Unified limiting form of graviton radiation at extreme energies

We derive the limiting form of graviton radiation in gravitational scattering at transplanckian energies ($E\gg M_P$) and small deflection angles. We show that --- owing to the graviton's spin 2 --- such limiting form unifies the soft- and Regge- regimes of emission, by covering a broad angular range, from forward fragmentation to deeply central region. The single-exchange emission amplitudes have a nice expression in terms of the transformation phases of helicity amplitudes under rotations. As a result, the multiple-exchange emission amplitudes can be resummed via an impact parameter $b$-space factorization theorem that takes into account all coherence effects. We then see the emergence of an energy spectrum of the emitted radiation which, being tuned on $\hbar/R \sim M_P^2/E \ll M_P$, is reminiscent of Hawking's radiation. Such a spectrum is much softer than the one na\"ively expected for increasing input energies and neatly solves a potential energy crisis. Furthermore, by including rescattering corrections in the (quantum) factorization formula, we are able to recover the classical limit and to find the corresponding quantum corrections. Perspectives for the extrapolation of such limiting radiation towards the classical collapse regime (where $b$ is of the order of the gravitational radius $R$) are also discussed.


I. INTRODUCTION
The thought experiment of trans-Planckian-energy gravitational scattering has been investigated, since the eighties [1][2][3][4][5][6][7], as a probe of quantum-gravity theories, mostly in connection with the problem of a possible loss of quantum coherence in a process leading classically to gravitational collapse. In an S-matrix framework such a loss would be associated with the breakdown of unitarity at sufficiently small impact parameters.
In the scattering regime of large energies ( ffiffi ffi s p ≫ M P ) but small deflection angles (i.e., in a regime far away from that of collapse), several authors proposed [1][2][3][4][5], on various grounds, an approximate semiclassical description, whose S-matrix exponentiates, at fixed impact parameter, an eikonal function of order α G ≡ Gs=ℏ ≫ 1, which is simply related to graviton exchanges at large impact parameters b ≫ R ≡ 2G ffiffi ffi s p . Such a description has its classical counterpart in the scattering of two Aichelburg-Sexl (AS) shock waves [8].
Starting from that leading eikonal approximation, the strategy followed in [6,7] consisted in a systematic study of subleading corrections to the eikonal phase, scattering angle, and time delays [9][10][11] in terms of the expansion parameter R 2 =b 2 (and l 2 s =b 2 if working within string theory). These corrections can be resummed, in principle, by solving a classical field theory and one can thus study the critical region b ∼ R where gravitational collapse is expected.
This program was carried out, neglecting string corrections and after a drastic truncation of the classical field theory due to Lipatov [12], in [13] (see also [14][15][16]). It was noted there that below some critical impact parameter value b c ∼ R (in good agreement with the expected classical critical value [17][18][19][20]), the S matrix-evaluated by taking UV-safe (regular), but possibly complex, solutions of the field equations-shows a unitarity deficit. This was confirmed, at the quantum level, by a tunneling interpretation of such restricted solutions [21][22][23]. The above results suggest that the lost information could possibly be recovered only through use of UV-sensitive solutions which, by definition, cannot be studied by the effective-action approach of [13] and remain to be investigated on the basis of the underlying (string) theory itself. It is also possible, of course, that the apparent loss of unitarity is caused instead by the drastic truncation made in [13] of Lipatov's effective field theory [12].
On the other hand, the parallel investigation of gravitational radiation associated with trans-Planckian scattering brought a worrisome surprise: even if such radiation is pretty soft, hqi ≃ ℏ=b being its typical transverse momentum, its rapidity density ∼α G is so large as to possibly endanger energy conservation [24,25], at least in the early naïve extrapolations of the available rapidity phase space [13,16]. Energy conservation can be enforced by hand, the result being that the flat low-energy spectrum (predicted by known zero-frequency-limit theorems [26]) extends up to a cutoff at ω ∼ b 2 =R 3 . But that would mean that a fraction Oð1Þ of the initial energy is emitted in gravitational radiation already at scattering angles Oðα −1=2 G Þ ≪ 1, something rather hard to accept.
This unexpected result prompted the study of the purely classical problem of gravitational bremsstrahlung in ultrarelativistic, small-angle gravitational scattering, a subject pioneered in the seventies by Peter D'Eath and collaborators [27,28] and by Kovacs and Thorne [29,30]. Those papers, however, were rather inconclusive about the ultrarelativistic limit (the method of Refs. [29,30], for instance, does not apply to scattering angles larger than m=E ≡ γ −1 , and thus, in particular, to our problem). Nonetheless, two groups of authors [31,32] managed to discuss directly the massless limit of the classical bremsstrahlung problem showing the absence of an energy crisis and the emergence of a characteristic frequency scale of order R −1 beyond which the emitted-energy spectrum is no longer flat (within the approximations used in [31] the spectrum decreases like ω −1 till the approximation breaks down at ω ∼ b 2 =R 3 ). These classical results called for a more careful investigation of the quantum problem.
And indeed the good surprise was that-after a careful account of matrix elements, phases, and coherence effects-the limiting form of such radiation for α G ≫ 1 takes a simple and elegant expression and has the unique feature of unifying two well-known limits of emission amplitudes: the soft and the Regge limit. As a consequence, besides reducing in a substantial way the total emittedenergy fraction, the spectrum drifts towards characteristic energies of order ℏ=R ∼ M 2 P =E ≪ M P , much smaller than those expected from the naïve Regge behavior, and reminiscent of Hawking's radiation [33] (see also [34]) from a black hole of mass E. That nice surprise, which we illustrate here in full detail, has been presented recently in a short paper [35].
We should note incidentally that, in a different but related investigation of trans-Planckian graviton production integrated over impact parameter, a similarly surprising feature was found (even more surprisingly by a tree-level calculation) in [36], the typical energy of the emitted gravitons being again of order ℏ=R, with a very large multiplicity of order s=M 2 P i.e. of a black hole entropy for M ∼ ffiffi ffi s p . The above list of surprises points in the direction of a more structural role of the gravitational radius in the radiation problem, rather than in the scattering amplitude calculation itself, so that approaching the collapse region at quantum level may actually be easier and more informative if made from the point of view of the radiation associated with the scattering process.
One may wonder what the deep reason is for all that. Here we show that our unified limiting form of radiation, at the first subleading level in the parameter R 2 =b 2 , is due to the dual role of the graviton spin two: on the one hand it determines, by multigraviton exchanges, the leading AS metric associated with the colliding particles as well as its radiative components at first subleading level; on the other hand, it also determines the transformation properties of the emission amplitudes for definite helicity final states. These, in turn, are closely connected to the emission currents themselves.
For the above reasons-after a brief introduction to eikonal scattering in Sec. II-we emphasize (Sec. III) the physical matrix elements of the relevant emission currents whose phases-due to the absence of collinear singularities in gravity-play a crucial role in both the soft and the Regge regimes. The unified form of graviton emission is then determined-at the single-exchange level-by matching the soft and Regge behaviors in all relevant angular regions, from nearly forward fragmentation to deeply central emission. The resulting expressions are just the Fourier transforms of two different components of the radiative metric tensor, which, however, yield identical results because of a transversality condition. The next step in the construction of the emission amplitudes is to resum the contributions of all the graviton exchanges that occur during eikonal scattering. This is done in Sec. IV, by establishing a b-factorization theorem for each single-exchange contribution, and by summing them up with the appropriate phases due to the dependence of the helicity amplitudes on the incidence direction. The outcome, already presented in [35], has a classical limit that resembles (but slightly differs from) the one of [31]. An important new result of this work is that, by also taking into account rescattering of the emitted gravitons all over the eikonal evolution, the classical limit of [31] is fully recovered together with some (or perhaps all) quantum corrections to it. This resummation yields a coherent average over incidence directions, up to the Einstein deflection angle Θ s ðbÞ ¼ 2R=b, providing important (de) coherence effects which tend to suppress frequencies of order ω > R −1 . The above procedure is finally generalized to multiple emissions by constructing the appropriate (unitary) coherent-state operator. The spectrum is then described and analyzed in Sec. V, both in frequency and in angular distribution. This is done, in this paper, by taking into account the incidence angle dependence only. Including rescattering effects, both at the classical and quantum level, is deferred to a later work. The ensuing perspectives for the development of the present method towards the classical collapse region (given in Sec. VI) are based on the new features of the resummation pointed out in this paper, which are typical of the emitted gravitational radiation associated with trans-Planckian scattering. Finally, a number of detailed calculations and useful remarks are left to the appendixes.

