Cyclic Mario worlds — colordecomposition for oneloop QCD
HJE
Cyclic Mario worlds  colordecomposition for oneloop QCD
Gregor Kalin 0 1
0 Box 516 , SE751 20 Uppsala , Sweden
1 Department of Physics and Astronomy, Uppsala University
We present a new color decomposition for QCD amplitudes at oneloop level as a generalization of the Del DucaDixonMaltoni and JohanssonOchirov decomposition at tree level. Starting from a minimal basis of planar primitive amplitudes we write down a color decomposition that is free of linear dependencies among appearing primitive amplitudes or color factors. The conjectured decomposition applies to any number of quark avors and is independent of the choice of gauge group and matter representation. The results also hold for higherdimensional or supersymmetric extensions of QCD. We provide expressions for any number of external quarkantiquark pairs and gluons.
Perturbative QCD; Scattering Amplitudes; Gauge Symmetry

1 Introduction
2
3
4
2.1
2.2
3.1
3.2
4.1
4.2
4.3
A basis for primitive oneloop amplitudes
Closed quark loop
Gluonic or mixed loop
Oneloop color decomposition
Color decomposition for a closed quark loop
Color decomposition for a mixed or gluonic loop
Conclusions and outlook
1
Introduction
New methods, like the (generalized) unitarity method and applications thereof [17{26],
recursive constructions [27{30], MHV rules [31] or integration technology [32{44] have
signi cantly improved the accessible region of NLO computations.
A standard technique in modern methods for computing tree level scattering
amplitudes in quantum chromodynamics (QCD) is the decomposition of the amplitudes into
purely kinematic primitive amplitudes and purely colordependent objects (color factors).
For treelevel processes the primitive, colorordered amplitudes are often computed using
recursion in terms of the number of legs [27{29, 31]. The decomposition into color and
kinematic parts is not unique  and it is not obvious how to nd the most useful and
compact formulae.
The standard SU(N ) tracebased color decomposition [45{49] at tree level does not
take advantage of all linear dependencies of primitive amplitudes and color factors, as the
sum goes over an overcomplete set of linear dependent primitive amplitudes and color
factors. The same holds for amplitudes with multiquark external states [50{54]. The
linear relations arise from the color algebra of gauge theory [55] and observed linear relations
between primitive amplitudes: the KleissKuijf (KK) [55] and the BernCarrascoJohansson
(BCJ) relations [56].
{ 1 {
These relations have been utilized to nd more compact color decompositions, rst
for the purely gluonic case by DelDucaDixonMaltoni (DDM) [57, 58] and later the
generalization to any number of external quarkantiquark pairs by Johansson and Ochirov
(JO) [59], which was proven by Melia [60] shortly thereafter.
At oneloop level color decompositions have been worked out for speci c cases [14,
58, 61{63], in a general tracebased setup [54, 64] including an overcomplete set of
primitive amplitudes and recently Ochirov and Page presented a general method to obtain the
full color dependence at loop level [65]. In this paper we propose a compact color
decomposition that eliminates linear dependencies of oneloop QCD amplitudes to general
multiplicity and number of external quarkantiquark pairs. The structure of the planar
diagrams contributing to a primitive amplitude and its corresponding color factor in the
color decomposition have a similar form as in the tree level case  the `Mario world'
diagrams [59, 60]. For oneloop diagrams internal lines carrying loop momentum form a cyclic
`ground level' for the Mario world and the external partons build up the `higher levels'
 mathematically the di erent parts in the tensor product of the Lie algebra. A basis of
oneloop primitive amplitudes is given by all such Mario world structures, for both cases
where either a closed quark or a gluonic loop is present in the diagrams contributing to
a primitive amplitude. For n external partons, whereof k are quarkantiquark pairs, the
size of the sets of primitive amplitudes considered in the color decomposition is (n
1)!=k!
for the case of a closed quark loop  or (n
2)!(n
k
1)=k! if the gauge group has
traceless Lie algebra generators  and 22k 1(n
1)!k!=(2k)! if at least one gluon carries
loop momentum.
We start by setting up the notation and review the treelevel color decomposition in
section 2. Section 3 de nes a basis of primitive oneloop amplitudes for the two cases where
either a closed quark loop or a mixed/gluonic loop is present in the amplitude. In what
follows we present a color decomposition to all multiplicities in section 4 together with a
pedagogical 5point example in section 4.3. We discuss the results and give an outlook in
section 5.
2
Tree level review and prerequisites
2Sn 2(f3;:::;ng)
We review the work by Johansson and Ochirov (JO) [59] about a new color decomposition
for massless QCD at tree level and introduce the necessary notation.
Del Duca, Dixon and Maltoni (DDM) [57, 58] presented a color decomposition for
gluonic amplitudes that removes the redundancy of KleissKuijf (KK) relations [55]  present
in the familiar tracebased SU(Nc) color decomposition [45, 46, 48, 49]. The decomposition
is written in terms of gauge group structure constants f~abc and primitive (colorordered)
amplitudes as
An
(0) = g
n 2 X f~a2a (3)b1 f~b1a (4)b2
f~bn 3a (n)a1 A(1; 2; );
where we de ne the structure constants in terms of the gauge group generators T a
(2.1)
(2.2)
f~abc = Tr([T a; T b]T c);
Tr(T aT b) = ab:
{ 2 {
The primitive amplitudes A(1; 2; ) are exactly the colorordered amplitudes appearing in
the tracebased color decomposition  and can be directly computed from planar diagrams
with the given external ordering using colorordered Feynman rules [66]. This
decomposition is valid for any choice of gauge group since it only relies on de ning properties of the
Lie algebra.
2.1
A basis for primitive QCD tree amplitudes
A generalization of the above color decomposition that includes any number of external
distinguishable quarkantiquark pairs builds on a basis of primitive tree amplitudes presented
by Melia [67, 68]. As in the purely gluonic case primitive amplitudes are de ned as the
sum over all planar diagrams with the given cyclic ordering of externals using colorordered
Feynman rules. For example we can diagrammatically represent a primitive amplitude as
quarks are linearly dependent under a generalization of KKrelations, rst observed in [62]
through nontrivial solutions of a linear system in terms of (Feynman) diagrams. Dyck
words [69, 70] allow for a compact description of a basis, i.e. a spanning set of linearly
independent primitive amplitudes. A Dyck word is a list of opening and closing brackets,
that are composed in a mathematically correct way. A pair of brackets corresponds to a
quarkantiquark pair, i.e. we identify an opening (closing) bracket with a quark (antiquark).
For n external partons we denote the set of Dyck words including all permutations of k
quarkantiquark pairs and n
2k insertions of gluons by Dyckn;k.
