\documentclass{PoS}

\usepackage{epsfig}

\title{Higgs boson production at LHC to NNLO accuracy and finite 
top quark mass effects}

\ShortTitle{Higgs boson production at LHC to NNLO accuracy and finite
top quark mass effects}

\author{\speaker{Matthias Steinhauser}%
       \thanks{This work was supported by the DFG through the SFB/TR~9
       ``Computational Particle Physics''. 
       Preprint Nos.: SFB/CPP-10-83, TTP10-37}\\
        KIT\\
        E-mail: \email{matthias.steinhauser@kit.edu}}

%\author{Another Author\\
%        Affiliation\\
%        E-mail: \email{...}}

\abstract{In this contribution we consider the production of a
Standard Model Higgs boson to next-to-next-to-leading order accuracy
taking into account a finite top quark mass. This result improves the
one based on the effective theory and is important for precise
predictions of Higgs boson production both at Tevatron and LHC.}

\FullConference{35th International Conference of High Energy Physics\\
                 July 22-28, 2010\\
                 Paris, France}

\begin{document}

\section{Introduction}

Calculations of higher orders in perturbation theory for the production
of a Standard Model Higgs boson have a long history. Already 20
years ago the next-to-leading (NLO) QCD corrections became
available~\cite{Dawson:1990zjDjouadi:1991tkaSpira:1995rr} and about
ten years ago also the NNLO corrections have been
computed~\cite{Harlander:2000mgHarlander:2002whAnastasiou:2002yzRavindran:2003um}
although only within the framework of an effective theory where it has been 
assumed that the top quark mass $M_t$ is much heavier than all other mass
scales involved in the process --- even the center-of-mass energy
which can be much larger then $M_t$.
It is thus very important to perform the NNLO calculation within full
QCD in order to obtain reliable predictions up to NNLO accuracy.
Recently two independent groups have completed this big 
enterprise~\cite{Pak:2009bx,Harlander:2009bw,Harlander:2009mq,Pak:2009dg,Harlander:2009my}.
In this contribution we briefly describe the calculation and results
obtained in Refs.~\cite{Pak:2009bx,Pak:2009dg}.\footnote{In
Refs.~\cite{Harlander:2009mq,Harlander:2009my} an expansion in 
$x = M_H^2/\hat{s}$ has been performed whereas in Ref.~\cite{Pak:2009dg} the
full $x$-dependence is kept.
Apart from that the results of~\cite{Harlander:2009mq,Harlander:2009my}
and~\cite{Pak:2009dg} agree.}

%-----

\section{Outline of the calculation}

Our result for the partonic cross section 
is based on the proper combination of two ingredients: the evaluation
in the limit of large center-of-mass energy and the asymptotic
expansion of the total cross section in inverse powers of $M_t$.
The leading contribution of the former, which is a constant at NLO and
a logarithm at NNLO, has been computed in Ref.~\cite{Marzani:2008az}. 
Due to an involved asymptotic expansion the latter 
is technically more challanging and requires a significant amount of
computer resourses in order to obtain several terms in the expansion.

\begin{figure}[b]
  \centering
  \begin{tabular}{ccc}
    \includegraphics[width=0.3\linewidth]{figs/part.nnlogg.logx.eps}
    &
    \includegraphics[width=0.3\linewidth]{figs/part.nnloqg.logx.eps}
    &
    \includegraphics[width=0.3\linewidth]{figs/part.nnloqb.logx.eps}
%    \\
%    (a) & (b) & (c)
  \end{tabular}
  \caption[]{\label{fig::part}Partonic NNLO cross sections for
    the $gg$, $qg$ and $q\bar{q}$
    channels (from left to right) as function of $x$ for $M_H=130$~GeV.
    Lines with longer dashes include higher order terms in $\rho$.
    The dotted lines correspond to the matched result (see
  Refs.~\cite{Pak:2009dg,Pak:2010mj}).} 
\end{figure}

