%Title: Two-loop O(alpha_s^2) MSSM corrections to the pole masses of heavy quarks.
%Author: A. Bednyakov, A.Onishchenko, V. Velizhanin, O. Veretin
%Published: <EM> Eur.Phys.J. C </EM> <B> 29 </B> (2003) 87-101. 
%hep-ph/0210258
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\def\ep              {\varepsilon }
\def\MS              {$\overline{\rm MS}$ }
\def\DR              {$\overline{\rm DR}$ }
\def\gl              {{\mathrm g}}
\def\Lag             {{\cal L}}
\def\beqaa           {\begin{eqnarray}}
\def\eeqaa           {\end{eqnarray}}
\def\demu            {\partial_{\mu}}
\def\st              {\ti t}
\def\sb              {\ti b}
\def\ti              {\tilde}
\def\sg              {\ti \gl}
\def\sq              {\ti q}
\def\beq             {\begin{equation}}
\def\eeq             {\end{equation}}
\def\sth             {\sin\t}
\def\cth             {\cos\t}
\def\sst             {\sin\theta_{\ti t}}
\def\cst             {\cos\theta_{\ti t}}
\def\ssb             {\sin\theta_{\ti b}}
\def\csb             {\cos\theta_{\ti b}}
\def\ttb             {\theta_{\ti b}}
\def\Oas             {{\cal O}(\alpha_{s})}
\def\msg             {m_{\sg}}
\def\a               {\alpha}
\newcommand{\msq}[1]   {m_{\ti q_{#1} }}
\newcommand{\msqp}[1]  {m_{\ti q'_{#1} }}
\newcommand{\mst}[1]   {m_{\ti t_{#1} }}
\newcommand{\msb}[1]   {m_{\ti b_{#1} }}
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\newcommand{\TeV}{\unskip\,\mathrm{TeV}}
\newcommand{\sla}[1]{#1 \!\!\!/}
\def\btd {{\phantom .}\bar{\!\!\!\!\!{\phantom a}\tilde d}}
\def\btu {\!\!{\phantom l}\bar{\!\!\!\!\!{\phantom m}\tilde u}}
\def\btq {\!\!{\phantom a}\bar{\!\!\!\!{\phantom a}\tilde q}}
\newcommand{\z}{&\hspace*{-8pt}}
\newcommand{\eps}{\varepsilon}
\newcommand{\e}{\varepsilon}
\newcommand{\G}{\Gamma}
\begin{document}

\begin{center}
{\Large \bf  Two-loop ${\cal O}(\alpha_s^2)$ MSSM corrections
to the pole masses of heavy quarks}
\\ \vspace*{5mm} A.~Bednyakov$^a$, A.~Onishchenko$^{b,c,d}$,
V.~Velizhanin$^e$, O.~Veretin$^b$
\end{center}

\begin{center}
a) Moscow State University, Moscow, Russia \\
\vspace*{0.5cm}
b) Institut f\"ur Teoretische Teilchenphysik \\
Universit\"at Karlsruhe, D-76128 Karlsruhe, Germany \\
\vspace*{0.5cm}
c) Institute for High Energy Physics, Protvino, Russia \\
\vspace*{0.5cm}
d) Institute for Theoretical and Experimental Physics, \\
Moscow, Russia \\
\vspace*{0.5cm}
e) Bogoliubov Laboratory of Theoretical Physics, \\
Joint Institute for Nuclear Research, Dubna, Russia
\end{center}


\vspace*{0.5cm}


\begin{abstract}
We present results for two-loop MSSM corrections to the relation
between pole and running masses of heavy quarks up to the order
${\cal O}(\alpha_s^2)$. The running masses are defined in \DR
scheme, usually  used in multiloop calculations in supersymmetric
theories. An analysis of the value of these corrections in
different regions of the parameter space is provided.
\end{abstract}


\section{Introduction}

Recently  a lot of theoretical and experimental efforts were spent
to find some new physics beyond that described by Standard Model
(SM). One of the most popular ways to extend the SM is to make our
world supersymmetric at some higher scale, then those currently
tested by experimental facilities over the world. A way to do it
minimally and at the same time to have renormalizable local
quantum filed theory is to consider  $N=1$ SUSY field theory. It
is obtained from the SM by replacing SM fields with groups of
fields, forming representations of $N=1$ SUSY algebra, and
introducing additional Higgs doublet, required by supersymmetry.
This way we obtain the Minimal Supersymmetric Standard Model
(MSSM) \cite{Haber:1984rc,Nilles:1983ge,Barbieri:xf}.

A huge theoretical research in this direction was made in last
years as well as parameter space of this model was heavily
constrained by experimentalists. Today people consider radiative
SUSY corrections to SM precision observables and try to observe
SUSY particles directly in the experiment. The precision with
which masses of heavy quarks are known at present, requires to
take into account in addition to leading one-loop SUSY corrections
also higher order corrections.

It is the aim of present article to calculate the ``leading''
${\cal O}(\alpha_s^2)$ MSSM corrections to the pole masses of
heavy quarks and to see how they affect the SUSY particle spectra,
obtained from renormalization group analysis with universal
boundary conditions at GUT scale. In fact we will be interested in
the relation between the pole mass of heavy quark and the running
heavy quark \DR mass. The latter is defined as a quantity computed
in dimensional reduction and renormalized minimally. \footnote{\DR
scheme \cite{Siegel}, which is similar to \MS scheme, is usually
used for calculation in supersymmetric models, because contrary to
\MS scheme it manifestly preserves supersymmetry, at least for not
very high loop orders \cite{Avdeev:1982xy}. The details about
renormalization scheme used will be given below.} By the "leading"
corrections we mean the corrections, which are mediated by strong
interactions and formally they should be dominant. However, as it
is known from one-loop calculations, in the case of $b$-quark
there exists large contribution from stop-chargino loop enhanced
by $t$-quark Yukawa coupling, $\tan\beta$ or by supersymmetric
Higgs mass parameter $\mu$. So, in this sense, our results for
$b$-quark are not complete. But, they can be used to see the
typical size of these corrections and already make some
conclusions on whether they should be accounted for in accurate
analysis or not.

To compute needed diagrams we use the large mass expansion
procedure and for the moment we restrict ourselves only by terms
up to ${\cal O}(m_b^2/M_{\rm SUSY}^2)$ order in a case of
$b$-quark and up to ${\cal O}(m_t^2/M_{\rm SUSY}^2)$ order in the
case of $t$-quark. It is a good approximation for $b$-quark.
However, as our numerical analysis shows, such approximation does
not work good in the case of $t$-quark and higher terms in large
mass expansion should be taken into account. We suppose to
calculate missing corrections from stop-chargino loops to
$b$-quark pole mass as well as to supply more terms in the
expansion in a relation between top pole and \DR masses in one of
our next papers.

The paper is organized as follows. In section 2 we define
quantities we want to compute and make comments on the choice of
our renormalization scheme. In section 3 we describe our model ---
the supersymmetric QCD (a subset of MSSM, relevant to the
calculation of $\alpha_s^n$ corrections) and present in detail the
renormalization procedure. Section 4 contains our results and
numerical analysis of the effect, produced by our correction on
supersymmetric particle spectra. The latter were computed with a
help of SoftSUSY program \cite{SoftSUSY} under universal boundary
conditions at GUT scale. And, finally, section 5 contains our
conclusion.



\section{Pole mass and choice of renormalization scheme}

First, we would like to note that there are several quark mass
definitions. This is mainly due to the fact, that quarks were not
ever observed as free particles. Therefore the definition of their
masses rely heavily on theoretical constructions. Different
definitions exist refering to different renormalization schemes
used in quantum field theories. We will mention only a few most
popular --- \MS mass $\overline m$, pole mass $M$ and \DR mass
$m$. In the nonsupersymmetric QCD the relations between these
three masses are known (to two-loops between \DR and pole masses
\cite{Avdeev:1997sz} and to two
\cite{Gray:1990yh,Fleischer:1998dw} and to three
\cite{Chetyrkin:1999ys,Melnikov:2000qh} loops between \MS and pole
masses) and in supersymmetric QCD case exists only one-loop
relation between \DR and pole masses of heavy quarks
\cite{Hempfling:1993kv,Hall:1993gn,SUSY94&Wright,Donini95,Pierce:1996zz}.
In this work we will establish the two-loop relation between \DR
and pole masses in supersymmetric QCD.

In \DR scheme particle masses and couplings depend explicitly on
renormalization scale $\mu$, which is taken in applications equal
to characteristic scale of a studied process. The renormalization
scale dependence of these quantities could be conveniently
described by renormalization group equations. They are known very
well for MSSM and supersymmetric QCD. It should be noted, that \DR
mass is a short distance quantity. That is, it is sensitive only
to short distance effects. To describe processes with the
characteristic scale of the order of quark mass itself a different
on-shell mass definition used and here pole mass of the particle
comes into play.

Pole mass of the particle is defined by the singularity of
corresponding two-point function. As explicit perturbative
calculations showed, the pole mass of quark is an infrared finite
and gauge invariant quantity. For this reason it is considered as
a physically meaningful quantity \cite{else}.  We will restrict
ourselves to perturbation theory only and will not analyze the
exact nature of the two-point function singularity which may
involve some nonperturbative dynamics. In the present paper we
employ the definition of the pole quark mass, where it equals to
the value of $p$, at which inverse quark propagator turns to zero.

The pole mass $M$ of a quark is defined as a formal solution for
$\hat p$ (in the Minkowski metric) at which the reciprocal of the
connected full propagator equals to zero:
\begin{equation}
\hat p - m - \Sigma(\hat p, m) = 0,
\label{pole}
\end{equation}
%
where $\Sigma(\hat p, m) = m \Sigma_1 (p^2,m)
+ (\hat p - m)\Sigma_2 (p^2,m)$ is the
one-particle-irreducible two-point function, $m$ may stand for the
bare or renormalized mass, $m_{\rm bare}$ or $m_{\rm ren}$,
depending on the prescription used in evaluating
$\Sigma$. The solution to eq.~(\ref{pole}) is sought order by order in
the perturbation theory. To two loops we have
%
\begin{equation}
M = m +\Sigma^{(1)}(m,m) +\Sigma^{(2)}(m,m)
+ \Sigma^{(1)}(m,m)~ \Sigma^{(1)'}(m,m) + {\cal O}(\Sigma^{(3)}),
\label{2loop}
\end{equation}
%
where $\Sigma^{(L)}$ is the $L$-loop contribution to $\Sigma$, and
the prime denotes the derivative with respect to the first
argument. In what follows, we will be interested in the relation
between pole quark mass $M$ and running \DR mass $m$, computed in
MSSM up to ${\cal O}(\alpha_s^2)$ order. Technically, to solve
this problem, we need to evaluate $\approx 200$ two-loop
propagator type diagrams involving many different mass scales (see
Fig.~\ref{twoloopdiag}). In general, it is quite complex problem to be solved
exactly. However, in our case all mass scales  can be divided in
two groups denoted by $m_{\rm soft}$ and $m_{\rm hard}$, such that
scales from $m_{\rm soft}$ much less then each of the scales from
$m_{\rm hard}$ group:

\begin{center}
\begin{tabular}{lcc}
& $m_{\rm soft}$ & $m_{\rm hard}$ \\
case of $b$-quark: & $m_b$ & $m_t$, $m_{\tilde q}$, $m_{\sg}$ \\
case of $t$-quark: & $m_t$ &  $m_{\tilde q}$, $m_{\sg}$
\end{tabular}
\end{center}
\noindent
Here $m_{\sg}$ is gluino mass, $m_{\tilde q}$ stands
for different squark masses. We also explicitly denoted
which mass scales belong to soft or hard groups in the cases
of $b$- and $t$-quarks.

  Given this mass hierarchy one can employ the large mass expansion
procedure to reduce evaluation of multi-scale two-loop integrals
to the calculation of single scale on-shell two-loop integrals,
two-loop tadpole integrals with two scales and products of
one-loop on-shell integrals and one-loop tadpole diagrams. To
compute two-loop single scale on-shell integrals we made use of
ONSHELL2 package \cite{onshell2}. For the evaluation of two-loop
tadpole diagrams with two scales the recurrence relations of
\cite{Davydychev} were used. Calculation of one-loop
self-energies, including their derivatives with respect to
momentum is an easy task and we will not make further comments on
it.

Now let us make some comments on the choice of the renormalization
scheme. Here we use the same renormalization prescription, as that
used by authors of \cite{Avdeev:1997sz}. To be more specific, we
use regularization by dimensional reduction, a modification of the
conventional dimensional regularization, originally proposed in
\cite{Siegel}. In this procedure the vector and spinor algebra in
the numerator of Feynman diagrams are four-dimensional, which is
the requirement imposed by supersymmetry. However, to make sense
from divergent momentum integrals we need to regularize them by
non-integer spacetime dimension $d=4-2\varepsilon$. In order for
quantum corrections not to break gauge invariance we need a
cancellation of squared momenta both in numerator and denominator
of Feynman integrals. Thus, the momenta should form a
$d$-dimensional subspace in four dimensions. As the momenta become
$d$-dimensional, four-vectors naturally split into true
$d$-vectors and so-called $\varepsilon$-scalars, obtained in the
process of dimensional reduction of original four-dimensional
lagrangian. The $\varepsilon$-scalars is nothing, but matter
fields. Their appearance is the only difference from the
conventional dimensional regularization.

We would like to note, that in general renormalization of
$\varepsilon$-scalars and their interactions is not identical to
that of vectors and so the original four-covariance may be spoiled
by quantum corrections. Moreover, quantum corrections may also
generate a mass for $\varepsilon$-scalars. There is an
arbitrariness in choosing the renormalization scheme for this mass
\cite{jones}. A consistent way is to choose the finite
$\varepsilon$-scalar mass counterterm so that the pole (and
renormalized) mass of the $\varepsilon$-scalars equals zero.  The
$\varepsilon$-scalar field renormalization is left minimal. Other
renormalizations are done minimally, that is by subtracting only
poles in $\varepsilon$.


\section{Definition of the model and renormalization procedure}

For our purposes here we do not need the complete MSSM but only
its part --- supersymmetric extension of QCD (SUSY QCD).
In the superfield formulation SUSY QCD lagrangian
consists of two parts: the rigid supersymmetric part and the part containing
soft supersymmetry breaking terms.
\begin{eqnarray}
{\cal L}_{\rm rigid} &=& \int d^2\theta \, \frac{1}{2}\,{\rm  Tr}
W^{\alpha}W_{\alpha} + \int d^2\bar{\theta} \, \frac{1}{2}\,{\rm  Tr}
\bar{W}_{\dot \alpha}\bar{W}^{\dot \alpha} \label{rigidlag} +
\int d^2\theta \,{\cal W} + \int d^2\bar{\theta} \, \bar{\cal W} \nonumber\\
&+& \int d^2\theta \,d^2\bar{\theta} \,
\left[\bar Q e^{2g_3T^a V^a}Q +
\bar U e^{-2g_3(T^a)^* V^a}U +
\bar D e^{-2g_3(T^a)^* V^a}D\right] , %+
\end{eqnarray}
%
where $Q=Q_L$, $U=U_R$, $D=D_R$,
$T^a=\lambda^a/2$ ($a$=1,...,8) with $\lambda^a$ being the Gell-Mann matrices
\footnote{Note that U and D are colour anti-triplets
      and the anti-colour generator is $-(T^a)^*$}
 and the gauge field strength tensors are
 $$W_{\alpha}=-\frac {1}{8g_3}\bar D^2e^{-2g_3 T^a V^a}D_\alpha
    e^{2g_3 T^a V^a},
   \ \ \ \bar W_{\dot \alpha}=-\frac {1}{8g_3} D^2e^{2g_3 T^a
V^a}\bar D_{\dot \alpha} e^{-2g_3 T^a V^a}$$
%

The superpotential of the ordinary SUSY QCD contains only
quadratic mass terms for chiral (antichiral) superfields.
Considering SUSY QCD as a part of MSSM, where all masses of
particles are generated through the Higgs mechanism, the
superpotential takes the following form
($\epsilon^{12}=\epsilon_{21}
=1$)
\begin{equation}
{\cal  W}=\epsilon^{ij}\left[y_d D H_1^i Q^j  +
y_u U H_2^j Q^i +\mu H_1^j H_2^i \right],\label{rigidsuppot}
\end{equation}
where $y_d$ and $y_u$ are Yukawa coupling constants carrying the
generation indices, which have been suppressed (as well as group
indices). In this work we will need the Higgs fields only to
generate masses of our particles and will not be interested in the
detail structure of MSSM Higgs sector.

To  perform the supersymmetry breaking, that satisfies the requirement of
"softness", we introduce a gluino mass term,
soft masses of  scalar superpartners of quarks and
soft trilinear couplings with Higgs fields,
which may be written in terms of  $N=1$ superfields, provided
one introduces external spurion superfields~\cite{spurion}
\begin{eqnarray}
{\cal L}_{\rm soft} &=&
-\frac 12\int d^2\theta \, 2 M_3\theta^2 {\rm Tr}W^{\alpha}W_{\alpha}
- \frac 12\int d^2\bar{\theta} \, 2\bar{M}_3\bar{\theta}^2 {\rm
Tr}\bar{W}_{\dot \alpha}\bar{W}^{\dot \alpha}  \nonumber \\
&-&\int d^2 \theta \, d^2\bar{\theta}
\left[M^2_{\tilde q}\bar Q e^{2g_3 T^a V^a} Q
+ M^2_{\tilde u}\bar U e^{-2g_3 (T^a)^* V^a} U
+ M^2_{\tilde d}\bar D
e^{-2g_3 (T^a)^* V^a} D \right] \theta^2 \bar{\theta}^2 \nonumber \\
&-&\int d^2\theta \,
    \epsilon^{ij}
\left[y_d A_d D H_1^i Q^j  +
y_u A_u U H_2^j Q^i \right]\theta^2 + {\rm h.c.}
\label{lagsoft}
\end{eqnarray}

