Hidden conformal symmetry in treelevel graviton scattering
Accepted: May
Hidden conformal symmetry in treelevel graviton
Florian Loebbert 0 1 2
Matin Mojaza 0 1
Jan Plefka 0 1 2
0 Am Muhlenberg 1 , 14476 Potsdam , Germany
1 Zum Gro en Windkanal 6 , 12489 Berlin , Germany
2 Institut fur Physik and IRIS Adlershof, HumboldtUniversitat zu Berlin
We argue that the scattering of gravitons in ordinary Einstein gravity possesses a hidden conformal symmetry at tree level in any number of dimensions. The presence of this conformal symmetry is indicated by the dilaton soft theorem in string theory, and it is reminiscent of the conformal invariance of gluon treelevel amplitudes in four dimensions. To motivate the underlying prescription, we demonstrate that formulating the conformal symmetry of gluon amplitudes in terms of momenta and polarization vectors requires manifest reversal and cyclic symmetry. Similarly, our formulation of the conformal symmetry of graviton amplitudes relies on a manifestly permutation symmetric form of the amplitude function.
Conformal and W Symmetry; Scattering Amplitudes

HJEP05(218)
1 Introduction
2 Poincare and conformal symmetry in momentum space
3 Conformal symmetry of YangMills amplitudes
4 Soft dilatons, conformal generators and graviton scattering
5 Conformal symmetry of graviton amplitudes
6 Summary and outlook
A Conformal generator on stripped amplitudes
B Computation of Mf3 0
C On gauge invariance of the dilaton soft theorem
D Conformal symmetry of twodilatontwograviton amplitude
within string theory, the KLT relations between open and closed string amplitudes [1]
allow for a representation of graviton treeamplitudes as products of colorstripped gluon
trees, also at the eld theory level. This representation of gravity as the \square" of
YangMills theory was lifted to an entirely new level through the colorkinematics duality of
Bern, Carrasco and Johansson (BCJ) [2, 3], which provides a concrete, yet still mysterious,
prescription for how YangMills amplitudes (at tree and looplevel) may be combined into
gravitational amplitudes upon replacing color degrees of freedom by kinematical ones.
Clearly symmetries play a decisive role in constraining the dynamics of quantum eld
theories and the Poincare invariance of scattering amplitudes is a builtinfeature of any
practical formalism to compute these: translational symmetry is guaranteed by the overall
momentum conserving deltafunction of the amplitude which in turn must be Lorentz
invariant modulo gauge transformations. Importantly, however, treelevel gluon amplitudes
in four dimensions are invariant under the larger group of conformal transformations. While
{ 1 {
this is a consequence of the classical conformal symmetry of the YangMills action, the
conformal symmetry of gluon treeamplitudes was rst fomulated by Witten [4], who used an
elegant representation of the conformal generators in spinorhelicity and twistor variables.
Given the close connection between the conformally invariant gluon treeamplitudes
and graviton trees, the natural question arises whether the existence of conformal
symmetry for the former leaves any imprint on the structure of the latter. Of course, from
an inspection of the EinsteinHilbert action, a naive conformal symmetry is immediately
ruled out due to the dimensionful gravitational coupling in d > 2. However, the existence
of hidden symmetries in quantum
eld theories, i.e. the appearance of symmetries at the
level of observables which are nonmanifest or nonexistent at the level of the action, has
been a recurrent theme in recent years  just as is the case for the colorkinematics
duality discussed above. So it might not be entirely misguided to explore the question of a
conformal symmetry of graviton treeamplitudes beyond d = 2.
In fact, it has been demonstrated that Einstein gravity in AdS space can be obtained
from conformal gravity [5{7]. However, a similar relation does not immediately carry over to
at space. Further clues towards a hidden symmetry of graviton amplitudes emerged from
the dicovery of novel subleading softgraviton theorems [8]. Extensive works of Strominger
et al. speculate about the existence of a hidden BMS symmetry [9, 10] for all massless
particle scattering processes in four dimensions. Along these lines, a recent prescription
maps gluon treeamplitudes in Minkowski space to the celestial sphere at in nity [11, 12].
The Lorentz symmetry of fourdimensional Minkowski space then acts as the 2d conformal
group on the celestial sphere.
While the status of this program is still inde nite, soft
theorems do indicate that conformal symmetry plays a role for gravity amplitudes. In fact,
the present work was largely motivated by the appearance of the conformal generators of
dilatations and special conformal transformations in the soft theorem for the string theory
dilaton eld, as derived in [13{18], and as especially pointed out in [19]. The above ndings
about the soft dilaton are formulated in terms of di erential operators in massless particle
momenta and polarizations. Since these variables obey onshell constraints, e.g. k2 = 0,
which generically do not commute with the action of derivatives, we are confronted with
an immediate puzzle concerning the interpretation of those results. In four dimensions, it
would be natural to translate the above statements into spinorhelicity space where the
onshell constraints are unambiguously resolved. However, this route seems not practical here
since we require a ddimensional treatment and a scaling dimension di erent from unity.1
We shall begin our discussion with an analysis of some general features of massless
scattering amplitudes and conformal symmetry. Employing a representation of the
dilatation and special conformal generators in terms of di erential operators in polarization and
momentum vectors (here referred to as momentum space)  and not in terms of helicity
spinors as done in [4]  the nonpreservation of the onshell constraints is demonstrated.
This is very subtle, as core properties of amplitudes such as gauge invariance are only
ful lled on the hypersurface of the onshell constraints. We expose the impact of these
HJEP05(218)
1Remember that using momentum spinors
and , treelevel YangMills amplitudes in four dimensions
are annihilated by the special conformal generator K =1 = @ @ [4] (up to collinear con gurations [20]).
{ 2 {
issues by performing a reanalysis of the conformal symmetry of YangMills amplitudes.
We then demonstrate that an analogue of the conformal invariance in four dimensions can
nevertheless be found, if an explicit cyclic and reversal symmetrization of the deltafunction
stripped amplitude is performed.
We move on to consider graviton scattering and explain how the string theory
softdilaton limit indicates the conformal invariance of graviton amplitudes in ordinary Einstein
gravity. A careful analysis suggests that the formulation of this conformal symmetry in
momentum space  in analogy to the YangMills case  requires a particular
representation of the graviton amplitude. While the performed analysis is very instructive, at this
stage of the paper our statements about the conformal symmetry are still conjectural.
In the nal part, we put our conjecture of the conformal symmetry of graviton
amplitudes to the test. We verify that, up to and including multiplicity six, treelevel graviton
amplitudes are annihilated by the generators of the conformal algebra. Here, the role of
manifest cyclic and reversal symmetry of YangMills amplitudes is taken by full
permutation symmetry. That is, in our momentum space formulation, special conformal invariance
is found only if we act on the graviton amplitude in a manifestly permutation symmetric
form. Said di erently, the special conformal generator maps the amplitude to the kernel
of the full permutation operator. When combined with dilatation and Poincare symmetry,
these observations imply the invariance of treelevel graviton amplitudes under the full
conformal algebra. We refer to this symmetry as hidden, since its presence seems
unexpected with regard to the dimensionful gravitational coupling. The hidden character of
the symmetry is emphasized by the fact that we require a multiplicity dependent scaling
dimension entering the conformal generators.
2
Poincare and conformal symmetry in momentum space
In momentum space, amplitudes of n massless particles are described by a function on the
support of an overall momentum conserving function:
(2.1)
(2.2)
(2.3)
An(k1; : : : ; kn) = (P )An(k1; : : : ; kn):
Here and throughout this work (P )
its argument P understood as P
(d)(P ) denotes the ddimensional function with
= Pin=1 ki . The function An is the socalled stripped
or simply stripped amplitude. We will consider massless states carrying (symmetric)
polarization tensors of the form " 1 s =
1
s , and An has the property of being linear
in " 1
s for each state, i.e.
An = "i i1 is A(ni;) i1 is = i
i1
i
is A(ni;) i1 is ;
where only the linear dependence on the ith polarization tensor was exposed. The momenta
and polarization vectors describing the scattering of massless particles with labels i =
1; : : : ; n have to obey the onmass shell and transversality conditions
ki ki = 0;
ki i = 0 ;
8i :
{ 3 {
We will refer to both of these sets of conditions as onshell conditions. Amplitudes of
polarized states described in terms of the i 's are additionally constrainted by gauge invariance;
they must be invariant under the gauge transformation i ! i + ki which is conveniently
written as
where we have introduced the generator of gauge transformations
WiAn = 0 ;
(2.4)
(2.5)
The above equation (2.4) is obeyed on the support of the function and onshell conditions.
We note that i is not a Lorentz vector (see e.g. [21] and the recent discussions in [22{25]).
Polarization tensors of physical states.
An elementary physical state should
correspond to an irreducible representation of the Lorentz group. However, for s even in (2.2) we
may also work with reducible tensor representations, as for instance done in string theory
for describing the physical modes of closed strings. For s = 2, the massless tensor product
state characterized by the polarization tensor
= i i describes a multiplet containing
both the graviton and a scalar component, the dilaton. The respective An;
represents in
this case a reducible stripped amplitude that becomes an amplitude for irreducible
representations, once it is contracted with either the graviton polarization tensor or the dilaton
projector [13, 26]. Speci cally, the symmetric tensor "i
= i i can be decomposed into a
traceless and a trace part "i
= "graviton(ki) + p
d 2 "dilaton(ki), where
"graviton (ki) = i i
p
d 2 "dilaton (ki) ; "dilaton (ki) = p
1
d 2
ki ki
ki ki : (2.6)
Here ki is an auxiliary momentum which obeys ki2 = 0 and ki ki = 1 and we have chosen the
normalizations such that (" " )dilaton = 1 and
i i = 0, we e ectively constrain "i
= "graviton.
"graviton = 0. Notice that by imposing
The stripped amplitude.
To consider An separately, i.e. to strip o the function, the
constraints from Poincare symmetry must be resolved. This can be achieved by eliminating
the momentum of one of the external states using momentum conservation (translational
invariance in coordinate space), as well as by imposing the constraints induced via the
onshell conditions of that state (see also [23, 27] for a detailed discussion). That is, by
choosing some state with label a the constraints are resolved by setting
ka =
n
X ki ;
i6=a
ka2 =
n
X
i6=j6=a
ki kj = 0 ;
a ka =
a
(2.7)
n
i6=a
X ki = 0 :
In any expression involving not the full amplitude distribution, but only the stripped
amplitude, it will be implicitly assumed that these constraints are enforced on An as well
as in the end of any manipulation applied to An. Of course the choice of the state a is
arbitrary leading to an ambiguity in the form of a stripped amplitude.
{ 4 {
In the following, we consider
i to be the same for all legs i, thus from hereon
i =
.
The action of the above generators Gi;
is realized via the standard tensor product:
2 fPi; Ji; Di; ; Ki; ; Sig on the whole amplitude
The dependence of the generators D
and K
on the conformal dimension
is stressed
since it will be of particular importance for us.
Kinematic hypersurface.
The above representation of the conformal generators does
not leave the kinematic constraint hypersurface of onshell scattering amplitudes invariant.
To be explicit, the generator K
does in general not commute with Wi, i.e.
[K ; Wi] =
nor does it commute with the onshell conditions ki2 = ki i = 0:
G
=
X Gi; :
n
i=1
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
Conformal transformations in momentum space.
The momentum space generators
of the conformal algebra acting on a single leg i of the amplitude read
P
i = ki ;
i
;
J
i
Ki; i
iSi ;
where for any fourvector X we employ the shorthand notation
Here S
i
is the `spin' operator which, for states with integer spin s whose polarization
tensor is symmetric and described by " 1 s =
s as in (2.2), can be written as
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Similarly, overall momentum conservation is generally not preserved:
[K ; ki2] = (d
2
2 )ki + 2 i Wi;
[K ; ki i] = i [(d
1
[K ; (P )] =
(d
D )
+ J
:
Hence, in general K
takes us o the constraint onshell surface in momentum space. For
completeness we note that [K ; i i] = 0.
The dilatation operator only su ers from not commuting with momentum conservation,
i.e. we have
[D ; (P )] =
d (P ):
{ 5 {
These commutators should, however, be considered on speci c amplitudes, where (some
of) the terms given above may vanish or cancel, as we will see in a moment in the case of
d = 4 YM amplitudes.
When acting on stripped amplitudes, it is useful to rewrite the conformal generators
in terms of di erential operators of the Lorentz invariant kinematical variables, ki kj , ki j
and i
j , for i 6= j, by use of the chain rule, see appendix A for explicit expressions. We
notice that such di erential operators were considered recently in [28] to describe relations
among amplitudes of di erent theories.
3
Conformal symmetry of YangMills amplitudes
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YangMills theory in four dimensions is classically conformally invariant. At the level of
scattering amplitudes, this was rst demonstrated in a manifestly fourdimensional
framework in [4] by using the spinorhelicity formalism with conformal generators represented
in spinor space. Importantly, in this formalism the onshell conditions (2.3) are
explicitly resolved.
Here we wish to understand how the conformal invariance of YM scattering amplitudes
manifests itself in d = 4 in the context of a general ddimensional treatment of scattering
amplitudes written in momentum space and how it relates to the onshell constraints.
Implications of representation de ciencies. In order to better understand the
implications of the onshell de ciency (2.15) of the special conformal generator K , we study
the conformal transformations of treelevel YM amplitudes, where we expect to see
invariance features when restricting the analysis to four dimensions. YM amplitudes An can be
decomposed into colorless partial amplitudes
X
P(2;:::;n)
An(1; : : : ; n) = (P ) gn 2
Tr[T a1
T an ]An(1; : : : ; n);
(3.1)
with T a the colorgroup generators and An denoting from here on a basis of
colordecomposed partial or simply partial amplitudes. The above sum runs over all noncyclic
permutations of the labels 1; : : : ; n, which can equivalently be expressed as a sum over
permutations with one label kept x.
In four dimensions, the classical scaling dimension of the YM
eld is
= 1.2
Independently of the spacetime dimension, the npoint stripped amplitude An is a homogeneous
function of the momenta of degree 4
n. Using this, it is easy to show that the dilatation
operator annihilates the YM amplitude at tree level in d = 4 and for
= 1, i.e.
D
An = 4
d + n(
d=4
1) An ==1 0;
(3.2)
where (2.16) and (3.1) were used, together with D =0An = (4
n)An. We note that
dilatation invariance is also obtained for the multiplicity dependent choice
n
= d 4 + 1 in
an arbitrary number of dimensions d.
2In d dimensions the canonical scaling dimension of the gluon is
theory, not all of the commutators in (2.12) and (2.15) vanish in four dimensions. Using
rst Lorentz invariance and linearity in the polarization vectors i , as well as onshell gauge
invariance WiAn = 0, which are generic ddimensional properties of An, we nd:
[K ; Wi]An =
[K ; ki2]An = (d
2
2 )ki An
(3.3)
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None of these commutators vanishes generically on the stripped amplitude, and in d = 4
with
= 1 only two out of four commutators vanish. Hence K =1 takes the amplitude
o the onshell surface in momentum space, even for d = 4. This leads to the question of
how conformal invariance in d = 4 is realized in momentum space.
Symmetrization prescription. Poincare invariance implies that the action of K
on
An takes the form3
K
An =
n
X
i=1
i Fi + X ki Gi;
n
i=1
where Fi and Gi are some Poincare invariant functions of the kinematic variables, which
inherit de nite homogeneity degrees in i and ki from the amplitude. The coe cients Fi
and Gi have the unwanted feature of being dependent on the ambiguity in resolving the
Poincare constraints, cf. (2.7), presumably as a consequence of (3.3). Nevertheless, in the
case of explicit lower point examples (to be discussed below) we observe that regardless of
this ambiguity
We interpret this as a feature of the underlying conformal symmetry of YM theory in d = 4,
where the ordinary scaling dimension of the gluon becomes
also notice that this relation is valid in any number of dimensions d, keeping
= 1 xed.
2
= d 2 = 1. However, we
On the other hand, the explicit examples also reveal that the naive application of K =1
to An does not in general give zero, i.e.
Does this mean that full conformal invariance of treelevel YM amplitudes cannot be seen
in momentum space?
3While
is not a Lorentz vector, it transforms into
+ f (p)p under a Lorentz transformation.
n
i=1
(P ) X ki Gi ==1 0 :
K =1 An =
At this point, it is useful to note that the stripped partial amplitudes An inherit cyclic
and reversal invariance from the color trace in (3.1) as follows:
An(1; 2; : : : ; n) = An(2; : : : ; n; 1);
An(1; 2; : : : ; n) = ( 1)nAn(n; : : : ; 2; 1):
In order to verify these symmetries, one generically has to resolve momentum conservation
and the onshell conditions, as prescribed by (2.7). However, since K
does not commute
with the onshell conditions, we cannot expect these symmetry properties to be preserved
by it. But since K
is a fully permutation symmetric di erential operator, it seems natural
that it should preserve the permutation symmetries of the amplitude. We can ensure this
by manifesting the cyclic and reversal symmetries of An by hand, and we denote this
manipulation by the symbol Cn, i.e.
Of course, the stripped amplitude and stripped symmetrized amplitude are identical on
the support of momentum conservation and onshell conditions, i.e.
In the explicit examples, to be discussed below, we nd that
Cn[An] = An :
K Cn[An] = Cn[K An] 6= K An ;
Cn[An] =
1
2n
X
[An(1; 2; : : : ; n) + ( 1)nAn(n; : : : ; 2; 1)] :
(3.8)
where the rst equality is the trivial statement that K , being permutation symmetric,
commutes with Cn, while the inequality shows that the naive application of K
on An does
not preserve the permutation properties of An. Remarkably, our explicit checks show that
for the speci c choice of resolving the Poincare constraints
we systematically nd for n = 3; 4; 5; 6 that
K =1 Cn[An] = 0 :
The details for each n are discussed below. At higher points (i.e. for n > 6) this claim
remains conjectural.
