# On non-perturbative unitarity in gravitational scattering

## Abstract

We argue that the tree-level graviton-scalar scattering in the Regge limit is unitarized by non-perturbative effects within General Relativity alone, that is without resorting to any extension thereof. At Planckian energy the back reaction of the incoming graviton on the background geometry produces a non-perturbative plane wave which softens the UV-behavior in turn. Our amplitude interpolates between the perturbative graviton-scalar scattering at low energy and scattering on a classical plane wave in the Regge limit that is bounded for all values of *s*.

## 1 Introduction

*s*,

*t*,

*u*are the usual Mandelstam variables and \(\kappa \) is the dimensionful gravitational coupling, grows without bound as

*s*increases at fixed

*t*. This state of affairs has given rise to an extensive activity in searching for a UV-completion of General Relativity (GR). String theory is one such complete theory whose legacy rests partly on the fact that it predicts an amplitude that is perturbatively unitary.

*s*but small

*t*(or Regge) limit can be reduced to a perturbative calculation on top of a plane wave as illustrated in Fig. 1.

## 2 Perturbative limit

*t*-channel diagram can then be calculated as follows: We first solve for the internal graviton \({\tilde{h}}\) around Minkowski background, \(\eta \), through

*h*. Next, we solve for the outgoing scalar field \({\tilde{\phi }}\) with the help of the Ansatz \(\phi =\phi _{(2)}+\lambda ^2 {\tilde{\phi }}\),

*g*. We note that (2) fixes \({\tilde{h}}\) only up to a solution of the homogeneous equation. The latter reproduces 3-particle (1 graviton) scattering amplitude upon substitution into (4). Note also that in (2) we can replace \( h_{(1)}\) by a wave packet since the equation for \({\tilde{h}}\) is linear in \( h_{(1)}\).

^{1}\(h=\lambda h_{(3)} +\lambda ^2 {\tilde{h}}\), we expand the background once again and get

*H*. Indeed, suppose that

*H*has momentum

*p*of order \(M_{Pl}\). Then

*H*cannot be treated as a perturbation of Minkowski space-time and back reaction on the geometry has to be taken into account. This can be done by replacing

*H*by a plane wave. In Einstein-Rosen coordinates [7] the plane wave metric reads

*h*). However, in the non-perturbative regime, Brinkmann coordinates [6] are more convenient, with

*H*this gives

*H*.

*H*. We first choose a polarization for

*H*by setting

*H*, come from the expansion of \(\epsilon _+\) and \(\varSigma _{ab}\) in (14). To continue we note that the deformation in \(\epsilon _+\) simply takes account of the fact that the transversality condition of \(h_{(3)}\) depends on

*H*, so that the \(\epsilon _+\) contribution is most naturally interpreted as a deformation of the 3-pt amplitude (16). The contribution at first order in

*H*to the connected four point function is then

*t*-channel contribution of the scalar-graviton into scalar-graviton scattering amplitude.

*s*and

*u*- channel contribution. Indeed, solving for the scalar wave equation in the plane wave background before and after interacting with \( {\tilde{h}}\) takes into account the interaction with

*H*. Furthermore, this should account for the contact interaction. Indeed, the last term in (14) gives an extra contribution

## 3 Non-perturbative calculation

*ingoing*boundary condition, \(\lim \limits _{x^+\rightarrow -\infty }(E_{(2)})^i_a(x^+)=\delta ^i_a\), and

*B*are of the form \(p_+ \varvec{\varSigma }(z)\), \(p_+ {\varvec{B}}(z)\) respectively . This allows us to extract the \(p_+\) dependence as

*s*. Using (26), the form (25) of the 4-point amplitude makes the unitarity of the amplitude at large

*s*manifest. Indeed, the integral in (25) is absolutely convergent for any value of

*u*(as we will see below). On the other hand for small

*s*(25) reduces to the perturbative amplitude (21). Thus the amplitude (25) is the non-perturbative, unitary completion of (1).

It is not hard to see that the terms in (9) containing \(\epsilon _+\) and \(\epsilon _a\) will similarly give a non-perturbative deformation of (16) preserving the unitarity of the latter.

We would like to stress, however, that the boundedness of the scattering amplitude does not imply that the total cross-section for the \((\phi (k),h_{\mu \nu }(p)\rightarrow \; \phi (l),h_{\mu \nu }(q))\) scattering is unitary since, due to the absence of momentum conservation, the integral over the outgoing momenta is not constrained. However, this feature is expected for scattering on an external potential. What our calculation shows then is that the question of unitarity of the gravitational four point scattering is actually not well posed. What we find is that at large center of mass energy, back reaction builds up an external field (the plane wave) so that at large *s* and small *t*, the four-point scattering is actually better described by a scattering off an external plane wave.

*u*with linear polarization. In this case we have

*s*(and also in

*u*).

The remaining terms in which lead to the 3-point amplitude (16) in the perturbative limit will also receive non-perturbative contributions upon replacing \(\phi _{(2)}\) and \(\phi _{(4)}\) by the exact solution in the plane wave background. It is not hard to see that that these are bounded in the large *s* limit.

## 4 Back reaction

*z*as a function of \({{\mathscr {E}}}\), that is,

*x*-direction in agreement with causality. On the other hand the pre-exponential factor accounting for (30) is not a coordinate artifact. It simply reflects the focusing of the geodesics which is a geometric property of plane waves [8].

## 5 Discussion

We have shown that the perturbative 4-particle amplitude evolves in the large *s*, small *t* limit into a non-perturbative expression involving a macroscopic non-linear plane wave, which is manifestly unitarity at the expense of smearing out the momentum conservation constraint. This picture is intuitively satisfactory, since due to backreaction, we expect that an energetic graviton sources a growing number of soft gravitons, eventually approaching a classical solution. Earlier approaches based on related ideas were proposed by ’t Hooft and others [2, 3] used a gravitational shock wave to represent an energetic scalar particle (in the geometrical optics approximation) and studied the propagation of a scalar field in that background. One might suggest that this setting should be related to ours via a boosted reference frame in which the incoming scalar \(\phi _{(2)}\) is energetic while the graviton \(h_{(1)}\) is perturbative. However, since the shock wave approximation [2] works only for point particle sources or superposition thereof [3], the matching is not clear. In fact the scenario of [2] does not have simple perturbative limit. On the other hand, at the calculational level some of our formulas are essentially identical to those in [13] but the physical interpretation is quite different.

Finally, we should emphasize our result does not yet allow us to conclude that GR unitarizes itself completely since large momentum transfer, where new physics usually arises, is not covered by our analysis. Other approaches which focus on the large *t* limit instead can be found in [4] for instance, where it is argued that black holes may unitarize the cross section in the large *t* limit. We have nothing new to say about that regime apart, perhaps, that in order to set up an experiment involving gravitons with large momentum transfer at least one of the ingoing gravitons must have energy of the order of \(M_{Pl}\) in which case our analysis becomes relevant. The same comment applies, of course, to scattering at transplanckian energies in string theory [14].

## Footnotes

- 1.
The choice of the linear contribution is fixed by the initial condition to have an ingoing graviton \(h=\lambda h_{(3)}\).

## Notes

### Acknowledgements

The authors would like to thank C. Gomez as well as Tomas Prochazka for discussions and, in particular, S. Mukhanov for substantial input. This work has received support from the Excellence Cluster ’Origins: From the Origin of the Universe to the First Building Blocks of Life’.

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