Long-distance singularities in multi-leg scattering amplitudes

Abstract

We report on the recent completion of the three-loop calculation of the soft anomalous dimension in massless gauge-theory scattering amplitudes. This brings the state-of-the-art knowledge of long-distance singularities in multi-leg QCD amplitudes with any number of massless particles to three loops. The result displays some novel features: this is the first time non-dipole corrections appear, which directly correlate the colour and kinematic degrees of freedom of four coloured partons. We find that non-dipole corrections appear at three loops also for three coloured partons, but these are independent of the kinematics. The final result is remarkably simple when expressed in terms of single-valued harmonic polylogarithms, and it satisfies several non-trivial constraints. In particular, it is consistent with the high-energy limit behaviour and it satisfies the expected factorization properties in two-particle collinear limits. Loops and Legs in Quantum Field Theory

Infrared singularities of QCD scattering amplitudes in the Regge limit to all orders

“…Infrared singularities of massless scattering amplitudes are now fully known, for general colour, kinematics and any number of partons, through three loops, owing to an explicit computation of the soft anomalous dimension at this order [26,27]. While through two loops infrared singularities are governed exclusively by a sum over colour dipoles formed by pairs of the hard-scattered partons [28][29][30][31], at three loops one encounters for the first time infrared singularities that are simultaneously sensitive to the colour and kinematics of three and four hard partons.…”

“…It has been shown that the soft anomalous dimension has a simple dipole structure up to two loops [28]. Corrections involving three and four partons arise starting at three loops, and a series of analyses has been performed in order to constrain their structure at three loops and beyond [29][30][31][35][36][37]; the complete correction at three loops was calculated recently [26,27].…”

“…A bootstrap procedure was then set up, which remarkably completely fixes the functional form of the non-dipole correction at three loops (up to an overall rational numerical factor) based on known information from the kinematic limits mentioned above, reproducing the result of the Feynman-diagram computation of refs. [26,27]. The prospects of extending this bootstrap procedure to higher loops provides an additional motivation to determining the soft anomalous dimension in the high-energy limit.…”

“…(4.4) ∆ (n) for n ≥ 3 accounts for multi-parton correlations. The three-loop correction ∆ (3) , correlating up to four hard partons, was calculated recently [26,27] for any number of partons in general kinematics. Specialising to 2 → 2 parton scattering in the high-energy limit, ref.…”

Wilson Lines and Webs in Higher-Order QCD

“…ref. [19] and references therein). The key quantity we wish to calculate is called the soft anomalous dimension Γ S , which controls the ultraviolet singularities of Wilson lines.…”