New worldsheet formulae for conformal supergravity amplitudes
HJE
New worldsheet formulae for conformal supergravity
Joseph A. Farrow 0 1
Arthur E. Lipstein 0 1
0 Durham , DH1 3LE , United Kingdom
1 Department of Mathematical Sciences, Durham University
We use 4d ambitwistor string theory to derive new worldsheet formulae for treelevel conformal supergravity amplitudes supported on re ned scattering equations. Unlike the worldsheet formulae for superYangMills or supergravity, the scattering equations for conformal supergravity are not in general re ned by MHV degree. Nevertheless, we obtain a concise worldsheet formula for any number of scalars and gravitons which we lift to a manifestly supersymmetric formula using four types of vertex operators. The theory also contains states with nonplane wave boundary conditions and we show that the corresponding amplitudes can be obtained from planewave amplitudes by applying momentum derivatives. Such derivatives are subtle to de ne since the formulae are intrinsically fourdimensional and onshell, so we develop a method for computing momentum derivatives of spinor variables.
Scattering Amplitudes; Conformal Field Models in String Theory; Supergrav

ity Models
1 Introduction
Review 2 3 4
5
B
C
D Nonplane wave examples
D.1 Nonplane wave scalar
D.2 Nonplane wave graviton
1
Introduction
Plane wave graviton multiplet scattering
3.1
Supersymmetric formula
Nonplane wave graviton multiplet scattering
Conclusion
A BRST quantization
Derivation of BerkovitsWitten formula
Momentum derivatives
Starting with the discovery of the ParkeTaylor formula for treelevel gluon amplitudes [
1
],
the study of scattering amplitudes has suggested intruiging new ways of formulating
quantum
eld theory. Building on the work of Nair [2], Berkovits and Witten [3, 4] proposed
a worldsheet model for N = 4 superYangMills (SYM) known as twistor string theory,
whose correlation functions give rise to an elegant formula for treelevel amplitudes in terms
of integrals over curves in twistor space [5]. It turns out however that twistor string theory
also contains conformal supergravity (CSGR) in its spectrum [6], so it is di cult to extend
this formula to looplevel. These ideas were then extended by Skinner to N = 8
supergravity (SUGRA) [
8
] following the discovery of a gravitational analogue of the ParkTaylor
formula by Hodges [7]. This framework was subsequently extended to a broad range of
theories after Cachazo, He, and Yuan (CHY) proposed worldsheet formulae for treelevel
scattering amplitudes of nonsupersymmetric gauge and gravitational theories in any
dimension [
9
]. These formulae are supported on solutions to the scattering equations, which
were previously discovered in the context of ordinary string theory [
10, 11
]. The worldsheet
theory underlying the CHY formulae was constructed by Mason and Skinner and is called
ambitwistor string theory [12]. In the case of 10d supergravity, this model is critical and
can be extended to looplevel [13, 14].
{ 1 {
Using a model known as 4d ambitwistor string theory, it is possible to obtain
manifestly supersymmetric formulae for treelevel scattering amplitudes in 4d N = 4 SYM and
N = 8 SUGRA supported on re ned scattering equations [15, 16]. In this approach the
scattering equations are split into two sets, which we refer to as left and right. In general,
the number of states in the left set is related to the Grassmann degree of the
superamplitude. Furthermore, in the case of SYM and SUGRA, it is also tied to the MHV degree
of the amplitude, which is related to the number of negative helicty supermultiplets being
scattered. In particular, an Nk 2MHV amplitude describes the scattering k negative
helicity multiplets. These formulae are simpler than twistor string formulae in that they do not
contain integrals over moduli of curves in twistor space, and are also simpler than the CHY
formulae in general dimensions in that the solutions to the re ned scattering equations are
split into di erent MHV sectors.1
Moreover, the formulae arising from 4d ambitwistor
string theory are intimately related to Grassmannian integral formulae for N = 4 SYM
and N = 8 SUGRA obtained using onshell diagrams [20{23].
Like the BerkovitsWitten twistor string, the 4d ambitwistor string for N = 4 SYM
contains CSGR in its spectrum, in particular a certain nonminimal version of N = 4
conformal supergravity (in the minimal version there is no coupling between the Weyl
tensor and scalar elds of the model [24]). Although CSGR is not unitary, it is nevertheless
of theoretical interest for several reasons. For example, it is renormalizable and can be made
UV
nite if coupled to N = 4 SYM [25, 26]. Furthermore, it is possible to obtain classical
Einstein gravity with cosomological constant by imposing Neumann boundary conditions
on conformal gravity [27] (see also [28, 29]), and this approach was used to deduce twistor
string formulae for scattering amplitudes of Einstein supergravity in at space [30, 31].
Given the large amount of symmetry in CSGR, we expect its scattering amplitudes
to have simple mathematical properties. The purpose of this paper is therefore to
investigate this structure using 4d ambitwistor string theory. CSGR amplitudes were previously
studied in [6, 32{36]. More recently, these amplitudes were shown to arise from taking the
doublecopy of superYangMills with a (DF )2 gauge theory [37]. An ambitwistor string
description of the (DF )2 theory was subsequently found in [38] and used to deduce a CHY
formula for conformal gravity amplitudes in general dimensions.
In this paper, we use 4d ambitwistor string theory to derive compact new worldsheet
formulae for CSGR amplitudes supported on re ned scattering equations. In contrast to
the worldsheet formulae for N = 4 SYM and N = 8 SUGRA, we nd that the number of
particles in the left set is not generally tied to the MHV degree. Nevertheless, the formulae
we obtain are very simple, allowing us to generalise previous results in many ways. For
example, we obtain a simple formula for scalargraviton amplitudes with any number of
particles in the left set. If only two particles are in the left set, this formula reduces to the
one previously derived by Berkovits and Witten. More generally, the formula can be readily
evaluated numerically and we match it against results obtained from Feynman diagrams
and double copy techniques developed in [37] up to 8 points with any number of particles
1In particular, the number of solutions to the re ned scattering equations is given by the Eulerian
number
n 3 for an npoint Nk 2MHV amplitude [17{19]. In contrast, the scattering equations in general
k 2
dimensions have (n
3)! solutions.
{ 2 {
in the left set. Moreover, we generalize this to a manifestly supersymmetric formula using
four types of vetex operators which describe states in either the left or right set and either
the positive or negative helicity graviton multiplet.
Since the equations of motion for conformal gravity are fourth order in derivatives,
they also admit nonplane wave solutions of the form A
xeik x. We nd that the vertex
operators for such states are very simple and give rise to scattering amplitudes which can
be obtained by taking momentum derivatives of plane wave amplitudes, and are therefore
wellde ned at least in a distributional sense. Vertex operators for nonplane wave states
were previously proposed in [33]. Whereas previous vertex operators were de ned only
for A2 = 0, our vertex operators are de ned for any A and appear to be more compact.
Nevertheless, computing nonplane wave amplitudes using 4d ambitwistor string theory
turns out to be subtle for several reasons. First of all, in order to compute worldsheet
correlators we introduce source terms in the path integral leading to deformed scattering
equations. The nal result is then obtained by taking functional derivatives with respect
to the sources and setting them to zero, which returns to the original scattering equations.
Second of all, since the formulae are manifestly fourdimensional and onshell, we develop
a prescription for taking momentum derivatives of spinor variables.
The structure of this paper is as follows. In section 2, we review some facts about
4d ambitwistor string theory that will be relevant in this paper. In section 3, we derive
worldsheet formulae for scattering amplitudes of graviton multiplet states with planewave
boundary conditions. In section 4, we generalize these formulae to nonplane wave states,
and in section 5 we present our conclusions and future directions. There are also four
appendices. In appendix A, we review the BRST quantization of 4d ambitwistor string
theory and show that the vertex operators presented in this paper are BRST invariant. In
appendix B, we show that our worldsheet formula for scalargraviton amplitudes reduces to
the BerkovitsWitten result when there are only two particles in the left set. In appendix C,
we describe a method for taking momentum derivatives of onshell variables. Finally, in
appendix D, we use this method to compute several examples of nonplane wave amplitudes.
2
Review
In this section, we will review the 4d ambitwistor string theory describing N = 4 SYM and
CSGR. The Lagrangian for this model is given by
(2.1)
where the worldsheet coordinate is a complex number
supertwistor space
L =
1
2
ZA = B
0
0 ~
1
I A
_ C ; WA = B ~ _ CA ;
@
~I
and Lj is the Lagrangian for a current algebra, the details of which will not be important.
Note that ; _ = 1; 2 are spinor indices and I = 1; 2; 3; 4 is an Rsymmetry index, and
{ 3 {
the corresponding worldsheet elds are bosonic and fermionic, repsectively. Furthermore,
recall that a 4d null momentum can be written in bispinor form as follows:
ki _ = i ~i_
where i labels the particle number. The worldsheet elds ; ~ can therefore be thought of
as parametrizing an onshell momentum, although their relation to the external spinors of
the amplitude will be made precise later on. The scattering amplitudes will ultimately be
expressed in terms of inner products hiji =
i j
twoindex LeviCivita symbol.
and [ij] = ~ _ ~
i j _ _ , where
_
is the
model have holomorphic conformal weight 12 . Note that the model has a GL(1) symmetry
Z;
1W . We will gauge this symmetry as well as the Virasoro symmetry
(which contains an SL(2) subgroup). The physical states of the model then correspond
to the BRST cohomology. In contrast to ordinary string theory, the spectrum of 4d
ambitwistor string only contains eld theory degrees of freedom, notably N = 4 SYM and
CSGR without an in nite tower of massive higherspin states. In appendix A we describe
the BRST quantization of this model in more detail.
Field theory scattering amplitudes are then obtained from correlation functions of
vertex operators corresponding to physical states. Each vertex operator is described by a
pair of complex numbers
= 1t (1; ), which correspond to homogeneous coordinates on
the Riemann sphere at treelevel. For example, an integrated vertex operator encoding the
N = 4 SYM multiplet (which consists of a gluon, six scalars, and eight fermions) has the
following form:
Vi( ) =
the vertex operator V~ by complex conjugating V and Fourier transforming back to
We de ne the left set L as the set of particles with V~ vertex operators, and the right set
R as those with V vertex operators. To compute an npoint Nk 2MHV superamplitude in
space.
N = 4 SYM is then obtained from the worldsheet correlator
DV~1 : : : V~kVk+1 : : : Vn
E
integrated over the locations of the vertex operators. Particular component amplitudes can
then be extracted by integrating out the appropriate
variables. At treelevel, we may take
all the vertex operators to be integrated and use the SL(2)
GL(1)
GL(2) symmetry to
x the coordinates of two vertex operators to be i = (1; 0) and
j = (0; 1). Note that
the remaining integral over worldsheet coordinates is localized by the delta functions in
the vertex operators which encode the scattering equations. The scattering equations are
then re ned according to how many particles are in the left and the right set.
{ 4 {
As we mentioned above, the 4d ambitwistor string also describes CSGR. The spectrum
of this theory contains the following graviton multiplets:
= h
1 2 3 4 + I J K
IJK + I J AIJ + I I +
+ =
+ 1 2 3 4 + I J K L
IJKL + I J AKL
IJKL + I JKL
IJKL + h+;
(2.2)
where h
refers to helicity
2,
IJK ; IJK
to helicity
3=2,
AIJ ; AIJ
to helicity
1,
I
; I to helicity
1=2, and
refer to spin0 states. Note that the spin1 states
above are distinct from the gluons of N = 4 SYM. Also note that the graviton multiplets
can have plane wave eik x boundary conditions or nonplane wave A
xeik x boundary
conditions because the equations of motion are fourth order in derivatives. We will present
vertex operators corresponding to the graviton multiplet states in the next sections and
demonstate their BRST invariance in appendix A. In contrast to the worldsheet formulae
for N
= 4 SYM and N
= 8 SUGRA, the scattering equations for CSGR are not in
general re ned by MHV degree since the left set can contain states from both graviton
multiplets. Note that the CSGR spectrum also contains gravitino multiplets consisting of
states with helicities
and planewave boundary conditions. The scattering
amplitudes for these states can be computed using the techniques we develop in this paper,
although we leave a detailed analysis for future work. For more details about spectrum of
CSGR in the context of the BerkovitsWitten model, see [6, 33].
3
Plane wave graviton multiplet scattering
In this section, we will consider scattering amplitudes for graviton multiplets with plane
wave boundary conditions in CSGR. First we derive a concise worldsheet formula for
scalargraviton amplitudes, and then we lift it to a supersymmetric formula. We denote
left set vertex operators with V~, and right set with V
. The scattering equations will then
be re ned by how many states are in the left set, which will not in general correspond to
the MHV degree.
The vertex operators describing gravitons and scalars are given by:
(3.1)
We verify the BRST invariance of these vertex operators in appendix A. Let
be the set
of positive/negative scalars, and G
be the set of positive/negative helicity gravitons, so
that G
Z dt h~i ~( )
Y
rg2G+
h+
Vrg
+
{ 5 {
integrated over the locations of the vertex operators (modulo GL(2)). The correlator can
be easily computed using the path integral formalism by combining the exponentials in the
vertex operators with the action to obtain the modi ed Lagrangian
~
tegrated out yielding delta functionals which localize the functional integrals over the
remaining elds onto solutions of the equations of motion
which are uniquely solved by
( r)
where (ij) = ( i
j ) =titj . The amplitude is then given by the following worldsheet
integral:
h;
An;jLj =
Z d
2 n
GL(2)
(SE)
Y
lg2G
Hlg
Y
~
Fl
Y
Fr
Y
rg2G+
~
Hrg
where d2 n
=
in=1d idti=ti3,
(SE) = Y 2 ~
l
For jLj = 2, the scattering equations have only one solution, and it can be found
analytically. As we show in appendix B, on the support of this solution (3.3) reduces to
the BerkovitsWitten result
i2 +[H+ j=1;j6=i hiji hixii hiyii
where
xi and yi are arbitrary reference spinors. For jLj > 2 we have veri ed (3.3) (and
its extension to include fermions and spin one states) numerically by matching it against
results obtained using Feynman diagrams and colorkinematics duality [37] up to eight
points with any number of particles in the left set.2 In order to do so, new techniques
were developed for numerically solving the scattering equations which will be reported on
in [39].
2We thank Henrik Johansson for providing numerical results derived from colorkinematics duality
against which to compare our worldsheet formula.
{ 6 {
(3.2)
(3.3)
(3.4)
(3.5)
The scalargraviton vertex operators in (3.1) can be lifted to the following supersymmetric
vertex operators:
V~ ( ) =
V~+( ) =
V ( ) =
V+( ) =
i 2j4 ~
i
where
the V~
These vertex operators encode all states in the positive/negative helicity graviton multiplets
(denoted with a
), which can occur in the left/right set (denoted with/without a tilde).
Hence, CSGR amplitudes are computed using four types of vertex operators, in contrast to
N = 4 SYM and N = 8 SUGRA which have only two types of vertex operators. Note that
vertex operators can be obtained by complex conjugating the V
vertex operators
and Fourier transforming back to
then obtained by computing a correlator of vertex operators integrated over the worldseet
(modulo GL(2)). As before, one can integrate out the ( ; ~) elds localizing ( ; ~) onto the
solutions in (3.2). In the supersymmetric case, we can similarly integrate out the
elds,
localizing the ~ eld onto the following solutions to the equations of motion:
t ~( ) =
X
r2R
r :
( r)
An npoint Nk 2MHV amplitude with jLj particles in the left set is then given by the
following worldsheet formula:
where k = j
j + j ~ j and
n
Ak;jLj =
Z d
2 n
GL(2)
(SE)
Hl
Y
l 2~
Y
l+2~+
~
Fl+
Y
Fr
Y
r+2 +
H~r+ ;
l2L
(SE) = Y 2j4 ~
l
long as jLj is preserved. For the special case where k = jLj, a formula in terms of integrals
over curves in twistor space was previously conjectured in [35], and it would be interesting
{ 7 {
(3.6)
to see how this formula is related to (3.7). Component amplitudes can be extracted by
integrating out the appropriate
variables. For example, the scalargraviton amplitudes
in (3.3) are obtained by integrating out ( l)4 for l 2 L, and setting ( r) = 0 for r 2 R. In
a similar way, one can also obtain component amplitudes with fermions and spin1 states.
4
Nonplane wave graviton multiplet scattering
The fourth order equations of motion for conformal gravity lead to a second set of graviton
multiplet states with boundary conditions A
xeik x. Note that if A k = 0, then this is
actually a solution to the second order wave equation. Following from this we nd that
vertex operators for nonplane wave states have a free vector index which will be contracted
We propose the following vertex operators describing nonplane wave gravitons
~ _
Vh ( ) =
V~ +_ ( ) =
V
_
( ) =
_
Vh+ ( ) =
Z dt
t
2
Z
Z
!
~ ( )~i_
t~( ) eith~( )ii
2 ( i
Since the ( ; ~) elds appear outside the exponentials, when computing correlation
functions we cannot simply combine them with the action and integrate them out as before.
On the other hand, if we add source terms for these elds, then we can compute a di erent
correlator where they only appear in the exponentials and obtain the original correlator
by taking functional derivatives with respect to the sources and setting them to zero
afterwards. In more detail, we add the following source terms to the Lagrangian:
Z
tJ ( 0)
0
; t~( ) =
~
r
( r)
+
Z
tJ ( 0)
0
:
(4.2)
which are uniquely solved by
X tl l (
l2L
and consider a correlator of vertex operators like the ones in (4.1), but without ( ; ~)
terms outside the exponenti als. This correlator can then be evaluated by combining the
exponentials of the vertex operators with the action and integrating out ( ; ~) giving rise
to delta functionals which localize the ( ; ~) elds onto solutions of the following equations
of motion:
X tr ~r (
r) + J~;
~
J
~ J
{ 8 {
r2R
X
r2R
( ) =
Z
0 ~( 0)
; ~( ) =
Z
0
( 0)
We can then restore the terms with ( ; ~) outside the exponentials by taking functional
derivatives with respect to (J; J~) and setting them to zero afterwards (note that after
setting the sources to zero, the scattering equations will no longer be deformed). From the
explicit form of ( ; ~) in (4.2), we conclude that correlators with nonplane wave vertex
operators can be evaluated by making the following substitutions for ( ; ~) outside of the
exponentials in (4.1):
These formulae are familiar from canonical quantization.
Note that the nonplane wave graviton vertex operators in (4.1) contain singular terms
which cancel out. In the rst vertex operator for example, a pole arises in the rst term
after making the replacement in (4.3), but this is precisely cancelled by the pole which
arises from
( ) in the second term. Indeed, looking at (3.2), we see that the residue of
the second pole is i, which precisely cancels the residue of the rst pole. We describe this
in more detail in appendix D, where we also work out some examples at n points with up
to two nonplane wave states, and show that amplitudes with nonplane wave states can
be obtained by acting on the planewave amplitudes with a momentum derivative for each
nonplane wave state. This could have been anticipated from the LSZ reduction formula
by noting that a nonplane wave solution can be written as a momentum derivative of a
planewave solution:
A x eik x = A
eik x;
where k is understood to be o shell prior to taking the derivative.
Since the amplitudes are manifestly 4d and onshell, we must de ne a prescription
for taking momentum derivatives of onshell quantities. We de ne such a prescription in
appendix C, and use it to derive the following formulae which are su cient to di erentiate
any little group invariant function of spinor brackets:
where
is an arbitrary spinor and
is a reference spinor which parametrizes an o shell
extension of the momentum k. Another subtlety about nonplane wave amplitudes is that
they can be expressed in many di erent ways. For example, using momentum conservation
to remove the dependence on the momentum of a particular leg, amplitudes with a single
nonplane wave state can be written with a derivative acting only on the
momentumconserving delta function, although the expressions we obtain from worldsheet calculations
will generally not be of this form for amplitudes with more than three legs. On the other
hand, amplitudes with nonplane wave states are wellde ned in a distributional sense. In
particular, if we multiply a nonplane wave amplitude by a test function, integrate over
{ 9 {
~ _
V
( ) =
~ _
V+ ( ) =
V
_
( ) =
V+ _ ( ) =
Z dt
t
2
Z
Z
tdt
Z dt
t
2
states in the left set.
5
Conclusion
momentum space, and perform integration by parts, then we are left with derivatives of
the test function times a planewave amplitude which is unambiguous.
Finally, let us point out that nonplane wave scalargraviton amplitudes can be
supersymmetrized using the following vertex operators, which are the nonplane wave analogues
of (3.6):
i
_
( )
!
2j4 ~
i
HJEP07(218)4
Once again, the superamplitude will depend on both the MHV degree and the number of
In this paper we investigate treelevel scattering amplitudes of graviton multiplets in CSGR
using 4d ambitwistor string theory. This model has the same spectrum as the
BerkovitsWitten twistor string (notably N = 4 SYM and a nonminimal version of N = 4 CSGR)
but gives rise to scattering amplitudes in the form of worldsheet integrals supported on
re ned scattering equations which are split into two sets, referred to as left and right. In
contrast to the 4d ambitwistor string formulae for N = 4 SYM and N = 8 SUGRA, we
nd that the scattering equations for CSGR are in general not re ned by MHV degree so
the amplitudes are labelled by the MHV degree as well as the size of the left set. On the
other hand, we are able to obtain very simple formulae for scattering amplitudes which
generalize previous results in several ways.
We obtain a compact formula describing the scattering of any number of scalars and
gravitons with any number of particles in the left set. If two particles are in the left set,
the worldsheet integrals can be solved analytically reproducing the results of Berkovits
and Witten. If more than two particles are in the left set, the worldsheet integrals can
be evaluated numerically and we match the results against those obtained using Feynman
diagrams and the double copy approach developed in [37] up to 8 points with any number
of particles in the left set. An explicit algorithm for numerically solving the scattering
equations and computing amplitudes with plane wave external states will be described
in [39]. Moreover we generalize the scalargraviton amplitudes to a supersymmetric formula
using four types of vertex operators which describe states in the left or right set and the
positive or negative helicity graviton multiplet.
Since the equations of motion for CSGR are fourth order in derivatives, there are
also graviton multiplets with nonplane wave boundary conditions of the form A
xeik x.
Amplitudes with such states are subtle to compute since this requires introducing sources
in the worldsheet path integral which lead to deformed scattering equations, as well as
developing a prescription for taking momentum derivatives of spinor variables. In the end
however, we show that the amplitudes can be obtained by acting on plane wave amplitudes
with momentum derivatives.
There are a number of interesting open questions:
Conformal symmetry is not manifest in our worldsheet formulae. As explained in [36],
this is not surprising since chosing plane wave external states singles out 2derivative
solutions to the 4derivative equations of motion, breaking conformal invariance. On
the other hand, the underlying theory has conformal symmetry so it would be
interesting to understand how it is realized at the level of amplitudes. Hidden conformal
symmetry of gravitational amplitudes was recently explored in [
40
], so it would be
interesting to see if the ideas developed in that paper can be applied to conformal
gravity.
As shown in [23], the 4d ambitwistor string formulae for N = 4 SYM and N = 8
SUGRA can be mapped into Grassmannian integral formulae which can be derived
from a completely di erent approach involving onshell diagrams. For N = 4 SYM,
these formulae suggest a new interpretation of the amplitudes as the volume of a
geometric object known as the Amplituhedron [
41
]. It would interesting to carry out
an analogous mapping for CSGR amplitudes and see if they have a similar geometric
interpretation.
A double copy construction has recently been proposed for CSGR [37], which involves
combining superYangMills with a certain nonsupersymmetric (DF )2 gauge theory,
and an ambitwistor string theory describing the latter in general dimensions was
proposed in [38]. It would be interesting to try to formulate the (DF )2 theory
using 4d ambitwistor string theory and obtain worldsheet formulae for the scattering
amplitudes supported on re ned scattering equations.
Classical Einstein gravity in dS4 can be obtained from conformal gravity by imposing
Neumann boundary conditions which
x external states to be of the BunchDavies
form (1 + ik ) e ik +i~k ~x, where
is the conformal time coordinate. These external
states have also been used to compute three and fourpoint de Sitter correlators using
Feynman diagrams and BCFW recursion and the results are consistent with
holography [42, 43]. Note that the BunchDavies state is essentially a linear combination
of planewave and nonplane wave states which are precisely of the form we have
studied in this paper. We therefore hope that the techniques developed in this paper
can be used to compute de Sitter correlators using worldsheet methods.3
In summary, we nd that 4d ambitwistor string theory reveals interesting new
mathematical structure in the scattering amplitudes of CSGR, which appears to be very di erent
3Note that correlators in dS and AdS are related by analytic continuation [45]. A twistor string formula
for N = 8 SUGRA in AdS4 was proposed in [
44
], although it is written in terms of external states which
make it di cult to relate it to results obtained using other methods.
from the structure previously found in N = 4 SYM and N = 8 SUGRA. We hope that
exploring the directions described above will lead to a deeper understanding of gravitational
amplitudes which can ultimately be applied to more realistic models.
Acknowledgments
We thank Simon Badger, Nathan Berkovits, and Paul Heslop for useful discussions, and
especially Tim Adamo for providing comments on the manuscript and Henrik Johansson
for sharing numerical results. AL is supported by the Royal Society as a Royal
Society University Research Fellowship holder, and JF is funded by EPSRC PhD scholarship
In this appendix, we will review how to BRST quantize the 4d ambitwistor string theory
in (2.1). After gauging the GL(1) symmetry (Z; W ) !
Z;
1W
and introducing a
where a is a worldsheet gauge eld and
L ! L + aZ
W + eTmatter;
1
2
Tmatter =
where Tj is the current algebra stress tensor.
Note that this action is a ( ; ) ghost system with holomorphic conformal weights
(1=2; 1=2). A general ( ; ) system with holomorphic conformal weights ( ; 1
) has the
stress tensor
T
=
where
=
1 for bosonic/fermionic statistics. The central charge can then be read o from the OPE of T with itself and is given by c = 2
We may gauge x e = a = 0 using the FadeevPopov procedure by introducing ghost
systems (b; c) and
~b; c~
with holomorphic conformal weights (2; 1) and (1; 0),
respectively. The stress tensor for the ghosts is then given by Tghost = Tbc + T~bc~ where
Using (A.1), the contribution of the ghosts to the central charge is cghost =
26
2 =
We then de ne the BRST charge Q as follows:
I
Q =
d (c (Tmatter + Tghost) + c~Z
W ) :
The key property that Q must satisfy is nilpotency, i.e. Q2 = 0. In order for Q to satisfy
this constraint, the total central charge must vanish. The (Z; W ) system has zero central
charge since the bosonic contributions cancel the fermionic ones, so the central charge of
the current algebra must be cj = +28.
The physical states of the theory correspond to the BRST cohomology. Hence, the
corresponding vertex operators must be Qclosed, i.e. fQ; Vg = 0. The condition of
Qclosure implies that the vertex operators must have holomorphic conformal weight w
and GL(1) weight qV = 0. The conformal and GL(1) weights may in turn be read o from
V = 1
the OPE of the vertex operator with T and Z
W :
T ( )V( 0) =
wV V( )
(
0)2 + : : : ; Z
W ( )V( 0) = qV V( )
0
+ : : :
where the ellipsis denote less singular terms.
Let us verify that the vertex operators considered in this paper are Qclosed. Since
the equations of motion for conformal gravity are fourth order in derivatives, the spectrum
contains plane wave states of the form eik x as well as nonplane wave states of the form
A xeik x. Moreover, in the ambitwistor string framework, vertex operators with opposite
helicity are simply complex conjugates of each other and are therefore naturally divided
into two sets, which we shall refer to as left and right sets (the scattering equations are
then re ned by the number of states in each set).
A planewave vertex operator in the right set is schematically of the form
2 ( i
t ( )) eit ( ) ~i ;
where ki = i ~i is the onshell momentum. Using the incidence relation adapted to
worldsheet elds ( ) = x
( ) (where x is a point in spacetime), we see that the exponential
reduces to a planewave on the support of the delta function. Let us therefore consider an
ansatz for a planewave vertex operator of the form
1. In practice, one can avoid a tedious (but straightforward) OPE calculation
using the following rules for computing conformal and GL(1) weights:
T : [Z] = [W ] =
Z
W : [Z] =
1
2
[ i] = h~ i
where weights of t are xed by the consistency condition that [tZ] = 0 (for vertex operators
in the left set, t will have opposite weights). We take external spinors have zero weight i.e.
i = 0. Applying these rules to the vertex operator in (A.2), we nd that
w
where wj is the conformal weight of the current algebra. Qclosure then implies that
and wj = 2
s, which implies that s
2.4 If s = 1, then the vertex operator reduces to
which describes a gluon in N = 4 SYM. For s = 2, the vertex operator describes a graviton
with plane wave boundary conditions:
where we discarded the current algebra since wj = 0.
To deduce the vertex operator for a scalar in the left set, consider the ansatz
1. Using the rules described above, one nds that
t ( )) eit ( ) ~i
(A.4)
ing ansatz:
following ansatz
0. If s = 1, then the vertex operator reduces to the gluon vertex operator
in (A.3), but if s = 0 it describes a scalar with planewave boundary conditions
Let us now turn our attention to nonplane wave states. Let us consider the
follow2 ( i
t ( )) eit ( ) ~i :
Following an analysis very similar to the one for (A.2), we nd that s =
= 2, so the
vertex operator reduces to that of a nonplane wave graviton. Similarly, we nd that the
A _
t ( )) eit ( ) ~i
must satisfy s = 0 and
=
1, and reduces to the vertex operator for a nonplane wave scalar.
4Note that if we do not impose the constraint qV = 0, i.e. we do not gauge the GL(1) symmetry of the
worldsheet theory, then vertex operators with higher spin appear to be allowed.
In this appendix, we will evaluate the worldsheet integral in (3.3) for the case jLj = 2. In
this case, the worldsheet integral is straightforward to evaluate analytically. Let us take
the left set to be L = f1; 2g and the right set to be R = f3; : : : ; ng. Using the GL(2)
symmetry to set
1 = (1; 0) and
2 = (0; 1), the delta functions encoding the re ned
scattering equations reduce to
Y 2 ~
l
l2L
tl ~ ( l) = h12i2 4(P );
2 ( r
t
r ( r)) =
(1r)2 (2r)2
h12i (12)
1
r
h21i
hr1i
2
r
hr2i
h21i ; r 2 R:
The remaining worldsheet integrals then localize onto the solution
r =
hr1i hr2i
h21i ; h21i ; r 2 R:
On the support of the re ned scattering equations, (3.3) then reduces to
An;jLj=2 = 4(P ) h12i2 Y (1r)2(2r)2
h;
Plugging in (B.1), we nd that the factor associated with each h+ leg is given by
where we de ne the gravitational inverse soft factor for leg j as follows:
r2R
X [rgr](1rg)2(2rg)2
(rgr) h12i (12)
=
X
[rgr] h1ri h2ri
r2R hrgri h1rgi h2rgi
=
rg1;n2 ;
where a and b are reference spinors. Using momentum conservation, it is possible to show
that the inverse soft factor is independent of the choice of reference spinors so we will just
refer to it as j;n. Similarly, we nd that the factor associated with each
+ leg is given by
(B.1)
(B.2)
(B.3)
(B.4)
(B.5)
HJEP07(218)4
where we checked the second equality numerically up to high multiplicity. Hence, we nd
that for jLj = 2, (3.3) reduces to the formula of Berkovits and Witten;
X
[rr0](rr0)(12)
r<r02R (l r)2(l r0)2 h12i
= l ;n;
We would like to be able to take derivatives of onshell quantities with respect to o
shell momenta. As written, this problem is not well speci ed as there are four degrees of
freedom in an o shell momentum p but only three degrees of freedom in the spinor variables
which parametrize an onshell momentum after removing the little group redundancy. We
work with real momenta such that ~ _ =
. To make the problem wellde ned, we will
introduce a fourth degree of freedom
in addition to the three degrees of freedom in the
spinor variables and write the o shell momentum in terms of ( ; ) as follows:
p
_
= ~ _
+
~
_
(C.1)
~_ is a reference null vector which encodes an o shell direction.
Inverting these equations to solve for (p) is simple, and we nd that (p) = 2pp2 . We
can also solve for (p) . To see this, contract with the spinor ~_ to arrive at ~_ p
_
= [ ~~] .
For real momenta it is clear that [ ~~] 2
=
such that [ ~~] = e i p
express it in terms of p as follows:
p, hence there must exist some phase (p)
p. Given that
is de ned only up to an arbitrary phase, we can
=
=
~
[ ~~]
:
1
pp0 + p3
p
1
p0 + p3 !
ip2 :
p
ei (p) ~_ p _ !
ei (p) ~_
p
1 ei (p) ~_ p _
2
p
p
~
_
p
+ i
_
p
+ i
(p) :
p
!
0
1
(p) =
1 (p)
2 h i
h i
=
=
h i
Note that for the choice = and = 0 we recover the wellknown expression
Di erentiating (C.2) with respect to the o shell momentum then gives
In general, di erentiating a function of spinor brackets which is not littlegroup
invariant will result in a @ (p) term. Let us therefore consider momentum derivatives of the
littlegroup invariants ~ _
a di erent external leg for example). We nd that
and
, where is an arbitrary spinor (which can come from
h i
2
h i2 [ ~~] ;
~
_
(C.2)
(C.3)
(C.4)
(C.5)
(C.6)
(C.7)
where in the second line we used the Schouten identity, and in the third equality we
used (C.4). Furthermore, using equation (C.1) we nd that
This calculation involves only derivatives of vectors with respect to vectors, and hence we
do not need equation (C.4). The result is
p
_
_
_
p
2
2
p
h i
~
= ~ _
.
o shell direction, i.e. _
= 0 and p
Note that the righthandside is a projection matrix which removes components along the
D
Nonplane wave examples
In this section, we will work out examples of scattering amplitudes for nonplane wave
states of the form A
xeik x using the vertex operators proposed in section 4, and use
the method described in appendix C to express them as momentum derivatives of plane
wave amplitudes.
D.1
Nonplane wave scalar
We will rst calculate an amplitude with two plane wave negative helicity gravitons, a
negative multiplet scalar with nonplane wave boundary conditions, and n
3 plane wave
positive helicity gravitons. We then de ne the left set to be L = f1; 2g, the right set R to
contain the remaining particles, and the set R0 = f4; : : : ng to be the set of positive helicity
gravitons. The vertex operators for these states can be found in (3.1) and (4.1).
After replacing ( ; ~) outside the exponential in the nonplane wave scalar vertex
operator and taking functional derivatives according to (4.3), we obtain the following
worldsheet formula:
A(h h
x h+ : : : h+) = A3 _
Z d2 n h12i2
GL(2) (12)2
Y
!
(SE):
(D.1)
The rst term comes from acting with the functional derivatives on the delta functions
brackets h~ ~ ( )i in the positivehelicity graviton vertex operators.
imposing the scattering equations, and the second term comes from acting on the spinor
We can evaluate the worldsheet integral analytically following the same procedure
described in appendix B. Using the GL(2) symmetry to x 1 = (1; 0) and 2 = (0; 1) and
converting the delta functions in the left set into a momentum conserving delta function,
we see that the remaining terms do not depend on ~1 or ~2. Furthermore, the Jacobian
from the scattering equation delta functions only contains angle brackets, so the @@~ will
act only on the momentum conserving delta function. We can then simplify this part of
the calculation as follows:
2
4(P ):
After some further simpli cation using the Schouten identity, we obtain
HJEP07(218)4
A(h h
x h+ : : : h+) = h12i4 A3 _
2R0
2R0
!
where the gravitational inverse soft factor
;n was de ned in (B.4).
by applying a momentum derivative to a planewave amplitude as follows:
Using the results in appendix C, it is not di cult to show that (D.3) can be obtained
A(h h
x h+ : : : h+) = h12i4 A3
Y
2R0
1
;n 4(P )A :
Clearly for n = 3, jR0j = 0 and the result holds. To show this for n > 3, let us compute
the momentum derivative of the gravitational inversesoft factor for leg j with respect to
particle i where i 6= j and assume that the reference spinors do not depend on i. We then
nd that
~j _
X [jk] hkai hkbi
k6=j hjki hjai hjbi
~j _
hjai hjbi hiji2 habi hjii i + hiai hbii j
;
~ _
i i
hjii
a hibi + [ji] hiai
_
hjii
b
!
(D.5)
where we used equations (C.5) and (C.7) and chose the reference spinor to be ~_ = ~ _ .
j
Setting i = , j = 3, a = 1, and b = 2, we see that the second term in (D.3) contains the
derivative of
;n, from which (D.4) follows.
D.2
Nonplane wave graviton
We now compute an amplitude with one negativehelicity nonplane wave graviton,
A(hx h h+ : : : h+). As outlined in section 4, the vertex operator for hx has divergences
(D.2)
(D.3)
(D.4)
which cancel, and we show the details of this here. Following the same steps as in the
previous section, we obtain the following worldsheet formula:
A(hx h h+ : : : h+) = A
+ X
r2R
Y
X
r06=r2R r002R
!
Y
where the rst term comes from acting with the functional and ordinary derivative in hx on
the delta functions imposing the scattering equations, the second term comes from acting
with the functional derivative on the spinor brackets
graviton vertex operators, and we have regulated divergent terms by taking a limit.
h~r ~ ( r)i in the positivehelicity
Cancelling the singular terms and carrying out the worldsheet integral as described in
appendix B then gives
A(hx h h+ : : : h+) = A
1 _
GL(2) (12)
(12)
Z d
2 n
+ X
r2R
h12i
~
1 r
_
(1r)
Y
X
r06=r2R r002R
r2R h1ri2 h2ri r02R;r06=r
Y
Using the results from appendix C to di erentiate the gravitational inverse soft factor as
we did in the previous section, we arrive at the nal result that
r2R
r;1n 2 4(P ) :
!
Following a similar calculation with two hx states we nd that
A(hx hx h+ : : : h+) = h12i4
Y
r2R
r;1n 2 4(P )
!
where A1;2 are vectors in the wavefunctions of particles 1,2.
Open Access.
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