%Title: Pseudo-scalar Higgs production at next-to-leading order SUSY-QCD
%Author: Robert V. Harlander and Franziska Hofmann
%Published: <EM> JHEP </EM> <B> 0603 </B> (2006) 050. 
%hep-ph/0507041

 
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\newcommand{\dreg}{{\abbrev DREG}}
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\title{Pseudo-scalar Higgs production at next-to-leading order SUSY-QCD}
\author{Robert V. Harlander and Franziska Hofmann\\
  {\it Institut f\"ur Theoretische Teilchenphysik,
  Universit\"at Karlsruhe\\
  D-76128 Karlsruhe, Germany}\\
    E-mail: \email{robert.harlander@cern.ch},
\email{fhofmann@particle.uni-karlsruhe.de}\\
}

\preprint{\hepph{0507041}, \sf TTP05--07, SFB/CPP-05-17 --- July 2005}

\abstract{
The production rate of the {\abbrev CP}-odd Higgs boson in
the Minimal Supersymmetric Standard Model is evaluated through
next-to-leading order in the strong coupling constant.  The divergent
integrals are regulated using Dimensional Reduction, with a
straightforward implementation of $\gamma_5$.  The well-known Standard
Model result is recovered in this scheme through a cancellation between
the quark and squark contributions as the masses of the supersymmetric
particles tend to infinity. }

\keywords{Higgs production, hadron collider, supersymmetry, next-to-leading order calculations}
\begin{document}
\setlength{\figwid}{7em}



\section{Introduction}\label{sec::Intro}
The \mssm{} predicts a fundamental {\abbrev CP}-odd scalar particle $A$,
commonly referred to as the pseudo-scalar Higgs boson. One of the most
important properties that distinguishes it from its {\abbrev CP}-even
analogues $h$ and $H$ is the absence of tree-level couplings to the
electro-weak gauge bosons $W$ and $Z$. Decay and production processes
through these particles, which have been shown to be extremely helpful
for {\abbrev CP}-even Higgs searches and studies, are thus very much
suppressed.  This leaves associated $t\bar tA$ and $b\bar bA$ production
as well as the loop-induced gluon fusion process as the most important
production modes of a pseudo-scalar Higgs boson at the {\abbrev LHC}
(for a recent review, see Ref.\,\cite{Djouadi:2005gij}).  In this paper,
we present the evaluation of the inclusive gluon fusion cross section
through \nlo{} in the strong coupling constant.

Despite the fact that the tree-level $gg\phi$ coupling vanishes
($\phi\in\{h,H,A\}$), gluon fusion in general has a comparatively large
cross section due to the high gluon luminosity at the {\abbrev LHC}, and
the large top-Yukawa coupling. In the limit where squarks and gluinos
are decoupled, {\abbrev QCD} corrections have been evaluated through
\nnlo{}~\cite{Harlander:2002wh,Harlander:2002vv,Anastasiou:2002wq,
Anastasiou:2002yz,Ravindran:2003um}; they have been shown to be
numerically significant but perturbatively well-behaved.

A precise determination of the gluon fusion cross section at the
{\abbrev LHC} could yield sensitivity to as yet undiscovered particles
that may mediate the $gg\phi$ coupling apart from the top and bottom
quarks. Within the \mssm{}, for example, top squarks can play a
significant role if they are lighter than around 400\,GeV (recall that
the Yukawa coupling for squarks is typically proportional to $m_q^2$
rather than $m_{\tilde q}^2$). In the case of {\abbrev CP}-even Higgs
bosons, such effects occur already at \lo{} (i.e., 1-loop). The \nlo{}
result was obtained in Ref.\,\cite{Harlander:2004tp} within an effective
theory for top, stop, and gluino masses much larger than the Higgs mass.

For the {\abbrev CP}-odd Higgs boson, however, squarks do not affect the
$ggA$ vertex at 1-loop level due to the structure of the $A\tilde
q\tilde q$ coupling, as will be shown below. The two-loop effects are
thus expected to have a larger influence than for the {\abbrev CP}-even
Higgs production. What adds to this is that, in the limit of large
$m_q$, the quark mediated contribution to the $ggA$ vertex does not
receive any \qcd{} corrections due to the Adler-Bardeen theorem. As we
will show, the only 2-loop \qcd{} effects to this coupling are due to
mixed gluino-quark-squark diagrams in this limit, leading to a
potentially increased sensitivity to the gluino mass. Of course, the
heavy quark limit is not applicable for the bottom mediated gluon-Higgs
coupling which does receive \qcd{}
corrections~\cite{Spira:1995rr,Spira:1997dg}.

The calculation involves a technical issue that deserves special
mention, namely the implementation of $\gamma_5$ within \dred{}.  Its
mathematically consistent and practically feasible formulation has been
a subject of interest for many years now~\cite{Siegel:1980qs} (for
recent developments concerning \dred{}, see
Ref.\,\cite{Smith:2004ck,Stockinger:2005gx}).  Here we adopt an approach
close to the prescription of
Refs.\,\cite{'tHooft:fi,Breitenlohner:1977hr}. An important consistency
check is obtained from the \sm{} limit: even for an infinitely heavy
\susy{} spectrum, the \susy{} diagrams give a non-vanishing contribution
which exactly cancels the 2-loop contributions arising from the \sm{}
diagrams in this scheme.  The well-known result of vanishing higher
order corrections to the quark-mediated $ggA$ coupling, as required by
the Adler-Bardeen theorem, is thus recovered in a non-trivial way.
Combining this observation with the considerations of
Ref.\,\cite{Stockinger:2005gx}, it seems likely that in a supersymmetric
theory the formal difficulties of \dred{} do not pose serious technical
problems in practical calculations~\cite{Siegel:1980qs}.

The outline of this paper is as follows: In \sct{sec::notation} we
introduce our notation and quote the \lo{} result for the partonic
process $gg\to A$. In \sct{sec::higher}, the effective Lagrangian
underlying our calculation is introduced and the treatment of $\gamma_5$
is discussed. \sct{sec::results} outlines the method of the calculation
and discusses the general structure of the result. It also provides the
analytic formulae in some limiting cases. The numerical influence of the
\nlo{} terms is discussed in \sct{sec::discussion}. In
\sct{sec::conclusions} our findings are summarized and an outlook on
possible extensions of this work is given.


\section{Notation and Leading Order Result}\label{sec::notation}

\subsection{Lagrangian}
We write the underlying Lagrangian in the following form:
\begin{equation}
\begin{split}
{\cal L} &= {\cal L}_{\rm QCD} + {\cal L}_{ qA}
+ {\cal L}_{\rm SQCD} + {\cal L}_{\tilde qA}\,,
\label{eq::lag}
\end{split}
\end{equation}
where
\begin{equation}
\begin{split}
{\cal L}_{qA} &= \sum_{q} \frac{m_q}{v}\,g_q^A\, \bar q\gamma_5 q
A\,,\qquad {\cal L}_{\tilde qA} = \sum_{q}\sum_{i=1}^{2}
\frac{m_q^2}{v}\,\tilde g_{q,ij}^A\, \tilde q_i\tilde q_j\,A\,.
\label{eq::lags}
\end{split}
\end{equation}
The sums $\sum_{q}$ run over all quark flavors.  $m_q$ is the mass of
quark $q$ and $v\approx 246$\,GeV the Higgs vacuum expectation value;
the coupling constants $g_q^A$ and $\tilde g_{q,ij}^A$ will be defined
below.  ${\cal L}_{\rm QCD}$ denotes the full {\abbrev QCD} Lagrangian
with six quark flavors, while ${\cal L}_{\rm QCD}+{\cal L}_{\rm SQCD}$
is the supersymmetric extension of ${\cal L}_{\rm QCD}$ within the
\mssm{}, i.e., ${\cal L}_{\rm SQCD}$ incorporates kinetic, mass, mixing,
and interaction terms of all the squarks and gluinos.  Since we will be
concerned with higher orders in the strong coupling $\alpha_s$ only, $A$
does not appear as a dynamical field and does not require a kinetic
term.

$\tilde q_1, \tilde q_2$ denote the squark mass eigenstates, related to
the chiral eigenstates $\tilde q_L, \tilde q_R$ (the superpartners of
the left- and the right-handed quark $q$) through\note{check}
\begin{equation}
\begin{split}
\left(
\begin{array}{c}
{\tilde q_1} \\
{\tilde q_2}
\end{array}
\right)
 &=
\left(
\begin{array}{cc}
\cos\theta_q & \sin\theta_q\\
-\sin\theta_q & \cos\theta_q
\end{array}
\right)
\left(
\begin{array}{c}
{\tilde q_L} \\
{\tilde q_R}
\end{array}
\right)\,.
\end{split}
\end{equation}
The coupling constants relevant for the following discussion
are\footnote{For a more complete collection of Feynman rules, see
Ref.\,\cite{Kraml-Eberl}, for example.}
\begin{equation}
\begin{split}
g_b^A &= \tan\beta\,,\qquad g_t^A = \cot\beta\,,\qquad
\tilde g_{t,11}^A = \tilde g_{t,22}^A = 0\,,\\
\tilde g_{t,12}^A &= - \tilde g_{t,21} = 
\frac{\mstop{1}^2 - \mstop{2}^2}{2m_t^2}\,\sin\theta_t\,\cot\beta
+ \frac{\muSUSY}{m_t}\,(\cot^2\beta +1 )\,.
\label{eq::couplings}
\end{split}
\end{equation}
The effect of quarks other than bottom and top can be neglected due to
their small Yukawa couplings; squark effects, on the other hand, are
typically suppressed by at least one power of $m_q/m_{\tilde q}$, so
that only top squarks will be considered in this paper.


\subsection{Hadronic cross section}
The cross section for the hadronic process $pp\to A+X$ at a
center-of-mass energy $s$ is determined by the formula\footnote{The
  modifications for $p\bar p$ collisions are obvious and shall not be
  pointed out explicitely in this paper.}
\begin{equation}
\begin{split}
\sigma(z) &= \sum_{i,j\in\{q,\bar q,g\}}
\int_z^1\dd x_1\int_{z/x_1}^1\dd
x_2\,\varphi_i(x_1)\varphi_j(x_2)\,\hat\sigma_{ij}
\left(\frac{z}{x_1x_2}\right)\,,
\qquad z:= \frac{M_A^2}{s}\,,
\label{eq::hadpart}
\end{split}
\end{equation}
where $\varphi_i(x)$ is the density of parton $i$ inside the proton.
$\hat\sigma_{ij}$ is the cross section for the process $ij\to A+X$,
where, as indicated in \eqn{eq::hadpart}, $i$ and $j$ are parton labels.
For our numerical analysis, we will use the {\abbrev MRST} parton
density sets throughout this paper~\cite{Martin:2001es,Martin:2003sk}.

Sample diagrams for $i\!\!=\!\!j\!\!=\!\!g$ 
and $i\!\!=\!\!g$, $j\!\!=\!\!q$ that contribute to the
inclusive Higgs production rate at \lo{} and \nlo{} are shown in
\fig{fig::lonlo}.


\FIGURE{
    \begin{tabular}{c}
      \begin{tabular}{cc}
      \includegraphics[bb=154 450 450
      660,width=8em]{triangle_1.ps} &
      \includegraphics[bb=154 450 450
      660,width=8em]{triangle_nlo_1.ps} \\
      (a) & (b)
      \end{tabular}
      \\[4em]
      \begin{tabular}{ccc}
      \includegraphics[bb=154 450 450
      660,width=8em]{triangle_nlo_real1_bw.ps} &
      \includegraphics[bb=154 450 450
      660,width=7.5em]{triangle_nlo_real2_bw.ps} &
      \includegraphics[bb=154 450 450
      660,width=7em]{triangle_nlo_real3_bw.ps} \\
      (c) & (d) & (e)
      \end{tabular}
      \\[4em]
      \begin{tabular}{cccc}
      \includegraphics[bb=154 450 450
      660,width=8em]{gluino2l1_2.ps} &
      \includegraphics[bb=154 450 450
      660,width=8em]{gluino2l2_1.ps} &
      \includegraphics[bb=154 450 450
      660,width=8em]{gluino2l3_2.ps} &
      \includegraphics[bb=154 450 450
      660,width=8em]{gluino2l4_2.ps}\\
      (f) & (g) & (h) & (i)
      \end{tabular}
    \end{tabular}
    \caption[]{\sloppy
      Diagrams contributing to the process $gg\to A+X$ in the
      \mssm{} at (a) \lo{} and (b)--(i) \nlo{}.
      According to the definition of the main text, (a)--(e) are referred
      to as ``\sm{} diagrams'', (f)--(i) as ``\susy{} diagrams''.
      The straight-curly line connecting quarks and squarks represents a
      gluino (denoted by $\tilde g$ in the text).
      \label{fig::lonlo}%
    }
}



\subsection{Standard Model limit and leading order result}
In the limit where all {\abbrev SUSY} masses tend to infinity (denoted
in the following by $M_{\rm SUSY}\to \infty$), the Lagrangian of
\eqn{eq::lag} reduces to
\begin{equation}
\begin{split}
{\cal L}^{\rm SM} &= {\cal L}_{\rm QCD} + {\cal L}_{qA}\,.
\label{eq::lagsm}
\end{split}
\end{equation}
This will be called the ``Standard Model'' limit in what follows,
despite the fact that the \sm{} does not contain a {\abbrev CP}-odd
Higgs boson. Consequently, diagrams without any squark and gluino lines
will be called ``\sm{} diagrams'' (e.g. \fig{fig::lonlo}\,(a)--(e)),
while ``\susy{} diagrams'' contain at least one propagator of a \susy{}
particle\footnote{In the following we will strictly distinguish between
\sm{}/\susy{} {\it diagrams} and \sm{}/\susy{} {\it results}: As it
turns out, the \sm{} diagrams do {\it not} give the \sm{} result in our
scheme.} (\fig{fig::lonlo}\,(f)--(i)).

Note that due to the antisymmetric structure of the $\tilde
g_{q,ij}^A$, \eqn{eq::couplings}, there are no \susy{} diagrams at the
one-loop level. The \lo{} result for the partonic process $gg\to A$ is
thus determined solely by diagrams of the type shown in
\fig{fig::lonlo}\,(a). It reads explicitely:
\begin{equation}
\begin{split}
\hat\sigma_{gg}(x) &= \sigma_{0}^A\,\delta(1-x) +
\order{\alpha_s^3}\,,\\
\sigma_{0}^A &= \frac{\pi}{256 v^2}\left(\api\right)^2\,
\bigg|\sum_{q\in\{t,b\}} {\cal A}_q(\tau_q)\bigg|^2\,,\\
{\cal
A}_q(\tau_q) &= g_q^A\, \tau_q f(\tau_q)\,,\qquad\tau_q =
\frac{4m_q^2}{M_A^2}\,,
\end{split}
\end{equation}
where
\begin{equation}
  \begin{split}
    f(\tau) = \left\{
    \begin{array}{ll}
      \displaystyle\arcsin^2\frac{1}{\sqrt{\tau}}\,, & \tau \geq 1\,,\\
      \displaystyle -\frac{1}{4}\left[
	\ln\frac{1+\sqrt{1-\tau}}{1-\sqrt{1-\tau}} - i\pi\right]^2\,, &
      \tau < 1\,.
    \end{array}
    \right.
    \label{eq::ftau}
  \end{split}
\end{equation}
Throughout this paper, $\alpha_s$ denotes the strong coupling constant,
renormalized in the $\msbar{}$ scheme for \qcd{} with five massless
quark flavors.



\section{Higher orders}\label{sec::higher}


\subsection{Standard Model result -- Effective Lagrangian}\label{sec::eff}
The \sm{} contributions, based on ${\cal L}_{\rm SM}$ of
\eqn{eq::lagsm}, are known through \nlo{} in terms of 1-dimensional
integral representations~\cite{Spira:1995rr}, implemented in the program
{\tt HIGLU}~\cite{Spira:1995mt}. They include both top and bottom quark
loops as shown in \fig{fig::lonlo}\,(a)--(e) ($q\in\{b,t\}$).

It has been shown in Ref.\,\cite{Spira:1995rr,Spira:1997dg} that the
\nlo{} top quark contributions are well approximated by the formula
\begin{equation}
\begin{split}
\sigma_t^{\rm NLO} &= K_\infty\,\sigma_t^{\rm LO}\,,\quad
\mbox{where}\qquad
K_\infty = \frac{\sigma_\infty^{\rm NLO}}{\sigma_\infty^{\rm LO}}\,.
\end{split}
\end{equation}
$\sigma_{\infty}$ is the cross section evaluated in an
effective theory, obtained by integrating out the top quark:
\begin{equation}
\begin{split}
{\cal L}_{\rm QCD} + {\cal L}_{qA} \stackrel{m_t\gg M_A}{\longrightarrow}
{\cal L}_{ggA}^{\rm SM} = 
    -\frac{\oddhiggs{}}{v}\left[\tilde C_1^{\rm SM}\, \topo_1 +
    \tilde C_{2}^{\rm SM}\, \topo_{2}\right]
    + {\cal L}_{\rm QCD}^{(5)}\,.
    \label{eq::leffsm}
\end{split}
\end{equation}
The operators are defined as
\begin{equation}
\begin{split}
 \topo_1 &= \frac{1}{4}
    \vep^{\mu\nu\alpha\beta}G^{a}_{\mu\nu} G^{a}_{\alpha\beta}\,,\qquad
    \topo_{2} = \sum_{q\neq t}\partial_\mu\left( \bar q
    \gamma^\mu\,\gamma_5 q\right)\,.  \label{eq::leff}
\end{split}
\end{equation}
Here, $\{q\} := \{d,u,s,c,b\}$ denotes the set of light (in our case
massless) quark fields and $G_{\mu\nu}^a$ the gluon field strength
tensor. ${\cal L}_{\rm QCD}^{(5)}$ is the Standard {\abbrev QCD}
Lagrangian with five quark flavors.

