^{1,2}

^{1}

^{1}

^{3}.

We consider an extension of the standard model by right-handed neutrinos and we argue that, under plausible assumptions, a neutrino mass of

The masses of the third generation electrically charged fermions are known to a fairly high precision: the top quark mass is

The huge hierarchy between the third generation charged fermion and neutrino masses, at least 9 orders of magnitude, suggests the existence of a different mechanism of mass generation for the neutral fermions, other than just a Yukawa coupling to the Higgs field. Arguably, the simplest and most economical framework to explain the differences between the electrically charged fermion masses and the neutrino masses is the seesaw mechanism

The seesaw model, on the other hand, cannot predict the concrete value of the neutrino masses, as the Yukawa couplings and the right-handed Majorana masses are

A large mass scale for the right-handed neutrinos could be generated, e.g., from a Yukawa interaction with a singlet scalar that takes a large vacuum expectation value

In the most minimal setup, only two mass scales are available: the Higgs mass parameter (or alternatively the electroweak symmetry breaking scale) and the Planck mass,

In this Letter, we argue that the seesaw model with Planck scale lepton number breaking can naturally generate, under simple and plausible assumptions, an

We consider for simplicity a model with one generation of lepton doublets,

If quantum effects were neglected, the model would predict the existence at low energies of a pseudo-Dirac neutrino pair, with masses

We note that when

An explicit calculation confirms this expectation. At the one-loop level, one finds corrections to the right-handed masses which are proportional to themselves

Two-loop diagram leading to the radiative generation of

At low energies the heavy neutrinos can be integrated out, leading to an effective neutrino mass

For perturbative values of

This discussion can be extended to the realistic case of three generations of lepton doublets. Here we just sketch the basic ideas and we defer a detailed discussion to a forthcoming publication

The neutrino Yukawa coupling

We present here the results for the scenario where at the cutoff scale

The active neutrino masses are generated by the seesaw mechanism. Similarly to the two-generation case, one obtains one neutrino mass eigenvalue with size

In contrast, the values of the two lighter active neutrino masses are not predicted, and strongly depend on the free parameters

This is illustrated in Fig.

Physical neutrino masses as a function of

Our assumption for a strongly hierarchical right-handed neutrino at the Planck scale is purely phenomenological. Nonetheless, it is interesting to speculate about the possible origin of such a structure. If the right-handed neutrino mass matrix is of gravitational origin, so that one can relate the lepton number breaking to physics at the Planck scale, it is natural to identify the overall right-handed neutrino mass scale with the Planck scale. Also, it has been argued that a gravitationally induced effective neutrino mass matrix would take a “democratic” form

In this framework, one (or two) of the right-handed masses are determined purely from quantum effects. Therefore, this framework renders a higher predictivity. More specifically, for the toy model with one lepton doublet and two right-handed neutrinos, the most general framework is defined by 5 free parameters at the cutoff scale: 2 moduli and 1 phase in the Yukawa coupling, and 2 right-handed masses. However, under the assumption that one of the right-handed masses vanishes at the cutoff scale (or is negligible compared to the radiatively induced contribution to the mass), all observables will depend, at most, on 3 free parameters, as one phase can be rotated away by a field redefinition. Notably, some observables may depend on even less parameters, such as the active neutrino mass, which only depends on two free parameters [cf. Eq.

In this work we have been motivated by generating dynamically the right-handed neutrino mass scale necessary to reproduce the neutrino mass inferred from oscillation experiments. However, a similar rationale can be applied in other frameworks to generate a mass scale for a fermion singlet. For instance, several works have advocated a keV mass sterile neutrino as dark matter candidate

Finally, we would like to emphasize that quantum effects on the right-handed neutrino masses can also be important in frameworks where the scale of lepton number breaking is below the Planck scale (as in some grand unification models) and/or in any framework where the tree-level right-handed mass hierarchy is very large. In this case, quantum effects induced by the heavier mass eigenvalues can radiatively induce masses for the lighter eigenvalues which can be much larger than the tree-level values, possibly affecting the phenomenology of the model.

We have considered an extension of the standard model by a Planck scale mass right-handed neutrino, motivated by the fact that the lepton symmetry is likely to be broken by gravitational effects at the Planck scale. We have argued that, in general, the masses of the lighter right-handed neutrinos are not protected by any symmetry and therefore they receive sizable contributions, possibly dominant, from quantum effects induced by the Planck-scale mass right-handed neutrino. Concretely, we have explicitly shown that an initially massless right-handed neutrino becomes massive due to two-loop effects. Furthermore, we have shown that when the heaviest right-handed neutrino interacts with the lepton doublets with an

This work has been partially supported by the DFG cluster of excellence EXC 153 “Origin and Structure of the Universe” and by the Collaborative Research Center SFB1258. T. T. acknowledges support from JSPS Fellowships for Research Abroad.