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%%%%%%%%%%%%      Generic Gravitational corrections     %%%%%%%%%%%%%
%%%%%%%%%%%%%%   to gauge coupling in SUSY SU(5) GUTs      %%%%%%%%%%
%%%%%%%%    by K.Huitu, Y.Kawamura, T.Kobayashi, K.Puolamaki  %%%%%%%
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\begin{document}
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{\normalsize
HIP-1999-52/TH\\
DPSU-99-6\\
August, 1999}\\
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\vspace{0.5cm}
\begin{center}

{\large \bf  Generic Gravitational Corrections 
to Gauge Couplings in SUSY $SU(5)$ GUTs}


\vspace{0.8cm}
 
{ Katri Huitu$^{{a}}$, Yoshiharu Kawamura$^{{c}}$, 
Tatsuo Kobayashi$^{{a,b}}$,   
Kai~Puolam\"{a}ki$^{{a}}$
}

\vspace{0.5cm}
$^a$ Helsinki Institute of Physics, 
FIN-00014 University of Helsinki, Finland \\
$^b$ Department of Physics, 
FIN-00014 University of Helsinki, Finland \\
and\\
$^c$ Department of Physics, Shinshu University,
Matsumoto 390-0802, Japan\\

\end{center}
\vspace{0.5cm}

\nopagebreak
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\begin{abstract}
We study non-universal corrections to the gauge couplings due to 
higher dimensional operators in supersymmetric $SU(5)$ grand 
unified theories.
The corrections are, in general, 
parametrized by three components originating from
{\bf 24}, {\bf 75} and {\bf 200} representations.
We consider the prediction of $\alpha_3(M_Z)$ 
along each {\bf 24}, {\bf 75} and {\bf 200} direction, 
and their linear combinations.
The magnitude of GUT scale and its effects on proton decay 
are discussed.
Non-SUSY case is also examined.

\end{abstract}
%PACS number(s):04.65.+e, 11.25.Mj, 12.60.Jv\\
%Keyword(s):SUSY GUT, coupling reduction,
%soft SUSY breaking parameters, radiative electroweak breaking
\vfill
\end{titlepage}
\pagestyle{plain}
\newpage
\def\thefootnote{\fnsymbol{footnote}}



1. The prediction of $\alpha_3(M_Z)$ from the 
precision data $\alpha$ and $\sin \theta_W$ 
is one of the strong motivations for supersymmetric 
grand unified theories (SUSY GUTs) \cite{SUSY-GUT}.
On the analysis of gauge coupling constants, several corrections
have been considered, e.g. 
threshold corrections \cite{guni,GUT-th}
due to superparticles at the weak scale 
and heavy particles around the GUT scale.
In addition, non-renormalizable interactions in gauge kinetic terms
can give corrections suppressed by the reduced Planck mass $M$
as an effect of quantum gravity \cite{g-nonuni} \footnote{Such 
corrections are important for the gauge coupling unification 
with extra dimensions, too \cite{HK}.}.
We call them gravitational corrections.
The gravitational corrections are not universal but 
proportional to group theoretical factors.
If the $F$-component of the Higgs field
has a non-vanishing vacuum expectation value (VEV), 
gauginos also receive a non-universal 
correction to their masses \cite{EKN}.


In the SUSY $SU(5)$ GUT, gravitational corrections are 
parametrized by three components originating from
{\bf 24}, {\bf 75} and {\bf 200} representations
because these (elementary and/or composite) fields can couple to 
the gauge multiplets in gauge kinetic terms.
Non-zero VEVs of $F$-component of these
fields lead to proper types of non-universal gaugino masses.
Recently, several phenomenological aspects of models with
such non-universal 
gaugino masses have been studied and interesting 
difference among models have been shown \cite{nonunigm} \footnote{
See also Ref.~\cite{GLMR}.}.
In our previous analysis, 
it is assumed that 
non-universal corrections to gauge couplings $\alpha_i$ $(i=1,2,3)$
are small enough for 
the $SU(5)$ breaking scale $M_U$ to be the ordinary unification scale 
$M_X = 2.0 \times 10^{16}$GeV.


In this paper, we study gravitational corrections to gauge couplings
based on the SUSY $SU(5)$ GUT.
We consider the prediction of $\alpha_3(M_Z)$ 
along each {\bf 24}, {\bf 75} and {\bf 200} direction, 
and their linear combination.
We discuss which value is allowed as $M_U$
in the presence of gravitational corrections 
and its phenomenological implication.
The non-SUSY case is also examined.


The gauge kinetic function is given by
\begin{eqnarray}
{\cal L}_{g.k.} &=& \sum_{\alpha, \beta} \int d^2\theta 
f_{\alpha\beta}(\Phi^I) W^\alpha W^\beta   +   H.c.
\nonumber \\
&=& -{1 \over 4} \sum_{\alpha, \beta}
Re f_{\alpha\beta}(\phi^I) F^\alpha_{\mu\nu} F^{\beta \mu\nu} 
\nonumber \\
&~& 
+ \sum_{\alpha, \beta, \alpha', \beta'} \sum_I F^I_{\alpha'\beta'} 
{\partial f_{\alpha\beta}(\phi^I) \over \partial \phi^I_{\alpha'\beta'}}
\lambda^\alpha \lambda^\beta   +   H.c. + \cdots
\label{gaugekinetic}
\end{eqnarray}
where $\alpha, \beta$ are indices related to gauge generators,
$\Phi^I$'s are chiral superfields and 
$\lambda^\alpha$'s are the $SU(5)$ gaugino fields.
The scalar and $F$-components of $\Phi^I$ are denoted by
$\phi^I$ and $F^I$, respectively.
The gauge multiplet is in the adjoint representation and 
the symmetric product of ${\bf 24} \times {\bf 24}$ is decomposed as 
\begin{equation}
( {\bf 24} \times {\bf 24} )_s = {\bf 1} + {\bf 24}+{\bf 75}+{\bf 200}.
\end{equation}
Hence the gauge kinetic function $f_{\alpha\beta}(\Phi^I)$ is
also decomposed as
\begin{eqnarray}
f_{\alpha\beta}(\Phi^I) &=& \sum_R f^{R}_{\alpha \beta}(\Phi^I) 
\label{f2}
\end{eqnarray}
where $f^{R}_{\alpha \beta}(\Phi^I)$ is a part of gauge kinetic functions 
which transforms as $R$-representation ($R={\bf 1},{\bf 24},{\bf 75},
{\bf 200}$).


After a breakdown of $SU(5)$ at $M_U$, a boundary condition (BC) 
of $\alpha_i$ is given by 
\begin{eqnarray}
\alpha_i^{-1}(M_U)=\alpha_U^{-1}(1+C_i)
\label{BC}
\end{eqnarray}
where $C_i$'s are non-universal factors which parametrize generic 
gravitational corrections such as
\footnote{The factors $1/2\sqrt{15}$, $1/6$ and $1/2\sqrt{21}$
come from that the normalization $Tr (T^a T^b) = \delta^{ab}/2$ of 
the $5 \times 5$, $10 \times 10$ and $15 \times 15$ matrices
representing ${\bf 24}$, ${\bf 75}$ and ${\bf 200}$.}
\begin{eqnarray}
(C_1,C_2,C_3)&=& {x_{\bf 24} \over 2\sqrt{15}}(-1,-3,2) 
+ {x_{\bf 75} \over 6}(-5,3,1) 
+ {x_{\bf 200} \over 2\sqrt{21}}(10,2,1) \nonumber \\
&=& x'_{\bf 24}(-1,-3,2) + x'_{\bf 75}(-5,3,1) 
+ x'_{\bf 200}(10,2,1) .
\label{Ci}
\end{eqnarray}
Here $x_R$'s are model-dependent quantities including 
the VEV of Higgs fields, and their order is 
supposed to be $O(M_U/M)$ or less.\\



2. Let us predict $\alpha_3(M_Z)$ using the experimental values 
$\alpha_1^{-1}(M_Z)=59.98$ and $\alpha_2^{-1}(M_Z)=29.57$ 
based on the assumption that 
the minimal supersymmetric standard model (MSSM) holds on
below $M_U$ and the SUSY $SU(5)$ GUT is realized above $M_U$.


In the case without gravitational corrections,
the following value is obtained
\begin{equation}
\alpha_3^{(0)}(M_Z) = 0.127
\label{alpha3-cal}
\end{equation}
based on solutions of one-loop renormalization group (RG) equations
with a common SUSY threshold $m_{SUSY} = M_Z$
\begin{eqnarray}
\alpha_i^{-1}(M_Z) = \alpha_U^{-1}  + 
{b_i \over 2\pi}\ln{M_U \over M_Z}
\label{RGE}
\end{eqnarray}
where $(b_1, b_2, b_3) = (33/5, 1, -3)$.
The experimental value of $\alpha_3(M_Z)$ is \cite{p-data}
\begin{equation}
\alpha_3(M_Z) = 0.119 \pm 0.002.
\label{alpha3-exp}
\end{equation}


There are several possibilities to explain the difference between values
(\ref{alpha3-cal}) and (\ref{alpha3-exp}), e.g.,
threshold corrections due to superparticles at the weak scale,
heavy particles around the GUT scale and
gravitational corrections.
Here we neglect non-universal threshold corrections
and pay attention to gravitational corrections
by the ${\bf 24}$, ${\bf 75}$ or
${\bf 200}$ Higgs field.
Before numerical calculations of two-loop RG equations, 
we estimate the magnitude of $x_R$ using analytical results
of one-loop RG equations with BC (\ref{BC}).
%\begin{eqnarray}
%\alpha_U^{-1}(1+C_i) = \alpha_i^{-1}(M_Z) - 
%{b_i \over 2\pi}\ln{M_U \over M_Z} .
%\label{RGE-gr}
%\end{eqnarray}
For small $x_R$, $\alpha_3(M_Z)$ receives the following 
correction,
\begin{equation}
\alpha_3(M_Z)= \alpha_3^{(0)}(M_Z)+\sum_i a_iC_i,
\label{1-loop1}
\end{equation}
where $a_1=-0.28$, $a_2=0.67$ and $a_3=-0.39$.
Using Eq. (\ref{Ci}), allowed regions for $x_R$ are estimated as
\begin{eqnarray}
&~& 0.019 \sileqq x_{\bf 24} \sileqq 0.031 , \quad 
-0.020 \sileqq x_{\bf 75} \sileqq -0.012 , \quad  
\nonumber \\
&~& 0.030 \sileqq x_{\bf 200} \sileqq 0.050
\label{xR}
\end{eqnarray}
for each contribution.


Now let us study corrections solving the two-loop RG equations 
numerically.
We take the top quark mass $m_t=175$ GeV and $\tan \beta =3$,
and assume the universal SUSY threshold $m_{SUSY}=1$ TeV.
The three curves in Fig. 1 correspond to predictions of $\alpha_3(M_Z)$
along the pure {\bf 24},  {\bf 75} and  {\bf 200} directions, respectively.
Here we use a notation $x$ instead of $x_R$.
Fig. 2 shows the magnitude of GUT scale for each case.
Thus we obtain a good agreement with the experiment in the region 
with $|x|\sileqq O(0.01)$ for the three pure directions.
This value is consistent with $x_R \sileqq O(M_U/M)$.
For small $x_R$, $a_i$'s in eq.(\ref{1-loop1}) are obtained 
as $a_1=-0.27$, $a_2=0.61$ and $a_3=-0.36$ at the two-loop level.
In addition, $\alpha_3(M_Z)$ receives the SUSY thereshold correction, 
$0.0039 \times (m_{SUSY}/1$ TeV).



\begin{center}
\input fig1.tex

Fig.1: $\alpha_3(M_Z)$ along the {\bf 24}, {\bf 75} and  {\bf 200} 
directions. 
\end{center}

\newpage
\begin{center}
\input fig2.tex

Fig.2: The GUT scale along the {\bf 24}, {\bf 75} and  {\bf 200} 
directions, where $T$ denotes $T=\log_{10}M_U$ [GeV]. 
\end{center}

We give comments on non-universal SUSY threshold corrections.
Under the assumption that the dominant contribution 
to gaugino masses comes from the VEV of
$F$-component of {\bf 24}, {\bf 75} or {\bf 200} Higgs field,
the ratio of gaugino mass magnitudes at the weak scale 
is given by \cite{EKN,nonunigm}
\begin{eqnarray}
M_1 : M_2 : M_3 &=& 0.4 : 0.8 : 2.9 ~~~~ (R={\bf 1}) , \nonumber \\
                &=& 0.2 : 1.2 : 2.9 ~~~~ (R={\bf 24}) , \nonumber \\
                &=& 2.1 : 2.5 : 2.9 ~~~~~~ (R={\bf 75}) , \nonumber \\
                &=& 4.1 : 1.6 : 2.9 ~~~~~~ (R={\bf 200}) . \nonumber 
\end{eqnarray}
This difference among gaugino masses leads to a small correction 
compared with the universal SUSY threshold.
For example,
$\alpha_3^{(0)}(M_Z)$ is raised by $0.001$ for $R={\bf 24}$.
Here we have assumed all soft scalar masses in the MSSM are equal to $M_3$.
Similarly, the case with ${\bf 75}$ Higgs condensation leads to a
tiny correction compared with the other two.


Detailed analysis shows that SUSY threshold corrections due to 
scalar masses could lead to sizable corrections \cite{guni}.
It is possible to deviate from $|x|\sileqq O(0.01)$ with a good agreement 
with the experimental value 
even in each pure direction case, although
the width of the good parameter region 
$\Delta x$ would be as narrow as those in Fig.1, 
i.e. $\Delta x = O(0.01)$.


The GUT scale threshold corrections due to heavy particles
are also important to the precise prediction of $\alpha_3(M_Z)$.
Since they depend on details of a GUT model, it would be difficult to
derive model-independent predictions.
We will discuss a model with {\bf 24} and {\bf 75} later.\\



3. We find that $|x|\sileqq O(0.01)$ and $M_U = 10^{16.1 \sim 16.2}$ GeV 
for the three pure directions 
without sizable non-universal threshold corrections.
Next let us explore a parameter region with a higher breaking scale.
It is expected that a higher GUT scale is realized
by some linear combination of the 
{\bf 24},  {\bf 75} and  {\bf 200} directions 
as we see from Figs.1 and 2.
Hereafter we consider only contributions from {\bf 24} and {\bf 75} Higgs
fields for simplicity.

First we estimate which value of $M_U$ is allowed
in the presence of gravitational corrections
based on one-loop RG analysis.
By using solutions of RG equations with the BC (\ref{BC}), 
the formula of $M_U$ is given by
\begin{eqnarray}
M_U = M_Z \cdot \exp \left( 2\pi 
((\alpha^{-1}_1 + \alpha^{-1}_2 + 2 \alpha^{-1}_3)(M_Z)
- 4 \alpha^{-1}_U) \over b_1 + b_2 + 2 b_3 \right)
\label{MU}
\end{eqnarray}
independent of $x'_{\bf 24}$ and $x'_{\bf 75}$.
The $x'_R$ $(R = {\bf 24}, {\bf 75})$ are written in terms of 
$\alpha_U$ and $M_U$ by
\begin{eqnarray}
x'_{R} = \alpha_U \sum_{i=1}^3 K_R^i (\alpha_i^{-1}(M_Z) - 
{b_i \over 2\pi}\ln{M_U \over M_Z})
\label{x'R}
\end{eqnarray}
where the matrix $K_R^i$ is given by
\begin{eqnarray}
K_R^i = \left(
\begin{array}{ccc}
-{1 / 18} & -{1 / 6} & {2 / 9}\\
-{5 / 36} & {1 / 12} & {1 / 18}
\end{array}
\right) .
\label{K}
\end{eqnarray}
Typical values of $\alpha_U^{-1}$, $x'_{\bf 24}$
and $x'_{\bf 75}$ are given in Table 1.
Here we use $\alpha_3^{-1}(M_Z) = 8.40$.
This result suggests that the GUT symmetry can be broken
down to the SM one $G_{SM}$ anywhere between $M_X$ and $M$.