The Wilson coefficients $\tilde C_1^{\rm SM}$ and $\tilde C_{2}^{\rm
SM}$ have been calculated through \nnlo{} in
Ref.\,\cite{Chetyrkin:1998mw}. $\tilde C_2^{\rm SM}$ contributes only at
\nnlo{} and shall not be discussed any further in this paper. The result
for $\tilde C_1^{\rm SM}$ is
\begin{equation}
\begin{split}
\tilde C_1^{\rm SM} &= -\cot\beta\frac{\alpha_s}{16\pi} +
\order{\alpha_s^4}\,,
\label{eq::c1sm}
\end{split}
\end{equation}
as it had been previously suggested in
Refs.\,\cite{Spira:1997dg,Kramer:1996iq} on the basis of the
Adler-Bardeen theorem~\cite{adlerbardeen}.

$\topo_1$ generates vertices which couple two and three gluons to the
pseudo-scalar Higgs (the $ggggA$-vertex vanishes due to the Jacobi
identity of the structure functions of SU(3)).  Sample diagrams that
contribute to the \nlo{} cross section in the effective theory of
\eqn{eq::leffsm} are shown in \fig{fig::process}. The full set has been
calculated through \nlo{} in Ref.\,\cite{Spira:1995rr}, and through \nnlo{} in
Ref.\,\cite{Harlander:2002vv,Anastasiou:2002wq,Ravindran:2003um}.


\FIGURE{%
    \begin{tabular}{c}
      \begin{tabular}{cc}
      \includegraphics[width=8em]{ggaeff0l1_bw.eps} &
      \includegraphics[width=8em]{ggaeff1l1_bw.eps} \\
      (a) & (b)\\[1em]
      \end{tabular}
      \\
      \begin{tabular}{ccc}
      \includegraphics[width=8em]{ggageff0l1_bw.eps} &
      \includegraphics[width=8em]{qgaqeff0l1_bw.eps} &
      \includegraphics[width=8em]{ggageff0l2_bw.eps} \\
      (c) & (d) & (e)
      \end{tabular}
    \end{tabular}
    \caption[]{\sloppy
      Diagrams contributing to the gluon fusion process in the effective
      theory at \nlo{}.
      \label{fig::process}
    }
}



\subsection{SUSY contributions}
We construct again an effective theory, this time by integrating out not
only the top quark, but also all the \susy{} particles. Since
the remaining degrees of freedom are the same as in the Standard Model
case of \sct{sec::eff}, the effective Lagrangian has exactly the same
form as in \eqn{eq::leffsm}, only with $\tilde C_1^{\rm SM}$ and $\tilde
C_2^{\rm SM}$ replaced by $\tilde C_1$ and $\tilde C_2$.

As in the \sm{} case, $\tilde C_2$ is zero through \nlo{}. It remains to
determine $\tilde C_1$ within the framework of \eqn{eq::lag} through
\nlo{} in $\alpha_s$.  The calculation of the \sm{} case in
Ref.\,\cite{Chetyrkin:1998mw} was done using dimensional regularization
(\dreg{}) as it is most convenient for loop calculations in {\abbrev
QCD}.  However, it is known that \dreg{} breaks \susy{} and thus the
formulae of Ref.\,\cite{Chetyrkin:1998mw} for the determination of
$\tilde C_1$ cannot be applied to the full Lagrangian of \eqn{eq::lag}
immediately.

A particularly subtle issue is the treatment of
$\gamma_5$. Ref.\,\cite{Chetyrkin:1998mw} adopted the method of
Ref.\,\cite{'tHooft:fi,Akyeampong:1973xi,Larin:1993tq} which requires
finite counter terms in order to restore gauge
invariance~\cite{Trueman:1979en}.  These counter terms are not known for
dimensionally regularized supersymmetry.

Therefore, we do not attempt to calculate the Feynman diagrams within
\dreg{}. Instead, we apply \dred{}~\cite{Siegel:1979wq} in order to
regularize the divergent integrals.  This is done by setting $D=4$ after
contracting all (Lorentz and spinor)
indices~\cite{Jack:1994bn,Jack:1997sr}. The loop integrals are
subsequently evaluated in $D=4-2\ep$ dimensions.

