Resonances

1. Resonances • - If cross section for muon pairs is • plotted one find the 1/s dependence • In the hadronic final state this trend • is broken by various strong peaks • Resonances: short lived states with • fixed mass, and well defined quantum • numbers  particles • -The exponential time dependence gives • the form of the resonance lineshape r,w J/Y s (cm2) 1 10 100

2. Resonances decay by strong interactions (lifetimes about • 10-23 s) • If a ground state is a member of an isospin multiplet, then • resonant states will form a corresponing multiplet too • Since resonances have very short lifetimes, they can only • be detected through their decay products: • p- + p n + X • A + B

3. Invariant mass of the particle is measured via masses of its • decay products: • A typical resonance peak • in K+K- invariant mass • distribution

4. - The wave function describing a decaying state is: • with ER = resonance energy and t = lifetime • - The Fourier transform gives: • The amplitude as a function of E is then: • K= constant, ER = central value of the energy of the state • But:

5. Spin • Suppose the initial-state particles are unpolarised. • Total number of final spin substates available is: • gf = (2sc+1)(2sd+1) • Total number of initial spin substates: gi = (2sa+1)(2sb+1) • One has to average the transition probability over all possible • initial states, all equally probable, and sum over all final states •  Multiply by factor gf/gi • All the so-called crossed reactions are • allowed as well, and described by the • same matrix-elements (but different • kinematic constraints)

6. The value of the peak cross-sectionsmaxcan be found using • arguments from wave optics: • With = wavelenght of scattered/scattering particle in cms • Including spin multiplicity factors, one gets the Breit-Wigner • formula: • sa and sb: spin s of the incident and target particles • J: spin of the resonant state

7. The resonant state c can decay in several modes. • “Elastic” channel: ca+b (by which the resonance was formed) • If state is formed through channel i and decays through channel j • Mean value of the Breit-Wigner shape is the mass of the resonance: • M=ER. G is the width of a resonance and is inverse mean lifetime of a particle at rest: G = 1/t To get cross-section for both formation and decay, multiply Breit-Wigner by a factor (Gel/G)2 To get cross-section for both formation and decay, multiply Breit-Wigner by a factor (Gi Gj /G)2

8. Mean value of the Breit-Wigner shape is the mass of the resonance: • M=ER. G is the width of a resonance and is inverse mean lifetime of a • particle at rest: G = 1/t

9. Internal quantum numbers of resonances are also derived From their decay products: X0 p+ + p- And for X0: B = 0; S = C = = T = 0; Q = 0  Y =0 and I3 = 0 • To determine whether I = 0, I =1 or I =2, searches for isospin multiplets have to be done. Example: r0(769) and r0(1700) both decay to p+p- pair and have isospin partners r+ and r-: p + p  p + r p + p0 For X0, by measuring angular distribution of the p+p- pair, the relative orbital angular momentum L can be determined  J=L ; P = P2p(-1)L = (-1)L ; C = (-1)L

10. Some excited states of pions: Resonances with B=0 are meson resonances, and with B=1 – baryon resonances Many baryon resonances can be produced in pion-nucleon scattering: Formation of a resonance R and its inclusive decay into a nucleon N

11. Peaks in the observed total cross section of the pp reaction Corresponds to resonances formation p scattering on proton

12. All resonances produced in pion-nucleon scattering have the same internal quantum numbers as the initial state: B = 1 ; S =C = = T = 0, and thus Y =1 and Q = I3 + 1/2 Possible isospins are I = ½ or I = 3/2, since for pion I = 1 and for nucleon I = ½ I = ½  N – resonances (N0, N+) I = 3/2  D-resonances (D-, D0, D+, D++) In the previous figure, the peak at ~1.2 GeV/c2 correspond to D0, D++ resonances: p+ + p  D++  p+ + p p- + p  D0  p- + p p0 + n

13. Fits by the Breit-Wigner formula show that both D0 and D++ • have approximately same mass of ~1232 MeV/c2 and width • ~120 MeV/c2 • Studies of angular distribution of decay products show that I(JP) = 3/2(3/2+) • Remaining members of the multiplet are also observed: • D-, D+ • There is no lighter state with these quantum numbers  D is • a ground state, although a resonance

14. The Z0 resonance The Z0 intermediate vector boson is responsible for mediating the neutral weak current interactions. MZ = 91 GeV,G = 2.5 GeV. The Z0, can decay to hadrons via pairs, into charged leptons e+e-,m+m-,t+t-or into neutral lepton pairs: The total width is the sum of the partial widths for each decay mode. The observedGgives for the number of flavours: Z0 Nn = 2.99  0.01

15. Quark diagrams • Convenient way of showing strong interaction processes: • Consider an example: • D++  p+ + p • The only 3-quark state consistent with D++ quantum number • is (uuu), while p = (uud) and p+ = (u ) • Arrow pointing to the right: particle, to the left, anti-particle • Time flows from left to right

16. Allowed resonance formation process: Formation and decay of D++ resonance in p+p scattering Hypothetical exotic resonance: Formation and decay of an exotic resonance Z++ in K+p elastic scattering

17. Quantum numbers of such a particle Z++ are exotic, moreover no resonance peaks in the corresponding cross-section: