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\documentclass{ws-procs9x6}


\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\nn}{\nonumber}
\newcommand{\MSb}{$\overline{\mbox{MS}}$}
\newcommand{\as}{\alpha_{\rm s}}
\newcommand{\DD}{{\cal D}}

\def\plus{{\!+\!}}
\def\minus{{\!-\!}}
\def\z#1{{\zeta_{#1}}}
\def\ca{{C_A}}
\def\cf{{C_F}}
\def\nf{{n^{}_{\! f}}}
\def\S(#1){{{S}_{#1}}}
\def\Ss(#1,#2){{{S}_{#1,#2}}}
\def\Sss(#1,#2,#3){{{S}_{#1,#2,#3}}}
\def\Ssss(#1,#2,#3,#4){{{S}_{#1,#2,#3,#4}}}
\def\Sssss(#1,#2,#3,#4,#5){{{S}_{#1,#2,#3,#4,#5}}}
\def\Npm{{{\bf N_{\pm}}}}
\def\Npmi{{{\bf N_{\pm i}}}}
\def\Nminus{{{\bf N_{-}}}}
\def\Nplus{{{\bf N_{+}}}}
\def\Nminustwo{{{\bf N_{-2}}}}
\def\Nplustwo{{{\bf N_{+2}}}}
\def\Nminusthree{{{\bf N_{-3}}}}
\def\Nplusthree{{{\bf N_{+3}}}}

\def\pqq(#1){p_{\rm{qq}}(#1)}
\def\Li(#1,#2){{\rm{Li}}_{#1}(#2)}
\def\H(#1,#2,#3,#4){{\rm{H}}_{#1,#2,#3}(#4)}



\begin{document}

\title{Deep-inelastic structure functions at NNLO and beyond}

\author{S. Moch}
\address{Deutsches Elektronensynchrotron DESY \\
Platanenallee 6, D--15738 Zeuthen, Germany}

\author{J.A.M. Vermaseren and A. Vogt}
\address{NIKHEF Theory Group \\
Kruislaan 409, 1098 SJ Amsterdam, The Netherlands}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% You may repeat \author \address as often as necessary      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\maketitle

\abstracts{We have calculated the fermionic contributions to the flavour non-singlet 
structure functions in deep-inelastic scattering (DIS) at third order of 
massless perturbative QCD. We obtain complete results for the
corresponding $\nf$-parts of the three-loop anomalous dimension and the three-loop coefficient functions 
for the structure functions $F_2$ and $F_L$. Our results agree with all 
partial and approximate results available in the literature. We discuss their implications for the 
threshold resummation at the next-to-next-to-leading logarithmic accuracy.}

\section{Introduction}
The calculation of perturbative QCD corrections for deep-inelastic 
structure functions is an important task.
The present and expected future experimental precision, 
for instance at HERA, calls for complete next-to-next-to-leading order
(NNLO) predictions.
These offer the possibility to determine the strong coupling constant 
$\alpha_s$ and to analyze the proton structure and its parton
content with unprecedented precision.
Knowledge of the latter is of particular importance for 
the analysis of hard scattering reactions at 
future LHC experiments.

At present, this level of accuracy is not yet possible, 
because the necessary anomalous dimensions governing 
the parton evolution at NNLO are not fully known. 
The two-loop coefficient functions of $F_2,F_3$ and
$F_L$ have been calculated some time 
ago~\cite{vanNeerven:1991nn,Zijlstra:1991qc,Zijlstra:1992kj,Zijlstra:1992qd,Moch:1999eb}, 
but for the corresponding three-loop anomalous dimensions 
$\gamma_{\rm pp'}^{(2)}$, only a finite number of fixed Mellin
moments~\cite{Larin:1994vu,Larin:1997wd,Retey:2000nq} 
are available.
As a first step towards the complete calculation, we have computed the 
fermionic three-loop contributions to the flavour non-singlet (NS) structure 
functions $F_2$ and $F_L$ in unpolarized electromagnetic deep-inelastic
scattering~\cite{Moch:2002sn}. 
Already these results have immediate consequences for threshold resummation of
soft gluons which we will discuss in the following. 

\section{Results}


We are considering only the non-singlet structure functions and the
method for the calculation of their moments closely 
follows ref.~\cite{Larin:1994vu}. Hence there is not much need to explain the 
physics of the method here again. Thus we will discuss only the differences 
introduced by the fact that we now compute all moments simultaneously as a 
function of the moment number $N$. 
%Since $N$ is not a fixed constant, we cannot do the calculation with the 
Since $N$ is not a fixed constant, we cannot resort to the techniques of 
ref.~\cite{Larin:1994vu}, where the Mincer program~\cite{Gorishnii:1989gt,%
Larin:1991fz} was used as the tool to solve the integrals. Instead, we will 
have to introduce new techniques. However, we can give $N$ a positive integer 
value at any point of the derivations and calculations, after which the Mincer 
program can be invoked to verify that the results are correct. From a practical 
point of view this is the most powerful feature of the Mellin-space approach, 
as it greatly simplifies the checking of all programs.

Similar to the fixed-$N$ computations of refs.~\cite
{Larin:1997wd,Retey:2000nq}, the diagrams are generated automatically with a 
special version of the diagram generator QGRAF~\cite{Nogueira:1991ex}.
For all the symbolic manipulations of the formulae we use the latest version of 
the program FORM~\cite{Vermaseren:2000nd}.
The calculation is performed in dimensional regularization~\cite
{'tHooft:1972fi,Bollini:1972ui,Ashmore:1972uj,Cicuta:1972jf} with 
$D=4-2\epsilon$. Hence the unrenormalized Mellin-space results will be 
functions of $\epsilon,\: N,$ and the values $\zeta_{3,\ldots, 5}$ of the 
Riemann $\zeta$-function. The renormalization is carried out in the \MSb-scheme
\cite{'tHooft:1973mm,Bardeen:1978yd} as described in ref.~\cite{Larin:1994vu}.

The major difference to the fixed-moment calculations is now that we
obtain much longer results due to 
the presence of the parameter $N$ in the answer. We have checked the 
correctness of each individual diagram for several values of $N$, by comparing 
with the results of a Mincer calculation. In addition we have compared the 
complete renormalized results with the results in the literature for the 
available values of $N$.

The perturbative expansion of the non-singlet coefficient functions
and anomalous dimensions can be written as
%
\bea
\label{eq:Cexp}
  C_{\,i,\rm ns}\left(\as,N\right) \! &= \!& \sum_{n=0}^\infty\,\: 
  \left(\frac{\as}{4\pi}\right)^{n}\: c^{\,(n)}_{\,i,\rm ns}(N) \:\: ,
\\ \label{eq:gexp}
 \gamma_{\,\rm{ns}}\left(\as,N\right) \, & = \!& \sum_{n=0}^\infty \, 
  \left(\frac{\as}{4\pi}\right)^{n+1} \gamma^{\,(n)}_{\,\rm ns}(N) 
\eea
%
with $i=2,L\,$ in eq.~(\ref{eq:Cexp}).

Our $N$-space results are expressed in terms of harmonic sums 
$S_{\vec{m}}(N)$. In the following all harmonic sums are understood to 
have the argument $N$, i.e., we employ the short-hand notation 
$S_{\vec{m}}\,\equiv\, S_{\vec{m}}(N)$. In addition we use operators 
$\Npm$ and $\Npmi$ which shift the argument $N$ of a given function by 
$\pm 1$ or a larger integer $i$,
%
\beq
\label{eq:shiftN}
  \Npm \, f(N) \: = \: f(N \pm 1) \:\: , \quad\quad
  \Npmi\, f(N) \: = \: f(N \pm i) \:\: .
\eeq

