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Executing Exact Hybrid Integration & Abel Summation with Error Bounds...
Computation completed flawlessly in 14431.5 seconds.
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Flavor Mix | J | q | x = (m_l+m_H)/(2 Sqrt[σ]) | β | nMax Used | Exact Abel Residue |R| +/- Error |
u (l-u) | 1 | 0 | 0.326139 | 1.12565 | 2000 | 0.17471 +/- 0.00098 |
u (l-u) | 2 | 0 | 0.326139 | 1.19169 | 2000 | 0.05933 +/- 0.00108 |
u (l-u) | 3 | 0 | 0.326139 | 1.22633 | 2000 | 0.03080 +/- 0.00034 |
u (l-u) | 4 | 0 | 0.326139 | 1.24919 | 10000 | 0.11545 +/- 0.01509 |
u (l-u) | 5 | 0 | 0.326139 | 1.26597 | 2000 | 0.02355 +/- 0.00049 |
u (l-u) | 6 | 0 | 0.326139 | 1.27909 | 2000 | 0.01756 +/- 0.00035 |
u (l-u) | 7 | 0 | 0.326139 | 1.28977 | 2000 | 0.01919 +/- 0.00060 |
u (l-u) | 8 | 0 | 0.326139 | 1.29873 | 2000 | 0.02734 +/- 0.00128 |
u (l-u) | 1 | -1 | 0.326139 | 1.05087 | 2000 | 0.45109 +/- 0.00067 |
u (l-u) | 2 | -1 | 0.326139 | 1.16534 | 2000 | 0.11739 +/- 0.00491 |
u (l-u) | 3 | -1 | 0.326139 | 1.21111 | 2000 | 0.03795 +/- 0.00038 |
u (l-u) | 4 | -1 | 0.326139 | 1.23875 | 2000 | 0.03081 +/- 0.00064 |
u (l-u) | 5 | -1 | 0.326139 | 1.25815 | 2000 | 0.03342 +/- 0.00110 |
u (l-u) | 6 | -1 | 0.326139 | 1.2729 | 10000 | 0.03659 +/- 0.00367 |
u (l-u) | 7 | -1 | 0.326139 | 1.28468 | 2000 | 0.01978 +/- 0.00086 |
u (l-u) | 8 | -1 | 0.326139 | 1.29444 | 2000 | 0.02812 +/- 0.00115 |
s (l-s) | 1 | 0 | 0.425659 | 0.952286 | 2000 | 0.22484 +/- 0.00025 |
s (l-s) | 2 | 0 | 0.425659 | 1.03932 | 2000 | 0.05657 +/- 0.00009 |
s (l-s) | 3 | 0 | 0.425659 | 1.08599 | 2000 | 0.02534 +/- 0.00005 |
s (l-s) | 4 | 0 | 0.425659 | 1.11712 | 2000 | 0.01498 +/- 0.00003 |
s (l-s) | 5 | 0 | 0.425659 | 1.14013 | 2000 | 0.01080 +/- 0.00002 |
s (l-s) | 6 | 0 | 0.425659 | 1.15819 | 2000 | 0.01042 +/- 0.00004 |
s (l-s) | 7 | 0 | 0.425659 | 1.17295 | 2000 | 0.01095 +/- 0.00046 |
s (l-s) | 8 | 0 | 0.425659 | 1.18537 | 2000 | 0.00697 +/- 0.00011 |
s (l-s) | 1 | -1 | 0.425659 | 0.857364 | 2000 | 0.82398 +/- 0.00057 |
s (l-s) | 2 | -1 | 0.425659 | 1.00427 | 2000 | 0.10093 +/- 0.00014 |
s (l-s) | 3 | -1 | 0.425659 | 1.06541 | 2000 | 0.03620 +/- 0.00006 |
s (l-s) | 4 | -1 | 0.425659 | 1.10288 | 2000 | 0.01896 +/- 0.00004 |
s (l-s) | 5 | -1 | 0.425659 | 1.12939 | 2000 | 0.01242 +/- 0.00003 |
