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\begin{document}


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\newabstract 

\begin{center}
{\large\bf Time-like splitting functions at NNLO in QCD}\\[0.5cm]
{\bf Sven-Olaf Moch}$^1$\\[0.3cm]
$^1$DESY, Platanenallee 6,  D-15738 Zeuthen, Germany \\
\end{center}

\noindent
We have employed relations between space-like and time-like deep-inelastic processes 
in perturbative QCD to calculate the next-to-next-to-leading order (NNLO) 
contributions to the time-like non-singlet and singlet quark-quark and gluon-gluon 
splitting functions for the scale dependence (evolution) of the 
parton fragmentation distributions $\,D_{\! f}^{\,h}(x,Q^2)$.
The evolution equations read
%
\begin{equation}
\label{eq:Devol}
  {d \over d \ln Q^2} \; D_{i}^{\,h} (x,Q^2) \:\: = \:\:
  \int_x^1 {dz \over z} \; P^{\,T}_{ji} \left( z,\alpha_{\rm s} (Q^2) \right)
  \:  D_{j}^{\,h} \Big( {x \over z},\, Q^2 \Big)\, ,
\end{equation}
%
where $x$ denotes the fraction of the momentum of the final-state parton $f$ carried by 
the outgoing hadron $h$, $\,Q^2$ is a time-like hard scale and 
summation over $\,j \, =\, q,\:\bar{q},\:g\,$ is understood. 
The time-like splitting functions $\,P^{\,T}_{ji}\,$ governing Eq.~(\ref{eq:Devol}) 
admit an expansion in powers of the strong coupling $\alpha_{\rm s}$, 
%
\begin{equation}
\label{eq:PTexp}
  P^{\,T}_{ji} \left( x,\alpha_{\rm s} (Q^2) \right) \:\: = \:\:
  a_{\rm s} \, P_{ji}^{(0)\,T}(x) \: + \: a_{\rm s}^{\:\!2} \, P_{ji}^{(1)\,T}(x)
  \: +\: a_{\rm s}^{\:\!3} P_{ji}^{(2)\,T}(x) \: +\: \ldots \:\: ,
\end{equation}
where we normalize to $\,a_{\rm s} \equiv \alpha_{\rm s}(Q^2)/ (4\pi)$.
The leading-order (LO) terms in Eq.~(\ref{eq:PTexp}) are identical to the 
space-like case of the initial-state parton distributions, 
which is the so-called Gribov-Lipatov relation~\cite{Gribov:1972ri}. 
Also at the next-to-leading order (NLO), the functions $\,P_{ji}^{(1)\,T}\!(x)$ 
are related to their space-like counterparts by a suitable analytic
continuation, see e.g.~\cite{Stratmann:1996hn}.
Following these ideas, we have derived at NNLO the 
three distinct non-singlet functions $P^{\,(2),T}_{\rm ns, \xi}$ (with $\xi = \pm,\rm v$)~\cite{Mitov:2006ic}
and the diagonal singlet quantities $P_{\rm qq}^{\,(2),T}$, $P_{\rm gg}^{\,(2),T}$~\cite{Moch:2007tx} 
from the corresponding space-like results~\cite{Moch:2004pa} and \cite{Vogt:2004mw} 
using two independent methods. 
One approach relied on mass factorization and the structure of the infrared singularities, the other 
implemented the concept of universal (kinematics independent) splitting functions as conjectured in~\cite{Dokshitzer:2005bf}.
The off-diagonal three-loop time-like quantities $P_{\rm gq}^{\,(2),T}$, $P_{\rm qg}^{\,(2),T}$ are 
presently still unknown.


{\footnotesize{
\begin{thebibliography}{99}
\bibitem{Gribov:1972ri}
V.N.~Gribov and L.N.~Lipatov,
Sov. J. Nucl. Phys. 15 (1972) 438, ibid.~675.
%% CITATION = SJNCA,15,438;%%
%% CITATION = SJNCA,15,675;%%

\bibitem{Stratmann:1996hn}
M.~Stratmann and W.~Vogelsang,
Nucl. Phys. B496 (1997) 41, {\tt hep-ph/9612250}.
%% CITATION = HEP-PH 9612250;%%

\bibitem{Mitov:2006ic}
A.~Mitov, S.~Moch and A.~Vogt,
Phys. Lett. B638 (2006) 61, {\tt hep-ph/0604053}.
%% CITATION = PHLTA,B638,61;%%

\bibitem{Moch:2007tx}
S.~Moch and A.~Vogt,
Phys. Lett. B659 (2008) 290, {\tt arXiv:0709.3899 [hep-ph]}.
%% CITATION = PHLTA,B659,290;%%

\bibitem{Moch:2004pa}
S. Moch, J.A.M. Vermaseren and A. Vogt,
Nucl. Phys. B688 (2004) 101, {\tt hep-ph/0403192}.
%% CITATION = HEP-PH 0403192;%%.

\bibitem{Vogt:2004mw}
A. Vogt, S. Moch and J.A.M. Vermaseren,
Nucl. Phys. B691 (2004) 129, {\tt hep-ph/0404111}.
%% CITATION = HEP-PH 0404111;%%.

\bibitem{Dokshitzer:2005bf}
Y.L.~Dokshitzer, G.~Marchesini and G.P.~Salam,
Phys. Lett. B634 (2006) 504, {\tt hep-ph/0511302}.
%% CITATION = HEP-PH 0511302;%%

\end{thebibliography}
}}


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