Physics Notes:  The Standard Model

 

Lecture 6: Feb 15, 2010                                                                 Back to PHY30

 

Before Class Questions

 

Is there rhyme or reason to particle masses?

Not really, but sometimes we can relate mass to something else we donÕt understand.   This reduces the number of parameters in the model.

There are about 19 parameters in the Standard Model including coupling constants, masses and a few more for the Higgs field.

Add in super symmetry and the number of parameters goes up to about 100.   However, the new parameters donÕt affect predictions for known particles.

Why is the electron magnetic moment/angular momentum not exactly 2?

DiracÕs original model predicted a value of 2.  The difference from that value is well understood as an interaction between the electron and virtual particles around it.  [This appears to be one of the major successes of QED]

In the last lecture the statement was made that when an electron is stopped (for instance at a screen), there is a chance that a photon is emitted.   If so, then when is a coupling constant applied?   Does it have something to do with acceleration of the particle?

An electron is constantly emitting and absorbing photons around it.  At rest, it is a perfect juggler, catching every photon it emits.  When you bump an electron hard enough you potentially prevent the re-capture of such a photon.   The chance that there is a photon to drop is the fine structure constant. 

[Now that IÕve asked this question, I have a followup.   Is there a spectrum of expected energies of the photons?   Would this spectrum be a consequence of the  rule?]

Why is deuteron stable given then instability of the neutron?

 

Short answer:  binding energy makes decay unfavorable

 

Lecture

 

SU(3) is our color symmetry.  Prof Susskind reiterated the notion that the symmetry has to be applied to all color carrying particles at the same time to leave physics unchanged.   This is similar to the notion that if one said that physics is unchanged by a rotation of coordinates that all particles have to have their coordinates rotated consistently for the physics to remain unchanged.    The color ÒcoordinatesÓ are transformed everywhere.

Repeat of our fermion table:

 

The SU(2) ÒFlavorÓ symmetry is between the pairs of particles  (u d) (c s) (t b) and lepton pairs.   [This symmetry isnÕt perfect because the masses and charges of the particles in the pairs are not the same.  The SU(3) symmetry does appear to be perfect]

If you add the photon, then you have the phase change on emission of the photon.

This gives us a product space:

SU(3) X SU(2) X U(1)

8 generators, 3 generators, phase

And we will build our gauge theory based on this product space.

W bosons were discovered relatively late.  They were postulated in the 60Õs and discovered in 1983?

Review of Coupling Constants

 

U(1)    e        

SU(3)     emission of gluon.   Probability  

SU(2)     Probability similar to electron 

            Why is this so weak?

 

Take the decay of the neutron:

What is the probability of this decay?  We can start with  .  One factor is for the emission of the W and one for the decay to electron and anti-neutrino.   This is not the whole story.   We have to include the propagator for the . 

Propagators

 

Propagators are functions of the proper time and the mass of the particle.

 

Where does the roll off happen?   Use dimensional analysis to approximate.

In units where ,

So   

Add back in the factors and you get

The heaver the boson, the shorter the distance over which it has appreciable probability.

 

In experiments, we usually know the momentums.  We usually donÕt know the positions.

We can relate momentum to position by Fourier Transform of momentum.   Momentum and position are complimentary .

We can re-express the propagator in terms of the momentum of the boson

 

And then our total amplitude for the process will be

 

The W particle has such a large mass that the denominator is usually dominated by the mass of the W.   In experiments starting in the 1950Õs higher values of K were explored, showing the above effect on the amplitude.

In the case of a photon, the mass is 0 and the value of K dominates the propagator.

In a beta decay (neutron->proton+electron+anti-neutrino), the momentum is small, so K can be ignored.

Process rates are controlled by coupling constants, masses of intermediate particles and available energies

How does available energy affect the rate?   The answer is that it is a tunneling process.   There is not enough energy to produce a W, but the final state is lower energy. 

A example of tunneling is the escape of an alpha particle from a nucleus:

 

 

On short time scales you can have an apparent violation of the escape energy requirement.   The lower the available energy, the shorter the time scales at which escape can happen and therefore the lower the amplitude.   [ It appears that the amplitude is an inverse exponential function of the height of the potential barrier.   I suppose that you could integrate across all time scales and sum the amplitudes for escape.   The amplitude would be 0 when is too large.]

Question:  Can you calculate the lifetime of the W?

Yes

 

Trying to observe virtual particles requires the energy of the photon used to make the observation be on the same order as the particle.   Such a photon obviously disturbs what you are trying to observe.

Suppose you have an oscillation:

 

If you hit the neutron hard enough with a photon, then the recapture of the W+ can be prevented, leaving a W+ to decay into a particle pair.  

If you hit a particle hard enough, anything can come out.

 

Question:  Why donÕt photons interact?

Photons can scatter by exchange of an electron positron pair.   The electron mass suppresses this interaction.

 

Spontaneous Symmetry Breaking

 

Suppose you have a round table with a fork placed between each pair of diners.   Once someone picks up a fork, the symmetry is broken and everyone else follows suit.

Suppose you have a bunch of coins in a lattice.  Each coin can either be heads or tails.  Further suppose that if neighbors have opposite states, then that costs energy.   The global state of lowest energy is either all heads or all tails.    A single coin in a known state of heads far away can be enough to trigger collapse to the all heads state.

Once our state is all heads, then local changes to tails cost energy.  The behavior of the system is then based on that ground state.   The symmetry is broken.

 

Explict symmetry breaking example:  suppose that heads has less energy than tails, then there would be only one ground state.

What is the difference between explicit and spontaneous?   With spontaneous symmetry breaking, you can get domain walls.   [A common example is magnetic domains in materials].   In the coin example, the left side of the table could be all heads and the right all tails.   [If one flipped any one coin on the domain boundary, the energy would increase because there would be more neighbors of the original state.   If one coin were flipped, then the change would heal]

My guess is that we are going to talk about the broken W SU(2) symmetry and the connection to the mass of the W bosons.  PhotonÕs and gluons are 0 mass, so one would expect their symmetries to be unbroken.