Three loop anomalous dimension of the non-singlet transversity operator in QCD

TL;DR

The authors compute the three-loop anomalous dimension for the flavour non-singlet transversity operator in QCD by evaluating fifteen fixed Mellin moments and reconstructing the full N-dependence using harmonic sums and maximal transcendentality, aided by the LLL algorithm. The resulting expressions are presented in both Mellin-N and Bjorken-x space, the latter via inverse Mellin transformation to obtain the corresponding splitting functions in terms of harmonic polylogarithms. This work extends the known lower-order results and provides complete three-loop non-singlet transversity evolution kernels, with cross-check against the N=16 moment and consistency with universal anomalous dimension structures in ${\mathcal N}=4$ SYM. The results enable precise phenomenological studies of transversity evolution and SIDIS/Drell-Yan processes at high perturbative accuracy.

Abstract

We calculate the three-loop anomalous dimension of the non-singlet transverse operator from N=1 to N=15. Using some guess we have reconstructed a general form of three-loop anomalous dimension for arbitrary Mellin moment N. Obtained result is transformed into Bjorken-x space by an inverse Mellin transformation. The final expressions are presented in both Mellin-N and Bjorken-x space.

Figures (2)

• Figure 1: The perturbative expansion of the anomalous dimension $\gamma_{\,{\mathrm {TR,ns}}}^{\, +}(N)$ for four flavours at $\alpha_s = 0.2$. In the right part the difference between our three-loop result and the corresponding result from Ref. Moch:2004pa is shown.
• Figure 2: The three-loop splitting function $P_{\mathrm {TR,ns}}^{\,(2) +}(x)$ for different numbers of active quarks, multiplied by $(1-x)$ as in Ref. Moch:2004pa. In the right part the difference between our three-loop result and the corresponding result from Ref. Moch:2004pa is shown.