II. TRANS-PLANCKIAN EIKONAL SCATTERING
Throughout this paper, as in [13], we restrict our attention to collisions in four-dimensional space-time and in the point-particle (or quantum field theory) limit. Consider first the elastic gravitational scattering p 1 þ p 2 → p 0 1 þ p 0 2 of two ultrarelativistic particles, with external momenta parametrized as describe both azimuth ϕ i and polar angles jΘ i j ≪ 1 of the corresponding 3-momentum with respect to the longitudinal z axis. This regime is characterized by a strong effective coupling α G ≡ Gs=ℏ ≫ 1 and was argued by several authors [1,2,4,6] to be described by an all-order leading approximation which has a semiclassical effective metric interpretation. The leading result for the S-matrix Sðb; EÞ in impact-parameter b ≡ J=E space has the eikonal form Sðb; EÞ ¼ exp½2iδ 0 ðb; EÞ; L being a factorized-and thus irrelevant-IR cutoff.
Corrections to the leading form (2.2) involve additional powers of the Newton constant G in two dimensionless combinations being the Planck length. Since α G ≫ 1 we can neglect completely the first kind of corrections. Furthermore, we can consider the latter within a perturbative framework since the impact parameter b is much larger than the gravitational radius R ≡ 2G ffiffi ffi s p .
In order to understand the scattering features implied by (2.2) we can compute the Q-space amplitude where the expression in the last line is obtained strictly speaking by extending the b integration up to small jbj ≲ R [1], where corrections may be large. But it is soon realized that the b integration in (2.4) is dominated by the saddlepoint which leads to the same expression for the amplitude, apart from an irrelevant Q-independent phase factor. The saddlepoint momentum transfer (2.5) comes from a large number hni ∼ α G of graviton exchanges ( Fig. 1), corresponding to single-hit momentum transfers hjq j ji ≃ ℏ=b which are small, with very small scattering angles jθ j j of order θ m ≃ ℏ=ðbEÞ. The overall scattering angle, though small for b ≫ R, is much larger than θ m and is jΘ s j ¼ 2R=b ¼ 2α G θ m , the Einstein deflection angle. In other words, every single hit is effectively described by the elastic amplitude which is in turn directly connected to the phase shift δ 0 : The relatively soft nature of trans-Planckian scattering just mentioned is also-according to [4]-the basis for its validity in the string-gravity framework. In fact, string The scattering amplitude of two trans-Planckian particles (solid lines) in the eikonal approximation. Dashed lines represent (Reggeized) graviton exchanges. The fast particles propagate on shell throughout the whole eikonal chain. The angles Θ j ≃ P j−1 i¼1 θ i denote the direction of particle 1 with respect to the z axis along the scattering process. theory yields exponentially suppressed amplitudes in the high-energy, fixed-angle limit [37] so that several softer hits may be preferred to a single hard one in the b ≫ R regime. Furthermore, this procedure can be generalized to multiloop contributions in which the amplitude, for each power of G, is enhanced by additional powers of s, due to the dominance of s-channel iteration in high-energy spin-2 exchange versus the t-channel one (which provides at most additional powers of log s). That is the mechanism by which the S matrix exponentiates an eikonal function (or operator) with the effective coupling α G ≡ Gs=ℏ and subleading contributions which are a power series in R 2 =b 2 (and/or l 2 s =b 2 ). Finally, the scattering is self sustained by the saddle point (2.5), so that string effects themselves may be small-and are calculable [4,6]-if b ≫ R; l s ≫ l P and even at arbitrary b if R ≪ l s .
Both the scattering angle (2.5) [and the S matrix (2. 2)] can be interpreted from the metric point of view [1] as the geodesic shift (and the quantum matching condition) of a fast particle in the AS metric [8] of the other.
More directly, the associated metric emerges from the calculation [9] of the longitudinal fields coupled to the incoming particles in the eikonal series, which turn out to be Such shock-wave expressions yield two AS metrics for the fast particles, as well as the corresponding time delay and trajectory shifts at leading level. When b decreases towards R ≫ l s , corrections to the eikonal and to the effective metric involving the R 2 =b 2 parameter have to be included, as well as graviton radiation, to which we now turn.

III. LIMITING FORM OF EMISSION FROM SINGLE-GRAVITON EXCHANGE
The basic emission process p 1 þ p 2 → p 0 1 þ p 0 2 þ q at tree level (Fig. 2) of a graviton of momentum q μ ∶q ¼ ℏωθ yields simple, and yet interesting, amplitudes in various angular regimes (Fig. 3) that we now consider, assuming a relatively soft emission energy ℏω ≪ E. Note that this restriction still allows for a huge graviton phase space, corresponding to classical frequencies potentially much larger than the characteristic scale R −1 , due to the large gravitational charge α G ≡ Gs=ℏ ≫ 1. We consider three regimes: (a) The regime jθ s j > jθj (where jθ s j ¼ jq s j=E is the single-hit scattering angle) is characterized by relatively small emission angles and subenergies. If scattering is due to a single exchange at impact parameter b, then jθ s j ∼ ℏ=Eb ≡ θ m andq is nearly collinear top 1 . In that region the amplitude is well described by external-line insertions, but turns out to be suppressed because of helicity conservation zeroes.
In this regime the subenergies reach the threshold of high-energy (Regge) behavior, still remaining in the validity region of external-line insertions, due to the condition jq s j ¼ Ejθ s j > ℏωjθj ¼ jqj which suppresses insertions on exchanged graviton lines. (c) Finally, in the regime jθ s j < ℏω E jθj the soft approximation breaks down in favor of the (high-energy) H-diagram amplitude [6] which contains internal-line insertions also [12].

A. Soft amplitudes in the Weinberg limit
In the soft regime (a∪b), the emission amplitude M ðλÞ soft of a graviton with momentum q and helicity (or polarization) λ can be expressed as the product of the elastic amplitude M el ¼ κ 2 s 2 =Q 2 and the external-line insertion factor J We are interested in the projections of the Weinberg current over states of definite positive/negative helicity, which can be conveniently defined by ð3:3Þ with ε 12 ¼ 1 and the − and þ signs in ϵ μ L corresponding to a graviton emission in the forward and backward hemisphere respectively.
By referring, for definiteness, to the forward hemisphere, we define the momentum transfers q 1ð2Þ ≡ p 1ð2Þ − p 0 1ð2Þ , q ¼ q 1 þ q 2 , and the scattering angle q 2 ≡ Eθ s , and restrict ourselves to the forward region jθj; jθ s j ≪ 1. Giving for ease of notation the results for a single helicity, a delicate but straightforward calculation (Appendix A) leads to the following explicit result in the c.m. frame with p 1 ¼ 0: where jQj can be unambiguously identified with jq 2 j in the (a) and (b) regions where Eq. (3.4) is justified. The simple expression (3.4) shows a 1=ω dependence, but no singularities at either θ ¼ 0 or θ ¼ θ s as we might have expected from the p i · q denominators occurring in (3.1). This is due to the helicity conservation zeros in the physical projections of the tensor numerators in (3.1). Therefore, there is no collinear enhancement of the amplitude in region (a) with respect to region (b), while we expect sizeable corrections to it in region (c), where internal insertions are important. The helicity phase transfer in Eq. (3.5) has a suggestive interpretation, made manifest by introducing a "z representation" (proven in Appendix A 3) as an integral between initial and final directions in the transverse z plane of the complex component of the Riemann tensor [31] in the AS metric of the incident particles. For our analysis we need to work both in momentum and in impact parameter space. We define b-space amplitudes, following the normalization convention 1 of [13] and [35], as The first behavior is typical of the IR amplitude, showing no singularities in the collinear (θ → 0) limit, and will be relevant for our final result also.
The second behavior in (3.13), after a simple integration in θ s , to which only the cos ϕ θ s sin ϕ θ term contributes, yields the result (where, for simplicity, we choose the x axis in the transverse plane to be aligned with b, that is ϕ θ ≡ ϕ θ − ϕ b ), which provides the most important term in the (b) region. We note that its maximum at ϕ θ ¼ π=2 is a reminder of the collinear zeroes.
As for region (c), we already noticed that the soft evaluation breaks down there in favor of the high-energy amplitude, and in Sec. III C we substantially improve our b-space amplitude in all regions by matching the soft and high-energy evaluations explicitly. The corresponding estimate, though yielding subleading corrections in region (b), will considerably change the bjqj > 1 behavior of Eq. (3.14).

