% GNUPLOT: LaTeX picture with Postscript
\setlength{\unitlength}{0.1bp}
\special{!
%!PS-Adobe-2.0
%%Creator: gnuplot
%%DocumentFonts: Helvetica
%%BoundingBox: 50 50 914 1130
%%Pages: (atend)
%%EndComments
/gnudict 40 dict def
gnudict begin
/Color false def
/Solid false def
/gnulinewidth 5.000 def
/vshift -33 def
/dl {10 mul} def
/hpt 31.5 def
/vpt 31.5 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
/DL { Color {setrgbcolor Solid {pop []} if 0 setdash }
 {pop pop pop Solid {pop []} if 0 setdash} ifelse } def
/BL { stroke gnulinewidth 2 mul setlinewidth } def
/AL { stroke gnulinewidth 2 div setlinewidth } def
/PL { stroke gnulinewidth setlinewidth } def
/LTb { BL [] 0 0 0 DL } def
/LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def
/LT0 { PL [] 0 1 0 DL } def
/LT1 { PL [4 dl 2 dl] 0 0 1 DL } def
/LT2 { PL [2 dl 3 dl] 1 0 0 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
/P { stroke [] 0 setdash
  currentlinewidth 2 div sub M
  0 currentlinewidth V stroke } def
/D { 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
  P } def
/A { stroke [] 0 setdash vpt sub M 0 vpt2 V
  currentpoint stroke M
  hpt neg vpt neg R hpt2 0 V stroke
  } def
/B { 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
  P } def
/C { 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
/T { 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
  P  } def
/S { 2 copy A C} def
end
%%EndProlog
}
\begin{picture}(4320,4628)(0,0)
\special{"
%%Page: 1 1
gnudict begin
gsave
50 50 translate
0.100 0.100 scale
0 setgray
/Helvetica findfont 100 scalefont setfont
newpath
-500.000000 -500.000000 translate
LTa
LTb
600 251 M
63 0 V
3474 0 R
-63 0 V
600 972 M
63 0 V
3474 0 R
-63 0 V
600 1693 M
63 0 V
3474 0 R
-63 0 V
600 2414 M
63 0 V
3474 0 R
-63 0 V
600 3135 M
63 0 V
3474 0 R
-63 0 V
600 3856 M
63 0 V
3474 0 R
-63 0 V
600 4577 M
63 0 V
3474 0 R
-63 0 V
806 251 M
0 63 V
0 4263 R
0 -63 V
1217 251 M
0 63 V
0 4263 R
0 -63 V
1628 251 M
0 63 V
0 4263 R
0 -63 V
2039 251 M
0 63 V
0 4263 R
0 -63 V
2451 251 M
0 63 V
0 4263 R
0 -63 V
2862 251 M
0 63 V
0 4263 R
0 -63 V
3273 251 M
0 63 V
0 4263 R
0 -63 V
3685 251 M
0 63 V
0 4263 R
0 -63 V
4096 251 M
0 63 V
0 4263 R
0 -63 V
600 251 M
3537 0 V
0 4326 V
-3537 0 V
600 251 L
LT0
2438 1289 D
1469 1534 D
3161 1520 D
1858 1347 D
2339 1159 D
1154 1318 D
3626 1390 D
3074 1289 D
1685 1275 D
3103 1332 D
1197 1433 D
1073 1376 D
1403 1491 D
1509 1361 D
865 1433 D
2192 1318 D
1135 1361 D
2075 1347 D
3329 1347 D
2717 1318 D
3085 1347 D
2773 1159 D
2320 1376 D
3154 1433 D
3047 1405 D
3569 1304 D
2772 1260 D
3435 1520 D
2732 1347 D
2831 1246 D
1923 1318 D
1075 1448 D
841 1996 D
1457 1419 D
964 1332 D
2582 1419 D
2341 1275 D
2776 1419 D
1614 1304 D
1424 1448 D
2356 1361 D
2199 1275 D
3236 1188 D
1794 1246 D
3165 1232 D
2413 1246 D
1063 1506 D
2366 1304 D
1455 1318 D
3465 1376 D
1033 1477 D
2631 1390 D
1598 1506 D
2648 1419 D
2656 1246 D
3289 1448 D
1425 1433 D
2952 1448 D
2813 1232 D
3546 1332 D
934 1621 D
2488 1174 D
1225 1520 D
3312 1332 D
817 1563 D
3444 1203 D
2610 1477 D
2212 1289 D
2360 1347 D
1801 1376 D
2055 1376 D
2306 1289 D
2917 1304 D
1802 1289 D
2827 1332 D
1462 1347 D
2658 1347 D
3316 1390 D
2403 1246 D
2846 1433 D
1500 1462 D
2777 1419 D
2468 1217 D
774 2400 D
3220 1275 D
2657 1318 D
1322 1592 D
2590 1332 D
947 1361 D
873 1751 D
3292 1174 D
714 1534 D
2736 1376 D
1232 1405 D
1640 1506 D
1923 1304 D
2672 1448 D
1463 1246 D
1847 1361 D
1911 1361 D
1963 1203 D
2736 1275 D
1385 1289 D
1351 1419 D
1744 1419 D
2589 1145 D
769 1534 D
1210 1246 D
1592 1506 D
2840 1419 D
3439 1405 D
1728 1289 D
2526 1289 D
3438 1232 D
885 1491 D
2446 1318 D
3120 1246 D
1270 1275 D
1089 1635 D
2810 1217 D
1289 1361 D
2484 1275 D
2756 1289 D
915 1433 D
2976 1188 D
3442 1376 D
1900 1419 D
1619 1405 D
2905 1462 D
2003 1332 D
1258 1520 D
2058 1275 D
3270 1376 D
1898 1188 D
1647 1433 D
2908 1318 D
2398 1188 D
2039 1145 D
2881 1419 D
1837 1477 D
3417 1232 D
1647 1289 D
964 1534 D
3204 1275 D
1428 1260 D
1529 1419 D
2175 1159 D
1263 1390 D
3551 1477 D
1533 1433 D
2214 1433 D
1834 1318 D
1379 1203 D
712 1289 D
3026 1332 D
2837 1232 D
2144 1332 D
2389 1390 D
710 1751 D
1962 1419 D
2125 1188 D
2899 1433 D
682 1679 D
690 1722 D
703 1780 D
715 1823 D
723 1866 D
732 1909 D
744 1953 D
752 1981 D
765 2039 D
777 2111 D
785 2154 D
797 1953 D
806 1895 D
908 1635 D
1011 1534 D
1114 1477 D
1217 1433 D
1320 1419 D
1423 1390 D
1525 1376 D
1628 1361 D
1731 1361 D
1834 1347 D
1937 1347 D
2039 1332 D
2142 1332 D
2245 1332 D
2348 1332 D
2451 1332 D
2554 1332 D
2656 1332 D
2759 1332 D
2862 1332 D
2965 1332 D
3068 1332 D
3170 1332 D
3273 1332 D
3376 1332 D
3479 1332 D
3582 1332 D
3685 1332 D
3787 1332 D
3890 1332 D
3993 1332 D
4096 1332 D
1230 1347 D
1227 1390 D
1996 1347 D
2705 1318 D
2941 1318 D
3655 1332 D
2222 1332 D
779 1996 D
3627 1332 D
2693 1318 D
1973 1318 D
3933 1332 D
2696 1332 D
2042 1318 D
872 1419 D
3632 1332 D
2108 1318 D
1978 1318 D
821 2111 D
2695 1318 D
2016 1318 D
2388 1318 D
1837 1318 D
1936 1318 D
2061 1318 D
1729 1332 D
1020 1405 D
2814 1332 D
1450 1347 D
3526 1332 D
1135 1491 D
2726 1318 D
3664 1332 D
2289 1318 D
1849 1318 D
2519 1332 D
3804 1332 D
3125 1332 D
2676 1318 D
2698 1318 D
1955 1318 D
3210 1332 D
1338 1347 D
2855 1318 D
1780 1332 D
1177 1347 D
1854 1347 D
3922 1332 D
2540 1318 D
3149 1332 D
1545 1405 D
2711 1318 D
1512 1332 D
3020 1318 D
2667 1318 D
3363 1332 D
790 2039 D
3097 1332 D
2796 1332 D
3313 1332 D
739 1751 D
2891 1332 D
1264 1347 D
1510 1347 D
1878 1318 D
1352 1332 D
704 1578 D
1014 1520 D
3162 1332 D
1652 1318 D
3567 1332 D
1303 1347 D
1010 1405 D
1595 1405 D
697 1837 D
1603 1361 D
1868 1318 D
2160 1318 D
3436 1332 D
3497 1332 D
3298 1332 D
1318 1347 D
1860 1318 D
2527 1318 D
1491 1332 D
1354 1332 D
2533 1318 D
3436 1332 D
3523 1332 D
1315 1405 D
2689 1318 D
2489 1318 D
3112 1332 D
2022 1318 D
2601 1318 D
1264 1347 D
2942 1318 D
2616 1318 D
1152 1376 D
3287 1332 D
2007 1318 D
1650 1318 D
3426 1332 D
3792 1332 D
2892 1332 D
1991 1318 D
1022 1664 D
2014 1318 D
3996 1332 D
2681 1318 D
2916 1332 D
1002 1578 D
3932 1332 D
2943 1332 D
1474 1332 D
3017 1332 D
3713 1332 D
1043 1361 D
2006 1332 D
1699 1318 D
3390 1332 D
2334 1332 D
2952 1318 D
1274 1419 D
1349 1477 D
1900 1332 D
2102 1332 D
2553 1318 D
966 1491 D
3717 1332 D
1897 1318 D
2515 1318 D
2843 1318 D
2506 1318 D
2016 1347 D
2867 1332 D
3129 1332 D
2421 1332 D
955 1419 D
3366 1332 D
3882 1332 D
1177 1390 D
1149 1361 D
2675 1332 D
3589 1332 D
1141 1347 D
3226 1332 D
1462 1419 D
3521 1332 D
3699 1332 D
2979 1318 D
3040 1318 D
3188 1332 D
1472 1419 D
1332 1361 D
3944 1332 D
4063 1332 D
1110 1361 D
1737 1318 D
4003 1332 D
3702 1332 D
3530 1332 D
1781 1318 D
1617 1347 D
3751 1332 D
2214 1318 D
2613 1318 D
3409 1332 D
3259 1332 D
2582 1318 D
850 1419 D
3256 1332 D
2499 1318 D
2692 1318 D
3750 1332 D
2181 1332 D
3536 1332 D
1640 1332 D
4041 1332 D
1985 1318 D
1695 1318 D
3751 1332 D
1997 1318 D
1155 1361 D
1501 1332 D
2352 1332 D
1840 1318 D
1959 1332 D
4011 1332 D
3719 1332 D
2352 1332 D
3297 1332 D
2731 1318 D
2723 1318 D
2091 1318 D
3830 1332 D
1976 1318 D
2030 1332 D
2521 1318 D
1529 1332 D
2848 1318 D
1282 1332 D
3840 1332 D
1016 1361 D
1424 1332 D
1629 1332 D
3850 1332 D
723 1693 D
2707 1318 D
3602 1332 D
3163 1332 D
1172 1347 D
2884 1318 D
803 1895 D
1040 1462 D
3769 1332 D
1270 1347 D
1426 1361 D
2016 1332 D
3723 1332 D
2375 1332 D
3805 1332 D
4023 1332 D
3569 1332 D
1242 1347 D
2806 1318 D
2473 1318 D
3034 1332 D
3682 1332 D
1174 1347 D
3024 1332 D
1459 1332 D
969 1491 D
2462 1332 D
2264 1318 D
2215 1347 D
2063 1318 D
2860 1318 D
2890 1318 D
2067 1304 D
2102 1318 D
1746 1318 D
3792 1332 D
3905 1332 D
2448 1318 D
2315 1318 D
2866 1318 D
1575 1332 D
1156 1347 D
818 1462 D
3583 1332 D
1450 1376 D
1492 1332 D
1516 1332 D
2783 1318 D
3525 1332 D
929 1405 D
1548 1405 D
1959 1332 D
2497 1318 D
3187 1332 D
2799 1318 D
694 1621 D
1568 1332 D
2363 1332 D
807 1578 D
2720 1318 D
3942 1332 D
2906 1318 D
1533 1390 D
3476 1332 D
963 1419 D
3336 1332 D
2895 1332 D
3399 1332 D
1870 1318 D
1441 1332 D
2139 1332 D
2735 1318 D
1256 1433 D
1378 1332 D
3638 1332 D
3526 1332 D
1814 1318 D
4047 1332 D
3388 1332 D
2554 1332 D
1263 1361 D
2886 1318 D
3598 1332 D
2764 1318 D
2475 1332 D
3753 1332 D
762 1895 D
1979 1318 D
2660 1318 D
3931 1332 D
1281 1462 D
1507 1332 D
1872 1318 D
2751 1318 D
1231 1361 D
3799 1332 D
3162 1332 D
2212 1332 D
3748 1332 D
3733 1332 D
748 1909 D
1856 1318 D
1807 1318 D
3167 1332 D
3313 1332 D
1183 1361 D
1995 1318 D
2544 1318 D
2023 1318 D
2131 1318 D
3966 1332 D
3876 1332 D
1201 1347 D
2247 1318 D
3205 1332 D
1425 1347 D
1046 1361 D
2767 1332 D
957 1448 D
1590 1332 D
3468 1332 D
3078 1318 D
2024 1347 D
3331 1332 D
982 1361 D
1516 1332 D
2702 1318 D
1099 1462 D
1896 1318 D
2195 1318 D
3120 1332 D
1119 1376 D
2956 1318 D
4061 1332 D
3714 1332 D
2648 1318 D
1944 1318 D
2524 1318 D
3230 1332 D
1683 1347 D
2193 1318 D
866 1433 D
2549 1318 D
3055 1332 D
3705 1332 D
3365 1332 D
1147 1433 D
2941 1318 D
3213 1332 D
3402 1332 D
3024 1318 D
3456 1332 D
1281 1361 D
1513 1332 D
3975 1332 D
1801 1318 D
1702 1332 D
2299 1318 D
3098 1332 D
3255 1332 D
2264 1332 D
2986 1318 D
1065 1448 D
3146 1318 D
1362 1361 D
3080 1332 D
3573 1332 D
1603 1332 D
1662 1318 D
1862 1332 D
1945 1318 D
2831 1318 D
2775 1318 D
1439 1332 D
3655 1332 D
1527 1332 D
1173 1376 D
2020 1318 D
1413 1347 D
3159 1332 D
2348 1332 D
1957 1318 D
846 2082 D
2551 1318 D
2473 1318 D
1350 1332 D
863 1419 D
1475 1332 D
3676 1332 D
1604 1332 D
880 1419 D
1569 1332 D
3827 1332 D
3898 1332 D
3157 1332 D
2878 1318 D
1465 1332 D
4027 1332 D
2470 1332 D
1603 1332 D
2982 1332 D
1464 1361 D
1550 1332 D
2912 1332 D
3824 1332 D
3400 1332 D
2136 1332 D
743 2126 D
3551 1332 D
2582 1332 D
1001 1707 D
3500 1332 D
3368 1332 D
2488 1318 D
3993 1332 D
2861 1318 D
LT1
806 1765 A
1011 1131 A
806 1433 M
0 664 V
775 1433 M
62 0 V
-62 664 R
62 0 V
1011 655 M
0 951 V
980 655 M
62 0 V
-62 951 R
62 0 V
stroke
grestore
end
showpage
}
\put(1052,684){\makebox(0,0)[l]{LEP}}
\put(847,2270){\makebox(0,0)[l]{LEP+SLD Average}}
\put(2862,2702){\makebox(0,0)[l]{$\mu > 0 \, GeV$}}
\put(2862,2991){\makebox(0,0)[l]{$m_t = 175 \pm 5 \, GeV$}}
\put(2862,3279){\makebox(0,0)[l]{$\tan\beta =2-30$}}
\put(2862,3568){\makebox(0,0)[l]{$A_0 =0-900 \, GeV$}}
\put(2862,3856){\makebox(0,0)[l]{$M_0 = 70-900\, GeV$}}
\put(2368,51){\makebox(0,0){$\vspace*{-0.2in}M_{1/2}\, (GeV)$}}
\put(100,2414){%
\special{ps: gsave currentpoint currentpoint translate
270 rotate neg exch neg exch translate}%
\makebox(0,0)[b]{\shortstack{$A_e$ }}%
\special{ps: currentpoint grestore moveto}%
}
\put(4096,151){\makebox(0,0){900}}
\put(3685,151){\makebox(0,0){800}}
\put(3273,151){\makebox(0,0){700}}
\put(2862,151){\makebox(0,0){600}}
\put(2451,151){\makebox(0,0){500}}
\put(2039,151){\makebox(0,0){400}}
\put(1628,151){\makebox(0,0){300}}
\put(1217,151){\makebox(0,0){200}}
\put(806,151){\makebox(0,0){100}}
\put(540,4577){\makebox(0,0)[r]{0.17}}
\put(540,3856){\makebox(0,0)[r]{0.165}}
\put(540,3135){\makebox(0,0)[r]{0.16}}
\put(540,2414){\makebox(0,0)[r]{0.155}}
\put(540,1693){\makebox(0,0)[r]{0.15}}
\put(540,972){\makebox(0,0)[r]{0.145}}
\put(540,251){\makebox(0,0)[r]{0.14}}
\end{picture}
% GNUPLOT: LaTeX picture with Postscript
\setlength{\unitlength}{0.1bp}
\special{!
%!PS-Adobe-2.0
%%Creator: gnuplot
%%DocumentFonts: Helvetica
%%BoundingBox: 50 50 914 655
%%Pages: (atend)
%%EndComments
/gnudict 40 dict def
gnudict begin
/Color false def
/Solid false def
/gnulinewidth 5.000 def
/vshift -33 def
/dl {10 mul} def
/hpt 31.5 def
/vpt 31.5 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
/DL { Color {setrgbcolor Solid {pop []} if 0 setdash }
 {pop pop pop Solid {pop []} if 0 setdash} ifelse } def
/BL { stroke gnulinewidth 2 mul setlinewidth } def
/AL { stroke gnulinewidth 2 div setlinewidth } def
/PL { stroke gnulinewidth setlinewidth } def
/LTb { BL [] 0 0 0 DL } def
/LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def
/LT0 { PL [] 0 1 0 DL } def
/LT1 { PL [4 dl 2 dl] 0 0 1 DL } def
/LT2 { PL [2 dl 3 dl] 1 0 0 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
/P { stroke [] 0 setdash
  currentlinewidth 2 div sub M
  0 currentlinewidth V stroke } def
/D { 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
  P } def
/A { stroke [] 0 setdash vpt sub M 0 vpt2 V
  currentpoint stroke M
  hpt neg vpt neg R hpt2 0 V stroke
  } def
/B { 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
  P } def
/C { 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
/T { 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
  P  } def
/S { 2 copy A C} def
end
%%EndProlog
}
\begin{picture}(4320,2592)(0,0)
\special{"
%%Page: 1 1
gnudict begin
gsave
50 50 translate
0.100 0.100 scale
0 setgray
/Helvetica findfont 100 scalefont setfont
newpath
-500.000000 -500.000000 translate
LTa
LTb
600 384 M
63 0 V
3474 0 R
-63 0 V
600 650 M
63 0 V
3474 0 R
-63 0 V
600 917 M
63 0 V
3474 0 R
-63 0 V
600 1183 M
63 0 V
3474 0 R
-63 0 V
600 1449 M
63 0 V
3474 0 R
-63 0 V
600 1716 M
63 0 V
3474 0 R
-63 0 V
600 1982 M
63 0 V
3474 0 R
-63 0 V
600 2248 M
63 0 V
3474 0 R
-63 0 V
600 2514 M
63 0 V
3474 0 R
-63 0 V
806 251 M
0 63 V
0 2227 R
0 -63 V
1217 251 M
0 63 V
0 2227 R
0 -63 V
1628 251 M
0 63 V
0 2227 R
0 -63 V
2039 251 M
0 63 V
0 2227 R
0 -63 V
2451 251 M
0 63 V
0 2227 R
0 -63 V
2862 251 M
0 63 V
0 2227 R
0 -63 V
3273 251 M
0 63 V
0 2227 R
0 -63 V
3685 251 M
0 63 V
0 2227 R
0 -63 V
4096 251 M
0 63 V
0 2227 R
0 -63 V
600 251 M
3537 0 V
0 2290 V
-3537 0 V
600 251 L
LT0
3161 1163 D
3626 1970 D
3074 731 D
3103 1483 D
3329 812 D
2717 1999 D
3085 606 D
2773 2099 D
3154 884 D
3047 1328 D
3569 949 D
3435 2033 D
2732 1964 D
2831 927 D
3236 1502 D
3165 1917 D
3465 1104 D
2656 1155 D
3289 1197 D
2952 1699 D
2813 2096 D
3546 1162 D
3312 1290 D
3444 1653 D
2917 2093 D
2658 1672 D
3316 1563 D
2777 2187 D
2468 1819 D
3220 1981 D
3292 1713 D
2736 1615 D
3439 1328 D
3438 792 D
3120 508 D
2810 1485 D
2976 343 D
3442 1221 D
2905 1939 D
3270 1984 D
2881 1762 D
3417 2148 D
3204 1595 D
3551 585 D
3026 1700 D
2837 1029 D
2899 970 D
2862 650 D
2965 650 D
3068 650 D
3170 650 D
3273 650 D
3376 650 D
3479 650 D
3582 650 D
3685 650 D
3787 650 D
3890 650 D
3993 650 D
4096 650 D
2705 2406 D
2941 1649 D
3655 600 D
3627 1992 D
3933 2212 D
3632 1246 D
2695 2089 D
2814 2485 D
3526 1525 D
3664 559 D
3804 1288 D
3125 636 D
2676 1418 D
2698 1630 D
3210 2130 D
2855 1586 D
3922 1520 D
2540 2070 D
3149 2508 D
2711 1731 D
3020 963 D
2667 1636 D
3363 1141 D
3097 1912 D
3313 1863 D
2891 354 D
3162 1673 D
3567 1966 D
3436 764 D
3497 1266 D
3298 1722 D
2533 2334 D
3436 939 D
3523 1068 D
2689 1877 D
3112 828 D
2601 2496 D
2942 1945 D
3287 1407 D
3426 1940 D
3792 2228 D
2892 2486 D
3996 1465 D
2681 2212 D
2916 464 D
3932 675 D
2943 2290 D
3017 1981 D
3713 1138 D
3390 914 D
2952 1696 D
3717 2235 D
2843 1192 D
2506 2012 D
2867 648 D
3129 1824 D
3366 330 D
3882 835 D
3589 1834 D
3226 399 D
3521 2133 D
3699 2136 D
2979 1755 D
3040 1725 D
3188 1704 D
3944 402 D
4063 1650 D
4003 1746 D
3702 1589 D
3530 1315 D
3751 605 D
2613 1886 D
3409 2283 D
3259 2462 D
2582 1706 D
3256 1702 D
2692 2318 D
3750 891 D
3536 337 D
4041 511 D
3751 532 D
4011 2156 D
3719 969 D
3297 522 D
2731 1096 D
2723 1734 D
3830 478 D
2848 1494 D
3840 750 D
3850 1227 D
2707 1633 D
3602 1988 D
3163 2308 D
2884 1466 D
3769 1831 D
3723 2028 D
3805 1976 D
4023 1696 D
3569 2303 D
2806 2356 D
2473 2401 D
3034 777 D
3682 752 D
3024 687 D
2860 1112 D
2890 770 D
3792 1880 D
3905 349 D
2866 1061 D
3583 840 D
2783 1727 D
3525 927 D
3187 1676 D
2799 2077 D
2720 1328 D
3942 1982 D
2906 2154 D
3476 1880 D
3336 2011 D
2895 694 D
3399 1157 D
2735 1503 D
3638 1915 D
3526 1622 D
4047 1082 D
3388 1721 D
2886 1395 D
3598 1423 D
2764 2030 D
3753 2387 D
2660 1659 D
3931 1603 D
2751 2291 D
3799 1670 D
3162 2174 D
3748 1101 D
3733 1481 D
3167 2237 D
3313 1027 D
3966 419 D
3876 1263 D
3205 818 D
3468 471 D
3078 947 D
3331 388 D
2702 2028 D
3120 769 D
2956 2094 D
4061 1112 D
3714 855 D
2648 1689 D
3230 2172 D
3055 383 D
3705 1981 D
3365 1743 D
2941 2093 D
3213 1937 D
3402 1171 D
3024 1265 D
3456 977 D
3975 1114 D
3098 800 D
3255 356 D
2986 1479 D
3146 1238 D
3080 2076 D
3573 1940 D
2831 1295 D
2775 1788 D
3655 1316 D
3159 2111 D
3676 2273 D
3827 2286 D
3898 1499 D
3157 338 D
2878 2251 D
4027 2009 D
2982 2505 D
2912 2352 D
3824 1299 D
3400 1170 D
3551 1633 D
3500 1187 D
3368 433 D
3993 1323 D
2861 1498 D
stroke
grestore
end
showpage
}
\put(806,1449){\makebox(0,0)[l]{$m_t = 175 \pm 5 \, GeV$}}
\put(806,2248){\makebox(0,0)[l]{LEP+SLD Average}}
\put(806,917){\makebox(0,0)[l]{$\mu > 0$}}
\put(806,1716){\makebox(0,0)[l]{$\alpha_s(M_Z)=0.119\pm 0.004$}}
\put(806,1982){\makebox(0,0)[l]{$s_l^2 = 0.23152\pm 0.00023$}}
\put(2368,51){\makebox(0,0){$\vspace*{-0.3in}M_{1/2} \, (GeV)$}}
\put(100,1396){%
\special{ps: gsave currentpoint currentpoint translate
270 rotate neg exch neg exch translate}%
\makebox(0,0)[b]{\shortstack{$\hspace*{-0.1in}M_0 \, (GeV)$}}%
\special{ps: currentpoint grestore moveto}%
}
\put(4096,151){\makebox(0,0){900}}
\put(3685,151){\makebox(0,0){800}}
\put(3273,151){\makebox(0,0){700}}
\put(2862,151){\makebox(0,0){600}}
\put(2451,151){\makebox(0,0){500}}
\put(2039,151){\makebox(0,0){400}}
\put(1628,151){\makebox(0,0){300}}
\put(1217,151){\makebox(0,0){200}}
\put(806,151){\makebox(0,0){100}}
\put(540,2514){\makebox(0,0)[r]{900}}
\put(540,2248){\makebox(0,0)[r]{800}}
\put(540,1982){\makebox(0,0)[r]{700}}
\put(540,1716){\makebox(0,0)[r]{600}}
\put(540,1449){\makebox(0,0)[r]{500}}
\put(540,1183){\makebox(0,0)[r]{400}}
\put(540,917){\makebox(0,0)[r]{300}}
\put(540,650){\makebox(0,0)[r]{200}}
\put(540,384){\makebox(0,0)[r]{100}}
\end{picture}
% GNUPLOT: LaTeX picture with Postscript
\setlength{\unitlength}{0.1bp}
\special{!
%!PS-Adobe-2.0
%%Creator: gnuplot
%%DocumentFonts: Helvetica
%%BoundingBox: 50 50 914 698
%%Pages: (atend)
%%EndComments
/gnudict 40 dict def
gnudict begin
/Color false def
/Solid false def
/gnulinewidth 5.000 def
/vshift -33 def
/dl {10 mul} def
/hpt 31.5 def
/vpt 31.5 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
/DL { Color {setrgbcolor Solid {pop []} if 0 setdash }
 {pop pop pop Solid {pop []} if 0 setdash} ifelse } def
/BL { stroke gnulinewidth 2 mul setlinewidth } def
/AL { stroke gnulinewidth 2 div setlinewidth } def
/PL { stroke gnulinewidth setlinewidth } def
/LTb { BL [] 0 0 0 DL } def
/LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def
/LT0 { PL [] 0 1 0 DL } def
/LT1 { PL [4 dl 2 dl] 0 0 1 DL } def
/LT2 { PL [2 dl 3 dl] 1 0 0 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
/P { stroke [] 0 setdash
  currentlinewidth 2 div sub M
  0 currentlinewidth V stroke } def
/D { 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
  P } def
/A { stroke [] 0 setdash vpt sub M 0 vpt2 V
  currentpoint stroke M
  hpt neg vpt neg R hpt2 0 V stroke
  } def
/B { 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
  P } def
/C { 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
/T { 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
  P  } def
/S { 2 copy A C} def
end
%%EndProlog
}
\begin{picture}(4320,2777)(0,0)
\special{"
%%Page: 1 1
gnudict begin
gsave
50 50 translate
0.100 0.100 scale
0 setgray
/Helvetica findfont 100 scalefont setfont
newpath
-500.000000 -500.000000 translate
LTa
LTb
600 251 M
63 0 V
3474 0 R
-63 0 V
600 476 M
63 0 V
3474 0 R
-63 0 V
600 701 M
63 0 V
3474 0 R
-63 0 V
600 926 M
63 0 V
3474 0 R
-63 0 V
600 1151 M
63 0 V
3474 0 R
-63 0 V
600 1376 M
63 0 V
3474 0 R
-63 0 V
600 1601 M
63 0 V
3474 0 R
-63 0 V
600 1826 M
63 0 V
3474 0 R
-63 0 V
600 2051 M
63 0 V
3474 0 R
-63 0 V
600 2276 M
63 0 V
3474 0 R
-63 0 V
600 2501 M
63 0 V
3474 0 R
-63 0 V
600 2726 M
63 0 V
3474 0 R
-63 0 V
754 251 M
0 63 V
0 2412 R
0 -63 V
1523 251 M
0 63 V
0 2412 R
0 -63 V
2292 251 M
0 63 V
0 2412 R
0 -63 V
3061 251 M
0 63 V
0 2412 R
0 -63 V
3829 251 M
0 63 V
0 2412 R
0 -63 V
600 251 M
3537 0 V
0 2475 V
-3537 0 V
600 251 L
LT0
3834 2563 M
180 0 V
600 2321 M
154 -68 V
154 -67 V
153 -67 V
154 -46 V
154 -67 V
154 -68 V
153 -67 V
154 -67 V
154 -68 V
154 -68 V
154 -67 V
153 -68 V
154 -67 V
154 -68 V
154 -67 V
154 -68 V
153 -67 V
154 -68 V
154 -67 V
154 -68 V
153 -67 V
154 -68 V
154 -90 V
LT1
3834 2463 M
180 0 V
600 2096 M
154 -68 V
154 -67 V
153 -68 V
154 -67 V
154 -68 V
154 -67 V
153 -90 V
154 -68 V
154 -67 V
154 -68 V
154 -67 V
153 -68 V
154 -90 V
154 -67 V
154 -68 V
154 -67 V
153 -68 V
154 -90 V
154 -67 V
154 -68 V
153 -90 V
154 -67 V
154 -68 V
LT2
3894 2363 D
1523 2096 D
3834 2363 M
180 0 V
-180 31 R
0 -62 V
180 62 R
0 -62 V
1523 1578 M
0 1036 V
1492 1578 M
62 0 V
-62 1036 R
62 0 V
stroke
grestore
end
showpage
}
\put(3774,2363){\makebox(0,0)[r]{${\small LEP+SLD}$}}
\put(3774,2463){\makebox(0,0)[r]{$A_0=M_0=M_{1/2}=200 \, GeV$}}
\put(3774,2563){\makebox(0,0)[r]{$A_0=M_0=M_{1/2}=600 \, GeV$}}
\put(2753,1601){\makebox(0,0)[l]{$M_h=115 \pm 7 \, GeV$}}
\put(754,926){\makebox(0,0)[l]{$\tan\beta =4 \,,\, \mu > 0 \, GeV$}}
\put(2753,476){\makebox(0,0)[l]{$M_h=108 \pm 5 \, GeV$}}
\put(2368,51){\makebox(0,0){$\vspace*{-0.2in}m_t\, (GeV)$}}
\put(100,1488){%
\special{ps: gsave currentpoint currentpoint translate
270 rotate neg exch neg exch translate}%
\makebox(0,0)[b]{\shortstack{$\hspace*{-0.2in}s^2_l$}}%
\special{ps: currentpoint grestore moveto}%
}
\put(3829,151){\makebox(0,0){190}}
\put(3061,151){\makebox(0,0){185}}
\put(2292,151){\makebox(0,0){180}}
\put(1523,151){\makebox(0,0){175}}
\put(754,151){\makebox(0,0){170}}
\put(540,2726){\makebox(0,0)[r]{0.2318}}
\put(540,2501){\makebox(0,0)[r]{0.2317}}
\put(540,2276){\makebox(0,0)[r]{0.2316}}
\put(540,2051){\makebox(0,0)[r]{0.2315}}
\put(540,1826){\makebox(0,0)[r]{0.2314}}
\put(540,1601){\makebox(0,0)[r]{0.2313}}
\put(540,1376){\makebox(0,0)[r]{0.2312}}
\put(540,1151){\makebox(0,0)[r]{0.2311}}
\put(540,926){\makebox(0,0)[r]{0.231}}
\put(540,701){\makebox(0,0)[r]{0.2309}}
\put(540,476){\makebox(0,0)[r]{0.2308}}
\put(540,251){\makebox(0,0)[r]{0.2307}}
\end{picture}
% GNUPLOT: LaTeX picture with Postscript
\setlength{\unitlength}{0.1bp}
\special{!
%!PS-Adobe-2.0
%%Creator: gnuplot
%%DocumentFonts: Helvetica
%%BoundingBox: 50 50 914 1130
%%Pages: (atend)
%%EndComments
/gnudict 40 dict def
gnudict begin
/Color false def
/Solid false def
/gnulinewidth 5.000 def
/vshift -33 def
/dl {10 mul} def
/hpt 31.5 def
/vpt 31.5 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
/DL { Color {setrgbcolor Solid {pop []} if 0 setdash }
 {pop pop pop Solid {pop []} if 0 setdash} ifelse } def
/BL { stroke gnulinewidth 2 mul setlinewidth } def
/AL { stroke gnulinewidth 2 div setlinewidth } def
/PL { stroke gnulinewidth setlinewidth } def
/LTb { BL [] 0 0 0 DL } def
/LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def
/LT0 { PL [] 0 1 0 DL } def
/LT1 { PL [4 dl 2 dl] 0 0 1 DL } def
/LT2 { PL [2 dl 3 dl] 1 0 0 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
/P { stroke [] 0 setdash
  currentlinewidth 2 div sub M
  0 currentlinewidth V stroke } def
/D { 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
  P } def
/A { stroke [] 0 setdash vpt sub M 0 vpt2 V
  currentpoint stroke M
  hpt neg vpt neg R hpt2 0 V stroke
  } def
/B { 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
  P } def
/C { 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
/T { 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
  P  } def
/S { 2 copy A C} def
end
%%EndProlog
}
\begin{picture}(4320,4628)(0,0)
\special{"
%%Page: 1 1
gnudict begin
gsave
50 50 translate
0.100 0.100 scale
0 setgray
/Helvetica findfont 100 scalefont setfont
newpath
-500.000000 -500.000000 translate
LTa
LTb
600 251 M
63 0 V
3474 0 R
-63 0 V
600 732 M
63 0 V
3474 0 R
-63 0 V
600 1212 M
63 0 V
3474 0 R
-63 0 V
600 1693 M
63 0 V
3474 0 R
-63 0 V
600 2174 M
63 0 V
3474 0 R
-63 0 V
600 2654 M
63 0 V
3474 0 R
-63 0 V
600 3135 M
63 0 V
3474 0 R
-63 0 V
600 3616 M
63 0 V
3474 0 R
-63 0 V
600 4096 M
63 0 V
3474 0 R
-63 0 V
600 4577 M
63 0 V
3474 0 R
-63 0 V
806 251 M
0 63 V
0 4263 R
0 -63 V
1217 251 M
0 63 V
0 4263 R
0 -63 V
1628 251 M
0 63 V
0 4263 R
0 -63 V
2039 251 M
0 63 V
0 4263 R
0 -63 V
2451 251 M
0 63 V
0 4263 R
0 -63 V
2862 251 M
0 63 V
0 4263 R
0 -63 V
3273 251 M
0 63 V
0 4263 R
0 -63 V
3685 251 M
0 63 V
0 4263 R
0 -63 V
4096 251 M
0 63 V
0 4263 R
0 -63 V
600 251 M
3537 0 V
0 4326 V
-3537 0 V
600 251 L
LT0
2438 1665 D
1469 1838 D
3161 1858 D
1858 1724 D
2339 1570 D
1154 1672 D
3626 1768 D
3074 1681 D
1685 1658 D
3103 1717 D
1197 1786 D
1073 1756 D
1403 1777 D
1509 1716 D
865 1753 D
2192 1690 D
1135 1724 D
2075 1708 D
3329 1726 D
2717 1692 D
3085 1693 D
2773 1567 D
2320 1735 D
3154 1803 D
3047 1748 D
3569 1688 D
2772 1656 D
3435 1856 D
2732 1726 D
2831 1643 D
1923 1694 D
1075 1814 D
841 2104 D
1457 1776 D
964 1804 D
2582 1778 D
2341 1655 D
2776 1786 D
1614 1649 D
1424 1800 D
2356 1740 D
2199 1660 D
3236 1596 D
1794 1644 D
3165 1631 D
2413 1636 D
1063 1807 D
2366 1684 D
1455 1696 D
3465 1725 D
1033 1853 D
2631 1757 D
1598 1844 D
2648 1783 D
2656 1613 D
3289 1800 D
1425 1787 D
2952 1809 D
2813 1629 D
3546 1712 D
934 1959 D
2488 1571 D
1225 1868 D
3312 1705 D
817 1886 D
3444 1611 D
2610 1835 D
2212 1668 D
2360 1717 D
1801 1735 D
2055 1725 D
2306 1674 D
2917 1688 D
1802 1648 D
2827 1708 D
1462 1720 D
2658 1718 D
3316 1762 D
2403 1639 D
2846 1802 D
1500 1804 D
2777 1776 D
2468 1614 D
774 2244 D
3220 1667 D
2657 1703 D
1322 1870 D
2590 1713 D
947 1731 D
873 1961 D
3292 1578 D
714 2113 D
2736 1755 D
1232 1764 D
1640 1839 D
1923 1682 D
2672 1809 D
1463 1640 D
1847 1736 D
1911 1703 D
1963 1580 D
2736 1660 D
1385 1674 D
1351 1747 D
1744 1765 D
2589 1556 D
769 1860 D
1210 1622 D
1592 1842 D
2840 1782 D
3439 1774 D
1728 1659 D
2526 1661 D
3438 1623 D
885 1832 D
2446 1697 D
3120 1647 D
1270 1639 D
1089 1900 D
2810 1621 D
1289 1739 D
2484 1664 D
2756 1675 D
915 1798 D
2976 1586 D
3442 1755 D
1900 1781 D
1619 1769 D
2905 1828 D
2003 1713 D
1258 1849 D
2058 1658 D
3270 1752 D
1898 1588 D
1647 1766 D
2908 1695 D
2398 1591 D
2039 1553 D
2881 1782 D
1837 1834 D
3417 1635 D
1647 1652 D
964 1847 D
3204 1665 D
1428 1615 D
1529 1739 D
2175 1568 D
1263 1751 D
3551 1838 D
1533 1795 D
2214 1788 D
1834 1692 D
1379 1583 D
712 2024 D
3026 1706 D
2837 1632 D
2144 1702 D
2389 1756 D
710 1994 D
1962 1778 D
2125 1595 D
2899 1794 D
682 1651 D
690 1629 D
703 1593 D
715 1553 D
723 1524 D
732 1494 D
744 1805 D
752 2059 D
765 2347 D
777 2609 D
785 2782 D
797 2239 D
806 2091 D
908 1873 D
1011 1805 D
1114 1768 D
1217 1745 D
1320 1730 D
1423 1718 D
1525 1710 D
1628 1705 D
1731 1700 D
1834 1697 D
1937 1694 D
2039 1693 D
2142 1692 D
2245 1691 D
2348 1691 D
2451 1691 D
2554 1691 D
2656 1692 D
2759 1693 D
2862 1693 D
2965 1694 D
3068 1695 D
3170 1696 D
3273 1697 D
3376 1698 D
3479 1699 D
3582 1701 D
3685 1702 D
3787 1703 D
3890 1705 D
3993 1706 D
4096 1707 D
1230 1711 D
1227 1725 D
1996 1695 D
2705 1696 D
2941 1695 D
3655 1702 D
2222 1689 D
779 2740 D
3627 1705 D
2693 1692 D
1973 1691 D
3933 1710 D
2696 1692 D
2042 1689 D
872 1796 D
3632 1703 D
2108 1688 D
1978 1688 D
821 2039 D
2695 1694 D
2016 1692 D
2388 1689 D
1837 1689 D
1936 1689 D
2061 1689 D
1729 1692 D
1020 1757 D
2814 1698 D
1450 1700 D
3526 1702 D
1135 1769 D
2726 1692 D
3664 1702 D
2289 1690 D
1849 1689 D
2519 1692 D
3804 1705 D
3125 1695 D
2676 1692 D
2698 1693 D
1955 1689 D
3210 1701 D
1338 1706 D
2855 1694 D
1780 1691 D
1177 1715 D
1854 1697 D
3922 1707 D
2540 1693 D
3149 1702 D
1545 1725 D
2711 1693 D
1512 1697 D
3020 1694 D
2667 1692 D
3363 1699 D
790 2505 D
3097 1698 D
2796 1693 D
3313 1701 D
739 2284 D
2891 1694 D
1264 1709 D
1510 1699 D
1878 1692 D
1352 1703 D
704 1328 D
1014 1797 D
3162 1698 D
1652 1694 D
3567 1705 D
1303 1706 D
1010 1755 D
1595 1722 D
697 1653 D
1603 1702 D
1868 1692 D
2160 1692 D
3436 1699 D
3497 1701 D
3298 1700 D
1318 1705 D
1860 1689 D
2527 1691 D
1491 1697 D
1354 1703 D
2533 1694 D
3436 1699 D
3523 1701 D
1315 1724 D
2689 1693 D
2489 1690 D
3112 1695 D
2022 1689 D
2601 1696 D
1264 1708 D
2942 1696 D
2616 1691 D
1152 1728 D
3287 1699 D
2007 1690 D
1650 1694 D
3426 1703 D
3792 1708 D
2892 1699 D
1991 1689 D
1022 1853 D
2014 1690 D
3996 1708 D
2681 1695 D
2916 1694 D
1002 1823 D
3932 1705 D
2943 1698 D
1474 1697 D
3017 1697 D
3713 1703 D
1043 1731 D
2006 1689 D
1699 1693 D
3390 1699 D
2334 1690 D
2952 1695 D
1274 1736 D
1349 1755 D
1900 1690 D
2102 1691 D
2553 1691 D
966 1799 D
3717 1707 D
1897 1689 D
2515 1691 D
2843 1693 D
2506 1692 D
2016 1699 D
2867 1693 D
3129 1698 D
2421 1692 D
955 1776 D
3366 1698 D
3882 1705 D
1177 1730 D
1149 1723 D
2675 1692 D
3589 1704 D
1141 1719 D
3226 1696 D
1462 1731 D
3521 1705 D
3699 1707 D
2979 1696 D
3040 1697 D
3188 1698 D
1472 1727 D
1332 1713 D
3944 1705 D
4063 1710 D
1110 1725 D
1737 1693 D
4003 1709 D
3702 1705 D
3530 1701 D
1781 1693 D
1617 1700 D
3751 1703 D
2214 1692 D
2613 1693 D
3409 1704 D
3259 1703 D
2582 1692 D
850 1806 D
3256 1699 D
2499 1690 D
2692 1695 D
3750 1703 D
2181 1689 D
3536 1700 D
1640 1693 D
4041 1706 D
1985 1692 D
1695 1693 D
3751 1703 D
1997 1689 D
1155 1724 D
1501 1696 D
2352 1691 D
1840 1691 D
1959 1693 D
4011 1711 D
3719 1703 D
2352 1690 D
3297 1697 D
2731 1692 D
2723 1693 D
2091 1688 D
3830 1704 D
1976 1689 D
2030 1690 D
2521 1692 D
1529 1695 D
2848 1693 D
1282 1706 D
3840 1705 D
1016 1735 D
1424 1700 D
1629 1693 D
3850 1705 D
723 2193 D
2707 1693 D
3602 1705 D
3163 1701 D
1172 1718 D
2884 1694 D
803 2119 D
1040 1773 D
3769 1706 D
1270 1708 D
1426 1705 D
2016 1692 D
3723 1707 D
2375 1691 D
3805 1707 D
4023 1709 D
3569 1706 D
1242 1709 D
2806 1697 D
2473 1694 D
3034 1694 D
3682 1702 D
1174 1717 D
3024 1694 D
1459 1698 D
969 1801 D
2462 1691 D
2264 1689 D
2215 1694 D
2063 1688 D
2860 1693 D
2890 1693 D
2067 1682 D
2102 1688 D
1746 1691 D
3792 1707 D
3905 1705 D
2448 1690 D
2315 1690 D
2866 1693 D
1575 1695 D
1156 1719 D
818 1849 D
3583 1701 D
1450 1710 D
1492 1697 D
1516 1696 D
2783 1694 D
3525 1700 D
929 1774 D
1548 1722 D
1959 1690 D
2497 1690 D
3187 1698 D
2799 1695 D
694 1557 D
1568 1696 D
2363 1691 D
807 1927 D
2720 1692 D
3942 1709 D
2906 1697 D
1533 1715 D
3476 1703 D
963 1774 D
3336 1702 D
2895 1693 D
3399 1699 D
1870 1689 D
1441 1699 D
2139 1692 D
2735 1693 D
1256 1743 D
1378 1701 D
3638 1705 D
3526 1703 D
1814 1690 D
4047 1707 D
3388 1701 D
2554 1691 D
1263 1714 D
2886 1694 D
3598 1703 D
2764 1695 D
2475 1692 D
3753 1709 D
762 2414 D
1979 1692 D
2660 1692 D
3931 1708 D
1281 1749 D
1507 1697 D
1872 1689 D
2751 1696 D
1231 1715 D
3799 1706 D
3162 1700 D
2212 1694 D
3748 1704 D
3733 1705 D
748 1979 D
1856 1691 D
1807 1692 D
3167 1701 D
3313 1699 D
1183 1718 D
1995 1691 D
2544 1691 D
2023 1689 D
2131 1691 D
3966 1705 D
3876 1706 D
1201 1715 D
2247 1691 D
3205 1696 D
1425 1700 D
1046 1736 D
2767 1693 D
957 1786 D
1590 1695 D
3468 1699 D
3078 1695 D
2024 1697 D
3331 1698 D
982 1743 D
1516 1696 D
2702 1694 D
1099 1767 D
1896 1691 D
2195 1690 D
3120 1695 D
1119 1729 D
2956 1697 D
4061 1708 D
3714 1703 D
2648 1692 D
1944 1691 D
2524 1690 D
3230 1701 D
1683 1696 D
2193 1689 D
866 1808 D
2549 1690 D
3055 1695 D
3705 1706 D
3365 1701 D
1147 1747 D
2941 1697 D
3213 1700 D
3402 1699 D
3024 1695 D
3456 1700 D
1281 1712 D
1513 1696 D
3975 1706 D
1801 1690 D
1702 1692 D
2299 1691 D
3098 1695 D
3255 1697 D
2264 1690 D
2986 1695 D
1065 1766 D
3146 1696 D
1362 1709 D
3080 1699 D
3573 1705 D
1603 1693 D
1662 1693 D
1862 1690 D
1945 1691 D
2831 1693 D
2775 1694 D
1439 1699 D
3655 1703 D
1527 1697 D
1173 1726 D
2020 1689 D
1413 1702 D
3159 1700 D
2348 1693 D
1957 1689 D
846 2070 D
2551 1691 D
2473 1690 D
1350 1703 D
863 1798 D
1475 1698 D
3676 1707 D
1604 1693 D
880 1796 D
1569 1694 D
3827 1709 D
3898 1707 D
3157 1696 D
2878 1697 D
1465 1698 D
4027 1711 D
2470 1692 D
1603 1693 D
2982 1700 D
1464 1704 D
1550 1695 D
2912 1698 D
3824 1705 D
3400 1699 D
2136 1689 D
743 1584 D
3551 1703 D
2582 1692 D
1001 1874 D
3500 1701 D
3368 1698 D
2488 1690 D
3993 1707 D
2861 1694 D
LT1
806 1823 A
1628 1717 A
806 1462 M
0 721 V
775 1462 M
62 0 V
-62 721 R
62 0 V
791 -894 R
0 856 V
-31 -856 R
62 0 V
-62 856 R
62 0 V
stroke
grestore
end
showpage
}
\put(2862,3135){\makebox(0,0)[l]{$\mu > 0 \, GeV$}}
\put(2862,3375){\makebox(0,0)[l]{$m_t = 175 \pm 5 \, GeV$}}
\put(2862,3616){\makebox(0,0)[l]{$\tan\beta =2-30$}}
\put(1669,1260){\makebox(0,0)[l]{CDF,UA2,D\O}}
\put(847,1453){\makebox(0,0)[l]{LEP}}
\put(2862,3856){\makebox(0,0)[l]{$A_0 =0-900 \, GeV$}}
\put(2862,4096){\makebox(0,0)[l]{$M_0 = 70-900\, GeV$}}
\put(2368,51){\makebox(0,0){$\vspace*{-0.2in}M_{1/2}\, (GeV)$}}
\put(100,2414){%
\special{ps: gsave currentpoint currentpoint translate
270 rotate neg exch neg exch translate}%
\makebox(0,0)[b]{\shortstack{$\hspace*{-0.3in}M_W \, (GeV)$ }}%
\special{ps: currentpoint grestore moveto}%
}
\put(4096,151){\makebox(0,0){900}}
\put(3685,151){\makebox(0,0){800}}
\put(3273,151){\makebox(0,0){700}}
\put(2862,151){\makebox(0,0){600}}
\put(2451,151){\makebox(0,0){500}}
\put(2039,151){\makebox(0,0){400}}
\put(1628,151){\makebox(0,0){300}}
\put(1217,151){\makebox(0,0){200}}
\put(806,151){\makebox(0,0){100}}
\put(540,4577){\makebox(0,0)[r]{81}}
\put(540,4096){\makebox(0,0)[r]{80.9}}
\put(540,3616){\makebox(0,0)[r]{80.8}}
\put(540,3135){\makebox(0,0)[r]{80.7}}
\put(540,2654){\makebox(0,0)[r]{80.6}}
\put(540,2174){\makebox(0,0)[r]{80.5}}
\put(540,1693){\makebox(0,0)[r]{80.4}}
\put(540,1212){\makebox(0,0)[r]{80.3}}
\put(540,732){\makebox(0,0)[r]{80.2}}
\put(540,251){\makebox(0,0)[r]{80.1}}
\end{picture}
% GNUPLOT: LaTeX picture with Postscript
\setlength{\unitlength}{0.1bp}
\special{!
%!PS-Adobe-2.0
%%Creator: gnuplot
%%DocumentFonts: Helvetica
%%BoundingBox: 50 50 914 698
%%Pages: (atend)
%%EndComments
/gnudict 40 dict def
gnudict begin
/Color false def
/Solid false def
/gnulinewidth 5.000 def
/vshift -33 def
/dl {10 mul} def
/hpt 31.5 def
/vpt 31.5 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
/DL { Color {setrgbcolor Solid {pop []} if 0 setdash }
 {pop pop pop Solid {pop []} if 0 setdash} ifelse } def
/BL { stroke gnulinewidth 2 mul setlinewidth } def
/AL { stroke gnulinewidth 2 div setlinewidth } def
/PL { stroke gnulinewidth setlinewidth } def
/LTb { BL [] 0 0 0 DL } def
/LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def
/LT0 { PL [] 0 1 0 DL } def
/LT1 { PL [4 dl 2 dl] 0 0 1 DL } def
/LT2 { PL [2 dl 3 dl] 1 0 0 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
/P { stroke [] 0 setdash
  currentlinewidth 2 div sub M
  0 currentlinewidth V stroke } def
/D { 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
  P } def
/A { stroke [] 0 setdash vpt sub M 0 vpt2 V
  currentpoint stroke M
  hpt neg vpt neg R hpt2 0 V stroke
  } def
/B { 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
  P } def
/C { 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
/T { 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
  P  } def
/S { 2 copy A C} def
end
%%EndProlog
}
\begin{picture}(4320,2777)(0,0)
\special{"
%%Page: 1 1
gnudict begin
gsave
50 50 translate
0.100 0.100 scale
0 setgray
/Helvetica findfont 100 scalefont setfont
newpath
-500.000000 -500.000000 translate
LTa
LTb
600 251 M
63 0 V
3474 0 R
-63 0 V
600 746 M
63 0 V
3474 0 R
-63 0 V
600 1241 M
63 0 V
3474 0 R
-63 0 V
600 1736 M
63 0 V
3474 0 R
-63 0 V
600 2231 M
63 0 V
3474 0 R
-63 0 V
600 2726 M
63 0 V
3474 0 R
-63 0 V
754 251 M
0 63 V
0 2412 R
0 -63 V
1523 251 M
0 63 V
0 2412 R
0 -63 V
2292 251 M
0 63 V
0 2412 R
0 -63 V
3061 251 M
0 63 V
0 2412 R
0 -63 V
3829 251 M
0 63 V
0 2412 R
0 -63 V
600 251 M
3537 0 V
0 2475 V
-3537 0 V
600 251 L
LT0
3889 2528 M
180 0 V
600 954 M
154 56 V
154 57 V
153 56 V
154 58 V
154 56 V
154 57 V
153 58 V
154 57 V
154 58 V
154 57 V
154 58 V
153 58 V
154 57 V
154 59 V
154 58 V
154 59 V
153 58 V
154 58 V
154 60 V
154 58 V
153 60 V
154 59 V
154 59 V
LT1
3889 2428 M
180 0 V
600 1054 M
154 58 V
154 60 V
153 58 V
154 60 V
154 59 V
154 59 V
153 61 V
154 60 V
154 60 V
154 61 V
154 61 V
153 62 V
154 61 V
154 61 V
154 63 V
154 62 V
153 62 V
154 63 V
154 63 V
154 64 V
153 63 V
154 63 V
154 65 V
LT2
3949 2328 D
1523 1508 D
2292 1290 D
3889 2328 M
180 0 V
-180 31 R
0 -62 V
180 62 R
0 -62 V
1523 766 M
0 1485 V
1492 766 M
62 0 V
-62 1485 R
62 0 V
2292 409 M
0 1763 V
2261 409 M
62 0 V
-62 1763 R
62 0 V
stroke
grestore
end
showpage
}
\put(3829,2328){\makebox(0,0)[r]{exp}}
\put(3829,2428){\makebox(0,0)[r]{$A_0=M_0=M_{1/2}=200 \, GeV$}}
\put(3829,2528){\makebox(0,0)[r]{$A_0=M_0=M_{1/2}=600 \, GeV$}}
\put(2369,2132){\makebox(0,0)[l]{{\small CDF,UA2,D\O}}}
\put(1600,2231){\makebox(0,0)[l]{{\small LEP}}}
\put(2753,746){\makebox(0,0)[l]{$\tan\beta =4 \,,\, \mu > 0 \, GeV$}}
\put(2368,51){\makebox(0,0){$\vspace*{-0.2in}m_t\, (GeV)$}}
\put(100,1488){%
\special{ps: gsave currentpoint currentpoint translate
270 rotate neg exch neg exch translate}%
\makebox(0,0)[b]{\shortstack{$\hspace*{-0.2in}M_W$}}%
\special{ps: currentpoint grestore moveto}%
}
\put(3829,151){\makebox(0,0){190}}
\put(3061,151){\makebox(0,0){185}}
\put(2292,151){\makebox(0,0){180}}
\put(1523,151){\makebox(0,0){175}}
\put(754,151){\makebox(0,0){170}}
\put(540,2726){\makebox(0,0)[r]{80.55}}
\put(540,2231){\makebox(0,0)[r]{80.5}}
\put(540,1736){\makebox(0,0)[r]{80.45}}
\put(540,1241){\makebox(0,0)[r]{80.4}}
\put(540,746){\makebox(0,0)[r]{80.35}}
\put(540,251){\makebox(0,0)[r]{80.3}}
\end{picture}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentstyle[psfig,aps,preprint]{revtex}
\topmargin -1.5cm
\textheight=22.5cm
\textwidth=16.5cm
\setlength{\oddsidemargin}{-.3cm}
\baselineskip=18pt
\parskip=4pt
\tightenlines
\pssilent
\newcommand{\th}{\Theta}
\newcommand{\pslash}{p\!\!\!/}
\newcommand{\hs}{\hat{s}^2}
\newcommand{\hc}{\hat{c}^2}
\newcommand{\ori}{\hspace{.1in}}
\newcommand{\fp}{f^{\prime}}
\newcommand{\ti}{\tilde}
%%%%%%%% Definition of \slash %%%%%%%%%%%%%%%
\makeatletter
\def\slash{\@ifnextchar[{\fmsl@sh}{\fmsl@sh[0mu]}}
\def\fmsl@sh[#1]#2{%
  \mathchoice
    {\@fmsl@sh\displaystyle{#1}{#2}}%
    {\@fmsl@sh\textstyle{#1}{#2}}%
    {\@fmsl@sh\scriptstyle{#1}{#2}}%
    {\@fmsl@sh\scriptscriptstyle{#1}{#2}}}
\def\@fmsl@sh#1#2#3{\m@th\ooalign{$\hfil#1\mkern#2/\hfil$\crcr$#1#3$}}
\makeatother
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\lsim{\mathrel{\rlap {\raise.5ex\hbox{$ < $}}
{\lower.5ex\hbox{$\sim$}}}}
\def\gsim{\mathrel{\rlap {\raise.5ex\hbox{$ > $}}
{\lower.5ex\hbox{$\sim$}}}}
\def\Im{{\rm Im}\, }
\def\Re{{\rm Re}\, }
%\def\np#1#2#3{{\it {Nucl. Phys.}} {\bf{B#1}} (#2) #3}
\def\np#1#2#3{{\it {Nucl. Phys.}} {\bf{B#1,}} #2 (#3)}
%\def\pl#1#2#3{{\it {Phys. Lett.}} {\bf{B#1}} (#2) #3}
\def\pl#1#2#3{{\it {Phys. Lett.}} {\bf{B#1,}} #2 (#3)}
%\def\prl#1#2#3{{\it {Phys. Rev. Lett. }}{\bf{#1}} (#2) #3}
\def\prl#1#2#3{{\it {Phys. Rev. Lett. }}{\bf{#1,}} #2 (#3)}
%\def\pr#1#2#3{{\it {Phys. Rev.}} {\bf{D#1}} (#2) #3}
\def\pr#1#2#3{{\it {Phys. Rev.}} {\bf{D#1,}} #2 (#3)}
%\def\prep#1#2#3{{\it {Phys. Rep.}} {\bf{#1}} (#2) #3}
\def\prep#1#2#3{{\it {Phys. Rep.}} {\bf{#1,}} #2 (#3)}
%\def\zp#1#2#3{{\it {Z. Phys.}} {\bf{C#1}} (#2) #3}
\def\zp#1#2#3{{\it {Z. Phys.}} {\bf{C#1,}} #2 (#3)}
\def\nl{\hfil\break}
\newcommand{\mpl}[2]{{\em Mod. Phys. Lett.}     {\bf A#1}, #2 }
%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\begin{titlepage}
\begin{flushright}