The reason why we point out the choice (3.11) is that K Cn[An] still bares some
sensitivity to the ambiguity in resolving the Poincare constraints, and (3.12) is not satis ed
for all choices.4 Understanding this feature is a nonlinear problem that requires further
investigation beyond the scope of this paper, which lies on graviton scattering. The point
4At least for n
5, by summing over all possible ways of resolving the constraints for a xed a in (2.7),
K annihilates the expression for any value of .
(3.7)
(3.9)
(3.10)
(3.12)
we want to make here is that by manifesting the cyclic and reversal symmetries of An, we
nd what seems to be the conformal symmetry of YM theory at the level of momentum
space amplitudes. It seems plausible that if one could manifest all symmetry properties of
An, such as the photon decoupling property, the KleissKuijf relations, and perhaps even
the BCJ relations, the sensitivity of these statements to the prescription of resolving (2.7)
disappears. In fact, this is what we observe in the graviton case to be discussed later, where
all symmetries of the amplitude can be easily implemented as full permutation symmetry.
The threepoint stripped YM amplitude takes the form
where we introduced the notation eij = i j and resolved the constraints from momentum
conservation by imposing k3 =
k1
k2 and ( 3 k1) =
( 2 k2). It is straightforward
to compute
which clearly does not vanish. Considering instead the cyclic and reversal symmetrized
form, which is readily obtained from the above expression,
C3[A3] = 12 1 (k2
k3) e23 + 12 2 (k3
k1) e31 + 12 3 (k1
k2) e12 ;
it is easily checked that for any value of
The threepoint amplitude is of course very special (e.g. it vanishes for real momenta).
However, as we will see next, the four, ve, and sixpoint amplitudes expose this symmetry
through the same procedure, but where
= 1 becomes a crucial choice.
The expression for A4 will not be provided here, but can be
straightforwardly computed from just four Feynman diagrams using textbook prescriptions.
The partial stripped amplitude A4 is obtained by imposing (3.11), speci cally k4 =
k1
k2
k3, s12 =
s13
s23 and 4 k1 =
4 (k2 + k3). We then nd
where Fi are some nonzero functions of the kinematic variables. For
proportional to ki contribute. By considering instead the cyclic and reversal symmetrized
6= 1, also terms
form we nd as conjectured
which holds only for
vanish, not only for
= 1. It moreover turns out that the coe cients of the ki here
= 1, but for any value of . This is, as in the threepoint example,
a special property at four points that we do not nd at higher points.
K =1 A4 =
i Fi 6= 0;
K =1 C4[A4] = 0 ;
4
X
i=1
{ 9 {
(3.14)
(3.15)
(3.16)
(3.17)
(3.18)
Curiously, this invariance is also seen when writing A4 in a manifestly gauge invariant
form. This form is obtained by employing the socalled t8tensor [29] and reads
A4 =
4
s12s23
t8; 1 1;:::; 4 4 k1 1
with sij = 2ki kj and t8 being a tensor with symmetry under exchange of any pairs of indices
f i ig and antisymmetry under the exchange i $
i in each pair, making it manifestly
gauge invariant. Here we emphasize that k4 =
(k1 + k2 + k3). In addition, the numerator
has manifest cyclic and reversal symmetry (in fact, it has full permutation symmetry), but
the denominator does not manifest these properties. Nevertheless, we nd that in this form
= 1. An additional, possibly related curiosity is that the
action of K
on A4 as given above is gauge invariant i
= 1. This can be shown using
only the symmetry properties of t8, i.e. without using onshell conditions. The absence of
a similarly compact expression for higher point amplitudes suggests these observations to
re ect a special symmetry at four points.
Fivepoint example.
Depending on parametrization, the vepoint YM amplitude
consists of some 400 individual terms.5 Imposing (3.11), we again
nd that while (3.5) is
satis ed, we still have
K =1 A5 =
By performing the cyclic and reversal symmetrization as in the previous examples, we then
nd as conjectured
(3.20)
(3.21)
This latter check has been performed numerically only and is satis ed i
= 1.
Sixpoint example.
Up to ve points, the YangMills tree amplitude in four dimensions
is either MHV (maximally helicity violating) or MHV. To ensure that the manifestation
of conformal symmetry, as found in the previous examples, is not just a special feature of
this helicity con guration, we have also explicitly checked the sixpoint case. Depending on
parametrization it consists of some 67000 individual terms. Also in this case, while we do
nd (3.5) to be satis ed, A6 de ned by imposing (3.11) is again generically not annihilated
by K =1. After manifesting the cyclic and reversal symmetries as in the previous examples,
we nevertheless nd our conjecture (3.12) to hold i
= 1. The checks here have all been
performed numerically.
4
Soft dilatons, conformal generators and graviton scattering
A hint on special conformal invariance of treelevel graviton amplitudes in eld theory is
observed when considering the soft behavior of massless closed strings in any string theory
5We thank C. Mafra and O. Schlotterer, as well as J. Bourjaily for providing us with explicit momentum
space expressions for YM treeamplitudes. The lower npoint espressions have been fully exposed in [30, 31].
(bosonic, heterotic, or in the NSNS sector of superstrings). These limits have recently
been studied in [15{18]. The results of those works, important to us, are the following.
Consider for simplicity scattering amplitudes in bosonic string theory using the operator
formalism (the same analysis and result applies in the RNS formalism of superstrings).
The npoint bosonic string amplitude of massless closed states carrying momenta ki and
polarizations i; i can be written as
with
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Mn0 = (P )Mfn 0 ;
Mfn 0 =
8
where P = Pin=1 ki, d is the ddimensional gravitational constant, and integration is over
the KobaNielsen variables zi, modulo SL(2; C) symmetry which is xed by
dVabc =
jza
d2zad2zbd2zc
zbj2jzb
zcj2jzc
zdj2
:
The points za; zb; zc can be xed to any point in the complex plane and the indices a; b; c
are any three from the set f1; : : : ; ng. The i
exponentiate the integrand (the i are thus also Gra mann).
, i are Gra mann variables introduced to
By explicitly calculating the integral over one of the zi in the limit where the
corresponding momentum is soft compared to the other momenta, it has been shown [15{17]
that the soft behavior of a symmetrically polarized, massless closed string state can be
described by the expression
Mn+01(k1; : : : ; kn; q) = (P + q) S Mfn 0 (k1; : : : ; kn) + O(q2) ;
where
J
i
(
S = d "q;
n
X
i=1
ki q
ki ki + i
k q ( iJi )
ki q
q q
2ki q
: Ji J
i : + O( 0q) ;
The soft, symmetrically polarized state carries momentum q with the polarization tensor
= q q . Normal ordering : : means that all derivatives act to the right. In bosonic
string theory, but not in superstring theory, an additional operator contributes to S at order
q (cf. [15]), which is proportional to the inverse string tension 0 and which is not of interest
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
here. The operators act on Mfn 0 , which is the integral representation in (4.2), but before
imposing momentum conservation, hence the tilde (see appendix B for further discussion).
This remark is important, because both sides of (4.4) involve the same (n + 1)point
function. Eq. (4.4) is therefore not a soft theorem in the usual sense, where amplitudes map
to amplitudes and this will be important to us. This `soft theorem' is instead stating that
the integration over the moduli of the soft state in Mn +01 can, for the rst three orders in
the qexpansion, be written as a di erential equation on Mfn 0 with the overall (n + 1)point
momentum conservation from Mn +01 being kept outside. We would like to understand to
what extent (4.4) can be interpreted in terms of amplitudes, in particular in the eld theory
limit where 0 ! 0.
It is useful to note that the theorem in (4.4) was also derived in eld theory from
onshell gauge invariance of Mn+1, under the assumption that no terms proportional to
"q appear6 up to O(q2), where instead of Mfn 0 an unknown, but in principle calculable,
npoint current with one leg o shell enters [19]. Consistency of the two di erent methods
of obtaining (4.4) thus indicates that there is a well de ned way of taking the 0 ! 0 limit of
Mfn 0 entering (4.4), but neither approach immediately yields relations among amplitudes.
For obtaining these, there are at least two ways to proceed: one is to analyze carefully
Mfn 0 , as it enters in (4.4), and its eld theory limit. We provide in appendix B such an
analysis in the simplest case of Mf3 0 , which clari es the problems and shows how the above
equations can be turned into a statement for a representation of the threepoint stripped
amplitude M3 0 . Another way is to commute the function through the soft operators to
obtain a statement on the level of amplitudes in analogy to the discussion of [32]. This is
the path we will take here. Crucial for the success of this approach is that after expanding
(P + q) as well as the prefactor Mfn+1 in powers of q, and after pulling the soft operator
through the resulting function (P ), all derivatives of the function cancel so that one
generically ends up with an expression including the npoint amplitude:
(P + q)Mn +01(k1; : : : ; kn; q) =
(P ) + q @P (P ) S Mfn 0 (k1; : : : ; kn) + O(q2)
h
i
= Se (P )Mn 0 (k1; : : : ; kn) + O(q2):
(4.7)
Because of npoint momentum conservation, Mfn 0 becomes the stripped scattering
amplitude Mn 0 in the last line. Note that the nontriviality in the above expression lies in the
existence of a di erential operator Se which satis es (4.7).
