Soft-Collinear Gravity and Soft Theorems Martin BenekePatrick Hagerand Robert Szafron Abstract This chapter reviews the construction of soft-collinear gravity the ef-

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Soft-Collinear Gravity and Soft Theorems

Martin Beneke,∗Patrick Hager†and Robert Szafron

Abstract This chapter reviews the construction of “soft-collinear gravity”, the ef-

fective ﬁeld theory which describes the interaction of collinear and soft gravitons

with matter (and themselves), to all orders in the soft-collinear power expansion,

focusing on the essential concepts. Among them are an emergent soft background

gauge symmetry, which lives on the light-like trajectories of energetic particles and

allows for a manifestly gauge-invariant representation of the interactions in terms of

a soft covariant derivative and the soft Riemann tensor, and a systematic treatment

of collinear interactions, which are absent at leading power in gravity. The gravita-

tional soft theorems are derived from soft-collinear gravity at the Lagrangian level.

The symmetries of the effective theory provide a transparent explanation of why soft

graviton emission is universal to sub-sub-leading power, but gauge boson emission

is not and suggest a physical interpretation of the form of the universal soft factors

in terms of the charges corresponding to the soft symmetries. The power counting

of soft-collinear gravity further provides an understanding of the structure of loop

corrections to the soft theorems.

Keywords

Gravitation, soft-collinear effective ﬁeld theory, effective Lagrangian, soft and

collinear divergences, soft theorem, power corrections, Einstein-Hilbert theory

M. Beneke, P. Hager

Physik Department T31, James-Franck-Straße 1, Technische Universit¨

at M¨

unchen, D–85748

Garching, Germany

R. Szafron

Department of Physics, Brookhaven National Laboratory, Upton, N.Y., 11973, U.S.A.

TUM-HEP-1425/22, 14 October 2022

∗Corresponding author

†Address after October 1st, 2022: PRISMA+Cluster of Excellence & Mainz Institute for Theo-

retical Physics, Johannes Gutenberg University, D–55099 Mainz, Germany

arXiv:2210.09336v1 [hep-th] 17 Oct 2022

2 Martin Beneke, Patrick Hager and Robert Szafron

“My reasons for now attacking this question are:

(1) Because I can. [...] (2) Because something

might go wrong and this would be interesting. Un-

fortunately, nothing does go wrong.”

S. Weinberg, Ref. [38]

1 Introduction

The gravitational force is widely perceived to be fundamentally different from the

gauge forces that govern the other microscopic interactions of the elementary parti-

cles. The gravitational interactions are inevitably non-renormalisable, calling for a

modiﬁcation at very short distances (or a non-trivial ultraviolet ﬁxed point). Their

underlying gauge symmetry is related to space-time transformations, in contrast

to the internal SU(3)×SU(2)×U(1) gauge symmetries operating on ﬁelds in rigid

Minkowski space-time.

Yet, from the low-energy perspective and applying the basic principles of quan-

tum ﬁeld theory, the Lagrangian of weak-ﬁeld gravity on Minkowski space follows

from the desire to construct a consistent theory for a massless spin-2 particle in very

much the same way as gauge theories do for the case of a massless spin-1 particle.

The universality and space-time symmetry of gravitation then arises from the re-

quirement that interacting massless ﬁelds with spin larger than 1

2must couple to a

conserved current, which is the energy-momentum tensor for spin-2. This motivates

a closer inspection of the relation between gauge theory and gravitational scattering

in Minkowski space.

The study of gravitational scattering amplitudes in quantised weak-ﬁeld gravity

has attracted much attention after the discovery of remarkable relations between

graviton and gluon scattering amplitudes [12,13], which state that tree-level ampli-

tudes of the former can be obtained by “squaring” Yang-Mills tree-level amplitudes

and replacing colour factors by kinematic ones. The simplest example is the three-

point amplitude in spinor-helicity notation (reviewed in [27]):

AYM(1−

a,2−

b,3+

c) = −gsfabc h12i3

h23ih31i

(1)

AEH(1−−,2−−,3++) = κ

2h12i6

h23i2h31i2.

Numerous extensions of such “double copy” or “colour-kinematics duality” rela-

tions have been found to different gauge/gravity theories, to non-trivial classical

backgrounds, and to the one-loop level.

