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Файл:Совр эксперимент на ускорителях (4 сем, мага) / Lecture-4-5_HEP_29_02-7_03_2024
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Scattering process
Target
Detector
incoming plane wave
February 29 - March 7, 2024
e
i k r
Modern accelerator physics, Lecture 4-5
outgoing plane and spherical waves
|
,
ɸ
)| = |
T
|
21
Scattering amplitude
For elastic scattering with
Scattering of particles with spin is more complicated
| k | = | k'| :
partial wave amplitudes
phase shift
February 29 - March 7, 2024
Modern accelerator physics, Lecture 4-5
22
Scattering amplitude
Inelasticity :
suppression of outgoing wave
Im T
1
Argand diagram :
February 29 - March 7, 2024
Modern accelerator physics, Lecture 4-5
2
2
Re T
23
Scattering amplitude
February 29 - March 7, 2024
Manley, Saleski, PRD 45, 4002
Example: pion - nucleon scattering
Modern accelerator physics, Lecture 4-5
24
iei
cot
2
Breit-Wigner formula (no spin)
Goal: to express the behaviour of the cross section near to a
resonance, i.e. when the scattering amplitudes goes through π/2
(spinless particles case)
iii
eee
f
i
sin
1
at resonance δ = π/2
power series expansion
resonance energy
assuming :
we obtain :
February 29 - March 7, 2024
R
2
E
)(cot
EE
R
25
d
EEEE
RR
dE
EEE
RR
E
R
1
)(
Ef
cot
Modern accelerator physics, Lecture 4-5
...........)(cot)()(cot)(cot
EEE
EE
R
2
0)(cot
2/
2/)(
R
iEEi
d
dE
Breit – Wigner
formula
Resonance
Using the Breit-Wigner formula, one obtains - for the case when a
given l is predominant :
1
2
2
el
)12(4)(
lE
EE
R
i
2
l
e
2
el
This is a quantum dependence on energy,
l
l
l
)12(4
i
2
that corresponds to a temporal dependence
of the state of the type :
ti
R
The Fourier transform of the decay law gives the E dependence :
2/
t
/*
t
)0()(
eItI
0 0
decay law of a particle
tiE
2/exp)0()0()(
iEteet
R
2/exp)0()()( iEiEtdtdtetE
R
2
4/
22
4/)(
February 29 - March 7, 2024
Modern accelerator physics, Lecture 4-5
26
Resonance with spin particles
K
In the case of an elastic resonance, the cross section is proportional
0
to the square modulus of this amplitude :
2
el
• This holds for elastic collisions of spinless particles. In general, if we
2/exp)(
)12(4)(
lE
iEiEtdtE
R
EE
R
2
iEE
2/)(
4/
22
R
4/)(
form a spin J resonance by making spin
J
February 29 - March 7, 2024
el
E
)(
Modern accelerator physics, Lecture 4-5
and
S
a
)12(4
)12()12(
22
EEss
Rba
particles collide:
S
b
4/
22
4/)(
27
Feynman diagrams
e
Space
+
e
Time
Expressions for matrix element :
T-channel S-channel
e
e
+
e
e
+
e
e
+
February 29 - March 7, 2024
Modern accelerator physics, Lecture 4-5
28
Feynman diagrams
Feynman rules: external lines
Write down the Feynman diagram(s) for the process
and label the momentum flow:
p’s for external lines, q’s for internal lines
Note that there are two flows:
particle / antiparticle
momentum
Now the components of the expression
External lines:
Electrons: initial state
Positrons: initial state
Photons: initial state
February 29 - March 7, 2024
final
state
final state
final state
Modern accelerator physics, Lecture 4-5
29
Feynman diagrams
Feynman rules: vertices and propagators
For
Internal lines:
each QED vertex:
momentum is “+” incoming, “-” outgoing from vertex
ge is the electromagnetic coupling
e+, e- propagator :
Photon propagator:
Indices match vertices / polarization:
Integral over internal momentum:
Finally: cancel the overall
February 29 - March 7, 2024
, what remains is: -i M
Modern accelerator physics, Lecture 4-5
30
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