K Matrix, Breit-Wigner, Riemann Sheets and Poles

Cesar Fernandez-Ramirez
Joint Physics Analysis Center & Jefferson Lab

E-mail: cesar@jlab.org

2015 International Summer Workshop on Reaction Theory,  June 8-15, 2015, Bloomington IN
Organized by Indiana University, Jefferson Lab and National Science Foundation

Notes

Scattering matrix, single channel

Analytic structure of the amplitude: 1st Riemann sheet

{}

Notebook definitions

Exit[];

<<NumericalDifferentialEquationAnalysis`;
Needs["NumericalCalculus`"]

SetDirectory[NotebookDirectory[]];
StyleList = {AbsoluteThickness[2.0], AbsolutePointSize[3], #} & /@ {Blue, Red, RGBColor[0,0,0], Purple};
SetOptions[Plot, Frame -> True, Axes -> {True, False}, PlotStyle -> StyleList, AspectRatio -> 0.8, FrameStyle -> Directive[13]];

ϵ=0.0000001;
epsilon=ϵ;

Basic Formulae

sth[m1_,m2_]:=(m1+ m2)^2;
q2[s_,m1_,m2_]:=(s-sth[m1,m2]);
qphys[s_,m1_,m2_]:=Sqrt[q2[s,m1,m2]];
qI[s_,m1_,m2_]:=I Sqrt[-q2[s,m1,m2]];

rhonorm[M_,m1_,m2_]:=If[M>sth[m1,m2],qphys[M,m1,m2],1];
rhophase[s_,m1_,m2_]:=If[s>sth[m1,m2],qphys[s,m1,m2],0];
rhoExact[s_,m1_,m2_]:=qI[s,m1,m2];

Analytic continuation of the “momentum”

mt1=0.140;
mt2=1.192;

iρ

GraphicsRow[{Plot[{Re[I*rhoExact[Res+I epsilon,mt1,mt2]],Re[I*rhoExact[Res-I epsilon,mt1,mt2]]},{Res,1,5}],Plot[{Im[I*rhoExact[Res+I epsilon,mt1,mt2]],Im[I*rhoExact[Res-I epsilon,mt1,mt2]]},{Res,1,5}],Plot[{Abs[rhoExact[Res+I epsilon,mt1,mt2]],Abs[rhoExact[Res-I epsilon,mt1,mt2]]},{Res,1,5}]},ImageSize->Full]

qfig1=Plot3D[Re[I*qI[Res+ I Ims,mt1,mt2]],{Res,1,5},{Ims,-1,1},ColorFunction->"Aquamarine"];
qfig2=Plot3D[Im[I*qI[Res+ I Ims,mt1,mt2]],{Res,1,5},{Ims,-1,0},ColorFunction->"Aquamarine"];
qfig3=Plot3D[Im[I*qI[Res+ I Ims,mt1,mt2]],{Res,1,5},{Ims,0,1},ColorFunction->"Aquamarine"];
qfig4=Plot3D[Im[I*qI[Res+ I Ims,mt1,mt2]],{Res,1,5},{Ims,0,1},ColorFunction->"Aquamarine"];
qfig5=Plot3D[Im[-I*qI[Res+ I Ims,mt1,mt2]],{Res,1,5},{Ims,-1,0},ColorFunction->"BrightBands",PlotRange->Automatic];

GraphicsRow[{Show[qfig1,PlotRange->Automatic,ImageSize->Medium],Show[qfig2,qfig3,PlotRange->Automatic,ImageSize->Medium],Show[qfig4,qfig5,PlotRange->Automatic,ImageSize->Medium]},ImageSize->Full]

Single Channel Breit-Wigner

Going to the 2nd Riemann Sheet

Breit-Wigner

DI0[s_,M_,m1_,m2_,x_]:=M^2-s -I* rhoExact[s,m1,m2]M x/rhonorm[M^2,m1,m2];
DII0[s_,M_,m1_,m2_,x_]:=M^2-  s+I*rhoExact[s,m1,m2] M x/rhonorm[M^2,m1,m2] ;
AI0[s_,M_,m1_,m2_,x_]:=M x/DI0[s,M,m1,m2,x]/rhonorm[M^2,m1,m2];
AII0[s_,M_,m1_,m2_,x_]:=AI0[s,M,m1,m2,x]/(  1+2 I rhoExact[s,m1,m2]  AI0[s,M,m1,m2,x] )

m1=0.140;
m2=1.192;
M=2.0;
x=0.2;

First Riemann sheet

Show[GraphicsRow[
{Plot[{Re[AI0[Res+ I epsilon,M,m1,m2,x]],Re[AI0[Res-I epsilon,M,m1,m2,x]]},{Res,1,5},PlotRange->{-0.5,0.5},GridLines->{{{sth[m1,m2],{Orange,Thick}}},{}}],Plot[{Im[AI0[Res+ I epsilon,M,m1,m2,x]],Im[AI0[Res-I epsilon,M,m1,m2,x]]},{Res,1,5},PlotRange->{-1,1},GridLines->{{{sth[m1,m2],{Orange,Thick}}},{}}],
Plot[{Abs[AI0[Res+ I epsilon,M,m1,m2,x]],Abs[AI0[Res-I epsilon,M,m1,m2,x]]},{Res,1,5},PlotRange->{0,1},GridLines->{{{sth[m1,m2],{Orange,Thick}}},{}}]}],ImageSize->Full]

f1=Plot3D[Im[AI0[Res+ I Ims,M,m1,m2,x]],{Res,1,5},{Ims,0,1},ColorFunction->"Aquamarine",PlotRange->{-2,2}];
f2=Plot3D[Im[AII0[Res+ I Ims,M,m1,m2,x]],{Res,1,5},{Ims,-1,0},ColorFunction->"BrightBands",PlotRange->{-2,2}];
f3=Plot3D[Im[AI0[Res+ I Ims,M,m1,m2,x]],{Res,1,5},{Ims,-1,0},ColorFunction->"Aquamarine"];
f4=Plot3D[Re[AI0[Res+ I Ims,M,m1,m2,x]],{Res,1,5},{Ims,-1,1},ColorFunction->"Aquamarine"];
f5=Plot3D[Re[AI0[Res+ I Ims,M,m1,m2,x]],{Res,1,5},{Ims,0,1},ColorFunction->"Aquamarine"];
f6=Plot3D[Re[AII0[Res+ I Ims,M,m1,m2,x]],{Res,1,5},{Ims,-1,0},ColorFunction->"BrightBands"];
f7=Plot3D[Abs[AI0[Res+ I Ims,M,m1,m2,x]],{Res,1,5},{Ims,0,1},ColorFunction->"Aquamarine",PlotRange->{0,8}];
f8=Plot3D[Abs[AII0[Res+ I Ims,M,m1,m2,x]],{Res,1,5},{Ims,-1,0},ColorFunction->"BrightBands",PlotRange->{0,8}];
f9=Plot3D[Abs[AI0[Res+ I Ims,M,m1,m2,x]],{Res,1,5},{Ims,-1,1},ColorFunction->"Aquamarine",PlotRange->{0,8}];

GraphicsRow[{Show[f4,PlotRange->Automatic,ImageSize->Medium],Show[f1,f3,PlotRange->Automatic,ImageSize->Medium],Show[f9,PlotRange->Automatic,ImageSize->Medium]},ImageSize->Full]

What is a Resonance:

A pole of the scattering amplitude in the unphysical Riemann sheets

First vs. Second Riemann Sheet

Show[GraphicsRow[
{Plot[{Re[AI0[Res+ I epsilon,M,m1,m2,x]],Re[AII0[Res-I epsilon,M,m1,m2,x]]},{Res,1,5},PlotRange->{-0.5,0.5},GridLines->{{{sth[m1,m2],{Orange,Thick}}},{}}],Plot[{Im[AI0[Res+ I epsilon,M,m1,m2,x]],Im[AII0[Res-I epsilon,M,m1,m2,x]]},{Res,1,5},PlotRange->{-1,1},GridLines->{{{sth[m1,m2],{Orange,Thick}}},{}}],
Plot[{Abs[AI0[Res+ I epsilon,M,m1,m2,x]],Abs[AII0[Res-I epsilon,M,m1,m2,x]]},{Res,1,5},PlotRange->{0,1},GridLines->{{{sth[m1,m2],{Orange,Thick}}},{}}]}],ImageSize->Full]

GraphicsRow[{Show[f5,f6,PlotRange->{-1,1},ImageSize->Medium],Show[f1,f2,PlotRange->{-1,1},ImageSize->Medium],Show[f7,f8,PlotRange->{0,8},ImageSize->Medium]},ImageSize->Full]

Pole search

DisplayForm[{"1st RS",a=NSolve[DI0[xs,M,m1,m2,x]==0,xs],"     ","2nd RS",b=NSolve[DII0[xs,M,m1,m2,x]==0,xs]}]

sp=Sqrt[xs/.b[[2]]]

Coupled channels Breit-Wigner

Coupled Channels

KMatrix[s_,M_,x1_,x2_]:=M/(M^2-s){{x1^2,x1 x2},{x1 x2,x2^2}};
rhoMatrix[rho1_,rho2_]:={{rho1,0},{0,rho2}};
Tmat=Dot[KMatrix[sth,Mth,xth1,xth2],Inverse[IdentityMatrix[2]-I*Dot[rhoMatrix[rhoth1,rhoth2],KMatrix[sth,Mth,xth1,xth2]]]];
{{Simplify[Tmat[[1,1]]],Simplify[Tmat[[1,2]]]},{Simplify[Tmat[[2,1]]],Simplify[Tmat[[2,2]]]}}// MatrixForm

DccI[s_,M_,m1_,m2_,m3_,m4_,x1_,x2_]:=M^2-s-I*M*rhoExact[s,m1,m2] x1^2/rhonorm[M^2,m1,m2]-I*M*rhoExact[s,m3,m4] x2^2/rhonorm[M^2,m3,m4];
DccII[s_,M_,m1_,m2_,m3_,m4_,x1_,x2_]:=M^2-s+I*M*rhoExact[s,m1,m2] x1^2/rhonorm[M^2,m1,m2]-I*M*rhoExact[s,m3,m4]x2^2/rhonorm[M^2,m3,m4];
DccIII[s_,M_,m1_,m2_,m3_,m4_,x1_,x2_]:=M^2-s+I*M*rhoExact[s,m1,m2] x1^2/rhonorm[M^2,m1,m2]+I*M*rhoExact[s,m3,m4]x2^2/rhonorm[M^2,m3,m4];
DccIV[s_,M_,m1_,m2_,m3_,m4_,x1_,x2_]:=M^2-s+I*M*rhoExact[s,m1,m2] x1^2/rhonorm[M^2,m1,m2]-I*M*rhoExact[s,m3,m4]x2^2/rhonorm[M^2,m3,m4];

TMatrixI[s_,M_,m1_,m2_,m3_,m4_,x1_,x2_]:=M/DccI[s,M,m1,m2,m3,m4,x1,x2]*{{x1^2/rhonorm[M^2,m1,m2],x1 x2/Sqrt[rhonorm[M^2,m1,m2] rhonorm[M^2,m3,m4]]},{x1 x2/Sqrt[rhonorm[M^2,m1,m2]rhonorm[M^2,m3,m4]],x2^2/rhonorm[M^2,m3,m4]}};
TMatrixII[s_,M_,m1_,m2_,m3_,m4_,x1_,x2_]:=M/DccII[s,M,m1,m2,m3,m4,x1,x2]*{{x1^2/rhonorm[M^2,m1,m2],x1 x2/Sqrt[rhonorm[M^2,m1,m2] rhonorm[M^2,m3,m4]]},{x1 x2/Sqrt[rhonorm[M^2,m1,m2]rhonorm[M^2,m3,m4]],x2^2/rhonorm[M^2,m3,m4]}};
TMatrixIII[s_,M_,m1_,m2_,m3_,m4_,x1_,x2_]:=M/DccIII[s,M,m1,m2,m3,m4,x1,x2]*{{x1^2/rhonorm[M^2,m1,m2],x1 x2/Sqrt[rhonorm[M^2,m1,m2] rhonorm[M^2,m3,m4]]},{x1 x2/Sqrt[rhonorm[M^2,m1,m2]rhonorm[M^2,m3,m4]],x2^2/rhonorm[M^2,m3,m4]}};
TMatrixIV[s_,M_,m1_,m2_,m3_,m4_,x1_,x2_]:=M/DccIV[s,M,m1,m2,m3,m4,x1,x2]*{{x1^2/rhonorm[M^2,m1,m2],x1 x2/Sqrt[rhonorm[M^2,m1,m2] rhonorm[M^2,m3,m4]]},{x1 x2/Sqrt[rhonorm[M^2,m1,m2]rhonorm[M^2,m3,m4]],x2^2/rhonorm[M^2,m3,m4]}};

1st Riemann sheet

m1=0.140;
m2=1.192;
m3=0.492;
m4=0.940;
M=1.8;
x1=0.45;
x2=0.25;

