Nucleon axial-vector form factor from threshold pion electroproduction: controlled finite-pion-mass corrections and the behavior of $G_A(Q^2)$ at multi-GeV momentum transfer

The isovector axial form factor $G_A(Q^2)$ of the nucleon governs the momentum dependence of the weak axial current and controls quasielastic neutrino scattering. In 1970, Nambu and Yoshimura extracted $G_A(q^2)$ up to $q^2\approx 7.3$ (GeV/c)$^2$ from the then-available threshold electropion-production data using the soft-pion low-energy theorem of current algebra and PCAC. Their analysis included model-based finite-pion-mass corrections (a covariant pole-dominance model) and specified vector-form-factor inputs (the standard dipole parametrization, "sufficient for our purpose"); what it left open was the model dependence and adequacy of those corrections — the paper itself suspected that the observed small-$q^2$ anomaly might be "a spurious effect caused by the inadequacy of our model and correction procedures" — together with the assumption, made explicit in footnote 14, that the dipole fit can be extrapolated to larger $q^2$. Later literature resolved the low-momentum-transfer side of these issues: dedicated threshold measurements found $M_A\approx 0.96$-$1.08$ GeV (contradicting the 1970 value $m_A=1.34$ GeV/c$^2$), and baryon chiral perturbation theory computed the corrections, explaining the long-standing electroproduction-versus-neutrino axial-radius discrepancy by a model-independent one-loop effect. What remains open is precisely the original program's high-momentum-transfer core: modern near-threshold electroproduction data reach only $Q^2=4.2$ GeV$^2$ in the charged-pion channel and $Q^2=4.5$ GeV$^2$ in the neutral-pion channel (CLAS), while the interpretation of the 1970s-era data extending to $q^2\approx 7.3$ (GeV/c)$^2$ remains model-dependent; the finite-pion-mass ('hard-pion') corrections to the low-energy theorem are model-dependent at the 20-30% level for $Q^2\sim 1$-$10$ GeV$^2$, and the modern determinations (z-expansion fits, two-component model fits, lattice QCD, light-cone sum rules) differ among themselves by 15-30% at a few GeV$^2$, leaving unresolved whether the pure dipole form remains an adequate description there. A controlled extraction of $G_A(Q^2)$ at multi-GeV momentum transfer from threshold pion electroproduction, with quantified corrections and uncertainties, and the resulting determination of the true $Q^2$-dependence of $G_A$ at large $Q^2$, remain unresolved.

Background

The weak axial current of the nucleon is characterized at low and moderate energies by the isovector axial form factor $G_A(Q^2)$, defined through the matrix element of the axial current between nucleon states, $\langle N(p')|j_\mu^{5,a}|N(p)\rangle = \bar{u}(p')\gamma_\mu\gamma_5 G_A(Q^2)\,\tau^a u(p)/2 + \dots$, where $Q^2=-q^2\ge 0$ is the space-like squared four-momentum transfer and $\tau^a$ the isospin Pauli matrices. Its value at $Q^2=0$ is the axial coupling $g_A\approx 1.27$, measured in nuclear beta decay; its $Q^2$-dependence controls quasielastic neutrino-nucleon scattering, the signal process of neutrino oscillation experiments, and is a basic piece of nucleon chiral structure.

PCAC (partially conserved axial-vector current — the statement that the divergence of the axial current is proportional to the pion field) combined with current algebra (the commutation relations satisfied by the components of the electromagnetic and weak currents) implies low-energy theorems, often called soft-pion theorems, for reactions in which a pion is emitted with vanishing four-momentum. Nambu, Lurie and Shrauner derived such a theorem for pion production induced by electromagnetic or weak currents: in the chiral limit of vanishing pion mass $m_\pi\to 0$, the electric dipole multipole amplitude $E_{0+}$ for pion production at exact threshold is fixed in terms of $G_A(Q^2)$ and the nucleon electromagnetic form factors. (Pion electroproduction is the reaction $e N\to e N\pi$ proceeding through a virtual photon $\gamma^*$; near threshold the amplitude is dominated by the two S-wave multipoles, transverse $E_{0+}$ and longitudinal $L_{0+}$, which can be separated by a Rosenbluth separation — measuring the cross section at fixed kinematics with varying virtual-photon polarization.) This theorem opened a route to the then poorly known $G_A(Q^2)$ from electron rather than neutrino experiments.

By convention $G_A(Q^2)$ is fitted with the dipole form $g_A(1+Q^2/M_A^2)^{-2}$ with a single axial mass $M_A\approx 1$ GeV from neutrino data. Nambu and Yoshimura (1970) applied the soft-pion theorem to the threshold electropion data then available (SLAC-MIT, DESY, and other measurements covering $q^2$ up to $7.3$ (GeV/c)$^2$), including a covariant pole model for the leading finite-$m_\pi$ corrections, and obtained a dipole fit with $m_A=1.34\pm 0.05$ GeV/c$^2$ fitted only to the high-$q^2$ data. They included model-based finite-$m_\pi$ corrections and specified vector-form-factor inputs, explicitly assumed (footnote 14 of that paper) that the dipole fit can be extrapolated to larger $q^2$, and left open the model dependence and adequacy of their correction procedure — the paper itself suspected that the observed small-$q^2$ flattening might be a spurious effect of that inadequacy — as well as the validation of the dipole extrapolation by future data.

