Twoloop supersymmetric QCD and halfmaximal supergravity amplitudes
HJE
Twoloop supersymmetric QCD and halfmaximal supergravity amplitudes
Henrik Johansson 0 1 2 3 5 6
Gregor Kalin 0 1 2 5 6
Gustav Mogull 0 1 2 4 5 6
Gauge Symmetry
0 The University of Edinburgh
1 Roslagstullsbacken 23 , 10691 Stockholm , Sweden
2 75108 Uppsala , Sweden
3 Nordita, Stockholm University and KTH Royal Institute of Technology
4 Higgs Centre for Theoretical Physics, School of Physics and Astronomy
5 Department of Physics and Astronomy, Uppsala University
6 Edinburgh EH9 3FD , Scotland , U.K
Using the duality between color and kinematics, we construct twoloop fourpoint scattering amplitudes in N = 2 superYangMills (SYM) theory coupled to Nf fundamental hypermultiplets. Our results are valid in D bound corresponds to sixdimensional chiral N close connection with N = 4 SYM theory  and, equivalently, sixdimensional N = (1; 1) SYM theory  we nd compact integrands with fourdimensional external vectors in both the maximallyhelicityviolating (MHV) and allchiralvector sectors. Via the doublecopy construction corresponding Ddimensional halfmaximal supergravity amplitudes with external graviton multiplets are obtained in the MHV and allchiral sectors. Appropriately tuning Nf enables us to consider both pure and mattercoupled supergravity, with arbitrary numbers of vector multiplets in D = 4. As a bonus, we obtain the integrands of the genuinely sixdimensional supergravities with N = (1; 1) and N = (2; 0) supersymmetry. Finally, we extract the potential ultraviolet divergence of halfmaximal supergravity
Scattering Amplitudes; Supergravity Models; Supersymmetric Gauge Theory

= (1; 0) SYM theory. By exploiting a
in D = 5
2 and show that it nontrivially cancels out as expected. Keywords: Scattering Amplitudes, Supergravity Models, Supersymmetric Gauge Theory, Gauge Symmetry
1 Introduction
Review 2 3
Four dimensions
Six dimensions
3.2.1
3.2.2
Cuts
3.3.1
3.3.2
2.1
2.2
3.1
3.2
3.3
4.1
4.2
4.3
4.4
4.5
4.6
5.1
5.2
4
Calculation of N = 2 SQCD numerators
Fourdimensional correspondence
Sixdimensional amplitudes
Oneloop example
Twoloop cuts
Master numerators
Ansatz construction
Symmetries and unitarity cuts
Additional constraints 4.4.1 4.4.2 4.4.3
Matterreversal symmetry
Twoterm identities
Matching with the N = 4 limit
Allchiral solutions
MHV solutions, bubbles and tadpoles
N = 0
5.1.1
5.1.2
N = 2
5.2.1
5.2.2
One loop
Two loops
One loop
Two loops
N = 4 construction
N = 2 construction
Colorkinematics duality with fundamental matter
Pure supergravities from the doublecopy prescription
Trees and cuts in N = 2 SQCD
{ 1 {
Introduction
It is by now well established that a variety of gauge and gravity theories are perturbatively
related through the socalled double copy. In terms of the asymptotic states a squaring
relation between gauge theory and gravity follows readily from representation theory; that
such a structure is preserved by the interactions is a remarkable fact rst brought to light
by the KawaiLewellenTye (KLT) relations [1] between open and closed string amplitudes.
More recently, the double copy has been understood to arise due to the
BernCarrascoJohansson (BCJ) duality between color and kinematics [2, 3] that is present in many
familiar gauge theories. This realization has opened the path towards constructions of
scattering amplitudes in a multitude of di erent gravity theories [3{10] starting from the
much simpler gaugetheory amplitudes.
The duality between color and kinematics refers to the observation that many familiar
gauge theories have a hidden kinematic structure that mirrors that of the gaugegroup color
structure [2, 3]. Amplitudes in, for example, pure YangMills or superYangMills (SYM)
theories in generic dimensions can be brought to forms where the kinematic numerators of
individual diagrams obey Jacobi relations in complete analogue with the color factors of
the same diagrams. A natural expectation is that one (or several) unknown kinematic Lie
algebras underlie the duality [11, 12]. At tree level the duality is known to be equivalent
to the existence of BCJ relations between partial amplitudes [2], which in turn have been
proven for pure SYM theories through a variety of di erent techniques [13{17]. As of
yet there is no proof of the duality at loop level, although a number of calculations have
established its presence up to four loops in the maximallysupersymmetric N = 4 SYM
theory [3, 18{23], up to two loops for pure YM [24], and at one loop in mattercoupled YM
theories with reduced supersymmetry [4{7, 25, 26].
Colorkinematics duality has been shown to be present in weaklycoupled quantum
chromodynamics (QCD) and generalizations thereof [6, 27]; this includes YangMills
theory coupled to massive quarks in any dimension, and corresponding supersymmetric
extensions. The BCJ amplitude relations for QCD were worked out in ref. [27] and proven in
ref. [28]. Using the duality for practical calculations in QCD phenomenology is a promising
avenue as it poses strong constraints on the diagrammatic form of an amplitude,
interweaving planar and nonplanar contributions to the point of trivializing certain steps of
the calculation [29{31]. In the limit of massless quarks the (super)QCD amplitudes are
a crucial ingredient in the doublecopy construction of pure N = 0; 1; 2; 3 supergravities
in four dimensions [6], as well as for the symmetric, magical and homogeneous N = 2
supergravities [9], and twin supergravities [
10
].
The double copy is most straightforwardly understood as replacing the color
factors in a gaugetheory amplitude by corresponding kinematic numerators [2, 3]. When
colorkinematics duality is present this replacement respects the Liealgebraic structure
of the amplitude, and furthermore enhances the gauge symmetry to di eomorphism
symmetry [32{34]. Additional enhanced global symmetries typically also arise [35{39]. The
resulting amplitudes are expressed in terms of Feynmanlike diagrams with two copies of
kinematical numerators, and describe scattering of spin2 states in some gravitational
the{ 2 {
ory. The double copy gives valid gravity amplitudes even if the kinematic numerators are
drawn from two di erent gauge theories [2], as long as the two theories satisfy the same type
of kinematic Lie algebra [7] (with at least one copy manifestly so [3, 40]). This exibility
of combining pairs of di erent gauge theories has given rise to a cornucopia of doublecopy
constructions [3{10, 19, 41{46].
More recently the double copy and colorkinematics duality has been observed to
extend to e ective theories such as the nonlinearsigma model (chiral Lagrangian), (Dirac)
BornInfeld, VolkovAkulov and special galileon theory [12, 47{51]. Likewise, the
double copy and duality show up in novel relations involving and interconnecting
stringand
eldtheory amplitudes and e ective theories [51{61]. The double copy has been
extended beyond perturbation theory to classical solutions involving black holes [62{65] and
gravitationalwave radiation [66{69].
In the work by Ochirov and one of the current authors [6] a detailed prescription was
given for removing unwanted axiondilatonlike states that appear in a naive doublecopy
construction of looplevel amplitudes of pure N = 0; 1; 2; 3 supergravities in D = 4
dimensions. The prescription calls for the introduction of compensating ghost states that are
obtained by tensoring fundamental matter multiplets of the two gauge theories entering
the double copy. For consistency of the double copy, the kinematic numerators of the gauge
theories need to obey Jacobi identities and commutation relations mirroring the
adjointand fundamentalrepresentation color algebra. The gauge theories are thus di erent
varieties of massless (super)QCD theories with a tunable parameter Nf corresponding the
number of quark
avors. Speci cally, the gravitational matter is turned into ghosts by
choosing Nf =
1 and Nf = 1 for the two gauge theories, respectively. The
prescription was shown to correctly reproduce the oneloop amplitudes in pure N = 0; 1; 2; 3; 4
supergravity.
In this paper, we obtain the twoloop amplitudes in N = 2 superQCD (SQCD) which
are needed for the computation of pure supergravity amplitudes at this loop order. Using
colorkinematics duality we compute the fourvector amplitude at two loops with
contributions from Nf internal hypermultiplets transforming in the fundamental representation.
The amplitudes are valid for any gauge group G and for any dimension D
6. Dimensional
regularization, in D = 4
2 dimensions, requires that the integrands are correct even for
D > 4 dimensions, and D = 6 is the maximal uplift where the theory exists. To de ne the
sixdimensional chiral theory, N = (1; 0) SYM coupled to hypers, we make use of its close
relationship to the maximally supersymmetric N = (1; 1) SYM theory, whose tree
amplitudes are conveniently written down using the sixdimensional version of spinorhelicity
notation [70].
Using the twoloop amplitudes in N = 2 SQCD, which obey colorkinematics
duality, we compute the (N = 2; Nf )
(N = 2; Nf = 1) double copy and obtain twoloop
amplitudes in N = 4 supergravity coupled to 2(Nf + 1) vector multiplets. For the choice
Nf =
1 we obtain pure N = 4 supergravity amplitudes. It should be noted that pure
N = 4 supergravity amplitudes can alternatively be obtained from a (N = 4)
(N = 0)
double copy [19, 41, 43, 45, 71{74], without the need of removing extraneous states.
Indeed, the twoloop fourpoint N = 4 supergravity amplitude was rst computed this way
in ref. [41], and with additional vector multiplets in ref. [73].
{ 3 {
Our construction of the halfmaximal supergravity amplitudes has several crucial
virtues compared to previous work. Obtaining the N = 2 SQCD amplitude in the process
is an obvious bonus; this amplitude is a stepping stone for computing the pure N = 2; 3
supergravity twoloop amplitudes, as explained in ref. [6]. Furthermore, in D = 6 the chiral
nature of the N = (1; 0) SYM theory allows for the doublecopy construction of two
inequivalent halfmaximal supergravity amplitudes: N = (1; 1) and N = (2; 0) supergravity.
Only the former is equivalent to the (N = 4)
(N = 0) construction. The latter theory,
pure N = (2; 0) supergravity, has a chiral gravitational anomaly that shows up at one
loop, thus rendering the doublecopy construction of the twoloop amplitude potentially
inconsistent in D = 6. However, it is known that the anomaly can be canceled by adding
21 selfdual N = (2; 0) tensor multiplets to the theory [75]. This corresponds to Nf = 10
in the current context.
We extract the ultraviolet (UV) divergences of the halfmaximal supergravity twoloop
integrals in D = 4 and D = 5. In D = 4 the integrals are manifestly free of divergences
by inspection of the power counting, whereas in D = 5 the individual integrals do diverge
but after summing over all contributions to the amplitude divergences cancel out; thus
exhibiting socalled enhanced UV cancellations [44, 74]. This nontrivially agrees with the
calculations of the UV divergence of the same amplitudes in ref. [73], and thus provides
an independent crosscheck of our construction. While we do not compute the D = 6
2
already at one loop, thus the twoloop amplitude has both 1= 2 and 1= poles [73].
divergences, we note that the pure N = (1; 1) theory should have a divergence starting
This paper is organized as follows: in section 2, we review colorkinematics duality
and the double copy with special attention on the fundamental matter case, and pure
supergravities. In section 3, we introduce the tree amplitudes and needed twoloop cuts for
N = 2 SQCD both in four and six dimensions. In section 4, the details of the calculation
of the twoloop numerators of SQCD are presented. In section 5, the corresponding
supergravity amplitudes are assembled and the UV divergence in
ve dimensions is computed.
Conclusions are presented in section 6 and the explicit N = 2 SQCD numerators and color
factors are given in appendix A.
2
Review
Here we review colorkinematics duality in supersymmetric YangMills (SYM) theory and
its extension to fundamental matter, following closely the discussion in ref. [6]. We also
show how the duality may be used to compute amplitudes in a wide variety of
(super)gravity theories using the double copy prescription.
2.1
Colorkinematics duality with fundamental matter
We begin by writing Lloop (supersymmetric) YangMills amplitudes with fundamental
matter as sums of trivalent graphs:
Am
(L) = iL 1gYmM+2L 2
X
cubic graphs i
Z
dLD` 1 nici ;
(2 )LD Si Di
(2.1)
{ 4 {
where gYM is the coupling. The structure of the gauge group G allows two types of
trivalent vertices: pureadjoint, and those with a particle in each of the adjoint, fundamental
tr(T aT b) = ab. Diagrammatically, these can be represented as
and antifundamental representations. The color factors ci may thus be expressed as
products of structure constants f~abc = tr([T a; T b]T c) and generators Tia, normalized such that
f~abc = c c
;
Tia = c a
¯
i
:
Si and Di are the usual symmetry factors and products of propagators respectively. Finally,
kinematic numerators ni collect kinematic information about the graphs.
