Transcript LHCC - uniud.it

- 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
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-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
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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
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- 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 )e dt   (0)  e
iEt
  

t    i  ER  E 
 2 

dt 
K
E  E R   i  2
K= constant, ER = central value of the energy of the state
But:  ( E )  ( E )
*
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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
abcd
• All the so-called crossed reactions are
allowed as well, and described by the
same matrix-elements (but different
kinematic constraints)
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ac b d
ad cb
ab cd
cd ab
• The value of the peak cross-section smax can be found using
arguments from wave optics:
s max  42 2J  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
2 sa  12 sb  1 E  ER 2  2
2

4

sa and sb: spin s of the incident and target particles
J: spin of the resonant state
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• The resonant state c can decay in several modes.
• “Elastic” channel: ca+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
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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
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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
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 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-,mm,tt or into neutral lepton pairs: n en e ,n mn m ,ntnt
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
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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
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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