%
% $Id: aipcheck.tex,v 1.4 2001/01/31 20:46:55 latex3 Exp $
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Testing for potential problems with this class
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newif\ifproblem
\newif\iftimesok

\typeout{***********************************************}
\typeout{*}
\typeout{* Testing if all files required for the aipproc}
\typeout{* class are available ...}
\typeout{*}
\typeout{***********************************************}

\typeout{*}
\typeout{* Looking for LaTeX2e ... }
\ifx\documentclass\undefined
 \typeout{*}
 \typeout{* Sorry this is a fatal error:}
 \typeout{*}
 \typeout{* The aipproc class can only be used with LaTeX2e which is}
 \typeout{* the standard LaTeX since 1994!}
 \typeout{*}
 \typeout{* Please make sure that your version of LaTeX is up-to-date}
 \typeout{* before attempting to use this class.}
 \typeout{*}
 \expandafter\stop
\else
 \typeout{* ... ok }
\fi

\typeout{*}
\typeout{* Looking for aipproc.cls ... }
\IfFileExists{aipproc.cls}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
     \typeout{*}
     \typeout{* Before you can use the aipproc class you have to unpack}
     \typeout{* it from the documented source.}
     \typeout{*}
     \typeout{* Run LaTeX on the file 'aipproc.ins', e.g.,}
     \typeout{*}
     \typeout{* \space\space latex aipproc.ins}
     \typeout{*}
     \typeout{* or whatever is necessary on your installation to process}
     \typeout{* a file with LaTeX. This should unpack a number of files for you:}
     \typeout{*}
     \typeout{* aipproc.cls \space and \space aip-*.clo}
     \typeout{*}
     \typeout{* After that retry processing this guide.}
     \typeout{*}
     \stop
}

\typeout{*}
\typeout{* Looking for fixltx2e.sty ... }
\IfFileExists{fixltx2e.sty}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found, trying fix2col.sty instead ... }
     \typeout{*}
     \IfFileExists{fix2col.sty}
	 {
	  \typeout{* ... ok }
	 }
	 {
	  \typeout{* ... not found! }
	  \typeout{*}
	  \typeout{* Sorry this is a fatal error:}
	  \typeout{*}
	  \typeout{* Your LaTeX distribution contains neither fixltx2e.sty}
	  \typeout{* nor fix2col.sty.}
	  \typeout{*}
	  \typeout{* This means that it is either too old or incompletely}
	  \typeout{* installed.}
	  \typeout{*}
	  \typeout{* fixltx2e.sty is part of the standard LaTeX distribution}
	  \typeout{* since 1999; fix2col.sty is an earlier version of this}
	  \typeout{* package.}
	  \typeout{*}
	  \typeout{* Best solution is to get the latest LaTeX distribution.}
	  \typeout{* If this is impossible for you, download fix2col.sty.}
	  \typeout{* You can get this software from a CTAN host.}
          \typeout{* Refer to http://www.tug.org to find such an archive on}
          \typeout{* the net.}
	  \typeout{*}
	  \typeout{* After you have updated your LaTeX distribution}
	  \typeout{* retry processing this guide.}
	  \stop
     }
}

\typeout{*}
\typeout{* Looking for fontenc.sty ... }
\IfFileExists{fontenc.sty}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
     \typeout{*}
     \typeout{* The fontenc package, which is part of standard LaTeX}
     \typeout{* (base distribution) has to be installed at the site to}
     \typeout{* run the aipproc class.}
     \typeout{*}
     \typeout{* The fact that it cannot be found either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
     \typeout{* improperly.}
     \typeout{*}
     \typeout{* Please make sure that your version of LaTeX is okay}
     \typeout{* before attempting to use this class. The LaTeX distribution}
     \typeout{* contains the file "ltxcheck.tex" which can be used to}
     \typeout{* test the basic functionality and integrity of your installation.}
     \typeout{*}
     \stop
    }

\typeout{*}
\typeout{* Looking for calc.sty ... }
\IfFileExists{calc.sty}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
     \typeout{*}
     \typeout{* The calc package, which is part of standard LaTeX}
     \typeout{* (tool distribution) has to be installed at the site}
     \typeout{* to run the aipproc class.}
     \typeout{*}
     \typeout{* The fact that it cannot be found either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
     \typeout{* only in parts.}
     \typeout{*}
     \typeout{* Please make sure that the tools distribution of LaTeX}
     \typeout{* is installed before attempting to use this class.}
     \typeout{*}
     \typeout{* (You might be able to get calc.sty separately for your}
     \typeout{* installation if you are unable to upgrade to a recent}
     \typeout{* distribution for some reason.)}
     \typeout{*}
     \stop
    }

\typeout{*}
\typeout{* Looking for varioref.sty ... }
\IfFileExists{varioref.sty}
    {
     \typeout{* ... ok }
     \gdef\variorefoptionifavailable{varioref,}
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Problem detected:}
     \typeout{*}
     \typeout{* The varioref package, which is part of standard LaTeX}
     \typeout{* (tool distribution) is not installed at this site.}
     \typeout{*}
     \typeout{* The fact that it cannot be found either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
     \typeout{* only in parts.}
     \typeout{*}
     \typeout{* You can use the aipproc class without this package but }
     \typeout{* you cannot make use of the options "varioref" or "nonvarioref".}
     \typeout{*}
     \typeout{* Please also note that the aipguide.tex documentation}
     \typeout{* normally uses the "varioref" option to show its}
     \typeout{* effects (which  will now fail).}
     \typeout{*}
     \typein{* Type <return> to continue ...}
     \problemtrue
     \gdef\variorefoptionifavailable{}
     \let\vpageref\pageref
     \let\vref\ref
    }

\typeout{*}
\typeout{* Looking for times.sty ... }
\IfFileExists{times.sty}
    {
     \begingroup
% load times and forget it immediately again
       \RequirePackage{times}
       \global\expandafter\let\csname ver@times.sty\endcsname\relax    
       \long\def\next{ptm}
       \ifx\rmdefault\next
         \typeout{* ... ok }
         \gdef\psnfssproblemoption{}
         \endgroup
         \timesoktrue
       \else
         \endgroup
     \typeout{* ... obsolete! }
     \typeout{*}
     \typeout{* Serious problem detected:}
     \typeout{*}
     \typeout{* The times package, which is part of standard LaTeX}
     \typeout{* (psnfss distribution) is obsolete at this site.}
     \typeout{*}
     \typeout{* The fact that it contains incorrect code either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
     \typeout{* only in parts with old files remaining!}
     \typeout{*}
     \typeout{* You can use the aipproc class without this package but}
     \typeout{* you have to specify the option "cmfonts" which result in}
     \typeout{* documents which are not conforming to the AIP layout specification!}
     \typeout{*}
     \typeout{* You can also try using the class in the following way:}
     \typeout{*}
     \typeout{* \space\space \string\documentclass[cmfonts]{aipproc}}
     \typeout{* \space\space \string\usepackage{times}}
     \typeout{* \space\space ...}
     \typeout{*}
     \typeout{* With luck this will result in Times Roman output but chances}
     \typeout{* are that you will get a larger number of error messages in}
     \typeout{* which case you have to remove the \string\usepackage declaration.}
     \typeout{*}
     \typein{* Type <return> to continue ...}
          \problemtrue
          \gdef\psnfssproblemoption{cmfonts}
          \def\textdegree{$^\circ$}            % used below but now
                                               % not setup
       \fi
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Serious problem detected:}
     \typeout{*}
     \typeout{* The times package, which is part of standard LaTeX}
     \typeout{* (psnfss distribution) can not be found.}
     \typeout{*}
     \typeout{* The fact that this package cannot be found either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
     \typeout{* only in parts!}
     \typeout{*}
     \typeout{* You can use the aipproc class without this package but }
     \typeout{* you have to specify the option "cmfonts" which result in}
     \typeout{* documents which are not conforming to the AIP layout specification!}
     \typeout{*}
     \typein{* Type <return> to continue ...}
     \problemtrue
     \gdef\psnfssproblemoption{cmfonts,}
    }

\iftimesok % don't bother testing other font options if times already
           % bad

\typeout{*}
\typeout{* Looking for t1ptm.fd or T1ptm.fd ... }
\IfFileExists{t1ptm.fd}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found, trying T1ptm.fd ... }
     \IfFileExists{T1ptm.fd}
          {
           \typeout{* ... ok }
          }
          {
           \typeout{* ... not found}
           \typeout{* Serious problem detected:}
           \typeout{*}
           \typeout{* The times package, which is part of standard LaTeX}
           \typeout{* (psnfss distribution) is available but the corresponding}
           \typeout{* .fd file (defining how to load Times Roman) is missing.}
           \typeout{*}
           \typeout{* The fact that this package is only partially installed}
           \typeout{* means that you LaTeX installation is unable to use Times}
           \typeout{* Roman fonts!}
           \typeout{*}
           \typeout{* You can use the aipproc class without this package but }
           \typeout{* you have to specify the option "cmfonts" which result in}
           \typeout{* documents which are not conforming to the AIP layout}
           \typeout{* specification!}
           \typeout{*}
           \typein{* Type <return> to continue ...}
           \problemtrue
           \timesokfalse
           \gdef\psnfssproblemoption{cmfonts,}
          }
    }

