Realistic radiative generation of first- and second-generation fermion masses: the completion problem of Weinberg's third-generation-only framework

In the Standard Model, charged-fermion masses are arbitrary Yukawa couplings spanning several orders of magnitude. Weinberg (Phys. Rev. D 101, 035020 (2020)) revisited an old proposal: only the third generation (top, bottom, tau) gets renormalizable tree-level masses, while second- and first-generation masses arise from one-loop and two-loop radiative corrections, implemented through a horizontal gauge symmetry (a symmetry acting across fermion generations rather than within one) $SO_L(3)\otimes SO_R(3)$ broken by a hidden scalar sector. His explicit construction fails: the loop formulas force rigid mass-ratio relations that contradict observation, no Cabibbo-Kobayashi-Maskawa (CKM) mixing arises, and generating mixing threatens the tree-level mass structure, induces excessive flavor-changing neutral currents, or leaves light pseudo-Goldstone bosons. He posed as an open problem the construction of a realistic theory of this type. Several recent anomaly-free gauged-flavor frameworks realize the tree/one-loop/two-loop structure with realistic charged-fermion spectra and quark mixing. However, each carries acknowledged residual tensions (strange- or down-quark masses 2.4-3 sigma off, new-physics scales pushed to 200 TeV-10^3 TeV, or a vectorlike fermion sector — new fermions whose left- and right-handed components transform identically under the gauge group — enlarged to the point that its free-parameter count matches earlier models), and Weinberg's own construction remains unrepaired. The surviving open core is a complete, clean, natural implementation satisfying all phenomenological constraints simultaneously.

Background

In the Standard Model (SM), each charged fermion mass is proportional to an arbitrary Yukawa coupling — the strength of its interaction with the Higgs field — and these couplings span more than five orders of magnitude, with a pronounced hierarchy: the third generation (top $t$, bottom $b$, tau $\tau$) is far heavier than the second ($c$, $s$, $\mu$), which is in turn far heavier than the first ($u$, $d$, $e$). One long-standing idea is that this hierarchy is dynamical: only the third generation has renormalizable (tree-level) Yukawa couplings, while the lighter generations acquire masses from radiative corrections — loop diagrams in which heavy particles are emitted and reabsorbed. A one-loop diagram naturally suppresses a mass by a factor of order $1/(16\pi^2)\approx 10^{-2}$ and a two-loop diagram by $1/(16\pi^2)^2\approx 10^{-4}$, matching the observed hierarchy. The idea dates to Weinberg's 1972 work on electromagnetic and weak masses and to horizontal-symmetry models of 1978-1979 (Barr and Zee; Wilczek and Zee; Yanagida), where a horizontal or family gauge symmetry acts across generations rather than within them; interest later lapsed, with the top-loop models of Dobrescu and Fox (2008) as a notable exception.

New physics is required: within the SM alone, if only the third generation is massive, an accidental chiral symmetry (one that emerges automatically from the renormalizable interactions rather than being imposed) $U(2)^5$ acting on the first two generations protects their masslessness to all loop orders. A new gauge interaction that treats the generations differently must break this symmetry. A further key requirement is finiteness: if a vanishing tree-level mass is forced by gauge symmetry, particle content, and renormalizability, then no counterterm exists to absorb ultraviolet divergences (the infinities of perturbation theory), so the loop-generated mass is finite and calculable.

Weinberg's 2020 model implements this with gauge group SM $\times\, SO_L(3)\otimes SO_R(3)$: the three generations of left-handed quark and lepton doublets form $(3,1)$ representations, the right-handed quarks and charged leptons form $(1,3)$ representations, and nine electroweak Higgs doublets $\Phi_{ia}$ form $(3,3)$ representations. For a suitable scalar potential, the stationary point has only $\langle\mathrm{Re}\,\Phi_{33}\rangle=\lambda\neq 0$, giving tree-level masses only to $t$, $b$, $\tau$. A hidden sector of electroweak-singlet scalars (no renormalizable couplings to fermions) breaks the remaining $SO_L(3)\otimes SO_R(3)$, making all new gauge bosons heavy. One-loop emission and absorption of the new gauge or scalar bosons then generates second-generation masses (finite, by the counterterm argument above), and two-loop diagrams generate first-generation masses.

