Abstract
Webs are sets of Feynman diagrams which manifest soft gluon exponentiation in gauge theory scattering amplitudes: individual webs contribute to the logarithm of the amplitude and their ultraviolet renormalization encodes its infrared structure. In this paper, we consider the particular class of boomerang webs, consisting of multiple gluon exchanges, but where at least one gluon has both of its endpoints on the same Wilson line. First, we use the replica trick to prove that diagrams involving self-energy insertions along the Wilson line do not contribute to the web, i.e. their exponentiated colour factor vanishes. Consequently boomerang webs effectively involve only integrals where boomerang gluons straddle one or more gluons that connect to other Wilson lines. Next we classify and calculate all boomerang webs involving semi-infinite non-lightlike Wilson lines up to three-loop order, including a detailed discussion of how to regulate and renormalize them. Furthermore, we show that they can be written using a basis of specific harmonic polylogarithms, that has been conjectured to be sufficient for expressing all multiple gluon exchange webs. However, boomerang webs differ from other gluon-exchange webs by featuring a lower and non-uniform transcendental weight. We cross-check our results by showing how certain boomerang webs can be determined by the so-called collinear reduction of previously calculated webs. Our results are a necessary ingredient of the soft anomalous dimension for non-lightlike Wilson lines at three loops.
Highlights
The structure of perturbative scattering amplitudes in non-Abelian gauge theories continues to be an important research area due to a wide range of phenomenological and formal applications
We classify and calculate all boomerang webs involving semi-infinite non-lightlike Wilson lines up to three-loop order, including a detailed discussion of how to regulate and renormalize them. We show that they can be written using a basis of specific harmonic polylogarithms, that has been conjectured to be sufficient for expressing all multiple gluon exchange webs
In calculating it one must make a distinction between the colour singlet case, relevant for example for an on-shell form factor, where the singularity structure is known in full to three loops, and the more complicated case of multi-leg scattering amplitudes, which is of interest here, where the soft anomalous dimension is matrix-valued in the space of possible colour flows in the underlying hard process
Summary
The structure of perturbative scattering amplitudes in non-Abelian gauge theories continues to be an important research area due to a wide range of phenomenological and formal applications. [55] (see [64] for a review and references [65, 66] for recent progress beyond three loops), in which webs are closed sets of diagrams related by permutations of gluon attachments on the Wilson lines Each such web is associated with a web mixing matrix describing how the colour and kinematic degrees of freedom are entangled in the logarithm of the amplitude. This greatly reduces the number of integrals that need to be evaluated and simplifies the work required to assemble all contributions Another important feature is that our final results can be written in terms of a special class of basis functions that have appeared already for MGEWs connecting four lines or fewer [44, 45], and that have been conjectured to hold for MGEWs more generally.
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