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 KK- 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 (2sc1)(2sd1)
• Total number of initial spin substates gi
  (2sa1)(2sb1)
• 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-section smax 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

7

• The resonant state c can decay in several modes.
• Elastic channel c?ab (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
• MER. 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
• MER. 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
  pp- pair and
• have isospin partners r and r-
• p? p ? p r?

p? p0
For X0, by measuring angular distribution of the
pp- pair, the relative orbital angular momentum
L can be determined ? JL P P2p(-1)L
(-1)L C (-1)L
10
Some excited states of pions Resonances
with B0 are meson resonances, and with B1
baryon resonances Many baryon resonances can
be produced in pion-nucleon scattering Forma
tion 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 ee-,mm-,tt- or into neutral lepton
pairs The total width is the sum of the partial
widths for each decay mode. The observed G gives
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 Formati
on and decay of D resonance in pp
scattering Hypothetical exotic resonance
Formation and decay of an exotic resonance Z in
Kp elastic scattering
17
Quantum numbers of such a particle Z are
exotic, moreover no resonance peaks in the
corresponding cross-section