Lectures 18 - PHY357: Resonances

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.)

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 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|²at 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, b³ is the threshold dependence for massive pions (139.6 MeV/c²), and F_p² 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

But

beam energy spread is greater than natural width --> observed peak cross section reduced by ~ G_resonance / G_beam
radiative tail because of higher energy e⁺e^- annihilations can radiate energy (probability ~a) to fall on resonance

	r⁰	w⁰	h⁰
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\|²at peak (if no interference)	38	1.3	-

Copyright 1998 David Bailey, University of Toronto