The onset of $\Lambda\Lambda$ hypernuclear binding

TL;DR

This study uses pionless EFT and stochastic variational methods to analyze the binding energies of light $ ext{Lambda} ext{Lambda}$ hypernuclei, assessing the conditions for their formation and stability in the strangeness S=-2 sector.

Contribution

It provides a systematic EFT-based calculation of light $ ext{Lambda} ext{Lambda}$ hypernuclei binding energies, clarifying the onset of binding and stability in this sector.

Findings

$^{3}_{ ext{Lambda} ext{Lambda}}$n and $^{4}_{ ext{Lambda} ext{Lambda}}$n are unbound.

Large $|a_{ ext{Lambda} ext{Lambda}}| extgreater 1.5$ fm needed for $^{4}_{ ext{Lambda} ext{Lambda}}$H binding.

$^{5}_{ ext{Lambda} ext{Lambda}}$H--$^{5}_{ ext{Lambda} ext{Lambda}}$He are stable with $ ext{Lambda}$ separation energy around 1 MeV.

Abstract

Binding energies of light, $A\leq 6$, $\Lambda\Lambda$ hypernuclei are calculated using the stochastic variational method in a pionless effective field theory (EFT) approach at leading order with the purpose of assessing critically the onset of binding in the strangeness S=-2 hadronic sector. The EFT input in this sector consists of (i) a $\Lambda\Lambda$ contact term constrained by the $\Lambda\Lambda$ scattering length $a_{\Lambda\Lambda}$, using a range of values compatible with $\Lambda\Lambda$ correlations observed in relativistic heavy ion collisions, and (ii) a $\Lambda\Lambda N$ contact term constrained by the only available $A\leq 6$ $\Lambda\Lambda$ hypernucler binding energy datum of $^{6}_{\Lambda\Lambda}$He. The recently debated neutral three-body and four-body systems $^{3}_{\Lambda\Lambda}$n and $^{4}_{\Lambda\Lambda}$n are found unbound by a wide margin. A relatively large value of $|a_{\Lambda\Lambda}| \gtrsim 1.5$ fm is needed to bind $^{4}_{\Lambda\Lambda}$H, thereby questioning its particle stability. In contrast, the particle stability of the $A=5$ $\Lambda\Lambda$ hypernuclear isodoublet $^{5}_{\Lambda\Lambda} $H--$^{5}_{\Lambda\Lambda}$He is robust, with $\Lambda$ separation energy of order 1 MeV.

Figures

Tables

Table 1. Table 1: Λ Λ \Lambda separation energies B Λ ( Λ ​ Λ A B_{\Lambda}(^{~{}~{}A}_{\Lambda\Lambda} Z) for A 𝐴 A =3–6, calculated using a Λ ​ Λ subscript 𝑎 Λ Λ a_{\Lambda\Lambda} = − 0.8 0.8 -0.8  fm, cutoff λ 𝜆 \lambda =4 fm -1 and the Alexander[B] Λ ​ N Λ 𝑁 \Lambda N interaction model  [ 24 ] . In each row a Λ ​ Λ ​ N Λ Λ 𝑁 \Lambda\Lambda N LEC was fitted to the underlined binding energy constraint.

Constraint (MeV)	${}_{\Lambda \Lambda }{}^{}{}_{}{}^3$n	${}_{\Lambda \Lambda }{}^{}{}_{}{}^4$n	${}_{\Lambda \Lambda }{}^{}{}_{}{}^4$H	${}_{\Lambda \Lambda }{}^{}{}_{}{}^5$H	${}_{\Lambda \Lambda }{}^{}{}_{}{}^6$He
$\Delta B_{\Lambda \Lambda }{(}_{\Lambda \Lambda }^6$He)=$\underset{¯}{0.67}$	–	–	–	1.21	3.28
$B_{\Lambda }{(}_{\Lambda \Lambda }^4$H)=$\underset{¯}{0.05}$	–	–	0.05	2.28	4.76
$B{(}_{\Lambda \Lambda }^4$n)=$\underset{¯}{0.10}$	–	0.10	0.86	4.89	7.89
$B{(}_{\Lambda \Lambda }^3$n)=$\underset{¯}{0.10}$	0.10	15.15	18.40	22.13	25.66

