Imagine I want to compute this
D†2Dα(Φ∗Φ)
where the D-s are super-covariant derivatives and Φ is a chiral superfield. Following the notation of this review http://arxiv.org/abs/hep-ph/9709356 on supersymmetry, the chiral covariant derivatives are (page 33)
Dα=∂∂θα−i(σμθ†)α∂μDα=−∂∂θα+i(θ†ˉσμ)α∂μ
D†˙α=∂∂θ†α−i(ˉσμθ)˙α∂μD†˙α=−∂∂θ†˙α+i(θσμ)˙α∂μ
let's now consider the following coordinate change
xμ→yμ=xμ+iθ†ˉσμθ
the review claims (page 34) that in these
y,θ,θ† coordinates the chiral covariant derivative are
Dα=∂∂θα−2i(σμθ†)α∂∂yμDα=−∂∂θα+2i(θ†ˉσμ)α∂∂yμ
D†˙α=∂∂θ†αD†˙α=−∂∂θ†˙α
we can as well go to coordinates
xμ→ˉyμ=xμ−iθ†ˉσμθ
where in page 35 it is claimed that the chiral covariant derivatives take the form
Dα=∂∂θαDα=−∂∂θα
D†˙α=∂∂θ†α−2i(ˉσμθ)˙α∂∂ˉyμD†˙α=−∂∂θ†˙α+2i(θσμ)˙α∂∂ˉyμ
so far so fine. It is obvious that for some computations the choice of the clever coordinates will make computations considerably less cumbersome. This is where my problems arise. The computation I want to do contains a chiral superfield, which can be expressed most compactly using the y,θ coordinates, whilst the antichiral superfield is better written using ˉy,θ†. My question is, is itlegitimate to use the chiral covariant derivatives is the ˉy,ˉθ form, even though what i wanna take the derivative of contains all yˉy,θ,θ†? even if the answer is no, what i the wisest way to perform my computation?