Partial Wave Amplitude Calculator

Paul M Eugenio
University of Massachusetts
31-Oct-97

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.

As of this date, calculation are limited to one-isobar decays such as
X-> rho pi,  X-> omega pi , X->a2 eta , ...

IMPORTANT!  In the Action menue->Evaluate Notebook

Blatt-Weisskopf centrifugal-barrier factors  F[L,p]
(See my note on generating barrier factors.)

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

N[Simplify[F[2,x]]]

D-functions

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]}]

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]]

Dfun2 is a 2 angle modified D function. The function exploits our choice
of phase conviention.

N[Dfun2c[1,-1,0,0.650474*Pi,0.708039]]

CG[l1_,m1_,l2_,m2_,l_,m_] :=ClebschGordan[{l1,m1},{l2,m2},{l,m}]

N[CG[2,0,1,-1,1,-1]]

Reflectivity Basis

TH[m_] := Switch[m,1,1/Sqrt[2],0,1/2,-1,0]

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])

Decay Amplitude in the Reflectivity Basis

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}]

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

Amplitude Calulations

Add you inputs below and Shift-Enter to calculate.

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=.