DisplayForm[{"1st threshold   ",sth[m1,m2],"GeV^2",Sqrt[sth[m1,m2]],"GeV","     ","2nd threshold  ",sth[m3,m4],"GeV^2",Sqrt[sth[m3,m4]],"GeV"}]

Show[GraphicsRow[
{Plot[{Re[TMatrixI[Res+ I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]],Re[TMatrixI[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]]},{Res,1,5},PlotRange->Automatic,GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}],Plot[{Im[TMatrixI[Res+ I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]],Im[TMatrixI[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]]},{Res,1,5},PlotRange->{-1.5,1.5},GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}],
Plot[{Abs[TMatrixI[Res+ I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]],Abs[TMatrixI[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]]},{Res,1,5},PlotRange->Automatic,GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}]}],ImageSize->Full]

Show[GraphicsRow[
{Plot[{Re[TMatrixI[Res+ I epsilon,M,m1,m2,m3,m4,x1,x2][[2,2]]],Re[TMatrixI[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[2,2]]]},{Res,1,5},PlotRange->Automatic,GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}],Plot[{Im[TMatrixI[Res+ I epsilon,M,m1,m2,m3,m4,x1,x2][[2,2]]],Im[TMatrixI[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[2,2]]]},{Res,1,5},PlotRange->{-0.6,0.6},GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}],
Plot[{Abs[TMatrixI[Res+ I epsilon,M,m1,m2,m3,m4,x1,x2][[2,2]]],Abs[TMatrixI[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[2,2]]]},{Res,1,5},PlotRange->Automatic,GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}]}],ImageSize->Full]

Show[GraphicsRow[
{Plot[{Re[TMatrixI[Res+ I epsilon,M,m1,m2,m3,m4,x1,x2][[1,2]]],Re[TMatrixI[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[1,2]]]},{Res,1,5},PlotRange->Automatic,GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}],Plot[{Im[TMatrixI[Res+ I epsilon,M,m1,m2,m3,m4,x1,x2][[1,2]]],Im[TMatrixI[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[1,2]]]},{Res,1,5},PlotRange->{-1,1},GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}],
Plot[{Abs[TMatrixI[Res+ I epsilon,M,m1,m2,m3,m4,x1,x2][[1,2]]],Abs[TMatrixI[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[1,2]]]},{Res,1,5},PlotRange->Automatic,GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}]}],ImageSize->Full]

We can see there are no poles in the 1st Riemann sheet

DisplayForm[{"1st RS",a=NSolve[DccI[xs,M,m1,m2,m3,m4,x1,x2]==0,xs]}]

ccIfig11rea=Plot3D[Re[TMatrixI[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,0,1},ColorFunction->"Aquamarine",PlotRange->Automatic];
ccIfig11reb=Plot3D[Re[TMatrixI[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,-1,0},ColorFunction->"Aquamarine",PlotRange->Automatic];
ccIfig11ima=Plot3D[Im[TMatrixI[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,0,1},ColorFunction->"Aquamarine",PlotRange->{-2,2}];
ccIfig11imb=Plot3D[Im[TMatrixI[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,-1,0},ColorFunction->"Aquamarine",PlotRange->{-2,2}];
ccIfig11absa=Plot3D[Abs[TMatrixI[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,0,1},ColorFunction->"Aquamarine",PlotRange->{0,2}];
ccIfig11absb=Plot3D[Abs[TMatrixI[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,-1,0},ColorFunction->"Aquamarine",PlotRange->{0,2}];

ccIfig22rea=Plot3D[Re[TMatrixI[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[2,2]]],{Res,1,5},{Ims,0,1},ColorFunction->"Aquamarine",PlotRange->Automatic];
ccIfig22reb=Plot3D[Re[TMatrixI[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[2,2]]],{Res,1,5},{Ims,-1,0},ColorFunction->"Aquamarine",PlotRange->Automatic];
ccIfig22ima=Plot3D[Im[TMatrixI[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[2,2]]],{Res,1,5},{Ims,0,1},ColorFunction->"Aquamarine",PlotRange->{-2,2}];
ccIfig22imb=Plot3D[Im[TMatrixI[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[2,2]]],{Res,1,5},{Ims,-1,0},ColorFunction->"Aquamarine",PlotRange->{-2,2}];
ccIfig22absa=Plot3D[Abs[TMatrixI[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[2,2]]],{Res,1,5},{Ims,0,1},ColorFunction->"Aquamarine",PlotRange->{0,2}];
ccIfig22absb=Plot3D[Abs[TMatrixI[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[2,2]]],{Res,1,5},{Ims,-1,0},ColorFunction->"Aquamarine",PlotRange->{0,2}];

ccIfig12rea=Plot3D[Re[TMatrixI[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,2]]],{Res,1,5},{Ims,0,1},ColorFunction->"Aquamarine",PlotRange->Automatic];
ccIfig12reb=Plot3D[Re[TMatrixI[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,2]]],{Res,1,5},{Ims,-1,0},ColorFunction->"Aquamarine",PlotRange->Automatic];
ccIfig12ima=Plot3D[Im[TMatrixI[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,2]]],{Res,1,5},{Ims,0,1},ColorFunction->"Aquamarine",PlotRange->Automatic];
ccIfig12imb=Plot3D[Im[TMatrixI[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,2]]],{Res,1,5},{Ims,-1,0},ColorFunction->"Aquamarine",PlotRange->Automatic];
ccIfig12absa=Plot3D[Abs[TMatrixI[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,2]]],{Res,1,5},{Ims,0,1},ColorFunction->"Aquamarine",PlotRange->Automatic];
ccIfig12absb=Plot3D[Abs[TMatrixI[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,2]]],{Res,1,5},{Ims,-1,0},ColorFunction->"Aquamarine",PlotRange->Automatic];

GraphicsRow[{Show[ccIfig11rea,ccIfig11reb,PlotRange->Automatic,ImageSize->Medium],Show[ccIfig11ima,ccIfig11imb,PlotRange->{-2,2},ImageSize->Medium],Show[ccIfig11absa,ccIfig11absb,PlotRange->{0,2},ImageSize->Medium]},ImageSize->Full]

Riemann sheets: 2^(Number of channels)

{}

Show[GraphicsRow[
{Plot[{Re[TMatrixI[Res+ I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]],Re[TMatrixII[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]],Re[TMatrixIII[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]]},{Res,1,5},PlotRange->Automatic,PlotStyle->{  {Blue,Thick},{Red,Dashed},{RGBColor[0,0,0],Dotted,Thick}},GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}],Plot[{Im[TMatrixI[Res+ I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]],Im[TMatrixII[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]],Im[TMatrixIII[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]]},{Res,1,5},PlotRange->{0,1.5},PlotStyle->{  {Blue,Thick},{Red,Dashed},{RGBColor[0,0,0],Dotted,Thick}},GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}],
Plot[{Abs[TMatrixI[Res+ I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]],Abs[TMatrixII[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]],Abs[TMatrixIII[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]]},{Res,1,5},PlotRange->{0,1.5},PlotStyle->{  {Blue,Thick},{Red,Dashed},{RGBColor[0,0,0],Dotted,Thick}},GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}]}],ImageSize->Full]

Let’s take a closer look to the thresholds region

Show[GraphicsRow[
{Plot[{Re[TMatrixI[Res+ I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]],Re[TMatrixII[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]],Re[TMatrixIII[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]]},{Res,sth[m1,m2]-0.2,sth[m3,m4]+0.2},PlotRange->Automatic,PlotStyle->{  {Blue,Thick},{Red,Dashed},{RGBColor[0,0,0],Dotted,Thick}},GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}],Plot[{Im[TMatrixI[Res+ I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]],Im[TMatrixII[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]],Im[TMatrixIII[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]]},{Res,sth[m1,m2]-0.2,sth[m3,m4]+0.2},PlotRange->Automatic,PlotStyle->{  {Blue,Thick},{Red,Dashed},{RGBColor[0,0,0],Dotted,Thick}},GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}],
Plot[{Abs[TMatrixI[Res+ I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]],Abs[TMatrixII[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]],Abs[TMatrixIII[Res- I epsilon,M,m1,m2,m3,m4,x1,x2][[1,1]]]},{Res,sth[m1,m2]-0.2,sth[m3,m4]+0.2},PlotRange->Automatic,PlotStyle->{  {Blue,Thick},{Red,Dashed},{RGBColor[0,0,0],Dotted,Thick}},GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}]}],ImageSize->Full]

