Building blocks of Cwebs in multiparton scattering amplitudes

The correlators of Wilson-line operators in non-abelian gauge theories are known to exponentiate, and their logarithms can be organised in terms of the collections of Feynman diagrams called Cwebs. The colour factors that appear in the logarithm correspond to completely connected diagrams and are determined by the web mixing matrices. In this article we introduce several new concepts: (a) Normal ordering of the diagrams of a Cweb, (b) Fused-Webs (c) Basis and Family of Cwebs. We use these ideas together with a Uniqueness theorem that we prove to arrive at an understanding of the diagonal blocks, and several null matrices that appear in the mixing matrices. We demonstrate using our formalism that, once the basis Cwebs present upto order $\alpha_{s}^{n}$ are determined, the number of exponentiated colour factors for several classes of Cwebs starting at order $\alpha_{s}^{n+1}$ can be predicted. We further provide complete results for the mixing matrices, to all orders in perturbation theory, for two special classes of Cwebs using our framework.


Introduction
The infrared (IR) structure of scattering amplitudes in gauge theories is an important object of study, and has a long history spanning almost a century [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. These structures are universal, that is they are independent of the hard scattering processes. The universality of these structures gives us remarkable all order insights of the perturbation theory. A recent review on the subject can be found in [18]. These studies have practical applications in the study of high energy scattering experiments at different colliders. The IR singularities that appear in the intermediate stages of calculations of the observables such as cross-sections cancel when the contributions of real emissions and virtual corrections are added. However, these singularities often leave their imprints in the form of large logarithms of kinematic invariants, which can make a fixed order result lose predictive power in certain kinematical regions. It is the universality of the IR structure which enables a summation of these large logarithms to all orders in perturbation theory and allows us to recover the predictive power in those kinematical regions [19][20][21]. Furthermore, a knowledge of these structures is also very helpful in organizing fixed order calculations. The cancellation of these IR singularities for complicated observables in colliders is not a trivial task, and using the universality of the IR singularities several efficient subtraction procedures for this purpose have been developed [22][23][24][25][26][27][28][29][30][31][32][33][34].
The factorization property of QCD in the IR limit enables us in studying these singular parts efficiently, without calculating the complicated hard parts. The soft function, that controls the IR singular parts in a scattering process, can be expressed in terms of matrix elements of Wilson line correlators [35,36]. These matrix elements also play an important role in QCD based effective theories [37][38][39]. The Wilson-line operators Φ(γ) evaluated on smooth space-time contours γ are defined as, (1.1) where A µ (x) = A µ a (x) T a is a non-abelian gauge field, and T a is a generator of the gauge algebra, which can be taken to belong to any desired representation, and P denotes path ordering of the gauge fields. If we restrict ourselves to multi-particle scattering amplitudes in gauge theories, then, we can write the soft function as, Here the Wilson lines are semi-infinite and point along the direction of the hard particle, that is, the smooth contours run along β k , the velocities of the particles involved in this scattering have limit from origin to ∞.
S n suffers from both ultra-violet (UV) and IR (soft) singularities, and requires renormalization. In dimensional regularization S n vanishes as it involves only scaleless integrals and thus, after renormalization it is given by its UV counterterms. The renormalized soft function obeys a renormalization group equation which leads to the following exponentiation: where Γ 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 important object of study on which we will focus in this article. The perturbative calculation of soft-anomalous dimension using renormalization group approach has a history of more than twenty years. Γ n was computed at one loop in [40] (see also [41]); at two loops in the massless case in [42,43], and in the massive case in [44][45][46][47][48]; finally, at three loops in the massless case in [49,50]. The calculation of soft anomalous dimension at four loops is an ongoing effort, and preliminary results are presented in [51][52][53][54][55][56][57][58][59][60][61].
The scale invariance of the soft function puts strong constraints on the form of Γ n . It was shown in [15,16,[62][63][64] that Γ n can only involve dipole correlations between the Wilson lines upto two loops; beyond two loops, quadrupole correlations can arise, which must depend on scale-invariant conformal cross ratios of the form ρ ijkl ≡ (β i ·β j β k ·β l )/(β i ·β k β j ·β l ): the first such correlations arise at three loops, with at least four Wilson lines, and were computed in [49,50]; further correlations may arise only in association with higher-order Casimir operators.
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, S n (γ i ) = exp W n (γ i ) , (1.4) where W n (γ i ) are known as webs, and can be directly computed using Feynman diagrams. Webs are defined as connected photon sub-diagrams in the abelian gauge theory, while in non-abelian gauge theory, for two Wilson line processes, webs are defined as two-line irreducible diagrams, that is, diagrams that remain connected upon cutting the Wilson lines [65][66][67]. In case of multi-parton scattering process, the webs in non-abelian gauge theory are defined as sets of diagrams that differ from each other by the order of gluon attachments on each Wilson line [68,69]. The kinematics and the colour factors of a diagram in a web mix among themselves, through a web mixing matrix, which can be determined using a replica trick algorithm [69,70].
Cwebs -a generalization of webs -which are a set of skeleton diagrams built out of connected gluon correlators attached to Wilson lines, and are closed under shuffles of the gluon attachments to each Wilson line were introduced in [71,72]. The mixing between diagrams of a web does not get altered upon replacing the diagrams by the corresponding skeleton diagrams. That is, the same mixing matrix describes the mixing of diagrams of a web and its corresponding Cweb.
The Cweb mixing matrices are central objects in the study of non-abelian exponentiation. An alternative approach of generating functionals was developed in [73][74][75]. There have been attempts to determine the web mixing matrices bypassing the replica trick algorithm. Combinatoric ideas such as partial order sets have been employed in constructing matrices for certain classes of Cwebs [76][77][78]. All prime dimensional mixing matrices were also constructed directly without using replica trick algorithm in [72].
In this article we have introduced several new ideas such as: (a) Normal ordering of the diagrams of a Cweb (b) Fused-Webs and (c) Basis and Family of Cwebs which prove extremely useful in making the structures present in the mixing matrices very transparent. We prove a Uniqueness theorem which together with the above ideas helps to determine the diagonal blocks of the mixing matrices of a Cweb. These ideas provide us with an ability to predict the rank of the mixing matrices or equivalently the number of independent exponentiated colour factors for several Cwebs.
This paper is structured as follows. In section 2, we review the known properties of the mixing matrices, and provide a Uniqueness theorem for Cweb mixing matrices. In section 3, we define an ordering among the diagrams of a Cweb, and describe the construction of Fused diagrams and Fused-Webs. These entities shed light on the different blocks of the mixing matrices. Further, we calculate the explicit forms of mixing matrices for two classes of Cwebs without using the replica trick in section 4. In section 5, we use Fused-Webs to calculate the diagonal blocks for three classes of Cwebs. Finally, we conclude our findings in section 6. Appendices A, B and C describe the replica trick, the application of Fused-Webs to provide the rank of mixing matrices, and the mixing matrices for the basis Cwebs present up to four loops, respectively.

Cweb mixing matrices: Properties and a Uniqueness theorem
We begin with the definition [71,72]   cr(r−2) . The remaining powers of g in eq. (2.1) arise from the attachments within the blobs and they do not enter into the counting of order of Cwebs.
Cwebs are the proper building blocks of the logarithm of Soft function; and are also useful in the organisation and counting of diagrammatic contributions at higher perturbative orders. The logarithm of the Soft function is a sum over all the Cwebs at each perturbative order: 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 d generates its exponentiated colour factor C,  Now, we will classify the diagrams based on their s values.

Classification of diagrams
We further classify irreducible diagrams into the following two categories: Completely entangled diagram: An irreducible diagram in which all the gluon correlators are entangled and thus none of the gluon correlators 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.
In this article we draw Wilson lines with an arrow whose tails are at the origin. The diagram in fig. (1a) is a reducible diagram. This diagram has three correlators, all of which can be sequentially shrunk to the origin in only one possible way and thus has s = 1. The diagrams in fig. ( if first k 1 diagrams in the Cweb have weight s 1 , followed by the k 2 diagrams with weight s 2 and so forth. We denote the corresponding mixing matrix for the Cweb as, The web mixing matrices are essential quantities for the determination of Wilson line correlators, and thus, the soft anomalous dimension matrix, and we now turn our focus on them.

