Dark Energy and Neutrino Flavor from the Weak Axion
Dynamical dark energy offers an alternative to a cosmological constant with distinct observational signatures. However, the small energy density scale, Hubble-sized mass, and Planckian excursions make simple models fine-tuned and unnatural. In this work, we show that a weak version of the axion, identified with the phase field of the anomalous $U(1)_{B+L}$ of the Standard Model, can generate the scale hierarchies expected for dark energy. The axion potential is controlled by sources of explicit baryon and lepton number violation and is radiatively stable. We show that the leading contribution comes from two inequivalent Weinberg operators, one $B+L$-conserving and one $B+L$-violating, which generate the axion potential. We propose a flavor selection rule based on a spontaneously broken $S_3$ permutation symmetry in the lepton sector that simultaneously removes the quadratic divergence and dominant temperature-dependent contributions. The resulting potential first appears at quartic order in the axion-dependent neutrino masses and, for the observed departure from tribimaximal mixing, its amplitude is parametrically close to the dark-energy density. The dominant uncertainty comes from $\delta_{\rm CP}$ and $\theta_{23}$, so experiments like Hyper-K and DUNE can directly test the model in the future. Cosmologically, the field behaves as thawing quintessence for $f$ close to $M_{pl}$, stays frozen by Hubble friction until late times and never enters the adiabatic regime.