aainstitutetext: Max-Planck-Institut für Physik, Werner-Heisenberg-Institut,
80805 München, Germanybbinstitutetext: Dipartimento di Fisica and Arnold-Regge
Center, Università di Torino,
and INFN, Sezione di Torino, Via Pietro Giuria 1, I-10125 Torino,
Italyccinstitutetext: The Institute of Mathematical Sciences, HBNI, IV Cross
Road, Taramani, Chennai 600113, India

# Soft-virtual correction and threshold resummation for $n$-colorless
particles to fourth order in QCD: Part II

Taushif Ahmed c Ajjath A. H c Pooja Mukherjee c V. Ravindran c and Aparna
Sankar<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present a systematic framework to study the leading threshold contributions
of the differential rapidity distribution for the production of any number of
colorless particles in the hadronic colliders. This has been achieved based on
the universality structure of the soft enhancements associated with the real
emissions, along with the factorization property of the differential cross
section and the renormalization group invariance. In this formalism, we
present a universal soft-collinear operator to compute the soft virtual
differential cross section for a generic $2\rightarrow n$ scattering process
up to next-to-next-to-next-to-next-to-leading order (N4LO) in perturbative
QCD. We also provide a universal operator to perform the threshold resummation
to next-to-next-to-next-to-leading logarithmic (N3LL) accuracy. We explicitly
present the approximate analytical results of the rapidity distributions at
N4LO and N3LL for the Higgs boson production through gluon fusion and bottom
quark annihilation, and also for the Drell-Yan production at the hadronic
collider.

††preprint: MPP-2020-176

##  1 Introduction

The upcoming era of high energy physics will confront a huge boom of data
driven by the upgraded run of the Large Hadron Collider (LHC). The High-
Luminosity LHC will also come into effect in a few years of time. In order to
fully exploit the increased quantity of data which will not only help to
detect a rare phenomena but also improve the precision, an enhanced amount of
effort by the theoretical physicists will be called for. In improving
theoretical precision, higher order quantum chromodynamics (QCD) and
electroweak (EW) corrections play an important role. Among different
observables, since the differential cross-section allows a wider range of
comparisons with the experimental data, over the past few decades several
attempts have been made to incorporate the higher order QCD and EW radiative
corrections to this observable. The topic of this article is concerning the
differential cross-section with respect to rapidity, in particular, we address
the question of computing the higher order QCD corrections to this observable
for any generic process at a hadron collider with all the final state
particles as colorless.

Despite its high importance, unlike the inclusive cross-section, the
differential rapidity distribution and its radiative corrections are computed
only for a limited number of scattering processes. The rapidity distributions
in Drell-Yan and of the scalar Higgs boson were computed to next-to-next-to-
leading order (NNLO) QCD in refs. Anastasiou:2003yy ; Anastasiou:2003ds  and
Anastasiou:2004xq , respectively. In case of the scalar Higgs boson produced
through gluon fusion, the next-to-NNLO (N3LO) QCD correction was incorporated
in ref. Dulat:2018bfe . Shortly before, it was approximated in ref.
Cieri:2018oms  in the formalism of $q_{T}$-subtraction. For the Higgs boson
production through bottom quark annihilation, it was computed to NNLO in ref.
Buehler:2012cu .

Needless to say, achieving a full QCD correction to any order is not easy and
with increasing perturbative order, the complexity level increase
substantially which often prevents us from achieving it. In absence of the
full QCD correction, it is often desirable to find an alternative method
averting the full complexity to capture the dominant contribution. In this
article, we discuss such a method, called soft-gluon or soft-virtual (SV)
approximation. In the soft limit, the momenta of all the real emission
diagrams are assumed to be infinitesimally small which leads to an all order
exponentiation of this contribution. In ref. Ravindran:2006bu ;
Ravindran:2007sv , a formalism to incorporate the soft-gluon contribution to
the rapidity distribution for the production of a colorless final state in
hadron collider is presented which, in this article, is extended to the case
of any number of final state colorless particles. The formalism is based on
QCD factorization, which dictates that the soft part of the real emission
diagrams factorizes from the hard contribution, and renormalization group (RG)
invariance. The factorized soft part is conjectured to fulfil a Sudakov type
differential equation with respect to the final state invariant mass square
and as a consequence, it is found to get exponentiated which not only provides
us with the fixed order result under soft limit but also enables us to perform
a resummation in the soft limit. For the production of arbitrary number of
colorless particles in hadronic collision, the soft part essentially remains
identical to the case of Sudakov type process since the real emission can only
takes place from the initial state partons. Using this idea, in this article,
we extend the formalism Ravindran:2006bu  to the case of $2\rightarrow n$
scattering, where $n$ denotes the number of final state colorless particles.
We show that combining the virtual matrix element that captures the process
dependence, the universal soft part and mass-factorization kernel in an
elegant way, we can calculate the SV differential rapidity distribution for
any generic $2\rightarrow n$ process. In addition, we also show how naturally
it leads to the threshold resummation for the same observable. Needless to
say, in some kinematic regions such as when partonic center-of-mass energy
becomes very close to threshold, the logarithms often become so large that it
endangers the validity of the perturbation theory and we are left with no
other choice but to perform an all order resummation in order to get a
reliable prediction.

In the literature, several results for the rapidity resummation employing
different methods are available. In ref. Mukherjee:2006uu , following the
conjecture given in Laenen:1992ey , the resummation of rapidity of $W^{\pm}$
gauge boson and in ref. Bolzoni:2006ky  of Drell-Yan are computed in Mellin-
Fourier (M-F) space. A detailed theoretical underpinnings and phenomenological
implications of threshold resummation of rapidity are examined in ref.
Bonvini:2010tp  emphasizing the role of prescriptions that take care of
diverging series at a given logarithmic accuracy. Our method belongs to a
category, so called direct QCD approach Catani:1989ne , which is based on ref.
Ravindran:2006bu ; Ravindran:2007sv ; Banerjee:2017cfc  resums the soft gluons
in two dimensional Mellin space (M-M). In Lustermans:2019cau , the merits of
different approaches are discussed in details.

Over the past decade, the formalism Ravindran:2006bu  for the $2\rightarrow 1$
scattering has been tested extensively through the computations of the
rapidity distribution of several Sudakov type process. In ref.
Ravindran:2006bu , the observable was computed partially at N3LO under the SV
approximation for the Higgs boson production through gluon fusion and for di-
lepton in Drell-Yan. Later, some of us completed the full SV correction at
N3LO in ref. Ahmed:2014uya  by incorporating the missing part from real
emission diagrams. One of the salient features of our formalism is that the
soft part that enters into the rapidity distribution is shown to be connected
to the respective part of inclusive cross-section through a very simple
relation involving gamma function of the dimensional regulator. This relation
was used to extract the required soft part from the respective quantity of the
SV cross-section Ahmed:2014cla ; Anastasiou:2014vaa . The SV rapidity
distribution at N3LO is also computed for the Higgs boson production through
bottom quark annihilation Ahmed:2014era . The threshold resummed results at
next-to-next-to-leading log (NNLL) for the Higgs boson Banerjee:2017cfc  and
Drell-Yan Banerjee:2018vvb  are also obtained. In this article, we extend the
formalism for computing the SV differential rapidity distribution and its
resummation to any $2\rightarrow n$ scattering process at the hadron collider
and we provide the results to next-to-N3LO (N4LO) and next-to-NNLL (N3LL) in
terms of generic quantities in ancillary file. We also provide the approximate
results of the rapidity distributions at N4LO and N3LL explicitly for the
Higgs boson production through gluon fusion and bottom quark annihilation, and
for the Drell-Yan production. The approximation comes from the unavailability
of the full four loop virtual matrix elements and the soft-collinear
distributions. The goal of this article is to present the formal methodology
of computing threshold rapidity corrections for any generic process of
$2\rightarrow n$ kind at hadron collider.

The paper is organized as follows: in section 2, we introduce the notion of
soft-virtual correction in the context of differential rapidity distribution
and then describe our formalism in details in section 2.1. The universality of
soft part leads us to define a quantity called the differential soft-collinear
operator that essentially captures the process independent part is also
introduced in section 2.1. In the next section 3, we extend our formalism to
incorporate the threshold resummation of the rapidity distribution. All the
results to N4LO and N3LL in terms of generic quantities are presented in
ancillary file in Mathematica format which can be used for any generic
$2\rightarrow n$ process once the corresponding virtual matrix element becomes
available provided the soft distribution function for Sudakov type process is
known.

##  2 Soft-virtual rapidity distribution

We begin by introducing the regime of soft gluon contribution to differential
rapidity distribution for the production of $n$-number of colorless particles
in hadron collisions within the framework of perturbative QCD. Our
prescription that we will develop subsequently to capture this contribution is
within the scope of QCD improved parton model where the collinear divergences
factorize to all orders in strong coupling constant
$\alpha_{s}=g_{s}^{2}/4\pi$. We consider a generic hadronic collision between
two hadrons $H_{1,(2)}$ having momentum $P_{1(2)}$ that produces a final state
consisting of $n$-number of colorless particles, denoted as $F_{i}(q_{i})$

$\displaystyle
H_{1}(P_{1})+H_{2}(P_{2})\rightarrow\sum_{i=1}^{n}F_{i}(q_{i})+X\,.$ (1)

Through the quantity $X$, we represent an inclusive hadronic state. $q_{i}$
stands for the momentum of corresponding colorless particle $F_{i}$. We denote
the invariant mass square of the final state by $q^{2}$ which is related to
the momenta $\\{q_{i}\\}$ through $q^{2}=\left(\sum_{i}q_{i}\right)^{2}$.
Without loss of generality, the rapidity of the final state invariant mass
system is defined as

$\displaystyle y\equiv\frac{1}{2}\ln\left(\frac{P_{2}\cdot q}{P_{1}\cdot
q}\right)\,.$ (2)

The differential rapidity distribution at the hadronic level can be written as

$\displaystyle\frac{d^{2}}{dydq^{2}}\sigma\left(\tau,q^{2},y\right)=\sigma_{B}(\tau,q^{2})W(\tau,q^{2},y)\quad\text{with}$
$\displaystyle
W=\sum_{a,b=q,{\bar{q}},g}\int_{0}^{1}dx_{1}\int_{0}^{1}dx_{2}f_{a}\left(x_{1},\mu_{F}^{2}\right)f_{b}\left(x_{2},\mu_{F}^{2}\right)\int_{0}^{1}dz\delta(\tau-
zx_{1}x_{2})$
$\displaystyle\qquad\qquad~{}~{}\times\int\left[dPS_{m}\right]\left|\overline{\cal
M}_{ab}\right|^{2}\delta\left(y-\frac{1}{2}\ln\left(\frac{P_{2}\cdot
q}{P_{1}\cdot q}\right)\right)\,.$ (3)

The dimensionless variables, $\tau$ and $z$, are respectively defined as the
ratios of invariant mass square of the final state to the square of hadronic
($S$) and partonic ($\hat{s}$) center-of-mass energies i.e.

$\displaystyle\tau\equiv\frac{q^{2}}{S}\quad\text{and}\quad
z\equiv\frac{q^{2}}{\hat{s}}\,.$ (4)

We denote the fraction of the initial state hadronic momentum carried by the
partons ($a,b$) that take part in the scattering at the partonic level as
$x_{1(2)}$, and these are constrained through the relation $\tau=zx_{1}x_{2}$
as reflected by the presence of the respective $\delta$-function in the
definition of $W$. $f_{a(b)}$ are the parton distribution functions (PDF)
renormalized at the factorization scale $\mu_{F}$. We represent the
renormalization scale through $\mu_{R}$. The mass-factorized scattering matrix
element is denoted through ${\overline{\cal M}}_{ab}$ containing an overline
to signify the sum and average over all the quantum numbers for the final and
initial state particles, respectively. The corresponding $m$-particle phase
space is $[dPS_{m}]$. Note that the numerical value of the integer $m$ depends
on the number of radiated partons which is solely controlled by the
perturbative order we are interested in. $\sigma_{B}$ in (2) is the leading
order contribution normalized by the delta function.

The scattering matrix element or equivalently the partonic differential
distribution can be perturbatively expanded in powers of strong coupling
constant $a_{s}(\mu_{R}^{2})\equiv\alpha_{s}(\mu_{R}^{2})/4\pi$. In this
article, we confine ourselves to the regime where the leading order (LO)
processes can only be initiated through color neutral quark or gluon channels,

$\displaystyle
q(p_{1})+{\bar{q}}(p_{2})\rightarrow\sum_{i=1}^{n}F_{i}(q_{i})\quad\text{and}\quad
g(p_{1})+g(p_{2})\rightarrow\sum_{i=1}^{n}F_{i}(q_{i})$ (5)

with the corresponding momenta $p_{1(2)}$. Moreover, we are interested in
computing the differential rapidity distribution only in the soft limit which
constrained all the partonic radiation to be only soft i.e. those can only
have nearly vanishing momenta. The goal of this article is to present a
prescription to compute the leading contribution of the differential rapidity
distribution in the soft limit which is commonly referred as soft-virtual (SV)
contribution. In order to define the soft limit for the rapidity distribution,
we choose to work with a set of symmetric scaling variables $x_{1(2)}^{0}$
instead of $y$ and $\tau$ which are related through

$\displaystyle
y\equiv\frac{1}{2}\ln\left(\frac{x_{1}^{0}}{x_{2}^{0}}\right)\quad\text{and}\quad\tau\equiv
x_{1}^{0}x_{2}^{0}\,.$ (6)

Note that unlike the inclusive cross-section, the choice of variables which
one needs to take in order to define the soft limit is not unique and as it
turns out, our choice of these new set of variables is crucial for our
prescription. In terms of these variables, the partonic contributions arising
from the subprocesses are found to depend on the ratios

$\displaystyle z_{i}\equiv\frac{x_{i}^{0}}{x_{i}}\,,$ (7)

which play the role of scaling variables at the partonic level. After
evaluating the $\delta$-function integration over $z$, the $W(\tau,q^{2},y)$
in (2) can be rewritten as

$\displaystyle W(x_{1}^{0},x_{2}^{0},q^{2})=$
$\displaystyle\sum_{a,b}\int_{x_{1}^{0}}^{1}\frac{dz_{1}}{z_{1}}\int_{x_{2}^{0}}^{1}\frac{dz_{2}}{z_{2}}f_{a}\left(\frac{x_{1}^{0}}{z_{1}},\mu_{F}^{2}\right)f_{b}\left(\frac{x_{2}^{0}}{z_{2}},\mu_{F}^{2}\right)$
$\displaystyle\times\Delta_{d,ab}\left(z_{1},z_{2},q^{2},\\{p_{j}\cdot
q_{k}\\},\mu_{F}^{2}\right)$ (8)

with the coefficient function

$\displaystyle\Delta_{d,ab}\left(z_{1},z_{2},q^{2},\\{p_{j}\cdot
q_{k}\\},\mu_{F}^{2}\right)=\int\left[dPS_{m}\right]\left|\overline{\cal
M}_{ab}\right|^{2}\delta\left(y-\frac{1}{2}\ln\left(\frac{P_{2}\cdot
q}{P_{1}\cdot q}\right)\right)\,.$ (9)

Being a scattering process containing $n$-number of final state colorless
particles, the partonic coefficient function does, in fact, depend on the
Mandelstam variables constructed out of all the independent external momenta
which is concisely denoted through $\\{p_{j}\cdot q_{k}\\}$. In order to find
the definition of soft limit in terms of the new partonic scaling variables,
we take the double Mellin moment of $W$ with respect to the variables
$N_{1(2)}$ which turns out to be

$\displaystyle W(N_{1},N_{2})$ $\displaystyle\equiv\int
dx_{1}^{0}(x_{1}^{0})^{N_{1}-1}\int
dx_{2}^{0}(x_{2}^{0})^{N_{2}-1}W(x_{1}^{0},x_{2}^{0})$
$\displaystyle=\sum_{ab}f_{a}(N_{1})f_{b}(N_{2})\Delta_{d,ab}(N_{1},N_{2})\,.$
(10)

All the quantities with functional dependence of $N_{1(2)}$ are in Mellin
space where the soft limit is defined by the simultaneous limit of
$N_{1(2)}\to\infty$. In terms of partonic scaling variables this condition
gets translated to $z_{1(2)}\to 1$. Note that we normalize the coefficient
function $\Delta_{d,ab}$ in such a way that at the leading order $W$ satisfies

$\displaystyle
W(x_{1}^{0},x_{2}^{0},q^{2},\mu_{R}^{2})=W_{B}\equiv\delta(1-x_{1}^{0})\delta(1-x_{2}^{0})\,.$
(11)

In the following section, we will present the prescription to calculate the
infrared safe SV differential rapidity distribution to N4LO in QCD for any
$2\rightarrow n$ scattering process which can be computed order by order in
perturbation theory:

$\displaystyle\Delta_{d,ab}^{\rm sv}\left(z_{1},z_{2},q^{2},\\{p_{j}\cdot
q_{k}\\},\mu_{F}^{2}\right)=a_{s}^{\lambda}(\mu_{R}^{2})\sum_{k=0}^{\infty}a_{s}^{k}(\mu_{R}^{2})\Delta_{d,ab}^{(k),{\rm
sv}}\left(z_{1},z_{2},q^{2},\\{p_{j}\cdot
q_{k}\\},\mu_{F}^{2},\mu_{R}^{2}\right)\,.$ (12)

$\lambda$ is the order of strong coupling constant at the leading order
partonic process.

###  2.1 Universal soft-collinear operator for SV rapidity distribution

In this section, we setup a framework to compute the soft-virtual corrections
to the rapidity distribution to all orders in strong coupling constant. The
infrared safe SV rapidity distribution can be obtained by combining the
ultraviolet (UV) renormalized virtual matrix elements with the soft gluon
contribution and performing appropriate mass factorization to get rid of
initial state collinear singularities. It is well-known that the combined soft
and collinear divergences, conveniently denoted as infrared (IR), in virtual
matrix elements factorize from the corresponding UV renormalized part to all
orders in perturbation theory and thereby in dimensional regularization we can
write

$\displaystyle{\cal M}_{ab,{\rm
fin}}\left(\\{p_{j}\\},\\{q_{k}\\},\mu_{R}^{2}\right)=\lim_{\epsilon\rightarrow
0}Z_{ab,\rm IR}^{-1}(q^{2},\mu_{R}^{2},\epsilon){\cal
M}_{ab}\left(\\{p_{j}\\},\\{q_{k}\\},\epsilon\right)$ (13)

with the space-time dimensions $D=4+\epsilon$. Without loss of generality, we
choose the renormalization scale to be equal to the scale of aforementioned
factorization which, of course, in general can be different. Upon multiplying
the renormalization constant $Z_{ab,{\rm IR}}$, the IR divergent part of the
UV renormalized matrix element ${\cal M}_{ab}$ gets compensated and we end up
with the finite part of the matrix element ${\cal M}_{ab,{\rm fin}}$. The
renormalization constant is a universal quantity as it is independent of the
details of the process, it only depends on the nature of external color
particles. It is fully independent of the number and nature of external
colorless particles. The universal structure of the IR divergences in
dimensional regularization, up to the single pole in $\epsilon$, were first
depicted in ref. Catani:1998bh  and subsequently in ref. Sterman:2002qn . For
scattering involving only two external colored particles, the universality of
the single pole at two-loop was established in terms of soft and collinear
anomalous dimensions in ref. Ravindran:2004mb  which was verified at three-
loop in ref. Moch:2005tm . An all order conjecture on the form of IR
divergences in QCD for any generic scattering process is depicted in terms of
soft anomalous dimension matrix in ref. Becher:2009cu ; Gardi:2009qi . We
provide the results of the renormalization constant to four loops in ancillary
file.

