Classical Observables using Exponentiated Spin factors: Electromagnetic Scattering

In [arXiv:1906.10100], the authors argued that the Newman-Janis algorithm on the space of classical solutions in general relativity and electromagnetism could be used in the space of scattering amplitudes to map an amplitude with external scalar states to an amplitude associated to the scattering of"infinite spin particles". The minimal coupling of these particles to the gravitational or Maxwell field is equivalent to the classical coupling of the Kerr black hole with linearized gravity or the so-called $\sqrt{\text{Kerr}}$ charged state with the electromagnetic field. The action of the Newman-Janis mapping on scattering amplitudes was then used to compute the linear impulse at first post-Minkowskian (1PM) order, via the Kosower, Maybee, O'Connell (KMOC) formalism. In this paper, we continue with the idea of using the Newman-Janis mapping on the space of scalar QED amplitudes to compute classical observables such as the radiative gauge field and the angular impulse. We show that for tree-level amplitudes, the Newman-Janis action can be reinterpreted as a dressing of the photon propagator. This turns out to be an efficient way to compute these classical observables. Along the way, we highlight a subtlety that arises in proving the conservation of angular momentum for scalar -$\sqrt{\text{Kerr}}$ scattering.


Introduction
The array of tools available at our disposal in computing observables in gravitational and electromagnetic scattering has seen a remarkable increase in the last few years.A central premise in this resurgence is the realization that on-shell amplitudes in gauge theories and gravity can be deployed to compute classical observables ranging from scattering angle to radiative flux in Post Minkowskian scattering .One of the most potent formalisms which uses the quantum S-matrix to generate classical observables was proposed in a seminal paper by Kosower, Maybee, and O'Connell (KMOC) [41][42][43][44][45].The KMOC formalism uses 2. We use the NJ shift to compute the angular impulse for the scalar and √ Kerr particles.We also highlight an important subtlety that underlies the computation of angular impulse for the spinning particle.This subtlety is crucially tied to the use of spin tensor (S µν ) as opposed to the spin pseudovector (a µ ) as the fundamental variable.These are related via the following duality relation a µ = 1 2m2 ϵ µνρσ p ν S ρσ . (1.1) 3. We show that to linear order in the initial spin parameter, the total angular impulse (of the scalar-√ Kerr system) is consistent with classical results [60] so long as the initial coherent state of the spinning particle is parametrized in terms of S µν and p µ .The plan of the paper is as follows.In section 2, we review the KMOC formalism used to compute classical observables from scattering amplitudes.In section 3, we give a brief review of the Newman-Janis algorithm used as a classical solution-generating technique and its manifestation at the level of three-point amplitudes.In section 4, we reformulate the NJ algorithm for the case of scattering amplitudes via a specific deformation of the photon polarisation data.We then use this version of the NJ algorithm to compute the radiation emitted by the scalar particle in section 5.In section 6 we compute the orbital angular impulse suffered by the scalar particle, the spin angular momentum change for the √ Kerr particle and highlight an important subtlety involved in computing the orbital angular momentum of the √ Kerr particle.We conclude by discussing some open questions in section 7.In the appendices we state our conventions, provide the classical calculations for the observables and evaluate some integrals that are used in the main text.

KMOC Formalism in a nut-shell
The KMOC formalism [41,42] is a framework that is used to compute classical observables from on-shell scattering amplitudes for large impact parameter scattering 3 .The procedure to compute classical observables is as follows: we start with an initial coherent state, compute the change in the expectation value of a self-adjoint quantum mechanical operator, and then take an appropriate classical limit to obtain the classical result.The main feature of the formalism is that the classical limit is taken before evaluating the full amplitude because of which the computation becomes significantly simpler.Additionally, radiation reaction effects are naturally inbuilt within the framework.For a short sample of the results obtained with the formalism, we refer the reader to [9,61,62].In this section, we shall highlight some of the features of the formalism relevant to us.
We start by describing the initial state, where ϕ i (p i )s are the minimum uncertainty wave packets (in momentum space) and ζ a i i are the coherent spin state wave function for the particles with the little group indices for the particles being denoted by a i .The wavepacket of the second particle is translated, with respect to the first particle's wavepacket, by a distance of b -the impact parameter.Since the initial particles are described by coherent states we have where σ 2 i is the variance and m i s are the masses of the particles.Here the expectation value of the momentum operator is with respect to the initial state in eq.(2.1).
The spin of a particle in quantum field theory is given by the expectation value of the Pauli-Lubanski vector [43], Hence, it is the expectation value of the above operator which gives the classical spin pseudovector, (2.5) The variance for the spin is also small, as for the momentum of the particle.For a more detailed construction of these wavefunctions, we refer the reader to [44].
We now move on to describe the construction of the classical observables.The basic idea is to compute the change in the expectation value of a quantum mechanical operator as this is what is relevant from a classical perspective.So we write where S = I + iT is the S-matrix.For the linear impulse, O A = P µ , the momentum operator, O A = J µν for angular impulse, O A = W µ /m for the spin kick and O A = A µ (x), the gauge field operator, from which we read off the radiation kernel.The expression in eq.(2.6) can be simplified using the on-shell completeness relation and unitarity of the S-matrix [41].
For the linear impulse, we get and for the orbital angular impulse, we have where we write the full 4-point amplitude as We provide a derivation of the above expression in Appendix B. We do not display the higher order contributions as in this work we will only be interested in calculating the observables to leading order in coupling.Similarly, for the gauge field, we can read off the radiation kernel [42] ⟨R The spin kick, to leading order in the coupling, is given by [43] ⟨∆a µ ⟩ = ∫ d4 q δ(2p (2.11) The classical limit is taken at the level of integrand, by expressing massless momenta in terms of their wave numbers (e.g.q i = qi ̵ h), appropriately rescaling the dimensionful couplings and keeping leading order terms in ̵ h.In QED, the dimensionful coupling is obtained by 4 e → e/ √ ̵ h.For spinning external states, in the classical limit the final state spin pseudovector can be written as where ω µν (p; q) is the infinitesimal boost parameter.With these expressions in hand, we can write down the classical limit of the linear impulse, the radiation kernel, and the spin kick, at leading order in the coupling.The expression for the former is Here ⟪f (p 1 , p 2 , q . ..)⟫ denotes the integration over the minimum uncertainty wave packets which localizes the momenta and spin onto their classical values.Similarly, for the radiation kernel, we get and for the spin kick, we get In all of the above expressions, we have taken out the ̵ h− scaling of the coupling constant.For the orbital angular impulse, we shall derive the corresponding expression in Section 6 as it is slightly more detailed.The KMOC formalism has been generalized to describe different types of scattering.For instance, in [42] the formalism has been extended to include incoming waves in the initial state.It has also been extended to include additional internal degrees of freedom like color charges in [63].Finally, it has been generalized to describe scattering in curved backgrounds [45,64].
3 The Newman-Janis algorithm for three-point amplitudes The Newman-Janis (NJ) algorithm has been known for a long time as a classical solutiongenerating technique, primarily used in the context of General relativity.In their original work [56], Newman and Janis showed that one can "derive" the Kerr metric from the Schwarzschild solution (when written in the so-called Kerr-Schild coordinates) by doing a complex transformation of the radial coordinate, with the parameter by which it transforms interpreted as the spin of the Kerr black hole solution.Interestingly, they observed that there exists a similar mapping between solutions of the free Maxwell's equations as well.In electrodynamics, the NJ algorithm generates the so-called √ Kerr field from the Coulombic field of charged point particle sitting at the origin [58].For static electromagnetic fields in vacuum, one can define the magnetostatic potential exactly as done for electrostatic solutions, since For a static point charge at the origin, then we have where ϕ and χ are the electrostatic and magnetostatic potential, respectively.Now, just as was done for the Kerr solution in GR, we do a complex transformation on the radial coordinate.We get Here ⃗ a is to be interpreted as the ring radius of the field, the radius at which there is a ring singularity.From the above expression, we can compute the √ Kerr electromagnetic field [58], In the recent past, there have been investigations in understanding the Newman-Janis algorithm in effective field theory (EFT).As shown in [65], the √ Kerr field in EFT can be thought of as being generated by a conserved current.The conserved current then defines a classical √ Kerr point particle with an infinite number of multipole moments described solely in terms of its mass (m), charge (q) and spin (a) [66].From the conserved current, we can compute the gauge field created by this configuration [67], where D µ n (m, q, a) are the multipole moments.This is the electromagnetic analog of the Kerr black hole, studied in [68].Remarkably, this recent understanding of √ Kerr field as a particle can also be understood, within the EFT framework [65], as the classical limit of the three-point amplitude of a massive spin -S particle interacting with a photon.We shall review this now.
Consider the three-point amplitude for a generic massive spin -S particle of mass m 2 "minimally"5 coupled to a photon.In the massive spinor helicity formalism [69], the amplitude is given by Here the massive spinor helicity variables |2⟩ and |2 ′ ⟩ are defined w.r.t incoming momentum p 2 and outgoing momentum p ′ 2 , respectively.The x−factor, which is a hallmark of a minimally coupled amplitude is defined via the photon polarization: x = 1 m 2 (ε + (q) ⋅ p 2 ).We now take the classical limit as described in the previous section.For the above amplitude, we replace q = ̵ hq and From the three particle kinematics, 2p 2 ⋅ q = − ̵ hq 2 , keeping the leading term in ̵ h, we obtain where we have suppressed all the SU(2) indices.Here s µ 2 is the Pauli-Lubanski pseudovector associated with the spin-S particle It was shown in [1] that if we take S → ∞, ̵ h → 0 such that S ̵ h = constant, then the above three-point amplitude exponentiates where a µ 2 = s µ 2 m 2 is the rescaled spin of the √ Kerr particle.We note that the classical limit of the massive spin -S particle has thus "spun" the three-point amplitude of a minimally coupled scalar (in the classical limit).This exponentiation is the realization of the Newman -Janis algorithm for three-point amplitudes, for scalars minimally coupled to the photon.

