Physics Notes: The Standard Model
Q: Suppose that our potential function (for the Higgs field example)
actually had multiple minima.
Things like charge are still quantized, so would this generate a family
of particles that differ only in their masses?
A: A good question, but the observed charges would still vary because of renormalization. We will get to this question eventually.
[This was my question. It seems like a muon electric charge
would be renormalized slightly differently as well, just due to the mass
difference.]
Q: How do we actually detect quarks?
A: We can hit a proton with a high-energy electron. The electron can hit one of the quarks in the proton and eject it, trailing a gluon flux tube behind. If there is low kinetic energy, then it will be pulled back, but if the kinetic energy is high, then it will stretch out, with the energy stored in the gluon taffy/rubber band growing linearly with the distance. At some point it is energetically favorable for the rubber band to break on a quark/anti-quark pair. [I suppose there are volunteer virtual quark pairs around that the flux tube could break on.]
The produced quark pairs are of the same type as the ejected quark, so hitting different quark types produces jets with different properties. The mesons furthest away from the proton are moving fastest. This is exactly what is seen in the lab.
There were also experiments at SLAC with electron positron collisions. These collisions create a high energy virtual photon from which many things can be produced.
Side note. It is not possible to create a single real photon from this collision because energy and momentum cannot be simultaneously conserved. This is easy to see in the 0 momentum frame for the e+ and e- . If a single photon were produced the total momentum would change to the momentum of the photon which is not 0.
What can be produced from this collision? Particle anti-particle pairs that have charge.
The amplitudes can now be computed with high accuracy for different collision energies.
The Nobel committee accepted meson jets produced at SLAC by collisions of electrons with protons as solid evidence for quarks.
Notice that there is no energy
from a uniform shift
of a space coordinate because F contains only derivatives.
If our Lagrangian was simply
Then our particle would be massless because F contains only derivitives.
For our field and potential function
we introduced the use of a covariant derivative.
Switching to our based
field with fixed f :
In these derivatives we can see that we have the covariant
transformation of with
rotation
. We can just set
and
move on with
.
After this we will drop the Ō
So
If is
shifted everywhere you have a mass.
The
term doesnÕt contribute because it
contains only derivatives.
[The shift in A would be a shift in
the field value. The positive
energy in small shifts away from 0 would create a restoring force and a
harmonic oscillator with ground state energy. Shifts along would not
affect F. This threw me a
bit because I was looking for something like
as the ŅshiftÓ based on the examples.]
Prof Susskind gave an example of a crystal where the lattice points are filled in with alternating charges. If performs a shift where the negative charges are uniformly shifted left and the positive charges are uniformly shifted right by a small amount, then this obviously adds energy.
In lecture 8 gauge bosons got their mass from the Higgs field. Now the question is how do fermions get their mass?
Some History:
Reflection appears to be a symmetry where you flip one of the
coordinates. (It could be about
any plane). This symmetry appears
to be valid for QED and quarks.
However, going back to experiments as early as the 40Õs it became clear
that reflection was not a good symmetry for the weak force.
Left and right-handed electrons are equivalent in QED. In the Weak decay that produces electrons, they always come out left-handed! There is something asymmetric.
Q: does this left-handedness also apply to quarks?
A: yes
There were a bunch of questions about handedness. A fundamental problem here is that electrons have mass and the right concept is something called chirality which is the same as handedness for massless particles, but is a more complicated notion for particles like the electron that have mass. Susskind wanted to avoid wasting time on the difference in this lecture.
[http://en.wikipedia.org/wiki/Chirality_(physics) : chiral phenomenon is one that is not identical to its mirror image (see Chirality). The spin of a particle may be used to define a handedness (aka chirality) for that particle. A
symmetry transformation between the two is called parity. The action of parity
acting on a Dirac fermion is called chiral symmetry.
An experiment on the weak decay of cobalt-60 nuclei carried out by Chien-Shiung Wu and collaborators in 1957
demonstrated that parity is not a symmetry of the universe.
]
We could imagine a world where left-handed electrons have charge, but the right-handed ones donÕt.
Lets use the Dirac Equations for left and right movers as a model.
is
a 4 element vector whoÕs elements correspond to the different states.
Only derivatives, so no
mass. If we add mass
The
matrix
interchanges (mixes) the 4 eqn.
Now letÕs split our left and right handed versions so we
have two equations (really 8 because of vector ).
(Directions reversed, so sign of space
reversed)
Now suppose that has charge but
does
not. [I presume the charge referred to here is the Weak
charge]
What breaks?
Charge conservation would break because complex rotations
(multiplication by would
fail to be a symmetry because
would not get a factor of
).
Note: Suppose that there was only ,
then if it had a mass, then you can bring it rest and reverse direction.
How could it get a mass? Add a coupling to a boson field with charge like the electron. [The simplified Higgs field was complex field in our example, which has a kind of quantized angular momentum in the field value, which Susskind identified with some type of charge]
Then the U(1) transformation would look like:
,
Suppose that is
our Higgs field.
where we transform
away. Then our equations become:
and
The mass of our fermion is then where ŅaÓ stands for the fermion of
interest. All fermions are
supposed to couple to the same Higgs field. The values
are
the Yukawa coupling constants and control the masses of different leptons and
quarks.
To get a feel for the size of the couplings, if the electron
is ~ 0.5MeV, then , for the top quark at 170GeV
.
The Higgs field isnÕt really frozen so our two right and terms would really be:
where the H field represents the
perturbation from the ground state.
From this we can see that the same
coupling
constants that determine the mass of leptons and quarks also govern the rate of
production of Higgs bosons.
We want to do experiments (with the LHC) to measure the
couplings to the Higgs boson. [I imagine that there is a connection to the earlier
discussion on the set of possible outcomes for electron positron
annihilation. By sweeping
the energy of collision up, higher mass outcomes get added to the set of
outcomes, eventually coming to outcomes that produce Higgs bosons and giving
enough information to pick out the individual coupling constants].
An interesting fact then is that all known particles get their mass this way, by coupling to the Higgs field.
Q: what about neutrinos?
A: We canÕt differentiate their velocity from c. There are no right-handed neutrinos [in the standard model]. Now we know that they do have a mass because we know that they oscillate between types. They do have a weak charge. Given a mass term, they may be able to oscillate between left and right handed (chirality) types. The anti particle of the neutrino is right handed. If they can mix, then because they are neutral they may form a particle known as a Majorana particle http://en.wikipedia.org/wiki/Majorana_particle, which is its own anti-particle.
All masses of known particles go back to f – Could other particles exist with higher mass? Sure – If a particle had a weak charge for both left and right movers, then it could independently have any mass. The natural mass scale for particles is >> f.
Small mass particles (all our known particles) get their mass via symmetry breaking.
Q: What is the energy of LHC compared to cosmic rays?
A: Cosmic rays can have energies as high at 1021 eV. The LHC gets some advantage by colliding two beams of particles, but the LHC energies are still less.