\section{Introduction}
\setcounter{equation}{0}\setcounter{footnote}{0}

The appearance of infinity in quantum field theory has long been one
of the annoying problems. Most of particle physicists would believe
that the ultimate theory, if it exits, should not contain any infinity
and needs no renormalization procedure. In the '80s, it was pointed
out that the requirement of no quadratic divergence leads a kind of
symmetry, supersymmetry \cite{2div}. Then it is interesting to
consider what symmetry appear due to the requirement of vanishing even
logarithmic divergence (vanishing $\beta$-functions) in supersymmetric
theories. Among these theories, $N=4$ and some $N=2$ theories have
zero $\beta$-functions, what is called finiteness, in all-order of
perturbation theory \cite{finitesusy} and they are believed that there
are duality symmetries in those theories \cite{duality}.  In this way,
imposing that there are no infinities in theories corresponds to very
important symmetries until now.  If that is the case, what happens in
$N=1$ theory from the requirement of finiteness?  In perturbative
region, there is the classification table of models in which gauge and
Yukawa couplings satisfy the conditions of finiteness in 1-loop order
\cite{table}. Moreover the all-order finiteness condition for these
couplings is also found \cite{all}. $\,\beta = 0\,$ is also strongly
related to the non-perturbative dynamics such as the electro-magnetic
duality transformation proposed by Seiberg \cite{seiberg}. However,
what (symmetry) corresponds to these finiteness conditions has not
been answered yet.

On the other hand from a viewpoint of low-energy phenomenology, $N=1$
supersymmetry which would come from the requirement of vanishing
quadratic divergences has provided us with many interesting
consequences. Therefore it will be surely important and become a first
step toward understanding of the meanings of finiteness to analyze
phenomenological results as a consequence of vanishing logarithmic
divergence.  The finiteness conditions stated in section 2 prohibit us
from applying them to U(1) gauge theory, and to MSSM.  Then we
apply these conditions to grand unified theory (GUT) and derive the
boundary conditions at GUT scale for couplings of low-energy theory
which we suppose to be MSSM. In particular in case of SU(5)
GUT models, many articles have obtained interesting results of the
fermion masses, superparticle masses, etc.\ \cite{fut,fut2,afut}.

In this paper we analyze this SU(5) model taking care of the following
two points which have not been included explicitly so far.
(1) Since we use the 2-loop order $\beta$-functions for
MSSM couplings we need to include the 1-loop order threshold
corrections \cite{weinberg}.  Especially, it is important in large
$\tan \beta$ model to include a SUSY threshold correction to bottom
quark mass $m_b$ which can be $20 \!\sim\! 40\,$\% of the uncorrected
value \cite{bot}.  (2) Furthermore in large $\tan \beta$ model, we
must check whether the Higgs potential can really generate the
radiative electroweak symmetry breaking \cite{sb} with the large value
of $\tan \beta$. To carry out this, it is necessary to take account of
a new parameter to the boundary condition for the Higgs mixing mass
parameter.

We briefly review the finiteness conditions in $N=1$ supersymmetric
gauge theory and its application the finite SU(5) model in section
2. In section 3 we consider the matching conditions of this finite
SU(5) model to MSSM and calculate low-energy predictions. The GUT and
SUSY threshold corrections which are characteristic to this model are
discussed in section 4. Section 5 is devoted to summary and
comments. The appendices contain the explicit tree-level form of
mass formulae and the SUSY threshold corrections to gauge couplings in
MSSM.

\vspace*{3mm}

\section{Finite SU(5) model}
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First we describe the all-order finiteness conditions for gauge
couplings and the couplings in superpotential sector.
We consider an anomaly free $N=1$ supersymmetric gauge theory based on
a simple gauge group $G$ with a gauge coupling $g$, and with the
superpotential

\begin{equation}
  W = \frac{1}{2} m_{ij}\Phi^i\Phi^j +
  \frac{1}{6}Y_{ijk}\Phi^i\Phi^j\Phi^k\,.
  \label{W}
\end{equation}
The 1-loop $\beta$-functions are given by
\begin{eqnarray}
  \beta_g^{\,(1)} &\!\! = \!\!& \frac{-g^3}{16\pi^2}\left( 3C_2(G) -
    \sum T(R_i) \right)\,, \\[2mm]
  \beta_{ij}^{\,(1)} &\!\! = \!\!& m_{ik}\gamma^{k\,(1)}_j +
  m_{jk}\gamma^{k\,(1)}_i\,, \\[2mm]
  \beta_{ijk}^{\,(1)} &\!\! = \!\!& Y_{ijl}\gamma^{l\,(1)}_k +
  Y_{jkl}\gamma^{l\,(1)}_i + Y_{kil}\gamma^{l\,(1)}_j\,, \\[3mm]
  && \hspace*{-5mm} \gamma^{i\,(1)}_j = \frac{1}{32\pi^2}\left (
    Y^{ikl} Y_{jkl}-4g^2C_2(R_i) \,\delta^i_j \right) \qquad
  ( Y^{ijk}\equiv Y_{ijk}^*)\,,
\end{eqnarray}
where $\gamma^{i\,(1)}_j$ are the 1-loop anomalous dimensions of the
field $\Phi^i$ and
\begin{eqnarray}
  \mbox{tr} (\,T^a T^b) = T(R)\delta^{ab}\,,\quad \sum_a \,T^aT^a =
  C_2(R)\underline{1} \;,\quad \sum_{c\,,\,d} f_{acd}f_{bcd} =
  C_2(G)\delta_{ab}\,,
\end{eqnarray}
$f_{abc}$ are the structure constants of the gauge group $G$.
Then the necessary and sufficient conditions for 1-loop
order finiteness\footnote{Interestingly enough, these conditions are
  necessary and sufficient
  for 2-loop order finiteness \cite{1to2}. }
(to contain no divergence) are $\beta_g^{\,(1)} = \gamma^{i\,(1)}_j =
0\,$;
\begin{eqnarray}
  3C_2(G) - \sum T(R_i) &\!\! = \!\!& 0 \,, \label{fc1-1} \\[2mm]
  Y^{ikl} Y_{jkl}-4g^2C_2(R_i) \,\delta^i_j  &\!\! = \!\!& 0\,.
  \label{fc1-2}
\end{eqnarray}
That is, the field contents satisfy the condition (\ref{fc1-1}) and
the conditions (\ref{fc1-2}) possess ($g$-expansion) solutions of the
form
\begin{eqnarray}
  \hspace*{1cm}Y_{ijk} = \rho_{ijk}\, g \,,\qquad\;\; \rho_{ijk} \in
  \bb{C}\;(\mbox{const.})\,.
\end{eqnarray}
The field contents which satisfy these conditions are listed in
Ref.$\,$\cite{table}.
In addition to the above conditions it is necessary to impose one more
condition so that this theory may have no divergence in all-order of
perturbation theory. This condition says that the solutions of
vanishing one-loop anomalous dimensions, $\gamma^{i\,(1)}_j = 0$, are
isolated and non-degenerate\footnote{By ``isolated'' and
  ``non-degenerate'', we mean that the solutions  cannot be multiple
  zeroes and are not parameterized either. }
when considered as solutions of vanishing one-loop Yukawa
$\beta$-functions, $\beta_{ijk}^{\,(1)} = 0$ \cite{all}.
Surprisingly enough, the conditions for all-order finiteness can be
expressed in term of the 1-loop order quantities ($\beta$-functions).
Therefore one may easily apply these conditions to definite models.

Next we discuss soft SUSY breaking sector. In general, this part of
potential takes the form (for simple gauge group $G$),
\begin{eqnarray}
  V_{soft} = {1\over 2} (\mu^2)_j^i \phi^*_i \phi^j + {1\over 2}
  B_{ij} \phi^i \phi^j + {1\over 6} h_{ijk} \phi^i \phi^j \phi^k +
  {1\over 2} M \lambda \lambda + {\rm h.c.}
\end{eqnarray}
where here and hereafter the fields $\phi^i$ in $V_{soft}$ denote the
scalar component of those superfields and $\lambda$ means gaugino.
For these parameters in $V_{soft}$,\ the (renormalization-group
invariant) finiteness conditions are found within only 2-loop order
\cite{soft} unlike for the dimensionless couplings.  These conditions
are given by
\begin{eqnarray}
  h_{ijk} &\!\! = \!\!& - M \,Y_{ijk}\,, \label{fc2-1} \\[2mm]
  (\mu^2)^i_j &\!\! = \!\!& \displaystyle\frac{1}{3} \,M
  M^*\delta^i_j\,,
  \label{fc2-2}
\end{eqnarray}
together with (\ref{fc1-1}), (\ref{fc1-2}).\footnote{The
  renormalization-group invariant relation for $B^{ij}$ \cite{soft} is
  not required by finiteness (within 2-loop level). } \
With these conditions, the theory has $\beta$-functions that all
vanish at least to 2-loop order. It is interesting to note that these
universal forms of finiteness condition are the same as those derived
from superstring or $N=1$ supergravity models \cite{soft}.

As a concrete example of phenomenological applications of the
finiteness conditions, we consider supersymmetric SU(5) GUT
models.  According to the classification tables in
Ref.$\,$\cite{table}, \ there exit only one field content which
fulfills the following requirements;
\vspace*{-1mm}
\begin{list}{$\cdot$}{}
\item It contains chiral three families (three $( {\bf {\bar
      5}}\,,\,{\bf 10} )$ sets). \vspace*{-3mm}
\item The remains of the contents are vector-like ones.
  \vspace*{-3mm}
\item It contains fields in an adjoint representation to break
  the GUT gauge group.
\end{list}
\vspace*{-1mm}
This model contains $( \bf
5\,,\,\bar{5}\,,\,10\,,\,\overline{10}\,,\,24  )$
with the multiplicities $( 4\,,\,7\,,\,3\,,\,0\,,\,1 )$. Then the
general superpotential for this contents is
\begin{eqnarray}
  W &\!\! = \!\!& {1\over 2} f_{ija} 10_i 10 _j H_a + {\bar f}_{ija}
  10_i {\bar 5}_j {\bar H}_a +{1\over 2} q_{ijk} 10_i
  {\bar 5}_j {\bar 5}_k \\[2mm]
  &&\hspace*{0.1cm} + {1\over 2} q^\prime_{ia b}
  10_i {\bar H}_a {\bar H}_b +f_{a b} {\bar H}_a \Sigma H_b + p
  \,\Sigma^3 + f_{i a} {\bar 5}_i \Sigma H_a + W_m  \nonumber
\end{eqnarray}
\begin{center}
  $(i,j = 1, 2, 3 \,,\quad a,b = 1, \cdots ,4)$
\end{center}
where $W_m$ contains the mass terms of $H_a\,,\,{\bar
  H}_a\,,\,\Sigma$. \
It is sufficient to impose following discrete symmetries in order to
get isolate and non-degenerate solutions and suppress the rapid
nucleon decay \cite{afut}.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
  \hline & $10_1$ & $10_2$ & $10_3$ & $\bar 5_1$ & $\bar 5_2$ & $\bar
  5_3$& $H_1$ & $H_2$ & $H_3$ & $H_4$ & $\Sigma$  \\ \hline
  $Z_7$ & 1 & 2 & 4 & 4 & 1 & 2 & 5 & 3 & 6 & 0 & 0 \\ \hline
  $Z_3$ & 1 & 2 & 0 & 0 & 0 & 0 & 1 & 2 & 0 & 0 & 0 \\ \hline
  $Z_2$ & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ \hline
\end{tabular}\\
\vspace*{0.5cm}
  Table 1 : The charges of the $Z_7\times Z_3\times Z_2$ (matter
  parity) \vspace*{0.1cm}

($\overline{H}_a$ has an opposite charge to that of $H_a$.)
\end{center}
With these symmetries, the superpotential is restricted to the form,
\begin{eqnarray}
  W = {1\over 2} f_{iii} 10_i 10 _i H_i + {\bar
    f}_{iii} 10_i {\bar 5}_i {\bar H}_i +f_{aa} {\bar H}_a
  \Sigma H_a + p \,\Sigma^3 + W_m
\label{sp}
\end{eqnarray}
and the unique solution which guarantees the all-order finiteness are
given by \cite{fut,fut2,afut}
\begin{eqnarray}
\begin{array}{c}
  f_{111} = f_{222} = f_{333} = \displaystyle{\sqrt{8\over 5}}\, g
  \,,~~~~~ {\bar f}_{111} = {\bar f}_{222} = {\bar
    f}_{333} = \displaystyle{\sqrt{6\over 5}}\, g \,,\\[4mm]
  \!\!\!\!f_{11} = f_{22} = f_{33} = 0 \,,~~~~~f_{44} = g\,,~~~~~p =
  \displaystyle{\sqrt{15\over 7}}\, g\,.
\end{array}
\label{sol1}
\end{eqnarray}

Then we apply the finiteness conditions to $V_{soft}$ to get final
expression of this SU(5) model. The general potential takes the form
\begin{eqnarray}
  V_{soft} &\!\! = \!\!& (m^2_5)_{ij} \bar{5}^\dagger_i \bar{5}_j +
  (m^2_{10})_{ij} 10^\dagger_i 10_j + (m^2_H)_{ab} H_a^\dagger H_b +
  (m^2_{\bar H})_{ab} \bar{H}_a^\dagger \bar{H}_b \nonumber \\[3mm]
  && \quad +\, m^2_{\Sigma} \Sigma^\dagger \Sigma + \biggl
  ( \;\frac{1}{2} M \lambda \lambda + B_{ab} \bar{H}_a H_b +
  B_{\Sigma} \Sigma \Sigma \label{pot} \\[2mm] &&
  \qquad  + \frac{h_{ijk}}{2} 10_i 10_j H_k + {\bar h}_{ijk} 10_i
    {\bar 5}_j {\bar H}_k + h_{ab} {\bar H}_a \Sigma H_b + h_p
    \,\Sigma^3 + \mbox{h.c.} \biggr)\,. \nonumber
\end{eqnarray}
Taking into account the form of the superpotential (\ref{sp}), we can
get the relations among the soft parameters for 2-loop order
finiteness from (\ref{fc2-1}) and (\ref{fc2-2})$\,$;
\begin{eqnarray}
  \begin{array}{c}
    h_{iii} = - M f_{iii} \,,\quad {\bar h}_{iii} = - M
    \bar{f}_{iii}\,,\quad h_{44} = - M f_{44} \,,\quad h_p = - M p\,,
    \\[5mm]
    (m^2_H)_{ab} = (m^2_{\bar H})_{ab} = \displaystyle{\frac{1}{3}}
    M^2 \delta_{ab}\,,\quad m^2_{\Sigma} = \displaystyle{\frac{1}{3}}
    M^2\,,\quad (m^2_{10})_{ij} = (m^2_5)_{ij} =
    \displaystyle{\frac{1}{3}} M^2 \delta_{ij}\,,
  \end{array}
  \label{sol2}
\end{eqnarray}
and all other elements are zero. This, together with the relations
(\ref{sol1}), provides us with the finite\footnote{All-order finite
  for gauge and Yukawa couplings, whereas (at least) 2-loop order
  finite for soft SUSY breaking parameters. } SU(5) model above GUT
scale.
Now, note that from the conditions of finiteness there are no
constraints for the B-parameters in the potential (\ref{pot}) as well
as the supersymmetric Higgs mass parameters.  These are to be
determined by the requirements of low-energy assumptions, which is the
task of the next section.

\section{Matching to MSSM and low-energy predictions}
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In this section, we analyze low-energy predictions of the finite
SU(5) model. This model which is supposed to break spontaneously to
MSSM at GUT scale $M_G$ casts the boundary conditions for the
couplings of MSSM from the finiteness conditions. Leaving GUT
threshold corrections to gauge and Yukawa couplings in the next
section, we first consider the matching of the parameters between MSSM
and the finite SU(5) at tree level.

The matching conditions for gauge couplings are trivial,
\begin{eqnarray}
  g_1(M_G) = g_2(M_G) = g_3(M_G) = g \,.
\label{gau}
\end{eqnarray}
where $g_1\,,\,g_2\,,\,g_3$ are gauge couplings of
$U(1)_Y\,,\,SU(2)_W\,,\,SU(3)_C$.

As for Yukawa couplings, one may think that from the solutions
(\ref{sol1}) a pair of light Higgs doublet which result from the
doublet-triplet splitting mechanism don't couple to any of the matter
fields $({\bf {\bar 5}\,,\,10})$.
However as mentioned in the end of section 2, the Higgs mass
parameters in $W_m$ are not constrained from finiteness and have a
freedom for tuning of mass parameters of $H \bar H$ to cause a Higgs
mixing at $M_G$. In this paper we take the nonzero couplings
of the Higgs doublets to only the third generation matter fields but
the Yukawa couplings for the first and second
generations are obtained in the same way \cite{softfut}.

After SU(5) symmetry breaking, we suppose the supersymmetric $H \bar
H$ mass terms take the form,
\begin{eqnarray}
  W'_m  &\!\! = \!\!& f_{44}\bar{H}_4 \left\langle\,
    \Sigma\,\right\rangle H_4 + M_1
  \bar{H}_1 H_1 + M_2 \bar{H}_2 H_2 + M_{ab} \bar{H}_a H_b
  \,,\quad (a,b = 3,4) \label{m12} \\[2mm]
  && \qquad\qquad\qquad M_1\,,\;M_2 \sim M_G \,. \nonumber
\end{eqnarray}
Substituting $\left\langle\, \Sigma\,\right\rangle = \omega\cdot
\mbox{diag}(2,2,2,-3,-3)$ into $W_m$,
the mass terms of $H_3$ and $H_4$ become as follows;
\begin{eqnarray}
  && \bar{H}_a^{(3)}M_{ab}^{(3)}H_b^{(3)} \equiv \bar{H}_a^{(3)}\left
    ( M_{ab} + \pmatrix{0& \cr &2\cr}\omega f_{44}
  \right)H_b^{(3)}\,,\\[2mm]
  &&\bar{H}_a^{(2)}M_{ab}^{(2)}H_b^{(2)} \equiv \bar{H}_a^{(2)}\left
    ( M_{ab} + \pmatrix{0& \cr &-3\cr}\omega f_{44}
  \right)H_b^{(2)}\,, \label{h2m} \\[2mm]
  && \quad\qquad H = \pmatrix{H^{(3)} \cr H^{(2)} \cr}\,, \qquad
  \bar{H} = \pmatrix{\bar{H}^{(3)} \cr \bar{H}^{(2)} \cr}\,.
\end{eqnarray}
We diagonalize $M^{(2)}$ of the pair of light Higgs doublets
\begin{eqnarray}
  &&\!\!H'_a = \pmatrix{\cos \theta & \sin \theta \cr -\sin \theta &
    \cos \theta \cr}H_a \,,\quad \bar{H}'_a = \pmatrix{\cos
    \bar{\theta} & \sin \bar{\theta} \cr -\sin \bar{\theta} & \cos
    \bar{\theta} \cr}\bar{H}_a \,,
\end{eqnarray}
\begin{eqnarray}
  &&\hspace*{-5mm}M^{(2)'} = \pmatrix{\cos \bar{\theta} & \sin
    \bar{\theta} \cr -\sin \bar{\theta} & \cos \bar{\theta}
    \cr}M^{(2)}\pmatrix{\cos \theta & -\sin \theta \cr \sin \theta &
    \cos \theta \cr} \equiv \pmatrix{\mu&0 \cr 0&\mu' \cr} \,,
  \label{h2m'} \\[3mm]
  &&\quad \qquad\qquad \mu \ll M_G  \; ( \sim
  M_W )\,,\quad  \mu' \sim  M_G
\end{eqnarray}
which leads to the triplet Higgs mass terms$\,$;
\begin{eqnarray}
  &&\hspace*{-1cm}M^{(3)'} = \pmatrix{\mu&0 \cr 0&\mu' \cr} + 5 \omega
  f_{44} \pmatrix{\sin \theta \sin \bar{\theta} &\cos \theta \sin
    \bar{\theta} \cr \sin \theta \cos \bar{\theta} & \cos \theta \cos
    \bar{\theta} \cr} \,.
\label{h3m'}
\end{eqnarray}
After this rotation, the Yukawa couplings of the third generations to
the light Higgs become
\begin{eqnarray}
  f_{333}' = f_{333}\cos \theta \,,\qquad \bar{f}_{333}' =
  \bar{f}_{333} \cos \bar{\theta} \,.
\label{rot}
\end{eqnarray}
{}From this, we can consider the following two separate cases:
\begin{list}{$\cdot$}{}
\item case (1) : \ $\cos \theta = \cos \bar{\theta} \,\sim\, 1$ \ \ \
  ( large $\tan\beta$ ). \vspace*{-2mm}
\item case (2) : \ $\cos \theta \sim 1\,,\, \cos \bar{\theta} \sim 0$
  \ ( small $\tan\beta$ ). \vspace*{-1mm}
\end{list}
In this paper, we use the next values for each case; (1) $\cos \theta
= \cos \bar{\theta} = 0.9856 \;( \sin \theta = \sin \bar{\theta} =
0.169 )$ and (2) $\cos \theta = 0.954\,,\,\cos \bar{\theta} = 0.03 \;
( \sin \theta = 0.3\,,\,\sin \bar{\theta} = 0.9995 )$.\footnote{For
  $\mu \sim 100$ GeV, $\mu' \sim 10^{16}$ GeV, we can take the mass
  parameters: \ (1) $M_{33} \sim 0.6\,,\,M_{34} = M_{43} \sim
  3.4\,,\,M_{44} \sim 20\,$ and (2) $M_{33} \sim 3.0\,,\,M_{34} \sim
  9.5\,,\,M_{43} \sim 0.1\,,\,M_{44} \sim 0.3\;\;(\times 10^{16}\,
  \mbox{GeV}\,)$. }
\ With the above assumptions the superpotential in MSSM take the form,
\begin{equation}
  W = y_t H Q_3\overline{t} + y_b \bar{H} Q_3\overline{b} +
  y_{\tau} \bar{H} L_3\overline{\tau} + \rho \bar{H} H \,.
\end{equation}
In the end, for these couplings the matching conditions at $M_G$ turn
out,
\begin{eqnarray}
  y_t (M_G) = f'_{333} = f_{333} \cos \theta \,,&\!\!\!\!\!& y_b(M_G)
  = y_\tau(M_G) = \bar{f}'_{333} = \bar{f}_{333} \cos \bar{\theta}\,,
  \label{yukawa} \\[3mm]
  &&\hspace*{-6mm}\rho \, (M_G) = \mu\,. \label{mu}
\end{eqnarray}

Finally, we consider the matching conditions in the soft SUSY breaking
sector.  In MSSM, a general form for this sector is
\begin{eqnarray}
  V_{soft} &\!\! = \!\!& m_1^2 \bar{H}^{\dagger}\bar{H} + m_2^2
  H^{\dagger}H + (m_3^2 \bar{H} H + \mbox{h.c.} ) \nonumber \\[3mm]
  && + \sum_{i=1,2,3} \left( m_{\tilde Q_i}^2 |Q_i|^2 + m_{\tilde
      L_i}^2 |L_i|^2 + m^2_{\tilde{u}_i} |\bar{u}_i |^2
    +m^2_{\tilde{d}_i} |\bar{d}_i |^2 + m^2_{\tilde{e}_i} |\bar{e}_i
    |^2\right)\nonumber \\[1.5mm]
  && \qquad + \left( h_t H Q_3\bar{t} + h_b \bar{H} Q_3\bar{b} +
    h_{\tau} \bar{H} L_3\bar{\tau} + \mbox{h.c.} \right) \\[2mm]
  && \qquad + \frac{1}{2} \Bigl( M_{\lambda_1} \lambda_1\lambda_1 +
  M_{\lambda_2} \lambda_2\lambda_2 + M_{\lambda_3} \lambda_3\lambda_3
  + \mbox{h.c.} \Bigr)\,. \nonumber
\end{eqnarray}
Taking into account the above Higgs rotation, the matching conditions
for these parameters (except for $m_3^2\,$) become as follows;
\begin{eqnarray}
  &&\hspace*{-8mm} h_t(M_G) = h_{333}\cos \theta \,,\quad h_b(M_G) =
  h_\tau(M_G) = \bar{h}_{333} \cos \bar{\theta}\,, \label{h}\\[2mm]
  &&\hspace*{0.2cm} m_{\tilde Q_i}^2(M_G) = m^2_{\tilde{u}_i}(M_G) =
  m^2_{\tilde{e}_i}(M_G)  = m^2_{10\,i} \,, \\[2mm]
  && \hspace*{1.2cm} m_{\tilde L_i}^2(M_G) = m^2_{\tilde{d}_i}(M_G) =
  m^2_{5\,i} \\[2mm]
  &&\hspace*{-0.1cm} m_1^2(M_G) = (m^2_{\bar{H}'})_{33}\,,\qquad
  m_2^2(M_G) = (m^2_{H'})_{33}\,, \\[2mm]
  &&\hspace*{0.3cm}M_{\lambda_1}(M_G) = M_{\lambda_2}(M_G) =
  M_{\lambda_3}(M_G) = M \,. \label{m}
\end{eqnarray}
As a consequence of finiteness, from (\ref{sol1}), (\ref{sol2}),
(\ref{gau}), (\ref{yukawa}), (\ref{mu}) and (\ref{h}) $\sim$ (\ref{m})
we obtain the boundary conditions for MSSM couplings at $M_G$,
\begin{eqnarray}
  &&\hspace*{2.5cm}g_1 = g_2 = g_3 = g\,, \label{bcg}\\[1mm]
  &&\hspace*{0.3cm} y_t = \sqrt{\frac{8}{5}}\, g \cos \theta \,,\qquad
  y_b = y_\tau = \sqrt{\frac{6}{5}}\, g \cos \bar{\theta}\,,
  \label{bcy}\\[1mm]
  &&\hspace*{-0.1cm}h_t = - \sqrt{\frac{8}{5}} M g \cos
  \theta\,,\qquad h_b = h_\tau = - \sqrt{\frac{6}{5}} M g \cos
  \bar{\theta}\,, \\[1mm]
  &&\hspace*{-1.5cm}m_{\tilde Q_i}^2 = m_{\tilde L_i}^2 =
  m^2_{\tilde{u}_i} = m^2_{\tilde{d}_i} = m^2_{\tilde{e}_i} = m_1^2 =
  m_2^2 = \frac{1}{3} M^2\,,\qquad (i = 1,2,3) \\[3mm]
  &&\hspace*{1.8cm}M_{\lambda_1} = M_{\lambda_2} = M_{\lambda_3} =
  M\,, \\[1mm]
  &&\hspace*{3.4cm}\rho = \mu \label{bcm}\,.
\end{eqnarray}

Furthermore we can determine a boundary condition for $m_3^2$ when we
consider to cause the doublet-triplet splitting in the soft SUSY
breaking sector as well as in $W'_m$. After SU(5) breaking, the soft
mass terms of $H^{(2)}_a$ become from (\ref{sol2}),
\begin{eqnarray}
  V_{soft}^{H^{(2)}} \,= \,\bar{H}_a^{(2)}\left( B_{ab} - M
    \pmatrix{0& \cr &-3\cr}\omega f_{44} \right)H_b^{(2)} \,.
\label{h2}
\end{eqnarray}
To complete the doublet-triplet splitting we should take \cite{fut2}
\begin{eqnarray}
  B_{ab} \,\simeq \,-M M_{ab}\,.
  \label{B}
\end{eqnarray}
Then from the equations (\ref{h2m}) and (\ref{h2m'}), we can see that
a pair of light Higgs doublet (scalar) actually survive down to the
low energy.
\begin{eqnarray}
  (\ref{h2}) &\!\! = \!\!& -M \bar{H}_a^{(2)}\left( M_{ab}
    +\pmatrix{0& \cr &-3\cr}\omega f_{44} \right)H_b^{(2)} \\[2mm]
  &\!\! = \!\!& -M \bar{H}^{(2)'}\pmatrix{\mu& 0 \cr 0 & \mu' \cr}
  H^{(2)'}\,.
\end{eqnarray}
Taking into account the uncertainty in the condition (\ref{B}), here
we introduce a free parameter $\delta m^2_3$ in the boundary condition
for $m_3^2\,$;
\begin{eqnarray}
  m^2_3(M_G) =  - M \mu + \delta m^2_3 \,,\qquad (\, |\delta m_3^2|
  \>\raisebox{0.4ex}{\mbox{$<$}}
  \hspace{-1.8ex}\raisebox{-0.8ex}{\mbox{$\sim$}}\> M^2_{SUSY})
  \label{m3}
\end{eqnarray}
which plays an important role in the following analyses\footnote{This
  parameter is necessary in order to cause the radiative symmetry
  breaking. }.
With the above boundary conditions (\ref{bcg}) $\sim$ (\ref{bcm}) and
(\ref{m3}) for MSSM couplings, we analyze the low-energy predictions
in each case (1), (2).  Although these conditions are universal and
very restrictive, we can find the parameter region where these
predictions are consistent with the low-energy experimental values.
When the threshold corrections are neglected, the analysis
procedure is as follows.
\\

\noindent$\bullet$ case (1) : ( large $\tan \beta$ model )
\setcounter{footnote}{0}