Both the pseudo-scalar vertex as well as the gluino-quark-squark
vertices involve $\gamma_5$ matrices. We anticommute these $\gamma_5$
matrices and use $\gamma_5^2=1$ until only one of them remains in the
Fermion trace. That one we replace
by~\cite{'tHooft:fi,Larin:1993tq,Chetyrkin:1998mw}
\begin{equation}
\begin{split}
\gamma_5 = \frac{i}{4!}\varepsilon_{\mu\nu\rho\sigma}
\gamma^{[\mu}
\gamma^{\nu}
\gamma^{\rho}
\gamma^{\sigma]}\,.
\end{split}
\end{equation}
It has been shown that the occurrence of the Levi-Civita symbol
$\varepsilon_{\mu\nu\rho\sigma}$ can lead to inconsistencies when
implemented in \dred{}~\cite{Siegel:1980qs}.  We circumvent them by
keeping the genuinely 4-dimensional Levi-Civita symbol
$\vep_{\mu\nu\rho\sigma}$ uncontracted until after the renormalization
procedure, in close analogy to
Refs.\,\cite{Larin:1993tq,Chetyrkin:1998mw}. As opposed to the
calculation in \dreg{}~\cite{Chetyrkin:1998mw}, however, working in
\dred{} by definition does not involve terms of $\order{\ep=2-D/2}$ that
require the above-mentioned finite counter terms in order to restore
gauge invariance.  Thus, the formulae of Ref.\,\cite{Chetyrkin:1998mw}
for projecting onto the coefficient function $\tilde C_1$ can be
translated to the \dred{} case by setting $D=4$ and ignoring the finite
renormalization of the pseudo-scalar current.\footnote{In the notation
of Ref.\,\cite{Chetyrkin:1998mw} this means $Z_5^p\equiv 1$.}

Clearly, the \sm{} diagrams alone will not yield the {\abbrev SM} result
of \eqn{eq::c1sm} in this way. However, we will explicitely demonstrate
that this result is recovered (through $\order{\alpha_s^3}$) by making
all the superpartner masses infinitely heavy {\it after} adding the
results for {\it all} diagrams.  This means that the {\abbrev SUSY}
diagrams lead to terms that do not vanish as $M_{\rm SUSY}\to
\infty$. Rather, they combine with the result for the {\abbrev SM}
diagrams to give the correct {\abbrev SM} result.

A rigorous proof of the validity of this approach through higher orders
is very desirable, of course, because it greatly simplifies precision
calculations in \susy{} models as they might be required by future
experimental data.



\section{Results}\label{sec::results}
The evaluation of the diagrams proceeds in complete analogy to
Ref.\,\cite{Harlander:2004tp}. At \nlo{}, one needs to evaluate massive
2-loop diagrams with vanishing external momenta. This is possible in a
fully analytic way with the help of the algorithm of
Ref.\,\cite{Davydychev:1992mt}\footnote{We are indepted to
M.~Steinhauser for providing us with his implementation of this
algorithm in the framework of {\tt MATAD}~\cite{Steinhauser:2000ry}.}. As
a cross check, we also calculated the diagrams in the limit $m_t\ll
\mstop{1}\ll \mstop{2}\ll \mgluino{}$ by using automated asymptotic
expansions~\cite{exp} and found agreement with the corresponding
expansion of the analytical result.

Subsequently, the bare coupling constant within the \susy{}-\qcd{}
theory is transformed to its renormalized 5-flavor \qcd{} expression in
the \msbar{} scheme as described in Ref.\,\cite{Harlander:2004tp}.

In \dred{}, and using the prescription for the treatment of $\gamma_5$
described above, we find the following contribution of all \sm{}
diagrams to the coefficient function:
\begin{equation}
\begin{split}
   \tilde C_1\big|_{\mbox{\scriptsize SM-dias}} &= -
   \frac{\alpha_s}{16\pi}\cot\beta\bigg\{
   1 - \frac{2}{3} \api \bigg\} + \order{\alpha_s^3}\,.
\end{split}
\end{equation}
As expected, it differs from the well-known result of
\eqn{eq::c1sm}. The \susy{} diagrams, on the other hand, add up to
\begin{equation}
\begin{split}
   \tilde C_1\big|_{\mbox{\scriptsize SUSY-dias}} &= -
   \frac{\alpha_s}{16\pi}\cot\beta\bigg\{ \frac{2}{3} \api \bigg\} +
   \order{\frac{m_t^2}{M^2}} + \order{\alpha_s^3}\,,\qquad
   M\in\{\mstop{},\mgluino\}\,,
\end{split}
\end{equation}
such that indeed the \sm{} limit is recovered as the \susy{} masses are
decoupled:
\begin{equation}
\begin{split}
\tilde C_1^{\rm SM} &= \lim_{M_{\rm SUSY}\to\infty}\left( \tilde
C_1\big|_{\mbox{\scriptsize SM-dias}} + \tilde C_1\big|_{
\mbox{\scriptsize SUSY-dias}}\right)\,.
\end{split}
\end{equation}
We are not aware of a calculation where this interplay between \dred{}
and $\gamma_5$ in a \susy{} theory has been observed before.

The general \nlo{} result for $\tilde C_1$ can then be cast into the
following form:
\begin{equation}
\begin{split}
\tilde C_1^{\rm SM} &= - \frac{\alpha_s}{16\pi}\cot\beta\bigg\{ 1 +
\api\,\frac{\muSUSY}{\mgluino}\left(\cot\beta + \tan\beta\right)\,
f(m_t,\mstop{1},\mstop{2},\mgluino)\bigg\} + \order{\alpha_s^3}\,.
\label{eq::c1res}
\end{split}
\end{equation}
A few observations may be worth pointing out:
\begin{itemize}
\item \eqn{eq::c1res} immediately shows that the \nlo{} corrections to
  the coefficient function will get more important w.r.t.\ the \lo{}
  term for large values of $\tan\beta$ and $\muSUSY$. On the other hand,
  large values of $\tan\beta$ increase the importance of bottom
  contributions even more, so that they usually obscure the squark
  effects. This will be shown explicitely for the {\abbrev SPS}\,1a
  scenario in \sct{sec::discussion}.
\item The squark mixing angle $\theta_t$, though present in individual
  diagrams, drops out in the final result.
\item Due to the absence of mass terms at \lo{}, the explicit form of
  $f$ is independent of the mass renormalization scheme.
\end{itemize}
The general result for $f(m_t,\mstop{1},\mstop{2},\mgluino)$ is too
voluminous to be displayed here. Instead, we implemented $\tilde C_1$ in
the program {\tt
evalcsusy.f}~\cite{Harlander:2004tp}\footnote{\label{foot::evalcsusy} The
program is available from {\tt
http://www-ttp.physik.uni-karlsruhe.de/Progdata/ttp04/ttp04-19.}} and
analyze its behavior numerically as shown in \sct{sec::discussion}.