B. Amplitude transformation
In order to compute the emission in the general case we need to establish an important point regarding the representation of the soft and Regge (see Sec. III C) singleexchange amplitudes. As already noted, the expression (3.5) is valid as it stands only if the initial direction (of the momentump 1 ) of the emitting particle is along the z axis, or forms with it a small angle jΘ i j ≪ θ m , the single-hit large angle threshold. But as shown in Sec. II, in the eikonal evolution the fast particles scatter on average by the angle jθ s ðbÞj ¼ 2R=b ¼ ðGsÞθ m ≫ θ m , and thus we need to compute the amplitudes in the case where the emission takes place with a generic incidence angle Θ i , possibly much larger than θ m .
Because of Lorentz invariance, we expect the Θ i dependence to occur through rotation scalars (which, in the small-angle kinematics, involve the differences Θ i − Θ j , the latter being angular 2-vectors of the fast particles), and a specific transformation phase also. The latter is in turn dependent on the definition of the helicity states jλ; q; Á Á Ái which is not uniquely determined. Since λ is a Lorentz invariant for the massless graviton, such a transformation phase is only allowed by the ambiguity in relating the q states to the z-axis state, due to the residual rotational invariance around the latter. For instance, Jacob and Wick (JW) [39] relate q to z by a standard rotation around the axis perpendicular to the hq; zi plane. Since such a definition is fully "body fixed" (that is, independent of external observables) we expect the JW amplitudes to be invariant for small rotations of the z axis around an axis perpendicular to it, because in such a limit the rotations involved will commute. 2 On the other hand, our helicity states are defined in terms of the physical polarizations in Eqs. (3.2)-(3.3), which are dependent not only on q μ , but also on the z axis, which occurs as an external variable in the T projection. We show in Appendix B that the ensuing relation to JW amplitudes is a simple multiplication by expðiλϕ θ Þ, where the azimuthal variable is generally Oð1Þ. Since the Θ i rotation acts on θ as the translation θ − Θ i for small polar angles, we expect the transformation phase to be nontrivial and given by exp½iλðϕ θ − ϕ θ−Θ i Þ, as fully proved in Appendix B and explicitly checked in Appendix A.
Therefore, in the forward region Θ i ; Θ f ; θ ≪ 1, Θ f being the outgoing direction of the (intermediate) fast particle, the momentum-space helicity amplitudes transform as (Appendix B) where λ is the helicity of the emitted graviton, λ ¼ −2 in our case. In b space, i.e., by Fourier transforming with respect to Q ¼ EðΘ f − Θ i Þ, one finds The expression (3.15) is easily argued for as a consistency requirement for the insertion of the Weinberg current on the double-exchange process (Fig. 4) in the soft limit. In fact, by the identity of Weinberg contributions ð3:17Þ which agrees with the direct calculation of Appendix A as well, from which it follows that (note that θ s ≃ Θ f − Θ i for the soft amplitude) C. Matching of soft amplitude with the Regge limit As soon as the rapidity interval 2Y b ¼ 2 logðEb=ℏÞ between p 0 1 and p 0 2 , and the relative rapidity Y b − y between p 0 1 and q become large, high-energy emission in the Regge limit becomes relevant, as predicted by the Lipatov vertex [12] (see Fig. 5) and the H diagram [6] [see Fig. 6(a)]. More precisely, for scattering due to singlegraviton exchange, the Regge limit is relevant in the region where the graviton is emitted at a relatively large angle, 1 ≫ jθj ≫ θ m ≡ ℏ=ðEbÞ (as already noted, jθ s j ∼ θ m in the single-hit case). This large-angle region comprises regions (b) and (c) discussed at the beginning of Sec. III and is particularly relevant for region (c) in which internal-line insertions are important.
Using the same kinematic notation as in Sec. III A, the Lipatov current is given by [12] (see also [6]) where q ⊥1 , q ⊥2 , q ⊥ denote transverse (vectorial) components to thep 1 direction (which of course coincide with q 1 , q 2 and q when p 1 ¼ Θ i ¼ 0) and the corresponding graviton emission amplitude (considering again a single helicity for definiteness) is note that −jq 1ð2Þ⊥ j 2 corresponds to the virtuality q μ 1ð2Þ q 1ð2Þμ in the Regge kinematics.
More quantitatively, in the c.m. frame with zero incidence angle (p 1 ¼ EΘ i ¼ 0) and in the forward region jθj; jθ s j ≪ 1, the amplitude takes the form [13] (see also Appendix A) (remember that q ¼ ωθ and q 2 ¼ Eθ s ). The corresponding amplitude in b space, according to the definition (3.7), is given by wherehðb; qÞ admits the integral representation ð3:23Þ and turns out to be equal to the Fourier transform with respect to q of the H-diagram field. The latter's expression in the space of the transverse coordinate x ¼ ðx 1 ; x 2 Þ of the emitted graviton is As is the case for the soft current (3.18), (3.22) is valid as it stands only if the initial p 1 direction is along the z axis. However, for a generic p 1 direction, the amplitude in the Regge limit transforms in the same way as the soft one in Eq. (3.16), that is (where EΘ i is the transverse part of the 4-momentum p 1 ), as directly proven in Appendix A.
To connect the small-angle (soft) and large-angle (Regge) regimes of the one graviton emission amplitude, it is convenient to rewrite Eq. (3.22) in terms of the (complex) variables θ ¼ q=ω and θ s ¼ q 2 =E: This expression differs from Eq. (3.8) by the replacement in the integrand of Eq. (3.26). By inspection, we see the important point that Regge and soft evaluations agree in region (b), in which the condition jθj ≫ jθ s j ≫ ω E jθj ensures that we are in the large-angle regime in the lhs with negligible internal insertions in the rhs, while Eq. (3.26) remains the only acceptable expression in region (c), where jθ s j < ω E jθj. Therefore, in order to get a reliable emission amplitude holding in all regions (a∪b∪c), we have to match the soft with the Regge evaluations. We start from the Fourier transform in Eq. (3.8) and we then add the difference of Regge and soft evaluations of Eq. (3.27) in region (c) and in part of region (b), the border being parametrized by the cutoff Δ c > 1 (see Fig. 3). Such difference has the form so that we get the expression ð3:30Þ If we then choose 1 ≪ Δ c ≪ E=ω the result (3.30) is weakly cutoff dependent and, in the Δ c → ∞ limit, is formally equal to the negative of the Fourier transform of the soft amplitude on the whole phase space, rescaled at By then using Eq. (3.8) we obtain the explicit form of the matched amplitude (with explicit ℏ dependence) where we have used the z representation of the helicity phases (3.6), by rescaling the z variable in the first term. The final result of Eq. (3.32)-derived on the basis of the soft-insertion formulas-is expressed in terms of the (ω-dependent) soft field in which the function Φ R turns out to be useful for the treatment of rescattering, too (Sec. IV B). Furthermore, for relatively large angles [θ ≫ θ m ∼ ℏ=ðEbÞ], Eq. (3.32) involves values of ℏωjzj=E ≲ θ m =θ which are uniformly small, and the expressions (3.33) can be replaced by their The latter quantities have a classical meaning, h s ðzÞ as a metric component (Sec. III D) and ΦðzÞ as a modulation function in the classical treatment of Ref. [31]. As a consequence, Eq. (3.32) takes the simpler form that will be mostly used in the following. Replacing h s ðzÞ by its ω-dependent form is needed if we want to treat the very-small-angle region and some quantum corrections also. From Eq. (3.35) we can see directly how the matching works. In fact, due to Eq. (3.32), the linear term (the log term) in Eq. (3.34) is in correspondence with external (internal) insertions of the emission current. In region (a), where z is pretty large, the linear term dominates and provides directly the soft limit. In region (b), the basic soft behavior (3.14) is reproduced but, with increasing values of bjqj, it is actually canceled by internal insertions in region (c), because in the small jzj limit the function ΦðzÞ is of order ∼Oðjzj 2 Þ. This is confirmed by the Regge representation (3.23) which shows, by inspection, a 1=ðbjqjÞ 2 behavior for bjqj ≫ 1.
To summarize, our matched amplitude (3.35), which, by construction, should be identical to the Regge one of Eq. (3.22) in region (c), is also a nice interpolation in (b∪c) and part of (a) with jθj > θ m . 3 For this reason we call Eq. (3.35) [Eq. (3.22)] the soft-based (Regge-based) representation of the same unified amplitude. Their identity can be directly proven by the equation which can be explicitly checked by switching to z; z Ã variables and integrating by parts. Equation (3.36) is in turn a direct consequence of the differential identity that is related in Sec. III D to a transversality condition of the radiative metric tensor. Our unified soft-Regge amplitude M has then, for a generic Θ i , the form where the Regge-based (soft-based) representation is used in the first (second) line. Equations (3.32)-(3.33) provide an improved small-angle description and some quantum corrections.

D. Radiative metric tensor
To complete the picture of single-exchange radiation, we recall the parallel calculation of radiative corrections to the metric fields and to the effective action [7,13]. At first subleading level this amounts to calculating the H-diagram fields δh and δa [Figs. 6(b) and 6(c)] occurring in the metric. By leaving aside time delays [9] we obtain [13] where, starting from a 0 in Eq. (2.8) we expand the profile function aðxÞ, to first order in R 2 =b 2 , and the field hðxÞ in the form (Fig. 6) andε μν are polarization tensor operators of the form, for instance, 4 3 The moderate-angle restriction becomes unimportant when the resummation of Sec. IV extends the collinear region up to Θ s ∼ R=b ≫ θ m . 4 With this prescription, the metric tensor (3.39) satisfies the transversality condition ∂ μ h μν ¼ 0. The polarization tensors differ from those of [13] by a factor of 1=2.
with similar ones of the LT polarization. Such results follow from a shock-wave solution of the effective-action equations of motion which expresses all metric components in terms of the basic scalar field hðxÞ ≡ 4j∂j 2 ϕðxÞ, where the explicit, single-valued form of ϕðxÞ, not to be confused with the modulating function ΦðzÞ, was found in [7] and is given by the single-valued ð3:43Þ From Eqs. (3.43) and (3.39), we obtain, in particular, the transverse plane metric components which are closely related to the fields h and h s introduced previously, because of the derivatives and are thus rooted in the spin-2 structure of the interaction. Furthermore, in parallel with the soft-based representation (3.35) for jet 1, we have by (3.45) the corresponding representation of the same amplitude in jet 2 Note however that, while hðxÞ ∼ h xx Ã is a scalar, the soft fields h s ðzÞ [h Ã s ð1 − zÞ] in jet 1 (jet 2) have J z ¼ 2ð−2Þ for rotations in the transverse plane, and in fact the opposite phase is factorized in Eq. (3.35) [Eq. (3.47)] which is relevant for jet 1 (jet 2) and helicity λ ¼ −2. In all cases the resulting amplitude will come out invariant for rotations around the z axis.
Finally, by starting from the soft-based representations just mentioned at Θ i ¼ 0 we can complete the symmetry x → b − x by constructing the helicity amplitudes for both jets at which relates opposite helicities with opposite 3-momentã q as a function of (3.48) can be checked in a straightforward way by using the explicit helicity projections of Appendix A, and is anyway a consequence of the helicity transformation properties of the soft fields in either jet.
We then conclude that the radiative metric components of Eq. (3.39) (based on the shock-wave solution of [7]) are consistent with the present single-exchange soft-Regge amplitudes and actually explain the unifying relationships by a transversality condition of the metric tensor. We see however that taking into account the helicity transformation phases is essential for completing the calculation of graviton emission, and superimposing single-exchange terms all along eikonal scattering.

IV. FACTORIZATION AND RESUMMATION
A. b factorization and matching So far we have considered the radiation associated to the single-graviton exchange contribution to the basic Planckian scattering process. But we know (Sec. II) that such a high-energy process is described by the eikonal resummation of a large number ∼Gs=ℏ of single hits, so that, for a given impact parameter value b, the scattering angle increases from θ m ¼ ℏ=ðEbÞ to jθ s j ¼ 2ðGs=ℏÞðℏ=EbÞ ¼ 2R=b, the Einstein deflection angle. This fact considerably enlarges the quasicollinear region with respect to θ s , which might be an important source of energy loss by radiation, so as to endanger energy conservation [25] unless explicitly enforced.
We start noticing that, despite the enlargement of the quasicollinear region, the external-line insertion amplitude (3.5) stays unchanged, being only dependent on the overall momentum transfer Q ¼ Eθ s of the process. Therefore, the damping of the collinear region and the large-angle behavior are both built in Eq. (3.12) for any values of θ s , as shown by Eq. (3.13).
One may wonder to what extent-or for which ω values-the external-line insertion method is tenable. There are two types of internal-line insertions: those on the fast, nearly on-shell particle lines of the eikonal iteration, and those on the exchanged-graviton lines. We now argue that both kinds of internal insertions can be taken into account using the soft/Regge matching strategy described in Sec. III C.
The first thing to notice is that fast-particle insertions are in fact already implicitly included in the soft approximation (3.5). In fact, in a general n-exchange contribution to eikonal scattering for each pair of propagating lines there are two pairs of insertions, one with the mass shell on the right (final) and one on the left (initial), whose currents are nearly equal (to order ℏω=E) and opposite in sign. Thus, the purely soft emission can be written in two equivalent ways, one as in Eq. (3.5) as a purely external line insertion, and the other as a sum of n contributions, where the insertion-still of the form (3.5)-is made internally on the fast lines which surround the ith exchanged graviton. Since in general these fast lines will have accumulated a nonnegligible transverse momentum, the ith internal insertion will be of the rotated form (3.18), with Θ i ¼ P i−1 j¼1 θ j . As for insertions on exchanged graviton lines, they are negligible if the emitted transverse momentum q ¼ ℏωθ is smaller than any of the exchanged graviton lines which, in the Regge limit, are all of order hq i i ∼ ℏ=b, thus leading to the condition bjqj ≪ ℏ. The latter is not surprising because precisely this parameter occurs in the subtraction term of Eq. (3.32) coming from the Regge estimate of region (c) for a single exchange. Fortunately this region-which is generally multidimensional for n exchanges-is most significant when both exchanged and emitted momenta are of the same order ℏω ≪ E. For double graviton exchange (Fig. 7), for instance, this corresponds to two diagrams, and more generally it allows one to count n diagrams per eikonal contribution in which we have to compute and add ΔM, i.e., the difference of Regge and soft amplitudes introduced in Sec. III C. Again, this has to be adjusted to take into account the direction of the fast legs, which is straightforward since the Regge and soft amplitudes acquire the same transformation phase.
In the end, this means that all internal-line insertionsfor fast particles and exchanged particles alike-can be accounted for by calculating n diagrams for the eikonal contribution with n exchanged gravitons, where the matched amplitude [Eqs. (3.32) and (3.35)] is inserted in turn in correspondence to the ith exchanged graviton, adjusting for the local incidence angle Θ i as in Eq. (3.18).
We recall at this point the soft-based representation of the matched amplitude (3.32), which, after adjusting for the incidence angle, acquires the form and is thus simply proportional to the translated Fourier transform (F.T.) of the soft field h s . We then use this representation for each "active" contribution. For n ¼ 2 (Fig. 7), for instance, we have q s1 ¼ Eθ 1 emitting with initial angle Θ 1 ¼ 0 and q s2 ¼ Eθ 2 with Θ 2 ¼ θ 1 , and we write, accordingly, We can see that the second active contribution, with nonzero incidence angle Θ 2 ¼ θ 1 has a translated θ dependence, which amounts to factorizing an eikonal with z-dependent argument. This generalized factorization can be extended to the general case with n > 2 exchanges, where however the θ translation involves Θ i ¼ P i−1 j¼1 θ j , i > 2, yielding higher powers of the eikonal with z-dependent argument. In formulas, we obtain, order by order, Furthermore, the sum in square brackets is given by the expression so that we finally get the factorized and resummed amplitude ð4:5Þ where we have expanded the logarithm in the exponent and neglected higher order terms in ℏ=Ebjθj. The latter can in principle be evaluated as quantum corrections to the basic formula of the last line. A more symmetric expression for the resummed amplitude (4.5) is obtained in the Breit frame (also called brickwall frame), where the initial and final transverse momenta are equal and opposite [i.e., AE 1 2 EΘ s ðbÞ]. We can reach the Breit frame by rotating the system of 1 2 Θ s ðbÞ. According to Eq. (4.1), this amounts to translating θ → θ − 1 2 Θ s ðbÞ, and at the end we obtain again the expression (4.5) but with ξ integrated over the symmetric interval ½−1=2; 1=2. In the following we often work in the Breit frame.
It is important to note that the z dependence in Eq. (4.3) adds corrections to the naïve factorization of δ 0 ðbÞ [13] which for any given n are of relative order ℏωhzi=E ∼ ℏ=ðbEjθjÞ and thus may appear to be negligible in the region jθj > θ m . However, this is not the case because of the counting factors of hni ∼ Gs=ℏ occurring in Eq. (4.4), which promote such corrections to order jΘ s ðbÞj=jθj making them essential for the physics of the radiation problem at frequencies ω ∼ R −1 . The effect of such corrections can be summarized by the introduction of the resummed field in the Breit frame with its resummation factor which is directly ωR dependent in the moderate-z form of the last line. Therefore, Eq. (4.5) can be summarized as

B. Rescattering corrections and classical limit
We have just seen that taking into account the sizeable incidence angles in multiple-exchange emission amplitudes provides important corrections to the naïve resummation formula which involve the effective coupling ωR ¼ Gs ℏ ℏω E and are thus essential for ωR ≃ Oð1Þ. One may wonder, at this point, whether additional corrections of relative order ℏω=E may be similarly enhanced by multiplicity factors, thus yielding important effects as well.
We do not have a complete answer to that question. We think however that kinematical corrections affecting incidence angles at relative order ℏω=E (and occurring in the currents' projections, Appendix A) are unimportant because they actually affect the factorization procedure at relative order Oðℏω=EÞ 2 , and are thus subleading. On the other hand, we argue that dynamical corrections due to rescattering of the emitted graviton are to be seriously considered, even though they are known [7] to be subleading for the calculation of the scattering amplitude of the fast particles themselves.
Indeed, consider for instance the contributions to the emission amplitude of the two graviton-exchange diagrams of Fig. 7. If the active exchange is #2, we just have to replace δ 0 ðbÞ by δ 0 ðjb − ℏω E bzjÞ because of the nontrivial incidence angle. But if the active exchange is #1, the next hit is a three-body one, which involves emitted graviton interactions also, as illustrated in Fig. 8 for an emitted graviton in jet 1 (top), rescattering with jet 2 (bottom). Therefore, the remaining δ 0 which was left uncorrected in Eq. (4.2) should be corrected also, by replacing it by where we note that the fast-particle gravitational charge has been decreased by an energy conservation effect of order ℏω=E, while the charge of the rescattering graviton is by itself of that order.
Since both replacements-due to incidence angles and rescattering alike-hold for any one of the single hits being considered [as in Eq. where we note the appearance, in the denominator, of the quantity Φ R introduced in Eq. (3.33) multiplied by 2iωR.
The first log in Φ R is due to the incidence angle while the second one is due to rescattering. But since −Φ R appears in the numerator if we upgrade h s ðzÞ to h s ðω; zÞ of Eq. (3.33), we simply get which is expressed as the algebraic sum of incidence angle and rescattering effects. We thus see, by inspection, that since Φ R ðω; zÞ → ΦðzÞ in the small ℏω=E limit, Eq. (4.10) reproduces the classical amplitude of Ref. [31] with the proper normalization according to our conventions 5 and helicity λ ¼ −2, ð4:11Þ where Θ s ðbÞ ¼ − 2R bb is, as usual, the fast-particle scattering angle.
We can also express the result (4.10) in a form similar to (4.5): showing explicitly how, for each value of ξ, the incidence angle effect depends on the evolution up to an incidence angle ξΘ s while rescattering depends on the complementary interval ð1 − ξÞΘ s of incidence angles. We can also say that the incidence angle dependence corresponds to the tilt in the fast-particle wavefront noted in [31], so that the rescattering counting is correctly reproduced by the simple overall subtraction in Eqs. (4.10)-(4.11). Furthermore, the residual ℏω=E dependence of the improved form (4.10) produces quantum corrections to the classical formula (4.11). It is amazing that the same function Φ R yields, on the one hand, the extension of the soft field to the small-angle part of region (a) and ensures, on the other hand, the rescattering corrections at quantum level.
In the following, we concentrate on the analysis of the result (4.5), which provides what we call the "geometrical" corrections due to scattering and emission with various incidence angles all over the eikonal evolution. The inclusion of rescattering corrections, leading to the classical result (4.11) and its quantum corrections (4.10), is deferred to a later work.

C. The resummed amplitude and its regimes
The amplitude 2iM (4.5) is directly normalized as the probability amplitude for the emission of a graviton in a scattering process occurring at impact parameter b. Its interpretation is that of a coherent average of the singleexchange amplitude over scattering angles ξΘ s ¼ −ξ 2R bb ranging from zero to Θ s .
The final result of our calculation in the geometrical approximation and in the Breit frame can also be expressed, more explicitly, in the form 2πz Ã2 e ibq·z sinðωRxÞ ωRx ΦðzÞ; ð4:13Þ where ΦðzÞ was defined in Eq. (3.34).
In order to understand the ω dependence of Eq. (4.13), it is convenient to rescale z → ωRz so as to write hω FIG. 8. Emission diagrams with subleading corrections. In (a) the eikonal scattering ∝ δðbÞ after graviton emission occurs with reduced energy E → E − ℏω. In (b) the graviton at x ¼ bz rescatters with the external particle at x ¼ b generating a term ∝ ℏωδðjb − bzjÞ. 5 In order to carry out the precise comparison one should keep in mind that R ¼ 4E ½32 ; Φ ¼ 1 8 Φ ½32 , and that scattering angles are defined with opposite sign conventions.
Therefore, we recover the soft limit in the form where we note the close relationship of 2M with the soft insertion factor in Eq. (3.8), evaluated-in the Breit frame-at scattering angle Θ s ðbÞ. The second line yields, in square brackets, the singularity-free expression sin ϕ θ− 1 2 Θ s ;θþ 1 2 Θ s and shows how the coupling jΘ s ðbÞj ¼ 2R=b is recovered. Furthermore, the cutoff jqjb < ℏ argued on the basis of the Φ-function behavior is consistent with the 1=ðjqjbÞ 2 behavior of the Regge form of the amplitude (3.23).
(2) ωR ≳ 1. In this regime, the amplitude starts feeling the decoherence factor ∼ 1 ωR due to the ξ average, which eventually dominates the large frequency spectrum of the energy-emission distribution (IV D) and establishes the key role of R. According to Eq. (4.14), resummation effects due to the sinx x factor are kinematically small in the region jθ x j ≫ jΘ s j because x ∼ jΘ s j jθ x j ≪ 1. Instead, in the region jθ x j ≲ jΘ s j they are important and tend to suppress the amplitude for ωR > 1. In order to see how, we anticipate from Eq. (4.24) the energy distribution formula of gravitational wave (GW) radiation so that it is instructive to look at the combination 2ωM in the limit ωR ≫ 1, in which x ∼ 1 ωR is supposed to be small, in order to avoid a higher power decrease.
Since we should have bωθ y y ∼ bωθ x x ∼ Oð1Þ, the condition x ∼ 1 ωR leads to a phase space in which x ≪ y ∼ Oð1Þ and θ y ∼ 1 bω ≪ θ x ∼ jΘ s j ¼ 2R b . In this region, in the Breit frame, we get the limit where we have setx ¼ ωRx and approximated Φðx; yÞ → Φð0; yÞ. We thus get a simple, factorized amplitude which-by performing the remaining integralshas the explicit form in both the forward and backward hemispheres. The above limiting amplitude is strongly collimated around jθ y j ∼ 1 bω ≪ 1 ωR (by which ϕ θ is very small), for any jθ x j < jΘ s j=2, that is in the ξ-averaging region. Furthermore, that distribution (confirmed numerically; see Fig. 11) comes from the transverse space region x ¼ 0, which becomes dominant at large ωR s.
In fact, we can easily calculate the contribution of (4.17) to the integrated distribution, that is where the coefficient comes from the θ y integral, in agreement with the dominant 1 ωR behavior of the spectrum to be discussed next.

D. Multigraviton emission and coherent-state operator
So far, we have considered single-graviton emission in the whole angular range. However, the extension to many gravitons by keeping the leading terms in the eikonal sense is pretty easy. For one emitted graviton we have factorized in b space one active (emitting) exchange out of n in n ways. Similarly for two gravitons we count nðn − 1Þ pairs of active exchanges emitting one graviton each, plus n exchanges which emit two gravitons, and so on. The first ones are independent and provide an exponential series for the single emission amplitudes we have just resummed, the second ones yield correlated emission for a pair of gravitons, and so on.
Resumming the independent emissions yields multiple emission amplitudes which are factorized in terms of the single-emission ones calculated so far. Virtual corrections are then incorporated by exponentiating both creation (a † λ ðqÞ) and destruction (a λ ðqÞ) operators of definite helicity λ (normalized to a wave-number δ-function commutator ½a λ ðqÞ; a † λ 0 ðq 0 Þ ¼ ℏ 3 δ 3 ðq −q 0 Þδ λλ 0 ), as follows, ð−q; q 3 Þ Ã is provided by Eq. (4.5) with a proper identification of variables. Since operators associated with opposite helicities commute, the above coherent-state operator is Abelian (and thus satisfies the Block-Nordsieck theorem) but describes both helicities, not only the IR singular longitudinal polarization.
The S matrix (4.20) is formally unitary because of the anti-Hermitian exponent, but needs a regularization because of the IR divergence mentioned before, due to the large distance behavior jh s ðzÞj ∼ jhðzÞj ∼ jzj −1 of the relevant fields. Because of the virtual real-emission cancellation, the regularization parameter can be taken to be Δω, the experimental frequency resolution (assumed to be much smaller than b −1 ), whose role will be further discussed in Sec. V. With that proviso, we can now provide the normal ordered form of Eq. (4.20) when acting on the initial state, which we identify as the graviton vacuum state j0i, as follows: is the no-emission probability, coming from the a; a † commutators. Due to the factorized structure of Eq. (4.21), it is straightforward to provide the full generating functional of inclusive distributions as functional of the fugacity z λ ðqÞ.
In particular, the unpolarized energy emission distribution of gravitational waves in the solid angle Ω and its multiplicity density are given by Both quantities are discussed in the next section.

A. Energy emission and multiplicity distributions
Starting from the soft/Regge emission amplitude in Eq. (4.13) we obtain, by Eq. (4.24), the multiplicity distribution in either jet J where the helicity sum provides the additional θ → −θ contribution, equivalent to a factor of 2 after angular integration. It is convenient to look first at the qualitative properties of the frequency dependence integrated over angles, by distinguishing the two regimes pointed out before.
(i) ωR ≪ 1. This is the infrared singular region originally looked at by Weinberg. The angular integration in jet 1 involves the two-dimensional vector q ¼ ℏωθ and the amplitude is dominated by its leading form (4.15). If bjqj=ℏ ≪ ωR (jθj ≪ Θ s ¼ 2R=b) the amplitude is ϕ dependent, but is independent of jθj because of the cancellation of the collinear singularities due to the helicity conservation zero, which has been extended to the whole region jθj < Θ s by our method. Therefore, the distribution acquires the form ∼const d 2 θ Θ 2 s ΘðΘ s − jθjÞ which effectively cuts off the bq integration at bjqj=ℏ ≥ ωR.
If instead ωR < bjqj=ℏ < bω we enter the intermediate angular region Θ s < jθj < 1 where (3.35) agrees with the basic form ∼ sin ϕ θ bjqj of (3.14) already noticed in [6], so that the integrated distribution is of the type More precisely, by Eq. (4.15), we get for the energyemission distribution where we have changed variables θ þ 1 2 Θ s → θ, integrated on both jets, and used the cutoff bωjθj < 1 due to the large bjqj suppression.
We thus find that the typical infrared distribution dω=ω is kept only in the tiny region ω < b −1 , with a rapidity plateau in the range jyj < Y s ¼ logð2=Θ s Þ ¼ logðb=RÞ, much restricted with respect to the full rapidity Y ¼ 2 logðEb=ℏÞ ∼ log s available in the single H-diagram evaluation. On the other hand, the corresponding smallω number density ðGs=πℏÞΘ 2 s agrees with that used in [6] for the calculation of the two-loop eikonal and with the zero-frequency limit (ZFL) of [26,38].
(ii) 1 < ωR < ðGs=ℏÞ. In this region we think it is tenable to assume the completeness of the q states, so that the spectrum, integrated over d 2 ðωθÞ of Eq. (5.1) and on both jets, is obtained by the Parseval identity in the form We note the typical IR behavior dω=ω of the number density which is present in this formulation also, and we also note the less typicalx ≡ ωRx dependence ð sinx x Þ 2 due to the coherent average over initial directions (4.6), occurring in the Fourier transform of the resummed field (4.7), which essentially acts as a cutoff Θð1 − ωRjxjÞ. Its action is ωR dependent, cuts off large values of jxj for ωR ≪ 1, and reduces the integration to small ones for ωR ≫ 1.
In particular, for ωR ≫ 1, the emitted-energy spectrum is considerably suppressed by our treatment of the collinear region, with respect to naïve H-diagram expectations. We get in fact from Eq. (5.4) for the emitted-energy fraction We see that the spectrum is decreasing like 1=ðωRÞ 2 for any fixed value of x, in front of an integral which is linearly divergent for x → 0. This means effectively a 1=ðωRÞ spectrum. More precisely by integrating the averaging factor in the small-x region we get, for ωR ≫ 1, the factor in front of the coefficient The latter is in agreement with the 1=ω behavior of Eq. (5.8) but with a slightly different coefficient ð2πÞ −1 ≃ 0.16 instead of ð2=3Þð1 − log 2Þ ≃ 0.20. We conclude that rescattering effects are somewhat important at large ωR, but do not change the qualitative 1=ωR behavior of the spectrum. A related important question is whether the emitted energy fraction (5.9) is limited by the quantum energy bound ω M < E=ℏ only, or instead should be cut off at the purely classical level. In such a case we would expect that the ωR distribution is further suppressed by higher order contributions to the Riemann tensor, yielding e.g. a ðωRÞ −2 behavior, or higher. An argument in favor of the classical cutoff, advocated in Ref. [31], is detailed in Appendix D.

B. Frequency and angular dependence
In this section we present plots of the resummed amplitude and of the corresponding radiated energy distribution obtained by numerical evaluation. In this way, besides verifying the asymptotic behaviors derived in Secs. IV C and VA, we can visualize the shape of such quantities in the transition region ωR ∼ 1. Furthermore, we can study the angular distribution of radiation and notice very peculiar features.
Our first task is to rewrite the resummed amplitude M given in Eq. (4.5) by means of a representation with the lowest number of integrals, having good convergence properties. It is possible to integrate the last line of Eq. (4.5) in d 2 z and express the result in terms of the special function strictly related to the exponential integral (and to the incomplete gamma function) E 1 ðzÞ ¼ Γð0; zÞ. We are thus left with a compact one-dimensional integral over ξ, where ½ξ min ; ξ max is ½0; 1 in the lab frame and ½−1=2; 1=2 in the Breit frame. The singularity at w ¼ 0, i.e., ξ ¼ θ=Θ s , is harmless, being integrable. From the previous expression it is clear that, apart from the prefactor ffiffiffiffiffi ffi α G p R, M depends only on ωR, jθ=Θ s j and ϕ θ .

Energy spectrum
Let us start by displaying the main features of the gravitational wave spectrum of Eq. (4.24) in the geometrical approximation of Eqs. (4.13) and (5.4). In Fig. 9(a) we plot (in log scale) the differential emitted energy with respect to ωR and jθ=Θ s j (i.e., after integration over the azimuthal angle ϕ). Figure 9(a) shows very clearly that the spectrum is dominated by a flat plateau (where kinematically accessible) whose shape can be easily explained as follows. The spectrum falls on the left (θ < Θ s ) because of phase space and the absence of collinear singularities. It also falls on the right when ωR ¼ bqΘ s =θ > Θ s =θ, since then bq > 1 [see Eq. (4.15)]. The last limitation (shaded region on the right) is due to the trivial kinematic bound θ < 1. As a result, for fixed ωR < 1 the length of plateau in log θ is logð1=ωRÞ while it disappears completely for ωR > 1.
This is the reason why the spectrum in ω shown in Fig. 9(b) (obtained by a further integration over the polar angle jθj and summing the contributions of the two hemispheres) shows two very distinct regimes: (i) ωR ≪ 1. In this regime the amplitude is well approximated by Eq. (5.3). We see that the really infrared regime holds only in the tiny region ω < 1=b ⇔ ωR < Θ s , with a rapidity plateau up to jyj < Y s ≡ logðb=RÞ [in Fig. 9(a) these are the deepest horizontal sections of the plateau which are limited on the right by the shaded region]. Such a rapidity plateau is much smaller than Y ¼ logðEb=ℏÞ, the rapidity range available in the single H-diagram emission. Correspondingly, the energy spectrum is flat, as one can see on the leftmost part of Fig. 9(b) for the lines with nonvanishing Θ s . On the other hand, here the small-ω number density in rapidity, ðGs=πÞΘ 2 s , agrees with the one used in [6] and with the ZFL of [26,38]. For larger values of ωR, as already noted, the length of the horizontal sections of the plateau decreases; therefore we observe a logarithmic decrease of the energy spectrum for ωR ≲ 1.
(ii) 1 < ωR < ω M R. This is the most interesting region of the spectrum, which in Fig. 9(b) exhibits the large ωR decrease ∼1=ωR, in perfect agreement with Eq. (5.8). This feature originates from graviton emission all along the eikonal chain, summarized in the resummation factor (4.6), which contains the effective coupling ωR and yields the decoherence effect for large ωR values which is exhibited in Fig. 9(b). It is important to note that curves for various values of Θ s (and thus of b) yield different ZFLs, as expected, but then coalesce in a common curve for ω ≳ 1=b, the blue curve in Fig. 9(b), which is therefore universal and corresponds to Eq. (5.4).
It is interesting to compare (Fig. 10) the blue curve of Fig. 9(b) of our geometrical approach with the classical counterpart of Ref. [31] in Eq. (5.10). We can see that the agreement is pretty good, even if the difference starts being important at large ωR values, suggesting that rescattering corrections [not included in Eq. (5.4)] and perhaps also quantum effects should be better evaluated in this region. For instance, the upper limit ω M quoted here, which occurs in the total emitted-energy fraction (5.9), is provided in any case by phase space, i.e., by the quantum frequency E=ℏ. But it is likely that, as advocated in Ref. [31] and illustrated in Appendix D, the classical theory will provide by itself a physical cutoff, of order ω M ∼ R −1 Θ −2 s .

Angular distributions
In order to have a picture of the full angular distribution of the radiation at various frequencies, in Fig. 11 we plot jMj in the hθ x =Θ s ; θ y =Θ s i-plane for some values of ωR; the maximum values of jMj are attained in the red regions, while vanishing values of the amplitude are shown in blue.
We see that for ωR ≪ 1 the emission is symmetric with respect to the symmetry axes of the process, and is spread in a rather wide region around the particles' directions, in particular at jϕ θ j ≃ π=2, in agreement with the sin ϕ q dependence of Eq. (4.15). Moving to larger values of ωR ∼ 1 the helicity amplitude shows an asymmetry with respect to the x axis. The symmetry is restored by the symmetrical behavior of the amplitude with opposite  helicity. We note also a progressive shrinkage of the emission in the region close to the particles' directions.
At large values of ωR ≫ 1 the effective support of the amplitude is just a thin strip around the interval θ ∈ ½0; Θ s whose width decreases as 1=ðωRÞ, also in this case in agreement with the analytic estimate (4.18). A more quantitative graphical representation of the azimuthal dependence of the amplitude is shown in Fig. 12, where we fix the polar angle jθj ¼ 1 2 jΘ s j and plot, for various values of ωR, the amplitude versus ϕ θ , rescaled by ωR. At small ωR ≪ 1 (dotted green curve) we see the expected j sin ϕ q j behavior of Eq. (4.15) (it becomes exactly j sin ϕ θ j for jθj much larger or much smaller than jΘ s j). At intermediate ωR ∼ 1 (dashed blue curve) the asymmetry in ϕ θ is evident, and the enhancement around ϕ θ ≃ 0 starts taking place. At large ωR ≫ 1 (solid red curve) the amplitude shows a narrow peak at ϕ θ ¼ 0, whose width and height are inversely proportional to ωR. At finite ϕ θ ≠ 0, the amplitude is more and more suppressed with increasing ωR, according to Eq. (4.18).
On one hand, the presence of such a narrow peak [which becomes of constant height after multiplication by ω; see Eq. (4.16)] explains the ðωRÞ −1 decrease of the radiated energy. On the other hand, it suggests that the radiation is sort of more and more confined along the trajectories of the fast particles with increasing R ∝ ffiffi ffi s p , at fixed b and ω.

C. Absorptive part and resummation effects
We are now in a position to discuss the total emission multiplicity which is related to the imaginary part of the resummed scattering amplitude and is also related to the no-emission probability P 0 ¼ e −hNi Θ s of Eq. (4.22). According to Eq. (4.6) its general expression is (Δω ¼ Oðb −1 Þ) where we have assumed Δω ¼ Oðb −1 Þ in order to have a reliable completeness of the q states.
In the first line of (5.15) we have made use of the general expression (4.6) of the resummed field valid for any α G , while in the second line we have considered the trans-Planckian limit α G ≫ 1, ωR fixed, which is the main interest of the present paper. Such two forms show very clearly that the estimate (5.15) for α G moderate to small is substantially different from the one in the trans-Planckian limit. In the first regime the resummation factor is a power series in α G , starting from 1 for α G → 0, limit in which (5.15) yields just the H-diagram result called 4Imδ 2 in [6]. As a consequence the ω values can go up to ω ∼ E=ℏ yielding a relatively large emitted energy and showing the ∼ log s dependence in rapidity used by Amati et al. [6] to hint at the real part of the amplitude from a dispersion relation. Furthermore, due to the logarithmic ω dependence of the resummation factor, the large-ω phase space is modified rather slowly by varying the α G value, thus suggesting an intermediate regime where the real part could be calculated also.
On the other hand in the trans-Planckian regime (α G ≫ 1, ωR fixed) of the present paper, the second line of (5.15) shows that values of ω ≳ OðR −1 Þ are substantially suppressed, thus leading to the reduced rapidity Y s ¼ 2 logðb=RÞ mentioned before, the subsequent resolution of the energy crisis, and the emergence of our Hawking-like radiation, which represents the main result of our investigation.
More precisely, in order to take into account arbitrarily small values of Δω in the trans-Planckian case, we distinguish the soft and large-frequency contributions as in Sec. VA by writing, to logarithmic accuracy, We see that the large-frequency integral (ω > R −1 ) yields just an R-independent constant rapidity Y > ≃ 0.56, while the soft one is determined by the reduced rapidity Y s ¼ logðb=RÞ, with the physical consequences mentioned before. The phase space for the left part will eventually disappear with increasing R. This suggests that in the extreme energy (and large angle) limit, the total emission multiplicity will become just proportional to α G with a coefficient of which Eq. (5.16) provides a provisional estimate.