Equations

$${\mathcal{L}}^{(\mathrm{LO})}=\sum _B{B}^{†}(i{\partial }_0+\frac{{\nabla }^2}{2M_B})B-{\mathcal{V}}_2-{\mathcal{V}}_3,$$

$$V_2=\sum _{IS}{C}_{\lambda }^{IS}\sum _{i<j}{\mathcal{P}}_{IS}(ij){\delta }_{\lambda }({\mathbf{r}}_{ij}),$$

$${\delta }_{\lambda }(\mathbf{r})={\left(\frac{\lambda }{2\sqrt{\pi }}\right)}^3\exp \left(-\frac{{\lambda }^2}{4}{\mathbf{r}}^2\right),$$

$$V_3=\sum _{\alpha IS}{D}_{\alpha \lambda }^{IS}\sum _{i<j<k}{\mathcal{Q}}_{IS}(ijk)\left(\sum _{\mathrm{cyc}}{\delta }_{\lambda }({\mathbf{r}}_{ij}){\delta }_{\lambda }({\mathbf{r}}_{jk})\right),$$

$$\frac{B_{\Lambda }(\lambda )}{B_{\Lambda }(\infty )}=\left[1+\alpha \frac{Q}{\lambda }+\beta {\left(\frac{Q}{\lambda }\right)}^2+\dots \right].$$

Full text

††journal: Physics Letters B

1 Introduction

Single-$\Lambda$ and double-$\Lambda$ ($\Lambda\Lambda$) hypernuclei provide a unique extension of nuclear physics into strange hadronic matter [1]. Whereas the behavior of a single $\Lambda$ hyperon in atomic nuclei has been deduced quantitatively by studying $\Lambda$ hypernuclei (${}_{\Lambda}^{\rm A}$Z) from $A$=3 to 208 [2], only three $\Lambda\Lambda$ hypernuclei (${}^{~{}\rm A}_{\Lambda\Lambda}$Z) are firmly established: the lightest known ${}^{~{}~{}6}_{\Lambda\Lambda}$He Nagara event [3] and two heavier ones, ${}^{10}_{\Lambda\Lambda}$Be and ${}^{13}_{\Lambda\Lambda}$B [4]. Remarkably, their binding energies come out consistently in shell-model calculations [5]. Few ambiguous emulsion events from KEK [6] and J-PARC [7] have also been reported. However, and perhaps more significant is the absence of any good data on the onset of $\Lambda\Lambda$ hypernuclear binding for $A<6$. In distinction from the heavier species, these very light $s$-shell species, if bound, could be more affected by microscopic strangeness ${\cal S}=-2$ dynamics. An obvious issue is the effect of a possible $\Xi N$ dominated $H$ dibaryon resonance some 20–30 MeV above the $\Lambda\Lambda$ threshold [8, 9] on $\Lambda\Lambda$ hypernuclear binding in general.

Several calculations of light $A<6$ $s$-shell $\Lambda\Lambda$ hypernuclei using $\Lambda\Lambda$ interactions fitted to ${}^{~{}~{}6}_{\Lambda\Lambda}$He suggest a fairly weak $\Lambda\Lambda$ interaction, with the onset of $\Lambda\Lambda$ hypernuclear binding defered to $A=4$. Indeed, a slightly bound $I=0$ ${}^{~{}~{}4}_{\Lambda\Lambda}$H(1+) was found in $\Lambda\Lambda pn$ four-body calculations by Nemura et al. [10, 11] but not in a four-body calculation by Filikhin and Gal [12] who nonetheless got it bound as a $\Lambda\Lambda d$ cluster. Unfortunately, the AGS-E906 counter experiment [13] searching for light $\Lambda\Lambda$ hypernuclei failed to provide conclusive evidence for the particle stability of ${}^{~{}~{}4}_{\Lambda\Lambda}$H [14, 15]. Interestingly, the neutral four-body system ${}^{~{}~{}4}_{\Lambda\Lambda}$n has been assigned in Ref. [15] to the main yet unexplained signal observed by AGS-E906. Recent few-body calculations of ${}^{~{}~{}4}_{\Lambda\Lambda}$n [16, 17] diverge on its particle stability, but since none was constrained by the ${}^{~{}~{}6}_{\Lambda\Lambda}$He binding energy datum, no firm conclusion can be drawn yet.

