Onset of $\eta$-nuclear binding in a pionless EFT approach

TL;DR

This paper investigates the formation of eta-nuclear bound states using a pionless EFT approach, revealing the minimal eta-nucleon scattering length needed for binding in helium isotopes.

Contribution

It introduces a pionless EFT framework with a self-consistent energy dependence treatment to study eta-nuclear binding, providing new insights into the minimal conditions for such states.

Findings

Binding in eta-3He requires Re a_etaN close to 1 fm.

Eta-4He can bind with Re a_etaN exceeding 0.7 fm.

Binding energies show weak dependence on the EFT regulator.

Abstract

$\eta NNN$ and $\eta NNNN$ bound states are explored in stochastic variational method (SVM) calculations within a pionless effective field theory (EFT) approach at leading order. The theoretical input consists of regulated $NN$ and $NNN$ contact terms, and a regulated energy dependent $\eta N$ contact term derived from coupled-channel models of the $N^{\ast}(1535)$ nucleon resonance plus a regulated $\eta NN$ contact term. A self consistency procedure is applied to deal with the energy dependence of the $\eta N$ subthreshold input, resulting in a weak dependence of the calculated $\eta$-nuclear binding energies on the EFT regulator. It is found, in terms of the $\eta N$ scattering length $a_{\eta N}$, that the onset of binding $\eta\,^3$He requires a minimal value of Re$\,a_{\eta N}$ close to 1 fm, yielding then a few MeV $\eta$ binding in $\eta\,^4$He. The onset of binding $\eta\,^4$He requires a lower value of Re$\,a_{\eta N}$, but exceeding 0.7 fm.

Tables

Table 1. Table 1: 4 He binding energies B ( 4 B(^{4} He) (in MeV) in LO π̸ italic-π̸ \not{\pi} EFT  SVM calculations, with LECs fitted to N ​ N 𝑁 𝑁 NN and N ​ N ​ N 𝑁 𝑁 𝑁 NNN low-energy data  [ 28 ] . The Λ → ∞ → Λ \Lambda\to\infty limit of B ( 4 B(^{4} He) was evaluated by using higher values of the scale Λ Λ \Lambda than listed here.

$\Lambda$ (fm${}_{}{}^{-1})$	2	4	6	8	$\to \infty$	exp.
$B{(}^4$He)	22.4	22.9	24.2	25.1	27.8$\pm$0.2	28.3

Table 2. Table 2: η 𝜂 \eta binding energies and widths (MeV) in the He isotopes from SVM calculations using η ​ N 𝜂 𝑁 \eta N potentials v η ​ N GW ​ ( E ) superscript subscript 𝑣 𝜂 𝑁 GW 𝐸 v_{\eta N}^{\rm GW}(E) with scale parameters Λ = 2 , 4 Λ 2 4 \Lambda=2,4  fm -1 , together with the corresponding self consistent values of the downward energy shift (in MeV) δ ​ s sc 𝛿 subscript 𝑠 sc \delta\sqrt{s_{\rm sc}} . The values of Γ ( η 3 \Gamma(\eta\,^{3} He) shown here outdate the erroneous, too large widths listed in Ref.  [ 13 ] .

	${\eta }^3$He			${\eta }^4$He
$\Lambda$ (fm${}_{}{}^{-1})$	$\delta \sqrt{s_{\mathrm{sc}}}$	$B_{\eta }$	${\Gamma }_{\eta }$	$\delta \sqrt{s_{\mathrm{sc}}}$	$B_{\eta }$	${\Gamma }_{\eta }$
2	$-$11.2	$-$0.16	0.24	$-$16.5	$-$0.15	0.34
4	$-$21.1	$+$0.30	1.46	$-$32.2	$+$1.54	2.82