Before moving further, we wish to iterate a well-known fact. For processes
involving only conserved operator, such as Drell-Yan, the coupling constant
renormalization is sufficient to get rid of all the UV divergences. However,
for other processes, such as the Higgs boson production in heavy quark
effective theory, an additional renormalization which often called the
operator renormalization is needed. This is a property inherent to the
operator itself.

In order to get the infrared safe and finite differential rapidity
distribution, we need to combine the UV renormalized virtual matrix element to
the real emission contributions in the soft limit and perform mass
factorization which ensures the removal of collinear singularities arising
from the initial state colored particles. Therefore, the universal nature of
IR divergences in virtual matrix element implies that the combined
contribution from the real emission diagrams and mass-factorization kernels
must exhibit the same universality. By employing the criteria of universal IR
structure and imposing the finiteness property of the rapidity distribution,
we develop the prescription to compute the rapidity distribution under SV
approximation for any generic $2\rightarrow n$ scattering process and present
the result in terms of universal quantities to N4LO QCD. Once the pure virtual
matrix element for any process of the kind under consideration becomes
available, our expression can immediately be employed to calculate the SV
rapidity distribution at that order in QCD.

In this article, we extend the prescription, which was introduced for the
Sudakov type process in ref. Ravindran:2006bu , to the case of $2\rightarrow
n$ scattering. We propose that the coefficient function for the rapidity
distribution in (12) can be written as a Mellin convolution of the pure
virtual contribution ${\cal F}$, soft-collinear distribution $\Phi_{d}$ and
mass-factorization kernel $\Gamma$, which read as

$\displaystyle\Delta_{d,a\bar{a}}^{\rm
sv}\left(z_{1},z_{2},q^{2},\\{p_{j}\cdot q_{k}\\},\mu_{F}^{2}\right)=$
$\displaystyle|{\cal M}^{(0)}_{a\overline{a}}|^{2}|{\cal
F}_{a\overline{a}}(\\{p_{j}\cdot
q_{k}\\},q^{2},\epsilon)|^{2}\delta(1-z_{1})\delta(1-z_{2})$
$\displaystyle\otimes{\cal
C}\exp\bigg{(}2\Phi_{d,a\overline{a}}\left(z_{1},z_{2},q^{2},\epsilon\right)-{\cal
C}\log\Gamma_{a\overline{a}}(z_{1},\mu_{F}^{2},\epsilon)\delta(1-z_{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\qquad-{\cal
C}\log\Gamma_{a\overline{a}}(z_{2},\mu_{F}^{2},\epsilon)\delta(1-z_{1})\bigg{)}\,.$
(14)

The pure virtual contribution is captured through the form factor ${\cal
F}_{a\bar{a}}$ that is defined as

$\displaystyle{\cal F}_{a\overline{a}}=1+\sum_{k=1}^{\infty}a_{s}^{k}{\cal
F}_{a\overline{a}}^{(k)}\equiv
1+\sum_{k=1}^{\infty}a_{s}^{k}\frac{\langle{\cal
M}_{a\overline{a}}^{(0)}|{\cal M}_{a\overline{a}}^{(k)}\rangle}{\langle{\cal
M}_{a\overline{a}}^{(0)}|{\cal M}_{a\overline{a}}^{(0)}\rangle}\,,$ (15)

where ${\cal M}_{a\overline{a}}^{(k)}$ represents the $k$-th order UV
renormalized matrix element of the underlying partonic level process
$a(p_{1})+\overline{a}(p_{2})\rightarrow\sum_{i=1}^{n}F_{i}(q_{i})$. The
symbol “${\cal C}$” stands for the convolution whose actions on a distribution
$g(z_{1},z_{2})$ is defined as

$\displaystyle{\cal
C}e^{g(z_{1},z_{2})}=\delta(1-z_{1})\delta(1-z_{2})+\frac{1}{1!}g(z_{1},z_{2})+\frac{1}{2!}\left(g\otimes
g\right)(z_{1},z_{2})+\cdots\,,$ (16)

where $\otimes$ denotes Mellin convolution. In this article in the context of
SV corrections, we encounter only $\delta(1-z_{i})$ and ${\cal D}_{j}(z_{i})$,
where

$\displaystyle{\cal
D}_{j}(z_{i})\equiv\left[\frac{\ln^{j}(1-z_{i})}{(1-z_{i})}\right]_{+}\,.$
(17)

The contribution from the real emission diagrams is contained in
$\Phi_{d,a\overline{a}}$ which we call as soft-collinear distribution. The
soft divergences arising from the real emission and virtual diagrams, which
are respectively encapsulated in $\Phi_{d}$ and ${\cal F}$, get cancelled. The
final state collinear singularity is guaranteed to go away, as dictated by
Kinoshita-Lee-Nauenberg (KLN) theorem, once the sum over all the final states
is performed. The mass factorization kernel takes care of the initial state
collinear singularities. As a result, the coefficient function $\Delta^{\rm
sv}_{d,a\bar{a}}$ in (2.1) becomes finite. By demanding the finiteness of this
quantity we can put a constraint on the soft-collinear distribution which
turns out to be a Sudakov type renormalization group (RG) equation. This has
profound implications which not only reveals a significant amount of insights
about the IR world but also it enables us to perform threshold resummation as
we will see in the next section. To be more precise, the solution of the RG
equation results an all order exponentiation of the soft-collinear
distribution. So, the whole job of computing the SV correction depends on our
ability to determine and explore the unknown distribution
$\Phi_{d,a\overline{a}}$ to which we now turn to111It is interesting to
compare the equation (2.1) to the corresponding one for computing the SV
cross-section which is proposed in another publication Ahmed:2020xxx ..

As we have discussed, the soft-collinear distribution essentially captures the
contribution arising from real emission diagrams which only can occur from
colored partons. Naturally, $\Phi_{d,a\overline{a}}$ for Sudakov form factor
i.e. $2\rightarrow 1$ and $2\rightarrow n$ scattering should essentially be
identical. The presence of more Mandelstam variables in the latter process
just makes it more involved in its kinematic dependence when it is expressed
in terms of $\\{p_{j}\cdot q_{k}\\}$. However, in terms of the total invariant
mass square of the final state colorless particles i.e. $q^{2}$, it has to be
exactly same as that of Sudakov process. In ref. Ravindran:2006bu , it was
conjectured to satisfy a integro-differential RG equation to all orders in QCD
coupling constant. The underlying reason behind this all order conjecture is
inspired by the akin integro-differential Sudakov equation Mueller:1979ih ;
Collins:1980ih ; Sen:1981sd  fulfilled by the form factor whose solution is
present explicitly to five loops order in massless QCD in refs. Moch:2005tm ;
Ravindran:2005vv ; Ravindran:2006cg ; Ahmed:2017gyt . By integrating the
differential rapidity distribution, we get the inclusive cross-section. Upon
taking the Mellin moment with respect to the same Mellin variable $N$ of this
relation we get

$\displaystyle\int_{0}^{1}dx_{1}^{0}\int_{0}^{1}dx_{2}^{0}(x_{1}^{0}x_{2}^{0})^{N-1}\frac{d\sigma}{dy}=\int_{0}^{1}d\tau\tau^{N-1}\sigma\,.$
(18)

By taking the limit $N\rightarrow\infty$ on both sides of this relation, we
can relate the soft-collinear distributions in rapidity and that of inclusive
cross-section. This is remarkable in a sense that given the soft-collinear
distribution for inclusive cross-section, we can automatically calculate it
for the rapidity. Since this is the only quantity that is unknown in
comparison to the ingredients for the computation of SV cross-section, we can
immediately calculate the SV rapidity distribution. The
$\Phi_{d,a\overline{a}}$ for the Sudakov form factor is determined to NNLO in
refs. Ravindran:2006bu  and in Ahmed:2014uya  at N3LO in QCD. In the current
article, for the first time we present the general analytical form of
$\Phi_{d,a\overline{a}}$ in terms of universal quantities at N4LO for any
generic $2\rightarrow n$ scattering. One of the most notable features of this
quantity is it satisfies the maximally non-Abelian property i.e.

$\displaystyle\Phi_{d,gg}=\frac{C_{A}}{C_{F}}\Phi_{d,q\bar{q}}\,,$ (19)

where the quadratic Casimirs in Adjoint and fundamental representations of
SU($n_{c}$) are denoted by $C_{A}=n_{c}$ and $C_{f}=(n_{c}^{2}-1)/2n_{c}$,
respectively. The property says that the soft-collinear distribution for quark
and gluon initiated processes which are respectively denoted by $q\bar{q}$ and
$gg$ are related by simple scaling of quadratic Casimirs. This essentially
signifies the universality of the real emission in the soft limit i.e. it is
independent of the details of the process, it solely depends on the nature of
the external partons. Needless to say, it is also quark flavour blind. The
relation in (19) was explicitly verified to NNLO in refs. Ravindran:2006bu
and at N3LO in Ahmed:2014uya . The flavour dependence of the
$\Phi_{d,a\bar{a}}$ was exploited in ref. Ahmed:2014era  in order to calculate
the SV rapidity distribution at N3LO for the Higgs boson production in bottom
quark annihilation. We expect the Casimir scaling to hold true to all orders
in perturbation theory since it originates entirely from the soft-collinear
part of the differential cross-section, and therefore it would indeed be
interesting to see whether truly it holds beyond N3LO with generalised Casimir
scaling Moch:2018wjh .

We decompose all the quantities into its singular (sing) and finite (fin)
parts as

$\displaystyle\Phi_{d,a\overline{a}}=\Phi_{d,a\overline{a},{\rm
sing}}+\Phi_{d,a\overline{a},{\rm fin}}\,,$
$\displaystyle\log\Gamma_{a\overline{a}}=\log\Gamma_{a\overline{a},{\rm
sing}}+\log\Gamma_{a\overline{a},{\rm fin}}\,,$ (20)

(2.1) can be recast into

$\displaystyle\Delta_{d,a\bar{a}}^{\rm sv}=$ $\displaystyle|{\cal
M}^{(0)}_{a\overline{a}}|^{2}|{\cal F}_{a{\overline{a}},{\rm
fin}}(\\{p_{j}\cdot
q_{k}\\},q^{2},\mu_{R}^{2})|^{2}\delta(1-z_{1})\delta(1-z_{2})$
$\displaystyle\otimes{\cal C}\exp\bigg{(}2\Phi_{d,{a\overline{a}},{\rm
fin}}\left(z_{1},z_{2},q^{2},\mu_{R}^{2}\right)-{\cal
C}\log\Gamma_{a{\overline{a}},{\rm
fin}}(z_{1},\mu_{F}^{2},\mu_{R}^{2})\delta(1-z_{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\qquad-{\cal
C}\log\Gamma_{a{\overline{a}},{\rm
fin}}(z_{2},\mu_{F}^{2},\mu_{R}^{2})\delta(1-z_{1})\bigg{)}\otimes\textbf{I}_{d,a{\overline{a}}}^{\rm
sv}\,,$ (21)

where

$\displaystyle\textbf{I}_{d,a\overline{a}}^{\rm sv}=$
$\displaystyle|Z_{a{\bar{a}},{\rm
IR}}(q^{2},\mu_{R}^{2},\epsilon)|^{2}\delta(1-z)\otimes\;{\cal
C}\exp\bigg{(}2\Phi_{d,{a\overline{a}},{\rm
sing}}\left(z_{1},z_{2},q^{2},\mu_{R}^{2},\epsilon\right)$
$\displaystyle-{\cal C}\log\Gamma_{d,a\overline{a},{\rm
sing}}(z_{1},\mu_{R}^{2},\epsilon)\delta(1-z_{2})-{\cal
C}\log\Gamma_{d,a\overline{a},{\rm
sing}}(z_{2},\mu_{R}^{2},\epsilon)\delta(1-z_{1})\bigg{)}\,.$ (22)

Through the decomposition of the quantities into singular and finite parts in
(2.1), we put together all the singular components of the rapidity
distribution into $\textbf{I}^{\rm sv}_{d,a\bar{a}}$ which must be unit
distribution $\delta(1-z_{1})\delta(1-z_{2})$ in order to get a finite
$\Delta^{\rm sv}_{d,a\bar{a}}$. In (2.1), the form factor and the leading
order matrix element are the only process dependent quantity. The remaining
part which comprises of the finite segments of the soft-collinear distribution
and mass factorization kernel is a process independent universal quantity
which we call as differential soft-collinear operator

$\displaystyle{\textbf{S}}_{d,a\overline{a}}(z_{1},z_{2},q^{2},\mu_{R}^{2},\mu_{F}^{2})\equiv$
$\displaystyle{\cal C}\exp\bigg{(}2\Phi_{d,{a\overline{a}},{\rm
fin}}\left(z_{1},z_{2},q^{2},\mu_{R}^{2}\right)-{\cal
C}\log\Gamma_{a{\overline{a}},{\rm
fin}}(z_{1},\mu_{R}^{2},\mu_{F}^{2})\delta(1-z_{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\qquad-{\cal
C}\log\Gamma_{a{\overline{a}},{\rm
fin}}(z_{2},\mu_{R}^{2},\mu_{F}^{2})\delta(1-z_{1})\bigg{)}\,.$ (23)

The expression of $\textbf{S}_{d,a\overline{a}}$ being process independent can
be used for any generic $2\rightarrow n$ scattering process. Since we are
confining our discussion to only those scattering processes with initial state
quark-antiquark pair of same flavours or a pair of gluon, we conveniently
rewrite the (2.1) as

$\displaystyle\Delta^{\rm sv}_{d,I}=|{\cal M}^{(0)}_{I}|^{2}|{\cal F}_{I,{\rm
fin}}|^{2}\delta(1-z_{1})\delta(1-z_{2})\otimes\textbf{S}_{d,I}$ (24)

where $I=q{\bar{q}},gg$. We can calculate the SV coefficient function for the
rapidity distribution order by order in perturbation theory by expanding it in
powers of $a_{s}$ according to (12). We provide the results of
$\textbf{S}_{d,I}$ and $\Delta_{d,I}^{\rm sv}$ for any generic process, which
can be expressed in terms of universal light-like cusp ($A^{I}$), eikonal or
soft ($f^{I}$) and virtual or collinear ($B^{I}$) anomalous dimensions along
with the process dependent form factors, in appendix A and B up to N4LO QCD,
respectively. Though the scale dependence can be restored employing the
renormalization group evolution, nonetheless for users’ convenience we provide
the results of $\Delta_{d,I}^{\rm sv}$ by keeping the explicit scale
dependence in ancillary files. Once the virtual matrix element becomes
available, one can directly use our generic results to calculate the SV
rapidity distribution for any generic process of the kind under consideration.
The anomalous dimensions are expanded in powers of $a_{s}(\mu_{R}^{2})$ as

$\displaystyle
X^{I}(\mu_{R}^{2})=\sum_{j=1}^{\infty}a_{s}^{j}(\mu_{R}^{2})X^{I}_{j}\,,$ (25)

where $X=A,B,f$. Thanks to recent calculations, the light-like cusp anomalous
dimensions are available to four loops Korchemsky:1987wg ; Moch:2004pa ;
Vogt:2004mw ; Henn:2019swt ; vonManteuffel:2020vjv  in QCD. The soft and
collinear anomalous dimensions can be extracted Ravindran:2004mb ; Moch:2005tm
from the quark and gluon collinear anomalous dimensions Becher:2006mr ;
Becher:2009qa  through the conjecture Ravindran:2004mb

$\displaystyle\gamma^{I}=2B^{I}+f^{I}$ (26)

to three loops. At four loop, only partial results are available in refs.
Davies:2016jie ; vonManteuffel:2020vjv ; Das:2019btv ; Das:2020adl .

We present the new results of $\Delta_{d,I}^{\rm sv}$ for $2\rightarrow 1$
processes such as the Drell-Yan and the Higgs boson productions through gluon
fusion as well as bottom quark annihilation at N4LO in appendix D. This has
been achieved by using the explicit results of the recently computed four loop
cusp anomalous dimension Korchemsky:1987wg ; Moch:2004pa ; Vogt:2004mw ;
Henn:2019swt ; vonManteuffel:2020vjv  along with the form factors for Drell-
Yan and the Higgs boson productions which are approximately available to
fourth order in QCD Henn:2016men ; Lee:2016ixa ; vonManteuffel:2016xki ;
Lee:2017mip ; vonManteuffel:2019wbj ; vonManteuffel:2020vjv . The previously
missing coefficients of $\delta(1-z_{1})\delta(1-z_{2})$ in $\Delta_{d,I}^{\rm
sv}$ at N4LO were due to the missing ${\cal O}(\epsilon^{0})$ results of form
factors and the soft gluon contributions at four loop. The full explicit
results of the quark and gluon form factors corresponding to Drell-Yan and the
Higgs boson production in gluon fusion are available at three loop to ${\cal
O}(\epsilon^{2})$ in ref. Gehrmann:2010ue ; Gehrmann:2010tu  which are also
required. The corresponding partial four loop form form factors are computed
in several articles over the past few years Henn:2016men ; Lee:2016ixa ;
vonManteuffel:2016xki ; Lee:2017mip ; vonManteuffel:2019wbj ;
vonManteuffel:2020vjv . For the case of Higgs production through bottom quark
annihilation, the three loop Gehrmann:2014vha ; Lee:2017mip  and partial four
loop results are available in ref. Lee:2017mip . Moreover, the one- and two-
loop results are needed to expand to ${\cal O}(\epsilon^{5})$ and ${\cal
O}(\epsilon^{3})$, respectively. Similarly for the coefficient which
contributes to the ${\cal O}(\epsilon^{0})$ part of four loop form factor, the
one-, two-, and three-loop results are needed to order ${\cal
O}(\epsilon^{6})$, ${\cal O}(\epsilon^{4})$ and ${\cal O}(\epsilon^{2})$,
respectively. The four loop explicit results which we present in this article
are still incomplete due to the unavailability of the full explicit results
for form factors as well as soft contributions resulting from the real
emission processes at four loop, so far these are the state-of-the-art
available results in the literature.

In ancillary file, the explicit expressions of all the anomalous dimensions
including the QCD $\beta$-functions to three loops can be found. In the
following section, we describe how our formulation of SV rapidity distribution
naturally leads us to the soft gluon resummation for any generic $2\rightarrow
n$ scattering process.

##  3 Threshold resummation and its universal soft-collinear operator

Here in this section, we develop the resummation formalism for the
differential distribution with respect to the rapidity variable $y$, for the
production of $n$-colorless particles which also paves the way for a wider
range of comparisons with the experiments. Earlier we have seen that there
exists an operator, $\textbf{S}_{d,I}$, referred as the differential soft-
collinear operator, which embeds the universality of all the soft enhancements
associated with the soft gluon emissions in the production of $n$-colorless
particles in the hadronic collision. The universality lies in the fact that
the operator, $\textbf{S}_{d,I}$ in (2.1), for the SV differential cross-
section depends only on the initial state partons and is completely
independent of the hard process under study. Besides being the process
independent operator, interestingly it also exhibits an exponential behaviour.
Recall that the threshold resummation Ahmed:2020xxx  relies on the fact that
the soft contribution exponentiates to all orders in perturbation theory,
owing to the Sudakov differential equation and the renormalization group
invariance. Following the same argument we proceed towards the resummation
formalism for differential cross-section as well.

