Gravitational Dyonic Amplitude at One-Loop and its Inconsistency with the Classical Impulse

The recent proposal [1911.06318, 2010.07861] of implementing electric-magnetic duality rotation at the level of perturbative scattering amplitudes and its generalisation to gravitational context where usual gravitational mass is rotated to the NUT parameter of the Taub-NUT spacetime opens up an interesting avenue for studying NUT-charged objects as dynamical entities, in contrast to the usual approach where NUT-charged objects are considered as a static background. We extend the tree-order analysis to one-loop order and find a disagreement between geodesic motion on Taub-NUT background and impulse computation of scattering amplitudes. As a by-product of our analysis, we find a relation between tidal response parameters and resonance excitation parameters in the language of quantum field theory scattering amplitudes.


Introduction
During the expedition for a solution of Einstein's equation having angular momentum, Taub-NUT spacetime had been discovered independently by Taub and Newman, Unti, and Tamburino [3,4]. The solution has an additional parameter mistaken for angular momentum at the time of discovery, the so-called NUT charge, and the spacetime can be considered as a simple generalisation of the Schwarzschild solution [5]. Interestingly, Taub-NUT solutions cannot be extrapolated or constructed from summing over higher order corrections of post-Newtonian expansion or modified gravitational theories [6][7][8], but it is one of the simplest, Hamilton-Jacobi soluble [9], and exact solutions of the vacuum Einstein equations. Although the spacetime with angular momentum has later been identified as the Kerr solution, the physical meaning and effects of NUT charges has still been studied, and the NUT parameter has later been identified as a source of locally rotating frame or gravitomagnetic force which becomes the Coriolis force in (post)-Newtonian regime [10,11], being the charge of dual supertranslation symmetry [1,12,13]. It was also found that angular momentum of Kerr spacetime and NUT parameter of Taub-NUT spacetime can coexist, leading to what is known as the Kerr-Taub-NUT spacetime [14].
Studies on physical implications of NUT charges can be found in the literature, especially in the context of astrophysical observations. The investigations based on geodesic analysis are: Lensing and atomic spectra [15], gravitational lensing [16,17], microlensing [18,19], particle acceleration [20,21], circular geodesics [22], and electromagnetic radiation from near horizon region in Kerr-Taub-NUT spacetime [23]. A complete analysis of geodesics, singularities, and causal structures of Taub-NUT spacetime has also been done recently [24]. Perfect fluids that can source NUT charge have been studied in ref. [25]. For a possibility of actual astrophysical objects having NUT charges, ref. [26] suggested the stellar-mass black hole-candidate X-ray binary, GRO J1655-40, as a possible NUT-charged object. Since horizons exist in Taub-NUT spacetime, black hole thermodynamics can be applied to it as well. This has been studied in recently papers such as refs. [27][28][29][30][31], although the thermodynamics of their Euclidean cousins have been studied much earlier [32].
In the aforementioned studies the NUT parameter was predominantly assigned to the static background, and to best of authors' knowledge any detailed study on consequences of dynamical objects (in the sense that they can be accelerated or decelerated by external influences) having NUT charges has not yet appeared. This is where the recent proposals of understanding the NUT charges as "electric-magnetic duality transformations" of graviton coupling in the context of scattering amplitudes [1,2] become interesting, as the proposal can be used to study dynamical properties of NUT-charged particles.
From the perspectives of an on-shell photon, electric and magnetic charges are distinguished by how they couple to distinct helicity modes of the photon; electric charges couple with the same sign, while magnetic charges couple with the opposite sign [49]. Reconciling the sign choices with CPT symmetry forces magnetic couplings to have imaginary coupling constants [49], and the phase rotation to the kinematic factor x introduced in ref. [1] can be understood as a phase rotation to the coupling constant that generates magnetic couplings from electric couplings. Pushing the analogy to gravitons, an ordinary mass would couple to both helicity modes of the gravitons with the same sign while a "magnetic mass" would couple with opposite signs. However, no known current could be found in Weinberg's analysis that would "magnetically" couple with the graviton and be consistent with CPT symmetry [49], and for this reason the possibility had been ruled out. Nevertheless, if such a coupling exists, it would correspond to the NUT parameter of (Kerr-)Taub-NUT spacetime [1,2], at least to leading Newton's constant order (G 1 ) in perturbation theory. The goal of present work is to examine this relation at next-to-leading order (G 2 ) in perturbation theory.
The manuscript is structured as follows. The motion of a test particle on Taub-NUT spacetime determined by general relativity will be reviewed in section 2 to the second order in Newton's constant, referred to as the second post-Minkowskian (2PM) order in the literature. The deflection angle 1 will be the main observable that will be compared with quantum field theory computations. The scattering amplitude computation of the equivalent perturbation order will be performed in section 3. In section 3.4 we find that the widely-used approximation tool, the eikonal approximation, is not valid at one-loop order when electric-magnetic duality transformation has been performed. Nonetheless, the imaginary part of the impulse computations cancel and a sensible deflection angle can be obtained. Based on a naïve expectation for the Compton amplitude, we obtain the deflection angle in section 3.5 and find a disagreement with the predictions of general relativity. Possible resolutions for closing the gap are considered in sections 3.5.2 and 3.5.3, which turn out to be insufficient for restoring coherence. As a by-product of our analysis, we obtain a relation between tidal response parameters and resonance excitation parameters, the equations (3.51) and (3.52). We conclude our study in section 4.

