Universe

We present a comprehensive sensitivity study of future CE$\nu$NS detectors, focusing on a cryogenic cesium iodide detector and a tonne-scale liquid argon one, currently being developed by the COHERENT Collaboration. These setups will enable precision measurements of the weak mixing angle at low energies and allow accurate extraction of the neutron nuclear distribution radius. We also demonstrate that next-generation detectors will place constraints on the neutrino charge radius comparable to or better than current global fits. In addition, we explore the sensitivity to non-standard neutrino electromagnetic properties, such as magnetic moments and millicharges, as well as new mediators. These findings reinforce the role of CE$\nu$NS experiments in the upcoming precision era, with future detectors playing a key role in advancing our understanding of neutrino interactions and electroweak physics at low energies.

This review examines the role of differential forms, Pfaffian systems, and hypersurfaces in general relativity. These mathematical constructions provide the essential tools for general relativity, in which the curvature of spacetime—described by the Einstein field equations—is most elegantly formulated using the Cartan calculus of differential forms. Another important subject in this discussion is the notion of conformal geometry, where the relevant invariants of a metric are characterized by Élie Cartan’s normal conformal connection. The previous analysis is then used to develop the null surface formulation (NSF) of general relativity, a radical framework that postulates the structure of light cones rather than the metric itself as the fundamental gravitational variable. Defined by a central Pfaffian system, this formulation allows the entire spacetime geometry to be reconstructed from a single scalar function, Z, whose level surfaces are null.

We study basis-independent structures in the Type-I seesaw mechanism for light Majorana neutrinos, assuming the canonical scenario with three heavy right-handed (sterile) neutrinos. Let $m_{\nu }$ denote the $3\times 3$ mass matrix of light neutrinos, obtained at tree level from heavy Majorana singlets with a diagonal mass matrix $D_N=\mathrm{diag}(M_1,M_2,M_3)$ and a Dirac matrix $m_D$. We show that all right actions F on the seesaw matrix that leave $m_{\nu }$ unchanged form the group . While oscillation data determine the PMNS matrix $U_{\mathrm{PMNS}}$ and the mass-squared splittings, they do not fix the F-class within G. We classify basis-invariant quantities into those that are class-blind (e.g., $det \eta$) and class-sensitive (e.g., $\mathrm{Tr}\eta$, $\mathrm{Tr}{\eta }^2$, an alignment measure, and CP-odd traces relevant to leptogenesis), where $\eta$ denotes the non-unitarity matrix of the light sector. We provide explicit formulas and both high-scale and GeV-scale benchmark examples that illustrate these invariant fingerprints and their scaling with $D_N$. This converts the degeneracy at fixed $m_{\nu }$ into measurable, basis-invariant fingerprints.