Axionless Solution to the Strong CP Problem -- two-zeros textures of the quark and lepton mass matrices and neutrino CP violation --

TL;DR

The paper presents a CP-invariant mechanism to solve the strong CP problem by constructing two-zero textures for the down-type quark mass matrix with a diagonal up-type matrix in a $T^2/Z_3$ orbifold. It identifies three viable textures, $A_1$–$A_3$, that reproduce CKM and charged-lepton masses while keeping $\bar{\theta}<10^{-10}$, and extends the construction to the neutrino sector to predict $\delta_{CP}$ and $m_{\beta\beta}$, with NH naturally realized and IH possible in certain Majorana-mass limits. It further analyzes two-right-handed-neutrino scenarios, showing NH can persist with tight $\delta_{CP}$ predictions (around $200^\circ$ or $250^\circ$) and $m_{\beta\beta}$ near a few meV, while IH becomes viable in specific limits, all consistent with a positive baryon asymmetry. The framework thereby links strong-CP, CKM, and leptonic CP violation within a single symmetric setup, yielding concrete, testable predictions for upcoming neutrino oscillation and $0\nu\beta\beta$ experiments.

Abstract

CP invariance is a very attractive solution to the strong CP problem in QCD. This solution requires the vanishing ${\rm arg}\,[{\rm det}\, M_d\, {\rm det} M_u]$, where the $M_d$ and $M_u$ are the mass matrices for the down- and up-type quarks. It happens if we have several zeros in the quark mass matrices. We proceed a systematic construction, in this paper, of two zeros textures for the down-type quark mass matrix while the mass matrix for the up-type quarks is always diagonal. We find only three types of the mass matrices can explain the observed CKM matrix, the masses of the quarks and the charged leptons and the small enough vacuum angle $θ< 10^{-10}$. We extend the mass construction to the neutrino sector and derive predictions on the CP violating parameter $δ_{CP}$ in the neutrino oscillation and the mass parameter $m_{ββ}$. It is extremely remarkable that the normal (NH) and inverted (IH) hierarchies in the neutrino masses are equally possible in the case where we introduce only two right-handed neutrinos $N$s. Furthermore, we have a strict prediction on the $δ_{CP} \simeq 200^\circ$ or $250^\circ$ in the NH case. If it is the case we can naturally explain the positive sign of the baryon asymmetry in the present universe.

Figures (8)

• Figure 1: The predicted distribution of $m_e/m_{\tau}$ by taking $k_e=3$ and $k_e'=1$ in the case of $\bm{A_1}$. The vertical red line denotes the central value of the observed one, and blue ones denote $\pm 10\%$ error-bars for eye guide.
• Figure 2: The predicted distribution of $m_{\mu}/m_{\tau}$ by taking $k_e=3$ and $k_e'=1$ in the case of $\bm{A_1}$. The vertical red line denotes the central value of the observed one, and blue ones denote $\pm 10\%$ error-bars for eye guide.
• Figure 3: The predicted $\delta_{CP}$ versus $m_{\beta\beta}$ for NH in the case of $\bm{A_1}$. The region between the horizontal red (blue) dashed-lines denotes $1\,(2)\sigma$ allowed one of $\delta_{CP}$ in NuFIT 6.0 (NH with SK atmospheric data) Esteban:2020cvm. The cyan and magenta regions denote the regions of $m_1\ll m_2$ and $m_1\lesssim m_2$, respectively.
• Figure 4: The predicted $\delta_{CP}$ versus $m_{\beta\beta}$ for NH in the case of $\bm{A_1}$ by putting the constraint the positive cosmological baryon number $Y_B>0$ for the case of $M_1< M_2$. The notations are same as in Fig. \ref{['fig:A1NH-mee-CP']}.
• Figure 5: The predicted $\delta_{CP}$ versus $m_{\beta\beta}$ for NH in the case of $A_1$ with infinite $M_1$. The region between the horizontal red (blue) dashed-lines denotes $1\,(2)\sigma$ allowed one of $\delta_{CP}$ in NuFIT 6.0 Esteban:2020cvm.