Abstract
Correlators of Wilson-line operators in non-abelian gauge theories are known to exponentiate, and their logarithms can be organised in terms of collections of Feynman diagrams called webs. In [1] we introduced the concept of Cweb, or correlator web, which is a set of skeleton diagrams built with connected gluon correlators, and we computed the mixing matrices for all Cwebs connecting four or five Wilson lines at four loops. Here we complete the evaluation of four-loop mixing matrices, presenting the results for all Cwebs connecting two and three Wilson lines. We observe that the conjuctured column sum rule is obeyed by all the mixing matrices that appear at four-loops. We also show how low-dimensional mixing matrices can be uniquely determined from their known combinatorial properties, and provide some all-order results for selected classes of mixing matrices. Our results complete the required colour building blocks for the calculation of the soft anomalous dimension matrix at four-loop order.
Highlights
To be somewhat more precise, gauge theory scattering amplitudes in the IR limit factorise into universal soft and collinear functions [15,16,17,18,19, 33], multiplying finite matching coefficients
In [1] we introduced the concept of Cweb, or correlator web, which is a set of skeleton diagrams built with connected gluon correlators, and we computed the mixing matrices for all Cwebs connecting four or five Wilson lines at four loops
The study of diagrammatic exponentiation of Wilson-line correlators has a long history, and has provided many important insights concerning the infrared structure of perturbative gauge amplitudes
Summary
We consider the two-line Cweb W2(1,0,1)(2, 4), which contains one twogluon correlator, no three-gluon correlators, and one four-gluon correlator; there are two gluon attachments on line 1 and four gluon attachments on line 2. Keeping in mind the orientation of the Wilson lines, the diagram portrayed in figure 7 is diagram C1 With this ordering, the column weights s(Ci) for the diagrams in the Cweb are collected in the vector. The mixing matrix, we find that the rank of the mixing matrix is rw = 7, which means that there will be 7 independent exponentiated colour factors for this Cweb, which are given by (Y C)1 = if acgf degf ebhTb1Ta1Tc2Td2Th2 + if acgf bdj f degTb1Ta1Tc2Tj2Te2 +if acgf cbmf degTb1Ta1Tm2 Td2Te2 − if abuf acgf degTu1 Tb2Tc2Td2Te2 ,. (Y C)2 = if acgf bdj f degTb1Ta1Tc2Tj2Te2 + if acgf cbmf degTb1Ta1Tm2 Td2Te2 −if abuf acgf degTu1 Tb2Tc2Td2Te2 ,. We observe that all the exponentiated colour factors correspond to completely connected Feynman diagrams, which verifies, as expected, the non-abelian exponentiation theorem [67]
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