In Ref.~\cite{Pak:2009dg} we have decided to consider the forward scattering
amplitudes and evaluate those imaginary parts which involve a cut of
the Higgs boson line.
In a first step we generate the diagrams and apply
subsequently the asymptotic expansion in the limit\footnote{$\hat{s}$ is the
  partonic center-of-mass energy.} $M_t^2\gg
\hat{s},M_H^2$, implemented in two independent programs.
This procedure factorizes the original triple-scale forward scattering functions
into massive vacuum integrals (with a single scale $M_t$) up to three
loops and four-point
one- and two-loop integrals dependent on $\hat{s}$ and $M_H$. 
As a result one obtains an expansion in $1/M_t$ which is valid for
$x = M_H^2/\hat{s} > x_{th} = M_H^2/(4M_t^2)$.
We match this result to a function $3 C_1  + ax$ (NLO) or
$- 9 C_2 \ln{x} + b$ (NNLO), where coefficients $C_1$ and $C_2$
are tabulated in Ref.~\cite{Marzani:2008az} and $a$, $b$ and
the matching point $x_m$ is chosen to provide the most ``natural''
smooth behaviour of the function.
We found that an $x_m$ such that the function and its first derivative match 
smoothly is a good choice at the NLO; at the NNLO, matching at $x_m = x_{th}/4$ 
produces reasonable results for $110~{\rm GeV} \le M_H \le {\rm 300}$~GeV and
is consistent with the region of $x$ where higher $\mathcal{O}(\rho^n)$
($\rho = M_H^2/M_t^2$)
corrections demonstrate good convergence. By varying the constants and
interpolating 
function shapes we have checked that the dependence of the hadronic
cross section on the exact details of the matching procedure is quite
small and that only the asymptotics near $x\to 0$ are important.
In the next Section we discuss the resulting partonic and hadronic
cross sections.

%-----

\section{Partonic and hadronic results}

We introduce the following notation for the partonic cross section
\begin{eqnarray}
  \hat{\sigma}_{ij\to H + X} &=& \hat{A}_{\rm LO} \left(
    \Delta_{ij}^{(0)} + \frac{\alpha_s}{\pi}~ \Delta_{ij}^{(1)}  
    + \left(\frac{\alpha_s}{\pi}\right)^2 \Delta_{ij}^{(2)} + \ldots
  \right)
  \,,\,\,\,
  \hat{A}_{\rm LO} \,\,=\,\, \frac{G_F~\alpha_s^2}{288\sqrt{2}\pi}
  f_0(\rho,0)
  \,,
  \label{eq::hatsigma}
\end{eqnarray}
where $ij$ denote one of the possible initial states:
$gg$, $qg$, $\bar{q}g$, $q\bar{q}$, $qq$, or $qq^\prime$, where $q$ and $q^\prime$
stand for (different) massless quark flavours. At NNLO the Higgs boson 
in the final state may be accompanied by zero, one or two gluons or 
light quarks. In general, the quantities $\Delta^{(k)}_{ij}$ depend on 
$x$ and $\rho$. Leading order mass dependence 
is then described by the function $f_0(\rho,0)$ which can be found 
in Ref.~\cite{Pak:2009bx}.
In Fig.~\ref{fig::part} we show the partonic NNLO
cross sections for the
numerically most important contributions $gg$, $qg$ and $q\bar{q}$
as function of $x$. Our final result obtained from the above
matching procedure is represented by the dotted lines.


The hadronic cross section is obtained by the convolution of
the partonic cross section $\hat{\sigma}_{ij\to H+X}$ 
with the corresponding parton distribution functions (PDFs).
In the following we present results for 
$pp$ collisions at the LHC peak energy $\sqrt{s} = 14~{\rm TeV}$.
We use the parton distribution function (PDF) set MSTW2008~\cite{Martin:2009iq}
and the $\alpha_s$ evolution at LO, NLO and NNLO when computing predictions
to the cross section at the corresponding order.