{\hskip.5cm}\\
\end{flushright}
\begin{centering}
\vspace{.3in}
%
{\bf  THE EFFECTIVE WEAK MIXING ANGLE IN THE MSSM
}\\
\vspace{2 cm}
{A. DEDES$^{1}$,
A. B. LAHANAS$^2$ and K. TAMVAKIS$^1$}\\
\vskip 1cm
{\it $^1$Division of Theoretical Physics,}\\
{\it University of Ioannina, GR-45110, Greece}\\ \vspace{0.5cm}
{\it $^2$Physics Department, Nuclear and Particle Physics Section
University of Athens, Athens 157 71, Greece}\\ \vspace{0.5cm}


\vspace{1.5cm}
\begin{abstract}
The predictions of the MSSM are discussed in the light of recent LEP and SLD 
precision data. The full supersymmetric one loop corrections to the effective 
weak mixing angle, experimentally determined in LEP and SLD experiments, are 
considered. It is demonstrated, both analytically and numerically, that,
 potentially 
dangerous, large logarithmic sparticle corrections are cancelled. The relative 
difference factor $\Delta k$ between the mixing angle defined as a ratio of
 couplings and the experimentally obtained angle is discussed. It is found 
 that $\Delta k$ is dominated by the oblique corrections, while the non-oblique 
 overall supersymmetric EW and SQCD corrections are negligible. The comparison 
 of the MSSM with radiative electroweak symmetry breaking to the LEP+SLD
 precision 
 data indicates that rather large values of the soft breaking parameter $M_{1/2}$ 
 in the region greater than 500 GeV are preferred.
\end{abstract}

\end{centering}
\vspace{.1in}








\vspace{1cm}
\begin{flushleft}

January 1998\\
\end{flushleft}
\hrule width 6.7cm \vskip.1mm{\small \small}
%
E-mails\,:\,adedes@cc.uoi.gr,
 alahanas@atlas.uoa.gr, tamvakis@cc.uoi.gr
%

\end{titlepage}





















\section { INTRODUCTION}

The electroweak mixing angle $\sin^2\theta_{W}$ , defined as a ratio of 
gauge couplings, provides a convenient means to test unification in
unified extensions of the Standard Model (SM) \cite{Lang}. This
quantity is not directly measured in experiments. Instead, LEP and SLD
studies employ an effective coupling  $\sin^2\theta^{lepton}_{eff}$ 
determined from on resonance asymmetries whose value is known with excellent
accuracy  \cite{LEP,altarelli,sirlin,gambino}.
 This effective mixing angle has been studied in detail in the
context of the SM at the one loop level in various renormalization schemes
with the dominant two loop heavy top contributions and three loop QCD
effects taken into account \cite{sirlin,gambino,all1,all2,djoua}.
Due to large cancellations between fermion and
boson contributions occurring at the one loop level,
in the $\overline{MS}$ scheme, these are the dominant
contributions to the difference 
 $ \sin^2\theta^{lepton}_{eff} -  \sin^2\theta_{W} \approx {\cal{O}}
  {(10^{-4})} $ which is less than the error quoted by the experimental
groups. Therefore, although conceptually different the two angles are
very close numerically.
The mixing angle  is sensitive on the values of the Higgs mass
$M_{H}$ and top mass $m_{t}$  through the quantities $\Delta {r_{W}}$
and $\Delta {\rho}$ and carries an uncertainty of about $.1 \%$
from its dependence on the electromagnetic coupling ${\alpha} (M_{Z}) $.
>From the predictions of $ \sin^2\theta^{lepton}_{eff} $
and $\Delta {\rho}$ one can draw useful theoretical conclusions concerning
the Higgs and W - boson masses having as inputs the Z - boson mass, the
value of the fine structure constant and the Fermi coupling constant which are
experimentally known to a high degree of accuracy.

