The Higgs Mechanism, Electroweak Theory, Fermion Masses and the Standard Model

Question 1

Spontaneous Symmetry Braking

Definition

Answer 1

• symmetry breaking refers to a reduction or total braking of symmetry
• spontaneous symmetry braking occurs when the Hamiltonian for a system has a particular symmetry but that symmetry is not shared by the ground state of the system

Question 2

Spontaneous Symmetry Braking

Ferromagnetism Example

Answer 2

• at high temperatures, a bar of ferromagnetic material will have no overall magnetism
• this is because the magnetic domains in the material are orientated in all different directions, since there is no preferred direction the Hamiltonian is system is symmetric under rotations
• but as the temperature falls, at a critical value, the domains begin to line up as this is a lower-energy state
• since the Hamiltonian is symmetric, there is no preferred direction for this to occur in but once it happens the state of the system is no longer symmetric it has ‘chosen’ a direction
• this is why such processes are described as spontaneous

Question 3

What is the Higgs mechanism?

Answer 3

-a way of spontaneously braking a symmetry

Question 4

Higgs Mechanism Toy Model

Outline

Answer 4

• consider a toy model of a scalar particle (or scalar field in the quantum regime) coupled to the electromagnetic field
• this requires the scalar to be complex-valued (since a real-valued scalar carries no current)
• and that there is a local U(1) symmetry
• the scalar is capable of self-interactions in which 4 such particles interact at a point

Question 5

Higgs Mechanism Toy Model

Equation of Motion

Answer 5

(∂² + μ²) = -λ(φφ)φ

-where μ is the scalar particles mass, λ is the coupling constant which describes the strength of self-interaction

Question 6

Higgs Mechanism Toy Model

Potential Energy Density

Answer 6

V = μ²φφ + λ(φφ)²

-where μ is the scalar particles mass, λ is the coupling constant which describes the strength of self-interaction

Question 7

Higgs Mechanism Toy Model

Stationary Points

Answer 7

-scalar particles are considered as quanta of the scalar field
-since the potential is symmetric about the origin only the magnitude of φ needs to be considered when finding stationary points
-differentiate V with respect to |φ|
=>
|φ|=0, a maximum and therefore unstable
OR
|φ| = √[-2μ²/4λ], minima
-note that λ can’t be negative as potential energy wouldn’t be bounded leading to |φ|->∞
-there are a ring of such minima around the origin

Question 8

Higgs Mechanism Toy Model

Ground State

Answer 8

• the ground state is in an antisymmetric configuration and the system will be forced into a state with a non-zero background value of φ
• since this background state singles out a particular direction, the U(1) symmetry we began with has been lost
• if we were to live in such a ground state, we would not ‘see’ this background since it is the same everywhere
• instead we would take φ’=φ-v where v is the background value of φ, the vacuum expectation value

Question 9

Higgs Mechanism Toy Model

Small Fluctuations Around Ground State

Answer 9

-for small variations in φ around the background value can parameterise:
φ = (v+h)e^(iξ)
-where h is the radial variation in φ and ξ gives the angular variation

Question 10

What is the electromagnetic current due to a scalar particle?

Answer 10

j^μ = qi(φ∂^μφ - φ∂^μφ) - 2q²A^μφ*φ

Question 11

Maxwell Equation for a Scalar Particle

Answer 11

-sub in current for scalar particle:

∂²A^μ - ∂^μ∂.A = qi(φ∂^μφ - φ∂^μφ) - 2q²A^μφ*φ

Question 12

Higgs Mechanism Toy Model

Maxwell Equation

Answer 12

-take Maxwell equation for a scalar particle
-sub in:
φ = (v+h)e^(iξ)
-since ξ only appears with a derivative: it is of the right form to act as a gauge transformation on the photon
-can ‘gauge away’ this object, effectively absorbing it into the photon
-leaves two interaction terms with residual scalar h on the RHS
-and an additional terms on the LHS for mass where m=qv√2

Question 13

Higgs Mechanism Toy Model

Mass Term for Photon?

Answer 13

• we find there is a mass term for the photon due to the constant background value of φ
• massive vector particles have an additional polarisation over massless particles so the photon appears to have gained a degree of freedom
• but this is ok, remember that the photon absorbed the ξ dependence so the number of degrees of freedom for the theory has remained the same
• the ξ has provided the longitudinal polarisation state
• this is just an example of the mechanism, photons don’t actually have a mass
• but the same mechanism is responsible for the masses of weak bosons

Question 14

Higgs Mechanism

What is the toy model for??

Answer 14

• it is the simplest version of the Higgs mechanism where we break a U(1) symmetry leaving no symmetry at all
• what we actually want to use the Higgs mechanism for is to break a U(2) symmetry such that we are left with the appropriate low energy theory

Question 15

Higgs Mechanism

Plan

Answer 15

• start with SU(2)lxU(1)y and break to a different SU(1), the electromagnetic symmetry, SU(1)em
• we only want to put left chiral components into the doublet since only left chiral objects interact through the weak force
• if we want independent left an right chiral parts, we need massless fermions
• this also solves the problem of different masses of electron and neutrino
• we will get mass back later, but still have the problem of different charges

Question 16

Electroweak Theory

Answer 16

• introduce two new charges hypercharge and weak isospin
• weak isospin distinguishes individual parts of a flavour doublet e.g. electron and neutrino, up and down
• hypercharge gives each member of a weak doublet the same isospin

Question 17

Weak Isospin

Answer 17

T
-behaves similarly to isospin but only acts on SU(2) doublets
T=0 for anything right chiral
T=1/2 for SU(2) doublet (left chiral)
-the third component of weak isospin, T3, is +1/2 for vl, ul,… and -1/2 for el, dl,…

