 4 Let us now look at the angular momenta of the massive particles. A similar inspec- Let us first inspect the three-point amplitude entering eq. ( To obtain eq. ( Our strategy is the following: we first consider the action of the exponentiated soft 2 for future investigation, but we will comment on it at the end of section M  ε µ ≥ 4 tion of the operators that the aboveMoreover, differential since operator the prefactor annihilateswe the conclude that scalar the three-point exponentialThe operator amplitude zeroth-order in the of third thegauge, term hence of soft the ( theorem last term drops as promised. expression in eq.appendix ( As the only place where where we used differential operator toexpress be well defined, we need to solve momentum conservation and The solution is thatthe in language this of case thek both previous exponentials section, give this the is exact the same fact contribution. that In one can act with the operator As stated in theas introduction, an two main exponential problems actingof arise on the when the denominator trying lower-point amplitude. tolast interpret The term eq. first in ( is eq. that ( one still gauge has invariance to sum over two exponentials, which would spoil the factorization of eq. ( with respect to the graviton factor acting onStrominger the soft three-point theorem. amplitude,the We Compton as have amplitude checked an that [ s all-order this extension agrees of with the the known Cachazo- versions of JHEP09(2019)056 , . ) ) . ε s 4 s 2 · k 2 i i · = 0, . 1 2.34 p | 1 s 1 (2.44) (2.45) (2.40) (2.41) (2.42) (2.43) | 4 2  p i ε  = 1 · | (2 µν 3 µν / 4  J 4 k ε 4 R J ν ε , . ε 4 ν · · (0) 3 µν s 4 ε · . p 4 2 2 J s ε µ ) , i p 2 ε p ν 4 M µ 4 s 4 · 4 k L 2 k − , m ε 12 i k i p (0) 3 · h µ i 1  4 | 3 =  k k   i 4 exp  ε µν )(2 µν → M 2 s exp 4 4 · 4 2 4 J J | s ε k ε ν ν 3] 2 1 a exp · 4 · | | · (0) 4 4 p I ε s 1 2 ε 2 s h 2 p | 2 h µ 2 s µ p | p M 4 4 i 2 4 2 h ] k 1 = k h a | i i )(2 I s | 4  2  =  k i  s implies · 1 µν 2 | ], can be constructed solely from this i 1 µν 4 exp J exp 4  ε p ν s ε J 3 2 4 0 it is evident, so we will focus on the 12 · 52 2 ν ε ) , | h (2 ε µν 4 3 · 4 p 2 µ ε k 4 p J  ε h 35 − 4 → µ · ε ν ·  3 k 4 · 2 µν 2 2 2 i 2 k = ε 4 ) p ) p i p J 4 µ ε  4 4  ( ν 4 k  · ε – 15 – k 4 2 k · · ) ε 2 4 i + 2 4 + ) is checked by repeating the computation for the µ p 2 exp k ε p  4 exp p  s · 3 · s k ( 2 k 2 | 2 i . Unitarity demands that the operator piece in ( 2 | µν a 1 2.35 p 4 p + 4 exp  I [2 J ( k is needed since the exponential operators act on different ε s h ν 2 s | · 2 s 4 2 ) | 4 + 2 a 1 + 4 ε 1 ] a k I m 1 , which is possible on the three-particle kinematics of the ε µ p a 2 I h · 4 p · I ) ] h = exp 4 . | 1 k 4 s a 1 → 3 k s 2 i ε I p , we also have =  p 2 k ) can be obtained from the all-order extension of the soft theo- 2.2 | s · i s · ( i 2  a I 2 1 i µν 1 I  i p = p 1 12 p 1 3 | J h 2 | 2.34 ( I ε exp (0) 3 ν 2 is the average momentum of the massive particle before and after h  p 3  ·  / s 2 ) is simplified to ε ) 2 ) p M µν 4 1 µ 2 (0) 3 4 µν 1 and s 3 k J m 4 p ε 4 0. In that limit the scalar part factors as 2 J k · ν ε 1 s ε · 2.36 ν M 4 i · 2 · − 1 m 4 = | 1 ε p → p  ε 1 2 1 p µ = h µ p ( 4 p ) 4 k 4  k i (0) 4 k exp ). Finally, the property ( i = ( · (0) 3  s  2 M 1 | p p M 2.36 [2 exp From the discussion of the previous section on the action of the angular-momentum exp s s 2 2 1 | 2 m h factorization channel: Here the insertion of bases. In orderchiral to basis, show as the in above section property, it is enough to write the left factor in the operator framework. Forpole the ( pole ( corresponding to the productinternal of momentum the by respective three-pointbehaves amplitudes. as Let us denote the rem ( opposite-helicity graviton 2.4 Factorization andIn soft view theorems of the exponentiation formulas, we now show how factorization is realized in this the scalar Compton amplitude, writtensoft e.g. factor: in [ This proves that eq. ( Hence we obtain where we recognize theso scalar there Weinberg soft is factor. no Recall contribution that from in the this other gauge graviton. As an easy check, we observe that operator on Moreover, our choicewhere of the referenceCompton spinor scattering. for Therefore, eq. ( JHEP09(2019)056 , . 4 0. ) , L , s s S ( 3 2 →  (2.48) (2.49) (2.46) (2.47) M 4 k .  · s ). Let us [32] 2 1 i µν 1 p 4 1] J | 14 ε 2.48 ν 2. For higher h 3] ·  4 | ε 1 1 + ≤ | µν µ p s 4 i 4 s 2 3 J h k
ε ν 42 i 3 · h ε  p µ [32] 3 i [13] k exp i 14  h 2 s  ) 2 4 s + 4 k 2 ε m i i · exp · 1 1 | s = 42 1 2 p h s s p |  2 2 ( i  ). For example, one can check that 1 µν i [13] | = 4 J = [2 ) is only valid up to 3] that appears at the fifth order by  41 ε ν s ) | s 4 2.29 2 · s 4 1 2 ih , we have recovered the extension of the i ε | 2 µν k 2 p ) follows from factorization of amplitudes 4 µ 2.49 1] · ) 4 J 24 m h 4 12 a ε ν h 3 − 4 h k to obtain the scattering angle at order I 4 1.1 · ε k [ i L ε 3] , s · 3 p | µ (0) 3  2 s 1 ] 1 4 )(2 | 2 – 16 – a p k [43] 4 4 M − I i k h | 4 exp  ·   s ) non-trivially extends the Cachazo-Strominger soft 3] + 2 | 2 | µν 1 p 1 i µν 4 a | ) = 2( J exp ε 3 I J 4 ν − 2.34 4 21 h h · ε s ν )(2 4 h s 2 3 4 · ε 1 | 2 , k  ε k ] µ p 2 I p µ a 4 · h 3 = I k s ] that the formula ( , p 1 | k appearing at higher orders in the soft expansion ( 2 i s 1 p 1 i 2  p 35 4  i m (  (2 ε 1 µν · | R , − 3 (0) 1 3 J (0) 3 = 2 the exponential is truncated only at the fourth order in the  4 exp ε p ν exp k = 3 · 4 M s s · µν 1 M ε ) k R 2 4 p , ]. We remark, however, that this expression completely hides the spin | 1 s µ J L · ε , (0) 3 = ( 4 ), that we used as a starting point of this section, in the limit 3 ν p (0) 3 · [2 1 4 35 k 2 R s p ε 1 i , M M 2 3 (0) M 2 µ p 1  2.36 4 m k 0 = M i → = exp  4 particles, written as a series in the angular momentum. Therefore, in our case the s k · 2 1 s | p exp −−−−−→ [2 s ) s It was pointed out in [ Let us remark that, in analogy to the three-point case, the exponential factor can be 2 s 2 | ( 4 1 2 m h M This is becauseangular for momentum, whereas onlyThis the extension second is order what wasby enables means guaranteed us of by in a the section Fourierthe soft transform contributions acting theorem. from directly contact on terms the at exponential. higher We spin leave orders the for study future of work. spins, one hasadding to contact eliminate terms. thethe spurious From contribution our pole from perspective, thisremark, spurious however, that pole our corresponds resulttheorem precisely ( in to the case of the Compton amplitude for minimally coupled spinning particles. that is given independence [ that we need here for the classical computation. which converts the Compton amplitude into the form of spin- statement of the subsubleading softof theorem massive ( particles with spin. brought into a compact form using identities like ( Here, using soft theorem ( The origin of the exponential soft factor in this case is nothing but the three-point amplitude Putting this together with the scalar piece we can write, for instance, On the other hand, we could have inserted the resolution of the identity in the right factor JHEP09(2019)056 2.1 (3.3) (3.4) (3.5) (3.6) (3.7) (3.1) (3.2) k. . ·  then the operator ia µν s − ε 2 S i ) we relate the two by · ν = 2 ε | p B µ , ρσ k S i αµ . ν and . − J p s µν s  µ α 2 is that the latter satisfies the J p k i ...ν ν ν 1 1 p exp | p ,ν µν . appearing in the soft theorem and µ 2 2 1 µνρσ ) T k 1 ε s  , µν m ε s 2 µν 2 ε · 1 S 2, which is self-dual. This means that J ...µ · µν m i ν ...ν − = 0, p m / 1  1 ] ε S p 2 µ 1 ε ν )( k µ ,ν + ε  ν αν · s ¯ · k σ k (see also appendix s − ε J p · µ 2 µ p [ µν α ...µ = k ...µ p ε 1 − p S 1 2.1 ( i iσ · µ ν µ δ ,µ ρσ  ε p = − ) p – 17 – 2 O J 2 µ 2 s ε ν k = 1 k = µν p ( m ...µ ε k − δ µ J 1 µν · 2 · k ν + ) ,µ σ ε = p a 2 π µ ε − µν k µνρσ µν S  ε = = J − 2 · ν acting on two massive states, represented by their polariza- i ε ) = (2 µν p m = µ k 2 J hOi k O − µν ( = J appearing in the expansion of µν µν T J ) µν ν k S p ( µ µν . Therefore, following section k 2 h C 1 m we have shown that the three-point and Compton amplitudes can be written ) becomes 2 ). Moreover, there is an obvious sign flip in the respective exponents, due to the 3.4 2.4 The key observation is that this operator acts on a chiral representation. That is, for Let us first show how to apply this definition to match the form of the stress-energy so eq. ( It can begraviton. checked To that compute this the generalized factor-of-two expectation relation value, we is will independent also need of to the consider helicity the of the On the three-point kinematics, one can show that negative helicity, if the statesis are algebraically built realized by from the spinors which implies that the soft operator reads, at the classical spin SSC ( sign difference between the differential andand algebraic appendix generators, as mentioned in section There is a subtleguide but us important in point the already followingdifference subsection present between on in the a angular this path momentum classical to operator matching the classical that scattering will angle. The crucial tion tensors, tensor of a single Kerr black hole that we derived in the introduction: 3.1 Linearized stress-energy tensorIn section of Kerr solution in an exponential form.tion value We of have also an motivated operator the definition of a generalized expecta- 3 Scattering angle as leading singularity JHEP09(2019)056 . s (3.8) (3.9) 2 (3.10) (3.11) (3.12a) (3.12b) [21] . s 2 s  i 2 i 1 | µν # 21 − ), — this time S j h , ε , s − ν ·   s ε

| 2 s p
1.10 µ i 2 k | µν i k . This leads to ε ih S i 1 mx [2 21 · | i ν h k 2 2 s  | ε 2 p )  |  µ | k , 2 µν k exp  s i ε  2.10 s 2 S s/ ih j · m 2 mx ν 2 k − , µν ε | p →∞  s  ε s µ S 2 = · k ν s =0 s i = lim ε p j X 1 + exp ⊗ 2 µ s [21] − ε  " 2 k  s s i 2 2 = | | →∞ lim →∞ − lim s s 2 2 [21] ) exp  h h ), as promised in eq. ( s 2 2 ( 2  ) and in the last line we extracted the x x 2 2 3.2 exp µν →∞ →∞ →∞ + s s s m 2 m S ε 2.30 ) is minus that of section + ν · ε = lim = lim = lim = = i ε · p s s s 2 s – 18 – µ s | 2 2 p k , ε i i s i ...ν ...ν 1 1 1 | | [12] 1

− s ,ν
# ,ν  | i  = 2( 1 j 1 ε limit of ( ε i [1  s µν 21 ) s i | h s ε exp J 1 ( 3 k | · ...ν projects out the anti-self-dual piece. However, we should ν ...ν 1 1 , in such a way that the amplitude already contains the ε ih p 2 →∞ mx ,ν µ s ,ν k →∞ s hM s µν | s k → ∞ s s/ m J 2 i 2.1 2 = lim ...µ ...µ s = lim − ν 1 →∞ ) is invariant with respect to the choice of the spinor basis as well. lim 1 s   ε s = s ,µ s j ,µ µ ) ) s to be outgoing, so ...µ s k  s ( 3 ...µ 1 ( 3 1 ⊗ exp 3.11 1 2 µ 1 ε s s µ 1 p =0 ε 2 M j M ε X s | s = s s 2 " . To that end we use the following representation of polarization tensors, ) h ) ...µ s s ...µ s ...µ ...µ 1 2 1 ( 1 ( 2 | 1 1 ε ,µ ε 2 ,µ →∞ ,µ ,µ 2 h · 2 s 2 2 ε ε ) ε ε s s s s s 2 = lim ( 1 2 2 2 →∞ ε s m m m m = lim Finally, we notice that the self-dual condition is natural when considering a definite- →∞ lim →∞ lim →∞ →∞ s lim lim s s s helicity coupling, e.g. keep in mind thatthat this coupling. is It just wouldparametrization an be of artifact interesting of section to ourcovariant-SSC find spin choice a tensor of built non-chiral chiral in. form, spinor analogous basis to to the describe vector Therefore, the GEV ( which recovers the Kerr gravitational couplingwith ( the SSC condition incorporated.can The also plus-helicity keep graviton the gives the minus same helicity GEV. and redo One the computation in the antichiral basis: always present at finitenot spin, truncate. it This is leads only to in the infinite-spin limit that the expansion does amplitude: Here we would like to emphasize a key point. Even though the exponential operator is where we have usedoperator as the a GEV. The same manipulation can be done for the three-point minus-helicity obtained as tensor products of the spin-1 polarization vectors ( where we now take product JHEP09(2019)056 . ) 0), v ) = k ( , spin (3.17) (3.19) (3.14) (3.15) (3.18) (3.20) (3.13) (3.16) a v > O (1 + m γ + 1 = p ] i , with 3 k . | p r k a 2 2 = [ p / v a | ) . r 2 4 m − [ 1 , and , and the other with p p , the relative velocity 2 1 − 1 ρσ b p + E − p √ S 1 = 3 ν b p p . p , = − 4 ε a , γ p = = ( · 2 b νρσ = 1 1 m µ a = a m p ε − p p 4 3 b 3 ε 2 b b p p − = 2 p · 1 m , m m · 2 b 2 4 a √ 2 2 p p p 1 k m p = 2 = = = = a √ µ b 2 3 − ], we can choose it as — each of which determines all the others, = γ 29 2 1 p p , and final momentum ) , x 1 γ, k ) b p – 19 – , k is the momentum transfer, v , t − 4 , a , p m ( = b k p 2 a a − 2 tot , and final momentum − ρσ a m p 3 ε m ) + m S (1 b p · ]. The total amplitude ν a 2 = — are related by γ ,s = b p p a 29 p + 2 s b s 2 2 = ( 4 p 2 b = 2 is the average momentum νρσ m ] i 3 µ √ m M a = k ε p | p 2 1 a 2 a r k + = , the total center-of-mass-frame energy and h p + p 1 | 2 a b s a m 1 a r x 2 p h m m m = − = = , initial momentum ]. Here a µ a 2 s = a tot 32 E p + , initial momentum ε a = b · s s m a p is the total momentum, and 2 = 0, it is convenient to fix the little-group scaling of the internal graviton (for √ , spin tot t b − p m = We consider the case, in the classical limit, in which the two particles’ rescaled spin vec- At 1 (between the inertial frames attached to the incoming momenta − a x tors are aligned with the system’s total angularscattering momentum. plane, They are and orthogonal tomomenta, are the see constant conserved. e.g. [ The scattering plane is defined containing all the tree-level one-graviton exchange). Following [ This implies and the corresponding relative Lorentzgiven factor fixed rest masses where The Mandelstam variable v is a function ofWe the define external as momenta usual and the external spin states (polarization tensors). following here the conventions of [ We now consider scattering(quantum number) of two massivemass spinning particles, one with mass 3.2 Kinematics and scattering angle for aligned spins JHEP09(2019)056 ]. In (3.22) (3.21) 53 ]. Com- , , ) 32 3 ) with one 25 G . ( ), the momen- 3.14 O  -channel residue t is trivially imple- + , we will find that µν 3.14 b i − t 2 ) ],  J ε b ) · G − ν ,s − ε b a 43 ) is the vectorial impact ), will appear as scalar k , s µ p ( 4 + k b − k i , 42 3.20 , 4 − − hM p ,  26 4 − p , →∞ ) differs only in that the aligned b 3 − exp lim p , ,s ( 3 a 2 b 2 a 3 4 ) s p p p x b x 3.21 b ( s · ) 3 ( k b + i s ˆ e ( 3 M  2 ˆ ) M ⊗ k µν a − 2 π ) ], the LS for the amplitude ( ε J ⊗ d + (2 · − ν ) 29 k ε a − , k Z µ – 20 – p 2 k p , k ]. The reason that this leads to classical effects is i 2 ∂ ∂b − p  , 2 31 ) 1 − p , ( exp 1 2 1 γv ) p p b p 2 b a 2 a ( ) to the spinning case with aligned spins, we find that its s by which both bodies are scattered in the center-of-mass ) x E x m ( 3 a . These amplitudes are now given in the exponential form a  θ s ˆ ( 3 2 b M 2 m Tree-level singularity for one-graviton exchange. , counted from the second particle to the first as in [ 3.21 ˆ b , the magnitudes of the vectors in ( m . In this “aligned-spin case”, up to order t M b + (2 2 a b  a p − t 1 m ) in the chiral basis. Summing over helicities, we have = = = and θ 2 2.33 ) b a Figure 2. ,s a is the generalized expectation value of the amplitude ( ] for aligned-spin scattering angles for binary black holes. a is integrated over the 2D scattering plane, and s i ( 4 ) b k 40 ) piece, which is dropped, is ultralocal after a Fourier transform [ ˆ , ,s ) and ( 0 M a t s ) = 2 sin ( ( 4 3 38 , ), similarly for O θ 2.32 ( k ( hM 36 O , O + 32 + θ 2 channel. Following sections 3.1graviton and exchange is 4.2 obtained of by [ minimal gluing two coupling, massive see higher-spin three-point figure by amplitudes eqs. at ( At 1PM or treeequivalent level, to one-graviton the exchange leading-singularity [ that prescription the reduces to a contrast to the one-loop case,mented from the the HCL fact defined that as the the computation leading is order done in under the support of the factorization tion of the applicabilityuse of here ( produces resultsof [ which are (quite nontrivially) fully consistent with3.3 the results First post-Minkowskian order tum transfer parameter with magnitude pared to the nonspinning/scalar case,spin this components version of ( parameters in the amplitude. While we do not claim to provide a first-principles deriva- where p the classical scattering angle frame, is given by the same relation as for the spinless case [ JHEP09(2019)056 ) } 0 ]. b 2 s , 32 → 1 , ) (3.26) (3.24) (3.25) 1 2 (3.23a) k and (3.23b) { , a s , = ). Finally, . a b s = (0 a  a ·
· k 3.19 ) . ˆ b p ) to reintroduce ˆ p a  a ) × · × b +  k k a k  a i i ) ) and ( a 2 b 1( + ( a − a  − )  a 3.18 b + 2 = +2 ( = b a γ · a σ a σ b a ˆ p a − a p γv ( ), γv ρ b ρ b b a × k . At tree level this is equivalent k × a m ν b m n 3.9 ν b m = log ( k a a p ˆ a p p i S ·

µ a µ a m n ) m m  p ˆ  p p b k ) gives  a b = ×  ∼ · σ + µνρσ k µνρσ exp 3.21 a i a k 2 i ), we obtain ρ i  i ) a k 2 k , and dividing by the normalization factor (  ), which in general contain both classical and v ν b ) valid on the three-point kinematics, we can − ~q p exp ×  µ a πG = +2 exp – 21 – 3.7 = p ˆ p 8 (1 2 k k ), as indicated by eqs. ( − → k  · k · v ], where the full tree-level amplitude for 2  π a b b µνρσ X d )

= 2 a a − 4 2 28 i 2 , ε ) and ( , γ 2 Z · (1 ) − log 2 b 27 b 3 2) γ 3.6 t  a ε m ∂ = +2 = ∂b 2 2 a ) − κ/ 2 = )( ∂ ∂b + ( v ) 2 m 2 µν µν a b a v a ε − − − ) for both spins in the aligned-spin configuration. This precisely  ε S S · ε a (denoted there by v (  /x · · ν − ν − 1 πG b k (1 /b ε ε b ε a   x ( b (1 p µ µ p b a k k = 8 (1  i i i X = 0, but at one loop the HCL is needed to drop further quantum 2 = 4   X X − t 2 a ) and is the unit vector in the direction of the relative momentum. Moreover, v v 2 2 GE hM = = +2 × ˆ v v p GE GE − ˆ p µν µν a b − − = = = ε J J (1 + ε · · γ − ν − ν ε ε b a tree in favor of the (classical) tensor structures µ µ p p = θ k k 2 b i i Finally, let us emphasize that, as stated in the introduction, this already differs from k − + /x = a quantum pieces, depending onvector. whether they This include is the preciselyt corresponding what power the of LS theto singles spin out set by the droppingcontributions the HCL from (quantum) the contraction LS, as we shall explain in the next subsection. matches the result for the 1PM aligned-spin binary-black-hole scattering angle found in [ the strategy implemented in e.g. [ was computed in the firstunder place. the Only COM then frame. it The wasthe evaluation expanded momentum of in transfer spin the effects NR requires limit tracking subleading orders in having used Inserting this into the scattering-angle formula ( (with the relative sign due to the direction of x restoring the prefactorarising of from the generalized expectation value as in eq. ( Here we used thethe on-shell Levi-Civita equality tensor andframe, thus to where expose therecall scalar that triple on products the three-point in helicity the factors center-of-mass satisfy the seemingly contradictory conditions go to infinity. Afterrewrite using the eqs. exponents ( in a form independent of the polarization vector: Here we will take the limit where both massive particles’ spin quantum numbers ( JHEP09(2019)056 ] 2. × 38 k > (3.27) a s and ) through a a · 3.14 ˆ p × ) and to all orders . 