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%
%       By Jean Orloff
%       Comments & suggestions by e-mail: ORLOFF@surya11.cern.ch
%       No modification of this file allowed if not e-sent to me.
%
% WHAT IS IT:
% psbox is a set of machine-independent TeX macros to
% 1) allow (Encapsulated) PostScript figure inclusion in all versions
%    of TeX (Plain, LaTeX) on all machines using a PostScript printer
% 2) facilitate the communication (e-mail, ftp, ...) of all the files
%    (text, macros, figs) needed to reproduce a TeX document by grouping
%    them together into a single, TeXable file.
% For more info, get the file pub/TeX/psbox/PSBOXALL.TEX by anonymous
%     ftp from cs.nyu.edu(=128.122.140.24)
%
% History:
%  1.34  \readfilename=final fix for all filename scans; try \psforptips
%  1.33: corrects \psnewinput for LaTeX (still fails if fname=a{b}c)
%  1.32: corrects \psfordvialw and adds .TEX to PSBOXALL(!)
%  1.31: adds \psfordvialw(?)
%  1.30: adds \splitfile & \joinfiles for multi-file management
%  1.24: fix error handling & add \psonlyboxes
%  1.22: makes \drawingBox \global for use in Phyzzx
%  1.21: accepts %%BoundingBox: (atend)
%  1.20: tries to add \psfordvitps for the TeXPS package.
%  1.10: adds \psforoztex, error handling...
%2345678 1 2345678 2 2345678 3 2345678 4 2345678 5 2345678 6 2345678 7 23456789
%
% Checking version no to avoid multiple loadings
\def\temp{1.34}%
\let\tempp=\relax
\expandafter\ifx\csname psboxversion\endcsname\relax
  \message{PSBOX(\temp) loading}%
\else
    \ifdim\temp cm>\psboxversion cm
      \message{PSBOX(\temp) loading}%
    \else
      \message{PSBOX(\psboxversion) is already loaded: I won't load
        PSBOX(\temp)!}%
      \let\temp=\psboxversion
      \let\tempp=\endinput
    \fi
\fi
\tempp
\let\psboxversion=\temp
\catcode`\@=11
% Every macro likes a little privacy...
%
%Trying to tame the variety of \special commands for Postscript: the
%  universal internal command \PSspeci@l##1##2 takes ##1 to be the
%  filename and ##2 to be the integer scale factor*1000 (as for usual
%   TeX \scale commands)
%
\def\psfortextures{%     For TeXtures on the Macintosh
%-----------------
\def\PSspeci@l##1##2{%
\special{illustration ##1\space scaled ##2}%
}}%
\def\psfordvitops{%      For the DVItoPS converter on IBM mainframes
%----------------
\def\PSspeci@l##1##2{%
\special{dvitops: import ##1\space \the\drawingwd \the\drawinght}%
}}%
\def\psfordvips{%      For DVIPS converter on VAX, UNIX and PC's
%--------------
\def\PSspeci@l##1##2{%
%    \special{/@scaleunit 1000 def}% never read dox without trying!
\d@my=0.1bp \d@mx=\drawingwd \divide\d@mx by\d@my% BUG! for large \drawingwd
\special{PSfile=##1\space llx=\psllx\space lly=\pslly\space%
urx=\psurx\space ury=\psury\space rwi=\number\d@mx
}}}%
\def\psforoztex{%        For the OzTeX shareware on the Macintosh
%--------------
\def\PSspeci@l##1##2{%
\special{##1 \space
      ##2 1000 div dup scale
      \number-\psllx\space \number-\pslly\space translate
}}}%
\def\psfordvitps{%       From the UNIX TeXPS package, vers.>3.12
%---------------
% Convert a dimension into the number \psn@sp (in scaled points)
\def\psdimt@n@sp##1{\d@mx=##1\relax\edef\psn@sp{\number\d@mx}}
\def\PSspeci@l##1##2{%
% psfig.psr contains the def of "startTexFig": if you can locate it
% and include the correct pathname, it should work
\special{dvitps: Include0 "psfig.psr"}% contains def of "startTexFig"
\psdimt@n@sp{\drawingwd}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\drawinght}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psllx bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\pslly bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psurx bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psury bp}
\special{dvitps: Literal "\psn@sp\space startTexFig\space"}
\special{dvitps: Include1 "##1"}
\special{dvitps: Literal "endTexFig\space"}
}}%
\def\psfordvialw{%   Try for dvialw, a UNIX public domain
%---------------
\def\PSspeci@l##1##2{
\special{language "PostScript",
position = "bottom left",
literal "  \psllx\space \pslly\space translate
  ##2 1000 div dup scale
  -\psllx\space -\pslly\space translate",
include "##1"}
}}%
\def\psforptips{%   For MS-DOS; LUOMA@brandeis.bitnet
%---------------
\def\PSspeci@l##1##2{{
\d@mx=\psurx bp
\advance \d@mx by -\psllx bp
\divide \d@mx by 1000\multiply\d@mx by \xscale
\incm{\d@mx}
\let\tmpx\dimincm
\d@my=\psury bp
\advance \d@my by -\pslly bp
\divide \d@my by 1000\multiply\d@my by \xscale
\incm{\d@my}
\let\tmpy\dimincm
\d@mx=-\psllx bp
\divide \d@mx by 1000\multiply\d@mx by \xscale
\d@my=-\pslly bp
\divide \d@my by 1000\multiply\d@my by \xscale
\at(\d@mx;\d@my){\special{ps:##1 x=\tmpx, y=\tmpy}}
}}}%
\def\psonlyboxes{%     Draft-like behaviour if none of the others works
%---------------
\def\PSspeci@l##1##2{%
\at(0cm;0cm){\boxit{\vbox to\drawinght
  {\vss\hbox to\drawingwd{\at(0cm;0cm){\hbox{({\tt##1})}}\hss}}}}
}}%
\def\psloc@lerr#1{%
\let\savedPSspeci@l=\PSspeci@l%
\def\PSspeci@l##1##2{%
\at(0cm;0cm){\boxit{\vbox to\drawinght
  {\vss\hbox to\drawingwd{\at(0cm;0cm){\hbox{({\tt##1}) #1}}\hss}}}}
\let\PSspeci@l=\savedPSspeci@l% restore normal output for other figs!
}}%
%\def\psfor...  add your own!
%
% Some common defs
%
\newread\pst@mpin
\newdimen\drawinght\newdimen\drawingwd
\newdimen\psxoffset\newdimen\psyoffset
\newbox\drawingBox
\newcount\xscale \newcount\yscale \newdimen\pscm\pscm=1cm
\newdimen\d@mx \newdimen\d@my
\newdimen\pswdincr \newdimen\pshtincr
\let\ps@nnotation=\relax
{\catcode`\|=0 |catcode`|\=12 |catcode`|%=12 |catcode`~=12
|catcode`#=12 |catcode`*=14
|xdef|backslashother{\}*
|xdef|percentother{%}*
|xdef|tildeother{~}*
|xdef|sharpother{#}*
}%
% useful to display special chars in \tt; fails for \,#,%
\def\R@moveMeaningHeader#1:->{}%
\def\uncatcode#1{%
\edef#1{\expandafter\R@moveMeaningHeader\meaning#1}}%
%
\def\execute#1{#1}% NOT stupid: cs in #1 are then identified BEFORE execution
\def\psm@keother#1{\catcode`#112\relax}% borrowed from latex
\def\executeinspecs#1{%
\execute{\begingroup\let\do\psm@keother\dospecials\catcode`\^^M=9#1\endgroup}}%
\def\@mpty{}%
% \if\matchin#1#2<=> \iftrue if #1 contains #2, <=>\iffalse otherwise:
% \if\matchexpin: idem, but #1 & #2 are first fully expanded (no \if
% inside!)
% \tmpa & \tmpb contain what's before and after the occurence of #2
\def\matchexpin#1#2{
  \fi%
%\message{(#1>#2)}
  \edef\tmpb{{#2}}%
  \expandafter\makem@tchtmp\tmpb%
  \edef\tmpa{#1}\edef\tmpb{#2}%
  \expandafter\expandafter\expandafter\m@tchtmp\expandafter\tmpa\tmpb\endm@tch%
  \if\match%
}%
\def\matchin#1#2{%
  \fi%
  \makem@tchtmp{#2}%
  \m@tchtmp#1#2\endm@tch%
  \if\match%
}%
\def\makem@tchtmp#1{\def\m@tchtmp##1#1##2\endm@tch{%
  \def\tmpa{##1}\def\tmpb{##2}\let\m@tchtmp=\relax%
  \ifx\tmpb\@mpty\def\match{YN}%
  \else\def\match{YY}\fi%
}}%
% converts any dimen in cm, with 1E-4 cm precision
\def\incm#1{{\psxoffset=1cm\d@my=#1
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \xdef\dimincm{\number\d@mx.}
  \advance\d@my by -\number\d@mx cm
  \multiply\d@my by 100
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \edef\dimincm{\dimincm\number\d@mx}
  \advance\d@my by -\number\d@mx cm
  \multiply\d@my by 100
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \xdef\dimincm{\dimincm\number\d@mx}
}}%
%
%  \ReadPSize{PSfilename} reads the dimensions of a PostScript drawing
%      and stores it in \drawinght(wd)
\newif\ifNotB@undingBox
\newhelp\PShelp{Proceed: you'll have a 5cm square blank box instead of
your graphics (Jean Orloff).}%
\def\s@tsize#1 #2 #3 #4\@ndsize{
  \def\psllx{#1}\def\pslly{#2}%
  \def\psurx{#3}\def\psury{#4}%  needed by a crazyness of dvips!
  \ifx\psurx\@mpty\NotB@undingBoxtrue% this is not a valid one!
  \else
    \drawinght=#4bp\advance\drawinght by-#2bp
    \drawingwd=#3bp\advance\drawingwd by-#1bp
%  !Units related by crazy factors as bp/pt=72.27/72 should be BANNED!
  \fi
  }%
\def\sc@nBBline#1:#2\@ndBBline{\edef\p@rameter{#1}\edef\v@lue{#2}}%
\def\g@bblefirstblank#1#2:{\ifx#1 \else#1\fi#2}%
{\catcode`\%=12
\xdef\B@undingBox{%%BoundingBox}}%
%% is not a true comment in PostScript, even if % is!
\def\ReadPSize#1{
 \readfilename#1\relax
 \let\PSfilename=\lastreadfilename
 \openin\pst@mpin=#1\relax
 \ifeof\pst@mpin \errhelp=\PShelp
   \errmessage{I haven't found your postscript file (\PSfilename)}%
   \psloc@lerr{was not found}%
   \s@tsize 0 0 142 142\@ndsize
   \closein\pst@mpin
 \else
% each entry in \GlobalInputList should be unique
   \if\matchexpin{\GlobalInputList}{, \lastreadfilename}%
   \else\xdef\GlobalInputList{\GlobalInputList, \lastreadfilename}%
     \immediate\write\psbj@inaux{\lastreadfilename,}%
   \fi%
   \loop
     \executeinspecs{\catcode`\ =10\global\read\pst@mpin to\n@xtline}%
     \ifeof\pst@mpin
       \errhelp=\PShelp
       \errmessage{(\PSfilename) is not an Encapsulated PostScript File:
           I could not find any \B@undingBox: line.}%
       \edef\v@lue{0 0 142 142:}%
       \psloc@lerr{is not an EPSFile}%
       \NotB@undingBoxfalse
     \else
       \expandafter\sc@nBBline\n@xtline:\@ndBBline
       \ifx\p@rameter\B@undingBox\NotB@undingBoxfalse
         \edef\t@mp{%
           \expandafter\g@bblefirstblank\v@lue\space\space\space}%
         \expandafter\s@tsize\t@mp\@ndsize
       \else\NotB@undingBoxtrue
       \fi
     \fi
   \ifNotB@undingBox\repeat
   \closein\pst@mpin
 \fi
\message{#1}%
}%
%
% \psboxto(xdim;ydim){psfilename}: you specify the dimensions and
%    TeX uniformly scales to fit the largest one. If xdim=0pt, the
%    scale is fully determined by ydim and vice versa.
%    Notice: psboxes are a real vboxes; couldn't take hbox otherwise all
%    indentation and all cr's would be interpreted as spaces (hugh!).
%
\def\psboxto(#1;#2)#3{\vbox{%
   \ReadPSize{#3}%
   \advance\pswdincr by \drawingwd
   \advance\pshtincr by \drawinght
   \divide\pswdincr by 1000
   \divide\pshtincr by 1000
   \d@mx=#1
   \ifdim\d@mx=0pt\xscale=1000
         \else \xscale=\d@mx \divide \xscale by \pswdincr\fi
   \d@my=#2
   \ifdim\d@my=0pt\yscale=1000
         \else \yscale=\d@my \divide \yscale by \pshtincr\fi
   \ifnum\yscale=1000
         \else\ifnum\xscale=1000\xscale=\yscale
                    \else\ifnum\yscale<\xscale\xscale=\yscale\fi
              \fi
   \fi
   \divide\drawingwd by1000 \multiply\drawingwd by\xscale
   \divide\drawinght by1000 \multiply\drawinght by\xscale
   \divide\psxoffset by1000 \multiply\psxoffset by\xscale
   \divide\psyoffset by1000 \multiply\psyoffset by\xscale
   \global\divide\pscm by 1000
   \global\multiply\pscm by\xscale
   \multiply\pswdincr by\xscale \multiply\pshtincr by\xscale
   \ifdim\d@mx=0pt\d@mx=\pswdincr\fi
   \ifdim\d@my=0pt\d@my=\pshtincr\fi
   \message{scaled \the\xscale}%
 \hbox to\d@mx{\hss\vbox to\d@my{\vss
   \global\setbox\drawingBox=\hbox to 0pt{\kern\psxoffset\vbox to 0pt{%
      \kern-\psyoffset
      \PSspeci@l{\PSfilename}{\the\xscale}%
      \vss}\hss\ps@nnotation}%
   \global\wd\drawingBox=\the\pswdincr
   \global\ht\drawingBox=\the\pshtincr
   \global\drawingwd=\pswdincr
   \global\drawinght=\pshtincr
   \baselineskip=0pt
   \copy\drawingBox
 \vss}\hss}%
  \global\psxoffset=0pt
  \global\psyoffset=0pt
  \global\pswdincr=0pt
  \global\pshtincr=0pt % These are local to one figure
  \global\pscm=1cm %should not be necessary
}}%
%
% \psboxscaled{scalefactor*1000}{PSfilename} allows to bypass the
%   rounding errors of TeX integer divisions for situations where the
%   TeX box should fit the original BoundingBox with a precision
%   better
%   than 1/1000.
%
\def\psboxscaled#1#2{\vbox{%
  \ReadPSize{#2}%
  \xscale=#1
  \message{scaled \the\xscale}%
  \divide\pswdincr by 1000 \multiply\pswdincr by \xscale
  \divide\pshtincr by 1000 \multiply\pshtincr by \xscale
  \divide\psxoffset by1000 \multiply\psxoffset by\xscale
  \divide\psyoffset by1000 \multiply\psyoffset by\xscale
  \divide\drawingwd by1000 \multiply\drawingwd by\xscale
  \divide\drawinght by1000 \multiply\drawinght by\xscale
  \global\divide\pscm by 1000
  \global\multiply\pscm by\xscale
  \global\setbox\drawingBox=\hbox to 0pt{\kern\psxoffset\vbox to 0pt{%
     \kern-\psyoffset
     \PSspeci@l{\PSfilename}{\the\xscale}%
     \vss}\hss\ps@nnotation}%
  \advance\pswdincr by \drawingwd
  \advance\pshtincr by \drawinght
  \global\wd\drawingBox=\the\pswdincr
  \global\ht\drawingBox=\the\pshtincr
  \global\drawingwd=\pswdincr
  \global\drawinght=\pshtincr
  \baselineskip=0pt
  \copy\drawingBox
  \global\psxoffset=0pt
  \global\psyoffset=0pt
  \global\pswdincr=0pt
  \global\pshtincr=0pt % These are local to one figure
  \global\pscm=1cm
}}%
%
%  \psbox{PSfilename} makes a TeX box having the minimal size to
%      enclose the picture
\def\psbox#1{\psboxscaled{1000}{#1}}%
%------------------------------------------------------
%  \joinfiles file1, file2, ...n \into joinedfilename .
%     makes one file out of many
%  \splitfile joinedfilename
%     the opposite
\newif\ifn@teof\n@teoftrue
\newif\ifc@ntrolline
\newif\ifmatch
\newread\j@insplitin
\newwrite\j@insplitout
\newwrite\psbj@inaux
\immediate\openout\psbj@inaux=psbjoin.aux
\immediate\write\psbj@inaux{\string\joinfiles}%
\immediate\write\psbj@inaux{\jobname,}%
%
% INPUT REDEFINITION
%
% works if #1 is a single character
\def\toother#1{\ifcat\relax#1\else\expandafter%
  \toother@ux\meaning#1\endtoother@ux\fi}%
\def\toother@ux#1 #2#3\endtoother@ux{\def\tmp{#3}%
  \ifx\tmp\@mpty\def\tmp{#2}\let\next=\relax%
  \else\def\next{\toother@ux#2#3\endtoother@ux}\fi%
\next}%
%
% \readfilename defs:
%
\let\readfilenamehook=\relax
\def\re@d{\expandafter\re@daux}% spares typing 10 \expandafter's...
\def\re@daux{\futurelet\nextchar\stopre@dtest}%
\def\re@dnext{\xdef\lastreadfilename{\lastreadfilename\nextchar}%
  \afterassignment\re@d\let\nextchar}%
\def\stopre@d{\egroup\readfilenamehook}%
\def\stopre@dtest{%
  \ifcat\nextchar\relax\let\nextread\stopre@d
  \else
    \ifcat\nextchar\space\def\nextread{%
      \afterassignment\stopre@d\chardef\nextchar=`}%
    \else\let\nextread=\re@dnext
      \toother\nextchar
      \edef\nextchar{\tmp}%
    \fi
  \fi\nextread}%
\def\readfilename{\bgroup%
  \let\\=\backslashother \let\%=\percentother \let\~=\tildeother
  \let\#=\sharpother \xdef\lastreadfilename{}%
  \re@d}%
%
% redefines \input using \readfilename
%
\xdef\GlobalInputList{\jobname}%
\def\psnewinput{%
  \def\readfilenamehook{% each entry in \GlobalInputList should be unique
    \if\matchexpin{\GlobalInputList}{, \lastreadfilename}%
    \else\xdef\GlobalInputList{\GlobalInputList, \lastreadfilename}%
      \immediate\write\psbj@inaux{\lastreadfilename,}%
    \fi%
    \ps@ldinput\lastreadfilename\relax%
    \let\readfilenamehook=\relax%
  }\readfilename%
}%
\expandafter\ifx\csname @@input\endcsname\relax    % then Plain
  \immediate\let\ps@ldinput=\input\def\input{\psnewinput}%
\else
  \immediate\let\ps@ldinput=\@@input
  \def\@@input{\psnewinput}%
\fi%
%
\def\nowarnopenout{%
 \def\warnopenout##1##2{%
   \readfilename##2\relax
   \message{\lastreadfilename}%
   \immediate\openout##1=\lastreadfilename\relax}}%
\def\warnopenout#1#2{%
 \readfilename#2\relax
 \def\t@mp{TrashMe,psbjoin.aux,psbjoint.tex,}\uncatcode\t@mp
 \if\matchexpin{\t@mp}{\lastreadfilename,}%
 \else
   \immediate\openin\pst@mpin=\lastreadfilename\relax
   \ifeof\pst@mpin
     \else
     \errhelp{If the content of this file is so precious to you, abort (ie
press x or e) and rename it before retrying.}%
     \errmessage{I'm just about to replace your file named \lastreadfilename}%
   \fi
   \immediate\closein\pst@mpin
 \fi
 \message{\lastreadfilename}%
 \immediate\openout#1=\lastreadfilename\relax}%
% % will have an unusual catcode below; use * instead
%\vbox
{\catcode`\%=12\catcode`\*=14
\gdef\splitfile#1{*
 \readfilename#1\relax
 \immediate\openin\j@insplitin=\lastreadfilename\relax
 \ifeof\j@insplitin
   \message{! I couldn't find and split \lastreadfilename!}*
 \else
   \immediate\openout\j@insplitout=TrashMe
   \message{< Splitting \lastreadfilename\space into}*
   \loop
     \ifeof\j@insplitin
       \immediate\closein\j@insplitin\n@teoffalse
     \else
       \n@teoftrue
       \executeinspecs{\global\read\j@insplitin to\spl@tinline\expandafter
         \ch@ckbeginnewfile\spl@tinline%Beginning-Of-File-Named:%\endcheck}*
       \ifc@ntrolline
       \else
         \toks0=\expandafter{\spl@tinline}*
         \immediate\write\j@insplitout{\the\toks0}*
       \fi
     \fi
   \ifn@teof\repeat
   \immediate\closeout\j@insplitout
 \fi\message{>}*
}*
\gdef\ch@ckbeginnewfile#1%Beginning-Of-File-Named:#2%#3\endcheck{*
 \def\t@mp{#1}*
 \ifx\@mpty\t@mp
   \def\t@mp{#3}*
   \ifx\@mpty\t@mp
     \global\c@ntrollinefalse
   \else
     \immediate\closeout\j@insplitout
     \warnopenout\j@insplitout{#2}*
     \global\c@ntrollinetrue
   \fi
 \else
   \global\c@ntrollinefalse
 \fi}*
\gdef\joinfiles#1\into#2{*
 \message{< Joining following files into}*
 \warnopenout\j@insplitout{#2}*
 \message{:}*
 {*
 \edef\w@##1{\immediate\write\j@insplitout{##1}}*
\w@{% This collection of files was produced with CERN psbox package}*
\w@{% To decompose and tex it:}*
\w@{%-save this with a filename CONTAINING ONLY LETTERS and a .TEX}*
\w@{% extension (say, JOINTFIL.TEX), in some uncrowded directory;}*
\w@{%-make sure you can \string\input\space psbox.tex (version>=1.3);}*
\w@{%  (else ftp cs.nyu.edu(=128.122.140.24):pub/TeX/psbox/, then get}*
\w@{%  and tex the file psboxall.tex; more info in psbREAD.ME)}*
\w@{%-tex JOINTFIL.TEX using Plain, or LaTeX, or whatever is needed by}*
\w@{%  the first file in the joining (after splitting JOINTFIL.TEX into}*
\w@{%  it's constituents, TeX will try to process it as it stands).}*
\w@{\string\input\space psbox.tex}*
\w@{\string\splitfile{\string\jobname}}*
\w@{\string\let\string\autojoin=\string\relax}*
}*
 \expandafter\tre@tfilelist#1, \endtre@t
 \immediate\closeout\j@insplitout
 \message{>}*
}*
\gdef\tre@tfilelist#1, #2\endtre@t{*
 \readfilename#1\relax
 \ifx\@mpty\lastreadfilename
 \else
   \immediate\openin\j@insplitin=\lastreadfilename\relax
   \ifeof\j@insplitin
     \errmessage{I couldn't find file \lastreadfilename}*
   \else
     \message{\lastreadfilename}*
     \immediate\write\j@insplitout{%Beginning-Of-File-Named:\lastreadfilename}*
     \executeinspecs{\global\read\j@insplitin to\oldj@ininline}*
     \loop
       \ifeof\j@insplitin\immediate\closein\j@insplitin\n@teoffalse
       \else\n@teoftrue
         \executeinspecs{\global\read\j@insplitin to\j@ininline}*
         \toks0=\expandafter{\oldj@ininline}*
         \let\oldj@ininline=\j@ininline
         \immediate\write\j@insplitout{\the\toks0}*
       \fi
     \ifn@teof
     \repeat
   \immediate\closein\j@insplitin
   \fi
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%% version 1.00 (May 03, 2001): put hep-th/0105037, submit PLB (No. 1473)


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\begin{document}

\begin{titlepage}

\noindent hep-th/0105037  \hspace*{\fill} KA-TP-13-2001

\vspace{2cm}
\begin{center}
{\Large \bf Causality and radiatively induced CPT violation }
\\[1cm]
C. Adam\footnote{E-mail address: adam@particle.uni-karlsruhe.de},
F.R. Klinkhamer\footnote{E-mail address:
frans.klinkhamer@physik.uni-karlsruhe.de}
\\[0.5ex]
{\it Institut f\"ur Theoretische Physik, Universit\"at Karlsruhe,
     D--76128 Karlsruhe, Germany}

\vspace{2cm}
{\bf Abstract} \\
\end{center}
We consider quantum electrodynamics with an additional Lorentz- and
CPT-violating
axial-vector term in the fermionic sector and
discuss the possibility that
radiative corrections  induce a  Lorentz- and CPT-violating
Chern--Simons-like term for the gauge field.  From the requirement
of causality and the assumed validity of
perturbation theory, we conclude that
the induced Chern--Simons-like term must be absent in the full
quantum field theory.


\vspace{2\baselineskip}
\begin{tabbing}
PACS \hspace{1.25em} \= : \hspace{0.25em} \=
                          11.15.Bt; 11.30.Qc; 11.30.Cp; 11.30.Er
                    \\[0.5ex]
Keywords       \> : \> Radiative corrections; Causality;
                       Lorentz noninvariance; CPT violation
\end{tabbing}
\vfill
\end{titlepage}



Over the last few years,  possible consequences of violations
of Lorentz and CPT invariance have been actively studied. The Standard
Model of known elementary particles and
interactions respects both of these symmetries.
Signals of Lorentz and CPT violation could, therefore,
be indicative of
new physics, e.g. quantum gravity \cite{Wal1}, superstrings
\cite{KoPo1}, or a new type of CPT anomaly within local quantum field
theory \cite{Kli1,Kli2}.

Both phenomenological consequences and theoretical implications of
Lorentz and CPT breaking have been studied in a series of papers
\cite{CFJ1}--\cite{KL01}. Moreover,
there have been extensive discussions on the possibility of
Lorentz- and CPT-symmetry breaking in the gauge field sector induced by
radiative corrections of an explicitly symmetry-breaking matter sector.
See Refs. \cite{CoKo2}--\cite{P-V01}, where
Ref. \cite{P-V01}, in particular, contains an extensive list of references.
In this Letter, we comment on this last question.

