#### Non-planar one-loop Parke-Taylor factors in the CHY approach for quadratic propagators

HJE
Non-planar one-loop Parke-Taylor factors in the CHY approach for quadratic propagators
Naser Ahmadiniaz 1 2 4
Humberto Gomez 1 2 3
Cristhiam Lopez-Arcos 1 2 3
Algebraic Geometry
0 theory. As a byproduct , we found
1 Campus Pampalinda , Calle 5 No. 62-00, Codigo postal 76001, Santiago de Cali , Colombia
2 61005 Gwangju , Korea
3 Universidad Santiago de Cali, Facultad de Ciencias Basicas
4 Center for Relativistic Laser Science, Institute for Basic Science , IBS
In this work we have studied the Kleiss-Kuijf relations for the recently introduced Parke-Taylor factors at one-loop in the CHY approach, that reproduce quadratic Feynman propagators. By doing this, we were able to identify the non-planar one-loop Parke-Taylor factors. In order to check that, in fact, these new factors can describe nonplanar amplitudes, we applied them to the bi-adjoint a new type of graphs that we called the non-planar CHY-graphs. These graphs encode all the information for the subleading order at one-loop, and there is not an equivalent of these in the Feynman formalism.
Scattering Amplitudes; Superstrings and Heterotic Strings; Di erential and
1 Introduction
2 Planar Parke-Taylor factors at one-loop 3 From KK to BDDK relations and non-planar Parke-Taylor factors at one-loop
KK relations for the partial one-loop Parke-Taylor factors
3.2 The BDDK relations for the one-loop Parke-Taylor factors
4.1
4.2
5.1
5.2
Four-point
Five-point
4 Bi-adjoint
3 scalar theory
Full bi-adjoint
5 Examples and non-planar CHY-graphs
6 General non-planar CHY-graphs
7 Discussions
A Four-point non-planar CHY-graph computation B Four-point amplitude from the Feynman rules 1 6
tudes, following Witten's work in 2003 [1], the proposal by Cachazo-He-Yuan (CHY) [2, 3]
o ers some advantages. The CHY formalism applies to several dimensions and also to a
large array of theories [4{7], that go even beyond eld theory [8{12]. The formalism is
based on the scattering equations (at tree-level)
Ea := X ka kb = 0;
b6=a
ab
ab := a
b
; a = 1; 2; : : : ; n;
with the a's denoting the local coordinates on the moduli space of n-punctured Riemann
spheres and ka2 = 0. To obtain the tree-level S-matrix we have to perform a contour integral
localized over solutions of these equations, i.e.
An =
Z
d tnree ItCreHeY( );
{ 1 {
where the integration measure, d tnree, is given by
d tnree =
Qn
a=1 d a
Vol (PSL(2; C))
Qn
( ij jk ki) ;
b6=i;j;k Eb
and the contour
is de ned by the n
3 independent scattering equations
Eb = 0; b 6= i; j; k :
A di erent integrand, ItCreHeY, describes a di erent theory. In the study of the scattering
equations at tree-level and the development of techniques for integration, many approaches
have been formulated [4, 13{36]. In particular, the method of integration that we use in
the calculations for the present paper was developed by one of the authors in [37], which
is called the
-algorithm.
The following step for the CHY formalism is going to loop corrections. A prescription
that allows to go to higher genus Riemann surfaces was developed in [38{42], which is called
the ambitwistor and pure spinor ambitwistor string theory. Another alternative approach
employing an elliptic curve was proposed in [43, 44] by one of the authors. Additionally,
in [45{47] they took an approach from tree-level, by introducing the forward limit with two
additional massive particles that played the role of the loop momenta.
The premise for the previously mentioned prescriptions at one-loop, is that from the
CHY formalism a new representation for the Feynman propagators arise, the so called
linear propagators, which look like (2` K + K2) 1 [41, 48]. Many interesting developments
in one-loop integrands identities (like the Kleiss-Kuijf (KK) identities [49]) and dualities
(e.g. the Bern-Carrasco-Johansson (BCJ) color-kinematics duality [50]) have been found
from the CHY approach, suported over the tree-level scattering equations with two extra
massive particles [51{53].
The linear propagators approach still leaves several open questions. First, there is no
direct way to relate some of the results with the ones from traditional eld theory, like the
BCJ numerators for example. Another question is about direct loop integration, and to see
if it is more e cient to compute these new integrals that appear, in comparison with many
well known and long time developed integration techniques for the traditional Feynman
propagators.
Currently, we are developing a program to obtain the traditional quadratic Feynman
propagators, (` + K) 2, directly from the CHY. The rst proposal came by one of the
authors in [54] for the 3 scalar theory. Following, two of the current authors in [55] presented
a reformulation for the one-loop Parke-Taylor factors that gives quadratic propagators, but
it was just made for leading (or planar) contributions. These new Parke-Taylor factors were
successfully tested in the massless bi-adjoint
extra particles come from splitting the extra two massive loop momenta (`+; ` ) into four
massless ones ((a1; b1); (b2; a2)). The splitting was motivated by taking an unitary cut on
a n-point Feynman diagram at two-loop (see gure 1), that leave us with a (n + 4)-point
{ 2 {
(1.3)
(1.4)
combine in order to give the loop momentum in the forward limit [54{56].
This work was motivated by the tree-level KK relations, which were found for gluons
at tree-level in 1988 [49]. Let us remind the algebraic relations that the partial amplitudes
satisfy
Atree( 1; : : : ; s; 1; 1; : : : ; r; n) = ( 1)s
Atree(1; ; n);
(1.5)
X
2OPf gf g
where the order preserving product (OP) merges the ordering
T into the ordering .
An one-loop version of the previous relations was found by Bern-Dixon-Dunbar-Kosower
in [57] (so they may be called the BDDK relations instead of the KK relations, since the
later are only at tree-level). Their outline for the proof relies on the structure constants, so
the result holds for any one-loop gauge theory amplitude where the external particles and
the particles circulating around the loop are both in the adjoint representation of SU(N).
This relation reads1
An;r
1 loop(1; : : : ; r
1 ; r; : : : ; n) = ( 1)r
X
2COPf gf g
An;1
1 loop( ):
(1.6)
The amplitudes we have on the left hand side belong to the non-planar sector of the
perturbative expansion. These amplitudes are far less understood than the planar ones,
starting with the notion of the integrand which is ambiguous. At one-loop order there is
no way to draw a Feynman diagram that could lead us to the integrand and we can only
make an incursion into them using the previous relations. From another point of view,
twistor methods like the ones from [58] apply only to planar diagrams, so the rst obstacle
to face from this perspective is the lack of duals for non-planar diagrams.
From a physics point of view, 't Hooft's large N limit [59] shows that the planar sector
of the amplitudes for a gauge theory with color group U(N ) (with xed 't Hooft coupling)
is dominating, so by going beyond and studying these non-planar interactions we can get a
better understanding of that limit, where the 't Hooft diagrams have a cylindrical topology.
1The cyclic ordering preserving product (COP) merges the cyclic permutations of f g = fr
1; : : : ; 1g
leaving the f g = fr; r + 1; : : : ; ng ordering
one at tree-level, and makes the proof for the relation a non-trivial task.
xed. This merging of the orderings is quite di erent from the
{ 3 {
In twistor string, a non-planar open string amplitude at one-loop (scattering of gluons) has
the topology of that of a closed string, so it is a cylinder (an annulus with vertex operators
on the two boundaries). This brings the inclusion of conformal graviton exchanges, which
may presents itself as a drawback to the formalism.
The idea in the present work is to understand the amplitudes in the non-planar
sector from the perspective of the CHY formalism. In this formalism, the ordering in the
amplitudes is directly associated with the Parke-Taylor factors, so our main focus will be
on those. Since our Parke-Taylor factors for the n-point one-loop amplitude come from
those at tree-level with (n + 4)-point, we will analyse the KK relations in order to extract
the BDDK relations for the amplitudes. At tree-level there is already a version of the KK
relations for the Parke-Taylor factors in the CHY formalism [52].
One important aspect to realize is that not all of the terms in the (n + 4)-point KK
relations contribute to what may be the n-point BDDK relations for the Parke-Taylor
factors. After identifying what contributes to one-loop there is a simple and powerful
result, a relation for the partial one-loop Parke-Taylor factors that involves the regular
shu e product for the orderings. The relations for the partial factors can be promoted
to relations for the full one-loop Parke-Taylor factors, this process involves taking cyclic
permutations and rearranging the partial factors. It is exactly after that promotion that
we nd the BDDK relations for them.2 In this way, we provide a new proof for the BDDK
relations for the amplitudes from a di erent approach. Along the path, a more general
type of relations appear, and for those we still don't have a direct physical interpretation.
It is known that the BDDK relations connect the non-planar amplitudes with the
planar ones, this will allow us to identify what would be the non-planar one-loop
ParkeTaylor factors, which are the building blocks for such amplitudes. Employing these new
objects in the construction of the CHY integrands, will lead to the identi cation of the
nonplanar CHY-graphs. This new type of graphs do not have an equivalent in the traditional
formalism and encode all the information for the non-planar order in a single expression,
they also provide a clear de nition of the integrand, at the level of the CYH formalism.
