Gauge Theory, Non-Abelian Gauge Theory and the Weak Interaction

Question 1

Global Transformation

Definition

Answer 1

-a transformation that is the same everywhere in spacetime

Question 2

How do we show that the Dirac equation is invariant under a global transformation?

Answer 2

-the most general transformation that can be applied to the spinor in the Dirac equation is:
Ψ -> e^(iα)Ψ
-where α is a constant
-sub in and show invariance

Question 3

Symmetry Group

Answer 3

• a set of transformations under which an equation remains invariant
• several such transformations in sequence still retains invariance

Question 4

Symmetry of the Dirac Equation
The U(1) Group

Answer 4

-the Dirace equation is invariant under the transformation:
Ψ -> e^(iα)Ψ
-here α can be any real number and the exponential can be any complex number of modulus 1
-several such transformations in sequence still retains invariance
-since this group has a continuous parameter labelling its elements, in addition to a group structure it also has the structure of a smooth manifold, a Lie group
-in particular, since the transformations are unitary and only have one degree of freedom, this group is known as the U(1) group

Question 5

Symmetry of the Dirac Equation

Global Symmetry

Answer 5

• since α is constant, we are applying the same transformation to Ψ at all spacetime points, the transformation is global
• we can say that the Dirac equation has global U(1) symmetry

Question 6

Local Transformation

Definition

Answer 6

• a transformation is local if it is a function of spacetime coordinates
• i.e. if the transformation applied is different at different points in spacetime

Question 7

Does the Dirac equation have local U(1) symmetry?

Answer 7

-apply transformation:
Ψ -> e^(iα(x))Ψ
-find that the U(1) symmetry in the Dirac equation IS NOT local

Question 8

Can the Dirac equation be modified to accommodate local U(1) symmetry?

Answer 8

-the reason that the Dirac equation is not locally U(1) symmetric is that the derivative ∂μΨ does not transform the same way as Ψ itself under a U(1) transformation
-if we replace this derivative with a derivative that does transform the same way, a ‘gauge-covariant’ derivative
-then we do have local U(1) symmetry is then:
Aμ’ = Aμ - 1/g ∂μ α

Question 9

What are the conditions for local U(1) symmetry of the Dirac equation?

Answer 9

-once the derivative has been switched for a gauge-covariant derivative, we do have local U(1) symmetry as long as:
Aμ’ = Aμ - 1/g ∂μ α
-that is, we can accommodate local U(1) symmetry in the Dirac equation as long as the fermion described couples to some other vector quantity with gauge symmetry
-e.g. coupled to photon

Question 10

Gauge-Covariant Derivative

Definition

Answer 10

Dμ = ∂μ + igAμ(x)
-here Aμ(x) is some vector and g is a constant
-we require:
DμΨ -> e^(iα(x))DμΨ
when
Ψ -> e^(iα(x))Ψ

Question 11

How can we produce gauge-covariant quantities?

Answer 11

-acting twice with a gauge-covariant derivative still produces something gauge-covariant:
DμDνΨ -> e^(iα(x))DμDνΨ
-any linear combination of such terms is also gauge-covariant, in particular the commutator [Dμ,Dν]

Question 12

Is the field-strength tensor gauge-covariant?

Answer 12

-yes, it can be shown that:
[Dμ,Dν] = igFμν
-this in turn means that the Maxwell equation can be included as an equation of motion for Aμ without breaking U(1) symmetry

Question 13

Does Aμ have mass?

Answer 13

• from the Proca equation we know that massive vector particles do not have gauge symmetry
• so Aμ is a massless vector field that is invariant under gauge transformations and obeys the Maxwell equation
• a photon matches this description, so we can accommodate local U(1) symmetry into the Dirac equation if and only if the fermion we wish to describe is coupled to an electromagnetic field

Question 14

Conserved Current for a Dirac Particle

Answer 14

j^μ = Ψ_ γμ Ψ

Question 15

Write down a fully local U(1) symmetric theory of a fermion and a vector particle

Answer 15

(iD/ - m)Ψ = 0
∂μ F^μν = qj^ν = q Ψ_ γμ Ψ
-these are (once again) the equations for quantum electrodynamics, this time arrived at using only the requirement of local U(1) symmetry

