Supersymmetric Lagrangian Transformation (Grassmann Numbers)

[Lonsdale]

I've been tasked with showing that a Lagrangian under a set of transformations changes by a time derivative. All has gone well, except I'm left with two remaining terms, that I am completely confident, aren't there by mistake (as the 16 terms that should be expected have all popped out with the correct signs etc).

The two terms, I believe, probably cancel but I don't just want to speculate. Anyway, the two terms that remain are,

$$W'' W' (ψ^*ε^* + εψ)$$

where W is the Superpotential (primes representing derivatives, w.r.t x) as a function of the spatial co-ordinate x. ψ represents the Superspace co-ordinate and epsilon represents the small change Grassmann parameter relevant to our transformation. Stars represent complex conjugates.

Now, I will also add that ψ is a Grassmann number so we can rewrite the terms such that,

$$W'' W' (ψ^*ε^* - ψε)$$

So, briefly put, do these two terms cancel one another out? If so, why?

(also, please do not delete my post for not following the template, I didn't find it necessary given that this is quite a small scale question)

Thanks for your time, Physics forum!

 

[Lonsdale]

Ok... after some thought I don't think they can cancel in this way (or the rest of what I have done wouldn't make sense).

Anyway, instead of writing everything out in full, I'm basically working the Lagrangian on page 5 of this source http://www.phys.columbia.edu/~kabat/susy/susyQM.pdf and using the transforms also on the bottom of page 6.

The terms above come from the final term of the Lagrangian, i.e.

$ψ^*ψW''$

The the 3rd and 7th terms when multiplying out the brackets. As I mentioned before, everything else canceled out in the calculation or was used to show that the change in the Lagrangian is a total time derivative, I was simply left with the aforementioned two terms.