\begin{table}

\caption{SUSY case}
\begin{center}
\begin{tabular}{|c|c|l|l|}
\hline
$M_U$ (GeV) &  $\alpha_U^{-1}$ & $x'_{\bf 24}$ 
& $x'_{\bf 75}$ \\
\hline\hline
$2 \times 10^{18}$ &  24.19 & 0.0322 & 0.0235 \\ \hline
$2 \times 10^{17}$ &  24.34 & 0.0139 & 0.0083 \\ \hline
$2 \times 10^{16}$ &  24.48 & $-0.0041$ & $-0.0067$ \\ \hline
\end{tabular}
\end{center}

\end{table}


Next we solve two-loop RG equations numerically with a common SUSY
threshold $m_{SUSY} = 1$TeV 
and calculate corrections for the linear combination of 
the {\bf 24} and {\bf 75} directions, i.e.,
$\langle {\bf 75} \rangle \cos \theta 
+ \langle {\bf 24} \rangle \sin \theta$.
Here GUT scale threshold corrections are not considered for simplicity.
Fig. 3 shows $\alpha_3(M_Z)$ against $\theta/\pi$ 
for $x=0.01, 0.05$ and 0.1 where we set $x=x_{\bf 24}=x_{\bf 75}$ and
Fig. 4 shows $M_U$ against $\theta/\pi$.
The correction to $\alpha_3(M_Z)$ is very small
at $\tan \theta \cong 1.4$ because of a cancellation between
contributions from ${\bf 24}$ and ${\bf 75}$.
The points with $\tan \theta = 1.4$ are denoted by 
the dotted vertical lines in Fig. 4. 
For $\sin \theta >0$, $M_U$ increases as $x$ does 
at $\tan \theta \cong 1.4$, although $\alpha_3(M_Z)$ does not change.
For example, $x=0.01, 0.05$ and $0.1$ correspond to $M_U=10^{16.2}$, 
$M_U=10^{16.5}$ and $M_U=10^{16.9}$ [GeV] at $\tan \theta =1.4$ 
and $\sin \theta >0$, while these values of $x$ 
correspond to $M_U=10^{16.1}$, $M_U=10^{15.9}$ and $M_U=10^{15.7}$ [GeV]
at $\tan \theta = 1.4$ and $\sin \theta <0$.
Such a relatively lower GUT scale is not realized naturally 
for $x > O(0.01)$
because the magnitude of $x$ is expected to be equal to or
smaller than $O(M_U/M)$.


\begin{center}
\input newfig3.tex

Fig.3: $\alpha_3(M_Z)$ along linear combinations of {\bf 24} and {\bf 75}
directions. 
\end{center}

\begin{center}
\input newfig4.tex

Fig.4: The GUT scale along linear combinations of {\bf 24} and {\bf 75}
directions, where $T$ denotes $T=\log_{10}M_X$ [GeV]. 
\end{center}

Similarly we can discuss the case with generic linear combination 
including a contribution from the ${\bf 200}$ Higgs field.
However, the model with the fundamental {\bf 200} Higgs particle has 
so large $\beta$-function coefficient that
the $SU(5)$ gauge coupling blows up near to 
the GUT scale.
For example, in a model with the fundamental {\bf 200} and {\bf 24}
Higgs particle and the minimal matter content,
the blowing-up energy scale $M_Y$ is obtained 
$M_Y/M_U=10^{0.74}=5.5$.
Thus, this model is not connected perturbatively with a theory 
at $M$.
There is a possibility that the {\bf 200} Higgs field is a composite one
made from fields with smaller representations.\\



4. Finally we discuss proton decay 
in the presence of gravitational corrections.
Here our purpose is to show qualitative features, how much 
gravitational corrections are important for discussions of proton decay.
Thus, we use only one-loop RG equations.
In order to carry out a precise analysis, 
it is necessary to consider two-loop effects, because 
two-loop effects of RG flows can be comparable to  
gravitational corrections.



For simplicity, we assume that particle contents of SUSY $SU(5)$ GUT
are $SU(5)$ gauge multiplet ${\bf 24}$, Higgs multiplets
${\bf 24}$ and ${\bf 75}$, and matter multiplets 
$N_g (\bar{\bf 5} + {\bf 10})$, ${\bf 5} + \bar{\bf 5}$
and extra matter multiplets.
Note that, in our usage, Higgs doublets in the MSSM belong to 
${\bf 5} + \bar{\bf 5}$
in matter multiplets and 
we assume that all extra matter multiplets acquire heavy masses
much bigger than $m_{SUSY}$.
The gauge symmetry $SU(5)$ is assumed to be broken down to $G_{SM}$ 
by a combination of 
VEVs of Higgs bosons ${\bf 24}$ and ${\bf 75}$.
In this case, would-be Nambu-Goldstone multiplets are a combination
of $({\bf 3}, {\bf 2})$ in ${\bf 24}$ and ${\bf 75}$
and a combination
of $(\bar{\bf 3}, {\bf 2})$ in ${\bf 24}$ and ${\bf 75}$.

Following the procedure in Ref.\cite{GUT-th}, we get the following relations
at one-loop level,
\begin{eqnarray}
&~& (3\alpha_2^{-1} - 2\alpha_3^{-1} - \alpha_1^{-1})(M_Z)
+ 12 \alpha_U^{-1}(x'_{\bf 24}-x'_{\bf 75}) \nonumber \\
&~& ~~~ = {1 \over 2\pi}\left({12 \over 5} \ln {\hat{M}_C \over M_Z}
 -2 \ln {m_{SUSY} \over M_Z} - {12 \over 5} \Delta_1\right) ,
\label{RGE-gr-th-1} \\
&~& (5\alpha_1^{-1} - 3\alpha_2^{-1} - 2\alpha_3^{-1})(M_Z)
+ 36 \alpha_U^{-1} x'_{\bf 75} \nonumber \\
&~& ~~~ = {1 \over 2\pi}\left(36 \ln {\hat{M}_U \over M_Z}
 +8 \ln {m_{SUSY} \over M_Z} + 36 \Delta_2 \right) 
\label{RGE-gr-th-2}
\end{eqnarray}
where we use solutions of RG equations of gauge couplings including 
a universal SUSY threshold and a GUT scale threshold correction.
Here $\hat{M}_C$ and $\hat{M}_U$ are an effective colored Higgs mass 
and GUT scale, respectively.
For example, in the minimal model $\hat{M}_C$ is 
the colored Higgs mass $M_{H_C}$ itself.
The effective GUT scale is given by
\begin{eqnarray}
\hat{M}_U = \left({M_V^2 M_{\bf 24} M_{\bf 75} \over M'} \right)^{1/3}
\label{GUT-scale}
\end{eqnarray}
where $M_V$ is $X$, $Y$ gauge boson mass, 
$M_{\bf 24}$ heavy ${\bf 24}$ Higgs mass, 
$M_{\bf 75}$ heavy ${\bf 75}$ Higgs mass
and $M'$ mass of orthogonal components to Nambu-Goldstone
multiplets.
The corrections $\Delta_{1,2}$ come from a mass splitting
among Higgs multiplets.
In a missing partner model, it is known that 
$\Delta_{1}$ is sizable,
e.g., $\Delta_{1} = \ln (1.7 \times 10^{4})$
and it can relax a constraint from proton decay\footnote{See the third
and fourth papers in Ref.\cite{GUT-th}.}.

Using relations (\ref{RGE-gr-th-1}) and (\ref{RGE-gr-th-2}),
we can estimate the magnitude of $\hat{M}_C$ and $\hat{M}_U$
as follows,
\begin{eqnarray}
\hat{M}_C &=& M_Z \cdot \left({m_{SUSY} \over M_Z}\right)^{5/6}
\nonumber \\
&~&
\times \exp \left({5\pi \over 6}((3\alpha_2^{-1} - 2\alpha_3^{-1} 
- \alpha_1^{-1})(M_Z) 
 + 12 \alpha_U^{-1}(x'_{\bf 24}-x'_{\bf 75})) 
+ \Delta_1 \right) \nonumber \\
&\sim& 3.2 \times 10^{15} \cdot \left({m_{SUSY} \over M_Z}\right)^{5/6}
\cdot \exp \left( 10 \pi \alpha_U^{-1}(x'_{\bf 24}-x'_{\bf 75}) 
+ \Delta_1 \right) 
\label{hatMC} \\
\hat{M}_U &=& M_Z \cdot \left({M_Z \over m_{SUSY}}\right)^{2/9}
\nonumber \\
&~& \times \exp \left({\pi \over 18}((5\alpha_1^{-1} - 3\alpha_2^{-1} 
- 2\alpha_3^{-1})(M_Z) + 36 \alpha_U^{-1} x'_{\bf 75}) 
- \Delta_2 \right) \nonumber \\
 &\sim& 4.8 \times 10^{16} \cdot \left({M_Z \over m_{SUSY}}\right)^{2/9}
\cdot \exp \left( 2\pi \alpha_U^{-1} x'_{\bf 75} - \Delta_2 \right)
\label{hatMU} 
\end{eqnarray}
where we use experimental data of $\alpha_i$.


SUSY-GUT models, in general, possess dangerous dimension-five and 
six operators to induce a rapid proton decay \cite{proton-decay}.
The dimension-five operators due to the exchange of colored Higgs boson
are suppressed only by a single power of
$\hat{M}_C$ in most cases.
The present lower bound of proton decay experiment suggests that
$\hat{M}_C$ is heavier than $O(10^{16})$GeV.
On the other hand, the dimension-six operators due to the exchange 
of $X$ and $Y$ gauge bosons
are suppressed by power of ${M}_V^2$.
Hence if $M_V$ is $O(10^{16})$GeV, nucleon lifetime can be longer than
$10^{34}$ years.
As we can see from Eqs. (\ref{hatMC}) and (\ref{hatMU}),
the magnitude of $\hat{M}_C$ and $M_V$ is sensitive to that of 
$\Delta_{1,2}$ and $\alpha_U^{-1} x'_R$.
It is important to get values or a relation between $\alpha_U^{-1} x'_R$
from other analysis.
%For example, if the mixing between several Higgs boson correlates 
%the mixing between their $F$-components,
%such a relation can be derived
%through precision measurement of gaugino masses.
Then we can obtain useful information on GUT scale mass spectrum
and scales such as $\hat{M}_C$ and $\hat{M}_U$
from precision measurement of sparticle masses
and RG analysis.


In the same way, we have analyzed a non-SUSY case with 
gravitational correction.
Under the assumption that particle contents of $SU(5)$ GUT
are $SU(5)$ gauge boson ${\bf 24}$, Higgs bosons
${\bf 24}$ and ${\bf 75}$, matter fermions 
$N_g (\bar{\bf 5} + {\bf 10})$, a fundamental representation Higgs 
${\bf 5}$ and extra matter fields, we get the following relations
at one-loop level,
\begin{eqnarray}
&~& (3\alpha_2^{-1} - 2\alpha_3^{-1} - \alpha_1^{-1})(M_Z)
+ 12 \alpha_U^{-1}(x'_{\bf 24}-x'_{\bf 75}) \nonumber \\
&~& ~~~~~~~~ 
= {1 \over 5\pi}\left( \ln {\hat{M}_C \over M_Z} - \Delta_1\right) ,
\label{RGE-gr-th-1-non} \\
&~& (5\alpha_1^{-1} - 3\alpha_2^{-1} - 2\alpha_3^{-1})(M_Z)
+ 36 \alpha_U^{-1} x'_{\bf 75} 
= {22 \over \pi}\left(\ln {\hat{M}_U \over M_Z}
+ \Delta_2 \right)  .
\label{RGE-gr-th-2-non}
\end{eqnarray}
Using relations (\ref{RGE-gr-th-1-non}) and (\ref{RGE-gr-th-2-non}),
we can estimate the magnitude of $\hat{M}_C$ and $\hat{M}_U$
as follows,
\begin{eqnarray}
\hat{M}_C &=& M_Z \cdot \exp \left(5\pi (3\alpha_2^{-1} - 2\alpha_3^{-1} 
- \alpha_1^{-1})(M_Z) 
 + 60 \pi \alpha_U^{-1}(x'_{\bf 24}-x'_{\bf 75}) 
+ \Delta_1 \right) \nonumber \\
&\sim& 10^{83} \cdot
\exp \left( 60 \pi \alpha_U^{-1}(x'_{\bf 24}-x'_{\bf 75}) 
+ \Delta_1 \right) 
\label{hatMC-non} \\
\hat{M}_U &=& M_Z \cdot 
 \exp \left({\pi \over 22}(5\alpha_1^{-1} - 3\alpha_2^{-1} 
- 2\alpha_3^{-1})(M_Z) + {18 \pi \over 11} \alpha_U^{-1} x'_{\bf 75} 
- \Delta_2 \right) \nonumber \\
 &\sim&  10^{14} \cdot 
 \exp \left( {18 \pi \over 11} \alpha_U^{-1} x'_{\bf 75} - \Delta_2 \right)
\label{hatMU-non} 
\end{eqnarray}
where we use experimental data of $\alpha_i$.

We require that the magnitude of $\hat{M}_U$ is bigger than $O(10^{16})$
to suppress a rapid proton decay by the exchange of $X$ and $Y$ 
gauge bosons.
Hence the magnitude of $x'_{\bf 75}$ is estimated as 
$x'_{\bf 75} = 0.024 \sim 0.044$ for $\hat{M}_U = 10^{16 \sim 18}$ GeV,
$\Delta_2 = 1$ and $\alpha_U = 1/45$.
%This value is consistent with the requirement 
%$x'_{\bf 75} \sileqq O(M_U/M)$.
Further we obtain a reasonable value as
$x'_{\bf 75} - x'_{\bf 24} \sim 0.019$
for $\hat{M}_C = 10^{16 \sim 18}$ GeV,
$\Delta_1 = 10$ and $\alpha_U = 1/45$.
Hence the non-SUSY $SU(5)$ GUT can revive in the presence of generic
gravitational corrections.\\



5. To summarize, we have studied non-universal corrections 
to gauge couplings due to higher dimensional operators 
along the three independent 
directions, {\bf 24}, {\bf 75} and {\bf 200} directions, and their 
linear combinations based on the SUSY $SU(5)$ GUT.
We have obtained a good agreement with the experimental values of 
$\alpha_i$ with $|x| \sileqq O(0.01)$ and $M_U = O(10^{16.1 \sim 16.2})$
GeV for the three pure directions.
A higher energy scale can be allowed as a breaking scale of $SU(5)$
in the presence of gravitational corrections by
a certain linear combination of contribution from 
{\bf 24} and {\bf 75} Higgs fields.
The constraints from the suppression of rapid proton decay
is sensitive to magnitude of $\Delta_{1,2}$ and $\alpha_U^{-1} x'_R$.
It is important to get values or a relation between $\alpha_U^{-1} x'_R$
from other analysis.
Then we can obtain useful information on GUT scale mass spectrum
and scales such as $\hat{M}_C$ and $\hat{M}_U$
from precision measurement of sparticle masses
and RG analysis.
The non-SUSY $SU(5)$ GUT can be revived in the presence of
gravitational corrections.



\section*{Acknowledgments}
This work was partially supported by the Academy of Finland under 
Project no. 163394. 
Y.K. acknowledges support by the Japanese Grant-in-Aid for Scientific 
Research ($\sharp$10740111) from the Ministry of
Education, Science and Culture.



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\end{document}








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