CHY formulae in 4d
Accepted: June
CHY formulae in 4d
Yong Zhang 0 1 2 3 4
0 CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics
1 Beijing 100875 , China
2 Department of Physics, Beijing Normal University
3 Beijing 100190 , China
4 Chinese Academy of Sciences
In this paper, we develop a rather general way to reduce integrands with polarization involved in the CachazoHeYuan formulae, such as the reduced Pfa an, its compacti cation and its squeezing, as well as the new object for F 3 amplitude. We prove that the reduced Pfa an vanishes unless evaluated on a certain set of solutions. It leads us to build up the 4d CHY formulae using spinors, which strains o tions. The supersymmetrization is straightforward and may provide a hint to understand ambitwistor string in 4d.
Scattering Amplitudes; String Field Theory

1 Introduction
2 4d CHY formulae
3
4
5
3.1
3.2
4.1
4.2
Discussion Reduced Pfa an in 4d for the k0 = k sector
Extension to all solution sectors
The vanishing of reduced Pfa an in other sectors
The new object Pn for higher dimension operator
A A conjecture on Jn;k0
B The reduced compacti ed Pfa an in 4d
C The reduced squeezed Pfa an
CachazoHeYuan (CHY) formulation, has been developed for a large variety of theories [1{
4]. It expresses treelevel Smatrix as an integral over the moduli space of Riemann spheres,
which are localized by a set of constraints, known as scattering equations [
1, 5, 6
]
Ea :=
X
b6=a a
sa b
b
= 0; for a = 1; 2; : : : ; n;
(1.1)
where sa b = (ka + kb)2 = 2ka kb, a denotes the position of the ath puncture and we denote
a b :=
a
b.It has been argued that what underpins the formulation is the ambitwistor
string theory [7{9].
The formulation has been inspired by Witten's revolutionary twistor string theory
for N = 4 superYangMills theory (SYM) in four dimensions [10], and in particular the
RoibanSpradlinVolovichWitten (RSVW) formulae for all tree amplitudes in the
theory [11]. Originally CHY discovered scattering equations in attempts to rewrite the
equations in the delta functions of RSVW formulae without using 4d spinor helicity variables [5],
thus by construction they reduce to RSVW equations in four dimensions. More precisely,
we have n 3 di erent sets of 4d equations, which are polynomial equations of degree
{ 1 {
d = 1; 2; : : : ; n 3. The n 3 sectors are labeled by k0 = d+1 = 2; : : : ; n 2, which coincide
with helicity sectors. A set of equations, which are completely equivalent to RSVW
equations, have been proposed in [12] based on ambitwistor string theory in four dimensions.
It turns out that they are more convenient for our purposes, and in particular for helicity
amplitudes. To write the equations in sector k0, we divide n particles into two sets of k0
and n k0 particles denoted as 0 and +0 respectively:
E _
b
~ _
b
p2+0
b p
t
b
X tp ~p_ = 0 for b 2
0;
Ep
p
tp
X tb b = 0 for p = +0;
Here the variables are 's and t's, which can be combined into n variables in C2, a =
t1a ( a; 1). The b p is the abbreviation of b
p. The 0 and +0 are arbitrary two sets of the
n external particles, with their length equal to k0 and n
k0 respectively. Di erent choices
just correspond to di erent link representation [
13, 14
], which share the same solution of
's. In this paper, we reserve
and + as the negative and positive helicity sets of external
particles and k the length of , i.e. the number of external particles of negative helicity. A
priori there is no relation between solution sector and helicity sector.
We refer the readers to [15] for the direct derivation of (1.2) from (1.1); in the same
paper, it has been shown that (1.2) is equivalent to RSVW equations, and one can freely
translate between the two forms. Each solution of (1.1) corresponds to a unique solution
f a; tag of (1.2) for some k0, with identical crossratios of the 's. For each k0, (1.2) have an
Eulerian number of solutions, En 3;k0 2, and the union of them for all sectors give (n 3)!
solutions of (1.1), with (n 3)! = Pkn0=22 En 3;k0 2 [5, 16].
It is highly nontrivial to reduce the localized integral measure of CHY formula, with
delta functions of (1.1), to that of 4d formula, with (1.2), for some k0 sector. The reduction
requires a sum over all sectors, and for each of them it results in a complicated conversion
factor that depends on k0. In addition, after we plug in spinorhelicity variables for e.g.
YangMills amplitudes, the CHY integrand behaves very di erently in di erent helicity
and solution sectors. As we will see, the Pfa an plays the role of \solution lter": it is
nonvanishing only on the solution sector that coincides with the helicity sector, which
is why we have a 4d formula for each helicity sector. What is even more interesting is
that in the right sector, the polarization part of the CHY integrand exactly cancels the
k0dependent conversion factor from the measure! Thus two complications cancel out, and
for YM we are left with a trivial ParkeTaylor factor in 4d.
Let's make the statement more precisely. For gauge theory and gravity, the most
important ingredient is a 2n
2n skew matrix
n
n :=
A
C
CT
B
;
Aa b =
; Ba b =
( ka kb a 6= b
a b
0
a = b
( aa bb a 6= b
0
a = b
; Ca b = < a b
8 a kb
: c6=a
P Ca c a = b
a 6= b
;
(1.3)
and we de ne its reduced Pfa an Pf 0 n := ( )a+b
Pfj njaa bb with 1 a<b n.
We try to factorize the Pf 0 n into two parts depending on particles of negative and
positive helicity respectively. Then we show in the right sector that is consistent to the
{ 2 {
helicity sector, each of the parts combines to a reduced determinant while in other sector
one of the part must vanish. That is,
Pf0 n k0 = kk0 det0 hk det0 h~n k ;
Here the two matrices, the k
k matrix hk and (n k)
introduced in [12] (see also [17, 18]) are given by
(n k) one h~n k essentially
hab = habi
and we de ne det0hk = det jhkjba=(tatb) (similarly for det0h~n k) where we use jhkjba to
denote the minor with any row a and column b deleted.
We rearrange the Pf0 n using some fundamental gauge invariant or almost gauge
invariant objects. It is either a (modi ed) trace of linearised
eld strength ornamented
~
in
with some 's or Caa. We view the 4d scattering equations (1.2) as a change of variables:
we refer to b2 0 , ~p2+0 and ta; a as \data" and the 4d scattering equations (1.2) as writing
b2 0 and
p2+0 in terms of the data. After plugging in this change of variables, the Caa
n directly reduces to object made up of spinors. What left to do is to deal with all
kind of trace. After all, somehow, we nd the reduced Pfa an reduces to the two reduced
determinants. This way of reduction is rather general: not only the reduced Pfa an, but
also many other integrands, such as the reduced compacti ed Pfa an used in EM, YMS,
DBI amplitudes or the new object Pn used in F 3; R2; R3 amplitudes are also related to
these two (extended) matrices. It may even be applied at loop level [19].
The paper is organized as follows. In section 2, we introduce the CHY formulae in
4d. In section 3, we study the reduction of Pfa ans to 4d for k0=k. First we see how
Pf n factorizes in 4d in a manifeslty gaugeinvairant way, which naturally leads to the
4d matrices hk and h~n k. Then we present the beautiful reduction of Pf 0 n, in a similar
but more nontrivial way. In section 4, we move to general case with arbitrary k0, which
requires generalized version of hkk0 and h~k0
n k matrices. We show that both Pf 0 n and
Pn reduce nicely into the generalized hk0 and h~k0 ; while Pf0 n directly vanishes when
k n k
k0 6= k, Pn does not and gives interesting formulae in 4d. The reduction of the reduced
compacti ed Pfa an and squeezed Pfa an is put in appendix B, C.
2
4d CHY formulae
We start with CHY formula for treelevel Smatrix of n massless particles:
Mn =
1
vol SL(2; C)
Z
n
Y d a
a=1
n
a=1
Y 0 (Ea) In(f ; k; : : :g) =
X
solutions
det0 n
In(f ; k; : : :g) ; (2.1)
where the precise de nition of the integral measure including delta functions can be found
in [
1
], and In is the CHY integrand de nes the theory. In the second equality one sums over
{ 3 {
(n 3)! solutions of (1.1), evaluated on the integrand and the Jacobian, which is de ned as
a reduced determinant:
det j nja b c
jp q rjja b cj
p q r
det 0 n :=
with
a b =
a b
sa2b ; for a 6= b ;
a a =
a b ;
X
b6=a
(2.2)
a b b c c a.
where the n
n matrix ( n), with entries f a bg :=
matrix; the rows p; q; r and columns a; b; c are deleted (corresponding to deleted equations
and variables, respectively), and we have two FadeevPopov factors, de ned as ja b cj :=
For gauge theory and gravity, the most important ingredients is the reduced Pfa an
Pf0 n given in (1.3). Many other integrands can be abtained by doing some operation on
it. The CHY integrand for npoint YangMills tree amplitudes reads
HJEP07(21)69
In
YM = Cn Pf0 n ;
Cn =
Tr(T I1 T I2
where Cn is the colordressed ParkeTaylor factor, with the sum over (n 1)! inequivalent
permutations.
The general 4d formula in solution sector k0 for npoint amplitudes reads:
Mn;k0 =
1
vol GL(2; C)
Z
n
Y d2 a
a=1
Y 0 2(Eb) 2(Ep) In f ; ; ~g =
pb22+00
X
k0 sec: sol:
In f ; ; ~g
Jn;k0
where d2 a := d a dttaa , and in addition to 4 deleted variables by GL(2), 4 redundant
equations in (1.2) are deleted which give overall delta functions for momentum conservation.
In the second equality, one sums over the Eulerian number, En 3;k0 2, solutions in sector
k0. The Jn;k0 is the Jacobian of the localized 2n
4 integrals
tm u;v v;w w;u hc di2
where we have chosen to eliminate tm; u; v; w and Eb_=c;d, with the FP factor hc di2 (for
The relation between the two jacobians is simple. Viewing (1.2) as a change of variables
Ep6=q;r the FP factor is [q r]2).
and plugging in it, we nd
det 0 n(fsab; ag)jk0 = Jn;k0 det0 hk0 det0 h~n k0 :
Here we don't need to plug in any solutions, but simply make a change of variables, so this
is really an equality between rational functions of the data, i.e. b's, ~p's, and a; ta's. The
two reduced determinants det0 hk0 and det0 h~n k0 can be thought as two resultants and are
divided by det 0 n as discussed in [20]. We nd that the quotient is just Jn;k0 . A conjecture
about the closed form of Jn;k0 is put in appendix A.
{ 4 {
(2.4)
(2.5)
(2.6)
review them here.
p 2 Sn
Thanks to (1.4), then for gluon amplitudes, the integrand is nothing but the
(colordressed) ParkeTaylor factor IYM = Cn. Di erent from (2.1), any tbtc tpptqq or tbbtpp with
n bc
b; c 2
objects could be a new 4d integrand, for example we add some tbbtpp to the InYM and we get
0 and p; q 2 +0 is GL(2; C) invariant and any known 4d integrand added with these
those for QCD in [21].
In this paper, we explicitly demonstrate the rst identity (1.4). Compared to this
identity, the second one (2.6) is a more boring one, as there is no polarization involved and just
kinematics reducing to 4 dimensions. One can check as many points as we want, without
any di culties (we have checked up to 50 points with all solution sectors numerically). A
proof based on direct inspection should be straightforward.
3
Reduced Pfa
an in 4d for the k0 = k sector
In this section, we show in a constructive way how the reduced Pfa an factorizes in four
dimensions for the solution sector that coincides with its helicity sector, k0 = k. We will
proceed in two steps: as a warm up, we show how it works for the vanishing Pfa an Pf n,
which factorizes into two vanishing determinants in 4d; then we apply it to the more
nontrivial case of the reduced Pfa an and show Pf 0 n = det0 h det0 h~. The reason for doing
so is that both Pf n and Pf0 n have similar expansions, as rst studied in [22], and we
From the de nition of Pfa an and thanks to the special structure of 2n
2n matrix
n, we can expand Pf n as a sum over n! permutations of labels 1; 2; : : : ; n, denoted as
Pf
n =
sgn(p) p =
sgn(p) I J
where sgn(p) denotes the signature of the permutation p and in the second equality, we use
the unique decomposition of any permutation p into disjoint cycles I; J;
; K given by
I = (a1a2
ai);
J = (b1b2
;
K = (c1c2
ck) ;
each
p is the product of its \cycle factors"
length of a cycle equals one, its cycle factor
K , which we de ne now. When the
(a) is given by the diagonal of Cmatrix:
X
p2Sn
(a) := Caa =
X
b6=a
a kb
ab
;
X
p2Sn
bj ) ;
I J
{ 5 {
and when the length exceeds one e.g. i > 1, the cycle factor is given by
I =
(a1a2 ai) := 21 tr(fa1 fa2
fai )
with
f
a
= ka a
a ka :
(3.4)
Here (a1a2a3 ai) = a1a2 a2a3
aia1 . The trace is over Lorentz indices and f
is the
linearized eld strengths of gluons. Note that the decomposition is manifestly gauge invariant:
for cycle factors with length more than 1 (3.4), the trace of f
is gauge invariant, while for
1cycles, (3.3), the factor is gauge invariant on the support of scattering equations (1.1).
(3.1)
(3.2)
(3.3)
1th; nth columns and rows have been deleted, the numerator of the cycle containing 1 and
1
WI = W[1a2 ai 1n] = 2 1 fa2 fa3
that the sum is taken over all p 2 Sn such that 1 is changed into n. There are (n
permutations in Sn so the sum consists of (n
The key observation in [22] allows us to expand the reduced Pfa an in terms of
building blocks, each of which is either the product of various closed cycles or an open
cycle involving the two deleted labels. Closed cycles have a very good property that they
will vanish unless all of their elements belong to same helicity. While the open cycle is much
tougher, as it's not gauge invariant individually (dependent on the gauge of the two deleted
particles) and wont't vanish when their elements come from di erent helicity sets. As a
warm up, we show in the rst subsection the Pfa an, which is the product of only closed
cycles [22], factorizes. Though the Pfa an equals zero, it very nontrivially factorizes into
determinants of two matrices. Also it is the naturel way to introduce the two matrices hk
and h~n k (1.5). In the next subsection we carefully deal with the open cycle and
nally
factorize the reduced Pfa an to two reduced determinants.
3.1
The Pfa an in 4d
an antiselfdual part: f
Let's start with the Pfa an, Pf n. In 4 dimension, f
reduces to a selfdual part and
f + respectively. An important property is that any two adjoint linearised strength elds
fb fp+ in the trace can exchange their place if the helicity of b; p are di erent, i.e.
!
f _ _ + _ _ f
. We denote these two parts as f
and
So we can always reduce those traces where particles of negative or positive helicity are
mixed each other to split ones which have a simple reduction in 4d. Then
fai ) = <
2 [aiai 1] [ai 1ai 2]
[a1ai] ;
8
>
>2 ha1a2i ha2a3i
>>:hb1b2i
; bx are all the particles of negative helicity from a1; a2;
; ai with their
ordering unchanged and similarly p1; p2;
; py are all the particles of positive helicity
from a1; a2;
; ai with their ordering unchanged. Note that tr (fa1 fa2
fai ) directly
vanishes if there is only one particle of negative helicity or only one particle of positive
{ 6 {
helicity in a1; a2;
; ai. However we see that the remaining case still e ectively vanish as
we always add up all permutations (see (3.1)) while
; pyg such that the ordering within fb1; b2;
fai ) in 4d in a remarkably simple way:
; pyg is preserved. Therefore, in the sum of (3.1), we can e ectively write
aig
aig
+ ;
(3.10)
Motivated by (3.10), we recall the o diagonal elements of the k
(n k)
(n k) one h~n k essentially introduced in [12] (see also [17, 18]):
k matrix hk and
hab = habi
ab
a 6= b; a; b 2
;
It is clear that when we have any cycle factor with length at least 2, it must be given by
the chain product of such o diagonal elements
(a1a2 ai) !
8
>
<
aig
aig
+ ;
(3.12)
To this point we have not used scattering equations and solution sectors in four dimensions.
The nontrivial part of the reduction concerns 1cycle, or the diagonal entries of Cmatrix.
Note that
(a) = Caa is only gauge invariant on the support of scattering equations, so
it is not surprising that to reduce it nicely one needs to use scattering equations in four
dimensions. Now we derive the explicit expression of Caa. When a 2
and because of the 4d scattering equations (1.2), we can make
Note that Caa depends on
the change of variables
~b_ = tb
X tp ~p_
p2+0
b p
X tb b
{ 7 {
In the last equality, we have collected the denominators together such that bp is canceled.
Now Caa factorizes into two factors
Caa =
0
X tbhabi
b6=a ta ab
1
A
p
X tatp[p ]
Similarly we can work out the case of a 2 + and a 2 +0
as ta P
p2+0
tpa~pp_ = ~a_ (3.14).
All gauge dependence is in the latter factor and it can be eliminated by scattering equations
(3.15)
(3.16)
(3.17)
Ca+a =
X
tb [ab]
:
b6=a; b2+0 ta ab
We rst discuss the k0 = k case and without loss of generality let's consider 0= , which
makes our discussion simpler. Then the above two cases are already enough here,
postponing other two cases in the following sections. Miraculously, Caa reduces to diagonal entries
of hk or h~n k [12] depending on the helicity:
haa = Caa =
X tb habi
b6=a ta ab
b2
a 2
;
h~aa = Ca+a =
a 2 + :
(3.18)
The important thing is that the diagonal entry is a linear combination of o diagonal
entries in that row/column. With these diagonal entries of hk or h~n k, the reduction for
(a1a2 ai) with i > 1 or i = 1 (for k0 = k) are both spelled out in one nice formula, (3.12).
We nd ha1a2 ha2a3
haia1 in (3.12) is just the ingredient of det hk,
det hk =
X sgn(q)hI1 hI2
q2Sk
hIs ;
with hI = h(a1a2 ai) = ha1a2 ha2a3
haia1 ;
(3.19)
where the sum is over all permutations of particles of negative helicity, i.e. q 2 Sk and
; Is are the cycles of the permutation q. Similarly works for h~a1a2 h~a2a3
~
haia1 .
Then, we see that Pf
n factorizes to two parts depending on particles of negative or
positive helicity respectively, with most of the terms vanishing and the surviving terms
combining to det hk or det h~n k,
Pf n k0=k = det hk det h~n k :
(3.20)
Obviously both det hk and det h~n k vanish since they both have a null vector; this is
consistent with the fact that Pf n vanishes due to the two null vectors.
{ 8 {
Now we turn to Pf0 n. Now we need to deal with the open cycle. Similarly, we can always
reduce these mixed open brackets into split one as any two adjoint linearised strength elds
fb fp+ in the kinematic numerator of open brackets 1
place if the helicity of b; p are di erent, i.e.
n can exchange their
1
demonstration we need to delete two columns and rows from negative and positive helicity
set respectively, so we assign 1
and n+. Using this property, we can always rearrange the
kinematic numerator in a split form with the ordering of particles of negative helicity and
the ordering of particles of positive helicity unchanged respectively. For example, with n>6,
1 f5+f2 f3 f6+f4 n+ = 1 f2 f5+f3 f6+f4 n+ =
= 1 f2 f3 f4 f5+f6+ n+
=
All 35 = 10 such kinematic numerators of open cycles whose ordering of negative and
positive particles between 1 and n are 2; 3; 4 and 5; 6 respectively equal to 1 f2 f3 f4 f5+f6+
+
n .
Further on, all such kinematic numerator can reduce to a product of some simple angle
brackets and square brackets as shown in the last equality. Here j ]; j i are the reference
of 1,n respectively.
For the general case with x particles of negative and y particles of positive helicity
x
between 1 and n, there are x+y cycles whose kinematic numerators are equal to those of
a certain split open cycles and they all reduce to a product of some simple angle brackets
and square brackets,
2 1 fb1 fb2 fbx fp+y fp+y 1 fp+1 n+ = h1b1ihb1b2i hbx 1bxihbx i
1
[
1
]hn i
[ py][pypy 1] [p2p1][p1n]
x
used the reversed ordering py; py 1;
; p1 for later convenience.
Here j ]; j i are the reference of 1,n respectively, i.e. 1 = j1[1i[ ]j ; n+ = jn]h j and we have
hn i
Since x+y such open brackets share same kinematic numerator, we try to combine
their denominators. They happen to be combined to the partial fraction identity (analogous
to KleissKuijf relations of amplitudes),
here f g means a2; a3;
; ai 1 and f g, f g means b1; b2;
spectively. f
g denotes the reverse ordering of the labels f g.
; bx, and p1; p2;
; py
re{ 9 {
Then ( )j j P
f g2OP(f g;f T g) [1a2 ai 1n] combines to
1 hbx i
2 1 fb1 fb2 fbx fp+y fp+y 1 fp+1 n+ = h1b1 hb1b2 hbx 1bx n i bxn
(1b1 bxnp1p2 py) h
:= h[1b1b2 bx]h~[np1p2 py]
only depend on bx or cy respectively.
reduces to hn i 1n
h1 i
hn i 1n
In the rst equality, we have plugged in (3.23). In the second equality, we have de ned
h[1b1b2 bx] as h1b1 hb1b2 hbx 1bx hn i bxn
hbx i and h~[np1p2 py] as h~np1 h~p1p2 hpy 1py [
1
] 1py
~
can treat 1 as b0 and if there is no particles of negative helicity between 1 and n, hn i bxn
h1 i ; similarly we can treat n as p0 and if there is no particles of positive
helicity between 1 and n, [
1
] 1py
[ n] . Note that these prefactors hn i bxn
[1a2 ai 1n] has particles with mixed helicity, h[1b1b2 bx] and h~[np1p2 py] do
have only particles of negative or positive helicity respectively. Adding that closed cycles
vanish unless all of their elements have same helicity, Pf 0 n decouples to two parts which
are dependent on particles of negative and positive helicity respectively,
0
X hI
I J
1
hJ A
sgn(r~) X h~[1p1 py]
X ~
hK
K L
~
hL
!
