Scattering of Spinning Black Holes from Exponentiated Soft Factors

Scattering of Spinning Black Holes from Exponentiated Soft Factors

Alfredo Guevara, Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, CanadaDepartment of Physics Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, CanadaCECs Valdivia & Departamento de Física, Universidad de Concepción, Casilla 160-C,
Concepción, Chile
   Alexander Ochirov, Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, CanadaDepartment of Physics Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, CanadaCECs Valdivia & Departamento de Física, Universidad de Concepción, Casilla 160-C,
Concepción, Chile
   and Justin Vines,, Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, CanadaDepartment of Physics Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, CanadaCECs Valdivia & Departamento de Física, Universidad de Concepción, Casilla 160-C,
Concepción, Chile

We provide evidence that the classical scattering of two spinning black holes is controlled by the soft expansion of exchanged gravitons. We show how an exponentiation of Cachazo-Strominger soft factors, acting on massive higher-spin amplitudes, can be used to find spin contributions to the aligned-spin scattering angle through one-loop order. The extraction of the classical limit is accomplished via the on-shell leading-singularity method and using massive spinor-helicity variables. The three-point amplitude for arbitrary-spin massive particles minimally coupled to gravity is expressed in an exponential form, and in the infinite-spin limit it matches the stress-energy tensor of the linearized Kerr solution. A four-point gravitational Compton amplitude is obtained from an extrapolated soft theorem, equivalent to gluing two exponential three-point amplitudes, and becomes itself an exponential operator. The construction uses these amplitudes to: 1) recover the known tree-level scattering angle at all orders in spin, 2) match previous computations of the one-loop scattering angle up to quadratic order in spin, 3) lead to new one-loop results through quartic order in spin. These connections map the computation of higher-multipole interactions into the study of deeper orders in the soft expansion.


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ETH Zürich, Institut für Theoretische Physik, Wolfgang-Pauli-Str. 27, 8093 Zürich, SwitzerlandMax Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam 14476, Germany

1 Introduction

In 2014 Cachazo and Strominger Cachazo:2014fwa () showed that the following universal relation holds for tree-level gravitational amplitudes in the soft limit


Here the soft momentum corresponds to an external graviton, and we have constructed its polarization tensor as . The sum is over the remaining external particles with momenta , and the operators correspond to their total angular momenta, whereas the first term in the equation is simply the standard Weinberg soft factor Weinberg:1965nx (). The realization Strominger:2013jfa () that soft theorems correspond to Ward identities for asymptotic symmetries at null infinity has led to many impressive developments He:2014laa (); Cachazo:2014fwa (); Cachazo:2014dia (); Kapec:2014opa (); Bern:2014vva (); Dumitrescu:2015fej (); Campiglia:2016hvg () in that area, see Strominger:2017zoo () for a recent review. Extensions of these relations to arbitrary subleading orders are known Campiglia:2018dyi (); Hamada:2018vrw (); Li:2018gnc (); Bern:2014vva () but are not universal and depend both on the matter content and the type of couplings considered Laddha:2017ygw (); Sen:2017xjn ().

In particular, a classical version of the soft theorem up to subsubleading order in has been used by Laddha and Sen Laddha:2018rle () to derive the spectrum of the radiated power in black-hole scattering with external soft graviton insertions. This relies on the remarkable fact that conservative and non-conservative effects of interacting black holes can be computed from scattering amplitudes for massive point-like particles Duff:1973zz (); BjerrumBohr:2002ks (); Neill:2013wsa (); Bjerrum-Bohr:2018xdl (). Moreover, rotating black holes admit a spin-multipole expansion in their effective potential, the order of which can be reproduced by scattering spin- minimally coupled particles exchanging gravitons Vaidya:2014kza (), as illustrated in figure 0(a).

Figure 1: (a) Four-point amplitude involving the exchange of soft gravitons, which leads to classical observables. The external massive states are interpreted as two black-hole sources.
(b) Comparison between the HCL and the non-relativistic limit in the COM frame Holstein:2008sx (); Holstein:2008sw (); Vaidya:2014kza (). Spin effects require subleading orders in the NR classical limit, but can be fully determined at the leading order in HCL through the soft expansion.