Spin dressing of the photon propagator
In this section, we interpret the exponentiation of the three-point amplitude, in eq.(3.8), as a "spin dressing" of the photon propagator.This is motivated by the simple observation that the three-point amplitude in eq.(3.8) can be written as where ε ′µ± (q, a 2 ) = ε µ± (q)e ±a 2 ⋅q .This was first observed in [59].Building on this observation, we move on to the construction of the four-point scattering amplitude involving a scalar particle of charge Q 1 and mass m 1 and a √ Kerr particle, mediated by photon.The incoming momenta for the particles are (p 1 , p 2 ) and the outgoing momenta are (p 1 +q, p 2 −q).Diagramatically, the four-point amplitude is represented in Figure 1.The amplitude is then where p ′ 1 = p 1 + ̵ hq, p ′ 2 = p 2 − ̵ hq and P µν ∶= ∑ h=± ε µ h ε ν −h .Using the three-point amplitudes in eq.(3.8), we note that eq.(4.2) can be written as where A µ 3,scalar is the three-point minimally coupled scalar amplitude.We now deform the internal photon projector P µν as follows Here we have used ε − = η µν and define the anti-symmetric part of the projector as − (q).Since the anti-symmetric part6 of the projector is ambiguous up to a residual gauge, we shall choose an expression for Π µν (q) that can be used in the computation of all the physical observables.We choose7 and substitute in equation eq.( 4.3) to obtain the amplitude with ϵ(p 1 , p 2 , a 2 , q) ∶= ϵ µνρσ p µ 1 p ν 2 a ρ 2 qσ .The amplitude depends on the external momenta of the scattering particles as well as the (classical) spin vector a µ .It is related to the spin tensor S µν via the dual relation, eq.(1.1).It is rather natural to interpret S µν as the independent spin tensor which can be thought of as an "intrinsic" spin angular momentum of a classical particle.In this case We will denote the projection of S µν 2 orthogonal to the time-like vector p µ 2 as S ⊥µν 2 .Thus we will interpret the spin pseudovector as a function of S ⊥µν 2 and p µ 2 .We will not explicitly indicate the dependence of a µ 2 on S µν 2 except in section 6, when we derive the angular impulse.
Using the above amplitude, the linear impulse for the scalar particle is Using the identities and rewriting the cosh(a 2 ⋅ q) and sinh(a 2 ⋅ q) terms as exponential functions, we obtain This is the expression obtained in [1].Hence, we see that the NJ algorithm, within the EFT for a √ Kerr particle, can also be interpreted as a deformation on the photon data rather than on the impact parameter as shown in [1].

