particle physics: introduction to the standard model ...· particle physics: introduction to the...

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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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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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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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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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 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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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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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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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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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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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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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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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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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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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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U(1) Gauge invariance:
Photon field
F = A A= (A + ) (A + )= A A + = = A A= F
Photon field ok
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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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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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Acceleration an

particle physics: introduction to the standard model ...· particle physics: introduction to the...

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