Oneloop ParkeTaylor factors for quadratic propagators from massless scattering equations
HJE
Oneloop ParkeTaylor factors for quadratic propagators from massless scattering equations
Humberto Gomez 0 1 2 3 4
Cristhiam LopezArcos 0 1 2 3
Pedro Talavera 0 1 2
0 Marti i Franques 1 , Barcelona 08028 , Spain
1 Campus Pampalinda , Calle 5 No. 6200, Codigo postal 76001, Santiago de Cali , Colombia
2 Blegdamsvej 17 , DK2100 Copenhagen , Denmark
3 Universidad Santiago de Cali, Facultad de Ciencias Basicas
4 Niels Bohr International Academy and Discovery Center, University of Copenhagen
In this paper we reconsider the CachazoHeYuan construction (CHY) of the so called scattering amplitudes at oneloop, in order to obtain quadratic propagators. In theories with colour ordering the key ingredient is the rede nition of the ParkeTaylor factors. After classifying all the possible oneloop CHYintegrands we conjecture a new oneloop amplitude for the massless Biadjoint 3 theory. The prescription directly reproduces the quadratic propagators of the traditional Feynman approach.
Scattering Amplitudes; Di erential and Algebraic Geometry; Superstrings

1 Introduction
2
3
4
5
6
7
8
4.1
4.2
5.1
5.2
5.3
of theories [4{7], even beyond eld theory [8, 9]. The main ingredient for this approach are
the treelevel scattering equations [2]
Ea :=
b6=a
X ka kb = 0;
ab
ab := a
b
;
a = 1; 2; : : : ; n;
where the a's denote punctures on the sphere. The treelevel Smatrix can be written in
terms of contour integrals localized over solutions of these equations on the moduli space
of npunctured Riemann spheres
An =
Z
d tnree
ItCreHeY( );
where the integration measure, d tnree, is given by
and the contour
is de ned by the n
3 independent scattering equations
d tnree =
Qn
Vol (PSL(2; C))
( ij jk ki)
Qn
b6=i;j;k Eb
Eb = 0; b 6= i; j; k :
The integrand, ItCreHeY, depends on the described theory. There are other approaches that
use the same moduli space [1, 10{12], but restricted to four dimensions.
There have been developed several methods to evaluate the integrals, from di erent
perspectives. Some approaches study the solutions to the scattering equations for particular
kinematics and/or dimensions [4, 13{19], others work with a polynomial form [20{29], or
formulating sets of integration rules [30{34]. A di erent approach was proposed in [35],
taking the double covered version of the sphere, the so called
algorithm, which we will
employ in this work.
A generalization for loop level of the CHY formalism has been made. The ambitwistor
and pure spinor ambitwistor worldsheet [36, 37] provided a prescription for a
generalization to higher genus Riemann surfaces [38{40]. A di erent approach was also developed
in [41{43], where the forward limit with two more massive particles, playing the role of
the loop momenta, were introduced. The scattering equations for massive particles were
already studied in [44, 45]. Another alternative approach using an elliptic curve was
developed in [46, 47].
The previous prescriptions give a new representation of the Feynman integrals with
propagators linear in loop momenta. In order to nd the equivalence with the usual
Feynman propagators, (` + K) 2, two additional steps must be taken: the rst one is the use
of partial fractions, and the second one is the shifting of loop momenta [39, 48].
Recently, one of the authors [49] proposed a di erent approach to obtain the quadratic
Feynman propagators directly from the CHYintegrands for the scalar
3 theory.1 The
motivation came by analysing a Riemann surface of genus two after an unitary cut, which look
exactly like a tree level diagram before the forward limit, but instead of the two massive
1There are some overlapping ideas with the recent paper published by Farrow and Lipstein [50].
{ 2 {
particles associated to the loop momenta there are four massless particles. This new
approach allows to work again with the scattering equations for massless particles, but at the
expenses of increasing the number to n + 4. In addition there is also the need to introduce
a new measure of integrations that guarantees the cut and then take the forward limit.
In the present work, we follow the line of thought of [49] and propose a reformulation
for the oneloop ParkeTaylor factors. Splitting the massive loop momenta (`+; ` ) into
the four massless ones ((a1; b1); (b2; a2)), we will have oneloop ParkeTaylor factors that
will enter into CHYintegrands to lead directly to the usual Feynman propagators. The
CHYintegrands in question are the ones for the Biadjoint
manifestly tadpolefree.2
Outline.
This paper is organized as follows. In section 2 we present our reformulation
of the oneloop ParkeTaylor factors (PT). The expression is written in terms of the
generalized holomorphic oneform on the Torus, !ia::jb. By exploiting algebraic identities we
formulate the Theorem 1: each term in the PT factors can be decomposed into terms
containing at least two !ia::jb factors, i.e. the PT factors are rearranged in an expansion
In section 3 we write, classify and match with their Feynman integrands counterparts,
some general type of CHYintegrands that can appear at oneloop level. Since we are
working with n + 4 massless particles, the contour integrals can be calculated using any of
the existing methods of integration. As we have mentioned already we employ the socalled
algorithm, with the choice of a new gauge xing, to solve them. This allows to
analytically evaluate arbitrary CHYintegrals using simple graphical rules. The classi cation is
made tracing the structures de ned in section 2. As will become clear each element inside
the partial amplitude have an unambigous correspondence with the elements of the
CHYgraphs, starting with the ngon, then following with the ones with tree level structures
attached to their corners.
Section 4 shows our proposal for the partial amplitude of the Biadjoint
3 theory at
oneloop with quadratic propagators: rst we give a simple review at tree level and oneloop
with linear propagators, then we propose our formula using our de nition for the PT factors.
In order to support our proposition in section 5 we perform explicitly the calculation
for the partial amplitudes of the three and fourpoint functions. We make an extensive use
of the results of previous sections. In particular we emphasize the direct interpretation of
the CHYintegrals in terms of Feynman diagrams. This mapping is codi ed in the following
equality at the integrand level
2N1+1 R d
sa1b1
R d tNre+e4 n 2
tree1 1 N
2As it will be explained, the number of !ia::jb is related with the polygon of the loop, for example, two
!ia::jb in the left integrand can only generate a bubble or a triangle.
{ 3 {
HJEP10(27)5
that constitutes one of the most important results of this work and it will be explained in
detail during the course of this paper.
In section 6 we comment on the issue of the externalleg bubble contributions.
Diagrams involved are singular and need to be regularized. Next section, 7, is for illustrative
purposes and is devoted to the i prescription and how to directly obtain it by dimension
reduction. Finally, in section 8 we conclude by summarizing our ndings.
For not disrupting the line of the paper more technical discussions have been
gathered in some appendices: in appendix A we explicitly show the su cient form of the
measure (3.5) to tackle the oneloop CHYintegral prescription. In particular how the
momenta combination it contains arrises. Proof of Theorem 1 is casted in appendix B where
it is discussed at length. Appendix C collects the relation between some techniques
developed across the paper and the linear propagator prescription. We conclude by probing an
statement of [47].
Before beginning section 2, we de ne the notation that is going to be used in the paper.
Notation. For convenience, in this paper we use the following notation
HJEP10(27)5
ij := i
j ;
!ia::jb :=
ab :
ia jb
integrands at twoloop. In addition, we de ne the ab's and !ia::jb's chains as
Note that !ia::jb are the generalization of the (1; 0)forms used in [51] to write the
CHY(1.5)
(1.6)
(i1; i2; : : : ; ip) := i1i2
(i1; i2; : : : ; ip)a!:b := !ia1::bi2
= !ia1::bi1
ip 1ip ipi1 ;
!a:b
ip 1:ip !iap::bi1
!a:b
ip 1:ip 1 !iap::bip ;
(i1 ; i2 ; : : : ; i!m; im+1; : : : ; in!; : : : ip )a:b := i1i2 i2i3
!iam:bim+1 im+1im+2
!ian:bin+1
ip 1ip ipi1 :
In order to have a graphical description for the CHYintegrands on a Riemann sphere
(CHYgraphs), it is useful to represent each a puncture as a vertex, the factor 1 as a
line and the factor ab as a dashed line that we call the antiline. Additionally, since we
often use the
algorithm3 [35] we introduce the color code given in gure 1 and 2 for a
ab
mnemonic understanding.
Finally, we introduce the momenta notation
kfa1;:::;amg = [a1; : : : ; am] :=
2
sa1:::am := kfa1;:::;amg;
ka1:::am :=
kai kaj :
m
X kai ;
i=1
m
X
ai<aj
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.
{ 4 {
unfixed massless puncture
massless puncture fixed by PSL(2,C)
massive puncture fixed by PSL(2,C)
massless puncture fixed by scale symmetry
similar to ones given in [39, 42, 43]. After that, we carry out some manipulation in order
to write algebraic identities that will be very convenient to perform computations using
the algorithm.
Before de ning the PT factor at oneloop, it is useful to remind that, at tree level in
the CHY approach, it is given by the expression
PTtree[ ] =
1
( 1; 2; : : : ; n)
;
where
is a generic ordering and n is the total number of particles.
Following the ideas presented in [42, 43], we formulate:
De nition. We de ne the ParkeTaylor factor at oneloop with ordering
as
PT1a1lo:ao2p[ ] :=
X
at least two !ia:1j:a2 's.
As it will be discussed later, the task of performing CHYintegrals using the PT de ned
in (2.2) is not simple. The di culty of these computations resides in the number of !ia:1j:a2 's,
more !ia:1j:a2 's imply that the singular solutions of the scattering equations do not contribute.
In fact, the minimum number of !ia:1j:a2 's so the CHYintegrals become simpler is two, as
we are going to explain in section 3. Nevertheless, it is always possible to manipulate
algebraically PT1a1lo:ao2p[ ] and to decompose it as a linear combination of terms that contain
in !ia:1j:a2 with two as its lowest power.
Theorem 1. The PT1a1lo:ao2p[ ] factor, which was de ned in (2.2), admits a power expansion
We shall only sketch some examples in order to illustrate this theorem, leaving the
complete, technical proof, together with the precise construction using the Schoutenlike
identity for the ij 's to the appendix B.
{ 5 {
(2.1)
(2.2)
The previous theorem allows to clarify, that the cancellations of the tadpoles in the
bi
3 theory can follow directly from an algebraic property of the oneloop PT factors
and not necessarily from the antisymmetry of the structure constant in the cubic vertex.
Before proceeding we shall introduce the following de nitions:
Dtay1p:ae2 0[1; : : : ; n]p ! := (1; : : : ; p) !1a:11:a2
!pa:1p:a2 + (2; : : : ; p + 1) !2a:12:a2
!pa+1:1a:2p+1
+
+ (n; 1; : : : ; p
with 1 < p
n;
Notice, that we have de ned the factors, Dtay1p:ae2 0, Dtay1p:ae2 I and Dtay1p:ae2 II, with the
particular ordering f1; 2; : : : ; ng, nevertheless, their de nitions for another ordering are
straightforward. These terms also carry the cyclic permutation invariance from the PT factor.
2.1
Examples
computation.
Twopoint
In this section we give some nontrivial examples in order to illustrate the above proposition.
The following identities are purely algebraic and they can be proven after a, somehow, direct
PT1a1lo:ao2p[1; 2] =
1
12 !2a:11:a2 +
1
21 !1a:12:a2 = PTtree[1; 2] Dtay1p:ae2 0[1; 2]2!;
where let us remind, Dtay1p:ae2 0[1; 2]2! = (1; 2) !1a:11:a2 !2a:12:a2 .
Threepoint
PT1a1lo:ao2p[1; 2; 3] =
1
12 23 !3a:11:a2 +
1
23 31 !1a:12:a2 +
1
31 12 !2a:13:a2
= PTtree[1; 2; 3] h 2Dtay1p:ae2 0[1; 2; 3]3! + Dtay1p:ae2 I[1; 2; 3]2!i
= PTtree[1; 2; 3] h Dtay1p:ae2 0[3; 2; 1]3! + Dtay1p:ae2 I[3; 2; 1]2!i :
{ 6 {
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
PT1a1lo:ao2p[1; 2; 3; 4]
Note that the (1=2) factor in Dtay1p:ae2 II[4; 3; 2; 1]2! comes from the fact there is a double
counting, i.e.
Dtay1p:ae2 II[4; 3; 2; 1]2! = (4 ; 3!; 2 ; 1!)a1:a2 + (3 ; 2!; 1 ; 4!)a1:a2
loop. Notice that the oneloop CHYintegral prescription given in [49] has the particular
structure4
In =
In :=
Z
dD`
(2 )D In ;
1
2n+1
Z
d
sa1b1
Z
where n is the number of massless external particles and the d tnr+ee4 is the tree level measure
de ned in [3]
d tnr+ee4 :=
EA :=
QnA+=41 d A
Vol (PSL(2; C))
xing PSL(2;C)
B=1
B6=A
n+4 kA kB = 0;
X
AB
with the identi cation
4For more details see appendix A.
fkn+1; kn+2; kn+3; kn+4g := fka1 ; ka2 ; kb1 ; kb2 g;
f n+1; n+2; n+3; n+4g := f a1 ; a2 ; b1 ; b2 g;
fEn+1; En+2; En+3; En+4g := fEa1 ; Ea2 ; Eb1 ; Eb2 g:
{ 7 {
(3.1)
(3.2)
(3.3)
(3.4)
n  p
=
1
2
(n p) anti  lines
2
n  p
2
Additionally, let us remind the measure, d , it is given by the expression5
Notice that in (3.2), without loss of generality, we have xed f 1; b1 ; b2 g and fE1; Eb1 ; Eb2 g.
This measure is introduced to take the forward limit of the four onshell loop momenta.
The motivation comes because momenta will appear combined in a particular way after
the integration of the
A's. In appendix A we give a more detailed explanation of how this
particular combination appears.
It is useful to recall that the integration over the A's variables is a contour integral,
therefore we do not worry to write the measure (2dD)`D .
which is localized over the solution of the scattering equations, i.e. EA = 0. In addition,
in this paper we are just interested to focus at the integrand level, in other words in In,
The classi cation is going to be made by taking into account the CHYintegrands
appearing in the Dtay:pbe
de nitions given in (2.3), (2.4) and (2.5). i.e.
ILa1:a2 ( ) = PTtree[1; 2; : : : ; n]
IRb1:b2 ( ) = PTtree[1; 2; : : : ; n]
Dtay1p:ae2 [n; n
12 !1b1:2:b2 :
1; : : : ; 1];
In order to not saturate the notation on the CHYgraphs, we introduce the de nition
given in gure 3.
3.1
One loop integrands classi cation
As it is going to be checked in section 5, the terms Dtay:pbe
there is a relation between them and Feynman diagrams. Basically, the number of !ia:1j:a2 's in
's have a physical meaning, as
IL
a1:a2 = PTtree[1; 2; : : : ; n]
Dtay1p:ae2 [n; n
corresponds to the number of legs attached to the loop in the Feynman diagram, and the
term in
IR
b1:b2 = PTtree[1; 2; : : : ; n]
ij !ib:1j:b2
with a single !ib:1j:b2 will be in charge of its ordering. In fact, the CHYintegrand, I(aLa11;:ab12;b2;a2)2 ,
IRb1:b2
can be represented as a linear combination of the three CHYgraphs given in gure 4.
In particular, in gure 5 we consider the simplest cases.
Therefore, the Dirac delta functions in (3.5) are well de ned.
5In this paper we are considering that the Ddimensional momentum space is real, i.e. ki 2 R
D 1;1.
{ 8 {
nC1HY
TREnE
the scale symmetry [35]. Clearly, we are using a di erent gauge than the one introduced
in [49]. This new gauge will allows us to work a la Feynman, i.e. each cut on the CHYgraph
becomes a Feynman propagator, quadratic in momenta.
The CHYgraphs in
gure 5 come from the following set of integrands
PTtree[1; : : : ; n] Dtay1p:ae2 0[n; : : : ; 1]n!
PTtree[1; : : : ; n] 12 !1b1:2:b2
PTtree[1; : : : ; n] Dtay1p:ae2 I[n; : : : ; 1](n 1)!
PTtree[1; : : : ; n] 12 !1b1:2:b2
:= I(I)
CHY[1; : : : ; n]1:2 +
;
PTtree[n; : : : ; 1] Dtay1p:ae2 I[1; : : : ; n](n 1)!
PTtree[1; : : : ; n] 12 !1b1:2:b2
:= I(CIHI)Y[1; : : : ; n]1:2 +
;
:= I(CIHIIY)[1; : : : ; n]1:2 +
;
where ellipsis stand for additional terms with the same structure6 as those given in the
graphs of gure 4.
Bearing in mind that the algorithm is able to solve the CHYgraphs up to an overall
sign the rst point to tackle is how to
x this ambiguity.7 Let us consider the integrands,
I(CIH)Y[1; : : : ; n]1:2; I(CIHI)Y[1; : : : ; n]1:2 and I(CIHIIY)[1; : : : ; n]1:2 above. Pulling out the common
factors they become
I(CIH)Y[1; : : : ; n]1:2 = ( 1) ( 1)n+1
I(CIHI)Y[1; : : : ; n]1:2 = ( 1) ( 1)n+1
I(CIHIIY)[1; : : : ; n]1:2 = ( 1) ( 1)n+1
"
"
"
2
2
2
an1a22
2
2
2
We claim that the terms in the square brackets have a direct representation in terms of
the CHYgraphs in gure 5.
In the following we make use of the
algorithm to perform the set of integrals
side the brackets in (3.7), ( 1)n+1, is checked numerically afterwards.8
I(CIH)Y[1; : : : ; n]1:2, I(CIHI)Y[1; : : : ; n]1:2 and I(CIHIIY)[1; : : : ; n]1:2. The conjectured overall sign
inProposition 1. There is an equality among the CHYintegral of (I), (3.7), and the oneloop
npoint Feynman integrand
I(CIH)Y[1; : : : ; n]1:2 :=
d
sa1b1
d tnr+ee4 I(I)
CHY[1; : : : ; n]1:2 = 3
Proof. The result follows from a direct calculation,
I(CIH)Y[1; : : : ; n]1:2 =
sa1b1
d tnr+ee4 I(I)
CHY[1; : : : ; n]1:2
=
=
a1b1 a2b2 b1b2
1
Qin=1 ia1 ia2
1b1 2b2 23
n1
6Sometimes, it is necessary to manipulate the integrand to obtain the CHYgraphs in gure 4.
7Notice that the overall sign can be xed by the technique developed in [33], where the antilines have
been considered.
( 3; E3)g, namely, the FaddeevPopov determinant becomes ( 12 23 31)2.
8For the numerical checking we have xed the punctures and scattering equations, f( 1; E1), ( 2; E2),
where in the last step we have mapped each element in the expression inside the bracket
to the CHYgraph: the rst factor is the box, namely the loop momenta sector, the second
one is in charge of connecting the loop momenta to all the external points, and the third
factor is the one that gives the ordering. With these identi cations all the integrands of
algotithm [35], we introduce
the \nearest neighbour" gauge xing. One can show that it is necessary to perform a total
of (n + 1)consecutive cuts, each one introducing a single propagator, like in the Feynman
diagrams. For instance, using the rules, there is only one non zero cut on the CHYgraph
in (3.9), as it is shown on left graph in (3.10).
this type will look alike but with the external points permuted.
In order to compute the R d tnr+ee4 integral within the
1
This rst cut gives the factor ka1b1
dubbed cut2. This cut is simple to compute and its result is given by
as it is shown on right graph in (3.10), see also
gure 3.
0
B
B
B
B
B
B
B
B
B
B
B
2) times this procedure we are led with
=
1
ka1b1
1
2
n
2
[
1
n 2
1
cut2
0
B
B
B
B
B
B
B
B
B
B
B
cut3
(3.10)
(3.11)
where clearly the resulting graph keeps the same form but with one less puncture. Iterating
and the resulting CHYgraph contains now a massive
puncture with momentum, ka1 + kb1 , which is connected with a double line to the puncture
1. By scale symmetry we x the nearest neighbour to 1 (next to its right), i.e. n, such
The resulting CHYgraph in (3.10) contains also only one non zero cut, which we have
1
ka1b1
cut(n+1)
1
C
C
2CC
C
C
A
(3.12)
This nal resulting graph has just one nonzero cut, as it is shown in (3.12), which we have
called \cut(n+1)" and its result by the rules is ka2b2
Summarizing, so far we have proven the equality
where the momentum conservation condition, Pin=1ki+ka1+ka2+kb1+kb2 = 0, has been used.
Carrying out the last integration, R d
sa1b1 , we are left with the following expression
HJEP10(27)5
1 Z
2n+1
d
sa1b1
2n+1
nally identi ed the algebraic expression with the pictorical oneloop
npoint Feynman integrand. This equality, which corresponds to our initial claim, has also
been checked numerically.
As a nal remark for this rst proposition, we can generalize the previous calculation
to the case of attaching CHY treelevel graphs instead of points as in (3.9). Schematically
it is
1
2N+1
d
sa1b1
Z
where N is the total number of external particles. The gluing process to join the CHY
treelevel graphs at the oneloop skeleton can be found in [46, 51]. These kind of graphs
can be solved in the same fashion as we did for the npoint case.
Proposition 2. The CHYintegral of (II), (3.7), identically vanishes.
1
2n+1
Z
d
I(CIHI)Y[1; : : : ; n]1:2 :=
sa1b1
d tnr+ee4 I(CIHI)Y[1; : : : ; n]1:2 = 0 :
(3.14)
1C1HY
TREE
2CHY
T2REE
Z
nCHY
TREEn
2n+1
"
n 3
cut1
1
2
Z
Z
where N is the total number of external particles. These type of graphs, with treelevel
CHYgraphs attached, can be solved in a similar way as we did for the case in (3.17).
Applying the
rules on the graph in (3.15) to compute the integral, R d tnr+ee4, one
obtains that there is just one nonzero cut
Furthermore all possible cuts on the resulting CHYgraph in (3.16) are zero, therefore
one can conclude that the integral, I(CIHI)Y[1; : : : ; n]1:2, vanishes, i.e.
I(CIHI)Y[1; : : : ; n]1:2 =
2n+1
Z
This calculation can be generalized to the case of attaching CHY treelevel graphs
instead of points. Schematically it is
=
=
2n+1
Z
Proof. As in the previous case, the proof is straightforward, but a little tedious.
1
BB n 3
1
1
2
1
2
11C1HY
TREE
To compute this integral we use the previous results together with the crossratio identity
It is straightforward to check that by multiplying the CHYintegrand (or graph) in (3.20)
times the above identity and using Proposition 2, the integral, I(CIHIIY)[1; : : : ; n]1:2, becomes
I(CIHIIY)[1; : : : ; n]1:2 =
sa1b1
d tnr+ee4 I(CIHIIY)[1; : : : ; n]1:2
1
1
2n+1
Z
d
=
=
I(CIH)Y[1; : : : ; n]1:2 + I(CIHI)Y[1; : : : ; n]1:2
I(CIH)Y[1; : : : ; n]1:2 ;
2
4
l
Proposition 3. There is a correspondence between the CHYintegral of (III), (3.7), and
the oneloop npoint Feynman integrand
1
2n+1
Z
I(CIHIIY)[1; : : : ; n]1:2 :=
d
sa1b1
d tnr+ee4 I(CIHIIY)[1; : : : ; n]1:2 = ( 1) 3
=
=
2n+1
2n+1
Z
1 Z
2n+1
"
d
d sa1b1 d tnr+ee4 ( 1)n+1
Z
Proof. The result follows from a direct calculation,
I(CIHIIY)[1; : : : ; n]1:2 =
sa1b1
d tnr+ee4 I(CIHIIY)[1; : : : ; n]1:2
1
2
and consequently
1
2n+1
d
sa1b1
Z
n3
This result can also be generalized for external CHY tree graphs. Schematically it is
1
2N+1
where N is the total number of external particles and again we can solve these type of
CHYgraphs like we have done in the previous cases.
Now that we have a classi cation for the type of integrands that will appear in our
calculations we can employ the results in the computation of the integrands for the
Biadjoint
The scattering amplitudes at tree level for the massless biadjoint
3 scalar theory are
given by the elements [4]
mtnree[ j ] :=
1
where
and the measure, d tnree, is given in (3.2).
In [4], it was shown that the integral, mtnree[ j ], is composed by the sum over all
the trivalent Feynman diagrams containing two planar embeddings, consistent with the
( ) ordering respectively.9 Speci cally mtnree[ j ] reduces to the sum over all elements
contained in the intersection among these two planar ordering. Schematically it can be
written as [4]
mtnree[ j ] = ( 1)n 3+n j mtnree[ j ] \ mtnree[ j ] ;
example, let us consider the element, mt4ree[1234j1243], it reads
where n j is determined by the number of ips between two permutations ; [4]. For
mt4ree[1234j1243] = mt4ree[1234j1234]\mt4ree[1243j1243] =
9There are many techniques to compute mtnree[ j ], such as ones given in [4, 18, 20, 24, 25, 29, 31{
33, 51, 52].
=
1
2
(4.1)
(4.2)
(4.3)
4
3
(4.4)
which is the right answer. In the previous example one can appreciate the advantage
of (4.3), since the calculation for both orderings is exactly the same but with external
legs permuted to the given ordering. Therefore, we only have to do the full calculation in
the canonical ordering (i.e. (1,2,. . . ,n)). This will prove extremely useful for the oneloop
calculations.
The CHY prescription to obtain the scattering integrands at oneloop, respecting the planar
orderings ( , ), is given by the elements
where
with
and the measure d 1n+lo2op
m1nloop[ j ] =
1
where, without loss of generality, we have xed f `+ ; ` ; 1g and fE`1+loop; E`1loop; E11loopg.
As it has been shown in [39, 42, 43], the CHYintegral in (4.5) reproduces the linear
propagators in the internal loop momentum ` , i.e. in the Qcut representation [53{55].
Therefore, as it is very well known, these results match with the traditional Feynman
propagators just after using the partial fraction identity and performing a shift in the loop
momentum.
4.2
A new proposal
In this section we propose a new CHY prescription for the Smatrix at oneloop of the
biadjoint
3 scalar theory, that takes into account the planar orderings. In this proposal
we will be able reproduce directly the quadratic propagators, in the same way as the
traditional Feynman approach.
Borrowing the line of reasoning in [49] and the prescription (4.5) we arrive at:
matrix elements at oneloop for the
3 biadjoint massless theory. We are going to present
HJEP10(27)5
several examples in order to support this statement.
Notice that despite the similarity among the prescriptions (4.5) and (4.9), there are
signi cant di erences between them: i) The total number of punctures do not match,
namely in (4.5) there are n + 2 punctures, out of which two are massive, while in (4.9)
there are n + 4 massless punctures. ii) The scattering equations are neither the same,
although in [41] it was shown that the massive scattering equations in (4.8) can be obtained
from (3.3) after dimensional reduction. Additionally, iii) the nal outcomes are di erent,
as it is going to be shown later, (4.5) produces linear propagators in ` , while (4.9) is able
to reproduce the quadratic propagators as the traditional Feynman approach.
After some manipulations I1CHloYop[ j ] in (4.10) becomes
(4.9)
(4.10)
(4.11)
I1CHloYop[ j ] := PT1a1lo:ao2p[ ]
=
X
X
2cyc( ) 2cyc( )
1
PT1b1l:obo2p[ ]
PTtree( 1; : : : ; n; a1; b1; b2; a2)
PTtree( 1; : : : ; n; b1; a1; a2; b2):
The integral
Z
As a consequence the integral, R d tnr+ee4 I1CHloYop[ j ], is just a sum over trivalent tree level
planar Feynman diagrams. However, this is not a very e cient and useful way to proceed
because there is a large number of singular Feynman diagrams that do not contribute to
the partial Amplitude, i.e. these diagrams must cancel out among them.
For example, consider the CHYintegrand
I1CHloYop[1; 2; 3j1; 2; 3] = PTtree(1; 2; 3; a1; b1; b2; a2) PTtree(1; 2; 3; b1; a1; a2; b2)+
(4.12)
where ellipsis stand for terms obtained under cyclic permutations. The CHYgraph for the
rst term of the expansion in (4.12) is represented on the left hand side of gure 6.
d t3r+ee4 PTtree(1; 2; 3; a1; b1; b2; a2)
PTtree(1; 2; 3; b1; a1; a2; b2) ;
contains the sum over all possible trivalent planar Feynman diagrams, [4, 29, 31, 35],
that have been depicted with a grey circle and blue lines in
gure 6(b). In particular,
in
gure 7(a) we give one example. Clearly, by performing the R d sa1b1 integral of this
diagram, one obtains a tadpole, such as it is shown in
gure 7(b). This singular diagram
must not contribute to the amplitude and therefore it cancels out with another one which
comes from the next contributions. This kind of analysis is tedious since the number of
tree level diagrams in (4.11) is large.
making the forward limit, i.e. by integrating R d (ka1 + kb1 )2.
In order to handle this group of cancelling diagrams without going through the detailed
analysis described above, we shall rely on the
ndings of sections 2 and 3 to classify the
Feynman diagrams at oneloop from the CHY approach. In the next section we are going
to show several examples where this new technology is applied and the conjecture over
M1nloop[ j ] will be checked.
