The Twin Higgs: Natural Electroweak Breaking from Mirror Symmetry
Z. Chacko
We present `twin Higgs models', simple realizations of the Higgs as a pseudo-Goldstone boson that protect the weak scale from radiative corrections up to scales of order 5 - 10 TeV. In the ultra-violet these theories have a discrete symmetry which interchanges each Standard Model particle with a corresponding particle which transforms under a twin or mirror Standard Model gauge group. In addition, the Higgs sector respects an approximate global SU(4) symmetry. When this global symmetry is broken, the discrete symmetry tightly constrains the form of corrections to the pseudo-Goldstone Higgs potential, allowing natural electroweak symmetry breaking. Precision electroweak constraints are satisfied by construction. These models demonstrate that, contrary to the conventional wisdom, stabilizing the weak scale does not require new light particles charged under the Standard Model gauge groups. %We present a simple model, the `twin Higgs model', based on mirror symmetry %where the Higgs is a pseudo-Goldstone boson and electroweak symmetry is %naturally broken. In the ultra-violet the theory has a global $SU(4)$ %symmetry. This symmetry is not exact - an $SU(2) \times SU(2)$ subgroup of %SU(4) is gauged. One of these $SU(2)$ gauge groups is to be identified with %SU(2$)_L$ of the Standard Model. The global symmetry is also softly broken %by the sector which generates the top quark Yukawa coupling, and does not %commute with hypercharge. However all these interactions respect a discrete %$Z_2$ mirror symmetry, which we label `twin parity', under which the two %SU(2) groups are interchanged. When global $SU(4)$ is broken to $SU(3)$ twin %parity tightly constrains the form of radiative corrections to the %pseudo-Goldstone Higgs potential, allowing natural electroweak symmetry %breaking. An immediate consequence of twin parity is the existence of a %mirror Standard Model with exactly the same particle content and gauge %groups, with every Standard Model particle having a `twin partner'. %Although the the smaller Yukawa couplings need not respect twin parity, %naturalness constrains the masses of most of the twin partners to be less %than a few hundred GeV. Potential mixing between the photon and its twin %imply that the twin partners may have tiny fractional electric charges and %dipole moments. Precision electroweak constraints are easily evaded. This %model demonstrates that it is possible to remove quadratic divergences from %the Higgs mass even if there is no new state charged under the Standard %Model gauge group with mass below 5 TeV.
hep-ph0506256.tex
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\begin{document}

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\pagestyle{plain}

\title{The Twin Higgs: Natural Electroweak Breaking from Mirror Symmetry}

\author{Z. Chacko}
\author{Hock-Seng Goh}
\affiliation{Department of Physics, University of Arizona, Tucson, AZ 85721}
\author{Roni Harnik}
\affiliation{
Department of Physics, University of California,
Berkeley, CA 94720 \\
Theoretical Physics Group,
Lawrence Berkeley National Laboratory, Berkeley, CA 94720}


%\date{\today}

\begin{abstract}

We present `twin Higgs models', simple realizations of the Higgs as a
pseudo-Goldstone boson that protect the weak scale from radiative corrections up to
scales of order 5 - 10 TeV. In the ultra-violet these theories have a discrete
symmetry which interchanges each Standard Model particle with a corresponding
particle which transforms under a twin or mirror Standard Model gauge group.  In
addition, the Higgs sector respects an approximate global SU(4) symmetry.  When this
global symmetry is broken, the discrete symmetry tightly constrains the form of
corrections to the pseudo-Goldstone Higgs potential, allowing natural electroweak
symmetry breaking.  Precision electroweak constraints are satisfied by construction.
These models demonstrate that, contrary to the conventional wisdom, stabilizing the
weak scale does not require new light particles charged under the Standard Model
gauge groups.

%We present a simple model, the `twin Higgs model', based on mirror symmetry
%where the Higgs is a pseudo-Goldstone boson and electroweak symmetry is
%naturally broken. In the ultra-violet the theory has a global $SU(4)$
%symmetry. This symmetry is not exact - an $SU(2) \times SU(2)$ subgroup of
%SU(4) is gauged. One of these $SU(2)$ gauge groups is to be identified with
%SU(2$)_L$ of the Standard Model. The global symmetry is also softly broken
%by the sector which generates the top quark Yukawa coupling, and does not
%commute with hypercharge.  However all these interactions respect a discrete
%$Z_2$ mirror symmetry, which we label `twin parity', under which the two
%SU(2) groups are interchanged. When global $SU(4)$ is broken to $SU(3)$ twin
%parity tightly constrains the form of radiative corrections to the
%pseudo-Goldstone Higgs potential, allowing natural electroweak symmetry
%breaking. An immediate consequence of twin parity is the existence of a
%mirror Standard Model with exactly the same particle content and gauge
%groups, with every Standard Model particle having a `twin partner'.
%Although the the smaller Yukawa couplings need not respect twin parity,
%naturalness constrains the masses of most of the twin partners to be less
%than a few hundred GeV. Potential mixing between the photon and its twin
%imply that the twin partners may have tiny fractional electric charges and
%dipole moments. Precision electroweak constraints are easily evaded.  This
%model demonstrates that it is possible to remove quadratic divergences from
%the Higgs mass even if there is no new state charged under the Standard
%Model gauge group with mass below 5 TeV.


\end{abstract}

\pacs{} \maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

In the Standard Model (SM) the weak scale is unstable under quantum corrections. This
suggests the existence of new physics at or close to a TeV that protects the Higgs
mass parameter of the SM against radiative corrections. While the exact form that
such new physics takes is unknown there are several interesting alternatives. One
possibility, first proposed in~\cite{GP, KG} is that the Higgs is naturally light
because it is the pseudo-Goldstone boson of an approximate global symmetry. This idea
has recently experienced a revival in the form of little Higgs
theories~{\cite{Little1, Little2}} (for a clear review and more references see
{\cite{Review}})  that protect the Higgs mass from radiative corrections up to scales
of order 5 - 10 TeV.

%The underlying concept behind
%little Higgs theories is the idea of `collective symmetry breaking' - the
%global symmetry is broken only when two or more couplings in the Lagrangian
%are non-vanishing.  This is a significant restriction on the form of the
%quantum corrections to the Higgs mass parameter, and can potentially give
%rise to natural electroweak symmetry breaking.

In this paper we propose a class of simple alternative realizations of the Higgs as a
pseudo-Goldstone boson that also protect the weak scale from radiative corrections up
to scales of order 5 - 10 TeV.
%%%%%%%%%%%%%
%We introduce another twin copy of the standard model.
%In the ultra-violet these theories have a discrete
%Z$_2$ symmetry which interchanges each Standard Model particle its corresponding 
%twin
%particle. In addition, the Higgs sector of the theory respects an approximate global 
%$SU(4)$ symmetry.
%%%%%%%%%%%%
In the ultra-violet these theories have a discrete
Z$_2$ symmetry which interchanges each Standard Model particle with a corresponding
particle which transforms under a twin or mirror Standard Model gauge group. In
addition, the
Higgs sector of the theory respects an approximate global $SU(4)$ symmetry.
%%%%%%%%%%%%
Although the weak and electromagnetic interactions, as well as the top Yukawa
coupling, violate the global symmetry they all respect the discrete interchange
symmetry.  When $SU(4)$ is broken to $SU(3)$, the discrete symmetry tightly
constrains the form of corrections to the pseudo-Goldstone Higgs potential, allowing
natural electroweak symmetry breaking.

