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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?

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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.

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