2006 High energy lecture 1 -...

IntroductionDirac equation

Quantization of FieldsGauge Symmetry

Spontaneous Gauge Symmetry BreakingStandard Model

Introduction to gauge theory2006 High energy lecture 1

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September 22, 2006

�©� �©� �&³ Introduction to gauge theory




Table of Contents

Introduction

Dirac equation

Quantization of Fields

Gauge Symmetry

Spontaneous Gauge Symmetry Breaking

Standard Model





ReferencesThe basic

References for quantum field theory

“Quark and Leptons” Halzen and Martin“Quantum Field Theory” Ryder“Quantum Field Theory” Mandl and Show“Gauge Theory of Elementary Particle Physics” Cheng and Li“Quantum Field Theory in a Nutshell” Zee“An Introduction to Quantum Field Theory” Peskin and Schroederand many more. . .





ReferencesThe basic

Introduction

The Standard Model (SM) is The basis of High Energy Physics.

SM is a local quantum gauge field theory with Spontaneous gaugesymmetry breaking mechanism a.k.a Higgs Mechanism.

Object of this lecture is to learn the basic concept of the gaugesymmetries and their breaking mechanism to understand SM.





ReferencesThe basic

The lecture will be a short introduction course to Quantum fieldtheory and gauge theory.

A modern approach to this subject is to use path integral andpropagator theory.

However, we will follow traditional Lagrangian approach. For thepath integral method, look for the references.

In elementary particle physics, we use the unit where ~ = c = 1.





Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality

Dirac equation

The classical field theory which describes EM field is consistentwith Special theory of relativity

but not with Quantum mechanics.

The Schrodinger equation describes low energy electrons in atombut it is not consistent with relativity.

Non-relativistic quantum mechanics cannot describe High energyparticle interactions.Need to combine quantum mechanics with special relativity.






Dirac equation

The classical field theory which describes EM field is consistentwith Special theory of relativitybut not with Quantum mechanics.








Dirac equation


The Schrodinger equation describes low energy electrons in atom

but it is not consistent with relativity.







Dirac equation









Dirac equation



Non-relativistic quantum mechanics cannot describe High energyparticle interactions.

Need to combine quantum mechanics with special relativity.






Dirac equation









In 1928, Dirac realized that the wave equation can be linear to thespace time derivative ∂µ ≡ ∂/∂xµ.

(iγµ∂µ −m)ψ = 0 (1)

Applying (iγµ∂µ −m) to (1) leads(1

2{γµ, γν}∂µ∂ν + m2

)ψ = 0 (2)

where {A,B} = AB + BA is anticommutator. If

{γµ, γν} = 2ηµν (3)

ηµν is the Minkowski metric η00 = 1, ηjj = −1 and otherwise zero.






Then the Dirac equation becomes

(∂2 + m2)ψ = 0

This is the same form as Klein-Gordon equation for the scalarfields.

(∂2 + m2)φ = 0

γµ satisfies Clifford algebra (3) can be written as 4× 4 matrices.






One representation of γµ satisfies (3) is

γ0 =

(I 00 −I

)γi =

(0 σi

−σi 0

)(4)

I is 2× 2 identity matrix and σi (i = 1, 2, 3) are Pauli matrices.

It is called Dirac basis.

Some useful notations:γµ ≡ ηµνγ

µ

6p ≡ γµpµ

e.g. Dirac equation(i 6∂ −m)ψ = 0






The matrixγ5 ≡ iγ0γ1γ2γ3

has the form in Dirac basis

γ5 =

(0 II 0

)(5)

and anticommute with γµ

{γ5, γµ} = 0

(γ5)† = γ5, (γ5)2 = 1






With the 6 matrices

σµν ≡ i

2[γµ, γν ]

{1, γµ, σµν , γµγ5, γ5} form a complete basis of 16 elements.

All 4× 4 matrices can be written as a linear combination of above16 matrices.

γµ can have different basis with the same physics.e.g. Weyl basis,

γ0 =

(0 II 0

), γi =

(0 σi

−σi 0

), γ5 =

(−I 00 I

)






If we transform the spinors to momentum space

ψ(x) =

∫d4p

(2π)4e−ipxψ(p)

The Dirac equation becomes

(γµpµ −m)ψ(p) = 0 (6)

Dirac spinor ψ can be divided into two 2-component spinors,

ψ =

(φχ

)






In Dirac basis, (γ0 − 1)ψ(p) = 0 in the rest frame pµ = (m,~0).

Only φ describes electron, which has two component.

For slowly moving electron, χ(p) is very small.






Lorentz transformation is defined as

Λ = e−12ωµνJµν

anti-symmetric ωµν = −ωνµ are 3 rotation and 3 boost parameters.

J ij are rotation generators and J0i are boost generators.

The coordinate xα transforms

x ′α = Λαβxβ






Spinors transforms under Lorentz transformation is

ψ′(x ′) = S(Λ)ψ(x)

whereS(Λ) = e−

i4ωµνσµν

AlsoSγνS−1 = Λν

µγµ

andS(Λ)† = γ0e

i4ωµνσµν

γ0






Define ψ = ψ†γ0 then

ψ(x)ψ(x) is invariant under Lorentz transformation (scalar).

ψ(x)γµψ(x) transform as Lorentz vector.

ψ(x)γ5ψ(x) transform as a pseudoscalar.

ψ(x)γ5γµψ(x) transform as Lorentz pseudovector.






Lagrangian L and Lagrangian density L is defined from the action

S =

∫dtL =

∫d4xL

In High energy physics(HEP) Lagrangian means Lagrangian densityL.

If the L is a function of field φ(x), the Euler-Lagrange eq. ofmotion should satisfied.

∂µ

(∂L

∂(∂µφ)

)− ∂L∂φ

= 0 (7)






Lagrangian of free Dirac field

L = ψ(iγµ∂µ −m)ψ (8)

From the eq. of motion

∂µ

(∂L

∂(∂µψ)

)− ∂L∂ψ

= 0

Dirac equation can be obtained

(iγµ∂µ −m)ψ = 0






Define chiral projection,

ψL(x) = PLψ(x) , ψR(x) = PRψ(x) .

The projection operators

PL =1− γ5

2, PR =

1 + γ5

2.

Then the Dirac spinor is sum of two chiral components

ψ(x) = ψL(x) + ψR(x)






In Weyl basis

γ5 =

(−I 00 I

).

Thus

ψ(x) =

(ψL

ψR

).

Some properties to notice

P2L = PL, P2

R = PR , PLPR = 0

γ5ψL = −ψL, γ5ψR = +ψR .






Dirac Lagrangian can be written in chiral components

L = ψ(iγµ∂µ −m)ψ

= ψLiγµ∂µψL + ψR iγµ∂µψR −m(ψLψR + ψRψL) (9)

If m = 0, ψL and ψR are independent and have additionalsymmetry

ψL −→ e iθLψL, ψR −→ e iθRψR ,

Weak interaction is called ‘chiral’ because it only interacts withleft-handed leptons.






Further readings

Check the references for Charge conjugation, Paritytransformation, CP and CPT .





Noether’s theoremScalar field quantizationDirac field quantization

Noether’s theorem

If a Lagrangian L with a field φa is invariant of a continuoustransformation φa −→ φa + δφa

0 = δL =δLδφa

δφa +δL

δ(∂µφa)δ(∂µφa) (10)

use the eq. of motion

δLδφa

= ∂µ

(δL

δ(∂µφa)

)(10) becomes

0 = ∂µ

(δL

δ(∂µφa)δφa

)�©� �©� �&³ Introduction to gauge theory





We define a current

Jµ ≡ δLδ(∂µφa)

δφa

Then∂µJµ = 0

We have a conserved current Jµ.

