Additional results on the four-loop flavour-singlet splitting functions in QCD

TL;DR

This paper extends the analytical computation of four-loop flavour-singlet splitting functions in QCD to N=22 moments, confirming previous approximations and moving closer to complete all-N expressions, with implications for collider physics.

Contribution

The authors provide extended analytical results for four-loop splitting functions in QCD, refining approximations and constraining the all-N form of non-rational contributions.

Findings

Numerical results agree with previous approximations.

Additional analytical constraints narrow down the form of splitting functions.

Updated approximations for different numbers of light flavours.

Abstract

We have extended our previous computations, performed analytically for a general gauge group, of the even-$N$ moments $\gamma_{\rm ik}^{\,(3)}(N)$ of the four-loop flavour-singlet splitting functions $P_{\rm ik}^{\,(3)}(x)$ to $N = 22$. The numerical QCD results perfectly agree with all predictions resulting from the approximations for $P_{\rm ik}^{\,(3)}(x)$ that we obtained before from the moments $N \leq 20 $ and endpoint constraints, confirming their reliability for collider-physics applications. Due to the additional analytical constraints provided by our new $N = 22$ results, we are now closing in on determining the all-$N$ forms of all non-rational ($\zeta$-function) contributions to $\gamma_{\rm ik}^{\,(3)}(N)$: only the $n_f^{0} \zeta_3$ parts of the quark-to-gluon (gq) and the $n_f^{1} \zeta_3$ parts of the gluon-to-gluon (qg) cases still need to be completed. Finally we extend our approximations for all $P_{\rm ik}^{\,(3)}(x)$ to $n_f^{} = 6$ light flavours and update those for $P_{\rm gq}^{\,(3)}(x)$ at lower $n_f^{}$.