The scattering angle from GR geodesic motion
The Taub-NUT metric is given as where M , N , and r h are ADM mass, NUT parameter(charge), and the location of the outer horizon, respectively. The Cauchy horizon lies outside the domain r ∈ [0, ∞) and therefore can be ignored for our purposes. Regularity of spacetime requires periodicity of 2π/N for the time-like t coordinates, but the condition can be loosened when dual supertranslation symmetry is promoted to a gauge symmetry [50]. The Euclidean version of the solutions known as Euclidean Taub-NUT and Taub-Bolt solutions, which appear in the context of gravitational instantons [32], are obtained from the above metric by a Wick rotation, t → −iτ and N → −iN , and their differences arise from allowed range for the radial coordinate r.

Killing fields, integrals of motion, and geodesics
Killing vector fields are the generators of the Taub-NUT spacetime isometry group, which in the coordinate patch of (2.1) take the following form [51].
Taking the inner product with the tangent vector ∂ λ gives the integrals of motion for the geodesics.
The relations can be inverted to find the tangent vector, The geodesic equation for a particle of unit mass then reduces to which can be further reduced to the one-dimensional radial equation, The parameters can be chosen without loss of generality such that the geodesic lies on an orbital cone 2 of θ = θ 0 determined as Unfortunately this coordinate system is inconvenient for the Hamilton-Jacobi analysis as the method of separation of variables cannot be applied. Doing a gauge transformation allows application of the method for geodesics lying on the orbital cone θ = θ 0 .

Gauge transformation and action-angle variables
The Hamilton-Jacobi analysis can be made more tractable by the gauge transformation, The metric (2.1) is then transformed to while the Killing fields (2.3) become (2.11) The old Killing fields (2.3) and the new Killing fields (2.11) are related by the relations, oc . (2.12) Integrals of motion from K oc are given as (2.13) 2 The conditions C > 0 and C = 0 correspond to linear and spiral scattering, respectively [24].
The tangent vector components 3 transform aṡ (2.14) and the geodesic equation reduces to We remind the reader that the geodesic lies on the orbital cone, θ = θ 0 , and the metric can be effectively considered as three-dimensional; The radial equation (2.7) becomeṡ The opening angle of the orbital cone in terms of the parameters of the geodesic, (2.8), needs to be expressed using new integrals of motion (2.13). We solve the pair of equations, for J ϕ and obtain with the sign choices J ϕ > 0, J > 0, N > 0, and cos θ 0 < 0. The opening angle θ 0 in terms of E, J and N is then given as (2.20)

Hamilton-Jacobi analysis on the orbital cone
The action functional in terms of action-angle variables on the orbital cone can be written using the effective three-dimensional metric.
where µ is the mass of the probe particle moving along the geodesic. The relativistic Hamilton-Jacobi equation is which coincides with the geodesic equation on the orbital cone.
Solving for the radial momentum P (r) gives (2.24)

Scattering at second post-Minkowskian order
Since we are interested in effects up to second post-Minkowskian order (2PM) or to second order in Newton's constant G, the expansion parameter G will be restored to GM and GN in this section. Standard Hamilton-Jacobi analysis can be applied to compute the angle variables. The change in the azimuthal angle during scattering, ∆ϕ, is where the r min is the largest positive root of the equation E 2 − V eff = 0. Expanding r min to G 2 order, When evaluating the integral (2.25) to G 2 order, simply expanding the integrand and lower/upper limits to G 2 order and effecting the integral mechanically will not give correct results. This is because the perturbations of integral limits can ruin the structure of the integral. Reparametrising the integration variable so that the integral limits are not perturbed will make the perturbation integral well-behaved.
The change in azimuthal angle during scattering is then given as Due to the presence of NUT parameter N , the geodesic motion lies on the orbital cone of θ = θ 0 and ∆ϕ should not be considered as the scattering angle. Instead, the scattering angle Φ and the deflection angle ϑ are given as cos Φ = cos 2 θ 0 + sin 2 θ 0 cos ∆ϕ, ϑ = Φ − π. (2.28) Solving the above equation to G 2 order yields the 2PM scattering angle where M 2 TN = M 2 + N 2 . In the Schwarzschild limit, N → 0, the orbital cone opening angle θ 0 becomes π/2 and geodesic motion becomes equatorial. In this limit ∆ϕ coincides with Φ.
The deflection angle ϑ is vectorial in this set-up, therefore we define the deflection angle vector ϑ asn out −n in . This is the main observable that will be compared with scattering amplitude source-probe limit computations.
where ∆θ 0 = θ 0 − π/2, ϑ ϕ = ∆ϕ − π. The deflection angle ϑ is given as the size of the deflection angle vector ϑ = ϑ . In amplitude computations, the incoming test particle will be considered as directed along the x-axis, which is not the incoming direction of the geodesic analysis. Therefore we rotate the reference frames to match the normalisations of amplitude computations. (2.32) The vectors and matrices can be explicitly evaluated. (2.33) The 2PM deflection angle vector in the transformed frame is then given as (2.34) The Schwarzschild limit can be recovered by the limit N → 0. The first component of the vector is directed along the direction of the incoming test particle, and is irrelevant for the perturbation order we are interested in. Note that while electric (M ) and magnetic (N ) masses deflect the particle in orthogonal directions at 1PM order (G 1 ), the deflections are aligned at 2PM order(G 2 ).

Momentum parametrisation and the holomorphic classical limit
A generic one-loop amplitude can be expanded on a basis of integrals known as scalar integrals. Unitarity methods [52,53] aim to compute the coefficients of the basis scalar integrals by examining analytic structures of the loop integral, often building up the integrand from lower perturbation order on-shell amplitudes. We resort to unitarity methods for the analysis of one-loop amplitude, as the analysis of refs. [1,2] was based on on-shell tree amplitudes whose continuation to off-shell kinematics 4 is unclear; the textbook method of constructing the loop integrand from Feynman rules requires off-shell data.
Classical physics is dictated by long-distance physics, or physics at large impact parameters. Since impact parameter b µ is the Fourier dual of transferred momentum q µ on the impact parameter space, the expansion in large impact parameter can be traded for an expansion in small transferred momentum. Dimensional analysis in Planck's constant shows that only the scalar box integrals and the scalar triangle integrals have the relevant dimensions to contribute to classical physics [54][55][56][57].
A computationally efficient means of probing the leading contributions is the holomorphic classical limit (HCL) [56], where transfer momentum q µ is complexified to a non-zero complex null vector. The subdominant q 2 contributions drop out in the computations as q µ is a null vector, but as q µ is non-zero certain information such as spin dependence can be reconstructed in this limit [58][59][60][61][62][63][64]. Cut momenta parametrisation will be given in this section for later reference; consult ref. [56] for derivation of the cut momenta parametrisations.
The parametrisation for external momenta is given as where spinor brackets are normalised by the conditions 5 , λη = [λη] = m a and λη = [λη] = m b . The limit β → 1 corresponds to the HCL, and following definitions for the variables will be used.
The momentum p 1 can be expanded in terms of unhatted spinors using above variables 6 .
For the b-topology triangle cut, figure 1, the cut momenta are given as (3.4) The b-topology contribution to the amplitude can be computed as where integration contour Γ for y encircles y = ∞, which can be traded for reading off y 0 coefficient of the expression's Laurent expansion at y = ∞ [65]. This scalar triangle integral contribution captures the | q| −1 ∼ r −2 dependence of the classical Hamiltonian in position space and | q| −1 ∼ b −1 dependence of the eikonal phase in impact parameter space.
The a-topology cut is defined as the mirror topology obtained by exchanging m a ↔ m b .