In the present work we study the light $A\leq 6$ $s$-shell $\Lambda\Lambda$ hypernuclei together with their nuclear and $\Lambda$ hypernuclear cores at leading-order (LO) $\not{\pi}$EFT. The $\not{\pi}$EFT approach was first applied to few-nucleon atomic nuclei in Refs. [18, 19] and recently also in lattice calculations of nuclei [20, 21, 22, 23] and to $s$-shell single-$\Lambda$ hypernuclei [24]. Focusing on $\not{\pi}$EFT applications to ${\cal S}=-2$ light systems, we note $\Lambda$-$\Lambda$-core LO calculations done for $A=4$ [25] and separately for $A=6$ [26], which therefore limits their predictive power. Among past non-EFT studies, the only work that covers all $s$-shell $\Lambda\Lambda$ hypernuclei is by Nemura et al. [11] who used simulated forms of outdated hard-core $YN$ and $YY$ Nijmegen potentials [27]. No chiral EFT ($\chi$EFT) calculations of $\Lambda\Lambda$ hypernuclei have been reported, although $\chi$EFT representations of the $\Lambda\Lambda$ interaction at LO [28] and NLO [29] do exist. Hence, the present LO $\not{\pi}$EFT work is the first comprehensive EFT application to light $\Lambda\Lambda$ hypernuclei, and could be generalized in principle to study multi-$\Lambda$ hypernuclei and strange hadronic matter.

The $\not{\pi}$EFT baryonic Lagrangian governing multi-$\Lambda$ hypernuclei requires, at LO, one $\Lambda\Lambda$ and one $\Lambda\Lambda N$ interaction terms beyond the interaction terms involved in the description of single-$\Lambda$ hypernuclei. We fit the new $\Lambda\Lambda$ contact term to a $\Lambda\Lambda$ scattering length value spanning a range of values, $-0.5$ fm to $-1.9$ fm, suggested by analyses of the $\Lambda\Lambda$ invariant mass spectrum [30] measured in the 12C$(K^{-},K^{+})$ reaction at the KEK-PS [31] and of $\Lambda\Lambda$ correlations [32] extracted from ultra-relativistic Au+Au collisions at the RHIC-STAR experiment [33]. The corresponding $\Lambda\Lambda$ interactions are weakly atractive, far from producing a $\Lambda\Lambda$ bound state. Recent analyses of LHC-ALICE experiments reach similar conclusions, but leave room also for a $\Lambda\Lambda$ bound state [34]. For each choice of $\Lambda\Lambda$ contact term we determine a $\Lambda\Lambda N$ three-body contact term, promoted to LO, by fitting to $\Delta B_{\Lambda\Lambda}(^{~{}~{}6}_{\Lambda\Lambda}$He)=$B_{\Lambda\Lambda}(^{~{}~{}6}_{\Lambda\Lambda}$He)$-2B_{\Lambda}(_{\Lambda}^{5}$He)=0.67$\pm$0.17 MeV. With such $\Lambda\Lambda$ and $\Lambda\Lambda N$ contact-term input our $\not{\pi}$EFT scheme exhibits renormalization scale invariance in the limit of point-like interactions, demonstrating that no $N\geq 4$ $N$-body contact term is required at LO, as first shown by Platter et al. [35] for a four-nucleon system. Applying this scheme to explore $A=3,4,5$ $\Lambda\Lambda$ hypernuclei, we find that unless $|a_{\Lambda\Lambda}|\gtrsim 1.5$ fm, ${}^{~{}~{}4}_{\Lambda\Lambda}$H is unlikely to be particle stable. The neutral systems ${}^{~{}~{}3}_{\Lambda\Lambda}$n and ${}^{~{}~{}4}_{\Lambda\Lambda}$n are found unstable by a wide margin. A robust particle stability is established for the ${}^{~{}~{}5}_{\Lambda\Lambda}$H–${}^{~{}~{}5}_{\Lambda\Lambda}$He $A=5$ isodoublet, with $\Lambda$ separation energy of order 1 MeV, providing further support for a recent J-PARC proposal P75 [36] to produce ${}^{~{}~{}5}_{\Lambda\Lambda}$H. Possible extensions of our work are briefly discussed in the concluding section.