Equations

$$\Psi =\sum _k{c}_k\mathcal{A}\left({\left[{\mathcal{Y}}_L^k(\hat{\mathbf{x}}){\chi }_S^k\right]}_{JM}{\xi }_{T{T}_z}^k\exp (-\frac{1}{2}{\mathbf{x}}^T{A}_k\mathbf{x})\right)$$

$$V_2(ij)=\left[c_S^{\Lambda }\frac{1}{4}(1-{\sigma }_i\cdot {\sigma }_j)+c_T^{\Lambda }\frac{1}{4}(3+{\sigma }_i\cdot {\sigma }_j)\right]{\delta }_{\Lambda }(r_{ij}),$$

$$V_3(ijk)=d_{NNN}^{\Lambda }{\delta }_{\Lambda }(r_{ij},r_{ik}),$$

$${\delta }_{\Lambda }(r_{ij})={\left(\frac{\Lambda }{2\sqrt{\pi }}\right)}^3\exp \left(-\frac{{\Lambda }^2}{4}{r}_{ij}^2\right),{\delta }_{\Lambda }(r_{ij},r_{ik})={\delta }_{\Lambda }(r_{ij}){\delta }_{\Lambda }(r_{ik}),$$

$$a_{\eta N}^{\mathrm{GW}}=0.96+i0.26a_{\eta N}^{\mathrm{CS}}=0.67+i0.20,$$

$$v_{\eta N}(E;r)=c_{\eta N}^{\Lambda }(E){\delta }_{\Lambda }(r),c_{\eta N}^{\Lambda }(E)=-\frac{4\pi }{2{\mu }_{\eta N}}{b}_{\eta N}^{\Lambda }(E),$$

$$⟨\delta \sqrt{s}⟩=-\frac{B}{A}-{\beta }_N\frac{1}{A}⟨T_N⟩+\frac{A-1}{A}{E}_{\eta }-{\xi }_A{\beta }_{\eta }{\left(\frac{A-1}{A}\right)}^2⟨T_{\eta }⟩,$$

$${\Gamma }_{\eta }=-2⟨{\Psi }_{g.s.}|\mathrm{Im}{V}_{\eta }|{\Psi }_{g.s.}⟩.$$

$$V_{\eta N_i{N}_j}=d_{\eta NN}^{\Lambda }{\delta }_{\Lambda }(r_{\eta N_i},r_{\eta N_j}).$$

Full text

††journal: Physics Letters B

1 Introduction

The $\eta N$ $s$-wave interaction near threshold, $E_{\rm th}(\eta N)=1487$ MeV, is attractive as realized first by coupling to the $\pi N$ channel [1] and subsequently confirmed, e.g. [2], by coupling the $\eta N$ channel to the entire set of meson-baryon channels with $0^{-}$ octet mesons and $\frac{1}{2}^{+}$ octet baryons, thereby generating dynamically the $N^{\ast}$(1535) $S_{11}$ resonance. The size of the resulting $\eta N$ energy dependent $s$-wave attraction, however, is strongly model dependent with values of the real part of the $\eta N$ scattering length as low as 0.2 fm [2] and up to nearly 1 fm in the $K$-matrix model of Green and Wycech (GW) [3] and in the recent Giessen coupled channels study [4]. Following the work of Ref. [1] it was soon realized that $\eta$-nuclear quasibound states might exist [5, 6] with widths determined by the scale of the imaginary part of the $\eta N$ scattering length. This imaginary part, due mostly to $\eta N\to\pi N$, is small in all models, between 0.2 to 0.3 fm. Nevertheless, no $\eta$-nuclear quasibound state has ever been established beyond doubt [7].

Recent optical-model calculations of such bound states [8, 9] using several energy-dependent $\eta N$ model amplitudes are summarized in Refs. [10, 11]. Whereas the appearance of $\eta$-nuclear bound states is robust in any of these $\eta N$ interaction models, the value of mass number $A$ at which binding begins is model dependent. Thus, the relatively strong $\eta N$ attraction in model GW even admits in such calculations a $1s_{\eta}$ bound state in 4He, with as low binding energy as 1.2 MeV and width of 2.3 MeV [11] calculated using a static 4He density. Unfortunately, the $\eta$-nucleus optical model approach is not trustable for as light nuclei as 4He, and genuine few-body calculations are required.

Photon- and hadron-induced reactions on nuclear targets provide useful constraints on possible $\eta$ bound states in very light nuclei, where according to a recent review [12] the most straightforward interpretation of the data is that $\eta d$ is unbound, $\eta\,^{3}$He is nearly or just bound, and $\eta\,^{4}$He is bound. Our previous few-body $\eta NN$ and $\eta NNN$ calculations [13], using the Minnesota [14] and Argonne AV4’ [15] $NN$ potentials, agree with this conjecture as far as the $\eta d$ and $\eta\,^{3}$He systems are concerned. A similar conclusion for $\eta\,^{3}$He has been reached recently by evaluating the $pd\to\eta\,^{3}$He near-threshold reaction [16]. And a recent WASA-at-COSY search for a possible $\eta\,^{4}$He bound state in the $dd\to\,{{}^{3}{\rm He}}N\pi$ reaction placed upper limits of a few nb on the production of a near-threshold bound state [17]. On the theoretical side, no precise few-body calculation of $\eta NNNN$ bound-state has ever been reported for $\eta\,^{4}$He.111A very recent preprint by Fix and Kolesnikov [18] reports on few-body calculations of the $\eta\,^{3}$He and $\eta\,^{4}$He scattering lengths, concluding that these systems are unbound.

