introduction to gauge theory - yonsei...

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Introduction to gauge theory

2008 High energy lecture 1

장 상 현

연세대학교

September 24, 2008

장 상 현 (연세대학교) Introduction to gauge theory September 24, 2008 1 / 72

Table of Contents

1 Introduction

2 Dirac equation

3 Quantization of Fields

4 Gauge Symmetry

5 Spontaneous Gauge Symmetry Breaking

6 Standard Model


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 Schroeder

and many more. . .


Introduction

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

SM is a local quantum gauge field theory with spontaneous gauge

symmetry breaking mechanism a.k.a. Higgs Mechanism.

Object of this lecture is to learn the basic concept of the gauge

symmetries and their breaking mechanism to understand SM.


The lecture will be a short introduction course to Quantum field theory

and gauge theory.

A modern approach to this subject is to use path integral and propagator

theory.

However, we will follow traditional Lagrangian approach. For the path

integral method, look for the references.

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

Mas, length, time, energy, momentum, temperatures can be measured in

“eV” or “eV−1” in this unit.


Paul Adrien Maurice Dirac, (1902 –

1984) was a British theoretical physicist.

Dirac made fundamental contributions to

the early development of both quantum

mechanics and quantum electrodynamics.

Among other discoveries, he formulated

the so-called Dirac equation, which de-

scribes the behavior of fermions and which

led to the prediction of the existence of an-

timatter.


Dirac equation

The classical field theory which describes EM field is consistent with

Special theory of relativity

but not with Quantum mechanics.

The Schrodinger equation describes low energy electrons in atom

but it is not consistent with relativity.

Non-relativistic quantum mechanics cannot describe High energy particle

interactions.

Need to combine quantum mechanics with special relativity.


In 1928, Dirac realized that the wave equation can be linear to the space

time derivative ∂µ ≡ ∂/∂xµ.

(iγµ∂µ −m)ψ = 0 (1)

Applying (iγµ∂µ −m) to (1) leads(12{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 scalar fields.

(∂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 ≡ ηµνγµ

6p ≡ γµpµ

e.g. Dirac equation

(i 6∂ −m)ψ = 0


The matrix

γ5 ≡ iγ0γ1γ2γ3

has the form in Dirac basis

γ5 =

(0 I

I 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 above 16

matrices.

γµ can have different basis with the same physics.

e.g. Weyl basis,

γ0 =

(0 I

I 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 =

(φ

χ

)


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)

where

S(Λ) = e−i4ωµνσµν

Also

SγνS−1 = Λνµγµ

and

S(Λ)† = γ0ei4ωµνσµνγ0


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 density L.

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

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

P 2L = PL, P

2R = PR, PLPR = 0

γ5ψL = −ψL, γ5ψR = +ψR.


Dirac Lagrangian can be written in chiral components

L = ψ(iγµ∂µ −m)ψ

= ψLiγµ∂µψL + ψRiγ

µ∂µψR −m(ψLψR + ψRψL) (9)

If m = 0, ψL and ψR are independent and have additional symmetry

ψL −→ eiθLψL, ψR −→ eiθRψR,

Weak interaction is called ‘chiral’ because it only interacts with

left-handed leptons.


Further readings

Check the references for Charge conjugation, Parity transformation, CP

and CPT .


Amalie Emmy Noether, (1882 – 1935)

was a German mathematician described

by Albert Einstein and others as the most

important woman in the history of math-

ematics, she revolutionized the theories

of rings, fields, and algebras.

She also is known for her contributions

to modern theoretical physics, especially

for the first Noether’s theorem which ex-

plains the connection between symmetry

in physics and conservation laws.


Noether’ theorem relates a continous symmetry to a conservation law.


Noether’s theorem

If a Lagrangian L with a field φa is invariant of a continuous

transformation φa −→ φa + δφa

0 = δL =δLδφa

δφa +δL

δ(∂µφa)δ(∂µφa) (10)

use the eq. of motionδLδφa

= ∂µ

(δL

δ(∂µφa)

)(10) becomes

0 = ∂µ

(δL

δ(∂µφa)δφa

)


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 of the

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→ eiθφ.


For free fermion

L = ψ(iγµ∂µ −m)ψ (14)

The Noether current

Jµ = ψγµψ (15)

corresponds the symmetry ψ → eiθψ.

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

This symmetry is called U(1) global symmetry, since θ is the same for 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)3

1√2E~p

(a(~p)e−ip·x + a(~p)†eip·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 of freedom.

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

φ(x)|0〉 =∫

d3p

(2π)3

12E~p

e−ip·x|~p〉

〈0|φ(x)|~p〉 = eip·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 satisfy anticommutation

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)3

1√2E~p

∑s

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

)ψ(x) =

∫d3p

(2π)3

1√2E~p

∑s

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

with −1 charge.

For instance Qe is the electric charge of electrons.

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


Maxwell equation

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

∂µFµν = 0 or ∂2Aν − ∂ν∂µAµ = 0 (18)

where

Fµν = ∂µAν − ∂νAµ

The Lagrangian for photon is

LMax = −14FµνF

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 − 14FµνF

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 invariance

The most significant property of the Lagrangian (20) is that it is invariant

under gauge transformation.

LMax = −14FµνF

µν is invariant under the transformation

Aµ(x)→ Aµ(x) +1e∂µΛ(x)

for any scalar function Λ(x)

While LDirac is invariant under

ψ(x)→ eiΛ(x)ψ(x)

only if Λ(x) = constant.


However the total Lagrangian with interaction term (20)

L = ψ(iγµDµ −m)ψ − 14FµνF

µν

and covariant Dirac equation (21)

[iγµ(∂µ − ieAµ)−m]ψ = 0

are invariant under local U(1) gauge symmetry.

ψ(x)→ eiΛ(x)ψ(x)

Aµ(x)→ Aµ(x) +1e∂µΛ(x)


The gauge boson mass term M2AA

µAµ is not invariant

under the gauge transformation

Aµ(x)→ Aµ(x) +1e∂µΛ(x).

Thus, the gauge invariant field Aµ should be massless.


Complex Scalar Field

The Lagrangian of a free complex scalar field

φ = (φ1 + iφ2)√

2 , φ∗ = (φ1 − iφ2)√

2

L = (∂µφ)(∂µφ∗)−m2φ∗φ (22)

is invariant under global gauge transformation

φ→ eiΛφ , 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)

where

δJµ = 2|φ|2∂µΛ

for

Aµ(x)→ Aµ(x) +1e∂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− 14FµνFµν (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)3

1√2E~p

3∑r=0

(ar(~p)εµr e

−ip·x + ar(~p)†εµ∗r eip·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


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 |k〉t = e−iHt|k〉0 (the

Hamiltonian H). Thus in the infinite time limit,

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. (e, e2, . . . ) from eAµψγµψ, e2AµA

µ|φ|2

The leading order term of (30) is proportional to G and higher order term

will be G2, G3, 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 energy at the

log scale.


Renormalizability

The leading order term is proportional to G.

If dimension [G] = a, roughly

the perturbation give G2Λ−2a contribution.

If a < 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] =32.

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) +14FµνFµν , (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 equal

probability.

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. Choose a

vacuum,

〈φ〉 = v =

√−µ2

2λand parametrize it as

φ(x) = ρ(x) exp[iθ(x)]

then

〈ρ〉 = v , 〈θ〉 = 0.


Insert φ(x) = (v + χ(x))eiθ(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 Mechanism

If we add gauge field

L = −14FµνFµν + |Dµφ|2 − µ2|φ|2 − λ|φ|4, (34)

= −14FµνFµν + 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→ eiΛφ (θ → θ + Λ), Aµ → Aµ +1e∂µΛ


If the symmetry is broken spontaneously,

〈φ〉 = v =

√−µ2

2λ

and insert ρ = v + χ to (35),

L = −14FµνFµν + (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 in addition 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 material act 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 and magnetic 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.


God Particle?

There were rumors that LHC, which started this month, can create

mini-black hole which eventually destroy the earth.


God Particle?

Instead of the end of the world, LHC experiment will (probably) discover

Higgs scalar (or so-called god-particle).


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