2 Application of $\not{\pi}$EFT to $\Lambda\Lambda$ hypernuclei

With $\Lambda\Lambda$ one-pion exchange forbidden by isospin invariance, the lowest mass pseudoscalar meson exchange is provided by a short range $\eta$ exchange ($\approx$0.4 fm) which is rather weak in SU(3) flavor. Pions appear in the $\Lambda\Lambda$ dynamics through excitation to fairly high-lying $\Sigma\Sigma$ intermediate states. Therefore, a reasonable choice of a $\not{\pi}$EFT breakup scale is 2$m_{\pi}$, same as argued for in our recent work on $\Lambda$ hypernuclei [24]. Excitation from $\Lambda\Lambda$ states to the considerably lower mass $\Xi N$ intermediate states requires a shorter range $K$ meson exchange which, together with other short-range exchanges, is accounted for implicitly by the chosen $\not{\pi}$EFT contact interactions. To provide a meaningful $\not{\pi}$EFT expansion parameter we note that since $\Delta B_{\Lambda\Lambda}(^{~{}~{}6}_{\Lambda\Lambda}$He) is less than 1 MeV, considerably smaller than $B_{\Lambda}(^{5}_{\Lambda}$He), a $\Lambda$ momentum scale $Q$ in ${}^{~{}~{}6}_{\Lambda\Lambda}$He may be approximated by that in ${}^{5}_{\Lambda}$He [24], namely $p_{\Lambda}\approx\sqrt{2M_{\Lambda}B_{\Lambda}}=83$ MeV/c, yielding a $\not{\pi}$EFT expansion parameter $(Q/2m_{\pi})\approx 0.3$ and LO accuracy of order $(Q/2m_{\pi})^{2}\approx 0.09$.

To construct the appropriate $\not{\pi}$EFT Lagrangian density at LO we follow our previous work on single-$\Lambda$ hypernuclei [24]:

[TABLE]

where $B=(N,\Lambda)$ and ${\cal V}_{2},{\cal V}_{3}$ consist of two-body and three-body $s$-wave contact interaction terms, each of which is associated with its own low-energy constant (LEC). These contact terms are shown diagrammatically in Fig. 1 and the corresponding LECs are listed alongside. Going from single-$\Lambda$ hypernuclei to multi-$\Lambda$ hypernuclei brings in one new $\Lambda\Lambda$ two-body LEC, $C_{5}$, and one new $\Lambda\Lambda N$ three-body LEC, $D_{5}$, each one labelled by the total Pauli-spin and isospin involved. This completes the set of LECs required to describe single-, double- and in general multi-$\Lambda$ hypernuclei at LO. Further contact terms, such as a three-body $\Lambda\Lambda\Lambda$ term, appear only at subleading orders.

Following the procedure applied in Ref. [18], the two-body contact interaction term ${\cal V}_{2}$ gives rise to a two-body potential

[TABLE]

where ${\cal P}_{IS}$ are projection operators on $s$-wave $NN,\Lambda N,\Lambda\Lambda$ pairs with isospin $I$ and spin $S$ values associated in Fig. 1 with two-body LECs. These LECs are fitted to low-energy two-body observables, e.g., to the corresponding $NN,\Lambda N,\Lambda\Lambda$ scattering lengths. The subscript $\lambda$ attached to $C^{IS}$ in Eq. (2) stands for a momentum cutoff introduced in a Gaussian form to regularize the zero-range contact terms:

[TABLE]

thereby smearing a zero-range (in the limit $\lambda\to\infty$) Dirac $\delta^{(3)}({\boldsymbol{r}})$ contact term over distances $\sim\lambda^{-1}$. The cutoff parameter $\lambda$ may be viewed as a scale parameter with respect to typical values of momenta $Q$. To make observables cutoff independent, the LECs must be properly renormalized. Truncating $\not{\pi}$EFT at LO and using values of $\lambda$ higher than the breakup scale of the theory (here $\approx$2$m_{\pi}$), observables acquire a residual dependence $O(Q/\lambda)$ which diminishes with increasing $\lambda$.

The three-body contact interaction, promoted to LO, gives rise to a three-body potential

[TABLE]

where ${\cal Q}_{IS}$ projects on $NNN$, $NN\Lambda$ and $\Lambda\Lambda N$ $s$-wave triplets with isospin $I$ and spin $S$ values associated in Fig. 1 with three-body LECs which are fitted to given binding energies. The subscript $\alpha$ distinguishes between the two $IS=\frac{1}{2}\frac{1}{2}$ $NNN$ and $\Lambda\Lambda N$ triplets marked in the figure.

Using two-body $V_{2}$ and three-body $V_{3}$ regularized contact interaction terms as described above, we solved the $A$-body Schrödinger equation variationally by expanding the wave function $\Psi$ in a correlated Gaussian basis using the SVM. For a comprehensive review of this method, see Ref. [38]. For a specific calculation of the three-body interaction matrix elements, see Ref. [39].

3 Results and discussion

We first discuss the case of ${}^{~{}~{}4}_{\Lambda\Lambda}$H, with $I=0$ and $J^{\pi}=1^{+}$, which following the brief discussion in the Introduction could signal the onset of $\Lambda\Lambda$ hypernuclear binding. For each of several given cutoff values $\lambda$ we searched for minimum values of $|a_{\Lambda\Lambda}|$, as a function of $\Delta B_{\Lambda\Lambda}(^{~{}~{}6}_{\Lambda\Lambda}$He), that would make ${}^{~{}~{}4}_{\Lambda\Lambda}$H particle stable. The choice of a specific value for this $\Delta B_{\Lambda\Lambda}$ determines the $\Lambda\Lambda N$ LEC necessary for the ${}^{~{}~{}4}_{\Lambda\Lambda}$H calculation, in addition to the $\Lambda\Lambda$ LEC determined by $a_{\Lambda\Lambda}$. The resulting values of $|a_{\Lambda\Lambda}|$ above which ${}^{~{}~{}4}_{\Lambda\Lambda}$H is particle stable are plotted in Fig. 2 as a function of $\Delta B_{\Lambda\Lambda}(^{~{}~{}6}_{\Lambda\Lambda}$He). Choosing sufficiently large values of the cutoff $\lambda$, say $\lambda\gtrsim$ 4 fm*-1*, for which convergence to the renormalization scale invariance limit $\lambda\to\infty$ is seen explicitly in the figure, one concludes that $|a_{\Lambda\Lambda}|$ needs to be larger than $\approx$1.5 fm to bind ${}^{~{}~{}4}_{\Lambda\Lambda}$H. A $\Lambda\Lambda$ scattering length of such size would make the $\Lambda\Lambda$ interaction almost as strong as the $\Lambda N$ interaction, whereas most theoretical constructions, e.g. recent Nijmegen models, suggest that it is considerably weaker, say $|a_{\Lambda\Lambda}|\approx 0.8$ fm [40]. For this reason we argue that ${}^{~{}~{}4}_{\Lambda\Lambda}$H is unlikely to be particle stable.

Using representative values $a_{\Lambda\Lambda}=-0.8$ fm and cutoff $\lambda$=4 fm*-1*, values for which according to Fig. 2 ${}^{~{}~{}4}_{\Lambda\Lambda}$H is particle unstable, one may reduce the repulsive $\Lambda\Lambda N$ LEC in order to make it particle stable. According to the first two rows in Table 1, this will overbind ${}^{~{}~{}6}_{\Lambda\Lambda}$He by $\approx$1.5 MeV. Reducing further the $\Lambda\Lambda N$ LEC one binds the neutral systems, first ${}^{~{}~{}4}_{\Lambda\Lambda}$n (third row) and then ${}^{~{}~{}3}_{\Lambda\Lambda}$n (fourth row), at a price of overbinding further ${}^{~{}~{}6}_{\Lambda\Lambda}$He. In fact, the particle stability of these $A=3,4$ neutral $\Lambda\Lambda$ systems is incompatible with the ${}^{~{}~{}6}_{\Lambda\Lambda}$He Nagara event binding energy datum for all values of cutoff $\lambda$ and scattering length $a_{\Lambda\Lambda}$ tested in Fig. 2. These results suggest quantitatively that the $A=3,4$ light neutral $\Lambda\Lambda$ hypernuclei are unbound within a large margin.