For practical purposes, it might be useful to provide the analytical
expression of the function $f$ for some limiting
cases. For example, if all masses are set equal,
\begin{equation}
\begin{split}
m_t=\mstop{1}=\mstop{2} = \mgluino =: M\,,
\end{split}
\end{equation}
we obtain
\begin{equation}
\begin{split}
    f(M,M,M,M) = \frac{5}{18} - \frac{9}{2}\,S_2 = -0.894176\ldots\,,
\end{split}
\end{equation}
where
$S_2 = 4\,{\rm Cl}_2\left(\pi/3\right)/(9\sqrt{3})$
has been inserted.

On the other hand, assuming 
\begin{equation}
\begin{split}
m_t=\mstop{1}=\mstop{2} =: M \ll \mgluino\,,
\label{eq::limtst<g}
\end{split}
\end{equation}
we find
\newcommand{\ltg}{l_{g}}
\begin{equation}
\begin{split}
 f(M,M,M,\mgluino) &= \frac{2}{3} + \frac{2}{3}\,\ltg
 + x_g \left( \frac{7}{6} -
 \frac{1}{3}\,\zeta_2 + \frac{13}{6}\,\ltg - \frac{1}{6}\,\ltg^2 \right)
 + \order{x_g^2}\,,\\
&x_g = \frac{M^2}{\mgluino^2}\,,\qquad \ltg = \ln x_g\,,\qquad
\zeta_2 = \frac{\pi^2}{6}\,.
\label{eq::mt<mg}
\end{split}
\end{equation}
Insertion into \eqn{eq::c1res} shows that, in contrast to scalar Higgs
production~\cite{Harlander:2003bb,Harlander:2003kf,Harlander:2004tp},
the result for $\tilde C_1$ is well-behaved as $\mgluino\to \infty$ at
finite $M$. This is because there are no squark contributions at \lo{}
which could affect the renormalization of the \nlo{} terms.

For completeness, let us also give the result for the somewhat
unphysical case
\begin{equation}
\begin{split}
m_t\gg \mstop{1}=\mstop{2} =\mgluino =: M\,,
\label{eq::limt<g}
\end{split}
\end{equation}
where $f$ assumes the form
\newcommand{\ltM}{l_{t}}
\begin{equation}
\begin{split}
f(m_t,M,M,M) &=  \frac{3}{2} + \frac{3}{2}\,\ltM 
+ x_t\left( \frac{11}{3} + 3\,\zeta_2 + \frac{29}{3}\,\ltM +
\frac{3}{2}\,\ltM^2 \right) + \order{x_t^2}\,,\\
&x_t = \frac{M^2}{m_t^2}\,,\qquad
l_{t} = \ln x_t\,.
\end{split}
\end{equation}


\section{Discussion}\label{sec::discussion}

\subsection[Coefficient function $\tilde C_1$]{%
  Coefficient function \bld{\tilde C_1}}

We will now discuss the numerical effect of the newly evaluated terms.
First, we investigate the dependence of $\tilde C_1$ on the squark and
gluino masses. To this aim, it is convenient to consider
the leading and the \nlo{} term separately:
\begin{equation}
\begin{split}
\tilde C_1 &= -\frac{\alpha_s}{16\pi}\left( \tilde c_1^{(0)} +
\frac{\alpha_s}{\pi} \,\tilde c_1^{(1)} \right) + \order{\alpha_s^3}\,.
\label{eq::c1pert}
\end{split}
\end{equation}
According to \eqn{eq::c1res},
\begin{equation}
\begin{split}
\tilde c_1^{(0)} &= \cot\beta\,.
\end{split}
\end{equation}

\fig{fig::c1mgluino} shows $\tilde c_1^{(1)}$ as a function of $\mgluino$
for various values of $\mstop{1}$ and $\mstop{2}$. In
\fig{fig::c1mstop2}, on the other hand, we consider it as a function of
$\mstop{2}$ and fix $\mgluino{}$ and $\mstop{1}$ at a few representative
values. Note that $\tilde c_1^{(1)}$ is symmetric in $\mstop{1}$ and
$\mstop{2}$ since it does not depend on $\theta_t$ as mentioned
above. In \fig{fig::c1mstop}, $\mstop{1}=\mstop{2}=\mstop{}$ is varied
for certain choices of $\mgluino$. And finally, in \fig{fig::sps1ac1} we
adopt the benchmark scenario {\abbrev SPS}1a of
Ref.\,\cite{Allanach:2002nj} and consider $\tilde c_1^{(0)}$ and $\tilde
c_1^{(1)}$ as functions of $m_{1/2}$.  In this case, the corresponding
low energy parameters can be evaluated with the help of a \susy{}
spectrum
calculator~\cite{Allanach:2001kg,Porod:2003um,Djouadi:2002ze}.\footnote{We
use {\tt SoftSusy}~\cite{Allanach:2001kg} in this paper.}

The general structure is quite similar in all figures: $\tilde c_1^{(1)}$
is of the order of 0.5 for moderate values of the \susy{} masses. As
they increase, $\tilde c_1^{(1)}$ tends to zero in agreement with the
general discussion above. In order to demonstrate this behavior more
clearly, \fig{fig::c1log} adopts the same parameters as
\fig{fig::c1mgluino}, but extends up to very large values of the gluino
mass.


