Quantifying the role of the $\Delta(1232)$ resonance in the quenching of Gamow–Teller and isovector M1 strength in nuclei

Bohr and Mottelson (1981) estimated, in a simple polarizability picture based on quark-model relations between the nucleon–nucleon and nucleon–$\Delta$ matrix elements of the spin–isospin operator $\sigma\tau$, that coupling to the $\Delta(1232)$ resonance quenches the effective isovector spin-dependent moments of nuclei: one effective $\sigma\tau$ interaction strength consistently described reduced M1 moments near $^{208}$Pb and the energies of the Gamow–Teller giant resonances, and the $\Delta$ was predicted to remove a substantial fraction of the summed Gamow–Teller and M1 strength. They left open the precise magnitude of the effect, the validity of the quark-model matrix-element ratios in a correlated nucleus, confirmation by complete summed-strength measurements across the nuclear chart, and the extension beyond the static (low-excitation-energy) limit. Four decades of work established the phenomenology — roughly half of the Ikeda sum-rule Gamow–Teller strength is missing within measured excitation windows, isoscalar spin-M1 strength is essentially unquenched while isovector spin-M1 is reduced, and ab initio calculations reproduce the observed quenching from many-body correlations plus two-body weak currents — but a controlled, quantitative, scheme-independent decomposition of the quenching into $\Delta$-isobar and nucleonic pieces, its mass and excitation-energy dependence, and full empirical closure of the strength sums remain unresolved and are actively debated.

Background

Gamow–Teller (GT) transitions are the simplest spin–isospin excitations of a nucleus: they are driven by the operator $\sigma\tau$, the product of the nucleon spin Pauli matrix $\sigma$ and the isospin Pauli matrix $\tau$ that distinguishes neutrons from protons. They dominate allowed nuclear $\beta$ decay and are measured in charge-exchange reactions such as $(p,n)$, $(n,p)$ and $(^3\mathrm{He},t)$. The same spin–isospin structure appears in the isovector spin part of the magnetic dipole (M1) operator, $g_s\,\sigma\tau$ (with $g_s$ the nucleon spin $g$-factor), which excites $1^+$ spin-flip modes electromagnetically or through forward-angle proton scattering $(p,p')$. For a nucleus of $N$ neutrons and $Z$ protons, the Ikeda sum rule fixes, model-independently, the net charge-exchange strength in any description built only from nucleons:

$$\sum B(\mathrm{GT}^-)-\sum B(\mathrm{GT}^+)=3(N-Z),$$

where $B(\mathrm{GT})$ is the squared GT matrix element.

Already in 1973, Wilkinson showed from accurately measured allowed $\beta$ decays and mirror-pair magnetic moments that observed spin–isospin matrix elements are systematically smaller than full-major-shell shell-model predictions computed with free-nucleon operators, by an average factor $\approx0.9$. After the discovery of giant GT strength in $(p,n)$ spectra (Doering et al. 1975) and the charge-exchange programs that followed (e.g. Gaarde et al. 1980), a sharper fact emerged: even after integrating over the giant resonance, only about half of the Ikeda sum-rule strength is found within the measured excitation windows. This deficit is the "quenching" problem. Two mechanisms were proposed. The first is coupling of the $\sigma\tau$ field to non-nucleonic degrees of freedom, in particular virtual excitation of the $\Delta(1232)$ resonance — the lightest baryon resonance, of mass $\approx1232\ \mathrm{MeV}$, spin $3/2$ and isospin $3/2$, carrying large spin–isospin strength roughly $300\ \mathrm{MeV}$ above the nucleon (Ericson, Figureau and Thévenet 1973; Oset and Rho 1979). A $\sigma\tau$ probe can promote a nucleon to a $\Delta$, leaving a hole (a "$\Delta$-hole" excitation); the coherent superposition of such excitations forms a $\Delta$-hole giant resonance near $300\ \mathrm{MeV}$ that removes strength from the nucleonic region. The second mechanism is purely nucleonic: many-body correlations — configuration mixing and the coupling of the one-particle–one-hole GT mode to two-particle–two-hole ($2p2h$) configurations — that fragment the strength into many weak levels, partly beyond experimental reach (Bertsch and Hamamoto 1982).

Bohr and Mottelson (1981) supplied the most quoted simple estimate of the first mechanism. They treated the nucleus as nucleons plus virtual $\Delta$'s, coupled by an effective $\sigma\tau\cdot\sigma\tau$ interaction of strength $\kappa$ (a Landau–Migdal-type effective force in the spin–isospin channel), and took the relative sizes of the $NN$, $N\Delta$ and $\Delta\Delta$ matrix elements of $\sigma\tau$ from the $SU(6)$-symmetric constituent-quark model — the picture in which the nucleon and the $\Delta$ are three-quark states sharing a totally symmetric spin–isospin quark wave function, which fixes ratios such as $\langle\Delta||\sigma\tau||N\rangle=(4\sqrt{2}/5)\,\langle N||\sigma\tau||N\rangle$. In the static limit $\hbar\omega\ll M_\Delta-M_N\approx300\ \mathrm{MeV}$, the effect of the virtual $\Delta$'s can be absorbed into renormalized nucleonic quantities, expressible through a static $\Delta$ polarizability $\chi_\Delta^{(0)}$ (the linear response of the $N\to\Delta$ sector to the applied $\sigma\tau$ field) of order $0.25$. With a single coupling $\kappa\approx29/A\ \mathrm{MeV}$ ($\approx140\ \mathrm{keV}$ for $^{208}\mathrm{Pb}$), they simultaneously described the reduced isovector spin $g$-factors seen in M1 data near $^{208}\mathrm{Pb}$ and the energies of the GT giant resonances, and concluded that the $\Delta$ removes a substantial fraction — of order one third to one half — of the summed isovector spin strength from the nucleonic region; for example, a $1^+$ spin-flip M1 mode in $^{208}\mathrm{Pb}$ was predicted near $7\ \mathrm{MeV}$ with strength of order $26\ (e\hbar/2Mc)^2$.