Radiation Kernel to all order in spin
In this section, we use the spin-dressed photon propagator (4.4) to compute the leading order radiative gauge field emitted by a scalar particle as it scatters in the background of a √ Kerr particle.The basic ingredient for computing the radiative field via KMOC formalism is the inelastic five-point amplitude as shown in Figure 2. We will only compute the radiation emitted from the scalar particle using the NJ algorithm.It is straightforward to compute the five-point amplitude when the photon is emitted from the scalar particle since, in this case, the complexity due to the spin is completely contained within the threepoint amplitude involving the √ Kerr particle.The other sub-amplitude needed to obtain The five-point amplitude appearing in the radiation kernel at leading order in coupling.
the full amplitude is then the ordinary scalar-Compton amplitude as indicated in Figure 3.Following the construction of the four-point massive amplitude in section 4, we use the deformed internal photon projector (4.4) to obtain the five-point amplitude as follows Here A µ 3,Scalar and A νδ 4,Scalar-Compton are the three-point massive scalar-photon and the scalar-Compton amplitude in scalar QED.To derive the amplitude, we'll be using the following momentum convention as indicated in Figure 3.There are three scattering diagrams.We start with diagram I in Figure 3.It is given by (5.4) Similarly, from diagram II, we obtain We use the usual Feynman rule for the scalar-photon four-point vertex in scalar QED theory to obtain the contribution from diagram III in Figure 3 A Next, we scale the massless momentum q µ 2 as ̵ hq µ 2 and collect the terms of O( ̵ h −2 ) needed to compute the leading order radiation kernel.But unlike the four-point case, individual diagrams in this tree-level amplitude contain superclassical terms.As expected, they cancel after summing up all the diagrams.To see this first of all we rewrite the massive propagator in diagram I as where we set k 2 = 0 and The second condition is due to one of the two on-shell delta functions present in the radiation kernel.Expanding the contribution from diagram I upto O( ̵ h −2 ), we find and Similarly, from diagram II, we get both O( ̵ h −2 ) and O( ̵ h −3 ) terms but the latter cancels with the contribution from I. (5.10) The O( ̵ h −2 ) terms from diagram III can be found trivially.We collect all the terms of (5.11) Using the formula of eq.(2.14), we obtain the radiation kernel as (5.12) This expression agrees with the result in (C.24), which is obtained using classical equations of motion.The radiation kernel for the scalar can also be implemented as a complex shift in the impact parameter space, just as it was done for linear impulse in [1].Consider the radiation kernel for scalar-scalar scattering in electrodynamics Inside this expression for the radiation kernel, we complexify b = b + ia 2 and re-write the expression as follows.
(5.14) with u 1 ⋅ u 2 = cosh w = γ.We can now factor out the overall e −iq⋅b in the second line where we use γ = cosh w and √ γ 2 − 1 = sinh w.We can now evaluate the derivatives with respect to initial velocity u 1 , with sinh w = βγ.Using the following identity, consistent with the two on-shell delta function constraints: u 2 ⋅ q = 0 and u 1 ⋅ q = k ⋅ u 1 , we find that and we recover the radiation kernel in scalar-√ Kerr scattering when the photon is emitted from the scalar particle in (5.12).
We note that to compute the total radiative flux (which includes the radiation emitted by the √ Kerr particle), we require the expression for the five-point amplitude where the incoming √ Kerr and scalar states are scattered into √ Kerr, scalar and a photon.A diagrammatic representation of this amplitude is in Fig. 4.
One avenue to compute the inelastic amplitude A 5 ( √ Kerr + scalar → √ Kerr + scalar + γ) is via an EFT computation of the Compton sub-amplitude which has been pursued extensively in the literature recently [10,59,60,.Hence a possible strategy to compute the leading order radiation kernel in the present case would be to use the Compton sub-amplitude to evaluate all the diagrams in Fig. 4.
However, a more natural route is to start with a gauge invariant bare Lagrangian of QED with √ Kerr charged matter and compute the five-point amplitude directly using the resulting Feynman rules.Given that the three-point coupling of √ Kerr with a photon is known, one can in principle use gauge invariance to fix all the higher point couplings.In [59], it was shown that just as in the case of scalars and fermions, a gauge-invariant Lagrangian which describes the minimal coupling of a √ Kerr particle with the electromagnetic field is simply the scalar QED lagrangian in which the gauge covariant derivative is replaced by a twisted covariant derivative, D (a) It would be rather natural to simply use this Lagrangian and compute the radiative field emitted during scalar− √ Kerr scattering.However as the author emphasizes in [59], such a Lagrangian is not consistent with all the Compton amplitudes, and as a result, it is unclear how it would lead to the correct answer for classical radiation.We stress that the computation of the complete radiation kernel emitted during scalar− √ Kerr scattering using the KMOC formalism has the potential to unravel the full power of the NJ algorithm.This will be pursued elsewhere [92].

Leading order angular impulse
In this section, we use the NJ algorithm to compute the leading order angular impulse ∆J µν for the scalar and √ Kerr particles.Angular impulse is an intriguing observable in D = 4 dimensions.Already for the scattering of scalar particles, it was shown in [93] that the net angular impulse of the particles (in a 2 → 2 scattering) does not add up to zero even at leading order in the coupling.The missing contribution is due to angular momentum stored in the late-time Coloumbic modes.In [93] this contribution was called electromagnetic scoot.We will denote the scoot as δ µν scalar-scoot , where the subscript indicates that the scoot has been computed for the case of scalar-scalar scattering.
The NJ algorithm offers a powerful tool to compute angular impulse for the scalar-√ Kerr system.We will denote the angular impulse for the scalar particle as ∆L µν 1 .It can be computed to all orders in spin using the NJ algorithm and the final result is given in eq.(6.17).The computation of total angular impulse (i.e.change in orbital angular momentum, ∆L µν 2 plus the change in spin angular momentum, ∆S ⊥µν 2 ) for the √ Kerr particle can also be done using the NJ algorithm.We will denote it as ∆J µν 2 .As noted below eq.(4.7), for the computation of the orbital angular impulse for the √ Kerr particle, we take S ⊥µν 2 as the independent variable and take a µ 2 (S ⊥ 2 , p 2 ) 8 .A result (in the integral form) for ∆L µν 2 and ∆S ⊥µν 2 appears in eq.(6.23) and (6.41), respectively.In principle, this completes the computation of angular impulse for the scalar-√ Kerr system to leading order in coupling.
To test our results, we compute the net angular impulse of the scattering particles and subject it to the conservation law.
Based on [93] we deduce that to leading order in the coupling, On general grounds, we expect that the entire contribution to the electromagnetic scoot is independent of the spin of the particles as it simply arises due to the late-time Coloumbic effects which do not depend on the spin,

.2)
8 A moment of reflection reveals that for orbital angular impulse of √ Kerr particle, the choice of a µ 2 versus S µν 2 as independent variable in A4 will produce inequivalent results as We thus expect that We verify the conservation to next to leading (i.e.linear) order in S µν 2 in a perturbative expansion which is valid when |a 2 | ≪ |b|.As we will argue, verifying conservation for finite spin |a 2 | ∼ |b| is rather subtle and will be pursued elsewhere [92].We start with the computation of the orbital angular impulse for the scalar -√ Kerr scattering, to leading order in coupling.

Orbital Angular Impulse
The leading order orbital angular impulse in the KMOC formalism is given by where p i 's are initial momenta and we denote the final momenta as pi = p i + ̵ hq i .Here A 4 (p 1 , p 2 → p1 , p2 ) is the four-point scalar− √ Kerr scattering amplitude given in (4.6).Since we express the amplitude as a function of (p i , qi ), we shall treat them as independent variables and consider the transformation (p i , pi ) → (p ′ i , q ′ i ) and then set p ′ i = p i to obtain the correct differential operator for the angular impulse.With we obtain the differential operators in new variables by treating p i = p i (p ′ i , q ′ i ) and pi = pi (p ′ i , q ′ i ).We find Using these transformations, we write the orbital angular impulse where we treat p ′ i = p i and q ′ i as independent variables9

