View source for Unveiling the Higgs mechanism to students

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 = \sqrt{{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 = \sqrt{{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