Shadow Celestial Amplitude Chi-Ming Chang12 Wei Cui21 Wen-Jie Ma21 Hongfei Shu21Hao Zou21

2025-05-03 0 0 609.99KB 35 页 10玖币

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Shadow Celestial Amplitude

Chi-Ming Chang1,2∗, Wei Cui2,1†, Wen-Jie Ma,2,1‡

Hongfei Shu2,1§,Hao Zou2,1¶

1Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China

2Yanqi Lake Beijing Institute of Mathematical Sciences and Applications (BIMSA),

Huairou District, Beijing 101408, P. R. China

Abstract

We study scattering amplitudes in the shadow conformal primary basis, which satisﬁes the

same deﬁning properties as the original conformal primary basis and has many advantages

over it. The shadow celestial amplitudes exhibit locality manifestly on the celestial sphere,

and behave like correlation functions in conformal ﬁeld theory under the operator product

expansion (OPE) limit. We study the OPE limits for three-point shadow celestial amplitude,

and general 2 →n−2 shadow celestial amplitudes from a large class of Feynman diagrams.

In particular, we compute the conformal block expansion of the s-channel four-point shadow

celestial amplitude of massless scalars at tree-level, and show that the expansion coeﬃcients

factorize as products of OPE coeﬃcients.

October 18, 2022

∗cmchang@tsinghua.edu.cn

†cwei@bimsa.cn

‡wenjia.ma@bimsa.cn

§shuphy124@gmail.com

¶hzou@bimsa.edu

arXiv:2210.04725v2 [hep-th] 17 Oct 2022

Contents

1 Introduction 1

2 Shadow conformal primary basis 4

2.1 Review on the celestial amplitudes . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Disadvantages of the celestial amplitudes . . . . . . . . . . . . . . . . . . . . 5

2.3 A diﬀerent conformal primary basis . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Translationsymmetry .............................. 8

3 OPE behaviour of the shadow celestial amplitudes 9

3.1 OPE analysis for three-point shadow celestial amplitudes . . . . . . . . . . . 10

3.2 OPE analysis from the generalized optical theorem . . . . . . . . . . . . . . 13

3.3 OPE analysis for a special class of Feynman diagrams . . . . . . . . . . . . . 14

3.4 OPE analysis for four-point shadow celestial amplitudes . . . . . . . . . . . . 15

4 Examples of shadow celestial amplitudes 17

4.1 Three-point shadow celestial amplitudes . . . . . . . . . . . . . . . . . . . . 17

4.2 Four-point shadow celestial amplitudes . . . . . . . . . . . . . . . . . . . . . 18

5 Outlook 22

A Preliminaries 24

A.1 The generalized optical theorem . . . . . . . . . . . . . . . . . . . . . . . . . 24

A.2 Shadow transformation and conformal partial waves . . . . . . . . . . . . . . 25

B The conformal integral 28

1 Introduction

Celestial holography is believed to be a concrete realization of holographic principles for quan-

tum gravity in asymptotically ﬂat spacetime (AFS) [1–5]. It relates scattering amplitudes of

a quantum ﬁeld theory or quantum gravity in a four-dimensional AFS to correlation func-

tions of a celestial conformal ﬁeld theory (CCFT) on the two-dimensional celestial sphere.

The Lorentz symmetry of the four-dimensional AFS is realized as the SL(2,C) conformal

symmetry of the celestial sphere. The goal of celestial holography is to study the scattering

amplitudes in AFS by using the techniques developed in conformal ﬁeld theory (CFT). One

of the most important achievements in celestial holography is recasting the soft theorems in

ﬂat space into Ward identities in two-dimensional CFTs. The currents associated to these

Ward identities generate asymptotic symmetries in the four-dimensional spacetime [6–16]

and can be re-organized into the w1+∞algebra [17–19].

To manifest the SL(2,C) Lorentz symmetry in the scattering amplitudes, one needs to

change the basis of asymptotic states from the standard plane-wave basis to the conformal

primary basis [20–23]. By deﬁnition, the conformal primary basis must satisfy the equations

of motion and transform covariantly under SL(2,C). The S-matrix elements in the confor-

mal primary basis are referred to as celestial amplitudes. The conformal primary basis that

is widely used in the literature for massless particles is built from the usual plane-wave basis

followed by a Mellin transformation. However, in this basis, the coordinates on the celestial

sphere relate directly to the solid angles of the ﬂat space momentum. Thus the correspond-

ing celestial amplitudes are highly constrained by four-dimensional kinematics and do not

take the standard form of CFT correlation functions. For example, the four-point celes-

tial amplitudes of massless scalars contain an unfamiliar delta-function δ(χ−¯χ) originated

from the momentum conservation. This distributional factor forces the celestial amplitude

to live on the equator of the celestial sphere. In addition, depending on the assignments of

the incoming and outgoing particles, the celestial amplitudes are only supported in disjoint

intervals on the equator.

Finally, in terms of the massless conformal primary basis, the celestial amplitudes do

not have proper conformal block expansion. This can be seen by looking at the s-channel

tree-level celestial amplitude of two incoming, two outgoing massless particles and one mas-

sive exchange particle. The imaginary part of the corresponding scattering amplitude in the

plane-wave basis is factorized into two three-point scattering amplitudes due to the opti-

cal theorem. This leads to a factorization in the conformal partial wave expansion of the

celestial amplitudes [24,25].1However, since the integration kernel in the conformal par-

tial wave expansion does not have poles located at the right half-plane, one does not get a

conformal block expansion by closing the contour. Again, in the literature, the studies of

conformal block expansion of celestial amplitudes are limited to the Klein space [28–31] or

three dimensional space [32,33].

To ﬁx these issues, we consider a diﬀerent set of conformal primary wavefunctions for

massless particles, which are in a diﬀerent branch of solutions to the two deﬁning properties

of conformal primary wavefunction [1], i.e. they satisfy the equations of motion and trans-

1Partial wave expansion of the celestial amplitudes is also studied in [26,27].

form covariantly under SL(2,C). It turns out that, up to a constant factor, these conformal

primary wavefunctions are equivalent to the shadow transformations [34,35] of the original

conformal primary wave functions.2We will refer to this basis as the shadow conformal

primary basis. Expanding the scattering amplitude in the shadow conformal primary basis,

we deﬁne the shadow celestial amplitudes, which can be obtained by performing the shadow

transformation of all external operators of the celestial amplitudes.3The shadow celestial

amplitudes resolve all the abovementioned issues and lead to the standard correlation func-

tions of CFTs. Speciﬁcally, the shadow celestial amplitudes of four massless particles no

longer have δ(χ−¯χ) and are deﬁned on the entire celestial sphere. In addition, the shadow

celestial amplitudes have well-behaved OPE limits. For scattering amplitudes with nexter-

nal massless real scalars, we consider the OPE limit by making the celestial coordinates of

the ﬁrst two incoming particles close to each other. For amplitudes from a large class of Feyn-

man diagrams, we ﬁnd that the shadow celestial amplitudes factorize as expected as n-point

correlation functions in a CFT. Using the generalized optical theorem, we also obtain the

correct factorization of the imaginary part of shadow celestial amplitudes for any Feynman

diagrams. What’s more, we compute the 4-point shadow celestial amplitudes involving four

external massless scalars and one exchange massive scalar and derive its conformal block

expansion in the 12 ↔34 channel. We ﬁnd that the coeﬃcients appearing in the conformal

block expansion give the correct OPE coeﬃcients obtained from the corresponding 3-point

shadow celestial amplitudes matching the results from the OPE analysis.

This paper is organized as follows. In Section 2, we review the celestial amplitudes, discuss

their disadvantages, and introduce the shadow celestial amplitudes. In Section 3, we analyze

the OPE limits of n-point shadow celestial amplitudes involving nexternal real massless

scalars and show that the OPE limits are well-deﬁned for the shadow celestial amplitudes.

In Section 4, we work out the conformal block expansion on the 4-point shadow celestial

amplitudes involving four external massless scalars and one exchange massive scalar. We

ﬁnd the complete agreement between the block expansion coeﬃcients and OPE coeﬃcients.

In Section 5, we conclude our work and point out a few future directions. In Appendix A,

we review the generalized optical theorem, the shadow transformation and conformal partial

waves, that will be used in later parts of this paper.

2The shadows of the conformal wave functions were previously studied in [20].

3Implementing the shadow transformation on one of the external operators of the celestial amplitude was

discussed in [36–38].

2 Shadow conformal primary basis

2.1 Review on the celestial amplitudes

Celestial amplitudes are obtained by expanding the position space amplitudes with respect

to the conformal primary wavefunctions [20] instead of the plane-waves, i.e.,4

A∆i(zi) = k+n

j=1 Zd4xj k

j=1

ϕ+

∆j(zj;xj) k+n

j=k+1

ϕ−

∆j(zj;xj)M(xi),(2.1)

where M(xj) is the scattering amplitude in position space. The conformal primary wave

functions ϕ±

∆(z;x) for massless and massive scalars with mass mare given by

ϕ±

∆(z;x) = Z+∞

dω ω∆−1e±iω ˆq·x−ϵω ,(2.2)

and

ϕ±

∆(z;x) = Zd3ˆp′

ˆp′0G∆(z, ¯z; ˆp′)e±imˆp′·x,(2.3)

respectively. Here zand ¯zare coordinates on celestial sphere and G∆(z, ¯z; ˆp) is the bulk-

to-boundary propagator. The coordinates (z, ¯z) on the celestial sphere are related to the

massless on-shell momenta qµthrough

qµ=ωˆqµ=ω(1 + z¯z, z + ¯z, −i(z−¯z),1−z¯z) (2.4)

and to the massive on-shell momenta pµthrough

pµ=mˆpµ=m

2y(1 + y2+z¯z, z + ¯z, −i(z−¯z),1−y2−z¯z).(2.5)

In terms of ˆpin (2.5), the bulk-to-boundary propagator takes the form 5

G∆(ˆq; ˆp′) = 1

(−ˆq·ˆp′)∆.(2.6)

4Throughout this paper, Oi(zi) should be understood as Oi(zi,¯zi). We use this abbreviation to simplify

the notation.

5In this paper, we use the most positive metric in four-dimensional ﬂat space, i.e.,gAB =

diag(−1,+1,+1,+1).

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摘要：

ShadowCelestialAmplitudeChi-MingChang1,2∗,WeiCui2,1†,Wen-JieMa,2,1‡HongfeiShu2,1§,HaoZou2,1¶1YauMathematicalSciencesCenter,TsinghuaUniversity,Beijing100084,China2YanqiLakeBeijingInstituteofMathematicalSciencesandApplications(BIMSA),HuairouDistrict,Beijing101408,P.R.ChinaAbstractWestudyscatteringampl...

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