非摄动统一性：引力散射的新视角

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"这篇论文探讨了引力散射中的非摄动统一性问题，主要集中在树级引力子-标量散射在Regge极限中的表现。作者Ivo Sachs和Tung Tran指出，无需借助广义相对论之外的任何扩展理论，仅通过非摄动效应就能实现引力散射的统一。在普朗克能量尺度上，引力子对背景几何的反作用产生非摄动的平面波，这一现象有助于缓解高能（UV）行为的尖锐度。论文的振幅分析显示，它在低能量扰动引力标量散射与Regge极限（对所有s值）中的经典平面波散射之间建立了联系。文章的主体部分首先引出了引力散射在量子场论中的挑战，特别是在高能行为时的单位arity问题。传统的摄动方法在高能量情况下可能会失效，因为引力相互作用的强度随着能量增加而增强，可能导致无限的截断反常。为了解决这个问题，作者提出了一种非摄动的视角，即在Regge极限中，引力子散射可以通过引力自身内在的非线性效应得到统一。接着，他们详细介绍了普朗克能量下背景几何的变化，这里的引力子不再被视为微扰背景的粒子，而是可以显著改变背景的实体。这种反作用产生的非摄动平面波改变了散射过程的动力学，从而软化了UV行为，避免了单位arity的破坏。这种现象提供了一种可能的途径，使得高能引力散射过程保持物理上可接受的行为。在数学表述上，作者展示了他们的振幅分析如何连接了低能扰动理论与高能的经典行为。这表明，尽管在不同能量尺度上，散射过程可能表现出截然不同的特性，但它们在非摄动框架下是连续的，并且在Regge极限中趋于一致。最后，论文讨论了这些发现对更广泛的物理学领域，特别是量子引力理论的潜在影响。非摄动统一性概念可能为理解引力在极端条件下的行为提供新的见解，同时挑战和指导未来量子引力理论的发展。由于文章是开放获取的，研究者们可以自由地访问并进一步研究这些概念，推动对引力相互作用本质的理解。参考文献：Eur.Phys.J.C(2019)79:914，DOI: 10.1140/epjc/s10052-019-7403-2。"

Eur. Phys. J. C (2019) 79:914

https://doi.org/10.1140/epjc/s10052-019-7403-2

Regular Article - Theoretical Physics

On non-perturbative unitarity in gravitational scattering

Ivo Sachs

1,a

, Tung Tran

2,b

Arnold Sommerfeld Center for Theoretical Physics, Ludwig Maximilian University of Munich, Theresienstr. 37, 80333 Munich, Germany

Albert Einstein Institute, Am Mühlenberg 1, 14476 Potsdam-Golm, Germany

Received: 29 September 2019 / Accepted: 11 October 2019 / Published online: 12 November 2019

Abstract We argue that the tree-level graviton-scalar scat-

tering in the Regge limit is unitarized by non-perturbative

effects within General Relativity alone, that is without resort-

ing to any extension thereof. At Planckian energy the back

reaction of the incoming graviton on the background geom-

etry 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

It is well known that perturbative scattering amplitudes

involving gravitons violate the unitarity bound at Planckian

energy even at tree-level. For instance, the scattering ampli-

tude of a graviton and a massless scalar ﬁeld is given by

[1]

tree

=−(ε

· ε

)

iκ

, (1)

where s, t, u are the usual Mandelstam variables and κ 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.

On the other hand, one may question whether the assump-

tion of asymptotic in- and out-states on which (1)isbuilt

holds for gravitons of Planckian energy since Gravity is a

non-linear theory whose coupling strength increases with

energy. One argument in favor of it is that a single graviton

can always be boosted to an inertial frame where its energy is

e-mail: ivo.sachs@physik.lmu.de

e-mails: tung.tran@physik.uni-muenchen.de; tung.tran@aei.mpg.de

small. However, for a two body scattering with large center

of mass (CoM) energy

√

s, there is no boost for which both

particles have small energy. Thus back-reaction will have to

be taken into account for at least one in-going particle. This

idea is not new. It was explored already many years ago by

’t Hooft [2] and others [3] who replaced an ingoing scalar

of transplanckian energy by a gravitational shock wave. One

may also interpret this back-reaction as a contribution to the

self completeness mechanism of gravity proposed by Dvali

and Gomez [4,5]. The starting point on which we base our

argument for a non-perturbative unitarization of (1) is similar

to [2] although the details are somewhat different. We per-

form a Lorentz boost such that the energy of the incomming

scalar is small while the incomming graviton has transplanck-

ian energy so that back reaction on geometry has to be taken

into account. Luckily, an exact solution to Einstein’s equa-

tion, accounting for the complete back reaction on geometry

is available in the form of a plane wave [6,7](c.f.[8–12]).

As a result, the non-perturbative generalization of (1)inthe

large s but small t (or Regge) limit can be reduced to a per-

turbative calculation on top of a plane wave as illustrated in

Fig. 1.

2 Perturbative limit

To see how this comes about let us ﬁrst recover the pertur-

bative amplitude (1)for(h

(1)

,φ

(2)

→ h

(3)

,φ

(4)

) in position

space. Without restricting the generality we make the follow-

ing momentum assignments

(1)

≡ p = (p

, 0, 0, 0), p

(2)

≡ k = (0, k

−

, 0, 0)

(3)

≡ q = (q

, q

−

, q

), p

(4)

≡ l = (l

, l

−

, l

In position space the t-channel diagram can then be calcu-

lated as follows: We ﬁrst solve for the internal graviton

around Minkowski background, η, through

123

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