We normalize the trivial leading-order (LO) coefficient function
and recover, of course, the well-known result for the LO anomalous 
dimension~\cite{Gross:1973rr,Georgi:1974sr}
%
\bea
  c^{\,(0)}_{\,2,{\rm ns}}(N) & \! = \! & 
%%START
%%L %%texc2qq0 =
         1
  %%
  %%
%%;
%%STOP
\:\: ,\label{eq:c2qq0}
\\
  \gamma^{\,(0)}_{\,\rm ns}(N) & \! = \! & 
%%START
%%L %%texgqq0 =
         \cf \* \big( 
            2 \* (\Nminus+\Nplus) \* \S(1) 
          - 3 
          \big)
  %%
  %%
%%;
%%STOP
\:\ .\label{eq:gqq0}
\eea
%

At next-to-next-to-leading order we have computed all 
fermionic contributions to the splitting function $\gamma^{\,(2)}_{\,
\rm ns}$,
%
\bea
\gamma^{\,(2)}_{\,\rm ns}(N) &=&  
%%START
%%L %%texgqq2 = 
 16\, \* \ca \* \cf \* \nf \* \bigg( 
            {3 \over 2} \* \z3
          - {5 \over 4}
          + {10 \over 9} \* \S(-3)
          - {10 \over 9} \* \S(3)
          + {4 \over 3} \* \Ss(1,-2)
          - {2 \over 3} \* \S(-4)
  %%
  %%
  \nonumber\\&& \mbox{}
          + 2 \* \Ss(1,1)
          - {25 \over 9} \* \S(2)
          + {257 \over 27} \* \S(1)
          - {2 \over 3} \* \Ss(-3,1)
       - \Nplus \*  \bigg[
            \Ss(2,1)
          - {2 \over 3} \* \Ss(3,1)
  %%
  %%
  \nonumber\\&& \mbox{}
          - {2 \over 3} \* \S(4)
          \bigg]
          + (1-\Nplus)  \*  \bigg[
            {23 \over 18} \* \S(3)
          - \S(2)
         \bigg]
       - (\Nminus+\Nplus)  \*   \bigg[
            \Ss(1,1)
  %%
  %%
  \nonumber\\&& \mbox{}
          + {1237 \over 216} \* \S(1)
          + {11 \over 18} \* \S(3)
          - {317 \over 108} \* \S(2)
          + {16 \over 9} \* \Ss(1,-2)
          - {2 \over 3} \* \Sss(1,-2,1)
          - {1 \over 3} \* \Ss(1,-3)
  %%
  %%
  \nonumber\\&& \mbox{}
          - {1 \over 2} \* \Ss(1,3)
          - {1 \over 2} \* \Ss(2,1)
          - {1 \over 3} \* \Ss(2,-2)
          + \S(1) \* \z3
          + {1 \over 2} \* \Ss(3,1)
           \bigg]
          \bigg)
+ 16\, \* \cf \* \nf^2 \* \bigg( 
            {17 \over 144}
  %%
  %%
  \:\:\: \nonumber\\&& \mbox{}
          - {13 \over 27} \* \S(1)
          + {2 \over 9} \* \S(2)
              + (\Nminus+\Nplus) \*  \bigg[
            {2 \over 9} \* \S(1)
          - {11 \over 54} \* \S(2)
          + {1 \over 18} \* \S(3)
          \bigg]
          \bigg)
  %%
  %%
  \:\:\: \nonumber\\&& \mbox{}
+  16\, \* \cf^2 \* \nf \* \bigg( 
            {23 \over 16}
          - {3 \over 2} \* \z3
          + {4 \over 3} \* \Ss(-3,1)
          - {59 \over 36} \* \S(2)
          + {4 \over 3} \* \S(-4)
          - {20 \over 9} \* \S(-3)
  %%
  %%
  \nonumber\\&& \mbox{}
          + {20 \over 9} \* \S(1)
          - {8 \over 3} \* \Ss(1,-2)
          - {8 \over 3} \* \Ss(1,1)
          - {4 \over 3} \* \Ss(1,2)
       + \Nplus \*  \bigg[
            {25 \over 9} \* \S(3)
          - {4 \over 3} \* \Ss(3,1)
          - {1 \over 3} \* \S(4)
          \bigg]
  %%
  %%
  \nonumber\\&& \mbox{}
               + (1-\Nplus)  \*  \bigg[
            {67 \over 36} \* \S(2)
          - {4 \over 3} \* \Ss(2,1)
          + {4 \over 3} \* \S(3)
          \bigg]
       + (\Nminus+\Nplus)  \*  \bigg[
            \S(1) \* \z3
  %%
  %%
  \nonumber\\&& \mbox{}
          - {325 \over 144} \* \S(1)
          - {2 \over 3} \* \Ss(1,-3)
          + {32 \over 9} \* \Ss(1,-2)
          - {4 \over 3} \* \Sss(1,-2,1)
          + {4 \over 3} \* \Ss(1,1)
          + {16 \over 9} \* \Ss(1,2)
  %%
  %%
  \nonumber\\&& \mbox{}
          - {4 \over 3} \* \Ss(1,3)
          + {11 \over 18} \* \S(2)
          - {2 \over 3} \* \Ss(2,-2)
          + {10 \over 9} \* \Ss(2,1)
          + {1 \over 2} \* \S(4)
          - {2 \over 3} \* \Ss(2,2)
          - {8 \over 9} \* \S(3)
           \bigg]
          \bigg)
  %%
  %%
  \nonumber\\&& \mbox{}
  %%
  %%
%%;
%%STOP
.\label{eq:gqq2}
\end{eqnarray}