The relevance of resummation of differential cross-section arises from the
fact that, in the limit $z_{1(2)}\rightarrow 1$, the logarithms of type
$\Big{(}a_{s}^{n}\log^{m_{1}}(1-z_{1})\log^{m_{2}}(1-z_{2})\Big{)}/\big{(}(1-z_{1})(1-z_{2})\big{)}$
for $m_{1}+m_{2}\leq 2(n-1)$, give rise to large contributions which could
potentially spoil the reliability of the perturbative series. Hence a
systematic way of exponentiating these large logarithms and resumming them to
all orders in perturbation theory becomes indispensable. In ref. Catani:1989ne
it was shown, in the context of differential distribution with respect to the
Feynman variable $x_{F}$, that the potential logarithms which give dominant
contributions in certain kinematic regions can be resummed to all orders in
perturbation theory in Mellin-Mellin (M-M) space approach. This approach was
also extended to rapidity distributions in the earlier works by one of the
authors of this paper (See Banerjee:2018vvb ; Banerjee:2017cfc  for details).
Note that this approach is different from the Mellin-Fourier (M-F) approach
Laenen:1992ey  proposed by Laenen & Sterman which was earlier discussed in the
introduction. In M-F formalism partonic cross-section is expressed in terms of
scaling variable $z$ and rapidity variable $y$ and then the threshold limit is
taken only for $z\rightarrow 1$ which resums delta ($\delta(1-z)$) and
distributions ($[\frac{\log(1-z)}{1-z}]_{+}$) in $z$, but for rapidity
variable $y$ only delta ($\delta(y)$) piece is resummed. In ref.
Banerjee:2018vvb , one of the authors of this paper has made a detailed
numerical comparison of M-M approach against the M-F approach and found that
both the approaches converges to a few percent correction to the fixed order
prediction at NNLL level. In the following we further extend the M-M approach
and derive the resummation formalism for the production of $n$-colorless
particles in a partonic collision.

Within the framework of M-M approach, both the partonic scaling variables
$z_{1(2)}$ are simultaneously taken to the threshold limit 1 and the
corresponding delta, $\delta(1-z_{i})$, and plus distributions,
$\Big{[}\frac{\ln^{m_{i}}(1-z_{i})}{1-z_{i}}\Big{]}_{+}$, are resummed to all
orders in perturbation theory. Due to the involvement of convolutions in the
$z_{1(2)}$ space, the resummation is performed in two dimensional Mellin space
where the differential cross-section is expressed in terms of simple normal
products along with the aforementioned exponential structure in form. In the
following we derive the generic formalism in terms of the Mellin variables
$N_{1}$ and $N_{2}$ corresponding to the $z_{1}$ and $z_{2}$ variables,
respectively. Hence the threshold limit $z_{1(2)}\rightarrow 1$ translate to
$N_{1(2)}\rightarrow\infty$ in Mellin space and the large logarithms
proportional to $\log N_{1(2)}$ are resummed to all orders in perturbation
theory.

To derive the all order behaviour of the SV differential cross-section,
$\Delta^{\mathrm{sv}}_{d,I}(z_{1},z_{2})$, in the two dimensional Mellin space
with $\bar{N}_{i}=N_{i}e^{\gamma_{E}}$, we begin with the Mellin moment of the
same, which takes the following form:

$\displaystyle\tilde{\Delta}^{\mathrm{sv}}_{d,I}(\bar{N}_{1},\bar{N}_{2})$
$\displaystyle=\bigg{[}\prod_{i=1,2}\int_{0}^{1}dz_{i}z_{i}^{N_{i}-1}\bigg{]}\Delta^{\mathrm{sv}}_{d,I}(\\{p_{j}\cdot
q_{k}\\},z_{1},z_{2})\,.$ (27)

$\gamma_{E}$ is the Euler-Mascheroni constant. In the previous section in
(2.1), $\Delta^{\mathrm{sv}}_{d,I}$ is decomposed into constituents
corresponding to the virtual as well as the soft-collinear real emission
contributions. Now in this section, we further decompose those contributions
into a process dependent and a process independent quantities. We denote the
process dependent coefficient $C_{d,0}^{I}$ in the context of $2\rightarrow n$
scattering process as,

$\displaystyle C_{d,0}^{I}\left(\\{p_{j}\cdot
q_{k}\\},q^{2},\mu_{F}^{2}\right)=|{\cal M}^{(0)}_{I}|^{2}|{\cal F}_{I,{\rm
fin}}(\\{p_{j}\cdot
q_{k}\\},q^{2},\mu_{R}^{2})|^{2}S_{res,\delta}^{d,I}(q^{2},\mu_{R}^{2},\mu_{F}^{2}).$
(28)

Here $C_{d,0}^{I}$ accounts for all the finite contributions coming from the
virtual corrections and the coefficients proportional to
$\delta(1-z_{1})\delta(1-z_{2})$ of the real emission contributions. Besides,
it also contains the finite part of the mass factorized kernel $\Gamma_{I,\rm
fin}$ in terms of $\log(\mu_{F}^{2}/\mu_{R}^{2})$ which results from the
coupling constant renormalization. The quantity $S^{d,I}_{res,\delta}$ which
we name as the differential soft-collinear operator for threshold resummation,
embeds the $\delta(1-z_{i})$ contributions from the soft distribution function
$\Phi_{d,I}$ and from $\Gamma_{I,\rm fin}$ in the following way:

$\displaystyle
S_{res,\delta}^{d,I}(q^{2},\mu_{R}^{2},\mu_{F}^{2})=\exp\left({2\Phi_{d,I,\delta}(q^{2},\mu_{R}^{2},\mu_{F}^{2})-2\log\Gamma_{{I,\delta}}}(\mu_{F}^{2})\right)\,.$
(29)

The subscript $\delta$ indicates $\delta(1-z_{1})\delta(1-z_{2})$ coefficients
of the aforementioned quantities. In a similar way, we denote the process
independent contributions to $\Delta_{d,I}^{\mathrm{sv}}$ as
$\Phi_{d,I}^{res}$ which comprises of the terms proportional to plus
distributions from $\Phi_{d,I}$ and $\Gamma_{I,\rm fin}$. Mathematically it
can be written as,

$\displaystyle\Phi^{res}_{d,I}(z_{1},z_{2},q^{2},\mu_{F}^{2})=~{}$
$\displaystyle 2\Phi_{d,I,{\cal D}}(z_{1},z_{2},q^{2})-{\cal
C}\log\Gamma_{I,{\cal D}}(z_{1},\mu_{F}^{2})\delta(1-z_{2})$
$\displaystyle-{\cal C}\log\Gamma_{I,{\cal
D}}(z_{2},\mu_{F}^{2})\delta(1-z_{1})\,,$ (30)

where the subscript ${\cal D}$ indicates the terms proportional to plus
distribution which includes, ${\cal D}_{i}(z_{1})\delta(1-z_{2})$ , ${\cal
D}_{i}(z_{2})\delta(1-z_{1})$, and ${\cal D}_{i}(z_{1}){\cal D}_{j}(z_{2})$.
Now following the approach given in ref. Ravindran:2006bu , the above equation
can be written in an integral form which is given as,

$\displaystyle\Phi^{res}_{d,I}(z_{1},z_{2},q^{2},\mu_{F}^{2})=$
$\displaystyle\Bigg{[}\delta(\overline{z}_{2})~{}\Bigg{(}{1\over\overline{z}_{1}}\Bigg{\\{}\int_{\mu_{F}^{2}}^{q^{2}\overline{z}_{1}}{d\lambda^{2}\over\lambda^{2}}A^{I}\left(a_{s}(\lambda^{2})\right)+D^{I}_{d}\left(a_{s}(q^{2}\overline{z}_{1})\right)\Bigg{\\}}\Bigg{)}_{+}$
$\displaystyle+{1\over
2}\Bigg{(}{1\over\overline{z}_{1}\overline{z}_{2}}\Bigg{\\{}A^{I}(a_{s}(z_{12}))+{dD^{I}_{d}(a_{s}(z_{12}))\over
d\ln z_{12}}\Bigg{\\}}\Bigg{)}_{+}+(z_{1}\leftrightarrow z_{2})\Bigg{]}\,,$
(31)

here the subscript + indicates the standard plus distribution and the other
constants are defined as $\bar{z}_{i}=(1-z_{i})$ and
$z_{12}=q^{2}\overline{z}_{1}\overline{z}_{2}$. The finite functions,
$D_{d}^{I}=\sum_{i=1}^{\infty}a_{s}^{i}D_{d,i}^{I}$, are related to the
threshold exponent $D^{I}_{i}$ of inclusive cross section owing to the
relation given in (18) ( See Ravindran:2006bu ; Banerjee:2017cfc  for more
details). For completeness, we provide the coefficients $A^{I}_{i}$ and
$D^{I}_{d,i}$ in the ancillary file.

Consequently the SV differential cross-section decomposes into a process
dependent and a process independent way and can be re-written in the following
form:

$\displaystyle\Delta^{\rm sv}_{d,I}\left(\\{p_{j}\cdot
q_{k}\\},z_{1},z_{2},q^{2},\mu_{F}^{2}\right)=$
$\displaystyle~{}C_{d_{,}0}^{I}\left(\\{p_{j}\cdot
q_{k}\\},q^{2},\mu_{F}^{2}\right)\delta(1-z_{1})\delta(1-z_{2})$
$\displaystyle\otimes{\cal
C}\exp(\Phi^{res}_{d,I}(z_{1},z_{2},q^{2},\mu_{F}^{2}))\otimes\textbf{I}_{d,I}^{\rm
sv}\,.$ (32)

Substituting (3) in (27) and after doing the two dimensional Mellin
transformation systematically, we obtain

$\displaystyle\tilde{\Delta}_{d,I}^{\mathrm{sv}}(\bar{N}_{1},\bar{N}_{2})=C_{d,0}^{I}\left(\\{p_{j}\cdot
q_{k}\\},q^{2},\mu_{F}^{2}\right)\exp\Big{(}\overline{G}_{d,0}^{I}(q^{2},\mu_{F}^{2})+\overline{G}_{d,\bar{N}}^{I}(q^{2},\mu_{F}^{2},\omega)\Big{)}\,,$
(33)

with $\omega=\beta_{0}a_{s}(\mu_{R}^{2})\ln(\bar{N}_{1}\bar{N}_{2})$. The
first coefficient of QCD $\beta$-function is denoted by
$\beta_{0}\equiv(11C_{A}-2n_{f})/3$, $n_{f}$ is the number of active light
quark flavours. Here, the decomposition in the exponent is done in such a way
that the coefficient $\overline{G}_{d,\bar{N}}^{I}$ contains $N_{1(2)}$
dependent terms, and the remaining ones are embedded in
$\overline{G}_{d,0}^{I}$. Besides this,
$\overline{G}_{d,\bar{N}}^{I}(q^{2},\mu_{F}^{2},\omega)$ also vanishes in the
limit $\omega\rightarrow 1$. Needless to say that both of these coefficients
has a universal structure in terms of the anomalous dimensions $A^{I}$ and
process independent coefficients $D^{I}$ and thus are dependent only on the
incoming partons. Further we combine the $N_{1(2)}$ independent coefficients
$\overline{G}_{d,0}^{I}$ with $C_{d,0}^{I}$ from (33) and define,

$\displaystyle{\bar{g}}_{d,0}^{I}(\\{p_{j}\cdot q_{k}\\},q^{2},\mu_{F}^{2})$
$\displaystyle=C_{d,0}^{I}\left(\\{p_{j}\cdot
q_{k}\\},q^{2},\mu_{F}^{2}\right)\exp\left(\overline{G}_{d,0}^{I}(q^{2},\mu_{F}^{2})\right)\,\par$
(34)

which can be expanded in terms of $a_{s}(\mu_{R}^{2})$ as,

$\displaystyle\overline{g}_{d,0}^{I}(\\{p_{j}\cdot
q_{k}\\},q^{2},\mu_{F}^{2})=\sum_{i=0}^{\infty}a_{s}^{i}(\mu_{R}^{2})\overline{g}_{d,0}^{I,i}(\\{p_{j}\cdot
q_{k}\\},q^{2},\mu_{F}^{2},\mu_{R}^{2})\,.$ (35)

From (34) it can be seen, that the coefficient ${\bar{g}}_{d,0}^{I}$ contains
finite contribution from virtual corrections, differential soft-collinear
operator for threshold resummation and $N$ independent terms coming from
Mellin transformation of plus distribution. Consequently, (33) gets modified
as,

$\displaystyle\tilde{\Delta}^{\mathrm{sv}}_{d,I}(\bar{N}_{1},\bar{N}_{2})=\overline{g}_{d,0}^{I}\left(\\{p_{j}\cdot
q_{k}\\},q^{2},\mu_{F}^{2}\right)\exp\Big{(}\overline{G}_{d,\bar{N}}^{I}\left(q^{2},\mu_{F}^{2},\omega\right)\Big{)}\,.$
(36)

where the exponent ${\overline{G}}_{d,\bar{N}}^{I}$ can be organized as a
resummed perturbation series in Mellin space as,

$\displaystyle\overline{G}^{I}_{d,\bar{N}}(q^{2},\mu_{F}^{2},\omega)={\overline{g}}^{I}_{d,1}(\omega)\ln(\bar{N}_{1}\bar{N}_{2})+\sum_{i=0}^{\infty}a_{s}^{i}(\mu_{R}^{2}){\overline{g}}^{I}_{d,i+2}(\omega,q^{2},\mu_{F}^{2},\mu_{R}^{2})\,.$
(37)

The explicit form in (36) when expanded till $k$-th order in powers of
$a_{s}(\mu_{R}^{2})$, gives the logarithmically enhanced contributions to the
fixed order results
$\tilde{\Delta}^{\mathrm{sv}}_{d,I}(\bar{N}_{1},\bar{N}_{2})$ up to the same
order. The successive terms in the above series given in (37) along with the
corresponding terms in (35) define the resummed accuracy as LL, NLL, NNLL,
N3LL and so on. In general for NkLL accuracy, terms up to
$\bar{g}_{d,{k+1}}^{I}$ must be included along with $\bar{g}_{d,0}^{I}$ up to
order $a_{s}^{k}(\mu_{R}^{2})$. The general expression for the coefficients
$\bar{g}_{d,0}^{I,i}$ and $\bar{g}_{d,i}^{I}$ up to $\mathrm{N^{3}LL}$ are
provided in the ancillary files.

The coefficients $\overline{G}^{I}_{d,\bar{N}}$ remains unaltered even for
$2\rightarrow n$ scattering process owing to its universality. However, the
process dependent coefficient function $\overline{g}_{d,0}^{I}$ changes for
the production of $n$-colorless particles due to the inclusion of process
specific form factor via (28) and (34). The results of these coefficients
appear as a product of $N_{1}$ and $N_{2}$ in the Mellin space, and all those
terms which are only function of $N_{1}$ or $N_{2}$ cancel internally.

We have also observed that the coefficients $\bar{g}_{d,0}^{I}$ and
$\overline{G}_{d,\bar{N}}^{I}$ coincides with their inclusive counterparts
$\bar{g}_{0}^{I}$ and $\overline{G}_{\bar{N}}^{I}$ respectively in the limit
$N_{1}\rightarrow N_{2}\rightarrow N$, provided the coefficients $D_{d}^{I}$
in (3) is expressed in terms of $D^{I}$ of inclusive soft distribution
function using the relation (18). Hence we infer that all the above
observations which hold true for $2\rightarrow 1$ scattering processes are
further extended and verified for any generic system of $n$-colorless
particles in the final state.

##  4 Conclusions and Outlook

In today’s era reducing the theoretical uncertainties remain one of the main
motivations for higher order radiative corrections. It is also particularly
relevant for constraining beyond SM scenarios and validation of the SM itself.
Among several observables, differential cross-sections allow a wider range of
comparisons with the experiment and hence several attempts were made in the
past for better theoretical understanding of the same. In this article, we
restrict ourselves to the discussion of differential rapidity distribution for
the production of $n$-number of colorless particles in the hadronic collision
within the realm of perturbative QCD.

Through this publication, we intend to present a systematic framework for the
study of soft-plus-virtual corrections to the differential distribution with
respect to the rapidity variable $y$, for the production of $n$-colorless
particles in the hadron collider. The infrared structure of rapidity
distribution which was earlier studied in ref. Ravindran:2006bu  for Sudakov
type processes is further extended to the case of $2\rightarrow n$ scattering.
We employ the universality of the soft enhancements associated with the real
emission diagrams. The main deviation from the Sudakov type formalism comes
from the virtual corrections where the kinematic dependence is much more
involved and hence these are now expressed in terms of scalar products of the
kind $\\{p_{j}\cdot q_{k}\\}$. The rest of the formalism relies on the
collinear factorization of the differential cross section, the renormalization
group invariance, universality of perturbative infrared structure of the
scattering amplitudes, and the process independence of the soft-collinear
distribution. Besides this, we also use an additional fact that the $N$-th
Mellin moment of the differential distribution has a relation with its
inclusive counterpart in the limit $N\rightarrow\infty$, as depicted through
(18). The mere use of this fact enables us to to get an all order relation
between the soft-collinear distribution of inclusive cross-section and that of
rapidity. Hence from the given quantity in inclusive part, we can determine it
for the rapidity and thereby avoid performing the explicit computation of the
real emission processes for rapidity distribution. The goal of this current
article is to present the general structure for the SV differential rapidity
distribution up to N4LO and also the resummed predictions till N3LL level in
QCD, which can be expressed in terms of universal anomalous dimensions along
with the process dependent virtual matrix elements. The former, which
comprises of process independent finite segments of soft-collinear
distribution and the mass factorized kernels, remains unaltered irrespective
of the number of colorless particles in the final states. Furthermore, the
soft-collinear distributions for the quark and gluon initiated processes are
found to be related to each other through simple quadratic Casimir scaling,
known as the maximally non-Abelian property.This is explicitly verified up to
N3LO. However, whether the validity will remain intact beyond this order with
generalized Casimir scaling, that will be an interesting thing to look into.

In summary, in order to obtain the fixed order as well as resummed prediction
for the differential rapidity distributions of a generic $n$-colorless final
states, one merely requires the form factor corresponding to the hard process
under study provided the soft-collinear distribution for Sudakov type process
is known. We present the analytical results for the fixed order up to
$\mathrm{N^{4}LO}$ and the resummed predictions up to $\mathrm{N^{3}LL}$ level
in the appendix for the scale choice of $\mu_{R}^{2}=\mu_{F}^{2}=q^{2}$ and
the same with the explicit scale dependence are provided in the ancillary
file.

## Acknowledgements

The algebraic computations have been done with the latest version of the
symbolic manipulation system Form Vermaseren:2000nd ; Ruijl:2017dtg . The work
of T.A. received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme, Novel
structures in scattering amplitudes (grant agreement No. 725110). We also
thank Goutam Das for useful discussions.

##  Appendix A Soft-collinear distribution for rapidity distribution

In this section, we present soft-collinear distribution ${\textbf{S}}_{d,I}$,
as defined in (2.1), in powers of $a_{s}(\mu_{R}^{2})$ up to N4LO. Expanding
the quantity in powers of $a_{s}$ as

$\displaystyle{\textbf{S}}_{d,I}(z_{1},z_{2},q^{2},\mu_{R}^{2},\mu_{F}^{2})=\delta(1-z_{1})\delta(1-z_{2})+\sum_{i=1}^{\infty}a_{s}^{i}(\mu_{R}^{2})\,{\textbf{S}}^{(i)}_{d,I}(z_{1},z_{2},q^{2},\mu_{R}^{2},\mu_{F}^{2})\,,$
(38)

we present the results for $\mu_{R}^{2}=\mu_{F}^{2}=q^{2}$. The result with
explicit scale dependence can be found from the ancillary file supplied with
the arXiv submission.

$\displaystyle\autobreak\par\par\textbf{S}_{d,I}^{(1)}={\cal
D}_{0}\overline{{\cal
D}}_{0}\bigg{\\{}\frac{1}{2}A^{I}_{1}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{1}\bigg{\\{}A^{I}_{1}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{0}\bigg{\\{}-f^{I}_{1}\bigg{\\}}\par+\delta(1-z_{1})\delta(1-z_{2})\bigg{\\{}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\bigg{\\}}+(z_{1}\leftrightarrow
z_{2})\,,\par$ $\displaystyle\autobreak\par\textbf{S}_{d,I}^{(2)}=\par{\cal
D}_{1}\overline{{\cal
D}}_{1}\bigg{\\{}\frac{3}{2}(A^{I}_{1})^{2}\bigg{\\}}\par+{\cal
D}_{0}\overline{{\cal
D}}_{2}\bigg{\\{}\frac{3}{2}(A^{I}_{1})^{2}\bigg{\\}}\par+{\cal
D}_{0}\overline{{\cal
D}}_{1}\bigg{\\{}-3A^{I}_{1}f^{I}_{1}-\beta_{0}A^{I}_{1}\bigg{\\}}\par+{\cal
D}_{0}\overline{{\cal
D}}_{0}\bigg{\\{}\frac{1}{2}(f^{I}_{1})^{2}+\frac{1}{2}A^{I}_{2}+A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-(A^{I}_{1})^{2}\zeta_{2}+\frac{1}{2}\beta_{0}f^{I}_{1}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{3}\bigg{\\{}\frac{1}{2}(A^{I}_{1})^{2}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{2}\bigg{\\{}-\frac{3}{2}A^{I}_{1}f^{I}_{1}-\frac{1}{2}\beta_{0}A^{I}_{1}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{1}\bigg{\\{}(f^{I}_{1})^{2}+A^{I}_{2}+2A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2(A^{I}_{1})^{2}\zeta_{2}+\beta_{0}f^{I}_{1}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{0}\bigg{\\{}-f^{I}_{2}-2f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+2A^{I}_{1}\zeta_{2}f^{I}_{1}+2(A^{I}_{1})^{2}\zeta_{3}-2\beta_{0}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\bigg{\\}}\par+\delta(1-z_{1})\delta(1-z_{2})\bigg{\\{}\frac{1}{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-\frac{1}{2}\zeta_{2}(f^{I}_{1})^{2}-A^{I}_{1}\zeta_{3}f^{I}_{1}+\frac{1}{5}(A^{I}_{1})^{2}\zeta_{2}^{2}+\beta_{0}\mathcal{\tilde{G}}^{I,(1)}_{d,2}\bigg{\\}}+(z_{1}\leftrightarrow
z_{2})\,,\par\par$
$\displaystyle\autobreak\par\textbf{S}_{d,I}^{(3)}=\par{\cal
D}_{2}\overline{{\cal
D}}_{2}\bigg{\\{}\frac{15}{8}(A^{I}_{1})^{3}\bigg{\\}}\par+{\cal
D}_{1}\overline{{\cal
D}}_{3}\bigg{\\{}\frac{5}{2}(A^{I}_{1})^{3}\bigg{\\}}\par+{\cal
D}_{1}\overline{{\cal
D}}_{2}\bigg{\\{}-\frac{15}{2}(A^{I}_{1})^{2}f^{I}_{1}-5\beta_{0}(A^{I}_{1})^{2}\bigg{\\}}\par+{\cal
D}_{1}\overline{{\cal
D}}_{1}\bigg{\\{}3A^{I}_{1}(f^{I}_{1})^{2}+3A^{I}_{1}A^{I}_{2}+3(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-6(A^{I}_{1})^{3}\zeta_{2}+5\beta_{0}A^{I}_{1}f^{I}_{1}+\beta_{0}^{2}A^{I}_{1}\bigg{\\}}\par+{\cal
D}_{0}\overline{{\cal
D}}_{4}\bigg{\\{}\frac{5}{8}(A^{I}_{1})^{3}\bigg{\\}}\par+{\cal
D}_{0}\overline{{\cal
D}}_{3}\bigg{\\{}-\frac{5}{2}(A^{I}_{1})^{2}f^{I}_{1}-\frac{5}{3}\beta_{0}(A^{I}_{1})^{2}\bigg{\\}}\par+{\cal
D}_{0}\overline{{\cal
D}}_{2}\bigg{\\{}3A^{I}_{1}(f^{I}_{1})^{2}+3A^{I}_{1}A^{I}_{2}+3(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-6(A^{I}_{1})^{3}\zeta_{2}+5\beta_{0}A^{I}_{1}f^{I}_{1}+\beta_{0}^{2}A^{I}_{1}\bigg{\\}}\par+{\cal
D}_{0}\overline{{\cal
D}}_{1}\bigg{\\{}-(f^{I}_{1})^{3}-3A^{I}_{2}f^{I}_{1}-3A^{I}_{1}f^{I}_{2}-6A^{I}_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+12(A^{I}_{1})^{2}\zeta_{2}f^{I}_{1}+10(A^{I}_{1})^{3}\zeta_{3}-\beta_{1}A^{I}_{1}-3\beta_{0}(f^{I}_{1})^{2}-2\beta_{0}A^{I}_{2}-8\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+6\beta_{0}(A^{I}_{1})^{2}\zeta_{2}-2\beta_{0}^{2}f^{I}_{1}\bigg{\\}}\par+{\cal
D}_{0}\overline{{\cal
D}}_{0}\bigg{\\{}f^{I}_{1}f^{I}_{2}+(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{1}{2}A^{I}_{3}+A^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{1}{2}A^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+A^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-\frac{5}{2}A^{I}_{1}\zeta_{2}(f^{I}_{1})^{2}-2A^{I}_{1}A^{I}_{2}\zeta_{2}-5(A^{I}_{1})^{2}\zeta_{3}f^{I}_{1}-2(A^{I}_{1})^{2}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+(A^{I}_{1})^{3}\zeta_{2}^{2}+\frac{1}{2}\beta_{1}f^{I}_{1}+\beta_{0}f^{I}_{2}+3\beta_{0}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-3\beta_{0}A^{I}_{1}\zeta_{2}f^{I}_{1}-3\beta_{0}(A^{I}_{1})^{2}\zeta_{3}+2\beta_{0}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{5}\bigg{\\{}\frac{1}{8}(A^{I}_{1})^{3}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{4}\bigg{\\{}-\frac{5}{8}(A^{I}_{1})^{2}f^{I}_{1}-\frac{5}{12}\beta_{0}(A^{I}_{1})^{2}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{3}\bigg{\\{}A^{I}_{1}(f^{I}_{1})^{2}+A^{I}_{1}A^{I}_{2}+(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2(A^{I}_{1})^{3}\zeta_{2}+\frac{5}{3}\beta_{0}A^{I}_{1}f^{I}_{1}+\frac{1}{3}\beta_{0}^{2}A^{I}_{1}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{2}\bigg{\\{}-\frac{1}{2}(f^{I}_{1})^{3}-\frac{3}{2}A^{I}_{2}f^{I}_{1}-\frac{3}{2}A^{I}_{1}f^{I}_{2}-3A^{I}_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+6(A^{I}_{1})^{2}\zeta_{2}f^{I}_{1}+5(A^{I}_{1})^{3}\zeta_{3}-\frac{1}{2}\beta_{1}A^{I}_{1}-\frac{3}{2}\beta_{0}(f^{I}_{1})^{2}-\beta_{0}A^{I}_{2}-4\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+3\beta_{0}(A^{I}_{1})^{2}\zeta_{2}-\beta_{0}^{2}f^{I}_{1}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{1}\bigg{\\{}2f^{I}_{1}f^{I}_{2}+2(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+A^{I}_{3}+2A^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+A^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+2A^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-5A^{I}_{1}\zeta_{2}(f^{I}_{1})^{2}-4A^{I}_{1}A^{I}_{2}\zeta_{2}-10(A^{I}_{1})^{2}\zeta_{3}f^{I}_{1}-4(A^{I}_{1})^{2}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+2(A^{I}_{1})^{3}\zeta_{2}^{2}+\beta_{1}f^{I}_{1}+2\beta_{0}f^{I}_{2}+6\beta_{0}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+2\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-6\beta_{0}A^{I}_{1}\zeta_{2}f^{I}_{1}-6\beta_{0}(A^{I}_{1})^{2}\zeta_{3}+4\beta_{0}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{0}\bigg{\\{}-f^{I}_{3}-2f^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-f^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-2f^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}+\zeta_{2}(f^{I}_{1})^{3}+2A^{I}_{2}\zeta_{2}f^{I}_{1}+4A^{I}_{1}\zeta_{3}(f^{I}_{1})^{2}+2A^{I}_{1}\zeta_{2}f^{I}_{2}+4A^{I}_{1}\zeta_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+4A^{I}_{1}A^{I}_{2}\zeta_{3}+4(A^{I}_{1})^{2}\zeta_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2(A^{I}_{1})^{2}\zeta_{2}^{2}f^{I}_{1}+6(A^{I}_{1})^{3}\zeta_{5}-8(A^{I}_{1})^{3}\zeta_{2}\zeta_{3}-2\beta_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2\beta_{0}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-4\beta_{0}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-2\beta_{0}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+2\beta_{0}\zeta_{2}(f^{I}_{1})^{2}+6\beta_{0}A^{I}_{1}\zeta_{3}f^{I}_{1}+4\beta_{0}A^{I}_{1}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-4\beta_{0}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}\bigg{\\}}\par+\delta(1-z_{1})\delta(1-z_{2})\bigg{\\{}\frac{1}{3}\mathcal{\tilde{G}}^{I,(3)}_{d,1}+\mathcal{\tilde{G}}^{I,(1)}_{d,1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+\frac{2}{3}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{3}-\frac{1}{3}\zeta_{3}(f^{I}_{1})^{3}-\zeta_{2}f^{I}_{1}f^{I}_{2}-\zeta_{2}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-A^{I}_{2}\zeta_{3}f^{I}_{1}-A^{I}_{1}\zeta_{3}f^{I}_{2}-2A^{I}_{1}\zeta_{3}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{2}{5}A^{I}_{1}\zeta_{2}^{2}(f^{I}_{1})^{2}+\frac{2}{5}A^{I}_{1}A^{I}_{2}\zeta_{2}^{2}-3(A^{I}_{1})^{2}\zeta_{5}f^{I}_{1}+4(A^{I}_{1})^{2}\zeta_{2}\zeta_{3}f^{I}_{1}+\frac{2}{5}(A^{I}_{1})^{2}\zeta_{2}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{5}{3}(A^{I}_{1})^{3}\zeta_{3}^{2}-\frac{4}{105}(A^{I}_{1})^{3}\zeta_{2}^{3}+\frac{2}{3}\beta_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+\frac{2}{3}\beta_{0}\mathcal{\tilde{G}}^{I,(2)}_{d,2}+2\beta_{0}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-\beta_{0}\zeta_{3}(f^{I}_{1})^{2}-2\beta_{0}\zeta_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2\beta_{0}A^{I}_{1}\zeta_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2\beta_{0}(A^{I}_{1})^{2}\zeta_{5}+2\beta_{0}(A^{I}_{1})^{2}\zeta_{2}\zeta_{3}+\frac{4}{3}\beta_{0}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,3}\bigg{\\}}+(z_{1}\leftrightarrow
z_{2})\,,\par$ $\displaystyle\autobreak\textbf{S}_{d,I}^{(4)}=\par{\cal
D}_{3}\overline{{\cal
D}}_{3}\bigg{\\{}\frac{35}{24}(A^{I}_{1})^{4}\bigg{\\}}\par+{\cal
D}_{2}\overline{{\cal
D}}_{4}\bigg{\\{}\frac{35}{16}(A^{I}_{1})^{4}\bigg{\\}}\par+{\cal
D}_{2}\overline{{\cal
D}}_{3}\bigg{\\{}-\frac{35}{4}(A^{I}_{1})^{3}f^{I}_{1}-\frac{35}{4}\beta_{0}(A^{I}_{1})^{3}\bigg{\\}}\par+{\cal
D}_{2}\overline{{\cal
D}}_{2}\bigg{\\{}\frac{45}{8}(A^{I}_{1})^{2}(f^{I}_{1})^{2}+\frac{45}{8}(A^{I}_{1})^{2}A^{I}_{2}+\frac{15}{4}(A^{I}_{1})^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-\frac{45}{4}(A^{I}_{1})^{4}\zeta_{2}+\frac{105}{8}\beta_{0}(A^{I}_{1})^{2}f^{I}_{1}+5\beta_{0}^{2}(A^{I}_{1})^{2}\bigg{\\}}\par+{\cal
D}_{1}\overline{{\cal
D}}_{5}\bigg{\\{}\frac{7}{8}(A^{I}_{1})^{4}\bigg{\\}}\par+{\cal
D}_{1}\overline{{\cal
D}}_{4}\bigg{\\{}-\frac{35}{8}(A^{I}_{1})^{3}f^{I}_{1}-\frac{35}{8}\beta_{0}(A^{I}_{1})^{3}\bigg{\\}}\par+{\cal
D}_{1}\overline{{\cal
D}}_{3}\bigg{\\{}\frac{15}{2}(A^{I}_{1})^{2}(f^{I}_{1})^{2}+\frac{15}{2}(A^{I}_{1})^{2}A^{I}_{2}+5(A^{I}_{1})^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-15(A^{I}_{1})^{4}\zeta_{2}+\frac{35}{2}\beta_{0}(A^{I}_{1})^{2}f^{I}_{1}+\frac{20}{3}\beta_{0}^{2}(A^{I}_{1})^{2}\bigg{\\}}\par+{\cal
D}_{1}\overline{{\cal
D}}_{2}\bigg{\\{}-5A^{I}_{1}(f^{I}_{1})^{3}-15A^{I}_{1}A^{I}_{2}f^{I}_{1}-\frac{15}{2}(A^{I}_{1})^{2}f^{I}_{2}-15(A^{I}_{1})^{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+45(A^{I}_{1})^{3}\zeta_{2}f^{I}_{1}+35(A^{I}_{1})^{4}\zeta_{3}-5\beta_{1}(A^{I}_{1})^{2}-20\beta_{0}A^{I}_{1}(f^{I}_{1})^{2}-15\beta_{0}A^{I}_{1}A^{I}_{2}-25\beta_{0}(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+40\beta_{0}(A^{I}_{1})^{3}\zeta_{2}-20\beta_{0}^{2}A^{I}_{1}f^{I}_{1}-3\beta_{0}^{3}A^{I}_{1}\bigg{\\}}\par+{\cal
D}_{1}\overline{{\cal
D}}_{1}\bigg{\\{}\frac{1}{2}(f^{I}_{1})^{4}+3A^{I}_{2}(f^{I}_{1})^{2}+\frac{3}{2}(A^{I}_{2})^{2}+6A^{I}_{1}f^{I}_{1}f^{I}_{2}+6A^{I}_{1}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+3A^{I}_{1}A^{I}_{3}+6A^{I}_{1}A^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{3}{2}(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+3(A^{I}_{1})^{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-\frac{39}{2}(A^{I}_{1})^{2}\zeta_{2}(f^{I}_{1})^{2}-18(A^{I}_{1})^{2}A^{I}_{2}\zeta_{2}-35(A^{I}_{1})^{3}\zeta_{3}f^{I}_{1}-12(A^{I}_{1})^{3}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+9(A^{I}_{1})^{4}\zeta_{2}^{2}+5\beta_{1}A^{I}_{1}f^{I}_{1}+3\beta_{0}(f^{I}_{1})^{3}+7\beta_{0}A^{I}_{2}f^{I}_{1}+8\beta_{0}A^{I}_{1}f^{I}_{2}+22\beta_{0}A^{I}_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+3\beta_{0}(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-40\beta_{0}(A^{I}_{1})^{2}\zeta_{2}f^{I}_{1}-34\beta_{0}(A^{I}_{1})^{3}\zeta_{3}+\frac{5}{2}\beta_{0}\beta_{1}A^{I}_{1}+\frac{11}{2}\beta_{0}^{2}(f^{I}_{1})^{2}+3\beta_{0}^{2}A^{I}_{2}+18\beta_{0}^{2}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-11\beta_{0}^{2}(A^{I}_{1})^{2}\zeta_{2}+3\beta_{0}^{3}f^{I}_{1}\bigg{\\}}\par+{\cal
D}_{0}\overline{{\cal
D}}_{6}\bigg{\\{}\frac{7}{48}(A^{I}_{1})^{4}\bigg{\\}}\par+{\cal
D}_{0}\overline{{\cal
D}}_{5}\bigg{\\{}-\frac{7}{8}(A^{I}_{1})^{3}f^{I}_{1}-\frac{7}{8}\beta_{0}(A^{I}_{1})^{3}\bigg{\\}}\par+{\cal
D}_{0}\overline{{\cal
D}}_{4}\bigg{\\{}\frac{15}{8}(A^{I}_{1})^{2}(f^{I}_{1})^{2}+\frac{15}{8}(A^{I}_{1})^{2}A^{I}_{2}+\frac{5}{4}(A^{I}_{1})^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-\frac{15}{4}(A^{I}_{1})^{4}\zeta_{2}+\frac{35}{8}\beta_{0}(A^{I}_{1})^{2}f^{I}_{1}+\frac{5}{3}\beta_{0}^{2}(A^{I}_{1})^{2}\bigg{\\}}\par+{\cal
D}_{0}\overline{{\cal
D}}_{3}\bigg{\\{}-\frac{5}{3}A^{I}_{1}(f^{I}_{1})^{3}-5A^{I}_{1}A^{I}_{2}f^{I}_{1}-\frac{5}{2}(A^{I}_{1})^{2}f^{I}_{2}-5(A^{I}_{1})^{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+15(A^{I}_{1})^{3}\zeta_{2}f^{I}_{1}+\frac{35}{3}(A^{I}_{1})^{4}\zeta_{3}-\frac{5}{3}\beta_{1}(A^{I}_{1})^{2}-\frac{20}{3}\beta_{0}A^{I}_{1}(f^{I}_{1})^{2}-5\beta_{0}A^{I}_{1}A^{I}_{2}-\frac{25}{3}\beta_{0}(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{40}{3}\beta_{0}(A^{I}_{1})^{3}\zeta_{2}-\frac{20}{3}\beta_{0}^{2}A^{I}_{1}f^{I}_{1}-\beta_{0}^{3}A^{I}_{1}\bigg{\\}}\par+{\cal
D}_{0}\overline{{\cal
D}}_{2}\bigg{\\{}\frac{1}{2}(f^{I}_{1})^{4}+3A^{I}_{2}(f^{I}_{1})^{2}+\frac{3}{2}(A^{I}_{2})^{2}+6A^{I}_{1}f^{I}_{1}f^{I}_{2}+6A^{I}_{1}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+3A^{I}_{1}A^{I}_{3}+6A^{I}_{1}A^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{3}{2}(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+3(A^{I}_{1})^{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-\frac{39}{2}(A^{I}_{1})^{2}\zeta_{2}(f^{I}_{1})^{2}-18(A^{I}_{1})^{2}A^{I}_{2}\zeta_{2}-35(A^{I}_{1})^{3}\zeta_{3}f^{I}_{1}-12(A^{I}_{1})^{3}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+9(A^{I}_{1})^{4}\zeta_{2}^{2}+5\beta_{1}A^{I}_{1}f^{I}_{1}+3\beta_{0}(f^{I}_{1})^{3}+7\beta_{0}A^{I}_{2}f^{I}_{1}+8\beta_{0}A^{I}_{1}f^{I}_{2}+22\beta_{0}A^{I}_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+3\beta_{0}(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-40\beta_{0}(A^{I}_{1})^{2}\zeta_{2}f^{I}_{1}-34\beta_{0}(A^{I}_{1})^{3}\zeta_{3}+\frac{5}{2}\beta_{0}\beta_{1}A^{I}_{1}+\frac{11}{2}\beta_{0}^{2}(f^{I}_{1})^{2}+3\beta_{0}^{2}A^{I}_{2}+18\beta_{0}^{2}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-11\beta_{0}^{2}(A^{I}_{1})^{2}\zeta_{2}+3\beta_{0}^{3}f^{I}_{1}\bigg{\\}}\par+{\cal
D}_{0}\overline{{\cal
D}}_{1}\bigg{\\{}-3(f^{I}_{1})^{2}f^{I}_{2}-2(f^{I}_{1})^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-3A^{I}_{3}f^{I}_{1}-3A^{I}_{2}f^{I}_{2}-6A^{I}_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-3A^{I}_{1}f^{I}_{3}-6A^{I}_{1}f^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-3A^{I}_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-6A^{I}_{1}f^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}+9A^{I}_{1}\zeta_{2}(f^{I}_{1})^{3}+24A^{I}_{1}A^{I}_{2}\zeta_{2}f^{I}_{1}+30(A^{I}_{1})^{2}\zeta_{3}(f^{I}_{1})^{2}+12(A^{I}_{1})^{2}\zeta_{2}f^{I}_{2}+24(A^{I}_{1})^{2}\zeta_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+30(A^{I}_{1})^{2}A^{I}_{2}\zeta_{3}+20(A^{I}_{1})^{3}\zeta_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-18(A^{I}_{1})^{3}\zeta_{2}^{2}f^{I}_{1}+42(A^{I}_{1})^{4}\zeta_{5}-60(A^{I}_{1})^{4}\zeta_{2}\zeta_{3}-\beta_{2}A^{I}_{1}-3\beta_{1}(f^{I}_{1})^{2}-2\beta_{1}A^{I}_{2}-8\beta_{1}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+6\beta_{1}(A^{I}_{1})^{2}\zeta_{2}-9\beta_{0}f^{I}_{1}f^{I}_{2}-12\beta_{0}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-3\beta_{0}A^{I}_{3}-10\beta_{0}A^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-7\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-14\beta_{0}A^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-6\beta_{0}A^{I}_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+31\beta_{0}A^{I}_{1}\zeta_{2}(f^{I}_{1})^{2}+18\beta_{0}A^{I}_{1}A^{I}_{2}\zeta_{2}+68\beta_{0}(A^{I}_{1})^{2}\zeta_{3}f^{I}_{1}+36\beta_{0}(A^{I}_{1})^{2}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-10\beta_{0}(A^{I}_{1})^{3}\zeta_{2}^{2}-5\beta_{0}\beta_{1}f^{I}_{1}-6\beta_{0}^{2}f^{I}_{2}-22\beta_{0}^{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-14\beta_{0}^{2}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+22\beta_{0}^{2}A^{I}_{1}\zeta_{2}f^{I}_{1}+22\beta_{0}^{2}(A^{I}_{1})^{2}\zeta_{3}-12\beta_{0}^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\bigg{\\}}\par+{\cal
D}_{0}\overline{{\cal
D}}_{0}\bigg{\\{}\frac{1}{2}(f^{I}_{2})^{2}+f^{I}_{1}f^{I}_{3}+2f^{I}_{1}f^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{1}{2}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+(f^{I}_{1})^{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-\frac{1}{2}\zeta_{2}(f^{I}_{1})^{4}+\frac{1}{2}A4+A^{I}_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{1}{2}A^{I}_{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+A^{I}_{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-\frac{5}{2}A^{I}_{2}\zeta_{2}(f^{I}_{1})^{2}-(A^{I}_{2})^{2}\zeta_{2}+\frac{1}{3}A^{I}_{1}\mathcal{\tilde{G}}^{I,(3)}_{d,1}+A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+\frac{2}{3}A^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{3}-\frac{10}{3}A^{I}_{1}\zeta_{3}(f^{I}_{1})^{3}-5A^{I}_{1}\zeta_{2}f^{I}_{1}f^{I}_{2}-5A^{I}_{1}\zeta_{2}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2A^{I}_{1}A^{I}_{3}\zeta_{2}-10A^{I}_{1}A^{I}_{2}\zeta_{3}f^{I}_{1}-4A^{I}_{1}A^{I}_{2}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-5(A^{I}_{1})^{2}\zeta_{3}f^{I}_{2}-10(A^{I}_{1})^{2}\zeta_{3}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-(A^{I}_{1})^{2}\zeta_{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-2(A^{I}_{1})^{2}\zeta_{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}+4(A^{I}_{1})^{2}\zeta_{2}^{2}(f^{I}_{1})^{2}+3(A^{I}_{1})^{2}A^{I}_{2}\zeta_{2}^{2}-21(A^{I}_{1})^{3}\zeta_{5}f^{I}_{1}+30(A^{I}_{1})^{3}\zeta_{2}\zeta_{3}f^{I}_{1}+2(A^{I}_{1})^{3}\zeta_{2}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{35}{3}(A^{I}_{1})^{4}\zeta_{3}^{2}-\frac{2}{3}(A^{I}_{1})^{4}\zeta_{2}^{3}+\frac{1}{2}\beta_{2}f^{I}_{1}+\beta_{1}f^{I}_{2}+3\beta_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{2}{3}\beta_{1}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-3\beta_{1}A^{I}_{1}\zeta_{2}f^{I}_{1}-3\beta_{1}(A^{I}_{1})^{2}\zeta_{3}+\frac{3}{2}\beta_{0}f^{I}_{3}+4\beta_{0}f^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{5}{2}\beta_{0}f^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+5\beta_{0}f^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}+\beta_{0}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-\frac{5}{2}\beta_{0}\zeta_{2}(f^{I}_{1})^{3}+\beta_{0}A^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-4\beta_{0}A^{I}_{2}\zeta_{2}f^{I}_{1}+\frac{2}{3}\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,2}+2\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-13\beta_{0}A^{I}_{1}\zeta_{3}(f^{I}_{1})^{2}-5\beta_{0}A^{I}_{1}\zeta_{2}f^{I}_{2}-16\beta_{0}A^{I}_{1}\zeta_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-9\beta_{0}A^{I}_{1}A^{I}_{2}\zeta_{3}-16\beta_{0}(A^{I}_{1})^{2}\zeta_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2\beta_{0}(A^{I}_{1})^{2}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+5\beta_{0}(A^{I}_{1})^{2}\zeta_{2}^{2}f^{I}_{1}-21\beta_{0}(A^{I}_{1})^{3}\zeta_{5}+26\beta_{0}(A^{I}_{1})^{3}\zeta_{2}\zeta_{3}+5\beta_{0}\beta_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+3\beta_{0}^{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+6\beta_{0}^{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}+5\beta_{0}^{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-3\beta_{0}^{2}\zeta_{2}(f^{I}_{1})^{2}+\frac{4}{3}\beta_{0}^{2}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,3}-11\beta_{0}^{2}A^{I}_{1}\zeta_{3}f^{I}_{1}-10\beta_{0}^{2}A^{I}_{1}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-\frac{4}{5}\beta_{0}^{2}(A^{I}_{1})^{2}\zeta_{2}^{2}+6\beta_{0}^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,2}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{7}\bigg{\\{}\frac{1}{48}(A^{I}_{1})^{4}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{6}\bigg{\\{}-\frac{7}{48}(A^{I}_{1})^{3}f^{I}_{1}-\frac{7}{48}\beta_{0}(A^{I}_{1})^{3}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{5}\bigg{\\{}\frac{3}{8}(A^{I}_{1})^{2}(f^{I}_{1})^{2}+\frac{3}{8}(A^{I}_{1})^{2}A^{I}_{2}+\frac{1}{4}(A^{I}_{1})^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-\frac{3}{4}(A^{I}_{1})^{4}\zeta_{2}+\frac{7}{8}\beta_{0}(A^{I}_{1})^{2}f^{I}_{1}+\frac{1}{3}\beta_{0}^{2}(A^{I}_{1})^{2}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{4}\bigg{\\{}-\frac{5}{12}A^{I}_{1}(f^{I}_{1})^{3}-\frac{5}{4}A^{I}_{1}A^{I}_{2}f^{I}_{1}-\frac{5}{8}(A^{I}_{1})^{2}f^{I}_{2}-\frac{5}{4}(A^{I}_{1})^{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{15}{4}(A^{I}_{1})^{3}\zeta_{2}f^{I}_{1}+\frac{35}{12}(A^{I}_{1})^{4}\zeta_{3}-\frac{5}{12}\beta_{1}(A^{I}_{1})^{2}-\frac{5}{3}\beta_{0}A^{I}_{1}(f^{I}_{1})^{2}-\frac{5}{4}\beta_{0}A^{I}_{1}A^{I}_{2}-\frac{25}{12}\beta_{0}(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{10}{3}\beta_{0}(A^{I}_{1})^{3}\zeta_{2}-\frac{5}{3}\beta_{0}^{2}A^{I}_{1}f^{I}_{1}-\frac{1}{4}\beta_{0}^{3}A^{I}_{1}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{3}\bigg{\\{}\frac{1}{6}(f^{I}_{1})^{4}+A^{I}_{2}(f^{I}_{1})^{2}+\frac{1}{2}(A^{I}_{2})^{2}+2A^{I}_{1}f^{I}_{1}f^{I}_{2}+2A^{I}_{1}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+A^{I}_{1}A^{I}_{3}+2A^{I}_{1}A^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{1}{2}(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+(A^{I}_{1})^{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-\frac{13}{2}(A^{I}_{1})^{2}\zeta_{2}(f^{I}_{1})^{2}-6(A^{I}_{1})^{2}A^{I}_{2}\zeta_{2}-\frac{35}{3}(A^{I}_{1})^{3}\zeta_{3}f^{I}_{1}-4(A^{I}_{1})^{3}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+3(A^{I}_{1})^{4}\zeta_{2}^{2}+\frac{5}{3}\beta_{1}A^{I}_{1}f^{I}_{1}+\beta_{0}(f^{I}_{1})^{3}+\frac{7}{3}\beta_{0}A^{I}_{2}f^{I}_{1}+\frac{8}{3}\beta_{0}A^{I}_{1}f^{I}_{2}+\frac{22}{3}\beta_{0}A^{I}_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\beta_{0}(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-\frac{40}{3}\beta_{0}(A^{I}_{1})^{2}\zeta_{2}f^{I}_{1}-\frac{34}{3}\beta_{0}(A^{I}_{1})^{3}\zeta_{3}+\frac{5}{6}\beta_{0}\beta_{1}A^{I}_{1}+\frac{11}{6}\beta_{0}^{2}(f^{I}_{1})^{2}+\beta_{0}^{2}A^{I}_{2}+6\beta_{0}^{2}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-\frac{11}{3}\beta_{0}^{2}(A^{I}_{1})^{2}\zeta_{2}+\beta_{0}^{3}f^{I}_{1}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{2}\bigg{\\{}-\frac{3}{2}(f^{I}_{1})^{2}f^{I}_{2}-(f^{I}_{1})^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-\frac{3}{2}A^{I}_{3}f^{I}_{1}-\frac{3}{2}A^{I}_{2}f^{I}_{2}-3A^{I}_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-\frac{3}{2}A^{I}_{1}f^{I}_{3}-3A^{I}_{1}f^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-\frac{3}{2}A^{I}_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-3A^{I}_{1}f^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}+\frac{9}{2}A^{I}_{1}\zeta_{2}(f^{I}_{1})^{3}+12A^{I}_{1}A^{I}_{2}\zeta_{2}f^{I}_{1}+15(A^{I}_{1})^{2}\zeta_{3}(f^{I}_{1})^{2}+6(A^{I}_{1})^{2}\zeta_{2}f^{I}_{2}+12(A^{I}_{1})^{2}\zeta_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+15(A^{I}_{1})^{2}A^{I}_{2}\zeta_{3}+10(A^{I}_{1})^{3}\zeta_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-9(A^{I}_{1})^{3}\zeta_{2}^{2}f^{I}_{1}+21(A^{I}_{1})^{4}\zeta_{5}-30(A^{I}_{1})^{4}\zeta_{2}\zeta_{3}-\frac{1}{2}\beta_{2}A^{I}_{1}-\frac{3}{2}\beta_{1}(f^{I}_{1})^{2}-\beta_{1}A^{I}_{2}-4\beta_{1}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+3\beta_{1}(A^{I}_{1})^{2}\zeta_{2}-\frac{9}{2}\beta_{0}f^{I}_{1}f^{I}_{2}-6\beta_{0}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-\frac{3}{2}\beta_{0}A^{I}_{3}-5\beta_{0}A^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-\frac{7}{2}\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-7\beta_{0}A^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-3\beta_{0}A^{I}_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+\frac{31}{2}\beta_{0}A^{I}_{1}\zeta_{2}(f^{I}_{1})^{2}+9\beta_{0}A^{I}_{1}A^{I}_{2}\zeta_{2}+34\beta_{0}(A^{I}_{1})^{2}\zeta_{3}f^{I}_{1}+18\beta_{0}(A^{I}_{1})^{2}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-5\beta_{0}(A^{I}_{1})^{3}\zeta_{2}^{2}-\frac{5}{2}\beta_{0}\beta_{1}f^{I}_{1}-3\beta_{0}^{2}f^{I}_{2}-11\beta_{0}^{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-7\beta_{0}^{2}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+11\beta_{0}^{2}A^{I}_{1}\zeta_{2}f^{I}_{1}+11\beta_{0}^{2}(A^{I}_{1})^{2}\zeta_{3}-6\beta_{0}^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{1}\bigg{\\{}(f^{I}_{2})^{2}+2f^{I}_{1}f^{I}_{3}+4f^{I}_{1}f^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+2(f^{I}_{1})^{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-\zeta_{2}(f^{I}_{1})^{4}+A4+2A^{I}_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+A^{I}_{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+2A^{I}_{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-5A^{I}_{2}\zeta_{2}(f^{I}_{1})^{2}-2(A^{I}_{2})^{2}\zeta_{2}+\frac{2}{3}A^{I}_{1}\mathcal{\tilde{G}}^{I,(3)}_{d,1}+2A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+\frac{4}{3}A^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{3}-\frac{20}{3}A^{I}_{1}\zeta_{3}(f^{I}_{1})^{3}-10A^{I}_{1}\zeta_{2}f^{I}_{1}f^{I}_{2}-10A^{I}_{1}\zeta_{2}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-4A^{I}_{1}A^{I}_{3}\zeta_{2}-20A^{I}_{1}A^{I}_{2}\zeta_{3}f^{I}_{1}-8A^{I}_{1}A^{I}_{2}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-10(A^{I}_{1})^{2}\zeta_{3}f^{I}_{2}-20(A^{I}_{1})^{2}\zeta_{3}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2(A^{I}_{1})^{2}\zeta_{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-4(A^{I}_{1})^{2}\zeta_{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}+8(A^{I}_{1})^{2}\zeta_{2}^{2}(f^{I}_{1})^{2}+6(A^{I}_{1})^{2}A^{I}_{2}\zeta_{2}^{2}-42(A^{I}_{1})^{3}\zeta_{5}f^{I}_{1}+60(A^{I}_{1})^{3}\zeta_{2}\zeta_{3}f^{I}_{1}+4(A^{I}_{1})^{3}\zeta_{2}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{70}{3}(A^{I}_{1})^{4}\zeta_{3}^{2}-\frac{4}{3}(A^{I}_{1})^{4}\zeta_{2}^{3}+\beta_{2}f^{I}_{1}+2\beta_{1}f^{I}_{2}+6\beta_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{4}{3}\beta_{1}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-6\beta_{1}A^{I}_{1}\zeta_{2}f^{I}_{1}-6\beta_{1}(A^{I}_{1})^{2}\zeta_{3}+3\beta_{0}f^{I}_{3}+8\beta_{0}f^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+5\beta_{0}f^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+10\beta_{0}f^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}+2\beta_{0}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-5\beta_{0}\zeta_{2}(f^{I}_{1})^{3}+2\beta_{0}A^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-8\beta_{0}A^{I}_{2}\zeta_{2}f^{I}_{1}+\frac{4}{3}\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,2}+4\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-26\beta_{0}A^{I}_{1}\zeta_{3}(f^{I}_{1})^{2}-10\beta_{0}A^{I}_{1}\zeta_{2}f^{I}_{2}-32\beta_{0}A^{I}_{1}\zeta_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-18\beta_{0}A^{I}_{1}A^{I}_{2}\zeta_{3}-32\beta_{0}(A^{I}_{1})^{2}\zeta_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-4\beta_{0}(A^{I}_{1})^{2}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+10\beta_{0}(A^{I}_{1})^{2}\zeta_{2}^{2}f^{I}_{1}-42\beta_{0}(A^{I}_{1})^{3}\zeta_{5}+52\beta_{0}(A^{I}_{1})^{3}\zeta_{2}\zeta_{3}+10\beta_{0}\beta_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+6\beta_{0}^{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+12\beta_{0}^{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}+10\beta_{0}^{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-6\beta_{0}^{2}\zeta_{2}(f^{I}_{1})^{2}+\frac{8}{3}\beta_{0}^{2}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,3}-22\beta_{0}^{2}A^{I}_{1}\zeta_{3}f^{I}_{1}-20\beta_{0}^{2}A^{I}_{1}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-\frac{8}{5}\beta_{0}^{2}(A^{I}_{1})^{2}\zeta_{2}^{2}+12\beta_{0}^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,2}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{0}\bigg{\\{}-f4-2f^{I}_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-f^{I}_{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-2f^{I}_{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-\frac{2}{3}f^{I}_{1}\mathcal{\tilde{G}}^{I,(3)}_{d,1}-2f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-\frac{4}{3}f^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{3}+\frac{2}{3}\zeta_{3}(f^{I}_{1})^{4}+3\zeta_{2}(f^{I}_{1})^{2}f^{I}_{2}+2\zeta_{2}(f^{I}_{1})^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+2A^{I}_{3}\zeta_{2}f^{I}_{1}+4A^{I}_{2}\zeta_{3}(f^{I}_{1})^{2}+2A^{I}_{2}\zeta_{2}f^{I}_{2}+4A^{I}_{2}\zeta_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+2(A^{I}_{2})^{2}\zeta_{3}+8A^{I}_{1}\zeta_{3}f^{I}_{1}f^{I}_{2}+8A^{I}_{1}\zeta_{3}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+2A^{I}_{1}\zeta_{2}f^{I}_{3}+4A^{I}_{1}\zeta_{2}f^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+2A^{I}_{1}\zeta_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+4A^{I}_{1}\zeta_{2}f^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-2A^{I}_{1}\zeta_{2}^{2}(f^{I}_{1})^{3}+4A^{I}_{1}A^{I}_{3}\zeta_{3}+8A^{I}_{1}A^{I}_{2}\zeta_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-4A^{I}_{1}A^{I}_{2}\zeta_{2}^{2}f^{I}_{1}+18(A^{I}_{1})^{2}\zeta_{5}(f^{I}_{1})^{2}+2(A^{I}_{1})^{2}\zeta_{3}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+4(A^{I}_{1})^{2}\zeta_{3}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-26(A^{I}_{1})^{2}\zeta_{2}\zeta_{3}(f^{I}_{1})^{2}-2(A^{I}_{1})^{2}\zeta_{2}^{2}f^{I}_{2}-4(A^{I}_{1})^{2}\zeta_{2}^{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+18(A^{I}_{1})^{2}A^{I}_{2}\zeta_{5}-24(A^{I}_{1})^{2}A^{I}_{2}\zeta_{2}\zeta_{3}+12(A^{I}_{1})^{3}\zeta_{5}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-\frac{70}{3}(A^{I}_{1})^{3}\zeta_{3}^{2}f^{I}_{1}-16(A^{I}_{1})^{3}\zeta_{2}\zeta_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{4}{3}(A^{I}_{1})^{3}\zeta_{2}^{3}f^{I}_{1}+30(A^{I}_{1})^{4}\zeta_{7}-36(A^{I}_{1})^{4}\zeta_{2}\zeta_{5}+12(A^{I}_{1})^{4}\zeta_{2}^{2}\zeta_{3}-2\beta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2\beta_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-4\beta_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-\frac{4}{3}\beta_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+2\beta_{1}\zeta_{2}(f^{I}_{1})^{2}+6\beta_{1}A^{I}_{1}\zeta_{3}f^{I}_{1}+4\beta_{1}A^{I}_{1}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2\beta_{0}\mathcal{\tilde{G}}^{I,(3)}_{d,1}-6\beta_{0}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-4\beta_{0}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{3}-2\beta_{0}f^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-\frac{4}{3}\beta_{0}f^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,2}-4\beta_{0}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+4\beta_{0}\zeta_{3}(f^{I}_{1})^{3}+6\beta_{0}\zeta_{2}f^{I}_{1}f^{I}_{2}+10\beta_{0}\zeta_{2}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+8\beta_{0}A^{I}_{2}\zeta_{3}f^{I}_{1}+4\beta_{0}A^{I}_{2}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+10\beta_{0}A^{I}_{1}\zeta_{3}f^{I}_{2}+28\beta_{0}A^{I}_{1}\zeta_{3}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+4\beta_{0}A^{I}_{1}\zeta_{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+8\beta_{0}A^{I}_{1}\zeta_{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}+4\beta_{0}A^{I}_{1}\zeta_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-4\beta_{0}A^{I}_{1}\zeta_{2}^{2}(f^{I}_{1})^{2}+42\beta_{0}(A^{I}_{1})^{2}\zeta_{5}f^{I}_{1}+4\beta_{0}(A^{I}_{1})^{2}\zeta_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-52\beta_{0}(A^{I}_{1})^{2}\zeta_{2}\zeta_{3}f^{I}_{1}-4\beta_{0}(A^{I}_{1})^{2}\zeta_{2}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-22\beta_{0}(A^{I}_{1})^{3}\zeta_{3}^{2}-8\beta_{0}\beta_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-4\beta_{0}^{2}\mathcal{\tilde{G}}^{I,(2)}_{d,2}-12\beta_{0}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-\frac{8}{3}\beta_{0}^{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,3}+6\beta_{0}^{2}\zeta_{3}(f^{I}_{1})^{2}+12\beta_{0}^{2}\zeta_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+20\beta_{0}^{2}A^{I}_{1}\zeta_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+8\beta_{0}^{2}A^{I}_{1}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+\frac{8}{5}\beta_{0}^{2}A^{I}_{1}\zeta_{2}^{2}f^{I}_{1}+16\beta_{0}^{2}(A^{I}_{1})^{2}\zeta_{5}-12\beta_{0}^{2}(A^{I}_{1})^{2}\zeta_{2}\zeta_{3}-8\beta_{0}^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,3}\bigg{\\}}\par+\delta(1-z_{1})\delta(1-z_{2})\bigg{\\{}\frac{1}{4}\mathcal{\tilde{G}}^{I,(4)}_{d,1}+\frac{1}{4}(\mathcal{\tilde{G}}^{I,(2)}_{d,1})^{2}+\frac{2}{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\mathcal{\tilde{G}}^{I,(3)}_{d,1}+(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+\frac{1}{3}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{4}-\zeta_{3}(f^{I}_{1})^{2}f^{I}_{2}-\frac{2}{3}\zeta_{3}(f^{I}_{1})^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-\frac{1}{2}\zeta_{2}(f^{I}_{2})^{2}-\zeta_{2}f^{I}_{1}f^{I}_{3}-2\zeta_{2}f^{I}_{1}f^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-\frac{1}{2}\zeta_{2}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-\zeta_{2}(f^{I}_{1})^{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}+\frac{3}{20}\zeta_{2}^{2}(f^{I}_{1})^{4}-A^{I}_{3}\zeta_{3}f^{I}_{1}-A^{I}_{2}\zeta_{3}f^{I}_{2}-2A^{I}_{2}\zeta_{3}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{2}{5}A^{I}_{2}\zeta_{2}^{2}(f^{I}_{1})^{2}+\frac{1}{5}(A^{I}_{2})^{2}\zeta_{2}^{2}-2A^{I}_{1}\zeta_{5}(f^{I}_{1})^{3}-A^{I}_{1}\zeta_{3}f^{I}_{3}-2A^{I}_{1}\zeta_{3}f^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-A^{I}_{1}\zeta_{3}f^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-2A^{I}_{1}\zeta_{3}f^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}+3A^{I}_{1}\zeta_{2}\zeta_{3}(f^{I}_{1})^{3}+\frac{4}{5}A^{I}_{1}\zeta_{2}^{2}f^{I}_{1}f^{I}_{2}+\frac{4}{5}A^{I}_{1}\zeta_{2}^{2}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{2}{5}A^{I}_{1}A^{I}_{3}\zeta_{2}^{2}-6A^{I}_{1}A^{I}_{2}\zeta_{5}f^{I}_{1}+8A^{I}_{1}A^{I}_{2}\zeta_{2}\zeta_{3}f^{I}_{1}+\frac{4}{5}A^{I}_{1}A^{I}_{2}\zeta_{2}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-3(A^{I}_{1})^{2}\zeta_{5}f^{I}_{2}-6(A^{I}_{1})^{2}\zeta_{5}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+5(A^{I}_{1})^{2}\zeta_{3}^{2}(f^{I}_{1})^{2}+4(A^{I}_{1})^{2}\zeta_{2}\zeta_{3}f^{I}_{2}+8(A^{I}_{1})^{2}\zeta_{2}\zeta_{3}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{1}{5}(A^{I}_{1})^{2}\zeta_{2}^{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+\frac{2}{5}(A^{I}_{1})^{2}\zeta_{2}^{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-\frac{11}{35}(A^{I}_{1})^{2}\zeta_{2}^{3}(f^{I}_{1})^{2}+5(A^{I}_{1})^{2}A^{I}_{2}\zeta_{3}^{2}-\frac{4}{35}(A^{I}_{1})^{2}A^{I}_{2}\zeta_{2}^{3}-15(A^{I}_{1})^{3}\zeta_{7}f^{I}_{1}+\frac{10}{3}(A^{I}_{1})^{3}\zeta_{3}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+18(A^{I}_{1})^{3}\zeta_{2}\zeta_{5}f^{I}_{1}-6(A^{I}_{1})^{3}\zeta_{2}^{2}\zeta_{3}f^{I}_{1}-\frac{8}{105}(A^{I}_{1})^{3}\zeta_{2}^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+14(A^{I}_{1})^{4}\zeta_{3}\zeta_{5}-10(A^{I}_{1})^{4}\zeta_{2}\zeta_{3}^{2}+\frac{1}{35}(A^{I}_{1})^{4}\zeta_{2}^{4}+\frac{1}{2}\beta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+\frac{1}{2}\beta_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,2}+\frac{4}{3}\beta_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-\beta_{1}\zeta_{3}(f^{I}_{1})^{2}-2\beta_{1}\zeta_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2\beta_{1}A^{I}_{1}\zeta_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2\beta_{1}(A^{I}_{1})^{2}\zeta_{5}+2\beta_{1}(A^{I}_{1})^{2}\zeta_{2}\zeta_{3}+\frac{1}{2}\beta_{0}\mathcal{\tilde{G}}^{I,(3)}_{d,2}+\beta_{0}\mathcal{\tilde{G}}^{I,(1)}_{d,2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+\frac{4}{3}\beta_{0}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\mathcal{\tilde{G}}^{I,(2)}_{d,2}+2\beta_{0}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-3\beta_{0}\zeta_{3}f^{I}_{1}f^{I}_{2}-4\beta_{0}\zeta_{3}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2\beta_{0}\zeta_{2}f^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2\beta_{0}\zeta_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-4\beta_{0}\zeta_{2}f^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-\beta_{0}\zeta_{2}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+\frac{2}{5}\beta_{0}\zeta_{2}^{2}(f^{I}_{1})^{3}-2\beta_{0}A^{I}_{2}\zeta_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-\frac{2}{5}\beta_{0}A^{I}_{2}\zeta_{2}^{2}f^{I}_{1}-8\beta_{0}A^{I}_{1}\zeta_{5}(f^{I}_{1})^{2}-2\beta_{0}A^{I}_{1}\zeta_{3}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-4\beta_{0}A^{I}_{1}\zeta_{3}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-2\beta_{0}A^{I}_{1}\zeta_{3}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+10\beta_{0}A^{I}_{1}\zeta_{2}\zeta_{3}(f^{I}_{1})^{2}+\frac{2}{5}\beta_{0}A^{I}_{1}\zeta_{2}^{2}f^{I}_{2}+\frac{8}{5}\beta_{0}A^{I}_{1}\zeta_{2}^{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-6\beta_{0}A^{I}_{1}A^{I}_{2}\zeta_{5}+6\beta_{0}A^{I}_{1}A^{I}_{2}\zeta_{2}\zeta_{3}-10\beta_{0}(A^{I}_{1})^{2}\zeta_{5}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+11\beta_{0}(A^{I}_{1})^{2}\zeta_{3}^{2}f^{I}_{1}+12\beta_{0}(A^{I}_{1})^{2}\zeta_{2}\zeta_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{2}{5}\beta_{0}(A^{I}_{1})^{2}\zeta_{2}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-15\beta_{0}(A^{I}_{1})^{3}\zeta_{7}+16\beta_{0}(A^{I}_{1})^{3}\zeta_{2}\zeta_{5}-\frac{16}{5}\beta_{0}(A^{I}_{1})^{3}\zeta_{2}^{2}\zeta_{3}+2\beta_{0}\beta_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,3}+\beta_{0}^{2}\mathcal{\tilde{G}}^{I,(2)}_{d,3}+\beta_{0}^{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,2})^{2}+\frac{8}{3}\beta_{0}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\mathcal{\tilde{G}}^{I,(1)}_{d,3}-6\beta_{0}^{2}\zeta_{3}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2\beta_{0}^{2}\zeta_{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-4\beta_{0}^{2}\zeta_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-\frac{3}{5}\beta_{0}^{2}\zeta_{2}^{2}(f^{I}_{1})^{2}-8\beta_{0}^{2}A^{I}_{1}\zeta_{5}f^{I}_{1}-4\beta_{0}^{2}A^{I}_{1}\zeta_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+6\beta_{0}^{2}A^{I}_{1}\zeta_{2}\zeta_{3}f^{I}_{1}+\frac{4}{5}\beta_{0}^{2}A^{I}_{1}\zeta_{2}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+3\beta_{0}^{2}(A^{I}_{1})^{2}\zeta_{3}^{2}+\frac{12}{35}\beta_{0}^{2}(A^{I}_{1})^{2}\zeta_{2}^{3}+2\beta_{0}^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,4}\bigg{\\}}+(z_{1}\leftrightarrow
z_{2}).\par$