In this case we have five free parameters, $g\,,\,M_G\,,\,M\,,\,\mu$
and $\delta m^2_3$.$\;$ Since we don't deal with the threshold
corrections in this section, we can treat $M_{SUSY}$ and $\tan \beta$
just like as free parameters in the procedure.  At first
we input $g\,,\,M_G\,,\,M_{SUSY}$ and $\tan \beta$, and run the
dimensionless couplings down to $M_Z$ by using 2-loop
$\beta$-functions \cite{2beta}.  Then we can tune these four input
parameters to reproduce the low-energy values which are consistent
with the experimental data \cite{exp},
\begin{eqnarray}
  \alpha_1(M_Z) &\!\! = \!\!& 0.01689 \pm 0.00005 \,,\\
  \alpha_2(M_Z) &\!\! = \!\!& 0.03322 \pm 0.00025 \,,\\
  \alpha_3(M_Z) &\!\! = \!\!& 0.12 \pm 0.01 \,,\\
  m_b(M_Z) = 3.1 \pm 0.4 \;\mbox{GeV}\,,&& m_\tau(M_Z) = 1.75 \pm 0.01
  \>\mbox{GeV} \,,\\
  \quad M_Z &\!\! = \!\!& 91.187\;\mbox{GeV} \,.
\end{eqnarray}
The dimensionfull parameters are not included in these
$\beta$-functions and we neglect the threshold corrections,
therefore the values of $M\,,\,\mu$ and $\delta m^2_3$ give no effects
to this tuning.
Next with the input value of $M$ in addition to the above parameter
set, we calculate the dimensionfull parameters at $M$ and tune $M$ so
that $M_{SUSY}$ in this set may be equal to a value of the following
quantity;
\begin{equation}
  M_{SB} \equiv \frac{1}{4}\left( 2 m_{{\tilde Q}_3}^2(M) + m_{{\tilde
        u}_3}^2(M) + m_{{\tilde d}_3}^2(M) \right)\,.
\end{equation}
Similarly, this adjustment of $M$ is independent of the input values
of $\mu$ and $\delta m^2_3$, since none of the $\beta$-functions of
other couplings contain $\rho$ and $m_3^2$ parameters.
In the last step, we tune $\mu$ and $\delta m^2_3$ so that the
low-energy Higgs potential actually realize the value of $\tan \beta$
in this parameter set and fulfill the constraints for radiative
electroweak symmetry breaking,
\begin{eqnarray}
  \tan^2 \beta &\!\! = \!\!& \frac{m_1^2 + \rho^2 +
    \frac{1}{2}M_Z^2}{m_2^2 + \rho^2 + \frac{1}{2}M_Z^2} \,, \\
  \sin 2\beta &\!\! = \!\!& \frac{ -2 m_3^2 }{m_1^2 + m_2^2 + 2\rho^2}
  \,.
\end{eqnarray}
At this stage, we should incorporate the 1-loop corrections to
the minimization of the Higgs potential in order to improve a
very sensitive dependence of VEV on the renormalization
point $Q$ \cite{higgs}.\footnote{To treat this more precisely, we
  should make use of the improving effective potential method
  \cite{bando}.}
\ The 1-loop corrected Higgs potential is
\begin{eqnarray}
  V = V^{(0)} + V^{(1)}\,,
\end{eqnarray}
\vspace*{-7mm}
\begin{eqnarray}
  V^{(0)} &\!\! = \!\!& ( m_1^2 + \rho^2 ) \bar{H}^\dagger \bar{H} +
  ( m_2^2 + \rho^2 ) H^\dagger H + ( m_3^2 \bar{H} H + \mbox{ h.c.} )
  \\[2mm]
  && \qquad\quad  +\; ( \mbox{D-terms} ) \,, \nonumber \\[2mm]
  V^{(1)} &\!\! = \!\!& \frac{1}{64\pi^2} \,\mbox{STr}\left[ {\cal
      M}^4\left ( \mbox{ln} \frac{{\cal M}^2}{Q^2} - \frac{3}{2}
    \right)\right]\,,
\end{eqnarray}
where ${\cal M}^2$ is a field-dependent mass-squared matrix and
$\mbox{STr} \,{\cal A} = \sum_j \,(-1)^{2j}\,(2j+1) \,\mbox{Tr} {\cal
  A}_j$ is a weighted supertrace. The explicit expression for ${\cal
  M}^2$ can be found in Ref.$\,$\cite{higgsexp}.
This exact 1-loop correction takes, however, a very complicated form.
So we here adopt the handy calculating method \cite{handy} which
incorporates the corrections only to the Higgs mass terms from
$V^{(1)}$.  This method is known to be almost enough to obtain
approximate form of the full 1-loop potential, because
the values of VEVs evaluated by this method are almost the
same as those obtained from the full 1-loop potential and the rapid
$Q$ dependence of the potential which mainly comes from that of the
running mass parameters becomes milder even in this method
\cite{handy}.

In this way, we can determine the input values of
$g\,,\,M_G\,,\,M\,,\,\mu$ and $\delta m^2_3\,$, and calculate the
low-energy predictions (gauge couplings and masses of fermions,
superparticles and Higgs etc.). \
We show the two type of example in Table 2, 3 (One is the parameters
set for the highest $M_{SB}$ (Table 2) and another is lowest
one. (table 3)$\,$) where $m_{\tilde x}$ are the superparticle masses
and $m_{H^\pm}\,,\,m_A\,,\,m_{H, h}$ are the Higgs scalar
masses which correspond to the charged, neutral CP-odd and neutral
CP-even ones, respectively. Their explicit tree level forms are given
in the appendix A.

The sparticle mass predictions are enough within the experimental
bounds \cite{exp}.  Note that, unlike the usual MSSM, the lightest
superparticle(LSP) is not a neutral one but
$\tau$-slepton\footnote{This problem may be avoided by considering an
  $R$-parity violating interaction $Q \bar d L$ which is needed for
  one of the possible interpretations of the high-$Q^2$ anomaly at
  HERA \cite{hera}.} because of the highly restrictive finiteness
conditions and the largeness of $y_b$ and $y_\tau$.
This property is characteristic to this finite SU(5) model with large
$\tan \beta$ and will be tested in future experiments.

\vspace*{7mm}

\noindent$\bullet$ case (2) : ( small $\tan \beta$ model )

In a same way as case (1), we need five free parameters for
radiative symmetry breaking in this case. Unlike the large $\tan
\beta$ models, there is no problem of the fine-tuning of Higgs mass
parameters (and large threshold correction to $m_b$) in small $\tan
\beta$ models. On the other hand, if we adopt the handy calculating
method to estimate the 1-loop corrections to the Higgs potential, we
should take care of another respect as mentioned in
Ref.\cite{handy}. This is that in this case the contributions to
D-term from $V^{(1)}$ near the flat direction ($\tan
\beta \sim 1$) are no longer small compared with that from the tree
level $V^{(0)}$. Therefore this method which includes only mass
corrections may become no more good approximation.  The contribution
to the quartic terms from $V^{(0)}$ is
\begin{equation}
 \sim  \frac{1}{8}( g_1^2 + g_2^2 )|v|^4 \cos^2 2 \beta \,,
  \label{flat0}
\end{equation}
and the typical one from $V^{(1)}$ is
\begin{equation}
  \sim \frac{1}{32 \pi^2} g_2^2 |v|^4 \,.
 \label{flat1}
\end{equation}
Therefore requiring that for instance, (\ref{flat1}) is within
$10\,$\% of (\ref{flat0}), we need
\begin{equation}
  \tan \beta \;\raisebox{0.2ex}{\mbox{$>$}}
  \hspace{-1.8ex}\raisebox{-1.0ex}{\mbox{$\sim$}}\; 1.35 \,.
\end{equation}
As for all other respects, we follow the analysis procedure in case
(1) and the representative result is shown in Table 4.

We can see from this, for example, that this model predicts that the
LSP is the lightest neutralino because $\mu$ can be taken smaller than
that in the large $\tan \beta$ case etc.\ and therefore this result
bears a close resemblance to the typical one in usual MSSM
\cite{mssm}.

\newpage

\begin{center}
  \renewcommand{\arraystretch}{1.25} \setlength{\doublerulesep}{1mm}
  \begin{tabular}{|c|c||c|c|} \hline
    \multicolumn{4}{|c|}{\large case (1) : example 1} \\
    \hline\hline $M_{GUT}$ & \multicolumn{2}{c}{ $1.246\times 10^{16}$
      ~(GeV)} & \\ \hline $\alpha_{GUT}$ & \multicolumn{2}{c}{ 0.0388
      } & \\ \hline $M$ & \multicolumn{2}{c}{ 612.315 ~(GeV)} & \\
    \hline $\mu$ & \multicolumn{2}{c}{ 1467.828 ~(GeV)} & \\ \hline
    $\delta m_3^2$ & \multicolumn{2}{c}{ $- (636.086)^2$ ~(GeV)} & \\
    \hline\hline $M_{SB}$ & 1000.0 ~(GeV)& $\tan \beta$ & 54.0 \\
    \hline \makebox[2cm]{ $ \alpha_1(M_Z)$} & \makebox[4cm]{0.016850}
    & \makebox[2cm]{ $m_t(M_Z)$ } & \makebox[4cm]{ 179.24 ~(GeV)} \\
    \hline $\alpha_2(M_Z)$ & 0.033333 & $m_b(M_Z)$ & 3.21 ~(GeV)\\
    \hline $\alpha_3(M_Z)$ & 0.112 & $m_\tau(M_Z)$ & 1.745 ~(GeV)\\
    \hline \hline $m_{\widetilde{t}_+}$ & 1064.3 ~(GeV)&
    $m_{\widetilde{u}_+}$ & 1243.4 ~
    (GeV)\\ \hline $m_{\widetilde{t}_-}$ & 892.4 ~(GeV)&
    $m_{\widetilde{u}_-}$ &
    1198.3 ~(GeV)\\ \hline $m_{\widetilde{b}_+}$ & 1051.2 ~(GeV)&
    $m_{\widetilde{d}_+}$ & 1245.8 ~(GeV)\\ \hline
    $m_{\widetilde{b}_-}$ & 915.1 ~
    (GeV)& $m_{\widetilde{d}_-}$ & 1193.4 ~ (GeV)\\ \hline
    $m_{\widetilde{\tau}_+}$
    & 516.2 ~(GeV)& $m_{\widetilde{e}_+}$ & 536.2 ~(GeV)\\ \hline
    $m_{\widetilde{\tau}_-}$ & 191.2 ~(GeV)& $m_{\widetilde{e}_-}$ &
    420.5 ~ (GeV)\\ \hline $m_{\widetilde{\nu}_\tau}$ & 478.5 ~(GeV)&
    $m_{\widetilde{\nu}_e}$ & 530.6 ~(GeV)\\ \hline
    $m_{\widetilde{\chi}_1^+}$ &
    837.5 ~(GeV)& $m_{\widetilde{\chi}_2^+}$ & 481.5 ~(GeV)\\ \hline
    $m_{\widetilde{\chi}_1^0}$ & 275.4 ~ (GeV)&
    $m_{\widetilde{\chi}_2^0}$ & 500.5
    ~(GeV)\\ \hline $m_{\widetilde{\chi}_3^0}$ & 794.9 ~(GeV)&
    $m_{\widetilde{\chi}_4^0}$ & 814.9 ~(GeV)\\ \hline $m_{H^{\pm}}$ &
    331.8 ~(GeV)& $m_{A}$ & 322.6 ~(GeV)\\ \hline $m_{H}$ & 322.7 ~
    (GeV)& $m_{h}$ & 89.0 ~(GeV)\\ \hline $M_{\lambda_3}$ & 1326.6 ~
    (GeV)& & \\ \hline $h_t$ & $-$ 853.6 ~ (GeV)& $h_b$ & $-$ 834.0 ~
    (GeV)\\ \hline $h_\tau$ & $-$ 107.1 ~ (GeV)& & \\ \hline
\end{tabular}
\vspace*{1cm}

Table 2 : {The low-energy predictions in case (1). ~(high $M_{SB}$)}
\end{center}

\begin{center}
  \renewcommand{\arraystretch}{1.25} \setlength{\doublerulesep}{1mm}
  \begin{tabular}{|c|c||c|c|} \hline
    \multicolumn{4}{|c|}{\large case (1) : example 2} \\
    \hline\hline $M_{GUT}$ & \multicolumn{2}{c}{ $1.116\times 10^{16}$
      ~(GeV)} & \\ \hline $\alpha_{GUT}$ & \multicolumn{2}{c}{ 0.0392
      } & \\ \hline $M$ & \multicolumn{2}{c}{ 473.385 ~(GeV)} & \\
    \hline $\mu$ & \multicolumn{2}{c}{ 1171.954 ~(GeV)} & \\ \hline
    $\delta m_3^2$ & \multicolumn{2}{c}{ $- (510.579)^2$ ~(GeV)} & \\
    \hline\hline $M_{SB}$ & 790.0 ~(GeV)& $\tan \beta$ & 54.0 \\
    \hline \makebox[2cm]{ $ \alpha_1(M_Z)$} & \makebox[4cm]{0.016930}
    & \makebox[2cm]{ $m_t(M_Z)$ } & \makebox[4cm]{ 178.9 ~(GeV)} \\
    \hline $\alpha_2(M_Z)$ & 0.033462 & $m_b(M_Z)$ & 3.20 ~(GeV)\\
    \hline $\alpha_3(M_Z)$ & 0.113 & $m_\tau(M_Z)$ & 1.745 ~(GeV)\\
    \hline \hline $m_{\widetilde{t}_+}$ & 858.0 ~(GeV)&
    $m_{\widetilde{u}_+}$ & 978.3 ~
    (GeV)\\ \hline $m_{\widetilde{t}_-}$ & 694.5 ~(GeV)&
    $m_{\widetilde{u}_-}$ &
    944.2 ~(GeV)\\ \hline $m_{\widetilde{b}_+}$ & 841.5 ~(GeV)&
    $m_{\widetilde{d}_+}$ & 981.3 ~(GeV)\\ \hline
    $m_{\widetilde{b}_-}$ & 708.2 ~
    (GeV)& $m_{\widetilde{d}_-}$ & 940.9 ~ (GeV)\\ \hline
    $m_{\widetilde{\tau}_+}$
    & 415.0 ~(GeV)& $m_{\widetilde{e}_+}$ & 416.5 ~(GeV)\\ \hline
    $m_{\widetilde{\tau}_-}$ & 111.5 ~(GeV)& $m_{\widetilde{e}_-}$ &
    326.4 ~ (GeV)\\ \hline $m_{\widetilde{\nu}_\tau}$ & 369.4 ~(GeV)&
    $m_{\widetilde{\nu}_e}$ & 409.2 ~(GeV)\\ \hline
    $m_{\widetilde{\chi}_1^+}$ &
    666.4 ~(GeV)& $m_{\widetilde{\chi}_2^+}$ & 367.9 ~(GeV)\\ \hline
    $m_{\widetilde{\chi}_1^0}$ & 215.0 ~ (GeV)&
    $m_{\widetilde{\chi}_2^0}$ & 391.1
    ~(GeV)\\ \hline $m_{\widetilde{\chi}_3^0}$ & 613.3 ~(GeV)&
    $m_{\widetilde{\chi}_4^0}$ & 638.6 ~(GeV)\\ \hline $m_{H^{\pm}}$ &
    261.8 ~(GeV)& $m_{A}$ & 250.1 ~(GeV)\\ \hline $m_{H}$ & 250.2 ~
    (GeV)& $m_{h}$ & 88.9 ~(GeV)\\ \hline $M_{\lambda_3}$ & 1042.7 ~
    (GeV)& & \\ \hline $h_t$ & $-$ 677.2 ~ (GeV)& $h_b$ & $-$ 662.5 ~
    (GeV)\\ \hline $h_\tau$ & $-$ 78.5 ~ (GeV)& & \\ \hline
\end{tabular}
\vspace*{1cm}

