{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 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}{CSTYLE "" -1 323 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Headi ng 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "War ning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE " Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {XPPEDIT 18 0 "pi^o;" "6#)%#piG%\"oG" } {TEXT -1 0 "" }{TEXT 257 35 " Decay Kinemetics, Maple Worksheet" }} {PARA 256 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 " This worksheet is part of the computational lab on the data analysis \+ of the Dalitz decay of the " }{XPPEDIT 18 0 "pi^o;" "6#)%#piG%\"oG" } {TEXT -1 413 " meson. Here we examine some of the kinematics of the de cay. We begin by assigning values to the various masses and other ki nematic quantities that don't change in the course of the calculations . In order to be able to do this analysis even without Maple, we also \+ use Maple to produce Fortran and C code from the Maple procedures. (Ma ybe soon we can also get Java output, although the C output is very si milar.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "with(codegen,fortran); with(codegen,C); " }{TEXT -1 31 "This loads t he code generators." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#%(fortranG" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7#%\"CG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "The Kroll-Wad a distribution function as a function of x and y, is " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "d^2*Gamma(x,y)/(dx*dy)" "6#*(%\"dG\"\"#-%&GammaG6$ %\"xG%\"yG\"\"\"*&%#dxGF+%#dyGF+!\"\"" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "No*(1-x)^3*(1+y^2+R/x)*(1+2*a*x)/x;" "6#*,%#NoG\"\"\"*$,&F%F%%\" xG!\"\"\"\"$F%,(F%F%*$%\"yG\"\"#F%*&%\"RGF%F(F)F%F%,&F%F%*(F.F%%\"aGF% F(F%F%F%F(F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 259 2 "x " }{TEXT -1 4 "and " }{TEXT 258 1 "y" }{TEXT -1 26 " are defi ned in the notes." }}{PARA 0 "" 0 "" {TEXT -1 31 "We can enter this in to Maple as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "No*(1 - x)^3 *(1 + y^2 + R/x)*(1 + 2*a*x)/x; " }{TEXT -1 18 "Enter this and see " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,%#NoG\"\"\",&F%F%%\"xG!\"\"\"\"$ ,(F%F%*$)%\"yG\"\"#F%F%*&%\"RGF%F'F(F%F%,&F%F%*(F.F%%\"aGF%F'F%F%F%F'F (" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "However, since we will want to evaluate this functi on over and over again, and since there are a whole bunch of constants to include," }}{PARA 0 "" 0 "" {TEXT -1 287 "we will define a procedu re to do the evaluation. The procedure can be called from anywhere on \+ this worksheet. If you are not familiar with Maple procedures, you ne ed only note that they are like a compiled language's subroutine or me thod, and that Maple interprets them in one big step:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 " Procedure dG_dxdy" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "dG_dxdy := proc(x,y) global a, alph a, R, eta, Mp0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "local Mp,Mn,Mpm, Me,BE,Etot,s,Ppi0,Epi0,beta,gam,MaxE,No;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Mp := 938.28: # proton mass" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "Mn := 939.57: # neutron mass" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Mp0 := 134.97: # pi-0 mass " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Mpm := 139.57: # pi-mi nus mass" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Me := 0.510999: \+ # electron mass" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "R := 4*(Me/M p0)^2: # magic ratio r" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "BE := \+ 0.4E-3: # binding E of pi - p system" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Etot := Mpm + Mp - BE: # total energy of pi + proton \+ at rest" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "s := Etot^2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Ppi0 := 0.5*sqrt( (s - (Mn + Mp0)^2) * (s \+ - (Mn - Mp0)^2)/S ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Epi0 := sq rt(Ppi0^2 + Mp0^2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "beta := Ppi0 /Epi0: # velocity of pi0/(speed of light)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "gam := 1/sqrt(1 - beta^2): # the usual gamma" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "MaxE := Mpm + Mp - Mn - BE: # max \+ ee E available in pi- p --> e+ e- n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "a := 0.03: # slope parameter (size)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "alpha := 1./137.: # fine structure constant" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eta := sqrt(1 - R/x):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "No := alpha/(4*Pi): # K-W normalization fact or" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " No*(1 - x)^3*(1 + y^2 + R/x) *(1 + 2*a*x)/x" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%(dG_dxdyGf*6$%\"xG%\"yG6/%#MpG%#MnG%$MpmG%#MeG %#BEG%%EtotG%\"sG%%Ppi0G%%Epi0G%%betaG%$gamG%%MaxEG%#NoG6\"F7C5>8$$\"& GQ*!\"#>8%$\"&dR*F=>%$Mp0G$\"&(\\8F=>8&$\"&dR\"F=>8'$\"'**4^!\"'>%\"RG ,$*&FK\"\"#FCF=\"\"%>8($FT!\"%>8),(FG\"\"\"F:FfnFV!\"\">8**$)FZFSFfn>8 +,$-%%sqrtG6#*(,&FinFfn*$),&F?FfnFCFfnFSFfnFgnFfn,&FinFfn*$),&F?FfnFCF gnFSFfnFgnFfn%\"SGFgn$\"\"&Fgn>8,-F`o6#,&*$)F]oFSFfnFfn*$)FCFSFfnFfn>8 -*&F]oFfnF_pFgn>8.*&FfnFfn-F`o6#,&FfnFfn*$)FhpFSFfnFgnFgn>8/,*FGFfnF:F fnF?FgnFVFgn>%\"aG$\"\"$F=>%&alphaG$\"+t+F*H(!