Abstract
We revisit the multi-loop structure of the anomalous-dimension matrix governing the infrared divergences of massless n-particle scattering amplitudes in non-abelian gauge theories. In particular, we derive its most general form at four-loop order, significantly simplifying corresponding expressions given previously. By carefully reevaluating the constraints imposed by two-particle collinear limits, we find that at four-loop order color structures involving {d}_R^{abcd} , the symmetrized trace of four group generators, appear along with cusp logarithms ln[μ2/(−sij)]. As a consequence, naive Casimir scaling of the cusp anomalous dimensions associated with the quark and gluon form factors is violated, while a generalized form of Casimir scaling still holds. Our results provide an important ingredient for resummations of large logarithms in n-jet cross sections with next-to-next-to-next-to leading logarithmic (N3LL) accuracy.
Highlights
Naive Casimir scaling of the cusp anomalous dimensions associated with the quark and gluon form factors is violated, while a generalized form of Casimir scaling still holds
Predictive power of this approach relies on the fact that the anomalous dimension is tightly constrained by the structure of the effective field theory: soft-collinear factorization implies that it is given by the sum of a soft and a collinear contribution, n
Given that there are no interactions among different collinear sectors of SCET [3,4,5,6], all non-trivial color and momentum dependence is encoded in the soft anomalous dimension Γs
Summary
Since the color structure of the collinear anomalous dimension is trivial, the hard anomalous dimension inherits the color structures of the soft anomalous dimension Γs({β}, μ) in (1.1). In a diagram in which cutting Wilson lines leads to two disconnected pieces, one can assign two different replicas I and J, but we can have I < J or J < I, each of which contributes according to (2.2) with a factor −1/2 to the exponent S. Repeating the exercise with four gluons, the maximum number which can arise at four-loop order, we obtain a linear combination of terms with three connected f abc symbols, corresponding to the last color structure in figure 2. Let us consider a more interesting example, in which two lines of a connected gluon cluster are attached to the same Wilson line, as depicted on the right-hand side of figure 3 This gives rise to the color structure EabDc TiaTibTic, where a and b connect to the same cluster and must be part of the same replica. A formal proof of this result has been put forward in [15] based on a generalized Baker-Campbell-Hausdorff formula
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