The developed non-planar CHY-graphs are applied to the massless Bi-adjoint
3 theory,
allowing us to explore the non-planar sector of the theory at one-loop, we will obtain all
the integrands which contribute to the amplitude at this quantum correction.
Outline.
The present work is organized as follows. In section 2 we review the one-loop
Parke-Taylor factors for planar corrections that were proposed in [55], with a few changes
in the notation and de ne the partial planar one-loop Parke-Taylor factor, which will be
studied in depth in the following sections.
Section 3 is divided in two parts. In the rst one we take the previously presented
partial Parke-Taylor factors, which are tree-level type ones, and apply the KK relations
on them in order to obtain the BDDK relations at one-loop. The development starts by
analysing the tree-level relations for n + 4 particles, from there it becomes clear that not all
of the terms contribute to the one-loop case, but a sector from them. With that analysis we
2The BDDK relations involve a more elaborated shu e product for the orderings, the COP product
that we mentioned before.
{ 4 {
nd the simplest one-loop case, from there we will go to the most general relation possible
at one-loop, this is possible by the realization of an algebraic relation that includes the
shu e product. The next part is where we nd the BDDK relations for the full one-loop
Parke-Taylor factors, these being a particular case which resemble the relations for gluon
amplitudes. We also nd some relations that generalize the BDDK ones.
In section 4 we employ the one-loop non-planar Parke-Taylor factors in the study of
one-loop amplitudes for the massless Bi-adjoint
3 theory. Since the CHY integrands for
this theory are given by the product of two Parke-Taylor factors, the contributions for
the one-loop amplitudes come from di erent sectors: the planar, the non-planar, and the
mixed one (planar; non-planar). The Bi-adjoint
3 theory plays a role in the double copy
between supersymmetric Yang-Mills and supergravity [60], also in color-kinematic relations
for scalar e ective eld theories [61]. Therefore, having and understanding the non-planar
sector has great value in the development of more relevant theories.
The non-planar CHY-graphs will be introduced in section 5 with some particular
worked out examples for the four-point and
ve-point cases. In section 6 we present the
general non-planar CHY-graphs for an arbitrary number of points. Finally, we present
our conclusions in section 7. Additionally, in order to make the paper self contained, we
provide the detailed calculations of the four-point amplitude from the CHY formalism in
appendix A and its exact equivalence with the results obtained from the standard method
based on the Feynman rules in appendix B.
Notation. For convenience, in this paper we use the following notation
(1.7)
(1.8)
ij := i
j ;
!ia::jb :=
ab :
ia jb
integrands at two-loop. In addition, we de ne the ab's and !ia::jb's chains as
Note that !ia::jb is the generalization of the (1; 0)-forms used in [56] to write the CHY
(i1; i2; : : : ; ip) := i1i2
(i1; i2; : : : ; ip)a!:b := !ia1::bi2
ip 1ip ipi1 ;
!a:b
ip 1:ip !iap::bi1 = !ia1::bi1
!a:b
ip 1:ip 1 !iap::bip ;
To have a graphical description for the CHY integrands on a Riemann sphere
(CHYgraphs), it is useful to represent each
a puncture as a vertex, the factor 1
as a line
ab
and the factor ab as a dashed line that we call the anti-line. Additionally, since we often
use the
-algorithm3 [37], then we introduce the color code given in
gure 2 and 3 for a
mnemonic understanding.
Finally, we introduce the momenta notation
m
i=1
{ 5 {
ka1;:::;am :=
X kai = [a1; : : : ; am];
2
sa1:::am := ka1;:::;am ;
s~a1:::am :=
kai kaj :
3It is useful to recall that the -algorithm xes four punctures, three of them by the PSL(2; C) symmetry
and the last one by the scale invariance.
m
X
ai<aj
unfixed massless puncture
massless puncture fixed by PSL(2,C)
massive puncture fixed by PSL(2,C)
massless puncture fixed by scale symmetry
CHY prescription. approach, it is given by the following
2
Planar Parke-Taylor factors at one-loop
In a previous work, we found a reformulation for the Parke-taylor factors that leads to
quadratic Feynman propagators by evaluating only massless scattering equations in the
Let us remind the expression for the Parke-Taylor factor at tree-level in the CHY
where \ " is a generic ordering and n is the total number of particles. In addition, it will
be useful to de ne a reduced Parke-Taylor factor, which does not involve all n particles, i.e.
PT[ ] =
1
( 1; 2; : : : ; n)
;
PcT[i1; : : : ; ip] :=
1
(i1; : : : ; ip)
;
(2.1)
(2.2)
where p < n.
As it was shown in [55], the planar one-loop Parke-Taylor factor (on the left-sector,
denoted by a-sector) for quadratic propagators is given by4
PT(a11):a2 [ ] := PT[ 1; 2; : : : ; n; a1; b1; b2; a2] + cyc( )
=
=
1
PcT[ 1; : : : ; n; a1; a2] :
!a1n::a21
(2.3)
4As it was explained in [54, 55], in order to compute scattering amplitudes at one-loop we must perform
the forward limit, ka1 =
ka2 (kb1 =
kb2 ). So, from the third line in (2.3), it is clear this de nition is
totally analog to the one given in [45, 46] by the expression
X
2cyc( )
PT(1)[ ] :=
PT[ 1; : : : ; n; +; ];
which is just able to reproduce linear propagators.
{ 6 {
It is useful to remember that the Parke-Taylor factor on the right-sector (or b-sector)
is de ned to be
PT(b11):b2 [ ] :=
PT(b11):b2 [ ], is always a function of the couples, (ka1 + kb1 ) and (ka2 + kb2 ).
This means, these type of integrals always produce quadratic propagators at one-loop by
the identi cation, (ka1 + kb1 ) =
(ka2 + kb2 ) = `.
Each one of the terms of the sum in (2.3) and (2.4) is called partial planar one-loop
Parke-Taylor factor which we denote them by pt(a11):a2 [i1; : : : ; ip] (pt(b11):b2 [i1; : : : ; ip]), namely
and for the b-sector we must just perform the replacement, (a1; a2) ! (b1; b2). Therefore,
for the rest of this work it is enough to work on the a-sector.
In [55], we found several algebraic manipulations on these Parke-Taylor factors that
allowed us to obtain the contributing planar diagrams for the bi-adjoint scalar
In order to go beyond the planar case, let us analyse a known relation between planar and
PT[ff1g f2; 3; 4; : : : ; n 1g; ng] := PT[1; 2; 3; 4; : : : ; n]+PT[2; 1; 3; 4; : : : ; n]
{ 7 {
(2.5)
(2.6)
(2.7)
where we have introduced the shu e product \
",5 just for a simple case of the rst
entry merging in the next entries up until the one at the (n-1) position, we will generalise
this to two lists of arbitrary size and even beyond two lists. This will set the root of our
developments towards the case of one-loop. Note that we are making the following abuse
in the notation
PT[A1 + A2 +
where Ai is an ordered list of n di erent elements.
The relation (3.1) will set the starting point for our analysis.
KK relations for the partial one-loop Parke-Taylor factors
Translating the relation (3.1) for the partial one-loop Parke-Taylor factors, which from a
technical point of view are still at tree-level, it can be written as
PT[1; 2; : : : ; n; a1; b1; b2; a2]+X PT[2; : : : ; i; 1; i+1; : : : ; n; a1; b1; b2; a2]+PT[2; : : : ; n; 1; a1; b1; b2; a2]
+PT[2; : : : ; n; a1; 1; b1; b2; a2]+PT[2; : : : ; n; a1; b1; 1; b2; a2] =
PT[2; : : : ; n; a1; b1; b2; 1; a2]:
(3.3)
Note that in the terms on the second line in (3.3) there is a splitting in the points
assigned to the loop momenta. Those terms are perfectly normal at tree-level, but at loop
level they would not lead to any quadratic Feynman propagator in the forward limit, in
other words, the identi cation, (ka1 + kb1 ) =
(ka2 + kb2 ) = `, is not well de ned on them.
In any case, we can analyse the terms where the splitting is not present.
Taking the sector of (3.3) where the loop momenta points are not split, we can nd a
di erent relation for the partial one-loop Parke-Taylor factors. The new relation is given by
PT[1; 2; 3; : : : ; n; a1; b1; b2; a2] + X PT[2; 3; : : : ; i; 1; i + 1; : : : ; n; a1; b1; b2; a2]
+ PT[2; 3; : : : ; n; 1; a1; b1; b2; a2] = !1a:11:a2 PcT[2; 3; : : : ; n; a1; b1; b2; a2]:
Again, with the introduction of the shu e product, this relation can be written in a
compact and more legible way as follows
pt(a11):a2 [f1g f2; : : : ; ng] := pt(a11):a2 [1; 2; : : : ; n]+pt(a11):a2 [2; 1; : : : ; n]+
+pt(a11):a2 [2; : : : ; n; 1]
= pt(a11):a2 [1] pt(a11):a2 [2; : : : ; n]:
The result from above can be generalized to lists of arbitrary size. In fact, we have been
able to
nd a more general relation for the partial planar one-loop Parke-Taylor factors,
which is given by the identity
General KK relations for the partial one-loop Parke-Taylor factors
pt(a11):a2 [fi1; i2; : : : ipg
fip+1; : : : iqg
5It is useful to remember the shu e product, f 1; : : : pg
f 1; : : : qg, has a total of (pp+!qq!)! terms.