Table 3 : {The low energy predictions in case (1). ~(low $M_{SB}$)}
\end{center}

\begin{center}
  \renewcommand{\arraystretch}{1.25} \setlength{\doublerulesep}{1mm}
  \begin{tabular}{|c|c||c|c|} \hline
    \multicolumn{4}{|c|}{\large case (2)} \\
    \hline\hline $M_{GUT}$ & \multicolumn{2}{c}{ $1.206\times 10^{16}$
      ~(GeV)} & \\ \hline $\alpha_{GUT}$ & \multicolumn{2}{c}{ 0.0392
      } & \\ \hline $M$ & \multicolumn{2}{c}{ 332.38 ~(GeV)} & \\
    \hline $\mu$ & \multicolumn{2}{c}{ 642.65 ~(GeV)} & \\ \hline
    $\delta m_3^2$ & \multicolumn{2}{c}{ $- (255.15)^2$ ~(GeV)} & \\
    \hline\hline $M_{SB}$ & 620.0 ~(GeV)& $\tan \beta$ & 3.1 \\ \hline
    \makebox[2cm]{ $ \alpha_1(M_Z)$} & \makebox[4cm]{0.016852} &
    \makebox[2cm]{ $m_t(M_Z)$ } & \makebox[4cm]{ 177.5 ~(GeV)} \\
    \hline $\alpha_2(M_Z)$ & 0.033119 & $m_b(M_Z)$ & 3.42 ~(GeV)\\
    \hline $\alpha_3(M_Z)$ & 0.110 & $m_\tau(M_Z)$ & 1.751 ~(GeV)\\
    \hline \hline $m_{\widetilde{t}_+}$ & 685.5 ~(GeV)&
    $m_{\widetilde{u}_+}$ & 690.8 ~
    (GeV)\\ \hline $m_{\widetilde{t}_-}$ & 453.8 ~(GeV)&
    $m_{\widetilde{u}_-}$ &
    667.4 ~(GeV)\\ \hline $m_{\widetilde{b}_+}$ & 665.8 ~(GeV)&
    $m_{\widetilde{d}_+}$ & 694.4 ~(GeV)\\ \hline
    $m_{\widetilde{b}_-}$ & 619.2 ~
    (GeV)& $m_{\widetilde{d}_-}$ & 665.8 ~ (GeV)\\ \hline
    $m_{\widetilde{\tau}_+}$
    & 293.6 ~(GeV)& $m_{\widetilde{e}_+}$ & 292.8 ~(GeV)\\ \hline
    $m_{\widetilde{\tau}_-}$ & 229.0 ~(GeV)& $m_{\widetilde{e}_-}$ &
    231.0 ~ (GeV)\\ \hline $m_{\widetilde{\nu}_\tau}$ & 284.4 ~(GeV)&
    $m_{\widetilde{\nu}_e}$ & 284.4 ~(GeV)\\ \hline
    $m_{\widetilde{\chi}_1^+}$ &
    559.0 ~(GeV)& $m_{\widetilde{\chi}_2^+}$ & 246.3 ~(GeV)\\ \hline
    $m_{\widetilde{\chi}_1^0}$ & 188.7 ~ (GeV)&
    $m_{\widetilde{\chi}_2^0}$ & 296.9
    ~(GeV)\\ \hline $m_{\widetilde{\chi}_3^0}$ & 411.5 ~(GeV)&
    $m_{\widetilde{\chi}_4^0}$ & 493.9 ~(GeV)\\ \hline $m_{H^{\pm}}$ &
    654.1 ~(GeV)& $m_{A}$ & 649.5 ~(GeV)\\ \hline $m_{H}$ & 651.6 ~
    (GeV)& $m_{h}$ & 72.0 ~(GeV)\\ \hline $M_{\lambda_3}$ & 734.5 ~
    (GeV)& & \\ \hline $h_t$ & $-$ 523.1 ~ (GeV)& $h_b$ & $-$ 63.2 ~
    (GeV)\\ \hline $h_\tau$ & $-$ 17.6 ~ (GeV)& & \\ \hline
\end{tabular}
\vspace*{1cm}

Table 4 : {The low energy predictions in case (2).}
\end{center}

\section{Threshold corrections}
\setcounter{equation}{0}\setcounter{footnote}{0}

In this section we address the threshold corrections at GUT and SUSY
scales. Two types of corrections are important$\,$; One is the GUT
corrections to gauge and Yukawa couplings \cite{gth,gsth} which may be
characteristic to this finite SU(5).\footnote{The GUT corrections to
  the other parameters (gaugino masses etc.) are so small that the
  following analysis is not affected too much \cite{gutth}.}
Another is the SUSY ones \cite{gsth,sth}, especially the corrections
to $m_b$ which are important in the large $\tan \beta$ model.
These corrections to the dimensionless couplings are thought to be
important in the sense that these parameters are now precisely
measured by experiments and so including these corrections may
restrict the models and their allowed parameter regions. However,
since the MSSM couplings are highly restricted at GUT scale by
considering the finiteness conditions and the low-energy physics, that
is, the experimental data and the constraints from the Higgs
potential, there is only a little room for varying the SUSY threshold
corrections except for these signs. Therefore we consider whether the
low-energy experimental values are consistently reproduced by tuning
the GUT threshold corrections. In this paper we neglect the 1-loop
corrections from electroweak gauge boson and top quark \cite{weak}
except for the important and large corrections to the mass of the
lightest CP-even Higgs boson \cite{higgsth}.

First, we discuss the 1-loop GUT threshold corrections.
These corrections to the standard gauge couplings are found to be
\begin{eqnarray}
  \frac{2\pi}{\alpha_i(\Lambda)} = \,\frac{2\pi}{\alpha_5}\, -
  \Delta_i^G(\Lambda)\,,
\end{eqnarray}
\begin{eqnarray}
  \Delta_1^G(\Lambda) &\!\! = \!\!&  \frac{5}{2} \ln \left
    ( \frac{M_V}{\Lambda} \right) -
  \frac{1}{4}\sum_{i = 1, 2} \ln \left( \frac{M_i}{\Lambda} \right)  -
  \frac{1}{10} \sum_{i = 3, 4} \ln \left( \frac{M_i}{\Lambda} \right)
  - \frac{3}{20} \ln \left( \frac{\mu'}{\Lambda} \right) ,\; \\[1mm]
  \Delta_2^G(\Lambda) &\!\! = \!\!&  \frac{3}{2} \ln \left
    ( \frac{M_V}{\Lambda} \right) -
  \frac{1}{2} \ln \left( \frac{M_\Sigma}{\Lambda} \right) -
  \frac{1}{4}\sum_{i = 1, 2} \ln \left( \frac{M_i}{\Lambda} \right) -
  \frac{1}{4} \ln \left( \frac{\mu'}{\Lambda} \right) \,,\\[1mm]
  \Delta_3^G(\Lambda) &\!\! = \!\!&  \ln \left( \frac{M_V}{\Lambda}
  \right) - \frac{4}{3} \ln \left( \frac{M_\Sigma}{\Lambda} \right) -
  \frac{1}{4}\sum_{i = 1, 2, 3, 4} \ln \left( \frac{M_i}{\Lambda}
  \right) \,,
\end{eqnarray}
where $M_V$ is the superheavy gauge bosons mass (which is
equivalent to that of the $(3,2,\pm\frac{5}{6})$ component of adjoint
Higgs $\Sigma\,$),~ $M_i$ and $\mu'$ are the masses of
superheavy parts of $H_a\,,\,{\bar H}_a$, and $M_\Sigma$ is the mass
of the color octet and SU(2) triplet component of $\Sigma$.
These mass parameters are defined by the conditions of SU(5)
symmetry breaking and the finiteness conditions as follows;
\begin{eqnarray}
  M_V &\!\! = \!\!&  5 \sqrt{2} g \omega = \frac{10}{3} \sqrt
  { \frac{14}{15} } M_{24}  \,,\\[1.5mm]
  M_\Sigma &\!\! = \!\!& 5 M_{24} \,,\\
  M_3 &\!\! = \!\!& 5 \omega f_{44} \sin \theta \sin {\bar \theta} =
  \frac{10}{3} \sqrt{ \frac{7}{15} } \sin \theta \sin {\bar \theta}\,
  M_{24} \,,\\
  M_4 &\!\! = \!\!& \mu' + 5 \omega f_{44} \cos \theta \cos {\bar
    \theta}  = \mu' + \frac{10}{3} \sqrt{ \frac{7}{15} } \cos \theta
  \cos {\bar \theta}\, M_{24} \,.
\end{eqnarray}
$M_1\,,\,M_2\,,\,\mu'$ and $\omega$ are defined in section 3
($\,$(\ref{m12}) and
(\ref{h2m'})$\,$). $\,M_{24}$ is the supersymmetric mass parameter of
$\Sigma$,
\begin{eqnarray}
  W_m = W'_m + M_{24} \,\Sigma^{\,2}.
\end{eqnarray}
Since we set $M_G = M_V = \Lambda$, there are three free parameters,
$M_1\,,\,M_2$ and $\mu'$ in the correction formulae. Furthermore since
$H_1$ and $H_2$ sector have same structures, hereafter without loss of
generality we set $M_1 = M_2 \equiv M_H$. For the example of case 1
(large $\tan \beta$) and case 2 (small $\tan \beta$), the allowed
region for $M_H$ and $\mu'$ which reproduce the experimental values of
the gauge couplings $\alpha_{1,2,3}$ are shown in Fig.1,$\,$2
including the SUSY threshold corrections to gauge couplings (appendix
B). These SUSY threshold corrections $\Delta^S_{1,2,3}$ are about $1
\sim 2$ \%. Therefore it is found that these can be cancelled by
$\Delta^G_{1,2,3}\,$. However
this region may be rather narrowed if the proton decay constraint
$(M_H\,,\,\mu'\>\raisebox{0.2ex}{\mbox{$>$}}
\hspace{-1.8ex}\raisebox{-1.0ex}{\mbox{$\sim$}}\> 10^{16}\,
\mbox{GeV})\,$ is considered.

\begin{figure}[htbp]
  \begin{center}
    \leavevmode
    \epsfxsize=10cm \epsfysize=8cm \ \epsfbox{fus-l.eps}
    \put(15,25){\large $\log_{10} M_H$}
    \put(-270,230){\large $\log_{10} \mu'$}
    \caption{ The allowed region for $M_H$ vs. $\mu'$. (large $\tan
      \beta$ case)}
  \end{center}
\end{figure}

\begin{figure}[htbp]
  \begin{center}
    \leavevmode
    \epsfxsize=10cm \epsfysize=8cm \ \epsfbox{fus-s.eps}
    \put(15,25){\large $\log_{10} M_H$}
    \put(-270,230){\large $\log_{10} \mu'$}
    \caption{ The allowed region for $M_H$ vs. $\mu'$. (small $\tan
      \beta$ case)}
  \end{center}
\end{figure}

Next we consider the GUT corrections to Yukawa couplings,
\begin{eqnarray}
  y_t^-( \Lambda ) &\!\! = \!\!& y_t^+ ( 1 + \Delta_t^G (\Lambda))\,,
  \\
  y_b^-( \Lambda ) &\!\! = \!\!& y_b^+ ( 1 + \Delta_b^G (\Lambda))\,,
  \\
  y_\tau^-( \Lambda ) &\!\! = \!\!& y_\tau^+ ( 1 + \Delta_\tau^G
  (\Lambda))\,,
\end{eqnarray}
\begin{eqnarray}
  16 \pi^2 \Delta_t^G(\Lambda) &\!\! = \!\!& - \frac{g^2}{2} \left( 5
    F(M_V^2,0) + 3
    F(M_3^2,M_V^2) \right) + \frac{3}{2} \left( y_t^{+2} \cos^2 \theta
    + y_b^{+2} \cos^2 {\bar \theta}
  \right) F(M_3^2,0) \nonumber \\
  && + \frac{3}{2} \left( y_t^{+2} \sin^2 \theta +
    y_b^{+2} \sin^2 {\bar \theta} \right) F(M_4^2,0) \nonumber \\[1mm]
  && + \frac{1}{2} f_{44}^2 \sin^2 \theta \sin^2 {\bar \theta} \left
    ( 3 F(M_3^2,M_V^2) + \frac{3}{2} F(M_\Sigma^2,0) + \frac{3}{10}
    F((0.2M_\Sigma)^2,0) \right) \nonumber \\[1mm]
 && + \frac{1}{2} f_{44}^2 \left( \cos^2 \theta \sin^2 {\bar \theta} +
   \sin^2 \theta \cos^2 {\bar \theta} \right) \left( 3
    F(M_4^2,M_V^2) \right. \nonumber \\[1mm]
  && \left. + \frac{3}{2} F(M_\Sigma^2,\mu'^2) + \frac{3}{10}
    F((0.2M_\Sigma)^2,\mu'^2) \right) \,,
\end{eqnarray}
\begin{eqnarray}
  16 \pi^2 \Delta_b^G(\Lambda) &\!\! = \!\!& - \frac{g^2}{2} \left( 5
    F(M_V^2,0) + 3 F(M_3^2,M_V^2) \right) +  \left( y_t^{+2} \cos^2
    \theta + \frac{3}{2} y_b^{+2} \cos^2 {\bar \theta}
  \right) F(M_3^2,0) \nonumber \\
  && + \left( y_t^{+2} \sin^2 \theta + \frac{3}{2}
    y_b^{+2} \sin^2 {\bar \theta} \right) F(M_4^2,0) \nonumber \\[1mm]
  && + \frac{1}{2} f_{44}^2 \sin^2 \theta \sin^2 {\bar \theta} \left
    ( 3 F(M_3^2,M_V^2) + \frac{3}{2} F(M_\Sigma^2,0) + \frac{3}{10}
    F((0.2M_\Sigma)^2,0) \right) \nonumber \\[1mm]
  && + \frac{1}{2} f_{44}^2 \left( \cos^2 \theta \sin^2 {\bar \theta} +
    \sin^2 \theta \cos^2 {\bar \theta} \right) \left( 3 F(M_4^2,M_V^2)
  \right. \nonumber \\[1mm]
  && \left. + \frac{3}{2} F(M_\Sigma^2,\mu'^2) + \frac{3}{10}
    F((0.2M_\Sigma)^2,\mu'^2) \right) \,,
\end{eqnarray}
\begin{eqnarray}
  16 \pi^2 \Delta_\tau^G(\Lambda) &\!\! = \!\!& - \frac{g^2}{2} \left
    ( 9 F(M_V^2,0) + 3 F(M_3^2,M_V^2) \right) +  \frac{3}{2} \left
    ( y_t^{+2} \cos^2 \theta + y_b^{+2} \cos^2 {\bar \theta}
  \right) F(M_3^2,0) \nonumber \\
  && +  \frac{3}{2} \left( y_t^{+2} \sin^2 \theta +
    y_b^{+2} \sin^2 {\bar \theta} \right) F(M_4^2,0) \nonumber \\[1mm]
  && + \frac{1}{2} f_{44}^2 \sin^2 \theta \sin^2 {\bar \theta} \left
    ( 3 F(M_3^2,M_V^2) + \frac{3}{2} F(M_\Sigma^2,0) + \frac{3}{10}
    F((0.2M_\Sigma)^2,0) \right) \nonumber \\[1mm]
 && + \frac{1}{2} f_{44}^2 \left( \cos^2 \theta \sin^2 {\bar \theta} +
   \sin^2 \theta \cos^2 {\bar \theta} \right) \left( 3
    F(M_4^2,M_V^2) \right. \nonumber \\[1mm]
  && \left. + \frac{3}{2} F(M_\Sigma^2,\mu'^2) + \frac{3}{10}
    F((0.2M_\Sigma)^2,\mu'^2) \right)
\end{eqnarray}
where the superscript $+$ denotes that these couplings are GUT scale
parameters. From the finiteness conditions (\ref{sol1}) we have,
\begin{eqnarray}
  y_t^+ = \sqrt{\frac{8}{5}}\; g\,,\qquad y_b^+ = \sqrt{\frac{6}{5}}\;
  g\,,\qquad f_{44} = g \,.
\end{eqnarray}
The threshold function $F(a,b)$ is defined as follows;
\begin{eqnarray}
  F(m^2_a,m^2_b) &\!\! = \!\!& \frac{1}{ m^2_a - m^2_b } \left( m^2_a
    \ln \left( \frac{m^2_a}{\Lambda^2} \right) - m^2_b \ln \left(
      \frac{m^2_b}{\Lambda^2} \right) \right) - 1 \,.
\end{eqnarray}
The typical values of these corrections to the low-energy fermion
masses and the bottom-tau ratio are shown in Fig.3,$\,$5.

\begin{figure}[htbp]
  \parbox{7cm}{
    \epsfxsize=7cm \epsfysize=7cm \ \epsfbox{top-l.eps}
    \put(-190,210){$\displaystyle{\frac{\Delta m_t(M_Z)}{m_t(M_Z)}}$}
    }
  \hspace*{8mm}
  \parbox{7cm}{
    \epsfxsize=7cm \epsfysize=7cm \ \epsfbox{bot-l.eps}
    \put(-190,210){$\displaystyle{\frac{\Delta m_b(M_Z)}{m_b(M_Z)}}$}
    }
\end{figure}

\begin{figure}[htbp]
  \vspace*{-10mm}
  \parbox{7cm}{
    \epsfxsize=7cm \epsfysize=7cm \ \epsfbox{tau-l.eps}
    \put(-185,205){$\displaystyle{\frac{\Delta
          m_\tau(M_Z)}{m_\tau(M_Z)}}$}
    }
  \hspace*{8mm}
  \parbox{7cm}{
    \epsfxsize=7cm \epsfysize=7cm \ \epsfbox{R-l.eps}
    \put(-200,210){$\displaystyle{\frac{\Delta
          R_{b/\tau}(M_Z)}{R_{b/\tau}(M_Z)}}$}
    }
  \vspace*{3mm}
  \caption{The GUT threshold corrections to fermion masses. (large
  $\tan \beta$ case) ~The \hspace*{1.7cm}solid and dashed lines
  indicate $M_H$ and $\mu'$ dependences.}
\end{figure}

\begin{figure}[htbp]
  \begin{center}
    \leavevmode
    \epsfxsize=10cm \epsfysize=8cm \ \epsfbox{Rat.eps}
    \put(15,25){\large $m_\tau(M_Z)$}
    \put(-270,230){\large $m_b(M_Z)$}
 \end{center}
 \vspace*{-5mm}
 \caption{ The typical values of $m_b(M_Z)$ and $m_\tau(M_Z)$ for the
 allowed parameter space in large $\tan \beta$ case.}
\end{figure}

We first investigate the large $\tan \beta$ case. In this case, the
SUSY threshold correction to $m_b(M_Z)$ is very large (about $25\,$\%)
for the contribution from gluino/squark and chargino/squark diagrams
due to large $\alpha_3$ and $y_t$, especially in models with the
universal soft SUSY breaking terms like this model \cite{bot,sth}. The
sign of this important correction, however, depends on that of the
supersymmetric Higgs mass parameter $\mu$ which can be easily
changed. This change gives only a slight effects to the low-energy
parameters and the SUSY threshold corrections. The SUSY correction to
$m_\tau(M_Z)$ is also important and about $5\,$\% which always has the
opposite sign to that to $m_b$ in large $\tan \beta$ case. Taking into
account the above facts, though $m_b(M_Z)$ has an experimental
uncertainty of about $15\,$\%, it is found from Fig.3 and 4 that in
this case we cannot predict the proper value of $m_b(M_Z)$.