Four decades of subsequent work transformed the framework but did not close the question. $\Delta$-hole calculations showed the magnitude to be sensitive to the effective nucleon–$\Delta$ spin–isospin interaction (the Landau–Migdal parameter $g'_\Delta$, whose value and assumed equality with the nucleonic $g'_N$ were uncertain; Arima et al. 1983). Shell-model analyses mapped the empirical systematics of the effective M1 and GT operators (Brown and Wildenthal 1983; Towner 1987; Osterfeld 1992), while large-space shell-model calculations argued that much of the "missing" strength is nucleonic fragmentation into unobservably weak states rather than genuine quenching (Caurier, Poves and Zuker 1995). Most recently, chiral effective field theory (EFT) — the systematic low-energy expansion of QCD in terms of nucleons, pions and optionally explicit $\Delta$'s — supplies two-body currents (2BC): corrections in which the weak or electromagnetic probe couples to a meson exchanged between two nucleons, with or without an intermediate $\Delta$ baryon. Ab initio (first-principles, microscopic-Hamiltonian) calculations including many-body correlations and chiral 2BC reproduce observed GT quenching and magnetic moments (Gysbers et al. 2019; Miyagi et al. 2024), but in the standard $\Delta$-less formulations the $\Delta$ contribution is absorbed into fitted low-energy constants, so the explicit $\Delta$ share that Bohr and Mottelson asked about is not isolated. Experimentally, sharp channel selectivity is now available — isoscalar spin-M1 strength is essentially unquenched while isovector spin-M1 in the same sd-shell nuclei is reduced by $\approx40\%$ (Matsubara et al. 2015) — and the RCNP charge-exchange program finds summed GT and spin-dipole strengths in medium-heavy nuclei at roughly half of the sum-rule limits within the $0$–$30\ \mathrm{MeV}$ nucleonic window (Yako et al. 2009; Ejiri 2025). What remains unsettled is the quantitative, scheme-independent decomposition of this reduction into $\Delta$-induced and nucleonic pieces, its dependence on mass number and excitation energy, and its empirical confirmation at the sum-rule level.

Problem Statement

Determine, quantitatively and within a controlled microscopic framework, the role of the $\Delta(1232)$ resonance in the effective spin-dependent (isovector M1 and Gamow–Teller) moments of nuclei — the question posed and left unresolved by Bohr and Mottelson's 1981 quark-model-based estimate. The following linked facets define the problem.

1. Magnitude and decomposition. Determine the fraction of the observed reduction of summed isovector spin strength (relative to the Ikeda sum rule $3(N-Z)$ and to unquenched shell-model and RPA (random-phase approximation) predictions) that is induced by virtual $N\leftrightarrow\Delta$ excitations — equivalently, by the $\Delta$-mediated parts of the spin–isospin response or of the two-body weak and electromagnetic currents — as distinct from nucleonic many-body correlations (configuration mixing, $2p2h$ spreading, particle-vibration coupling) and from non-$\Delta$ two-body currents. The decomposition must be well defined at the level of the theory used (e.g., chiral EFT with explicit $\Delta$ degrees of freedom versus the $\Delta$-less theory, with the fate of the resorbed low-energy constants made explicit), and must state whether the $\Delta$ share is representable by a nucleus-independent effective reduction of the one-body $\sigma\tau$ coupling, as in Bohr and Mottelson's $\chi_\Delta^{(0)}\approx0.25$ or Ejiri's $K_\Delta\approx0.7$; these values are candidate estimates to be tested, not established results.

2. Matrix-element input. Test, within many-body calculations that include short-range correlations and configuration mixing, the $SU(6)$ quark-model relations between the $NN$, $N\Delta$ and $\Delta\Delta$ matrix elements of $\sigma\tau$ on which the original estimate rests, together with the associated effective nucleon–$\Delta$ spin–isospin interaction strengths; or replace them by consistently determined couplings with quantified uncertainties.

3. Empirical closure. Establish, by complete measurements of summed GT and M1 strengths extending to high excitation energy with quantified continuum, background and reaction-model (unit-cross-section, multipole-decomposition) systematics, whether the summed strengths across the nuclear chart saturate the nucleon-only sum rules or are universally reduced, and by how much; isolate the isovector spin-flip quenching and its mass dependence from light nuclei to the heavy double-beta-decay nuclei.

4. Dynamical regime. Extend the mapping between the $N+\Delta$ description and the effective nucleon-only description beyond the static, low-frequency limit $\hbar\omega\ll M_\Delta-M_N$: determine how the effective spin–isospin interaction and the quenching factor vary with excitation energy and momentum transfer in the giant-resonance region and above.

The objective is a consistent quantitative account — theoretical and empirical, with uncertainties — of the $\Delta$-induced share of the quenching of isovector spin-dependent nuclear strength, across nuclei and excitation energies. The empirical facet is assessed from the summed GT and M1 strength measurements reported in the cited references, within their measured excitation-energy windows and quoted systematics.

References

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