Load Data

Import[FileNameJoin[{NotebookDirectory[],"HepLib.m"}]]

ResJ0=C2M@Import[FileNameJoin[{NotebookDirectory[],"J0.txt"}]];
ResJ1=C2M@Import[FileNameJoin[{NotebookDirectory[],"J1.txt"}]];
ResJ2=C2M@Import[FileNameJoin[{NotebookDirectory[],"J2.txt"}]];

(*需要除以4，(Subscript[对应上面的Φ, π])*)
ResJ0=ResJ0/4;
ResJ1=ResJ1/4;
ResJ2=ResJ2/4;

R0=Collect[Plus@@ResJ0,{C33,C66},Factor]
Variables[%]

Plus@@ResJ1//Simplify
Variables[%]

R2=Plus@@ResJ2//Simplify
Variables[%]

R0

RR0=-I R0/.{gs->(Sqrt[απ 4]),u->x,v->y};

TEX

Factor@Coefficient[RR0,C33 S11]/.{1-x->X,x-1->-X,1-y->Y,y-1->-Y,4 x y-3x-3y+2->Expand[4 x y-3x-3y+2/.{x->1-X,y->1-Y}]}

Factor@Coefficient[RR0,C66 S00]/.{1-x->X,x-1->-X,1-y->Y,y-1->-Y,4 x y-3x-3y+2->Expand[4 x y-3x-3y+2/.{x->1-X,y->1-Y}]}

HepLib

Coefficient[RR0,C33 S11]/.{m->1,απ->1}
(4 u v-3 u-3 v+2)(4 u v-u-v) (((2 u-1))^2)(((2 v-1))^2)%//M2C;
(*Export[FileNameJoin[{NotebookDirectory[],"J033.txt"}],%]*)

Coefficient[RR0,C66 S00]/.{m->1,απ->1}
(4 u v-3 u-3 v+2)(4 u v-u-v) (((2 u-1))^2)(((2 v-1))^2)%//M2C;
(*Export[FileNameJoin[{NotebookDirectory[],"J066.txt"}],%]*)

R2

RR2=-I R2/.{gs->(Sqrt[απ 4]),u->x,v->y,(k-2 p).lim1002->3k.lim1002,(p-2 k).lim1002->-3k.lim1002,p.lim1001->-k.lim1001};

TEX

Factor@Coefficient[RR2,C33 S11]/.{1-x->X,x-1->-X,1-y->Y,y-1->-Y,4 x y-3x-3y+2->Expand[4 x y-3x-3y+2/.{x->1-X,y->1-Y}]}

Factor@Coefficient[RR2,C66 S00]/.{1-x->X,x-1->-X,1-y->Y,y-1->-Y,4 x y-3x-3y+2->Expand[4 x y-3x-3y+2/.{x->1-X,y->1-Y}]}

HepLib

Coefficient[(RR2/((3 k.lim1001 k.lim1002+4 (m^2) lim1002.lim1001)))//Cancel,C33 S11]/.{m->1,απ->1}
(4 u v-3 u-3 v+2)(4 u v-u-v) (((2 u-1))^2)(((2 v-1))^2)%//M2C;
(*Export[FileNameJoin[{NotebookDirectory[],"J233.txt"}],%]*)

Gegenbauer

Table[GegenbauerC[n,3/2,x],{n,0,8}]//Simplify//M2C

Load Conv

Conv-J033

ConvJ033=Import[FileNameJoin[{NotebookDirectory[],"Conv-J033.txt"}]]//C2M//ChopVE

ConvJ033-Transpose@ConvJ033//SimplifyVE//ChopVE

Conv-J066

ConvJ066=Import[FileNameJoin[{NotebookDirectory[],"Conv-J066.txt"}]]//C2M//ChopVE

ConvJ066-Transpose@ConvJ066//SimplifyVE//ChopVE

Conv-J233

ConvJ233=Import[FileNameJoin[{NotebookDirectory[],"Conv-J233.txt"}]]//C2M//ChopVE

ConvJ233-Transpose@ConvJ233//SimplifyVE//ChopVE

TEX

mns=Flatten[Table[{m,n},{m,1,8+1},{n,m,8+1}],1];

mnsEven=Cases[mns,p_/;EvenQ[p[[1]]+p[[2]]]];

mns//Length
mnsEven//Length

If[OddQ[Length@mns],AppendTo[mns,{1,1}]];
If[OddQ[Length@mnsEven],AppendTo[mnsEven,{1,1}]];

mns//Length
mnsEven//Length

Convs={
ConvJ033/.VE[v_,_]:>v/.Complex[r_,i_]:>Complex[N[r,4],N[i,4]],
ConvJ066/.VE[v_,_]:>v/.Complex[r_,i_]:>Complex[N[r,4],N[i,4]],
ConvJ233/.VE[v_,_]:>v/.Complex[r_,i_]:>Complex[N[r,4],N[i,4]]
};

sConv=Map[Function[mn,
tt=ToString[mn-1]<>" & "<>StringJoin@@Riffle[Map[("$"<>ToString[#]<>"$")&,{
Convs[[1,mn[[1]],mn[[2]]]],Convs[[2,mn[[1]],mn[[2]]]],Convs[[3,mn[[1]],mn[[2]]]]
}]," & "];
StringReplace[tt,{"{"->"(","}"->")","I"->"i"}]
],mns]

sConv=Map[Function[mn,
tt=ToString[mn-1]<>" & "<>StringJoin@@Riffle[Map[("$"<>ToString[#]<>"$")&,{
Convs[[1,mn[[1]],mn[[2]]]],Convs[[2,mn[[1]],mn[[2]]]],Convs[[3,mn[[1]],mn[[2]]]]
}]," & "];
StringReplace[tt,{"{"->"(","}"->")","I"->"i"}]
],mnsEven]

StringJoin[Table[sConv[[i]]<>" &\n "<>sConv[[i+Length@sConv/2]]<>"\\\\\n",{i,Length@sConv/2}]]

RunDec & MaTeX

(*Please install MaTeX and RunDec*)

<<MQgraf`MaTeX`

<<RunDec`

NumDef={asMz->0.1182,Mz->91.1876,Mt->173.21,Mb->4.7,Mc->1.6,muc->1.2,mub->3.97,Mtau->1.77686};

Clear[As]
As[muR_]:=As[muR]=AsRunDec[asMz/.NumDef,Mz/.NumDef,muR,2]

As[2 1.5]

As[2 4.8]

Input Parameters

OO[0,33,I]=0.0347;
OO[0,33,II]=0.0187;

OO[0,36,I]=0.0211;
OO[0,36,II]=-0.0161;

OO[0,66,I]=0.0128;
OO[0,66,II]=-0.0161;

OO[2,33,I]=0.0144;
OO[2,33,II]=0.0126;

Clear[aπ,aK];

aπ[0,_]=1;
aK[0,_]=1;

aπ[2,RQCD]=0.116;
aπ[_,RQCD]=0;

aπ[2,LPC]=0.258;
aπ[4,LPC]=0.122;
aπ[_,LPC]=0;

aK[1,RQCD]=-0.0525;
aK[2,RQCD]=0.106;
aK[_,RQCD]=0;

aK[1,LPC]=-0.108;
aK[2,LPC]=0.170;
aK[3,LPC]=-0.043;
aK[4,LPC]=0.073;
aK[_,LPC]=0;

γ0[n_]:=2 (4/3)(4Sum[(1/i),{i,1,n+1}]-3-(2/((n+1)(n+1))))

γ[n_]:=With[{b0=((11 3-2 3)/3)},((γ0[n])/(2b0))]

aπ[n_,μ_,m_]:=aπ[n,m](((((As[μ])/(As[2]))))^(γ[n]))
aK[n_,μ_,m_]:=aK[n,m](((((As[μ])/(As[2]))))^(γ[n]))

fπ=0.131;
fK=0.160;

Γtot=0.12;

Rbc=With[{mc=1.5,mb=4.8,vb=(Sqrt[0.1]),vc=(Sqrt[0.3])},(((((mb  vb)/(mc vc))))^9)]

mc=1.5;
mb=4.8;

Decay Width

Clear[DwJ]
DwJ[j_,μ_:2mc,aM_:LPC,iM_:I,f_:fπ,a_:aπ,mass_:mc,rbc_:1]:=Module[{AA},
If[j==0,AA=(((π As[μ]))^3) (f^2) (1/(mass^6))Sum[(O3/4)a[m-1,μ,aM] a[n-1,μ,aM] ConvJ033[[m,n]]+(O6/4)a[m-1,μ,aM] a[n-1,μ,aM] ConvJ066[[m,n]]/.VE[v_,_]:>v,{m,1,9},{n,1,9}];
((Expand[AA (AA/.Complex[r_,i_]:>Complex[r,-i])])/(8π))/.{
(O3^2)->rbc OO[0,33,iM],(O6^2)->rbc OO[0,66,iM],O3 O6->rbc OO[0,36,iM]
},
AA=(((π As[μ]))^3) (f^2) (1/(mass^8)) Sum[(O3/4)a[m-1,μ,aM] a[n-1,μ,aM] ConvJ233[[m,n]]/.VE[v_,_]:>v,{m,1,9},{n,1,9}];
((Expand[(96(mass^4)) AA (AA/.Complex[r_,i_]:>Complex[r,-i])])/((2 2+1) 8π))/.{
(O3^2)->rbc OO[0,33,iM],(O6^2)->rbc OO[0,66,iM],O3 O6->rbc OO[0,36,iM]
}
]//Re]

T4c

@J=0

DwJ[0,2mc,LPC,I,fπ,aπ]
%/Γtot

DwJ[0,2mc,LPC,II,fπ,aπ]
%/Γtot

DwJ[0,2mc,RQCD,I,fπ,aπ]
%/Γtot

DwJ[0,2mc,RQCD,II,fπ,aπ]
%/Γtot

@J=2

DwJ[2,2mc,LPC,I,fπ,aπ]
%/Γtot

DwJ[2,2mc,LPC,II,fπ,aπ]
%/Γtot

DwJ[2,2mc,RQCD,I,fπ,aπ]
%/Γtot

DwJ[2,2mc,RQCD,II,fπ,aπ]
%/Γtot

Plots & Tables

SetOptions[MaTeX,Magnification->1,FontSize->13]
Unprotect[PdfImageResolution];
PdfImageResolution=300

Plots

Plot[{
(10^9) DwJ[0,μ,LPC,I,fπ,aπ],
(10^9) DwJ[0,μ,LPC,II,fπ,aπ],
(10^9) DwJ[0,μ,RQCD,I,fπ,aπ],
(10^9) DwJ[0,μ,RQCD,II,fπ,aπ]
},{μ,mc,4mc},Frame->True,PlotRange->{{mc,6},{-1,2}},
FrameLabel->{MaTeX["\\mu (\\rm GeV)"],MaTeX["\\Gamma(T^{(0)}_{4c}\\to\\pi^+\\pi^-)(\\times10^{-9}{\\rm GeV})"]},
Epilog->{
{Red,Dashed,Line[{{3.75,1.7},{4.4,1.7}}]},Text[MaTeX["\\mbox{LPC Model-I}"],{4.5,1.7},{-1,0}],
{Blue,Dashed,Line[{{3.75,1.5},{4.4,1.5}}]},Text[MaTeX["\\mbox{LPC Model-II}"],{4.5,1.5},{-1,0}],
{Red,Dotted,Line[{{3.75,1.3},{4.4,1.3}}]},Text[MaTeX["\\mbox{RQCD Model-I}"],{4.5,1.3},{-1,0}],
{Blue,Dotted,Line[{{3.75,1.1},{4.4,1.1}}]},Text[MaTeX["\\mbox{RQCD Model-II}"],{4.5,1.1},{-1,0}]
},
PlotStyle->{{Red,Dashed},{Blue,Dashed},{Red,Dotted},{Blue,Dotted}}]

Plot[{
(10^9) DwJ[0,μ,LPC,I,fK,aK],
(10^9) DwJ[0,μ,LPC,II,fK,aK],
(10^9) DwJ[0,μ,RQCD,I,fK,aK],
(10^9) DwJ[0,μ,RQCD,II,fK,aK]
},{μ,mc,4mc},Frame->True,PlotRange->{{mc,6},{-1,2}},
FrameLabel->{MaTeX["\\mu (\\rm GeV)"],MaTeX["\\Gamma(T^{(0)}_{4c}\\to K^+K^-)(\\times10^{-9}{\\rm GeV})"]},
Epilog->{
{Red,Dashed,Line[{{3.75,1.7},{4.4,1.7}}]},Text[MaTeX["\\mbox{LPC Model-I}"],{4.5,1.7},{-1,0}],
{Blue,Dashed,Line[{{3.75,1.5},{4.4,1.5}}]},Text[MaTeX["\\mbox{LPC Model-II}"],{4.5,1.5},{-1,0}],
{Red,Dotted,Line[{{3.75,1.3},{4.4,1.3}}]},Text[MaTeX["\\mbox{RQCD Model-I}"],{4.5,1.3},{-1,0}],
{Blue,Dotted,Line[{{3.75,1.1},{4.4,1.1}}]},Text[MaTeX["\\mbox{RQCD Model-II}"],{4.5,1.1},{-1,0}]
},
PlotStyle->{{Red,Dashed},{Blue,Dashed},{Red,Dotted},{Blue,Dotted}}]

Plot[{
(10^9) DwJ[2,μ,LPC,I,fπ,aπ],
(10^9) DwJ[2,μ,LPC,II,fπ,aπ],
(10^9) DwJ[2,μ,RQCD,I,fπ,aπ],
(10^9) DwJ[2,μ,RQCD,II,fπ,aπ]
},{μ,mc,4mc},Frame->True,PlotRange->{{mc,6},{-0.1,1}},
FrameLabel->{MaTeX["\\mu (\\rm GeV)"],MaTeX["\\Gamma(T^{(2)}_{4c}\\to\\pi^+\\pi^-)(\\times10^{-9}{\\rm GeV})"]},
Epilog->{
{Red,Dashed,Line[{{3.75,0.8},{4.4,0.8}}]},Text[MaTeX["\\mbox{LPC Model-I}"],{4.5,0.8},{-1,0}],
{Blue,Dashed,Line[{{3.75,0.7},{4.4,0.7}}]},Text[MaTeX["\\mbox{LPC Model-II}"],{4.5,0.7},{-1,0}],
{Red,Dotted,Line[{{3.75,0.6},{4.4,0.6}}]},Text[MaTeX["\\mbox{RQCD Model-I}"],{4.5,0.6},{-1,0}],
{Blue,Dotted,Line[{{3.75,0.5},{4.4,0.5}}]},Text[MaTeX["\\mbox{RQCD Model-II}"],{4.5,0.5},{-1,0}]
},
PlotStyle->{{Red,Dashed},{Blue,Dashed},{Red,Dotted},{Blue,Dotted}}]

Plot[{
(10^9) DwJ[2,μ,LPC,I,fK,aK],
(10^9) DwJ[2,μ,LPC,II,fK,aK],
(10^9) DwJ[2,μ,RQCD,I,fK,aK],
(10^9) DwJ[2,μ,RQCD,II,fK,aK]
},{μ,mc,4mc},Frame->True,PlotRange->{{mc,6},{-0.1,1}},
FrameLabel->{MaTeX["\\mu (\\rm GeV)"],MaTeX["\\Gamma(T^{(2)}_{4c}\\to K^+K^-)(\\times10^{-9}{\\rm GeV})"]},
Epilog->{
{Red,Dashed,Line[{{3.75,0.8},{4.4,0.8}}]},Text[MaTeX["\\mbox{LPC Model-I}"],{4.5,0.8},{-1,0}],
{Blue,Dashed,Line[{{3.75,0.7},{4.4,0.7}}]},Text[MaTeX["\\mbox{LPC Model-II}"],{4.5,0.7},{-1,0}],
{Red,Dotted,Line[{{3.75,0.6},{4.4,0.6}}]},Text[MaTeX["\\mbox{RQCD Model-I}"],{4.5,0.6},{-1,0}],
{Blue,Dotted,Line[{{3.75,0.5},{4.4,0.5}}]},Text[MaTeX["\\mbox{RQCD Model-II}"],{4.5,0.5},{-1,0}]
},
PlotStyle->{{Red,Dashed},{Blue,Dashed},{Red,Dotted},{Blue,Dotted}}]

Tables

N2[ex_]:=N[ex,2]

J=0

{DwJ[0,2mc,LPC,I,fπ,aπ],DwJ[0,2mc,LPC,II,fπ,aπ],DwJ[0,2mc,RQCD,I,fπ,aπ],DwJ[0,2mc,RQCD,II,fπ,aπ]};
Rationalize[%,0]//N2
Rationalize[%%/Γtot,0]//N2

{DwJ[0,2mc,LPC,I,fK,aK],DwJ[0,2mc,LPC,II,fK,aK],DwJ[0,2mc,RQCD,I,fK,aK],DwJ[0,2mc,RQCD,II,fK,aK]};
Rationalize[%,0]//N2
Rationalize[%%/Γtot,0]//N2

J=2

{DwJ[2,2mc,LPC,I,fπ,aπ],DwJ[2,2mc,LPC,II,fπ,aπ],DwJ[2,2mc,RQCD,I,fπ,aπ],DwJ[2,2mc,RQCD,II,fπ,aπ]};
Rationalize[%,0]//N2
Rationalize[%%/Γtot,0]//N2

{DwJ[2,2mc,LPC,I,fK,aK],DwJ[2,2mc,LPC,II,fK,aK],DwJ[2,2mc,RQCD,I,fK,aK],DwJ[2,2mc,RQCD,II,fK,aK]};
Rationalize[%,0]//N2
Rationalize[%%/Γtot,0]//N2

J=0/J=2 @ (Subscript[T, 4b])

{DwJ[0,2mb,LPC,I,fπ,aπ,mb,Rbc],DwJ[0,2mb,LPC,II,fπ,aπ,mb,Rbc],DwJ[0,2mb,RQCD,I,fπ,aπ,mb,Rbc],DwJ[0,2mb,RQCD,II,fπ,aπ,mb,Rbc]};
Rationalize[%,0]//N2

{DwJ[0,2mb,LPC,I,fK,aK,mb,Rbc],DwJ[0,2mb,LPC,II,fK,aK,mb,Rbc],DwJ[0,2mb,RQCD,I,fK,aK,mb,Rbc],DwJ[0,2mb,RQCD,II,fK,aK,mb,Rbc]};
Rationalize[%,0]//N2

{DwJ[2,2mb,LPC,I,fπ,aπ,mb,Rbc],DwJ[2,2mb,LPC,II,fπ,aπ,mb,Rbc],DwJ[2,2mb,RQCD,I,fπ,aπ,mb,Rbc],DwJ[2,2mb,RQCD,II,fπ,aπ,mb,Rbc]};
Rationalize[%,0]//N2

{DwJ[2,2mb,LPC,I,fK,aK,mb,Rbc],DwJ[2,2mb,LPC,II,fK,aK,mb,Rbc],DwJ[2,2mb,RQCD,I,fK,aK,mb,Rbc],DwJ[2,2mb,RQCD,II,fK,aK,mb,Rbc]};
Rationalize[%,0]//N2