We have subjected our results to a number of checks. First of all, we have 
calculated some lower even moments in an arbitrary covariant gauge with 
the Mincer program~\cite{Gorishnii:1989gt,Larin:1991fz}, keeping the 
gauge parameter $\xi$ in the gluon propagator. All dependence on $\xi$ 
does cancel in the final results. 
Secondly the $\nf^{\!\!\!2}$-contribution to $\gamma_{\,\rm ns}^{\,(2)}$
is known from the work of Gracey \cite{Gracey:1994nn} and we agree 
with his result. Furthermore the coefficients of $\ln^{\,k} N$, 
$k = 3,\ldots 5$, of $c^{\,(3)}_{\,2,\rm ns}(N)$ agree with the 
prediction of the soft-gluon resummation \cite{Vogt:1999xa}.
Finally, we have checked the result of each individual diagram for 
several integer values of $N$ by comparing with the results of a Mincer 
calculation. Thus, as the strongest check, our results reproduce the 
fixed even moments $N=2,\dots,14$ computed in 
refs.~\cite{Larin:1994vu,Larin:1997wd,Retey:2000nq}. 


\section{Threshold resummation}

It is well known that perturbative QCD corrections to structure
functions receive large logarithmic corrections, which originate from 
the emission of soft gluons. These corrections are relevant at large
values of the scaling variable $x$ (in Mellin space at large values of
the Mellin moment $N$) and can be resummed to all orders in
perturbation theory. It is interesting to investigate the implications 
of our three-loop results~\cite{Moch:2002sn} for the threshold 
exponentiation~\cite{Sterman:1987aj,Catani:1989ne,Catani:1991rp}
at next-to-next-to-leading logarithmic (NNLL) accuracy~\cite{Vogt:2000ci}.