Finally, note that the method used in [42, 43] could be applied here with a small
variation: the o shell momenta, ` and ` , are split into two onshell momenta, (ka1 ; kb1 )
and (ka2 ; kb2 ), as is shown in gure 6.
5
Examples
In the following section we will use all the previous results to calculate the particular cases
n = 3 and n = 4. Since n = 3 is simpler, it contains only two possible orderings, we will
show all the contributions by direct calculation of the integrands in (2.7).
Before giving the examples, it is useful to introduce the following notation
:= 3TRFEEEY
+ 3TRFEEEY
+
+ 3TRFEEEY l
3FEY
TREE
2FEY
TREE
4FEY
TREE
1FEY
TREE
5FEY
TREE
nFEY
TREE
6FEY
TREE
2TRFEEEY l
4FEY
TREE
1FEY
TREE
5FEY
TREE
nFEY
TREE
6FEY
TREE
2FEY
TREE
4FEY
TREE
1FEY
TREE
5FEY
TREE
l nTRFEEEY
6FEY
TREE
1
1
(b)
1
l
2
3
2
3
2FEY
TREE
4FEY
TREE
1FEY
TREE
5FEY
TREE
nFEY
TREE
6FEY
TREE
:
(5.1)
ated to the CHYgraph (The grey circle means the sum over all possible trivalent vertices.).
2
3
2
2
Ω sa1b1
2
2
1
1
(a)
1
2
3
2
1
2
1
5.1
In this case there are just two partial amplitudes, one coming from
expressions for PT1a1lo:ao2p[1; 2; 3], one written in terms of the Dtay:pbe
and another from M31loop[1; 2; 3j3; 2; 1]. In addition, as inferred from (2.7), we have two
's from the ordering
(1,2,3), and the other in terms of the ordering (3,2,1). In the following we disentangle each
Let us start with the rst CHYintegrand
I1CHloYop[1; 2; 3j1; 2; 3] =
2
1
2
3
3
+ 0
+ cyc(1; 2; 3)
:
+ cyc(1; 2; 3)]
where we have introduced the shorthand notation
I(CIH)Y[i; (j; k)]i:j := ( 1) ( 1)
I(CIH)Y[(k; i); j]i:j := ( 1) ( 1)
"
"
2
2
In terms of the CHYgraphs one has
d
sa1b1
That can be simpli ed further using (3.22)
Z
d t3r+ee4<
>
d
1
2
1
2
1
2
1
2
sa1b1
3 + 0
1
2
3
8
>
>
>
>
>
>
>
2 3
1
3
1
2
=
=
1
s23
1
s13
9
>
>
>
>
>
>
>
>
;
>
>
1
2
1
(ka2 + kb2 )2 (k1 + ka1 + kb1 )2
(5.6)
1
(ka2 + kb2 )2 (k1 + k3 + ka1 + kb1 )2
: (5.7)
1
24
1
24
sa1b1
sa1b1
Z
Z
d t3r+ee4 0
d t3r+ee4 0
over R d
ka1
2
) (D)(kb2 + kb1
external vectors11 (
Notice that by performing the integral R d , i.e. at the forward limit, the momentum
conservation condition becomes, k1 + k2 + k3 = 0 (s12 = s13 = s23 = 0), and the
expressions (5.6), (5.7) are ill de ned. This fact indicates that these type of terms need some
regularization.10 One way to obtain a well de ned result by integrating (5.6) and (5.7)
is to regularize the forward limit condition, i.e. instead to consider the measure,
dDka2 dDkb2
(D)(ka2 +ka1 ) (D)(kb2 +kb1 ), we allow for the measure dDka2 dDkb2
(D)(ka2 +
2 ), where
is an in nitesimal vector ( 2
0) orthogonal to the
ki = 0). Using this new measure the momentum conservation
condition becomes, k1 + k2 + k3 =
(s12 = s13 = s23 = 2), and now we are able to integrate (5.6)
and (5.7). Considering the leading order term one has
From the general result in section 3.1 the rst integral reduces to
1 Z
24
while the rest of terms can be cast in the general form depicted in gure 4(a), and their
computations is totally similar to the one presented in section 3.1
HJEP10(27)5
s13 (ka2 + kb2 )2 (k1 + k3 + ka1 + kb1 )2
=
=
1
s23
1
s13
2
l
1
`2 (` + k1)2 =
1
`2 (` + k1 + k3)2 =
10Notice that in the linear propagator approach these kind of diagrams are absent [42, 43], see section 6.
11Let us recall this condition depends of the dimension of the momentum space.
3
;
1
3
3
2
; (5.8)
l
1
:
Finally, adding all the partial results one gets
which is the expected answer, and has also been checked numerically. In section 6 we will
brie y discuss about the externalleg bubbles contributions in M31loop[ j ].
To compute the next partial amplitude we use the second expression for PT1a1lo:ao2p[1; 2; 3]
in (2.7). The CHYintegrand for this case is
= I(CIH)Y[3; 2; 1]3:2 +I(CIHIIY)[3; 2; 1]3:2 +I(CIH)Y[3; (2; 1)]3:2 +I(CIH)Y[(1; 3); 2]3:2
= I(CIH)Y[3; (2; 1)]3:2 +I(CIH)Y[(1; 3); 2]3:2 +cyc(3; 2; 1) ;
where in the rst line we used the inversion property of the PT factors, and in the third
one the result, I(CIHIIY)[3; 2; 1]3:2 =
I(CIH)Y[3; 2; 1]3:2 + (zero after integration), given in (3.22).
Translating to CHYgraphs the amplitude becomes,
M31loop[1; 2; 3j3; 2; 1] =
d
sa1b1
M31loop[1; 2; 3j3; 2; 1] =
>
0
>
>
>
>
8
>
>
>
>
>
:
>
>
8
<
>
>
>
>
>
>
:3
>
>
>
>
>
>
>
>
;
;
The integrals entering in (5.11) were already computed in the example above, (5.6){(5.8).
Collecting then we can write the total result as
which has been checked numerically.
Hitherto we have found the Feynman diagrams expansion for the canonical ordering,
M31loop[1; 2; 3j1; 2; 3] and its opposite ordering, M31loop[1; 2; 3j3; 2; 1], in (5.9), (5.12)
respectively. With those it is now straightforward to verify the relation
( 1) M31loop[1; 2; 3j1; 2; 3] \ M31loop[3; 2; 1j3; 2; 1] = M31loop[1; 2; 3j3; 2; 1] ;
(5.13)
(5.10)
(5.11)
9
>
>
>
>
>
>
>
>
>
>
;
(5.12)
this paper as
which is the oneloop equivalent to (4.3) [4]. In [42] a general relation at oneloop was
conjectured, but in the Qcut representation, i.e. using the prescription presented in section 4.1.
Although we have not got a general proof for (5.13), we have a strong numerical evidence
that the conjecture formulated in [42] can be extended to the new proposal formulated in
M1nloop[ j ] = ( 1)n j M1nloop[ j ] \ M1nloop[ j ] :
In this case we will have to deal with three di erent independent orderings,
We will calculate explicitly the rst one and rely heavily in the use of (5.14) in order to
infer the rest of them.
