Four-loop non-singlet splitting functions in the planar limit and beyond

Abstract

We present the next-to-next-to-next-to-leading order (N 3 LO) contributions to the non-singlet splitting functions for both parton distribution and fragmentation functions in perturbative QCD. The exact expressions are derived for the terms contributing in the limit of a large number of colours. For the remaining contributions, approximations are provided that are sufficient for all collider-physics applications. From their threshold limits we derive analytical and high-accuracy numerical results, respectively, for all contributions to the four-loop cusp anomalous dimension for quarks, including the terms proportional to quartic Casimir operators. We briefly illustrate the numerical size of the four-loop corrections, and the remarkable renormalization-scale stability of the N 3 LO results, for the evolution of the non-singlet parton distribution and the fragmentation functions. Our results appear to provide a first point of contact of four-loop QCD calculations and the so-called wrapping corrections to anomalous dimensions in N = 4 super Yang-Mills theory.

The principle of maximal transcendentality and the four-loop collinear anomalous dimension

“…In a conformal theory, this "eikonal bypass" only requires [60] knowledge of the virtual anomalous dimension, the coefficient of δ(1 − x) in the DGLAP kernel. The virtual anomalous dimension in large-N c QCD was computed recently at four loops [41], and we can make use of its leading transcendental part to do the conversion. Once we have the eikonal quantity, we use its non-abelian exponentiation property [61,62], which means that it is "maximally non-abelian".…”

“…However, there is a workaround, the eikonal bypass discussed in the introduction, which involves converting the non-eikonal quark collinear anomalous dimension to an eikonal (Wilson line) quantity [60], with the help of the recent four-loop result for the DGLAP kernels in the large N c limit of QCD [41]. In particular, we need the coefficient of δ(1 − x) in this result, the virtual anomalous dimension.…”

“…[41], the twist-two anomalous dimensions or DGLAP kernels were computed in the large N c limit of QCD to four loops. In the limit that x → 1, as in eq.…”

“…The next-to-leading term as x → 1 is the coefficient of δ(1 − x), sometimes called the virtual anomalous dimension, or B δ , or just B. The large-N c , leading transcendentality terms in B for quarks are given by [41,80]:…”

“…We do so by leveraging two recent four-loop computations in QCD in the large N c limit [40,41], as well as the principle of maximal transcendentality [42][43][44][45]. This principle states that for suitable quantities, such as the BFKL and DGLAP kernels, the result in N = 4 SYM can be obtained from that in QCD by converting the fermion representation…”