Properties of mixing matrices
General all order properties of the mixing matrices were first observed in [69], and were proven in [79]. Further, a conjecture regarding the columns of the mixing matrices was proposed in [80].
Below we list down these properties.
1. Idempotence: These matrices are idempotent and act as projection operators: Thus their eigenvalues can only be either 0 or 1, which further implies that their trace is equal to their rank.

Non-abelian exponentiation:
The general non-abelian exponentiation theorem [81] states that the colour factors that survive the above projection by R are the ones that are associated with a diagram that has only one gluon correlator.
3. Row sum rule: The elements of web mixing matrices obey the row sum rule . Column sum rule: The mixing matrices obey the following column sum conjecture: The idempotence of mixing matrices implies that R projects onto only those combinations of kinematic factors that do not contain ultraviolet sub-divergences 1 . For the case of two Wilson lines, the absence of sub-divergences was proved in [65][66][67]. However, for more than two Wilson lines, the all order proof for column sum is not available, although this property has been verified upto four loops [71,72,80,81]. The connection of the column-sum rule to UV sub-divergences is explained in a coordinate-space picture in [35]. In coordinate-space, UV divergences arise from short distances between interaction vertices, and, thus the 'shrinkable' correlators are naturally associated to UV sub-divergences, eq. (2.9) guarantees that these correlators are projected out of the webs.

Uniqueness Theorem
A careful survey of the elements of the large number of web mixing matrices available in the literature -at two [69], three [81], and four loops [71,72] -gives an impression of repeating structures. One is tempted to find some organising principle that could remove the veil from these structures and possibly find the building blocks of these matrices. Towards this goal we prove the following Uniqueness theorem: Uniqueness: For a given column weight vector S = {s(d 1 ), s(d 2 ), . . . , s(d n )} with all s(d i ) = 0, the mixing matrix is unique.
An important consequence of uniqueness is that, if mixing matrix of a Cweb at some order, that has a column weight vector S with only non-zero entries, is known, then we can without any further work write down the mixing matrix of another Cweb that appears at the same or higher perturbative order if it has the same weight vector S.

The proof
In the diagram, shown in fig  4. Discard Cwebs that are given by the product of two or more disconnected lower-order webs.
2 These Cwebs were studied in detail in [82], using the results of [83].

5.
In a massless theory, discard all self-energy Cwebs, where all gluon lines attach to the same Wilson line, as they vanish as a consequence of the eikonal Feynman rules. 6. Discard Cwebs that have been generated by the procedure more than once.
Cwebs generated from the above algorithm that admit same possible shuffles of the attachments on each of the Wilson lines present in the parent diagram, and no possible shuffles on any additional Wilson line that has been introduced, form a family, f . Further, we call a Cweb that belongs to a family a member of that family. The weight vector S is the same for all members of the family as shuffles on the Wilson lines remain the same. Thus a family 3 will be denoted by f (S). Furthermore, since the mixing matrix of a Cweb is completely determined by the shuffles on the Wilson lines, there is a unique mixing matrix for every family.       diagrams, but it generates Cwebs at one order higher that belong to the same family.

Diagrams Sequences s-factors
The third step will produce Cwebs that contain only one diagram, or have been already generated by first and second steps of the algorithm. For example the Cweb shown in fig. (8e) is same as that in fig. (8f). Therefore, for the proof of theorem it is sufficient to analyse only the first two categories.
We conclude, thus, that only the first step of the algorithm can produce a new basis Cweb. Crucially, it is evident that if the steps of the algorithm generate a Cweb with a given weight vector S with all non-zero entries, then it is a member of a unique family. Recall that each family has a unique mixing matrix and this completes the proof of the Uniqueness theorem.

Application at four loops
We will now show the utility of the Uniqueness theorem for Cwebs that appear at four loops. In and R(1 6 ) are given by  all the Cwebs that have reducible diagrams, without any further work. The explicit forms of the mixing matrices for new basis webs that appear at four loops can be found in the appendix C.
We will see in later sections that the matrices of basis Cwebs form the building blocks of the matrices corresponding to those Cwebs that contain irreducible diagrams. We would like to

Normal ordering and Fused-Webs
Having said all that we had for the Cwebs which contain only reducible diagrams in the previous section, we now turn our attention to those Cwebs which have one or more irreducible diagrams.
We will describe here how the basis matrices present upto order α n s appear in a general mixing matrix of these Cwebs, at orders higher than α n s .

Normal ordering
We begin by ordering the diagrams of a Cweb in such a way that the irreducible diagrams appear before the reducible diagrams. Using the fact that the exponentiated colour factors of reducible diagrams are independent of the irreducible diagrams in a Cweb [69], we immediately find that upon ordering in this manner, the general structure of a n × n mixing matrix for a Cweb with l diagrams having s = 0, and m diagrams having s = 0 becomes, where l + m = n . We write this more compactly as where A and D are square matrices of order l × l, and m × m corresponding to irreducible and reducible diagrams respectively; O is null matrix of order m × l, and B is a matrix of order l × m.
More layers of structure get unveiled if the diagrams are further ordered as That is we arrange the diagrams such that the first k diagrams are completely entangled, followed by (l − k) partially entangled, further followed by reducible diagrams which appear in ascending order of their s-factors. We define this order of diagrams of a Cweb as Normal order, and the corresponding mixing matrix to be Normal ordered.
Consider W The replica trick algorithm which determines the explicit form of the mixing matrices works in a fashion that it can disentangle two entangled correlators, however, the converse is not possible.
This implies that the ECF of d 1 in (3.3), for example, is of the following form (see eq. (A.4))  In general for a completely entangled diagram d i in (3.3), the ECF is given by, That is, the ECF for a completely entangled diagram consists of colour of the partially entangled diagrams and reducible diagrams along with its own colour. A careful examination of the replica trick algorithm reveals that the ECF of a partially entangled diagram will not contain the colours of the completely entangled diagrams.
Thus, if a Cweb contains l irreducible diagrams, out of which k are completely entangled, and (l −k) are partially entangled, then the Normal ordering puts the matrix A in the following general form where I is an identity matrix of order k that corresponds to completely entangled diagrams, A L is a square matrix of order (l − k) corresponding to partially entangled diagrams, O is null matrix of From the foregoing discussion we conclude that the general structure of the mixing matrix R after Normal ordering is We further explore the structure of matrices A and D in the next subsections.

Diagonal block D of a mixing matrix R
Recall that the matrix D gives mixing between the reducible diagrams of a Cweb. Now, we prove that D obeys all the properties of the mixing matrices that are known to date: • Idempotence: Using R 2 = R, eq. (3.2) gives That is, D is idempotent.
• Row sum rule: Applying row sum rule to the general form given in eq. (3.1), the rows associated with the reducible diagrams give, where d ij are the elements of matrix D given in eq. (3.1). That is the block D also satisfies zero row-sum rule.
• Column sum rule: Applying column sum rule on the explicit form of the mixing matrix given in eq. (3.1) and remembering that the first l diagrams have s = 0, the columns corresponding to the reducible diagrams give, Thus, the matrix D satisfies the zero column-sum rule.
Hence, we conclude that D satisfies the known properties of a web mixing matrix with S D = As we have proved that D satisfies the properties of a mixing matrix, and has its own column weight vector S D , we can use the Uniqueness theorem stated in section 2 to write the explicit form of D, provided S D is column weight vector for a known family of Cwebs f (S D ).

Structure of the block A: Fused-Webs
In this section, we explore the structure of the block A of mixing matrices that corresponds to mixing of the irreducible diagrams of a Cweb. In each of these irreducible diagrams, there are at least two correlators which are entangled. We call, an entangled piece, the group of correlators for which dragging any correlator will drag all the other correlators to the origin. Next we introduce the concepts of Fused diagrams and Fused-Webs which will prove to be very useful in understanding the structure of block A. The mixing matrix of a Fused-Web is given by D fict . However, note that, even though the mixing matrix D fict satisfies all the properties of a web mixing matrix that are known to date, Fused-Web is not really a Cweb as not all the diagrams of the fictitious Cweb are part of it.
We need to ensure that the idea of Fused-Webs is consistent with the replica trick algorithm. Towards this end, recall that the replica trick algorithm determines the replica ordered colour factors R C(d) h corresponding to each hierarchy h for all the diagram in a Cweb (see eq. (A.4)). Given a hierarchy h the replica ordering operator R disentangles the correlators of entangled piece if the replica number associated with the correlators of entangled piece are different. Thus, an entangled piece remains entangled only when the correlators are associated with same replica number. Further, the operator R can never entangle two correlators that were not entangled to begin with. Therefore, in Fused diagrams, it is legitimate to replace these replica variables by a single replica variable.
We can distil the above discussion into an algorithm to determine the diagonal blocks of the matrix A using Fused-Webs. The steps of the algorithm will be explained in detail afterwards with the help of an explicit example. In the next subsection, we show an explicit example of a Cweb, where we apply the above steps to calculate the diagonal blocks of the matrix A.

Application of Fused-Web
Let us study the Cweb W  • Second type appears in the partially entangled diagram, shown in fig. (13B), where the threepoint correlator is entangled with the red two-point correlator. This kind of entangled piece   fig. (13b). Similarly, Fused diagram corresponding to d 6 having the order of attachments {CBDA} can also be obtained.

Diagrams Sequences s-factors
• The third and final kind of entangled piece appears in diagram shown in fig. (13C), where the three-point correlator is entangled with the blue two-point correlator, whose Fused diagram is shown in fig. (13c). This kind also belongs to two diagrams having the order of attachments We discuss below the Fused-Webs generated by the three distinct entangled pieces in order. If Normal ordered diagrams are further ordered such that the partially entangled diagram d 6 appears next to d 3 , and d 5 next to d 4 , then the mixing matrix for this Cweb will be, Here X is null matrix of order two for this case. However, it is not true in general as the action of replica ordering operator on a diagram with one kind distinct entangled piece maps it to that of another kind. From the foregoing discussion it is evident that, the diagonal blocks of any web mixing matrix will always be the matrices of basis Cwebs, except for the class mentioned in the end of the section 2.2.
The explicit form of the mixing matrix of the Cweb W (2,1) 4 (1, 1, 1, 4) after ordering the diagrams in the fashion mentioned above is which agrees with eq. (3.13). The D block of this mixing matrix is R(1 6 ) given in appendix C. The above example shows how the basis matrices appear in a mixing matrix of a Cweb whose column weight vector has one or more zero entries.

Direct construction of two special classes of Cwebs
Attempts have been made to understand the structure of mixing matrices and ideas from Combinatorics such as posets have proven useful in making all order predictions for certain special classes of webs [76][77][78]. A systematic approach towards unravelling these structures was initiated in [72], where all the prime dimensional mixing matrices at any order in perturbation theory were obtained using the known properties of the matrices that are listed in section 2. Now, armed with the learnings of sections 2 and 3 we push this program further. Our starting point is eq. (3.2), which we rewrite below: We can utilize this result to determine the rank without knowing the full form of R. In this article, we only utilize the idempotence of A and D, however, the third relation in eq. (4.1) provides additional constraints on the elements of mixing matrices which will be explored in the future. 6 At one loop the mixing matrix is identity matrix; at two loops there is only one basis Cweb, R(1 2 ); at three loops we get two additional basis Cwebs with matrices R(1 6 ) and R(1 2 , 2 2 ). All these matrices can appear as D and diagonal blocks of A at four-loops and beyond.
In section 5 we will present all the Cwebs at four loops that contain in their respective D blocks, the matrices corresponding to the Basis Cwebs upto three loops. Before that we will first take up two special classes S = {0, 0, . . . , 0, 1 1 }, and S = {0, 0, . . . , 0, 1 2 } in turn as we can completely determine their corresponding mixing matrices. 6 We thank the anonymous referee for pointing out these additional conditions. Though, we can not predict all the matrix elements of R for this class in general, we can completely construct R for a subclass which has only two identical gluon correlators as shown in fig. (14).
These Cwebs are present only at even loop orders and their all the l − 1 irreducible diagrams are completely entangled. As we have seen in earlier sections the completely entangled diagrams will give an identity matrix of order l − 1; the mixing matrix for this subclass, thus, reduces to The remaining elements are fixed to by applying zero-row sum rule: We now have, for this subclass, the complete mixing matrix R and its rank without using the replica Note that there are two Cwebs W 2 (2, 2) has two diagrams, thus, it will have two dimensional matrix R of the form above, while W (0,2) 2 (2, 4) has six diagrams, and will have the corresponding six dimensional matrix. These results agree with the explicit matrices for these webs obtained in [69,72].
The next Cwebs belonging to this subclass will occur at six loops in the perturbation theory.
We predict that at six loops there will be two Cwebs: W (0,0,2) 2 (4, 4), and W (0,0,2) 2 (2, 6), having eighteen and twenty diagrams respectively, and their corresponding mixing matrices will be of the form eq. (4.8). The number of Cwebs of this subclass at order α n s , where n is an even integer is given by The above sequence is known as Non-negative Integers Repeated [85]. The elements of the sequence are 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, · · · . and it consists two distinct gluon correlators. We call these as Octopus-Pair Cwebs. It is easy to see that all the diagrams of this Cweb are completely entangled except two which are reducible. We obtain the two reducible diagrams when all blue coloured gluons are placed either above or below the red gluons on the Wilson lines. Furthermore, note that for each of these two diagrams there is only one way to sequentially shrink the two correlators to the hard interaction vertex and hence, s-factors for the reducible diagrams are equal to one. Therefore, after normal ordering the column weight The block D of the mixing matrix is fixed to be R (1 2 ) by the Uniqueness theorem.