4 a k a ] and emphasized in ( = 0 O ). We finally expand in 2 35 5 a that remains in the loop , )) aligned-spin scattering a )) 2 fits very nicely into the ( y 2 y 29 ( 4 3 G O ≤ ` p p ( = 0 and recovers the three- a ℓ − O 2 s 4 k : p ( 3 , ] for computing its classical piece. The LS = 0 2 29 )) 4 3 y – 22 – k k ( ` − 3 p ( 1 2 p p Triangle leading-singularity configuration. . This is done on the support of the Holomorphic Classical 3 , 2 b ] that for the spinless case the Compton amplitude for identical m 54 , ) = y Figure 3. 31 , ( 2 ` 29 is the rescaled spin of the particle that appears in the Compton amplitude, a a the Compton amplitude needs the introduction of contact terms for is needed, together with three-point interactions. The derivation is thus valid which accounts for a null momentum transfer ]. The computation of the possible contributions to the LS from contact terms is the spin of other particle. As explained already in [ 2.3 2.3 11 in the exponential form of the three and four point amplitudes entering the triangle b 40 , where a b Our strategy is to identify the spin-multipole-coupling operators b The name “Holomorphic Classical Limit” is due to the external momenta being complex at that point. a a · 11 ˆ p leading singularity, see figure Limit, be computed directly once theresums multipole all operators orders have in been bothspins identified. spins, and but The find is final perfect not formula agreement justifiedand with starting the [ at linear- and quadratic-order-in-spinarising results in of [ the higher-spin Compton amplitude is left for future work. in and section Nevertheless, the exponential structureFourier found transform already and for leads to a compact formula for the scattering function, which can It was argued in [ helicities leads to nowill be classical proven contribution. somewhere else.section This This fact implies that is only(to also the describe opposite-helicity true minimally case coupled for treated elementary arbitrary in particles) at spin, least as up to the triangle leading singularityis proposed now in given [ by aintegration contour after integral cutting for the a three single propagators complex of variable figure 3.4 Second post-Minkowskian order In this section weangle. derive a It compact is obtained form from for the the one-loop 2PM version (or of the four-point amplitude ( JHEP09(2019)056 , ]. ) + . 4 29 k  (3.28) (3.29) (3.30) b 2 − t m , , `,  4  p O , µν − b )  , ( + J − 3 ) t is the leading- + 4 b ε ν t  b · s k − 3 t − ( 3 in the expansion. − 3 ε b m − LS − p √ µ ˆ s √ m 3 M √  s k )] ,`, . The loop momenta, 2 2 i 4 |  and the scaling of the O y p ` s − | h 2 b O − s k  ( 2 h m ) + ] i given by eq. (3.17) of [ y ` accounts for a simultaneous | → ∞ (1 + ) + k . Here Γ (0) 3 | | exp y v y y ) − | k ⊗ M − 3 = − ) 2.1 (1 k or  | y − 3 k (1 + − k  k | [2 µν h a k `,  − γ y J − 4 = h 1 y a ε ν 2 − | `, y · − 4 2 µν 1 , b y 2 1 m ε − | 3 − J − 3 p µ , p ε ν 4 ( 3 · = = − 3 ) k p 3 ε | | b ] = i ( s p ). In turn, this fixes the little-group scaling 3 4 µ 3 ( 3 3  k k k | (0) 3 h h k ˆ 1 i M in-between the three-point amplitudes to denote – 23 – 3.18 p exp M | − | ⊗ ) 4 ` 4 )  k − 4 )] − ih h 4 2 2 ` . The soft expansion in ,k | . We can now insert the exponential expressions (for ) y , , , k exp 4 2 + 3 2 + k 3 y   ⊗ t t ,k − b b , k  2 (1+ − − 2 p v m m (1 p and µν √ √ a 3 − − , , − J − 4 3   y 1 , ε ν k  πy · 1 − 4 p [2 O O 2 2 b ( 1 ε p t ( p µ m ) dy 4 (0) 4 a ) + ) + k s  . = 0 the momentum transfer reads LS ( 4 i y y t M Γ O t ˆ  Z − M 2 + πy dy t γ 2 t exp is fixed by the condition ( πy ](1 + ](1 3 b dy − 2 | k k − LS × y | | m k Γ √ 2 a 1 2 1 2 √ h LS 2 Z Γ v t m ], Z 4 9 = 4 k − 2 ] = ] = t | i 2) 4 4 3 iκ 4 √ − k k b k 2) and evaluate the scalar pieces, obtaining | | 2) κ/ − 3 √ With the previous consideration, the above operator formula in the infinite spin limit is Before proceeding to compute the GEV, let us clarify an important point. Recall that Recall that at Let us first recap the triangle leading singularity, also introducing a more economic ( m b k i ≤ = κ/ 8 h ( m a i 8 expansion are not spoiledthe in introduction the of infinite contact spinthe terms limit. exponent, is which only The appears needed reason as to is a cancel pole that the in at spurious thefourth-order arbitrary pole exact amplitude spin, coming in only the from at expansion fifth of order. the left exponential and fully exact in the expansion to leading orders in in the tree-level case theThe exponential infinite spin operator limit was did truncated notfor alter at the promoting order lower orders such 2 in finite the numberholds exponential but for of simply the terms accounted Compton to amplitude, a that full is, series. the first We five assume orders such reproducing condition the still exponential spinors of both internal gravitons s together with their correspondingHere spinors, we will are only functions need of the following limits: where we have inserted the operatoroperator multiplication, insingularity the contour that same can sense be as obtained at in either section expansion in both powers of spin. formulation of it. It consists ofwith a the contour Compton integral obtained amplitude. by gluing Our three-point starting amplitudes point is the expression point kinematics studied in section JHEP09(2019)056 and 2 (3.39) (3.35) (3.36) (3.38) (3.31) (3.32) (3.34) (3.37) (3.33) p , , 1  p b a · . , up to a scale ˆ p 1 k − × ) − 4 for k ε 2 · ) − ε y 1 y as the classical operator p ] i 2 ( b 0 . k | a  r k · → 1 ) (1 + lim t 2 i ˆ p p 2[ | y = 0 kinematics. In fact, using r − × √ =0 [ t . t 1  k