The chiral superfields have the following component fields
representations
\begin{eqnarray}
Q&=&\tilde q_L-i\,\theta\sigma^\mu
\bar\theta\partial_\mu\tilde q_L-\frac14
\theta\theta\bar\theta\bar\theta
\partial^\mu\partial_\mu\tilde q_L+\sqrt{2}\theta q_L-\frac{i}{\sqrt{2}}
\theta\theta\bar\theta\bar\sigma^\mu\partial_\mu q_L+
\theta\theta F_q \,,\\
U&=& \btu_R-i\,\theta\sigma^\mu
\bar\theta\partial_\mu \btu_R-\frac14
\theta\theta\bar\theta\bar\theta
\partial^\mu\partial_\mu \btu_R+\sqrt{2}\theta u_R-
\frac{i}{\sqrt{2}}
\theta\theta\bar\theta\bar\sigma^\mu\partial_\mu u_R+
\theta\theta F_u \,,\\
D&=& \btd_R-i\,\theta\sigma^\mu
\bar\theta\partial_\mu \btd_R -\frac14
\theta\theta\bar\theta\bar\theta
\partial^\mu\partial_\mu \btd_R +\sqrt{2}\theta d_R-
\frac{i}{\sqrt{2}}
\theta\theta\bar\theta\bar\sigma^\mu\partial_\mu d_R+
\theta\theta F_d
\end{eqnarray}
%
and for the vector superfields in Wess-Zumino gauge we have
\begin{eqnarray}
V^a&=&\theta\sigma^\mu\bar\theta
G^a_\mu+\theta\theta\bar\theta{\bar{\tilde \gl}}^a+\bar\theta\bar\theta\theta{\tilde \gl}^a
+\frac12\theta\theta\bar\theta\bar\theta D^a.
\end{eqnarray}
Then, rewriting everything in the component fields, the lagrangian
of SUSY QCD takes the form ($q_R=(u_R,d_R)$)
%
\begin{eqnarray}
{\cal L}&=&-\frac 14 G^a_{\mu\nu}G^{a\;\mu\nu}
+i{\bar {\tilde \gl}}^a \bar \sigma^\mu D_\mu {\tilde \gl}^a
+i\bar q_R \bar \sigma^\mu D_\mu q_R
+i\bar q_L \bar \sigma^\mu D_\mu q_L\nonumber\\
&+& \left(D^\mu{\tilde q}_R\right)^\dagger D_\mu  {\tilde q}_R
+\left(D^\mu{\tilde q}_L\right)^\dagger D_\mu  {\tilde q}_L
-g_3\btq_R T^a D^a {\tilde q}_R
+g_3\btq_L T^a D^a {\tilde q}_L
\nonumber\\
&+&\sqrt{2}  g_3{q}_R T^a {\tilde \gl}^a {\tilde q}_R
-\sqrt{2}  g_3{\bar q}_L T^a \bar{\tilde \gl}^a {\tilde q}_L
+\sqrt{2}  g_3\btq_R T^a \bar{\tilde \gl}^a { \bar q_R }
-\sqrt{2}  g_3\btq_L T^a {{\tilde \gl}}^a q_L
\nonumber\\
&+&\frac 12 D^a D^a
+\frac 12 D^i D^i
+\frac 12 D' D'
+g_2\btq_L \frac{\sigma^i}{2} D^i {\tilde q}_L
+g_2\bar h_j \frac{\sigma^i}{2} D^i h_j
+g_1\bar h^0_j \frac{{Y}_{h_j}}{2} D' h^0_j
\nonumber\\
&+&g_1\btq_L \frac{{Y}_{{\tilde q}_L}}{2} D' {\tilde q}_L
-g_1\btq_R \frac{{Y}_{{\tilde q}_R}}{2} D' {\tilde q}_R
+ \bar F_{h^0_1}F_{h^0_1}
+ \bar F_{h^0_2}F_{h^0_2}
+ \bar F_{q_L}F_{q_L}
+ \bar F_{q_R}F_{q_R}
\nonumber\\
&-&\left[\mu h^0_1 F_{h^0_2}
+\mu F_{h^0_1} h^0_2
-y_d \btd_R F_{h^0_1} \tilde d_L
-y_u \btu_R F_{h^0_2} \tilde u_L\right .
\nonumber\\
&-&\left . y_d(F_{d_R} h^0_1 \tilde d_L+ \btd_R h^0_1 F_{d_L})
-y_u(F_{u_R} h^0_2 \tilde u_L+ \btu_R h^0_2 F_{u_L})\right .
\nonumber\\
&+&\left . y_d d_R h_1^0 d_L
+y_u u_R h_2^0 u_L
+y_d A_d {\tilde d}_R h_1^0 {\tilde d}_L
+y_u A_u {\tilde u}_R h_2^0 {\tilde u}_L
+ {\rm h.c.} \right]
\nonumber\\
&-&\frac{M_3}{2}{\tilde \gl}^a {\tilde \gl}^a
-\frac{M_3}{2}\bar{\tilde \gl}^a \bar{\tilde \gl}^a
-{M}^2_{\tilde q}\btq_L{\tilde q}_L
-{M}^2_{\tilde u}\btu_R{\tilde u}_R
-{M}^2_{\tilde d}\btd_R{\tilde d}_R
\nonumber,\end{eqnarray}
%
where gauge covariant derivative and field strength are defined as
\begin{eqnarray}
D_\mu & = & \partial_\mu+i g_3 T^a G^a_\mu \,, \nonumber\\
G^a_{\mu\nu} & = & \partial_\mu G^a_\nu - \partial_\nu G^a_\mu
-g_3 f^{abc}G^b_\mu G^c_\nu \,. \nonumber
\end{eqnarray}
%
Here we also added auxiliary $SU(2)\times U(1)$ fields $D^i$ and $D'$,
which are necessary to describe correctly the mass generation mechanism.
$\sigma^i$ denote  Pauli matrices and $Y_q$ is a hypercharge of a particle $q$.
Eliminating auxiliary fields $F_q$, $\bar F_q$, $D^a$, $D^i$
and $D'$ and introducing four-component spinor notation
similar to~\cite{Haber:1984rc,Kuroda} one gets
%
\begin{eqnarray}
{\cal L}&=&-\frac 14 G^a_{\mu\nu}G^{a\;\mu\nu}
+\frac {i}{2}{\bar {\tilde \gl}}^a \gamma^\mu D_\mu {\tilde \gl}^a
+i\bar q\gamma^\mu D_\mu q
+\left(D^\mu{\tilde q}_R\right)^\dagger D_\mu  {\tilde q}_R
+ \left(D^\mu{\tilde q}_L\right)^\dagger D_\mu  {\tilde q}_L
\nonumber\\
&+&\sqrt{2}g_3{\bar q}P_L T^a {\tilde \gl}^a {\tilde q}_R
-\sqrt{2}g_3{\bar q}P_R T^a {\tilde \gl}^a {\tilde q}_L
+\sqrt{2}g_3\btq_R T^a \bar{\tilde \gl}^a P_R q
-\sqrt{2}g_3\btq_L T^a \bar{\tilde \gl}^a P_L q
\nonumber\\
&-&\frac{g_3^2}{2}(\btq_L T^a {\tilde q}_L
 -\btq_R T^a {\tilde q}_R)
 (\btq_L T^a {\tilde q}_L
 -\btq_R T^a {\tilde q}_R)
-\frac{M_3}{2}\bar{\tilde \gl}^a {\tilde \gl}^a
+{\cal L}_{M} \,,
\label{lagsqnd} \\
{\cal L}_{M}&=&
-{M}^2_{\tilde q}\btq_L{\tilde q}_L
-{M}^2_{\tilde u}\btu_R{\tilde u}_R
-{M}^2_{\tilde d}\btd_R{\tilde d}_R
- y_d h^0_1 \bar d d
- y_u h^0_2 \bar u u
\label{lagsqndm} \nonumber\\
&-&\left(\frac{g_2^2}{2} I^3_{\tilde q_L}\btq_L\tilde q_L
- \frac{g_1^2}{4} {Y}_{\tilde q_L}\btq_L\tilde q_L
+ \frac{g_1^2}{4} {Y}_{\tilde q_R}\btq_R\tilde q_R\right)
\left(\bar h^0_1h^0_1 - \bar h^0_2h^0_2 \right)
\nonumber\\
&-& y_d^2\left(h^0_1\tilde d_L\right)^\dagger h^0_1\tilde d_L
- y_d^2 \bar h^0_1 h^0_1 \btd_R \tilde d_R
- y_u^2\left(h^0_2\tilde u_L\right)^\dagger h^0_2\tilde u_L
- y_u^2 \bar h^0_2 h^0_2 \btu_R \tilde u_R
\nonumber\\
&-&
\left[
y_d h^0_1A_d \btd_R {\tilde d}_L
+y_u h^0_2 A_u \btu_R {\tilde u}_L
-y_d\mu h^0_2 \btd_R\tilde d_L
-y_u\mu h^0_1 \btu_R\tilde u_L + {\rm h.c} \right]
\label{lagsqndmnot}
\end{eqnarray}
%
where $P_L=(1-\gamma_5)/2$ and $P_R=(1+\gamma_5)/2$ are the chiral
and antichiral projectors.


After Higgsing and taking into account that
\begin{eqnarray}
&&m_d= y_d v_1,\ \ \ \ \ m_u= y_u v_2,\ \ \ \ \
\tan \beta =\frac{v_2}{v_1}, \ \ \  \ \
\sin^2\theta_W=\frac{g_1^2}{g_1^2+g_2^2},
\nonumber\\
&&m_Z^2=\frac 12 (g_1^2+g_2^2)(v_1^2+v_2^2)
=\frac 12 \frac{g_1^2}{\sin^2 \theta_W}(v_1^2+v_2^2),\nonumber\\
&&\cos 2\beta = \frac{v_1^2-v_2^2}{v_1^2+v_2^2},\ \ \ \ \
e_q=\frac{Y_q}{2}+I^3_q
\nonumber,
\end{eqnarray}
it is convenient to rewrite ${\cal L}_M$ from eq.~(\ref{lagsqndm})
in the following form \cite{Ellis:1983ed}
\begin{equation}
{\cal L}_M=-m_q \bar q q
- m^2_{{\tilde q}_L} \btq_L {\tilde q}_L
- m^2_{{\tilde u}_R} \btu_R {\tilde u}_R
- m^2_{{\tilde d}_R} \btd_R {\tilde d}_R
- a_q m_q \btq_L {\tilde q}_R
- a_q m_q \btq_R {\tilde q}_L
\end{equation}
with
\begin{eqnarray}
&&  \msq{L}^2 = M^2_{\ti q}
 + m_Z^2 \cos 2\beta\,(I^3_{{\tilde q}_L} - e_q\sin^2\theta_W) + m_q^2 \,,
\nonumber\\
&&m_{{\tilde u}_{R}}^2=M^2_{\ti u}+e_u\,m_Z^2\cos 2\beta\,\sin^2\theta_W+
m_u^2 \,,
\qquad a_u = A_u - \mu \, \cot\beta
\nonumber\\
&& m_{{\tilde d}_{R}}^2=M^2_{\ti d}+e_d\,m_Z^2\cos 2\beta\,\sin^2\theta_W+
m_d^2 \,,
\qquad a_d = A_d- \mu \,  \tan\beta
\nonumber
\end{eqnarray}
where $e_q$ and $I^3_q$ are the electric charge and the third
component of the weak isospin of a particle $q$, $m_q$ is the mass
of the partner quark and $\mu$ is a Higgs mass parameter. $M_{\ti
q}$, $M_{\ti u}$, and $M_{\ti d}$ are soft SUSY breaking masses,
$A_u$ and $A_d$ are trilinear couplings as in eq.~(\ref{lagsoft}).
Family indices have been suppressed.

Now we must diagonalize the mass matrix ${\cal M}^2_{\tilde q}$ of
 squarks to determine the physical mass eigenstates
\begin{equation}
  {\cal M}_{\sq}^2 =
  \left( \begin{array}{cc}  \msq{L}^2 & a_q m_q \\
                            a_q m_q   & \msq{R}^2
  \end{array}\right) \;=\;
  ({\cal R}^{\sq})^\dagger \left( \begin{array}{cc}  \msq{1}^2 & 0 \\
                                          0         & \msq{2}^2
                \end{array}\right) {\cal R}^{\sq} \,.
\label{msqmat}
\end{equation}

According to eq.~(\ref{msqmat}) ${\cal M}_{\sq}^2$ is diagonalized by an
unitary matrix ${\cal R}^{\sq}$. Assuming that CP violating phases
occur only in the CKM matrix, we choose ${\cal R}^{\sq}$ to be real.
The weak eigenstates $\sq_L^{}$ and $\sq_R^{}$ are
related thus to their mass eigenstates $\sq_1^{}$ and $\sq_2^{}$ by
%
\begin{equation}
  {\sq_1^{} \choose \sq_2^{}} = {\cal R}^{\sq}\,{\sq_L^{} \choose \sq_R^{}},
  \hspace{8mm}
{\cal R}^{\sq}=\left(\begin{array}{rr}\cos\theta_{\sq} &\sin\theta_{\sq} \\
       -\sin\theta_{\sq} & \cos\theta_{\sq} \end{array}\right) ,
\label{squarkrotation}
\end{equation}
with $\theta_{\sq}$ being the squark mixing angle.
The mass eigenvalues are given by
\begin{equation}
  m^2_{\sq_{1\!,2}} = \frac12 \left( \msq{L}^2 + \msq{R}^2
  \mp \sqrt{(\msq{L}^2 - \msq{R}^2)^2 + 4\, a^2_q m_q^2 } \,\right).
%\label{eq:sqmasseigenvalues}
\end{equation}
By convention, we choose $\sq_1^{}$ to be the lightest mass
eigenstate. Notice, that $\msq{1}\leq\msq{L,R}\leq\msq{2}$.

