%Paper: 
%From: leandros@cfata3.harvard.edu (leandros perivolaropoulos)
%Date: Tue, 22 Dec 92 15:39:38 -0500


%%                              JNL.TEX
%%
%%                This is JNL.TEX Version 0.3 as of 6/12/85.
%%
%%      This is a set of TeX 82 macros designed to produce scientific
%%      papers with a minimum of fuss and using as much of plain.tex as
%%      possible.  The user need only know what is in the TeXbook, and
%%      the macros under ``user definitions'' below.  Also, the user
%%      definitions are intended to be as simple as possible, so that
%%      the user may change them as desired.


%%
%%  Font definitions suitable for the IMAGEN (Written by Tony Kennedy)
%%

%  Define a whole menagerie of pseudo-12pt fonts

\font\twelverm=cmr10 scaled 1200    \font\twelvei=cmmi10 scaled 1200
\font\twelvesy=cmsy10 scaled 1200   \font\twelveex=cmex10 scaled 1200
\font\twelvebf=cmbx10 scaled 1200   \font\twelvesl=cmsl10 scaled 1200
\font\twelvett=cmtt10 scaled 1200   \font\twelveit=cmti10 scaled 1200

\skewchar\twelvei='177   \skewchar\twelvesy='60

%  Define \...point macros to change fonts and spacings consistently

\def\twelvepoint{\normalbaselineskip=12.4pt
  \abovedisplayskip 12.4pt plus 3pt minus 9pt
  \belowdisplayskip 12.4pt plus 3pt minus 9pt
  \abovedisplayshortskip 0pt plus 3pt
  \belowdisplayshortskip 7.2pt plus 3pt minus 4pt
  \smallskipamount=3.6pt plus1.2pt minus1.2pt
  \medskipamount=7.2pt plus2.4pt minus2.4pt
  \bigskipamount=14.4pt plus4.8pt minus4.8pt
  \def\rm{\fam0\twelverm}          \def\it{\fam\itfam\twelveit}%
  \def\sl{\fam\slfam\twelvesl}     \def\bf{\fam\bffam\twelvebf}%
  \def\mit{\fam 1}                 \def\cal{\fam 2}%
  \def\tt{\twelvett}
  \textfont0=\twelverm   \scriptfont0=\tenrm   \scriptscriptfont0=\sevenrm
  \textfont1=\twelvei    \scriptfont1=\teni    \scriptscriptfont1=\seveni
  \textfont2=\twelvesy   \scriptfont2=\tensy   \scriptscriptfont2=\sevensy
  \textfont3=\twelveex   \scriptfont3=\twelveex  \scriptscriptfont3=\twelveex
  \textfont\itfam=\twelveit
  \textfont\slfam=\twelvesl
  \textfont\bffam=\twelvebf \scriptfont\bffam=\tenbf
  \scriptscriptfont\bffam=\sevenbf
  \normalbaselines\rm}

%       tenpoint

\def\tenpoint{\normalbaselineskip=12pt
  \abovedisplayskip 12pt plus 3pt minus 9pt
  \belowdisplayskip 12pt plus 3pt minus 9pt
  \abovedisplayshortskip 0pt plus 3pt
  \belowdisplayshortskip 7pt plus 3pt minus 4pt
  \smallskipamount=3pt plus1pt minus1pt
  \medskipamount=6pt plus2pt minus2pt
  \bigskipamount=12pt plus4pt minus4pt
  \def\rm{\fam0\tenrm}          \def\it{\fam\itfam\tenit}%
  \def\sl{\fam\slfam\tensl}     \def\bf{\fam\bffam\tenbf}%
  \def\smc{\tensmc}             \def\mit{\fam 1}%
  \def\cal{\fam 2}%
  \textfont0=\tenrm   \scriptfont0=\sevenrm   \scriptscriptfont0=\fiverm
  \textfont1=\teni    \scriptfont1=\seveni    \scriptscriptfont1=\fivei
  \textfont2=\tensy   \scriptfont2=\sevensy   \scriptscriptfont2=\fivesy
  \textfont3=\tenex   \scriptfont3=\tenex     \scriptscriptfont3=\tenex
  \textfont\itfam=\tenit
  \textfont\slfam=\tensl
  \textfont\bffam=\tenbf \scriptfont\bffam=\sevenbf
  \scriptscriptfont\bffam=\fivebf
  \normalbaselines\rm}

%%
%%      Various internal macros
%%

\def\beginlinemode{\endmode
  \begingroup\parskip=0pt \obeylines\def\\{\par}\def\endmode{\par\endgroup}}
\def\beginparmode{\endmode
  \begingroup \def\endmode{\par\endgroup}}
\let\endmode=\par
{\obeylines\gdef\
{}}
\def\singlespace{\baselineskip=\normalbaselineskip}
\def\oneandathirdspace{\baselineskip=\normalbaselineskip
  \multiply\baselineskip by 4 \divide\baselineskip by 3}
\def\oneandahalfspace{\baselineskip=\normalbaselineskip
  \multiply\baselineskip by 3 \divide\baselineskip by 2}
\def\doublespace{\baselineskip=\normalbaselineskip \multiply\baselineskip by 2}
\def\triplespace{\baselineskip=\normalbaselineskip \multiply\baselineskip by 3}
\newcount\firstpageno
\firstpageno=2
\footline={\ifnum\pageno<\firstpageno{\hfil}%
\else{\hfil\twelverm\folio\hfil}\fi}
\let\rawfootnote=\footnote              % We must set the footnote style
\def\footnote#1#2{{\rm\singlespace\parindent=0pt\rawfootnote{#1}{#2}}}
\def\raggedcenter{\leftskip=4em plus 12em \rightskip=\leftskip
  \parindent=0pt \parfillskip=0pt \spaceskip=.3333em \xspaceskip=.5em
  \pretolerance=9999 \tolerance=9999
  \hyphenpenalty=9999 \exhyphenpenalty=9999 }
\def\dateline{\rightline{\ifcase\month\or
  January\or February\or March\or April\or May\or June\or
  July\or August\or September\or October\or November\or December\fi
  \space\number\year}}
\def\received{\vskip 3pt plus 0.2fill
 \centerline{\sl (Received\space\ifcase\month\or
  January\or February\or March\or April\or May\or June\or
  July\or August\or September\or October\or November\or December\fi
  \qquad, \number\year)}}

%%
%%      Page layout, margins, font and spacing (feel free to change)
%%

\hsize=6.5truein
%\hoffset=1truein
\vsize=8.9truein
%\voffset=1truein
\parskip=\medskipamount
\twelvepoint            % selects twelvepoint fonts (cf. \tenpoint)
\doublespace            % selects double spacing for main part of paper (cf.
                        %       \singlespace, \oneandahalfspace)
\overfullrule=0pt       % delete the nasty little black boxes for overfull box

%%
%%      The user definitions for major parts of a paper (feel free to change)
%%

\def\preprintno#1{
 \rightline{\rm #1}}    % Preprint number at upper right of title page

\def\title                      %  Title on title page
  {\null\vskip 3pt plus 0.2fill
   \beginlinemode \doublespace \raggedcenter \bf}

\def\author                     %  Author(s) name(s)  on title page
  {\vskip 3pt plus 0.2fill \beginlinemode
   \singlespace \raggedcenter}

\def\affil                      % Affiliations (can intermix with \author)
  {\vskip 3pt plus 0.1fill \beginlinemode
   \oneandahalfspace \raggedcenter \sl}

\def\abstract                   % Begin abstract
  {\vskip 3pt plus 0.3fill \beginparmode
   \doublespace \narrower ABSTRACT: }

\def\endtitlepage               % End title page, begin body of paper
  {\endpage                     %       This subsumes \body
   \body}

\def\body                       % Begin text body;  can be used to end
  {\beginparmode}               % \title, \author, \affil, \abstract,
                                % \reference, or \figurecaption modes

\def\head#1{                    % Head;  NOTE enclose the text in {}
  \filbreak\vskip 0.5truein     %  e.g., \head{I. Introduction}
  {\immediate\write16{#1}
   \raggedcenter \uppercase{#1}\par}
   \nobreak\vskip 0.25truein\nobreak}

\def\subhead#1{                 % Subhead;  NOTE enclose the text in {}
  \vskip 0.25truein             % e.g., \subhead{A. History of the Problem}
  {\raggedcenter #1 \par}
   \nobreak\vskip 0.25truein\nobreak}

\def\refto#1{$|{#1}$}           % For references in text as superscript

\def\references                 % Begin references -- basic format is Phys Rev
  {\subhead{References}         % I.e., volume, page, year (space after commas)
   \beginparmode
   \frenchspacing \parindent=0pt \leftskip=1truecm
   \parskip=8pt plus 3pt \everypar{\hangindent=\parindent}}

\gdef\refis#1{\indent\hbox to 0pt{\hss#1.~}}    % Ref list numbers.

\gdef\journal#1, #2, #3, 1#4#5#6{               % Journal reference.  Comma set
    {\sl #1~}{\bf #2}, #3, (1#4#5#6)}           % off: name, vol, page, year

\def\refstylenp{                % Nucl Phys(or Phys Lett) ref style: V, Y, P
  \gdef\refto##1{ [##1]}                                % Reference in text []
  \gdef\refis##1{\indent\hbox to 0pt{\hss##1)~}}        % Ref list numbers)
  \gdef\journal##1, ##2, ##3, ##4 {                     % Journal reference
     {\sl ##1~}{\bf ##2~}(##3) ##4 }}

\def\refstyleprnp{              % Input like pr, output like np!!
  \gdef\refto##1{ [##1]}                                % Reference in text []
  \gdef\refis##1{\indent\hbox to 0pt{\hss##1)~}}        % Ref list numbers)
  \gdef\journal##1, ##2, ##3, 1##4##5##6{               % Journal reference
    {\sl ##1~}{\bf ##2~}(1##4##5##6) ##3}}

\def\pr{\journal Phys. Rev., }

\def\pra{\journal Phys. Rev. A, }

\def\prb{\journal Phys. Rev. B, }

\def\prc{\journal Phys. Rev. C, }

\def\prd{\journal Phys. Rev. D, }

\def\prl{\journal Phys. Rev. Lett., }

\def\jmp{\journal J. Math. Phys., }

\def\rmp{\journal Rev. Mod. Phys., }

\def\cmp{\journal Comm. Math. Phys., }

\def\np{\journal Nucl. Phys., }

\def\pl{\journal Phys. Lett., }

\def\endreferences{\body}

\def\figurecaptions             % Begin figure captions
  { \beginparmode
   \subhead{Figure Captions}
%  \parskip=24pt plus 3pt \everypar={\hangindent=4em}
}

\def\endfigurecaptions{\body}

\def\endpage                    %  Eject a page
  {\vfill\eject}

\def\endpaper                   %  Ways to say goodbye
  {\endmode\vfill\supereject}
\def\endjnl
  {\endpaper}
\def\endit
  {\endpaper\end}



%%
%%      Various little user definitions
%%

\def\ref#1{Ref. #1}                     %       for inline references
\def\Ref#1{Ref. #1}                     %       ditto
\def\fig#1{Fig. #1}
\def\Equation#1{Equation (#1)}          % For citation of equation numbers
\def\Equations#1{Equations (#1)}        %       ditto
\def\Eq#1{Eq. (#1)}                     %       ditto
\def\eq#1{Eq. (#1)}                     %       ditto
\def\Eqs#1{Eqs. (#1)}                   %       ditto
\def\eqs#1{Eqs. (#1)}                   %       ditto
\def\frac#1#2{{\textstyle{#1 \over #2}}}
\def\half{{\textstyle{ 1\over 2}}}
\def\eg{{\it e.g.,\ }}
\def\Eg{{\it E.g.,\ }}
\def\ie{{\it i.e.,\ }}
\def\Ie{{\it I.e.,\ }}
\def\etal{{\it et al.}}
\def\etc{{\it etc.}}
\def\via{{\it via}}
\def\sla{\raise.15ex\hbox{$/$}\kern-.57em}
\def\leaderfill{\leaders\hbox to 1em{\hss.\hss}\hfill}
\def\twiddle{\lower.9ex\rlap{$\kern-.1em\scriptstyle\sim$}}
\def\bigtwiddle{\lower1.ex\rlap{$\sim$}}
\def\gtwid{\mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}}}
\def\ltwid{\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}}}
\def\square{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt
   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt}
\def\ucsb{Department of Physics\\University of California\\
Santa Barbara CA 93106-9530}
\def\ucsd{Department of Physics\\University of California\\
La Jolla, CA 92093}
\def\begintable{\offinterlineskip\hrule}
\def\tablespace{height 2pt&\omit&&\omit&&\omit&\cr}
\def\tablerule{\tablespace\noalign{\hrule}\tablespace}
\def\endtable{\hrule}
\def\prim{{\scriptscriptstyle{\prime}}}
\def\comp{{\rm C}\llap{\vrule height7.1pt width1pt depth-.4pt\phantom t}}
\def\fint{\rlap{$\biggl\rfloor$}\biggl\lceil}
\def\slash#1{\rlap{$#1$}\thinspace /}
\def\m@th{\mathsurround=0pt }
\def\leftrightarrowfill{$\m@th \mathord\leftarrow \mkern-6mu
 \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill
 \mkern-6mu \mathord\rightarrow$}
\def\overleftrightarrow#1{\vbox{\ialign{##\crcr
     \leftrightarrowfill\crcr\noalign{\kern-1pt\nointerlineskip}
     $\hfil\displaystyle{#1}\hfil$\crcr}}}

%% *********** New stuff follows *******************

\font\titlefont=cmr10 scaled\magstep3
\def\uof{Department of Physics\\University of Florida\\
Gainesville, FL 32608}
\def\grant{This research was supported in part by the
Institute for Fundamental Theory and by DOE contract
DE-FG05-86-ER40272.}

\def\oneandfourfifthsspace{\baselineskip=\normalbaselineskip
  \multiply\baselineskip by 9 \divide\baselineskip by 5}

\def\martinstyletitle                      %  Title on title page
  {\null\vskip 3pt plus 0.2fill
   \beginlinemode \doublespace \raggedcenter \titlefont}

\font\twelvesc=cmcsc10 scaled 1200

\def\author                     %  Author(s) name(s)  on title page
  {\vskip 3pt plus 0.2fill \beginlinemode
   \singlespace \raggedcenter\twelvesc}






%%
%%      AmSTeX compatability definitions
%%
%%      To run a TeX file originally intended for AmSTeX, only small changes
%%      should be necessary (I hope).  Use the line \input jnl at the start.
%%      Remove the lines \input amstex, \documentstyle{itpjnl} at the
%%      beginning;  also remove all the page layout stuff (\parindent=1cm,
%%      \hsize=5.28125in etc.)  The page layout is now done automatically.
%%      Also OMIT the qualifier \magnification=1200 when you IMPRINT the
%%      .dvi file.  (\TagsOnRight is harmless, you can take it out or leave
%%      it in.)  I believe most AmSTeX will work with no change.  One problem
%%      is \footnote, which is a little different in that it now needs to
%%      have an explicit asterisk *  (or whatever) included, like this:
%%              \footnote*{Text winds up at bottom of page.}
%%      This is discussed on p. 116 of the TeXbook.  IGNORE the AmSTeX
%%      documentation (if you can call it that);  refer to the TeXbook.
%%
%%      Note that many commands in AmSTeX have their equivalents in the
%%      TeXbook, perhaps with different names and slightly differing
%%      usage. E.g., the old \align in AmSTeX is replaced by \eqalign
%%      (p. 190) and \aligntag is replaced by \eqalignno (p. 192).
%%      \align and \aligntag still work, but I recommend that you use
%%      \eqalign and \eqalignno in documents run under jnl.
%%
%%      See me if you have any problems  -- Doug.
%%

\def\TagsOnRight{}
\def\topmatter{}
\def\endtitle{\body}
\def\endauthor{\body}
\def\endaffil{\body}

\def\heading                            % Heading
  {\vskip 0.5truein plus 0.1truein      % e.g., \heading I. NOTES \endheading
   \beginparmode \def\\{\par} \parskip=0pt \singlespace \raggedcenter}

\def\endheading
  {\par\nobreak\vskip 0.25truein\nobreak\beginparmode}

\def\subheading                         % Subheading
  {\vskip 0.25truein plus 0.1truein     % e.g., \subheading{A. The Problem}
   \beginlinemode \singlespace \parskip=0pt \def\\{\par}\raggedcenter}

\def\endsubheading
  {\par\nobreak\vskip 0.25truein\nobreak\beginparmode}

\def\tag#1$${\eqno(#1)$$}

\def\align#1$${\eqalign{#1}$$}

\def\endalign{\cr}

\def\aligntag#1$${\gdef\tag##1\\{&(##1)\cr}\eqalignno{#1\\}$$
  \gdef\tag##1$${\eqno(##1)$$}}

\def\endaligntag{}

\def\binom#1#2{{#1 \choose #2}}

\def\stack#1#2{{#1 \atop #2}}

\def\overset #1\to#2{{\mathop{#2}\limits^{#1}}}
\def\underset#1\to#2{{\let\next=#1\mathpalette\undersetpalette#2}}
\def\undersetpalette#1#2{\vtop{\baselineskip0pt
\ialign{$\mathsurround=0pt #1\hfil##\hfil$\crcr#2\crcr\next\crcr}}}

\def\enddocument{\endit}

%%
%%      Various little user definitions
%%

\def\ref#1{Ref.~#1}                     %       for inline references
\def\Ref#1{Ref.~#1}                     %       ditto
\def\[#1]{[\cite{#1}]}
\def\cite#1{{#1}}
%%\def\Equation#1{Equation~(#1)}                % For citation of equation numb
%%\def\Equations#1{Equations~(#1)}      %       ditto
%%\def\Eq#1{Eq.~(#1)}                   %       ditto
%%\def\Eqs#1{Eqs.~(#1)}                 %       ditto
\def\(#1){(\call{#1})}
\def\call#1{{#1}}
\def\taghead#1{}
\def\frac#1#2{{#1 \over #2}}
\def\half{{\frac 12}}
\def\third{{\frac 13}}
\def\fourth{{\frac 14}}
\def\12{{1\over2}}
\def\eg{{\it e.g.,\ }}
\def\Eg{{\it E.g.,\ }}
\def\ie{{\it i.e.,\ }}
\def\Ie{{\it I.e.,\ }}
\def\etal{{\it et al.\ }}
\def\etc{{\it etc.\ }}
\def\via{{\it via\ }}
\def\cf{{\sl cf.\ }}
\def\sla{\raise.15ex\hbox{$/$}\kern-.57em}
\def\leaderfill{\leaders\hbox to 1em{\hss.\hss}\hfill}
\def\twiddle{\lower.9ex\rlap{$\kern-.1em\scriptstyle\sim$}}
\def\bigtwiddle{\lower1.ex\rlap{$\sim$}}
\def\gtwid{\mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}}}
\def\ltwid{\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}}}
\def\square{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt
   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt}
\def\tdot#1{\mathord{\mathop{#1}\limits^{\kern2pt\ldots}}}
\def\super#1{$^{#1}$}
\def\pmb#1{\setbox0=\hbox{#1}%
  \kern-.025em\copy0\kern-\wd0
  \kern  .05em\copy0\kern-\wd0
  \kern-.025em\raise.0433em\box0 }
\def\qed{\vrule height 1.2ex width 0.5em}
\def\const{{\rm const}}
\def\itp{Institute for Theoretical Physics}
\def\itpucsb{\itp\\University of California\\Santa Barbara, %
California 93106-9530}
\def\itpgrant{This research was supported in part by the National
Science Foundation under Grant No.~PHY82-17853,
supplemented by funds from the National Aeronautics and Space
Administration, at the University of California at Santa Barbara.}






%%  EQNORDER.TEX   11/05/85 Doug E.
%%
%% This macro package is intended for use with JNL.
%% It will automatically order and sort the equations in a paper
%% by order of appearance.  To use, say \input eqnorder
%% after \input jnl (and after all definitions of \eqno etc.,
%% but before any use of \eqno etc.  Use \() to cite equations
%% in the text.  Use \eqno() or \tag to put the numbers on displayed
%% equations; or use &() with \eqalignno{} as explained in the TeXBOOK.
%%
%%      EQNORDER depends on the macro \() to refer to equations in the
%% text; use it as Equation \() or Eq. \() or Eqs. \(), etc.
%% EQNORDER also contains a macro \call{} which can be used to refer
%% equations; e.g., ``Equation \call{19} blah...'' will produce
%%  output ``Equation 19 blah''.  Multiple citations must be separated
%% by commas.  E.g., \(24,26,27) and \call{3,7} are legal.  A sequence
%% of equation numbers can be referred to by, e.g., \(3-7) which means
%% the same as (3,4,5,6,7).
%%
%% Equation ``numbers'' can actually be any alphanumeric string;
%% e.g., equation \tag Schroedinger $$ can be referred to by
%% \(Schroedinger).  In fact, if you expect to renumber the equations,
%% it is actually easier and less confusing to tag them with names
%% rather than numbers.
%%
%% There is one big rule:  You cannot refer to an equation before you
%% display it.  There is a limited loophole:  You can refer to the
%% first, second or third equation number just below where you are
%% as \(+1), \(+2), or \(+3).  In the same way, the equation number
%% just above can be called \(0), and the three preceding numbers
%% \(-1), \(-2), \(-3).
%%
%% TeX keeps the equation number as a count \tagnumber.  \tagnumber
%% is initially 0, and it is incremented by 1 just BEFORE it is used
%% to tag a displayed equation.  You are free to reset \tagnumber,
%% which you can do just by writing e.g. \tagnumber=23.
%%
%% If you label a displayed equation with a null number, \tag $$
%% or \eqno() or &(), an incremented \tagnumber will be generated for
%% the equation, but the only way to refer to that equation is
%% via the \(+n) or \(0) or \(-n) notation.
%%
%% In long papers, the author often numbers the equations anew
%% in each section in the style \tag 6.1 $$, \tag 6.2 $$, and so
%% forth;  the equations are then referred to by \(6.1) etc.
%% One reason for doing this is to minimize chaos when equations
%% have to be renumbered --- but this is what EQNORDER already does!
%% If you still want to use such a style, just declare \taghead{6.}
%% for example at the beginning of Section 6.  The effect of \taghead
%% is to reset \tagnumber to 0, and to save the argument of \taghead
%% to so that it can be put in front of each equation number in the
%% output.
%%
%% Sometimes a sequence of displayed equations is labelled with
%% the same number (e.g., 25) and then sublabeled a,b,c,d...  Use
%% the form \tag 25 a$$, \tag 25 b$$,... or \eqno(25 a)$$,
%% \eqno(25 b)$$,... to put the numbers on such a sequence;  note
%% the space.  Such equations can be referred to in the text either
%% as \(25) or \(25 a).  Also such constructions as \(25 a,25 b,26)
%% are legal.  Again, note carefully the position of the space.  The
%% effect of the space is to mark the end of the equation number that
%% TeX keeps track of;  the following string (a or b or ...) is just
%% put out without modification.  Thus constructions like \tag 25 ' $$
%% and \tag 25 '' $$ are legal.
%%
%% If you have your own pet macros to call equations such as, e.g.,
%% \def\eqnpet#1{($#1$))}, you can bring it to the attention of
%% REFORDER so all \eqnpet's will be properly calld simply by
%% declaring ``\callall\eqnpet'' after \eqnpet is defined and
%% after \input eqnorder.  This has the effect of redefining the macro
%% as e.g., \def\eqnpet#1{($\call{#1}$)}.  (Such \callall'ed macros
%% must have exactly one argument #1, as in \eqnpet.)

\catcode`@=11
\newcount\tagnumber\tagnumber=0

\immediate\newwrite\eqnfile
\newif\if@qnfile\@qnfilefalse
\def\write@qn#1{}
\def\writenew@qn#1{}
\def\w@rnwrite#1{\write@qn{#1}\message{#1}}
\def\@rrwrite#1{\write@qn{#1}\errmessage{#1}}

\def\taghead#1{\gdef\t@ghead{#1}\global\tagnumber=0}
\def\t@ghead{}

\expandafter\def\csname @qnnum-3\endcsname
  {{\t@ghead\advance\tagnumber by -3\relax\number\tagnumber}}
\expandafter\def\csname @qnnum-2\endcsname
  {{\t@ghead\advance\tagnumber by -2\relax\number\tagnumber}}
\expandafter\def\csname @qnnum-1\endcsname
  {{\t@ghead\advance\tagnumber by -1\relax\number\tagnumber}}
\expandafter\def\csname @qnnum0\endcsname
  {\t@ghead\number\tagnumber}
\expandafter\def\csname @qnnum+1\endcsname
  {{\t@ghead\advance\tagnumber by 1\relax\number\tagnumber}}
\expandafter\def\csname @qnnum+2\endcsname
  {{\t@ghead\advance\tagnumber by 2\relax\number\tagnumber}}
\expandafter\def\csname @qnnum+3\endcsname
  {{\t@ghead\advance\tagnumber by 3\relax\number\tagnumber}}

\def\equationfile{%
  \@qnfiletrue\immediate\openout\eqnfile=\jobname.eqn%
  \def\write@qn##1{\if@qnfile\immediate\write\eqnfile{##1}\fi}
  \def\writenew@qn##1{\if@qnfile\immediate\write\eqnfile
    {\noexpand\tag{##1} = (\t@ghead\number\tagnumber)}\fi}
}

\def\callall#1{\xdef#1##1{#1{\noexpand\call{##1}}}}
\def\call#1{\each@rg\callr@nge{#1}}

\def\each@rg#1#2{{\let\thecsname=#1\expandafter\first@rg#2,\end,}}
\def\first@rg#1,{\thecsname{#1}\apply@rg}
\def\apply@rg#1,{\ifx\end#1\let\next=\relax%
\else,\thecsname{#1}\let\next=\apply@rg\fi\next}

\def\callr@nge#1{\calldor@nge#1-\end-}
\def\callr@ngeat#1\end-{#1}
\def\calldor@nge#1-#2-{\ifx\end#2\@qneatspace#1 %
  \else\calll@@p{#1}{#2}\callr@ngeat\fi}
\def\calll@@p#1#2{\ifnum#1>#2{\@rrwrite{Equation range #1-#2\space is bad.}
\errhelp{If you call a series of equations by the notation M-N, then M and
N must be integers, and N must be greater than or equal to M.}}\else %
{\count0=#1\count1=#2\advance\count1 by1\relax\expandafter\@qncall\the\count0,%
  \loop\advance\count0 by1\relax%
    \ifnum\count0<\count1,\expandafter\@qncall\the\count0,%
  \repeat}\fi}

\def\@qneatspace#1#2 {\@qncall#1#2,}
\def\@qncall#1,{\ifunc@lled{#1}{\def\next{#1}\ifx\next\empty\else
  \w@rnwrite{Equation number \noexpand\(>>#1<<) has not been defined yet.}
  >>#1<<\fi}\else\csname @qnnum#1\endcsname\fi}

\let\eqnono=\eqno
\def\eqno(#1){\tag#1}
\def\tag#1$${\eqnono(\displayt@g#1 )$$}

\def\aligntag#1\endaligntag
  $${\gdef\tag##1\\{&(##1 )\cr}\eqalignno{#1\\}$$
  \gdef\tag##1$${\eqnono(\displayt@g##1 )$$}}

\let\eqalignnono=\eqalignno

\def\eqalignno#1{\displ@y \tabskip\centering
  \halign to\displaywidth{\hfil$\displaystyle{##}$\tabskip\z@skip
    &$\displaystyle{{}##}$\hfil\tabskip\centering
    &\llap{$\displayt@gpar##$}\tabskip\z@skip\crcr
    #1\crcr}}

\def\displayt@gpar(#1){(\displayt@g#1 )}

\def\displayt@g#1 {\rm\ifunc@lled{#1}\global\advance\tagnumber by1
        {\def\next{#1}\ifx\next\empty\else\expandafter
        \xdef\csname @qnnum#1\endcsname{\t@ghead\number\tagnumber}\fi}%
  \writenew@qn{#1}\t@ghead\number\tagnumber\else
        {\edef\next{\t@ghead\number\tagnumber}%
        \expandafter\ifx\csname @qnnum#1\endcsname\next\else
        \w@rnwrite{Equation \noexpand\tag{#1} is a duplicate number.}\fi}%
  \csname @qnnum#1\endcsname\fi}

\def\ifunc@lled#1{\expandafter\ifx\csname @qnnum#1\endcsname\relax}

\let\@qnend=\end\gdef\end{\if@qnfile
\immediate\write16{Equation numbers written on []\jobname.EQN.}\fi\@qnend}

\catcode`@=12

%% DEBUG
%%\def\see#1 {\expandafter\show\csname#1\endcsname}



%%  REFORDER.TEX   6/7/85 Doug E.
%%     (mods: 3/25/87 R.G.Palmer)
%%
%% This macro package is intended for use with JNL.
%% It will automatically order and sort the references in a paper
%% by order of first citation.(!!)  To use, say \input reforder
%% after \input jnl (and after all definitions of \refto etc.,
%% in particular after any use of the \refstyleXX macros),
%% but before any use of \refto etc.  Use \refto{} (or \ref{} and
%% \Ref{}) to cite references in the text.  Use \refis{} to supply
%% the references,  SKIP A LINE after each reference.  Open the
%% reference listing with \references and close it with \endreferences.
%%
%% REFORDER depends on the
%% JNL macros \refto{}, \ref{}, \Ref{} to identify citation of references.
%% REFORDER also contains a macro \cite{} which can be used to cite
%% references; e.g., ``Reference \cite{19} blah...'' will produce
%%  output ``Reference 19 blah''.  Multiple citations can be separated
%% by commas.  E.g., \refto{24,26,27} and \cite{3,7}
%% are legal.  Also legal is \refto{3-7}, which expands to mean the same
%% as \refto{3,4,5,6,7}.  Reference ``numbers'' can in general be any
%% alphanumeric string; e.g. BjorkenAndDrell is perfectly OK used in
%% the form\ref{BjorkenAndDrell};  such strings should contain no blanks.
%%
%% If you have your own pet macros to cite references such as, e.g.,
%% \def\referpet#1{$^(#1)$)}, you can bring it to the attention of
%% REFORDER so all \referpet's will be properly cited simply by
%% declaring ``\citeall\referpet'' once near the beginning, after
%% \referpet is defined and
%% after \input reforder.  This has the effect of redefining the macro
%% as e.g., \def\referpet#1{$^(\cite{#1})$}.  (Such \citeall'ed macros
%% must have exactly one argument #1, as in \referpet.)  See e.g.,
%% the end of this file where \refto, \ref and \Ref are \citeall'ed.
%%
%% REFORDER depends on the macro \refis{} to supply each reference.
%%      \refis{} can be used to supply a reference anywhere in the paper
%% after its first citation.  The macro \endreferences actually triggers
%% sorting and listing of references.  Skip a line after a reference
%% listing (or, alternatively, end each listing in \par).
%%
%% Use \ignoreuncited after \input reforder if you wish to ignore
%% references that are supplied but not cited.  This is particularly
%% useful if you maintain a master file of references (each supplied
%% with \refis{}) but only use a subset of these in a given paper.
%%      Include your reference file (with \input) between \references
%%      and \endreferences.
%%
%% Use \referencefile after \input reforder if you want an ordered
%%      source listing of the references in file <name>.ref
%%
%% The \reftorange macro can be used to produce a superscript
%% reference range, like $^{10-15}$.  (The \refto macro always
%% lists the references one by one, even for e.g. \refto{10-15}).
%% Use e.g. \reftorange{10}{11-14}{15} -- the references in the
%% middle group are cited but only 10-15 appears in the text.
%%      Note that \reftorange does NOT check for increasing order.

\catcode`@=11
\newcount\r@fcount \r@fcount=0
\newcount\r@fcurr
\immediate\newwrite\reffile
\newif\ifr@ffile\r@ffilefalse
\def\w@rnwrite#1{\ifr@ffile\immediate\write\reffile{#1}\fi\message{#1}}

\def\writer@f#1>>{}
\def\referencefile{%     Stuff to write .REF file
  \r@ffiletrue\immediate\openout\reffile=\jobname.ref%
  \def\writer@f##1>>{\ifr@ffile\immediate\write\reffile%
    {\noexpand\refis{##1} = \csname r@fnum##1\endcsname = %
     \expandafter\expandafter\expandafter\strip@t\expandafter%
     \meaning\csname r@ftext\csname r@fnum##1\endcsname\endcsname}\fi}%
  \def\strip@t##1>>{}}
\let\referencelist=\referencefile

\def\citeall#1{\xdef#1##1{#1{\noexpand\cite{##1}}}}
\def\cite#1{\each@rg\citer@nge{#1}} % Variable No. of args, separated by

\def\each@rg#1#2{{\let\thecsname=#1\expandafter\first@rg#2,\end,}}
\def\first@rg#1,{\thecsname{#1}\apply@rg} % each@ag is a general purpose
\def\apply@rg#1,{\ifx\end#1\let\next=\relax%   variable no. of arg. macro.
\else,\thecsname{#1}\let\next=\apply@rg\fi\next}% args separated by commas

\def\citer@nge#1{\citedor@nge#1-\end-} % Check for M-N range (M and N numbers)
\def\citer@ngeat#1\end-{#1}
\def\citedor@nge#1-#2-{\ifx\end#2\r@featspace#1 % Single argument
  \else\citel@@p{#1}{#2}\citer@ngeat\fi} % M-N range of arguments
\def\citel@@p#1#2{\ifnum#1>#2{\errmessage{Reference range #1-#2\space is bad.}%
    \errhelp{If you cite a series of references by the notation M-N, then M and
    N must be integers, and N must be greater than or equal to M.}}\else%
 {\count0=#1\count1=#2\advance\count1 by1\relax\expandafter\r@fcite\the\count0,
  \loop\advance\count0 by1\relax%   Loop from M to N
    \ifnum\count0<\count1,\expandafter\r@fcite\the\count0,%
  \repeat}\fi}

\def\r@featspace#1#2 {\r@fcite#1#2,} % Eat spaces at beginning or end of arg
\def\r@fcite#1,{\ifuncit@d{#1}%    Cite individual reference
    \newr@f{#1}%
    \expandafter\gdef\csname r@ftext\number\r@fcount\endcsname%
                     {\message{Reference #1 to be supplied.}%
                      \writer@f#1>>#1 to be supplied.\par}%
 \fi%
 \csname r@fnum#1\endcsname}
\def\ifuncit@d#1{\expandafter\ifx\csname r@fnum#1\endcsname\relax}%
\def\newr@f#1{\global\advance\r@fcount by1%
    \expandafter\xdef\csname r@fnum#1\endcsname{\number\r@fcount}}

\let\r@fis=\refis   % Save old \refis, redefine
\def\refis#1#2#3\par{\ifuncit@d{#1}%      Use two params #2 #3 to strip blank
   \newr@f{#1}%
   \w@rnwrite{Reference #1=\number\r@fcount\space is not cited up to now.}\fi%
  \expandafter\gdef\csname r@ftext\csname r@fnum#1\endcsname\endcsname%
  {\writer@f#1>>#2#3\par}}

\def\ignoreuncited{%   redefine \refis if ignoring uncited references
   \def\refis##1##2##3\par{\ifuncit@d{##1}%
    \else\expandafter\gdef\csname r@ftext\csname r@fnum##1\endcsname\endcsname%
     {\writer@f##1>>##2##3\par}\fi}}

\def\r@ferr{\endreferences\errmessage{I was expecting to see
\noexpand\endreferences before now;  I have inserted it here.}}
\let\r@ferences=\references
\def\references{\r@ferences\def\endmode{\r@ferr\par\endgroup}}

\let\endr@ferences=\endreferences
\def\endreferences{\r@fcurr=0%    Save old \endreferences, redefine
  {\loop\ifnum\r@fcurr<\r@fcount%   Loop over refnum and produce text
    \advance\r@fcurr by 1\relax\expandafter\r@fis\expandafter{\number\r@fcurr}%
    \csname r@ftext\number\r@fcurr\endcsname%
  \repeat}\gdef\r@ferr{}\endr@ferences}

% Save old \endpaper, redefine it to write parting message.

