Physics Notes: Supersymmetry, Grand Unification, and String Theory

Lecture 9: May 31, 2010 Back to PHY31

Topics: Supersymmetry breaking, GUTs

Review of Symmetry Breaking

A good analog for the type of symmetry breaking that occurs in Supersymmetry happens in ferromagnets. Imagine that you are a small creature inside of a ferromagnet. In a simple model of a ferromagnet each individual atom (magnet) has an orientation and interacts with the immediate neighbors. From an energy point of view, two neighbors are in a low energy state if their directions are aligned and a high energy state if their directions are opposing.

Imagine that an atom as a particle is a spin j particle. Then there are 2j+1 states corresponding to all integer spins whose absolute value is less than or equal to j. The complementary + and – spin states have the same energy by symmetry and one could imagine that they all have the same energy.

Now put the atom in a magnetic field B oriented along the Z axis, then the energy levels are split because they interact differently with the B field. This splitting is known as the Zeeman effect.

Inside a ferromagnet where the magnets are aligned an atom could have states that normally have the same energy level split into different energies.

How do you tell if the symmetry was broken explicitly or spontaneously from inside of the crystal? An explicit case might be a magnetic field set up outside of the ferromagnet. Otherwise, the magnet orientations may have just lined up in some arbitrary direction to get to a low energy state. From inside the ferromagnet, what you can do is to disturb the magnets in a region, varying their orientation across a region. Then you let go. The result in the spontaneous case would be waves of orientation change propagating out. These are Goldstone bosons, which are massless (energy goes to 0 when momentum goes to 0).

These Goldstone bosons are responsible for giving mass to gauge bosons.

If you apply a symmetry operation like rotation to the vacuum, then if you get:

Then the vacuum state is symmetric. If instead you get:

GB stands for Goldstone Boson

Then the vacuum has some kind of spontaneous symmetry breaking

Supersymmetry Math for Symmetry Breaking

Lets start with a constrained superfield:

which has an associated variable substitution that makes it easier to build Supersymmetric Lagrangians:

Which lets us write the superfield in terms of y and , simplifying the calculations

After translating back from y to x, the Taylor’s series expansion generates derivative terms and we pick out the terms that have all of the Grassmann variables (because the integral of other terms is 0).

Suppose we have a Lagrangian like

Then the terms of interest are pretty simple to pick out:

The first term can be recognized as a derivative with respect to .

[I’m not clear on the second term]

[perhaps ]

The second term is recognizable as a second derivative.

Now F is a peculiar thing. The equations of motion have no derivatives. We may as well solve them and eliminate F.

We should have

and

Substituting back into the Lagrangian:

[It is interesting how recasting the equations in terms of derivatives with respect to has simplified the equations]

One possibility is that at the minimum

Implying that the vacuum state is symmetric and F=0.

Another possibility is that the minimum is not zero, implying that F is not 0. If F is not zero in the vacuum state, then application of the symmetry operator Q would rotate it into other components. In this case F behaves like a magnetic field on atomic states, splitting the energy levels of fermion and boson states. Fermion and boson superpartners would not have the same mass.

Supersymmetry and Gravity

One more implication – if the ground state has energy, this is not good for gravity, which depends on energy density.

In a ferromagnet, rotation of the magnetic field orientation from place to place gives rise to a goldstone boson.

Local supersymmetric transforms will add a massless fermion called a goldstino”

In supergravity theories gravity makes a contribution to the potential

[not sure if I have this right]

Which can nearly bring the vacuum energy back down to 0. Otherwise the implied vacuum energy would be too big and the universe should have expanded much much further than we observe.

Some Supersymmetry Jargon

In the ferromagnet and atom example the atom would be called a “sector” of the theory. The field of the ferromagnet would be another sector.

Note: If the field is very strong, then the atom is distorted – very stretched out.

If the field is weak, the atomic energy levels are only slightly split. The atom’s change in behavior would almost be unnoticeable. In this case the Ferromagnet would correspond to a “hidden sector”.

Spin states

A massive particle of spin J has 2J + 1 states. The massless photon has spin states of . Where does the other state come from for the gauge bosons come from. It comes from the goldstone boson.

The same thing happens in supersymmetry. The massless goldstino has two spin states like the photon. [There is some connection here to the gravitino, which is a spin 3/2 particle, but my notes are unclear as is my memory]

GUTs

The standard model is a composite model

Color Weak EM

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

This group is a subgroup of SU(5). In fact, SU(5) is the simplest such group with SU(3) X SU(2) X SU(1) as a subgroup.

Reminder: SU(5) has a representation as the set of 5x5 matrices such that

The group can be completely described in terms of it’s generators

is Hermitian, and

How many such generators are there?

The diagonal is real and if we count each upper triangle entry as two parameters because they are complex, then we have all the parameters because the values of the lower triangle are determined by the upper one. This yields 25. In general for SU(n) we have n²-1 generators, leaving out the identity.

Generators for SU(5)

So there are 24 independent generators for SU(5). We want to map the known transforms onto the matrix representations. If we start with a block diagonal form, then the blocks will not interact.

To add the U(1) generator we add one new matrix with trace 0 that distinguishes the two blocks.

SU(5) adds 24 new gauge bosons.

We are talking about the left handed particles because the right handed ones don’t have the SU(2) component – coupling with the weak force.

Instead of talking about the left and right handed electron and their anti-particles, it is more convenient to work with the left handed electron and the right handed positron, and their anti-particles.