Octopus-Pair Cwebs with
Next, we determine the block A. As all the irreducible diagrams are completely entangled, the matrix A is an identity matrix of order (n − 2), that is, Thus the mixing matrix for this type of Cwebs with n diagrams is constrained to the following form: The rank of R (1 2 ) is one and the rank of A is equal to n − 2, the rank of the mixing matrix is thus, that is, these Cwebs have n − 1 independent exponentiated colour factors.
The next step is to determine c i and b i using the properties of the mixing matrices. The row-sum rule imposes (4.13) and using idempotence property of mixing matrices we get (4.14) With this we have uniquely fixed all the elements of the mixing matrix for this class of Cwebs, and it is then given by A subclass of the webs shown in fig. (15) was considered in [72]. We observe that eleven nonprime dimensional, and four prime dimensional mixing matrices belong to this class of Cweb for massless Wilson lines at four loops that were calculated in [71,72]. These are listed in table (5).

Determining the number of independent ECFs for Cwebs
In the previous section we showed how our formalism of Fused-Webs together with the Uniqueness theorem of section 2 can completely determine the two classes of Cwebs to all orders in perturbation theory. Now, in this section we will use Fused-Web formalism and the Uniqueness theorem to construct the diagonal blocks of mixing matrices of a Cweb at order α n+1 s using basis Cwebs that appear upto order α n s . We have seen that there are three basis Cwebs shown in fig. (3), and table 3 with the mixing matrices R(1 2 ), R(1 6 ) and R(1 2 , 2 2 ) that appear upto three loops for massless Wilson lines. We can use these as the building blocks and construct three sets of Cwebs that contain these mixing matrices as their respective D blocks. These three classes of Cwebs have weight vectors {0, 0, . . . , 0, 1 2 }, S = {0, 0, . . . , 0, 1 6 }, and S = {0, 0, . . . , 0, 1 2 , 2 As a demonstration of the concept we will pick a Cweb at four loops for each of these three classes and apply the Fused-Webs formalism to construct the diagonal blocks of A, which will again correspond to the three basis Cwebs. This will allow us to predict the rank and thus the total number of independent exponentiated colour factors that these Cwebs have without going through the complete computation by a code based replica trick algorithm.

Number of Diagrams
Cweb Perturbative order Mixing matrix R rank r(R)  . This matrix appears first at three loops, and was first presented in [81]. The rank of this mixing matrix is two, which tells us that the rank of mixing matrix for this class of Cwebs will be,

