Physics Notes:  The Standard Model

Lecture 10: Mar 15, 2010                                                              Back to PHY30

 

Questions Before Class

Q: Renormalization is used in statistical mechanics too.  Is it the same thing and could you review the concept?   What about the inclusion of general relativity breaks renormalization?

Renormalization relates to the degrees of freedom of the system.   Systems have both short and long wavelength modes.

In statistical mechanics you have a large collection of objects with spin or other degrees of freedom.   In this case you can smooth out the spin [or vibration] modes, treating them as a continuous field.

Renormalization is a [modeling] trick.  Experiments donŐt access/observer the very short wavelengths.   However, the dynamics of the short wave length parts of the system may have an effect on the dynamics of the longer wavelengths.

Consider the following system of springs and weights

 

On the left we have a low frequency sub-system and on the right we have a high frequency one.  The idea is to solve the high frequency system in the background of the low frequency behavior.  Then the high frequency behavior is added back into the low frequency solution.

Another example is a small child swinging their legs rapidly while a swing goes back and forth.   The low frequency behavior of the swing will be governed by the mass of the child, length of chain and so on.   The kicking of the child can be modeled as yielding small oscillations at a higher frequency, which can then be added back to the low frequency solution.

This is called renormalization.

The problem with including general relativity is that the fluctuations of space mess up the formulation in such a way that the high frequency fluctuations canŐt be decoupled from the low frequency ones.

Recap Question:  Could you review the use of group theory and comment on the connection to spectroscopy of hadrons?

One of the origins of group theory is that with respect to the masses of the proton and neutron the up and down quarks have nearly the same very low mass.   If you ignore electric charge,  is an SU(2) symmetry.   When you swap u for d you swap (or rotate) protons into neutrons and neutrons into protons.   mesons would also rotate in the same way.   This symmetry is broken in 2 ways:  the masses are not the same [u:1.5-3.3Mev, d:3.5-6Mev], and there is an obvious charge difference.

Isospin is a symmetry that is today almost considered an accident.   When you add the strange quark [70-130MeV] you can form an approximate SU(3) symmetry  with u, d, and s as the basis vectors of the space, and start organizing particles according to the multiplets of SU(3).   SU(3) has 8 generators.  Other quarks like the charm quark are too heavy to make this useful.

Sometimes different phenomenon have the same group structure.  Spin is an SU(2) group, as is the weak charge.   While the phenomenon are different, the mathematics are similar.

 

Next Quarter

 

We will cover a key puzzle:  The Gauge Hierarchy Problem

Which is a motivation for SuperSymmetry.

Unification with SU(5) which has subgroups SU(3), SU(2), U(1).   In this formulation quarks and leptons are part of the same multiplet.

 

More on the Mass of Fermions

 

We havenŐt built the Lagrangian for the Dirac Equation yet.  

The change to convert a Lagragian is simple, just multiply by

 

Q:  What is the ŇbarÓ notation

This notation lets us write our equations more concisely.

Rewriting our Lagrangian with  we have:

Now we replace the combinations of with a new set of matrices, which are called the gamma matrices.

There is one more

 

The eigenvectors of  correspond to an observable ŇchiralityÓ, which is a kind of handedness.

Using the notation we can re-express the Lagrangian

       i goes from 1 to 3

Using the indices, we can compact this to

 

The mass term exchanges Chirality.  Expanding in terms of :

The first term absorbs an L and produces a R and the second absorbs an R and produced an L.

 

The W Boson

 

This particle is purely left-handed and is only emitted by left handed particles.

Hence, left-handed particles have weak charge and right-handed ones donŐt.

The Higgs has weak charge so we have a doublet.   [Left-handed Higgs have weak charge and right handed ones donŐt]

  [Maybe we should be using  here instead of H]

Remember that , so the field H is a perturbation around , the minimum of the Mexican hat potential function  [add link].

Replacing the field , we get

           

This leaves us with left to right couplings yielding a mass and couplings to the Higgs boson.  An important fact is that both  sides of this expression have the same coupling constant.  This means that the production rate of Higgs bosons should be governed by the same coupling constant that gives our particles mass.   This is something that could be tested at LHC if it can demonstrate the existence of the Higgs boson.

 

An estimate of f  is about 200GeV.    We can then get the coupling constants from the masses of the particles.

What About the Higgs Itself?

 

The potential energy function for the field is the Mexican hat potential.

 

At the origin the potential function is dominated by the  term and looks like an upside down parabola.   The field at the origin is not stable.   To add a minimum the simplest change is to add a term. 

 

Where is the minimum?

