%Title: Precise predictions for Higgs cross sections at the Large Hadron Collider 
%Author: Robert Harlander
%Published: <EM> Nucl.Phys.Proc.Suppl. </EM> <B> 135 </B> (2004) 30-34. 
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% TTP04-12
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% Contribution for Loops & Legs, April 25-30, 2004, Zinnowitz, Germany.
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\title{Precise predictions for Higgs cross sections at the Large
Hadron Collider}

\author{
\vspace{-10em}
\begin{flushright}
{\it Loops \& Legs 2004}, April 25-30, Zinnowitz, Germany --- TTP04-12\\[4em]
\end{flushright}
\vspace*{3.8em}
Robert Harlander\address[TTP]{%
	Institut f\"ur Theoretische Teilchenphysik\\
	Universit\"at Karlsruhe\\
	D-76128 Karlsruhe, Germany}
        \thanks{Supported by {\it Deutsche Forschungsgemeinschaft},
	  contract HA\,2990/2-1}}
       

\runtitle{Higgs cross sections at the LHC}
\runauthor{R.\ Harlander}

\begin{document}

%- }}}
%- {{{ abstract:

\begin{abstract}
The status of the theoretical predictions for the most important Higgs
production processes at the Large Hadron Collider is reviewed. Both the
Standard Model and neutral Higgs production in the \mssm{} are
considered.
\end{abstract}

\maketitle

%- }}}
%- {{{ Intro:

\section{Introduction}
%
Within the last few years, the inclusive cross sections for the dominant
Higgs production mechanisms at hadron colliders have become under very
good theoretical control. This is to a large part due to the development
of new techniques for the evaluation of radiative corrections. These
precise results have brought confidence into the -- in some cases very
large -- theoretical K-factors. They should now be (and already are)
used for experimental analyses concerning the discovery potential of a
Higgs boson or measurements of Higgs couplings and other properties.

We will start our overview with the four dominant production modes in
the Standard Model (\sm{}): Gluon fusion, weak boson fusion, associated
$t\bar tH$ production, and Higgs Strahlung. Then we will discuss some
aspects of Higgs production in the Minimal Supersymmetric Standard Model
(\mssm{}), in particular the effect of squark loops in the gluon fusion
process, and the associated Higgs production with bottom quark pairs.

%- }}}
%- {{{ Gluon Fusion:

\section{Gluon Fusion}
Gluon fusion is the Higgs production process with the largest cross
section at the \lhc{} (and also at the Tevatron).  The next-to-leading
order (\nlo{}) corrections are very large, amounting up to 70\%--100\%
of the leading order (\lo{}) result~\cite{gghnloeff,gghnlo}. They are
extremely well approximated by the limit $m_t\gg M_H$, even at Higgs
boson masses that are far beyond this limit (for a detailed discussion,
see Ref.\,\cite{Harlander:2003xy}).

Nevertheless, even in this ``heavy-top limit,'' the
next-to-next-to-leading order (\nnlo{}) calculation was not performed
before the development of new techniques for the evaluation of loop and
phase space integrals.  For example, the corrections from virtual gluon
exchange could be evaluated for the first time~\cite{Harlander:2000mg}
with the help of new equivalence relations among Feynman
integrals\,\cite{Baikov:2000jg}.

For the phase space integrals of the double-gluon emission
contributions, two new techniques were introduced:
The method of ``expansion and
inversion''~\cite{Harlander:2002wh,Kilgore:2002sk,Harlander:2002vv},
and the method of ``generalized Cutkosky rules''~\cite{Anastasiou:2002yz}.
The contributions from real radiation in gluon fusion were later
confirmed\,\cite{Ravindran:2003um} by extending the integral tables of
Ref.\,\cite{Hamberg:1991np} (see also Ref.\,\cite{Ravindran:2004cx}).

The \nnlo{} prediction for gluon fusion is shown in \fig{fig::ggh},
together with the lower order results. This figure gives the desired
confidence in the perturbative prediction of this process.  The bands
indicate the variation of the result when changing the unphysical
factorization and renormalization scales $\mu_F$ and $\mu_R$ by a factor
of two around the Higgs mass.

