††institutetext: Department of Physics,
Faculty of Science and Engineering,
Swansea University,
Swansea, SA2 8PP, UK

# Identities amongst the Two Loop Partial Amplitudes of Yang-Mills theory

David C. Dunbar<EMAIL_ADDRESS>
###### Abstract

We characterise possible identities among the two-loop partial amplitudes of
gluon scattering in Yang-Mills theory. We use known amplitudes in an
exhaustive search to identify potential new relations. We find two candidate
relations which may extend to all-$n$ amplitudes.

###### Keywords:

NNLO computations

##  1 Introduction

The scattering amplitudes of gluons within a pure $SU(N_{c})$ gauge theory are
important from a phenomenological viewpoint where there is considerable demand
for new predictions particularly at “Next-to-Next-to-Leading Order” (NNLO)
(Amoroso:2020lgh, ). Also, amplitudes are the custodians of the symmetries of
the theory and as such are important theoretical objects encapsulating
information on the symmetries and properties of the theory.

The amplitudes for gluon scattering are functions both of the kinematic
variables of the scattered particles but also depend upon their gauge charges
or color. Given this, it is often convenient and informative to expand the
full amplitude in terms of color structures $C^{\lambda}$ multiplying partial
amplitudes which contain the kinematic dependence

${\cal A}_{n}^{(\ell)}=\sum_{\lambda}{A}_{n:\lambda}^{(\ell)}C^{\lambda}\,,$
(1)

where $\ell$ denotes the loop order of the $n$-point amplitude.

An important expansion is where the color factors are products of the trace of
color matrices. Gauge invariance implies that not all the partial amplitudes
are independent but that there may be relations amongst them possibly allowing
less computational effort to determine the full color amplitude. In particular
“decoupling identities” which are obtained by extending the gauge group to
$U(N_{c})$ and looking at the relations necessary for the $U(1)$ gauge boson
to decouple. However, these do not exhaust the relations amongst partial
amplitude and, for example, there exist Kleiss-Kuijf relations (Kleiss:1988ne,
) among the tree amplitudes which coincide with the decoupling identities for
low $n$ but are beyond decoupling identities in general. Also, at one loop the
subleading in color partial amplitudes can be expressed in terms of the
leading (Bern:1994zx, ): this identity again coinciding with decoupling for
low $n$. In ref. (Naculich:2011ep, ) and (Edison:2011ta, ; Edison:2012fn, ) it
was shown that further identities among the two-loop amplitudes for four and
five point amplitudes were obtained by using iterative methods assuming a
three-point diagrammatic expansion.

In this paper we wish to explore the space of possible linear identities by
examining the known amplitudes among two-loop amplitudes. Specifically these
are expressions of the form

$\sum_{\lambda}a_{\lambda}{A}_{n:\lambda}^{(2)}=0\;,$ (2)

where the $a_{\lambda}$ are pure numbers not dependent upon the kinematic
variables or helicity of the outgoing states.

Unfortunately there are very few two-loop amplitudes in pure Yang-Mills for
whom analytic forms are known which we can use to test and examine potential
identities. Amplitudes are organised according to the helicity of the external
gluon: we use the convention that all states are outgoing. Only for the four
point amplitude are all partial amplitudes known for all helicity states
(Glover:2001af, ; Bern:2002tk, ). At five-point the leading in color
amplitudes are known in analytic form from a series of papers. Initially, due
to its simplicity, the first calculation was for the special helicity
configuration where all the (outgoing) helicities are positive (or all
negative) which we describe as the “all-plus” amplitude (Gehrmann:2015bfy, ;
Dunbar:2016aux, ). These two loop amplitudes are considerably simpler in
functional form than a general amplitude because the tree amplitude vanishes
and consequently the one-loop amplitude is a pure rational function
(Bern:1993qk, ). Subsequently the leading in color single minus amplitude
(Badger:2018enw, ) and the remaining leading in color partial amplitudes
(Abreu:2019odu, ) have been computed.

Beyond the four-point amplitude and the leading in color at five point, only
amplitudes with the all-plus helicity configuration are know. Specifically
these are known for five (Badger:2019djh, ; Dunbar:2019fcq, ) and six gluons.
(Dalgleish:2020mof, ). Additionally a specific partial amplitude is known for
$n$ points. (Dunbar:2020wdh, ).

Using the all-plus five and six-point known amplitudes we can verify the
expected relations and determine if, and how many, further linear relations
exist among the partial amplitudes. Although relations identified for a
particular helicity can only be conjectured to extend to all helicities we can
definitely say there are no further identities beyond those satisfied by the
all-plus amplitude.

##  2 Color Decompositions

In this section we will examine the color decomposition in detail. We will
start with a review of tree and one loop decompositions before discussing the
two-loop amplitudes.

In a Yang-Mills gauge theory with the gluons lying in the adjoint
representation a $n$-gluon amplitude may be expanded in the gauge coupling
constant,

${\cal A}_{n}=g^{n-2}\sum_{\ell\geq 0}a^{\ell}{\cal A}_{n}^{(\ell)}\;,$ (3)

where $a=g^{2}e^{-\gamma_{E}\epsilon}/(4\pi)^{2-\epsilon}$ and ${\cal
A}_{n}^{(\ell)}$ is the $\ell$ loop amplitude which will be expanded in color
structures as in eqn. (2).

Our starting point is the color trace decomposition of a $n$-point gluon
scattering amplitude at tree, $\ell=0$, level within a $SU(N_{c})$ or
$U(N_{c})$ gauge theory. The full amplitude can be decomposed into partial
amplitudes in a process which separates color and kinematics

$\displaystyle{\cal A}_{n}^{(0)}(1,2,3,\cdots,n)$ $\displaystyle=$
$\displaystyle\sum_{S_{n}/Z_{n}}\,{\rm Tr}[T^{a_{1}}T^{a_{2}}\cdots
T^{a_{n}}]A_{n}^{(0)}(a_{1},a_{2},\cdots,a_{n}),$ (4)

where the amplitude is expressed in terms of the color matrices $T^{a_{i}}$
rather than the structure constants of the group. The partial amplitudes
$A_{n}^{(0)}(a_{1},a_{2},\cdots,a_{n})$ are cyclically symmetric but not fully
crossing symmetric. They are however fully gauge invariant since for
sufficiently large $N_{c}$ the external gluons can have ”color” choices which
reduces the sum to a single term. If, we can use the cyclic symmetry to choose
$a_{1}=1$, then the sum over permutations is over the $(n-1)!$ permutations of
$(2,\cdots,n)$. Additionally, the partial amplitudes are reflection symmetric

$\displaystyle{A}_{n}^{(0)}(1,2,3,\cdots,n)$ $\displaystyle=$
$\displaystyle(-1)^{n}A_{n}^{(0)}(n,\cdots,3,2,1)\;,$ (5)

so there are $(n-1)!/2$ individual partial amplitudes. The above decomposition
can be derived from field theory (Cvitanovic:1980bu, ; Mangano:1990by, ;
Mangano:1988kk, ) but it naturally arises in string theory and consequently in
field theory (Kosower:1987ic, ; Kosower:1988kh, ).

Not all the partial amplitudes are independent but are related by gauge
properties. One route is to note that the expansion also applies to a
$U(N_{c})$ gauge group but must vanish if one (or more) of the external gluons
carry a $U(1)$ charge. Demanding this produces a decoupling identity. For
example setting leg $1$ to be $U(1)$ and extracting the coefficient of $\,{\rm
Tr}[T^{2}T^{3}\cdots T^{n}]$ implies that

$A_{n}^{(0)}(1,2,3,\cdots,n)+A_{n}^{(0)}(2,1,3,\cdots,n)+\cdots
A_{n}^{(0)}(2,3,\cdots,1,n)=0.$ (6)

The expansion in terms of color traces is not unique and other expansions in
terms of color structures exist. In particular there is an expansion in terms
of the structure constants (DelDuca:1999rs, )

$\displaystyle{\cal A}_{n}^{(0)}(1,2,3,\cdots,n)$ $\displaystyle=$
$\displaystyle\sum_{S_{n-2}}f^{1a_{2}b_{1}}f^{b_{1}a_{3}b_{2}}\cdots
f^{b_{n-1}a_{n-1}n}A_{n}^{(0)}(1,a_{2},\cdots,a_{n-1},n),$ (7)

where we have selected legs $1$ and $n$ and the summation is over the $(n-2)!$
permutations $(a_{2},a_{3},\cdots,a_{n-1})$ of the remaining legs
$2,\cdots,n-1$.

The partial amplitudes in eqn. (7) are the same as in eqn. (4). To equate the
two expressions and to show this requires the following identity among the
partial amplitudes,

$A_{n}^{(0)}(1,\\{\alpha\\},n,\\{\beta\\})=(-1)^{|\beta|}\sum_{\sigma\in
OP(\alpha,\beta^{T})}A_{n}^{(0)}(1,\\{\sigma\\},n)$ (8)

where $\alpha$ and $\beta$ are some sets of the remaining indices i.e.
$\\{\alpha\\}=\\{a_{2},\cdots a_{p}\\}$ and $\\{\beta\\}=\\{a_{p+1},\cdots
a_{n-1}\\}$. The summation is over the order permutations of $\alpha$ and
$\beta^{T}$. That is permutations of the union of the sets where the ordering
of $\alpha$ and $\beta^{T}$ are preserved. The summation contains
$(r+s)!/r!s!$ terms where $r,s$ are the number of indices in the sets $\alpha$
and $\beta$ respectively. In appendix A, specific examples of order
permutations are given.

These identities, known as Kleiss-Kuijf relations (Kleiss:1988ne, ), overlap
with the decoupling identities but contain for $n\geq 7$ more information. For
$n\leq 6$ decoupling identities are sufficient to prove eqn. (8) but for $n>6$
the rank of the system of decoupling identities in not large enough to prove
eqn. (8) (although for $n=7$ the rank is just insufficient). We can see this
from table 1 where we have evaluated the number of independent conditions
(rank) obtained by all possible decoupling equations. For $n=6$ this rank of
36 is exactly that needed to express the 60 $A_{6}^{(0)}(1,\sigma)$ in terms
of the 24 $A_{6}^{(0)}(1,\sigma,n)$. However for $n=7$ there is (just)
insufficient information with the disparity increasing with $n$. In practice,
at $n=7$ we cannot find decoupling relations which express

$A^{(0)}_{7}(1,a,b,c,7,d,e)$ (9)

in terms of $A_{7}^{(0)}(1,\sigma,7)$ although the combination

$A^{(0)}_{7}(1,a,b,c,7,d,e)+A^{(0)}_{7}(1,a,b,7,c,d,e)$ (10)

may be.

Beyond $n=7$, it is the $A^{(0)}_{n}(1,S_{1},n,S_{2})$ where the length of
both $S_{i}$ is at least two and one has length greater than two which the
decoupling identities fail to determine.

The partial amplitudes $A_{n}^{(0)}(1,a_{2},\cdots,a_{n-1},n)$ form a minimal
set in that there are no further linear relations between them valid for all
helicities as can be verified by examining the known tree amplitudes.

n |  $\\#$ of $A_{n}^{(0)}(1,\sigma)$ | Rank of Decoupling System |  $\\#$ of $A_{n}^{(0)}(1,\sigma,n)$ |  $\\#$ of extra relations
---|---|---|---|---
6 | 60 | 36 | 24 | 0
7 | 360 | 239 | 120 | 1
8 | 2520 | 1696 | 720 | 104
Table 1: This table enumerates the rank of the system of decoupling identities
and demonstrates this is not enough to determine eqn. (8) for $n>6$. The final
column shows the number of extra relations contained in the Kleiss-Kuijf
relations beyond the decoupling identities.

We are interested in this article in relations between amplitudes with
constant coefficients. For tree amplitudes there are further relations if we
allow the coefficients which depend upon kinematic factors but which are
independent of helicity. These rely upon the Bern, Carrasco and Johansson
relation (BCJ) for gauge theories (Bern:2008qj, ). These can reduce the
$(n-2)!$ basis to a $(n-3)!$ basis for tree amplitudes (Bjerrum-Bohr:2009ulz,
).

At one-loop the decomposition in terms of color traces contains both single
trace terms and double trace terms (Bern:1990ux, )

$\displaystyle{\cal A}_{n}^{(1)}(1,2,$ $\displaystyle 3$
$\displaystyle,\cdots,n)=\sum_{S_{n}/Z_{n}}N_{c}\,{\rm
Tr}[T^{a_{1}}T^{a_{2}}\cdots
T^{a_{n}}]A_{n:1}^{(1)}(a_{1},a_{2},\cdots,a_{n})$
$\displaystyle+\sum_{r}\sum_{P_{n:r}}$ $\displaystyle\hskip-15.0pt\,{\rm
Tr}[T^{a_{1}}\cdots T^{a_{r-1}}]\,{\rm Tr}[T^{a_{r}}\cdots
T^{a_{n}}]A_{n:r}^{(1)}(a_{1},a_{2},\cdots,a_{r-1};a_{r},\cdots,a_{n})\;.$
(11)

The single trace term has a factor of $N_{c}$ and is thus referred to as the
leading-in-color (or planar) contribution. The summation over $r$ for a
$SU(N_{c})$ theory is $r=3,\cdots[n/2]$. For a $U(N_{c})$ theory the summation
is over $r=2,\cdots[n/2]$ with the $r=2$ term
$A_{n:2}^{(1)}(a_{1};a_{2},\cdots,a_{n})$ which has color structure $\,{\rm
Tr}(T^{a_{1}})\,{\rm Tr}(T^{a_{2}}\cdots T^{a_{n}})$. We refer to partial
amplitudes such as $A_{n:2}^{(1)}$ as a $U(N_{c})$ specific amplitude.

For the one-loop amplitude the double trace terms are not independent but can
be expressed in terms of the leading

$A_{n:r}^{(1)}(a_{1},a_{2},\cdots,a_{r-1};a_{r},\cdots,a_{n})=(-1)^{r}\sum_{\sigma\in
COP\\{\alpha\\}\\{\beta^{T}\\}}A_{n:1}^{(1)}(\sigma)$ (12)

where $\alpha=\\{a_{1},\cdots a_{r-1}\\}$ and $\beta=\\{a_{r}\cdots a_{n}\\}$.
The summation is over the ordered permutations as before but factoring out
equivalent permutations due to cyclic symmetry (see appendix A for examples).
This relation allows the double trace terms to be derived from the leading in
color terms only. This reduces the number of functional forms to be computed
in a calculation considerably.

There are some analogues between the one-loop relation eqn. (12) and the tree
relation eqn. (8). This relation can be obtained from the decoupling equations
for $n\leq 5$ but beyond $n=5$ the decoupling equations are not sufficient:
e.g. see (Feng:2011fja, ). Explicitly, at $n=6$, decoupling identities
determine the combination

$A_{6:4}^{(1)}(a_{1},a_{2},a_{3};a_{4},a_{5},a_{6})+A_{6:4}^{(1)}(a_{1},a_{3},a_{2};a_{4},a_{5},a_{6})$
(13)

but in themselves do not determine the individual terms.

The relation (12) can be shown in multiple ways. It is a natural relation when
Yang-Mills theory is viewed as the low energy limit of open string theory
(Bern:1994zx, ). In open string theory, the gauge content of a $U(N_{c})$
theory is carried by the Chan-Paton factors at the string ends which carry the
color charge of quarks and anti-quarks respectively. The amplitude is given by
an integration over all possible world sheets. External adjoint states are
obtained by inserting a vertex operator with corresponding color matrix to a
boundary. The equivalent for one-loop is a world sheet with two boundaries.
The contribution to a term

$\,{\rm Tr}[T^{a_{1}}\cdots T^{a_{r-1}}]\,{\rm Tr}[T^{a_{r}}\cdots T^{a_{n}}]$
(14)

arise from where states $a_{1}$ to $a_{r-1}$ are attached to a single boundary
and states $a_{r}$ to $a_{n}$ are attached to the other. In ref (Bern:1994zx,
) this diagrammatic view was developed into a proof of the relation (12). In
ref (Feng:2011fja, ) the relation was derived using unitarity methods
(Bern:1994zx, ; Bern:1994cg, ) together with the Kleiss-Kuif relations.

Note that the leading in color terms do not satisfy the Kleis-Kuif relation
(8) in themselves. The simplest decoupling identity being

$A_{n:2}^{(1)}(1;2,3,\cdots,n)+A^{(1)}_{n:1}(1,2,3,\cdots,n)+A_{n:1}^{(1)}(2,1,3,\cdots,n)+\cdots
A_{n:1}^{(1)}(2,\cdots,1,n)=0.$ (15)

This is similar to the tree decoupling but with the additional $A_{n:2}^{(2)}$
term. The decoupling identities will then imply (at least for $n\leq 6$)

$A_{n:1}^{(1)}(1,\\{\alpha\\},n,\\{\beta\\})=(-1)^{|\beta|}\sum_{\sigma\in
OP(\alpha,\beta^{T})}A_{n:1}^{(1)}(1,\\{\sigma\\},n)+\sum A_{n:2}^{(1)}$ (16)

so unless the functions $A_{n:2}^{(1)}$ are zero eqn. (8) is not satisfied by
the $A_{n:1}^{(1)}$.

The explicit form for $A_{n:2}^{(1)}$ of the all-plus helicity amplitude is

$\displaystyle A_{n:2}^{(1)}(1^{+};2^{+},3^{+},\cdots,n^{+})$ $\displaystyle=$
$\displaystyle-i\frac{1}{\langle 2\,3\rangle\langle 3\,4\rangle\cdots\langle
n\,2\rangle}\sum_{2\leq i<j\leq n}[1\,i]\langle i\,j\rangle[j\,1]$ (17)

which is explicitly non-vanishing in eqn. (16). This expression is from
reference (Dunbar:2019fcq, ). Although we can regard the $U(N_{c})$ specific
amplitudes as un-physical they are gauge invariant and as such appear as
building blocks in other amplitudes -this particular partial amplitude is also
the one-loop amplitude between a single photon and $n-1$ gluons (all of
positive helicity) due to a scalar or fermion loop for which an earlier
explicit analytic form exists (Bern:1993qk, ). The leading in color all-plus
amplitude is related to the $N=4$ MHV amplitude by a dimension shifting of the
loop integrals (Bern:1996ja, ; Britto:2020crg, ). Note that the expression in
eqn. (17) matches that obtained from eqn. (12) but only after significant
simplification. Analytic expressions derived from the decoupling and other
such identities are often inefficient.

We now turn to the topic of this letter. A general two-loop amplitude for the
scattering of $n$ gluons in a pure $SU(N_{c})$ or $U(N_{c})$ gauge theory may
be expanded in a color trace basis as

$\displaystyle{\cal
A}_{n}^{(2)}(1,2,\cdots,n)=N_{c}^{2}\sum_{S_{n}/\mathcal{P}_{n:1}}\mathop{\hbox{\rm
Tr}}\nolimits[T^{a_{1}}T^{a_{2}}\cdots
T^{a_{n}}]A_{n:1}^{(2)}(a_{1},a_{2},\cdots,a_{n})$ (18) $\displaystyle+$
$\displaystyle
N_{c}\sum_{r=2}^{[n/2]+1}\sum_{S_{n}/\mathcal{P}_{n:r}}\mathop{\hbox{\rm
Tr}}\nolimits[T^{a_{1}}T^{a_{2}}\cdots T^{a_{r-1}}]\mathop{\hbox{\rm
Tr}}\nolimits[T^{b_{r}}\cdots
T^{b_{n}}]A_{n:r}^{(2)}(a_{1},a_{2},\cdots,a_{r-1};b_{r},\cdots,b_{n})$
$\displaystyle+$
$\displaystyle\sum_{s=1}^{[n/3]}\sum_{t=s}^{[(n-s)/2]}\sum_{S_{n}/\mathcal{P}_{n:s,t}}\mathop{\hbox{\rm
Tr}}\nolimits[T^{a_{1}}\cdots T^{a_{s}}]\mathop{\hbox{\rm
Tr}}\nolimits[T^{b_{s+1}}\cdots T^{b_{s+t}}]\mathop{\hbox{\rm
Tr}}\nolimits[T^{c_{s+t+1}}\cdots T^{c_{n}}]$ $\displaystyle\hskip
199.16928pt\times
A_{n:s,t}^{(2)}(a_{1},\cdots,a_{s};b_{s+1},\cdots,b_{s+t};c_{s+t+1},\cdots,c_{n})$
$\displaystyle+$ $\displaystyle\sum_{S_{n}/\mathcal{P}_{n:1}}\mathop{\hbox{\rm
Tr}}\nolimits[T^{a_{1}}T^{a_{2}}\cdots
T^{a_{n}}]A_{n:1B}^{(2)}(a_{1},a_{2},\cdots,a_{n})\,.$

The partial amplitudes multiplying any trace of color matrices are cyclically
symmetric in the indices within the trace. The expression has single, double
and triple trace terms. In string theory these would arise from surfaces with
three boundaries. The is also a single trace term $A_{n:1B}^{(2)}$ which is
sub-sub leading in powers of $N_{c}$. This arises in string theory from a
separate two-loop surface with a single boundary (Dunbar:2020wdh, ).

The summations simply count each color structure exactly once. Specifically,
when the sets are of different lengths ($r-1\neq\frac{n}{2}$, $s\neq t$,
$t\neq\frac{n-s}{2}$ and $3s\neq m,n$) the sets $\mathcal{P}_{n:\lambda}$ are

$\displaystyle\mathcal{P}_{n:1}$ $\displaystyle=Z_{n}(a_{1},\cdots,a_{n}),$
$\displaystyle\mathcal{P}_{n:r}$
$\displaystyle=Z_{r-1}(a_{1},\cdots,a_{r-1})\times
Z_{n+1-r}(a_{r},\cdots,a_{n}),\;\;\ r>1,r-1\neq n+1-r$
$\displaystyle\mathcal{P}_{n:s,t}$
$\displaystyle=Z_{s}(a_{1},\cdots,a_{s})\times
Z_{t}(a_{s+1},\cdots,a_{s+t})\times Z_{n-s-t}(a_{s+t+1},\cdots,a_{n})\,.$ (19)

When the sets have equal lengths, to avoid double counting

$\displaystyle\mathcal{P}_{2m:m+1}$
$\displaystyle=Z_{m}(a_{1},\cdots,a_{m})\times
Z_{m}(a_{m+1},\cdots,a_{2m})\times Z_{2},$ (20)
$\displaystyle\mathcal{P}_{n:s,s}$
$\displaystyle=Z_{s}(a_{1},\cdots,a_{s})\times
Z_{s}(a_{s+1},\cdots,a_{2s})\times Z_{n-2s}(a_{2s+1},\cdots,a_{n})\times
Z_{2},$ $\displaystyle\mathcal{P}_{3m:m,m}$
$\displaystyle=Z_{m}(a_{1},\cdots,a_{m})\times
Z_{m}(a_{m+1},\cdots,a_{2m})\times Z_{m}(a_{2m+1},\cdots,a_{3m})\times S_{3},$
$\displaystyle\mathcal{P}_{2m:2s,m-s}$
$\displaystyle=Z_{2s}(a_{1},\cdots,a_{2s})\times
Z_{m-s}(a_{2s+1},\cdots,a_{s+m})\times Z_{m-s}(a_{s+m+1},\cdots,a_{2m})\times
Z_{2}\,.$

The partial amplitudes are

$A_{n:r}^{(2)}\;\;r=1\cdots[n/2]\;\;\;\
A_{n:r,s}^{(2)}\;\;r,s=1\cdots[n/3],r\leq s$ (21)

of which the $A_{n:2}^{(2)}$ and $A_{n:1,s}^{(2)}$ are the $U(N_{c})$ specific
functions.

The two-loop expansion is an expansion in powers of $N_{c}$. Decoupling
identities do involve different powers of $N_{c}$ since if we set

$T^{1}\longrightarrow T^{U(1)}=\frac{1}{\sqrt{N_{c}}}I$ (22)

then

$N_{c}^{2}\mathop{\hbox{\rm Tr}}\nolimits\left[T^{1}T^{a_{2}}\cdots
T^{a_{n}}\right]\longrightarrow N_{c}^{3/2}\mathop{\hbox{\rm
Tr}}\nolimits\left[T^{a_{2}}\cdots T^{a_{n}}\right]$ (23)

and

$N_{c}^{1}\mathop{\hbox{\rm Tr}}\nolimits\left[T^{1}\right]\mathop{\hbox{\rm
Tr}}\nolimits\left[T^{a_{2}}\cdots T^{a_{n}}\right]\longrightarrow
N_{c}^{3/2}\left[T^{a_{2}}\cdots T^{a_{n}}\right]$ (24)

and equating the coefficient of $\mathop{\hbox{\rm
Tr}}\nolimits\left[T^{a_{2}}\cdots T^{a_{n}}\right]$ gives a decoupling
identity amongst terms of different order in $N_{c}$. However, decoupling
identities do not relate the single trace amplitudes $A_{n:1B}^{(2)}$ to the
other amplitudes. Instead they obey decoupling identities among themselves
identical to those satisfied by the tree amplitudes $A_{n}^{(0)}$.
Consequently, for $n\leq 6$ these are guaranteed to obey the same relation
shown in eqn. (8)

Decoupling identities can be used to express the $U(N_{c})$ specific
amplitudes in terms of the In terms of $SU(N_{c})$ partial amplitudes,. For
example, from the above we have

$A_{n:2}^{(2)}(1;2,3,\cdots,n)+A_{n:1}^{(2)}(1,2,3,\cdots,n)+A_{n:1}^{(2)}(2,1,3,\cdots,n)+\cdots+A_{n:1}^{(2)}(2,\cdots,1,n)=0\,.$
(25)

This allows $A_{n:2}^{(2)}$ to be expressed in terms of the $A_{n:1}^{(2)}$.
For the $SU(N_{c})$ amplitudes decoupling identities are relatively limited
only determining the triple trace partial amplitudes $A_{n:2,r}^{(2)}$.

In total, after setting $T^{1}$ to be a $U(1)$,

$\bullet$ the coefficient of $\,{\rm Tr}(T^{2}\cdots T^{n})$ fixes
$A_{n:2}^{(2)}$ as a sum of $A_{n:1}^{(2)}$ amplitudes

$\bullet$ the coefficient of $\,{\rm Tr}(T^{2}\cdots T^{s-1})\,{\rm
Tr}(T^{s}\cdots T^{n})$ fixes $A_{n:1,s}^{(2)}$ is terms of $A_{n:s}^{(2)}$
and $A_{n:s+1}^{(2)}$

$\bullet$ the coefficient of $\,{\rm Tr}(T^{2})\,{\rm Tr}(T^{3}\cdots
T^{s+2})\,{\rm Tr}(T^{s+3}\cdots T^{n})$ enables the $SU(N_{c})$ functions the
$A_{n:2,s}^{(2)}$ to be fixed as

$\displaystyle A_{n:2,s}^{(2)}(1,2\;;3\cdots s+3;s+4\cdots n)=$
$\displaystyle-\sum A_{n:1,s}^{(2)}(2;1,3,\cdots s+3;s+4\cdots n)$
$\displaystyle-\ \sum$ $\displaystyle A_{n:1,s+1}^{(2)}(2;3\cdots
s+3;1,s+4\cdots n)$ (26) $\displaystyle=-\hskip-28.45274pt\sum_{\sigma\in
COP(\\{3,\cdots s+3\\},\\{1\\})}A_{n:1,s}^{(2)}(2;\sigma;s+4,\cdots n)$
$\displaystyle-\hskip-25.0pt\sum_{\sigma\in COP(\\{s+4,\cdots
n\\},\\{1\\})}A_{n:1,s}^{(2)}(2;3,\cdots s+3;\sigma)$

where the summation denotes summing over the different locations leg one may
appear. This expression is likely to be the simplest but the $U(N_{c})$
specific functions can be substituted to leave an expression purely in terms
of the $SU(N_{c})$ functions.

The first triple trace term which cannot be determined from decoupling
identities will be the nine-point partial amplitude $A_{9:3,3}^{(2)}$ for
which decoupling identities only determine the combination

$A_{9:3,3}^{(2)}(1,2,3;4,5,6;7,8,9)+A_{9:3,3}^{(2)}(1,3,2;4,5,6;7,8,9)$ (27)

In the following sections we review the known identities for $n=5$ and review
and explore possible new identities for $n=6$.

##  3 Identities among Five point partial amplitudes

In this section we will review the known identities amongst the five point
partial amplitudes.

The five point partial amplitudes in the color-trace basis expansion are

$A_{5:1}^{(2)},A_{5:2}^{(2)},A_{5:3}^{(2)},A_{5:1,1}^{(2)},A_{5:1,2}^{(2)}\hbox{
and }A_{5:1B}^{(2)}$ (28)

of which the three

$A_{5:1}^{(2)}\;(12),A_{5:3}^{(2)}\;(15)\hbox{ and }A_{5:1B}^{(2)};(12)$ (29)

are the $SU(N_{c})$ functions. The numbers in brackets indicate the number of
independent amplitudes of each type. The $A_{5:1,2}^{(2)}$ vanish since
$A_{5:1,2}^{(2)}(a;b,c;d,e)=-A_{5:1,2}^{(2)}(a;c,b;e,d)=-A_{5:1,2}^{(2)}(a;b,c;d,e)$.

We now review the known identities among these partial amplitude. Firstly,
taking the decoupling identities among the $A_{5:1B}^{(2)}$. For these the
decoupling identity system has rank 6 which allows a Kleiss-Kuijf relation
leaving the six $A_{5:1B}^{(2)}(1,a,b,c,5)$ as independent functions. For the
remaining functions the decoupling identities only determine the form of the
specifically $U(N_{c})$ functions $A_{5:2}^{(2)}$ and $A_{5:1,1}^{(2)}$ in
terms of $A_{5:3}^{(2)}$ and $A_{5:1}^{(2)}$ but place no further constraints
upon these functions.

In ref (Edison:2011ta, ) using iteration and color kinematic duality a further
6 relations were found. Unlike decoupling identities these involved both the
$A_{5:1B}^{(2)}$ and other partial amplitudes and indeed allowed for a
solution for $A_{5:1B}^{(2)}$ in terms of $A_{5:1}^{(2)}$ and $A_{5:3}^{(2)}$.

$\displaystyle A_{5:1B}^{(2)}(1,2,3,4,5)$ $\displaystyle=$ $\displaystyle-
A_{5:1}^{(2)}(1,2,4,3,5)+2A_{5:1}^{(2)}(1,2,5,3,4)+A_{5:1}^{(2)}(1,2,5,4,3)$
(30) $\displaystyle-
A_{5:1}^{(2)}(1,3,2,4,5)+2A_{5:1}^{(2)}(1,3,4,2,5)-5A_{5:1}^{(2)}(1,3,5,2,4)$
$\displaystyle-2A_{5:1}^{(2)}(1,3,5,4,2)+2A_{5:1}^{(2)}(1,4,2,3,5)+A_{5:1}^{(2)}(1,4,3,2,5)$
$\displaystyle+2A_{5:1}^{(2)}(1,4,5,2,3)+A_{5:1}^{(2)}(1,4,5,3,2)$
$\displaystyle-$
$\displaystyle\frac{1}{2}\sum_{Z_{5}(1,2,3,4,5)}\bigg{(}A_{5:3}^{(2)}(1,2;3,4,5)-A_{5:3}^{(2)}(1,3;2,4,5)\biggr{)}\,.$

which reduces the number of functional forms necessary to calculate from three
to two.

This expression is rather strange in several respects. Firstly, it contains
factors which are a rather unnatural and secondly the only leading in color
term $A_{5:1}^{(2)}$ missing is $A_{5:1}^{(2)}(1,2,3,4,5)$. I.e. the only term
missing is that with the same ordering of legs as the $A_{5:1B}^{(2)}$. Part
of the structure is driven by symmetry. Assuming that $A^{(2)}_{5:1B}$ can be
expressed in terms of $A^{(2)}_{5:1}$ and $A^{(2)}_{5:3}$ and demanding the
reflection and cyclic symmetries of $A_{5:1B}^{(2)}$ would imply

$\displaystyle A_{5:1B}^{(2)}(1,2,3,4,5)$ $\displaystyle=$ $\displaystyle
a_{1}(A_{5:1}^{(2)}(1,2,3,4,5))+a_{2}(A_{5:1}^{(2)}(1,3,5,2,4))$
$\displaystyle+a_{3}(A_{5:1}^{(2)}(1,2,4,3,5)-A_{5:1}^{(2)}(1,2,5,4,3)$
$\displaystyle\hskip-14.22636pt+A_{5:1}^{(2)}(1,3,2,4,5)-A_{5:1}^{(2)}(1,4,3,2,5)-A_{5:1}^{(2)}(1,4,5,3,2))$
$\displaystyle\hskip-14.22636pt+a_{4}(A_{5:1}^{(2)}(1,2,5,3,4)+A_{5:1}^{(2)}(1,3,4,2,5)$
$\displaystyle\hskip-14.22636pt-
A_{5:1}^{(2)}(1,3,5,4,2)+A_{5:1}^{(2)}(1,4,2,3,5)+A_{5:1}^{(2)}(1,4,5,2,3))$
$\displaystyle+a_{5}\sum_{Z_{5}(1,2,3,4,5)}A_{5:3}^{(2)}(1,2;3,4,5)$
$\displaystyle+a_{6}\sum_{Z_{5}(1,2,3,4,5)}A_{5:3}^{(2)}(1,3;2,4,5)$ (31)

which matches the correct term with $a_{1}=0$, $a_{5}=-5$ etc.

The expression has been written in terms of the $SU(N_{c})$ amplitudes
$A_{5:1}^{(2)}$ and $A_{5:3}^{(2)}.$ It can be rewritten in terms of the
$U(N_{c})$ specific amplitudes in several alternate forms:

$\displaystyle A_{5:1B}^{(2)}(1,2,3,4,5)$ $\displaystyle=$
$\displaystyle\sum_{Z_{5}(1,2,3,4,5)}\bigg{(}A_{5:2}^{(2)}(1;3,5,2,4)-\frac{1}{2}A_{5:3}^{(2)}(1,2;3,4,5)+\frac{1}{2}A_{5:3}^{(2)}(1,3;2,4,5)\biggr{)}\,.$
or $\displaystyle A_{5:1B}^{(2)}(1,2,3,4,5)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{Z_{5}(1,2,3,4,5)}\bigg{(}A_{5:1,1}^{(2)}(1,2;3,4,5)-A_{5:1,1}^{(2)}(1,3;2,4,5)\bigg{)}$
(32)

which has more natural factors.

Part of the purpose of this article will be an exhaustive search for possible
relation like (30) for $n>5$.

##  4 Identities among Six point partial amplitudes

In the expansion of the two loop amplitude (18) at six-point there are the
following partial amplitudes

$A_{6:1}^{(2)},A_{6:2}^{(2)},A_{6,3}^{(2)},A_{6:4}^{(2)},A_{6:1,1}^{(2)},A_{6:1,2}^{(2)},A_{6:2,2}^{(2)}\hbox{
and }A_{6:1B}^{(2)}$ (33)

Of these the $SU(N_{c})$ amplitudes are

$A_{6:1}^{(2)}\;(60),A_{6:3}^{(2)}\;(45),A_{6:4}^{(2)}\;(20),A_{6:2,2}^{(2)}\;(15)\;\hbox{
and }A_{6:1B}^{(2)}\;(60)$ (34)

with the number of independent amplitudes after applying cyclic and reflection
symmetry given in brackets. Decoupling identities imply that the
$A_{6:2,2}^{(2)}$ can be expressed in terms of the others via (2) leaving four
partial amplitudes which need to be computed for an $SU(N_{c})$ amplitude.

We will be exploring possible linear identities between these partial
amplitudes. These will fit into three groups

1\. Those obtained by decoupling identities. This system of identities has
rank 207. Once the $U(N_{c})$ specific partial amplitudes have been solved for
the system of decoupling identities has rank sixty-six (66). There split into
two distinct groups. Thirty-six (36) exclusively involve $A^{(2)}_{6:1B}$
whilst the other thirty (30) involve purely the remaining $SU(N_{c})$
amplitudes. The thirty six identities can be used to reduce the
$A^{(2)}_{6:1B}$ amplitudes to the independent 24
$A^{(2)}_{6:1B}(1,a,b,c,d,6)$.

2\. Those obtained by iterative methods as Edison and Naculich not in 1. (14).
These are different from the decoupling identities in that they involve both
the $A^{(2)}_{6:1B}$ and the others. However numerically there are not enough
to solve for the 24 $A^{(2)}_{6:1B}$ .

3\. Identities which are satisfied by the partial amplitudes of the two-loop
all-plus amplitude. It is the third type we will be looking for. In
particular, we will investigate whether the number of necessary partial
amplitudes may be reduced from four.

Identities of types 1. and 2. are guaranteed to be satisfied for all
helicities. Identities of type 3 are not so guaranteed however they set an
envelope on possible all-helicity identities

First we examine the information given by the decoupling identities after
solving for the $U(N_{c})$ specific functions. For six points, among the
$SU(N_{c})$ the remaining decoupling identities form a system of rank 30. This
determines the partial amplitude $A^{(2)}_{6:2,2}$ via

$\displaystyle A^{(2)}_{6:2,2}(a,b;r,s;m,n)$ $\displaystyle=$ $\displaystyle
A_{6:3}^{sym}(r,s;a,b,m,n)+A_{6:3}^{sym}(m,n;a,b,r,s)$ (35) $\displaystyle+$
$\displaystyle A_{6:4}^{sym}(a,m,n;b,r,s)+A_{6:4}^{sym}(b,m,n;a,r,s)$

where we define the combinations

$A_{6:4}^{sym}(a,b,c;r,s,t)\equiv
A_{6:4}^{(2)}(a,b,c;r,s,t)+A_{6:4}^{(2)}(a,b,c;r,t,s)$ (36)

and

$A_{6:3}^{sym}(a,b;r,s,t,u)\equiv
A^{(2)}_{6:3}(a,b;r,s,t,u)+A^{(2)}_{6:3}(a,b;r,t,s,u)+A^{(2)}_{6:3}(a,b;r,s,u,t)$
(37)

The decoupling identities can be used to solve for the combination
$A_{6:4}^{sym}$ with

$A_{6:4}^{sym}(a,b,c;r,s,t)=\frac{1}{2}\sum_{\sigma}A_{6:1}^{(2)}(\sigma)-\frac{1}{2}\left(A^{sym}_{6:3}(a,b;c,r,s,t)+A^{sym}_{6:3}(a,c;b,r,s,t)+A^{sym}_{6:3}(b,c;a,r,s,t)\right)$
(38)

where the summation is over the entire 60 $A_{6:1}^{(2)}$. This amounts to 10
independent constraints. Note this expression is not symmetric between the
first and second triple of indices so we obtain the identities among the
$A_{6:3}^{(2)}$

$\displaystyle
A^{sym}_{6:3}(a,b;c,r,s,t)+A^{sym}_{6:3}(b,c;a,r,s,t)+A^{sym}_{6:3}(c,a,;b,r,s,t)$
$\displaystyle=A^{sym}_{6:3}(r,s;t,a,b,c)+A^{sym}_{6:3}(s,t;r,a,b,c)+A^{sym}_{6:3}(t,r;sa,b,c)$
(39)

This set of (10) identities has rank 5 so eqn. (38) and eqn. (39) are the
final consistency constraints among the partial amplitudes determined by the
decoupling identities and as such are satisfied for all helicity
configurations.

We can now examine possible identities in the amplitudes beyond these. We
expect at least 14 from the results of ref. (Edison:2012fn, ). Our methodology
is to consider relations,

$\sum c_{\lambda,i}R^{(2)}_{6:\lambda}(\sigma_{i})=0$ (40)

where the $R^{(2)}_{6:\lambda}$ are the rational parts of the six-point all-
plus amplitude and to evaluate that as sufficient kinematic points with
rational momenta that the system can be solved analytically for rational
constants $c_{\lambda,i}$. We again comment that this method will identify any
possible identities but these are only potential identities which may or may
not extend to all amplitudes. We use the rational terms calculated in ref.
(Dunbar:2019fcq, ) which are include in appendix C. The rational terms are
only part of the amplitude. After identifying possible relations we check
whether these also apply to the polylogarithmic parts- these are also
available in ref. (Dunbar:2019fcq, ).

In total, we find a further 20 identities among the rational terms of partial
amplitudes. I.e. 6 beyond that of (Edison:2012fn, ). An obvious initial
comment is that these will not be enough to solve for the 24 $A_{6:1B}^{(2)}$.
In fact, they split into 14 identities involving $A_{6:1B}^{(2)}$ plus six
which only involve the others. Before looking at the new relations we briefly
comment upon the 14 involving $A_{6:1B}^{(2)}$.

In particular, if we consider the 24 $A_{6:1B}^{(2)}$ then the rank of the
identities involving these is only 14 as recognised in identified in ref
(Edison:2012fn, ) and confirmed by our studies of the rational parts of the
all-plus so it is not possible to solve for these. Instead the possible
identities involve the combinations

$\displaystyle A_{6:1B}^{sym_{1}}$ $\displaystyle\equiv$
$\displaystyle\sum_{sym(b,c,d)}A_{6:1B}^{(2)}(1,a,b,c,d,6)$ $\displaystyle
A_{6:1B}^{sym_{2}}$ $\displaystyle\equiv$
$\displaystyle\sum_{sym(a,b,c)}A_{6:1B}^{(2)}(1,a,b,c,d,6)$ $\displaystyle
A_{6:1B}^{sym_{3}}$ $\displaystyle\equiv$ $\displaystyle
A_{6:1B}^{(2)}(1,a,b,c,d,6)+A_{6:1B}^{(2)}(1,a,c,b,d,6)\ $
$\displaystyle+A_{6:1B}^{(2)}(1,d,b,c,a,6)+A_{6:1B}^{(2)}(1,d,c,b,a,6)$
$\displaystyle A_{6:1B}^{sym_{4}}$ $\displaystyle\equiv$ $\displaystyle
A_{6:1B}^{(2)}(1,a,b,c,d,6)+A_{6:1B}^{(2)}(1,a,b,d,c,6)+A_{6:1B}^{(2)}(1,c,a,b,d,6)$
(41) $\displaystyle+$ $\displaystyle
A_{6:1B}^{(2)}(1,d,a,b,c,6)+A_{6:1B}^{(2)}(1,c,d,a,b,6)+A_{6:1B}^{(2)}(1,d,c,a,b,6)$

and act as consistecy contraints. For these we can solve for these
combinations in terms of the other $SU(N_{c})$ functions

$A_{6:1B}^{sym_{i}}=\sum_{r,=1,3,4}\sum_{i}c_{r,i}A^{(2)}_{6:r}(\sigma_{i})$
(42)

but these don’t allow a solution for an individual $A^{(2)}_{6:1B}$. The
expressions of the RHS are subject to manipulation using eqs (38) and (39). A
specific version are given in appendix B.

After implementing these relations there remain only six relations amongst the
rational terms of the all-plus amplitude. This amplitude, of course, consists
of more than rational terms. If we also demand that the relations apply to the
polylogarithmic parts also these six relations are reduced to a single
relation which can be expressed

$\displaystyle A_{6:3}^{sym}(a,b;r,s,t,u)=$ $\displaystyle
A_{6:4}^{sym}(a,b,r;s,t,u)+A_{6:4}^{sym}(a,b,s;r,t,u)$ (43) $\displaystyle+$
$\displaystyle A_{6:4}^{sym}(a,b,t;r,s,u)+A_{6:4}^{sym}(a,b,u;r,s,t)\;.$

Although this looks like multiple relations by choosing different values for
$(a,b,c,d,e,f)$ in fact it only adds one: the difference between two choices
of this relation lies within the space spanned by the decoupling identities.

We can use (43) to obtain a new expression for $A_{6:2,2}^{(2)}$ which we can
express as,

$\displaystyle A_{6:2,2}^{(2)}($ $\displaystyle a,b$
$\displaystyle;e,f;r,s)=\frac{1}{2}\sum_{\sigma\in
COP(\\{e,f\\},\\{r,s\\})}A_{6:3}^{(2)}(a,b;\sigma)$ (44) $\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{\sigma\in
COP(\\{a,b\\},\\{r,s\\})}A_{6:3}^{(2)}(e,f;\sigma)+\frac{1}{2}\sum_{\sigma\in
COP(\\{a,b\\},\\{e,f\\})}A_{6:3}^{(2)}(r,s;\sigma)\;.$

This form of $A_{6:2,2}^{(2)}$ is different from that of eqn. (35) and is not
a consequence of decoupling. It relies upon the relation (43) -or is
equivalent to it.

Equations (44) and (43) are only proven to apply to the all-plus amplitude but
can be conjectured to apply more generally : this will be explored in the next
section.

In summary, we have confirmed the expected results for identities between the
amplitudes and have identified that there is a maximum of a single potential
relation beyond these. Although only a single relation this hints at more
general relations at higher points. This is reminiscent of the Kleiss-Kuif
relations at tree level. We explore a possible extension of this to higher
points in the next section.

##  5 Triple Trace Term

Up to eight points, the triple trace terms of the two loop amplitude are
determined from the decoupling identities as discussed in eqn. (2). However,
there is a further speculative relationship for the triple trace which we
propose for all $n$,

$\displaystyle A_{n:r,s}^{(2)}(\alpha;\beta;\gamma)$ $\displaystyle=$
$\displaystyle\frac{(-1)^{|\gamma|}}{2}\sum_{\sigma\in
COP(\alpha,\gamma^{T})}A_{n:s+1}^{(2)}(\beta;\sigma)+\frac{(-1)^{|\beta|}}{2}\sum_{\sigma\in
COP(\gamma,\beta^{T})}A_{n:r+1}^{(2)}(\alpha;\sigma)$ (45) $\displaystyle+$
$\displaystyle\frac{(-1)^{|\alpha|}}{2}\sum_{\sigma\in
COP(\beta,\alpha^{T})}A_{n:n-r-s+1}^{(2)}(\gamma;\sigma)$

provided $|\alpha|,|\beta|,|\gamma|>1$, which expresses the triple trace in
terms of double trace terms.

This relationship lies outside the relationships of the decoupling identities.
It is speculative but is satisfied for

$\alpha=\\{\alpha_{1},\alpha_{2}\\},\;\beta=\\{\beta_{1},\beta_{2}\\},\;$ (46)

for the all-plus six point amplitude, as seen in the previous section., where
$\gamma=\\{\gamma_{1},\gamma_{2}\\}$.

The seven-point all plus amplitude has also been calculated (Dalgleish, ). We
have checked that the rational parts of this specific helicity amplitude where
$\gamma=\\{\gamma_{1},\gamma_{2},\gamma_{3}\\}$ also satisfies this relation.
For $n=7$, after solving for the $U(N_{c})$ specific partial amplitudes and
$A_{7:2:2}^{(2)}$ the decoupling identities impose consistency conditions
among the $SU(N_{c})$ partial functions (excluding $A_{n:7B}^{(2)}$) with rank
105. Relation (45) lies outside these identities. Unlike $n=6$ it is not a
single additional relation but adds 35 to the rank of identities.

The expression is robust against collinear limits. For sets of length two,
there is no distinction between the set and its transpose due to cyclic
symmetry so our six and seven point amplitudes do not determine this. However,
we have chosen the ordering so that the amplitude satisfies reflection
symmetry for all $n$. Similarly, the overall factors may have a more general
form when the sets have arbitrary length.

This relation has corrections if any of the traces has length one. If two have
length one, $\alpha=\\{\alpha_{1}\\},\beta=\\{\beta_{1}\\}$ the following
variant holds

$\displaystyle A_{n:1,1}^{(2)}(\alpha;\beta;\gamma)$ $\displaystyle=$
$\displaystyle\frac{(-1)^{|\alpha|}}{2}\sum_{\sigma\in
COP(\gamma,\alpha^{T})}A_{n:2}^{(2)}(\beta;\sigma)+\frac{(-1)^{|\beta|}}{2}\sum_{\sigma\in
COP(\gamma,\beta^{T})}A_{n:2}^{(2)}(\alpha;\sigma)$ (47) $\displaystyle+$
$\displaystyle{(-1)^{|\alpha|}}\sum_{\sigma\in
COP(\beta,\alpha^{T})}A_{n:3}^{(2)}(\sigma;\gamma)\;,$

whilst if only one has length one, $\alpha=\\{\alpha_{1}\\}$,
$\beta=\\{\beta_{1},\cdots,\beta_{s-1}\\}$
$\gamma=\\{\gamma_{1},\cdots,\gamma_{t-1}\\}$

$\displaystyle A_{n:1,s-1}^{(2)}(\alpha;\beta;\gamma)=-\sum_{\sigma\in
COP(\gamma,\alpha^{T})}A_{n:s}^{(2)}(\beta;\sigma)$
$\displaystyle-\sum_{\sigma\in
COP(\beta,\alpha^{T})}A_{n:s+1}^{(2)}(\sigma;\gamma)\;.$ (48)

(If $s=t$, $\sigma$ and $\gamma$ as interchanged in the final term.) Unlike
eqn. (45), eqs. (47) and (48) can be derived from decoupling identities.

##  6 $A_{n:1B}^{(2)}$

There exists a conjectured form of $A_{n:1B}^{(2)}$ for the all-plus
amplitude. (Dunbar:2020wdh, )

$A_{n:1B}^{(2)}(1^{+},2^{+},\cdots n^{+})$ (49)

with closed expressions for the rational part of this amplitude. These are
included in appendix D. We use this amplitude to check for relations. This
partial amplitude has been verified for six and seven points amplitudes and
has the correct symmetries, factorisations and collinear limits. For eight and
nine point the expression has been verified numerically (Kosower:2022bfv, ).

We have verified, up to $n=12$ that these specific helicity amplitudes satisfy
Kleiss-Kuif type relations,

$A_{n:1B}^{(2)}(1,\\{\alpha\\},n,\\{\beta\\})=(-1)^{|\beta|}\sum_{\sigma\in
OP(\alpha,\beta^{T})}A_{n:1B}^{(2)}(1,\\{\sigma\\},n)\;.$ (50)

Secondly, we have checked there are no further relations solely amongst these
partial amplitudes. Specifically there are no constants $c_{\sigma}$
satisfying

$\sum_{\sigma}c_{\sigma}A_{n:1B}^{(2)}(1,\sigma(2,\cdots n-1),n)=0\;.$ (51)

So these are no relations for the all plus this rules out any possible
helicity independent relations.

It is very tempting to conjecture that equation (50) holds for all-$n$ and all
helicities but we have limited data at this point. The relation in eqn. (50)
could even suggest a relationship between the four and three trace terms of
three loop Yang-mills - however without any possible checks at this point.

##  7 Conclusions

This article has explored possible relationships among the partial amplitudes
of two-loop gluon scattering. This has essentially been an “experimental”
style search using the known analytic forms of partial amplitudes. This
approach is exhaustive : no potential relations will be missed. This search is
limited by the very small number of amplitudes which are known in closed
analytic form. Nonetheless we can draw some conclusions and make some
conjectures. In particular, on the negative side, we can confirm that the
relations such as eqn. (30) does not extend beyond the five point case.
However on the positive side we have relationships of eqn. (45) and eqn (50)
which we present as plausible conjectures.

We thank Adam Dalgleish, Warren Perkins and Joe Strong for helpful comments
and access to the unpublished seven point rational terms. This work was
supported by the UKRI Science and Technology Facilities Council (STFC)
Consolidated Grant No. ST/T000813/1.

##  Appendix A Permutation Sums

In the expressions of this article we have summations which are over sets
formed by the merger of two sets. In particular we have the sum over ordered
products $OP(\alpha,\beta)$ and the summation over the cyclic ordered
permutations $COP(\alpha,\beta)$ where, due to cyclic symmetry, cyclically
equivalent permutations are factored out In this appendix we illustrate these
by some examples. For sets $\alpha_{r}$ and $\beta_{s}$ of length $r$ and $s$
respectively these sets contain the following number of elements

$\displaystyle|OP(\alpha_{r},\beta_{s})|$ $\displaystyle=$
$\displaystyle\frac{(r+s)!}{(r)!(s)!}$ (52)
$\displaystyle|COP(\alpha_{r},\beta_{s})|$ $\displaystyle=$
$\displaystyle\frac{(r+s-1)!}{(r-1)!(s-1)!}\;.$ (53)

As examples, for $\alpha=\\{1\\}$, $\beta=\\{2,3,4,5\\}$,

$\displaystyle OP(\alpha,\beta)$ $\displaystyle=$
$\displaystyle\Bigl{\\{}\\{1,2,3,4,5\\},\\{2,1,3,4,5\\},\\{2,3,1,4,5\\},\\{2,3,4,1,5\\},\\{2,3,4,5,1\\}\Bigr{\\}}$
$\displaystyle COP(\alpha,\beta)$ $\displaystyle=$
$\displaystyle\Bigl{\\{}\\{1,2,3,4,5\\},\\{2,1,3,4,5\\},\\{2,3,1,4,5\\},\\{2,3,4,1,5\\}\Bigr{\\}}$
(54)

where $\\{2,3,4,5,1\\},$ is omitted from $COP(\alpha,\beta)$ since it is
cyclically equivalent to $\\{1,2,3,4,5\\},$.

For $\alpha=\\{1,2\\}$, $\beta=\\{3,4,5\\}$,

$\displaystyle
OP(\alpha,\beta)=\Bigl{\\{}\\{1,2,3,4,5\\},\\{1,3,2,4,5\\},\\{1,3,4,2,5\\},\\{1,3,4,5,2\\},$
$\displaystyle\\{3,1,2,4,5\\},\\{3,1,4,2,5\\},\\{3,1,4,5,2\\},\\{3,4,1,2,5\\},$
$\displaystyle\\{3,4,1,5,2\\},\\{3,4,5,1,2\\}\Bigr{\\}}$ (55)

and

$\displaystyle
COP(\alpha,\beta)=\Bigl{\\{}\\{1,2,3,4,5\\},\\{1,2,4,5,3\\},\\{1,2,5,3,4\\},\\{1,3,2,4,5\\},$
$\displaystyle\\{1,4,2,5,3\\},\\{1,5,2,3,4\\},\\{1,3,4,2,5\\},\
\\{1,4,5,2,3\\},$
$\displaystyle\\{1,5,3,2,4\\},\\{1,3,4,5,2\\},\\{1,4,5,3,2\\},\\{1,5,3,4,2\\}\Bigr{\\}}\;.$
(56)

We can use this notation to re-express some of the formulae in this article.
For example eqn. (6)

$\sum_{\sigma\in COP(\\{1\\},\\{2,3,\cdots n\\}}A^{(0)}_{n}(\sigma)=0$ (57)

and eqn. (15) is

$A^{(1)}_{n:2}(1;2,3,\cdots n)+\sum_{\sigma\in COP(\\{1\\},\\{2,3,\cdots
n\\}}A^{(1)}_{n:1}(\sigma)=0\;.$ (58)

##  Appendix B Explicit Form for Constraints involving $A_{6:1B}^{(2)}$.

We provide here an explicit realisation of the constraints upon the six-point
amplitudes involving $A_{6:1B}^{(2)}$ as in eqn. (42),

$\displaystyle A_{6:1B}^{sym_{1}}$ $\displaystyle\equiv$
$\displaystyle\sum_{sym(b,c,d)}A_{6:1B}^{(2)}(1,2,b,c,d,6)=\sum_{P(b,c,d)}\biggl{(}A_{6:4}^{sym}(1,b,c;2,d,6)$
(59) $\displaystyle+$ $\displaystyle
A_{6:3}^{(2)}(1,b;2,c,6,d)/2-A_{6:3}^{(2)}(b,c;1,2,6,d)/2-A_{6:3}^{(2)}(1,b;1,6,2,d)/2$
$\displaystyle-$ $\displaystyle
A_{6:1}^{(2)}(1,b,2,c,d,6)-A_{6:1}^{(2)}(1,2,b,c,6,d)+3A_{6:1}^{(2)}(1,2,6,b,c,d)$
$\displaystyle+$ $\displaystyle
3A_{6:1}^{(2)}(1,b2,6,c,d)+3A_{6:1}^{(2)}(1,b,c,,d,2,6)+3A_{6:1}^{(2)}(1,b,c,2,6,d)\biggr{)}\;,$

$\displaystyle A_{6:1B}^{sym_{3}}$ $\displaystyle\equiv$ $\displaystyle
A_{6:1B}^{(2)}(1,a,b,c,d,6)+A_{6:1B}^{(2)}(1,a,c,b,d,6)\
+A_{6:1B}^{(2)}(1,d,b,c,a,6)+A_{6:1B}^{(2)}(1,d,c,b,a,6)$ (60)
$\displaystyle=$ $\displaystyle-
A_{6:3}^{(2)}({1,6;a,b,d,c})-A_{6:3}^{(2)}({b,c;1,a,6,d})-A_{6:4}^{sym}({1,a,d;b,c,6})-A_{6:4}^{sym}({1,b,c;a,d,6})$
$\displaystyle-$ $\displaystyle
A_{6:4}^{sym}({1,a,6;b,c,d})-A_{6:4}^{sym}({1,b,6;a,c,d})-A_{6:4}^{sym}({1,c,6;a,b,d})-A_{6:4}^{sym}({1,d,6;a,b,c})$
$\displaystyle+$ $\displaystyle\sum_{b\leftrightarrow
c}\Biggl{(}A_{6:3}^{(2)}({1,b;a,c,6,d})+A_{6:3}^{(2)}({1,b;a,d,c,6})+A_{6:3}^{(2)}({b,6;1,a,d,c})+A_{6:3}^{(2)}({b,6;1,c,a,d})\Biggr{)}$
$\displaystyle+$ $\displaystyle\sum_{a\leftrightarrow d,b\leftrightarrow
c}\Biggl{(}A_{6:4}^{sym}(1,a,b;c,d,6)+A_{6:1}^{(2)}({1,a,b,c,d,6})+A_{6:1}^{(2)}({1,a,b,6,d,c})+A_{6:1}^{(2)}({1,a,6,b,d,c})$
$\displaystyle-3A_{6:1}^{(2)}({1,a,d,b,c,6})-3A_{6:1}^{(2)}({1,a,d,b,6,c})-3A_{6:1}^{(2)}({1,a,d,6,b,c})$
$\displaystyle-3A_{6:1}^{(2)}({1,b,a,d,c,6})-3A_{6:1}^{(2)}({1,b,c,a,d,6})-3A_{6:1}^{(2)}({1,b,a,d,6,c})\Biggr{)},$

