#### Pions as gluons in higher dimensions

Revised: March
Pions as gluons in higher dimensions
Cli ord Cheung 0 1 3 6 7 8 9
Grant N. Remmen 0 1 3 4 5 7 8 9
Chia-Hsien Shen 0 1 2 3 7 8 9
Congkao Wen 0 1 2 3 6 7 8 9
0 Berkeley , CA 94720 , U.S.A
1 University of California , Berkeley, CA 94720 , U.S.A
2 Mani L. Bhaumik Institute for Theoretical Physics
3 California Institute of Technology , Pasadena, CA 91125 , U.S.A
4 Berkeley Center for Theoretical Physics, Department of Physics
5 Theoretical Physics Group, Lawrence Berkeley National Laboratory
6 Walter Burke Institute for Theoretical Physics
7 Department of Physics and Astronomy, UCLA , Los Angeles, CA 90095 , U.S.A
8 Thus, all factors of @ Z
9 have separated o terms proportional to @ Z
We derive the nonlinear sigma model as a peculiar dimensional reduction of Yang-Mills theory. In this framework, pions are reformulated as higher-dimensional gluons arranged in a kinematic con guration that only probes cubic interactions. This procedure yields a purely cubic action for the nonlinear sigma model that exhibits a symmetry enforcing color-kinematics duality. Remarkably, the associated kinematic algebra originates directly from the Poincare algebra in higher dimensions. Applying the same construction to gravity yields a new quartic action for Born-Infeld theory and, applied once more, a cubic action for the special Galileon theory. Since the nonlinear sigma model and special Galileon are subtly encoded in the cubic sectors of Yang-Mills theory and gravity, respectively, their double copy relationship is automatic.
E ective Field Theories; Sigma Models; Scattering Amplitudes; Space-Time
Symmetries
Amplitudes preamble
From gluons to pions
Unifying relations for amplitudes
Transmutation as special kinematics
4
From gravitons to photons and galileons
Dimensional reduction to the nonlinear sigma model
Color-ordered formulation
Kinematic algebra as Poincare algebra
Dimensional reduction to Born-Infeld theory
Dimensional reduction to the special galileon theory
Origin of the double copy
1 Introduction
2
3
2.1
2.2
3.1
3.2
3.3
4.1
4.2
4.3
5
Conclusions
1
Introduction
Recent work [
1
] has demonstrated how gravity encodes a uni ed description of Yang-Mills
(YM) theory, the nonlinear sigma model (NLSM), Born-Infeld (BI) theory, and the special
Galileon (SG) theory [2{4], as originally anticipated in the context of the
Cachazo-HeYuan formalism [3, 5, 6]. In particular, the tree-level S-matrices of these theories can be
\transmuted" from that of gravity via simple operators that act as di erentials on the
space of kinematic invariants.
In this paper, we argue that the amplitudes construction derived in ref. [
1
] is
equivalent to a peculiar version of dimensional reduction and can be implemented at the level
of the action. Physically, our construction recasts pions as gluons in a special kinematic
con guration in higher dimensions, thus reformulating the NLSM in d dimensions as a
particular dimensional reduction of YM theory in 2d + 1 dimensions. The resulting
description coincides precisely with one recently proposed in ref. [7], where the NLSM action
is comprised purely of cubic interactions exhibiting an explicit symmetry that maintains
color-kinematics duality [8].
Furthermore, by applying our dimensional reduction to gravity in 2d+1 dimensions, we
obtain a new action for BI theory in d dimensions. In this representation, the interaction
vertices truncate at quartic order. Applying this operation again to BI then yields the
cubic double copy action for SG proposed in ref. [7], which is term-by-term the square of
the NLSM action previously mentioned.
{ 1 {
This e ect o ers some insight into the physical origins of double copy relations [8{10].
Since the cubic sector of gravity is trivially the square of that of YM theory, the double
copy relationship is inherited by the SG and NLSM. This is reminiscent of the manifestation
of the double copy in self-dual YM and gravity [11], but applicable in general spacetime
dimension. Remarkably, by deriving the NLSM action in ref. [7] directly from YM theory,
we learn that the associated kinematic algebra is actually a direct descendant of the
higherdimensional Poincare algebra.
tudes, they display some unconventional traits that di erentiate them from the standard
action formulations of the quantum
eld theories we consider. In particular, these actions
are typically taken to be functions of a single physical eld, so properties like Bose
symmetry and S-matrix unitarity are obvious. However, as discovered in ref. [7], the new NLSM
action that makes the double copy relationship explicit involves more than one type of eld:
there are additional auxiliary elds present that obscure the underlying Bose symmetry and
S-matrix unitarity (e.g., tree-level factorization). The usual NLSM tree amplitudes are
reproduced as a speci c choice of external states in this new formulation. The auxiliary elds
in our actions | and amplitudes going beyond this prescribed choice of external states |
do not have any clear physical signi cance. Accordingly, the action representations we
derive in this paper are physical in the sense that they reproduce the correct tree-level
scattering amplitudes when our prescribed choices of external states are made.
The construction of alternative tree-level representations of quantum eld theories with auxiliary states has helped in understanding the double copy and simplifying the perturbation theory [7, 12].
As in the case of the double copy itself, the question of
whether this construction extends to loop order is nontrivial and will likely involve the
introduction of ghost elds, so we leave this question for future work. When restricted to
the external states that are relevant to pion scattering, ref. [
1
] proved that both properties
are present at the level of amplitudes using on-shell recursion relations [13{15], though a
more direct physical understanding is still missing. In the present paper, we design the
special type of dimensional reduction precisely to realize the transmutation in ref. [
1
], so
permutation invariance and unitarity follow from the proof therein.
The remainder of this paper is organized as follows. In section 2, we summarize the
results of ref. [
1
], which de ned a set of unifying relations connecting scattering amplitudes
across a spectrum of theories.
We then discuss the action-level representation of this operation for the NLSM in section 3, followed by its implications for color-kinematics duality. Finally, we apply this construction to the gravity action to derive BI theory and the SG in section 4 and conclude in section 5.
2
Amplitudes preamble
In this section, we review the mechanics of transmutation at the level of
scattering amplitudes [
1
] and show how it is equivalent to a certain implementation of
dimensional reduction.
{ 2 {
(2.1)
2.1
Consider a tree-level color-ordered scattering amplitude in YM theory. As proven in ref. [
1
],
gluons can be transmuted into pions via a simple di erential operation,
where pipj , piej , and eiej are Lorentz invariant products of the momenta and polarization
vectors and the YM color structure on the left-hand side is mapped to the NLSM
avor
structure on the right-hand side. As required by little group covariance, the transmutation
operator e ectively strips o
all polarization vectors in order to generate an amplitude of
scalars. The very same transmutation operator also converts tree-level amplitudes of BI
photons into those of SG scalars,
n 1 0
Y
(2.2)
Crucially, eqs. (2.1) and (2.2) apply to amplitudes in any representation, provided they
are written as a function of kinematic invariants in general spacetime dimension. This is
possible because the transmutation operators are precisely engineered to be invariant under
reshu ing of terms via total momentum conservation and on-shell conditions [
1
].
Note that the right-hand sides of eqs. (2.1) and (2.2) are manifestly cyclic and
permutation invariant, respectively, while the left-hand sides are not. This feature is generic:
while transmutation selects two special legs, chosen here to be 1 and n, the nal answer is
independent of this choice. As we will see, the absence of manifest cyclic and permutation
invariance will persist at the action level.
In ref. [
1
] it was shown how transmutation also applies to gravity | or more precisely,
the low-energy e ective eld theory of the closed string, which describes gravity coupled to
a dilaton and two-form gauge eld. Throughout, we will for brevity refer to this multiplet
of states collectively as the \extended graviton."1 The extended graviton amplitudes are
a function of non-symmetric tensor polarizations, e
= e e , and are the natural output
of various \gravity = gauge2" relations arising from the BCJ [8] and KLT [17]
constructions. Transmuting the extended graviton amplitude yields the scattering amplitude of
BI photons,
n 1 0
Y
(2.3)
Here the transmutation operator only strips o the barred polarizations, so the resulting
expression is still a function of the unbarred polarizations labeling the external BI photons.
Combined with eq. (2.2), eq. (2.3) shows that applying the transmutation twice to an
extended graviton amplitude leads to that of SG.
1The theory of gravity coupled to a dilaton and a two-form gauge eld has several aliases, including
\N = 0 supergravity" and the theory of the \fat graviton" [16].
The transmutation procedure outlined above is actually equivalent to a certain variation of
dimensional reduction. To understand why, we rst examine the case of pions transmuted
from gluons, as described in eq. (2.1). With the bene t of hindsight, let us de ne a theory of
(2d + 1)-dimensional gluons dimensionally reduced to a d-dimensional subspace on which
the external momenta have support. The (2d + 1)-dimensional momentum vector for a
massless gluon is
PiM = (pi ; 0; 0);
(2.4)
HJEP04(218)9
expressed in block form where the rst and third entries are d-dimensional and the middle
entry is one-dimensional. Throughout, we use calligraphic indices to label the full (2d +
1)dimensional space and, Greek indices to label both sets of d-dimensional spaces. It is
important to point out that this latter choice of indices is simply a convenient abuse of
notation; we do not identify the two d-dimensional spaces.
By inspection, we see that eq. (2.1) is equivalent to the following choice of external
polarizations,
E1M = EnM = (0; 1; 0)
and
EiM = (pi ; 0; i pi )
for
i 6= 1; n:
(2.5)
This is merely a choice of polarization and the two d-dimensional spaces remain independent
spacetime directions. In order to verify this claim it su ces to compute the kinematic
invariants corresponding to eqs. (2.4) and (2.5). For example, the invariants built purely
from momenta are
Meanwhile, since the polarizations of legs 1 and n are orthogonal to all other legs, we
nd that
PiPj = pipj :
E1En = 1;
{ 4 {
while EiEj = 0 for all other combinations due to crucial factors of the imaginary number i
in eq. (2.5). Finally, the invariants constructed from polarizations and momenta are
PiEj = pipj
for
j 2= f1; ng;
with PiE1 = PiEn = 0. Hence, this choice of external kinematics implements precisely the
di erential operator in eq. (2.1). To obtain this result, it was important that the gluon
amplitude is linear in each of the polarization vectors.
The choice of kinematics in eqs. (2.4) and (2.5) describes a dimensional reduction from
2d + 1 dimensions down to d dimensions. Physically, legs 1 and n are polarized in their own
exclusive extra dimension, while legs 2 through n
1 describe polarizations residing in the d-dimensional subspaces that are proportional to the physical d-dimensional momentum. In subsequent sections, we translate this special kinematic con guration into an operation at the level of the action.
(2.6)
(2.7)
(2.8)
Let us now apply the dimensional reduction described in the previous section to derive the
NLSM from YM theory. For YM theory in 2d + 1 dimensions, the Lagrangian is
1
4
LYM =
Tr FMN F MN
+ LGF
with
MAN
valued under a normalization convention where
in units where the gauge coupling g = 2 and the gluon elds AM = AM
p
1
2
LGF =
so the full action is equal to
1
Tr T aT b
= ab
and
[T a; T b] = i 2 f abcT c:
p
For simplicity we implement Feyman gauge by choosing
In what follows, we prove how the YM action reduces to the NLSM action in ref. [7] on
the dimensional reduction corresponding to eq. (2.5).
3.1
Dimensional reduction to the nonlinear sigma model
According to eq. (2.5), the gluon eld AM is split into the component elds
which without loss of generality can be parameterized by
AM = XM + YM + ZM
;
XM = p (X ; 0; iX )
YM = (0; Y; 0)
ZM = p (Z ; 0; +iZ );
p
a T a are
adjoint(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
{ 5 {
2Note that both d-dimensional spacetime factors are separately in Lorentzian mostly-plus signature while
the single extra dimension is spatial.
(3.8)
(3.9)
(3.10)
(3.11)
so the square of the gluon eld is
By construction, XM and ZM have the form of polarizations of opposite helicity. As a
result, we obtain the useful identities
and similarly for all analogous expressions involving derivatives on elds. Finally, we note
that, in accordance with eq. (2.4), the elds are polarized in the full (2d + 1)-dimensional
space but only carry momentum in the rst d spacetime dimensions, so
Expanding the YM action in terms of X , Y , and Z , we obtain
AMAM = X Z
+ Z X
+ Y 2:
XMX M = ZMZM = 0
XMYM = ZMYM = 0
LYM = L(Y2M) + L(Y3M) + L(Y4M);
where the terms at each power in elds are
L(Y2M) = Tr X
Z
+
Y
Y
1
2
+ fX
$ Z g
(3.12)
L(Y4M) = Tr [X ; Z ] [Z ; X ] + [X ; X ] [Z ; Z ] + 2 [X ; Y ] [Z ; Y ] :
Remarkably, one can consistently drop the majority of terms in the action (3.12) because
they do not actually contribute to tree-level pion scattering. This truncation is possible as
a consequence of two important simpli cations, which we now discuss.
