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\begin{document}
\begin{titlepage}
\begin{flushright}
 \\
\end{flushright}

\vspace*{0.8cm}
\begin{center}
\vspace*{0.5cm}
{\LARGE \bf Big Corrections from a Little Higgs} \\
\vspace*{1.5cm}

\mbox{\bf
{Csaba Cs\'aki}$^{a}$, {Jay Hubisz}$^{a}$,
{Graham D. Kribs}$^{b}$,}\\
\mbox{\bf {Patrick Meade}$^{a}$, {and John Terning}$^{c}$} \\

\vspace*{0.8cm}

$^{a}$ {\it Newman Laboratory of Elementary Particle Physics, \\
Cornell University, Ithaca, NY 14853} \\
\vspace*{0.1cm}
$^{b}$ {\it Department of Physics, University of Wisconsin, Madison, WI 53706} \\
\vspace*{0.1cm}
$^{c}$ {\it Theory Division T-8, Los Alamos National Laboratory, Los Alamos, 
NM 87545} \\
\vspace*{0.8cm}
{\tt  csaki@mail.lns.cornell.edu, hubisz@mail.lns.cornell.edu,
kribs@physics.wisc.edu, meade@mail.lns.cornell.edu, terning@lanl.gov}
\end{center}

\vspace*{1cm}

\begin{abstract}
\vskip 3pt
\noindent 
We calculate the tree-level expressions for the electroweak
precision observables in the $SU(5)/SO(5)$ littlest Higgs model. 
The source for these corrections are the exchange of heavy gauge
bosons, explicit corrections due to non-linear $\sigma$-model dynamics
and a triplet Higgs VEV.
Weak isospin violating contributions are
present because there is no custodial $SU(2)$ global symmetry.
The bulk of these weak isospin violating corrections arise from
heavy gauge boson exchange while a smaller contribution
comes from the triplet Higgs VEV.
A global fit is performed to the experimental data and we find that 
throughout the parameter space the symmetry breaking scale is bounded by
$f > 4$ TeV at 95\% C.L.  Stronger 
bounds on $f$ are found for generic choices 
of the high energy gauge couplings.  We find that even in the best 
case scenario one would need fine tuning of less than a percent 
to get a Higgs mass as light as 200 GeV.

\end{abstract}



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\section{Introduction}
\label{sec:intro}
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Recently there has been excitement generated by the 
revival~\cite{little1,littlest,littlestmoose,Witek,littlepheno,Sekhar}
of the idea that the Higgs is a pseudo-Goldstone boson
\cite{HiggsPseudo,KaplanGeorgi,KaplanGeorgiSU2}. 
New models have been generated 
that provide
plausible realizations of this scenario with the feature that
they cancel all 
quadratically
divergent contributions to the Higgs mass at one-loop. For a recent
review see~\cite{Martin}. These 
``little'' Higgs models were
originally motivated~\cite{little1} by extra dimensional theories where the
Higgs is an extra component of the gauge fields~\cite{Manton,extra1,extra2,
extra3,extra4,extra5}, however the simplest little Higgs 
models~\cite{littlest,littlestmoose,Witek} do not retain 
any resemblance to extra dimensional theories.
At first glance these models seem 
to allow a cutoff as large as 10 TeV without more than 10\% fine tuning in
the Higgs mass squared.  In particular a littlest Higgs model has
been proposed which is based on breaking an $SU(5)$ symmetry down to  
$SO(5)$. When two $SU(2)$ subgroups (as well as two non-orthogonal 
$U(1)$'s) of the $SU(5)$ are gauged they
are broken down to a diagonal $SU(2)_L\times U(1)_Y$ which can be 
identified with
the electroweak interaction gauge group of the standard model (SM).  
The main idea 
behind the little Higgs models is that there are enough symmetries
in the theory that the simultaneous introduction
of two separate symmetry breaking terms is needed to force the Higgs to be a 
pseudo-Goldstone boson 
rather than an exact Goldstone boson. For example in the gauge sector of
the $SU(5)/SO(5)$ model  
both $SU(2)$ gauge interactions are required to give a mass term to
the Higgs, thus the 
quadratically divergent mass contributions must be proportional to
powers of both $SU(2)$ gauge couplings and hence can only appear at
two-loop order. 

Since the weak interaction gauge group
is a mixture of two different gauge groups there are a variety
of corrections to the predictions for electroweak observables.  This 
type of mixing correction is well known from previous extensions of the
SM such as the ununified model \cite{Georgiununified} and the 
$SU(3)$ electroweak model 
\cite{DimKap}.  
Current precision electroweak data place constraints 
on the masses of the heavy analogues of the $W$ and $Z$ to be of order
2-10 TeV depending on the model \cite{CST,CEKT}.
However in the little Higgs models the 
corrections are more dangerous  since a
``custodial'' $SU(2)$ symmetry is not automatically enforced as it is in the
SM (as well as the alternative models mentioned above)
or in other words weak isopsin is violated. (The importance of 
custodial $SU(2)$ was recently
emphasized in~\cite{Sekhar}.)
These weak isospin violating effects come
both from heavy gauge boson exchange and to a lesser extent
from the presence of a triplet Higgs VEV.
In addition to these effects there are further corrections arising
from the fact that in little Higgs models the Higgs is part of a 
non-linear $\sigma$ model  rather than a linear $\sigma$ model as in 
the SM.  

In the little Higgs
models raising the masses of the heavy gauge bosons means that log 
divergent terms become large and hence that the fine-tuning becomes
more severe.  
A similar effect happens with the masses of the fermions that cancel the 
(even larger) top loop, which can make the situation even worse than one 
might have expected.  
For sufficiently large masses the large fine-tuning needed means that the 
model fails to address the original motivation that inspired it.
In general there are two ways to try to satisfy the precision electroweak 
bounds in these models. First one can take one of the two $SU(2)$  
gauge couplings 
to be large which reduces the direct coupling of quarks and leptons to the
heavy gauge boson; we will see that this approach turns out to be 
unfavorable since it maximizes the mass mixing between heavy and light
gauge bosons which thus requires that the scale where the two gauge 
groups break to $SU(2)_L$ to be large. This is problematic since it 
raises the heavy
gauge boson and fermion masses and thus increases the fine-tuning as 
described above.
The second approach is to tune the two  couplings (especially the
 two $U(1)$ couplings) 
to be equal, then the mass 
mixing effects vanish and the bounds on the high 
breaking scale are driven by the weak-isospin breaking effects in four-fermion
interactions,  which again result
in a strong bound on the breaking scale. 
In fact the correction to the weak isopsin violating $\Delta \rho_* =
\alpha T$ parameter is independent of the choice of the gauge couplings
and  can be brought to an acceptably small value only by raising
the symmetry breaking scale $f$.


In this paper we calculate the corrections to electroweak  observables in
the littlest Higgs model.  We restrict ourselves to tree-level effects. 
We then perform a global fit to the precision electroweak
data which allows us to quantify the bounds on the masses of the heavy
gauge bosons and fermions and thus also quantify the required amount of 
fine-tuning. To illustrate the importance of weak isospin breaking
we perform fits with and without the 
$SU(2)_L$ triplet\footnote{The appearance of the Higgs triplet 
is not an essential
part of little Higgs models, for example the $SU(6)/Sp(6)$ model 
considered in~\cite{Witek} does not have such a particle in its spectrum,
instead it has two Higgs doublets and an extra singlet.} VEV.
We find that artificially setting the triplet VEV to zero does not improve the
situation, since there is still isospin breaking from heavy gauge boson 
exchange.

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\section{The Littlest Higgs Model at Tree-Level}
\label{spectrum-sec}
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We consider the little Higgs model based on the 
non-linear $\sigma$ model describing 
an \break 
$SU(5)/SO(5)$ symmetry breaking~\cite{littlest}. 
This symmetry breaking can be thought of as
originating from
a VEV of a symmetric tensor of the $SU(5)$ global symmetry. A convenient basis
for this breaking is characterized by the direction $\Sigma_0$ for 
expectation value for
the symmetric tensor of the form
\begin{equation}
\Sigma_0 =\left( \begin{array}{ccccc} &&&1& \\&&&&1\\ &&1\\1\\&1 \end{array}
\right).
\end{equation}
The Goldstone fluctuations are then described by the pion fields $\Pi =
\pi^a X^a$, where the $X^a$ are the broken generators of $SU(5)$. The 
non-linear sigma model field is then 
\begin{equation}
\Sigma (x) = e^{i\Pi/f} \Sigma_0 e^{i \Pi^T/f}=e^{2i\Pi/f} \Sigma_0.
\end{equation}
where $f$ is the scale of the VEV that accomplishes the breaking.
An 
$[SU(2)\times U(1)]^2$ subgroup\footnote{Note that the two $U(1)$ 
generators
are not orthogonal and thus may have kinetic mixing terms, which will 
imply additional corrections to electroweak observables.  Such kinetic
mixing terms are not generated at one-loop in the effective theory, but
may be generated by physics above the cut-off if there are heavy 
particles charged under both U(1) groups.   Here we will assume
these effects are absent.}  of 
the $SU(5)$ global symmetry 
is gauged, where the generators of the gauged symmetries are given by
\begin{eqnarray}
&Q_1^a=\left( \begin{array}{ccc} \sigma^a/2 &0 & 0 \\ 
0 & 0 & 0\\ 0 & 0 & 0
\end{array}\right), \ \ \ &Y_1=
{\rm diag}(-3,-3,2,2,2)/10\nonumber \\
&Q_2^a=\left( \begin{array}{ccc} 0 & 0 & 0\\
0 & 0 & 0 \\ 
0 &0&-\sigma^{a*}/2\end{array} \right), 
& Y_2={\rm diag}(-2,-2,-2,3,3)/10~,
\end{eqnarray}
where $\sigma^a$ are the Pauli $\sigma$ matrices. The $Q^a$'s are $5 \times 5$
matrices written in terms of $2 \times 2$, 1, and $2 \times 2$ blocks.
The Goldstone boson matrix $\Pi$, 
in terms of the uneaten fields, is then given by
\begin{equation}
\Pi = \left( \begin{array}{ccc} 
0 & \frac{H^\dagger}{\sqrt{2}} & \phi^\dagger
\\ \frac{H}{\sqrt{2}}& 0 & \frac{H^*}{\sqrt{2}}\\ \phi 
&\frac{H^T}{\sqrt{2}} & 0
\end{array}\right),
\end{equation}
where $H$ is the little Higgs doublet $(h^+,h^0)$ and $\phi$ is a 
complex triplet 
Higgs, forming a symmetric tensor $\phi_{ij}$. 
This triplet should have a very small expectation value 
($\cal{O}$(GeV)) in order
to not give too large a contribution to the $T$ parameter. 




We will write the gauge couplings of the $SU(2)$'s as $g_1$ and  $g_2$,
and similarly for the $U(1)$'s: $g_1^\prime$  and $g_2^\prime$. 
We can assume that the quarks and leptons have their usual quantum numbers 
under $SU(2)_L$ 
$\times$ U(1)$_Y$, but they are assigned under the first $SU(2) \times 
U(1)$
gauge groups.

The kinetic energy term of the non-linear $\sigma$ model is
\beq
\frac{f^2}{8} {\rm Tr} D_\mu \Sigma (D^\mu \Sigma)^\dagger
\eeq
where
\beq
D_\mu \Sigma = \partial_\mu \Sigma - i \sum_j \left[ g_j W_j^a (Q_j^a 
\Sigma
+ \Sigma Q_j^{aT} )- B_j( Y_j \Sigma + \Sigma Y_j)\right]~.
\eeq

Thus at the scale of symmetry breaking $f$ 
(neglecting for the moment the Higgs VEV)
the gauge bosons of the four groups mix to form the 
the light electroweak gauge bosons and heavy partners. 
In the ($W_1^a$, $W_2^a$) basis (for $a=1,2,3$)
the mass matrix is:
\beq
\frac{f^2}{4} \left(\begin{array}{cc} g_1^2  &  -g_1 g_2  \\  
-g_1 g_2  &g_2^2   \end{array}\right)
\eeq
Thus the light and heavy mass eigenstates  are:
\beq
W_L^a &=& s W_1^a + c W_2^a\\
W_H^a &=& -c W_1^a + s W_2^a
\eeq
with masses
\beq
M_{W_L} &=& 0 \\
M_{W_H} &=& \sqrt{g_1^2 + g_2^2} \frac {f}{2}
\eeq
where
\beq
s= \frac{g_2}{\sqrt{g_1^2+g_2^{2}}},\ \ \ \ 
c=\frac{g_1}{\sqrt{g_1^2+g_2^{2}}} ~.
\eeq

The $SU(2)$ singlet gauge bosons arise
from the U(1) gauge bosons $B_1$ 
and 
the $B_2$.    
The mass matrix in the ($B_1$,$B_2$) basis 
at the high scale is
\beq
\frac{f^2}{20} \left(\begin{array}{cc} g_1^{\prime 2} &
   -g_1^\prime  g_2^\prime  \\  
 - g_1^\prime  g_2^\prime  
&g_2^{\prime 2} \end{array}\right)
\eeq
Thus the light and heavy mass eigenstates  are:
\beq
B_L^a &=& s^\prime B_1 + c^\prime B_2\\
B_H^a &=& -c^\prime B_1 + s^\prime B_2
\eeq
with masses
\beq
M_{B_L} &=& 0 \\
M_{B_H} &=& \sqrt{ g_1^{\prime 2} + g_2^{\prime 2}} \frac {f}{\sqrt{20}}
\eeq
where
\beq
s^\prime= \frac{g_2^\prime}{\sqrt{ g_1^{\prime 2}+g_2^{\prime 2}}},\ \ \ \ 
c^\prime=\frac{g_1^\prime}{\sqrt{g_1^{\prime ^2}+g_2^{\prime 2}}} ~.
\eeq


The effective gauge couplings of the $SU(2)_L$ $\times$ U(1)$_Y$ groups are:
\beq
g &=&  g_1 s\\
g^\prime &=& g_1^\prime s^\prime
~.
\eeq
Assuming that the fermions transform only under the $SU(2)_1\times U(1)_1$
gauge group,
the coupling of $W^a_H$ $(B_H)$ to quarks and leptons is 
$- g_1 c$ $(- g_1^\prime  c^\prime)$.