On the other hand in small $\tan \beta$ case, the SUSY threshold
corrections $\Delta^S_{t,\,b, \tau}$ are all about a few
percents. Therefore we may tune $\Delta^G(M_H\,,\,\mu')$ so that
the low-energy values may be properly reproduced. However in this
case, since $y_{ b, \,\tau}$ are small the ratio of bottom and tau
masses, $R_{b/\tau}$, becomes large and to make matters worse the
correction $\Delta^S_b$ always makes a positive contribution which is
independent of the sign of $\mu$-parameter. Then the experimental
bound $R_{b/\tau}(M_Z) \>\raisebox{0.4ex}{\mbox{$<$}}
\hspace{-1.8ex}\raisebox{-0.8ex}{\mbox{$\sim$}}\> 2.0$ highly
constrains the parameter spaces. In Fig.2, by considering these
experimental constraints, only the left and above region of the gray
one is allowed. An example in this allowed parameter space is
shown in Table 5 in which the 1-loop corrections to the mass of the
lightest Higgs boson, $m_h$, are also included. However it seems that
only very narrow parameter regions are left even in this case due to
the correction to $m_b$ and $m_\tau$.

\begin{figure}[htbp]
  \parbox{7cm}{
    \epsfxsize=7cm \epsfysize=7cm \ \epsfbox{top-s.eps}
    \put(-190,210){$\displaystyle{\frac{\Delta m_t(M_Z)}{m_t(M_Z)}}$}
    }
  \hspace*{8mm}
  \parbox{7cm}{
    \epsfxsize=7cm \epsfysize=7cm \ \epsfbox{bot-s.eps}
    \put(-190,210){$\displaystyle{\frac{\Delta m_b(M_Z)}{m_b(M_Z)}}$}
    }
\end{figure}

\begin{figure}[htbp]
  \vspace*{-10mm}
  \parbox{7cm}{
    \epsfxsize=7cm \epsfysize=7cm \ \epsfbox{tau-s.eps}
    \put(-185,205){$\displaystyle{\frac{\Delta
          m_\tau(M_Z)}{m_\tau(M_Z)}}$}
    }
  \hspace*{8mm}
  \parbox{7cm}{
    \epsfxsize=7cm \epsfysize=7cm \ \epsfbox{R-s.eps}
    \put(-200,210){$\displaystyle{\frac{\Delta
          R_{b/\tau}(M_Z)}{R_{b/\tau}(M_Z)}}$}
  }
  \vspace*{3mm}
  \caption{The GUT threshold corrections to fermion masses. (small
  $\tan \beta$ case) ~The \hspace*{1.7cm}solid and dashed lines
  indicate $M_H$ and $\mu'$ dependences.}
\end{figure}

\begin{center}
  \renewcommand{\arraystretch}{1.25} \setlength{\doublerulesep}{1mm}
  \begin{tabular}{|c|c||c|c|} \hline
    \multicolumn{4}{|c|}{\large case (2) with threshold corrections} \\
    \hline\hline $M_{GUT}$ & \multicolumn{2}{c}{ $1.206\times 10^{16}$
      ~(GeV)} & \\ \hline $\alpha_{GUT}$ & \multicolumn{2}{c}{ 0.0392
      } & \\ \hline $M$ & \multicolumn{2}{c}{ 332.38 ~(GeV)} & \\
    \hline $\mu$ & \multicolumn{2}{c}{ 642.65 ~(GeV)} & \\ \hline
    $\delta m_3^2$ & \multicolumn{2}{c}{ $- (255.15)^2$ ~(GeV)} & \\
    \hline\hline $M_{SB}$ & 620.0 ~(GeV)& $\tan \beta$ & 3.1 \\ \hline
    \makebox[2cm]{ $ \alpha_1(M_Z)$} & \makebox[4cm]{0.016888} &
    \makebox[2cm]{ $m_t(M_Z)$ } & \makebox[4cm]{ 180.5 ~(GeV)} \\
    \hline $\alpha_2(M_Z)$ & 0.033014 & $m_b(M_Z)$ & 3.49 ~(GeV)\\
    \hline $\alpha_3(M_Z)$ & 0.115 & $m_\tau(M_Z)$ & 1.747 ~(GeV)\\
    \hline \hline $m_{\widetilde{t}_+}$ & 694.7 ~(GeV)&
    $m_{\widetilde{u}_+}$ & 700.2 ~
    (GeV)\\ \hline $m_{\widetilde{t}_-}$ & 461.5 ~(GeV)&
    $m_{\widetilde{u}_-}$ &
    677.2 ~(GeV)\\ \hline $m_{\widetilde{b}_+}$ & 675.6 ~(GeV)&
    $m_{\widetilde{d}_+}$ & 703.7 ~(GeV)\\ \hline
    $m_{\widetilde{b}_-}$ & 627.8 ~
    (GeV)& $m_{\widetilde{d}_-}$ & 675.6 ~ (GeV)\\ \hline
    $m_{\widetilde{\tau}_+}$
    & 293.7 ~(GeV)& $m_{\widetilde{e}_+}$ & 292.9 ~(GeV)\\ \hline
    $m_{\widetilde{\tau}_-}$ & 229.0 ~(GeV)& $m_{\widetilde{e}_-}$ &
    231.1 ~ (GeV)\\ \hline $m_{\widetilde{\nu}_\tau}$ & 284.5 ~(GeV)&
    $m_{\widetilde{\nu}_e}$ & 284.5 ~(GeV)\\ \hline
    $m_{\widetilde{\chi}_1^+}$ &
    560.8 ~(GeV)& $m_{\widetilde{\chi}_2^+}$ & 245.9 ~(GeV)\\ \hline
    $m_{\widetilde{\chi}_1^0}$ & 188.2 ~ (GeV)&
    $m_{\widetilde{\chi}_2^0}$ & 296.4
    ~(GeV)\\ \hline $m_{\widetilde{\chi}_3^0}$ & 411.8 ~(GeV)&
    $m_{\widetilde{\chi}_4^0}$ & 495.1 ~(GeV)\\ \hline $m_{H^{\pm}}$ &
    650.7 ~(GeV)& $m_{A}$ & 646.1 ~(GeV)\\ \hline $m_{H}$ & 648.2 ~
    (GeV)& $m_{h}$ & 94.8 ~(GeV)\\ \hline $M_{\lambda_3}$ & 743.7 ~
    (GeV)& & \\ \hline $h_t$ & $-$ 537.3 ~ (GeV)& $h_b$ & $-$ 64.9 ~
    (GeV)\\ \hline $h_\tau$ & $-$ 17.6 ~ (GeV)& & \\ \hline
\end{tabular}
\vspace*{5mm}

\begin{list}{}{}
\item \hspace*{-7mm} Table 5 : The low energy predictions in case (2)
  including GUT and SUSY threshold corrections. ($M_H = 0.4 \times
  10^{16}$ GeV and $\mu' = 2.5 \times 10^{16}$ GeV)
\end{list}
\end{center}

\section{Summary and Comments}
\setcounter{equation}{0}\setcounter{footnote}{0}

We discussed the SU(5) model with the finiteness conditions. In
this model $\beta$-functions of gauge and Yukawa couplings (and
supersymmetric mass parameters) are zero in all-order of perturbation
theory and the all other $\beta$-functions are zero in, at least,
two-loop order. Especially we analyzed the low energy taking into
account the problem in the Higgs potential. That is, we checked
whether the Higgs potential actually fulfill the both constraints of
large $\tan \beta$ and radiative electroweak symmetry breaking
including the 1-loop corrections to the Higgs mass parameters from
heavy $(\sim M_{SUSY})$ sector. As a result, it is found that without
breaking the finiteness conditions a free parameter is needed to
satisfy these constraints.  We also estimated the GUT and SUSY
threshold corrections to the dimensionless couplings in this
model. Including these corrections left us very small
available parameter spaces in this model. \ In this paper, we
discussed the particular form of the Higgs mass matrix, and yet it is
interesting problem to analyze a more general form of this matrix in
order to investigate the proton decay constraint, light fermion
masses, the CP-violation and so on.
The alternative way to construct the realistic and restricted (GUT)
model is the coupling constant reduction method \cite{ccrgut} based on
renormalization-group invariant relations among couplings which are
the solutions of the so-called reduction equations \cite{ccr}. Though
with these relations the models is not necessarily finite, one can
reduce the number of free parameters in models and increase the
predictive power as well as models with finiteness
conditions. Moreover the application of this method to the soft SUSY
breaking sector in ordinary minimal SU(5) model leads non-universal
boundary conditions for soft mass parameters at GUT scale. Therefore
one may improve the problem of the too large value of the SUSY
threshold correction to $m_b$ in large $\tan \beta$ model unlike the
finite SU(5) model.  In either way, the success or failure of the
models and the determination of the allowed parameter region entirely
depend on the near future experiments.

\section*{Acknowledgments}

We wish to express our deep gratitude to M.Bando for many helpful
discussions and encouragements in writing this paper. We would like to
thank to J.Sato for useful comments on the manuscript. We are also
grateful to T.Kugo, N.Maekawa and K.Shizuya for useful comments on the
analysis.

\vspace*{5mm}
\noindent
{\bf Note added:} After completing this paper, we became aware of the
papers by D.R.T.\ Jones et al.\ in Refs.\ \cite{fut,2beta}.\ \
We would like to thank D.R.T.\ Jones for informing us of their works.

\vspace*{3cm}

\renewcommand{\thesection}{\Alph{section}}
\setcounter{section}{1}
\setcounter{equation}{0}\setcounter{footnote}{0}

{\Large\bf \hspace*{-7mm}Appendix A}{\large\sl \hspace*{1ex} tree
  level mass formulae in MSSM}
\\

In this appendix we express the tree level mass formulae for MSSM
particles \cite{mssm}. In the following formulae, we neglect the
Yukawa couplings for the first and second generations.
\\

\noindent $\bullet$ sfermion masses for the third generation
\begin{eqnarray}
  M_{\tilde{t}_\pm}^2 &\!\! = \!\!&  \frac{1}{2} \left( m^2_{\tilde
      Q_3} + m^2_{\tilde t} + 2 m_t^2 + \frac{1}{2} M_Z^2 \cos 2 \beta
  \right. \\[1mm] && \left. \pm \sqrt{ \left ( m^2_{\tilde Q_3} -
        m^2_{\tilde t} + \left( \frac{1}{2} - \frac{4}{3} \sin^2
          \theta_W \right) M_Z^2 \cos 2 \beta \right)^2 + 4 |m_t|^2
      \left| A_t - \rho \cot \beta \right|^2 } \,\right)\,, \nonumber
  \\[1mm]
  M_{\tilde{b}_\pm}^2 &\!\! = \!\!&  \frac{1}{2} \left( m^2_{\tilde
      Q_3} + m^2_{\tilde b} + 2 m_b^2 - \frac{1}{2} M_Z^2 \cos 2 \beta
  \right. \\[1mm] && \left. \pm \sqrt{ \left( m^2_{\tilde Q_3} -
        m^2_{\tilde b} - \left( \frac{1}{2} - \frac{2}{3} \sin^2
          \theta_W \right) M_Z^2 \cos 2 \beta \right)^2 + 4 |m_b|^2
      \left| A_b - \rho \tan \beta \right|^2 } \,\right)\,, \nonumber
  \\[1mm]
  M_{\tilde{\tau}_\pm}^2 &\!\! = \!\!&  \frac{1}{2} \left( m^2_{\tilde
      L_3} + m^2_{\tilde \tau} + 2 m_\tau^2 - \frac{1}{2} M_Z^2 \cos 2
    \beta \right. \\[1mm] && \left. \pm \sqrt{ \left( m^2_{\tilde L_3}
        - m^2_{\tilde \tau} - \left( \frac{1}{2} - 2 \sin^2 \theta_W
        \right) M_Z^2 \cos 2 \beta \right)^2 + 4 |m_\tau|^2
      \left| A_\tau - \rho \tan \beta \right|^2 }
    \,\right)\,. \nonumber
\end{eqnarray}

\noindent $\bullet$ sfermion masses for the first and second
generations
\begin{eqnarray}
  M_{\tilde{u}_+}^2 &\!\! = \!\!&  m^2_{\tilde Q_{1, 2}} + \left
    ( \frac{1}{2} - \frac{2}{3} \sin^2 \theta_W \right) M_Z^2 \cos 2
  \beta \,,\\[1mm]
  M_{\tilde{u}_-}^2 &\!\! = \!\!&  m^2_{\tilde u_{1, 2}} + \frac{2}{3}
  \sin^2 \theta_W M_Z^2 \cos 2 \beta \,,\\[1mm]
  M_{\tilde{d}_+}^2 &\!\! = \!\!&  m^2_{\tilde Q_{1, 2}} - \left
    ( \frac{1}{2} - \frac{1}{3} \sin^2 \theta_W \right) M_Z^2 \cos 2
  \beta \,,\\[1mm]
  M_{\tilde{d}_-}^2 &\!\! = \!\!&  m^2_{\tilde d_{1, 2}} - \frac{1}{3}
  \sin^2 \theta_W  M_Z^2 \cos 2 \beta \,,\\[1mm]
  M_{\tilde{e}_+}^2 &\!\! = \!\!&  m^2_{\tilde L_{1, 2}} - \left
    ( \frac{1}{2} - \sin^2 \theta_W \right) M_Z^2 \cos 2 \beta
  \,,\\[1mm]
  M_{\tilde{e}_-}^2 &\!\! = \!\!&  m^2_{\tilde e_{1, 2}} - \sin^2
  \theta_W M_Z^2 \cos 2 \beta \,.
\end{eqnarray}

\noindent $\bullet$ chargino masses
\begin{eqnarray}
  m^2_{\tilde{\chi}_{1, 2}} &\!\! = \!\!& \frac{1}{2} \left
    ( (M_{\lambda_2}^2+\rho^2+2M_W^2) \pm
    \sqrt{(M_{\lambda_2}^2+\rho^2+2M_W^2)^2-4(M_{\lambda_2}
      \rho-M_W^2\sin 2\beta)^2} \right) \,.  \nonumber \\[1mm]
  \
\end{eqnarray}

\noindent $\bullet$ neutralino masses

The neutralino mass terms are :
\begin{eqnarray}
  {\cal L}_{\mbox{nm}} \,=\, - \frac{1}{2} \pmatrix{\widetilde{B}_L &
    \widetilde{W}^3_L & \widetilde{H}^0_{1L} & \widetilde{H}^0_{2L}
    \cr} M_n \pmatrix{\widetilde{B}_L \cr \widetilde{W}^3_L \cr
    \widetilde{H}^0_{1L} \cr \widetilde{H}^0_{2L} \cr} +
  \mbox{h.c.}\,,
\end{eqnarray}
where
\begin{eqnarray}
  \hspace*{-7mm}M_n = \pmatrix{M_{\lambda_1} & 0 & -M_Z \sin \theta_W
    \cos \beta & M_Z \sin \theta_W \sin \beta \cr 0 & M_{\lambda_2} &
    M_Z \cos \theta_W \cos \beta & - M_Z \cos \theta_W \sin \beta \cr
    -M_Z \sin \theta_W \cos \beta & M_Z \cos \theta_W \cos \beta & 0 &
    - \rho \cr M_Z \sin \theta_W \sin \beta & - M_Z \cos \theta_W \sin
    \beta & -\rho & 0 \cr }~\,.
\end{eqnarray}
\\

For $M_{SUSY} \gg M_Z\,$, $\,$the eigenvalues of $M_n$ are given by
\begin{eqnarray}
  m_{\widetilde{\chi}^0_1} &\!\! = \!\!& M_{\lambda_1} + \frac{M_Z^2
    \sin^2 \theta_W}{M_{\lambda_1}^2 - \rho^2} \left( M_{\lambda_1} +
    \rho \sin 2\beta \right) \,,\\[1mm]
  m_{\widetilde{\chi}^0_2} &\!\! = \!\!& M_{\lambda_2} + \frac{M_Z^2
    \cos^2 \theta_W}{M_{\lambda2}^2 - \rho^2} \left( M_{\lambda_2} +
    \rho \sin 2\beta \right)\,,\\[1mm]
  m_{\widetilde{\chi}^0_3} &\!\! = \!\!& \rho + \frac{M_Z^2 ( 1+ \sin
    2\beta ) }{ 2 ( \rho - M_{\lambda_1} ) ( \rho - M_{\lambda_2} ) }
  \left( \rho - M_{\lambda_1} \cos^2 \theta_W -
    M_{\lambda_2} \sin^2 \theta_W \right) \,,\\[1mm]
  m_{\widetilde{\chi}^0_4} &\!\! = \!\!& \rho + \frac{M_Z^2 ( 1- \sin
    2\beta ) }{ 2 ( \rho + M_{\lambda_1} ) ( \rho + M_{\lambda_2} ) }
  \left( \rho + M_{\lambda_1} \cos^2 \theta_W + M_{\lambda_2}
    \sin^2 \theta_W \right) \,.
\end{eqnarray}
\\

\noindent $\bullet$ Higgs scalar masses
\begin{eqnarray}
  m^2_{H^\pm} \!&\!\! = \!\!& m_1^2 + m_2^2 + 2 \rho^2 + M_W^2
  \,,\\[2mm]
  m_{A}^2 \,&\!\! = \!\!& m_1^2 + m_2^2 + 2\rho^2 \,,\\[2mm]
  m^2_{H} \,&\!\! = \!\!& \frac{1}{2} \left( m^2_{A} + M_Z^2 + \sqrt
    { \left ( m^2_{A} + M_Z^2 \right)^2 - 4 m^2_{A} M_Z^2 \cos^2
      2\beta} \right) \,,\\[2mm]
  m^2_{h} \,&\!\! = \!\!& \frac{1}{2} \left( m^2_{A} + M_Z^2 - \sqrt
    { \left( m^2_{A} + M_Z^2 \right)^2 - 4 m^2_{A} M_Z^2 \cos^2
      2\beta} \right) \,.
\end{eqnarray}

\vspace*{1.5cm}

\setcounter{section}{2}
\setcounter{equation}{0}\setcounter{footnote}{0}

{\Large\bf \hspace*{-7mm}Appendix B}{\large\sl \hspace*{1ex} SUSY
  threshold corrections to gauge couplings}
\\

In this appendix we present the explicit forms of the SUSY threshold
corrections to the MSSM gauge couplings \cite{gsth,sth} in the same
approximation as appendix A. For the explicit forms for the Yukawa
coupling thresholds, see Ref.\cite{gsth,sth}.

\begin{eqnarray}
  \frac{2\pi}{\alpha^-_i(\Lambda)} \>=\>
  \frac{2\pi}{\alpha^+_i(\Lambda)} - \Delta_i^S(\Lambda) -
  \Delta_i^{DR}\,,
\end{eqnarray}
\begin{eqnarray}
  \Delta_1^S(\Lambda) &\!\! = \!\!& \frac{1}{15} \ln \left
    ( \frac{M_{\tilde{t}_+}}{\Lambda} \right) + \frac{4}{15} \ln \left
    ( \frac{M_{\tilde{t}_-}}{\Lambda} \right) - \frac{1}{5} \sin^2
  \theta_{\tilde t} \,\ln \left
    ( \frac{M_{\tilde{t}_+}}{M_{\tilde{t}_-}} \right) \nonumber
  \\[1mm]
  && - \frac{1}{30} \ln \left( \frac{M_{\tilde{b}_+}}{\Lambda} \right)
  + \frac{1}{15} \ln \left( \frac{M_{\tilde{b}_-}}{\Lambda} \right) -
  \frac{1}{10} \sin^2  \theta_{\tilde b} \ln \left
    ( \frac{M_{\tilde{b}_+}}{M_{\tilde{b}_-}} \right) \nonumber
  \\[1mm]
  && + \frac{1}{10} \ln \left( \frac{M_{\tilde{\tau}_+}}{\Lambda}
  \right) + \frac{1}{5} \ln \left( \frac{M_{\tilde{\tau}_-}}{\Lambda}
  \right) - \frac{1}{10} \sin^2 \theta_{\tilde \tau} \ln \left
    ( \frac{M_{\tilde{\tau}_+}}{M_{\tilde{\tau}_-}} \right) \nonumber
  \\[1mm]
  && + \sum_{i = 1, 2} \left\{ + \frac{1}{15} \ln \left
      ( \frac{M_{\tilde{u}_{L i}}}{\Lambda} \right) + \frac{4}{15} \ln
    \left( \frac{M_{\tilde{u}_{R i}}}{\Lambda} \right) - \frac{1}{30}
    \ln \left( \frac{M_{\tilde{d}_{L i}}}{\Lambda} \right)
  \right. \nonumber \\[1mm]
  && \left. + \frac{1}{15} \ln \left( \frac{M_{\tilde{d}_{R
            i}}}{\Lambda} \right) + \frac{1}{10} \ln \left
      ( \frac{M_{\tilde{e}_{L i}}}{\Lambda} \right) + \frac{1}{5} \ln
    \left( \frac{M_{\tilde{e}_{R i}}}{\Lambda}
    \right)  \right\} \nonumber \\[1mm]
  && + \frac{2}{5} \ln \left( \frac{m_{\tilde{\chi}_2}}{\Lambda}
  \right) - \frac{1}{5} \left( \sin^2 \theta_L + \sin^2 \theta_R
  \right) \ln \left( \frac{m_{{\tilde \chi}_2}}{m_{{\tilde \chi}_1}}
  \right)\,, \\[1mm]
  \Delta_2^S(\Lambda) &\!\! = \!\!& \frac{1}{3} \ln \left
    ( \frac{M_{\tilde{t}_+}}{\Lambda} \right) - \frac{1}{3} \sin^2
  \theta_{\tilde t} \,\ln \left
    ( \frac{M_{\tilde{t}_+}}{M_{\tilde{t}_-}} \right) + \frac{1}{3}
  \sum_{i = 1, 2} \ln \left( \frac{M_{\tilde{u}_{L i}}}{\Lambda}
  \right) \nonumber \\[1mm]
  && + \frac{1}{6} \ln \left( \frac{M_{\tilde{b}_+}}{\Lambda} \right)
  - \frac{1}{6} \sin^2 \theta_{\tilde b} \,\ln \left
    ( \frac{M_{\tilde{b}_+}}{M_{\tilde{b}_-}} \right) + \frac{1}{6}
  \sum_{i = 1, 2} \ln \left( \frac{M_{\tilde{d}_{L i}}}{\Lambda}
  \right) \nonumber \\[1mm]
  && + \frac{1}{6} \ln \left( \frac{M_{\tilde{\tau}_+}}{\Lambda}
  \right) -  \frac{1}{6} \sin^2 \theta_{\tilde \tau} \,\ln \left
    ( \frac{M_{\tilde{\tau}_+}}{M_{\tilde{\tau}_-}} \right) +
  \frac{1}{6} \sum_{i = 1, 2} \ln \left( \frac{M_{\tilde{e}_{L
          i}}}{\Lambda} \right)    \nonumber \\[1mm]
  && + \frac{4}{3} \ln \left( \frac{m_{\tilde{\chi}_1}}{\Lambda}
  \right) + \frac{2}{3} \ln \left( \frac{m_{\tilde{\chi}_2}}{\Lambda}
  \right) + \frac{1}{3} \left( \sin^2 \theta_L + \sin^2 \theta_R
  \right) \ln \left( \frac{m_{{\tilde \chi}_2}}{m_{{\tilde \chi}_1}}
  \right)\,, \\[1mm]
  \Delta_3^S(\Lambda) &\!\! = \!\!& 2 \ln \left
    ( \frac{M_{\lambda_3}}{\Lambda} \right) + \frac{1}{6} \sum_{i =
    \pm} \left\{ \ln \left( \frac{M_{\tilde{t}_i}}{\Lambda} \right) +
    \ln \left( \frac{M_{\tilde{b}_i}}{\Lambda} \right) \right\}
  \nonumber \\[1mm]
  && + \frac{1}{6} \sum_{i = 1, 2} \left\{ \ln \left
      ( \frac{M_{\tilde{u}_{L i}}}{\Lambda} \right) + \ln \left
      ( \frac{M_{\tilde{u}_{R i}}}{\Lambda} \right)+ \ln \left
      ( \frac{M_{\tilde{d}_{L i}}}{\Lambda} \right) + \ln
      \left( \frac{M_{\tilde{d}_{R i}}}{\Lambda} \right) \right\},
\end{eqnarray}
\vspace*{-2mm}
\begin{eqnarray}
  \hspace*{-3mm} \Delta_i^{DR} \,=\, -\, \frac{C_2(G_i)}{12 \pi} \,=\,
  \cases{ \displaystyle{- \frac{1}{4 \pi}} \hspace*{7mm}
    ( \;\mbox{for}\; SU(3)_C \,) \vspace*{1.5mm} \cr \vspace*{3mm}
    \displaystyle{- \frac{1}{6 \pi}} \hspace*{7mm} ( \;\mbox{for}\;
    SU(2)_W \,) \cr \hspace*{3mm} 0  \hspace*{10.5mm} ( \;\mbox{for}\;
    U(1)_Y\, ) \cr }
\end{eqnarray}
where the squark and chargino mixing angles are given by
\begin{eqnarray}
  && \hspace*{-2cm} \tan 2 \theta_{\tilde t} \,=\, \frac{2| m_t \left
      ( A_t - \mu \cot \beta \right) |}{ m^2_{{\tilde Q}_3} -
    m^2_{\tilde t} + \left( \frac{1}{2} - \frac{4}{3} \sin^2 \theta_W
    \right) M_Z^2 \cos 2 \beta } \,, \\[1mm]
  && \hspace*{-2cm} \tan 2 \theta_{\tilde b} \,=\, \frac{2| m_b \left
      ( A_b - \mu \tan \beta \right) |}{ m^2_{{\tilde Q}_3} -
    m^2_{\tilde b} - \left( \frac{1}{2} - \frac{2}{3} \sin^2 \theta_W
    \right) M_Z^2 \cos 2 \beta } \,, \\[1mm]
  && \hspace*{-2cm} \tan 2 \theta_{\tilde \tau} \,=\, \frac{2| m_\tau
    \left( A_\tau - \mu \tan \beta \right) |}{ m^2_{{\tilde L}_3} -
    m^2_{\tilde \tau} - \left( \frac{1}{2} - 2 \sin^2 \theta_W \right)
    M_Z^2 \cos 2 \beta } \,,\\[1mm]
  && \hspace*{-2cm} \tan 2 \theta_L \,=\, \frac{ 2\sqrt{2} M_W \left
      ( M_{\lambda_2} \cos \beta + \mu \sin \beta \right)}
  { M_{\lambda_2}^2 - \mu^2 - 2 M_W^2 \cos 2\beta } \,, \\[2mm]
  && \hspace*{-2cm} \tan 2\theta_R \,=\, \frac{ 2\sqrt{2} M_W \left
      ( M_{\lambda_2} \sin \beta + \mu \cos \beta \right)}
  { M_{\lambda_2}^2 - \mu^2 + 2 M_W^2 \cos 2\beta } \,.
\end{eqnarray}

\vspace*{2cm}
\setlength{\baselineskip}{15.5pt}


\section{\label{intro}Introduction}
The theoretical understanding of the damping mechanism believed to govern longitudinal magnetization relaxation in metallic ferromagnets, originally due to Kambersk\'y \cite{Kambersky1970,Kambersky1976}, has in recent years resulted in quantitative damping estimates for realistic transition metal band structures \cite{Steiauf2005, Gilmore2007, Gilmore2010}. Although of great interest where engineering of damping is desired \cite{Ralph2008}, these calculations  remain largely uncompared to experimental data. Kambersk\'y damping may be characterized by the so-called Gilbert damping constant $\alpha$ in the Landau-Lifshitz-Gilbert macrospin torque equation of motion, and formally describes how the spin-orbit interaction in itinerant electron systems results in damping of magnetization dynamics \cite{Kambersky1976}. \citet{Schoen2016} have reported that $\alpha$ is minimized for Co-Fe alloy compositions at which the density-of-states at the Fermi level is minimized, in reasonable agreement with Kambersk\'y model predictions \cite{Mankovsky2013}. Furthermore, half-metallic, or near half-metallic ferromagnets such as full-Heusler compounds have been predicted to demonstrate an ultralow Kambersk\'y $\alpha$ ($\leq 10^{-3}$) due to their spin-resolved band structure near the Fermi level \cite{Liu2009}. Finally, anisotropy of the Kambersk\'y damping in single crystals has been predicted, which is more robust for Fermi surfaces with single-band character \cite{Gilmore2010,Qu2014}.

The Gilbert damping constant is often reported through measurements of the ferromagnetic resonance (FMR) linewidth $\Delta H$, which may be expressed as a sum of individual contributions
\begin{equation}\label{LWtotal}
\Delta H = \frac{2\alpha f}{\gamma}+\Delta H_0+\Delta H_{TMS},
\end{equation}
where the first term is the Gilbert damping linewidth ($f$ is the FMR frequency, $\gamma$ is the gyromagnetic ratio), $\Delta H_0$ is a frequency-independent inhomogeneous broadening, and $\Delta H_{TMS}$ represents an extrinsic two-magnon scattering (TMS) linewidth contribution \cite{Arias1999,McMichael2004} that is, in general, a nonlinear function of frequency. In recent years it has been realized that TMS linewidths are pervasive for the conventional in-plane geometry of thin film FMR measurements, requiring either the perpendicular-to-plane FMR geometry \cite{Schoen2015} (for which TMS processes are suppressed) or sufficiently broadband measurements \cite{Woltersdorf2004} to extract the bare Gilbert $\alpha$. For instance, recent FMR linewidth studies on Heusler compounds have reported distinct TMS linewidths \cite{Mizukami2009,Qiao2016}, which challenged simple inference of the Gilbert $\alpha$.

In this article, we present FMR linewidth measurements for epitaxial Heusler thin films for all principal orientations of the magnetization with respect to the symmetry axes. For the in-plane configuration, large and anisotropic TMS-dominated linewidths are observed.  In the perpendicular-to-plane configuration, for which the TMS process is inactive \cite{Arias1999}, the Gilbert $\alpha$ and inhomogeneous broadening are measured. We find evidence of a low ($\sim$10$^{-3}$) Gilbert $\alpha$ in these Heusler thin films, accompanied by a large and anisotropic TMS contribution to the linewdith for in-plane magnetization. We conclude by discussing the interplay of low Gilbert $\alpha$ and large TMS, and we emphasize the nature by which the TMS may conceal the presence of anisotropic Kambersk\'y $\alpha$.
\section{Samples}\label{samples}
The Heusler alloy films used for these measurements were grown by molecular beam epitaxy (MBE) by co-evaporation of elemental sources in ultrahigh vacuum (UHV). The MgO(001) substrates were annealed at 700~$^{\circ}$C in UHV followed by growth of a 20 nm thick MgO buffer layer by e-beam evaporation at a substrate temperature of 630~$^{\circ}$C. The 10 nm thick Co$_2$MnAl and Co$_2$MnSi films were grown on the MgO buffer layers at room temperature and then annealed at 600~$^{\circ}$C for 15 minutes \textit{in situ} in order to improve crystalline order and surface morphology. The 24 nm thick Co$_2$FeAl sample was grown using the same MgO substrate and buffer layer preparation, but at a substrate temperature of 250~$^{\circ}$C with no post-growth anneal. Reflection high energy electron diffraction (RHEED) was monitored during and after growth of all samples and confirmed the expected epitaxial relationship of MgO(001)$\langle$110$\rangle$ $\vert\vert$ Heusler(001)$\langle$100$\rangle$. X-ray diffraction (XRD) demonstrated the existence of a single phase of (001)-oriented Heusler, along with the presence of the (002) reflection, confirming at least B2 ordering in all cases. In addition, for the Co$_2$MnSi film only, the ($111$) reflection was observed, indicating L2$_1$ ordering [see Fig. \ref{fig:1}(a)]. All of the films were capped with several nm of e-beam evaporated AlOx for passivation prior to atmospheric exposure. The effective magnetization for the 24~nm thick Co$_2$FeAl film was determined from anomalous Hall effect saturation field to be 1200~emu/cm$^3$, which is consistent with measurements of Ref.\ \cite{Cui2014} for L2$_1$ or B2-ordered films, along with 990~emu/cm$^3$ and 930~emu/cm$^3$ for the Co$_2$MnSi and Co$_2$MnAl films, respectively. Hereafter, we will refer to the Co$_2$MnSi(10~nm)/MgO as the ``CMS" film, the Co$_2$MnAl(10~nm)/MgO film as the ``CMA" film, and the Co$_2$FeAl(24~nm)/MgO film as the ``CFA'' film.
\section{\label{sec:experiment}Experiment}
Broadband FMR linewidth measurements were performed at room temperature with a coplanar waveguide (CPW) transmission setup, similar to that discussed in detail in Refs.\ \cite{Kalarickal2006, Montoya2014}, placed between the pole faces of an electromagnet. A cleaved piece of the sample ($\sim$2~mm$\times$1~mm) was placed face-down over the centerline of the CPW. A rectifying diode was used to detect the transmitted microwave power, and a $\sim$100~Hz magnetic field modulation was used for lock-in detection of the transmitted power, resulting in a signal $\propto d\chi/dH$ (where $\chi$ is the film dynamic magnetic susceptibility). The excitation frequency could be varied from 0-50 GHz, and a microwave power near 0~dBm was typically used. It was verified that all measurements discussed in this article were in the small precession cone angle, linear regime. The orientation of the applied magnetic field could be rotated to arbitrary angle in the film plane (IP), or applied perpendicular to the film plane (PP). We emphasize again that TMS contributions are suppressed in the PP configuration \cite{McMichael2004}. The resonance fields were fit as a function of applied frequency in order to extract various magnetic properties of the films.

The magnetic free energy per unit volume used to generate the resonance conditions for these samples is given by
\begin{equation}\label{eq:FT}
  F_\mathbf{M} = - \mathbf{M} \cdot \mathbf{H} + K_1 \sin^2 \phi \cos^2 \phi + 2 \pi M_{eff}^2 \cos^2 \theta,
\end{equation}
where $\mathbf{H}$ is the applied field, $\phi$ and $\theta$ are the azimuthal and polar angles of the magnetization, respectively, $K_1$ is a first order in-plane cubic anisotropy constant, and $4 \pi M_{eff}$ is the PP saturation field, which includes the usual demagnetization energy and a first order uniaxial anisotropy due to interfacial effects. The parameters obtained by fitting to Eq.\ \ref{eq:FT} are shown in Table \ref{tab:magneticparameters}. The uncertainty in these parameters was estimated by measuring a range of different sample pieces, and using the standard deviation of the values as the error bar. The long-range inhomogeneity characteristic of epitaxial samples makes this a more accurate estimate of the uncertainty than the fitting error. The magnetic-field-swept FMR lineshapes were fit to the derivative of Lorentzian functions \cite{Montoya2014} in order to extract the full-width at half-maximum linewidths $\Delta H$ [magnetic field units, Fig.\ \ref{fig:1}(b)], which are the focus of this article.
\begin{figure}
\includegraphics{figure1_v14.pdf}
\caption{(a) Wide-angle x-ray diffraction $\phi$-scans of $\langle202\rangle$ (blue) and $\langle111\rangle$ (red) peaks for the CMS film. (b) Typical derivative susceptibility lineshapes for these samples at different microwave excitation frequencies. The fits are shown as solid lines. (c) In-plane hysteresis loops for CFA obtained with a vibrating-sample magnetometer (VSM). (d) Atomic force microscopy (AFM) image of surface topography for CFA. RMS roughness is 0.2~nm.}
\label{fig:1}
\end{figure}
\begin{table*}
    \caption{Summary of the magnetic properties extracted from the dependence of the resonance field on applied frequency for both field in-plane ($\vert\vert$) and field perpendicular-to-plane ($\perp$) configurations, along with the Gilbert $\alpha$ and inhomogeneous broadening from the perpendicular-to-plane configuration. $2 K_1/M_s$ and $4 \pi M_{eff}$ are the in-plane and perpendicular-to-plane anisotropy fields, respectively (see Eq.\ \ref{eq:FT}), and $g$ is the Land\'e $g$-factor.}
    \label{tab:magneticparameters}
    \begin{ruledtabular}
    \begin{tabular}{l c c c c c c c}
       \\[-9pt]
       Sample & $2 K_1/M_s$ (Oe)  & $4\pi M_{eff}^{\vert\vert}$ (kOe) & $4\pi M_{eff}^{\perp}$ (kOe) & $g^{\vert\vert}$ & $g^{\perp}$ & $\alpha_{001} (\times 10^{-3})$ & $\Delta H_{0}$ (Oe) \\[4pt]
        \hline
        \\[-6pt]
        CMS & 280 & 12.3 & 13.3 & 2.04 & 2.04 & $1.5 \pm 0.1$ & $9 \pm 1$\\
        CMA & 35 & 11.3 & 11.7 & 2.06 & 2.08 & $1.8 \pm 0.2$ & $12 \pm 3$\\
        CFA & 230 & 15.1 & 15.5 & 2.06 & 2.07 & $<$ 0.8 & $100 \pm 6$\\
        CFA 500~$^\circ$C anneal & N/A & N/A & 15.1 & N/A & 2.07 & $1.1 \pm 0.1$ & $45 \pm 1$
    \end{tabular}
    \end{ruledtabular}
\end{table*}
The maximum resonant frequency was determined by the maximum magnetic field that could be applied for both IP and PP electromagnet configurations, which was 10.6~kOe and 29~kOe, respectively. For the IP measurement, the angle of the applied field in the plane of the film was varied to determine the in-plane magnetocrystalline anisotropy of our samples, which was fourfold-symmetric for the three films characterized in this article. The anisotropy was confirmed using vibrating-sample magnetometry (VSM) measurements, an example of which is shown in Fig.\ \ref{fig:1}(c), which shows IP easy and hard axis hysteresis loops for the CFA film. For the PP measurement, alignment was verified to within $\sim$0.1$^{\circ}$ to ensure magnetization saturation just above the PP anisotropy field, thus minimizing field-dragging contributions to the linewidth.
\begin{figure}
\includegraphics{OOPlinewidths_v8.pdf}
\caption{Linewidths as a function of frequency with the field applied perpendicular to plane, for which two-magnon scattering is inactive.  The black squares are data for the CMS film, the red circles are for the CMA film, and the blue triangles are for the CFA film. In addition, linewidths are shown for a CFA film that was annealed at 500~$^{\circ}$C \textit{ex situ} (magenta diamonds). Corresponding linear fits are shown along with the extracted Gilbert damping factor $\alpha$. The blue dashed lines indicate an upper bound of $\alpha_{001} = 8\times 10^{-4}$ and a lower bound of $\alpha_{001} = 0$ for CFA.}
\label{fig:PP_summary}
\end{figure}
\section{Results and Analysis}\label{data}
\begin{figure}
\includegraphics{angledependence_v9.pdf}
\caption{Azimuthal angular dependence of the linewidths (left ordinate, blue circles) and resonance fields (right ordinate, black squares) for (a) CMS, (b) CMA, and (c) CFA. The excitation frequency was 20 GHz for CMS, 15 GHz for CMA, and 20 GHz for CFA. The solid lines are sinusoidal fits.}
\label{fig:angledependence}
\end{figure}
\subsection{\label{PP}Perpendicular-to-plane linewidths}
First we discuss the results of the PP measurement.  As stated in Sec. \ref{sec:experiment}, the TMS extrinsic broadening mechanism is suppressed when the magnetization is normal to the plane of the film. We can thus fit our data to Eq. \ref{LWtotal} with $\Delta H_{TMS} = 0$, greatly simplifying the extraction of the Gilbert damping constant $\alpha$ and the inhomogeneous broadening $\Delta H_0$. Prior knowledge of $\Delta H_0$ is particularly important for constraining the analysis of the IP measurements, as we shall discuss.

The dependence of $\Delta H$ on frequency for the CMS, CMA, and CFA films in the PP configuration is summarized in Fig.\ \ref{fig:PP_summary}, in which fits to Eq.\ \ref{LWtotal} are shown with $\Delta H_{TMS}$ set to zero. For the CMS film, $\alpha_{001} = (1.5\pm0.1)\times 10^{-3}$ and $\Delta H_0 = 9$~Oe, while for the CMA film $\alpha_{001} = (1.8\pm0.2)\times 10^{-3}$ and $\Delta H_0 = 12$~Oe. Co$_2$MnSi$_{2/3}$Al$_{1/3}$/MgO and Co$_2$MnSi$_{1/3}$Al$_{2/3}$/MgO films (both 10~nm thick) were also measured, with Gilbert damping values of $\alpha_{001} = (1.8\pm0.2)\times 10^{-3}$ and $\alpha_{001} = (1.5\pm0.1)\times 10^{-3}$, respectively (not shown). For CFA, we obtained a damping value of $\alpha_{001} = 3\times 10^{-4}$  with an upper bound of $\alpha_{001} < 8\times 10^{-4}$ and $\Delta H_0 = 100$~Oe. These fit parameters are also contained in Table \ref{tab:magneticparameters}. The source of the large inhomogeneous broadening for the CFA film is unclear: AFM measurements [Fig.\ \ref{fig:1}(d)] along with XRD indicate that the film is both crystalline and smooth. Note that the range of frequencies shown in Fig.\ \ref{fig:PP_summary} are largely governed by considerations involving the Kittel equation \cite{Kittel1948}: measurements below 10~GHz were not used due to the increasing influence of slight misalignment on $\Delta H$ (through field-dragging) for resonant fields just above the saturation value. A piece of the CFA sample was annealed at 500~$^{\circ}$C \textit{ex situ}, which reduced the inhomogenoeus broadening to $\sim$45 Oe (still a relatively large value) and increased the Gilbert damping to $\alpha_{001} = 1.1\times 10^{-3}$ (similar behavior in CFA was seen in Ref.\ \cite{Kumar2017}). The constraint of $\alpha_{001} < 8\times 10^{-4}$ is among the lowest of reported Gilbert damping constants for metallic ferromagnets, but the $\alpha\sim 10^{-4}$ range is not unexpected based on Kambersk\'y model calculations performed for similar full-Heusler compounds \cite{Liu2009} or other recent experimental reports \cite{Oogane2018,Guillemard2019}. It should be noted that \citet{Schoen2016} have recently reported $\alpha = 5\times 10^{-4}$ for Co$_{25}$Fe$_{75}$ thin films, where spin pumping and radiative damping contributions were subtracted from the raw measurement. Spin pumping contributions to the intrinsic damping are not significant in our films, as no heavy-metal seed layers have been used and the films have thicknesses of 10~nm or greater. For the radiative damping contribution \cite{Schoen2015} in the geometry of our CPW and sample, we calculate contributions $\alpha_{\text{rad}}\lesssim 1 \times 10^{-4}$, which is below the uncertainty in our damping fit parameter.
\begin{figure}
\includegraphics{alllinewidths_v4.pdf}
\caption{Linewidths along all three principal directions for CMS (a), CMA (b), and CFA (c). Heusler crystalline axes are labeled by $\langle$100$\rangle$ (black), $\langle$110$\rangle$ (red), and [001] (blue). In all three cases, $\langle$110$\rangle$ is the in-plane easy axis and $\langle100\rangle$ is the in-plane hard axis. The corresponding fits are shown as the solid curves, where the in-plane linewidths are fit using Eq.\ \ref{TMS_LW} and the out-of-plane linewidths are fit to the Gilbert damping model. The fit parameters are given in Table \ref{tab:TMSparameters}.}
\label{fig:alllinewidths}
\end{figure}
\subsection{\label{IP}In-plane linewidths}
With the intrinsic damping and inhomogeneous broadening characterized by the PP measurement, we turn our attention to the IP linewidth measurements, for which TMS contributions are present. For hard-axis measurements, frequencies $\lesssim5$~GHz were not used due to the influence of slight magnetic field misalignment on the linewidths. For easy-axis measurements, the lower limit is determined by the zero-field FMR frequency. Figure \ref{fig:angledependence} shows the dependences of the resonance fields and linewidths on the angle of the in-plane field. An important observation seen in Fig.\ \ref{fig:angledependence} is that the linewidth extrema are commensurate with those of the resonance fields and therefore the magnetocrystalline anisotropy energy.  This rules out field-dragging and mosaicity contributions to the linewidth, which can occur when the resonance field depends strongly on angle \cite{Zakeri2007}. We note that similar IP angular dependence of the FMR linewidth, which was attributed to an anisotropic TMS mechanism caused by a rectangular array of misfit dislocations, has been reported by \citet{Kurebayashi2013} and Woltersdorf and Heinrich \cite{Woltersdorf2004} for epitaxial Fe/GaAs(001) ultrathin films.