The subsequent history splits by momentum transfer. At low $Q^2\lesssim 1$ GeV$^2$, dedicated threshold experiments (Frascati, Daresbury/DESY, Mainz) established $M_A\approx 0.96$-$1.08$ GeV, and baryon chiral perturbation theory (CHPT — the low-energy effective field theory of QCD organized as an expansion in $m_\pi$ and small momenta) computed the corrections to the soft-pion theorem systematically; in particular it explained the long-standing $\sim 5\%$ axial-radius discrepancy between electroproduction- and neutrino-based extractions by a model-independent one-loop effect arising because the chiral limit and the photon point do not commute. At multi-GeV $Q^2$, however, the chiral expansion fails: the limits $m_\pi\to 0$ and $Q^2\to\infty$ do not commute — at large momentum transfer the emitted pion cannot be soft with respect to both the initial and the final nucleon — so the low-energy theorem acquires hard-pion corrections that no small parameter controls; estimates place them at the 20-30% level for $Q^2\sim 1$-$10$ GeV$^2$. Modern theory addresses this region with light-cone sum rules (LCSR — a QCD technique expressing baryon form factors through nucleon distribution amplitudes and perturbative kernels) and with model-independent z-expansion parametrizations (a conformal-map expansion in a variable $z(Q^2)$ that does not presuppose a functional shape), while lattice QCD computes $G_A$ directly at discrete $Q^2$. Experimentally, CLAS has measured near-threshold $\gamma^* p\to n\pi^+$ up to $Q^2=4.2$ GeV$^2$, and the neutral-pion channel near threshold up to $Q^2=4.5$ GeV$^2$ (Khetarpal et al. 2013, which accesses generalized rather than axial form factors), but the axial form factor extracted there still rests on the uncontrolled soft-pion relation, and the determinations in this region still differ among themselves by 15-30% at a few GeV$^2$, leaving the adequacy of the pure dipole form an open question. The unresolved target is a controlled determination of $G_A(Q^2)$ at multi-GeV momentum transfer from threshold pion electroproduction, and through it the true large-$Q^2$ behavior of the axial form factor.

Problem Statement

This is the surviving high-momentum-transfer core of the question left open by Nambu and Yoshimura (1970), whose low-momentum-transfer part has since been resolved (refs. 9 and 10): determine the isovector axial form factor $G_A(Q^2)$ of the nucleon from near-threshold charged-pion electroproduction ($e p\to e n\pi^+$; the near-threshold $\pi^0$ channel is a related route whose existing analysis accesses generalized rather than axial form factors) at multi-GeV momentum transfer — the region $Q^2\gtrsim 1$ GeV$^2$ up to the charged-pion CLAS coverage, $Q^2=4.2$ GeV$^2$ — and settle, in the sense of the source paper's footnote 14, whether the extracted parametrization of $G_A$ remains valid at $q^2$ beyond that coverage.

Concretely, the research target is:

1. Establish a controlled treatment of the finite-pion-mass (hard-pion) corrections to the soft-pion low-energy theorem that relates the threshold electric-dipole amplitude $E_{0+}$ to $G_A(Q^2)$ and the nucleon vector form factors — valid at $Q^2\gtrsim 1$ GeV$^2$ where the chiral expansion fails and current estimates of these corrections are model-dependent at the $20$-$30\%$ level — or provide an equivalent controlled extraction framework, consistent with chiral perturbation theory in the low-$Q^2$ region where the latter is reliable. Here 'controlled' means that the residual model dependence of these corrections is identified, quantified, and propagated into the extracted $G_A(Q^2)$ rather than left as an uncontrolled systematic, with the quoted $20$-$30\%$ spread taken as the current baseline to be quantified rather than assumed away; and an alternative framework is 'equivalent' when it addresses the same region and input data with a comparable quantification of that residual model dependence.

2. Extract $G_A(Q^2)$ (or its parametrization) from threshold or near-threshold electroproduction data over the multi-GeV region $Q^2\gtrsim 1$ GeV$^2$ up to the charged-pion CLAS coverage, $Q^2=4.2$ GeV$^2$ (the neutral-pion CLAS coverage and the model-dependent 1970s-era data are context, not part of the required extraction) — modern CLAS measurements reach $Q^2=4.2$ GeV$^2$ in the charged-pion channel and $Q^2=4.5$ GeV$^2$ in the neutral-pion channel, while the 1970s-era data extending to $q^2\approx 7.3$ (GeV/c)$^2$ depend for their interpretation on the source's model-dependent correction procedures — with radiative and higher-wave corrections (P-wave multipoles and the tail of the $\Delta(1232)$, the first excited nucleon resonance) and the residual model dependence of the finite-pion-mass corrections propagated.

3. Determine the $Q^2$-dependence of $G_A$ at large momentum transfer — in particular whether the dipole form $g_A(1+Q^2/M_A^2)^{-2}$ remains an adequate description or a more general (for example z-expansion) parametrization is required — and thereby validate or refute the source paper's footnote-14 assumption that the dipole fit can be extrapolated to $q^2$ beyond the data coverage, together with its stated consequence of a $2.7\times$ enhancement of the asymptotic quasielastic neutrino-nucleon cross section. Comparison with independent neutrino-scattering data is the test anticipated by the source itself; lattice-QCD and light-cone sum rule determinations provide further context, carrying different and partly correlated systematics.

The 1970 assumption that a dipole fit can simply be extrapolated to larger $q^2$ remains unvalidated: no modern precision threshold or near-threshold electroproduction data exist above $Q^2=4.5$ GeV$^2$, the interpretation of the 1970s-era measurements reaching $q^2\approx 7.3$ (GeV/c)$^2$ rests on the source's own model-dependent correction procedures, and the extractions in the modern $1$-$4.5$ GeV$^2$ region still rest on the uncontrolled soft-pion relation.

References

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