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Multifield Galileons and higher codimension branes
Kurt Hinterbichler,
*
Mark Trodden,
†
and Daniel Wesley
‡
Center for Particle Cosmology, Department of Physics and Astronomy,
University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
(Received 30 August 2010; published 7 December 2010)
In the decoupling limit, the Dvali-Gabadadze-Porrati model reduces to the theory of a scalar field ,
with interactions including a specific cubic self-interaction—the Galileon term. This term, and its quartic
and quintic generalizations, can be thought of as arising from a probe 3-brane in a five-dimensional bulk
with Lovelock terms on the brane and in the bulk. We study multifield generalizations of the Galileon and
extend this probe-brane view to higher codimensions. We derive an extremely restrictive theory of
multiple Galileon fields, interacting through a quartic term controlled by a single coupling, and trace its
origin to the induced brane terms coming from Lovelock invariants in the higher codimension bulk. We
explore some properties of this theory, finding de Sitter like self-accelerating solutions. These solutions
have ghosts if and only if the flat space theory does not have ghosts. Finally, we prove a general
nonrenormalization theorem: multifield Galileons are not renormalized quantum mechanically to any loop
in perturbation theory.
DOI: 10.1103/PhysRevD.82.124018 PACS numbers: 04.50.Kd, 11.25.Uv
I. INTRODUCTION
A particularly fruitful way of extending both the stan-
dard models of particle physics and cosmology is the
hypothesis of extra spatial dimensions beyond the three
that manifest themselves in everyday physics. Historically,
such ideas have provided a tantalizing possibility of uni-
fying the basic forces through the geometry and topology
of the extra-dimensional manifold and, in recent years,
have been the basis for attempts to tackle the hierarchy
problem. In this latter incarnation, a crucial insight has
been the realization that different forces may operate in
different dimensionalities by confining the standard model
particles to a 3 þ 1-dimensional submanifold—the
brane—while gravity probes the entire spacetime—the
bulk—due to the equivalence principle. Such constructions
allow, among other unusual features, for infinite extra
dimensions, in contrast to the more usual compactified
theories.
In the case of a single extra dimension, a further refine-
ment was introduced in [1], where a separate induced
gravity term was introduced on the brane. The resulting
4 þ1-dimensional action
S ¼
M
3
5
2
Z
d
5
x
ffiffiffiffiffiffiffiffi
G
p
R½Gþ
M
2
4
2
Z
d
4
x
ffiffiffiffiffiffiffi
g
p
R½g (1)
is known as the DGP (Dvali-Gabadadze-Porrati) model
and yields a rich and dramatic phenomenology, with, for
example, a branch of four-dimensional cosmological solu-
tions, which self-accelerate at late times, and a set of
predictions for upcoming missions, which will perform
local tests of gravity.
It is possible to derive a four-dimensional effective
action for the DGP model by integrating out the bulk. It
has been claimed [2,3] that a decoupling limit for DGP
exists, in which the four-dimentional effective action
reduces to a theory of a single scalar , representing
the position of the brane in the extra dimension, with a
cubic self-interaction term ð@Þ
2
h (though this claim
is not without controversy, see for example [4]). This term
has the properties that its field equations are second order
(despite the fact that the Lagrangian is higher order),
which is important for avoiding ghosts. It is also invariant
(up to a total derivative) under the following Galilean
transformation:
ðxÞ!ðxÞþc þb
x
; (2)
with c and b
constants.
These properties are interesting in their own right, and
terms that generalize the cubic DGP term studied (without
considering a possible higher-dimensional origin) in [5]
are referred to as Galileons. Requiring the invariance (2)
forces the equations of motion to contain at least two
derivatives acting on each field, and there exists a set of
terms that lead to such a form with exactly two derivatives
on each field (in fact, the absence of ghosts in a nonlinear
regime demands that there be at most two derivatives on
each field). These are the terms that were classified in [5]
and take the schematic form
L
n
@@ð@
2
Þ
n2
; (3)
*
kurthi@physics.upenn.edu
†
trodden@physics.upenn.edu
‡
dwes@sas.upenn.edu
PHYSICAL REVIEW D 82, 124018 (2010)
1550-7998=2010=82(12)=124018(15) 124018-1 Ó 2010 The American Physical Society
with suitable Lorentz contractions and dimensionful coef-
ficients. In d spacetime dimensions there are d such terms,
corresponding to n ¼ 2; ...;dþ 1. The n ¼ 2 term is just
the usual kinetic term ð@Þ
2
, the n ¼ 3 case is the DGP
term ð@ Þ
2
h, and the higher terms generalize these.
