How to Verify SUSY Invariance for Kinetic Terms in Wess and Bagger?

[need_help]

Hello, i try to solve the exercise in Wess and bagger. chapter 7, exercise 8. (its on page 50)
I compute the transformation laws correctly(its on the book) and check the susy invariant for
pure yang-mills term. (terms relate with W and W ¯)
However, during the calculating susy invariant for the rescaling (V →2gV) kinetic term, i got trouble.
what i want to check is
Φ + e 2gV Φ| = F∗ F − |D m A| ^2 − i ¯ ψ¯ σ m ˜ D m ψ + i √ 2g(A ∗ T a ψλ a − ¯ λ a T a A ¯ ψ) + gD a T a A ∗ A
is invariant under susy transformation.

I know this term is invariant under susy, but my calculation mess up and couldn't get right answer.
Is there anyone can show the details for this calculations? or give some calculation tips for this variations? I try to collect the D term and F term to make it vanish but still the other term survives...

 

[LightHero]

To check the susy invariance of the kinetic term, you have to take the variation of each part under the susy transformation. For the first part, F*F, the variation is given by:δ(F∗F) = 2F∗ δF + δF∗FFor the second part, |Dm A| ^2, the variation is given by:δ(|Dm A| ^2) = 2(Dm A) δ(Dm A) + δ(Dm A) (Dm A)For the third part, i ¯ ψ¯ σ m ˜ D m ψ, the variation is given by:δ(i ¯ ψ¯ σ m ˜ D m ψ) = i ¯ ψ¯ σ m ˜ δ(D m ψ) + δ(¯ ψ¯) σ m ˜ D m ψFor the fourth part, i √ 2g(A ∗ T a ψλ a − ¯ λ a T a A ¯ ψ), the variation is given by:δ(i √ 2g(A ∗ T a ψλ a − ¯ λ a T a A ¯ ψ)) = i √ 2g(A ∗ T a δ(ψλ a) − δ(¯ λ a) T a A ¯ ψ)For the fifth part, gD a T a A ∗ A, the variation is given by:δ(gD a T a A ∗ A) = gD a T a (δA ∗ A + A ∗ δA)By combining all the variations together and using the susy transformations for the fields, you can show that the kinetic term is indeed invariant under susy transformation.