In the framework of supersymmetric extensions of the SM \cite{NHK}
 the situation changes
since $\sin^2\theta_{W}$ as well as $ \sin^2\theta^{lepton}_{eff} $
receive contributions from the superparticles in addition to ordinary
particles. Coupling unification at the GUT scale in conjunction with
experimental data for the strong coupling constant at $M_{Z}$ and radiative
breaking of the Electroweak Symmetry impose stringent constraints on the
extracted value for $\sin^2\theta_{W}$. However $\sin^2\theta_{W}$
is plagued by large logarithms  $\log(M_Z/M_S)$, where $M_S$ is the
effective supersymmetry breaking scale\footnote{See 
for instance P. Chankowski, Z. Plucienic and S. Pokorski in 
ref.\cite{polon}.}. Unlike $\sin^2\theta_{W}$ the
experimentally determined $ \sin^2\theta^{lepton}_{eff} $  is not plagued
by such potentially dangerous large logarithms due to decoupling.
Therefore, the difference of the two angles is not numerically small
any more and $\sin^2\theta_{W}$ cannot be directly used for comparison
with experimental data. Thus, in supersymmetric theories the precise relation
between the two angles is highly demanded.
The non-decoupled supersymmetric corrections to $\sin^2\theta^{lepton}_{eff}$
are expected to be small of order $(M_Z/M_S)^2$.
However small these contributions may be,
they are of particular importance, since the
experimental accuracy is very high, and these corrections can be larger than
the SM corrections occurring beyond the one loop order.
Moreover the effect of the one loop supersymmetric corrections
may not be necessarily suppressed in some sectors, such as the
neutralino and chargino sectors, which are
characterized by a relatively small effective supersymmetry breaking scale for
particular inputs of the soft SUSY breaking parameters.
Motivated by this we undertake a complete one loop
study of the supersymmetric corrections to the effective
mixing angle in the context of the MSSM which is the simplest supersymmetric
extension of the Standard Theory.

Although there are several studies \cite{polon} in literature concerning
the value of the weak mixing angle $\sin^2\theta_{W}$ in the MSSM and other
unified supersymmetric extensions of the SM,
only a few have tackled the
problem of calculating the complete supersymmetric corrections
${\cal {O}} {(M_Z/M_S)^2}$ to
the experimentally measured angle $ \sin^2\theta^{lepton}_{eff} $.
 In ref.\cite{Bagger}  the effective mixing angle is calculated in 
particular cases and the 
decoupling of large logarithms is numerically shown.
In that calculation all the one-loop corrections, including the non-universal
supersymmetric vertex and external fermion corrections , for the leptonic
effective mixing angle were considered. The non-universal corrections were
found to be small.
In other studies \cite{Hollik1,finnell,kolda},
the serious constraints 
imposed by unification and radiative electroweak symmetry breaking
\cite{Tamvakis} have not been considered. Instead the MSSM parameters are
considered as free parameters chosen in the optimal way to improve
the observed deficiencies of the SM in describing the data.
 
In the present article we show explicitly how the cancellation 
of potentially dangerous logs takes place and perform a systematic
numerical study by scanning the entire parameter space having as our main
outputs the effective weak mixing angle, the values of the on Z - resonance
asymmetries measured in experiments, as well as the value of the strong
coupling constant at $M_Z$. In each case we also give the theoretical
prediction for the W - boson mass through its relation to the parameter rho
and the weak mixing angle.

It is perhaps worth noting that non-universal
corrections, claimed to be small, are dominated by large logs.
These logs cancel at the end, as expected. Nevertheless, their presence
dictates that non-decoupled terms of order $(M_Z/M_S)^2$ may be of the
same order of magnitude as the corresponding terms stemming from
the universal corrections and cannot be a priori omitted.
Knowing from other studies that universal corrections tend to
decrease the value of the effective mixing angle by almost six
standard deviations from the experimental central value it is
important to see what is the effect of the non-universal
contributions. We take into account
all constraints from unification and radiative
EW symmetry breaking. These constraints, along with the
experimental bounds for the strong coupling constant and
$\sin^2\theta^{lept}_{eff}$, may restrict further the allowed parameter space. 

\section{FORMULATION OF THE PROBLEM}

The value of the weak mixing angle, defined as the ratio of
the gauge couplings, is  
%%%%%%%%%%%%%%
\begin{equation}
\hs(Q) = \frac{\hat{g}^{\prime 2}(Q)}{\hat{g}^2(Q)+\hat{g}^{\prime 2}(Q)}  \;,
\label{sinratio}
\end{equation}
%%%%%%%%%%%%%%%%%
where $\hat{g}$ and $\hat{g}^\prime$ are the $SU(2)$ and $U(1)_Y$
gauge couplings. Throughout this paper the hat refers to 
renormalized quantities in the modified $\overline{DR}$ scheme
\cite{Siegel,Martin}.
These couplings are running in the sense that they
depend on the scale Q. Particularly for the electroweak processes,
Q is chosen to be $M_Z$. There are many sources for the determination
of the $\hs$. From muon decay, for instance, and knowing that 
$M_Z=91.1867 \pm 0.0020 GeV$, $\alpha_{EM}=1/137.036$ and 
$G_F=1.16639(1)\times10^{-5}\, GeV^{-2}$, we get in the ($\overline{DR}$)
scheme
%%%%%%%%%%%%%%
\begin{equation}
\hs \hc=\frac{\pi\, \alpha_{EM}\, 
}{\sqrt{2}\,M_Z^2\,G_F\,
(1-\Delta \hat{\alpha})\,\hat{\rho}\,(1-\Delta \hat{r}_W)}\; ,
\label{sindr}
\end{equation}
%%%%%%%%%%%%%%%%%
where 
%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\hat{\rho}^{-1}&=&1-\Delta \hat{\rho}=
1-\frac{\Pi_{ZZ}(M_Z^2)}{M_Z^2}+\frac{\Pi_{WW}(M_W^2)}
{M_W^2}\label{ro}\;,\\[3mm]
\Delta \hat {r}_W&=&\frac{\Pi_{WW}(0)-\Pi_{WW}(M_W^2)}{M_W^2}+
\hat{\delta}_{VB}\label{drw}\;,\\[3mm]
\hat{\alpha}&=&\frac{\alpha_{EM}}{1-\Delta{\hat{\alpha}}}
\label{da}\;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%
$\Pi$'s are the transverse gauge bosons self energies 
evaluated in the $\overline{DR}$ scheme.
Explicit forms for these self energies can be obtained from
ref.\cite{Bagger}.
The weak mixing angle obtained from (\ref{sindr})
although it plays a crucial role in the analysis of
grand unification, it is {\it not}
an experimental quantity. Actually, it is obtained after
fitting experimental observations with
$\alpha_{EM}$ and $G_F$ as accurately known
parameters (for more details {\it see} ref.\cite{Peskin}).
The radiative corrections on $\hs$ involve two subtleties:
{\it i}) the renormalization scheme dependence\footnote{
We are working on the modified $\overline{DR}$ scheme of ref.\cite{Martin}
which preserves supersymmetry up to two-loops.}
and {\it ii}) the 
dependence on the mass of the top quark, Higgs masses and superparticle
masses which depend on the 
supersymmetric breaking parameters $M_{1/2}$, $M_0$, 
and $A_0$.

%\vspace*{0.4in}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\centerline{\input{sinhat.ps}}
%\begin{center}
%\footnotesize{{\bf Figure 1:}
% The values of the running weak mixing angle $\hat{s}^2$ at
%$M_Z$ in the $\overline{DR}$ scheme,
% as it is obtained from the ratio of gauge couplings for various
%input universal soft gaugino masses $M_{1/2}$
% which are taken at $M_{GUT}$.
%$\hat{s}^2(M_Z)$ starts strongly increasing
% for $M_{1/2}>>M_Z$ or $M_{1/2}<<M_Z$
%due to the non-decoupling of SUSY-particles
% from its expression (\ref{sindr}).}
%\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%


 As we can see from  Fig.1, 
$\hs$ takes large values
when we increase  the masses of the soft breaking parameters.
In other words, the soft breaking parameters do not decouple
from $\hs$. This
is due to the fact that the net effect of the contributions
(\ref{ro}),(\ref{drw}),(\ref{da}) to (\ref{sindr}),
 contains large logarithms
of the form $\log(\frac{M_{SUSY}}{M_Z})$. 
On the other hand, the LEP collaborations \cite{LEP} employ an
{\it effective weak mixing angle} $\sin^2\theta^{f}_{eff}
\equiv s^2_f$, first introduced
by the authors of ref.\cite{sirlin}, which is not plagued by large logarithms
due to decoupling.
It is a common belief among
GUT theorists that these two angles $\hs$ and $s_f^2$, although
different conceptually,  are very close numerically \cite{gambino}. 
Nevertheless, this is not true in the MSSM since there are 
large logarithmic dependencies of the weak mixing angle $\hs$.

The tree level Lagrangian associated with the $Zf\overline f$
can be written in the form
%%%%%%%%%%%%%%
\begin{equation}
{\cal L}_{tree}^{Zf\overline f}=\frac{\hat{e}}{2\hat{c}\hat{s}}
Z_\mu \overline{f} \gamma^\mu \left [ \left ( T_3^f -2 \hs Q^f 
\right ) - \gamma_5 T_3^f \right ] f \;,
\label{ltree}
\end{equation}
%%%%%%%%%%%%
where $Q^f$ is the electric charge and $T_3^f$ is the third component
of isospin of the fermions $f$. Electroweak corrections in (\ref{ltree})
yield the effective Lagrangian
%%%%%%%%%%%%%%
\begin{equation}
{\cal L}_{eff}^{Zf\overline f}=(\,\sqrt{2}\, G_F 
\, M_Z\,) \, \rho_f^{1/2} Z_\mu \overline{f} \gamma^\mu 
\left [ \left ( T_3^f -2 \hs \hat{k}_f Q^f \right )
 - \gamma_5 T_3^f \right ] f \;,
\label{leff}
\end{equation}
%%%%%%%%%%%%%%%%
which is relevant to study Neutral Current processes on the Z - resonance.
Then, the effective weak mixing angle is simply defined from (\ref{leff}) as
%%%%%%%%%%%%%%
\begin{equation}
s_f^2 \equiv \hs \hat{k}_f = \hs ( 1 + \Delta \hat{k}_f )\;.
\label{sineff}
\end{equation}
%%%%%%%%%%%%%%%%%
The angle $s_f^2$ can be compared directly
with experiment while $\hs$ can be predicted from a Grand
Unification analysis. The LEP and SLD average gives the value
$0.23152\pm 0.00023$ \cite{altarelli} for the  $s_l^2 \equiv
sin^2\theta^{lept}_{eff}$. Since 
%%%%%%%%%%%%%%%%%
\begin{eqnarray}
c_f^2 = \hc \left ( 1 - \frac{\hs}{\hc} \Delta k_f \right )\;,
\nonumber 
\end{eqnarray}
%%%%%%%%%%%%%
one obtains by making use of equations (\ref{sindr}) and 
(\ref{sineff})
%%%%%%%%%%%%%%
\begin{equation}
s^2_{f}c^2_{f}=\frac{\pi\, \alpha_{EM}\, (1+\Delta \hat{k}_f)\,
(1-\frac {\hs}{\hc}\,\Delta  \hat{k}_f)}{\sqrt{2}\,M_Z^2\,G_F\,
(1-\Delta \hat{\alpha})\,\hat{\rho}\,(1-\Delta \hat{r}_W)}\;,
\label{sef}
\end{equation}
%%%%%%%%%%%%%%%%%
where 
%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\Delta \hat{k}_f&=&\frac{\hat{c}}{\hat{s}}\,\frac{\Pi_{Z\gamma}(M_Z^2)-
\Pi_{Z\gamma}(0)}{M_Z^2}+ \frac{\hat{\alpha} \hc}{\pi \hs}\log(\frac{M_W^2}
{M_Z^2}) - \frac{\hat{\alpha}}{4\pi \hs} V_f (M_Z^2) + \delta k_f^{SUSY} \;.
\label{dkapaf}
\end{eqnarray}
%%%%%%%%%%%%%%%
The function $V_f(M_Z^2)$ can be obtained from 
ref.\cite{sirlin}\footnote{In the case f=bottom,
 the important top quark corrections
to $Zb\bar{b}$ vertex should be added to $V_f$.}. 
$\delta k_f^{SUSY}$ denotes the non-universal supersymmetric self energies
and vertex corrections to $s_f^2$. 

In order to study MSSM (or SM) corrections to $s_f^2$, we
need calculate first the $Z$ and $W$ gauge boson self
energy corrections which contribute to 
$\hat\rho$ and $\Delta \hat r_W$. Our expressions agree with those of
ref.\cite{Bagger}\footnote{To match our conventions with those of ref.$\cite{Bagger}$ we have to
replace their matrices by the following: $N\rightarrow {\cal{O}}^T$, and
$U\rightarrow U^{*}$.} and \cite{Rociek}.  
We need also calculate the $Z-\gamma$ propagator corrections, 
the wave function renormalization of external fermions as well as the 
$Z \overline{f} f$ vertex corrections which contribute to $\Delta \hat k_f$.
The supersymmetric contributions to last two were found to be negligible,
 for the leptonic case,
in the minimal supergravity model studied in ref.\cite{Bagger}.
Including all these corrections in (\ref{sef}), we expect that
the effective weak mixing angle
$s_f^2$, does not suffer from potentially large logarithms,
$\sim log(\frac{M^2_{SUSY}}{M_Z^2})$.

At this point we should say that when the electroweak symmetry
is broken by radiative corrections, the value of the parameter $\mu$, which
specifies the mixing of the two Higgs multiplets within the superpotential,
turns out to be  of the order of the supersymmetry breaking scale
in most of the parameter space. Under these circumstances it is not only
the large logarithms $log(\frac{M^2_{SUSY}}{M_Z^2})$ which should be
cancelled but also logarithms involving the parameter $\mu$.


\section{Decoupling of $log(M_{SUSY}/M_Z)$ in the effective mixing angle }

In this section we will first show how the potentially dangerous 
$\sim log(\frac{M_{1,2}}{M_Z}), log(\frac{\mu}{M_Z})$
from the contributions of the neutralinos and charginos are cancelled in
the expression for ${\sin^2}\theta^{f}_{eff}$ when the soft SUSY breaking
parameter $M_{1/2}$ is large ($M_{SUSY}^2>>M_Z^2$).

There are three sources of large logarithms which affect the value of the
weak mixing angle $\sin^2\theta^{f}_{eff}$  :\\
i)   Gauge boson self energies which feed large logs to the quantities
$\Delta \hat r_W$, $\hat\rho$ and $\Delta \hat k_f$.  \\
ii)  Vertex, external wave function renormalizations and box corrections
to muon decay which affect $\Delta \hat r_W$ through $\delta_{VB}^{SUSY}$. \\
iii) Non-universal vertex and external fermion corrections to 
$Z \overline{f} f$ coupling which affects $\Delta \hat k_f$. \\
We shall see that the corrections
 (i) are cancelled against large logs stemming from the
electromagnetic coupling ${\hat{\alpha}}(M_Z)$. The rest, (ii) and (iii),
are cancelled against themselves.

In order to prove the cancellation of the large $log(M_{SUSY}/M_Z)$ terms
among the dimensionless quantities $\Delta \hat r_W$, $\hat\rho$,
$\Delta \hat k_f$ and ${\hat{\alpha}}(M_Z)$, through which
$\sin^2\theta^{f}_{eff}$ is defined, it suffices
to ignore the electroweak
symmetry breaking effects {\it e.g} $<H_1^o>=<H_2^o>=0$. In this
case the masses of charginos and neutralinos take
the simple form
%%%%%%%%%%%%%
\begin{eqnarray}
m_{{\chi}_i^o}&=&M_1\,\,\,,\,\,\,M_2\,\,\,,\,\,\,\mu\,\,\,,\,\,\,-\mu \;,\\
m_{{\chi}_i^{+}}&=&M_2\,\,\,,\,\,\,\mu\;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%

\vspace*{1cm}
{\bf{i) \underline{Vector boson self energy corrections }}}

\vspace*{.4cm}
 The contributions  from the chargino/neutralino sector to the
vector bosons self energies are
%%%%%%%%%%%%%
%%%%%%%%%%%%%
\begin{eqnarray}
\Pi_{ZZ}^{\chi_i^o/\chi_i^+}&=&\frac{\hat{g}^2}{16\pi^2}
\Biggl \{\frac{1}{2}
H(\mu,\mu)\left [\frac{1}{\hat{c}^2}+4\left (\hat{c}-\frac{1}{2\hat{c}}
\right )^2\right ]+
\mu^2B_0(\mu,\mu)\left [\frac{1}{\hat{c}^2}+4\left 
(\hat{c}-\frac{1}{2\hat{c}}\right )^2 \right ] 
\nonumber \\[2mm] &+& 
2 \hat{c}^2H(M_2,M_2)+4\hat{c}^2M_2^2B_0(M_2,M_2)\Biggr \}\;,\\[5mm]
\Pi_{WW}^{\chi_i^o/\chi_i^+}&=&\frac{\hat{g}^2}{16\pi^2}\Biggl [H(\mu,\mu)+
2\mu^2B_0(\mu,\mu)+2H(M_2,M_2)+4M_2^2B_0(M_2,M_2)\Biggr ]\;,\\[5mm]
\Pi_{Z\gamma}^{\chi_i^o/\chi_i^+}&=&\frac{ \hat{e}
 \hat{g} \hat{c}_{2\theta_W}}
{16\pi^2\hat{c}}
\Biggl [4\tilde{B}_{22}(\mu,\mu)+p^2B_0(\mu,\mu)\Biggr ]\nonumber\\[2mm] &+&
\frac{2 \hat{e}
 \hat{g} \hat{c}}{16\pi^2}\Biggl [4\tilde{B}_{22}(M_2,M_2)+p^2
B_0(M_2,M_2)\Biggr ]\;,
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%
where $\hat{g}=\frac{\hat{e}
}{\hat{s}}$ is the running $\overline{DR}$ $SU(2)$ gauge coupling.

In order to calculate the dependence of $s_f^2$ on $M_{1,2} /\mu $ 
we make use of eqs.\ref{ro},\ref{drw},\ref{da} and
reduce all functions appearing in the expressions for the two point
functions above in terms of the basic integrals $A_0, B_0$ (see Appendix B).
Isolating the logarithmic dependencies on $M_{1,2} /\mu $ we find that,
%%%%%%%%%%%%%
\begin{eqnarray}
\Delta
 \hat{r}_W&=&\frac{\hat{\alpha}}{4\pi}
\frac{2}{3\hs}\Biggl [1+2\log\biggl (\frac{M_2^2}{Q^2}\biggr )+
\log\biggl(\frac{\mu^2}
{Q^2}\biggr ) \Biggr]
\;,\\[3mm]
\Delta
 \hat{\rho}&=&\frac{\hat{\alpha}}{4\pi}
\frac{2}{9\hc}
\Biggl [1+2 \hat{c}_{2\theta_W}+6\hc \log\biggl (\frac{M_2^2}{Q^2}\biggr )
%\nonumber \\[1.5mm] &+&3
+3 \hat{c}_{2\theta_W}\log\biggl (\frac{\mu^2}{Q^2}\biggr )\Biggr ]
\;,\\[3mm]
\Delta
 \hat{k}_f&=&-\frac{\hat{\alpha}}{4\pi}
\frac{2 \cot\theta_W}{9\hs}\Biggl [\hat{s}_{2\theta_W}+
\hat{c}_{2\theta_W}\tan\theta_W\nonumber \\[1.5mm] &+&
3\hat{s}_{2\theta_W}
\log\biggl (\frac{M_2^2}{Q^2}\biggr )+3\hat{c}_{2 \theta_W}\tan\theta_W
\log\biggl (\frac{\mu^2}{Q^2}\biggr )\Biggr ]\;,\\[3mm]
\Delta
 \hat{\alpha}&=&-\frac{\hat{\alpha}}{3\pi}\Biggl [\log\biggl (\frac{M_2^2}
{Q^2}\biggr )+\log\biggl (\frac{\mu^2}{Q^2}\biggr )\Biggr ]\;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%
The angle ${\hat{\theta}}_W$ is the weak mixing angle defined through
 ratios of
couplings in the $\overline{DR}$ scheme and 
$\theta_W$ is the on shell mixing angle defined by
$ \sin^2\theta_W = 1-M_W^2/M^2_Z$. In the equations above
$\hat{c}_{2\theta_W} \equiv \cos(2{\hat{\theta}}_W)$,
$\hat{s}_{2\theta_W} \equiv \sin(2{\hat{\theta}}_W)$
with similar definitions for ${c}_{2\theta_W}, {s}_{2\theta_W}$.

Plugging in all that into (\ref{sef}),
 we find that $s^2_{eff}c^2_{eff}$ is corrected
as 
\begin{equation}
\Delta (s^2_{eff}c^2_{eff})=\frac{\pi\, \alpha_{EM}}
{\sqrt{2}\,M_Z^2\,G_F}\,\,\frac{8}{9}(\frac{\hat{\alpha}}{4\pi})\;,
%\delta_{\hat{s}_o^2\hat{c}_o^2}^{M_2}
%=\frac{4}{9}(\frac{\hat{\alpha}}{4\pi})\,\,\,+``non-oblique''
\end{equation}
%%%%%%%%%%%%%%%%%%
which at one loop order is independent of large logs.
It must be noted that this
result is also independent of the sign of $\mu$.
However this finite correction vanishes when the next to
leading terms in the expansion of $B_0$ are considered.


%\subsection{Vertex+Box Corrections}
\vspace*{1cm}
{\bf{ii) \underline{Vertex and Box Corrections from muon decay }}}

\vspace*{.4cm}
The non-universal contribution to $\Delta\hat{r}_W$, which
contains vertex and box as well as external wave
function renormalization corrections, is divided into two parts
%%%%%%%%%%%%
\begin{equation}
\delta_{VB}=\delta_{VB}^{SM}+\delta_{VB}^{SUSY}\;.
\end{equation}
%%%%%%%%%%%%%%%%
The Standard model part appears in ref. $\cite{sirlin} $.  
The supersymmetric contributions can
be found in refs. $\cite{Bagger}$,$\cite{Pokorski}$. We reproduce the
results of ref. $\cite{Bagger}$ for the wave - function and vertex corrections
here,
%%%%%%%%%%%%
\begin{equation}
\delta_{VB}^{SUSY}=-\frac{\hs\hc}{2\pi \hat{\alpha}}M_Z^2{\cal R}e a_1
+(\delta \upsilon_e+\frac{1}{2}\delta Z_e+ \frac{1}{2}\delta Z_{\nu_e})+
(\delta \upsilon_{\mu}+\frac{1}{2}\delta Z_{\mu}+
\frac{1}{2}\delta Z_{\nu_{\mu}})\;,
\label{dvb}
\end{equation}
%%%%%%%%%%%%%%%%
where the wave-function and vertex corrections are
\begin{equation}
16\pi^2~\delta Z_{\nu_e} \ =\ -\ \sum_{i=1}^2
\left|a_{\tilde\chi_i^+\nu_e\tilde e_L}\right|^2
B_1(0,m_{\tilde\chi^+_i},m_{\tilde e_L}) - \sum_{j=1}^4
\left|a_{\tilde\chi_j^0\nu_e\tilde\nu_e}\right|^2
B_1(0,m_{\tilde\chi^0_j},m_{\tilde\nu_e})~,
\end{equation}
%
\begin{equation}
16\pi^2~\delta Z_e\ =\ -\ \sum_{i=1}^2
\left|a_{\tilde\chi_i^+e\tilde\nu_e}\right|^2
B_1(0,m_{\tilde\chi^+_i},m_{\tilde\nu_e}) \ -\ \sum_{j=1}^4
\left|a_{\tilde\chi_j^0e\tilde e_L}\right|^2
B_1(0,m_{\tilde\chi^0_j},m_{\tilde e_L})~,
\end{equation}
%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
16\pi^2~\delta v_e &=& \sum_{i=1}^2\sum_{j=1}^4
a_{\tilde\chi_i^+\nu_e\tilde e_L}a^*_{\tilde\chi_j^0e\tilde e_L}
\ \Biggl\{ - \ {\sqrt{2}\over g}a_{\tilde\chi_j^0\tilde\chi_i^+W}
m_{\tilde\chi_i^+} m_{\tilde\chi_j^0} \,C_0(m_{\tilde
e_L},m_{\tilde\chi_i^+},m_{\tilde\chi_j^0}) \nonumber\\
&&\qquad\qquad+ \ {1\over\sqrt2g}b_{\tilde\chi_j^0\tilde\chi_i^+W}
\ \biggl[\,B_0(0,m_{\tilde\chi_i^+},m_{\tilde\chi_j^0}) + m_{\tilde
e_L}^2\,C_0(m_{\tilde e_L},m_{\tilde\chi_i^+},m_{\tilde\chi_j^0}) -
{1\over2}\,\biggr] \Biggr\}\nonumber\\ &-& \sum_{i=1}^2\sum_{j=1}^4
a_{\tilde\chi_i^+e\tilde\nu_e}a_{\tilde\chi_j^0\nu_e\tilde\nu_e}
\ \Biggl\{ - \ {\sqrt2\over g}b_{\tilde\chi_j^0\tilde\chi_i^+W}
m_{\tilde\chi_i^+} m_{\tilde\chi_j^0} \,C_0(m_{\tilde\nu_e},
m_{\tilde\chi_i^+}, m_{\tilde\chi_j^0}) \nonumber\\ &&\qquad\qquad+
\ {1\over\sqrt2g} a_{\tilde\chi_j^0\tilde\chi_i^+W}
\ \biggl[\,B_0(0,m_{\tilde\chi_i^+},m_{\tilde\chi_j^0}) +
m_{\tilde\nu_e}^2 \, C_0(m_{\tilde\nu_e}, m_{\tilde\chi_i^+},
m_{\tilde\chi_j^0}) - {1\over2} \,\biggr]\Biggr\} \nonumber\\ &+&
{1\over2}\sum_{j=1}^4 a_{\tilde\chi_j^0e\tilde e_L}^*
a_{\tilde\chi_j^0\nu_e\tilde\nu_e} \ \biggl[\,B_0(0,m_{\tilde
e_L},m_{\tilde\nu_e}) +
m_{\tilde\chi_j^0}^2\,C_0(m_{\tilde\chi_j^0},m_{\tilde
e_L},m_{\tilde\nu_e}) + {1\over2} \,\biggr]~,
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%
and the non-vanishing couplings are given by
{\footnote{ To conform with the notation of ref. \cite{Bagger}
we use the couplings
$a_{\tilde\chi_a^0\tilde\chi_i^+W} \equiv  g {\cal P}^L_{a i}$,
$b_{\tilde\chi_a^0\tilde\chi_i^+W} \equiv  g {\cal P}^R_{a i} $. 
$ {\cal P}^L_{a i}$ and ${\cal P}^R_{a i}$ are given in the
Appendix A ( see Eqs. A.13 ).
Also the  lepton, slepton, chargino (or neutralino) couplings in the
equations (25-28) differ in sign
from those given in A.20.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
a_{\tilde{\chi}^{+}_1\nu_e\tilde{e}_L} &=&
a_{\tilde{\chi}^{+}_1 e \tilde{\nu}_e}=\frac{\hat{e}}{\hat{s}}\;, \\
a_{\tilde{\chi}^{0}_1\nu_e\tilde{\nu}_e}&=&
a_{\tilde{\chi}^{0}_1 e \tilde{e}_L}=-\frac{\hat{e}}{\sqrt{2}
\hat{c}}\;,\\
a_{\tilde{\chi}^{0}_2\nu_e\tilde{\nu}_e}&=&
- \,
a_{\tilde{\chi}^{0}_2e \tilde{e}_L}=\frac{\hat{e}}{\sqrt{2}\hat{s}}\;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%
In all expressions above the functions $B_{0,1}, C_0$ are considered with 
vanishing momenta squared and their analytic expressions in terms of
the masses involved are given in Appendix B. 
%{\footnote{our $B_1$ differs
%in sign from that used in ref. \cite{Bagger}.}.
We recall that we have ignored
EW symmetry breaking effects so that $m_{{\tilde\chi_1}^0}=M_1$,
$m_{{\tilde\chi_2}^0}=M_2$ and $m_{\tilde{\nu}_e}=m_{\tilde{e}_L}=
M_{\tilde{L}}$. We have compared these results with
those of Ref.$\cite{Pokorski}$ and we have found agreement.
Dangerous large log corrections are contained only in the second and
third part of the eq.(\ref{dvb}). For these terms we obtain, 
%%%%%%%%%%%%%%%%%
\begin{eqnarray}
& &\left ( \delta \upsilon_e +\frac{1}{2}\delta Z_e+
 \frac{1}{2}\delta Z_{\nu_e} \right )
= -\frac{1}{\hs}\, \left (\frac{\hat{\alpha}}{4\pi} \right )
\Biggl \{ 2 M_2^2 C_0(M_{\tilde{L}},M_2,M_2) 
\nonumber \\[2mm] &-&  {M_{\tilde{L}}^2} C_0(M_{\tilde{L}},M_2,M_2)
+\frac{1}{4} M_2^2 C_0(M_2,M_{\tilde{L}},M_{\tilde{L}})
-\frac{1}{4} M_1^2 \frac{\hs}{\hc} C_0(M_1,M_{\tilde{L}},M_{\tilde{L}})
\nonumber \\[2mm] &-& 
 B_0(0,M_2,M_2) + \frac{1}{4}\left (1-\frac{\hs}{\hc}
\right ) B_0(0,M_{\tilde{L}},M_{\tilde{L}}) + \frac{1}{2} +
\frac{1}{8} \left ( 1-\frac{\hs}{\hc} \right )
\nonumber \\[2mm] &+& 
\frac{3}{2} B_1(0,M_2,M_{\tilde{L}}) + \frac{\hs}{2\hc}
B_1(0,M_1,M_{\tilde{L}}) \Biggr \} \;.
\label{cdvb}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%
Using Eqs. (B7-B10) of Appendix B, we find that the expression above
involves no large logarithms. Also as said before 
the first term ($\Re\alpha_1$) in eq. (\ref{dvb})
contains finite parts which go as $\sim \frac{M_Z}{M_{SUSY}} $.
Thus no large logarithmic terms arising from the wave function and
vertex corrections of the muon decay and
the decoupling of large logarithms in $s_f^2$ appear.


\vspace*{1cm}
{\bf{iii) \underline{Non-universal corrections to $\Delta\hat{k}_f$}}} 
 
%\subsubsection{ Formulation }
\vspace*{.4cm} 
The $Zf\overline{f}$ vertex corrections can be written as
%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
i \ori {\frac{\hat{e}}{2 \hat{s} \hat{c}}} \; {\gamma^{\mu}} \;
( \, F_V^{(f)} \, - \, \gamma_5  F_A^{(f)} )\;,
\label{dkf}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%
where  $ F_V^{(f)} , F_A^{(f)} $ denote the vector and axial couplings
respectively. Incorporating the tree level couplings 
we can write this vertex in a slightly different form as
{\footnote {We follow the notation of Ref.\cite{finnell} which will be useful
in what will follow.}}
%%%%%%%%%%%%%%%%%%
\begin{equation}
i \ori {\frac{\hat{e}}{\hat{s} \hat{c}}} \; {\gamma^{\mu}} \;
( \, u'_L {\cal P_L} + u'_R {\cal P_R} )\;,
\end{equation}
%%%%%%%%%%%%%%%%%%
where
%%%%%%%%%%%%%%%%%
\begin{eqnarray}
u'_L &=& u_L + \frac{F^{(f)}_L}{16\pi^2}\;,\\[2mm]
u'_R &=& u_R + \frac{F^{(f)}_R}{16\pi^2}\;.
\end{eqnarray}
%%%%%%%%%%%%%%
In the equations above $u_L, u_R$ are the tree level left and right handed
couplings respectively related to the vector $v_f$ and axial $a_f$ tree level
couplings by $v_f=u_L+u_R$, $a_f=u_L-u_R$. $ F_{L,R}^{(f)}$ denote
the corresponding one loop corrections to the aforementioned couplings,  
with the coefficient $1/{16\pi^2}$ factored out for
convenience. These are related to $ F_V^{(f)} , F_A^{(f)} $ of eq. (30) by
\begin{eqnarray}
F^{(f)}_V &\equiv&  \frac{1}{16\pi^2} ({F^{(f)}_L}+{F^{(f)}_R})\;, \\[2mm]
F^{(f)}_A &\equiv&  \frac{1}{16\pi^2} ({F^{(f)}_L}-{F^{(f)}_R})  \;.
\end{eqnarray}
%%%%%%%%%%%%%%
As a result the corrections to $\Delta\hat{k}_f$ are given by
%%%%%%%%%%%%%%
\begin{equation}
\Delta\hat{k}_f = - \frac{1}{16\pi^2}\;
\frac{1}{\hs Q_f (u_L-u_R)}\; (\, u_L F_R^{(f)} - u_R F_L^{(f)} \,)\;,
\end{equation}
%%%%%%%%%%%%%%%
and are equivalent to the well known expression
%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\Delta{\hat{k}_f} \ori = \ori - {\frac{1}{2 {\hat{s}}^2 Q_f}} \ori
( \; F_V^{(f)} \; - {\frac{v_f}{a_f}} \;  F_A^{(f)} )\;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%
%The relations between vector and axial couplings are
%\begin{eqnarray}
%F^{(f)}_V &\equiv& u_L + u_R + \frac{F^{(f)}_L +F^{(f)}_R}{16\pi^2} \\
%F^{(f)}_A &\equiv& u_L - u_R + \frac{F^{(f)}_L -F^{(f)}_R}{16\pi^2} \\
%\end{eqnarray}
%%%%%%%%%%%%%%

In eq. (\ref{dkapaf}) we have denoted by $\delta k_f^{SUSY}$ the
supersymmetric contributions to $\Delta \hat{k}_f$.
Here we consider the example of the decoupling of large logs
in $\delta k_f^{SUSY}$ in
the case where the fermion $f$ stands for a ``down" quark denoted by $b$ 
being in the same
isospin multiplet with the ``up" quark denoted by $t$. In this case
we have
%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
u_L &=& -\frac{1}{2} + \frac{1}{3}\hs \;, \\ 
u_R &=& \frac{1}{3}\hs \;, \\
Q_f &=& -\frac{1}{3}\;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%
The cases of other fermion species are treated in a similar manner.
In what follows we will consider only the chargino corrections to vertices
and external fermion lines. The decoupling of large logarithmic terms
arising from 
the neutralinos exchanges proceeds in exactly the same manner.

We will first discuss the self energy corrections to $Zb\bar{b}$
vertex. From the  diagrams of the Figure 2a, we obtain
{\footnote{ The functions $b_1, c_0$ used throughout this section
which are defined below should not be confused with the Passarino-Veltman
functions \cite{passarino} which are commonly denoted by capital letters.
These are
actually the reduced Passarino-Veltman functions \cite{ahn} defined as
%%%%%%%%%%%%%%%%%
\begin{eqnarray} 
b_1(m_1,m_2,q) &\equiv& \int_{0}^{1} dx\; x \log 
\frac{x m_1^2+(1-x) m_2^2-q^2 x (1-x) -i \epsilon}{Q^2}\;, \nonumber \\[1mm] 
c_0(m_1,m_2,m_3) &\equiv& \int_0^1 dx
\int_0^{1-x}dy \log \frac{(1-x-y) m_1^2+x m_2^2+y m_3^2-
(1-x-y)(x+y) m_b^2-x y P^2}{Q^2}\;. \nonumber
\end{eqnarray}
}}
, in an obvious notation,
%%%%%%%%%%%%%%%%
\begin{eqnarray}
F_L^{(b)} &=& \sum_{i=1,2}\sum_{j=1,2}\; b_1(m_{\ti{t}_j},
m_{\ti{\chi}_i},m_b)\;
u_L\;|a_{ij}^{b\ti{t}\ti{\chi}}|^2 \;, \\[2mm]
F_R^{(b)} &=& \sum_{i=1,2}\sum_{j=1,2}\; b_1(m_{\ti{t}_j},
m_{\ti{\chi}_i},m_b)\;
u_R\;|b_{ij}^{b\ti{t}\ti{\chi}}|^2 \;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%
 
%%%%%%%%%%%%%%%
%\vspace{.25in}
%\centerline{\hbox{\psfig{figure=fig1a.ps,width=2.5in}
%\psfig{figure=fig1b.ps,width=2.5in}}}
%\begin{center}
%\footnotesize{{\bf Figure 2 :} External fermion corrections to the 
%$Zb\bar{b}$ vertex.}
%\end{center}
%%%%%%%%%%%%%%%%%%%%%
 
 
 
 
 
>From the Appendix A (see the discussion following eqs. A.19) 
we get $a_{11}^{b\ti{t}\ti{\chi}}=g$,
$a_{22}^{b\ti{t}\ti{\chi}}=-h_t$ and $b_{21}^{b\ti{t}\ti{\chi}}=-h_b$. 
All other couplings vanish when the electroweak symmetry breaking effects are
ignored. Thus we get,
%%%%%%%%%%%%%%%%
\begin{eqnarray}
F_L^{(b)} &=& u_L \biggl [\; g^2 b_1(m_{\ti{t}_L},M_2,m_b)+
h_t^2 b_1(m_{\ti{t}_R},\mu,m_b)\;\biggr ]\;, \\[2mm]
F_R^{(b)} &=& u_R \biggl [\; h_b^2 b_1(m_{\ti{t}_L},\mu,m_b)\;\biggr ]\;.
\end{eqnarray}
%%%%%%%%%%%%%%%%
 
 
 
 
%\subsubsection{ Vertex corrections to $Zb\bar{b}$ vertex }

On the other hand, from the first triangle graph of Figure 2b we obtain,
%{\footnote{ $c_0(m_1,m_2,m_3)=\int_0^1 dx
%\int_0^{1-x}dy \log (\frac{(1-x-y) m_1^2+x m_2^2+y m_3^2-
%(1-x-y)(x+y) m_b^2-x y P^2}{Q^2})$}}
 
 
%%%%%%%%%%%%%%%
%\vspace{.25in}
%\centerline{\psfig{figure=fig2.ps,width=3in}}
%\begin{center}
%\footnotesize{{\bf Figure 3 :} Squark and chargino contribution to
%the $Zb\bar{b}$ vertex.}
%\end{center}
%%%%%%%%%%%%%%%%%%%%%
 
 
%%%%%%%%%%%%%%%%
\begin{eqnarray}
F_L^{(b)} &=& \sum_{i,j,k=1,2}\;
c_0(m_{\ti{\chi}_k},m_{\ti{t}_i},m_{\ti{t}_j})\;
\left (\frac{2}{3}\hs\delta_{ij}-\frac{1}{2} K_{i1}^{\ti{t} *}
K_{j1}^{\ti{t}}\right )\;a_{ki}^{b\ti{t}\ti{\chi}}
a_{kj}^{b\ti{t}\ti{\chi} *}\;, \\[2mm]
F_R^{(b)} &=& \sum_{i,j,k=1,2}\;
c_0(m_{\ti{\chi}_k},m_{\ti{t}_i},m_{\ti{t}_j})\;
\left (\frac{2}{3}\hs\delta_{ij}-\frac{1}{2} K_{i1}^{\ti{t} *}
K_{j1}^{\ti{t}}\right )\;b_{ki}^{b\ti{t}\ti{\chi}}
b_{kj}^{b\ti{t}\ti{\chi} *}\;,
\end{eqnarray}
%%%%%%%%%%%%%%%%%%
which, when the electroweak effects are ignored, have the following form,  
%%%%%%%%%%%%%%%%
\begin{eqnarray}
F_L^{(b)} &=& \left (\frac{2}{3}\hs-\frac{1}{2}\right )\, g^2
c_0(M_2,m_{\ti{t}_L},m_{\ti{t}_L})+\frac{2}{3}\,\hs h_t^2
c_0(\mu,m_{\ti{t}_R},m_{\ti{t}_R})\;,\\[2mm]
F_R^{(b)} &=&\left (\frac{2}{3}\hs-\frac{1}{2}\right )\, h_b^2
c_0(\mu,m_{\ti{t}_L},m_{\ti{t}_L})\;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%
 
The calculation of the second diagram of Figure 2b gives
{\footnote{
%\begin{eqnarray}
$[c_2,c_6](m_1,m_2,m_3)=\int_0^1 dx\int_0^{1-x}[1,x]\;
\frac{1}{(1-x-y) m_1^2+x m_2^2+y m_3^2-(1-x-y) (x+y) m_b^2
-x y P^2-i\epsilon}\;.$
%\end{eqnarray}
}}
 
 
%%%%%%%%%%%%%%%%%%%%
%\vspace{.25in}
%\centerline{\psfig{figure=fig3.ps,width=3in}}
%\begin{center}
%\footnotesize{{\bf Figure 4 :} Chargino and squark contribution to
%the $Zb\bar{b}$ vertex.}
%\end{center}
%%%%%%%%%%%%%%%%%
 
%%%%%%%%%%%%%%%%
\begin{eqnarray}
F_L^{(b)} &=&-\sum_{i,j,k=1,2}\;
\Biggl \{\;\biggl [{P^2} c_6(m_{\ti{t}_k},m_{\ti{\chi}_i},
m_{\ti{\chi}_j})-\frac{1}{2}-c_0(m_{\ti{t}_k},m_{\ti{\chi}_i},
m_{\ti{\chi}_j})\;\biggr ]{\cal A^L}_{ij}\nonumber \\[2mm] &+&
{m_{\ti{\chi}_i} m_{\ti{\chi}_j}}\;
c_2(m_{\ti{t}_k},m_{\ti{\chi}_i},m_{\ti{\chi}_j})\; {\cal A^R}_{ij}\Biggr \}
a_{ik}^{b\ti{t}\ti{\chi}}a_{jk}^{b\ti{t}\ti{\chi} *}\;, \\[2mm]
%%%%%%%%%%%%%%%
F_R^{(b)} &=&-\sum_{i,j,k=1,2}\;
\Biggl \{\;\biggl [{P^2} c_6(m_{\ti{t}_k},m_{\ti{\chi}_i},
m_{\ti{\chi}_j})-\frac{1}{2}-c_0(m_{\ti{t}_k},m_{\ti{\chi}_i},
m_{\ti{\chi}_j})\;\biggr ]{\cal A^R}_{ij}\nonumber \\[2mm] &+&
{m_{\ti{\chi}_i} m_{\ti{\chi}_j}}\;
c_2(m_{\ti{t}_k},m_{\ti{\chi}_i},m_{\ti{\chi}_j})\; {\cal A^L}_{ij}\Biggr \}
b_{ik}^{b\ti{t}\ti{\chi}}b_{jk}^{b\ti{t}\ti{\chi} *}\;, 
\end{eqnarray}
%%%%%%%%%%%%%%%%%
where $P$ is the momentum carried by the Z - boson. 
The couplings ${{\cal A}^L}_{ij},{{\cal A}^R}_{ij}$ can be read from
Appendix A (see Eqs. A.17). In the absence of electroweak symmetry breaking
effects the only non-vanishing couplings are
%%%%%%%%%%%
\begin{eqnarray}
{{\cal A}^L}_{11} &=& \hc={{\cal A}^R}_{11}\;,\\[2mm]
{{\cal A}^L}_{22} &=& \hc-\frac{1}{2}={{\cal A}^R}_{22}\;.
\end{eqnarray}
%%%%%%%%%%%%%%%%
Thus, we obtain
%%%%%%%%%%%%%

%%%%%%%%%%%%%
\begin{eqnarray}
F_L^{(b)} &=& g^2 \hc c_0(m_{\ti{t}_L},M_2,M_2)+
h_t^2 \; \left (\hc-\frac{1}{2}\right )\; c_0(m_{\ti{t}_R},\mu,\mu)\;,\\[2mm]
F_R^{(b)} &=& h_b^2\;\left (\hc-\frac{1}{2}\right )\;c_0(m_{\ti{t}_L},\mu,\mu)
\;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%
 
 
%\subsubsection{ The Decoupling of Large Logs }
 
Summing up the diagrams of  Figures 2a and 2b we get
%%%%%%%%%%%
\begin{eqnarray}
F_L^{(b)} &=& \left (-\frac{1}{2}+\frac{1}{3} \hs \right)
\left [\; g^2 b_1(m_{\ti{t}_L},M_2,m_b)+
h_t^2 b_1(m_{\ti{t}_R},\mu,m_b)\;\right ] \nonumber \\[2mm] &+&
\left (\frac{2}{3}\hs-\frac{1}{2}\right ) g^2
c_0(M_2,m_{\ti{t}_L},m_{\ti{t}_L})+\frac{2}{3}\hs h_t^2
c_0(\mu,m_{\ti{t}_R},m_{\ti{t}_R}) \nonumber \\[2mm] &+&
g^2 \hc c_0(m_{\ti{t}_L},M_2,M_2)+
h_t^2 \; \left (\hc-\frac{1}{2}\right )\; c_0(m_{\ti{t}_R},\mu,\mu)\;,\\[4mm]
%%%%%%%%%%%%%%%%%%%%%%%%%
F_R^{(b)} &=& \frac{1}{3}\hs
  h_b^2 b_1(m_{\ti{t}_L},\mu,m_b)\;
+
\left (\frac{2}{3}\hs-\frac{1}{2}\right )h_b^2\,
c_0(\mu,m_{\ti{t}_L},m_{\ti{t}_L})
\nonumber \\[2mm] &+&
h_b^2\;\left (\hc-\frac{1}{2}\right )\;c_0(m_{\ti{t}_L},\mu,\mu)\;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%
 
In the limit of $q^2=m_b^2\simeq 0$, $P^2= M_Z^2\rightarrow 0$
or $M_Z^2<<M_{SUSY}^2$, the following useful relations hold,
%%%%%%%%%%%
\begin{eqnarray}
c_0(m_1,m_2,m_3)&=&b_1(m_2,m_1,0)\;, \\[2mm]
c_0(m_1,m_2,m_3)-b_1(m_1,m_2,0)&=&\frac{1}{m_1^2-m_2^2} \;
\left [m_1^2 m_2^2 \log\left (\frac{m_1^2}{m_2^2}\right )
-\frac{1}{2}\left (m_1^2+m_2^2\right )\;\right ]\;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%

Using these we have for the expressions for $F^{(b)}_{L,R}$ above 
%%%%%%%%%%%%%
\begin{equation}
F_L^{(b)}= h_t^2\; O\left (\frac{m_{\ti{t}_R}^2}{\mu^2}\right )+
g^2\; O\left (\frac{m_{\ti{t}_L}^2}{M_2^2}\right )\;,
\end{equation}
%%%%%%%%%%%%%%%%
and
%%%%%%%%%%%
%%%%%%%%%%%%%
\begin{equation}
F_R^{(b)}=h_b^2 \; O\left (\frac{m_{\ti{t}_L}^2}{\mu^2}\right )\;,
\end{equation}
%%%%%%%%%%%%%%
which is independent of large logs and the decoupling of
terms $\log(\frac{M_{SUSY}}{M_Z})$ is manifest.

So far we have considered the cancellation of potentially large logarithms
involving the soft SUSY breaking scale $M_{1/2}$ and the mixing parameter
$\mu$ which arise from the neutralino and chargino sectors when
$M_{1/2}>>M_Z$. A similar analysis can be repeated for the corresponding
contributions of the squark and slepton sectors, whose masses depend also on
the soft SUSY breaking parameters $M_0$, when $M_0$ gets large.
We have carried out
such an analysis and found that the decoupling of large logarithms does indeed
occur when these parameters get large values. It is not necessary to
 present the details
of such a calculation here. We merely state that large logarithms arising
from the vector boson self energy corrections which contribute to the
quantities
$\Delta \hat r_W$, $\hat\rho$ and $\Delta \hat k_f$ cancel against those
from ${\hat{\alpha}}(M_Z)$. Also, the large log contributions 
from the muon decay amplitude,
which affect the effective mixing angle through $\delta_{VB}^{SUSY}$, 
cancel among themselves.
As for the large logarithmic contributions to the weak mixing angle 
from the non-universal corrections to the factor
$\Delta \hat k_f$, these are found to be cancelled in exactly the same
way as in the case of the neutralinos and charginos\footnote{The
logarithmic corrections of the Higgs sector to the $Zb\bar{b}$
vertex and external $b$ lines are cancelled in exactly the same 
manner.}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%    SQCD part %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{1cm}
{\bf{iv) \underline{SQCD corrections to $\Delta\hat{k}_f$}}}

\vspace*{.4cm} 
 The last corrections to be considered are the SQCD non-universal corrections
\cite{sqcdrefs}
which, due the largeness of the strong coupling constant, are,
naively, expected to
yield contributions larger than those of the electroweak sector. This case
is of relevance only when the external fermions in the $Zf\overline f$ vertex
are quark fields and is of particular interest for the bottom case whose
measurement of the Forward / Backward asymmetry ${\cal A}_b^{FB}$ 
yields the most precise individual measurement at LEP.

The one loop correction to $Zq\overline q$ vertex ({\it see} Figure 2c)
where two squarks, which
are coupled to the Z - boson, and a gluino are exchanged yields for the
Left and Right handed couplings defined in Eqs. (30)-(33),
%%%%%%%%%%%%%%%%
\begin{eqnarray}
F_L^{(q)} &=& {\frac{16}{3}} \; (4 \pi {\alpha}_s ) \;
\sum_{i=1,2}\sum_{j=1,2}\;
{K^{\tilde q}}^{\star}_{j1} K^{\tilde q}_{i1}  \;  A_{\tilde q}^{ji} \;
{C_{24}}(m_q^2,M_Z^2,m_q^2; M_{\tilde g}^2,m_{\tilde {q_i}}^2,
m_{\tilde {q_j}}^2) \;, \\[2mm]
F_R^{(q)} &=& {\frac{16}{3}} \; (4 \pi {\alpha}_s ) \;
\sum_{i=1,2}\sum_{j=1,2}\;
{K^{\tilde q}}^{\star}_{j2} K^{\tilde q}_{i2}  \;  A_{\tilde q}^{ji} \;
{C_{24}}(m_q^2,M_Z^2,m_q^2; M_{\tilde g}^2,m_{\tilde {q_i}}^2,
m_{\tilde {q_j}}^2)  \;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%
In these, the coupling $ A_{\tilde q}^{ji}$ is given by
%%%%%%%%%%%%%%%%
\begin{eqnarray}
A_{\tilde q}^{ji} \; = \;
u_L \; {K^{\tilde q}}^{\star}_{j1} K^{\tilde q}_{i1} +
u_R \; {K^{\tilde q}}^{\star}_{j2} K^{\tilde q}_{i2}\;,  \nonumber
\end{eqnarray}
%%%%%%%%%%%%%%%%%%
with $K^{\tilde q}_{ij}$ the matrix diagonalizing the squark ${\tilde q} $
mass matrix. The function $C_{24}$, with momenta and masses as shown,
is the coefficient of  $g_{\mu \nu}$ in the tensor three point integral
( This is denoted by $C_{20}$ in ref. \cite{Hollik2} ). The contribution
of ${F_{L,R}}^{(q)}$ to the form factor $\Delta\hat{k}_q$ is free of large
logarithms. In order to understand this consider the case of vanishing
quark mass $m_q$. In this case the matrix $K^{\tilde q}_{ij}$ becomes the
unit matrix.  It is easy to see that the contribution to
$\Delta\hat{k}_q$, as this is read from Eq. (36), is proportional to the
difference
\begin{eqnarray}
{C_{24}}(m_q^2,M_Z^2,m_q^2; M_{\tilde g}^2,m_{\tilde {q_1}}^2,
m_{\tilde {q_1}}^2)  - 
{C_{24}}(m_q^2,M_Z^2,m_q^2; M_{\tilde g}^2,m_{\tilde {q_2}}^2,
m_{\tilde {q_2}}^2) \;.  \nonumber
\end{eqnarray}
%%%%%%%%%%%%%%%%%%
In this difference the leading log terms cancel each other. Note that
 it would 
vanish if the left and right handed squark fields happened to
be degenerate in mass. Due to their mass splitting however
the result is not vanishing but at any rate small.
In general the SQCD vertex
corrections turn out to be smaller than the corresponding
electroweak corrections, as we have verified numerically.

 As for the external quark contributions ({\it see} Figure 2d) we find 
%%%%%%%%%%%%%%%%
\begin{eqnarray}
F_L^{(q)} &=& {\frac{8}{3}} \; (4 \pi {\alpha}_s ) \; u_L \;
\left [ c^2 \; {B_1}(m_q^2,M_{\tilde g}^2,m_{\tilde {q_1}}^2) +
  s^2 \; {B_1}(m_q^2,M_{\tilde g}^2,m_{\tilde {q_2}}^2)\right ]\;, \\
F_R^{(q)} &=& {\frac{8}{3}} \; (4 \pi {\alpha}_s ) \; u_R \;
\left [ s^2 \; {B_1}(m_q^2,M_{\tilde g}^2,m_{\tilde {q_1}}^2) +
  c^2 \; {B_1}(m_q^2,M_{\tilde g}^2,m_{\tilde {q_2}}^2)\right ] \;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%
In the equation above
$c \equiv K^{\tilde q}_{11}, s \equiv K^{\tilde q}_{12} $. Their contribution
to $\Delta\hat{k}_q$ is free of large logarithms and
small due to cancellations of the leading terms exactly as
in the case of the vertex corrections discussed previously. In fact in the
limit of vanishing quark mass the self energy corrections to 
$\Delta\hat{k}_q$ is proportional to the difference
\begin{eqnarray}
{B_1}(m_q^2,M_{\tilde g}^2,m_{\tilde {q_1}}^2) -
 {B_1}(m_q^2,M_{\tilde g}^2,m_{\tilde {q_2}}^2) \;. \nonumber
\end{eqnarray}
%%%%%%%%%%%%%%%%%%
which vanishes when the squark masses are equal. Therefore,
following the same arguments as in the vertex case, we are led to the
conclusion that SQCD contributions from the 
external quark lines are small. 

Besides the cancellations discussed above which lead to relatively small
SQCD vertex and external fermion corrections, these two contributions 
tend to cancel each other since they contribute with opposite signs. This
results in very small overall SQCD corrections to $\Delta\hat{k}_q$
almost one to two orders of magnitude smaller than the
corresponding non-universal electroweak corrections. We shall come back to
this point later when discussing our numerical results.


In the following section we shall discuss our numerical results
concerning the predictions of the MSSM for the effective mixing angle and
asymmetries. We will also
present the corresponding theoretical predictions for the mass
of the W - boson through its connection to the parameter rho and the
effective mixing angle.



