Precision Measurements of $\as$ at HERA\footnote{Talk given in the XIII International Workshop on Deep Inelastic Scattering, April $27^{th}$ - May $1^{st}$, 2005, Madison, Wisconsin USA.}
Claudia Glasman\footnote{Ram\'on y Cajal Fellow.}
The precision measurements of the strong coupling constant, $\as$, and its energy-scale dependence carried out at HERA by the H1 and ZEUS Collaborations are reviewed. An average value of \\ \centerline{ $\overline\asz=0.1186\pm 0.0011\ ({exp.})\pm 0.0050\ ({th.})$}\\ is obtained from these measurements. The combined HERA determinations of the energy-scale dependence of $\as$ clearly show the running of $\as$ from jet data alone and are in agreement with the running of the coupling as predicted by QCD.
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
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hfs-glasman-proc.tex
\input{aipcheck}

\documentclass[sort&compress,final]
  {aipproc}

\usepackage{epsfig}

\layoutstyle{6x9}

%definitions
\def\as{\alpha_s}
\def\asz{\as(\mz)}
\def\mz{M_Z}
\def\etjet{E_T^{\rm jet}}
\def\colab#1{#1 Coll.}
\def\etal{et al.}
\def\Journal#1#2#3#4{{#1} {#2} (#3) #4}
\def\NIM{{\em Nucl. Instr. and Meth.} {\bf A}}
\def\NPB{{\em Nucl. Phys.} {\bf B}}
\def\PLB{{\em Phys. Lett.}  {\bf B}}
\def\PRL{{\em Phys. Rev. Lett.}}
\def\PRD{{\em Phys. Rev.} {\bf D}}
\def\PRX{{\em Phys. Rev.}}
\def\PRE{{\em Phys. Rep.}}
\def\ZPC{{\em Z. Phys.} {\bf C}}
\def\ZPX{{\em Z. Phys.}}
\def\EPC{{\em Eur. Phys. Jour.} {\bf C}}
\def\EPD{{\em Eur. Phys. Jour. direct} {\bf C}}
\def\CPC{{\em Comp. Phys. Comm.}}
\def\JPG{{\em J. Phys.} {\bf G}}
\def\PS{{\em Phys. Scr.}}
\def\JHEP{JHEP}

\begin{document}

\title{Precision Measurements of $\as$ at HERA\footnote{Talk given in
    the XIII International Workshop on Deep Inelastic Scattering,
    April $27^{\rm th}$ - May $1^{\rm st}$, 2005, Madison, Wisconsin
    USA.}}

\classification{12.38.Qk}
\keywords{strong coupling constant}

\author{Claudia Glasman\footnote{Ram\'on y Cajal Fellow.}}
{address={Universidad Aut\'onoma de Madrid}
}

\begin{abstract}
The precision measurements of the strong coupling constant, $\as$, and
its energy-scale dependence carried out at HERA by the H1 and ZEUS
Collaborations are reviewed. An average value of \\
\centerline{
$\overline\asz=0.1186\pm 0.0011\ ({\rm exp.})\pm 0.0050\ ({\rm th.})$}\\
is obtained from these measurements. 
The combined HERA determinations of the energy-scale dependence
of $\as$ clearly show the running of $\as$ from jet data alone and are
in agreement with the running of the coupling as predicted by QCD.
\end{abstract}

\maketitle

\section{Introduction}

The strong coupling constant, $\as$, is one of the fundamental
parameters of QCD. However, its value is not predicted by the theory
and must be determined by experiment. Many precise and consistent
determinations of $\as$ from diverse phenomena underlie the success of
perturbative QCD (pQCD). At HERA, $\as$ has been determined from many
observables, which include jet cross sections and structure functions,
by the H1 and ZEUS Collaborations. All the available
determinations~\cite{dijetzeus,incdiszeus,incdish1,nlofitzeus,nlofitzeusn,nlofith1,subnczeus,subcczeus,incgpzeus,multijetzeus,shanczeus}
are shown in Fig.~\ref{fig1}a. They are in good agreement with each
other and are consistent with the current world average
($\overline\asz^{WA}=0.1182\pm 0.0027$~\cite{wa}). These
determinations, most of which come from observables which involve jet
algorithms, lead to determinations of $\as$ some of which are as
precise as those from more inclusive measurements. The uncertainty in
these determinations is dominated by the theoretical contributions,
which amount to $4\%$ for jet cross sections and fits of structure
functions and $8\%$ for the internal structure of jets, whereas the
experimental uncertainties amount to $\sim 3\%$.

%Figure 1
\begin{figure}[h]
\setlength{\unitlength}{1.0cm}
\begin{picture} (10.0,8.0)
\put (-2.0,0.0){\epsfig{figure=dis_shape237col.eps,width=8.0cm}}
\put (5.0,-0.5){\epsfig{figure=dis_shape30col.eps,width=8.0cm}}
\put (-2.5,6.0){\bf\small (a)}
\put (5.0,6.0){\bf\small (b)}
\end{picture}
\caption{(a) Summary of $\asz$ determinations at HERA compared with
the world average; (b) $\asz$ determinations at HERA compared with
the HERA average.
\label{fig1}}
\end{figure}

\section{Averaging the $\as$ determinations from HERA}

To make a proper average of all these diverse measurements, the
correlation among the different determinations has to be taken into
account. The experimental contribution to the uncertainty due to that
of the energy scale of the jets, which is the dominant source in
the jet measurements, is correlated among the determinations
from each experiment. On the theoretical side, the uncertainty
coming from the proton parton distribution functions (PDFs) is
certainly correlated whereas that coming from the hadronisation
corrections is only partially correlated. The uncertainty coming from
the terms beyond NLO is correlated up to a certain, a priori unknown,
degree; since these uncertainties are dominant, special care must be
taken in the treatment of these uncertainties when making an average
of the HERA determinations.

Several methods have been used to obtain an average value of $\as$ from
the HERA measurements and its uncertainty. Using a naive method in
which all uncertainties are assumed to be uncorrelated, the average
value and its uncertainty are:

\vspace{0.1cm}
\centerline{$\overline\asz=0.1188\pm 0.0020\ ({\rm ZEUS+H1}).$}
\vspace{0.1cm}

The second method used is that developed by
M. Schmelling~\cite{schmelling} to average correlated data when
correlations are present but hard to quantify. 
In this method, an error-weighted average and an
optimised correlation error were obtained from the error covariance
matrix by assuming an overall correlation factor between the total
errors of all measurements; the overall factor was determined by the
condition that the overall $\chi^2/{\rm dof}$ is equal to unity.
First, an error-weighted average was done separately for
the ZEUS and H1 measurements, and then the two averages were
combined:

\vspace{0.1cm}
\centerline{$\overline\asz=0.1196\pm 0.0060\ ({\rm ZEUS})\hspace{1cm}{\rm and}\hspace{1cm}\overline\asz=0.1166\pm 0.0053\ ({\rm H1}),$}
\vspace{0.1cm}

\centerline{$\overline\asz=0.1188\pm 0.0057\ ({\rm ZEUS+H1}).$}
\vspace{0.1cm}

The averages from the ZEUS and H1 determinations are compatible within
the uncertainties. The uncertainty of the combined average is 
$\sim 5\%$. 
This procedure gives rise to relatively large uncertainties when there
are large correlations among some of the measurements, as it is the
case here. To overcome this effect, the method has been repeated by
restricting to the most accurate measurements~\cite{bethke}. The
result of applying the procedure to those measurements with a total
error
$\Delta\asz<0.006$~\cite{incdiszeus,incdish1,nlofitzeusn,nlofith1,incgpzeus}
is:

\vspace{0.1cm}
\centerline{$\overline\asz=0.1192\pm 0.0047\ (\Delta\as^i<0.006)\ ({\rm ZEUS+H1}).$}
\vspace{0.1cm}

Finally, a more reliable, but conservative, approach has been used in
which the known correlations from the determinations of $\as$ coming
from the same experiment were taken into account (``correlation
method''). The theoretical uncertainties arising from the terms beyond
NLO were assumed to be (conservatively) fully correlated.
Error-weighted averages were obtained separately for the ZEUS and H1
measurements:

\vspace{0.1cm}
\centerline{$\overline\asz=0.1200\pm 0.0023\ ({\rm exp.})_{-0.0049}^{+0.0058}\ ({\rm th.})\ ({\rm ZEUS}),$}
\vspace{0.1cm}

\centerline{$\overline\asz=0.1160\pm 0.0016\ ({\rm exp.})_{-0.0049}^{+0.0048}\ ({\rm th.})\ ({\rm H1}).$}
\vspace{0.1cm}

A HERA average was obtained by using the error-weighted average method
on the ZEUS and H1 averages, assuming the experimental uncertainties
to be uncorrelated and taking the overall theoretical uncertainty as the
linear average of its contribution in each experiment. As a result, the
average of the HERA measurements and its uncertainty are:

\vspace{0.1cm}
\centerline{$\overline\asz=0.1186\pm 0.0011\ ({\rm exp.})\pm 0.0050\ ({\rm th.})\ ({\rm ZEUS+H1}).$}
\vspace{0.1cm}

This average, together with the individual values considered, is shown
in Fig.~\ref{fig1}b. It is found to be in good agreement with the
current world average, which does not include any of these
determinations. The results of applying Schmelling's and the
correlation methods are very similar, giving confidence on the average
obtained and its estimated uncertainty.

\section{Energy-scale dependence of $\as$}

The H1 and ZEUS Collaborations have tested the pQCD prediction for the
energy-scale dependence of the strong coupling constant by determining
$\as$ from the measured differential jet cross sections at different
$\etjet$~\cite{dijetzeus,incdiszeus,incdish1,incgpzeus}.
Figure~\ref{fig2}a shows the determinations of the energy-scale
dependence of $\as$ as a function of $\etjet$ from H1 and ZEUS. The
determinations are consistent with the running of $\as$ as predicted
by pQCD over a large range in $\etjet$.

The determinations of $\as(\etjet)$ from H1 and ZEUS at similar
$\etjet$ have been combined using the correlation method explained
above. The combined HERA determinations of the energy-scale dependence
of $\as$ are shown in Fig.~\ref{fig2}b, in which the running of $\as$
from HERA jet data alone is clearly observed.

%Figure 2
\begin{figure}[h]
\setlength{\unitlength}{1.0cm}
\begin{picture} (10.0,6.0)
\put (-3.0,-1.0){\epsfig{figure=dis_shape47col.eps,width=8.0cm}}
\put (5.0,-1.0){\epsfig{figure=dis_shape4813col.eps,width=8.0cm}}
\put (3.6,4.6){\bf\small (a)}
\put (11.6,4.6){\bf\small (b)}
\end{picture}
\caption{(a) $\as$ as a function of $\etjet$ from H1 and ZEUS; (b)
  combined $\as$ as a function of $\etjet$ from HERA jet data. In both
  figures, the QCD prediction for the running of $\as$ is also
  shown. In (a), the inner error bars display the statistical
  uncertainties and the outer error bars display the systematic and
  theoretical uncertainties added in quadrature. In (b), the inner
  (outer) error bars show the experimental (theoretical) uncertainties.
\label{fig2}}
\end{figure}

\section{Summary}

A comprehensive average of $\as$ and its energy-scale dependence from
HERA data has been performed taking into account the known
correlations in each experiment and assuming conservatively that the
theoretical uncertainties arising from the terms beyond NLO are fully
correlated. The HERA average is

\vspace{0.1cm}
\centerline{$\overline\asz=0.1186\pm 0.0011\ ({\rm exp.})\pm 0.0050\ ({\rm th.}).$}
\vspace{0.1cm}
 
The experimental uncertainty of this average is $\sim 0.9\%$ and the
theoretical uncertainty amounts to $\sim 4\%$.
There is still room for improvement when the next-to-NLO (NNLO)
calculations needed for the determination of the PDFs are included and
when the NNLO calculations needed for jet-based observables are
finished.

\begin{theacknowledgments}
This work has been carried out in collaboration with Juan Terr\'on.
I would like to thank my colleagues from H1 and ZEUS for their help in
the preparation of this report. 
\end{theacknowledgments}


\begin{thebibliography}{99}

\bibitem{dijetzeus} 
\colab{ZEUS}, J. Breitweg \etal, \Journal{\PLB}{507}{2001}{70}.

\bibitem{incdiszeus} 
\colab{ZEUS}, S. Chekanov \etal, \Journal{\PLB}{547}{2002}{164}.

\bibitem{incdish1} 
\colab{H1}, C. Adloff \etal, \Journal{\EPC}{19}{2001}{289}.

\bibitem{nlofitzeus} 
\colab{ZEUS}, S. Chekanov \etal, \Journal{\PRD}{67}{2003}{012007}.

\bibitem{nlofitzeusn} 
\colab{ZEUS}, S. Chekanov \etal, DESY 05-050 (March 2005), hep-ph/0503274.

\bibitem{nlofith1} 
\colab{H1}, C. Adloff \etal, \Journal{\EPC}{21}{2001}{33}.

\bibitem{subnczeus} 
\colab{ZEUS}, S. Chekanov \etal, \Journal{\PLB}{558}{2003}{41}.

\bibitem{subcczeus} 
\colab{ZEUS}, S. Chekanov \etal, \Journal{\EPC}{31}{2003}{149}.

\bibitem{incgpzeus} 
\colab{ZEUS}, S. Chekanov \etal, \Journal{\PLB}{560}{2003}{7}.

\bibitem{multijetzeus} 
\colab{ZEUS}, S. Chekanov \etal, DESY 05-019 (January 2005), hep-ex/0502007.

\bibitem{shanczeus} 
\colab{ZEUS}, S. Chekanov \etal, \Journal{\NPB}{700}{2004}{3}.

\bibitem{wa}
S. Bethke, hep-ex/0407021.

\bibitem{schmelling}
M. Schmelling, \Journal{\PS}{51}{1995}{676}.

\bibitem{bethke}
S. Bethke, \Journal{\JPG}{26}{2000}{R27}.

\end{thebibliography}

\end{document}

hep-ex0506035.extracted_bib
\begin{thebibliography}{99}

\bibitem{dijetzeus} 
\colab{ZEUS}, J. Breitweg \etal, \Journal{\PLB}{507}{2001}{70}.

\bibitem{incdiszeus} 
\colab{ZEUS}, S. Chekanov \etal, \Journal{\PLB}{547}{2002}{164}.

\bibitem{incdish1} 
\colab{H1}, C. Adloff \etal, \Journal{\EPC}{19}{2001}{289}.

\bibitem{nlofitzeus} 
\colab{ZEUS}, S. Chekanov \etal, \Journal{\PRD}{67}{2003}{012007}.

\bibitem{nlofitzeusn} 
\colab{ZEUS}, S. Chekanov \etal, DESY 05-050 (March 2005), hep-ph/0503274.

\bibitem{nlofith1} 
\colab{H1}, C. Adloff \etal, \Journal{\EPC}{21}{2001}{33}.

\bibitem{subnczeus} 
\colab{ZEUS}, S. Chekanov \etal, \Journal{\PLB}{558}{2003}{41}.

\bibitem{subcczeus} 
\colab{ZEUS}, S. Chekanov \etal, \Journal{\EPC}{31}{2003}{149}.

\bibitem{incgpzeus} 
\colab{ZEUS}, S. Chekanov \etal, \Journal{\PLB}{560}{2003}{7}.

\bibitem{multijetzeus} 
\colab{ZEUS}, S. Chekanov \etal, DESY 05-019 (January 2005), hep-ex/0502007.

\bibitem{shanczeus} 
\colab{ZEUS}, S. Chekanov \etal, \Journal{\NPB}{700}{2004}{3}.

\bibitem{wa}
S. Bethke, hep-ex/0407021.

\bibitem{schmelling}
M. Schmelling, \Journal{\PS}{51}{1995}{676}.

\bibitem{bethke}
S. Bethke, \Journal{\JPG}{26}{2000}{R27}.

\end{thebibliography}