The present work reports for the first time on precise few-body $\eta NNNN$ calculations in which the Stochastic Variational Method (SVM) is applied to $\eta$ plus few-nucleon Hamiltonians constructed in Leading Order (LO) within a Pionless Effective Field Theory. While this $\not{\pi}$EFT approach has been applied before to few-nucleon systems, e.g. [19, 20], and more recently in lattice-nuclei calculations [21, 22], it is extended here to include constituent pseudoscalar mesons for which pion exchange with nucleons is forbidden by parity conservation of the strong interactions. In particular, the single $\eta N$ contact term required in LO is provided by the $\eta N$ $s$-wave scattering amplitude $F_{\eta N}(E_{\rm sc})$ at a subthreshold energy $E_{\rm sc}$ derived self consistently within the few-body calculation, as practised in our previous work [13]. This is demonstrated in two $\eta N$ interaction models, GW [3] and CS (Cieplý–Smejkal [23]), which exhibit strong energy dependence of $F_{\eta N}(E)$ arising from the proximity of the $N^{\ast}$(1535) resonance. The results of the few-body calculations reported in the present work suggest that the onset of binding $\eta\,^{3}$He requires a value of Re$\,a_{\eta N}$ close to 1 fm, whereas the onset of binding $\eta\,^{4}$He requires a somewhat weaker $\eta N$ interaction with Re$\,a_{\eta N}$ exceeding 0.7 fm.

2 Methodology

Here we outline the methodology of the present work, including the few-body SVM used, the choice of $NN$, $NNN$, $\eta N$ and $\eta NN$ $\not{\pi}$EFT regulated contact terms, and the self consistent treatment of the energy-dependent $\eta N$ term.

2.1 SVM calculations

SVM calculations were introduced in the mid seventies to few-body nuclear problems [24], and used extensively with correlated Gaussian bases since the mid 1990s [25]. The SVM was benchmarked together with six other few-body methods in calculating the 4He binding energy [26]. Correlated Gaussian trial wavefunctions in this method are written as

[TABLE]

where the summation index $k$ runs with linear variational parameters $c_{k}$ on all possible values of the total spin $S$ and the total orbital angular momentum $L$, as well as on all possible intermediate coupling schemes, $\chi_{S}$ and $\xi_{T}$ stand for spin and isospin functions of the $N$-particle system, respectively, ${\cal Y}_{L}$ is the orbital part of $\Psi$ formed by coupling successively spherical harmonics in the $(N-1)$ relative coordinates of which the vector ${\bf x}$ is made, and $\cal A$ antisymmetrizes over nucleons. The matrix $A_{k}$ introduces $N(N-1)/2$ nonlinear variational parameters which are chosen stochastically. For a comprehensive review see Ref. [27].

2.2 Pionless EFT nuclear interactions

Here we follow a $\not{\pi}$EFT approach at LO. To this order the nuclear interaction consists of two-body and three-body contact (zero-range) terms,

[TABLE]

[TABLE]

where these contact terms are smeared by using normalized-to-one Gaussians with a regulating momentum-space scale (cut-off) parameter $\Lambda$:

[TABLE]

with $\delta_{\Lambda}(r_{ij})$ in the zero-range limit $\Lambda\to\infty$ becoming a Dirac $\delta({\bf r}_{ij})$ function. For a given value of the scale parameter $\Lambda$, two-body low-energy constants (LEC) $c^{\Lambda}_{S}$ and $c^{\Lambda}_{T}$ are fitted to the $S=0$ $pn$ scattering length and to the $S=1$ deuteron binding energy $B(d)$, respectively, and a three-body LEC $d_{NNN}^{\Lambda}$ is fitted to $B(^{3}$H). Following Ref. [28] a small corrective proton-proton contact term, with LEC $c^{\Lambda}_{pp}$, is introduced together with the Coulomb interaction between protons to reproduce the 3He binding energy. These two-body and three-body LECs are listed in Ref. [28] where $c^{\Lambda}_{pp}$ was found to effectively adjust $c^{\Lambda}_{S}$ by less than 0.1% over the full $\Lambda$-range tested in the present work. The 4He calculated binding energy $B(^{4}$He) provides then a check on how reasonable this LO $\not{\pi}$EFT version is. This is demonstrated in Table 1 for four representative values of $\Lambda$. The calculated values of $B(^{4}$He) depend only moderately on the scale parameter $\Lambda$, exhibiting renormalization scale invariance by approaching in the limit $\Lambda\to\infty$ a finite value 27.8$\pm$0.2 MeV which compares well with $B_{\rm exp}(^{4}$He)=28.3 MeV, despite the fact that only LO contributions are accounted for in this $\not{\pi}$EFT version.

2.3 Pionless EFT $\eta N$ interactions

Parity conservation forbids pion exchange in the $\eta N$ interaction, suggesting thereby that a $\not{\pi}$EFT approach may be justified. With spin and isospin zero for the $\eta$ meson, a single two-body $\eta N$ contact term is needed at LO. Below we derive the corresponding LEC from the $\eta N$ $s$-wave scattering amplitude $F_{\eta N}(E)$ calculated in two meson-baryon coupled-channel interaction models, GW [3] and CS [23], and shown in Fig. 1. Whereas the GW model used in our previous work [13] is an on-shell $K$-matrix model that considers $\eta N\leftrightarrow\pi N$ coupling, the CS model is a meson-baryon multi-channel chirally motivated model in which the $\eta N$ interaction is extremely short ranged and practically momentum independent below the momentum breakdown scale specified in the next paragraph. Both models capture the main features of the underlying $N^{\ast}(1535)$ resonance which peaks about 50 MeV above the $\eta N$ threshold energy $E_{\rm th}=1487$ MeV and generates considerable energy dependence of $F_{\eta N}$ near threshold. In particular, both Re$\,F_{\eta N}(E)$ and Im$\,F_{\eta N}(E)$, which at threshold are given by the scattering lengths (in fm)

[TABLE]

decrease monotonically in these models upon going into the subthreshold region while displaying considerable model dependence.

The $\eta N$ scattering lengths listed above are of order 1 fm or less, much smaller than the $NN$ scattering lengths whose large size justifies the use of $\not{\pi}$EFT in light nuclei. For $\eta N$ interactions as weak as implied by this size of $a_{\eta N}$, and with no $\eta N$ bound or virtual state expected, the $\eta N$ scattering length alone does not provide a meaningful criterion of fitting into an EFT approach. Alternatively, we estimate $p_{\eta}R\lesssim\frac{\pi}{2}$ for the momentum $p_{\eta}$ of a weakly bound $\eta$-nuclear state in a square well of radius $R$. With $R=2$ fm or a bit larger for the He isotopes, we get $p_{\eta}\lesssim 150$ MeV/c ($\approx$0.76 fm*-1*). Since the lowest-mass allowed meson exchange in pseudoscalar meson interaction with octet baryons is owing to vector mesons, with a typical mass of $m_{\rho}=770$ MeV, this range of $\eta$-nuclear momenta can be accommodated comfortably within the nuclear LO $\not{\pi}$EFT approach of the preceding subsection. The $\not{\pi}$EFT small parameter associated with this momentum breakdown scale of $Q^{\rho}_{\rm high}\approx m_{\rho}=3.9$ fm*-1* is given by $(p_{\eta}/Q^{\rho}_{\rm high})^{2}\approx 0.04$.

As in previous work [13], and in order to account for the energy dependence inherent in the meson-baryon coupled channel dynamical generation of the $N^{\ast}(1535)$ resonance, we construct energy-dependent local potentials $v_{\eta N}(E)$ that produce the $\eta N$ energy dependent scattering amplitude $F_{\eta N}(E)$ below threshold in models GW and CS. For a given $\eta N$ interaction model, the on-shell scattering amplitude $F_{\eta N}(E)$ serves as a single datum to which LO $\not{\pi}$EFT two-body regulated contact terms of the form

[TABLE]

are fitted. Here $\delta_{\Lambda}$ is a regulating normalized-to-one Gaussian with scale parameter $\Lambda$, as per Eq. (4), and $c^{\Lambda}_{\eta N}(E)$ is an energy dependent LEC conveniently related through the $\eta N$ reduced mass $\mu_{\eta N}$ to a strength function $b^{\Lambda}_{\eta N}(E)$ of length dimension. The specific value $c^{\Lambda}_{\eta N}(E_{\rm sc})$ of this LEC for a given cut-off $\Lambda$ is determined self consistently in the $\eta$-nuclear SVM calculation as detailed in the next subsection. By using the same value of $\Lambda$ in all $NN$, $NNN$, $\eta N$ and $\eta NN$ regulating Gaussians we reach a consistent extension of the nuclear $\not{\pi}$EFT to a combined $\eta$-nuclear $\not{\pi}$EFT approach.222Along with the $NNN$ LEC that averts a Thomas collapse of the $NNN$ system, an $\eta NN$ LEC $d_{\eta NN}^{\Lambda}$ is needed to avert $\eta NN$ collapse. Given no $\eta NN$ datum, a rough estimate $d_{\eta NN}^{\Lambda}=d_{NNN}^{\Lambda}$ is made here; see Appendix A: Erratum to the PLB published version.

In order to study the renormalization scale invariance of our few-body $\eta$-nuclear results, as shown for the purely nuclear case of $B(^{4}$He) in Table 1, we discuss below $\not{\pi}$EFT calculations done for several representative values of the scale parameter, $\Lambda=2,4,6,8$ fm*-1*. The last two values, clearly, exceed the momentum breakdown scale $Q^{\rho}_{\rm high}\approx 3.9$ fm*-1* of the underlying $N^{\ast}(1535)$ resonance model for the $\eta N$ interaction, or even more so the lower momentum breakdown scale $q^{\rho}_{\rm high}\approx 3.0$ fm*-1* set by excitation of vector meson degrees of freedom absent in the underlying $N^{\ast}(1535)$ dynamical models considered here, such as the $\rho$ meson produced at threshold in the strong pion exchange reaction $\pi N\to\rho N$ with $p^{\rho}_{\rm th}=586$ MeV/c. Finally, the model dependence of the LO $\eta N$ contact term introduced by studying two quite different $\eta N$ interaction models, GW and CS, leaves little motivation to go at present beyond LO. Hence, discussion of higher orders in $\not{\pi}$EFT is left to future work.

For a given value of $\Lambda$, the subthreshold values of the complex strength parameter $b^{\Lambda}_{\eta N}(E)$ in Eq. (6) were fitted to the complex phase shifts derived from the subthreshold scattering amplitudes $F_{\eta N}(E)$ in models GW and CS. The resulting values of the strength parameter $b^{\Lambda}_{\eta N}(E)$ for $\eta N$ subthreshold energies in model GW, shown in Fig. 2, fall off monotonically for both real and imaginary parts in going deeper below threshold, except for small kinks near threshold that reflect the threshold cusp of Re $F_{\eta N}(E)$ at $E_{\rm th}$ in Fig. 1. Similar curves for $b^{\Lambda}_{\eta N}(E)$ are obtained in model CS, with values smaller uniformly for both real and imaginary parts than model GW yields, in accordance with the relative strength of the generating scattering amplitudes $F_{\eta N}(E)$ shown in Fig. 1. We note that increasing $\Lambda$ leads to weaker strengths $b^{\Lambda}_{\eta N}(E)$ and also to a weaker energy dependence. Inspecting Fig. 2, one also notes that Im b^{\Lambda}_{\eta N}(E)$$\llRe $b^{\Lambda}_{\eta N}(E)$, which justifies treating Im $v_{\eta N}$ perturbatively in the present calculations.

2.4 Energy dependence

To determine the $\eta N$ subthreshold energy at which $v_{\eta N}(E)$ is evaluated as input to the $\eta$-nuclear few-body calculations reported below, we denote the shift away from threshold by $\delta\sqrt{s}\equiv\sqrt{s}-\sqrt{s_{\rm th}}$, expressing it in terms of output expectation values [13]:

[TABLE]

where $\beta_{N(\eta)}\equiv m_{N(\eta)}/(m_{N}+m_{\eta})$, $\xi_{A}\equiv Am_{N}/(Am_{N}+m_{\eta})$, $T_{N}$ and $T_{\eta}$ are the nuclear and $\eta$ kinetic energy operators evaluated in terms of internal Jacobi coordinates, with $T=T_{N}+T_{\eta}$ the total intrinsic kinetic energy of the system, $B$ is the total binding energy of the $\eta$-nuclear few-body system and $E_{\eta}=\langle\Psi|(H-H_{N})|\Psi\rangle$, where $H_{N}$ is the Hamiltonian of the purely nuclear part in its own cm frame, and the total Hamiltonian $H$ is evaluated in the overall cm frame. The imaginary, absorptive part of the $\eta N$ interaction is suppressed in this discussion. Noting that $(A-1)\langle T_{N:N}\rangle$ in Eq. (7) of Ref. [13] equals $\langle T_{N}\rangle$ here, Eq. (7) coincides with the former equation apart from a kinematical factor $\xi_{A}$ introduced here to make correspondence with the $\eta$-nuclear, last Jacobi coordinate with which $T_{\eta}$ is associated. Requiring that the expectation value $\langle\delta\sqrt{s}\rangle$ on the l.h.s. of Eq. (7), as derived from the solution of the Schroedinger equation, agrees with the input value $\delta\sqrt{s}$ for $v_{\eta N}(E)$, this equation defines a self-consistency cycle in our few-body $\eta$-nuclear calculations. Since each one of the four terms on the r.h.s. of Eq. (7) is negative, the self consistent energy shift $\delta\sqrt{s_{\rm sc}}$ is also negative, with size exceeding a minimum nonzero value obtained from the first two terms in the limit of vanishing $\eta$ binding. Eq. (7) in the limit $A\gg 1$ coincides with the nuclear-matter expression used in Refs. [8, 9] for calculating $\eta$-nuclear quasibound states.

Fig. 3 demonstrates how the self consistency requirement works in actual calculations. The three curves plotted in each panel are obtained by interpolating a sequence of calculated $\eta\,^{3}$He bound-state energies (squares) and the corresponding expectation values $\langle\delta\sqrt{s}\rangle$ (circles) from Eq. (7) for $A=3$, as a function of the input $\delta\sqrt{s}$ to the energy argument $E_{\rm th}+\delta\sqrt{s}$ of $v_{\eta N}^{\rm GW}$, for two choices of the momentum scale parameter $\Lambda=2,\,4$ fm*-1*. The dashed vertical line marks the self consistent value of $\delta\sqrt{s}$ at which the outcome bound-state energy $E(\eta\,{{}^{3}{\rm He}})$ is evaluated, and the dashed horizontal line marks the 3He core energy $E(^{3}$He). Note that the self consistent value $E_{\rm sc}(\eta\,{{}^{3}{\rm He}})$ is higher than $E(^{3}$He) in the left panel for $\Lambda=2$ fm*-1*, while it is lower than $E(^{3}$He) in the right panel for $\Lambda=4$ fm*-1*. This means that, correspondingly, $\eta\,^{3}$He is slightly unbound for $\Lambda=2$ fm*-1* while slightly bound for $\Lambda=4$ fm*-1*. We note furthermore that for threshold values $v_{\eta N}^{\rm GW}(E_{\rm th})$, i.e. $\delta\sqrt{s}=0$, $\eta\,^{3}$He is bound in both cases (and also if the often used but unfortunately unfounded self consistency requirement [29] $\delta\sqrt{s}=E_{\eta}$ is imposed). Finally, the upper curve in Fig. 3 shows the expectation value $\langle H_{N}\rangle$ of the nuclear core energy, which is bounded from below by the 3He core energy $E(^{3}$He) marked by the dashed horizontal line.

3 Results and Discussion

Separation energies $B_{\eta}\equiv B(\eta\,{{}^{A}{\rm He}})-B(^{A}$He) (often called $\eta$ binding energies) of the $\eta\,^{A}$He isotopes with $A=3,4$ were calculated self consistently in the SVM using LO $\not{\pi}$EFT $NN$, $NNN$ and $\eta NN$ regulated contact terms introduced in Eqs. (2)–(4) and Footnote 2, and a regulated $\eta N$ energy dependent contact term specified by Eq. (6), with scale parameters $\Lambda=2,4,6,8$ fm*-1*. Two $\eta N$ coupled-channels models were used, GW [3] and CS [23]. The CS $\eta N$ interaction was found by far too weak to bind $\eta\,^{3}$He, and only by a fraction of MeV short of binding $\eta\,^{4}$He. The binding energies $B_{\eta}$ were evaluated using real Hamiltonians in which Im $v_{\eta N}$ was disregarded. Restoring Im $v_{\eta N}$ in optical model calculations was found particularly important for near-threshold bound states, lowering their calculated $B_{\eta}$ by 0.2$\pm$0.1 MeV. The $\eta$-nuclear widths $\Gamma_{\eta}$ were calculated with wavefunctions $\Psi_{\rm g.s.}$ generated by these real Hamiltonians:

[TABLE]

Here, $V_{\eta}$ sums over all pairwise $\eta N$ interactions. Since $|{\rm Im}\,V_{\eta}|\ll|{\rm Re}\,V_{\eta}|$, this is a reasonable approximation.

Binding energies $B_{\eta}$ and widths $\Gamma_{\eta}$ resulting from these self consistent calculations are listed in Table 2 for the $\eta N$ potentials $v_{\eta N}^{\rm GW}(E)$ with scale parameters $\Lambda=2,4$ fm*-1*, values for which our self consistency procedure was demonstrated in Fig. 3. Higher values of $\Lambda$ exceed by far the momentum breakdown scale $q^{\rho}_{\rm high}$ introduced in our previous work [13] for $N^{\ast}$(1535) resonance meson-baryon models in which the $\eta N$ scattering amplitude $F_{\eta N}$ is determined. Taken literally, this would mean that the GW $\eta N$ interaction hardly binds $\eta\,^{3}$He, if at all, and is likely to bind slightly $\eta\,^{4}$He, with $B_{\eta}$ of order 1 MeV.

Sequences of calculated values of $\eta$ binding energy $B_{\eta}$ using $\eta N$ potentials $v_{\eta N}^{\rm GW}(E)$ are shown for $\eta\,^{3}$He and $\eta\,^{4}$He in Figs. 4 and 5, respectively, as a function of $\Lambda^{-1}$. The figures demonstrate that the larger $\Lambda$, the larger is the resulting $\eta$ binding energy $B_{\eta}$ in spite of a similar increase in the value of $-\delta\sqrt{s_{\rm sc}}$ in self consistent calculations which implies a weaker $\eta N$ potential strength $b(E_{\rm sc})$. The dependence of $B_{\eta}$ on $\Lambda$ is weak for $\eta\,^{3}$He and moderate for $\eta\,^{4}$He in these calculations. For $\Lambda\geq 4$ fm*-1*, $B_{\eta}$ varies linearly in $\Lambda^{-1}$, with an average error of 50 keV for $\eta\,^{3}$He and 300 keV for $\eta\,^{4}$He, and with twice these errors upon extrapolating $\Lambda\to\infty$. In contrast, for calculations done at the $\eta N$ threshold, i.e. $\delta\sqrt{s}$=0, the resulting values of $B_{\eta}$ shown in the figures depend strongly on $\Lambda$ with almost perfect linear dependence on $\Lambda^{-1}$ for $\Lambda\geq 4$ fm*-1*.

Interestingly, Figs. 4 and 5 also suggest that $B_{\eta}$ assumes a finite value $B_{\eta}^{\Lambda\to\infty}$ in the limit of point $\eta N$ interaction. This follows directly from the introduction of a stabilizing $\eta NN$ LEC $d_{\eta NN}^{\Lambda}$ analogous to the $NNN$ LEC $d_{NNN}^{\Lambda}$ of Eq. (3). Here we used a value $d_{\eta NN}^{\Lambda}=d_{NNN}^{\Lambda}$. The sensitivity of our results to this choice is studied in Appendix A. Generally, a Thomas collapse of three-body systems is averted in $\not{\pi}$EFT by promoting a non-derivative three-body contact term from N2LO to LO.