Concretely, we start from the following Lagrangian density:
\beq \label{eq:lagrangian}
{\cal L}=\bar \psi \, \left(i\dsla -e\, \Asla -m -\gaf\, \bsla \right)\, \psi
-{\textstyle\frac{1}{4}}\, F_{\mu\nu}F^{\mu\nu} +{\cal L}_{\rm gf}\, ,
\eeq
and use units for which $c$ $=$ $\hbar$ $=$ $1$.
Here, $A_\mu$ is an Abelian gauge potential with field strength tensor
$F_{\mu\nu} \equiv \partial_\mu A_\nu - \partial_\nu A_\mu$, $\psi$ a
4-component Dirac spinor with nonzero
mass $m$ and $e$ the gauge coupling constant.
As usual, the slash of a vector stands for a
contraction with the Dirac matrices, $\Asla \equiv \gamma^\mu A_\mu$.
Further, the quantity $b_\mu$ is a constant, prescribed ``four-vector''
with the dimension of mass.
A gauge fixing term ${\cal L}_{\rm gf}$ has also been added to the
Lagrangian  (\ref{eq:lagrangian}), since we are interested in the full quantum
field theory with dynamical fields $\psi$ and $A_\mu$.
The Lagrangian  (\ref{eq:lagrangian})
is thus a generalized version of one-flavor quantum electrodynamics
(QED) \cite{JR76}, where the generalization is
provided by the CPT- and Lorentz-breaking term
$-\bar \psi \,\gaf\, \bsla \,\psi$.

Let us assume for the moment that we are interested in
processes which involve only photons in the initial and final
states. Such processes may be described by an effective action which is
obtained from Eq. (\ref{eq:lagrangian}) by integrating out the fermionic
degrees of freedom,
\beq \label{eq:eff-action}
S^{\;\mathrm{eff}}=\int {\rm d}^4 x \,\left(
-{\textstyle\frac{1}{4}}\, F_{\mu\nu}F^{\mu\nu} +{\cal L}_{\rm gf}
+\ln  \frac{{\rm det} (i\dsla -e\, \Asla -m -\gaf\, \bsla )
}{{\rm det}( i\dsla  -m -\gaf\, \bsla ) }\, \right) \, .
\eeq
In the limit $b_\mu \to 0$, the effective action (\ref{eq:eff-action}) is
given by the standard effective gauge field action of one-flavor QED \cite{JR76},
whereas for $b_\mu \ne 0$ additional terms will be present.

An example of such an additional term, quadratic in the gauge field, would be a
Chern--Simons-like term (see Refs. \cite{CFJ1,AK01} and references quoted therein),
\beq \label{eq:CS-term}
S_{\rm CS-like}=
\int {\rm d}^4 x \, {\textstyle\frac{1}{4}}
\, k_\mu\, \epsilon^{\mu\nu\rho\sigma} A_\nu F_{\rho\sigma} \, .
\eeq
Here, the quantity $k_\mu$ is a fixed ``four-vector'' with the dimension
of mass.
This Chern--Simons-like term (\ref{eq:CS-term}) has the same behavior
under Lorentz and CPT transformations as the term proportional to
$b_\mu$ in the Lagrangian  (\ref{eq:lagrangian}). It can, therefore, be
induced by radiative corrections (concretely, by vacuum
polarization). This implies that $S_{\rm CS-like}$ appears in the
effective action (\ref{eq:eff-action}) with $k_\mu$ related to $b_\mu$,
\beq \label{eq:coeff}
k_\mu =\zeta \, b_\mu \, ,
\eeq
where $\zeta$ is a numerical coefficient that has to be determined.
(Terms of higher order in $b_\mu$ turn out to be absent, as will
be discussed below.)

The question of how to determine the correct value for this constant $\zeta$
has given rise to a lively debate in the recent literature.
The authors of Ref. \cite{CoKo2} observed that a
calculation which is perturbative in $b_\mu$ leads to a finite but
undetermined value for $\zeta$. The authors of Ref. \cite{CoGl1}, on the
other hand, found a vanishing value for $\zeta$ by requiring a gauge-invariant
regularization of the axial current, but this requirement has
been criticized as being too restrictive from a physical point
\cite{JaKo1}. In fact, the authors of Ref.
\cite{JaKo1} again concluded that the value
of $\zeta$ remains undetermined. Further calculations in different
regularization schemes have since been performed, giving
a number of different values for the coefficient $\zeta$.
See, for example, Refs. \cite{ChOh99}--\cite{P-V01}.

In most of the above-quoted articles, the gauge potential is either
explicitly treated as an external field or, at least, no use is made of
the fact that $A_\mu$ is a dynamical quantized field. The full quantum
field theory (\ref{eq:lagrangian}) was studied in Ref.
\cite{Bon01}, where it was
argued that a correct implementation of gauge invariance via Ward
identities requires a zero value for $\zeta $. However, this conclusion was
again criticized as being too restrictive \cite{P-V01}.

The debate reviewed in the last two paragraphs
mainly focused on the issue of gauge invariance.
It would, therefore, be desirable to have an independent line of reasoning,
based on another fundamental property of quantum field theory.
The purpose of this Letter is to determine the value of $\zeta$ from
the requirement of {\em microcausality} of the quantum field theory defined
by the Lagrangian (\ref{eq:lagrangian}).

Let us, first, recall some results from the literature on
the constant $\zeta$ as defined by Eq. (\ref{eq:coeff}).
One result is that $\zeta$ is a numerical
constant that does not depend on the symmetry-breaking four-vector
$b_\mu$, provided the term proportional to
$b_\mu$ in the theory (\ref{eq:lagrangian}) can be treated
perturbatively. Then, the leading contribution to the \CS-like term
(\ref{eq:CS-term}), which is linear in $b_\mu$, comes from a naively
linearly divergent diagram  (which is why the contribution is ambiguous,
even though it turns out to be finite).
All possible higher-order contributions to the
\CS-like term come from convergent diagrams. These contributions are,
therefore,
unique and arguments based on the classical symmetries
of the theory remain valid. From the gauge invariance of the classical
axial current one may conclude that these higher contributions to the
coefficient $\zeta$ are, in fact, zero. For a more detailed discussion, see,
in particular, Refs. \cite{JaKo1,P-V01}.

Within perturbation theory,
$\zeta$ does not depend on whether $b_\mu$ is spacelike or
timelike. This observation will be important in the sequel, because the
effective theory (\ref{eq:eff-action}) with a Chern--Simons-like term
(\ref{eq:CS-term}) included will turn out to behave rather differently for
spacelike and timelike $k_\mu$.

 From now on, we assume that both $b_\mu$ and $k_\mu$ are purely
timelike, that is, $b_\mu =(b_0 ,0,0,0)$ and $k_\mu =(k_0 ,0,0,0)$. With
this choice, we will find that microcausality can be maintained for
the original quantum field theory (\ref{eq:lagrangian}) and, therefore,
for the effective field theory (\ref{eq:eff-action}).
It has, however, been demonstrated in Ref. \cite{AK01} that microcausality is
violated for a theory with a Chern--Simons-like term (\ref{eq:CS-term})
and purely timelike $k_\mu =(k_0 ,0,0,0)$.
This implies that the coefficient $\zeta$ in Eq. (\ref{eq:coeff}) should be
chosen equal to zero. We like to emphasize at this point that our
argument is valid only for a dynamical, quantized gauge field. It does
not apply to the external field problem.

Let us then start by investigating the causality behavior of the
\emph{original} theory (\ref{eq:lagrangian}) with  $b_\mu =(b_0 ,0,0,0)$.
For completeness, we should mention that this theory may have some unusual
properties. As pointed out in Section IV B of Ref. \cite{KL01}, a high-energy
fermion can, for example, decay by emitting a virtual photon which creates a
fermion-antifermion pair ($f$ $\rightarrow$ $f+\gamma$ $\rightarrow$
$f+\bar{f}+f$).
It is, however, not clear that this instability affects the consistency of the
theory. For now,
we simply assume the validity of the usual perturbation theory methods
\cite{JR76}.
The Lagrangian (\ref{eq:lagrangian}) consists of
three pieces that have to be investigated separately.

First, the Maxwell and gauge-fixing part of
the Lagrangian density (\ref{eq:lagrangian}),
\beq
{\cal L}_{A-\mathrm{kin}} = -{\textstyle\frac{1}{4}}\,
        F_{\mu\nu}F^{\mu\nu} +{\cal L}_{\rm gf}\, ,
\eeq
is known to respect the requirements of causality \cite{JR76}.
Classically, the two polarization modes of the electromagnetic plane wave
travel with constant speed $c=1$. In the quantized Maxwell theory,
microcausality holds for physical, gauge-invariant operators (e.g. the
transverse components of the gauge potential or the electric and
magnetic fields). Concretely, the commutator of two physical operators
vanishes for spacelike separations.

Second, the fermionic part of the Lagrangian density (\ref{eq:lagrangian}),
\beq \label{eq:fermion-lagr}
{\cal L}_{\psi-\mathrm{kin}} =\bar \psi \, (i\dsla -m -\gaf\, \bsla )\, \psi\, ,
\eeq
is more difficult. Fortunately, the causal behavior of this
quadratic Lagrangian for purely timelike $b_\mu$ has already
been determined in Ref. \cite{CoKo1}. For the metric
$g_{\mu\nu} ={\rm diag} (-1,1,1,1)$,
the fermionic Lagrangian (\ref{eq:fermion-lagr}) leads to the dispersion
relation
\beq \label{eq:fermi-disp}
(p^2 -b^2 +m^2 )^2 +4p^2 b^2 -4(p\cdot b)^2 =0\, ,
\eeq
with the solutions
\beq
p_0^2 =\omega_\pm^2 \equiv |\vec p \,|^2 +m^2 +b_0^2 \pm 2\,|b_0|\,|\vec
p\, | \,,
\eeq
for the case of purely timelike $b_\mu=(b_0 ,0,0,0)$.

The dispersion relation (\ref{eq:fermi-disp}) is shown in Fig. 1 for
the values $b_0$ $=$ $m$ $=$ $1/\sqrt{2}$
(more realistic would, of course, be $|b_0|$ $<<$ $m$).
Two remarks are in order. First, the dispersion relation
(\ref{eq:fermi-disp}) has four real roots $p_0 =q_i, \, i=1\ldots
\, 4$, for arbitrary values of $|\vec p\,|$. Second, the group velocities,
\beq
\vec v_{\rm g}^{\; \pm}
\equiv \frac{\partial \omega_\pm}{\partial \vec p}
=       \frac {\vec p}{|\vec p\, |}\;
\frac{|\vec p\, |\pm |b_0|}{\sqrt{(|\vec p\, | \pm |b_0|)^2 +m^2}} \le 1\, ,
\eeq
do not exceed the constant velocity of light $c$,
which is equal to 1 in our units.
This classical reasoning indicates that the causal structure of
the theory (\ref{eq:fermion-lagr}) remains unchanged by the inclusion of
a purely timelike $b_\mu$.

In the corresponding
quantum field theory, the issue of microcausality is determined by the
anti-commutator function $i S(x-y) \equiv \{ \psi (x),\bar \psi (y)\}$,
which has to vanish for spacelike separations $(x-y)^2 >0$ in order to
maintain microcausality. This anti-commutator has been calculated
in Ref. \cite{CoKo1} for a purely timelike $b_\mu =(b_0 ,0,0,0)$.
The result is that $S(x-y)$ may be expressed like
\beq
S(x-y) = (i\dsla -\gaf \bsla +m ) (i\dsla +\gaf \bsla +m )
(i\dsla +\gaf \bsla -m ) \, \Delta (x-y) \, ,
\eeq
where $\Delta (x)$ is defined as
\beq
\Delta (x)= i\oint_{C}\frac{{\rm d}p_0}{2\pi}\int \frac{{\rm d}^3
  p}{(2\pi)^3} \;\frac{e^{ip\cdot x}}{
(p^2 -b^2 +m^2 )^2 +4p^2 b^2 -4(p\cdot b)^2 } \; .
\eeq
Here, the integration contour $C$ encircles all four poles in the
counter-clockwise direction. For $b_\mu =(b_0 ,0,0,0)$, the integration
can be performed explicitly  \cite{CoKo1}:
\beq
\Delta (x) = -\frac{i}{8\pi} \, \frac{x^0}{|x^0|} \,
             \frac{\sin b_0 |\vec  x|}{b_0 |\vec x|} \, J_0 (m\sqrt{-x^2})\, ,
\eeq
for timelike separation $x^2 <0$ ($J_0$ is the zero-order Bessel function), and
\beq
\Delta (x) =0 \, ,
\eeq
for spacelike separation $x^2 >0$. Hence, microcausality is
maintained for the theory (\ref{eq:fermion-lagr}) with purely timelike $b_\mu$.

The last part of the original Lagrangian density (\ref{eq:lagrangian})
that has to
be studied is the interaction term,
\beq \label{eq:int-term}
{\cal L}_{A\psi-\mathrm{int}}=-e \, \bar \psi  \, \Asla  \, \psi \, .
\eeq
This term is strictly local, so that causality holds at the classical level.
However, the composite operator $J_\mu = \bar \psi \gamma_\mu
\psi$ is singular in the quantum theory and needs
regularization. Treating the interaction term perturbatively,
some calculations may then lead to ambiguous results. These ambiguities
are fixed by the symmetries of the theory and by the appropriate
physical conditions. One physical condition is certainly that causality
should continue to hold for the full quantum field theory, if at all
possible.
Therefore, microcausality is a requirement at the level of
the perturbatively defined quantum field theory (\ref{eq:lagrangian}).

Next, we turn to the causality behavior of the \emph{effective}
theory (\ref{eq:eff-action}), with the Chern--Simons-like term
(\ref{eq:CS-term}) included. This problem has already been studied
in Ref. \cite{CFJ1} for the classical field theory and
in Ref. \cite{AK01} for the quantum theory.
Here, we only need to review the results.
The quadratic part of the effective action reads
\beq \label{eq:quad-eff-action}
S^{\;\mathrm{eff}}_{A-\mathrm{kin}}
=\int \rmd ^4 x \, \left( \,{\textstyle \frac{1}{2}}\,
A_\mu \left(g^{\mu\nu}\Box -\partial^\mu \partial^\nu
- \, \epsilon^{\mu\nu\rho\sigma}k_\rho
\partial_\sigma \right) A_\nu + {\cal L}_{\rm gf} \,\right) \; ,
\eeq
with $k_\mu =(k_0 ,0,0,0)$. Ignoring the gauge-fixing part for the
moment, one finds the dispersion relation \cite{CFJ1}
\beq \label{eq:boson-disp}
p^4 + k^2 p^2 -(k\cdot p)^2 =0 \; ,
\eeq
with the solutions
\beq \label{eq:frequency-time}
p_0^2 =  \omega_\pm^2 \equiv |\vec p\, |^2 \pm |k_0| |\vec p\, | \; .
\eeq

The dispersion relation (\ref{eq:boson-disp}) is shown in Fig. 2.
It is obvious that there is no separation
into positive and negative frequency parts.
In addition,
the group velocities of both degrees of freedom may become arbitrarily large.
For the plus sign in Eq. (\ref{eq:frequency-time}), one has, for example,
\beq
\vec v_{\rm g}^{\; +} \equiv
\frac{\partial \omega_+}{\partial \vec p}=\frac{\vec p}{|\vec p\, |} \;
\frac{2|\vec p\, |+|k_0|}{2\sqrt{|\vec p\, |^2 +|k_0||\vec p\, |}} \; ,
\eeq
which becomes arbitrarily large for arbitrarily small $|\vec p\, |$,
as long as $|k_0|$ $\neq$ $0$.
For the minus sign in Eq. (\ref{eq:frequency-time}), the energy even
becomes imaginary at low momenta $|\vec p\, |<|k_0|$.
In the quantized theory, these low-momentum modes
would grow exponentially with time (instead of evolving unitarily).
It is, therefore, mandatory to exclude
these modes from the theory in order to maintain unitarity.
This, in turn, leads to a violation of
microcausality in the theory (\ref{eq:quad-eff-action}),
as has been shown in Section 5.2 of Ref. \cite{AK01}.
For a purely spacelike $k_\mu$, on the other hand, there is no violation
of microcausality, see Section 5.1 of Ref. \cite{AK01}.

To summarize, we have studied the issue of microcausality for the quantum field
theory (\ref{eq:lagrangian}). We have found that both the purely bosonic part
(Maxwell theory) and the purely fermionic part (Dirac theory with
an additional $b_\mu$-term) maintain
microcausality.
Therefore, the free theory (\ref{eq:lagrangian}) without interaction
term (\ref{eq:int-term}) is causal.
If we now impose the requirement that microcausality
continues to hold for the full quantum field theory
(\ref{eq:lagrangian}) with the interaction term (\ref{eq:int-term}) included,
we conclude that the Chern--Simons-like term
(\ref{eq:CS-term}) must be absent in the effective action
(\ref{eq:eff-action}) for purely timelike $b_\mu$.
The reason is that microcausality would be violated by
the Chern--Simons-like term for purely timelike
$k_\mu$, at least within the realm of unitary perturbation theory.
For general timelike $b_\mu$, there always exists a coordinate
system for which $b_\mu$ is purely timelike. Therefore, our argument
holds also for general timelike $b_\mu$.

For spacelike $b_\mu$, we assume moreover that a
perturbative expansion in $b_\mu$ is possible for sufficiently small
values of $b_\mu$, that is, the coupling constants in the effective gauge field
action (\ref{eq:eff-action}) can be expanded in powers of $b_\mu$.
(The effective expansion parameter is $b^2/m^2$, on dimensional grounds.)
With this additional assumption, the Chern--Simons-like term (3) must also be absent in
the effective action (\ref{eq:eff-action}) for sufficiently small spacelike
$b_\mu$.

In other words, the ambiguity of
the value of the coefficient $\zeta$, which is present in a naive
perturbative treatment,
is fixed by the requirement of microcausality of the full quantum field
theory (\ref{eq:lagrangian}). The unique value for the coefficient $\zeta$
of Eq. (\ref{eq:coeff}) which preserves microcausality is
\beq
\zeta=0 \, .
\eeq
We emphasize once more
that the calculations were performed for the specific
choice of a purely timelike $b_\mu$. But, within perturbation theory,
the value of the coefficient $\zeta$
does not depend on $b_\mu$, so that $\zeta=0$ holds for arbitrary $b_\mu$.
Note also that the well-known Pauli--Villars and dimensional regularization
methods do indeed give $\zeta=0$, at least for small enough $b^2/m^2$;
cf. Refs. \cite{CoKo2,Bon01}.

As a consequence, the Lorentz and CPT violation introduced by the
$b_\mu$ term in the Lagrangian (\ref{eq:lagrangian}) does not appear in
the effective action (\ref{eq:eff-action}) via the quadratic local
Chern--Simons-like term (\ref{eq:CS-term}). It is, of course, still possible that
Lorentz and CPT violation is induced in the effective action
(\ref{eq:eff-action}) by higher-order local terms or even nonlocal terms.

At this moment, it may be of interest to mention a recent calculation of
the induced Chern--Simons-like term (\ref{eq:CS-term}) at
finite temperature \cite{CGS01}, which agrees
with our general argument.
Irrespective of the ambiguity of the coefficient $\zeta$ at zero
temperature, the authors of Ref. \cite{CGS01} find
a unique and finite additional contribution to the induced
Chern--Simons-like term at finite temperature. This
finite-temperature contribution
is nonzero for the spacelike component of $b_\mu$ :
$k_i^{\mathrm{finite}-T} =16 \,
F(\xi ) \, b_i $, with $\xi \equiv m/2\pi T$ and the function $F(\xi )$ as
calculated in  Ref. \cite{CGS01}. (Note that $F$ vanishes as the
temperature $T$ goes to zero.)  On the other hand, no extra Chern--Simons-like
term is induced for the time component $b_0$ : $k_0^{\mathrm{finite}-T} =0$.
The heat bath is, of course, a further source
of explicit violation of Lorentz invariance, which explains
the different results for timelike and spacelike directions. But these
finite-temperature results are precisely what one expects from the general
requirement of causality. A Chern--Simons-like term with a purely
spacelike $k_i$ does  not lead to a violation of microcausality
\cite{AK01} and its presence is compatible with the requirement
of causality. On the other hand, a nonzero induced $k_0$ would violate
causality and must therefore be absent.

