(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "X Window System Mathematica Notebook Front End Version 2.2"; X11StandardFontEncoding; fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, fontName, "times"; fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, fontName, "times"; fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, fontName, "times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, fontName, "times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, fontName, "times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, fontName, "times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, fontName, "times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, 12, fontName, "courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, fontName, "courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, 12, fontName, "courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, 12, fontName, "courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, 12, fontName, "courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, fontName, "courier"; fontset = name, inactive, noPageBreakInGroup, nohscroll, preserveAspect, M7, italic, B65535, 10, fontName, "times"; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 12, fontName, "times"; fontset = leftheader, 12, fontName, "times"; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, 12, fontName, "times"; fontset = leftfooter, 12, fontName, "times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "courier"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times";paletteColors = 128; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] Partial Wave Amplitude Calculator :[font = subsubtitle; inactive; preserveAspect; plain; bold; italic] Paul M Eugenio University of Massachusetts 31-Oct-97 :[font = input; preserveAspect] :[font = text; inactive; preserveAspect; center; plain; bold] This Mathematica package calculates amplitudes in the same spirit of the E852 code "gamp". The formulas follow that of S.U. Chungs's Formulas for Partial Wave Analysis Ver. II but differ in formula # 3.2, where both this package and the PWA code use 2 BIG D functions and not 1 D function and 1 d function. :[font = text; inactive; preserveAspect; center] As of this date, calculation are limited to one-isobar decays such as X-> rho pi, X-> omega pi , X->a2 eta , ... :[font = text; inactive; preserveAspect; fontColorRed = 65535; fontSize = 14; fontName = "Rock"] IMPORTANT! In the Action menue->Evaluate Notebook :[font = subtitle; inactive; preserveAspect; startGroup] Blatt-Weisskopf centrifugal-barrier factors F[L,p] (See my note on generating barrier factors.) ;[s] 2:0,0;53,1;96,-1; 2:1,0,0 ,times,1,18,0,0,0;1,0,0 ,times,1,12,0,0,0; :[font = input; preserveAspect; startGroup] h1[n_,z_] := (-I)^(n+1)*Exp[I*z]/z* Sum[I^s/(s!(2*z)^s) (n +s)!/(n-s)! ,{s,0,n}] NFactor[l_] := Switch[l, 0,1, 1,2, 2,13, 3,277, 4,12746, 5,99881, 6,118394977] F[l_,q_] := Sqrt[NFactor[l]/(q^2 h1[l,q] ComplexExpand[Conjugate[h1[l,q]]])] Pr = 0.1973 :[font = output; output; inactive; preserveAspect; endGroup] 0.1973 ;[o] 0.1973 :[font = input; preserveAspect; startGroup] N[Simplify[F[2,x]]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 3.605551275463989*(x^4/(9. + 3.*x^2 + x^4))^(1/2) ;[o] 4 x 3.60555 Sqrt[---------------] 2 4 9. + 3. x + x :[font = subtitle; inactive; preserveAspect; startGroup] D-functions :[font = input; preserveAspect] d[l_,mp_,m_,theta_] := (-1)^(mp-m) * Sqrt[(l+mp)! (l-mp)! (l+m)! (l-m)!] * Sum[( (-1)^k Cos[theta/2]^(2l-(mp-m+2k)) * Sin[theta/2]^(mp-m+2k) )/( (l-mp-k)! (l+m-k)! * (mp-m+k)! k! ) ,{k,Max[0,m-mp],Min[l-mp,l+m]}] :[font = input; preserveAspect] Dfun[l_,mp_,m_,alpha_,beta_,gamma_] := Exp[-I mp alpha] d[l,mp,m,beta] Exp[-I mp gamma] Dfunc[l_,mp_,m_,alpha_,beta_,gamma_] := Conjugate[Exp[-I mp alpha] d[l,mp,m,beta] Exp[-I mp gamma]] Dfun2[l_,mp_,m_,phi_,theta_] := Exp[-I mp phi] d[l,mp,m,theta] Dfun2c[l_,mp_,m_,phi_,theta_] := Conjugate[Exp[-I mp phi] d[l,mp,m,theta]] :[font = text; inactive; preserveAspect; center] Dfun2 is a 2 angle modified D function. The function exploits our choice of phase conviention. :[font = input; preserveAspect; startGroup] N[Dfun2c[1,-1,0,0.650474*Pi,0.708039]] :[font = output; output; inactive; preserveAspect; endGroup] -0.2093836360069349 - 0.4094301451694883*I ;[o] -0.209384 - 0.40943 I :[font = input; preserveAspect] CG[l1_,m1_,l2_,m2_,l_,m_] :=ClebschGordan[{l1,m1},{l2,m2},{l,m}] :[font = input; preserveAspect; startGroup] N[CG[2,0,1,-1,1,-1]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 0.3162277660168379 ;[o] 0.316228 :[font = subtitle; inactive; preserveAspect; startGroup] Reflectivity Basis :[font = input; preserveAspect] TH[m_] := Switch[m,1,1/Sqrt[2],0,1/2,-1,0] :[font = input; preserveAspect] reDfunc[J_,m_,lam_,phi_,theta_,P_,eps_] := TH[m]*( Dfun2c[J,m,lam,phi,theta] - eps P (-1)^(J-m) * Dfun2c[J,-m,lam,phi,theta]) :[font = input; preserveAspect; endGroup] :[font = subtitle; inactive; preserveAspect; startGroup] Decay Amplitude in the Reflectivity Basis :[font = input; preserveAspect] Amp[J_,m_,l_,s_,phi_,theta_,phih_,thetah_,P_,eps_] := Sqrt[2*l+1]*Sqrt[2*s+1]*F[l,p/Pr]*F[s,q/Pr]*BW* Sum[reDfunc[J,m,lam,phi,theta,P,eps] * Dfun2c[s,lam,0,phih,thetah] CG[l,0,s,lam,J,lam], {lam,-s,s}] :[font = text; inactive; preserveAspect; center] For Omega decays remove the Barrier Factor F[s,q/Pr] and use the normalized lambda factor as the BW factor. For helicity angles use the normal to the omega decay plane as the analyzer. All inputs for the calculation are available in the gamp debug print out ;[s] 2:0,0;189,1;261,-1; 2:1,0,0 ,times,0,12,0,0,0;1,0,0 ,times,1,12,0,0,65535; :[font = input; preserveAspect; endGroup] :[font = subtitle; inactive; preserveAspect; startGroup] Amplitude Calulations :[font = input; preserveAspect; startGroup] Add you inputs below and Shift-Enter to calculate. :[font = message; inactive; preserveAspect; endGroup] Syntax::sntxi: Incomplete expression. :[font = input; preserveAspect; startGroup] J=1 P=+1 M=1 eps=-1 l=2 s=1 theta=2.77 phi=0.69*Pi thetah=0.708 phih=0.65*Pi BW=0.1861 p=0.0641353 q= 0.002 Print[Amplitude is] N[Amp[J,M,l,s,phi,theta,phih,thetah,P,eps]] J=. P=. M=. eps=. l=. s=. theta=. phi=. thetah=. phih=. BW=. p=. :[font = output; output; inactive; preserveAspect] 1 ;[o] 1 :[font = output; output; inactive; preserveAspect] 1 ;[o] 1 :[font = output; output; inactive; preserveAspect] 1 ;[o] 1 :[font = output; output; inactive; preserveAspect] -1 ;[o] -1 :[font = output; output; inactive; preserveAspect] 2 ;[o] 2 :[font = output; output; inactive; preserveAspect] 1 ;[o] 1 :[font = output; output; inactive; preserveAspect] 2.77 ;[o] 2.77 :[font = output; output; inactive; preserveAspect] 0.69*Pi ;[o] 0.69 Pi :[font = output; output; inactive; preserveAspect] 0.708 ;[o] 0.708 :[font = output; output; inactive; preserveAspect] 0.65*Pi ;[o] 0.65 Pi :[font = output; output; inactive; preserveAspect] 0.1861 ;[o] 0.1861 :[font = output; output; inactive; preserveAspect] 0.06413530000000001 ;[o] 0.0641353 :[font = output; output; inactive; preserveAspect] 0.002 ;[o] 0.002 :[font = print; inactive; preserveAspect] Amplitude is :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] 1.788634373988184*10^-20 + 0.0002259748092892823*I ;[o] -20 1.78863 10 + 0.000225975 I ^*)