Status Report of NNLO QCD Calculations
Michael Klasen\footnote{Permanent address: Laboratoire de Physique Subatomique et de Cosmologie, Universit\'e Joseph Fourier/CNRS-IN2P3, 53 Avenue des Martyrs, 38026 Grenoble, France}
We review recent progress in next-to-next-to-leading order (NNLO) perturbative QCD calculations with special emphasis on results ready for phenomenological applications. Important examples are new results on structure functions and jet or Higgs boson production. In addition, we describe new calculational techniques based on twistors and their potential for efficient calculations of multiparticle amplitudes.
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}
	  {
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paper.tex
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\documentclass[
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%%  ,draft            % use draft while you are working on the paper
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\begin{document}

\title{Status Report of NNLO QCD Calculations}

\classification{12.38.Bx}
\keywords      {Perturbative QCD calculations}

\author{Michael Klasen\footnote{Permanent address:
  Laboratoire de Physique Subatomique et de Cosmologie,
  Universit\'e Joseph Fourier/CNRS-IN2P3, 53 Avenue des Martyrs,
  38026 Grenoble, France}}{address={Institute for Nuclear Theory,
  University of Washington, Box 351550, Seattle, WA 98195-1550, USA}
}


\begin{abstract}
 We review recent progress in next-to-next-to-leading order (NNLO)
 perturbative QCD calculations with special emphasis on results ready for
 phenomenological applications. Important examples are new results on
 structure functions and jet or Higgs boson production. In addition, we
 describe new calculational techniques based on twistors and their potential
 for efficient calculations of multiparticle amplitudes.
\end{abstract}

\maketitle

\vspace*{-9cm}
\noindent LPSC 05-054 \\
INT-ACK 05-038 \\
hep-ph/0506252
\vspace*{ 7cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% MAINMATTER
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

Despite the fact that the theory of strong interactions, quantum
chromodynamics (QCD), is today a well-established part of the Standard Model
of particle physics, it continues to be an extremely active field of
experimental and theoretical research. This is as true for non-perturbative
(and often lattice) determinations of meson and baryon spectra, decay
constants, or form factors for electric dipole moments or $B$-meson decays,
as it is for perturbative calculations. Today, the experimental precision of
high-energy collider data always requires calculations beyond the leading
order (LO) in the strong coupling constant, $\alpha_s(\mu)$, and often even
beyond the next-to-leading order (NLO) for reliable comparisons and
determinations of unknown Standard Model parameters.

After a relatively slow start due to large technical difficulties in the
mid-1990s, QCD calculations at next-to-next-to-leading order (NNLO) have
recently attained their goals for several phenomenological applications. The
aim of this Report is therefore to review recent progress in this field with
special emphasis on phenomenologically relevant results, including structure
functions, jet, and Higgs boson production. Very recently, twistor methods
have received particular attention, as they may offer a route to efficient
multiparticle calculations, even beyond tree-level. They will therefore be
briefly discussed, before we present our conclusions.

\section{Structure Functions}

Since its start-up in 1992, the DESY $ep$ collider HERA has provided a
wealth of data on the structure functions of the proton, and thus its parton
densities (PDFs), in a large region of Bjorken-$x$ and $Q^2$ \cite{Behnke}.
The PDFs represent at the same time an important ingredient in the
search for new physics, that is currently underway at the Fermilab Tevatron
and is an important research goal at the CERN LHC \cite{Roeck}. In
particular, QCD uncertainties in PDF determinations, {\it e.g.} from
renormalization and factorization scale and scheme \cite{Klasen:1996yk,%
Martin:2004ir} variations, have to be under control before disagreement
between theory and experiment can be interpreted as evidence for new
physics.

It is thus fortunate that, after completion of the three-loop singlet
splitting functions (obtained from the $1/\epsilon$-poles in dimensional
regularization) and longitudinal coefficient function (obtained from the
finite terms) \cite{Vogt:2004mw}, a
full NNLO calculation of the structure functions $F_{1,2}(x,Q^2)$ (and thus
also of $F_L = F_2 - 2 x F_1)$, is now available. These results have been
obtained using a large variety of techniques, such as application of the
optical theorem, transformation to Mellin space, mapping of diagrams with
composite topologies to basic building blocks, and integration by parts.
They also relied on recent advances in mathematics (harmonic sums) and
computer algebra (QGRAF, FORM). Fortunately, the results can also be applied
to other physical observables such as photon structure functions and total
cross sections in $e^+e^-$-annihilation.

While the numerical effects of the NNLO QCD corrections are well visible in
splitting functions and coefficient functions {\em separately} and show
clear differences in normalization, although not in shape, from earlier
leading-$\ln(x)$ estimates (see Fig.\ \ref{fig:1}), the
%
\begin{figure}[h]
  \includegraphics[width=.29\textwidth]{splitting}
  \includegraphics[width=.24\textwidth]{coefficient}
  \caption{\label{fig:1}Splitting functions (left) and longitudinal
  coefficient functions (right) at NNLO \cite{Vogt:2004mw}.}
\end{figure}
%
total effect in the {\em convoluted} structure functions is only
important for very small or large $x$ and small $Q^2$. For this reason, the
NNLO corrections have not yet been included in the global CTEQ analysis, as
the authors find sufficient stability of their fit at NLO
\cite{Huston:2005jm}, whereas the MRST collaboration have now implemented
the exact NNLO splitting functions, but find minor effects w.r.t.\ a
DIS-scheme motivated variation of the input gluon density
\cite{Martin:2004ir} or an earlier analysis using approximate NNLO splitting
functions \cite{Martin:2002dr}.

\section{Jet production in electron-positron annihilation}

Calculations of observables less inclusive than deep inelastic scattering
(DIS) structure functions and total $e^+e^-$ cross sections do not allow for
a straight-forward application of the optical theorem, as they require
multiparticle cuts in the three-loop photon propagator. For this reason,
NNLO jet calculations in $e^+e^-$-annihilation have been lagging behind,
although recently
considerable progress has also been achieved in this field. The interference
terms of $1\to3$ tree-level and two-loop amplitudes are now known, as are
the single soft and/or collinear regions of $1\to4$ squared one-loop
amplitudes. What remains to be calculated are the maximally double-soft and
triple-collinear regions of $1\to5$ squared tree-level amplitudes. Here,
antenna functions have successfully been used for quark-gluon and
gluon-gluon final states \cite{Gehrmann-DeRidder:2005hi}.

A first preliminary numerical result has been obtained for the NNLO
contribution to the $C_F^2$ color class of the average thrust
\cite{Gehrmann-DeRidder:2004xe}
\begin{eqnarray*}
 \langle 1-T \rangle &=& \int (1-T)\frac{1}{\sigma_0}\frac{d\sigma}{dT}
 ~~=~~ C_F \left[ \left(\frac{\alpha_s}{2\pi}\right) A +
 \left(\frac{\alpha_s}{2\pi}\right)^2 B+\left(\frac{\alpha_s}{2\pi}\right)^3
 C + \ldots\right],
\end{eqnarray*}
where $A=1.57$ and $B=32.3$ were known and now $C(C_F^2)=-20.4 \pm 4$.
The full result will be important for reliable estimates of higher-twist
effects in $e^+e^-$ event shapes. Furthermore, analytic continuation allows
to apply the results obtained for three-jet production in
$e^+e^-$-annihilation also to dijet production in DIS.

\section{Higgs boson production at hadron colliders}

The inclusive NNLO cross section for light/heavy scalar and pseudo-scalar
Higgs production in $gg$ scattering at hadron colliders has been known for
several years, but the relevance of these calculations has been limited by
the need for experimental cuts on associated photons, jets or $b$-quarks.
Fully differential cross sections, such as a recent NNLO calculation for
Higgs plus dijet production \cite{Anastasiou:2004xq},
allow for transverse energy vetos on the additional jets and thus for a
suppression of the QCD background in the diphoton (or other) Higgs decay
channels (see Fig.\ \ref{fig:2}).
%
\begin{figure}[h]
  \includegraphics[width=.40\textwidth]{higgs}
  \caption{\label{fig:2}Higgs production at the LHC with diphoton decay and
  no additional hard jets \cite{Anastasiou:2004xq}.}
\end{figure}
%

\section{Jet hadroproduction and twistor methods}

For hadron-hadron scattering, the two-loop $2\to2$ parton amplitudes
have been known for some time, and the result for polarized $qq\to qq$
scattering has recently been confirmed using supersymmetric methods
\cite{DeFreitas:2004tk}. Since the soft/collinear singular regions of
one-loop $2\to3$ amplitudes are also known, recent work has focused on
multiple soft/collinear emissions in tree-level multiparton amplitudes and
on extending the dipole subtraction formalism to NNLO, where two
alternatives to previously known methods have been proposed
\cite{Frixione:2004is,Somogyi:2005xz}. Although the subtraction terms are in
principle known, the singular regions often overlap and need to be
disentangled to avoid double-counting. Furthermore, phase space
factorization and integration of the singular matrix elements still needs to
be resolved. For this reason, further progress towards a full NNLO
calculation for jet hadroproduction has been slow, and numerical integration
methods using sector decomposition \cite{Binoth:2004jv} may have to be
developped further.

Multiparticle amplitudes may be calculated efficiently by relating
perturbative gauge theory to the $D$-instanton expansion for a topological
string in twistor space. While originally motivated by
$N=4$ super Yang-Mills theory, this approach has been shown to work at
tree-level for generic QCD amplitudes. $N=4$ supersymmetric and
cut-constructible one-loop diagrams have also been calculated, and
extensions to non-supersymmetric theories and two-loop amplitudes are
underway \cite{Dixon}.

\section{Conclusion}

Although this brief Report is necessarily far from complete, it should be
clear that NNLO QCD calculations represent a theoretically challenging,
phenomenologically important, and fast-moving field of research allowing
for interesting advances in the fields of mathematics, computer science
and last, but not least, elementary particle physics.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% BACKMATTER
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{theacknowledgments}
 The author thanks the working group convenors for the kind invitation, the
 DoE's INT at the University of Washington for kind hospitality and partial
 financial support during preparation of this work, and L.\ Dixon and T.\
 Teubner for useful discussions.
\end{theacknowledgments}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% The following lines show an example how to produce a bibliography
%% without the help of the BibTeX program. This could be used instead
%% of the above.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{9}

\bibitem{Behnke}
O.~Behnke and E.~Gallo for the H1 and ZEUS Collaborations, these
proceedings.

\bibitem{Roeck}
A.~de~Roeck for the HERA-LHC Workshop, these proceedings.

%\cite{Klasen:1996yk}
\bibitem{Klasen:1996yk}
M.~Klasen and G.~Kramer,
%``Large transverse momentum jet production and DIS distributions of the
%proton,''
Phys.\ Lett.\ B {\bf 386}, 384 (1996).
%[arXiv:hep-ph/9605210].
%%CITATION = HEP-PH 9605210;%%

%\cite{Martin:2004ir}
\bibitem{Martin:2004ir}
A.~D.~Martin, R.~G.~Roberts, W.~J.~Stirling and R.~S.~Thorne,
%``Physical gluons and high-E(T) jets,''
Phys.\ Lett.\ B {\bf 604}, 61 (2004).
%[arXiv:hep-ph/0410230].
%%CITATION = HEP-PH 0410230;%%

%\cite{Vogt:2004mw}
\bibitem{Vogt:2004mw}
S.~Moch, J.~A.~M.~Vermaseren and A.~Vogt,
%``The three-loop splitting functions in QCD: The singlet case,''
Nucl.\ Phys.\ B {\bf 691}, 129 (2004);
%[arXiv:hep-ph/0404111].
%%CITATION = HEP-PH 0404111;%%
%
%\cite{Moch:2004xu}
%\bibitem{Moch:2004xu}
%S.~Moch, J.~A.~M.~Vermaseren and A.~Vogt,
%``The longitudinal structure function at the third order,''
Phys.\ Lett.\ B {\bf 606}, 123 (2005);
%[arXiv:hep-ph/0411112].
%%CITATION = HEP-PH 0411112;%%
%
%\cite{Vermaseren:2005qc}
%\bibitem{Vermaseren:2005qc}
%J.~A.~M.~Vermaseren, A.~Vogt and S.~Moch,
%``The third-order QCD corrections to deep-inelastic scattering by photon
%exchange,''
%arXiv:
hep-ph/0504242.
%%CITATION = HEP-PH 0504242;%%

%\cite{Huston:2005jm}
\bibitem{Huston:2005jm}
J.~Huston, J.~Pumplin, D.~Stump and W.~K.~Tung,
%``Stability of NLO global analysis and implications for hadron collider
%physics,''
%arXiv:
hep-ph/0502080.
%%CITATION = HEP-PH 0502080;%%

%\cite{Martin:2002dr}
\bibitem{Martin:2002dr}
A.~D.~Martin, R.~G.~Roberts, W.~J.~Stirling and R.~S.~Thorne,
%``NNLO global parton analysis,''
Phys.\ Lett.\ B {\bf 531}, 216 (2002).
%[arXiv:hep-ph/0201127].
%%CITATION = HEP-PH 0201127;%%

%\cite{Gehrmann-DeRidder:2005hi}
\bibitem{Gehrmann-DeRidder:2005hi}
A.~Gehrmann-De Ridder, T.~Gehrmann and E.~W.~N.~Glover,
%``Quark-gluon antenna functions from neutralino decay,''
Phys.\ Lett.\ B {\bf 612}, 36 (2005);
%[arXiv:hep-ph/0501291].
%%CITATION = HEP-PH 0501291;%%
%
%\cite{Gehrmann-DeRidder:2005aw}
%\bibitem{Gehrmann-DeRidder:2005aw}
%A.~Gehrmann-De Ridder, T.~Gehrmann and E.~W.~N.~Glover,
%``Gluon gluon antenna functions from Higgs boson decay,''
%Phys.\ Lett.\ B 
{\bf 612}, 49 (2005);
%[arXiv:hep-ph/0502110].
%%CITATION = HEP-PH 0502110;%%
%
%\cite{Ridder:2005cm}
%\bibitem{Ridder:2005cm}
%A.~Gehrmann-De Ridder, T.~Gehrmann and E.~W.~N.~Glover,
%``Antenna subtraction at NNLO,''
%arXiv:
hep-ph/0505111.
%%CITATION = HEP-PH 0505111;%%

%\cite{Gehrmann-DeRidder:2004xe}
\bibitem{Gehrmann-DeRidder:2004xe}
A.~Gehrmann-De Ridder, T.~Gehrmann and E.~W.~N.~Glover,
%``Infrared structure of e+ e- $\to$ 3jets at NNLO: The C(F)**2 contribution,''
Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 135}, 97 (2004).
%[arXiv:hep-ph/0407023].
%%CITATION = HEP-PH 0407023;%%

%\cite{Anastasiou:2004xq}
\bibitem{Anastasiou:2004xq}
C.~Anastasiou, K.~Melnikov and F.~Petriello,
%``Higgs boson production at hadron colliders: Differential cross sections
%through next-to-next-to-leading order,''
%arXiv:
hep-ph/0409088
%%CITATION = HEP-PH 0409088;%%
and
%\cite{Anastasiou:2005qj}
%\bibitem{Anastasiou:2005qj}
%C.~Anastasiou, K.~Melnikov and F.~Petriello,
%``Fully differential Higgs boson production and the di-photon signal through
%next-to-next-to-leading order,''
%arXiv:
hep-ph/0501130.
%%CITATION = HEP-PH 0501130;%%

%\cite{DeFreitas:2004tk}
\bibitem{DeFreitas:2004tk}
A.~De Freitas and Z.~Bern,
%``Two-loop helicity amplitudes for quark-quark scattering in QCD and  gluino
%gluino scattering in supersymmetric Yang-Mills theory,''
JHEP {\bf 0409}, 039 (2004).
%[arXiv:hep-ph/0409007].
%%CITATION = HEP-PH 0409007;%%

%\cite{Frixione:2004is}
\bibitem{Frixione:2004is}
S.~Frixione and M.~Grazzini,
%``Subtraction at NNLO,''
%arXiv:
hep-ph/0411399.
%%CITATION = HEP-PH 0411399;%%

%\cite{Somogyi:2005xz}
\bibitem{Somogyi:2005xz}
G.~Somogyi, Z.~Trocsanyi and V.~Del Duca,
%``Matching of singly- and doubly-unresolved limits of tree-level QCD squared
%matrix elements,''
%arXiv:
hep-ph/0502226.
%%CITATION = HEP-PH 0502226;%%

%\cite{Binoth:2004jv}
\bibitem{Binoth:2004jv}
T.~Binoth and G.~Heinrich,
%``Numerical evaluation of phase space integrals by sector decomposition,''
Nucl.\ Phys.\ B {\bf 693}, 134 (2004).
%[arXiv:hep-ph/0402265].
%%CITATION = HEP-PH 0402265;%%

\bibitem{Dixon}
L.~Dixon, these proceedings.

\end{thebibliography}

%\endinput

\end{document}

%%
%% End of file `template-6s.tex'.

hep-ph0506252.extracted_bib
\begin{thebibliography}{9}

\bibitem{Behnke}
O.~Behnke and E.~Gallo for the H1 and ZEUS Collaborations, these
proceedings.

\bibitem{Roeck}
A.~de~Roeck for the HERA-LHC Workshop, these proceedings.

%\cite{Klasen:1996yk}
\bibitem{Klasen:1996yk}
M.~Klasen and G.~Kramer,
%``Large transverse momentum jet production and DIS distributions of the
%proton,''
Phys.\ Lett.\ B {\bf 386}, 384 (1996).
%[arXiv:hep-ph/9605210].
%%CITATION = HEP-PH 9605210;%%

%\cite{Martin:2004ir}
\bibitem{Martin:2004ir}
A.~D.~Martin, R.~G.~Roberts, W.~J.~Stirling and R.~S.~Thorne,
%``Physical gluons and high-E(T) jets,''
Phys.\ Lett.\ B {\bf 604}, 61 (2004).
%[arXiv:hep-ph/0410230].
%%CITATION = HEP-PH 0410230;%%

%\cite{Vogt:2004mw}
\bibitem{Vogt:2004mw}
S.~Moch, J.~A.~M.~Vermaseren and A.~Vogt,
%``The three-loop splitting functions in QCD: The singlet case,''
Nucl.\ Phys.\ B {\bf 691}, 129 (2004);
%[arXiv:hep-ph/0404111].
%%CITATION = HEP-PH 0404111;%%
%
%\cite{Moch:2004xu}
%\bibitem{Moch:2004xu}
%S.~Moch, J.~A.~M.~Vermaseren and A.~Vogt,
%``The longitudinal structure function at the third order,''
Phys.\ Lett.\ B {\bf 606}, 123 (2005);
%[arXiv:hep-ph/0411112].
%%CITATION = HEP-PH 0411112;%%
%
%\cite{Vermaseren:2005qc}
%\bibitem{Vermaseren:2005qc}
%J.~A.~M.~Vermaseren, A.~Vogt and S.~Moch,
%``The third-order QCD corrections to deep-inelastic scattering by photon
%exchange,''
%arXiv:
hep-ph/0504242.
%%CITATION = HEP-PH 0504242;%%

%\cite{Huston:2005jm}
\bibitem{Huston:2005jm}
J.~Huston, J.~Pumplin, D.~Stump and W.~K.~Tung,
%``Stability of NLO global analysis and implications for hadron collider
%physics,''
%arXiv:
hep-ph/0502080.
%%CITATION = HEP-PH 0502080;%%

%\cite{Martin:2002dr}
\bibitem{Martin:2002dr}
A.~D.~Martin, R.~G.~Roberts, W.~J.~Stirling and R.~S.~Thorne,
%``NNLO global parton analysis,''
Phys.\ Lett.\ B {\bf 531}, 216 (2002).
%[arXiv:hep-ph/0201127].
%%CITATION = HEP-PH 0201127;%%

%\cite{Gehrmann-DeRidder:2005hi}
\bibitem{Gehrmann-DeRidder:2005hi}
A.~Gehrmann-De Ridder, T.~Gehrmann and E.~W.~N.~Glover,
%``Quark-gluon antenna functions from neutralino decay,''
Phys.\ Lett.\ B {\bf 612}, 36 (2005);
%[arXiv:hep-ph/0501291].
%%CITATION = HEP-PH 0501291;%%
%
%\cite{Gehrmann-DeRidder:2005aw}
%\bibitem{Gehrmann-DeRidder:2005aw}
%A.~Gehrmann-De Ridder, T.~Gehrmann and E.~W.~N.~Glover,
%``Gluon gluon antenna functions from Higgs boson decay,''
%Phys.\ Lett.\ B 
{\bf 612}, 49 (2005);
%[arXiv:hep-ph/0502110].
%%CITATION = HEP-PH 0502110;%%
%
%\cite{Ridder:2005cm}
%\bibitem{Ridder:2005cm}
%A.~Gehrmann-De Ridder, T.~Gehrmann and E.~W.~N.~Glover,
%``Antenna subtraction at NNLO,''
%arXiv:
hep-ph/0505111.
%%CITATION = HEP-PH 0505111;%%

%\cite{Gehrmann-DeRidder:2004xe}
\bibitem{Gehrmann-DeRidder:2004xe}
A.~Gehrmann-De Ridder, T.~Gehrmann and E.~W.~N.~Glover,
%``Infrared structure of e+ e- $\to$ 3jets at NNLO: The C(F)**2 contribution,''
Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 135}, 97 (2004).
%[arXiv:hep-ph/0407023].
%%CITATION = HEP-PH 0407023;%%

%\cite{Anastasiou:2004xq}
\bibitem{Anastasiou:2004xq}
C.~Anastasiou, K.~Melnikov and F.~Petriello,
%``Higgs boson production at hadron colliders: Differential cross sections
%through next-to-next-to-leading order,''
%arXiv:
hep-ph/0409088
%%CITATION = HEP-PH 0409088;%%
and
%\cite{Anastasiou:2005qj}
%\bibitem{Anastasiou:2005qj}
%C.~Anastasiou, K.~Melnikov and F.~Petriello,
%``Fully differential Higgs boson production and the di-photon signal through
%next-to-next-to-leading order,''
%arXiv:
hep-ph/0501130.
%%CITATION = HEP-PH 0501130;%%

%\cite{DeFreitas:2004tk}
\bibitem{DeFreitas:2004tk}
A.~De Freitas and Z.~Bern,
%``Two-loop helicity amplitudes for quark-quark scattering in QCD and  gluino
%gluino scattering in supersymmetric Yang-Mills theory,''
JHEP {\bf 0409}, 039 (2004).
%[arXiv:hep-ph/0409007].
%%CITATION = HEP-PH 0409007;%%

%\cite{Frixione:2004is}
\bibitem{Frixione:2004is}
S.~Frixione and M.~Grazzini,
%``Subtraction at NNLO,''
%arXiv:
hep-ph/0411399.
%%CITATION = HEP-PH 0411399;%%

%\cite{Somogyi:2005xz}
\bibitem{Somogyi:2005xz}
G.~Somogyi, Z.~Trocsanyi and V.~Del Duca,
%``Matching of singly- and doubly-unresolved limits of tree-level QCD squared
%matrix elements,''
%arXiv:
hep-ph/0502226.
%%CITATION = HEP-PH 0502226;%%

%\cite{Binoth:2004jv}
\bibitem{Binoth:2004jv}
T.~Binoth and G.~Heinrich,
%``Numerical evaluation of phase space integrals by sector decomposition,''
Nucl.\ Phys.\ B {\bf 693}, 134 (2004).
%[arXiv:hep-ph/0402265].
%%CITATION = HEP-PH 0402265;%%

\bibitem{Dixon}
L.~Dixon, these proceedings.

\end{thebibliography}