Independent Research · United Kingdom
Independent Researcher, United Kingdom
Dated: 21 January 2026
Abstract
The charged fermion mass spectrum and leading Cabibbo–Kobayashi–Maskawa matrix elements are derived from a small set of geometrically fixed moduli in a Type IIB string compactification. Three moduli values, previously stabilized by fixed-point and entropy arguments, control all hierarchical structures: a bulk Kähler modulus τb = 220, a small-cycle Kähler modulus τs = 12.4, and a complex structure modulus U∗ fixing the lepton-sector suppression coefficient κ = π/Im(U∗) = 2.85. Quark and lepton Yukawa hierarchies arise from powers of the universal ratio ε = τs/τb = 0.056, with integer or fractional weights determined by localization geometry. Localized fermions exhibit exponential suppression governed by κ, while bulk-extended fields undergo a power-law suppression controlled by the same coefficient. The electron mass is shown to arise from a bulk normalization effect, yielding me/mτ = ε2.85 without introducing new parameters. Tree-level mass ratios reproduce observed hierarchies within 13%; inclusion of standard Quantum Chromodynamics running eliminates the down-sector discrepancy entirely, yielding agreement at the few-percent level across all sectors. The framework provides a unified geometric origin of charged fermion masses and quark mixing, tightly linked to vacuum energy stabilization.
The origin of fermion mass hierarchies and quark mixing angles remains one of the central open problems in high-energy physics. While the Standard Model accommodates these quantities through arbitrary Yukawa couplings, ultraviolet-complete theories are expected to constrain or determine them from more fundamental principles. String theory compactifications offer a natural arena in which hierarchical structures may emerge from geometry, topology, and moduli stabilization.
This work presents a derivation of the charged fermion spectrum within a Type IIB string compactification, using only moduli values fixed independently by vacuum energy and entropy considerations [1]. The construction does not introduce continuous flavor parameters and relies solely on discrete localization data and Kähler normalization effects. A central result is that a single geometric coefficient controls both exponential and power-law suppressions, depending on whether a fermion wavefunction is localized on a small cycle or extends into the bulk geometry.
The analysis relies on three moduli values that are treated as fixed inputs, derived elsewhere from fixed-point stabilization of the scalar potential:
These values define a universal small parameter
No additional continuous parameters are introduced in the flavor sector.
Localized matter fields on magnetized D7-branes acquire Yukawa couplings suppressed by overlap integrals of their internal wavefunctions. For fields localized on the small cycle, this suppression is exponential and controlled by the complex structure modulus:
This relation governs the muon-to-tau mass ratio and fixes the coefficient κ = 2.85.
Bulk-extended fields, by contrast, are normalized by the Kähler metric rather than by exponential localization. Supersymmetric Kähler analyses show that bulk fields scale as
while localized fields scale as
As a result, bulk-extended fermions acquire power-law suppressions in ε rather than exponential ones. Importantly, the exponent of this power law is fixed by the same coefficient κ that controls the localized exponential suppression.
The charged lepton hierarchy illustrates the transition between suppression regimes. The muon and tau are localized on the same matter curve, yielding
in excellent agreement with observation.
The electron wavefunction extends into the bulk geometry. Its mass is therefore suppressed by a power of ε rather than by an exponential:
This reproduces the observed ratio me/mτ = 2.9 × 10−4 at the few-percent level, without introducing an independent parameter. The appearance of the same coefficient κ in both relations reflects a geometric unity between localized and bulk suppression mechanisms.
Quark masses are organized by integer modular weights nf, determined by flux and localization data. At tree level,
A representative assignment consistent with the geometry is:
The resulting bare hierarchies reproduce the correct ordering and scale of quark masses. Residual O(1) factors arise from Kähler normalization and are expected from the geometry. Including standard Quantum Chromodynamics running from the string scale to low energies removes the remaining down-sector discrepancy, yielding agreement at the few-percent level.
Quark mixing angles follow from the same geometric expansion parameter. Identifying the Wolfenstein parameter as λ = ε1/2 gives
where A = 0.82 and ρ, η are the usual Wolfenstein parameters. These values agree with observation at the 5–12% level. A derivation of ρ and η from complex structure phases is left for future work.
The charged fermion spectrum emerges from a small set of fixed geometric inputs. Exponential and power-law suppressions are not independent mechanisms but two manifestations of the same underlying coefficient, determined by the complex structure modulus. The electron mass, often treated as anomalously small, is shown to follow naturally from bulk Kähler normalization without introducing additional parameters.
The framework tightly links flavor hierarchies to vacuum energy stabilization and moduli geometry. While the present analysis is performed at tree level, radiative corrections are well understood and do not spoil the geometric structure.
A unified geometric origin of charged fermion masses and quark mixing has been presented within a Type IIB compactification. Three fixed moduli values determine all hierarchical structures, with no continuous flavor parameters. The results suggest a deep connection between moduli stabilization, vacuum energy, and flavor physics, and motivate further investigation of complex structure phases and neutrino masses within the same framework.