The CHYintegrand for the partial amplitude M41loop[1; 2; 3; 4j1; 2; 3; 4] reads
After expanding the terms we obtain several cancellations due to the identity,
I(CIHIIY)[ ] =
I(CIH)Y[ ], which was proven in (3.22) in the simplest case. After a rather
cumbersome calculation the above amplitude can be cast in terms of CHYgraphs as
M41loop[1; 2; 3; 4j1; 2; 3; 4] =
d
sa1b1
Z
d t4r+ee4 <
2
8
>
>
>
>
>
>
>
>
>
>
:
+ 0
1
2
1
2
4
3
4
3
All the integrals in (5.16) can be computed using the Propositions 1, 2, 3 and the technology
described in their proofs. The result is, following the same order: one box, four triangles,
two bubbles and the externalleg bubbles The latter must be regularized. In terms of
I1CHloYop[1; 2; 3; 4j1; 2; 3; 4] =
PTtree[1; 2; 3; 4]PTtree[1; 2; 3; 4]
3Dtay1p:ae2 0[1; 2; 3; 4]4! + 2Dtay1p:ae2 I[1; 2; 3; 4]3!
12 !1b1:2:b2 + 23 !2b1:3:b2 + 34 !3b1:4:b2 + 41 !4b1:1:b2 :
1
2
1
2
4
4
3
1
2
1
2
1
2
3
4
Feynman diagrams one has
2
1
2
4
3
4
2
4
3
0
B1
B
B
B
4
2
1
C
C
A
1
3
where the grey circle in the third graph inside of the bracket means the sum over all possible
trivalent planar diagrams. This result was checked numerically.
In order to calculate analytically the next contributions we are going to use the
conjecture in (5.14)
M41loop[1; 2; 3; 4j4; 3; 2; 1] = ( 1) M41loop[1; 2; 3; 4j1; 2; 3; 4] \ M41loop[4; 3; 2; 1j4; 3; 2; 1]
M41loop[1; 2; 3; 4j1; 2; 4; 3] = ( 1) M41loop[1; 2; 3; 4j1; 2; 3; 4] \ M41loop[1; 2; 4; 3j1; 2; 4; 3]
+ cyc(1; 2; 3; 4)CC ;
1
3
2
4
3
1
=
=
1
2
1
2
4
3
4
3
1
2
1
2
4
3
4
3
0
B1
B
B
B
4
4
2
3
1
3
2
These results were checked numerically concluding that, in fact, the conjecture in (5.14)
works perfectly.
With this procedure we can easily go to higher point cases, since we can always
construct the CHYgraphs for M1nloop[1j1], where 1 means canonical ordering, and know how
to calculate all the CHYintegrals.
General structure of M1Nloop[1N j1N ]
From the CHYgraphs representation found for M31loop[13j13] and M41loop[14j14] in (5.4)
and (5.16) respectively, where 1N means the canonical ordering (1; 2; : : : ; N ), it is direct to
obtain the general expression for M1Nloop[1N j1N ]. To be precise, up to global sign, one has
M1Nloop[1N j1N ] =
1
2N+1
Z
d
sa1b1
Z
X Gchy1a;::2:;N [[i]] + cyc(1; : : : ; N ) ;
+ cyc(1; 2; 3; 4)C
3
+ cyc(1; 2; 3; 4)CC ;
where the set, Gchy1a;::2:;N , is de ned as
8
>
being Gchy1n;:::;N [[i]] an element in Gchy1n;:::2;N . For instance
2
Gchy11;2;3;4 = <
1
Gchy1n;:::2;N := <
All possible CHYgraphs with the form n 2
;
(5.21)
Therefore, it now is clear that (5.20) is in agreement with (5.4) and (5.16).
Using the same techniques developed in the proof of Proposition 1, we can compute
the CHYintegral for a generic element in Gchy1n;:::2;N . Thus, we assert the following general
result at the integrand level
1
2
1 1 N
2 2 3
n
9
>
>
>
>
>
>
=
>
>
>
>
>
>
;
2
1
9
>
>
>
=
:
4
>
3 >;
>
=
N
Pn
(5.23)
8
>
>
>
>
>
>
:
1
2
4
tree2
2 3
(5.22)
HJEP10(27)5
1
2N+1
Z
d
Z
sa1b1
d tNre+e4 n 2
where the grey circles mean the sum over all possible trivalent planar diagrams and the
symbol, \Pn", means the loop circle is a regular polygon of n edges, where we de ne P2 as
integrands that contribute to M1Nloop[ j ].
Finally, using the general results given in (5.20) and the \map" in (5.23), the
structure of M1Nloop[1N j1N ] becomes simple. In [56], we also analyse the general case for the
6
Externalleg bubbles
As it is known, in the linear propagators formalism the externalleg bubbles vanish. To
illustrate that, let us consider the following Feynman integrand
l
1
=
where we have applied the partial fraction identity, A1B = A (B A) + B (A B) . Performing
1 1
a shifting over the loop momentum variable in the second term in (6.1), i.e. ` ! `~ = ` + ki,
one obtains12
k2) :
i
Since we are interested on the external leg bubbles,13 we assume that ki is an external
massless onshell particle and therefore (6.2) vanishes. This is the reason, in principle, why
in the linear propagator formalism the externalleg bubbles do not appear [43]. In [42]
it was also argued that the externalleg bubbles contribution must be regularized. In the
proposal we present in this work, the externalleg bubbles appear in a natural way and been
also singular their contribution need to be regularize. Notice that from the map (5.23),
it is straightforward to identify the externalleg bubbles contributions in M1Nloop[1N j1N ].
This contribution is given by the expression
sa1b1
Z
where Gchy01;::E:;NLB is de ned as
Gchy01;::E:;NLB := <
0
8
>
>
>
>
>
>
:
2
1
2 2
j=1
N
; 0
Gchy10;:::;N , and this is the generator of the externalleg bubbles (ELB).
Finally, we believe perhaps it would be interesting to nd a regularization method in
the CHY coordinates.
7
Feynman i prescription
(6.2)
3
(6.3)
(7.1)
d tNre+e4 4
X Gchy01;::E:;NLB[[j]] + cyc(1; : : : ; N )5 ;
1 N
2
9
3 >>>
=
>
>
>
;
:
In order to obtain the full form of the traditional Feynman propagators, we show that
our proposal is able to reproduce the Feynman i . This term can be included at each
propagator in a pragmatic and simple way, by dimensional reduction in the momenta of
the auxiliary punctures, i.e. (ka1 ; ka2 ; kb1 ; kb2 ).
than the rest of the kinematic data, namely,
Let us consider that the momenta of the auxiliary punctures has one more dimension
KaM1;2 := (ka1;2 ; kaD1+;21);
KbM1;2 := (kb1;2 ; kbD1;+21);
kM := (k ; 0);
`M := (` ; 0);
= 1; : : : ; D ;
= 1; : : : n ;
where D is the number of physical dimensions and n is the number of external particles.
The forward limit measure, d , is modi ed in the following way
d := dD(ka1 +kb1 ) (D)(ka1 +kb1 `)dD+1Ka2 dD+1Kb2
(D+1)(Ka2 +Ka1 ) (D+1)(Kb2 +Kb1 ):
12We are assuming that the integration measure over ` is invariant under this transformation.
13On this example we are considering that the propagator, K12 , is regularized.
After performing all integrals, the propagators become
1
(Ka1 + Kb1 + P )2 =
1
1
where P M = (p ; 0) is a momentum vector given by the sum of external momenta, i.e
P M = kM1 +
kMi = (k 1 +
Finally, with the identi cation, 2 kaD1+1 k
b1
+ k i ; 0) := (p ; 0), therefore (Ka1 + Kb1 ) P = (ka1 + kb1 ) p.
D+1 = i , we obtain the Feynman propagators in
full form.
8
Along the line of reasoning introduced in [49], we have proposed a reformulation for the
oneloop ParkeTaylor factors given in [39, 42, 43]. Exploiting the algebraic (Schoutenlike)
identity between the ij 's in the PT factors, we were able to show that they can be expanded
in such a way that no tadpoles integrands will appear later, so for biadjoint
3 theory
this cancellation is not necessarily related to the antisymmetry of the structure constant
in the cubic vertex. The construction also allowed us to build and classify systematically
all the contributing CHYintegrals to the oneloop npoint case.
These new PT factors were used to calculate the partial amplitudes for the biadjoint
3 theory at oneloop. It can be seen that the prescription presents advantages over the
previous ones:
In this approach the CHYintegrals are supported over n + 4 massless scattering
equations only, where n is the total number of external particles. Thus, all the known
techniques to compute these type of integrals can be used [4, 18, 20, 24, 25, 29, 31{
33, 51, 52].
It gives directly the quadratic Feynman propagators, unlike the already known
prescriptions, which must apply the partial fractions identity, namely, their results are
written in the Qcut language [53].
The corresponding CHYgraphs are well suited to be easily solved using the
algorithm allowing to calculate the integrands for higher points cases in a
similar fashion to Feynman diagrams. In addition, with the technology developed in this
work, we are able to compute the CHYintegrals directly in the forward limit, up to
externalleg bubble con gurations. The reason is because the singular solutions of
the scattering equations do not contribute.
In addition, it is straightforward to note that if one takes all the permutations instead
of the cyclic ones, i.e,
IL
a1:a2 =
X
2Sn 1
1
where
it with, IR
1 := 1 and Sn 1 is the set of all permutations of f2; 3; : : : ; ng, and by integrating
b1:b2 = PT1b1l:obo2p[1; 2; : : : ; n], one obtains the ngon with the canonical ordering,
which is a consequence of Proposition 1. Therefore, we can say that the calculations in [49]
are a particular case of our prescription.
At this time, most of computations performed at loop level using the CHY prescription
have been obtained in the Qcut language [38{43, 46{48, 51, 57{59], i.e. linear propagators
in the loop momenta. Additionally, it is very well known that the PT factors is one of the
most important ingredient to de ne several types of theories in the CHY approach, most
notably YangMills and Einstein gravity (from the KawaiLewellenTye (KLT) relations
point of view).
We are con dent that by using the new formulation of the PT factors at oneloop, as it
has been proposed in this work, we will be able to extend our ideas beyond the biadjoint
3 case, in particular for the YangMills theory (see appendix A and eq. (A.13)), Recently,
many works about the BernCarrascoJohansson (BCJ) duality and KLT kernel in the CHY
context have been published [60{67]. In particular, at oneloop all found results have been
written in the Qcuts representation [62, 63]. Thus, following the lines of the new proposal
developed here, it would be very interesting to obtain result in terms of the conventional
propagators, (` + K) 2, and to compare with the technologies presented in [62, 63, 68{70].
Moreover, extensions to higher loops are being developed [56]. In addition, it would
be fascinating to found the origin of this new prescription or its relationship with the
Ambitwistor string theory [36, 71{73].
Acknowledgments
H.G. would like to thank to E. BjerrumBohr, J. Bourjaily, and P. Damgaard for discussions.
H.G. is very grateful to the Niels Bohr Institute  University of Copenhagen for hospitality
and partial nancial support during this work. We thank to S. Mizera and P. Damgaard
for useful comments. The work of H.G. is supported by USC grant DGICOCEINNo
935621115N22. P.T. is partially supported by MINECO grant FPA201676005C21P.
A
CHYintegrands at oneloop
In this appendix we will give a simple way to construct CHYintegrands that have a
particular dependence on the loop momenta after integration, i.e. they come as couples, (ka1 +kb1 )
As it was mentioned previously, the proposal given in this paper follows the idea
presented in [49]. The main idea that motivated the oneloop calculation in [49] is that the
and (ka2 + kb2 ).
CHYintegral
1
2n+1
Z
dDka2 dDkb2 (D)(ka2 + ka1 ) (D)(kb2 + kb1 )
Z
1
2
1
1
5
2
2
4
(A.1)
3 + per(1; : : : ; n)
by the integral, R dD(ka1 + kb1 ) (D)(ka1 + kb1
is just the unitary cut of the twoloop diagram [51], at the integrand level, as it has been
represented above. Therefore, in order to obtain a oneloop integrand, we multiply by the
factor, (ka1 + kb1 )2, and we make the identi cation, ka1 + kb1 = `. This process is performed
`). This would be a simple explanation
why the measure, d
sa1b1 , is introduced in that particular way. Note that the (A.1)
CHYintegrand is a generalization for the one found in [39, 47]
Z
6
which is the one that reproduces only linear propagators.
Generalizing the (A.1) idea, our proposal is
In :=
1
2n+1
Z
So, a natural question is: what must be the form of the integrands, fIL
to obtain a function of the couples, (ka1 + kb1 ) and (ka2 + kb2 ) ? Before giving an answer
of this question, it is useful to remind our oneloop ParkeTaylor factors construction.
a1:a2 ; IR
b1:b2 g, in order
In [42, 43], the planar oneloop ParkeTaylor factors for linear propagators were
presented, and they can be written like
PT1loop[ ] :=
1
(`+; ` )
X
1
Following the previous proposal and the generalization of (A.2) given in (A.1), in this paper
we proposed the planar ParkeTaylor factors at oneloop for quadratic propagators as
PT1a1l:oao2p[ ]
X
X
1
1
n 1 n
n 1 n
It is not obvious that by using these integrands we will obtain a functional dependence of
the loop momenta like (ka1 + kb1 ) and (ka2 + kb2 ), nevertheless, it is not di cult to show
that this turn out to be the case. First of all, as in (4.11), it is straightforward to check
that the integrands in (A.4) can be written as
Each term in the (A.5) sums is called a partial planar oneloop ParkeTaylor factor, and
we denote them as
PT1Lloop[ ] := PTtree[ 1; : : : ; n; a1; b1; b2; a2] =
PT1Rloop[ ] := PTtree[ 1; : : : ; n; b1; a1; a2; b2] =
(a1; b1; b2; a2) ( 1; : : : ; n; a1; a2)
(a1; b1; b2; a2) ( 1; : : : ; n; b1; b2)
(a1; a2)
(b1; b2)
PT1Rloop, for two generic orderings14
and , it can be represented as
Next, by taking the CHYintegral of the product of these two factors, PT1Lloop and
1
1
α1
(A.5)
:
;
(A.6)
(A.7)
β
2
βn
;
(A.8)
(A.9)
(A.10)
Z
d tnr+ee4 PT1Lloop[ 1; : : : ; n]
PT1Rloop[ 1; : : : ; n]
α
1
β
1
α
2
αn
\
2
2
1
1
1
1
β
1
β
2
β
2
2
α
2
αn
\
2
2
1
1
1
1
where we have used the intersection property [4] from section 4, and the grey circles mean
the sum over all possible trivalent planar diagrams. Clearly, in (A.7) we have shown that
the CHYintegral, R d tnr+ee4 PT1loop[ ]
L
R
PT1loop[ ], is in fact a function of the two o shell
momenta which come from the combinations of four onshell momenta: (ka1 + kb1) and
(ka2 +kb2). This implies that the whole construction developed in this paper is well de ned,
i.e. the three types of CHYintegrands:
ILa1:a2( )
ILa1:a2( )
ILa1:a2( )
IRb1:b2( )
IRb1:b2( )
IRb1:b2( )
= PT1Lloop[ ]
PT1Rloop[ ] ;
PT1a1l:oao2p[ ]
;
14Note that this is a general case of the example shown in (4.12).
give a functional dependence of the loop momenta like (ka1 + kb1 ) and (ka2 + kb2 ). In
addition, from the identities
X
2Sn 1
X
2Sn 1
PT1Lloop[ 1; : : : ; n 1; n] =
PT1Rloop[ 1; : : : ; n 1; n] =
where Sn 1 is the group of (n
1)permutations, the CHYintegrand in (A.1),
;
gives also a functional dependence of (ka1 + kb1 ) and (ka2 + kb2 ).
