The theoretical description of double beta decay modes requires the use of techniques which originate in various branches of physics, like: a) nuclear interactions, b) nuclear structure models and methods, c) electroweak decays, d) elementary particle physics models and related symmetries. In at first glance different field, that is the search for the understanding of the composition of the dark matter in the Universe, did appear questions which are closely related to the double beta decay studies, both experimentally and theoretically.

The paper is devoted to the exploration of the possibilities which the assumption about the existence of a neutral-bosonic- complex scalar field, called the axion, offers to the electroweak sector of the adopted theory of particles and fields, as well as to the The conventional Higgs mechanism does not give mass to neutrinos, like it does with the rest of the particles. Some 50 years ago R.D. Peccei and H.R. Quinn [1,2] have proposed the existence of the axion in order to explain for the suppression of the neutron electric-dipole moment, and introduced a global U(1) symmetry called the Peccei-Quinn (PQ) symmetry. In this picture, the physical vacuum contains some background fields $\mathsf{\Phi }$, being the axion one of these. In the simplest version the axion is a neutral scalar complex field which acquires a non-vanishing vacuum expectation value due to its associated double well potential [1,2].

Shortly after the Big Bang, the temperature was high and the PQ symmetry was manifest. When the temperature fell low enough a phase transition occurred and the PQ field acquired a non-zero vacuum expectation value ${\langle \mathsf{\Phi }\rangle }_0$, leading to a spontaneous symmetry breaking at an unknown energy scale $f_a$ [2].

The coupling of the axion to the gluons and to quarks suppresses the neutron electric-dipole moment, providing a solution to the strong CP problem. In extended scenarios of the couplings the axion could also interact with pairs of photons and pairs of baryons. In addition to their role in cosmology, with reference to the dark matter composition [3], axions may play a role in neutrino physics [4], because the coupling of neutrinos with axions could provide a mechanism to explain for non-zero neutrino masses. In analogy with the conventional Higgs’s mechanism, the addition of a neutral scalar field with of a non-zero vacuum expectation value in a quadratic plus quartic potential provides a mass term when coupled to neutrinos, in the same way the non-zero vacuum expectation value of the Higgs’s boson gives mass to other particles. Naturally, the axion-neutrino coupling is not responsible for the neutrinoless double beta decay mode, rather it gives mass to the neutrino, but the dynamics of the double beta decay includes other non-less important ingredients, as well, both from the nuclear structure side, like availability in phase space, the inclusion of nucleon and nuclear correlations and the knowledge of the microscopic structure of participant nuclear states and their energy differences, and the extension of the Standard Model Lagrangian by the inclusion of right handed currents and their couplings [5].

With this motivation in mind, we have explored the consequences of the coupling between axions and neutrinos, in order to compare the neutrino mass values resulting from this coupling with the upper limits to the neutrino mass determined from the non-observation of the neutrinoless double beta decay. We have introduced a Lagrangian describing the neutrino-axion coupling and calculated the neutrino mass insertion. Next, we have reviewed the essentials of the formalism of neutrinoless double beta ($0\nu \beta \beta$) decay to make a connection with the axion-neutrino picture and compared the results from both scenarios to investigate the compatibility between them.

We start from the Lagrangian [6,7] which describes the derivative coupling between neutrinos ($\psi$) and axions ($\mathsf{\Phi }$). By separating spatial and temporal derivatives, the Lagrangian is split up in the following terms:

Following the argument by Peccei and Quinn [1,2], the axion field acquires a non-zero vacuum expectation value, ${\langle \mathsf{\Phi }\rangle }_0$, in presence of the potential [6,7] By imposing the condition one gets the solutions and Thus, the axion scalar field is written and with it we get for the Lagrangian, written in natural units: Therefore, at lowest order in the neutrino-axion interaction, we introduce the correspondence since ${\psi }^{†}\psi$ is the neutrino density.

To calculate the contributions to the neutrino mass coming from the spin-dependent term of the Lagrangian, we write, for the transition amplitude [8] where $\overrightarrow{\mathbf{S}}$ is acting on the fermionic sector.

For spin-up and spin-down neutrino states, we get: and respectively, where for short $\mathsf{\Phi }=\mathsf{\Phi }(\overrightarrow{x},t)$ in the above equations. These expressions are model-dependent since they would imply the knowledge of the spatial dependence of $\mathsf{\Phi }(\overrightarrow{x},t)$.

For the sake of completeness, as an example of spatial DM distributions, if we adopt for the axions a directional Gaussian parallel to the neutrino incoming direction (arbitrarily chosen in the z-direction), there will not be a spin-flip term contributing to the amplitude $\langle f|\overrightarrow{\nabla }\mathsf{\Phi }.\overrightarrow{\mathbf{S}}|i\rangle$, while the spin-up contribution will then look like [8,9,10] li and for spin-down one we have

In Equations (13) to (14), for the factor $\langle f|\overrightarrow{\nabla }\mathsf{\Phi }.\overrightarrow{\mathbf{S}}|i\rangle$, ${\mathcal{N}}_i$ and ${\mathcal{N}}_f$ are the normalization factors of the initial and final neutrino states, respectively. The quantities with primes define the energy and components of the neutrino momentum in the final state.

The neutrino propagator $S\left(p\right)$ is analytically expressed as [11]

After evaluating $\Sigma \left(p\right)$ on shell, we have finally obtained the one-loop correction to the neutrino mass due to the interaction with axions. It is written as The explicit forms of ${\Sigma }_p$ and ${\Sigma }_m$ are listed in [12].

The physical mass of the neutrino can be computed as

The derivation of the previous equations involved the ordering of higher order corrections to the propagator, as well as the definition of a criteria to determine the strength of the coupling $g_a$ for each proposed mass scale $m_a$. The details have been presented in Refs. [8,9,10,11,12].

As a first step, we have computed the maximum value for the neutrino mass considering the limits imposed by the experimental limits of the non-observation of the neutrinoless double beta decay. We have performed the calculation by taking different values of the effective weak coupling constants and the neutrino mass, and constrained their ranges by using the information coming from the non-observation of the neutrinoless double beta decay, as discussed in Section 2.1. The maximum value for the electron neutrino mass is approximately $0.3$ eV and corresponds to $<\lambda >=<\eta >=0$. Afterwards, we have calculated one-loop corrections to the neutrino-mass propagator as a function of the axion mass and of the coupling constant $g_a$. We have performed two different analysis, namely i)by fixing the value of ${\tilde{m}}_{\nu li}$ at the zero-order neutrino mass, and, ii)by fixing the value of ${\tilde{m}}_{\nu }$ at the maximum value allowed by the non-observation of the neutrinoless double beta decay.

We have calculated the neutrino mass for the axion mass $m_a$ in the range ${10}^{-10}\mathrm{eV}-0.1$ eV. In Figure 1 we show some results of the calculations. The one-loop corrections to the neutrino mass depend on both $g_a^2$ and $m_a^2$. Therefore, one can study the effects on the neutrino mass, $m_{\nu }$, induced by the variation of $g_a$ and $m_a$. In spite of the rather involved structure exhibited by the propagator’s corrections, the results show a smooth trend, for $m_{\nu }$ as a function of $m_a$.

The double beta decay, in its neutrinoless mode, is perhaps one of the most rich process from the point of view of the physics involved. Its observation will definitively demonstrate that the current view of the electroweak interactions must be changed drastically. It will also determine the future of large scale experiments, like ATLAS in CERN, in the search of new generations of mediators of the weak interactions, both in the bosonic and fermionic sectors of the theory. From the point of view of nuclear structure models it will also set a very selective criteria for the adoption or rejection of nuclear hamiltonians, nuclear coupling schemes and methods to determine nuclear wave functions. Last but not least it will demonstrate the validity of Majorana’s theory about the neutrino.

As an alternative way of thinking about double beta decay and neutrino properties we have extended the notion of the U(1) symmetry breaking in the axion sector of a Lagrangian which includes the coupling of neutrinos and axions. The zero order mass insertion, of the neutrino propagator, was corrected by adding one-loop terms depending on the neutrino and axion momenta. By combining the results of these corrections with current limits on the non-observation of the neutrinoless double beta decay, one can determine exclusion and allowed regions in the parametric space sustained by the strength of the axion-neutrino interaction, the axion mass and the couplings corresponding to left and right handed sectors of the electroweak currents. The results are compatible with neutrino masses smaller than few tenths of eV.

I have shared with Sabin Stoica many years of fruitful discussions in subjects of common interest, like several approaches to the nuclear structure components of the single and double beta decay models. In all cases he has shown himself in a very direct and gently mood. He did contribute largely to the success of a conference dedicated to double beta decay studies and did contribute too by his continuous presence in journals. Thanks Sabin for all of these and we hope you shall continue with these activities in the years to come.