HJEP09(217)
The algebraic structure of the gauge group gives rise to linear relationships between
the color factors. In the adjoint representation these are Jacobi identities between the
structure constants,1
f~a1a2bf~ba3a4 = f~a4a1bf~ba2a3
f~a2a4bf~ba3a1 ;
4
3
1
2
c
4
3
and in the fundamental representation they are commutation relations,
Ti1{2 f~ba3a4 = Tia13Tja{42
b
Tia14Tja{32 = [T a3 ; T a4 ]i1{2 ;
1
2
:
1
2
;
c
c
4
3
4
3
a
b
1
2
= c
4
3
1
2
= c
1
2
c
4
3
ci = cj
ck:
The identities lead to relationships between the color factors of the form
Furthermore, the ip of two legs at a pureadjoint vertex leads to an overall sign ip,
f~abc =
f~acb. The same behavior can be implemented when swapping fundamental with
antifundamental legs by de ning new generators T{aj =
Tja{.
Kinematic numerators ni satisfying colorkinematics duality  we shall refer to these
as colordual  obey the same linear relations as their respective color factors. We
demand that
ni = nj
ni !
nk
ni
,
ci = cj
ci !
ci:
ck;
The second identity incorporates a sign change for a ip of two legs at a given cubic vertex.
The existence of a set of colordual numerators is generally nontrivial. At tree level,
their existence has been proven in Ddimensional pure (super)YangMills theories using
an inversion formula between numerators and colorordered amplitudes [76, 77].
This
relies on the partial amplitudes satisfying BCJ relations [2, 13{15]. For (super)YangMills
theories with arbitrary fundamental matter, where less is known, the corresponding BCJ
relations [6] have been proven for QCD [28].
1This may also be viewed as a commutation relation in the adjoint representation (Taadj)bc = f~bac.
{ 5 {
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
When including matter multiplets it is also useful to consider the twoterm identity
c
4
3
1
2
=? c
4
3
1
2
(2.7)
for indistinguishable matter multiplets. Depending on the gaugegroup representation, this
may or may not hold. Although fundamental representations of generic gauge groups do
not obey this relation, the generator of U(1) does, as well as generators of particular tensor
representations of U(Nc). It turns out that it is possible to nd colordual numerators that
ful ll these twoterm identities. However, the twoterm identities are not necessary for the
doublecopy prescription [3] that we will now discuss.
2.2
Pure supergravities from the doublecopy prescription
Our main motivation for nding colordual numerators of gaugetheory amplitudes is the
associated doublecopy prescription: this o ers a simple way to obtain gravity amplitudes
in a variety of (supersymmetric) theories. The inclusion of fundamentalmatter multiplets
widens the class of gravity amplitudes obtainable in this way; in particular, it enables the
doublecopy construction of pure N < 4 supergravity amplitudes, including N = 0 Einstein
gravity [6]. In this paper we will mostly be interested in pure N
= 4 (halfmaximal)
supergravity at two loops; however, the approach that we now review also applies more
generally.
The basic setup is as follows. Take a pair of N ; M
2 SYM multiplets, VN and
VM0, both of which are nonchiral  for instance, the former should include 2N chiral and
2N antichiral onshell states. Then analyze their double copy: one recovers a factorizable
graviton multiplet of the form
eq. (2.8) one assigns opposite statistics to
X, X in eq. (2.9) are ghosts.
N (but not to 0M), which in turn means that
In this paper N = M = 2; the gaugetheory matter multiplets are hypermultiplets
(hypers). The complex matter
N (and the conjugate
N
) belong to the fundamental
(antifundamental) representation of the gauge group G. In order to cancel the matter in
The usual numerator double copy is modi ed when concealing unwanted matter states.
Starting from a colordual presentation of the SYM amplitude (2.1) one replaces the color
factors ci by numerators of the second copy as usual. However, one also now includes
an appropriate sign that enforces the opposite statistics of internal hyper loops, and the
numerator is conjugated (reversing matter arrows),
ci ! ( 1)jijn0i ;
{ 6 {
where jij is the number of matter loops in the given diagram. It is useful to generalize
the above replacement by promoting ( 1) be a tunable parameter, Nf , which counts the
number of fundamental matter multiplets in the rst SYM copy (setting Nf = 1 in the
second copy),
ci ! (Nf )jijn0i :
In the supergravity theory, the number of complexi ed matter multiplets is then NX = (1+
Nf ) where, depending on the amount of supersymmetry, X is either a complexi ed vector,
hyper, chiral multiplet or complex scalar. For instance, when Nf = 0 the factorizable
double copy is recovered (we take 00 = 1).
After the double copy is performed, we can assemble terms on common denominators
and thus consider (super)gravity numerators given by
where all numerators sharing the same propagator Di are summed over, i.e. all possible
routings of internal vector and complex matter multiplets. The bar on top of the second
numerator implies a reversal of hyper loop directions, i.e. conjugating the hypermultiplets.
It is su cient that only one of the two numerator copies, ni or n0i, is colordual. We are
left with complete (super)gravity amplitudes:
Mm
(L) = iL 1
2
m+2L 2
X
consists of N = 2 SYM coupled to supersymmetric matter multiplets (hypermultiplets) in
the fundamental representation of a generic gauge group G. For simplicity, we consider the
limit of massless hypers.2 N = 2 SQCD's particle content is similar to that of N = 4 SYM,
so we refrain from describing its full Lagrangian; instead, we project its tree amplitudes out
of the wellknown ones in N = 4 SYM. We then proceed to calculate generalized unitarity
cuts. As our intention is to evaluate cuts in D = 4
2 dimensions, we also consider
sixdimensional trees. These are carefully extracted from the N = (1; 1) SYM theory 
the sixdimensional uplift of fourdimensional N = 4 SYM.
The onshell vector multiplet of N = 4 SYM contains 2N = 16 states:
VN =4( I ) = A+ + I I+ +
2
1 I J 'IJ +
1
3! IJKL I J K L + 1 2 3 4A ;
(3.1)
where we have introduced chiral superspace coordinates I with SU(4) Rsymmetry indices
fI; J; : : :g. The N = 4 multiplet is CPT selfconjugate and thus it is not chiral. The
2Amplitudes with massive hypers can be recovered from the massless case by considering the D = 6
version of the theory and reinterpreting the extradimensional momenta of the hypers as a mass.
{ 7 {
VN =2( I ) = VN =2 + 3 4
V N =2;
VN =2( I ) = A+ + I I+ + 1 2'12;
V N =2( I ) = '34 + I34J I J + 1 2A ;
VN =4 = VN =2 +
N =2 +
N =2;
N =2( I ) = ( 3+ + I 'I3 + 1 2 4 ) 3;
N =2( I ) = ( 4+ + I 'I4
where the SU(2) indices I; J = 1; 2 are inherited from SU(4). As is obvious, this is a subset
of the full N = 4 multiplet:
where we have introduced a hypermultiplet and its conjugate parametrized as
apparent chirality is an artifact of the notation; it can equally well be expressed in terms
of antichiral superspace coordinates I :
VN =4( I ) = 1 2 3 4A+ +
+ A :
The onshell spectrum of N
= 2 SYM contains 2
2N = 8 states, which can be
packaged into one chiral VN =2 and one antichiral V N =2 multiplet, each with four states.
As explained in ref. [6], it is convenient to combine them into a single nonchiral multiplet
VN =2 using the alreadyintroduced superspace coordinates:
One can easily check that these elements make up the full N = 4 spectrum (3.1).
The particle content of N
= 2 SQCD is the same as that of ordinary N
= 4
SYM, except that the hypermultiplet
N =2 ( N =2) transforms in the fundamental
(antifundamental) representation of the gauge group G. We also generalize to Nf =
avors
by attaching
avor indices f ; ; : : :g to the hypers, i.e. ( N =2) , ( N =2) . The di erent
representation a ects the color structure of the amplitudes and the avor adds more
structure; however, when considering colorordered tree amplitudes with identically avored
hypers the distinction between N = 2 SQCD and N = 4 SYM is insigni cant. The former
amplitudes are obtainable from the latter by decomposing according to the above N = 2
multiplets, as we shall now demonstrate. For multi avor N = 2 SQCD tree amplitudes
the procedure to extract them from the N = 4 ones is more complicated, although some
tricks have been developed for this purpose [78, 79].
3.1
Four dimensions
SYM [80, 81]:3
To calculate fourdimensional tree amplitudes we start from the colorstripped
ParkeTaylor formula for maximallyhelicityviolating (MHV) scattering of n states in N = 4
hVN =4VN =4
VN =4iMHV = i
8(Q)
h12ih23i
hn1i
;
(3.6)
3Here we adopt the usual spinorhelicity notation, see for example [82, 83].
{ 8 {
(3.2)
(3.3a)
(3.3b)
(3.3c)
(3.4)
(3.5a)
(3.5b)
where for N supersymmetries the supersymmetric delta function is
2N (Q) = Y
Xhi ji iI jI :
N
n
I=1 i<j
By inspection of the hypers (3.5),
N =2 states are identi ed by a single 3 factor and
states carry
4. In the full N = 2 multiplet VN =2 (3.3) a factor of 3 4 identi es a state
belonging to V N =2; anything else belongs to V
N =2. This enables us to project out the
relevant MHV amplitudes in N = 2 SQCD, giving colorordered amplitudes
where the ellipsis represents insertions of VN =2. We will only use tree amplitudes with up
to ve external legs, so MHV and MHV are su cient.
3.2
Six dimensions
The sixdimensional uplift of fourdimensional N = 4 SYM is N = (1; 1) SYM; an onshell
superspace for this theory was introduced by Dennen, Huang and Siegel (DHS) [84]. The
DHS formalism uses a pair of Grassmann variables, a and ~a_ , carrying littlegroup indices
in the two factors of SU(2)
SU(2)
SO(4). The full nonchiral onshell multiplet is
VN =(1;1)( a; ~a_ ) =
+
a
a + ~a_ ~a_ + 0( )2 + gaa_ a ~a_ + 0(~)2
+ ~a_ ( )2 ~a_ +
a a(~)2 + ( )2(~)2;
(3.9)
change the handedness of (anti)chiral spinors a ( ~a_ ).
where ( )2 = 12 a a, (~)2 = 12 ~a_ ~a_ . Note that in six dimensions CPT conjugation does not
3.2.1
Fourdimensional correspondence
The correspondence between fourdimensional N
= 4 SYM and sixdimensional N
=
(1; 1) SYM is most easily seen using a nonchiral superspace construction of the N = 4
multiplet [85]. Two chiral and two antichiral superspace coordinates are used to capture
N = 4 SYM's 16 onshell states: we choose 1
superspaces is done with halfFourier transforms:
, 2
, 3 and
4
. Switching between the
Z
chiral( I ) =
d 2d 3 e 2 2+ 3 3
nonchiral( I=1;4; I=2;3):
In terms of the nonchiral superspace, a valid mapping to six dimensions is [86]
a $ ( 1; 2);
~
a_
$ ( 3; 4):
{ 9 {
(3.7)
N =2
(3.8a)
(3.8b)
(3.8c)
(3.10)
(3.11)
A+
ψ+
I
ϕIJ
ψI
−
A
−
η
I
ηJ
ηK
ηL
η˜
g21˙
χ˜ ˙
1
¯
χ˜ ˙
1
HJEP09(217)
(3.12)
and ~ act as raising and lowering operators in the weight space [87] as
denoted by solid black arrows. Redcolored states belong to the vector multiplet; green and
bluecolored states belong to the two parts of the D = 6, N = (1; 0) hypermultiplet. The dotted gray
arrows show how 6D states are projected to the fourdimensional weight space.
This allows us to identify onshell states between the fourdimensional N = 4 and
sixdimensional N = (1; 1) multiplets. We do this for the N = 2 vector (3.3) and matter (3.5)
multiplets separately; the results are illustrated in
gure 1.