Soft graviton theorem. The eld theory limit of (4.4) consistently reproduces the
treelevel soft theorem of the graviton [21, 33], including the recently discovered subsubleading
term [8, 32, 34] for any number of dimensions d. This can be easily seen, by rst noting that
for the graviton, the normal ordering symbol in the operator S in (4.5) can be removed
due to the tensor properties of "graviton, yielding immediately the known form for the
graviton soft operator, and secondly by noting that this operator has been shown to have
6This assumption holds at treelevel to all orders, in fact, as a corollary of the KLT relations [1], which
state that the amplitude factorizes into two copies of YangMills amplitudes.
the property [32]
(P + q)Sgraviton = Sgraviton (P ) + O(q2) :
(4.8)
Hence, the problem of understanding (4.4) in terms of amplitudes is immediately solved in
the case of the graviton and, in this case, (4.7) holds for Se = S.
Soft dilaton theorem. The soft behavior of the dilaton is obtained by replacing the
polarization tensor of the soft state with the projector (cf. [13, 26] and the discussion
around (2.6))
"dilaton = p
1
d
2
(
q q
q q ) ; for q2 = 0 and q q = 1:
(4.9)
HJEP05(218)
Assuming for simplicity that all hard states are either gravitons or dilatons, such that we
can take i ! i, the `soft theorem' in (4.4) takes the form [15{17]:
(P )S + q @P (P ) S 0 + (SW + SV ) (P )iMfn 0 + O(q2);
where the coe cients of (P ) and 0(P ) are given by the local operators
S = 2
D =0 + q K =0;
S 0 = 2
D =0;
as well as the nonlocal operators
n
X q
SV =
(Si Si;
(4.12)
Here Wi is the generator of gauge transformations (2.5). To obtain this result, momentum
conservation was used7 as well as the onshell conditions q2 = 0, ki2 = ki
i = 0 and Lorentz
invariance of Mfn 0 . The operator SW was moved to the left of the function by making
use of the following, easily checked relation:
(P + q)SW = SW (P ) + O(q2):
Notice that SW contains the operator @ki Wi which annihilates manifestly gauge invariant
expressions. For the contributions to (4.10) where Mfn 0 is directly multiplied by (P ), the
integral expression de ned in (4.2) becomes the proper stripped amplitude, i.e.
(SW + SV ) (P )Mfn 0 = (SW + SV )Mn0 :
From an analysis of Feynmann diagrams, these terms must correspond to diagrams of the
type in gure 1, due to the propagatorpole at ki q = 0, as will be further discussed below.
7The factor 2 in the soft operator S arises by use of momentum conservation from the leading q 1 term
in (4.4); i.e. the projector (4.9) on the leading term gives
2 (P + q)P q = 2 (P + q)q q = 2 (P + q). This
can also be obtained by rst expanding the function
2 (P +q)P q =
2 (P )P q
2(q @P (P ))P q. The
rst term is zero because P = 0, while for the second term we can use the identity P @P (P ) =
(P ),
leading to the same result.
(4.11)
(4.13)
(4.14)
The splitting into SW and SV is done, because as we will argue in a moment, we can
e ectively set
SW Mn0 = 0 + O(q2):
(4.15)
Remarkably, the operators D =0 and K =0 appearing in (4.11) after the projection
onto the dilaton, are exactly the conformal generators de ned in (2.8) for
= 0, as rst
pointed out in [19]. This gives a rst glimpse at the role played by the conformal algebra
in the context of graviton scattering, on which we will elaborate below.
The leading part of the above soft theorem was already understood in works dating
back to the 1970s in relation to scale renormalization in string theory [13, 14]. The full
soft behavior (4.10) was derived in [15{17] for di erent string theory setups and in [19]
in eld theory. As shown in [16], the expression (4.10) is universal; i.e. it describes the
soft behavior of the dilaton in any string theory, meaning that the operators of order 0q
in (4.5) vanish for the dilaton. There, however, the operator SW was immediately dropped,
but we wish here to scrutinize the arguments for this.
The operator SW . If one assumes that the amplitude Mn0 takes a manifestly gauge
invariant form, on which the action of Wi = ki @ i is identically zero (i.e. without using
onshell constraints), the operator SW can be immediately dropped. However, such a
representation of the amplitude is generically not known, and the argument seems not
particularly useful for us. Let us therefore give an alternative argument for why this term
should drop out in (4.10). Notice that we can write
n
X q
i=1 ki q
SW (P )Mfn 0 + O(q2) = SW Mn0 + O(q2) =
where we used that (P )Mfn 0 =
Mn0 , as well as the expansion of the momentum shift
in the form
i (ki + q) @ i Mn0 (k1; : : : ; ki + q; : : : ; kn) + O(q2) = 0 + O(q2):
(4.17)
n
X q
lines while gravitons correspond to wavy lines. The soft particle carries momentum q.
The nal zero follows from onshell gauge invariance of the amplitude Mn0 , which does
not necessitate manifest gauge invariance. Notice, however, that the rewriting in (4.17)
relies on the addition of an in nite number of terms of higher order in q which complete
the original expression to the full gauge invariance condition. The potential danger of such
a resummation is that the orderbyorder gauge invariance in the qexpansion is spoiled.
This is further discussed in appendix C. In consequence, we will assume that the operator
SW can be dropped in (4.10), and this turns out to be consistent with our observations in
the subsequent section 5.8
The operator SV . The operator SV comprises only derivatives acting on the
polarization vectors i, and hence the quadratic dependence of Mn 0 on i can be exploited to
rewrite these terms more transparently as
n
X
i=1
q q
2ki q
S
i Si;
(P )
i i Mn;0i;
(4.18)
i=1
= (P ) Xn q q ( i i) +
ki q
(q
i
)
2
Mn;0i; :
In eld theory, we interpret the above terms as onshell emissions from external states (see
also [16]). Since for a graviton we have g
g = 0 and for a dilaton i i
term corresponds to the decay of an internal graviton to two external dilatons, one of them
soft (see
gure 2a). The second term on the other hand corresponds to the emission of
, the rst
a hard graviton from the soft dilaton leg (see
gure 2b). If the hard external states are
taken to be ingoing, instead of outgoing, the diagram shows that a soft dilaton turns a
hard graviton into a hard dilaton. Thus, depending on whether the ith state is a graviton
or dilaton, only one of these terms contributes. For clarity we consider from here on the
case where all n hard states are gravitons. In that case the soft theorem in (4.10) can, as
8Note that if all reference vectors used to de ne the polarization vectors are chosen to be equal and such
that i q = 0, the operator SW vanishes without employing any resummation. The same applies to the
operator SV .
just argued, be rewritten in the form
i
(4.19)
+ p d
ki q
(P )Mn 0 (k1; : : : ; (ki); : : : ; kn) + O(q2) ;
where in the second line we sum over npoint amplitudes with the ith state being a dilaton
rather than a graviton. This expression points at a peculiarity: in the eld theory limit,
treelevel scattering amplitudes of gravitons with an odd number of dilatons vanish. The
reason for this can be understood from the lowenergy action of the string, which shows
(in the Einstein frame) that the dilaton couples only quadratically to gravitons:
Slowenergy =
d x
d p
g R
(4.20)
Since also the B eld couples only quadratically to the graviton, scattering processes
involving only gravitons and dilatons do not involve virtual B elds at tree level.
Translating this Z2 symmetry to generic treelevel scattering processes involving
dilatons and gravitons, we conclude that only amplitudes with an even number of external
dilatons are nonzero.9 Furthermore, at tree level pure graviton scattering is completely
described by the EinsteinHilbert term in the action. Therefore, taking the limit 0 ! 0
of (4.19) the contribution from the operator SV vanishes, and we end up with the
consistency condition
h
0 =
(P )(2
D =0)i l0i!m0 Mfn 0 + O(q2) ; (4.21)
where it should be understood that all "i
= i i are taken to be symmetric, traceless
polarization tensors corresponding to n gravitons. This line of arguments applies, however,
also to the case where the n hard states comprise both gravitons and an even number of
dilatons, and thus tracelessness of "i is not necessary for an even subset of the set of n
polarization tensors.
Consequences for graviton scattering.
The obstacles for intepreting (4.21) in terms
of eld theory scattering amplitudes are the facts that the formal expression Mfn is not
multiplied by the momentum function, and moreover that the equation includes a derivative
of this function at order q1 (cf. the discussion around (4.7)).
There is, however, an immediate corollary of (4.21): at leading order q0, this equation
reduces to
(P ) [2
D =0] l0i!m0 Mfn 0 = (P ) [2
D =0] Mn = 0;
(4.22)
where Mn is the stripped npoint graviton amplitude in eld theory. To obtain the stripped
amplitude, the commutator (2.16) was tacitly, but importantly, used. This corollary states
9In four dimensions, these statements hold even at loop level as a consequence of U(1) charge
conservation, because the dilaton forms a complex U(1) multiplet together with the B eld which in four dimensions
is dual to a pseudoscalar eld.