Question 16

Global Colour Symmetry

Answer 16

• if we were to replace all the red quarks with blue and vice versa, there would be no effect on the universe
• there is a global symmetry in how the colours are assigned to quarks
• we can take this further, replacing all blue quarks with a superposition as long as a corresponding transformation is made to the other colours to retain orthonormality
• in fact we can transform the ‘vector’ of quark colours (r,b,g) with some 3x3 matric M as long as it retains orthonormality, i.e. as long as M is unitary
• if we also have det(M)=1, these transformations form a group, SU(3)

Question 17

Unitary Matrix

Definition

Answer 17

M† = M^(-1)

Question 18

How many degrees of freedom are there in choosing M?

Answer 18

-3x3 matrix with complex entries => 18 real numbers
-BUT
-unitary => 9 entries fixed
-det(M)=1 => 1 entry fixed
=> 8 degrees of freedom

Question 19

Infinitessimal Transformations

Answer 19

-since the 8 degrees of freedom for M are continuous parameters αi, we can find transformatino matrices that are only infinitesimally removed from the identity
-in fact, we can find 8 such independent matrices, Ti
-a general infinitesimal transformation can then be constructed as a linear combination of these:
𝛿M = 𝛿 αi Ti

Question 20

Constructing Matrices from the Infinitesimal Transformations

Description

Answer 20

• to construct any matrix in the group, choose a ‘direction’ in the space group of transformations and a ‘distance’ from the identity
• we can construct a finite transformation from repeated application of infinitesimal transformations

Question 21

Constructing Matrices from the Infinitesimal Transformations

M(α1)

Answer 21

-assuming αi=0 for i=2,…,8 we can see the transformation M(α1) can be written:
M = lim ( 1 + α1/N)^N
= e^(α1T1)
(where the limit is taken as N->∞)
-since each factor represents an infinitesimal change in the T1 direction
-this limit gives the exponential function

Question 22

Constructing Matrices from the Infinitesimal Transformations
any SU(3) matrix

Answer 22

-any member of SU(3)can be written as:
M = e^(i αiTi)
-where the repeated indices indicate summation
-the 8 independent infinitesimal transformation matrices are called generators for the group, and must by linearly independent, Hermitian and traceless

Question 23

Generators and Group Structure

Answer 23

-since the entire symmetry group is generated by the generator matrices, most of the structure of the group can be determined by looking at how they interact with each other
-looking at the commutators of the generators we define a set of structure constants, fijk, for the group such that:
[Ti,Tj] = 2i fijk Tk

Question 24

Albeian and Non-Albeian

Definition

Answer 24

• if the structure constant of a group are non-zero, the group elements will not generally commute with each other, in this case the group is said to be non-Abelian
• Abelian groups are, by definition, groups in which group operations commute

Question 25

Choice of Generators

Answer 25

• as long as the matrices meet the criteria ( linear independence, Hermitian, traceless) they may be chosen arbitrarily
• each choice of generators is a representation e.g. the Gell-Mann matrices can be used to represent the generators of the SU(3) group

Question 26

Pauli Matrices and SU(2)

Answer 26

• the Pauli matrices are possible set of generators for the SU(2) group
• restricting only to T1, T2 and T3 of the Gell-Mann matrices would give SU(2) transformations in the first two components of the vector

Question 27

Dirac equation for non-interacting quarks?

Answer 27

-if quarks did not interact, we could write down the Dirac equation for each colour as:
(i∂/ - m)𝜱 = 0
-where 𝜱 is a 3x1 vector with entries ψr, ψg, ψb
-this is invariant under transformation:
𝜱 -> m𝜱 = e^(iαiTi)𝜱
-hence we have invariance under global SU(3) symmetry
-but NOT local SU(3) symmetry

Question 28

Introducing local SU(3) symmetry to the Dirac equation for non-interacting quarks

