Sphalerons with Two Higgs Doublets \footnote{\lowercase{\uppercase{T}alk given at {\em \uppercase{S}trong and \uppercase{E}lectroweak \uppercase{M}atter}, \uppercase{M}arseille, 14-17 \uppercase{J}une 2000}}
Mark Hindmarsh and Jackie Grant
We report on work studying the properties of the sphaleron in models of the electroweak interactions with two Higgs doublets in as model-independent a way as possible: by exploring the physical parameter space described by the masses and mixing angles of the Higgs particles. If one of the Higgs particles is heavy, there can be several sphaleron solutions, distinguished by their properties under parity and the behaviour of the Higgs field at the origin. In general, these solutions are not spherically symmetric, although the departure from spherical symmetry is small. %\vspace{24pt} SUSX-TH-01-001
procs4xxx.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% ws-p8-50x6-00.tex : 20-11-97
%% This Latex2e file rewritten from various sources for use in the
%% preparation of the (smaller [8.50''x6.00'']) single-column proceedings 
%% Volume, latest version by R. Sankaran with acknowledgements to Susan 
%% Hezlet and Lukas Nellen. Please comments to:rsanka@wspc.com.sg
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\documentclass{ws-p8-50x6-00}


% Greek letter abbreviations

\def\al{\alpha}  
\def\be{\beta} 
\def\ga{\gamma}
\def\de{\delta}
\def\ep{\epsilon}
\def\ze{\zeta}
\def\et{\eta}
\def\th{\theta}
\def\io{\iota}
\def\ka{\kappa}
\def\la{\lambda}
\def\rh{\rho}
\def\si{\sigma}
\def\ta{\tau}
\def\up{\upsilon}
\def\ph{\phi}
\def\ch{\chi}
\def\ps{\psi}
\def\om{\omega}
\def\De{\Delta}
\def\Ga{\Gamma}
\def\Th{\Theta}
\def\La{\Lambda}
\def\Si{\Sigma}
\def\Up{\Upsilon}
\def\Ph{\Phi}
\def\Ch{\Chi}
\def\Ps{\Psi}
\def\Om{\Omega}

\def\varep{\varepsilon}


% Boldface math letters
\def\bg{{\mathbf{g}}}
\def\bj{{\mathbf{j}}}
\def\bk{{\mathbf{k}}}
\def\bn{{\mathbf{n}}}
\def\bp{{\mathbf{p}}}
\def\bq{{\mathbf{q}}}
\def\br{{\mathbf{r}}}
\def\bu{{\mathbf{u}}}
\def\bv{{\mathbf{v}}}
\def\bx{{\mathbf{x}}}
\def\by{{\mathbf{y}}}
\def\bz{{\mathbf{z}}}
\def\bA{{\mathbf{A}}}
\def\bB{{\mathbf{B}}}
\def\bE{{\mathbf{E}}}
\def\bF{{\mathbf{F}}}
\def\bR{{\mathbf{R}}}
\def\bal{\mbox{\boldmath $\al$}}
\def\bbe{\mbox{\boldmath $\be$}}
\def\bsi{\mbox{\boldmath $\si$}}
\def\bna{\mbox{\boldmath $\na$}}
\def\bdot{\mbox{\boldmath $\cdot$}}
\newcommand{\bsp}[2]{\mbox{\boldmath $#1\cdot#2$}}

% Caligraphic letters
\def\N{{\cal N}}
\def\L{{\mathcal{L}}}
\def\H{{\mathcal{H}}}
\def\P{{\mathcal{P}}}

% Environment abbreviations

\newcommand{\ben}{\begin{equation}}
\newcommand{\een}{\end{equation}}
\renewcommand{\bea}{\begin{eqnarray}}
\renewcommand{\eea}{\end{eqnarray}}
\newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}}
\newcommand{\bi}{\begin{itemize}}
\newcommand{\ei}{\end{itemize}}

% New numbering commands

%\renewcommand{\theequation}{\thesection.\arabic{equation}}
%\renewcommand{\thefigure}{\thesection.\arabic{figure}}
%\renewcommand{\thetable}{\thesection.\arabic{table}}


% Symbols etc.

