17 August 2026

For fixed Standard Model (SM) non-Abelian representations of 15 chiral fermions with arbitrary hypercharges, anomaly cancellation admits the usual assignment and a distinct nonstandard solution. In the latter, the colored exotic quark and exotic lepton weak doublets and exotic lepton singlet have zero hypercharge, whereas the two colored exotic quark singlets carry opposite hypercharges $-q$ and $q$. The exotic neutral lepton singlets naturally play the role of the heavy neutrinos. The minimal model with two exotic lepton copies gives a rank-two seesaw, the minimal seesaw model, with one massless active neutrino and predicts $m_{ββ} = 1.2 \text{-} 4.1 \, \mathrm{meV}$ for normal ordering or $15.9\text{-}48.9 \,\mathrm{meV}$ for inverted ordering. In a direct SM realization, generating exotic quark masses through the SM Higgs mechanism forces the exotic quarks to carry electric charges $\pm 1/2$. A separate $\mathrm{SU}(2)_{L'}$ realization of the nonstandard solution can allow exotic quarks from several TeV to $10\,\mathrm{TeV}$ with order-one Yukawa couplings. In this model, the charged exotic quarks and leptons carry electric charges $\pm q$. In both cases, the lightest exotic quark and lepton are stable, but suitable choices of their charges and masses can satisfy experimental constraints.

Fermionic field types in arbitrary signature are often inferred from complex Spin representations alone, which obscures distinctions associated with real structures, disconnected spacetime reflections, and action-level bilinear constraints. We give a convention-explicit classification that treats the full real Clifford algebra and its even subalgebra separately, thereby distinguishing pinors from spinors and Pin-equivariant Majorana-type structures from structures that are only Spin-equivariant. Adjoint, complex-conjugation, and transposition intertwiners are combined with elementary and collective reflection lifts into a unified sign calculus. The resulting local Clifford classification is organised into thirty-two periodicity sectors determined by the Bott class and the total dimension modulo eight, without including theory-dependent gauge, anomaly, or global constraints. For each sector, we identify the available Dirac, Weyl, Majorana, symplectic Majorana, Majorana-Weyl, and symplectic Majorana-Weyl fields. The existence of an invariant pairing is distinguished from intertwiner compatibility, Grassmann and chiral selection rules, covariant index contraction, and Hermiticity of local bilinear operators.

We compute the one-loop effects of vector-like fermions (VLFs) on the Higgs-strahlung cross-section, the flagship precision observable of future $e^+e^-$ Higgs factories such as FCC-ee. We consider four benchmark extensions of the Standard Model (SM): two in which a VL quark (VLQ) or VL lepton (VLL) doublet, together with its singlet partners, couples to the Higgs boson through purely internal Yukawa interactions, and two in which a single VL singlet mixes directly with the third-generation charged lepton or neutrino. Working throughout in the on-shell renormalization scheme and imposing perturbativity bounds on the relevant couplings under renormalization-group (RG) running, we compute both the universal self-energy and the non-universal vertex contributions to the Higgs-strahlung amplitude at one loop, together with the oblique parameters $S$ and $T$ and the $H\toγγ$ rate as complementary cross-checks. In the two Yukawa-driven scenarios, we find that a sizable fraction of the parameter space still allowed by current LHC searches produces shifts in $σ(ZH)$ at or above the per-mille sensitivity envisioned for FCC-ee, offering a radiative probe of these couplings. In the two mixing-driven scenarios, existing electroweak-precision bounds on the relevant mixing angles already preclude an observable effect, although we characterize the underlying loop dynamics in full generality for future reference. These results identify the VL Yukawa couplings, rather than any residual mixing with the SM fermions, as the more promising target for Higgs-strahlung precision measurements at a future $e^+e^-$ collider.

The Shimura curve of discriminant $D$ for $D=14, 15$ is uniformized by a subgroup of an arithmetic quadrilateral Fuchsian group $(2, 2, 2, q)$, where $q=4, 6$. We relate the generator of the ring of quaternionic modular forms on this Shimura curve to explicit Heun functions for the quadrilateral group. We also discuss how the Picard-Fuchs equation of the associated family of abelian surfaces has solutions that are modular forms on $X^{D}(1) / W_{D}$, where $W_D$ is the full group of Atkin-Lehner involutions. This leads us to completely describe the rational exceptional sets of the associated Heun functions, and the algebraic values attained by the Heun function on these points, for example $${\rm He}\left( 81, \frac{1}{2}; \frac{1}{3}, \frac{1}{6}, \frac{1}{2}, \frac{1}{2};-\frac{729}{112}\right)= \left( \frac{2^2 \cdot 3^3 \cdot 5^3}{7^5} \right)^{\frac{1}{6}}$$.

A classical class number relation of Hurwitz expresses the Fourier coefficients of the product of a unary theta function and the class number generating function. Here, we establish an infinite family of analogous class number relations obtained by replacing the unary quadratic form $m^2$ by positive-definite binary quadratic forms. These identities involve Cohen's generalized class numbers and depend only on the genus of the underlying quadratic form. For this, we construct a genus-dependent level-lowering operator.

We study the sensitivity of IceCube to blazar-boosted dark matter in a fermionic dark matter model with a massive vector mediator coupling to quarks. To this aim, we compute the diffuse flux arising from a sample of 324 blazars with proton spectra inferred from multiwavelength observations, adopting conservative dark matter spike profiles around the central supermassive black holes and consistently accounting for attenuation effects during propagation through the Earth. The dark matter-nucleon scattering cross section is evaluated by including elastic, resonant single pion production, and deep inelastic contributions, with particular emphasis on resonant single-pion production channels in order to smoothly cover the transition between the elastic and deep inelastic regimes. Using IceCube neutrino data, we derive constraints on the parameter space of the model and show that this detection strategy can surpass the sensitivity of conventional direct-detection experiments for dark matter masses below $\sim 1$ GeV. We find that the signal is dominated by deep inelastic scattering and is therefore more sensitive to comparatively heavy mediators, while resonance processes provide a reduction of the event rate, reaching up to about $9\%$ near the experimental threshold. Our results demonstrate that IceCube constitutes a powerful probe of sub-GeV dark matter scenarios through the observation of blazar-boosted dark matter.

Rare kaon decays provide sensitive probes of the flavour structure of the Standard Model and of possible new physics. We perform a global analysis incorporating recent experimental results and updated Standard Model predictions, including the latest measurement of $K^+ \to π^+ ν\barν$ and lepton flavour universality observables in $K^+ \to π^+ \ell^+\ell^-$. The fit favours a best-fit point close to the Standard Model, while a second local minimum remains phenomenologically relevant. We define benchmark scenarios associated with these two regions and investigate the prospective sensitivity of NA62 and KOTO-II to the new physics parameter space. We consider projected measurements of $K^+ \to π^+ ν\barν$ and lepton flavour universality observables in $K^+ \to π^+ \ell^+\ell^-$ at NA62, and of $K_L \to π^0 ν\barν$, $K_L \to π^0 e^+e^-$, and $K_L \to π^0 μ^+μ^-$ at KOTO-II. We find that KOTO-II has significant potential to probe and discriminate between the viable new physics scenarios, with NA62 providing complementary sensitivity.