# Unveiling the Higgs mechanism to students

From Jonathan Gardner's Physics Notebook

Notes:

- Charged particle between two charged plates.
- Energy due to electric potential V:

- <math>U = qV\;</math>

- Energy due to electric field itself:

- <math>U = {\epsilon_0 \over 2} E^2 \mathcal{V}\;</math>

- Combine:

- <math>U = qV + {\epsilon_0 \over 2}E^2 \mathcal{V}\;</math>

- Divide by volume to get energy for a particular point

- <math>u = {U \over \mathcal{V}} = {q \over \mathcal{V}} V + {\epsilon_0 \over 2}E^2\;</math>

- Comment: Energy caused by:
- charge interacting with field potential
- field interacting with itself

- Note symmetry

- Note this is true for ALL conservative forces
- source of field times potential of that field
- a field times another field, possibly itself.

- Introduce Relativity:

- <math>u = {q \over \mathcal{V}}V + {\epsilon_0 \over 2}E^2 + {mc^2 \over \mathcal{V}}\;</math>

- Note that symmetry is broken!
- Mass is source of gravity
- But gravitational potential is nowhere to be found?!?
- What in the world? Our beautiful universe is now no longer beautiful.
- It looks like m is doing something by itself, while q needs the field / potential.

- We're going to "fix" this by breaking it.

- Note minimum energy: charge, field, mass all 0.
- This is important: We are going to see something where the things are not zero to get minimum energy.

3. Introducing the Higgs field

- Let's drop <math>mc^2</math>. We're going to try and get it back by introducing a new field.

- <math>u = {q \over \mathcal{V}} V + {\epsilon_0 \over 2}E^2\;</math>

- Let's invent a new field, <math>\phi</math> with its potential <math>\Phi</math>.
- It interacts with E, itself, and some new source "a".

- <math>u = {q \over \mathcal{V}} V + {\epsilon_0 \over 2}E^2 + {a \over \mathcal{V}} \Phi + gE\phi + g'\phi^2\;</math>

- Symmetry is preserved!

- Explain:
- q and <math>{\epsilon_0 \over 2}</math> are "coupling constants", describe how tightly the things interact.
- <math>{1 \over \mathcal{V}}</math> is the density of the particle in the volume, rather, the matter.
- a, g, and g' are the coupling constants of the particle, the electric-higgs interaction, and the higgs-higgs interaction.
- Note g' is not the derivative of g!

- Let's substitute to show the symmetry:
- <math>\mathcal{P} = {1 \over \mathcal{V}}</math>
- <math>F_p</math> for the potential of a field F
- <math>c_i</math> the coupling constants.

- <math>u = c_1 \mathcal{P} E_p + c_2 E E + c_3 \mathcal{P}\phi_p + c_4 E \phi + c_5 \phi \phi\;</math>

- Beautiful!

- Note: When all the c's are 0, the energy is at a minimum.

- Let's add a <math>\phi^4</math> term.
- energy density of <math>phi</math> is now:

- <math>u &= -g'\phi^2 + \phi^4</math>

- Where is the minimum?

- <math>\phi_0 = \sqrtTemplate:G' \over 2 \ne 0</math>

- Draw graph!

- We used to define a vacuum as "no fields, no particles".
- Let's redefine it as "minimum energy", ie, we sucked all the energy out we can.
- That means even if you remove everything, if you can make the energy even lower by adding a field or a particle, then you need to.

- Note! minimum energy is where <math>\phi = \sqrtTemplate:G' /over 2</math>
- So, in a vacuum, <math>\phi</math> is not zero!

- Our new energy density:
- Note: g' < 0!

- <math>u = {q \over \mathcal{V}} V + {\epsilon_0 \over 2}E^2 + {a \over \mathcal{V}} \Phi + gE\phi - |g'|\phi^2 | \phi^4\;</math>

- Note that adding <math>\phi^4</math> doesn't really break the symmetry. It is still the product of 2, or rather, 2 or more fields.
- In a vacuum, there is a Higgs field!

4. Higgs Boson

- Let's rewrite <math>\phi = \phi_0 + \eta</math>
- <math>\phi_0</math> is minimum field, the vacuum field, the field that exists when you have nothing.
- <math>\eta</math> is now the changing bit, ranges from 0 to negative to very big as we increase. We call this the Higgs field.

- Now we need to rewrite <math>\Phi = \Phi_0 + \Phi_1</math>
- <math>\Phi_0</math> is potential of Higgs field at its minimum
- <math>\Phi_1</math> is the changing bit.

- Now we have:

- <math>u = {q \over \mathcal{V}} V + {\epsilon_0 \over 2}E^2 + {a \over \mathcal{V}} (\Phi_0 + \Phi_1) + gE(\phi_0 + \eta) - |g'|(\phi_0 + \eta)^2 + (\phi_0 + \eta)^4\;</math>

- Let's talk about 'a'.
- Now we have
*something*for the mass to interact in a vacuum (because we've redefined what a vacuum is.) - We want:

<math>{mc^2 \over \mathcal{V}} = {a \over \mathcal{V}} (\Phi_0 + \Phi_1)</math>

- So we get:

<math>m = {a \over c^2} \Phi_0</math>

- Progress!

- But now we have this leftover term:

- <math>{a \over \mathcal{V}}\Phi_1</math>

- That represents the interaction of a particle with the field <math>\eta</math>.
- We can calculate a:

- <math>a = {mc^2 \over \Phi_0}</math>

- Same story for the Electric Field
- If g is not 0, then we have a mass for the Electric Field. Meaning, if we have an Electric Field with no charge or masses present, then the Electric Field -Higgs Field interaction looks like a mass-Higgs interaction. We know this isn't the case, Electric Fields do not have mass, so we must set g = 0.

- The final two terms can be expanded:

- <math>

\begin{align} -|g'|\phi^2 + \phi^4

&= -|g'|(\phi_0 + \eta)^2 + (\phi_0 + \eta)^4 \\ &= -|g'|(\phi_0^2 + 2\phi_0\eta + \eta^2) + (\phi_0^4 + 4\phi_0^3\eta + 6\phi_0^2\eta_2 + 4\phi_0^3\eta + \eta^4) \\ &= -|g'|\phi_0^2 -2|g'|\phi_0\eta -|g'|\eta^2 + \phi_0^4 + 4\phi_0^3\eta + 6\phi_0^2\eta_2 + 4\phi_0^3\eta + \eta^4 \\

\end{align} </math>

- <math>-|g'|\phi_0^2</math> is a constant, so it's irrelevant. That is, we can change the energy up or down everywhere and it doesn't mean a thing.
- <math>-2|g'|\phi_0\eta</math> is a mass term for the Higgs field.
- <math>-|g'|\eta^2</math> is the Higgs field interacting with itself.
- THe higher order terms are just the Higgs field interacting with itself or the vacuum. They are really combinations of the lower-order terms.

- Review:
- We said that the vacuum is not where nothing exists, but where the energy is minimized.
- We introduced a new field, the Higgs field.
- The Higgs field doesn't interact with the electric field.
- The Higgs field does interact with itself and with the vacuum.
- The Higgs field does interact with mass.

- This is not a very good description of what's going on
- It is the best you are going to get without doing Quantum Field Theory.
- It is a pretty accurate