Particle Physics: Basic Concepts and the Standard Model, Study notes of Particle Physics

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Oxana Smirnova

Particle Physics Department

2001 Spring Semester

Lund University

Particle Physics experimental insight

• e

-^ e →^

W

+W

-^ →

μν

qq

I. Basic concepts

➠ Particle physics studies the elementary “building blocks” of matter and interactions between them.

➠ Matter consists of particles and fields.

➠ Particles interact via forces caused by fields.

➠ Forces are being carried by specific particles, called gauge [‘gejdz] bosons.

Forces of nature:

1. gravitational

2. weak

3. electromagnetic

4. strong

The Standard Model

➠ Electromagnetic and weak forces can be described by a single theory ⇒ the “Electroweak Theory” was developed in 1960s (Glashow, Weinberg, Salam).

➠ Theory of strong interactions appeared in 1970s: “Quantum Chromodynamics” (QCD).

➠ The “Standard Model” (SM) combines both.

Main postulates of SM:

1. Basic constituents of matter are quarks and leptons (spin 1/2).

2. They interact by means of gauge bosons (spin 1).

3. Quarks and leptons are subdivided into 3 generations.

SM does not explain neither appearance of the mass nor the reason for existence of 3 generations.

Figure 1: The Standard Model Chart

Units and dimensions

➠ The energy is measured in electron-volts :

1 eV ≈ 1.602 ✕ 10 -19^ J

1 keV = 10^3 eV; 1 MeV = 10^6 eV; 1 GeV = 10^9 eV

The Planck constant (reduced) is then:

≡ h / 2 π = 6.582 ✕ 10 -22^ Mev s

and the “conversion constant” is:

c = 197.327 ✕ 10 -15^ MeV m

➠ For simplicity, the natural units are used:

= 1 and c = 1

so that the unit of mass is eV/c^2 , and the unit of momentum is eV/c

h

_

h

_

h

_

Antiparticles

➠ Particles are described by a wavefunction:

is the coordinate vector, - momentum vector, E

and t are energy and time.

For relativistic particles, E 2 =p 2 +m 2 , and the Shrödinger equation (2) is replaced by the Klein-Gordon equation (3):

Ψ ( x t , ) = Ne i^ (^ px^ – Et )

x p

i ∂ t

∂ Ψ ( x t , ) 1 2m

= – ------- ∇^2 Ψ ( x t , )

t 2

2

∂

• ∂ Ψ = –∇^2 Ψ^ ( x t , ) + m 2 Ψ ( x t , )

Dirac-Pauli representation of matrices α i and β:

Here I is 2✕2 unit matrix and σ i are Pauli matrices:

Also possible is Weyl representation:

α i

0 σ i σ i 0 











 = β I^0  (^) 0 – I 

 

 

σ 1 0 1  (^) 1 0 

 

  = σ 2 0 – i  (^) i 0 

 

  = σ 3 1 0  (^) 0 – 1 

 

 

α i

• σ i 0 0 σ i 











 = β 0 I  (^) I 0 

 

 

Dirac’s picture of vacuum

The “hole” created by the appearance of the electron with a positive energy is interpreted as the presence of electron’s antiparticle with the opposite charge.

➠ Every charged particle has the antiparticle of the same mass and opposite charge.

Figure 3: Fermions in Dirac’s representation.

Feynman diagrams

In 1940s, R.Feynman developed a diagram technique for representing processes in particle physics.

Main assumptions and requirements:

❖ Time runs from left to right

❖ Arrow directed towards the right indicates a particle, and otherwise - antiparticle

❖ At every vertex, momentum, angular momentum and charge are conserved (but not energy)

❖ Particles are usually denoted with solid lines, and gauge bosons - with helices or dashed lines

Figure 5: A Feynman diagram example

Virtual processes: a) e-^ → e-^ + γ b) γ + e-^ → e-

c) e+^ → e +^ + γ d) γ + e+^ → e+

e) e +^ + e -^ → γ f) γ → e+^ + e -

g) vacuum → e+^ + e -^ + γ h) e +^ + e -^ + γ → vacuum

Figure 6: Feynman diagrams for basic processes involving electron, positron and photon

❖ Number of vertices in a diagram is called its order.

❖ Each vertex has an associated probability proportional to a coupling constant , usually denoted as “α”. In discussed processes this constant is

❖ For the real processes, a diagram of order n

gives a contribution to probability of order α n.

Provided sufficiently small α, high order contributions to many real processes can be neglected, allowing rather precise calculations of probability amplitudes of physical processes.

α em e^

2 4 πε 0

------------^1 137

= ≈^ ---------

Diagrams which differ only by time-ordering are usually implied by drawing only one of them

This kind of process implies 3!=6 different time orderings

(a) (b)

Figure 9: Lowest order contributions to e +^ e -^ → γγ

Figure 10: Lowest order of the process e +^ e -^ → γγγ

Exchange of a massive boson

In the rest frame of particle A:

where , ,

,

From this one can estimate the maximum distance over which X can propagate before being absorbed:

, and this energy violation

can exist only for a period of time ∆ t ≈ /∆ E , hence

the range of the interaction is

r ≈ R ≡ / MX c

Figure 12: Exchange of a massive particle X

A E ( 0 , p 0 ) → A E ( (^) A , p ) + X E ( (^) x , – p )

E 0 = M (^) A p 0 = ( 0 0 0 , , )

E (^) A = p 2 + M (^) A^2 E (^) X = p 2 + M (^) X^2

∆ E = EX + E (^) A – M (^) A ≥ M (^) X

h

_

h

_

➠ For a massless exchanged particle, the interaction has an infinite range (e.g., electromagnetic)

➠ In case of a very heavy exchanged particle (e.g., a W boson in weak interaction), the interaction can be approximated by a zero-range , or point interaction :

RW = /MW = /(80.4 GeV/c^2 ) ≈ 2 ✕ 10 -18^ m

Considering particle X as an electrostatic potential V(r), the Klein-Gordon equation (3) for it will look like

Figure 13: Point interaction as a result of

M x → ∞

h

_

h

_

∇^2 V r^ ( )^1 r 2

----- ∂ ∂ r

------- (^) r 2 ∂ V ∂ r

 -------^ =^ MX

2 = V r ( )