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\begin{document}

\pagestyle{empty} \begin{titlepage}
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\title{\Large \bf Universal Extra Dimensions and \\ the Higgs Boson Mass
\\ [1.3cm]}

\author{\normalsize
%\small
\bf \hspace*{-.3cm} Thomas Appelquist,
Ho-Ung Yee
 \\ \\ {\small {\it
\vspace*{-5cm}
Department of Physics, Yale University, New
Haven, CT 06520, USA \footnote{e-mail: thomas.appelquist@yale.edu, \, ho-ung.yee@yale.edu}
}}\\
 }

\date{ } \maketitle

\vspace*{-7.9cm}
\noindent \makebox[12.7cm][l]{\small \hspace*{-.2cm}
 {\small YCTP-P10-02 } \\
\makebox[11.8cm][l]{\small \hspace*{-.2cm} November 2, 2002 }
{\small } \\

 \vspace*{10.5cm}

  \begin{abstract}
{\small
We study the combined constraints on the compactification scale $1/R$ and
the Higgs mass $m_H$ in the standard model with one or two universal extra
dimensions. Focusing on precision measurements and employing the
Peskin-Takeuchi $S$ and $T$ parameters, we analyze the allowed region
in the $(m_H, 1/R)$ parameter space consistent with current
experiments. For this purpose, we calculate complete one-loop KK mode
contributions to $S$, $T$, and $U$, and also estimate the contributions from
physics above the cutoff of the higher-dimensional standard model. A compactification scale
$1/R$ as low as $250\,{\rm GeV}$ and significantly extended regions of $m_H$
are found to be consistent with current precision data.
}
\end{abstract}

\vfill \end{titlepage}

\baselineskip=18pt \pagestyle{plain} \setcounter{page}{1}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%55
\section{Introduction} \setcounter{equation}{0}

In models with universal extra dimensions (UED's) in which all the standard model fields propagate,
bounds on the compactification scale, $1/R$, have been estimated from precision experiments to be as low as $\sim\,300\,{\rm GeV}$ \cite{Appelquist:2000nn,Agashe:2001xt}.
This would lead to an exciting phenomenology in the next generation of collider
experiments \cite{Cheng:2002ab,Rizzo:2001sd,Macesanu:2002db,Petriello:2002uu}. Above the compactification scale,
the effective theory becomes a higher dimensional field theory whose equivalent description in 4D consists of
the standard model fields and towers of their KK partners whose interactions are very similar to those in the standard model.
Because the effective theory above the compactification scale (the higher dimensional standard model) breaks down at the scale
$M_s$, where the theory becomes nonperturbative, the towers of KK particles must be cut off at this
scale in an appropriate way. The unknown physics above $M_s$ can be described by operators of higher mass dimension whose
coefficients can be estimated.

To obtain the standard-model chiral fermions from the corresponding extra dimensional fermion fields, the higher dimensional
standard model is compactified on an orbifold
to mod out the unwanted chirality by orbifold boundary conditions. For a single (two) universal extra dimension(s),
this is $S^1/Z_2$ ( $T^2/Z_2$ ) \cite{Ponton:2001hq}.
The interactions involving nonzero KK particles are largely determined by the bulk lagrangian in terms of the higher dimensional standard model,
while the effects from possible terms localized at the orbifold fixed points are relatively volume-suppressed.
The KK particles enter various quantum corrections to give contributions to precision measurements. Studies of
their effects on the precision electroweak measurements in terms of $S$ and $T$ parameters \cite{Appelquist:2000nn}, on the flavor
changing process $b\,\rightarrow\,s\,+\gamma$ \cite{Agashe:2001xt}, and on the anomalous muon magnetic moment \cite{Agashe:2001ra,Appelquist:2001jz}
have shown that these effects are consistent with current precision experiments if $1/R$ is above a few hundred ${\rm GeV}$.
The cosmic relic density of the lightest KK particle as a dark matter candidate is also of the right order of magnitude \cite{Servant:2002aq},
and its direct or indirect detection is within the reach of future experiments \cite{Cheng:2002ej,Hooper:2002gs,Servant:2002hb,Majumdar:2002mw}.

In this paper, we address the effects of the new physics above $1/R$ on the combined constraints for the Higgs mass and $1/R$.
%Without direct observations,
Current knowledge of the Higgs mass has been inferred from its contributions to the electroweak precision observables.
Because the new physics in terms of KK partners and higher dimension operators representing physics above $M_s$ also contributes to these
observables, the constraints on the Higgs mass can be significantly altered in the UED framework. %\cite{Chivukula:1999az}.
The effects on the precision observables from non zero KK modes depend on both
$1/R$ and the Higgs mass, $m_H$ (through KK Higgs particles), while the standard model (the zero modes) contributions are functions of $m_H$ alone. We therefore
analyze the allowed region in the $(m_H,1/R)$ parameter space consistent with the current precision measurements.
%The results in this paper can be viewed in two ways. Once we know the compactification scale from future collider experiments or direct/indirect
%detection of the KK dark matter, we would get a prediction on the Higgs mass. On the other hand, if the Higgs mass is directly measured, our results will
%provide us with information on $1/R$. Hopefully in the case where we know both of them, we might be able to test the viability of the UED theories.

Current precision electroweak experiments are sensitive to new-physics corrections to fermion-gauge boson vertices and
gauge-boson propagators. The most sensitive fermion-gauge boson vertex is the $Zb\bar{b}$ vertex.
Contributions to it were analyzed in Ref.\cite{Appelquist:2000nn}. The dominant contribution comes from
loops with KK top-bottom doublets:
\be
\delta g_{L}^{b} \sim \frac{\alpha}{4\pi} \frac{m_t^2}{M_j^2}\quad,  %\,\,\,{\rm or} \,\,\,\frac{\alpha}{4\pi} \frac{m_{W,Z}^2}{M_j^2}
\label{vertex}
\ee
where $M_j\,=\,\sqrt{j_1^2+\cdots+j_\delta^2}/R$, and $j=(j_1,\cdots,j_\delta)$ is a set of indices of KK levels in $\delta$ extra dimensions.
It was noted there that these corrections are less important than the Peskin-Takeuchi $S$ and $T$ parameters \cite{Peskin:1990zt} in constraining UED theories
for the phenomenologically interesting region of $1/R \gg m_t$.
%(For an analysis of the vertex correction in the process $b\,\rightarrow\,s\,+\,\gamma$, see \cite{Agashe:2001xt}.)
%Moreover, for vertices with external light fermions, the above dominant term is simply absent.
We therefore focus on the Peskin-Takeuchi parameters. 
%the gauge-boson propagator corrections in terms of the Peskin-Takeuchi
%$S,T$ and $U$ parameters \cite{Peskin:1990zt}.
% postponing more complete treatment including full vertex corrections to the future.

We consider two possibilities; a single universal extra dimension on $S^1/Z_2$ and two universal extra dimensions on $T^2/Z_2$.
In the case of a single extra dimension, the cutoff effects from physics above $M_s$ are estimated to be negligible and we can do a reliable
calculation of the contributions from KK modes alone. This is not the case for the model with two extra dimensions.
The UED theory on $T^2/Z_2$ is a particularly interesting model because it points to 
three generations \cite{Dobrescu:2001ae} (See also \cite{Fabbrichesi:2001fx}),
and can explain the longevity of protons \cite{Appelquist:2001mj}. 
The neutrino oscillation data can also be accomodated within this model \cite{Appelquist:2002ft}.
However, the sums over the KK particle contributions to precision observables are logarithmically divergent with two extra dimensions,
and effects from above the cutoff $M_s$
must be included. We estimate these effects using higher dimension operators, which makes the analysis only qualitative,
but we can still extract useful information from the results.

In the next section, we describe the calculation of $S,T$ and $U$ from one-loop diagrams with KK particles, and a subtlety involved in this calculation.
In section 3, we estimate the contributions to $S$ and $T$ from physics above the cutoff $M_s$. Sections 4 and 5 are devoted to the details of the analysis
with both a single extra dimension on $S^1/Z_2$ and two extra dimensions on $T^2/Z_2$. We summarize and conclude in section 6.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{KK-mode contributions to the $S,T$ and $U$ parameters} \setcounter{equation}{0}

%\subsection{Calculations of $S,T$ and $U$ from a given $j$th KK level}

In the analysis of Ref.\cite{Appelquist:2000nn}, it was argued that
the dominant contributions to $S$ and $T$ come from KK modes of the top-bottom quark doublet:
\be
T^t_j\,\sim\,\frac{1}{\alpha}\frac{3m_t^2}{8\pi^2 v^2}\frac{2}{3}\frac{m_t^2}{M_j^2}\quad,\qquad
S^t_j\,\sim \,\frac{1}{6\pi}\frac{m_t^2}{M_j^2}\quad.
\ee
%For a Higgs mass less than $250 \,{\rm GeV}$,
It was shown that the constraint from $T$ is stronger than that from $S$.
The $U$ parameter is numerically much smaller than $S$ and $T$, thus much less important in constraining UED theories.
An important premise in Ref.\cite{Appelquist:2000nn} was that the Higgs mass, $m_H$, is lighter than $250\,{\rm GeV}$.
% it was fixed at $200\,{\rm GeV}$ in that analysis.

If the Higgs mass $m_H$ is large, however, the contributions from the standard model Higgs and its higher KK modes become important
and eventually dominate over the KK quark contributions. %This is because the effects from the standard model Higgs and its KK partners
%have a logarithmic dependence on $m_H$.
A key point is that the Higgs contribution to $T$ is negative, which
is opposite to the KK quark contribution. (For $S$, both KK quarks and KK Higgs contributions are positive.)
Thus, the two contributions can compensate each other to relax the $T$ constraints, allowing
an extended region in the $(m_H,1/R)$ parameter space. Moreover, a large $m_H$ can also bring important constraints from $S$, requiring
a combined $S$ and $T$ analysis rather than separate ones.
It is thus important to do a more complete analysis allowing for the possibility of a large Higgs mass.

We calculate complete one-loop corrections from a given $j$th KK level of the standard model fields (with a single Higgs doublet)
to gauge-boson self energies:
$\Pi^j_{WW}, \Pi^j_{ZZ}, \Pi^j_{\gamma \gamma}$ and $\Pi^j_{Z \gamma}$ (See the appendix). Here $j$ represents a positive
integer for one extra dimension or a set of $\delta$ non negative integers in the case of $\delta$ extra dimensions.
The total contribution from extra dimensions will be the sum over $j$.
In the large KK mass limit $M_j \gg m_t,m_H$, the
contributions to $S,T$ and $U$ parameters are proportional to $\frac{m_t^2
}{M_j^2},\,{\rm or}\,\frac{m_H^2}{M_j^2}$.  In one extra dimension, there is one KK mode for each positive interger $j$, and the sum converges.
However, in two or more extra dimensions, there are degenerate KK modes having the same $M_j$, which makes the sum divergent. 
With two extra dimensions, the cutoff sensitivity is logarithmic.
In our calculation of $S,T$ and $U$, we use the tree-level formula for the masses of KK particles neglecting
corrections from one-loop gauge interactions and boundary terms localized at the orbifold fixed points \cite{Cheng:2002iz}.
This is justified because these are of one-loop order and the shifts due to them, which are already of one-loop order, are two-loop effects.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{A subtlety in the calculation} \setcounter{equation}{0}
Before presenting our results, we discuss a subtlety in the calculation.
The conventional definition of the T parameter is 
\be
\alpha(m_Z)\,T\,\,\equiv\,\,\frac{\Pi_{WW}(p^2=0)}{m_W^2}-\frac{\Pi_{ZZ}(p^2=0)}{m_Z^2}\quad,
\label{defT} 
\ee 
where the $\Pi$ functions are the gauge-boson self energies
arising from new, non-standard-model physics, and $\alpha(m_Z)\approx 1/128$. When the non-standard-model
physics is "oblique" (entering dominantly through the gauge-boson self
energies), this definition corresponds directly to a physical measurement.
An example is provided by a loop of KK modes of the standard-model fermions
such as the top quark. In general, the new physics can also contribute
through vertex corrections and box diagrams. An example of this is provided
by one-loop corrections involving KK gauge bosons. All the pieces must then be
combined to insure a finite and gauge invariant (physical) result. Indeed,
we find in our calculation that the one-loop divergences in $T$ as defined above, arising from KK gauge bosons and KK Higgs bosons, do not
cancel, although $S$ and $U$, as conventionally defined, turn out to be
finite and well defined.

The one-loop contributions to $\Pi_{WW}$ and $\Pi_{ZZ}$ are listed in the appendix. The computation has been done in Feynman gauge.
From the tabulation, one can see that in this gauge, $T$ is UV-divergent at the one-loop level, and that the divergence 
arises from graphs involving loops of KK gauge bosons and KK Higgs bosons.
This indicates that there should be a non-vanishing counterterm for the $T$ parameter of (\ref{defT}).
% Counterterms are generally scheme and gauge dependent and the finite 
%$T$ parameter after adding a counterterm would not be a physical, scheme and gauge invariant observable. 
%However, it turns out that the counterterm for the $T$ parameter does not have a scheme dependence (though gauge dependence remains.).
It can be shown that, because of the constraints from gauge symmetry, 
the counterterm for $T$ is determined by the $A_\mu Z^\mu$-counterterm at the one-loop level.
Once we fix the $A_\mu Z^\mu$-counterterm, corresponding to photon-$Z$ mass mixing, cancelling $\Pi_{Z\gamma}(0)$ to ensure a massless photon propagator,
the counterterm for $T$ is completely determined in terms of $\Pi_{Z\gamma}(0)$. 

As a result, in the basis in which the photon-$Z$ mass matrix is diagonal through one-loop, it can be shown that the modified $T$
parameter including the counterterm takes the form
\be
\alpha(m_Z)\,\tilde{T}\,\,\equiv\,\,\frac{\Pi_{WW}(0)}{m_W^2}-\frac{\Pi_{ZZ}(0)}{m_Z^2}
-2\,{\rm cos}\theta_w\,{\rm sin}\theta_w\,\frac{\Pi_{Z\gamma}(0)}{m_W^2}\quad.
\label{defTnew} 
\ee 
It can be checked explicitly from the appendix that this expression is UV-finite. As the finiteness originates from
a certain relation between counterterms determined by gauge symmetry, it is true in any gauge.
%Because of the physical definition of the counterterm, it is also the case that this expression is independent of any conventional renormalization schemes.
Of course, $\tilde{T}$ is not, in general, a gauge-invariant, physical observable unless it is combined with vertex corrections and box diagrams.
%(Because we are now using counterterms, the relevant counterterms must be added to vertices, too. We assume that this has been done 
%when we mention vertex corrections later.)

The important observation, however, is that the contribution of the 
Higgs-boson KK modes
to the vertices and box diagrams are negligible at the one-loop level since 
they are
suppressed by small Yukawa couplings when they
couple to the light external fermions. Thus the dominant contributions to 
$\tilde{T}$
when $m_H$ and $m_t$ are large compared to the gauge-boson masses, must by
themselves be gauge invariant. It is straightforward to determine these from the appendix. For a $j$th KK level,
\bear
\tilde{T}^j_{\rm KK Higgs}\approx\frac{1}{4\pi}\frac{1}{c_w^2}
\,f_T^{\rm KK Higgs}\bigg(\frac{m_H^2}{M_j^2}\bigg)
\quad, \qquad
\tilde{T}^j_{\rm \,\,KK \,top\,\,}\approx\frac{1}{\alpha(m_Z)}\frac{3\,m_t^2}{8\pi^2v^2}
\,f_T^{\rm \,\,KK \,top\,\,}\bigg(\frac{m_t^2}{M_j^2}\bigg)
\quad,
\label{KKhigtop}
\eear
where $v=246\,{\rm GeV}$ is the VEV of the zero mode Higgs boson and
\bear
f_T^{\rm KK Higgs}(z)&=&\frac{5}{8}-\frac{1}{4z}+\bigg(-\frac{3}{4}-\frac{1}{2z}+\frac{1}{4z^2}\bigg)\log(1+z) \quad , \nonumber \\
f_T^{\rm \,\,KK \,top\,\,}(z)&=&1-\frac{2}{z}+\frac{2}{z^2}\log(1+z)\quad . 
\label{ftnsforT}
\eear
%The important observation is that non-oblique KK gauge-boson contributions to vertices and $\tilde{T}$ are numerically
%much smaller than the effects induced by KK Higgs and KK top quarks through their contributions to gauge-boson propagators.
%(For purely oblique corrections, we have $\Pi_{Z\gamma}(0)=0$ and $\tilde{T}$ becomes the original $T$ parameter.)
Note that the KK Higgs boson contributions to $\tilde{T}$ are negative, while the contributions from the KK top quarks are positive. 

\begin{figure}[t]
\begin{center}
\scalebox{1}[1]{\includegraphics{SSSM.eps}\includegraphics{TTSM.eps}}%\includegraphics{UUSM.eps}}
\par
\vskip-2.0cm{}
\end{center}
\caption{\small Contributions to $S$ and $T$ from the standard model(the zero modes) and $j$'th KK levels of one UED compactified on $S^1/Z_2$.
Here, $1/R=400 \,{\rm GeV}\,,\,m_{H}^{\rm ref}=115\,{\rm GeV}$ and $m_t=174\,{\rm GeV}$.}
\label{kkmodes}
\end{figure}
%compared to contributions from KK Higgs or KK top quarks;
%\be
%\frac{\alpha}{4\pi}\,\frac{m_H^2}{M_j^2}\quad {\rm when}\quad \,m_H \ll M_j\,,\qquad{\rm or} \qquad\frac{\lambda_t^2}{16\pi^2}\,\frac{m_t^2}{M_j^2}\quad,
%\ee
%where $\lambda_t\,\sim\,1$ is the top Yukawa coupling. 
%For large Higgs masses of interest that we are focusing here,
By contrast, the typical size of one-loop KK gauge-boson contributions to $\alpha\tilde{T}$, vertex corrections, or box diagrams is of order
\be
\frac{\alpha}{4\pi}\,\frac{m_W^2}{M_j^2}\quad.
\ee
Clearly these are negligible compared to the contributions (\ref{KKhigtop}) when $m_H^2,\,m_t^2\,\gg\,m_W^2$.
This leads us, to a good approximation, to neglect them
and to focus on the dominant, gauge-invariant, oblique contributions (\ref{KKhigtop}) in our numerical analysis.
 
By the same reasoning, the (gauge-dependent) KK gauge-boson contributions to $S$ and $U$ can be neglected. From the appendix, one can write down the
dominant, gauge-invariant expressions for $S$, which are similar to (\ref{KKhigtop}), arising from KK Higgs bosons and KK top quarks:
\bear
S^j_{\rm KK Higgs}\approx\frac{1}{4\pi}
\,f_S^{\rm KK Higgs}\bigg(\frac{m_H^2}{M_j^2}\bigg)
\quad, \qquad
S^j_{\rm \,\,KK \,top\,\,}\approx\frac{1}{4\pi}
\,f_S^{\rm \,\,KK \,top\,\,}\bigg(\frac{m_t^2}{M_j^2}\bigg)
\quad, 
\label{KKhigtopS}
\eear
where
\bear
f_S^{\rm KK Higgs}(z)&=&-\frac{5}{18}+\frac{2}{3z}+\frac{2}{3z^2}+\bigg(\,\frac{1}{3}-\frac{1}{z^2}-\frac{2}{3z^3}\bigg)\log(1+z) \quad , \nonumber \\
f_S^{\rm \,\,KK \,top\,\,}(z)&=&\frac{2z}{1+z}-\frac{4}{3}\,\log(1+z) \quad . 
\label{ftnsforS}
\eear
These, together with (\ref{KKhigtop}), are the basis of our numerical calculations.


%This justifies our use of $\tilde{T}$ as a meaningful
%parameter which can be treated as the conventional $T$ parameter.
%one exception to this is $Zb\bar{b}$ vertex where we have a contribution from KK top quarks. However, as pointed out after (\ref{vertex}),
%this contribution may safely be neglected when constraining UED theories from precision measurements.

In Fig.\ref{kkmodes}, we show contributions to $S$ and $T$ from different $j$th levels in terms of the Higgs mass, for a
representative value of $1/R$, in the case of a single extra dimension on $S^1/Z_2$. 
We also include the standard model contributions from the Higgs (zero mode) after fixing
the reference Higgs mass at $115\, {\rm GeV}$. The contributions from higher KK levels become small rapidly,
consistent with the decoupling behavior \cite{Appelquist:tg}. % for heavy particles \cite{Appelquist:tg}.
%We also see from Fig.\ref{kkmodes} that $U$ is indeed small enough to be neglected.
Results for the case of two extra dimensions on $T^2/Z_2$
exhibit similar behavior for each $j$th level, though we must take into account degeneracy when summing them.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Contributions to $S$ and $T$ from physics above $M_s$}

Because our effective theory breaks down at the cutoff scale $M_s$,
we also estimate the contributions from physics above this scale by examining the relevant
local operators of higher mass dimension, whose coefficients incorporate unknown physics above $M_s$. To find the operators that give
direct tree-level contributions to $S$ and $T$, it is convenient to use the matrix notation for the Higgs fields,
\be
\CM\equiv \left(\,i\sigma^2 \CH^*\,,\,\CH\,\right)=\left(\ba{ccc}h^{0*} & h^+ \\ -h^{+*} & h^0 \ea \right).
\ee
Here, the $\CM$, $\CH$ and all calligraphic fields in the following are the fields in $(4+\delta)$ dimensions, whereas the
corresponding roman letters will represent the 4 dimensional zero modes after KK decomposition.
The $SU(2)_L \times U(1)_Y$ gauge rotation is $\CM\,\rightarrow\,U_L(x)\CM e^{-i\alpha (x)\sigma^3}$ and the covariant derivative is
\be
D_{\alpha}\CM\,=\,\partial_{\alpha}\CM\,+\,i\,\hat{g}\,\CW_{\alpha}^{\,a}\,\frac{\sigma^a}{2}\,\CM\,-\,i\,\hat{g}'\,\CB_{\alpha}\,\CM\,\frac{\sigma^3}{2}\quad,
\ee
where $\CW_{\alpha}^{\,a},\,\CB_{\alpha}$ are the gauge fields in $(4+\delta)$ dimensions and $\hat{g},\,\hat{g}'$
are the corresponding $(4+\delta)$-dimensional gauge couplings whose mass dimension is $-\frac{\delta}{2}$. The mass dimension of
$\CW_{\alpha}^{\,a},\,\CB_{\alpha}$ and $\CM \,(\,\CH)$ is $(1+\frac{\delta}{2})$.
The gauge invariance
dictates that the Higgs potential up to quartic order (i.e. up to mass dimension $(4+2\delta)$ ) depends only on $\frac{1}{2}{\rm Tr}[\CM^\dagger\CM]=\CH^\dagger\CH$,
which implies the enlarged $SU(2)_L \times SU(2)_R$ symmetry,
\be
\CM\,\rightarrow\,U_L\CM U_R \quad.
\ee
After the zero mode Higgs field gets a VEV, $<M >=\frac{v}{\sqrt{2}}\,\mathbf{1}$, $v=246\,{\rm GeV}$, this symmetry is broken down to the diagonal custodial
$SU(2)_C$ which protects $T$ at tree level. Hypercharge interactions violate custodial $SU(2)_C$, inducing
nonzero $T$ at loop level.

When we consider operators of higher mass dimension, however, the gauge invariance can no longer prevent operators that violate
the custodial symmetry. There is one independent, custodial symmetry-violating operator of the lowest mass dimension\footnote{Other possible operators can be shown to be equivalent to (\ref{Toperator}) up to additive custodial-symmetric operators.}
$(6+2\delta)$:
\bear
&&c_1\cdot\frac{\hat{\lambda}}{2^2\cdot2!\cdot M_s^2}\,{\rm Tr}[\sigma^3(D_\alpha\CM)^\dagger\CM]\cdot{\rm Tr}[\sigma^3(D^\alpha\CM)^\dagger\CM]\quad, \nonumber \\
&&=c_1\cdot\frac{\hat{\lambda}}{2^2\cdot2!\cdot M_s^2}\,
\big(\CH^\dagger \stackrel{\leftrightarrow}{D_\alpha}\CH\big)\big(\CH^\dagger \stackrel{\leftrightarrow}{D^\alpha}\CH\big)\quad,
\label{Toperator}
\eear
where $\alpha=1,\dots,(4+\delta)$. We have extracted the $(4+\delta)$-dimensional Higgs self coupling $\hat{\lambda}$, of mass dimension $-\delta$,
which appears in the quartic interaction between four Higgs fields,
\be
\CL_{(4+\delta)}\quad\supset\quad\frac{\hat{\lambda}}{2!}\big(\CH^\dagger\CH)^2=\frac{\hat{\lambda}}{2^2\cdot 2!}\big({\rm Tr}[\CM^\dagger\CM]\big)^2\quad,
\label{quartic}
\ee
expecting that $\hat{\lambda}$ reflects the strength of the underlying dynamics responsible for
similar kinds of four-Higgs interactions. Except for the custodial symmetry violation, the operator (\ref{Toperator}) simply has two more
derivatives than (\ref{quartic}) and we have pulled out all the expected factors (including various numerical counting factors)
in writing (\ref{Toperator}). We then expect that
$c_1$ should be a constant no larger than of order unity. If there is a suppression of the custodial symmetry violation,
$c_1$ will be small compared to unity. 

After KK decomposition, the relevant 4D operator from (\ref{Toperator}) is obtained after replacing $(4+\delta)$-dimensional fields with
the corresponding 4D zero modes,
\be
\CM\,(\,\CH\,)\,\rightarrow\,\frac{\sqrt{2}}{(2\pi R)^{\delta /2}}\,M\,(\,H\,) \quad,
\label{subs}
\ee
and integrating over the extra $\delta$ dimensions, $\int d^{\delta}y=(2\pi R)^{\delta}/2$. (The factor 2 is from the $Z_2$ orbifold).
Also replacing $\hat{\lambda}$ with the 4D Higgs self coupling $\lambda$,
\be
\hat{\lambda}=\frac{(2\pi R)^{\delta}}{2}\,\lambda\quad,
\ee
the resulting 4D operator is
\bear
&&c_1\cdot\frac{\lambda}{2^2\cdot2!\cdot M_s^2}\,{\rm Tr}[\sigma^3(D_\mu M)^\dagger M]\cdot{\rm Tr}[\sigma^3(D^\mu M)^\dagger M] \quad,\nonumber \\
&&=c_1\cdot\frac{\lambda}{2^2\cdot2!\cdot M_s^2}\,
\big(H^\dagger \stackrel{\leftrightarrow}{D_\mu}H\big)\big(H^\dagger \stackrel{\leftrightarrow}{D^\mu}H\big)\quad .
\label{4dToperator}
\eear
Note that $(2\pi R)$ factors have dropped out in the expression (\ref{4dToperator}).
The contribution to $T$ from physics above $M_s$ can be estimated from (\ref{4dToperator}) to be
\be
T^{UV}=c_1\cdot\frac{\lambda}{2^2\cdot2!\cdot M_s^2}\cdot\frac{2\,v^2}{\alpha(m_Z)}=c_1\cdot\frac{\lambda\,v^2}{4M_s^2\cdot\alpha(m_Z)}=
c_1\cdot\frac{m_H^2}{4M_s^2\cdot\alpha(m_Z)} \quad .
\label{deltaT}
\ee
This result will be used in later sections when we estimate all the contributions to $T$ in one or two extra dimensions.

We next discuss the $S$ parameter. From the definition of $S$,
\be
S=-\frac{16\pi}{g\cdot g'}\,\frac{d}{dq^2}\Pi_{3Y}(q^2)\bigg|_{q^2=0}\quad,
\ee
where $g$ and $g'$ are the 4D gauge coupling constants of $SU(2)_L\times U(1)_Y$,
it is clear that we need an operator that couples the $SU(2)_L$ gauge field $\CW^3$ with
the hypercharge gauge field $\CB$ to describe the contributions to $S$ from physics above $M_s$.
It is not difficult to find the operator with the lowest mass dimension \cite{Appelquist:1993ka},
\be
-c_2\cdot\frac{\hat{g}\,\hat{g}'}{2^2\cdot 2!\cdot M_s^2}\,\,\CB_{\alpha\beta}\,\,{\rm Tr}\big[\,\CM\sigma^3\CM^\dagger\,\CW^{\alpha\beta}\,\big]\quad .
\label{Soperator}
\ee
For each field strength, $\CW_{\alpha\beta}$ and $\CB_{\alpha\beta}$, we have included a counting factor of $\frac{1}{2}$.
We have also pulled out the $(4+\delta)$-dimensional
gauge couplings, $\hat{g}$ and $\hat{g}'$, expecting that the $\CW$ and $\CB$ fields naturally couple to the underlying dynamics
that generates (\ref{Soperator}) with the strength of gauge couplings.
Having done this, we expect $c_2$ to be a constant of order unity.
The corresponding 4D operator from (\ref{Soperator}), after the substitutions (\ref{subs}) and
\be
\big(\CW_{\alpha\beta},\,\CB_{\alpha\beta}\big)\,\rightarrow\,\frac{\sqrt{2}}{(2\pi R)^{\delta /2}}\,\big(W_{\mu\nu},\,B_{\mu\nu}\big)
\quad,\qquad \big(\hat{g},\,\hat{g}'\big)\,=\,\frac{(2\pi R)^{\delta /2}}{\sqrt{2}}\,\big(g,\,g'\big)\quad,
\ee
and the volume integration $\int d^{\delta}y=(2\pi R)^{\delta}/2$, is
\be
-c_2\cdot\frac{g\,g'}{2^2\cdot 2!\cdot M_s^2}\,\,B_{\mu\nu}\,\,{\rm Tr}\big[\,M\sigma^3M^\dagger\,W^{\mu\nu}\,\big]\quad.
\label{4dSoperator}
\ee
This gives the following estimate of $S$ from physics above the cutoff scale:
\be
S^{UV}=c_2\cdot\frac{2\pi\,v^2}{M_s^2}\quad .
\label{deltaS}
\ee
Note again that the final result doesn't depend explicitly on the number of extra dimensions nor the compactification scale, $1/R$.
This estimate will also be useful later when we discuss the case of one or two extra dimensions.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{One universal extra dimension on $S^1/Z_2$} \setcounter{equation}{0}