Weight counting. First of all, let us determine which interaction vertices actually enter
into the tree-level Feynman diagrams for pion scattering. According to eq. (2.5), pion
scattering corresponds to higher-dimensional gluon scattering where legs 1 and n are Y
particles and all other states are longitudinally-polarized Z
particles. In particular, the
latter states all have polarizations proportional to their respective momenta p . At the
level of the amplitude, we have
A( 1; 2; : : : ; n 1; n) = A(Y1; Z2; : : : ; Zn 1; Yn);
(3.13)
which incidentally matches the prescription proposed in ref. [7].
As it turns out, since the external states are restricted to longitudinal Z states and a
pair of Y states, this severely limits which interactions can contribute to the amplitude. By
drawing tree-level Feynman diagrams explicitly, it becomes obvious that none of the quartic
interactions can appear. Since we are interested in the NLSM, it is desirable to further
simplify the action in order to make the color-kinematics duality manifest, as in ref. [7].
{ 6 {
To systematically enumerate which interactions in eq. (3.12) can appear in a tree-level
scattering amplitude for the NLSM according to the external states speci ed by eq. (3.13),
we de ne a \pseudo-helicity" for each external particle type,
h, where n is the total number of particles in the operator and h
is the sum of all pseudo-helicities in the operator. At tree level, there is a simple addition
rule for the weights. This is because the weights satisfy w[A] = w[AL] + w[AR]
2 for an
counting in general, see ref. [18]. The weight of each component eld is
so each term in the action in eq. (3.12) has weight
+2. However, since the
pion scattering amplitude contains all Z
states except for a pair of Y states, the target
amplitude has weight w = +2. This implies that pion amplitudes only receive contributions
from w = +2 interactions, so it is consistent to entirely drop all terms in the Lagrangian
with weight w > +2. The resulting truncated action is eq. (3.12) with all the quartic terms
and half the cubic terms dropped,
LNLSM = Tr X
Z + Y
: (3.17)
This action is similar but not yet equal to the NLSM action proposed in ref. [7].
Transverse condition.
To establish complete equivalence requires a second simpli cation of the action that arises from certain transverse properties of the elds. First, we rewrite eq. (3.17), up to total derivatives, as
LNLSM = Tr X
Z
+
Y
Y + i X
[Z ; Z ] + Z [Y; @ Y ]
(3.18)
where we have de ned the eld strength for the X
eld,
X
(3.19)
The action in eq. (3.18) di ers from that of ref. [7] by terms proportional to the longitudinal
component, @ Z . As we now show, these terms are always projected out of tree-level pion
amplitudes and can be consistently dropped. To understand why, consider a factor of @ Z
that appears in an interaction contributing to a Feynman diagram. If the Z
eld contracts
into an external state, then it vanishes by the on-shell conditions. On the other hand, if the
eld is contracted with an internal propagator, then the o -diagonal structure of the
kinetic term links this eld to the X
eld of some internal vertex. According to eq. (3.18),
all interaction vertices that involve X
are either a function of the eld strength X
or
eld strength X
simply zeroes out
this longitudinal contribution. In the latter, the internal vertex is also proportional to the
longitudinal component @ Z , so we can then apply the same logic from the beginning.
ultimately terminate at an external leg or on an X
eld strength.
can be consistently dropped from
the action, thus establishing the equivalence of eq. (3.18) with the result of ref. [7], which
was originally derived from scattering amplitudes rather than dimensional reduction.
In terms of Feynman diagrams, the perturbation expansion for the action in eq. (3.18)
is drastically simpler than that of the conventional representation of the NLSM action,
where U = exp(i aT a=f ) and f is the pion decay constant. The exponential form of
the nonlinear eld generates an in nite tower of higher- and higher-order interactions that
contribute unnecessary complexity to the Feynman diagrammatic expansion, and obscures
the color-kinematics duality in NLSM [19{24]. In contrast, the NLSM representation in
eq. (3.18) is purely cubic and manifests the color-kinematics duality inherits from YM.
Note that the pion decay constant is absorbed into the normalization of the longitudinal
polarizations of the Z external states.
3.2
Color-ordered formulation
For future reference, we summarize here the color-ordered Feynman rules derived from the
NLSM action in eq. (3.18). Since we are in Feynman gauge, the propagators take the
simple form,
(3.20)
(3.21)
(3.22)
X1μ
Z
μ
1
Y
Zμ
where the X and Z
elds are conjugate particles. The three-particle Feynman vertices are
Y
X⌫
=
=
i
i
p2
p2
;
Z⌫
2
Z
⇢
3
Y2
Y3
=
2i (p1
p
1
)
=
i (p2
p3 );
{ 8 {
which are far simpler to implement than Feynman rules in the conventional approach to
perturbation theory in the NLSM [25, 26]. Recall the NLSM amplitude is given by the
states chosen in eq. (3.13).