Here the quark coefficient function for $F^{}_{2}$ can, up to terms 
which vanish for $N \to \infty$, be written as
%
\bea
\label{eq:csoft}
  C_{\,2}(\as, N) \: =\: 
  (1 + a_{\rm s\,} g_{01}^{} + a_{\rm s\,}^2 g_{02}^{} + \ldots ) \: 
  \exp \, [\,G^N(Q^2)  \, ] \, ,
\eea
%
where the resummation exponent $G^N$ can be expanded as 
%
\begin{equation}
\label{eq:GNexp}
  G^N(Q^2) \: =\:  
  L\, g_1^{}(a_{\rm s\,}L) + g_2^{}(a_{\rm s\,}L) + 
  a_{\rm s}\, g_3^{}(a_{\rm s\,}L) + \ldots\, 
\end{equation}
%
with $a_{\rm s} = \as/(4\pi)$ and $L = \ln N$. The functions 
$g_l^{}$ depend on universal coefficients $A_{\,i\,\leq\, l}$ and 
$B_{\,i\,\leq\, l\!-\!1}$ 
and process-dependent parameters $D_{\,i\,\leq\, l\!-\!1}^{\,\rm
DIS}$ (see e.g. ref.~\cite{Vogt:2000ci} for the precise definitions of the 
functions $g_{1,2,3}^{}$).

In a physical picture, the resummation of the perturbative expansion
for $C_{\,2}$ rests upon the refactorization of 
$C_{\,2}$ (valid in the threshold region of phase
space) into separate functions of the jet-like, soft, and off-shell
quanta that contribute to its quantum corrections. 
Each of the functions organizes large corrections
corresponding to a particular region of phase space~\cite{Contopanagos:1997nh}.

To NNLL accuracy, the function $g_3$ involves the new coefficients $A_3$,
$B_2$ and $D_2^{\,\rm DIS}$. These coefficients can be fixed by
expanding eq.~(\ref{eq:csoft}) in powers of $\as$ and comparing to the 
result of the full fixed-order calculation~\cite{Moch:2002sn}. 
In the \MSb\ scheme, the parameter $A_3$ is simply the coefficient 
of $\ln N$ in $\gamma_{\,\rm ns}^{(2)}(N)$ or, equivalently, 
of $1/(1-x)_+$ in $P_{\,\rm ns}^{(2)}(x)$. It reads
%
\bea
\lefteqn{
   A_3\: = \: (1178.8 \pm 11.5) }
\\ 
&&\nonumber
   + C_A C_F \nf\, \left[ - \frac{836}{27} + \frac{160}{9}\:\z2 
               - \frac{112}{3}\:\z3 \right]
   + C_F^{\,2} \nf\, \left[ -  \frac{110}{3}  + 32\:\z3 \right]
\\ 
&&\nonumber
   + C_F \nf^{\!\!2} \left[ - \frac{16}{27}\,\right] \:\: ,
\eea
%
where $C_A=3$, $C_F=4/3$ and $\nf$ is the number of light (massless)
flavours. The estimate of the non-fermionic part~\cite{Vogt:2000ci} is based on 
the approximations of $P_{\,\rm ns}^{(2)}(x)$ constructed in 
ref.~\cite{vanNeerven:2000wp} using the first six even-integer 
moments~\cite{Retey:2000nq} and its small-$x$ limit~\cite{Blumlein:1996jp}. 
The exact fermionic part has been obtained independently in 
refs.~\cite{Moch:2002sn,Berger:2002sv}, while the $\nf^{\! 2}$ contribution is 
already known from ref.~\cite{Gracey:1994nn}.

The complete results for $B_2$ and $D_2^{\,\rm DIS}$ can be 
inferred from fermionic result of the three-loop coefficient function 
$c_{\,2,\rm ns}^{\,(3)}$ in ref.~\cite{Moch:2002sn}, yielding
%
\bea
\label{eq:B2D2}
  B_2 \: & =\! &
  C_F^{\,2}  \left[ -\frac{3}{2} - 24\,\zeta_3 + 12\,\zeta_2 \right] 
  \: +\: C_F C_A  \left[ - \frac{3155}{54} + 40\,\zeta_3
                 + \frac{44}{3}\,\zeta_2 \right] 
  \\ & & \mbox{} + \:
  C_F \nf\,  \left[ \frac{247}{27} - \frac{8}{3}\,\zeta_2 \right] 
  \:\: ,   \nonumber\\
  D_2^{\,\rm DIS} \!\! & =\! & 0 \:\: .
\eea
%
As a matter of fact, the contribution to $c_{\,2,\rm ns}^{\,(3)}$ involves
only a linear combination, $\beta_0 (B_2 + 2\,D_2^{\,\rm DIS})$, with 
$\beta_0$ being the one-loop coefficient of the QCD $\beta$-function.
However, the different combination $B_2 + D_2^{\,\rm DIS}$ has been determined 
in ref.\cite{Vogt:2000ci} by comparing the expansion of eq.~(\ref{eq:csoft})
to the two-loop coefficient function $c_{\,2,\rm ns}^{\,(2)}$ of 
ref.~\cite{Matsuura:1989sm}.
Thus, $B_2$ and $D_2^{\,\rm DIS}$ can be disentangled. 
It is interesting to observe the vanishing of $D_1^{\,\rm DIS}$ and
$D_2^{\,\rm DIS}$, for which an all-order generalization 
has been proposed in ref.~\cite{Forte:2002ni,Gardi:2002xm}.
This is in contrast to the Drell-Yan process, 
where the functions $D_l^{\,\rm DY}$ are generally different 
from zero. 
For instance, $D_2^{\,\rm DY}$ has been derived in ref.~\cite{Vogt:2000ci}. 

Let us briefly illustrate numerically for large $N$ the improvement due to the NNLL 
corrections for the soft gluon exponent $G^N$ of deep-inelastic scattering.
In fig.~1 on the left, we show the resummation exponent $G^N(Q^2)$ of
eq.~(\ref{eq:GNexp}). Here, we choose $\mu_r^2 = \mu_f^2 = 
Q^2$, $\nf = 4$ and $\as = 0.2$, which corresponds to scales between 
about 25 and 50 GeV$^2$, depending on the precise value of $\as(M_Z^2)$. 
In fig.~1 on the right, we display the convolution with a schematic, 
but typical input evaluated with the so-called `minimal-prescription' 
contour~\cite{Catani:1996yz}. 
It is obvious from both figures that knowledge of the leading
logarithmic (LL) and next-to-leading logarithmic (NLL) 
terms~\cite{Sterman:1987aj,Catani:1989ne,Catani:1991rp} alone,
i.e., those enhanced by factors $\ln N (\alpha_s 
\ln N)^n$ and $(\alpha_s \ln N)^n$, 
is not sufficient for reliably determining the function $G^N$
and its impact after convolution even for rather large values of $N$ and
$x$. 

The NNLL corrections discussed here are rather small over a wide
range. This indicates that the soft-gluon exponent 
$G^N(Q^2)$ stabilizes and that the soft-gluon effects on the 
\MSb\ quark coefficient function can be reliably estimated.
Recall, however, that the NNLL corrections are large for the `physical
kernel' governing the scaling violations of the non-singlet structure
function $F_{2,\rm ns}$ \cite{vanNeerven:2001pe}.


\begin{figure}[!thb]
\vspace*{0.4cm}
\begin{center}
\centerline{
\epsfxsize=6.2cm\epsfbox{GN.ps}
\epsfxsize=6.2cm\epsfbox{GNconv.ps}
}
\vspace*{-0.2cm}
\caption{\label{fig:one}
Left: 
 The LL, NLL and NNLL approximations for the resummation
 exponent $G^N(Q^2)$ in eq.~(\ref{eq:GNexp}) at $\mu_r^2 = \mu_f^2 = Q^2$ 
 for $\as(Q^2) = 0.2$ and four flavours. 
Right: These results convoluted with a typical input shape 
 $xf = x^{1/2}(1-x)^3$.}
\end{center}
\vspace*{-0.5cm}
\end{figure}


%\bibliography{../Bib/refs}
%\bibliographystyle{../Bib/h-physrev3}

\begin{thebibliography}{10}

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\end{document}
