In the minimal SO(10) grand unified theory with Higgs fields in the 10H10_H, 16H+16‾H16_H+\overline{16}_H, and adjoint 45H45_H representations, the heavy right-handed-neutrino Majorana mass vanishes at tree level and is generated only at two loops (Witten, 1980). The literal 1980 minimal model is excluded by its own charged-fermion mass relations and by modern neutrino and cosmological data, but the mechanism-level question it raised remains open: does Witten's two-loop radiative right-handed-neutrino mass admit a quantitatively viable realization in minimal SO(10) grand unification, with the full two-loop integrals evaluated and all constraints — gauge coupling unification, proton decay limits, the seesaw scale, large lepton mixing, the light-neutrino mass scale, and, where claimed, thermal leptogenesis — satisfied simultaneously? The literature from the 1980 original through the split-supersymmetric SO(10) revival and the 2014–2025 minimal flipped SU(5) program addresses the mechanism in these settings without settling the minimal SO(10) question. A recurring obstacle is the dichotomy between gauge unification constraints — which push the B−LB-L breaking scale below the unification scale in non-supersymmetric models or cancel the loop in supersymmetric ones — and the absolute size of the two-loop mass scale.

Background

Grand unified theories (GUTs) embed the Standard Model gauge group into a single simple group, so that the strong, weak, and electromagnetic couplings merge into one at a very high energy scale, of order 101510^{15}–101610^{16} GeV. The gauge group SO(10) is special because each fermion generation — the quarks, the charged leptons, the neutrino, and one additional particle, the right-handed neutrino νR\nu_R — fits into a single 16-dimensional spinor representation. A vacuum expectation value (VEV), the nonzero value a quantum field can settle into in empty space, of a Higgs field in the spinor representation 16H16_H (together with its conjugate 16‾H\overline{16}_H) breaks SO(10) and in particular breaks the symmetry under B−LB-L, the difference of baryon and lepton numbers, which is what allows the heavy neutrino to acquire a Majorana mass: a mass term that pairs νR\nu_R with itself rather than with an antiparticle, forbidden in the Standard Model. In the product of two fermion spinors, 16⊗1616 \otimes 16, the component containing the Standard-Model singlet νRνR\nu_R\nu_R lies only in the 126-dimensional representation, so in a model whose Higgs fields are the 10H10_H (generating fermion Dirac masses), the 16H+16‾H16_H+\overline{16}_H, and the 45H45_H (the adjoint, the representation in which the gauge bosons themselves transform), the νR\nu_R Majorana mass MRM_R vanishes at tree level.

Witten (1980) showed that MRM_R is nevertheless generated radiatively at two loops: the diagram contains one Higgs-1010 propagator and two adjoint gauge vertices, whose group-theory structure 10⊗45⊗4510 \otimes 45 \otimes 45 contains a 126; two insertions of the 16H16_H VEV, supplying the B−LB-L breaking; and one mixing insertion between the Higgs 1010 and 1616 with angle ε\varepsilon, generated by a trilinear 10⋅16⋅1610 \cdot 16 \cdot 16 coupling. A discrete symmetry could forbid that trilinear coupling, in which case the mass is absent to all orders. Light neutrino masses then follow from the seesaw relation mν≈mq2/MRm_\nu \approx m_q^2/M_R, with the neutrino Dirac mass equal to the up-quark mass mqm_q. For unification scale M∼1015M \sim 10^{15} GeV and ε∼1/10\varepsilon \sim 1/10, the mechanism gives MR∼107mqM_R \sim 10^7 m_q, i.e. MR∼105M_R \sim 10^5–10610^6 GeV for the electron family and ∼1010\sim 10^{10}–101110^{11} GeV for the tau family, corresponding to light masses mνe∼1m_{\nu_e} \sim 1 eV, mνμ∼100m_{\nu_\mu} \sim 100 eV, mντ∼1m_{\nu_\tau} \sim 1–1010 keV — the last exceeding by two orders of magnitude the cosmological upper bound on the sum Σmν\Sigma m_\nu of the light-neutrino masses. The paper itself states that the simplest model is ruled out because the relations md=mem_d = m_e and ms=mμm_s = m_\mu forced by a single 10H10_H are not satisfied, and that without knowing all the superheavy masses and couplings entering the two-loop diagram the actual neutrino masses cannot be predicted.

