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

Abstract

We have extended our previous computations, performed analytically for a general gauge group, of the even-$N$ moments $γ_{\rm ik}^{\,(3)}(N)$ of the four-loop flavour-singlet splitting functions $P_{\mathrm{ik}}^{(3)}(x)$ to $N=22$. The numerical QCD results perfectly agree with all predictions resulting from the approximations for $P_{\mathrm{ik}}^{(3)}(x)$ that we obtained before from the moments $N\le 20$ 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 ($\hat{I}\P$-function) contributions to $γ_{\rm ik}^{\,(3)}(N)$: only the $n_f^0\hat{I}{\P }_3$ parts of the quark-to-gluon (gq) and the $n_f^1\hat{I}{\P }_3$ parts of the gluon-to-gluon (qg) cases still need to be completed. Finally we extend our approximations for all $P_{\mathrm{ik}}^{(3)}(x)$ to $n_f^{}=6$ light flavours and update those for $P_{\mathrm{gq}}^{(3)}(x)$ at lower $n_f^{}$.

13 pages, LaTeX, 8 ancillary files (FORM files with results and FORTRAN subroutines)