4 Conclusion

To summarize, we have presented genuine few-body SVM calculations of $\eta NNN$ ($\eta\,^{3}$He) bound states and, for the first time, also $\eta NNNN$ ($\eta\,^{4}$He) bound states, using LO $\not{\pi}$EFT interactions where the $\eta N$ interaction contact term was derived in coupled channels studies of the $N^{\ast}(1535)$ nucleon resonance. Special care was taken of the energy dependence of the input $\eta N$ subthreshold scattering amplitude by using a self consistency procedure. The present results exhibit renormalization scale invariance of the calculated $\eta$ binding energies owing to the introduction of a repulsive $\eta NN$ contact interaction. For physically motivated values of $\Lambda$, the onset of $\eta\,^{3}$He binding occurs for Re$\,a_{\eta N}$ close to 1 fm, as in model GW [3], consistently with our previous hyperspherical-basis $\eta NNN$ calculations [13]. The onset of $\eta\,^{4}$He binding requires a lower value of Re $a_{\eta N}$, exceeding however 0.7 fm; it is comfortably satisfied in model GW but not in model CS [23]. Further dedicated experimental searches for $\eta\,^{4}$He bound states are desirable in order to confirm the recent negative report from WASA-at-COSY [17] which, taken at face value, implies that Re $a_{\eta N}\lesssim 0.7$ fm. Similar results and conclusions hold valid in SVM calculations using non-EFT realistic nuclear models [14, 15] augmented by the same $\eta N$ interaction models used here, and will be reported elsewhere.

Acknowledgments

We thank Jiří Mareš and Martin Schaefer for useful discussions on related matters. This work was supported in part (NB) by the Israel Science Foundation grant 1308/16, in part (NB, BB) by Pazi Fund grants, and in part (EF, AG) by the EU initiative FP7, Hadron-Physics3, under the SPHERE and LEANNIS cooperation programs.

Appendix A: Erratum [30] to “Onset of $\eta$-nuclear

binding in a pionless EFT approach” [31]

A three-body $\eta NN$ force was inadvertently overlooked in the potential model description and discussion in Ref. [31]. In the actual calculations, however, the LO interaction between the $\eta$ and the nucleons was composed of the $\eta N$ term discussed here in Sect. 2.3, supplemented by an $\eta NN$ term

[TABLE]

In this expression, $\delta_{\Lambda}(r_{\eta N_{i}},r_{\eta N_{j}})$ is a product of normalized pairwise Gaussians $\delta_{\Lambda}(r_{\eta N_{i}})$ and $\delta_{\Lambda}(r_{\eta N_{j}})$, with range parameter inversely proportional to the momentum-scale parameter $\Lambda$, as defined by Eq. (4) here. For the results presented in this paper, the low energy constant (LEC) $d_{\eta NN}^{\Lambda}$ was set equal to the nuclear $NNN$ LEC $d_{NNN}^{\Lambda}$. Setting $d_{\eta NN}^{\Lambda}=0$, the $\eta$-deuteron ($\eta\,d$) system, and therefore any $\eta$-nucleus system, would collapse as $\Lambda\rightarrow\infty$.

The parameter $d^{\Lambda}_{\eta NN}$ is a free parameter to be fixed by experimental data. In the absence of such data one may estimate its value using the nuclear $NNN$ LEC, $d^{\Lambda}_{\eta NN}=d^{\Lambda}_{NNN}$, as done here [31], or to set a bound on its value accepting that $\eta\,d$ is unbound [12], i.e. set $d^{\Lambda}_{\eta NN}$ so that $B_{\eta}(\eta\,d)=0$. To check the sensitivity of the present results [31] to these distinct choices of $d^{\Lambda}_{\eta NN}$, we show in Figs. 6 and 7 $\eta$ separation energies $B_{\eta}$ in $\eta\,^{3}$He and $\eta\,^{4}$He, respectively, calculated using $\eta N$ potentials $v_{\eta N}^{\rm GW}(E)$ under these two choices of $d^{\Lambda}_{\eta NN}$. Figs. 6 and 7 update the original Figs. 4 and 5 here [31].

Figures 6 and 7 demonstrate that the two choices made for the three-body $\eta NN$ LEC yield practically identical values of $B_{\eta}$ in the limit $\Lambda\rightarrow\infty$. For values f $\Lambda$ near the physical breakdown scale $\Lambda\approx 4$ fm*-1*, however, $B_{\eta}$ differs by about 0.7 MeV for $\eta\,^{3}$He and 2 MeV for $\eta\,^{4}$He between the two choices applied in self consistent calculations (lower group of curves). Since $\eta\,d$ is unbound [12], the choice marked in dashed lines in both figures is likely to somewhat overestimate $B_{\eta}$. Nevertheless, these $\eta$ separation energies are in good agreement with the non-EFT $B_{\eta}$ values calculated recently using the same two-body energy dependent $\eta N$ interaction [32].