0 = q K =0 (P ) + q @P (P ) (2
0 = q K =0 (P )
n
d 2 X
n
i=1
d)i l0i!m0 Mfn 0 :
l0i!m0 Mfn 0
#
n
i=1
= q K
= dn2 (P ) l0i!m0 Mfn 0
where in the second line we introduce a nontrivial scaling parameter
=
d
n
2
:
Hence, assuming that a representation of Mfn 0 exists for which
(4.23)
(4.24)
(4.25)
(4.26)
(4.27)
(4.28)
= d 2
n
(4.29)
(4.30)
that for any number of external states n, eld theory treelevel graviton amplitudes are
homogeneous functions of degree 2 in the momenta. This is indeed a wellknown fact. In
ref. [19] this consistency condition was also discussed from a string theory perspective.
At subleading order, the above equation (4.21) reduces to
0 =
h
D =0)i l0i!m0 Mfn 0 :
In order to turn Mfn 0 into the stripped amplitude, we pull the momentum conserving
function through the special conformal generator using the commutation relation (2.15)
modulo Lorentz invariance of Mfn 0 :
We may rewrite this in the following suggestive way:
we arrive at the statement
(P )
n
i=1
K
Mn = 0 ;
for
=
d
n
2
:
Here, Mn = (P )Mn is the ngraviton eld theory scattering amplitude. We may translate
this into an invariance statement for the stripped amplitude by noting that for
we have
K
Mn = (P )K
Mn = (P )K
Mn ;
where the corollary (4.22) was used. Hence, we nd that K
stripped graviton amplitude:
= dn2 also annihilates the
(P )K
Mn = 0 ;
for
=
d
n
2
:
It follows that all generators of the conformal algebra annihilate the treelevel graviton
amplitude Mn provided we are working on a representation of Mfn 0 which obeys (4.27).
In appendix B our explicitly derived result for Mf3 0 indeed takes a form that obeys (4.27).
In the following section we will argue that, at least for the stripped
eld theory
amplitude Mn, such a representation can be obtained by making the permutation symmetry of
graviton scattering manifest.
Conformal symmetry of graviton amplitudes
In this section we study the relations derived in the previous section on explicit eld
theory amplitudes.
We would like to understand when (4.27) and (4.30) are satis ed.
Our study of conformal transformations of YangMills amplitudes in momentum space
exposed the necessity to manifest physical properties of the stripped amplitude, namely
cyclic permutation and reversal symmetry, in order to observe the invariance properties of
amplitudes in four dimensions. The analogous property for graviton amplitudes is Bose
or full permutation symmetry. The importance of manifesting this symmetry was also
recently noticed in [27, 35].
Interestingly, it turns out that by manifesting full permutation symmetry of the
stripped graviton amplitudes, the identity (4.27) as well as the implication of special
conformal symmetry (4.30) are satis ed  at least for three, four, ve, and six external gravitons.
This is reminiscent of the YangMills case. Additionally, we observe that the ambiguities
arising from stripping o the function become irrelevant when acting on the manifestly
permutation symmetric amplitude. We will detail these results in the following.
Constructing graviton amplitudes from YangMills amplitudes.
Treelevel
graviton amplitudes can be constructed from YangMills amplitudes in various ways. One way is
through the socalled KLT relations [1], which relate closed string amplitudes to a
doublecopy of open string amplitudes through a momentum kernel with a wellde ned eld theory
limit (for a brief review on the eld theory relations, see e.g. [36, 37]). In this work, the
relations at three, four, ve and six points will be used, to be precise:
M3(1; 2; 3) = iA3(1; 2; 3)A3(1; 2; 3) ;
M4(1; 2; 3; 4) =
is12A4(1; 2; 3; 4)A4(1; 2; 4; 3) ;
M5(1; 2; 3; 4; 5) = is12s34A5(1; 2; 3; 4; 5)A5(2; 1; 4; 3; 5)
+ is13s24A5(1; 3; 2; 4; 5)A5(3; 1; 4; 2; 5) ;
M6(1; 2; 3; 4; 5; 6) =
is12s45A6(1; 2; 3; 4; 5; 6) s35A6(2; 1; 5; 3; 4; 6)
+ (s34 + s35)A6(2; 1; 5; 4; 3; 6) + P(2; 3; 4) :
h
i
Here sij = 2ki kj , and An and An are the colordecomposed partial amplitudes as de ned
in (3.1), with the bar on An indicating that the polarization vectors can be distinguished,
i.e. in An we take i ! i. Being interested in graviton amplitudes, however, we need to
set i ! i, or simply An ! An. The term P(2; 3; 4) indicates a sum over all permutations
of the indices 2; 3 and 4. The graviton amplitudes are then given by
Mn = (P )
d n 2
2
Mn ;
with
d denoting the ddimensional gravitational constant. As remarked, the polarization
vectors in the two copies of YangMills amplitudes are identi ed with the graviton
polarization tensor by
"g
with
= 0. The stripped amplitude Mn is obtained
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
Here f denotes a function of n momenta and polarization vectors.10 As for Cn in the
YangMills case (3.9), we have of course that
Pn[Mn] = Mn ;
where by equality we mean on resolving the Poincare constraints as prescribed by (2.7) (as
in all previous discussions).
Conformal invariance of graviton amplitudes.
We would like to establish explicitly
whether treelevel graviton amplitudes can be considered conformally invariant in the sense
described in the previous section. Let us rst point out some general features. As in (3.4),
Poincare invariance constrains the action of K
on Mn to the form
Pn[f (k1; 1; : : : ; kn; n)] =
f (k1; 1; : : : ; kn; n):
1
n!
X
P(1;:::;n)
(5.6)
(5.7)
(5.8)
(5.9)
(5.10)
(5.11)
by implementing the relations (2.7). The amplitude is fully permutation symmetric, i.e.
as a consequence of Bose symmetry. We stress that this identity is generically not manifest,
but holds modulo momentum conservation and onshell conditions. For our purposes,
however, it is useful to make this symmetry manifest at the level of the stripped amplitude
by explicitly permutation symmetrizing it. The notation Pn[Mn] will be used to indicate
this manipulation:
HJEP05(218)
K
Mn =
n
i=1
(P ) X ki Gi /
:
n
X
i=1
n
i=1
n
X
i=1
In all cases to be discussed below, we curiously nd that
Hence, for
= 0 the second sum in (5.9) vanishes, reminiscent of what happens for
YangMills amplitudes for
= 1, cf. (3.5). While in both cases we lack a rigorous understanding
of this feature, we notice that
= 0 is the classical scaling dimension of a graviton in d = 2,
where gravity is known to be conformal. Rephrasing the above, we thus have generically11
K =0Mn =
Pin=11 ki the dependence of f on kn will be trivial before symmetrization.
The symmetrization Pn in n variables, however, reintroduces kn.
11Note that if all reference vectors used to de ne the polarization vectors are chosen to be equal and such
that i q = 0 holds, we nd q K =0Mn = 0: See Footnote 8 in this context.
This similarity to the d = 4 YangMills case (cf. e.g. (3.6)) suggests that the above nontrivial
functions Fi arise from a de ciency of preserving permutation symmetry. In fact, it turns
out that in the studied examples up to (and including) six points we have
K
Pn[Mn] = 0 ;
for any value of . In particular, this implies12
K =0 Pn[Mn] = 0;
and
ki Pn[Mn] = 0 :
n
Hence, Pn[Mn] furnishes a representation of the eld theory amplitude which obeys the
condition (4.27) in the previous section 4. As discussed in that section, choosing the
peculiar value
HJEP05(218)
=
d
n
2
;
the above results together with (cf. (4.22))
establish our conjecture of full conformal invariance of treelevel graviton amplitudes in
ordinary Einstein gravity. We have veri ed these statements analytically and for generic
spacetime dimension d up to multiplicity four. For mulitplicity ve and six we have checked
the above conjecture (5.13) by numerical evaluation of the coe cients Fi and Gi in (5.9)
using fourdimensional kinematical data generated by S@M [38]. These results are
insensitive to the ambiguities in obtaining the stripped amplitude by resolving the Poincare
constraints as described in (2.7). The ndependence of the scaling dimension
in (5.14)
suggests that these results do not follow from the Lagrangian description of gravity in a
straightforward manner.
Threepoint example.
At three points the KLT momentum kernel is unity, and the
threepoint graviton amplitude is simply given by the square of the colorordered
YangMills amplitude:
M3
(P )A3(1; 2; 3)2 :
The stripped graviton amplitude thus reads M3 = A3(1; 2; 3)2, and we again resolve
Poincare symmetry as prescribed by (2.7). Now, using the expression in (3.13) for A3,
one observes that
K
M3 = 4(1 +
)A3 K
A3 ;
which coincidentally vanishes for
=
1, but not otherwise. If instead of A3(1; 2; 3)2
we consider P3[A23] or C3[A3]2 (notice that the latter is also permutation symmetric), we
observe that (5.13) holds for any
.