Although the smaller Yukawa couplings need not respect the discrete symmetry,
naturalness constrains the masses of most of the twin/mirror partners not to exceed a
few hundred GeV.  Precision electroweak constraints are satisfied by construction,
since although these new particles may be very light, they do not transform under the
SM gauge groups. This is in contrast to little Higgs theories where these constraints
are often a severe problem {\cite{Csaba}}.

%This does not preclude the photon and the
%mirror photon from having a very small mixing, and therefore the mirror particles
%may
%have tiny fractional electric charges.

We illustrate the basic idea by way of a simple example where the global
symmetry is realized linearly.  Consider a complex scalar field, $H$, that
transforms as a fundamental under a global $SU(4)$ symmetry. The potential
for this field is given by
\begin{equation}
    V(H)= -m^2 H^\dagger H + {\lambda}{}(H^\dagger H)^2\,.
\end{equation}
Since the mass squared of $H$ is negative it will develop a VEV, $\langle |H|
\rangle = m/\sqrt{2\lambda}\equiv f$, that breaks $SU(4)\to SU(3)$ yielding 7
massless Nambu-Goldstone bosons.  We now break the $SU(4)$ explicitly by
gauging an $SU(2)_A\times SU(2)_B$ subgroup. The field $H$ transfoms
as~$\left({ H_A , H_B} \right)$ where $H_A$ is a doublet under $SU(2)_A$ and
$H_B$ is a doublet under $SU(2)_B$. At the end of the day we will identify
$SU(2)_A$ with $SU(2)_L$ of the SM.
Since $SU(4)$ is now broken explicitly, the would-be Goldstones
pick up a mass that is proportional to the explicit breaking.
Specifically, gauge loops contribute a quadratically divergent mass to the
components of $H$ as
\begin{equation}
    \Delta V= \frac{9 g_A^2 \Lambda^2}{64\pi^2}H_A^\dagger H_A
        + \frac{9 g_B^2 \Lambda^2}{64\pi^2}H_B^\dagger H_B +\ldots\,,
\end{equation}
a loop factor below the cutoff $\Lambda$ of the theory.  The
mechanism in our model hinges on the following simple observation. Suppose we
now impose an additional $Z_2$ symmetry, which we label `twin parity', which
interchanges $H_A$ and $H_B$ and also interchanges the gauge bosons of
$SU(2)_A$ with those of $SU(2)_B$.
%\begin{equation}
%   H_A\longleftrightarrow H_B\, \;\;\;\;
%        SU(2)_A\longleftrightarrow SU(2)_B\,.
%\end{equation}
This symmetry forces the two gauge couplings to be equal, $g_A=g_B\equiv g$.
The gauge contribution to the mass of $H$ is now
\begin{equation}
    \Delta V= \frac{9 g^2 \Lambda^2}{64\pi^2}
    (H_A^\dagger H_A+H_B^\dagger H_B)=
    \frac{9 g^2 \Lambda^2}{64\pi^2} H^\dagger H
\end{equation}
which is invariant under $SU(4)$ and therefore does not contribute a mass to the
Goldstones.  In other words, imposing twin parity constrains the
quadratically divergent mass terms to have an $SU(4)$ invariant form.
The Goldstones are therefore completely insensitive to quadratic divergences
from gauge loops.

Gauge loops will however contribute a logarithmically divergent term to the potential
that is not $SU(4)$ symmetric and has the general form $ \kappa \left(|H_A|^4 +
|H_B|^4 \right)$ where $\kappa$ is of order $g^4/16 \pi^2 {\rm log} \left(\Lambda/ gf
\right)$. Provided $\Lambda$ is not very much larger than $f$ this leads to the
would-be Goldstones acquiring a mass of order $g^2 f/4 \pi$ which is of order the
weak scale for $f$ of order a TeV.  Notice that we could have obtained exactly the
same result by imposing `mirror parity' - invariance under $t \rightarrow t$,
$\vec{x} \rightarrow - \vec{x}$ along with the interchange of every particle in
sector A with its CP conjugate in B.

%In particular, examining the the
%Coleman-Weinberg potential one finds that an $SU(4)$-breaking quartic is generated
%\begin{eqnarray}
%\label{gaugequartic}
%   \Delta V_{CW}&=&
%   \frac{3g^2}{128\pi^2} (H_A^\dagger H_A)^2
%   \left(\log\frac{M_{W_A}^2}{\Lambda^2} -\frac{3}{2}\right) \\
%   &+& \frac{3g^2}{128\pi^2}  (H_B^\dagger H_B)^2
%   \left(\log\frac{M_{W_B}^2}{\Lambda^2} -\frac{3}{2}\right)+ \ldots
%   \nonumber
%   \end{eqnarray}
%where $W_A$ and $W_B$ are the gauge bosons of the $A$ and $B$ groups.
%This approach to solving the little hierarchy problem an immediate
%advantage. Because the symmetry that protects the Higgs is actually
%discrete there are no new gauge bosons that mix with the SM gauge bosons at
%tree level, as EW precision tests strongly suggest.

This approach to stabilizing the weak scale against quantum corrections from gauge
loops can be generalized to include all the other interactions in the SM. To do this,
we gauge two copies of the SM, A and B, with our SM being SM$_{\rm A}$.
We can then extend the discrete symmetry in either of the two ways below.
\begin{itemize}
\item
Interchange every SM$_{\rm A}$ particle with the corresponding particle in 
SM$_{\rm B}$.
\item
Impose $t \rightarrow t$, $\vec{x} \rightarrow - \vec{x}$ along with the interchange 
of every SM$_{\rm A}$ particle with its CP conjugate in SM$_{\rm B}$.
\end{itemize}
These symmetries, while similar, are distinct. Each one relates the gauge and Yukawa
interactions in the A sector to those in the B sector. While the former is a simple
generalization of twin parity which we label `twin symmetry', the latter extends
mirror parity to the familiar mirror symmetry {\cite{mirror}}.
Either choice of the discrete symmetry ensures that any quadratically divergent
contribution to the Higgs mass has a form $\propto \Lambda^2 (|H_A|^2 + |H_B|^2)$
which is harmless due to its accidental $SU(4)$ symmetry. Although quantum
corrections to the quartic are in general not $SU(4)$ invariant, once again these
only lead to logarithmically divergent contributions to the mass of the
pseudo-Goldstone Higgs field, allowing for a natural hierarchy between $f$ and the
weak scale.

In both the twin and mirror symmetric cases the only renormalizable interactions
between the SM sector and the hidden sector allowed by gauge invariance are the Higgs
quartic, which is assumed to have an $SU(4)$ invariant form at the cut-off $\Lambda$,
and a mixing term between the hypercharge gauge boson and its partner, which we
neglect for the present discussion and will return to later.

