Classical Spin Gravitational Compton Scattering

We introduce a novel bootstrap method for classical Compton scattering amplitudes involving two massless gluon/graviton particles and two arbitrary-spin infinite-mass particles in a heavy-mass effective field theory context. Using a suitable ansatz, we deduce new and explicit classical spin results for gluon four and five-point infinite mass processes that exhibit exponentiated three-point factorizations to all orders in spin and feature no spurious poles. We discuss the generalization of our bootstrap to higher multiplicities and summarize future potential applications.


Introduction
Observations of gravitational waves from black hole mergers have fast evolved into an exciting avenue for testing Einstein's classical theory.To facilitate this, a modern and efficient computational program inspired by various practical quantum field theory implementations of general relativity, see, for instance, [1][2][3][4][5][6][7], was developed in [8,9] prompted by the state of current theoretical analysis and precision requirements of observations [10].It has made quantum scattering amplitudes for superheavy black holes -taken as point particles -a powerful resource for delivering much-needed analytic precision in fully relativistic post-Minkowskian frameworks and promises new computational efficiency, particularly in analytical regions of the near-merger regimes.
For addressing classical spins, it is customary to operate in the context of a Lagrangian formalism that involves higher spin particles systematized in expansions with increasing multi-pole interactions as pioneered in [69].Unfortunately, the formalism behind this is relatively elaborate, and derived amplitudes usually contain intricate redundancies.In this presentation, we take an alternative route and bootstrap covariant arbitrary-spin amplitudes from factorization limits.
Our main focus here will be classical spin Compton scattering amplitudes, where an infinite mass spinning particle interacts with gluons.As we explain, we can extend such amplitudes into results for gravitational Compton scattering amplitudes utilizing the double-copy [70][71][72].The rationale for mainly considering gravitational Compton scattering amplitudes in the infinite mass limit is that such amplitudes (in the limit where the mass is infinite) are convenient building blocks in generalized unitarity approaches [25-28, 73, 74] that identify the gravitational classical radial action computing from an integrand determined by generalized unitarity and constrained by delta functions [23,24,26].
In the bootstrap method, we use minimal physical input constraints such as the special exponentiated minimally coupled Kerr black hole three-point amplitude [34][35][36]42], and we will introduce a convenient ansatz for the amplitude that avoids spurious poles and is fixed uniquely (up to polynomial contact terms) from imposing factorization identities.We find support for our classical amplitudes by comparing them to results derived at four-point linear and quadratic spin order [50,75].Beyond quadratic spin order, we stress that our formalism requires additional contact deformations to describe Kerr black hole scattering.We will return to this point in the conclusion.
We arrange this paper as follows: in section 2, we will quickly review infinite mass Compton gluon and graviton tree amplitudes to establish a convenient and efficient framework for computation.Then we will introduce our new bootstrap method and demonstrate the calculation of four and five-point classical amplitudes in section 3.An essential inspiration in the bootstrap method is the formalism [76].Finally, we will outline a generalization of the proposed computational technology in section 4 and discuss possible applications and questions for the future in section 5.

Classical Compton amplitudes
The tree amplitudes considered in this paper involve an infinity mass and spin particle with finite classical ring radius a = s/m (s → ∞, m → ∞), which interacts with an arbitrary n − 2 number of gluons in the context of a heavy-mass effective field theory (HEFT) [26,49,[76][77][78][79][80][81].We depict the processes considered in the following way: Here p 1 , . . ., p n−2 denotes momenta of gluons while pn−1 and pn are momenta for the heavy spinning particle.In the heavy mass limit the velocity v is fixed by the on-shell condition v • v = 1 and v • q = 0 which follows from p2 n = p2 n−1 = (mv − q) 2 = m 2 .We define q ≡ (p 1 + . . .+ p n−2 ).

Three-point amplitude
We start with the infinite mass three-point Compton amplitude, which we define as follows (using shorthand notation for momenta) (2. 2) It can conveniently be written in terms of the three-point color-kinematic numerator We will in the following focus on the minimally coupled three-point tree amplitude that takes a closed exponentiated form for arbitrary spin and double copies to a Kerr graviton amplitude as shown in [34][35][36]42] We can rewrite the exponential by separating the even and odd powers in the spin variable and using the identity where the .= indicates that the equality is satisfied when contracted with ε 1 .In the above expressions ε i denotes gluon polarizations and S µν ≡ −mǫ µνρσ v ρ a σ .Since the singularity in the second term of Eq. (2.5) is removable, we can define an analytic extension that is holomorphic everywhere in the complex plane by considering the function Thus we can define which defines a three-point numerator which has the advantage of manifestly featuring no pole terms in a We note that gauge invariance is manifest in Eq. (2.8) from the anti-symmetric property of the spin tensor and the three point on-shell condition v • p 1 = 0, which imply that w 1 • p 1 = 0. We conclude this section by noting that three-point gravity amplitudes can be computed from the double copy using a product of one scalar gluon numerator, N 0 (1, v), and one arbitrary spin numerator factor, N a (1, v) [82], (2.9)

Classical gravity amplitude decomposition in the heavy mass limit
Before we discuss the four-point gravity amplitude, we will review some material related to convenient amplitude decompositions in the heavy mass limit.We will here refer to the discussion in [26] (see also [27]), and start by noting that it is convenient to decompose gravity amplitudes in the following way when considering classical black hole scattering perturbatively: The first term in this diagrammatical expression corresponds to a velocity cut imposed by a δ-function constraint.This contribution is of order O(m 3 ) 1 .The second term, which we will refer to as the classical tree amplitude, is of order O(m 2 ) (O( 0 )).Since the first term is an iteration in the infinite mass limit, the following section focuses solely on computing the second term of (2.10).In Appendix A, we will discuss calculating the first contribution directly when considering a finite heavy mass.
One can follow the same reasoning at five-point, and focus only on the last term of the following expansion: (2.11) 3 Spinning Compton Amplitudes

Four-point Amplitude
We will now consider how to compute the classical four-point infinity-mass amplitude and start by writing the amplitude in terms of color-kinematic massless pole numerator factors (we define the shorthand notation p 12 ≡ p 1 + p 2 ): Here N a (1, 2, v) and N a (2, 1, v) are numerators, and in the last step, we have used a commutator form N a ([1, 2], v) to write the amplitude in shorthand form.The colorkinematic numerator N a ([1, 2], v) has a massive pole in v • p 1 [83], and thus it follows from its factorization behavior that we have which we will rewrite as where This separates the color-kinematic numerator into a − with a pole, and an analytic and finite part Now factorizing the numerator for p2 12 → 0 we see that where we can write the three-point Yang-Mills numerator as, see e.g.[76], We consider an ansatz form to deduce N ′ a ([1, 2], v) at four points.We first introduce the function G 2 (x 1 ; x 2 ) (a generalization of the function G 1 (x 1 )) where one again can prove that all apparent singularities are removable 2 We then work out which types of terms we can have in the ansatz considering the above functions and evaluate the factorization in Eq. (3.4).We find that any ansatz piece has to be of the following universal form (depending on the power in the spin variable), even where we have defined x i ≡ p i • a and x i...j ≡ p i...j • a.We follow the logic that we can only have a product of at most two G-functions since we only have two gluons at four points, and we insist on preserving the scaling in the spin variable of the three-point amplitudes by requiring that factors of a present in the denominators of the G functions cancel out with field strength monomial terms in the ansatz terms.Using the property of symmetry under parity (a → −a) of the G-functions, it is furthermore easy to infer that only products G 1 (x 1 )G 1 (x 2 ) are possible for the even part.For the odd part, one can have either G 1 (x 12 ) or G 2 (x 1 ; x 2 ) with a factor of S or ... • S • X • a, respectively, with X denoting a field strength tensor product.
Thus, we consider all the 14 possible independent monomials in the ansatz and bootstrap its form using the fact that all massless factorizations are consistent.
Thereby we entirely fix the factorization behavior in Eq. (3.4), which leads to the expression: To fix the factorization, we sum over gluon states using Although the form Eq. (3.8) is it is not manifestly anti-symmetric, one can straightforwardly check its anti-symmetry using the relation: Using the double copy relations, it follows that we have the classical spinning gravitational Compton amplitude: where: As mentioned in the introduction, the expression for the Compton amplitude in gravity matches the recent literature [50,68,75] up to quadratic order in spin.

Five-point Compton scattering
We will now consider the five-point Compton amplitude.We can write this in the form again expressed in terms of color-kinematic numerators solely organized in terms of the massless poles.Compared to the four-point case, we now have three massive poles in the color-kinematic numerators and thus arrive at the following factorization behavior: Again as we did for the four-point case, we use factorization to recast the expression into one where we can identify the finite piece In this case, we find that it can be composed of the following building blocks (extending the logic from the four-point case) where {i, j, k} are elements in the possible permutations of the set {1, 2, 3} and As we discussed in the four-point case, we have some redundancy since the following type of identities hold The ansatz for the finite piece at five points has 981 independent (non-redundant) terms.Again we fix it completely by imposing the massless cut conditions, which leads to the following unique solution: where the spin-even part is and the odd spin part is and we define F µν ij ≡ F µσ i F ν jσ .The appearance of cosh(x i ) arises from the linear combination of the G-functions depicted in Eq. (4.2).We have checked that the colorkinematic numerator satisfies the following manifest (see refs.[84,85]) crossing symmetry condition: where the operator I is an identity operation and P (τ ) denotes a cyclic permutation of gluon labels, i.e., P (321) f (1, 2, 3) ≡ f (3, 1, 2).
Finally, we note that the classical spinning gravitational Compton amplitude can be obtained from the double copy: where the five-point numerator for the scalar case is [76]: 4 Building blocks for an all-multiplicity amplitude ansatz Considering the four and five-point numerators, we will now make some general remarks about extensions of the formalism to all-multiplicity Compton amplitudes for massive classical spinning particles.Starting with amplitude representations with numerators organized in massless poles similar to the lower point examples, we can again decompose the color-kinematic numerators into pieces with massive poles and a finite term.We will focus mainly on bootstrapping the finite contribution.Similar to the lower point demonstrations, one can set up a convenient ansatz based on a generalization of the G 1 , G 2 , and G 3 functions.One can prove in all generality that the function Here τ is the {2, • • • , r} and we have a summation over all the partitions of τ into subsets τ a and τ b , including the empty set.It is easy to show that the following generic recursive relation holds These functions also obey the following surprising relations: The next step is to constrain the form of the finite term for the numerator.Again as in the lower point cases, to preserve the scaling in the spin variable of the threepoint amplitudes, we require that the factors of a present in the denominators of the G functions cancel out with the elements in the monomials.In particular, if we define the degree of an object, e.g.
by its inverse scale order, (i.e., the number of factors of 1/λ when taking a → λa and λ → i∞).Thus, for a general number of points, the following general ansatz for contributions is conjectured: where M (r) denotes a product of functions G i (x) such that the total degree is r, and P (r) is a product of monomials of the arbitrary scalar products p • Z • a with total degree −r (X, Y, Z all denote products of field strength tensors).The building blocks Eq. (4.5) are all the possible combinations that respect the scaling in a and that include the correct factors of v.These terms can be further constrained by considering parity requirements, so that for the even(odd) part, the ansatz should be even(odd) under the transformation a → −a.
After constructing the ansatz, the free parameters are fixed by the massless cut conditions: The four and five points results indicate that our bootstrap procedure leads to a universal arbitrary-spin infinite-mass amplitude determined solely from exponentiated threepoint amplitudes and factorization constraints.We expect this to be the case at any multiplicity.

Conclusion
This paper discusses a new systematic bootstrap procedure for the computation of the leading in mass classical Compton scattering of arbitrary spinning infinity mass particles and gluons/gravitons that avoid spurious poles and agrees with linear and quadratic results published in [50,75].We have demonstrated this procedure by deducing new explicit classical results for four and five-point processes and discussed all multiplicity generalizations.
Our results for four-and five-point scattering are suitable for concrete computations beyond this paper's scope; in particular, it opens an avenue for efficient calculation of the gravitational attraction of spinning black holes.For instance, by considering the following type of diagrammatic contributions illustrated below: we can compute non-spinning observables at the tree and one-loop post-Minkowskian order, see e.g.[26,76,83] for examples using heavy-mass effective field theory or [27,28] using an analogous diagrammatic formulation in terms of velocity cuts.
Preliminary work on generalizations of the formalism presented suggests matching results beyond quadratic order (including Kerr, with the introduction of appropriate contact deformations) is feasible.It is beyond the scope of this paper, and we plan to return to it in a forthcoming publication.
Another question is working out a connection between the numerators explored with three legs fixed and the formalism with numerators with two legs fixed used discussed in [86,87] using momentum kernel identities that link the two formalisms [88].While we have only computed the leading in mass part of the Compton amplitude, it is interesting to extend the results of this paper to the total amplitude, considering that the spinning particle has finite mass.To handle such amplitudes, one has to go beyond the simple factorization procedure regarded in the infinite mass limit.The reason is for high spin fields, where the mass is finite, there are different non-trivial effects from the complete spin propagator, which is a consequence of the massless particles interacting with the spinning particle and thereby altering its angular momentum.We will discuss this question further in appendix A.
In order to see that this amplitude has the correct factorization behaviour on the massive cut, we first take the limit p 4 • p 1 → 0 in Eq. (A.13), and only then perform an expansion in 1/m, obtaining the following result: M a (1, 2, 3, 4) This is consistent with our reasoning in section 2.2 since, taking into account the regulators iε, the decomposition of the amplitude contains a term of order O(m 3 ) where the massive cut is imposed by a δ-function.For example, the bottom graph in Eq. (5.1) is deduced from (A.17)

B Removable singularities in the G functions
We will here present the proof that the G n (x 1 ; x 2 , • • •, x n ) in Eq. (4.1) functions only feature removable singularities, i.e. when expanding these functions as power series around x 1 = ... = x n = 0, all poles are removable, leading to a finite result at x 1 = ... = x n = 0. We will prove this by induction.For n = 2, it is straightforward to check this property simply by performing a Taylor expansion and seeing that the singular term cancels out.