(3.4)
(3.5)
{ 8 {
One particular case is to take all the lists with size 1, a straightforward calculation
gives the result
pt(a11):a2 [f1g f2g f3g
fng] = pt(a11):a2 [1] pt(a11):a2 [2]
1 := 1 and Sn 1 is the set of all permutations of f2; 3; : : : ; ng. Let us recall that
the expression in (3.6) reproduces the symmetrized n-gon at one-loop.
Another particular case is by considering only two lists, taking the canonical ordering
we have the following
pt(a11):a2 [f1; 2; : : : pg
fp + 1; : : : ng] = pt(a11):a2 [1; 2; : : : p]
pt(a11):a2 [p + 1; : : : n];
(3.7)
we obtain a similar structure as the BDDK relation between the planar and non-planar
amplitudes at one-loop [57, 62], but the shu e is di erent, since we are not dealing with full
one-loop Parke-Taylor factors yet. We de ne the partial non-planar one-loop Parke-Taylor
factors as
pt(a11):a2 [i1; i2; : : : ; ip j ip+1; : : : ; n] := pt(a11):a2 [fi1; i2; : : : ipg
fip+1; : : : ing]
(3.8)
= pt(a11):a2 [i1; i2; : : : ; ip]
pt(a11):a2 [ip+1; : : : ; in]:
Taking the sum over the corresponding permutations we build the full non-planar one-loop
Parke-Taylor factors 6 as
PT(a11):a2 [ 1; : : : ; p j p+1; : : : ; n] :=
=
1
1
(a1; b1; b2; a2) 2cyc( )
(a1; b1; b2; a2) 2cyc( )
X
where f 1; : : : ; pg and f p+1; : : : ; ng are two di erent generic orderings such that
f 1; : : : ; pg \ f p+1; : : : ; ng = ; and f 1; : : : ; pg [ f p+1; : : : ; ng = f1; 2; : : : ; ng.
The partial and full non-planar one-loop Parke-Taylor factors de ned in (3.8) and (3.9)
can be generalized, now for an arbitrary number of lists, in the following way
pt(a11):a2 [i1; : : : ; ip j ip+1; : : : ; iq j
:= pt(a11):a2 [fi1; : : : ; ipg
j im; : : : ; in]
fip+1; : : : ; iqg
6Note that the partial non-planar one-loop Parke-Taylor factors for quadratic propagators can be
written as
1
(a1; b1; b2; a2) pt(a11):a2 [1; 2; : : : ; p j n; n 1; : : : ; p + 1]
= ( 1)n p
This expression is totally similar to the one given for non-planar linear propagators in [47, 52], where
a1 ! + and a2 !
and we can write the full version in terms of these like
PT(a11):a2 [ 1; : : : ; p j p+1; : : : ; q j
j m; : : : ; n]
:=
1
where f 1; : : : ; pg; f p+1; : : : ; ng; : : :f m; : : : ; ng, are di erent generic orderings such that
f 1; : : : ; pg [ f p+1; : : : ; ng [
[ f p+1; : : : ; ng = f1; 2; : : : ; ng and they are disjoint,
meaning they have no element in common. Although these generalizations are well de ned
in the CHY side, we still do not have an understanding of what would be their physical
meaning. So far we have worked with tree-like expressions so the KK relations still applied,
in the following subsection we shall see the one-loop behaviour of the relations.
3.2
The BDDK relations for the one-loop Parke-Taylor factors
In the previous section we found general relations for the partial one-loop Parke-Taylor
factors, these are a sector of the KK relations. In the following section our interest lies in
nding BDDK relations that involve the full one-loop Parke-Taylor factors de ned in (2.3)
and (3.9).
To obtain full one-loop Parke-Taylor factors on the right hand side of (3.9) we have to
expand the sum and collect the terms related by a cyclic permutation. The result can be
arranged again in a sum as follows
PT(a11):a2 [ 1; : : : ; p j p+1; : : : ; n] =
PT(a11):a2 [ 1; : : : ; n];
(3.12)
X
2cyc( ) = n
where cyc( )
= n
fcyc( )
f p+1; : : : ; n 1g; ng. As an example, for the non-planar
ordering [1; 2j3; 4; 5] we have the relation
PT(a11):a2 [1; 2j3; 4; 5] = PT(a11):a2 [1; 2; 3; 4; 5] + PT(a11):a2 [1; 3; 2; 4; 5] + PT(a11):a2 [1; 3; 4; 2; 5]
+ PT(a11):a2 [2; 1; 3; 4; 5] + PT(a11):a2 [3; 1; 2; 4; 5] + PT(a11):a2 [3; 1; 4; 2; 5]
+ PT(a11):a2 [2; 3; 1; 4; 5] + PT(a11):a2 [3; 4; 1; 2; 5] + PT(a11):a2 [2; 3; 4; 1; 5]
+ PT(a11):a2 [3; 2; 4; 1; 5] + PT(a11):a2 [3; 4; 2; 1; 5] + PT(a11):a2 [3; 2; 1; 4; 5]:
(3.13)
This product for two orderings has been usually denoted in the literature as
COPf Tgf g, where
T is the reversed ordering of . The proof of this result for
gluons subamplitudes can be found in [57], where they approach it from the string theory and
eld theory sides, here we have obtained it as a consequence of the regular shu e product
that appears at the tree-level KK relations.
Finding the relation between the full one-loop Parke-Taylor allows us to see more
clearly the bridge between CHY and the subamplitudes from the traditional eld theory
approach. Another important point is that a result in [55] shows an expansion of the
planar Parke-Taylor factors at one-loop in terms of !'s, which displays an advantage from
a computational point of view.
Going to the more general case in (3.11), the same analysis can be performed to nd
a relation with the full one-loop Parke-Taylor factors. After expanding and collecting the
terms we are left with
General BDDK relations for the full one-loop Parke-Taylor factors
PT(a11):a2[ 1; : : : ; p j p+1; : : : ; q j
j m; : : : ; n] =
P
2cyc( ) cyc( ) ::: = n
PT(a11):a2[ 1; : : : ; n];
where the product of orderings follows the same de nition given after (3.12). Here even
the simplest non-trivial example (that could be [1; 2j3; 4j5; 6]) gives a considerable number
of terms to t on a single page.
4
Bi-adjoint
3 scalar theory
Having found the non-planar Parke-Taylor factors in the previous section, we will apply
them to the simplest non-trivial theory we can build from them, the Bi-adjoint
3 scalar
theory. These new factors will allow us to dive directly into the non-planar sector of the
theory. In the CHY prescription we will have a clearly de ned integrand for a non-planar
amplitude, something that does not happen in the traditional formalism, and opens a new
gate to explore these interactions.
Here we propose a non-planar CHY prescription for the S-matrix at one-loop for the
bi-adjoint
3 scalar theory. By construction, with our Parke-Taylor factors we will be able
reproduce directly the quadratic propagators, which are the important ones if one wants
to study the behaviour of the singularities in the non-planar sector, although the analysis
of singularities is beyond the scope of the present work.
4.1
Full bi-adjoint
3 amplitude at one-loop
Along the line of reasoning of [47] and with the partial Parke-Taylor factors de ned
in section 2, we de ne the full bi-adjoint
3 amplitude at one-loop with
avor group
Tr(T i 1 T i 2
T i n T i a1 T i a2 )
1 Z
2n+1
d tn+4 (a1; b1; b2; a2) PcT[ 1; : : : ; n; a1 ; a2 ]
(a1; b1; b2; a2) PcT[ 1; : : : ; n; b1 ; b2 ]
and d tn+4, are given by the expressions7 [3, 54]
d := dD(ka1 +kb1 ) (D)(ka1 +kb1 `) dDka2 dDkb2
(D)(ka2 +ka1 ) (D)(kb2 +kb1 );
Therefore, the Dirac delta functions in (4.2) are well de ned.
7In this paper we are considering that the D-dimensional momentum space is real, i.e. ki 2 R
D 1;1.
(4.1)
(4.2)
and
d tn+4 :=
QnA+=41 d A
Vol (PSL(2; C))
xing PSL(2;C)
contour is de ned by the massless scattering equations
EA :=
B=1
B6=A
nX+4 kA kB = 0;
AB
where we are making the following identi cations
Finally, without loss of generality, note that in (4.3) we have
xed f 1; b1 ; b2 g and
fE1; Eb1 ; Eb2 g.
It is very important to remark that the proposal given in (4.1) is well de ned. After
performing the contour integral, R d tn+4 , the result obtained has a functional dependence
of the loop momenta couples8 like (ka1 + kb1 ) and (ka2 + kb2 ), therefore, the Dirac delta
functions in d
can be carried out without any inconvenience.
As it has been argued in [47], since we are looking for the forward limit with the
measure (D)(ka1 + ka2 )
(D)(kb1 + kb2 ) in (4.2), then it requires that when summing
over the U(N ) (and U(N~ )) degrees of freedom of the two internal particles they must be
identi ed. Being more precise, we must introduce the sum
where N 2 (N~ 2) is the dimension of U(N ) (U(N~ )). Now, by using the identities9
A = 1; 2; : : : ; n + 4;
with
8It is guaranteed by the KK relations in (3.7) and the footnote 6. Since the partial non-planar one-loop
Parke-Taylor factors can always be written as a linear combination of the partial planar one-loop
ParkeTaylor factors then, as it was explained in the appendix A in [55], this implies the CHY-integral is always
a function of the momenta (ka1 + kb1 ) and (ka2 + kb2 ).
9The second line in (4.7) is known as the re ection identity, which is simple to check for the adjoint
representation.
m1n loop = 4
(N N~ )
[ j ; j ] are given by
as Mn
and mn
1 loop
(1 NP;NP)
(1 P;P)
where P(NP) means planar (non-planar), mn
[ ; ] is the same amplitude de ned in [55]
[ j ], and the non-planar contributions, mn
(1 P;NP)
(1 NP;P)
[ ; j ] (mn
[ j ; ])
q=1 2Sn p=Zn p; 2Sn q=Zn q
2Sp=Zp; 2Sq=Zq
X
(a1; a2) (b1; b2)
(a1; b1; b2; a2)2
X
CHY
I(1 P;NP)[ 1; : : : ; n ; 1; : : : ; pj p+1; : : : ; n] :=
Finally, the amplitude, mn
[ j ; ], is de ned in a similar way to the one given
m(n1 P;NP)[ 1; : : : ; n ; 1; : : : ; pj p+1; : : : ; n] :=
1
2n+1
Z
dD`
Z
d
sa1b1
d
t
n+4
CHY
I(1 P;NP)[ 1; : : : ; n ; 1; : : : ; pj p+1; : : : ; n];
m(n1 NP;NP)[ 1; : : : ; pj p+1; : : : ; n ; 1; : : : ; qj q+1; : : : ; n] :=
1
2n+1
Z
dD`
Z
d
d
t
n+4
CHY
I(1 NP;NP)[ 1; : : : ; pj p+1; : : : ; n ; 1; : : : ; qj q+1; : : : ; n];
PcT[ 1; : : : ; n; a1; a2]
PcT[ 1; : : : ; q; b1; q+1; : : : ; n; b2];
CHY
I(1 NP;NP)[ 1; : : : ; pj p+1; : : : ; n ; 1; : : : ; qj q+1; : : : ; n] :=
(a1; a2) (b1; b2)
(a1; b1; b2; a2)2
PcT[ 1; : : : ; p; a1; p+1; : : : ; n; a2]
PcT[ 1; : : : ; q; b1; q+1; : : : ; n; b2]:
Since the bi-adjoint theory comes with double trace, we have mixed sectors which are
not entirely non-planar. Our focus here will be in the de nition of the integrands for such
sectors and how they are related to the planar sector.
In the following sections we are going to focus on these non-planar amplitudes, we will
give a few examples and de ne the non-planar CHY-graphs.
In the previous section we have de ned the non-planar 3 amplitudes at one-loop. In [55],
a technology to deal with this type CHY structures and graphs was developed, one of the
ideas to remember from there is that by integrating a CHY integrand we can obtain several
Feynman integrands and therefore all the information about the possible interactions at a
certain order. For the planar case the relation CHY-graphs/Feynman-diagrams was mainly
one-to-one, no on the non-planar case we will see that is not the case any more, but again a
non-planar CHY diagram contains all the information for the interactions at that level. The
new integrands that we have to deal with are I(C1HY
P;NP)[ ; j ] and I(C1HY
NP;NP)[ j ; j ], so
let us see how the look by using the Parke-Taylor factors we found in 3.
From the identity in footnote 6, it is straightforward to see
I(C1HYP;NP)[ 1; : : : ; n ; 1; : : : ; pj p+1; : : : ; n]
= PT(a11):a2 [ 1; : : : ; n] PT(b11):b2 [ 1; : : : ; p j n; n 1; : : : ; p+1];
I(C1HYNP;NP)[ 1; : : : ; pj p+1; : : : ; n ; 1; : : : ; qj q+1; : : : ; n]
= PT(a11):a2 [ 1; : : : ; p j n; n 1; : : : ; p+1] PT(b11):b2 [ 1; : : : ; q j n; n 1; : : : ; q+1]:
So, for convenience we de ne:
for bi-adjoint 3 scalar theory are de ned as
De nition. The non-planar partial amplitudes, M(n1 P;NP)[ ; j ] and M(n1 NP;NP)[ j ; j ],
(4.13)
(4.14)
(4.15)
(4.16)
M(n1 P;NP)[ 1; : : : ; n ; 1 : : : ; p j p+1; : : : ; n]
:= m(n1 P;NP)[ 1; : : : ; n ; 1 : : : ; p j n; : : : ; p+1];
M(n1 NP;NP)[ 1; : : : ; pj p+1; : : : ; n ; 1 : : : ; q j q+1; : : : ; n]
:= m(n1 NP;NP)[ 1; : : : ; pj n; : : : ; p+1 ; 1 : : : ; q j n; : : : ; q+1]:
In a similar way we de ne M(n1 NP;P)[ j ; ].
with the eld theory results.
5
Examples and non-planar CHY-graphs
In the next section we will present some examples, which can be directly compared
Let us now translate the integrands we just introduced into their corresponding
CHYgraphs in order to get more information from them. In this section we compute some
particular examples for lower number of points, n = 4; 5 by using the new proposal given
in section 4.2. Moving forward, we will de ne and analyse the more general structure of
the non-planar CHY-graphs at one-loop in a simple way, which will be presented in the
next section.
Before giving the examples, it is useful to introduce the line notation10
"m" anti- lines
1
1
2
2
;
where the number of anti-lines \m" is equal to, m = # lines 4. Finally, in order to obtain
a more compact notation, we bring in the following de nitions
HJEP05(218)
n l
= [tree 1; tree 2; : : : ; tree !n] + [tree 2; : : : ; tree n; tree !1] +
+ [tree n; tree 1; : : : ; tree (n 1)] ;
!ia:1j:a2 , while for PT(b11):b2 [
5.1
where the arrow over the bracket, [: : !:] , means the transit of the loop momentum \`" and
\tree i" is a generic Feynman diagram at tree level. For simplicity in the notation we omit
the integral, R dD`.
Finally, in order to obtain a correspondence between the CHY-graphs for non-planar
bi-adjoint scalar theory and Feynman diagrams at one-loop, we follow the same procedure
performed in [55], i.e. we will carry out a power expansion11 of PT(a11):a2 [
] in terms of
] we will use its original de nition.
First of all, we analyse the simplest example, the four-point computation.12
Let us consider the NP; P contribution, which is given by the expression
M(41 NP;P)[1; 2j3; 4 ; 1; 2; 3; 4] =
sa1b1
1
Z
24+1
Z
d
d t4+4 PT(a11):a2 [1; 2j3; 4]
PT(b11):b2 [1; 2; 3; 4]:
(5.4)
10This line is the same square de ned in [55].
11Note that the lowest power of that expansion for the non-planar Parke-Taylor factors is four, it is a
consequence of the Theorem 1 in [55].
12In appendix B we carry out the four-point computation using the Feynman rules.
(5.1)
;
(5.2)
!
(5.3)
These kind of graphs were already computed in [55] and the answer is
Now, we compute the NP; NP part of the four-point case. Let us consider the amplitude
M(41 NP;NP)[1; 2j3; 4 ; 1; 2j3; 4]
=
=
1 Z
24+1
1 Z
25
d
sa1b1
Z
Z
1
where we have used the same identity as in the above example. This CHY-integral was
computed by one of the authors in [54] and the result is
d t4+4 PT(a11):a2 [1; 2j3; 4] PT(b11):b2 [1; 2j3; 4]
d t4+4 (a1; b1; b2; a2)2 (!1a:11:a2 !2a:12:a2 !3a:13:a2 !4a:14:a2 ) (!1b1:1:b2 !2b1:2:b2 !3b1:3:b2 !4b1:4:b2 ) ;
has been shown in [55], it is simple to draw the CHY-graphs
M(41 NP;NP)[1; 2j3; 4 ; 1; 2j3; 4] = 2
4
+ per(2; 3; 4) :
(5.8)
Next, in order to compare this example with the previous one and to obtain more
information from it, we use the same procedure as in (5.5). Thus, on the a-sector we apply the
identity, PT(a11):a2 [1; 2j3; 4] = (a1; b1; b2; a2) 1
!1a:11:a2 !2a:12:a2 !3a:13:a2 !4a:14:a2 , but, on the b-sector
we use the de nition given in (3.9), therefore we have the graph expansion
M(41 NP;NP)[1; 2 j 3; 4 ; 1; 2 j 3; 4] = 25
1 Z
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
1
3
83
>
>
>
>
>
<
>
>
>
>
>
:
4
1
3
1
2
(5.5)
The CHY-graphs obtained above are a new type of graphs, which will be called13 the
nonplanar CHY-graphs (or butter y graphs). Using the
-algorithm developed by one of the
authors in [37], we can compute this butter y graph in a simple way, and the result is
(see appendix A)
1 Z
25
d
sa1b1
Z
= [1; 2; 3; !4] + [1; 3; 2; !4] + [1; 3; 4; !2] + [3; 1; 2; !4] + [3; 1; 4; !2] + [3; 4; 1; !2]
This interesting result is generalized in section 6. Certainly, by summing the four butter y
graphs from (5.9), one obtains the same answer as in (5.8).
Notice that we have not used the Parke-Taylor identities found in section 3.2 in the
above examples, for instance
PT(a11):a2 [1; 2j3; 4] = PT(a11):a2 [1; 2; 3; 4] + PT(a11):a2 [1; 3; 2; 4] + PT(a11):a2 [3; 1; 2; 4]
(5.11)
+ PT(a11):a2 [2; 1; 3; 4] + PT(a11):a2 [2; 3; 1; 4] + PT(a11):a2 [3; 2; 1; 4] :
We could do that and then to apply the technology developed in [55]. However, this
procedure is longer and tedious, we would also lose the correspondence between the
nonplanar CHY-graphs and Feynman diagrams at one-loop.
What the previous paragraph is actually telling us, is that we have found a nice way
to embed the BDDK identities into a single expression, all this from the relations we found
at the level of the Parke-Taylor factors in section 3. From the CHY approach, we can
nd all the interactions that are involved in the non-planar sector directly by solving the
corresponding CHY integrand. Since the 4-point case is so simple and just showed us one
type of interaction (the non-planar CHY-graph gave just boxes), let us take a look to a
nontrivial example, the 5-point case, where more interactions shall appear.
13Note that the graph in (5.9) is like a double copy of the graph in (5.5) without the connection between
b1 and b2 .
HJEP05(218)
In this section we will consider a less trivial example, the ve-point case computations. This
example is richer than the previous one, here we will see di erent interactions appearing
in the non-planar sector. Diagrams with trees will be attached to the loop will appear
now, but those tree diagrams should satisfy certain conditions imposed by the structure of
the gauge group, all of this now embedded into the non-planar Parke-Taylor factors (this
argument will become clear in the next section). The rst contributing integrand we will
take into consideration is the mixed non-planar-planar case, it is given by
sa1b1
1
Z
25+1
Z
d
From the results found in [55], it is straightforward to check the identity
we obtain the graph expansion
d
5
3
4 +
sa1b1
Z
+
1
2
5
3
4
(5.12)
(5.13)
i
9
>
>
>
>
>
>
=
>
>
>
>
>
>
;
(5.14)
+ cyc ;
here \cyc" stands for cyclic permutations of the labels, (1; 2; 3; 4; 5), but by keeping the
connection among them. Note that the graphs in (5.14) are totally similar to the ones
obtained in [55], equation (5:3). In fact, the only new graph is the second one, which is a
generalization of the graph in proposition 3 of [55]. Following the same procedure that was
applied there, we multiply this graph by the cross-ratio identity, 1 =
therefore the second term in (5.13) becomes
53 !3a:15:a2 + 5a1 3a2 ,
3a1 5a2
34 45 !5a:13:a2
1
2
1
2
5
3
5
3
4 +
d
1
2
1
2
5
3
5
3
4
sa1b1
+
1
3
4
1
Z
25+1
Z
d
=
4
+
(5.16)
The second resulting graph is a generalization of the one given in proposition 2 of [55].
This graph is simple to compute using the -algorithm and it vanishes. So, the M(51 NP;P)
amplitude can now be written as
M(51 NP;P)[1; 2j3; 4; 5 ; 1; 2; 3; 4; 5]
1
3
4
where all these graphs were already mapped to Feynman diagrams in [55]. Thus, the nal
5
+ 2
5
+ 2
5
4
:
(5.18)
Next, we compute the NP; NP contribution to the ve-point case. Let us consider the
amplitude
sa1b1
Such as it was done previously, we use the expansion given in (5.13) for PT(a11):a2 [1; 2j3; 4; 5]
1 Z
1
2
d
5
+
sa1b1
Z
+
5
3
4
and the original de nition for PT(b11):b2 [1; 2j3; 4; 5] written in (3.9), i.e.
3
4
3
4
Thus, one obtains the graph expansion
The rst, third and fourth CHY-graphs are a generalization of the butter y graph that
was found in the above section, equation (5.9). On the other hand, the second CHY-graph
is a combination between the butter y graph and the graph in proposition 3 of [55]. So,
by multiplying this graph by the cross-ratio identity, 1 =
term in (5.20) becomes
and the graph turns into
!5a:15:a2
1 =
=
5
+ 5
:
(5.24)
Like it happened in the previous example, the second resulting graph is a mixing between
the butter y graph and the one given in proposition 2 of [55]. The result for this graph is
1
2
3
4
1
2
again zero. Therefore, we can now write the M(51 NP;NP) amplitude as
sa1b1
Certainly, we have been able to rewrite the M(51 NP;NP) amplitude just in terms of butter y
graphs (non-planar CHY-graphs), in the same way as it was made for M(41 NP;NP). The
computation of these graphs is completely similar to the one performed in (5.10) which
leads to
=
4
1
2
1
2
1
2
= 4
=
4
5
3
4
Z
Z
Z
8
>
>
>
>
>
>
<
= [1; 2; f3; 5g; !4] + [1; f3; 5g; 2; !4] + [1; f3; 5g; 4; !2] +
d
5
+
2
5
l
l
l
1
Z
1
2
5
l
l
l
1
1
2
1
2
2
3 +
4
3
5
+ 2
!
3 + 2
2
l
l
l
1
1
4
1
4 5
= [1; 2; 3; f5; 4!g] + [1; 3; 2f5; 4!g] + [1; 3; f5; 4g; !2] +
= [1; 2] [3; f5; 4!g] ;
where we have denoted f3; 5g, f5; 4g and f4; 3g the tree level sector. Therefore, the nal
answer is given by
M(51 NP;NP)[1; 2j3; 4; 5 ; 1; 2j3; 4; 5] = S[1; 2j3; 5; 4] + S[1; 2j5; 4; 3] + S[1; 2j4; 3; 5]
S[2; 1j3; 5; 4] + S[2; 1j5; 4; 3] + S[2; 1j4; 3; 5] ; (5.29)
where we have de ned S[a1; a2ja3; a4; a5] as
S[a1; a2ja3; a4; a5]
[a1; a2] [a3; a4; a!5] + [a1; a2] [fa3; a4g; a!5] + [a1; a2] [a3; fa4; a5!g] :
It is simple to check the total number of Feynman diagrams in (5.29) is,14 60 pentagons
+72 boxes = 132, while in the CHY representation we only have 18 CHY-graphs
(equation (5.25)).
the amplitudes, M(41 NP;P)
found from the intersection15
Finally, looking at the results obtained in this section, it is interesting to note that
[1; 2j3; 4 ; 1; 2; 3; 4] and M(51 NP;P)[1; 2j3; 4; 5 ; 1; 2; 3; 4; 5], can be
M(41 NP;P)[1; 2j3; 4; 1; 2; 3; 4] = M(41 NP;NP)[1; 2j3; 4; 1; 2j3; 4]\M(41 P;P)[1; 2; 3; 4; 1; 2; 3; 4];
such as it was done in the planar case, [46, 55]. Although we do not have a formal proof,
there are evidences that the previous intersection relation could be applied to higher number
of points, therefore we conjecture the following general relation, up to an overall sign
M(n1 NP;NP)[ j ; j ] = M(n1 NP;NP)[ j ; j ] \ M(n1 NP;NP)[ j ; j ] :
(5.31)
where the ordered lists,
= f 1; : : : ; ig,
= f 1; : : : ; j g,
= f 1; : : : ; mg, satisfy the conditions,
= ; and
[
=
= f 1; : : : ; kg and
[
= f1; 2; : : : ; ng.
In the following section we will generalize the previous results to an arbitrary number of
points. Additionally, it is useful to remind that all computations were checked numerically.
6
General non-planar CHY-graphs
We have found a new type of CHY-graphs at one-loop, the non-planar CHY-graphs (or
butter y graphs ). Those type of graphs could be identi ed as a generalization of the planar
case obtained in [55]. In addition, let us recall that when the CHY-integrand associated
to a planar CHY-graph is integrated, its result is a sum of Feynman diagrams at one-loop.
This fact is summarized by the equality 1
2N+1
Z
d
sa1b1
Z
N
=
T n
Pn
T 2
;
(6.1)
14Under the equivalence relation given by the loop momentum shifting in (5.3), there are 12 nonequivalent
pentagons and 18 nonequivalent boxes. In addition, note that these 12 nonequivalent pentagons are in perfect
15This equality is given up to an overall sign.
agreement with the expansion given in (3.13) for PT(a11):a2 [1; 2j3; 4; 5].
where the grey circles mean the sum over all possible trivalent planar diagrams16 (Ti),
the symbol \Pn" in the loop circle denotes a regular polygon of n-edges and \N " is total
number of particles.