\fi


\newcommand\CheckFDFile[3]{%
  \typeout{*}
  \typeout{* Looking for #1#3.fd or #2#3.fd ... }
  \IfFileExists{#1#3.fd}
    {
     \typeout{* ... ok }
    }
    {
     \IfFileExists{#2#3.fd}
      {
       \typeout{* ... ok }
      }
      {\problemtrue
       \typeout{* ... not found! }
      }
    }
}

\iftimesok % don't bother testing other font options if Times already bad

\typeout{*}
\typeout{* Looking for mathptm.sty ... }
\IfFileExists{mathptm.sty}
    {
     \typeout{* ... ok }
     \CheckFDFile{ot1}{OT1}{ptmcm}
     \CheckFDFile{oml}{OML}{ptmcm}
     \CheckFDFile{oms}{OMS}{pzccm}
     \CheckFDFile{omx}{OMX}{psycm}
     \ifproblem
      \typeout{*}
      \typeout{* Problem detected:}
      \typeout{*}
      \typeout{* The mathptm package, which is part of standard LaTeX}
      \typeout{* (psnfss distribution) was found but some or all of its}
      \typeout{* support files describing which fonts to load are missing!}
      \typeout{*}
      \typeout{*}
      \typeout{* The fact that this package is only partially installed}
      \typeout{* means that the mathptm package cannot be used!}
      \typeout{*}
      \typeout{* You can use the aipproc class without this package but }
      \typeout{* you have to specify the option "nomathfonts" so that}
      \typeout{* math formulas will be typeset using Computer Modern.}
      \typeout{*}
      \typein{* Type <return> to continue ...}
      \problemtrue
      \gdef\psnfssproblemoption{nomathfonts,}
     \fi
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Problem detected:}
     \typeout{*}
     \typeout{* The mathptm package, which is part of standard LaTeX}
     \typeout{* (psnfss distribution) can not be found.}
     \typeout{*}
     \typeout{* The fact that this package cannot be found either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
     \typeout{* only in parts!}
     \typeout{*}
     \typeout{* You can use the aipproc class without this package but }
     \typeout{* you have to specify the option "nomathfonts" so that}
     \typeout{* math formulas will be typeset using Computer Modern.}
     \typeout{*}
     \typein{* Type <return> to continue ...}
     \problemtrue
     \gdef\psnfssproblemoption{nomathfonts,}
    }

\typeout{*}
\typeout{* Looking for mathtime.sty ... }
\IfFileExists{mathtime.sty}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* The mathime package can not be found.}
     \typeout{*}
     \typeout{* This is not a serious problem because this package is}
     \typeout{* only of interest if you own the commerical MathTime fonts.}
     \typeout{*}
     \typeout{* You can use the aipproc class without this package but }
     \typeout{* you cannot use the "mathtime" option of the class.}
     \typeout{*}
     \typein{* Type <return> to continue ...}
     \problemtrue
    }
\else
\fi % iftimesok

\typeout{*}
\typeout{* Looking for graphicx.sty ... }
\IfFileExists{graphicx.sty}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Problem detected:}
     \typeout{*}
     \typeout{* The graphics package, which is part of standard LaTeX}
     \typeout{* (graphics distribution) can not be found.}
     \typeout{*}
     \typeout{* The fact that this package cannot be found either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
     \typeout{* only in parts!}
     \typeout{*}
     \typeout{* You can use the aipproc class without this package but }
     \typeout{* you cannot use commands like \protect\includegraphics
                or \protect\resizebox}
     \typeout{* in this case.}
     \typeout{*}
     \typeout{* Please note that you will get a further error message below}
     \typeout{* about: "graphicx.sty not found" because the class will try}
     \typeout{* to load this package! Type return in response to that error.}
     \typeout{*}
     \typeout{* As a result the illustrations in aipguide will look strange.}
     \typeout{*}
     \typein{* Type <return> to continue ...}

