Novel Transversity Properties in SIDIS
Leonard Gamberg
We consider a rescattering mechanism to calculate a leading twist $T$-odd pion fragmentation function, a candidate for filtering the transversity properties of the nucleon. We evaluate the single spin azimuthal asymmetry for a transversely polarized target in semi-inclusive deep inelastic scattering (for HERMES kinematics) and the double $T$-odd $\cos2\phi$ asymmetry in this framework.
aipcheck.tex
%
% $Id: aipcheck.tex,v 1.7 2002/01/11 19:50:46 frank Exp $
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Testing for potential problems with this class
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newif\ifproblem
\newif\iftimesok

\makeatletter
\def\IfStandaloneCheck{\def\next{aipcheck}
  \edef\currjob{\jobname}
  \edef\next{\meaning\next}
  \edef\currjob{\meaning\currjob}
  \ifx\currjob\next
    \expandafter\@firstoftwo
  \else
    \expandafter\@secondoftwo
  \fi
}
\makeatother

\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


\def\next#1/#2/#3\next{#1#2}
\typeout{*}
\typeout{* Testing that LaTeX2e is not too old ... }
\ifnum\expandafter\next\fmtversion\next<199612 \relax
 \typeout{* ... what a vintage! }
 \typeout{*}
 \typeout{* Sorry this is a fatal error:}
 \typeout{*}
 \typeout{* The aipproc class can only be used with a recent version}
 \typeout{* of LaTeX2e. Your version is dated \fmtversion\space --- but}
 \typeout{* at least the 1996/12/01 version is required!}
 \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
 \ifnum\expandafter\next\fmtversion\next<199806 \relax
   \typeout{* ... probably ok }
   \typeout{*}
   \typeout{* Your version of LaTeX2e is quite old --- the aipproc class}
   \typeout{* hasn't been tested with your release.}
   \typeout{*}
   \typeout{* We believe that it will probably work, but if you encounter}
   \typeout{* problems you will need upgrade your installation.}
   \typeout{*}
   \typein{* Type <return> to continue ...}
   \problemtrue
 \else
   \typeout{* ... ok }
 \fi
\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 aipxfm.sty ... }
\IfFileExists{aipxfm.sty}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
     \typeout{*}
     \typeout{* The aipxfm.sty file which is part of the aipproc distribution}
     \typeout{* must be installed in a directory which is searched by LaTeX.}
     \typeout{*}
     \typeout{* Please install this file and retry.}
     \typeout{*}
     \stop
}

\typeout{*}
\typeout{* Looking for aip-8s.clo ... }
\IfFileExists{aip-8s.clo}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
     \typeout{*}
     \typeout{* The aip-8s.clo file which is part of the aipproc distribution}
     \typeout{* must be installed in a directory which is searched by LaTeX.}
     \typeout{*}
     \typeout{* Please install this file and retry.}
     \typeout{*}
     \stop
}

\typeout{*}
\typeout{* Looking for aip-8d.clo ... }
\IfFileExists{aip-8d.clo}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
     \typeout{*}
     \typeout{* The aip-8d.clo file which is part of the aipproc distribution}
     \typeout{* must be installed in a directory which is searched by LaTeX.}
     \typeout{*}
     \typeout{* Please install this file and retry.}
     \typeout{*}
     \stop
}


\typeout{*}
\typeout{* Looking for aip-6s.clo ... }
\IfFileExists{aip-6s.clo}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
     \typeout{*}
     \typeout{* The aip-6s.clo file which is part of the aipproc distribution}
     \typeout{* must be installed in a directory which is searched by LaTeX.}
     \typeout{*}
     \typeout{* Please install this file and retry.}
     \typeout{*}
     \stop
}


\typeout{*}
\typeout{* Looking for aip-arlo.clo ... }
\IfFileExists{aip-arlo.clo}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
     \typeout{*}
     \typeout{* The aip-arlo.clo file which is part of the aipproc distribution}
     \typeout{* must be installed in a directory which is searched by LaTeX.}
     \typeout{*}
     \typeout{* Please install this file and retry.}
     \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.ctan.org and search for "fix2col".}
	  \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