We can make an importan remark here. The tree diagrams attached to the loop satisfy
two conditions: 1) they cannot have particles from the two orderings, and 2) one tree cannot
contain all the particles belonging to one ordering. This is known from the structure of the
gauge group, but here it appears from the Parke-Taylor factors. From the expression (3.9)
we can apply an expansion in ! for each individual ordering, like we showed in [55], this
will give us the contributing type of diagrams, e ectively satisfying the previous conditions.
From the butter y graphs obtained in (5.9) and (5.25), we generalize the planar
CHYgraph in (6.1) to the non-planar case. Additionally, by using the -algorithm, it is
straightforward to compute this new kind of graph. Thus, a general butter y graph and its result
in terms of Feynman diagrams is given by the expression
HJEP05(218)
1
2N+1
Z
N
where, as in (6.1), the grey circles mean the sum over all possible trivalent planar diagrams
(Ti), \Pn" denotes a regular polygon of n-edges and \N " is the total number of particles.
Finally, using the results found in [55] and the CHY-graphs representation obtained
for M41 NP:NP[1; 2j3; 4 ; 1; 2j3; 4] and M51 NP:NP[1; 2j3; 4; 5 ; 1; 2j3; 4; 5] in (5.9) and (5.25)
respectively, we formulate a general expression for MN
1
NP;NP[ j ; j ]. To be precise, up
16As it is very well known, at tree-level there is a map between the CHY-graphs and Feynman diagrams
given by
Z
d tn
n
1
1
2
;
where the grey circle means the sum over all possible trivalent planar diagrams with the ordering
(1; 2; : : : ; n).
1 2
T p
5
2
3
4
= n
= N- 1
T n
( "
+
n
p
2
Pn
#
= [T1; : : : ; Tp] [Tp+1; : : : ; Tn!] ;
+
)
T p+1
1 terms ;
T p
(6.3)
(6.2)
8
>
>
>
>
>
>
>
>
>
NPchy(a:bj ) :=
All possible non-planar CHY graphs with the form
being NPchy(a:bj )[[i]] the i-th element in NPchy(a:bj ). For instance
NPchy(31:3;2;3j4;5;6) = <
6
NPchy(31:2;2;3j4;5;6) = <
8
>
>
>
>
>
>
:
>
>
8
>
>>6
>
>
>
>
>
>
:
>
>
>
4
5
4
5
1
1
2
2
9
>
>
>
>
6
3
;
3 ;
4
5
to an overall sign, we propose the following expression
M1N NP:NP[ j ; j ] =
1
2N+1
Z
the set, NPchy(a:bj ), as
where the ordered lists
and
are given by,
j j and j j are the lengths of the lists, i.e. j j = p and j j = N
= f1; 2; : : : ; pg and
= fp+1; p+2; : : : ; N g,
p, and we have de ned
+
1
2
3
(6.4)
9
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
;
(6.5)
(6.6)
9
>
>
>
>
>
>
α - sector
1
1
2 3
a
3
:
(6.7)
This is clear that (6.4) is in agreement with (5.9) and (5.25). We can now summarise the results obtained in this work in the following and nal section.
7
Discussions
In this paper we continued the program to obtain one-loop quadratic Feynman propagators
directly from CHY prescription, now into the non-planar sector of the amplitudes. Studying
the one-loop Parke-Taylor factors presented in [55], we found the well known KK relations
for amplitudes in momentum space, now in the context of functions of
variables in
CHY formalism. These relations allowed a generalization, and from then on we found the
BDDK relations at one-loop level, that leaded us also to obtain a generalization to the
multi-trace one-loop Parke-Taylor factors, which as a special case have the double-trace
one, called the non-planar one-loop Parke-Taylor factors.
The BDDK relations have been traditionally used to obtain the subleading order
contributions at one-loop in terms of leading ones. What we have done here, is exploit the
BDDK relations and as a consequence of them obtain a remarkable result, the non-planar
CHY-graphs. There is not an equivalent of these graphs in the traditional formalism and
among the advantages they o er we have:
The BDDK relations are already embedded on them, so we only need to integrate
their corresponding integrand to obtain all the contributing interactions in the
nonplanar sector.
The subleading order can now be written directly in terms of these graphs, since they
encode all the information for this order in a fewer number of graphs, o ering a more
compact presentation in comparison to Feynman diagrams.
Easier computation and generalization for higher number of points.
Using intersections between graphs the amplitudes for mixed orderings can be
obtained in a straightforward way.
We applied the non-planar CHY-graphs in the study of the bi-adjoint scalar theory at
one-loop. We have found all there is to know, as far as we understand, for this theory at
that loop order, at the level of Feynman integrands with on-shell external particles.
There are several directions to move on from the developments we have done in this
paper. We are ready to apply all the technology we have developed in the study of the
Yang-Mills theory and gravity at one-loop with quadratic Feynman propagators. Since the
Parke-Taylor factors enter into the CHY integrand for Yang-Mills like
HJEP05(218)
I(1 P)(1; 2; : : : ; n) = PT(a11):a2 [1; 2; : : : ; n]
YM
na2;b2j 1 njb1;a1
where na2;b2j 1
njb1;a1 are the BCJ numerators, which must be found (see [51, 52] for the
linear representation), we can apply the developments of this work directly by replacing
the planar Parke-Taylor factor by the non-planar one, obtaining the integrand for the
non-planar sector of Yang-Mills like
YM
I(1 NP)(1; 2; : : : ; i; i + 1; : : : ; n)
= PT(a11):a2 [1; 2; : : : ; i; i + 1; : : : ; n]
na2;b2j 1 njb1;a1
This can be done in a straightforward manner, because the Pfa an, P
pt(b11):b2 [ 1;:::; n]
(a1;b1;b2;a2) , does not have anything to do with the ordering.
The BDDK relations we have found here play an important role in a version of the BCJ
duality we are currently working on. The study of the BCJ duality have already been done
for the case of linear propagators [52, 53], so it will be very interesting comparison to make.
2Sn na2;b2j 1 njb1;a1
X
2Sn
X
2Sn
pt(b11):b2 [ 1; : : : ; n] ;
Another important direction, is going to higher number of loops for the Parke-Taylor
factors. There, we could study the relations that may exist among them, that is another
work in progress. There is not much study done in this part at the moment.
Since we have found a way to encode all the one-loop information in a fewer number
of CHY-graphs(integrands), it would be desirable to be able, or understand the viability,
of performing the loop integration before the contour integration in the
variable, this
could lead to a new and more compact way of calculating loop corrections, and could lead to
a better understanding of the singularities in the non-planar sector, a topic not completely
understood, except for some particular cases.
Acknowledgments
We would like to thank E. Bjerrum-Bohr for reading the manuscript and discussions. H.G.
would like to thank to J. Bourjaily, and P. Damgaard for discussions. H.G. is very grateful
to the Niels Bohr Institute | University of Copenhagen for hospitality. The work of N.A.
was supported by IBS (Institute for Basic Science) under grant No. IBS-R012-D1. The
work of H.G. is supported by USC grant DGI-COCEIN-No 935-621115-N22.
A
Four-point non-planar CHY-graph computation
In this appendix we will work out an example using the -algorithm in order to compute
the non-planar CHY-graphs. Since that the
-algorithm is a graphical method we can
omit the integral R d t4+4.
The simplest non-trivial example is the four-point case computation obtained in (5.9).
Applying the -rules presented in [37], we identify that there are three non-zero cuts given by
HJEP05(218)
Note that in this paper we have used this gauge xing for all graphs, which was very useful in the planar case [55].
However, for the non-planar case this gauge is not as
e cient. Although it is not evident at rst sight, the cuts in (A.1) generate non-trivial
CHY-subgraphs with spurious poles. Thus, we are going to choose a new gauge xing
which produces CHY-subgraphs with only physical poles.
Let us x the punctures f 1
; 2
; 3g by PSL(2; C), and the puncture f 4g by scaling
symmetry. Under this gauge xing, the non-planar graph becomes
3
4
3
4
1
2
1
2
=
3
4
=
3
4
1
2
1
2
+
+
3
4
3
4
1
2
1
2
+
+
3
4
3
4
The above cuts are straightforward to be computed and give
1
2
1
2
1
2
(ka1 +kb1 +k1 +k2)2
(ka1 +kb1 +k3 +k4)2
(ka1 +kb1 +k1 +k3)2
0
B
B
B
B
B
B
B
B
B
B
B
B
0
B3
CC ; (A.3)
CC ; (A.4)
given by
Note that the CHY-subgraphs obtained in (A.3) and (A.4) have the same structure as the
one given in (6.1), which was computed in datail in [55]. So, the nal answer for cut-1 and
1
2
1
2
+
B
B
B
B
B
B
:
1
1C
A
0
B4
1
C
C
C
C
A
1
C
C
C
C
A
]
:
1
:
cut
cut
1 =
2 =
2
5
2
5
sa2b2 sa1b1 s1a1b1 s12a1b1 s123a1b1
sa2b2 sa1b1 s3a1b1 s34a1b1 s341a1b1
On the other hand, the subgraphs in (A.5) have the same form as the ones studied in [54].
