Are there two types of D-term and two types of F-term in SUSY?

Question

I've noticed that one can obtain D-terms either by integrating a vector superfield (the vector multiplet) over superspace or by integrating a Kahler potential over superspace. In both cases we get functions of the D-auxiliary field (albeit different functions) that contribute to the Lagrangian density. A subsequent integration over space gives the action contribution.

Similary we can get an F-term either by integrating a chiral superfield over superspace or by integrating a superpotential over superspace. These just lead to different functions of the F-auxiliary field for the Lagrangian density contribution. Again, integration over space then gives the action contribution.

So it seems to me that we have two different types of D-term and two different types of F-term depending on whether we choose to write our lagrangian density in terms of superfields or in terms of Kahler/Superpotentials.

Is it correct to say that it is strictly one or the other? In other words we couldn't have a Lagrangian density containing a vector superfield as well as a Kahler potential, thus resulting in two separate D-terms when these are integrated over superspace.

Is there some relation between the superfield approach and the potential approach that I'm missing? Why use the two seperate approaches at all?

Answer 1

The D-term is the last term in the Taylor expansion of a vector superfield over fermionic coordinates, $D \theta^1\theta^2\bar\theta^1\bar\theta^2$. Similarly, the F-term is the "middle" term $F\theta^1\theta^2$ which only contains the unbarred fermionic variables. Chiral superfields only depend on these coordinates (half of the superspace) so they're the last terms, too.

Chiral superfields have F-terms in them. Vector superfields have D-term in them. Kähler parts of the action are composite vector superfields so they also have D-terms. Similarly, the superpotential part of the action is a composite chiral superfield so it has F-terms.

The D-terms and F-terms that define the potentials and dynamics are those obtained from the Kähler potential and the superpotential in the Lagrangian. The D-terms and F-terms talked about when particular fields are discussed are the D-terms and F-terms of these particular fields. There can't ever be any confusion. One never sums or has to distinguish "two types of F-terms" or "two types of D–terms". There are as many types of D-terms or F-terms as the number of elementary or composite vector superfields or chiral superfields one may invent.