Note that the color-ordered formulation naturally arises from YM action in the
Gervais
Neveu gauge [27],
which is dimensionally reduced to
1
N + AMAN AMA
N
;
(3.23)
1
2
(3.24)
HJEP04(218)9
up to terms that may be consistently dropped as a consequence of weight counting or the
transverse condition discussed previously.
For the sake of completeness, we also remind the readers that the tree-level pion
amplitudes are reproduced by the amplitudes in eq. (3.13). According to the special kinematics
in eq. (2.5), the d-dimensional polarizations for Z particles are chosen as the longitudinal
mode, Z = p . Note that the choice of longitudinal polarization does not contradict with
the transverse condition discussed earlier. The transverse condition applies to the
irrelwhich has nothing to do with the choice of
polarization Z
.
3.3
Kinematic algebra as Poincare algebra
As emphasized in ref. [7], the Feynman diagrams associated with the NLSM action in
eq. (3.17) automatically satisfy the Jacobi identities and are thus manifestly compliant with
color-kinematics duality. Remarkably, the Jacobi identities are enforced by a symmetry of
the NLSM action,
1
1
1
X
Z
0
0
Y Y
Y Z
0
Z
Z
1
C
A
1
C
A
Z
C ;
A
X
X . In particular, the Noether current conservation equations
for these symmetries are literally equal to the Jacobi identities for kinematic numerators,
modulo terms that vanish under the transverse conditions discussed earlier.
As noted in ref. [7], the Z transformations are simply Poincare transformations acting
on the d-dimensional subspace. This is obvious if we identify Z = a + b x , where a is
{ 9 {
(3.25)
a constant vector labeling translations and b is a constant antisymmetric matrix labeling
rotations and boosts.
But what of the remaining symmetries, X and Y ? By recasting the NLSM as a
dimensional reduction of YM, we learn that these symmetries have a geometric origin |
namely, Lorentz boosts in higher dimensions! Concretely, consider a matrix parameterizing
a Lorentz transformation acting on the extra-dimensional space,
MN = B 0
0 0
0
0
0
p
i 2 Y
0
p
i 2 Y
These transformations act rigidly on the indices of elds and do not involve derivatives
because there are no momenta
owing in the extra dimensions. The extra-dimensional
Lorentz transformation shifts the gluon eld by AM ! AM + ~
AM
, which in terms of the
component elds is
1
1
0
0
0 X
0 X
Z
Z
X
X
(X
0
(X
Y Y
Y Y
Z ) 1
C
A
Z )
1
A
Z ) C :
At present, these symmetries still di er from eq. (3.25). However, as discussed earlier, the
NLSM is de ned by a truncated version of the YM action. As a result of the truncation,
a symmetry in YM is not guaranteed to be a symmetry of the NLSM.
Nevertheless, a close descendant of the extra-dimensional Lorentz symmetry is still
preserved under weight truncation. To see why, recall that the original YM action can
be partitioned into two weight sectors, LYM = LYM
eq. (3.27) shift the weights of the component elds by
(w=2) + LYM
(w=4). The transformations in
wh~X i
= 0; 2
and
wh~Y i
=
1;
so extra-dimensional Lorentz transformations mix terms of di erent weight. In order to
determine the component of the Lorentz transformation that leaves LYM
simply drop all transformations that are an invariance only with the help of LYM
transformations can never be symmetries of the truncated action. On the other hand, for
transformations that shift the weights strictly negatively, it is impossible for any variation
(w=2) invariant, we
(w=4). These
of LYM
(w=4) to ever cancel a variation in LYM
in weight. Thus, for a negative shift in weight, LYM
(w=2) because these terms are already separated
(w=2) will be itself invariant. Truncating
the symmetry transformation in eq. (3.27) down to terms that shift the weight by
1
and
2, we obtain the color-kinematics symmetry of the NLSM shown in eq. (3.25). In
summary, color-kinematics duality in the NLSM arises from a higher-dimensional spacetime
symmetry of YM theory.
