In evaluating the vacuum structure of quantum field theories you need to find the minima of the effective potential including perturbative and nonperturbative corrections where possible.
In supersymmetric theories, you often see the claim that the Kähler potential is the suitable quantity of interest (as the superpotential does not receive quantum corrections).
For simplicity, let's consider just the case of a single chiral superfield:
$\Phi(x,\theta)=\phi(x)+\theta^\alpha\psi_\alpha(x) + \theta^2 f(x)$
and its complex conjugate. The low-energy action functional that includes the Kähler and superpotential is
$$
S[\bar\Phi,\Phi] = \int\!\!\!\mathrm{d}^8z\;K(\bar\Phi,\Phi)
+ \int\!\!\!\mathrm{d}^6z\;W(\Phi) + \int\!\!\!\mathrm{d}^6\bar{z}\;\bar{W}(\bar\Phi)
$$
Keeping only the scalar fields and no spacetime derivatives, the components are
$$\begin{align}
S[\bar\Phi,\Phi]\big|_{\text{eff.pot.}} = &\int\!\!\!\mathrm{d}^4x\Big(\bar{f}f\,\frac{\partial^2K(\bar\phi,\phi)}{\partial\phi\partial{\bar\phi}} + f\,W'(\phi) + \bar{f}\, W(\phi)\Big) \\
\xrightarrow{f\to f(\phi)}
-\!&\int\!\!\!\mathrm{d}^4x\Big(\frac{\partial^2K(\bar\phi,\phi)}{\partial\phi\partial{\bar\phi}}\Big)^{-1}|W'(\phi)|^2
=: -\!\int\!\!\!\mathrm{d}^4x \ V(\bar\phi,\phi)
\end{align}$$
where in the second line we solve the (simple) equations of motion for the auxiliary field.
The vacua are then the minuma of the effective potential $V(\bar\phi,\phi)$.
However, if you read the old (up to mid 80s) literature on supersymmetry they calculate the effective potential using all of the scalars in the theory, i.e. the Coleman-Weinberg type effective potential using the background/external fields $\Phi(x,\theta)=\phi(x) + \theta^2 f(x)$. This leads to an effective potential
$U(\bar\phi,\phi,\bar{f},f)$ which is more than quadratic in the auxiliary fields, so clearly not equivalent to calculating just the Kähler potential. The equivalent superfield object is the Kähler potential + auxiliary fields' potential, as defined in "Supersymmetric effective potential: Superfield approach" (or here). It can be written as
$$
S[\bar\Phi,\Phi] = \int\!\!\!\mathrm{d}^8z\;\big(K(\bar\Phi,\Phi) + F(\bar\Phi,\Phi,D^2\Phi,\bar{D}^2\bar{\Phi})\big)
+ \int\!\!\!\mathrm{d}^6z\;W(\Phi) + \int\!\!\!\mathrm{d}^6\bar{z}\;\bar{W}(\bar\Phi)
$$
where
$F(\bar\Phi,\Phi,D^2\Phi,\bar{D}^2\bar{\Phi})$ is at least cubic in $D^2\Phi,\bar{D}^2\bar{\Phi}$.
The projection to low-energy scalar components of the above gives the effective potential $U(\bar\phi,\phi,\bar{f},f)$ that is in general non-polynomial in the auxiliary fields and so clearly harder to calculate and work with than the quadratic result given above.
So my question is: when did this shift to calculating only the Kähler potential happen and is there a good reason you can ignore the corrections of higher order in the auxiliary fields?
Best Answer
Indeed, your question has nothing to do with the distinction between 1PI and Wilsonian. The answer is that the terms which contain nontrivial dependence on $D^2\Phi$ are to be dropped if the breaking of supersymmetry is small compared to the natural ("supersymmetric") mass scale in the problem. You can see this by noting that the effective potential has to be of the form $f^2 F(f/M^2)$ where $f$ is the SUSY breaking scale and $M$ is some supersymmetric scale (which can the VEV of some modulus, too). Another way to see this is that terms with more powers of $D^2\Phi$ have a higher engineering dimension and thus have to be divided by some SUSY scale, so their effect disappears as f/M^2->0.
In some physical scenarios these corrections could be important, but since in dynamical models having rigorous control over the physics usually entails having SUSY-breaking as a small effect, in most of the literature these terms are dropped.