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%% subsection 4.3 Squark masses [slac-pub-7180-0-0-4-3 in slac-pub-7180-0-0-4: slac-pub-7180-0-0-4-4]
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\subsection{\usemenu{slac-pub-7180::context::slac-pub-7180-0-0-4-3}{Squark masses}}\label{subsection::slac-pub-7180-0-0-4-3}
\begin{figure}[t]
\epsfysize=1.5in \epsffile[-70 405 590 575]{F14.ps}
\begin{center}
\parbox{5.5in}{
\caption[]{\small (a) Full one-loop corrections to the first generation
squark mass, $m_{\tilde u_L}$, versus the ratio $m_{\tilde
g}/m_{\tilde u_L}$. (b) The difference between the full corrections
and the approximation in the text, versus $m_{\tilde g}/m_{\tilde
u_L}$. (These are essentially the electroweak corrections.)\label{lsq}}}
\end{center}
\end{figure}
The first two generations of squarks receive QCD \cite{32} and
electroweak corrections. However, it is a very good approximation
to ignore the electroweak graphs, since the dominant corrections
come from gluon/squark and gluino/quark loops. Neglecting the
quark masses, these corrections are as follows,
%
\begin{equation}
m_{\tilde q}^2\ =\ \hat m_{\tilde q}^2(Q) \left[\,1 + \left({\Delta
m_{\tilde q}^2 \over m_{\tilde q}^2}\right)\,\right]\ ,
\end{equation}
%
where
%
\begin{eqnarray}
\left( {\Delta m_{\tilde q}^2 \over m_{\tilde q}^2} \right) &=&
\ {g^2_3 \over 6\pi^2} \ \bigg[\,2 B_1(m_{\tilde q},m_{\tilde q},0) +
{A_0(m_{\tilde g})\over m_{\tilde q}^2} - (1 - x) B_0(m_{\tilde q},
m_{\tilde g},0)\, \bigg] \nonumber \\ &=&\ {g^2_3 \over 6\pi^2}
\bigg[\,1 + 3x + (x-1)^2\ln|x-1| - x^2\ln x\ + 2x\ln\left({Q^2\over
m_{\tilde q}^2}\right)\,\bigg]~,
\label{QCDcorr}
\end{eqnarray}
%
and $x=m_{\tilde g}^2/m_{\tilde q}^2$.
For the case of universal boundary conditions the gluino mass is less
than or roughly equal to the squark mass, so the correction
(\docLink{slac-pub-7180-0-0-4.tcx}[QCDcorr]{34}) is essentially finite at $Q = m_{\tilde q}$. From
Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[lsq]{14} we see that it varies from around 1\% for $x \ll 1$ to
between 4 and 5\% for $x \simeq 1$. We also see that the electroweak
corrections are small, less than 0.5\%.
\begin{figure}[t]
\epsfysize=2.5in \epsffile[-140 220 0 535]{F15.ps}
\begin{center}
\parbox{5.5in}{
\caption[]{\small (a) The corrections to the heavy top squark mass
versus its mass. (b) The difference between the full one-loop heavy
top squark mass and the approximation, Eq.~(\docLink{slac-pub-7180-0-0-4.tcx}[top sq app]{35}). (c)
Same as (a), for ${\tilde b}_1$. (d) Same as (a), for ${\tilde b}_2$.
\label{tb}}}
\end{center}
\end{figure}
The third generation squark masses receive Yukawa corrections on the
order of, and opposite in sign to, the QCD corrections. In
Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[tb]{15} we show the full corrections to the third generation
heavy squark masses. As usual, the tree-level masses are
defined in terms of the gauge-boson and quark pole masses, as well as
the soft masses $M_Q(M_Q)$, $M_U(M_U)$, and $M_D(M_D)$. The tree-level
mass matrices also contain $\tan\beta(M_Z)$, $\mu(\mu)$, and $A_i({\rm
max} (|A_i|,M_Z))$, where $A_i$ denotes the top or bottom $A$-term.
(Our convention for the third generation squarks is to associate the
subscript 1 with the mostly left-handed squark. Since the light top
squark is predominantly right-handed, its mass is denoted $m_{\tilde
t_2}$.)
