The last week of the semester was dedicated to Feynman rules in superspace. Before we go into that, it is good to review and set our conventions in superspace.
A remark: as in any quantum field theory analysis, here we will see the need for regularization. Now we need to to preserve the supersymmetry of the theory and the task of finding a regulator that obeys this criteria is non-trivial. Nevertheless, there exist such regularization schemes, namely dimensional reduction.
In any case, as action we will take the Wess-Zumino model coupled to super Yang-Mills. Of course this theory only has simple supersymmetry, but actually this is one of the only cases where superspace methods are useful. I do not want to rule out superspace methods for extended supersymmetry since one never knows what one might end up doing for his/her Ph.D. thesis. 😉 The action has the form:
The prepotential is
In four-dimensional Minkowski spacetime the Lorentz group is doubly covered by . We will label spinors with a Weyl index . The supercovariant derivatives, (giving objects that are covariant under supersymmetric transformations) are defined as follows:
Now we set the convention for raising and lowering Weyl indices (which actually we do not follow in the definition of the supercovariant derivatives): Weyl indices are raised with the two-dimensional antisymmetric symbol in the ¨from north-west to south-east¨ fashion. Namely,
The bars on spinors with dotted indices will be omited in what follows (they are redundant anyway). For example, we have the anticommutators for the supercovariant derivatives:
Now we move on to integration. We start with
Similarly for the dotted coordinates:
This expression follows from the single integration:
The integral of anticommuting variables can be written in terms of the derivative.
This will be useful later. Total supercovariant derivatives integrate to zero:
Here we have ignore boundary terms. Integration over all superspace can be expressed as
Now we turn to chiral superfields. A superfield is a function that is defined on superspace (i.e. it has dependence on the spacetime coordinates and the supercoordinates also). A chiral superfield satisfies the constraint:
Either or but not both.
With chiral superfields one can do wonders. For example, sticking to the first choice for constraint defining a chiral superfield, we have the property
This is useful when re-writing chiral integrals as integrals over the whole supercoordinates:
We have forgotten about Dirac delta functions for supercoordinates! For a single supercoordinate we define:
In particular, for the case of the identity function we get
This result can be generalized to the full superspace integral:
(Writing the integral as derivatives as discussed above.) The solution to this equation is
We can mention some properties related to this delta function: