ach.tex
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%%Title: ach.tex
%%Creator: gnuplot 3.7 patchlevel 1
%%CreationDate: Fri Dec 27 17:29:53 2002
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gnudict begin
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/dl {10 mul} def
/hpt_ 31.5 def
/vpt_ 31.5 def
/hpt hpt_ def
/vpt vpt_ def
/M {moveto} bind def
/L {lineto} bind def
/R {rmoveto} bind def
/V {rlineto} bind def
/vpt2 vpt 2 mul def
/hpt2 hpt 2 mul def
/Lshow { currentpoint stroke M
  0 vshift R show } def
/Rshow { currentpoint stroke M
  dup stringwidth pop neg vshift R show } def
/Cshow { currentpoint stroke M
  dup stringwidth pop -2 div vshift R show } def
/UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def
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/DL { Color {setrgbcolor Solid {pop []} if 0 setdash }
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/LT1 { PL [4 dl 2 dl] 0 1 0 DL } def
/LT2 { PL [2 dl 3 dl] 0 0 1 DL } def
/LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def
/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def
/LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def
/LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def
/LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def
/LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def
/Pnt { stroke [] 0 setdash
   gsave 1 setlinecap M 0 0 V stroke grestore } def
/Dia { stroke [] 0 setdash 2 copy vpt add M
  hpt neg vpt neg V hpt vpt neg V
  hpt vpt V hpt neg vpt V closepath stroke
  Pnt } def
/Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V
  currentpoint stroke M
  hpt neg vpt neg R hpt2 0 V stroke
  } def
/Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M
  0 vpt2 neg V hpt2 0 V 0 vpt2 V
  hpt2 neg 0 V closepath stroke
  Pnt } def
/Crs { stroke [] 0 setdash exch hpt sub exch vpt add M
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/TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M
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  hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V closepath stroke
  Pnt  } def
/Star { 2 copy Pls Crs } def
/BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M
  0 vpt2 neg V  hpt2 0 V  0 vpt2 V
  hpt2 neg 0 V  closepath fill } def
/TriUF { stroke [] 0 setdash vpt 1.12 mul add M
  hpt neg vpt -1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V closepath fill } def
/TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt -1.62 mul V closepath stroke
  Pnt  } def
/TriDF { stroke [] 0 setdash vpt 1.12 mul sub M
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  hpt 2 mul 0 V
  hpt neg vpt -1.62 mul V closepath fill} def
/DiaF { stroke [] 0 setdash vpt add M
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  hpt vpt V hpt neg vpt V closepath fill } def
/Pent { stroke [] 0 setdash 2 copy gsave
  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath stroke grestore Pnt } def
/PentF { stroke [] 0 setdash gsave
  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath fill grestore } def
/Circle { stroke [] 0 setdash 2 copy
  hpt 0 360 arc stroke Pnt } def
/CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def
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/C1 { BL [] 0 setdash 2 copy        moveto
       2 copy  vpt 0 90 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C2 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 90 180 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C3 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 0 180 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C4 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 180 270 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C5 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 0 90 arc
       2 copy moveto
       2 copy  vpt 180 270 arc closepath fill
               vpt 0 360 arc } bind def
/C6 { BL [] 0 setdash 2 copy moveto
      2 copy  vpt 90 270 arc closepath fill
              vpt 0 360 arc closepath } bind def
/C7 { BL [] 0 setdash 2 copy moveto
      2 copy  vpt 0 270 arc closepath fill
              vpt 0 360 arc closepath } bind def
/C8 { BL [] 0 setdash 2 copy moveto
      2 copy vpt 270 360 arc closepath fill
              vpt 0 360 arc closepath } bind def
/C9 { BL [] 0 setdash 2 copy moveto
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              vpt 0 360 arc closepath } bind def
/C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill
       2 copy moveto
       2 copy vpt 90 180 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C11 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 0 180 arc closepath fill
       2 copy moveto
       2 copy  vpt 270 360 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C12 { BL [] 0 setdash 2 copy moveto
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               vpt 0 360 arc closepath } bind def
/C13 { BL [] 0 setdash  2 copy moveto
       2 copy  vpt 0 90 arc closepath fill
       2 copy moveto
       2 copy  vpt 180 360 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C14 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 90 360 arc closepath fill
               vpt 0 360 arc } bind def
/C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill
               vpt 0 360 arc closepath } bind def
/Rec   { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto
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/Square { dup Rec } bind def
/Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def
/S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def
/S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def
/S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def
/S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def
/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
/S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill
       exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def
/S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill
       2 copy vpt Square fill
       Bsquare } bind def
/S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def
/S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def
/S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill
       Bsquare } bind def
/S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill
       Bsquare } bind def
/S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def
/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill
       2 copy vpt Square fill Bsquare } bind def
/S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill
       2 copy exch vpt sub exch vpt Square fill Bsquare } bind def
/S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def
/D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def
/D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def
/D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def
/D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def
/D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def
/D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def
/D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def
/D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def
/D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def
/D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def
/D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def
/D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def
/D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def
/D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def
/D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def
/D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def
/DiaE { stroke [] 0 setdash vpt add M
  hpt neg vpt neg V hpt vpt neg V
  hpt vpt V hpt neg vpt V closepath stroke } def
/BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M
  0 vpt2 neg V hpt2 0 V 0 vpt2 V
  hpt2 neg 0 V closepath stroke } def
/TriUE { stroke [] 0 setdash vpt 1.12 mul add M
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/TriDE { stroke [] 0 setdash vpt 1.12 mul sub M
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  hpt 2 mul 0 V
  hpt neg vpt -1.62 mul V closepath stroke } def
/PentE { stroke [] 0 setdash gsave
  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath stroke grestore } def
/CircE { stroke [] 0 setdash 
  hpt 0 360 arc stroke } def
/Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def
/DiaW { stroke [] 0 setdash vpt add M
  hpt neg vpt neg V hpt vpt neg V
  hpt vpt V hpt neg vpt V Opaque stroke } def
/BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M
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  hpt2 neg 0 V Opaque stroke } def
/TriUW { stroke [] 0 setdash vpt 1.12 mul add M
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  hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V Opaque stroke } def
/TriDW { stroke [] 0 setdash vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt -1.62 mul V Opaque stroke } def
/PentW { stroke [] 0 setdash gsave
  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  Opaque stroke grestore } def
/CircW { stroke [] 0 setdash 
  hpt 0 360 arc Opaque stroke } def
/BoxFill { gsave Rec 1 setgray fill grestore } def
end
%%EndProlog
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34 -2 V
33 -1 V
33 -1 V
34 0 V
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1.000 UL
LT2
550 843 M
263 0 V
450 2060 M
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33 -1 V
33 -1 V
34 -1 V
33 -2 V
33 -2 V
34 -3 V
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33 -4 V
33 -3 V
34 -3 V
33 -3 V
33 -2 V
34 -2 V
33 -1 V
33 -1 V
34 -1 V
33 0 V
1.000 UL
LT3
550 706 M
263 0 V
450 2060 M
33 0 V
34 -1 V
33 -2 V
33 -2 V
34 -3 V
33 -3 V
33 -4 V
34 -5 V
33 -5 V
33 -5 V
34 -7 V
33 -7 V
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34 -17 V
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34 -5 V
33 -5 V
33 -4 V
34 -3 V
33 -3 V
33 -2 V
34 -1 V
33 0 V
1.000 UL
LT4
550 569 M
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450 2060 M
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33 -2 V
33 -2 V
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33 -4 V
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33 -6 V
33 -6 V
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34 -7 V
33 -5 V
33 -5 V
34 -4 V
33 -3 V
33 -2 V
34 -2 V
33 0 V
1.000 UL
LT5
550 432 M
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450 2060 M
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34 -2 V
33 -1 V
33 -3 V
34 -4 V
33 -4 V
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34 -6 V
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achZ.tex
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%%Title: achZ.tex
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LT2
550 1602 M
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450 1004 M
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LT3
550 1484 M
263 0 V
450 1004 M
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33 -11 V
34 -11 V
33 -11 V
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33 -11 V
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33 -10 V
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33 -9 V
34 -9 V
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34 -2 V
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LT4
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34 6 V
33 7 V
33 6 V
34 5 V
33 6 V
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afb.tex
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%!PS-Adobe-2.0 EPSF-2.0
%%Title: afb.tex
%%Creator: gnuplot 3.7 patchlevel 1
%%CreationDate: Fri Dec 27 17:31:16 2002
%%DocumentFonts: 
%%BoundingBox: 0 0 360 216
%%Orientation: Landscape
%%EndComments
/gnudict 256 dict def
gnudict begin
/Color true def
/Solid false def
/gnulinewidth 5.000 def
/userlinewidth gnulinewidth def
/vshift -33 def
/dl {10 mul} def
/hpt_ 31.5 def
/vpt_ 31.5 def
/hpt hpt_ def
/vpt vpt_ def
/M {moveto} bind def
/L {lineto} bind def
/R {rmoveto} bind def
/V {rlineto} bind def
/vpt2 vpt 2 mul def
/hpt2 hpt 2 mul def
/Lshow { currentpoint stroke M
  0 vshift R show } def
/Rshow { currentpoint stroke M
  dup stringwidth pop neg vshift R show } def
/Cshow { currentpoint stroke M
  dup stringwidth pop -2 div vshift R show } def
/UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def
  /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def
/DL { Color {setrgbcolor Solid {pop []} if 0 setdash }
 {pop pop pop Solid {pop []} if 0 setdash} ifelse } def
/BL { stroke userlinewidth 2 mul setlinewidth } def
/AL { stroke userlinewidth 2 div setlinewidth } def
/UL { dup gnulinewidth mul /userlinewidth exch def
      10 mul /udl exch def } def
/PL { stroke userlinewidth setlinewidth } def
/LTb { BL [] 0 0 0 DL } def
/LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def
/LT0 { PL [] 1 0 0 DL } def
/LT1 { PL [4 dl 2 dl] 0 1 0 DL } def
/LT2 { PL [2 dl 3 dl] 0 0 1 DL } def
/LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def
/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def
/LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def
/LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def
/LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def
/LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def
/Pnt { stroke [] 0 setdash
   gsave 1 setlinecap M 0 0 V stroke grestore } def
/Dia { stroke [] 0 setdash 2 copy vpt add M
  hpt neg vpt neg V hpt vpt neg V
  hpt vpt V hpt neg vpt V closepath stroke
  Pnt } def
/Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V
  currentpoint stroke M
  hpt neg vpt neg R hpt2 0 V stroke
  } def
/Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M
  0 vpt2 neg V hpt2 0 V 0 vpt2 V
  hpt2 neg 0 V closepath stroke
  Pnt } def