s (l-s) | 6 | -1 | 0.425659 | 1.14965 | 2000 | 0.00996 +/- 0.00002 |
s (l-s) | 7 | -1 | 0.425659 | 1.16592 | 2000 | 0.01381 +/- 0.00060 |
s (l-s) | 8 | -1 | 0.425659 | 1.17941 | 2000 | 0.00820 +/- 0.00018 |
c (l-c) | 1 | 0 | 2.08153 | 0.44573 | 2000 | 29.22913 +/- 0.49300 |
c (l-c) | 2 | 0 | 2.08153 | 0.536772 | 10000 | 10.70256 +/- 0.83746 |
c (l-c) | 3 | 0 | 2.08153 | 0.593889 | 10000 | 15.29128 +/- 2.79154 |
c (l-c) | 4 | 0 | 2.08153 | 0.63564 | 10000 | 8.66310 +/- 1.19829 |
c (l-c) | 5 | 0 | 2.08153 | 0.6685 | 10000 | 20.40966 +/- 29.86264 |
c (l-c) | 6 | 0 | 2.08153 | 0.695538 | 10000 | 5.66384 +/- 6.97218 |
c (l-c) | 7 | 0 | 2.08153 | 0.718464 | 10000 | 2.41482 +/- 1.16802 |
c (l-c) | 8 | 0 | 2.08153 | 0.738329 | 10000 | 1.55612 +/- 1.03071 |
c (l-c) | 1 | -1 | 2.08153 | 0.365159 | 10000 | 137.37655 +/- 6.87261 |
c (l-c) | 2 | -1 | 2.08153 | 0.497849 | 10000 | 18.43320 +/- 1.05364 |
c (l-c) | 3 | -1 | 2.08153 | 0.567917 | 10000 | 10.23571 +/- 0.72426 |
c (l-c) | 4 | -1 | 2.08153 | 0.616158 | 10000 | 6.47116 +/- 0.68617 |
c (l-c) | 5 | -1 | 2.08153 | 0.652944 | 10000 | 3.62384 +/- 0.97230 |
c (l-c) | 6 | -1 | 2.08153 | 0.682619 | 10000 | 3.27840 +/- 2.54209 |
c (l-c) | 7 | -1 | 2.08153 | 0.707439 | 10000 | 2.75578 +/- 1.40931 |
c (l-c) | 8 | -1 | 2.08153 | 0.72873 | 10000 | 2.10658 +/- 1.92340 |
b (l-b) | 1 | 0 | 6.24221 | 0.228783 | 2000 | 683.01621 +/- 21.97888 |
b (l-b) | 2 | 0 | 6.24221 | 0.284484 | 10000 | 336.65477 +/- 26.58266 |
b (l-b) | 3 | 0 | 6.24221 | 0.3222 | 10000 | 165.82324 +/- 12.55448 |
b (l-b) | 4 | 0 | 6.24221 | 0.351361 | 10000 | 67.73609 +/- 7.94506 |
b (l-b) | 5 | 0 | 6.24221 | 0.375365 | 10000 | 84.66291 +/- 17.62700 |
b (l-b) | 6 | 0 | 6.24221 | 0.39587 | 10000 | 91.21325 +/- 24.63045 |
b (l-b) | 7 | 0 | 6.24221 | 0.413824 | 10000 | 34.39822 +/- 7.89685 |
b (l-b) | 8 | 0 | 6.24221 | 0.429824 | 10000 | 13.14867 +/- 2.00038 |
b (l-b) | 1 | -1 | 6.24221 | 0.183142 | 2000 | 21668.79705 +/- 463.18084 |
b (l-b) | 2 | -1 | 6.24221 | 0.260062 | 10000 | 1568.42682 +/- 74.63724 |
b (l-b) | 3 | -1 | 6.24221 | 0.304755 | 10000 | 203.42554 +/- 28.63065 |
b (l-b) | 4 | -1 | 6.24221 | 0.337576 | 10000 | 71.62477 +/- 7.69153 |
b (l-b) | 5 | -1 | 6.24221 | 0.363881 | 10000 | 141.80662 +/- 33.02195 |
b (l-b) | 6 | -1 | 6.24221 | 0.385985 | 10000 | 19.68734 +/- 2.01014 |
b (l-b) | 7 | -1 | 6.24221 | 0.405123 | 10000 | 24.04580 +/- 4.67663 |
b (l-b) | 8 | -1 | 6.24221 | 0.422039 | 10000 | 238.18551 +/- 91.07630 |
In[]:=
(*=========================================================================*)(*8.FASTPOST-PROCESSING:CalculateExactMasses&NormalizedCrossSections*)(*=========================================================================*)Mth[β_,x_]:=sqrtsigma*(Kcal[β,x]*(β+Sin[2*β]/2)+2*x*Cos[β]);(*Step8A:Calculaterawcrosssectionsanderrorsforallstates*)rawCrossSections=Map[Module[{flavor,J,q,x,β,finalNMax,bareIntegral,bareError,massPred,Kphys,crossSection,crossSectionErr},{flavor,J,q,x,β,finalNMax,bareIntegral,bareError}=#;massPred=Mth[β,x];Kphys=Kcal[β,x]/sqrtsigma;crossSection=(If[q==0,8*Pi*Sec[β]^2,4*Pi*Tan[β]^2]*bareIntegral)/Kphys;crossSectionErr=(If[q==0,8*Pi*Sec[β]^2,4*Pi*Tan[β]^2]*bareError)/Kphys;{flavor,J,q,x,β,massPred,crossSection,crossSectionErr}]&,results];(*Step8B:Extracttheρ(770)crosssection(flavor"u (l-u)",J=1,q=-1)*)rhoRef=SelectFirst[rawCrossSections,#[[1]]=="u (l-u)"&&#[[2]]==1&&#[[3]]==-1&];If[MissingQ[rhoRef],Print["[!] Warning: ρ(770) state not found in results. Normalizing by 15.9046."];sigmaRho=15.904594841351724`;,sigmaRho=rhoRef[[7]];];(*Step8C:Normalizeallcrosssectionsandformattheoutputstrings*)crossSectionResults=Map[Module[{flavor,J,q,x,β,massPred,crossSection,crossSectionErr,csStr},{flavor,J,q,x,β,massPred,crossSection,crossSectionErr}=#;csStr=ToString[NumberForm[crossSection/sigmaRho,{10,5}]]<>" +/- "<>ToString[NumberForm[crossSectionErr/sigmaRho,{10,5}]];{flavor,J,q,x,β,massPred,csStr}]&,rawCrossSections];Print["\n==========================================================================="];Print[" EXACT MASS SPECTRUM & NORMALIZED PARTIAL CROSS-SECTIONS (Phi_* = 0)"];Print["==========================================================================="];Print[" σ_J values are normalized to σ_ρ(770) = ",sigmaRho," GeV^-2"];headerCS={"Flavor","J","q","x","β","M_th (GeV)","σ_J / σ_ρ +/- Error"};formattedTableCS=Prepend[crossSectionResults,headerCS];Grid[formattedTableCS,Frame->All,Alignment->Center,Spacings->{2,1}]
===========================================================================
EXACT MASS SPECTRUM & NORMALIZED PARTIAL CROSS-SECTIONS (Phi_* = 0)
===========================================================================
σ_J values are normalized to σ_ρ(770) = 5.99184 GeV^-2
Out[]=
Flavor | J | q | x | β | M_th (GeV) | σ_J / σ_ρ +/- Error |
u (l-u) | 1 | 0 | 0.326139 | 1.12565 | 1.16949 | 0.98892 +/- 0.00557 |
u (l-u) | 2 | 0 | 0.326139 | 1.19169 | 1.58753 | 0.32626 +/- 0.00594 |
u (l-u) | 3 | 0 | 0.326139 | 1.22633 | 1.91151 | 0.16716 +/- 0.00187 |
u (l-u) | 4 | 0 | 0.326139 | 1.24919 | 2.18598 | 0.62167 +/- 0.08123 |
u (l-u) | 5 | 0 | 0.326139 | 1.26597 | 2.42854 | 0.12611 +/- 0.00264 |
u (l-u) | 6 | 0 | 0.326139 | 1.27909 | 2.64829 | 0.09365 +/- 0.00189 |
u (l-u) | 7 | 0 | 0.326139 | 1.28977 | 2.85068 | 0.10204 +/- 0.00321 |
u (l-u) | 8 | 0 | 0.326139 | 1.29873 | 3.0393 | 0.14501 +/- 0.00678 |
u (l-u) | 1 | -1 | 0.326139 | 1.05087 | 0.879096 | 1.00000 +/- 0.00149 |
u (l-u) | 2 | -1 | 0.326139 | 1.16534 | 1.39556 | 0.27556 +/- 0.01153 |
u (l-u) | 3 | -1 | 0.326139 | 1.21111 | 1.75742 | 0.09075 +/- 0.00091 |
u (l-u) | 4 | -1 | 0.326139 | 1.23875 | 2.05354 | 0.07439 +/- 0.00156 |
u (l-u) | 5 | -1 | 0.326139 | 1.25815 | 2.31056 | 0.08122 +/- 0.00268 |
u (l-u) | 6 | -1 | 0.326139 | 1.2729 | 2.54087 | 0.08934 +/- 0.00895 |
u (l-u) | 7 | -1 | 0.326139 | 1.28468 | 2.7514 | 0.04848 +/- 0.00211 |
u (l-u) | 8 | -1 | 0.326139 | 1.29444 | 2.94654 | 0.06911 +/- 0.00282 |
s (l-s) | 1 | 0 | 0.425659 | 0.952286 | 1.21319 | 0.68989 +/- 0.00077 |
s (l-s) | 2 | 0 | 0.425659 | 1.03932 | 1.62547 | 0.16406 +/- 0.00026 |
s (l-s) | 3 | 0 | 0.425659 | 1.08599 | 1.94629 | 0.07159 +/- 0.00013 |
s (l-s) | 4 | 0 | 0.425659 | 1.11712 | 2.21863 | 0.04167 +/- 0.00008 |
s (l-s) | 5 | 0 | 0.425659 | 1.14013 | 2.4596 | 0.02971 +/- 0.00006 |
s (l-s) | 6 | 0 | 0.425659 | 1.15819 | 2.67809 | 0.02843 +/- 0.00012 |
s (l-s) | 7 | 0 | 0.425659 | 1.17295 | 2.87946 | 0.02970 +/- 0.00125 |
s (l-s) | 8 | 0 | 0.425659 | 1.18537 | 3.06721 | 0.01881 +/- 0.00029 |
s (l-s) | 1 | -1 | 0.425659 | 0.857364 | 0.928839 | 0.77876 +/- 0.00054 |
s (l-s) | 2 | -1 | 0.425659 | 1.00427 | 1.43584 | 0.10645 +/- 0.00015 |
s (l-s) | 3 | -1 | 0.425659 | 1.06541 | 1.79361 | 0.03959 +/- 0.00007 |
s (l-s) | 4 | -1 | 0.425659 | 1.10288 | 2.08717 | 0.02115 +/- 0.00004 |
s (l-s) | 5 | -1 | 0.425659 | 1.12939 | 2.34237 | 0.01404 +/- 0.00003 |
s (l-s) | 6 | -1 | 0.425659 | 1.14965 | 2.57127 | 0.01136 +/- 0.00003 |
s (l-s) | 7 | -1 | 0.425659 | 1.16592 | 2.78067 | 0.01586 +/- 0.00069 |
s (l-s) | 8 | -1 | 0.425659 | 1.17941 | 2.97486 | 0.00948 +/- 0.00021 |
c (l-c) | 1 | 0 | 2.08153 | 0.44573 | 2.329 | 28.66200 +/- 0.48343 |
c (l-c) | 2 | 0 | 2.08153 | 0.536772 | 2.6581 | 8.84795 +/- 0.69234 |
c (l-c) | 3 | 0 | 2.08153 | 0.593889 | 2.92541 | 11.55210 +/- 2.10892 |
c (l-c) | 4 | 0 | 2.08153 | 0.63564 | 3.15816 | 6.16875 +/- 0.85327 |
c (l-c) | 5 | 0 | 2.08153 | 0.6685 | 3.36773 | 13.92078 +/- 20.36835 |
c (l-c) | 6 | 0 | 2.08153 | 0.695538 | 3.56027 | 3.73658 +/- 4.59974 |
c (l-c) | 7 | 0 | 2.08153 | 0.718464 | 3.73954 | 1.55093 +/- 0.75017 |
c (l-c) | 8 | 0 | 2.08153 | 0.738329 | 3.90808 | 0.97742 +/- 0.64740 |
c (l-c) | 1 | -1 | 2.08153 | 0.365159 | 2.11494 | 10.36944 +/- 0.51876 |
c (l-c) | 2 | -1 | 2.08153 | 0.497849 | 2.50441 | 1.86064 +/- 0.10635 |
c (l-c) | 3 | -1 | 2.08153 | 0.567917 | 2.79716 | 1.16374 +/- 0.08234 |
c (l-c) | 4 | -1 | 2.08153 | 0.616158 | 3.04521 | 0.79049 +/- 0.08382 |
c (l-c) | 5 | -1 | 2.08153 | 0.652944 | 3.26538 | 0.46536 +/- 0.12486 |
c (l-c) | 6 | -1 | 2.08153 | 0.682619 | 3.46586 | 0.43715 +/- 0.33897 |
c (l-c) | 7 | -1 | 2.08153 | 0.707439 | 3.65138 | 0.37856 +/- 0.19360 |
c (l-c) | 8 | -1 | 2.08153 | 0.72873 | 3.82502 | 0.29652 +/- 0.27074 |
b (l-b) | 1 | 0 | 6.24221 | 0.228783 | 5.62966 | 422.22810 +/- 13.58694 |
b (l-b) | 2 | 0 | 6.24221 | 0.284484 | 5.87462 | 168.16847 +/- 13.27878 |
b (l-b) | 3 | 0 | 6.24221 | 0.3222 | 6.07799 | 73.41825 +/- 5.55849 |
b (l-b) | 4 | 0 | 6.24221 | 0.351361 | 6.25798 | 27.59199 +/- 3.23638 |
b (l-b) | 5 | 0 | 6.24221 | 0.375365 | 6.42218 | 32.37648 +/- 6.74085 |
b (l-b) | 6 | 0 | 6.24221 | 0.39587 | 6.57468 | 33.16284 +/- 8.95501 |
b (l-b) | 7 | 0 | 6.24221 | 0.413824 | 6.71802 | 11.99309 +/- 2.75327 |
b (l-b) | 8 | 0 | 6.24221 | 0.429824 | 6.85388 | 4.42376 +/- 0.67301 |
b (l-b) | 1 | -1 | 6.24221 | 0.183142 | 5.4739 | 276.63442 +/- 5.91319 |
b (l-b) | 2 | -1 | 6.24221 | 0.260062 | 5.75944 | 28.27149 +/- 1.34536 |
b (l-b) | 3 | -1 | 6.24221 | 0.304755 | 5.97995 | 4.27885 +/- 0.60222 |
b (l-b) | 4 | -1 | 6.24221 | 0.337576 | 6.17031 | 1.66291 +/- 0.17857 |
b (l-b) | 5 | -1 | 6.24221 | 0.363881 | 6.34175 | 3.53788 +/- 0.82385 |
b (l-b) | 6 | -1 | 6.24221 | 0.385985 | 6.49971 | 0.51956 +/- 0.05305 |
b (l-b) | 7 | -1 | 6.24221 | 0.405123 | 6.64738 | 0.66435 +/- 0.12921 |
b (l-b) | 8 | -1 | 6.24221 | 0.422039 | 6.7868 | 6.83931 +/- 2.61518 |