:
Here we have explicitly written out the open cycles to emphasis them.
; I;
the cycles of permutations r of negative helicity particles except 1 and ; K;
; L are the
cycles of permutations r~ of positive helicity except n.
For example, with 1 2 3+4+,
Pf0 4 =
h[
1
]h(2) + h[12]
h~[4]h~(3) + h~[43] ;
with 1 2 3 4+5+,
h[
1
]h(2)h(3) + h[
1
]h(23) + h[12]h(3) + h[132] + h[
13
]h(2) + h[123]
h~[5]h~(4) + h~[54] :
Without the loss of generality, we let
= f1; 2;
; kg and + = fk+1; k+2;
We try to prove the two parts in (3.26) combine to two reduced determinants of matrices hk,
~
hn k respectively, de ned as det0 hk =
+ as hk has a null vector (t1; t2;
tbtc
det jhkjttbc ; det0 h~n k =
tq
det jh~n kjtp with b; c 2
tptq
; tk) and h~n k has a null vector (tk+1; tk+2;
Note that
det jhkjb1x = ( )
x det jhkj11
hbxc!h1c
= ( )
x
X
r2Sk 1
sgn(r) h(bx )
hI
hJ :
hbxc!h1c
Here r is any permutation of particles of negative helicity except 1, and (bx
); I;
are the cycles of r. c can be anyone of 1; 2;
h~np1 h~p1p2 hpy 1py [
1
] 1py
~
we write the rst part in the r.h.s. of (3.26) as a sum over all possible bx, i.e. bx = 1; 2;
This equality can also be seen by collecting terms with the same prefactor hn i bxn
0

{z
0
; J are the cycles of permutations of particles of negative
helicity except 1 and bx. Then each term of the summation in r.h.s. of the above
equation equals det jhkjb1x up to a prefactor. Summing over all possible bx, i.e. bx = 1; 2;
k,
gives the left parenthesis of r.h.s. in (3.26). Similar derivations leads to the right
parenthesis. Then
0 k
X
10
n
X
We insert ttn1ttbbxx in every term of the rst sum of above equation and ttn1ttppyy in every term
of the second sum, which doesn't change the value of Pf 0 n. Then
0 k
X hbx itbx tn det jhkjb1x 10
X
py=k+1
[ py]t1tpy det jh~n kjpny 1
; k reduce to det0 hk, all det jh~n kjpny
with py = k +
All gauge dependence of particle 1 and n combine to one factor respectively and on the
support of 4d scattering equation (1.2),
the two prefactors before the determinants in (3.34) reduce to 1 respectively. Then we get
tn
X tp ~p_ = ~1_ ;
1 p
Pf0 n k0=k = det0 hk det0 h~n k :
(3.30)
(3.31)
(3.32)
(3.34)
(3.35)
(3.36)
Extension to all solution sectors
We have arrived at (3.32) without using the explicitly form of 1length cycle, i.e. Caa.
When extended to all solution sectors, those cycles whose length are longer than 1 don't
change, while the 1length cycles change to Caa with the solutions of k0 sectors plugged in.
That is, we need to enhance the origin two matrices to hkk0 and h~k0
n k with their diagonal
entries depending on the solution sector k0 while the o diagonal entries unchanged. The
expression of Caa with a 2
is true even when k0 6= k,
0 has been given in (3.16). Note that this expression
haa =
Caa =
Now we derive the expression of Caa with a not consistent in helicity sector and solution
sector. When a 2
but a 2= 0, we have
Caa =
1
[a ]
X
p2+0; p6=a
HJEP07(21)69
(4.1)
(4.2)
(4.3)
(4.4)
After we plug in the changes of variables (3.14), unlike (3.15), terms with a 2 +0 and p = a
both contribute.
The rst term on the r.h.s. also factorizes into two parts following the trick used in (3.15), (3.16),
ab
tp[p ]
X
p2+0; p6=a ta ap[a ] A = 0 ;
2
ba
1
while it vanishes as shown in the last equality because the part in the rst parenthesis
vanishes on the support of 4d scattering equation (1.2), (note that a 2 +0)
ta
a b
then we see that Caa only has contribution from the term of p = a, and we obtain
Consequently, we have
By a parity transformation, we can directly obtain h~aa
Caa =
haa =
X
b<c
but a 2= 0 ;
a 2 + and a 2 +0
a 2 + but a 2= +0 :
(4.8)
(4.5)
(4.6)
(4.7)
When k0 = k, these extended matrices come back to their original ones. When k0 6= k,
one of det hkk, det h~k
n k must vanish. Further more, when k0 < k, after deleting appropriate
the h~k
row and column of hkk, the determinant of the remaining matrix still vanishes, so does
n k when k0 > k, which results in the vanishing of Pf 0 n in k0 6= k sectors. We
will discuss this in section 4.1. Some integrands receive the contribution from the k0 6= k
sectors, such as Pn, which will be discussed in section 4.2.
4.1
The vanishing of reduced Pfa an in other sectors
We start from the equation (3.32). Note that we have got this by deleting the 1th and
nth rows and columns of
n. We can also delete other rows and columns to get a similar
expression.
What's more, along the demonstration of (3.5) to (3.32), we don't use the
scattering equations (1.2), in other words (3.32) is true for any solutions. After (3.32), the
scattering equation is used and we demonstrate (3.36). Now we move to other solution
sectors. Without loss of generality, let's consider
= f1; 2; : : : ; kg while 0 = f1; 2; : : : ; k0g,
which makes our discussion simpler. Then when k0 < k, further on, we can also and always
delete (k0 + 1)th and nth column and row instead of the of 1th and nth ones, then the
reduced Pfa an becomes
0 k
X
10
n
X
Notice that we still have to calculate the determinants of series of matrices. Instead of both
summations in r.h.s. of (4.9) being combined to simple factors as shown in (3.34), we show
that the determinants of matrices jhkk0 jbkx0+1 with bx = 1; 2;
vanish identically.
; k in the rst summation
These matrices all come from the original matrix hkk0 with the (k0 + 1)th column deleted
and the 1th; 2th;
; kth row deleted respectively. An important observation is that the rst
k0 columns of these matrices are linearly dependent as
; k0th columns of anyone of matrices jhkk0 jbkx0+1 with
These equations come from two totally di erent origins as a
fact that the diagonal elements haa are a linear combination of some o diagonal entries
as shown in (4.1) with some appropriate coe cients.
While for the cases of a > k0, note that a now belongs to the set +0, and the validity
of (4.11) come from the change of variables (3.14). What we need here is the cases of
a = k0 + 1; k0 + 2;
; k,
Obviously after we act a on both sides of above equation, both sides vanish, that is
for a = k0 + 1; k0 + 2;
; k
(4.12)
for a = k0 + 1; k0 + 2;
; k
(4.13)
ta
X tb b =
a b
k0
b=1
a
ab
k0
b=1
X tbhab =
X tb habi = 0
After understanding (4.10), now it is easy to understand the vanishing of all matrices
jhkk0 jbkx0+1 with bx = 1; 2;
a = 1; 2;
; k0, and then add 2th; 3th;
; k0 th column to the 1th column; in this way we
obtain a new 1th column whose entries all equal to zero because of (4.10). Since we just
do some fundemantal operation on these matrices and we obtain a column with all entries
equal to zero, all determinants of these matrices vanish.
When k0 < k, the determinants of matrices in the rst summation in (4.9) vanish;
when k0 > k, the determinants of matrices in the second summation in (4.9) vanish. Pf0 n
only receives the contribution of k0 = k sector, then we proved the identity (1.4) given in
; k. We take a multiple of the ath row of these matrice by ta for
the introduction.
In the reduction of Pf0 n, we reorganize the Pfa an using some fundamental
(almost) gauge invariant objects and then deal with these objects, nally we reconstruct the
reduced Pfa an using det hk and det h~n k. This procedure is quite general. The reduced
Pfa an can be thought as putting two kinematic of the deleted particles in higher
dimensions and the Lorentz contraction of them and anything else vanish unless contraction of
them each other equal to 1.1 Similarly, we can put polarizations of m pairs of particles in
higher dimension and then we get the reduced compacti ed Pfa an, which is the integrand
of EinsteinMaxwell, EinsteinMaxwellscalar, YangMillsscalar, BornInfeld amplitudes.
This can be viewed as the \fancy reduced Pfa an" as we may meet several open brackets
in each term of the expansion. Besides, the valid solution sector for the reduced compacted
Pfa an is shifted to k0 = k + m sector with 2m the number of the particles whose
polarization are set in higher dimension. These are presented in the Appdendix B. Further
more, the reduced squeezed Pfa an, which is the integrand of EinsteinYangMills, can be
obtained by some combination of the reduced compacti ed Pfa an. So all these methods
can be applied in the reduced squeezed Pfa an, seen in Appdendix C.
There are some integrands that can't be organized as a matrix, let alone its Pfa an,
such as the Pn used in F 3; R2; R3 amplitudes. We can still reduce it into objects related
to det hk and det h~n k and make some properties manifest. Besides, it receives the
contribution from several sectors and one need to add up all of these to get the corresponding
amplitudes, as discussed in the following subsection.
4.2
The new object Pn for higher dimension operator
As shown in [21], as a generalization of the reduced Pfa an in YangMills theory, Pn is a
new, gaugeinvariant object that leads to gluon amplitudes with a single insertion of F 3, and
gravity amplitudes by KawaiLewellenTye relations. When reduced to four dimensions for
given helicities, this new object vanishes for any solution of scattering equations on which
the reduced Pfa an is nonvanishing. This intriguing behavior in four dimensions explains
the vanishing of graviton helicity amplitudes produced by the GaussBonnet R2 term, and
provides a scatteringequation origin of the decomposition into selfdual and antiselfdual
parts for F 3 and R3 amplitudes.
No matter what Pn is, it must be gauge invariant. It's most natural to start from
the expansion of Pfa an in a manifest gauge invariant way (3.1). Reorganize these gauge
invariant objects according to their length and we de ne the minimal gauge invariant and
guage invariant objects P as
(4.14)
(4.15)
Pi1 i2 ir :=
X
jI1j=i1;jI2j=i2; ;jIrj=ir
I1 I2
Ir ;
i
2
ir.
Then Pfa an can be written as
Pf n =
X ( )n mPi1i2 im = 0 ;
i
1Here is some subtlety. We have to complex k1; kn set in higher dimensions such that they dotting
anything else equal to 0, while they dotting each other equal to 1. It is just a mathematics trick after all
nothing about k1; kn changes beyond the reduced Pfa an.
Decorated with some appropriate coe cient, P can be used to build up some unknown
CHY integrands, such as the new integrand is de ned as
Pn =
X
1 i1i1+i2i+2 +im=n
im n
( )
n m (Ni>1 + c) Pi1 i2 im ;
where Ni>1 denotes the number of indices in i1; i2;
; im which are larger than 1, or the
number of cycles with length at least 2; c is just any constant because we can add any
multiplet of (4.15) without changing the answer.
In 4d, Pf n reduce to the determinants of hkk0 and h~kn0 k. Similarly we de ne
HJEP07(21)69
Hi1i2 i` =
X
jI1j=i1;jI2j=i2; ;jItj=i`
hI1 hI2
hI` ;
+ i` = k and the convention i1
i
2
as a sum of H and similarly works det h~k0 ,
i`. Then det hkk0 can be rewritten
det hkk0 =
X( )k `Hi1i2 i` :
fig`k
n k
det h~kn0 k =
X
f~ig`n~ k
( )
n k `~H~~i1~i2 ~i`~ ;
fig`k
where we have introduced shorthand notation for the summation range, fig`k means i1 +
i2 + : : : i` = k and i1
i
2
i` and similarly for f~ig`n~ k
.
Further, similar to the de nition of (4.16), we de ne two auxiliary objects Hkk0 and H~nk0 k,
Hkk0 =
X( )k `Ni>1Hi1i2 i` ;
~k0
Hn k =
( )
n k `~N~i>1H~i1~i2 ~i`~ :
~
where each mix summation over i and ~i decouples to two independent summation over i
and over ~i respectively. Then
Pn k0 = Hkk0 det h~kn0 k + det hkk0 H~nk0 k :
X
f~ig`n~ k
10
10
X
f~ig`n~ k
X
f~ig`n~ k
(4.16)
(4.17)
(4.18)
(4.19)
(4.20)
(4.21)
(4.22)
Thanks to (3.12), P reduces to several products H and H~ in 4d,
Pi1i2 im =
j;~j
X Hj1j2 j` H~~j1~j2 ~j`~ ;
Here the sum is over all distinct partition of i1i2
im into two parts j1j2
j` and
~j1~j2
~j`~, with j1 + j2 +
+ j` = k and ~j1 + ~j2 +
+ ~j`~ = n
into two parts Nj>1 and N~j>1 (set c = 0) which depend on
k. Dividing Ni>1 in (4.16)
and + sets respectively, Pn
reduces to:
n k `~H~~i1~i2 ~i`~AC
0
fig`k
0
fig`k
k0 < k, InF 3 reads
det hkk0 H~n k, which gives the antiselfdual amplitude of those theory.
When k0 = k, both terms in the r.h.s. of (4.22) vanish, which is orthogonal to Pf 0 n and
answers the vanishing of R2 theory which is a GaussBonnet term in 4 dimensions. When
k0 < k, the second term vanishes and Pn reduces to Hk det h~kn0 k, which gives the selfdual
amplitude of F 3; R3 theory etc. When k0 > k, the rst term vanishes and Pn reduces to
For example, the integrand for F 3 theory reads In
F 3 = Cn Pn. In 4 dimension, when
IF 3
n
= Cn Hkk0 det h~k0
det0 hk0 det0 h~n k0
n k
(4.23)
HJEP07(21)69
Here det0hk0 det0 h~n k0 comes from the transition of two forms of scattering equations as
shown in (2.6).