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\section{The Low-energy Effective Action}
\label{effective-sec}
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We now construct the effective theory below the mass scale of the heavy
gauge bosons.  Integrating out $W^a_H$ and $B_H$ induces  additional 
operators in the effective theory. These operators modify the usual
relations between the standard model parameters, and therefore their
coefficients can be constrained from electroweak precision measurements.
There are three types of operators that will be relevant for us: four-fermion
interactions,
corrections of the coupling of the $SU(2)_L \times U(1)_Y$ gauge
bosons to their corresponding currents, and operators that are quadratic 
in the $SU(2)_L \times U(1)_Y$ gauge fields.  
For simplicity we will work in a unitary gauge and
only keep track of the $h \equiv {\rm Re}\ h^0$ component of the Higgs field.

Exchanges of the  heavy $W^a_H$ and $B_H$ gauge bosons give the 
following operators which are quadratic in the light gauge fields:
\begin{eqnarray}
{\cal L}_{2 {\rm mix}}&=&
-\frac{g^2  (s^2-c^2)^2 }{8 f^2} W^{a \mu}_L W^{a}_{\mu L}
h^{4}  
- \frac{5 g^2  (s^{\prime 2}-c^{\prime 2})^2} {8 f^2} W^{3 \mu}_L 
W^{3}_{\mu L} h^{4} \nonumber \\ && 
- \frac{g^{\prime 2}  (s^2-c^2)^2} {8 f^2} B^{\mu}_L B_{\mu L}  h^{4}
- \frac{5 g^{\prime 2}  (s^{\prime 2}-c^{\prime 2})^2} {8 f^2} B^{\mu}_L 
B_{\mu L}  h^{4}
\\&&
+ \frac{g g^{\prime}  (s^2-c^2)^2} {4 f^2} W^{3\mu}_L B_{\mu L}  h^{4}
\label{operators1}
\end{eqnarray}
For example, the first term arises in the following way. The kinetic term of
the little Higgs field $(D_\mu \Sigma) D^\mu \Sigma^\dagger$ contains 
the coupling
\beq
{\cal L}_{\tilde{W}^2 h^{2}}=
\frac{g_1 g_2}{4}  W^a_{1 \mu} W_2^{a \mu} h^{2}~.
\eeq
Expressing $W_1$ and $W_2$ in terms of $W_L$ and $W_H$ we obtain a 
coupling 
between the heavy and light gauge bosons of the form 
\beq
{\cal L}_{W_L W_H h^{2}}=-\frac{g_1 g_2 (g_1^2- 
g_2^2)}{4(g_1^2+g_2^2)}  W^a_{\mu L}W^{a \mu}_H {h}^2~.
\eeq 
The first term in ${\cal L}_{2 {\rm mix}}$ then arises by integrating out 
the
heavy gauge boson $W^{a \mu}_H$ by taking its equation of
motion and expressing it in terms of the light fields.

In addition there are terms in the effective theory that are quadratic in 
light gauge fields and 
quartic in Higgs fields simply because the model is a non-linear $\sigma$
model as well as couplings to the $SU(2)_L$ triplet $\phi$:
\begin{eqnarray}
{\cal L}_{2 {\rm nl}\sigma}&=&
\frac{g^2}{4 f^2} W^{a \mu}_L W^{a}_{\mu L} h^{4}  
+ \frac{g^{\prime 2} }
{4 f^2} B^{\mu}_L B_{\mu L} h^{4}
-\frac{g g^\prime}{2 f^2} B^{\mu}_L W^{3}_{\mu L} h^{4}~,  \nonumber \\
{\cal L}_{2 \phi}
&=& \frac{g^2}{2} W^{a \mu}_L W^{a}_{\mu L} \phi^{2}  +
\frac{g^2}{2} W^{3 \mu}_L W^{3}_{\mu L} \phi^{2}
+ g^{\prime 2}  B^{\mu}_L B_{\mu L} \phi^{2}
-2 g g^\prime B^{\mu}_L W^{3}_{\mu L} \phi^{2}~.
\label{operators2}
\end{eqnarray}
We will only keep terms to order $\phi^2$ since $\langle \phi \rangle$
is phenomenologically
required to be small and is parametrically of order $v^2/f$ so corrections of
order $h^4/(v^2 f^2)$ and $\phi^2/v^2$ are actually the same order in
the $v/f$ expansion. 


The operators in (\ref{operators1}) and (\ref{operators2}) 
give corrections to the light
gauge boson masses after the Higgs  gets a VEV. Thus
after $h$ and $\phi$ get VEVs:
\beq
\langle h \rangle&=& \frac{v}{\sqrt{2}}~,\\
\langle \phi \rangle &=& v^\prime~,
\eeq
and including the effects of the higher dimension operators 
(\ref{operators1},\ref{operators2}), we find
that the mass of the $W$ is
\beq
M_W^2 &=&  g^2  \frac{v^2}{4}\left(1
+\frac{  (s^4+6s^2 c^2+c^4) v^2}{ 4  f^2}
+4\frac{v^{\prime 2}}{v^2} \right)~.
\eeq
Similarly, the mass of the $Z$ is
\beq
M_Z^2 &=&(g^2 + g^{\prime 2})\frac{v^2}{4} 
\left(1+\frac{ ( s^4 +6s^2 c^2+c^4)v^2}{4 f^2}-\frac{5(s^{\prime 
2}-c^{\prime 2})^2 v^2}{4 f^2}
+ 8\frac{v^{\prime 2}}{v^2} \right)~.
\label{Zmass}
\eeq

In addition, exchanges of $W^a_H$ and $B_H$ give
corrections to the coupling of the $SU(2)_L\times U(1)_Y$ gauge
bosons to their corresponding currents and additional four-fermion operators:
\beq
&&{\cal L}_{\rm c}=
g W^a_{L \mu} J^{a \mu}\left(1+ 
  \frac{c^2(s^2-c^2) h^{2}}{ f^2}\right)
+  g^\prime B_{L \mu} J^\mu_{Y}
\left(1- \frac{5 c^{\prime 2} (s^{\prime 4}-c^{\prime 4}) 
h^{2}}{f^2}\right)
\nonumber \\
&&+
g  W^3_{L \mu}J^\mu_{Y} \frac{5 (s^{\prime 4}-c^{\prime 4})h^{2}}{f^2}
-  g^\prime B_{L \mu} J^{3 \mu} 
 \frac{c^2(s^2-c^2) h^{2}}{f^2}
-J_\mu^aJ^{a\mu} \frac{2 c^4}{f^2}-J_\mu^Y  J^{Y\mu}\frac{10 c^{\prime 4}}{f^2}
\label{currents}
\eeq

Using this expression we can now evaluate the effective Fermi coupling
$G_F$ in this theory. The simplest way to obtain the answer for this is by
integrating out the $W_L$ bosons from the theory by adding the $W$ mass term
to (\ref{currents}). The expression we obtain for the effective four-fermion 
operator is
\begin{equation}
-\frac{g^2}{2M_W^2} J^{+\mu}J^-_\mu 
\left[1+\frac{c^2(s^2-c^2) v^2}{f^2}\right]-
J^{+\mu}J^-_\mu \frac{2 c^4}{f^2}=-2 \sqrt{2} G_F 
J^{+\mu}J^-_\mu,
\end{equation}
where $J^\pm=\frac{1}{2}(J^1\pm iJ^2)$.
Plugging in the correction to the $W$ mass we obtain that $G_F$ in this model
is corrected by 
\begin{equation}
\label{treeGF}
\frac{1}{G_F}= \sqrt{2}v^2\left(1+\frac{v^2}{4f^2}
+4\frac{v^{\prime 2}}{v^2}\right).
\end{equation}