Noether’s Theorem

A conserved current is associated with a continuous symmetry ofthe Lagrangian.






Charge is defined as

Q =

∫d3xJ0 =

∫d3x

δLδ(∂0φa)

δφa. (11)

Since dQ/dt = 0, the charge is conserved.

π(x) ≡ δLδ(∂0φa)

is canonical momentum (density) corresponding to φa.






Free scalar field Lagrangian

L = |∂µφ|2 −m2|φ|2 (12)

of Klein-Gordon equation of motion

(∂2 + m2)φ = 0.

The Noether current

Jµ = i [(∂µφ∗)φ− φ∗(∂µφ)], (13)

corresponds the symmetry φ→ e iθφ.






For free fermion

L = ψ(iγµ∂µ −m)ψ (14)

The Noether current

Jµ = ψγµψ (15)

corresponds the symmetry ψ → e iθψ.

∂µJµ = (∂µψ)γµψ + ψγµ∂µψ = (imψ)γµψ + ψγµ(−imψ) = 0

This symmetry is called U(1) global symmetry, since θ is the samefor any space-time x .






Quantization of scalar field

For free scalar field, the canonical momentum is π = φ.

Being a “quantum” field theory requires:

1. φ(x) and π(x) becomes operator

2. and they satisfy canonical commutator relation.

[φ(~x , t), π(~y , t)] = iδ(3)(~x − ~y)

[φ(x), φ(y)] = [π(x), π(y)] = 0






If E~p =√|~p|2 + m2,

φ(x) =

∫d3p

(2π)31√2E~p

(a(~p)e−ip·x + a(~p)†e ip·x

)∣∣∣p0=E~p

π(x) = ∂0φ(x) (16)

[a(~p), a(~p′)†] = (2π)3δ(3)(~p − ~p′), (17)

a(~p)† creates one particle state from vacuum |0〉

|~p〉 =√

2E~pa(~p)†|0〉

a(~p) destroys vacuum a(~p)|0〉 = 0.






Quantum field is a harmonic oscillator with continuous degree offreedom.

φ(x) acting on vacuum, create a particle at x .

φ(x)|0〉 =

∫d3p

(2π)31

2E~pe−ip·x |~p〉

〈0|φ(x)|~p〉 = e ip·x is free particle wave function.






Dirac field quantization

For free Dirac field, canonical momentum is

π =δL

δ(∂0ψ)= iψ†.

Not like the scalar case, fermion field should satisfyanticommutation relation

{ψa(~x , t), ψ†b(~y , t)} = δ(3)(~x − ~y)δab

{ψa(~x , t), ψb(~y , t)} = {ψ†a(~x , t), ψ†b(~y , t)} = 0

a, b are spinor components






We can write the field operators

ψ(x) =

∫d3p

(2π)31√2E~p

∑s

(b(p, s)u(p, s)e−ip·x + d†(p, s)v(p, s)e ip·x

)ψ(x) =

∫d3p

(2π)31√2E~p

∑s

(d(p, s)v(p, s)e−ip·x + b†(p, s)u(p, s)e ip·x

)s = 1, 2 is spin index.

{b(p, s), b†(p′, s ′)} = {d(p, s), d†(p′, s ′)} = (2π)3δ(3)(~p − ~p′)δss′






Both b(p, s) and d(p, s) annihilate vacuum

b(p, s)|0〉 = d(p, s)|0〉 = 0

b†(p, s) and d†(p, s) creates particle with energy momentum p

but they are charge conjugated state with each other.

We define b†(p, s) creates a fermion and

d†(p, s) creates an anti-fermion.






From the Noether current Jµ = ψγµψ, there is a conserved charge

Q =

∫d3xψ†(x)ψ(x) =

∫d3p

(2π)3

∑s

(b†(p, s)b(p, s)− d†(p, s)d(p, s)

)b†(p, s) creates a fermion with +1 charge and d†(p, s) creates afermion with −1 charge.

For instance Qe is the electric charge of electrons.

U(1) symmetry must be related with electric charge conservation!





Maxwell equationGauge invarianceComplex Scalar FieldGauge field quantization

Maxwell equation

The Maxwell equation is eq. of motion for photon field Aµ(x)

∂µFµν = 0 or ∂2Aν − ∂ν∂µAµ = 0 (18)

whereFµν = ∂µAν − ∂νAµ

The Lagrangian for photon is

LMax = −1

4FµνF

µν (19)






Lint = eAµψγµψ is a covariant interaction term between Dirac and

Maxwell field where e is a coupling constant.

Then the combination of electro and fermion Lagrangian is

L = LDirac + LMax + Lint

= ψ(iγµDµ −m)ψ − 1

4FµνF

µν (20)

Dµ = ∂µ − ieAµ is a covariant derivative.

Then the Dirac equation with electromagnetic interaction is

[iγµ(∂µ − ieAµ)−m]ψ = 0 (21)






Gauge invarianceThe most significant property of the Lagrangian (20) is that it isinvariant under gauge transformation.

LMax = −14FµνF

µν is invariant under the transformation

Aµ(x) → Aµ(x) +1

e∂µΛ(x)

for any scalar function Λ(x)

While LDirac is invariant under

ψ(x) → e iΛ(x)ψ(x)

only if Λ(x) = constant.






However the total Lagrangian with interaction term (20)

L = ψ(iγµDµ −m)ψ − 1

4FµνF

µν

and covariant Dirac equation (21)

[iγµ(∂µ − ieAµ)−m]ψ = 0

is invariant under local U(1) gauge symmetry.

ψ(x) → e iΛ(x)ψ(x)

Aµ(x) → Aµ(x) +1

e∂µΛ(x)






The gauge boson mass term M2AAµAµ is not invariant

under the gauge transformation

Aµ(x) → Aµ(x) +1

e∂µΛ(x).

Thus, the gauge invariant field Aµ should be massless.






Complex Scalar Field

φ = (φ1 + iφ2)√

2 , φ∗ = (φ1 − iφ2)√

2

The Lagrangian of a free complex scalar field

L = (∂µφ)(∂µφ∗)−m2φ∗φ (22)

is invariant under global gauge transformation

φ→ e iΛφ , φ∗ → e−iΛφ∗ ,

where Λ is a real constant.

However, it is not invariant for local gauge Λ(x)






For small Λ(x), the local gauge transformation can be written as

φ→ φ+ iΛ(x)φ ,

∂µφ→ ∂µφ+ iΛ(x)(∂µφ) + i(∂µΛ(x))φ .

Then Euler-Lagrange equation leads (10)

δL = Jµ∂µΛ(x) (23)

where the conserved current is

Jµ = i [(∂µφ∗)φ− φ∗(∂µφ)].






As the case of fermions, we can add the interaction therm

Lint = −eJµAµ (24)

between scalar field and gauge field. Then

δLint = −e(δJµ)Aµ − Jµ∂µΛ, (25)

whereδJµ = 2|φ|2∂µΛ

for

Aµ(x) → Aµ(x) +1

e∂µΛ(x).






To cancel the extra term −e(δJµ)Aµ, we must add

Lext = e2AµAµ|φ|2

The total Lagrangian

Lscalar = (Dµφ)(Dµφ)∗ −m2φ∗φ− 1

4FµνFµν (26)

is local U(1) gauge symmetric, where

Dµφ = (∂µ − ieAµ)φ

is a covariant derivative.






With the gauge invariant Lagrangian (20), the total Lagrangian

L = ψ(iγµDµ −mf )ψ + (Dµφ)(Dµφ)∗ −m2sφ

∗φ− 1

4FµνFµν (27)

gives a complete description of the world with

1. a local U(1) symmetric charged scalar with mass ms ,

2. a local U(1) symmetric charged fermion with mass mf ,

3. a local U(1) symmetric neutral massless gauge boson(photon).

Or, in one sentence, Quantum electrodynamics (QED).






For a free (non-interacting) gauge field ( E~p =√|~p|2 + m2 )

Aµ(x)=

∫d3p

(2π)31√2E~p

3∑r=0

(ar (~p)εµr e−ip·x + ar (~p)†εµ∗r e ip·x

)(28)

εµ: polarization vector, r : indices of polarization.

[ar (~p), as(~p′)†] = (2π)3δrsδ

(3)(~p − ~p′), (29)

is a quantization condition for a photon field,

where one photon state is |p〉 ∝ ar (~p)†|0〉 and ar (~p)|0〉 = 0.

Photon is neutral since it is a real field





RenormalizationCut-off of infinityCoupling constantsrenormalizable termSpontaneous symmetry breakingAnderson-Higgs Mechanism

Renormalization

Only a rough description of renormalization will be presented.

Read the Field Theory references for the details of this subject.






If a system of particles |{ki}〉in scatters to |{pf }〉out

{ki}, {pf } are set of initial and final momenta of particles.

The matrix element M is defined as

out〈{pf }|{ki}〉in = (2π)(4)δ(∑

ki −∑

pf ) · iM

Without going through the details, the scattering cross section is

dσ ∝M2.






M can be expanded with time-evolution, i.e. the Hamiltonian.

out〈{pf }|{ki}〉in = limT→∞

〈{pf }|e−iH(2T )|{ki}〉 (30)

Amplitude of scattering can be expanded with interaction terms.

Interaction term of fields are proportional to coupling constant G .

e.g. (mf ,m2s , e, . . . )

The leading order term of (30) is proportional to G and higherorder term will be G 2, G 3, etc.

This is a basic idea of perturbation expansion.






Higher order term in M contains momentum integrals.

And the integral diverges with p →∞.

It could be fatal problem of the field theory itself.

The solution that the theorist found(?) is very simple.






Cut it off!






Integrate momentum p to finite cut-off Λ to make the theory finite.

By summing up the perturbation series and re-normalizing it,

we can obtain the physical values.

The final result can depend on at most log(Λ).

The physical parameters (couplings, masses) varies with the energyat the log scale.






Renormalizability

The leading order term is proportional to G .

If dimension [G ] = x , roughly

the perturbation give G 2Λ−2x contribution.

If x < 0, the sum depends strongly on Λ and renormalization fails.

[G ] > 0 is a condition for renormalizable interaction.






From [L] = 4 and [x ] = −1, [∂µ] = [m] = 1,

We obtain for scalar fields

[φ] = [Aµ] = 1,

and for fermion fields

[ψ] =3

2.

Therefore, [e] = 0.






Any term with dimension more than 4 need a coupling [G ] < 0

and is not renormalizable.

Only gauge invariant dimension 4 term is |φ|4.

The renormalizable Lagrangian with U(1) gauge symmetry is

L = ψ(iγµDµ −mf )ψ + |Dµφ|2 − V (|φ|2) +1

4FµνFµν , (31)

where

V (|φ|2) = µ2|φ|2 + λ|φ|4. (32)






The minimum of potential (32) is at

φ = 0 and |φ|2 = −µ2

2λ

For λ > 0 and real mass µ, φ = φ∗ = 0 is the absolute minimum.

For µ2 < 0, the potential has more than one minimum.






If q = |φ|, there is two minimum in the potential.‘






Since φ = (φ1 + iφ2)√

2, V (φ1, φ2) has a shape of Mexican hat






Spontaneous Symmetry Breaking

What will happen, if there are more than one ground state?

Like coin flipping, system can choose each ground state with equalprobability.

Even after the system select a specific ground state,

the Lagrangian has the symmetry.

However, the solution itself does not have a symmetry, anymore.