\section{Numerical Analysis and Results}

For a given set of pole masses $m_t^{pole}$, $m_b^{pole}$, $m_{\tau}^{pole}$
we define the $\overline{DR}$ Yukawa couplings at $M_Z$.
To start with, we set a test value
for the $\hs$ ({\it i.e} $\hs=0.2315$)  and we define the 
$\overline{DR}$ gauge couplings $\hat{g}_1$ and $\hat{g}_2$ at $M_Z$. The
numerical output is independent of the starting value for $\hs$. For $\hs$ 
around the value given above the number of iterations needed for convergence
is minimized.
Then we use the 2-loop Renormalization Group equations 
\cite{2loop} to run up to
the scale $M_{GUT}$ where $\hat{g}_1$ and $\hat{g}_2$ meet. At 
$M_{GUT}$ we impose the unification condition
%%%%%%%%%%%%%%%
\begin{equation}
g_{GUT} \equiv \hat{g}_1 = \hat{g}_2 = \hat{g}_3 \;.
\end{equation}
%%%%%%%%%%%%%%%
Assuming universal boundary conditions for the soft breaking 
parameters $M_0$, $M_{1/2}$ and $A_0$, we run down to $M_Z$ and find the
couplings and 
the soft masses at $M_Z$ which are inputs for the self energies of the
gauge bosons, wave functions and vertex corrections and they
define the new $\hs$. The whole procedure is iterated until
convergence is reached satisfying the full one loop minimization
conditions in order to have radiative symmetry breaking observing the 
experimental bounds on supersymmetric particles.
For the calculation of the one loop integrals encountered we have made use of
the {\tt FF} library  \cite{olden}.
The conversion
of the ``theoretical'' $\hs$ to the experimental $s_f^2$ through
eq.(\ref{sineff})  gives our basic output : the effective weak
mixing angle $s_f^2$. In addition, the value of the strong QCD
coupling, as it is calculated in the $\overline {MS}$ scheme at $M_Z$, is among
our outputs \cite{dedes2}. Note that we
have used as inputs the parameters  $\alpha_{EM}$, $M_Z$ 
and $G_F$ which are experimentally known to a high degree of accuracy,
as well the masses of leptons and quarks.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% DISCUSSION ON DK's%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The factor $\Delta \hat k_f$ needed to pass from the $\hat{s}^2(M_Z)$ to
the effective angle $s_f^2(M_Z)$ receives universal corrections, from the
${\gamma} - Z$ propagator, and non-universal corrections arising from
vertices and external wave function renormalizations. We find that
the non-universal Electroweak supersymmetric corrections are very small.
Although separately vertex and external fermion corrections are large they
cancel each other yielding contributions almost two orders of magnitude
smaller than the rest of the electroweak corrections. The non-universal
$SQCD$ contributions although a priori expected to to be larger than the
Electroweak corrections turn out to be even smaller. The reason for this was
explained in the previous section. In fact they are found to be one to two
orders of magnitude smaller than the corresponding electroweak corrections.
We conclude therefore that at the present level of accuracy one can
safely ignore the supersymmetric non-universal corrections to the factors
$\Delta \hat k_f$. The situation is very clearly depicted in Table I where
for some characteristic input values we give the contributions of the various
sectors to $\Delta \hat k_f$, as well as their total contributions, and also
the corresponding predictions for the 
values of the effective mixing angle and the asymmetries. Concerning the
values displayed in Table I, in a representative case, a few
additional remarks are in order: \\
i) The bulk of the supersymmetric corrections to $\Delta \hat k_f$
is carried by the universal corrections which are sizable, due to
their dependence on large logarithmic terms. These cancel similar terms in
$\hat{s}^2$. \\
ii) The contribution of Higgses, which is small, mimics that of the Standard
Model with a mass in the vicinity of $\simeq 100 GeV$. \\
iii) Gauge and Higgs boson contributions tend to cancel large universal
contributions of matter fermions. Concerning the gauge boson contributions
note that they are different for the different fermion species $l,c,b$.
This is due to the fact that their non-universal corrections depend on the
charge and weak isospin assignments of the external fermions and on
the mass of the top for when the external fermion is a bottom.  \\
iv) The slepton universal corrections are suppressed relative to their
corresponding squark contributions. This is due to the following reason.
The couplings of the left and right handed sleptons to the neutral
Z - boson depend on the angle $\hat{s}^2$ and would be exactly opposite if
$\hat{s}^2$ happened to be $\frac{1}{4}$. Thus their 
contributions to the
${\gamma} - Z$ propagator would be exactly opposite if their
masses were equal leading to a vanishing slepton contribution.The fact that
$\hat{s}^2 \simeq .23$ is close to $\frac{1}{4}$ in conjunction with the fact
that the left and right handed sleptons are characterized by small mass
splittings leads to the conclusion that universal slepton contributions to
$\Delta \hat k_f$ are small.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%% END of discussion on DK's%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In Figure 3, we display the effective weak mixing angle $s_l^2(M_Z)$,
 obtained from
the vertex $Z-l^+ - l^-$,  and the weak mixing
angle $\hat{s}^2(M_Z)$  as  functions of the soft gaugino
mass parameter $M_{1/2}$, the soft  parameter $M_0$, $A_0$ as well as
the parameters  $\tan\beta$ and $m_t$ inside the region which is indicated
in the figure.
Non-universal supersymmetric vertex and external fermion corrections
have been taken into account.
%The experimental values extracted from LEP, $s_l^2(M_Z)=0.23199 \pm 0.00028$,
%and SLD, $s_l^2(M_Z)=0.23055 \pm 0.00041$ 
%\cite{altarelli}, are also indicated in
% Figure 3 for comparison.
As is well known there is a discrepancy between the LEP and SLD
 experimental values of $s^2_{eff}$.
 The LEP average $s^2_{eff}=0.23199 \pm 0.00028$
 differs   by $2.9\sigma$ from the SLD value $s^2_{eff}=0.23055 \pm 0.00041$
obtained from the single measurement of left-right asymmetry \cite{altarelli}.
The LEP+SLD average value is $s^2_{eff}=0.23152+0.00023$.
We observe that $\hat{s}^2(M_Z)$
 takes on the ``theoretical'' value $\hat{s}^2=0.2377$ for
$M_{1/2}=900 \, GeV$ and  becomes larger and larger due to the fact
that it contains large
logarithms.
Manifest cancellation of large logarithmic terms is obtained in the
extracted value of the effective weak mixing angle as we have analytically
demonstrated in the previous chapter.
%Exploring the whole parameter space by giving arbitrary
%values in the plane $M_0$-$M_{1/2}$, we
%found that the effective weak mixing angle achieves its experimental LEP
%value quoted  above in a large range of the input values on the soft 
%parameter space.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In Figure 3,
the dispersion of the values of
$s^2_l(M_Z)$ in the lower region of $M_{1/2}$ is caused by the
presence of the finite parts of order 
${\cal O}\left (M_Z/M_{SUSY}\right)$ in the expression (\ref{sineff}),
which become very important when $M_{1/2}\sim M_0\sim M_Z$ (case which
is preferred by SLD data) and 
 contribute
positively to $\Delta k$.
When $M_{1/2}\rightarrow 900 \; GeV$
(case which is rather preferred by LEP data) then $s^2_l(M_Z)\rightarrow
0.23145$ independently of the value of $M_0$. It must be noted that,
when $M_{1/2}=M_Z$ and $M_0\simeq 200\,GeV$,
 the values of the two angles are equal, {\it i.e.}
$s_l^2(M_Z)=\hat{s}^2(M_Z)=0.2310$.  We have also explored  the case
where the Higgs mixing parameter is negative ($\mu < 0$). In this case,
 as $M_{1/2}$ tends to larger
values $M_{1/2}\rightarrow 900 \, GeV$,   $s_l^2(M_Z)$
approaches 
the value $0.23145$ which means that, for large values of $M_{1/2}$, 
$s_l^2(M_Z)$ is independent of the sign of $\mu$ as it is expected
from the decoupling shown in chapter III. 
The sign of $\mu$ does not affect either  the $\hat{s}^2(M_Z)$ value
for large $M_{1/2}$.
 The effect of the sign
appears in the lower values of $M_{1/2}$. In the region $M_{1/2}\rightarrow
M_Z$, the value of $\Delta k_l$ ($l=e,\,\mu,\,\tau$)
 is always negative in the case $\mu < 0$
 and thus $s^2_l < \hat{s}^2$.
There is no possibility of equality between the two angles 
in this case. 
The largest value of $s_l^2(M_Z)=0.2315$
($\hat{s}^2(M_Z)\rightarrow 0.238$)
 is reached when
$M_{1/2}\rightarrow 1200\, GeV$. Just above this value
 no radiative symmetry breaking occurs. 
The lowest value of $s_l^2(M_Z)=0.2305$
($\hat{s}^2(M_Z)=0.2302$), for $\mu >0$, and
$s_l^2(M_Z)=0.2309$ ($\hat{s}^2(M_Z)=0.2316$) for $\mu<0$,
 is bounded by the new experimental limit on the chargino
mass which is around $\sim 84-86\, GeV$ \cite{Tata}.

%As is well known there is a discrepancy between the LEP and SLD
% experimental values of $s^2_f$. The LEP average $s^2_f=0.23199 \pm 0.00028$
% differs   by $2.9\sigma$ from the SLD value $s^2_f=0.23055 \pm 0.00041$
%obtained from the single measurement of left-right asymmetry \cite{altarelli}.
%The LEP+SLD average value is $s^2_f=0.23152+0.00023$.
In Figure 4, we plot
the values of $s^2_f$ for the fermions $f=c,b$. 
In the large SUSY breaking  limit, where all superparticles
are quite massive ($M_{1/2}\rightarrow 900 \, GeV$),
we obtain for the central values  $s^2_b=0.2330$ and $s^2_c=0.2314$. 
In the light limit, $M_{1/2}\simeq
M_0 \simeq M_Z$, they  take on the values,
  $s^2_b=0.2298$ and $s^2_c=0.2308$.
The main effect in the extracted values of the effective angle $s_l^2$
 is coming
dominantly from
the variation of $M_{1/2}$ and secondly from $M_0$.
If $M_{1/2}$ is kept constant, the variation
of $M_0$ from 100 to 900 GeV, changes  $s^2_f$ by +0.0005. In addition, the 
effect of $A_0$ on the effective angle is negligible. The effect of the
independent parameter $\tan\beta$ is also negligible if it remains in the
region $\tan\beta \simeq 5-28$.
Large loop corrections to the $b$-Yukawa coupling (or to the bottom pole mass),
which are proportional to
the term $\mu \tan\beta$, 
affect  the
obtained values of $s^2_f$ in the large $\tan\beta$ region \cite{Dedes3}.

There is a strong correlation of the output value of the effective weak mixing
angle with the top quark mass as it is shown in Figure 5. In this
Figure, we have chosen two characteristic sets of input values  $A_0=M_0=
M_{1/2}=600 \, GeV$ and  $A_0=M_0=M_{1/2}=200 \, GeV$.  It is
clear that the first case is most preferable if one assumes the LEP+SLD
data, where $s_l^2=0.23152 \pm 0.00023$. The present combined
CDF/D{\O} \cite{top} result for $m_t=175 \pm 5 \, GeV$,  
is 
also compatible with the first case.
Radiatively corrected light Higgs boson masses are also shown in Figure 5.
 Figures 6 and 7 display  the 
range of predictions for the mass of the W-gauge boson in the MSSM. As
one can see, the W-mass is in  agreement with the presently 
experimentally observed value, $M_W=80.427 \pm 0.075$
($M_W=80.405 \pm 0.089$) GeV obtained from LEP (CDF,UA2,D{\O}) experiments
\cite{LEP} for rather low (high) values of $M_{1/2}$ 
 in the region of $m_t=175 \pm 5$ GeV.
Variation of $m_t$ equal to $+5 \, GeV$ leads to variation of $M_W$ equal to
$+0.032 \, GeV$ while the effect on  $s_l^2$ is $-0.00017$.

The {\it left-right} asymmetries are given by the effective Lagrangian 
(\ref{leff}) with
%%%%%%%%%%%%%%%%%%
\begin{equation}
A_{LR}^f \ =\ {\cal A}^f \ =\ \frac{2\, \upsilon_{eff}^f / a_{eff}^f}
{1+\left ( \upsilon_{eff}^f / a_{eff}^f \right )^2 } \;,
\label{aet}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%
where
%%%%%%%%%%%%%%%
\begin{eqnarray}
\upsilon_{eff}^f \ &=& \ T_3^f - 2\, s^2_f \, Q^f \;, \nonumber \\[2mm]
a_{eff}^f \ &=& \ T_3^f \;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%
As it is depicted in Figure 8, the MSSM prediction for ${\cal A}^e$ agrees
with the LEP+SLD average value (${\cal A}^e = 0.1505 \pm 0.0023$) when
both $M_{1/2}$ and $M_0$
take on values around $M_Z$. In the heavy limit (large $M_{1/2}$),
 the 
MSSM agrees with the LEP value ${\cal A}^e = 0.1461 \pm 0.0033$. 
Note that as $M_{1/2}\rightarrow 900 \, GeV$, the value of ${\cal A}^e$ tends
asymptotically (which means that large logarithms have been decoupled
from the expression (\ref{aet}) ) to the value $0.1476$ corresponding
to $s^2_l \simeq 0.23145$ ({\it see} also Figure 4).

In the results shown in Figures 3-8, we have not considered the constraint
resulting from the experimental value of $\alpha_s$.
In Figure 9, we have plotted the acceptable values
of the soft breaking parameters $M_{1/2}$ and $M_0$\footnote{
We examine the region where $A_0\,,\,M_0\,,\,M_{1/2}\lesssim 900$ GeV.},
which are compatible with 
the  LEP+SLD ($\alpha_s=0.119
\pm 0.004$,  $s^2_{eff}=0.23152 \pm 0.00023$) \cite{altarelli}
 and the CDF/D{\O}
($m_t=175 \pm 5 \, GeV$) \cite{top} data.
The trillinear
soft couplings as well as the parameter $\tan\beta(M_Z)$ are taken
arbitrarily in the region ($0-900 \, GeV$) and ($2-30$), respectively.
 As we observe 
from  Figure 9, MSSM with radiative EW breaking
is valid in the region $M_{1/2} \gtrsim 500 \, GeV$ and
$M_0 \gtrsim 70 \, GeV$ \footnote{
The requirement that
the LSP is neutral puts this bound on $M_0$.}.
In this region, the physical gluino mass is above
$1 \, TeV$, the LSP (one of the neutralinos)
 is $ \gtrsim 200 \, GeV$, the chargino masses are
$m_{\tilde{\chi}_{1,2}} \gtrsim 650\, ,\, 370 \, GeV$, the stop masses are
$m_{\tilde{t}_{1,2}} \gtrsim 1000\, ,\, 790 \, GeV$, the sbottom masses are
$m_{\tilde{b}_{1,2}} \gtrsim 1000\, ,\, 960 \, GeV$, the slepton masses are
$m_{\tilde{\tau}_{1,2}} \gtrsim 340\, ,\, 210 \, GeV$, the sneutrinos are
$m_{\tilde{\nu}} \gtrsim 330 , GeV$ 
and the radiative 1-loop corrected Higgs masses
are $M_h, M_{A,H,H\pm} \gtrsim 108 \,,\, 780 \, GeV$, respectively.
Thus, we conclude that
the recent LEP+SLD and CDF/D{\O} data analysis
 favours the MSSM with radiative symmetry breaking
only  in the heavy limit of the sparticle masses.








\section{Conclusions}

We have considered the supersymmetric one loop corrections to the
effective mixing angle $s_f^2$ which is experimentally determined in LEP and
SLD experiments from measurements of on resonance left/right and
forward/backward asymmetries. This effective angle differs from the
corresponding mixing angle $\hs$ defined as the ratio of couplings which is
useful
to test unification of couplings in unified schemes encompassing the
Standard Model. The difference of the two angles, while very small in
the Standard Model, is substantial in supersymmetric extensions of it 
due to large logarithmic $log(\frac{M^2_{SUSY}}{M_Z^2})$ dependences
of $\hs$. Thus, although $\hs$ is a useful theoretical tool to test
the unification of couplings, it is not the proper quantity to compare with
experimental data which have already reached a high degree of accuracy.
Therefore, the relation between the two definitions is of utmost importance
for phenomenological studies of supersymmetric extensions of the Standard
Model.

In this article we have calculated all corrections to the factor
$\Delta{k_f}$ relating the two angles $s_f^2$ and $\hs$ including the
non-universal corrections from vertices and external fermions.
While $\Delta{k_f}$ is plagued by large logarithms in the limit where 
the supersymmetry breaking scale is large, the effective weak mixing angle
does not suffer from such large logarithms. 
In fact, we have proven that there are no dangerous logarithmic corrections
$log(\frac{M^2_{1/2}}{M_Z^2})$ from the chargino/neutralino sector
to the effective weak mixing angle. The decoupling of large logarithms
involving the Higgsino mixing parameter $\mu$, which in the constrained MSSM
with radiative symmetry breaking,
is large, is obtained in the same manner.
The cancellation of potentially dangerous terms also holds for the
contributions of the squark and slepton sector. The
cancellation of the  $log(\frac{M^2_{SUSY}}{M_Z^2})$ terms in the
$\overline{DR}$ scheme had been shown only numerically in previous studies.

It must be noted that there are large logarithmic terms 
in the    ``non-oblique'' supersymmetric
wave function renormalization of external fermions and vertex corrections
of the vertex $Zf\overline{f}$. Nevertheless, we have analytically proven  
that they get decoupled from $\Delta{k_f}$ and, hence, from
the effective weak mixing angle itself. In addition to
the analytical results described in  chapter III,
we have also displayed representative numerical results in Table I in two
particular cases of the MSSM.

We have also presented analytically, the decoupling of the
large logarithmic terms from $s^2_f$ in the case of the non-universal
SQCD corrections. Besides the self-cancellations of
this terms from the relevant diagrams Fig.2c and Fig.2d, there are
additional cancellations from the summation of these diagrams due
to their opposite sign. 
We have found that these corrections are very small and 
could be safely ignored from the analysis in the present experimental 
accuracy.

We have further proceeded to a numerical study of the one loop corrected
effective mixing angle having as inputs the values of $\alpha_{EM}$, $M_Z$,
the Fermi coupling constant $G_F$ and the experimental values for the
fermion masses. Assuming coupling constant unification and radiative
breaking of the electroweak symmetry we have scanned the  
soft SUSY breaking parametric space and given theoretical predictions for the
value of the effective mixing angles, the value of the strong coupling
constant at $M_Z$ and the value of the W - boson mass as this is determined
from the parameter $\rho$ and the effective weak mixing angle.
We find that the large logarithmic corrections of the
form $log (\frac{M_{SUSY}^2}{M_Z^2})$ indeed get decoupled  from the 
extracted value of the effective weak mixing angle in the region of
large $M_{1/2}$ and $M_0$ (Figure 3) following our 
analytical calculations. The predicted MSSM values of the effective
angles are in agreement with the LEP+SLD data (Figure 4)
 as well as
with the new CDF/D{\O} \cite{top}
 results for the top mass $m_t=175 \pm 5 \, GeV$ 
(Figure 5)
 in the region where
all superparticles are quite massive. In this region, MSSM predicts 
values of the W-gauge boson mass which are
 in  agreement  with the new \cite{LEP}
CDF,UA2,{D\O} average value $80.405 \pm 0.089 \, GeV$ (Figures 6,7).
Large logarithms are also decoupled from the {\it left-right} asymmetry value
${\cal A}^e$. MSSM seems to prefer the experimental LEP value of 
${\cal A}^e$, rather than
the average value from LEP+SLD (Figure 8). Finally, values
of $M_{1/2}$ which are greater than $500 \, GeV$ are favoured by
the MSSM if one assumes the present LEP and CDF/D{\O} data for
$s_l^2$, $\alpha_s$ and $m_t$ (Figure 9).

\vspace*{0.1cm}
{\noindent\bf Note Added :} 


\vspace{.2cm}
After submitting this article for publication we became aware of the paper
by P. Chankowski and S. Pokorski \cite{Pok3} where corrections to the
leptonic mixing angle and predictions for the W boson mass are presented.



\vspace*{0.1cm}
{\noindent\bf Acknowledgements} 

\vspace{.2cm}
The authors wish to thank Peggy Kouroumalou who collaborated in the early stages 
of this work. A.D. and K.T. acknowledge financial support from the 
research program $\Pi{\rm ENE}\Delta$-95
of the Greek Ministry of Science and Technology. A.B.L. and K. T. acknowledge
 support from
the TMR network ``Beyond the Standard Model", ERBFMRXCT-960090. 
A. B. L. acknowledges 
support from the Human Capital and Mobility program CHRX-CT93-0319.



\newpage
{\noindent\bf Appendix A: Quick reference to neutralino/chargino
and their interactions}
\vspace{.7cm}
\setcounter{equation}{0}
\renewcommand{\theequation}{A.\arabic{equation}}


In the $\tilde{B}$, $\tilde{W}^{(3)}$, 
$i \tilde{H}_{1}^0$, $i \tilde{H}_{2}^0$, basis 
the neutralino mass matrix is
%%%%%%%%%%%%%%%%%%%
\begin{equation}
{\cal M}_N \ =\ \left(\begin{array}{cccc} M_1 & 0 &
g^\prime \frac{\upsilon cos\beta}{2} &
-g^\prime \frac{\upsilon sin\beta}{2}
\\[1mm] 0 & M_2 & -g \frac{\upsilon cos\beta}{2} &
g \frac{\upsilon sin\beta}{2}
\\[1mm] g^\prime \frac{\upsilon cos\beta}{2}&
-g \frac{\upsilon cos\beta}{2} & 0 & -\mu \\[1mm]
-g^\prime \frac{\upsilon sin\beta}{2}
&g \frac{\upsilon sin\beta}{2} & -\mu & 0
\end{array} \right)\ .\label{mchi0}
\end{equation}
%%%%%%%%%%%%%%
The mass eigenstates (${\tilde \chi}_{1,2,3,4}^0$)
 of neutralino mass matrix ${\cal M}_N$ are
written as 
%%%%%%%%%%%%%%%%%%%
\begin{equation}
{\cal O} \,\,  \left ( \begin{array}{c} {\tilde \chi}_1^0 \\
{\tilde \chi}_2^0 \\ {\tilde \chi}_3^0 \\ {\tilde \chi}_4^0
\end{array} \right )
\ =\ \left ( \begin{array}{c} \tilde{B} \\ \tilde{W}^{(3)} \\
i \tilde{H}_{1}^0 \\ i \tilde{H}_{2}^0
\end{array} \right )\;.
\end{equation}
%%%%%%%%%%%%%%%%%%%
and 
%%%%%%%%%%%%
\begin{equation}
{\cal O}^T {\cal M}_N {\cal O}\ ={\rm Diag} \left (
m_{{\tilde \chi}^0_1},m_{{\tilde \chi}^0_2}, m_{{\tilde \chi}^0_3},
m_{{\tilde \chi}^0_4} \right ) \;,
\end{equation}
%%%%%%%%%%%%
where ${\cal O}$ is a real orthogonal matrix. Note that when 
electroweak breaking effects are ignored ${\cal O}$ can get
the form
%%%%%%%%%%%%%%%%%%%
\begin{equation}
{\cal O} \ =\ \left ( \begin{array}{cc} {\bf 1_2} & {\bf 0_2} \\[1mm]
{\bf 0_2} & \begin{array}{cc} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
-\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{array} \end{array} 
\right ) \;.
\end{equation}
%%%%%%%%%%%%%%%%%

The chargino mass matrix can be obtained from the following
Lagrangian mass terms
%%%%%%%%%%%%
\begin{equation}
{\cal L}^{mass}_{charginos} \ =\ 
- \left ( \tilde{W}^- , {i \tilde{H}_{1}^-} \right ) {\cal M}_c 
\left (\begin{array}{c} \tilde{W}^+ \\ {i \tilde{H}_{2}^+} \end{array}
\right ) \, + \, (h.c) \;,
\end{equation}
%%%%%%%%
where we have defined $\tilde{W}^\pm \equiv \frac{ \tilde{W}^{(1)}
\mp i \tilde{W}^{(2)} }{\sqrt{2}}$ and 
%%%%%%%%%
\begin{equation}
{\cal M}_C \ =\ \left (\begin{array}{cc} M_2 &
 -g \frac{\upsilon sin\beta}{\sqrt{2}} \\[1mm]
- g \frac{\upsilon cos\beta}{\sqrt{2}} & \mu \end{array}
\right )\; .
\label{chmat}
\end{equation}
%%%%%%%%%%%%
Diagonalization of this matrix gives
%%%%%%%%
\begin{equation}
U {\cal M}_c V^\dagger \ =\ \left (\begin{array}{cc} m_{\tilde{\chi}_1}
& 0 \\[1mm]
0 & m_{\tilde{\chi}_2} \end{array} \right ) \; .
\end{equation}
%%%%%%%%%%%%%%%%%%%
Thus,
%%%%%%%%%%%%
\begin{equation}
{\cal L}^{mass}_{charginos} \ =\ -m_{\tilde{\chi}_1} 
\bar{\tilde{\chi}_1} \tilde{\chi}_1 -  m_{\tilde{\chi}_2}
\bar{\tilde{\chi}_2} \tilde{\chi}_2 \; .
\end{equation}
%%%%%%%%%%%%%%
The Dirac chargino states $\tilde{\chi}_{1,2}$ are given by
%%%%%%%%%%%%%%%
\begin{equation}
\tilde{\chi}_1 \equiv \left (\begin{array}{c} \lambda_1^+ \\
\bar{\lambda}_1^- \end{array} \right ) \,\, , \,\,
\tilde{\chi}_2 \equiv \left (\begin{array}{c} \lambda_2^+ \\
\bar{\lambda}_2^- \end{array} \right ) \;.
\end{equation}
%%%%%%%%%%%%%%%%%
The two component Weyl spinors $\lambda^\pm_{1,2}$ are related
to $\tilde{W}^\pm$, ${i \tilde{H}_{1}^-}$, ${i \tilde{H}_{2}^+}$ by
%%%%%%%%%%%%%%%
\begin{equation}
V \left ( \begin{array}{c} \tilde{W}^+ \\ {i \tilde{H}_{2}^+}
\end{array} \right ) \equiv \left (\begin{array}{c}
\lambda_1^+ \\ \lambda_2^+ \end{array} \right ) \,\, , \,\,
\left ( \tilde{W}^- \, , \, {i \tilde{H}_{1}^-} \right ) U^\dagger
 \equiv \left ( \lambda_1^- \, , \, \lambda_2^- \right ) \;.
\end{equation}
%%%%%%%%%%%%%%%%

The gauge interactions of charginos and neutralinos can
be read from the following Lagrangian\footnote{
In our notation $\hat{e} \equiv $electron's charge just
opposite to that used in ref. \cite{Hollik2}.}
%%%%%%%%%%%%%%%
\begin{equation}
{\cal L} \ =\ \hat{g} \left ( W^+_\mu J_-^\mu + W_\mu^- J_+^\mu 
\right ) + \hat{e} A_\mu J_{em}^\mu +\frac{\hat{e}}{\hat{s} \hat{c}}
Z_\mu J_Z^\mu \;.
\end{equation}
%%%%%%%%%%%%%%%%%
Also,
%%%%%%%%%%%%%
\begin{equation}
\left (\begin{array}{c} Z_\mu \\[1mm] A_\mu \end{array} \right )
\ =\ \left ( \begin{array}{cc} \hat{c} & \hat{s} \\[1mm]
-\hat{s} & \hat{c} \end{array} \right ) \, 
\left ( \begin{array}{c} W_\mu^{(3)} \\[1mm] B_\mu \end{array}
\right ) \;.
\end{equation}
%%%%%%%%%%%%%%%%%
The currents $J_+^\mu$, $J_{em}^\mu$ and $J_Z^\mu$ are given by
%%%%%%%%%%%%%%%%%
\begin{equation}
J_+^\mu \equiv \bar {{\tilde \chi}^0_a}  \gamma^\mu \left [
{\cal P}_L {\cal P}^L_{a i}+ 
{\cal P}_R {\cal P}^R_{a i}  \right ]
\tilde{\chi}_i \;\; a=1...4,\;\; i=1,2 \;,
\end{equation}
%%%%%%%%%%%%%%
where
${\cal P}_{L,R} = \frac{1 \mp \gamma_5 }{2}$ and
%%%%%%%%%%%%%%%
\begin{eqnarray}
& &{\cal P}^L_{a i}\equiv +\frac{1}{\sqrt{2}} {\cal O}_{4 a} V^{*}_{i 2}
- {\cal O}_{2 a} V^{*}_{i 1}\;,\nonumber \\[2mm]
& &{\cal P}^R_{a i}\equiv -\frac{1}{\sqrt{2}} {\cal O}_{3 a} U^{*}_{i 2}
- {\cal O}_{2 a} U^{*}_{i 1} \;.
\end{eqnarray}
%%%%%%%%%%%%%% 
The electromagnetic current $J_{em}^\mu$ is
%%%%%%%%%%%%
\begin{equation}
J_{em}^\mu \ =\ \bar{\tilde\chi}_1 \gamma^\mu \tilde{\chi}_1 +
\bar{\tilde\chi}_2 \gamma^\mu \tilde{\chi}_2 \;.
\end{equation}
%%%%%%%%%%%%%%%%%%%
Finally, the neutral current $J_Z^\mu$ can be read from
%%%%%%%%%%%%%%%
\begin{equation}
J^\mu_Z \equiv \bar  {{\tilde \chi}_i}  \gamma^\mu \left [
{\cal P}_L {\cal A}^L_{i j} + {\cal P}_R {\cal A}^R_{i j} \right ]
\tilde{\chi}_j + \frac{1}{2} \bar {{\tilde \chi}^0_a} \gamma^\mu \left [
{\cal P}_L {\cal B}^L_{a b} + {\cal P}_R {\cal B}^R_{a b} \right ]
{{\tilde \chi}^0_b} \;,
\end{equation}
%%%%%%%%%%%%%%%%%%%%%
with
%%%%%%%%%%%%%%
\begin{eqnarray}
{\cal A}^L_{i j} &=& \hc \delta_{i j} -\frac{1}{2} V_{i 2} V^{*}_{j 2}
\;,\nonumber \\
{\cal A}^R_{i j} &=& \hc \delta_{i j} -\frac{1}{2} U_{i 2} U^{*}_{j 2}
\;,\nonumber \\
{\cal B}^L_{a b} &=& \frac{1}{2} \left ( {\cal O}_{3 a} {\cal O}_{3 b} -
{\cal O}_{4 a} {\cal O}_{4 b} \right )
\;,\nonumber \\
{\cal B}^R_{a b} &=& - {\cal B}^L_{a b} \;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%
Note that since ${\cal B}^R_{a b} = - {\cal B}^L_{a b}$ the neutralino
contribution to $J_Z^\mu$ can be cast into the form
%%%%%%%%%%%%%%%%%%%
\begin{equation}
J_Z^\mu \ =\ -\frac{1}{2} {\cal B}^L_{a b} \left ( \bar{{\tilde \chi}^0_a}
\gamma^\mu \gamma^5 {{\tilde \chi}^0_b} \right ) \;.
\end{equation}
%%%%%%%%%%%%%




For the supersymmetric external fermion corrections we need know the
chargino and neutralino couplings to fermions and sfermions.
The relevant chargino couplings are given by the following
Lagrangian terms
%%%%%%%%%%%%%
\begin{equation}
{\cal L} = i \; {\bar{\tilde {\chi}}}_{i}^{c} \;
 ({\cal P}_L \, a^{ {f^{\prime}}   {\tilde f} }_{ij} +
{\cal P}_R \, b^{ {f^{\prime}}   {\tilde f} }_{ij}) \,  {f^\prime } \,
 {\tilde f}_{j}^{*}
\, + \,
i \; {\bar{\tilde {\chi}}}_{i} \;
 ({\cal P}_L \, a^{f {\tilde f}^{\prime} }_{ij} +
{\cal P}_R \, b^{f {\tilde f}^{\prime} }_{ij}) \,  {f} \,
 {\tilde f}_{j}^{\prime *} \, + \, (h.c) \;.
\end{equation}
In this, ${\chi}_{i} \; (i=1,2)$ are the positively charged charginos  and 
${\chi}_{i}^{c}$ the corresponding charge conjugate states having 
opposite charge. $f\, , \,{f}^{\prime}$ are ``up" and ``down"
fermions, quarks or leptons, while ${\tilde f}_i\, , \,{\tilde f}_i^{\prime}$
are the corresponding sfermion mass eigenstates.
The left and right-handed couplings appearing above are given by
%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
a^{{f^{\prime}}{\tilde f} }_{ij} \ori &= \ori
g V_{i1}^{*} \, K^{\tilde f}_{j1} - h_{f} V_{i2}^{*}  K^{\tilde f}_{j2} \ori ,
\ori  &b^{ {f^{\prime}} {\tilde f} }_{ij} \ori =
 \ori -h_{f ^\prime} \,  U_{i2}^{*} K^{\tilde f}_{j1}  \;,  \nonumber  \\
a^{f {\tilde f}^{\prime} }_{ij} \ori &= \ori
g U_{i1} \, K^{{\tilde f}^{\prime}}_{j1} + h_{f^\prime} \, U_{i2}
  K^{{\tilde f}^{\prime}}_{j2} \ori , \ori
&b^{f {\tilde f}^{\prime} }_{ij} \ori = \ori
h_{f } \,  V_{i2} K^{{\tilde f}^{\prime}}_{j1} \;.  \nonumber
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%
In the equation above $h_{f } \, , \, h_{f ^\prime} $ are the Yukawa
couplings of the up and down fermions respectively. The matrices
$K^{\tilde{f},{\tilde f}^\prime} $ which diagonalize the sfermion mass matrices become the
unit matrices in the absence of left-right sfermion mixings.
For the electron and muon family the lepton masses are taken to be
vanishing in the case that mixings do not occur. In addition the 
right-handed couplings, are zero.  \newline 
The corresponding neutralino couplings are given by
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
{\cal L} = i \; {\bar  {{\tilde \chi}^0_a}} \;
 ({\cal P}_L \, a^{ {f}   {\tilde f} }_{aj} +
{\cal P}_R \, b^{ {f}   {\tilde f} }_{aj}) \,  {f} \,
 {\tilde f}_{j}^{*}
\, + \,
i \; {\bar{{\tilde \chi}^0_a}} \;
 ({\cal P}_L \, a^{{f^{\prime}} {\tilde f}^{\prime} }_{aj} +
{\cal P}_R \, b^{{f^{\prime}} {\tilde f}^{\prime} }_{aj}) \, {f^{\prime}} \,
 {\tilde f}_{j}^{\prime *} \, + \, (h.c) \;.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%
The left and right-handed couplings for the up fermions, sfermions
are given by
%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
a^{{f}{\tilde f}}_{aj} \ori &= &\ori
{\sqrt{2}} \, ( g {T^{3}_f}{O_{2a}}+{g^{\prime}}{\frac{Y_f}{2}} \,{O_{1a}})
\, {K^{f}_{j1}} \ori + \ori h_{f} \, {O_{4a}}\,  {K^{f}_{j2}}  \ori , \ori
\nonumber \\
b^{{f}{\tilde f} }_{aj} \ori & = & \ori
{\sqrt{2}} \, (-{g^{\prime}}{\frac{Y_{f^c}}{2}} \,{O_{1a}}) \, {K^{f}_{j2}}
\ori - \ori h_{f} \, {O_{4a}}\,  {K^{f}_{j1}} \;,  \nonumber  
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%
while those for the down fermions and sfermions are given by
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
a^{{\fp}{\tilde \fp}}_{aj} \ori &= &\ori
{\sqrt{2}} \,
( g {T^{3}_{\fp}}{O_{2a}}+{g^{\prime}}{\frac{Y_\fp}{2}} \,{O_{1a}})
\, {K^{\fp}_{j1}} \ori - \ori h_{\fp} \, {O_{3a}}\,  {K^{\fp}_{j2}}  \ori , \ori
\nonumber \\
b^{{\fp}{\tilde \fp} }_{aj} \ori & = & \ori
{\sqrt{2}} \, (-{g^{\prime}}{\frac{Y^{\prime}_{f^c}}{2}} \,{O_{1a}})
 \, {K^{\fp}_{j2}}
\ori + \ori h_{\fp} \, {O_{3a}}\,  {K^{\fp}_{j1}}\;.   \nonumber  
\end{eqnarray}