/Crs { stroke [] 0 setdash exch hpt sub exch vpt add M
  hpt2 vpt2 neg V currentpoint stroke M
  hpt2 neg 0 R hpt2 vpt2 V stroke } def
/TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M
  hpt neg vpt -1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V closepath stroke
  Pnt  } def
/Star { 2 copy Pls Crs } def
/BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M
  0 vpt2 neg V  hpt2 0 V  0 vpt2 V
  hpt2 neg 0 V  closepath fill } def
/TriUF { stroke [] 0 setdash vpt 1.12 mul add M
  hpt neg vpt -1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V closepath fill } def
/TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt -1.62 mul V closepath stroke
  Pnt  } def
/TriDF { stroke [] 0 setdash vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt -1.62 mul V closepath fill} def
/DiaF { stroke [] 0 setdash vpt add M
  hpt neg vpt neg V hpt vpt neg V
  hpt vpt V hpt neg vpt V closepath fill } def
/Pent { stroke [] 0 setdash 2 copy gsave
  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath stroke grestore Pnt } def
/PentF { stroke [] 0 setdash gsave
  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath fill grestore } def
/Circle { stroke [] 0 setdash 2 copy
  hpt 0 360 arc stroke Pnt } def
/CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def
/C0 { BL [] 0 setdash 2 copy moveto vpt 90 450  arc } bind def
/C1 { BL [] 0 setdash 2 copy        moveto
       2 copy  vpt 0 90 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C2 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 90 180 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C3 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 0 180 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C4 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 180 270 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C5 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 0 90 arc
       2 copy moveto
       2 copy  vpt 180 270 arc closepath fill
               vpt 0 360 arc } bind def
/C6 { BL [] 0 setdash 2 copy moveto
      2 copy  vpt 90 270 arc closepath fill
              vpt 0 360 arc closepath } bind def
/C7 { BL [] 0 setdash 2 copy moveto
      2 copy  vpt 0 270 arc closepath fill
              vpt 0 360 arc closepath } bind def
/C8 { BL [] 0 setdash 2 copy moveto
      2 copy vpt 270 360 arc closepath fill
              vpt 0 360 arc closepath } bind def
/C9 { BL [] 0 setdash 2 copy moveto
      2 copy  vpt 270 450 arc closepath fill
              vpt 0 360 arc closepath } bind def
/C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill
       2 copy moveto
       2 copy vpt 90 180 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C11 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 0 180 arc closepath fill
       2 copy moveto
       2 copy  vpt 270 360 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C12 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 180 360 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C13 { BL [] 0 setdash  2 copy moveto
       2 copy  vpt 0 90 arc closepath fill
       2 copy moveto
       2 copy  vpt 180 360 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C14 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 90 360 arc closepath fill
               vpt 0 360 arc } bind def
/C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill
               vpt 0 360 arc closepath } bind def
/Rec   { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto
       neg 0 rlineto closepath } bind def
/Square { dup Rec } bind def
/Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def
/S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def
/S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def
/S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def
/S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def
/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
/S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill
       exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def
/S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill
       2 copy vpt Square fill
       Bsquare } bind def
/S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def
/S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def
/S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill
       Bsquare } bind def
/S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill
       Bsquare } bind def
/S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def
/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill
       2 copy vpt Square fill Bsquare } bind def
/S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill
       2 copy exch vpt sub exch vpt Square fill Bsquare } bind def
/S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def
/D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def
/D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def
/D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def
/D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def
/D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def
/D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def
/D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def
/D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def
/D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def
/D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def
/D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def
/D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def
/D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def
/D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def
/D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def
/D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def
/DiaE { stroke [] 0 setdash vpt add M
  hpt neg vpt neg V hpt vpt neg V
  hpt vpt V hpt neg vpt V closepath stroke } def
/BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M
  0 vpt2 neg V hpt2 0 V 0 vpt2 V
  hpt2 neg 0 V closepath stroke } def
/TriUE { stroke [] 0 setdash vpt 1.12 mul add M
  hpt neg vpt -1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V closepath stroke } def
/TriDE { stroke [] 0 setdash vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt -1.62 mul V closepath stroke } def
/PentE { stroke [] 0 setdash gsave
  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath stroke grestore } def
/CircE { stroke [] 0 setdash 
  hpt 0 360 arc stroke } def
/Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def
/DiaW { stroke [] 0 setdash vpt add M
  hpt neg vpt neg V hpt vpt neg V
  hpt vpt V hpt neg vpt V Opaque stroke } def
/BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M
  0 vpt2 neg V hpt2 0 V 0 vpt2 V
  hpt2 neg 0 V Opaque stroke } def
/TriUW { stroke [] 0 setdash vpt 1.12 mul add M
  hpt neg vpt -1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V Opaque stroke } def
/TriDW { stroke [] 0 setdash vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt -1.62 mul V Opaque stroke } def
/PentW { stroke [] 0 setdash gsave
  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  Opaque stroke grestore } def
/CircW { stroke [] 0 setdash 
  hpt 0 360 arc Opaque stroke } def
/BoxFill { gsave Rec 1 setgray fill grestore } def
end
%%EndProlog
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450 803 M
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1.000 UL
LT2
3087 1555 M
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450 970 M
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LT3
3087 1418 M
263 0 V
450 1261 M
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34 -23 V
33 -23 V
1.000 UL
LT5
3087 1144 M
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33 -29 V
33 -28 V
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33 -30 V
33 -29 V
34 -30 V
33 -31 V
33 -30 V
34 -31 V
33 -31 V
33 -31 V
34 -32 V
33 -31 V
33 -32 V
34 -32 V
33 -32 V
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stroke
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end
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afbZ.tex
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%%Title: afbZ.tex
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LTb
450 300 M
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450 593 M
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450 887 M
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1.000 UL
LT1
3087 1720 M
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450 925 M
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34 -1 V
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1.000 UL
LT2
3087 1602 M
263 0 V
450 648 M
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33 4 V
34 4 V
33 5 V
33 4 V
34 5 V
33 4 V
33 5 V
34 4 V
33 5 V
33 5 V
34 4 V
33 5 V
1.000 UL
LT3
3087 1484 M
263 0 V
450 1317 M
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33 -3 V
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33 -3 V
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33 -4 V
33 -4 V
34 -4 V
33 -5 V
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34 -4 V
33 -5 V
33 -5 V
34 -5 V
33 -5 V
33 -5 V
34 -5 V
33 -5 V
33 -6 V
34 -5 V
33 -6 V
33 -6 V
34 -6 V
33 -6 V
33 -6 V
34 -6 V
33 -6 V
33 -6 V
34 -7 V
33 -7 V
33 -6 V
34 -7 V
33 -7 V
33 -7 V
34 -7 V
33 -7 V
33 -7 V
34 -8 V
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34 -8 V
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34 -8 V
33 -8 V
33 -8 V
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33 -8 V
34 -8 V
33 -9 V
33 -8 V
34 -8 V
33 -8 V
33 -8 V
34 -8 V
33 -9 V
33 -8 V
34 -8 V
33 -8 V
1.000 UL
LT4
3087 1366 M
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450 972 M
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33 -1 V
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33 -1 V
33 -2 V
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33 -1 V
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33 -1 V
34 -2 V
33 -1 V
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33 -1 V
33 -2 V
34 -2 V
33 -1 V
33 -2 V
34 -1 V
33 -2 V
33 -2 V
34 -2 V
33 -1 V
33 -2 V
34 -2 V
33 -1 V
1.000 UL
LT5
3087 1248 M
263 0 V
450 367 M
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34 0 V
33 0 V
33 1 V
34 0 V
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33 1 V
34 0 V
33 1 V
33 1 V
34 1 V
33 2 V
33 1 V
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33 2 V
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34 2 V
33 2 V
33 2 V
34 2 V
33 2 V
33 2 V
34 2 V
33 3 V
33 3 V
34 2 V
33 3 V
33 3 V
34 3 V
33 4 V
33 3 V
34 4 V
33 4 V
33 3 V
34 4 V
33 5 V
33 4 V
34 4 V
33 5 V
33 4 V
34 5 V
33 5 V
33 6 V
34 5 V
33 6 V
33 5 V
34 6 V
33 6 V
33 6 V
34 7 V
33 6 V
33 7 V
34 7 V
33 7 V
33 7 V
34 7 V
33 8 V
33 8 V
34 8 V
33 8 V
33 8 V
34 8 V
33 9 V
33 9 V
34 8 V
33 9 V
33 10 V
34 9 V
33 9 V
33 10 V
34 10 V
33 10 V
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alr.tex
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alrZ.tex
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aud.tex
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audZ.tex
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diffcs.tex
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%%Title: diffcs.tex
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top5arxiv2.tex
\documentstyle[12pt,times]{article}
\begin{document}
\def \beq{\begin{equation}}
\def \eeq{\end{equation}}
\def \bea{\begin{eqnarray}}
\def \eea{\end{eqnarray}}
\def \ee {e^+e^-}
\def \ra {\rightarrow}
\def \g {\gamma}
\def \cvg {c_v^{\gamma}}
\def \cag {c_a^{\gamma}}
\def \cvz {c_v^Z}
\def \caz {c_a^Z}
\def \cdg {c_d^{\gamma}}
\def \cdz {c_d^Z}
\def \cdgz {c_d^{\gamma,Z}}
\def \t{$t\;$}
\def \toverline{$\overline t$}
\def \tbar{$\overline t\;$}
\def \el{E_l}
\def \ttt{\theta_t}
\def \tht{\theta_t}
\def \thl{\theta_l}
\def \thetal{\theta_l}
\def \phil{\phi_l}
\def \CP{$CP$}
\def \f{\frac}
\def \o{\overline}
\def \n{\noindent}
\def \half{\frac{1}{2}}
\def \Re{{\rm Re}}
\def \re{{\rm Re}}
\def \Im{{\rm Im}}
\def \im{{\rm Im}}
\def \bb{\bibitem}
\begin{flushright}
\end{flushright}
\begin{center}
{\large \bf \boldmath Single decay-lepton angular distributions in polarized
$e^+e^- \ra t\o{t}$ and simple angular asymmetries as a measure of 
$CP$-violating top dipole couplings
} 
\vskip .5cm 
{\large Saurabh D. Rindani\footnote{Email address: \tt saurabh@prl.ernet.in}} 
\vskip .25cm 
{\it Theory Group, Physical Research Laboratory \\
Navrangpura, Ahmedabad 380009, India} 
\vskip 1cm 
{\small \bf Abstract}
\end{center}
\vskip .2cm
\def \tt {t\overline {t}}
\begin{quote}
{\small
In the presence of an electric dipole coupling of $\tt$
to a photon, and an analogous ``weak" dipole coupling to the $Z$,
$CP$ violation in the process $\ee \ra \tt$ results in modified polarization
of the top and anti-top. This polarization can be analyzed by
studying the angular distributions of decay charged
leptons when the top or anti-top decays leptonically. 
Analytic expressions are presented for these distributions when
either $t$ or $\overline t$ decays leptonically, including ${\cal O}
(\alpha_s)$ QCD corrections in the soft-gluon approximation. The angular
distributions are insensitive to anomalous interactions in top decay. 
Two types
of simple $CP$-violating polar-angle asymmetries and two azimuthal asymmetries,
which do not need the
full reconstruction of the $t$ or $\o t$, are studied.  
Independent 90\% CL limits that may be obtained on the real and
imaginary parts of the electric and weak dipole couplings at a linear collider
operating at $\sqrt{s}=500$ GeV with integrated luminosity 500 fb$^{-1}$ and 
also at $\sqrt{s}=1000$ GeV with integrated luminosity 1000 fb$^{-1}$ have 
been evaluated. 
The effect of longitudinal electron and/or positron beam polarizations 
has been included. 
}
\end{quote}