5
In CHY representation, the fundamental gauge invariant objects are quite common, either
Caa or the trace of some linearised eld strength together with some 's. In this paper, we
nd a rather general way to reduce this gauge invariant objects into that made up of spinors
using 4d scattering equations. Particularly, we show how the reduced Pfa an reduces to
some determinants and why it vanishes on the support of most solutions. This explains
why only some particular solutions contribute to the YM or GR amplitudes according to
their helicity structure in 4d and provides a basis for dividing the solutions of scattering
equations into MHV,NMHV,
,MHV sectors which contributes to corresponding YM or
GR amplitudes respectively, also seen in [23].
We extend this methods to reduced compacti ed Pfa an where some polarizations are
set in higher dimensions, which is building block for EM, EMS, YMS or DBI amplitudes.
We give the explicit reduction results of the compacti ed Pfa an up to 3 pairs of particles
whose polarizations are set in higher dimension and provides the general way to get the
reduction with arbitrary pairs. Another interesting integrand with polarization involved is
the reduced squeezed Pfa an, which is the building block of EYM theory [3, 4, 24, 25]. As
it can be expressed by some combination of the reduced compacti ed Pfa an, its reduction
in 4d is directly obtained from that of the reduced compacti ed Pfa an. The results of
one and two gluon color traces are explicitly presented in the appendix C.
Even some integrands which can't be organized as a matrix, let alone its Pfa an,
such as the new integrand Pn used in F 3; R2; R3 theory, also can be enclosed in this
procedure. We decompose Pn to some fundamental gauge invariant objects,reduce these
fundamental gauge invariant objects rst and then organize them into a compacted form,
which apparently shows most information of the Pn, and explains the orthogonality of F 3
and YM amplitudes, the vanishing of GaussBonnet term R2 and the selfdual and
antiselfdual amplitudes of F 3 or R3 amplitudes in 4d. In fact, we use these properties to guess
what the compacted form in 4d of Pn should be, then x the coe cient of the fundamental
gauge invariant objects and
nally get the Pn. This is quite general to
integrand of an unknown theory. Even when the scattering equations or
nd the CHY
dependence of
the entries in the matrix
n has been changed, their CHY integrand are very likely to be
decomposed into some Caa or trace like fundamental gauge invariant objects. And we can
reduce these objects rst, organize them into a form manifest showing some properties the
theory requires and
nally con rm their CHY integrand. This can even be applied at loop
level, as shown in [19, 26].
Instead of reducing all kinds of integrands in 4d, we now turn to the general 4d CHY
formulae. After overcoming the obstacles to reduce integrands with polarization involved,
the calculation of CHY formulae becomes much simpler. We develop the 4d CHY formulae
to directly calculate the amplitude of the some theory. The reduced Pfa an behaves like
a solution lter, making the building of 4d CHY formulae natural. As if the general CHY
formulae has been reduced to 4d CHY formulae and the number of solutions decrease from
(n
3)! to En 3;k0 2.
We have discussed the reduction of the reduced compacti ed Pfa an and squeezed
Pfa an in appendix B, C and discussed how the valid solution sector shift from the helicity
sector. The more polarizations there are, the more e cient our procedure is. Even when
there is no polarization involved, and the reduced compacti ed Pfa an totally reduce to
Pf0An times something, our reduction procedure still holds and it tells us only the k0 = n=2
solution sector contributes. This means CHY formulae of some e ective eld theory such
as BornInfeld, DiracBornInfeld, NonLinear Sigma Model, Special Galileon theories with
Pf0An acting as CHY integrand also reduce to a set of 4d CHY formulae. Even for some
theories that receive the contribution of several solution sectors such as those with Pn
acting as CHY integrand, the physical meaning of 4d CHY formulae is also apparent: the
contribution from the k0 < k sectors gives selfdual amplitude and that of k0 > k gives
antiselfdual amplitudes.
Many good properties shared by CHY formulae are still inherited by the 4d CHY
formulae. The soft limits has been discussed in [15, 27{30]. Factorisation should also be
easy to study. Not only the CHY formulae have a simple representation in 4d, 4d CHY
formulae can also help us understand the CHY formulae in general dimension. Besides,
the supersymmetrization of the 4d CHY formulae is directly and we just need to replace
the ~
a with f~a j aAg in the scattering equation in (2.4) as shown in [15]. This way we can
involve fermions in CHY formulae, for example we use SYM amplitudes to build up QCD
amplitudes as shown [31]. In the same paper, we use two set of spinors to describe the
massive Higgs, which has been generalized to calculate form factors [32, 33].
We tentatively study whether the solutions divide beyond in 4d. Especially we hope
something interesting come out in 6 dimensions where we also have a good spinor
representation [34{36] and some nice result of CHY formulae in 6d comes out. We can treat a
massive particle in 4d as a massless particles in 6d, especial the massive loop particles in
4d. Up to now, our result is negative and we didn't nd the solutions of scattering equation
divide again in other dimension.
CHY formulae has been extended to 1loop level, as discussed in [37, 38]. It has been
known that what underpins the CHY formulae is ambitwistor string. And ambitwistor
string theory has been extended to 1loop level, as shown in [7, 39, 40]. However we nd
the solutions at 1loop level don't divide into several sectors again in SYM or SUGRA
theory. CHY formulae has singular solutions at 1loop, how about 4d CHY formulae and
how does it contribute to the divergence bubble or tadpole? Also it is interesting to check
whether the integrand still factorizes to two objects that depend on particles of negative
helicity or positive helicity respectively. Also it will help us to build the general 1loop
CHY integrand.
As discussed in [41], also it is very useful to study the positivity of the jacobian or
integrand in 4d CHY formulae. (2.6) is a useful identity as it link several objects. As shown
in [41], det 0 n(fsab; ag) is positive at the positive region. We know that det0 hk0 det0 h~n k0
is exactly the result of Pf 0 n with k0 external particles of negative helicity. If the 4d
jacobian Jn;k0 is also positive at the positive region, it strongly supports that the YM
amplitude is also positive in some regions.
Acknowledgments
Yong Zhang thanks S. He and F. Teng very much for discussions. Y.Z.'s research is partly
supported by NSFC Grants No. 11235003.
A
A conjecture on Jn;k0
In this section, without making confusion, we denote a b =: (ab).
We have strong
evidence to support the following conjecture,
tatb
Y
1 y k0 2
(bxcx)hbxcxi(pyqy)[pyqy](dxyryx)2CC
Here the sum is over all b1<c1; b2<c2;
; bn k0 2<cn k0 2;
b1; c1; b2; c2;
;
bn k0 2; cn k0 2 2
+0. The xth row of matrix d is
0 and p1<q1; p2<q2;
; pk0 2<qk0 2; p1; q1; p2; q2;
; pk0 2; qk0 2 2
0nfbx; cxg for 1 x n k0 2 and the yth row of matrix
r is +0nfpy; qyg for 1 y k0 2. When k0=2 or n
numerically checked this formula up to 9 points in all solution sectors, and 15 points in
2, (dxyryx)2 reduces to 1. We have
k0 = 3 sector.
Here are some examples.
For the MHV solution sector, it can be analytically proved that
Jn;2 =
n
a=1
(bc)n 4hbcin 4
Q t2 Q (bp)2(cp)2
a
p2+0
Qn
a=1 ta
Here fb; cg =
entries are 2
with a compensate tb3tc3(bc)2hbci2 , the minor becomes a \diagonal" matrix whose diagonal
2 matrices and their determinants can be easily calculated out as (pb)2(pc)2t3p
(bc)hbci
for p = +0, then we get (A.2).
0. After deleting the 4 columns and rows about b; c of the matrix in (2.5)
1
A
(A.1)
(A.2)
For the NMHV solution sector,
J6;3 = n
Here d; r is a particle label as the breviate of d11; r11 and fb; c; dg = 0, fp; q; rg = +0.
J7;3 = n
(A.3)
(A.4)
Here d1; d2 is a particle label as the breviate of d11; d21 and fb1; c1; d11g = 0,fb2; c2; d21g =
0. r1; r2 is a particle label as the breviate of r11; r12 and fp; q; r11; r12g = +0. Note that
this restrain doesn't
x r11; r12 totally as r11; r12 can exchange their value. However it
doesn't a ect the value of J7;3 as we always sum over all b1 < c1; b2 < c2.
Jn;3 = n
X
Y
(bp)2 b<c;p<q 1 x n 5
1
(bxcx)hbxcxi(pq)[pq](dxrx)2A
(A.5)
Here dx is a particle label as the breviate of dx1 and fbx; cx; dxg = 0. rx is a particle label
as the breviate of r1x and fr11; r12;
r1;n 5g = +0nfp; qg.
For NNMHV solution sector,
J8;4 = n
a=1
1
Q
(d11r11)2(d21r12)2(d21r12)2(d22r22)2 :
(A.6)
Here fb1; c1; d11; d12g =
0, fb2; c2; d21; d22g = 0, fp1; q1; r11; r12g = +0,fp2; q2; r21; r22g =
+0. Note that this restrain doesn't
x d11; d12 totally as d11; d12 can exchange their value
neither does r11; r12. In most cases, it doesn't a ect the value of J8;4 however a few cases
do rely on particular ordering of d11; d12 or r11; r12, leaving a further study to x the nal
expression of Jn;k0.
B
The reduced compacti ed Pfa an in 4d
In the main text, we have discussed the reduced Pfa an which we delete 1th and nth
rows and columns of the matrix
n. Then we introduce the open cycle to reduce it into
two reduced determinants. Also we can e ectively think that we set the momenta of the
particles 1; n in higher dimension and they dotting everything equal to zero unless they
dotting themselves equal to 1 to make the complement 1 . Then we can still decompose
the reduced Pfa an into some (modi ed) closed cycles. One closed cycle must contain
1; n both or vanish if it just contains one of them, and then it reduces to open cycle as
k1; kn are set in higher dimension. This can be extended to other cases as now we set the
1n
polarisation of some pairs of particles in higher dimension. We call it reduced compacti ed
Pfa an which is the building block for EM, EMS, YMS amplitudes, as discussed in [4].
Then we can use the similar trick to reduce the reduced compacti ed Pfa an in 4d.
We donate the set of particles whose polarisation are set in higher dimension as h.
Obviously, the length of set h must be even. Further on, we let the polarisation of particles
in h be anyone of an orthogonal bases and they dotting each other equal to 1 or 0, donated
as IaIb . Then the compacti ed Pfa an can be think of as being deleted the rows and
columns of h in the last n rows and columns of the matrix
complement it with a Pfa an. That is,
n donated as Pf0j njffhhgg and
with
Pf0 n;m = Pf0j njfhg
fhg Pf[X ]h
Pf[X ]h =
X
Here 2m is length of the set h. First we considering the case with m = 1, that is only
one pair of particles donated as e1; e2 that needs dimension reduction. For simplicity, we
also delete the rows and columns of e1; e2 in the rst n rows and columns to satisfy the
mass dimension, i.e. we e ectively set the momenta of e1; e2 in higher dimension. Then
unless they contain and only contain both e1; e2. Then this cycle, which equals to
in the expansion of the reduced compacti ed Pfa an, all cylces that contain e1; e2 vIae1nIies2h
factor out, leaving all other cycles normal as if e1; e2 not existed. They factor into two
determinants of two matrices in 4d, just like the factorisation of Pfa an in (3.20), with
e1e2 ,
the diagonal elements equal to Caa plugging a certain solution sector k0, as expressed
in (4.1), (4.7), (4.8). One of two determinants of these two matrices will vanish trivially
unless k0 = k + 1. That is, we need to assign e1; e2 to into two sets, for example, we let
0 =
[ fe1g and +0 = + [ fe2g. Then the reduced compacti ed Pfa an with only one
pair of particles needing dimension reduction reduces to
Pf0 n;1 k0 = k+1;k0 2
det jhk0 jee11 det jh~n k0 jee22 Ie1 Ie2 :
The expression of hab with a; b 2
Here we can extend the de nition domain from
to
enclose he1b or h~e2b, though it is not important as such entries will always been deleted from
the matrices hk0 and h~n k0 in the above equation. In this case, the exchange of e1 $ e2
will a ect the expression of the diagonal elements of hk0 and h~n k0 but it won't a ect the
nal result. For later convenience, we write the above equation in a slightly di erent way,
and h~ab with a; b 2 + are given in (3.11), (4.1), (4.8).
[ fe1g and from + to + [ fe2g, to
Pf0 n;1 k0 = k+1;k0
det jhk0 jee11 det jh~n k0 jee22 Pf[X ]h :
Now we consider the case with m = 2, i.e. 4 particles donated as e1; e2; e3; e4 need
dimension reduction. There are be 3 perfect matching to make pairs in the expansion of
Pf[X ]h, as shown in (B.2). We can take these perfect individually and at last add them
1
e1e2
1
e1e2
(B.3)
(B.4)
up. For example, we consider a perfect matching that e1; e2 a pair and e3; e4 a pair. Still
we e ectively set the momenta of e1; e2 in higher dimension and then
The left pair e3; e4 must be adjoint and enclosed in one cycle and it reduces to an open
cycle similar to (3.6) with the polarisations on the ends replaced by kinematics as
e1e2
Ie1 Ie2 factors out.
tr feh4 fa3 fa4
Here feh just means the polarisation of particle e is set in higher dimension and fa means
when reduced to 4 dimension, the helicity of particle a can be negative or positive. Also
any two adjoint linearised strength elds fb fp+ in the trace can exchange their place if the
helicity of b; p are di erent. So we use this property to split the kinematic numerator of
this open cycle in respect of negative and positive helicity rst, then use the partial fraction
identity to spilt the denominator, and nally cut the open cycle into two closed cycles,
fe1; e2g a pair and fe3; e4g a pair gives
k+2;k0 2
1
e1e2
0
X
bx2fe3g[
10
X
py2fe4g[+
Here heb = hebi , h~ep = [ep] . Compared to (3.25), we have replaced the 1 = j1[1i[ ]j or
eb ep
hbx i [ py]
n+ = jn]h j by ke = jei[ej, and then the prefactor hn i bxn or [
1
] 1py
hn i
are replaced by
Then followed by other closed cycles, similar to (3.32), the perfect matching with
1
e1bx
hbxe4 det jhk0 je1e3 A@
~ e2py
hpye3 det jh~n k0 je2e4 A
Ie1 Ie2 Ie3 Ie4 :
(B.6)
(B.7)
The factor k+2;k0 come from the fact that we must assign e1; e2 in di erent sets and e3; e4 in
di erent sets, for example
0 =
[fe1; e3g and +0 = +[fe2; e4g. In this case, the exchange
of e1 $ e2 or e3 $ e4 doesn't a ect the
nal result. The exchange of fe1; e2g $ fe3; e4g
does't a ect the
nal result.
The other two perfect matching can be think of as the
exchange of fe1; e2; e3; e4g $ fe1; e3; e2; e4g and fe1; e2; e3; e4g $ fe1; e4; e2; e3g. However
we have a cleverer choice that each perfect matching in (B.2) must share the same coe cient
Pf0j
compacti ed Pfa an with two pairs of particles needing dimension reduction reduces to
n;2jffhhgg. We can read this coe cient Pf 0j
n;2jffhhgg from (B.7) and then the reduced
e1bx
hbxe4 det jhk0 je1e3 A@
h~pye3 det jh~n k0 jee22ep4y A
0
X
bx2fe3g[
10
X
py2fe4g[+
1
e1e2
e3e4 Pf[X ]h :
(B.8)
One can change the fe1; e2; e3; e4g by any other permutation and it won't change the nal
result. They just di erent representation of Pf 0j n;2jffhhgg as the choice of prefactor in (B.6)
is rather arbitrary: one choose hbxe3 as prefactor as well as he4b1 , so does h~pye4 and h~p1e3 .
Now we consider the case with m = 3, i.e. 6 particles donated as e1; e2; e3; e4; e5; e6 need
dimension and then
dimension reduction. Still we consider the perfect matching with fe1; e2g a pair,fe3; e4g
a pair and fe5; e6g a pair rst. Still we e ectively set the kinematics of e1; e2 in higher
Ie1 Ie2 factors out. The left two pairs fe3; e4g and fe5; e6g must be
e1e2
adjoint in the trace respectively, or vanish. However these two pairs can be enclose in
two di erent cycles or in a common cycle. The former case is simpler. We split the open
cycles, cut it into modi ed open cycles and rearrange them together with other closed
cycles into determinants. That is, those of the expansion of Pf 0 n;2 k0 that contain such
X
hpye3 h~qwe5 det jh~n k0 jee22ep4yeq6w CCC Ie1 Ie2 Ie3 Ie4 Ie5 Ie6 :
~
(B.9)
The factor k+3;k0 come from the fact that we must assign each pair of particles in this
perfect matching into di erent sets, for example
0 =
[ fe1; e3; e5g and +0 = + [
fe2; e4; e6g. In this case, the exchange of particles in each pair doesn't a ect the nal
result. The exchange of di erent pairs also does't a ect the nal result.
However the case where fe3; e4g and fe5; e6g are enclosed in a common cycle also
contribute and we need to deal with it more carefully. We nd that the equation (B.5) can
be extended to
tr feh4 fa13
fai1 feh5 feh6 fa2
3
h
faj2 fe3
or even more general form. Still any two adjoint linearised strength
trace can exchange their place if the helicity of b; p are di erent. However this time the
exchanging is blocked by feh. So we treat the region feh4 fa1
3
fai1 feh5 and feh6 fa2
3
faj2 feh3
individually and split the kinematic numerator of these region separately, followed by the
splitting of denominators using partial fraction identity. Finally,
elds fb fp+ in the
Here we have given the general form with arbitrary pairs of particles in h enclosed in a
e b
e p
. And feh 4 is the linearised strength elds next to
feh 3 . It could just be feh 2
and at this case e 3 e 2 reduces to 1.
Compared to (B.6), here comes out a factor ee 33 ee 24 . We can think there is also a factor
e3e4 = 1 in equation (B.6). However there is something di erent essentially. For example,
in the former case, tr(feh3 feh4 ) tr(feh5 feh6 ) , e3 groups with e4 and e5 groups with e6, while in
the later case, tr(feh4
feh3 ) , e3 groups with e6 and e4 groups with e5. In the former
case, the group pairs are consistent with the perfect matching pairs and the exchanging of
particles in the same perfect matching pair is identical, while in the latter cases, the group
pairs are not consistent with the perfect matching pairs and the exchanging of particles
in the same perfect matching pair is two di erent contributions. However it is still not
tough. The reduction of those that contain such cycles tr(feh4
feh5 feh6
(e4 e5e6 e3)
feh3 ) in the expansion
of Pf0 n;2 k0 can be got from (B.9) by exchanging e5 $ e3 if ignoring the factor e 3 e 2
e 3 e 4
and the reduction of those that contain such cycles
(e4
Pf0 n;2 k0 can be got from that of tr(feh4 feh5 feh6 feh3 ) by exchanging e5 $ e6. Taking all
the perfect matching into consideration, then the reduced compacti ed Pfa an with three
e5e6
e3)
pairs of particles needing dimension reduction reduces to
(e4 e6e5 e3)
tr(feh4 feh6 feh5 feh3 ) in the expansion of
Pf0 n;3 k0 = k+3;k0 BBB e4e5 e6e3 BB
B
e1e2
hpye5 h~qwe3 det jh~n k0 jee22ep4yeq6w CCC+ e5 $ e3
~
e5 $ e6 CCPf[X ]h :
0
B
B
0
B
B
B
B
X
B@qpwy22ffee46gg[[++
C
C
A
1
C
C
A
!
The minus before the exchanging of e5 $ e6 comes from
absorbed in Pf[X ]h. When it comes to the cases with m
e6e5 =
e5e6 and
3, there are no more new objects coming out and just some more calculation and we can always reduce the reduced compacti ed Pfa an into some determinants.
C
The reduced squeezed Pfa
an
We write the reduced compacti ed Pfa an (B.1), (B.2) in a slightly di erent way,
Pf0 n;m =
X
1
!C
C
C
A
(B.12)
e5e6 is
Here we use Pfj jH;a1;b1;a2;b2; am 1;bm 1;H to show what column and rows are left in the
matrix
dimension as H. Here Tr(T Ia1 T Ib1 ) can be seen as a twogluon ParkeTaylor factor. As
n and we donate the set of particles whose polarisation are not set in higher
shown in [4], it can be extended to a ParkeTaylor factor with arbitrary number of gluons,
Ca1;a2; ;as =
X
!2Ss=Zs
Tr T I!(a1) T I!(a2)
T I!(as)
!(a1)!(a2) !(a2)!(a3)
!(as)!(a1)
:
(C.2)
We denote the set of gluons as g and the subsets sharing in the same color trace as
Tr1; Tr2;
; Trm. Then the half integrand for EYM are given by
HJEP07(21)69
CTr1
CTrm
sgn(fa; bg) a1;b1
am 1;bm 1 Pfj jH;a1;b1;a2;b2; am 1;bm 1;H :
(C.3)
(C.4)
(C.5)
X
a1<b12Tr1
am 1<bm 12Trm 1
with single gluon color trace,
and that of double gluon color traces
The reduction of Pfj jH;a1;b1;a2;b2; am 1;bm 1;H can be obtained from the above section
(the explicit form with m = 1; 2; 3 are given in (B.4), (B.8), (B.12) except removing Pf[X ]h
Here we present the reduction of the reduced squeezed Pfa an for EYM amplitudes
Cgdet[h]H det hh~i
H+
CTr1 CTr2
0
X
X
e1<e22Tr1 bx2fe1g[H
10
bx
X
py2fe2g[H+
h~pye1 det hh~i
H+;e2 e2
py1
A e21e2 :
Open Access.
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