Calculated values of the $\Lambda$ separation energy $B_{\Lambda}(^{~{}~{}5}_{\Lambda\Lambda}$H) are shown in Fig. 3. Several representative values of the $\Lambda\Lambda$ scattering length were used: $a_{\Lambda\Lambda}$=$-0.5,-0.8,-1.9$ fm, spanning a broad range of values suggested in $\Lambda\Lambda$ correlation studies [30, 32] of experimental spectra mentioned in the Introduction. Again, the choice of $a_{\Lambda\Lambda}$ determines the one $\Lambda\Lambda$ LEC required at LO, while the $\Lambda\Lambda N$ LEC was fitted to the $\Delta B_{\Lambda\Lambda}(^{~{}~{}6}_{\Lambda\Lambda}$He)=0.67$\pm$0.17 MeV datum. For the $\Lambda N$ interaction terms we generally used the Alexander[B] $\Lambda N$ model [24], with its $\Lambda N$ $C_{3}$ and $C_{4}$ LECs (see Fig. 1) corresponding to the scattering lengths $a_{s}$=$-1.8$ fm and $a_{t}$=$-1.6$ fm, respectively, from [41]. For cutoff $\lambda$=4 fm*-1* we also used three other $\Lambda N$ interaction models from Ref. [24]: $\chi$LO [42], $\chi$NLO [43] and NSC97f [44], demonstrating that the $\Lambda N$ model dependence is rather weak when it comes to $\Lambda$ separation energies in double-$\Lambda$ hypernuclei, provided $B_{\Lambda}$ values of single-$\Lambda$ hypernuclei for $A<5$ are fitted to generate the necessary $\Lambda NN$ LECs.111This is reminiscent of the weak dependence of $\Lambda$ separation energies in single-$\Lambda$ hypernuclei on the input $NN$ interaction, found in few-body calculations by Nogga et al. [45]. Calculated values of $B_{\Lambda}(^{5}_{\Lambda}$He), compatible with those from Ref. [24] are also shown in the figure, demonstrating the suitability of the input $\Lambda N$ model. One observes that ${}^{~{}~{}5}_{\Lambda\Lambda}$H comes out particle stable over a broad range of finite cutoff values used in the calculations. This is not the case for ${}^{~{}~{}4}_{\Lambda\Lambda}$H which, as discussed above, is unbound with respect to ${}^{3}_{\Lambda}$H for most of the permissible parameter space.

The calculated $B_{\Lambda}$ values shown in Fig. 3 exhibit renormalization scale invariance in the limit of $\lambda\to\infty$. To figure out the associated $B_{\Lambda}(\lambda\to\infty)$ values, we extrapolated $B_{\Lambda}(\lambda)$ for $\lambda\geq 4$ fm*-1* using a power series in the small parameter $Q/\lambda$:

[TABLE]

The corresponding extrapolation curves are shown by dotted lines in Fig. 3, converging at asymptotic values $B_{\Lambda}(\infty)$ given with their extrapolated uncertainties by the gray horizontal bands in the figure. ${}^{~{}~{}5}_{\Lambda\Lambda}$H remains particle stable in this limit with $\Lambda$ separation energy $B_{\Lambda}(\infty)=1.14\pm 0.01^{+0.44}_{-0.26}$ MeV, where the first uncertainty is due to extrapolating by use of Eq. (5) and the second one is due to the $a_{\Lambda\Lambda}$ and $B_{\Lambda}$ uncertainties.

The $\Lambda$ separation energies $B_{\Lambda}(^{~{}~{}5}_{\Lambda\Lambda}$H) studied above are correlated with those of ${}^{~{}~{}6}_{\Lambda\Lambda}$He in a way reminiscent of the Tjon line correlation between binding energies calculated for 3H and 4He [46]. This is shown in Fig. 4 by the linear dependence of $B_{\Lambda}(^{~{}~{}5}_{\Lambda\Lambda}$H), for two given values of the cutoff $\lambda$, on the value assumed for $\Delta B_{\Lambda\Lambda}(^{~{}~{}6}_{\Lambda\Lambda}$He), which was varied for this purpose around the ‘physical’ value 0.67$\pm$0.17 MeV. We note that the cutoff dependence of this correlation is very weak. The hypernuclear correlation noted here is generated by variation of the $\Lambda\Lambda N$ LEC which is derived from $\Delta B_{\Lambda\Lambda}(^{~{}~{}6}_{\Lambda\Lambda}$He). This is similar to the origin and realization of Tjon-line correlations in nuclear physics, where many-body contact interaction terms beyond three-body terms do not appear at LO [35]. However unlike other physics applications where Tjon lines were shown to hold, its appearance here does not require proximity to the unitary limit.