\FIGURE{%
  \psfrag{C}{ $\tilde{c}_1^{(1)}$}
  \psfrag{Mgluino}{ $m_{\tilde{g}}$/GeV}
  \includegraphics[width=0.7\textwidth]{abb11_1.eps}
\caption[]{\label{fig::c1mgluino}$\tilde c_1^{(1)}$ as a function of the
  gluino mass for $m_t=178$\,GeV, $\muSUSY = 150$\,GeV, $\tan\beta=10$,
  and $(\mstop{1},\mstop{2}) =
  (200,200)/(200,400)/(200,600)/(400,600)$\,GeV [long/\ldots/short
  dashes].}  }


\FIGURE{%
  \psfrag{C}{ $\tilde{c}_1^{(1)}$}
  \psfrag{Mstop2}{ $\mstop{2}$/GeV}
  \includegraphics[width=0.7\textwidth]{abb11_2.eps}
\caption[]{\label{fig::c1mstop2}$\tilde c_1^{(1)}$ as a function of
  $\mstop{2}$ for $m_t=178$\,GeV, $\muSUSY = 150$\,GeV, $\tan\beta=10$,
  $\mgluino = 400$\,GeV, and $\mstop{1} =
  200/300/400/600$\,GeV [long/\ldots/short
  dashes].}  }


\FIGURE{%
  \psfrag{C}{ $\tilde{c}_1^{(1)}$}
  \psfrag{M12}{ $\mstop{}$/GeV}
  \includegraphics[width=0.7\textwidth]{abb11_3.eps}
\caption[]{\label{fig::c1mstop}$\tilde c_1^{(1)}$ as a function of
  $\mstop{}\equiv \mstop{1}=\mstop{2}$ for $m_t=178$\,GeV, $\muSUSY =
  150$\,GeV, $\tan\beta=10$, $\mgluino = 200/400/600/800$\,GeV
  [long/\ldots/short dashes].}  }


\FIGURE{%
    \begin{tabular}{c}
      \includegraphics[bb=110 240 490 560,width=.7\textwidth]{%
	sps1ac1.ps}
    \end{tabular}
    \parbox{.9\textwidth}{
      \caption[]{\label{fig::sps1ac1}\sloppy
	\lo{} and \nlo{} contribution to the coefficient function
	$\tilde C_1$ as defined in \eqn{eq::c1pert} for the {\abbrev
	SPS}1a scenario.
        }}
}


\FIGURE{%
  \psfrag{C}{ $\tilde{c}_1^{(1)}$}
  \psfrag{Mgluino}{ $m_{\tilde{g}}$/GeV}
  \includegraphics[width=0.7\textwidth]{Logplot3.eps}
\caption[]{\label{fig::c1log}Same as \fig{fig::c1mgluino}, but extended
  abscissa in order to demonstrate the asymptotic
  behavior of $\tilde c_1^{(1)}$.}  }


\subsection{Cross section}
The inclusive \nlo{} hadronic cross section for pseudo-scalar Higgs
production receives contributions from the subprocesses $gg\to A(+g)$,
$qg\to Aq$, and $q\bar q\to Ag$.  At this order, squarks
only affect the process $gg\to A$, because of the antisymmetric structure
of the ${\tilde q}_i{\tilde q}_jA$ coupling, cf.\
Eqs.\,(\ref{eq::lags}), (\ref{eq::couplings}).
Recalling the notation of \eqn{eq::hadpart}, we write
\begin{equation}
\begin{split}
\hat\sigma_{gg}(x)
 &= \hat\sigma_{tb}(x)
+ \Delta\hat\sigma_{\tilde tt}(x)
+ \Delta\hat\sigma_{\tilde tb}(x)
+ \order{\alpha_s^4}\,,
\end{split}
\end{equation}
where $\hat\sigma_{tb}$ denotes the contributions arising from the top and
bottom mediated gluon-Higgs couplings. It can be evaluated through
\nlo{} for arbitrary top, bottom, and Higgs masses with the help of the
{\tt FORTRAN} program {\tt HIGLU}. The same is true for the $qg$
sub-process (see \fig{fig::lonlo}\,(e)).

$\Delta\hat\sigma_{\tilde tt}$ and $\Delta\hat\sigma_{\tilde tb}$ are the
effects arising from the interference of stop-induced with top- and
bottom-induced amplitudes:
\begin{equation}
\begin{split}
\Delta\hat\sigma_{\tilde tq} \sim \Re\left({\cal M}_{\tilde t}^{(1)\ast}
{\cal M}_q^{(0)}\right)\,,\qquad q\in\{b,t\}\,.
\end{split}
\end{equation}
${\cal M}_q^{(0)}$ is expressed in terms of ${\cal A}_q$ from
\eqn{eq::ftau}, while for ${\cal M}_{\tilde t}^{(1)}$, we use the
expression evaluated in the effective theory to obtain
\begin{equation}
\begin{split}
\Delta\hat\sigma_{\tilde tq}(x) &=\frac{\pi}{128v^2}
 \left(\api\right)^3\Re\left(\tilde c_1^{(1)}\,{\cal
 A}_q(\tau_q)\right)\,
 \delta(1-x)\,.
\end{split}
\end{equation}
Note that ${\cal M}_{\tilde t}^{(1)}$ has a branch cut at
$M_A=2m_t$. Thus, we expect our result to be valid for $M_A<2m_t$.
Recall, however, that in the \sm{} case, the heavy top limit still
provides an excellent approximation for Higgs masses much larger than
$2m_t$~\cite{Spira:1995rr,Spira:1997dg,Harlander:2003xy}.