#7>%$etaG-F`o6#,&FfnFfn* &FPFfn9$FgnFgn>80,$*&FjqFfn%#PiGFgn#FfnFT*,FfrFfn,&FfnFfnFdrFgnFhq,(Ff nFfn*$)9%FSFfnFfnFcrFfnFfn,&FfnFfn*(FSFfnFfqFfnFdrFfnFfnFfnFdrFgnF76'F fqFjqFPF_rFCF7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " \+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 " To see what this \+ this procedure does, we just enter it:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "dG_dxdy(x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,%#PiG!\"\",&\"\"\"F(%\"xGF&\"\"$,(F(F(*$ )%\"yG\"\"#F(F(*&$\"+'HwNt&!#9F(F)F&F(F(,&F(F(*&$\"\"'!\"#F(F)F(F(F(F) F&$\"+=v\"[#=!#7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "To see a typi cal value of " }{XPPEDIT 18 0 "d^2*Gamma(x,y)/(dx*dy)" "6#*(%\"dG\"\"# -%&GammaG6$%\"xG%\"yG\"\"\"*&%#dxGF+%#dyGF+!\"\"" }{TEXT -1 52 " we n eed only enter it with floating point numbers:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "dG_dxdy(0.1,0.1); " }{TEXT -1 4 "test" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%%#PiG!\"\"$\"+qPU _8!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "To generate fortran code we need only use the p rocedure name as an argument to " }{TEXT 260 7 "fortran" }{TEXT -1 4 " or " }{TEXT 261 1 "C" }{TEXT -1 11 " functions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "fortran (dG_dxdy); " }{TEXT -1 21 "generate Fortran code" }} {PARA 6 "" 1 "" {TEXT -1 43 " doubleprecision function dG_dxdy(x, y)" }}{PARA 6 "" 1 "" {TEXT -1 23 " doubleprecision x" }}{PARA 6 "" 1 "" {TEXT -1 23 " doubleprecision y" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 24 " doubleprecision BE" }} {PARA 6 "" 1 "" {TEXT -1 26 " doubleprecision Epi0" }}{PARA 6 "" 1 "" {TEXT -1 26 " doubleprecision Etot" }}{PARA 6 "" 1 "" {TEXT -1 26 " doubleprecision MaxE" }}{PARA 6 "" 1 "" {TEXT -1 24 " \+ doubleprecision Me" }}{PARA 6 "" 1 "" {TEXT -1 24 " doublepreci sion Mn" }}{PARA 6 "" 1 "" {TEXT -1 24 " doubleprecision Mp" }} {PARA 6 "" 1 "" {TEXT -1 25 " doubleprecision Mpm" }}{PARA 6 "" 1 "" {TEXT -1 24 " doubleprecision No" }}{PARA 6 "" 1 "" {TEXT -1 26 " doubleprecision Ppi0" }}{PARA 6 "" 1 "" {TEXT -1 26 " \+ doubleprecision beta" }}{PARA 6 "" 1 "" {TEXT -1 25 " doublepre cision gam" }}{PARA 6 "" 1 "" {TEXT -1 23 " doubleprecision s" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 37 " com mon/global/a,alpha,R,eta,Mp0" }}{PARA 6 "" 1 "" {TEXT -1 23 " dou bleprecision a" }}{PARA 6 "" 1 "" {TEXT -1 27 " doubleprecision a lpha" }}{PARA 6 "" 1 "" {TEXT -1 23 " doubleprecision R" }}{PARA 6 "" 1 "" {TEXT -1 25 " doubleprecision eta" }}{PARA 6 "" 1 "" {TEXT -1 25 " doubleprecision Mp0" }}{PARA 6 "" 1 "" {TEXT -1 0 " " }}{PARA 6 "" 1 "" {TEXT -1 22 " Mp = 0.93828D3" }}{PARA 6 "" 1 "" {TEXT -1 22 " Mn = 0.93957D3" }}{PARA 6 "" 1 "" {TEXT -1 23 " Mp0 = 0.13497D3" }}{PARA 6 "" 1 "" {TEXT -1 23 " Mp m = 0.13957D3" }}{PARA 6 "" 1 "" {TEXT -1 23 " Me = 0.510999D0 " }}{PARA 6 "" 1 "" {TEXT -1 26 " R = 4*Me**2/Mp0**2" }}{PARA 6 "" 1 "" {TEXT -1 19 " BE = 0.4D-3" }}{PARA 6 "" 1 "" {TEXT -1 24 " Etot = Mpm+Mp-BE" }}{PARA 6 "" 1 "" {TEXT -1 19 " \+ s = Etot**2" }}{PARA 6 "" 1 "" {TEXT -1 60 " Ppi0 = 0.5D0*sqr t((s-(Mn+Mp0)**2)*(s-(Mn-Mp0)**2)/S)" }}{PARA 6 "" 1 "" {TEXT -1 35 " \+ Epi0 = sqrt(Ppi0**2+Mp0**2)" }}{PARA 6 "" 1 "" {TEXT -1 24 " \+ beta = Ppi0/Epi0" }}{PARA 6 "" 1 "" {TEXT -1 31 " gam = 1/ sqrt(1-beta**2)" }}{PARA 6 "" 1 "" {TEXT -1 27 " MaxE = Mpm+Mp- Mn-BE" }}{PARA 6 "" 1 "" {TEXT -1 18 " a = 0.3D-1" }}{PARA 6 " " 1 "" {TEXT -1 31 " alpha = 0.7299270073D-2" }}{PARA 6 "" 1 " " {TEXT -1 25 " eta = sqrt(1-R/x)" }}{PARA 6 "" 1 "" {TEXT -1 41 " No = alpha/0.3141592653589793D1/4" }}{PARA 6 "" 1 "" {TEXT -1 54 " dG_dxdy = No*(1-x)**3*(1+y**2+R/x)*(1+2*a*x)/x" } }{PARA 6 "" 1 "" {TEXT -1 14 " return" }}{PARA 6 "" 1 "" {TEXT -1 9 " end" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 197 "As you can see, this does look li ke real code, but it does sort of get in the way. So from now on we wi ll place a # symbol (the comment command) in front of the code generat ion commands, like this:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " #C(dG_dxdy); " }{TEXT -1 15 "generate C code" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "If you want to generate some code, you need only remove the comment (go ahead and try it)." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Let's now get back to the Kroll-Wada distributi on function " }{XPPEDIT 18 0 "d^2*Gamma(x,y)/(dx*dy)" "6#*(%\"dG\"\"#- %&GammaG6$%\"xG%\"yG\"\"\"*&%#dxGF+%#dyGF+!\"\"" }{TEXT -1 58 " . As i ndicated in the writeup, the range of allowed x is " }}{PARA 257 "" 0 "" {TEXT -1 11 " R< x < 1, " }}{PARA 0 "" 0 "" {TEXT -1 25 "and that o f allowed y is " }}{PARA 257 "" 0 "" {TEXT -1 1 "-" }{XPPEDIT 18 0 "et a" "6#%$etaG" }{TEXT -1 8 " < y < +" }{XPPEDIT 18 0 "eta;" "6#%$etaG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 16 "where " }} {PARA 257 "" 0 "" {XPPEDIT 18 0 "eta = sqrt(1-R/x);" "6#/%$etaG-%%sqrt G6#,&\"\"\"F)*&%\"RGF)%\"xG!\"\"F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Since the procedure we have just enterred to calculate " } {XPPEDIT 18 0 "d^2*Gamma(x,y)/(dx*dy)" "6#*(%\"dG\"\"#-%&GammaG6$%\"xG %\"yG\"\"\"*&%#dxGF+%#dyGF+!\"\"" }{TEXT -1 18 " evaluates R and " } {XPPEDIT 18 0 "eta" "6#%$etaG" }{TEXT -1 47 ", and since we have made \+ the variable R and eta" }}{PARA 0 "" 0 "" {TEXT -1 80 "global, we can \+ determine their values from anywhere in this worksheet. They are:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "R; eta;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'HwNt&!