PACS Nos: 14.65.Ha, 11.30.Er

\vskip .5cm
\section{Introduction}
An $e^+e^-$  linear collider operating at centre-of-mass (cm)
energy of 500 GeV or higher and with an integrated luminosity of several
hundred inverse femtobarns should be able to study with precision
various properties of the top quark. The possibility of setting up such a
collider is under consideration at a number of places at the moment,
particularly, JLC (Joint Linear Collider) in Japan \cite{JLC}, TESLA in Germany 
\cite{tesla}, and NLC
(Next Linear Collider) in the U.S.A. \cite{snow}

While the standard model (SM) predicts $CP$ violation outside
the \mbox{$K$-,}
$D$- and $B$-meson systems to be unobservably small,
in some extensions of SM, $CP$ violation might be considerably
enhanced, especially in the presence of a heavy top quark.  In
particular,
$CP$-violating electric dipole form factor of the top quark, and
the analogous $CP$-violating ``weak" dipole form factor in the $\tt$
coupling
to $Z$, could be enhanced.  These $CP$-violating form factors could
be
determined in a model-independent way at high energy $\ee$ linear
colliders,
where $\ee \ra \tt$ would proceed through virtual $\g$ and $Z$
exchange.

Since a heavy top quark with a mass of the order of 175 GeV
decays before it hadronizes \cite{heavytop}, it has
been suggested \cite{toppol} that top polarization asymmetry in
$\ee \ra \tt$ can be used to determine the $CP$-violating dipole
form factors, since polarization information would be retained in
the decay product distribution.  There have been several proposals
in which the $CP$-violating dipole couplings could be measured in
decay momentum correlations or asymmetries  with or without beam
polarization. For a review see \cite{sonirev}.

In this context it is important to note that top polarization can
only be studied using top decay. Therefore, for the information
from decay distributions to reflect correctly top polarization,
the decay amplitudes for various top polarization states have to
be known accurately. In particular, if there are any anomalous
effects in the decay process, they have to be known accurately.
Better still, the decay distributions chosen for the study have
to be insensitive to anomalous effects in the decay process. The
single-lepton angular distributions that we discuss in this work
satisfy the latter condition -- they accurately reflect the
polarization of the top quark resulting from the production
process, while one can continue to use SM in the decay process.

It has been found that one-loop QCD corrections to the process $\ee \ra \tt$
can be as high as 30\% for cm energy $\sqrt{s}=500$ GeV \cite{kodaira}. It is
therefore important to examine the effect of these QCD corrections in the decay
lepton distributions \cite{sga}, and their consequent effect on the measurement
of \CP-violating couplings.

In this paper we re-visit some suggestions made
\cite{pou1,pou2,pou3} for the measurement of top dipole moments in
$\ee \ra \tt$ using angular asymmetries of the charged lepton
produced in the semi-leptonic decay of one of $t$ and \tbar, while
the other decays hadronically. The purpose is to highlight 
certain features of the proposal which have become more significant in the
light of recent developments, and to update the numerical results. The 
improvements included in this update are several.
Firstly, ${\cal O}(\alpha_s)$ QCD corrections in the
soft-gluon approximation have been now been included.
Secondly, a
simplification used in earlier work \cite{pou1,pou2,pou3}, that of neglecting
\CP~ violation in top decay, has been dispensed with in the
light of recent work \cite{hioki,sdr1}. It turns out that for
angular asymmetries of the charged lepton considered here, \CP~
violation in the decay (or for that matter even arbitrary
\CP-conserving modifications of the $tbW$ vertex) has no effect,
if the $b$-quark mass is neglected. 
Finally, there is now a better idea of
luminosities possible at a future linear collider.
Together with
updated values of beam polarization now considered feasible, the
estimates of possible limits on dipole moments would be more
realistic.
Thus the estimates in earlier work have been improved upon and put on sounder
theoretical footing.  

Earlier proposals have considered a variety of \CP-violating
observables, with varying sensitivities. These include, in addition to angular
asymmetries, also vector and tensor correlations \cite{bern,cuy}, 
and expectation values of 
optimal variables \cite{atwood}. (For a discussion on
relative sensitivities of some variables, see \cite{lietti}). We
have chosen certain angular asymmetries here which have some
advantages over others, even though they may not be the most
sensitive ones. The advantages are: (i) Our asymmetries are in the
laboratory frame, making them directly observable. (ii) They
depend on final state momenta, rather than on top polarization.
Polarization is measured only indirectly through the decay
distributions. We therefore concentrate only on actual
decay-lepton distributions, which are the simplest to observe.
(iii) The observables we choose either do not depend on precise
determination of energy and momentum of top quarks, or, in case of
azimuthal asymmetries of the lepton, depend minimally on the top
momentum direction for the sake of defining the coordinate axes.
This has the advantage of higher accuracy. (iv) As stated before,
leptonic angular distribution is free from background from \CP~
violation in top decay, and gives a direct handle on anomalous
couplings in top production. (v) The polar-angle asymmetries we
consider can be obtained in analytical form, which is useful for
making quick computations. It is possible to get analytical forms
for certain azimuthal asymmetries as well, provided no angular
cuts are imposed. (vi) The asymmetries considered here are rather
simple conceptually, and hopefully, also from the practical
measurement point of view.

Our single-lepton asymmetries have another obvious advantage, that
since either $t$ or $\o{t}$ is allowed to decay hadronically,
there is a gain in statistics, as compared to double-lepton
asymmetries.

Our results are based on fully analytical calculation of single
lepton distributions in the production and subsequent decay of
$\tt$. We present fully differential angular distribution as well as the
distribution in the polar angle of the lepton with respect to the
beam direction in the centre-of-mass (cm) frame for arbitrary
longitudinal beam polarizations. These distributions for 
SM were first obtained by Arens and Sehgal
\cite{arens}. Distributions including the effect of \CP~ violation
only in production were obtained in \cite{pou2,pou3}, whereas,
with all anomalous effects included in the $\gamma t\overline{t}$
and $Zt\overline{t}$ vertices, as well as decay $tbW$ vertex were
obtained in \cite{hioki,sdr1}. Angular distributions in SM with
${\cal O}(\alpha_s)$ QCD corrections in the soft-gluon
approximation were obtained in \cite{sga}. The distributions
including anomalous effects in both top production and decay, and
including ${\cal O}(\alpha_s)$ QCD corrections in the soft-gluon
approximation are presented here for the first time. While QCD corrections to
$e^+e^- \ra t\o{t}$ are substantial, to the extent of about 30\% at $\sqrt{s} =
500$ GeV, their effect on leptonic angular distributions is much smaller
\cite{sga}. The main effect on the results will be to the sensitivity, through
the $1/\sqrt{N}$ factor, where $N$ is the number of events.
A part of this work was reported in \cite{kek}.

The rest of the paper is organized as follows. In Sec. 2, we
describe the cal\-cula\-tion of the decay-lepton angular
dist\-ribu\-tion from a decaying  $t$ or $\o t$ in $\ee \ra \tt$.
In Sec. 3 we describe $CP$-violating asymmetries. Numerical
results are presented in Sec. 4, and Sec. 5 contains our
conclusions. The Appendix contains certain expressions which are
too lengthy to be put in the main text.

\section{Calculation of  lepton  angular distributions}

We describe in this section the calculation of $l^+\; (l^-)$
distribution in
$\ee \ra \tt$ and the subsequent decay $t \ra b l^+ \nu_l\;
(\overline{t} \ra
\overline{b} l^- \overline{\nu_l})$.  We adopt the narrow-width
approximation for
$t$ and $\overline{t}$, as well as for $W^{\pm}$ produced in
$t,\;\overline{t}$
decay.

  We assume the top quark couplings to $\g$ and $Z$ to be
given by the vertex factor  $ie\Gamma_\mu^j$, where
\beq
\Gamma_\mu^j\;=\;c_v^j\,\g_\mu\;+\;c_a^j\,\g_\mu\,\g_5\;+
\;\f{c_d^j}{2\,m_t}\,i\g_5\,
(p_t\,-\,p_{\overline{t}})_{\mu},\;\;j\;=\;\g,Z,
\eeq
with
\bea
\cvg&=&\f{2}{3},\:\;\;\cag\;=\;0, \nonumber \\
\cvz&=&\f {\left(\f{1}{4}-\f{2}{3} \,x_W\right)}
{\sqrt{x_W\,(1-x_W)}},
 \\
\caz&=&-\f{1}{4\sqrt{x_W\,(1-x_W)}}, \nonumber \eea and
$x_W=sin^2\theta_W$, $\theta_W$ being the weak mixing angle. In
addition to the SM couplings $c^{{\g},Z}_{v,a}$ we have introduced
the $CP$-violating electric and weak dipole form factors,
$e\cdg/m_t$ and $e\cdz/m_t$, which are assumed small.  Use has
also been made of the Dirac equation in rewriting the usual dipole
coupling $\sigma_{\mu\nu}(p_t+p_{\overline{t}})^{\nu}\g_5$ as
$i\g_5(p_t-p_{\overline{t}})_{\mu}$, dropping small corrections to
the vector and axial-vector couplings. We will work in the
approximation in which we keep only linear terms in $\cdg$ and
$\cdz$. Addition of other \CP-conserving form factors will not
change our results in the linear approximation.

To include ${\cal O}(\alpha_s)$ corrections in the soft-gluon approximation (SGA), we need to modify the above vertices, as explained in \cite{sga}.
These modified vertices are given by
\beq\label{gvert}
\Gamma_{\mu}^{\gamma} =  c_v^{\gamma} \gamma_{\mu} + \left[ c_M^{\gamma}
 +
i\g_5\, c_d^{\gamma} \right]\,
\frac{(p_t - p_{\overline{t}})_{\mu}}{2 m_t} ,
\eeq
\beq\label{Zvert}
\Gamma_{\mu}^{Z} =  c_v^{Z} \gamma_{\mu} +  c_a^{Z}
\gamma_{\mu}\gamma_5
  + \left[ c_M^{Z}+
i\g_5 \, c_d^Z\right]\, \frac{(p_t - p_{\overline{t}})_{\mu}}{2
m_t} , \eeq where \beq c_v^{\gamma} = \f{2}{3} (1+A), \eeq \beq
c_v^Z = \frac{1}{\sin\theta_W \cos\theta_W} \left( \frac{1}{4} -
\frac{2}{3} \sin^2\theta_W\right) (1+A), \eeq \beq c_a^{\gamma}=0,
\eeq \beq c_a^Z = \frac{1}{\sin\theta_W \cos\theta_W}
\left(-\frac{1}{4}\right) (1+A+2 B), \eeq \beq c_M^{\gamma} =
\frac{2}{3} B, \eeq \beq c_M^Z = \frac{1}{\sin\theta_W
\cos\theta_W} \left( \frac{1}{4} - \frac{2}{3}
\sin^2\theta_W\right) B. \eeq The form factors $A$ and $B$ are
given to order $\alpha_s$ in SGA (see, for example, \cite{kodaira,tung} by \bea\label{A} {\rm
Re} A &=& \hat{\alpha}_s \left[ \left( \frac{1+\beta^2}{\beta}
\log \frac{1+\beta}{1-\beta} - 2\right)\log\frac{4 \omega^2_{\rm
max}}{m_t^2} - 4 \right.\nonumber \\ && \left. +
\frac{2+3\beta^2}{\beta}\log\frac{1+\beta}{1-\beta} +
\frac{1+\beta^2}{\beta} \left\{ \log\frac{1-\beta}{1+\beta}\left(
3 \log\frac{2\beta}{1+\beta}\right.\right.\right. \nonumber \\ &&
\left.\left.\left. + \log\frac{2\beta}{1-\beta} \right) + 4 {\rm
Li}_2 \left(\frac{1-\beta}{1+\beta}\right) +
\frac{1}{3}\pi^2\right\}\right], \eea \beq\label{B} {\rm Re}
B=\hat{\alpha}_s \frac{1-\beta^2}{\beta} \log
\frac{1+\beta}{1-\beta}, \eeq \beq {\rm Im} B = - \hat{\alpha}_s
\pi \frac{1-\beta^2}{\beta}, \eeq where
$\hat{\alpha}_s=\alpha_s/(3\pi)$, $\beta = \sqrt{1-4 m_t^2/s}$,
and Li$_2$ is the Spence function. ${\rm Re} A$ in eq. (\ref{A})
contains the effective form factor for a cut-off $\omega_{\rm
max}$ on the gluon energy after the infrared singularities have
been cancelled between the virtual- and soft-gluon contributions
in the on-shell renormalization scheme. Only the real part of the
form factor $A$ has been given, because the contribution of the
imaginary part is proportional to the $Z$ width, and hence
negligibly small \cite{kodaira,ravi}. The imaginary part of $B$,
however, contributes to azimuthal distributions.

The helicity amplitudes for $\ee \ra \g^*,Z^* \ra \tt$ in the cm
frame, including $\cdgz$ and $c_M^{\g ,Z}$ couplings, have been
given in \cite{asymm} (see also Kane {\it et al.}, ref.
\cite{toppol}).

We write the contribution of a general $tbW$ vertex to $t$ and \tbar decays
as
\bea\label{tbw}
\Gamma^{\mu}_{tbW}& =& -\f{g}{\sqrt{2}}V_{tb} \o{u}(p_b)
\left[\gamma^{\mu}(f_{1L}
P_L+f_{1R}P_R)\f{}{} \right. \nonumber \\
&& \left. - \f{i}{m_W} \sigma^{\mu\nu} (p_t - p_b)_{\nu}
 (f_{2L}P_L+f_{2R}P_R) \right] u(p_t),
\eea
\bea\label{tbarbw}
\o{\Gamma}^{\mu}_{tbW}& =& -\f{g}{\sqrt{2}}V_{tb}^* \o{v}(p_{\o{t}})
\left[\gamma^{\mu}(\o{f}_{1L} P_L + \o{f}_{1R} P_R )\f{}{} \right. \nonumber \\
&& \left. -
\f{i}{m_W} \sigma^{\mu\nu} (p_{\o{t}} - p_{\o{b}})_{\nu} (\o{f}_{2L}P_L+\o{f}
_{2R}P_R) \right]
v(p_{\o{b}}),
\eea
where $P_{L,R} =\half (1\pm \gamma_5)$, and $V_{tb}$ is the
Cabibbo-Kobayashi-Maskawa matrix element, which we take to be equal to one.
If \CP~ is conserved, the form factors $f$ above obey the relations
\beq
f_{1L}=\o{f}_{1L};\;\; f_{1R}=\o{f}_{1R},
\eeq
and
\beq
f_{2L}=\o{f}_{2R};\;\; f_{2R}=\o{f}_{2L}.
\eeq
Like $c_d^{\gamma}$ and $c_d^Z$ above, we will also treat $f_{2L,R}$ and
$\o{f}_{2L,R}$ as small, and retain only terms linear in them.
For the form factors $f_{1L}$ and $\o{f}_{1L}$, we retain their SM values,
viz., $f_{1L} = \o{f}_{1L} = 1$. $f_{1R}$ and $\o{f}_{1R}$ do not contibute in
the limit of vanishing $b$ mass, which is used here. Also, $f_{2L}$ and
$\o{f}_{2R}$ drop out in this limit.

The
helicity amplitudes for
\[t \ra b W^+,\;
\;W^+ \ra l^+ \nu_l\]
and\[\overline{t} \ra \overline{b}W^-,\;\; W^- \ra
l^-\overline{\nu_l}\]
in the respective rest frames of $t$, $\overline{t}$,
in the limit that all masses
except the top mass are neglected,
are given in ref. \cite{sdr1}.

Combining the production and decay amplitudes in the narrow-width
approximation for $t,\overline{t},W^+,W^-$, and using appropriate
Lorentz
boosts to calculate everything in the $\ee$ cm frame, we get the
$l^+$
and $l^-$ angular distributions for the case of $e^-$, $e^+$ with
polarization
$P_e$, $P_{\o e}$ to be:
\bea\label{triple}
\lefteqn{\f{d^3\sigma^{\pm}}{d\cos\ttt d\cos\thl d\phil} = \f{3\alpha^2\beta
m_t^2}{8s^{2}}B_tB_{\o{t}} \f{1}{(1-\beta\cos\theta_{tl})^3}}
\nonumber\\
&\times &\left[{\cal  A}^{\pm} (1-\beta\cos\theta_{tl})
+{\cal B}^{\pm} (cos\theta_{tl} - \beta )\right. \nonumber \\
&&\left.
+{\cal  C}^{\pm} (1-\beta^2)\sin\tht\sin\thl (\cos\tht\cos\phil -
\sin\tht\cot\thl )\right. \nonumber \\
&&\left. +{\cal D}^{\pm} (1-\beta^2) \sin\tht\sin\thl\sin\phil \right],
\eea
where $\sigma^+$ and $\sigma^-$ refer respectively to $l^+$ and
$l^-$ distributions, with the same notation for the kinematic
variables of
particles and antiparticles.  Thus, $\theta_t$, is the polar angle of
$t$  (or \tbar), and $\el,\;\thetal,\;\phil$ are the energy, polar
angle and azimuthal angle of $l^+$ (or $l^-$).
All the angles are now in the cm frame, with the $z$ axis chosen
along the $e^-$ momentum, and the $x$ axis chosen in the plane
containing the $e^-$ and $t$ directions.
$\theta_{tl}$ is the angle between
the $t$ and $l^+$ directions (or $\overline t$ and $l^-$ directions).
$\beta$
is the $t$ (or $\o t$) velocity: \(\beta=\sqrt{1-4m_t^2/s}\), and
$\gamma = 1/\sqrt{1-\beta^2}$. $B_t$ and $B_{\o t}$ are respectively the
branching ratios of $t$ and
$\o t$ into the final states being considered.

The coefficients ${\cal A}^{\pm}$, ${\cal B}^{\pm}$, ${\cal C}^{\pm}$ and ${\cal D}^{\pm}$ are
given  by
\bea
{\cal A}^{\pm} & = & A_0 \pm A_1 \cos\tht + A_2 \cos^2\tht , \\
{\cal B}^{\pm} & = & B_0^{\pm} \pm B_1 \cos\tht + B_2^{\pm} \cos^2\tht , \\
{\cal C}^{\pm} & = & \pm C_0^{\pm} + C_1^{\pm} \cos\tht , \\
{\cal D}^{\pm} & = & \pm D_0^{\pm} + D_1^{\pm} \cos\tht ,
\eea
The quantities $A_i$, $B_i^{\pm}$, $C_i^{\pm}$ and $D_i^{\pm}$ occurring
in the above
equation are functions of the masses, $s$, the degrees of $e$ and $\o
e$  polarization ($P_e$ and $P_{\o e}$), and the coupling constants.
They are listed in the Appendix.

It should be emphasized that, as shown in \cite{hioki,sdr1}, the distribution
in (\ref{triple}) does not depend on anomalous effects in the $tbW$ vertices 
(\ref{tbw}) and (\ref{tbarbw}). 
In the limit of small $b$-quark mass, and in the linear
approximation, the effect of all
possible form factors in the angular distributions is the same overall
factor which appears in the total width. Consequently, this factor
cancels out with another appearing in the denominator \cite{hioki,sdr1}.
As a consequence, the angular distributions are totally independent of any
anomalous effects, \CP violating or \CP conserving, in the decay
vertex.  
Thus even ${\cal O}(\alpha_s)$ QCD corrections
to the $tbW$ vertices would not be felt in (\ref{triple}).

To obtain the single-differential polar-angle distribution, we integrate over 
$\phi$ from 0 to $2 \pi$, and finally over cos$\,\theta_t$
from $-1$ to +1.  The final result is

\bea\label{cldist}
\lefteqn{
\f{d\sigma^{\pm}}{d\cos\theta_l}=\frac{3\pi\alpha^2}{32s}
B_t B_{\overline t} \beta
\left\{4A_0
\mp 2A_1\left(\f{1-\beta^2}{\beta^2} \log\f{1+\beta}{1-\beta}-
\f{2}{\beta}\right) \cos\theta_l \right.}\nonumber \\
&&\left. + 2A_2 \left(
\f{1-\beta^2}{\beta^3}\log\f{1+\beta}{1-\beta}
(1-3\cos^2\theta_l) \right. \right. \nonumber \\
&& \left. \left. - \f{2}{\beta^2} (1-3\cos^2\theta_l-\beta^2+2 \beta^2
\cos^2\theta_l) \right) \right. \nonumber \\
&&\left. \pm 2B_1\f{1-\beta^2}{\beta^2}  \left(
\f{1}{\beta}\log\f{1+\beta}{1-\beta} -
2 \right) \cos\theta_l \right. \nonumber \\
&&  \left. + B_2^{\pm} \f{1-\beta^2}{\beta^3}
\left( \f{\beta^2-3}{\beta} \log\f{1+\beta}{1-\beta} + 6 \right)
(1-3\cos^2\theta_l) \right. \nonumber \\
&&\left. \pm
2C_0^{\pm}\f{1-\beta^2}{\beta^2} \left( \f{1-\beta^2}{\beta}
\log\f{1+\beta}{1-\beta} - 2 \right) \cos\theta_l
\right. \nonumber \\
&& \left. - C_1^{\pm}\f{1-\beta^2}{\beta^3} \left( \f{3(1-\beta^2)}{\beta}
\log\f{1+\beta}{1-\beta} -2(3-2\beta^2)\right) (1-3
\cos^2\theta_l) \right\}.
\eea
This is the same expression as in \cite{pou2} and \cite{sdr1}. However, the
significance of the functions $A_i$, $B_i$, $C_i$ and $D_i$ is different in
each case.

We now proceed to a discussion of \CP-odd asymmetries resulting from the use of
the above distributions.

\section{$CP$-violating angular asymmetries}

We will work with two different types of asymmetries, one which does not
depend on the azimuthal angles
angle of the decay lepton, so that the azimuthal angle is fully integrated over,
and the
other dependent on the azimuthal angle. In all cases, we assume a cut-off of
$\theta_0$ on the forward and backward directions of the charged lepton.
Some cut-off on the forward and backward angles is certainly needed from an
experimental point of view; we furthermore exploit the cut-off to optimize the
sensitivity.

In the first case, namely polar asymmetries,
we define two independent \CP-violating asymmetries, which depend on
different
linear combinations of Im$\cdg$ and Im$\cdz$ . (It is not
possible to define \CP-odd
 quantities which determine Re$\cdgz$ using single-lepton polar
distributions, as can
be seen from the expression for the \CP-odd combination
$\frac{d\sigma^+}{d\cos\theta_l}(\theta_l)
-\frac{d\sigma^-}{d\cos\theta_l}(\pi-\theta_l)$). One is simply the
total
lepton-charge asymmetry, with a cut-off of $\theta_0$ on the forward
and
backward directions:
\beq\label{ach}
A_{ch}(\theta_0)=\frac{
{\displaystyle      \int_{\theta_0}^{\pi-\theta_0}}d\theta_l
{\displaystyle          \left( \frac{d\sigma^+}{d\theta_l}
        -   \frac{d\sigma^-}{d\theta_l}\right)}}
{
{\displaystyle      \int_{\theta_0}^{\pi-\theta_0}}d\theta_l
{\displaystyle          \left( \frac{d\sigma^+}{d\theta_l} +
\frac{d\sigma^-}{d\theta_l}\right)}}.
\eeq
The other is the lept\-onic forward-backward asy\-mmetry com\-bined
with charge asy\-mmetry, again with the angles within $\theta_0$ of
the forward and back\-ward directions excluded:
\beq\label{afb}
A_{fb}(\theta_0)= \frac{ {\displaystyle
\int_{\theta_0}^{\frac{\pi}{2}}}d\theta_l {\displaystyle
\left( \frac{d\sigma^+}{d\theta_l} +
\frac{d\sigma^-}{d\theta_l}\right)} {\displaystyle
- \int^{\pi-\theta_0}_{\frac{\pi}{2}}}d\theta_l {\displaystyle
\left( \frac{d\sigma^+}{d\theta_l} +    \frac{d\sigma^-}{d\theta_l}
\right)}}
{
{\displaystyle      \int_{\theta_0}^{\pi-\theta_0}}d\theta_l
{\displaystyle          \left( \frac{d\sigma^+}{d\theta_l} +
\frac{d\sigma^-}{d\theta_l}\right)}}.
\eeq

Analytic expressions for both these aymmetries may be easily obtained using
(\ref{cldist}), and are not displayed here explicitly.

We note the fact that ${\cal A}_{ch}(\theta_0)$ vanishes for
$\theta_0=0$, since then it is simply the asymmetry between
the total rates of production of $l^+$ and $l^-$. It then vanishes so long
as \CP-violation in decay does not contribute.
${\cal A}_{fb}(\theta_0)$, however, is nonzero for
$\theta_0=0$. 
This implies that the $CP$-violating charge asymmetry
does not exist unless a cut-off is imposed on the lepton production
angle. ${\cal A}_{fb}(\theta_0)$, however, is nonzero for
$\theta_0=0$.

We now define angular asymmetries of the second type, which depend on the
range of the
azimuthal angle $\phi_l$ of the charged lepton. These are called the up-down
and left-right asymmetries, and depend respectively on the real and imaginary
parts of the dipole couplings.

The up-down asymmetry is defined by
\beq\label{aud}
A_{ud}(\theta_0)=\f{1}{2\,\sigma (\theta_0)}\int_{\theta_0}^{\pi-\theta_0}
\left[
\f{d\,\sigma^+_{\rm up} } {d\,\theta_l}
-\f{d\,\sigma^+_{\rm down} } {d\,\theta_l}
+
\f{d\,\sigma^-_{\rm up}} {d\,\theta_l}
-\f{d\,\sigma^-_{\rm down} } {d\,\theta_l}
\right] {d\,\theta_l} ,
\eeq
where
\beq\label{sigma0}
\sigma (\theta_0) = \int_{\theta_0}^{\pi-\theta_0} \f{d\,\sigma }
{d\,\theta_l}d\,\theta_l
\eeq
is the SM cross section for the semi-leptonic final state, with a forward and
backward cut-off of $\theta_0$ on $\thl$.
Here up/down refers to
$(p_{l^{\pm}})_y\;\raisebox{-1.0ex}{$\stackrel{\textstyle>}{<}$}\;0,\;
\:(p_{l^{\pm}})_y$ being the $y$
component of $\vec{p}_{l^{\pm}}$ with respect to a coordinate system
chosen in the $e^+\,e^-$ center-of-mass (cm) frame so that the
$z$-axis is along $\vec{p}_e$, and the $y$-axis is along
$\vec{p}_e\,\times\,\vec{p}_t$.  The $t\bar{t}$ production
plane is thus the $xz$ plane.  Thus, ``up" refers to the range $0<\phi_l<\pi$,
and ``down" refers to the range $\pi<\phi_l<2\pi$.

The left-right asymmetry is defined by
\beq\label{alr}
A_{lr}(\theta_0)=\f{1}{2\,\sigma (\theta_0)}\int_{\theta_0}^{\pi-\theta_0}
\left[
\f{d\,\sigma^+_{\rm left} } {d\,\theta_l}
-\f{d\,\sigma^+_{\rm right} } {d\,\theta_l}
+
\f{d\,\sigma^-_{\rm left}} {d\,\theta_l}
-\f{d\,\sigma^-_{\rm right} } {d\,\theta_l}
\right] {d\,\theta_l} ,
\eeq
Here left/right refers to
$(p_{l^{\pm}})_x\;\raisebox{-1.0ex}{$\stackrel{\textstyle>}{<}$}\;0,\;
\:(p_{l^{\pm}})_x$ being the $x$
component of $\vec{p}_{l^{\pm}}$ with respect to the coordinate system
system defined above.
Thus, ``left" refers to the range $-\pi /2<\phi_l<\pi /2$,
and ``right" refers to the range $\pi /2<\phi_l<3\pi /2$.

Analytic expressions for the up-down and left-right symmetry are not available
for nonzero cut-off in $\theta_l$. Hence, the angular integrations have been
done numerically in what follows.

These azimuthal asymmetries with a different choice of axes were discussed
in \cite{asymm,pou1}, without a cut-off $\theta_0$. 
Two other asymmetries were defined in \cite{pou1}, which helped to disentangle
the two dipole couplings from each other. However, we do not discuss these
here. Instead, we will assume that the electron beam polarization can be made
to change sign to give additional observable quantities to enable this
disentanglement.

All these asymmetries are a measure of $CP$ violation in the unpolarized
case and in the case when polarization is present, but
$P_e=-P_{\overline{e}}$.  When $P_e\neq -P_{\overline{e}}$, the
initial state is not invariant under $CP$, and therefore
$CP$-invariant interactions can contribute to the asymmetries.
However, to leading order in $\alpha$, these $CP$-invariant
contributions vanish in the limit $m_e=0$.  Order-$\alpha$ collinear
helicity-flip photon emission can give a $CP$-even contribution.
However, this background has been estimated in \cite{back}, and found to be
negligible for certain \CP-odd correlations for the kind of luminosities
under consideration. It has also been estimated for $A_{fb}$ and $A_{ch}$, and
again found negligible \cite{thesis}. The background is zero in the case of
$A_{ud}$ \cite{thesis}. It is expected that the background will also be 
negligible for $A_{lr}$ thought it has not been calculated explicitly. 

\section{Numerical Results}

In this section we describe results for the calculation
of 90\% confidence level (CL) limits that could be put on Re$\cdgz$ and 
Im$\cdgz$ using the asymmetries described in the previous sections.

We look at only semileptonic final states. That is to say, when $t$
decays leptonically, we assume $\o t$ decays hadronically, and {\it
vice versa}. We sum over the electron and muon decay channels. Thus,
$B_tB_{\o t}$ is taken to be $2/3\times2/9$.

We have considered unpolarized beams, as well as the case when the electron
beam has a longitudinal polarization of 80\%, 
either left-handed or right-handed. We
have also considered the possibility of two runs for identical time-spans
with the polarization reversed in the second run. The positron beam is assumed
to be unpolarized. Later on we discuss the results in the case when the
positron beam is also polarized. 

We assume an integrated luminosity of 500 fb$^{-1}$ for a cm energy of 500 GeV,
and an integrated luminosity of 1000 fb$^{-1}$ for a cm energy of 1000 GeV. The
limits for different integrated luminosities
can easily be obtained by scaling appropriately the
limits presented here by the inverse square root of the factor by which 
the luminosity 
is scaled. We comment later on on the results for a cm energy of 800 GeV
with an integrated luminosity of 800 GeV. 

We use the parameters $\alpha = 1/128$, $\alpha_s(m_Z^2)=0.118$, $m_Z= 91.187$
GeV, $m_W=80.41$ GeV, $m_t=175$ GeV and $\sin^2\theta_W=0.2315$. We have used,
following \cite{kodaira}, a gluon energy cut-off of 
$\omega_{\rm max}=(\sqrt{s}-2m_t)/5$. While qualitative results would be
insensitive, exact
quantitative results would of course depend on the choice of cut-off.

Fig. \ref{cs.fig}  shows the SM cross section $\sigma (\theta_0)$, defined in
eq. (\ref{sigma0}), for 
$t$ or $\o{t}$ production, followed by its semileptonic decay, with a cut-off
$\theta_0$ on the lepton polar angle, plotted against $\theta_0$ for the two 
choices of $\sqrt{s}$ and for different electron beam polarizations.

\begin{figure}[ptb]
\begin{center}
\input{diffcs.tex}
\caption{\small
The  SM cross
section for decay leptons in the process $e^+e^- \rightarrow t\overline{t}$
plotted as a function of the cut-off $\theta_0$ on the lepton polar angle 
in the forward and backward directions for $e^-$ beam longitudinal
polarizations $P_e=-0.8,0,+0.8$ and for values of total cm energy 
$\sqrt{s}= 500$
GeV and $\sqrt{s}=1000$ GeV.}
\label{cs.fig}
\end{center}
\end{figure}

Fig. \ref{ach.fig} shows the asymmetry $A_{ch}$ defined in eq.~(\ref{ach}) 
arising when either of the (imaginary parts of) electric and weak dipole 
couplings takes the value 1, the other taking the value 0, plotted as a
function of the cut-off $\theta_0$, for the polarized and unpolarized cases,
for two different cm energies. Fig. \ref{afb.fig} is the corresponding figure
for $A_{fb}$ defined in eq.~(\ref{afb}). 

\begin{figure}[ptb]
\begin{center}
\input{ach.tex}
\input{achZ.tex}
\vskip -.3cm
\caption{\small
The  asymmetry $A_{ch}$ defined in the text, for Im~$c_d^{\g}=1$, Im~$c_d^Z=0$
(upper figure), and for Im~$c_d^{\g}=0$, Im~$c_d^Z=1$ (lower figure),
plotted as a function of the cut-off $\theta_0$ on the lepton polar angle
in the forward and backward directions for $e^-$ beam longitudinal
polarizations $P_e=-0.8,0,+0.8$ and for values of total cm energy
$\sqrt{s}= 500$
GeV and $\sqrt{s}=1000$ GeV.}
\label{ach.fig}
\end{center}
\end{figure}

%\begin{figure}[ptb]
%\begin{center}
%\input{achZ.tex}
%\vskip .3cm
%\caption{\small
%The  asymmetry $A_{ch}$ defined in the text, for Im~$c_d^{\g}=0$ and 
%Im~$c_d^Z=1$,
%plotted as a function of the cut-off $\theta_0$ on the lepton polar angle
%in the forward and backward directions for $e^-$ beam longitudinal
%polarizations $P_e=-0.8,0,+0.8$ and for values of total cm energy
%$\sqrt{s}= 500$
%GeV and $\sqrt{s}=1000$ GeV.}
%\label{achZ.fig}
%\end{center}
%\end{figure}

\begin{figure}[ptb]
\begin{center}
\input{afb.tex}
\input{afbZ.tex}
\vskip -.3cm
\caption{\small
The  asymmetry $A_{fb}$ defined in the text, for Im~$c_d^{\g}=1$, Im~$c_d^Z=0$
(upper figure), and for  Im~$c_d^{\g}=0$, Im~$c_d^Z=1$ (lower figure),
plotted as a function of the cut-off $\theta_0$ on the lepton polar angle
in the forward and backward directions for $e^-$ beam longitudinal
polarizations $P_e=-0.8,0,+0.8$ and for values of total cm energy
$\sqrt{s}= 500$
GeV and $\sqrt{s}=1000$ GeV.}
\label{afb.fig}
\end{center}
\end{figure}

%\begin{figure}[ptb]
%\begin{center}
%\input{afbZ.tex}
%\vskip .3cm
%\caption{\small
%The  asymmetry $A_{fb}$ defined in the text, for Im~$c_d^{\g}=0$ and
%Im~$c_d^Z=1$,
%plotted as a function of the cut-off $\theta_0$ on the lepton polar angle
%in the forward and backward directions for $e^-$ beam longitudinal
%polarizations $P_e=-0.8,0,+0.8$ and for values of total cm energy
%$\sqrt{s}= 500$
%GeV and $\sqrt{s}=1000$ GeV.}
%\label{afbZ.fig}
%\end{center}
%\end{figure}
 
Similarly, the asymmetries $A_{ud}$ from eq.~(\ref{aud}) and $A_{lr}$ from
eq.~(\ref{alr}), which depend respectively on the real and imaginary parts of
$c_d^{\g,Z}$, are shown in Figs. \ref{aud.fig} and \ref{alr.fig}. Again, only
one of the couplings take a nonzero value, in this case 0.1, 
while the others are vanishing.

\begin{figure}[ptb]
\begin{center}
\input{aud.tex}
\input{audZ.tex}
\vskip -.3cm
\caption{\small
The  asymmetry $A_{ud}$ defined in the text, for Re~$c_d^{\g}=0.1$,
Re~$c_d^Z=0$ (upper figure), and for Re~$c_d^{\g}=0$, Re~$c_d^Z=0.1$ (lower
figure),
plotted as a function of the cut-off $\theta_0$ on the lepton polar angle
in the forward and backward directions for $e^-$ beam longitudinal
polarizations $P_e=-0.8,0,+0.8$ and for values of total cm energy
$\sqrt{s}= 500$
GeV and $\sqrt{s}=1000$ GeV.}
\label{aud.fig}
\end{center}
\end{figure}

%\begin{figure}[ptb]
%\begin{center}
%\input{audZ.tex}
%\vskip .3cm
%\caption{\small
%The  asymmetry $A_{ud}$ defined in the text, for Re~$c_d^{\g}=0$,
%Re~$c_d^Z=0.1$, Im~$c_d^{\g}=0$ and Im~$c_d^Z=0$
%plotted as a function of the cut-off $\theta_0$ on the lepton polar angle
%in the forward and backward directions for $e^-$ beam longitudinal
%polarizations $P_e=-0.8,0,+0.8$ and for values of total cm energy
%$\sqrt{s}= 500$ GeV and $\sqrt{s}=1000$ GeV.}
%\label{audZ.fig}
%\end{center}
%\end{figure}

\begin{figure}[ptb]
\begin{center}
\input{alr.tex}
\input{alrZ.tex}
\vskip -.3cm
\caption{\small
The  asymmetry $A_{lr}$ defined in the text, 
for Im~$c_d^{\g}=0.1$,
Im~$c_d^Z=0$ (upper figure), and for Im~$c_d^{\g}=0$, Im~$c_d^Z=0.1$ (lower
figure),
plotted as a function of the cut-off $\theta_0$ on the lepton polar angle
in the forward and backward directions for $e^-$ beam longitlrinal
polarizations $P_e=-0.8,0,+0.8$ and for values of total cm energy
$\sqrt{s}= 500$
GeV and $\sqrt{s}=1000$ GeV.}
\label{alr.fig}
\end{center}
\end{figure}

%\begin{figure}[ptb]
%\begin{center}
%\input{alrZ.tex}
%\vskip .3cm
%\caption{\small
%The  asymmetry $A_{lr}$ defined in the text, for Re~$c_d^{\g}=0$,
%Re~$c_d^Z=0$, Im~$c_d^{\g}=0$ and Im~$c_d^Z=0.1$
%plotted as a function of the cut-off $\theta_0$ on the lepton polar angle
%in the forward and backward directions for $e^-$ beam longitlrinal
%polarizations $P_e=-0.8,0,+0.8$ and for values of total cm energy
%$\sqrt{s}= 500$ GeV and $\sqrt{s}=1000$ GeV.}
%\label{alrZ.fig}
%\end{center}
%\end{figure}


Tables 1-5
 show the results on the limits obtainable for each of these
possibilities. In all cases, the value of the cut-off $\theta_0$ has been
chosen to get the best sensitivity for that specific item. In case of
$A_{fb}$, the sensitivity is maximum for $\theta_0=0$. In that case, the 
cut-off has been arbitrarily chosen to be 10$^\circ$. 

\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
&&
\multicolumn{3}{|c|}{$A_{ch}$}&\multicolumn{3}{|c|}{$A_{fb}$}\\
\cline{3-8}
$\sqrt{s}$ (GeV) &$P_e$ & $\theta_0$ & Im$c_d^{\gamma}$ &
 Im$c_d^Z$ &$\theta_0$ & Im$c_d^{\gamma}$ & Im$c_d^Z$ \\
\hline
    &  0    &64$^{\circ}$ & $ 0.053$ & $ 0.31$ & $10^{\circ}$ & 0.054 & 0.60\\
500 &$+0.8$ &64$^{\circ}$ & $ 0.052$ & 0.13   & $10^{\circ}$ & 0.049 &$ 0.11$\\
    &$-0.8$ &63$^{\circ}$ & $ 0.053$ & $ 0.092$ & $10^{\circ}$ & 0.059 & 0.11 \\
\hline
    & 0      &64$^{\circ}$ & $ 0.029$ & $ 0.18$ & $10^{\circ}$ &0.032&0.36 \\
1000& $+0.8$ &64$^{\circ}$ & $ 0.028$ & 0.074 &   $10^{\circ}$ &0.029&$ 0.069$\\
    & $-0.8$ &64$^{\circ}$ & $ 0.028$ &$ 0.051$&  $10^{\circ}$ &0.034&$ 0.063$\\
\hline
\end{tabular}
\caption{Individual 90\% CL limits on dipole couplings obtainable from
$A_{ch}$ and
$A_{fb}$ for $\sqrt{s}=500$ GeV with integrated luminosity 500 fb$^{-1}$
and for $\sqrt{s}=1000$ GeV with integrated luminosity 1000 fb$^{-1}$ for
different electron beam polarizations $P_e$. Cut-off $\theta_0$ is chosen to
optimize the sensitivity.}
\end{center}
\end{table}

In Table 1 we give the 90\% confidence level (CL) limits that can be 
obtained on Im$c_d^{\g}$ and
Im$c_d^Z$, assuming one of them to be nonzero, the other taken to be vanishing.
The limit is defined as the value of Im$\cdg$ or Im$\cdz$ for which the
corresponding asymmetry $A_{ch}$ or $A_{fb}$ becomes equal to $1.64/\sqrt{N}$,
where $N$ is the total number of events.

\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$\sqrt{s}$ (GeV)  & $\theta_0$ & Im$c_d^{\gamma}$ & Im$c_d^Z$\\
\hline
500 & 40$^{\circ}$ & 0.37 & 2.6\\
1000& 40$^{\circ}$ & 0.20 & 1.5 \\
\hline
\end{tabular}
\caption{Simultaneous 90\% CL limits on dipole couplings obtainable from
$A_{ch}$ and
$A_{fb}$ for $\sqrt{s}=500$ GeV with integrated luminosity 500 fb$^{-1}$
and for $\sqrt{s}=1000$ GeV with integrated luminosity 1000 fb$^{-1}$ for
unpolarized beams. Cut-off $\theta_0$ is chosen to
optimize the sensitivity.}
\end{center}
\end{table}

Table 2 shows possible 90\% CL limits for the unpolarized case, when results
from $A_{ch}$ and $A_{fb}$ are combined.
The idea is that each asymmetry measures a different linear
combination of Im$c_d^{\g}$ and
Im$c_d^Z$. So a null result for the two asymmetries will correspond to two
different bands of regions allowed at 90\% CL in the space of Im$c_d^{\g}$ and
Im$c_d^Z$. The overlapping region of the two bands leads to the limits given in
Table 2. In this case, for 90\% CL, the asymmetry is required to be
$2.15/\sqrt{N}$, corresponding to two degrees of freedom. Incidentally, the same
procedure followed for $P_e=\pm 0.8$ gives much worse limits. 

\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
&
\multicolumn{3}{|c|}{$A_{ch}$}&\multicolumn{3}{|c|}{$A_{fb}$}\\
\cline{2-7}
$\sqrt{s}$ (GeV)&  $\theta_0$ & Im$c_d^{\gamma}$ &
 Im$c_d^Z$ &$\theta_0$ & Im$c_d^{\gamma}$ & Im$c_d^Z$ \\
\hline
500 & 64$^{\circ}$ & 0.090& 0.19 & $10^{\circ}$ & 0.091 & 0.19 \\
1000& 64$^{\circ}$ & 0.049& 0.10 &$10^{\circ}$ & 0.053 & 0.11 \\
\hline
\end{tabular}
\caption{Simultaneous limits on dipole couplings combining data from
polarizations $P_e=0.8$ and $P_e=-0.8$, using separately $A_{ch}$ and
$A_{fb}$ for $\sqrt{s}=500$ GeV with integrated luminosity 500 fb$^{-1}$
and for $\sqrt{s}=1000$ GeV with integrated luminosity 1000 fb$^{-1}$.
Cut-off $\theta_0$ is chosen to optimize the sensitivity.}
\end{center}
\end{table}

Similarly, using one of the two asymmetries, but two different polarizations of
the electron beam, one can get two bands in the parameter plane, which give
simultaneous limits on the dipole couplings. The results for electron
polarizations $P_e=\pm 0.8$ are given in Table 3 for each of the asymmetries
$A_{ch}$ and $A_{fb}$.