(3.27)
(3.28)
The construction described above can be applied straightforwardly to gravity. However, as
discussed in ref. [
1
], the natural theory to which to apply transmutation is the low-energy
e ective theory of the closed string. The action, SG =
coupled to an antisymmetric two-form BMN and a dilaton , with interactions given by
dDx LG, describes a metric g
MN
LG = p
g
2
2
R
1
2(D
where 2 = 32 G. In the conventional picture, one expands the graviton in perturbations,
HJEP04(218)9
HMN
(2) =
LG
(3) =
LG
(4) =
2 HMN
4 HMN HRN
8 HMN HRS @T HRN
8 HMN HMR@N HST
As we will explain, terms of higher order will be not be needed for our analysis.
g
MN =
MN + h
MN
;
as distinct elds. For our purposes, however, it will be
contreating h
MN
, BMN
, and
by a general tensor HMN
venient to repackage the degrees of freedom into a single extended graviton eld described
. An action of this form was derived in ref. [12] in the context
of pure gravity, but in fact also reproduces all extended graviton amplitudes as well.
Unfortunately, the associated propagator deviates from the simple 1=p2 Feynman propagator
form, so we will not consider the action of ref. [12] further here.
An action-level version of transmutation requires an extended graviton action expressed
a general ansatz for an e ective eld theory of the extended graviton HMN
in terms of HMN with a simple propagator going as 1=p2. To derive such an action, we build
and constrain its
coe cients to match known tree-level amplitudes constructed from the KLT relations [17].
Among the family of resulting actions, we choose the remaining free coe cients to simplify
our results by reducing the number of terms in the Lagrangian. In natural units where
= 1, our resulting extended graviton action is
LG = L(G2) + L(G3) + L(G4) +
;
where the terms at each order are
1
1
1
1
(4.2)
(4.3)
(4.4)
Next, let us implement the procedure described in the above sections to derive the BI
action from the extended graviton action in eqs. (4.3) and (4.4). We take the unbarred
and barred indices in the extended graviton action to run over d dimensions and 2d + 1
dimensions, respectively, so the extended graviton eld is a d
where each component eld is given by
L(B2I) = X
+ fY Y
! X
L(B4I) =
1
16
Z
Z
Y
+
1
p Z
These de nitions enforce a similar nilpotency condition as before, and likewise for terms involving derivatives. We also note that the derivatives only have support on the d-dimensional subspace, so
and
Plugging into the extended graviton action in eq. (4.4), we obtain a new action for the
BI theory,
LBI = L(B2I) + L(B3I) + L(B4I);
where the terms at each power are given by
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
Z
Z
Z
Z
+ fY Y
! X
Here we have dropped all interactions at quintic order and higher because they can be
truncated by a weight counting argument that will be discussed shortly. Moreover, we
because they can be discarded due to an
analogue of the transverse conditions discussed earlier.
p Z
2
)
):
1
8
Weight counting.
Our earlier weight counting arguments are straightforwardly generalized to the case of gravity. Since the dimensional reduction is only applied to the barred indices, the weights are de ned in the same way as in eq. (3.15). Following eq. (2.3), the tree-level BI amplitude is
proportional to e p . Note that these external states are simply the tensor product of BI
photon polarizations e with the Y and Z external states for the NLSM in eq. (3.13). Since
the BI amplitude has uniform weight w = +2, we can truncate the action by dropping all
terms with weight w > +2.
The extended gravity interactions take the schematic form
(3)
LG
and
(4)
LG
where we ignore all index structure except the barred or unbarred nature of the derivatives.
Let us consider the possible index structures and their weights in turn. Since the barred
derivative only lives in the rst d dimensions, we nd that
HMR
w HMR
This implies that the extended graviton eld contracting with the derivative has weight
(4.12)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
On the other hand, the nilpotency of XM and ZM implies that
which in turn xes the weight
HMRH
N R
X
Z
From eqs. (4.14) and (4.16), we conclude that the weights for cubic and quartic order are
w LG
h (3)i
+2
and
w LG
h (4)i
+2:
can thus be dropped.