  For the mixing angle $\theta_{\sq}$ we require $0\leq\theta_{\sq} <\pi$.
Then one can write
\begin{equation}
  \cos\theta_{\sq} =
  \frac{- a_q m_q}{\sqrt{(\msq{L}^2-\msq{1}^2)^2 + a_q^2 m_q^2}},
  \hspace{8mm}
  \sin\theta_{\sq} =
  \frac{\msq{L}^2-\msq{1}^2}
       {\sqrt{(\msq{L}^2-\msq{1}^2)^2 + a_q^2 m_q^2}}.
\end{equation}
Moreover, $|\cos\theta_{\sq}| > \frac{1}{\sqrt 2}\,$ if
$\msq{L}<\msq{R}$ and
$|\cos\theta_{\sq}| < \frac{1}{\sqrt 2}\,$ if $\msq{R}<\msq{L}$.

Making transition in eq.~(\ref{lagsqnd}) from ${\tilde q}_R$ and
${\tilde q}_L$ to ${\tilde q}_1$ and ${\tilde q}_2$ with the help
of the matrix from eq.~(\ref{squarkrotation}) we get for
quark-gluino-squark trilinear and four-squarks vertices
complicated expressions due to squark mixing, which we do not
present here. To produce input amplitudes of the diagrams needed
in our calculations we made use of Mathematica package FeynArts
\cite{FeynArts}, which have SUSY QCD as part of MSSM model
\cite{FAMSSM}. The Feynman diagrams contributing to quarks
self-energies in one and two loop order are shown in
Fig.~\ref{twoloopdiag}.

\begin{center}
\begin{figure}[t]%\vspace*{3cm}
\hspace*{1.5cm}
\includegraphics[scale=0.72]{diag1.eps}
\caption{Feynman diagrams for ${\cal O}(\alpha_s^2)$ contribution
to quark self-energies
(last two diagrams at the one loop level of order ${\cal O}(\alpha_s)$).
The diagrams in the first line are the pure QCD contribution.}
\label{twoloopdiag}
\end{figure}
\end{center}

\vspace*{-12mm} Note again, that doing these calculations we work
in the minimal subtraction scheme \MS adopted to supersymmetry ---
dimensional reduction with minimal subtraction \DR. To obtain a
relation between pole and \DR quark masses we need to know  \DR
renormalization constants for quark, gluino, squark,
$\varepsilon$-scalar masses,
gauge and Yukawa charges%
    \footnote{By Yukawa charge we mean
              the Yukawa coupling of $\varepsilon$-scalar to fermions}
as well as renormalization of squark mixing angle. We will
describe evaluation of each of these quantities in next
subsections.


\subsection{Renormalization of QCD sector}

First we have to renormalize the charge and the masses of quarks
as well as fields. The relations between bare and renormalized
parameters are given by
%
\begin{eqnarray}
  \alpha_{s,0} &=& \mu^{\varepsilon/2} Z_{\alpha_s} \alpha_s \,, \\
  m_{q,0} &=& Z_{m_q} m_q \,, \\
  G_0 &=& Z_G^{1/2} G \,, \\
  q_0 &=& Z_q^{1/2} q \,,
\end{eqnarray}
%
where $\mu$ is a renormalization scale.

  The charge renormalization constant can be obtained from the
renormalization constants of the fields and vertex with the use of
the following relation:
%
\begin{eqnarray}
  Z_{\alpha_s} = Z^2_{qGq}\, Z^{-2}_{q} \, Z^{-1}_G \,, \label{renqgq}
\end{eqnarray}
%
where $Z_q$ and $Z_G$ correspond to renormalizations of quark and
gluon fields and $Z_{qGq}$ renormalizes the quark-gluon
interaction vertex.

One can also renormalize the gauge parameter%
    \footnote{
              Gauge parameter $\xi$ is defined in such a way that $\xi=0$ and
              $\xi=1$ correspond to Landau and Feynman gauge respectively.
             }
with the help of introduced above constant $Z_G$
%
\begin{equation}
\xi_0= Z_G\xi \,.
\label{xirenorm}
\end{equation}
%
However in the gauge invariant quantities (like e.g. the pole
mass) the gauge parameter drops out already from the bare
expression and renormalization (\ref{xirenorm}) is not needed.

  Our results for renormalization constants
in \DR scheme read
\begin{eqnarray}
Z_{m_q} &=& 1 - C_F\frac{\alpha_s}{2\pi}  \frac{1}{\varepsilon}
   - C_F\left(\frac{\alpha_s}{4\pi} \right)^2
\left( \left(6- 3C_A - 2C_F
%\frac{11}{2} C_A - 2t n_f+\frac{9}{2} C_F
       \right)
 \frac{1}{\varepsilon^2}
    + \left(-6 + 3C_A - 2C_F
%\frac{97}{12} C_A-\frac{5}{3} t n_f+\frac{3}{4} C_F
      \right)
 \frac{1}{\varepsilon} \right) ,  \label{zm2} \\
Z_{q} & = & 1 - \frac{\alpha_s}{4\pi}C_F
\left( \xi + 1\right)\frac{1}{\varepsilon}\label{zquark} , \\
Z_{G} & = & 1 - \frac{\alpha_s}{4\pi}
\left( 6 - \frac{3-\xi}{2} C_A \right) \frac{1}{\varepsilon} , \label{zbg}\\
Z_{qGq} & = & 1 - \frac{\alpha_s}{4\pi}
       \left( (1 +\xi) C_F
    + \frac{3+\xi}{4} C_A \right) \frac{1}{\varepsilon} , \label{zqgq}\\
Z_{\alpha_s} &=&1-\frac{\alpha_s}{4\pi} \left( 3 C_A-6 \right)
    \frac{1}{\varepsilon} .  \label{zg}
\end{eqnarray}
For $SU(N_c)$ we have $C_F=(N_c^2-1)/(2N_c)$ and $C_A=N_c$. Here
we explicitly put the number of quark flavors to six.

There are several possibilities to check the correctness of our
results. The coupling renormalization constant (\ref{zg}) can be
compared with that, obtained from the one-loop MSSM beta function
for $SU(3)$ group. We have also checked, that in the case of
ordinary QCD we recover renormalization constants given in
\cite{Avdeev:1997sz}. The 2-loop \DR quark mass renormalization
constant (\ref{zm2}) has been obtained from eq. (\ref{2loop}) and
contains poles in $\varepsilon$ remaining after all other
renormalizations were done. Here we mean, that one should perform
one-loop renormalizations of strong coupling constant and particle
masses. A subtle point is the renormalization of Yukawa
interaction of $\eps$-scalars and quarks. In SUSY QCD discussed
here, one do not need to introduce separate coupling constant $Y$
for this interaction, as supersymmetry insures that it
renormalizes in the same way as strong coupling
constant\footnote{See section about $\eps$-scalar sector
renormalization for details}. However, to make our renormalization
procedure as general as possible ( and also applicable in a case
of nonsupersymmetric QCD for checks) we introduce additional
independent constant $Y$ and renormalize it separately. So, here
we follow renormalization procedure described in
\cite{Avdeev:1997sz}. Renormalizing a physical quantity, like pole
mass, we ignore field renormalization constants, which is valid
procedure, provided we are evaluating the contributions of
counterterms for the whole sum of diagrams. $1/\varepsilon^2$
poles in $Z_{m_q}$ can be checked with the use of renormalization
group technique. Indeed, let us write
%
\begin{eqnarray}
m_0 &=& m \left(1+\frac{\alpha_s}{4\pi}\frac{Z^{(1,1)}}{\ep}
+ \left( \frac{\alpha_s}{4\pi} \right)^2 \frac{Z^{(2,1)}}{\ep}
+ \left( \frac{\alpha_s}{4\pi} \right)^2 \frac{Z^{(2,2)}}{\ep^2}
\right) \nonumber \\
&=& m \left(
1+\frac{z^{(1)}}{\ep}+\frac{z^{(2)}}{\ep^2} + \ldots
\right),
\end{eqnarray}
%
where
$
z^{(n)} = \sum_{l=1}^\infty(\alpha_s/4\pi)^l Z^{(l,n)} $. Now
using renormalization group equation\footnote{In fact $m_0$ is not
renormalization group invariant in our renormalization scheme.
However, to check leading poles in $\eps$ we can treat it as RG
invariant, as different prescriptions used to renormalize
$\eps$-scalar sector give different results only at subleading
order in $\eps$. This point will be discussed in detail in section
concerning $\eps$-scalar sector renormalization.}
%
\begin{eqnarray}
0 = \mu^2\frac{d}{d\mu^2} m_0 \nonumber
\end{eqnarray}
%
we obtain
%
\begin{eqnarray}
\gamma &=& \frac{1}{2}g\frac{\partial}{\partial g}z^{(1)} \,, \nonumber \\
\left(\gamma + \beta_g\frac{\partial}{\partial g} +
\sum_i\gamma_i m_i \frac{\partial}{\partial m_i}
\right)z^{(n)} &=& \frac{1}{2}g\frac{\partial}{\partial g}z^{(n+1)} \,,
\end{eqnarray}
%
($g$ is related to $\alpha_s$ via $g^2/4\pi=\alpha_s$). The
formula presented are general and also applicable in a theory with
spontaneously broken symmetry \cite{Jegerlehner:2001fb}. In the
case of SUSY QCD from the previous formula it follows that
%
\begin{eqnarray}
2Z^{(2,2)} = \left(Z^{(1,1)}\right)^2 +2\beta_g^{(1)}Z^{(1,1)}.
\end{eqnarray}
%
Substituting into this relation the expressions for  $Z^{(1,1)} =
-2C_F$ and $\beta_g^{(1)} = -\frac{1}{2}(3C_A-6)$ we get the same
result for $1/\ep^2$ poles as in eq.~(\ref{zm2}).