Therefore, we have found an answer for the question formulated previously, to obtain
of the factors, PT1Lloop and PT1Rloop, respectively.
a CHYintegral that is able to give a functional dependence of the momenta, (ka1 + kb1 )
and (ka2 + kb2 ), the integrands, (a1;b1;b2;a2) and (a1b;b11:b;b22(;a)2) , must be a linear combination
ILa1:a2 ( ) IR
Finally, from the ideas given in this appendix, in order to reproduce the planar
contribution at oneloop (with quadratic propagators) for YangMills theory, we propose the
following prescription [56]
A1YMloop(1; 2; : : : ; n) =
Z
Z d
sa1b1
2n+1
na2;b2j 1 njb1;a1
Z
d tnr+ee4 PT1a1l:oao2p[1; 2; : : : ; n]
(or PT1Rloop), as it is claimed in this appendix.
where na2;b2j 1
njb1;a1 are the BCJ numerators, which must be found (see [62, 63] for
linear representation). Additionally, some progress towards the construction of the partial
nonplanar oneloop ParkeTaylor factors is in development. In [56], we have shown that
those nonplanar ParkeTaylor factors can be written as a linear combination of PT1Lloop
B
Proof of the oneloop ParkeTaylor factor expansion
The path we found to write (2.2) as a sum of terms with a minimum number of two
!ia:1j:a2 's was not a straightforward one, but it allowed us to see that the space for the
oneloop CHY diagrams with a
xed number of external points is bigger than the one
of Feynman diagrams. We have seen that the diagrams we have encounter so far can be
written in terms of the diagrams we computed in section 3, even diagrams that cannot be
solved using the
algorithm, which happens to be the case for the original ParkeTaylor
factor. Our proof will be supported only in the use of the Schoutenlike identity
(A.11)
(A.12)
1 =
ac bd
ad bc ;
ab cd
(B.1)
which give us a crossratio to relate the diagrams algebraically, without the use of the
scattering equations.
Our starting point is the expression (2.2) for the oneloop ParkeTaylor factor with
ordering , it can be rewritten as follows
PT1a1lo:ao2p[ ] := PTtree[ ]
n 1
!a1n::a21 :
X
Since the proof works the same way for any ordering, we can take
= (12 : : : n), then
HJEP10(27)5
(B.2)
(B.4)
One thing to attempt would be to solve the diagram corresponding to the rst term
and then take cyclic permutations of the result, but it leads to singular cuts, so we cannot
apply the
algorithm. Since the PTtree is a global factor, we can perform our analysis
! and then apply the cyclic permutations to get the whole PT1loop.
12!1a:12:a2 =
12 a1a2 :
1a1 2a2
We want to nd terms with the higher order of !'s, since we are missing 2n 2 factors in
the denominator, we use the crossratios (B.1) to obtain them. Now we have (B.4) times \1"
12 a1a2
1a1 2a2
23 a1a2 + 2a2 3a1
2a1 3a2
34 a1a2 + 3a2 4a1
3a1 4a2
n1 a1a2 + na2 1a1 : (B.5)
na1 1a2
Expanding all the products and performing the sum over cyclic permutations will give
us more than just the term with n !'s, it will give all the correct terms15 down to !2,
but we will still have terms linear in !. Actually, those linear terms belong to the inverse
ordering. Schematically, the expansion now looks like this
PT1a1lo:ao2p[1; 2; : : : ; n] = PTtree[1; 2; : : : ; n] n
: : : )(n)(!! : : : !)(n)
(
1n
+ (n
+ 2
1)
: : : !)(n)(!! : : : !)(n 2) + : : : + : : :
( !)(2)( !)(2)
PT1a1lo:ao2p[n; n
(B.6)
write PT1a1lo:ao2p[n; n
diagram.
where the round brackets mean closed cycles, and the super indices on them mean the
number of factors inside. All the terms inside the square brackets belong to the (12. . . ,n)
ordering, so we put a super index 1 on them. A somehow unexpected result, is that we can
1; : : : ; 1] as a sum of all the terms on the square brackets (i.e. from
its inverse order), but with all the coe cients equal to 1. The expression we have, again
15By correct we mean that each one of these terms will give the integrand for a contributing Feynman
1; : : : ; 1] = PTtree[1; 2; : : : ; n] (
: : : )(n)(!! : : : !)(n)
: : : !)(n)(!! : : : !)(n 2) + : : : + : : :
+ ( !)(2)( !)(2)
1n
:
To prove the previous relation we apply the inverse procedure with the Schouten like
cancel out the denominators will appear and we will arrive to the PT1a1lo:ao2p[n; n
identity, we dismantle the numerator by mixing the
ij 's with the
a1a2 's, factors that
Replacing (B.7) in (B.6) we will have an expression with no linear terms in !. Its
coe cients are the only ones modi ed
PT1a1lo:ao2p[1; 2; : : : ; n] = PTtree[1; 2; : : : ; n] (n
1)
(
: : : )(n)(!! : : : !)(n)
+ (n
+ 1
2)
: : : !)(n)(!! : : : !)(n 2) + : : : + : : :
( !)(2)( !)(2)
1n
:
Now this oneloop ParkeTaylor factor will enter into CHY integrands that can be easily
solved using the
algorithm, these give also the correct contributions for the biadjoint
3 scalar theory.
C
From quadratic to linear propagators in the CHYgraphs
The computational techniques developed in this work can be applied to the linear
propagators approach as well.
CHYgraphs, meaning
Schematically, the CHYgraphs that lead to linear propagators can be obtained from
the ones related to quadratic ones just by replacing the box loop by a \line loop" in the
n2
1
2
l
n3
l+
l(n p 1) anti lines
where, `+ =
` := `, and ` is the o shell loop momentum, `2 6= 0. For example, the
general CHYgraph given in (3.9) becomes
1
2
3
:
(B.7)
(B.8)
(C.1)
(C.2)
Here we apply the nearest neighbour gauge xing to compute the CHYgraph on the left
hand side of (C.2), which was described in detail in the proof of Proposition 1, where it
was applied to calculate the CHYgraph on the right hand side. Following exactly the same
steps we obtain the following result
Z
2
1
`2][(`+k1 +kn)2
`2][(` + k1 + kn + kn 1)2
`2]
[(` + Pn
i;i6=2 ki)2
which is linear in ` and where the measure d 1n+lo2op was de ned in (4.7).
Finally, if the result found in (C.3) is summed over all possible permutations, it provides
a proof to the conjecture proposed in [47], equation (7.18).
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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