Correlator webs of massive multiparton amplitudes at four loops: A study of
boomerang webs

Neelima Agarwal$\,{}^{a}$, Sourav Pal$\,{}^{b}$, Aditya Srivastav$\,{}^{c}$,
Anurag Tripathi$\,{}^{c}$

$\,{}^{a}$Department of Physics, Chaitanya Bharathi Institute of Technology,
Gandipet, Hyderabad, Telangana State 500075, India
$\,{}^{b}$Theoretical Physics Division, Physical Research Laboratory,
Navrangpura, Ahmedabad 380009, India
$\,{}^{c}$Department of Physics, Indian Institute of Technology Hyderabad,
Kandi, Sangareddy, Telangana State 502284, India

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Abstract: Logarithm of the soft function can be organized into sets of Feynman
diagrams known as Cwebs. We introduced a new formalism in [1], that allows to
determine several of the building blocks of Cweb mixing matrices without
explicit computations.  In [2] we used this formalism to obtain the diagonal
blocks of four general classes of Cwebs to all orders in perturbation theory
which also covered all the four loop Boomerang Cwebs connecting four Wilson
lines. In this work we present complete mixing matrices and exponentiated
colour factors for Boomerang Cwebs at four loops that connect three and four
Wilson lines. Also, we present a more efficient version of the algorithm of
generating Cwebs that was presented in [3]. This new algorithm has been used
to generate the Cwebs in the present work.

###### Contents

  1. 1 Introduction
  2. 2 Cwebs and web mixing matrices
  3. 3 An improved algorithm to generate Cwebs
  4. 4 A brief description of the code CwebGen 2.0
  5. 5 Boomerang Cwebs at Four loops
  6. 6 Conclusions
  7. 7 Appendix
  8. A Replica trick and mixing matrices
  9. B Calculation of $s$-factor in different representations
  10. C Boomerang Cwebs at Four loops
    1. C.1 Boomerang Cwebs connecting four Wilson lines
    2. C.2 Boomerang Cwebs connecting three Wilson lines

##  1 Introduction

The study of infrared (IR) singularity structure of scattering amplitudes in
non-abelian gauge theories has a rich and long history [4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. A comprehensive review of the IR
singularities of gauge theories is elucidated in a recent review article [21].
The universal structure of IR singularities of gauge theories, that is their
independence of relevant hard scattering process helps one to study their all
order structure in perturbation theory. The structure of IR singularities not
only helps us in understanding gauge theories to all orders in the
perturbation theory, but also remains important for phenomenological
applications. These studies are relevant to the high energy scattering
experiments at different colliders. The calculations of physical observables
such as scattering cross section and decay rates suffer from these IR
divergences which eventually cancel upon adding the real emission contribution
to the virtual correction at a given perturbative order. This cancellation
results in large logarithms of kinematic invariants that damage the predictive
power of fixed order calculations in certain kinematical regions. Again the
universality property of IR singularity provides a way of summing these
logarithms to all order in perturbation and helps in recovering the
predictions in those regions. The intricate cancellation of IR poles between
real and virtual contributions for observables relevant to colliders is a non-
trivial task for which several subtraction procedures [22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35] are developed.

The QCD factorization theorem enables us in studying the singular parts of
scattering amplitudes involving massless gauge bosons, without calculating the
complicated hard part. The soft function, which captures the IR singular parts
of a scattering amplitude, is expressed as the correlators of Wilson line
operators. The usual Wilson-line operators $\Phi(\zeta)$ evaluated on smooth
space-time contours $\zeta$ are defined as,

$\displaystyle\Phi\left(\zeta\right)\,\equiv\,\mathcal{P}\exp\left[{\rm
i}g\\!\int_{\zeta}dx\cdot{\bf A}(x)\right]\,.$ (1)

Here ${\bf A}^{\mu}(x)=A^{\mu}_{a}(x)\,{\bf T}^{a}$ is a non-abelian gauge
field, and ${\bf T}^{a}$ is a generator of the gauge algebra, which can be
taken to belong to any desired representation, and $\mathcal{P}$ denotes path
ordering of the gauge fields. The soft function in general can be expressed
as,

$\displaystyle{\cal
S}_{n}\left(\zeta_{i}\right)\,\equiv\,\bra{0}\prod_{k=1}^{n}\Phi\left(\zeta_{k}\right)\ket{0}\,.$
(2)

These Wilson lines are semi-infinite Wilson lines along the direction of the
hard particle. Thus, one can write the soft function as,

$\displaystyle{\cal
S}_{n}\Big{(}\beta_{i}\cdot\beta_{j},\alpha_{s}(\mu^{2}),\epsilon\Big{)}\,\equiv\,\bra{0}\prod_{k=1}^{n}\Phi_{\beta_{k}}\left(\infty,0\right)\ket{0},\quad\Phi_{\beta}\left(\infty,0\right)\,\equiv\,\mathcal{P}\exp\left[{\rm
i}g\\!\int_{0}^{\infty}d\lambda\,\beta\cdot{\bf A}(\lambda\beta)\right]\,.$
(3)

The object $\mathcal{S}_{n}$ suffers from both UV and IR (soft) singularities,
and thus needs renormalization. In dimensional regularization $d=4-2\epsilon$,
higher order terms of unrenormalized soft function $\mathcal{S}_{n}$ vanish,
giving $\mathcal{S}_{n}=1$. The simple argument for this is that the soft
function is made out of Wilson line correlators, which involve only scaleless
integrals. Thus, the renormalized soft function
$\mathcal{S}_{n}^{(\text{ren})}$ contains pure UV counterterms.

The renormalized soft function obey a renormalization group equation and
solving this equation results in an exponentiation of the form,

$\displaystyle\mathcal{S}_{n}^{(\text{ren})}\Big{(}\beta_{i}\cdot\beta_{j},\alpha_{s}(\mu^{2}),\epsilon\Big{)}\,=\,\mathcal{P}\exp\left[-\frac{1}{2}\int_{0}^{\mu^{2}}\frac{d\lambda^{2}}{\lambda^{2}}\,{\bf\Gamma}_{n}\Big{(}\beta_{i}\cdot\beta_{j},\alpha_{s}(\lambda^{2}),\epsilon\Big{)}\right]\,,$
(4)

where ${\bf\Gamma}_{n}$ is known as the soft anomalous dimension. In case of
processes involving multi-parton scatterings, the soft anomalous dimension is
a matrix, which is an interesting theoretical object to study, and is our main
focus in this article. The renormalization group approach has been used for
more than two decades to calculate the soft-anomalous dimension. One-loop
calculations for ${\bf\Gamma}_{n}$ were performed in [36, 37, 38, 39], while
two-loop calculations were done for both the massless case in [40, 41] and the
massive case in [42, 43, 44, 45, 46, 47, 48, 49]. The three-loop calculations
were finally carried out for the massless case in [50, 51]. The calculation of
soft anomalous dimension at four loops is an ongoing effort. Several studies
in this direction are available in the literature in [52, 53, 54, 55, 56, 57,
58, 59, 60, 61, 62, 63, 64, 65, 66]. The kinematic dependence of the soft
anomalous dimension for scatterings involving only massless lines is
restricted due to the constraints discussed in [18, 67, 68, 19]. However,
these constraints do not hold true for scatterings involving massive
particles. The state-of-the-art knowledge for soft anomalous dimension is
known upto two loops for scatterings involving massive particle [42, 43, 44,
45, 46, 47, 48, 49], and at three-loop for one massive Wilson line [69].

An alternative approach to determine the exponent of the soft function is
through diagrammatic exponentiation. In terms of Feynman diagrams, the soft
function has the form,

$\displaystyle{\cal S}_{n}\left(\gamma_{i}\right)\,=\,\exp\Big{[}{\cal
W}_{n}\left(\gamma_{i}\right)\Big{]}\,,$ (5)

where ${\cal W}_{n}\left(\gamma_{i}\right)$ is known as webs, and can be
directly computed using Feynman diagrams. Webs were first defined as two-line
irreducible diagrams for scattering involving two Wilson lines [70, 71, 72].
In case of multiparton scattering process, webs in non-abelian gauge theory
[73, 74] are defined as the sets of diagrams that differ among each other by
the order of gluon attachments on each Wilson lines. A generalization of a web
called Cweb was introduced in [3, 75].

The state-of-the-art studies for massive multiparton webs at three loops [76],
for massless webs upto three loops [74, 77, 78, 79], at four loops [3, 75],
and partially at five loops [80] advance significantly. The kinematics and the
colour factors of diagrams in a Cweb mix among themselves, through a web
mixing matrix.

These web mixing matrices are crucial objects in the study of non-abelian
exponentiation, and the general method to calculate them is by applying a
well-known replica trick algorithm [74, 81]; an alternative approach of
generating functionals was developed in [82, 83, 84]. The web mixing matrices
are combinatorial objects and have also been studied from the viewpoint of
combinatorial mathematics using posets [85, 86, 87]. Further, a novel method
of calculating the diagonal blocks — using several new concepts such as Normal
Ordering, Uniqueness theorem and Fused-Webs — of the mixing matrices has been
developed in [1] and has been applied for a certain classes of webs in [2].

Recently, Boomerang webs are introduced in [76] to calculate the soft
anomalous dimension at three loops for scattering processes involving massive
particles, such as top quark, whose mass cannot be ignored, in several QCD
processes. Boomerang webs are defined as webs, which contain at least one
gluon propagator whose both ends attach to the same massive Wilson line. Note
that, Boomerang webs vanish identically in the case of massless Wilson lines.
Following the definition of Cwebs, Boomerang Cwebs are defined in [2] as the
Cwebs that contain at least one two-point gluon correlator whose both ends are
attached to the same massive Wilson line. In [76], the authors have computed
the web mixing matrices, their exponentiated colour factors, and kinematics
for all three-loop Boomerang webs. Recently, the diagonal blocks of mixing
matrices for four-loop Boomerang Cwebs connecting four Wilson lines are
presented in [2] using the concept of Fused-Webs [1] and combinatorial
properties of the Cwebs. However, to determine the exponentiated colour
factors and the kinematic contribution of these Cwebs one needs to calculate
the explicit form of the mixing matrices.

In this article we present the explicit results of the mixing matrices for all
the Boomerang Cwebs that connect three and four massive Wilson lines. To
enumerate these Cwebs uniquely, we have modified the algorithm of enumerating
Cwebs [3, 75] and implemented it in a Mathematica code. Further, we have
modified the older version of an in-house Mathematica code used in [3, 75] to
calculate the mixing matrices for Cwebs using the replica-trick algorithm.
These modifications are incorporated in our in-house Mathematica Code CwebGen
2.0.

The rest of the paper is structured as follows. In section 2, we review Cwebs
and the properties of web mixing matrices. In the section 3 we give the
details of the modified algorithm to generate Cwebs at a given perturbative
order and its comparison with the older versions. Section 4 discusses the
working of CwebGen 2.0. In section 5, we describe the calculation of mixing
matrices for two Boomerang Cwebs at four loops. Finally in section 6 we
conclude our findings and give a future outlook on our results. In ancillary
file Boomerang.nb, we provide the explicit form of the mixing matrices and the
corresponding column weight vector for all Boomerang Cwebs that are considered
in this article.

Figure 1: An all order four-point connected gluon correlator is shown. The
blob on the left hand side contains all higher order corrections an expansion
of which is given on the right hand side.

##  2 Cwebs and web mixing matrices

For the convenience of the reader, we collect here some of the definitions
that are used in this article.

#### Definitions

  * Web: A set of diagrams closed under shuffles of the gluon attachments — originating from different gluon propagators or three-point or four-point gluon vertices — on each Wilson line.

  * Cweb: A set of diagrams closed under shuffles of the gluon attachments — originating from different gluon correlators (see fig. (1) for an example) — on each Wilson line.

  * Multi-pronged-boomerang correlator: An $m$-point gluon correlator ($m\geq 2$) whose all ends attach to one Wilson line. The contributions of these for massless Wilson lines vanish identically.

  * Boomerang Cweb: A Cweb that contains at least one multi-pronged-boomerang correlator with $m=2$ and none with $m>2$.

  * Weight factor ($s$-factor): The weight factor $s(d)$ for a diagram $d$ is defined as the number of ways in which the gluon correlators can be sequentially shrunk to their common origin.

  * Column weight vector:  We can construct a column weight vector out of the $s$-factors for a Cweb with $n$ diagrams as

$\displaystyle{\color[rgb]{0.3,0.3,0.3}S=\\{s(d_{1}),s(d_{2}),\ldots,s(d_{n})\\}}.$
(6)

  * ${\color[rgb]{0.3,0.3,0.3}\mathbf{W_{n}^{(c_{2},\ldots,c_{p})}(k_{1},\ldots,k_{n})}}$:  This is how we denote a Cweb constructed out of $c_{m}$ $m$-point connected gluon correlators ($m=2,\ldots,p$), where $k_{i}$ denotes number of attachments on $i^{\text{th}}$ Wilson line.

Note that the perturbative expansion for an $m$-point connected gluon
correlator starts at ${\cal O}(g^{m-2})$, while each attachment to a Wilson
line carries a further power of $g$, the perturbative expansion for a Cweb can
be written as

$W_{n}^{(c_{2},\ldots,c_{p})}(k_{1},\ldots,k_{n})\,=\,g^{\,\sum_{i=1}^{n}k_{i}\,+\,\sum_{r=2}^{p}c_{r}(r-2)}\,\sum_{j=0}^{\infty}\,W_{n,\,j}^{(c_{2},\ldots,c_{p})}(k_{1},\ldots,k_{n})\,g^{2j}\,,$
(7)

which defines the perturbative coefficients
$W_{n,\,j}^{(c_{2},\ldots,c_{p})}(k_{1},\ldots,k_{n})$. Here the perturbative
order of this Cweb is
$g^{\,\sum_{i=1}^{n}k_{i}\,+\,\sum_{r=2}^{p}c_{r}(r-2)}$.

Cwebs are the proper building blocks of the logarithm of the Soft function —
they are useful in the organization and counting of diagrammatic contributions
at higher perturbative orders [3, 75]. The logarithm of the Soft function is a
sum over all the Cwebs at each perturbative order:

$\displaystyle{\cal S}\,=\,\exp\left[\sum_{W}\sum_{d,d^{\prime}\in
W}{K}(d)\,R_{W}(d,d^{\prime})\,C(d^{\prime})\right]\,.$ (8)

The $d$ here denotes a diagram in a Cweb $W$ and its corresponding kinematic
and colour factor are denoted by ${K}(d)$ and $C(d)$. The action of web mixing
matrix $R_{W}$ on the colour of a diagram $C(d)$ generates its exponentiated
colour factor $\widetilde{C}(d)$,

$\displaystyle\widetilde{C}(d)\,=\,\sum_{d^{\prime}\in
W}R_{W}(d,d^{\prime})\,C(d^{\prime})\,.$ (9)

The contribution of colour and kinematic factors to a web $W$ can be arranged
in a more transparent manner if we diagonalize the mixing matrix $R$:

$\displaystyle
W=\left(K^{T}Y^{-1}\right)YRY^{-1}\left(YC\right)\,=\,\sum_{j=1}^{r}\left(K^{T}Y^{-1}\right)_{j}\left(YC\right)_{j}\,,$
(10)

where $Y$ is the diagonalizing matrix and $YRY^{-1}\equiv\mathcal{D}$ is the
diagonal matrix that we get; furthermore we have arranged $\mathcal{D}_{jj}=1$
for $1\leq j\leq r$. $\left(YC\right)_{j}$ are also referred to as
exponentiated colour factors and the corresponding kinematic factors are
$\left(K^{T}Y^{-1}\right)_{j}$. In this article we will present the
exponentiated colour factors in terms of $\left(YC\right)_{j}$.

To understand the structure of the soft anomalous dimension, the study of the
web mixing matrices is crucial. We will begin this study by listing down the
properties that they are known to obey [73, 74, 88, 77, 86, 87, 85, 1] which
we list down below.

#### Properties of mixing matrices

  1. 1.

Idempotence: A Cweb mixing matrix is idempotent, that is, $R^{2}=R$. This
implies that: (a) the eigenvalues of $R$ can either be 0 or 1, (b) trace
$\textrm{tr}(R)$ is equal to its rank, and (c) it acts as a projection
operator; acting on its right, it give completely connected exponentiated
colour factors.

  2. 2.

Zero row sum rule: The entries of $R$ obey the zero row sum rule
$\sum_{d^{\prime}}R(d,d^{\prime})=0$. This ensures that while acting on the
vector of kinematic factors, the web mixing matrix implements cancellation of
the leading divergences among the diagrams of the Cweb.

  3. 3.

Column sum rule: Along with these general properties, the mixing matrices also
obey a conjectured column-sum rule of the form

$\displaystyle\sum_{d}s(d)R(d,d^{\prime})=0\,.$ (11)

  4. 4.

Uniqueness: For a given column weight vector
$S=\\{s(d_{1}),s(d_{2}),\ldots,s(d_{k})\\}$ with all $s(d_{i})\neq 0$, the
mixing matrix is unique.

In section 5, we will verify these properties for four loop Boomerang Cwebs.

In the next section, we describe a recursive algorithm that generates Cwebs
present at $\mathcal{O}(g^{2l+2})$ using Cwebs at $\mathcal{O}(g^{2l})$.

##  3 An improved algorithm to generate Cwebs

An algorithm to generate Cwebs at $l+1$ loops using Cwebs at $l$ loops was
presented in [3] in which some of the present authors were also involved. This
was later used in [75] to obtain the Cwebs that are present at four loops. Now
we will present (for ease of reading) the original algorithm below and then
spell out a modified and more efficient version of the algorithm, which is
implemented in a subroutine of in-house Mathematica code CwebGen 2.0.

#### Original algorithm

The original algorithm presented in [3, 75] to generate Cwebs at $l+1$ loops
using Cwebs at $l$ loops is reproduced below:

  1. 1.

Add a two-gluon connected correlator connecting any two Wilson lines
(including Wilson lines that had no attachments at lower orders).

  2. 2.

Connect an existing $m$-point correlator to any Wilson line (again, including
Wilson lines with no attachments at lower orders), turning it into an
$(m+1)$-point correlator.

  3. 3.

Connect an existing $m$-point correlator to an existing $n$-point correlator,
resulting in an $(n+m)$-point correlator.

  4. 4.

Discard the duplicate Cwebs.

(a)

(b)

(c)

(d)

Figure 2:  Representative diagrams of Cwebs appearing at two loops connecting
light like Wilson lines. Cwebs (a), and (b) belong to $\\{W_{2}^{2}\\}$, and
$\\{W_{3}^{2}\\}$ has Cwebs (c), and (d)

.

As discussed in [3, 21], one of the major drawback of the above algorithm is
that it generates multiple copies of a Cweb and they need to be discarded
before starting the calculation of mixing matrices at a given perturbative
order.

#### New algorithm

We find that it is sufficient to use Cwebs at $l$ loops connecting $m$ Wilson
lines to generate all Cwebs at $l+1$ loops connecting the same number of
Wilson lines; also the third step of the original algorithm generates the
Cwebs that are already generated using the first and second steps. Based on
these, an improved and efficient algorithm to generate Cwebs recursively is
given below:

  1. 1.

To generate Cwebs at $l+1$ loops connecting at most $l+1$ Wilson lines
starting from a Cweb at $l$ loops connecting $m$ ($1\leq m\leq l+1$) lines

    1. (a)

Connect any two existing Wilson lines by introducing a two-point gluon
correlator (for massive Wilson lines a two point correlator can also be
attached to a single line),

    2. (b)

Connect any existing $k$-point gluon correlator to an existing Wilson line.

  2. 2.

To generate Cwebs at $l+1$ loops connecting the highest number ($l+2$) of
Wilson lines, the following steps need to be applied to Cwebs at $l$ loops
connecting the highest number $(l+1)$ of lines allowed at this order:

    1. (a)

Connect a new Wilson line to any of the existing lines by introducing a two-
point gluon correlator.

    2. (b)

Connect any existing $k$-point gluon correlator to a new Wilson line.

  3. 3.

Discard the duplicate Cwebs.

Let us now explain how the above algorithm works and also contrast it with the
older version of the algorithm. Let us denote the set of all Cwebs,
$\\{W_{n}^{(c_{2},\ldots,c_{p})}(k_{1},\ldots,k_{n})\\}$ that are present $l$
loops connecting $n$ lines by $\\{W_{n}^{l}\\}$. To generate
$\\{W_{n}^{l+1}\\}$ the earlier version of the algorithm required
implementation of operations on both $\\{W_{n}^{l}\\}$ and
$\\{W_{n-1}^{l}\\}$. In contrast, the new algorithm requires operations only
on $\\{W_{n}^{l}\\}$ — the Cwebs in $\\{W_{n}^{l+1}\\}$ generated from
$\\{W_{n-1}^{l}\\}$ need an extra gluon that can connect the existing lines or
blobs with a new Wilson line, however, we can simply start with
$\\{W_{n}^{l}\\}$ and attach a gluon to the existing lines or the blobs of the
correlators in all possible ways to generate $\\{W_{n}^{l+1}\\}$.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: Cwebs (webs) of $\\{W_{3}^{3}\\}$ generated using the non recursive
algorithm

We apply above reasoning to generate three loop Cwebs $\\{W_{3}^{3}\\}$ using
two loop Cwebs $\\{W_{3}^{2}\\}$. Cwebs at two loops, as shown in fig. (2)
fall into two sets: $\\{W_{2}^{2}\\}$ containing Cwebs (a) and (b), and
$\\{W_{3}^{2}\\}$ containing Cwebs (c) and (d). All the Cwebs at three-loops
connecting three lines [78, 3] are shown in the fig. (3). The same Cwebs are
generated if we use the algorithm on $\\{W_{3}^{2}\\}$. In the fig. (3) we
have indicated by dashed boxes the Cwebs that are contained in
$\\{W_{3}^{2}\\}$ in fig. (2); note that each of them has a box that contains
one of the Cwebs in $\\{W_{3}^{2}\\}$.

In a similar fashion Cwebs in the set $\\{W_{2}^{2}\\}$ are present in
$\\{W_{3}^{3}\\}$ except those in figs. (3b), and (3e). This example, thus,
provides a justification for our above algorithm.