\vspace*{1cm}
%\newpage
{\noindent\bf Appendix B: Passarino - Veltman functions }
\vspace{.7cm}
\setcounter{equation}{0}
\renewcommand{\theequation}{B.\arabic{equation}}

%%%%%%%%%%%%%%%%%%
All functions appearing in the propagator corrections in 
the main text can be expressed in terms of the basic Passarino - Veltman
integrals $A_0, B_0$ in the following way
%%%%%%%%%%%%%%%%%%%
\begin{equation}
A_0(m)= m^2 (\frac{1}{\hat{\epsilon}} + 1 - \ln\frac{m^2}{Q^2} )\;,
\end{equation}
%%%%%%%%%%%%%%%%%
\begin{equation}
B_0(p,m_1,m_2)=\frac{1}{\hat{\epsilon}}
-\int_0^1 dx \ln\frac{(1-x)m_1^2+x m_2^2
-x (1-x)p^2-i\epsilon}{Q^2}\;,
\end{equation}
%%%%%%%%%%%%%%%%%
where $\frac{1}{\hat{\epsilon}}=\frac{1}{\epsilon}-\gamma_E+\ln 4\pi$.
This reduction can be done with the following identities (in what follows
we made use of these functions only)
%%%%%%%%%%%%%
\begin{eqnarray}
H(p,m_1,m_2)&=&4\tilde{B}_{22}(p,m_1,m_2)+
(p^2-m_1^2-m_2^2)B_0(p,m_1,m_2)\;,\\[3mm]
\tilde{B}_{22}(p,m,m)&=&-\frac{1}{12}p^2 B_0(p,m,m)-\frac{1}{18}p^2
+\frac{1}{3}\left [m^2+m^2 B_0(p,m,m)-A_0\right ]\;,\\[3mm]
\tilde{B}_{22}(0,m,m)&=&0\;,\\[3mm]
B_0[p,m,m]&\stackrel{m^2>>p^2}{\longrightarrow}&-\ln(m^2/Q^2)\;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%
For the vertex and box corrections to muon decay we need the
functions $B_0, B_1, C_0$
at zero momenta. The following relations are useful in order to express
the contributions to $\delta^{SUSY}_{VB}$ in terms of the
masses of the particles in the loop

\begin{eqnarray}
B_0(0,m_1,m_2) \ =\ {1\over\hat\epsilon} + 1 + \ln\left(Q^2\over
m_2^2\right) + {m_1^2\over m_1^2-m_2^2} \ln\left(m_2^2\over
m_1^2\right)~,
%\nonumber
\end{eqnarray}
%
\begin{eqnarray}
B_1(0,m_1,m_2) \ =\ {1\over2}\biggl[{1\over\hat\epsilon} + 1 +
\ln\left(Q^2\over m_2^2\right) + \left({m_1^2\over m_1^2 -
m_2^2}\right)^2\ln\left({m_2^2\over m_1^2}\right) +
{1\over2}\left({m_1^2+m_2^2\over m_1^2-m_2^2}\right)\biggr]~,
%\nonumber
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
C_0( m_1,m_2,m_3)\ =\ - \int_0^1 dx \int_0^{1-x} dy\, \frac{1}
{m_1^2 x +m_2^2 y +m_3^2 (1-x-y)}\;,
%\nonumber
\end{eqnarray}
%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
C_0(m_1,m_2,m_2)\ =\ \frac{1}{m_1^2 -m_2^2} + \frac{m_1^2}{\left ( m_1^2
- m_2^2 \right )^2} \ln \left ( \frac{m_2^2}{m_1^2} \right ) \;.
%\nonumber
\end{eqnarray}
%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}

\bibitem{Lang}  see for instance, P. Langacker in {\it ``Precision Experiments,
Grand Unification and Compositeness"}, invited talk SUSY 95, Palaiseau
 France, May 1995, NSF-ITP-95-140, UPR-0683T.

\bibitem{LEP}
R. M. Barnett {\it al.}, \pr{54}{1}{1996}, 
and 1997 off-year partial update for the 1998 edition available on 
    the PDG WWW pages ({\tt URL: http://pdg.lbl.gov/}). 

%LEP Electroweak Working Group and the SLD Heavy Flavour Group
%CERN-PRE/96-183,  contribution
%to the 28th International Conference on High Energy
%Physics, Warsaw, Poland (1996).

\bibitem{altarelli}
G. Altarelli, R. Barbieri and F. Caravaglios, e-print .

\bibitem{sirlin}
G. Degrassi and A. Sirlin, \np{352}{342}{1991}.


\bibitem{gambino}
P. Gambino and A. Sirlin, \pr{49}{1160}{1994}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{all1}
W. J. Marciano and J. L. Rosner, \prl{65}{2963}{1990} and Erratum {\it ibid.}
{\bf 68}, 898 (1992); 

G. Degrassi, S. Fanchiotti and A. Sirlin, \np{351}{49}{1991}; 

B. A. Kniehl and A. Sirlin, \pr{47}{883}{1993}; 

S. Fanchiotti, B. A. Kniehl and A. Sirlin, \pr{48}{307}{1993}; 

B. A. Kniehl, presented at Tennessee International
Symposium on Radiative Corrections: 
Status and Outlook, Gatlinburg, TN, 27 Jun - 1 Jul 1994; 

A. Sirlin, \pl{348}{201}{1995}, addendum-{\it ibid} 
{\bf B352}, 498 (1995);

B. A. Kniehl and A. Sirlin, \np{458}{35}{1996}; 

P. Gambino, {\it Acta. Phys. Polon.} {\bf B27}, 3671 (1996);

G. Degrassi, P. Gambino and A. Sirlin, \pl{394}{188}{1997}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{all2}
R. Barbieri, M. Beccaria, P. Ciafaloni, G. Curci and A. Vicere,
\pl{288}{95}{1992} ; \np{409}{105}{1993};

J. Fleischer, O. V. Tarasov and F. Jegerlehner, \pl{319}{249}{1993};

G. Buchalla and A. J. Buras, \np{398}{285}{1993};

L. Andeev et al, \pl{336}{560}{1994}, E: {\it {ibid}}.
{\bf{B349}} 597 (1995); 

K. G. Cheturkin, J. H. K{\"{u}}hn and M. Steinhauser,
\prl{75}{3394}{1995};

K. G. Cheturkin, J. H. K{\"{u}}hn and A. Kwiatkowski, in:
{\it {Precision Calculations for the Z resonance}}, CERN 95-03,eds.
D. Bardin, W. Hollik, G. Passarino, S. Peris, A. Santamaria,
CERN-TH-95-21 (1995);

G. Degrassi, P. Gambino and A. Vicini, \pl{383}{219}{1996}. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{djoua}
A. Djouadi and C. Verzegnassi, \pl{195}{265}{1987};

A. Djouadi, Nuovo Cimento {\bf{100A}} (357) 1988;

B. A. Kniehl, \np{347}{86}{1990};

A. Djouadi and P. Gambino, \pr{49}{3499 and 4705}{1994}, ibid.
{\bf{D51}} 218 (1995), E: {\it {ibid}}. {\bf{D53}} 4111 (1996). 

%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{NHK}H. P. Nilles, {\it Phys. Rep. }
{\bf 110} (1984)1;\\
H. E. Haber and G. L. Kane,
{\it Phys. Rep. }{\bf 117} (1985)75;\\
 A. B. Lahanas and D. V. Nanopoulos, {\it Phys. Rep.}
{\bf 145} (1987)1.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{polon}
M. Drees, K. Hagiwara, and A. Yamada, \pr{45}{1725}{1992}; 

G. Altarelli, R. Barbieri, and F. Caravaglios, \pl{314}{357}{1993}; 

P. Langacker and N. Polonski, \pr{47}{4028}{1993}; {\bf{D52}} 3081 (1995); 

N. Polonski, Report No. UPR-0641-T, e-print ;

D. Garcia and J. Sola, \pl{354}{335}{1995};

G.L. Kane, R.G. Stuart and J.D. Wells, \pl{354}{350}{1995};

P. H. Chankowski, Z. Plucienik and S. Pokorski, \np{439}{23}{1995}; 

P.H. Chankowski and S. Pokorski, \pl{366}{188}{1996};

P. H. Chankowski, Z. Plucienik, S. Pokorski and C. E. Vayonakis,
\pl{358}{264}{1995};

D. Garcia and J. Sol\`{a}, \mpl{9}{211}(1994);

D. Garcia, R.A. Jim\'{e}nez and J. Sol\`{a}, \pl{347}{309}{1995},
{\it ibid.} {\bf B347} 321 (1995), and Erratum {\it ibid}
 {\bf B351} 602 (1995);

D.M. Pierce and J. Erler, e-print , talk
presented at the 5th International Conference on Physics Beyond the
Standard Model, Balholm (1997);

P.H. Chankowski, J. Ellis and S. Pokorski, e-print ;

W.~de Boer {\it et al.}, e-print ;

D.M. Pierce and J. Erler, e-print .
%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Bagger}J. Bagger, K. Matchev, D. Pierce and R. Zhang, 
\np{491}{3}{1997}.

\bibitem{Rociek}P. Chankowski et al, \np{417}{101}{1994};

P. Chankowski, S. Pokorski and J. Rosiek, \np{423}{437}{1994}.

\bibitem{Hollik1}
W. de Boer, A. Dabelstein, W. Hollik, W. Mosle, U. Schwickerath,
\zp{75}{627}{1997}; \\
W. Hollik in: {\it { Electroweak Precision Observables in the MSSM
and Global Analysis of Precision Data}}, Talk at the International
workshop on Quantum Effects in the MSSM, Barcelona 9-23 September 1997.

\bibitem{finnell}
M. Boulware and D. Finnell, \pr{44}{2054}{1991}.

\bibitem{kolda}
J.D. Wells, C. Kolda and G.L. Kane, \pl{338}{219}{1994};

M. Drees, R.M. Godbole, M. Guchait, S. Raychaudhuri, D.P. Roy,
\pr{54}{5598}{1996};

P.H. Chankowski and S. Pokorski, \np{475}{3}{1996}.


\bibitem{Tamvakis}J. Ellis, J. Hagelin, D.V. Nanopoulos and K. Tamvakis,
\pl{125}{1983}{275};

L. E. Ibanez and G. G. Ross, \pl{110}{215}{1982};

L. Alvarez-Gaume, J. Polchinski and M. Wise, \np{221}{495}{1983};

%J. Ellis, A.B. Lahanas, D.V. Nanopoulos and K. Tamvakis,
%\pl{134}{429}{1983}.

\bibitem{Siegel}
W. Siegel, \pl{84}{193}{1979}; 

 D. M. Capper, D. R. T. Jones and
P. van Nieuwenhuizen,   \np{167}{479}{1980}; 

I. Antoniadis, C. Kounnas and K. Tamvakis, \pl{119}{377}{1982};

S. P. Martin and M. T. Vaughn, Phys. Lett. B318(1993)331.


\bibitem{Martin}
I. Jack, D. R. T. Jones, S. P. Martin, M. T. Vaughn and
Y. Yamada, \pr{50}{5481}{1994}.




%\bibitem{NHK}H. P. Nilles, {\it Phys. Rep. }
%{\bf 110} (1984)1;\\
%H. E. Haber and G. L. Kane,
%{\it Phys. Rep. }{\bf 117} (1985)75;\\
% A. B. Lahanas and D. V. Nanopoulos, {\it Phys. Rep.}
%{\bf 145} (1987)1.


\bibitem{dedes2}
A. Dedes, A. B. Lahanas, J. Rizos and K. Tamvakis, \pr{55}{2955}{1997}.

\bibitem{Peskin}
M. E. Peskin and D. V. Schroeder, {\it An Introduction to Quantum
Field Theory}, Addison-Wesley, 1995.


%\bibitem{LEPII}
%The LEP Collaborations ALEPH, DELPHI, L3, OPAL, the LEP
%Electroweak Working Group and the SLD Heavy Flavour 
%Working Group, CERN-PRE/96-183,  contributed
%to the 28th International Conference on High Energy
%Physics, Warsaw, Poland (1996).


\bibitem{Pokorski}P.H. Chankowski, A. Dabelstein, W. Hollik, 
W.M. Mosle, S. Pokorski and J. Rosiek, \np{417}{101}{1994}.

\bibitem{passarino}
G. Passarino and M. Veltman, \np{160}{151}{1979}.

\bibitem{ahn}
C. Ahn, B. Lynn, M. Peskin and S. Selipsky, \np{309}{221}{1988}.

\bibitem{2loop}
S. P. Martin and M. T. Vaughn, \pr{50}{2282}{1994};

Y. Yamada, \pr{50}{3537}{1994};

I. Jack and D. R. T. Jones, \pl{333}{372}{1994}.

\bibitem{olden}
G. J. van Oldenborgh, {\it {Comput. Phys. Commun. }} {\bf {66}} 
1 (1991).

\bibitem{Tata}
X. Tata, Lectures given at 9th Jorge
Andre Swieca Summer School: Particles and Fields, Sao Paulo, Brazil, 
16-28 Feb 1997, e-print .

\bibitem{Dedes3}
A. Dedes and K. Tamvakis, \pr{56}{1496}{1997}.

\bibitem{top}

D{\O} Collaboration: S. Abachi {\it et al.}, \prl{79}{1197}{1997};

CDF Collaboration: F. Abe {\it et al.}, \prl{79}{1992}{1997}.

\bibitem{Hollik2}
W. Hollik in {\it {Renormalization of the Standard Model}}, published in
 {\bf {"Precision Tests of the Standard Model"}}, World Scientific, ed.
 P. Langacker.

\bibitem{sqcdrefs}
L. Clavelli, P.W. Coulter and L.R. Surguladze, e-print .

The two loop QCD corrections to the scalar quark contributions to the
$\rho$ parameter is the subject of:

A. Djouadi, P. Gambino, S. Heinemeyer, W. Hollik, C. J\"{u}nger and
G. Weiglein, \prl{78}{3626}{1997};

A. Djouadi, e-print ;

A. Djouadi, P. Gambino, S. Heinemeyer, W. Hollik, C. J\"{u}nger and
G. Weiglein, e-print ;

G. Weiglein, 
Talk given at 21st
International School of Theoretical Physics (USTRON 97), 
Ustron, Poland, 19-24 Sep 1997, e-print .

\bibitem{Pok3}
P. Chankowski and S. Pokorski ,  to appear 
in ``Perspectives in supersymmetry" edited by G.L. Kane,
World Scientific, 1997. 

\end{thebibliography}


%%%%%%%%%%%%%%%%%%%%%%%%%% TABLE %%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%\draft
\mediumtext
\begin{table}
\caption{Partial and total contributions to ${\Delta k}_f$, 
($f=lepton,charm,bottom$), for two sets of inputs shown at the top.
Also shown are the predictions for the effective weak mixing angles and the
asymmetries. In the first five rows we display the universal contributions
to $10^3 \; \times \;{\Delta k}$ of squarks ($\tilde q$),
sleptons($\tilde l$), Neutralinos and Charginos (${\tilde Z},{\tilde C}$),
ordinary fermions and Higgses %$\; \; \;  $
(The number shown in the middle below the
"charm" column refers to "lepton" and "bottom" as well). In the next five rows
we display the contributions of gauge bosons as well as the supersymmetric
$Electroweak \; (EW)$ and $SQCD$ vertex
and external fermion wave function renormalization corrections
to $10^3 \; \times \;{\Delta k}$. }
%%%%%%
\begin{tabular}{ccccccccc}
 &$M_0=200$&$M_{1/2}=200$&$A_0=200$& &
 &$M_0=400$&$M_{1/2}=400$&$A_0=500$ \\
 &$m_t=175$&$tanb=4$&$\mu > 0$&  &
 &$m_t=175$&$tanb=4$&$\mu > 0$  \\
\tableline
\\
 &$lepton$ &$charm$ &$bottom$& &
 &$lepton$ &$charm$ &$bottom$ \\   \\
\tableline
\\
${\tilde q}$& &-6.6206 & && & &-8.8188 & \\
${\tilde l}$& &-0.3123 & && & &-0.3791 & \\
${\tilde Z},{\tilde C}$& &-4.4892 & && & &-9.6138 & \\
$Fermions$& &4.6573 & && & &4.5103 & \\
$Higgs$& &-0.7312 & && & &-1.0113 & \\  \\
$Gauge$&-3.1782  &-3.6272 &2.2911  &&
       &-3.1128   &-3.5572 &2.2822  \\ \\
$Vertex(EW)$&1.2324&3.5581 &12.7641  &&
       &2.2825   &4.8539 &18.1519  \\
$Wave(EW)$&-1.3209&-3.5208 &-12.9598 &&
       &-2.2973   &-4.8422 &-18.0059\\   \\
$Vertex(SQCD)$& - &0.2110 &-1.1129  &&
       & -  &0.2361 &-1.1166 \\
$Wave(SQCD)$&- &-0.2103 &1.1012 &&
       & -  &-0.2359 &1.1135\\
\tableline
\\ 
%$Total$ & & & && & & & \\
${\Delta k}\;(\times \; 10^2 ) $ &-1.0763 &-1.1085 &-0.5412 &&
&-1.8440 &-1.8858 &-1.2888  \\
%\tableline
${sin^2} {\theta}_f$& 0.23134 & 0.23126 &0.23259 && &0.23145
& 0.23135 & 0.23276\\
${\cal A}_{LR}^f$& 0.1485 & 0.6684 &0.9348 && &0.1476 &0.6681 & 0.9347\\
${\cal A}_{FB}^f$& 0.0165 & 0.0744 &0.1041 && &0.0163 & 0.0740 & 0.1035\\

\end{tabular}
\end{table}
%%%%%%%%%%%%%%%%%%%%%% End of TABLE I%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%   FIGURES %%%%%%%%%%%%%%%%%%%%%%%
%\newpage

\vspace*{1.5in}
%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\input{sinhat.ps}}
\begin{center}
\footnotesize{{\bf Figure 1:}
 The values of the running weak mixing angle $\hat{s}^2$ at
$M_Z$ in the $\overline{DR}$ scheme 
defined as a ratio of gauge couplings for various
input universal soft gaugino masses $M_{1/2}$
 for particular input of $M_0$ , $A$ , $\tan \beta$ and $m_{t}$ . 
 The strong dependence of $\hat{s}^2$ on $M_{1/2}$ near $M_Z$ is due to 
 the presence of sparticle thresholds.}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newpage
%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%

%\vspace*{1in}

\centerline{\hbox{\psfig{figure=fig1a.ps,width=2.5in}
\psfig{figure=fig1b.ps,width=2.5in}}}
\begin{center}
\footnotesize{{\bf Figure 2a :} Self energy chargino and squark 
 corrections to the
$Zb\bar{b}$ vertex.}
\end{center}
%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%
%\vspace*{1in}
\centerline{\psfig{figure=fig2.ps,width=2.5in}}
%\begin{center}
%%%%%%%%%%%%%%%%%%%%
\vspace*{0.4in}
\centerline{\psfig{figure=fig3.ps,width=2.5in}}
\begin{center}
\footnotesize{{\bf Figure 2b :} Chargino and squark contributions to
the $Zb\bar{b}$ vertex.}
\end{center}
%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%
%\vspace*{1in}
\centerline{\psfig{figure=figqcdver.ps,width=2.5in}}
\begin{center}
\footnotesize{{\bf Figure 2c :} Supersymmetric QCD corrections to
the $Zq\bar{q}$ vertex from gluino and squark contributions.}
\end{center}
%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%
\centerline{\hbox{\psfig{figure=figqcdwfa.ps,width=2.5in}
\psfig{figure=figqcdwfb.ps,width=2.5in}}}
\begin{center}
\footnotesize{{\bf Figure 2d :} Self energy
gluino and squark contributions to the
$Zq\bar{q}$ vertex.}
\end{center}
%%%%%%%%%%%%%%%%%%%%%




\vspace*{0.25in}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\input{sinem12+.ps}}
\begin{center}
\footnotesize{{\bf Figure 3 :}
 The effective weak mixing angle $s_l^2(M_Z)$ in comparison to the
weak mixing angle $\hat{s}^2(M_Z)$ versus $M_{1/2}$ when the
soft parameters $M_0$, $A_0$ vary in the indicated regions.
The width of each branch is due mainly to the variation on $M_0$,
for low $M_{1/2}<200$ GeV, and to the variation of the top mass
for $M_{1/2}>200$ GeV.
The effect of the variation of $A_0=0-900$ GeV and of $\tan\beta=5-28$ 
on $s_l^2$ is negligible.
The error bar show the measured value of $s_l^2=0.23152 \pm 0.00023$,
 obtained at LEP and SLD. 
 The MSSM value is in agreement with the LEP+SLD data for  the bulk of the values
in the soft parameter space.}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%





\vspace*{0.25in}
%%%%%%%%%%%%%%%%%%%%
\centerline{\input{sinall+.ps}}
\begin{center}
\footnotesize{{\bf Figure 4 :} The effective weak mixing angles 
$s^2_c$ and $s^2_b$ . In the region
 $M_{1/2} \rightarrow 900 \, GeV$, the two angles are
separated from each other. 
The dispersion of points around the central value for $M_{1/2}>200$ GeV
is due to the variation of the top mass. For the limiting behaviour 
to be more clearly exhibited, in this  figure
and in figures 3,7 and 8 we do not display the dispersion of points for
$M_{1/2}>800$ GeV.}
% and they agree with the LEP average value
%s^2_f=0.23199 \pm 0.00028$.}
\end{center}
%%%%%%%%%%%%%%%%%%%%%

\newpage

\vspace*{2.25in}
%%%%%%%%%%%%%%%%%%%%
\centerline{\input{mt+.ps}}
\begin{center}
\footnotesize{{\bf Figure 5 :} MSSM predictions for $s^2_l$ as a function
of $m_t$ for two different characteristic values of the
soft breaking parameters. The corresponding values of the light Higgs mass
and their errors due to the variation of $m_t$ are indicated.}
\end{center}
%%%%%%%%%%%%%%%%%%%%%


\newpage

\vspace*{2.25in}
%%%%%%%%%%%%%%%%%%%%
\centerline{\input{mwmt+.ps}}
\begin{center}
\footnotesize{{\bf Figure 6 :} MSSM predictions for physical mass of the
W-boson as a funcion of $m_t$ for the same inputs as in Figure 5.}
% Values around $M_0=M_{1/2}=300 \, GeV$ are
%preferred. }
\end{center}
%%%%%%%%%%%%%%%%%%%%%

\vspace*{0.25in}
%%%%%%%%%%%%%%%%%%%%
\centerline{\input{mw+.ps}}
\begin{center}
\footnotesize{{\bf Figure 7 :}  MSSM prediction for the mass
 of the W-gauge boson 
 as a function of the independent soft parameters $M_{1/2}$, $M_0$ and $A_0$.  
 The  experimental value $M_W=80.427 \pm 0.075$ ($M_W=80.405 \pm 0.089$) GeV
obtained at LEP (CDF,UA2,D{\O})
 is shown for comparison. }
\end{center}
%%%%%%%%%%%%%%%%%%%%%

\vspace*{0.25in}
%%%%%%%%%%%%%%%%%%%%
\centerline{\input{ae+.ps}}
\begin{center}
\footnotesize{{\bf Figure 8 :} The  left-right asymmetry $A_e$ in the
MSSM as a function
of $M_{1/2}$ when we vary  $M_0$, $A_0$, $\tan\beta$ and $m_t$.}
\end{center}
%%%%%%%%%%%%%%%%%%%%%



\vspace*{2in}
%%%%%%%%%%%%%%%%%%%%
\centerline{\input{all+.ps}}
\begin{center}
\footnotesize{{\bf Figure 9 :} Acceptable values in the $M_{1/2}$-$M_0$
plane according to the LEP+SLD data.
 The values of $\tan\beta$ and $A_0$ are taken 
 in the region $2-30$ and $0-900$ GeV, respectively. Only large
values of $M_{1/2}$ are acceptable.}
\end{center}
%%%%%%%%%%%%%%%%%%%%%



 
\end{document}









% GNUPLOT: LaTeX picture with Postscript
\setlength{\unitlength}{0.1bp}
\special{!
%!PS-Adobe-2.0
%%Creator: gnuplot
%%DocumentFonts: Helvetica
%%BoundingBox: 50 50 914 1130
%%Pages: (atend)
%%EndComments
/gnudict 40 dict def
gnudict begin
/Color false def
/Solid false def
/gnulinewidth 5.000 def
/vshift -33 def
/dl {10 mul} def
/hpt 31.5 def
/vpt 31.5 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
/DL { Color {setrgbcolor Solid {pop []} if 0 setdash }
 {pop pop pop Solid {pop []} if 0 setdash} ifelse } def
/BL { stroke gnulinewidth 2 mul setlinewidth } def
/AL { stroke gnulinewidth 2 div setlinewidth } def
/PL { stroke gnulinewidth setlinewidth } def
/LTb { BL [] 0 0 0 DL } def
/LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def
/LT0 { PL [] 0 1 0 DL } def
/LT1 { PL [4 dl 2 dl] 0 0 1 DL } def
/LT2 { PL [2 dl 3 dl] 1 0 0 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
/P { stroke [] 0 setdash
  currentlinewidth 2 div sub M
  0 currentlinewidth V stroke } def
/D { 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
  P } def
/A { stroke [] 0 setdash vpt sub M 0 vpt2 V
  currentpoint stroke M
  hpt neg vpt neg R hpt2 0 V stroke
  } def
/B { 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
  P } def
/C { 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
/T { 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
  P  } def
/S { 2 copy A C} def
end
%%EndProlog
}
\begin{picture}(4320,4628)(0,0)
\special{"
%%Page: 1 1
gnudict begin
gsave
50 50 translate
0.100 0.100 scale
0 setgray
/Helvetica findfont 100 scalefont setfont
newpath
-500.000000 -500.000000 translate
LTa
LTb
600 251 M
63 0 V
3474 0 R
-63 0 V
600 684 M
63 0 V
3474 0 R
-63 0 V
600 1116 M
63 0 V
3474 0 R
-63 0 V
600 1549 M
63 0 V
3474 0 R
-63 0 V
600 1981 M
63 0 V
3474 0 R
-63 0 V
600 2414 M
63 0 V
3474 0 R
-63 0 V
600 2847 M
63 0 V
3474 0 R
-63 0 V
600 3279 M
63 0 V
3474 0 R
-63 0 V
600 3712 M
63 0 V
3474 0 R
-63 0 V
600 4144 M
63 0 V
3474 0 R
-63 0 V
600 4577 M
63 0 V
3474 0 R
-63 0 V
806 251 M
0 63 V
0 4263 R
0 -63 V
1217 251 M
0 63 V
0 4263 R
0 -63 V
1628 251 M
0 63 V
0 4263 R
0 -63 V
2039 251 M
0 63 V
0 4263 R
0 -63 V
2451 251 M
0 63 V
0 4263 R
0 -63 V
2862 251 M
0 63 V
0 4263 R
0 -63 V
3273 251 M
0 63 V
0 4263 R
0 -63 V
3685 251 M
0 63 V
0 4263 R
0 -63 V
4096 251 M
0 63 V
0 4263 R
0 -63 V
600 251 M
3537 0 V
0 4326 V
-3537 0 V
600 251 L
LT0
2982 2674 D
2438 3219 D
1469 2933 D
3161 3037 D
1858 3253 D
2339 3954 D
1154 2976 D
3626 3617 D
3074 3271 D
1685 3322 D
3103 3297 D
1197 3011 D
1073 3028 D
1403 2829 D
1509 3037 D
865 2431 D
2192 3132 D
1135 2968 D
2075 3366 D
3329 3348 D
2717 3089 D
3085 3106 D
2773 3729 D
2320 3063 D
3154 3651 D
3047 3098 D
3569 3245 D
2772 3625 D
3435 3037 D
2732 3210 D
2831 3288 D
1923 3279 D
1075 2976 D
841 2405 D
1457 3020 D
964 2907 D
2582 3089 D
2341 3123 D
2776 4041 D
1614 3028 D
1424 3106 D
2356 3123 D
2199 3132 D
3236 4058 D
1794 3694 D
3165 3703 D
2413 3314 D
1063 2353 D
2366 3720 D
1455 3098 D
3465 3080 D
1033 2803 D
2631 3253 D
1598 3002 D
2648 3132 D
2656 3115 D
3289 3054 D
1425 3452 D
2952 3184 D
2813 3842 D
3546 3634 D
934 2864 D
2488 3175 D
1225 3141 D
3312 3089 D
817 2821 D
3444 3331 D
2610 3175 D
2212 3106 D
2360 3253 D
1801 3184 D
2055 3072 D
2306 3184 D
2917 3556 D
1802 3054 D
2827 3271 D
1462 3366 D
2658 3080 D
3316 3098 D
2403 3123 D
2846 3201 D
1500 3037 D
2777 3054 D
2468 3115 D
774 580 D
3220 3859 D
2657 3660 D
1322 2864 D
2590 3902 D
947 2907 D
873 2466 D
3292 3374 D
714 2475 D
2736 3504 D
1232 2994 D
1640 3271 D
1923 3634 D
2672 3539 D
1463 3582 D
1847 3210 D
1911 3028 D
1963 3219 D
2736 3279 D
1385 3089 D
1351 2942 D
1744 3037 D
2589 3729 D
769 2855 D
1210 3002 D
1592 3149 D
2840 3720 D
3439 3348 D
1728 3193 D
2526 3080 D
3438 3141 D
885 2674 D
2446 4032 D
3120 3418 D
1270 3020 D
1089 2743 D
2810 3470 D
1289 3054 D
2484 3184 D
2756 3703 D
915 2648 D
2976 3175 D
3442 3547 D
1900 3262 D
1619 3072 D
2905 3487 D
2003 3106 D
1258 2942 D
2058 3686 D
3270 3383 D
1898 3115 D
1647 3002 D
2908 3123 D
2398 3271 D
2039 3807 D
2881 3098 D
1837 3521 D
3417 4162 D
1647 3123 D
964 2760 D
3204 3409 D
1428 2994 D
1529 2985 D
2175 3746 D
1263 3037 D
3551 3132 D
1533 3513 D
2214 3046 D
1834 3201 D
1379 3037 D
712 3002 D
3026 3141 D
2837 3184 D
2144 3080 D
2389 3167 D
710 2440 D
1962 3175 D
2125 3158 D
2899 3123 D
682 2561 D
690 2544 D
703 2518 D
715 2492 D
723 2466 D
732 2449 D
744 2423 D
752 2414 D
765 2388 D
777 2353 D
785 2327 D
797 2466 D
806 2509 D
908 2734 D
1011 2829 D
1114 2890 D
1217 2933 D
1320 2959 D
1423 2985 D
1525 3002 D
1628 3020 D
1731 3028 D
1834 3046 D
1937 3054 D
2039 3054 D
2142 3063 D
2245 3063 D
2348 3072 D
2451 3072 D
2554 3072 D
2656 3080 D
2759 3080 D
2862 3080 D
2965 3080 D
3068 3080 D
3170 3080 D
3273 3080 D
3376 3080 D
3479 3080 D
3582 3080 D
3685 3080 D
3787 3080 D
3890 3080 D
3993 3080 D
4096 3080 D
1230 3020 D
1227 2976 D
1996 3054 D
2705 3080 D
2941 3080 D
3655 3080 D
2222 3072 D
779 2414 D
3627 3080 D
2693 3080 D
1973 3072 D
3933 3080 D
2696 3080 D
2042 3072 D
872 2942 D
3632 3080 D
2108 3072 D
1978 3072 D
821 2397 D
2695 3080 D
2016 3072 D
2388 3080 D
1837 3063 D
1936 3072 D
2061 3072 D
1729 3054 D
1020 2942 D
2814 3080 D
1450 3028 D
3526 3080 D
1135 2890 D
2726 3080 D
3664 3080 D
2289 3080 D
1849 3063 D
2519 3072 D
3804 3080 D
3125 3080 D
2676 3080 D
2698 3080 D
1955 3072 D
3210 3080 D
1338 3020 D
2855 3080 D
1780 3054 D
1177 3020 D
1854 3046 D
3922 3080 D
2540 3080 D
3149 3080 D
1545 2976 D
2711 3080 D
1512 3054 D
3020 3080 D
2667 3080 D
3363 3080 D
790 2405 D
3097 3080 D
2796 3080 D
3313 3080 D
739 2622 D
2891 3080 D
1264 3028 D
1510 3028 D
1878 3072 D
1352 3046 D
704 2760 D
1014 2847 D
3162 3080 D
1652 3063 D
3567 3080 D
1303 3028 D
1010 2950 D
1595 2985 D
697 2466 D
1603 3028 D
1868 3072 D
2160 3072 D
3436 3080 D
3497 3080 D
3298 3080 D
1318 3028 D
1860 3063 D
2527 3080 D
1491 3046 D
1354 3046 D
2533 3080 D
3436 3080 D
3523 3080 D
1315 2976 D
2689 3080 D
2489 3080 D
3112 3080 D
2022 3072 D
2601 3080 D
1264 3028 D
2942 3080 D
2616 3080 D
1152 2976 D
3287 3080 D
2007 3072 D
1650 3063 D
3426 3080 D
3792 3080 D
2892 3080 D
1991 3072 D
1022 2734 D
2014 3072 D
3996 3080 D
2681 3080 D
2916 3080 D
1002 2795 D
3932 3080 D
2943 3080 D
1474 3046 D
3017 3080 D
3713 3080 D
1043 3011 D
2006 3063 D
1699 3063 D
3390 3080 D
2334 3072 D
2952 3080 D
1274 2950 D
1349 2916 D
1900 3063 D
2102 3063 D
2553 3080 D
966 2855 D
3717 3080 D
1897 3072 D
2515 3080 D
2843 3080 D
2506 3080 D
2016 3046 D
2867 3080 D
3129 3080 D
2421 3072 D
955 2924 D
3366 3080 D
3882 3080 D
1177 2968 D
1149 2994 D
2675 3080 D
3589 3080 D
1141 3020 D
3226 3080 D
1462 2968 D
3521 3080 D
3699 3080 D
2979 3080 D
3040 3080 D
3188 3080 D
1472 2976 D
1332 3002 D
3944 3080 D
4063 3080 D
1110 3002 D
1737 3063 D
4003 3080 D
3702 3080 D
3530 3080 D
1781 3063 D
1617 3028 D
3751 3080 D
2214 3080 D
2613 3080 D
3409 3080 D
3259 3080 D
2582 3080 D
850 2942 D
3256 3080 D
2499 3080 D
2692 3080 D
3750 3080 D
2181 3072 D
3536 3080 D
1640 3054 D
4041 3080 D
1985 3072 D
1695 3063 D
3751 3080 D
1997 3072 D
1155 2994 D
1501 3046 D
2352 3072 D
1840 3072 D
1959 3054 D
4011 3080 D
3719 3080 D
2352 3072 D
3297 3080 D
2731 3080 D
2723 3080 D
2091 3072 D
3830 3080 D
1976 3072 D
2030 3063 D
2521 3080 D
1529 3046 D
2848 3080 D
1282 3037 D
3840 3080 D
1016 3002 D
1424 3037 D
1629 3054 D
3850 3080 D
723 2700 D
2707 3080 D
3602 3080 D
3163 3080 D
1172 3011 D
2884 3080 D
803 2509 D
1040 2890 D
3769 3080 D
1270 3028 D
1426 3020 D
2016 3063 D
3723 3080 D
2375 3072 D
3805 3080 D
4023 3080 D
3569 3080 D
1242 3037 D
2806 3080 D
2473 3080 D
3034 3080 D
3682 3080 D
1174 3011 D
3024 3080 D
1459 3037 D
969 2847 D
2462 3072 D
2264 3080 D
2215 3063 D
2063 3072 D
2860 3080 D
2890 3080 D
2067 3089 D
2102 3072 D
1746 3063 D
3792 3080 D
3905 3080 D
2448 3080 D
2315 3080 D
2866 3080 D
1575 3054 D
1156 3011 D
818 2907 D
3583 3080 D
1450 3002 D
1492 3046 D
1516 3046 D
2783 3080 D
3525 3080 D
929 2942 D
1548 2985 D
1959 3063 D
2497 3080 D
3187 3080 D
2799 3080 D
694 2630 D
1568 3054 D
2363 3072 D
807 2769 D
2720 3080 D
3942 3080 D
2906 3080 D
1533 2994 D
3476 3080 D
963 2924 D
3336 3080 D
2895 3080 D
3399 3080 D
1870 3063 D
1441 3054 D
2139 3063 D
2735 3080 D
1256 2933 D
1378 3046 D
3638 3080 D
3526 3080 D
1814 3063 D
4047 3080 D
3388 3080 D
2554 3072 D
1263 3002 D
2886 3080 D
3598 3080 D
2764 3080 D
2475 3072 D
3753 3080 D
762 2483 D
1979 3072 D
2660 3080 D
3931 3080 D
1281 2924 D
1507 3037 D
1872 3063 D
2751 3080 D
1231 3002 D
3799 3080 D
3162 3080 D
2212 3063 D
3748 3080 D
3733 3080 D
748 2457 D
1856 3072 D
1807 3063 D
3167 3080 D
3313 3080 D
1183 3011 D
1995 3072 D
2544 3080 D
2023 3072 D
2131 3072 D
3966 3080 D
3876 3080 D
1201 3011 D
2247 3080 D
3205 3080 D
1425 3037 D
1046 2994 D
2767 3080 D
957 2890 D
1590 3063 D
3468 3080 D
3078 3080 D
2024 3046 D
3331 3080 D
982 2994 D
1516 3054 D
2702 3080 D
1099 2890 D
1896 3072 D
2195 3080 D
3120 3080 D
1119 2985 D
2956 3080 D
4061 3080 D
3714 3080 D
2648 3080 D
1944 3072 D
2524 3080 D
3230 3080 D
1683 3037 D
2193 3072 D
866 2924 D
2549 3080 D
3055 3080 D
3705 3080 D
3365 3080 D
1147 2933 D
2941 3080 D
3213 3080 D
3402 3080 D
3024 3080 D
3456 3080 D
1281 3011 D
1513 3046 D
3975 3080 D
1801 3063 D
1702 3054 D
2299 3080 D
3098 3080 D
3255 3080 D
2264 3072 D
2986 3080 D
1065 2899 D
3146 3080 D
1362 3011 D
3080 3080 D
3573 3080 D
1603 3046 D
1662 3063 D
1862 3063 D
1945 3072 D
2831 3080 D
2775 3080 D
1439 3046 D
3655 3080 D
1527 3054 D
1173 2985 D
2020 3072 D
1413 3028 D
3159 3080 D
2348 3063 D
1957 3072 D
846 2414 D
2551 3080 D
2473 3080 D
1350 3046 D
863 2942 D
1475 3054 D
3676 3080 D
1604 3054 D
880 2933 D
1569 3046 D
3827 3080 D
3898 3080 D
3157 3080 D
2878 3080 D
1465 3037 D
4027 3080 D
2470 3072 D
1603 3054 D
2982 3080 D
1464 3020 D
1550 3054 D
2912 3080 D
3824 3080 D
3400 3080 D
2136 3072 D
743 2319 D
3551 3080 D
2582 3072 D
1001 2700 D
3500 3080 D
3368 3080 D
2488 3080 D
3993 3080 D
2861 3080 D
LT1
2982 2574 A
2438 1895 A
1469 1722 A
3161 1722 A
1858 1852 A
2339 1990 A
1154 1895 A
3626 1808 A
3074 1886 A
1685 1912 A
3103 1860 A
1197 1800 A
1073 1834 A
1403 1756 A
1509 1843 A
865 1852 A
2192 1869 A
1135 1860 A
2075 1852 A
3329 1843 A
2717 1878 A
3085 1843 A
2773 1999 A
2320 1826 A
3154 1782 A
3047 1800 A
3569 1886 A
2772 1912 A
3435 1722 A
2732 1852 A
2831 1921 A
1923 1878 A
1075 1791 A
841 1480 A
1457 1800 A
964 1895 A
2582 1800 A
2341 1912 A
2776 1791 A
1614 1886 A
1424 1791 A
2356 1834 A
2199 1904 A
3236 1973 A
1794 1929 A
3165 1938 A
2413 1929 A
1063 1756 A
2366 1886 A
1455 1886 A
3465 1826 A
1033 1774 A
2631 1817 A
1598 1730 A
2648 1791 A
2656 1929 A
3289 1765 A
1425 1800 A
2952 1774 A
2813 1938 A
3546 1860 A
934 1670 A
2488 1981 A
1225 1722 A
3312 1860 A
817 1713 A
3444 1955 A
2610 1748 A
2212 1886 A
2360 1852 A
1801 1834 A
2055 1826 A
2306 1895 A
2917 1886 A
1802 1895 A
2827 1860 A
1462 1852 A
2658 1852 A
3316 1808 A
2403 1929 A
2846 1774 A
1500 1774 A
2777 1791 A
2468 1947 A
774 1298 A
3220 1904 A
2657 1869 A
1322 1687 A
2590 1860 A
947 1869 A
873 1627 A
3292 1981 A
714 1748 A
2736 1826 A
1232 1817 A
1640 1739 A
1923 1886 A
2672 1774 A
1463 1938 A
1847 1843 A
1911 1843 A
1963 1964 A
2736 1904 A
1385 1904 A
1351 1808 A
1744 1800 A
2589 1999 A
769 1713 A
1210 1947 A
1592 1739 A
2840 1800 A
3439 1808 A
1728 1904 A
2526 1886 A
3438 1938 A
885 1782 A
2446 1869 A
3120 1921 A
1270 1929 A
1089 1670 A
2810 1947 A
1289 1852 A
2484 1904 A
2756 1895 A
915 1808 A
2976 1964 A
3442 1826 A
1900 1800 A
1619 1808 A
2905 1756 A
2003 1860 A
1258 1730 A
2058 1904 A
3270 1826 A
1898 1973 A
1647 1791 A
2908 1869 A
2398 1973 A
2039 2007 A
2881 1791 A
1837 1748 A
3417 1938 A
1647 1895 A
964 1748 A
3204 1904 A
1428 1921 A
1529 1808 A
2175 1999 A
1263 1826 A
3551 1739 A
1533 1791 A
2214 1782 A
1834 1878 A
1379 1973 A
712 1947 A
3026 1860 A
2837 1929 A
2144 1860 A
2389 1817 A
710 1618 A
1962 1791 A
2125 1973 A
2899 1782 A
682 1722 A
690 1696 A
703 1661 A
715 1618 A
723 1592 A
732 1557 A
744 1531 A
752 1506 A
765 1471 A
777 1428 A
785 1402 A
797 1514 A
806 1549 A
908 1696 A
1011 1748 A
1114 1782 A
1217 1800 A
1320 1817 A
1423 1826 A
1525 1834 A
1628 1843 A
1731 1852 A
1834 1852 A
1937 1852 A
2039 1860 A
2142 1860 A
2245 1860 A
2348 1860 A
2451 1860 A
2554 1860 A
2656 1860 A
2759 1860 A
2862 1860 A
2965 1860 A
3068 1860 A
3170 1860 A
3273 1860 A
3376 1860 A
3479 1860 A
3582 1860 A
3685 1860 A
3787 1860 A
3890 1860 A
3993 1852 A
4096 1852 A
1230 1860 A
1227 1834 A
1996 1852 A
2705 1869 A
2941 1869 A
3655 1860 A
2222 1869 A
779 1454 A
3627 1860 A
2693 1869 A
1973 1869 A
3933 1852 A
2696 1860 A
2042 1869 A
872 1808 A
3632 1860 A
2108 1869 A
1978 1869 A
821 1445 A
2695 1869 A
2016 1869 A
2388 1869 A
1837 1869 A
1936 1869 A
2061 1869 A
1729 1860 A
1020 1826 A
2814 1869 A
1450 1860 A
3526 1860 A
1135 1774 A
2726 1869 A
3664 1860 A
2289 1869 A
1849 1869 A
2519 1860 A
3804 1860 A
3125 1860 A
2676 1869 A
2698 1869 A
1955 1869 A
3210 1860 A
1338 1860 A
2855 1869 A
1780 1869 A
1177 1860 A
1854 1852 A
3922 1860 A
2540 1869 A
3149 1860 A
1545 1808 A
2711 1869 A
1512 1869 A
3020 1869 A
2667 1869 A
3363 1860 A
790 1462 A
3097 1860 A
2796 1860 A
3313 1860 A
739 1583 A
2891 1860 A
1264 1860 A
1510 1860 A
1878 1869 A
1352 1869 A
704 1705 A
1014 1765 A
3162 1860 A
1652 1869 A
3567 1860 A
1303 1860 A
1010 1826 A
1595 1808 A
697 1635 A
1603 1852 A
1868 1869 A
2160 1869 A
3436 1860 A
3497 1860 A
3298 1860 A
1318 1860 A
1860 1869 A
2527 1869 A
1491 1869 A
1354 1869 A
2533 1869 A
3436 1860 A
3523 1860 A
1315 1826 A
2689 1869 A
2489 1869 A
3112 1860 A
2022 1869 A
2601 1869 A
1264 1860 A
2942 1869 A
2616 1869 A
1152 1843 A
3287 1860 A
2007 1869 A
1650 1869 A
3426 1860 A
3792 1860 A
2892 1860 A
1991 1869 A
1022 1661 A
2014 1869 A
3996 1860 A
2681 1869 A
2916 1860 A
1002 1722 A
3932 1860 A
2943 1860 A
1474 1869 A
3017 1869 A
3713 1860 A
1043 1852 A
2006 1869 A
1699 1869 A
3390 1860 A
2334 1860 A
2952 1869 A
1274 1808 A
1349 1765 A
1900 1869 A
2102 1860 A
2553 1869 A
966 1782 A
3717 1860 A
1897 1869 A
2515 1869 A
2843 1869 A
2506 1869 A
2016 1843 A
2867 1860 A
3129 1860 A
2421 1860 A
955 1817 A
3366 1860 A
3882 1860 A
1177 1834 A
1149 1852 A
2675 1860 A
3589 1860 A
1141 1860 A
3226 1860 A
1462 1800 A
3521 1860 A
3699 1860 A
2979 1869 A
3040 1869 A
3188 1860 A
1472 1808 A
1332 1843 A
3944 1860 A
4063 1852 A
1110 1852 A
1737 1869 A
4003 1852 A
3702 1860 A
3530 1860 A
1781 1869 A
1617 1852 A
3751 1860 A
2214 1869 A
2613 1869 A
3409 1860 A
3259 1860 A
2582 1869 A
850 1808 A
3256 1860 A
2499 1869 A
2692 1869 A
3750 1860 A
2181 1869 A
3536 1860 A
1640 1869 A
4041 1852 A
1985 1869 A
1695 1869 A
3751 1860 A
1997 1869 A
1155 1852 A
1501 1869 A
2352 1860 A
1840 1869 A
1959 1860 A
4011 1852 A
3719 1860 A
2352 1869 A
3297 1860 A
2731 1869 A
2723 1869 A
2091 1869 A
3830 1860 A
1976 1869 A
2030 1869 A
2521 1869 A
1529 1869 A
2848 1869 A
1282 1860 A
3840 1860 A
1016 1852 A
1424 1860 A
1629 1869 A
3850 1860 A
723 1601 A
2707 1869 A
3602 1860 A
3163 1860 A
1172 1852 A
2884 1869 A
803 1549 A
1040 1791 A
3769 1860 A
1270 1860 A
1426 1852 A
2016 1860 A
3723 1860 A
2375 1860 A
3805 1860 A
4023 1852 A
3569 1860 A
1242 1860 A
2806 1869 A
2473 1869 A
3034 1860 A
3682 1860 A
1174 1852 A
3024 1860 A
1459 1869 A
969 1774 A
2462 1860 A
2264 1869 A
2215 1852 A
2063 1869 A
2860 1869 A
2890 1860 A
2067 1886 A
2102 1869 A
1746 1869 A
3792 1860 A
3905 1860 A
2448 1869 A
2315 1869 A
2866 1869 A
1575 1869 A
1156 1852 A
818 1782 A
3583 1860 A
1450 1843 A
1492 1869 A
1516 1869 A
2783 1869 A
3525 1860 A
929 1826 A
1548 1817 A
1959 1869 A
2497 1869 A
3187 1860 A
2799 1869 A
694 1739 A
1568 1869 A
2363 1860 A
807 1713 A
2720 1869 A
3942 1852 A
2906 1869 A
1533 1826 A
3476 1860 A
963 1817 A
3336 1860 A
2895 1860 A
3399 1860 A
1870 1869 A
1441 1869 A
2139 1860 A
2735 1869 A
1256 1800 A
1378 1869 A
3638 1860 A
3526 1860 A
1814 1869 A
4047 1852 A
3388 1860 A
2554 1860 A
1263 1852 A
2886 1869 A
3598 1860 A
2764 1869 A
2475 1860 A
3753 1852 A
762 1523 A
1979 1869 A
2660 1869 A
3931 1860 A
1281 1782 A
1507 1860 A
1872 1869 A
2751 1869 A
1231 1852 A
3799 1860 A
3162 1860 A
2212 1852 A
3748 1860 A
3733 1860 A
748 1540 A
1856 1869 A
1807 1869 A
3167 1860 A
3313 1860 A
1183 1852 A
1995 1869 A
2544 1869 A
2023 1869 A
2131 1869 A
3966 1860 A
3876 1860 A
1201 1860 A
2247 1869 A
3205 1860 A
1425 1860 A
1046 1843 A
2767 1860 A
957 1800 A
1590 1869 A
3468 1860 A
3078 1860 A
2024 1852 A
3331 1860 A
982 1843 A
1516 1869 A
2702 1869 A
1099 1791 A
1896 1869 A
2195 1869 A
3120 1860 A
1119 1843 A
2956 1869 A
4061 1852 A
3714 1860 A
2648 1869 A
1944 1869 A
2524 1869 A
3230 1860 A
1683 1860 A
2193 1869 A
866 1800 A
2549 1869 A
3055 1860 A
3705 1860 A
3365 1860 A
1147 1808 A
2941 1869 A
3213 1860 A
3402 1860 A
3024 1869 A
3456 1860 A
1281 1852 A
1513 1869 A
3975 1860 A
1801 1869 A
1702 1869 A
2299 1869 A
3098 1860 A
3255 1860 A
2264 1860 A
2986 1869 A
1065 1800 A
3146 1860 A
1362 1852 A
3080 1860 A
3573 1860 A
1603 1869 A
1662 1869 A
1862 1869 A
1945 1869 A
2831 1869 A
2775 1869 A
1439 1869 A
3655 1860 A
1527 1869 A
1173 1843 A
2020 1869 A
1413 1860 A
3159 1860 A
2348 1860 A
1957 1869 A
846 1445 A
2551 1869 A
2473 1869 A
1350 1869 A
863 1808 A
1475 1869 A
3676 1860 A
1604 1869 A
880 1808 A
1569 1869 A
3827 1852 A
3898 1860 A
3157 1860 A
2878 1869 A
1465 1869 A
4027 1852 A
2470 1860 A
1603 1869 A
2982 1860 A
1464 1852 A
1550 1869 A
2912 1860 A
3824 1860 A
3400 1860 A
2136 1869 A
743 1445 A
3551 1860 A
2582 1860 A
1001 1635 A
3500 1860 A
3368 1860 A
2488 1869 A
3993 1860 A
2861 1869 A
stroke
grestore
end
showpage
}
\put(2862,2574){\makebox(0,0)[r]{$f=c$}}
\put(2862,2674){\makebox(0,0)[r]{$f=b$}}
\put(2862,511){\makebox(0,0)[l]{$\mu > 0 \, GeV$}}
\put(2862,770){\makebox(0,0)[l]{$m_t = 175 \pm 5 \, GeV$}}
\put(2862,943){\makebox(0,0)[l]{$\tan\beta =2-30$}}
\put(2862,1116){\makebox(0,0)[l]{$A_0 =0-900 \, GeV$}}
\put(2862,1289){\makebox(0,0)[l]{$M_0 = 70-900\, GeV$}}
\put(2368,51){\makebox(0,0){$\vspace*{-0.2in}M_{1/2}\, (GeV)$}}
\put(100,2414){%
\special{ps: gsave currentpoint currentpoint translate
270 rotate neg exch neg exch translate}%
\makebox(0,0)[b]{\shortstack{$\hspace*{-0.3in}s^2_f (M_Z)$}}%
\special{ps: currentpoint grestore moveto}%
}
\put(4096,151){\makebox(0,0){900}}
\put(3685,151){\makebox(0,0){800}}
\put(3273,151){\makebox(0,0){700}}
\put(2862,151){\makebox(0,0){600}}
\put(2451,151){\makebox(0,0){500}}
\put(2039,151){\makebox(0,0){400}}
\put(1628,151){\makebox(0,0){300}}
\put(1217,151){\makebox(0,0){200}}
\put(806,151){\makebox(0,0){100}}
\put(540,4577){\makebox(0,0)[r]{0.2345}}
\put(540,4144){\makebox(0,0)[r]{0.234}}
\put(540,3712){\makebox(0,0)[r]{0.2335}}
\put(540,3279){\makebox(0,0)[r]{0.233}}
\put(540,2847){\makebox(0,0)[r]{0.2325}}
\put(540,2414){\makebox(0,0)[r]{0.232}}
\put(540,1981){\makebox(0,0)[r]{0.2315}}
\put(540,1549){\makebox(0,0)[r]{0.231}}
\put(540,1116){\makebox(0,0)[r]{0.2305}}
\put(540,684){\makebox(0,0)[r]{0.23}}
\put(540,251){\makebox(0,0)[r]{0.2295}}
\end{picture}
% GNUPLOT: LaTeX picture with Postscript
\setlength{\unitlength}{0.1bp}
\special{!
%!PS-Adobe-2.0
%%Creator: gnuplot
%%DocumentFonts: Helvetica
%%BoundingBox: 50 50 914 1130
%%Pages: (atend)
%%EndComments
/gnudict 40 dict def
gnudict begin
/Color false def
/Solid false def
/gnulinewidth 5.000 def
/vshift -33 def
/dl {10 mul} def
/hpt 31.5 def
/vpt 31.5 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
/DL { Color {setrgbcolor Solid {pop []} if 0 setdash }
 {pop pop pop Solid {pop []} if 0 setdash} ifelse } def
/BL { stroke gnulinewidth 2 mul setlinewidth } def
/AL { stroke gnulinewidth 2 div setlinewidth } def
/PL { stroke gnulinewidth setlinewidth } def
/LTb { BL [] 0 0 0 DL } def
/LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def
/LT0 { PL [] 0 1 0 DL } def
/LT1 { PL [4 dl 2 dl] 0 0 1 DL } def
/LT2 { PL [2 dl 3 dl] 1 0 0 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
/P { stroke [] 0 setdash
  currentlinewidth 2 div sub M
  0 currentlinewidth V stroke } def
/D { 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
  P } def
/A { stroke [] 0 setdash vpt sub M 0 vpt2 V
  currentpoint stroke M
  hpt neg vpt neg R hpt2 0 V stroke
  } def
/B { 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
  P } def
/C { 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
/T { 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
  P  } def
/S { 2 copy A C} def
end
%%EndProlog
}
\begin{picture}(4320,4628)(0,0)
\special{"
%%Page: 1 1
gnudict begin
gsave
50 50 translate
0.100 0.100 scale
0 setgray
/Helvetica findfont 100 scalefont setfont
newpath
-500.000000 -500.000000 translate
LTa
600 251 M
0 4326 V
LTb
600 251 M
63 0 V
3474 0 R
-63 0 V
600 792 M
63 0 V
3474 0 R
-63 0 V
600 1333 M
63 0 V
3474 0 R
-63 0 V
600 1873 M
63 0 V
3474 0 R
-63 0 V
600 2414 M
63 0 V
3474 0 R
-63 0 V
600 2955 M
63 0 V
3474 0 R
-63 0 V
600 3496 M
63 0 V
3474 0 R
-63 0 V
600 4036 M
63 0 V
3474 0 R
-63 0 V
600 4577 M
63 0 V
3474 0 R
-63 0 V
600 251 M
0 63 V
0 4263 R
0 -63 V
993 251 M
0 63 V
0 4263 R
0 -63 V
1386 251 M
0 63 V
0 4263 R
0 -63 V
1779 251 M
0 63 V
0 4263 R
0 -63 V
2172 251 M
0 63 V
0 4263 R
0 -63 V
2565 251 M
0 63 V
0 4263 R
0 -63 V
2958 251 M
0 63 V
0 4263 R
0 -63 V
3351 251 M
0 63 V
0 4263 R
0 -63 V
3744 251 M
0 63 V
0 4263 R
0 -63 V
4137 251 M
0 63 V
0 4263 R
0 -63 V
600 251 M
3537 0 V
0 4326 V
-3537 0 V
600 251 L
LT0
1113 4036 D
2552 3636 D
1627 2674 D
3243 3966 D
1998 3268 D
2458 3636 D
1326 2349 D
3688 4188 D
3160 4031 D
1834 3149 D
3188 4025 D
1367 2522 D
1248 2473 D
1564 2598 D
1665 2868 D
1050 1619 D
2318 3452 D
1308 2387 D
2206 3425 D
3404 4107 D
2820 3852 D
3171 4052 D
2873 3901 D
2440 3577 D
3237 4025 D
3134 4020 D
3634 4253 D
2872 3847 D
3505 4112 D
2834 3825 D
2928 3934 D
2060 3344 D
1250 2365 D
1027 1003 D
1615 2803 D
1144 1879 D
2690 3723 D
2460 3636 D
2875 3798 D
1765 2993 D
1584 2814 D
2474 3642 D
2324 3474 D
3316 4139 D
1938 3236 D
3247 4090 D
2529 3642 D
1239 2052 D
2484 3620 D
1614 2906 D
3535 4182 D
1210 2176 D
2738 3750 D
1750 2917 D
2754 3750 D
2762 3842 D
3366 4047 D
1585 2803 D
3044 3906 D
2911 3890 D
3611 4204 D
1116 1971 D
2601 3750 D
1393 2603 D
3388 4123 D
1004 1484 D
3514 4188 D
2717 3733 D
2337 3566 D
2478 3587 D
1944 3133 D
2187 3463 D
2427 3604 D
3010 3955 D
1945 3149 D
2924 3879 D
1620 2830 D
2763 3788 D
3392 4063 D
2519 3685 D
2943 3869 D
1656 2901 D
2877 3858 D
2582 3760 D
962 592 D
3300 4085 D
2762 3793 D
1487 2436 D
2698 3744 D
1128 2100 D
1057 1370 D
3369 4150 D
905 1197 D
2837 3804 D
1401 2582 D
1790 2911 D
2061 3279 D
2776 3760 D
1621 2911 D
1988 3225 D
2049 3231 D
2099 3322 D
2838 3842 D
1546 2820 D
1514 2571 D
1890 3101 D
2697 3847 D
958 1592 D
1379 2479 D
1745 2922 D
2937 3852 D
3510 4139 D
1874 3106 D
2637 3750 D
3508 4188 D
1069 1738 D
2560 3647 D
3204 4101 D
1437 2587 D
1264 2084 D
2908 3890 D
1455 2619 D
2597 3701 D
2857 3836 D
1097 1987 D
3066 4020 D
3513 4155 D
2039 3285 D
1770 3009 D
2999 3890 D
2137 3377 D
1426 2479 D
2190 3371 D
3348 4047 D
2036 3382 D
1797 3003 D
3002 3912 D
2514 3712 D
2172 3463 D
2976 3852 D
1979 3220 D
3488 4177 D
1797 3014 D
1144 1922 D
3285 4063 D
1588 2722 D
1684 2803 D
2302 3604 D
1430 2565 D
3616 4171 D
1688 2939 D
2339 3485 D
1976 3209 D
1541 2755 D
903 1727 D
3115 3961 D
2934 3928 D
2272 3447 D
2506 3577 D
902 1197 D
2098 3247 D
2253 3528 D
2993 3885 D
875 938 D
883 905 D
895 857 D
907 797 D
914 754 D
922 705 D
934 640 D
942 592 D
954 505 D
965 413 D
973 343 D
985 743 D
993 873 D
1091 1538 D
1190 1884 D
1288 2138 D
1386 2349 D
1484 2528 D
1583 2684 D
1681 2825 D
1779 2949 D
1877 3063 D
1976 3166 D
2074 3263 D
2172 3355 D
2270 3436 D
2369 3517 D
2467 3593 D
2565 3658 D
2663 3728 D
2762 3788 D
2860 3847 D
2958 3906 D
3056 3955 D
3155 4009 D
3253 4058 D
3351 4107 D
3449 4150 D
3548 4193 D
3646 4236 D
3744 4280 D
3842 4317 D
3941 4355 D
4039 4388 D
4137 4426 D
1398 2679 D
1396 2446 D
2131 3322 D
2807 3858 D
3034 3939 D
3716 4269 D
2346 3495 D
967 451 D
3689 4236 D
2796 3798 D
2108 3420 D
3981 4350 D
2799 3809 D
2175 3409 D
1057 2041 D
3694 4236 D
2237 3447 D
2113 3328 D
1008 857 D
2798 3836 D
2149 3458 D
2505 3620 D
1979 3225 D
2073 3322 D
2192 3436 D
1876 3090 D
1197 2149 D
2912 3917 D
1609 2825 D
3592 4193 D
1307 2171 D
2828 3820 D
3724 4274 D
2410 3593 D
1990 3236 D
2631 3717 D
3858 4301 D
3209 4036 D
2780 3793 D
2801 3815 D
2091 3355 D
3291 4074 D
1502 2701 D
2951 3896 D
1924 3144 D
1348 2641 D
1995 3187 D
3971 4344 D
2651 3750 D
3232 4063 D
1699 2841 D
2814 3825 D
1668 3052 D
3109 3977 D
2772 3798 D
3436 4128 D
978 548 D
3182 4020 D
2895 3879 D
3389 4112 D
930 878 D
2986 3934 D
1431 2711 D
1666 2857 D
2018 3371 D
1515 2906 D
896 1176 D
1192 1906 D
3244 4042 D
1802 3187 D
3631 4215 D
1469 2728 D
1188 2176 D
1748 2906 D
889 813 D
1755 2928 D
2009 3349 D
2287 3555 D
3507 4171 D
3565 4182 D
3375 4101 D
1482 2744 D
2001 3263 D
2638 3723 D
1648 2960 D
1517 2895 D
2643 3766 D
3507 4166 D
3590 4198 D
1480 2538 D
2793 3820 D
2601 3685 D
3197 4025 D
2155 3404 D
2709 3815 D
1431 2765 D
3034 3950 D
2723 3755 D
1324 2398 D
3364 4096 D
2141 3414 D
1799 3176 D
3497 4161 D
3847 4301 D
2986 3950 D
2125 3371 D
1200 1825 D
2147 3409 D
4041 4366 D
2785 3836 D
3009 3944 D
1181 1825 D
3981 4366 D
3035 3966 D
1632 2928 D
3106 3988 D
3772 4269 D
1220 2490 D
2140 3333 D
1847 3220 D
3463 4144 D
2453 3577 D
3044 3944 D
1440 2446 D
1512 2544 D
2039 3247 D
2231 3404 D
2663 3739 D
1146 1862 D
3775 4274 D
2036 3301 D
2626 3717 D
2940 3879 D
2618 3728 D
2150 3349 D
2962 3906 D
3213 4031 D
2537 3647 D
1135 2046 D
3440 4161 D
3933 4344 D
1348 2371 D
1321 2479 D
2780 3804 D
3653 4220 D
1313 2603 D
3306 4096 D
1621 2728 D
3588 4198 D
3758 4269 D
3070 3961 D
3128 3988 D
3270 4052 D
1629 2738 D
1496 2609 D
3992 4382 D
4106 4388 D
1283 2501 D
1883 3263 D
4048 4372 D
3761 4263 D
3596 4198 D
1925 3295 D
1768 2944 D
3808 4307 D
2339 3582 D
2720 3782 D
3480 4161 D
3338 4107 D
2690 3755 D
1035 2035 D
3334 4085 D
2611 3685 D
2795 3847 D
3806 4290 D
2307 3468 D
3602 4231 D
1790 3074 D
4085 4415 D
2120 3447 D
1843 3203 D
3807 4307 D
2131 3371 D
1327 2452 D
1658 2987 D
2471 3593 D
1982 3301 D
2095 3285 D
4056 4377 D
3777 4280 D
2471 3587 D
3373 4123 D
2833 3820 D
2825 3831 D
2221 3420 D
3883 4339 D
2111 3360 D
2163 3349 D
2632 3728 D
1685 2993 D
2944 3890 D
1448 2782 D
3892 4328 D
1194 2463 D
1584 2830 D
1779 3112 D
3902 4317 D
914 1095 D
2810 3820 D
3665 4226 D
3245 4063 D
1343 2576 D
2979 3906 D
991 851 D
1217 2025 D
3825 4285 D
1436 2738 D
1586 2744 D
2150 3333 D
3781 4274 D
2492 3609 D
3859 4301 D
4068 4377 D
3633 4220 D
1410 2771 D
2905 3906 D
2586 3739 D
3122 3988 D
3742 4274 D
1345 2587 D
3112 3988 D
1617 2879 D
1149 1852 D
2576 3669 D
2387 3560 D
2339 3506 D
2194 3398 D
2956 3890 D
2985 3912 D
2198 3490 D
2232 3414 D
1892 3214 D
3847 4296 D
3955 4372 D
2562 3658 D
2436 3582 D
2962 3896 D
1728 3085 D
1327 2565 D
1005 1852 D
3647 4231 D
1609 2738 D
1649 2944 D
1672 2966 D
2883 3863 D
3591 4204 D
1111 2106 D
1703 2847 D
2095 3295 D
2609 3685 D
3269 4052 D
2897 3885 D
886 992 D
1722 3122 D
2481 3598 D
995 1414 D
2822 3820 D
3990 4350 D
3000 3939 D
1688 2825 D
3544 4177 D
1143 2052 D
3411 4123 D
2989 3917 D
3471 4144 D
2010 3268 D
1600 3003 D
2267 3436 D
2836 3831 D
1423 2409 D
1540 2890 D
3700 4242 D
3593 4193 D
1957 3220 D
4090 4393 D
3461 4139 D
2664 3723 D
1430 2565 D
2981 3906 D
3661 4220 D
2864 3869 D
2588 3679 D
3809 4290 D
951 613 D
2114 3447 D
2765 3793 D
3979 4344 D
1448 2441 D
1663 2884 D
2012 3274 D
2852 3874 D
1399 2565 D
3854 4296 D
3245 4058 D
2337 3501 D
3804 4285 D
3790 4274 D
938 646 D
1996 3312 D
1950 3285 D
3250 4063 D
3389 4107 D
1354 2560 D
2130 3431 D
2654 3717 D
2156 3409 D
2260 3517 D
4013 4393 D
3927 4328 D
1371 2603 D
2370 3577 D
3286 4069 D
1585 2814 D
1223 2398 D
2868 3852 D
1137 1949 D
1742 3139 D
3537 4198 D
3165 4004 D
2157 3349 D
3406 4144 D
1161 2382 D
1672 3047 D
2805 3836 D
1273 2122 D
2034 3355 D
2321 3533 D
3204 4031 D
1292 2409 D
3048 3961 D
4103 4399 D
3772 4280 D
2753 3788 D
2080 3393 D
2635 3706 D
3309 4085 D
1832 3025 D
2319 3479 D
1051 1954 D
2659 3723 D
3142 4015 D
3764 4263 D
3438 4128 D
1320 2252 D
3033 3955 D
3293 4069 D
3474 4144 D
3113 3971 D
3525 4171 D
1448 2598 D
1669 2976 D
4021 4366 D
1944 3193 D
1850 3090 D
2420 3609 D
3184 4020 D
3333 4112 D
2387 3528 D
3077 3955 D
1241 2084 D
3230 4031 D
1525 2663 D
3167 4020 D
3637 4215 D
1755 3003 D
1812 3166 D
2002 3220 D
2082 3382 D
2928 3874 D
2875 3863 D
1599 2901 D
3716 4247 D
1682 3090 D
1344 2425 D
2154 3393 D
1573 2776 D
3242 4052 D
2466 3598 D
2093 3349 D
1032 1019 D
2660 3733 D
2587 3669 D
1513 2879 D
1048 2052 D
1632 3030 D
3736 4263 D
1756 3052 D
1064 2014 D
1723 2971 D
3880 4312 D
3948 4334 D
3240 4069 D
2974 3934 D
1623 2884 D
4071 4377 D
2584 3679 D
1755 3057 D
3073 3993 D
1622 2782 D
1704 3020 D
3006 3955 D
3877 4312 D
3472 4144 D
2264 3436 D
933 554 D
3616 4204 D
2690 3755 D
1180 1743 D
3568 4188 D
3441 4155 D
2601 3679 D
4039 4372 D
2957 3896 D
LT1
1113 3936 A
2552 1062 A
1627 949 A
3243 954 A
1998 1035 A
2458 1122 A
1326 1046 A
3688 1013 A
3160 1057 A
1834 1073 A
3188 1040 A
1367 992 A
1248 1019 A
1564 970 A
1665 1024 A
1050 992 A
2318 1046 A
1308 1024 A
2206 1035 A
3404 1035 A
2820 1051 A
3171 1035 A
2873 1127 A
2440 1019 A
3237 992 A
3134 1008 A
3634 1057 A
2872 1073 A
3505 954 A
2834 1035 A
2928 1084 A
2060 1051 A
1250 986 A
1027 727 A
1615 997 A
1144 1040 A
2690 1003 A
2460 1073 A
2875 997 A
1765 1051 A
1584 992 A
2474 1024 A
2324 1068 A
3316 1111 A
1938 1078 A
3247 1089 A
2529 1084 A
1239 954 A
2484 1057 A
1614 1051 A
3535 1024 A
1210 976 A
2738 1013 A
1750 959 A
2754 1003 A
2762 1084 A
3366 986 A
1585 997 A
3044 986 A
2911 1089 A
3611 1046 A
1116 905 A
2601 1116 A
1393 949 A
3388 1040 A
1004 932 A
3514 1100 A
2717 970 A
2337 1062 A
2478 1035 A
1944 1019 A
2187 1019 A
2427 1062 A
3010 1057 A
1945 1062 A
2924 1040 A
1620 1030 A
2763 1035 A
3392 1013 A
2519 1084 A
2943 992 A
1656 981 A
2877 997 A
2582 1095 A
962 532 A
3300 1068 A
2762 1046 A
1487 922 A
2698 1040 A
1128 1030 A
1057 840 A
3369 1122 A
905 943 A
2837 1019 A
1401 1008 A
1790 959 A
2061 1051 A
2776 986 A
1621 1084 A
1988 1024 A
2049 1030 A
2099 1105 A
2838 1073 A
1546 1062 A
1514 997 A
1890 997 A
2697 1132 A
958 943 A
1379 1084 A
1745 959 A
2937 1003 A
3510 1008 A
1874 1062 A
2637 1057 A
3508 1089 A
1069 965 A
2560 1046 A
3204 1078 A
1437 1073 A
1264 900 A
2908 1095 A
1455 1024 A
2597 1068 A
2857 1062 A
1097 992 A
3066 1111 A
3513 1019 A
2039 1003 A
1770 1003 A
2999 976 A
2137 1040 A
1426 949 A
2190 1068 A
3348 1019 A
2036 1105 A
1797 997 A
3002 1046 A
2514 1111 A
2172 1132 A
2976 997 A
1979 970 A
3488 1089 A
1797 1062 A
1144 949 A
3285 1068 A
1588 1073 A
1684 1003 A
2302 1127 A
1430 1013 A
3616 970 A
1688 992 A
2339 992 A
1976 1046 A
1541 1100 A
903 1062 A
3115 1046 A
2934 1089 A
2272 1040 A
2506 1013 A
902 846 A
2098 997 A
2253 1105 A
2993 992 A
875 873 A
883 857 A
895 830 A
907 803 A
914 786 A
922 765 A
934 743 A
942 727 A
954 705 A
965 673 A
973 651 A
985 743 A
993 770 A
1091 900 A
1190 943 A
1288 970 A
1386 992 A
1484 1003 A
1583 1013 A
1681 1019 A
1779 1024 A
1877 1030 A
1976 1035 A
2074 1035 A
2172 1040 A
2270 1040 A
2369 1040 A
2467 1040 A
2565 1040 A
2663 1046 A
2762 1046 A
2860 1046 A
2958 1046 A
3056 1046 A
3155 1046 A
3253 1046 A
3351 1046 A
3449 1040 A
3548 1040 A
3646 1040 A
3744 1040 A
3842 1040 A
3941 1040 A
4039 1040 A
4137 1040 A
1398 1035 A
1396 1013 A
2131 1035 A
2807 1046 A
3034 1046 A
3716 1040 A
2346 1046 A
967 721 A
3689 1040 A
2796 1046 A
2108 1046 A
3981 1040 A
2799 1046 A
2175 1046 A
1057 1003 A
3694 1040 A
2237 1046 A
2113 1046 A
1008 667 A
2798 1046 A
2149 1046 A
2505 1046 A
1979 1046 A
2073 1046 A
2192 1046 A
1876 1040 A
1197 1003 A
2912 1046 A
1609 1035 A
3592 1040 A
1307 970 A
2828 1046 A
3724 1040 A
2410 1046 A
1990 1046 A
2631 1040 A
3858 1040 A
3209 1046 A
2780 1046 A
2801 1046 A
2091 1046 A
3291 1040 A
1502 1035 A
2951 1046 A
1924 1040 A
1348 1035 A
1995 1035 A
3971 1040 A
2651 1046 A
3232 1040 A
1699 1003 A
2814 1046 A
1668 1040 A
3109 1046 A
2772 1046 A
3436 1046 A
978 705 A
3182 1046 A
2895 1040 A
3389 1040 A
930 840 A
2986 1040 A
1431 1035 A
1666 1035 A
2018 1046 A
1515 1040 A
896 922 A
1192 954 A
3244 1046 A
1802 1046 A
3631 1040 A
1469 1035 A
1188 1008 A
1748 1008 A
889 803 A
1755 1030 A
2009 1046 A
2287 1046 A
3507 1040 A
3565 1040 A
3375 1046 A
1482 1035 A
2001 1046 A
2638 1046 A
1648 1040 A
1517 1040 A
2643 1046 A
3507 1040 A
3590 1040 A
1480 1008 A
2793 1046 A
2601 1046 A
3197 1046 A
2155 1046 A
2709 1046 A
1431 1035 A
3034 1046 A
2723 1046 A
1324 1019 A
3364 1046 A
2141 1046 A
1799 1046 A
3497 1040 A
3847 1040 A
2986 1046 A
2125 1046 A
1200 884 A
2147 1046 A
4041 1040 A
2785 1046 A
3009 1046 A
1181 927 A
3981 1040 A
3035 1046 A
1632 1040 A
3106 1046 A
3772 1040 A
1220 1030 A
2140 1046 A
1847 1046 A
3463 1046 A
2453 1046 A
3044 1046 A
1440 997 A
1512 976 A
2039 1040 A
2231 1040 A
2663 1046 A
1146 970 A
3775 1040 A
2036 1046 A
2626 1046 A
2940 1046 A
2618 1046 A
2150 1030 A
2962 1046 A
3213 1046 A
2537 1040 A
1135 997 A
3440 1040 A
3933 1040 A
1348 1013 A
1321 1024 A
2780 1046 A
3653 1040 A
1313 1035 A
3306 1040 A
1621 997 A
3588 1040 A
3758 1040 A
3070 1046 A
3128 1046 A
3270 1046 A
1629 1003 A
1496 1024 A
3992 1040 A
4106 1040 A
1283 1030 A
1883 1046 A
4048 1040 A
3761 1040 A
3596 1040 A
1925 1046 A
1768 1030 A
3808 1040 A
2339 1046 A
2720 1046 A
3480 1040 A
3338 1040 A
2690 1046 A
1035 1003 A
3334 1046 A
2611 1046 A
2795 1046 A
3806 1040 A
2307 1046 A
3602 1040 A
1790 1046 A
4085 1040 A
2120 1046 A
1843 1046 A
3807 1040 A
2131 1046 A
1327 1024 A
1658 1040 A
2471 1040 A
1982 1046 A
2095 1040 A
4056 1040 A
3777 1040 A
2471 1046 A
3373 1040 A
2833 1046 A
2825 1046 A
2221 1046 A
3883 1040 A
2111 1046 A
2163 1040 A
2632 1046 A
1685 1040 A
2944 1046 A
1448 1040 A
3892 1040 A
1194 1030 A
1584 1040 A
1779 1046 A
3902 1040 A
914 867 A
2810 1046 A
3665 1040 A
3245 1040 A
1343 1030 A
2979 1046 A
991 770 A
1217 981 A
3825 1040 A
1436 1035 A
1586 1030 A
2150 1040 A
3781 1040 A
2492 1046 A
3859 1040 A
4068 1040 A
3633 1040 A
1410 1035 A
2905 1046 A
2586 1046 A
3122 1046 A
3742 1040 A
1345 1030 A
3112 1046 A
1617 1040 A
1149 965 A
2576 1040 A
2387 1046 A
2339 1035 A
2194 1046 A
2956 1046 A
2985 1046 A
2198 1051 A
2232 1046 A
1892 1046 A
3847 1040 A
3955 1040 A
2562 1046 A
2436 1046 A
2962 1046 A
1728 1046 A
1327 1030 A
1005 981 A
3647 1040 A
1609 1024 A
1649 1040 A
1672 1040 A
2883 1046 A
3591 1040 A
1111 1008 A
1703 1008 A
2095 1040 A
2609 1046 A
3269 1046 A
2897 1046 A
886 900 A
1722 1046 A
2481 1040 A
995 927 A
2822 1046 A
3990 1040 A
3000 1046 A
1688 1013 A
3544 1040 A
1143 997 A
3411 1040 A
2989 1046 A
3471 1046 A
2010 1046 A
1600 1040 A
2267 1040 A
2836 1046 A
1423 992 A
1540 1040 A
3700 1040 A
3593 1040 A
1957 1046 A
4090 1040 A
3461 1040 A
2664 1046 A
1430 1024 A
2981 1046 A
3661 1040 A
2864 1046 A
2588 1040 A
3809 1040 A
951 776 A
2114 1046 A
2765 1046 A
3979 1040 A
1448 981 A
1663 1040 A
2012 1046 A
2852 1046 A
1399 1030 A
3854 1040 A
3245 1046 A
2337 1040 A
3804 1040 A
3790 1040 A
938 765 A
1996 1046 A
1950 1046 A
3250 1040 A
3389 1040 A
1354 1030 A
2130 1046 A
2654 1046 A
2156 1046 A
2260 1046 A
4013 1040 A
3927 1040 A
1371 1030 A
2370 1046 A
3286 1046 A
1585 1035 A
1223 1024 A
2868 1046 A
1137 986 A
1742 1046 A
3537 1040 A
3165 1046 A
2157 1035 A
3406 1040 A
1161 1024 A
1672 1040 A
2805 1046 A
1273 976 A
2034 1046 A
2321 1046 A
3204 1046 A
1292 1024 A
3048 1046 A
4103 1040 A
3772 1040 A
2753 1046 A
2080 1046 A
2635 1046 A
3309 1040 A
1832 1035 A
2319 1046 A
1051 992 A
2659 1046 A
3142 1046 A
3764 1040 A
3438 1040 A
1320 992 A
3033 1046 A
3293 1046 A
3474 1046 A
3113 1046 A
3525 1040 A
1448 1030 A
1669 1040 A
4021 1040 A
1944 1046 A
1850 1040 A
2420 1046 A
3184 1046 A
3333 1040 A
2387 1040 A
3077 1046 A
1241 981 A
3230 1046 A
1525 1030 A
3167 1046 A
3637 1040 A
1755 1040 A
1812 1046 A
2002 1046 A
2082 1046 A
2928 1046 A
2875 1046 A
1599 1040 A
3716 1040 A
1682 1046 A
1344 1019 A
2154 1046 A
1573 1035 A
3242 1046 A
2466 1040 A
2093 1046 A
1032 684 A
2660 1046 A
2587 1046 A
1513 1040 A
1048 1003 A
1632 1040 A
3736 1040 A
1756 1046 A
1064 997 A
1723 1040 A
3880 1040 A
3948 1040 A
3240 1040 A
2974 1046 A
1623 1040 A
4071 1040 A
2584 1040 A
1755 1046 A
3073 1046 A
1622 1030 A
1704 1040 A
3006 1046 A
3877 1040 A
3472 1046 A
2264 1046 A
933 662 A
3616 1040 A
2690 1040 A
1180 862 A
3568 1040 A
3441 1040 A
2601 1046 A
4039 1040 A
2957 1046 A
LT2
1113 3836 B
679 1073 B
1053 3836 M
180 0 V
-180 31 R
0 -62 V
180 62 R
0 -62 V
679 949 M
0 248 V
648 949 M
62 0 V
-62 248 R
62 0 V
stroke
grestore
end
showpage
}
\put(993,3836){\makebox(0,0)[r]{exp.}}
\put(993,3936){\makebox(0,0)[r]{$s^2_l$}}
\put(993,4036){\makebox(0,0)[r]{$\hat{s}^2$}}
\put(718,1278){\makebox(0,0)[l]{{\small LEP+SLD}}}
\put(2958,1873){\makebox(0,0)[l]{$\mu > 0 \, GeV$}}
\put(2958,2144){\makebox(0,0)[l]{$m_t = 175 \pm 5 \, GeV$}}
\put(2958,2414){\makebox(0,0)[l]{$\tan\beta =2-30$}}
\put(2958,2684){\makebox(0,0)[l]{$A_0 = 0-900 \, GeV$}}
\put(2958,2955){\makebox(0,0)[l]{$M_0 = 70-900\, GeV$}}
\put(2368,51){\makebox(0,0){$\vspace*{-0.2in}M_{1/2}\, (GeV)$}}
\put(100,2414){%
\special{ps: gsave currentpoint currentpoint translate
270 rotate neg exch neg exch translate}%
\makebox(0,0)[b]{\shortstack{ }}%
\special{ps: currentpoint grestore moveto}%
}
\put(4137,151){\makebox(0,0){900}}
\put(3744,151){\makebox(0,0){800}}
\put(3351,151){\makebox(0,0){700}}
\put(2958,151){\makebox(0,0){600}}
\put(2565,151){\makebox(0,0){500}}
\put(2172,151){\makebox(0,0){400}}
\put(1779,151){\makebox(0,0){300}}
\put(1386,151){\makebox(0,0){200}}
\put(993,151){\makebox(0,0){100}}
\put(600,151){\makebox(0,0){0}}
\put(540,4577){\makebox(0,0)[r]{0.238}}
\put(540,4036){\makebox(0,0)[r]{0.237}}
\put(540,3496){\makebox(0,0)[r]{0.236}}
\put(540,2955){\makebox(0,0)[r]{0.235}}
\put(540,2414){\makebox(0,0)[r]{0.234}}
\put(540,1873){\makebox(0,0)[r]{0.233}}
\put(540,1333){\makebox(0,0)[r]{0.232}}
\put(540,792){\makebox(0,0)[r]{0.231}}
\put(540,251){\makebox(0,0)[r]{0.23}}
\end{picture}
% GNUPLOT: LaTeX picture with Postscript
\setlength{\unitlength}{0.1bp}
\special{!
%!PS-Adobe-2.0
%%Creator: gnuplot
%%DocumentFonts: Helvetica
%%BoundingBox: 50 50 770 554
%%Pages: (atend)
%%EndComments
/gnudict 40 dict def
gnudict begin
/Color false def
/Solid false def
/gnulinewidth 5.000 def
/vshift -33 def
/dl {10 mul} def
/hpt 31.5 def
/vpt 31.5 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
/DL { Color {setrgbcolor Solid {pop []} if 0 setdash }
 {pop pop pop Solid {pop []} if 0 setdash} ifelse } def
/BL { stroke gnulinewidth 2 mul setlinewidth } def
/AL { stroke gnulinewidth 2 div setlinewidth } def
/PL { stroke gnulinewidth setlinewidth } def
/LTb { BL [] 0 0 0 DL } def
/LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def
/LT0 { PL [] 0 1 0 DL } def
/LT1 { PL [4 dl 2 dl] 0 0 1 DL } def
/LT2 { PL [2 dl 3 dl] 1 0 0 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
/P { stroke [] 0 setdash
  currentlinewidth 2 div sub M
  0 currentlinewidth V stroke } def
/D { 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
  P } def
/A { stroke [] 0 setdash vpt sub M 0 vpt2 V
  currentpoint stroke M
  hpt neg vpt neg R hpt2 0 V stroke
  } def
/B { 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
  P } def
/C { 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
/T { 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
  P  } def
/S { 2 copy A C} def
end
%%EndProlog
}
\begin{picture}(3600,2160)(0,0)
\special{"
%%Page: 1 1
gnudict begin
gsave
50 50 translate
0.100 0.100 scale
0 setgray
/Helvetica findfont 100 scalefont setfont
newpath
-500.000000 -500.000000 translate
LTa
LTb
600 251 M
63 0 V
2754 0 R
-63 0 V
600 483 M
63 0 V
2754 0 R
-63 0 V
600 716 M
63 0 V
2754 0 R
-63 0 V
600 948 M
63 0 V
2754 0 R
-63 0 V
600 1180 M
63 0 V
2754 0 R
-63 0 V
600 1412 M
63 0 V
2754 0 R
-63 0 V
600 1645 M
63 0 V
2754 0 R
-63 0 V
600 1877 M
63 0 V
2754 0 R
-63 0 V
600 2109 M
63 0 V
2754 0 R
-63 0 V
702 251 M
0 63 V
0 1795 R
0 -63 V
1041 251 M
0 63 V
0 1795 R
0 -63 V
1381 251 M
0 63 V
0 1795 R
0 -63 V
1720 251 M
0 63 V
0 1795 R
0 -63 V
2059 251 M
0 63 V
0 1795 R
0 -63 V
2399 251 M
0 63 V
0 1795 R
0 -63 V
2738 251 M
0 63 V
0 1795 R
0 -63 V
3078 251 M
0 63 V
0 1795 R
0 -63 V
3417 251 M
0 63 V
0 1795 R
0 -63 V
600 251 M
2817 0 V
0 1858 V
-2817 0 V
600 251 L
LT0
600 546 M
7 -14 V
10 -21 V
10 -25 V
7 -19 V
7 -21 V
10 -28 V
7 -21 V
10 -37 V
10 -39 V
7 -31 V
10 172 V
7 56 V
85 286 V
85 148 V
84 110 V
85 90 V
85 77 V
85 67 V
85 61 V
85 53 V
84 49 V
85 44 V
85 42 V
85 39 V
85 35 V
85 35 V
85 32 V
84 28 V
85 30 V
85 26 V
85 25 V
85 26 V
85 21 V
85 23 V
84 21 V
85 21 V
85 19 V
85 18 V
85 19 V
85 18 V
84 17 V
85 16 V
85 14 V
85 16 V
stroke
grestore
end
showpage
}
\put(2059,483){\makebox(0,0)[l]{$m_t \ =\ 174\,GeV$}}
\put(2059,716){\makebox(0,0)[l]{$\tan\beta\ =\ 4$}}
\put(2059,948){\makebox(0,0)[l]{$M_0\ =\ A_0 \ =\ 200\, GeV$}}
\put(2008,51){\makebox(0,0){$\vspace*{-0.2in}M_{1/2}\, (GeV)$}}
\put(100,1180){%
\special{ps: gsave currentpoint currentpoint translate
270 rotate neg exch neg exch translate}%
\makebox(0,0)[b]{\shortstack{$\hspace*{-0.2in} \hat{s}^2 (M_Z)$}}%
\special{ps: currentpoint grestore moveto}%
}
\put(3417,151){\makebox(0,0){900}}
\put(3078,151){\makebox(0,0){800}}
\put(2738,151){\makebox(0,0){700}}
\put(2399,151){\makebox(0,0){600}}
\put(2059,151){\makebox(0,0){500}}
\put(1720,151){\makebox(0,0){400}}
\put(1381,151){\makebox(0,0){300}}
\put(1041,151){\makebox(0,0){200}}
\put(702,151){\makebox(0,0){100}}
\put(540,2109){\makebox(0,0)[r]{0.238}}
\put(540,1877){\makebox(0,0)[r]{0.237}}
\put(540,1645){\makebox(0,0)[r]{0.236}}
\put(540,1412){\makebox(0,0)[r]{0.235}}
\put(540,1180){\makebox(0,0)[r]{0.234}}
\put(540,948){\makebox(0,0)[r]{0.233}}
\put(540,716){\makebox(0,0)[r]{0.232}}
\put(540,483){\makebox(0,0)[r]{0.231}}
\put(540,251){\makebox(0,0)[r]{0.23}}
\end{picture}

