The 1972 Georgi–Glashow paper "Spontaneously Broken Gauge Symmetry and Elementary Particle Masses" asked how approximate relations among fermion masses can arise in renormalizable gauge theories whose gauge symmetry is broken only spontaneously, classified zeroth-order mass relations into symmetry-protected, representation-content, and dynamical types, constructed an SU(2)×U(1) model in which the electron mass vanishes at zeroth order and arises as a finite two-loop effect of order α m_μ, and proved a vacuum-alignment theorem stating that unbroken-subgroup mass relations persist under small (radiative) perturbations unless extra massless non-Goldstone scalars exist. The problem posed here is the surviving open core of that program: whether a phenomenologically viable renormalizable gauge theory can be built in which small fermion masses are generated radiatively without dependence on parameters that cannot be measured or constrained. No accepted solution is known, and the question remains an active model-building program, now one strand of the fermion-flavor problem.

Background

By 1972 the electroweak theory built on a spontaneously broken gauge symmetry — a symmetry whose transformations vary over spacetime and whose breaking is realized by the vacuum rather than by explicit symmetry-violating terms in the Lagrangian — had become the leading theory of weak and electromagnetic interactions. Weinberg's "A Model of Leptons" is the archetypal SU(2)×U(1) example. 't Hooft and B. W. Lee had shown that such theories are renormalizable, meaning that ultraviolet divergences can be absorbed into a finite number of redefined parameters (counterterms); explicit ('manifest') breaking of a gauge symmetry — symmetry-violating terms inserted directly into the Lagrangian — by contrast, destroys renormalizability. When a continuous symmetry is broken spontaneously, massless Goldstone bosons — scalar modes unavoidably associated with symmetry breaking by the vacuum — appear, and in a gauge theory they are absorbed by the gauge vector bosons (spin-one force carriers), which thereby acquire masses (the Higgs mechanism), leaving remnant physical scalar particles (Higgs bosons, called 'spinless mesons' in the 1972 literature).

Many observed fermion masses obey approximate relations: the Gell-Mann–Okubo formula (a linear relation among the masses of the eight baryons of the approximate SU(3) flavor symmetry of the strong interactions), the near-degeneracy of proton and neutron (isospin), and the smallness of the electron-to-muon mass ratio. The conventional explanation assumed a hierarchy of interactions in which the stronger interactions have the more extensive symmetry. Georgi and Glashow observed that this explanation is unavailable in spontaneously broken gauge theories, since manifest breaking is forbidden, and asked where approximate mass formulas can come from in such theories. Following Weinberg's contemporaneous analysis of the naturalness of mixing angles and small masses in renormalizable theories, they defined a fermion mass to be calculable when it need not appear as an independent renormalization counterterm — the zeroth-order mass matrix then receives contributions only from gauge-invariant bare masses and from Yukawa couplings to scalar fields with nonzero vacuum expectation values. They classified zeroth-order (tree-level) mass relations into three types: type (1), enforced by a subgroup of the gauge group left unbroken by the vacuum; type (2), enforced by the representation content of the scalar multiplets, i.e. by charge-selection rules that forbid the corresponding Yukawa couplings; and type (3), dynamical accidents of the scalar potential. In an SU(3) model with the scalars in the adjoint (octet) representation they showed that the most general renormalizable potential aligns the vacuum so as to preserve isospin and hypercharge, and that octet selection rules reproduce the Gell-Mann–Okubo formula at zeroth order as a type-(2) relation. In an SU(2)×U(1) chiral model with left- and right-handed lepton doublets (lepton pairs paired by the SU(2) gauge symmetry, whose left- and right-handed chirality components transform differently), selection rules force the electron mass to vanish at zeroth order, and two-loop diagrams with internal vector bosons generate a finite electron mass of order αmμ\alpha m_{\mu}, where α\alpha is the fine-structure constant — the canonical example of a naturally small, radiatively generated fermion mass. Weinberg's companion paper "Electromagnetic and weak masses" supplied the finiteness mechanism: weak-interaction diagrams cancel the ultraviolet infinities of the electromagnetic self-mass diagrams, so the radiative mass is finite and calculable.

The effective potential — the potential whose extrema determine the true vacuum once quantum corrections are included — can qualitatively change the vacuum structure. In an appendix, Georgi and Glashow proved a vacuum-alignment theorem: if the zeroth-order vacuum leaves a subalgebra of the symmetry group unbroken, then the relations enforced by that unbroken subgroup persist under any sufficiently small, symmetry-invariant perturbation of the Lagrangian (such as higher-order radiative corrections), provided the theory contains no massless spinless particles other than the Goldstone modes. Contemporaneously, Coleman and Weinberg showed in massless scalar electrodynamics that the one-loop effective potential generates a new symmetry-breaking minimum far from the classical extremum, with a mass scale not present in the classical Lagrangian ('dimensional transmutation'). The interplay between that phenomenon and the Georgi–Glashow theorem, and the regime with extra massless non-Goldstone scalars — which produces pseudo-Goldstone bosons, scalars that are massless only accidentally at tree level and acquire small radiative masses — was clarified shortly afterwards in E. Weinberg's thesis and in the Georgi–Pais theorems.

Problem Statement

The problem is the surviving core of the questions raised by the Georgi–Glashow (1972) framework: in a renormalizable gauge theory whose gauge symmetry is broken only spontaneously, construct a mechanism that generates phenomenologically important small fermion masses — the electron mass being the original example — such that (a) the mass vanishes at zeroth order by a symmetry of the theory (protection by an unbroken subgroup, or by the representation content of the scalar sector) and first arises at a well-defined order of perturbation theory as a finite, calculable quantity requiring no independent renormalization counterterm; and (b) the numerical value of the radiatively generated mass depends only on parameters fixed by accessible experimental data, rather than — as in the 1972–1973 two-loop electron-mass models, whose results depend on the masses of unobserved vector bosons and other free parameters — on quantities that cannot be measured or constrained. A viable construction therefore consists of a definite model (gauge group, fermion and scalar representations, charge assignments, scalar potential and its vacuum-alignment pattern) in which the target mass vanishes at zeroth order by an unbroken-subgroup or representation-content selection rule and first arises at the first nonvanishing loop order as a finite, calculable quantity requiring no independent mass counterterm, whose generated mass depends only on parameters fixed by accessible experimental data, and which is phenomenologically viable — the radiatively generated mass or masses matching the observed value or values, and the model satisfying existing flavor-changing, CP-violating and collider bounds. The question is settled either by such a construction or, on equal terms, by the complementary no-go result: a proof that within the class of renormalizable gauge theories whose gauge symmetry is broken only spontaneously — the class the construction route ranges over, with any additional assumptions stated explicitly — a radiatively generated fermion mass cannot be simultaneously finite and calculable and dependent only on experimentally accessible parameters. The companion questions raised by the same paper — whether radiative corrections can create new extrema of the scalar effective potential far from the zeroth-order extrema, and the fate of the unbroken-subgroup persistence theorem when extra massless non-Goldstone scalar directions exist — belong to the same scientific objective (the radiative stability of the zeroth-order mass pattern) but were substantially settled between 1973 and 1977.

References

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