\let\r@fend=\endpaper\gdef\endpaper{\ifr@ffile
\immediate\write16{Cross References written on []\jobname.REF.}\fi\r@fend}

\catcode`@=12

\def\reftorange#1#2#3{$^{\cite{#1}-\setbox0=\hbox{\cite{#2}}\cite{#3}}$}

\citeall\refto  % These macros will generate citations
\citeall\ref  %
\citeall\Ref  %


%\input jnl.tex
%\input reforder.tex
%
%\input [tanmay.texing]/reforder.tex
%\input /home/tanmay/.texing/reforder.tex
\vglue 0. truein
\title
{
Detailed Stability Analysis of Electroweak Strings
}
\author
{Margaret James}
\affil
{
D.A.M.T.P., Silver Street,
University of Cambridge, Cambridge, CB39EW, U.K.
}
\smallskip
\author
{Leandros Perivolaropoulos}
\affil
{
Division of Theoretical Astrophysics,
Harvard-Smithsonian Center for Astrophysics,
60 Garden Street, Cambridge, MA 02138.
}
\smallskip
\author
{Tanmay Vachaspati}
\affil
{
Tufts Institute of Cosmology, Department of Physics and Astronomy,
Tufts University, Medford, MA 02155.
}

\abstract
%\doublespace

We give a detailed stability analysis
of the Z-string in the standard
electroweak model. We identify the mode that determines the
stability of the string and numerically map the region of parameter
space where the string is stable. For $\sin^2 \theta_W
= 0.23$, we find that the strings are unstable for a
Higgs mass larger than 23GeV. Given the latest constraints
on the Higgs mass from LEP, this shows that, if the standard
electroweak model is realized in Nature, the existing vortex solutions
are unstable.


\beginsection{1. Introduction}

Recent studies have shown that the stability of topological defects may persist
when they are embedded in theories where they are not topologically stable.
Such ``embedded defects'' are exact solutions of the equations of motion in
the theories where they are embedded and can be dynamically stable.

A typical example of such a defect is the
semilocal string [1,2],
an embedding of the Nielsen-Olesen vortex in a ``semilocal'' model with
$SU(2)_{global}\times U(1)_{local}\rightarrow U(1)_{global}$
symmetry breaking. The semilocal string has been shown to be stable for a
finite parameter sector [3,4]
even though the vacuum manifold in the semilocal
model is $S^3$. In fact, since $\pi_1(S^3)=1$, the stability of the
semilocal string is dynamical rather than topological.
The crucial reason for this stability is that for a certain
parameter region, the
increase in gradient energy necessary for the string to decay
is more than the
corresponding decrease in
potential energy. Therefore, for that
parameter region, the decay would increase the total energy and is not
favored energetically.

The obvious generalization of the semilocal model is the electroweak model
in which the $SU(2)$ symmetry becomes gauged: $SU(2)_L \times U(1)_Y
\rightarrow U(1)_{em}$. It is possible to show that
the Nielsen-Olesen vortex may be embedded in the electroweak model
i.e. the electroweak equations of motion with a vortex-like ansatz
reduce to the Nielsen-Olesen equations [5,6].
Two such embedded vortex solutions - the so-called $\tau -$
and $Z-$ strings - are known [7]. Here we shall only consider the
$Z-$string as the $\tau -$string is expected to be unstable
for all values of the parameters [7].

The phenomenological successes of the standard electroweak
model [8] make the question of stability of the electroweak string
a very interesting and important one. A stable electroweak string would
for the first time in particle physics provide a macroscopic, stable
coherent state at low enough energies to be accessible to particle physics
experiments. It is this question of stability [9] that we
are addressing in detail in this paper. In particular, we construct a map
of the parameter space of the standard electroweak model showing the
range of parameters where the $Z-$string is stable and where it is
unstable. Here we give the details of the calculation as well as
the physical reasoning for the simplifications that occur in this
originally highly complicated problem. A brief exposition of the
main results we derive here may be found in Ref.[15].

The structure of the paper is the following: in the next section we give
a review of the electroweak string showing that the electroweak
equations of motion reduce to the Nielsen-Olesen equations for a
particular vortex ansatz. In section 3 we show how can the stability
problem of the electroweak string be reduced to the eigenvalue problem
of a single Schroedinger-like eigenvalue problem.
This is a non-trivial simplification since the initial system
of coupled perturbations involves twenty coupled degrees of freedom
which after tedious manipulations not only decouple but also reduce to
a single eigenvalue equation. Finally in section 4
we solve this eigenvalue problem and construct a map showing the
parameter sector corresponding to stability.


\beginsection{2. Review of Electroweak Strings}
\taghead{2.}

We consider static bosonic field configurations in the Weinberg-Salam model:
there is no time dependence and we choose a gauge where the zero components of
the gauge fields are set
to zero. The energy functional is given by
$$
E = \int d^3 x\left [
          \fourth G_{ij} ^a G_{ij} ^a + \fourth F_{Bij} F_{Bij}
          + (D_j \phi ) ^{\dag} (D_j \phi ) +
            \lambda (\phi ^{\dag} \phi - \eta ^2 /2 )^2
              \right ]
\eqno (2.1)
$$

The Weinberg
angle is given by tan $\theta_w={g'/g}$. The masses of the
W-boson, the Z-boson and Higgs boson are, respectively,
$$
M_W={1\over
2}g\eta ,\qquad\ M_Z={1\over 2}\alpha\eta ,\qquad
M_H=\sqrt{2\lambda}\eta\ ,
\eqno(2.2)
$$
where
$$
\alpha \equiv \sqrt{ g^2 + {g'} ^2 }
\eqno(2.3)
$$
The time-independent field equations are
$$
D_j F^a_{ij}=- {\textstyle {1\over2}}ig(\phi^{\dagger}
\ \tau^a D_i \phi - (D_i \phi)^{\dagger} \ \tau^a \phi)
\eqno(2.4)
$$
$$
\partial_jf_{ij}=- {\textstyle{1\over2}}ig'\left(\phi^{\dagger}D_i\phi-(D_i
\phi)^{\dagger}\phi\right)
\eqno(2.5)
$$
$$
D_iD_i\phi=
       2\lambda(\phi^{\dagger}\phi-{\textstyle{1\over 2}}\eta^2)\phi\ ,
\eqno(2.6)
$$
The symbols are in the standard notation
defined in Ref. 10.
In addition, we recall the usual mixing formula:
$$
Z^\mu \equiv cos\theta_W W^{\mu 3} - sin\theta_W B^\mu \  ,
\ \ \ \
A^\mu \equiv sin\theta_W W^{\mu 3} + cos\theta_W B^\mu \  ,
\eqno (2.7)
$$

The vortex solution extremising (2.1) is given by [5,6]:
$$
\eqalign {
W^{\mu 1} = 0 & = W^{\mu 2} = A^\mu , \ \ \
Z^\mu = [ A^\mu ] _{NO} = - {{v_{NO} (r)} \over r} {\hat e}_\theta
\cr
&
\phi = f_{NO} (r) e^{im \theta } \Phi \ , \ \ \
\Phi \equiv \pmatrix{0\cr 1\cr}
\cr
}
\eqno (2.8)
$$
where, the coordinates $r$ and $\theta$ are polar coordinates
in the $xy-$plane. The integer $m$ is the winding number of the
vortex and, here, we shall restrict ourselves to the case
$m=1$. The
subscript $NO$ on the functions $f$ and
$A^\mu$ means
that they are identical to the corresponding functions found
by Nielsen and Olesen [11] for the usual Abelian-Higgs string.
On substituting eq. \(2.8) into the equations of motion they reduce to
$$
f'' + {{f'} \over r} - \left ( 1- {\alpha \over 2} v \right ) ^2
                             {f \over {r^2}}
   - 2 \lambda \left ( f^2 - {{\eta ^2} \over 2} \right ) f = 0
\eqno (2.9)
$$
$$
v'' - {{v'} \over r} + \alpha \left ( 1 - {\alpha \over 2} v \right ) f^2 = 0
\eqno (2.10)
$$
where primes denote differentiation with respect to $r$
and the subscripts $NO$ have been dropped for convenience. These are solved
together with the
boundary conditions:
$$
f(0) = 0 = v(0), \ \ \  f(\infty ) = {\eta \over {\sqrt{2}}} , \ \ \
v(\infty ) = {2 \over \alpha}
\eqno (2.11)
$$
The string solutions resulting from these equations have been studied
previously by several authors in
a lot of detail. A sample of these papers may be found in the
collection of Ref. 12.