\subsection{Gluino mass renormalization}

\begin{center}
\begin{figure}[t]
\hspace*{5cm}
\includegraphics[scale=1]{diag2.eps}
\caption{One-loop diagrams  contributing to gluino mass counterterm.}
\label{diaggluino}
\end{figure}
\end{center}
\vspace*{-8mm}
%

The renormalization of gluino mass is performed in a way similar to quark
mass renormalization. Relevant diagrams with gluino-gluon and
quark-squark loops are shown in Fig.\ref{diaggluino}.
Explicit calculations give the following results for gluino
mass
$m_{\sg,0} = Z_{\msg} m_{\sg}$,
gluino wave function renormalization constants
$Z_{\sg}$
as well as
for gluino-gluon-gluino vertex renormalization constant $Z_{\sg G \sq}$
%
\begin{eqnarray}
Z_{\msg} & = & 1+ \frac{\alpha_s}{4\pi}
             \left( 6 - 3 C_A \right) \frac{1}{\varepsilon} , \\
Z_{\sg}  & = & 1 - \frac{\alpha_s}{4\pi}
   \left( \xi C_A + 6  \right) \frac{1}{\varepsilon} , \label{zgluino} \\
Z_{\sg G \sg} & = & 1 - \frac{\alpha_s}{4\pi} \left( 6
    + \frac{3+5\xi}{4} C_A \right) \frac{1}{\varepsilon} , \label{zglggl}.
%\label{Zgluino}
\end{eqnarray}

   From eqs.~(\ref{zbg}), (\ref{zgluino}) and (\ref{zglggl})
one can extract the gauge coupling renormalization constant
\begin{equation}
     Z_{\alpha_s}   =   Z_{\sg G \sg}^2 Z_{\sg}^{-2} Z_{G}^{-1},
     \label{renglggl}
\end{equation}
which, of course, coincide with eq.~(\ref{zg}). This serves as an
additional check. The result for gluino mass renormalization
constant can be verified also with help of known gluino mass
beta-function.


\subsection{Squark sector renormalization}

To discuss the renormalization of squark sector let us start with
the lagrangian written in terms of physical states. The bare
lagrangian in terms of bare parameters and bare fields is given by
%
\footnote{We don't include bare lagrangian for four-squark interaction because
counterterms associated with the four-squark vertices
will give contribution of order ${\cal O}(\alpha_s^3)$.}
%
\begin{eqnarray}
{\cal L}^{\rm  bare}_{\sq} & =  & \demu \btq_i \demu \sq_i
- \msq{i}^2 \btq_i \sq_i + \Lag^{\rm bare}_{q \sg \sq} ,\label{lagbaresq}\\
%\hspace*{2cm}
{\cal L}^{\rm bare}_{q \sg \sq}  &= & - \sqrt{2} g_3
\bar{q} \left( P_R \cos \theta_{\sq} -
  P_L \sin \theta_{\sq} \right)T^a  \sg^a \sq_1 \nonumber\\
& +& \sqrt{2} g_3 \bar{q} \left(P_R \sin \theta_{\sq}
+ P_L \cos \theta_{\sq} \right)T^a \sg^a \sq_2 \nonumber\\
%
& - & \sqrt{2} g_3 \btq_1 T^a
\bar{\sg}^a \left( P_L \cos \theta_{\sq} -
  P_R \sin \theta_{\sq} \right) q  \nonumber\\
& + &\sqrt{2} g_3 \btq_2 T^a
\bar{\sg}^a \left(P_L \sin \theta_{\sq}
+ P_R \cos \theta_{\sq} \right) q \label{lagbareqglsq}
%\label{baresquark}
\end{eqnarray}
%
We can rewrite this lagrangian in terms of renormalized physical
quantities and counterterms
\begin{eqnarray}
\label{lagrenormalized}
 {\cal L}^{\rm  bare}_{\sq} & =  & {\cal L}_{\sq} + \delta
{\cal
L}_{\sq} \\
\delta {\cal L}_{\sq} & =  & \delta Z_{\tilde{q}} (\demu \btq_i
\demu \sq_i - \msq{i}^2 \btq_i \sq_i)
 - \delta \msq{i}^2 \btq_i \sq_i
 -\delta { \tilde m_{12}}^2 (\btq_2 \sq_1+\btq_1 \sq_2)
 + \delta Z_{q \sg \sq} {\cal L}_{q \sg \sq}
  \label{ctlagbaresq}
\end{eqnarray}
where ${\cal L}_{\sq}$ and ${\cal L}_{q \sg \sq}$ have the same
form as (\ref{lagbaresq}) and (\ref{lagbareqglsq}). It should be
noticed, that this set of counterterms is sufficient to cancel all
rising divergencies. The $\delta {\tilde m_{12}^2}$ counterterm is
used to render non-diagonal squark self-energies ${\tilde
\Sigma}_{\sq_i\sq_j}~(i\neq j)$ finite. In practical calculations,
it is convenient to trade $\delta {\tilde m_{12}^2}$ for the
counterterm for squark mixing angle. Indeed, we can diagonalize
the lagrangian (\ref{lagrenormalized}) with the following field
redefinitions
\begin{equation}
\left(
\begin{array}{c}
{\tilde q'}_1\\[3mm]
{\tilde q'}_2
\end{array}
\right)= \left( \begin{array}{cc}
{1} & {\delta\theta_{\sq}} \\[3mm]
{-\delta\theta_{\sq}} & {1}
\end{array}\right)
\left(
\begin{array}{c}
{\tilde q}_1\\[3mm]
{\tilde q}_2
\end{array}
\right)
\end{equation}
where we have taken into account that $\delta\theta_{\sq}
\sim{\cal O}(\alpha_s)$, $\cos\left(\delta\theta_{\sq}\right)
=1+{\cal O}(\alpha_s^2)$ and
$\sin\left(\delta\theta_{\sq}\right)=\delta\theta_{\sq}+{\cal
O}(\alpha_s^3)$
%
Demanding that the new mass matrix of squarks
\begin{equation}
%  \left[{\cal M}_{\sq}^2\right]^{(1)} =
\left( \begin{array}{cc}
{1} & {\delta\theta_{\sq}} \\[3mm]
{-\delta\theta_{\sq}} & {1}
\end{array}\right)
\left( \begin{array}{cc} m_{\tilde{q}_1}^2+\delta Z_{\tilde{q}}
m_{\tilde{q}_1}^2 + \delta m_{\tilde{q}_1}^2 &
\delta {\tilde m_{12}^2} \\[3mm]
\delta {\tilde m_{12}^2} & m_{\tilde{q}_2}^2+\delta Z_{\tilde{q}}
m_{\tilde{q}_2}^2 + \delta m_{\tilde{q}_2}^2
\end{array}\right)
\left( \begin{array}{cc}
{1} & {-\delta\theta_{\sq}} \\[3mm]
{\delta\theta_{\sq}} & {1}
\end{array}\right) \,.
\label{diagmmmol}
\end{equation}
must be diagonal and expanding eq.~(\ref{diagmmmol}) in gauge
coupling constant $\alpha_s$ one gets
\begin{equation}
\delta\theta_{\sq}=\frac{\delta {\tilde m_{12}^2}}
{m^2_{\sq_1}-m^2_{\sq_2}} =
\frac{({\tilde \Sigma}^{(1)}_{\sq_1\sq_2})_{\rm div}}
{m^2_{\sq_1}-m^2_{\sq_2}} ,
\end{equation}
where subscript ``div'' stands for the $1/\varepsilon$ part of expression.
%%%AVB
Rewriting eq.~(\ref{lagrenormalized}) in terms of new fields
${\tilde q'}_i$ we come to the following renormalization
prescription for the squark sector
\begin{eqnarray}
    m_{\sq,0}^2 &=& m_{\tilde{q}_i}^2+ \delta m_{\tilde{q}_i}^2 = Z_{m^2_{\tilde{q}_i}} m_{\tilde{q}_i}^2 ,\\
    \tilde{q}_{i,0} &=& (1+ {1\over 2} Z_{\tilde{q}} )\tilde{q}_i =  Z_{\tilde{q}}^{1/2} \tilde{q}_i  \\
    \theta_{\tilde{q},0} &=& \theta_{\tilde{q}} + \delta \theta_{\tilde{q}} = Z_{\theta_{\tilde{q}}} \theta_{\tilde{q}}
\end{eqnarray}

\begin{center}
\begin{figure}[t]%\vspace*{3cm}
\hspace*{2cm}
\includegraphics[scale=1]{diag3.eps}
\caption{One-loop diagrams contributing to the squark mass
and squark mixing angle counterterm.}
\label{squarksdiag}
\end{figure}
\end{center}

\vspace*{-8mm} Explicit calculation of squark self-energies gives
(see diagrams in Fig.~\ref{squarksdiag}) the following expression
for the renormalization of squark mixing angle
\begin{equation}
\delta\theta_{\sq}=C_F\frac{\alpha_s}{4\pi}
\left( -\frac{\sin \left(4 \theta_{\sq}\right)}{2}
+\frac{4 m_q\msg\cos\left(2\theta_{\sq}\right)}{m_{\sq_1}^2-m_{\sq_2}^2}
\right) \frac{1}{\varepsilon} .
\label{deltatheta}
\end{equation}
%
Note, that renormalization (\ref{deltatheta}) of the mixing angle
is minimal. It is the correct choice, if one wants to have final
result expressed in terms of running \DR sparticle masses.

