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- 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 s (cm2) Resonances r,w J/Y - 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 s 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 M. Cobal, PIF 2003 A+B -Invariant mass of the particle is measured via masses of its decay products: W 2 ( E A E B ) 2 ( p A pB ) 2 2 2 E p M2 A typical resonance peak in K+K- invariant mass distribution M. Cobal, PIF 2003 - The wave function describing a decaying state is: (t ) (0)e t iw R t 2t e (0)e t iER 2 with ER = resonance energy and t = lifetime - The Fourier transform gives: g w (t )eiwt dt 0 The amplitude as a function of E is then: ( E ) (t )eiEtdt (0) e t i E R E 2 dt K E E R i 2 K= constant, ER = central value of the energy of the state But: ( E ) ( E ) * M. Cobal, PIF 2003 s s max 2 4 E E 2 2 R 4 • 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 abcd • All the so-called crossed reactions are allowed as well, and described by the same matrix-elements (but different kinematic constraints) M. Cobal, PIF 2003 ac b d ad cb ab cd cd ab • The value of the peak cross-section smax can be found using arguments from wave optics: s max 4 2 2 J 1 With = wavelenght of scattered/scattering particle in cms • Including spin multiplicity factors, one gets the Breit-Wigner formula: 2 4 2 J 1 4 s 2sa 12sb 1 E ER 2 2 2 4 sa and sb: spin s of the incident and target particles J: spin of the resonant state M. Cobal, PIF 2003 • The resonant state c can decay in several modes. • “Elastic” channel: ca+b (by which the resonance was formed) To get cross-section for both formation and decay, multiply Breit-Wigner by a factor (el/)2 • If state is formed through channel i and decays through channel j To get cross-section for both formation and decay, multiply Breit-Wigner by a factor (i j /)2 • Mean value of the Breit-Wigner shape is the mass of the resonance: M=ER. is the width of a resonance and is inverse mean lifetime of a particle at rest: = 1/t M. Cobal, PIF 2003 N (W ) K (W W0 ) 2 2 / 4 • Mean value of the Breit-Wigner shape is the mass of the resonance: M=ER. is the width of a resonance and is inverse mean lifetime of a particle at rest: = 1/t M. Cobal, PIF 2003 Internal quantum numbers of resonances are also derived From their decay products: X0 + + - And for X0: ~ B = 0; S = C = B = 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 pair and have isospin partners r+ and r-: + p p + r + 0 For X0, by measuring angular distribution of the +- pair, the relative orbital angular momentum L can be determined J=L ; P = P2(-1)L = (-1)L ; C = (-1)L 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 M. Cobal, PIF 2003 Peaks in the observed total cross section of the p reaction Corresponds to resonances formation scattering on proton 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 = B = 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 D++ + + p - + p D0 - + p 0 + n M. Cobal, PIF 2003 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 M. Cobal, PIF 2003 The Z0 resonance The Z0 intermediate vector boson is responsible for mediating the neutral weak current interactions. Z0 MZ = 91 GeV, = 2.5 GeV. The Z0, can decay to hadrons via qq pairs, into charged leptons e+e-,mm,tt or into neutral lepton pairs: n en e ,n mn m ,n tn t The total width is the sum of the partial widths for each decay mode. The observed gives for the number of flavours: Nn = 2.99 0.01 M. Cobal, PIF 2003 Quark diagrams • Convenient way of showing strong interaction processes: Consider an example: D++ + + p The only 3-quark state consistent with D++ quantum number is (uuu), while p = (uud) and + = (u ) d Arrow pointing to the right: particle, to the left, anti-particle Time flows from left to right 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 K+p elastic scattering M. Cobal, PIF 2003 Quantum numbers of such a particle Z++ are exotic, moreover no resonance peaks in the corresponding cross-section: M. Cobal, PIF 2003