Finally, to fix all SM parameters we need to identify the 
photon and the neutral-current couplings from (\ref{currents}):
\begin{eqnarray}
&&{\cal L}_{\rm nc}=
e A_{\mu} J^\mu_{Q}
+ \frac{e}{s_W c_W}Z_\mu\left[ J^{3 \mu}
\left(1
+  \frac{c^2(s^2-c^2)h^{2}}{f^2}
+\frac{5c^{\prime 
2}(s^{\prime 2}-c^{\prime 2})h^{2}}{f^2}\right) \right.
\nonumber  \\ && \left. - J^{\mu}_Q\left(s_w^2 +  
5  \frac{c'^2(s'^2-c'^2)h^{2}}{f^2}
\right)\right] 
-(J^3-J_Q)^\mu  (J^3-J_Q)_{\mu}\frac{10 c^{\prime 4}}{f^2}-\frac{2c^4}{f^2}
J^{3 \mu}J_\mu^3~ . \label{neutral}
\end{eqnarray}
Here $e= gg'/\sqrt{g^2+g^{\prime 2}}$ as in the standard model, 
thus there is no
correction to the expression of the electric charge $e$ 
compared to the SM. Also $s_W^2$ in the above expression is the SM value
of the tree-level weak mixing angle $s_W^2=g'^2/(g^2+g'^2)$.
Similarly to the evaluation of the 
effective $G_F$ we can calculate the low-energy effective four-fermion 
interactions from the neutral currents. The result we obtain is
\begin{eqnarray}
{\cal L}_{NC} =&& -\frac{g^2+g'^2}{2 M_Z^2} \left[ 
J_3-s_W^2 J_Q+\frac{v^2}{2f^2} 
(c^2(s^2-c^2) J_3
-5 c'^2 (c'^2-s'^2) J_Y)\right]^2 
\nonumber \\ && -\frac{10c'^4}{f^2} J_Y^2
 -\frac{2c^4}{f^2} J_3^2.
\label{NC}
\end{eqnarray}
where the $M_Z$ is the physical $Z$ mass given in (\ref{Zmass}). 

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\section{The Contributions to Electroweak Observables}
\label{observables-sec}
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To relate the model parameters to observables 
we use $\alpha(M_z)$, $G_F$, and $M_Z$ as input parameters.
We then use the 
standard definition of the weak mixing angle 
$\sin \theta_0$ from the $Z$ pole \cite{Sformulas},
\beq
\label{s2}
\sin^2 \theta_0 \cos^2 \theta_0 &=& \frac{\pi \alpha(M_Z^2)}{\sqrt{2} G_F M_Z^2}~,\\
\sin^2 \theta_0&=&0.23105 \pm 0.00008~,
\eeq
where~\cite{ErlerLang} 
$\alpha(M_Z^2)^{-1}=128.92\pm 0.03$ is the running SM fine-structure
constant evaluated at $M_Z$.
We can relate this measured value with the bare value $s_W^2$ 
in this class of models, by using the expressions
 \beq
s_0^2\equiv \sin^2  \theta_0 &=& s_W^2 +\delta s_W^2 
= s_W^2 -\frac{s_W^2 c_W^2}{c_W^2-s_W^2}
\left[ \frac{\delta G_F}{G_F}+\frac{\delta M_Z^2}{M_Z^2} \right]
\nonumber \\ &=& s_W^2-
\frac{s_W^2 c_W^2}{c_W^2-s_W^2}\left[4\Delta'+
\Delta \left(-\frac{5}{4} +c^2(1-c^2)+5 c'^2(1-c'^2)\right)\right],
\label{rens2}
\eeq
where we have defined 
\beq
\Delta \equiv \frac{v^2}{f^2}, \qquad 
\Delta'\equiv \frac{v'^2}{v^2}~.
\eeq
Also, we have the simple result that the
running couplings defined by Kennedy and Lynn~\cite{Lynn}
which appear in $Z$-pole 
asymmetries are the same as
the bare couplings:
\beq
s^2_*(M_Z^2)=s_W^2, \ \ \ 
e^2_*(M_Z^2)=e^2 ~.
\eeq

In order to compare to experiments, we can relate our corrections of
the neutral-current couplings to the generalized modifications of the 
$Z$ couplings as defined by Burgess et al. \cite{Burgess},
\begin{equation}
{\cal L}=\frac{e}{s_Wc_W} \sum_i \bar{f}_i \gamma^\mu 
\left( (g_L^{f,SM}+\delta \tilde{g}_L^{ff}) P_L + 
(g_R^{f,SM}+\delta \tilde{g}_R^{ff}) P_R \right) f_i 
Z_\mu,  \end{equation}
where $P_{L,R}$ are left and right projectors, and
\begin{equation}
\frac{1}{s_W c_W} = \frac{1}{s_0 c_0} 
\left[ 1 - 2\Delta'
-\frac{\Delta}{2}\left(-\frac{5}{4} + c^2(1-c^2) 
+ 5c'^2(1-c'^2)\right) \right] \; .
\end{equation}
 From (\ref{neutral}) we obtain that
\begin{equation}
\delta \tilde{g}^{ff}=\Delta 
\left[\frac{5t_3^f}{2} (c'^2-2c'^4) +\frac{t_3^f}{2}(c^2-2c^4)-q^f 
\frac{5}{2} (c'^2-2 c'^4)\right].
\end{equation}
For the individual couplings this implies
\begin{eqnarray}
\delta \tilde{g}_L^{uu}&=& \frac{\Delta}{12} \left[3c^2(1-2c^2)-5c'^2(1-2c'^2)
\right],
\qquad\>\> \delta \tilde{g}_R^{uu}= 
  -\frac{5 \Delta}{3} c'^2 (1-2c'^2), \nonumber \\
\delta \tilde{g}_L^{dd}&=&
  \frac{\Delta}{12}\left[ -3c^2(1-2c^2)-5c'^2(1-2c'^2)\right],
\quad  \delta \tilde{g}_R^{dd}= 
 \frac{5\Delta}{6}c'^2(1-2c'^2), \nonumber \\
\delta \tilde{g}_L^{ee}&=& 
  -\frac{\Delta}{4}\left[c^2(1-2c^2)-5c'^2(1-2c'^2)\right], 
\qquad \delta \tilde{g}_R^{ee}= 
  -\frac{5\Delta}{2} c'^2(1-2c'^2), \nonumber \\
\delta \tilde{g}_L^{\nu\nu}&=&
  -\frac{\Delta}{4}\left[-c^2(1-2c^2)-5c'^2(1-2c'^2)\right],
\end{eqnarray}
where $\delta \tilde{g}^{\mu\mu}=\delta \tilde{g}^{\tau\tau}=
\delta \tilde{g}^{ee}$, and similarly 
$\delta \tilde{g}^{tt}=\delta \tilde{g}^{cc}=\delta \tilde{g}^{uu}$, 
$\delta \tilde{g}^{bb}=\delta \tilde{g}^{ss}=\delta \tilde{g}^{dd}$. 

In order to calculate the corrections to the low-energy precision 
observables for neutrino scattering and for atomic parity 
violation we need to write the low-energy 
effective neutral current interaction (\ref{NC}) in
the form 
\begin{equation}
-\frac{4 G_F}{\sqrt{2}} \rho_* (J_3-s_*^2(0) J_Q)^2 +\alpha J_Q^2.
\end{equation}
Here $s_*^2(0)$ is the low-energy value of the effective Weinberg angle,
different from $s_*^2 (M_Z^2)$. The last term proportional to $\alpha$ will not contribute to any of the
low-energy processes we are constraining, therefore what one needs to do
is to express $\rho_*$ and $s_*^2$ in terms of our variables 
$\Delta,\Delta' ,c,c'$.
We find the following expressions:
\begin{eqnarray}
&& \rho_* = 1-4\Delta' +\frac{5}{4} \Delta \nonumber \\
&& s_*^2(0)
= s_W^2-\frac{\Delta}{2}\left[ s_W^2 ( c^2+5 c'^2)-5 c'^2\right],
\label{rhostar}
\end{eqnarray}
where $s_W^2$ has to be expressed in terms of $s_0^2$ using (\ref{rens2}).
Note that $s_*^2(0) \ne s_*^2(M_Z)$ due to the corrections to four-fermion
interactions from heavy gauge boson exchange.
The low-energy observables can then be expressed using the relations
\begin{eqnarray}
&g_L^2= \rho_*^2\left[\frac{1}{2}-s_*^2(0)+\frac{5}{9} s_*^4(0)\right]
~,\ \ \ \ & 
g_{eV}(\nu e \rightarrow \nu e) = 2\rho_*\left[s_*^2(0) - \frac{1}{4}\right]~,
\nonumber \\  
&g_R^2=\rho_*^2 \frac{5}{9} s_*^4(0), &  
g_{eA}(\nu e \rightarrow \nu e) = -\frac{\rho_*}{2}~, \nonumber \\
& Q_W(Z,N)=-\rho_* \left[ N-(1-4 s_*^2(0)) Z\right]~.
\end{eqnarray}
In the Appendix we calculate the shifts in the electroweak precision
observables in terms of the parameters $c,c',\Delta$  and $\Delta'$ 
defined above.

Until now we have treated $\Delta$ and $\Delta'$ as independent variables.
However the triplet VEV is not independent from the Higgs VEV, since it 
arises from the same operators that give rise to the quartic scalar potential
of the little Higgs~\cite{littlest}. 
The leading terms are the quadratically divergent
pieces in the Coleman-Weinberg (CW) potential (which by construction does 
not contribute to the little Higgs mass) and their tree-level counter terms.
For example the gauge boson contribution to the CW potential is
\begin{equation}
\frac{\Lambda^2}{16 \pi^2} {\rm Tr} M_V^2 (\Sigma ),
\end{equation}
where $M_V^2$ is the gauge boson mass matrix in an arbitrary $\Sigma$ 
background. For example the first 3$\times$3 block of the gauge boson
mass matrix (corresponding to the first $SU(2)$ gauge group) is~\cite{littlest}
\begin{equation}
M_V^{2\ ab}=g_1^2 {\rm Tr} (Q_1^a\Sigma+\Sigma Q_1^{a\, T})(\Sigma^{\dagger} 
Q_1^b+Q_1^{b\, T} \Sigma^\dagger ).
\end{equation}
Evaluating the full expression for the gauge boson contributions 
results in a potential (to cubic order) of the form
\begin{equation}
V_{GB}= a f^2\left[ (g_1^2+g_1'^2) \left| \phi_{ij}+\frac{i}{4f}( h_i h_j+
h_j h_i)\right|^2+ (g_2^2+g_2'^2) \left| \phi_{ij}-\frac{i}{4f}( h_i h_j+
h_j h_i)\right|^2\right],
\label{CW1}
\end{equation}
where $a$ is a constant of order one determined by the relative
size of the tree-level and loop induced terms, and we have used 
$\Lambda \sim 4\pi f$. Similarly, the fermion loops contribute
\begin{equation}
-a' \lambda_1^2 f^2\left| \phi_{ij}+\frac{i}{4f}( h_i h_j+
h_j h_i)\right|^2,
\label{CW2}
\end{equation}
where $\lambda_1$ (and $\lambda_2$) are the Yukawa couplings and mass terms
\begin{equation}
{\cal L}_{fermion}= \lambda_1 (q_3'h +f \tilde{t})u_3^c +\lambda_2 f \tilde{t}
\tilde{t}^c.
\end{equation}
Here $\tilde{t},\tilde{t}^c$ are the extra vector-like color triplet 
fermions necessary to cancel the quadratic divergences to the little Higgs
mass from the top loops, and the physical right-handed top is
$(\lambda_2 q_3'-\lambda_1 \tilde{t})/\sqrt{\lambda_1^2+\lambda_2^2}$ 
with a Yukawa coupling $\lambda_1\lambda_2/\sqrt{\lambda_1^2+\lambda_2^2}$.
The mass of the heavy fermion is $\sqrt{\lambda_1^2+\lambda_2^2}f$.

Thus one can see from (\ref{CW1}) and (\ref{CW2}) that there must be a triplet
VEV of order $v' \sim v^2/f$ as we have assumed before. In terms of the
parameters $a$, $a'$,and $\lambda_1$ the triplet VEV is given by
\begin{equation}
v'=-\frac{i v^2}{4f} \frac{a(g_1^2+g_1'^2-g_2^2-g_2'^2)-
a'\lambda_1^2}{a(g_1^2+g_1'^2+g_2^2+g_2'^2)-
a'\lambda_1^2}.
\label{tripletvev}
\end{equation}
However, since the terms (\ref{CW1}) and (\ref{CW2}) are also responsible
for the quartic scalar coupling of the little Higgs, one can eliminate the
parameters $a',\lambda_1$ from the expression for the
triplet VEV. The quartic scalar coupling
is given by~\cite{littlest}
\begin{equation}
\label{lambda}
\lambda =\frac{[a(g_1^2+g_1'^2)-a'\lambda_1^2][a(g_2^2+g_2'^2)]}{a(g_1^2+g_1'^2+g_2^2+g_2'^2)-a'\lambda_1^2},
\end{equation}
and thus we obtain that 
\begin{equation}
\Delta' \equiv \frac{v'^2}{v^2} = \frac{\Delta}{16} \left[ 
\frac{2 \lambda}{a(g_2^2+g_2'^2)}-1\right]^2~.
\label{deltaprime}
\end{equation}
For a Higgs mass of order 200 GeV, $\lambda\approx 1/3$ 
(at tree-level).
The above formula can then be used for the fit with  reasonable
values of the coefficient $a$.  There is one further constraint on
the parameters of the model.  In ref. \cite{littlest} it was shown
that in order to have a positive mass squared for the triplet
one must have:
\beq
a(g_1^2+g_1'^2+g_2^2+g_2'^2)>a'\lambda_1^2
\eeq
which is equivalent (using our previous constraint (\ref{lambda})) to requiring
\beq
a > \frac{\lambda}{g_2^2+g_2'^2}~.
\label{abound}
\eeq
If the triplet mass squared is negative it   implies that the
triplet gets a VEV of order $f$, which impossible to reconcile with
electroweak data.
This constraint will prove to be important since it excludes the region
of large triplet VEVs. From Eq. (\ref{deltaprime}) we see that it 
enforces
\begin{equation}
\Delta' < \frac{\Delta}{16}~.
\label{deltaprimebound}
\end{equation}
Actually the bound on $a$ is more severe than Eq. (\ref{abound})
due to the non-observation of the triplet scalar at LEP, but  
Eq. (\ref{abound}) will be sufficient for our purposes.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results and Interpretation}
\label{results-sec}
\setcounter{equation}{0}
\setcounter{footnote}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Since the parameter $a$ is expected to be ${\cal O}(1)$, we will consider
fixed values of $a$ in the range 0.1 - 2.
However to begin the discussion of our results we will artificially set
the triplet VEV to zero.
This not only makes the analysis and interpretation simpler it also
contains the essential physics that constrains the model.
We performed a three parameter global fit 
(as described in \cite{RSfit}) to the 21 precision 
electroweak observables given in Table~\ref{table}. 
The best fit was found to be for $c \simeq 1$, 
$c' \simeq 0.32$, and $f\approx8.9$ 
TeV\@, with a $\chi^2$ per degree
of freedom ($21-3=18$):
\begin{equation}
\frac{\chi^2_{\rm best}}{(\mbox{d.\ of f.})} \simeq 1.54
\end{equation}
that is slightly worse than the fit to the SM,
\begin{equation}
\frac{\chi^2_{\rm SM}}{(\mbox{d.\ of f.})} \simeq 1.38 \; .
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\epsfxsize=0.75\textwidth
\centerline{\epsfbox{limitbands689599.eps}}
\caption{The region of parameters excluded to 68\%, 95\%, and 99\% C.L.
is shown as a function of $c'$.  The parameter $c$ was allowed
to vary between $0.1 < c < 0.995$ for each $c'$ to give the least 
restrictive bound on $f$.  (See also Fig.~\ref{contour-fig}.)}
\label{limit-fig}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
First consider the region of parameters relevant to the model.
To ensure the high energy gauge couplings $g_{1,2},g'_{1,2}$ are
not strongly coupled, the angles $c = g/g_2$, $s = g/g_1$ 
$c' = g'/g'_2$, $s' = g'/g'_1$ cannot be too small.  We 
conservatively allow for $c,s,c',s' > 0.1$, or equivalently
$0.1 < c,c' < 0.995$.  We allow $f$ to take on any value  
(although for small enough $f$ there will be constraints from
direct production of $B_H$).  The general procedure we used is to
systematically step through values of $c$ and $c'$, finding the
lowest value of $f$ that leads to a shift in the $\chi^2$
corresponding to the 68\%, 95\%, and 99\% confidence level (C.L.).
For a three-parameter fit, this corresponds to a $\Delta \chi^2$
of about $3.5$, $7.8$, $11.3$ from the minimum, respectively.  
Globally, for any choice of high energy gauge couplings, we find
\begin{equation}
f > (4.1, \, 4.6, \, 5.2) \; {\rm TeV} \quad {\rm at} \quad 
(99\%, \, 95\%, 68\%) \; {\rm C.L.}
\end{equation}
We used $m_h = 115$ GeV, and verified that the bound is not lowered
for larger values of the Higgs mass.
Of course these bounds are saturated only for very specific values
of the gauge couplings.  The bound on $f$ is perhaps best illustrated as
a function of $c'$, which we do in Fig.~\ref{limit-fig}.  The shaded 
area below the lines shows the region of parameter space excluded by 
precision electroweak data.  Note that 
we numerically found the value of $c$ that gave the 
\emph{least restrictive} bound on $f$ for every $c'$.  For a specific choice
of $c$ the bound on $f$ can be stronger.  
This is shown in Fig.~\ref{contour-fig}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\epsfxsize=0.75\textwidth
\centerline{\epsfbox{lhlimit.eps}}
\caption{The region of parameters excluded to 95\% C.L.
is shown as a function of $c'$.  The region below the contours is excluded
to 95\% C.L. for $c$ equal to $0.1$ (solid), $0.5$ (dotted), 
$0.7$ (dashed), $0.99$ (dot-dashed). The shaded region is excluded
for any choice of $c$.  }
\label{contour-fig}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where we show contours of the 95\% excluded region for fixed $c$
while $c'$ was allowed to vary.  This figure makes it clear that
the least restrictive bound is obtained for different values of
$c$ as $c'$ is varied.  The shaded region is identical to 
the 95\% C.L. region shown in Fig.~\ref{limit-fig}, illustrating
how the exclusion regions in Fig.~\ref{limit-fig} were obtained.

While we fit to 21 observables, inevitably certain 
observables are more sensitive to the new physics.
To gain some insight into the main 
observables leading to the bounds shown in 
Figs.~\ref{limit-fig}--\ref{contour-fig}, we show in 
Fig.~\ref{observables-fig} the deviation of $\Gamma_Z$, $A^b_{FB}$, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\centerline{
\epsfxsize=0.55\textwidth
\epsfbox{fixed0.1.eps}
\hfill
\epsfxsize=0.55\textwidth
\epsfbox{fixed0.99.eps}}
\caption{The difference between the predicted and the experimentally 
measured values for four observables in standard deviations (the pull).  
In both figures the decay constant was fixed to $f=4$ TeV\@.  The figure
on the left (right) has a fixed $c=0.1$ ($c=0.99$); other values
interpolate between these two figures.  Notice that the observable
that gives the largest contribution varies depending on $c'$.}
\label{observables-fig}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$A_{LR}$, and $Q_W$ from the experimental value (the pull) as a 
function of $c'$ for fixed $f = 4$ TeV and two choices of 
$c = 0.1$ and $0.99$.  A set of parameters is typically ruled out 
by the global fit once a single observable has a pull 
greater than of order $\pm 4$.  The variation of $A^b_{FB}$ and 
$A_{LR}$ also explains the appearance in 
Figs.~\ref{limit-fig}--\ref{contour-fig} of a rise in the bound
on $f$ for small $c'$ (where $A^b_{FB}$ is important) large
$c'$ (where many observables are important) and the bump in the 
middle (where $A_{LR}$ is important). Note that the region of large
$c^\prime$ corresponds to the $U(1)_1$ gauge coupling 
(the gauge coupling of the quarks and leptons) getting strong.


We can now move on to discuss the fits done for reasonable values of the
parameter $a$. We have redone the fit with $a=(0.1,0.5,1,2)$ and
the results are displayed in Fig. \ref{C-fig}. For generic couplings
we see that the 95\% C.L. bound is well above 4 TeV.
It is clear from the plots
that only with a very small value of $a$ does
the triplet VEV have a significant effect on the fit. 
We can see from Eq. (\ref{deltaprime}) that small values of $a$ enhance
the triplet VEV and thus the improvement in the fit comes from an accidental
cancellation between the two sources of weak isopsin breaking.
As pointed out in \cite{littlest} the VEV of the triplet Higgs is naturally
expected to be small, and we  see from Fig. \ref{C-fig}
that it is generically a sub-leading 
effect. However, by fine tuning the
coupling $a$ to be unnaturally small we can make the the triplet
VEV un-naturally large and force a cancellation between the two 
independent sources of weak isospin breaking for a narrow range of gauge
couplings. The amount of cancellation which is possible is however
limited by Eq. (\ref{abound}), and the region of parameter space
corresponding to a maximal cancellation is in the disallowed region
where the triplet mass squared is negative.  This effect cuts off
the small $a$ region. Imposing this constraint, it turns out
that the weakest bound arises for $a=0.17$, where the
bound on $f$ becomes  4.0 TeV. The easiest way to obtain this bound is to 
do the fit imposing
$\Delta^\prime = \Delta/16$, shown in Fig. \ref{b-fig}.  
In general the bound on $f$ could be stronger
than what is found by this method, because the point corresponding to the
bound might be eliminated by imposing the constraint of a positive triplet 
mass squared.  It turns out that the bounding point is not excluded and thus
does indeed provide the best bound. We also find a bound on the mass of 
the heavy gauge bosons: $M_{B_H}>650$ GeV and $M_{W_H}> 2.7$ TeV.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\epsfxsize=0.75\textwidth
\centerline{\epsfbox{CC.eps}}
\caption{The region of parameters excluded to 95\% C.L.
is shown as a function of $c'$.  The four figures correspond
to $a=0.1,0.5,1,2$ corresponding to top-left, top-right, bottom-left, and
bottom-right. The region below the contours is excluded
to 95\% C.L. for $c$ equal to $0.1$ (solid), $0.5$ (dotted), 
$0.7$ (dashed), $0.99$ (dot-dashed). The heavy solid line displays the bound
from Fig. 2. The shaded region corresponds to 
the extension of the excluded  region obtained by requiring a positive triplet
mass squared. }
\label{C-fig}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\epsfxsize=0.75\textwidth
\centerline{\epsfbox{limitb.eps}}
\caption{The region of parameters excluded to 95\% C.L.
is shown below the lines. The solid curve is for $\Delta^\prime=0$
as in Fig. 1, while the dashed curve is for $\Delta^\prime=\Delta/16$.
The weakest bound at $c^\prime=0.29$ corresponds to $a=0.17$ 
(using Eq.(\ref{deltaprime})). }
\label{b-fig}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Given the above bound on the scale $f$ we can quantify the amount of 
fine-tuning in the model using the fine-tuning measure proposed 
in \cite{littlest}.
The contribution to the Higgs mass squared from the heavy partner of the top 
(with mass $m^\prime$) is
\beq
- \frac{3 \lambda_t^2}{4 \pi^2} m^{\prime 2} \log \frac{4 \pi f}{m^\prime}~.
\eeq
It was shown in ref. \cite{littlest} that $m^\prime > 2 f$,
so with our bound on $f$ we have $m^\prime > 8$ TeV, 
which for a 200 GeV Higgs implies a 
fine-tuning of 0.4\%.




The generic
reason for obtaining relatively strong constraints on the symmetry
breaking scale in this model is that weak isospin is violated. 
In the SM there is an $SU(2)$ global symmetry (called ``custodial'' $SU(2)$)
which protects the $\rho_*$ parameter from large corrections. These 
corrections in the SM can only come from custodial $SU(2)$ violating
interactions like hypercharge and Yukawa couplings. 
The importance of custodial $SU(2)$ was noted in the early literature 
on composite Higgs models~\cite{KaplanGeorgi}, and recently
emphasized again in~\cite{Sekhar}. 
However, in the littlest Higgs model there is no custodial symmetry
since the embedding of $SU(2)_L$ into $SO(5)$ breaks all non-Abelian
global symmetries.  This is because the embedding of $SU(2)_L$ into
$SO(5)$ is chosen to be ${\bf 5} \rightarrow {\bf 2} + {\bf 2} + {\bf 1}$.
Indeed, we find that the $\rho_*$ 
parameter in (\ref{rhostar}) gets corrections of order
$v^2/f^2$ independently of the mixing angles (and even in the absence of
a triplet VEV), and therefore this 
correction on its own would yield a constraint of $f\geq 3.6$ TeV from
the 95\% CL bound on the $T=\alpha^{-1}\rho_*< 0.75$ parameter
\cite{ErlerLang}, therefore
it is not hard to understand the bounds found above.
A composite Higgs model with a custodial $SU(2)$ was proposed by 
in ref.~\cite{KaplanGeorgiSU2}, however that model is not 
a little Higgs model since it has uncanceled one-loop divergences. 




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
\label{conclusions-sec}
\setcounter{equation}{0}
\setcounter{footnote}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We have calculated the electroweak precision constraints on the littlest 
Higgs model, incorporating corrections 
resulting from heavy gauge boson exchange, a triplet VEV, and nonlinear
$\sigma$ model interactions.  Using a 
global fit to 21 observables, we found that generically throughout 
the parameter space the smallest symmetry breaking scale consistent 
with present experimental measurements is well above 4 TeV
and for particular parameters the bound is 
$f > 4.0$ TeV at 95\% C.L\@, which implies that the Higgs mass
squared is tuned to 0.4\%.
This bound arises for a specific choice of the 
high energy gauge couplings, roughly $a=0.17$, $c = g/g_2 \sim 0.99$,
and $c' = g'/g'_2 \sim 0.3$.    The origin of the strong constraints on this
model is the absence of a custodial $SU(2)$ symmetry, leading to large
contributions to $\rho_* = \alpha T$, even in the absence of a
triplet Higgs VEV.  To the best of our knowledge,
no little Higgs model constructed to date has a custodial $SU(2)$
symmetry,
suggesting that similarly strong constraints are expected in
other little Higgs models.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgments}

G.D.K. and J.T. thank the particle theory group at Cornell University for 
a very pleasant visit where this work was completed.
We also thank Nima Arkani-Hamed, Sekhar Chivukula, 
Ann Nelson, Martin Schmaltz, Elizabeth Simmons, and Witold Skiba
for helpful discussions and the Aspen Center for Physics where this project
originated.
The research of C.C., J.H., and P.M. 
is supported in part by the NSF under grant
 and in part by the DOE OJI grant DE-FG02-01ER41206. 
The research of G.D.K. is supported by the US Department of Energy
under contract DE-FG02-95ER40896.
The research of J.T. is supported
by the US Department of Energy under contract W-7405-ENG-36.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Appendix A:  Predictions for  Electroweak Observables}
\renewcommand{\theequation}{A.\arabic{equation}}
\setcounter{equation}{0}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this appendix we give the predictions for the shifts in 
the electroweak precision observables due to new tree-level physics beyond the
SM in the $SU(5)/SO(5)$ littlest Higgs model.
The electroweak observables depend on three parameters,
$c,c'$ and $\Delta$.  Using the results given in~\cite{Sformulas,Burgess} 
we find the following results:

\begin{eqnarray}
\Gamma_Z &=& \left( \Gamma_Z \right)_{SM} \left[1 +\Delta (1.7-0.23 c^2-0.89 c^4-3.8 c'^2+0.83 c'^4) -5.4 \Delta' \right] 
 \nonumber \\
R_e &=& \left( R_e \right)_{SM} \left[1 +\Delta (0.34 -0.18 c^2+0.08 c^4-3.0 c'^2+4.6 c'^4) -1.1 \Delta' \right] 
 \nonumber \\
R_\mu &=& \left( R_\mu \right)_{SM} \left[1 +\Delta (0.34 -0.18 c^2+0.08 c^4-3.0c'^2+4.6 c'^4) -1.1 \Delta' \right]
 \nonumber \\
R_\tau &=& \left( R_\tau \right)_{SM} \left[1 +\Delta (0.34 -0.18 c^2+0.08 c^4-3.0c'^2+4.6 c'^4) -1.1 \Delta' \right]
 \nonumber \\
\sigma_h &=& \left( \sigma_h \right)_{SM} \left[1 +\Delta (-0.04 +0.02 c^2+0.01 c^4+0.31 c'^2-0.48 c'^4) +0.12 \Delta' \right]
 \nonumber \\
R_b &=& \left( R_b \right)_{SM} \left[ 1 +\Delta (-0.08+0.04 c^2-0.02 c^4+0.66 c'^2-1.0 c'^4) +0.24 \Delta' \right] 
 \nonumber \\ 
R_c &=& \left( R_c \right)_{SM} \left[1 +\Delta (0.15-0.08 c^2+0.04 c^4
-1.3 c'^2+1.9 c'^4) -0.47 \Delta' \right] 
 \nonumber \\
A_{FB}^e &=& \left( A_{FB}^e \right)_{SM}  +\Delta(0.73-0.38 c^2 +0.18 c^4
-6.4 c'^2+9.8 c'^4) -2.3 \Delta' \nonumber \\
A_{FB}^\mu &=& \left( A_{FB}^\mu \right)_{SM}  +\Delta(0.73-0.38 c^2 +0.18 c^4
-6.4 c'^2+9.8 c'^4)-2.3 \Delta' \nonumber \\
A_{FB}^\tau &=& \left( A_{FB}^\tau \right)_{SM}+\Delta(0.73-0.38 c^2 +0.18 c^4
-6.4 c'^2+9.8 c'^4)-2.3 \Delta' \nonumber \\
A_{\tau}(P_\tau) &=& \left( A_{\tau}(P_\tau) \right)_{SM} +\Delta (3.2 -1.7 c^2+0.78 c^4-28 c'^2+43 c'^4) -10 \Delta'
\nonumber \\
A_{e}(P_\tau) &=& \left( A_{e}(P_\tau) \right)_{SM} +\Delta (3.2 -1.7 c^2+0.78 c^4-28 c'^2+43 c'^4) -10 \Delta'
\nonumber \\
A_{FB}^b &=& \left( A_{FB}^b \right)_{SM} +\Delta (2.3 -1.2 c^2+0.54 c^4-20 c'^2+30 c'^4) -7.2 \Delta' \nonumber \\
A_{FB}^c &=& \left( A_{FB}^c \right)_{SM} +\Delta (1.8 -0.91 c^2+0.42 c^4
-15 c'^2+23 c'^4) -5.6 \Delta' \nonumber \\
A_{LR} &=& \left( A_{LR} \right)_{SM} +\Delta (3.2 -1.7 c^2+0.78 c^4-28 c'^2+43 c'^4) -10 \Delta' \nonumber \\
M_W &=& \left( M_W \right)_{SM} \left[1+\Delta(0.89-0.21c^2+0.21c^4-2.3 c'^2+1.1 c'^4)-2.9 \Delta' \right] \nonumber \\
g_L^2(\nu N \rightarrow \nu X) &=& 
\left( g_L^2(\nu N \rightarrow \nu X) \right)_{SM} +
\Delta(1.1 -0.16 c^2  + 0.25 c^4  -2.7 c'^2  + 
1.2 c'^4 ) \nonumber \\ && -3.4 \Delta'  \nonumber \\
g_R^2(\nu N \rightarrow \nu X) &=& 
\left( g_R^2(\nu N \rightarrow \nu X) \right)_{SM} 
+ \Delta (-0.032 + 0.055 c^2  - 0.085 c^4  +
0.92 c'^2  \nonumber \\ && - 0.42 c'^4 ) +0.10 \Delta'  \nonumber \\
g_{eV}(\nu e \rightarrow \nu e) &=& 
\left( g_{eV}(\nu e \rightarrow \nu e) \right)_{SM}+ 
\Delta (-0.87 + 0.43 c^2  - 0.66 c^4  +7.1 c'^2  - 
3.3 c'^4 )\nonumber \\ &&+2.8 \Delta'  \nonumber \\
g_{eA}(\nu e \rightarrow \nu e) &=& 
\left( g_{eA}(\nu e \rightarrow \nu e) \right)_{SM} -\frac{5\Delta}{8}
+2 \Delta' \nonumber \\
Q_W(Cs) &=& \left( Q_W(Cs) \right)_{SM}+ \Delta 
(-1.5 - 47 c^2  + 73 c^4  
-786 c'^2  + 363 c'^4)+4.7 \Delta' \nonumber \\
\end{eqnarray}
We also give in Table (\ref{table}) 
the experimental data \cite{ErlerLang,LEPEWG}
and the SM predictions used for our fit.
%\TABLE[htbp]{
\begin{table}[!htp]
\begin{center}
\begin{tabular}{|c|c|c|}\hline
Quantity & Experiment & SM($m_h=115$ GeV) \\ \hline 
$\Gamma_Z$ & 2.4952 $\pm$ 0.0023 & 2.4965 \\
$R_e$ & 20.804 $\pm$ 0.050 & 20.744 \\
$R_\mu$ & 20.785 $\pm$ 0.033 & 20.744 \\ 
$R_\tau$ & 20.764 $\pm$ 0.045 & 20.744 \\ 
$\sigma_h$ & 41.541 $\pm$ 0.037 & 41.480 \\ 
$R_b$ & 0.2165 $\pm$ 0.00065 & 0.2157  \\
$R_c$ & 0.1719 $\pm$ 0.0031 & 0.1723 \\
$A_{FB}^e$ & 0.0145 $\pm$ 0.0025 & 0.0163 \\
$A_{FB}^\mu$ & 0.0169 $\pm$ 0.0013 & 0.0163 \\
$A_{FB}^\tau$ & 0.0188 $\pm$ 0.0017 & 0.0163 \\
$A_{\tau}(P_\tau)$ & 0.1439 $\pm$ 0.0043 & 0.1475 \\ 
$A_{e}(P_\tau)$ & 0.1498 $\pm$ 0.0048 & 0.1475 \\
$A_{FB}^b$ & 0.0994 $\pm$ 0.0017 & 0.1034 \\
$A_{FB}^c$ & 0.0685 $\pm$ 0.0034 & 0.0739 \\
$A_{LR}$ & 0.1513 $\pm$ 0.0021 & 0.1475 \\
$M_W$ & 80.450 $\pm$ 0.034 & 80.389 \\
$g_L^2(\nu N \rightarrow \nu X)$ & 0.3020 $\pm$ 0.0019 & 0.3039 \\
$g_R^2(\nu N \rightarrow \nu X)$ & 0.0315 $\pm$ 0.0016 & 0.0301 \\
$g_{eA}(\nu e \rightarrow \nu e)$ & -0.507 $\pm$ 0.014 & -0.5065 \\
$g_{eV}(\nu e \rightarrow \nu e)$ & -0.040 $\pm$ 0.015 & -0.0397 \\
$Q_W(Cs)$ & -72.65 $\pm$ 0.44 & -73.11 \\ 
\hline
\end{tabular}
\end{center}
\caption{The experimental results~\cite{ErlerLang,LEPEWG}
and the SM predictions for the various
electroweak precision observables used for the fit. The SM predictions 
are for $m_h=115$ GeV and $\alpha_s=0.12$ and  
calculated~\cite{Erler} using GAPP~\cite{GAPP}.}
\label{table}
\end{table}
%}



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\end{document}











