Lopsided texture compatible with thermal leptogenesis in partially composite Pati--Salam unification

In this paper, we consider a lopsided flavor texture compatible with thermal leptogenesis in partially composite Pati--Salam unification. The Davidson--Ibarra bound $M_{\nu R1} \gtrsim 10^9$ GeV for the successful thermal leptogenesis can be recast to the Froggatt--Nielsen (FN) charge of the lopsided texture. We found the FN charge $n_{\nu1}$ of the lightest right-handed neutrino $\nu_{R1}$ can not be larger than a upper bound, $n_{\nu1} \lesssim 4.5$. From the viewpoint of unification, the FN charges of the neutrinos $n_{\nu i}$ should be the same to that of other SM fermions. Then, two cases $n_{\nu i} = n_{qi} = (3,2,0)$ and $ n_{\nu i} = n_{l i} = (n+1,n,n)$ are considered. Observations of PS model shows that the case of $n=0$, $n_{li} = n_{di} = (1,0,0)$ will be the simplest realization. To decrease the FN charges of these fermions from the GUT invariant FN charges $n_{qi} = (3,2,0)$, we utilize the partial compositeness. In this picture, the hierarchies of Yukawa matrices are a consequence of mixings between massless chiral fermions $f_{L}, f'_{R}$ and massive vector fermions $F_{L,R}, F'_{L,R}$. This is induced by the linear mixing terms $\lambda^{f} \bar f_{L} F_{R}$ and $\lambda^{f'} \bar F'_{L} f'_{R}$. As a result of the partial compositeness, the decreases of FN charges require fine-tunings between mass and Yukawa matrices either for the increases of $\lambda^{f, f'}$ or for the decreases of $M_{F,F'}$. Therefore, the case for $n=2$ and $n_{di} = n_{li} = (3,2,2)$, which requires only increases of FN charges will be appropriate to build a natural model.