These terms have appeared in various contexts apart
from DGP; for example, the n ¼ 4, 5 terms seem to appear
in the decoupling limit of an interesting interacting theory
of Lorentz invariant massive gravity [6]. They have been
generalized to curved space [7,8], identified as possible
ghost-free modifications of gravity and cosmology
[5,9–13], and used to build alternatives to inflation [14]
and dark energy [15,16].
Another remarkable fact, which we will prove for a more
general multifield model in Sec. VI , is that the L
n
terms
above do not get renormalized upon loop corrections, so
that their classical values can be trusted quantum mechani-
cally. Also, from an effective field theory point of view,
there can exist regimes in which only these Galileon terms
are important.
It is natural to consider whether the successes of the
DGP model can be extended and improved in models in
which the bulk has higher codimension, and whether the
drawbacks of the five-dimensional approach, such as the
ghost problem in the accelerating branch, might be ame-
liorated in such a setting. Since our understanding of the
complexities of the DGP model has arisen primarily
through the development of a four-dimensional effective
theory in a decoupling limit, one might hope to achieve a
similar understanding of theories with larger codimension.
This is the aim of this paper.
We do not consider the full higher codimension DGP or
a decoupling limit thereof. Instead, we are interested in
generalizing the Galileon actions to multiple fields and
exploring the probe-brane-world view of these terms, ex-
tending the work of [17] on the single-field case. The
theory which emerges from the brane construction in co-
dimension N has an internal SOðNÞ symmetry in addition
to the Galilean symmetry. This is extremely restrictive, and
in four dimensions it turns out that there is a single non-
linear term compatible with it. This makes for a fascinating
four-dimensional field description: a scalar field theory
with a single allowed coupling, which receives no quantum
corrections.
II. SINGLE-FIELD GALILEONS AND
GENERALIZATIONS
In codimension one, the decoupling limit of DGP con-
sists of a four-dimensional effective theory of gravity
coupled to a single scalar field , representing the bending
mode of the brane in the fifth dimension. The field self-
interaction includes a cubic self-interaction ð@Þ
2
h,
which has the following two properties:
(1) The field equations are second order,
(2) The terms are invariant up to a total derivative under
the internal Galilean transformations ! þc þ
b
x
, where c, b
are arbitrary real constants.
In [5], this was generalized, and all possible Lagrangian
terms for a single scalar with these two properties were
classified in all dimensions. They are called Galileon
terms, and there exists a single Galileon Lagrangian
at each order in , where ‘‘order’’ refers to the number
of copies of that appear in the term. For n 1, the
(n þ 1)th order Galileon Lagrangian is
L
nþ1
¼ n
1
1
2
2
n
n
ð@
1
@
1
@
2
@
2
@
n
@
n
Þ; (4)
where
1
1
2
2
n
n
1
n!
X
p
ð1Þ
p
1
pð
1
Þ
2
pð
2
Þ
n
pð
n
Þ
: (5)
The sum in (5) is over all permutations of the indices,
with ð1Þ
p
the sign of the permutation. The tensor (5)is
antisymmetric in the indices, antisymmetric the in-
dices, and symmetric under interchange of any , pair
with any other. These Lagrangians are unique up to total
derivatives and overall constants. Because of the antisym-
metry requirement on , only the first n of these Galileons
are nontrivial in n dimensions. In addition, the tadpole term
is Galilean-invariant, and we therefore include it as the
first-order Galileon.
Thus, at the first few orders, we have
L
1
¼ ;
L
2
¼½
2
;
L
3
¼½
2
½½
3
;
L
4
¼
1
2
½
2
½
2
½
3
½þ½
4
1
2
½
2
½
2
;
L
5
¼
1
6
½
2
½
3
1
2
½
3
½
2
þ½
4
½½
5
þ
1
3
½
2
½
3
1
2
½
2
½½
2
þ
1
2
½
3
½
2
: (6)
We have used the notation for the matrix of partials
@
@
, and ½
n
Trð
n
Þ, e.g. ½¼h,
½
2
¼@
@
@
@
, and ½
n
@
n2
@, i.e.
½
2
¼@
@
, ½
3
¼@
@
@
@
. The above
terms are the only ones which are nonvanishing in four
dimensions. The second is the standard kinetic term for a
scalar, while the third is the DGP Lagrangian (up to a
total derivative).
KURT HINTERBICHLER, MARK TRODDEN, AND DANIEL WESLEY PHYSICAL REVIEW D 82, 124018 (2010)
124018-2
The equations of motion derived from (4) are
E
nþ1
L
nþ1
¼nðn þ1Þ
1
1
2
2
n
n
ð@
1
@
1
@
2
@
2
@
n
@
n
Þ¼0 (7)
and are second order, as advertised.
