12 August 2026

We identify two commensurate structures in the measured quark and lepton mixing data, one multiplicative and one angular, and their exact relation. With the hierarchy parameter $\varepsilon\equiv|V_{ub}|^{3/10}=0.1869\pm0.0018$, the measured Fritzsch--Xing (FX) angles satisfy $\sinθ_u=\varepsilon^{26/18}$, $\sinθ_d=\varepsilon^{17/18}$, and $\sinθ=\varepsilon^{34/18}$ with coefficients within $2.5\%$ of unity, and the phase is maximal, $φ_{\rm FX}=92.3^\circ\pm2.7^\circ$ against $π/2$. Unit coefficients reproduce the remaining moduli at the percent level and fix the derived Jarlskog invariant, $J=\varepsilon^{111/18}\sinφ_{\rm FX}$, matching the measured $3.08(14)\times10^{-5}$. The rephasing-invariant image of this structure is a unitarity triangle quantized on the $π/24$ lattice, a digital clock, $(α,β,γ)=(12,3,9)\times7.5^\circ$, consistent with data at $1σ$ through $α\simeqφ_{\rm FX}$ and the theorem $\tanβ\simeq\sqrt{\varepsilon}$; degree-level $γ$ confronts $67.5^\circ$--$68^\circ$. For the leptons, where only the angular structure can act, its PMNS analog, formed from columns $1$ and $3$, free of Majorana phases, on the same lattice requires $δ_{\rm CP}\simeq296^\circ$ in the upper octant or $244^\circ$ in the lower, testable at DUNE and Hyper-Kamiokande. The quark and lepton triangles share $β=22.5^\circ$, measured for quarks, predicted for leptons, and differ by a transfer of two lattice units, with corollary $δ_{\rm CP}\simeq-δ\simeq295^\circ$. The same $\varepsilon$ sets the neutrino mass ratio, $r\equiv m_2/m_3=\varepsilon^{19/18}$, and the PMNS first row follows in powers of $r$, $(r^{1/9},\,r^{1/3},\,\tfrac{\sqrt{3}}{2}r)$, both relations at $0.2σ$ and testable at JUNO. The analysis is confined to the parameterization level; no dynamics is assumed.

Contrary to the common lore that observable lepton-number violation (LNV) is inevitably suppressed by tiny neutrino masses, we identify a class of seesaw models in which arbitrarily large LNV can naturally coexist with sub-eV neutrino masses. We construct a symmetry-protected texture-zero structure in the neutrino Yukawa couplings and heavy Majorana mass matrix that gives rise to the required accidental symmetry, thereby protecting the light neutrinos from acquiring mass even in the presence of arbitrarily large LNV in the heavy sector. Small neutrino masses arise naturally from lifting the texture-zero structure while preserving the underlying symmetry. The resulting framework offers a rich and experimentally accessible phenomenology, predicting Heavy Neutral Leptons with sizeable active-sterile mixing over a broad range of experimentally accessible masses, giving rise to observable LNV signatures at collider and intensity-frontier experiments.

A comprehensive numerical analysis of the latest germanium based CE$ν$NS data from the COHERENT Ge-mini and TEXONO experiments has been conducted to test the Standard Model (SM) and search for new physics. By combining CE$ν$NS and E$ν$NS signals with a consistent treatment of detector effects and systematic uncertainties, we obtain a low energy determination of the weak mixing angle from COHERENT Ge-mini, in agreement with the SM prediction. We derive novel constraints on neutrino electromagnetic properties, including the magnetic moment, millicharge, charge radius, and anapole moment, with TEXONO providing particularly competitive bounds. Inclusion of E$ν$NS, significantly improves the sensitivity to the neutrino millicharge by up to three orders of magnitude. We also investigated light scalar and vector mediators, finding striking complementarity between reactor and stopped pion sources across different mediator mass regimes. Finally, we put bounds on sterile neutral leptons production through transition dipole, scalar, and vector portals, probing masses from the sub MeV to tens of MeV scale. Our results demonstrate that current germanium based CE$ν$NS experiments provide a powerful low energy laboratory for precision electroweak tests and complementary probes of a broad class of physics beyond the SM.

This is a review paper intended to illustrate a few critical issues concerning Dirac, Weyl and Majorana fermions and their differences. The first part consists in basic introductions to fermions, and more in detail to fermions in 4d, focusing in particular in what differentiate the three type of fermions: chirality, helicity, mass, field equations, properties under discrete symmetry transformations, Actions, Observables. On this basis we tackle a series of challenging and sometime controversial problems: the non-existence of Weyl fermion propagators, and the ways to circumvent it; the regularizations for Weyl fermion amplitudes, in particular the appropriateness of using the PV regularization; the definition of a functional integral for Weyl fermion, the difference between Weyl and massless Majorana fermions and the difference between the Dirac and Majorana mass terms. Most notably we come to the conclusion that the trick of replacing the non-existing Weyl fermion propagator with a massless Dirac one, although it may yield in some cases correct results, is flawed at the very origin by a logical loophole. We then show which is the correct way to proceed in this case. We consider also the topic of applying the Wick rotation at the classical Action level for Weyl fermions and conclude that this procedure leads to a nonequivalent theory. Finally, although anomalies are not the central focus here, we have deemed it useful to summarily review the relation, and its cohomological basis, that exists between the lack of a Weyl fermion propagator (non-invertibility of the Weyl-Dirac operator) and the appearance of dangerous anomalies in the theory.

Primordial black holes (PBHs) form through several distinct mechanisms, each imprinting a characteristic mass function that a single-mass treatment risks erasing. Building on the FRISBHEE code, we introduce FRISHBEE, implementing four mass functions--log-normal, power-law, critical collapse, metric preheating--in a multimodal framework, with spin modeled via Gaussian and Fishbach profiles. Crucially, each channel's weight is derived from the primordial collapse probability rather than fitted, tying the mass function to the power-spectrum amplitude--a link between inflationary model-building and CMB observables. Solving the Friedmann-Boltzmann equations for $M_{\rm PBH}^{\rm in}=10^{7}$~g in the pre-BBN window, we compute $ΔN_{\rm eff}$ for five distributions, three spin configurations, and three weightings. Extended mass functions enhance $ΔN_{\rm eff}$ over the monochromatic benchmark by factors of $\sim\!1.03$ (critical collapse) to $\sim\!1.84$ (log-normal), preserving the hierarchy $LN > MM > PL > MP > CC$ across the mass window $\{10^{5},10^{7},10^{8}\}$~g. Near-extremal spin ($a_{\star}=0.99$) and spin-2 dark radiation invert this trend: superradiant spin loss averages away faster in broad distributions than in a monochromatic population, letting the monochromatic approximation overestimate $ΔN_{\rm eff}$. Against CMB-S4 and Simons Observatory sensitivities ($σ\simeq 0.06$ and $0.05$), the monochromatic benchmark is undetectable for scalar dark radiation (sDR $=0$), while log-normal, power-law, and multimodal distributions reach $\sim\!1.6$--$1.7σ$; for spin-2 dark radiation the signal drops over an order of magnitude, with none detectable. The formation channel thus governs the magnitude and observational accessibility of the Hawking signal, establishing $ΔN_{\rm eff}$ as a discriminant between single- and multi-channel PBH formation scenarios.

Hadronically decaying particles with lifetimes $τ\gtrsim0.02\,{\rm s}$ are excluded by Big Bang Nucleosynthesis almost independently of their abundance: the mesons from their decays convert protons into neutrons faster than the reverse, and helium is overproduced. For heavy neutral leptons (HNLs), this blankets the couplings the upcoming accelerator searches can reach. We propose a scenario with large lepton flavor asymmetries in which HNLs evade it. A Dirac HNL decays into $π^+$ and its antiparticle into $π^-$, so an asymmetry between the two populations injects an excess of $π^+$, which converts the neutrons back to their standard abundance, while a cancellation between the asymmetries and the HNL decays keeps the shift in $N_{\rm eff}$ small. This opens the parameter space with $m_l+m_π\lesssim m_N\lesssim1\,{\rm GeV}$ and $τ_N\lesssim1\,{\rm s}$, including a substantial region within the reach of SHiP. A discovery there, combined with the BBN and CMB observables, would fix the primordial flavor asymmetries up to one remaining direction. The relic neutrino background and a possibly first-order cosmic QCD transition, a source of gravitational waves, may probe that direction, and with it the lepton asymmetry of the Universe.

We explore the $U(1)_{(B-L)_3}$ gauge symmetry under which only third generation fermions are charged. The cancelation of the $[U(1)_{(B-L)_3}]^3$ anomaly requires a chiral fermion, $χ_L$, singlet under the gauge group but charged under $U(1)_{(B-L)_3}$. This chiral fermion can play the role of thermal dark matter with a relic abundance set via the freeze-out scenario through the annihilation to pairs of the third generation fermions. In the quark mass basis, the new gauge boson will have off-diagonal couplings to the $b$ and $s$ quarks which can lead to a new contribution to $B^+\to K^+ν_τ\barν_τ$. We discuss the relevant bounds from the LHC searches for a new gauge boson, dark matter searches, electroweak precision data, $Υ$ decay, the CKM matrix and the $B_s^0-\bar{B}_s^0$ mixing. We entertain the possibility of explaining the $B^+\to K^+ν\barν$ excess recently reported by Belle II within this model.

In the context of type IIB superstring compactification, we demonstrate the assisted fibre inflation proposal for a four-field model realized using the orientifold of a tetra-quadric Calabi-Yau (CY) threefold. This CY threefold has an underlying permutation symmetry $S_4$ and belongs to both the list of CY threefolds, namely the Kreuzer-Skarke (KS) database as well as the Complete Intersection Calabi-Yau (CICY) database. After fixing the overall volume modulus of the CY threefold using the framework of perturbative large volume scenario, there are three Kähler moduli which remain flat and assist in driving fibre inflation via sub-leading corrections. In a particular benchmark model, we show that the effective inflaton shift of around $5.7$ M$_p$, as needed for driving fibre inflation, can be successfully shared by three inflaton moduli which need to be individually shifted by nearly $2.2$ M$_p$ only ! The main motivation for the work is to show that the effective large field excursions of the inflaton field is possible without the need of pushing the individual volume moduli towards a large super-Planckian excursion or close to the boundary of the Kähler cone which may create various subsequent challenges for the effective field theory and supergravity approximations, especially in Swiss-Cheese based models of fibre inflation.