The symbols ${\tilde{\cal G}}^{,I,j}_{d,k}$ that appear in the aforementioned
soft-collinear distributions are provided explicitly in the ancillary files
with the arXiv submission.

##  Appendix B Soft-virtual partonic rapidity distribution

We expand the $\Delta^{\rm sv}_{d,I}$, as defined in (24), in powers of
$a_{s}(\mu_{R}^{2})$ through

$\displaystyle\Delta^{\rm sv}_{d,I}\left(\\{p_{j}\cdot
q_{k}\\},z_{1},z_{2},q^{2},\mu_{F}^{2}\right)$
$\displaystyle=\delta(1-z_{1})\delta(1-z_{2})|{\cal M}^{(0)}_{I,{\rm
fin}}|^{2}$
$\displaystyle+\sum_{i=1}^{\infty}a_{s}^{i}(\mu_{R}^{2})\Delta^{{\rm
sv},(i)}_{d,I}\left(\\{p_{j}\cdot
q_{k}\\},z_{1},z_{2},q^{2},\mu_{F}^{2},\mu_{R}^{2}\right).$ (40)

Here, we present $\Delta^{\rm sv}_{d,I}$ to fourth order for the specific
scale choice $\mu_{F}^{2}=\mu_{R}^{2}=q^{2}$. The results with explicit
dependence on $\mu_{R}$ and $\mu_{F}$ are provided in the ancillary files
supplied with the arXiv submission.