To discuss the numerical effect of our calculation we decompose the prediction
of the total cross section into its LO, NLO and NNLO contributions
%\begin{eqnarray}
$  \sigma_{pp^\prime\to H+X}(s) =   
  \sigma^{\rm LO} + \delta\sigma^{\rm NLO} + \delta\sigma^{\rm NNLO}
$
%  \,,
%  \label{eq::deltasigma}
%\end{eqnarray}
and denote the heavy top quark approximation with an additional
subscript $\infty$.
In the following we present the numerically most important
contribution from the $gg$ channel. In Figs.~\ref{fig::opt1_nnlo}(a)--(c) we
show the NNLO contribution to the hadronic cross section,
$\delta\sigma^{\rm NNLO}$, normalized to the infinite top quark
mass result where in each case the three lines correspond to the
inclusion of terms of order $\rho^0$ (short dashes), $\rho^1$ and
$\rho^2$ (long dashes). The difference among the three plots is that in~(a), 
the exact LO top quark mass dependence
is factored out as in Eq.~(\ref{eq::hatsigma}), while in~(b) the partonic
cross sections both in numerator and denominator are strictly expanded in $\rho$.
Finally, in~(c) we expand $\hat{A}_{LO}$ in the numerator
but keep it exact in the denominator.

For the fully expanded option~(b) one observes for $M_H=300$~GeV corrections
up to 40\% originating from the linear $\rho$ term which further increase to
almost 60\% after including the $\rho^2$ term.
However, when the exact leading-order top quark mass dependence is factored
out (case~(a)), the corrections amount to at most 8\%.
Considering the fact that the NNLO terms contribute about 10\% 
of the total NNLO cross section we conclude that the 
top quark mass suppressed terms at NNLO alter the prediction by less than
1\%. This justifies the use of the heavy top mass approximation for the
evaluation of the NNLO hadronic cross section.
The latter conclusion is also obtained from Fig.~\ref{fig::opt1_nnlo}(c).
It is interesting to remark that the slight deviation of the
$\rho^0$-curve in (a) is an effect of the matching procedure which is
not present in the denominator of the ordinate axis. Furthermore,
panel~(c) indicates that the infinite-top quark mass result (with
factored exact LO result)
approximates the exact result (including finite top quark mass
effects) to a few percent level.
Let us also stress that the matched result obtained from the different
approximation in $\rho$ only leads to slightly different hadronic
contributions. The difference in the three curves in
Fig.~\ref{fig::opt1_nnlo} essentially comes from the top quark mass
corrections to the $\delta$-function part of the partonic cross section. 

\begin{figure}[t]
  \centering
  \begin{tabular}{ccc}
    \includegraphics[width=0.3\linewidth]{figs/hadr.gg2.eps}
    &
    \includegraphics[width=0.3\linewidth]{figs/hadr.gg2notf.eps}
    &
    \includegraphics[width=0.3\linewidth]{figs/hadr.gg2notf2.eps}
    \\ (a) & (b) & (c)
  \end{tabular}
  \caption[]{\label{fig::opt1_nnlo}(a), (b) and (c):
    Ratio of the NNLO hadronic cross section 
    ($gg$ contribution) including successive higher orders in $1/M_t$ normalized
    to the infinite top quark mass result. In (a) the exact LO mass
    dependence is factorized both in the numerator and denominator.
    In (b) numerator and denominator are expanded in $\rho$, and 
    in (c) only the numerator is expanded.}
\end{figure}

%-----

{\it Acknowledgments:} I would like to thank A. Pak and M. Rogal for a fruitful
collaboration on the subjects presented in this contribution.

\begin{thebibliography}{99}
%  \input{ichep10_ggh_ref.tex}
%
% ichep10_ggh_ref.tex -- generated by sortref-2.3.5  
% ((C) R. Harlander, http://www.robert-harlander.de/software/)
% on Thu Sep  9 21:12:12 CEST 2010
%

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\end{thebibliography}

\end{document}

