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JHEP06(2017)064

Published for SISSA by Springer

Received: April 4, 2017

Accepted: May 1, 2017

Published: June 13, 2017

Spontaneously broken Yang-Mills-Einstein

supergravities as double copies

Marco Chiodaroli,

a,d

Murat G¨unaydin,

Henrik Johansson

c,d,e

and Radu Roiban

b,f

Max-Planck-Institut f¨ur Gravitationsphysik, Albert-Einstein-Institut,

Am M¨uhlenberg 1, 14476 Potsdam, Germany

Institute for Gravitation and the Cosmos, The Pennsylvania State University,

University Park PA 16802, U.S.A.

Theory Division, Physics Department, CERN,

CH-1211 Geneva 23, Switzerland

Department of Physics and Astronomy, Uppsala University,

SE-75108 Uppsala, Sweden

Nordita, KTH Royal Institute of Technology and Stockholm University,

Roslagstullsbacken 23, SE-10691 Stockholm, Sweden

Kavli Institute for Theoretical Physics, University of California,

Santa Barbara, CA 93106-4030, U.S.A.

E-mail: marco.chiodaroli@physics.uu.se, mgunaydin@psu.edu,

henrik.johansson@physics.uu.se, adu@phys.psu.edu

Abstract: Color/kinematics duality and the double-copy construction have proved to be

systematic tools for gaining new insight into gravitational theories. Extending our ear-

lier work, in this paper we introduce new double-copy constructions for large classes of

spontaneously-broken Yang-Mills-Einstein theories with adjoint Higgs ﬁelds. One gauge-

theory copy entering the construction is a spontaneously-broken (super-)Yang-Mills theory,

while the other copy is a bosonic Yang-Mills-scalar theory with trilinear scalar interactions

that display an explicitly-broken global symmetry. We show that the kinematic numera-

tors of these gauge theories can be made to obey color/kinematics duality by exhibiting

particular additional Lie-algebraic relations. We discuss in detail explicit examples with

N = 2 supersymmetry, focusing on Yang-Mills-Einstein supergravity theories belonging to

the generic Jordan family in four and ﬁve dimensions, and identify the map between the

supergravity and double-copy ﬁelds and parameters. We also brieﬂy discuss the application

of our results to N = 4 supergravity theories. The constructions are illustrated by explicit

examples of tree-level and one-loop scattering amplitudes.

Keywords: Extended Supersymmetry, Higgs Physics, Scattering Amplitudes, Supergrav-

ity Models

ArXiv ePrint: 1511.01740

Open Access,

 The Authors.

Article funded by SCOAP

doi:10.1007/JHEP06(2017)064

JHEP06(2017)064

1 Introduction

Einstein’s theory of gravity and spontaneously-broken gauge theory are two of the pillars of

our current understanding of the known fundamental interactions of Nature. While super-

symmetric ﬁeld theories that combine gravitational interactions and spontaneous symmetry

breaking have been studied extensively at the Lagrangian level, the perturbative S matrices

of these theories have largely been unexplored.

Modern work on scattering amplitudes in matter-coupled gravitational theories has

been largely focused on pure supergravities and on cases in which additional matter consists

of abelian vectors (i.e. Maxwell-Einstein supergravities) or fermion/scalar ﬁelds. A key tool

has been the double-copy construction [1, 2], which has led to a dramatic simpliﬁcation of

perturbative calculations. For example, explicit expressions of one-, two-, three- and four-

loop amplitudes have been obtained for N = 4, N = 5 and N = 8 supergravities in refs. [2–

14]. For the case of N ≤ 4, one-loop four-point superamplitudes have been obtained for the

generic Jordan family of N = 2 Maxwell-Einstein supergravity (MESG) theories [15–17],

for pure supergravities with N ≤ 4 [15, 17, 18], and for orbifolds thereof [15, 19].

The double-copy construction assumes the existence of presentations of gauge-theory

scattering amplitudes that exhibit color/kinematics duality. The duality states that, in

an amplitude’s Feynman-like diagrammatic expansion, one can ﬁnd numerator factors

that obey Lie-algebraic kinematic relations mirroring the relations satisﬁed by the cor-

responding gauge-group color factors. Once found, the numerators may play the role of

these color factors in any gauge theory amplitude, and upon substitution one obtains valid

gravitational amplitudes. There is by now extensive evidence for the duality and for the

double-copy construction in wide classes of Yang-Mills (YM) theories and in the associated

(super)gravity theories. Examples where color/kinematics duality has been demonstrated

include: pure super-Yang-Mills (SYM) theories [1, 2, 20–22], SYM theories with adjoint

matter [15, 17, 18], self-dual Yang-Mills theory [23, 24], QCD and super-QCD [25, 26], YM

coupled to φ

theory [27], and YM theory extended by a higher-dimensional operator [28].

It has also been observed that the duality is not limited to YM gauge theories, but it also

applies to certain Chern-Simons-matter theories [29–31], as well as to the non-linear sigma

model/chiral Lagrangian [32] and to the closed (heterotic) sector of string theory [33].

Amplitudes in Maxwell-Einstein supergravities are obtained by a double-copy con-

struction of the form (pure SYM)⊗(YM coupled to scalars). Subgroups of the global

symmetries of Maxwell-Einstein supergravities can be gauged.

In the resulting theories

some of the vector ﬁelds become gauge ﬁelds of the chosen gauge group and transform in its

adjoint representation. Therefore, the only subgroups of the global symmetry that can be

gauged are those whose adjoint representation is smaller than the number of vector ﬁelds

that transform non-trivially under the global symmetry group. In ﬁve dimensions, gauging

only a subgroup of the global symmetry group in N = 2 Maxwell-Einstein supergravity

theories does not introduce a potential for the scalar ﬁelds and hence the resulting theory

is guaranteed to have a Minkowski vacuum state.

While gauging part of the R-symmetry group is very interesting, here we will focus on gaugings that

only aﬀect the other global symmetries.

– 1 –

JHEP06(2017)064

The double-copy construction of a wide class of Yang-Mills-Einstein supergravity

(YMESG) theories was given in [27], where it was shown that one of the two gauge-

theory factors is a pure SYM theory, and the other is a bosonic YM theory coupled to

scalars that transforms in the adjoint representation of both the gauge group and a global

symmetry group. The latter theory has trilinear φ

couplings, and hence we refer to it as

YM + φ

theory. Through the double-copy construction, the global symmetry of the non-

supersymmetric gauge-theory factor becomes a local symmetry, and the trilinear scalar

couplings generate the minimal couplings of the corresponding gauge ﬁelds. The gravi-

tational supersymmetry is directly inherited from the SYM theory, thus accommodating

N = 1, 2, 4 YMESG theories and N = 0 Yang-Mills-Einstein (YME) theories. Earlier

work [34] introduced the same type of construction for single-trace tree-level YME ampli-

tudes. Recent work on YME amplitudes takes several diﬀerent approaches, see refs. [35–41].

It is essential to explore the validity of the double-copy construction away from the

origin of the moduli space. In particular, a natural and physically-motivated extension

is to consider cases in which the supergravity gauge symmetry is spontaneously broken

through the Brout-Englert-Higgs mechanism. We will present such an extension in the

present paper. As a key result, we ﬁnd that one of the two gauge-theory factors is the

spontaneously-broken pure SYM theory (or, alternatively stated, the Coulomb branch of

pure SYM theory), while the other is a particular massive deformation that explicitly

breaks the global symmetries of the YM + φ

theory.

Identifying the relation between asymptotic states of the supergravity theory and the

corresponding states of the gauge-theory factors is an important aspect of the double-

copy construction. For gauge theories with only adjoint ﬁelds, the double copy gives

a supergravity state for every tensor product of gauge-theory states (not counting the

degeneracy of the representation). In cases in which the gauge-theory matter transforms

in non-adjoint representations of the gauge group, the double-copy construction allows for

better tuning of the matter content of the gravitational theory, since only certain tensor

products of the gauge-theory matter are allowed.

In ref. [19] color/kinematics duality was extended to non-adjoint representations in

the context of orbifolds of N = 4 SYM, and the associated double copies were found to

be matter-coupled supergravity theories. The construction required that: (1) the gauge

groups of the two gauge theories should be identiﬁed, and (2) supergravity states corre-

spond to gauge-invariant bilinears that can be formed out of the gauge-theory states. This

construction correlates gauge- and global-group representations appearing in the resulting

gauge theories.

In ref. [17] color/kinematics duality was extended to theories with ﬁelds in the funda-

mental representation and used to construct pure N ≤ 4 supergravity theories as well as

matter-coupled theories. In this construction, the necessary condition for the double copy

to be valid is that the kinematic matter-dependent numerators obey the same relations

as the corresponding color factors with fundamental representations. Upon replacing the

color factors with kinematic numerator factors one similarly obtains a double copy that

correlates the representations of the states of the two gauge-theory sides.

For the double-copy constructions of supergravity theories with spontaneously-broken

gauge symmetry, the identiﬁcation of the asymptotic states will follow closely the non-

– 2 –

JHEP06(2017)064

adjoint or fundamental cases. However, the details of the kinematic algebra obeyed by

the numerators will diﬀer substantially compared to previous situations. The kinematic

Jacobi identities and commutation relations will be extended by additional identities which

are inherited from the Jacobi relations of the theory with unbroken gauge symmetry. We

stress that our construction works well with — but does not require — supersymmetry,

and similarly works in all dimensions in which the theories are deﬁned, as it is expected

for color/kinematics duality.

The paper is organized as follows. In section 2 we review color/kinematics duality, and

identify matter-coupled gauge theories with ﬁelds in several diﬀerent representations of the

gauge group and speciﬁc cubic and quartic couplings which obey the duality. We extend

color/kinematics duality and the double-copy construction to massive ﬁeld theories, as well

as to ﬁeld theories with spontaneously-broken gauge symmetry, paying close attention to

the construction of asymptotic states. In particular subsection 2.6 discusses extensions of

the double-copy construction and contains our main results of this generalization.

In section 4 we review, from the Lagrangian perspective, the Higgs mechanism in four-

and ﬁve-dimensional N = 2 Yang-Mills-Einstein supergravities. Such theories are uniquely

speciﬁed by their cubic interactions and provide simple examples of our construction. In

particular, we identify the four-dimensional symplectic frame in which the amplitudes from

the spontaneously-broken Yang-Mills-Einstein supergravity Lagrangian reproduce the ones

from the double-copy construction.

In section 5, we compute tree-level scattering amplitudes in the gauge theories dis-

cussed in section 3 and in the supergravity theories discussed in section 4. We ﬁnd the

constraints imposed by color/kinematics duality on the cubic and quartic couplings of the

gauge theories, identify the precise map between supergravity states and gauge-invariant

billinears of gauge-theory states, and give the relation between the gauge-theory and su-

pergravity parameters.

In section 6, we discuss loop-level calculations in theories formulated in the earlier

sections. Section 7, discusses brieﬂy spontaneously-broken N = 4 Yang-Mills-Einstein

supergravity theories. We review the bosonic part of their Lagrangians in ﬁve dimensions

and discuss how their amplitudes can be obtained through the double-copy construction

with a straightforward extension of the results obtained for N = 2 theories.

2 Color/kinematics duality and double copy

In this section, we review the color/kinematics duality applied to gauge theories that have

ﬁelds in complex representations of the gauge group. Giving concrete examples, we write

down Lagrangians of several gauge theories where the duality should be present. We then

spontaneously (and explicitly) break the symmetries of these theories, and in the process

generalize color/kinematics duality to such situations. Finally, we give the double-copy

prescription for spontaneously- and explicitly-broken theories.

– 3 –

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