 

If one had a particle where L & R both had weak charge, then it would be easy for that particle to have a mass.  [Or a particle that had no weak charge at all, then you wouldnŐt have to worry about the left-right coupling of a mass term.]

 

The thinking is that the natural mass scale of particles that donŐt get their mass from a Higgs mechanism is very high - so high that we canŐt detect such particles with experiments.   The natural mass scale would be up around the Planck mass or the unification mass, which is 13 orders of magnitude higher than the mass of the W/Z or the Higgs which should be similar.

Q: Are their Higgs like mechanisms in condensed matter physics?

Yes – you could treat Helium as a fundamental particle and create a field for it.  Symmetry breaking in a Bose Condensate would lead to superfluidity.   If the atoms were charged then it would make a Higgs like field and give mass to photons that would couple to the charge.

We have also talked about the fact that photons get a mass inside of a superconductor.  The Higgs like field would be field of Cooper Pairs which would be charge -2 bosons.  [The internal field variables could be the separation between the pairs of electrons with the direction of the vector between them taking the place of the angle in our simplified Higgs model.   There should be a natural value of  , the separation, which is greater than 0 where there is minimum energy.   Cooper pairs would have charge so the photon obviously couples to them.]  The mass given to the photon in this case is tiny.

Renormalization

 

When we measure charge we are usually measuring forces at substantial separations and inferring the charge from CoulombŐs law.

This law is good for relatively stationary charges in a vacuum.

In a material or in air the result is different, a little less – why?

Suppose two opposite charges are inserted into a conductor. 

 

If we move the charges lose and closer together, we can get them inside the spacing of the atoms and then the charges are no longer canceled.   The effective charges depend on the distance.   This is called a ŇrunningÓ charge.

           

 

Now suppose that our charges are embedded in a dielectric where charges are locally confined but can move small distances.   In this case the charges create small dipoles around them.

 

If you draw a sphere around either of the two charges you will find that the orientation of the dipoles reduces the charge imbalance inside of that sphere, leading to an effective reduction in the observed charge (via the force between the charges).   Again, we have a running charge.  It is different from the conductor case because the shielding effect does not send the running charge to 0 at a distance, but it does reduce it.   When you approach inside of the size of the dipoles, the shielding goes away and e(r) returns to the standard value.

The same thing happens in QFT [presumably for any charge type, because you will have particle/anti-particle charge carriers in the vacuum.]   For electric charge virtual pairs perform the shielding.  The electric field of the charges orients the virtual pairs – they are polarized.   A real vacuum is a dielectric.   The  in CoulombŐs law is not the bare electric charge.   Unlike a material dielectric, the virtual electron positron pairs have a spectrum of energies.   High energy corresponds to small size.   The result is that the electric charge is a running charge that continues to climb logarithmically with .   These effects become apparent on the scale of the Compton wavelength.   A key point is that a logarithmically growing function is not bounded.   At some point it is expected that there is some new physics that cuts this growth off and that there is a real ŇbareÓ electron charge.   A speculation is that this cutoff is somewhere around the Planck length.

As we have seen, the strong force behaves differently because of the gluon-gluon interaction, yielding a potential function that grows linearly with distance.   The Weak force is in between.

A Remarkable Fact

 

If one plots the running charges of the electric, weak and strong forces against inverse distance you get the following:

 

The strong, weak, and electromagnetic forces coupling constants all cross at very nearly the same point.  Gravity crosses a couple of orders of magnitude later.  There is something very deep here that seems to indicate that all forces are connected.

 

Gravity at Small Distances

 

Why should gravity become stronger at small distances?

This will be a rough argument.    The force between two masses is:

   For particles at rest.

 

More accurately, energy gravitates and the m terms should be replaced

So

 

If a particle is localized to a very small distance, then the momentum spread must be large, and therefore the expectation value for momentum must be large.

  and the energy is a function of P and m, but P will swamp the rest energy so

  and hence the force becomes

 

[Interesting fact in this line is that the rest mass of the particle becomes swamped by the momentum, which is controlled by the distance – the mass of the particle doesnŐt really matter]

Where do the gravitational and electromagnetic forces become equal?

 

   (Susskind stuck in the c h-bar for dimensional consistency)

 

The e2 is now unit-less and 1, so we can drop it. 

   [SusskindŐs result is missing the G, but I donŐt see how it disappeared]

 

This is basically the Plank length.

The big puzzle here is why from the Higgs field is so small when the Planck length (corresponding to a very high energy/mass) is so small.   [This appears to be the Gauge Hierarchy Problem]