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,G 8(***!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "So we see that R is a very small number a nd that x will be between 0 (really R) and 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 " Examination of \+ " }{XPPEDIT 18 0 "d*Gamma(x)/dx;" "6#*(%\"dG\"\"\"-%&GammaG6#%\"xGF%%# dxG!\"\"" }{TEXT -1 18 ", Procedure dG_dx" }}{PARA 0 "" 0 "" {TEXT -1 59 "Rather than measure the electron distribution for specific " } {TEXT 262 1 "x" }{TEXT -1 6 " and " }{TEXT 263 1 "y" }{TEXT -1 65 " v alues for the electrons, we can also measure it for a specific " } {TEXT 264 1 "x" }{TEXT -1 30 " value regardless of what the " }{TEXT 265 1 "y" }{TEXT -1 82 " value is. We obtain the theoretical form expe cted for the distribution over just " }{TEXT 266 1 "x" }{TEXT -1 69 " \+ values by integrating the Kroll-Wada distribution over all possible " }{TEXT 267 1 "y" }{TEXT -1 8 " values:" }}{PARA 257 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "d*Gamma(x)/dx;" "6#*(%\"dG\"\"\"-%&GammaG6#% \"xGF%%#dxG!\"\"" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "Int(d^2*Gamma(x)/ dxdy,x = -eta .. eta);" "6#-%$IntG6$*(%\"dG\"\"#-%&GammaG6#%\"xG\"\"\" %%dxdyG!\"\"/F,;,$%$etaGF/F3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 74 "This integral is easy to perform and is given by the Maple proc edure dG_dx" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "dG_dx := pro c(x) (alpha/(3*Pi))*(1/x)*(1 - x)^3*(2 + R/x) * sqrt(1 - R/x) end; \+ " }{TEXT -1 13 "enter and see" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dG_dxGf*6#%\"xG6\"F(F(,$*.%&alphaG\"\"\"%#PiG!\"\"9$ F.,&F,F,F/F.\"\"$,&\"\"#F,*&%\"RGF,F/F.F,F,-%%sqrtG6#,&F,F,F4F.F,#F,F1 F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "#fortran(dG_dx);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "#C(dG_dx);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "To see what this looks like as a function of " }{TEXT 268 1 "x" } {TEXT -1 25 ", we make a semilog plot:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "with(plots, logplot);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#%(logplotG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot( dG_dx(x), x=R..0.1, title=`linear Kroll-Wada distribution`);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7ap7$$\"3. +++'HwNt&!#A$\"\"!F,7$$\"3UwJT[\"GTD\"!#@$\"3_Rl&f[av6\"!#;7$$\"3i_jAn ')*[$>F0$\"3(=A^Z*\\z0x!#<7$$\"3cG&Rg=pch#F0$\"3q8(fjtT;!eF97$$\"3/0F& [qRkH$F0$\"31n@Znv`PYF97$$\"3_\")emB-@xRF0$\"3pJKC*)Q-eQF97$$\"3Xd!zCu !)zl%F0$\"3m%Q,R#)H5I$F97$$\"3QLAHh7vQ`F0$\"3-lyKP\"*o$)GF97$$\"3')4a5 !y@&>gF0$\"3^(z]NBS&fDF97$$\"3L'e=*)H#H+nF0$\"3DXP,]1g+BF97$$\"3\"GwJx \"G1\"Q(F0$\"3.\"=CK`S!*3#F97$$\"3@Q\\aOL$=1)F0$\"31/(popsH\">F97$$\"3 o9\"e`&QgU()F0$\"3@P=>YA3hIfol9\"F^p$\"3y1.Bl6-X8F97$$\"3r*RU\\kXY@\"F^p$\"3IgRU)[ 5&p7F97$$\"3D?\"F97$$\"3zM]q[(*z]8F^p$\"3U() \\*R3q79\"F97$$\"3L_je+o()=9F^p$\"3*[8?Oagj3\"F97$$\"3()pwY_Q&p[\"F^p$ \"3E#oW\")ypk.\"F97$$\"3j()*[V!4.b:F^p$\"3)*QN]!o,%4**!#=7$$\"3Q0.Bcz5 B;F^p$\"3aVs(QrSA\\*Fir7$$\"3#Hi6\"3]=\"p\"F^p$\"3^V0*3W='3\"*Fir7$$\" 3@eU(=6Rt#=F^p$\"3iOf^'G@qU)Fir7$$\"3^$*oj:K\\j>F^p$\"3]eaBbG&)RyFir7$ $\"3-H&*R>tk*4#F^p$\"31$)G*Qlm(GtFir7$$\"35k@;B9!eB#F^p$\"3eE:pJgYHF^p$\"3WCJ%)*Ql'4_Fir7$$\"3t@o`%3PN=$F^p$\"3%GxaIk* f=[Fir7$$\"3K]TA:\\SdOF^p$\"3arARf&G%)=%Fir7$$\"3Oz9\"fus78%F^p$\"3Qvl 565z-PFir7$$\"3Li'3=o'=kYF^p$\"3Qe![Tx6XF$Fir7$$\"3=Yeq<15(>&F^p$\"39m ppfg0MHFir7$$\"3.IIg`X,IdF^p$\"3lWIxLW#pl#Fir7$$\"3,8-]*[GHE'F^p$\"3Ua \"oH[apU#Fir7$$\"3rnHGg8#eL(F^p$\"3x<*38d;`1#Fir7$$\"3a@d1JUr3%)F^p$\" 3enB^*fqfz\"Fir7$$\"385mvU,Va5!#>$\"3))>7cJ%**HU\"Fir7$$\"3Bp6H8kU_7Fg x$\"3%R>^EWj3>\"Fir7$$\"3iL\"QI3SuX\"Fgx$\"3+\"RS'y(*)p,\"Fir7$$\"3()H bc8bYp;Fgx$\"3q(>%)Q-G6#))Fgx7$$\"3$RB[e(4\"3)=Fgx$\"3u<$)>]A`zxFgx7$$ \"3/haYhd?)4#Fgx$\"3t0)Rog^s#pFgx7$$\"3?jv'p?(o*G#Fgx$\"3D+l$\\qp2J'Fg x7$$\"3'4\"fMi*\\_]#Fgx$\"3cfLe@hnHdFgx7$$\"3(\\o3\"Ryp@FFgx$\"3\"omH( Gk'*Q_Fgx7$$\"3%p(e=vaGIHFgx$\"3;mQ+Qa![$[Fgx7$$\"37DmzJIq>JFgx$\"3Y>] x71s9XFgx7$$\"3GC^YF)Q\\M$Fgx$\"3*f&e4MmT\"=%Fgx7$$\"3k+Vi!4Vd`$Fgx$\" 3:f+\"4r)QKRFgx7$$\"3IFyR3QpdPFgx$\"3t]x:*GQYn$Fgx7$$\"3j>#HpKFgx7$$\"3=nR#)\\l' \\P%Fgx$\"3am*>q)p$e4$Fgx7$$\"3Dv!\\!*GI\"*e%Fgx$\"3G+@KSkdJHFgx7$$\"3 _!fTjX*z&y%Fgx$\"3W#Hs3wdPz#Fgx7$$\"3k,#[BsLz*\\Fgx$\"3#ziN=pPtl#Fgx7$ $\"3G\\(zag\"G=_Fgx$\"351;m2-YFDFgx7$$\"3j8\")\\&3&45aFgx$\"3loQxr!zIU #Fgx7$$\"3#p%*>))*)esh&Fgx$\"3fq%>dW;%=BFgx7$$\"3!)>^(*)Qz7$eFgx$\"3`G ;`Et;=AFgx7$$\"3u\"GNXoc1/'Fgx$\"3EJhd([Jq7#Fgx7$$\"3=!ppR)*RKC'Fgx$\" 3EbC24(HZ/#Fgx7$$\"3`xX[)>u\"okFgx$\"3/AfbFBXf>Fgx7$$\"3&yazXf(GqmFgx$ \"3'[AP`z)y()=Fgx7$$\"3WO::&y'3')oFgx$\"3Lz`x*=tf\"=Fgx7$$\"3gjf(e#Hj \"3(Fgx$\"3a59y@nsa>3lT&H(Fgx$\"3i'G\"*z$od\"p\"Fgx7$$\" 3O)>2')>ql\\(Fgx$\"3h$Hbqa%\\N;Fgx7$$\"3AEC*[wLoq(Fgx$\"3k$*\\.\"4]+e \"Fgx7$$\"3\"**3(3O!3d()Fgx$\"3)e4*\\n)>OM\"Fgx7$$\"3;,Crzg C]*)Fgx$\"3s2R#\\&**G18Fgx7$$\"32hPC#zN;<*Fgx$\"3fyxpL=[l7Fgx7$$\"3#4t P.R\\'p$*Fgx$\"3+R`c_PlI7Fgx7$$\"3*\\dPg=t2e*Fgx$\"3/&3F=\\V^>\"Fgx7$$ \"3m_;J(fXGy*Fgx$\"3U>O3Tqii6Fgx7$$\"3/+++++++5Fir$\"3&o@Ihv'=H6Fgx-%' COLOURG6&%$RGBG$\"#5!\"\"F+F+-%&TITLEG6#%?linear~Kroll-Wada~distributi onG-%+AXESLABELSG6$Q\"x6\"Q!F`gl-%%VIEWG6$;$\"+'HwNt&!