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

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Gauge theories

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 theories1

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,


Lagrangians in relativistic fields

Lagrangians in relativistic fields

  • Particles in classical mechanics and relativistic fields (ħ=c=1)

  • Euler Lagrange equations in field theory

Lagrangian

density


Klein gordon lagrangian for scalar field

Klein-Gordon Lagrangian for scalar field

  • 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


Dirac lagrangian for spinor field

Dirac Lagrangian for Spinor field

  • 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


Proca lagrangian for vector field

Proca Lagrangian for Vector field

  • 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


Maxwell lagrangian for massless vector field with source j

Maxwell Lagrangian for Massless vector field with Source Jμ

  • Suppose

  • The Euler-Lagrange equations yield

  • It follows (continuity equation)


Gauge theories2

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,


2 local gauge invariance

2.Local Gauge Invariance

  • 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 ! )


Gauge theories

  • 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”


Gauge theories

  • 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 


Gauge theories

  • 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 theories3

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 transformations of higher rank

Gauge Transformations of Higher Rank

  • 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


3 yang mills theory

3.Yang-Mills theory

  • 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.


3 yang mills theory1

3.Yang-Mills theory

  • 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


Gauge theories

  • 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


Gauge theories

Transformation

for

  • is not invariant under local SU(2) gauge transformations

  • Introduce vector fields,

    and covariant derivative

  • The resulting Lagrangian is


Gauge theories

  • 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


Gauge theories

  • 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 theories4

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,


4 chromodynamics

4.Chromodynamics

  • The free Lagrangian for a particular flavor

  • Use matrix representation

  • Here, the 3 masses are identical so


Gauge theories

  • 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


Gauge theories

  • is not invar. under local SU(3) transformation, so seek

  • Introduces 8 vector (gauge) fields

  • So

  • The resulting Lagrangian is

Lots of algebra


Gauge theories

(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


Gauge theories

“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 theories5

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,


5 the mass term

5.The mass term

  • 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


Gauge theories

  • 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.


Gauge theories

  • 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


Gauge theories

  • 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.


Gauge theories

[ graph of U(Φ)]

  • The third and fourth terms represent couplings of the form


Gauge theories6

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,


6 spontaneous symmetry breaking

6.Spontaneous symmetry-breaking

  • 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


Gauge theories

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


Gauge theories

  • [ 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


Gauge theories

  • 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!)


Gauge theories

  • 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 theories7

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,


7 the higgs mechanism

7.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.


Gauge theories

  • Replace partials in with covariant derivatives, etc.

  • Thus

  • Define the new fields

  • Lagrangian becomes


Gauge theories

  • 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


Gauge theories

  • We could choose a gauge in which so that

    This would make

    In this particular gauge, therefore

(Eliminates Goldstone Boson !)


Gauge theories

  • 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)


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