We note that $a_{\Lambda\Lambda}$ includes implicitly the coupling of the $\Lambda\Lambda$ channel to the higher mass $I$=$S$=0 $\Xi N$ and $\Sigma\Sigma$ channels. However, beginning with ${}^{~{}~{}6}_{\Lambda\Lambda}$He the coupling to the relatively low-lying $\Xi N$ channel is partially Pauli blocked (with the formed nucleon excluded from the $s$ shell). It could be argued then that the reference value of $\Delta B_{\Lambda\Lambda}(^{~{}~{}6}_{\Lambda\Lambda}$He) used in this work has to be somewhat increased in order to account for the blocked states which are included effectively in the present LO application of $\not{\pi}$EFT to $\Lambda\Lambda$ hypernuclei. The coupled-channel calculations by Vidaña et al. [50] suggest an increase of $\approx$0.25 MeV which according to Fig. 4 would increase $B_{\Lambda}(^{~{}~{}5}_{\Lambda\Lambda}$H) by roughly 0.15 MeV and ${}^{~{}~{}4}_{\Lambda\Lambda}$H, had it been bound, by no more than 0.03 MeV. Interestingly, recent HAL QCD Collaboration studies based on LQCD find a vanishingly small $I=0$ ${}^{1}S_{0}$ $\Lambda\Lambda-\Xi N$ coupling potential for $r\gtrsim 0.6$ fm [51]. Consequences of this exremely weak coupling, as well as the one derived at NLO within $\chi$EFT [29], on few-body $\Lambda\Lambda$ hypernuclei should be studied in future NLO calculations.

4 Summary and outlook

The focus in this first comprehensive $\not{\pi}$EFT application to light $\Lambda\Lambda$ hypernuclei was to study the onset of binding in the ${\cal S}=-2$ hadronic sector by constraining $\Delta B_{\Lambda\Lambda}(^{~{}~{}6}_{\Lambda\Lambda}$He) to the most recent value 0.67$\pm$0.17 MeV [4] assigned to the Nagara event [3]. We varied the value assumed for the $\Lambda\Lambda$ scattering length $a_{\Lambda\Lambda}$ over a range of values suggested in Refs. [30, 32] by analyzing $\Lambda\Lambda$ correlations observed in several high-energy production reactions, but barring an unlikely $\Lambda\Lambda$ bound state [8, 9, 52]. Our results suggest with little model dependence that both members of the $A=5$ isodoublet pair, ${}^{~{}~{}5}_{\Lambda\Lambda}$H and ${}^{~{}~{}5}_{\Lambda\Lambda}$He, are particle stable. Of the $A=4$ $\Lambda\Lambda$ hypernuclei, the particle stability of the $I=0$ ${}^{~{}~{}4}_{\Lambda\Lambda}$H(1+) requires values of $|a_{\Lambda\Lambda}|\gtrsim 1.5$ fm, which are unlikely in our opinion. The $I=1$ excited state ${}^{~{}~{}4}_{\Lambda\Lambda}$H(0+), or its isospin analog state ${}^{~{}~{}4}_{\Lambda\Lambda}$n are far from being bound; if any of these were established experimentally, the soundness of the Nagara event would suffer a serious setback.

Extensions of the present LO work should consider explicit $\Lambda\Lambda$-$\Xi N$-$\Sigma\Sigma$ coupling in the ${}^{1}S_{0}$ channel or, at least, address momentum dependent $\Lambda\Lambda$ interaction components generated in NLO EFT through effective-range ($r_{\Lambda\Lambda}$) contributions. We note that no conclusive determination of $r_{\Lambda\Lambda}$ exists yet because of the scarce and inaccurate hyperon-hyperon (mostly $\Xi^{-}p$) scattering and reaction data available in the $\approx$25 MeV interval between the $\Lambda\Lambda$ and $\Xi N$ thresholds. For example, small values of $r_{\Lambda\Lambda}$ between 0.3 to 0.8 fm were derived from such data in the LO $\chi$EFT work of the Jülich-Bonn group [28] using values of $a_{\Lambda\Lambda}$ about $-$1.5 fm. In contrast, large values of $r_{\Lambda\Lambda}$ between 5 to 7 fm were derived from the same data in the NLO $\chi$EFT work of the Jülich-Bonn-Munich group [29] using values of $a_{\Lambda\Lambda}$ about $-$0.65 fm. This dichotomy is apparent also for the Nijmegen soft core potentials listed in Table I of Ref. [30] and would have to be considered in future studies of $\Lambda\Lambda$ hypernuclei.

Acknowledgments

The work of L.C. and N.B. was supported by the Pazy Foundation and by the Israel Science Foundation Grant No. 1308/16. The work of M.S. and J.M. was supported by the Czech Science Foundation GACR Grant No. 19-19640S.