Dirac mass matrix textures and the lightest right-handed neutrino mass scale in Type I seesaw leptogenesis

The type I seesaw mechanism is one of the leading proposed explanations for how neutrinos acquire their tiny masses. However, the mass scale of the undiscovered right-handed neutrinos required by this mechanism remains undetermined. Assuming vanilla leptogenesis in the two-flavor regime, we work backwards to find the required general textures of the Dirac mass matrix from which we determine the mass of the lightest right-handed neutrino to be around $10^9 {\rm GeV}$ to $10^{12} {\rm GeV}$.

💡 Research Summary

The paper investigates the relationship between the Dirac neutrino mass matrix textures and the mass scale of the lightest right‑handed (RH) neutrino in the context of type I seesaw leptogenesis. Starting from the well‑known type I seesaw formula (M_\nu = -M_D M_R^{-1} M_D^T), the authors assume a hierarchical spectrum for the heavy RH neutrinos ((M_1 \ll M_2 \ll M_3)) and focus on “vanilla” thermal leptogenesis, i.e. the standard Boltzmann equations without additional model‑specific effects.

Leptogenesis proceeds through the CP‑violating decays of the lightest RH neutrino (N_1) into lepton doublets (L_\alpha) and the Higgs boson. The CP asymmetry for each lepton flavour (\epsilon_{1\alpha}) is expressed in terms of the Dirac mass matrix elements (Eqs. 3–7). The authors distinguish three flavour regimes, determined by the magnitude of (M_1): the unflavoured regime ((M_1 \gtrsim 10^{12},\text{GeV})), the two‑flavour regime ((10^9,\text{GeV} \lesssim M_1 \lesssim 10^{12},\text{GeV})), and the three‑flavour regime ((M_1 \lesssim 10^{9},\text{GeV})). In the two‑flavour regime the tau Yukawa interactions are in equilibrium while the electron and muon flavours remain indistinguishable, leading to a specific form for the generated lepton asymmetry (Eq. 9).

The central idea of the work is to reverse‑engineer the Dirac mass matrix such that leptogenesis must occur in the two‑flavour regime. To achieve this, three conditions are imposed on the matrix: (i) the effective wash‑out parameters for the three flavours are equal, (\tilde m_{1e} = \tilde m_{1\mu} = \tilde m_{1\tau}), which translates into equal magnitudes for the first column entries of (M_D); (ii) the sum of the CP asymmetries vanishes, (\epsilon_{1e} + \epsilon_{1\mu} + \epsilon_{1\tau}=0); and (iii) either (\epsilon_{1e} + \epsilon_{1\mu}=0) or (\epsilon_{1\tau}=0). Condition (i) forces the ratios (r_{\mu 1}= \rho, r_{e1}) and (r_{\tau 1}= \sigma, r_{e1}) with (\rho,\sigma = \pm 1).

The authors then analyse the CP‑asymmetry expressions and discover that the vanishing of the sum in (ii) can be ensured if the scalar products ((\mathbf{s}_J!\cdot!\mathbf{r}_J)(\mathbf{c}_J!\cdot!\mathbf{r}_J)=0) for each heavy neutrino index (J). Here (\mathbf{s}J) and (\mathbf{c}J) are vectors built from the sines and cosines of the phase differences (\theta{\alpha J} - \theta{e1}), while (\mathbf{r}J) encodes the real coefficients (r{\alpha J}) with the signs (\rho,\sigma). The condition reduces to either (\mathbf{s}_J!\cdot!\mathbf{r}J=0) or (\mathbf{c}J!\cdot!\mathbf{r}J=0). Solving these linear relations yields six distinct families of Dirac mass matrices, labelled (M_I) through (M{VI}) (Eqs. 34–39). Each family is characterised by a common overall factor (r{e1} e^{i\theta{e1}}) and by linear combinations of sines (or cosines) of the phase differences multiplied by arbitrary real coefficients (\alpha_J,\beta_J).

Explicit formulas for the CP asymmetries (\epsilon_{1\alpha}) for each texture are derived (Eqs. 40–45 and 43–45). The authors use the large‑mass approximations (F(x) \simeq -3/(2\sqrt{x})) and (G(x) \simeq -1/x) together with an empirical efficiency factor (\kappa(\tilde m)) (Eq. 12) to argue that the product (\epsilon_{1\alpha}\kappa(\tilde m_{1\alpha})) can naturally generate the observed baryon asymmetry (Y_B \simeq 8.7\times10^{-11}) without fine‑tuning.

In Section 3.2 the authors compare their general results with a previously studied specific texture (Eq. 58), showing that the earlier matrix is a special case of their (M_{II}) family after suitable phase redefinitions.

Finally, to satisfy the additional condition (iii) the authors impose further phase relations (Eqs. 52–57). For example, in the (M_I) texture they require (\theta_{e2}-\theta_{e1}=n\pi), (\theta_{\tau2}-\theta_{\tau1}=n\pi) and a relation between (\alpha_2) and (\beta_2) that involves sines of the phase differences. Similar constraints are listed for the other five families. When these constraints are fulfilled, the generated lepton asymmetry resides exclusively in the two‑flavour regime, which, by definition, forces the lightest RH neutrino mass to lie in the interval (10^9;\text{GeV} \lesssim M_1 \lesssim 10^{12};\text{GeV}).

The paper concludes that the six identified Dirac mass matrix textures provide a model‑independent way to infer the RH neutrino mass scale from flavour‑dependent leptogenesis requirements. The authors acknowledge several limitations: the assumption of a strongly hierarchical RH spectrum, the neglect of possible contributions from (N_2) and (N_3) decays, the absence of a detailed fit to low‑energy neutrino oscillation data, and the omission of temperature‑dependent flavour transition effects. Nonetheless, the work offers a clear analytic framework linking texture zeros and phase alignments in (M_D) to the viable leptogenesis regime, and it suggests that future experimental searches for heavy neutrinos should target the (10^9)–(10^{12}) GeV window.