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\begin{document}

\preprint{\hbox {January 2003} }

\draft
\title{Robustness and Predictivity of 4 TeV Unification}
\author{\bf Paul H. Frampton, Ryan M. Rohm and Tomo Takahashi}
\address{Department of Physics and Astronomy,\\
University of North Carolina, Chapel Hill, NC  27599.}
\maketitle
\date{\today}


\begin{abstract}
The stability of the predictions of two of the standard model
parameters, $\alpha_3(M_Z)$ and $\sin^2 \theta(M_Z)$,
in a $M_U \sim 4$ TeV unification model is examined.
It is concluded that varying the unification scale
between $M_U \simeq 2.5$ TeV and $M_U \simeq 5$ TeV 
leaves robust all predictions within reasonable bounds. 
Choosing $M_U = 3.8 \pm 0.4$ TeV gives, at lowest order,
accurate predictions at $M_Z$. Flavor-changing effects
and deviations from precision electroweak data are discussed. 
\end{abstract}
\pacs{}

\newpage

\bigskip
\bigskip

\noindent {\it Introduction}

\bigskip

\noindent One of the principal motivations for extending
the standard model is the GUT gauge hierarchy between the weak
scale and the grand unification or GUT scale.
A related concern, not addressed here, is the Planck hierarchy 
between the weak scale and the Planck scale; the model
we consider has flat spacetime, vanishing Newton's constant
and infinite Planck scale.

The most popular solution of the GUT hierarchy is low-energy
supersymmetry\cite{DG,DRW,ADF,ADFFL} where the three gauge couplings
$\alpha_i(\mu)$ (i = 1,2,3) run logarithmically 
from $\mu = M_Z \sim 91$ GeV, where they are known, up to 
$M_{GUT} \sim 2 \times 10^{16}$ GeV, where they coincide 
with impressive accuracy.

In a recently-proposed model\cite{PHF02}, grand unification occurs
differently. The three couplings run from $\mu = M_Z$
up to a lower unification scale $M_U \sim 4$ TeV,
at which scale the theory is embedded in a larger
gauge group $G \equiv SU(3)^{12}$. The $SU(3)$ gauge couplings
$\alpha_j(\mu)$ (j=1-12) are all equal at $\mu = M_U$.
The embedding of the standard model gauge group
in the larger gauge group $G$ provides a group-theoretical
explanation for the different values of $\alpha_i(M_U)$.

This low-scale unification model also has a top-down inspiration
from string theory through the AdS/CFT correspondence
\cite{M,GKP,W} arising from consideration of a Type IIB 
superstring in d = 10 dimensional
spacetime compactified on $AdS_5 \times S^5$.
Using a finite group 
$\Gamma = Z_{12}$ in an abelian orbifold $AdS_5 \times S^5/\Gamma$ 
gives a quiver gauge theory\cite{PHF99} with gauge group
$SU(N)^{12}$ either with no supersymmetry
${\cal N} = 0$ \cite{PHF02} or
with ${\cal N} = 1$ supersymmetry\cite{FK}

Several issues were left open in \cite{PHF02}:
robustness of the predictions under variations
of the scale $M_U$ (conversely, the accuracy of the
predictions at $\mu = M_Z$); 
the size of flavor-changing effects, and
the consistency of the additional states around $M \sim M_U$
with constraints imposed by precision low-energy data.
In this article we shall address all of these issues.

\bigskip
\bigskip

\noindent {\it Robustness of Predictions to Variation in $M_U$}

\bigskip

\noindent The calculations of \cite{PHF02} were done in the one-loop
approximation to the renormalization group equations
without threshold effects.
Because the couplings remain weak and the
scales $M_U$ and $M_Z$ are relatively close, this is self-consistent. Other
corrections
due to non-perturbative effects, and the effects of
large extra dimensions, are outside of the scope of this paper. 
In one sense the robustness of this TeV-scale
unification is almost self-evident, in that it follows from the weakness
of the coupling constants in the evolution from $M_Z$ to $M_U$.
That is, in order to define the theory at $M_U$,
one must combine the effects of
threshold corrections ( due to O($\alpha(M_U)$)
mass splittings )
and potential corrections from redefinitions
of the coupling constants and the unification scale.
We can then {\it impose} the coupling constant relations at $M_U$
as renormalization conditions and this is valid
to the extent that higher order corrections do
not destabilize the vacuum state.

We shall approach the comparison with data in two
different but almost equivalent ways. The first
is ``bottom-up", where we use as input the
requirement that the values of $\alpha_3(\mu)/\alpha_2(\mu)$ and
$\sin^2 \theta (\mu)$ are expected to be $5/2$
and $1/4$, respectively, at $\mu = M_U$.
Using the experimental ranges allowed for
$\sin^2 \theta (M_Z) = 0.23113 \pm 0.00015$,
$\alpha_3 (M_Z) = 0.1172 \pm 0.0020$ and
$\alpha_{em}^{-1} (M_Z) = 127.934 \pm 0.027$
from \cite{PDG} we have plotted in Figure 1
the values of $\sin^2 \theta (M_U)$
(vertical axis) and $\alpha_3 (M_U) / \alpha_2(M_U)$
(horizontal axis) for a range of $M_U$ between 1.5 TeV
and 8 TeV.
Allowing a maximum discrepancy of $\pm 1\%$ in
$\sin^2 \theta (M_U)$ and 
$\pm 4\%$ in $\alpha_3 (M_U) / \alpha_2 (M_U)$
as reasonable estimates of corrections, we deduce that
the unification scale $M_U$ may vary
between 2.5 TeV and 5 TeV. Thus the theory is
robust in the sense that uncertainty in the
renormalization group equations does not effect the
existence of unification. 


\bigskip
\bigskip

\noindent {\it Accuracy of Predictions at $\mu = M_Z$}

\bigskip

\noindent Alternatively, to test of predictivity 
we fix the unification values at $M_U$ of
$\sin^2 \theta(M_U) = 1/4$ and $\alpha_3 (M_U) / 
\alpha_2 (M_U) = 5/2$ and compute the
resultant predictions at the scale $\mu = M_Z$.
The results are shown for $\sin^2 \theta (M_Z)$
in Fig. 2 with the allowed range\cite{PDG}
$\alpha_3 (M_Z) = 0.1172 \pm 0.0020$. The precise
data on $\sin^2 (M_Z)$ are indicated in Fig. 2 
demonstrating that the model makes correct
predictions for $\sin^2 \theta (M_Z)$.
Similarly, in Fig 3, there is a plot of the
prediction for $\alpha_3 (M_Z)$ versus
$M_U$ with $\sin^2 \theta(M_Z)$ held
within the allowed empirical range. 
The two quantities plotted in Figs 2 and 3
are consistent for similar ranges of $M_U$:
both $\sin^2 \theta(M_Z)$ and $\alpha_3(M_Z)$ 
are within the empirical limits
if $M_U = 3.8 \pm 0.4$ TeV.

\bigskip
\bigskip

\noindent {\it Flavor-Changing Effects}

\bigskip

\noindent The model has many additional gauge bosons
at the unification scale, including neutral $Z^{'}$'s,
which could mediate flavor-changing processes
on which there are strong empirical upper limits.
A detailed analysis will require specific identification
of the light families and quark flavors with
the chiral fermions appearing in the quiver diagram
for the model. We can make only the general
observation that the lower bound on a $Z^{'}$
coupling like the standard $Z$
is quoted as $M(Z^{'}) < 1.5$ TeV \cite{PDG};
this is safely below the $M_U$ values considered here
which could be identified with the mass of new gauge bosons.
This suggests that flavor-changing
processes are under control in the model, but
this issue will require more careful analysis when
a specific identification of the quark states
is carried out.

\bigskip
\bigskip

\noindent {\it Consistency with Precision Electroweak Data}

\bigskip

\noindent Since there are many new states predicted 
at the unification scale $\sim 4$ TeV, there is a danger
of being ruled out by precision low energy data.
This issue is conveniently studied in terms
of the parameters $S$ and $T$ introduced in \cite{Peskin},
designed to measure departure from the predictions
of the standard model.
Concerning $T$, if the new $SU(2)$ doublets are
mass-degenerate and hence do not violate a custodial
$SU(2)$ symmetry, they do not contribute $T$.
This provides a constraint on the spectrum
of new states.
According to \cite{Peskin}, a multiplet of degenerate
heavy chiral fermions gives a contribution to $S$:

\begin{equation}
S = C \sum_i \left( t_{3L}(i) - t_{3R}(i) \right)^2 / 3 \pi
\label{capitalS}
\end{equation} 
where $t_{3L,R}$ is the third component of weak isopspin
of the left- and right- handed component of
fermion $i$ and $C$ is the number of colors.
In the present model, the additional fermions are non-chiral
and fall into vector-like multiplets and so do not
contribute
to $S$.
Provided that the extra isospin multiplets 
at the unification scale $M_U$ are sufficiently
mass-degenerate, therefore, there is no conflict
with precision data at low energy.

\bigskip
\bigskip

\noindent {\it Discussion}

\bigskip

\noindent The plots we have presented clarify the accuracy
of the predictions of this TeV unification scheme for
the precision values accurately measured at the Z-pole.
The predictivity is as accurate for $\sin^2 \theta$ as
it is for supersymmetric GUT models\cite{DG,DRW,ADF,ADFFL}. 
There is, in addition, an accurate prediction for $\alpha_3$
which is used merely as input in SusyGUT models.

At the same time, the accuracy of the predictions remains robust
if we allow the unification scale to vary
from about 2.5 TeV to 5 TeV. 

In conclusion, since this model ameliorates the GUT hierarchy
problem and naturally accommodates three families, it
provides a viable alternative to the widely-studied
GUT models which unify by logarithmic evolution
of couplings up to much higher scales. 


\bigskip
\bigskip
\bigskip
\bigskip

\noindent {\it Acknowledgements}

\bigskip

\noindent This work was supported in part by the
US Department of Energy under
Grant No. DE-FG02-97ER-41036.

\newpage




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\end{thebibliography}

\newpage

\bigskip
\bigskip

\noindent \underline{\bf Figure Captions}

\bigskip

\noindent Fig. 1.

\noindent Plot of $\sin^2 \theta (M_U)$ versus $\alpha_3(M_U)/\alpha_2 (M_U)$
for various choices of $M_U$.

\bigskip

\noindent Fig.2.

\noindent Plot of $\sin^2 \theta(M_Z)$ versus $M_U$ in TeV, assuming
$\sin^2 \theta(M_U) = 1/4$ and $\alpha_3 / \alpha_2 (M_U) = 5/2$.

\bigskip

\noindent Fig.3.

\noindent Plot of $\alpha_3 (M_Z)$ versus $M_U$ in TeV, assuming
$\sin^2 \theta(M_U) = 1/4$ and $\alpha_3 / \alpha_2 (M_U) = 5/2$.

\newpage


\begin{figure}

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\begin{figure}

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