1. M. Cobal, PIF 2003 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 1 10 100  (cm 2 ) J/  ,,

2. M. Cobal, PIF 2003 -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. M. Cobal, PIF 2003 -Invariant mass of the particle is measured via masses of its decay products: A typical resonance peak in K + K - invariant mass distribution

4. M. Cobal, PIF 2003 - The wave function describing a decaying state is: with E R = resonance energy and  = lifetime - The Fourier transform gives: The amplitude as a function of E is then: K= constant, E R = central value of the energy of the state But:

5. M. Cobal, PIF 2003 Spin Suppose the initial-state particles are unpolarised. Total number of final spin substates available is: g f = (2s c +1)(2s d +1) Total number of initial spin substates: g i = (2s a +1)(2s b +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 g f /g i All the so-called crossed reactions are allowed as well, and described by the same matrix-elements (but different kinematic constraints)

6. M. Cobal, PIF 2003 The value of the peak cross-section  max can 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: s a and s b : spin s of the incident and target particles J: spin of the resonant state

7. M. Cobal, PIF 2003 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=E R.  is the width of a resonance and is inverse mean lifetime of a particle at rest:  = 1/  To get cross-section for both formation and decay, multiply Breit-Wigner by a factor (  el /  ) 2 To get cross-section for both formation and decay, multiply Breit-Wigner by a factor (  i  j /  ) 2

8. M. Cobal, PIF 2003 Mean value of the Breit-Wigner shape is the mass of the resonance: M=E R.  is the width of a resonance and is inverse mean lifetime of a particle at rest:  = 1/ 

9. M. Cobal, PIF 2003  Internal quantum numbers of resonances are also derived From their decay products: X 0   + +  - And for X 0 : B = 0; S = C = = T = 0; Q = 0  Y =0 and I 3 = 0  To determine whether I = 0, I =1 or I =2, searches for isospin multiplets have to be done. Example:  0 (769) and  0 (1700) both decay to  pair and have isospin partners  + and  - :   + p  p +     +  0 For X 0, by measuring angular distribution of the  +  - pair, the relative orbital angular momentum L can be determined  J=L ; P = P 2  (-1) L = (-1) L ; C = (-1) L

10. M. Cobal, PIF 2003 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. M. Cobal, PIF 2003 Peaks in the observed total cross section of the   p reaction Corresponds to resonances formation   scattering on proton

12. M. Cobal, PIF 2003 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 = I 3 + 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 (N 0, N + ) I = 3/2   -resonances (  -,  0,  +,  ++ ) In the previous figure, the peak at ~1.2 GeV/c2 correspond to  0,  ++ resonances:  + + p   ++   + + p  - + p   0   - + p  0 + n

13. M. Cobal, PIF 2003  Fits by the Breit-Wigner formula show that both  0 and  ++ have approximately same mass of ~1232 MeV/c 2 and width ~120 MeV/c 2  Studies of angular distribution of decay products show that I(J P ) = 3/2(3/2 + )  Remaining members of the multiplet are also observed:  -,  + There is no lighter state with these quantum numbers   is a ground state, although a resonance

14. M. Cobal, PIF 2003 The Z 0 intermediate vector boson is responsible for mediating the neutral weak current interactions. M Z = 91 GeV,  = 2.5 GeV. The Z 0, can decay to hadrons via pairs, into charged leptons e+e-,  or into neutral lepton pairs: The total width is the sum of the partial widths for each decay mode. The observed  gives for the number of flavours: N = 2.99  0.01 Z0Z0 The Z 0 resonance

15. M. Cobal, PIF 2003 Quark diagrams Convenient way of showing strong interaction processes: Consider an example:  ++   + + p The only 3-quark state consistent with  ++ quantum number is (uuu), while p = (uud) and  + = (u ) Arrow pointing to the right: particle, to the left, anti-particle Time flows from left to right

16. M. Cobal, PIF 2003 Allowed resonance formation process: Formation and decay of D++ resonance in  +p scattering Hypothetical exotic resonance: Formation and decay of an exotic resonance Z++ in  +p elastic scattering

17. M. Cobal, PIF 2003 Quantum numbers of such a particle Z++ are exotic, moreover no resonance peaks in the corresponding cross-section: