In 1959 Steven Weinberg proposed what he described as seemingly the only test of the Feynman–Gell-Mann conserved-vector-current (CVC) theory for muon interactions completely free of nuclear-structure ambiguities: compare the rate of a 0⁺→0⁺ muon-capture transition, μ⁻ + A(0⁺, T=0) → ν + B(0⁺, T=1), with the cross section for inelastic electron scattering to the isobaric-analog state A* at the same momentum transfer q2≈mμ2q^2 \approx m_\mu^2 — a ratio in which the unknown nuclear form factor cancels and the axial current cannot contribute. The only suitable level known then was the 0⁺, T=1 state of ²⁸Al at 972.4 keV; identifying further suitable levels was left open. The later literature settled the motivation but not the test: CVC was confirmed by other channels and is structural in the Standard Model; the realized A = 28 program instead measured the axial-dominated 1⁺, 2201 keV Gamow–Teller transition (ending in the "²⁸Si puzzle"); the capture half of the 0⁺ test exists only as three imprecise, mutually discrepant γ-yield measurements; the required longitudinal electron-scattering form factor was never measured; and no screening of additional 0⁺, T=1 candidate levels exists. The problem — execute the form-factor-free ratio test and establish the level inventory — remains open.

Background

Ordinary muon capture (OMC) is the semileptonic weak process μ⁻ + (A,Z) → ν_μ + (A,Z−1): a negative muon stopped in matter forms a muonic atom, cascades to its 1s orbit, and is captured by the nucleus, converting a proton into a neutron. Because the emitted neutrino carries roughly 100 MeV, the process probes the nuclear weak current at a squared momentum transfer q2≈mμ2q^2 \approx m_\mu^2 (mμm_\mu is the muon mass, about 105 MeV/c2c^2) — two orders of magnitude above nuclear β decay.

In 1958 Feynman and Gell-Mann proposed the conserved vector current (CVC) hypothesis: the vector part of the hadronic weak current is, together with the electromagnetic current, a member of an isospin triplet. Isospin is the near-exact nuclear symmetry treating the proton and neutron as two states of one particle; CVC therefore fixes the weak vector form factors of a nuclear transition from electromagnetic data under an isospin rotation. Testing this structure inside nuclei was hampered by ignorance of nuclear wavefunctions: any capture-rate prediction folds the weak current together with an unknown nuclear transition matrix element.

Weinberg's 1959 observation was that for one class of transitions this ambiguity cancels exactly. Let A be a light even-even nucleus (which then has a 0⁺, T = 0 ground state, an isospin singlet) and B a 0⁺, T = 1 excited state of the odd-odd neighbor. In a 0⁺→0⁺ transition — in β-decay language a Fermi transition, driven purely by the nuclear charge operator — the axial-vector part of the weak current cannot contribute by angular-momentum and parity selection. The μ⁻-capture rate to B and the cross section for inelastic electron scattering A(e,e′)A*, where A* is the isobaric-analog state of B (the T = 1 member of the same isospin multiplet inside A), then contain the same nuclear form factor F(q2)F(q^2) — the Fourier transform of the transition charge density — related by isospin symmetry. If the electron energy and scattering angle are chosen so that the electron-scattering momentum transfer equals the capture momentum transfer, the ratio of capture rate to scattering cross section is a known function of kinematics alone, independent of F(q2)F(q^2), normalized by the superallowed ftft value (the precisely measured comparative half-life of the analogous 0⁺→0⁺ nuclear β decay; "superallowed" denotes the fastest, unmixed Fermi transitions). Weinberg wrote that this prediction "seems to be the only test of the Feynman–Gell-Mann theory for muon interactions that is completely free of theoretical ambiguities arising from our ignorance of the details of nuclear structure," and noted that extending it to non-zero-spin transitions would require a reliable theory of the axial-vector contributions to muon capture, then unavailable.

The experimental shape of the problem follows from the construction. Odd-odd nuclei essentially never have 0⁺ ground states, so B is an excited state whose population must be detected through its de-excitation γ-rays; the rate is further suppressed because isospin conservation forces F(0)=0F(0) = 0; and the test is best performed in as light a nucleus as possible, to avoid departures from isospin conservation, from the Born approximation (the one-photon-exchange approximation underlying the electron-scattering analysis), and from overlapping nuclear levels. In 1959 the only suitable level known to Weinberg was in ²⁸Al — by today's evaluated data, the 0⁺, T = 1 state at 972.4 keV, de-exciting by a 941.7 keV electric-quadrupole (E2) γ ray — and the closing parenthetical of his experimental discussion, "The only suitable nuclear level known to the author occurs in Al²⁸", left the identification of further suitable levels open.