#### Comparison of older and newer Mathematica implementations

The in-house Mathematica code CwebGen 2.0 that we have used to generate the
results reported in this paper incorporates a new subroutine that is based on
the more efficient algorithm of generating Cwebs presented above. With this
new algorithm we have reproduced the known results of Cwebs at three [78] and
four loops [3, 75] where multi-pronged-boomerang correlators do not appear,
which provides a check on our implementation. This algorithm is completely
general and applicable at any perturbative order. As mentioned in the
algorithm, certain Cwebs (colour factors) are generated multiple times and we
discard the duplicates to obtain the unique ones. At four loops, for Cwebs
without multi-pronged-boomerang correlator, the new algorithm generates $150$
Cwebs as compared to $226$ Cwebs that are generated by original algorithm;
after discarding the duplicates only $60$ remain [3, 75]. The new algorithm
generates $2.5$ times of the unique Cwebs, whereas the previous algorithm
generates $3.8$ times. Thus, this new algorithm increases the efficiency by
around $35\%$.

In this work, we have used the subroutine where the colour factors of one
diagram from each three-loop Boomerang Cwebs are given as inputs to generate
colour factors of Boomerang Cwebs at four loops. The older algorithm when
applied on each of the $9$ Boomerang Cwebs present at three loops, generates
$95$ Boomerang Cwebs, whereas, the new algorithm generates $71$. After
discarding the duplicates only $45$ remain at four loops. Thus the new
algorithm increases the efficiency in finding unique Cwebs by $25\%$. As we go
beyond four loops, the number of Cwebs at lower orders increase, thus we
expect that new algorithm will become more useful in finding Cwebs at higher
orders.

##  4 A brief description of the code CwebGen 2.0

One of the most powerful techniques in the combinatorial problems in physics,
which involves exponentiation, is the replica trick [89]. For Wilson line
correlators, the replica trick algorithm was developed in [74, 81]. The same
replica trick was adopted in [3, 75] for the calculation of the mixing
matrices for four-loop Cwebs. The details of replica trick — discussed in
appendix A — will be used in the calculation of mixing matrices for Boomerang
Cwebs at four loops.

We present below the algorithm of current version of the in-house Mathematica
code CwebGen 2.0, that incorporates replica trick algorithm and is a
significantly improved version of the codes that were used in [3, 75]. This
code has been used in this work to obtain the Cweb mixing matrices.

  * •

The code begins by generating the colour of one of the Cweb diagrams; however,
it is different from the usual colour assignment as we also assign replica
variables $\\{i,j,k,\ldots\\}$ to each of the gluon correlators that are
present in the diagram.

  * •

A subroutine then shuffles the attachments on each of the Wilson lines
belonging to different correlators, and correspondingly different replica
variables, to generate all the diagrams of the Cweb.

  * •

Next hierarchies between the replica variables are generated. For example, for
a Cweb with two correlators, with replica variables $i$ and $j$, the code
generates hierarchies $\\{h\\}=\\{i=j,i>j,i<j\\}$. Then number of distinct
replica variables $N_{r}$ for each hierarchy is calculated, for example, for
hierarchy $i=j$, $N_{r}=1$, whereas, for $i>j$, and $i<j$, $N_{r}=2$. Using
this, $M_{N_{r}}(h)$ is obtained using eq. (44).

  * Note that previous versions of the code could generate all the possible hierarchies only if the number of replica variables was less than or equal to 4. CwebGen 2.0 can generate the hierarchies for any number of replica variables.

  * •

Another subroutine then generates replica ordered colour factors
$\textbf{R}\big{[}C(d)\big{|}h\big{]}$ for each of the hierarchies (see
appendix A) for every diagram of the Cweb. The code then generates the data
for each diagram in the Cweb (as given in the Table 1 of [74]) and gives the
mixing matrix $R_{W}$. A diagonalizing matrix $Y_{W}$ is constructed to
diagonalize $R_{W}$ using its right eigenvectors. $Y_{W}$ then acts on the
column vector containing the colour factors of each of the diagrams of the
Cweb, and then gives the corresponding independent exponentiate colour
factors.

The runtime of the code depends on the number of correlators or equivalently
number of replica variables. The computation time for four loops is larger as
compared to three loops, since the maximum number of hierarchies at four loops
is 75 as opposed to 13 at three-loops. In addition to large number of
hierarchies, the total number of diagrams in any Boomerang Cwebs is larger
than that for Cwebs involving no multi-pronged-boomerang correlator, at a
given perturbative order.

The algorithm and its implementation in two older versions of the same code
can be found in [3, 75]. To make a comparison of the present version with the
previous versions, we have calculated the mixing matrix of
$\text{W}_{5}^{(4)}{(1,1,1,1,4)}$ appearing at four loops [3]. The largest
dimension mixing matrix is $24\times 24$ and it took $7$ days in the first
version [3], upon subsequent improvement it took $6.4$ hours [75]. The same
calculation by CwebGen 2.0 takes only $1.54$ seconds. Thus, the latest version
of code is almost $15000$ times faster as compared to the previous versions.

To test the efficiency of our code, we choose a class of Cwebs
$W^{n}_{n+1}(1,1,\ldots n)$ whose mixing matrices have dimension $n!$, where
$n$ is the perturbative order. Further, the Cwebs of this class have the
largest dimension of mixing matrices at a given perturbative order connecting
maximum number of lines. We run a benchmark test on the computation time for
the two loop Cweb that belongs to this category. We also provide the
computation times for the Cwebs of this category for three and four loops,
which are already known previously in the literature. To test the capability
of our code CwebGen 2.0 in handling higher orders, we have computed mixing
matrices of this category appearing at five and six loops. The computation
times of all these mixing matrices are provided in table (1). It is evident
that CwebGen 2.0 is not only well-suited for four-loop but is also capable of
handling higher loops.

loops | Dimension | Computation time
---|---|---
2 | 2 | 0.02 s
3 | 6 | 0.07 s
4 | 24 | 1.07 s
5 | 120 | 6.05 m
6 | 720 | 2.79 h
Table 1: Benchmark tests of CwebGen 2.0. All these tests are performed in a
computer with Intel (R) Xeon (R) CPU E5-2697 v4 @ 2.30GHz and 64GB RAM.

In the next section, we provide the results for two Boomerang Cwebs that are
present at four loops and the results for the remaining Cwebs will be
presented in the appendix C.

##  5 Boomerang Cwebs at Four loops

Using CwebGen 2.0 and discarding the duplicates we get 8 and 20 Boomerang
Cwebs connecting four and three Wilson lines respectively. In [2], we had
obtained the diagonal blocks of mixing matrices for all the 8 Boomerang Cwebs
that are present at four loops using the formalism introduced in [1].
Additionally, we were also able to make predictions for the diagonal blocks of
4 classes of Cwebs to all orders in $\alpha_{s}$. Of course, we could not
obtain the complete mixing matrices and exponentiated colour factors in that
work which is the subject matter of this paper.

Before going into the details of the calculation, let us briefly review the
concepts which will be useful in understanding the structure of the mixing
matrices presented in this paper. The mixing matrices of Cwebs have a general
structure, which is manifest, when the diagrams of these Cwebs are put in a
Normal order. This structure was first discovered in [1].

##### Normal Order:

First the diagrams of a Cweb are classified according to their $s$-factors: a
diagram $d$ is reducible if $s(d)\neq 0$, whereas it is irreducible if
$s(d)=0$. Further irreducible diagrams are classified into the following two
categories.

Completely entangled diagram: An irreducible diagram in which all the gluon
correlators are entangled and thus none of them can be independently shrunk to
the origin.

Partially entangled diagram: An irreducible diagram which has at least one
gluon correlator which is not entangled with the other correlators.
Normal Ordering of diagrams in a Cweb with $n$ diagrams, is then, defined as
the order in which first $k$ diagrams are completely entangled, followed by
$(l-k)$ partially entangled diagrams, then followed by reducible diagrams in
ascending order of their $s$-factors. The mixing matrix $R$ then takes the
following general form [1],

$\displaystyle R=\left(\begin{array}[]{cc}A_{l\times l}&B_{l\times m}\\\
O_{m\times l}&D_{m\times
m}\end{array}\right)\,=\left(\begin{array}[]{c|c}\begin{array}[]{cc}{I}_{k\times
k}&(A_{U})_{k\times(l-k)}\\\ {O}_{(l-k)\times
k}&\quad\,\,(A_{L})_{(l-k)\times(l-k)}\end{array}&B_{l\times m}\\\
\hline\cr{O}_{m\times l}&D_{m\times m}\end{array}\right)\,,$ (18)

where $l+m=n$. Here A and D are square matrices of order $l$, and $m$
corresponding to irreducible and reducible diagrams, respectively; O is null
matrix of order $m\times l$. Additionally $D$ itself is a mixing matrix as
proved in [1]. Further $A$ has a sub-structure — it contains an identity
matrix $I$ of order $k$ that corresponds to completely entangled diagrams, a
square matrix $A_{L}$ of order $(l-k)$ corresponding to partially entangled
diagrams, a null matrix $O$ of order $(l-k)\times k$, and a matrix $A_{U}$ of
order $k\times(l-k)$.

Above structure follows from the replica trick algorithm which can always
disentangle two correlators, however can never entangle them. Therefore, the
exponentiated colour factors of reducible diagrams do not contain the colour
of the irreducible diagrams; and those of completely entangled diagrams do not
contain the colour of the partially entangled diagrams. The diagrams of all
the Cwebs considered in this paper are Normal Ordered and thus the mixing
matrices of them have the form given in eq. (18).

We now present the results for two Cwebs: one connecting four Wilson lines,
and another connecting three Wilson lines. In each case we will present (i)
Column weight vector $S$, (ii) Mixing Matrix $R$, (iii) Exponentiated colour
factors $(YC)$.

####  $\text{W}\,_{4}^{(1,0,1)}\,(3,1,1,1)$: a four-line Boomerang Cweb

Boomerang Cweb $\text{W}\,_{4}^{(1,0,1)}\,(3,1,1,1)$, shown in fig (4),
connects four Wilson lines with one multi-pronged-boomerang correlator (m=2)
and a four point gluon correlator. This Cweb has three possible shuffles on
Wilson line $1$. The shuffle of gluon attachments from the two correlators
generates three diagrams as shown in fig. (4); thus the order of mixing matrix
for this Cweb is three. To proceed we choose the order of the diagrams as in
the fig. (4). This order is labeled using the order of attachments on line 1:
$C_{1}\,=\,\\{ABK\\}$, $C_{2}\,=\,\\{AKB\\}$ and $C_{3}\,=\,\\{BAK\\}$ .

(a)

(b)

(c)

Figure 4: Diagrams of Boomerang Cweb $\text{W}\,_{4}^{(1,0,1)}\,(3,1,1,1)$

The mixing matrix $R$ and the diagonalizing matrix $Y$ has been calculated
using CwebGen 2.0 and we get,

$\displaystyle R\,=\,\dfrac{1}{2}\left(\begin{array}[]{ccc}2&-1&-1\\\
0&\;\;\,1&-1\\\ 0&-1&\;\;\,1\\\ \end{array}\right)\,,\qquad\qquad\qquad
Y\,=\,\left(\begin{array}[]{ccc}1&-1&0\\\ 0&-1&1\\\ 0&\;\;\,1&1\\\
\end{array}\right)\,.$ (25)

This matches with the result obtained in [2].

The column weight vector $S$ for this Cweb can be calculated by determining
the $s$-factors [77] for each diagram. The $s$-factor for $C_{1}$ is zero as
one can not shrink any of the two correlators to the origin of Wilson lines.
In the diagram $C_{2}$ one can shrink the multi-pronged-boomerang correlator
first then the four point correlator, thus there is only one way to shrink all
the correlators to origin, making $s=1$. The $s$-factor for $C_{3}$ is also
one except the fact that here the multi-pronged-boomerang correlator has to be
shrunk after the four point correlator. Thus the column weight vector becomes
$S=\\{0,1,1\\}\,.$

The matrix $R$ satisfies all the known properties of mixing matrix: it is
idempotent, $R^{2}=R$, satisfies the zero row sum rule as in each row of
matrix sum of entries is zero, and, the conjectured column sum rule. The rank
of this mixing matrix is two as there are only two independent rows in $R$. We
obtain the independent exponentiated colour factors given as,

$\displaystyle(YC)_{1}\,=\,0\,,\qquad(YC)_{2}\,=\,-i\,f^{abn}f^{bch}f^{deh}\,\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{n}\mathbf{T}_{2}^{c}\mathbf{T}_{3}^{d}\mathbf{T}_{4}^{e}\,.$
(26)

Let us define

${N}_{ecf}=\textrm{number of non-vanishing independent ECFs}.$ (27)

For the above Cweb ${N}_{ecf}=1$, whereas the rank of the mixing matrix
$r(R)=2$. The vanishing of one of the independent ECFs can be understood in
terms of $\widetilde{C}$. Applying $R$ on the $C$ we get,

$\displaystyle\widetilde{C}_{1}=\dfrac{1}{2}\left(2C_{1}-C_{2}-C_{3}\right),\qquad\widetilde{C}_{2}=\dfrac{1}{2}\left(C_{2}-C_{3}\right),\qquad\widetilde{C}_{3}=\dfrac{1}{2}\left(C_{3}-C_{2}\right)\,.$
(28)

The colour factors of diagrams $C_{2}$ and $C_{3}$ are identical; the only
difference is due to the placement of the Boomerang in each of the diagrams,
however, it contributes the same factor $C_{A}$. Hence the exponentiated
colour factors $\widetilde{C}_{2}$ and $\widetilde{C}_{3}$ in the above
equations vanish. This makes ECFs for certain diagrams vanish for Boomerang
Cwebs and it holds true to all orders in the perturbation theory [76]. Recall
that when a Cweb is Normal Ordered, the block $D$ gives mixing between the
reducible diagrams. All these diagrams contain Boomerang that give $C_{A}$
factors, and thus their ECFs vanish. That is,

$\displaystyle{N}_{ecf}$ $\displaystyle=\text{rank}(R)\hskip
14.22636pt\text{Cwebs without {\color[rgb]{0,0,0}multi-pronged-boomerang
correlators}},$ $\displaystyle{N}_{ecf}$ $\displaystyle<\text{rank}(R)\hskip
14.22636pt\text{Cwebs}\;\text{{\color[rgb]{0,0,0} containing multi-pronged-
boomerang correlators}.}$ (29)

The results presented in the appendix also exhibit this property. The mixing
matrices and exponentiated colour factors for remaining seven Boomerang Cwebs
that connect four Wilson lines are given in appendix C.1.

####  $\text{W}\,_{3,\text{I}}^{(2,1)}(3,3,1)$: a three-line Boomerang Cweb

In $\text{W}\,_{3,\text{I}}^{(2,1)}(3,3,1)$, shown in fig. (5), one multi-
pronged-boomerang correlator (m=2), a two-point and a three-point gluon
correlator connect the three lines. The subscript “I” indicates that there are
more than one Cwebs with the same attachment and correlator content. The
shuffle of attachments generates nine diagrams; the order of diagrams is given
in table 2 along with their $s$-factors.

Figure 5: Boomerang Cweb $\text{W}\,_{3,\text{I}}^{(2,1)}(3,3,1)$

Diagrams | Sequences | s-factors
---|---|---
$C_{1}$ | $\\{\\{ABF\\},\\{GHL\\}\\}$ | 0
$C_{2}$ | $\\{\\{AFB\\},\\{GHL\\}\\}$ | 0
$C_{3}$ | $\\{\\{ABF\\},\\{GLH\\}\\}$ | 0
$C_{4}$ | $\\{\\{ABF\\},\\{HGL\\}\\}$ | 0
$C_{5}$ | $\\{\\{BAF\\},\\{GHL\\}\\}$ | 0

Diagrams | Sequences | s-factors
---|---|---
$C_{6}$ | $\\{\\{AFB\\},\\{GLH\\}\\}$ | 1
$C_{7}$ | $\\{\\{BAF\\},\\{HGL\\}\\}$ | 1
$C_{8}$ | $\\{\\{AFB\\},\\{HGL\\}\\}$ | 2
$C_{9}$ | $\\{\\{BAF\\},\\{GLH\\}\\}$ | 2

Table 2: Order of diagrams of Cweb $\text{W}\,_{3,\text{I}}^{(2,1)}(3,3,1)$
and their $s$-factors

Using these $s$-factors we obtain the column weight vector $S$:

$\displaystyle S\;=\;\\{0,0,0,0,0,1,1,2,2\\}\,.$ (30)

Using CwebGen 2.0 we obtain

$\displaystyle
R=\frac{1}{6}\left(\begin{array}[]{ccccccccc}6&-3&-3&-3&-3&2&2&1&1\\\
0&3&0&0&-3&-1&2&-2&1\\\ 0&0&3&-3&0&-1&2&1&-2\\\ 0&0&-3&3&0&2&-1&-2&1\\\
0&-3&0&0&3&2&-1&1&-2\\\ 0&0&0&0&0&2&2&-2&-2\\\ 0&0&0&0&0&2&2&-2&-2\\\
0&0&0&0&0&-1&-1&1&1\\\ 0&0&0&0&0&-1&-1&1&1\\\ \end{array}\right)\,.$ (40)

This satisfies all the known properties listed in section 2. This Cweb has
four independent exponentiated colour factors, two of which vanish due to the
presence of a Boomerang. The remaining two ECFs are,

$\displaystyle{(YC)}_{3}$
$\displaystyle=i\,f^{edk}f^{abn}f^{bcd}\,\textbf{T}_{1}^{a}\textbf{T}_{1}^{n}\textbf{T}_{2}^{c}\textbf{T}_{2}^{k}\textbf{T}_{3}^{e}+i\,f^{eck}\,f^{abn}\,f^{bcd}\,\textbf{T}_{1}^{a}\,\textbf{T}_{1}^{n}\,\textbf{T}_{2}^{k}\,\textbf{T}_{2}^{d}\,\textbf{T}_{3}^{e}\,$
$\displaystyle{(YC)}_{4}$
$\displaystyle=i\,f^{edk}\,f^{abn}\,f^{bcd}\,\textbf{T}_{1}^{a}\,\textbf{T}_{1}^{n}\,\textbf{T}_{2}^{c}\,\textbf{T}_{2}^{k}\,\textbf{T}_{3}^{e}\,.$

The results for all the twenty Boomerang Cwebs that connect three Wilson lines
at four loops are given in appendix C.2.

##  6 Conclusions

In this article we study Boomerang Cwebs that connect three and four Wilson
lines at four loops which are important ingredients in the studies of
scattering involving massive particles.

To enumerate Cwebs at a given perturbative order, we have introduced an
improved version of the algorithm that was developed in [3]. We presented some
details of the current version of the in-house Mathematica code CwebGen 2.0,
that incorporates this new algorithm, and the replica trick, and it is a
significantly faster version of the codes that were used in [3, 75].

Figure 6: Exponentiated colour factor obtained from all the Boomerang Cwebs

We have computed the mixing matrices, the diagonalizing matrices and the
exponentiated colour factors for all these Cwebs. We have verified that our
results match with the predictions of the diagonal blocks presented in [2]. We
found that the general structure of exponentiated colour factor of all the
twenty eight Boomerang Cwebs, shown in fig. (6), is same as the general
structure of ECFs for the Cwebs without the multi-pronged-boomerang
correlators at four loops [3, 75]. This is an artifact of the Cwebs with
multi-pronged-boomerang correlator, for which quadratic Casimir $C_{A}$ (the
colour of a Boomerang) is absent from all the exponentiated colour factors.
This is in agreement with the calculations of Boomerang Cwebs at three loops
[76]. We found that the ECFs corresponding to the $D$ block of mixing matrices
vanish which is consistent with [76]. The exponentiated colour factors have
long expressions for Cwebs connecting three Wilson lines and thus we refrain
from presenting it in this article. However, they can be obtained from the
authors upon request as FORM files.

The interplay of colour and kinematics in the study of soft anomalous
dimension has been an interesting object of study and it often puts
constraints on the general structure of the anomalous dimension. It will be an
interesting study to determine the constraints on the massive soft anomalous
dimension based on the exponentiated colour factors presented in this article.

## Acknowledgments

NA, SP, AT would like to thank Prof. Lorenzo Magnea for collaboration on the
related earlier projects. AS would like to thank CSIR, Govt. of India, for a
SRF Fellowship (09/1001(0075)/2020-EMR-I). A part of the computing was
performed in the TDP resources at the Physical Research Laboratory (PRL).

##  7 Appendix

##  Appendix A Replica trick and mixing matrices

In this appendix we briefly discuss the replica trick algorithm, which will be
used in the calculation of the mixing matrices for Boomerang Cwebs at four
loops. To start with, we consider the path integral of the Wilson line
correlators as,

$\displaystyle\mathcal{S}_{n}(\gamma_{i})=\,\int\mathcal{D}A_{\mu}^{a}\,\exp(iS(A_{\mu}^{a}))\prod_{k=1}^{n}\phi_{k}(\gamma_{k})=\exp[\mathcal{W}_{n}(\gamma_{i})]\,,$
(41)

where $S(A_{\mu}^{a})$ is the classical action of the gauge fields. In order
to proceed with the replica trick algorithm, one introduces $N_{r}$ non-
interacting identical copies of each gluon field $A_{\mu}$, which means, we
replace each $A_{\mu}$ by $A_{\mu}^{i}$, where, $i=1,\ldots,N_{r}$. Now, for
each replica, we associate a copy of each Wilson line, thereby, replacing each
Wilson line by a product of $N_{r}$ Wilson lines. Thus, in the replicated
theory, the path integral of the Wilson line correlator can then be written
as,

$\displaystyle{\cal S}_{n}^{\,{\rm
repl.}}\left(\gamma_{i}\right)\,=\,\Big{[}{\cal
S}_{n}\left(\gamma_{i}\right)\Big{]}^{N_{r}}\,=\,\exp\Big{[}N_{r}\,{\cal
W}_{n}(\gamma_{i})\Big{]}\,=\,{\bf 1}+N_{r}\,{\cal W}_{n}(\gamma_{i})+{\cal
O}(N_{r}^{2})\,.$ (42)

Now, using this equation, one can calculate $\mathcal{W}_{n}$ by calculating
$\mathcal{O}(N_{r})$ terms of the Wilson line correlator in the replicated
theory. The method of replicas involves five steps, which are summarized
below.

  1. -

Associate a replica number to each connected gluon correlator in a Cweb.

  2. -

Define a replica ordering operator R, which acts on the colour generators on
each Wilson line and order them according to their replica numbers. Thus, if
$\textbf{T}_{i}$ denotes a colour generator for a correlator belonging to
replica number $i$, then action of R on $\textbf{T}_{i}\textbf{T}_{j}$
preserves the order for $i\leq j$, and reverses the order for $i>j$. Thus,
replica ordered colour factor for a diagram in a Cweb will always be a diagram
of the same Cweb.

  3. -

The next step in order to calculate the exponentiated colour factors, one
needs to find the hierarchies between the replica numbers present in a Cweb.
If a Cweb has $m$ connected pieces, we call hierarchies $h(m)$. $h(m)$ are
known as Bell number or Fubini number [90] in the number theory and
combinatorics. The first few Fubini numbers are given by
$h(m)=\\{1,1,3,13,75,541\\}$ for $m=0,1,2,3,4,5$. At four loops, the highest
number of correlator in a Cweb is $m_{\text{max}}=4$, which corresponds to
$h_{\text{max}}=75$.

  4. -

The next object is to calculate $M_{N_{r}}(h)$, which counts the number of
appearances of a particular hierarchy in the presence of $N_{r}$ replicas. For
a given hierarchy $h$, which contains $n_{r}(h)$ distinct replicas, the
multiplicity $M_{N_{r}}(h)$ is given by,

$\displaystyle
M_{N_{r}}(h)\,=\,\frac{N_{r}!}{\big{(}N_{r}-n_{r}(h)\big{)}!\,\,n_{r}(h)!}\,.$
(43)

  5. -

The exponentiated colour factor for a diagram $D$ is then given by,

$\displaystyle C_{N_{r}}^{\,{\rm
repl.}}(D)\,=\,\sum_{h}M_{N_{r}}(h)\,\textbf{R}\big{[}C(D)\big{|}h\big{]}\,,$
(44)

where $\textbf{R}\big{[}C(D)\big{|}h$ is the replica ordered colour factor of
diagram $D$, for hierarchy $h$. Finally, the exponentiated colour factor for
diagram $D$ is computed by extracting the coefficient of $\mathcal{O}(N_{r})$
terms of the above equation.

##  Appendix B Calculation of $s$-factor in different representations

In this section, we demonstrate the computation of $s$-factors by shrinking
correlators in two different representations. Let us consider the Cweb
$W^{(1,0,1)}_{5}(2,1,1,1,1)$. Here, in fig. 7a the hard interaction vertex is
located at the point where tails of all the Wilson lines meet. The same
diagram is drawn in fig. 7b where the arrows on the Wilson lines are used to
indicate the hard vertex: the tail ends of each of the Wilson lines meet at
the hard vertex.

(a)  (b)

Figure 7: Two different representations of Cweb $W^{(1,0,1)}_{5}(2,1,1,1,1)$

In fig. (7a) we first shrink the four gluon correlator to the hard interaction
vertex, followed by the two gluon correlator this gives $s=1$ for this
diagram. Now we describe the procedure for finding the $s$-factors when the
diagrams are drawn as in fig. (7b). Pull each of the gluon correlators to the
tail ends of the associated Wilson lines without dragging the attachments from
other correlators. For example in fig. (7b), we can pull the four gluon
correlator to the tails of attached Wilson lines (1, 2, 3 and 4) followed by
the pulling of two gluon correlator to the tails of Wilson lines 1 and 5. This
is the only way we can sequentially pull these gluon correlators without
dragging other correlators in the process, giving $s=1$. Following the same
steps one can compute the $s$-factors for any diagram in any of the
representations.

##  Appendix C Boomerang Cwebs at Four loops

###  C.1 Boomerang Cwebs connecting four Wilson lines

  * 1.

$\textbf{W}\,_{4}^{(4)}(3,1,3,1)$

This Cweb is made up with four two-gluon correlator. It contains eighteen
diagrams, one of which is displayed below. The table below shows the chosen
order of the eighteen shuffles of the gluon attachments, their corresponding
$s$-factors.
Diagrams Sequences s-factors $C_{1}$ $\\{\\{ACK\\},\\{FHD\\}\\}$ 0 $C_{2}$
$\\{\\{ACK\\},\\{FDH\\}\\}$ 0 $C_{3}$ $\\{\\{ACK\\},\\{HFD\\}\\}$ 0 $C_{4}$
$\\{\\{ACK\\},\\{HDF\\}\\}$ 0 $C_{5}$ $\\{\\{ACK\\},\\{DFH\\}\\}$ 0 $C_{6}$
$\\{\\{ACK\\},\\{DHF\\}\\}$ 0 $C_{7}$ $\\{\\{AKC\\},\\{HFD\\}\\}$ 1 $C_{8}$
$\\{\\{AKC\\},\\{HDF\\}\\}$ 1 $C_{9}$ $\\{\\{CAK\\},\\{FDH\\}\\}$ 1 $C_{10}$
$\\{\\{CAK\\},\\{DFH\\}\\}$ 1 $C_{11}$ $\\{\\{AKC\\},\\{FHD\\}\\}$ 2 $C_{12}$
$\\{\\{AKC\\},\\{FDH\\}\\}$ 2 $C_{13}$ $\\{\\{AKC\\},\\{DFH\\}\\}$ 2 $C_{14}$
$\\{\\{AKC\\},\\{DHF\\}\\}$ 2 $C_{15}$ $\\{\\{CAK\\},\\{FHD\\}\\}$ 2 $C_{16}$
$\\{\\{CAK\\},\\{HFD\\}\\}$ 2 $C_{17}$ $\\{\\{CAK\\},\\{HDF\\}\\}$ 2 $C_{18}$
$\\{\\{CAK\\},\\{DHF\\}\\}$ 2

The mixing matrix, and the diagonal matrix are given by,

$\displaystyle\begin{split}R=\frac{1}{12}\scalebox{0.9}{\mbox{$\displaystyle\left(\begin{array}[]{cccccccccccccccccc}4&-2&-2&-2&4&-2&1&-3&-1&1&2&-3&1&2&0&1&-1&0\\\
-2&4&-2&4&-2&-2&-3&1&1&-1&2&1&-3&2&0&-1&1&0\\\
-2&-2&4&-2&-2&4&1&1&1&1&-2&1&1&-2&-2&1&1&-2\\\
-2&4&-2&4&-2&-2&-1&1&1&-3&0&1&-1&0&2&-3&1&2\\\
4&-2&-2&-2&4&-2&1&-1&-3&1&0&-1&1&0&2&1&-3&2\\\
-2&-2&4&-2&-2&4&1&1&1&1&-2&1&1&-2&-2&1&1&-2\\\
0&0&0&0&0&0&3&-1&1&-3&-2&-1&3&-2&2&-3&1&2\\\
0&0&0&0&0&0&-1&3&-3&1&-2&3&-1&-2&2&1&-3&2\\\
0&0&0&0&0&0&1&-3&3&-1&2&-3&1&2&-2&-1&3&-2\\\
0&0&0&0&0&0&-3&1&-1&3&2&1&-3&2&-2&3&-1&-2\\\
0&0&0&0&0&0&-1&-1&1&1&2&-1&-1&2&-2&1&1&-2\\\
0&0&0&0&0&0&-1&1&-1&1&0&1&-1&0&0&1&-1&0\\\
0&0&0&0&0&0&1&-1&1&-1&0&-1&1&0&0&-1&1&0\\\
0&0&0&0&0&0&-1&-1&1&1&2&-1&-1&2&-2&1&1&-2\\\
0&0&0&0&0&0&1&1&-1&-1&-2&1&1&-2&2&-1&-1&2\\\
0&0&0&0&0&0&-1&1&-1&1&0&1&-1&0&0&1&-1&0\\\
0&0&0&0&0&0&1&-1&1&-1&0&-1&1&0&0&-1&1&0\\\
0&0&0&0&0&0&1&1&-1&-1&-2&1&1&-2&2&-1&-1&2\\\
\end{array}\right)$}}\,,\end{split}$ $\displaystyle\mathcal{D}\,=\,$
$\displaystyle(\mathbf{1}_{4},0)\,.$ (45)

The exponentiated colour factors are given by,

$\displaystyle(YC)_{1}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle(YC)_{2}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle(YC)_{3}$ $\displaystyle=$ $\displaystyle
if^{acn}f^{bkg}f^{cdk}\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{n}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{g}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{4}$ $\displaystyle=$ $\displaystyle
if^{acn}f^{bdr}f^{crm}\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{n}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{m}\mathbf{T}_{4}^{d}\,.$
(46)

  * 2.

$\textbf{W}\,_{4}^{(4)}(4,1,2,1)$

This is a Boomerang Cweb which is made up with four two-gluon correlators. It
has twenty-four diagrams, and one of them is shown below. The table below
shows chosen order of shuffles and their corresponding $s$-factors.

Diagrams | Sequences | s-factors
---|---|---
$C_{1}$ | $\\{\\{ABKC\\},\\{FD\\}\\}$ | 0
$C_{2}$ | $\\{\\{ABKC\\},\\{DF\\}\\}$ | 0
$C_{3}$ | $\\{\\{ABCK\\},\\{FD\\}\\}$ | 0
$C_{4}$ | $\\{\\{ABCK\\},\\{DF\\}\\}$ | 0
$C_{5}$ | $\\{\\{ACKB\\},\\{FD\\}\\}$ | 0
$C_{6}$ | $\\{\\{ACKB\\},\\{DF\\}\\}$ | 0
$C_{7}$ | $\\{\\{ACBK\\},\\{FD\\}\\}$ | 0
$C_{8}$ | $\\{\\{ACBK\\},\\{DF\\}\\}$ | 0
$C_{9}$ | $\\{\\{BACK\\},\\{FD\\}\\}$ | 0
$C_{10}$ | $\\{\\{BACK\\},\\{DF\\}\\}$ | 0
$C_{11}$ | $\\{\\{CABK\\},\\{FD\\}\\}$ | 0
$C_{12}$ | $\\{\\{CABK\\},\\{DF\\}\\}$ | 0
$C_{13}$ | $\\{\\{AKBC\\},\\{FD\\}\\}$ | 1
$C_{14}$ | $\\{\\{BAKC\\},\\{FD\\}\\}$ | 1
$C_{15}$ | $\\{\\{CAKB\\},\\{DF\\}\\}$ | 1
$C_{16}$ | $\\{\\{CBAK\\},\\{DF\\}\\}$ | 1
$C_{17}$ | $\\{\\{AKBC\\},\\{DF\\}\\}$ | 2
$C_{18}$ | $\\{\\{AKCB\\},\\{FD\\}\\}$ | 2
$C_{19}$ | $\\{\\{AKCB\\},\\{DF\\}\\}$ | 2
$C_{20}$ | $\\{\\{BAKC\\},\\{DF\\}\\}$ | 2
$C_{21}$ | $\\{\\{BCAK\\},\\{FD\\}\\}$ | 2
$C_{22}$ | $\\{\\{BCAK\\},\\{DF\\}\\}$ | 2
$C_{23}$ | $\\{\\{CAKB\\},\\{FD\\}\\}$ | 2
$C_{24}$ | $\\{\\{CBAK\\},\\{FD\\}\\}$ | 2

The mixing matrix and the diagonal matrix for this Cweb are given by,

$\displaystyle\begin{split}R=\frac{1}{12}\scalebox{0.8}{\mbox{$\displaystyle\left(\begin{array}[]{cccccccccccccccccccccccc}4&-4&0&0&0&0&0&0&0&0&-4&4&-1&-1&-3&-3&1&-2&2&1&-2&2&3&3\\\
-2&2&0&0&0&0&0&0&0&0&2&-2&1&1&1&1&-1&0&0&-1&0&0&-1&-1\\\
-2&2&6&-6&-4&4&0&0&-2&2&-4&4&-1&1&-3&-3&1&2&-2&-1&-2&2&3&3\\\
4&-4&-6&6&2&-2&0&0&4&-4&2&-2&1&-3&1&1&-1&-2&2&3&0&0&-1&-1\\\
0&0&0&0&2&-2&0&0&-2&2&0&0&-1&1&1&1&1&0&0&-1&2&-2&-1&-1\\\
0&0&0&0&-4&4&0&0&4&-4&0&0&1&-3&-1&-1&-1&2&-2&3&-2&2&1&1\\\
-2&2&0&0&-4&4&6&-6&-2&2&-4&4&1&1&-3&1&-1&0&0&-1&2&-2&3&-1\\\
4&-4&0&0&2&-2&-6&6&4&-4&2&-2&-3&-3&1&-1&3&2&-2&3&-2&2&-1&1\\\
0&0&0&0&-4&4&0&0&4&-4&0&0&-1&-1&-3&1&1&2&-2&1&-2&2&3&-1\\\
0&0&0&0&2&-2&0&0&-2&2&0&0&1&1&1&-1&-1&-2&2&-1&0&0&-1&1\\\
-2&2&0&0&0&0&0&0&0&0&2&-2&1&1&1&1&-1&0&0&-1&0&0&-1&-1\\\
4&-4&0&0&0&0&0&0&0&0&-4&4&-3&-3&-1&-1&3&2&-2&3&2&-2&1&1\\\
0&0&0&0&0&0&0&0&0&0&0&0&3&-1&1&-3&-3&-2&2&1&-2&2&-1&3\\\
0&0&0&0&0&0&0&0&0&0&0&0&-1&3&-3&1&1&-2&2&-3&-2&2&3&-1\\\
0&0&0&0&0&0&0&0&0&0&0&0&1&-3&3&-1&-1&2&-2&3&2&-2&-3&1\\\
0&0&0&0&0&0&0&0&0&0&0&0&-3&1&-1&3&3&2&-2&-1&2&-2&1&-3\\\
0&0&0&0&0&0&0&0&0&0&0&0&-1&1&-1&1&1&0&0&-1&0&0&1&-1\\\
0&0&0&0&0&0&0&0&0&0&0&0&-1&-1&1&1&1&2&-2&1&2&-2&-1&-1\\\
0&0&0&0&0&0&0&0&0&0&0&0&1&1&-1&-1&-1&-2&2&-1&-2&2&1&1\\\
0&0&0&0&0&0&0&0&0&0&0&0&1&-1&1&-1&-1&0&0&1&0&0&-1&1\\\
0&0&0&0&0&0&0&0&0&0&0&0&-1&-1&1&1&1&2&-2&1&2&-2&-1&-1\\\
0&0&0&0&0&0&0&0&0&0&0&0&1&1&-1&-1&-1&-2&2&-1&-2&2&1&1\\\
0&0&0&0&0&0&0&0&0&0&0&0&-1&1&-1&1&1&0&0&-1&0&0&1&-1\\\
0&0&0&0&0&0&0&0&0&0&0&0&1&-1&1&-1&-1&0&0&1&0&0&-1&1\\\
\end{array}\right)$}}\end{split}$ (47)
$\displaystyle\mathcal{D}\,=\,(\mathbf{1}_{6},0)$ (48)

Thus, six independent exponentiated colour factors are given by,

$\displaystyle(YC)_{1}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle(YC)_{2}$ $\displaystyle=$ $\displaystyle
if^{abg}f^{cdk}f^{cng}\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{n}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{k}\mathbf{T}_{4}^{d}\,,$
$\displaystyle+\;if^{abg}f^{acn}f^{cdk}\mathbf{T}_{1}^{n}\mathbf{T}_{1}^{g}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{k}\mathbf{T}_{4}^{d}\,,;$
$\displaystyle(YC)_{3}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle(YC)_{4}$ $\displaystyle=$ $\displaystyle
i\,f^{abn}f^{ack}f^{cdm}\,\mathbf{T}_{1}^{n}\mathbf{T}_{1}^{k}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{m}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{5}$ $\displaystyle=$ $\displaystyle
i\,f^{ang}f^{bcn}f^{cdk}\,\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{g}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{k}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{6}$ $\displaystyle=$ $\displaystyle
i\,f^{ack}f^{bkg}f^{cdr}\,\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{g}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{r}\mathbf{T}_{4}^{d}\,.$
(49)

  * 3.

$\textbf{W}\,_{4}^{(4)}\,(3,1,2,2)$

This Cweb has twelve diagrams, one of them is shown below. The table shows the
chosen order of shuffle and the corresponding $s$-factors.
Diagrams Sequences s-factors $C_{1}$ $\\{\\{ADK\\},\\{CH\\},\\{EF\\}\\}$ 0
$C_{2}$ $\\{\\{ADK\\},\\{CH\\},\\{FE\\}\\}$ 0 $C_{3}$
$\\{\\{ADK\\},\\{HC\\},\\{EF\\}\\}$ 0 $C_{4}$
$\\{\\{ADK\\},\\{HC\\},\\{FE\\}\\}$ 0 $C_{5}$
$\\{\\{AKD\\},\\{HC\\},\\{FE\\}\\}$ 1 $C_{6}$
$\\{\\{DAK\\},\\{CH\\},\\{EF\\}\\}$ 1 $C_{7}$
$\\{\\{AKD\\},\\{CH\\},\\{EF\\}\\}$ 2 $C_{8}$
$\\{\\{AKD\\},\\{HC\\},\\{EF\\}\\}$ 2 $C_{9}$
$\\{\\{DAK\\},\\{CH\\},\\{FE\\}\\}$ 2 $C_{10}$
$\\{\\{DAK\\},\\{HC\\},\\{FE\\}\\}$ 2 $C_{11}$
$\\{\\{AKD\\},\\{CH\\},\\{FE\\}\\}$ 4 $C_{12}$
$\\{\\{DAK\\},\\{HC\\},\\{EF\\}\\}$ 4

The mixing matrix, and the diagonal matrix are given by,

$\displaystyle\begin{split}R=\,\frac{1}{12}\left(\begin{array}[]{cccccccccccc}4&-4&-4&4&-3&-1&-3&3&1&-1&3&1\\\
-2&2&2&-2&1&1&1&-1&-1&1&-1&-1\\\ -2&2&2&-2&1&1&1&-1&-1&1&-1&-1\\\
4&-4&-4&4&-1&-3&-1&1&3&-3&1&3\\\ 0&0&0&0&3&-3&3&-3&3&-3&-3&3\\\
0&0&0&0&-3&3&-3&3&-3&3&3&-3\\\ 0&0&0&0&1&-1&1&-1&1&-1&-1&1\\\
0&0&0&0&-1&1&-1&1&-1&1&1&-1\\\ 0&0&0&0&1&-1&1&-1&1&-1&-1&1\\\
0&0&0&0&-1&1&-1&1&-1&1&1&-1\\\ 0&0&0&0&-1&1&-1&1&-1&1&1&-1\\\
0&0&0&0&1&-1&1&-1&1&-1&-1&1\\\
\end{array}\right),\,\end{split}\mathcal{D}\,=\,(\mathbf{1}_{2},0)\,.$

The rank of the mixing matrix is 2, which corresponds to the following
exponentiated colour factors,

$\displaystyle(YC)_{1}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle(YC)_{2}$ $\displaystyle=$ $\displaystyle
i\,f^{adg}f^{bcn}f^{cdk}\,\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{g}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{k}\mathbf{T}_{4}^{n}\,.$
(50)

  * 4.

$\textbf{W}\,_{4}^{(2,1)}\,(3,1,2,1)$

This is first of two Cwebs which has the same correlator and attachment
content. It has six diagrams, one of which is shown below. The table below
shows the chosen order of shuffles, and their corresponding $s$-factors.
Diagrams Sequences s-factors $C_{1}$ $\\{\\{ABK\\},\\{DE\\}\\}$ 0 $C_{2}$
$\\{\\{ABK\\},\\{ED\\}\\}$ 0 $C_{3}$ $\\{\\{AKB\\},\\{DE\\}\\}$ 1 $C_{4}$
$\\{\\{BAK\\},\\{DE\\}\\}$ 1 $C_{5}$ $\\{\\{BAK\\},\\{ED\\}\\}$ 2 $C_{6}$
$\\{\\{AKB\\},\\{ED\\}\\}$ 2

The mixing matrix, and the diagonal matrix are given by,

$\displaystyle\begin{split}R=&\,\frac{1}{6}\left(\begin{array}[]{cccccc}3&-3&-1&2&1&-2\\\
-3&3&2&-1&-2&1\\\ 0&0&2&2&-2&-2\\\ 0&0&2&2&-2&-2\\\ 0&0&-1&-1&1&1\\\
0&0&-1&-1&1&1\\\
\end{array}\right)\,,\mathcal{D}\,=\,(\mathbf{1}_{2},0)\,.\end{split}$ (51)

Finally, the exponentiated colour factors are given by,

$\displaystyle(YC)_{1}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle(YC)_{2}$ $\displaystyle=$ $\displaystyle
i\,f^{abn}f^{bcd}f^{dme}\,\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{n}\mathbf{T}_{2}^{c}\mathbf{T}_{3}^{m}\mathbf{T}_{4}^{e}\,.$
(52)

  * 5.

$\textbf{W}\,_{4}^{(2,1)}(3,2,1,1)$

This is the second Cweb with same correlator and attachment content. It has
six diagrams, one of them is shown below. The table shows the chosen order of
shuffle and their corresponding $s$-factors.
Diagrams Sequences s-factors $C_{1}$ $\\{\\{ABK\\}\\{CF\\}\\}$ 0 $C_{2}$
$\\{\\{ABK\\}\\{FC\\}\\}$ 0 $C_{3}$ $\\{\\{AKB\\}\\{FC\\}\\}$ 1 $C_{4}$
$\\{\\{BAK\\}\\{FC\\}\\}$ 1 $C_{5}$ $\\{\\{BAK\\}\\{CF\\}\\}$ 2 $C_{6}$
$\\{\\{AKB\\}\\{CF\\}\\}$ 2

The mixing matrix and the diagonal matrix for this Cweb are given by,

$\displaystyle\begin{split}R=\frac{1}{6}\left(\begin{array}[]{cccccc}3&-3&2&-1&-2&1\\\
-3&3&-1&2&1&-2\\\ 0&0&2&2&-2&-2\\\ 0&0&2&2&-2&-2\\\ 0&0&-1&-1&1&1\\\
0&0&-1&-1&1&1\\\
\end{array}\right),\,\mathcal{D}\,=\,(\mathbf{1}_{2},0)\,.\end{split}$ (53)

Finally, the exponentiated colour factors are given by,

$\displaystyle(YC)_{1}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle(YC)_{2}$ $\displaystyle=$ $\displaystyle
i\,f^{abk}f^{abg}f^{cde}\,\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{g}\mathbf{T}_{2}^{k}\mathbf{T}_{3}^{d}\mathbf{T}_{4}^{e}\,.$
(54)

  * 6.

$\textbf{W}\,_{4}^{(2,1)}\,(4,1,1,1)$

This Cweb is made up with one three-gluon correlator, and two two-gluon
correlators. This has 12 diagrams, one of them is shown below. The table shows
the chosen order shuffle on Wilson line $1$, and their corresponding
$s$-factors.
Diagrams Sequences s-factors $C_{1}$ $\\{\\{ABCK\\}\\}$ 0 $C_{2}$
$\\{\\{ACBK\\}\\}$ 0 $C_{3}$ $\\{\\{ABKC\\}\\}$ 0 $C_{4}$ $\\{\\{ACKB\\}\\}$ 0
$C_{5}$ $\\{\\{BACK\\}\\}$ 0 $C_{6}$ $\\{\\{CABK\\}\\}$ 0 $C_{7}$
$\\{\\{AKBC\\}\\}$ 1 $C_{8}$ $\\{\\{AKCB\\}\\}$ 1 $C_{9}$ $\\{\\{BAKC\\}\\}$ 1
$C_{10}$ $\\{\\{BCAK\\}\\}$ 1 $C_{11}$ $\\{\\{CAKB\\}\\}$ 1 $C_{12}$
$\\{\\{CBAK\\}\\}$ 1

The $R$, and $D$ matrices are given by,

$\displaystyle\begin{split}R=\frac{1}{6}\left(\begin{array}[]{cccccccccccc}6&0&-3&-3&-3&-3&-1&2&2&-1&2&2\\\
0&6&-3&-3&-3&-3&2&-1&2&2&2&-1\\\ 0&0&3&0&0&-3&-1&-1&-1&-1&2&2\\\
0&0&0&3&-3&0&-1&-1&2&2&-1&-1\\\ 0&0&0&-3&3&0&-1&2&-1&-1&2&-1\\\
0&0&-3&0&0&3&2&-1&2&-1&-1&-1\\\ 0&0&0&0&0&0&2&-1&-1&-1&-1&2\\\
0&0&0&0&0&0&-1&2&-1&2&-1&-1\\\ 0&0&0&0&0&0&-1&-1&2&-1&2&-1\\\
0&0&0&0&0&0&-1&2&-1&2&-1&-1\\\ 0&0&0&0&0&0&-1&-1&2&-1&2&-1\\\
0&0&0&0&0&0&2&-1&-1&-1&-1&2\\\
\end{array}\right),\,\end{split}\mathcal{D}\,=\,(\mathbf{1}_{6},0)\,.$ (55)

Finally, the exponentiated colour factors are given by,

$\displaystyle(YC)_{1}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle(YC)_{2}$ $\displaystyle=$
$\displaystyle-i\,f^{abg}f^{cde}f^{ckg}\,\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{k}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{d}\mathbf{T}_{4}^{e}\,$
$\displaystyle-i\,f^{abg}f^{ack}f^{cde}\,\mathbf{T}_{1}^{k}\mathbf{T}_{1}^{g}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{d}\mathbf{T}_{4}^{e},$
$\displaystyle(YC)_{3}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle(YC)_{4}$ $\displaystyle=$
$\displaystyle-i\,f^{abg}f^{ack}f^{cde}\,\mathbf{T}_{1}^{g}\mathbf{T}_{1}^{k}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{d}\mathbf{T}_{4}^{e}\,,$
$\displaystyle(YC)_{5}$ $\displaystyle=$ $\displaystyle
i\,f^{akg}f^{bcg}f^{cde}\,\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{k}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{d}\mathbf{T}_{4}^{e}\,,$
$\displaystyle(YC)_{6}$ $\displaystyle=$
$\displaystyle-i\,f^{ack}f^{bkg}f^{cde}\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{g}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{d}\mathbf{T}_{4}^{e}\,.$
(56)

  * 7.

$\textbf{W}\,_{4}^{(1,0,1)}\,(3,1,1,1)$

This Cweb has one four-gluon correlator and one two-gluon correlator. It has
three diagrams, one of them is shown below. The figure below shows one of the
three diagrams for this Cweb, and the table gives chosen order of shuffles and
their corresponding $s$-factors.
Diagrams Sequences s-factors $C_{1}$ $\\{\\{ABK\\}\\}$ 0 $C_{2}$
$\\{\\{AKB\\}\\}$ 1 $C_{3}$ $\\{\\{BAK\\}\\}$ 1

The $R$, and $D$ matrices are given by,

$\displaystyle\begin{split}R=\frac{1}{2}\left(\begin{array}[]{ccc}2&-1&-1\\\
0&1&-1\\\ 0&-1&1\\\
\end{array}\right),\,\mathcal{D}\,=\,&(\mathbf{1}_{2},0)\,.\end{split}$ (57)

This mixing matrix agrees with the universal form of $3\times 3$ mixing
matrices computed in [75]. Finally, the exponentiated colour factors are given
by,

$\displaystyle(YC)_{1}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle(YC)_{2}$ $\displaystyle=$
$\displaystyle-i\,f^{abn}f^{bch}f^{deh}\,\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{n}\mathbf{T}_{2}^{c}\mathbf{T}_{3}^{d}\mathbf{T}_{4}^{e}\,.$
(58)

  * 8.

$\textbf{W}\,_{4}^{(4)}(5,1,1,1)$

This is the largest Boomerang Cweb that can connect four Wilson lines. It has
sixty diagrams, one of them is shown below. The following table shows the
chosen order of shuffle and their corresponding. We refrain to present the
mixing matrix fo this Cweb here due to its larger dimension, however it can be
found in the ancillary file Boomerang.nb.

Diagrams | Sequences | s-factors
---|---|---
$C_{1}$ | $\\{\\{ABCDK\\}\\}$ | 0
$C_{2}$ | $\\{\\{ABCDK\\}\\}$ | 0
$C_{3}$ | $\\{\\{ACBDK\\}\\}$ | 0
$C_{4}$ | $\\{\\{ACDBK\\}\\}$ | 0
$C_{5}$ | $\\{\\{ADBCK\\}\\}$ | 0
$C_{6}$ | $\\{\\{ADCBK\\}\\}$ | 0
$C_{7}$ | $\\{\\{ABKCD\\}\\}$ | 0
$C_{8}$ | $\\{\\{ABCDC\\}\\}$ | 0
$C_{9}$ | $\\{\\{ABCKD\\}\\}$ | 0
$C_{10}$ | $\\{\\{ABDKC\\}\\}$ | 0
$C_{11}$ | $\\{\\{ACKBD\\}\\}$ | 0
$C_{12}$ | $\\{\\{ACKDB\\}\\}$ | 0
$C_{13}$ | $\\{\\{ACBKD\\}\\}$ | 0
$C_{14}$ | $\\{\\{ACDKB\\}\\}$ | 0
$C_{15}$ | $\\{\\{ADKBC\\}\\}$ | 0
$C_{16}$ | $\\{\\{ADKCB\\}\\}$ | 0
$C_{17}$ | $\\{\\{ADBKC\\}\\}$ | 0
$C_{18}$ | $\\{\\{ADCKB\\}\\}$ | 0
$C_{19}$ | $\\{\\{BACKD\\}\\}$ | 0
$C_{20}$ | $\\{\\{BACDK\\}\\}$ | 0
$C_{21}$ | $\\{\\{BADKC\\}\\}$ | 0
$C_{22}$ | $\\{\\{BADCK\\}\\}$ | 0
$C_{23}$ | $\\{\\{BCADK\\}\\}$ | 0
$C_{24}$ | $\\{\\{BDACK\\}\\}$ | 0
$C_{25}$ | $\\{\\{CABKD\\}\\}$ | 0
$C_{26}$ | $\\{\\{CABDK\\}\\}$ | 0
$C_{27}$ | $\\{\\{CADKB\\}\\}$ | 0
$C_{28}$ | $\\{\\{CADBK\\}\\}$ | 0
$C_{29}$ | $\\{\\{CBADK\\}\\}$ | 0
$C_{30}$ | $\\{\\{CDABK\\}\\}$ | 0

Diagrams | Sequences | s-factors
---|---|---
$C_{31}$ | $\\{\\{DABKC\\}\\}$ | 0
$C_{32}$ | $\\{\\{DABCK\\}\\}$ | 0
$C_{33}$ | $\\{\\{DACKB\\}\\}$ | 0
$C_{34}$ | $\\{\\{DACBK\\}\\}$ | 0
$C_{35}$ | $\\{\\{DBACK\\}\\}$ | 0
$C_{36}$ | $\\{\\{DCABK\\}\\}$ | 0
$C_{37}$ | $\\{\\{AKBCD\\}\\}$ | 1
$C_{38}$ | $\\{\\{AKBDC\\}\\}$ | 1
$C_{39}$ | $\\{\\{AKCBD\\}\\}$ | 1
$C_{40}$ | $\\{\\{AKCDB\\}\\}$ | 1
$C_{41}$ | $\\{\\{AKDBC\\}\\}$ | 1
$C_{42}$ | $\\{\\{AKDCB\\}\\}$ | 1
$C_{43}$ | $\\{\\{BAKCD\\}\\}$ | 1
$C_{44}$ | $\\{\\{BAKDC\\}\\}$ | 1
$C_{45}$ | $\\{\\{BCAKD\\}\\}$ | 1
$C_{46}$ | $\\{\\{BCDAK\\}\\}$ | 1
$C_{47}$ | $\\{\\{BDAKC\\}\\}$ | 1
$C_{48}$ | $\\{\\{BDCAK\\}\\}$ | 1
$C_{49}$ | $\\{\\{CAKBD\\}\\}$ | 1
$C_{50}$ | $\\{\\{CAKDB\\}\\}$ | 1
$C_{51}$ | $\\{\\{CBAKD\\}\\}$ | 1
$C_{52}$ | $\\{\\{CBDAK\\}\\}$ | 1
$C_{53}$ | $\\{\\{CDAKB\\}\\}$ | 1
$C_{54}$ | $\\{\\{CDBAK\\}\\}$ | 1
$C_{55}$ | $\\{\\{DAKBC\\}\\}$ | 1
$C_{56}$ | $\\{\\{DAKCB\\}\\}$ | 1
$C_{57}$ | $\\{\\{DBAKC\\}\\}$ | 1
$C_{58}$ | $\\{\\{DBCAK\\}\\}$ | 1
$C_{59}$ | $\\{\\{DCAKB\\}\\}$ | 1
$C_{60}$ | $\\{\\{DCBAK\\}\\}$ | 1

The diagonal matrix for this Cweb is given by,

$\displaystyle\mathcal{D}\,=\,(\mathbf{1}_{24},0)$ (59)

There are twenty-four independent exponentiated colour factors for this Cweb
given as

$\displaystyle(YC)_{1}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle(YC)_{2}$ $\displaystyle=$
$\displaystyle-i\,f^{abg}f^{cdk}f^{nkg}\,\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{n}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}-i\,f^{abg}f^{ank}f^{cdk}\,\mathbf{T}_{1}^{n}\mathbf{T}_{1}^{g}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{3}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle(YC)_{4}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle(YC)_{5}$ $\displaystyle=$ $\displaystyle
i\,f^{acg}f^{bdk}f^{nkg}\,\mathbf{T}_{1}^{n}\mathbf{T}_{1}^{a}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}+i\,f^{acg}f^{ank}f^{bdk}\,\mathbf{T}_{1}^{g}\mathbf{T}_{1}^{n}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{6}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle(YC)_{7}$ $\displaystyle=$ $\displaystyle
i\,f^{ack}f^{bng}f^{dnk}\,\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{g}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}-i\,f^{ack}f^{adg}f^{bng}\,\mathbf{T}_{1}^{n}\mathbf{T}_{1}^{k}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}$
$\displaystyle-i\,f^{abg}f^{ack}f^{dnk}\,\mathbf{T}_{1}^{g}\mathbf{T}_{1}^{n}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}-i\,f^{ack}f^{adg}f^{bnk}\,\mathbf{T}_{1}^{g}\mathbf{T}_{1}^{n}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{8}$ $\displaystyle=$ $\displaystyle
i\,f^{abg}f^{cnk}f^{dng}\,\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{k}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}-i\,f^{abg}f^{adn}f^{adz}\,\mathbf{T}_{1}^{n}\mathbf{T}_{1}^{k}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}$
$\displaystyle-i\,f^{abg}f^{ack}f^{dng}\,\mathbf{T}_{1}^{k}\mathbf{T}_{1}^{n}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}+i\,f^{abg}f^{adn}f^{cnk}\,\mathbf{T}_{1}^{k}\mathbf{T}_{1}^{g}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{9}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle(YC)_{10}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle(YC)_{11}$ $\displaystyle=$
$\displaystyle-i\,f^{adn}f^{akg}f^{bcg}\,\mathbf{T}_{1}^{k}\mathbf{T}_{1}^{n}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{12}$ $\displaystyle=$
$\displaystyle-i\,f^{abg}f^{adk}f^{cng}\,\mathbf{T}_{1}^{n}\mathbf{T}_{1}^{k}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}-i\,f^{adk}f^{ang}f^{bcg}\,\mathbf{T}_{1}^{n}\mathbf{T}_{1}^{k}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{13}$ $\displaystyle=$ $\displaystyle
i\,f^{ang}f^{bkg}f^{cdk}\,\mathbf{T}_{1}^{n}\mathbf{T}_{1}^{a}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{14}$ $\displaystyle=$
$\displaystyle-i\,f^{akg}f^{bng}f^{cdk}\,\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{n}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{15}$ $\displaystyle=$
$\displaystyle-i\,f^{abk}f^{bdg}f^{cng}\,\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{k}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{16}$ $\displaystyle=$ $\displaystyle
i\,f^{akg}f^{bdg}f^{cnk}\,\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{n}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{17}$ $\displaystyle=$ $\displaystyle
i\,f^{adn}f^{bkg}f^{cnk}\,\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{g}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{18}$ $\displaystyle=$ $\displaystyle
i\,f^{adn}f^{bnk}f^{adz}\,\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{g}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{19}$ $\displaystyle=$
$\displaystyle-i\,f^{acg}f^{adk}f^{bng}\,\mathbf{T}_{1}^{n}\mathbf{T}_{1}^{k}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}-i\,f^{acg}f^{ank}f^{bdk}\,\mathbf{T}_{1}^{g}\mathbf{T}_{1}^{n}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{20}$ $\displaystyle=$
$\displaystyle-i\,f^{acg}f^{adk}f^{bng}\,\mathbf{T}_{1}^{n}\mathbf{T}_{1}^{k}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}-i\,f^{acg}f^{adk}f^{bnk}\,\mathbf{T}_{1}^{g}\mathbf{T}_{1}^{n}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{21}$ $\displaystyle=$ $\displaystyle
i\,f^{adz}f^{ank}f^{bck}\,\mathbf{T}_{1}^{n}\mathbf{T}_{1}^{z}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}-i\,f^{ank}f^{bck}f^{dny}\,\mathbf{T}_{1}^{y}\mathbf{T}_{1}^{a}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{22}$ $\displaystyle=$
$\displaystyle-i\,f^{abg}f^{cdk}f^{nkg}\,\mathbf{T}_{1}^{a}\mathbf{T}_{1}^{n}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}-i\,f^{abg}f^{ack}f^{dnk}\,\mathbf{T}_{1}^{n}\mathbf{T}_{1}^{g}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}$
$\displaystyle-i\,f^{abg}f^{ank}f^{cdk}\,\mathbf{T}_{1}^{n}\mathbf{T}_{1}^{g}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}-i\,f^{abg}f^{ack}f^{dng}\,\mathbf{T}_{1}^{k}\mathbf{T}_{1}^{n}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{23}$ $\displaystyle=$
$\displaystyle-i\,f^{abn}f^{adk}f^{ckg}\,\mathbf{T}_{1}^{n}\mathbf{T}_{1}^{g}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}+i\,f^{abn}f^{akg}f^{cdk}\,\mathbf{T}_{1}^{n}\mathbf{T}_{1}^{g}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}\,,$
$\displaystyle(YC)_{24}$ $\displaystyle=$
$\displaystyle-i\,f^{abg}f^{adk}f^{cng}\,\mathbf{T}_{1}^{n}\mathbf{T}_{1}^{k}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}-i\,f^{adk}f^{ang}f^{bcg}\,\mathbf{T}_{1}^{n}\mathbf{T}_{1}^{k}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}$
(60)
$\displaystyle-i\,f^{abg}f^{adk}f^{cnk}\,\mathbf{T}_{1}^{g}\mathbf{T}_{1}^{n}\mathbf{T}_{2}^{b}\mathbf{T}_{3}^{c}\mathbf{T}_{4}^{d}\,.$

###  C.2 Boomerang Cwebs connecting three Wilson lines

  * 1.

$\textbf{W}\,_{3}^{(4)}(4,1,3)$
This is the first of the two Cwebs with same correlator and attachment
content. It has thirty-six diagrams, one of them is displayed below. The table
shows the chosen order of shuffles and their corresponding $s$-factors. We do
not to present the mixing matrix fo this Cweb here due to its larger
dimension, however it can be found in the ancillary file Boomerang.nb.

Diagrams | Sequences | s
---|---|---
$C_{1}$ | $\\{\\{ABFG\\},\\{LMP\\}\\}$ | 0
$C_{2}$ | $\\{\\{ABGF\\},\\{LMP\\}\\}$ | 0
$C_{3}$ | $\\{\\{BAFG\\},\\{LMP\\}\\}$ | 0
$C_{4}$ | $\\{\\{BAGF\\},\\{PML\\}\\}$ | 0
$C_{5}$ | $\\{\\{BAGF\\},\\{LMP\\}\\}$ | 0
$C_{6}$ | $\\{\\{ABFG\\},\\{LPM\\}\\}$ | 0
$C_{7}$ | $\\{\\{ABFG\\},\\{MLP\\}\\}$ | 0
$C_{8}$ | $\\{\\{ABGF\\},\\{PLM\\}\\}$ | 0
$C_{9}$ | $\\{\\{ABGF\\},\\{PML\\}\\}$ | 0
$C_{10}$ | $\\{\\{ABGF\\},\\{LPM\\}\\}$ | 0
$C_{11}$ | $\\{\\{ABGF\\},\\{MPL\\}\\}$ | 0
$C_{12}$ | $\\{\\{ABGF\\},\\{MLP\\}\\}$ | 0
$C_{13}$ | $\\{\\{AGBF\\},\\{LPM\\}\\}$ | 0
$C_{14}$ | $\\{\\{AGBF\\},\\{LMP\\}\\}$ | 0
$C_{15}$ | $\\{\\{AGBF\\},\\{MLP\\}\\}$ | 0
$C_{16}$ | $\\{\\{BAFG\\},\\{PLM\\}\\}$ | 0
$C_{17}$ | $\\{\\{BAFG\\},\\{PML\\}\\}$ | 0
$C_{18}$ | $\\{\\{BAFG\\},\\{LPM\\}\\}$ | 0

Diagrams | Sequences | s
---|---|---
$C_{19}$ | $\\{\\{BAFG\\},\\{MPL\\}\\}$ | 0
$C_{20}$ | $\\{\\{BAFG\\},\\{MLP\\}\\}$ | 0
$C_{21}$ | $\\{\\{BAGF\\},\\{PLM\\}\\}$ | 0
$C_{22}$ | $\\{\\{BAGF\\},\\{LPM\\}\\}$ | 0
$C_{23}$ | $\\{\\{BAGF\\},\\{MPL\\}\\}$ | 0
$C_{24}$ | $\\{\\{BAGF\\},\\{MLP\\}\\}$ | 0
$C_{25}$ | $\\{\\{BFAG\\},\\{LPM\\}\\}$ | 0
$C_{26}$ | $\\{\\{BFAG\\},\\{LMP\\}\\}$ | 0
$C_{27}$ | $\\{\\{BFAG\\},\\{MLP\\}\\}$ | 0
$C_{28}$ | $\\{\\{ABFG\\},\\{PLM\\}\\}$ | 1
$C_{29}$ | $\\{\\{ABFG\\},\\{MPL\\}\\}$ | 1
$C_{30}$ | $\\{\\{AGBF\\},\\{PML\\}\\}$ | 1
$C_{31}$ | $\\{\\{AGBF\\},\\{MPL\\}\\}$ | 1
$C_{32}$ | $\\{\\{BFAG\\},\\{PLM\\}\\}$ | 1
$C_{33}$ | $\\{\\{BFAG\\},\\{PML\\}\\}$ | 1
$C_{34}$ | $\\{\\{ABFG\\},\\{PML\\}\\}$ | 2
$C_{35}$ | $\\{\\{AGBF\\},\\{PLM\\}\\}$ | 2
$C_{36}$ | $\\{\\{BFAG\\},\\{MPL\\}\\}$ | 2

The diagonal matrix is given as

$\displaystyle\mathcal{D}\,=\,(\mathbf{1}_{16},0)$ (61)

  * 2.

$\textbf{W}\,_{3}^{(4)}(3,4,1)$
This is the second Cweb with same correlator and attachment content. The
shuffle of correlators on Wilson line 1 and 2 generates thirty-six diagrams,
out of which one is shown below. The tables show the chosen order of shuffle
and the corresponding $s$-factors. We do not to present the mixing matrix fo
this Cweb here due to its larger dimension, however it can be found in the
ancillary file Boomerang.nb.

Diagrams | Sequences | s-factors
---|---|---
$C_{1}$ | $\\{\\{ABF\\},\\{GHPL\\}\\}$ | 0
$C_{2}$ | $\\{\\{ABF\\},\\{GPHL\\}\\}$ | 0
$C_{3}$ | $\\{\\{AFB\\},\\{HGPL\\}\\}$ | 0
$C_{4}$ | $\\{\\{AFB\\},\\{GHLP\\}\\}$ | 0
$C_{5}$ | $\\{\\{AFB\\},\\{GHPL\\}\\}$ | 0
$C_{6}$ | $\\{\\{AFB\\},\\{GPHL\\}\\}$ | 0
$C_{7}$ | $\\{\\{AFB\\},\\{GPLH\\}\\}$ | 0
$C_{8}$ | $\\{\\{AFB\\},\\{PGHL\\}\\}$ | 0
$C_{9}$ | $\\{\\{ABF\\},\\{HGLP\\}\\}$ | 0
$C_{10}$ | $\\{\\{ABF\\},\\{HGPL\\}\\}$ | 0
$C_{11}$ | $\\{\\{ABF\\},\\{HPGL\\}\\}$ | 0
$C_{12}$ | $\\{\\{ABF\\},\\{GHLP\\}\\}$ | 0
$C_{13}$ | $\\{\\{ABF\\},\\{GLHP\\}\\}$ | 0
$C_{14}$ | $\\{\\{ABF\\},\\{GLPH\\}\\}$ | 0
$C_{15}$ | $\\{\\{ABF\\},\\{GPLH\\}\\}$ | 0
$C_{16}$ | $\\{\\{ABF\\},\\{PHGL\\}\\}$ | 0
$C_{17}$ | $\\{\\{ABF\\},\\{PGHL\\}\\}$ | 0
$C_{18}$ | $\\{\\{ABF\\},\\{PGLH\\}\\}$ | 0

Diagrams | Sequences | s-factors
---|---|---
$C_{19}$ | $\\{\\{BAF\\},\\{HGPL\\}\\}$ | 0
$C_{20}$ | $\\{\\{BAF\\},\\{GHLP\\}\\}$ | 0
$C_{21}$ | $\\{\\{BAF\\},\\{GHPL\\}\\}$ | 0
$C_{22}$ | $\\{\\{BAF\\},\\{GPHL\\}\\}$ | 0
$C_{23}$ | $\\{\\{BAF\\},\\{GPLH\\}\\}$ | 0
$C_{24}$ | $\\{\\{BAF\\},\\{PGHL\\}\\}$ | 0
$C_{25}$ | $\\{\\{AFB\\},\\{HGLP\\}\\}$ | 1
$C_{26}$ | $\\{\\{AFB\\},\\{HPGL\\}\\}$ | 1
$C_{27}$ | $\\{\\{BAF\\},\\{GLPH\\}\\}$ | 1
$C_{28}$ | $\\{\\{BAF\\},\\{PGLH\\}\\}$ | 1
$C_{29}$ | $\\{\\{AFB\\},\\{GLHP\\}\\}$ | 2
$C_{30}$ | $\\{\\{AFB\\},\\{GLPH\\}\\}$ | 2
$C_{31}$ | $\\{\\{AFB\\},\\{PHGL\\}\\}$ | 2
$C_{32}$ | $\\{\\{AFB\\},\\{PGLH\\}\\}$ | 2
$C_{33}$ | $\\{\\{BAF\\},\\{HGLP\\}\\}$ | 2
$C_{34}$ | $\\{\\{BAF\\},\\{HPGL\\}\\}$ | 2
$C_{35}$ | $\\{\\{BAF\\},\\{GLHP\\}\\}$ | 2
$C_{36}$ | $\\{\\{BAF\\},\\{PHGL\\}\\}$ | 2

The diagonal matrix for this Cweb is given by,

$\displaystyle\mathcal{D}\,=\,(\mathbf{1}_{12},0)$ (62)

  * 3.

$\textbf{W}\,_{3}^{(4)}(4,2,2)$
This is a Cweb made out of four two-gluon correlators, and has 48 diagrams. We
present one of the diagrams below. The table gives the chosen order of shuffle
and their corresponding $s$-factors. We do not to present the mixing matrix fo
this Cweb here due to its larger dimension, however it can be found in the
ancillary file Boomerang.nb.

Diagrams | Sequences | s-factors
---|---|---
$C_{1}$ | $\\{\\{ABKC\\},\\{DE\\},\\{FH\\}\\}$ | 0
$C_{2}$ | $\\{\\{ABCK\\},\\{ED\\},\\{HF\\}\\}$ | 0
$C_{3}$ | $\\{\\{ABCK\\},\\{DE\\},\\{FH\\}\\}$ | 0
$C_{4}$ | $\\{\\{ACKB\\},\\{ED\\},\\{HF\\}\\}$ | 0
$C_{5}$ | $\\{\\{ACBK\\},\\{ED\\},\\{HF\\}\\}$ | 0
$C_{6}$ | $\\{\\{ACBK\\},\\{DE\\},\\{FH\\}\\}$ | 0
$C_{7}$ | $\\{\\{BAKC\\},\\{DE\\},\\{FH\\}\\}$ | 0
$C_{8}$ | $\\{\\{BACK\\},\\{DE\\},\\{FH\\}\\}$ | 0
$C_{9}$ | $\\{\\{CAKB\\},\\{ED\\},\\{HF\\}\\}$ | 0
$C_{10}$ | $\\{\\{CABK\\},\\{ED\\},\\{HF\\}\\}$ | 0
$C_{11}$ | $\\{\\{AKBC\\},\\{DE\\},\\{FH\\}\\}$ | 0
$C_{12}$ | $\\{\\{AKCB\\},\\{ED\\},\\{HF\\}\\}$ | 0
$C_{13}$ | $\\{\\{ABKC\\},\\{ED\\},\\{FH\\}\\}$ | 0
$C_{14}$ | $\\{\\{ABKC\\},\\{ED\\},\\{HF\\}\\}$ | 0
$C_{15}$ | $\\{\\{ABKC\\},\\{DE\\},\\{HF\\}\\}$ | 0
$C_{16}$ | $\\{\\{ABCK\\},\\{ED\\},\\{FH\\}\\}$ | 0
$C_{17}$ | $\\{\\{ABCK\\},\\{DE\\},\\{HF\\}\\}$ | 0
$C_{18}$ | $\\{\\{ACKB\\},\\{ED\\},\\{FH\\}\\}$ | 0
$C_{19}$ | $\\{\\{ACKB\\},\\{DE\\},\\{FH\\}\\}$ | 0
$C_{20}$ | $\\{\\{ACKB\\},\\{DE\\},\\{HF\\}\\}$ | 0
$C_{21}$ | $\\{\\{ACBK\\},\\{ED\\},\\{FH\\}\\}$ | 0
$C_{22}$ | $\\{\\{ACBK\\},\\{DE\\},\\{HF\\}\\}$ | 0
$C_{23}$ | $\\{\\{BACK\\},\\{ED\\},\\{FH\\}\\}$ | 0
$C_{24}$ | $\\{\\{BACK\\},\\{ED\\},\\{HF\\}\\}$ | 0

Diagrams | Sequences | s-factors
---|---|---
$C_{25}$ | $\\{\\{BACK\\},\\{DE\\},\\{HF\\}\\}$ | 0
$C_{26}$ | $\\{\\{BCAK\\},\\{DE\\},\\{FH\\}\\}$ | 0
$C_{27}$ | $\\{\\{CABK\\},\\{ED\\},\\{FH\\}\\}$ | 0
$C_{28}$ | $\\{\\{CABK\\},\\{DE\\},\\{FH\\}\\}$ | 0
$C_{29}$ | $\\{\\{CABK\\},\\{DE\\},\\{HF\\}\\}$ | 0
$C_{30}$ | $\\{\\{CBAK\\},\\{ED\\},\\{HF\\}\\}$ | 0
$C_{31}$ | $\\{\\{AKBC\\},\\{ED\\},\\{FH\\}\\}$ | 1
$C_{32}$ | $\\{\\{AKBC\\},\\{ED\\},\\{HF\\}\\}$ | 1
$C_{33}$ | $\\{\\{AKCB\\},\\{ED\\},\\{FH\\}\\}$ | 1
$C_{34}$ | $\\{\\{AKCB\\},\\{DE\\},\\{FH\\}\\}$ | 1
$C_{35}$ | $\\{\\{BAKC\\},\\{ED\\},\\{FH\\}\\}$ | 1
$C_{36}$ | $\\{\\{BAKC\\},\\{DE\\},\\{HF\\}\\}$ | 1
$C_{37}$ | $\\{\\{BCAK\\},\\{ED\\},\\{HF\\}\\}$ | 1
$C_{38}$ | $\\{\\{BCAK\\},\\{DE\\},\\{HF\\}\\}$ | 1
$C_{39}$ | $\\{\\{CAKB\\},\\{ED\\},\\{FH\\}\\}$ | 1
$C_{40}$ | $\\{\\{CAKB\\},\\{DE\\},\\{HF\\}\\}$ | 1
$C_{41}$ | $\\{\\{CBAK\\},\\{DE\\},\\{FH\\}\\}$ | 1
$C_{42}$ | $\\{\\{CBAK\\},\\{DE\\},\\{HF\\}\\}$ | 1
$C_{43}$ | $\\{\\{AKBC\\},\\{DE\\},\\{HF\\}\\}$ | 2
$C_{44}$ | $\\{\\{AKCB\\},\\{DE\\},\\{HF\\}\\}$ | 2
$C_{45}$ | $\\{\\{BAKC\\},\\{ED\\},\\{HF\\}\\}$ | 2
$C_{46}$ | $\\{\\{BCAK\\},\\{ED\\},\\{FH\\}\\}$ | 2
$C_{47}$ | $\\{\\{CAKB\\},\\{DE\\},\\{FH\\}\\}$ | 2
$C_{48}$ | $\\{\\{CBAK\\},\\{ED\\},\\{FH\\}\\}$ | 2

The diagonal matrix for this Cweb is given by,

$\displaystyle\mathcal{D}\,=\,(\mathbf{1}_{22},0)$ (63)

  * 4.

$\textbf{W}\,_{3,\text{I}}^{(4)}(3,2,3)$

This is a Cweb made out of four two-gluon correlators, and has 18 diagrams. We
present one of the diagrams below. The table gives the chosen order of shuffle
and their corresponding $s$-factors.

Diagrams | Sequences | s-factors
---|---|---
$C_{1}$ | $\\{\\{ABF\\},\\{GH\\},\\{LMP\\}\\}$ | 0
$C_{2}$ | $\\{\\{AFB\\},\\{GH\\},\\{MLP\\}\\}$ | 0
$C_{3}$ | $\\{\\{AFB\\},\\{GH\\},\\{LMP\\}\\}$ | 0
$C_{4}$ | $\\{\\{AFB\\},\\{GH\\},\\{LPM\\}\\}$ | 0
$C_{5}$ | $\\{\\{ABF\\},\\{GH\\},\\{MPL\\}\\}$ | 0
$C_{6}$ | $\\{\\{ABF\\},\\{GH\\},\\{MLP\\}\\}$ | 0
$C_{7}$ | $\\{\\{ABF\\},\\{GH\\},\\{PML\\}\\}$ | 0
$C_{8}$ | $\\{\\{ABF\\},\\{GH\\},\\{PLM\\}\\}$ | 0
$C_{9}$ | $\\{\\{ABF\\},\\{GH\\},\\{LPM\\}\\}$ | 0
$C_{10}$ | $\\{\\{BAF\\},\\{GH\\},\\{MLP\\}\\}$ | 0
$C_{11}$ | $\\{\\{BAF\\},\\{GH\\},\\{LMP\\}\\}$ | 0
$C_{12}$ | $\\{\\{BAF\\},\\{GH\\},\\{LPM\\}\\}$ | 0
$C_{13}$ | $\\{\\{AFB\\},\\{GH\\},\\{MPL\\}\\}$ | 1
$C_{14}$ | $\\{\\{BAF\\},\\{GH\\},\\{PLM\\}\\}$ | 1
$C_{15}$ | $\\{\\{AFB\\},\\{GH\\},\\{PML\\}\\}$ | 2
$C_{16}$ | $\\{\\{AFB\\},\\{GH\\},\\{PLM\\}\\}$ | 2
$C_{17}$ | $\\{\\{BAF\\},\\{GH\\},\\{MPL\\}\\}$ | 2
$C_{18}$ | $\\{\\{BAF\\},\\{GH\\},\\{PML\\}\\}$ | 2

The mixing and diagonalizing matrices are given as

$\displaystyle\begin{split}R=&\,\frac{1}{12}\left(\begin{array}[]{cccccccccccccccccc}12&4&-6&2&2&-6&-4&2&-6&2&-6&4&-2&-2&2&0&0&2\\\
0&4&0&-4&0&0&0&0&0&-4&0&4&-2&-6&-4&6&2&4\\\
0&-2&6&-4&0&0&0&0&0&2&-6&4&0&-2&-2&2&0&2\\\
0&-2&0&2&0&0&0&0&0&2&0&-2&2&2&0&-2&-2&0\\\
0&0&0&0&2&0&-4&2&0&0&0&0&0&-2&0&0&-2&4\\\
0&-2&0&2&-4&6&-4&8&-6&-4&0&4&2&-6&0&-2&2&4\\\
0&0&0&0&-4&0&8&-4&0&0&0&0&2&2&-4&2&2&-4\\\
0&0&0&0&2&0&-4&2&0&0&0&0&-2&0&4&-2&0&0\\\
0&4&0&-4&8&-6&-4&-4&6&2&0&-2&-6&2&4&2&-2&0\\\
0&-2&0&2&0&0&0&0&0&2&0&-2&2&2&0&-2&-2&0\\\
0&4&-6&2&0&0&0&0&0&-4&6&-2&-2&0&2&0&2&-2\\\
0&4&0&-4&0&0&0&0&0&-4&0&4&-6&-2&4&2&6&-4\\\
0&0&0&0&0&0&0&0&0&0&0&0&2&-2&-4&2&-2&4\\\
0&0&0&0&0&0&0&0&0&0&0&0&-2&2&4&-2&2&-4\\\
0&0&0&0&0&0&0&0&0&0&0&0&-2&2&4&-2&2&-4\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&2&-2&-4&2&-2&4\\\ \end{array}\right)\\\
\mathcal{D}\,=\,&\,(\mathbf{1}_{6},0)\end{split}$ (64)

  * 5.

$\textbf{W}\,_{3,\text{II}}^{(4)}(3,2,3)$

This is the second Cweb with same correlator and attachment content. It has
eighteen diagrams, one of them is shown below. The table shows the chosen
order of shuffle and their corresponding $s$-factors.

Diagrams | Sequences | s-factors
---|---|---
$C_{1}$ | $\\{\\{APB\\},\\{FG\\},\\{HLM\\}\\}$ | 0
$C_{2}$ | $\\{\\{APB\\},\\{GF\\},\\{HLM\\}\\}$ | 0
$C_{3}$ | $\\{\\{ABP\\},\\{FG\\},\\{LHM\\}\\}$ | 0
$C_{4}$ | $\\{\\{ABP\\},\\{FG\\},\\{HLM\\}\\}$ | 0
$C_{5}$ | $\\{\\{ABP\\},\\{FG\\},\\{HML\\}\\}$ | 0
$C_{6}$ | $\\{\\{ABP\\},\\{GF\\},\\{LHM\\}\\}$ | 0
$C_{7}$ | $\\{\\{ABP\\},\\{GF\\},\\{HLM\\}\\}$ | 0
$C_{8}$ | $\\{\\{ABP\\},\\{GF\\},\\{HML\\}\\}$ | 0
$C_{9}$ | $\\{\\{BAP\\},\\{FG\\},\\{HLM\\}\\}$ | 0
$C_{10}$ | $\\{\\{BAP\\},\\{GF\\},\\{HLM\\}\\}$ | 0
$C_{11}$ | $\\{\\{APB\\},\\{FG\\},\\{LHM\\}\\}$ | 1
$C_{12}$ | $\\{\\{BAP\\},\\{GF\\},\\{HML\\}\\}$ | 1
$C_{13}$ | $\\{\\{APB\\},\\{FG\\},\\{HML\\}\\}$ | 2
$C_{14}$ | $\\{\\{APB\\},\\{GF\\},\\{HML\\}\\}$ | 2
$C_{15}$ | $\\{\\{BAP\\},\\{FG\\},\\{LHM\\}\\}$ | 2
$C_{16}$ | $\\{\\{BAP\\},\\{GF\\},\\{LHM\\}\\}$ | 2
$C_{17}$ | $\\{\\{APB\\},\\{GF\\},\\{LHM\\}\\}$ | 4
$C_{18}$ | $\\{\\{BAP\\},\\{FG\\},\\{HML\\}\\}$ | 4

The mixing and diagonalizing matrices are given as

$\displaystyle\begin{split}R=\frac{1}{12}\left(\begin{array}[]{cccccccccccccccccc}4&-4&0&0&0&0&0&0&-4&4&-1&-3&-3&3&1&-1&1&3\\\
-2&2&0&0&0&0&0&0&2&-2&1&1&1&-1&-1&1&-1&-1\\\
0&0&4&0&-4&-4&0&4&0&0&-1&-3&1&-1&-3&3&1&3\\\
-2&2&-2&6&-4&2&-6&4&-4&4&1&-3&1&-1&1&-1&-1&3\\\
0&0&-2&0&2&2&0&-2&0&0&1&1&-1&1&1&-1&-1&-1\\\
0&0&-2&0&2&2&0&-2&0&0&1&1&-1&1&1&-1&-1&-1\\\
4&-4&4&-6&2&-4&6&-2&2&-2&-3&1&-1&1&-1&1&3&-1\\\
0&0&4&0&-4&-4&0&4&0&0&-3&-1&3&-3&-1&1&3&1\\\
-2&2&0&0&0&0&0&0&2&-2&1&1&1&-1&-1&1&-1&-1\\\
4&-4&0&0&0&0&0&0&-4&4&-3&-1&-1&1&3&-3&3&1\\\
0&0&0&0&0&0&0&0&0&0&3&-3&-3&3&-3&3&-3&3\\\
0&0&0&0&0&0&0&0&0&0&-3&3&3&-3&3&-3&3&-3\\\
0&0&0&0&0&0&0&0&0&0&-1&1&1&-1&1&-1&1&-1\\\
0&0&0&0&0&0&0&0&0&0&1&-1&-1&1&-1&1&-1&1\\\
0&0&0&0&0&0&0&0&0&0&-1&1&1&-1&1&-1&1&-1\\\
0&0&0&0&0&0&0&0&0&0&1&-1&-1&1&-1&1&-1&1\\\
0&0&0&0&0&0&0&0&0&0&-1&1&1&-1&1&-1&1&-1\\\
0&0&0&0&0&0&0&0&0&0&1&-1&-1&1&-1&1&-1&1\\\ \end{array}\right)\end{split}$
$\displaystyle\mathcal{D}\,=\,(\mathbf{1}_{4},0)$ (65)

  * 6.

$\textbf{W}\,_{3,\text{I}}^{(4)}(5,1,2)$

This Cweb has four two point correlator including two boomerang on line 1. It
has thirty diagrams, one of them is shown below. The table shows the chosen
order of shuffle and their corresponding $s$-factors. We do not to present the
mixing matrix fo this Cweb here due to its larger dimension, however it can be
found in the ancillary file Boomerang.nb.

Diagrams | Sequences | s
---|---|---
$C_{1}$ | $\\{\\{ABFGH\\},\\{MP\\}\\}$ | 0
$C_{2}$ | $\\{\\{ABFGH\\},\\{PM\\}\\}$ | 0
$C_{3}$ | $\\{\\{AFBHG\\},\\{MP\\}\\}$ | 0
$C_{4}$ | $\\{\\{AFBHG\\},\\{PM\\}\\}$ | 0
$C_{5}$ | $\\{\\{AFBGH\\},\\{MP\\}\\}$ | 0
$C_{6}$ | $\\{\\{AFBGH\\},\\{PM\\}\\}$ | 0
$C_{7}$ | $\\{\\{AFHBG\\},\\{MP\\}\\}$ | 0
$C_{8}$ | $\\{\\{AFHBG\\},\\{PM\\}\\}$ | 0
$C_{9}$ | $\\{\\{AFHGB\\},\\{MP\\}\\}$ | 0
$C_{10}$ | $\\{\\{AFHGB\\},\\{PM\\}\\}$ | 0
$C_{11}$ | $\\{\\{AFGBH\\},\\{MP\\}\\}$ | 0
$C_{12}$ | $\\{\\{AFGBH\\},\\{PM\\}\\}$ | 0
$C_{13}$ | $\\{\\{AFGHB\\},\\{MP\\}\\}$ | 0
$C_{14}$ | $\\{\\{AFGHB\\},\\{PM\\}\\}$ | 0
$C_{15}$ | $\\{\\{AGBFH\\},\\{MP\\}\\}$ | 0

Diagrams | Sequences | s
---|---|---
$C_{16}$ | $\\{\\{AGBFH\\},\\{PM\\}\\}$ | 0
$C_{17}$ | $\\{\\{AGFBH\\},\\{MP\\}\\}$ | 0
$C_{18}$ | $\\{\\{AGFBH\\},\\{PM\\}\\}$ | 0
$C_{19}$ | $\\{\\{AGFHB\\},\\{MP\\}\\}$ | 0
$C_{20}$ | $\\{\\{AGFHB\\},\\{PM\\}\\}$ | 0
$C_{21}$ | $\\{\\{GAFBH\\},\\{MP\\}\\}$ | 0
$C_{22}$ | $\\{\\{GAFBH\\},\\{PM\\}\\}$ | 0
$C_{23}$ | $\\{\\{GAFHB\\},\\{MP\\}\\}$ | 0
$C_{24}$ | $\\{\\{GAFHB\\},\\{PM\\}\\}$ | 0
$C_{25}$ | $\\{\\{ABFHG\\},\\{MP\\}\\}$ | 1
$C_{26}$ | $\\{\\{GABFH\\},\\{PM\\}\\}$ | 1
$C_{27}$ | $\\{\\{ABFHG\\},\\{PM\\}\\}$ | 2
$C_{28}$ | $\\{\\{ABGFH\\},\\{MP\\}\\}$ | 2
$C_{29}$ | $\\{\\{ABGFH\\},\\{PM\\}\\}$ | 2
$C_{30}$ | $\\{\\{GABFH\\},\\{MP\\}\\}$ | 2

The diagonalizing matrix is given as

$\displaystyle\mathcal{D}\,=\,(\mathbf{1}_{10},0)$ (66)

  * 7.