     \gdef\resizebox##1##2{}
     \gdef\includegraphics{\textbf{graphics package missing:}}
     \problemtrue
    }

\typeout{*}
\typeout{* Looking for textcomp.sty ... }
\IfFileExists{textcomp.sty}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Problem detected:}
     \typeout{*}
     \typeout{* The textcomp package, which is part of standard LaTeX}
     \typeout{* (base distribution) can not be found.}
     \typeout{*}
     \typeout{* The fact that this package cannot be found either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
     \typeout{* only in parts!}
     \typeout{*}
     \typeout{* You can use the aipproc class without this package but }
     \typeout{* you will always get the error: "textcomp.sty not found"}
     \typeout{* because the class will try to load this package!}
     \typeout{* Type return in response to that error.}
     \typeout{*}
     \typein{* Type <return> to continue ...}

     \def\textdegree{$^\circ$}         % used below but now
                                       % not set up
     \problemtrue
    }


\typeout{*}
\typeout{* Looking for url.sty ... }
\IfFileExists{url.sty}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Problem detected:}
     \typeout{*}
     \typeout{* The url package, which should be part of a good LaTeX}
     \typeout{* distribution, can not be found.}
     \typeout{*}
     \typeout{* Without this package you will not be able to use the \string\url}
     \typeout{* command. Try to download this package from a CTAN  host.}
     \typeout{* Refer to http://www.tug.org to find such an archive on}
     \typeout{* the net.}
     \typeout{*}
     \typein{* Type <return> to continue ...}

     \problemtrue
    }



\typeout{*}
\typeout{* Looking for natbib.sty ... }
\IfFileExists{natbib.sty}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Serious problem detected:}
     \typeout{*}
     \typeout{* The natbib package, which should be part of a good LaTeX}
     \typeout{* distribution, can not be found.}
     \typeout{*}
     \typeout{* Without this package you will not be able to use certain}
     \typeout{* citation styles. See the aipguide documentation!}
     \typeout{*}
     \typeout{* Especially the layout for ARLO requires this package!}
     \typeout{*}
     \typeout{* Try to download this package from a CTAN  host.}
     \typeout{* Refer to http://www.tug.org to find such an archive on}
     \typeout{* the net.}
     \typeout{*}
     \typein{* Type <return> to continue ...}

     \problemtrue
    }


\typeout{*}
\typeout{* ... finished testing}
\typeout{*}
\ifproblem
\typeout{* The tests have reveiled some problems in your TeX installation.}
\typeout{*}
\typeout{* Please review the above comments carefully and read the file}
\typeout{* README for further information.}
\typeout{*}
\typeout{*****************************************************************}
\typein{* Type <return> to continue ...}
\else
\typeout{****************************************************************}
\typeout{*}
\typeout{* The tests have reveiled no problems in your TeX installation.}
\typeout{*}
\typeout{****************************************************************}
\fi


% if this file is run standalone stop otherwise continue
\def\next{aipcheck}
\edef\currjob{\jobname}
\edef\next{\meaning\next}
\edef\currjob{\meaning\currjob}
\ifx\currjob\next
  \expandafter\stop
\fi

\endinput
%%%=======================================================
%%% =
%%% = Sample file for the class "aipproc"
%%% =
%%%=======================================================

\input{aipcheck.tex}

%\typeout{}
%\typeout{**********************************************}
%\typeout{*}
%\typeout{* Here you can type in any options for the class.}
%\typeout{* A full list of supported options is described in}
%\typeout{* the aipguide documentation.}
%\typeout{*}
%\typeout{* Or you may simply press the <RETURN> key in which}
%\typeout{* case the file is processed with the standard settings.}
%\typeout{*}
%\typeout{**********************************************}
%\typeout{}
%\typein[\selectedoptions]{Any options to the class?}
%\ifx\empty\selectedoptions
  \def\selectedoptions{final}
%\fi

\documentclass[
   \selectedoptions
  ]
  {aipproc}

\newcommand\sample{%
 Infandum, regina, iubes renovare dolorem, Troianas ut opes et
 lamen\-ta\-bile regnum cruerint Danai; quaeque ipse miserrima vidi, et
 quorum pars magna fui. Quis talia fando Myrmidonum Dolopumve aut duri
 miles Ulixi temperet a lacrimis? \smallsample
}

