The source paper showed that in 4d N=1{\cal N}=1 type IIA Calabi–Yau orientifolds with intersecting D6-branes, tiny Dirac neutrino masses mν≃gν⟨H⟩m_\nu \simeq g_\nu\langle H\rangle can arise without fine-tuned Yukawas: the right-handed neutrino kinetic term scales as Kνν≃1/gν2K_{\nu\nu} \simeq 1/g_\nu^2, where gν≃10−14−10−12g_\nu \simeq 10^{-14}-10^{-12} is the gauge coupling of the massive U(1)νU(1)_\nu under which the νR\nu_R are charged, fixing simultaneously the string scale Ms≃10−700M_s \simeq 10-700 TeV, two large hidden dimensions felt only by the νR\nu_R sector at 0.1−5000.1-500 eV, and a tower of νR\nu_R-like states. However, the demonstration used only semi-local, illustrative quivers. The open problem is to realize this mechanism in a complete, Ramond–Ramond (RR) tadpole-free type II orientifold compactification with a Standard-Model-like spectrum — including a concrete stabilization of the modulus UU at u≃gν−2u \simeq g_\nu^{-2} and of the remaining compact dimensions, verification that Euclidean D2-brane instantons do not generate unwanted νR\nu_R Majorana masses, an analysis determining whether a μHuHd\mu H_uH_d term is generated or absent and whether an acceptable Higgs/Higgsino sector survives, and a check of baryon stability against higher-dimensional operators at the low string scale.

Background

Whether neutrinos are Dirac or Majorana particles, and why their masses are so tiny, is unresolved. Dirac masses require Yukawa couplings Yν≃10−12−10−13Y_\nu \simeq 10^{-12}-10^{-13} (relative to the Higgs vacuum expectation value ⟨H0⟩\langle H_0\rangle), which look fine-tuned; the see-saw mechanism avoids this by making neutrinos Majorana with heavy right-handed partners. It is an old question how either option is realized in string theory.

A standard laboratory is 4d N=1\mathcal{N}=1 type IIA string theory compactified on a Calabi–Yau threefold orientifold (a quotient of the compact space by a symmetry combined with world-sheet parity reversal, producing orientifold O6-planes), with D6-branes (extended objects filling four-dimensional spacetime and wrapping three-dimensional cycles of the compact space) intersecting at angles. Chiral Standard-Model-like spectra arise at the intersections, with gauge group U(3)×U(2)×U(1)nU(3)\times U(2)\times U(1)^n; the U(1)U(1) factors beyond hypercharge acquire Stueckelberg masses through B∧FB\wedge F couplings to two-forms BB from the Ramond–Ramond sector, so that e.g. lepton number is a massive gauge symmetry. The perturbative spectrum always includes massless right-handed neutrinos νR\nu_R, but Euclidean D2-brane instantons (E2-branes wrapping internal three-cycles and intersecting the νR\nu_R-localizing branes) can generate exponentially suppressed νR\nu_R Majorana masses if their fermionic zero modes permit; searches in large classes of semi-realistic orientifolds found no example with the required zero-mode structure.

The source paper (Casas–Ibáñez–Marchesano, JHEP 01 (2025) 083) takes the Dirac route, building on the companion result (same authors, arXiv:2403.09775) that at infinite distance in complex-structure moduli space Yukawa couplings scale as Y≃u−rY \simeq u^{-r} with 1/4≤r≤11/4 \le r \le 1, while towers of light states carrying the quantum numbers of the massless quarks and leptons — called gonions — descend below the string scale. In a limit where a single complex-structure modulus UU grows (u→∞u \to \infty), two compact dimensions become large and are felt only by the νR\nu_R sector, the right-handed-neutrino field metric grows as Kνν≃uK_{\nu\nu} \simeq u, and the neutrino Yukawa obeys Yν≃u−1/2≃gνY_\nu \simeq u^{-1/2} \simeq g_\nu, where gνg_\nu is the coupling of the massive U(1)νU(1)_\nu under which the νR\nu_R are charged. Fixing gν≃10−14−10−12g_\nu \simeq 10^{-14}-10^{-12} to match oscillation data then fixes all scales at once: mν≃gν⟨H⟩m_\nu \simeq g_\nu \langle H\rangle, a νR\nu_R-gonion and Kaluza–Klein/winding tower at m0≃gν2MP≃0.1−500m_0 \simeq g_\nu^2 M_{\rm P} \simeq 0.1-500 eV, and a string scale Ms≃gνMP≃10−700M_s \simeq g_\nu M_{\rm P} \simeq 10-700 TeV, with a light U(1)νU(1)_\nu vector boson of mass 10−3−0.110^{-3}-0.1 eV or heavier. Swampland arguments reinforce the Dirac option: the non-supersymmetric Anti-de-Sitter instability conjecture (the conjecture that stable non-supersymmetric AdS vacua are inconsistent with quantum gravity), applied to the Standard Model on a circle, requires the lightest neutrino to be Dirac and light, mν,1≲a Λcc1/4m_{\nu,1} \lesssim a\,\Lambda_{\rm cc}^{1/4} with aa of order one and Λcc\Lambda_{\rm cc} the cosmological constant.

The demonstration, however, is semi-local: the explicit five-stack SM-like quiver is an illustrative toroidal setup, not a complete Calabi–Yau compactification with verified Ramond–Ramond tadpole cancellation (the homological condition that total D-brane charge cancels the orientifold charge, required for consistency). The paper explicitly leaves open the stabilization mechanism fixing UU at u≃gν−2u \simeq g_\nu^{-2} and the other four compact dimensions, the E2-instanton analysis (Majorana masses, and the fate of the μHuHd\mu H_uH_d term that U(1)νU(1)_\nu forbids perturbatively), baryon stability once operators of dimension higher than five are included, and a dedicated study of the experimental constraints on the weakly coupled U(1)νU(1)_\nu vector boson, whose coupling structure differs from standard dark-photon or B−LB-L benchmarks.

Problem Statement

Construct a complete, globally consistent 4d N=1\mathcal{N}=1 type IIA Calabi–Yau orientifold compactification with intersecting D6-branes (or a demonstrably equivalent realization in another duality frame of the string landscape) whose Ramond–Ramond tadpole cancellation and full massless spectrum are verified, which carries a Standard-Model-like chiral spectrum, and in which all conditions of the small-Dirac-neutrino-mass mechanism of the source paper hold simultaneously:

  1. There is an asymptotic regime of a complex-structure modulus UU (u→∞u \to \infty) in which exactly two compact dimensions become large and are felt only by the νR\nu_R (right-handed neutrino) sector, the neutrino Yukawa couplings scale as Yν≃u−1/2≃gνY_\nu \simeq u^{-1/2} \simeq g_\nu with gν≃10−14−10−12g_\nu \simeq 10^{-14}-10^{-12} so that mν≃gν⟨H⟩m_\nu \simeq g_\nu \langle H\rangle (with ⟨H⟩\langle H\rangle the Higgs vacuum expectation value) matches neutrino-oscillation data, while the quark and charged-lepton Yukawa couplings and the Standard-Model gauge couplings remain unsuppressed.

  2. A concrete dynamical mechanism stabilizes UU at u≃gν−2u \simeq g_\nu^{-2} — together with the remaining compact dimensions — without destabilizing this hierarchy, with the induced string scale Ms≃gνMP≃10−700M_s \simeq g_\nu M_{\rm P} \simeq 10-700 TeV (where MPM_{\rm P} is the reduced Planck mass) and the tower of νR\nu_R-gonion states (light string excitations carrying νR\nu_R quantum numbers) and Kaluza–Klein/winding modes at m0≃gν2MP≃0.1−500m_0 \simeq g_\nu^2 M_{\rm P} \simeq 0.1-500 eV.

  3. The non-perturbative sector is controlled in the complete geometry: Euclidean D2-brane instanton zero modes are analyzed, showing that induced νR\nu_R Majorana masses remain sufficiently suppressed to preserve the Dirac character; it is determined whether a μHuHd\mu H_u H_d term (the supersymmetric Higgsino mass term, forbidden perturbatively by U(1)νU(1)_\nu in the five-stack quiver) is generated or absent, and whether an acceptable Higgs/Higgsino sector survives; and baryon stability is checked against higher-dimensional operators at the low string scale.

The systematic experimental-constraint analysis of the light U(1)νU(1)_\nu vector boson and the collider/astrophysical/dark-matter phenomenology of the predicted spectrum are natural follow-ups identified by the source, but they are separable studies and do not belong to the completion boundary of this construction problem.

References

  1. Gonzalo F. Casas, Luis E. Ibáñez, Fernando Marchesano, 'On small Dirac Neutrino Masses in String Theory', JHEP 01 (2025) 083, DOI 10.1007/JHEP01(2025)083, arXiv:2406.14609, https://arxiv.org/abs/2406.14609
  2. Gonzalo F. Casas, Luis E. Ibáñez, Fernando Marchesano, 'Yukawa Couplings at Infinite Distance and Swampland Towers in Chiral Theories', JHEP 09 (2024) 170, DOI 10.1007/JHEP09(2024)170, arXiv:2403.09775, https://arxiv.org/abs/2403.09775
  3. Gonzalo F. Casas, Luis E. Ibáñez, 'Moduli Self-Fixing', JHEP 06 (2026) 033, DOI 10.1007/JHEP06(2026)033, arXiv:2507.05345, https://arxiv.org/abs/2507.05345
  4. Mudassar Sabir, Adeel Mansha, Tianjun Li, Zhi-Wei Wang, 'Fermion Masses and Mixings in String Theory with Dirac Neutrinos', Phys. Rev. D 111 (2025) L071702, DOI 10.1103/PhysRevD.111.L071702, arXiv:2407.19458, https://arxiv.org/abs/2407.19458
  5. Min-Seok Seo, 'Species scale associated with Weinberg operator and bound on Majorana neutrino mass', JHEP 12 (2025) 001, DOI 10.1007/JHEP12(2025)001, arXiv:2501.02682, https://arxiv.org/abs/2501.02682
  6. Jonathan J. Heckman, Cumrun Vafa, Timo Weigand, Fengjun Xu, 'Dark Dimension and the Grand Unification of Forces', Phys. Rev. D 111 (2025) 046014, DOI 10.1103/PhysRevD.111.046014, arXiv:2409.01405, https://arxiv.org/abs/2409.01405
  7. P. S. Bhupal Dev, Pedro A. N. Machado, Ivan Martinez-Soler, 'Pseudo-Dirac Neutrinos and Relic Neutrino Matter Effect on the High-energy Neutrino Flavor Composition', Phys. Lett. B 862 (2025) 139306, DOI 10.1016/j.physletb.2025.139306, arXiv:2406.18507, https://arxiv.org/abs/2406.18507