The impulse formula
Define pre-eikonal phase as are the normalised measure and delta constraint in momentums space. While the above integral kernel is widely used to compute eikonal amplitudes, the terminology pre-eikonal phase has been adopted because diagrams that do not contribute to the eikonal phase will also be effected by this kernel. In the COM frame with small momentum mismatch q µ ∼ w µ 1 ∼ 0, the delta constraints can be simplified as where E real = √ s is the COM frame total energy and p com is the spatial momentum in the COM frame. The above integral is performed in the Euclidean 2-plane dual to the impact parameter space b. Note that therefore the normalisation factor for the deflection angle is given as [62] 1 The impulse formula eq.(4.31) and eq.(4.32) of ref. [66] can be approximated to when production of extra particles is neglected, i.e. X = ∅, and the full expression is assumed to be dominated by small momentum mismatch q µ ∼ w µ 1 ∼ 0. This approximation is valid up to G 2 order in the classical limit, and the impulse up to this order can be written as; where momenta of hard particles have been suppressed and the superscript on A denotes loop-order of the amplitude; e.g. A (n) is the n-loop amplitude. Scaling the impulse by | p com | gives the deflection angle, which is reliable up to G 2 order 7 .
The master integral for (3.8) is where n = 2−2ǫ is the dimensional regulator for UV/IR divergences. The integral is convergent for 0 < m + n < 3/2. Differentiating by m, The reliability relies on two facts; (1) extra particle production can only contribute at G 3 or higher orders, and (2) the small angle approximation 2 sin(θ/2) ≃ θ is exact up to θ 2 order, where leading scaling is θ ∝ G. Following results will be used in the computations.

The impulse at tree order
This analysis has already been presented in ref. [1], which has been included for completeness. The on-shell three-point amplitudes are given as [48] A + φφh = is the coupling to positive(negative) helicity graviton, and x is the kinematic factor carrying little group weights of the graviton.
is the positive helicity polarisation vector for a massless spin-1 particle with momentum p 3 . The phase rotation ϕ generates magnetic coupling [1]. The m a m b → m a m b tree amplitude can be obtained by gluing the on-shell three-point amplitudes, and when particle a is subject to phase rotation ϕ the amplitude is given as;  The sign difference comes from q 2 = − q 2 ↔ log |bµ| 2π . The deflection angle from (pre-)eikonal phase χ is given as; where imaginary part of the impulse has been re-interpreted as i √ σ 2 − 1q µ = ±ǫ µ (u 1 , u 3 , q), with sign choice fixed by matching to the Newtonian limit [1]. This result can be matched to general relativity predictions (2.34) using the parameter mappings of table 2, yielding consistent with (2.34) to this order.

The one-loop amplitude : Boxes and eikonal (non-)exponentiation
The box topology contribution to the one-loop amplitude in the HCL is; iA where σ = p 1 ·p 2 mam b > 1 has been applied to the imaginary part of the scalar integrals. The scalar integral coefficients can be obtained by imposing the fourth cut condition on the triplecut solutions (3.4). Adding up, the logarithmic terms in square brackets cancel and only the imaginary part of the scalar integrals survive. (3.28) The pre-eikonal phase is computed as; where only the leading b dependence has been kept. Setting the phase rotation zero ϕ = 0, which is consistent with exponentiation of the eikonal phase; This relation obviously breaks down when non-zero phase rotation ϕ = 0 is introduced, rendering eikonal approximation for dyonic amplitudes questionable. However, their impulse contributions cancel each other, Therefore, (3.33) Thus, only the triangle contributions contribute to the classical deflection angle.
3.5 The one-loop amplitude : Triangles

Analysis from the naïve Compton amplitude
The gravitational Compton amplitude is needed when evaluating the triangle contribution for the loop amplitude, as can be seen on the left side of figure 1. The gravitational Compton amplitudes are given as [48,58,59,67,68] , where Mandelstam invariants are defined as 8 s = (p 1 + p 2 ) 2 , t = (p 1 + p 4 ) 2 , and u = (p 1 + p 3 ) 2 . As in (3.19), NUT charges can be generated by phase rotation e 2iϕ (e −2iϕ ) for each positive(negative) helicity graviton [1], which results in the expressions, .
As noted in ref. [67], these two amplitudes do not satisfy the t-channel factorisation conditions. This is the reason they are referred to as naïve Compton amplitudes. Nevertheless, the impulse and the deflection angle will be analysed using the naïve amplitudes (3.35) and the corrections due to the difference (3.39) will be analysed in the following section.
Combining (3.4), (3.19), and (3.35) into (3.5), it can be shown that to leading order in HCL. Summing over all possible intermediate graviton states is effectively equivalent to summing over these two terms and doubling the result. Summing over a-topology by exchanging m a ↔ m b , the G 2 -order deflection angle is then given as There are a few notable points to these results.
1. The (−t) −1/2 dependence solely comes from the scalar triangle integral I △ (b) . This means the residue integral dy y A 4 A 3 A 3 must scale as (β − 1) 0 to be consistent with the geodesic computations. The three-point amplitudes A 3 can be shown to scale as (β −1) 0 through direct computations, therefore the condition can be refined to (β − 1) 0 scaling on the Compton amplitude A 4 . 3. The b-topology contribution (3.36) scales as ∝ m 2 a m 3 b . Put differently, incorrectness of the Compton amplitude for NUT-charged particles is inconsequential for backgroundprobe analysis where NUT-charged particle acts as the background, because the massive propagator on the heavier side must be cut 9 .
For completeness, possible resolutions for closing the gap between geodesic motion computations and amplitude results will be surveyed, to no avail.