\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
&&
\multicolumn{3}{|c|}{$A_{ud}$}&\multicolumn{3}{|c|}{$A_{lr}$}\\
\cline{3-8}
$\sqrt{s}$ (GeV) &$P_e$ & $\theta_0$ & Re$c_d^{\gamma}$ &
 Re$c_d^Z$ &$\theta_0$ & Im$c_d^{\gamma}$ & Im$c_d^Z$ \\
\hline
    &  0    &25$^{\circ}$ & $ 0.066$ & $ 0.022$ & $30^{\circ}$ & 0.015 & 0.088\\
500 &$+0.8$ &30$^{\circ}$ & $ 0.019$ & 0.023  & $35^{\circ}$ & 0.015 & 0.038\\
    &$-0.8$ &25$^{\circ}$ & $ 0.015$ & 0.020 & $30^{\circ}$ & 0.015 &0.026\\
\hline
    & 0      &30$^{\circ}$ & $ 0.029$ & $ 0.0096$ & $60^{\circ}$ &0.021&0.13 \\
1000& $+0.8$ &35$^{\circ}$ &  0.0082 & 0.010 &   $60^{\circ}$ &0.021&$ 0.055$\\
    & $-0.8$ &30$^{\circ}$ &  0.0066& 0.0089 &  $60^{\circ}$ &0.021&$ 0.038$\\
\hline
\end{tabular}
\caption{Individual 90\% CL limits on dipole couplings obtainable from
$A_{ud}$ and
$A_{lr}$ for $\sqrt{s}=500$ GeV with integrated luminosity 500 fb$^{-1}$
and for $\sqrt{s}=1000$ GeV with integrated luminosity 1000 fb$^{-1}$ for
different electron beam polarizations $P_e$. Cut-off $\theta_0$ is chosen to
optimize the sensitivity.}
\end{center}
\end{table}

Table 4 lists the 90\% CL limits which may be obtained on the real and
imaginary parts of the dipole couplings using $A_{ud}$ and $A_{lr}$, assuming
one of the couplings to be nonzero at a time.

\begin{table}\label{t5}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
&
\multicolumn{3}{|c|}{$A_{ud}$}&\multicolumn{3}{|c|}{$A_{lr}$}\\
\cline{2-7}
$\sqrt{s}$ (GeV)&  $\theta_0$ & Re$c_d^{\gamma}$ &
 Re$c_d^Z$ &$\theta_0$ & Im$c_d^{\gamma}$ & Im$c_d^Z$ \\
\hline
500 & 25$^{\circ}$ & 0.022& 0.029& $35^{\circ}$ & 0.020& 0.041\\
1000& 30$^{\circ}$ & 0.0097& 0.013 &$60^{\circ}$ & 0.028 & 0.059 \\
\hline
\end{tabular}
\caption{Simultaneous limits on dipole couplings combining data from
polarizations $P_e=0.8$ and $P_e=-0.8$, using separately $A_{ud}$ and
$A_{lr}$ for $\sqrt{s}=500$ GeV with integrated luminosity 500 fb$^{-1}$
and for $\sqrt{s}=1000$ GeV with integrated luminosity 1000 fb$^{-1}$. Cut-off
$\theta_0$ is chosen to optimize the sensitivity.}
\end{center}
\end{table}

Table 5 shows simultaneous limits on Re$c_d^{\g}$ and Re$c_d^Z$ obtainable 
from combining the data on $A_{ud}$ for $P_e=+0.8$ and $P_e=-0.8$, 
and similarly,
limits on Im$c_d^{\g}$ and Im$c_d^Z$ from data on $A_{lr}$ for the two
polarizations.

\section{Conclusions and Discussion}

We have presented in analytic form  the single-lepton angular
distribution in the production and subsequent decay of $\tt$ in the
presence of electric and weak dipole form factors of the top quark, 
including ${\cal O}(\alpha_s)$ QCD corrections in SGA. Anomalous contributions
to the $tbW$ decay vertex do not affect these distributions. 
We have also included effects of longitudinal electron and positron 
beam polarizations. We have then obtained analytic expressions for certain 
simple $CP$-violating polar-angle  
asymmetries, specially chosen so that they do not require the
reconstruction of the  $t$ or $\o t$ directions or energies. 
We have also evaluated numercially azimuthal asymmetries which need minimal
information on the $t$ or \tbar momentum direction alone.
We have
analyzed these asymmetries to obtain simultaneous 90\% CL limits on
the electric and weak dipole couplings which
would be possible at future linear $\ee$ collider 
operating at
$\sqrt{s}= 500$ GeV with an integrated luminosity of 500 fb$^{-1}$, and at 
$\sqrt{s}= 1000$ GeV with an integrated luminosity of 1000 fb$^{-1}$.
We assume electron beam polarization of $\pm$ 80\%, while the positron beam 
is unpolarized. The results are presented in Figs. 1-5 and Tables 1-5.

In general, simultaneous 90\% CL limts on $\cdg$ and $\cdz$ 
which can be obtained
with the polarized 500 GeV option are of the order of 0.1--0.2,
corresponding to dipole moments of about $(1-2)\times 10^{-17} e$ cm, 
if the asymmetries $A_{ch}$
or $A_{fb}$ are used. The limits improve by a factor of 4 to 6 if the azimuthal
asymetries $A_{ud}$ or $A_{lr}$ are used. However, putting in a top detection
efficiency factor of 10\% in the case of azimuthal asymmetries, where top
direction needs to be determined, would bring down these limts to the same
level of $(1-2)\times 10^{-17} e$ cm.

For $\sqrt{s}=1000$ GeV and an integrated luminosity of 1000 fb$^{-1}$, the
limits obtainable would be better by a factor of 3 or 4 in each case, bringing
them to the level of $(2-3)\times 10^{-18} e$ cm in the best cases.

Though we have not presented detailed results here, a numerical evaluation of
possible limits has been carried out for other possibilities, like (i) a 
slightly higher
electron beam polarization of 0.9, (ii) positron beam
polarized to the extent of 0.6, in addition to polarized electron beam, a
possibility envisaged in the context of TESLA, (iii) a beam energy of 800 GeV,
with an integrated luminosity of 800 fb$^{-1}$. The conclusions are as follows.

An increase in the electron polarization from 0.8 to 0.9 (with the positrons
unpolarized) improves the sensitivity by about 30 to 50\% in case of polar-angle
asymmetries $A_{ch}$ and $A_{fb}$, and to a lesser extent, 10 to 15\% in the
case of the measurement of Re$\cdg$ and Im$\cdz$ by azimuthal asymmetries. 

Including longitudinal positron polarization of 0.6 (always opposite in sign to
the polarization of the electron) improves the sensitivity in all cases by
about 20 to 30\%.

We conclude that it is probably worthwhile from the top dipole coupling point
of view to improve the electron polarization
by a small amount rather than invest in a new or difficult technology to
achieve a high positron polarization.

The improvement in sensitivity in going from cm energy of 800 GeV to 1000 GeV,
with a simultaneous increase in integrated luminosity from 800 fb$^{-1}$ 
to 1000 fb$^{-1}$, is about 5 to 10\% in the case of polar-angle asymmetries,
and 20 to 25\% in case of $A_{ud}$. However, the sensitivity {\em worsens} 
in the case of measurement using $A_{lr}$, by about 10\% or so.

Our general conclusion is that the sensitivity to the measurement of
individual dipole couplings Re$\cdg$ and Im$\cdz$ 
is improved considerably if the electron beam is
polarized, a situation which might easily obtain at linear colliders.
As a consequence, simultaneous  limits on all the couplings are improved by
beam polarization. 

The theoretical predictions for $c_d^{\g,Z}$ are at the level of
$10^{-2}-10^{-3} $, as for example, in the neutral-Higgs-exchange and
supersymmetric models of \CP~ violation \cite{bern,asymm,new,sonirev}. 
In other  models, like the charged-Higgs-exchange \cite{sonirev} 
or third-generation 
leptoquark models \cite{pou4}, the prediction are even lower.  Hence
the measurements suggested here at the 500 GeV option 
cannot exclude these modes at the 90\% C.L.  It will be necessary to use the
1000 GeV option with a suitable luminosity to test at least some of the models.

It is necessary to repeat this study including experimental detection 
efficiencies.
Given an overall efficiency, we could still get an
idea of the limits on the dipole couplings by scaling them as the inverse square
root of the efficiency.

We have not included a cut-off on decay-lepton energies which may be required
from a practical point of view. However, our results are perfectly valid if the
cut-off is reasonably small. For example, for $\sqrt{s}=500$ GeV, the minimum
lepton energy allowed kinematically is about 7.5 GeV. So a cut-off below that
would need no modification of the results.

Contact $\ee\tt$ interactions violating \CP~ have
been ignored in this work. They should be taken into account for a complete
treatment of \CP~ violation in $\ee \ra \tt$.

We have restricted ourselves to energies in the $t\o{t}$ continuum. Studies in
the threshold region are also interesting and have been investigated in
\cite{sumino}.

%\newpage
\vskip 1cm

\noindent{\Large \bf Appendix}
\vskip .5in

The expressions for $A_i$, $B_i$, $C_i$ and $D_i$ occurring in
equation (8) are listed below. They include  to first order the form factors
$\cdg$ and $\cdz$, as well as $c_M^{\g}$ and $c_M^Z$. Terms containing products
of $\cdgz$ with $c_M^{\g,Z}$ have been dropped. It is also understood that
terms proportional to products of $A$ or $B$ (which are of order $\alpha_s$)
and $\cdg$ or $\cdz$ have to omitted in the calculations.