Section VII of the paper spells out why this is not realistic. When the new bosons are heavy, the one-loop formulas force rigid mass-ratio relations: gauge-boson dominance gives $m_c/m_t = m_s/m_b = m_\mu/m_\tau$ (Eq. 61), charged-scalar dominance gives $m_c/m_b = m_s/m_t$ (Eq. 62), and neutral-scalar dominance gives $m_c/m_t^3 = m_s/m_b^3 = m_\mu/m_\tau^3$ (Eq. 63) — all in conflict with the observed spectrum. The construction also produces no CKM (Cabibbo-Kobayashi-Maskawa) quark mixing angles; generating mixing requires abandoning the assumption that certain scalar-potential coefficients ($\sigma_N$, $\sigma_{NN'}$) vanish, which protected the naturalness of the vacuum and the third-generation-only tree masses, and raises the risk of flavor-changing neutral currents (FCNC) — flavor-violating processes that are absent at tree level in the SM and are tightly bounded experimentally. Finally, the accidental symmetries of the scalar potential (the reflection $R$ and hidden-sector sign symmetries) cannot simply be left unbroken: without inter-sector couplings a massless Goldstone boson would remain, and with them the couplings must be strong enough to keep the resulting pseudo-Goldstone boson (an approximately massless scalar left over when an approximate symmetry breaks) heavy enough to have escaped detection. Weinberg concluded by hoping the models would provoke new ideas for a realistic theory in which radiative corrections account for the masses of the first and second generations, together with guidance in dealing with the problems that will arise in such a theory.

Problem Statement

Construct a complete, realistic renormalizable extension of the Standard Model in which, in the tree approximation, only the third generation of quarks and charged leptons (top, bottom, tau) acquires mass through renormalizable Yukawa couplings, while the second- and first-generation masses arise respectively from one-loop and two-loop radiative corrections, and in which all of the following are simultaneously satisfied:

(i) the predicted charged-fermion mass spectrum reproduces the observed masses within experimental and lattice-QCD uncertainties (lattice QCD being the numerical simulation of the strong interactions on a discretized space-time grid, which supplies the standard determinations of the light-quark masses) — including the very different mass-ratio patterns of up-type quarks, down-type quarks, and charged leptons — rather than the rigid universal ratios forced by Weinberg's original construction (his Eqs. (61)-(63));

(ii) the observed Cabibbo-Kobayashi-Maskawa quark mixing, including CP violation, emerges from the loop-generated mass matrices without destabilizing the scalar vacuum structure that enforces the third-generation-only tree masses;

(iii) the accidental symmetries of the construction (such as Weinberg's reflection $R$ and the hidden-sector sign symmetries) are broken without leaving any pseudo-Goldstone boson light enough to have been detected;

(iv) all new gauge and scalar bosons are heavy enough to have escaped direct searches and to satisfy experimental bounds on flavor-changing neutral currents, while the loop-generated masses retain their observed magnitudes;

(v) the third-generation-only tree-level mass structure and the vacuum pattern are natural — their protection follows from the gauge symmetry and the particle content rather than from fine-tuned cancellations among the scalar-potential coefficients.

Such a theory may be realized within Weinberg's specific $SO_L(3)\otimes SO_R(3)$ framework — in particular, does any choice of hidden-sector scalar content and inter-sector coupling coefficients repair the wrong mass-ratio predictions while keeping the third-generation tree-level masses and vacuum pattern natural? — or in another gauge-theoretic framework realizing the same tree/one-loop/two-loop structure. This is the program-level question posed in the concluding section of Weinberg's 2020 paper: a realistic theory in which radiative corrections account for the masses of the first and second generations of quarks and leptons, together with guidance in dealing with the problems that will arise in such a theory. The single scientific objective uniting the requested results is a dynamically consistent model whose loop-suppression structure explains the observed generational hierarchy; components (i)-(v) are the jointly required phenomenological faces of that one objective, not independent deliverables.

References

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