  Summarizing the result of calculation the renormalization constants
for the squark sector read
%
\begin{eqnarray}
Z_{\sq}&=&1-C_F\frac{\alpha_s}{4\pi}\left(\xi-1\right)\frac{1}{\varepsilon} ,
\label{zsq} \\
Z_{m^2_{\tilde{q}_1}} &=& 1 +  \frac{C_F}{m^2_{\tilde{q}_1}}
\frac{\alpha_s}{4\pi }
   \Biggl( \left(m_{\sq_2}^2-m_{\sq_1}^2\right) \sin^2 2 \theta_{\sq}
+4 \left(m_q \msg \sin 2 \theta_{\sq}- m_q^2 - \msg^2\right) \Biggr)
\frac{1}{\varepsilon} , \\
Z_{m^2_{\tilde{q}_2}}  & = &  1 + \frac{C_F}{m^2_{\tilde{q}_2}}
\frac{\alpha_s}{4\pi}
    \Biggl( \left(m_{\sq_1}^2-m_{\sq_2}^2\right) \sin^2 2 \theta_{\sq}
-4 \left(m_q \msg \sin 2 \theta_{\sq} + m_q^2 + \msg^2\right) \Biggr)
\frac{1}{\varepsilon} ,\\
Z_{q \sg \sq} & = & 1 - \frac{\alpha_s}{4\pi}
        \left(\xi C_F + \frac{3+\xi}{2} C_A \right)\frac{1}{\varepsilon} ,
\label{zqglsq} \\
Z_{\theta_{\tilde{q}}} &=& 1+
\frac{ \delta\theta_{\tilde{q}}}{\theta_{\tilde{q}}} .
\end{eqnarray}
%
Using eqs. (\ref{zquark}), (\ref{zgluino}), (\ref{zsq}) and (\ref{zqglsq})
one can also get gauge coupling renormalization constant
$Z_{\alpha_s}$ from quark-gluino-squark vertex
%
\begin{equation}
 Z_{\alpha_s}   =   Z_{q\sg \sq}^2 Z_{q}^{-1} Z_{\sg}^{-1} Z_{\sq}^{-1}
\end{equation}
%
which, due to supersymmetry, of cause coincides with the results
of eqs.~(\ref{renqgq}) and (\ref{renglggl}).

\begin{center}
\begin{figure}[t]
\hspace*{0.8cm}
\includegraphics[scale=0.75]{diag4.eps}
\caption{One-loop diagrams contributing to the
$\varepsilon$-scalar wave function and mass counterterms}
\label{escalardiag2}
\end{figure}
\end{center}
\vspace*{-12mm}

\subsection{$\eps$-scalar sector renormalization}

The renormalization of $\eps$-scalar sector in SUSY QCD goes along
the same lines as for the ordinary QCD in dimensional reduction.
Here we will remind the tricks used to obtain corresponding
renormalization constants and write down their values in SUSY QCD.
The part of SUSY QCD lagrangian containing $\eps$-scalar fields
could be obtained through the dimensional reduction of lagrangian
from eq.(\ref{lagsqnd}). As a result we have
\begin{eqnarray}
{\cal L}_{\eps} &=& \frac{1}{2}\left(D^{\mu}G_{\sigma}\right)^\dag
D_{\mu}G_{\sigma} - g_3\bar q\gamma_{\sigma}T^aG^a_{\sigma}q
-\frac{1}{4}g_3^2f^{a b c}f^{a d e}G^b_{\sigma}G^c_{\sigma '}
G^d_{\sigma}G^e_{\sigma '} \nonumber \\
&&-\frac{1}{2}g_3\bar{\sg}\gamma_{\sigma}T^aG^a_{\sigma}{\sg}
+g_3^2{\btq} T^aT^bG^a_{\sigma}G^b_{\sigma}{\sq}.
\end{eqnarray}
Here $G_{\sigma (\sigma ')}$ denote $\eps$-scalar fields and
indices $\sigma$, $\sigma '$ belong to $2\eps$ subspace.

The standard way is to use Feynman rules, derived from this
lagrangian to account for contribution of $\eps$-scalars. However,
it turns out that for some problems one can use far more simple
arguments to get a desired solution. For example, we are
interested both in full and separate contributions of vectors and
$\eps$-scalars to the quantity, like Green function or amplitude,
without external vector indices. So, vectors and $\eps$-scalars
fields are located on internal lines only. Then to solve this
problem one should
\begin{enumerate}
\item perform $d$-dimensional vector and spinor algebra \item for
full contribution - replace $d$ in the numerator of scalarized
expression with 4; \\
for $\eps$-scalar contribution - replace $d^n$ with
$4^n-(4-2\eps)^n$.
\end{enumerate}
When we are dealing with a Green function containing external
vector indices and want to extract a contribution of either
external vector or $\eps$-scalar fields, a corresponding projector
on $4-2\varepsilon$ or $2\varepsilon$ subspaces should be
constructed. Internal $\eps$-scalars and vectors are treated along
the same way as described above. At this point one should keep in
mind, that particle momenta are always orthogonal to
$2\varepsilon$ subspace. For example, in the case of
$\varepsilon$-scalar propagator (see Fig.\ref{escalardiag2}) we
use as a projector $\frac{1}{2\eps}g^{\mu\nu}$ ($\eps$-scalar
propagator is always diagonal - see notice above) and after
contraction of external indices replace $d^n$ with
$4^n-(4-2\eps)^n$. At the same moment, evaluating corrections to
the interaction between $\eps$-scalar and quarks (see
Fig.\ref{escalardiag3}), required for the determination of Yukawa
charge renormalization, one should follow the same steps as in a
case of $\eps$-scalar propagator with the only difference that now
projector is given by
\begin{eqnarray}
{\rm Tr}\left( \frac{1}{8C_FC_A\eps}T^a\gamma^{\mu}
 \Gamma_{\mu}
\right),
\end{eqnarray}
where $\Gamma^{\mu}$ is an interaction vertex of gluon field with
quarks, computed in SUSY QCD without separating $\epsilon$-scalar
contributions  but using 4-dimensional vector and spinor algebras.
\begin{center}
\begin{figure}[t]%\vspace*{3cm}
\hspace*{2cm}
\includegraphics[scale=1]{diag5.eps}
\caption{One-loop diagrams for $\eps$-scalar Yukawa coupling
renormalization constant.} \label{escalardiag3}
\end{figure}
\end{center}
\vspace*{-12mm}
At the one-loop order the $\eps$-scalar field renormalization
constant is given by
%
\begin{eqnarray}
Z_s = 1 - \frac{\alpha_s}{4\pi} \Bigl( 6-C_A(2-\xi)  \Bigr)
    \frac{1}{\varepsilon} .
\end{eqnarray}
%
For the one-loop Yukawa charge renormalization constant,
describing the interaction  of $\epsilon$-scalar with fermions in
SUSY QCD we have
%
\begin{eqnarray}
Z_Y = Z^2_{qsq}Z_q^{-2}Z_s^{-1} =
1 - \frac{\alpha_s}{4\pi}\left(3C_A-6\right)\frac{1}{\varepsilon} .
\end{eqnarray}
%

  Note, that in SUSY QCD, contrary to the case of non-supersymmetric QCD,
it coincides with the gauge charge renormalization constant. It is due
to supersymmetry, which now protects tree-level coincidence of
the $\varepsilon$-scalar and vector coupling constants.

The one-loop nonminimal mass counterterm for $\varepsilon$-scalars reads
\begin{eqnarray}
\delta m_s^2 = -\frac{\alpha_s}{4\pi}\left(
\sum_{n_f}\left(\frac{m_f^2}{\mu^2}\right)^{\!\!-\varepsilon}
\!\!\!\!m_f^2\left[\frac{2}{\varepsilon}+2\right]
-\sum_{n_{\tilde f}}\left(\frac{m_{\tilde f}^2}{\mu^2}\right)^{\!\!-\varepsilon}
\!\!\!\!m_{\tilde f}^2\left[\frac{1}{\varepsilon}+1\right]
+\left(\frac{m_{\sg}^2}{\mu^2}\right)^{\!\!-\varepsilon}
\!\!\!\!m_{\sg}^2\left[\frac{2}{\varepsilon}+2\right]
\right) .
\end{eqnarray}
It is necessary to insure that the pole mass of
$\varepsilon$-scalar is zero. Here $n_f$ denotes sum over
different quark flavours and $n_{\tilde f}$ used to denote sum
over different squarks. It is just this counterterm, that cancels
$m_{{\rm hard}}^2$ terms occurring in two-loop diagrams and
ensures decoupling of large scale physics for physical quantity
like quark pole mass. A way to renormalize $\eps$-scalar mass
nonminimally was first proposed in \cite{Avdeev:1997sz} and shown
to be a consistent renormalization procedure. In fact, there are
two theoretically admissible prescriptions to deal with
$\eps$-scalar mass renormalization. One is to renormalize
$\eps$-scalar mass minimally \cite{jones} and another one is the
total subtraction of loop corrections to the $\eps$-scalar mass
\cite{Avdeev:1997sz}. In the first approach $\eps$-scalar mass
becomes a running quantity and physical quantities, like pole mass
in our case, will in general depend on it together with other
physical renormalized parameters. However, $\eps$-scalar mass is
not a physical quantity and it is more correct to treat it as an
artifact of our regularization and renormalization schemes. From
this point of view the second approach is more natural, if  we are
going to follow as close as possible the original idea of
dimensional reduction suggesting the zero tree-level mass for both
$\eps$-scalars and gauge bosons. Of course all renormalization
schemes are equivalent and could be related via final
recalculation of parameters. An introduction of nonminimal
subtraction for $\eps$-scalar mass leads to the fact that bare
quark mass is not anymore a renormalization group invariant. The
point is that the physical quark mass depends on two mutually
correlated bare masses: quark mass itself and $\eps$-scalar mass.
Both these quantities are $\mu$ dependent and only the physical
pole mass is RG invariant. For precisely this reason the RG
equations for the bare quark mass should be written with great
care.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Results of calculation}

Here we present the results of our calculation. Corrections
to the pole masses are parametrized as follows
%
\begin{equation}
\frac{M_{\rm pole}}{m} = 1 + \left( \frac{\Delta m}{m} \right),
\label{PM}
\end{equation}
%
where $\Delta m$ starts from the one-loop order.

Through
calculations we reproduced all known results about radiative
corrections to the pole masses of bottom and top quarks: one-loop
MSSM
\cite{Hempfling:1993kv,Hall:1993gn,SUSY94&Wright,Donini95,Pierce:1996zz}
and two loop QCD \cite{Gray:1990yh,Fleischer:1998dw} (in \MS) and
\cite{Avdeev:1997sz} (in \DR) corrections.
As an additional check the renormalization group analysis can be
applied to confirm independently the $1/\varepsilon$ and $1/\varepsilon^2$
terms. We get complete agreement between our pole terms
and those from RG for \DR quark mass renormalization constant. And
the last but not least: calculation were performed in the linear
gauge with a free QCD gauge parameter. The correction to the pole
mass (\ref{PM}) is however appears to be gauge independent.