Cwebs with
We show an explicit example of this class using the Cweb W (2,1) 3 (2, 2, 3) shown in fig. (16), present at four loops.  D-block of mixing matrix for this Cwebs is R(1 6 ). Now we will use the procedure of Fused-Webs described in section (3.3) to find the diagonal blocks of A. The diagrams, d 1 , d 2 , d 3 , and d 4 , given in table (6), are completely entangled diagrams. Each of these form a Fused-Web with single diagram, therefore we get the identity matrix of order four. There is only one kind of entangled piece in the partially entangled diagrams d 5 , and d 6 .  The entangled piece is shown in fig. (17a) The diagonal blocks of the mixing matrix for this Cweb is given by, Therefore, the rank of the mixing matrix for Cweb W The explicit calculation of the mixing matrix for this Cweb was presented in [72], and is given by,  corresponding to S = {1 2 , 2 2 }, and has rank one. Following the same procedure as described above, we can write the rank of the mixing matrix for this class as, We illustrate the procedure of using Fused-diagrams for this class of Cweb using the W  This Cweb has six diagrams, whose shuffles are given in the table (  (1, 1, 2, 3) Thus, block D corresponds to R(1 2 , 2 2 ). The block diagonals of A, can be determined by identifying the entangled pieces. There is only one kind of entangled piece, which is present in d 1 , and d 2 , which is displayed in fig. (19a), which corresponds to a Fused diagram shown in fig. (19b). Now, as this entangled piece appears only in diagrams d 1 , and d 2 of this Cweb, given in table (7), thus, both of them are partially entangled diagrams, and hence, there is no completely entangled diagram.
Thus, we do not have any identity matrix in block A. this Fused-Web is R(1 2 ). Therefore, block A is equal R(1 2 ), and its rank is 1.
The diagonal blocks of the mixing matrix for this Cweb are given by, Hence, the rank of mixing matrix for this Cweb is, The explicit form of mixing matrix was calculated using replica trick in [72], given by, The diagonal blocks of the mixing matrix predicted from our procedure using Fused-Webs and the Uniqueness theorem agrees with the explicit form calculated in [72] using the replica trick. Results for other four-loop Cwebs of this class are presented in appendix B.2.

Summary and outlook
The logarithm of the Soft function can be expressed as a sum over Cwebs that can be written down in terms of Feynman diagrams. This exponentiation of the Cwebs allows us to make predictions of the IR structures in the multiparton scattering amplitudes to all orders in the perturbation theory.
The diagrams of a Cweb mix via a mixing matrix such that they select only colour factors that correspond to fully connected diagrams. The mixing matrices have been studied extensively in the literature.
In this article we have developed a new formalism that allows us to predict the number of We have proved a Uniqueness theorem which states that, for a given column weight vector S = {s(d 1 ), s(d 2 ), . . . , s(d n )} with all s(d i ) = 0, the mixing matrix is unique. We have also introduced the concept of Fused-Webs which has helped us determine the diagonal blocks of the mixing matrices that correspond to the mixing between the irreducible diagrams of a Cweb. Together these ideas provide us with an ability to predict the rank of the mixing matrices or equivalently the number of independent exponentiated colour factors that are present for a given Cweb.
Using our formalism we can predict, without doing the explicit calculations using the replica trick algorithm, the explicit form of mixing matrices of 26 out of 60 Cwebs present at four loops connecting massless Wilson lines using the matrices from two and three loops which is 43% of total number of Cwebs present at four loops. Using Fused-Webs we can further predict the diagonal blocks of 9 mixing matrices at four loops. Thus, in total, we can predict the rank of 35 mixing matrices at four loops which is 58% of the total Cwebs without using the replica trick. All the predictions match with the known results presented in [71,72].
It would be interesting to see if this framework can be expanded further to provide more understanding of the structures present in the mixing matrices. It would also be interesting to see the implications of this formalism on the kinematic structure of the Cwebs.

Acknowledgement
NA, SP and AT would like to thank Lorenzo Magnea for collaboration on earlier projects on Cwebs.
SP would like to thank MoE, Govt. of India, for an SRF fellowship, AS would like to thank CSIR, Govt. of India, for a JRF fellowship (09/1001(0075)/2020-EMR-I).

A Replica trick
One of the powerful techniques in the combinatorial problems in physics, which involves exponentiation is the replica trick [86]. For Wilson line correlators, the replica trick algorithm was developed in [69,70]. The same replica trick was adopted in [71,72] for the calculation of the mixing matrices for four-loop Cwebs. Here, we briefly discuss the replica trick algorithm, which was used in the calculation of the mixing matrices for Cwebs at four loops. To start with, we consider the path integral of the Wilson line correlators as, where S(A a µ ) 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 µ , which means, we replace each A µ by A i µ , where, i = 1, . . . , 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, Now, using this equation, one can calculate W n by calculating O(N r ) terms of the Wilson line correlator in the replicated theory. The method of replicas involves five steps, which are summarized below.
-Associate a replica number to each connected gluon correlator in a Cweb.
-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 T i denotes a colour generator for a correlator belonging to replica number i, then action of R on T i T j preserves the order for i ≤ 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.
-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.
-The exponentiated colour factor for a diagram d is then given by,