Here we present a complementary picture to the one of Laddha:2018rle () for the conservative sector (i.e. with no external gravitons) focusing on spinning black holes. It was shown by one of the authors in Guevara:2017csg () that the classical (-independent) piece of the spin- amplitude can be extracted from a covariant Holomorphic Classical Limit (HCL), which set the external kinematics such that the momentum transfer between the massive sources is null. On the support of the leading-singularity (LS) construction Cachazo:2017jef (), which drops parts, the condition reduces the amplitude to a purely classical expansion in spin multipoles of the form , where carries the intrinsic angular momentum of the black hole (see figure 0(b)). This precisely matches the soft expansion once the momentum transfer is recognized as the graviton momentum and the classical spin vector is identified with the angular momentum of the matter particles. On the classical side, these amplitudes have been shown to reproduce the effective post-Newtonian (PN) potential associated to the collision of two rotating black holes Vaidya:2014kza (); Holstein:2008sx (); Guevara:2017csg ().

To see the soft expansion more explicitly, consider the energy-momentum tensor of a single linearized Kerr black hole, which has recently been written down in an exponential form by one of the authors Vines:2017hyw ():


where , and is the rescaled spin vector of the black hole. The magnitude is exactly the radius of its ring singularity. Here we have performed a Fourier transform of the worldline formulas (18) and (32a) of Vines:2017hyw (). Now, the interaction vertex between a graviton and a massive source corresponds to the contraction . After taking the graviton to be on-shell and replacing by , this becomes


where we have used the support of the delta functions. This expression can be written in a simple form by introducing the spin tensor


satisfying , after which it becomes


The term inside the parentheses is precisely the exponential completion of the expansion in eq. (1). Note that the prefactor corresponds to the contribution of the energy-momentum tensor of the linearized Schwarzschild solution Damour:2016gwp ().

Even though the fact that classical gravitational quantities can be reproduced from QFT computations has been known for a long time, the precise conceptual foundations of the matching are still lacking.111Very recent progress on relating classical observables to quantum amplitudes has been made in Kosower:2018adc (). The goal of one of the authors in Guevara:2017csg () was simply to show the agreement of the LS method with the previous computations of Holstein:2008sw (); Holstein:2008sx (); Vaidya:2014kza (). Moreover, in Guevara:2017csg () the new massive spinor-helicity variables of Arkani-Hamed, Huang and Huang Arkani-Hamed:2017jhn () were implemented to construct operators carrying spin multipoles. These operators were then matched, trough a change of basis, to those constructed in Holstein:2008sw (); Holstein:2008sx (); Vaidya:2014kza () in terms of polarization vectors and Dirac spinors, enabling a systematic translation between the LS and the standard QFT amplitude in the limit. It is only after computing the effective potential from this amplitude that one matches the post-Newtonian potential of general relativity.

The computation of the classical piece of the amplitude was made direct, through the leading singularity, for arbitrary spin and all orders in the center-of-mass energy . Both the tree-level and one-loop versions of this computation correspond to a single order in the post-Minkowskian (PM) expansion (see e.g. recent discussion in Damour:2016gwp (); Bini:2017xzy (); Vines:2017hyw (); Damour:2017zjx (); Bini:2018ywr (); Bjerrum-Bohr:2018xdl (); Cheung:2018wkq (); Vines:2018gqi () and many more references therein), i.e. at a fixed power of . However, the explicit match to the standard QFT amplitude was only performed up to spin-1 and leading order in (which corresponds to the standard PN expansion). Moreover, the computation of the PN effective potential through the Born approximation suffers some problems Holstein:2008sx (); Neill:2013wsa (). Such potential is not gauge-invariant, i.e. not an observable, and can undergo canonical and non-canonical transformations that become cumbersome when spin is considered as part of the phase space. Moreover, at one loop the Born approximation itself requires the subtraction of tree-level pieces and suffers from some (apparent) inconsistencies already at spin-1 Holstein:2008sw (). For these reasons a more direct conversion from the LS into a gravitational observable is evidently needed. Very recently, a direct approach was proposed in the amplitudes setup to evaluate the scattering angle of classical general relativity Bjerrum-Bohr:2018xdl (), i.e. the deflection angle of two massive particles in the large-impact-parameter regime. It was demonstrated that for scalar particles the scattering angle computed by Westphal Westpfahl:1985 () can be obtained via a simple 2D Fourier transform of the classical limit of the amplitude.