\newcommand\smallsample{%
 Et iam nox umida caelo praecipitat,
 suadentque cadentia sidera somnos.
 Sed si tantus amor casus cognoscere
 nostros et breviter Troiae supremum audire laborem, quamquam animus
 meminisse horret, luctuque refugit, incipiam.
}

%\typeout{}
%\typeout{**********************************************}
%\typeout{*}
%\typeout{* Type in the layout style name to be used:}
%\typeout{*}
%\typeout{* `6x9' for AIP proceedings 6x9 inch single column}
%\typeout{* `8x11single' for AIP proceedings 8x11 inch single column}
%\typeout{* `8x11double' for AIP proceedings 8x11 inch double column}
%\typeout{* `arlo' for Acoustics Research Letters Online}
%\typeout{*}
%\typeout{* ATTENTION:}
%\typeout{*}
%\typeout{* \space Typesetting the guide with different styles might
%                  produce}
%\typeout{* \space errors if the .aux file is kept between runs!}
%\typeout{*}
%\typeout{**********************************************}
%\typeout{}
%\typein[\selectedlayoutstyle]{What layout?}
%\layoutstyle\selectedlayoutstyle
\layoutstyle{6x9}

\SetInternalRegister\hbadness{8000} % pseudo latin isn't breaking very well :-)

% We need something to show special stuff for ARLO only.
%
\newcommand\doingARLO[2][]{%
  \ifx\mmref\undefined #1\else #2\fi
}
  
\begin{document}

%
%ESTAS SON ALGUNAS DE MIS PROPIAS DEFINICIONES !!!
%
\def\bsig{\mbox{\boldmath $\sigma$}}                          
\def\bsig{\mbox{\boldmath $\Sigma$}}
\def\bgam{\mbox{\boldmath $\gamma$}}
\def\bgam{\mbox{\boldmath $\Gamma$}}
\def\bphi{\mbox{\boldmath $\phi$}}
\def\bphi{\mbox{\boldmath $\Phi$}}
\def\btau{\mbox{\boldmath $\tau$}}
\def\btau{\mbox{\boldmath $\Tau$}}
\def\btau{\mbox{\boldmath $\partial$}}
\def\Delc{{\Delta}_{\circ}}
\def\bp{\mid {\bf p} \mid}
\def\al{\alpha}
\def\bet{\beta}
\def\gam{\gamma}
\def\del{\delta}
\def\Del{\Delta}
\def\te{\theta}
\def\nua{{\nu}_{\alpha}}
\def\nui{{\nu}_i}
\def\nuj{{\nu}_j}
\def\nue{{\nu}_e}
\def\num{{\nu}_{\mu}}
\def\nut{{\nu}_{\tau}}
\def\2te{2{\theta}}
\def\chic#1{{\scriptscriptstyle #1}}
\def\chicl{{\chic L}}
\def\lam{\lambda}
\def\SU{SU(2)_{\chic L} \otimes U(1)_{\chic Y}}
\def\Lam{\Lambda}
\def\sig{\sigma}
\def\'#1{\ifx#1i\accent19\i\else\accent19#1\fi}
\def\O{\Omega}
\def\o{\omega}
\def\s{\sigma}
\def\D{\Delta}
\def\d{\delta}
\def\df{\rm d}
\def\8{\infty}
\def\ld{\lambda}
\def\eps{\epsilon}
\def\ref#1{$^{#1}$}
\def\chicl{{\chic L}}
\def\lam{\lambda}
\def\SU{SU(2)_{\chic L} \otimes U(1)_{\chic Y}}
\def\Lam{\Lambda}
\def\sig{\sigma}
\def\'#1{\ifx#1i\accent19\i\else\accent19#1\fi}
\def\O{\Omega}
\def\o{\omega}
\def\s{\sigma}
\def\D{\Delta}
\def\d{\delta}
\def\df{\rm d}
\def\8{\infty}
\def\ld{\lambda}
\def\eps{\epsilon}
\def\ref#1{$^{#1}$}
%\def\'#1{\ifx#1i\accent19\i\else\accent19#1\fi}

\title 
      []
      {Dirac Neutrino Anapole Moment}

\classification{13.15.+g, 13.10.+q, 13.40.Gp, 25.30.Pt}
\keywords{anapole moment of neutrino; charge radius of neutrino; 
electromagnetic properties of neutrino}