The computation of these subgraphs is straightforward and the total result of the cut-3 is
cut
3 =
2
5
sa2b2 sa1b1 s13a1b1
1
s1a1b1
+
1
s3a1b1
1
s132a1b1
+
1
s134a1b1
Adding the cuts, (cut
1) + (cut
2) + (cut
3), we obtain the nal result for this
fourpoint non-planar CHY-graph, namely
1
2
=
+
2
5
1
1
1
Therefore, by carrying out the integral, 215 R d sa1b1 , it is trivial to check the previous
expression becomes (5.10).
Finally, note that this gauge xing can be applied to higher number of points.
Φbb!
2
k4
k1
Φc3c! = fabcf˜a!b!c!
k2
k3
in which we have the following Lie algebra
L =
1
2
aa0 +
1 f abcf~a0b0c0 aa0 bb0 cc0 ;
[T a; T b] = if abcT c
Four-point amplitude from the Feynman rules
In this appendix we calculate the four-point amplitude from the standard Feynman rules
to compare with our
nal results based on the master formulas we obtained in previous
sections. We have the following Lagrangian for the bi-adjoint scalar eld theory
with two sets of generators fT ag and fT~ag and their corresponding structure constants,
f abc and f~abc [32]. In this theory we only have the following three-vertex to be used for
constructing tree- and loop-level diagrams, see [4] for some examples in tree-level amplitudes.
In this paper we are interested in one-loop amplitudes in bi-adjoint scalar
eld theory,
especially its non-planar contributions. The rst one-loop diagram with non-planar
counterpart in this theory is the four-point amplitude which will be constructed using the above
three-vertex to compare with the CHY results in section 5.1. After sewing four copies of
the three-vertex in gure 4 we get the irreducible contribution to the four-point amplitude
depicted in gure 5 which leads to the following representation
3!
D
Z
4
X ki
i=1
dDl
N2
X
b;c;d;e=1
1
(2 )D l2[l + k2]2[l + k2 + k3]2[l
k1]2
f ca1b f ba2d f da3e f ea4c
f~c0a01b0 f~b0a02d0 f~d0a03e0 f~e0a04c0
N~ 2
X
b0;c0;d0;e0=1
:
(B.1)
(B.2)
(B.3)
HJEP05(218)
and
N~ 2
X
f~c0a01b0 f~b0a02d0 f~d0a03e0 f~e0a04c0 = 2N~ tr(T~a01 T~a02 T~a03 T~a04 ) + 2tr(T~a01 T~a02 )tr(T~a03 T~a04 )
+ 2tr(T~a01 T~a03 )tr(T~a02 T~a04 ) + 2tr(T~a01 T~a04 )tr(T~a02 T~a03 ) ;
where N 2 (N~ 2) is the dimension of U(N ) (U(N~ )). Now if we exclude the fully planar part
of (B.3) the rest of the amplitude gives
A0Irr[k1; k2; k3; k4] =
D
4
X ki T01234 Z
i=1
dDl
1
(2 )D l2[l + k2]2[l + k2 + k3]2[l
k1]2
T01234 = T01234 P;NP + T01234 NP;P + T01234 NP;NP ;
T1234 P;NP = 4N tr(T a1 T a2 T a3 T a4 )ntr(T~a1 T~a2 )tr(T~a3 T~a4 ) + tr(T~a1 T~a3 )tr(T~a2 T~a4 )
After using identities in (4.7) for the natural ordering of the external scalars in the loop
(canonical ordering: 1234) we get
X
b;c;d;e
= 2N tr(T a1 T a2 T a3 T a4 ) + 2tr(T a1 T a2 )tr(T a3 T a4 )
+ 2tr(T a1 T a3 )tr(T a2 T a4 ) + 2tr(T a1 T a4 )tr(T a2 T a3 ) ;
(B.4)
represents the mixed planar-nonplanar (P; NP) part,
T1234 NP;P = 4N~ tr(T~a1 T~a2 T~a3 T~a4 )ntr(T a1 T a2 )tr(T a3 T a4 ) + tr(T a1 T a3 )tr(T a2 T a4 )
for the mixed nonplanar-planar (NP; P) part and
T1234 NP;NP = 4ntr(T a1 T a2 )tr(T a3 T a4 )+tr(T a1 T a3 )tr(T a2 T a4 )+tr(T a1 T a4 )tr(T a2 T a3 )o
n
tr(T~a1 T~a2 )tr(T~a3 T~a4 )+tr(T~a1 T~a3 )tr(T~a2 T~a4 )+tr(T~a1 T~a4 )tr(T~a2 T~a3 )o ;
+ tr(T~a1 T~a4 )tr(T~a2 T~a3 )o ;
(B.8)
+ tr(T a1 T a4 )tr(T a2 T a3 )o ;
(B.9)
for the nonplanar-nonplanar (NP; NP) contribution of the color structure. In total we
have six inequivalent diagrams for the one-loop four-point amplitude with the following
orderings
S4=Z4 = nf1234g; f1243g; f1324g; f1342g; f1423g; f1432go ;
(B.11)
which only the rst ordering has been shown in (B.3). The full four-point amplitude
(irreducible contribution) is the sum of all the orderings in (B.11), which can be written
as (A0Irr[i; j; k; l]
A0Irr[ki; kj ; kk; kl])
A0Irr = A0Irr[1; 2; 3; 4] + A0Irr[1; 2; 4; 3] + A0Irr[1; 3; 2; 4]
+ A0Irr[1; 3; 4; 3] + A0Irr[1; 4; 2; 3] + A0Irr[1; 4; 3; 2] :
(B.12)
Note that the T1234 NP;NP is invariant under exchanging any two external scalars, which
indicates its appearance for all six orderings.
In the following we present the CHY results from (4.8) for this amplitude to compare
with the results from the Feynman rules in (B.12). m4
for the color decomposition and one for the momentum integral. If we exclude the planar
contribution (which has been discussed in [55]) in m4
1 loop we have:
1 loop in (4.8) has two parts, one
I(1 P;NP) = 4(N ) X
X
3
p=1 2S4=Z4
X
2Sp=Zp
2S4 p=Z4 p
= 4N tr(T a1 T a2 T a3 T a4 )
nTr(T~a1 T~a2 ) Tr(T~a3 T~a4 )m(41 P;NP)[1; 2; 3; 4 ; 1; 2j3; 4]
+Tr(T~a1 T~a3 ) Tr(T~a2 T~a4 )m(41 P;NP)[1; 2; 3; 4 ; 1; 3j2; 4]
+Tr(T~a1 T~a4 ) Tr(T~a2 T~a3 )m(41 P;NP)[1; 2; 3; 4 ; 1; 4j2; 3]
o
+ ve more permutations ;
which coincides with the color structure of (B.6) after replacing T01234 with T01234 P;NP
and considering other permutations in (B.12). Now the loop integral appearing in (5.6)
(after some momentum shifts) corresponds to what we have in (B.6). We also used the
fact that
M(41 P;NP)[1; 2; 3; 4; 1; 2j3; 4]
M(41 P;NP)[1; 2; 3; 4; 1; 3j2; 4]
M(41 P;NP)[1; 2; 3; 4; 1; 4j2; 3] :
The same discussion holds for the NP; P part. { 30 { (B.13) (B.14)
Now, the most nontrivial part is the NP; NP contribution. For this piece of the
amplitude from (4.8) we get
I(1 NP;NP) = 4 X
3
p=1
q=1 2S4 2p=SZp4=Zpp;; 22SS4q=Zqq=Z4 q
X
+m(41 NP;NP)[1; 2j3; 4; 13j24]tr(T~a1 T~a3 )tr(T~a2 T~a4 )
+m(41 NP;NP)[1; 3j2; 4; 13j24]tr(T~a1 T~a3 )tr(T~a2 T~a4 )
+m(41 NP;NP)[1; 4j2; 3; 13j24]tr(T~a1 T~a3 )tr(T~a2 T~a4 )
We have already calculated the momentum integrals appearing here in (5.8), it contains
all the six inequivalent momentum integrals we have for the four-point amplitude, which
means all m(41 NP;NP)[
] in (B.15) are equivalent. After factorizing the momentum
integral, (B.15) is exactly what we have for the NP; NP contribution from the standard
computation in T1234 NP;NP after considering all permutations and momentum integrals in (B.12).
To summarize, in this appendix we have shown the exact equivalence for the non-planar
contribution to the four-point amplitude between our formula in the CHY side (4.8) and
the one from the standard method based on the Feynman rules for the bi-adjoint scalar eld
theory in (B.12). The same discussion applies to higher order amplitudes at one-loop level.
Open Access.
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
[1] E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math.
Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].
[2] F. Cachazo, S. He and E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye
orthogonality, Phys. Rev. D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE].
[3] F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles in Arbitrary Dimensions,
Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE].
[4] F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and
Gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].