Here we will show that the natural extension of the scattering angle, for aligned spins as in Vines:2017hyw (); Vines:2018gqi (), can be computed with spinning particles directly from the LS. The building blocks needed for this computation are the three-point amplitude and the Compton amplitude for massive spinning particles interacting with soft gravitons. We will use the soft expansion with respect to the internal gravitons to write the building blocks in an exponentiated form, which fits naturally into the Fourier transform leading to the first and second post-Minkowskian (1PM and 2PM) scattering angles in a resummed form.

Summary of Results

In section 2.2 we show that the three-point scattering amplitude between two massive particles of spin and one graviton is given by


where the exponential operator is generated by the angular momentum , as appearing in the soft theorem (1). This operator acts naturally on the product states and , which are constructed from the new spinor-helicity variables introduced by Arkani-Hamed, Huang and Huang Arkani-Hamed:2017jhn (). Denoting the operator by we write this as


where corresponds to the amplitude for a massive scalar emitting a graviton. In section 2.3 we extend this result to the distinct-helicity Compton amplitude, showing that,


up to corrections of fifth order in (appearing only for ). In the operator form, and can be chosen from either particle three or four, which simply amounts to a change of basis. The soft theorem (1) in this case is extrapolated in an exponential form, and corresponds to the simple statement of factorization of the Compton amplitudes into three-point amplitudes given by eq. (7) and its plus-helicity version.

The formulas (7) and (8) are the two bulding blocks needed to compute the scattering angle. In order to recover the classical observables we introduce and compute the generalized expectation value (GEV)


Here we focus on integer-spin particles for simplicity, therefore we use polarization tensors for spin-. We first show that, with ,


where on the LHS is the linearized stress-energy tensor of the Kerr black hole (5). We then construct the aligned-spin scattering angle as in Kabat:1992tb (); Akhoury:2013yua (); Bjerrum-Bohr:2018xdl (),


(see section 3.2 for definitions of the prefactors). Here corresponds to the 4-pt amplitude of figure 0(a), with masses and and spins and . We compute this amplitude at both tree and one-loop levels using the LS proposed in Guevara:2017csg (). The Fourier transform can be performed using the exponential forms (7)-(8). We find the following expression for the aligned-spin scattering angle as a function of the masses and , the rescaled spins (ring-radii, intrinsic angular momenta per mass) and , the relative velocity at infinity , and the (proper) impact parameter ,

where with , and

This agrees with previous classical computations for two spinning black holes performed up to spin-squared order in Bini:2018ywr (); Vines:2018gqi (), and resums those contributions in a compact form, including higher orders. We have indicated in (12c) that this expression is valid up to quartic order in one of the spins (but to all orders in the other spin), according to the minimally coupled higher-spin amplitudes.

2 Multipole expansion of three- and four-point amplitudes

2.1 Massive spin-1 matter

We start our discussion of the multipole expansion by dissecting the case of graviton emission by two massive vector fields. The corresponding three-particle amplitude reads222We omit the constant-coupling prefactors in front of tree-level amplitudes, we use .


where is the average momentum of the spin-1 particle before and after the graviton emission and the polarization tensor of the graviton (with momentum ) is split into two massless polarization vectors. The derivation of eq. (13) from the Proca action is detailed in appendix A, which also motivates that the term involving can be thought of as an angular-momentum contribution to the scattering. In other words, we are tempted to interpret the combination as being (proportional to) the classical spin tensor.

However, we now face our first challenge: as explained in Holstein:2008sw (); Holstein:2008sx (); Vaidya:2014kza (), the spin-1 amplitude contains up to quadrupole interactions, i.e. quadratic in spin, whereas only the linear piece is apparent in eq. (13). To rewrite this contribution in terms of multipoles, we can use a redefined spin tensor


It is introduced in appendix B via a two-particle expectation value/matrix element, which we call the generalized expectation value (GEV)


Here is constructed as an angular-momentum operator shifted in such a way that its GEV satisfies the Fokker-Tulczyjew covariant spin supplementary condition (SSC) Fokker:1929 (); Tulczyjew:1959 ()


In this paper we find this condition to be crucial for the matching to the rotating-black-hole computation of Vines:2017hyw (), as the classical spin tensor  (4) satisfies the above SSC by definition. The purpose of this SSC is to constrain the mass-dipole components  of the spin tensor of an object to vanish in its rest frame. In a classical setting it puts the reference point for the intrinsic spin of an spatially extended object at its rest-frame center of mass.

Inserting this spin tensor in eq. (17), we rewrite the above amplitude as


where for further convenience we also expressed scalar products by a helicity variable first introduced in ArkaniHamed:2008gz ()


Now, in the GEV of the amplitude,


we recognize the dipole coupling of eq. (5) as the term linear in both and . Indeed, particles with spin couple naturally to the field-strength tensor of the graviton , analogously to the magnetic dipole moment .333We thank Yu-tin Huang for emphasizing to us the analogy to the electromagnetic Zeeman coupling. Following the non-relativistic limit, the third term was identified in Holstein:2008sw (); Holstein:2008sx (); Vaidya:2014kza (); Guevara:2017csg () to be the quadrupole interaction for spin-1. It may seem a priori puzzling that the interaction is regarded as the square of . This is because the statement is true at the levels of spin operators, but not at the level of the (generalized) expectation values, i.e. . In order to expose the exponential structure described in the introduction and construct such spin operators at any order, we are going to recast the multipole expansion in terms of spinor-helicity variables.

2.1.1 Spinor-helicity recap

This subsection can be skipped if the reader is familiar with the massive spinor-helicity formalism of Arkani-Hamed, Huang and Huang Arkani-Hamed:2017jhn (),444The spinor-helicity conventions used in the present paper are detailed in the latest arXiv version of Ochirov:2018uyq (). which is well suited to describe scattering amplitudes for massive particles with spin. Much like its massless counterpart, this formalism allows to construct all of the scattering kinematics from basic spinors that transform covariantly with respect to the little group of the associated particle. The massive little group is , so the Pauli-matrix map from two-spinors to momenta


involves a contraction of the indices (not to be confused with the spinorial indices and ). This is in contrast to the massless case, where the little group is , so its index is naturally hidden inside the complex nature of massless two-spinors


Now just as and are convenient to built massless polarization vectors (23), we can use the massive spinors and to construct spin- external wavefunctions. For instance, massive polarization vectors are explicitly


where the symmetrized little-group indices represent the physical spin-projection numbers with respect to a spin quantization axis, as chosen by the massive spinor basis. Note that the dotted and undotted spinor indices themselves must always be contracted and do not represent a quantum number.

Let us also point out here that the massless polarization vectors and hence the associated helicity variable (18) can be written in terms of massless spinors as


where is independent of the reference momentum on the three-point on-shell kinematics.

2.1.2 Spin-1 amplitude in spinor-helicity variables

We can now obtain concrete spinor-helicity expressions for the amplitude (13). Choosing he polarization of the graviton to be negative, we have


where we have reduced all and the chiral spinor basis of and using the following identities for the three-point kinematics,555The transition between the chiral spinors and the antichiral ones is always possible Arkani-Hamed:2017jhn () via the Dirac equations and .


We also use for henceforth, i.e. it carries helicity unless stated otherwise. From eq. (24) we can see that going to the chiral spinor basis has both an advantage and a disadvantage. On the one hand, the multipole expansion becomes transparent in the sense that the spin order of a term is identified by the leading power of . On the other hand, the exponential structure of the vector basis is spoiled by a shift by higher multipole terms. However, this is just an artifact of the chiral basis, and we should see that the answer obtained from the generalized expectation value is the same.

The main advantage of the spinor-helicity variables for what we wish to achieve in this paper is that now we can switch to spinor tensors and , as representations of the massive-particle states 1 and 2. Introducing the symbol for the symmetrized tensor product, we can rewrite eq. (24a) as


Here the operators have their lower indices symmetrized, i.e. , and the notation assumes that the reader keeps in mind the spins associated with each momentum. Combining all the terms in eq. (24) into the amplitude, we obtain


Now in the multipole expansion of the Kerr stress-energy tensor (5), the quadrupole operator is of the simple form , whereas in our amplitude (17) it has the form . One then could wonder if in some sense the latter is the square of . We know show that this is precisely the case if the angular momentum is realized as a differential operator.

In appendix C we construct the differential form of the angular-momentum operator in momentum space starting from its definition


which involves the standard orbital piece and the “intrinsic” contribution dependent on spin. This operator admits a much simpler realization in terms of spinor variables, similar to the one derived in Witten:2003nn () for the massless case. For a massive particle of momentum we find that the differential operator for the total angular momentum is given by


We can now act with the operator on the product state . For the negative helicity of the graviton, we have


Applying the spinor differential operator above we find666More explicitly, we have with similar manipulations for higher powers.


Although it is the differential operator that is realized by the soft theorem, its algebraic form is very easy to obtain on three-particle kinematics. Indeed, if we take a tensor-product version of the standard chiral generator and use it as an algebraic realization of , it is direct to check that it acts in the same way as the differential operator above:


These identities allow us to reinterpret the last two terms in the amplitude formula (27) as the non-zero powers of this dipole operator acting on the on the state :


and rewrite the amplitude as


It is now clear that these terms are (1) precisely the differential operators of the soft expansion (1) and (2) the scalar, spin dipole and quadrupole interactions in the expansion of the Kerr energy momentum tensor (5). In this way, we interpret the three terms in the amplitude (27) as the multipole contributions with respect to the chiral spinor basis, despite the fact that they do not equal the multipoles in eq. (17) individually. Furthermore, as the operator annihilates the spin-1 state for , the three terms can be obtained from an exponential


It can be checked explicitly that acting with the operator on the state yields the same result, i.e. in this sense the operator is Hermitian. On the other hand, choosing the other helicity of the graviton will yield the parity conjugated version of eq. (35), where the parity-odd terms in the exponential switch sign, that is


In the next section we extend this procedure to arbitrary spin. Let us point out that the explicit amplitude can be brought into a compact form by changing the spinor basis. In fact, the three-point identities (25) imply that the amplitude formula (27) collapses into


However, let us stress that this form completely hides the spin structure that was already explicit in the vector form (17). The purpose of the insertion of the differential operators is precisely to extract the spin-dependent pieces from the minimal coupling (37), which will then be matched to the Kerr black hole.

2.2 Exponential form of three-particle amplitude

In this section we generalize the previous discussion to arbitrary spin. The starting point in this case is the three-point amplitudes for massive matter minimally coupled to gravity in the little-group sense Arkani-Hamed:2017jhn ():


As explained in the previous section, in such a compact form all the dependence on the spin tensor is completely hidden. In order to restore it, we need to write the minus-helicity amplitude in the chiral basis


Here we have taken advantage of the symmetrized tensor product that enables us to perform the binomial expansion (we have omitted the identity factors in the tensor product). Even though this already corresponds to an expansion in the “spin operator” of Guevara:2017csg (), here we recast this into exponential form by inserting the differential angular momentum operator


Indeed, it is easy to generalize the formulae (31) to product states of spin-, namely


In other words, in general the operator (40) is nilpotent of order .777Interestingly, due to its property (41) the spinorial differential operator (40) can be regarded as a ladder operator for a spin- representation. Of course, this also admits an algebraic realization, which is the trivial extension of the formula (32). From this we can derive the formal relations888For , eq. (42) corresponds to the operator used in Guevara:2017csg () to perform the matching with the standard QFT amplitude. We note, however, that the classical quantity matches the quantity used in Guevara:2017csg () only when the spin tensor satisfies the SSC (16), as can be seen by squaring both terms.


