A predictive calculation of the electron–muon mass ratio: a finite radiative electron mass without unmeasured parameters in a unified gauge theory

The electron–muon mass ratio $m_e/m_\mu \approx 1/207 \sim \alpha$ (where $\alpha$ is the electromagnetic fine-structure constant) is an unexplained input of the Standard Model: both lepton masses are free Yukawa parameters. Georgi and Glashow (1973) showed that in unified electroweak gauge theories whose symmetry forbids a tree-level electron mass and its counterterm, the electron mass is a finite, calculable one-loop effect of order $\alpha m_\mu$; but in their explicit SU(3)×SU(3)×SU(3) and SU(3)×U(1) models the predicted value depends on unmeasured parameters — masses of unobserved vector bosons, mixing angles, and a heavy-lepton mass $m_X$ — so the approach does not yield a clean, parameter-free prediction of the electron–muon mass ratio. The open problem is to construct a phenomenologically viable renormalizable unified gauge theory in which the electron mass is finite and calculable and the electron–muon mass ratio is predicted without such unmeasured-parameter dependence, ideally parameter-free from measured constants such as $\alpha$, $m_\mu$, and the weak mixing angle. No accepted parameter-free prediction of $m_e/m_\mu$ exists to date; the charged-lepton piece of the flavor puzzle remains open.

Background

In the Standard Model of particle physics, charged-lepton masses arise from Yukawa couplings — free parameters of the Lagrangian that couple each lepton to the Higgs field — so the electron–muon mass ratio $m_e/m_\mu \approx 1/207$ is an input, not a prediction. The smallness of this ratio, comparable to the electromagnetic fine-structure constant $\alpha \approx 1/137$, has long suggested that the electron mass might be "entirely electromagnetic in origin." In ordinary renormalized perturbation theory this idea fails: a fermion's radiative self-mass is either exactly zero, protected by some unbroken symmetry, or divergent, in which case the infinity is absorbed into a counterterm (a parameter added to the Lagrangian whose value is fixed by a renormalization condition), leaving the physical mass arbitrary.

Renormalizable spontaneously broken gauge theories changed the situation. In such theories the gauge symmetry is broken not by explicit mass terms but by scalar fields acquiring nonzero vacuum expectation values, and every divergence is cancelled by a counterterm already allowed by the gauge symmetry. Weinberg's 1967 SU(2)×U(1) model of weak and electromagnetic interactions is the prototype. Weinberg (1972) then observed that if the representation content of the fields forbids a zeroth-order fermion mass, the corresponding mass counterterm is forbidden as well, so the mass generated entirely by higher-order (radiative) effects is finite and calculable; Georgi and Pais (1974) systematized such "calculability" conditions, distinguishing symmetry protections of several kinds (representation content, accidental symmetries of the scalar potential such as in Weinberg's pseudo-Goldstone-boson analysis, and constraints imposed by renormalizability).

Georgi and Glashow (1973) applied these ideas to the electron–muon mass ratio. They first showed that the direct attempt fails in Weinberg's chiral SU(3)×SU(3) model — in which the known leptons form a Konopinski–Mahmoud triplet $(\mu^+, \nu, e^-)$, an older assignment that treats the muon as the antiparticle of a heavy electron: the same one-loop diagrams (exchange of a doubly charged massive vector boson $W$ of electromagnetic-strength coupling through a muon line, which would give $m_e \sim \alpha m_\mu$) also force a vacuum expectation value on a scalar coupled to the electron, which reintroduces an electron-mass counterterm and hence an arbitrary mass. They then constructed two models that evade this: an SU(3)×SU(3)×SU(3) gauge model, and an SU(3)×U(1) model with an additional heavy charged lepton $X$. In both, the electron is massless at tree level and acquires a finite, calculable mass of order $\alpha m_\mu$ at one loop.

The remaining difficulty is predictivity. In both models the computed electron mass depends on masses of unobserved superheavy vector bosons, on mixing angles, and, in the SU(3)×U(1) case, on the heavy-lepton mass $m_X$, which must be of order a few hundred GeV or more; the SU(3) gauge structures also carry gauge anomalies (quantum violations of classical symmetries that make a theory inconsistent unless they cancel) and are difficult to extend to hadrons. The authors concluded that a parameter-free prediction of the electron–muon mass ratio "certainly seems unlikely" within these ideas, though one might "at least... do slightly better" (for example by eliminating the need for more than one gauge coupling constant, at a prohibitive cost in extra fields), and they expressed the hope that their ideas would be useful "in the search for better models."

In the modern Standard Model the electron and muon Yukawa couplings remain free parameters, and explaining the pattern of fermion masses — the "flavor problem" — is one of the field's central open questions. The radiative-mass program initiated by the 1973 paper (light fermion masses as loop effects) continues to the present day without having produced an accepted parameter-free prediction of $m_e/m_\mu$; the best-known partial result is due to Barr and Zee (1978), who obtained

$\frac{m_e}{m_\mu} = N\,\frac{\alpha}{\pi\sin^2\theta_W}$

where $\theta_W$ is the weak mixing angle (a measured parameter fixing the relative strengths of the neutral weak and electromagnetic interactions) and $N$ is a model-dependent pure number of order unity.

Problem Statement

Construct a renormalizable unified gauge theory of weak and electromagnetic interactions (possibly embedded in a larger unification), consistent with established phenomenology — in particular electroweak precision data, neutral-current constraints, and bounds on lepton-flavor violation such as $\mu\to e\gamma$ — in which:

(i) the gauge symmetry forbids both a tree-level electron mass and an electron-mass counterterm, so that the electron mass is a finite, calculable radiative effect of order $\alpha m_\mu$, with at most the muon mass appearing as a lepton-mass counterterm; and

(ii) the predicted electron–muon mass ratio $m_e/m_\mu$ agrees with the measured value $m_e/m_\mu \approx 1/207$ and is determined by measured quantities (such as $\alpha$, $m_\mu$, and the weak mixing angle $\theta_W$) together with pure numbers uniquely fixed by the model's symmetries, with no adjustable model choices and no dependence on the masses of unobserved vector bosons, mixing angles, or heavy-lepton masses beyond the reach of experiment.

The search for such a theory requires specifying the gauge group and matter content, giving the symmetry argument that no electron-mass counterterm is allowed, exhibiting the loop computation of the electron mass with its residual parameter dependence quantified, and confronting the model with phenomenological constraints. This is the 'search for better models' posed by the concluding section of Georgi and Glashow (1973); the single overarching objective is a predictive dynamical account of the electron–muon mass ratio, of which the model construction, the calculability proof, the loop calculation, and the phenomenological checks are jointly necessary components.

References

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