Physics Notes: Supersymmetry, Grand Unification, and String Theory

 

 

Lecture 3: April 12, 2010                                                               Back to PHY31

 

 

This lecture was short because of a seminar on the SLAC X-ray laser that started 1 hour into the normal period of Professor SusskindÕs lecture.    The notes are therefore also short.   The main topic was more development of the notion of propagators.   We covered a bit of propagators last quarter .

Renormalization of Mass

 

Feynman diagrams have several uses.   One is to model scattering.   They can also help with computation of the ÒeffectiveÓ Lagrangian.   We can remove complexity by using an energy or momentum cutoff.   These are nearly equivalent notions of cutoff, different only in the rest mass contribution to energy, which is often very small.

Feynman diagrams contain propagators, which represent transporting a particle from one space-time location to another.

 

We integrate from a point back to nearby that point.    For very small proper time separations, the integral diverges and this is why we establish a cutoff.   The integration is stopped inside of a distance limit.   We can alternatively integrate across momentum space staying under a maximum momentum.   [The argument for such cutoffs seems to be that at these limits, new physics appears which prevents the infinities from being real.   We donÕt know what that new physics is, but we will just integrate across the domain we think we know and hope for the best. ]

We have three basic particle types we have to worry about.  Spin 0, Spin ½ and Spin 1.

The propagator represents a creation operation at Y and an annihilation operation at X.  In general, all components of the field are represented.

      For a scalar spin-0 particle

   For a fermion.  

In the fermion case both i and j are iterated through the components of the field, so the propagator is actually a matrix.

A general sense of what the propagator has to look like when evaluated can be had from dimensional analysis.

Remembering that the Action is dimensionless if we set , we have

So  must have units

 

From the first term we can see that we have

So

And therefore  must be dimensionless.

The Feynman graph representing the loop has two factors of  and two factors of  from the propagator yielding units of  which happens to match the units of the mass term coefficient .   Given this we can expect the propagator to have the form:

 

Where K is some constant from the integration containing like things.    At high momentum (or equivalently, small distance) you can ignore the mass and this form is pretty good.   Remember that:

 

If , then the amplitude drops off rapidly.   Something like

 

Applying this to graph for the  term, which looks like

 

 

The graph on the right adds a correction to the effective mass.   However, it makes an infinite contribution as goes to 0.   This is where we assume a fuzzyness on a scale of  which is something like the Planck scale.   We either integrate over spacetime and limit the interval to be far enough from zero, or we integrate across momentum space and cap the momentum.

The renormalization contribution to the effective mass can be larger than the base mass term .

 

Fermions

 

Start with the action for a fermion:

It is easiest to figure out the units of the propagator from the mass term, but obviously you get the same answer either way.

If  and mass has units of , then it must be the case that

So the propagator would have units of.   Complicating things is that the field has 4 components, so the actual form of the propagator looks like

 

This propagator is actually a matrix coupling the components of , in the same way as the mass term.

The Sign of Fermion Loops

 

A pair of loops, either fermion or boson, will have a positive contribution.   If the two loops are near each other, then there are two cases, either the particles are exchanged in the loops or they arenÕt.    For bosons, these two cases just add to the amplitude.   For fermions, when we exchange indistinguishable particles, we have to multiply by a phase of -1 before adding.  [There is an excellent discussion of this in the Dirac lecture by Feynman].   

This alternate case where we exchange fermions must therefore have a contribution that is the negative of two separate fermion loops, which has the same contribution as two boson loops.   But this is a single fermion loop!   The implication is that individual fermion loops give a negative contribution.

If we suppose that bosons had a fermion partner with the same charge and mass, then we could imagine that the divergence in the renormalized mass would cancel out.

Fermion Loop Contribution to Scalar Renormalization

 

We can also have graphs with 3 edges connecting to a vertex.    Two of these can be used to construct a second order graph using a fermion/anti-fermion pair loop to contribute to the renormalization of a scalar mass [Higgs].

We still have to integrate over separations larger than .   We can hold one point fixed and integrate over the displacement to the other point.

 

The minus sign comes from the fact that this is a fermion loop.   Adding this to our previous contribution we get

We still need a miracle to get .

This kind of miracle is what is posited in Supersymmetry.