\begin{eqnarray}
A_0 & = & 2 \left\{  (2-\beta^2) \left[ 2\vert c_v^{\g}\vert ^2 +
2(r_L+r_R){\rm Re}(c_v^{\g}c_v^{Z*}) + (r_L^2+r_R^2)\vert c_v^Z\vert ^2 \right]
\right.  \nonumber \\
&& \left.
+  \beta^2 (r_L^2+r_R^2)\vert c_a^Z\vert ^2
-2\beta^2\left[2{\rm Re}(c_v^{\g}c_M^{\g *})
\right. \right. \nonumber \\
&& \left. \left.
+(r_L+r_R)\Re (c_v^{\g}c_M^{Z*}+c_v^Zc_M^{\g *})
+ (r_L^2+r_R^2)\Re (c_v^Zc_M^{Z*})\right]
\right. (1- P_e P_{\overline e}) \nonumber \\
&& +  \left. (2-\beta^2) \left[
2(r_L-r_R)\Re (c_v^{\g}c_v^{Z*})
+ (r_L^2-r_R^2)\vert c_v^Z\vert ^2 \right]
\right.
\nonumber \\
& &  \left.
  +  \beta^2 (r_L^2-r_R^2)\vert c_a^Z\vert ^2
- 2\beta^2
\left[(r_L-r_R)\Re (c_v^{\g}c_M^{Z*}+c_v^Zc_M^{\g *})
\right. \right.  \nonumber \\
&& \left. \left.
 + (r_L^2 - r_R^2)\Re (c_v^Zc_M^{Z *})
\right] \right\} (P_{\overline e}-P_e), \nonumber \\
%
A_1&=& -8 \beta \Re \left( c_a^{Z*} \left\{
\left[(r_L-r_R)c_v^{\g} + (r_L^2-r_R^2)c_v^Z \right]  (1-
P_eP_{\overline e}) \right. \right. \nonumber \\ && \left. \left. +
\left[(r_L+r_R)c_v^{\g} + (r_L^2+r_R^2)c_v^Z \right] (P_{\overline
e}-P_e) \right\} \right),  \nonumber\\
%
A_2&=& 2 \beta^2 \left\{ \left[
2\vert c_v^{\g}\vert ^2 + 4\Re (c_v^{\g}c_M^{\g *})
+ 2(r_L+r_R)\Re (c_v^{\g}c_v^{Z*} +c_v^{\g}c_M^{Z*} + c_v^Zc_M^{\g *})
\right.\right. \nonumber \\
&&\left. \left.
+(r_L^2+r_R^2)\left(
\vert c_v^Z\vert ^2 + \vert c_a^Z\vert ^2 +2 \Re (c_v^Zc_M^{Z*}) \right)
 \right]  (1- P_e P_{\overline e})
\right. \nonumber \\ && \left. +  \left[ 2(r_L-r_R)\Re (c_v^{\g}c_v^{Z*}
+c_v^{\g}c_M^{Z*}
+ c_v^Z c_M^{\g*}) \right. \right. \nonumber \\
&& \left.\left. 
+(r_L^2-r_R^2)\left( \vert c_v^Z\vert ^2 + \vert c_a^Z\vert ^2
+ 2\Re (c_v^Zc_M^{Z*}) \right) \right]
(P_{\overline e}-P_e) \right\}, \nonumber\\
%
B_0^{\pm}&=& 4\beta
\left\{ \left(\Re c_v^{\g} + r_L\Re c_v^Z\right) \left(r_L \Re c_a^Z \mp {\rm
Im}c_d^{\g} \mp r_L {\rm Im}c_d^Z \right) (1-P_e)(1+P_{\overline e})
\right. \nonumber \\ 
&& \left. +  \left(\Re c_v^{\g} + r_R\Re c_v^Z\right)
\left(r_R \Re c_a^Z \mp {\rm Im}c_d^{\g} \mp r_R {\rm Im}c_d^Z \right)
(1+P_e)(1-P_{\overline e}) \right\}, \nonumber \\
%
B_1&=& -4 \left\{
\left[\vert c_v^{\g}+r_Lc_v^Z\vert ^2+ \beta^2 r_L^2 \vert c_a^Z\vert ^2 \right]
(1-P_e)(1+P_{\overline e}) \right. \nonumber \\ && \left.  - 
\left[\vert c_v^{\g}+r_Rc_v^Z\vert ^2+ \beta^2 r_R^2 \vert c_a^Z\vert ^2 \right]
(1+P_e)(1-P_{\overline e}) \right\}, \nonumber \\
%
B_2^{\pm}&=& 4\beta
\left\{  \left(\Re c_v^{\g} + r_L\Re c_v^Z\right) \left(r_L \Re c_a^Z \pm {\rm
Im}c_d^{\g} \pm r_L {\rm Im}c_d^Z \right) (1-P_e)(1+P_{\overline e})
\right. \nonumber \\ && \left.+ \left(\Re c_v^{\g} + r_R\Re c_v^Z\right)
\left(r_R\Re  c_a^Z \pm {\rm Im}c_d^{\g} \pm r_R {\rm Im}c_d^Z \right)
(1+P_e)(1-P_{\overline e}) \right\}, \nonumber\\
%
C_0^{\pm}&=&4
\left\{ \left[ \vert c_v^{\g} + r_Lc_v^Z\vert ^2
- \beta^2 \gamma^2 \left( \Re c_v^{\g} + r_L \Re c_v^Z \right)\left( \Re
c_M^{\g} + r_L \Re c_M^Z \right) \right. \right. \nonumber \\
&&\left. \left.   \pm \beta^2
\gamma^2 r_L \Re c_a^Z \left( {\rm Im} c_d^{\g} + {\rm Im} c_d^Z
r_L \right) \right] (1-P_e)(1+P_{\overline e}) \right. \nonumber \\
&& \left. - \left[ \vert c_v^{\g} + r_Rc_v^Z\vert ^2
- \beta^2 \gamma^2 \left( \Re c_v^{\g} + r_R \Re c_v^Z \right)\left( \Re
c_M^{\g} + r_R \Re c_M^Z \right) \right. \right. \nonumber \\
&& \left. \left. \pm \beta^2
\gamma^2 r_R \Re c_a^Z \left( {\rm Im} c_d^{\g} + {\rm Im} c_d^Z
r_R \right) \right] (1+P_e)(1-P_{\overline e}) \right\}, \nonumber\\
%
C_1^{\pm}&=&- 4\beta \left\{ \left[ \left(\Re c_v^{\g} +
r_L\Re c_v^Z\right) \left(r_L \Re c_a^Z \pm \gamma^2 {\rm Im}c_d^{\g} \pm r_L
\gamma^2 {\rm Im}c_d^Z \right) \right. \right. \nonumber \\ &&\left. \left.
- \beta^2 \gamma^2 r_L \Re c_a^Z\left(\Re c_M^{\g} +r_L \Re c_M^Z) \right)
\right] (1-P_e)(1+P_{\overline e}) \right.
\nonumber \\
&& \left.+ \left[ \left(\Re c_v^{\g} + r_R\Re c_v^Z\right)
\left(r_R \Re c_a^Z \pm \gamma^2 {\rm Im}c_d^{\g} \pm r_R
\gamma^2 {\rm Im}c_d^Z \right) \right. \right. \nonumber \\ && \left. \left.
- \beta^2 \gamma^2 r_R \Re c_a^Z\left(\Re c_M^{\g} +r_R \Re c_M^Z) \right)
\right] (1+P_e)(1-P_{\overline e})
\right\},\nonumber \\
%
D_0^{\pm}&=& 4 \beta \left\{ \left[
\Im \left[\left( (c_V^{\g} +
r_L c_v^Z) -\beta^2 \gamma^2 ( c_M^{\g} + r_L c_M^Z)\right) r_L c_a^{Z*}\right]  \right. \right.
\nonumber \\ && \left. \left.
\mp  \gamma^2 \left(\Re c_v^{\g} +
r_L\Re c_v^Z\right) \left( {\rm Re}c_d^{\g} + r_L {\rm Re}c_d^Z \right)
\right]
(1-P_e)(1+P_{\overline e}) \right. \nonumber \\ && \left.
-\left[ \Im \left[\left( (c_V^{\g} +
r_R c_v^Z) -\beta^2 \gamma^2 ( c_M^{\g} + r_R c_M^Z)\right) r_R c_a^{Z*}\right]  \right.\right.
\nonumber \\ &&\left. \left.
\mp \gamma^2 \left(\Re c_v^{\g} + r_R\Re c_v^Z\right) \left({\rm Re}c_d^{\g} 
+ r_R {\rm Re}c_d^Z \right)\right]
 (1+P_e)(1-P_{\overline e}) \right\}, \nonumber \\
%
D_1^{\pm}&=& 4 \beta^2  \gamma^2 \left\{ \left[ (\Re c_v^{\g} + r_L \Re c_v^Z)(\Im
c_M^{\g}  \right. \right. \nonumber \\
&&\left. \left. + r_L \Im c_M^Z) \pm r_L \Re c_a^Z \left( {\rm Re}c_d^{\g} +
r_L {\rm Re} c_d^Z \right)\right] (1-P_e)(1+P_{\overline e})
\right. \nonumber \\
&& \left.+\left[ (\Re c_v^{\g} + r_R \Re c_v^Z)(\Im c_M^{\g} + r_R \Im c_M^Z) 
\right.\right.
\nonumber \\
&& \left.\left. \pm r_R \Re c_a^Z \left( {\rm Re}c_d^{\g} 
+ r_R {\rm Re} c_d^Z \right)\right]
(1+P_e)(1-P_{\overline e})
\right\}. \nonumber
\end{eqnarray}
The relations
\[
r_L =  \f{(\half - x_W)}{(1-m_Z^2/s)\sqrt{x_W (1-x_W)}} 
\]
and 
\[
r_R =  \f{- x_W}{(1-m_Z^2/s)\sqrt{x_W (1-x_W)}} 
\]
are used in writing the above equation.

\noindent {\bf Note added in proof}
\vskip .2cm
The result shown in references \cite{hioki} and \cite{sdr1}, 
that the lepton angular 
distributions do not depend on anomalous couplings occurring in top 
decay, has now been shown to be valid even when the $b$-quark mass is 
not neglected, see B. Grzadkowski and Z. Hioki, Phys. Lett. B {\bf 557}, 
55 (2000).

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%%CITATION = HEP-PH 9708332;%%
\bb{sumino} 
M.~Jezabek, T.~Nagano and Y.~Sumino,
%``Probe of CP violation in e+ e- $\to$ t anti-t near threshold,''
Phys.\ Rev.\ D {\bf 62}, 014034 (2000)
[arXiv:hep-ph/0001322];
%%CITATION = HEP-PH 0001322;%%
Y.~Sumino,
%``Probe of CP violation in e+ e- $\to$ t anti-t near threshold,''
arXiv:hep-ph/0007326.
%%CITATION = HEP-PH 0007326;%%
\end{thebibliography}


\end{document}

hep-ph0304046.extracted_bib
\begin{thebibliography}{99}
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S.~M.~Lietti and H.~Murayama,
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%%CITATION = HEP-PH 0001304;%%
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%%CITATION = NUPHA,B393,46;%%
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%%CITATION = HEP-PH 0105318;%%
%``CP violation in open t anti-t production at a linear collider,'' 
to appear
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%%CITATION = HEP-PH 0202045;%%
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M.~M.~Tung, J.~Bernab\' eu and J.~Pe\~ narrocha,
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%%CITATION = HEP-PH 9809411;%%
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B.~Ananthanarayan and S.~D.~Rindani,
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\bb{thesis} P. Poulose, Ph.D. thesis (1997) submitted to Gujarat University
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%%CITATION = NONE;%%
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A.~Bartl, E.~Christova and W.~Majerotto,
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%%CITATION = HEP-PH 9708332;%%
\bb{sumino} 
M.~Jezabek, T.~Nagano and Y.~Sumino,
%``Probe of CP violation in e+ e- $\to$ t anti-t near threshold,''
Phys.\ Rev.\ D {\bf 62}, 014034 (2000)
[arXiv:hep-ph/0001322];
%%CITATION = HEP-PH 0001322;%%
Y.~Sumino,
%``Probe of CP violation in e+ e- $\to$ t anti-t near threshold,''
arXiv:hep-ph/0007326.
%%CITATION = HEP-PH 0007326;%%
\end{thebibliography}