Resummation of soft gluon effects were shown to contribute less than
6\%~\cite{Catani:2003zt}, thus confirming the stability of the \nnlo{}
prediction.

\begin{figure}
\begin{center}
\includegraphics[bb=80 300 465 560,width=20em]{figs/sigsnnlo14.ps}
\caption[]{\label{fig::ggh}\lo{}, \nlo{}, and \nnlo{} prediction for
  Higgs production in gluon fusion in the
  \sm{}~\cite{Harlander:2002wh,Anastasiou:2002yz,Ravindran:2003um}.}
\end{center}
\end{figure}

Electroweak corrections in the large-$m_t$ limit give a very small
effect of below 1\%~\cite{Djouadi:1994ge}. Recently, electroweak
corrections arising from light fermions were evaluated and found to be
quite sizable around $M_H = 2M_W$~\cite{Aglietti:2004nj}.


%- }}}
%- {{{ Weak Boson Fusion:

\section{Weak Boson Fusion}
This process has been shown to be extremely useful both for Higgs
discovery at the \lhc{} in almost all of the relevant mass range, as
well as for measurements in the Higgs sector, in particular Higgs
couplings and {\abbrev CP} properties of the Higgs boson(s) (see
Ref.\,\cite{Plehn:2001nj} and references therein).

Among the welcome features of this process is the smallness of the
{\abbrev QCD} corrections to it, and their technical simplicity at
\nlo{}. The latter results from the absence of pentagon diagrams due to
their vanishing color factor. Thus, all of the \nlo{} corrections are
analogous to Deep Inelastic Scattering.

The \nlo{} total cross section for weak boson fusion has thus been known
for a long time~\cite{Han:1991ia}. Only recently, however, has a fully
differential partonic \nlo{} Monte Carlo program become
available~\cite{Figy:2003nv}. Differential analyses are particularly
important for this process because the tagging of the forward-scattered
jets is essential for an identification of the {\abbrev WBF} signal.

%- }}}
%- {{{ Associated ttH production:

\section{Associated \bld{t\bar tH} production}
Even though the cross section for the production of a Higgs with a
$t\bar t$ pair is rather small, this mode is very attractive because of
its remarkable signature: four $b$ quarks in association with two $W$
bosons. In addition, this process is directly sensitive to the
top--Yukawa coupling.

The calculation of the {\abbrev NLO} corrections is rather
challenging.  One of the obstacles are massive pentagon diagrams.
In fact, the $t\bar tH$ processes triggered the development of
new, efficient algorithms for the evaluation of this class of
integrals, avoiding the occurrence of spurious singularities in the
Gram determinants~\cite{Denner:2002ii}.

From the theoretical point of view, the \nlo{} terms have a very
satisfactory effect~\cite{tthnloshort}. The variation of the cross
section prediction with renormalization and factorization scale
reduces drastically, as one can see in \fig{fig::tthscale}. The overall
effect on the total rate is of the order of +20\% at the central scale of
$\mu_R = \mu_F = m_t + M_H/2$.

\begin{figure}
\begin{center}
  \includegraphics[bb=40 280 560 620,width=18em]{figs/ttH_sc_lhc.eps}
\caption[]{\label{fig::tthscale}\sloppy Variation of the inclusive cross
  section for $pp\to t\bar tH$ at the \lhc{} with $\mu = \mu_R = \mu_F$
  (from the first reference in Ref.\,\cite{tthnloshort}).  }
\end{center}
\end{figure}

%- }}}
%- {{{ Higgs Strahlung:

