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Gauge Theories

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Gauge Theories

Lagrangians in relativistic field theory

Local gauge invariance

Yang-Mills theory

Chromodynamics

The mass term

Spontaneous symmetry-breaking

The Higgs mechanism

These notes are not original, and rely heavily on information and examples in the texts “Introduction to Particle Physics” by David Griffiths,

Gauge Theories

Lagrangians in relativistic field theory

Local gauge invariance

Yang-Mills theory

Chromodynamics

The mass term

Spontaneous symmetry-breaking

The Higgs mechanism

These notes are not original, and rely heavily on information and examples in the texts “Introduction to Particle Physics” by David Griffiths,

- Particles in classical mechanics and relativistic fields (ħ=c=1)
- Euler Lagrange equations in field theory

Lagrangian

density

- Let
- Then
so

- This is the Klein-Gordon equation.
- Since the field is a scalar, it describes a particle of spin 0 and mass m

and

- Let
Then

so

This is the Dirac equation for a particle of spin 1/2 and mass m

NOTE - is a 4-dimensional field (a spinor)

and

- Introduce
- Let
Then

so

This is Proca equation for a particle of spin 1 and mass m

NOTE – A is a 4-vector field

and

- Suppose
- The Euler-Lagrange equations yield
- It follows (continuity equation)

Gauge Theories

Lagrangians in relativistic field theory

Local gauge invariance

Yang-Mills theory

Chromodynamics

The mass term

Spontaneous symmetry-breaking

The Higgs mechanism

These notes are not original, and rely heavily on information and examples in the texts “Introduction to Particle Physics” by David Griffiths,

- The Dirac Lagrangian is invariant under the transformation
(global gauge trans. belonging to U(1) group)

- However, if the phase depends upon position in space-time,
(local gauge transformation)

- Is Dirac Lagrangian invariant under local gauge transformation?
( NO ! )

- More convenient to replace (x) by
so, under

- We can add something to LDirac to make it invariant under this local gauge transformation.
where

New (vector) “gauge field”

- However, the full Lagrangian must also include a “free term” for the gauge field. Consider Proca Lagrangian (vector field)
- Note that is invariant but is not.
Evidently gauge field must be massless (mA=0)

- So, we arrive at the Lagrangian:
with

Dirac Fermions mass

Mass m, charge q

Maxwell (E/M) field

Photons (m=0)

Interaction

between A and J

- The difference between global and local gauge trans. Arises from the term
- We can arrive at the same Lagrangian by replacing each partial derivative in the original Lagrangian with a “covariant derivative”
and every by and requiring that the new field transform under the gauge transformation as

So,

BUT

Gauge Theories

Lagrangians in relativistic field theory

Local gauge invariance

Yang-Mills theory

Chromodynamics

The mass term

Spontaneous symmetry-breaking

The Higgs mechanism

- The most general form of gauge transformation is
where is a unitary matrix and H is Hermitian.

are the 3 Pauli

Matrices.

liare the 8

Gell-Mann

matrices

- This was orginally proposed by Yang and Mills to describe a world with neutrons and protons thought of as point-like particles of spin ½ and with similar masses.

- Suppose we have two spin ½ fields, ψ1 and ψ2
- Using a matrix representation in which we combine the two
- We can then write the Lagrangian as
- If then this becomes

“Looks” like the Dirac Lagrangian

for a single particle of mass m

- This is NOT the Dirac Lagrangian for a single particle, however, but for the doublet state =(1, 2), each i with its normal 4 spinor components.
- Gauge transformations must now be introduced as 2x2 matrices, members of the SU(2) group, of the form:
where

I is the 2x2 unit matrix

- Here, we consider, a global SU(2) gauge transformation
- Define local SU(2) gauge transformation

and

Transformation

for

- is not invariant under local SU(2) gauge transformations
- Introduce vector fields,
and covariant derivative

- The resulting Lagrangian is

- Each of the 3 Aμ fields requires its own free Lagrangian
(Proca mass term is excluded by local gauge invariance.)

- The complete Yang-Mills Lagrangian
- (describes two equal-mass Dirac fields in interaction with three massless vector gauge fields.)
- The Dirac fields generate three currents

- NOTE – in carrying out the algebra involved, we also find it necessary to re-define each vector (gauge) field tensor as
AND, the gauge transformation of the field is

“Interaction term”

Gauge Theories

Lagrangians in relativistic field theory

Local gauge invariance

Yang-Mills theory

Chromodynamics

The mass term

Spontaneous symmetry-breaking

The Higgs mechanism

- The free Lagrangian for a particular flavor
- Use matrix representation
- Here, the 3 masses are identical so

- Now we need SU(3) global transformation (3x3 matrix)
where

and we only consider = 0

- The local SU(3) gauge transformation we use is
where

- is not invar. under local SU(3) transformation, so seek
- Introduces 8 vector (gauge) fields
- So
- The resulting Lagrangian is

Lots of algebra

(NOTE – massless, vector field

and F is 8-vector of field tensors).

- now we add the free gluon Lagrangian
- The complete Lagrangian for Chromodynamics is then
- Dirac fields constitute eight color 4-currents
carried by the 8 mass-less, vector gluons

Quarks

Free gluon field

Interaction of

Quarks with

gluon field

“Interaction term”

- NOTE – in carrying out the algebra involved, we also find it necessary to re-define each vector (gauge) field tensor as
AND, the gauge transformation of the field is

as for the Yang-Mills theory,

EXCEPT we have to define 8-dimensional vector product:

Gauge Theories

Lagrangians in relativistic field theory

Local gauge invariance

Yang-Mills theory

Chromodynamics

The mass term

Spontaneous symmetry-breaking

The Higgs mechanism

- The principle of local gauge invariance works beautifully for the strong and E.M. interactions.
- The application to weak interactions was stymied because gauge fields have to be massless.
- Can we make gauge theory to accommodate massive gauge fields?
Yes, by using spontaneous symmetry-breaking and the Higgs mechanism.

- Suppose

- If we expand the exponential
the second term looks like the mass term in the K.G. Lagrangian with

The higher-order terms represent couplings, of the form

This is not supposed to be a realistic theory

In general, the “mass term” is second order in a field.

- To identify how a mass term in a Lagrangian may be disguised, we pick out the term proportional to Φ2 in
This (second term) looks like mass, and the third term like an interaction.

- BUT, if that is the mass term, then m is imaginary (!!)
- Feynman calculus comes from a perturbation about the ground state (vacuum), treating the fields as fluctuations about that state: Φ=0
But for the Lagrangian above, Φ=0 is NOT the ground state.

To determine the true ground state, write Lagrangian as

- so,
and the minimum in U occurs at

- Introduce a new field variable
In terms of η

Now second term is a mass term, with the correct sign.

[ graph of U(Φ)]

- The third and fourth terms represent couplings of the form

Gauge Theories

Lagrangians in relativistic field theory

Local gauge invariance

Yang-Mills theory

Chromodynamics

The mass term

Spontaneous symmetry-breaking

The Higgs mechanism

- From the mass term, the original Lagrangian is even in Φ
- The reformulated Lagrangian is not even in η
(the symmetry has been broken)

- It happened because the vacuum (either of the two ground states) does not share the symmetry of the Lagrangian

NOT

Symmetric about

here OR here

Symmetric

about here

U

1

2

- For example, the Lagrangian with “spontaneously broken” continuous symmetry
(it is, in fact, invariant under rotations in Φ1Φ2 space SO(2)symmetry)where,

The condition for a minimum is, therefore, that

We may as well pick,

“Mexican Hat”

symmetry

- [ Spontaneous
symmetry breaking

in a plastic strip ]

- [ The potential
function ]

“Spontaneous”

Choice of the

solution x

A single point

on the circle

U(1 , 2 )

2

x

1

Circle of minima

- Introduce new fields
- Rewriting the Lagrangian in terms of new variables,
- The first term is a free K.G. Lagrangian for the field η
the second term is a free Lagrangian for the field ξ

Goldstone Boson

(unwanted!)

- The third term defines five couplings
- In this form, the Lagrangian doesn’t look symmetrical at all
(the symmetry has been broken by the selection of a particular vacuum state)

- One of the fields (ξ) is automatically mass-less

Gauge Theories

Lagrangians in relativistic field theory

Local gauge invariance

Yang-Mills theory

Chromodynamics

The mass term

Spontaneous symmetry-breaking

The Higgs mechanism

- If we combine the two real fields into a single complex field
- The rotational [SO(2)] symmetry that was spontaneously broken becomes invariant under U(1) phase transformation.
- We can make the system invar. under local gauge trans.

- Replace partials in with covariant derivatives, etc.
- Thus
- Define the new fields
- Lagrangian becomes

- The first line describes a scalar particle
and a massless Goldstone boson (ξ)

- The second line describes the free gauge field Aμ,
it has acquired a mass

- Term in curly brackets specifies various coupling of ξ, η, Aμ
- We still have the unwanted Goldstone boson (ξ)
as interaction, it leads to a vertex of the form

- We could choose a gauge in which so that
This would make

In this particular gauge, therefore

(Eliminates Goldstone Boson !)

- We have eliminated the Goldstone boson and the offending term in . We are then left with a single massive scalar η(the Higgs particle) and massive gauge field Aμ
- A mass-less vector field carries two degrees of freedom (transverse polarizations). When Aμacquires mass, it picks up a third degree of freedom (longitudinal polarization)
Q: where did this extra degree of freedom come from?

A: it came from the Goldstone boson, which meanwhile disappeared from the theory.

The gauge field “ate” the Goldstone boson, thereby acquiring

both a mass and a third polarization state (Higgs mechanism)