Compton hot-fix and impulse non-fix
The naïve amplitude (3.35) is obviously wrong. As argued in the previous section, closing this loophole is unlikely to close the gap between geodesic computations and amplitude results. Nontheless, let us analyse the effects of naïve amplitude corrections to the impulse.
Although the naïve amplitudes (3.35) do not satisfy the t-channel factorisation conditions, they do satisfy the factorisation conditions on the s-and u-channels. This implies that the difference between the correct amplitude and the naïve amplitude can only have poles on the t-channel, where the residue on the t-channel is proportional to the product of on-shell three-point amplitudes, and the sum over exchanged t-channel graviton states has been suppressed. Although the exact expression for the difference (3.39) is not available, the exactness of the expression is unimportant as leading HCL contributions dominate classical physics; when studying classical contributions from the b-topology cut of figure 1, the leading behaviour of the expression (3.39) is all that matters 10 .
The Mandelstam invariant t for the Compton amplitude inside the b-topology cut is so the pole contributions dominate in the HCL β → 1. Therefore, the difference of the real amplitude and the naïve amplitude can be approximated as products of on-shell three-point amplitudes sitting on the t-channel pole.
(3.42) 9 If the Compton amplitude A φahhφa scales as ∝ m 3 a /m b , it would provide a loophole to this argument. Whether manifestation of Lorentz covariance breaking (the Dirac-Misner strings) can generate such a scaling will be left for future work. 10 If triangle coefficient computation and HCL commutes; look at section 3.5.3 for a counterexample. However, all computations seem to imply that noncommutativity arises at subleading (β − 1) orders.
where the superscript η denotes the helicity of the exchanged graviton and c η is the proportionality constant for that exchange contribution 11 . The on-shell three-point amplitudes can be approximated as for the on-shell three-point amplitude of NUT-charged particle coupling to a graviton, and for the on-shell three-graviton amplitude. While the amplitude (3.43) can be evaluated at the strict HCL, evaluating the amplitude (3.44) requires some care; the amplitude is first evaluated at momentum conservation violating kinematics where only k 3 + k 4 is taken to the HCL, and then the resulting expression is taken to the HCL. The reason for taking such a convoluted sequence of limits is because the strict HCL is the collinear limit k 3 ∝ k 4 ; A hhh = 0 at HCL.
Direct computation of the HCL approximation (3.42) to the difference (3.39) yields Depending on helicity, the three-point amplitudes of the b-topology cut scales as y ±2 . This means augmentation to the Compton amplitude must scale as y 0 or y ±4 to contribute to the triangle coefficient (3.36), but (3.45) do not have the required scaling; fixing the naïve Compton amplitude will not fix the mismatch of scattering angle computations.

Exciting internal structures and NUT-exciting results
As argued in appendix A, there is a strong tension in treating electrically charged and magnetically charged particles simultaneously as particles without internal structures, at least for abelian gauge theory. Although the same argument does not apply to gravity, the argument alludes to the possibility that NUT-charged particles cannot be structureless. This motivates the following question; will the disagreement between geodesic computations and amplitude computations be resolved when effects of NUT-charged particle's internal structures are taken into account? 11 This approximation relies on the observation that any polynomial Lorentz invariants constructed from the momenta and spinors of external kinematics satisfying the little group weight constraints of gravitons vanish in the HCL. An obvious loophole is when spinors parametrising Dirac-Misner string degrees of freedom can be used to form Lorentz invariants; e.g. the invariant [k3ξ] 4 [k4ξ] 4 for A ++ will not vanish in the HCL when the spinor parametrising the Dirac-Misner string |ξ] is independent of HCL, unless |ξ] ∝ |λ]. One way to model internal structure excitations of the particle is to consider transmutation of quantum numbers such as mass or spin. This process can be captured by on-shell threepoint amplitudes.