To further study the anisotropy of the IP $\Delta H$ in our films, we have measured $\Delta H$ at the angles corresponding to the extrema of $H_{FMR}$ (and $\Delta H$) in Fig.\ \ref{fig:angledependence} over a range of frequencies. These data are shown in Fig.\ \ref{fig:alllinewidths}, along with the PP ([001]) measurements for each sample. A distinguishing feature of the data shown in Fig.\ \ref{fig:alllinewidths} is the significant deviation between IP and PP linewidths in all but one case (CMS$\langle100\rangle$). Large and nonlinear frequency dependence of the IP linewidths is strongly suggestive of an active TMS linewidth broadening mechanism. In the presence of TMS, careful analysis is required to separate the Gilbert damping from the TMS linewidth contributions. We therefore describe the TMS mechanism in more detail in the following section in order to analyze the IP linewidths in Fig.\ \ref{fig:alllinewidths} and extract the Gilbert damping.
\begin{figure}
    \includegraphics{TMStheory_v6.pdf}
    \caption{(a) Two-magnon scattering linewidth contribution for values of  Gilbert damping $\alpha = 10^{-2}, 5 \times 10^{-3}, 10^{-3},$ and $10^{-4}$. The inset shows magnon dispersions for an applied field of $H$~=~1~kOe. (b) Contours of the degenerate mode wavenumber $\mathbf{q}_{2M}$ in the film plane as a function of wavevector angle relative to the magnetization for $f_{FMR}$ = 16, 24, and 32~GHz. The dashed circle indicates the wavenumber of a defect with size $\xi$ = 100~nm.}
\label{fig:theorycartoon}
\end{figure}
 \subsection{Two-magnon scattering model}\label{TMS}
The TMS mechanism leads to a characteristic nonlinear frequency dependence of $\Delta H$ \cite{Arias1999,McMichael2004}. In Fig.\ \ref{fig:alllinewidths}, the IP $\Delta H$ is not a linear function of frequency, but possesses the ``knee" behavior characteristic of the frequency dependence of linewidths dominated by the TMS mechanism. We have fit our data to the TMS model described by \citet{McMichael2004}, in which the TMS linewidth $\Delta H_{TMS}$ is given by \cite{Krivosik2007,Kalarickal2008}
\begin{equation}\label{TMS_LW}
\Delta H_{TMS} = \frac{\gamma^2\xi^2H'^2}{df/dH\lvert_{f_{FMR}}}\int\Gamma_{0\mathbf{q}}C_{\mathbf{q}}(\xi)\delta_\alpha(\omega-\omega_{\mathbf{q}})d^2q,
\end{equation}
where $\Gamma_{0\mathbf{q}}$ is the defect-mediated interaction term between magnons at wavevector $0$ and $\mathbf{q}$, $C_{\mathbf{q}}(\xi) = (1+(q\xi)^2)^{-3/2}$ is the correlation function of the magnetic system with correlation length $\xi$, and $H'$ is the magnitude of the characteristic inhomogeneity (units of magnetic field). The $\delta_\alpha$-function in Eq.\ \ref{TMS_LW} selects only the magnon scattering channels that conserve energy. In the limit of zero intrinsic damping, it is identical to the Dirac delta function, but for finite $\alpha$ it is replaced by a Lorentzian function of width $\delta\omega = (2\alpha \omega/\gamma)d\omega/dH$. The magnon dispersion relation determining $\omega_{\mathbf{q}}$ is the usual Damon-Eshbach thin film result \cite{Eshbach1960,Krivosik2007} with the addition of magnetocrystalline anisotropy stiffness field terms extracted from the dependence of the resonance field on the applied frequency for the IP configuration. The film thickness $d$ affects the states available for two-magnon scattering through the dispersion relation, namely, the linear term which gives rise to negative group velocity for small $q$ ($\propto - q d$). The IP FMR linewidth data shown in Fig.\ \ref{fig:alllinewidths} were fit to Eq.\ \ref{LWtotal} (with Eq.\ \ref{TMS_LW} used to evaluate $\Delta H_{TMS}$) with $\xi$, $\alpha$, and $H'$ as fitting parameters (shown in Table \ref{tab:TMSparameters}). The correlation length $\xi$ remains approximately constant for different in-plane directions, while the strength $H'$ is larger for the $\langle100\rangle$ directions in the CMA and CFA samples and the $\langle110\rangle$ directions in the CMS sample. Some degree of uncertainty results from this fitting procedure, because for linewidth data collected over a limited frequency range, $\xi$ and $\alpha$ are not completely decoupled as fitting parameters. In absolute terms, however, the largest systematic errors come from the exchange stiffness, which is not well-known.  The error bars given in Table \ref{tab:TMSparameters} were calculated by varying the exchange stiffness over the range $400~\textrm{meV}~\textrm{\AA}^2$ to $800~\textrm{meV}~\textrm{\AA}^2$, and recording the change in the fit parameters. This range of values was chosen based on previous Brillouin light scattering measurements of the exchange stiffness in similar Heusler compounds \cite{Kubota2009,Gaier2009}. In addition, we note that in Eq.\ \ref{LWtotal} $\Delta H_0$ is taken to be isotropic, with the value given by the PP linewidth measurements shown in Fig.\ \ref{fig:PP_summary}. Although certain realizations of inhomogeneity may result in an anisotropic $\Delta H_0$ (see Ref.\ \cite{Woltersdorf2004} for a good discussion), doing so here would only serve to create an additional fitting parameter.
\subsection{Effect of low intrinsic damping}\label{alphadependence}
The effect of low intrinsic damping on the two-magnon linewidth can be seen in Fig.\ \ref{fig:theorycartoon}(a). As $\alpha$ decreases, with all other parameters fixed, $\Delta H_{TMS}$ steadily increases and becomes increasingly nonlinear (and eventually nonmonotonic) with frequency. In particular, a ``knee" in the frequency dependence becomes more pronounced for low damping (see e.g.\ Fig.\ \ref{fig:theorycartoon}(a) curve for $\alpha = 10^{-4}$). The physics giving rise to the knee behavior is illustrated in Fig.\ \ref{fig:theorycartoon}(b). The TMS process scatters magnons from zero to non-zero wavevector at small $q$.  There is assuemd to be sufficient disorder to allow for the momentum $q$ to be transferred to the magnon system.  There will always be, however, a length scale $\xi$ below which the disorder decreases, so that the film becomes effectively more uniform at large wavevectors. The corresponding FMR frequencies are those for which the contours of constant frequency (the figure eights in Fig.~\ref{fig:theorycartoon}) in $q$-space have extrema at $q \sim \xi^{-1}$. The TMS rate is also determined by the interplay of the magnon density of states, the effective area in $q$-space occupied by the modes that conserve energy, and the Gilbert damping. The knee behavior is more pronounced for low $\alpha$ due to the increased weight of the van Hove singularity coming from the tips of the figure eights, in the integrand of Eq.\ \ref{TMS_LW}. Although a larger window of energies, set by the width of $\delta_\alpha$, is available for larger $\alpha$, this smears out the singularity in the magnon density of states, removing the sharp knee in the TMS linewidth as a function of frequency. The PP measurement confirms that all of these epitaxial Heusler films lie within the range $\alpha < 2 \times 10^{-3}$. Ferromagnetic films with ultralow $\alpha$ are therefore increasingly prone to large TMS linewidths (particularly for metals with large $M_s$). The TMS linewidths will also constitute a larger fraction of the total linewidth due to a smaller contribution from the Gilbert damping. In practice, this is why experimental reports \cite{Schoen2016,Oogane2018,Guillemard2019} of ultralow $\alpha$ have almost all utilized the PP geometry.
\subsection{Discussion}\label{discussion}
\begin{table}
    \caption{Summary of the fitting parameters used to fit the in-plane data of Fig.\ \ref{fig:alllinewidths} (black squares and red circles) to Eqs.\ \ref{LWtotal} and\ \ref{TMS_LW}. CFA refers to the unannealed Co$_2$FeAl sample.}
    \label{tab:TMSparameters}
    \begin{ruledtabular}
    \begin{tabular}{l r r r}
       \\[-8pt]
       Sample (Field Direction) & $\alpha~(\times 10^{-3})$ & $\xi$ (nm) & $H^{\prime}$ (Oe) \\[4pt]
        \hline
        \\[-6pt]
        CMS$\langle$110$\rangle$ & $1.6 \pm 0.2$ & $40 \pm 25$ & $55 \pm 30$\\
        CMS$\langle100\rangle$ & $1.5 \pm 0.1$ & $40 \pm 25$ & $30 \pm 15$\\
        CMA$\langle$110$\rangle$ & $3.1 \pm 0.2$ & $70 \pm 20$ & $30 \pm 5$\\
        CMA$\langle100\rangle$ & $4.7 \pm 0.4$ & $55 \pm 10$ & $90 \pm 5$\\
        CFA$\langle$110$\rangle$ & $2.0 \pm 0.3$ & $20 \pm 10$ & $175 \pm 60$\\
        CFA$\langle100\rangle$ & N/A & N/A & N/A
    \end{tabular}
    \end{ruledtabular}
\end{table}
The results of the IP linewidth fits to Eqs.\ \ref{LWtotal} and \ref{TMS_LW} are summarized in Table \ref{tab:TMSparameters}. In the case of CMS, the high-frequency slopes in Fig.\ \ref{fig:alllinewidths}(a) approach the same value along each direction, as would be expected when the frequency is large enough for the TMS wavevector to exceed the inverse of any defect correlation length.  In this limit, $\alpha$ is isotropic (within error limits).

Next, we discuss the CMA IP data shown in Fig.\ \ref{fig:alllinewidths}(b) and Table \ref{tab:TMSparameters}. It is clear from this figure that a good fit can be obtained along both $\langle100\rangle$  and $\langle110\rangle$ directions. In Table \ref{tab:TMSparameters} it can be seen that the value of the defect correlation length $\xi$ is approximately the same along both directions. However, the values of $\alpha$ we obtain from fitting to Eqs.\ \ref{LWtotal} and \ref{TMS_LW} do not agree well with the PP value of $\alpha_{001} = 1.8 \times 10^{-3}$ (Fig.\ \ref{fig:PP_summary}). Anisotropic values of $\alpha$ have been both predicted \cite{Gilmore2010,Qu2014} and observed \cite{Chen2018}, and an anisotropic $\alpha$ is possibly the explanation of our best-fit results. The in-plane $\langle100\rangle$ and [001] directions are equivalent in the bulk, so the anisotropy would necessarily be due to an interface anisotropy energy \cite{Chen2018} or perhaps a tetragonal distortion due to strain \cite{Li2019}.

Finally, we discuss the CFA linewidths shown in Fig.\ \ref{fig:alllinewidths}(c) and Table \ref{tab:TMSparameters}. This sample has by far the largest two-magnon scattering contribution, which is likely related to the anomalously large inhomogeneous broadening and low intrinsic damping [see Fig.\ \ref{fig:theorycartoon}(a)] observed in the PP measurement. A good fit of the data was obtained when the field was applied along the $\langle110\rangle$ direction. Notably, the IP $\langle110\rangle$ best fit value of $2.1\times 10^{-3}$ is nearly a factor of 3 larger than the $\alpha_{001}$ upper bound on the same sample (Table \ref{tab:magneticparameters}), strongly suggesting an anisotropic Gilbert $\alpha$. A striking anisotropy in the IP linewidth was revealed upon rotating the magnetization to the $\langle100\rangle$ orientation. For the $\langle100\rangle$ case, which yielded the largest TMS linewidths measured in this family of films, we were not able to fit the data to Eq.\ \ref{TMS_LW} using a set of physically reasonable input parameters. We believe that this is related to the consideration that higher order terms in the inhomogeneous magnetic energy (see Ref.\ \cite{Krivosik2007}) need to be taken into account. Another reason why this may be the case is that the model of \citet{McMichael2004} assumes the inhomogeneities to be grain-like, whereas the samples are epitaxial [see Fig.\ \ref{fig:1}(a)]. Atomic force microscopy images of these samples [Fig.\ \ref{fig:1}(d)] imply that grains, if they exist, are much larger than the defect correlation lengths listed in Table \ref{tab:TMSparameters}, which are of order 10's of nm. We also note that there does not appear to be a correlation between the strength of two-magnon scattering $H'$ and the cubic anisotropy field $2K_1/M_s$, which would be expected for grain-induced two-magnon scattering.
\section{Summary and Conclusion}\label{summary}
We conclude by discussing the successes and limitations of the \citet{McMichael2004} model in analyzing our epitaxial Heusler film FMR linewidth data. We have shown that two-magnon scattering is the extrinsic linewidth-broadening mechanism in our samples. Any model which takes this as its starting point will predict much of the qualitative behavior we observe, such as the knee in the frequency dependence and the large linewidths IP for low $\alpha$ films. The TMS model used in this article (for the purpose of separating TMS and Gilbert linewidth contributions) is, however, only as accurate as its representation of the inhomogeneous magnetic field and the underlying assumption for the functional form of $C_{\mathbf{q}}(\xi)$. Grain-like defects are assumed, which essentially give a random magnetocrystalline anisotropy field. We did not, however, explicitly observe grains in our samples with AFM, at least below lengthscales of $\sim$10~$\mu$m [Fig.\ \ref{fig:1}(d)]. Misfit dislocations, a much more likely candidate in our opinion, would cause an effective inhomogeneous magnetic field which could have a more complicated spatial profile and therefore lead to anisotropic two-magnon scattering (see Ref.\ \cite{Woltersdorf2004}). The perturbative nature of the model also brings its own limitations, and we believe that the CFA$\langle100\rangle$ data, for which we cannot obtain a satisfactory fit, are exemplary of a breakdown in the model for strong TMS. Future work should go into methods of treating the two-magnon scattering differently based on the type of crystalline defects present, which will in turn allow for a more reliable extraction of the Gilbert damping $\alpha$ and facilitate the observation of anisotropic Gilbert damping, enabling quantitative comparison to first-principles calculations.

Regardless of the limitations of the model, we emphasize three critical observations drawn from the linewidth measurements presented in this article. First, in all cases we observe large and anisotropic TMS linewidth contributions, which imply inhomogeneity correlation lengthscales of order tens-to-hundreds of nanometers. The microscopic origin of these inhomogeneities is the subject of ongoing work, but are likely caused by arrays of misfit dislocations \cite{Woltersdorf2004}. The relatively large lengthscale of these defects may cause them to be easily overlooked in epitaxial film characterization techniques such as XRD and cross-sectional HAADF-STEM, but they still strongly influence magnetization dynamics. These defects and their influence on the FMR linewidth through TMS complicate direct observation of Kambersk\'y's model for anisotropic and (in the case of Heusler compounds) ultralow intrinsic damping in metallic ferromagnets. Second, we observed low intrinsic damping through our PP measurement, which was $<2\times10^{-3}$ for all of our samples. Finally, we have presented the mechanism by which FMR linewidths in ultralow damping films are particularly likely to be enhanced by TMS, the anisotropy of which may dominate any underlying anisotropic Kambersk\'y damping.

This work was supported by NSF under DMR-1708287 and by SMART, a center funded by nCORE, a Semiconductor Research Corporation program sponsored by NIST. The sample growth was supported by the DOE under DE-SC0014388 and the development of the growth process by the Vannevar Bush Faculty Fellowship (ONR N00014-15-1-2845). Parts of this work were carried out in the Characterization Facility, University of Minnesota, which receives partial support from NSF through the MRSEC program.