Physics Letters B 763 (2016) 365–369
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Exact solutions for the biadjoint scalar field
C.D. White
a,b
a
School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, Scotland, UK
b
Centre for Research in String Theory, Queen Mary University of London, London E1 4NS, UK
a r t i c l e i n f o a b s t r a c t
Article history:
Received
20 June 2016
Accepted
24 October 2016
Available
online 26 October 2016
Editor:
M. Cveti
ˇ
c
Biadjoint scalar theories are novel field theories that arise in the study of non-abelian gauge and gravity
amplitudes. In this short paper, we present exact nonperturbative solutions of the field equations, and
compare their properties with monopole-like solutions in non-abelian gauge theory. Our results may pave
the way for nonperturbative studies of the double copy.
© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
There has recently been much attention on the relationship be-
tween
different types of field theory. A notable example is the
double copy of refs. [1–3], which relates perturbative scattering am-
plitudes
in non-abelian gauge and gravity theories. At tree-level
this has a string theoretic explanation from the well-known KLT re-
lations [4].
However, the double copy remains a conjecture at loop
level, where it has been tested up to four loops [2,5–26], to all or-
ders
in certain kinematic limits [8,27–31], and in the self-dual sec-
tor [32].
Arelated programme of work relating amplitudes in dif-
ferent
theories has arisen from the CHY equations of refs. [33–36].
These express tree-level amplitudes in gauge and gravity theories
in terms of an abstract integral representation involving punctures
on the Riemann sphere, which is itself reminiscent of string the-
ory:
indeed, the CHY equations can be obtained from ambitwistor
string theory [37–39], which framework has recently been used to
extend the former to one-loop level [40,41].
All
of the above work focuses on perturbative properties of the
respective field theories. If the double copy is truly correct, it must
also be possible to relate exact classical solutions of gauge theory
and gravity, where these are known. First steps in this direction
were undertaken in ref. [42], which found an infinite class of clas-
sical
solutions in General Relativity (namely stationary Kerr–Schild
metrics), possessing well-defined counterparts in a gauge theory.
Examples include the Schwarzschild and Kerr black holes, which
give rise to pointlike and rotating distributions of charge in the
gauge theory respectively. One may generalise the former case to
a pointlike dyon in the gauge theory, which double copies to the
Taub-NUT solution in GR [43]. More recently, ref. [44] examined an
E-mail address: Christopher.White@glasgow.ac.uk.
arbitrarily accelerating (and radiating) particle in gauge theory and
gravity, and further work has examined whether the source terms
in both theories are physical [45].
Both
the double copy and the CHY equations relate solutions
in well-known gauge theories and gravity. This is not the whole
story, however, in that one also finds more exotic theories whose
amplitudes are related to gauge theory. In this paper we focus on
biadjoint scalar theories, containing a field =
aa
T
a
˜
T
a
, where
aa
∈ R, and {T
a
} and {
˜
T
a
} are generators of two (possibly dif-
ferent)
Lie algebras:
[T
a
, T
b
]=if
abc
T
c
, [
˜
T
a
,
˜
T
b
]=i
˜
f
abc
˜
T
c
. (1)
We write the Lagrangian defining the theory as
L =
1
2
∂
μ
aa
∂
μ
aa
+
y
3
f
abc
˜
f
a
b
c
aa
bb
cc
, (2)
which gives rise to the equation of motion
∂
2
aa
− yf
abc
˜
f
a
b
c
bb
cc
= 0. (3)
This is a novel theory, which we may think of as describing a scalar
field with two types of charge. Although such theories seem not to
be directly applicable to nature, they nevertheless have an impor-
tance
of their own. Firstly, they underly the structure of gauge and
gravity theories, in both the double copy and CHY approaches. For
example, ref. [46] explains how to use biadjoint theories as build-
ing
blocks for constructing amplitude numerators in gauge theory,
in such a way as to make the double copy manifest. Secondly,
the field equation of eq. (3) was crucial in ref. [42] for arguing
that the classical double copy considered there was related to the
BCJ story for perturbative amplitudes. Finally, scalar theories are
often simpler than gauge or gravity theories. Thus, one may po-
tentially
use exact solutions of the former to investigate solutions
http://dx.doi.org/10.1016/j.physletb.2016.10.052
0370-2693/
© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.