$\textbf{W}\,_{3,\text{II}}^{(4)}(5,1,2)$

This is second Cweb with same correlator content as previous one. It has sixty
diagrams, one of them is shown below. The table shows chosen order of shuffle
and their corresponding $s$-factors. We do not to present the mixing matrix fo
this Cweb here due to its larger dimension, however it can be found in the
ancillary file Boomerang.nb.

Diagrams | Sequences | s
---|---|---
$C_{1}$ | $\\{\\{ABFGH\\},\\{MP\\}\\}$ | 0
$C_{2}$ | $\\{\\{ABHGF\\},\\{MP\\}\\}$ | 0
$C_{3}$ | $\\{\\{ABGFH\\},\\{MP\\}\\}$ | 0
$C_{4}$ | $\\{\\{ABGHF\\},\\{MP\\}\\}$ | 0
$C_{5}$ | $\\{\\{AGBFH\\},\\{MP\\}\\}$ | 0
$C_{6}$ | $\\{\\{AGBHF\\},\\{MP\\}\\}$ | 0
$C_{7}$ | $\\{\\{BAFGH\\},\\{MP\\}\\}$ | 0
$C_{8}$ | $\\{\\{BAHGF\\},\\{PM\\}\\}$ | 0
$C_{9}$ | $\\{\\{BAHGF\\},\\{MP\\}\\}$ | 0
$C_{10}$ | $\\{\\{BAGFH\\},\\{MP\\}\\}$ | 0
$C_{11}$ | $\\{\\{BAGHF\\},\\{PM\\}\\}$ | 0
$C_{12}$ | $\\{\\{BAGHF\\},\\{MP\\}\\}$ | 0
$C_{13}$ | $\\{\\{BGAFH\\},\\{MP\\}\\}$ | 0
$C_{14}$ | $\\{\\{BGAHF\\},\\{PM\\}\\}$ | 0
$C_{15}$ | $\\{\\{BGAHF\\},\\{MP\\}\\}$ | 0
$C_{16}$ | $\\{\\{ABFHG\\},\\{MP\\}\\}$ | 0
$C_{17}$ | $\\{\\{ABHFG\\},\\{PM\\}\\}$ | 0
$C_{18}$ | $\\{\\{ABHFG\\},\\{MP\\}\\}$ | 0
$C_{19}$ | $\\{\\{ABHGF\\},\\{PM\\}\\}$ | 0
$C_{20}$ | $\\{\\{ABGFH\\},\\{PM\\}\\}$ | 0
$C_{21}$ | $\\{\\{ABGHF\\},\\{PM\\}\\}$ | 0
$C_{22}$ | $\\{\\{AHBFG\\},\\{MP\\}\\}$ | 0
$C_{23}$ | $\\{\\{AHBGF\\},\\{PM\\}\\}$ | 0
$C_{24}$ | $\\{\\{AHBGF\\},\\{MP\\}\\}$ | 0
$C_{25}$ | $\\{\\{AHGBF\\},\\{MP\\}\\}$ | 0
$C_{26}$ | $\\{\\{AGBHF\\},\\{PM\\}\\}$ | 0
$C_{27}$ | $\\{\\{AGHBF\\},\\{MP\\}\\}$ | 0
$C_{28}$ | $\\{\\{BAFHG\\},\\{PM\\}\\}$ | 0
$C_{29}$ | $\\{\\{BAFHG\\},\\{MP\\}\\}$ | 0
$C_{30}$ | $\\{\\{BAFGH\\},\\{PM\\}\\}$ | 0
Diagrams | Sequences | s
---|---|---
$C_{31}$ | $\\{\\{BAHFG\\},\\{PM\\}\\}$ | 0
$C_{32}$ | $\\{\\{BAHFG\\},\\{MP\\}\\}$ | 0
$C_{33}$ | $\\{\\{BAGFH\\},\\{PM\\}\\}$ | 0
$C_{34}$ | $\\{\\{BFAHG\\},\\{MP\\}\\}$ | 0
$C_{35}$ | $\\{\\{BFAGH\\},\\{MP\\}\\}$ | 0
$C_{36}$ | $\\{\\{BFGAH\\},\\{MP\\}\\}$ | 0
$C_{37}$ | $\\{\\{BGAFH\\},\\{PM\\}\\}$ | 0
$C_{38}$ | $\\{\\{BGFAH\\},\\{PM\\}\\}$ | 0
$C_{39}$ | $\\{\\{BGFAH\\},\\{MP\\}\\}$ | 0
$C_{40}$ | $\\{\\{GABFH\\},\\{MP\\}\\}$ | 0
$C_{41}$ | $\\{\\{GABHF\\},\\{PM\\}\\}$ | 0
$C_{42}$ | $\\{\\{GABHF\\},\\{MP\\}\\}$ | 0
$C_{43}$ | $\\{\\{GAHBF\\},\\{MP\\}\\}$ | 0
$C_{44}$ | $\\{\\{GBAFH\\},\\{PM\\}\\}$ | 0
$C_{45}$ | $\\{\\{GBAFH\\},\\{MP\\}\\}$ | 0
$C_{46}$ | $\\{\\{GBAHF\\},\\{PM\\}\\}$ | 0
$C_{47}$ | $\\{\\{GBAHF\\},\\{MP\\}\\}$ | 0
$C_{48}$ | $\\{\\{GBFAH\\},\\{MP\\}\\}$ | 0
$C_{49}$ | $\\{\\{ABFHG\\},\\{PM\\}\\}$ | 1
$C_{50}$ | $\\{\\{ABFGH\\},\\{PM\\}\\}$ | 1
$C_{51}$ | $\\{\\{AHBFG\\},\\{PM\\}\\}$ | 1
$C_{52}$ | $\\{\\{AHGBF\\},\\{PM\\}\\}$ | 1
$C_{53}$ | $\\{\\{AGBFH\\},\\{PM\\}\\}$ | 1
$C_{54}$ | $\\{\\{AGHBF\\},\\{PM\\}\\}$ | 1
$C_{55}$ | $\\{\\{BFAHG\\},\\{PM\\}\\}$ | 1
$C_{56}$ | $\\{\\{BFAGH\\},\\{PM\\}\\}$ | 1
$C_{57}$ | $\\{\\{BFGAH\\},\\{PM\\}\\}$ | 1
$C_{58}$ | $\\{\\{GABFH\\},\\{PM\\}\\}$ | 1
$C_{59}$ | $\\{\\{GAHBF\\},\\{PM\\}\\}$ | 1
$C_{60}$ | $\\{\\{GBFAH\\},\\{PM\\}\\}$ | 1

Diagonalizing matrix is given as

$\displaystyle\mathcal{D}\,=\,(\mathbf{1}_{32},0)$ (67)

  * 8.

$\textbf{W}\,_{3,\text{I}}^{(2,1)}(4,2,1)$
This is the first Cweb that connects three Wilson lines and is made out of a
three-gluon correlator and 2 two-gluon correlators. We present one
representative diagram of this Cweb. The table shows the chosen order of
shuffles and their corresponding $s$-factors.
Diagrams Sequences s-factors $C_{1}$ $\\{\\{ABFG\\},\\{HL\\}\\}$ 0 $C_{2}$
$\\{\\{ABFG\\},\\{LH\\}\\}$ 0 $C_{3}$ $\\{\\{ABGF\\},\\{HL\\}\\}$ 0 $C_{4}$
$\\{\\{ABGF\\},\\{LH\\}\\}$ 0 $C_{5}$ $\\{\\{BAFG\\},\\{HL\\}\\}$ 0 $C_{6}$
$\\{\\{BAFG\\},\\{LH\\}\\}$ 0 $C_{7}$ $\\{\\{BAGF\\},\\{HL\\}\\}$ 0 $C_{8}$
$\\{\\{BAGF\\},\\{LH\\}\\}$ 0 $C_{9}$ $\\{\\{AFBG\\},\\{LH\\}\\}$ 1 $C_{10}$
$\\{\\{BGAF\\},\\{HL\\}\\}$ 1 $C_{11}$ $\\{\\{AFBG\\},\\{HL\\}\\}$ 2 $C_{12}$
$\\{\\{BGAF\\},\\{LH\\}\\}$ 2

The mixing matrix and the diagonal matrix for this Cweb are given by,

$\displaystyle\begin{split}R=&\frac{1}{6}\left(\begin{array}[]{cccccccccccc}3&-3&0&0&0&0&0&0&2&-1&-2&1\\\
-3&3&0&0&0&0&0&0&-1&2&1&-2\\\ 0&0&3&-3&0&0&0&0&2&-1&-2&1\\\
0&0&-3&3&0&0&0&0&-1&2&1&-2\\\ 0&0&0&0&3&-3&0&0&2&-1&-2&1\\\
0&0&0&0&-3&3&0&0&-1&2&1&-2\\\ 0&0&0&0&0&0&3&-3&2&-1&-2&1\\\
0&0&0&0&0&0&-3&3&-1&2&1&-2\\\ 0&0&0&0&0&0&0&0&2&2&-2&-2\\\
0&0&0&0&0&0&0&0&2&2&-2&-2\\\ 0&0&0&0&0&0&0&0&-1&-1&1&1\\\
0&0&0&0&0&0&0&0&-1&-1&1&1\\\
\end{array}\right)\end{split},\mathcal{D}\,=\,(\mathbf{1}_{5},0)\,.$ (68)

  * 9.

$\textbf{W}\,_{3,\text{II}}^{(2,1)}(4,2,1)$
This is the second Cweb with same correlator and attachment content. It has
twenty four diagrams, one of them is shown below. The table lists all possible
shuffles and their corresponding $s$-factors.
Diagrams Sequences s-factors $C_{1}$ $\\{\\{ABFG\\},\\{HL\\}\\}$ 0 $C_{2}$
$\\{\\{ABGF\\},\\{LH\\}\\}$ 0 $C_{3}$ $\\{\\{ABGF\\},\\{HL\\}\\}$ 0 $C_{4}$
$\\{\\{AGFB\\},\\{LH\\}\\}$ 0 $C_{5}$ $\\{\\{AGBF\\},\\{LH\\}\\}$ 0 $C_{6}$
$\\{\\{AGBF\\},\\{HL\\}\\}$ 0 $C_{7}$ $\\{\\{BAFG\\},\\{HL\\}\\}$ 0 $C_{8}$
$\\{\\{BAGF\\},\\{HL\\}\\}$ 0 $C_{9}$ $\\{\\{GAFB\\},\\{LH\\}\\}$ 0 $C_{10}$
$\\{\\{GABF\\},\\{LH\\}\\}$ 0 $C_{11}$ $\\{\\{AFBG\\},\\{HL\\}\\}$ 0 $C_{12}$
$\\{\\{AFGB\\},\\{LH\\}\\}$ 0 $C_{13}$ $\\{\\{ABFG\\},\\{LH\\}\\}$ 0 $C_{14}$
$\\{\\{AGFB\\},\\{HL\\}\\}$ 0 $C_{15}$ $\\{\\{BAGF\\},\\{LH\\}\\}$ 0 $C_{16}$
$\\{\\{BGAF\\},\\{HL\\}\\}$ 0 $C_{17}$ $\\{\\{GABF\\},\\{HL\\}\\}$ 0 $C_{18}$
$\\{\\{GBAF\\},\\{LH\\}\\}$ 0 $C_{19}$ $\\{\\{AFBG\\},\\{LH\\}\\}$ 1 $C_{20}$
$\\{\\{AFGB\\},\\{HL\\}\\}$ 1 $C_{21}$ $\\{\\{BAFG\\},\\{LH\\}\\}$ 1 $C_{22}$
$\\{\\{BGAF\\},\\{LH\\}\\}$ 1 $C_{23}$ $\\{\\{GAFB\\},\\{HL\\}\\}$ 1 $C_{24}$
$\\{\\{GBAF\\},\\{HL\\}\\}$ 1
The mixing matrix $R$, and the diagonal matrix $D$ are given by,

$\displaystyle\begin{split}R=\scalebox{0.9}{\mbox{$\displaystyle\frac{1}{6}\left(\begin{array}[]{cccccccccccccccccccccccc}6&0&0&0&0&0&0&0&0&0&-3&0&-3&0&0&-3&-3&0&2&-1&-1&2&2&2\\\
0&6&0&0&0&0&0&0&0&0&0&0&-3&-3&-3&0&-3&0&-1&2&2&-1&2&2\\\
0&0&6&0&0&0&0&0&0&0&-3&0&-3&-3&-3&-3&-3&0&2&2&2&2&2&2\\\
0&0&0&6&0&0&0&0&0&0&0&-3&0&-3&-3&0&0&-3&-1&2&2&2&-1&2\\\
0&0&0&0&6&0&0&0&0&0&0&-3&-3&-3&-3&0&-3&-3&2&2&2&2&2&2\\\
0&0&0&0&0&6&0&0&0&0&0&0&-3&-3&-3&0&-3&0&2&-1&2&2&2&-1\\\
0&0&0&0&0&0&6&0&0&0&-3&0&0&0&0&-3&0&0&2&-1&-4&2&2&-1\\\
0&0&0&0&0&0&0&6&0&0&-3&0&0&-3&-3&-3&0&0&2&2&-1&2&2&-1\\\
0&0&0&0&0&0&0&0&6&0&0&-3&0&0&0&0&0&-3&-1&2&2&-1&-4&2\\\
0&0&0&0&0&0&0&0&0&6&0&-3&-3&0&0&0&-3&-3&2&2&2&-1&-1&2\\\
0&0&0&0&0&0&0&0&0&0&3&0&0&0&0&-3&0&0&-1&-1&-1&2&-1&2\\\
0&0&0&0&0&0&0&0&0&0&0&3&0&0&0&0&0&-3&-1&-1&-1&2&-1&2\\\
0&0&0&0&0&0&0&0&0&0&0&0&3&0&0&0&-3&0&-1&-1&-1&-1&2&2\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&3&-3&0&0&0&-1&-1&2&2&-1&-1\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&-3&3&0&0&0&-1&2&-1&-1&2&-1\\\
0&0&0&0&0&0&0&0&0&0&-3&0&0&0&0&3&0&0&2&2&-1&-1&-1&-1\\\
0&0&0&0&0&0&0&0&0&0&0&0&-3&0&0&0&3&0&2&-1&2&-1&-1&-1\\\
0&0&0&0&0&0&0&0&0&0&0&-3&0&0&0&0&0&3&2&2&-1&-1&-1&-1\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&2&-1&-1&-1&-1&2\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&-1&2&-1&2&-1&-1\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&-1&-1&2&-1&2&-1\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&-1&2&-1&2&-1&-1\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&-1&-1&2&-1&2&-1\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&2&-1&-1&-1&-1&2\\\
\end{array}\right)$}}\end{split}$
$\displaystyle\mathcal{D}\,=\,(\mathbf{1}_{16},0)$ (69)

  * 10.

$\textbf{W}\,_{3,\text{III}}^{(2,1)}(4,2,1)$
This is the fourth Cweb with same correlator and attachment content. It has
twelve diagrams, one of them is displayed below.
Diagrams Sequences s-factors $C_{1}$ $\\{\\{ABFA\\},\\{HL\\}\\}$ 0 $C_{2}$
$\\{\\{AFBA\\},\\{HL\\}\\}$ 0 $C_{3}$ $\\{\\{ABAF\\},\\{HL\\}\\}$ 0 $C_{4}$
$\\{\\{AFAB\\},\\{HL\\}\\}$ 0 $C_{5}$ $\\{\\{BAFA\\},\\{HL\\}\\}$ 0 $C_{6}$
$\\{\\{FABA\\},\\{HL\\}\\}$ 0 $C_{7}$ $\\{\\{AABF\\},\\{HL\\}\\}$ 1 $C_{8}$
$\\{\\{AAFB\\},\\{HL\\}\\}$ 1 $C_{9}$ $\\{\\{BAAF\\},\\{HL\\}\\}$ 1 $C_{10}$
$\\{\\{BFAA\\},\\{HL\\}\\}$ 1 $C_{11}$ $\\{\\{FAAB\\},\\{HL\\}\\}$ 1 $C_{12}$
$\\{\\{FBAA\\},\\{HL\\}\\}$ 1

The mixing matrix, and the diagonal matrix are given by,

$\displaystyle\begin{split}R=\frac{1}{6}\left(\begin{array}[]{cccccccccccc}6&0&-3&-3&-3&-3&-1&2&2&-1&2&2\\\
0&6&-3&-3&-3&-3&2&-1&2&2&2&-1\\\ 0&0&3&0&0&-3&-1&-1&-1&-1&2&2\\\
0&0&0&3&-3&0&-1&-1&2&2&-1&-1\\\ 0&0&0&-3&3&0&-1&2&-1&-1&2&-1\\\
0&0&-3&0&0&3&2&-1&2&-1&-1&-1\\\ 0&0&0&0&0&0&2&-1&-1&-1&-1&2\\\
0&0&0&0&0&0&-1&2&-1&2&-1&-1\\\ 0&0&0&0&0&0&-1&-1&2&-1&2&-1\\\
0&0&0&0&0&0&-1&2&-1&2&-1&-1\\\ 0&0&0&0&0&0&-1&-1&2&-1&2&-1\\\
0&0&0&0&0&0&2&-1&-1&-1&-1&2\\\
\end{array}\right),\,\mathcal{D}\,=\,(\mathbf{1}_{6},0)\,.\end{split}$ (70)

  * 11.

$\textbf{W}\,_{3,\text{I}}^{(2,1)}(3,2,2)$
This Cweb also has twelve diagrams, and one of them is shown below. The table
shows twelve shuffles and their corresponding $s$-factors.
Diagrams Sequences s-factors $C_{1}$ $\\{\\{ABF\\},\\{GH\\},\\{LM\\}\\}$ 0
$C_{2}$ $\\{\\{ABF\\},\\{HG\\},\\{ML\\}\\}$ 0 $C_{3}$
$\\{\\{AFB\\},\\{GH\\},\\{LM\\}\\}$ 0 $C_{4}$
$\\{\\{AFB\\},\\{HG\\},\\{ML\\}\\}$ 0 $C_{5}$
$\\{\\{ABF\\},\\{GH\\},\\{ML\\}\\}$ 0 $C_{6}$
$\\{\\{ABF\\},\\{HG\\},\\{LM\\}\\}$ 0 $C_{7}$
$\\{\\{BAF\\},\\{GH\\},\\{LM\\}\\}$ 0 $C_{8}$
$\\{\\{BAF\\},\\{HG\\},\\{ML\\}\\}$ 0 $C_{9}$
$\\{\\{AFB\\},\\{GH\\},\\{ML\\}\\}$ 1 $C_{10}$
$\\{\\{BAF\\},\\{HG\\},\\{LM\\}\\}$ 1 $C_{11}$
$\\{\\{AFB\\},\\{HG\\},\\{LM\\}\\}$ 2 $C_{12}$
$\\{\\{BAF\\},\\{GH\\},\\{ML\\}\\}$ 2

The mixing matrix, and the diagonal matrix are given by,

$\displaystyle\begin{split}R=&\frac{1}{6}\left(\begin{array}[]{cccccccccccc}6&0&-3&0&-3&-3&-3&0&2&2&1&1\\\
0&6&0&-3&-3&-3&0&-3&2&2&1&1\\\ 0&0&3&0&0&0&-3&0&-1&2&-2&1\\\
0&0&0&3&0&0&0&-3&-1&2&-2&1\\\ 0&0&0&0&3&-3&0&0&-1&2&1&-2\\\
0&0&0&0&-3&3&0&0&2&-1&-2&1\\\ 0&0&-3&0&0&0&3&0&2&-1&1&-2\\\
0&0&0&-3&0&0&0&3&2&-1&1&-2\\\ 0&0&0&0&0&0&0&0&2&2&-2&-2\\\
0&0&0&0&0&0&0&0&2&2&-2&-2\\\ 0&0&0&0&0&0&0&0&-1&-1&1&1\\\
0&0&0&0&0&0&0&0&-1&-1&1&1\\\ \end{array}\right)\end{split}$
$\displaystyle\mathcal{D}\,=\,$ $\displaystyle(\mathbf{1}_{6},0)$ (71)

  * 12.

$\textbf{W}\,_{3,\text{II}}^{(2,1)}(3,2,2)$
This Cweb has six diagrams, one of which is shown below. The table shows the
chosen order of shuffles and their corresponding $s$-factors.
Diagrams Sequences s-factors $C_{1}$ $\\{\\{ABF\\},\\{GH\\},\\{LM\\}\\}$ 0
$C_{2}$ $\\{\\{ABF\\},\\{HG\\},\\{LM\\}\\}$ 0 $C_{3}$
$\\{\\{AFB\\},\\{GH\\},\\{LM\\}\\}$ 1 $C_{4}$
$\\{\\{BAF\\},\\{HG\\},\\{LM\\}\\}$ 1 $C_{5}$
$\\{\\{AFB\\},\\{HG\\},\\{LM\\}\\}$ 2 $C_{6}$
$\\{\\{BAF\\},\\{GH\\},\\{LM\\}\\}$ 2

The $R$ and $D$ matrices are given by,

$\displaystyle\begin{split}R=&\frac{1}{6}\left(\begin{array}[]{cccccc}3&-3&-1&2&1&-2\\\
-3&3&2&-1&-2&1\\\ 0&0&2&2&-2&-2\\\ 0&0&2&2&-2&-2\\\ 0&0&-1&-1&1&1\\\
0&0&-1&-1&1&1\\\
\end{array}\right)\end{split},\mathcal{D}\,=\,(\mathbf{1}_{2},0)\,.$ (72)

  * 13.