Therefore, we can rewrite eq. (39) as an exponential


where we note that the exponential expansion, albeit valid to all orders, becomes trivial at order . It can be read from eq. (39) that the spin operator of Guevara:2017csg () corresponds precisely to . Moreover, in the formal limit the exponential can be realized as a linear operator that does not truncate! However, let us stress that even for finite spins the exponential operator in


is still present and can be mapped to classical observables such as the scattering angle. This framework will be particularly useful at order , since the arbitrary spin version (and hence the limit) of the Compton amplitude is not yet known.

The transition to the positive helicity should amount to exchanging angle brackets with square brackets. However, this procedure maps the massless polarization vectors to minus each other (see eq. (128)), while the field-strength-like combination


does not have a relative minus sign between the helicities. The combination thus develops an additional minus sign upon a helicity flip, as in eqs. (35) and (36):


The form (44) makes explicit the fact that the higher-spin amplitude is non-local Arkani-Hamed:2017jhn (). However, despite the appearance of the factor in the denominator, the exponential factor is gauge-invariant due to the three-particle kinematics. We further recognize in the argument of the exponential the same structure as the one appearing in the Cachazo-Strominger soft theorem. In fact, as will be made explicit in the next section, the extended soft factor of Cachazo and Strominger is just an instance of a three-point amplitude of higher-spin particles. The poles present in the extended soft factor (1) simply arise when gluing these three-point amplitudes.

The formula (44) is our first main result. Note that this holds for the full three-point amplitude with no classical limit whatsoever. This formula matches precisely the Kerr energy-momentum tensor (5), with corresponding to the scalar piece (the Schwarsczhild case). In section 3 we will use this compact form to compute the scattering angle of two Kerr black holes at linear order in .

2.3 Exponential form of gravitational Compton amplitude

The task of this section is to extend the construction presented in the previous one to the Compton amplitude, without the support of three particle kinematics. In particular, we will show that for the distinct-helicity amplitude the following holds


Here the momentum  and the polarization vector  in the exponential operator can be associated to either of the two gravitons. The only difference comes from the choice of the spinor basis. Explicitly, we have


Analogously, can be associated to either of the massive particles. As we explain later, the polarization vectors may be chosen such that , so the denominator is also universal.

The importance of this amplitude (as opposed to the same-helicity case) is that it controls the classical contribution at order , as was shown directly in Guevara:2017csg (); Bjerrum-Bohr:2018xdl (). In Guevara:2017csg () the classical piece was argued to lead to the correct 2PN potential after a Fourier transform. In the new approach of Bjerrum-Bohr:2018xdl () the classical contribution in the spinless case was identified by computing the scattering angle. In section 3 we will use the Compton amplitude as an input for computing the scattering angle with spin up to order , agreeing with previously known results at order . We will see that this exponential form is extremely suitable for the computation of the latter as a Fourier transform.

Our strategy is the following: we first consider the action of the exponentiated soft factor acting on the three-point amplitude, as an all order extension of the Cachazo-Strominger soft theorem. We have checked that this agrees with the known versions of the Compton amplitude Arkani-Hamed:2017jhn (); Bjerrum-Bohr:2017dxw (), at least for . We leave the problem of for future investigation, but we will comment on its origin at the end of this section.

The proof of eq. (47) starts by considering an all-order extension of the soft expansion (1) with respect to the graviton :


As stated in the introduction, two main problems arise when trying to interpret eq. (1) as an exponential acting on the lower-point amplitude. The first is that gauge invariance of the denominator is not guaranteed. Here we simply fix , so the last term in eq. (49) vanishes, as we will show in a moment. The second problem is that one still has to sum over two exponentials, which would spoil the factorization of eq. (47). The solution is that in this case both exponentials give the exact same contribution. In the language of the previous section, this is the fact that one can act with the operator