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

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 $n_f^{}$.