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 γ i k ( 3 ) ( N ) of the four-loop flavour-singlet splitting functions P i k ( 3 ) ( x ) to N = 22 . The numerical QCD results perfectly agree with all predictions resulting from the approximations for P i k ( 3 ) ( x ) that we obtained before from the moments N ≤ 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 ( ζ -function) contributions to γ i k ( 3 ) ( N ) : only the n f 0 ζ 3 parts of the quark-to-gluon (gq) and the n f 1 ζ 3 parts of the gluon-to-gluon (gg) cases still need to be completed. Finally we extend our approximations for all P i k ( 3 ) ( x ) to n f = 6 light flavours and update those for P g q ( 3 ) ( x ) at lower n f .