%Title: Pseudo-scalar Higgs production at next-to-leading order SUSY-QCD
%Author: Robert V. Harlander and Franziska Hofmann
%Published: * JHEP * ** 0603 ** (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