$\textbf{W}\,_{3,\text{I}}^{(2,1)}(3,3,1)$
This is the first Cweb with same correlator and attachment content. It Cweb
has nine diagrams, one of them is shown below. The table shows chosen order of
shuffle and their corresponding $s$-factors.
Diagrams Sequences s-factors $C_{1}$ $\\{\\{ABF\\},\\{GHL\\}\\}$ 0 $C_{2}$
$\\{\\{AFB\\},\\{GHL\\}\\}$ 0 $C_{3}$ $\\{\\{ABF\\},\\{GLH\\}\\}$ 0 $C_{4}$
$\\{\\{ABF\\},\\{HGL\\}\\}$ 0 $C_{5}$ $\\{\\{BAF\\},\\{GHL\\}\\}$ 0 $C_{6}$
$\\{\\{AFB\\},\\{GLH\\}\\}$ 1 $C_{7}$ $\\{\\{BAF\\},\\{HGL\\}\\}$ 1 $C_{8}$
$\\{\\{AFB\\},\\{HGL\\}\\}$ 2 $C_{9}$ $\\{\\{BAF\\},\\{GLH\\}\\}$ 2
The $R$ and $D$ matrices are given by,

$\displaystyle\begin{split}R=\frac{1}{6}\left(\begin{array}[]{ccccccccc}6&-3&-3&-3&-3&2&2&1&1\\\
0&3&0&0&-3&-1&2&-2&1\\\ 0&0&3&-3&0&-1&2&1&-2\\\ 0&0&-3&3&0&2&-1&-2&1\\\
0&-3&0&0&3&2&-1&1&-2\\\ 0&0&0&0&0&2&2&-2&-2\\\ 0&0&0&0&0&2&2&-2&-2\\\
0&0&0&0&0&-1&-1&1&1\\\ 0&0&0&0&0&-1&-1&1&1\\\
\end{array}\right),\end{split}\mathcal{D}\,=\,(\mathbf{1}_{4},0)\,.$ (73)

  * 14.

$\textbf{W}\,_{3,\text{II}}^{(2,1)}(3,3,1)$
This is the second Cweb with same correlator and attachment content. It has
nine diagrams, out which one is shown below. The table shows chosen order of
all possible shuffles and their corresponding $s$-factors.

Diagrams | Sequences | s-factors
---|---|---
$C_{1}$ | $\\{\\{ABF\\},\\{GHL\\}\\}$ | 0
$C_{2}$ | $\\{\\{AFB\\},\\{GHL\\}\\}$ | 0
$C_{3}$ | $\\{\\{ABF\\},\\{HGL\\}\\}$ | 0
$C_{4}$ | $\\{\\{ABF\\},\\{GLH\\}\\}$ | 0
$C_{5}$ | $\\{\\{BAF\\},\\{GHL\\}\\}$ | 0
$C_{6}$ | $\\{\\{AFB\\},\\{HGL\\}\\}$ | 1
$C_{7}$ | $\\{\\{AFB\\},\\{GLH\\}\\}$ | 1
$C_{8}$ | $\\{\\{BAF\\},\\{HGL\\}\\}$ | 2
$C_{9}$ | $\\{\\{BAF\\},\\{GLH\\}\\}$ | 2

The $R$, and $D$ matrices are given by,

$\displaystyle\begin{split}R=\frac{1}{6}\left(\begin{array}[]{ccccccccc}6&-3&-3&-3&-3&2&2&1&1\\\
0&3&0&0&-3&-1&2&-2&1\\\ 0&0&3&-3&0&-1&2&1&-2\\\ 0&0&-3&3&0&2&-1&-2&1\\\
0&-3&0&0&3&2&-1&1&-2\\\ 0&0&0&0&0&2&2&-2&-2\\\ 0&0&0&0&0&2&2&-2&-2\\\
0&0&0&0&0&-1&-1&1&1\\\ 0&0&0&0&0&-1&-1&1&1\\\
\end{array}\right),\,\mathcal{D}\,=\,(\mathbf{1}_{4},0)\end{split}$ (74)

  * 15.

$\textbf{W}\,_{3,\text{III}}^{(2,1)}(3,3,1)$
This is the third Cweb with same correlator and attachment content. It has
nine diagrams, out which one is shown below. The table shows chosen order of
all possible shuffles and their corresponding $s$-factors.
Diagrams Sequences s-factors $C_{1}$ $\\{\\{ABF\\},\\{GHL\\}\\}$ 0 $C_{2}$
$\\{\\{AFB\\},\\{GHL\\}\\}$ 0 $C_{3}$ $\\{\\{ABF\\},\\{GLH\\}\\}$ 0 $C_{4}$
$\\{\\{ABF\\},\\{HGL\\}\\}$ 0 $C_{5}$ $\\{\\{BAF\\},\\{GHL\\}\\}$ 0 $C_{6}$
$\\{\\{AFB\\},\\{HGL\\}\\}$ 1 $C_{7}$ $\\{\\{BAF\\},\\{GLH\\}\\}$ 1 $C_{8}$
$\\{\\{AFB\\},\\{GLH\\}\\}$ 2 $C_{9}$ $\\{\\{BAF\\},\\{HGL\\}\\}$ 2
The mixing matrix and the diagonal matrix are given by,

$\displaystyle\begin{split}R=\frac{1}{6}\left(\begin{array}[]{ccccccccc}6&-3&-3&-3&-3&2&2&1&1\\\
0&3&0&0&-3&-1&2&-2&1\\\ 0&0&3&-3&0&2&-1&-2&1\\\ 0&0&-3&3&0&-1&2&1&-2\\\
0&-3&0&0&3&2&-1&1&-2\\\ 0&0&0&0&0&2&2&-2&-2\\\ 0&0&0&0&0&2&2&-2&-2\\\
0&0&0&0&0&-1&-1&1&1\\\ 0&0&0&0&0&-1&-1&1&1\\\
\end{array}\right),\,\mathcal{D}\,=\,(\mathbf{1}_{4},0).\end{split}$ (75)

  * 16.

$\textbf{W}\,_{3}^{(1,0,1)}(3,2,1)$
This Cweb has three diagrams, one which is displayed below. The mixing matrix
for this particular Cweb agrees with the general form of any prime dimensional
mixing matrices [75]. The table shows the chosen order of shuffles and their
corresponding $s$-factors.
Diagrams Sequences s-factors $C_{1}$ $\\{\\{ABF\\}\\}$ 0 $C_{2}$
$\\{\\{AFB\\}\\}$ 1 $C_{3}$ $\\{\\{FAB\\}\\}$ 1
The $R$ and $D$ matrices are given by,

$\displaystyle\begin{split}R=\frac{1}{2}\left(\begin{array}[]{ccc}2&-1&-1\\\
0&1&-1\\\ 0&-1&1\\\
\end{array}\right),\,\mathcal{D}\,=\,(\mathbf{1}_{2},0)\,.\end{split}$ (76)

  * 17.

$\textbf{W}\,_{3,\text{I}}^{(2,1)}(5,1,1)$
This is the first Cweb with same correlator and attachment content. It has
fifteen diagrams, one of which is shown below. The table shows the chosen
order of shuffle and their corresponding $s$-factors.
Diagrams Sequences s-factors $C_{1}$ $\\{\\{ABFGH\\}\\}$ 0 $C_{2}$
$\\{\\{ABHGF\\}\\}$ 0 $C_{3}$ $\\{\\{ABGFH\\}\\}$ 0 $C_{4}$
$\\{\\{ABGHF\\}\\}$ 0 $C_{5}$ $\\{\\{AGBFH\\}\\}$ 0 $C_{6}$
$\\{\\{AGBHF\\}\\}$ 0 $C_{7}$ $\\{\\{AFBGH\\}\\}$ 0 $C_{8}$
$\\{\\{ABFHG\\}\\}$ 0 $C_{9}$ $\\{\\{ABHFG\\}\\}$ 0 $C_{10}$
$\\{\\{AGFBH\\}\\}$ 0 $C_{11}$ $\\{\\{GABFH\\}\\}$ 0 $C_{12}$
$\\{\\{GABHF\\}\\}$ 0 $C_{13}$ $\\{\\{AFBHG\\}\\}$ 1 $C_{14}$
$\\{\\{AFGBH\\}\\}$ 1 $C_{15}$ $\\{\\{GAFBH\\}\\}$ 1

The mixing matrix and the diagonal matrix for this Cweb are given by,

$\displaystyle\begin{split}R=&\frac{1}{12}\left(\begin{array}[]{ccccccccccccccc}12&0&0&0&0&0&-6&-6&0&-6&-6&0&2&2&8\\\
0&12&0&0&0&0&-6&0&-6&-6&0&-6&2&2&8\\\ 0&0&12&0&0&0&-12&-6&0&-12&-6&0&8&8&8\\\
0&0&0&12&0&0&-12&0&-6&-12&0&-6&8&8&8\\\ 0&0&0&0&12&0&-6&-6&0&-6&-6&0&8&2&2\\\
0&0&0&0&0&12&-6&0&-6&-6&0&-6&8&2&2\\\ 0&0&0&0&0&0&6&0&0&-6&0&0&-4&2&2\\\
0&0&0&0&0&0&0&6&0&0&-6&0&-4&-4&8\\\ 0&0&0&0&0&0&0&0&6&0&0&-6&-4&-4&8\\\
0&0&0&0&0&0&-6&0&0&6&0&0&2&2&-4\\\ 0&0&0&0&0&0&0&-6&0&0&6&0&8&-4&-4\\\
0&0&0&0&0&0&0&0&-6&0&0&6&8&-4&-4\\\ 0&0&0&0&0&0&0&0&0&0&0&0&2&-4&2\\\
0&0&0&0&0&0&0&0&0&0&0&0&-4&8&-4\\\ 0&0&0&0&0&0&0&0&0&0&0&0&2&-4&2\\\
\end{array}\right)\,\\\ \mathcal{D}\,=\,&(\mathbf{1}_{10},0)\end{split}$ (77)

  * 18.

$\textbf{W}\,_{3,\text{II}}^{(2,1)}(5,1,1)$
This is the second Cweb with same correlator and attachment content. It has
thirty diagrams, one of which is shown below. The table gives the chosen order
of thirty shuffles and their corresponding $s$-factors. We do not to present
the mixing matrix fo this Cweb here due to its larger dimension, however it
can be found in the ancillary file Boomerang.nb.

Diagrams | Sequences | s-factors
---|---|---
$C_{1}$ | $\\{\\{ABFG\\}\\}$ | 0
$C_{2}$ | $\\{\\{ABFH\\}\\}$ | 0
$C_{3}$ | $\\{\\{AFGB\\}\\}$ | 0
$C_{4}$ | $\\{\\{AFBG\\}\\}$ | 0
$C_{5}$ | $\\{\\{AFBH\\}\\}$ | 0
$C_{6}$ | $\\{\\{AFHB\\}\\}$ | 0
$C_{7}$ | $\\{\\{FAGB\\}\\}$ | 0
$C_{8}$ | $\\{\\{FABG\\}\\}$ | 0
$C_{9}$ | $\\{\\{FABH\\}\\}$ | 0
$C_{10}$ | $\\{\\{FAHB\\}\\}$ | 0
$C_{11}$ | $\\{\\{FBAG\\}\\}$ | 0
$C_{12}$ | $\\{\\{FBAH\\}\\}$ | 0
$C_{13}$ | $\\{\\{AGFB\\}\\}$ | 0
$C_{14}$ | $\\{\\{ABGF\\}\\}$ | 0
$C_{15}$ | $\\{\\{AFGH\\}\\}$ | 0

Diagrams | Sequences | s-factors
---|---|---
$C_{16}$ | $\\{\\{AFHG\\}\\}$ | 0
$C_{17}$ | $\\{\\{BAFG\\}\\}$ | 0
$C_{18}$ | $\\{\\{BAFH\\}\\}$ | 0
$C_{19}$ | $\\{\\{BFAG\\}\\}$ | 0
$C_{20}$ | $\\{\\{BFAH\\}\\}$ | 0
$C_{21}$ | $\\{\\{FAGH\\}\\}$ | 0
$C_{22}$ | $\\{\\{FAHG\\}\\}$ | 0
$C_{23}$ | $\\{\\{FBHA\\}\\}$ | 0
$C_{24}$ | $\\{\\{FHAB\\}\\}$ | 0
$C_{25}$ | $\\{\\{AGBF\\}\\}$ | 1
$C_{26}$ | $\\{\\{AGFH\\}\\}$ | 1
$C_{27}$ | $\\{\\{BAGF\\}\\}$ | 1
$C_{28}$ | $\\{\\{BFHA\\}\\}$ | 1
$C_{29}$ | $\\{\\{FHAG\\}\\}$ | 1
$C_{30}$ | $\\{\\{FHBA\\}\\}$ | 1

The diagonal matrix for this Cweb is given by,

$\displaystyle\mathcal{D}\,=\,(\mathbf{1}_{20},0)$ (78)

  * 19.

$\textbf{W}\,_{3}^{(1,0,1)}(4,1,1)$
This is a Cweb made out of one four-gluon correlator and a two-gluon
correlator. It has six diagrams, one of them is displayed below. The table
shows the chosen order of shuffles and their corresponding $s$-factors.
Diagrams Sequences s-factors $C_{1}$ $\\{\\{ABFG\\}\\}$ 0 $C_{2}$
$\\{\\{ABGF\\}\\}$ 0 $C_{3}$ $\\{\\{BAFG\\}\\}$ 0 $C_{4}$ $\\{\\{BAGF\\}\\}$ 0
$C_{5}$ $\\{\\{BGAF\\}\\}$ 1 $C_{6}$ $\\{\\{AFBG\\}\\}$ 1

The $R$, and $D$ matrices are given by,

$\displaystyle\begin{split}R=\frac{1}{2}\left(\begin{array}[]{cccccc}2&0&0&0&-1&-1\\\
0&2&0&0&-1&-1\\\ 0&0&2&0&-1&-1\\\ 0&0&0&2&-1&-1\\\ 0&0&0&0&1&-1\\\
0&0&0&0&-1&1\\\
\end{array}\right),\,\mathcal{D}\,=\,(\mathbf{1}_{5},0)\end{split}$ (79)

  * 20.

$\textbf{W}\,_{3}^{(4)}(6,1,1)$
This is the largest Cweb, it has ninety diagrams. The table shows the chosen
order of shuffle and their corresponding $s$-factors. We do not to present the
mixing matrix fo this Cweb here due to its larger dimension, however it can be
found in the ancillary file Boomerang.nb.

Diagrams | Sequences | s-factors
---|---|---
$C_{1}$ | $\\{\\{ABFGHL\\}\\}$ | 0
$C_{2}$ | $\\{\\{ABFHGL\\}\\}$ | 0
$C_{3}$ | $\\{\\{ABLGHF\\}\\}$ | 0
$C_{4}$ | $\\{\\{ABLHGF\\}\\}$ | 0
$C_{5}$ | $\\{\\{ABGFHL\\}\\}$ | 0
$C_{6}$ | $\\{\\{ABGLHF\\}\\}$ | 0
$C_{7}$ | $\\{\\{ABGHFL\\}\\}$ | 0
$C_{8}$ | $\\{\\{ABGHLF\\}\\}$ | 0
$C_{9}$ | $\\{\\{ABHFGL\\}\\}$ | 0
$C_{10}$ | $\\{\\{ABHLGF\\}\\}$ | 0
$C_{11}$ | $\\{\\{ABHGFL\\}\\}$ | 0
$C_{12}$ | $\\{\\{ABHGLF\\}\\}$ | 0
$C_{13}$ | $\\{\\{AGBFHL\\}\\}$ | 0
$C_{14}$ | $\\{\\{AGBLHF\\}\\}$ | 0
$C_{15}$ | $\\{\\{AGBHFL\\}\\}$ | 0
$C_{16}$ | $\\{\\{AGBHLF\\}\\}$ | 0
$C_{17}$ | $\\{\\{AGHBFL\\}\\}$ | 0
$C_{18}$ | $\\{\\{AGHBLF\\}\\}$ | 0
$C_{19}$ | $\\{\\{AHBFGL\\}\\}$ | 0
$C_{20}$ | $\\{\\{AHBLGF\\}\\}$ | 0
$C_{21}$ | $\\{\\{AHBGFL\\}\\}$ | 0
$C_{22}$ | $\\{\\{AHBGLF\\}\\}$ | 0
$C_{23}$ | $\\{\\{AHGBLF\\}\\}$ | 0
$C_{24}$ | $\\{\\{AFBGLH\\}\\}$ | 0
$C_{25}$ | $\\{\\{AFBGHL\\}\\}$ | 0
$C_{26}$ | $\\{\\{AFBHLG\\}\\}$ | 0
$C_{27}$ | $\\{\\{AFBHGL\\}\\}$ | 0
$C_{28}$ | $\\{\\{AFGBHL\\}\\}$ | 0
$C_{29}$ | $\\{\\{AFHBGL\\}\\}$ | 0
$C_{30}$ | $\\{\\{ABFLGH\\}\\}$ | 0

Diagrams | Sequences | s-factors
---|---|---
$C_{31}$ | $\\{\\{ABFLHG\\}\\}$ | 0
$C_{32}$ | $\\{\\{ABFGLH\\}\\}$ | 0
$C_{33}$ | $\\{\\{ABFHLG\\}\\}$ | 0
$C_{34}$ | $\\{\\{ABLFGH\\}\\}$ | 0
$C_{35}$ | $\\{\\{ABLFHG\\}\\}$ | 0
$C_{36}$ | $\\{\\{ABLGFH\\}\\}$ | 0
$C_{37}$ | $\\{\\{ABLHFG\\}\\}$ | 0
$C_{38}$ | $\\{\\{ABGFLH\\}\\}$ | 0
$C_{39}$ | $\\{\\{ABGLFH\\}\\}$ | 0
$C_{40}$ | $\\{\\{ABHFLG\\}\\}$ | 0
$C_{41}$ | $\\{\\{ABHLFG\\}\\}$ | 0
$C_{42}$ | $\\{\\{AGFBLH\\}\\}$ | 0
$C_{43}$ | $\\{\\{AGFBHL\\}\\}$ | 0
$C_{44}$ | $\\{\\{AGFHBL\\}\\}$ | 0
$C_{45}$ | $\\{\\{AGBFLH\\}\\}$ | 0
$C_{46}$ | $\\{\\{AGBLFH\\}\\}$ | 0
$C_{47}$ | $\\{\\{AGHFBL\\}\\}$ | 0
$C_{48}$ | $\\{\\{AHFBLG\\}\\}$ | 0
$C_{49}$ | $\\{\\{AHFBGL\\}\\}$ | 0
$C_{50}$ | $\\{\\{AHFGBL\\}\\}$ | 0
$C_{51}$ | $\\{\\{AHBFLG\\}\\}$ | 0
$C_{52}$ | $\\{\\{AHBLFG\\}\\}$ | 0
$C_{53}$ | $\\{\\{AHGFBL\\}\\}$ | 0
$C_{54}$ | $\\{\\{AHGBFL\\}\\}$ | 0
$C_{55}$ | $\\{\\{GAFBHL\\}\\}$ | 0
$C_{56}$ | $\\{\\{GABFLH\\}\\}$ | 0
$C_{57}$ | $\\{\\{GABFHL\\}\\}$ | 0
$C_{58}$ | $\\{\\{GABLFH\\}\\}$ | 0
$C_{59}$ | $\\{\\{GABLHF\\}\\}$ | 0
$C_{60}$ | $\\{\\{GABHFL\\}\\}$ | 0

Diagrams | Sequences | s-factors
---|---|---
$C_{61}$ | $\\{\\{GABHLF\\}\\}$ | 0
$C_{62}$ | $\\{\\{GAHFBL\\}\\}$ | 0
$C_{63}$ | $\\{\\{GAHBFL\\}\\}$ | 0
$C_{64}$ | $\\{\\{GAHBLF\\}\\}$ | 0
$C_{65}$ | $\\{\\{GHABFL\\}\\}$ | 0
$C_{66}$ | $\\{\\{GHABLF\\}\\}$ | 0
$C_{67}$ | $\\{\\{HAFBGL\\}\\}$ | 0
$C_{68}$ | $\\{\\{HABFLG\\}\\}$ | 0
$C_{69}$ | $\\{\\{HABFGL\\}\\}$ | 0
$C_{70}$ | $\\{\\{HABLFG\\}\\}$ | 0
$C_{71}$ | $\\{\\{HABLGF\\}\\}$ | 0
$C_{72}$ | $\\{\\{HABGFL\\}\\}$ | 0
$C_{73}$ | $\\{\\{HABGLF\\}\\}$ | 0
$C_{74}$ | $\\{\\{HAGFBL\\}\\}$ | 0
$C_{75}$ | $\\{\\{HAGBFL\\}\\}$ | 0
$C_{76}$ | $\\{\\{HAGBLF\\}\\}$ | 0
$C_{77}$ | $\\{\\{HGABFL\\}\\}$ | 0
$C_{78}$ | $\\{\\{HGABLF\\}\\}$ | 0
$C_{79}$ | $\\{\\{AFBLGH\\}\\}$ | 1
$C_{80}$ | $\\{\\{AFBLHG\\}\\}$ | 1
$C_{81}$ | $\\{\\{AFGBLH\\}\\}$ | 1
$C_{82}$ | $\\{\\{AFGHBL\\}\\}$ | 1
$C_{83}$ | $\\{\\{AFHBLG\\}\\}$ | 1
$C_{84}$ | $\\{\\{AFHGBL\\}\\}$ | 1
$C_{85}$ | $\\{\\{GAFBLH\\}\\}$ | 1
$C_{86}$ | $\\{\\{GAFHBL\\}\\}$ | 1
$C_{87}$ | $\\{\\{GHAFBL\\}\\}$ | 1
$C_{88}$ | $\\{\\{HAFBLG\\}\\}$ | 1
$C_{89}$ | $\\{\\{HAFGBL\\}\\}$ | 1
$C_{90}$ | $\\{\\{HGAFBL\\}\\}$ | 1

Diagonalizing matrix is given as

$\displaystyle\mathcal{D}\,=\,(\mathbf{1}_{50},0)$ (80)