Consider the Dyck word `()(()())'. For n = 12 external states, including k = 4
quarkantiquark pairs, we can for example make an assignment of partons of the form
( ) ( ( ) ( ) )
121345657884 2 Dyck12;4:
We assume here and in the following that all quarkantiquark pairs have distinct avor 
the one avor case can be recovered by summing over all combinations of quarkantiquark
pairings (see e.g. [54, 71]). The numbers 1, 4, 5 and 8 label quarkantiquark pairs, whereas
2, 3, 6 and 7 mark inserted gluons at various places. Note that the quark i and the
corresponding antiquark i have di erent momenta even though they carry the same number.
A basis of primitive amplitudes is then simply given by
Bn;k = nA(0)(1; ; 1) j
0
2 Dyckn 2;k 1 ;
o
4
5
2
4
1
(2.4)
(2.5)
1
2
3
6
5
1
(2.6)
(2.8)
HJEP04(218)
diagram describes the color factor C(0)(1; 2; 3; 4; 3; 2; 5; 6; 5; 1). It corresponds to the primitive
amplitude given in (2.3).
where one quarkantiquark pair is xed at the beginning and the end. The size of this
basis has been shown to be (n
2)!=k!. This basis turns out to be useful to generalize the
DDM color decomposition. We discuss the full tree level color decomposition of QCD in
the next section.
2.2
Color decomposition of QCD tree amplitudes
A tracebased color decomposition for QCD for a SU(Nc) gauge group leads to rather
complex expressions [14, 54, 61{64]. The JO color decomposition [59] provides a structured
form which holds for any number of external partons and for any gauge group.
Given Melia's basis of primitive amplitudes, the JO decomposition proposes a form of
the color factors C(0) such that the full amplitude (for a xed number of external particles n
with k quarkantiquark pairs) is recovered by
A(n0;k) = g
n 2 X C(0)(1; ; 1)A(0)(1; ; 1):
2Dyckn 2;k 1
The result is most easily understood in a diagrammatic way. The color diagrams that
contribute to a color factor have a particular structure, which has been named `Mario world'
due to its similarity with the virtual world of a certain arcade game [72].1 An example
is given in gure 1. The color factor of a single graph is determined only considering the
color part of the Feynman rules. We use the conventions
b
f~abc =
a
c
;
Tia =
i
a
a

;
Tsai =

i
=
Tia :
(2.7)
The notation of fundamental generators with ipped indices Tsai
Tia is used to
introduce an arti cial antisymmetry for the fundamental generators, similar to the adjoint
generators (Taadj)bc
f~bac:
1We use the term Mario world diagram for a single color diagram as well as for diagrams representing
a color factor (a sum of color diagrams) which we write using composite vertices as explained below.
f~cab =
Tai =
f~bac;
Tia:
{ 4 {
la =
=
l
X 1
s=1 
a
These identities applied to subparts of a color diagram lead to linear relations between
color factors. In our case the number of linearly independent color diagrams matches the
size of the basis of primitive amplitudes for xed number of external states n and number
of quarkantiquark pairs k. This means that this color decomposition is minimal in the
sense that the sum (2.6) is over a minimal set of (independent) color factors and primitive
amplitudes and it is justi ed to call (2.5) a basis.
The last ingredient needed is an operation on color diagrams introduced by JO. We
present a mathematical as well as a diagrammatic notation to de ne the operator
1
z a
T
{z
l
1
1
{
1
}
l =
+
+
+
+
;
(2.13)
Color factors of di erent diagrams are related by the Jacobi identity and the de ning
commutation relation, inherited from the coloralgebra of the gauge group. The identities
for the adjoint and the fundamental representation are given by
f~dacf~cbe
f~dbcf~cae = f~abcf~dce;
Tiaj Tjbk
Tibj Tjak = f~abcTick:
The relations in terms of diagrams are
with l the nestedness of the gluon line (i.e. how many quark lines are below the given gluon
line) with adjoint index a. There is a di erence to the notation used in [59]: we insert a
general object T
a which stands for either T a or Tsa depending on the orientation of the
quark line this object connects to. This notation is independent of the chosen quark line
directions and will be especially useful for the oneloop discussion. For our choice of quark
line directions in the tree level case we will always have that T
a = T a.2
The operator allows for a compact de nition of the color factors in the decomposition (2.6)
C(0)(1; ; 1) = f1j j1g q!jqagg;
2Note that [59] uses a di erent convention where the lowest quark line points in the opposite direction.
They correspondingly insert Tsa at the lowest level s = 1.
{ 5 {
=
=
s
}
;
:
(2.9)
(2.10)
(2.11)
(2.12)
(2.14)
HJEP04(218)
where we use the bracket notation to indicate fundamental color indices and ag is the adjoint
index of the external gluon g. Note that the nestedness of a quark line also includes the line
itself. A precise de nition of the object T a introduced before, and the bracket notation, is
fijT a1
fijT a1
T am jjg = fijT a1
T am jjg = fijT a1
T am jjg = (T a1
T am jjg = ( 1)m(T am
T am )ij ;
T a1 )ji;
which implements the correct contraction of fundamental indices depending on the
orientation of the quark line. A bracket with nestedness l acts on the part of the tensor product
with the same nestedness. The evaluation is most easily done by resolving brackets from
inside to outside, using
fkjT a1
i
f j
fkj(T a1
= fij
= fkjT a1
T ar )
(T b1
T ar jlg(T b1
T ar jlgfij
(T b1
T bs )
T bs )
T bs )
l
j g
j
j g
j
j g
j
j g
(2.15)
(2.16)
recursively.
Diagrammatically a color factor C(0)(1; ; 1) is given by a Mario world diagram with
the appropriate
quarkantiquark pair.
base line 1
1, and external ordering and nesting according to the Dyck word . The
purely gluonic case (i.e. the DDM color decomposition) can be recovered by de ning
0a = Taadj that acts on the `zeroth level' of the Mario world diagram, i.e. simply contracting
0a operators and xing two gluons at the start and the end instead of a
A number of examples of this color decomposition can be found in the original
paper [59]. This completes the QCD tree level color decomposition for any number of
external gluons and quarkantiquark pairs. We discuss the oneloop extension in the rest of
these notes.
3
A basis for primitive oneloop amplitudes
We begin by de ning primitive oneloop QCD amplitudes. These primitive amplitudes are
planar and colorordered. A basis of independent primitive amplitudes is constructed using
Dyck words, and builds up a minimal color decomposition, given the corresponding color
factors  similar to the tree level results reviewed above.
We de ne primitive oneloop amplitudes An;k following [61], where n is the number of
external partons and k counts the quarkantiquark pairs, such that n
2k is the number
of external gluons. Primitive oneloop amplitudes are purely kinematic objects analogous
to colorordered tree level amplitudes. They are built from planar diagrams with a
xed
(cyclic) ordering of external particles  and for amplitudes including quarks (external or
in the loop) a xed routing of these elements, i.e. a primitive amplitude speci es on which
side a quark line, a gluon or a loop lies with respect to all quark lines. Additionally we
distinguish amplitudes that have a closed fermion loop from ones that have at least one
gluon line carrying loop momentum. Primitive amplitudes can be directly computed using
colorordered Feynman rules (see e.g. [66]), summing up all the planar graphs with the
{ 6 {
speci ed routing of quark lines. Modern methods like generalized unitary [17, 18, 73{76],
(looplevel) BCFW [28, 29, 77{81] or Qcuts [82, 83] simplify this computation for QCD
as well as for supersymmetric extensions thereof.
Note that for massless theories diagrams with a bubble on an external leg or a tadpole
of the form
come with an illde ned vanishing propagator. From a purely diagrammatic point of view
we observe that the inclusion of external bubbles and tadpoles with a closed quark loop pose
no problem to the colordecomposition and can formally be included. A gauge group with
vanishing traces of corresponding Lie algebra generators Tr(T a) = 0,3 e.g. a semisimple
Lie algebra, induces vanishing color factors for diagrams containing a quark tadpole and so
these contributions can be ignored. In the following we will though mostly give attention
to the more general case assuming the presence of diagrams with quark tadpoles, validating
the discussion for any gauge group. In the case that the trace of every generator of the
Lie algebra is zero the results obtained here can be slightly simpli ed as we will remark
accordingly. For gluonic tadpoles the color decomposition of the form proposed no longer
holds on a formal level. Imposing that the color factors of such diagrams vanish due to the
antisymmetry of the structure constants xes their behavior and we can safely ignore them.
An example for a 10point oneloop primitive amplitude can be represented as
The superscript qL[1L; 2R; 3L; 5R] speci es the type of loop (q for a closed quark loop or g
for a loop that contains at least one gluonic line) and its properties. If the loop is a quark
loop the superscript qL=R speci es on which side the external particles lie with respect to
the loop line direction  equivalently one could specify the orientation of the loop. The
rest of the arguments of the superscript specify on which side the quark loop lies with
respect to the quark line with the given number, i.e. for this example the quark loop lies on
the left side of rst quark line, on the right side of the second loops line and so on. There is
no need to specify the side on which each quark line or gluon lies with respect to all other
quark lines since this is clear by the external ordering, if one considers only nonvanishing
primitive amplitudes.
3This constraint for example appears for theories whose gauge group contains factors of U(1), for which
it has been shown that they cannot be consistently coupled to gravity unless all U(1) generators are
traceless [84].
{ 7 {
AqL[1L;2R;3L;5R](1; 1; 2; 3; 3; 4; 2; 5; 5; 6) = 4
A primitive amplitude is computed by summing up all planar diagrams that have
the given external ordering and quark line routing. In the language of the diagrammatic
representation (3.1) this is achieved by drawing all possible planar graphs on the annulus
that is bounded by the loop line on the inside and the dashed line carrying the
external partons on the outside. Planarity allows for a uniform treatment of loopmomentum
assignment [85]. Using dualspace coordinates pi
xi
xi 1 [86] one de nes the loop
momentum via `
x0
xn which corresponds to the convention that the loop momentum
ows between the last and the rst external leg on the annulus
From all of these primitive amplitudes it is possible to pick a subset, that we will
call a basis, in the sense that they form a maximal set of linearly independent elements,
and we can express the full colordressed amplitudes in terms of these objects and color
factors only. Linear relations between primitive amplitudes can be found by expressing
primitive amplitudes in terms of planar diagrams  assuming that distinct diagrams are
independent  and reducing the emerging system of equations. This general algorithm
to nd a set of linearly independent primitive amplitudes is described in detail in [62]. A
basis obtained in this way is in general not unique.
It turns out that the number of linearly independent primitive amplitudes is the same
as the number of independent (with respect to Jacobi identities and commutation relations)
color factors of diagrams of the same process as expected for a basis of kinematic objects.
In this sense the color decomposition presented below is over a minimal set of primitive
amplitudes and color factors as in the tree level case. We check the matching of these
numbers explicitly up to at least seven points for all possible combinations of external
particles and every loop type. Note that for a gauge group with traceless algebra generators
Tr(T a) = 0 the number of independent color factors is reduced since diagrams containing
a quark tadpole have a vanishing color structure. The number of independent primitive
amplitudes shrinks by the same amount when diagrams with a tadpole that is formed by
a quark line  corresponding to exactly these vanishing color factors  are ignored.
We discuss our choice of basis for the case of a closed quark loop separately from the
case of a mixed or purely gluonic loop. The special case of a purely gluonic amplitude we
will treat separately. Note that the above example (3.1) of a primitive amplitude is not a
basis element for our choice of basis.
3.1
Closed quark loop
As for the case of tree level amplitudes, a basis for primitive oneloop amplitudes is most
conveniently presented using Dyck words [67] and the Mario world structure. In contrast
{ 8 {
to the tree level case one does not x a quark line as the base of the Mario world diagram,
instead the quark loop plays the role of this line. Since we do not x any external particles
we need the notion of cyclicity for Dyck words. We de ne the set of cyclically distinct
Dyck words including gluon insertions and permutations of quarkantiquark pairs by
Dyckn;k = Dyckn;k=fcyclic transformationsg:
(3.3)
A basis element is represented by such a cyclic Dyck word. For example, a possible
assignment of quark line numbers and insertions of gluons for the Dyck word `(())()' is
12321454 2 Dyckn;k which represents a primitive amplitude that diagrammatically has
HJEP04(218)
the form
AqR[1L2L4L](12321454) =
1
1 ;
(3.4)
4
2
where 1, 2 and 4 label quark lines, and 3 and 5 gluon insertions. We choose the quark loop
to lie on the left side of every quark line, and every quark line to the right of the loop line.
As will be clear when discussing the color decomposition, this choice of quark routings
allows us to recover the tree level conventions when cutting through the quark loop. This
information fully speci es a primitive amplitude.
A complete basis for xed n and k is given by all cyclic Dyck words:
Bn;k = nAqL[iLji2fquark linesg]( ) j
q
2 Dyckn;k :
o
The size of this basis can be computed using a recursive description of Dyck words. The
number of Dyck words with k brackets without any labeling or gluon insertion is given by
the Catalan number Ck = (2k)!=((k + 1)!k!). In order to keep track of cyclicity we demand
that the rst quark is always assigned to the rst bracket or to one of the brackets inside
the rst closed substructure (i.e. between the rst opening and its corresponding closing
bracket). An explicit expression for the number of labeled cyclic Dyck words (without gluon
insertions) is the number of all possible Dyck words inside the rst closed substructure Cs
of length s, times the number of (noncyclic) Dyck words this structure can be followed
by Ck s 1. Including the permutation of the all except the
rst quarkantiquark pair
(k
1)!, this statement is expressed as
where the factor (s + 1) counts all possible assignments of the rst quarkantiquark pair
inside the rst closed substructure. The number of available slots for the rst gluon
insertion is given by 2k and is enlarged by one for every further gluon. The size of the basis is
4
2
(3.5)
(3.6)
{ 9 {
k n n
0
1
2
3
4
5
of independent color factors for amplitudes with a closed quark loop. For the cases listed explicit
checks have been done for the size of the basis, the number of independent color factors and the
color decomposition. The variable n counts the total number of external particles and k denotes
the number of quarkantiquark pairs. The numbers are explicitly given by (n
1)!=k!.
HJEP04(218)
then computed to be
jDyckn;kj =
(2k
1)!
k!
(2k)(2k + 1)
(n
1) =
(3.7)
(n
k!
1)!
:
The size of the basis is the same as the number of independent color factors, as we check
explicitly for the cases listed in table 1.
For the case of a gauge group with traceless Lie algebra generators the size of the basis
is reduced to (n
2)!(n
k
1)=k!, since we ignore contributions from diagrams with
quark tadpoles. A basis is obtained by removing primitive amplitudes where all externals
lie `inside' a single quarkantiquark pair, i.e. primitive amplitudes corresponding to Dyck
words of the form `(
)', where the last bracket closes the rst one. This prescription can
directly be applied in the discussion of the color decomposition below to obtain a formula
for a decomposition that is minimal for this case.
3.2
Gluonic or mixed loop
We rst treat the case where at least one external quarkantiquark pair is present. As
in the closed quark loop case we start with the notion of cyclic Dyck words, with the
di erence that one has to additionally consider a modi ed scheme for the assignment of
quark/antiquark to opening/closing brackets. We use square brackets to indicate
quarkantiquark pairs with inverted assignment of quark and antiquark. We use the conventions
that we always ip the assignment of all quark lines of a whole substructure of a Dyck word
starting from the outermost level. Consider for example `(()())[[[]]]()' where we ipped the
quarkantiquark assignment for every bracket in the second substructure.
A speci c basis element is denoted by such a modi ed cyclic Dyck word. For example,
given the modi ed Dyck word `()[[]]' with n = 8 and k = 3 we can assign/insert particle
labels in the following way:
Ag[1L2R3R](11234325) = 4
1 ;
(3.8)
3
3
2
2
5
1
(3.9)
(3.10)
HJEP04(218)
where we again
x the loop to lie on the left hand side of normaldirected quark lines
and on the right hand side of inverted quark lines. In this example the quark lines with
label 2 and 3 are inverted in the sense that the antiquark appears before the quark in the
Dyck word.
A possible choice of basis for primitive amplitudes is then given by all modi ed cyclic
Dyck words, where we restrict the modi ed Dyck words to never have a square bracket in
the start (i.e. we do not invert the directions of quark lines in the rst substructure). We
call the corresponding set mDyckn;k such that we can write the basis as
Bn;k>0 = nAg[r( )]( )
g
2 mDyckn;k ;
o
where r represents the quark line routing that the modi ed Dyck word
induces. The
basis contains 22k 1(n
1)!k!=(2k)! elements, where again n is the total number of external
partons and k is the number of quarkantiquark pairs. This can be seen by considering the
number of (unlabeled) Dyck words with possible re ection of all substructures z(k), which
is determined by the recursion relation
k 1
s=0
where Cs is the Catalan number that counts the number of Dyck words. The recursion
works as follows: we count all possible Dyck words inside the rst closed substructure times
all possible Dyck words with re ections that can follow. The recursion relation is solved
k
by z(k) = 2k as one can check explicitly.
To nd the size of the basis we do a similar counting as in the quark case, replacing
the factor Ck s 1 by z(k
s
1):
jmDyckn;kj =
s
1)(s + 1) (k
1)! (2k)(2k + 1)
(n

gluon i{nzsertions
1)
}
(3.11)
independent color factors for amplitudes with a gluonic or mixed loop. For the cases listed explicit
checks have been done for the size of the basis, the number of independent color factors and the
color decomposition. The variable n counts the total number of external particles and k denotes
the number of quarkantiquark pairs. The numbers are explicitly given by 22k 1(n
1)!k!=(2k)!.
HJEP04(218)
The special case where no external quarkantiquark pair is present has already been
described in [58]. In that case the basis consists of all permutations of the external gluons
up to cyclicity, and additionally re ection
g
Bn;0 = fAg(1; ) j
2 Sn 1=Z2g :
(3.12)
Again, the size of the basis is the same as the number of independent color factors for all
combinations of external partons. We explicitly check this for the cases listed in table 2.
We also note that the substructures, i.e. a closed subword at top level of the Dyck word,
are treelevel basis elements, up to a possible global inversion of the quark lines inside the
whole substructure.
The primitive amplitudes appearing in the two bases are su cient to write down a color
decomposition of any oneloop amplitude. In the next section we describe a decomposition
that includes exactly these bases and a set of linearly independent color factors. From
the independence of the color factors in the decomposition it follows that in anomalyfree
theories the primitive amplitudes in the basis are gaugeinvariant quantities. Since one can
make di erent choices of fermion routings we conclude that any planar primitive amplitude
is gaugeinvariant.
4
Oneloop color decomposition
This section presents the main results of this work, a conjecture for the color decomposition
of oneloop QCD amplitudes. The decomposition uses the basis above and is minimal in
the sense that the size of the basis is exactly the same as the number of independent color
factors. We treat the case of a closed quark loop again separately from the rest. The full
amplitudes for a xed number of external particles n, whereof k are quarkantiquark pairs,
is then obtained by summing up contributions from both cases
A(n1;k) = An;k + Agn;k;
q
(4.1)
where the superscript q denotes contributions with a closed quark loops and g contributions
with a mixed or purely gluonic loop.