[5] F. Cachazo, S. He and E.Y. Yuan, Scattering Equations and Matrices: From Einstein To
Yang-Mills, DBI and NLSM, JHEP 07 (2015) 149 [arXiv:1412.3479] [INSPIRE].
[6] F. Cachazo, S. He and E.Y. Yuan, Einstein-Yang-Mills Scattering Amplitudes From
Scattering Equations, JHEP 01 (2015) 121 [arXiv:1409.8256] [INSPIRE].
170 [arXiv:1604.03893] [INSPIRE].
[8] H. Gomez and E.Y. Yuan, N-point tree-level scattering amplitude in the new Berkovits`
string, JHEP 04 (2014) 046 [arXiv:1312.5485] [INSPIRE].
[9] S. Mizera and G. Zhang, A String Deformation of the Parke-Taylor Factor, Phys. Rev. D 96
[11] T. Azevedo and O.T. Engelund, Ambitwistor formulations of R2 gravity and (DF )2 gauge
theories, JHEP 11 (2017) 052 [arXiv:1707.02192] [INSPIRE].
strings, JHEP 10 (2017) 216 [arXiv:1707.08840] [INSPIRE].
[12] T. Azevedo and R.L. Jusinskas, Connecting the ambitwistor and the sectorized heterotic
[13] F. Cachazo, S. He and E.Y. Yuan, Scattering in Three Dimensions from Rational Maps,
JHEP 10 (2013) 141 [arXiv:1306.2962] [INSPIRE].
(2014) 215402 [arXiv:1312.7743] [INSPIRE].
[14] C. Kalousios, Massless scattering at special kinematics as Jacobi polynomials, J. Phys. A 47
[15] C.S. Lam, Permutation Symmetry of the Scattering Equations, Phys. Rev. D 91 (2015)
[16] L. Dolan and P. Goddard, The Polynomial Form of the Scattering Equations, JHEP 07
045019 [arXiv:1410.8184] [INSPIRE].
(2014) 029 [arXiv:1402.7374] [INSPIRE].
[arXiv:1505.03571] [INSPIRE].
[17] F. Cachazo and H. Gomez, Computation of Contour Integrals on M0;n, JHEP 04 (2016) 108
[18] C. Kalousios, Scattering equations, generating functions and all massless ve point tree
amplitudes, JHEP 05 (2015) 054 [arXiv:1502.07711] [INSPIRE].
[19] L. Dolan and P. Goddard, General Solution of the Scattering Equations, JHEP 10 (2016)
149 [arXiv:1511.09441] [INSPIRE].
[20] R. Huang, J. Rao, B. Feng and Y.-H. He, An Algebraic Approach to the Scattering
Equations, JHEP 12 (2015) 056 [arXiv:1509.04483] [INSPIRE].
[21] C. Cardona and C. Kalousios, Elimination and recursions in the scattering equations, Phys.
Lett. B 756 (2016) 180 [arXiv:1511.05915] [INSPIRE].
(2016) 178 [arXiv:1509.08908] [INSPIRE].
D 93 (2016) 105009 [arXiv:1509.08897] [INSPIRE].
[22] C. Cardona and C. Kalousios, Comments on the evaluation of massless scattering, JHEP 01
[23] M. S gaard and Y. Zhang, Scattering Equations and Global Duality of Residues, Phys. Rev.
[24] C. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily and P.H. Damgaard, Scattering
Equations and Feynman Diagrams, JHEP 09 (2015) 136 [arXiv:1507.00997] [INSPIRE].
[25] C. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily and P.H. Damgaard, Integration Rules
for Scattering Equations, JHEP 09 (2015) 129 [arXiv:1506.06137] [INSPIRE].
[26] F. Cachazo and G. Zhang, Minimal Basis in Four Dimensions and Scalar Blocks,
arXiv:1601.06305 [INSPIRE].
[27] S. He, Z. Liu and J.-B. Wu, Scattering Equations, Twistor-string Formulas and Double-soft
Limits in Four Dimensions, JHEP 07 (2016) 060 [arXiv:1604.02834] [INSPIRE].
[28] F. Cachazo, S. Mizera and G. Zhang, Scattering Equations: Real Solutions and Particles on
a Line, JHEP 03 (2017) 151 [arXiv:1609.00008] [INSPIRE].
[29] J. Bosma, M. S gaard and Y. Zhang, The Polynomial Form of the Scattering Equations is an
H-Basis, Phys. Rev. D 94 (2016) 041701 [arXiv:1605.08431] [INSPIRE].
[30] M. Zlotnikov, Polynomial reduction and evaluation of tree- and loop-level CHY amplitudes,
JHEP 08 (2016) 143 [arXiv:1605.08758] [INSPIRE].
[31] T. Wang, G. Chen, Y.-K.E. Cheung and F. Xu, A di erential operator for integrating
one-loop scattering equations, JHEP 01 (2017) 028 [arXiv:1609.07621] [INSPIRE].
[32] C.R. Mafra, Berends-Giele recursion for double-color-ordered amplitudes, JHEP 07 (2016)
080 [arXiv:1603.09731] [INSPIRE].
[33] R. Huang, B. Feng, M.-x. Luo and C.-J. Zhu, Feynman Rules of Higher-order Poles in CHY
Construction, JHEP 06 (2016) 013 [arXiv:1604.07314] [INSPIRE].
[34] C. Cardona, B. Feng, H. Gomez and R. Huang, Cross-ratio Identities and Higher-order Poles
of CHY-integrand, JHEP 09 (2016) 133 [arXiv:1606.00670] [INSPIRE].
[35] T. Wang, G. Chen, Y.-K.E. Cheung and F. Xu, A Combinatoric Shortcut to Evaluate
CHY-forms, JHEP 06 (2017) 015 [arXiv:1701.06488] [INSPIRE].
[36] K. Zhou, J. Rao and B. Feng, Derivation of Feynman Rules for Higher Order Poles Using
Cross-ratio Identities in CHY Construction, JHEP 06 (2017) 091 [arXiv:1705.04783]
[INSPIRE].
scattering equations, JHEP 06 (2016) 101 [arXiv:1604.05373] [INSPIRE].
[38] L. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP 07 (2014)
048 [arXiv:1311.2564] [INSPIRE].
[arXiv:1311.4156] [INSPIRE].
[39] N. Berkovits, In nite Tension Limit of the Pure Spinor Superstring, JHEP 03 (2014) 017
[40] Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Loop Integrands for Scattering Amplitudes
from the Riemann Sphere, Phys. Rev. Lett. 115 (2015) 121603 [arXiv:1507.00321]
[INSPIRE].
[41] Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, One-loop amplitudes on the Riemann
sphere, JHEP 03 (2016) 114 [arXiv:1511.06315] [INSPIRE].
[42] T. Adamo and E. Casali, Scattering equations, supergravity integrands and pure spinors,
JHEP 05 (2015) 120 [arXiv:1502.06826] [INSPIRE].
[43] C. Cardona and H. Gomez, CHY-Graphs on a Torus, JHEP 10 (2016) 116
[arXiv:1607.01871] [INSPIRE].
[arXiv:1605.01446] [INSPIRE].
[44] C. Cardona and H. Gomez, Elliptic scattering equations, JHEP 06 (2016) 094
Integration Rules for Loop Scattering Equations, JHEP 11 (2015) 080 [arXiv:1508.03627]
Amplitudes, JHEP 08 (2016) 008 [arXiv:1512.05001] [INSPIRE].
supergravity integrands, JHEP 04 (2015) 013 [arXiv:1412.3787] [INSPIRE].
Level, Phys. Rev. Lett. 118 (2017) 161601 [arXiv:1612.00417] [INSPIRE].
gauge theories and gravity, Nucl. Phys. B 930 (2018) 328 [arXiv:1706.00640] [INSPIRE].
03 (2018) 068 [arXiv:1711.09923] [INSPIRE].
Rev. D 95 (2017) 106006 [arXiv:1703.04714] [INSPIRE].
propagators from massless scattering equations, JHEP 10 (2017) 175 [arXiv:1707.08584]
03 (2017) 092 [arXiv:1612.06854] [INSPIRE].
[7] F. Cachazo , P. Cha and S. Mizera , Extensions of Theories from Soft Limits , JHEP 06 ( 2016 ) [45] C. Baadsgaard , N.E.J. Bjerrum-Bohr , J.L. Bourjaily , P.H. Damgaard and B. Feng , [46] S. He and E.Y. Yuan , One-loop Scattering Equations and Amplitudes from Forward Limit , Phys. Rev. D 92 ( 2015 ) 105004 [arXiv: 1508 .06027] [INSPIRE]. [47] F. Cachazo , S. He and E.Y. Yuan , One-Loop Corrections from Higher Dimensional Tree [49] R. Kleiss and H. Kuijf , Multi-Gluon Cross -sections and Five Jet Production at Hadron Colliders, Nucl . Phys. B 312 ( 1989 ) 616 [INSPIRE]. [50] Z. Bern , J.J.M. Carrasco and H. Johansson , New Relations for Gauge-Theory Amplitudes , Phys. Rev. D 78 ( 2008 ) 085011 [arXiv: 0805 .3993] [INSPIRE].