At this point, it is useful to note the symmetries of the string configuration.
Firstly it is axially symmetric i.e. it is invariant under the action of the
symmetry operator generated by the generalised angular momentum operator $$
K_z\ =\ L_z\ +\ S_z\ + \ I_z\ . \eqno(2.12) $$ $L_z$ and $S_z$ are the usual
orbital and spin pieces respectively of the spatial angular momentum operator.
Explicitly $${L_z}=-i{\partial\over \partial\theta}\ {\bf
1}\qquad\left({S_z}{\vec a}\right)_j=-i\epsilon_{3jk}{\vec a}_k\ {\bf 1}\
,\eqno(2.13)$$ where $\vec a$ is any vector field and ${\bf 1}$ is the $2\times
2$ unit matrix. . Note that $S_z$ annihilates the scalar Higgs field. $I_z$ is
composed of a $U(1)$ generator, $Y$ and an $SU(2)$ generator, $T^3$

$$ I_z= -\half( Y-T^3)\ . \eqno(2.14) $$
$Y$ and $T^3$ act on the Higgs field on the left. $Y$ annihilates the gauge
field and $T^3$ acts via a commutator bracket.

The configuration has two further symmetries. It is invariant under the
combination of reflection in the x-axis and complex conjugation. Also it is
invariant under the action of the global $U(2)$ gauge tranformation given
by $${\overline U}=\pmatrix{-1&0\cr 0&1\cr}\eqno(2.15)$$ Our eventual expansion
 of the
perturbations will be in Fourier modes but we shall point out some connections
between these and the eigenfunctions of the above commuting symmetry operators.

%\tanmay

\beginsection{3. Stability of Electroweak Strings}
\taghead{3.}

 The vortex solution given in eq. \(2.8) is not topologically stable. This
means
that any field configuration can be continuously deformed to the vacuum. Hence
we are investigating the metastability of the vortex solution i.e.
whether it is a {\it local} maximum or minimum in configuration space. We
consider infinitesimal perturbations of the vortex configuration and ask if the
variation in the energy is positive or negative.

Let us write
$$
\phi = \pmatrix{\phi_1\cr \phi_{NO} + \phi_2\cr}
\eqno (3.1)
$$
$$
Z^\mu = Z_{NO} ^\mu + \delta Z^\mu
\eqno (3.2)
$$
$$
T^1 \equiv diag( - cos2\theta_W , 1 ),
\eqno (3.3)
$$
and,
$$
{\bf d} _j \equiv ( \partial _j {\bf 1} + i\half \alpha T^1 Z_j ) \  .
\eqno (3.4)
$$


The perturbations can depend on the $z-$coordinate and the $z-$components
of the vector fields can be non-zero also. However, since the vortex
solution has translational invariance along the $z-$direction, it is
easy to see that the $z-$dependence in the perturbations can be ignored
and the $z-$components of the gauge fields can be set to zero.
This follows from (2.1) where
the relevant $z-$dependent terms in the integrand are:
$$
          \half G_{i3} ^a G_{i3} ^a + \fourth F_{Bi3} F_{Bi3}
          + (D_3 \phi ) ^{\dag} (D_3 \phi )
\eqno (3.5)
$$
This contribution to the energy
is strictly non-negative and is minimized (that is, made to vanish)
by setting the $z-$components of the gauge fields
to zero and also considering the perturbations to be independent of the
$z-$coordinate. For this reason,
we shall drop all reference to the $z-$coordinate
in the calculations below and it will be understood that the energy is
actually the energy {\it per unit length} of the string.

Now we calculate the energy of the perturbed configuration
discarding terms of cubic and higher
order in the infinitesimal perturbations. We find,
$$
E = ( E_{NO} + \delta E_{NO} ) + E_1 + E_c + E_W
\eqno (3.6)
$$
where,
$E_{NO}$ is the energy of the Nielsen-Olesen string and $\delta E_{NO}$ is
the energy variation due to the perturbations $\phi_2$ and $\delta Z^\mu$.
The variation $E_1$ is due to the perturbation $\phi_1$ in the upper
component of the Higgs field:
$$
E_{1} = \int d^2 x \left [
       |{\bar d}_j \phi_1 |^2 + 2 \lambda ( f^2 - \eta^2 /2 ) | \phi_1 |^2
                         \right ] \  ,
\eqno (3.7)
$$
where,
$$
{\bar d} _j \equiv \partial _j - i{\alpha \over 2} cos(2\theta_W ) Z_j \  .
\eqno (3.8)
$$
The contribution from the $\phi$ and
${\vec W}^{\bar a}$ [13] interaction is:
$$
E_c = cos\theta_W \int d^2 x J_j ^{\bar a} W_j ^{\bar a}
\eqno (3.9)
$$
$$
J_j ^{\bar a} \equiv \half i \alpha \left [
                     \phi^{\dag} \tau^{\bar a} {\bf d}_j \phi
         - ( {\bf d}_j \phi ) ^{\dag} \tau^{\bar a} \phi \right ]
\eqno (3.10)
$$
and the energy in the ${\vec W}^{\bar a}$ and $\vec A$ bosons is
$$
\eqalign {
E_W \equiv
&
           \int d^2 x \biggl [
            \gamma {\vec W} ^1 \times {\vec W} ^2 \cdot
                                                 \vec \nabla \times \vec Z
+ \half |\vec \nabla \times {\vec W} ^1
                  + \gamma {\vec W} ^2 \times \vec Z | ^2
\cr &\cr
&+
\half |\vec \nabla \times {\vec W} ^2
                  + \gamma \vec Z \times {\vec W} ^1 | ^2
+ \fourth g^2 f^2 ( {\vec W } ^{\bar a} ) ^2
+ \half ( \vec \nabla \times \vec A )^2
                     \biggr ] \  .
\cr}
\eqno (3.11)
$$
where, $\gamma \equiv g cos\theta_W$. It may be remarked that
the $f$ and $\vec Z$ fields in eqs. (3.7)-(3.11)
are the unperturbed fields of the string since
we are only keeping up to quadratic terms in the infinitesimal
quantities.

Firstly we note that the perturbations of the fields that make up the string do
not couple to the other available perturbations. i.e. the perturbations in the
fields $f$ and $v$ only occur inside the variation $\delta E_{NO}$. We can
understand this as follows: the perturbation of the string solution has
${\overline U}=1$, where ${\overline U}$ is given by
eq. \(2.15)., whereas the other perturbations have
${\overline U}=-1$. Now,
since we know that the Nielsen-Olesen string with unit winding number is stable
to perturbations for any values of the parameters then necessarily, $\delta E
_{NO} \ge 0$ and the perturbations $\phi_2$ and $\delta Z^\mu$ cannot
destabilize the vortex. Then, we are justified in ignoring these perturbations
and setting $\delta E _{NO} = 0$. Also we note that the ${\vec A}$ boson only
appears in the last term of eq. \(3.11) and obviously makes a positive
contribution so we can set ${\vec A}$ to zero.


%\tanmay: 13 February 1992

We now consider an expansion of the remaining perturbations in Fourier modes.
This gives,
$$
\phi _1 = \chi (r) e^{im\theta}
\eqno (3.12)
$$
for the $m^{th}$ mode where $m$ is any integer. For the gauge fields
we have,
$$
{\vec W}^1 = \left [
\left \{ {\bar f}_{1} (r) cos(n\theta ) + f_{1} sin(n\theta ) \right \}
{\hat e} _r +
{1 \over r} \left \{
           - {\bar h}_{1} sin(n\theta ) + h_{1} cos(n \theta ) \right \}
{\hat e} _\theta
\right ]
\eqno (3.13)
$$
$$
{\vec W}^2 = \left [
\left \{ - {\bar f}_{2} (r) sin(n\theta ) + f_{2} cos(n\theta ) \right \}
{\hat e} _r +
{1 \over r} \left \{
           {\bar h}_{2} cos(n\theta ) + h_{2} sin(n \theta ) \right \}
{\hat e} _\theta
\right ]
\eqno (3.14)
$$
for the $n^{th}$ mode where $n$ is a non-negative integer.
Inserting the expressions for the $m^{th}$ mode of $\phi_1$ and the
$n^{th}$ mode of the ${\vec W}^{\bar a}$ fields
in the energy functional gives:
$$
E_1 =
       2\pi \int dr \  r \left [
           {\chi '}^2 + \left \{
              {1 \over {r^2}}
            \left ( m + { \alpha \over 2} cos 2\theta_W v \right ) ^2 +
             2 \lambda \left ( f^2 - {{\eta ^2} \over 2} \right )
                     \right \} \chi ^2 \right ]
\eqno (3.15)
$$
$$
\eqalign{
E_c =  \delta_{\pm n,1-m} \pi \alpha cos\theta_W
          \int dr r
&
\biggl [ -(f\chi ' - \chi f') (f_2 \mp f_1 )
\cr
&
             - {f \over {r^2}} \chi \left \{
                \mp n + 2 - {\alpha \over 2} (1-cos2\theta_W ) v \right \}
               (h_2 \pm h_1 )
\biggr ]
\cr
}
\eqno (3.16)
$$
for $n \ne 0$. In the case when $n=0$, the expression for $E_c$ is equal to the
above expression multiplied by a factor of 2 and
with $f_1$ and $h_2$ set equal to zero. Finally,
$$
\eqalign{
E_W = \pi \int {{dr} \over r}
&
\biggl [
                   \gamma ( f_2 h_1 - f_1 h_2 ) v'
\cr
&
+
\half \left |- n f_1 + {h_1} ' - \gamma v f_2 \right | ^2 +
\half \left | n f_2 + {h_2} ' + \gamma v f_1 \right | ^2
\cr
&
+
\fourth g^2 f^2
[ r^2 ( {f_1} ^2 + {f_2} ^2 ) + {h_1}^2 + {h_2}^2 ] \biggr ] +
( f_{\bar a} \rightarrow {\bar f} _{\bar a} ,
  h_{\bar a} \rightarrow {\bar h} _{\bar a} ) \  .
\cr }
\eqno (3.17)
$$
for $n \ne 0$. In the case when $n=0$, $E_W$ is given by (3.17) multiplied
by a factor of 2 and with $f_1$ and $h_2$ set equal to zero.

