The universality of these structures gives us remarkable all-order insights from perturbation theory. An alternative approach to determine the exponent of the soft function is by diagrammatic exponentiation. We prove a Uniqueness Theorem which together with the above ideas helps to determine the diagonal blocks of the mixing matrices of a Cweb.
In section 2 we discuss the known properties of the mixture matrices and provide a uniqueness theorem for Cweb mixture matrices. In general, all order properties of the mixture matrices were first observed in [69] and proven in [79]. Any rearrangement of the attachments on and across the Wilson lines, keeping the number of Wilson lines fixed, produces a different diagram that is still reducible.
The weight vector S is the same for all members of the family since shuffles on the Wilson lines remain the same. Similarly, all the Cwebs shown in figure 5 have the same fluctuations possible on each of the Wilson lines and therefore belong to the same family f(16).
JHEP06(2022)020
Normal ordering
We start by arranging the diagrams of a Cweb so that the irreducible diagrams appear before the reducible diagrams. Taking advantage of the fact that the exponentiated color factors of reducible diagrams are independent of the irreducible diagrams in a Cweb [69], we immediately discover that when arranging in this way the general structure of a n×n. Of the three irreducible diagrams, d3 and d6 are partially entangled, as the black correlator can be scaled down to the origin independently of the other correlators.
The replica trick algorithm, which determines the explicit form of the mixture matrices, works in a way that it can separate two entangled correlators, but the reverse is not possible. This implies that the ECF of d1 in (3.3), for example, is of the following form (see Eq. In general, for a completely tangled diagram di in (3.3), the ECF is given by,. 3.5) That is, The ECF of a fully entangled diagram consists of the color of the partially entangled diagrams and diminutive diagrams along with its own color.
A close inspection of the trick replica algorithm reveals that the ECF of a partially complex diagram will not contain the colors of fully complex diagrams. From the previous discussion, we conclude that the general structure of the mixing matrix R is under normal ordering.
Diagonal block D of a mixing matrix R
Structure of the block A: Fused-Webs
Here the diagrams in (a), (b) and (c) have an entangled piece consisting of the blue and the red gluon correlators with two and three Wilson lines. Therefore, the tangled part of diagram (a) is replaced by a two-point Fused correlator; and that of diagrams (b), and (c) with a three-point Fused correlator, yields the Fused diagrams (d), (e), and (f), respectively. We can distill 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 later with the help of a clear example. The number of these fully entangled diagrams in a Cweb is the order of the identity matrix that appears in the normal order confusion matrix of the Cweb. Identify the separate entangled parts that appear in the partially entangled diagrams and obtain the fused diagram for each of them.
Shuffle the appendices of each Wilson line in a Fused diagram to generate the corresponding fictitious Cweb whose reducible diagrams will form the associated Fused-Web. Order the diagrams of the Cweb attached to a Fused-Web so that they appear next to each other. Repeat steps 3, 4, and 5 for all distinct entangled pieces in a Cweb to compute the diagonal blocks of the matrix A.
In the next subsection we show an explicit example of a Cweb, applying the above steps to calculate the diagonal blocks of the matrix A.
Application of Fused-Web
The explicit form of the mixing matrix of the Cweb W after ordering the diagrams in the manner mentioned above is. We have also established the structure of the diagonal blocks of A using the idea of Fused-Webs in section 3. The above form is sufficient to enable us to obtain the rank of the mixing matrix Ronce we know the diagonal blocks of A, and D which are basically the mixing matrices of base Cwebs.
The rank of R(12) is one and the rank of A is equal ton−2, the rank of the mixture matrix is thus,. The next step is to determine ci and bi using the properties of the mixture matrices. With this, we have uniquely fixed all the elements of the mixing matrix for this class of Cwebs, and it is then given by .
We have fully solved this class—the mixing matrix is given in Eq. 4.15) and there are no -1 independent color factors expressed if there are Cweb diagrams. The explicit computation of the mixing matrix for this Cweb is presented in [72], and is given by,. Following the same procedure as described above, we can write the order of the mixing matrix for this class as, .
Therefore, blockA is equal to R(12), and its rank is 1. The diagonal blocks of the mixing matrix for this Cweb are given by, R. The diagonal blocks of the mixing matrix are predicted from our procedure using Fused-Webs and the Uniqueness theorem corresponds to the explicit form calculated in [72] using the replica trick. So, 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.
The same replica trick was used in [71, 72] for computing the mixing matrices of four-loop Cwebs. Here we briefly discuss the replica-trick algorithm that was used in the computation of the mixing matrices for Cwebs at four loops. This table classifies the irreducible diagrams of the Cweb according to the entangled pieces and provides Fused-Webs with the corresponding mixing matrices for the Cweb.
Therefore, the mixture matrices of the Fused-Webs for this Cweb, present on the diagonal blocks of A, are given as. The number of exponentiated color factors is the rank of the mixing matrix, which is given as.
W (2,1) 3,III (2, 2, 3)
In this class Cwebs, the D-block of mixing matrices is R(12,22), the full form of which is given in appendix C. The rank for this class of matrices will therefore be, This Cweb, shown in figure 32, has six diagrams, four of which are reducible, and the remaining two are partially entangled. The sequence of diagrams in the Cweb given in table 20 is chosen so that diagrams with the same type of tangle appear together.
Therefore, mixing matrices of Fused-Webs for this Cweb, present on the diagonal blocks of A, are given as, . This Cweb, shown in Figure 34, has eight diagrams, four of which are reducible and the remaining four are partially entangled. The order of diagrams in Cweb given in Table 22 is chosen so that diagrams with the same kind of tangled pieces appear together.
This Cweb, shown in Figure 36, has six diagrams, four of which are reducible and the remaining two are partially entangled. The order of diagrams in Cweb given in Table 24 is chosen so that diagrams with the same kind of tangled pieces appear together. Therefore, mixing matrices of Fused-Webs for this Cweb, present on the diagonal blocks of A, are given as, . and the rank of A is.
The number of color factors expressed for this Cweb is the rank of R, thus using Eq. B.29). In this appendix, we present the mixing matrices used as a basis for writing D, and the diagonal blocks of A. The unique mixing matrix for this type of Cweb first appears in two loops and has the form,.
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, . The mixing matrix for this type of Cweb first appears at four loops and has
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution, and reproduction in any medium, provided the original author(s) and source are credited. Sen, Asymptotic behavior of the wide-angle on-shell quark scattering amplitudes in non-Abelian gauge theories, Phys.