## References

  * [1] N. Agarwal, S. Pal, A. Srivastav and A. Tripathi,  _Building blocks of Cwebs in multiparton scattering amplitudes_ ,  JHEP 06, 020 (2022),  doi:10.1007/JHEP06(2022)020,  [2204.05936].
  * [2] N. Agarwal, S. Pal, A. Srivastav and A. Tripathi,  _Deciphering colour building blocks of massive multiparton amplitudes at 4-loops and beyond_ ,  JHEP 02, 258 (2023),  doi:10.1007/JHEP02(2023)258,  [2212.06610].
  * [3] N. Agarwal, A. Danish, L. Magnea, S. Pal and A. Tripathi,  _Multiparton webs beyond three loops_ ,  JHEP 05, 128 (2020),  doi:10.1007/JHEP05(2020)128,  [2003.09714].
  * [4] F. Bloch and A. Nordsieck,  _Note on the Radiation Field of the electron_ ,  Phys. Rev. 52, 54 (1937),  doi:10.1103/PhysRev.52.54.
  * [5] V. V. Sudakov,  _Vertex parts at very high-energies in quantum electrodynamics_ ,  Sov. Phys. JETP 3, 65 (1956),  [Zh. Eksp. Teor. Fiz.30,87(1956)].
  * [6] D. R. Yennie, S. C. Frautschi and H. Suura,  _The infrared divergence phenomena and high-energy processes_ ,  Annals Phys. 13, 379 (1961),  doi:10.1016/0003-4916(61)90151-8.
  * [7] T. Kinoshita,  _Mass singularities of Feynman amplitudes_ ,  J. Math. Phys. 3, 650 (1962),  doi:10.1063/1.1724268.
  * [8] T. D. Lee and M. Nauenberg,  _Degenerate Systems and Mass Singularities_ ,  Phys. Rev. 133, B1549 (1964),  doi:10.1103/PhysRev.133.B1549,  [,25(1964)].
  * [9] G. Grammer, Jr. and D. R. Yennie,  _Improved treatment for the infrared divergence problem in quantum electrodynamics_ ,  Phys. Rev. D8, 4332 (1973),  doi:10.1103/PhysRevD.8.4332.
  * [10] A. H. Mueller,  _On the Asymptotic Behavior of the Sudakov Form-factor_ ,  Phys. Rev. D20, 2037 (1979),  doi:10.1103/PhysRevD.20.2037.
  * [11] J. C. Collins,  _Algorithm to Compute Corrections to the Sudakov Form-factor_ ,  Phys. Rev. D22, 1478 (1980),  doi:10.1103/PhysRevD.22.1478.
  * [12] A. Sen,  _Asymptotic Behavior of the Sudakov Form-Factor in QCD_ ,  Phys. Rev. D24, 3281 (1981),  doi:10.1103/PhysRevD.24.3281.
  * [13] A. Sen,  _Asymptotic Behavior of the Wide Angle On-Shell Quark Scattering Amplitudes in Nonabelian Gauge Theories_ ,  Phys. Rev. D28, 860 (1983),  doi:10.1103/PhysRevD.28.860.
  * [14] G. P. Korchemsky and A. V. Radyushkin,  _Renormalization of the Wilson Loops Beyond the Leading Order_ ,  Nucl. Phys. B283, 342 (1987),  doi:10.1016/0550-3213(87)90277-X.
  * [15] G. P. Korchemsky,  _Sudakov Form-factor in QCD_ ,  Phys. Lett. B220, 629 (1989),  doi:10.1016/0370-2693(89)90799-5.
  * [16] L. Magnea and G. F. Sterman,  _Analytic continuation of the Sudakov form-factor in QCD_ ,  Phys. Rev. D42, 4222 (1990),  doi:10.1103/PhysRevD.42.4222.
  * [17] L. J. Dixon, L. Magnea and G. F. Sterman,  _Universal structure of subleading infrared poles in gauge theory amplitudes_ ,  JHEP 08, 022 (2008),  doi:10.1088/1126-6708/2008/08/022,  [0805.3515].
  * [18] E. Gardi and L. Magnea,  _Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes_ ,  JHEP 03, 079 (2009),  doi:10.1088/1126-6708/2009/03/079,  [0901.1091].
  * [19] T. Becher and M. Neubert,  _On the Structure of Infrared Singularities of Gauge-Theory Amplitudes_ ,  JHEP 06, 081 (2009),  doi:10.1088/1126-6708/2009/06/081,10.1007/JHEP11(2013)024,  [Erratum: JHEP11,024(2013)],  [0903.1126].
  * [20] I. Feige and M. D. Schwartz,  _Hard-Soft-Collinear Factorization to All Orders_ ,  Phys. Rev. D90(10), 105020 (2014),  doi:10.1103/PhysRevD.90.105020,  [1403.6472].
  * [21] N. Agarwal, L. Magnea, C. Signorile-Signorile and A. Tripathi,  _The infrared structure of perturbative gauge theories_ ,  Phys. Rept. 994, 1 (2023),  doi:10.1016/j.physrep.2022.10.001,  [2112.07099].
  * [22] A. Gehrmann-De Ridder, T. Gehrmann and E. W. N. Glover,  _Antenna subtraction at NNLO_ ,  JHEP 09, 056 (2005),  doi:10.1088/1126-6708/2005/09/056,  [hep-ph/0505111].
  * [23] G. Somogyi, Z. Trocsanyi and V. Del Duca,  _Matching of singly- and doubly-unresolved limits of tree-level QCD squared matrix elements_ ,  JHEP 06, 024 (2005),  doi:10.1088/1126-6708/2005/06/024,  [hep-ph/0502226].
  * [24] S. Catani and M. Grazzini,  _An NNLO subtraction formalism in hadron collisions and its application to Higgs boson production at the LHC_ ,  Phys. Rev. Lett. 98, 222002 (2007),  doi:10.1103/PhysRevLett.98.222002,  [hep-ph/0703012].
  * [25] M. Czakon,  _A novel subtraction scheme for double-real radiation at NNLO_ ,  Phys. Lett. B693, 259 (2010),  doi:10.1016/j.physletb.2010.08.036,  [1005.0274].
  * [26] R. Boughezal, C. Focke, X. Liu and F. Petriello,  _$W$ -boson production in association with a jet at next-to-next-to-leading order in perturbative QCD_,  Phys. Rev. Lett. 115(6), 062002 (2015),  doi:10.1103/PhysRevLett.115.062002,  [1504.02131].
  * [27] G. F. R. Sborlini, F. Driencourt-Mangin and G. Rodrigo,  _Four-dimensional unsubtraction with massive particles_ ,  JHEP 10, 162 (2016),  doi:10.1007/JHEP10(2016)162,  [1608.01584].
  * [28] F. Caola, K. Melnikov and R. Rontsch,  _Nested soft-collinear subtractions in NNLO QCD computations_ ,  Eur. Phys. J. C77(4), 248 (2017),  doi:10.1140/epjc/s10052-017-4774-0,  [1702.01352].
  * [29] F. Herzog,  _Geometric IR subtraction for final state real radiation_ ,  JHEP 08, 006 (2018),  doi:10.1007/JHEP08(2018)006,  [1804.07949].
  * [30] L. Magnea, E. Maina, G. Pelliccioli, C. Signorile-Signorile, P. Torrielli and S. Uccirati,  _Local analytic sector subtraction at NNLO_ ,  JHEP 12, 107 (2018),  doi:10.1007/JHEP06(2019)013,10.1007/JHEP12(2018)107,  [Erratum: JHEP06,013(2019)],  [1806.09570].
  * [31] L. Magnea, E. Maina, G. Pelliccioli, C. Signorile-Signorile, P. Torrielli and S. Uccirati,  _Factorisation and Subtraction beyond NLO_ ,  JHEP 12, 062 (2018),  doi:10.1007/JHEP12(2018)062,  [1809.05444].
  * [32] Z. Capatti, V. Hirschi, A. Pelloni and B. Ruijl,  _Local Unitarity: a representation of differential cross-sections that is locally free of infrared singularities at any order_ ,  JHEP 04, 104 (2021),  doi:10.1007/JHEP04(2021)104,  [2010.01068].
  * [33] L. Magnea, C. Milloy, C. Signorile-Signorile, P. Torrielli and S. Uccirati,  _Strongly-ordered infrared limits for subtraction counterterms from factorisation_ ,  PoS LL2022, 075 (2022),  doi:10.22323/1.416.0075,  [2209.06102].
  * [34] G. Bertolotti, L. Magnea, G. Pelliccioli, A. Ratti, C. Signorile-Signorile, P. Torrielli and S. Uccirati,  _NNLO subtraction for any massless final state: a complete analytic expression_ (2022),  [2212.11190].
  * [35] G. Bertolotti, L. Magnea, G. Pelliccioli, A. Ratti, C. Signorile-Signorile, P. Torrielli and S. Uccirati,  _Towards the automation of the Local Analytic Sector Subtraction_ ,  PoS LL2022, 056 (2022),  doi:10.22323/1.416.0056.
  * [36] N. Kidonakis and G. F. Sterman,  _Subleading logarithms in QCD hard scattering_ ,  Phys. Lett. B 387, 867 (1996),  doi:10.1016/0370-2693(96)01080-5.
  * [37] N. Kidonakis and G. F. Sterman,  _Resummation for QCD hard scattering_ ,  Nucl. Phys. B 505, 321 (1997),  doi:10.1016/S0550-3213(97)00506-3,  [hep-ph/9705234].
  * [38] N. Kidonakis, G. Oderda and G. F. Sterman,  _Evolution of color exchange in QCD hard scattering_ ,  Nucl. Phys. B531, 365 (1998),  doi:10.1016/S0550-3213(98)00441-6,  [hep-ph/9803241].
  * [39] I. A. Korchemskaya and G. P. Korchemsky,  _High-energy scattering in QCD and cross singularities of Wilson loops_ ,  Nucl. Phys. B437, 127 (1995),  doi:10.1016/0550-3213(94)00553-Q,  [hep-ph/9409446].
  * [40] S. M. Aybat, L. J. Dixon and G. F. Sterman,  _The Two-loop anomalous dimension matrix for soft gluon exchange_ ,  Phys. Rev. Lett. 97, 072001 (2006),  doi:10.1103/PhysRevLett.97.072001,  [hep-ph/0606254].
  * [41] S. M. Aybat, L. J. Dixon and G. F. Sterman,  _The Two-loop soft anomalous dimension matrix and resummation at next-to-next-to leading pole_ ,  Phys. Rev. D74, 074004 (2006),  doi:10.1103/PhysRevD.74.074004,  [hep-ph/0607309].
  * [42] A. Mitov, G. F. Sterman and I. Sung,  _The Massive Soft Anomalous Dimension Matrix at Two Loops_ ,  Phys. Rev. D79, 094015 (2009),  doi:10.1103/PhysRevD.79.094015,  [0903.3241].
  * [43] A. Ferroglia, M. Neubert, B. D. Pecjak and L. L. Yang,  _Two-loop divergences of scattering amplitudes with massive partons_ ,  Phys. Rev. Lett. 103, 201601 (2009),  doi:10.1103/PhysRevLett.103.201601,  [0907.4791].
  * [44] A. Ferroglia, M. Neubert, B. D. Pecjak and L. L. Yang,  _Two-loop divergences of massive scattering amplitudes in non-abelian gauge theories_ ,  JHEP 11, 062 (2009),  doi:10.1088/1126-6708/2009/11/062,  [0908.3676].
  * [45] N. Kidonakis,  _Two-loop soft anomalous dimensions and NNLL resummation for heavy quark production_ ,  Phys. Rev. Lett. 102, 232003 (2009),  doi:10.1103/PhysRevLett.102.232003,  [0903.2561].
  * [46] N. Kidonakis,  _NNLL resummation for s-channel single top quark production_ ,  Phys. Rev. D 81, 054028 (2010),  doi:10.1103/PhysRevD.81.054028,  [1001.5034].
  * [47] N. Kidonakis,  _Two-loop soft anomalous dimensions for single top quark associated production with a $W^{-}$ or $H^{-}$_,  Phys. Rev. D 82, 054018 (2010),  doi:10.1103/PhysRevD.82.054018,  [1005.4451].
  * [48] N. Kidonakis,  _Next-to-next-to-leading-order collinear and soft gluon corrections for t-channel single top quark production_ ,  Phys. Rev. D 83, 091503 (2011),  doi:10.1103/PhysRevD.83.091503,  [1103.2792].
  * [49] Y.-T. Chien, M. D. Schwartz, D. Simmons-Duffin and I. W. Stewart,  _Jet Physics from Static Charges in AdS_ ,  Phys. Rev. D85, 045010 (2012),  doi:10.1103/PhysRevD.85.045010,  [1109.6010].
  * [50] O. Almelid, C. Duhr and E. Gardi,  _Three-loop corrections to the soft anomalous dimension in multileg scattering_ ,  Phys. Rev. Lett. 117(17), 172002 (2016),  doi:10.1103/PhysRevLett.117.172002,  [1507.00047].
  * [51] O. Almelid, C. Duhr, E. Gardi, A. McLeod and C. D. White,  _Bootstrapping the QCD soft anomalous dimension_ ,  JHEP 09, 073 (2017),  doi:10.1007/JHEP09(2017)073,  [1706.10162].
  * [52] T. Becher and M. Neubert,  _Infrared singularities of scattering amplitudes and N 3LL resummation for $n$-jet processes_,  JHEP 01, 025 (2020),  doi:10.1007/JHEP01(2020)025,  [1908.11379].
  * [53] G. Falcioni, E. Gardi, C. Milloy and L. Vernazza,  _Climbing three-Reggeon ladders: four-loop amplitudes in the high-energy limit in full colour_ ,  Phys. Rev. D 103, L111501 (2021),  doi:10.1103/PhysRevD.103.L111501,  [2012.00613].
  * [54] G. Falcioni, E. Gardi, N. Maher, C. Milloy and L. Vernazza,  _Scattering amplitudes in the Regge limit and the soft anomalous dimension through four loops_ ,  JHEP 03, 053 (2022),  doi:10.1007/JHEP03(2022)053,  [2111.10664].
  * [55] G. Falcioni, E. Gardi, N. Maher, C. Milloy and L. Vernazza,  _Two-parton scattering in the high-energy limit: climbing two- and three-Reggeon ladders_ ,  SciPost Phys. Proc. 7, 007 (2022),  doi:10.21468/SciPostPhysProc.7.007,  [2111.05166].
  * [56] S. Catani, D. Colferai and A. Torrini,  _Triple (and quadruple) soft-gluon radiation in QCD hard scattering_ ,  JHEP 01, 118 (2020),  doi:10.1007/JHEP01(2020)118,  [1908.01616].
  * [57] S. Moch, B. Ruijl, T. Ueda, J. A. M. Vermaseren and A. Vogt,  _Four-Loop Non-Singlet Splitting Functions in the Planar Limit and Beyond_ ,  JHEP 10, 041 (2017),  doi:10.1007/JHEP10(2017)041,  [1707.08315].
  * [58] V. Ahrens, M. Neubert and L. Vernazza,  _Structure of Infrared Singularities of Gauge-Theory Amplitudes at Three and Four Loops_ ,  JHEP 09, 138 (2012),  doi:10.1007/JHEP09(2012)138,  [1208.4847].
  * [59] S. Moch, B. Ruijl, T. Ueda, J. A. M. Vermaseren and A. Vogt,  _On quartic colour factors in splitting functions and the gluon cusp anomalous dimension_ ,  Phys. Lett. B782, 627 (2018),  doi:10.1016/j.physletb.2018.06.017,  [1805.09638].
  * [60] K. G. Chetyrkin, G. Falcioni, F. Herzog and J. A. M. Vermaseren,  _Five-loop renormalisation of QCD in covariant gauges_ ,  JHEP 10, 179 (2017),  doi:10.1007/JHEP12(2017)006,10.3204/PUBDB-2018-02123,10.1007/JHEP10(2017)179,  [Addendum: JHEP12,006(2017)],  [1709.08541].
  * [61] G. Das, S.-O. Moch and A. Vogt,  _Soft corrections to inclusive deep-inelastic scattering at four loops and beyond_ ,  JHEP 03, 116 (2020),  doi:10.1007/JHEP03(2020)116,  [1912.12920].
  * [62] G. Das, S. Moch and A. Vogt,  _Approximate four-loop QCD corrections to the Higgs-boson production cross section_ ,  Phys. Lett. B 807, 135546 (2020),  doi:10.1016/j.physletb.2020.135546,  [2004.00563].
  * [63] A. von Manteuffel, E. Panzer and R. M. Schabinger,  _Cusp and collinear anomalous dimensions in four-loop QCD from form factors_ ,  Phys. Rev. Lett. 124(16), 162001 (2020),  doi:10.1103/PhysRevLett.124.162001,  [2002.04617].
  * [64] J. M. Henn, G. P. Korchemsky and B. Mistlberger,  _The full four-loop cusp anomalous dimension in $\mathcal{N}=4$ super Yang-Mills and QCD_,  JHEP 04, 018 (2020),  doi:10.1007/JHEP04(2020)018,  [1911.10174].
  * [65] C. Duhr, B. Mistlberger and G. Vita,  _Soft integrals and soft anomalous dimensions at N 3LO and beyond_,  JHEP 09, 155 (2022),  doi:10.1007/JHEP09(2022)155,  [2205.04493].
  * [66] N. Kidonakis,  _Four-loop massive cusp anomalous dimension in QCD: A calculation from asymptotics_ ,  Phys. Rev. D 107(5), 054006 (2023),  doi:10.1103/PhysRevD.107.054006,  [2301.05972].
  * [67] E. Gardi and L. Magnea,  _Infrared singularities in QCD amplitudes_ ,  Nuovo Cim. C32N5-6, 137 (2009),  doi:10.1393/ncc/i2010-10528-x,  [Frascati Phys. Ser.50,137(2010)],  [0908.3273].
  * [68] T. Becher and M. Neubert,  _Infrared singularities of scattering amplitudes in perturbative QCD_ ,  Phys. Rev. Lett. 102, 162001 (2009),  doi:10.1103/PhysRevLett.102.162001,10.1103/PhysRevLett.111.199905,  [Erratum: Phys. Rev. Lett.111,no.19,199905(2013)],  [0901.0722].
  * [69] Z. L. Liu and N. Schalch,  _Infrared Singularities of Multileg QCD Amplitudes with a Massive Parton at Three Loops_ ,  Phys. Rev. Lett. 129(23), 232001 (2022),  doi:10.1103/PhysRevLett.129.232001,  [2207.02864].
  * [70] G. F. Sterman,  _Infrared divergences in perturbative QCD_ ,  AIP Conf. Proc. 74, 22 (1981),  doi:10.1063/1.33099.
  * [71] J. G. M. Gatheral,  _Exponentiation of Eikonal Cross-sections in Nonabelian Gauge Theories_ ,  Phys. Lett. 133B, 90 (1983),  doi:10.1016/0370-2693(83)90112-0.
  * [72] J. Frenkel and J. C. Taylor,  _Nonabelian eikonal exponentiation_ ,  Nucl. Phys. B246, 231 (1984),  doi:10.1016/0550-3213(84)90294-3.
  * [73] A. Mitov, G. Sterman and I. Sung,  _Diagrammatic Exponentiation for Products of Wilson Lines_ ,  Phys. Rev. D82, 096010 (2010),  doi:10.1103/PhysRevD.82.096010,  [1008.0099].
  * [74] E. Gardi, E. Laenen, G. Stavenga and C. D. White,  _Webs in multiparton scattering using the replica trick_ ,  JHEP 11, 155 (2010),  doi:10.1007/JHEP11(2010)155,  [1008.0098].
  * [75] N. Agarwal, L. Magnea, S. Pal and A. Tripathi,  _Cwebs beyond three loops in multiparton amplitudes_ ,  JHEP 03, 188 (2021),  doi:10.1007/JHEP03(2021)188,  [2102.03598].
  * [76] E. Gardi, M. Harley, R. Lodin, M. Palusa, J. M. Smillie, C. D. White and S. Yeomans,  _Boomerang webs up to three-loop order_ ,  JHEP 12, 018 (2021),  doi:10.1007/JHEP12(2021)018,  [2110.01685].
  * [77] E. Gardi, J. M. Smillie and C. D. White,  _On the renormalization of multiparton webs_ ,  JHEP 09, 114 (2011),  doi:10.1007/JHEP09(2011)114,  [1108.1357].
  * [78] E. Gardi, J. M. Smillie and C. D. White,  _The Non-Abelian Exponentiation theorem for multiple Wilson lines_ ,  JHEP 06, 088 (2013),  doi:10.1007/JHEP06(2013)088,  [1304.7040].
  * [79] E. Gardi,  _From Webs to Polylogarithms_ ,  JHEP 04, 044 (2014),  doi:10.1007/JHEP04(2014)044,  [1310.5268].
  * [80] S. Mishra, S. Pal, A. Srivastav and A. Tripathi,  _Multiparton Cwebs at five loops_ (2023),  [2305.17452].
  * [81] E. Laenen, G. Stavenga and C. D. White,  _Path integral approach to eikonal and next-to-eikonal exponentiation_ ,  JHEP 03, 054 (2009),  doi:10.1088/1126-6708/2009/03/054,  [0811.2067].
  * [82] A. A. Vladimirov,  _Exponentiation for products of Wilson lines within the generating function approach_ ,  JHEP 06, 120 (2015),  doi:10.1007/JHEP06(2015)120,  [1501.03316].
  * [83] A. A. Vladimirov,  _Generating function for web diagrams_ ,  Phys. Rev. D90(6), 066007 (2014),  doi:10.1103/PhysRevD.90.066007,  [1406.6253].
  * [84] A. Vladimirov,  _Structure of rapidity divergences in multi-parton scattering soft factors_ ,  JHEP 04, 045 (2018),  doi:10.1007/JHEP04(2018)045,  [1707.07606].
  * [85] M. Dukes, E. Gardi, H. McAslan, D. J. Scott and C. D. White,  _Webs and Posets_ ,  JHEP 01, 024 (2014),  doi:10.1007/JHEP01(2014)024,  [1310.3127].
  * [86] M. Dukes, E. Gardi, E. Steingrimsson and C. D. White,  _Web worlds, web-colouring matrices, and web-mixing matrices_ ,  J. Comb. Theory Ser. A120, 1012 (2013),  doi:10.1016/j.jcta.2013.02.001,  [1301.6576].
  * [87] M. Dukes and C. D. White,  _Web matrices: structural properties and generating combinatorial identities_ (2016),  [1603.01589].
  * [88] E. Gardi and C. D. White,  _General properties of multiparton webs: Proofs from combinatorics_ ,  JHEP 03, 079 (2011),  doi:10.1007/JHEP03(2011)079,  [1102.0756].
  * [89] M. Mézard, G. Parisi and M. Virasoro,  _Spin glass theory and beyond_ ,  World Scientific Lecture Notes in Physics 9 (1987).
  * [90] https://oeis.org/A000670  .