\documentstyle[times,pramana,epsf,floats]{ias}
\begin{document}
\mark{{Littlest Higgs...}{P Poulose}}
\title{Littlest Higgs Model and W Pair Production at ILC}

\author{P. Poulose}
\address{Indian Institute of Technology Guwahati,
Guwahati - 781 039, India}
\keywords{EWSB, Littlest Higgs, W Polarization Fraction}
\pacs{}
\abstract{
Among the viable alternatives to the Standard Higgs Mechanism is the recently
proposed Little Higgs models.  The advantage here is that the model has an
elementary light neutral scalar particle, which arises dynamically as
against its ad hoc introduction in the Standard Model. The model also avoids 
hierarchy problem.  We have investigated the W pair production at ILC to
study the Littlest Higgs model using different observables. Specifically,
polarization fraction of W boson is expected to be measured very accurately
at ILC.  We use this to put limit on the scale parameter, f in the model.
}

\maketitle
\def \be{\begin{eqnarray}}
\def \ee{\end{eqnarray}}
\section{Introduction}

Study of electroweak symmetry breaking mechanism is one of the central 
problems of particle physics today. 
The SM Higgs mechanism is simple, but ad hoc-ly introduces an elementary
scalar particle in the theory.  Mass of 
the higgs particle is expected, from theoretical and experimental 
considerations, to be in the range of $10^2$ GeV.  But at one-loop level 
higgs mass square acquire quadratically diverging corrections; 
a difficulty known as hierarchy problem.
Models including supersymmetric extensions of the Standard Model, 
composite higgs models, strongly interacting EW scenario, higgs-less models, 
little higgs models, etc. suggest different ways to take care of the 
hierarchy problem.  Of these, we will be concerned here with the 
little higgs models \cite{lh:intro}.  
Unlike supersymmetric theories, little higgs theories
work on the gauge sector of the theory.  An appealing aspect of the scenario
is that the scalars are not introduced ad hoc-ly, and the symmetry breaking is 
generated dynamically.

Although, LHC is expected to investigate the 
electroweak symmetry breaking (EWSB) mechanism, it will require 
%will be visible at the upcoming Large Hadron Collider (LHC). But LHC 
%may not be able to distinguish between different models, which require
%detailed analysis of couplings, etc. The proposed International Linear
%Collider (ILC) will have a 
the clean environment of the proposed International Linear Collider (ILC) 
to study the details and, for example,
distinguish between different scenarios.
%
%perform detailed studies of various couplings, making it a suitable
%machine to differentiate different models. EWSB leaves its signature in 
%the gauge sector, for the massive gauge bosons get their longitudinal 
%components from the symmetry breaking sector. 
%ILC is expected to produce a large number of gauge boson pairs, 
%where precision measurements are going to scrutinize their couplings, 
%and other properties in detail.

In this write-up we will discuss W pair production at ILC
within the framework of one version
of the little higgs models, known as the littlest higgs models \cite{lsth}. 
In the next section we will 
introduce the littlest higgs model. In section 3 we will discuss 
the process $e^+e^-\rightarrow W^+W^-$ and the observables that are
sensitive to the littlest higgs model. 

\section{The Little Higgs Models}

The scenario is analogous to the description of {low energy
	hadronic interactions} by a non-linear realization of the chiral
	symmetry
	 {$SU(2)_L\times SU(2)_R$ broken down to 
	 $SU(2)_I$}\hskip 15mm at energy scale $f$.
	 {Pions} ($\pi$) are taken to be the Nambu Goldstone Bosons (NGB) 
	of the symmetry breaking, the their dynamics 
can be defined by a non-linear sigma model.

In {Little Higgs} models \cite{lh:intro}, 
similarly, we consider the non-linear realization of some global symmetry 
{$G$ broken down to $H$}. The Nambu-Goldstone Bosons (NGB) of the symmetry
breaking are candidate Higgs field.  In a specific model, called Littlest 
Higgs (LH) model\cite{lsth} $G\equiv SU(5)$ is broken down to 
$H\equiv SO(5)$. This leaves us $24-10=14$ NGB's.  
Interaction of NGB's are described by non-linear sigma model, which 
is an effective theory valid below the cut off
{$\Lambda \sim 4\pi\:f$}.
To identify some of these NGB's as Higgs particles we gauge a subgroup 
of $SU(5)$. In the original version of the littlest higgs model 
{$\left[SU_1(2)\times U_1(1)\right]\times 
\left[SU_2(2)\times U_2(1)\right]$} $\subset SU(5)$ is gauged, which 
is broken down to the Standard Model(SM) gauge group
{$SU(2)_L\times U(1)_Y$}.
Under the SM gauge group, the 14 NGB's transform as
$({\bf 1},0)+({\bf 3},0)+({\bf 2},\frac{1}{2})
+({\bf 3},1)$. 
SM gauge bosons, $\vec{W}_L^\mu$ and ${B}_L^\mu$  remain massless at 
this stage, while the gauge bosons belonging to the broken sector,
$\vec{W}_H^\mu$ and ${B}_H^\mu$, become massive after absorbing 
the singlet and the real triplet (with hypercharge, $Y=0$).
The doublet NGB field has all the correct quantum numbers to 
be identified as the standard higgs doublet. At tree level, they have only
derivative couplings. Quantum corrections at one-loop level will generate
Coleman-Weinberg potential with quadratic and quartic terms.
Gauge symmetry is constructed such that,
in the absence of any one (original) gauge interaction, 
higgs is massless at all orders. This also ensures that quadratically 
divergent contributions to the mass square term at one-loop level are 
canceled between the two gauge bosons from the two sectors. 
Logarithmically divergent terms contribute to the potential.
	In order to avoid quadratic divergence due to 
	top quark loop, a pair of (weak-singlet) Weyl quark 
	{ $U_L,\; U_R$} is introduced, which mix with the ordinary left-
and right- quarks to give the mass eigenstates.
Here again, it is arranged such that the quadratic divergence coming from
the standard top quark is canceled by its heavy counterpart, and the 
logarithmically diverging part adds to the Coleman-Weinberg potential.
The triplet left over NGB's 
also add logarithmically diverging quadratic terms to the potential.
 Presence of triplet in the loop also generates quartic terms in the potential.
The higgs potential thus generated 
breaks the { $SU(2)_L\times U(1)_Y$} symmetry spontaneously.
At the same time, higgs mass is protected from acquiring quadratically
divergent mass corrections at one loop.

Precision electroweak measurements constrain this model to have 
$f\:>\: 4\; TeV$, leaving the cut-off $\Lambda\: >\:12\;TeV$, thwarting 
the original motivation of solving hierarchy problem.
% This means, 
%to get a light higgs ($M_H\sim 200 \;GeV$) requires large fine
%tuning !
There are variations that avoid this difficulty; two among which 
are {\em (i)} Introduce T-Parity \cite{lsth:T}, 
{\em (ii)} Change the gauge sector \cite{lsth:G}.  
In the second approach, which we will be concerned here with 
the gauge group considered is { $SU_1(2)\times SU_2(2)\times U_Y(1)$}, 
which is broken down to the standard { $SU_L(2)\times U_Y(1)$}. 
Situation is similar to the earlier case, but without $B_H$. 
%This idea is motivated by the 
%fact that in the earlier case, mass of the heavy $U(1)$ gauge boson 
%is given by
%$M(B_H)\sim 0.16\:f$. Inorder to have heavy enough $B_H$ to avoid large 
%corrections to precision electroweak observables, we needed large $f$.
In this model we have
three heavy gauge bosons; $W_H^\pm$ and $Z_H$, 
in addition to the standard $W^\pm,\;Z$ and $A$. Masses of the new
gauge bosons are given by 
\begin{equation}
M_{W_H}=M_{Z_H}=\frac{g}{\sin 2\theta}\;f,
\end{equation}
where 
{$g = \frac{g_1g_2}{\sqrt{g_1^2+g_2^2}}$} is the  standard $SU(2)_L$ coupling 
and { $\theta$} is the mixing angle between $SU_1(2)$ and $SU_2(2)$. 
Electroweak precision measurements allow $ f>1\;\;TeV$ with 
$\cos\theta\sim 1/3 $. 
Some of the phenomenological studies of little higgs models in the context
of LHC and ILC are listed in reference \cite{lh:pheno}.

\section{The Process: $e^+e^-\rightarrow W^+W^-$}
We will now consider the effect of this scenario in { $W$} pair production at a 
high energy linear {  $e^+e^-$} collider.
Apart from the standard channels, this process gets contribution from 
a $Z_H$ mediated $s$-channel. Along with this, there could also be 
differences between the SM predictions and the littlest higgs
model predictions through changed couplings.
The SM gauge couplings ($g_{WW\gamma},\;\;g_{WWZ}$) 
and the $W$ couplings to the fermions are unchanged, but the
fermionic couplings of the standard model $Z$ boson pick up an 
additional contribution of the order of $v^2/f^2$. The vector and axial 
vector couplings of $Z$ to the electrons are given respectively by
\begin{equation}
c^v_{eeZ}=\frac{g}{2 c_W}\:\left[
        { \left(-\frac{1}{2}+2x_W\right)}+\frac{v^2}{f^2}\:
        \frac{\sin 4\theta}{8}\right]
\end{equation}
\begin{equation}
c^a_{eeZ}=\frac{g}{2 c_W}\:\left[{  \frac{1}{2}}-\frac{v^2}{f^2}\;
        \frac{\sin 4\theta}{16}\cot\theta\right].
\end{equation}
Couplings of the heavy $Z_H$ with $W$ and electrons are given by
\begin{equation}
g_{WWZ_H}=\frac{g v^2}{8f^2}\:\sin 4\theta,
\end{equation}
\begin{equation}
c^v_{eeZ_H}=\frac{-g}{4}\:\cot\theta, \;\;\;\;
c^a_{eeZ_H}=\frac{g}{4}\:\cot\theta 
\end{equation}
We will now present our results. In the first place, we study the 
deviation of total cross section from the SM value.
In figure \ref{fig:sigma} (left) we plot 
the total cross section against the centre of mass energy. 
In order to get an estimate of how far we can probe the scale $f$ at ILC,
we consider two c.m. energy values, 500 GeV and 800 GeV, and assume 
a luminosity of 1 ab$^-{1}$. At
1$ \sigma$ level we can probe $f$ up to 6 TeV at both energies.
\begin{figure}[htbp]
\epsfysize=7cm \epsfxsize=7cm
\centerline{\epsfbox{
/home/poulose/Talks/June2006/figure/Fin-sg_roots.ps} 
\epsfysize=7cm \epsfxsize=7cm
\epsfbox{
/home/poulose/Talks/June2006/figure/Fin-f_roots.ps} 
}
\caption{Unpolarized cross section (Left) and 
Polarization fractions (Right) against centre of mass energy. 
Solid line corresponds 
to SM, while the dashed line corresponds to the Littlest Higgs model. 
}
\label{fig:sigma}
\end{figure}
LEP has measured the fractional cross section of the polarized 
$W$'s very precisely \cite{WPfrac}. 
AT ILC this precision is expected to be even better. Defining the
polarization fractions as
\begin{equation}
f^0=\frac{\sigma(e^+e^-\rightarrow W^+W^-_L)}{\sigma_{unpol}},\;\;\;\;
f^T=\frac{\sigma(e^+e^-\rightarrow W^+W^-_T)}{\sigma_{unpol}}
\end{equation}
where $L$ refers to the longitudinal polarization and 
$T=\pm$ refers to the transverse polarizations, we plot them in 
figure \ref{fig:sigma} (right). We find that the longitudinal fraction is
changed from 3.8\% to 4.4\%, and from 1.9\% to 4.0\% at centre of mass
energies 500 GeV and 800 GeV respectively.

%\begin{figure}[htbp]
%\epsfysize=7cm \epsfxsize=8cm
%\centerline{\epsfbox{
%/home/poulose/Talks/June2006/figure/Fin-f_roots.ps}}
%\caption{Cross section agains centre of mass energy. Solid line corresponds 
%to SM, while the dashed line corresponds to the Littlest Higgs model. 
%}
%\label{fig:fracs}
%\end{figure}

It is clear that beam polarization can be used to switch off the 
dominant $t$-channel production to which LH model does not add any 
new contribution.  This will enhance the sensitivity considerably. 
One could also consider different observable, including
forward-backward asymmetry to probe the fermionic couplings of the gauge 
bosons.  Work on this is in progress.

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\end{thebibliography}
\end{document}

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