\author{L. G. Cabral-Rosetti$^a$, M. Moreno$^b$ and A. Rosado$^{b,c}$}
{address={$^a${\it Instituto de Ciencias Nucleares, Universidad Nacional 
Autnoma de Mxico (ICN-UNAM),\\
Circuito Exterior, C.U., Apartado Postal 70-543, 94510 Mxico, D.F.,  
Mxico.}\\
{$^b${\it Instituto de Fsica, Universidad Nacional Autnoma de Mxico, 
(IF-UNAM).\\
Circuito de la Investigacin Cientfica, C.U., Apartado Postal 
20-364, 01000 Mxico, D.F.,  Mxico.}\\
$^c${\it Instituto de Fsica, Benemrita Universidad Autnoma 
de Puebla, (IF-BUAP).\\
Apartado Postal J-48, Colonia San Manuel, Puebla, Puebla. 72570, Mxico.}
}\\
},
  email={luis@nuclecu.unam.mx, matias@fenix.ifisicacu.unam.mx, 
rosado@fenix.ifisicacu.unam.mx},
  thanks={}
}
%\fi

% \copyrightholder{Acoustical Scociety of America}
\copyrightyear  {2001}

\begin{abstract}
We calculate the Dirac neutrino anapole moment ($a_{\nu_l}$) in the context 
of the Standard Model (SM) making use of the Dirac form factor 
$F_{\chic D}(q^2)$ introduced recently by J. Bernabu, L. G. Cabral-Rosetti,
J. Papavassiliou y J. Vidal by using the Pinch Technique (PT) 
formalism, working in two different gauge-fixing schemes ($R_\xi$ guage and 
the electroweak BFM), at the one loop level. We show that the neutrino 
anapole form factor $F_{\chic A}(q^2)$ and Dirac form factor 
$F_{\chic D}(q^2)$ are related as follows: $F_{\chic A}(q^2) = 
\frac{1}{q^2} F_{\chic D}(q^2)$. Hence, the Dirac neutrino charge radius 
$\langle r^2_{\nu_l} \rangle$ and the anapole moment satisfy the simple 
relation $a_{\nu_l} = \frac{1}{6} \langle r^2_{\nu_l} \rangle$. 
Therefore, we show that the anapole moment (as the charge radius) of 
the neutrino is a physical quantity, which only gets contribution from 
the proper neutrino electromagnetic vertex (in electroweak BFM), and that 
$a_{\nu_l}$ is of the order $10^{-34} \, cm^2$.\ \footnote{Contribution to
the Proceedings of the {\it VIII Mexican Workshop on Particles and Fields} 
of the {\it Divisi\'on de Part{\'\i}culas y Campos de la Sociedad Mexicana de 
F{\'\i}sica} ({\bf DPyC-SMF}), Cd. de Zacatecas, Zacatecas, M\'exico 
November 14-20, 2001.}
\end{abstract}

\date{\today}

\maketitle

\section{Introduction}
In 1987, M. Abak and C. 
Aydin \cite{abak} calculated $a_{\nu_l}$ in the context of the Standard 
Model of the electroweak interaction (SM) \cite{weinberg}, using
the 't Hooft-Feynman gauge \cite{hooft} and conclude that $a_{\nu_l}$ is
too small to be measured. In 1992, A. Gngora-T and R. G. Stuart, 
defined the charge radius and anapole moment of a free fermion as being 
its vector and axial-vector contact interactions with an external 
electromagnetic current and they got, at one loop in the SM, a finite 
and gauge invariant expression for these quantities \cite{gongora}. In 
1987, H. Czyz {\it et al.} \cite{czyz} discussed the anapole moment of 
charged leptons in the context of the SM and showed that this quantity 
is gauge dependent and therefore is not a physical quantity in this 
model. In 1991, M. J. Musolf and B. R. Holstein analized the neutrino
matrix element of the electromagnetic current for low $q^2$ and showed
that the anapole moment is just $\frac{1}{6}$ times the charge radius
\cite{musolf}. In 2000, A. Rosado presented a calculation of the neutrino 
anapole moment in the linear $R_\xi$ gauge at the one loop level in the 
context of the SM and showed explicitly that it is an infinite and gauge 
dependent quantity in this model. He also introduced, the electroweak 
anapole moment $a_{\nu_l}^{\chic {EW}}$ through the elastic scattering
$\nu_l l'$ which is finite, gauge independent, and independent of the 
lepton $l'$ used to define it \cite{rosado1, rosado2}.

