Photon-Parton Splitting Functions at the Next-to-Next-to-Leading Order of QCD

Abstract

We have calculated the splitting functions governing the evolution of the unpolarized parton distributions of the photon at the next-to-next-to-leading order (NNLO) of massless perturbative QCD. The results, presented here mainly in terms of compact and accurate parametrizations, are consistent with our previous approximations based on the lowest six even-integer Mellin moments. Consequently the NNLO corrections are small in both the MSbar and the DIS_gamma factorization schemes at momentum fractions x >~ 0.1.

Figures (2)

• Figure 1: The exact results for the $a_{\rm em} a_{\rm s}^2$ photon-quark (left) and photon-gluon (right) splitting functions (multiplied by $x$) in the $\overline{\hbox{MS}}$ scheme, compared with the (dashed) estimated error bands based on the lowest six even-integer moments Moch:2001im.
• Figure 2: The perturbative expansion (\ref{['Pgexp']}) of the photon-quark (left) and photon-gluon (right, multiplied by $x$) splitting functions in the DIS$_{\gamma}$ factorization scheme for typical values of $\alpha_{\rm s}$ and $n_{\rm f}^{}$. Note the very different scales of the two graphs.