Answer 28

-introduce a covariant derivative Dμ such that:
Dμ 𝜱 -> e^(iαiTi) Dμ 𝜱
-so let:
Dμ = ∂μ + i gs Aμ^i Ti
-where gs is the strong force coupling constant and repeated i index indicates summation over i=1,…,8
-construct the commutator [Dμ,Dν]
-for the earlier case of the photon, [Aμ,Aν] vanished, but in this case it is non-zero and we have to define Gμν^i
-we now have a theory of 3 quark colours that interact with 8 independent massless vector particles with gauge invariance, these are the gluons and we have produced the strong interaction

Question 29

Equation of Motion for Gluons

Answer 29

Dμ G^iμν = j^νi
-here j^νi is the source term or current for the ith gluon just as the electric current j^μ was the source for photons in the EM case
-and
G^iμν = ∂μAν^i - ∂νAμ^i - 2gsf^ijkAμ^jAν^k
-this equation of motion for gluons tells us that gluons interact among themselves

Question 30

Why do gluons interact with each other?

Answer 30

• gluons themselves carry colour currents (or colour charges)
• in fact, it is the colour of gluons that differentiates the eight independent bosons
• we are free to assign these colours however we want as long as the eight gluons are linearly independent
• BUT since the eight independent gluons arise from the eight generators of the gauge group, in specifying a representation for the generators we have also fixed the gluons

Question 31

Colour Factor

Answer 31

-since multiple gluons could play the same role in an interaction, when computing amplitudes in Feynman diagrams, it is necessary to sum over all possibilities
-since each of the possibilities gives a very similar diagram, we calculate one and then multiply by the colour factor:
CF = 1/2 Σ C1C2
-where the sum is over the possible diagrams
-C1 and C2 are the factors due to colour arising at each vertex in the diagram

Question 32

Interactions Between Colours

Answer 32

-like colours repel and different colours attract
-the strength of attraction between different colours is half the strength of repulsion between between like colours
-can also show that the attraction between e.g. red and anti-red is equal in magnitude to the repulsion between red and red
-the only thing as attractive as anti-red to a red quark is a combination of blue and green
-once all three colours are combined or a colour is combined with its anti-colour we find they are in a colour singlet state:
1/√3 (rr_ + bb_ + gg_)
-from the outside, the singlet state appears to have no colour charge so does not interact via the strong force, this is precisely the reason that quarks form mesons and baryons
-there are also interactions that swap colours

Question 33

SU(3) Symmetries of the Standard Model

Answer 33

• the standard model appears to have two SU(3) symmetries, one for flavour acting on the triplet u,d,s and one for colour acting on r,b,g
• but there is an important difference between them
• the colour SU(3) is an exact symmetry in the sense that red, blue and green quarks really are indistinguishable in terms of their interactions
• the flavour SU(3) symmetry is only approximate since the three quarks involved have different electric charges and masses so are distinguishable from one another

Question 34

Flavour Symmetry

Answer 34

• the fact that SU(3) symmetry is a reasonably good approximation is evident in the fact that only the hadrons with light quarks (u,d,s) are arranged as representations of SU(3)
• we know now that there are additional quarks c,b,t but the mass difference between these and the light quarks means that any approximation to SU(4) or higher symmetry group is poor
• this is why we consider light-quark hadrons to be composed of linear combinations of light quarks whereas hadrons with c quarks are considered as having a definite quark content
• the fact that flavour symmetry is approximate does not mean that it isn’t useful, in fact as we probe higher and higher energies, the mass of individual quarks becomes negligible compared with the energy scale and the approximation imprpves

Question 35

Complications of the Weak Interaction for a Gauge Symmetry Approach

Answer 35

1) the weak interaction is chiral, it is found experimentally to couple to left-chiral spinors (or left-chiral components of spinors)
2) bonsons that mediate the weak force are massive and arguments for EM and SNF dictate that gauge bosons must be massless
3) the different particles that can interact with each other through the weak force are very different from each other e.g. the electron and the neutrino are very different with different charge and mass but still interact via the weak force

Question 36

How to overcome the complications of the weak interaction?

Answer 36

• the solution is to notice that at very high energies, the electron and the neutrino do begin to behave very similarly
• the strong force becomes the dominant interaction, and neither of these particles participate in it
• if we were only aware of the electron and neutrino through very high-energy experiments, we might well say they look approximately identical
• so, we assume there is an SU(2) symmetry at high energy (or short distance scales) that gets spontaneously broken at lower energies