To study the numerical effects, we consider the {\abbrev SPS}1a
scenario.  In a first step, the bottom quark Yukawa coupling will be
switched off for illustrational purposes. \fig{fig::sps1aktop} then
shows the \nlo{} K-factor with and without the top-stop interference
term $\Delta\hat\sigma_{\tilde tt}$.  The effect of the \susy{}
contributions is a decrease in the \nlo{} cross section by up to 10\%
for $M_A< 2m_t$, which is indeed significantly larger than for the case
of scalar Higgs production~\cite{Harlander:2004tp} due to the absence of
higher order \sm{} corrections to $\tilde C_1$.

Taking into account bottom quark effects overshadows most of this shift
as shown in \fig{fig::sps1ak}. On the one hand, this is due to the large
size of the pure bottom quark contributions. On the other hand, the
top-stop and bottom-stop interference terms are almost equal in
magnitude, but opposite in sign, as illustrated in \fig{fig::sps1abt}.
This means that, like in the case of scalar Higgs production, the higher
order corrections are accounted for fairly precisely by rescaling the
\lo{} result with the \sm{} K-factor. 

Finally, \fig{fig::sps1asig} shows the inclusive cross section through
\nlo{} for the {\abbrev SPS}\,1a scenario, including effects of top and
bottom quarks as well as top squarks.\footnote{The singularity at
$M_A=2m_t$ in Figs.\,\ref{fig::sps1aktop}, \ref{fig::sps1ak}, and
\ref{fig::sps1asig} is due to the threshold behavior of the \nlo{} quark
contribution~\cite{Spira:1995rr,Spira:1997dg}.}


\FIGURE{%
    \begin{tabular}{c}
      \includegraphics[bb=110 240 490 560,width=.7\textwidth]{%
	sps1aktop.ps}
    \end{tabular}
    \parbox{.9\textwidth}{
      \caption[]{\label{fig::sps1aktop}\sloppy \nlo{} K-factor, taking
	into account only top quark loops (dashed) and adding the
	top-stop interference contributions (solid).  }}
}


\FIGURE{%
    \begin{tabular}{c}
      \includegraphics[bb=110 240 490 560,width=.7\textwidth]{%
	sps1ak.ps}
    \end{tabular}
    \parbox{.9\textwidth}{
      \caption[]{\label{fig::sps1ak}\sloppy
	K-factor, taking into account top- and bottom-quark loops
	(dashed), and adding the top-stop and bottom-stop interference
	contributions (solid).
        }}
}


\FIGURE{%
    \begin{tabular}{c}
      \psfrag{entry1}{ $(\sigma_{tb} + \Delta\sigma_{\tilde tt})/\sigma_{tb}$}
      \psfrag{entry2}{ $(\sigma_{tb} + \Delta\sigma_{\tilde tb})/\sigma_{tb}$}
      \psfrag{entry3}{ $(\sigma_{tb} + \Delta\sigma_{\tilde tb} +
      \Delta\sigma_{\tilde tt})/\sigma_{tb}$}
      \includegraphics[bb=110 240 490 560,width=.7\textwidth]{%
	sps1abtx.ps}
    \end{tabular}
    \parbox{.9\textwidth}{
      \caption[]{\label{fig::sps1abt}\sloppy
	Relative effect of the top-stop (dotted) and bottom-stop (dashed)
	interference terms, as well as their sum (solid).
        }}
}


\FIGURE{%
    \begin{tabular}{c}
      \includegraphics[bb=110 240 490 560,width=.7\textwidth]{%
	sps1asig.ps}
    \end{tabular}
    \parbox{.9\textwidth}{
      \caption[]{\label{fig::sps1asig}\sloppy
	Total cross section including top, bottom, and stop effects
	through \lo{} (dashed) and \nlo{} (solid).
        }}
}



\section{Conclusions}\label{sec::conclusions}

The corrections to the effective Higgs-gluon coupling for pseudo-scalar
Higgs bosons have been evaluated in the \mssm{} through first order in
the strong coupling constant $\alpha_s$. The numerical effects were
studied within a specific \susy{} scenario ({\abbrev SPS}\,1a); further
studies can be performed easily using the publicly available numerical
routine {\tt evalcsusy.f}~(see footnote on page \pageref{foot::evalcsusy}).

The calculation also addresses a technical issue, since it involves the
$\gamma_5$ matrix in a non-trivial way. In analogy to
Ref.\,\cite{Chetyrkin:1998mw}, we avoided the contraction of the
Levi-Civita symbol with $D$ dimensional quantities. We argued that the
calculation in \dred{} does not require finite counter terms as opposed
to the \dreg{} approach, provided the underlying theory is
supersymmetric. Indeed, we observe a non-trivial cancellation of
regularization scheme dependent terms among the \sm{} and \susy{}
diagrams, such that the limit of an infinitely heavy \susy{} spectrum is
consistent with the well-known \sm{} result.

It would be interesting to investigate these observations in more
detail, in particular to prove the validity of our implementation of
$\gamma_5$ within \dred{} in a rigorous way. Further corroboration could
be obtained from its application at second order $\alpha_s$. This
corresponds to the evaluation of $\tilde C_1$ at three loops for which
the technical tools are in principle available~\cite{Harlander:1998dq}.

From the phenomenological point of view, one could calculate the
photonic decay rate of the {\abbrev CP}-odd Higgs boson in a very
similar fashion.  There, however, one does not need to rely strictly on
the effective Lagrangian, but could evaluate the first few terms of a
Taylor expansion in $M_A^2/M^2$ along the lines of
Ref.\,\cite{Steinhauser:1996wy}, where $M\in\{m_t,\mstop{},\mgluino\}$.


\paragraph{Acknowledgments.}
Our thanks go to K.~Chetyrkin, A.~Djouadi, and J.~K\"uhn for
encouragement and inspiring conversations. We are grateful to
M.~Steinhauser for enlightening discussions concerning the treatment of
$\gamma_5$, and D.~St\"ockinger for his constructive comments on the
manuscript.




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\end{document}