) 1 y − ) v ) , (1 + − 4 − 4 ˆ ε v µν a , ε · = + ) = 2 | | · J = exp − ] 3 ] (1 + 1 4 4 ν 4 1 ε 4 k, p ). We find k a − 4 y k y · =0  k p ( h 2 ˆ h ε t 2 2 3 ] ) we can identify this factor with ( ] 3 )

r k  µ 3 0 p r [ k 6= 0. This means that in the limit we have ] × µν y i 4 b √ γm | ( 2 − [ k y 4 3.29 4 ] | → k a J ε / y 2 lim t 4 4 k − 3.19 2 · 2 | =0 − ν r k t √ µν = 1 b ε 3 √ =0 √ r k γm (1 + p J  µ 2[ t p − – 24 – | 2 ) = ν −
k µν r a √ ε [ can be taken as the vector dependence contributes to the contour integral. = µν − a − = 2 J µν µ a = − 4 ) ε J ν 3 3 − y ˆ k ε · J − 4 y − 4 ν k k − 4 y ˆ · ε ν − 4 ε 1 ) = = ε − 4 ˆ 4 ε = 0 we recover three-particle kinematics for µ 1 p ˆ ε 4 − 4 µ ( p t for µ 4 k ε =0 (1 + 1 y 4 . Using eq. ( k t = 0, i.e. · i − 3

r k − 1 4 ) ε  − p ε − 4 ( ·  ˆ ε = = 0 3 · → k lim 1 t µν a p − 4 J ( ε = exp = 0. In order to evaluate, it we will need the following trick. First ν · − 4 t ε  1 µ using p 4 µν b k − 3 J 0 ε 2.3 ν 6= 0 the numerator is gauge invariant, hence we can write · − 3 → lim ε t t 3 µ p 3 k i − ] is some reference spinor such that [  r | Now, recall that the left exponential corresponds to the Compton amplitude and was exp . This means that the combination is independent of the choice of On the other hand, recallk that at The limit can be evaluated directly using eq. ( where and which is singular at note that at that will enter the GEV, whereas the fixed in section where the polarization vector that cancels. We have again identified exponential factor on the right can be obtained straight at we find of the right exponential. Let us now proceed to evaluate the exponents of both. The JHEP09(2019)056 . 4 } ε  R · , b ) 1 i a = a p · − (3.43) (3.44) (3.42) (3.40) (3.41) | z a (3.45a) ˆ p , z − × ×  . {| b a z ˆ k p a = a 2 · )( 1 · y 5 v ˆ p + vz y − . LS ˆ z p  b × 2 1) − ) − a k 2 1 + − z × − 1 y i iz z k ↔ ( − ) − − vz ) 2 a ( a v a 2 / b a 3 y a (1 + a a · · . At the same time the v 1) ˆ p × ˆ p (1 + ∞ − × and − ˆ 2 p , × (1 + k y ) 2 b z 1 v = k 2 y z − v  m ( )

vz − − z 2 2 − a − b a y y y vy a − πi ↔ h dz ) terms) of our contour integral as 2 2 2 z 2 (1 · 1 0 v a , we have vz t i √ γm k − − k LS ( 2 m i − (1 + − 2 Γ 2 − )  O v = Z z is induced by the unphysical pole y y  y × 1 . in 4 − ) 2 −  1 va exp /v − (1 y − y exp LS 1 + y | b µν ε a 2 − − 2 2 ) reads = · k k i J ε 4 / | = 1 za 2 1 ) 1 + · 1 3 ) as in the previous section in order to compute − ν = π − d − p z ε 1 2 – 25 –

− ( 1) 4 vz  , for reasons we will explain in a moment. Then 3.21 µ p = a k 1 b y k Z a

3.24 − z − /v 2 2 = 0 are mapped to  exp 2 1 × v 4 4 / / ]. Here let us unify both descriptions by means of the vz ) z (1 ) ) ) 3 3 4 y 2 ( 2 v − 31 2 )] =0 − y 1) 1) γ vz vz t 2 v z 2 1 4 to account for the HCL difference between a triangle R > ) y  − − − − 2 and − πi 2 2 (1 + − y dz µν (1 + a 2 2 z z (1 (1 b − 4 ) v ( ( J ) and using − ∞ ˆ ε (1 + 2 y ν LS za · v − 4 is obtained by exchanging can be chosen as a contour around zero or infinity. This inversion Γ πi πi ˆ (1 ε = dz dz − 1 − Z 2 2 3 − µ . − 3.36 p y 4 LS θ y 3 b b (1 y LS LS 4 k

πy Γ Γ t m 2 2 [2 −  4 Z Z 2 a / − ) 3 m dy = = ∂ ∂ √ ∂b ∂b 1) 2 vz LS 4 4 b b G , where Γ µν − a − v v . 2 Z − 4 m m J 2 2 2 θ ε π 2 ν z (1 · 4 − 4 t ( E E γ + ε 2 2 3 b 1 − − / µ p πi θ 4 m dz √ 2 = 2 a k πG πG 2 i 0 v LS m / = = 9 Let us now discuss the choice of contour Γ Attaching the same normalization ( 4 Γ → 2 lim t / Z iκ θ hM angle is where we have specialized to aligned spins. The total one-loop contribution to the scattering contributions. The essential singularity at in the exponential expansion.for We some take large the but contourthe around finite contribution infinity radius, to to the be scattering Γ angle ( which now incorporates thehave second also helicity inserted assignment a forintegral factor the and exchanged of its gravitons. leadingthe singularity. We massive propagators Note that inside the the branch Compton cut amplitude singularity and is does induced not by lead to classical Both contours around polynomial structure gets reduced tocut at most in quadratic, the at integral. the cost We of now introducing have a the branch one-loop triangle contribution as As already explained, Γ accounts for a parity conjugationinvariance of of the the amplitude, triangle and diagram the [ change equivalence of follows from variables parity the GEV, we write the leading order (i.e. dropping − Putting all together in ( JHEP09(2019)056 is cl and O . The 1). In ]. ∞ , (3.45b) 1 /v 58 = , − ( as given by . Neverthe- z = 1 ]. a 39 ∈ O , a z 38 z 25 0. We now show ) yields the explicit → . b a a 3.44 ) simply grabs the pole b va as va 3.43 + b = would disappear at every order, as well, the contour integral can → ∞ − − + + z z z + ] to construct such quantities in an eco- . ], so it would be interesting to see if these and R < z 31 + , 31 + z z 29 = , z b z together with the branch cut at a – 26 – ], as well as the conjectural one-loop quadratic-in- , respectively. This can be seen by noticing that + − /v b by demanding 38 and a z a 0. This is the reason we consider a contour at finite va − ∞ = 1 z + . Interestingly, this matches their effective counterpart, → and z = and on next-to-next-to-leading-order post-Newtonian re- b bv ) resums part of the contributions from both + z 0 the poles at 2 hOi z , a = G and a = → a ], based on results from the exact quadrupolar test-black-hole ), so that, as long as − 3.45a b cl z ) in the introductory summary. Let us stress that the formu- ∞ 40 as O , a + 3.43 = a + a 1.12 /v z z 1 ) can only be expected to be valid up to fourth order in → in eq. ( ]. An extension of the GEV may be needed to incorporate time dependence, 1.12 − z is distinguished from /v 37 ]. , 1 + , by ensuring the consistency of the small-spin expansion. If we were to take and drops the branch cut contribution together with the pole at z 32 and 57 into poles located at ] expanded to order ) and ( , ∞ ∞ R > 55 56 /v = 3.44 The natural desired extension of the leading-singularity method is the computation