Orbital angular impulse of the scalar particle
For the scalar particle, we do integration by parts on the second term in the first line in eq.(6.7) and integrate over q2 to obtain with where A 4 is given in eq.(4.6).As we will suppress the explicit dependence of a µ 2 on p µ 2 as in the radiation kernel derivation.It is straightforward to evaluate the expression in ∆L µν 1,I .Since p 1 and q are independent, we get Evaluation of ∆L µν 1,II on the other hand needs a more careful analysis since it involves derivative of the on-shell delta function.In order to simplify this, we shall decompose the momentum qµ along p 1,2 and in the transverse direction where the coefficients are given by with x 1,2 ∶= (p 1,2 ⋅ q).Due to this change of variables, the measure transforms as follows In terms of x 1,2 and q⊥ variables, we rewrite where we have done the x 2 integral and used b ⋅ p 1,2 = 0. Next, we perform an integration by parts in x 1 .At this stage, we note that from eq. ( 6.11) we can write Therefore any first order derivative of 1 q2 w.r.t x 1 or x 2 vanishes due to the on-shell delta function constraints: where . Summing the two expressions ∆L µν 1,I and ∆L µν 1,II , we obtain the orbital angular impulse of the scalar particle In Appendix C.2 we have verified the result in classical theory.
We use the results of the integrals from appendix D and obtain the orbital angular impulse as follows Here we have defined where Π ν ρ is the projector into the plane orthogonal to both u 1 and u 2 [43], with Πa 2 = √ Πa 2 ⋅ Πa 2 and b = √ −b 2 .In eq.( 6.18), µ 1 is the infrared (IR) cut-off.Note that the spin dependent terms in first line are not IR divergent as it involves derivative over b and the scalar term contributes to the electromagnetic scoot.
The angular impulse for the scalar particle to linear order in spin written in terms of 2 ) . (6.21)

Kerr particle
The integral expression for the leading order orbital angular impulse of √ Kerr particle can be written as, We use the formula for linear impulse to rewrite the above integral as follows where The evaluation of ∆L µν 2,I is rather subtle as for any function f (a 2 , q) we obtain terms in- where we have used using the dual relation (1.1).The derivation of ∆L µν 2,I (and hence ∆L µν 2 ) (for |a 2 | ∼ |b|) will be pursued elsewhere [92].In this paper, we simply evaluate the orbital angular impulse to linear order in S ⊥µν 2 .Using the dual relation (1.1), we obtain where µ 2 is the IR cutoff.This matches with the result obtained in classical theory given in Appendix C.3.

Spin Angular Impulse
The computation of the spin angular impulse ∆S ⊥µν 2 is rather straightforward via the NJ algorithm.Using the inverse of the dual relation in eq.(1.1), S ⊥µν = ϵ µνρσ p ρ a σ (6.27) we obtain [94] ∆S ⊥µν 2 = ϵ µνρσ ∆p 2ρ a 2σ + ϵ µνρσ p 2ρ ∆a 2σ , (6.28) where ∆a µ 2 is known as the spin kick.We note that although S ⊥µν 2 is the fundamental spin degree of freedom, the NJ algorithm lets us directly compute the spin kick which can then be used to deduce ∆S ⊥µν 2 .Since the linear impulse doesn't receive any radiative contribution at leading order in the coupling, the expression for the linear impulse ∆p µ 2 is exactly opposite to ∆p µ 1 , derived in section 4 and it is given by where we rewrite the sinh(a 2 ⋅ q) term using the identity We study the leading order spin kick using the following formula [43] The commutator in the first term is defined in the SU(2) little group space.The SU(2) indices are left implicit under the double angle bracket notation, explained in section 2. Note that, the formula (6.31) appears to be non-uniform in the order of ̵ h, however, the commutator term also includes an additional factor of ̵ h and it is given by where we display the SU(2) indices (I, J, K).From hereafter, we shall drop the SU(2) indices and the double angle bracket notation altogether.Using (6.32), we get Next we consider the following commutator where we defined Y ∶= ( cosh (a 2 ⋅q) (a 2 ⋅q) − sinh (a 2 ⋅q) (a 2 ⋅q) 2 ).Using eq.A.5 it can be shown that, to get Next, we use the identity where we have set Substituting various commutator expressions, we obtain the leading order spin kick as where we make use of the identity This matches with the spin kick obtained in [65] using classical equations of motion.Plugging these expressions into (6.28),we obtain the following expression for spin angular impulse Using the integral results, we get It can be verified that the RHS of eq.(6.42) is ∆S ⊥µν 2 as it satisfies the SSC constraint to leading order in the coupling.

