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Particle Physics: Introduction to the Standard ModelQuantum Electrodynamics (I)

Frdric Machefertfrederic@cern.ch

Laboratoire de lacclrateur linaire (CNRS)

Cours de lcole Normale Suprieure24, rue Lhomond, Paris

January 19th, 2017

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Quantum Field TheoryThe Lagrangian

The Feynman RulesExample of processes

Acceleration and Detection

Part II

Quantum Electrodynamics (I)

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Quantum Field TheoryThe Lagrangian

The Feynman RulesExample of processes

Acceleration and Detection

1 Quantum Field TheoryWhy do we need quantum field theory ?Special relativity and quantum field theoryDiagram ordersA brief recipe...

2 The Lagrangian

3 The Feynman Rules

4 Example of processesMoeller ScatteringBhabha

5 Acceleration and DetectionAccelerationDetection

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Quantum Field TheoryThe Lagrangian

The Feynman RulesExample of processes

Acceleration and Detection

Why do we need quantum field theory ?Special relativity and quantum field theoryDiagram ordersA brief recipe...

The History

The history

Introduction of particles (oo)

Particle-Wave dualism (deBroglie wave length)

Particles are fields in a quantum field theory

1941: Stueckelberg proposes to interpret electron lines going back intime as positrons

end of 1940s: Feynman, Tomonaga, Schwinger et al developrenormalization theory

anomalous magnetic moment predicted (not today)

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Quantum Field TheoryThe Lagrangian

The Feynman RulesExample of processes

Acceleration and Detection

Why do we need quantum field theory ?Special relativity and quantum field theoryDiagram ordersA brief recipe...

Why do we need quantum field theory ?

from E = mc2 to quantum field theory

The Einstein equation makes a relation between energy and mass

E = mc2

This means that if there is enough energy, we can create a particle with agiven mass m.However, due to conservation laws, it will most probably be necessary toproduce twice the particles mass (particle and antiparticle).Hence

Particle number is not fixed

The types of particles present is not fixed

This is in direct conflict with nonrelativistic quantum mechanics and forexample the Schrdinger equation that treats a constant number of particlesof a certain type.

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Quantum Field TheoryThe Lagrangian

The Feynman RulesExample of processes

Acceleration and Detection

Why do we need quantum field theory ?Special relativity and quantum field theoryDiagram ordersA brief recipe...

Attempts to incorporate special relativity in Quantum mechanics

Quantum mechanics and special relativity

Schrdinger equation contained first order time derivative and second orderspace derivatives

~2

2m2

x2+ V = i~

t

Not compatible with special relativity (E2 = p2 + m2).First attempt consisted to promote the time derivative to the second order.This resulted in the Klein Gordon equation :

1c22

t2

2

x2=

m2c2

~2

But this leads to funny features:

Negative presence probabilities,

negative energy solutions

Dirac solved the problems by reducing the spatial derivative power.Resulted in the Dirac equation.

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Quantum Field TheoryThe Lagrangian

The Feynman RulesExample of processes

Acceleration and Detection

Quantum Field Theory in a nutshell

e

t

Leading Order (LO) diagram isthe simplest diagram

The electron is on-shell(p2 = m2e), no interactions

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Quantum Field TheoryThe Lagrangian

The Feynman RulesExample of processes

Acceleration and Detection

e

t

NLO (next-to-leading order)diagram

Process not allowed in classicalmechanics

Heisenberg: Et 1 process allowed for reabsorptionafter t 1/E

Quantum mechanics: add alldiagrams, but that would alsoinclude N = Each vertex is an interaction andeach interaction has a strength(|M|2 = 1/137)Perturbation theory withSommerfeld convergence

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Quantum Field TheoryThe Lagrangian

The Feynman RulesExample of processes

Acceleration and Detection

Rough recipe for the Feynman calculations

Process calculation

Construct the Lagrangian of Free Fields

Introduce interactions via the minimal substitution scheme

Derive Feynman rules

Construct (ALL) Feynman diagrams of the process

Apply Feynman rules

Some aspects are not part of these lectures, but will sketch the ideas

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Quantum Field TheoryThe Lagrangian

The Feynman RulesExample of processes

Acceleration and Detection

Remember the particle zoo

treat only the carrier of theinteraction

as well as the e

(

uLdL

) (

cLsL

) (

tLbL

)

