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Following a semiclassical eikonal approach—justified at transplanckian energies order by order in the deflection angle

The recent discovery of gravitational waves (GW) in black hole and neutron-star mergers

In particle physics, gravitational scattering of light particles or strings at extremely high (i.e., transplanckian) energies has been considered since the late eighties

In particular, the emergence of an effective generalized uncertainty principle (GUP) holding in string theory has been pointed out

At such energies,

Another emerging property of transplanckian gravitational scattering is a sort of “antiscaling” law by which the higher the center-of-mass energy, the softer the characteristic energy of the final particles. This property has been seen both in the string-size-dominated regime

More recently, the low-frequency gravitational bremsstrahlung spectrum has also been investigated

The purpose of the present paper is to illustrate the essentials of the eikonal model just mentioned, and then to focus on the derivation of soft-graviton features, in order to see whether they are affected by the

We should notice from start that, in our approach, we shall mostly refer to scattering at fixed impact parameter

An important goal of the paper is to show (Sec.

Given such a regular behavior of the resumed amplitude, the study of soft limits is straightforward and based on the simple form of the resummed radiation amplitude in the classical limit given in Secs.

With the aim of being as much as possible self-contained the rest of the paper is organized as follows: In Sec.

In this section we summarize the ideas and assumptions introduced in

Throughout this paper, as in

Boldface symbols denote transverse vectors.

Strictly speaking, if

This regime is characterized by a strong effective coupling

Corrections to the leading form

In order to understand the scattering features implied by

The scattering amplitude of two transplanckian 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

In other words, every single hit is effectively described by the elastic amplitude

Here we use a cutoff regularization of IR

The relatively soft nature of transplanckian scattering just mentioned is also—according to

Both the scattering angle

More directly, the associated metric emerges from the calculation

We start, in the ACV framework, from the irreducible (possibly resummed

Consider now, at tree level, the emission of a graviton with energy

Center-of-mass view of the collision at impact parameter

Weinberg’s external insertion recipe factorizes in

Single-exchange emission diagram in

We note that the phase difference in

Inserting Eq.

Furthermore, it was shown in

The superscript

We notice that Eq.

By then replacing

The long distance features of the Coulomb-like interaction mentioned before at leading level

Due to the exponentiation of the

The H diagram providing the first subleading correction to the eikonal phase.

At one-loop level, starting from Eq.

The above result for

We are grateful to Pierre Vanhove for having brought this reference to our attention.

At two-loop level the situation is more involved because the H-diagram predicts

That divergence is actually to be expected in the imaginary part, in order to compensate a similar divergence of virtual corrections, so as to yield a finite total emission probability. The trouble would be if the divergence of

Fortunately ACV were able to show that the IR divergence cancels out in

This relatively simple derivation, basically a recollection of

So far, following

Indeed, we meet immediately a possible problem at the single-graviton exchange level. The amplitude (say, for helicity

We note that the expected soft behavior

In other words, here we stress the point that the single-exchange amplitude is very sensitive to the IR in the span

This feature can be ascribed to the fact that the single-exchange amplitude in

We note that, because of

Finally, we resum the independent emissions of many gravitons whose amplitudes are factorized in terms of the emission factor

Given

In this section we will study the gravitational radiation spectrum

Since we shall not use anymore complex notation for transverse vectors, from now on we denote the modulus of a transverse vector with the corresponding nonboldface symbol, e.g.,

On the other hand, in the

For

For

After recasting Eq.

We start by defining

We then split the

Picture of the polar and azimuthal angles in the transverse plane.

Since

We thus see that there is a logarithmic enhancement of the nominal

By then leaving

Note that the amplitudes for

Inspection of the

On the other hand,

Because of the forward-backward symmetry of the process, graviton radiation in the backward hemisphere occurs at the same rate. In practice, in the small-angle kinematics,

Enhanced subleading corrections come entirely from the last term in Eq.

We may now collect all terms in

Before moving on to a discussion of the spectrum at generic values of

The same conclusions can be drawn by recalling that the two helicity amplitudes

Furthermore, we notice that a similar resummation can be performed on the next-to-leading (NL) log amplitude

In order to study the small

On the other hand, for

It is also clear that the leading contribution comes from the

Our

B. Sahoo and A. Sen, private communication. One of us (GV) would like to thank Ashoke Sen for several discussions about how the first subleading correction contributes to different polarizations.

which can be seen as a confirmation of their recipe and as a way to fix the scale of theLet us now go back to

In conclusion we can write

Before proceeding further let us note again (see the above discussion of the small

We are making here the implicit assumption that the

The above differential spectrum is supposedly accurate at

In this subsection we present numerical results that can be obtained by direct numerical integration of the full eikonal model

First of all we want to asses the validity of our approximations, which we use to derive the main features of the radiation in the infrared region

Actually, we plot a “reduced” spectrum with the kinematical factor

(a) The (reduced) graviton frequency spectrum against

We analyze next the properties of the frequency spectra. We note their common logarithmic decrease (already pointed out in

This peculiar feature is due to the subleading terms of the amplitude. In fact the leading spectrum decreases monotonically in the whole

Behavior of the spectrum in the soft limit

Actually, by fitting the exact spectrum with the function

In order to confirm the robustness of the

To summarize, we believe that our model provides strong evidence for the structure of the subleading coefficients in the soft limit of graviton emission amplitudes, with terms of order

In this paper we have developed our previous work on the spectrum of gravitational waves emitted in the high-energy gravitational scattering of massless particles at leading order in the deflection angle

Remarkably, the spectra obtained in this “classical” limit exhibit a break in the spectrum at the characteristic “Hawking-frequency” scale

In this work we have reconsidered carefully the low-frequency part of the spectrum,

It seems instead that the more conventional infrared divergences can be tamed through the usual Block-Nordsieck procedure, or, alternatively, by using appropriate coherent states (or the Fadeev-Kulish procedure

The advantage of the eikonal approach pursued in this paper is that it leads directly to a singularity-free result and to an unambiguous determination of the logarithmically enhanced contributions to the spectrum, including the determination of the scale inside the logs. The way our approach avoids the infinities is conceptually very simple. The infinite gravitational Coulomb phase, as already remarked by Weinberg in 1965

At subleading order there is a correction to the ZFL of relative order

At sub-sub-leading order there is instead a

It would be interesting to see how these results extend to physically more interesting cases e.g.: (i) to smaller impact parameters (i.e., larger deflection angles) up to (and beyond?) the regime of inspiral; and/or, (ii) to arbitrary masses and energies of the two colliding particles.

We would like to thank the Galileo Galilei Institute for hospitality during most of our collaboration meetings. One of us (G. V.) would like to thank Andrea Addazi and Massimo Bianchi for useful discussions about the relation between this work and Ref.