#9$\"\"\"Fgfl%(D EFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "What you should see is a function that has a very steep peak near " }{TEXT 269 2 "x=" }{TEXT -1 84 "0. Since experiment al measurements are always some average over some small range of " } {TEXT 270 1 "x" }{TEXT -1 101 " values, it is very hard to measure a f unction that falls off this rapidly and be sure of the single " } {TEXT 271 1 "x" }{TEXT -1 120 " value you are at. To get a better feel for what we need to measure, let's look at this same function on a se milog plot:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "logplot(dG_d x(x), x=R..0.1, title=`semilog Kroll-Wada distribution`); \+ " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$ 7`p7$$\"3UwJT[\"GTD\"!#@$\"+1(o#[5!\"*7$$\"3i_jAn')*[$>F*$\"+5XgF*$\"+2'>;3%F37$$\"3L'e=*)H#H+nF*$\"+ \"Q7%=OF37$$\"3\"GwJx\"G1\"Q(F*$\"+l'o%*>$F37$$\"3@Q\\aOL$=1)F*$\"+;x3 3hIfol9\"Fho$\"+%=\"H(G\"F37$$\"3r*RU\\kXY@\"Fho$\"+JHOO5F37$$\"3DFho$!+%)3#p0\"F37 $$\"3-H&*R>tk*4#Fho$!+f5p\\8F37$$\"35k@;B9!eB#Fho$!+BA>C;F37$$\"3hGe]Q `tsCFho$!+gE`k?F37$$\"38$\\\\QDp'4FFho$!+@g'[Y#F37$$\"3ldJ>pJgYHFho$!+ +$GF37$$\"3t@o`%3PN=$Fho$!+f:zqJF37$$\"3K]TA:\\SdOFho$!+y))[zPF37$$ \"3Oz9\"fus78%Fho$!+#*zq9VF37$$\"3Li'3=o'=kYFho$!+QW`[[F37$$\"3=Yeq<15 (>&Fho$!+BfJD`F37$$\"3.IIg`X,IdFho$!+tz?cdF37$$\"3,8-]*[GHE'Fho$!+(o$Q \\hF37$$\"3rnHGg8#eL(Fho$!+.P8]oF37$$\"3a@d1JUr3%)Fho$!+qx+duF37$$\"38 5mvU,Va5!#>$!+KF&zY)F37$$\"3Bp6H8kU_7Fbx$!+m.QT#*F37$$\"3iL\"QI3SuX\"F bx$!+=T$o#**F37$$\"3()Hbc8bYp;Fbx$!+veZa5F-7$$\"3$RB[e(4\"3)=Fbx$!+:l/ 46F-7$$\"3/haYhd?)4#Fbx$!+Q!R%f6F-7$$\"3?jv'p?(o*G#Fbx$!+ow\"**>\"F-7$ $\"3'4\"fMi*\\_]#Fbx$!+E*p=C\"F-7$$\"3(\\o3\"Ryp@FFbx$!+%Qa2G\"F-7$$\" 3%p(e=vaGIHFbx$!+)*4i:8F-7$$\"37DmzJIq>JFbx$!+?\"p`M\"F-7$$\"3GC^YF)Q \\M$Fbx$!+dlny8F-7$$\"3k+Vi!4Vd`$Fbx$!+fNM09F-7$$\"3IFyR3QpdPFbx$!+/ay M9F-7$$\"3j>#HA4:F-7$$\"3Dv!\\!*GI\"*e%Fbx$!+x()*G`\"F-7$$\" 3_!fTjX*z&y%Fbx$!+`7\"Qb\"F-7$$\"3k,#[BsLz*\\Fbx$!+_Kbv:F-7$$\"3G\\(za g\"G=_Fbx$!+tcJ(f\"F-7$$\"3j8\")\\&3&45aFbx$!+8Cj:;F-7$$\"3#p%*>))*)es h&Fbx$!+^&3[j\"F-7$$\"3!)>^(*)Qz7$eFbx$!+'p0Sl\"F-7$$\"3u\"GNXoc1/'Fbx $!+\"3EAn\"F-7$$\"3=!ppR)*RKC'Fbx$!+$4k$*o\"F-7$$\"3`xX[)>u\"okFbx$!+) Hlyq\"F-7$$\"3&yazXf(GqmFbx$!+'fYSs\"F-7$$\"3WO::&y'3')oFbx$!+n0*3u\"F -7$$\"3gjf(e#Hj\"3(Fbx$!+50zb>3lT&H(Fbx$!+q#32')>ql\\(Fbx$!+**3N'y\"F-7$$\"3AEC*[wLoq(Fbx$!+X\"H8!=F-7$$\"3\"**3 (3O!3d()Fbx$!+$ eB<(=F-7$$\"3;,CrzgC]*)Fbx$!+8/'R)=F-7$$\"32hPC#zN;<*Fbx$!+&3Wx*=F-7$$ \"3#4tP.R\\'p$*Fbx$!+>T')4>F-7$$\"3*\\dPg=t2e*Fbx$!+\\*zD#>F-7$$\"3m_; J(fXGy*Fbx$!+!efX$>F-7$$\"3/+++++++5!#=$!+CUBZ>F--%'COLOURG6&%$RGBG$\" #5!\"\"$\"\"!FeflFdfl-%&TITLEG6#%@semilog~Kroll-Wada~distributionG-%+A XESLABELSG6$Q\"x6\"Q!F^gl-%%VIEWG6$;$\"+'HwNt&!#9$\"\"\"Fcfl%(DEFAULTG -%*AXESTICKSG6$Figl7O/$!\"#FeflQ&.1e-1F^gl/$!+/+(*)p\"F-F_gl/$!+X(yG_ \"F-F_gl/$!+4+%zR\"F-F_gl/$!+'**H5I\"F-F_gl/$!+]([=A\"F-F_gl/$!+g>!\\: \"F-F_gl/$!+8+\"p4\"F-F_gl/$!+\"\\dd/\"F-F_gl/$FcflFeflQ#.1F^gl/$!+V+q *)pF3F_gl/$!+^uyG_F3F_gl/$!+(3+%zRF3F_gl/$!+f**H5IF3F_gl/$!+(\\([=AF3F _gl/$!++'>!\\:F3F_gl/$!*H,5p*F3F_gl/$!*2\\dd%F3F_gl/FdflQ#1.F^gl/$\"+d **H5IF3F_gl/$\"+\\D@rZF3F_gl/$\"+8**f?gF3F_gl/$\"+T+q*)pF3F_gl/$\"+.D^ \"y(F3F_gl/$\"++/)4X)F3F_gl/$\"+r)**3.*F3F_gl/$\"+$4DCa*F3F_gl/$FhglFe flQ%.1e2F^gl/$\"+'**H5I\"F-F_gl/$\"+b77x9F-F_gl/$\"+\"**f?g\"F-F_gl/$ \"+/+(*)p\"F-F_gl/$\"+]7:yF-F_gl/$ \"+4DCa>F-F_gl/$\"\"#FeflQ%.1e3F^gl/$\"+'**H5I#F-F_gl/$\"+b77xCF-F_gl/ $\"+\"**f?g#F-F_gl/$\"+/+(*)p#F-F_gl/$\"+]7:yFF-F_gl/$\"+S!)4XGF-F_gl/ $\"+()**3.HF-F_gl/$\"+4DCaHF-F_gl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 " So even for this small range of " }{TEXT 272 1 "x" }{TEXT -1 89 " values, there i s a four-order-of-magnitude drop off, with a singularity at the origin . " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 74 "T ry, in the Maple commands above, plotting this over different ranges \+ of " }{TEXT 273 1 "x" }{TEXT -1 17 " (especially for " }{TEXT 274 1 "x " }{TEXT -1 34 " near 0) to observe its behavior." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The effect of the pion si ze on the distribution is obtained by multiplying the distribution by \+ the " }{TEXT 278 4 "form" }{TEXT -1 1 " " }{TEXT 277 6 "factor" } {TEXT -1 1 " " }{TEXT 275 9 "(1+2ax), " }{TEXT -1 6 "where " }{TEXT 276 1 "a" }{TEXT -1 123 " is related to the size of the pion. The next plot shows the distribution with and without the effect of the form f actor. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "a; " }{TEXT -1 28 "Check \+ what value we have set" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"$!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "logplot(\{dG_dx(x), dG_dx (x)*(1 + 2*a*x)\}, x=.5...(0.8));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"3++++++++]!#=$!+oW=*R$!\"*7$ $\"3g****\\i9Rl]F*$!+sR\"=U$F-7$$\"33+]PC#)GA^F*$!+A3jTMF-7$$\"3;++v$e ui=&F*$!+Y]1kMF-7$$\"3&)***\\i3&o]_F*$!+3Q\"o[$F-7$$\"3#)**\\(oX*y9`F* $!+evi4NF-7$$\"3f+]P9CAu`F*$!+m,%4`$F-7$$\"3f**\\P*zhdV&F*$!+k*yJb$F-7 $$\"3++]P>fS*\\&F*$!+@\"pjd$F-7$$\"34+](=$f%Gc&F*$!+$Q'o*f$F-7$$\"3:++ ]#y,\"GcF*$!+v,*Qi$F-7$$\"3M++Dr\"zbo&F*$!+.2SXOF-7$$\"3s****\\(4&G]dF *$!+x/%)pOF-7$$\"3()****\\7nD:eF*$!+\"HHYp$F-7$$\"3#3++D!*oy(eF*$!+PPw =PF-7$$\"3a+]PpnsMfF*$!+hw*3u$F-7$$\"3o++]siL-gF*$!+7w\\nPF-7$$\"3y*** ****Q5'fgF*$!+V#z-z$F-7$$\"3w**\\P/QBEhF*$!+3z2 &)\\'F*$!+R!4M(RF-7$$\"3?+]P>:mklF*$!+]x[-SF-7$$\"3=+]iv&QAi'F*$!++w:G SF-7$$\"3G++vtLU%o'F*$!+91FcSF-7$$\"3L+++bjm[nF*$!+*Hcd3%F-7$$\"3N++vy b^6oF*$!+Vk0:TF-7$$\"3k+]PMaKsoF*$!+?C&Q9%F-7$$\"3=++D6W%)RpF*$!+]OOwT F-7$$\"3Y+++:K^+qF*$!+><31UF-7$$\"30++]7,HlqF*$!+7_OQUF-7$$\"3')**\\P4 w)R7(F*$!+G\\8oUF-7$$\"3<++]x%f\")=(F*$!+Y$p7I%F-7$$\"3B+]P/-a[sF*$!+I R.LVF-7$$\"3q+](=Yb;J(F*$!+Zi(oO%F-7$$\"34++]i@OttF*$!+zii+WF-7$$\"3_+ ]PfL'zV(F*$!+`\")pOWF-7$$\"33,++!