Multipole Expansion
of CGLN-amplitudes

Graphics, packages and global definitions

SetOptions[{Plot,ListPlot,ListLinePlot,ParametricPlot,ContourPlot,Plot3D,Graphics},​​BaseStyle{13,FontFamily"Times",Italic}];​​SetOptions[{Plot,ListLinePlot},PlotStyle{{Blue,Thickness[0.005]},{Red,Thickness[0.005]},{Black,Thickness[0.005]},{Green//Darker,Thickness[0.005]}}];​​SetOptions[ListPlot,PlotStyle{{Blue,PointSize[0.010]},{Red,PointSize[0.010]}}];​​SetOptions[$FrontEnd,"DynamicUpdating"True]​​
Needs["NumericalCalculus`"]
Needs["ErrorBarPlots`"]​​(*Needs["NonlinearRegression`"]*)​​SetDirectory[NotebookDirectory[]]
C:\Fredy\K-Lambda-2021\L+P PWA\Final results\Complete set of observables
Off[NIntegrate::"slwcon"]​​Off[NIntegrate::"ncvb"]
rndg[μ_,σ_]:=RandomReal[NormalDistribution[μ,σ]]
P
n_
[x_]:=LegendreP[n,x]
P1s[x_]=
P
1
'[x];P2s[x_]=
P
2
'[x];P3s[x_]=
P
3
'[x];P4s[x_]=
P
4
'[x];P5s[x_]=
P
5
'[x];P6s[x_]=
P
6
'[x];P7s[x_]=
P
7
'[x];​​P1ss[x_]=
P
1
''[x];P2ss[x_]=
P
2
''[x];P3ss[x_]=
P
3
''[x];P4ss[x_]=
P
4
''[x];P5ss[x_]=
P
5
''[x];P6ss[x_]=
P
6
''[x];P7ss[x_]=
P
7
''[x];
νW1[t0_]:=νk[kl[W1],t0](*pionthreshold*)​​νW2[t0_]:=νk[kl[W2],t0](*etathreshold*)​​νW4[t0_]:=νk[kl[W4],t0]​​νW3[t0_]:=Module[{ang},ang=If[t0<tthrG,180,0];​​W3=w/.FindRoot[t[w,ang]t0,{w,1.50}];νk[kl[W3],t0]]
pf1:=PaddedForm[#,{5,1}]&​​pf2:=PaddedForm[#,{5,2}]&​​pf3:=PaddedForm[#,{5,3}]&​​pf4:=PaddedForm[#,{5,4}]&​​pf5:=PaddedForm[#,{5,5}]&​​pf6:=PaddedForm[#,{6,6}]&​​pf8:=PaddedForm[#,{10,8}]&​​pf10:=PaddedForm[#,{10,10}]&

Multipole Expansion

◼

Expanding CGLN amplitudes

P
l_
[x_]:=LegendreP[l,x]
F1[x_]:=
Lmax
∑
ℓ=0
(ℓMp[ℓ]+Ep[ℓ])
P
ℓ+1
'[x]+((ℓ+1)Mm[ℓ]+Em[ℓ])
P
ℓ-1
'[x]​​F2[x_]:=
Lmax
∑
ℓ=1
((ℓ+1)Mp[ℓ]+ℓMm[ℓ])
P
ℓ
'[x]​​F3[x_]:=
Lmax
∑
ℓ=1
(-Mp[ℓ]+Ep[ℓ])
P
ℓ+1
''[x]+(Mm[ℓ]+Em[ℓ])
P
ℓ-1
''[x]​​F4[x_]:=
Lmax
∑
ℓ=2
(Mp[ℓ]-Ep[ℓ]-Mm[ℓ]-Em[ℓ])
P
ℓ
''[x]
F1[x]/.xCos[θ]​​F2[x]/.xCos[θ]​​F3[x]/.xCos[θ]​​F4[x]/.xCos[θ]
0
0
0
0
◼

Expanding CGLN amplitudes

BGReplacement={Ep[0]0.8+0.75,Mm[1]0.7+0.65,Ep[1]0.6+0.55,Mp[1]0.5+0.45}​​BGReplacementReandIm={u10.8,u20.7,u30.6,u40.5,v10.75,v20.65,v30.55,v40.45}
{Ep[0]0.8+0.75,Mm[1]0.7+0.65,Ep[1]0.6+0.55,Mp[1]0.5+0.45}
{u10.8,u20.7,u30.6,u40.5,v10.75,v20.65,v30.55,v40.45}
◼

Expanding CGLN amplitudes

F1[x]/.xCos[θ]​​F2[x]/.xCos[θ]​​F3[x]/.xCos[θ]​​F4[x]/.xCos[θ]
Ep[0]+3Cos[θ](Ep[1]+Mp[1])
Mm[1]+2Mp[1]
3(Ep[1]-Mp[1])
0
CGLNF1[θ_]:=F1[Cos[θ]]​​CGLNF1[θ]​​CGLNF2[θ_]:=F2[Cos[θ]]​​CGLNF2[θ]​​CGLNF3[θ_]:=F3[Cos[θ]]​​CGLNF3[θ]​​CGLNF4[θ_]:=F4[Cos[θ]]​​CGLNF4[θ]
Ep[0]+3Cos[θ](Ep[1]+Mp[1])
Mm[1]+2Mp[1]
3(Ep[1]-Mp[1])
0
CGLNF1fin[x_]:=Collect[Refine[CGLNF1[x]/.{Ep[0]u1+v1,Mm[1]u2+v2,Ep[1]u3+v3,Mp[1]u4+v4},{u1,u2,u3,u4,v1,v2,v3,v4}∈Reals]//Simplify,Cos[x]]//ComplexExpand​​CGLNF1fin[Θ]​​CGLNF1fin[0.1]/.BGReplacementReandIm​​​​CGLNF2fin[x_]:=Collect[Refine[CGLNF2[x]/.{Ep[0]u1+v1,Mm[1]u2+v2,Ep[1]u3+v3,Mp[1]u4+v4},{u1,u2,u3,u4,v1,v2,v3,v4}∈Reals]//Simplify,Cos[x]]//ComplexExpand​​CGLNF2fin[Θ]​​CGLNF2fin[0.1]/.BGReplacementReandIm​​​​​​CGLNF3fin[x_]:=Collect[Refine[CGLNF3[x]/.{Ep[0]u1+v1,Mm[1]u2+v2,Ep[1]u3+v3,Mp[1]u4+v4},{u1,u2,u3,u4,v1,v2,v3,v4}∈Reals]//Simplify,Cos[x]]//ComplexExpand​​CGLNF3fin[Θ]​​CGLNF3fin[0.1]/.BGReplacementReandIm​​​​​​CGLNF4fin[x_]:=Collect[Refine[CGLNF4[x]/.{Ep[0]u1+v1,Mm[1]u2+v2,Ep[1]u3+v3,Mp[1]u4+v4},{u1,u2,u3,u4,v1,v2,v3,v4}∈Reals]//Simplify,Cos[x]]//ComplexExpand​​CGLNF4fin[Θ]​​CGLNF4fin[0.1]/.BGReplacementReandIm​​​​
◼

​
For L=1 it has a form

◼

Let us evaluate conditions for the fixed b1 phase
(the final result are three conditions Eq1CGLNF1, Eq2CGLNF1, Eq3CGLNF1)

◼

Finding explicit form of complex conjugate

◼

Calculating differential cross section

◼

Perform the inverse procedure
(fit differential cross section with 3 constraints, and try to restore the input which generated the numeric data)