(

eLeL

) (

LL

) (

L L

)

uR cR tRdR sR bReR R R

g

W, Z

H

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Quantum Field TheoryThe Lagrangian

The Feynman RulesExample of processes

Acceleration and Detection

Lagrangian field theory

The Lagrangian and the Action

The Lagrangian is defined by

L = T V

The action is the time integration of the Lagrangian, S =

Ldt . This is afunctional: its argument is a function and it returns a number.Assuming that the Action should be minimal

S =

t2

t1

L(q, q)dt with S = 0

(the qi(t) being the generalized coordinates)leads to the Euler-Lagrange equation

d

dt

L

qi Lqi

= 0

The familiar equations of motion can be obtained from this equation.

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Quantum Field TheoryThe Lagrangian

The Feynman RulesExample of processes

Acceleration and Detection

Lagrangian field theory

Lagrangian density

Lagrangian formalism is now applied to fields, which are functions ofspacetime (x , t). The Lagrangian is, in the continuous case, the spaceintegration of the Lagrangian density.

L = T V =

Ld3x

and the action becomes

S =

Ldt =

Ld4x

Typically,L = L(, )

From a Lagrangian density and the Euler-Lagrange equation, equationsgoverning the evolution of particles (i.e. fields) can be derived.

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Quantum Field TheoryThe Lagrangian

The Feynman RulesExample of processes

Acceleration and Detection

The photon

MAXWELL equations:

F(x) = j(x)

F(x) = 0

with the photon field tensor:

F(x) = A(x) A(x)

A being the usual vector potential,A = (, ~A) and j(x) the currentdensity.

Fermions

Schrdinger equation is i~t

= H,

with H = ~22m2 + V . H should be

chosen to satisfy special relativy,H = c~ (i~) + mc2.i and are actually 4 4 matrices,0 = and i = iThe Dirac equation is obtained:

(i m)(x) = 0

leading to: (x)(i m)(x) with = 0 = T

0

The free Lagrangian (L0)

L0 = 14

F(x)F(x) + (x)(i m)(x)

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Quantum Field TheoryThe Lagrangian

The Feynman RulesExample of processes

Acceleration and Detection

Minimal Substitution

i i + eA(x)(x)i(x)

(x)(i + eA(x))(x)= (x)i(x) + e(x)

A(x)(x)

leads to a coupling between photon and fermion fields:

Interaction Lagrangian L

L = jA = e(x)A(x)(x)

the negative sign for j = e(x)(x)

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Quantum Field TheoryThe Lagrangian

The Feynman RulesExample of processes

Acceleration and Detection

Dirac equation for adjoint spinor

i m = 0 i() m = 0 i()() m = 0 i()() m = 0 i()() m = 0 i() m = 0i()

+ m = 0

EM current conserved

j = [e]

= e() e Dirac= ime + iem Dirac adjoint= 0

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Quantum Field TheoryThe Lagrangian

The Feynman RulesExample of processes

Acceleration and Detection

Gauge Invariance

Invariance of the Lagrangian under local U(1) transformationsor: why should physics depend on the location ?

A A + (x)(x) exp (ie(x))(x)

L0 + L = L LLocal gauge invariance under a U(1) gauge symmetry (1929 Weyl)if 6= f (x) it is a global U(1) symmetry.

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Quantum Field TheoryThe Lagrangian

The Feynman RulesExample of processes

Acceleration and Detection

U(1) Gauge invariance:

Photon field

F = A A= (A + ) (A + )= A A + = = A A= F

Photon field ok

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Quantum Field TheoryThe Lagrangian

The Feynman RulesExample of processes

Acceleration and Detection

Fermion field

(i m)

= 0(i m)

exp (ie)0(i m)( exp (ie))= exp (ie)(i m)( exp (ie))= exp (ie)i() exp (ie)+ exp (ie)i exp (ie))+ exp (ie)(m) exp (ie)= i() + exp (ie)iieexp (ie) e()+ (m)= (i m) e()

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Quantum Field TheoryThe Lagrangian

The Feynman RulesExample of processes

Acceleration and Detection

Interaction

eA(x)= e exp (ie)(A + ) exp (ie)= e(A + )= eA + e

()

Interaction term combined with fermion field (ie) okgauge invariance of the fermion field cries for the introduction of a gaugeboson!

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Quantum Field TheoryThe Lagrangian

The Feynman RulesExample of processes

Acceleration an