*>=+vF*$!+y:>sWF-7$$\"3Z++DE&4Qc(F*$! +'=$H4XF-7$$\"3i+]P%>5pi(F*$!+y^$pa%F-7$$\"3-,++bJ*[o(F*$!+G$4Be%F-7$$ \"3(4+]7<[8v(F*$!+/(HQi%F-7$$\"3L,++Ijy5yF*$!+S9!>m%F-7$$\"3^+]P/)fT(y F*$!+/e_.ZF-7$$\"3&4+DcI;[$zF*$!+PLUWZF-7$$\"3U+++++++!)F*$!+;/g*y%F-- %'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F`[lF_[l-F$6$7S7$F($!+#p@?T$F-7$F/$!+ )f;[V$F-7$F4$!+5txaMF-7$F9$!+mKPxMF-7$F>$!+rZG+NF-7$FC$!+I/EBNF-7$FH$! +%4BZa$F-7$FM$!+9s6nNF-7$FR$!+WzY!f$F-7$FW$!+2_%Rh$F-7$Ffn$!+;NJQOF-7$ F[o$!++*o*fOF-7$F`o$!+\"prXo$F-7$Feo$!+FT_4PF-7$Fjo$!+(>;Qt$F-7$F_p$!+ 4K4cPF-7$Fdp$!+WK'Gy$F-7$Fip$!+1*)y0QF-7$F^q$!+`]vKQF-7$Fcq$!+KH!p&QF- 7$Fhq$!+1-r$)QF-7$F]r$!+]Tb4RF-7$Fbr$!+rK'o$RF-7$Fgr$!+NMEiRF-7$F\\s$! +t3-!*RF-7$Fas$!+OaE>SF-7$Ffs$!+3'z]/%F-7$F[t$!+T%[L2%F-7$F`t$!+V]*H5% F-7$Fet$!+cDXKTF-7$Fjt$!+U2ShTF-7$F_u$!+543%>%F-7$Fdu$!+@2&RA%F-7$Fiu$ !+thRcUF-7$F^v$!+(f7jG%F-7$Fcv$!+Ztg>VF-7$Fhv$!+RF_^VF-7$F]w$!+PE_&Q%F -7$Fbw$!+zmU>WF-7$Fgw$!+C(fcX%F-7$F\\x$!+A$38\\%F-7$Fax$!+g&o&GXF-7$Ff x$!+6yOmXF-7$F[y$!+:k)=g%F-7$F`y$!+(HsNk%F-7$Fey$!+0?z\"o%F-7$Fjy$!+zS dBZF-7$F_z$!+;DikZF-7$Fdz$!+,<'*4[F--Fiz6&F[[lF_[lF\\[lF_[l-%+AXESLABE LSG6$Q\"x6\"Q!F]el-%%VIEWG6$;$\"\"&F^[l$\"\")F^[l%(DEFAULTG-%*AXESTICK SG6$Fgel7=/$!\"&F`[lQ&1e-05F]el/$!+/+(*)p%F-F^el/$!+X(yG_%F-F^el/$!+4+ %zR%F-F^el/$!+'**H5I%F-Q&5e-05F]el/$!+]([=A%F-F^el/$!+g>!\\:%F-F^el/$! +8+\"p4%F-F^el/$!+\"\\dd/%F-F^el/$!\"%F`[lQ&.1e-3F]el/$!+/+(*)p$F-F^el /$!+X(yG_$F-F^el/$!+4+%zR$F-F^el/$!+'**H5I$F-Q&.5e-3F]el/$!+]([=A$F-F^ el/$!+g>!\\:$F-F^el/$!+8+\"p4$F-F^el/$!+\"\\dd/$F-F^el/$!\"$F`[lQ&.1e- 2F]el/$!+/+(*)p#F-F^el/$!+X(yG_#F-F^el/$!+4+%zR#F-F^el/$!+'**H5I#F-Q&. 5e-2F]el/$!+]([=A#F-F^el/$!+g>!\\:#F-F^el/$!+8+\"p4#F-F^el/$!+\"\\dd/# F-F^el" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 19 "The green curve is " }{XPPEDIT 18 0 "d*Gamma(x)/dx;" "6#*(%\"dG \"\"\"-%&GammaG6#%\"xGF%%#dxG!\"\"" }{TEXT -1 202 " without the correc tion for the pion's size, while the red curve plots contains the corre ction for the pion's finite size. As you can see, this is not a big ef fect. Because we want to deduce a value of " }{TEXT 279 1 "a" }{TEXT -1 114 " from experimental data, this small difference that it will be hard to deduce a statistically meaningful value of " }{TEXT 280 1 "a " }{TEXT -1 29 " from the experimental data. " }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 78 "Although the effect we wish to measure appears to be slight, if the value of " }{TEXT 282 1 "a" }{TEXT -1 110 " were larger, the effect would be easier to measu re. Issue the plotting command again using the prompts below " }{TEXT 281 0 "" }{TEXT -1 13 "with a value " }{TEXT 283 1 "a" }{TEXT -1 54 " \+ = 0.1, and see if that would provide a better signal." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 284 14 "Examination of" }{TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*Gamma(x,y)/(dx*dy);" "6#*(% \"dG\"\"#-%&GammaG6$%\"xG%\"yG\"\"\"*&%#dxGF+%#dyGF+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Next we look at 3D plot of the full distr ibution " }{XPPEDIT 18 0 "d^2*Gamma(x,y)/(dx*dy);" "6#*(%\"dG\"\"#-%& GammaG6$%\"xG%\"yG\"\"\"*&%#dxGF+%#dyGF+!\"\"" }{TEXT -1 15 " versus both " }{TEXT 285 1 "x" }{TEXT -1 5 " and " }{TEXT 286 1 "y" }{TEXT -1 2 ". 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FFFFB3FEFFFC3E0D3E47935D3B3E328BC5176-%+AXESLABELSG6%%\"xG%\"yG%!G" 1 2 0 1 10 0 2 1 1 3 2 1.000000 41.000000 47.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 332 "Click on the plot so that a box appears around it and s o that the graphical interface is activated. Now you can click buttons to add labels, axes, etc. Find the button that gives you labels for t he axes (so you know which end is up). Try ``grabing'' the graph and r otating it with the mouse, and thereby see how the dependences on " } {TEXT 287 1 "x" }{TEXT -1 5 " and " }{TEXT 288 1 "y" }{TEXT -1 21 " ar e quite different." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 34 " Procedures for ymax, and ymin." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Imposing li mits on the energy and the angle between the " }{TEXT 289 2 "e+" } {TEXT -1 5 " and " }{TEXT 290 3 "e- " }{TEXT -1 51 " (the ``opening'' \+ angle) restricts the values for " }{TEXT 291 1 "x" }{TEXT -1 5 " and \+ " }{TEXT 292 1 "y" }{TEXT -1 123 ". The equations follows from relati vistic kinematics and can be found on p. 112 in Peter Gumplinger's the sis. The maximum " }{TEXT 293 1 "y" }{TEXT -1 7 " value " }{TEXT 294 5 "ymax " }{TEXT -1 26 "is given by the procedure:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ymax := p roc(x) local Emin, Et, Pt, ym;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "E min := 20.: # Your choice. The units are MeV." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Et := (1 + x)*Mp0/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Pt := (1 - x)*Mp0/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ym := sqrt(Et^2 - 4*Emin*(Et - Emin))/Pt;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "if ym>1. then return(1) else return(ym) fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%ymaxGf*6 #%\"xG6&%%EminG%#EtG%#PtG%#ymG6\"F-C'>8$$\"#?\"\"!>8%,$*&,&\"\"\"F99$F 9F9%$Mp0GF9#F9\"\"#>8&,$*&,&F9F9F:!\"\"F9F;F9F<>8'*&-%%sqrtG6#,&*$)F5F =F9F9*(\"\"%F9F0F9,&F5F9F0FCF9FCF9F?FC@%2$F9F3FEOF9OFEF-F-F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 " The value for the minimum " }{TEXT 295 1 "y" }{TEXT -1 33 " va lue is given by the procedure " }{TEXT 296 4 "ymin" }{TEXT -1 1 ":" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " ymin := proc(x) global Mp02 ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Theta_Min := 140.: # Units ar e degrees" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "stm := sin(Theta_Min*P i/180);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "stm := evalf(stm);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ctm := cos(Theta_Min*Pi/180);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "ctm := evalf(ctm);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Mp02 := Mp0^2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "A := ((Mp02 + x)/(Mp02 - x))^2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "B := 8*Mp02*x/((1 - ctm)*(Mp02 - x)^2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "C := A - B;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ymi n := sqrt(abs(C));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "if C<0 then r eturn(0) else return(ymin) fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "en d;" }}{PARA 7 "" 1 "" {TEXT -1 70 "Warning, `Theta_Min` is implicitly \+ declared local to procedure `ymin`\n" }}{PARA 7 "" 1 "" {TEXT -1 64 "W arning, `stm` is implicitly declared local to procedure `ymin`\n" }} {PARA 7 "" 1 "" {TEXT -1 64 "Warning, `ctm` is implicitly declared loc al to procedure `ymin`\n" }}{PARA 7 "" 1 "" {TEXT -1 62 "Warning, `A` \+ is implicitly declared local to procedure `ymin`\n" }}{PARA 7 "" 1 "" {TEXT -1 62 "Warning, `B` is implicitly declared local to procedure `y min`\n" }}{PARA 7 "" 1 "" {TEXT -1 65 "Warning, `ymin` is implicitly d eclared local to procedure `ymin`\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #>%%yminGf*6#%\"xG6(%*Theta_MinG%$stmG%$ctmG%\"AG%\"BGF$6\"F.C->8$$\"$ S\"\"\"!>8%-%$sinG6#,$*&F1\"\"\"%#PiGF<#F<\"$!=>F6-%&evalfG6#F6>8&-%$c osGF9>FE-FB6#FE>%%Mp02G*$)%$Mp0G\"\"#F<>8'*&,&FLF<9$F8(,$**FLF_%(codegenG%\"CG,&FRF8)- %%sqrtG6#-%$absG6#Fjn@%2FjnF4OF4OF_oF.6#FLF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "#f ortran(ymax); " }{TEXT -1 21 "generate Fortran code " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "#C(ymax);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "So as we make measurements for different " }{TEXT 297 1 "x" } {TEXT -1 42 " values, there will be a limited range of " }{TEXT 298 1 "y" }{TEXT -1 56 " values that are allowed. To see this relation, we p lot:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "R;" }{TEXT -1 79 " \+ First check on \+ R value" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'HwNt&!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "plot(\{'ymin(x*Mp02)','-ymin(x*Mp0 2)','ymax(x)','-ymax(x)'\}, x = R..1, labels=[`x (squared sum of 4-mo menta)`, `Allowed y (energy difference)`], title=`Max and Min y vs x`) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6)-%'CURVESG6 $7do7$$\"+'HwNt&!#9$\"+u![)****!#57$$\"+'e&*F-7$$\"+2%HIX \"F-$\"+]#G$e%*F-7$$\"+5V;l;F-$\"+fqOV$*F-7$$\"+%*)=m(=F-$\"+ULy:#*F-7 $$\"+Cf7%4#F-$\"+u8'*o!*F-7$$\"+Niq&G#F-$\"+t%RX#*)F-7$$\"+*G!Q,DF-$\" +DDFU()F-7$$\"+A*Rzr#F-$\"+e)fY`)F-7$$\"+c_jEHF-$\"+#f`oI)F-7$$\"+71:; JF-$\"+M+$=2)F-7$$\"+,F]TLF-$\"+b()**\\xF-7$$\"+![0C`$F-$\"+jv9LuF-7$$ \"+&zqWv$F-$\"+\"GL(**pF-7$$\"+Ei.^RF-$\"+$\\82a'F-7$$\"+e'*pmTF-$\"+' >`c#fF-7$$\"+^4QpUF-$\"+h7nxbF-7$$\"+XA1sVF-$\"+)okA=&F-7$$\"+-%*>zWF- $\"+eO\"Rq%F-7$$\"+hlL'e%F-$\"+b%HC8%F-7$$\"+;>s%o%F-$\"+P0J$[$F-7$$\" +ss5$y%F-$\"+Y;a@EF-7$$\"+Yxj4[F-$\"+)4kHK#F-7$$\"+@#oh$[F-$\"+^(3M(>F -7$$\"+&p)pi[F-$\"+')>LQ:F-7$$\"+q\"H#*)[F-$\"+HX)3,*F17$$\"+2W\\-\\F- $\"+C;_m:F17$$\"+X'fd\"\\F-$\"\"!Fhu7$$\"+#)[-H\\F-Fgu7$$\"+>,HU\\F-Fg u7$$\"+%f?)o\\F-Fgu7$$\"+n5N&*\\F-Fgu7$$\"+%*=e0^F-Fgu7$$\"+>F\"e@&F-F gu7$$\"+O_s2aF-Fgu7$$\"+7g*\\h&F-Fgu7$$\"+0q7HeF-Fgu7$$\"+1ChQgF-Fgu7$ $\"+-.ITiF-Fgu7$$\"+b1NmkF-Fgu7$$\"+0%o&omF-Fgu7$$\"+lE\")F-Fgu7$$\"+$\\NSL)F-Fgu7$$\"+?^6Y&)F-Fgu7$$\"+? 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ITiFddlF\\]l7$$\"3\\+++b1NmkFddlF\\]l7$$\"3Z+++0%o&omFddlF\\]l7$$\"3I+ ++lE\")FddlF\\]l7$$\"3Y+++$\\NSL)FddlF\\]l7$$\"3h* *****>^6Y&)FddlF\\]l7$$\"3&*******>'Qkv)FddlF\\]l7$$\"3++++uSq\\*)Fddl F\\]l7$$\"3f*****R437<*FddlF\\]l7$$\"3u******HRKp$*FddlF\\]l7$$\"3l*** **Htc0e*FddlF\\]l7$$\"3r*****zZLFy*FddlF\\]l7$F\\]lF\\]l-F`]l6&Fb]lFc] lFc]lFgu-F$6$7Z7$F_dl$!3\"4Miu#QctSFddl7$Ffdl$!3CZ5=QI;(Q%Fddl7$F\\el$ !30?yt\\#>;n%Fddl7$Fael$!3OV'e$pNE0]Fddl7$Ffel$!3Fz&eHW/oN&Fddl7$F[fl$ !33Yb;w$>Ms&Fddl7$F`fl$!3\">RJP))4$zgFddl7$Fefl$!3Psr8*p)>lkFddl7$Fjfl $!3?H'y#[VE%)oFddl7$F_gl$!3?8IEIzwBtFddl7$Fdgl$!31+BW(z!Q+yFddl7$Figl$ !3OAllowed~y~(energy~difference)G-%%VIEWG6$;F(F\\]l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "As set in the procedures for " }{TEXT 299 4 "ymin" }{TEXT -1 5 " and " }{TEXT 300 4 "ymax" }{TEXT -1 130 ", this plot corresponds to a minimum energy of 20 MeV and a mi nimum opening angle of 140 degrees. Note that the allowed region of " }{TEXT 301 1 "x" }{TEXT -1 5 " and " }{TEXT 302 1 "y" }{TEXT -1 50 " \+ lying between the straight lines corresponds to " }{TEXT 303 4 "ymax" }{TEXT -1 81 " determined by the minimum energy requirement. The curve d lines are the loci of (" }{TEXT 304 3 "x,y" }{TEXT -1 290 ") values \+ corresponding to a fixed opening angle. As the opening angle gets lar ger, these curves move to the right. Since our detectors are relative ly small, all accepted events must lie in the initial crescent between maximum and minimum opening angles. In summary, the maximum value of " }{TEXT 305 1 "x" }{TEXT -1 108 " is limited by the maximum opening \+ angle. The minimum value is given by the intersection of the two curv es." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 248 "T hese energies and angles are calculated in the pi-zero rest frame. Th e transformation to the lab frame distorts the picture somewhat, but i t can't change an angle by more than 11.3 degrees, so the above descri ption is still qualitatively correct." }}{PARA 0 "" 0 "" {TEXT -1 2 " \+ " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 306 10 "Decay Rate" }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "d^2*Gamma(x,y)/(dx*dy);" "6#*(%\"dG\"\"#-%&GammaG6$ %\"xG%\"yG\"\"\"*&%#dxGF+%#dyGF+!\"\"" }{TEXT -1 1 " " }{TEXT 307 26 " with Kinematic Constraints" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Let us now look at the decay rate " }{XPPEDIT 18 0 "d^2*Gamma(x,y)/(dx*dy);" "6#*(%\"dG\"\"#-%&GammaG6$%\"xG%\"yG\" \"\"*&%#dxGF+%#dyGF+!\"\"" }{TEXT -1 72 " within the limits imposed by energy and momentum conservation (allowed " }{TEXT 308 1 "x" }{TEXT -1 5 " and " }{TEXT 309 1 "y" }{TEXT -1 40 " values). We do that with \+ the procedure " }{TEXT 310 4 "xsec" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "xsec := proc(x, y) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "outline := Heaviside(ymax(x) - abs(y)) * Heaviside(ab s(y) - ymin(x*Mp02));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dG_dxdy(x, y)*outline(x,y);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " end;" }}{PARA 7 "" 1 "" {TEXT -1 68 "Warning, `outline` is implicitly declared local to procedure `xsec`\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%xsecGf*6$ %\"xG%\"yG6#%(outlineG6\"F+C$>8$*&-%*HeavisideG6#,&-%%ymaxG6#9$\"\"\"- %$absG6#9%!\"\"F8-F16#,&F9F8-%%yminG6#*&F7F8%%Mp02GF8F=F8*&-%(dG_dxdyG 6$F7F " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "#fortran(xsec); #C(xsec); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "R; " }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 68 " \+ Check Value" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'Hw Nt&!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot3d('xsec(x,y) ', x = R..(.07), y = -1..1);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6$-%%GRIDG6%;$\"+'HwNt&!#9$\"\"(!\"#;$!\"\"\"\"!$\"\" \"F07;7;$F0F0F5F5F5F5F5F5F5F5F5F5F5F5F5F5F5F5F5F5F5F5F5F5F5F5F4F4F4F4F 4F4F4F4F4F4F4F4F4F4F4F4F4F4F4F4F4F4F4F4-%+AXESLABELSG6%%\"xG%\"yGQ!6\" " 1 2 0 1 10 0 2 1 1 2 2 1.000000 46.000000 45.000000 0 0 "Curve 1" }} }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 80 "Again, make sure to select the graph and add labels to \+ tell the variables apart." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 311 23 "y-integrated decay rate " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 211 "We now explore the y-integra ted decay rate to study the effect of the x cut (the exclusion of cert ain x values) on the statistical accuracy of the result. First the in tegrated structure-independent decay rate, " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "Gamma = Int(d*Gamma(x)/dx,x = rr .. 1);" "6#/%&GammaG-% $IntG6$*(%\"dG\"\"\"-F$6#%\"xGF*%#dxG!\"\"/F-;%#rrGF*" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "int(dG_dx(x), x = R..1.);" }{TEXT -1 43 " \+ Could evaluate with Simpson too" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +A`V&=\"!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "dG_dx(x); " }{TEXT -1 35 "what is this function analytically?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,%#PiG!\"\"%\"xGF&,&\"\"\"F)F'F&\"\"$,&\"\"#F)*&$\"+ 'HwNt&!#9F)F'F&F)F),&F)F)*&$\"+'HwNt&F0F)F'F&F&#F)F,$\"+C+4LC!#7" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "int(dG_dx(x),x); " }{TEXT -1 25 "analytic integral, a mess" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$ ***&,&%\"xG$\"++++]7\"\"%$\"*Pqp;(\"\"!!\"\"\"\"\"F'F.#F/\"\"#F'!\"#,. *&),&*$)F'F1F/F(*&$F,F-F/F'F/F.#\"\"$F1F/)F'Fu$*\"#JF/F5F/F'F/F/F/*&F&F/F'F/#F .F1$\"+;l3/8!#o" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "This is the theoretical branching ratio for Dalitz decay relative to ordinary pi-zero decay. It is in \+ good agreement with the measured branching ratio. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Note (by trying differe nt values for " }{TEXT 312 1 "a" }{TEXT -1 61 ") that the form factor \+ has very little effect on this number." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "a:=1; a; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 " int(dG_dx(x), x = R..1.);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+A`V&=\"!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 313 23 "y-integrated decay r ate" }{TEXT -1 1 " " }{TEXT 314 20 "using Simpson's Rule" }}{PARA 0 " " 0 "" {TEXT -1 167 "We now evaluate the same integral using a Maple p rocedure that implements the Simpson Rule. You should get a similar as before, but now you can also (instead) use the " }{TEXT 315 7 "codege n" }{TEXT -1 66 " command to obtasn a compiled language code to do the integration." }}{SECT 0 {PARA 4 "" 0 "" {TEXT 316 14 "Simpson's Rule " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "Simpson := proc (f, a, b) local N, ans, h,n; \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "N := 61; #odd number of integra tion points" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ans := 0.; #in itialization" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "h := (b-a)/(N-1.); \+ # interval size" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "for n from 2 to N-1 by 2 do ans := evalf(ans + 4.