4.1
The only missing part for the color decomposition is the de nition of the color factors
associated to an element in the basis of primitive amplitudes that will allow us to write
q
An;k = g
n X Cq( )Aq( ):
We drop the superscript for the quark line routings since these are xed by the conventions
HJEP04(218)
described in the above section. For this case the color factors can be obtained by a simple
modi cation of the tree level prescription: in contrast to the tree level case the quark loop
plays the role of the
xed external quarkantiquark pair at the base of the Mario world
diagram. For example, a graphical representation of the color factor associated to the
primitive amplitude in eq. (3.4) is given by
Cq(12321454) =
1
2
3
2
Formally the color factors are conveniently written using the bracket notation from the
tree level case
Cq( ) = fij jig q!jqagg;
where we implicitly sum over the fundamental color index i. Note that l counts the
nestedness of the parton including the loop line.
As an example, we work out the color factor (4.3), dropping the gluon line with label
3 for simplicity. First we plug in the replacement rules and expand the
operators
In the next step we use the evaluation rule for the tensor product and the fundamental
(4.2)
(4.4)
(4.5)
brackets (2.16) to arrive at
= fijf2jT cj2gf1jT
f1jT bj1gf4jT dj4gfijT bT cT dT a5 jig + f1jT bj1gf4jT dT a5 j4gfijT bT cT djig
+ f1jT bT cj1gf4jT dj4gfijT bT dT a5 jig + f1jT bT cj1gf4jT dT a5 j4gfijT bT djig :
Using (2.15) we can write the result in a more convenient notation
= (T c)i2{2 Ti1{1 Tid4{4 Tr(T bT cT dT a5 ) + Ti1{i (T dT a5 )i4{4 Tr(T bT cT d)
b b
+ (T bT c)i1{1 (T d)i4{4 Tr(T bT dT a5 ) + (T bT c)i1{1 (T dT a5 )i4{4 Tr(T bT d) ;
where the traces appear due to the summation over the fundamental index i. This
expression is valid for any choice of the gauge group. A similar treatment can be done for
every color factor appearing in the decomposition. A Mathematica implementation of this
algorithm is provided in an ancillary le.
Note that cutting through the quark loop of the Mario world representation of a color
factor directly recovers a tree level color factor with n + 2 external particles including k + 1
quarkantiquark pairs. Conversely, closing the base quark line of a tree level color factor
produces a looplevel color object.
4.2
Color decomposition for a mixed or gluonic loop
We rst discuss the case where at least one quarkantiquark pair is present. As before we
write the color decomposition as
g
An;k>0 = g
n X Cg( )Ag( ):
The color factors Cg( ) can be given in a recursive way, reusing results from the tree level
case. The graphical notation allows for a short description: the outermost quark lines in the
Dyck word are integrated into the mixed loop. Inside of each of these quark lines the color
factor takes the form of a treelevel color factor. Gluons that are in the outermost level
are directly connected to the loop. Consider the primitive amplitude in eq. (3.8). With
the given prescription one can immediately write down the corresponding color factor in a
(4.6)
(4.7)
(4.8)
HJEP04(218)
diagrammatic form
To nd an expression for the color factors in terms of the tensor product structure as before
we make use of the `zeroth level' of the tensor product that we introduced for the purely
formally de ne the color factors
0a = Taadj. We introduce replacement rules that will allow us to
(4.10)
(4.11)
(4.12)
R = < qclosing
>
>
>
>
8qopening
>>>qopening
>
>
>
>
:
>>> qclosing
! fqjT
b
! jbgfqjT b;
! jqg;
! T bjqgfbj
g !
bg
l
lb 1; l > 1
l = 1
l > 1 ;
l = 1
where qopening stands for either a quark or an antiquark that corresponds to a opening
bracket and similar for the `closing' superscript. The outermost quark lines receive a
contribution to the level zero tensor product using the bracket notation with an adjoint
index b. Note that the replacement rules for quarks at nestedness level l = 1 should in
principle receive an additional minus sign since the gluon line connects from below. This
minus sign cancels out between the opening and closing bracket for every quarkantiquark
pair at this level.
The formula for the color factors then simply reads
where a sum over the adjoint index a of the gluon loop line is implicit. Brackets with
adjoint indices are evaluated using the rules
and extensions thereof for more insertions for Tadj. To exemplify this rule we explicitly
work out the above color factor (4.9). First we use the replacement rules (4.10) and expand
the tensor product inside the
operators
Cg(11234325)
= fajjbgf1jT bT cj1gfcjjdgf2jT df3j(T e
e1) 2a4 j3gT f j2gff j 0a5 jag
= abf1jT T j g
b c 1 cdf2jT df3j(T e
= ab cdf1jT bT cj1gf2jT df3jT
e
T e)(1
T a4 + T a4
1)j3gT f j2gff jTaad5jjag
T eT a4 + T eT a4
T ej3gT f j2gf~fa5a:
(4.13)
Cg( ) = faj jagjR;
fajjbg = ab;
fajTabdjjcg = f~abc;
We can now use the evaluation rule for the tensor product and the fundamental
brackets (2.16)
Cg(11234325)
Cg(11234325)
= f1jT aT cj1gf2jf3jT ej3gT cT eT a4 T f + f3jT eT a4 j3gT cT eT f j2gf~fa5a
= f1jT T j g f2jT cT eT a4 T f j2gf3jT ej3g + f2jT cT eT f j2gf3jT eT a4 j3g f~fa5a:
a c 1
(4.14)
We use (2.15), replacing T
a ! Tsa for the quark lines 2 and 3 with inverted direction
= f1jT aT cj1g f2jTscTseTsa4 Tsf j2gf3jTsej3g + f2jTscTseTsf j2gf3jTseTsa4 3
j g f~fa5a
=
(T aT c)i1{1 (T f T a4 T eT c)i2{2 Tie3{3 + (T f T eT c)i2{2 (T a4 T d)i3{3 f~fa5a:
(4.15)
This result is valid for any choice of gauge group. A Mathematica implementation of this
algorithm is provided in an ancillary le.
The color decomposition for the special case of a purely gluonic amplitude [57] is
obtained by changing the sum in (4.8) to
g
An>2;0 = g
n X Cg(1; )Ag(1; ):
2Sn 1(f2;:::;ng)=Z2
g
A2;0 =
g
2
2
Cg(1; 2)Ag(1; 2):
For the very special case of the twopoint purely gluonic amplitude the color decomposition
comes with an additional factor 1=2
The factor 1=2 is a relict from the symmetry factor of the only contributing diagram. The
de nition of color factors in the purely gluonic case is the same as for the generic case.
4.3
We discuss a complete example for the color decomposition of a 5point, oneloop amplitude
with two quarkantiquark pairs, labeled by 1 and 2, and an external gluon with number
3. For the two types of loop we will rst work out the basis and then write down the
corresponding color factors  using the graphical notation  that make up the full color
decomposition.
of a single gluon:
Let us give the basis of primitive amplitudes in the case of in internal closed quark loop
according to the prescription (3.5) in terms of labeled cyclic Dyck words with the insertion
q
B5;2 =
Aq(13122); Aq(11322); Aq(11232); Aq(11223);
Aq(13221); Aq(12321); Aq(12231); Aq(12213);
Aq(23112); Aq(21312); Aq(21132); Aq(21123) :
where we organized the quark lines such that every line has a xed quark structure and
contains all four possible insertions of the gluon. We also dropped the superscript that
(4.16)
(4.17)
(4.18)
encodes the quark routing since it is the same for all of these amplitudes. The corresponding
color factors in the graphical notation are
1
1
1
1
3
3
where the remaining four color factors can be obtained by a permutation of the quark lines
1 $ 2 of the last four color factors in (4.19).
The primitive amplitudes in (4.18) and the color factors (4.19) contain already the
complete information for this sector of the amplitudes and can be assembled to obtain A5;2
according to (4.2).
The basis of primitive amplitudes with a mixed or gluonic loop contains 16 elements. As
before we organize the primitive amplitudes line by line according to their quark structure
B5;2 = nAg[1L;2L](13122); Ag[1L;2L](11322); Ag[1L;2L](11232); Ag[1L;2L](11223);
g
Ag[1L;2R](13122); Ag[1L;2R](11322); Ag[1L;2R](11232); Ag[1L;2R](11223);
Ag[1L;2L](13221); Ag[1L;2L](12321); Ag[1L;2L](12231); Ag[1L;2L](12213);
Ag[1L;2L](23112); Ag[1L;2L](21312); Ag[1L;2L](21132); Ag[1L;2L](21123) :
o
The second line contains the case where the quark line 2 is inverted. This also implies
a change of the side on which the gluon loop lies with respect to this line, hence the
(4.19)
HJEP04(218)
q
(4.20)
superscript 2R. The corresponding color factors according to (4.11) have the graphical form
1
1
1
1
1
1
3
where the remaining four color factors can again be obtained by a permutation of the quark
lines 1 $ 2.
g
The second part of the full amplitude A5;2 is computed from the primitive
amplitudes (4.20) and the color factors (4.21) using the decomposition sum in (4.8). This
completes the computation of the full oneloop amplitudes in terms of primitive amplitudes
and color factors.
5
Conclusions and outlook
We have explicitly given a basis of planar primitive oneloop amplitudes su cient for
recovering any colordressed oneloop QCD amplitude. The purely kinematic primitive
amplitudes are gaugeinvariant (for theories with no gauge anomalies) and are computed
using colorordered Feynman rules. The number of independent primitive amplitudes for
multiplicity n is in the general case given by (n 1)!=k!+22k 1(n 1)!k!=(2k)! where k counts
the number of quarkantiquark pairs. The rst summand is reduced to (n 2)!(n k 1)=k!
for the case of a gauge group with traceless generators. The two contributions are split
into amplitudes that contain a closed quark loop and amplitudes that have at least one
gluon carrying loop momentum.
The conjectured color decomposition includes a minimal set of linearly independent
primitive amplitudes (under all relations with constant coe cients) and color factors
independent under the Jacobi identity and the commutation relation. The results are
independent of the choice of the gauge group and the number of quark avors, and are applicable
to massless and massive QCD as well as supersymmetric extensions thereof.
Highprecision computations of multiple jet events observed at LHC require a high
multiplicity of external partons. The new color decomposition has the advantage of being
analytically compact and dampens the factorial growth in the number of highermultiplicity
primitive amplitudes. It has as such the potential to improve the e ciency of
phenomenological QCD computations.
Compared to results obtained with techniques for a SU(N ) gauge group [54, 61, 62] we
circumvent the step of assembling primitive amplitudes into partial amplitudes and instead
provide a direct formula to obtain the full colordressed amplitude in terms of primitive
amplitudes. In the former constructions a given primitive amplitude contributes in general
to several partial amplitudes and appears as such several times with a di erent color factor
in the color decomposition. The formulae given here collect all these terms inside one single
color factor for every primitive amplitude in the basis. We have checked that our results
match the full amplitudes obtained in [54, 62] at four points.
We provide a Mathematica implementation for the computation of color factors given
a (modi ed) Dyck word and algorithms that produce the basis of primitive amplitudes in
terms of Dyck words. The ancillary le is attached to the arXiv submission and contains
the examples explicitly worked out in this paper.
For theories with colorkinematics duality [56, 87] there exist further relations between
primitive amplitudes, called BCJ relations. At tree level a color decomposition using this
even smaller subset of independent amplitudes is described in [59]. These relations have
also been understood to arise due to a colorfactor symmetry [88, 89]. Subsequently we
expect the BCJ relations to further reduce of the size of the basis of primitive amplitudes at
oneloop [90{100]. There is also the hope that a systematic treatment of the color algebra
leads to an improved understanding of the kinematic algebra underlying QCD through the
colorkinematics duality.
The `Mario world' structure of the results raises the hope of higherloop generalizations.
A remaining challenge is to properly de ne gaugeinvariant primitive amplitudes for
nonplanar graphs and a basis that induces a compact color decomposition. A simple test case
at twoloops is for example N = 2 supersymmetric QCD [101].
Acknowledgments
I would like to thank Henrik Johansson for a lot of enlightening discussions on the topic and
feedback on the manuscript. I am also grateful to Konstantina Polydorou for comments
on the draft of this paper. The research is supported by the Swedish Research Council
under grant 62120145722, the Knut and Alice Wallenberg Foundation under grant KAW
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
HJEP04(218)
any medium, provided the original author(s) and source are credited.
Lett. 66 (1991) 1669 [INSPIRE].
[arXiv:0704.1835] [INSPIRE].
[2] R. Britto, F. Cachazo and B. Feng, Generalized unitarity and oneloop amplitudes in N = 4
superYangMills, Nucl. Phys. B 725 (2005) 275 [hepth/0412103] [INSPIRE].
111 [hepph/0612277] [INSPIRE].
[7] W.T. Giele, Z. Kunszt and K. Melnikov, Full oneloop amplitudes from tree amplitudes,
JHEP 04 (2008) 049 [arXiv:0801.2237] [INSPIRE].
[8] W.T. Giele and G. Zanderighi, On the Numerical Evaluation of OneLoop Amplitudes: The
Gluonic Case, JHEP 06 (2008) 038 [arXiv:0805.2152] [INSPIRE].
[9] R.K. Ellis, W.T. Giele, Z. Kunszt and K. Melnikov, Masses, fermions and generalized
Ddimensional unitarity, Nucl. Phys. B 822 (2009) 270 [arXiv:0806.3467] [INSPIRE].
[10] R.K. Ellis, W.T. Giele, Z. Kunszt, K. Melnikov and G. Zanderighi, Oneloop amplitudes for
W + 3 jet production in hadron collisions, JHEP 01 (2009) 012 [arXiv:0810.2762]
[11] M. Assadsolimani, S. Becker and S. Weinzierl, A Simple formula for the infrared singular
part of the integrand of oneloop QCD amplitudes, Phys. Rev. D 81 (2010) 094002
[arXiv:0912.1680] [INSPIRE].
(2010) 013 [arXiv:1010.4187] [INSPIRE].
[12] S. Becker, C. Reuschle and S. Weinzierl, Numerical NLO QCD calculations, JHEP 12
[13] S. Becker, C. Reuschle and S. Weinzierl, E ciency Improvements for the Numerical
Computation of NLO Corrections, JHEP 07 (2012) 090 [arXiv:1205.2096] [INSPIRE].
[arXiv:1105.4319] [INSPIRE].
(2013) 014025 [arXiv:1304.1253] [INSPIRE].
[arXiv:1505.00567] [INSPIRE].
HJEP04(218)
[18] Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes
into loop amplitudes, Nucl. Phys. B 435 (1995) 59 [hepph/9409265] [INSPIRE].
[19] Z. Bern, L.J. Dixon and D.A. Kosower, Progress in one loop QCD computations, Ann. Rev.
Nucl. Part. Sci. 46 (1996) 109 [hepph/9602280] [INSPIRE].
[20] R.K. Ellis, W.T. Giele and Z. Kunszt, A Numerical Unitarity Formalism for Evaluating
OneLoop Amplitudes, JHEP 03 (2008) 003 [arXiv:0708.2398] [INSPIRE].
[21] C.F. Berger et al., An Automated Implementation of OnShell Methods for OneLoop
Amplitudes, Phys. Rev. D 78 (2008) 036003 [arXiv:0803.4180] [INSPIRE].
[22] G. Ossola, C.G. Papadopoulos and R. Pittau, CutTools: A Program implementing the OPP
reduction method to compute oneloop amplitudes, JHEP 03 (2008) 042 [arXiv:0711.3596]
[23] P. Mastrolia, G. Ossola, C.G. Papadopoulos and R. Pittau, Optimizing the Reduction of
OneLoop Amplitudes, JHEP 06 (2008) 030 [arXiv:0803.3964] [INSPIRE].
[24] P. Mastrolia, G. Ossola, T. Reiter and F. Tramontano, Scattering AMplitudes from
Unitaritybased Reduction Algorithm at the Integrandlevel, JHEP 08 (2010) 080
[arXiv:1006.0710] [INSPIRE].
[25] S. Badger, B. Biedermann and P. Uwer, NGluon: A Package to Calculate Oneloop
Multigluon Amplitudes, Comput. Phys. Commun. 182 (2011) 1674 [arXiv:1011.2900]
[26] V. Hirschi, R. Frederix, S. Frixione, M.V. Garzelli, F. Maltoni and R. Pittau, Automation
of oneloop QCD corrections, JHEP 05 (2011) 044 [arXiv:1103.0621] [INSPIRE].
[27] F.A. Berends and W.T. Giele, Recursive Calculations for Processes with n Gluons, Nucl.
Phys. B 306 (1988) 759 [INSPIRE].
[28] R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons,
Nucl. Phys. B 715 (2005) 499 [hepth/0412308] [INSPIRE].
[29] R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of treelevel recursion relation
in YangMills theory, Phys. Rev. Lett. 94 (2005) 181602 [hepth/0501052] [INSPIRE].
[30] C.R. Mafra, O. Schlotterer, S. Stieberger and D. Tsimpis, A recursive method for SYM
npoint tree amplitudes, Phys. Rev. D 83 (2011) 126012 [arXiv:1012.3981] [INSPIRE].
[31] F. Cachazo, P. Svrcek and E. Witten, MHV vertices and tree amplitudes in gauge theory,
JHEP 09 (2004) 006 [hepth/0403047] [INSPIRE].
[32] G.J. van Oldenborgh and J.A.M. Vermaseren, New Algorithms for One Loop Integrals, Z.
Phys. C 46 (1990) 425 [INSPIRE].
[33] T. Gehrmann and E. Remiddi, Di erential equations for two loop four point functions,
Nucl. Phys. B 580 (2000) 485 [hepph/9912329] [INSPIRE].
Nucl. Phys. B 646 (2002) 220 [hepph/0207004] [INSPIRE].
[34] C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD,
[hepph/0306192] [INSPIRE].
HJEP04(218)
[hepph/0511176] [INSPIRE].
functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].
[40] C. Anastasiou, C. Duhr, F. Dulat and B. Mistlberger, Soft triplereal radiation for Higgs
production at N3LO, JHEP 07 (2013) 003 [arXiv:1302.4379] [INSPIRE].
[41] F. Caola, J.M. Henn, K. Melnikov and V.A. Smirnov, Nonplanar master integrals for the
production of two o shell vector bosons in collisions of massless partons, JHEP 09 (2014)
043 [arXiv:1404.5590] [INSPIRE].
[42] J.M. Henn, Multiloop integrals made simple: applications to QCD processes, in Proceedings,
49th Rencontres de Moriond on QCD and High Energy Interactions, La Thuile, Italy,
March 22{29, 2014, pp. 289{292 (2014) [arXiv:1405.3683] [INSPIRE].
153001 [arXiv:1412.2296] [INSPIRE].
[44] Y. Zhang, Lecture Notes on Multiloop Integral Reduction and Applied Algebraic Geometry,
2016, arXiv:1612.02249, http://inspirehep.net/record/1502112/ les/arXiv:1612.02249.pdf
[INSPIRE].
[INSPIRE].
[45] M.L. Mangano, The Color Structure of Gluon Emission, Nucl. Phys. B 309 (1988) 461
[46] F.A. Berends and W. Giele, The Six Gluon Process as an Example of WeylVan Der
Waerden Spinor Calculus, Nucl. Phys. B 294 (1987) 700 [INSPIRE].
[47] D. Kosower, B.H. Lee and V.P. Nair, Multi gluon scattering: a string based calculation,
Phys. Lett. B 201 (1988) 85 [INSPIRE].
298 (1988) 653 [INSPIRE].
Nucl. Phys. B 362 (1991) 389 [INSPIRE].