D. Towards large-angle resummation
It is of obvious interest to try to extend the radiation treatment presented here to the extreme energy region R ∼ b where the scattering angle Θ s ðbÞ becomes of order unity or larger, and a classical gravitational collapse may take place. By following the path led by [13] and mentioned before, we encounter two kinds of effects: (i) those due to the evaluation of the elastic eikonal function δ ∼ α G fðR 2 =b 2 ; l 2 P =b 2 Þ, which becomes a strong-coupling series showing perhaps some critical singularity at b ¼ R, as in the reduced-action model [7]; and (ii) those arising from the ξ averaging, that is the coherent sum of δexchange emissions at the radiation level that we have just emphasized for δ ¼ δ 0 as the origin of the key role of R in the energy-emission spectrum.
By focusing on the second kind of effects, we may consider the first one as simply the source of some structure in δðbÞ and in the related semiclassical trajectories [9] that will show up in the ξ averaging also. Therefore, the new features of elastic scattering in the strong coupling regime for R ∼ b will provide new effects at the radiation level. A nice picture of the present situation is exhibited in Fig. 11(d), in which the frequencies ω > 1=R send a last signal before being suppressed. This supposedly essential message emphasizes the present span 2R=b of the incidence angles ξΘ s , and the impact-parameter directionb. Both parameters are expected to change with increasing R, because the semiclassical trajectory is likely to approach a quasibound shape and the question is how much that change will affect, by the ξ averaging, the emerging radiation.
To provide an example, the impact-parameter directionb is expected to rotate by following the trajectory during time evolution, and thus it is possible (though not obvious) that thex andŷ directions will be mixed by the ξ averaging. If that is the case, the y variable will be, on the average, proportional to x and, as a consequence, the modulation function ΦðzÞ will be small and of order jzj 2 , thus acquiring a cylindrical symmetry and implying that the distribution (5.5) is of type 1=ðωRÞ 2 [and not 1=ðωRÞ]. That behavior would yield a faster suppression when approaching R ∼ b and would automatically provide a cutoff in the energy fraction (5.9).
It is of course important to establish whether such a sizeable change of the emerging radiation will really occur or not. This is nontrivial, however, because it requires a formal description of the ξ averaging for higher orders in the δ-exchange emission also. It would appear, though, that looking at the ξ averaging by keeping the semiclassical trajectory standpoint may produce some changes, but smooth ones, with no real hints of information loss.
If we now switch to the full quantum level, it is clear that-besides modifying the ξ averaging by single-δ emission-we have to modify multiple emissions also by introducing correlations in the coherent state (4.20) by the procedure of Sec. IV D. The simplest one concerns the two-graviton emission amplitude at order G 3 s 2 , an example of which is given in the diagram of Fig. 13(a), which introduces quadratic terms ∼a 2 , a †2 in the exponent of (4.20). At the same time such a diagram occurs in the corrections to the elastic eikonal exemplified in Figs. 13(b) and 13(c), which contain further powers of s because of s-channel iteration, and are thus of second order in the effective coupling R 2 =b 2 . The calculation of 13(a) can be devised by following the lines of Sec. III in the various soft and Regge regions, so as to evaluate the correlation term.
Upgrading the present method to include the steps just described looks therefore within reach. It may shed light on the existence of a large ω M cutoff, of possibly (higher order) classical origin or quantum mechanical one. Going much further however seems very hard, because treating both polarizations at higher orders of the effective-action expansion is a fully two-dimensional problem, unlike the reduced action model with one polarization in the axisymmetric case investigated in [13]. We nevertheless hope that, even just at the next order, the present approach may provide us with some global insight on the interplay of radiation and scattering in the strongcoupling regime.

VI. SUMMARY AND PERSPECTIVES
We have investigated, in this paper, the peculiar features of the graviton radiation associated to gravitational particle scattering at energies much larger than the Planck mass. That scattering, at small deflection angles, is described by a semiclassical S matrix in b space, which exponentiates the eikonal function δðbÞ, of order α G ≡ Gs=ℏ ≫ 1 and expressed as a power series in R 2 =b 2 , R ≫ l P being the gravitational radius of the system (Sec. II).
We find here that the ensuing radiation is expressed as a superposition of a large number ∼α G of single-hit emission processes, each one being derived in a high-energy form which unifies the soft and Regge limits in the whole angular range (Sec. III). Combining the large emission number (Gs=ℏ ≫ 1) with the relatively small emitted energy (ℏω=E ≪ 1) produces in the emission amplitudes the effective coupling ωR which tunes the resulting spectrum on the inverse gravitational radius (Sec. IV). For that reason the emerging radiation is Hawking-like-that is with characteristic energies ∼ℏ=R which decrease for increasing input energies, even in the small-deflection angle regime in which the S matrix is explicitly unitary. In fact, as a consequence of coherence effects in the superposition just mentioned, our unified amplitudes are found to have a surprisingly simple interference pattern in ωR, suppressing large frequencies ω ≫ 1=R and reducing the radiated energy fraction to order Θ 2 s (Sec. V). Finally, we generalize the (quantum) factorization method of single-hit emissions to include rescatterings of the emitted graviton and to resum them (Sec. IV B). The ensuing emission amplitude neatly agrees with Ref. [31] in the classical limit and also includes the mentioned quantum effects in a simple and elegant way, thus calling for further investigation in the near future.
The ultimate goal of our thought experiment beyond the Planck scale is actually to reach large scattering angles and the extreme-energy region R ≳ b where a classical gravitational collapse may take place. Amati et al. proceeded a long way towards that goal from the scattering amplitude stand point in the reduced action model [13,23]. In such a truncated model they found that the S Matrix, as functional of the UV-safe solutions, shows an impact-parameter singularity in the classical collapse region, thus causing a unitarity deficit that they were unable to circumvent by lack of information on the associated radiation (and on short distances).
We stress the point that, from the radiation point of view, we are better off with the method presented here. In fact, we have just summarized two steps: the first one yields the emission amplitudes for the single-hit process of δ 0 exchange [and corrections thereof in δðbÞ, Sec. V D]; the second one performs their superposition all along the eikonal deflection, with its interference pattern. The latter may in turn feed back on higher order corrections to the scattering amplitude itself. Therefore, by applying the present method to an improved eikonal function, we could possibly provide the radiation features given those of scattering and vice versa, by thus estimating the exchange of information between them. We hope on this basis to be able to approach the classical collapse region in a smoother way, and to test in a more direct way the features to be expected from a unitary evolution of the system.

ACKNOWLEDGMENTS
We wish to thank the Galileo Galilei Institute for Theoretical Physics and the Kavli Institute for Theoretical Physics, University of California, Santa Barbara (research supported in part by the National Science Foundation under Grant No. NSF PHY11-25915) where part of this work was carried out.

APPENDIX A: PHYSICAL PROJECTIONS OF THE WEINBERG AND LIPATOV CURRENTS
In this section we calculate the explicit projections of the Weinberg and Lipatov currents over physical helicity states, proving in particular Eqs. (3.4), (3.18) and (3.21), the transformation law (3.15), and the symmetry relations (3.10) and (3.48).
We work in the gauges specified in Eqs. (3.2)-(3.3) which differ from the one used in [6,13] because the gauge vector ϵ μ L has the subtraction −q μ =jqj (q μ =jqj) in the forward (backward) emission case. Such subtraction is allowed by current conservation 6 and is devised to suppress the longitudinal projections of external momenta in jet 2 (jet 1) which are oppositely directed. For a generic tensor current J μν we define in terms of the basic complex vectors (note the nonstandard in the forward jet, where we have q 3 ¼ ω ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − jθj 2 p > 0, and 6 The Lipatov current is exactly conserved, while the Weinberg current is conserved up to corrections of order Oðω=EÞ which we neglect throughout the paper. in the backward one, where we send q 3 →q 3 ¼ −q 3 < 0, leaving the other q μ components unchanged. We then see by inspection that, in the small-angle kinematics, the vectors (A2) [(A3)] have negligible longitudinal projection on jet 2 (jet 1) momenta, thereby making the corresponding contributions to the currents negligible altogether, because transverse projections are anyway of order Oðθ 2 i Þ due to the lack of collinear enhancement in the opposite jet.

Forward hemisphere
We need to calculate the typical scalar products (restricting ourselves for cleanness of notation to the negative helicity projection) which are given in complex notation in terms of Θ i ≡ Θ 1 , that is thep 1 incidence angle in the general parametrization (2.1). By using Eq. (A1), the p 1 contribution to the Weinberg current is then Here we note the lack of collinear singularity due to the cancellation of the squared numerator with the denominator in Eq. (A4). The contribution from p 1 0 is analogous, with just the replacement Θ 1 → Θ 0 1 , while p 2 and p 2 0 give negligible contributions in this gauge and hemisphere, as explained before.
Therefore, introducing the coupling κ and adding up the relevant terms we get Here we have used the approximate relation neglecting the momentum conservation corrections of order Oðω=EÞ, which is allowed in regions ðaÞ þ ðbÞ where the Weinberg current is relevant and jqj=jq 2 j is small.
For the Lipatov current (including for convenience the denominator associated to the q μ 1 , q μ 2 virtualities), the negative helicity projection is given by [see Eq. (3.19)], where we have Here we have used current conservation to replace q μ 1 − q μ 2 → −2q μ 2 and q ⊥1 , q ⊥2 and q ⊥ denote transverse (vectorial) components with respect to thep 1 direction (and q ⊥1 , q ⊥2 and q ⊥ , the corresponding complex versions), which are related to q 1 , q 2 and q by a rotation of angle jΘ i j of the reference frame. Note in particular that since we are considering a forward emission, q μ 2 has practically no longitudinal component, while q μ 1 has; taking also into account that q μ ¼ q 1μ þ q 2μ , this implies Now, rewriting q ⊥1 ¼ q ⊥ − q ⊥2 , using q ⊥ ¼ ωðθ − Θ i Þ and taking into account Eq. (A12), we can rewrite Eq. (A11) in the form By substituting expression (A13) in Eq. (A9) we finally get which proves (by setting Θ i ¼ 0) Eq. (3.21) of the text and the transformation law to general Θ i (3.15) for the Lipatov current. We note that the would-be singularities at q ⊥1 ; q ⊥2 ¼ 0 have been canceled due to Eq. (A13) and replaced by the phase difference in Eq. (A14), which also reduces the q ⊥ ¼ 0 singularity to a linear integrable (in two dimensions) one.

Backward hemisphere
In this case the jet 2 is characterized by an incidence angleΘ i ≡ Θ 2 (which in the center-of-mass framẽ p 1 þp 2 ¼ 0 is simply provided byΘ i ¼ −Θ 1 ¼ −Θ i ) and by the fact that the emitted q μ has a negativeẑ component,q 3 ¼ −ω ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − jθj 2 p < 0. By using the backward helicity vectors (A3) we obtain and, as a consequence Since jet 1 is switched off, the Weinberg current is simply where we used the relation and neglected, as in the forward case, the Oðω=EÞ corrections in the ðaÞ þ ðbÞ regions. We note that Eq. (A17) can be recast in a form transforming like the complex conjugate (or opposite helicity) of (A7) by the replacement θ → −θ, to yield This proves Eq. (3.9) and yields the basis for the relationship (3.10), in which we also use the F.T. with respect to q 2 ¼ Eθ s þ q.
All that is left is to obtain the helicity projections of the Lipatov current in the backward hemisphere. Using Eqs. (3.19) and (A1) we obtain where we used the fact that q μ 1 has negligible longitudinal components for an emission in jet 2 and defined (in complex notation) The vector current projections are where we performed the algebra along the lines explained before. We thus obtain At Θ i ¼ 0 the above result agrees with that in the forward jet by the trivial replacement q 1 ¼ q − q 2 , meaning that the Regge limit yields the same form of the amplitude in either jet. On the other hand, at nonzero Θ i , and by the replacement q → −q; q 1 → −q 2 , it yields the helicity relation by finally using the transformationq ⊥2 → −q ⊥ þ q 2 in the F.T. under the same replacement, we prove Eq. (3.48) of the text.

z representation
A generic phase difference of the form e 2iϕ θ − e 2iϕ θ 0 , with θ, θ 0 generic 2-vectors, of the kind that appears in the physical projections of both the Weinberg (3.8) and the Regge (3.22) amplitudes, can be conveniently rewritten in integral form: where A is an arbitrary scale (the integration measure d 2 z=z Ã2 in the integral is scale invariant). This is easily verified by performing first the azimuthal, then the radial integration in d 2 z. We get (setting for simplicity A ¼ 1) where in the last step we used the scale invariance of the integration measure, the standard integration formula R J 2 ðxÞ x ¼ − J 1 ðxÞ x and the fact that J 1 ðxÞ ∼ x=2 near x ¼ 0. Equation (A25) is thus proven.
where a prime means "rotated by R" and we have used Eq. (B5) in the second step.
In the case of small polar angles (θ; Θ i ; Θ f ≪ 1) we can write q ¼ ωðθ; 1Þ, p i ¼ EðΘ i ; 1Þ etc., and the previous rotation R amounts just to a translation of −Θ i on the angular variables: q 0 ¼ ωðθ − Θ i ; 1Þ, p 0 1 ¼ Eð−Θ i ; 1Þ and so on. (2) The last quantity in Eq. (B7) would be just M ð0Þ ðΘ f − Θ i ; θ − Θ i Þ, if only p 1 (and not p 0 1 ) would appear "inside" the polarization ϵ Ã . But ϵðp 0 1 ; q 0 Þ, being orthogonal to q 0 , is obtained by applying to ϵðp 1 ; q 0 Þ a rotation around q 0 : where α is a suitable (possibly large) rotation angle, and we have exploited the fact that a polarization tensor of helicity λ acquires a phase factor under rotations. To sum up, It remains to compute the rotation angle α. This is easily derived by rotating the original system by the (small) angle −θ in such a way that is put on the z axis so that the hq 00 i ⊥ plane becomes the transverse plane, as depicted in Fig. 14. Explicitly, It is now evident that the azimuth of p 00 1 in the hq 00 i ⊥ plane is φ p 00 1 ¼ ϕ −θ ¼ ϕ θ þ π and that of p 00 i is φ p 00 i ¼ ϕ Θ i −θ ¼ ϕ θ−Θ i þ π, and their difference is just the sought rotation angle because the ϵ T s are orthogonal to the projections of p i and p 1 on the hqi ⊥ plane. The same reasoning applies in the case of backward emission, e.g., withq ¼ ωðθ; −1Þ directed in the opposite direction with respect to Fig. 14. In this case the rotation angle needed to align q along −ẑ is just θ ¼ −θ. Furthermore, the azimuthal angle α between the polarization vectors must be counted in the opposite direction, because the thumb of the right hand now points towards the negative z direction. In conclusioñ

Relation with Jacob-Wick conventions
The relation of our helicity amplitudes with those defined by JW [39] can be understood by comparing in the two frameworks the choice of the polarization vectors for a generic graviton 3-momentum q with polar angle θ and azimuth ϕ.
Let q ¼ RðωẑÞ, where, according to JW conventions, R is the rotation matrix of angle θ and axis alongẑ × q (thus belonging to the transverse plane). Such a matrix R is conveniently written in terms of the usual Euler angles ðα; β; γÞ ¼ ðϕ; θ; −ϕÞ, so that it can be represented as the product of three rotations along the y and z axis: JW define a reference helicity state when the particle (here the graviton) has momentum ωẑ along the positive z axis. This means that they implicitly fix a pair of polarization vectors orthogonal to ωẑ, i.e., ϵ 1 ¼ŷ and ϵ 2 ¼ −x, so as to build the right-hand orthogonal basis fωẑ; ϵ 1 ; ϵ 2 g. The transformation of the helicity state in Eq. (6) of [39]  corresponds to rotate the graviton momentum and the polarization vectors with the matrix R of Eq. (B12), in such a way that the right-hand basis adapted to q is fq; Rϵ 1 ; Rϵ 2 g.
On the other hand, our convention (3.3) of the polarization vectors requires ϵ T to be orthogonal to both q andẑ, and it is easy to see that where the rotation matrix R ϕ;θ;0 differs from that of JW by the vanishing of the last Euler angle ϕ, which does not affect the action ωẑ → q, but changes the orientation of the polarization vectors in the hqi ⊥ plane.
In practice, our right-hand basis fq; ϵ T ; ϵ L g is obtained by applying to the JW reference basis fωẑ; ϵ 1 ; ϵ 2 g the rotation R ϕ;θ;0 of Eq. (B13). As a consequence, our polarization vectors are rotated by an angle þϕ around q with respect to those of JW. It follows that 2 −1=2 ϵ AE ¼ R q ðϕÞϵ JW AE ¼ e ∓iϕ ϵ JW AE while for the helicity amplitudes (involving contractions with ϵ μνÃ AE ) we have M λ ðqÞ ¼ e iλϕ M JW λ ðqÞ: ðB14Þ Let us now rederive the amplitude transformation phase of Eqs. (B9)-(B10). In the JW conventions, the amplitudes are invariant under rotations bringing ωẑ ↔ q, provided the rotation axis is in the transverse plane. In the case of small emission angles θ ≪ 1, adopting now the two-dimensional angular notations, such rotations are translations in the transverse components of forward momenta like q, p i , p f , as explained in the previous subsections B 1-B 2. Therefore, starting with the amplitude M 1 − e 2iϕ 12 ðq 1 þ q 2 Þ 2 e −i½q 1 ·xþq 2 ·ðx−bÞ ; ðC1Þ where ϕ ij ≡ ϕ i − ϕ j .
By denoting with A ≡ jxj and B ≡ jx − bj the moduli of the external vectors (see Fig. 15) appearing in the last exponent, and by explicitly writing out the various azimuthal angles, we rewrite h in the form ð1 − e 2iϕ 12 Þe −iðq 1 A cos ϕ A1 þq 2 B cos ϕ 2B Þ ðq 1 þ q 2 e iϕ 12 Þðq 1 þ q 2 e −iϕ 12 Þ ; ðC2Þ where q i ≡ jq i j and ϕ A (ϕ B ) is the azimuthal angle of the two-dimensional vector x (x − b). Since ϕ AB ¼ ϕ A1 þ ϕ 12 þ ϕ 2B , the integrations over ϕ 1 and ϕ 2 actually provide a double convolution, which can be diagonalized by a Fourier transform. In practice, by computing the partial waves with respect to the angle ϕ AB , we obtain since only the pole at z ¼ −q < =q > is enclosed by the contour.
The azimuthal integral for m ≤ −2, after the change of variable z → 1=z, keeps its original structure, determining the (anti)symmetry property Φ m ¼ −Φ −m−2 . Note also that Φ m is symmetric in the exchange q 1 ↔ q 2 .
Let us then proceed with the computation of h m ðA; BÞ for m ≥ 0. We have 2π 2 h m ðA;BÞ ¼ By expressing the q 2 variable in terms of the ratio ρ ≡ q 2 =q 1 , the q 1 integration reduces to the orthogonality relation for Bessel functions: ΘðB − AÞ þ fA ↔ Bg: ðC7Þ The h field can now be obtained by summing the Fourier series where we introduced the complex numbers A ≡ Ae iϕ A and B ≡ Be iϕ B . It turns out that the square brackets in the last equation are equal, and we finally obtain The components h TT and h LT correspond to the real and imaginary part of h, respectively, and read (ϕ A ¼ ϕ xb ) Some remarks are in order: (i) The final form confirms the UV-safe solution of the differential equation (2.15) of [13]. (ii) The simple expression of the solution in the rhs of Eq. (C9) has the same form of the phase factors in the integral representation (C1) coming from H-diagram vertices, evaluated at the angle ϕ AB ¼ ϕ x;x−b . In particular, the h TT component has a geometrical significance, embodied in the relation

Calculation of the h field in momentum space
The two-dimensional Fourier transform of the complex field hðb; xÞ with respect to the transverse variable x is given byh ðb; qÞ ≡ Z d 2 xe iq·x hðb; xÞ In fact, by replacing hðb; xÞ with the integral representation (C1), the integration in x just provides a delta function δ 2 ðq − q 1 − q 2 Þ which is then used to perform the integration in q 1 ¼ q − q 2 , according to Eqs. (2.11)-(2.12) of [13]. By introducing the complex variables q ≡ q x þ iq y ; the angular factors can be written in rational form: