Gravitational Bremsstrahlung from Reverse Unitarity
Enrico Herrmann, Julio Parra-Martinez, Michael S. Ruf, Mao Zeng
CCALT-TH-2021-003, FR-PHENO-2021-02, OUTP-21-02P
Gravitational Bremsstrahlung from Reverse Unitarity
Enrico Herrmann, Julio Parra-Martinez, Michael S. Ruf, and Mao Zeng Mani L. Bhaumik Institute for Theoretical Physics,UCLA Department of Physics and Astronomy, Los Angeles, CA 90095, USA Walter Burke Institute for Theoretical Physics,California Institute of Technology, Pasadena, CA 91125, USA Physikalisches Institut, Albert-Ludwigs Universit¨at Freiburg, D-79104 Freiburg, Germany Rudolf Peierls Centre for Theoretical Physics,University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
We compute the total radiated momentum carried by gravitational waves during the scatteringof two spinless black holes at the lowest order in Newton’s constant, O ( G ), and all orders in ve-locity. By analytic continuation into the bound state regime, we obtain the O ( G ) energy loss inelliptic orbits. This provides an essential step towards the complete understanding of the third-post-Minkowskian binary dynamics. We employ the formalism of Kosower, Maybee, and O’Connell(KMOC) which relates classical observables to quantum scattering amplitudes and derive the rele-vant integrands using generalized unitarity. The subsequent phase-space integrations are performedvia the reverse unitarity method familiar from collider physics, using differential equations to obtainthe exact velocity dependence from near-static boundary conditions. Introduction.
There has been enormous progress inapplying scattering amplitude tools, such as general-ized unitarity [1–3] and the double copy [4–9], togetherwith effective field theory ideas [10–12], to the classicalrelativistic two-body problem, geared towards applica-tions for current and future gravitational wave detectors[13, 14]. Such techniques have produced new results forthe dynamics of spinless [12, 15–28] and spinning [29–42] black holes, including finite-size effects [43–51]. Withexceptions [52–58], this effort has focused on the con-servative dynamics, described by a two-body Hamilto-nian [12, 59], or the scattering gravitational waveform[60–64]. In this letter we use amplitude methods to com-pute a radiative observable for a bound binary system ingeneral relativity. We do so by first calculating the mo-mentum emitted in the form of gravitational waves dur-ing the scattering of two spinless black holes at O ( G )and all orders in velocity. By analytic continuation fromscattering to bound kinematics [65–67] we obtain the O ( G ) energy loss in an elliptic orbit of a binary system.We use Kosower, Maybee, and O’Connell’s (KMOC)[68] formalism to express classical observables directly interms of scattering amplitudes and their unitarity cuts.Recently, this formalism has been used [69] to understandclassical soft radiation [70–74]. By focusing on an inclu-sive observable, involving a sum over final states of thescattering event, we avoid the need for detailed knowl-edge of the gravitational waveform and subtleties arisingfrom infrared divergences in its phase. In a well definedsense, observables calculated in the KMOC formalism areanalogous to inclusive cross sections in collider physics.Taking this analogy seriously allows us to import cru-cial technology developed in the particle-physics context.Concretely, we use generalized unitarity to construct theloop integrands; we employ (canonical) differential equa-tions [75–79] adapted to the post-Minkowskian expansionin classical gravity [80], together with the reverse unitar-ity method [81–84] for the phase-space integration. KMOC formalism.
In this work, we compute classicalgravitational observables in the KMOC formalism [68].The basic idea of this approach is to set up a gedankenexperiment for the scattering of two wavepackets, widelyseparated by an impact parameter b µ , and measure thechange in an observable O , with corresponding quantumoperator O , between in and out states∆ O = (cid:104) out | O | out (cid:105) − (cid:104) in | O | in (cid:105) . (1)Since the out state is related to the in state via the timeevolution operator, i.e. the S -matrix, | out (cid:105) = S | in (cid:105) , we canwrite the change in the observable in terms of the scat-tering amplitude M = − i( S − p µi is given as mo-mentum difference of particle i by measuring operator P i and the radiated momentum ∆ R µ by measuring R µ .We are interested in classical observables. This corre-sponds to the regime where the Compton wavelength ofthe particles representing the black holes is the smallestlength scale in the problem (we point to Ref. [68] for a de-tailed discussion of this limit). For us, it suffices to statethat we are interested in regions of external kinematicswhere the massive particle momenta p i scale like O (1)in the classical counting and the four-momentum trans-fer q , as well as the graviton loop variables (cid:96) i scale like O ( (cid:126) ). Employing the terminology from the “method ofregions” [85], the classical (cid:126) expansion is then equivalentto the so-called soft expansion. This classical countingwill play a crucial role when constructing loop integrandsand evaluating the corresponding integrals.In the classical limit, the dependence on the shape ofthe wavepackets drops out and one arrives at [86]∆ O = (cid:90) ˆd D q ˆ δ ( − p · q ) ˆ δ (2 p · q ) e i b · q ( I O, v + I O, r ) (2)where, borrowing language from collider observables, onecan define virtual and real kernels, I O , v and I O , r , which a r X i v : . [ h e p - t h ] J a n respectively depend on the virtual amplitude, and its uni-tarity cuts including a phase-space integration akin tothose appearing in cross sections. The observable of in-terest is specified by a corresponding measurement func-tion . In the KMOC formalism, this amounts to a nu-merator insertion or differential operator acting on thecomponent amplitudes in Eq. (2).In this letter, we focus on the radiated momentum,∆ R µ . As explained in Ref. [68], this observable only re-ceives real contributions, and the corresponding kernelexpressed in terms of scattering amplitudes I µR, r = (cid:88) X (cid:90) dΦ X (cid:96) µX × p − p − q − p + qp M M ∗ ‘ X − p + ‘ − p + ‘ = (cid:88) X (cid:90) dΦ X ( (cid:96) , (cid:96) , { (cid:96) X } ) (cid:96) µX (3) × M X ( p , p , − p + (cid:96) , − p + (cid:96) , { (cid:96) X } ) × M ∗ X ( {− (cid:96) X } , p − (cid:96) , p − (cid:96) , − p − q, − p + q ) , is given by a sum over unitarity cuts featuring the ex-change of sets of messengers , X , in our case gravitons,including the empty set. As usual, such unitarity cuts in-volve an integral over the n -point Lorentz invariant phasespace (dΦ n ). The measuring function for ∆ R µ is encodedin the insertion of (cid:96) µX , representing the total momentumcarried by the messengers. We use an “all-outgoing” con-vention for the momenta in the scattering amplitudes andmostly-minus signature metric. Eq. (3) is closely relatedto the textbook formula for the radiated momentum [87]∆ R µ = (cid:90) dΦ ( k ) k µ h ∗ νρ ( k ) h νρ ( k ) , (4)in terms of the momentum-space gravitational waveform, h µν ( k ). However, the calculation of the waveform re-quires computing multi-scale integrals depending on dif-ferent components of k µ , whereas, as we will argue below,the direct computation of the multi-particle phase-spaceintegral involves simpler functions of a single scale.Although Eq. (3) is valid beyond perturbation theory,in this work, we will expand it perturbatively in GE cm /b ,where G is Newton’s constant, and E cm the center ofmass energy. The first contribution to ∆ R µ arises at O ( G ), or third post-Minkowskian (3PM) order, sinceBremsstrahlung of finite energy gravitons can only oc-cur once one of the black holes is deflected due to itsgravitational interaction with the other. Integrands from generalized unitarity.
