Physics Notes: Supersymmetry, Grand Unification, and String Theory

 

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Lecture 7: May 17, 2010                                                                Back to PHY31

 

Topics:  Superfields, Preserving Supersymmetry in Lagrangians

 

Questions

 

Q:  What is the mass of the lightest Supersymmetric partner?

It needs to be < 1TeV to be a good dark matter particle. 

Q:  CanÕt Fermilab reach 1TeV

If you were banging two 1TeV electrons together, then yes.   If you are banging protons together, then each quark can be thought of as carrying 1/3 TeV, with messy collisions that rarely extract all the energy.

Q: What was the motivation for Supersymmetry?

It turned up in the math of String Theory as formulated in approximately 1972, but it was not recognized at first.   The first string theory had bosons.   Later when fermions were added, Supersymmetry was added as a side effect but not recognized.

Bruno Zumino found it in the math of string theory.   In 1980 the connection was made to the renormalization problem.

Lecture

 

LetÕs return to a model of fermions and bosons at rest.

 

We have our standard creation and annihilation operators

              create and annihilate a boson

              create and annihilate a fermion

 

Boson Field and Lagrangian

           

The boson field will be .

Out of it we construct a Lagrangian, which could look like:

           

 

From this we can use the Euler-Lagrange equation to get the equations of motion.

           

 

Which we recognize as a harmonic oscillator of frequency .

 

Fermion Field and Lagrangian

 

The fermion field will be .

If we create a simple Lagrangian

           

Then the equation of motion would look like

 

Which is information free!    Now if we treat fermion fields as Grassmann numbers we get a different result.

           

So

           

Which at least says something.

For bosons was the restoring force.   For fermions we have to split into left and right-handed fields with the mass term coupling them together.

           

If this were a boson field, the result would be trivial.   Treating the field variables as Grassmann numbers changes the story.

 

  

 

The minus sign courtesy of swapping  left over .

Likewise:

           

 

Simplifying, we have:

           

Now we can combine the Lagrangians:

           

Note that there are no interactions between the boson and fermion fields.

Review of Boson/Fermion Symmetry Operations

 

[Note the change here to use in the definitions.  I suspect the reason for this is that the norm of a complex Grassmann variable  is .  So putting in the here makes this definition of the transforms match the coordinate transform definition, which uses Grassmann variables.]

 

  Delete fermion, add boson

    Delete fermion, add boson

             

                              [Lecture 5 shows how to derive this]

                           

           

This last bit we can recognize as the mass times the population count of the bosons and fermions.   This is the total energy.

The other commutation rules involving Q and H are:

           

           

               - Whatever measures, it is conserved.

 are supercharges.   Q can be used as a basis for states.

The Coordinate Transform Version of Q

 

Note:  time and space derivatives of fields are anti-Hermitian.   You should think of them as pure imaginary when conjugating an expression involving them.

We add two conjugate Grassmann variables   and to our normal coordinate .

We can then define  in terms of them.

           

We can then show that [see derivation in lecture 5]

           

A Superfield

 

We can make a Lagrangian out of our superfield in something like the usual way with squares and derivatives of the field.   The Lagrangian

 

will also have a TaylorÕs series expansion:

From this we can form the action:

           

When you integrate it out with respect to  and , you always get the highest order term from the Lagrangian.   Check the integration rules from Lecture 6.

Now we are ready to try the coordinate transform:

           

           

When you have a small displacement you simply add the sum of the derivatives        times the displacements.   [Small is a funny word here when Grassmann variables are involved.]

           

                       

 

             

              

Building Lagrangians That Preserve Supersymmetry

 

When we write Lagrangians for relativistic theories, we have a formalism with tensors that results in a Lagrangian that is a scalar – the same in all reference frames.    This automatically preserves all the relativistic symmetries.

We need the equivalent for Supersymmetry.

First – what works and what doesnÕt work?

You can square a superfield:

           

                       

                                   

                                   

                                   

Sorting variables into the same order and combining:

                       

This Lagrangian term is ok because it is built out of which we already know has the symmetry.   What about the kinetic terms?     We canÕt just use a time derivative because that will break the symmetry.

[Why is that?

Suppose:

           

                 

                 

The part that will contribute to the action will be the  part, so we donÕt need to get the rest.

           

Rearranging:

           

           

           

Interesting – each pair is a pair of conjugates, if we subtract in this order, then we get -2 times the imaginary part, so we have a pure imaginary number.    Normally, with  we would have the product of two imaginary numbers from the time derivatives, which would be real.   Something is already broken – courtesy of the Grassmann variables!

End – back to lecture]

 

[I think the idea in this next section is that if you have a constraint operation D that commutes with the symmetry generators, that you can derive from it a variable substitution, expressing the equivalent constraint.   This makes me think of covariant derivatives like in gauge invariance or in GR]

Suppose that you have a symmetry – for example a rotational symmetry for vector field

A symmetry has a set of generators – just one in this case

Suppose we have a linear constraint operator D (containing derivatives + stuff) such that:

           

This constraint can be thought of as a constraint on the input arguments x and y, specifying a relationship between them that makes the constraint true

Since has rotational symmetry, we would like to show that

               which corresponds to a small rotation

Which would mean that our constraint holds after transform:

           

If D and R commute or anti-commute, then:

So if D and R (or our symmetry generators in general) commute, then application of D will preserve the symmetry.

In Supersymmetry the symmetry generators are , , and .

Let

               and  

 

You can check that the anti-commutator here.

 

We require the constraint to be invariant to transformation by Q.

                minus sign from anti-commutator

A superfield that obeys this constraint is known as a Chiral superfield.

The corresponding variable substitution is:

             [I tried it this way and didnÕt get cancelations]

 [This next bit is my experiment -

             

[I should really change the last term from using D to avoid confusion]

And we write out the superfield in terms of , , and  .

Expanding each of the terms with a Taylor series:

           

Now apply

     

     

And the conjugate is then:

Now we can combine them to form a term for a Lagrangian.

                     

                      

Assuming that this is a term in the Lagrangian the  terms would survive in the action, leaving:

           

Order changes sign

           

           

           

Ok, so at least our result is real.   IÕm not sure quite what this means.   It is very strange that  disappeared.   It seems like the constraint would also tell us that is zero, so:

           

End of experiment]

The punch line is that with Lagrangians formed this way that divergences disappear.