I am following some notes on supersymmetry and I need some clarification.
Chapter fice deals wuth sypersymmetric Lagrangians and the superpotential is introduced. It is stated that the interaction terms of chiral superfields are given by
Lint=∫d2θW(Φ)+h.c.
where
W is the superpotential and
Φ is a chiral superfield.
Φ=ϕ+√2θψ+iθσμˉθ∂μϕ−θθF−i√2θθ∂μψσμˉθ−14θθˉθˉθ∂2ϕ
in page 81, it is said that this integral is easy if we notice that we can expand the superpotential
W(Φ)=W(ϕ)+√2∂W∂ϕθψ−θθ(∂W∂ϕF+12∂2W∂ϕ∂ϕψψ)
so
Lint=−∂W∂ϕF−12∂2W∂ϕ∂ϕψψ+h.c.
all this looks very good from afar. Nonetheless, let's take the specific case
W=MΦ2
where
M is some mass constant and try to perform
∂W∂ϕ=2MΦ∂Φ∂ϕ=2M∂∂ϕ(ϕ+√2θψ+iθσμˉθ∂μϕ−θθF−i√2θθ∂μψσμˉθ−14θθˉθˉθ∂2ϕ)
here goes my question, when taking the partial derivative of either
Ψ or
W respect to
ϕ,
which are the other variables? Do I need to consider
Ψ(ϕ,ψ,F)? what do I do then with
∂μϕ and
∂2ϕ?
This post imported from StackExchange Physics at 2015-06-02 11:46 (UTC), posted by SE-user silvrfück