%For example, all one needs to do to cancel the quadratic divergences from the
%top is to introduce a $Z_2$ twin for the top and couple it to the twin Higgs
%such that $Z_2$ is respected.  This way, every quadratically divergent
%contribution to an $|H_A|^2$ mass will be complemented by an equal
%contribution to $|H_B|^2$ in the twin sector such that the quadratically
%sensitive mass only appears in the $SU(4)$ invariant combination
%$|H_A|^2+|H_B|^2$ that does not involve the Goldstones.  In this case the top
%loops from both sectors will introduce an $SU(4)$ breaking quartic in the
%spirit of the gauge contribution in Equation~(\ref{gaugequartic}).

At one loop the largest contribution to the pseudo-Goldstone Higgs potential arises
from the top Yukawa coupling, and is logarithmically sensitive to the cutoff.
%giving a logarthmically divergent correction
%to the mass which is of
%order $y_t f /4\pi$ when $\Lambda$ is not far above $f$.
%As we shall see, twin symmetry alone is not quite
%enough to stabilize the weak scale against radiative corrections up to 10 TeV
%in the presence of this effect, which is enhanced by group theory factors.
%However, one can improve on this
However, in the twin symmetric case it is straightforward to make this
contribution finite.
One possible approach is to enlarge the approximate global symmetry of the
top Yukawa coupling to $SU(6)\times SU(4) \times U(1)$ with the $(SU(3)_c\times
SU(2)\times U(1))_{A,B}$ subgroups being gauged.
%Twin symmetry is then a discrete subgroup
%of the larger global symmetry.
We do this by introducing the following chiral fermions
\begin{eqnarray}
    Q_L&=&(\mathbf{6}, \mathbf{\bar 4}) \nonumber\\
    &=& (\mathbf{3}, \mathbf{2} ;\mathbf{1}, \mathbf{1} ) +
    (\mathbf{1}, \mathbf{1} ;\mathbf{3}, \mathbf{2} ) +
    (\mathbf{3}, \mathbf{1} ;\mathbf{1}, \mathbf{2} ) +
    (\mathbf{1}, \mathbf{2} ;\mathbf{3}, \mathbf{1} ) \nonumber\\
    &\equiv& q_A + q_B + \tilde q_A + \tilde q_B\nonumber\\
    T_R&=&(\mathbf{\bar 6}, \mathbf{1})\nonumber\\
    &=&(\mathbf{\bar 3}, \mathbf{1} ;\mathbf{1}, \mathbf{1} ) +
    (\mathbf{1}, \mathbf{1} ;\mathbf{\bar 3}, \mathbf{1} )
    \equiv t_A + t_B
\end{eqnarray}
which transform as shown
under $SU(6)\times SU(4)$ and under $\left[SU(3)\times SU(2)\right]^2$,
where we have suppressed the hypercharge quantum numbers.  One can then write
an $SU(4)$ invariant Yukawa coupling
\begin{equation}
\label{topmodule}
    y H Q_L T_R + \mbox{h.c.}
\end{equation}
The $SU(4)$ symmetric matter content contains exotic left handed quarks,
$\tilde q_{A,B}$ that are charged under color of one sector and the weak
group of the twin, and vice versa. We introduce additional fermions with
opposite charge assignment, $\tilde q^c_{A,B}$ with which the exotic quarks
can get a~$Z_2$ symmetric mass
$M\left( \tilde q_A^c \tilde q_A +\tilde q_B^c \tilde q_B \right)\,. $
The mass parameter $M$ is the only source of $SU(4)$ breaking in the top
sector, and it only breaks this symmetry softly.
%This has the benefit of
%completely removing the sensitivity of the potential for the Goldstone to the
%cutoff.
The top contribution to the Higgs potential in this model will then be
finite at one loop.

%The effective potential given to the Goldstones from the top and gauge
%sectors is not sufficient to yield correct electroweak symmetry breaking. We
%must thus assume that the scalar's self-interaction also violates $SU(4)$ to
%some extent. We thus add a scalar quartic
%\begin{equation}
%   \label{quartic}
%   V\supset \lambda' \left(H_A^\dagger H_A\right)^2\,.
%\end{equation}
%This interaction has the benefit that it does not give the Higgs a mass at
%tree level, however, it does violate $Z_2$ and will thus introduce a
%quadratic divergence. We will therefore require $\lambda'$ to be somewhat
%small. One could have added various other quartic operators, but we found
%that the quartic given in equation~(\ref{quartic}) is phenomenologically
%preferred. Choosing this quartic rather than another is, however, technically
%natural.

%This is because the beta function of the dimensionless quartic cannot depend
%on the dimensionfull soft mass, and thus only involves $SU(4)$ preserving
%physics. The contribution of the top sector to the Goldstone mass will thus
%be finite\footnote{It is interesting to note that the cutoff insensitivity is
%lost if the soft masses do not respect $Z_2$. In this case a logarithmically
%divergent mass squared is generated in the linear model.}, as will be shown
%explicitly in the next section.




%Implementing this mechanism in the Higgs interaction sector requires some
%care. Any potential for the golds tones will have to be a function of an
%$SU(4)$ breaking operator. The only such operator is $|H_A|^2-|H_B|^2$, since
%any other gauge invariant operator can be expressed in terms of this operator
%and the $SU(4)$ invariant $|H_A|^2+|H_B|^2$. Furthermore, the operator above
%is odd under $Z_2$ and thus the lowest order operator we can write down is
%\begin{equation}
%   V\supset \lambda' \left(|H_A|^2-|H_B|^2\right)^2
%\end{equation}
%As is the case in the gauge and top sectors, the quadratic divergences
%generated by these operators do not affect the Goldstone because of the $Z_@$
%symmetry.  This operator will an $O(1)$ generate a quartic for the Goldstone
%but, as we will see in the next section,
%it will also generate a finite contribution to the Goldstone mass
%that is unacceptably large.
%However, we can also add a soft $Z_2$ breaking term to the potential
%\begin{equation}
%   \mu (|H_A|^2-|H_B|^2)
%\end{equation}
%This will also contribute to the mass of the Goldstone at the same order as
%the quartic term and the two contributions can cancel partially. This
%situation is similar to \cite{simplest} where a soft global symmetry breaking
%parameter (for a $U(1)$ in that case) was added to cancel a contribution to
%the Higgs mass that is too large. The parameter $\mu$ is radiatively stable
%since it is the only $Z_2$ breaking parameter. Furthermore, since it is soft,
%it does not introduce quadratic divergences to the Higgs potential.

%In the rest of this letter we will elaborate a more concrete model and show
%how such a model is realized non-linearly. blah blah.

