The Higgs Mechanism, Electroweak Theory, Fermion Masses and the Standard Model Flashcards
Spontaneous Symmetry Braking
Definition
- 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
Spontaneous Symmetry Braking
Ferromagnetism Example
- 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
What is the Higgs mechanism?
-a way of spontaneously braking a symmetry
Higgs Mechanism Toy Model
Outline
- 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
Higgs Mechanism Toy Model
Equation of Motion
(∂² + μ²) = -λ(φφ)φ
-where μ is the scalar particles mass, λ is the coupling constant which describes the strength of self-interaction
Higgs Mechanism Toy Model
Potential Energy Density
V = μ²φφ + λ(φφ)²
-where μ is the scalar particles mass, λ is the coupling constant which describes the strength of self-interaction
Higgs Mechanism Toy Model
Stationary Points
-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
Higgs Mechanism Toy Model
Ground State
- 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
Higgs Mechanism Toy Model
Small Fluctuations Around Ground State
-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
What is the electromagnetic current due to a scalar particle?
j^μ = qi(φ∂^μφ - φ∂^μφ) - 2q²A^μφ*φ
Maxwell Equation for a Scalar Particle
-sub in current for scalar particle:
∂²A^μ - ∂^μ∂.A = qi(φ∂^μφ - φ∂^μφ) - 2q²A^μφ*φ
Higgs Mechanism Toy Model
Maxwell Equation
-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
Higgs Mechanism Toy Model
Mass Term for Photon?
- 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
Higgs Mechanism
What is the toy model for??
- 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
Higgs Mechanism
Plan
- 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
Electroweak Theory
- 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
Weak Isospin
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,…
Hypercharge
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
Higgs Mechanism
Doublet
- 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
Higgs Mechanism
Gauge Bosons
-have gauge bosons W1,W2,W3 from SU(2) and B from U(1)y
Higgs Mechanism
Observable Particles
-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
Higgs Mechanism
Weak Mixing Angle
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
Electromagnetic Coupling Constant
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?!
Why are weak interactions not stronger than electromagnetic?
- 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
Higgs Mechanism
The Higgs Boson
- 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
Higgs Mechanism
Fermion Mass Problem
- 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
Higgs Mechanism
Energy Scale
- 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
Higgs Mechanism
Giving Fermions Mass
Interaction Equation
-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
Higgs Mechanism
Giving Fermions Mass
Vacuum Expectation Value
-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
Higgs Mechanism
Mass and Strength of Interaction with the Higgs Boson
-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
The Standard Model
Description
-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
The Standard Model
Fermion Representaitons
-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
The Standard Model
Right Chiral Neutrino
-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