12On the amplitude which only depends on Lorentz invariants, the vanishing of the summed
momentum derivative, i.e. Pin=1 @ki Pn[Mn] = 0, can be alternatively stated in terms of the two conditions
(5.12)
(5.13)
(5.14)
(5.15)
(5.16)
(5.17)
At four points starting from the stripped amplitude Mn, we nd
by the exact same procedure as in the threepoint case that (5.12) and (5.13) are
fullled. We have additionally, like in the case of YangMills theory, also studied the relations
starting from the manifestly gauge invariant expression for Mn obtained by using the
expression (3.19) for A4 in the KLT relations, which in terms of the stripped amplitude reads
M4 = ( i)
16
s12s23s13
(t8; 1 1;:::; 4 4 k1 1
This expression has the virtue of being manifestly gauge invariant. It is additionally
manifestly permutation symmetric in the labels 1; 2; 3 upon replacement of k4 =
(k1 + k2 + k4).
It can be readily checked using this expression that (5.12) and (5.13) are also ful lled in
this special case.
In appendix D we provide in addition an explicit demonstration of (5.12) in the case
of twodilaton+twograviton scattering, where this equation is also expected to hold (cf.
the discussion below (4.21)).
Five and sixpoint checks.
At ve and sixpoints we nd again by the same
procedure as detailed in the threepoint case that the relations (5.12) and (5.13) are ful lled.
The details should by now be clear, but let us make a remark about the computational
complexity in observing (5.12) and (5.13): at n = 6, the amplitude M6 contains, in
expanded form and before any reduction, roughly 108 terms. Permutation symmetrizing
increases this number by a factor of 5! (since the KLT expression is already symmetric in
three labels), while the action of K
boosts the number of terms at least by two orders of
magnitude, ending naively with some
1012 terms, that in the end sum to zero.
6
Summary and outlook
The aim of the present paper was to identify the consequences of the string theory soft
dilaton limit for graviton scattering. We found that the indications for a conformal symmetry
can indeed be turned into an invariance statement of treelevel graviton amplitudes under
the full conformal algebra, though at the cost of a multiplicity dependent scaling dimension.
Moreover, the formulation of the symmetry in terms of di erential operators in momenta
and polarization vectors, which does not leave the onshell constraint surface invariant,
hinges on the full symmetrization of the graviton amplitude. Dropping the symmetrization
prescription, we observe similarities to the special conformal symmetry of the
(unsymmetrized) YangMills amplitude in four dimensions, but no full conformal symmetry.
Let us summarize our observations in some more detail. Clearly, gluon and graviton
amplitudes are both invariant under Poincare symmetry. Invariance of the full
amplitude (including the momentum conserving function) under dilatations generated by D
requires to have
YM :
=
+ 1;
d
4
n
Gravity :
=
d
n
2
:
(6.1)
These values for
become multiplicity independent in the spacetime dimensions d = 4
and d = 2, respecively, for which the conformal symmetry of the respective theories is well
known. The most involved analysis concerns the generator K
of special conformal
transformations. Poincare invariance implies that the action of this generator on the stripped
YangMills or gravity amplitude, here collectively denoted by An, takes the form
X
i=1
i=1
(6.2)
In the case of YangMills theory we nd that Gi = 0 for
= 1. In the case of gravity we
nd Gi = 0 for
= 0.
The special conformal generator commutes with the momentum conserving function
only for the values (6.1) of , which imply dilatation invariance of the full amplitude.
The nonvanishing of the coe cients Fi in (6.2) appears to be related to the
incompatibility of K
with the onshell constraints  at least for the conformal YangMills theory in
4d. We thus make the physical symmetries of the amplitudes manifest, i.e. cyclic/reversal
or full permutation symmetry, respectively. In the case of YangMills theory in d = 4 and
for
= 1, we then indeed
nd Fi = 0, if the constraints from momentum conservation
are resolved as prescribed by (3.11). We note that, somewhat unsatisfactorily, di erent
ways of resolving these constraints do not always give this result. We attribute the latter
observation to the fact that not all symmetries of the YM amplitude are manifest after
cyclization and reversal.
For the case of fully permutation symmetrized graviton amplitudes, however, this
ambiguity seems to play no role and we nd Fi = 0 for any value of
and d. Since in the
gravity case manifest symmetrization leads to invariance under the special conformal
generator independently of the value of
, we may choose the scaling dimension as in (6.1),
which guarantees also dilatation invariance and a vanishing commutator of K
with (P ).
Hence, for this multiplicity dependent choice of the scaling dimension we observe full
conformal symmetry of treelevel graviton amplitudes, if the amplitudes are in a manifestly
permutation symmetric form.
The observations of this paper lead to a tower of followup questions. The most
pressing point is the meaning of the conformal properties found here. Do they represent
curious coincidences or is there a deeper signi cance behind them? In particular, it would
be important to overcome the speci c symmetrization prescription employed here, which is
due to the use of momentum and polarization variables. Formulating our observations in
a form that manifestly preserves the onshell constraints should distill the physical content
of these statements.
Another point is whether the nding of a conformal symmetry of treelevel graviton
amplitudes in any dimension may also be transferred to the YangMills case, once we
manifest all symmetries of the amplitude and allow for a multiplicity dependent scaling
dimension. If this would indeed be the case, it would emphasize the need for an interpretation
of our results.
A natural question is whether the above conformal symmetry of graviton amplitudes
can be deduced from the EinsteinHilbert action. In order to approach this problem, a
convenient starting point might be the simpli cation of this action as expressed in [39, 40],
which employs only cubic interactions (see also [41, 42]), or by considering the twistor action
of [43]. In fact, it should be very enlightening to make connection to twistor methods for
gravity amplitudes (see [43, 44] and references therein). Intriguingly, these approaches were
motivated by Maldacena's embedding of treelevel Einstein gravity into conformal gravity
in curved space [6] (see also [7]), as well as by Hodges' determinant expression for MHV
graviton amplitudes making Bose symmetry manifest [45]. In relation to our observations
in pure Einstein gravity, it would also be interesting to understand how the conformal
symmetry of amplitudes in conformal (super)gravity is realized, see e.g. [46{48]. The latter
analysis might help in translating our statements into spinorhelicity or twistor variables.
We speculate that BCFW recursion relations [49, 50], valid for treelevel graviton
amplitudes [51, 52], could provide another path towards a proof of the conformal properties
presented in this paper. In fact, permutation symmetry seems also to play a pivotal role
in the recursive constructibility of graviton treeamplitudes [35].
Recently, treelevel gluon amplitudes were mapped to correlators on the 2d celestial
sphere [11, 12]. Here, the twodimensional conformal symmetry of these correlators
originates from the Lorentz symmetry of the 4d gluon amplitudes. It would be interesting
to understand which role the 4d conformal symmetry plays on the celestial sphere and to
extend this analysis to the graviton case.
Our motivation to look for conformal properties of graviton amplitudes was largely
based on a detailed analysis of the string theory soft dilaton limit derived in [15{18]. It
would be important to see whether one can deduce similar consequences of soft limits
for eld theory graviton amplitudes at loop order, and whether the conformal generators
also play a role there. A
rst approach could be to consider perturbative EinsteinHilbert
gravity and try to apply the conformal symmetry generators to the oneloop fourpoint
graviton amplitude, which for fourdimensional helicities has been derived already in [53].
For generic helicity con gurations, however, the respective amplitudes are divergent in four
dimensions. Nevertheless, it would be interesting to see whether the conformal properties
persist for the
nite contributions or for d 6= 4. Alternatively, it would be interesting to
de ne a nite remainder function by factoring out the divergent pieces and to see whether
such a quantity has conformal properties (e.g. in the sense of [54]). Exploring these
directions should reveal whether the observed conformal invariance at tree level re ects a
generic hidden symmetry of graviton scattering or not.
Another interesting direction would be to turn the logic of this paper around: assuming
conformal symmetry of graviton treeamplitudes, what are the implications for single or
multiple soft graviton limits? Considering the analysis of [55] concerning a similar question
in the YangMills case, strong constraints may be expected. Similarly one may wonder
which implications the observed conformal invariance has on the KLT relations or the
colorkinematics duality. Does it constrain the KLT kernel or the BCJ kinematic numerators?
And how can our results be seen in the new formalism of Cachazo, He and Yuan [56] for
YangMills and gravity amplitudes?
On the other hand, these modern formulations of gravity amplitudes can be taken as a
motivation to search for the imprints of the conformal symmetry of gluon scattering in their
gravitational double copy. Intriguingly, we note that in the maximally supersymmetric
extension of 4d YangMills theory, the N = 4 super YangMills model, this conformal
symmetry is in fact not only lifted to a N = 4 superconformal one, but rather to an in nite
dimensional symmetry algebra known as the Yangian [57]. The latter is in turn related to
a dual superconformal symmetry [58], and persist also in supersymmetric gauge theories
in d = 3; 6 and 10 dimensions [59{62]. In fact, these symmetry structures are constructive,
in the sense that they determine the form of the treelevel scattering amplitudes [20, 63].
One may thus wonder whether these extended conformal symmetries leave an imprint in
their respective supergravity double copies.
We hope that the present work o ers new grounds for understanding some of the many
mysterious features seen in recent years in gravity and YangMills theory.
Acknowledgments
MM would like to thank Yegor Korovin for collaboration at the early stages of this work,
as well as Henrik Johansson for clarifying communication on the eld theory properties of
dilaton scattering. We are grateful to Niklas Beisert, Paolo Di Vecchia, Henrik
Johansson, Ra aele Marotta, Tristan McLoughlin, Karapet Mkrtchyan and Oliver Schlotterer for
helpful discussions and for comments on our manuscript. The work of FL is funded by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) 
Projektnummer 363895012.
A
Conformal generator on stripped amplitudes
We rewrite the action of D
and K , given in (2.8), on stripped amplitudes in terms of
di erential operators of the kinematic invariants:
HJEP05(218)
sij = pi pj ;
wij = i pj ;
eij = i j :
Notice that sij is here di erent by a factor of two from the main text. The npoint stripped
amplitude can be considered a function of these invariants as follows:
An = An(si<j;j ; wi6=j;j ; ei<j;j )
8i; j = 1; : : : ; n :
Notice that we are assuming the amplitude to be a function of onshell variables only, i.e.
sii = wii = eii = 0. The latter assumption eii = 0 is a property of the graviton, but does
not change the more general KLT amplitudes, involving gravitons, dilatons and B elds,
since eii do not enter there.
We introduce the di erential operators
and we identify
correct counting.
similar to the property of sij and eij . This is important for taking into account the
for
X = fs; w; eg ;
and
(A.1)
(A.2)
(A.3)
(A.4)
ing (2.8) and using the chain rule:
D An =
i
With these ingredients it is straightforward to derive the following relations
considerX h
i;j6=i
X
i;j6=i;l6=i
X
i;j6=i
i
An
i
B
Computation of Mf3
In this appendix we study Mfn 0 entering in (4.4), where overall momentum conservation
involves n + 1 lightlike momenta, for the simplest case of n = 3.
With n + 1point momentum conservation, instead of npoint momentum conservation,
Mfn 0 depends in principle also on how the measure dVabc in (4.2) is xed (the SL(2; C)
Mobius symmetry). This means that (4.4) has the potential technical problem of not being
wellde ned for a generic choice of dVabc. On the other hand, (4.4) says that there must be
a way to
x dVabc that makes the righthand side of the expression wellde ned through
order q, since the the lefthand side can equally well be calculated by rst integrating and
then expanding in q. To understand this better, we consider the simplest case of n = 3
with momentum conservation P + q = k1 + k2 + k3 + q = 0, with ki2 = q2 = 0.
For brevity, denote the integrand in (4.2) by In = InLInR, where L and R indicate
the holomorphic, respectively antiholomorphic parts, and introduce the compact notation
i = i i , as well as the rescaling K
i =
2 ki and Q
q 0
2 n
X
i<j
q 0 q , thus
n
2
i Kj
3
zj 5 :
(A.5)
(A.6)
(B.1)
(B.2)
I
nL = Y(zi
n
i<j
zj )KiKj exp 4
(zi
i
zj )2 + X
j
i6=j i
z
For n = 3 there is no integration to be done, because all moduli are completely xed by
the Mobius symmetry. However, this is in principle only well de ned for P = Pn
i=1 ki = 0.
Here we instead study the integration under the constraint P + q = 0. To see the problem
and possible resolution, let us study the textbook method of doing this calculation. The
integration is xed by
dV123 =
jz1
d2z1d2z2d2z3
z2j2jz2
z3j2jz3
z1j2
:
The standard choice for xing the three points is at z = 0; 1; 1. Since there is no integration
to be done, we can just as well consider only one holomorphic part (which is equivalent
to studying the openstring threepoint case) and then multiply with the antiholomorphic
part in the end. After performing the Gra mann integration, the holomorphic part reads
(denoting i j
eij):
"
e12
3 K1 +
z1
3 K2
z3
z2
Z Qi3=1 dzi Z n
Y d i InL =
i=1
z1)1+K1K2(z2
z3)1+K2K3(z3
z1)1+K1K3
1 K2 +
z2
1 K3
2 K1 +
z1
2 K3
1 K2 +
z2
z3
3 K1 +
z1
z3
3 K2
z1
z2
1 K3
z3
#
:
Choosing z1 = 0; z2 = 1; z3 = z ! 1, we get
Expanding and neglecting terms of O(1=z) we get
= (1
z)1+K2K3(z)1+K1K3" e12
3 K1
(1)2
3 K2
1
1 K2
3 K1
1
z
1 K3
3 K2
z
1
#
:
= (1
z
z
z
1 K2
1
1 K3
"
z)K2K3(z)K1K3 e12( 3 K1) z e12 3 (K1 + K2)
e13 ( 2 K1)
1
z
z e23 ( 1 K2)
z) ( 1 K2) ( 2 K1) ( 3 K1) + z ( 1 K2) ( 2 K1) ( 3 K2)
( 1 K2) ( 2 K3) ( 3 K1) +
( 1 K2) ( 2 K3) ( 3 K2)
= e(K1+K2) K3 ln(z) e12( 3 K1) z e12 3 (K1 + K2)
e13 ( 2 K1) + e23 ( 1 K2) ( 1 K2) ( 2 K1) ( 3 K1)
+ z ( 1 K2) ( 2 K1) 3 (K1 + K2) ( 1 K2) ( 2 K3) 3 (K1 + K2)
+ ( 1 K3) ( 2 K1) 3 (K1 + K2) + O(1=z) :
( 1 K3) ( 2 K1) ( 3 K1) + ( 1 K3) ( 2 K1) ( 3 K2) + O(1=z)
#
z
1
z
#
(B.3)
(B.4)
(B.5)
Here we used momentum conservation to get the cyclic permutation symmetric form in the
rst line, which is nothing but the threepoint open string expression in the bosonic string.
The second and third line shows the deviation for nonzero Q.
Consider the rst line of (B.6) only. If we cyclically symmetrize z1; z2; z3 over f0; 1; 1g
when
xing the Mobius symmetry, we would, because of momentum conservation and
Q2 = 0, obtain
where A3 0 is equal to
z Q (K1+K2+K3)A3 0 (1; 2; 3) = z0A3 0 (1; 2; 3) = A3 0 (1; 2; 3) ;
A3 0 (1; 2; 3) = e12( 3 K1) + e23 ( 1 K2) + e13 ( 2 K3) + ( 1 K2) ( 2 K3) ( 3 K1) :
This shows that for wellde nedness at z = 1 it is here necessary to keep cyclic permutation
symmetry, when performing the integration by
xing the points.
Next, let us consider the terms with the factor z in the third line of (B.6) aftercyclically
permutation symmetrizing the result:
Clearly this is not wellde ned at z = 1 if K1 + K2 + K3 6= 0. Let us impose momentum
conservation K1 + K2 + K3 =
Q (as well as on shell conditions) to make this more clear:
= z Q K3 e12( 3 K1) + e23 ( 1 K2) + e13 ( 2 K3) + ( 1 K2) ( 2 K3) ( 3 K1)
+ O(1=z) :
(B.6)
(B.7)
(B.8)
(B.9)
(B.10)
(B.11)
1 Q K3 ( 3 Q) [e12
+ z1 Q K2 ( 2 Q) [e31
+ z1 Q K1 ( 1 Q) [e23
( 1 K2) ( 2 K1)]
( 3 K1) ( 1 K3)]
( 2 K3) ( 3 K2)] :
A3 0 (1; 2; 3) =
A3 0 (3; 2; 1) :
A3 should on top of cyclic symmetry also satisfy the reversal symmetry
We can ensure this by also imposing this reversal symmetry, when xing the Mobius
symmetry. This would send the above terms to
1 Q K3 ( 3 Q) [e12
+ z1 Q K2 ( 2 Q) [e31
+ z1 Q K1 ( 1 Q) [e23
( 1 K2) ( 2 K1)]
z
1 Q K1 ( 1 Q) [e32
( 3 K2) ( 2 K3)]
( 3 K1) ( 1 K3)]
( 2 K3) ( 3 K2)]
z
z
1 Q K2 ( 2 Q) [e13
1 Q K3 ( 3 Q) [e21
( 1 K3) ( 3 K1)]
( 2 K1) ( 1 K2)]
We have thus shown that the integration is wellde ned when the momentum is conserved
up to a lightlike deformation, here parametrized by Q, by preserving the symmetry
properties of the integrand when xing the Mobius symmetry.