The symmetry is broken spontaneously.






The spontaneous symmetry breaking of U(1)

If the scalar field has imaginary mass,

it can have continuous (Mexican hat shape) ground state. Choosea vacuum,

〈φ〉 = v =

√−µ2

2λ

and parametrize it as

φ(x) = ρ(x) exp[iθ(x)]

then〈ρ〉 = v , 〈θ〉 = 0.






Insert φ(x) = (v + χ(x))e iθ(x) to Lagrangian

L = |∂φ|2 − µ2|φ|2 − λ|φ|4, (33)

= (∂χ)2 − λv4 − 4λv2χ2 − 4λvχ3 − λχ4 + (v + χ)2(∂θ)2.

χ has a real mass and θ is massless.

θ is called Nambu-Goldstone boson.

(33) does not have global U(1) symmetry.

It is broken spontaneously.






Goldstone theorem

Whenever a continuous symmetry is spontaneously broken,

a massless (Nambu-Goldstone) boson emerge.

Any degree of freedom moves along with the flat direction

does not have a mass.

No mass term m2φ2






Anderson-Higgs MechanismIf we add gauge field

L = −1

4FµνFµν + |Dµφ|2 − µ2|φ|2 − λ|φ|4, (34)

= −1

4FµνFµν + e2ρ2(Bµ)2 + (∂ρ)2 − µ2ρ2 − λρ4.

where Bµ = Aµ − (1/e)∂µθ and

Fµν = ∂µAν − ∂νAµ = ∂µBν − ∂νBµ

are invariant under U(1) transformation,

φ→ e iΛφ (θ → θ + Λ), Aµ → Aµ +1

e∂µΛ






If the symmetry is broken spontaneously,

〈φ〉 = v =

√−µ2

2λ

and insert ρ = v + χ to (35),

L = −1

4FµνFµν + (ev)2(Bµ)2 + e2(2vχ+ χ2)(Bµ)2

+(∂χ)2 − 4λv2χ2 − 4λvχ3 − λχ4 − λv4 (35)

There is no Goldstone boson θ, while Bµ gains a mass M =√

2ev .






In the Anderson-Higgs (also, Landau-Ginzburg-Kibble) mechanism,

the massless degree of freedom is eaten by gauge field.

As a consequence, the gauge field becomes massive.

Since the massive photon has an extra degree of freedom inaddition to two polarizations,

the total degree of freedom does not change.






Ferromagnetism

The Hamiltonian of Ferromagnet has rotational symmetry of spin,

A spin can point any direction which is global SO(3) symmetry.

If spin aligns one direction, SO(3) is spontaneously broken to

SO(2): a symmetry of rotation around spin direction.

Since a continuous symmetry is spontaneously broken,

there exists Goldstone mode called spin wave.






Superconductivity

When temperature went down SO(2) which is equivalent to U(1)is also broken spontaneously.

At low temperature a pair of electron in superconducting materialact like a boson (Higgs scalar).

This is the same case as we discussed, local U(1) gauge symmetry.

There is no Goldstone mode, but the photon becomes massive.

Which explains why electric force becomes short-ranged andmagnetic field cannot penetrate in superconducting material.





The Standard Model

I will close this lecture with a comment about the Standard Model.

The Standard Model(SM) is a SU(2)L × U(1)Y local gauge theory

with 6 quarks and leptons as a basis.

SU(2)L is non-Abelian gauge symmetry, which is rather

complicated than Abelian gauge group U(1)

But the basic concept is the same.





In SM, instead of U(1), SU(2) is broken by Higgs mechanism,

As a result, there exist three massive gauge bosons W±,Z .

Also the quarks and leptons which are chiral field to SU(2)L

and originally massless, obtain the masses.

SM is extremely successful, both in theory and experiments.

Except, we have not seen Higgs scalar, yet.





Read the references on Quantum field theory!


2006 High energy lecture 1 -...

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