$\displaystyle\autobreak\par\par{\Delta}^{\text{sv},(1)}_{d,I}=\par|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{0}\overline{{\cal
D}}_{0}\bigg{\\{}\frac{1}{2}A^{I}_{1}\bigg{\\}}\par+\delta(1-z_{1})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\overline{{\cal
D}}_{1}\bigg{\\{}A^{I}_{1}\bigg{\\}}\par+\delta(1-z_{1})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\overline{{\cal
D}}_{0}\bigg{\\{}-f^{I}_{1}\bigg{\\}}\par+\delta(1-z_{1})\delta(1-z_{2})\bigg{\\{}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+\delta(1-z_{1})\delta(1-z_{2})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\bigg{\\{}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\bigg{\\}}+(z_{1}\leftrightarrow
z_{2})\,,\par$
$\displaystyle\autobreak\par{\Delta}^{\text{sv},(2)}_{d,I}=\par\par+{\cal
D}_{0}\overline{{\cal D}}_{0}\bigg{\\{}A^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{1}\overline{{\cal
D}}_{1}\bigg{\\{}\frac{3}{2}(A^{I}_{1})^{2}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{0}\overline{{\cal
D}}_{2}\bigg{\\{}\frac{3}{2}(A^{I}_{1})^{2}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{0}\overline{{\cal
D}}_{1}\bigg{\\{}-3A^{I}_{1}f^{I}_{1}-\beta_{0}A^{I}_{1}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{0}\overline{{\cal
D}}_{0}\bigg{\\{}\frac{1}{2}(f^{I}_{1})^{2}+\frac{1}{2}A^{I}_{2}+A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-(A^{I}_{1})^{2}\zeta_{2}+\frac{1}{2}\beta_{0}f^{I}_{1}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{1}\bigg{\\{}2A^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{0}\bigg{\\{}-2f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+\delta(1-z_{1})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\overline{{\cal
D}}_{3}\bigg{\\{}\frac{1}{2}(A^{I}_{1})^{2}\bigg{\\}}\par+\delta(1-z_{1})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\overline{{\cal
D}}_{2}\bigg{\\{}-\frac{3}{2}A^{I}_{1}f^{I}_{1}-\frac{1}{2}\beta_{0}A^{I}_{1}\bigg{\\}}\par+\delta(1-z_{1})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\overline{{\cal
D}}_{1}\bigg{\\{}(f^{I}_{1})^{2}+A^{I}_{2}+2A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2(A^{I}_{1})^{2}\zeta_{2}+\beta_{0}f^{I}_{1}\bigg{\\}}\par+\delta(1-z_{1})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\overline{{\cal
D}}_{0}\bigg{\\{}-f^{I}_{2}-2f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+2A^{I}_{1}\zeta_{2}f^{I}_{1}+2(A^{I}_{1})^{2}\zeta_{3}-2\beta_{0}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\bigg{\\}}\par+\delta(1-z_{1})\delta(1-z_{2})\bigg{\\{}\frac{1}{2}{\cal
M}^{(1,1)}_{I,\text{fin}}+{\cal
M}^{(0,2)}_{I,\text{fin}}+2\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+\delta(1-z_{1})\delta(1-z_{2})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\bigg{\\{}\frac{1}{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-\frac{1}{2}\zeta_{2}(f^{I}_{1})^{2}-A^{I}_{1}\zeta_{3}f^{I}_{1}+\frac{1}{5}(A^{I}_{1})^{2}\zeta_{2}^{2}+\beta_{0}\mathcal{\tilde{G}}^{I,(1)}_{d,2}\bigg{\\}}+(z_{1}\leftrightarrow
z_{2})\,,\par$
$\displaystyle\autobreak{\Delta}^{\text{sv},(3)}_{d,I}=\par\par+{\cal
D}_{1}\overline{{\cal D}}_{1}\bigg{\\{}3(A^{I}_{1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+{\cal D}_{0}\overline{{\cal
D}}_{2}\bigg{\\{}3(A^{I}_{1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+{\cal D}_{0}\overline{{\cal
D}}_{1}\bigg{\\{}-6A^{I}_{1}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}-2\beta_{0}A^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+{\cal D}_{0}\overline{{\cal
D}}_{0}\bigg{\\{}(f^{I}_{1})^{2}{\cal M}^{(0,1)}_{I,\text{fin}}+A^{I}_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+\frac{1}{2}A^{I}_{1}{\cal
M}^{(1,1)}_{I,\text{fin}}+A^{I}_{1}{\cal
M}^{(0,2)}_{I,\text{fin}}+2A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}-2(A^{I}_{1})^{2}\zeta_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+\beta_{0}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{2}\overline{{\cal
D}}_{2}\bigg{\\{}\frac{15}{8}(A^{I}_{1})^{3}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{1}\overline{{\cal
D}}_{3}\bigg{\\{}\frac{5}{2}(A^{I}_{1})^{3}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{1}\overline{{\cal
D}}_{2}\bigg{\\{}-\frac{15}{2}(A^{I}_{1})^{2}f^{I}_{1}-5\beta_{0}(A^{I}_{1})^{2}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{1}\overline{{\cal
D}}_{1}\bigg{\\{}3A^{I}_{1}(f^{I}_{1})^{2}+3A^{I}_{1}A^{I}_{2}+3(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-6(A^{I}_{1})^{3}\zeta_{2}+5\beta_{0}A^{I}_{1}f^{I}_{1}+\beta_{0}^{2}A^{I}_{1}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{0}\overline{{\cal
D}}_{4}\bigg{\\{}\frac{5}{8}(A^{I}_{1})^{3}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{0}\overline{{\cal
D}}_{3}\bigg{\\{}-\frac{5}{2}(A^{I}_{1})^{2}f^{I}_{1}-\frac{5}{3}\beta_{0}(A^{I}_{1})^{2}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{0}\overline{{\cal
D}}_{2}\bigg{\\{}3A^{I}_{1}(f^{I}_{1})^{2}+3A^{I}_{1}A^{I}_{2}+3(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-6(A^{I}_{1})^{3}\zeta_{2}+5\beta_{0}A^{I}_{1}f^{I}_{1}+\beta_{0}^{2}A^{I}_{1}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{0}\overline{{\cal
D}}_{1}\bigg{\\{}-(f^{I}_{1})^{3}-3A^{I}_{2}f^{I}_{1}-3A^{I}_{1}f^{I}_{2}-6A^{I}_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+12(A^{I}_{1})^{2}\zeta_{2}f^{I}_{1}+10(A^{I}_{1})^{3}\zeta_{3}-\beta_{1}A^{I}_{1}-3\beta_{0}(f^{I}_{1})^{2}-2\beta_{0}A^{I}_{2}-8\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+6\beta_{0}(A^{I}_{1})^{2}\zeta_{2}-2\beta_{0}^{2}f^{I}_{1}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{0}\overline{{\cal
D}}_{0}\bigg{\\{}f^{I}_{1}f^{I}_{2}+(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{1}{2}A^{I}_{3}+A^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{1}{2}A^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+A^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-\frac{5}{2}A^{I}_{1}\zeta_{2}(f^{I}_{1})^{2}-2A^{I}_{1}A^{I}_{2}\zeta_{2}-5(A^{I}_{1})^{2}\zeta_{3}f^{I}_{1}-2(A^{I}_{1})^{2}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+(A^{I}_{1})^{3}\zeta_{2}^{2}+\frac{1}{2}\beta_{1}f^{I}_{1}+\beta_{0}f^{I}_{2}+3\beta_{0}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-3\beta_{0}A^{I}_{1}\zeta_{2}f^{I}_{1}-3\beta_{0}(A^{I}_{1})^{2}\zeta_{3}+2\beta_{0}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{3}\bigg{\\{}(A^{I}_{1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{2}\bigg{\\{}-3A^{I}_{1}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}-\beta_{0}A^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{1}\bigg{\\{}2(f^{I}_{1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+2A^{I}_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+A^{I}_{1}{\cal
M}^{(1,1)}_{I,\text{fin}}+2A^{I}_{1}{\cal
M}^{(0,2)}_{I,\text{fin}}+4A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}-4(A^{I}_{1})^{2}\zeta_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+2\beta_{0}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{0}\bigg{\\{}-2f^{I}_{2}{\cal M}^{(0,1)}_{I,\text{fin}}-f^{I}_{1}{\cal
M}^{(1,1)}_{I,\text{fin}}-2f^{I}_{1}{\cal
M}^{(0,2)}_{I,\text{fin}}-4f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}+4A^{I}_{1}\zeta_{2}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}+4(A^{I}_{1})^{2}\zeta_{3}{\cal
M}^{(0,1)}_{I,\text{fin}}-4\beta_{0}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+\delta(1-z_{1})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\overline{{\cal
D}}_{5}\bigg{\\{}\frac{1}{8}(A^{I}_{1})^{3}\bigg{\\}}\par+\delta(1-z_{1})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\overline{{\cal
D}}_{4}\bigg{\\{}-\frac{5}{8}(A^{I}_{1})^{2}f^{I}_{1}-\frac{5}{12}\beta_{0}(A^{I}_{1})^{2}\bigg{\\}}\par+\delta(1-z_{1})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\overline{{\cal
D}}_{3}\bigg{\\{}A^{I}_{1}(f^{I}_{1})^{2}+A^{I}_{1}A^{I}_{2}+(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2(A^{I}_{1})^{3}\zeta_{2}+\frac{5}{3}\beta_{0}A^{I}_{1}f^{I}_{1}+\frac{1}{3}\beta_{0}^{2}A^{I}_{1}\bigg{\\}}\par+\delta(1-z_{1})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\overline{{\cal
D}}_{2}\bigg{\\{}-\frac{1}{2}(f^{I}_{1})^{3}-\frac{3}{2}A^{I}_{2}f^{I}_{1}-\frac{3}{2}A^{I}_{1}f^{I}_{2}-3A^{I}_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+6(A^{I}_{1})^{2}\zeta_{2}f^{I}_{1}+5(A^{I}_{1})^{3}\zeta_{3}-\frac{1}{2}\beta_{1}A^{I}_{1}-\frac{3}{2}\beta_{0}(f^{I}_{1})^{2}-\beta_{0}A^{I}_{2}-4\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+3\beta_{0}(A^{I}_{1})^{2}\zeta_{2}-\beta_{0}^{2}f^{I}_{1}\bigg{\\}}\par+\delta(1-z_{1})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\overline{{\cal
D}}_{1}\bigg{\\{}2f^{I}_{1}f^{I}_{2}+2(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+A^{I}_{3}+2A^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+A^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+2A^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-5A^{I}_{1}\zeta_{2}(f^{I}_{1})^{2}-4A^{I}_{1}A^{I}_{2}\zeta_{2}-10(A^{I}_{1})^{2}\zeta_{3}f^{I}_{1}-4(A^{I}_{1})^{2}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+2(A^{I}_{1})^{3}\zeta_{2}^{2}+\beta_{1}f^{I}_{1}+2\beta_{0}f^{I}_{2}+6\beta_{0}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+2\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-6\beta_{0}A^{I}_{1}\zeta_{2}f^{I}_{1}-6\beta_{0}(A^{I}_{1})^{2}\zeta_{3}+4\beta_{0}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\bigg{\\}}\par+\delta(1-z_{1})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\overline{{\cal
D}}_{0}\bigg{\\{}-f^{I}_{3}-2f^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-f^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-2f^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}+\zeta_{2}(f^{I}_{1})^{3}+2A^{I}_{2}\zeta_{2}f^{I}_{1}+4A^{I}_{1}\zeta_{3}(f^{I}_{1})^{2}+2A^{I}_{1}\zeta_{2}f^{I}_{2}+4A^{I}_{1}\zeta_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+4A^{I}_{1}A^{I}_{2}\zeta_{3}+4(A^{I}_{1})^{2}\zeta_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2(A^{I}_{1})^{2}\zeta_{2}^{2}f^{I}_{1}+6(A^{I}_{1})^{3}\zeta_{5}-8(A^{I}_{1})^{3}\zeta_{2}\zeta_{3}-2\beta_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2\beta_{0}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-4\beta_{0}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-2\beta_{0}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+2\beta_{0}\zeta_{2}(f^{I}_{1})^{2}+6\beta_{0}A^{I}_{1}\zeta_{3}f^{I}_{1}+4\beta_{0}A^{I}_{1}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-4\beta_{0}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}\bigg{\\}}\par+\delta(1-z_{1})\delta(1-z_{2})\bigg{\\{}{\cal
M}^{(1,2)}_{I,\text{fin}}+{\cal
M}^{(0,3)}_{I,\text{fin}}+\mathcal{\tilde{G}}^{I,(2)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}+\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(1,1)}_{I,\text{fin}}+2\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,2)}_{I,\text{fin}}+2(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}-\zeta_{2}(f^{I}_{1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}-2A^{I}_{1}\zeta_{3}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}+\frac{2}{5}(A^{I}_{1})^{2}\zeta_{2}^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+2\beta_{0}\mathcal{\tilde{G}}^{I,(1)}_{d,2}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+\delta(1-z_{1})\delta(1-z_{2})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\bigg{\\{}\frac{1}{3}\mathcal{\tilde{G}}^{I,(3)}_{d,1}+\mathcal{\tilde{G}}^{I,(1)}_{d,1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+\frac{2}{3}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{3}-\frac{1}{3}\zeta_{3}(f^{I}_{1})^{3}-\zeta_{2}f^{I}_{1}f^{I}_{2}-\zeta_{2}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-A^{I}_{2}\zeta_{3}f^{I}_{1}-A^{I}_{1}\zeta_{3}f^{I}_{2}-2A^{I}_{1}\zeta_{3}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{2}{5}A^{I}_{1}\zeta_{2}^{2}(f^{I}_{1})^{2}+\frac{2}{5}A^{I}_{1}A^{I}_{2}\zeta_{2}^{2}-3(A^{I}_{1})^{2}\zeta_{5}f^{I}_{1}+4(A^{I}_{1})^{2}\zeta_{2}\zeta_{3}f^{I}_{1}+\frac{2}{5}(A^{I}_{1})^{2}\zeta_{2}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{5}{3}(A^{I}_{1})^{3}\zeta_{3}^{2}-\frac{4}{105}(A^{I}_{1})^{3}\zeta_{2}^{3}+\frac{2}{3}\beta_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+\frac{2}{3}\beta_{0}\mathcal{\tilde{G}}^{I,(2)}_{d,2}+2\beta_{0}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-\beta_{0}\zeta_{3}(f^{I}_{1})^{2}-2\beta_{0}\zeta_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2\beta_{0}A^{I}_{1}\zeta_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2\beta_{0}(A^{I}_{1})^{2}\zeta_{5}+2\beta_{0}(A^{I}_{1})^{2}\zeta_{2}\zeta_{3}+\frac{4}{3}\beta_{0}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,3}\bigg{\\}}+(z_{1}\leftrightarrow
z_{2})\,,\par\par$
$\displaystyle\autobreak{\Delta}^{\text{sv},(4)}_{d,I}=\par\par+{\cal
D}_{2}\overline{{\cal D}}_{2}\bigg{\\{}\frac{15}{4}(A^{I}_{1})^{3}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+{\cal D}_{1}\overline{{\cal
D}}_{3}\bigg{\\{}5(A^{I}_{1})^{3}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+{\cal D}_{1}\overline{{\cal
D}}_{2}\bigg{\\{}-15(A^{I}_{1})^{2}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}-10\beta_{0}(A^{I}_{1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+{\cal D}_{1}\overline{{\cal
D}}_{1}\bigg{\\{}6A^{I}_{1}(f^{I}_{1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+6A^{I}_{1}A^{I}_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+\frac{3}{2}(A^{I}_{1})^{2}{\cal
M}^{(1,1)}_{I,\text{fin}}+3(A^{I}_{1})^{2}{\cal
M}^{(0,2)}_{I,\text{fin}}+6(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}-12(A^{I}_{1})^{3}\zeta_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+10\beta_{0}A^{I}_{1}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}+2\beta_{0}^{2}A^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+{\cal D}_{0}\overline{{\cal
D}}_{4}\bigg{\\{}\frac{5}{4}(A^{I}_{1})^{3}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+{\cal D}_{0}\overline{{\cal
D}}_{3}\bigg{\\{}-5(A^{I}_{1})^{2}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}-\frac{10}{3}\beta_{0}(A^{I}_{1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+{\cal D}_{0}\overline{{\cal
D}}_{2}\bigg{\\{}6A^{I}_{1}(f^{I}_{1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+6A^{I}_{1}A^{I}_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+\frac{3}{2}(A^{I}_{1})^{2}{\cal
M}^{(1,1)}_{I,\text{fin}}+3(A^{I}_{1})^{2}{\cal
M}^{(0,2)}_{I,\text{fin}}+6(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}-12(A^{I}_{1})^{3}\zeta_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+10\beta_{0}A^{I}_{1}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}+2\beta_{0}^{2}A^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+{\cal D}_{0}\overline{{\cal
D}}_{1}\bigg{\\{}-2(f^{I}_{1})^{3}{\cal
M}^{(0,1)}_{I,\text{fin}}-6A^{I}_{2}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}-6A^{I}_{1}f^{I}_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}-3A^{I}_{1}f^{I}_{1}{\cal
M}^{(1,1)}_{I,\text{fin}}-6A^{I}_{1}f^{I}_{1}{\cal
M}^{(0,2)}_{I,\text{fin}}-12A^{I}_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}+24(A^{I}_{1})^{2}\zeta_{2}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}+20(A^{I}_{1})^{3}\zeta_{3}{\cal
M}^{(0,1)}_{I,\text{fin}}-2\beta_{1}A^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}-6\beta_{0}(f^{I}_{1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}-4\beta_{0}A^{I}_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}-\beta_{0}A^{I}_{1}{\cal
M}^{(1,1)}_{I,\text{fin}}-2\beta_{0}A^{I}_{1}{\cal
M}^{(0,2)}_{I,\text{fin}}-16\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}+12\beta_{0}(A^{I}_{1})^{2}\zeta_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}-4\beta_{0}^{2}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+{\cal D}_{0}\overline{{\cal
D}}_{0}\bigg{\\{}2f^{I}_{1}f^{I}_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+\frac{1}{2}(f^{I}_{1})^{2}{\cal
M}^{(1,1)}_{I,\text{fin}}+(f^{I}_{1})^{2}{\cal
M}^{(0,2)}_{I,\text{fin}}+2(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}+A^{I}_{3}{\cal
M}^{(0,1)}_{I,\text{fin}}+\frac{1}{2}A^{I}_{2}{\cal
M}^{(1,1)}_{I,\text{fin}}+A^{I}_{2}{\cal
M}^{(0,2)}_{I,\text{fin}}+2A^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}+A^{I}_{1}{\cal
M}^{(1,2)}_{I,\text{fin}}+A^{I}_{1}{\cal
M}^{(0,3)}_{I,\text{fin}}+A^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}+A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(1,1)}_{I,\text{fin}}+2A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,2)}_{I,\text{fin}}+2A^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}-5A^{I}_{1}\zeta_{2}(f^{I}_{1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}-4A^{I}_{1}A^{I}_{2}\zeta_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}-10(A^{I}_{1})^{2}\zeta_{3}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}-(A^{I}_{1})^{2}\zeta_{2}{\cal
M}^{(1,1)}_{I,\text{fin}}-2(A^{I}_{1})^{2}\zeta_{2}{\cal
M}^{(0,2)}_{I,\text{fin}}-4(A^{I}_{1})^{2}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}+2(A^{I}_{1})^{3}\zeta_{2}^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+\beta_{1}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}+2\beta_{0}f^{I}_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+\frac{1}{2}\beta_{0}f^{I}_{1}{\cal
M}^{(1,1)}_{I,\text{fin}}+\beta_{0}f^{I}_{1}{\cal
M}^{(0,2)}_{I,\text{fin}}+6\beta_{0}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}+2\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}{\cal
M}^{(0,1)}_{I,\text{fin}}-6\beta_{0}A^{I}_{1}\zeta_{2}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}-6\beta_{0}(A^{I}_{1})^{2}\zeta_{3}{\cal
M}^{(0,1)}_{I,\text{fin}}+4\beta_{0}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{3}\overline{{\cal
D}}_{3}\bigg{\\{}\frac{35}{24}(A^{I}_{1})^{4}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{2}\overline{{\cal
D}}_{4}\bigg{\\{}\frac{35}{16}(A^{I}_{1})^{4}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{2}\overline{{\cal
D}}_{3}\bigg{\\{}-\frac{35}{4}(A^{I}_{1})^{3}f^{I}_{1}-\frac{35}{4}\beta_{0}(A^{I}_{1})^{3}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{2}\overline{{\cal
D}}_{2}\bigg{\\{}\frac{45}{8}(A^{I}_{1})^{2}(f^{I}_{1})^{2}+\frac{45}{8}(A^{I}_{1})^{2}A^{I}_{2}+\frac{15}{4}(A^{I}_{1})^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-\frac{45}{4}(A^{I}_{1})^{4}\zeta_{2}+\frac{105}{8}\beta_{0}(A^{I}_{1})^{2}f^{I}_{1}+5\beta_{0}^{2}(A^{I}_{1})^{2}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{1}\overline{{\cal