\def\math{\mathsurround 0pt}
\def\oversim#1#2{\lower.5pt\vbox{\baselineskip0pt \lineskip-.5pt
        \ialign{$\math#1\hfil##\hfil$\crcr#2\crcr{\scriptstyle\sim}\crcr}}}
\def\lap{\mathrel{\mathpalette\oversim {\scriptstyle <}}}
\def\gap{\mathrel{\mathpalette\oversim {\scriptstyle >}}} 

\def\dbar{{{\,\mathchar'26}\mkern-9mu\!d\,}}
\def\debar{{{\,\mathchar'26}\mkern-9mu\delta}}

\def\no#1{\mbox{\boldmath $:$} #1 \mbox{\boldmath $:$}}

\def\pa{\partial}

\def\half{\frac{1}{2}}
\def\na{\nabla}
\def\ap{\approx}
\def\vp{\varphi}
\def\pt{\propto}
\def\tr{\mathop{\rm tr}\nolimits}
\newcommand{\ket}[1]{|#1\rangle}
\newcommand{\bra}[1]{\langle#1|}
\newcommand{\braket}[2]{\langle#1|#2\rangle}
\newcommand{\sla}[1]{\not\!#1}


\def\omk{{\omega_{\bk}}}
\def\omkp{{\omega_{\bk}'}}
\def\omp{\omega_{\mathbf{p}}}
\def\id{{\mathsf 1}}

\def\prll{\parallel}

\def\Re{\mathrm{Re}}
\def\Im{\mathrm{Im}}


\begin{document}

\title{Sphalerons with Two Higgs Doublets
\footnote{\lowercase{\uppercase{T}alk given at {\em \uppercase{S}trong and 
\uppercase{E}lectroweak 
\uppercase{M}atter}, \uppercase{M}arseille, 14-17 \uppercase{J}une 2000}}}

\author{Mark Hindmarsh and Jackie Grant}

\address{Centre for Theoretical Physics\\
University of Sussex\\
Falmer, Brighton BN1 9QJ\\
U.K.\\
{E-mail: m.b.hindmarsh@sussex.ac.uk, j.j.grant@sussex.ac.uk}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% You may repeat \author \address as often as necessary      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\maketitle

\abstracts{
We report on work studying the properties of the sphaleron in 
models of the electroweak interactions with two Higgs doublets 
in as model-independent a way as possible: by
exploring the physical parameter space described by the masses and 
mixing angles of the Higgs particles.  If one of the Higgs particles 
is heavy, there can be several sphaleron solutions, distinguished by their
properties under parity and the behaviour of the Higgs field at the origin. 
In general, these solutions are not spherically symmetric, although the
departure from spherical symmetry is small.
%\vspace{24pt}
SUSX-TH-01-001}
%\end{abstract}



\section{Introduction}

One of the major unsolved problems in particle cosmology is accounting for the
baryon asymmetry of the Universe. This asymmetry is 
usually expressed in terms of the parameter
$\eta$, defined as the ratio between the baryon number density $n_B$ and the
entropy density $s$: $\eta = n_B/s \sim 10^{-10}$.  Sakharov~\cite{Sak67} laid
down the framework for any explanation: the theory of baryogenesis 
must contain B violation;
C and CP violation; and a departure from thermal equilibrium.  All these
conditions are met by the Standard Model~\cite{KuzRubSha85}
and its extensions, and so there is
considerable optimism that the origin of the baryon asymmetry can be found in
physics accessible at current and planned accelerators (see
\cite{Baryo} for reviews).

Current attention is focused on the Minimal Supersymmetric Standard Model, 
where there are many sources of CP violation over and above the CKM
matrix~\cite{CPMSSM},
and the phase transition can be first order for Higgs masses up to 120
GeV, providing the right-handed stop is very light and the
left-handed stop very massive~\cite{PhTranMSSM}.

B violation is provided by sphalerons~\cite{KliMan84}, at a rate 
$\Ga_s \simeq
\exp(-E_s(T)/T)$, where $E_s(T)$ is the energy of the
sphaleron at temperature $T$.  This rate must not be so large that the baryon
asymmetry is removed behind the bubble wall, and this condition can be
translated into a lower bound on the sphaleron mass
%~\cite{Sha86}
\( E_s(T_c)/T_c \gap 45.\)
Thus it is clear that successful baryogenesis requires a careful
calculation of the sphaleron properties.

