Determine the remaining all-N ζ3 contributions to γ_qg^(3)(N) and γ_gq^(3)(N)

Determine the all-N analytic expressions for the ζ3 contributions to the four-loop flavour-singlet anomalous dimensions γ_qg^(3)(N) and γ_gq^(3)(N)—the Mellin moments of the quark-to-gluon and gluon-to-quark splitting functions P_qg^(3)(x) and P_gq^(3)(x)—specifically for the quadratic-Casimir × n_f color-factor components of γ_qg^(3)(N) and for the n_f^2 components of γ_gq^(3)(N), which are currently not known at all values of N.

Background

The paper extends previous computations of even-N moments of the four-loop flavour-singlet splitting functions to N=22 and uses these results to constrain and verify x-space approximations relevant for collider phenomenology. The authors also derive new all-N results for many ζ-function contributions to the anomalous dimensions.

Despite these advances, certain color-factor components of the ζ3 contributions to the off-diagonal anomalous dimensions γ_qg3(N) and γ_gq3(N) remain undetermined at all N. Completing these components would further clarify the all-N structure of the four-loop splitting functions and support efforts toward fully determining them.

References

With the help of the new results at N=22, most of the ζ-function contributions to the flavour-singlet anomalous dimensions are now known at all N, only the quadratic-Casimir n_f parts of γqg{(3)}(N)|{ζ3} and n_f2 parts of γgq{(3)}(N)|{ζ3} are not known yet.

— Additional results on the four-loop flavour-singlet splitting functions in QCD  (2512.10783 - Falcioni et al., 11 Dec 2025) in Summary (end of main text)