B Cwebs with Fused-Webs
In this appendix, we show the direct construction of the diagonal blocks of web mixing matrices, using Fused-Webs.     This table provides the Fused-Webs with the associated mixing matrices for the Cweb. For example, in this Cweb, there are ten completely entangled diagrams, whose order of attachments can be read of from table 8. This then form the Fused-Webs with single diagram, whose mixing matrix is identity matrix of order ten.
Further, there are four distinct entangled pieces appearing in partially entangled diagrams. Each entangled piece is associated with two diagram of Cweb, whose Fused diagrams form a Fused-Web.
Thus there are four Fused-Webs with their corresponding mixing matrices.
The order of diagrams in the Cweb given in table 8, is chosen such that diagrams with same kind of entangled piece appear together. Therefore, mixing matrices of the Fused-Webs for this Cweb, present on the diagonal blocks of A, are given as,      The procedure Fused-Webs is applied to this Cweb, which results in  The order of diagrams in the Cweb given in table 10, is chosen such that diagrams with same kind of entangled piece appear together. Therefore, mixing matrices of the Fused-Webs for this Cweb, present on the diagonal blocks of A, are given as,    (1, 2, 4) The order of diagrams in the Cweb given in table 12, is chosen such that diagrams with same kind of entangled piece appear together. Therefore, mixing matrices of the Fused-Webs for this Cweb, present on the diagonal blocks of A, are given as,

Diagrams Sequences s-factors
Diagrams Sequences s-factors the rank of A is, (B.14) Thus the rank of A is, The number of exponentiated colour factors for this Cweb is,   The procedure of Fused-Webs is applied to this Cweb, which results in  The order of diagrams in the Cweb given in table 18, is chosen such that diagrams with same kind of entangled piece appear together. Therefore, mixing matrices of the Fused-Webs for this Cweb, present on the diagonal blocks of A, are given as, The rank of A is then, The number of exponentiated colour factors is the rank of the mixing matrix which is given as B.2 Fused-Webs of Cweb having S = {0, 0, · · · , 0, 1 2 , 2 2 } In this class of Cwebs, the D-block of mixing matrices is R(1 2 , 2 2 ), whose full form is given in appendix C. Thus the rank for this class of matrices will be, Hence, the diagonals blocks of sub-matrix A determines the rank for this class of Cwebs.         The procedure of Fused-Webs is applied to this Cweb, which results in  The order of diagrams in the Cweb given in table 22, is chosen such that diagrams with same kind of entangled piece appear together. Therefore, mixing matrices of the Fused-Webs for this Cweb, present on the diagonal blocks of A, is given as,   (1, 1, 2, 3) (1, 1, 2, 3)

W
The procedure developed in section 3.3, is applied to this Cweb, which results in  (1, 1, 2, 3) The order of diagrams in the Cweb given in table 24, is chosen such that diagrams with same kind of entangled piece appear together. Therefore, mixing matrices of the Fused-Webs for this Cweb, present on the diagonal blocks of A, is given as,

C Mixing matrices for basis Cwebs
In this appendix, we present the mixing matrices used as a basis to write down D, and the diagonal blocks of A.

Mixing matrix for S = {1 2 }
The unique mixing matrix for this type of Cweb first appears at two loops, and has the form, This matrix was constructed directly using the known properties in [72].

2.
Mixing matrix for S = {1 6 } The mixing matrix for this type of Cweb first appears at three loops, and has the form, The explicit form of this matrix was computed directly in [76], using the idea of posets.

Mixing matrix for S = {1 24 }
The mixing matrix for this type of Cweb first appears at four loops, and has the form,

Mixing matrix for
The mixing matrix for this type of Cweb first appears at three loops, and has the form, The mixing matrix for this type of Cweb first appears at four loops, and has the form, (C.5)

6.
Mixing matrix for S = {1 2 , 2 4 , 4 2 } The mixing matrix for this type of Cweb first appears at four loops, and has the form,