With an eyetowards more general observables, at fixed order in G , in-stead of computing both real and virtual contributions inEq. (2) separately, we obtain the integrand for the virtual amplitudes and then take appropriate cuts and insert (cid:96) µX from Eq. (3) to obtain the real contribution relevant for FIG. 1. Generalized unitarity cuts relevant for the radiatedmomentum. Shaded blobs denote tree-level amplitudes, visi-ble legs are on shell, and we exclude any phase-space integrals. the radiated momentum. The virtual integrand for two-to-two scattering of massive scalars is derived by gener-alized unitarity [1–3] in the same fashion as described inRefs. [18, 20]. Unlike for the conservative integrand ob-tained in previous works [18, 20], in order to construct thescattering amplitude in the full soft region we are forcedto include additional terms that are necessary, once radi-ation is taken into account. However, in order to captureall terms relevant for the radiated momentum in the clas-sical limit, it suffices to match the set of cuts in Fig. 1,the first of which is familiar from the conservative sec-tor [18, 20]. Relative to the three particle cut in Eq. (3),these have the advantage that only four-particle ampli-tudes are involved, and some quantum contributions areautomatically dropped. We compute these unitarity cutsby sewing tree-level amplitudes in D = 4 − (cid:15) dimensionstaking advantage of the additional simplifications of gen-eralized gauge invariance [88, 89].To find a diagrammatic representation of the integrandthat matches the unitarity cuts in Fig. 1, we write anansatz in terms of the cubic diagrams in Fig. 2 withkinematic numerators. The first five graphs were alreadypresent in the conservative result, however, since we arenow dealing with additional cuts, their numerators mightslightly change. Furthermore, there are three new graphswhich contribute to the radiated momentum computa-tion. Each kinematic numerator of a graph Γ in Fig. 2 iswritten as a polynomial of Lorentz products of the inde-pendent external momenta p i and the loop momenta (cid:96) j up to mass dimension twelve. In addition, we impose thediagram symmetries on the ansatz, as well as the mini-mum (cid:126) counting dictated by the number of three-gravitonvertices in the graph. Matching this ansatz against thecuts in Fig. 1 then determines the unknown parametersin the ansatz. All that is left to do is to evaluate thethree-particle cut of the resulting integrand, perform thephase-space integrals in Eq. (3), and Fourier transform(c.f. Eq. (2)) to impact parameter space. Soft expansion and reverse unitarity.
In order toefficiently evaluate the phase-space integrals appearingin the KMOC kernel (3) we are inspired by the enor-mous progress in cross-section calculations in a colliderphysics setting where similar real contributions appearand are handled on equal footing to the virtual ones via reverse unitarity [81], see also e.g. Refs. [82–84]. In thereverse unitarity setup, one replaces on-shell delta func-tions (and their n -th derivatives in intermediate steps ofour calculations) that appear in phase-space integrals by FIG. 2. Cubic diagrams relevant for the radiated momentum. the difference of propagators with varying i ε prescription2 π i( − n n ! δ ( n ) ( z ) = 1( z − i ε ) n +1 − z + i ε ) n +1 , (5)which allows us to employ standard tools for loop in-tegrals like dimensional regularization, integration-by-parts (IBP) identities [90], and (canonical) differentialequations [75–79] to evaluate a minimal set of master in-tegrals . For all practical purposes, we can treat any on-shell delta function as a propagator which significantlysimplifies our computations and circumvents the difficul-ties in having to evaluate integrals containing derivativesof delta functions that would otherwise appear.In fact, we calculate soft integrals , obtained by expand-ing the original integrals in Fig. (2) in the limit where thegravitons are much softer than the matter lines (due tothe (cid:126) scaling of momenta assigned below Eq. (1)). In-verse graviton propagators, (cid:96) i , are unchanged whereasinverse matter propagators, ( (cid:96) i + p j ) − m , are expandedinto linearized expressions, 2 (cid:96) i · p j . Within reverse uni-tarity, the phase-space delta functions are treated on thesame footing and are likewise expanded. The resultingsoft integrals are homogeneous in the masses and momen-tum transfer with the scale given by dimensional analysis,which we commonly strip from our expressions.The construction of the differential equations for thesoft master integrals (cid:126) I has been discussed in Ref. [80]d (cid:126) I ( y ) = A ( y ) (cid:126) I ( y ) , (6)where y = σ + O ( q ) and σ = p · p / ( m m ) is therelativistic Lorentz factor. The O ( q ) shift relating y and σ is a technicality of the soft expansion and detailedin Ref. [80]. Since IBP relations are agnostic to the i ε prescription, the connection matrix A is identical for cutand virtual integrals. This allows to directly import thecanonical basis constructed in Ref. [80].The complete list of master integrals that survive onthe triple cut relevant for the radiated momentum is il-lustrated in Fig. 3. The cubic ladder-diagram with fourmatter propagators is notably missing from the masterintegrals, but will be present in the calculation of theradiation reaction on the matter lines [91]. The defini-tion of cut integrals involving raised propagator powersis well established (see e.g. Ref. [92]); one practical def-inition is that integration-by-parts reduction can alwaysreduce such integrals to integrals without raised propa-gator powers where the meaning of cuts is clear.All but the fourth master integral have a single s-channel Cutkosky cut (the triple cut shown in the Fig. 3) FIG. 3. Master integrals relevant for the radiated momentum.The dashed line indicates the cut; double lines, cut or uncut,are linearized propagators, and a dot ( • ) indicates a squaredpropagator, corresponding to the n =2 case of Eq. (5). so that the phase-space integral is simply equal to twicethe imaginary part of the virtual integrals (without cut);obtained in Refs. [91, 93] by solving differential equa-tions found in Ref. [80] but with boundary conditionsevaluated in the soft rather than potential region. Thefourth cut integral in Fig. 3, however, has to be calcu-lated directly using reverse unitarity and the followingdifferential equation [94]dd x − x x = 1 x , (7)where we changed variables y ≡ (1+ x ) / (2 x ) , Using thetools described above we have computed the radiatedmomentum at O ( G ) from the real kernel I µR,r in Eq. (3)and the subsequent Fourier transform to impact param-eter space outlined in Eq. (2) with the following result∆ R µ = G m m | b | u µ + u µ σ + 1 E ( σ ) + O ( G ) , (9)where u µi = p µi /m i , we restrict to D = 4, and define E ( σ ) π = f + f log (cid:18) σ +12 (cid:19) + f σ arcsinh (cid:113) σ − √ σ − , (10) f = 210 σ − σ +339 σ − σ +3148 σ − σ +115148 ( σ − / ,f = − σ + 60 σ − σ + 76 σ − √ σ − , (11) f = (cid:0) σ − (cid:1) (cid:0) σ − σ + 11 (cid:1) σ − / . Eq. (9) has the expected homogeneous mass dependence,which, as pointed out in Refs. [67, 95], implies that theresult is fixed by the probe limit m (cid:28) m . Note thatthe result in Eq. (9) is purely longitudinal and yields theenergy radiated as gravitational waves. In the center-of-mass frame it is given by∆ E hyp = ( p + p ) · ∆ R | p + p | = G M ν | b | h ( ν, σ ) E ( σ )+ O ( G ) . (12)We define h ( ν, σ ) ≡ (cid:112) ν ( σ − ν ≡ m m /M , and the total mass M ≡ ( m + m ).The contribution to the above energy loss from each blackhole is inversely proportional to its mass. From the scat-tering (hyperbolic motion) result in the c.m. frame ofEq. (12), one can obtain the energy loss for an ellipticorbit via analytic continuation [65–67]∆ E ell ( σ, J ) = ∆ E hyp ( σ, J ) − ∆ E hyp ( σ, − J ) , (13)which requires writing the energy loss in terms of theangular momentum J = b M ν √ σ − /h ( σ, ν ) and ana-lytically continuing the result from the physical region σ > σ < 1, where σ is relatedto the dimensionless binding energy E = h ( ν,σ ) − ν < E ell ( σ, J ) = G M ν (1 − σ ) J h ( ν, σ ) (cid:101) E ell ( σ ) + O ( G ) . (14)We define an analogous rescaled function (cid:101) E ell ( σ ) (cid:101) E ell ( σ ) π = (cid:101) f − (cid:101) f log (cid:18) σ +12 (cid:19) + (cid:101) f σ arcsin (cid:113) − σ √ − σ , (15)with (cid:101) f i = 2 f i , and f i given in Eq. (11) subject to theadditional replacement ( σ − n → (1 − σ ) n for odd in-tegers n . Note that the elliptic orbit energy loss presentedin Eq. (14) has the expected simplified ν dependence ob-served by [67] that is inherited from the analytic contin-uation of the hyperbolic result. Cross-checks. Our result for the energy loss for scat-tering black holes can be expanded in small velocity v = √ σ − /σ E ( σ ) π = 3715 v + 2393840 v + 6170310080 v + 3131839354816 v + O ( v ) , (16)and compared to known post-Newtonian (PN) data. Thefirst three terms in Eq. (16) are found to agree with theresult known up to 2PN [67, 95, 96]. We can also compare the energy loss for elliptic orbits in Eq. (14) for small ve-locities to the 3PN accurate results for the instantaneousenergy flux integrated over an orbit from Refs. [96–104]in the large eccentricity limit, i.e. to leading order in large J . The velocity expansion is equivalent to Eq. (16) (upto a factor of 2) since it is controlled by the same ana-lytic function and we find perfect agreement where ourresults overlap. Note that the full 3PN flux includes tail(or hereditary) contributions at 1.5, 2.5 and 3PN orderwhich are known analytically only in the limit of smalleccentricity [105–107]. The agreement of our result withthe instantaneous part suggests that the tail contribu-tions must be sub-leading at large eccentricity.Going back to the hyperbolic orbit, instead of the low-velocity result, we can also compare the ultra-relativisticlimit σ → ∞ of Eq. (10). The apparent logarithmicdivergence cancels and one finds E ( σ ) = 358 π (1 + 2 log 2) σ + O (cid:0) σ (cid:1) . (17)This can be compared to the prediction by Kovacs andThorne [95], based upon the numerical probe calculationby Peters [108]. Both expressions agree structurally, butdisagree in the numerical coefficient. We note that ourresults are only valid for σ (cid:28) ( GE cm /b ) − , beyond whichperturbation theory breaks down. This is evidentiated bythe fact that using Eq. (17) one would conclude that atlarge enough σ the radiated energy exceeds the incomingenergy, which is nonsense. This can be interpreted assignaling the necessity of accounting for destructive in-terference in multi-graviton emission, which cuts off thespectrum of gravitational waves at high-frequency [109],as explained in Refs. [110–112].In addition, we have compared our result in Eq. (12)with the coefficient of the tail term in the O ( G ) radialaction of Ref. [28], which is proportional to ∆ E [67, 113,114], finding full agreement. (Non)-Universality. At O ( G ) it has been shown thatthe gravitational deflection angle has universal proper-ties in the ultra-relativistic limit [52, 55, 56, 115]. Wehave also computed the radiated momentum in N = 8supergravity [116] (for BPS angle φ = π/ f = 8 σ ( σ − / , f = − σ √ σ − , f = 16 σ ( σ − σ − / . (18)As in pure gravity, the ultra-relativistic limit of the ra-diated momentum is controlled by the combinations f and − f + f / φ ).Although the limit does not coincide with Eq. (17) in itsrational prefactor (35/8 vs. 8), we note that the ratio ofthe logarithmic (log 2) and non-logarithmic contributionsappears universal, i.e.,lim σ →∞ − f + f f (cid:12)(cid:12)(cid:12) N =8 = lim σ →∞ − f + f f (cid:12)(cid:12)(cid:12) GR = 2 . (19) Conclusions. In this letter we report our computationof the radiated energy emitted in gravitational waves dur-ing the scattering of two spinless black holes in generalrelativity to leading order in Newton’s constant and allorders in velocity. Furthermore, we obtain the radiatedenergy in elliptic orbits by analytic continuation from thescattering problem [67]. Expanding our results in smallvelocity we find perfect agreement with the known PNdata [95, 108]. In the high-energy regime, we agree withthe kinematic dependence described in Ref. [95] but dis-agree with their numerical coefficient.Besides the radiated momentum discussed here, theKMOC formalism can also be used to calculate the trans-verse impulse on individual particles, which yields the de-flection angle [18, 20, 22, 23], including radiation reaction[56, 58, 115]. This computation is more involved than theone presented here, because it requires the full virtualsoft amplitude, so we defer its discussion. The tools de-scribed here can be directly used to compute observablesfor spinning [118], and charged [119] black holes. Fur-thermore, in retaining the collider-physics analogy, andby restricting the integration over phase-space, one can imagine computing differential observables, such as theradiated energy spectrum, analogous to e.g. rapidity dis-tributions (see Ref. [82]). We leave the discussion of suchobservables to future work. Acknowledgments: We thank Zvi Bern, Chia-HsienShen, Radu Roiban, and Mikhail Solon for helpful com-ments and collaboration on related projects, and Clif-ford Cheung and Rafael Porto for discussions and com-ments on the draft. We are also grateful to Paolo DiVecchia, Carlo Heissenberg, Rodolfo Russo, and GabrieleVeneziano for stimulating discussions regarding radiativeeffects and cross-checks of various soft master integrals.E.H. thanks Lance Dixon and Bernhard Mistlberger fordiscussions about the reverse unitarity method. E.H.is supported by the U.S. Department of Energy (DOE)under Award Number DE-SC0009937. J.P.-M. is sup-ported by the U.S. Department of Energy (DOE) un-der Award Number DE-SC0011632. M.S.R.’s work isfunded by the German Research Foundation (DFG)within the Research Training Group GRK 2044. M.Z.’swork is funded by the U.K. Royal Society through GrantURF \ R1 \ [1] Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower,Nucl. Phys. 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