{}From Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[tb]{15} we see that the heavy top squark mass receives
corrections in the range $-5$ to 2\%, while the bottom squark masses
receive corrections mostly in the 0 to 3\% range. We note that in
none of these cases does the leading logarithm approximation work
well: as is the case for all the squarks and sleptons, these
corrections are essentially non-logarithmic. (The light top squark
mass does receive some substantial logarithmic corrections, but they
are generally not larger than the finite corrections.)
We will now present our approximation for the top squark mass matrix.
We will derive our approximation for the case of the light top squark,
but it also works quite well for the heavy top squark (see
Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[tb]{15}(b)). The mass of the light top squark receives
potentially large additive corrections proportional to the the strong
coupling and the top and bottom Yukawa couplings. We approximate the
corrections to $m_{\tilde t_2}$ by neglecting $g, \ g'$ and the Yukawa
couplings of the first two generations. We also neglect all quark
masses except $m_t$, which eliminates all sfermion mixing except for
that of the top squarks.
We neglect the mixing of charginos and neutralinos, so the two heavy
neutralinos and the heavy chargino all have mass $|\mu|$. We also make
the approximations $m_h = M_Z$ and $m_H = m_{H^+} = m_A$. Finally, we
set $p=0$ in the $B$-functions if any of the other arguments is much
bigger than $m_{{\tilde t}_2}$.
This gives the following expressions for the one-loop corrections to
the top squark mass matrix,
%
\begin{equation}
{\cal M}_{\tilde t}^2\ =\ \hat {\cal M}_{\tilde t}^2(Q)\ +\ \pmatrix{
\Delta M^2_{LL} & \Delta M^2_{LR} \cr \Delta M^2_{LR} & \Delta
M^2_{RR}}\ .\label{top sq app}
\end{equation}
%
The $\Delta M^2$ entries are as follows:
%
\begin{eqnarray}
\Delta M^2_{LL} & =& \ {g^2_3 \over 6\pi^2}\ \bigg\{ 2 m_{{\tilde
t}_2}^2 \left[ c_t^2 B_1(m_{{\tilde t}_2},m_{{\tilde t}_1},0) \ +
\ s_t^2 B_1(m_{{\tilde t}_2},m_{{\tilde t}_2},0) \right] \nonumber \\ &&
\qquad +\ A_0(m_{\tilde g})\ +\ A_0(m_t) \ -\ ( m_{{\tilde t}_2}^2 -
m_{\tilde g}^2 - m_t^2) B_0(0,m_{\tilde g},m_t) \bigg\} \nonumber \\
&-&{1\over16\pi^2} \bigg[ \lambda_t^2 s_t^2 A_0(m_{{\tilde t}_1})\ +
\ \lambda_b^2 A_0(m_{\tilde b}) \nonumber \\ && \qquad -\ 2 (\lambda_t^2 +
\lambda_b^2) A_0(\mu) \ +\ (\lambda_t^2 c_{\beta}^2 + \lambda_b^2
s_{\beta}^2) A_0(m_A) \bigg] \nonumber \\ &-& {\lambda_t^2\over32\pi^2}
\bigg[ \Lambda(\theta_t,\beta) B_0(0,m_{{\tilde t}_1},m_A) \ +
\ \Lambda(\theta_t-{\pi\over2},\beta)B_0(0,0,m_A) \nonumber \\ & &
\qquad +\ \Lambda(\theta_t,\beta-{\pi\over2}) B_0(0,m_{{\tilde
t}_1},0) \ +\ \Lambda(\theta_t-{\pi\over2},\beta-{\pi\over2})
B_0(m_{{\tilde t}_2},m_{{\tilde t}_2},m_Z) \bigg] \nonumber \\ &-&
{1\over16\pi^2} \bigg[ \left( \lambda_t^2 m_t^2 c_\beta^2 \ +
\ \lambda_b^2 (\mu c_\beta - A_b s_\beta)^2 \right) B_0(0,m_{\tilde
b},m_A) \nonumber \\ && \qquad + \ \left( \lambda_t^2 m_t^2 s_\beta^2
\ +\ \lambda_b^2 (\mu s_\beta + A_b c_\beta)^2 \right) B_0(0,m_{\tilde
b},0) \bigg] \\
%
%
\Delta M^2_{LR} &=& \ - {g^2_3 \over 6\pi^2} c_ts_t \left[
(m_{{\tilde t}_1}^2 + m_{{\tilde t}_2}^2) B_0(m_{{\tilde t}_2},
m_{{\tilde t}_1},0) \ +\ 2 m_{{\tilde t}_2}^2
B_0(m_{{\tilde t}_2},m_{{\tilde t}_2},0) \right]
\nonumber \\ &-& {g^2_3 \over 3\pi^2} m_t m_{\tilde g}
B_0(0,m_t,m_{\tilde g}) \ -\ {3\lambda_t^2\over16\pi^2} c_t s_t
A_0(m_{{\tilde t}_1}) \nonumber \\ &-& {\lambda_t^2\over32\pi^2} \bigg[
\Omega( \theta_t,\beta) B_0(0,m_{{\tilde t}_1},m_A) \ +
\ \Omega(-\theta_t,\beta) B_0(0,0,m_A) \nonumber \\ && \qquad +
\ \Omega( \theta_t,{\pi\over2}+\beta)B_0(0,m_{{\tilde t}_1},0) \ +
\ \Omega(-\theta_t,{\pi\over2}+\beta)B_0(m_{{\tilde t}_2},
m_{{\tilde t}_2},M_Z) \bigg] \nonumber \\
&-&{1\over16\pi^2} \bigg[ -\biggl(
\lambda_t^2 m_t c_\beta (\mu s_\beta - A_t c_\beta) \ +\ \lambda_b^2
m_t s_\beta (\mu c_\beta - A_b s_\beta) \biggr)
B_0(0,m_{\tilde b},m_A)\nonumber\\
&& \qquad +\ \lambda_t^2 m_t s_\beta (\mu c_\beta
+ A_t s_\beta) B_0(0,m_{\tilde b} ,0) \bigg] \\
%
%
\Delta M^2_{RR} &=& \ {g^2_3 \over 6\pi^2}\ \bigg\{ 2 m_{{\tilde
t}_2}^2 \left[ s_t^2 B_1(m_{{\tilde t}_2},m_{{\tilde t}_1},0) \ +
\ c_t^2 B_1(m_{{\tilde t}_2},m_{{\tilde t}_2},0) \right] \nonumber \\ &&
\qquad +\ A_0(m_{\tilde g})\ + \ A_0(m_t) \ -\ ( m_{{\tilde t}_2}^2 -
m_{\tilde g}^2 - m_t^2) B_0(0,m_{\tilde g},m_t) \bigg\} \nonumber \\
&-&{\lambda_t^2\over16\pi^2} \left[ c_t^2 A_0(m_{{\tilde t}_1})\ +
\ A_0(m_{\tilde b}) \ -\ 4 A_0(\mu) \ +\ 2 c_{\beta}^2 A_0(m_A) \right]
\nonumber \\ &-& {\lambda_t^2\over32\pi^2} \bigg[
\Lambda({\pi\over2}-\theta_t,\beta) B_0(0,m_{{\tilde t}_1},m_A) \ +
\ \Lambda(-\theta_t,\beta)B_0(0,0,m_A) \nonumber \\ & & \qquad +
\ \Lambda({\pi\over2}-\theta_t,\beta-{\pi\over2})B_0(0,m_{{\tilde
t}_1},0) \ +\ \Lambda(-\theta_t,\beta-{\pi\over2}) B_0(m_{{\tilde
t}_2},m_{{\tilde t}_2},m_Z) \bigg] \nonumber \\ &-& {1\over16\pi^2}
\bigg[ \left( \lambda_b^2 m_t^2 s_\beta^2 +\lambda_t^2 (\mu s_\beta -
A_t c_\beta)^2 \right) B_0(0,m_{\tilde b},m_A) \nonumber \\ && \qquad
+ \ \lambda_t^2 (\mu c_\beta + A_t s_\beta)^2 B_0(0,m_{\tilde b},0)
\bigg] \ .
\label{stop self}
\end{eqnarray}
%
We have defined the two functions
%
\begin{eqnarray}
\Lambda(\theta_t,\beta) & = & \left( 2 m_t \cos \beta \cos\theta_t\ -
\ (\mu \sin \beta - A_t \cos \beta ) \sin\theta_t\right)^2 \nonumber \\
&&\qquad +\ (\mu \sin \beta - A_t \cos \beta )^2 \sin^2\theta_t \\
\Omega(\theta_t,\beta) & = & 2 m_t^2 \cos^2 \beta \sin 2\theta_t
\ -\ 2 m_t \cos\beta (\mu \sin \beta - A_t \cos \beta ) \ .
\end{eqnarray}
%
Note that the running mass matrix $\hat{\cal M}^2_t(Q)$ in
Eq.~(\docLink{slac-pub-7180-0-0-4.tcx}[top sq app]{35}) contains the soft masses $M_Q$ and $M_U$
(as well as $\mu$, $A_t$, etc.) at some common scale, $Q$. In the
limit $\lambda_b \rightarrow 0$, these expressions are equivalent
to the results of Ref.~\cite{10}, with certain external
momenta set to zero.
\begin{figure}[t]
\epsfysize=2.5in \epsffile[-140 220 0 535]{F16.ps}
\begin{center}
\parbox{5.5in}{
\caption[]{\small (a) The full one-loop light top
squark mass, versus the tree-level mass (in GeV). On the x-axis we
plot sign$(m_{\tilde t_2}^2)|m_{\tilde t_2}^2|^{1/2}$, so a negative
tree-level mass corresponds to $m_{\tilde t_2}^2 < 0$. (b) The
difference between the full correction and the approximation in the
text, versus the one-loop mass. (c) The light top squark mass
at one loop, versus $A_0/M_{\tilde q}$. The solid line corresponds to
point (I) in the text, the dashed to point (II). (d) The running and
one-loop light top squark mass versus the renormalization scale $Q$,
for the choice of parameters (I) (solid) and (II) (dashed) in the
text, with $A_0/M_{\tilde q}=-1.83$ and $-2.2$, respectively. The
running mass curves each have points where $m_{\tilde t_2}^2$ becomes
negative. In these cases we plot the signed square-root of the
mass-squared, as in (a).
\label{stop}}}
\end{center}
\end{figure}
These approximations depend on the mass of the light top squark.
Normally, one would take it to be the tree-level mass. For the case
at hand, however, the choice is more subtle because for very light top
squarks, the radiative corrections can be quite large. In fact, the
radiative correction can change the top squark mass squared from
negative to positive. Therefore we shall take $m_{\tilde t_2}$ to be
the one-loop pole mass, which we find by iteration. (We find the
one-loop top squark mixing angle by iteration as well.) We show the
light top squark one-loop pole mass versus the tree-level mass in
Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[stop]{16}(a), where the tree-level mass is the eigenvalue of the
mass matrix which contains the running parameters $M_U^2$ and $M_Q^2$
evaluated at their own scale (or $M_Z$, whichever is larger; the
tree-level mass matrix also contains the top quark and $Z$-boson pole
masses, $\tan\beta(M_Z)$, $\mu(\mu)$, and $A_t({\rm max}(|A_t|,M_Z))$.
In Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[stop]{16}(b) we see that our approximation for the light top
squark mass holds to within 10 GeV.
With the present unification assumptions, a top squark with mass less
than $M_Z$ requires that the RR term in the mass matrix be small and
that the LR element, proportional to the $A$-term, be large. The
light top squark mass results from a cancellation between the diagonal
and off-diagonal terms, which requires a fine tuning. We illustrate
this in Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[stop]{16}(c), where we plot the light top squark
one-loop mass versus $A_0/M_{\tilde q}$. On the same plot we show the
curves corresponding to two choices of parameters, (I) $\tan\beta=20,
\ M_0=500$ GeV, $M_{1/2}=100$ GeV, and $\mu<0$, and (II) $\tan\beta=5,
\ M_0=100$ GeV, $M_{1/2}=200$ GeV, and $\mu>0$. Whether at tree-level
or one-loop, the parameter $A_0$ must be tuned to one part in 75
to obtain a light top squark mass below 50 GeV.
We see from Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[stop]{16}(a) that the light top squark mass-squared
can be raised from $-$(100 GeV)$^2$ at tree-level to over (100
GeV)$^2$ at one loop. For such large corrections, it is important to
keep in mind that two-loop effects might be important. The size of
these effects can be estimated by the scale dependence of the one-loop
mass. In Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[stop]{16}(d) we show the scale dependence at the
points (I) and (II), with $A_0=-985$ GeV and $A_0=-907$ GeV,
respectively. We see that as the renormalization scale increases from
100 to 1000 GeV, the running masses vary over a wide range. In
contrast, the scale dependence of the one-loop masses is quite mild.