Unequal masses
The possibilities enumerated above exhausts all kinematicly allowed gauge-invariant on-shell three-point amplitudes as demonstrated in appendix B. Introduction of these processes affect the b-topology cut, figure 1, through alterations in the Compton amplitude or in the pair of three-point amplitudes. The equal mass case m φ = m φ ′ = m will be studied first, where cut momentum parametrisation (3.4) can still be used.
The modifications to the Compton amplitude from transmutation processes will always accompany s-or u-channel poles, therefore their primary effect appears on the four-particle cut; their effect on the three-particle cut is subleading in (β − 1) compared to the naïve answer (3.36). As an explicit demonstration, when approximations to the modifications of the Compton amplitude similar to the previous section are adopted, computation of b-topology contributions to A △ (b) from intermediate spin-1 (spin-2) particle contains additional powers of (β − 1) and yields the scaling ∝ q 0 (∝ q 2 ), which are contact interactions not affecting the long-distance physics 12 . For this reason, only the modifications to the pair of three-point amplitudes appearing on the b-topology cut, figure 1, will be studied in detail for the equal mass case.
For the equal helicity exchange channel, the sum over intermediate states for the pair of three-point amplitude yields which implies dy y A ++ A − A − = 0; the equal helicity exchange channel still does not contribute to the b-topology contribution A △ (b) . On the other hand, for the mixed helicity exchange channel the same computations lead to . An interesting observation point is that for s ′ = 2 transmutations, the modification to the b-topology contribution ∆A △ (b) scales as ∝ |q| 3 ∼ r −6 , which is known to be the 1/r dependence for tidal effect corrections [69]. While the resulting σ-dependence for m φ = m φ ′ case is inconsistent with the results of ref. [69], the conclusions are different for the case of m φ = m φ ′ .
For unequal mass case m φ ′ = m φ , the Compton amplitudes are modified by inclusion of φ ′ exchange on the s-and u-channels as where the three-point amplitudes (B.3) have been glued on the channels as in ref. [48] to arrive at the results. The absolute square of the coupling |γ ′ | 2 is related to the characteristic decay rate of resonance excitations and m 2 φ ′ −m 2 φ ≃ 2m φ ∆m is related to the characteristic frequency of the resonance excitations ω = ∆m.
Interestingly, the order of taking the HCL and effecting the dy y integral matters. When constituent tree amplitudes are taken to the HCL before dy y integration, the modifications to b-topology contribution are computed as which scales as ∝ |q| 3 ∼ r −6 and can be recast as where lowest order expansion in ∆m a = m a ′ − m a has been performed on the last line. This has an interpretation as tidal effects; comparing with eq.(4) of ref. [69], the tidal response coefficients λ and η can be related to γ ′ and ∆m by 13 which in turn relates them to the worldline tidal operator coefficients through eq.(12) of the same reference.
The above equation relates the parameters of dissipation through resonance excitations to that of tidal responses, which is reminiscent of Kramers-Kronig relations.
On the other hand, performing the dy y integration and then taking the HCL results in which has the wanted scaling ∝ |q| −1 ∼ r −2 . However (3.53) cannot resolve the discrepancy between geodesic computations and amplitude computations as the mass scaling is ∝ m 2 a m 3 b , similar to naïve Compton amplitude impulse computation (3.36); if the particle a having internal structures is considered as the background, then this contribution is suppressed by the mass ratio m b /m a compared to the a-topology contribution.
When the massive propagator of the φ ′ particle is cut as in figure 1 with the condition m φ ′ = m φ , the leading small q 2 contributions to ∆A △ (b) are computed as 14 (3.54) The interpretation of this result is unclear: Although the overall coefficient (∆m b /m b ) 6 highly suppresses the result, the above scales as |q| −3 ∼ log r which actually grows at long distances. If the limit m b ′ → m b is taken before the limit q 2 → 0 then the resulting cut computation reduces to (3.47) instead, therefore it is unlikely that computations leading to the above result are wrong. How to correctly interpret this result will be left for a future work. 13 If there are more than one resonance modes, the relation becomes a sum over different resonance modes. A quantum mechanical version of the relation has been considered in ref. [70].
14 The cut momentum parametrisations (3.4) are no longer valid, and the new cut conditions k 2 3 = k 2 4 = L 2 − m 2 b ′ = 0 were solved to arrive at the result (3.54).
In conclusion, exciting the NUT charges, modelled as transmutations to distinct particle species, cannot close the gap between geodesic computations of general relativity and amplitude computations at one-loop. While the results are not exciting from the viewpoint of original motivations, the relation (3.51) between internal excitation parameters and tidal deformation parameters obtained from (3.50) is an interesting by-product of the analysis that calls for future studies.