In this work, we calculate the neutrino anapole moment $a_{\nu_l}$ in 
the context of the SM of the electroweak interactions $\SU$ making use 
of the Dirac form factor introduced by J. Bernabu, L. G. Cabral-Rosetti, 
J. Papavassiliou y J. Vidal \cite{bernabeu}. This electromagnetic form 
factor was obtained through the process 
$e^+ e^- \rightarrow \nu_l \bar{\nu_l}$ at the one loop level, 
by using the Pinch Technique (PT) formalism \cite{JMC, CJM, JPP, DAS, wat}, 
working in two different gauge-fixing schemes ($R_\xi$ guage and the 
electroweak BFM), at the one loop level, in the context of SM and becomes 
for arbitrary momentum 
transfer $q^2$: finite, independent on the gauge-fixing parameter, on 
the gauge-fixing scheme employed, on the Higgs and quark sector of the 
theory, and on the properties of the charged lepton used to define it. 
This paper is organized as follows. In the next section,  we show that the
neutrino anapole form factor $F_A(q^2)$ and $F_D(q^2)$ are related as 
follows: $F_{\chic A}(q^2) = \frac{1}{q^2} F_{\chic D}(q^2)$. Hence, the 
neutrino charge radius $\langle r^2_{\nu_l} \rangle$ and the anapole 
moment satisfy the simple relation 
$a_{\nu_l} = \frac{1}{6}\langle r^2_{\nu_l} \rangle$.
Therefore, we show that the anapole moment (as the charge radius) of 
the neutrino is a physical quantity. Also in this section, we give the 
expression of the anapole form factor and the numerical values of the 
anapole moment for the three different neutrino species. Finally, we 
present our conclusions.

\begin{figure}
\includegraphics[height=.19\textheight]{Figura1.eps}
\caption{
Only two proper vertex diagrams contribute to the neutrino charge radius 
$\langle r^2_{\nu_l} \rangle$ and therefore neutrino anapole form factor 
$F_{\chic A}(q^2)$ at the one-loop level after that the massive gauge 
cancellation, which takes place when the Pinch Technique is used, in the 
electroweak Background Field Method.}
\label{vertex}
\end{figure}

\section{The neutrino anapole moment}

The matrix element of the electromagnetic current in the frame of the SM
can be expressed at the lowest order in $\alpha$, {\it for all} $q^2$, where
$q=p-p'$, in terms of only one form factor $F_{\chic D}(q^2)$ as
\cite{chung, mohapatra, cabral1, C3}

\begin{equation}
{\cal M}_\mu = F_{\chic D}(q^2) \bar u_{\nu_l} (p') \gamma_\mu
(1 - \gamma_5) u_{\nu_l} (p).
\end{equation}

\noindent 
In Fig. 1, we show only the two one-loop diagrams, which contribute to
the Dirac form factor, after that the massive gauge cancellation, which
takes place when the PT is used \cite{bernabeu}, in the electroweak 
Background Field Method \cite{BFM}. For a massless Dirac neutrino we can 
rewrite Eq. (1), for all $q^2$, as follows

\begin{equation}
{\cal M}_\mu = \bar u_{\nu_l} (p') \{ \gamma_\mu f_1(q^2) 
- \gamma_\lambda \gamma_5
[{g^{\lambda}}_{\mu} q^2 - q^{\lambda} q_{\mu}] f_3(q^2) \} u_{\nu_l} (p)
\end{equation}

\noindent 
where\cite{marshak} $f_1(q^2) = F_{\chic D}(q^2)$ and 
$f_3(q^2) = F_{\chic A}(q^2) = \frac{1}{q^2}F_{\chic D}(q^2)$ are the Dirac and anapole form 
factor of the neutrino, respectively, with

\begin{equation}
f_1(0) = F_{\chic D}(0) = 0,
\end{equation}

\begin{equation}
\langle r^2_{\nu_l} \rangle = -\ 6\  \frac{\partial f_1(q^2)}{\partial q^2}
\Bigg |_{q^2=0} \hspace{2mm} = -\ 6\  
\frac{\partial F_{\chic D}(q^2)}{\partial q^2}
\Bigg |_{q^2=0},
\end{equation}

\noindent 
and

\begin{equation}
a_{\nu_l} = f_3(0) = \frac{F_{\chic D}(q^2)}{q^2} \Bigg |_{q^2=0}
\hspace{2mm} = \frac{\partial F_{\chic D}(q^2)}{\partial q^2} 
\Bigg |_{q^2=0}.
\end{equation}

\noindent 
That is,

\begin{equation}
a_{\nu_l} = \frac{1}{6}\ \langle r^2_{\nu_l} \rangle.
\end{equation}

\noindent 
Finally, according to the results given for $F_{\chic D}(q^2)$ in 
Eq. (7.10) of Ref. \cite{bernabeu}, 

\begin{equation}
\begin{array}{c}
\displaystyle
F_{\chic D} (q^2) = -\, \frac{\alpha e}{8 \pi s_{\chic W}^2} 
\Bigg\{1 + \Bigg( \frac{1}{2} + \frac{M^2_{\chic W}}{q^2} \Bigg)
\Big[ B_0 (q^2; m_l^2, m_l^2) 
- B_0 (q^2; M^2_{\chic W}, M_{\chic W}^2) \Big]
\\[0.5cm]
\displaystyle
+\, M_{\chic W}^2 \Bigg( 2 + \frac{M^2_{\chic W}}{q^2} \Bigg)
C_0 (0, q^2, 0; m_l^2, M_{\chic W}^2, M^2_{\chic W})
+\, \frac{(q^2 + M_{\chic W}^2)^2}{q^2}
C_0 (0, q^2, 0; M^2_{\chic W}, m_l^2, m_l^2)
\Bigg\}\, .
\end{array}
\end{equation}

\noindent 
Hence, we conclude that the anapole moment, as the charge radius,
of the neutrino is a physical quantity, which only gets contribution 
from the proper neutrino electromagnetic vertex Fig.~(1). Taking into 
account the relations among scalar, two-points $B_0$ and three-points
$C_0$ Passarino-Veltman functions reported in the Refs. 
\cite{cabral1, C3, cabral2} we get

\begin{equation}
a_{\nu_l} = \frac{G_F}{24 \sqrt{2} \pi^2} \left \{ 3 - 2 \, \log
\Bigg ( \frac{m_l^2}{M_W^2} \Bigg ) \right \},
\end{equation}

\noindent 
The numerical evaluation of the above expression for the three
different neutrino species yields: $a_{\nu_e} = 6.8 \times 10^{-34} \, cm^2$, 
$a_{\nu_\mu} = 4.0 \times 10^{-34} \, cm^2$ and 
$a_{\nu_\tau} = 2.5 \times 10^{-34} \, cm^2$.

\section{Conclusions}

The main goal in this paper has been to calculate the anapole moment of 
the neutrino, making use of the Dirac form factor introduced by J. 
Bernabu, L. G. Cabral-Rosetti, J. Papavassiliou y J. Vidal 
\cite{bernabeu}. This form factor was obtained 
through the physical process $e^+ e^- \rightarrow \nu_l \bar{\nu_l}$ 
working in the linear $R_\xi$ gauge and using the electroweak BFM in the 
context of the standard model of the electroweak interactions, at the one 
loop level. We showed in frame of the SM, that the neutrino anapole form 
factor $F_{\chic A}(q^2)$ and the Dirac form factor $F_{\chic D}(q^2)$ are 
related as follows: $F_{\chic A}(q^2) = \frac{1}{q^2} F_{\chic D}(q^2)$. 
Therefore the neutrino anapole moment $a_{\nu_l}$ and the neutrino charge 
radius $\langle r^2_{\nu_l} \rangle$ satisfy the simple 

relation $a_{\nu_l} = \frac{1}{6}\langle r^2_{\nu_l} \rangle$. Hence, we 
showed that the anapole moment (as the charge radius) of the neutrino 
becomes a physical quantity, which has the following properties: (i)
it only gets contribution from the proper neutrino electromagnetic 
vertex, (ii) it is finite, (iii) it is independent on the gauge-fixing 
parameter, (iv) it is independent on the gauge-fixing scheme employed, 
(v) it does not depend on the Higgs or quark sector of the theory, (vi) 
it does not depend on the properties of the charged lepton used to define it. 
The numerical values of $a_{\nu_l}$ for the three different neutrinos are 
of the order $10^{-34} \, cm^2$.

\begin{theacknowledgments}
This work was supported in part by the {\it Programa de Apoyo a Proyectos de 
Investigacin e Inovacin Tecnolgica} ({\bf PAPIIT}) de la 
{\bf DGAPA-UNAM} {\it No. de Proyecto}: {\sc IN109001}. 
\end{theacknowledgments}

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\end{document}