graviton. Since w HMRHN R
h
Because extended gravity is a two-derivative theory, all higher-order interaction terms
share the same derivative structures as eq. (4.12) except with more powers of the extended
i > 0, terms at quintic order and higher have w > +2 and
+
p Z
X
X
Z
Z
Z
Z
Z
Z
Z
X
+
Z
X
);
1
p Z
As before, we can exploit the transverse properties of the elds to eliminate even more terms. Up to total derivatives, the action in eq. (4.10) is equal to where we have de ned the right-index eld strength for X ,
X
Crucially, the eld X
only appears in the action through its eld strength X
terms proportional to @ Z . This allows us to apply an argument similar to that in
that contributes to a Feynman diagram will
ultimately be projected to zero on an external leg or attached to an internal vertex. Since all
internal vertices involving X
depend only on the eld strength X
or are proportional
, these longitudinal contributions are always eventually zeroed out. The resulting
BI action in eq. (4.18) also agrees with an action-level double copy construction combining
YM theory and the NLSM [28]. Let us contrast the quartic representation of BI action in eq. (4.18) with the canonical representation of BI action arising from brane-localized gauge elds,
LBI =
T
q
det(
+ 2
0F );
(4.20)
where the determinant structure induces an in nite tower of interactions. As before, all of
the dimensionful coupling constants in our new BI action are absorbed into the
normalization of the longitudinal polarizations. Similar to the NLSM, our formulation does not
manifest permutation invariance and unitarity, though these are still present in scattering
amplitudes, as proved in ref. [
1
]. However, thanks to its nite interactions, it is
tremendously simpler to calculate amplitudes in this action. In ref. [29], a simpli cation of the
BI action was constructed using auxiliary
elds and a setup speci c to certain spacetime
dimensions. In contrast, our formulation is valid in arbitrary spacetime dimension and the
construction follows directly from our analogous treatment of the NLSM; it may therefore
also o er some insight for the double copy structure of BI theory.
4.2
Dimensional reduction to the special galileon theory
Last but not least, we apply action-level transmutation again to BI theory to obtain an
action for the SG theory. This is equivalent to a double dimensional reduction of the
extended graviton action, taking the unbarred and barred indices of the extended graviton
into a (2d + 1)
to both run over 2d + 1 dimensions. We then decompose the extended graviton eld HMM
where each contribution is
HMM
= XMM + YMM + ZMM
;
XMM
=
YMM
ZMM
=
0 0 0 0 1
= B 0 Y 0 C
A
X
0
iX
0
1
B
0
1
B
Z
0
+iZ
0
0
0
0
0
iX
0
X
0
Z
0 +iZ
1
C
A
1
C :
A
The components not shown all enter in pairs in the action so they can be consistently
dropped from the action provided we are interested in tree-level amplitudes only involving
the states represented above. These de nitions again imply a nilpotency condition,
XMRXNR = ZMRZNR = XRMX RN
= ZRMZRN = 0
XMRYNR = ZMRYNR = XRMY N
R
= ZRMY N
R = 0;
and likewise for structures with additional derivatives. Using the weight-counting
arguments provided at the end of this section, we can expand the extended gravity action in
components and truncate, yielding the action for the SG,
(4.21)
(4.22)
where each term is given by
L(S2G) = X
L(S3G) =
1
2
Z
+
Y
Y
1
2
LSG = L(S2G) + L(S3G) ;
1
4
Z
):
Z
Z
+ Z
X
g
As we will show, all possible quartic interactions in the extended graviton action can
be consistently dropped due to the weight counting argument presented in the subsequent
discussion. Moreover, all quintic and higher-order interactions can also be dropped because
the SG action is equivalent to a transmutation of BI action, which itself originates from
the extended graviton action truncated to quartic order.
Weight counting. From eqs. (2.2) and (2.3), we see that the SG amplitude is given by
corresponding to a pair of Y states with all other external states given by Z
particles
whose polarizations are longitudinal and thus proportional to p p . Note that these
external states are the \square" of the Y and Z external states for the NLSM in eq. (3.13).
Since dimensional reduction is applied to both barred and unbarred indices, it is natural
to promote the weight into a two-component vector, (w; w) = (n
h; n
h), where h and
h correspond to the pseudo-helicity for the unbarred and barred indices. For each state,
we have
w[X
For an amplitude on its factorization channel, A
ALAR, their weights are related by
w[A] = w[AL] + w[AR]
2 and w[A] = w[AL] + w[AR]
2, so we conclude that the tree
level SG amplitude has weight (w; w) = (+2; +2). From the schematic form in eq. (4.12),
YM. As we learned in the NLSM, these interactions have w = +4, which can be truncated,
the quartic and higher interactions of the extended graviton action dimensionally reduce
to terms with either w > +2 or w > +2, so they can be consistently dropped.
Transverse conditions.
Next, let us consider the transverse properties of the elds in the SG action.
De ning an analogue of the Riemann tensor as in ref. [7],
X
(4.28)
our nal form for the SG action becomes
HJEP04(218)9
1
2
1
4
LSG = X
Z
+
Y
Y
X
Z
Z
+ Z
);
), the eld X
appears in the action only in the form of X
. By an argument exactly analogous to
) can be
dropped. As was shown in ref. [7], eq. (4.29) can also be obtained from eq. (3.17) via the
action-level double copy.
symmetry [30],
The cubic SG action in eq. (4.29) is substantially simpler than the canonical
formulation of the SG action, which describes a scalar invariant under an extended shift
where a, b , and c
are a constant scalar, vector, and traceless symmetric tensor,
respectively. In four dimensions, the canonical form of the SG action is
LSG =
1
(4.29)
(4.30)
(4.31)
while in d dimensions there is a tower of even-point interactions at all valences less than or
equal to d+1. In contrast, the SG action in eq. (4.29) is purely cubic for general dimension.