\subsection{one-loop result}
The corrections to the bottom and top quark pole masses at one
loop are well known for a long time for QCD as well as for MSSM.
Here we reproduce them only up to terms of order
${\cal O}({m_{\rm soft}^2}/{m_{\rm hard}^2})$
in large mass expansion
procedure. For pure QCD we have:
\begin{equation}
\left({\Delta m_b\over m_b}\right)^{\rm QCD}=
C_F\frac{\alpha_s}{4\pi}\left[5 -3 \ln
\left(\frac{m^2_b}{\mu^2}\right)\right]
\end{equation}
and squark-gluino MSSM contribution is given by
\begin{eqnarray}
\left({\Delta m_b\over m_b}\right)^{\tilde b {\tilde {\mathrm
g}}}&=& C_F\frac{\alpha_s}{8\pi}\left[-3 + \frac{m^2_{\tilde
b_1}}{ m^2_{\tilde b_1}-m^2_{\tilde {\mathrm g}}} +
\frac{m^2_{\tilde b_2}}{ m^2_{\tilde b_2}-m^2_{\tilde {\mathrm
g}}} + 2 \ln \left(\frac{m^2_{\tilde {\mathrm
g}}}{\mu^2}\right)\right . \nonumber \\
% +
%  \left( 1 + \frac{{m^2_{\tilde b_1}}^2}
%  {2\,({ -m^2_{\tilde {\mathrm g}} + m^2_{\tilde b_1} ) }^2} -
%  \frac{m^2_{\tilde b_1}}{-m^2_{\tilde {\mathrm g}} + m^2_{\tilde b_1}} +
%     \frac{{m^2_{\tilde b_2}}^2}
% {2\,{( -m^2_{\tilde {\mathrm g}} + m^2_{\tilde b_2}) }^2} -
%    \frac{m^2_{\tilde b_2}}{-m^2_{\tilde {\mathrm g}} + m^2_{\tilde b_2}}
%\right .\nonumber \\ &&
%+ \left .
% \frac{m_{\tilde {\mathrm g}}\,m^2_{\tilde b_1}\,\sin (2\theta_{\tilde b})}
%{m_{b}\,\left( -m^2_{\tilde {\mathrm g}} + m^2_{\tilde b_1} \right)}-
%\frac{m_{\tilde {\mathrm g}}\,m^2_{\tilde b_2}\,\sin (2\theta_{\tilde b})}
%      {m_{b}\,\left( -m^2_{\tilde {\mathrm g}} + m^2_{\tilde b_2} \right)
%          } \right) \,\ln (m^2_{\tilde {\mathrm g}}) \nonumber \\ &&
&+&\left . \frac{m^2_{\tilde b_1}}{ m^2_{\tilde b_1}-m^2_{\tilde
{\mathrm g}}}
  \left(2 - \frac{{m^2_{\tilde b_1}}}
   {{m^2_{\tilde b_1}-m^2_{\tilde {\mathrm g}}}}  -
 2\sin (2\theta_{\tilde b})\frac{m_{\tilde {\mathrm g}}}{m_{b} } \right)
\,\ln \left(\frac{m^2_{\tilde b_1}}{m^2_{\tilde {\mathrm
g}}}\right) \right . \nonumber \\ &+& \left .
  \frac{m^2_{\tilde b_2}}{m^2_{\tilde b_2}-m^2_{\tilde {\mathrm g}}}
  \left(2- \frac{{m^2_{\tilde b_2}}}
{{m^2_{\tilde b_2}-m^2_{\tilde {\mathrm g}} }} + 2\sin
(2\theta_{\tilde b}) \frac{m_{\tilde {\mathrm g}}} {m_{b}} \right)
\,\ln\left(\frac{m^2_{\tilde b_2}}{m^2_{\tilde {\mathrm
g}}}\right)\right]
\end{eqnarray}
which coincide with the results of
\cite{Hempfling:1993kv,Hall:1993gn,SUSY94&Wright,Donini95,Pierce:1996zz}.
To obtain similar expressions for the case of top one should
perform  obvious substitution $b\rightarrow t$. Results for
squark-chargino and higgs-quarks loops also known, but we
interesting in this paper only in ${\cal O}(\alpha_s)$ and ${\cal
O}(\alpha_s^2)$ correction.

\subsection{Two-loop result}

The two-loop QCD result in $\overline{\rm DR}$-scheme for $b$-quark
(with four light quarks)
\begin{eqnarray}
&&\left(\frac{\Delta m_b}{m_b}\right)^{\rm QCD}= C_F
  \left(\frac{\alpha_s}{4\pi} \right)^2
\left \{ - \frac{623}{18}   -
       8\,\zeta_2 + {\ln^2 \left(\frac{m_b^2}{m_t^2} \right) } -
       \frac{13}{3}\ln \left(\frac{m_b^2}{m_t^2}
\right) \right . \nonumber\\ &&+
       C_F\,\left[ - \frac{59}{8}   +
          30\,\zeta_2 - 48\,\ln (2)\,\zeta_2 +
          12\,\zeta_3 +
          \frac{3}{2} \ln \left(\frac{m_b^2}{\mu^2} \right)
            + \frac{9}{2}{ \ln^2 \left(\frac{m_b^2}{\mu^2}
\right) } \right] \nonumber\\ &&+       C_A\,\left[
\frac{1093}{24} - 8\,\zeta_2 +
          24\,\ln (2)\,\zeta_2 - 6\,\zeta_3 -
          \frac{179}{6} \ln \left(\frac{m_b^2}{\mu^2} \right)
           + \frac{11}{2}{ \ln^2 \left(\frac{m_b^2}{\mu^2}
\right) } \right] \nonumber\\ &&+
      \left . 26\, \ln \left(\frac{m_b^2}{\mu^2} \right)  -
       6\,{ \ln^2 \left(\frac{m_b^2}{\mu^2} \right) } \right \}
\end{eqnarray}
%
and for $t$-quark (with five light quarks)
%
\begin{eqnarray}
&&\left(\frac{\Delta m_t}{m_t}\right)^{\rm QCD}=
   C_F \left( \frac{\alpha_s}{4\pi} \right)^2
   \left \{ - 43   -
       12\,\zeta_2  +
       26\, \ln \left(\frac{m_t^2}{\mu^2} \right)  -
       6\,{ \ln^2 \left(\frac{m_t^2}{\mu^2} \right) }
\right . \nonumber\\ &&+
       C_F\,\left[ - \frac{59}{8}   +
          30\,\zeta_2 - 48\,\ln (2)\,\zeta_2 +
          12\,\zeta_3 +
          \frac{3}{2} \ln \left(\frac{m_t^2}{\mu^2} \right)
            + \frac{9}{2}{ \ln^2 \left(\frac{m_t^2}{\mu^2}
\right) } \right] \nonumber\\ &&+ \left .      C_A\,\left[
\frac{1093}{24} - 8\,\zeta_2 +
          24\,\ln (2)\,\zeta_2 - 6\,\zeta_3 -
          \frac{179}{6} \ln \left(\frac{m_t^2}{\mu^2} \right)
           + \frac{11}{2}{ \ln^2 \left(\frac{m_t^2}{\mu^2}
\right) } \right]  \right \}
\end{eqnarray}
coincide with the first terms of the expansion in $m_b/m_t$ from
\cite{Avdeev:1997sz}

We find very simple answer for one particular limit.
Namely, in the case, when $m_{{\tilde t}_1}=m_{{\tilde
t}_2}=m_{{\tilde b}_1}=m_{{\tilde b}_2}= m_{{\tilde
\gl}}=m_{{\tilde u}_{1,2}}=m_{{\tilde d}_{1,2}}= m_{{\rm SUSY}}=M$,
the expression looks like
%
\begin{eqnarray}
&&\left(\frac{\Delta m_q}{m_q}\right)^{\rm MSSM}= \left(
\frac{\alpha_s}{4\pi} \right)^2
 \left\{ C_F^2 \left[\frac{23}{24} - \frac{13}{6}
\ln\left(\frac{M^2}{\mu^2}\right) +
\frac{1}{2}\ln^2\left(\frac{M^2}{\mu^2}\right) -
   3\ln\left(\frac{M^2}{\mu^2}\right)
   \ln\left(\frac{m_q^2}{\mu^2}\right)\right]
\right.\nonumber\\ &&+ \left.C_F\left[\frac{47}{3} +
20\ln\left(\frac{M^2}{\mu^2}\right) +
6\ln^2\left(\frac{M^2}{\mu^2}\right) -
6\ln\left(\frac{M^2}{\mu^2}\right)\ln\left(\frac{m_q^2}{\mu^2}\right)
\right] \right.\\ &&+ \left.C_F
C_A\left[\frac{175}{72}+\frac{41}{6}\ln\left(\frac{M^2}{\mu^2}
\right) - \frac{1}{2}\ln^2\left(\frac{M^2}{\mu^2}\right) -
2\ln\left(\frac{M^2}{\mu^2}\right)
\ln\left(\frac{m_q^2}{\mu^2}\right) \right]\right\}\nonumber
\end{eqnarray}


Since the complete formulae\footnote{They could be found in source archive
file for this paper, submitted to HEP database }
are too large to be presented here,
we present only the influence of our results
on the masses of $b$- and $t$-quark and the spectra of
supersymmetric particles, obtained as solutions of RG equations.
We incorporated our formulae into SoftSUSY code \cite{SoftSUSY}
and analysed the difference in the predictions for particle masses
obtained with and without our corrections to $b$- and $t$-quark
pole masses.
\begin{equation}
\Delta_p=%\frac{\Delta m_p}{m_p}=
\frac{m_p^{\mathrm
{CODE}}-m_p^{\mathrm {CODE+Corrections}}} {m_p^{\mathrm
{CODE}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;},
\end{equation}
where $p$ stands for quarks or different supersymmetric
particles.