1
The first few orders of the equations of motion are
E
1
¼ 1; (14)
E
2
¼2½; (15)
E
3
¼3ð½
2
½
2
Þ; (16)
E
4
¼2ð½
3
þ 2½
3
3½½
2
Þ; (17)
E
5
¼
5
6
ð½
4
6½
4
þ8½½
3
6½
2
½
2
þ3½
2
2
Þ: (18)
By adding a total derivative, and by using the following
identity for the symbol in L
nþ1
1
1
...
n
n
¼
1
n
ð
1
1
2
2
...
n
n
1
2
2
1
2
3
...
n
n
þþð1Þ
n
1
n
2
1
...
n
n1
Þ; (19)
the Galileon Lagrangians can be brought into a (sometimes
more useful) different form, which illustrates that the
(n þ 1)th order Lagrangian is just ð@Þ
2
times the nth
order equations of motion
L
nþ1
¼
nþ1
2nðn1Þ
ð@Þ
2
E
n
n1
2
@
1
½ð@Þ
2
1
1
n1
n1
@
1
@
2
@
2
@
n1
@
n1
: (20)
From the simplified form (20), we can see that L
3
, for
example, takes the usual Galileon form ð@Þ
2
h.
These Galileon actions can be generalized to the multi-
field case, where there is a multiplet
I
of fields.
2
The
action in this case can be written
L
nþ1
¼ S
I
1
I
2
I
nþ1
1
1
2
2
n
n
ð
I
nþ1
@
1
@
1
I
1
@
2
@
2
I
2
@
n
@
n
I
n
Þ; (21)
with S
I
1
I
2
I
nþ1
a symmetric constant tensor. This is invari-
ant under individual Galilean transformations for each field
I
!
I
þ c
I
þ b
I
x
, and the equations of motion are
second order
E
I
L
I
¼ðn þ 1ÞS
II
1
I
2
I
n
1
1
2
2
n
n
ð@
1
@
1
I
1
@
2
@
2
I
2
@
n
@
n
I
n
Þ: (22)
The theory containing these Galilean-invariant operators
is not renormalizable, i.e. it is an effective field theory with
a cutoff , above which some UV completion is required.
As was mentioned in the introduction, the L
n
terms above
do not get renormalized upon loop corrections, so that their
classical values can be trusted quantum mechanically (see
Sec. VI). The structure of the one-loop effective action (in
3 þ 1 dimensions) is, schematically
3
[3],
X
m
4
þ
2
@
2
þ @
4
log
@
2
2
@@
3
m
: (23)
One should consider quantum effects within the effec-
tive theory, since there are other operators of the same
dimension that might compete with the Galileon terms.
However, there can exist interesting regimes where
1
Beyond their second-order nature, these Lagrangians possess
a number of other interesting properties. Under the shift sym-
metry ! þ , the Noether current is
j
nþ1
¼nðnþ1Þ
1
2
2
n
n
ð@
1
@
2
@
2
@
n
@
n
Þ: (8)
Shift symmetry implies that the equations of motion are equiva-
lent to the conservation of this current
E
nþ1
¼@
j
nþ1
: (9)
However, the Noether current itself can also be written as a
derivative
j
nþ1
¼ @
j
nþ1
; (10)
where there are many possibilities for j
nþ1
, two examples of
which are
j
nþ1
¼ nðn þ 1Þ
2
2
n
n
ð@
2
@
2
@
n
@
n
Þ; (11)
j
nþ1
¼nðn þ 1Þ
2
2
n
n
ð@
2
@
2
@
3
@
3
@
n
@
n
Þ: (12)
Thus the equations of motion can in fact be written as a double
total derivative
E
nþ1
¼@
@
j
nþ1
: (13)
2
As we put the finishing touches to this paper, several preprints
appeared which also discuss generalizations to the Galileons
[18–20].
3
Strictly speaking, quantum effects calculable solely within
the effective theory are only those associated with log-
divergences. Power divergences are regularization dependent
and depend upon some UV completion or matching condition.
In dimensional regularization with minimal subtraction, they do
not even show up, corresponding to making a special and
optimistic assumption about the UV completion, i.e. that
power-law divergences are precisely cancelled somehow by
the UV contributions. However, it is important to stress that
the conclusions about the Galileon Lagrangian are true even in
the presence of generic power divergences, i.e. even with a
generic UV completion.
MULTIFIELD GALILEONS AND HIGHER CODIMENSION ... PHYSICAL REVIEW D 82, 124018 (2010)
124018-3