By inspection, the SG action in eq. (4.29) is obtained by squaring all of the terms in the
NLSM action in eq. (3.18). This is an action-level manifestation of the double copy [7].
Our prescription for dimensional reduction actually trivializes the origin of the double copy
structure, by the following argument. Our discussion of weight counting reveals that pion
scattering is encoded within the cubic sector of YM theory. However, the purely cubic
topologies of YM theory automatically satisfy kinematic Jacobi identities up to contact
terms coming from the quartic vertices. This is because one can always probe a maximal
factorization channel on which every propagator is on shell. In this limit, the only
contributions to amplitudes are the cubic diagrams, so these contributions necessarily satisfy the
kinematic Jacobi identities up to terms involving contact terms. However, since all quartic
terms are eliminated by the choice of external states corresponding to pion scattering, the
mismatch from the kinematic Jacobi identities is eliminated and the resulting cubic action
automatically satis es them. We thus conclude that since the cubic sector of YM double
copies into the cubic sector of gravity and these coincide with the NLSM and the SG, the
actions that result from our dimensional reduction automatically manifest the double copy.
5
Conclusions
In this paper, we have proposed a variation of dimensional reduction that excises the NLSM
from YM theory as well as BI theory and the SG theory from the extended graviton action.
This operation is essentially an action-level incarnation of the transmutation operation on
scattering amplitudes derived in ref. [
1
]. These relations reveal the origin of the kinematic
algebra of the NLSM as the higher-dimensional Poincare invariance of an underlying YM
theory. Remarkably, the NLSM and SG arise from purely cubic interactions in YM and
gravity, while BI arises from only the cubic and quartic interactions of gravity. Since the
cubic sector of YM theory automatically double copies into gravity, the same is trivially
true for the NLSM to the SG. Note that the theories obtained here | the NLSM, BI theory,
and the SG theory | precisely coincide with the exceptional theories studied in ref. [31]
argued to be the natural e ective eld theory analogues of YM theory and gravity.
Our results suggest a number of directions for future work. One avenue is to derive
action-level versions of the other transmutation operations presented in ref. [
1
]. For
instance, one expects an action-level operation that sends gravity to YM theory. While this
is naturally accomplished by Kaluza-Klein reduction, the simplicity of the S-matrix
mapping suggests that something more minimal is possible. Such a realization may teach us
new structures of YM theory, such as color-kinematics duality.
Another direction deserving of further study is higher loop order in perturbation theory.
Since ref. [
1
] derived unifying relations for tree-level scattering amplitudes, the procedure
for dimensional reduction derived here is only guaranteed to reproduce amplitudes at tree
level. As is also the case for the double copy construction, matching at higher loop order
will likely involve additional structure. It would also be interesting to study the loop-level
amplitudes computed from the actions presented here and to compare them with known
results in the NLSM, BI theory, and the SG theory.
Last but not least, pions are famously known to be related to gluons through the
Goldstone boson equivalence theorem. Although the (2d + 1)-dimensional transmutation
is proven in ref. [
1
] by modern S-matrix techniques, it would be illuminating to show the
connection to the Goldstone boson equivalence theorem. Such a relation would also o er
new insights into the nature of transmutation.
Acknowledgments
We thank Andres Luna, John Joseph M. Carrasco, Song He, and Yu-tin Huang for helpful
discussions. C.C. is supported by a Sloan Research Fellowship and C.C., C.-H.S., and C.W.
are supported in part by a DOE Early Career Award under Grant No. DE-SC0010255 and
by the NSF under Grant No.
NSF PHY-1125915.
G.N.R. was supported at Caltech
by a Hertz Graduate Fellowship and a NSF Graduate Research Fellowship under Grant
No. DGE-1144469 and is currently supported at University of California, Berkeley by
the Miller Institute for Basic Research in Science. C.-H.S. is also supported by Mani L.
Bhaumik Institute for Theoretical Physics. This material is based upon work supported
by the U.S. Department of Energy, O ce of Science, O ce of High Energy Physics, under
Award Number DE-SC0011632.
Open Access.
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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