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 $n_{\nu 1}$ of the lightest right-handed neutrino ${\nu }_{R1}$ can not be larger than a upper bound, $n_{\nu1} \lesssim 4.5$\mathrm{.}Fromtheviewpointofunification,theFNchargesoftheneutrinos$n_{\nu i}$shouldbethesametothatofotherSMfermions\mathrm{.}Then,twocases$n_{\nu i} = n_{qi} = (3,2,0)$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 $f_{L}, f’{R}$andmassivevectorfermions$F{L,R}, F’{L,R}$. 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 ${\lambda }^{f,f^{\mathrm{\prime }}}$ or for the decreases of $M_{F,F^{\mathrm{\prime }}}$. 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.