One physical consequence of a nonvanishing spacelike $k_\mu$ is the
birefringence of light {\it in vacuo}. Observations of distant
radio galaxies severely restrict the possible values of the $k_\mu$
parameter: $|k_\mu| \lsim 10^{-33}$eV, see Refs. \cite{CFJ1,WPC}
and references quoted therein.
If $\zeta$ had a unique nonzero value of order $\alpha$ $\equiv$ $e^2/4\pi$,
this bound would translate into a strong upper bound on the
Lorentz- and CPT violating parameter $b_\mu$ for the fermion sector, as
pointed out in Refs. \cite{CoKo2,CoGl1,JaKo1}. Our result $\zeta =0$
implies that no such strong experimental upper bound on $b_\mu$ exists,
leaving room for a comparatively large Lorentz and CPT violation in the
fermionic matter sector (provided the $b_\mu$-like terms do not
affect the consistency of the theory).

If it should turn out that, after all, $k_\mu$ has a tiny nonzero value,
then the result $\zeta =0$
implies that this nonzero value cannot be traced back to a
symmetry-breaking term in the matter sector for Dirac fermions, at least for
a Lagrangian like Eq. (\ref{eq:lagrangian}).
A nonzero Chern--Simons-like term (\ref{eq:CS-term}) could then
be the low-energy remnant of a more fundamental theory like superstring theory,
via a mechanism that is not yet understood in detail \cite{KoPo1}. Or,
it could be induced by Weyl fermions over a nontrivial spacetime
manifold via the CPT anomaly \cite{Kli1,Kli2}.

In fact, it has been shown in Ref. \cite{Kli1}
that certain chiral gauge theories,
defined over a four-dimensional spacetime manifold
with a compact separable dimension and appropriate spin structure, give
rise to a CPT anomaly precisely of the form of the Chern--Simons-like term
(\ref{eq:CS-term}).
(An example of an appropriate chiral gauge theory would be the well-known $SO(10)$
grand-unified theory with three families of quarks and leptons or even the
Standard Model with an additional assumption, see Section 5 of Ref. \cite{Kli1}.)
In this case, the magnitude of $k_\mu$ is given by the
inverse of the length of the compact dimension, thereby naturally
accounting for the required smallness of $|k_\mu |$. The direction of
$k_\mu$ is given by the direction of the compact dimension,
so that the different causality behavior of
timelike and spacelike $k_\mu$  does not come unexpected
\cite{AK01}. Indeed, a spacetime manifold with a compact timelike dimension
contains closed timelike curves and has causality problems
already at the macroscopic level.

\section*{Acknowledgements}
We thank R. Jackiw and M. Perez-Victoria for useful comments.

\begin{thebibliography}{99}
\bibitem{Wal1}
R.M. Wald, Phys. Rev. D 21 (1980) 2742.
\bibitem{KoPo1}
V.A. Kosteleck\'{y},  R. Potting, Nucl. Phys. B 359 (1991) 545.
\bibitem{Kli1}
F.R. Klinkhamer, Nucl. Phys. B 578 (2000) 277.
\bibitem{Kli2}
F.R. Klinkhamer,  J. Nishimura, Phys. Rev. D 63 (2001) 097701.
\bibitem{CFJ1}
S.M. Carroll, G.B. Field,  R. Jackiw, Phys. Rev. D 41 (1990) 1231.
\bibitem{CoKo1}
D. Colladay,  V.A. Kosteleck\'{y}, Phys. Rev. D 55 (1997) 6760.
\bibitem{CoKo2}
D. Colladay,  V.A. Kosteleck\'{y}, Phys. Rev. D 58 (1998) 116002.
\bibitem{CoGl1}
S. Coleman,  S.L. Glashow, Phys. Rev. D 59 (1999) 116008.
\bibitem{KL01}
V.A. Kosteleck\'{y}, R. Lehnert, Phys. Rev. D 63 (2001) 065008.
\bibitem{JaKo1}
R. Jackiw,  V.A. Kosteleck\'{y}, Phys. Rev. Lett. 82 (1999) 3572.
\bibitem{ChOh99}
J.M. Chung, P. Oh, Phys. Rev. D 60 (1999) 067702.
\bibitem{Chen99}
W.F. Chen, Phys. Rev. D 60 (1999) 085007.
\bibitem{P-V99}
M. Perez-Victoria, Phys. Rev. Lett. 83 (1999) 2518.
\bibitem{Chan99}
L.H. Chan,  arXiv:hep-ph/9907349.
\bibitem{Bon01}
G. Bonneau, Nucl. Phys. B 593 (2001) 398.
\bibitem{CCGF}
M. Chaichian, W.F. Chen,  R. Gonzales Felipe, Phys. Lett. B 503 (2001) 215.
\bibitem{ChCh01}
J.M. Chung, B.K. Chung, Phys. Rev. D 63 (2001) 105015.
\bibitem{P-V01}
M. Perez-Victoria, J. High Energy Phys. 04 (2001) 032.
\bibitem{JR76}
J.M. Jauch,  F. Rohrlich, \emph{The Theory of Photons and Electrons},
2nd ed., Springer, New York, 1976.
\bibitem{AK01}
C. Adam, F.R. Klinkhamer, Nucl. Phys. B in press, arXiv:hep-ph/0101087.
\bibitem{CGS01}
L. Cervi, L. Griguolo, D. Seminara, arXiv:hep-th/0104022.
\bibitem{WPC}
J.F.L. Wardle, R.A. Perley, M.H. Cohen, Phys. Rev. Lett. 79 (1997) 1801.

\end{thebibliography}

\newpage
\input psbox.tex
\begin{figure}
$$ \psboxscaled{1000}{radindCSfig1.ps} $$
\caption{The dispersion relation (\ref{eq:fermi-disp})
in the $(p_0,|\vec p\, |)$ halfplane for purely timelike parameter
$b_\mu =(m,0,0,0)$ and fermion mass $m=1/\sqrt{2}$.
}
\end{figure}

\begin{figure}
$$ \psboxscaled{1000}{radindCSfig2.ps} $$
\caption{The dispersion relation (\ref{eq:boson-disp}) in the
$(\mathrm{Re}\, p_0,|\vec p\, |)$ and $(\mathrm{Im}\, p_0,|\vec p\, |)$
halfplanes for purely timelike
\CS~parameter $k_\mu = (1,0,0,0)$, with broken (solid)
curves corresponding to the plus (minus) sign in Eq. (\ref{eq:frequency-time}).}
\end{figure}

\end{document}




%% for hep-th also include psbox.tex.

\\
Title      : Causality and radiatively induced CPT violation
Author     : C. Adam and F.R. Klinkhamer
Comments   : LaTeX, 12 pages, 2 PS figures, v3: to appear in PLB
Report-no  : KA-TP-13-2001
\\
We consider quantum electrodynamics with an additional Lorentz- and
CPT-violating axial-vector term in the fermionic sector and discuss the
possibility that radiative corrections  induce a  Lorentz- and CPT-violating
Chern-Simons-like term for the gauge field.  From the requirement of causality
and the assumed validity of perturbation theory, we conclude that the induced
Chern-Simons-like term must be absent in the full quantum field theory.
\\