* f(a+h*(n-1) )) od; #loop for \+ even points" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for n from 3 to N-2 \+ by 2 do ans := evalf(ans + 2.* f(a+h*(n-1))) od; #loop for even" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "ans := evalf((ans + f(a)+f(b))*h/ 3.); #endpoint 1 & N plus h/3" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "en d;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(SimpsonGf*6%%\"fG%\"aG%\"bG6& %\"NG%$ansG%\"hG%\"nG6\"F/C(>8$\"#h>8%$\"\"!F7>8&*&,&9&\"\"\"9%!\"\"F= ,&F2F=$F=F7F?F??(8'\"\"#FD,&F2F=F=F?%%trueG>F5-%&evalfG6#,&F5F=*&$\"\" %F7F=-9$6#,&F>F=*&F9F=,&FCF=F=F?F=F=F=F=?(FC\"\"$FD,&F2F=FDF?FF>F5-FI6 #,&F5F=*&$FDF7F=FOF=F=>F5-FI6#,$*&,(F5F=-FP6#F>F=-FP6#F " 0 "" {MPLTEXT 1 0 19 "#fortran(Si mpson); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "#C(Simpson);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "First we test this with " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "int(sin(x),x = 0 .. Pi) = 2;" "6#/-%$intG6$-%$sinG6#%\" xG/F*;\"\"!%#PiG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Si mpson(sin,0.,Pi); " }{TEXT -1 24 "test, should be about 2." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#3+++#!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Now we check that R is still defined:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "R; #check size of R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'HwNt&!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 178 "Since R is so close to 0, and since the integrand is singular \+ at zero, we break the integral up into two pieces, with the first piec e near the origin (where most of the area is):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 58 "Simpson(dG_dx,R,0.006)+ Simpson(dG_dx,0.006,1. ); " }{TEXT -1 46 "use two intervals since such rapid a fall \+ off." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+%ftW;\"!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "So while this value is not exactl y the same as what Maple (somehow) computed, it is accurate enough for estimates." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 317 21 "Integrated Deca y Rate" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Int(x*d*Gamma(x)/dx,x = x_min .. x_max);" "6#-%$IntG6$**%\"xG\"\"\"%\"dGF(-%&GammaG6#F'F(%#dxG!\"\" /F';%&x_minG%&x_maxG" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "The integrated, weighted decay rate " }{XPPEDIT 18 0 "Int(x*d* Gamma(x)/dx,x = x_min .. x_max);" "6#-%$IntG6$**%\"xG\"\"\"%\"dGF(-%&G ammaG6#F'F(%#dxG!\"\"/F';%&x_minG%&x_maxG" }{TEXT -1 153 " is useful \+ as ??. We calculate it with a procedure that simply evaluates the int egral, either with Maple's built-in function, or with our Simpson rule :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "IntwtRate := proc (xmin, xmax);" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "int(dG_dx(x)*x, x = xmin..xmax)" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Intw tRateGf*6$%%xminG%%xmaxG6\"F)F)-%$intG6$*&-%&dG_dxG6#%\"xG\"\"\"F1F2/F 1;9$9%F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "IntwtRateS := proc (xmin, xmax);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f :=(x)->x*dG_dx(x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Sim pson(f, xmin, 3*xmin)+Simpson(f, 3*xmin,xmax)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 7 "" 1 "" {TEXT -1 68 "Warning, `f` is i mplicitly declared local to procedure `IntwtRateS`\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+IntwtRateSGf*6$%%xminG%%xmaxG6#%\"fG6\"F+C$>8$f*6 #%\"xGF+6$%)operatorG%&arrowGF+*&9$\"\"\"-%&dG_dxG6#F6F7F+F+F+,&-%(Sim psonG6%F.F6,$F6\"\"$F7-F=6%F.F?9%F7F+F+F+" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "As we can s ee by calculation, it is" }}{PARA 0 "" 0 "" {TEXT -1 17 "relatively sm all:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "IntRate(R,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%(I ntRateG6$$\"+'HwNt&!#9\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "IntRateS(R,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%)IntRateSG6$ $\"+'HwNt&!#9\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 392 "Since it is hard to measure a sma ll number with any precision, a good experimentalist is always on the \+ lookout for some related quantity that can be measured more reliably. \+ Often the ratio of two quantities may provide a more meaningful signal than either quantity by itself. So let's examine the ratio of the int egrated weighted distribution to that of the integrated unweighted dis tribution" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "Rat := int(x*dG_dx(x),x \+ = xmin .. 1)/int(dG_dx(x),x = xmin .. 1);" "6#>%$RatG*&-%$intG6$*&%\"x G\"\"\"-%&dG_dxG6#F*F+/F*;%%xminGF+F+-F'6$-F-6#F*/F*;F1F+!\"\"" }} {PARA 0 "" 0 "" {TEXT -1 111 "We assign this to a function in Maple, p lot it up, and then note that a nice signal appearsfor large values of " }{TEXT 318 4 "xmin" }{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Rat := (xmin) -> IntwtRate(xmin,1.)/(int(dG_dx(x), x= xmin..1.));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$RatGf*6#%%xminG6\"6$ %)operatorG%&arrowGF(*&-%*IntwtRateG6$9$$\"\"\"\"\"!F2-%$intG6$-%&dG_d xG6#%\"xG/F:;F0F1!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot(Rat(xmin), xm in=R..1.);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-% 'CURVESG6$7^p7$$\"3G+++]xd%Q(!#@$\"3Z!)[A([SUj%!#>7$$\"3#******pyz&>9! 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