The expression (B.6) contains additional terms depending on Q; i.e. all terms in the
second line of (B.6). In the end, we want only to consider the expression up to O(q2), so
we can simply set the prefactor z Q K3 = 1 + O(q) for those terms. Then one
the remaining terms all vanish up to O(q2). This is easy to see for the term of order 00,
nds that
which after cyclic and reversal symmetrization reads
e13 ( 2 Q) + e12 ( 3 Q) + e23 ( 1 Q)
e31 ( 2 Q)
e32 ( 1 Q)
e21 ( 3 Q) = 0 :
(B.12)
For the terms of order 0, rst notice that by using momentum conservation and onshell
conditions, we have the identity:
( 1 K3) ( 3 K1) ( 2 Q)
( 3 Q) (( 1 K2) ( 2 K1)
( 1 Q) ( 3 K1) ( 2 Q)
( 3 Q) (( 1 K2) ( 2 Q)
( 1 Q) ( 2 K1)) :
(B.13)
After the manipulation, the second term in the rst line is zero identically. The terms in
the last line are of order q2. Consider the rst term, which we now cyclic and reversal
( 1 K3) ( 3 K1) ( 2 Q)
( 2 K1) ( 1 K2) ( 3 Q)
( 3 K2) ( 2 K3) ( 1 Q)
+ ( 1 K3) ( 3 K1) ( 2 Q) + ( 2 K3) ( 3 K2) ( 1 Q) + ( 1 K2) ( 2 K1) ( 3 Q)
= 0 :
The terms add up to zero, and thus there are no terms linear in q entering in the cyclic
and reversal symmetrized expression. Finally, let us rewrite the terms of order q2 so that
they are manifestly cyclic and reversal symmetric. It is easiest to rst rewrite them once
again using momentum and onshell conditions,
( 1 Q) ( 2 Q) ( 3 K1)
( 2 Q) ( 3 Q) ( 1 K2)
( 3 Q) ( 1 Q) ( 2 K3) ;
which is already cyclic symmetric. Manifesting also reversal symmetry nally gives:
( 1 Q) ( 2 Q) 3 K12
( 2 Q) ( 3 Q) 1 K23
( 3 Q) ( 1 Q) 2 K31;
where Kij = Ki 2Kj .
preserved in xing the Mobius symmetry, thus reads
Mf3 0 = d
A3 0 (1; 2; 3)
A3 0 (3; 2; 1)
p 0
The
nal result of the integration, where cyclic and reversal symmetry have been
0
2
p q q ( 1 2 3 k12 + 2 3 1 k23 + 3 1 2 k31)
#2
;
= p 0
d
C3[A3 0 ]2 + O( 0q2)
(B.14)
(B.15)
(B.16)
(B.17)
where A3 0 is given in (B.8) and kij = ki 2kj . Notice that the leading term of order q0 upon
squaring the bracket is a manifestly permutation symmetric expression for M3 0 , i.e. C3[A3 0 ]2
(cf. below (5.17)). Notice also that in the eld theory limit 0 ! 0 or up to order q2, one
exactly gets C3[A3]2, respectively C3[A3 0 ]2, up to normalization. Thus, C3[A3 0 ]2 represents
one consistent input in (4.4) for calculating the soft limit of the dilaton in scattering
processes with three other massless closed strings, as a consequence of ensuring cancellation
of all large z dependences. As discussed after (5.17), this expression consistently gives the
correct, vanishing eld theory limit of the soft dilaton scattering with three gravitons.
C
On gauge invariance of the dilaton soft theorem
HJEP05(218)
In this appendix we discuss how gauge invariance is ensured in the expressions (4.4)
and (4.10) at order q. For this, we consider the commutator of the soft operators with
The orderq operators in (4.4) can be expressed as
S(1) =
q
2
X q q
It can be checked that the two terms separately commute with Wi. The second term is
easy to check due to the linearity of the operators
[Wi ; 2" q q L
i Si ] =
2" q
(ki q)
i
S
i ki
[
ki[ @ i] ; (C.7)
a b was used. The sum of the three commutators
[Wi ; " q q L
i Li ] = " q
i Si ] = " q
(ki q)
(ki q)
where the notation a[ b ] = a b
vanishes, such that
The rst term is more involved. To understand the cancellations taking place, let us
consider each component of the operator
" q Ji q Ji
= " q q (Li L
i + Si S
+ 2Li Si ) ;
where in the last term we used the symmetric contractions in
and
to sum two terms.
Here Li and Si are given by
L
S
i
Using that q "
= 0, one then nds the following commutators:
i
Li ki
[
(C.1)
(C.2)
(C.3)
(C.4)
(C.5)
(C.6)
(C.8)
Notice that symmetry and transversality of "
were used to obtain these relations, but
tracelessness was not necessary.
Now, consider the soft dilaton operator at order q:
Sd(1il)aton = q K0 + X
i=1
q q
2ki q
S
i Si;
The dilaton projector used to obtain this operator, is symmetric and transverse, thus it
follows from the preceding discussion that the operator in this form also commutes with Wi.
To disentangle the di erent contributions to this operator, we need to understand how
HJEP05(218)
the di erent parts commute with Wi. First, notice that we can rewrite
ki q
S
i Si;
+ 12 i i @2i ; (C.10)
where q2 = 0 was used. Notice that the rst term involving [2
acting on graviton amplitudes, because of their quadratic dependence on i. It is instructive
to not yet use this property. We consider the commutator of each term:
Wi ;
ki q
q
Wi ; ki q
i
Wi ; ki q
i q
ki q
2
2
2
ki q
2
i 2
i
Wi ;
ki q
i :
Apart from the second commutator, where ki i = 0 was used, the vanishing of the sum of
terms is rather entangled, and are at this level not separable. However, we are considering
these operators and commutators on graviton amplitudes, and thus it is only really relevant
to understand the e ective commutators on the stripped amplitude Mn =
i i Mn(i;) .
Assuming the following properties for Mn(i;) :
ki Mn(i;)
= ki Mn(i;)
valid for treelevel graviton amplitudes, we nd
Wi ;
2ki q
i
2
2
i
Mn = 0;
Wi ;
q
ki q
{ 30 {
(C.11)
(C.12)
(C.13)
(C.14)
(C.15)
(C.16)
(C.17)
(C.18)
(C.19)
(C.20)
This explains why we can consider the terms in (C.10) separately from the rest of the
terms in (C.9). The last two commutators, however, do not separately vanish, unless Mn
is manifestly gauge invariant. Instead, the sum ensures commutation:
ki q
Wi ; q K0
This exposes a potential problem in separating the contribution of those two operators in
the soft theorem, as done in the main text. The issue, however, also appears for YangMills
amplitudes in d = 4, where the commutator has exactly the same de ciency, i.e.
d=4
Wi ; K
An =
We have argued in the main text that the operator which is nonlocal in q in (C.21) is
not a real order q contribution to the soft behavior of the dilaton as it can be resummed
to zero. Regarding the gauge invariance, this makes sense if we can replace that zero with
another zero, which also ensures gauge invariance. This is, in fact, possible, i.e.
HJEP05(218)
The new replacing term obviously annihilates the graviton amplitude, since
[Wi ; q K0 + (2
(2
for any i = 1; : : : ; n. Since now the replacing term is local in q, we can even disregard
it, i.e.
h
Wi ; K0 + (2
Mn = 0 :
An equivalent relation exists in the YangMills case by replacing the factor 2 with 1 (in
d = 4 and for
= 1). We have thus argued that K0 e ectively commutes with Wi in
this generalized sense. Thus it seems to be possible to disentangle the two contributions
in (C.21), as done in the main text. It is plausible that the de ciency of K
diately annihilating YangMills amplitudes in d = 4, or K0 not immediately annihilating
graviton amplitudes for any d, is related to the subtleties exposed here.
=1 not
immeD
Conformal symmetry of twodilatontwograviton amplitude
By replacing two polarization tensors with two dilaton projectors in the KLT expression for
the fourgraviton eld theory amplitude, one gets the twodilatontwograviton eld theory
amplitude. The corresponding stripped amplitude can be expressed as
M2 2g =
16 (tk1
3k2
4k2
stu
4) 2
;
where labels 1 and 2 denote the dilatons and label 3 and 4 denote the gravitons. The
Mandelstam variables here read
s = k1 k2 ; t = k2 k3 ; u =
s
t :
(C.21)
(C.22)
(C.23)
(C.24)
(C.25)
(D.1)
(D.2)
Notice that the function has been stripped o according to (2.7) by imposing the
constraints k4 =
k3, k3
4 =
(k1 + k2)
4, and k1 k3 =
k1 k2
k2 k3.
This expression is manifestly permutation symmetric in labels 1 and 2, but not in
3 and 4. This lack of manifest permutation symmetry, shows up also when we consider
the action
with
K0 M2 2g = 3 F3 + 4 F4;
F3 =
F4 = 32 k2
t
u
3
4
3k2
4k2
3k2
s
4k2
;
;
(D.3)
(D.4)
(D.5)
i.e. the righthand side of (D.3) is not vanishing, nor is it permutation symmetric (not even
on upon imposing momentum conservation and onshell conditions). But if we enforce
permutation symmetry on (D.3) it is easy to see that
K0 M2 2g(1; 2; 3; 4) + M2 2g(1; 2; 4; 3) = K0 M2 2g +
K^0 M2 2g 3$4
= 0 :
(D.6)
One can think of enforcing permutation symmetry simply as a means to restore the
symmetry which is otherwise `lost' due to momentumconservation not commuting with K0 .
Since K0 does commute with permutation symmetrization, we could also have started with
manifesting the 3 $ 4 permutation symmetry in M2 2g.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 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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