Kahler potental terms linear in visible sector fields

[SUSY]

Hello,

I have a question regarding the expansion of the Kahler potential in visible sector fields $C^{\alpha}$:

It is usually said that the Kahler potential can be expanded as follows: $$K = K_{hid}(\phi,\phi^*) + K_{\bar{\alpha} \beta}(\phi,\phi^*) C^{*\bar{\alpha}} C^{\beta} + \frac{1}{2} (Z_{\alpha \beta}(\phi,\phi^*) C^{\alpha} C^{\beta} + h.c.) + ...$$
where the $\phi$ are the hidden fields.

I was wondering why there are no terms linear in $C^{\alpha}$, i.e. why there are no terms
$$P_{\alpha}(\phi,\phi^*) C^{\alpha} + h.c. \subset K$$

I always thought the Kahler potential should be assumed as general as possible and that would include such terms. Can someone tell me why they are usually assumed to be absent? Are there papers about that specific question that I could consult?

Thank you very much,
SUSYAs an edit:
One often finds terms such as $\propto \frac{1}{S + S^*}$ for some stringfield $S$. Again, such terms are absent for the visible fields, i.e. there are no terms $\propto \frac{1}{C^{\alpha} + C^{*\bar{\alpha}}}$ and I can't seem to understand why that should be so...

 

[fzero]

Let's recall that the visible sector contains those fields that are charged under a gauge group that contains the Standard Model interactions, while the hidden sector does not directly participate in the Standard Model interactions. The two sectors can interact together via gravity or perhaps some very high energy gauge interaction. This means that the visible sector fields have conserved charges that the hidden sector fields do not.

A term linear in a visible sector field will generally not conserve these charges, so such terms are forbidden. Some terms like the $C^\alpha C^\beta$ one that you wrote down could also be forbidden, but would be allowed if that particular field only had a $\mathbb{Z}_2$ charge. Similarly, a non-perturbative term like $1/\mathrm{Re}(C^\alpha)$ would also be forbidden. Some hidden sector or string fields (in particular the dilaton) might not have such conserved charges, so $1/\mathrm{Re}(S)$ could be allowed. A term proportional to $C^\alpha C^{*\alpha}$ would tend to be invariant under the SM charges, so can be generated perturbatively via any messenger interactions.