∆S ⊥µν
2 p 2ν + S ⊥µν 2 ∆p 2ν = 0 .( At linear order in S ⊥µν 2 , the spin angular impulse can be evaluated. 2 ) (6.44) This expression is in agreement with the result of [60], which was derived using the classical equations of motion.
We now have all the expressions to compute the total angular impulse, ∆J µν in eq.( 6.1) for the the scalar-√ Kerr scattering to linear order in spin.This is given by the sum of eqs.(6.21),(6.26)and (6.44).We obtain the following result where The IR cutoffs are related to the proper times of the two particles via µ 2 µ 1 = τ 1 τ 2 [20,95].Hence, the total angular momentum for scalar -√ Kerr scattering is conserved, to linear order in spin.The study of conservation of angular momentum to all orders in spin is under investigation [92].

Discussion
Building upon the synthesis of the Newman-Janis (NJ) algorithm with the KMOC formalism in [1], in this work, we have used the NJ algorithm to compute classical observables beyond the linear impulse for electromagnetic scattering involving √ Kerr particles.As is well known, the real power of the NJ algorithm lies in all orders in spin results for classical observables.We hope that by combining the on-shell methods along with the NJ algorithm one can build "loop integrands" associated with the scattering of √ Kerr particles such that our analysis can be extended beyond leading order in the coupling.For some early attempts in this direction, see [28].
Our main idea is to simply re-interpret the three-point coupling involving √ Kerr as a "spin dressing" of the photon polarisation data while computing higher point amplitudes.Our work shows that the resulting "spin-dressed" photon propagator is particularly useful for constructing the five-point amplitude where the photon is emitted from the external scalar state.We used this five-point amplitude to derive the radiation emitted by the scalar to all orders in the spin and found perfect agreement with its calculation using equations of motion.
We then used the four-point amplitude computed using the spin-dressed photon propagator to compute the angular impulse of the scalar as well as √ Kerr particles.Here we encountered a subtlety.We used the spin tensor S µν 2 instead of the spin-vector a µ 2 as the fundamental spin degree of freedom.With this choice, we found that the result for the angular impulse of the √ Kerr particle is consistent with the angular momentum conservation.As explained in Section 6 the reason for this choice becomes evident if one recalls that the calculation of the angular impulse via the KMOC formalism involves the expectation value of differential operators acting on amplitudes.Choosing the spin to be parametrized by either a 2 or S 2 leads to different results as a consequence.To leading order in S 2 (valid as long as |a 2 | ≪ |b|) we have checked that our results are consistent with [60].
Using the NJ algorithm, we also calculated the leading-order orbital angular impulse of a scalar particle to all order in the spin of the √ Kerr particle.In addition we gave a closed form expression for the total angular impulse of the √ Kerr particle to leading order in spin.An all-order-in spin evaluation of the total angular momentum of the √ Kerr particle is beyond the scope of this paper and will be pursued in [92].Our broader goal is to compute classical gravitational observables for the Kerr black hole.In this context, the double copy will be an important tool [96].For conservative observables (at leading order in coupling), the double copy of the three-point amplitudes is the only ingredient that is needed.It has been observed in [69] that the three-point amplitude for a massive spin -S minimally coupled to gravity can be obtained by simply squaring the 'x' factors in eq.(3.5).Using this in [1], the 1PM linear impulse for the scalar-Kerr system was obtained.It will be interesting to use this double copy to study the angular impulse for the same system and check the conservation of angular momentum.However, to study radiation from gravitational scattering involving Kerr black holes, we need the double copy of the non-abelian counterpart of the √ Kerr solution.Since we do not have a consistent bare Lagrangian for the latter we leave the question of computing gravitational radiation for Kerr black holes from amplitudes, to future work.In the meantime, it might be instructive to study gravitational radiation from Kerr black holes directly using the equations of motion derived in [87].However, recently the waveform for the scattering of a Kerr 1 -Kerr 2 black hole system has been computed in both the perturbative spin parameter to O(a 4 ), the highest order to which the answer is unambiguous [97].The case of a Schwarzschild black hole scattering with a Kerr black hole has also been studied in [86] and the waveform has been computed for the Schwarzschild black hole to all orders in spin and for the Kerr black hole to O(a 4 ).The former case is the gravitational analogue of the radiative gauge field in eq.(5.12).Additionally, the waveform for this system has also been computed to NLO in [98] to linear order in spin.Finally, it is also important to note that in [99], the waveform for the Kerr 1 -Kerr 2 system has been computed to all orders in spin for both the black holes.These result will be interesting to compare with, once we have a result from our NJ perspective.
We are interested only in the leading order term and hence, we concentrate only on the first term.Plugging in the expression for the initial state in eq.(2.1), we get Here we have suppressed the little group indices a i .Now, we use to write the first term in terms of the differential operator, we use the hermiticity of the orbital angular momentum operator i.e Plugging the last equation and eq.(B.3) into eq.(B.2), we get By relabelling from r i = p i + q i , we get where ⟨∆L µν i ⟩ denotes the integration over the wave functions.This is the same expression in eq.(2.8),where we have substituted the defnition of the 4-point scattering amplitude

C Classical calculations
In this appendix, we present the classical calculations of angular impulse and the radiation kernel for scalar in scalar-√ Kerr scattering to leading order in coupling.

C.1 Equations of motion
We present all the equations of motion that will be used to derive the physical observables discussed in the main text.The equation of motion for the scalar particle in scalar-√ Kerr scattering [65] is where F µν + (x + ia 2 ) is the self-dual part of the electromagnetic field strength of the √ Kerr particle.The self dual and anti-self dual field strengths are defined w.r.t to Minkowski metric as follows Note that, the self dual and anti-self dual fields are related to each other via complex conjugation 10 Therefore we can rewrite the real part of the self dual field strength as We then use the definition (C.2) to express the equation of motion as follows Note that, the field strength appearing here is due to a charged scalar particle!We now use the following expression for the field strength in momentum space to substitute in (C.6) to obtain an expression for ṗµ 1 to all order in spin a µ We assume that (for all cn be real)

C.2 Orbital angular impulse for the scalar particle
Classically, the orbital angular impulse is defined as where p µ = m ẋµ .Now using the parametrization of the classical trajectory of particle 1 x µ 1 (τ ) = u µ 1 τ , the orbital angular momentum impulse to leading order in coupling is dL µν We use the following identity from appendix A.5: to rewrite (C.8) as follows to derive the LO expression for the orbital angular impulse for the scalar particle (a 2 , p 2 , q) + (p 1 ∧ q) µν ϵ(a 2 , q, p 1 , p 2 where the coefficients are given in equation (6.11).Again, we can do the x 2 = p 2 ⋅ q integral in the above integral and write x 1 τ m 1 ) [ cosh(a 2 ⋅ q){(p 1 ⋅ p 2 )(p 1 ∧ q) µν − (p 1 ⋅ q)(p 1 ∧ p 2 ) µν } + i sinh(a 2 ⋅ q) a 2 ⋅ q {x 1 p [µ 1 ϵ ν] (a 2 , p 2 , q) + (p 1 ∧ q) µν ϵ(a 2 , q, p 1 , p 2 )}] , (C. 15) where x 1 = p 1 ⋅ q.Integrating by parts in x 1 variable and then completing the τ integral, we obtain The orbital angular impulse for the √ Kerr particle is given by Using the equation of motion for the √ Kerr particle to linear order in spin

Figure 1 :
Figure 1: The four-point scalar-√ Kerr amplitude with photon exchange.Here 'a 2 ' denotes the rescaled spin of the √ Kerr particle.

Figure 3 :
Figure 3: Diagrams contributing to the tree level five-particle amplitude with a photon emitted from the scalar particle.

Figure 4 :
Figure 4: Diagrams contributing to the tree level five-point amplitude with a photon emitted from the √ Kerr particle.