\section{A Non-Linear Realization}\label{nonlinear}

We now construct a realistic twin symmetric model that implements these symmetries
non-linearly.
The linear model we have been working with should be considered merely one
possibility for a UV-completion of the non-linear one, and others may well
exist.
%The construction of such a theory is well known and is reviewed
%in~\cite{Schmaltz:2005ky}, for example.
The pseudo-Goldstone fields of the non-linear model are those which survive
after integrating out the radial mode of the field $H$ in the linear model.
We parametrize these degrees of freedom as
\begin{eqnarray}
\label{pseudo}
H = \exp(\frac{i}{f} {h}^a t^a) \pmatrix{ 0 \cr 0 \cr 0 \cr f }
\equiv \pmatrix{ 0 \cr 0 \cr 0 \cr f }
+ i \pmatrix{ h^1 \cr h^2 \cr h^3 \cr h^0 } + \ldots
\end{eqnarray}
where $h^{1,\ldots,3}$ are complex and $h^0$ is real.  In general the
effective theory for these fields will contain all of the operators allowed
by the non-linearly realized $SU(4)$ symmetry, suppressed by the cutoff scale
$\Lambda$. Assuming the theory is strongly coupled at the cutoff we can
estimate $\Lambda\sim 4 \pi f$.  However, any potential for the
pseudo-Goldstone fields can only emerge from those interactions which violate
the global $SU(4)$ symmetry, specifically their gauge and Yukawa couplings. In
particular the electroweak gauge interactions and the top Yukawa contribute
the most to the pseudo-Goldstone potential and must therefore be studied in
detail. We will thus calculate the contributions to the one loop
Coleman-Weinberg (CW) potential~\cite{CW} from these couplings. At one loop
the gauge and top sectors contribute separately, simplifying the
calculation.

As before, we gauge two copies of the SM, A and B.
The vev $f$
breaks $SU(2)_B\times U(1)_B$ down to a single $U(1)$, giving $W_B$ and $Z_B$
masses of order $gf$.
The $SU(2)_A$ doublet $h^T\equiv (h^1, h^2)$ is left uneaten and
is identified as the SM Higgs. The couplings of the pseudo-Goldstone
fields to the $SU(2) \times U(1)$ gauge fields and their mirror partners are
given by expanding out $H = (H_A, H_B)$ in terms of the pseudo-Goldstones as
given by eq. ({\ref{pseudo}}) in the interaction
\begin{equation}
\left|\left(\partial_\mu + ig W_{\mu, A}
+ \frac{i}{2} g' B_{\mu, A}\right)H_A \right|^2
+ \left( A \rightarrow B \right)\,.
\end{equation}
A simple way of calculating the effective potential is to calculate the
vacuum energy as a function of the field dependent masses of all of the
fields in the theory.
In the absence of
quadratic divergences this leads to the formula
\begin{equation}
    V_{CW} = \pm \frac{1}{64\pi^2} \sum_i {M_i^4}
    \left(\log\frac{\Lambda^2}{M_i^2} + \frac{3}{2}\right)
\end{equation}
where the sum is over all degrees of freedom, the sign being negative for
bosons and positive for fermions. In evaluating this sum the higher order
terms in the expansion eq. ({\ref{pseudo}}) are often useful, giving
\begin{eqnarray}
        \label{expand}
        H_A^\dagger H_A &=& h^\dagger h - \frac{(h^\dagger h)^2}{3f^2} +\ldots
        \nonumber\\
        H_B^\dagger H_B &=& f^2 - h^\dagger h + \frac{(h^\dagger h)^2}{3f^2}
        -\ldots
\end{eqnarray}
This expansion manifests the fact that the $Z_2$ invariant $(|H_A|^2 +
|H_B|^2)$ is independent of the Higgs.
%One can also conveniently re-sum these
%two operators to all orders in $h$ as a sine squared and a cosine squared.
Writing the Higgs potential in the form
\begin{equation}
\label{potential}
        V(h)= m_h^2 h^\dagger h + \lambda_h (h^\dagger h)^2 + \ldots
\end{equation}
we find that the contribution to the Higgs mass term from the gauge sector is
%\begin{eqnarray}
%        m_h^2=
%        &-& \frac{6g^4 f^2}{128\pi^2}
%   \left( \log\frac{g^2 f^2}{2 \Lambda^2} -\frac{1}{2}\right) \\
%        &-& \frac{3(g^2+g'^2)^2 f^2}{128\pi^2}
%   \left( \log\frac{(g^2 + g'^2) f^2}{2 \Lambda^2} -\frac{1}{2}\right)\,.
%   \nonumber
%\end{eqnarray}
\begin{eqnarray}
\label{gauge}
        m_h^2
    &=& \frac{6 g^2 M^2_{W_B}}{64\pi^2 }
    \left( \log\frac{ \Lambda^2} {M^2_{W_B}} + 1 \right) \\
    &+& \frac{3 (g^2+g'^2) M^2_{Z_B}}{64\pi^2 }
    \left( \log\frac{ \Lambda^2} {M^2_{Z_B}} + 1 \right) \,
    \nonumber
\end{eqnarray}
where
$M_{W_B}^2= g^2 f^2/2$ and $M_{Z_B}^2= (g^2+g'^2)f^2/2 \,$.
Eq.~(\ref{gauge}) holds if electromagnetism in the twin sector is an unbroken
gauge symmetry as in the SM. However it is also possible that QED
in the twin sector is a broken symmetry and that the twin photon has a
mass.  This could arise if, for example, the hypercharge gauge boson in the
twin sector has a mass $M_{B}$ which softly breaks the twin symmetry.
We do not specify a dynamical origin for this mass since it is technically
natural for the dynamics which generate it to lie at scales above the
cutoff $\Lambda$. In the limit that $M^2_B \gg g'^2 f^2$ the second term in
eq.~({\ref{gauge}}) becomes approximately
\begin{equation}
% \left[
 \frac{3
  g^2 M^2_{W_B}}{64\pi^2 }
        \left( \log\frac{ \Lambda^2} {M^2_{W_B}} +1 \right) +
        \frac{3 g'^2 M^2_{B}}{64 \pi^2}
        \left( \log\frac{\Lambda^2}{M_{B}^2} +1 \right)
% \right]
\end{equation}
The contribution to the Higgs quartic from this sector is small and can be neglected.

We now turn to the top sector. The couplings of the pseudo-Goldstone fields
to the top quark are obtained by expanding out $H$ as in eq. ({\ref{pseudo}})
in the $SU(4) \times SU(6)$ invariant interaction ($y H Q_L T_R +
\mbox{h.c.}$) of eq.  ({\ref{topmodule}}). The $h$ dependent masses of the
fields in the top sector are determined from this and from the $SU(4)$ breaking
mass term, and can be expressed as
\begin{eqnarray}
       m^2_{t_A}=\frac{y^2 M^2}{M^2 +y^2f^2} h^\dagger h &\qquad&
       m^2_{T_A}= M^2 +y^2f^2 \nonumber\\
       m^2_{t_B}= y^2 f^2 \qquad\qquad &\qquad&
       m^2_{T_B}= M^2
\end{eqnarray}
to leading order in $|h|^2$, where we have asumed for simplicity that $y$ is real.
This leads to the following contributions to the Higgs potential of
eq.~({\ref{potential}}).
\begin{eqnarray}
        m_h^2&=&
        \frac{3}{8\pi^2}\frac{y^2 M^2}{M^2-y^2f^2}
%       &\times&
        \left( M^2\log\frac{m_{T_A}^2}{m^2_{T_B}}
        -y^2f^2 \log\frac{m^2_{T_A}}{m^2_{t_B}}\right)\,,
        \nonumber\\
        \lambda_h &=& -\frac{m_h^2}{3f^2} +
        \frac{3}{16 \pi^2}\frac{y^4 M^4}{(M^2+y^2f^2)^2}
        \log\frac{m^2_{T_A}}{m^2_{t_A}} \nonumber\\
        &+& \frac{3}{16\pi^2}
        \frac{y^4M^4 (M^2+y^2f^2)}{(M^2-y^2f^2)^3}
        \log\frac{m^2_{T_B}}{m^2_{t_B}} \nonumber\\
        &-& \frac{3}{32\pi^2}
        \left[\frac{4 y^4 M^4}{(M^2-y^2f^2)^2}
        +\frac{y^4 M^4}{(M^2+y^2f^2)^2}\right]
\end{eqnarray}

In order to get the correct pattern of electroweak symmetry breaking we also add to
the theory a `$\mu$ term' that softly breaks the discrete $Z_2$ twin symmetry. This
term takes the form $\mu^2 H_A^\dagger H_A$ and contributes to $m_h^2$ and
$\lambda_h$ as dictated by eq. ({\ref{expand}}). In addition, since the smaller
Yukawa couplings do not contribute significantly to the Higgs potential, we do not
require them to respect the discrete symmetry. In this non-linear model, the absence
of quadratically divergent contributions to the Higgs mass can be understood as a
consequence of cancellations between the familiar SM loop corrections and new loop
corrections that arise from the (mostly non-renormalizable) couplings of the Higgs to
the twin sector.

%This may be reproduced in the linear model of the previous section if one naively
%take $m^2\sim\Lambda^2$ and $\lambda~(4\pi)^2$.

%For example, the field dependent
%mass matrix in our top sector is
%\begin{equation}
%   \label{topmass}
%   \begin{array}{c}
%       \pmatrix{ \bar q_A &  \bar{\tilde q}_A }
%   \\ \\  \end{array}
%   \pmatrix{ y H_A & 0  \cr
%          y H_B & M }
%          \pmatrix{t_A \cr \tilde q^c_A }
%          +\mbox{h.c} + \ A\leftrightarrow B
%\end{equation}
%Once $H_A$ and $H_B$ get vevs of $v$ and $f$ respectively, the mass
%spectrum for the top fermions to leading order in~$v^2/f^2$ is
%\begin{eqnarray}
%   m^2_{t_A}=\frac{y^2 M^2}{M^2 +y^2f^2}v^2 &\qquad&
%   m^2_{T_A}= M^2 +y^2f^2 \nonumber\\
%   m^2_{t_B}= y^2 f^2 \qquad\qquad &\qquad&
%   m^2_{T_B}= M^2
%\end{eqnarray}
%Identifying the lightest state, $t_A$, as our top quark determines the Yukawa
%coupling $y$ to be
%\begin{equation}
%   y=\frac{M}{\sqrt{M^2-f^2}},
%\end{equation}
%which requires $M>f$.
%
%It is straight forward to find the mass-squared matrix and its eigenvalues,
%$M^2_i$, which are functions of $H_{A,B}$.  The CW potential for Weyl
%fermions is potential is simply
%\begin{equation}
%   V_{CW}=-\frac{1}{16\pi^2} \sum_i \frac{M_i^4}{4}
%   \left( \log\frac{M_i^2}{\Lambda} -\frac{3}{2} \right)
%\end{equation}
%The quadratic part of the CW potential is absent because $SU(4)$ is broken
%softly, tough even if this were not the case, the quadratic divergence would
%not affect the Higgs because of $Z_2$.
%Plugging the expansion of Equation~(\ref{expand}) we can get a the Higgs
%potential that has the form
%\begin{equation}
%   V(h)= m_h^2 h^\dagger h + \lambda_h (h^\dagger h)^2 + \ldots
%\end{equation}
%The top sector contributes
%\begin{eqnarray}
%   m_h^2&=&
%   \frac{3}{8\pi^2}\frac{y^2 M^2}{M^2-y^2f^2}
%   &\times&
%   \left( M^2\log\frac{m_{T_A}^2}{m^2_{T_B}}
%   -y^2f^2 \log\frac{m^2_{T_A}}{m^2_{t_B}}\right)\,,
%   \nonumber\\
%   \lambda_h &=& -\frac{m_h^2}{3f^2} +
%   \frac{3}{16 \pi^2}\frac{y^4 M^4}{(M^2+y^2f^2)^2}
%   \log\frac{m^2_{T_A}}{m^2_{t_A}} \nonumber\\
%   &+& \frac{3}{16\pi^2}
%   \frac{y^4M^4 (M^2+y^2f^2)}{(M^2-y^2f^2)^3}
%   \log\frac{m^2_{T_B}}{m^2_{t_B}} \nonumber\\
%   &-& \frac{1}{32\pi^2}
%   \left[\frac{18 y^4 M^4}{(M^2-y^2f^2)^2}
%   +\frac{9y^4 M^4}{(M^2+y^2f^2)^2}\right]
%\end{eqnarray}
%Since the Yukawa coupling $y$ is set by the top mass, there are only two free
%parameters in this potential, the vev $f$ and the soft mass $M$. In
%Figure~\ref{plot-top} we show the dependence of $\sqrt{-m_h^2}$ on the soft
%mass for $f=1$ TeV. The mass parameter varies between 215 and 400 GeV as $M$
%goes up to the cutoff $\Lambda\sim4\pi f$. The steep rise near $M\sim 1$ TeV
%is due to the fact that $y$ diverges as $M->f$. the minimum contribution to
%$|m_h|$ is for $M=\sqrt{2}f$.
%
%\begin{figure}[tb]
%\includegraphics[width=7.5cm]{massparameter}
%\caption{The negative mass squared parameter induced by top sector
%   loops as a function of the $SU(4)$ breaking soft mass $M$ for $f=1$
%   TeV.}
%\label{plot-top}
%\end{figure}
%
%As we will discuss in the next section, we may wish to consider the case
%where the twin photon is massive for phenomenological reasons. We will assume
%that this happens by some additional scalar, $\phi_B$, that is charged under
%$QED_B$ gets a vev. We will leave the dynamics that cause it to get a vev
%unspecified, but can take its vev to be naturally at the cutoff. The twin
%hypercharge gauge boson, $B_B$, will pick up a mass of $M^2_{B_B}= g'^2 \langle
%\phi_B \rangle^2 /2$. Since this breaks the twin $Z_2$ this will introduce a
%divergent contribution to the Higgs mass that is quadratic in the vev
% $\langle \phi_B \rangle$
%\begin{equation}
%   \Delta m_h^2= \frac{3g'^4 \langle \phi_B \rangle^2}{128\pi^2}
%   \left( \log\frac{2\Lambda^2}{g'^2\langle \phi_B \rangle ^2}
%   -\frac{1}{2}\right)
%\end{equation}
%\begin{equation}
%   \Delta m_h^2= \frac{3M^4_{B_B}}{32\pi^2}
%   \left( \log\frac{\Lambda^2}{M_{B_B}^2}
%   -\frac{1}{8}\right)
%\end{equation}
%When $\langle \phi_B \rangle$ is taken to the cutoff the gauge contribution
%to $m_h^2$ is roughly +(150 GeV)$^2$.
%The quartic we introduced in Equation~(\ref{quartic}) will contribute
%quadratically to $m_h^2$
%\begin{equation}
%   \Delta m_h \sim \frac{\lambda'}{16\pi^2} \Lambda^2\,,
%\end{equation}
%where the sign of this contribution is determined by unknown UV physics.
%Since we have a discrete choice we will choose this sign to be positive which
%will partially cancel the negative mass squared generated by the top sector.
%
%This cancellation will introduce a mild fine tuning in our model.  To estimate
%the amount of fine tuning we first set the parameters $f$ and $M$ which set
%the top sector potential. As a representative value we choose $f=1$ TeV which
%allows the cutoff to be above 10 TeV which is safe for precision EW
%measurements.  This potential dominates the contribution from the gauge
%sector which can be ignored for the sake of fine tuning
%estimate\footnote{This is the conservative choice, since the gauge sector
%also partially cancels the top contributions.}. We can then choose the
%quartic parameter $\lambda'$ that gives the correct electroweak symmetry
%breaking. The value of the added quartic that give this is in the range
%0.04--0.14 for values of $M$ between 1.6 and 12 TeV. The mass of the Higgs
%grows with the quartic, and so this range of parameters give Higgs masses
%between 114 GeV and 195 GeV. Once a set of parameters is chosen, one can
%measure the fine tuning as the sensitivity of the electroweak scale to the
%input parameters. In our case the highest sensitivity is to the quartic
%$\lambda'$, and so the fine tuning is
%\begin{equation}
%   \Delta^{-1}=\frac{\lambda'}{M_z^2}
%   \frac{\partial M_z^2}{\partial \lambda'}\,.
%\end{equation}
%The fine tuning in the range of parameters discussed above is roughly 16\%
%for a Higgs mass of 114 GeV and 13\% for a Higgs mass of 195 GeV.
%
%As we include the gauge contribution to the potential the possible range of
%the Higgs mass is somewhat smaller, depending whether and at what scale
%$U(1)_{QED_B}$ is broken. If QED$_B$ is unbroken the highest possible Higgs
%mass is 185 GeV, but if QED$_B$ is broken at $\langle \phi_B \rangle$ the
%highest Higgs mass is 135 GeV, both of which involve a mild fine tuning of
%13-14\%.

\section{Phenomenology}

%The absence of light fields charged under any of the SM gauge groups
%implies that the phenomenology of such models is strikingly distinct from that of
%any
%of the other proposed solutions to the hierarchy problem.

For phenomenological purposes we divide twin symmetric models into two classes -
those where the top sector is extended as in eq.~({\ref{topmodule}}), and those where
it is not.  As we now explain, the experimental constraints in these two cases are
different. In the first case the exotic quarks $\tilde q_{A,B}$ and $\tilde
q^c_{A,B}$, which are charged under both $U(1)_A$ and $U(1)_B$, lead to kinetic
mixing between the photon and its twin partner at one loop {\cite{Holdom}}. Since the
experimental constraints on such mixing are very severe the twin photon must be
heavy. In the second case, however, there are no particles charged under both sets of
gauge groups, and a preliminary analysis does not reveal any non-zero contribution to
the kinetic mixing term up to three loop order. In this scenario it may therefore be
phenomenologically allowed for the twin photon to be massless, provided a kinetic
mixing term is not present at the cutoff. The mirror symmetric model shares the same
phenomenology as the twin symmetric model without the extended top sector.

We now study each of these two scenarios in more detail, starting with theories with
the extended top sector. In this case the strongest bound arises from the requirement
that the twin neutrinos (and the twin photon itself) not contribute significantly to
the energy density of the universe at the time of Big Bang
Nucleosynthesis~(BBN)~{\cite{CG,Davidson}}. This constraint can be satisfied if the
following two conditions are met. 
\begin{itemize} 
\item There is large
entropy production during the QCD phase transition, significantly more 
than during the corresponding
transition in the twin sector.  
\item The two sectors are not in thermal equilibrium
at any time after the QCD phase transition.  
\end{itemize} 
Since the dynamics of the QCD phase transition is expected to be sensitive to the
number of light quarks and their masses, which are not constrained to be the same in
the two sectors, it is certainly plausible that the first condition is satisfied.
What about the second? If the mixing term is zero at the cutoff and is only generated
at the one-loop level through the exchange of the exotic quarks, the two sectors will
not be in equilibrium below a few hundred MeV provided the twin photon mass $M_{B}$
is larger than a few hundred GeV. In such a scenario the twin electron cannot go out 
of
the bath by annihillating into photons once the temperature falls below it's mass,
as in the SM. Instead the twin electron must be extremely light so as not to
contribute too much to the energy density of the universe at late times. We expect
that twin baryons will constitute some or all of the dark matter in the universe,
depending on the baryon asymmetry in the mirror sector.

Although this model predicts the existence of new light twin states, the fact that
these particles are not charged under the SM gauge group implies that it may not be
easy to test. In particular, precision electroweak constraints are easily satisfied.
However, one possibility is to look for invisible decays of the SM Higgs into twin
fermions {\cite{Foot}}. The relevant vertex arises from substituting the expansion
eq. ({\ref{pseudo}}) into the Yukawa coupling of $H_B$ to twin fermions.  The
branching ratio for invisible Higgs decays is of order $|\langle{h}\rangle/f|^2$.


%The only significant renormalizable interaction
%between the SM fields and their mirror partners is through the Higgs quartic term.
%This term implies sizeable interactions between the longitudinal components of the
%SM
%W bosons and their mirror partners that can potentially be probed in colliders. We
%leave a detailed study of this for future work.

We now estimate the fine-tuning in this class of models for two sets of
parameters.  For $f = 800$ GeV, $\Lambda \sim 4 \pi f \approx 10$ TeV, $M =
6.0$ TeV, $M_B = 1$ TeV we find that in order to obtain the SM
values of $M_W$ and $M_Z$ we need the soft $Z_2$ breaking parameter $\mu
\approx$ 240 GeV. The Higgs mass is then about 120 GeV. Estimating the fine-tuning
as $\partial \; {\rm log} M_Z^2 / \partial \; {\rm log} \mu^2 $ we find that
it is of order 13 \% (1 in 8). For $f = 500$ GeV, $\Lambda \sim 4 \pi f
\approx 6$ TeV, $M = 5.5$ TeV, $M_B = 1$ TeV we find the soft $Z_2$ breaking
parameter $\mu$ needs to be around 150 GeV.  The Higgs mass is again around
120 GeV and the fine-tuning 38\% (1 in 3).
%%%
These results are sumarized in Table~\ref{table-sumary}.
%%%
This shows that these models stabilize the weak scale up to 5-10 TeV.


Let us now turn to mirror symmetric models (and twin symmetric models without the
extended top sector). We are specifically interested in the scenario where the mirror
photon is massless, since it appears current experimental bounds cannot exclude this
possibility. This class of models also predict new light mirror fermions. These may
now have tiny fractional electric charges if the kinetic mixing term between the
photon and its mirror partner is very small but non-zero.  Apart from this, the
phenomenological implications are expected to be similar to the case with the
extended top sector.  We now estimate the fine-tuning for one specific parameter
choice. Note that the formulas of the previous section generalize to the case without
the extended top sector when the limit $M \rightarrow \Lambda$ is taken, up to finite
terms.  For $f = 800$ GeV, $\Lambda \sim 4 \pi f \approx 10$ TeV we find that the
Higgs mass is 166 GeV and the fine-tuning is $\sim$11\% (1 in 9). For $f= 500$ GeV,
$\Lambda\sim 4\pi f \approx 6$ TeV we get a Higgs mass of 153 GeV with a fine tuning
of 31\% (1 in 3). This shows that this class of models also stabilizes the weak scale
up to 5-10 TeV. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{table}[tbh]
\begin{tabular}{|c|c|c|c|c|c|c|} \hline $\Lambda${\tiny (TeV)} & $f${\tiny (GeV)} &
$M${\tiny (TeV)} & $M_B${\tiny (TeV)} & $\mu${\tiny(GeV)} & $m_{h}${\tiny (GeV)} &
Tuning \\ \hline 10 & 800 & 6 & 1 & 239 & 122 & 0.134 \\ 6 & 500 & 5.5 & 1 & 145 &
121 & 0.378 \\ 10 & 800 & --- & 0 & 355 & 166 & 0.112 \\ 6 & 500 & --- & 0 & 203 &
153 & 0.307 \\ \hline \end{tabular} \caption{ A sumary of the Higgs mass and fine
tuning, $\partial \; {\rm log} M_Z^2 / \partial \; {\rm log} \mu^2 $, for four sample
points of parameter space. The upper two lines are for twin symmetric models with the
extended top sector. The lower two lines are for minimal $Z_2$ models
without the extended top sector.} \label{table-sumary} \end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\section{Conclusions}

In summary we have constructed a new class of models where the Higgs emerges
as a pseudo-Goldstone whose mass is protected against radiative corrections
up to scales of order 5-10 TeV. These theories demonstrate that, contrary to
the conventional wisdom, stabilizing the weak scale does not require new light
particles transforming under the SM gauge groups.
\\

%\acknowledgments
\noindent
{\bf Acknowledgments --} 
Z.C thanks M. Luty, N. Okada and W.D. Toussaint for discussions. Z.C and H.S.G are
supported by the NSF under grant PHY-0408954.  R.H is supported in part by the DOE
under contract DE-AC03-76SF00098 and in part by NSF grant PHY-0098840.



\begin{thebibliography}{99}

%\cite{Georgi:1974yw}
\bibitem{GP}
H.~Georgi and A.~Pais,
%``Calculability And Naturalness In Gauge Theories,''
Phys.\ Rev.\ D {\bf 10}, 539 (1974);
%%CITATION = PHRVA,D10,539;%%
%\cite{Georgi:1975tz}
%\bibitem{Georgi:1975tz}
%H.~Georgi and A.~Pais,
%``Vacuum Symmetry And The Pseudogoldstone Phenomenon,''
Phys.\ Rev.\ D {\bf 12}, 508 (1975).
%%CITATION = PHRVA,D12,508;%%

%\cite{Kaplan:1983fs}
\bibitem{KG}
D.~B.~Kaplan and H.~Georgi,
%``SU(2) X U(1) Breaking By Vacuum Misalignment,''
Phys.\ Lett.\ B {\bf 136}, 183 (1984);
%%CITATION = PHLTA,B136,183;%%
%\cite{Kaplan:1983sm}
%\bibitem{Kaplan:1983sm}
D.~B.~Kaplan, H.~Georgi and S.~Dimopoulos,
%``Composite Higgs Scalars,''
Phys.\ Lett.\ B {\bf 136}, 187 (1984);
%%CITATION = PHLTA,B136,187;%%
%\cite{Georgi:1984af}
%\bibitem{Georgi:1984af}
H.~Georgi and D.~B.~Kaplan,
%``Composite Higgs And Custodial SU(2),''
Phys.\ Lett.\ B {\bf 145}, 216 (1984).
%%CITATION = PHLTA,B145,216;%%


%\cite{Arkani-Hamed:2001nc}
\bibitem{Little1}
N.~Arkani-Hamed, A.~G.~Cohen and H.~Georgi,
Phys.\ Lett.\ B {\bf 513}, 232 (2001)

%\cite{Arkani-Hamed:2002qx}
\bibitem{Little2}
 N.~Arkani-Hamed, A.~G.~Cohen, E.~Katz, A.~E.~Nelson, T.~Gregoire and J.~G.~Wacker,
 JHEP {\bf 0208}, 021 (2002);
 N.~Arkani-Hamed, A.~G.~Cohen, E.~Katz and A.~E.~Nelson,
 JHEP {\bf 0207}, 034 (2002);
 T.~Gregoire and J.~G.~Wacker,
 JHEP {\bf 0208}, 019 (2002);
 I.~Low, W.~Skiba and D.~Smith,
 Phys.\ Rev.\ D {\bf 66}, 072001 (2002)
 D.~E.~Kaplan and M.~Schmaltz,
 JHEP {\bf 0310}, 039 (2003).

%\cite{Schmaltz:2005ky}
\bibitem{Review}
M.~Schmaltz and D.~Tucker-Smith,
[arXiv:hep-ph/0502182].

\bibitem{Csaba}
 C.~Csaki, J.~Hubisz, G.~D.~Kribs, P.~Meade and J.~Terning,
 Phys.\ Rev.\ D {\bf 67}, 115002 (2003);
 Phys.\ Rev.\ D {\bf 68}, 035009 (2003);
 J.~L.~Hewett, F.~J.~Petriello and T.~G.~Rizzo,
 JHEP {\bf 0310}, 062 (2003).

\bibitem{mirror}
%\cite{Lee:1956qn}
  T.~D.~Lee and C.~N.~Yang,
  %``Question Of Parity Conservation In Weak Interactions,''
  Phys.\ Rev.\  {\bf 104}, 254 (1956);
  I.~Yu.~Kobzarev, L.~B.~Okun and I.~Ya.~Pomeranchuk,
  [Yad.\ Fiz.\ {\bf 3}, 1154 (1966)];
%\cite{Pavsic:1974rq}
  M.~Pavsic,
  %``External Inversion, Internal Inversion, And Reflection Invariance,''
  Int.\ J.\ Theor.\ Phys.\  {\bf 9}, 229 (1974).
%\cite{Blinnikov:1982eh}
  S.~I.~Blinnikov and M.~Y.~Khlopov,
  %``On Possible Effects Of 'Mirror' Particles,''
  [Yad.\ Fiz.\  {\bf 36}, 809 (1982)].
  %%CITATION = SJNCA,36,472;%%


%\cite{Coleman:1973jx}
\bibitem{CW}
  S.~R.~Coleman and E.~Weinberg,
  %``Radiative Corrections As The Origin Of Spontaneous Symmetry Breaking,''
  Phys.\ Rev.\ D {\bf 7}, 1888 (1973).
  %%CITATION = PHRVA,D7,1888;%%


%\cite{Holdom:1986eq}
\bibitem{Holdom}
  B.~Holdom,
  %``Searching For Epsilon Charges And A New U(1),''
  Phys.\ Lett.\ B {\bf 178}, 65 (1986).
  %%CITATION = PHLTA,B178,65;%%

%\cite{Carlson:1987si}
\bibitem{CG}
E.~D.~Carlson and S.~L.~Glashow,
%``Nucleosynthesis Versus The Mirror Universe,''
Phys.\ Lett.\ B {\bf 193}, 168 (1987).
%%CITATION = PHLTA,B193,168;%%

%\cite{Davidson:2000hf}
\bibitem{Davidson}
  For a summary of various bounds see S.~Davidson, S.~Hannestad and G.~Raffelt,
  %``Updated bounds on milli-charged particles,''
  JHEP {\bf 0005}, 003 (2000).
%  [arXiv:hep-ph/0001179].
%  %%CITATION = HEP-PH 0001179;%%


%\cite{Foot:1991bp}
\bibitem{Foot}
  R.~Foot, H.~Lew and R.~R.~Volkas,
  %``A Model with fundamental improper space-time symmetries,''
  Phys.\ Lett.\ B {\bf 272}, 67 (1991);
  %%CITATION = PHLTA,B272,67;%%
%\cite{Foot:1991py}
%  R.~Foot, H.~Lew and R.~R.~Volkas,
  %``Possible consequences of parity conservation,''
  Mod.\ Phys.\ Lett.\ A {\bf 7}, 2567 (1992).
  %%CITATION = MPLAE,A7,2567;%%



\end{thebibliography}


\end{document}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

hep-ph0506256.extracted_bib
\begin{thebibliography}{99}

%\cite{Georgi:1974yw}
\bibitem{GP}
H.~Georgi and A.~Pais,
%``Calculability And Naturalness In Gauge Theories,''
Phys.\ Rev.\ D {\bf 10}, 539 (1974);
%%CITATION = PHRVA,D10,539;%%
%\cite{Georgi:1975tz}
%\bibitem{Georgi:1975tz}
%H.~Georgi and A.~Pais,
%``Vacuum Symmetry And The Pseudogoldstone Phenomenon,''
Phys.\ Rev.\ D {\bf 12}, 508 (1975).
%%CITATION = PHRVA,D12,508;%%

%\cite{Kaplan:1983fs}
\bibitem{KG}
D.~B.~Kaplan and H.~Georgi,
%``SU(2) X U(1) Breaking By Vacuum Misalignment,''
Phys.\ Lett.\ B {\bf 136}, 183 (1984);
%%CITATION = PHLTA,B136,183;%%
%\cite{Kaplan:1983sm}
%\bibitem{Kaplan:1983sm}
D.~B.~Kaplan, H.~Georgi and S.~Dimopoulos,
%``Composite Higgs Scalars,''
Phys.\ Lett.\ B {\bf 136}, 187 (1984);
%%CITATION = PHLTA,B136,187;%%
%\cite{Georgi:1984af}
%\bibitem{Georgi:1984af}
H.~Georgi and D.~B.~Kaplan,
%``Composite Higgs And Custodial SU(2),''
Phys.\ Lett.\ B {\bf 145}, 216 (1984).
%%CITATION = PHLTA,B145,216;%%


%\cite{Arkani-Hamed:2001nc}
\bibitem{Little1}
N.~Arkani-Hamed, A.~G.~Cohen and H.~Georgi,
Phys.\ Lett.\ B {\bf 513}, 232 (2001)

%\cite{Arkani-Hamed:2002qx}
\bibitem{Little2}
 N.~Arkani-Hamed, A.~G.~Cohen, E.~Katz, A.~E.~Nelson, T.~Gregoire and J.~G.~Wacker,
 JHEP {\bf 0208}, 021 (2002);
 N.~Arkani-Hamed, A.~G.~Cohen, E.~Katz and A.~E.~Nelson,
 JHEP {\bf 0207}, 034 (2002);
 T.~Gregoire and J.~G.~Wacker,
 JHEP {\bf 0208}, 019 (2002);
 I.~Low, W.~Skiba and D.~Smith,
 Phys.\ Rev.\ D {\bf 66}, 072001 (2002)
 D.~E.~Kaplan and M.~Schmaltz,
 JHEP {\bf 0310}, 039 (2003).

%\cite{Schmaltz:2005ky}
\bibitem{Review}
M.~Schmaltz and D.~Tucker-Smith,
[arXiv:hep-ph/0502182].

\bibitem{Csaba}
 C.~Csaki, J.~Hubisz, G.~D.~Kribs, P.~Meade and J.~Terning,
 Phys.\ Rev.\ D {\bf 67}, 115002 (2003);
 Phys.\ Rev.\ D {\bf 68}, 035009 (2003);
 J.~L.~Hewett, F.~J.~Petriello and T.~G.~Rizzo,
 JHEP {\bf 0310}, 062 (2003).

\bibitem{mirror}
%\cite{Lee:1956qn}
  T.~D.~Lee and C.~N.~Yang,
  %``Question Of Parity Conservation In Weak Interactions,''
  Phys.\ Rev.\  {\bf 104}, 254 (1956);
  I.~Yu.~Kobzarev, L.~B.~Okun and I.~Ya.~Pomeranchuk,
  [Yad.\ Fiz.\ {\bf 3}, 1154 (1966)];
%\cite{Pavsic:1974rq}
  M.~Pavsic,
  %``External Inversion, Internal Inversion, And Reflection Invariance,''
  Int.\ J.\ Theor.\ Phys.\  {\bf 9}, 229 (1974).
%\cite{Blinnikov:1982eh}
  S.~I.~Blinnikov and M.~Y.~Khlopov,
  %``On Possible Effects Of 'Mirror' Particles,''
  [Yad.\ Fiz.\  {\bf 36}, 809 (1982)].
  %%CITATION = SJNCA,36,472;%%


%\cite{Coleman:1973jx}
\bibitem{CW}
  S.~R.~Coleman and E.~Weinberg,
  %``Radiative Corrections As The Origin Of Spontaneous Symmetry Breaking,''
  Phys.\ Rev.\ D {\bf 7}, 1888 (1973).
  %%CITATION = PHRVA,D7,1888;%%


%\cite{Holdom:1986eq}
\bibitem{Holdom}
  B.~Holdom,
  %``Searching For Epsilon Charges And A New U(1),''
  Phys.\ Lett.\ B {\bf 178}, 65 (1986).
  %%CITATION = PHLTA,B178,65;%%

%\cite{Carlson:1987si}
\bibitem{CG}
E.~D.~Carlson and S.~L.~Glashow,
%``Nucleosynthesis Versus The Mirror Universe,''
Phys.\ Lett.\ B {\bf 193}, 168 (1987).
%%CITATION = PHLTA,B193,168;%%

%\cite{Davidson:2000hf}
\bibitem{Davidson}
  For a summary of various bounds see S.~Davidson, S.~Hannestad and G.~Raffelt,
  %``Updated bounds on milli-charged particles,''
  JHEP {\bf 0005}, 003 (2000).
%  [arXiv:hep-ph/0001179].
%  %%CITATION = HEP-PH 0001179;%%


%\cite{Foot:1991bp}
\bibitem{Foot}
  R.~Foot, H.~Lew and R.~R.~Volkas,
  %``A Model with fundamental improper space-time symmetries,''
  Phys.\ Lett.\ B {\bf 272}, 67 (1991);
  %%CITATION = PHLTA,B272,67;%%
%\cite{Foot:1991py}
%  R.~Foot, H.~Lew and R.~R.~Volkas,
  %``Possible consequences of parity conservation,''
  Mod.\ Phys.\ Lett.\ A {\bf 7}, 2567 (1992).
  %%CITATION = MPLAE,A7,2567;%%



\end{thebibliography}