In the case of one extra dimension, the sum over the KK contributions to $S, T$ and $U$ is convergent. Thus we can obtain reliable results
if the convergence is fast enough so that the cutoff effects on the KK sum are insignificant.
We see from Fig.\ref{kkmodes} that the contributions from higher KK levels become small rapidly.
The error of summing only up to the 11'th KK level is estimated to be less than 1\%.
The cutoff $M_s$ is estimated to be $\sim 30\cdot 1/R$ \cite{Appelquist:2000nn}, implying that the cutoff is irrelevant for the KK sum.
%In the following, when we refer to
%the KK contributions, we will mean the total sum over $j$.
Because the standard model also contributes to
the oblique parameters as we change the Higgs mass from $m_{H}^{\rm ref}=115 \,{\rm GeV}$, we must include those in the final $S,T$ and $U$
calculation.

Although the KK sum is insensitive to the cutoff, it is important to check explicitly that
the cutoff effects in terms of higher dimension operators are indeed negligible. From (\ref{deltaT}) and (\ref{deltaS}),
their size can be read conveniently from the following expression:
\bear
&&T^{UV}=c_1\cdot\,1.6\times 10^{-2}\Bigg(\frac{m_H}{200\,{\rm GeV}}\Bigg)^2\Bigg(\frac{300\,{\rm GeV}}{1/R}\Bigg)^2\Bigg(\frac{30}{M_sR}\Bigg)^2\quad,\nonumber \\
&&S^{UV}=c_2\cdot\,4.7\times10^{-3}\Bigg(\frac{300\,{\rm GeV}}{1/R}\Bigg)^2\Bigg(\frac{30}{M_sR}\Bigg)^2\quad.
\label{5dst}
\eear
The current constraints on the magnitude of $S$ and $T$ from the precision measurements are roughly $0.2$.
With $c_2$ being of order unity, $S^{UV}$ is sufficiently small to be neglected in the total $S$ contributions.
However, $c_1$ of order unity would give a sizable $T^{UV}$ if the Higgs mass is much larger than $200\, {\rm GeV}$.
Thus, we could lose the predictability of $T$ in the region of large Higgs mass, even if the KK sum converges.
To extract reliable predictions from the KK sum alone, we may need to have a naturally smaller $c_1$ than of order unity.
We next argue that we indeed expect $c_1$ to be as small as 0.1. Then, $T^{UV}$ can be safely neglected in the range of Higgs mass
discussed in this paper.

The key observation is that $M_s\sim 30\cdot 1/R$ is the scale where 5D QCD coupling becomes nonperturbative, while
the electroweak sector remains perturbative and is still described by the effective 5D standard model.
Because the Higgs fields are QCD-neutral, couplings to the quark sector must be invoked to generate the custodial symmetry-violating operator (\ref{Toperator}).
The largest such coupling is the top Yukawa coupling.
The 5D top Yukawa coupling, $\hat{\lambda}_t$ has the mass dimension $-1/2$, and the dimensionless loop expansion parameter in 5D
is given by
\be
\frac{\hat{\lambda}_t^2 \,M_s}{24\,\pi^3}\,=\,\frac{(\pi R\lambda_t^2) M_s}{24\,\pi^3}\,\sim\,\frac{\pi R\,M_s}{24\,\pi^3}\,\sim\,\frac{30}{24\,\pi^2}
\,\sim\,0.13 \quad .
\ee
where $\lambda_t\,\sim\,1$ is the 4D top Yukawa coupling and we have used the relation $\hat{\lambda}_t=\sqrt{\pi R}\,\lambda_t$. The factor
$24\,\pi^3$ is from the 5D momentum integration.
This indicates that the top Yukawa coupling of the Higgs fields to the quarks is still perturbative at the scale $M_s$, and $c_1$ can be expected
to contain this factor.
At the scale where the electroweak sector becomes nonperturbative, which is somewhat higher than $M_s$,
additional contributions to (\ref{Toperator}) will be generated by strong electroweak dynamics, possibly without any approximate custodial symmetry,
but then the suppression scale is higher than $M_s$, which again makes $c_1\,\lae\,0.1$ . By contrast, there is
no obvious reason to expect $c_2$ to be smaller than of order unity.
With these estimates, $T^{UV}$ and $S^{UV}$ from (\ref{5dst}) are small enough to be neglected in calculating $S$ and $T$ contributions.


\begin{figure}[t]
\begin{center}
\scalebox{1}[1]{\includegraphics{contour.eps}}
\par
\vskip-2.0cm{}
\end{center}
\caption{\small Some contours of total $S,T$ from the standard model and its higher KK modes in the 5D UED model on $S^1/Z_2$.
Here $m_{H}^{\rm ref}=115\,{\rm GeV}\,,\,m_t=174\,{\rm GeV}$. Up to 11 KK levels are included. The vertical line is the
direct search limit $m_H \ge 114\,{\rm GeV}$ (95\%C.L.) \cite{Heister:2001kr}.}
\label{contours}
\end{figure}
Having seen that the KK sum is reliable, we now analyze the consequences of the KK contributions to $S$ and $T$ by considering
the current combined $(S,T)$ constraints from the elecroweak precision measurements.
It is helpful first to see how the total $S$ and $T$ vary in the $(m_H,1/R)$ parameter space to get a rough idea of
how the constraints from $S$ and $T$ shape the allowed region in the $(m_H,1/R)$ parameter space.
In Fig.\ref{contours}, we show a contour plot of some values of total $S$ and $T$ contributions
from the standard model and its higher KK modes in the $(m_H,1/R)$ parameter space.
We also include the direct-search limit of $m_H \ge 114 \,{\rm GeV}$(95\% confidence level (C.L.)) \cite{Heister:2001kr}.
%The current constraint on $S$ and $T$ separately from the global fit of precision data is $|\,T| \le 0.20\,,\, |\,S|\le 0.15 \,\,({\rm 95\%C.L.})$
%for $m_{H}^{\rm ref}=115\,{\rm GeV}$ \cite{Erler:ig}.
Because of a compensation between positive KK quark contribution and negative KK Higgs contribution to $T$, we see that
as $m_H$ increases, the lower bound on $1/R$ from $T$ is relaxed. 
%If we consider only the $T$ constraint
%as in Ref.\cite{Appelquist:2000nn} using $|\,T| \le 0.20\,\,(\,{\rm 95\%C.L.},\,m_{H}^{\rm ref}=115\,{\rm GeV}$ \cite{Erler:ig}),
%the lower bound on $1/R$ could be $\sim 200 \,{\rm GeV}$ for $m_H\sim 400\,{\rm GeV}$.
%This may be compared to the previous bound of $1/R\gae 300 \,{\rm GeV}$ assuming $m_H \sim 200 \,{\rm GeV}$ \cite{Appelquist:2000nn}.
For even larger $m_H$, large positive contributions to $S$ from the Higgs KK modes make the region excluded. 
%Note that negative $S$ contours are below the $T=0.2$ line, so there is no constraint from large negative $S$ region. This is not true for negative $T$.
When $1/R$ is larger than $\sim 450 \,{\rm GeV}$, the constraint that $T$ may not be large and negative sets an upper bound on $m_H$.
This can be understood from the fact that the Higgs sector gives negative contributions to $T$ as in the usual standard
model. %But, this upper bound is stronger than the standard model case because we are including higher KK Higgs modes in addition
%to the zero mode standard model Higgs.

\begin{figure}[t]
\begin{center}
\scalebox{1}[1]{\includegraphics{90percent.eps}}
\par
\vskip-2.0cm{}
\end{center}
\caption{\small The 90\% C.L. allowed region in the 5D UED model on $S^1/Z_2$. Up to 11 KK levels are included.
Also shown is the direct search limit $m_H \ge 114\,{\rm GeV}$.}
\label{90percent}
\end{figure}
Because the constraints on the $S$ and $T$ parameters have a strong correlation \cite{Erler:ig}, separate $S$ and $T$ constraints
are incomplete. We therefore consider the current combined $(S,T)$ constraints to find the allowed region in the $(m_H,1/R)$ parameter space.
To find a (90\%) confidence level region, we analyze $\Delta \chi^2$ contours in the $(m_H,1/R)$ parameter space.
For this purpose, we may think of the $(m_H,1/R)$ parameters as a change of variables from $(S,T)$ because the number of
fitting parameters is two in both cases. Thus, we can simply use the $\Delta \chi^2$ contours
in the $(S,T)$ plane, for example, in Ref.\cite{Erler:ig}.
The resulting 90\% C.L. allowed region is shown in Fig.\ref{90percent}.
The region of smaller $1/R$ and larger $m_H$ than would be allowed from separate $S$ and $T$ constraints appears as a consequence
of the correlation between the $S$ and $T$ constraints.
%because $S$ and $T$ are both positive in this region,
%thus positively correlated.
The boundary of the region away from the tip is largely determined by $T$ constraints.
For $m_H\sim 800 \,{\rm GeV}$, even $1/R\sim 250\,{\rm GeV}$ is possible, and this should be testable
in the next collider experiments \cite{Cheng:2002ab,Rizzo:2001sd}.
%But it clearly opens an interesting possibility of seeing them soon \cite{Cheng:2002ab}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Two universal extra dimensions on $T^2/Z_2$} \setcounter{equation}{0}

In the case of one extra dimension, the KK contributions to $S,T$ and $U$ converge rapidly before
encountering the cutoff $M_s$, and the contributions from physics above $M_s$ are sufficiently small to be neglected.
Thus, practically the presence of $M_s$ is not significant. 
However, the KK sum diverges logarithmically in the 6D standard 
model, so we cannot
expect a reliable estimate from only summing the KK modes. A possible 
procedure is
to sum the KK modes up to the cutoff of the 6D model and then, as described 
in section 3,
to represent the physics beyond the cutoff by an appropriate operator. A 
problem with this
procedure is that while each term in the KK sum maintains 4D gauge invariance, the truncated sum  
is not expected
to respect the the full 6D gauge invariance upon which the 6D standard 
model is based \footnote{H.-U.Y. thanks Takemichi Okui for
discussions of this point at TASI 2002.}. As
noted below, however, the natural cutoff on the effective 6D theory is at 
about the fifth or sixth
KK level. With successive terms falling like $1/j$ and with the high energy 
contribution
represented by a 6D-gauge-invariant operator, we expect the lack of 6D gauge 
invariance to be relatively small - perhaps no more than a $20\%$ effect. We adopt this procedure 
with the
understanding that unlike the 5D case, only rough estimates are being 
provided in
six dimensions.

The most stringent estimate on $M_s$ in 6D comes from the naturalness
of the Higgs mass under quadratically divergent radiative corrections \cite{Appelquist:2002ft}. For a valid effective-theory description,
the six dimensional Higgs mass parameter $\hat{M}_H$ (the coefficient of the quadratic term of the 6D Higgs field)
should be below $M_s$, but at the same time, it shouldn't be small compared to the one-loop radiative correction on naturalness grounds:
\be
M_s \,\,>\,\,\hat{M}_H\,\,\gae\,\,\delta \hat{M}_H\,\sim\,\sqrt{\frac{\hat{\lambda}M_s^2}{128\pi^3}}\,M_s\quad,
\ee
where $\hat{\lambda}$ is the Higgs self coupling in 6D. The factor of $128\pi^3$ arises from the six dimensional momentum integral.
This gives the following relation involving the Higgs VEV $v=246\,{\rm GeV}$:
\be
v=\bigg[\pi\,R\,M_s\,(\hat{\lambda}M_s^2)^{-1/2}\bigg]\hat{M}_H\,\,\gae\,\,\frac{1}{\sqrt{128\pi}}(R\,M_s)^2\,R^{-1}\quad.
\ee
Taking $1/R$ of a few hundred ${\rm GeV}$ gives $R\,M_s\,\sim\,5$.
This result is similar to an estimate using the renormalization group analysis of both gauge couplings and Higgs self
coupling \cite{Arkani-Hamed:2000hv} showing that $M_s$ should be around five times of the compactification scale.
We therefore take $M_s\,\sim\,5/R$ in the following.

The contributions to $S$ and $T$ from physics above $M_s$, estimated in (\ref{deltaT}) and (\ref{deltaS}), can be written as
\bear
&&T^{UV}=c_1\cdot\,0.57\,\Bigg(\frac{m_H}{200\,{\rm GeV}}\Bigg)^2\Bigg(\frac{300\,{\rm GeV}}{1/R}\Bigg)^2\Bigg(\frac{5}{M_sR}\Bigg)^2\quad,\nonumber \\
&&S^{UV}=c_2\cdot\,0.17\,\Bigg(\frac{300\,{\rm GeV}}{1/R}\Bigg)^2\Bigg(\frac{5}{M_sR}\Bigg)^2\quad.
\label{6dst}
\eear
As in the 5D case, we expect $c_2$ to be a parameter of order unity. But in
contrast to 5D, there may be no good reason to anticipate that $c_1$ should
be less than unity. The reason is that in
6D, the scale at which the electroweak interactions (including the Yukawa
couplings to the top quark and other fermions) become strong is not much
above $M_s$, the scale at which 6D QCD becomes strong. Thus,
the breaking of custodial symmetry encoded in the operator (\ref{Toperator}) may 
be near-maximal. Since these estimates are crude, however, we
will allow in the estimates below for both maximal breaking of
custodial symmetry ($c_1 \approx 1$) as well as the presence of some
suppression of this breaking ($c_1 \approx 0.1$).

\begin{figure}[t]
\begin{center}
\scalebox{1}[1]{\includegraphics{6DC2+1.eps} \includegraphics{6DC2-1.eps}}
\par
\vskip-2.0cm{}
\end{center}
\caption{\small The 90\% C.L. allowed regions for several values of $c_1$ and $c_2$ in the 6D UED model on $T^2/Z_2$.
Also shown is the direct search limit $m_H \ge 114\,{\rm GeV}$.}
\label{6D90percent}
\end{figure}
In Fig.\ref{6D90percent}, we show several 90\% C.L. allowed regions
taking $c_1=\pm \,1$ or $\pm \,0.1$, and $c_2=\pm \,1$.
As mentioned above, the contribution from physics below $M_s$ is estimated by summing KK contributions up to $M_s\,=5/R$.
The plot shows very different characteristic features for different signs and magnitudes of $c_1$ and $c_2$.
It should be taken only to indicate possibilities, though, because of the uncertainty in the estimates of $c_1$ and $c_2$.
  
  
First, consider the case $c_1=\,\pm 0.1$. In this case, the contributions to $T$ from physics above $M_s$ do not affect the shapes of the regions significantly.
However, an important dependence on the sign of $c_2$
appears.
For negative $c_2$ (the right figure), a region of larger Higgs mass can be allowed, compared to the case of
positive $c_2$ (the left figure). This can be understood from the fact that this region is constrained by large positive
KK contributions to $S$ as can be seen in Fig.\ref{contours}. With negative $c_2$, the contribution from physics above $M_s$ can cancel the KK
contributions in this region, relieving the constraint from $S$.
Since no such cancellation is involved when $c_2=\,1$, 
the left figure may describe a more generic allowed region for the case $c_1=\,\pm 0.1$.

In the (perhaps more likely) case $c_1=\,\pm 1$, the contributions to $T$ from physics 
above the cutoff have significant effects on the shape of the regions,
while the contributions to $S$ from above the cutoff play a lesser role in determining the allowed regions. When $c_1=\,+1$,
a region where both $m_H$ and $1/R$ are large can be allowed, because the large negative total
contributions to $T$ from KK modes with large $m_H$ can be compensated by the positive UV contribution to $T$.
With $c_1=\,-1$, Higgs masses lighter than $\sim 400\,{\rm GeV}$ are preferred. %which is similar to the case of the minimal 4D standard model. 
This is because
the negative UV contribution to $T$ can then be cancelled by the dominant positive contributions to $T$ from the KK top-quark doublets.
Higgs masses heavier than $\sim 400 \,{\rm GeV}$ are excluded in the $c_1=\,-1$ case because they too give a negative total KK contribution to $T$.
%, making worse the constraint from the $T$ parameter.

Although we can't extract precise information from the plots of Fig.\ref{6D90percent}, due to the uncertainty in the signs and magnitudes of the coefficients $c_1$ and $c_2$,
the above results do tell us that the possibility of a large Higgs mass and a relatively small compactification scale is not excluded in 6D UED theories.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions} \setcounter{equation}{0}

The discovery of additional spatial dimensions accessible to the
standard-model
fields (universal extra dimensions) would be a spectacular realization of physics beyond the standard
model.
The first study \cite{Appelquist:2000nn} of the constraint on the compactification scale, $1/R$, from precision electroweak measurements
in theories of universal extra dimensions
gave the bound $1/R\,\gae\,300\,{\rm GeV}$. But an assumption of that analysis was that the Higgs mass $m_H$ is less than $250\,{\rm GeV}$.
In this paper, we have considered the precision electroweak constraints in terms of the $S$ and $T$ parameters
without assuming that $m_H \lae250\,{\rm GeV}$.


%Without direct observations, the current knowledge of the Higgs mass comes from the comparison of the precision measurements with loop corrections
%involving both the standard model and possible new physics. Unless the contributions from new physics are negligible compared to those from the standard
%model, we should include the new physics effects on the precision measurements and this could lead to the possibility of a larger Higgs mass than in
%the standard model alone \cite{Chivukula:1999az}. In theories with universal extra dimensions, the effects coming from the new physics
%above the compactification scale are in fact sensitively dependent on the Higgs mass. %Therefore,
%it is natural to let the Higgs mass vary in a more complete analysis of the constraints on the universal extra dimension scenarios.


We have shown that current precision measurements, when analyzed
with both the compactification scale $1/R$ and the Higgs mass $m_H$ taken
to be free parameters, lead to a lower bound on $1/R$ that is quite
sensitive to
$m_H$ and can be as low as $~ 250 \,{\rm GeV}$. This becomes possible if $m_H$
is larger than allowed in the minimal standard model -- as large as $~800 \,
{\rm GeV}$.
Equivalently, in the presence of low-scale universal extra dimensions,
precision
measurements allow a considerably larger $m_H$ than in the framework of the
minimal standard model. The main reason for this is that the negative contributions to the $T$ parameter from the Higgs boson and its KK partners
can be cancelled by the positive contributions from KK top quarks.

A light compactification scale would have important
consequences for the possibility of direct detection of KK particles in the next collider experiments \cite{Cheng:2002ab,Rizzo:2001sd,Macesanu:2002ew}. 
%\cite{Servant:2002aq,Cheng:2002ej,Hooper:2002gs,Servant:2002hb,Majumdar:2002mw},
%and KK effects on the production and decay of the Higgs particle can be
%significant and
%depend strongly on the Higgs mass \cite{Petriello:2002uu}.
The KK dark matter density \cite{Servant:2002aq}
and its direct or indirect detection \cite{Cheng:2002ej,Hooper:2002gs,Servant:2002hb,Majumdar:2002mw} are
sensitive both
to the compactification scale and to the Higgs mass through the rates of
the Higgs-mediated
processes. It would be interesting to reanalyze them in the allowed
$(m_H,1/R)$ parameter
region obtained in this paper.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
 {\bf Acknowledgements:} \
 We would like to thank Bogdan Dobrescu for many critical comments in the early stages of this work, and Tatsu Takeuchi for an important 
 discussion on section 2.2.
 We also thank Hsin-Chia Cheng and Eduardo Ponton for helpful discussions. This work was supported by DOE under contract DE-FG02-92ER-40704.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Appendix : Summary of one-loop KK contributions to gauge-boson self energies}
\renewcommand{\theequation}{A.\arabic{equation}}
\setcounter{equation}{0}

In this appendix, we summarize the calculation of one-loop diagrams with intermediate KK particles for the (zero mode) gauge-boson propagators.
We introduce the higher dimensional analog of the $R_{\xi}$ gauge with $\xi=1$, in which extra dimensional components of gauge
bosons can be treated as 4D scalar fields without any mixed kinetic terms with 4D components. Because KK number is conserved at vertices and
the external lines are zero modes, all KK particles in one-loop diagrams are in the same $j$th level.
We group the diagrams into five classes such that quadratic divergences cancel within a class.\\\\
(a) Loops with KK quarks of the third generation\\
(b) Loops with KK gauge bosons, in which at least one internal line is a 4D component and loops with KK ghosts\\
(c) Loops with KK gauge bosons with extra dimensional components (should be multiplied by $\delta$, the number of extra dimensions)\\
(d) Loops with KK particles from the Higgs sector\\
(e) Loops with one KK gauge boson and one KK particle from the Higgs sector\\\\
In the following, $s_w \equiv \sin\theta_w$, $c_w \equiv \cos\theta_w$, $E \equiv \frac{2}{\varepsilon}-\gamma+\log4\pi$, and
\bear
\Delta^2_j(m_1^2,m_2^2,x) \equiv M_j^2-x(1-x)p^2+(1-x)m_1^2+x\,m_2^2 % \nonumber
\eear
where $M_j^2 \equiv \big(\frac{j}{R}\big)^2$.

{\bf (1) $WW$ self energy}
\bear
\Pi^{j(a)}_{WW}(p^2)&=&\frac{\alpha}{4\pi}\frac{-6}{s_w^2}\int^1_0 dx\,\Big(E-\log\Delta^2_j(0,m_t^2,x)\Big)\Big(2x(1-x)p^2-x\,m_t^2\Big)\nonumber\\
\Pi^{j(b)}_{WW}(p^2)&=&\frac{\alpha}{4\pi}\frac{c_w^2}{s_w^2}\int^1_0 dx\,
\Big(E-\log\Delta^2_j(m_W^2,m_Z^2,x)\Big) \nonumber \\
&&\qquad\qquad\qquad\cdot\Big(2(-4x^2+4x+1)p^2+(3-4x)m_Z^2+(4x-1)m_W^2\Big)\nonumber \\
&+&\frac{\alpha}{4\pi}\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_W^2,0,x)\Big)\Big(2(-4x^2+4x+1)p^2+(4x-1)m_W^2\Big)\nonumber \\
\Pi^{j(c)}_{WW}(p^2)&=&\frac{\alpha}{4\pi}\frac{c_w^2}{s_w^2}\int^1_0 dx\,
\Big(E-\log\Delta^2_j(m_W^2,m_Z^2,x)\Big) \nonumber \\
&&\qquad\qquad\qquad\cdot\Big(-(4x^2-4x+1)p^2+(1-2x)m_Z^2+(2x-1)m_W^2\Big)\nonumber \\
&+&\frac{\alpha}{4\pi}\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_W^2,0,x)\Big)\Big(-(4x^2-4x+1)p^2+(2x-1)m_W^2\Big)\nonumber \\
\Pi^{j(d)}_{WW}(p^2)&=&\frac{\alpha}{4\pi}\frac{1}{4s_w^2}\int^1_0 dx\,
\Big(E-\log\Delta^2_j(m_W^2,m_Z^2,x)\Big) \nonumber \\
&&\qquad\qquad\qquad\cdot\Big(-(4x^2-4x+1)p^2+(1-2x)m_Z^2+(2x-1)m_W^2\Big)\nonumber \\
&+&\frac{\alpha}{4\pi}\frac{1}{4s_w^2}\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_W^2,m_H^2,x)\Big) \nonumber \\
&&\qquad\qquad\qquad\cdot\Big(-(4x^2-4x+1)p^2+(1-2x)m_H^2+(2x-1)m_W^2\Big)\nonumber \\
\Pi^{j(e)}_{WW}(p^2)&=&\frac{\alpha}{4\pi}\frac{-1}{s_w^2}\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_W^2,m_H^2,x)\Big)m_W^2 \nonumber \\
&+&\frac{\alpha}{4\pi}(-s_w^2)\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_W^2,m_Z^2,x)\Big)m_Z^2 \nonumber \\
&+&\frac{\alpha}{4\pi}(-1)\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_W^2,0,x)\Big)m_W^2 \nonumber
\eear

{\bf (2) $ZZ$ self energy}
\bear
\Pi^{j(a)}_{ZZ}(p^2)&=&\frac{\alpha}{4\pi}\frac{-3 + 8 s_w^2-\frac{32}{3}s_w^4}{s_w^2c_w^2}
\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_t^2,m_t^2,x)\Big)\Big(2x(1-x)p^2\Big)\nonumber\\
&+&\frac{\alpha}{4\pi}\frac{3}{s_w^2c_w^2}\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_t^2,m_t^2,x)\Big)m_t^2\nonumber \\
&+&\frac{\alpha}{4\pi}\frac{-3 + 4 s_w^2-\frac{8}{3}s_w^4}{s_w^2c_w^2}
\int^1_0 dx\,\Big(E-\log\Delta^2_j(0,0,x)\Big)\Big(2x(1-x)p^2\Big)\nonumber\\
\Pi^{j(b)}_{ZZ}(p^2)&=&\frac{\alpha}{4\pi}\frac{c_w^2}{s_w^2}\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_W^2,m_W^2,x)\Big)\Big((-8x^2+14x-1)p^2+2m_W^2\Big)\nonumber\\
\Pi^{j(c)}_{ZZ}(p^2)&=&\frac{\alpha}{4\pi}\frac{2c_w^2}{s_w^2}\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_W^2,m_W^2,x)\Big)\Big((-2x^2+3x-1)p^2\Big)\nonumber\\
\Pi^{j(d)}_{ZZ}(p^2)&=&\frac{\alpha}{4\pi}\frac{1}{4s_w^2c_w^2}\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_Z^2,m_H^2,x)\Big)\nonumber \\
&&\qquad\qquad\qquad\cdot\Big((-4x^2+4x-1)p^2+(1-2x)m_H^2+(2x-1)m_Z^2\Big)\nonumber \\
&+&\frac{\alpha}{4\pi}\frac{(c_w^2-s_w^2)^2}{2s_w^2c_w^2}\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_W^2,m_W^2,x)\Big)\Big((-2x^2+3x-1)p^2\Big)\nonumber\\
\Pi^{j(e)}_{ZZ}(p^2)&=&\frac{\alpha}{4\pi}\frac{-1}{s_w^2c_w^2}\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_Z^2,m_H^2,x)\Big)m_Z^2 \nonumber \\
&+&\frac{\alpha}{4\pi}(-2s_w^2)\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_W^2,m_W^2,x)\Big)m_Z^2 \nonumber
\eear

{\bf (3) $Z\gamma$ self energy}
\bear
\Pi^{j(a)}_{Z\gamma}(p^2)&=&\frac{\alpha}{4\pi}\frac{-4+\frac{32}{3}s_w^2}{s_wc_w}
\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_t^2,m_t^2,x)\Big)\Big(2x(1-x)p^2\Big)\nonumber\\
&+&\frac{\alpha}{4\pi}\frac{-2+\frac{8}{3}s_w^2}{s_wc_w}
\int^1_0 dx\,\Big(E-\log\Delta^2_j(0,0,x)\Big)\Big(2x(1-x)p^2\Big)\nonumber\\
\Pi^{j(b)}_{Z\gamma}(p^2)&=&\frac{\alpha}{4\pi}\frac{c_w}{s_w}
\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_W^2,m_W^2,x)\Big)\Big((-8x^2+14x-1)p^2+2m_W^2\Big)\nonumber\\
\Pi^{j(c)}_{Z\gamma}(p^2)&=&\frac{\alpha}{4\pi}\frac{2c_w}{s_w}
\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_W^2,m_W^2,x)\Big)\Big((-2x^2+3x-1)p^2\Big)\nonumber\\
\Pi^{j(d)}_{Z\gamma}(p^2)&=&\frac{\alpha}{4\pi}\frac{c_w^2-s_w^2}{s_wc_w}
\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_W^2,m_W^2,x)\Big)\Big((-2x^2+3x-1)p^2\Big)\nonumber\\
\Pi^{j(e)}_{Z\gamma}(p^2)&=&\frac{\alpha}{4\pi}\frac{2s_w}{c_w}
\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_W^2,m_W^2,x)\Big)m_W^2\nonumber
\eear

{\bf (4) $\gamma\gamma$ self energy}
\bear
\Pi^{j(a)}_{\gamma\gamma}(p^2)&=&\frac{\alpha}{4\pi}\frac{-32}{3}
\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_t^2,m_t^2,x)\Big)\Big(2x(1-x)p^2\Big)\nonumber\\
&+&\frac{\alpha}{4\pi}\frac{-8}{3}
\int^1_0 dx\,\Big(E-\log\Delta^2_j(0,0,x)\Big)\Big(2x(1-x)p^2\Big)\nonumber\\
\Pi^{j(b)}_{\gamma\gamma}(p^2)&=&\frac{\alpha}{4\pi}
\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_W^2,m_W^2,x)\Big)\Big((-8x^2+14x-1)p^2+2m_W^2\Big)\nonumber\\
\Pi^{j(c)}_{\gamma\gamma}(p^2)&=&\frac{\alpha}{4\pi}
\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_W^2,m_W^2,x)\Big)\Big(2(-2x^2+3x-1)p^2\Big)\nonumber\\
\Pi^{j(d)}_{\gamma\gamma}(p^2)&=&\frac{\alpha}{4\pi}
\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_W^2,m_W^2,x)\Big)\Big(2(-2x^2+3x-1)p^2\Big)\nonumber\\
\Pi^{j(e)}_{\gamma\gamma}(p^2)&=&\frac{\alpha}{4\pi}
\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_W^2,m_W^2,x)\Big)\big(-2m_W^2\big)\nonumber
\eear






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 \end{thebibliography} \vfil \end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%GARBAGES

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Because our effective description in terms of the 6D SM breaks down above $M_s$,
we can only estimate the effects
on $S,T$ and $U$ from the physics above $M_s$ by dimensional analysis......
In the usual 4D langauge, the largest effects on $S,T$ come
from the following effective operators in the Higgs sector \cite{Buchmuller:1985jz},
\be
{\cal L}_4 \,\supset\,-\frac{a}{2!\,\Lambda^2}\{[D_{\mu},D_{\nu}]H \}^{\dagger}
\{[D^{\mu},D^{\nu}]H\}\,+\,\frac{b\,\kappa^2}{2!\,\Lambda^2}
(H^{\dagger}\stackrel{\leftrightarrow}{\raisebox{0.1mm}{$D^{\mu}$}}H)(H^{\dagger}
\stackrel{\leftrightarrow}{\raisebox{0.1mm}{$D$}}_{\mu}H)
\ee
($a,b$ are dimensionless coefficients, $\kappa$ is the (strong) coupling constant of unknown dynamics, and $\Lambda$
is the corresponding scale of the unknown physics.)
Their contributions to $S,T$ are, at tree level,
\be
\Delta S=\frac{4\pi a v^2}{\Lambda^2}\,\,,\,\,\Delta T=\frac{b \,\kappa^2 v^2}{\alpha(M_Z) \Lambda^2}
\ee
,where $\alpha(M_Z)=1/129$ is the EM coupling constant at the scale $M_Z$.
Because we have the 6D effective theory below $M_s$, these operators should have come from the corresponding 6D operators,
\be
{\cal L}_6 \,\supset\,-\frac{1}{2!\,M_s^2}\{[D_{M},D_{N}]\hat{H} \}^{\dagger}
\{[D^{M},D^{N}]\hat{H}\}\,+\,\frac{\hat{\kappa}^2}{2!\,M_s^4}
(\hat{H}^{\dagger}\stackrel{\leftrightarrow}{\raisebox{0.1mm}{$D^M$}}\hat{H})(\hat{H}^{\dagger}
\stackrel{\leftrightarrow}{\raisebox{0.1mm}{$D$}}_{M}\hat{H})
\label{6doperator}
\ee
($M,N$ are six dimensional indices. We omitted assumed order one coefficients for simplicity.)
Here, $\hat{H}$ is the 6D Higgs field and $\hat{\kappa}^2\,\sim\, 128\pi^3$ is the estimated strong coupling in 6D.
After KK decomposition, this gives,
\be
\frac{a}{\Lambda^2}\,\sim\,\frac{1}{M_s^2}\,\,,\,\,\,\,\frac{b\,\kappa^2}{\Lambda^2}\,\sim\,\frac{\hat{\kappa}^2}{2\pi^2\,M_s^2(R\,M_s)^2}
\ee
Thus, contributions to $S,T$ from physics above $M_s$ are estimated to be
\be
\Delta S\,\sim\,\frac{4\pi v^2}{M_s^2}\,\,,\,\,\,\,\Delta T\,\sim\,\frac{64\pi\, v^2}{\alpha(M_Z) M_s^2(R\,M_s)^2}
\ee
Inserting $M_s\,\sim\, 2 \,{\rm TeV}$ gives $\Delta S\sim 0.2\,,\,\Delta T\sim 15.7$, which are comparable to or bigger than the results of KK sum
up to $M_s$, which
makes effective KK sum results completely
unreliable. (Although $\Delta T$ could be estimated less assuming custodial symmetry \cite{Chivukula:1999az}, we expect the same conclusion.)
On the contrary, this can be taken as a problem for the 6D SM because naive estimate for $T$ contribution is too large and we need to
have some mechanism to suppress the dimensionless coefficient of the second operator in (\ref{6doperator}).
In any case, this means that there can be potentially large effects on $S,T$ and we cannot exclude the possibility of low $1/R$ or
higher $m_H$ than expected from SM alone.
%%%%%%%%%%%%%%%%%%%%%%
Although it is perfectly reasonable to take $c_1$ of order one, it is an interesting possibility to assume an approximate
custodial symmetry above $M_s$ and to see how much the possible suppression for $c_1$ is.
A useful estimate of the size of this suppression can be inferred from the size of the custodial violating top Yukawa coupling,
\be
\CL_6\quad \supset\quad \hat{\lambda_t}\,\bar{\CQ_3}\,\CU_3 \,i\sigma^2\,\CH^*\quad,
\label{topyukawa}
\ee
where $\CQ_3,\,\CU_3$ are the 3'rd generation quark doublet and top singlet, and $\hat{\lambda_t}$ is the 6D top Yukawa coupling whose size is,
\be
\hat{\lambda_t}=(\sqrt{2}\,\pi\,R)\,\lambda_t\,\sim\,(\sqrt{2}\,\pi\,R)\quad.
\ee
($\lambda_t$ is the 4D top Yukawa coupling which is of order 1.)
If the custodial symmetry were broken as strongly as possible, the natural size of $\hat{\lambda_t}$ would be $\sqrt{128\pi^3}/M_s$, assuming that
top Yukawa coupling becomes strong at the similar scale as $M_s$.
Thus, the suppression factor due to the approximate custodial symmetry is $(\sqrt{2}\pi \,R\,M_s)/\sqrt{128 \pi^3}\,\sim\,1/3$.
Because (\ref{topyukawa}) has $I=1$ under the custodial isospin and the operator for $T^{UV}$ has $I=2$, the suppression for $c_1$
should be $(1/3)^2\,\sim\,0.1$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Muon $(g-2)$ plot...
\begin{figure}[t]
\begin{center}
\scalebox{1}[1]{\includegraphics{e+e-basedmuon.eps}\includegraphics{taubasedmuon.eps}}
\par
\vskip-2.0cm{}
\end{center}
\caption{\small 68\% and 95\% C.L. allowed regions for the 6D UED on $T^2/Z_2$ from the recent muon $(g-2)$ data. The center curve is the $\chi_{min}^2$ curve.
$c_B$ and $c_W$ are the coefficients of the relevant higher dimension operators in 6D.}
\label{muonfit}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this paper, we have analyzed the allowed region in the $(m_H,1/R)$ parameter space consistent with current precision measurements
in terms of the $S$ and $T$ parameters.
We have done a complete calculation of the corrections from the KK particles to the gauge-boson propagators for this purpose.
In the case of one extra dimension on $S^1/Z_2$, the effects from physics above the compactification scale
are reliably calculable and we obtained an interesting region (Fig.\ref{90percent}) allowed by current experiments.
The allowed Higgs mass was found to be as large as $550 \,{\rm GeV}$ in this case. In the case of two extra dimensions on $T^2/Z_2$,
the calculations suffer from the uncertainty of cutoff effects from unknown physics above $M_s$, the scale where our effective theory
in terms of a higher dimensional standard model breaks down. Depending on the signs and magnitudes of these contributions, the
constraints on the range of allowed Higgs mass can be very different, but again, they clearly show that a large Higgs mass
is a possibility in the UED theories. The lower bound on $1/R$ depends sensitively on $m_H$ and can be as low as $200\,{\rm GeV}$ in both one and two
extra dimensions.

Certainly, a large $m_H$ of $\sim\,550 \,{\rm GeV}$ or a small compactification scale of $\sim 200\,{\rm GeV}$ can have interesting implications
on Higgs phenomenology of the UED theories \cite{Petriello:2002uu} and the possibility of the KK dark matter
and its detection prospect \cite{Servant:2002aq,Cheng:2002ej,Hooper:2002gs,Servant:2002hb}. It was shown in Ref.\cite{Petriello:2002uu} that
the KK effects on the production and decay of the Higgs particle can be significant and very much depend on the Higgs mass.
The KK dark matter density \cite{Servant:2002aq} and its direct or indirect detections \cite{Cheng:2002ej,Hooper:2002gs,Servant:2002hb}
are sensitive both to the compactification scale through the mass of the KK particles, and to the Higgs mass through the rates of the Higgs-mediated
processes. It would be interesting to reanalyze them in the allowed $(m_H,1/R)$ parameter region we obtained in this paper.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This phenomenon also occurs in the usual standard-model radiative
corrections and has been studied before. \cite{Kennedy:1988sn}...more...
Non-oblique effects on both gauge-boson propagators and fermion vertices (and box diagrams), when they are taken separately, 
are not gauge invariant, physical observables. However, for the case of $R_\xi$ gauges, 
there is a way to regroup these into 'oblique' and 'vertex' parts such that they are independent of both cutoff and $\xi$ parameter.
Of course, this separations are not unique and, in general, we must choose one procedure and combine 'oblique' and 'vertex' parts
at the end to obtain physically observable quantities.

We choose the procedure of Kennedy and Lynn \cite{Kennedy:1988sn}. 
They showed that it is possible to separate
the non-oblique, vertex corrections into two parts. One part
mimics the oblique corrections in that it is universal, that is, independent
of the external fermion species. The other depends on external fermion
species and leads, for example, to the estimate (\ref{vertex}) of the
$Zb\bar{b}$ vertex correction. We absorb the universal piece into gauge-boson propagator corrections.
Kennedy and Lynn 
observed that it is closely related to the fact that, in general, $\Pi_{Z
\gamma}(0)\,\ne\,0$, implying one-loop mass mixing between the photon and
the Z boson, signaling the need to rediagonalize the gauge-boson mass matrix
arising from the Higgs sector. The arbitrariness of the universal piece is fixed by the requirement that
it precisely cancels $Pi_{Z\gamma}(0)$ ensuring the massless photon propagator.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The universal contribution is in general
%cutoff dependent and may be incorporated into a modified and physical
%definition of $T$, as well as $S$ and $U$. As it turns out, this procedure gives
%finite, cutoff-independent and $R_\xi$ gauge-independent answers for $S,\,T$ and $U$.
%But, of course, this




%The one-loop corrections include a mixing term corresponding to an $A_{\mu}Z^{\mu}$ - counterterm
%describing an admixture of $Z^{\mu}$ in the new, massless eigenvector.
%The full set of counterterms at this level then includes \cite{Kennedy:1988sn} that removes
%$\Pi_{Z\gamma}(0)$ entirely, maintaining the masslessness of the photon.
%along with
%counterterms that remove the divergence in $T^j$.
%For finite, well-defined quantum effects, like the $T$ parameter, counterterm contributions must be absent. This requirement
%is satisfied by symmetries of the theory that impose certain relations between conterterms.
%In the presence of an $A_{\mu}Z^{\mu}$ - counterterm, it turns out that the definition (\ref{defT}) must be modified to have a conterterm cancellation.

%Quantum loops induce corrections to the tree-level
%mass matrix. (Because the mass matrix depends on the coupling constants, the renormalization of the coupling constants must
%be taken care of, too. Kennedy and Lynn \cite{Kennedy:1988sn} carefully did this and noted that the final result is independent of gauge choice.)
%The eigenvectors of the quantum corrected mass matrix are not the eigenvectors of the tree-level one; % and the mismatch is given by $\Pi_{Z \gamma}(0)\,\ne\,0$.
%To satisfy the massless photon
%condition, there must be an $A_{\mu}Z^{\mu}$-counterterm to cancel $\Pi_{Z \gamma}(0)$, and this term arises
%when the renormalized $A_{\mu}$ and $Z_{\mu}$ fields are not the eigenvectors of the bare mass matrix.
%Because naive
%tree-level relations between gauge-boson masses like $\rho=1$ come from the bare lagrangian, this mismatch between true and bare eigenvectors
%is expected to contribute to various quantum corrected relations.

The Kennedy-Lynn procedure may be summarized in the following simple rule of thumb
\cite{Kennedy:1988sn},\footnote{Kennedy and Lynn missed the third rule, but
it is straightforward to derive it {\it from their paper}.}

(1) Subtract   $\Pi_{Z\gamma}(0)$ from $\Pi_{Z \gamma}$ by hand.

(2) Add   $-2\,{\rm csc}\,\theta_{W}\,\Pi_{Z\gamma}(0)$ to $\Pi_{ZZ}$.

(3) Add   $-\,{\rm csc}\,\theta_{W}\,\frac{<\stackrel{\rightarrow}{I^2}-I_3^2>_0}{<I_3^2>_0}\,\Pi_{Z \gamma}(0)$ to $\Pi_{WW}$.

(4) Use these in the definitions of $S,\, T$ and $U$.\\
Here, $<\cdots>_0$ is the VEV of the Higgs scalar(s) and $\stackrel{\rightarrow}{I}$ is the weak isospin.
After this modification, we indeed obtain a finite answer for the $T$ parameter, while the finiteness of $S$ and $U$ is intact.

The important point is that....

%The fore-mentioned universal piece in vertex corrections is unambiguously chosen
%such that it precisely cancels 1-loop $\Pi_{Z\gamma}(0)$ when combined with 1-loop $\Pi$ functions (Rule (1)), ensuring to have a massless photon.
%Having correctly diagonalized $\Pi$ functions, we can use Peskin-Takeuchi definitions for $S,\,T$ and $U$ to represent corrections to
%physical observables due to gauge-boson propagator corrections that are independent of external fermion species. 
%Remaining vertex corrections after removing the universal
%piece according to the above procedure are expected to be negligible as estimated in (\ref{vertex}).

Although the fact $\Pi_{Z \gamma}(0)\,\ne\,0$ seems to contradict the conventional assumption of electromagnetic Ward identity, the naive Ward identity
leading to $\Pi_{Z \gamma}(0)\,=\,0$ is not true if the $U(1)$ gauge group
emerges from a nonabelian gauge group through spontaneous breaking. This is because the gauge fixing and the ghost sector
of the original nonabelian gauge theory violate
naive $U(1)$ gauge symmetry explicitly. %This is natural because the gauge fixing and
%the corresponding ghost term break nonabelian gauge symmetry explicitly and $U(1)$ was part
%of the original gauge group. This doesn't mean that we don't have $U(1)$ at low energy. Ghost fields are unphysical and integrated out
%at low energy because the low energy {\it abelian} $U(1)$ doesn't need ghosts. Actually, we need ghosts only in nonabelian case, thus
%integrating out ghosts is intimately related to integrating out $W^{\pm}$ and $Z$ fields. At high energy, we have a nonabelian
%theory with a ghost sector, which violates gauge symmetry explicitly.
The naive Ward identity must be replaced by the corresponding
nonabelian version, the Slavnov-Taylor identity, which involves ghosts in an essential way.
%The proof of masslessness of the photon
%will not be a simple matter in this case, but we know that it must be true.
%In low energy effective theory, we have a pure $U(1)$ theory
%with ghosts and massive spin 1 gauge bosons integrated out and the Ward identity is restored.
Indeed, ghost loops and massive broken gauge-boson loops generate $\Pi_{Z \gamma}(0)\,\ne\,0$ in the standard model. %but the
%photon should remain massless to all orders.
In our model, for each KK level of the massive $W^{\pm}$ and $Z$, there are
also ghost KK modes, and they, too, generate $\Pi_{Z \gamma}(0)\,\ne\,0$.
%The (zero mode) photon remains massless just as in the standard model even
%though the naive Ward identity looks violated. The proof of this fact using
%Slavnov-Taylor type identities will not be provided in this paper.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
{\bf Decoupling behavior of KK contributions}

According to the decoupling theorem \cite{Appelquist:tg}, heavy KK particle contributions to
the finite, well defined predictions of QFT should go to zero when their masses approach infinity. We now check this for the
case of gauge-boson propagator corrections. The calculations in the above show that one-loop KK contributions
have the following universal structure,
\bear
&&\int^1_0 dx\,\Big(E-\log\Delta^2_j(m_1^2,m_2^2,x)\Big)F(x) \\
&=&\int ^1_0 dx\,\bigg(E-\log\Big(M_j^2-x(1-x)p^2+(1-x)m_1^2+x\,m_2^2\Big)\bigg)F(x) \quad.\nonumber
\eear
Pulling out $\log M_j^2$ from the logarithm, we have,
\be
\int ^1_0 dx\,\bigg(\Big(E-\log M_j^2\Big)-\log\Big(1-x(1-x)\frac{p^2}{M_j^2}+(1-x)\frac{m_1^2}{M_j^2}+x\,\frac{m_2^2}{M_j^2}\Big)\bigg)F(x)\quad.
\ee
Note that the remaining logarithmic term shows decoupling when $M_j^2\to \infty$, thus if the term $\Big(E-\log M_j^2\Big)$
disappeared in the physical predictions at low energy, we would have proven the decoupling behavior. But, $\varepsilon$-pole, $E$, should
indeed disappear from finite, well-defined predictions of QFT from renormalizability and $E$ always comes in the above combination.
Thus, decoupling is established. % proved from the renormalizability of our theory.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bear
\alpha\tilde{T}^j_{\rm KK Higgs}&\approx&\frac{\alpha}{16\pi}\frac{1}{c_w^2}
\,\int^1_0 dx\,\Bigg[\frac{m_H^2}{M_j^2+x\,m_H^2}\,(2x-1)(x-1)+\,\log\bigg(1+x\,\frac{m_H^2}{M_j^2}
\bigg)\,(2x-5)\Bigg]\quad, \nonumber \\
\alpha\tilde{T}^j_{\rm \,\,KK \,top\,\,}&\approx&\frac{3\,m_t^2}{4\pi^2v^2}\Bigg[\log\bigg(1+\frac{m_t^2}{M_j^2}\bigg)-2\,\int^1_0 dx\,\log\bigg(1+x\,\frac{m_t^2}{M_j^2}\bigg)
\,x\Bigg] \quad,
\label{KKhigtop}
\eear

\bear
\alpha S^j_{\rm KK Higgs}&\approx&\frac{\alpha}{4\pi}
\,\int^1_0 dx\,\Bigg[\frac{m_H^2}{M_j^2+x\,m_H^2}\,x(x-1)(2x-1)+\,\log\bigg(1+x\,\frac{m_H^2}{M_j^2}
\bigg)\,(4x^2-4x+1)\Bigg]\,\,, \nonumber \\
\alpha S^j_{\rm \,\,KK \,top\,\,}&\approx&\frac{\alpha}{2\pi}\,\Bigg[\,\frac{m_t^2}{M_j^2+m_t^2}\,-\,\frac{2}{3}\,\log\bigg(1+\frac{m_t^2}{M_j^2}\bigg)
\Bigg] \quad.
\label{KKhigtoptoS}
\eear


