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

s

1 10 100

(cm

2 )

J/,

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, thenresonant states will form a corresponing multiplet too

-Since resonances have very short lifetimes, they can onlybe detected through their decay products:

p- + p n + X A + B

M. Cobal, PIF 2003

-Invariant mass of the particle is measured via masses of itsdecay products:

A typical resonance peakin K+K- invariant mass distribution

222

222 )()(

MpE

ppEEW BABA

M. Cobal, PIF 2003

- The wave function describing a decaying state is:

with ER = resonance energy and = 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:

22 )0()0()( RR

iEttti eeet

dtetg ti

0

)(

2)0()()( 2

iEE

KdtedtetE

R

EEitiEt R

)()(* EE

4

422

2

max

REE

M. Cobal, PIF 2003

• 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) badc

dcba

bcda

dbca

dcba

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:

sa and sb: spin s of the incident and target particles J: spin of the resonant state

4

41212

12422

22

Rba EEss

J

124 2max J

M. Cobal, PIF 2003

• The resonant state c can decay in several modes.

• “Elastic” channel: ca+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. 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, multiplyBreit-Wigner by a factor (el/)2

To get cross-section for both formation and decay, multiply Breit-Wigner by a factor (i j /)2

M. Cobal, PIF 2003

4/22)0()(

WW

KWN

• 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/

M. Cobal, PIF 2003

Internal quantum numbers of resonances are also derived From their decay products:

X0 + + -

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: 0(769) and 0(1700) both decay to pair and have isospin partners + and -:

+ p p +

B~

+ 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-nucleonscattering:

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 reactionCorresponds 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 = = 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 -resonances (-, 0, +, ++)

In the previous figure, the peak at ~1.2 GeV/c2 correspond to 0, ++ resonances:

+ + p ++ + + p- + p 0 - + p

B~

0 + n

M. Cobal, PIF 2003

Fits by the Breit-Wigner formula show that both 0 and ++ 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: -, +

There is no lighter state with these quantum numbers is a ground state, although a resonance

M. Cobal, PIF 2003

The Z0 intermediate vector boson is responsible for mediating the neutral weak current interactions.

MZ = 91 GeV, = 2.5 GeV.The Z0, can decay to hadrons via pairs, into charged leptonse+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

,,ee

Z0

qq

The Z0 resonance

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 )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 +p elastic scattering

M. Cobal, PIF 2003

Quantum numbers of such a particle Z++ are exotic, moreoverno resonance peaks in the corresponding cross-section: