# Correction to fermion mass in Supergravity mediated supersymmetry-breaking

I recently came across this paper about axino (fermion superparter of axion) mass in supergravity: https://doi.org/10.1016/0370-2693(92)90547-H

In this paper, they think of a specific superpotential (they consider two cases. I focus on the first one.) that is separated into visible sector and hidden sector. Also they assume minimal form of the Kahler potential. With the superpotential and Kahler potential defined, we can define a dimensionless Kahler function $G$.

$$G=\left|z\right|^{2}+\left|\phi\right|^{2}+\left|\phi^{\prime}\right|^{2}+\left|X\right|^{2}+\ln\left|h\left(z\right)+g\left(\phi,\:\phi^{\prime},\:X\right)\right|^{2}$$

$$g\left(\phi,\,\phi^{\prime},\,X\right)=\lambda\left(\phi\phi^{\prime}-f^{2}\right)X$$

(Here, $z$ is the hidden sector field, and $h(z)$ is the hidden sector superpotential which is not specified. $g$ is the visible sector superpotential. $\phi$ and $\phi^\prime$ are the chiral multiplets with opposite PQ charge, and $X$ is a PQ singlet.)

With this $G$, we can compute the fermion mass matrix.

QCD axion is a pseudo-Nambu-Goldstone boson and it has very small mass (usually $\mu$eV scale) from non-perturbative effects of QCD (I don't want to discuss this point in detail). If we ignore this effect, axion is massless at tree level. If supersymmetry is not broke, axino should be massless at tree level also.

Now here is their argument in the paper: If we calculate the axino mass from the fermion mass matrix using $G$, the axino mass is about gravitino mass.

It seems that from SUGRA mediation effect (at least in the first example they showed), mass of the axino gets huge correction. I am very new to supergravity, and I wonder how all the other fermions in the Standard model are safe from this effect. I also noticed that there has been many efforts to understand axino mass in supergravity. But they don't mention about the mass correction of other fermions other than axino. I guess I am missing a point.

Is there a way to understand how the other fermion masses (such as Standard model fermions) are safe from this kind of effect?

First of all, I would be very surprised if the axion is massless at tree-level as you say. Since the hidden sector $z$ breaks supersymmetry, its scalar acquires a VEV (vacuum expectation value). If you calculate the scalar potential $V$ (see eq. (2) in the paper you cited), calculate $m_X^2= \frac{d^2 V}{dX d\bar X}$, and replace $z$ with its VEV, I am convinced that you will find something non-zero.
The fermion mass matrix is given by eq. (3) in the paper. Again, you should replace the fields with their VEVs (for which one would need to specify the function $h(z)$). If you calculate the masses for the fermions of $X$ and the $\phi$, you will see that they are massive as well, and probably of the same order as the gravitino mass.
I am a bit confused on why you call 'SUGRA mediation' an affect. In gravity mediated supersymmetry breaking, there is a visible sector and a hidden sector $z$. The scalar potential for the hidden sector is constructed to have a non-trivial minimum that breaks supersymmetry, and the scalar field $z$ acquires a VEV. Since in supergravity, there are Planck suppressed couplings between the hidden sector and the visible sector, you can expand the potential in $X \bar X$ and find that the potential has a mass term depending on $z$. Let me emphasise that this is a tree-level mass, and not an 'effect'. In contrast, gravity mediated supersymmetry breaking usually comes together with anomaly mediated supersymmetry breaking which gives mass corrections at 1-loop level to the gauginos.
Side note #2: The model above does not yet contain Standard Model fermions. If you want to couple Standard Model fields $\chi_1, \chi_2, h$ to this model, you can include them in the function $g$ by adding a term proportional to $\chi_1 \chi_2 h$, where $h$ is a Higgs. Since the fermion mass is proportional to the derivatives of $G$, and the VEVs of the SM fields vanish (the Higgs VEV also vanishes here, but its mass will becomes tachyonic after running down the renormalization group equations to allow for a Higgs mechanism at low energy), the fermion masses will vanish too.