of It is clear that a more precise definition is needed for the generalized expectation value With this contour prescription, evaluating the integral in eq. ( and → ∞ z = 1 + + such as what occurs with classical momentum deflection orhigher spin orders, holonomy both [ in loopsties and were powers computed of for gravitational spin. theoriescan Examples in [ of be higher-loop also leading applied singulari- to compute classical observables. On the other hand, extending the that we used. Our constructiontwo can particle be states thought as in the theof average scattering a of amplitude, an classical operator which observable isas mapped computed for to instance the inconstant expectation [ the value worldline formalism, in the case where the operator In this work we have presentedservative a classical new gravitational connection observables, betweenholes. extended in soft particular This theorems extends for and the scattering con- approachnomic initiated of way in through spinning [ leading black singularities.the It extraction also complements of the classical general results picture from regarding on-shell methods, provided e.g. in [ limit [ sults [ 4 Discussion las ( less, they condense non-trivial informationsimple for contour the integral. scattering We anglelinear-in-spin have up classical checked to computation that of that these [ orderspin results into expression precisely a given match in the [ one-loop z radius be evaluated from the poles at results given by eq. ( leaving poles only at that case, the leading-singularityat prescription in the integralnon-expanded ( expression ( z The root that the appropriate leading singularityz in the contour integralan is expansion given around by the residues at where JHEP09(2019)056 4 ], S 59 and ), which ]. i → ∞ ε ], thus de- · s 61 , 35 p 60 particles lead to s -pole level, or up to s , specifically matching 2 2 ) spin-orbit effects which S 1 ) to higher orders. As we S ∝ ( ], hence one could hope that 1.1 O ) in higher orders would then 10 i , , etc. ε 9 3 · S p ∝ , yielding the S ∝ – 27 – ], where this choice arises naturally from a BCFW ] that amplitudes for massive spin- 62 particle minimally coupled to gravity. This correlation 27 s ]. Reinserting powers of ( ] with explicit calculations at leading post-Newtonian orders, 1 ], by means of a BCFW argument, that in the MHV sector of 27 ), requires decreasing the powers of the numerator ( 62 µν i is the body’s intrinsic angular momentum: J ν S ε µ for a massive spin- k , where s → ∞ 2 S Note that the quadrupolestops level being corresponds universal. to the order ata which spin-3/2 the particle adds soft a theorem black-hole octupole a spin-1/2 particle adds only aare dipole universal (body-independent) in gravity; a spin-1 particle further addsthe a quadrupole spin-induced of quadrupole a spinning black hole when constructed with minimal coupling. a scalar particle corresponds to a monopole (with no higher multipoles); s 0, and to what extent we should expect it to hold. It was found in [ It was already pointed out in [ • • • • → the scattering particlesimpose [ additional constraints thatexponentiating the go soft beyond factor, these aThis very conservation specific is laws. choice precisely of what the Therefore,deformation. polarization is when A vectors second done is problem required. in that we [ dealt with here is the sum over different particles, A general statementevident for problems gravity for the amplitudeshave naive is extrapolation seen, however of increasing the still thecombination formula missing. powers ( ( of angular Theregenerates momentum, are unphysical encoded poles. a in few Moreover,to the the fundamental gauge-invariant first conservation two laws orders enjoy corresponding gauge to invariance the thanks linear and angular momenta of at one-loop order. Itbetween is, classical however, black not holes yet~ clear and why minimally we coupled should quantum expect particles this with correspondence gravity amplitudes there is also a natural exponential completion of the soft theorem. limit was shown by Vaidyacorresponding [ to the nonrelativisticlevel. limits of In this tree-level paper, amplitudes, werelativistically, up have to provided to further all the evidence orders that spin-2 in this or spin correspondence at holds, tree fully level, and for at least the first few orders in spin The complete spin-multipole series of a black hole is seemingly obtained by taking the a classical potential for bodiesstars. with spin-induced multipoles The such amplitudes asorder black match holes or the neutron classical potential up to the 2 deeper orders in theboth soft on expansion. the More matter precisely, contentsuch it and problem is the is coupling known tractable that toscribing gravity at these black [ orders least holes. depend for matterand Our it minimally methodology would coupled clearly be to resembles desirable gravity to a [ formally soft implement bootstrap it approach via recursion [ relations [ range of validity in powers of spin is now clearly related to the problem of understanding JHEP09(2019)056 ] ]. 20 – 74 . We , (A.2) (A.1) 1 − 17 , 73 n , 65 M 15 , ], as explored 14 69 – 65 . , and then inspecting µ = 0 → ∇ µν N µ T µ ∂ ∂ , µ A µ A 2 2 ) starting from the massive spin-1 La- m 2.1 + L ⇒ µν F µν – 28 – . Let us, however, take an alternative route of η ) µν L F − g µν 1 4 σ − ∂g √ − ]) and classical observables arising from massive ampli- A ( ν ∂ = ∂ 64 g , − 2 L µσ √ 63 F . In order to compute the minimal cubic vertex to gravity, one − = µ = A µν ν T µν ∂ N T ] that has been known in QED for a long time. The latter one has − ν 72 – A µ ∂ 70 , = 20 , µν 2 F ] from the point of view of soft theorems. Finally, it would be also interesting to A textbook application of Noether’s theorem for translations yields the following tensor An obvious question which arises from this construction is whether it is possible to 22 computing the energy-momentum tensorprocedure directly will in explicitly flat identifythe the particle. space. contribution The of the reason intrinsic is angular that momentum this of where needs to the extractcan the be energy-momentum done tensor by sourced covariantizing by thisthe this action, metric i.e. field. variation, by In promoting principle, this A Three-point amplitude withHere spin-1 matter we compute thegrangian three-point amplitude ( of Innovation, Science and Economicthrough Development Canada the and Ministry by of thethe Research, Province of Innovation European Ontario and Union’s Science. HorizonSklodowska-Curie AO 2020 grant has research agreement received and 746138. funding innovation from programme under the Marie the gravitational and gaugeare couplings grateful of to massive the particles organizerswas in completed. of private the AG correspondence. workshop thanks “QCD We kindwork Meets hospitality was from Gravity initiated, IV”, the and where AlbertPerimeter this Einstein CONICYT Institute work Institute, project is where supported 21151647 this by for the financial Government support. of Canada Research through at the Department Acknowledgments We would like toHuang, thank Ben Nima Maybee, Matin Arkani-Hamed, Mojaza,sions. Fabi´anBautista, Donal Freddy O’Connell, We Cachazo, and are Yu-tin Jan very Steinhoff for grateful useful to discus- Yu-tin Huang in particular for clarifying some aspects of tudes. The natural candidate forin such [ a connection issee radiative a effects link [ betweengences the exponentiation [ presented hererecently and appeared the in exponentiation the of computation IR of diver- tail effects from the EFT perspective [ showed that in the cases ofloop, interest these for two computing problems the can scattering be angle overcome at by tree a level judicious andestablish choice one of a the link polarization vectors. between(or BMS at the symmetries black studied hole horizon at [ null/spatial infinity [ which destroys the realization of the exponential as an overall factor acting on JHEP09(2019)056 µν N . we λ µν T S (A.5) (A.6) (A.8) (A.3) (A.4) 2) B (A.7a) (A.7b) + ↔ , λ µν . , τ 2) L + (1 A  = 0 ↔ = 0, we obtain ) µν στ 2 3 Σ µν ε p · T λσ · + (1 µ µν 2  ∂ , ) iF Σ p . τ · 3  ] 1 ε A = ν 2 = , which is solved by ): p · , which in appendix ε τ ] that will help us identify 1 ] λµ στ 3 )( µ 2 ν µ ε τ [ 1 A 3 p Σ ε δ λ µν A.2 . ε ⇒ τ ε )( ·  µ ν S 1 [ · 2 νσ 3 νσ 1 λ ε 1 ε η iε p F ε · ∂ µν,σ µ ) ν λµ ( 1 1 2 3 − Σ 6= 0), therefore its orbital angular S p p µν − ( ) = 2 ν τ 2 = h σ − ) δ ] τ 2 λ µλ ν µν − 2 N , fails to give the correct three-point 2. Its momentum-space version in the − A ε ∂ ε L correspond to the massive spin-1 matter ) µσ T · B λ · ( ) µ ν 2 [ ∂ ν µν η i 2 2 N κ/ ∂ µ νλ [ ε − λ ∂ ε ( ε µν i µν T S · − · ∂ B Σ 1 Σ = 1 i µν λ · · = σ ε + ε ε 1 ∂ 1 − and A τ )( ε ε )( ν 3 1 – 29 – λµ ν ε ] may be adjusted to yield a symmetric energy- = )( ε ε ∂ λ µν · 3 · µν,σ S 1 ε 1 µσ  76 p · p , ( λ µν 1 2 F 1 ( is not conserved. Let us fix that by generalizing   p ) 75 ) ( [ µν ε = 3  λµ h N · ε ν 1 ,B T 3 · = p ν ε µνρ 2 λµ ν µ x p B ) comes from 3 ( λµ ν B µν ,S − ip B T = 2( λ A.8 3 λν µν ∂ N λ µν → − → h T S M + µ belong to the massless graviton. Putting the above terms together λ τ σ − x ∂ 3 A A µν N ε − ν = T µν στ ∂ = Σ = µσ ρσ λ µν νµ F N µν L F T ) µν T are the Lorentz generators Σ h − νρ and integrating by parts, we obtain the gravitational interaction vertex h µν µν µ N µν ∂ T lacks symmetry in its indices (notice e.g. be symmetric now yields the condition ( Contracting the resulting energy-momentum tensor with a traceless symmetric gravi- h i − µν µν N The second term ininterpret eq. as ( a spin expectationgravitational value, interaction. so it can be regarded as the spin contribution to the where the transverse polarization vectors and two copies of and using the three-pointthe on-shell amplitude kinematic conditions where we suppress thescattering coupling-constant amplitude factor gives the following contributions: ton the spin contribution inside theT three-point amplitude. Imposing that the corrected tensor Here Σ where the Belinfante tensor momentum tensor matching the gravitational one.to Lorentz To do transformations. that, The we conservation applythen of Noether’s implies the theorem total angular momentum amplitude, as opposedT to the onemomentum obtained from covariantization.to a larger The class reason of tensors is that that are all conserved due to eq. ( Its contraction with an on-shell graviton, JHEP09(2019)056 12 (B.3) (B.4) (B.5) (B.1) (B.2) from A ) is a le- , , ) ] ˆ p µ τ B.4 δ νσ . E,P ) η ˆ p − ) = ( ν τ P,E δ ( , E,~p 1 m µσ . η ] [ 6= 0 = = ( ν 1 i 1 ε ) µ p µ ε 2 s µ p [ = · ε ∗ 2 ) with respect to the average · ∗ 2 τ ). Although eq. ( 1 ε iε ε 2.4 2 ( , 2.4 ] of spinors for a massive momentum µν,σ / ] it is desirable to consider a spin ) , , = ,
2 Σ ν 2 ε 32 50 τ 1 , ε · = 2 = 1 ε = 3 ) 1 τ 35 b b 1 b ε 1 ε ( ε = = + · / · ] µν,σ k ν 2 ) is to introduce a generalized expectation ∗ 2 , ] Σ ε 2 ε ν p σ µ p , a , a , a = 0, this spin tensor immediately satisfies the [ 1 ε + ( ∗ 2 0 µ p µ p µ ε s s p [ iε ν 1 – 30 – ε ∗ p ) to the case of two different states (one incoming 2 − ε · ) = iε        − 2 p i ε B.1 = 1 · = | = 2 i ρ k p 1 ( µν | ) µν τ p 12 in the vector representation. Due to the transversality of 2 ε Σ S ab p h i | τ 2 ε p 2 · ]). This ambiguity allows the spin tensor to be transformed µν ε h − · µν,σ νλ 77 ∗ p = = Σ Σ ε 2 of the massive particle before and after graviton emission: · / µν ∗ pσ 12 µν 12 ) ε 2 S pab S ε p µ is the momentum transfer. However, the spin tensor is intrinsically ( = p and one outgoing with − . The starting point is the one-particle expectation value of the angular- 2 i 1 p 1 p ), the one-particle spin quantization is explicitly µνλρ we consider all momenta incoming, we suppress the conjugation sign p | p ˆ  i p 2.1 ). For instance, in the helicity basis [ − 2 p µν | m 1 1 2 p = ( Σ p h E,P | 2.10 p = p − h ab p = ) = ( i ). = µ k a h 2.4 E,~p µν p m Now a natural way to extend eq. ( S The conjugation rule between the incoming and outgoing states in the massive spinor-helicity for- = ( 12 µ p reference point for the intrinsic angular momentumto of its a overall spatially orbital extended momentum body about (asby opposed the a origin) frame is change at (see its center e.g. of [ mass, but it getsmalism shifted amounts tolation lowering in and eq. raising the ( little-group indices, as indicated by the completeness re- where ambiguous, as the separationmomentum between the is orbital relativistically and intrinsic frame-dependent. pieces of In the total a angular classical setting, for instance, the Noether’s theorem. Now intensor a that classical satisfies computation thegitimate [ spin definition, supplementary it does conditionmomentum not ( satisy the covariant SSC ( Since in section and rewrite the above as which is the (normalized) angular momentum contribution obtained in appendix with momentum value such that it gives one for a unit operator: where for now weused suppress the the Lorentz generators spin-projection/little-group Σ the labels both of massive the polarization vectors, states.SSC ( We also Here we construct the spinmatics of tensor section for a massivemomentum spin-1 operator particle in for the the quantum-mechanical three-particle sense: kine- B Spin tensor for spin-1 matter JHEP09(2019)056 (B.6) (C.6) (C.3) (C.4) (C.5) (C.1) (C.2) 2. The , / ]  ] k ν |

˙ µ β 2 µ σ γ ε | ) below. σ k µ 1 accounts for the h ˙ ε αγ ∂ µν · , ν ∂k ν, is antisymmetric in J r  = k β ¯ σ ) ˙ λ β µ − . Finally, we note that ˜ λ ∂ + ( k ˙ αβ ˙ k β µν,στ ∂ to accommodate for the 1 ] ν γ µ, ε ν α , ν σ ¯ ( ˙ ) σ ∂ r # 2 ˜ λ 1 2 ˙ ∂x ˙ ε αγ β , · µ αβ ˜ µ, λ [ ) as the covariant-SSC one. τ = ∂  ¯ k σ ∂ ∂ ˙ ( ix ˙ α ∂p β + ˙ B.4 i α ˜ 4 λ , also valid for ∂ ] ˙ µ ). Namely, the action of spinorial β [ σ σ ∂ ˙ = 2 α p for some vector µν, p = 51  2 ¯ µν, τ σ ] ˙ C.3 ) β 1 ˙ ρ again. Since Σ ν α ˙ β Σ α m pos. µν r ˜ λ σ . Adjusting ∂ µν, σ ] µ ) is [ p µν [ µν, ∂λ − ν Σ + p ¯ ] σ α , = ( ν 2 − β C.2 µ ε λ ] ) and ( ∂ µ ν  ∂ = [ 1 ∂/∂p ∂λ i coincides with that of the vectorial one. ∂k – 31 – ε ∂ µ ρ ˙ [ ∂p β C.2 2 ,L µ p p α ˜ λ µ k  [ = 2 , ˙ µν β µν 2
µν, β µ α ∼ S ip ε L σ is not invertible for massless particles, but we can still µν σ ˙ i · γβ for a null momentum transfer α J + 1 1 2 µ ˙ µν µ, λ β = 2 ε 2 k ¯ ν σ β L " µν = ˙ γ m − σ µν L ν ˙ α = α µ L σ = µ = α } → = ∂ ] and σ ˙ µν 2 ∂k α ν − p 0 we retrieve the spin tensor ( ˜ λ J p µ α µν = , µν µ ˙ J γβ [ ˙ , where the difference ), which can be more concisely written in spinor indices as α β ] S → λ ∂k 12 ∂λ ν, ν λ S ¯ σ k r { ˙ α,β ˙ C.3 λ γ = µ α [ Let us warm up with the case of a massless p µ α p J α 2 σ 2 ∂ m + i ∂λ 4 + µν = S µν 12 α S ), we obtain → µν, β = 2.4 σ µν The generator ( µν S S to check the consistencygenerator between on a eqs. function ( of momentum that the spinor map use the chain rule where the matrices are the left-handed and right-handed representations of the Lorentz-group algebra. Note Massless case. spinorial version of the angular momentum ( in which we encounter theboth Lorentz pairs generators Σ of indices,differential we and algebraic notice operators, the subtle difference in signs between the actions of the in terms of theof spinor-helicity the variables. orbital The piece starting point is the momentum-space form where we have used that in the classical limit C Angular-momentum operator Here we consider the total angular momentum relative shift between SSC ( as JHEP09(2019)056 . } = ˙ b β  ˜ µ λ a ) , p ˙ (C.7) (C.8) (C.9) β a α (C.13) (C.10) (C.11) (C.12) ∂ ˜ λ λ ∂ { . ] a α ˙ ( ˙ β ˜ λ β . Of course, ˙ ε α ˙ γ , αβ , so it can be . ˜ is the same as λ ˙  α ˙ γ µ   ˙ ] for half-integer γ b β ∂ + k ˜ ∂p λ αγ ˙ β and ... ) incorporates both  . 50 ˙ ab , α ˙ βa γ   α | + − λ a α 35 k a λ , C.6 ) ) ˙ [ ) γ λ ˙ s s ˙ ˙ α αβ γ β s s β 2 2 | ˙ ˙ i ∂  ] α a α a , µ, r = k ) [ ˜ ˜ λ λ ∂λ ¯ | σ s βγ α 1 1 ˙ s ˙ r k 2 β β a  α 1 2 i ˙ [ | − − a α ( ˙ α α s s k q ˜ s s λ 2 2 | λ − p [ α 1 α a α a ˙  α ε 2 − | [ λ λ i = s i q s 2 √ [ ˙ a α a α ) is precisely that of the algebraic ··· ··· − λ ˜ λ = 2 αβ ∂ = 2 1 i 2  ∂ ˙ ˙ 2 = β α a ˙ i ˙ γ C.6 2 ··· α ] ˜ λ γ 2 ˙ ˙ α α ) ˙ 1 i 2 α,β − α ˙ ˙ √ β
a α q k α µν, [ 2 ˜ λ , ¯ σ ˙ 1 α α,β 1 2 µ a + α ˙ ( α a α µν, β ˙ ∂ γ λ ˜ λ but rather of its spinors ∂p σ γ εS – 32 – 2 h ˙ s ) 1 ( β a ,J 1 µ ˙ a 1 β s/ ˜ , ε a λ ( k m   ( α ˙ 2 ˙ β α α ˙ λ a β ˙ | µ α α,β λ i ˜ k λ α = σ + ∂ ⇒ [ ∂ β S 1 2 s α r k ˙ 1 ˙ i β h α − ˙ α α s r ] ε =  | ν τ s h µν, δ 2 2 ...α µ  1 µ ¯ σ = ( tensors are parametrized in terms of massive spinor-helicity [ √ ˙ ∂ α ˙ a 2 ...a α s ∂p ˙ γ 1 1 s iε ˜ λ s/ − γ α a = m µ ε 2 ε αa is a first-order differential operator, it distributes when acting on ˙ ˙ α + β = 2 + α ∂p = ∂λ ε µν ˙ τ α,β βa s J α ˙ ∂ α = s J ∂λ It is direct to generalize the above discussion to massive momenta µν,σ s 2 αa ...α Σ α ]. The angular-momentum operator in the space of massive spinors ). Finally, the action on polarization tensors can be tested to be a Lorentz 1 ∂ σ ˙ α , which constitutes the intrinsic angular momentum ...a ε ∂λ µν, β 1 78 1 a α σ C.2 µν = ε 2 [ αa / τ ] µν ) λ a and naturally expands into the left- and right-handed Lorentz generators: J   p µν | s 2 µ = a Therefore, we conclude that the spinorial differential operator ( εS σ | ( ··· a 1 µν a p J ε transformation. The spin- variables as with an obvious extensionspins. by an Indeed, additional since factor of Dirac spinor [ it is again easythat to of check eq. that ( the action on a function of This operator is by construction invariant under the little group SU(2). Using the chain rule the orbital and intrinsic contributions, so itMassive is case. the total angular-momentumh operator. is given by Specializing to the negative-helicity case for concreteness, we indeed find Here the lastdiscarded term in is a a physical amplitude. gauge contribution explicitly proportional to an integer spin shouldshow not that by itself the depend actiongenerator on Σ of the the auxiliary differential spinors. operator Fortunately, we ( can of the massless particle.spinor-helicity For variables, instance, when we write the polarization tensors in terms of we do not regard them as functions of has more information than its momentum-space counterpart, as it cares about the helicity JHEP09(2019)056 ]. ] Phys. 96 ] 166 , SPIRE Phys. ] ]. IN , [ Phys. Rev. , ]. (2011) 060 SPIRE ]. , IN Phys. Rev. (2014) 152 05 [ Phys. Rev. , , 07 SPIRE ]. SPIRE ]. IN arXiv:1702.03934 ]. 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