\section{Higgs Strahlung~\cite{Brein:2004ue}}
%
At \lo{}, the production of a Higgs boson in association with a vector
boson, $p p \to VH+X, (V=W,Z)$ proceeds through $q\bar{q}$ annihilation,
$q\bar{q}' \to V^* \to VH$.  The \nlo{} \qcd{} corrections coincide with
those to the Drell-Yan process and increase the cross section by about
30\%~\cite{Han:1991ia}.  Beyond \nlo{}, the \qcd{} corrections to $VH$
production differ from those to the Drell-Yan process by contributions
where the Higgs boson couples to a heavy fermion loop.  The impact of
these additional terms is, however, expected to be small in
general~\cite{Dicus:1985wx}.  Moreover, for $ZH$ production the
one-loop-induced process $gg\to ZH$ contributes at \nnlo{}.  The \nnlo{}
corrections corresponding to the Drell-Yan mechanism as well as the
$gg\to ZH$ contribution have been calculated in
Ref.\,\cite{Brein:2003wg}.  These \nnlo{} corrections further increase
the cross section by the order of 5--10\%.  Most importantly, a
successive reduction of the scale dependence is observed when going from
\lo{} to \nlo{} to \nnlo{}. The respective scale uncertainties are about
10\%, 5\%, and 2\% at the \lhc{}.  At this level of accuracy,
electroweak corrections become significant and need to be included to
further improve the theoretical prediction.  In
Ref.\,\cite{Ciccolini:2003jy}, the electroweak ${\cal O}(\alpha)$
corrections have been calculated; they turn out to be negative and about
$-5$\% or $-10$\% depending on whether the weak couplings are derived
from $G_\mu$ or $\alpha(M_Z^2)$, respectively.

A combined analyses of \qcd{} and electroweak effects has been
presented in Ref.\,\cite{Brein:2004ue}. \fig{fig::higgsstrahlung} shows
the sum of the cross sections for $W^+H$ and $W^-H$ production at the
\lhc{}, including \lo{}, \nlo{}, and \nnlo{} \qcd{} effects, as well as
the $\order{\alpha}$ electroweak corrections. The inclusion of all
higher order effects makes Higgs Strahlung by now the most precise Higgs
production process at hadron colliders. The remaining uncertainties are
estimated to be of the order of 5--10\%.

\begin{figure}
\begin{center}
\includegraphics[bb=80 300 465 560,width=20em]{figs/ew-qcd-w-lhc-.ps}
\caption[]{\label{fig::higgsstrahlung}K-factor for $WH$ production at
  the \lhc{} after inclusion of the \nnlo{} \qcd{} and electroweak
  $\order{\alpha}$ corrections. The bands reflect the variation of the
  cross section upon variation of $\mu_F$ and $\mu_R$ within a factor of
  three around $M_H+M_W$.}
\end{center}
\end{figure}


%- }}}
%- {{{ SUSY:

\section{Supersymmetry}

In the \mssm{}, several aspects
have to be taken into account when considering the production of a
neutral Higgs boson: For example, for large $\tan\beta$, the bottom
quark can contribute significantly to the gluon-Higgs coupling.  This
effect has been studied at \nlo{} in Ref.~\cite{gghnlo}.  A large value
of $\tan\beta$ also brings in the associated production of a Higgs boson
with bottom quarks as one of the main production mechanisms. This will
be discussed in \sct{sec::bbh}, while the influence of supersymmetric
particles to the gluon fusion process will be discussed in
\sct{sec::squarks}.  The production of pseudo-scalar Higgs bosons
through gluon fusion for small $\tan\beta$ is known through
\nnlo{}~\cite{Harlander:2002vv,Anastasiou:2002wq}, with very similar
conclusions as in the case of scalar Higgs production.


%- {{{ Squarks:

\subsection{Squarks}\label{sec::squarks}
%
Motivated by its success in the \sm{} case, one can construct an
effective Lagrangian for the gluon-Higgs interaction also in the
\mssm{}. This corresponds to integrating out the top quark and all
supersymmetric particles. Within the framework of this effective
Lagrangian, all calculations can be performed in standard five-flavor
{\abbrev QCD}. They can thus be taken over from the \sm{} case.

At sufficiently small values of $\tan\beta$, the dominant effects to the
gluon-Higgs coupling from \mssm{} particles are due to top-squarks.  In
comparison to the \sm{} particles, this contribution goes like
$(m_t/m_{\tilde t}^2)^2$, however, with a possible enhancement for large
absolute values of the trilinear coupling $A_t$. For moderate values of
$A_t$, however, only top squarks with $m_{\tilde t}\lesssim 400$\,GeV
give a sizable effect.

At \nlo{}, the \susy{} relation between the top and the stop Yukawa
coupling requires to include also gluino effects in order to arrive at
finite results.  In a very simplifying limit without squark mixing, the
\nlo{} corrections were found to be very similar to the \sm{}
case~\cite{Harlander:2003bb} (see also Ref.\,\cite{Dawson:1996xz}). A
more general analysis of the \susy{} parameter space will be presented in
Ref.\,\cite{gghsusy}.

The dominant corrections to the Higgs production cross section originate
from real gluon emission. Thus, it is possible to derive a rather
precise estimate of the \nnlo{} terms based on the \nnlo{} result in the
\sm{}~\cite{Harlander:2002wh,Anastasiou:2002yz,Ravindran:2003um} and the
\nlo{} effective Higgs-gluon coupling~\cite{Harlander:2003bb}. In this
way, the reduced scale uncertainty of the \nnlo{} in the \sm{} directly
carries forward to the supersymmetric case. The result is shown in
\fig{fig::squarksnnlo}, details can be found
in~\reference{Harlander:2003kf}.

\begin{figure}
\begin{center}
\includegraphics[bb=80 300 465 560,width=20em]{figs/kss14nnlo.ps}
\caption[]{\label{fig::squarksnnlo}K-factor for gluon fusion at the
  \lhc{} in the \sm{} (dashed blue) and including squark effects (solid
  red) at \nlo{} and \nnlo{}.  The width of the dark shaded \nnlo{} band
  is an estimate of the uncertainty due to neglected \nnlo{} terms to
  the gluon-Higgs coupling. The light shaded band is the uncertainty
  induced by scale variation at \nnlo{}. From
  Ref.\,\cite{Harlander:2003kf}.}
\end{center}
\end{figure}


%- }}}
%- {{{ bbH:

\subsection{Associated \bld{b\bar bH} production}\label{sec::bbh}
%
There are two complementary approaches to calculate the inclusive cross
section for this process: The so-called four-flavor scheme, assuming
$gg\to b\bar bH$ as the \lo{} partonic process, and the five-flavor
scheme, where $b\bar b\to H$ is the \lo{} contribution.  The latter
resums collinear logarithms $\sim \ln(m_b/M_H)$ arising from small
transverse bottom momenta $p_{T,b}$. However, the \lo{} of the
five-flavor scheme neglects contributions from large $p_{T,b}$;
they are re-introduced through higher order {\abbrev QCD}
corrections. In fact, a \nnlo{} calculation~\cite{Harlander:2003ai}
consistently combines the all-order resummation of the low-$p_{T,b}$
region with the \lo{} contributions from large $p_{T,b}$.

Recently, the \nlo{} corrections for the four-flavor scheme became
available. The total cross section is in good agreement with the result
from the five-flavor scheme for small values of $M_\phi$. For larger
values, the agreement becomes worse as it is expected because the
collinear logarithms become numerically more important. For more
details, see Ref.\,\cite{kraemer}.

\begin{figure}
\begin{center}
\includegraphics[bb=50 280 560 630,width=18em]{figs/lhc_0b_mh-.ps}
\end{center}
\caption[]{Total cross section for $pp\to b\bar b\phi+X$ in the four-
  and the five-flavor approach ($gg\to bbh$~\cite{bbhnlo} [light shaded]
  and $b\bar b\to h$~\cite{Harlander:2003ai} [dark shaded],
  respectively). The width of the bands corresponds to the theoretical
  uncertainty as estimated from scale variation. From
  \cite{Campbell:2004pu}.}
\end{figure}

%- }}}

%- }}}
%- {{{ Conclusions, Acknowledgments:

\section{Conclusions}
%
We have reviewed the theoretical status of the most important (neutral)
Higgs production processes in the \sm{} and the \mssm{}. All of them are
under very good theoretical control due to recent advances in the
evaluation of radiative corrections. The precision for the inclusive
cross section is in all cases at least of the order of the expected
experimental precision.

\paragraph{Acknowledgments}
It is a pleasure to thank O.~Brein, A.~Djouadi, W.~Kilgore, and
M.~Steinhauser for fruitful collaboration on some of the topics
discussed above. Furthermore, I would like to thank the organizers for
the invitation and the pleasant atmosphere during the workshop.

%- }}}
%- {{{ bibliography:

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%- }}}

\end{document}