D}}_{5}\bigg{\\{}\frac{7}{8}(A^{I}_{1})^{4}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{1}\overline{{\cal
D}}_{4}\bigg{\\{}-\frac{35}{8}(A^{I}_{1})^{3}f^{I}_{1}-\frac{35}{8}\beta_{0}(A^{I}_{1})^{3}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{1}\overline{{\cal
D}}_{3}\bigg{\\{}\frac{15}{2}(A^{I}_{1})^{2}(f^{I}_{1})^{2}+\frac{15}{2}(A^{I}_{1})^{2}A^{I}_{2}+5(A^{I}_{1})^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-15(A^{I}_{1})^{4}\zeta_{2}+\frac{35}{2}\beta_{0}(A^{I}_{1})^{2}f^{I}_{1}+\frac{20}{3}\beta_{0}^{2}(A^{I}_{1})^{2}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{1}\overline{{\cal
D}}_{2}\bigg{\\{}-5A^{I}_{1}(f^{I}_{1})^{3}-15A^{I}_{1}A^{I}_{2}f^{I}_{1}-\frac{15}{2}(A^{I}_{1})^{2}f^{I}_{2}-15(A^{I}_{1})^{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+45(A^{I}_{1})^{3}\zeta_{2}f^{I}_{1}+35(A^{I}_{1})^{4}\zeta_{3}-5\beta_{1}(A^{I}_{1})^{2}-20\beta_{0}A^{I}_{1}(f^{I}_{1})^{2}-15\beta_{0}A^{I}_{1}A^{I}_{2}-25\beta_{0}(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+40\beta_{0}(A^{I}_{1})^{3}\zeta_{2}-20\beta_{0}^{2}A^{I}_{1}f^{I}_{1}-3\beta_{0}^{3}A^{I}_{1}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{1}\overline{{\cal
D}}_{1}\bigg{\\{}\frac{1}{2}(f^{I}_{1})^{4}+3A^{I}_{2}(f^{I}_{1})^{2}+\frac{3}{2}(A^{I}_{2})^{2}+6A^{I}_{1}f^{I}_{1}f^{I}_{2}+6A^{I}_{1}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+3A^{I}_{1}A^{I}_{3}+6A^{I}_{1}A^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{3}{2}(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+3(A^{I}_{1})^{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-\frac{39}{2}(A^{I}_{1})^{2}\zeta_{2}(f^{I}_{1})^{2}-18(A^{I}_{1})^{2}A^{I}_{2}\zeta_{2}-35(A^{I}_{1})^{3}\zeta_{3}f^{I}_{1}-12(A^{I}_{1})^{3}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+9(A^{I}_{1})^{4}\zeta_{2}^{2}+5\beta_{1}A^{I}_{1}f^{I}_{1}+3\beta_{0}(f^{I}_{1})^{3}+7\beta_{0}A^{I}_{2}f^{I}_{1}+8\beta_{0}A^{I}_{1}f^{I}_{2}+22\beta_{0}A^{I}_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+3\beta_{0}(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-40\beta_{0}(A^{I}_{1})^{2}\zeta_{2}f^{I}_{1}-34\beta_{0}(A^{I}_{1})^{3}\zeta_{3}+\frac{5}{2}\beta_{0}\beta_{1}A^{I}_{1}+\frac{11}{2}\beta_{0}^{2}(f^{I}_{1})^{2}+3\beta_{0}^{2}A^{I}_{2}+18\beta_{0}^{2}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-11\beta_{0}^{2}(A^{I}_{1})^{2}\zeta_{2}+3\beta_{0}^{3}f^{I}_{1}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{0}\overline{{\cal
D}}_{6}\bigg{\\{}\frac{7}{48}(A^{I}_{1})^{4}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{0}\overline{{\cal
D}}_{5}\bigg{\\{}-\frac{7}{8}(A^{I}_{1})^{3}f^{I}_{1}-\frac{7}{8}\beta_{0}(A^{I}_{1})^{3}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{0}\overline{{\cal
D}}_{4}\bigg{\\{}\frac{15}{8}(A^{I}_{1})^{2}(f^{I}_{1})^{2}+\frac{15}{8}(A^{I}_{1})^{2}A^{I}_{2}+\frac{5}{4}(A^{I}_{1})^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-\frac{15}{4}(A^{I}_{1})^{4}\zeta_{2}+\frac{35}{8}\beta_{0}(A^{I}_{1})^{2}f^{I}_{1}+\frac{5}{3}\beta_{0}^{2}(A^{I}_{1})^{2}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{0}\overline{{\cal
D}}_{3}\bigg{\\{}-\frac{5}{3}A^{I}_{1}(f^{I}_{1})^{3}-5A^{I}_{1}A^{I}_{2}f^{I}_{1}-\frac{5}{2}(A^{I}_{1})^{2}f^{I}_{2}-5(A^{I}_{1})^{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+15(A^{I}_{1})^{3}\zeta_{2}f^{I}_{1}+\frac{35}{3}(A^{I}_{1})^{4}\zeta_{3}-\frac{5}{3}\beta_{1}(A^{I}_{1})^{2}-\frac{20}{3}\beta_{0}A^{I}_{1}(f^{I}_{1})^{2}-5\beta_{0}A^{I}_{1}A^{I}_{2}-\frac{25}{3}\beta_{0}(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{40}{3}\beta_{0}(A^{I}_{1})^{3}\zeta_{2}-\frac{20}{3}\beta_{0}^{2}A^{I}_{1}f^{I}_{1}-\beta_{0}^{3}A^{I}_{1}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{0}\overline{{\cal
D}}_{2}\bigg{\\{}\frac{1}{2}(f^{I}_{1})^{4}+3A^{I}_{2}(f^{I}_{1})^{2}+\frac{3}{2}(A^{I}_{2})^{2}+6A^{I}_{1}f^{I}_{1}f^{I}_{2}+6A^{I}_{1}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+3A^{I}_{1}A^{I}_{3}+6A^{I}_{1}A^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{3}{2}(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+3(A^{I}_{1})^{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-\frac{39}{2}(A^{I}_{1})^{2}\zeta_{2}(f^{I}_{1})^{2}-18(A^{I}_{1})^{2}A^{I}_{2}\zeta_{2}-35(A^{I}_{1})^{3}\zeta_{3}f^{I}_{1}-12(A^{I}_{1})^{3}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+9(A^{I}_{1})^{4}\zeta_{2}^{2}+5\beta_{1}A^{I}_{1}f^{I}_{1}+3\beta_{0}(f^{I}_{1})^{3}+7\beta_{0}A^{I}_{2}f^{I}_{1}+8\beta_{0}A^{I}_{1}f^{I}_{2}+22\beta_{0}A^{I}_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+3\beta_{0}(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-40\beta_{0}(A^{I}_{1})^{2}\zeta_{2}f^{I}_{1}-34\beta_{0}(A^{I}_{1})^{3}\zeta_{3}+\frac{5}{2}\beta_{0}\beta_{1}A^{I}_{1}+\frac{11}{2}\beta_{0}^{2}(f^{I}_{1})^{2}+3\beta_{0}^{2}A^{I}_{2}+18\beta_{0}^{2}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-11\beta_{0}^{2}(A^{I}_{1})^{2}\zeta_{2}+3\beta_{0}^{3}f^{I}_{1}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{0}\overline{{\cal
D}}_{1}\bigg{\\{}-3(f^{I}_{1})^{2}f^{I}_{2}-2(f^{I}_{1})^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-3A^{I}_{3}f^{I}_{1}-3A^{I}_{2}f^{I}_{2}-6A^{I}_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-3A^{I}_{1}f^{I}_{3}-6A^{I}_{1}f^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-3A^{I}_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-6A^{I}_{1}f^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}+9A^{I}_{1}\zeta_{2}(f^{I}_{1})^{3}+24A^{I}_{1}A^{I}_{2}\zeta_{2}f^{I}_{1}+30(A^{I}_{1})^{2}\zeta_{3}(f^{I}_{1})^{2}+12(A^{I}_{1})^{2}\zeta_{2}f^{I}_{2}+24(A^{I}_{1})^{2}\zeta_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+30(A^{I}_{1})^{2}A^{I}_{2}\zeta_{3}+20(A^{I}_{1})^{3}\zeta_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-18(A^{I}_{1})^{3}\zeta_{2}^{2}f^{I}_{1}+42(A^{I}_{1})^{4}\zeta_{5}-60(A^{I}_{1})^{4}\zeta_{2}\zeta_{3}-\beta_{2}A^{I}_{1}-3\beta_{1}(f^{I}_{1})^{2}-2\beta_{1}A^{I}_{2}-8\beta_{1}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+6\beta_{1}(A^{I}_{1})^{2}\zeta_{2}-9\beta_{0}f^{I}_{1}f^{I}_{2}-12\beta_{0}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-3\beta_{0}A^{I}_{3}-10\beta_{0}A^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-7\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-14\beta_{0}A^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-6\beta_{0}A^{I}_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+31\beta_{0}A^{I}_{1}\zeta_{2}(f^{I}_{1})^{2}+18\beta_{0}A^{I}_{1}A^{I}_{2}\zeta_{2}+68\beta_{0}(A^{I}_{1})^{2}\zeta_{3}f^{I}_{1}+36\beta_{0}(A^{I}_{1})^{2}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-10\beta_{0}(A^{I}_{1})^{3}\zeta_{2}^{2}-5\beta_{0}\beta_{1}f^{I}_{1}-6\beta_{0}^{2}f^{I}_{2}-22\beta_{0}^{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-14\beta_{0}^{2}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+22\beta_{0}^{2}A^{I}_{1}\zeta_{2}f^{I}_{1}+22\beta_{0}^{2}(A^{I}_{1})^{2}\zeta_{3}-12\beta_{0}^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\bigg{\\}}\par+|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}{\cal D}_{0}\overline{{\cal
D}}_{0}\bigg{\\{}\frac{1}{2}(f^{I}_{2})^{2}+f^{I}_{1}f^{I}_{3}+2f^{I}_{1}f^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{1}{2}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+(f^{I}_{1})^{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-\frac{1}{2}\zeta_{2}(f^{I}_{1})^{4}+\frac{1}{2}A4+A^{I}_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{1}{2}A^{I}_{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+A^{I}_{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}-\frac{5}{2}A^{I}_{2}\zeta_{2}(f^{I}_{1})^{2}-(A^{I}_{2})^{2}\zeta_{2}+\frac{1}{3}A^{I}_{1}\mathcal{\tilde{G}}^{I,(3)}_{d,1}+A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+\frac{2}{3}A^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{3}-\frac{10}{3}A^{I}_{1}\zeta_{3}(f^{I}_{1})^{3}-5A^{I}_{1}\zeta_{2}f^{I}_{1}f^{I}_{2}-5A^{I}_{1}\zeta_{2}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2A^{I}_{1}A^{I}_{3}\zeta_{2}-10A^{I}_{1}A^{I}_{2}\zeta_{3}f^{I}_{1}-4A^{I}_{1}A^{I}_{2}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-5(A^{I}_{1})^{2}\zeta_{3}f^{I}_{2}-10(A^{I}_{1})^{2}\zeta_{3}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-(A^{I}_{1})^{2}\zeta_{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}-2(A^{I}_{1})^{2}\zeta_{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}+4(A^{I}_{1})^{2}\zeta_{2}^{2}(f^{I}_{1})^{2}+3(A^{I}_{1})^{2}A^{I}_{2}\zeta_{2}^{2}-21(A^{I}_{1})^{3}\zeta_{5}f^{I}_{1}+30(A^{I}_{1})^{3}\zeta_{2}\zeta_{3}f^{I}_{1}+2(A^{I}_{1})^{3}\zeta_{2}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{35}{3}(A^{I}_{1})^{4}\zeta_{3}^{2}-\frac{2}{3}(A^{I}_{1})^{4}\zeta_{2}^{3}+\frac{1}{2}\beta_{2}f^{I}_{1}+\beta_{1}f^{I}_{2}+3\beta_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{2}{3}\beta_{1}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-3\beta_{1}A^{I}_{1}\zeta_{2}f^{I}_{1}-3\beta_{1}(A^{I}_{1})^{2}\zeta_{3}+\frac{3}{2}\beta_{0}f^{I}_{3}+4\beta_{0}f^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{5}{2}\beta_{0}f^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+5\beta_{0}f^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}+\beta_{0}(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-\frac{5}{2}\beta_{0}\zeta_{2}(f^{I}_{1})^{3}+\beta_{0}A^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-4\beta_{0}A^{I}_{2}\zeta_{2}f^{I}_{1}+\frac{2}{3}\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,2}+2\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-13\beta_{0}A^{I}_{1}\zeta_{3}(f^{I}_{1})^{2}-5\beta_{0}A^{I}_{1}\zeta_{2}f^{I}_{2}-16\beta_{0}A^{I}_{1}\zeta_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-9\beta_{0}A^{I}_{1}A^{I}_{2}\zeta_{3}-16\beta_{0}(A^{I}_{1})^{2}\zeta_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-2\beta_{0}(A^{I}_{1})^{2}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}+5\beta_{0}(A^{I}_{1})^{2}\zeta_{2}^{2}f^{I}_{1}-21\beta_{0}(A^{I}_{1})^{3}\zeta_{5}+26\beta_{0}(A^{I}_{1})^{3}\zeta_{2}\zeta_{3}+5\beta_{0}\beta_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+3\beta_{0}^{2}\mathcal{\tilde{G}}^{I,(2)}_{d,1}+6\beta_{0}^{2}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}+5\beta_{0}^{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}-3\beta_{0}^{2}\zeta_{2}(f^{I}_{1})^{2}+\frac{4}{3}\beta_{0}^{2}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,3}-11\beta_{0}^{2}A^{I}_{1}\zeta_{3}f^{I}_{1}-10\beta_{0}^{2}A^{I}_{1}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-\frac{4}{5}\beta_{0}^{2}(A^{I}_{1})^{2}\zeta_{2}^{2}+6\beta_{0}^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,2}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{5}\bigg{\\{}\frac{1}{4}(A^{I}_{1})^{3}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{4}\bigg{\\{}-\frac{5}{4}(A^{I}_{1})^{2}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}-\frac{5}{6}\beta_{0}(A^{I}_{1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{3}\bigg{\\{}2A^{I}_{1}(f^{I}_{1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+2A^{I}_{1}A^{I}_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+\frac{1}{2}(A^{I}_{1})^{2}{\cal
M}^{(1,1)}_{I,\text{fin}}+(A^{I}_{1})^{2}{\cal
M}^{(0,2)}_{I,\text{fin}}+2(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}-4(A^{I}_{1})^{3}\zeta_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+\frac{10}{3}\beta_{0}A^{I}_{1}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}+\frac{2}{3}\beta_{0}^{2}A^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{2}\bigg{\\{}-(f^{I}_{1})^{3}{\cal
M}^{(0,1)}_{I,\text{fin}}-3A^{I}_{2}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}-3A^{I}_{1}f^{I}_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}-\frac{3}{2}A^{I}_{1}f^{I}_{1}{\cal
M}^{(1,1)}_{I,\text{fin}}-3A^{I}_{1}f^{I}_{1}{\cal
M}^{(0,2)}_{I,\text{fin}}-6A^{I}_{1}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}+12(A^{I}_{1})^{2}\zeta_{2}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}+10(A^{I}_{1})^{3}\zeta_{3}{\cal
M}^{(0,1)}_{I,\text{fin}}-\beta_{1}A^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}-3\beta_{0}(f^{I}_{1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}-2\beta_{0}A^{I}_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}-\frac{1}{2}\beta_{0}A^{I}_{1}{\cal
M}^{(1,1)}_{I,\text{fin}}-\beta_{0}A^{I}_{1}{\cal
M}^{(0,2)}_{I,\text{fin}}-8\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}+6\beta_{0}(A^{I}_{1})^{2}\zeta_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}-2\beta_{0}^{2}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{1}\bigg{\\{}4f^{I}_{1}f^{I}_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+(f^{I}_{1})^{2}{\cal
M}^{(1,1)}_{I,\text{fin}}+2(f^{I}_{1})^{2}{\cal
M}^{(0,2)}_{I,\text{fin}}+4(f^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}+2A^{I}_{3}{\cal
M}^{(0,1)}_{I,\text{fin}}+A^{I}_{2}{\cal
M}^{(1,1)}_{I,\text{fin}}+2A^{I}_{2}{\cal
M}^{(0,2)}_{I,\text{fin}}+4A^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}+2A^{I}_{1}{\cal
M}^{(1,2)}_{I,\text{fin}}+2A^{I}_{1}{\cal
M}^{(0,3)}_{I,\text{fin}}+2A^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}+2A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(1,1)}_{I,\text{fin}}+4A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,2)}_{I,\text{fin}}+4A^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}-10A^{I}_{1}\zeta_{2}(f^{I}_{1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}-8A^{I}_{1}A^{I}_{2}\zeta_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}-20(A^{I}_{1})^{2}\zeta_{3}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}-2(A^{I}_{1})^{2}\zeta_{2}{\cal
M}^{(1,1)}_{I,\text{fin}}-4(A^{I}_{1})^{2}\zeta_{2}{\cal
M}^{(0,2)}_{I,\text{fin}}-8(A^{I}_{1})^{2}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}+4(A^{I}_{1})^{3}\zeta_{2}^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+2\beta_{1}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}+4\beta_{0}f^{I}_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+\beta_{0}f^{I}_{1}{\cal
M}^{(1,1)}_{I,\text{fin}}+2\beta_{0}f^{I}_{1}{\cal
M}^{(0,2)}_{I,\text{fin}}+12\beta_{0}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}+4\beta_{0}A^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}{\cal
M}^{(0,1)}_{I,\text{fin}}-12\beta_{0}A^{I}_{1}\zeta_{2}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}-12\beta_{0}(A^{I}_{1})^{2}\zeta_{3}{\cal
M}^{(0,1)}_{I,\text{fin}}+8\beta_{0}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+\delta(1-z_{1})\overline{{\cal
D}}_{0}\bigg{\\{}-2f^{I}_{3}{\cal M}^{(0,1)}_{I,\text{fin}}-f^{I}_{2}{\cal
M}^{(1,1)}_{I,\text{fin}}-2f^{I}_{2}{\cal
M}^{(0,2)}_{I,\text{fin}}-4f^{I}_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}-2f^{I}_{1}{\cal
M}^{(1,2)}_{I,\text{fin}}-2f^{I}_{1}{\cal
M}^{(0,3)}_{I,\text{fin}}-2f^{I}_{1}\mathcal{\tilde{G}}^{I,(2)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}-2f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(1,1)}_{I,\text{fin}}-4f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,2)}_{I,\text{fin}}-4f^{I}_{1}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+2\zeta_{2}(f^{I}_{1})^{3}{\cal
M}^{(0,1)}_{I,\text{fin}}+4A^{I}_{2}\zeta_{2}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}+8A^{I}_{1}\zeta_{3}(f^{I}_{1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+4A^{I}_{1}\zeta_{2}f^{I}_{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+2A^{I}_{1}\zeta_{2}f^{I}_{1}{\cal
M}^{(1,1)}_{I,\text{fin}}+4A^{I}_{1}\zeta_{2}f^{I}_{1}{\cal
M}^{(0,2)}_{I,\text{fin}}+8A^{I}_{1}\zeta_{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}+8A^{I}_{1}A^{I}_{2}\zeta_{3}{\cal
M}^{(0,1)}_{I,\text{fin}}+2(A^{I}_{1})^{2}\zeta_{3}{\cal
M}^{(1,1)}_{I,\text{fin}}+4(A^{I}_{1})^{2}\zeta_{3}{\cal
M}^{(0,2)}_{I,\text{fin}}+8(A^{I}_{1})^{2}\zeta_{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}-4(A^{I}_{1})^{2}\zeta_{2}^{2}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}+12(A^{I}_{1})^{3}\zeta_{5}{\cal
M}^{(0,1)}_{I,\text{fin}}-16(A^{I}_{1})^{3}\zeta_{2}\zeta_{3}{\cal
M}^{(0,1)}_{I,\text{fin}}-4\beta_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}-4\beta_{0}\mathcal{\tilde{G}}^{I,(2)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}-2\beta_{0}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(1,1)}_{I,\text{fin}}-4\beta_{0}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,2)}_{I,\text{fin}}-8\beta_{0}(\mathcal{\tilde{G}}^{I,(1)}_{d,1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}-4\beta_{0}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,2}{\cal
M}^{(0,1)}_{I,\text{fin}}+4\beta_{0}\zeta_{2}(f^{I}_{1})^{2}{\cal
M}^{(0,1)}_{I,\text{fin}}+12\beta_{0}A^{I}_{1}\zeta_{3}f^{I}_{1}{\cal
M}^{(0,1)}_{I,\text{fin}}+8\beta_{0}A^{I}_{1}\zeta_{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}{\cal
M}^{(0,1)}_{I,\text{fin}}-8\beta_{0}^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,2}{\cal
M}^{(0,1)}_{I,\text{fin}}\bigg{\\}}\par+\delta(1-z_{1})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\overline{{\cal
D}}_{7}\bigg{\\{}\frac{1}{48}(A^{I}_{1})^{4}\bigg{\\}}\par+\delta(1-z_{1})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\overline{{\cal
D}}_{6}\bigg{\\{}-\frac{7}{48}(A^{I}_{1})^{3}f^{I}_{1}-\frac{7}{48}\beta_{0}(A^{I}_{1})^{3}\bigg{\\}}\par+\delta(1-z_{1})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\overline{{\cal
D}}_{5}\bigg{\\{}\frac{3}{8}(A^{I}_{1})^{2}(f^{I}_{1})^{2}+\frac{3}{8}(A^{I}_{1})^{2}A^{I}_{2}+\frac{1}{4}(A^{I}_{1})^{3}\mathcal{\tilde{G}}^{I,(1)}_{d,1}-\frac{3}{4}(A^{I}_{1})^{4}\zeta_{2}+\frac{7}{8}\beta_{0}(A^{I}_{1})^{2}f^{I}_{1}+\frac{1}{3}\beta_{0}^{2}(A^{I}_{1})^{2}\bigg{\\}}\par+\delta(1-z_{1})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\overline{{\cal
D}}_{4}\bigg{\\{}-\frac{5}{12}A^{I}_{1}(f^{I}_{1})^{3}-\frac{5}{4}A^{I}_{1}A^{I}_{2}f^{I}_{1}-\frac{5}{8}(A^{I}_{1})^{2}f^{I}_{2}-\frac{5}{4}(A^{I}_{1})^{2}f^{I}_{1}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{15}{4}(A^{I}_{1})^{3}\zeta_{2}f^{I}_{1}+\frac{35}{12}(A^{I}_{1})^{4}\zeta_{3}-\frac{5}{12}\beta_{1}(A^{I}_{1})^{2}-\frac{5}{3}\beta_{0}A^{I}_{1}(f^{I}_{1})^{2}-\frac{5}{4}\beta_{0}A^{I}_{1}A^{I}_{2}-\frac{25}{12}\beta_{0}(A^{I}_{1})^{2}\mathcal{\tilde{G}}^{I,(1)}_{d,1}+\frac{10}{3}\beta_{0}(A^{I}_{1})^{3}\zeta_{2}-\frac{5}{3}\beta_{0}^{2}A^{I}_{1}f^{I}_{1}-\frac{1}{4}\beta_{0}^{3}A^{I}_{1}\bigg{\\}}\par+\delta(1-z_{1})|{\cal
M}^{(0)}_{I,\text{fin}}|^{2}\overline{{\cal