Conclusion
In this study the classical impulse at the second post-Minkowskian order (G 2 ) was analysed from Hamilton-Jacobi analysis of general relativity on Taub-NUT spacetime and quantum field theory one-loop amplitude calculations. Although the computations agreed at 1PM, the computations were shown to disagree at 2PM. Possible modifications to one-loop amplitude computations that could resolve the disagreement had been analysed, but all considered scenarios had failed to close the gap.
There are a few loopholes to the amplitude analysis, one being possible modifications to the amplitudes due to presence of Dirac-Misner strings. As mentioned in footnote 9, the argument that only the b-topology cut of figure 1 with m b ≫ m a is relevant for 2PM scattering angle analysis in the background-probe limit is based on the expectation that the Compton amplitude A φ b hhφ b , which appears in the a-topology cut, knows nothing about the mass of particle a. If the degrees of freedom parametrising the Dirac-Misner string attached to particle b somehow knows about m a then the argument based on (3.36) that a-topology contribution is negligible would break down. The approximations to the difference of correct and naïve Compton amplitude (3.42) also contains a similar loophole as explained in footnote 11.
It is also possible that application of unitarity methods found in the literature need generalisations when magnetic charges are involved. For example, it is unclear how pairwise little group introduced in ref. [71], which demands introduction of a new quantum number (pairwise helicity) for each dyonic pair of particles, can be applied to unitarity methods. When unitarity methods are applied to loop effects for electric charge-monopole scattering, the cuts can be performed in a way that all resulting tree diagrams only involve one type of charges. This results in an awkward situation where none of the constituent subdiagrams require pairwise helicity but the diagram as a whole does.
It would be an interesting exercise to ponder the ramifications of the relation (3.51) and its generalisations. Black holes of general relativity in four space-time dimensions are known to be inert to tidal forces, expressed in the form of vanishing Love numbers [72]. The relation implies that vanishing of tidal Love numbers poses a constraint on quantum mechanical excitations of the black hole. We also remark that the relation is similar in spirit to the Fermi four-fermion interactions being completed to weak interactions of the standard model.

Acknowledgments
We would like to thank to Yu-Tin Huang for technical discussions, Seok Kim for helpful suggestions, and our Ph.D. advisors, Sangmin Lee and Nakwoo Kim, for feedbacks. JK and MS also thank for the hospitality of Kyung Hee University and Seoul National University during their respective visits. JK was supported in part by the Science and Technology Facilities Council (

A The elementarity tension between electric charges and monopoles
It is well known that Lagrangian field theory description of a system including electrically charged particles and magnetically charged particles as elementary degrees of freedom in its spectrum simultaneously can only be attained at the price of (1) manifest Lorentz invariance or (2) manifest local description [73][74][75][76][77]. Nevertheless, the following question may be asked; is it eligible to describe electrically charged particles and magnetically charged particles as "elementary particles" in an effective theory whose cut-off is much higher than the particles' masses? Here, "elementary particle" refers to particles whose internal structures can be ignored and regarded as point particles. For example, in sufficiently low energy experiments protons and pions can be regarded as "elementary particles".
A simple argument shows this is unlikely. The classical radius r c is the scale where point particle description of the body breaks down in the classical theory, and the Compton wavelength λ C is the scale where classical description of the body breaks down and quantum theory must be introduced. If r c > λ C , there is an intermediate length scale L, r c > L > λ C , such that its non-point-particle-like structure must become manifest because it cannot be clouded by quantum effects. Therefore, for a particle to be described as an "elementary particle" its Compton wavelength must be greater than its classical radius; λ C > r c . A slightly stronger condition λ C ≫ r c cannot be imposed on electric and magnetic charges simultaneously due to Dirac-Schwinger-Zwanziger quantisation conditions.
Ignoring O(1) numerical coefficients, the classical radius and the Compton wavelength of an electrically charged particle of mass m and unit charge e is given as; Likewise, the ratio for a magnetically charged particle of unit charge is given as r c /λ C ≃ g 2 / c = α g . The least restrictive Dirac-Schwinger-Zwanziger quantisation condition demands eg ≃ O(1), which translates to α e α g ≃ O(1); the "elementary particle" conditions α e ≪ 1 and α g ≪ 1 are not compatible with the quantisation conditions, unless the effective theory cut-off scale Λ EFT screens the classical radius and allows "elementary particle" descriptions.
In the latter case, the condition Λ EFT < m must be satisfied by the heavier of the charges 15 .
Note that the "classical radius" for gravitational interactions is also a scale where Newtonian gravity becomes inadequate; the radius can be defined as the scale where Newtonian gravitational self-energy becomes comparable to its rest mass. Up to O(1) numerical coefficients, which is the scale of the body's Schwarzschild radius; general relativity must be introduced at this scale.
B All on-shell three-point amplitudes corresponding to transmutation All kinematicly allowed gauge-invariant on-shell three-point amplitudes of the type A φφ ′ h are enumerated in this appendix, which are special cases of the classification initiated in ref. [48]. The massive spinor-helicity variables of ref. [48] is used for massive spinning particles and conventions for the variables follow that of ref. [58].
For sake of simplicity, only the coupling to positive helicity graviton will be considered. The coupling to negative helicity can be achieved by exchanging 16  where |3] and |3 are spinors for the graviton leg and the exponent shows how many times each spinor appears on the numerator/denominator of the amplitude.

B.1 Equal mass
When m φ = m φ ′ = m, the kinematic factor x of (3.20) introduced in ref. [48] can be used to provide additional little group weight for the graviton leg. There are a total of three possibilities.
In the above expression, the transmuted particle φ ′ has been given the leg number 2 and the couplings γ i 's have been scaled to be dimensionless using κ and m.

B.2 Unequal mass
The kinematic factor x of (3.20) is gauge-invariant only if m φ = m φ ′ , where gauge-invariance denotes invariance of the expression under the gauge transform ε µ (p 3 ) ∼ ε µ (p 3 ) + cp µ 3 . Gauge invariance enforces the condition p 1 · p 3 = 0, which in turn enforces p 2 1 = p 2 2 ; the existence of x-factor demands 17 m φ = m φ ′ . This means four more spinors must be introduced to form a Lorentz invariant from four |3]α spinors of the graviton leg. The only possibility is to introduce four spinors of the φ ′ leg, meaning the transmuted particle φ ′ must have spin s ′ = 2.
This result is consistent with angular momentum conservation and can be understood as a gravitational analogue of the selection rule ∆L = ±1 for dipole transitions of the hydrogen atom; when m φ = m ′ φ , a real process of heavier of the two decaying into the lighter of the two by emitting a graviton is allowed.
Assuming m φ > m ′ φ , the decay process φ → φ ′ h can be analysed at the rest frame of φ. In this frame the total angular momentum of the initial state φ is zero, which must be conserved after the decay. Since orbital angular momentum of the final state φ ′ h cannot contribute to the total angular momentum directed along the direction of the emitted graviton, the spins of the decay products must sum up to zero. The graviton has spin 2 along this direction, so the decay product φ ′ also must have spin 2 along this direction with opposite signs. Extension of the analysis for the case m φ < m ′ φ is straightforward.

C Dyonic generalisation of effective one-body formalism
In Newtonian mechanics, the two-body problem can be reduced to a one-body problem using reduced mass. The effective one-body (EOB) formalism is an attempt to generalise the reduction to relativistic systems, where the two-body motion is mapped to an effective onebody geodesic motion on a background metric [79]. The formalism often reduces to finding a map between the real energy E real of the two-body system and the effective energy E eff of the one-body geodesic motion, which smoothly interpolates between background-probe limit and comparable mass case. In the extreme mass-ratio limit the effective metric often reduces to known black hole solutions. To find the energy map, one first sets the action-angle variables of the real two-body system and the effective one-body system to be the same, and then inverts the relation between energy and action-angle variables to obtain a relation between the energy of the real system and the energy of the effective system.
We will attempt a generalisation of effective one-body formalism for scattering states [80] to cases involving "magnetic mass", using amplitude computations as the input for real dynamics. The dynamics obtained from amplitudes defies an interpretation as the motion on a Taub-NUT background with backreaction of the probe particle taken into account, as the extreme mass-ratio limit is not the geodesic motion on a Taub-NUT background. Therefore, or, equivalently, (C.10) The effective deflection angle can be obtained as (C.11)

C.3 The 2PM mapping
Using β 1 = 1 from first order matching, the second order deflection angle with undetermined parameters {β} of the effective metric is given as (C.18) In the Schwarzschild limit, the effective deflection angle reduces to This is in contrast to the second order result from amplitude computations, which does not depend on the phase ϕ but only on the inertial masses. (C.20) Apparently (C.20) cannot be reproduced from (C.18) by adjusting the β 2 and β 3 , but can be matched to (C. 19) under absence of NUT charges. It seems our expectation of finding a consistent effective metric of EOB formalism that matches amplitude computations cannot be met by the generalisation considered.

(C.21)
It is suffice to solve for the zeros of the functions to be the same, u r,0 = u e,0 . The two solutions, β 1 = 1 and β 2 = 1, correspond to the second order post-Minkowskian expansion of the Schwarzschild solution. This gives us the same energy map (C.17) obtained from first order analysis.