%\CheckFDFile{ot1}{OT1}{ot1ztmcm}
%\CheckFDFile{oml}{OML}{omlztmcm}
%\CheckFDFile{oms}{OMS}{omsztmcm}
%\CheckFDFile{omx}{OMX}{omxztmcm}

\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,}
     \else
      \typeout{*}
      \typeout{* Looking for mathptmx.sty ... }
      \IfFileExists{mathptmx.sty}
       {
        \typeout{* ... ok }
        \CheckFDFile{ot1}{OT1}{ztmcm}
        \CheckFDFile{oml}{OML}{ztmcm}
        \CheckFDFile{oms}{OMS}{ztmcm}
        \CheckFDFile{omx}{OMX}{ztmcm}
        \ifproblem
	  \typeout{*}
	  \typeout{* Problem detected:}
	  \typeout{*}
	  \typeout{* The mathptmx 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 mathptmx package cannot be used!}
	  \typeout{*}
	  \typeout{* You can use the aipproc class without this package but }
	  \typeout{* you have to specify the option "mathptm" (no x) so that}
	  \typeout{* math formulas use the older version with upright greek letters.}
	  \typeout{*}
	  \typein{* Type <return> to continue ...}
	  \problemtrue
	  \gdef\psnfssproblemoption{mathptm,}
        \fi
       }
       {
	\typeout{* ... not found! }
	\typeout{*}
	\typeout{* Problem detected:}
	\typeout{*}
	\typeout{* The mathptmx package, which is part of standard LaTeX}
	\typeout{* (psnfss distribution) can not be found.}
	\typeout{*}
	\typeout{* This is unfortunate but not a disaster as the older}
	\typeout{* version of the package "mathptm" (no x) seems to exist.}
	\typeout{*}
	\typeout{* You can use the aipproc class without this package but }
	\typeout{* you have to specify the option "mathptm" so that}
	\typeout{* math formulas use the older version with upright greek letters.}
	\typeout{*}
	\typein{* Type <return> to continue ...}
	\problemtrue
	\gdef\psnfssproblemoption{mathptm,}
       }
      \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.ctan.org and search for "url".}
     \typeout{*}
     \typein{* Type <return> to continue ...}

     \problemtrue
    }


\makeatletter

\typeout{*}
\typeout{* Looking for natbib.sty ... }
\IfFileExists{natbib.sty}
    {
     \IfStandaloneCheck
       {\begingroup
        \let\@listi\relax
        \let\thebibliography\@empty
        \let\bibstyle\@empty
        \RequirePackage{natbib}
        \@ifpackagelater{natbib}{1999/05/29}
	  {
           \typeout{* ... ok }
	  }{
           \typeout{* ... might be too old! }
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\begin{document}

\title{Novel Transversity Properties in SIDIS}


\author{Leonard Gamberg}{
  address={Division of Science,
Penn State-Berks Lehigh Valley College, Reading, PA 19610,
USA}
}

\author{Gary R. Goldstein}{
  address={Department of Physics
          and Astronomy, Tufts University,
           Medford, MA 02155, USA}
}


\author{Karo A. Oganessyan}{
 address={INFN-Laboratori Nazionali di Frascati, Enrico Fermi 40,
  I-00044 Frascati, Italy}
  ,altaddress={DESY, Notkestrasse 85, 22603 Hamburg, Germany}
}


\begin{abstract}
We consider a  rescattering mechanism to calculate a leading 
twist $T$-odd pion  
fragmentation function, a candidate for filtering the transversity 
properties of the nucleon. We evaluate the single spin azimuthal asymmetry  
for a transversely polarized target in semi-inclusive deep inelastic 
scattering (for HERMES kinematics) and  the double  
$T$-odd $\cos2\phi$ asymmetry in this framework.
\end{abstract}

\maketitle

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%% MAINMATTER
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\subsubsection{Introduction}
\vskip-0.35cm
The transversity distribution, $h_1$  which 
measures the probability to find a transversely polarized quark in the 
transversely polarized nucleon, is as important for the description 
of the internal nucleon spin structure as the more familiar 
helicity distribution function, $g_1$. However, it still remains 
unmeasured, unlike the spin-average and helicity distribution 
functions, which are known experimentally and extensively modeled 
theoretically. The difficulty is that $h_1$ is chiral odd, 
and consequently suppressed in 
inclusive deep inelastic scattering (DIS) processes~\cite{jaffe91}; it has 
to be accompanied by a second chiral-odd quantity.  
Semi-inclusive deep inelastic scattering (SIDIS) on
polarized nucleons is one of 
several~\cite{SMC,HERMES,STAR} promising methods  proposed to access
transversity. It relies on just such a quantity
the so called Collins fragmentation function~\cite{cnpb93}, which  
correlates the transverse spin of the fragmenting quark to the transverse 
momentum of the produced hadron. Beside being chiral-odd, this fragmentation 
function is also time-reversal odd ($T$-odd)~\cite{boer,ANSL} 
which makes its 
calculation challenging.  
In this context, the  non-zero single spin asymmetries in  recent  
measurements~\cite{HERMES} may
signal the existence of a non-trivial $T$-odd effects which are
intimately tied to our understanding of transversity.
Here we  explore~\cite{gambH} a  one-gluon exchange 
mechanism, for the fragmentation 
of a transversely polarized quark into a spinless
hadron  similar to the approach we 
applied~\cite{gamb_gold_ogan1,gamb_gold_ogan2} 
to the distribution of the transversely polarized quarks in 
the both unpolarized and transversely polarized nucleons. 
The non-perturbative information about the quark content of the target 
and the fragmentation of quarks into hadrons 
in SIDIS is encoded in 
the general form of the factorized cross sections in terms of  
the quark distributions $\Phi(p)$ and  fragmentation functions $\Delta(k)$, 
 entering the hadronic tensor.  
To leading 
order in $1/Q^2$~\cite{mulders2} the fragmentation functions are projected from
\bea
\Delta(k,P_h) \hskip -.10cm=\hskip -.10cm
\sum_X \int 
\frac{d\xi^+d^2 \xi_\perp}{2z\,(2\pi)^3} \ 
e^{ik\cdot \xi} \,\langle 0\ 
\vert 
{\cal G}_{{\scriptscriptstyle{[\xi^+,-\infty]} } } 
\psi (\xi) 
\vert X;P_h\rangle\langle X;P_h\vert \overline \psi(0) 
{\cal G}^\dagger_{{\scriptscriptstyle{[0,-\infty]}}}
\vert 0 \rangle \Big|_{\xi^- = 0}.
\label{FF} 
\eea
Here $k$  quark 
fragmenting momenta and $P_h$ is the
fragmented hadron momenta. 
The path ordered exponential along the light like 
direction $\xi^-$ is 
$${\cal G}_{[\xi^-,\infty]}={\cal P}
\exp{\left(-ig\int_{\xi^-}^\infty d\xi^- A^+(\xi)\right)}.
$$
In non-singular gauges~\cite{cplb,ji}, the gauge link   gives rise
to initial and final  state interactions which in turn 
provide a mechanism to generate leading  twist $T$-odd 
contributions to both the distribution and {\em fragmentation}
functions.  
The joint product of these functions enter
novel azimuthal asymmetries and single spin asymmetries (SSAs) 
that have been reported in the 
literature~\cite{bhs,gold_gamb,gamb_gold_ogan1,gamb_gold_ogan2}.
 Such an analysis was recently applied to the $T$-odd 
$f_{1T}^\perp$~\cite{bhs,ji,sivers} 
and $h_1^\perp$~\cite{gold_gamb,gamb_gold_ogan1,gamb_gold_ogan2} 
distribution functions in addition to 
$T$-odd baryon fragmentation functions~\cite{metz}.
We apply~\cite{gambH} an analogous procedure to generate 
the $T$-odd pion fragmentation 
function, $H_1^\perp(z)$ (see also~\cite{bac_metz}). 
\subsubsection{Pion Fragmentation Function}
\vskip -.35cm
The leading order  contributions 
to the $T$-odd  fragmentation functions 
come from the first non-trivial term in expanding
 the path ordered gauge link operator.
The corresponding Feynman rules are those 
for interactions between an eikonalized struck
quark and the remaining target~\cite{col82} depicted in Fig.~\ref{analyze}.  
In modeling the highly off-shell
fragmenting quark we adopt a minimal spectator~\cite{hood} approach.
We couple the on-shell spectator, as a quark  
interacting with the produced pion 
through a Gaussian distribution
in the transverse momentum dependence of the quark-spectator-pion 
vertex ~\cite{gamb_gold_ogan2,gambH} in order to address
the $\log$ divergence
arising in  the moments of fragmentation functions. 
The leading order (in $1/Q$) one loop contribution 
which arises in the limit that the
virtual photon's
momentum becomes large
corresponding to the rescattering
of the initial state quark depicted  in Fig.~\ref{analyze}.
The resulting  twist 2, $T$-odd  
contribution  the  fragmentation function~\cite{gambH}
projected from $\Tr\left(\gamma^-\gamma^\perp\gamma_5\Delta\right)$ is
$$H_1^\perp(z,k_\perp) 
\hspace{-.10cm}=\hspace{-.10cm}
\frac{{\N^\prime}^2 f^2g^2}{(2\pi)^4}\frac{1}{4z}\frac{(1-z)}{z}
\frac{m}{\Lambda^\prime(k^2_\perp)}
\frac{M_\pi}{k_\perp^2}
e^{{\scriptstyle{-b^\prime\left(k^2_\perp- \Lambda^\prime(0)\right)}}}
\left[\Gamma(0,{\scriptstyle{b\Lambda^\prime(0)}})
\hspace{-.10cm}-\hspace{-.10cm}
\Gamma(0,{\scriptstyle{b^\prime\Lambda^\prime(k^2_\perp)}})\right] ,
$$
$\Gamma(0,z)$ is the incomplete gamma function and 
$\Lambda^\prime(k^2_\perp)=k_\perp^2 +\frac{1-z}{z^2}M_\pi^2+ 
\frac{\mu^2}{z} -\frac{1-z}{z} m^2$. 
The average $<k^2_{\perp}>=1/b^\prime$ is a regulating scale 
which we fit to the expression for the integrated unpolarized 
fragmentation function 
$$
D_1(z)=\frac{{\N^\prime}^2 f^2}{4(2\pi)^2}
\frac{1}{z}\frac{\left(1-z\right)}{z}
\{\frac{m^2-\Lambda^\prime(0)}{\Lambda^\prime(0)}
-\left[2b^\prime\left(m^2-\Lambda^\prime(0)\right)-1\right]
e^{2b^\prime\Lambda^\prime(0)}
\Gamma(0,{\scriptstyle{2b^\prime\Lambda^\prime(0)}})\}
$$
which is in  good agreement with the
distribution of Ref.~\cite{kretz}.
\begin{figure}
\includegraphics[height=4.0cm]{collins_asy2.eps}
\includegraphics[width=5.0cm,height=4.0 cm]{ap2.eps}
\caption{\label{analyze} 
Left Panel: $h_1^\perp\star H_1^\perp$ $\cos 2\phi$ 
asymmetry. 
Right Panel: 
The weighted analyzing power
$H_1^{\perp (1)}(z)/D_1(z)$ as a function of $z$.}
\end{figure}
In Fig.~\ref{analyze} the weighted 
the analyzing power, $H_1^{\perp (1)}(z)/ D_1(z)$, is displayed. 
The resulting behavior is similar to a previous model ansatz 
proposed by Collins and calculated in Ref.~\cite{kotz}.
The $\cos2\phi$ asymmetry of SIDIS is projected out of the cross section 
and depends on a leading double $T$-odd product, 
\bea
\langle \frac{\vert P^2_{h{\perp}} \vert}{M M_\pi} \cos2\phi \rangle
{\scriptscriptstyle_{UU}}&=& 
\frac{{8(1-y)} \sum_q e^2_q h^{\perp(1)}_1(x) z^2 H^{\perp(1)}_1(z)}
{{(1+{(1-y)}^2)}  \sum_q e^2_q f_1(x) D_1(z)}.
\label{ASY_cos} 
\eea
$UU$ indicates unpolarized beam and 
target  and $h_1^{\perp (1)}(z)$ is the weighted
moment of the distribution  
function~\cite{boer,gamb_gold_ogan2}.
For a transversely polarized target 
nucleon, the $\sin(\phi+\phi_s)$ asymmetry\cite{cnpb93,mulders2} 
can be similarly obtained yielding, the convolution of two chiral-odd 
structures,
\bea
\langle \frac{P_{h\perp}}{M_\pi}
\sin(\phi+\phi_s) \rangle_{\scriptscriptstyle UT}
&=& 
\big|S_T\big|\frac{2(1-y) \sum_q e^2_q h_1(x) z H^{\perp(1)}_1(z)}
{{(1+{(1-y)}^2)}  \sum_q e^2_q f_1(x) D_1(z)}.
\label{ASY_sin}
\eea
\begin{figure}
\includegraphics[height=4.0 cm,width=3.7cm]{ut_x.eps}
\includegraphics[height=4.0 cm,width=3.7cm]{ut_z.eps}
\includegraphics[height=4.0 cm,width=3.7cm]{cos2_diq_x.eps}
\includegraphics[height=4.0 cm,width=3.7cm]{cos2_diq_z.eps}
\caption{\label{collins}   The 
\protect{${\langle \sin(\phi+\phi_s) 
\rangle}_{\scriptscriptstyle UT}$ } asymmetry for $\pi^+$ production 
as  a function of  $x$ and $z$. 
The \protect{${\langle \cos2 \phi \rangle}_{\scriptscriptstyle UU}$} 
asymmetry for \protect{$\pi^+$} production 
as a function of \protect{$x$} and  \protect{$z$}.}
\end{figure}
The variable range to coincides with the HERMES 
kinematics~\cite{gambH}.  In Fig.~\ref{collins} the 
asymmetry of Eq.~(\ref{ASY_sin}) 
for $\pi^+$ production on a proton target is presented 
as a function of $x$ and $z$, respectively 
indicating approximately a  
$10-15\%$  $P_{h\perp}/M_{\pi}$ weighted $\sin(\phi+\phi_s)$  asymmetry.
Also, in Fig.~\ref{ASY_sin} the  $P^2_{h\perp}/(M M_{\pi})$ weighted 
$\cos 2\phi$ asymmetry of Eqs.~(\ref{ASY_cos}) 
for $\pi^+$ production on an unpolarized proton target is presented as 
a function of $x$ and $z$, respectively indicating 
a few percent effect.  
\vskip-1cm
\subsubsection{Conclusion}
\vskip-0.35cm
A mechanism to generate the $T$-odd Collins 
fragmentation function that is derived from the gauge link
has been considered. This approach complements 
 the approach that was employed to 
generate the $T$-odd distribution functions,    $f_{1T}^\perp$ and $h_1^\perp$
that fuel the Sivers and $\cos 2\phi$ asymmetries. 
The derivation of $H_1^\perp$ 
is consistent with the observation that intrinsic transverse quark momenta and 
angular momentum conservation are intimately tied with studies of 
transversity. 
Furthermore, this approach is interesting in that it does not suffer from
the possible cancellation of the Collins effect cited in~\cite{jin_jaffe}.
This effect is generated in the 
non-trivial phase associated with the gauge link 
operator~\cite{cplb,ji,metz,gamb_gold_ogan1,gamb_gold_ogan2,pij}.
We have evaluated the
analyzing power and predicted
the $P_{h\perp}/M_{\pi}$ weighted $\sin(\phi+\phi_S)$ asymmetry
at HERMES energies. Additionally, we 
predict that  there is a non-trivial $\cos2\phi$ asymmetry associated with the
asymmetric distributions of transversely polarized quarks inside
unpolarized hadrons. 
Generalizing from these model calculations, it is clear that initial and 
final state  interactions can account for  leading twist 
$T$-odd contributions to SSAs.  
Using rescattering as a mechanism to generate 
$T$-odd distribution and fragmentation functions 
opens a new window into the theory and phenomenology of transversity in 
hard processes.
\vspace*{-0.75cm}

\subsubsection{Acknowledgments}
\vskip-0.4cm
{\footnotesize
L.G. thanks the organizers of CIPANP for the invitation to present
this work. 
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\bb{99}
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Phys. Rev D {\bf 67}, 054003  (2003).



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