07 September 2026

Much of the literature on neutrinoless double beta ($0νββ$) decay with light new-physics mediators focuses on invisible missing energy. We investigate the scenario in which a massive Majoron-like particle $φ$ is produced on-shell during $0νββ$ decay and subsequently decays into visible final states after travelling a macroscopic distance. Specifically, we analyze the displaced energy deposition from $φ$ decays into a photon pair ($γγ$), a photon and a dark photon ($γγ_D$) and an electron-positron pair ($e^+e^-$). We demonstrate that the displaced decays modify the expected visible energy spectra and provide novel, distinct experimental signatures at current and upcoming $0νββ$ experiments, with the promise of improving the sensitivity of the standard invisible Majoron searches in $0νββ$ decay by more than two orders of magnitude. The relevant effective couplings can naturally arise in well-motivated ultraviolet-complete scenarios that conventional $0νββ$ decay searches cannot probe.

We introduce a new method for studying mod-$\ell$ congruences between eigenforms through the modular representation theory of $\mathrm{PGL}_2(\mathbb{F}_p)$. When $p\equiv \pm 1 \pmod{\ell}$, we use this theory to construct and describe extra structures on spaces of modular forms with $Γ_0(p^2)$-level at $p$ and a fixed mod-$\ell$ Galois representation. The structural results we obtain can be viewed as a refinement of classical level-raising theorems since they not only allow us to prove the existence of congruences, but also to count the number of such congruences. Our methods work equally well in the residually irreducible and residually reducible cases, allowing us to prove several new instances of congruences between Eisenstein series and cuspforms (as well as independently rederiving classical results of Mazur and more recent results of Lang--Wake). Notably, our approach proves these results without computing constant terms of Eisenstein series, without Galois deformation theory and $R=\mathbb{T}$ theorems, and without the Jacquet--Langlands correspondence.

We study the future prospects for precision measurements of purely leptonic charged-current kaon and pion decays. In particular, we consider the single ratios $R_{Kπ}^{\ell=μ,e} = Γ(K \to \ellν) / Γ(π\to \ellν)$, for which many experimental uncertainties cancel, as well as the double ratio $R_{Kπ}^e / R_{Kπ}^μ$, which provides a particularly sensitive probe of new physics. This study is timely in light of two converging developments. On the experimental side, a new conceptual design for a dedicated setup based on a slow-extracted proton beam and a multi-year data-taking program, is expected to achieve a precision at the $10^{-4}$ level. On the theory side, recent lattice-QCD calculations have reached a comparable level of accuracy. Finally, we assess the impact of these next-generation measurements on searches for physics beyond the Standard Model and show that they can provide constraints that are either competitive with or complementary to existing bounds.