Question 18

Hypercharge

Answer 18

Y = 2(q-T3)

-left and right chiral parts have different values of Y and T3 but both compnents of an SU(2) doublet have the same Y

Question 19

Higgs Mechanism

Doublet

Answer 19

• introduce doublet of complex scalars with Y=+1 and weak isospin 1/2
• i.e. a doublet under SU(2) symmetry and also transforms under new U(1) symmetry SU(1)y

Question 20

Higgs Mechanism

Gauge Bosons

Answer 20

-have gauge bosons W1,W2,W3 from SU(2) and B from U(1)y

Question 21

Higgs Mechanism

Observable Particles

Answer 21

-give scalars a vacuum expectation value that breaks symmetry, this leaves a U(1) symmetry that is not U(1)y
-and observable particles with mass eigenstates:
W± = W1 ± iW2, same mass
Zo = cos(θw)W3 + sin(θw)B, larger mass
-and one massless boson corresponding to the remaining U(1) symmetry:
A = -sin(θw)W3 + cos(θw)B
-so A is a photon and the U(1) symmetry us electromagnetic, U(1)em

Question 22

Higgs Mechanism

Weak Mixing Angle

Answer 22

θw ~ 28’

• notice the electromagnetic and weak interactions are two sides of the same theory
• this is why unified interactions are known as the electroweak theory

Question 23

Electromagnetic Coupling Constant

Answer 23

e = gw’ cos(θw) = gw sin(θw)

• where gw’ is the weak coupling constant and gw is the U(1) hypercharge coupling constant
• note this implies gw>e i.e. weak interactions are stronger than electromagnetic?!

Question 24

Why are weak interactions not stronger than electromagnetic?

Answer 24

• weak interactions are not inherently weak, gw>e
• their apparent weakness is a result of the large mass of the exchange particles that mediate them
• if the energy scale is much less than the W± mass, then the amplitude for the process is suppressed due to the propagator ∝ 1/[p²-Mw²]
• so if the exchange particle mass Mw is large, the propagator is small and the process is unlikely

Question 25

Higgs Mechanism

The Higgs Boson

Answer 25

• at high energy, the scalar doublet has 12 DoF and the four gauge bosons W1, W2, W3 and B are massless
• at low energy, there are three massive gauge bosons which have stolen DoF from the scalar leaving 1DoF
• this corresponds to 1 real scalar particle, the Higgs boson
• the Higgs boson is what is left over after the Higgs field gives the gauge bosons their mass

Question 26

Higgs Mechanism

Fermion Mass Problem

Answer 26

• so far have constucted a theory of weak interactions that accounts for their left-chiral nature and allows for massive gauge bosons
• and solves the problem of weakly coupled particles having different properties
• but we observe fermions to be massless where as the theory so far assumes them to be massless

Question 27

Higgs Mechanism

Energy Scale

Answer 27

• at high energy, above the electroweak breaking scale, the two particles in the weak doublet have the same properties (same hyper charge and both massless)
• below the electroweak breaking scale, their properties are different but this doesn’t matter since there is no longer any symmetry that says they should be the same

Question 28

Higgs Mechanism
Giving Fermions Mass
Interaction Equation

Answer 28

-consider a model consisting of one left-chiral and one right-chiral fermion, fl and fr as well as a complex scalar, φ
-Dirac equation for these particles:
i∂/ fl = 0 and i∂/ fr = 0
-what is the most general interaction we could have?
-try
i∂/ fl = λφ fl
-where λ is the coupling constant
-this breaks Lorentz symmetry, LHS is right chiral, RHS is left chiral
-try
i∂/ fl = λφ fl^c
-both sides now right chiral, BUT transform differently under U(1)y
-only option is:
i∂/ fl = λφ fr

Question 29

Higgs Mechanism
Giving Fermions Mass
Vacuum Expectation Value

Answer 29

-have
i∂/ fl = λφ fr
-give scalar field, φ, a vacuum expectation value:
φ = (v+h) e^(iξ)
-gauge away ξ 
=> φ = (v+h)
-sub in:
i∂/ fl = λv fr + λr fr
-the h terms allow the fermions to interact with the new field while the v terms are of the right form to give the fermions mass

Question 30

Higgs Mechanism

Mass and Strength of Interaction with the Higgs Boson

Answer 30

-as a fermion travels along, it interacts with the Higgs field
-each time it interacts it switches to its opposite chiral state causing a slight change in direction
-the particle follows a zig zag path at the speed of light since it is massless
-when the particle is observed on a macroscopic scale the zig zags can’t be seen it just appears to be travelling along a straight path but slower
m∝λ
-higher strength of interaction with the Higgs field means more zig zagging so slower linear motion, so greater mass => heavier particles interact more strongly with the Higgs boson

Question 31

The Standard Model

Description

Answer 31

-consists of a gauge group and Poincare group and some fields that form representations of the two groups:
SU(3)c x SU(2)l x U(1)y
-where c indicates colour, l indicates left chiral and y indicates hypercharge

Question 32

The Standard Model

Fermion Representaitons

Answer 32

-fermion representations for one generation:
Ql = (3,2) +1/3
ur = (3,1) +4/3
dr = (3,1) -2/3
Ll = (1,2) -1
er = (1,1) -2
-where L means leptons

Question 33

The Standard Model

Right Chiral Neutrino

Answer 33

-note that the right chiral neutrino is missing from the standard model, with discovery of neutrino oscillations we now know that neutrinos have mass so must include right chiral component:
νr = (1,1) 0