%\section{Sphalerons in History}

Here we report on work on sphalerons in the two-doublet Higgs model (2DHM) in
which we study the properties of sphalerons in as general a set of
realistic models as possible.  In doing so we try to express parameter space in
terms of physical quantities: Higgs masses and mixing angles,
which helps us
avoid regions of parameter space which have already been ruled out by LEP.  

Previous work on sphalerons in 2DHMs
\cite{KasPecZha91,BacTinTom96,MorOakQui96,Kle98} has restricted either the
Higgs potential or the ansatz in some way. Our potential is restricted only by
a softly broken discrete symmetry imposed to minimize flavour-changing neutral
currents (FCNCs). Our ansatz is the most general spherically symmetric one, including
possible C and P violating field configurations~\cite{Yaf89}.

We firstly check our results against the existing literature, principally BTT
\cite{BacTinTom96}, who found a new P-violating ``relative winding'' (RW)
sphaleron, specific to multi-doublet models, albeit at $M_A=M_{H^\pm}=0$. 
This is distinguished from Yaffe's 
P-violating deformed sphaleron~\cite{Yaf89} or
``bisphaleron'' by a difference in the
behaviour of each of the two Higgs fields at the origin.
 We then reexamine the sphaleron in a more realistic part
of parameter space, where $M_A$ and $M_{H^\pm}$ are above their experimental
bounds.  
%The preliminary indications are that the RW sphaleron appears at yet
%larger Higgs masses (at least when mixing is neglected).  
We reiterate the
point made in~\cite{GraHin98} that introducing Higgs sector CP violation makes a significant
difference to the sphaleron mass (between ten and fifteen percent),
and may significantly change  bounds on the
Higgs mass from electroweak baryogenesis.

\section{Two Higgs doublet electroweak theory}

The
most general quartic potential for 2DHMs has 14 parameters, only
one of which, the Higgs vacuum expectation value, $\upsilon$, is known.  However, we are
aided by the observation~\cite{HHG} that FCNCs can
be suppressed by imposing a softly-broken discrete symmetry 
$\phi_{1(2)} \rightarrow +(-) \phi_{1(2)}$, %$\phi_{2} \rightarrow -\phi_{2}$, 
and results
in a potential with 10 real parameters.  One of these %parameters 
may be removed
by a phase redefinition of the fields; the vacuum configuration is then
entirely real, and CP violation is contained in one term 
\( 
2\chi_{2}\left(\Re( \phi_{1}^{\dagger}\phi_{2})-\frac{\upsilon_{1}\upsilon_{2}}{2}\right)
\Im(\phi_{1}^{\dagger}\phi_{2})\).
Ignoring couplings to other fields, when $\chi_2 = 0$ there is a discrete
symmetry
%\ben
%\label{e:Cxm_phi}
\(\phi_{\alpha}\rightarrow{-i}\sigma_{2}\phi_{\alpha}^{\ast}\),
%\een
which sends $\Im(\phi_{1}^{\dagger}\phi_{2})\to
-\Im(\phi_{1}^{\dagger}\phi_{2})$.  This can be identified as %charge conjugation 
C invariance.  

Following~\cite{GraHin98} we
determine as many as possible of the nine parameters in the
potential from physical ones. The physical parameters at hand are the four
masses of the Higgs particles, the three mixing angles of the neutral Higgses,
one of which, $\theta_{CP}$, is the only CP violating physical parameter, ($\theta_{CP}$ 
mixes the CP even and CP odd neutral Higgs sector),
and $\upsilon$. 

We further check that the potential 
for these sets of physical parameters is always bounded from below.
%This
%leaves one undetermined parameter, which we choose to be $\la_3$, the
%coefficient of the term \(
%2\left(\phi_{1}^{\dagger}\phi_{1}-\frac{\upsilon_{1}^2}{2}\right)
%\left(\phi_{2}^{\dagger}\phi_{2}-\frac{\upsilon_{2}^2}{2}\right)\).
%The conversion between the parameters of Eq.\  \ref{e:pot} and these masses 
%can be found in~\cite{GraHin98}
%\footnote{There are two typographical errors in \ref{GraHin98} which will be
%corrected in a future publication~\cite{GraHin00}}.
%swapped cos and sin in Eq.\ 
%\ref{e:lambda1} and Eq.\  \ref{e:lambda2}, 
%and a sign error in Eq.\  \ref{e:chi1}.}

%It should be noted that although the vacuum is guaranteed to be stable with
%real Higgs masses, there are extra conditions which arise from the
%condition that the potential be bounded from below, and also that the vacuum 
% be a
%global minimum.  The first condition is easily taken into account and all the
%parameter space here results in bounded potentials.  We have also checked
%numerically that (\ref{e:minimum}) is the global minimum. .

\section{Sphaleron ansatz and numerical methods}

The most general static spherically symmetric ansatz is, in the radial gauge
%\(W \hat{r} = 0\)
\bea
\phi_{\alpha}=%\frac{\upsilon_{\alpha}}{\sqrt{2}}
(F_{\alpha}+{i}G_{\alpha}\hat{x}^a\sigma^{a})
\left(\begin{array}{c}0\\1\end{array}\right), ~~
%\\
%W_0&=&\frac{1}{g}A_0\hat{x}^a\frac{\sigma^a}{2i}\nonumber\\
W^a_i=%\frac{1}{g}
\left[\frac{(1+\beta)}{r}\varepsilon_{aim}\hat{x}_m
      +\frac{\alpha}{r}(\delta_{ai}-\hat{x}_a\hat{x}_i)\right]. \label{e:ansatz}
%      +A_1\hat{x}_a\hat{x}_i\right]\frac{\sigma^a}{2i}.
\eea
Here, the subscript $\alpha=1,2$, and $F_\al=a_\al+ib_\al$ and $G_\al=c_\al+id_\al$
are complex functions.
%Static energy functional:
%\[
%E_s  = E_s[\al(r),\be(r),a_\al(r),c_\al(r)]
%\]
The boundary conditions can be most easily expressed in terms of the functions
$\chi$, $\Psi$, $H_\al$, $h_\al$, and $\Th_\al$, defined by
\bea
-\be + i \al  =  %\sqrt{2} 
\chi e^{ i\Psi}, ~~~
%\\
a_{\al} + i c_{\al} = %2 \frac{\upsilon_{\al}}{\upsilon} 
H_{\al} e^{i \Th_{\al}}, ~~~
%\quad
b_{\al} + i d_{\al} =  %2 \frac{\upsilon_{\al}}{\upsilon} 
h_{\al} e^ {i \Th_{\al}},
\eea
and one can show that as $r\to 0$, {either} $H_\al^2+h_\al^2  \to 0$ 
or $\Theta_1 \to \Psi/2 + n_1\pi$ and $\Theta_2  \to \Psi/2 + n_2 \pi$, ($n_1,n_2 \in {\mathbf Z}$).
%It is 
These boundary conditions 
%which 
distinguish between the various types of
sphaleron solution:  the ordinary sphaleron has  $H_\al^2+h_\al^2  \to 0 $ as
$r\to 0$, the bisphaleron has non-vanishing Higgs fields with $n_1
= n_2$, and the RW sphaleron non-vanishing Higgs fields with
$n_1 \ne n_2$. These integers represent the winding of the Higgs field around
spheres of constant $|\phi_\al|$, although only their difference has any gauge
invariant meaning.

Note that the ansatz is potentially inconsistent, as $\Im(\phi_1^{\dagger}\phi_2) \propto \hat{x}_3$,
%\ben
%\Im(\phi_1^{\dagger}\phi_2)=\hat{x}_3 (F^*_1G_2 - G^*_2F_1)\up_1\up_2/2, 
%\een
 a point which has not been noted before. In practice, the non-spherically
symmetric parts of the static energy functional, $E[f_A]$, contribute less than 1\% of the
total, and so we are justified in assuming the fields of the ansatz, $f_A$, are a function only of 
the radial co-ordinate $r$ and then integrating over $\hat{x}_3$.
%ignoring deviations. {\bf please change}

%\section{Numerical method}
We look for solutions to $E[f_A]$
%where $f_A$ denotes the 10 fields of the ansatz \ref{e:ansatz}.
%We use a 
using
a Newton method~\cite{Yaf89}
which is an efficient way of finding
extrema (and not just minima). The method can be briefly characterised as 
updating the fields $f_A$ by 
%an amount 
$\delta f_A$, given by the solution of 
${E}^{''} \delta f = - {E}^{'}$, where the primes denote functional differentialion 
with respect to $f_A$.
%The method can be briefly characterised as
%updating the fields $f_A$ by an amount $\delta f_A$, given by the solution of 
%\ben
%\label{e:Newt_raph}
%\frac{\delta^{2} { E} }{\delta f_B \delta f_A}  \delta f_B = -\frac{\delta {E} }{\delta f_A} 
%\een
%which we can abbreviate
%as ${E}^{''} \delta f = - {E}^{'}$. Provided ${E}^{''}$ has no
%zero eigenvalues, the equation has a unique solution, subject to 
%boundary conditions.
A particular advantage to using this method is that because we calculate ${E}^{''}$, it is straightforward to
get the curvature eigenvalues, and therefore to check that the solution really is the lowest energy
unstable solution.

%%Most of our results were obtained by discretising Eq.\ (\ref{e:Newt_raph})
%on a radial lattice with 51 points, uniform in the coordinate $
%s  =  \ln[(1 + \mu r)/({1 + r})]/{\ln |C|}$, where $\mu =
%\max(M_h,M_H,M_A,M_{H^\pm})/{M_w}$ and $C = ({1+ \mu R})/({1+ R})$, with the
%maximum radius $R$ taken to be $20M^{-1}_W$.

\section{Results}


\begin{figure}[t]
\epsfxsize=6cm
\epsfbox{energy3.eps}
\epsfxsize=6cm
\epsfbox{chern3.eps}
\caption{\label{f:EC3} 
Contours of sphaleron energy ($M_W/\al_W$)
and Chern-Simons number. 
% There was no mixing, and 
%$M_A = 241$ Gev, $M_{H^\pm}=161$ Gev, $\tan\be=6$, $\la_3=-0.05$.
% The solid lines in the 
%energy plot represent the relative winding sphaleron, while the dashed lines
%are for the ordinary sphaleron.
}
\end{figure}

\begin{figure}[t]
\epsfxsize=6cm
\epsfbox{first_eig3.eps}
\epsfxsize=6cm
\epsfbox{second_eig3.eps}
\caption{\label{f:Eig3} 
Eigenvalues ($M_w$) of the sphaleron solutions as a function of 
the CP even Higgs masses $M_h$ and $M_H$.  There was no mixing, and 
$M_A = 241$ Gev, $M_{H^\pm}=161$ Gev, $\tan\be=6$, $\la_3=-0.05$. 
The solid lines in the 
most negative eigenvalue
plot represent the relative winding sphaleron, while the dashed lines
are for the ordinary sphaleron, for the dotted region the potential 
was unbounded from below.}
\end{figure}

We first checked our method and code against the results of BTT
\cite{BacTinTom96}, and Yaffe\cite{Yaf89}  finding agreement in the energy of better than 1 part in
$10^3$ for a wide range of parameters.  We also measured the Chern-Simons
numbers, $n_{CS}$, of the solutions that they discovered and determined that they were near
1/2, but not exactly 1/2 as with the sphaleron solution. Further they appeared 
in P conjugate pairs
with $n_{CS}$ of the pair adding to exactly one. 
We then looked at more realistic values of $M_A$ and $M_{H^\pm}$, with results
that are displayed in Figs.\ \ref{f:EC3} and \ref{f:Eig3}. 
% We show
%contours of energy, Chern-Simons number, and eigenvalues for the ordinary and
%RW sphaleron, as a function of the CP even Higgs masses $M_h$ and $M_H$.
%  We
%chose zero CP violation and zero mixing between the CP even Higgses,
%$\tan\be=6$, $M_A=241$ Gev, $M_{H^\pm}=161$ Gev, and $\la_3=-0.05$.
Note first of all the well-known feature that the sphaleron mass
depends mainly on $M_h$.  Secondly, for increasing
$M_H$, the curvature matrix of the sphaleron
develops a second negative eigenvalue (Fig. \ref{f:Eig3}, right),
signalling the
appearance of a pair of RW sphalerons.  The lower of the two 
$n_{CS}$ is plotted on the right
Fig.\ \ref{f:EC3}.  
The departure from $n_{CS} = 1/2$ 
is 
small, as is the difference in energy between the RW and ordinary
sphalerons for the Higgs masses we examined, however the most negative curvature 
eigenvalue of the RW sphaleron can be double that of the sphaleron.
More detailed results and discussion of their significance 
are reserved for a future publication~\cite{GraHin00}.

%Finally, although not displayed here, we calculated the size of the
%non-spherically symmetric contributions to the energy density, which appear as
%dipole and quadrupole moments in the energy density.  For the solutions we have
%examined these contributions are always less than 1\% of the total energy
%density. {\bf yukky end}




\section*{Acknowledgments}
MH and JG are supported by PPARC.
This work was conducted on the SGI Origin platform using COSMOS
Consortium facilities, funded by HEFCE, PPARC and SGI.
We also acknowledge computing support from
the Sussex High Performance Computing Initiative.



\begin{thebibliography}{99}

\bibitem{Sak67} A.D. Sakharov, {\it JETP Lett.} {\bf 6} 24 (1967).
\bibitem{KuzRubSha85} V.A. Kuzmin, V.A. Rubakov, and M.E. Shaposhnikov, 
{\it Phys. Lett. B} \textbf{155} 36 (1985).
\bibitem{Baryo} V.A Rubakov and M.E. Shaposhnikov, Phys. Usp. {\bf 39}, 
461 (1996)  [{hep-ph/9603208}];
%M. Trodden, {\it Rev. Mod. Phys.} {\bf 71} 1463-1500 (1999) 
% [{hep-ph/9803479}]; A. Riotto, {\it Theories of Baryogenesis},
%Summer School in High Energy Physics and Cosmology, Trieste, (1998)
%[hep-ph/9807454]
%\bibitem{HueNel96} P. Huet and A. Nelson, (1996)
%\bibitem{Car+97} M. Carena et al (1997)
%\bibitem{CliKai00} J. Cline and K. Kainulainen, (2000)
\bibitem{CPMSSM} P. Huet and A.E. Nelson, {\it Phys.
Rev.} {\bf D53} 4578 (1996) [hep-ph/9506477];
M. Carena, M. Quiros and C.E.M. Wagner, {\it Phys. Lett.} {\bf B390} 6919
(1997) [hep-ph/9603420];  J. Cline, M.Joyce and K. Kainulainen,
{\it JHEP} {\bf 0007} 018 (2000)
%\bibitem{EspDeC96} J.R. Espinosa and B. de Carlos, (2000)
%\bibitem{LaiRum98} M. Laine and K. Rummukainen (1998)
%\bibitem{CliMoo98} J. Cline and G. Moore (1998)
\bibitem{PhTranMSSM} B. de Carlos and J.R.Espinosa, {\it Phys. Lett.} {\bf
B407} 12 (1997);
M. Laine and K. Rummukainen, {\it Nucl. Phys.} {\bf B545} 141 
(1999) [hep-ph/9811369]; J.M. Cline and G.D. Moore, {\it Phys. Rev. Lett.} {\bf
81} 3315 (1998) [hep-ph/9806354]
%\bibitem{Kaj+98} K. Kajantie, M. Laine, K.Rummukainen, 
%	M. Shaposhnikov, and M. Tsypin, {hep-ph/9809435}  
%\bibitem{tHo76} G. 't Hooft, Phys. Rev. Lett. {\bf 37} 8 (1976).
%\bibitem{CohKapNel91} A. Cohen, D. Kaplan and A. Nelson, (1991)
%\\bibitem{Sha86} M. Shaposhnikov, (1986)
%\bibitem{Sha87} M. Shaposhnikov, (1987)
%\bibitem{DasHasNev74} R. Daschen, B. Haslacher and A. Neveu, (1974)
%\bibitem{Son80} V. Soni, (1980)
%\bibitem{SphOrig} R. Daschen, B. Haslacher and A. Neveu, (1974);
%V. Soni, (1980)
\bibitem{KliMan84} F.R. Klinkhamer and N.S. Manton, {\it Phys.
Rev.} \textbf{D30} 2212 (1984). 
\bibitem{KasPecZha91} B. Kastening, R.D. Peccei, and X. Zhang, {\it Phys.
Lett.} \textbf{B266} 413 (1991).
\bibitem{BacTinTom96} C. Bachas, P. Tinyakov, and T.N. Tomaras, {\it Phys.
Lett.} \textbf{B385} 237 (1996), [hep-ph/9606348]
\bibitem{MorOakQui96} J.M. Moreno, D. Oaknin and M. Quiros,
{\it Nucl. Phys.} {\bf B483} 267 (1996) [hep-ph/9695387]
\bibitem{Kle98} B. Kleihaus, {\it Mod. Phys. Lett.} {\bf A14}
1431 (1999) [hep-ph/9808295]
%\bibitem{HigMas} C.A. Caso {\it et al.}, European Physical Journal {\bf C3} 1
%	 (1998).
\bibitem{Yaf89} L. Yaffe, {\it Phys. Rev.} {\bf D40} 3463 (1989)
\bibitem{GraHin98} J. Grant and M. Hindmarsh, {\it Phy. Rev.} {\bf D59} 116014
(1999) [hep-ph/9811289]
%\bibitem{Man95} N. Manton, in ``Formation and Interactions of 
%	Topological Defects'', NATO ASI Series, eds. R. Brandenberger and 
%	 Davis, (Plenum, New York, 1995).
%\bibitem{AkiKikYan88} T. Akiba, H. Kikuchi, and T. Yanagida,  Phys. Rev. D
%\textbf{38} 1937 (1988).
%\bibitem{NolKun95}
%G. Nolte and J. Kunz, Phys. Rev. D \textbf{51} 3061 (1995), hep-ph/9409445
\bibitem{HHG} J.F. Gunion, H.E. Haber, G. Kane, and S. Dawson,  
	``The Higgs Hunter's Guide'' (Addison-Wesley, Redwood City, 1990).
\bibitem{GraHin00} J. Grant and M. Hindmarsh, in preparation (2000).
%\bibitem{RatYaf88}
%B. Ratra and L.G. Yaffe, \emph{} Phys. Lett. B \textbf{205} 57 (1988)
%\bibitem{NumRecC} W. Press, S. Teukolsky, W. Veterling, and B. Flannery,  
%``{Numerical Recipes in C}''  (Cambridge Univ. Press, Cambridge, 1992).

\end{thebibliography}

\end{document}



hep-ph0101096.extracted_bib
\begin{thebibliography}{99}

\bibitem{Sak67} A.D. Sakharov, {\it JETP Lett.} {\bf 6} 24 (1967).
\bibitem{KuzRubSha85} V.A. Kuzmin, V.A. Rubakov, and M.E. Shaposhnikov, 
{\it Phys. Lett. B} \textbf{155} 36 (1985).
\bibitem{Baryo} V.A Rubakov and M.E. Shaposhnikov, Phys. Usp. {\bf 39}, 
461 (1996)  [{hep-ph/9603208}];
%M. Trodden, {\it Rev. Mod. Phys.} {\bf 71} 1463-1500 (1999) 
% [{hep-ph/9803479}]; A. Riotto, {\it Theories of Baryogenesis},
%Summer School in High Energy Physics and Cosmology, Trieste, (1998)
%[hep-ph/9807454]
%\bibitem{HueNel96} P. Huet and A. Nelson, (1996)
%\bibitem{Car+97} M. Carena et al (1997)
%\bibitem{CliKai00} J. Cline and K. Kainulainen, (2000)
\bibitem{CPMSSM} P. Huet and A.E. Nelson, {\it Phys.
Rev.} {\bf D53} 4578 (1996) [hep-ph/9506477];
M. Carena, M. Quiros and C.E.M. Wagner, {\it Phys. Lett.} {\bf B390} 6919
(1997) [hep-ph/9603420];  J. Cline, M.Joyce and K. Kainulainen,
{\it JHEP} {\bf 0007} 018 (2000)
%\bibitem{EspDeC96} J.R. Espinosa and B. de Carlos, (2000)
%\bibitem{LaiRum98} M. Laine and K. Rummukainen (1998)
%\bibitem{CliMoo98} J. Cline and G. Moore (1998)
\bibitem{PhTranMSSM} B. de Carlos and J.R.Espinosa, {\it Phys. Lett.} {\bf
B407} 12 (1997);
M. Laine and K. Rummukainen, {\it Nucl. Phys.} {\bf B545} 141 
(1999) [hep-ph/9811369]; J.M. Cline and G.D. Moore, {\it Phys. Rev. Lett.} {\bf
81} 3315 (1998) [hep-ph/9806354]
%\bibitem{Kaj+98} K. Kajantie, M. Laine, K.Rummukainen, 
%	M. Shaposhnikov, and M. Tsypin, {hep-ph/9809435}  
%\bibitem{tHo76} G. 't Hooft, Phys. Rev. Lett. {\bf 37} 8 (1976).
%\bibitem{CohKapNel91} A. Cohen, D. Kaplan and A. Nelson, (1991)
%\\bibitem{Sha86} M. Shaposhnikov, (1986)
%\bibitem{Sha87} M. Shaposhnikov, (1987)
%\bibitem{DasHasNev74} R. Daschen, B. Haslacher and A. Neveu, (1974)
%\bibitem{Son80} V. Soni, (1980)
%\bibitem{SphOrig} R. Daschen, B. Haslacher and A. Neveu, (1974);
%V. Soni, (1980)
\bibitem{KliMan84} F.R. Klinkhamer and N.S. Manton, {\it Phys.
Rev.} \textbf{D30} 2212 (1984). 
\bibitem{KasPecZha91} B. Kastening, R.D. Peccei, and X. Zhang, {\it Phys.
Lett.} \textbf{B266} 413 (1991).
\bibitem{BacTinTom96} C. Bachas, P. Tinyakov, and T.N. Tomaras, {\it Phys.
Lett.} \textbf{B385} 237 (1996), [hep-ph/9606348]
\bibitem{MorOakQui96} J.M. Moreno, D. Oaknin and M. Quiros,
{\it Nucl. Phys.} {\bf B483} 267 (1996) [hep-ph/9695387]
\bibitem{Kle98} B. Kleihaus, {\it Mod. Phys. Lett.} {\bf A14}
1431 (1999) [hep-ph/9808295]
%\bibitem{HigMas} C.A. Caso {\it et al.}, European Physical Journal {\bf C3} 1
%	 (1998).
\bibitem{Yaf89} L. Yaffe, {\it Phys. Rev.} {\bf D40} 3463 (1989)
\bibitem{GraHin98} J. Grant and M. Hindmarsh, {\it Phy. Rev.} {\bf D59} 116014
(1999) [hep-ph/9811289]
%\bibitem{Man95} N. Manton, in ``Formation and Interactions of 
%	Topological Defects'', NATO ASI Series, eds. R. Brandenberger and 
%	 Davis, (Plenum, New York, 1995).
%\bibitem{AkiKikYan88} T. Akiba, H. Kikuchi, and T. Yanagida,  Phys. Rev. D
%\textbf{38} 1937 (1988).
%\bibitem{NolKun95}
%G. Nolte and J. Kunz, Phys. Rev. D \textbf{51} 3061 (1995), hep-ph/9409445
\bibitem{HHG} J.F. Gunion, H.E. Haber, G. Kane, and S. Dawson,  
	``The Higgs Hunter's Guide'' (Addison-Wesley, Redwood City, 1990).
\bibitem{GraHin00} J. Grant and M. Hindmarsh, in preparation (2000).
%\bibitem{RatYaf88}
%B. Ratra and L.G. Yaffe, \emph{} Phys. Lett. B \textbf{205} 57 (1988)
%\bibitem{NumRecC} W. Press, S. Teukolsky, W. Veterling, and B. Flannery,  
%``{Numerical Recipes in C}''  (Cambridge Univ. Press, Cambridge, 1992).

\end{thebibliography}