[48] M.L. Mangano, S.J. Parke and Z. Xu, Duality and MultiGluon Scattering, Nucl. Phys. B
[49] Z. Bern and D.A. Kosower, Color decomposition of one loop amplitudes in gauge theories,
[50] P. Cvitanovic, P.G. Lauwers and P.N. Scharbach, Gauge Invariance Structure of Quantum
Chromodynamics, Nucl. Phys. B 186 (1981) 165 [INSPIRE].
(1991) 301 [hepth/0509223] [INSPIRE].
[54] C. Reuschle and S. Weinzierl, Decomposition of oneloop QCD amplitudes into primitive
amplitudes based on shu e relations, Phys. Rev. D 88 (2013) 105020 [arXiv:1310.0413]
[INSPIRE].
[55] R. Kleiss and H. Kuijf, MultiGluon Crosssections and Five Jet Production at Hadron
Colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE].
Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].
[56] Z. Bern, J.J.M. Carrasco and H. Johansson, New Relations for GaugeTheory Amplitudes,
[57] V. Del Duca, A. Frizzo and F. Maltoni, Factorization of tree QCD amplitudes in the
highenergy limit and in the collinear limit, Nucl. Phys. B 568 (2000) 211
[hepph/9909464] [INSPIRE].
[58] 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].
[59] H. Johansson and A. Ochirov, ColorKinematics Duality for QCD Amplitudes, JHEP 01
(2016) 170 [arXiv:1507.00332] [INSPIRE].
[arXiv:1509.03297] [INSPIRE].
[60] T. Melia, Proof of a new colour decomposition for QCD amplitudes, JHEP 12 (2015) 107
[61] Z. Bern, L.J. Dixon and D.A. Kosower, One loop corrections to two quark three gluon
amplitudes, Nucl. Phys. B 437 (1995) 259 [hepph/9409393] [INSPIRE].
[62] H. Ita and K. Ozeren, Colour Decompositions of Multiquark Oneloop QCD Amplitudes,
JHEP 02 (2012) 118 [arXiv:1111.4193] [INSPIRE].
[63] S. Badger, B. Biedermann, P. Uwer and V. Yundin, Numerical evaluation of virtual
corrections to multijet production in massless QCD, Comput. Phys. Commun. 184 (2013)
1981 [arXiv:1209.0100] [INSPIRE].
[64] T. Schuster, Color ordering in QCD, Phys. Rev. D 89 (2014) 105022 [arXiv:1311.6296]
[INSPIRE].
(2017) 100 [arXiv:1612.04366] [INSPIRE].
[65] A. Ochirov and B. Page, Full Colour for Loop Amplitudes in YangMills Theory, JHEP 02
[66] L.J. Dixon, Calculating scattering amplitudes e ciently, in QCD and beyond. Proceedings,
Theoretical Advanced Study Institute in Elementary Particle Physics (TASI95), Boulder,
U.S.A., June 4{30, 1995, pp. 539{584 (1996) [hepph/9601359] [INSPIRE].
[67] T. Melia, Dyck words and multiquark primitive amplitudes, Phys. Rev. D 88 (2013) 014020
[arXiv:1304.7809] [INSPIRE].
Mathematics 225 (2000) 121.
[68] T. Melia, Dyck words and multiquark amplitudes, PoS(RADCOR 2013)031.
[69] P. Duchon, On the enumeration and generation of generalized dyck words, Discrete
[arXiv:1312.0599] [INSPIRE].
HJEP04(218)
[76] R. Britto, Loop Amplitudes in Gauge Theories: Modern Analytic Approaches, J. Phys. A
44 (2011) 454006 [arXiv:1012.4493] [INSPIRE].
Cuts, JHEP 12 (2008) 067 [arXiv:0810.2964] [INSPIRE].
[77] E.W. Nigel Glover and C. Williams, OneLoop Gluonic Amplitudes from Single Unitarity
[78] I. Bierenbaum, S. Catani, P. Draggiotis and G. Rodrigo, A TreeLoop Duality Relation at
Two Loops and Beyond, JHEP 10 (2010) 073 [arXiv:1007.0194] [INSPIRE].
[79] H. Elvang, D.Z. Freedman and M. Kiermaier, Integrands for QCD rational terms and
N = 4 SYM from massive CSW rules, JHEP 06 (2012) 015 [arXiv:1111.0635] [INSPIRE].
[80] S. CaronHuot, Loops and trees, JHEP 05 (2011) 080 [arXiv:1007.3224] [INSPIRE].
[81] R.H. Boels, On BCFW shifts of integrands and integrals, JHEP 11 (2010) 113
[arXiv:1008.3101] [INSPIRE].
[82] C. Baadsgaard, N.E.J. BjerrumBohr, J.L. Bourjaily, S. CaronHuot, P.H. Damgaard and
B. Feng, New Representations of the Perturbative Smatrix, Phys. Rev. Lett. 116 (2016)
061601 [arXiv:1509.02169] [INSPIRE].
[83] R. Huang, Q. Jin, J. Rao, K. Zhou and B. Feng, The Qcut Representation of Oneloop
Integrands and Unitarity Cut Method, JHEP 03 (2016) 057 [arXiv:1512.02860] [INSPIRE].
[INSPIRE].
[85] N. ArkaniHamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local Integrals for Planar
Scattering Amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [INSPIRE].
[86] J.M. Drummond, J. Henn, V.A. Smirnov and E. Sokatchev, Magic identities for conformal
fourpoint integrals, JHEP 01 (2007) 064 [hepth/0607160] [INSPIRE].
[87] 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].
[88] R.W. Brown and S.G. Naculich, Colorfactor symmetry and BCJ relations for QCD
amplitudes, JHEP 11 (2016) 060 [arXiv:1608.05291] [INSPIRE].
[89] R.W. Brown and S.G. Naculich, BCJ relations from a new symmetry of gaugetheory
amplitudes, JHEP 10 (2016) 130 [arXiv:1608.04387] [INSPIRE].
[90] R.H. Boels, R.S. Isermann, R. Monteiro and D. O'Connell, ColourKinematics Duality for
OneLoop Rational Amplitudes, JHEP 04 (2013) 107 [arXiv:1301.4165] [INSPIRE].
HJEP04(218)
gauge theories and gravity, Nucl. Phys. B 930 (2018) 328 [arXiv:1706.00640] [INSPIRE].
JHEP 10 (2017) 105 [arXiv:1707.05775] [INSPIRE].
amplitudes, arXiv:1711.09104 [INSPIRE].
JHEP 03 (2018) 068 [arXiv:1711.09923] [INSPIRE].
integral relations, JHEP 12 (2017) 122 [arXiv:1710.11010] [INSPIRE].
supergravity amplitudes, JHEP 09 (2017) 019 [arXiv:1706.09381] [INSPIRE].