Whether this two-loop mechanism — as opposed to the tree-level alternative with a Higgs 126 — can be quantitatively realized in a minimal grand unified theory survived the demise of the literal 1980 model. Its modern study proceeds in two main frameworks. In supersymmetric SO(10) with split supersymmetry (scalar particles at the unification scale, their fermionic partners near a TeV), Bajc and Senjanović revived the mechanism with extended Yukawa sectors (10H+10H′10_H+10'_H or 10H+120H10_H+120_H), obtaining large lepton mixing angles — contrasting with the small Cabibbo–Kobayashi–Maskawa (CKM) quark mixing angles, the measured pattern of quark flavor mixing — and, in the 10H+120H10_H+120_H case, degenerate (nearly equal-mass) light neutrinos. In the flipped SU(5) unification — the group SU(5)×U(1)SU(5) \times U(1) with the electric-charge assignments of certain SU(5) representations exchanged — Arbeláez Rodríguez, Kolešová, and Malinský transplanted the mechanism; Harries, Malinský, and Zdráhal fully evaluated the two-loop integrals and derived correlated proton-decay and neutrino-mass predictions; and Fonseca, Malinský, Miřátský, and Zdráhal, together with Malinský, Fonseca, Miřátský, and Zdráhal, studied whether thermal leptogenesis — generation of the cosmic matter–antimatter asymmetry by heavy-neutrino decays — can work when the heavy-neutrino mass scale itself is only a two-loop effect. A recurring obstacle, documented by Arbeláez et al., is the dichotomy between gauge unification, which in non-supersymmetric models drives the B−LB-L breaking scale well below the unification scale and thus suppresses the loop, while supersymmetric settings tend to cancel it, and the required absolute size of the induced mass scale. A third strand analyses the same minimal content without the loop: the vacuum and symmetry-breaking structure of the non-supersymmetric 45H⊕16H45_H\oplus16_H theory (ref 13), and an effective minimal SO(10) with higher-dimensional operators, whose two-loop gauge running and thresholds force the intermediate scale close to the unification scale and constrain the proton-decay lifetime together with the absolute neutrino mass scale (ref 14). Both work in the effective description of this content rather than by evaluating Witten's two-loop diagram, so they constrain the class without settling the mechanism's viability.

Problem Statement

The original 1980 question — whether the minimal O(10) grand unified theory predicts acceptable neutrino masses through the two-loop radiative right-handed-neutrino Majorana mass — is closed negatively: that model fails its own charged-fermion mass relations, and its light-neutrino masses and mixing pattern conflict with modern data. The surviving open core, as framed by the subsequent literature, is: does Witten's two-loop radiative right-handed-neutrino mass mechanism admit a quantitatively viable realization in minimal SO(10) grand unification — that is, in an SO(10) theory whose Higgs sector consists of the 10H10_H, the 16H+16‾H16_H+\overline{16}_H and the adjoint 45H45_H, optionally augmented by the extra Yukawa representations 10H′10'_H and/or 120H120_H, with no 126H126_H-type field (so that the right-handed-neutrino Majorana mass arises only radiatively) and no additional gauge factors — one whose right-handed-neutrino Majorana mass MRM_R is simultaneously consistent with gauge coupling unification and perturbativity, proton decay limits, the observed large lepton mixing, and light-neutrino masses compatible with oscillation data and the cosmological bound on Σmν\Sigma m_\nu (and, where invoked, with thermal leptogenesis)? Supersymmetric (including split-supersymmetric) and non-supersymmetric realizations of this content are both in scope. The question concerns minimal SO(10) constructions with this Higgs content and no additional gauge factors; the tree-level 126-type seesaw, TeV-scale radiative models with additional dark-sector particles, and realizations of the mechanism in minimal flipped SU(5) (refs 8–11) are different classes of model rather than instances of this one.

References

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