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%% subsection 4.2 Neutralino and Chargino Masses [slac-pub-7180-0-0-4-2 in slac-pub-7180-0-0-4: slac-pub-7180-0-0-4-3]
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\subsection{\usemenu{slac-pub-7180::context::slac-pub-7180-0-0-4-2}{Neutralino and Chargino Masses}}\label{subsection::slac-pub-7180-0-0-4-2}
The complete set of corrections to the neutralino and chargino masses
\cite{31,8} is given in Appendix D. In this section we present a
set of approximations to these corrections. These approximations are
more involved than those discussed above because there are no color
corrections that would dominate the results.
Our approximation is as follows. We start by assuming that $|\mu|
\,>\,M_1,M_2, M_Z.$ (We find that $M_Z^2/\mu^2$ and $M_2^2/\mu^2$ are
less than 0.53; see Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[m vs msq]{8}). We work with the
undiagonalized tree-level (chargino or neutralino) mass matrix, and
correct the diagonal entries only, that is, the parameters $M_1$,
$M_2$, and $\mu$. This approximation neglects the {\em corrections}
to the off-diagonal entries of the mass matrices, which leads to an
error of order $(\alpha/4\pi)M^2_Z/\mu^2$ in the masses.
We simplify our expressions by setting all loop masses and external
momenta to their diagonal values, i.e. we set $m_{\tilde\chi_1^0} =
M_1$, etc. We neglect all Yukawa couplings except $\lambda_t$ and
$\lambda_b$. We also ignore the mixings of the charginos and
neutralinos in the radiative corrections. This also leads to an error
of order $(\alpha/4\pi)M_Z^2/\mu^2$ in our final result.
We further simplify our expressions by setting all quark masses to
zero, and by assuming that all squarks are degenerate with mass
$M_{Q_1}$, and that all sleptons are degenerate as well with mass
$M_{L_1}$. We also take the Higgs masses to be $m_h = M_Z$ and $m_H =
m_{H^+} = m_A$. This means that we also neglect terms of order
$(\alpha/4\pi)M_Z^2/m_A^2$.
In this limit, the dominant correction to $M_1$ comes from
quark/squark, chargino/charged-Higgs and neutralino/neutral-Higgs
loops. We find
%
\begin{eqnarray}
\left({\Delta M_1 \over M_1}\right)\ &=& \ -{g'^2 \over 16
\pi^2}\,\bigg\{ \,11 B_1(M_1,0,M_{Q_1}) + 9 B_1(M_1,0,M_{L_1})
\nonumber \\ &&\ +\ {\mu\over M_1} \sin(2\beta)\ \bigg( \,
B_0(M_1,\mu,m_A) - B_0(M_1,\mu,M_Z)\,\bigg) \nonumber \\ &&\ +
\ B_1(M_1,\mu,m_A) + B_1(M_1,\mu,M_Z) \, \bigg\}\ .
\label{M1a}
\end{eqnarray}
%
Since $M_Z,\ M_1 \ll \mu$, we can simplify this
expression by setting $M_Z = M_1 = 0$ inside the $B$ functions. This
gives
%
\begin{eqnarray}
\left({\Delta M_1 \over M_1}\right)\ &=&\ {g'^2 \over 32 \pi^2}\,
\bigg\{\,11 \theta_{M_1 M_{Q_1}} + 9 \theta_{M_1 M_{L_1}} +
\theta_{M_1 \mu M_Z} - 2B_1(0,\mu,m_A) \nonumber \\ &&\ +\ {2 \mu\over
M_1} \sin(2\beta)\ \bigg(\,B_0(0,\mu,0)\,-\,B_0(0,\mu,m_A)\,\bigg)\ -
\ {23 \over 2}\, \bigg\}\ ,
\label{M1b}
\end{eqnarray}
%
where $B_0(0,m_1,m_2)$ and $B_1(0,m_1,m_2)$ were defined in
(\docLink{slac-pub-7180-0-0-3.tcx}[b0(0)]{9}) and (\docLink{slac-pub-7180-0-0-3.tcx}[b1(0)]{10}), and $\theta_{m_1 \ldots m_2} \equiv
\ln(M^2/Q^2)$ with $M^2 = {\rm max}(m_1^2,...,m_2^2)$. The form of
the finite corrections depends on the assumed hierarchy in the
low-energy spectrum, but the leading logarithms are always correctly
given by the $\theta$ terms. We set the first subscript of a $\theta$
term, $m_1$, equal to the external momentum. Note that when the
renormalization scale equals the external momentum, $Q=m_1$, the theta
function reduces to the familiar form, $\theta_{m_1 m_2} =
\ln(m^2_2/Q^2)\,\theta(m^2_2 - Q^2)$.