First let us inspect $E_1$. Here the negative contributions can come
from the term proportional to $\chi ^2$. The coefficient of $\chi ^2$ is
composed
of two terms: the second term comes from  comes from the potential part and is
negative while the first term comes from the kinetic part and is always
positive and is smallest when $m=0$, at least in
the region near the center of the string where an instability is most
likely to develop. That is, the $m=0$ mode is the ``most dangerous'' mode.

Next we inspect $E_W$. Here the analysis is less obvious. Yet one can see
that the only term that can be negative is the first term that arises from
the term ${\vec W} ^1 \times {\vec W} ^2 \cdot \vec \nabla \times \vec Z$
in eq. (3.11). For this to contribute at the center of the string, the
vector fields must not vanish there. But the only mode that need not vanish
at the center is the $n=1$ mode; all other modes have to vanish at the
center if they are to be single-valued and finite. Therefore the only mode
that can give negative contributions at the center of the string is the
$n=1$ mode. Hence, we restrict ourselves to considering the
$m=0, n=1$ mode.

There is also a generally accepted idea which leads us to expect the least
stable mode to be the $m=0$, $n=1$ mode. As noted, the original configuration
has $K_z=0$ and ${\overline U}=1$ . The $m=0$ mode is also
$K_z=0$ with ${\overline U}=-1$. Turning our attention to the gauge fields
since
$$K_z\ e^{in\phi}\tau_\pm=(n\pm 1)\ e^{in\phi}\tau_\pm\ ,\eqno(3.18)$$ where
$\tau_\pm=\tau_1\pm i\tau_2$, we see that taking the combinations $f_1 - f_2$
 and
$h_1 + h_2$ we get $K_z=0$ and
${\overline U}=-1$. Intuitively we expect the least stable mode to have
$K_z=0$ since as $\bigl\vert K_z\bigr\vert$ increases one gets an increasing
 centrifugal barrier. The
other combinations $f_1 + f_2$ and $h_1 - h_2$ are a superposition of $K_z=\pm
2$ and we will see they decouple and drop out of our analysis.

Also note that
the barred and unbarred perturbations decouple. ( Under the combined operation
of reflection in the x-axis and complex
conjugation the barred and unbarred variables have
opposite sign.) Furthermore, the stability problem
in the barred variables is contained within the problem of the unbarred
variables (all we need to do is to set $\phi_1 = 0$). Therefore, it is
sufficient to consider only the unbarred functions.

We will now systematically simplify the expression for the energy variation.
After a lot of algebra, we obtain the first step:
\medskip

$$\eqalign{{\delta E\over 2\pi} = &
       \int dr \  r \biggl [
          \left( 1- {{(grf)^2} \over 2{P_+}} \right) {\chi '}^2 +
               M^2 \chi ^2 \biggr ]
\cr
&\ \cr
       &+ \int {dr \over 2r} \biggl [
           {A_{\pm}} {\xi_{\pm} '}^2 +
               S_{\pm} {\xi_{\pm} }^2 \biggr ]
\cr
&\ \cr
 &+ \alpha cos\theta_W \int dr \  r
         \biggl [{\{(1-\gamma v)\xi_+'+\gamma v'\xi_+\} \over P_+}(f\chi
'-f'\chi)-
           {f\chi \over {r^2}} (1-\alpha sin^2 \theta_W v ) \xi_+
              \biggr ]
\cr &\ \cr
& +T_+(F_-,\chi,\xi_+)+T_-(F_+,\xi_-)\cr}
\eqno (3.19)
$$
where,
$${T_+(F_-,\chi,\xi_+)=\int {dr\over 2r}\biggl [
   {\sqrt{P_+}} F_- +
   { {\{ (1-\gamma v) \xi_+ ' + \gamma v' \xi _+ \} } \over
       {\sqrt{P_+}}}
- \alpha cos\theta_W r^2 {{(f\chi ' - f' \chi )} \over {\sqrt{P_+}}}
                     \biggr ] ^2
}\eqno (3.20)
$$
$$T_-(F_+,\xi_-)=\int {dr \over 2r}\biggl [
          {\sqrt{P_-}} F_+ +
   { {\{ (1+\gamma v) \xi_- ' - \gamma v' \xi _- \} } \over
       {\sqrt{P_-}}}\biggr]^2
\eqno(3.21)
$$
$$P_{\pm} = (1 \mp \gamma v )^2 + {g^2 r^2 f^2\over 2}
\eqno (3.22)
$$
$$
F_{\pm} = {{f_2 \pm f_1} \over {2}}
\eqno (3.23)
$$
$$
\xi_{\pm} = {{h_2 \pm h_1} \over 2}
\eqno (3.24)
$$
$$
A_\pm (r) = {{g^2 r^2 f^2} \over {2P_\pm (r)}}
\eqno (3.25)
$$
\
$$
M^2 = {{\alpha ^2} \over {4 r^2}} cos^2 2\theta_W v^2 +
             2 \lambda \left ( f^2 - {{\eta ^2} \over 2} \right )
      - {{(grf')^2} \over 2{P_+}}  -
{1 \over r} {d \over {dr}} \left ( {{g^2 r^3 f f'} \over 2{P_+}} \right )
\eqno (3.26)
$$
\

$$
S_\pm (r) =  {g^2 f^2\over 2} - {{\gamma ^2 {v'}^2} \over {P_\pm (r)}} \pm
    r {{d \  } \over {dr}} \left [
            \gamma {{v'} \over r} {{(1 \mp \gamma v )} \over {P_\pm (r)}}
                          \right ] \  .
\eqno (3.27)
$$


As expected the problem in $\chi, F_-$ and $\xi_+$ has decoupled from the
problem in $F_+$ and $\xi_-$. Furthermore the only terms containing $F_-$ and
$F_+$ are $T_-$ and $T_+$ respectively and since these are whole squares
they can be set to zero. This then leaves us with a problem in  $\chi$
and $\xi_+$ and a problem in just $\xi_-$.

We first discuss the $\xi_-$ problem. We conjecture that the relevant
potential, $S_-$ is positive for any values of the parameters $\beta$
and $\cos\theta_W$ (where $\beta=8\lambda/g^2$). We motivate this by
considering
the asymptotic behaviour as $r\rightarrow 0$ and as $r\rightarrow \infty$: in
both these limits $S_-$ is always positive. Our conjecture is therefore
reasonable since the function is basically exponential, modulated by
polynomials. (We have checked it numerically for many pairs of parameters.)

Hence our analysis is reduced to considering perturbations in $\xi_+$ and
$\chi$ alone. We express the change in the energy in the form of an eigenvalue
problem:
$$
\delta E[ \chi , \xi_+ ] = 2\pi \int dr \  r
              ( \chi , \xi _+ ) {\bf O} \pmatrix{\chi\cr \xi_+\cr}
\eqno (3.28)
$$
where, {\bf O} is a $2\times 2$ matrix differential operator.

It is now useful to identify the form of the perturbations $\chi$
and $\xi_+$ that are pure gauge transformations of the string
configuration.
It is easy to see that perturbations of the form
$$
\delta\phi=ig\psi\phi_0,{\ {\delta W}_i=-iD_{0i}\psi} \  ,
\eqno (3.29)
$$
where $\psi$ is a real {\it L(SU(2))}
valued function and the $0$ subscript denotes the unperturbed
fields, is an infinitesimal gauge transformation of the
original
vortex solution. (In (3.29), $W_i$ represents
$\vec \tau \cdot \vec W _i$.) If we now require that these
purely gauge perturbations do not affect the
string configuration itself, then we can only have
$$
\psi=s(r)\pmatrix{0&ie^{-i\phi}\cr -ie^{i\phi}&0\cr}
\eqno (3.30)
$$
where $s(r)$ is any smooth function. This means that
perturbations given by
$$
\pmatrix{\chi\cr\xi_+}\ =\ s(r)\pmatrix{-gf\cr 2(1-\gamma v)\cr}
\eqno(3.31)
$$
are pure gauge perturbations that do not affect the string configuration.
Therefore, such perturbations cannot contribute to the energy variation
and must be annihilated
by ${\bf O}$. Then, in the two-dimensional
space of $(\chi , \xi_+ )$ perturbations, we can choose a basis in
which one direction is pure gauge and is given by (3.31) and the other
orthogonal direction is the direction of physical perturbations.
The physical mode is,
$$
\zeta\ =\ (1-\gamma v)\chi\ +\ {gf\over 2}\xi_+\quad .
\eqno(3.32)
$$

It was a good check on our algebra that on eliminating $\xi_+$ in terms of
$\zeta$ and $\chi$ in eq. (3.28)
the functional reduces to one depending only on
$\zeta$:
$$
\delta E[ \zeta ] = 2\pi \int dr \  r
              \zeta  {\overline O} \zeta
\eqno (3.33)
$$
where ${\overline O}$ is the differential operator
$${\overline O} = - {1 \over r} {d \over {dr}} \left (
             {r \over {P_+}} {d \over {dr}} \right ) + U(r)
\eqno (3.34)$$
and
$$ U(r) = {{{f'}^2} \over {P_+ f^2}} + {{2 S_+} \over {g^2 r^2 f^2}} +
          {1 \over r} {d \over {dr}} \biggl (
                     {{r f'} \over {P_+ f}} \biggr ) \  .\eqno (3.35)$$

To summarise, the question of stability reduces to asking if the operator
${\overline O}$ has negative eigenvalues in its spectrum. That is, whether the
eigenvalue $\omega$ of the Schrodinger equation,
$$ {\overline O} \zeta =\omega\zeta \  , \eqno (3.36) $$
can be negative. The eigenfunction $\zeta$ must also
satisfy the boundary conditions $\zeta (r=0) = 1$ and $\zeta \rightarrow c$
($c$ is some constant) as $r \rightarrow\infty$.

In this way we have reduced the stability analysis down to one Schrodinger
equation which we will solve numerically.

%\tanmay: 17 February 1992.

\beginsection{4. Numerical Analysis}
\taghead{4.}

We solve eq. \(3.36) together with the Nielson-Olesen eqs. (2.9) and (2.10).
However, it is convenient to work with rescaled dimensionless variables.
Hence, we define
$$
P \equiv {{\sqrt{2}} \over \eta} f , \ \ \ \
V \equiv {\alpha \over 2} v , \ \ \ \
R \equiv {{\alpha \eta} \over {2\sqrt{2}}} r \  .
\eqno (4.1)
$$
In terms of these dimensionless variables, the Nielsen-Olesen equations
\(2.9) and \(2.10) become,
$$
P'' + {{P'} \over R} - \left ( 1- V \right ) ^2
                             {P \over {R^2}}
   + \beta ( 1 - P^2 ) P = 0
\eqno (4.2)
$$
$$
V'' - {{V'} \over R} + 2 ( 1 - V ) P^2 = 0
\eqno (4.3)
$$
where primes now denote differentiation with respect to $R$. The functions
$P$ and $V$ also satisfy the boundary conditions:
$$
P(0) = 0 = V(0), \ \ \  P(\infty ) = 1 , \ \ \
V(\infty ) = 1
\eqno (4.4)
$$
The problem now has only two
free parameters: $\cos{\theta_W}$ and $\beta$.
This may be seen by rescaling fields and coordinates in the operator
$\bar{O}$ as in (4.1). With these rescalings the quantities $S_+$ and $P_+$ in
the eigenvalue problem (3.36) get replaced by: $$ S_+^* = {P^2\over 2} - cos^2
\theta_W {{{V'}^2} \over {P^*_+}}
       - {R \over 2} {d \over {dR}} \left \{
          {{V'} \over R} {{(n - 2 cos^2 \theta_W V ) } \over {P^*_+}}
                                   \right \}
$$
where,
$$
P^*_+ = (1 - 2 cos^2 \theta _W V )^2 + 2 cos^2 \theta _W R^2 P^2
$$

The rescaled eigenvalue problem was
solved by using a fifth order Runge-Kutta algorithm.
We kept $\beta$ fixed and found $\theta_w$ for which the
lowest eigenvalue changes sign. We repeated this procedure for several
values of $\beta$ and found the corresponding values of critical parameters
$(\sqrt{\beta}, \sin^2 \theta_w)$.
The above method was used to scan the range $0.07 \leq \beta \leq 1.0$.
Lower values of $\beta$ make the numerical analysis
fairly intensive since then there are two widely different scales in the
problem corresponding to the two widely different masses.
Our results are shown in Fig. 1  where we
plot the critical values of $\sqrt{\beta}$ (the ratio of the Higgs mass
to the $Z$
mass) versus the corresponding values of $\sin^2 \theta_w$.  In sector III, on
the right-hand side of the data line,
equation (3.36) had no negative eigenvalues implying
string stability.
Thus we may distinguish three sectors in Fig. 1:
sector I where the electroweak strings are unstable, sector III where strings
are stable, and,
the presently unexplored region shown as sector II
($\beta < 0.07$ or $m_H < 24 GeV$).
It is evident that the
physically realized values: $\sin^2 \theta_w=0.23$ and
$\sqrt{\beta}=m_H / m_Z > 0.62$
(see Ref. 14) lie entirely inside sector I.
This brings us to the main result of this paper: if the standard electroweak
model is the physically realized model, then the existing
vortex solutions in the bare model are unstable.

\beginsection{5. Outlook}

We have rigorously established that the electroweak model admits stable
vortex solutions for a certain range of parameters. This result is
exciting in that it makes the possibility of observing coherent states
in particle physics closer to reality. On the other hand, it is somewhat
dissappointing that the values of the parameters that Nature has
actually chosen are such that the vortex is unstable. However, this
still does not mean that the vortex will be unstable in the real world
since our analysis only applies to the bare electroweak model. In the
context of the early universe, for example, we must do a stability
analysis at high temperatures. One should also consider the possibility
that Nature has chosen
an extension of the standard model where the Higgs potential is more
complicated. In such circumstances, the stability issue would have
to be readdressed.

\bigskip

\beginsection {\it{Acknowledgements:}}

We would like to thank Miguel Ortiz for his help with the numerical
calculations. This work was supported by a CfA postdoctoral fellowship
(L.P.) and by the NSF(T.V.). M.J. acknowledges a research studentship from
SERC.


\vfill
\eject
{\centerline {References}}
\parindent 0 pt

1. T. Vachaspati and A. Ach\'ucarro, Phys. Rev. D {\bf 44},
3067 (1991).

2. G. W. Gibbons, M. Ortiz, F. Ruiz-Ruiz and T. Samols,
DAMTP preprint (1992); J. Preskill, Cal Tech preprint (1992).

3. M. Hindmarsh, Phys. Rev. Lett. {\bf 68}, 1263 (1992).

4. A. Ach\'ucarro, K. Kuijken, L. Perivolaropoulos and
T. Vachaspati, Nucl. Phys. B, to be published.

5. T. Vachaspati, Phys. Rev. Lett. {\bf 68}, 1977 (1992).

6. T. Vachaspati, Tufts preprint (1992).

7. T. Vachaspati and M. Barriola, Tufts preprint (1992).

8. S. Weinberg, Phys. Rev. Lett. {\bf 19}, 1264 (1967);
A. Salam in ``Elementary Particle Theory'', ed. N. Svarthholm, Stockholm:
Almqvist, Forlag AB, pg 367.

9. We are only considering the stability under small
perturbations. Hence, the word ``stability'' should always be understood
as ``meta-stability''.

10. J. C. Taylor, ``Gauge Theories of Weak Interactions'',
Cambridge University Press, 1976.

11. H. B. Nielsen and P. Olesen, Nucl. Phys. B{\bf{61}}, 45 (1973).

12. ``Solitons and Particles'', ed. C. Rebbi and G. Soliani,
World Scientific, 1984.

13. Barred indices always range from 1 to 2.

14. J. Steinberger, Phys. Rep. {\bf 203}, 345 (1991).

15. M. James, L. Perivolaropoulos and T. Vachaspati, to appear in Phys. Rev.
{\bf D}
(rapid communication).

\vfill
\eject

\head{Figure Captions}


 A map of parameter space showing the results of the stability analysis.
Sector I contains unstable strings, sector III contains stable strings
and we have not explored sector II. We also indicate the physically
allowed range of parameter space. The data from LEP constrains the Higgs
mass to $M_H >53 GeV$ which implies $\sqrt{\beta} > 0.62 GeV$ and the observed
$\sin^2 \theta_W$ is 0.23.


\endjnl
\end


