Isoscalar Monopole Resonance of the Alpha Particle: A Prism to Nuclear Hamiltonians

We present an ab initio study of the isoscalar monopole excitations of ${}^4\mathrm{He}$ using different realistic nuclear interactions, including modern effective field theory potentials. In particular we concentrate on the transition form factor $F_{\mathcal{M}}$ to the narrow $0^{+}$ resonance close to threshold. $F_{\mathcal{M}}$ exhibits a strong potential model dependence, and can serve as a kind of prism to distinguish among different nuclear force models. Compared to the measurements obtained from inelastic electron scattering off ${}^4\mathrm{He}$, one finds that the state-of-the-art theoretical transition form factors are at variance with experimental data, especially in the case of effective field theory potentials. We discuss some possible reasons for such a discrepancy, which still remains a puzzle.

Figure 1

Theoretical transition form factor $|F_{\mathcal{M}}(q^2){|}^2$ with $G_E^n=0$ calculated with various force models: $\mathrm{AV}18+\mathrm{UIX}$ (full line), ${\mathrm{N}}^3\mathrm{LO}+{\mathrm{N}}^2\mathrm{LO}$ (red band), result from Ref. 17 (dot-dashed line). Data from Frosch et al. [10], Walcher [4], and Köbschall et al. [11].

Figure 2

Elastic form factor $|F_{\mathrm{el}}(q^2)|$ of ${}^4\mathrm{He}$ calculated with various force models: $\mathrm{AV}18+\mathrm{UIX}$ (full line), ${\mathrm{N}}^3\mathrm{LO}+{\mathrm{N}}^2\mathrm{LO}$ (red band), result from Ref. 21 with AV18+UIX (dot-dashed line). Data from Frosch et al. [36].

Figure 3

(a) ${\mathcal{S}}_{\mathcal{M}}^{\mathrm{bg}}(q,\omega )$ at $q=1.5{\mathrm{fm}}^{-1}$ for $\mathrm{AV}18+\mathrm{UIX}$ obtained with different values of $f_R$ (see text): $f_R=|F_{\mathcal{M}}(q){|}^2$ (full line), $f_R=0.99|F_{\mathcal{M}}(q){|}^2$ (dashed line), $f_R=1.01|F_{\mathcal{M}}(q){|}^2$ (dot-dashed line). (b) Nonresonant background contribution ${\mathcal{S}}_{\mathcal{M}}^{\mathrm{bg}}(q,\omega )$: AV18+UIX (full line), ${\mathrm{N}}^2\mathrm{LO}+{\mathrm{N}}^3\mathrm{LO}$ (red band).