In[]:=
Fitted String Tension (Pions): σ = 0.280905
Fitted String Tension (Rhos): σ = 0.235102
Fitted String Tension (Kaons): σ = 0.320037
Kaon Flavor Mass Shift:
2
Δm
= 0.187252
Out[]=
In[]:=
(*Clearandinitialize*)ClearAll[a,eps,J,s,q,spectralEq,trajectoryEq,jSol,sSol,sAsEps,jAsEps];​​​​(*1.DefinetheSpectralEquationandsolveforJasafunctionofa*)​​(*Usea=1/2-epstoapproachthepolefrombelow*)​​spectralEq=4*Pi*Cot[Pi*a]==(1-2*a)/(a*(1-a))+1/(a+24*(2*J+q));​​jSol=Solve[spectralEq,J][[1]];​​jAsEps[eps_]:=J/.jSol/.a->(1/2-eps);​​​​(*2.DefinetheTrajectoryEquationandsolveforsasafunctionofa*)​​trajectoryEq=J+q/2==(Sin[Pi*a]^4/(2*Pi^2*a*(1-a)))*s-a/48;​​sSol=Solve[trajectoryEq,s][[1]];​​sAsEps[eps_]:=s/.sSol/.J->jAsEps[eps]/.a->(1/2-eps);​​​​(*3.SeriesExpansioninepsaround0*)​​(*Order4issufficienttoisolatethe1/s^2correctionB2*)​​epsOrder=4;​​jSeries=Series[jAsEps[eps],{eps,0,epsOrder}]//Normal;​​sSeries=Series[sAsEps[eps],{eps,0,epsOrder}]//Normal;​​​​Print["--- Formal Expansions ---"];​​Print["J(eps) = ",jSeries];​​Print["s(eps) = ",sSeries];​​​​(*4.Inverts(eps)tofindeps(s)fors->Infinity*)​​(*WeuseInverseSeriestofindthemappingfromstoeps*)​​epsOfS=(InverseSeries[Series[sAsEps[eps],{eps,0,epsOrder}],sVar]/.sVar->s)//Normal;​​Print["--- Exact Asymptotic Regge Trajectory eps(s) ---"];​​epsOfS//Simplify​​(*5.FinalResult:Substituteeps(s)intoJ(eps)tofindJ(s)*)​​jOfS=jSeries/.eps->epsOfS//Series[#,{s,Infinity,3}]&//Normal;​​​​Print["--- Exact Asymptotic Regge Trajectory J(s) ---"];​​jOfS//FullSimplify
--- Formal Expansions ---
J(eps) =
1
192eps(-2+
2
π
)
+
eps(72-48
2
π
+11
4
π
)
576
2
(-2+
2
π
)
-
3
eps
2
π
(-1440+240
2
π
-12
4
π
+
6
π
)
8640
3
(-2+
2
π
)
+
1
96
(-1-48q)
s(eps) =
2
π
384eps(-2+
2
π
)
+
eps
2
π
(48-24
2
π
+5
4
π
)
1152
2
(-2+
2
π
)
+
3
eps
2
π
(1440-1440
2
π
+1110
4
π
-324
6
π
+37
8
π
)
8640
3
(-2+
2
π
)
--- Exact Asymptotic Regge Trajectory eps(s) ---
Out[]=
(
2
π
(
8
π
(2160-2160
2
π
+1065
4
π
-254
6
π
+27
8
π
)+184320
4
π
3
(-2+
2
π
)
(48-24
2
π
+5
4
π
)
2
s
+81537269760
6
(-2+
2
π
)
4
s
))(31310311587840
7
(-2+
2
π
)
5
s
)
--- Exact Asymptotic Regge Trajectory J(s) ---
Out[]=
-
1
96
-
q
2
+
6
π
(48-24
2
π
+5
4
π
)
32614907904
4
(-2+
2
π
)
3
s
+
2
π
36864(-2+
2
π
)s
+
2s
2
π
In[]:=
N
2
π
36864(-2+
2
π
)

Out[]=
0.0000340208