Resonances
When a system of two interacting particles have the same quantum numbers (invariant mass, spin, charge, …) as a particle, they can form that particle as a resonance. The maximum total coupling strength for a given particle is the unitarity limit, so if the resonant particle can couple to more than one final state, then the branching ratios and cross sections for a particular final state are proportional to the coupling strength.
Breit Wigner Resonance Scattering Cross Section
(The Breit-Wigner energy distribution is simply the Fourier transform of an exponential decay time distribution.)
p^{+}p total cross section: D^{++}(1236)
Mass: M = 1232 MeV Full Width: G=120 MeV
Spin: 3/2
Branching Ratio : BR(D^{++}Æ p^{+}p)=99.5%
p^{+}p Æp^{+}p cross section at peak of resonance:
e^{+} e^{-} -->r^{0},w^{0} -->p^{+} p^{-} (F&H Figure 10.20)
r^{0} | w^{0} | h^{0} | |
Mass: | 769 MeV | 782 MeV | 547 MeV |
Full Width: | G=151 MeV | 8.4 MeV | 0.0012 MeV |
Spin^{Parity}: | 1^{-} | 1^{-} | 0^{-} |
Branching Ratios : | |||
BR(p^{+} p^{-}) | ~100% | 2.2% | - |
BR(e^{ +} e^{-}) | 4.5x10^{-5} | 7.2x10^{-5} | 10^{-10} (theory) |
|F_{p}|^{2}at peak (if no interference) | 38 | 1.3 | - |
The non-resonant QED (Quantum Electrodynamic) cross section for the production of pions is
where p/3s is the cross section for massless point pions, b^{3} is the threshold dependence for massive pions (139.6 MeV/c^{2}), and F_{p}^{2} is the form factor for real composite pions.
The h meson is not observed because it is a pseudoscalar meson and it cannot mix with a single virtual photon to mediate a decay into an electron-positron pair.
e^{+} e^{-} -->f -->K^{+} K^{-} (F&H Figure 10.21)
Mass: M(f) = 1019.4 MeV M(K^{±})=493.7 MeV
Full Width: G=4.41 MeV
Spin: 1^{-}
Branching Ratio : BR(f--> e^{ +}e^{-})=3.1x10^{-4},
BR(f--> K^{ +}K^{-})=50%
Cross section:
s_{peak}=2.2mb (xphase
space)
e^{+} e^{-} --> J/y Æhadrons (F&H Figure 10.23)
Mass: M(J/y) = 3097 MeV
Full Width: G=0.07 MeV
Spin: 1^{-}
Branching Ratio : BR(J/y --> e^{ +}e^{-})=6.9%,
BR(J/y --> hadrons)=86%
Cross section: s_{peak}=90mb
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