Starting with the N
= 2 vector multiplet (3.3), we switch to the fourdimensional
nonchiral superspace and then use the coordinate mapping (3.11):
Z
VN =2( a; ~a_ ) =
d 3d 2e 2 2+ 3 3
VN =2( I )
= ~1_ ( 2+
1'12 + 2A+
( 1; 2)! a;( 3; 4)!~a_
_
1 2 1+) + ~2(
1
1A
+ 2'34
This clearly forms part of VN =(1;1): for instance,
2+ = ~1_ and '12 = g11_ . As it contains
a sixdimensional vector and a pair of antichiral spinors, we identify it as the onshell
multiplet of N = (1; 0) SYM [88]:
Hypermultiplets are transformed by the same procedure:
VN =(1;0)( a; ~a_ ) = ~a_ ~a_ + gaa_ a ~a_ + ~a_ ( )2 ~a_ :
N =2( a; ~a_ ) = '23 + 1
N =2( a; ~a_ ) = ~1_ ~2_ ('24
4
1
+
Each of these contains a chiral spinor and a pair of scalars, so we recognize them as the
two parts of the N = (1; 0) hypermultiplet:
N =(1;0)( a; ~a_ ) =
N =(1;0)( a; ~a_ ) = 0(~)2 +
+
a
a + 0( )2;
a
a(~)2 + ( )2(~)2:
The three multiplets make up VN =(1;1) (3.9) as expected. Notice that the above procedure
was not unique: we could instead have chosen to identify VN =2 with VN =(0;1), which carries
the chiral spinors
a and
a
. The choice was made when we picked the speci c mapping
between four and sixdimensional states (3.11).4
Amplitudes in this theory are given in terms of the sixdimensional spinorhelicity
formalism [70]. For each particle i they depend on the chiral and antichiral spinors, i and ~i,
and the supermomenta,
q
iA = i
A;a
i;a;
~qi;A = ~i;A;a_ ~i :
a_
(no sum over i)
(3.16)
HJEP09(217)
The indices fA; B; : : :g belong to SU*(4)  the spin group of SO(1,5) Lorentz symmetry.
For notational convenience we de ne the two fourbrackets:
hia; jb; kc; ldi
[ia_ ; jb_ ; kc_; ld_]
ABCD i
j
A;a B;b C;c D;d ;
k
l
ABCD ~i;A;a_ ~j;B;b_ ~k;C;c_ ~l;D;d_ :
The fourpoint colorordered tree amplitude of N = (1; 1) SYM is [84]
hVN =(1;1)VN =(1;1)VN =(1;1)VN =(1;1)i
D=6 =
i
4(Pi qi) 4(Pi ~qi) ;
st
where s = (p1 + p2)2 and t = (p2 + p3)2; the fermionic delta functions are
4
4
X qi =
X ~qi =
i
i
1
1
X
4! i;j;k;l
X
4! i;j;k;l
hia; jb; kc; ldi i;a j;b k;c l;d;
ia_ ; jb_ ; kc_; ld_ ~ia_ ~jb_ ~kc_ ~ld_:
This amplitude is invariant under qi $ ~qi  a re ection of the nonchiral nature of
N = (1; 1) SYM. In six dimensions there is no notion of helicity sectors (like NkMHV
sectors in four dimensions) because rotations of the little group SO(4)
SU(2)
SU(2)
mix them. The above superamplitude is therefore a single expression that uni es all the
fourdimensional helicity sectors.
We can now write down sixdimensional equivalents of the fourdimensional N = 2
SQCD trees (3.8). States are identi ed using the ~a_ variables: we project these out of
4(Pi ~qi) in the fourpoint amplitude (3.18). With four external legs the colorordered
amplitudes are
hV1V2V3V4i
h 1 2; V3V4i
h 1 2;
3;
D=6 =
D=6 =
4i
D=6 =
i
i
i
st
st
4(Pi qi) [1a_ ; 2b_ ; 3c_; 4d_]~1a_ ~
2b_ ~3c_ ~4d_;
4(Pi qi) [21_ ; 22_ ; 3c_; 4d_]~21 ~2 ~
_ 2_ 3c_ ~4d_;
st
4(Pi qi) [21_ ; 22_ ; 31_ ; 32_ ]~21 ~2 ~
_ _ _ _
2 31 ~32;
4For instance, in ref. [85] the states are instead identi ed as a $ ( 1; 4), ~a_ $ ( 3; 2).
(3.17a)
(3.17b)
(3.18)
(3.19a)
(3.19b)
(3.20a)
(3.20b)
(3.20c)
where f ; ; : : :g are avor indices. These amplitudes are not invariant under qi $ ~qi: our
use of the VN =(1;0) multiplet implies that ~qi encodes information about the external states.
In principle, we will also need
vepoint amplitudes when computing threeparticle cuts of
the twoloop amplitude. However, the vepoint N = (1; 1) SYM tree amplitude [84] is less
compact and, as we will discuss in section 4.3, it turns out that these cuts do not provide
any new physical information compared to the fourdimensional ones given the constraints
on our ansatze.
When the momenta are taken in a fourdimensional subspace the sixdimensional trees
should reproduce their fourdimensional counterparts (3.8). This is easily con rmed: write
the sixdimensional spinors in terms of fourdimensional ones [70],
a =
m = `4
i `5;
me = `4 + i `5:
5For a good review of both four and sixdimensional unitarity techniques see ref. [90].
where here the Greek letters denote SL(2,C) spinor indices (not to be confused with the
previouslyintroduced
avor indices, the distinction should be clear from the context).
Then substitute into the sixdimensional tree amplitudes (3.20). Finally, switch to
fourdimensional fermionic coordinates using the halfFourier transform (3.10) and the state
mapping (3.11) on each of the external legs. A helpful intermediate relationship is [85]
Z
4
i=1
Y d i;2 exp
4
X
i=1
i;2 i
2
2
4
4
X qi
i
3
a!( 1; 2)5 =
~a_ !( 3; 4)
h34i
which exchanges the fermionic delta functions. This procedure gives all external helicity
sectors in four dimensions, but at four points only the MHV is nonzero.
3.3
Cuts
In the next section we will use ansatze to construct colordual numerators for N = 2 SQCD.
The physical input for these will come from unitarity cuts. In four dimensions the cuts
are calculable using the MHV tree amplitudes (3.8)  the techniques involved are fairly
standard and we will not repeat them here. In six dimensions we follow ref. [89]; however,
the tree amplitudes (3.20) create some new subtleties that we will now address.5
As we always take external states in a fourdimensional subspace, the key to this
approach is nding sixdimensional spinor solutions for the onshell loop momenta in terms of
fourdimensional external spinors. We write the sixdimensional massless onshell condition
as a fourdimensional massive condition:
`2 = `
2
(`4)2
(`5)2 = `
2
m me = 0;
where ` is the part of ` living in the fourdimensional subspace. The complex masses are
related to the fth and sixth components of `:
(3.21)
(3.22)
(3.23)
(3.24)
Now considering ` as a massive onshell fourmomentum we write it in terms of two pairs
of fourdimensional spinors, ( ; ~) and ( ; ~), as:
` _ =
~ _ +
mm
e
2
~ _ :
Appropriate sixdimensional spinors are then [87]
A
a =
im
e
h i
~ _
By taking `4 = `5 = 0, and therefore m = m = 0, we recover the fourdimensional
e
We now illustrate the cutting method by calculating the oneloop quadruple cut in gure 2.
Without loss of generality we set Nf = 1; powers of Nf can be restored at any point by
counting the number of hyper loops. The quadruple cut integrand is then
Cut
1
2
=
Z
2
i=1
Y d2 li d2 ~li (l2
p3)2A(0)
L
(l1
p1)2A(R0):
(3.27)
4
3
1
2
i
i
s
(3.25)
(3.26)
HJEP09(217)
(3.28a)
(3.28b)
(3.29)
A(0) =
L
A(R0) =
s(l2
s(l1
4(Pi2L qi)
4(Pi2R qi)
p3)2 [(l1)1_ ; (l1)2_ ; 3c_; 4d_]~l1 l1 3c_ ~4d_;
1_ ~2_ ~
p1)2 [(l2)1_ ; (l2)2_ ; 1a_ ; 2b_ ]~l2 l2 1a_ ~2b_ ;
1_ ~2_ ~
and, with ` = l1 = p1 + p2 + l2, the onshell conditions are `2 = (`
p1)2 = (`
p1
p2)2 =
(` + p4)2 = 0. Putting the pieces together, and summing over the supersymmetric states,
we obtain the cut for internal hypers:
Cut
4
3
=
4(Pi qi) [(l2) 1_; (l2) 2_; 1a_ ; 2b_ ][(l1)1_ ; (l1)2_ ; 3c_; 4d_]~1a_ ~
2b_ ~3c_ ~4d_:
This expression can now be converted to external spinors and loop momenta using the
previouslygiven expressions. The full details on how to work this out may be found in
ref. [89]; we simply quote the nal results below.
The external states are purely fourdimensional, and thus through the halfFourier
transform (3.10) we may split up the cut into the contributions from sectors of di erent
powers of the fourdimensional Grassmann variables iI
. Unlike the maximallysupersymmetric
case, the N = 2 SYM loop amplitudes will not have automaticallyvanishing integrands
when all external states are picked from the chiral vector multiplet VN =2  thus, besides
the MHV sector we will also have an \allchiral" helicity sector (as well as an
\allantichiral" sector that is trivially obtained by CPT conjugation). A further sector would be
Instead of starting from the (rather complicated) threepoint sixdimensional amplitudes
we study factorization limits of the simpler fourpoint amplitudes. Using (3.20b),
taken on shell. To compute it, we rst write it as a twoparticle cut involving a pair of fourpoint
tree amplitudes  as indicated by the transparent blobs  and then we multiply by the inverse
poles exposed inside these blobs. The quadruple cut limit is then nite.
the double copy.
The cut (3.29) then becomes
the onechiralvector case (and its CPT conjugate), but we will not consider it here. All
sectors are consistent with N = 2 supersymmetry, but the allchiral integrand will only
give rise to O( ) contributions in D = 4
2 dimensions. Nevertheless, the integrand of the allchiral sector is important as it gives nonvanishing supergravity amplitudes through We will come back to the allchiral case shortly, but for now consider the MHV sector.
4
CutMHV 3
1
2
= mm
e
s
where we have introduced the variables
ij =
h12ih34i
to identify contributions from external legs i, j belonging to the antichiral vector multiplet
V N =2.
The sixdimensional cut contains all the information to be formally reinterpreted as
a Ddimensional cut. Strictly speaking, it is only valid for D
6 since the theory does
not exist beyond six dimensions, but we may write it in a dimensionallyagnostic form.
Parametrizing the dimension as D = 4
2 , the loop momentum is conveniently split
into the four and extradimensional parts as ` = ` + . As external momenta are pi are
purely fourdimensional, we expect that
appears in the Lorentzinvariant combination
agrees with the Ddimensional fundamental box numerator found in ref. [6].
2 =
> 0. Indeed, we may eliminate m and me in favor of 2 = mem. The cut then
Now consider the allchiral sector, which is identi ed by all external states belonging
to the chiral VN =2 multiplet. In this case the cut (3.29) becomes
(3.30)
(3.31)
(3.32)
4
Cutallchiral 3
1
2
=
[12][34]
h12ih34i
m2 4(Q):
propagators are understood as being taken on shell.
Interestingly, rotational symmetry in the extradimensional direction has been broken.
Naively, we cannot eliminate the complex variable m2 in favor of the rotational invariant
2 (as we did in the MHV sector). This is because the external states probe the
extradimensional space and pick out a chiral direction. Indeed, the component amplitudes in
the allchiral sector always have either scalars or fermions (of the vector multiplet) on
the external legs, and these particles are secretly six dimensional (e.g. the scalars are the
extradimensional gluons). There is no allgluon amplitude in the allchiral sector, and
thus there is no component amplitude in this sector that involves only fourdimensional
external states.
We can avoid the issue with complex momenta by instead choosing to work with a
vedimensional embedding corresponding to m = me. This should be su cient to parametrize
the physical content of the dimensionallyregulated theory. In ve dimensions the spinors
belong to USp(2,2)
SO(1,4), which means that they can be made to satisfy a reality condition (see e.g. recent work [91]):
2 is an element of the sixdimensional Cli ord algebra. This e ectively
identi es the littlegroup SU(2) labels a and a_. One can then write
2 = m2 = me2, which
The twoloop cuts that we wish to calculate are displayed in gures 3 and 4. As the
oneloop feature of complex extradimensional momenta related to chiral states remains, at
two loops we choose to keep all the degrees of freedom of the sixdimensional momenta.
The integrands presented will therefore be unavoidably chiral. We will continue to use the
complex m and me variables, which we generalize to mi and mei for di erent loop momenta
`i = `i + i. These are related to the extradimensional parts of the loop momenta by
ij = me(imj);
( i; j ) = i me[imj];
(3.34)
where ij =
i
j > 0. Again, for the fourpoint amplitude the issue of complex
momenta is only relevant to the allchiral sector, while the MHV sector will also have a
chiral dependence introduced through the antisymmetric combination ( 1; 2). As we
exposed propagators are understood as being taken on shell. Notice that cuts (b) and (c) di er by
the routing of fundamental hypermultiplets on the righthand side.
shall see, a nonchiral Ddimensional amplitude will emerge in halfmaximal supergravity
when the double copy is performed between the amplitudes of a chiral and an antichiral
gauge theory.
The procedure for calculating the twoloop cuts is exactly the same as we have outlined
at one loop. A minor distinction is that we avoid excessive analytical manipulations of the
cuts: for the purposes of tting an ansatze in the next section we found it su cient to
compute the cuts numerically on various phasespace points.
4
Calculation of N
= 2 SQCD numerators
In this section we describe the computation of colordual N = 2 SQCD numerators. All
external states are taken from the vector multiplet VN =2 (3.3). Without loss of generality,
we consider colordual numerators with a single hypermultiplet avor (Nf = 1); powers of
Nf are straightforward to restore at the end by counting the number of hyper loops in a
given diagram. We compute both the allchiralvector and MHV externalhelicity sectors,
and as the discussion for the two sectors is mostly the same, we present them alongside
each other. The full list of diagrams involved in our allchiral and MHV solutions is given
in
gure 5. Expressions for their numerators are presented in section 4.5 and appendix A
respectively. The solutions are also provided in ancillary les attached to the arXiv version
of this paper.
In the MHV sector we provide two alternative solutions. The rst of these includes
nonzero numerators corresponding to bubbleonexternalleg and tadpole diagrams  diagrams
(17){(24) in
gure 5. All of these diagrams have propagators of the form 1=p2
1=0 that
are illde ned in the onshell limit (unless amputated away). Their appearance in the
solution follow from certain desirable but auxiliary relations that we choose to impose on
the colordual numerators, making them easier to nd through an ansatz. As we shall
(10)
(15)
(20)
HJEP09(217)
(1)
(6)
(11)
(2)
(7)
(12)
(16)
(17)
(18)
(19)
(21)
(22)
(23)
(24)
The eight master graphs that we choose to work with are (1){(5), (13), (19) and (22).
tadpoles and external bubbles are dropped from the nal amplitudes it is useful to consider them
at intermediate steps of the calculation.
see, contributions from these diagrams either vanish upon integration or can be dropped
because the physical unitarity cuts are insensitive to them. However, as they potentially
can give nonvanishing contributions to the ultraviolet (UV) divergences in N = 2 SQCD,
we also provide an alternative colordual solution in which such terms are manifestly absent.
The two solutions may be found in separate ancillary les.
To aid the discussion we generally represent numerators pictorially (as we have already
done for color factors and cuts). For example, the doublebox numerator is represented as
n1(1234; `1; `2) = n
4 ←ℓ2ℓ1→ 1
3
2
:
(4.1)
Similar correspondences are made for the other diagrams in gure 5.
(3)
(8)
(13)
4.1
We begin by identifying a basis of master numerators for twoloop fourpoint diagrams and
for external vector multiplets. All other numerators, called descendants, are expressed as
linear combinations of the masters using Jacobi identities. For instance, doubletriangle
numerators are given as di erences of double boxes:
Considering the combined equation system of all all Jacobi (2.3) and commutation (2.4)
relations, it is straightforward to
nd eight masters: two with no internal matter content,
three with a single matter loop and three with two matter loops. The same choice is made
for both the allchiral and MHV sectors: these are speci ed in gure 5. Our task is to
calculate expressions for the numerators of these diagrams.
The reduction of a given numerator to a linear combination of masters is generally not
unique. Additional numerator relations therefore give a rst set of consistency constraints
for the masters. Further constraints come from (i) the requirement that all numerators
satisfy the symmetries of their color factors and (ii) that the overall integrand matches the
unitarity cuts given in
gures 3 and 4. In the MHV sector, we satisfy these requirements
by
nding suitable ansatze for the master numerators and solving for constraints on the
relevant coe cients. In the allchiral sector this is unnecessary: the system is su ciently
simple that the solution can simply be postulated by knowledge of the maximal cuts.
4.2
Ansatz construction
Our procedure for constructing ansatze for the eight MHVsector master numerators is
a straightforward twoloop extension of the discussion in ref. [6]. The ansatz for each
master numerator is of the same form. The basic building blocks include the Mandelstam
invariants sij = (pi + pj )2 together with contractions of loop momenta `i (i = 1; 2) and
external momenta pi (i = 1; : : : ; 4). As in section 3.3.1 we decompose Ddimensional loop
momenta into their four and extradimensional parts as `i = `i + i. The following set of
kinematic variables is then su cient:
M = sij ; `is; `it; `iu; `i `j ; ij ;
where
`
t
i
ij =
i
j :
`
s
i
2`i (p1 + p2);
u
i
2`i (p2 + p3); `
2`i (p3 + p1);
Note that M contains 14 independent objects after accounting for u =
s
t, where
s = s12, t = s23 and u = s13. We also include contractions with the fourdimensional
LeviCivita tensor:
(4.3)
(4.4)
=
(1; 2; 3; `1); (1; 2; 3; `2); (1; 2; `1; `2); (2; 3; `1; `2); (3; 1; `1; `2) ;
(4.5)
which acts only on the fourdimensional parts of the loop momenta. To be speci c about
normalizations, given some momenta ki, we de ne (k1; k2; k3; k4)
Det(ki ).
The last ingredient is the antisymmetric combination ( 1; 2)  we saw earlier how
it arises from the chiral nature of the N
extradimensional echo of the sixdimensional LeviCivita tensor:
= (1; 0) theory (see section 3.3.2). It is an
where vi are some fourdimensional vectors, such as external momenta and polarizations
(their speci c form is not important since they cancel out in the above ratio). It satis es
HJEP09(217)
We label external states using fourdimensional variables: in section 3 we introduced
variables ij that carry the correct helicity weight for all MHV helicity con gurations (3.31).
The indices i and j label the legs belonging to the antichiral part of the vector multiplet
V N =2, i.e. those containing negativehelicity gluons. With these building blocks, su cient
ansatze for the master numerators are
ni(1234; `1; `2) =
X
The objects cim;jk, dim;jnk and eim;jk are free parameters to be solved for; M (N) denotes the set
of monomials of engineering dimension 2N built from the set M in eq. (4.3):
M (N) =
( N
Y z
i=1
i zi 2 M
)
:
Note that there exist nonlinear relations between the monomials in M (N) and the
LeviCivita objects in . In principle, we could use these to slightly reduce the size of the ansatz;
however, the bene t in doing so is marginal. Furthermore, working with an overdetermined
set of building blocks is helpful when searching for a compact form of the integrand.
Finally, we should comment on the fact that the kinematic numerators we choose to
work with have poles of the form
jk=sj2k. Because we choose to work with gaugeinvariant
building blocks (as opposed to polarizationdependent numerators) the price to pay is
some nonlocality in the numerators.6 While the speci c form of the nonlocality is not a
priori obvious we nd that the observation of ref. [6] carries over to two loops: the MHV
numerators have at worst jk=sj4k N poles, where N is the number of supersymmetries.
4.3
Symmetries and unitarity cuts
Graph symmetries, or automorphisms, can be used to obtain relations between
numerators with di erent externalleg orderings and di erently parametrized loop momenta. For
6One reason for this is that the number of gaugeinvariant local functions one can write down is strongly
dependent on the engineering dimension. If one wants to have access to su ciently many of these while
keeping the overall dimension small one is forced to compensate with momentum factors in the denominator.
!
(4.7)
(4.8)
and promoting them to ghosts. More generally, by using Nf hypermultiplet avors in one
of the gaugetheory copies (and Nf = 1 for the other copy), i.e. considering N = 2 SQCD,
this construction is generalizable to NV = 2(1 + Nf ) vector multiplets.
Dimensional regularization requires us to construct the loop integrands in D > 4
dimensions. It is therefore convenient to consider a sixdimensional uplift of the
fourdimensional double copy (5.15):
HN =(1;1) = VN =(1;0)
VN =(0;1);
where HN =(1;1) is the multiplet of N = (1; 1) supergravity (its onshell particle content is
described in ref. [88]). The precise form of HN =(1;1) will not concern us here; it is enough
for us to know that, upon dimensional reduction to four dimensions, it gives the desired
HN =4 and VN =4 multiplets. A similar sixdimensional interpretation works for the double
copy of hypermultiplets:
It is well known that the uplift of N = 4 supergravity to six dimensions is not unique.
Besides the N = (1; 1) theory, there exists the chiral N = (2; 0) supergravity (see ref. [88]
for further details), which has the following state content in the factorizable double copy:
The extra selfdual tensor multiplet TN =(2;0) can be removed (or more can be added) by
exploiting that the same multiplet appears in the double copy of the matter multiplets:
(5.17)
(5.18)
(5.19)
(5.20)
TN =(2;0) =
N =(1;0)
N =(1;0):
By removing the bar on the second factor we emphasize that the halfhypers should
transform in a pseudoreal representation in the gauge theories. This avoids overcounting the
number of tensor multiplets that are obtained in the double copy.
In terms of the sixdimensional gaugetheory numerators, doublecopying to obtain
pure N = (1; 1)type supergravities is simple: multiply the N = (1; 0) SQCD numerators
by those corresponding to the chiralconjugate theory, obtained by swapping mi $
(which reverses the sign of the LeviCivita invariant ( i; j )). For the N = (2; 0)type
supergravities simply square the N = (1; 0) SQCD numerators. For vedimensional loop
momenta, the two alternative double copies become identical as mi = mei and ( i; j ) = 0.
Since the external states of our numerators carry fourdimensional momenta, they
mei
can be properly identi ed using the fourdimensional double copy (5.15). The chiral and
antichiral supergravity multiplets, H
N =4 and HN =4, are associated with the chiral and
antichiral N = 2 multiplets, VN =2 and V N =2, respectively:
HN =4 = VN =2
VN =2;
HN =4 = V N =2
V N =2:
(5.21)
The cross terms between external states VN =2 and V N =2 give the extra two VN =4 multiplets
that should be manually truncated away in the pure fourdimensional theory.
In terms of the helicity sectors of SQCD the double copy can be performed in each
sector separately since the crossterms between the sectors should integrate to zero. This
vanishing is necessary in order for the gravitational Rsymmetry SU(4) to emerge out of
the Rsymmetry of the two gaugetheory factors SU(2)
SU(2). Thus the allchiral and
MHV sectors of N = 4 supergravity can be isolated by considering colordual numerators
of N = 2 SQCD belonging to the allchiral and MHV sectors respectively. At two loops,
these are the sets of numerators provided in the previous section.
In the allchiral sector the only nonzero oneloop numerators are boxes. The relevant
N = (1; 1)type double copy gives the following halfmaximal supergravity numerator in
generic dimensions:
HJEP09(217)
+ (Ds
numerator in six dimensions:8
where we have replaced Nf using NV = 2(1 + Nf ) and NV = Ds
4. The modulussquare
notation of the numerators indicates multiplication between the chiral and chiralconjugate
numerators: these are related by mi $ mei and ( i; j ) !
states chiral conjugation is the same as complex conjugation). Pure theories in D = 4; 5; 6
( i; j ) (for real external
dimensions can be obtained by setting Ds = D. After plugging in the N
numerators given earlier (4.15) into eq. (5.22), and using
numerator (5.10) of the (N = 0)
(N = 4) construction.
2 = mem, one recovers the box
The N
= (2; 0)type double copy gives the following N
= (2; 0) supergravity box
N [N =(2;0) SG]
4
3
1
2
=
n
4
3
1
2
2
+ (NT
1) n
1
2
2
;
(5.23)
where the modulus square is replaced by an ordinary square and the parameter NT =
1 + 2Nf counts the number of selfdual N = (2; 0) tensor multiplets in the sixdimensional
theory (with NT = 0 being the pure theory). In general, for any diagram and at any loop
order, we can obtain the N = (2; 0) supergravity numerators from the N = (1; 1) ones by
replacing the modulus square with an ordinary square and replacing (Ds 6) with (NT
1),
as exempli ed above.
In the MHV sector the oneloop amplitude of halfmaximal gravity obtained from
SQCD has already been considered by Ochirov and one of the present authors [6]. In this
case the triangle and bubble diagrams will also contribute to the amplitude.
5.2.2
Two loops
At two loops there are more nonzero numerators to consider (see ancillary les for the
assembled supergravity numerators). For instance, the double copy of the doublebox
8The Grassmannodd parameters iI should not be literally squared in eq. (5.23) or eq. (5.22); instead
I
the Rsymmetry index should be shifted i ! i
I+4 in one of the copies.
4
3
4
3
3
4 ←ℓ2ℓ1→ 1 2
+ (Ds
2
=
numerators gives
given earlier (5.13) using ij = me(imj).
They are found using similar double copies:
0
4 ←ℓ2ℓ1→ 1 2
4 ←ℓ2ℓ1→ 1 21
+ n
+ n
3
2
3
2
3
2
where the last two terms are related by relabelling and reparametrization of the states
and momenta. It is a straightforward exercise to show that, in the allchiral sector, this
matches the doublebox numerator (5.14a) found by the (N = 0)
(N = 4) construction:
one assembles the combinations of mi and mei into the extradimensional function F1( 1; 2)
HJEP09(217)
In the allchiral sector, there are only two more nonzero supergravity numerators.
2
;
(5.25)
21
A ;
(5.26)
(5.24)
A ;
(5.27)
21
A ;
(5.28)
n
+ (Ds
ℓ2ւ ℓ1→ 1
3
2
2
+ n 4
ℓ2ւ ℓ1→ 1
3
2
+ n 4
ℓ2ւ ℓ1→ 1
3
2
2
+ (Ds
6)2 n
4 տℓ2ℓ1ր 1 2
3
2
where, again, some of the terms are related by relabelling and reparametrization. In the
allchiral sector, these expressions perfectly match the (N = 0)
(N = 4) double copy
given in eqs. (5.14b) and (5.14c).
Identical formulas apply for the MHVsector supergravity numerators, of which there
0
ℓ1→ 1
2
2
3
+ n 4ℓ2ւ
ℓ1→ 1
2
2
3
+ n 4ℓ2ւ
ℓ1→ 1
2
3
4 3
n ℓ↓2ℓ1ր 1 2
+ (Ds
6)
2
+ n ℓ↓2ℓ1ր 1 2
;
are two more:
ℓ1→ 1
=
2
3
2
N [N =4 SG] ℓ↓2ℓ1ր 1 2
As already explained, we drop the remaining bubbleonexternalleg and tadpole diagrams
since they integrate to zero in dimensional regularization (see section 4.6).
Pure halfmaximal supergravity numerators (including N = (1; 1) but not N = (2; 0))
are obtained by setting Ds = D. The N = (2; 0) supergravity twoloop numerators can
be obtained from the above formulas by replacing the modulus square with an ordinary
square and (Ds
6) with (NT
1).
Enhanced ultraviolet cancellations in D = 5
2 the twoloop pure halfmaximal supergravity amplitude is known to be
UV nite for all external helicity con gurations. This was demonstrated in ref. [72] as
an example of an enhanced cancellation. While a potentially valid counterterm seems to
exist, recent arguments con rm that halfmaximal D = 5 supergravity is
nite at two
loops [72, 99{101]. Using the obtained pure supergravity amplitude (5.2) we con rm this
enhanced UV cancelation. In the allchiral sector, the only UVdivergent integrals are the
planar and nonplanar double boxes, divergences for which may be found in ref. [72]. It
is a simple exercise to show that these contributions, when substituted into the twoloop
supergravity amplitude (5.2), give an overall cancellation.
In the MHV sector, where there is a wider range of nontrivial integrals involved, the
UV calculation provides a useful check on our results. Following the procedure outlined
in ref. [102], we consider the limit of small external momenta with respect to the loop
momenta jpij
j`j j in the integrand (5.2). This is formally achieved by taking pi !
pi
for a small parameter , keeping only the leading term. The resulting integrand is then
reduced to a sum of vacuumlike integrands of the form [103]
1
1
Z dD`1dD`2
i D2 2 ( `21) 1 ( `22) 2 ( (`1 + `2 + k)2) 3
= HD( 1; 2; 3)( k )
2 D 1 2 3 ; (5.29)
where we have introduced a scale k2 to regulate intermediate IR divergences and
HD( 1; 2; 3)
D
2
D
2
2
Coe cients of these integrals depend only on the external momenta. When D = 5
there are no subdivergences to subtract so the leading UV behavior in each integral is at
most O( 1).
`i `j using9
This requires us to eliminate all loopmomentum dependence in the numerators. First
we remove ( 1; 2) using ( 1; 2)2 =
212; any odd powers integrate to zero. All
remaining loopmomentum dependence can then be converted to contractions of the form
`i `j !
`i `j
D
:
(5.31)
Contractions `i `j can then be converted to inverse propagators, which shifts the i indices
in eq. (5.29). Similar reductions hold for higher tensor ranks; the relevant identities follow
from Lorentz invariance.
9This prescription also works for extradimensional components. We write ij = ~ `i `j where ~ is
the extradimensional part of the metric,
=
~ .
We present the MHVsector UV divergences diagram by diagram. Diagrams containing
tadpoles or bubbles on external legs are dropped since they vanish for dimensional reasons
in gravity (the integrals evaluate to positive powers of the momentum or mass used to
regulate the infrared singularities, implying that no logarithmicallydependent UV poles
survive). For the remaining diagrams, considering the numerators of the rst MHV solution
(see appendix A.1), we obtain the following UV divergences:
(29Ds
142)
210
3
Ds)
70
N [N =4 SG] 4ℓ2ւ ℓ1→ 1
3
I
I
1
1
1
(4 )5
5 2
I
I
(4 )5
5 2
212 + 123 + 124 + 223 + 224 + 324 + O( 0);
(18
23Ds)
105
( 122 + 324)
2(4 + 5Ds)
105
( 123 + 124 + 223 + 224)
+ O( 0);
2
2
3
2
2
=
=
=
= O( 0);
=
(5.32a)
(5.32b)
(5.32c)
(5.32d)
(5.32e)
212 + 123 + 124 + 223 + 224 + 324 + O( 0):
These integrals cancel among themselves when substituted into the assembled
amplitude (5.2), and after summing over permutations of external legs. Using the numerators of
the second MHV solution (see appendix A.2), we arrive at the same vanishing result. This
con rms that the UV divergence is absent in D = 5 dimensions.
6
Conclusions and outlook
In this paper we have computed the twoloop integrand of fourvector scattering in N = 2
SQCD in a colordual form. This provides the rst example of colordual numerators in a
twoloop amplitude containing Nf fundamental matter multiplets.
The calculation builds on and extends the previous oneloop work by Ochirov and one
of the present authors [6]. Through the doublecopy construction, the colordual N = 2
SQCD numerators can be recycled into the construction of pure and mattercoupled N = 4
supergravity amplitudes. In particular, we have considered numerators belonging to the
MHV and allchiral sectors of N
= 2 SQCD, where, in the latter, all external states
belong to the chiral VN =2 multiplet (3.3b). The latter are nonzero before (and zero after)
integration, but they are needed as the double copy gives nonvanishing gravity amplitudes
in the allchiral sector. Indeed, this peculiar behavior of the allchiral sector is needed in
order to give correct U(1)anomalous amplitudes in N = 4 supergravity [92], while not
introducing any corresponding anomalies in the N = 2 gauge theory [95].
We found the twoloop numerators by tting ansatze to physical data from generalized
unitarity cuts and utilizing kinematic Jacobi relations and graph symmetries. Furthermore,
we used the fact that there is a close connection between the states of N
and N = 4 SYM after accounting for simple di erences in gaugegroup representation
and
avor. For the colordual numerators of N
= 2 SQCD we identi ed the separate
contributions from the vector multiplets and the fundamental hypermultiplets. We then
constrained them by demanding that appropriate linear combinations of the numerators
sum up to their known N = 4 SYM counterparts. For this to work seamlessly, we amended
the kinematic numerator relations with a speci c (auxiliary) twoterm identity (2.7) valid
in the limit Nf ! 1. This enabled us to nd unique colordual numerators in both the
MHV and allchiral helicity sectors.
Somewhat surprisingly, the twoloop MHV solution found using the above rules
includes nonzero numerators corresponding to bubbleonexternalleg and tadpole diagrams.
Heuristic expectations from power counting of individual diagrams suggest that these
diagrams should be absent [6, 20, 25, 104, 105]; however, this expectation seems not to
be compatible with the twoterm identity. The appearance of cubic tadpoles and external
bubbles is potentially troublesome since they have singular propagators; nevertheless, these
diagrams vanish after integration in the massless theories and can thus be dropped.
Since general expectations from N = 2 supersymmetry suggest that colordual
numerators should exist where the singular bubbleonexternalleg and tadpole diagrams are
absent [6], we searched and found such a solution that di ers only slightly from the rst
one. This second solution does not obey the auxiliary twoterm identity (which is optional
from the point of view of colorkinematics duality) but it has somewhat better
diagrambydiagram and loopbyloop UV power counting than the rst solution. This potentially
makes it better suited for studies of UV behavior in supergravity theories constructed out
of double copies involving N = 2 SQCD.
The need to use dimensional regularization for the amplitudes required us to de ne
the gauge and gravity theories in D = 4
2 dimensions. Speci cally, it is necessary
to precisely know the integrand in D > 4 dimensions: this led us to consider the
sixdimensional spinorhelicity formalism [70] and the corresponding onshell superspace [84].
In six dimensions the N = 2 SQCD theory lifts to a chiral N = (1; 0) SYM theory with
fundamental hypers. The chirality has important consequences: in the allchiralvector
sector we encounter numerators depending on complexi ed extradimensional momenta,
and in the MHV sector it was necessary to introduce ( 1; 2) to account for a dependence
on the sixdimensional LeviCivita tensor.
However, this chiral dependence canceled upon doublecoping chiral and antichiral
sets of numerators corresponding to an N = (1; 0)
N = (0; 1) construction: we
successfully obtained D
6dimensional N = 4 supergravity numerators from the N = (1; 1)
supergravity theory. Alternatively, chiral N = (2; 0) supergravity amplitudes in six
dimensions could be obtained from our numerators using the N = (1; 0)
N = (1; 0) double
SYM
N = (0; 0)
N = (1; 0)
N = (1; 1)
N = (1; 0)
N = (1; 0)
N = (1; 1)
N = (1; 1)
(0; 0)
(0; 0)
(0; 0)
(0; 1)
(1; 0)
(1; 0)
(1; 1)
haba_ b_
Bab
Ba_ b_
HN =(1;0)
TN =(1;0)
HN =(1;1)
HN =(1;1)
HN =(2;1)
HN =(2;2)
TN =(2;0)
(
Bab
Aaa_
)
TN =(1;0)
VN =(1;0)
VN =(1;1)
TN =(2;0)
{
{
{
)
(
Ba_ b_
Aaa_
{
{
VN =(1;0)
VN =(1;1)
pure supergravities in six dimensions for various amounts of supersymmetry. The multiplets are:
graviton H, tensor T , vector V, and hyper/chiral . In the nonsupersymmetric case there are two
natural choices for the matter double copy, leading to either vectors or tensors and scalars; however,
neither perfectly matches the matter content in the V
it is always possible to nd matching matter states in the V
V product. For the supersymmetric cases
V and
products, implying the
latter can be used as ghosts. For the (1,0) and (2,0) supergravities, the matter double copy can be
done using a halfhypermultiplet in a pseudoreal representation [9] (instead of fundamental); this
gives a selfdual tensor multiplet.
copy. In ve dimensions, both constructions reduce to the same expressions corresponding
to amplitudes of D = 5 halfmaximal supergravity. We explicitly calculated the UV
divergences of the twoloop supergravity integrals in D = 5
2 dimensions, and showed that
the 1= poles nontrivially cancel out in the full amplitude.
An important aspect of the double copy when moving between dimensions is that
the precise details for constructing the pure supergravities is sensitive to the dimension
D and chirality of the theory. In four dimensions, we have to subtract two unwanted
internal matter multiplets to arrive at pure supergravity, whereas in six dimensions the
pure N = (1; 1) theory is factorizable (i.e. the double copy of N = (1; 0)
N = (0; 1)
SYM exactly produces the pure theory). In contrast, pure N = (2; 0) supergravity is
nonfactorizable, and thus to obtain it one needs to subtract out exactly one selfdual N = (2; 0)
tensor multiplet from the N = (1; 0)
N = (1; 0) double copy. This is straightforwardly
done for our twoloop numerators using ghosts of type
(and dropping the contributions
of states
to avoid overcount). Similarly, for pure N = (1; 0) supergravity one needs
to subtract out a single N = (1; 0) tensor multiplet from the N = (1; 0)
N = (0; 0) double
copy. Table 1 lists the multiplet decomposition of di erent sixdimensional double copies.
As can be seen from this table, except for the nonsupersymmetric case, the strategy
for obtaining pure sixdimensional supergravities is a direct generalization of the
fourdimensional situation [6]: the states in the matterantimatter tensor product can be used
as ghosts to remove unwanted states in the vectorvector tensor product.
HJEP09(217)
Several extensions of this work are possible. Using the obtained integrands a next step
is to complete the integration of the twoloop amplitudes in N = 2 SQCD and halfmaximal
supergravities in various dimensions. A particularly interesting case to study is the
superconformal boundary point of SQCD, where color and avor are balanced Nf = 2Nc. Both
leading and subleading Nc contributions can be obtained from the current integrand, since
all nonplanar diagrams are included. Previous fourpoint one and twoloop results in this
theory include the works of refs. [106{110].
Finding colordual numerators with furtherreduced supersymmetry is an important
task this would enable calculations of twoloop pure N < 4 supergravity amplitudes. As
these supergravities are not factorizable, a conventional doublecopy approach involving
only adjointrepresentation particles is not possible.
The N
= 4 SYM multiplet and
tree amplitudes could be further decomposed into lowersupersymmetric pieces in order to
provide input for the calculation (see related work [111]). An interesting challenge would
be to obtain twoloop pure Einstein gravity amplitudes using the double copy, removing the
unwanted dilaton and axion trough the ghost prescription of ref. [6], in order to recalculate
the UV divergence
rst found by Goro
and Sagnotti [112{114]. Recently, the role of
evanescent operators for the twoloop UV divergence was clari ed in ref. [115] (see also
refs. [116, 117]); it would be interesting to examine how such contributions enter into a
doublecopy construction of the twoloop amplitude.
Finally, a potentially challenging but rewarding future task is to consider the
threeloop amplitudes in N = 2 SQCD using the same building blocks for the numerators as
used in the current work, and imposing colorkinematics duality. This calculation would
be a stepping stone towards obtaining the pure N < 4 supergravity amplitudes at three
loops, and a natural direction to take if one wants to elucidate the UV behavior of these
theories.
Acknowledgments
The authors would like thank Simon Badger, Marco Chiodaroli, Oluf Engelund, Alexander
Ochirov, Donal O'Connell and Tiziano Peraro for useful discussions and for collaborations
on related work. The research is supported by the Swedish Research Council under grant
62120145722, the Knut and Alice Wallenberg Foundation under grant KAW 2013.0235,
and the Ragnar Soderberg Foundation under grant S1/16. The research of G.M. is also
supported by an STFC Studentship ST/K501980/1.
A
Twoloop MHV solutions
We present two distinct versions of the colordual fourpoint twoloop N = 2 SQCD
amplitude in the MHV sector, manifesting di erent properties and auxiliary constraints on the
numerators. The following shorthand notations are adopted:
i
`i = `i + i ;
2`i (p1 + p2) ;
t
i
ij =
2`i (p2 + p3) ; `
u
i
`3
2`i (p3 + p1) ;
`1 + `2 ;
(A.1)
i
j ;
where pi are fourdimensional external momenta and `i are internal Ddimensional loop
momenta. Without loss of generality we set Nf = 1 in the numerators; powers of Nf can be
restored by counting the number of hyper loops in each diagram. The symmetry factors Si
used in eq. (2.1) are given by the number of permutations of the internal lines that leaves
the graph invariant. The symmetry factor is 2 for the graphs 14, 17, 20 and 23 in gure 5;
all other graphs have symmetry factor 1.
The following two subsections list the numerators for the two MHV solutions discussed
in the paper; the same expressions, together with color and symmetry factors, are also
given in machinereadable ancillary les submitted to arXiv. We also provide a similar le
containing the twoloop allchiral solution discussed in section 4.5:
HJEP09(217)
First twoloop MHV solution: solMHV1.txt,
Second twoloop MHV solution: solMHV2.txt, Allchiral twoloop solution: solAllChiral.txt.
n
+
n1
4 ←ℓ2ℓ1→ 1
3
2
=
n2
n
4 ←ℓ2ℓ1→ 1
3
2
=
A.1
First solution
The rst MHV solution incorporates: matterreversal symmetry (section 4.4.1), twoterm
identities (section 4.4.2) and matching with the N = 4 limit (section 4.4.3). There are 19
nonzero numerators labelled according to gure 5:
2( 13 + 22) ( 12 + 34) + 2i ( 1; 2)( 12
34)
u (`s2)2
(`t2)2 + (`2u)2 + `s1`s3
`t1`t3 + `1 `3
u u
`t3`2u + `t2`3u
+ 2s `1u`3u + (`2u)2 + 2su2
4su(`1 `3 + `22 + 13 + 22)
t $ u
132+u2 24
+ `t3 (1; 2; 3; `1) + `t2 (1; 2; 3; `2) + t (1; 2; `1; `2) 4i( 14t2 23) ;
12( 12 + 34) + i ( 1; 2)( 12
34)
+ u `s1`s2 + (`1u
`t1)(`t2 + `2u) + 2s `1u`2u
2u(`1 `2 + 12)
t $ u
t $ u
+ `2u (1; 2; 3; `1) + u (1; 2; `1; `2) 22ii(( 11u342 24)
134+u2 24
2 =
ℓ1→ 1
3
2 =
ℓ2ւ
ℓ1→ 1
3
2
=
s $ u
s $ t
s $ u
s $ t
n3 n
3
2
13( 12 + 34) i ( 1; 2)( 12
34)
t $ u
t $ u
t2
23);
`s1(`t2 + `2u) (`t1 + `1u)`s2 + 4s`2 `3 12+ 34
2s
`2 `3 + 41s `s1(`t2 + `2u) (`t1 + `1u)`s2 ( 12 + 34)
s2
t2
i ( 1; 2)( 12 34 + 13 24 14 + 23);
2s2 + (`t1 + `1u)`s2
`s1(`t2 + `2u) 4s(`2 `3 + 13 + 22) 12+ 34
2s
t $ u
u2
4i( 14 23)
t2
2u
2t
( 13 + 24)
( 14 + 23)
ℓ2ւ
3
= n
`s1(`t2 + `2u) 4s(`2 `3 + 23)
t $ u
24)
23);
4t2
s2
4 12( 12 + 34) + 4i ( 1; 2)( 12
34)
t $ u
+ u `s1`s2 + (`1u `t1)(`t2 + `2u) + 2s `1u`2u 2u(`1 `2 + 12) 13+ 24
u2
+ (`2u + 2p3 `2) (1;2;3;`1) + 2p1 `1 (1;2;3;`2)
+ (`t2 + 2p3 `2) (1;2;3;`1) + 2p2 `1 (1;2;3;`2)
t2
u2
=
2 12( 12 + 34) 2i ( 1; 2)( 12
34)
`2u (1;2;3;`1) + u (1;2;`1;`2)
t $ u
= n
4 ←ℓ2ℓ1→ 1
3
2
t $ u
u2
t2
23);
n2;
n13 n
ℓ2ℓ1ր 1 2
ℓ2ℓ1ր 1 2
4
4
←ℓ2րℓ1 1
3
n14 n ↓
3 = 4`2 `3( 12 + 34 + 13 + 24 + 14 + 23);
n16 n ↓
3 = 2`2 `3( 12 + 34 + 13 + 24 + 14 + 23);
n17 n 4
2 = 4`2 (`2 p4)( 12 + 34 + 13 + 24 + 14 + 23);
n18 n 4 ←ℓ2րℓ1 1 2 = 2`2 (p4 `2)( 12 + 34 + 13 + 24 + 14 + 23);
3
4 3 = 4`1 `2( 12 + 34 + 13 + 24 + 14 + 23);
ℓ1 1
n23 n ℓ↓2 4ր 3 2 = 8p4 `2( 12 + 34 + 13 + 24 + 14 + 23);
n24 n ℓ↓2 4ր 3 2 = 4`2 p4( 12 + 34 + 13 + 24 + 14 + 23):
A.2 Second solution
Another possible presentation of the twoloop colordual N = 2 SQCD amplitude in the
MHV sector is found by demanding that singular diagrams corresponding to graphs with
tadpoles or bubbles on external legs vanish in the onshell limit. We keep two nonsingular
bubbleonexternalleg graphs (17 and 18): these numerators contain factors of p24 that
cancel the singular 1=p24 propagators coming from the external lines. This solution consists
of 18 nonvanishing colordual numerators:
n1 n
4 ←ℓ2ℓ1→ 1
3
2 =
(`s1 `s 2`12 2`22 + s 6 13 6 22) 12+3 34 + 2i ( 1; 2)( 12 34)
2
`t2 + `2u)(`t2 + `2u) + 2u2(`s1
`s2 2`12 2`22) 75 136+u2 24
t $ u
3
146+t2 23
+ (`1u + 2p3 `2) (1;2;3;`1) + (`2u + 2p1 `1) (1;2;3;`2) + s (1;3;`1;`2) 4i( 1u32 24)
+ `t3 (1;2;3;`1) + `t2 (1;2;3;`2) + t (1;2;`1;`2) 4i( 14t2 23);
4 ←ℓ2ℓ1→ 1
2 =
+" 15u `s1`s2 + (`1u `t1)(`t2 + `2u) + 4u2 `s
2
+ s 8u2 + 30`1u`2u
t $ u
`s1 + 2`12 + 2`22
#
1630+u224
1640+t223
t $ u
2
n4 n 4ℓ2ւ
3
2 =
n5 n 4ℓ2ւ
n6 n 4ℓ2ւ
2 =
2 =
2 =
4s 5`s1 `s2 + 10`12 2`22 + 30 13
12+ 34 i ( 1; 2)( 12
30
15u `s1`s3 + (`1u `t1)(`t3 + `3u) + 2u2(5`s1 + `s2
10`12 + 2`22)
s 8u2
t $ u
30`1u`3u + 60u(`1 `3 + 13)
`3u (1;2;3;`1) + u (1;2;`1;`2) 2i( 13 24)
u2
t $ u
t2
23);
HJEP09(217)
3 (`t1 + `1u)`s2
`s1(`t2 + `2u)
4s(p4 `1 + 2p4 `2 + `1 `2) 12+ 34
6s
( 12 + 13 + 14 + 23 + 24 + 34);
p4 (`1 `2) 2`1 `2 ( 12 + 13 + 14 + 23 + 24 + 34);
+ 3 (`t1 + `1u)`s2
`s1(`t2 + `2u)
4s p4 `1 + 2p4 `2) + `1 `2 12+ 34
12s
`t1 (1;2;3;`2)
s2
t2
s $ u
s $ t
s $ u
s $ t
34)
#
60u2
60t2
6u
6t
12u
12t
2s2 + 3 (`t1 + `1u)`s2
+ 2s `s1 + `t2 + `2u
3
2
ℓ2ւ
6
+6
6
4
2
+6
4
2
+6
4
(`t1)2 + (`1u)2 + (`s2)2
(`t2)2 + (`2u)2
+ `s1`s2
`t1`t2 + ` `
1u 2u + `s1`2u
u s
` `
1 2
+ 2u2 `
2(p4 `1 + `1 `3 + `22)
t $ u
6s
u2
t2
34)
s1 s2 + (`1u
`t1)(`t2 + `2u)
u2 6`s1 + 2`1u + 2`t1
+ `2u (1;2;3;`1) + u (1;2;`1;`2)
ℓ1→ 1
3
2
+ s 8s
t $ u
=
3`s1
15 `s1(`t2 + `2u)
(`t1 + `1u)`s2
8s 6`s1 2`t1 2`1u + 3`s2 `t2 `2u + 8`12 16`1 `2 4`22 60 12
12+ 34
+ s 8u2 + 30`1u`2u
t $ u
23);
5(`t1 + `1u + 2(`t2 + `2u)) 4`12 + 20`2 `3 + 60 23
15u `s2(`s3
u
`1)
s1 2u + `2u`3u
u2 3`s1 + 5(`t2 + `2u + `t3 + `u) + 4`12
3
20`2 `3
+ s 8u2
30`2u`3u + 60u(`2 `3 + 23)
t $ u
3
7 13+ 24
5 60u2
60t2
i ( 1; 2) ( 13
24)
23);
3
14+ 23
60t2
12+ 34
60s
s2
2 =
n12 n
2 =
t $ u
t $ u
t $ u
t $ u
t $ u
t2 23);
t2 23);
34)
+ u `s1`s2 + (`1u `t1)(`t2 + `2u) + 2s `1u`2u 2u(`1 `2 + 12) 13u+2 24
34)
`2u (1;2;3;`1) + u (1;2;`1;`2) 44ii(( 11u342 24)
34)
13t+2 24
4i( 1u32 24)
n16 n ℓ↓2ℓ1ր 14 23 = 31`12( 12 + 13 + 14 + 23 + 24 + 34);
n17 n 4 ←ℓ2րℓ1 1 2 = 34p42 ( 12 + 13 + 14 + 23 + 24 + 34);
3
n18 n 4 ←ℓ2րℓ1 1 2 = 23p24 ( 12 + 13 + 14 + 23 + 24 + 34):
3
A.3 Color
The color factors corresponding to the N = 2 SQCD diagrams in gure 5 are given here in
terms of structure constants f~abc = tr([Ta;Tb]Tc) and traces over generators Ta,
normalc1 = f~a1bcf~a2bdf~a3ef f~a4egf~df hf~gch;
c2 = tr(T a1 T a2 T bT a3 T a4 T b);
c3 = f~a3bcf~a4db tr(T a1 T a2 T cT d);
c13 = tr(T a1 T a2 T b) tr(T a3 T a4 T b);
c14 = f~a1bcf~a2dbf~a3edf~a4f ef~f ghf~gch;
c15 = tr(T a1 T a2 T a3 T a4 T aT a);
c4 = f~a1bcf~a2bdf~a3def~a4f gf~eghf~f ch; c16 = f~a1bcf~a2dbf~a3edf~a4f e tr(T cT f );
c5 = tr(T a1 T a2 T a3 T bT a4 T b);
c6 = f~a4ab tr(T a1 T a2 T a3 T bT a);
c7 = f~a1bcf~a2dbf~a3ed tr(T a4 T cT e);
c17 = f~a1bcf~a2dbf~a3edf~a4f gf~ehcf~hgf ;
c18 = f~a1bcf~a2dbf~a3edf~ef c tr(T a4 T f );
c19 = tr(T a1 T a2 T a3 T b) tr(T a4 T b);
c8 = f~a1bcf~a2dbf~a3ef f~a4ghf~dhef~gcf ; c20 = 0;
c9 = f~a3bc tr(T a1 T a2 T bT a4 T c);
c10 = f~a1bcf~a2db tr(T a3 T dT a4 T c);
c21 = f~a1bcf~a2dbf~a3edf~a4f ef~f gc tr(T g);
c22 = tr(T a1 T a2 T a3 T a4 T b) tr(T b);
c11 = f~a1bcf~a2dbf~a3ef f~a4gef~dhcf~hf g; c23 = 0;
c12 = f~a3bcf~a4dbf~ecd tr(T a1 T a2 T e);
c24 = f~a1bcf~a2dbf~a3edf~a4f gf~egc tr(T f ):
(A.2)
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
[1] H. Kawai, D.C. Lewellen and S.H.H. Tye, A Relation Between Tree Amplitudes of Closed
and Open Strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].
Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].
[2] Z. Bern, J.J.M. Carrasco and H. Johansson, New Relations for GaugeTheory Amplitudes,
[3] Z. Bern, J.J.M. Carrasco and H. Johansson, Perturbative Quantum Gravity as a Double
Copy of Gauge Theory, Phys. Rev. Lett. 105 (2010) 061602 [arXiv:1004.0476] [INSPIRE].
[4] J.J.M. Carrasco, M. Chiodaroli, M. Gunaydin and R. Roiban, Oneloop fourpoint
amplitudes in pure and mattercoupled N
4 supergravity, JHEP 03 (2013) 056
[arXiv:1212.1146] [INSPIRE].
[5] M. Chiodaroli, Q. Jin and R. Roiban, Color/kinematics duality for general abelian orbifolds
of N = 4 super YangMills theory, JHEP 01 (2014) 152 [arXiv:1311.3600] [INSPIRE].
[6] H. Johansson and A. Ochirov, Pure Gravities via ColorKinematics Duality for
Fundamental Matter, JHEP 11 (2015) 046 [arXiv:1407.4772] [INSPIRE].
[7] M. Chiodaroli, M. Gunaydin, H. Johansson and R. Roiban, Scattering amplitudes in N = 2
MaxwellEinstein and YangMills/Einstein supergravity, JHEP 01 (2015) 081
[arXiv:1408.0764] [INSPIRE].
[8] M. Chiodaroli, M. Gunaydin, H. Johansson and R. Roiban, Spontaneously Broken
YangMillsEinstein Supergravities as Double Copies, JHEP 06 (2017) 064
[arXiv:1511.01740] [INSPIRE].
[9] M. Chiodaroli, M. Gunaydin, H. Johansson and R. Roiban, Complete construction of
magical, symmetric and homogeneous N = 2 supergravities as double copies of gauge
07 (2011) 007 [arXiv:1105.2565] [INSPIRE].
[11] R. Monteiro and D. O'Connell, The Kinematic Algebra From the SelfDual Sector, JHEP
[12] C. Cheung and C.H. Shen, Symmetry for FlavorKinematics Duality from an Action, Phys.
Rev. Lett. 118 (2017) 121601 [arXiv:1612.00868] [INSPIRE].
[13] N.E.J. BjerrumBohr, P.H. Damgaard and P. Vanhove, Minimal Basis for Gauge Theory
Amplitudes, Phys. Rev. Lett. 103 (2009) 161602 [arXiv:0907.1425] [INSPIRE].
[14] S. Stieberger, Open & Closed vs. Pure Open String Disk Amplitudes, arXiv:0907.2211
[15] B. Feng, R. Huang and Y. Jia, Gauge Amplitude Identities by Onshell Recursion Relation
in Smatrix Program, Phys. Lett. B 695 (2011) 350 [arXiv:1004.3417] [INSPIRE].
[16] Y.X. Chen, Y.J. Du and B. Feng, A Proof of the Explicit Minimalbasis Expansion of Tree
Amplitudes in Gauge Field Theory, JHEP 02 (2011) 112 [arXiv:1101.0009] [INSPIRE].
[17] L.A. Barreiro and R. Medina, RNS derivation of Npoint disk amplitudes from the revisited
Smatrix approach, Nucl. Phys. B 886 (2014) 870 [arXiv:1310.5942] [INSPIRE].
[18] J.J. Carrasco and H. Johansson, FivePoint Amplitudes in N = 4 superYangMills Theory
and N = 8 Supergravity, Phys. Rev. D 85 (2012) 025006 [arXiv:1106.4711] [INSPIRE].
[19] Z. Bern, C. BoucherVeronneau and H. Johansson, N
4 Supergravity Amplitudes from
Gauge Theory at One Loop, Phys. Rev. D 84 (2011) 105035 [arXiv:1107.1935] [INSPIRE].
[20] Z. Bern, J.J.M. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, Simplifying Multiloop
Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Amplitudes, Phys.
Rev. D 85 (2012) 105014 [arXiv:1201.5366] [INSPIRE].
[21] N.E.J. BjerrumBohr, T. Dennen, R. Monteiro and D. O'Connell, Integrand Oxidation and
OneLoop ColourDual Numerators in N = 4 Gauge Theory, JHEP 07 (2013) 092
[arXiv:1303.2913] [INSPIRE].
[22] S. He, R. Monteiro and O. Schlotterer, Stringinspired BCJ numerators for oneloop MHV
amplitudes, JHEP 01 (2016) 171 [arXiv:1507.06288] [INSPIRE].
[23] C.R. Mafra and O. Schlotterer, Twoloop
vepoint amplitudes of super YangMills and
supergravity in pure spinor superspace, JHEP 10 (2015) 124 [arXiv:1505.02746] [INSPIRE].
[24] Z. Bern, S. Davies, T. Dennen, Y.t. Huang and J. Nohle, ColorKinematics Duality for
Pure YangMills and Gravity at One and Two Loops, Phys. Rev. D 92 (2015) 045041
[arXiv:1303.6605] [INSPIRE].
[25] J. Nohle, ColorKinematics Duality in OneLoop FourGluon Amplitudes with Matter,
Phys. Rev. D 90 (2014) 025020 [arXiv:1309.7416] [INSPIRE].
[26] M. Berg, I. Buchberger and O. Schlotterer, From maximal to minimal supersymmetry in
string loop amplitudes, JHEP 04 (2017) 163 [arXiv:1603.05262] [INSPIRE].
[27] H. Johansson and A. Ochirov, ColorKinematics Duality for QCD Amplitudes, JHEP 01
(2016) 170 [arXiv:1507.00332] [INSPIRE].
[28] L. de la Cruz, A. Kniss and S. Weinzierl, Proof of the fundamental BCJ relations for QCD
amplitudes, JHEP 09 (2015) 197 [arXiv:1508.01432] [INSPIRE].
[29] S. Badger, G. Mogull, A. Ochirov and D. O'Connell, A Complete TwoLoop, FiveGluon
Helicity Amplitude in YangMills Theory, JHEP 10 (2015) 064 [arXiv:1507.08797]
Loops, JHEP 12 (2015) 135 [arXiv:1511.06652] [INSPIRE].
[31] D. Chester, BernCarrascoJohansson relations for oneloop QCD integral coe cients,
Phys. Rev. D 93 (2016) 065047 [arXiv:1601.00235] [INSPIRE].
supergravities, arXiv:1607.04129 [INSPIRE].
[33] M. Chiodaroli, M. Gunaydin, H. Johansson and R. Roiban, Explicit Formulae for
YangMillsEinstein Amplitudes from the Double Copy, JHEP 07 (2017) 002
[arXiv:1703.00421] [INSPIRE].
Gauge Invariance, arXiv:1612.02797 [INSPIRE].
[34] N. ArkaniHamed, L. Rodina and J. Trnka, Locality and Unitarity from Singularities and
[35] L. Borsten, M.J. Du , L.J. Hughes and S. Nagy, Magic Square from YangMills Squared,
Phys. Rev. Lett. 112 (2014) 131601 [arXiv:1301.4176] [INSPIRE].
[36] A. Anastasiou, L. Borsten, M.J. Du , L.J. Hughes and S. Nagy, Super YangMills, division
algebras and triality, JHEP 08 (2014) 080 [arXiv:1309.0546] [INSPIRE].
[37] A. Anastasiou, L. Borsten, M.J. Du , L.J. Hughes and S. Nagy, A magic pyramid of
supergravities, JHEP 04 (2014) 178 [arXiv:1312.6523] [INSPIRE].
[38] A. Anastasiou, L. Borsten, M.J. Du , L.J. Hughes and S. Nagy, YangMills origin of
gravitational symmetries, Phys. Rev. Lett. 113 (2014) 231606 [arXiv:1408.4434] [INSPIRE].
[39] A. Anastasiou, L. Borsten, M.J. Hughes and S. Nagy, Global symmetries of YangMills
squared in various dimensions, JHEP 01 (2016) 148 [arXiv:1502.05359] [INSPIRE].
[40] Z. Bern, T. Dennen, Y.t. Huang and M. Kiermaier, Gravity as the Square of Gauge
Theory, Phys. Rev. D 82 (2010) 065003 [arXiv:1004.0693] [INSPIRE].
[41] C. BoucherVeronneau and L.J. Dixon, N
4 Supergravity Amplitudes from Gauge Theory
at Two Loops, JHEP 12 (2011) 046 [arXiv:1110.1132] [INSPIRE].
[42] J. Broedel and L.J. Dixon, Colorkinematics duality and doublecopy construction for
amplitudes from higherdimension operators, JHEP 10 (2012) 091 [arXiv:1208.0876]
[43] Z. Bern, S. Davies, T. Dennen, A.V. Smirnov and V.A. Smirnov, Ultraviolet Properties of
N = 4 Supergravity at Four Loops, Phys. Rev. Lett. 111 (2013) 231302 [arXiv:1309.2498]
[44] Z. Bern, S. Davies and T. Dennen, Enhanced ultraviolet cancellations in N = 5 supergravity
at four loops, Phys. Rev. D 90 (2014) 105011 [arXiv:1409.3089] [INSPIRE].
[45] Z. Bern, S. Davies and T. Dennen, The Ultraviolet Critical Dimension of HalfMaximal
Supergravity at Three Loops, arXiv:1412.2441 [INSPIRE].
181602 [arXiv:1701.02519] [INSPIRE].
[46] Z. Bern, J.J. Carrasco, W.M. Chen, H. Johansson and R. Roiban, Gravity Amplitudes as
Generalized Double Copies of GaugeTheory Amplitudes, Phys. Rev. Lett. 118 (2017)
[48] C. Cheung, K. Kampf, J. Novotny and J. Trnka, E ective Field Theories from Soft Limits
[49] F. Cachazo, S. He and E.Y. Yuan, Scattering Equations and Matrices: From Einstein To
YangMills, DBI and NLSM, JHEP 07 (2015) 149 [arXiv:1412.3479] [INSPIRE].
[50] Y.J. Du and C.H. Fu, Explicit BCJ numerators of nonlinear simga model, JHEP 09
(2016) 174 [arXiv:1606.05846] [INSPIRE].
[51] J.J.M. Carrasco, C.R. Mafra and O. Schlotterer, Abelian Ztheory: NLSM amplitudes and
'corrections from the open string, JHEP 06 (2017) 093 [arXiv:1608.02569] [INSPIRE].
[52] C.R. Mafra, O. Schlotterer and S. Stieberger, Complete NPoint Superstring Disk
Amplitude II. Amplitude and Hypergeometric Function Structure, Nucl. Phys. B 873 (2013)
461 [arXiv:1106.2646] [INSPIRE].
[53] C.R. Mafra and O. Schlotterer, The Structure of nPoint OneLoop Open Superstring
Amplitudes, JHEP 08 (2014) 099 [arXiv:1203.6215] [INSPIRE].
[54] J. Broedel, O. Schlotterer and S. Stieberger, Polylogarithms, Multiple Zeta Values and
Superstring Amplitudes, Fortsch. Phys. 61 (2013) 812 [arXiv:1304.7267] [INSPIRE].
[55] Y.t. Huang, O. Schlotterer and C. Wen, Universality in string interactions, JHEP 09
(2016) 155 [arXiv:1602.01674] [INSPIRE].
the open string, arXiv:1612.06446 [INSPIRE].
[56] J.J.M. Carrasco, C.R. Mafra and O. Schlotterer, Semiabelian Ztheory: NLSM+ 3 from
arXiv:1705.03025 [INSPIRE].
(2014) 056 [arXiv:1410.0239] [INSPIRE].
[57] P. Tourkine and P. Vanhove, Higherloop amplitude monodromy relations in string and
gauge theory, Phys. Rev. Lett. 117 (2016) 211601 [arXiv:1608.01665] [INSPIRE].
[58] S. Hohenegger and S. Stieberger, Monodromy Relations in HigherLoop String Amplitudes,
arXiv:1702.04963 [INSPIRE].
[59] S. He and O. Schlotterer, New Relations for GaugeTheory and Gravity Amplitudes at Loop
Level, Phys. Rev. Lett. 118 (2017) 161601 [arXiv:1612.00417] [INSPIRE].
[60] S. He, O. Schlotterer and Y. Zhang, New BCJ representations for oneloop amplitudes in
gauge theories and gravity, arXiv:1706.00640 [INSPIRE].
[61] C. Cheung, C.H. Shen and C. Wen, Unifying Relations for Scattering Amplitudes,
[62] R. Monteiro, D. O'Connell and C.D. White, Black holes and the double copy, JHEP 12
[63] A. Luna, R. Monteiro, D. O'Connell and C.D. White, The classical double copy for
TaubNUT spacetime, Phys. Lett. B 750 (2015) 272 [arXiv:1507.01869] [INSPIRE].
[64] A.K. Ridgway and M.B. Wise, Static Spherically Symmetric KerrSchild Metrics and
Implications for the Classical Double Copy, Phys. Rev. D 94 (2016) 044023
[arXiv:1512.02243] [INSPIRE].
[65] G. Cardoso, S. Nagy and S. Nampuri, Multicentered N = 2 BPS black holes: a double copy
description, JHEP 04 (2017) 037 [arXiv:1611.04409] [INSPIRE].
Bremsstrahlung and accelerating black holes, JHEP 06 (2016) 023 [arXiv:1603.05737]
[arXiv:1611.07508] [INSPIRE].
[67] A. Luna et al., Perturbative spacetimes from YangMills theory, JHEP 04 (2017) 069
[68] W.D. Goldberger and A.K. Ridgway, Radiation and the classical double copy for color
charges, Phys. Rev. D 95 (2017) 125010 [arXiv:1611.03493] [INSPIRE].
[69] W.D. Goldberger, S.G. Prabhu and J.O. Thompson, Classical gluon and graviton radiation
from the biadjoint scalar double copy, arXiv:1705.09263 [INSPIRE].
[70] C. Cheung and D. O'Connell, Amplitudes and SpinorHelicity in Six Dimensions, JHEP 07
(2009) 075 [arXiv:0902.0981] [INSPIRE].
[71] Z. Bern, S. Davies, T. Dennen and Y.t. Huang, Absence of ThreeLoop FourPoint
Divergences in N = 4 Supergravity, Phys. Rev. Lett. 108 (2012) 201301 [arXiv:1202.3423]
[72] Z. Bern, S. Davies, T. Dennen and Y.t. Huang, Ultraviolet Cancellations in HalfMaximal
Supergravity as a Consequence of the DoubleCopy Structure, Phys. Rev. D 86 (2012)
105014 [arXiv:1209.2472] [INSPIRE].
[73] Z. Bern, S. Davies and T. Dennen, The Ultraviolet Structure of HalfMaximal Supergravity
with Matter Multiplets at Two and Three Loops, Phys. Rev. D 88 (2013) 065007
[arXiv:1305.4876] [INSPIRE].
[74] Z. Bern, M. Enciso, J. ParraMartinez and M. Zeng, Manifesting enhanced cancellations in
supergravity: integrands versus integrals, JHEP 05 (2017) 137 [arXiv:1703.08927]
Phys. B 303 (1988) 286 [INSPIRE].
[75] N. Seiberg, Observations on the Moduli Space of Superconformal Field Theories, Nucl.
[76] M. Kiermaier, Gravity as the Square of Gauge Theory, talk at Amplitudes 2010, Queen
Mary University of London, London U.K. (2010), http://www.strings.ph.qmul.ac.uk/ theory/Amplitudes2010/. [77] N.E.J. BjerrumBohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, The Momentum
Kernel of Gauge and Gravity Theories, JHEP 01 (2011) 001 [arXiv:1010.3933] [INSPIRE].
[78] V. Del Duca, L.J. Dixon and F. Maltoni, New color decompositions for gauge amplitudes at
tree and loop level, Nucl. Phys. B 571 (2000) 51 [hepph/9910563] [INSPIRE].
[79] T. Melia, Getting more avor out of one avor QCD, Phys. Rev. D 89 (2014) 074012
[arXiv:1312.0599] [INSPIRE].
(1986) 2459 [INSPIRE].
(1988) 215 [INSPIRE].
[80] S.J. Parke and T.R. Taylor, An Amplitude for n Gluon Scattering, Phys. Rev. Lett. 56
[81] V.P. Nair, A Current Algebra for Some Gauge Theory Amplitudes, Phys. Lett. B 214
[82] L.J. Dixon, Calculating scattering amplitudes e ciently, hepph/9601359 [INSPIRE].
[83] H. Elvang and Y.t. Huang, Scattering Amplitudes in Gauge Theory and Gravity,
Cambridge University Press, Cambridge U.K. (2015). [84] T. Dennen, Y.t. Huang and W. Siegel, Supertwistor space for 6D maximal super
YangMills, JHEP 04 (2010) 127 [arXiv:0910.2688] [INSPIRE].
[87] R. Boels, Covariant representation theory of the Poincare algebra and some of its
extensions, JHEP 01 (2010) 010 [arXiv:0908.0738] [INSPIRE].
[88] B. de Wit, Supergravity, in Proceedings of Euro Summer School on Unity of Fundamental
Physics: Gravity, Gauge Theory and Strings, 76th session, Les Houches France (2001)
[hepth/0212245] [INSPIRE].
[arXiv:1103.1869] [INSPIRE].
[89] Z. Bern, J.J. Carrasco, T. Dennen, Y.t. Huang and H. Ita, Generalized Unitarity and
SixDimensional Helicity, Phys. Rev. D 83 (2011) 085022 [arXiv:1010.0494] [INSPIRE].
[90] Z. Bern and Y.t. Huang, Basics of Generalized Unitarity, J. Phys. A 44 (2011) 454003
[91] S. Badger, C. Br nnumHansen, F. Buciuni and D. O'Connell, A unitarity compatible
approach to oneloop amplitudes with massive fermions, JHEP 06 (2017) 141
[arXiv:1703.05734] [INSPIRE].
[92] J.J.M. Carrasco, R. Kallosh, R. Roiban and A.A. Tseytlin, On the U(1) duality anomaly
and the Smatrix of N = 4 supergravity, JHEP 07 (2013) 029 [arXiv:1303.6219] [INSPIRE].
[93] M.B. Green, J.H. Schwarz and L. Brink, N = 4 YangMills and N = 8 Supergravity as
Limits of String Theories, Nucl. Phys. B 198 (1982) 474 [INSPIRE].
[94] Z. Bern and A.G. Morgan, Massive loop amplitudes from unitarity, Nucl. Phys. B 467
[95] D.Z. Freedman, R. Kallosh, D. Murli, A. Van Proeyen and Y. Yamada, Absence of U(1)
5 Supergravities, JHEP 05 (2017) 067
(1996) 479 [hepph/9511336] [INSPIRE].
Anomalous Superamplitudes in N
[arXiv:1703.03879] [INSPIRE].
QCD amplitudes and coupling shifts, Phys. Rev. D 66 (2002) 085002 [hepph/0202271]
[97] S. Badger, H. Frellesvig and Y. Zhang, A TwoLoop FiveGluon Helicity Amplitude in
QCD, JHEP 12 (2013) 045 [arXiv:1310.1051] [INSPIRE].
05 (2014) 136 [arXiv:1312.1326] [INSPIRE].
[98] A. Ochirov and P. Tourkine, BCJ duality and double copy in the closed string sector, JHEP
[99] G. Bossard, P.S. Howe, K.S. Stelle and P. Vanhove, The vanishing volume of D = 4
superspace, Class. Quant. Grav. 28 (2011) 215005 [arXiv:1105.6087] [INSPIRE].
[100] G. Bossard, P.S. Howe and K.S. Stelle, Anomalies and divergences in N = 4 supergravity,
Phys. Lett. B 719 (2013) 424 [arXiv:1212.0841] [INSPIRE].
[101] G. Bossard, P.S. Howe and K.S. Stelle, Invariants and divergences in halfmaximal
supergravity theories, JHEP 07 (2013) 117 [arXiv:1304.7753] [INSPIRE].
FourLoop FourPoint Amplitude in N = 4 superYangMills Theory, Phys. Rev. D 82
(2010) 125040 [arXiv:1008.3327] [INSPIRE].
[hepph/0307297] [INSPIRE].
Factors, JHEP 02 (2013) 063 [arXiv:1211.7028] [INSPIRE].
supersymmetric YangMills theory, Phys. Rev. Lett. 117 (2016) 271602 [arXiv:1610.02394]
SQCD and in N = 4 SYM at one loop, JHEP 08 (2008) 033 [arXiv:0805.4190] [INSPIRE].
Coupling, talk at Workshop on Gauge and String Theory, ETH, Zurich Switzerland (2008),
09 (2014) 017 [Erratum ibid. 1502 (2015) 022] [arXiv:1406.7283] [INSPIRE].
SCQCD, Phys. Lett. B 747 (2015) 325 [arXiv:1502.07614] [INSPIRE].
266 (1986) 709 [INSPIRE].
Alter Ultraviolet Divergences in Quantum Gravity without Physical Consequences, Phys.
theories , Phys. Rev. Lett . 117 ( 2016 ) 011603 [arXiv: 1512 .09130] [INSPIRE].
[10] A. Anastasiou et al., Twin supergravities from YangMills theory squared , Phys. Rev. D 96 [47] G. Chen and Y.J. Du , Amplitude Relations in Nonlinear model , JHEP 01 ( 2014 ) 061 of Scattering Amplitudes , Phys. Rev. Lett . 114 ( 2015 ) 221602 [arXiv: 1412 .4095] [INSPIRE].
[66] A. Luna , R. Monteiro , I. Nicholson , D. O'Connell and C.D. White , The double copy: [85] Y.t. Huang, NonChiral Smatrix of N = 4 Super YangMills , arXiv: 1104 . 2021 [INSPIRE]. [86] H. Elvang , Y.t. Huang and C. Peng , Onshell superamplitudes in N < 4 SYM , JHEP 09 [102] Z. Bern , J.J.M. Carrasco , L.J. Dixon , H. Johansson and R. Roiban , The Complete [103] A.G. Grozin , Lectures on multiloop calculations, Int. J. Mod. Phys. A 19 ( 2004 ) 473 [104] R.H. Boels , B.A. Kniehl , O.V. Tarasov and G. Yang , Colorkinematic Duality for Form [108] R. Andree and D. Young , Wilson Loops in N = 2 Superconformal YangMills Theory ,