Soft-Collinear Gravity and Soft Theorems 3

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k+p"

Fig. 1 Emission of a boson with momentum kfrom an external energetic particle with momen-

tum p.

Much can be learnt by looking at the behaviour of quantum amplitudes in the

infrared (IR). When an energetic, massless particle with momentum pµemits an-

other massless particle with momentum kµ, see Figure 1, the internal propagator

1/(p+k)2becomes singular in the soft limit kµ→0, and in the collinear limit

pµkkµ. When such conﬁgurations are integrated over the phase space of the emit-

ted particle, or appear inside loops, the result is a logarithmic divergence. It has been

recognised from the early days of quantum ﬁeld theory that the soft and collinear

limits exhibit universal, process-independent features [16,28,30] and that a precise

deﬁnition of quantum-mechanical observables is required to obtain sensible, IR-

ﬁnite results. The study of these limits in quantum electrodynamics and non-abelian

gauge theories accordingly has a long history.

In QCD, which is strongly interacting in the infrared, understanding the soft and

collinear limit is of paramount importance to make predictions for high-energy scat-

tering, and culminates in powerful factorisation theorems [23]. In this way, one iso-

lates the infrared physics in some well-deﬁned functions, while leaving the more

process-dependent features for perturbative calculations. Given the presence of sev-

eral momentum scales, it is logical to apply effective Lagrangians to capture the IR

physics, especially as Lagrangians are better suited than amplitudes to uncover the

gauge-invariance and recursive structure of multi-scale problems. Soft-collinear ef-

fective theory (SCET) [2,3,5,6] has therefore emerged as an important conceptual

and calculational tool for factorisation in gauge theories. It is designed to precisely

reproduce Feynman amplitudes in their soft and collinear limits. Moreover, at least

in principle, it can do so beyond the leading power in the expansion in small scale

ratios. Although the case of high-energy gravitational scattering appears to be of less

practical relevance, in view of the above mentioned relations between graviton and

Yang-Mills scattering amplitudes, which is presently not understood at Lagrangian

level, it is suggestive to apply effective Lagrangian techniques to at least the soft

and collinear limits of graviton amplitudes—which is the subject of this chapter.

These limits already exhibit interesting similarities and differences between

massless spin-1 and spin-2 particles (gauge bosons and gravitons, respectively) cou-

pled to matter. It has been noted long ago [38] that the “eikonal” or leading soft limit

of gravity is very similar to gauge theory. The long-wavelength radiation “sees” only

the direction of motion (classical trajectory) and charge of energetic particles. Thus,

in the eikonal approximation, the amplitude for radiating a single soft graviton from

energetic particles with momenta pµ

iemerging from a hard scattering process,

4 Martin Beneke, Patrick Hager and Robert Szafron

Arad(pi;k) = κ

2∑

pµ

ipν

iεµν (k)

pi·kA(pi),(2)

is obtained from its gauge-theory correspondent,

Arad(pi;k) = −gs∑

pi·εa(k)

pi·kA(pi),(3)

by simply replacing the gauge charge (generator) by the gravitational charge, mo-

mentum, ta

i→pν

iand adjusting the coupling gsand polarisation vector εa

µ(k)to the

gravitational coupling, κ=√32πGN, and polarisation tensor, εµν (k). Since eikon-

alised propagators 1/(pi·k)are closely related to semi-inﬁnite Wilson line oper-

ators, we expect soft graviton Wilson lines to play a similar role for soft graviton

physics [32,40] as they do in gauge theories. Eqs. (2), (3) represent the leading

terms in the so-called soft theorems, to which we shall return in a later section of

this chapter.

The collinear limit of graviton amplitudes is, however, very different from the

one of gauge amplitudes. In fact, in gravity, collinear enhancements and singulari-

ties are absent altogether. As a consequence, even if the gravitational coupling was

not minuscule, i.e. near Planckian scattering energies, energetic particles do not pro-

duce gravitational jets, which in QCD constitute the most visible footprints of the

non-abelian charges of the quarks and gluons. The absence of collinear singulari-

ties for graviton emission was ﬁrst shown in [38] in the simultaneous eikonal limit.

Weinberg also noted that it would be rather troublesome, if this was not the case,

since it would prevent the existence of massless particles with gravitational charges,

that is, any non-vanishing four momentum. However, while massless particles with

gauge charges do not exist in Nature, and hence there is no conﬂict with the exis-

tence of collinear singularities in gauge theories, there are massless particles which

gravitate, such as the photons and the gravitons themselves.

There is a simple qualitative explanation for the absence of collinear graviton

singularities based on the classical radiation pattern [11]. When an energetic parti-

cle with virtuality much less than its three-momentum squared p2emits a graviton

with momentum kwith small angle θbetween pand k, the near mass-shell singu-

larity of the emitting particle propagator 1/(|p||k|(1−cosθ)) yields a factor θ−2

for the splitting amplitude. Quantising the radiation ﬁeld in the spherical basis with

single-particle states |kjm;λi, where λdenotes helicity (±2 for gravitons and ±1

for gauge bosons) and jm the angular momentum quantum numbers with respect

to the quantisation axis p, this implies that the emitted graviton must be in a state

|kj0;λi, where m=0 due to angular momentum and helicity conservation. The an-

gular dependence of this state is given by the spin-weighted spherical harmonic or

Wigner function Dj

±λ,0(k)∝sin|λ|θ

2, which tends to zero as θ|λ|in the θ→0 limit.

Thus, the splitting amplitude has no singularity in the collinear limit for graviton

emission (λ=±2) in contrast to the case of gauge bosons.

The above argument refers to the physical polarisation states of the graviton and

thus does not cover the properties of individual Feynman amplitudes in general,

Soft-Collinear Gravity and Soft Theorems 5

in particular in covariant gauges, which do have collinear divergences. The formal

demonstration of the absence of collinear divergences without the restriction to the

eikonal limit adopted in [38] has been presented only relatively recently [1] with

diagrammatic factorisation methods. This fact is made evident in the construction of

the soft-collinear effective Lagrangian for gravity (“soft-collinear gravity”) [11]: the

leading effective Lagrangian describing collinear graviton self-interactions and their

interactions with matter is a free theory. This motivates the investigation of collinear

gravitational physics at sub-leading order in the collinear expansion, where it is non-

trivial, and naturally leads to the systematic construction of soft-collinear effective

gravity beyond the leading power in both, the collinear and soft limits [10].

The present chapter starts with a review of basic ideas and methods for soft-

collinear Lagrangians, assuming no prior familiarity with the subject. We then pro-

vide a technically light-weight discussion of soft-collinear gravity, focusing on the

exposition of the principles of the construction, the structure and emergent symme-

tries of the result at the expense of many technical details, for which we refer to

[10]. By deﬁnition, soft-collinear gravity builds an extension of the so-called soft

theorems to all orders in the loop and soft expansion. It is nevertheless of interest

to rederive them from the effective Lagrangian [9]. In the last section of this chap-

ter, we brieﬂy cover how this provides an understanding of why in gravity the soft

theorem extends to the next-to-next-to-soft order (but does not in gauge theory) and

how the form of the universal terms is related to the (emergent) soft gauge symme-

tries of the effective Lagrangian. The chapter concludes with a discussion of loop

corrections to the soft theorem.

2 Basic ideas and concepts

The following section sets up the notation and introduces a number of concepts that

arise in effective ﬁeld theory (EFT) and SCET in particular.

2.1 Perturbative gravity

The full theory, from which SCET gravity is constructed, is the Einstein-Hilbert

theory with action

SEH =−2

κ2Zd4x√−gR ,(4)

coupled to matter, here a minimally-coupled scalar ﬁeld ϕin the curved space-time

with metric tensor gµν . The matter part is described by the action

Sϕ=Zd4x√−g1

2gµν ∂µϕ∂νϕ,(5)

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Soft-CollinearGravityandSoftTheoremsMartinBeneke,PatrickHagerandRobertSzafronAbstractThischapterreviewstheconstructionofsoft-collineargravity,theef-fectiveeldtheorywhichdescribestheinteractionofcollinearandsoftgravitonswithmatter(andthemselves),toallordersinthesoft-collinearpowerexpansion,focus...

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