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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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Modern accelerator
physics
Lecture 4-5
Fundamentals of High Energy Physics
MEPhI, February 29 - March 7, 2024
Alexander Malinin
gm GeV eV
Introduction
q
q
q
q
q
Elementary particles = basic constituents of matter
Particles are point-like (excluding the bound states)
To break matter into its smallest pieces, need high energy
Elementary particle physics = high energy physics
Present highest energy achieved (with accelerators):
LHC (2015) proton beams 6.5 TeV + 6.5 TeV = 13 TeV
q
Theoretical discussions on the unification of basic forces
has reached the Planck energy scale:
February 29 - March 7, 2024
c
G
N
1/ 2
5 19 28
10 10 10
Modern accelerator physics, Lecture 4-5
2
Quantum numbers
Fundamental particles of the Standard Model
February 29 - March 7, 2024
Modern accelerator physics, Lecture 4-5
3
Example: Charmonium family
As the mass of the charmed quark is quite large, the velocities of c and c
in a bound state are small enough that many important features of these
states can be described using non-relativistic potential models.
February 29 - March 7, 2024
Modern accelerator physics, Lecture 4-5
~
4
Aharonov- Bohm effect
February 29 - March 7, 2024
Modern accelerator physics, Lecture 4-5
5
Minkowski space
Four-vector :
For example, for a particle:
),(),,,(
aaaaaaA
03210
),( pEp
Minkowski pseudo-euclidean metric
)()(
babababababaAB
scalar product:
0033221100
Lorentz transformations for uniform translatory motion along
the x axis: β=v
'
a
0
'
a
1
'
a
2
'
a
3
/
with vx velocity, γ=1/(1-
c
x
00
00
0100
1000
a
a
a
a
0
1
2
3
2)1/2
β
a
a
aa
2
3
10
aa
10
February 29 - March 7, 2024
Modern accelerator physics, Lecture 4-5
6
6
Minkowski space
The Lorentzian scalars:
00
2
0
2
0
2
0
The Lorentz Boost :
0'0
vppp
002
LL
xxxxxxx
liketimex
likelightx
likespacex
The Special-Relativity spacetime :
xxyxyxyx
pp
TT
'
February 29 - March 7, 2024
LL
0'
vppp
Modern accelerator physics, Lecture 4-5
7
7
Scattering rate
Colliding particle beams:
N
In this case, the relative velocity between the two particles
determines the flux:
= Na (Va+ Vb)
ɸ
a
Rate = Na (Va+ Vb) Nb σ , where: σ is the cross section.
February 29 - March 7, 2024
a
V
a
N
Modern accelerator physics, Lecture 4-5
b
V
b
8
Fermi’s Golden Rule
Fermi’s Golden Rule states: that the probability of a transition in
quantum mechanics is given by the product of:
• The absolute value of the matrix element (aka amplitude) squared
• The available density of states.
Typically a decay of a particle into states with lighter product masses
has more “phase space” and more likely to occur.
February 29 - March 7, 2024
Modern accelerator physics, Lecture 4-5
9
Scattering amplitude
February 29 - March 7, 2024
Modern accelerator physics, Lecture 4-5
10
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