and

$\displaystyle A_{6:1B}^{sym_{4}}$ $\displaystyle\equiv$ $\displaystyle
A_{6:1B}^{(2)}(1,2,3,4,5,6)+A_{6:1B}^{(2)}(1,2,3,5,4,6)+A_{6:1B}^{(2)}(1,4,2,3,5,6)$
(61)
$\displaystyle+A_{6:1B}^{(2)}(1,5,2,3,4,6)+A_{6:1B}^{(2)}(1,4,5,2,3,6)+A_{6:1B}^{(2)}(1,5,4,2,3,6)$
$\displaystyle=$
$\displaystyle-A^{(2)}_{6:3}({1,2;3,4,6,5})-A^{(2)}_{6:3}({1,6;2,4,3,5})-A^{(2)}_{6:3}({2,3;1,4,6,5})$
$\displaystyle-A^{(2)}_{6:3}({3,6;1,4,2,5})-A^{(2)}_{6:3}({4,5;1,2,6,3})-A^{(2)}_{6:3}({4,5;1,3,2,6})$
$\displaystyle+$ $\displaystyle\sum_{a\leftrightarrow
b}\Biggl{(}2A^{(2)}_{6:4}({1,3,a;2,b,6})+2A^{(2)}_{6:4}({1,3,a;2,6,b})+A^{(2)}_{6:4}({1,2,a;3,6,b})+A^{(2)}_{6:4}({1,a,6;2,3,b})$
$\displaystyle-A^{(2)}_{6:4}({1,2,3;a,b,6})-A^{(2)}_{6:4}({1,2,6;3,a,b})-A^{(2)}_{6:4}({1,3,6;2,a,b})-A^{(2)}_{6:4}({1,a,b;2,3,6})$
$\displaystyle+A^{(2)}_{6:1}({1,a,2,b,3,6})+A^{(2)}_{6:1}({1,a,2,3,6,b})+A^{(2)}_{6:1}({1,2,3,a,6,b})+A^{(2)}_{6:1}({1,2,a,3,b,6})$
$\displaystyle-A^{(2)}_{6:1}({1,a,2,3,b,6})-A^{(2)}_{6:1}({1,a,b,2,3,6})-A^{(2)}_{6:1}({1,2,3,a,b,6})$
$\displaystyle-A^{(2)}_{6:1}({1,2,3,6,a,b})+-A^{(2)}_{6:1}({1,2,a,3,6,b})+-A^{(2)}_{6:1}({1,2,a,b,3,6})$
$\displaystyle+6A^{(2)}_{6:1}({1,3,a,2,6,b})+6A^{(2)}_{6:1}({1,3,a,6,2,b})$
$\displaystyle+2A^{(2)}_{6:1}({1,2,a,b,6,3})+2A^{(2)}_{6:1}({1,2,a,6,3,b})+2A^{(2)}_{6:1}({1,2,a,6,b,3})+2A^{(2)}_{6:1}({1,2,6,3,a,b})$
$\displaystyle+2A^{(2)}_{6:1}({1,2,6,a,3,b})+2A^{(2)}_{6:1}({1,3,2,a,b,6})+2A^{(2)}_{6:1}({1,3,2,a,6,b})+2A^{(2)}_{6:1}({1,3,2,b,a,6})$
$\displaystyle+2A^{(2)}_{6:1}({1,3,a,2,b,6})+2A^{(2)}_{6:1}({1,3,6,a,2,b})+2A^{(2)}_{6:1}({1,a,2,6,3,b})+2A^{(2)}_{6:1}({1,a,3,2,b,6})$
$\displaystyle+2A^{(2)}_{6:1}({1,a,3,2,6,b})+2A^{(2)}_{6:1}({1,a,3,b,2,6})+2A^{(2)}_{6:1}({1,a,b,3,2,6}))$
$\displaystyle-2A^{(2)}_{6:1}({1,2,6,a,b,3})-2A^{(2)}_{6:1}({1,3,2,a,b,6})$
$\displaystyle-2A^{(2)}_{6:1}({1,3,2,6,a,b})-2A^{(2)}_{6:1}({1,3,a,b,2,6})-2A^{(2)}_{6:1}({1,3,6,2,a,b})\Biggr{)}\;.$

There are 14 independent identities. No single identity of the above completes
this but for example the identities (60) and (61) combined do.

##  Appendix C Rational Terms of Two Loop All plus six-point amplitude
amplitude

The IR singular structure of a color partial amplitude is determined by
general theorems (Catani:1998bh, ). Consequently we can split the amplitude
into singular terms $U^{(2)}_{n:\lambda}$ and finite terms and further split
the finite terms into the polylogarithmic parts $P^{(2)}_{n:\lambda}$ and the
rational parts $R_{n:\lambda}^{(2)}$

$\displaystyle A^{(2)}_{n:\lambda}=$ $\displaystyle
U^{(2)}_{n:\lambda}+P^{(2)}_{n:\lambda}+R_{n:\lambda}^{(2)}\;.+{\mathcal{O}}(\epsilon)\,.$
(62)

As the all-plus tree amplitude vanishes, $U^{(2)}_{n:\lambda}$ simplifies
considerably and is at worst $1/\epsilon^{2}$ (Kunszt:1994np, ;
Dunbar:2019fcq, ). The partial amplitudes of the six-point all-plus amplitude
have all been computed. The leading in color term was calculated first in
(Dunbar:2016gjb, ) and subsequently confirmed by Badger et.al.
(Badger:2016ozq, ). It was later presented in an alternative form
(Dunbar:2017nfy, ). The remaining partial amplitudes were computed in
(Dalgleish:2020mof, ). We use the rational part of these amplitudes to test
relationships. These are presented here for completeness.

$\mathbf{R_{6:1}^{(2)}}$

$\displaystyle R_{6:1}^{(2)}(a,b,c,d,e,f)$
$\displaystyle=\frac{i}{9}\sum_{\mathcal{P}_{6:1}}\frac{G_{6:1}^{1}+G_{6:1}^{2}+G_{6:1}^{3}+G_{6:1}^{4}+G_{6:1}^{5}}{\langle
a\,b\rangle\langle b\,c\rangle\langle c\,d\rangle\langle d\,e\rangle\langle
e\,f\rangle\langle f\,a\rangle}$ (63) where $\displaystyle
G_{6:1}^{1}(a,b,c,d,e,f)$ $\displaystyle=\frac{s_{cd}s_{df}\langle
f|a\,P_{abc}|e\rangle}{\langle f\,e\rangle\,t_{abc}}+\frac{s_{ac}s_{cd}\langle
a|f\,P_{def}|b\rangle}{\langle a\,b\rangle\,t_{def}},$ $\displaystyle
G_{6:1}^{2}(a,b,c,d,e,f)$ $\displaystyle=\frac{[a\,b][e\,f]}{\langle
a\,b\rangle\langle e\,f\rangle}\langle a\,e\rangle^{2}\langle
b\,f\rangle^{2}+\frac{1}{2}\frac{[f\,a][c\,d]}{\langle f\,a\rangle\langle
c\,d\rangle}\langle a\,c\rangle^{2}\langle d\,f\rangle^{2},$ $\displaystyle
G_{6:1}^{3}(a,b,c,d,e,f)$ $\displaystyle=\frac{s_{df}\langle
f\,a\rangle\langle c\,d\rangle[a\,c][d\,f]}{t_{abc}},$ $\displaystyle
G_{6:1}^{4}(a,b,c,d,e,f)$ $\displaystyle=\frac{\langle a|be|f\rangle
t_{abc}}{\langle a\,f\rangle}$ $\displaystyle G_{6:1}^{5}(a,b,c,d,e,f)$
$\displaystyle=s_{fa}s_{bc}+s_{ac}s_{be}+\frac{5}{2}s_{af}s_{cd}-8[a|bcf|a\rangle-8[a|cde|a\rangle-\frac{1}{2}[a|cdf|a\rangle-\frac{11}{2}[b|cef|b\rangle$
(64)

$\mathbf{R_{6:3}^{(2)}}$

$\displaystyle R_{6:3}^{(2)}(a,b;c,d,e,f)$
$\displaystyle=\sum_{\mathcal{P}_{6:3}}\Bigg{[}\frac{i}{3}\Big{(}H^{1}_{6:3}(a,b,c,d,e,f)-H^{1}_{6:3}(a,b,c,d,f,e)\Big{)}$
$\displaystyle+\frac{i}{3}\frac{\Big{(}G_{6:3}^{2}(a,b,c,d,e,f)+G_{6:3}^{3}(a,b,c,d,e,f)+G_{6:3}^{4}(a,b,c,d,e,f)\Big{)}}{\langle
a\,b\rangle\langle b\,c\rangle\langle c\,a\rangle\langle d\,e\rangle\langle
e\,f\rangle\langle f\,d\rangle}$
$\displaystyle+\frac{i}{12}\frac{G_{6:3}^{5}(a,b,c,d,e,f)}{\langle
a\,b\rangle\langle b\,c\rangle\langle c\,d\rangle\langle d\,e\rangle\langle
e\,f\rangle\langle f\,a\rangle}\Bigg{]}$ (65)

where

$\displaystyle H_{6:3}^{1}(a,b,c,d,e,f)$
$\displaystyle=\frac{G_{6:3}^{1}(a,b,c,d,e,f)}{\langle a\,b\rangle\langle
b\,c\rangle\langle c\,d\rangle^{2}\langle d\,e\rangle\langle
e\,f\rangle\langle f\,a\rangle}+\frac{[c\,d]}{\langle
c\,d\rangle^{2}}\frac{\langle c\,f\rangle\langle
d\,b\rangle[b|f|d\rangle}{\langle a\,b\rangle\langle a\,f\rangle\langle
b\,f\rangle\langle d\,e\rangle\langle e\,f\rangle}$ $\displaystyle
G_{6:3}^{1}(a,b,c,d,e,f)$ $\displaystyle=s_{ce}\langle c|bf|d\rangle-
s_{cf}\langle c|be|d\rangle$ $\displaystyle G_{6:3}^{2}(a,b,c,d,e,f)$
$\displaystyle=\frac{[d|P_{def}b|a]\langle
d|fP_{def}|a\rangle+s_{de}[f|cbd|f\rangle+[b|df|e]\langle
b|cP_{abc}|e\rangle}{t_{def}}$ $\displaystyle G_{6:3}^{3}(a,b,c,d,e,f)$
$\displaystyle=-\frac{s_{df}\langle d|fb|c\rangle[c|P_{abc}|e\rangle}{\langle
d\,e\rangle t_{def}}-\frac{s_{de}\langle f|db|c\rangle[c|d|e\rangle}{\langle
e\,f\rangle t_{def}}$ $\displaystyle G_{6:3}^{4}(a,b,c,d,e,f)$
$\displaystyle=-s_{bd}s_{de}-[a|bde|a\rangle+[b|cde|b\rangle-[a|bdf|a\rangle$
$\displaystyle+[b|cdf|b\rangle+[b|cef|b\rangle-[b|def|b\rangle$ $\displaystyle
G_{6:3}^{5}(a,b,c,d,e,f)$
$\displaystyle=-4s_{ac}^{2}+2s_{ab}s_{ad}-2s_{ac}s_{ad}+2s_{ab}s_{ae}-2s_{ac}s_{ae}+2s_{bd}^{2}-2s_{be}^{2}+2s_{bf}^{2}$
$\displaystyle-8s_{ac}s_{cd}+4s_{bc}s_{cd}+12s_{bd}s_{cd}+6s_{cd}^{2}-8s_{ac}s_{ce}+12s_{bc}s_{ce}+16s_{bd}s_{ce}$
$\displaystyle+4s_{be}s_{ce}+8s_{cd}s_{ce}+2s_{ce}^{2}+2s_{cf}^{2}-8s_{ac}s_{de}-4s_{ad}s_{de}-4s_{bc}s_{de}+4s_{cd}s_{de}$
$\displaystyle+4s_{ce}s_{de}-8[a|bce|a\rangle-39[a|bcf|a\rangle-18[a|bdf|a\rangle+2[a|bef|a\rangle$
$\displaystyle-10[a|cdf|a\rangle-2[a|cef|a\rangle-4[a|def|a\rangle+8[b|cde|b\rangle-4[b|cdf|b\rangle$
$\displaystyle-4[b|cef|b\rangle-4[b|def|b\rangle-4[c|def|c\rangle$ (66)

$\mathbf{R_{6:4}^{(2)}}$

$\displaystyle R_{6:4}^{(2)}(a,b,c,d,e,f)$
$\displaystyle=\frac{i}{36}\sum_{\mathcal{P}_{6:4}}\Bigg{[}\frac{\Big{(}G_{6:4}^{1}(a,b,c,d,e,f)+G_{6:4}^{2}(a,b,c,d,e,f)\Big{)}}{\langle
a\,b\rangle\langle b\,c\rangle\langle c\,a\rangle\langle d\,e\rangle\langle
e\,f\rangle\langle f\,d\rangle}$ $\displaystyle\hskip
36.98866pt+12\frac{\Big{(}G^{3}_{6:4}(a,b,c,d,e,f)+G^{4}_{6:4}(a,b,c,d,e,f)\Big{)}}{\langle
a\,b\rangle\langle c\,d\rangle\langle d\,e\rangle\langle e\,f\rangle\langle
f\,c\rangle}\Bigg{]},$ (67)

where

$\displaystyle G^{1}_{6:4}(a,b,c,d,e,f)$ $\displaystyle=\frac{4\,\langle
e|P_{abc}a|b\rangle[e|dP_{abc}|b]}{t_{abc}},$ $\displaystyle
G^{2}_{6:4}(a,b,c,d,e,f)$
$\displaystyle=s_{ad}^{2}+106\,s_{ab}s_{ad}+102\,[a|bcd|a\rangle-4\,[a|bde|a\rangle-4\,[a|dbe|a\rangle,$
$\displaystyle G^{3}_{6:4}(a,b,c,d,e,f)$ $\displaystyle=-\frac{[a\,b]}{\langle
a\,b\rangle}\Bigl{(}\langle a|cd|b\rangle+\langle a|ef|b\rangle\Bigr{)},$
$\displaystyle G^{4}_{6:4}(a,b,c,d,e,f)$ $\displaystyle=[a|cd|b]+[a|ef|b].$
(68)

$\mathbf{R_{6:2,2}^{(2)}}$

$\displaystyle R_{6:2,2}^{(2)}(a,b;c,d;e,f)$
$\displaystyle=\sum_{\mathcal{P}_{6:2,2}}i\,\frac{G_{6:2,2}^{1}(a,b,c,d,e,f)+G_{6:2,2}^{2}(a,b,c,d,e,f)}{\langle
a\,b\rangle\langle b\,c\rangle\langle c\,a\rangle\langle d\,e\rangle\langle
e\,f\rangle\langle f\,d\rangle},$ (69)

where

$\displaystyle G_{6:2,2}^{1}(a,b,c,d,e,f)$ $\displaystyle=\frac{\langle
b|P_{abc}f|d\rangle[b|cP_{abc}|d]}{t_{abc}},$ $\displaystyle
G_{6:2,2}^{2}(a,b,c,d,e,f)$ $\displaystyle=s_{ad}[e|P_{bc}|e\rangle-
s_{ac}[e|P_{fa}|e\rangle-s_{af}s_{ae}-s_{ae}s_{cd}.$ (70)

##  Appendix D All $n$ form of $R_{n:1B}^{(2)}(1^{+},2^{+},\cdots n^{+})$

There exists a conjectured form of $A_{n:1B}^{(2)}$ for the all-plus
amplitude. (Dunbar:2020wdh, ). We use this amplitude to check the relation of
eqn.(50). The amplitude has been verified for six and seven points amplitudes
and has the correct symmetries, factorisations and collinear limits. For eight
and nine point the expression has been verified numerically (Kosower:2022bfv,
). The amplitude is split into its constituents as in eqn. (62). The rational
term is split into two parts

$\displaystyle R^{(2)}_{n:1B}(1^{+},2^{+},\cdots,n^{+})$
$\displaystyle=R^{(2)}_{n:1B_{1}}(1^{+},2^{+},\cdots,n^{+})+R^{(2)}_{n:1B_{2}}(1^{+},2^{+},\cdots,n^{+})$
(71)

where

$R^{(2)}_{n:1B_{1}}(1^{+},2^{+},\cdots,n^{+})=-2i\,\text{C}_{\text{PT}}(1,2,\cdots,n-1,n)\times\sum_{1\leq
i<j<k<l\leq n}\epsilon(i,j,k,l)$ (72) $\displaystyle
R^{(2)}_{n:1B_{2}}(1^{+},2^{+},\cdots,n^{+})=4i\sum_{r=1}^{n-4}\sum_{s=r+4}^{n}$
$\displaystyle\sum_{i=r+1}^{s-2}\sum_{j=i+1}^{s-1}\epsilon(\\{1,\cdots,r\\},j,i,\\{s,\cdots,n\\})(-1)^{i-j+1}\times\sum_{\alpha\in
S_{r,s,i,j}}\text{C}_{\text{PT}}(\\{\alpha_{S_{r,s,i,j}}\\})\,.$ (73)

We have defined

$\epsilon(\\{a_{1},a_{2},\cdots,a_{m}\\},b,c,\\{d_{1},d_{2},\cdots,d_{p}\\})\equiv\sum_{i=1}^{m}\sum_{j=1}^{p}\epsilon(a_{i},b,c,d_{j})\,,$
(74) $C_{PT}(a_{1},a_{2},a_{3},\cdots,a_{n})\equiv{1\over\langle
a_{1}\,a_{2}\rangle\langle a_{2}\,a_{3}\rangle\cdots\langle
a_{n}\,a_{1}\rangle}\equiv{1\over Cy(a_{1},a_{2},a_{3},\cdots,a_{n})}\,.$ (75)

To define $S_{r,s,i,j}$ we divide the list of indices,

$\displaystyle\\{1,2,3,\cdots,n\\}$
$\displaystyle=\\{1,\cdots,r;r+1,\cdots,i-1;i;i+1,\cdots,j-1;j;j+1,\cdots,s-1;s,\cdots,n\\}$
$\displaystyle\equiv\\{1,\cdots r,\\}\oplus S_{1}\oplus\\{i\\}\oplus
S_{2}\oplus\\{j\\}\oplus S_{3}\oplus\\{s,\cdots,n\\}$ (76)

with

$S_{1}=\\{r+1,\cdots,i-1\\},\;\;\;S_{2}=\\{i+1,\cdots,j-1\\},\;\;\
S_{3}=\\{j+1,\cdots,s-1\\}\,.$ (77)

The sets $S_{i}$ may be null. Then

$S_{r,s,i,j}=Mer(S_{1},\bar{S}_{2},S_{3})$ (78)

where $\bar{S}_{2}$ is the reverse of $S_{2}$ and
$Mer(S_{1},\bar{S}_{2},S_{3})$ is the set of all mergers of the three sets
which respect the ordering within the $S_{i}$ and

$\alpha_{S_{r,s,i,j}}=\\{1,\cdots,r\\}\oplus\\{j\\}\oplus\alpha\oplus\\{i\\}\oplus\\{s,\cdots,n\\}\;.$
(79)

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