The subsequent literature settled the motivation but not the test. CVC was confirmed through finite-momentum-transfer comparisons of β decay, muon capture, and electron scattering in the A = 12 nuclear triad (Kubodera and Kim 1973; Hwang and Primakoff 1977) and is structural in the electroweak Standard Model. The needed axial-vector capture theory was supplied within a few years (Morita and Fujii 1960; Foldy and Walecka 1965), and the non-zero-spin extension was realized — but aimed at a different target: the induced pseudoscalar coupling gPg_P (the form factor encoding the partially-conserved-axial-current, PCAC, pion-pole enhancement of the axial current at the capture momentum transfer). On ²⁸Si that program measures the 1⁺, 2201 keV Gamow–Teller transition (Gamow–Teller being the axial spin–isospin analog of the Fermi transition). Its world-average helicity-amplitude ratio — the ratio of the two spin-helicity amplitudes of the capture transition, extracted from γ-neutrino correlation experiments — is X=0.554±0.042X = 0.554 \pm 0.042, implying gP/gA=1.0±1.2g_P/g_A = 1.0 \pm 1.2 against the PCAC/chiral prediction of about 6.4: the "²⁸Si puzzle," attributed by Gorringe and Fearing (2004) to model sensitivity of the axial matrix element, i.e. exactly the nuclear-structure ambiguity the 0⁺ channel was designed to avoid. Meanwhile the capture half of the 0⁺ test exists only as three mutually ~2σ γ-yield measurements (2.0(3)%, 2.96(36)%, and 2.30(11)% per capture, with a cascade-subtracted direct yield of 0.5(2)%), the required longitudinal charge (C0 — the Coulomb, or charge, multipole of the electron-scattering response, as opposed to its transverse magnetic part) electron-scattering form factor at q≈100q \approx 100 MeV/cc was never measured (the high-resolution ²⁸Si electron-scattering literature, e.g. Lüttge et al. 1996, concerns transverse isovector M1, i.e. magnetic-dipole, transitions), and no screened inventory of additional 0⁺, T = 1 candidates exists — only scattered flags in evaluated data, such as the ⁴⁰K 0⁺ level at 1643.6 keV, which ENSDF (the Evaluated Nuclear Structure Data File of the US National Nuclear Data Center) marks as populated in ⁴⁰Ca muon capture but comments as "anti-analog," leaving its isospin assignment unsettled.

Problem Statement

Establish Weinberg's 0⁺→0⁺ form-factor-free ratio test of the vector weak current in muon capture, and determine the inventory of nuclear levels on which it can be performed. For the known candidate transition — μ⁻ + ²⁸Si(0⁺, T = 0, ground state) → ν_μ + ²⁸Al(0⁺, T = 1, 972.4 keV), the level Weinberg identified as the only one known to him — this requires, as jointly supporting one objective:

(1) The capture half: the partial capture rate for the 0⁺, 972.4 keV state, via its 941.7 keV de-excitation γ ray, with the cascade feeding from higher levels subtracted and quantified.

(2) The electron-scattering half: the longitudinal charge (C0, the Coulomb or charge multipole of the electron-scattering response, as opposed to its transverse magnetic part) inelastic electron-scattering cross section — or equivalently the form factor — for the 0⁺, T = 1 isobaric-analog state in ²⁸Si, at momentum transfer matched to the capture momentum transfer q2≈mμ2q^2 \approx m_\mu^2, with absolute normalization.

(3) The comparison: form the ratio of (1) to (2), normalize by the superallowed ftft value, propagate uncertainties, and confront it with the CVC prediction (Eq. (6) of the 1959 Letter), including whatever radiative, relativistic, Coulomb-distortion, and isospin-breaking corrections are needed at the achieved precision.

(4) The level inventory: screen the light nuclei for additional 0⁺, T = 1 excited states of odd-odd nuclei reachable from stable even-even T = 0 ground states, with well-separated levels and clean de-excitation γ rays; settle borderline isospin assignments (for example, whether the ⁴⁰K 0⁺, 1643.6 keV level populated in ⁴⁰Ca muon capture is the analog or an anti-analog state); and state the expected rates and detectability of the candidates.

The test on the one known level is the joint subject of (1)–(3); (4) extends the same test to further levels, which the source's closing parenthetical left open. The original 1959 motivation — whether the Feynman–Gell-Mann CVC structure holds — has been settled by other channels and is not part of what remains open; the open problem is the unexecuted test itself and the level inventory it asked for, which together would deliver the only muon-capture transition in a complex nucleus governed purely by the vector current.

The ratio in (3) is a measured-versus-predicted comparison against Eq. (6) of the 1959 Letter, so the C0 form factor entering (2) must be an empirical form factor at the matched momentum transfer itself; a model-calculated form factor does not serve, because the design of the test is that the form factor cancels between (1) and (2). Absent an empirical determination of the C0 form factor at the matched momentum transfer, the comparison in (3) can be given only as a bound or as the sensitivity that the missing data would have to reach.

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