In Fig.\ref{fig:tanb} we present the variation of $\Delta_p$ as a
function of $\tan \beta$ for $m_0=400\GeV$, $m_{1/2}=400\GeV$,
$A_0=0$ and $M_{GUT}=1.9\times 10^{16}\GeV$. We plotted $\Delta_p$
for five different cases:

$\bullet$ {\bf Bottom 1L} - difference between results obtained
with code including and excluding MSSM one loop corrections to the
pole masses of $b$-quarks

$\bullet$ {\bf Top 1L} - the same as above for $t$-quarks

$\bullet$ {\bf Bottom} - difference between results obtained with
code containing two loop MSSM ${\cal O}(\alpha_s^2)$ correction to
$b$-quark together with MSSM one loop corrections to the pole
masses of $b$- and $t$-quarks and results obtained with code
containing MSSM one loop corrections to the pole masses of b- and
$t$-quarks

$\bullet$ {\bf Top} - the same as above for $t$-quarks

$\bullet$ {\bf All} - difference between results obtained with
code containing  two loop MSSM ${\cal O}(\alpha_s^2)$ correction
together with MSSM one loop corrections to the pole masses of b-
and $t$-quarks and results obtained with code without any MSSM
(one loop and two loop) corrections to the pole masses of b- and
$t$-quarks.

\begin{figure}[h]
\psfig{figure=htmass.eps,height=4.5cm,width=8.5cm}
\psfig{figure=htmb.eps,height=4.5cm,width=8.5cm}
\psfig{figure=htmt.eps,height=4.5cm,width=8.5cm}
\psfig{figure=htmgl.eps,height=4.5cm,width=8.5cm}
\psfig{figure=htmh0.eps,height=4.5cm,width=8.5cm}
\psfig{figure=htmstau2.eps,height=4.5cm,width=8.5cm}
\psfig{figure=htmsb1.eps,height=4.5cm,width=8.5cm}
\psfig{figure=htmst2.eps,height=4.5cm,width=8.5cm}
\psfig{figure=htmsu1.eps,height=4.5cm,width=8.5cm}~~~~~~~
\psfig{figure=htmchi01.eps,height=4.5cm,width=8.5cm}
\caption{$\Delta_p$ as a function of $\tan \beta$. In the first
picture we plotted a specter of masses} \label{fig:tanb}
\end{figure}

From this figure one may see, that two loop correction for the
relation between $b$-quark pole and \DR masses gives sizeable
contribution only to the solution obtained with SoftSUSY program
for \DR $b$-quark mass at $M_Z$ and almost do not influence
spectra of SUSY particles. However, if one consider large
$\tan\beta$ it becomes impotent for other particle spectra and
thus should be taken into account in accurate RG analysis. Note,
that our two-loop SUSY QCD correction for a wide range of
parameter space is always less, then similar 1-loop contribution.
The two-loop MSSM correction for $t$-quark pole mass, on the other
hand, contributes to almost all SUSY particle masses, obtained as
a solution of renormalization group equations with universal
boundary conditions at GUT scale. This 2-loop correction greater
then 1-loop MSSM contribution almost everywhere in MSSM parameter
space. This could be attributed to potentially large corrections
coming from higher terms in ${m_t}/{M_{SUSY}}$
expansion.


To see the dependence of $\Delta_p$ on $m_0$, $m_{1/2}$ and $A_0$ we use %so called
a set of benchmark points and parameter lines in the MSSM
parameter space from \cite{BDEGMOPW,aix,modelline,SPS} which
corresponds to different scenarios in the search for Supersymmetry
at present and future colliders. While a detailed scan over the
more-than-hundred-dimensional parameter space of the MSSM is
clearly not practicable, even a sampling of the three-
(four-)dimensional parameter space of $m_0$, $m_{1/2}$ and $A_0$
($\tan \beta$) is beyond the present capabilities for
phenomenological studies, especially when one tries to simulate
experimental signatures of supersymmetric particles within
detectors. For this reason one often resorts to specific benchmark
scenarios, i.e.\ one studies only specific parameter points or at
best samples a one-dimensional parameter space (the latter is
sometimes called a model line~\cite{modelline}), which exhibit
specific characteristics of the MSSM parameter space.

Some recent proposals for SUSY benchmark scenarios may be found in
\cite{BDEGMOPW,aix,SPS}. We refer to the ``Snowmass Points and
Slopes'' (SPS)~\cite{modelline,SPS}, which  consist of model lines
(``slopes''), i.e. continuous sets of parameters depending on one
dimensionful parameter and specific benchmark points, where each
model line goes through one of the benchmark points.

In Fig.\ref{fig:linea} we present the results for $\Delta_p$ as a
function defined along {\bf Model Line A} $$ m_0 = -A_0 = 0.4 \,
m_{1/2}, \quad m_{1/2} \mbox{ varies}, \quad \tan\beta = 10, \quad
\mu > 0 . $$ where benchmark point is $$ m_0 = 100 \GeV, \quad
m_{1/2} = 250 \GeV, \quad A_0 = -100 \GeV, \quad \tan\beta = 10,
\quad \mu > 0 .
$$
Here and below $\mu$ is the supersymetric Higgs mass parameter
\begin{figure}[h]
\psfig{figure=hamass.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hamb.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hamt.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hamgl.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hamh0.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hamstau2.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hamsb1.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hamst2.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hamsu1.eps,height=4.5cm,width=8.5cm}~~~~~~~
\psfig{figure=hamchi01.eps,height=4.5cm,width=8.5cm}
\caption{$\Delta_p$ as a function defined along {\bf Model Line A}
from corrections on $m_{1/2}$. In the first picture we plotted a
specter of masses } \label{fig:linea}
\end{figure}

As one can easily see from Fig.\ref{fig:linea}, increasing
$m_{1/2}$ at fixed $\tan\beta$ 2-loop SUSY QCD correction to
$b$-quark pole mass almost do not contribute to heavy
supersymmetric particle spectra. On the other hand, our correction
to $t$-quark pole mass is more important at small values of
$m_{1/2}$, where it becomes even larger then corresponding 1-loop
contribution.

In Fig.\ref{fig:linec} we present the results for $\Delta_p$
defined as a function along {\bf Model Line C} $$ m_0 = m_{1/2},
\quad m_{1/2} \mbox{ varies}, \quad A_0=0, \quad \tan\beta = 35,
\quad \mu > 0 . $$ where benchmark point is $$ m_0 = 300 \GeV,
\quad m_{1/2} = 300 \GeV, \quad A_0 = 0 \GeV, \quad \tan\beta =
35, \quad \mu > 0 . $$
\begin{figure}[h]
\psfig{figure=hcmass.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hcmb.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hcmt.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hcmgl.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hcmh0.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hcmstau2.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hcmsb1.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hcmst2.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hcmsu1.eps,height=4.5cm,width=8.5cm}~~~~~~~
\psfig{figure=hcmchi01.eps,height=4.5cm,width=8.5cm}
\caption{$\Delta_p$ as a function defined along {\bf Model Line C}
from corrections on $m_{1/2}$. In the first picture we plotted a
specter of masses} \label{fig:linec}
\end{figure}

Situation with Fig.\ref{fig:linec} is analogous to that of
Fig.\ref{fig:linea}, even if we are increasing $\tan\beta$ from 10
up to 35. An importance of our corrections increases only at very
large values of $\tan\beta$ (50 and more), as one can see from
Fig.\ref{fig:tanb} .

For such large $\tan \beta$ we just used some particular values
$m_0$ and $m_{1/2}$ from allowed region from~\cite{Ellis:2001ms}.
We use next parametrization $$ m_0 = m_{1/2}, \quad m_{1/2} \mbox{
varies}, \quad A_0=0, \quad \tan\beta = 55, \quad \mu > 0 .
$$ The results for $\Delta_p$ are shown in Fig.\ref{fig:linem} and we see, that in
this case our 2-loop correction to $b$-quark pole mass gives
sizeable contribution to the masses of gluino and sbottoms.

\begin{figure}[h]
\psfig{figure=hmmass.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hmmb.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hmmt.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hmmgl.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hmmh0.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hmmstau2.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hmmsb1.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hmmst2.eps,height=4.5cm,width=8.5cm}
\psfig{figure=hmmsu1.eps,height=4.5cm,width=8.5cm}~~~~~~~
\psfig{figure=hmmchi01.eps,height=4.5cm,width=8.5cm}
\caption{$\Delta_p$ at large value of $\tan\beta=55$ as a function
of $m_{1/2}$. In the first picture we plotted a specter of masses}
\label{fig:linem}
\end{figure}




\section{Conclusion}

We presented results for two-loop MSSM corrections to the relation
between pole and running masses of heavy quarks up to ${\cal
O}(\alpha_s^2)$ order. We provided a detail analysis of the value
of these corrections in different regions of parameter space and
discussed their impact on SUSY particle spectra. We would like to
note, that our results presented here may be also included in the
codes of programs like Isajet\cite{isajet}, SuSpect\cite{suspect}
and used for predictive phenomenological analyses like
\cite{Ellis:2001ms}, \cite{deBoer:1995fr}, \cite{Blazek:1996yv},
\cite{Baer:2002gm}. In one of our next papers we suppose to
calculate missing corrections from stop-chargino loops to
$b$-quark pole mass as well as supply more terms in the expansion
in a relation between top pole and \DR masses.


The authors would like to thank K.Chetyrkin, M.Kalmykov, D.Kazakov
and V.Smirnov for fruitful discussions and multiple comments. This
work of A.O. and O.V. was supported by DFG-Forschergruppe
"Quantenfeldtheorie, Computeralgebra and Monte-Carlo-Simulation"
(contract FOR 264/2-1) and by BMBF under grant No 05HT9VKB0.
Financial support for A.B. and V.V. from RFBR grants \#
02-02-16889 and \# 00-15-96691 is kindly acknowledged.

%\newpage

\begin{thebibliography}{**}

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%%CITATION = PHRVA,D49,6168;%%

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%\cite{Allanach:2002nj}
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%%CITATION = HEP-PH 0202233;%%

%\cite{Ellis:2001ms}
\bibitem{Ellis:2001ms}
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%%CITATION = HEP-PH 0102098;%%
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%%CITATION = HEP-PH 0107316;%%

%\cite{Baer:1999sp}
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%\bibitem{Djouadi:1998di}
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%%CITATION = HEP-PH 9901246;%%

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%%CITATION = HEP-PH 9603350;%%
%\cite{deBoer:1994mp}
%\bibitem{deBoer:1994mp}
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%%CITATION = HEP-PH 9405419;%%
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\bibitem{Baer:2002gm}
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%\cite{Jegerlehner:2001fb}
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%%CITATION = HEP-PH 0105304;%%

\end{thebibliography}

\end{document}