The leading logarithmic corrections are easy to read from
Eq.~(\docLink{slac-pub-7180-0-0-4.tcx}[M1b]{26}).\footnote{The logarithmic part of $-2B_1(0,\mu,m_A)$
in Eq.~(\docLink{slac-pub-7180-0-0-4.tcx}[M1b]{26}) is given by $\theta_{M_1\mu m_A}$.} Note that the
terms proportional to $\sin(2\beta)$ are enhanced by the ratio
$\mu/M_1$. These finite terms are completely missed in the
run-and-match approach because they do not contribute to the beta
function.
In a similar way, we approximate the corrections to $M_2$ from
quark/squark and Higgs loops. They are
%
\begin{eqnarray}
\left({\Delta M_2 \over M_2 }\right)\ &=&\ -\ {g^2 \over 16
\pi^2}\,\bigg\{ \,9 B_1(M_2,0,M_{Q_1}) + 3 B_1(M_2,0,M_{L_1})
\nonumber \\ &&\ +\ {\mu\over M_2} \sin(2\beta)\ \bigg(\,
B_0(M_2,\mu,m_A) - B_0(M_2,\mu,M_Z)\,\bigg) \nonumber \\ &&\ +
\ B_1(M_2,\mu,m_A) + B_1(M_2,\mu,M_Z) \, \bigg\}\ .
\label{mchi02_1}
\end{eqnarray}
%
Setting $M_Z = M_2 = 0$ inside the $B$ functions, we find
%
\begin{eqnarray}
\left({\Delta M_2 \over M_2 }\right)\ &=&\ {g^2 \over 32 \pi^2}\,
\bigg\{\,9 \theta_{M_2M_{Q_1}} + 3 \theta_{M_2M_{L_1}}\ +
\ \theta_{M_2\mu M_Z} - 2B_1(0,\mu,m_A) \nonumber \\ &&\ +\ {2 \mu\over
M_2} \sin(2\beta)\ \bigg(\,B_0(0,\mu,0) \,-\, B_0(0,\mu,m_A) \,\bigg)
\ -\ {15 \over 2}\, \bigg\}\ .
\label{mchi02_1_2}
\end{eqnarray}
%
There are additional corrections to $M_2$ from gauge boson loops.
Because $M_2$ enters both the chargino and the neutralino mass
matrices, the corrections differ slightly for the two cases. However,
to the order of interest, it suffices to use the neutralino result,
%
\begin{equation}
\left({\Delta M_2 \over M_2 } \right)^{\rm gauge} \ =\ {g^2\over 4
\pi^2} \bigg\{\, 2 B_0(M_2,M_2,M_W) - B_1(M_2,M_2,M_W) \,\bigg\}\ .
\end{equation}
%
Because $M_2$ is of order $M_W$, one must use the full $B$ functions
in this expression. Alternatively, one can use the following
empirical fit which works to better than 1\%,
%
\begin{eqnarray}
\left({\Delta M_2 \over M_2 } \right)^{\rm gauge} &=& -\ {g^2\over 4
\pi^2} \bigg\{\,{3\over2} \theta_{M_2M_W} + \theta(M_W-M_2) \left(
1.57 {M_2 \over M_W} - 1.85 \right) \nonumber \\ && -
\ \theta(M_2-M_W)\, \left[ \, 0.54 \ln \left({M_2 \over M_W} -
0.8\right) + 1.15 \,\right] \,\bigg\} \ .
\end{eqnarray}
%
The corrections to $\mu$ are obtained in a similar manner. In the
limit $g'^2 \ll g^2$, we find
%
\begin{eqnarray}
\label{deltamu}
\left({\Delta \mu \over \mu }\right)\ &=& \ -\ {3\over 32 \pi^2}
\bigg[\, (\lambda_b^2 + \lambda_t^2) B_1(\mu,0,M_{Q_3}) + \lambda_t^2
B_1(\mu,0,M_{U_3}) + \lambda_b^2 B_1(\mu,0,M_{D_3})\, \bigg]\\ &&\ -
\ {3g^2 \over 64\pi^2} \bigg[ B_1(\mu,M_2,m_A) + B_1(\mu,M_2,M_Z) +2
B_1(\mu,\mu,M_Z) - 4 B_0(\mu,\mu,M_Z)\, \bigg] \ .\nonumber
\end{eqnarray}
%
As above, we set $M_Z = M_2 = 0$ inside the $B$ function, in which
case (\docLink{slac-pub-7180-0-0-4.tcx}[deltamu]{31}) reduces to
%
\begin{eqnarray}
\left({\Delta \mu \over \mu }\right)\ &=& \ -\ {3\over 32 \pi^2}
\bigg[\, (\lambda_b^2 + \lambda_t^2) B_1(\mu,0,M_{Q_3}) + \lambda_t^2
B_1(\mu,0,M_{U_3}) + \lambda_b^2 B_1(\mu,0,M_{D_3})\, \bigg] \nonumber
\\ &&\ +\ {3g^2 \over 64\pi^2} \bigg[ {1\over 2} \theta_{\mu M_2M_Z}
\, - \, 3 \theta_{\mu M_Z} \,-\,B_1(\mu,0,m_A) \, +\,4\,\bigg]\ .
\end{eqnarray}
%
The expression for $B_1(p,0,m)$ is given in Eq.~(\docLink{slac-pub-7180-0-0-4.tcx}[b1_p0m2]{24}).
\begin{figure}[t]
\epsfysize=2.5in \epsffile[-115 220 600 535]{F11.ps}
\begin{center}
\parbox{5.5in}{
\caption[]{\small Corrections to the lightest neutralino mass, as in
Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[gl]{10}.
\label{n1}}}
\end{center}
\end{figure}
In Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[n1]{11} we show the corrections to the lightest neutralino
mass. In Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[n1]{11}(a) we show the full correction in percent,
with the tree-level mass defined as the eigenvalue of the mass matrix,
where the running parameters $M_1,\ M_2,$ and $\mu$ are evaluated at
their own scale. (The tree-level mass matrices also contain
$\tan\beta$ at $M_Z$ and the $W$- and $Z$-boson pole masses.) The
one-loop masses have negligible scale dependence.
As usual, the full corrections are made up of logarithmic and finite
pieces. The logarithmic corrections are shown in Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[n1]{11}(b) and
the finite corrections are shown in Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[n1]{11}(c). Note that the
finite corrections can be more than half as large as the logarithmic
corrections. Indeed, the finite corrections can be larger than 5\% in
the small $M_1$ region, primarily because of the Higgsino-loop term
proportional to $\mu$. In Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[n1]{11}(d) we show the difference
between the full one-loop result and our approximation. Here, and in
the following two figures, the logarithmic corrections
[Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[n1]{11}(b)] include an explicit sum over the soft squark and
slepton masses, while the approximations [Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[n1]{11}(d)] use a
single soft squark or slepton mass.
\begin{figure}[t]
\epsfysize=2.5in \epsffile[-140 220 0 535]{F12.ps}
\begin{center}
\parbox{5.5in}{
\caption[]{\small Corrections to the lightest chargino mass, as in
Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[gl]{10}.
\label{c1}}}
\end{center}
\end{figure}
Figure \docLink{slac-pub-7180-0-0-4.tcx}[c1]{12} shows, similarly, the corrections to the lightest
chargino mass. Again there is a term proportional to $\mu$ which
dominates the finite corrections when $M_2$ is small. In this region,
the finite corrections can be as large as 10\%. In Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[c1]{12}(d)
we show that the difference between our approximate correction and the
full one-loop mass is less than 2\%. These corrections are
quantitatively similar to the corrections to the second-lightest
neutralino mass.
The corrections to the heavy chargino mass are shown in Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[c2]{13}.
These corrections are less than a few percent, as are the corrections
for the two heaviest neutralino masses. The logarithmic corrections
are in the range 0 to 2.5\%, and the finite corrections are in the
range 0 to $-3\%$. Figure \docLink{slac-pub-7180-0-0-4.tcx}[c2]{13}(d) shows that our approximation
for the heavy chargino mass generally holds to better than 0.5\%. Our
approximation also works to typically better than 1\% for the two
heaviest neutralino masses, but it can be off by nearly 2\%.
\begin{figure}[t]
\epsfysize=2.5in \epsffile[-123 220 17 535]{F13.ps}
\begin{center}
\parbox{5.5in}{
\caption[]{\small Corrections to the heaviest chargino mass, as in
Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[gl]{10}.
\label{c2}}}
\end{center}
\end{figure}