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JHEP02(2018)095

Published for SISSA by Springer

Received: June 5, 2017

Revised: November 14, 2017

Accepted: February 8, 2018

Published: February 15, 2018

Unifying relations for scattering amplitudes

Cliﬀord Cheung,

Chia-Hsien Shen

and Congkao Wen

a,b

Walter Burke Institute for Theoretical Physics, California Institute of Technology,

Pasadena, CA 91125, U.S.A.

Mani L. Bhaumik Institute for Theoretical Physics,

Department of Physics and Astronomy, UCLA,

Los Angeles, CA 90095, U.S.A.

E-mail: clifford.cheung@caltech.edu, chshen@caltech.edu,

cwen@caltech.edu

Abstract: We derive new amplitudes relations revealing a hidden unity among a wide-

ranging variety of theories in arbitrary spacetime dimensions. Our results rely on a set of

Lorentz invariant diﬀerential operators which transmute physical tree-level scattering am-

plitudes into new ones. By transmuting the amplitudes of gravity coupled to a dilaton and

two-form, we generate all the amplitudes of Einstein-Yang-Mills theory, Dirac-Born-Inﬁeld

theory, special Galileon, nonlinear sigma model, and biadjoint scalar theory. Transmuta-

tion also relates amplitudes in string theory and its variants. As a corollary, celebrated

aspects of gluon and graviton scattering like color-kinematics duality, the KLT relations,

and the CHY construction are inherited traits of the transmuted amplitudes. Transmu-

tation recasts the Adler zero as a trivial consequence of the Weinberg soft theorem and

implies new subleading soft theorems for certain scalar theories.

Keywords: Eﬀective Field Theories, Scattering Amplitudes

ArXiv ePrint: 1705.03025

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP02(2018)095

JHEP02(2018)095

question then arises: why these theories? Imbued with symmetries of disparate origin and

character, these theories do not obviously conform to any cohesive organizing principle that

would place them on equal footing in the eyes of the S-matrix.

In this paper, we oﬀer an explanation for the peculiar universality of these structures.

Our results follow from a set of simple unifying relations which “transmute” the tree-level

scattering amplitudes of certain theories into those of others. By transmuting the S-matrix

of the mother of all theories — gravity coupled to a dilaton and two-form, i.e., “extended

gravity” — we beget the S-matrices of all the theories previously mentioned, thus revealing

their many shared traits as congenital.

Transmutation is by design independent of the particular form in which an ampli-

tude happens to be represented. To achieve this, we study amplitudes as formal functions

of Lorentz invariant products of the external on-shell kinematic data: e

, p

, and p

where e

and p

denotes polarization and momentum of particle i, respectively, and i 6= j.

1,2

In this language, it is straightforward to construct a basis of gauge invariant operators which

preserve on-shell kinematics. Perhaps unsurprisingly, these operators automatically trans-

mute gauge invariant scattering amplitudes into new ones. Our roster of transmutation

operators is summarized as follows:

• The trace operator T

= ∂

reduces the spin of particles i and j by one unit and

places them within a new color trace structure. This operator transmutes gravitons

into photons, gluons into biadjoint scalars, and BI photons into DBI scalars.

• The insertion operator T

ijk

= ∂

− ∂

reduces the spin of particle j by one unit

and inserts it between particles i and k within a color trace structure. This operator

transmutes gravitons into gluons, gluons into biadjoint scalars, and BI photons into

pions, at the level of color-ordered amplitudes of the resulting theories, which we will

properly deﬁne in section 3.

• The longitudinal operator L

∂

reduces the spin of particle i by one

unit while converting it to a longitudinal mode. Necessarily applied to all particles

at once, this operator transmutes gravitons into BI photons, gluons into pions, and

BI photons into SG scalars.

These transmutation operators form the building blocks of the uniﬁed web of theories

depicted in ﬁgure 1. Theory space is partitioned into three triangular regions corresponding

to the nearest of kin to extended gravity, BI, and YM. The corners of the web denote all of

the theories previously discussed, while the edge and bulk regions denote hybrid theories.

Remarkably, these triangular regions are not independent — due to KLT [1] and BCJ [2],

extended gravity can be recast as the double copy of YM, and BI as the product of YM

So as not to complicate our notation we omit the usual “·” denoting Lorentz-invariant contractions.

In the extended gravity, all the degrees of freedom can be incorporated into a single polarization tensor

µν

. It can written as a product of two copies of YM polarizations, e

µν

= e

˜e

. The usual pure graviton

corresponds to the symmetric and traceless component of e

µν

. Since the transmutation only acts on one of

the copies, we focus on the un-tilded copy without loss of generality.

– 2 –

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