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.)

 Decays and Partial Widths

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- -->r0,w0 -->p+ p-        (F&H Figure 10.20)

r0 w0 h0
Mass: 769 MeV782 MeV 547 MeV
Full Width: G=151 MeV 8.4 MeV 0.0012 MeV
SpinParity: 1- 1- 0-
Branching Ratios :
    BR(p+ p-) ~100% 2.2% -
    BR(e + e-) 4.5x10-5 7.2x10-5 10-10 (theory)
|Fp|2at peak (if no      interference) 381.3 -

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, b3 is the threshold dependence for massive pions (139.6 MeV/c2), and Fp2 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: speak=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: speak=90mb


Copyright 1998 David Bailey, University of Toronto