Riemann Sheets

ccIIfig11reb=Plot3D[Re[TMatrixII[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,sth[m1,m2],sth[m3,m4]},{Ims,-1,0},PlotRange->{-2,2},ColorFunction->"Rainbow"];
ccIIfig11imb=Plot3D[Im[TMatrixII[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,sth[m1,m2],sth[m3,m4]},{Ims,-1,0},PlotRange->{-2,2},ColorFunction->"Rainbow"];
ccIIfig11absb=Plot3D[Abs[TMatrixII[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,sth[m1,m2],sth[m3,m4]},{Ims,-1,0},PlotRange->{0,2},ColorFunction->"Rainbow"];

ccIIIfig11reb=Plot3D[Re[TMatrixIII[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,sth[m3,m4],5},{Ims,-1,0},PlotRange->{-2,2},ColorFunction->"BrightBands"];
ccIIIfig11imb=Plot3D[Im[TMatrixIII[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,sth[m3,m4],5},{Ims,-1,0},PlotRange->{-2,2},ColorFunction->"BrightBands"];
ccIIIfig11absb=Plot3D[Abs[TMatrixIII[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,sth[m3,m4],5},{Ims,-1,0},PlotRange->{0,2},ColorFunction->"BrightBands"];

ccIVfig11reb=Plot3D[Re[TMatrixIV[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,sth[m1,m2],5},{Ims,-1,0},PlotRange->{-2,2},ColorFunction->"DeepSeaColors"];
ccIVfig11imb=Plot3D[Im[TMatrixIV[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,sth[m1,m2],5},{Ims,-1,0},PlotRange->{-2,2},ColorFunction->"DeepSeaColors"];
ccIVfig11absb=Plot3D[Abs[TMatrixIV[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,sth[m1,m2],5},{Ims,-1,0},PlotRange->{0,2},ColorFunction->"DeepSeaColors"];

GraphicsRow[{Show[ccIfig11rea,ccIIfig11reb,ccIIIfig11reb,PlotRange->{-2,2},ImageSize->Medium],Show[ccIfig11ima,ccIIfig11imb,ccIIIfig11imb,PlotRange->{-2,2},ImageSize->Medium],Show[ccIfig11absa,ccIIfig11absb,ccIIIfig11absb,PlotRange->{0,2},ImageSize->Medium]},ImageSize->Full]

Poles in Unphysical Riemann Sheets

DisplayForm[{"2nd RS",a=NSolve[DccII[xs,M,m1,m2,m3,m4,x1,x2]==0,xs]}]

spII=Sqrt[xs/.a[[2]]]

DisplayForm[{"3rd RS",b=NSolve[DccIII[xs,M,m1,m2,m3,m4,x1,x2]==0,xs]}]

spIII=Sqrt[xs/.b[[2]]]

DisplayForm[{"4th RS",c=NSolve[DccIV[xs,M,m1,m2,m3,m4,x1,x2]==0,xs]}]

spIV=Sqrt[xs/.c[[2]]]

Closer look to the different Riemann sheets for the 11 channel

BccIIfig11rea=Plot3D[Re[TMatrixII[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,0,1},PlotRange->{-2,2},ColorFunction->"Rainbow"];
BccIIfig11ima=Plot3D[Im[TMatrixII[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,0,1},PlotRange->{-2,2},ColorFunction->"Rainbow"];
BccIIfig11absa=Plot3D[Abs[TMatrixII[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,0,1},PlotRange->{0,2},ColorFunction->"Rainbow"];
BccIIfig11reb=Plot3D[Re[TMatrixII[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,-1,0},PlotRange->{-2,2},ColorFunction->"Rainbow"];
BccIIfig11imb=Plot3D[Im[TMatrixII[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,-1,0},PlotRange->{-2,2},ColorFunction->"Rainbow"];
BccIIfig11absb=Plot3D[Abs[TMatrixII[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,-1,0},PlotRange->{0,2},ColorFunction->"Rainbow"];

BccIIIfig11rea=Plot3D[Re[TMatrixIII[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,0,1},PlotRange->{-2,2},ColorFunction->"BrightBands"];
BccIIIfig11ima=Plot3D[Im[TMatrixIII[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,0,1},PlotRange->{-2,2},ColorFunction->"BrightBands"];
BccIIIfig11absa=Plot3D[Abs[TMatrixIII[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,0,1},PlotRange->{0,2},ColorFunction->"BrightBands"];
BccIIIfig11reb=Plot3D[Re[TMatrixIII[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,-1,0},PlotRange->{-2,2},ColorFunction->"BrightBands"];
BccIIIfig11imb=Plot3D[Im[TMatrixIII[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,-1,0},PlotRange->{-2,2},ColorFunction->"BrightBands"];
BccIIIfig11absb=Plot3D[Abs[TMatrixIII[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,-1,0},PlotRange->{0,2},ColorFunction->"BrightBands"];

BccIVfig11rea=Plot3D[Re[TMatrixIV[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,0,1},PlotRange->{-2,2},ColorFunction->"DeepSeaColors"];
BccIVfig11ima=Plot3D[Im[TMatrixIV[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,0,1},PlotRange->{-2,2},ColorFunction->"DeepSeaColors"];
BccIVfig11absa=Plot3D[Abs[TMatrixIV[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,0,1},PlotRange->{0,2},ColorFunction->"DeepSeaColors"];
BccIVfig11reb=Plot3D[Re[TMatrixIV[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,-1,0},PlotRange->{-2,2},ColorFunction->"DeepSeaColors"];
BccIVfig11imb=Plot3D[Im[TMatrixIV[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,-1,0},PlotRange->{-2,2},ColorFunction->"DeepSeaColors"];
BccIVfig11absb=Plot3D[Abs[TMatrixIV[Res+ I Ims,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,1,5},{Ims,-1,0},PlotRange->{0,2},ColorFunction->"DeepSeaColors"];

GraphicsRow[{Show[ccIfig11rea,BccIIfig11reb,PlotRange->{-2,2},ImageSize->Medium],Show[ccIfig11ima,BccIIfig11imb,PlotRange->{-2,2},ImageSize->Medium],Show[ccIfig11absa,BccIIfig11absb,PlotRange->{0,2},ImageSize->Medium]},ImageSize->Full]

GraphicsRow[{Show[ccIfig11rea,BccIIIfig11reb,PlotRange->{-2,2},ImageSize->Medium],Show[ccIfig11ima,BccIIIfig11imb,PlotRange->{-2,2},ImageSize->Medium],Show[ccIfig11absa,BccIIIfig11absb,PlotRange->{0,2},ImageSize->Medium]},ImageSize->Full]

GraphicsRow[{Show[ccIfig11rea,BccIVfig11reb,PlotRange->{-2,2},ImageSize->Medium],Show[ccIfig11ima,BccIVfig11imb,PlotRange->{-2,2},ImageSize->Medium],Show[ccIfig11absa,BccIVfig11absb,PlotRange->{0,2},ImageSize->Medium]},ImageSize->Full]

Adding the phase space -> Physics

qphysphase[s_,m1_,m2_]:=If[Re[s]>sth[m1,m2],Sqrt[q2[s,m1,m2]],0];
PhysicalAmplitude[s_,M_,m1_,m2_,m3_,m4_,x1_,x2_]:={{ qphysphase[s,m1,m2] x1^2 /rhonorm[M^2,m1,m2], Sqrt[qphysphase[s,m1,m2]qphysphase[s,m3,m4]] x1 x2/Sqrt[rhonorm[M^2,m1,m2] rhonorm[M^2,m3,m4]]},{ Sqrt[qphysphase[s,m1,m2]qphysphase[s,m3,m4]] x1 x2/Sqrt[rhonorm[M^2,m1,m2] rhonorm[M^2,m3,m4]],  qphysphase[s,m3,m4] x2^2/rhonorm[M^2,m3,m4] }}*M/DccI[s+I*epsilon*0.1,M,m1,m2,m3,m4,x1,x2];
PhysicalAmplitudeD[s_,M_,m1_,m2_,m3_,m4_,x1_,x2_]:={{ qphysphase[s,m1,m2] x1^2 /rhonorm[M^2,m1,m2], Sqrt[qphysphase[s,m1,m2]qphysphase[s,m3,m4]] x1 x2/Sqrt[rhonorm[M^2,m1,m2] rhonorm[M^2,m3,m4]]},{ Sqrt[qphysphase[s,m1,m2]qphysphase[s,m3,m4]] x1 x2/Sqrt[rhonorm[M^2,m1,m2] rhonorm[M^2,m3,m4]],  qphysphase[s,m3,m4] x2^2/rhonorm[M^2,m3,m4] }}*M/DccI[s-I*epsilon*0.1,M,m1,m2,m3,m4,x1,x2];

Show[GraphicsRow[
{Plot[{Re[PhysicalAmplitude[Res,M,m1,m2,m3,m4,x1,x2][[1,1]]],Re[PhysicalAmplitude[Res,M,m1,m2,m3,m4,x1,x2][[2,2]]],Re[PhysicalAmplitude[Res,M,m1,m2,m3,m4,x1,x2][[1,2]]]},{Res,1,5},PlotRange->{-1,1},GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}],Plot[{Im[PhysicalAmplitude[Res,M,m1,m2,m3,m4,x1,x2][[1,1]]],Im[PhysicalAmplitude[Res,M,m1,m2,m3,m4,x1,x2][[2,2]]],Im[PhysicalAmplitude[Res,M,m1,m2,m3,m4,x1,x2][[1,2]]]},{Res,1,5},PlotRange->{-0.5,1},GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}],
Plot[{Abs[PhysicalAmplitude[Res,M,m1,m2,m3,m4,x1,x2][[1,1]]],Abs[PhysicalAmplitude[Res,M,m1,m2,m3,m4,x1,x2][[2,2]]],Abs[PhysicalAmplitude[Res,M,m1,m2,m3,m4,x1,x2][[1,2]]]},{Res,1,5},PlotRange->{0,1},GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}]}],ImageSize->Full]

Closer look to the thresholds

Show[GraphicsRow[{Plot[{Re[PhysicalAmplitude[Res,M,m1,m2,m3,m4,x1,x2][[1,1]]],Re[PhysicalAmplitude[Res,M,m1,m2,m3,m4,x1,x2][[2,2]]],Re[PhysicalAmplitude[Res,M,m1,m2,m3,m4,x1,x2][[1,2]]]},{Res,sth[m1,m2]-0.2,sth[m3,m4]+0.2},PlotRange->{-1,1},GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}],Plot[{Im[PhysicalAmplitude[Res,M,m1,m2,m3,m4,x1,x2][[1,1]]],Im[PhysicalAmplitude[Res,M,m1,m2,m3,m4,x1,x2][[2,2]]],Im[PhysicalAmplitude[Res,M,m1,m2,m3,m4,x1,x2][[1,2]]]},{Res,sth[m1,m2]-0.2,sth[m3,m4]+0.2},PlotRange->{-0.5,1},GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}],Plot[{Abs[PhysicalAmplitude[Res,M,m1,m2,m3,m4,x1,x2][[1,1]]],Abs[PhysicalAmplitude[Res,M,m1,m2,m3,m4,x1,x2][[2,2]]],Abs[PhysicalAmplitude[Res,M,m1,m2,m3,m4,x1,x2][[1,2]]]},{Res,sth[m1,m2]-0.2,sth[m3,m4]+0.2},PlotRange->{0,1},GridLines->{{{sth[m1,m2],{Orange,Thick}},{sth[m3,m4],{Orange,Thick}}},{}}]}],ImageSize->Full]

S2[s_,M_,m1_,m2_,m3_,m4_,x1_,x2_]:=Dot[ConjugateTranspose[IdentityMatrix[2]+2 I PhysicalAmplitude[s,M,m1,m2,m3,m4,x1,x2]],IdentityMatrix[2]+2 I PhysicalAmplitude[s,M,m1,m2,m3,m4,x1,x2] ]

utest1 = Plot[Re[S2[Res,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,sth[m1,m2],5}];utest2 = Plot[Re[S2[Res,M,m1,m2,m3,m4,x1,x2][[1,2]]],{Res,sth[m1,m2],5}];
utest3 = Plot[Re[S2[Res,M,m1,m2,m3,m4,x1,x2][[2,2]]],{Res,sth[m1,m2],5}];
utest4 = Plot[Im[S2[Res,M,m1,m2,m3,m4,x1,x2][[1,1]]],{Res,sth[m1,m2],5}];utest5 = Plot[Im[S2[Res,M,m1,m2,m3,m4,x1,x2][[1,2]]],{Res,sth[m1,m2],5}];
utest6 = Plot[Im[S2[Res,M,m1,m2,m3,m4,x1,x2][[2,2]]],{Res,sth[m1,m2],5}];

GraphicsRow[{Show[utest1,PlotRange->{0.95,1.05},ImageSize->Medium],Show[utest2,PlotRange->{-0.05,0.05},ImageSize->Medium],Show[utest3,PlotRange->{0.95,1.05},ImageSize->Medium]},ImageSize->Full]

GraphicsRow[{Show[utest4,PlotRange->{-0.05,0.05},ImageSize->Medium],Show[utest5,PlotRange->{-0.05,0.05},ImageSize->Medium],Show[utest6,PlotRange->{-0.05,0.05},ImageSize->Medium]},ImageSize->Full]