Is there a bound Lambda-n-n ?

The HypHI Collaboration at GSI argued recently for a (Lambda-n-n) bound state from the observation of its two-body (t + pi-) weak-decay mode. We derive constraints from several hypernuclear systems, in particular from the A=4 hypernuclei with full consideration of (Lambda N<-->Sigma N) coupling, to rule out a bound (Lambda-n-n).


Introduction
The lightest established Λ hypernucleus is known since the early days of hypernuclear physics to be 3 Λ H T =0 , in which the Λ hyperon is weakly bound to the T = 0 deuteron core, with ground-state (g.s.) separation energy B Λ ( 3 Λ H)=0.13±0.05 MeV and spin-parity J P = 1 2 + . There is no evidence for a bound spin-flip partner with J P = 3 2 + . For a brief review on related results deduced from past emulsion studies of light hypernuclei, see Ref. [1].
As for 3 Λ H T =1 , given the very weak binding of the Λ hyperon in the T = 0 g.s., and that the T = 1 NN system is unbound, it is unlikely to be particle stable against decay to Λ + p + n. Similarly, assuming charge independence, Λnn is not expected to be particle stable. As early as 1959 just six years following the discovery of the first Λ hypernucleus, it was concluded by Downs and Dalitz upon performing variational calculations of both T = 0, 1 ΛNN systems that the isotriplet ( 3 Λ n, 3 Λ H T =1 , 3 Λ He) hypernuclei do not form bound states [2]. This issue was revisited in Refs. [3,4,5] using various versions of Nijmegen hyperon-nucleon (Y N) potentials within ΛNN Faddeev equations for states with total orbital angular momentum L = 0 and all possible values of total angular momentum J and isospin T . Again, no Λnn bound state was found in any of these studies as long as 3 Λ H T =0 (J P = 1 2 + ) was only slightly bound. Similar conclusions were reached in Refs. [6,7,8] based on chiral constituent quark model Y N interactions, and in Ref. [9] based on recently constructed NLO chiral EFT Y N interactions [10]. Note that ΛN ↔ ΣN coupling was fully implemented in the more recent 3 Λ n studies [4,5,6,7,8,9]. A more general discussion of stability vs. instability for 3 Λ n in the context of neutral hypernuclei with strangeness −1 and −2 has been given very recently in Ref. [11].
A claim for particle stability of 3 Λ n has been made recently by the HypHI Collaboration [12] observing a signal in the t + π − invariant mass distribution following the bombardment of a fixed graphite target by 6 Li projectiles at 2A GeV in the GSI laboratory. The binding energy of the conjectured weakly decaying 3 Λ n is 0.5±1.1±2.2 MeV, with a large standard deviation σ=5.4±1.4 MeV. As noted above, there is unanimous theoretical consensus based on ΛNN bound-state calculations that 3 Λ n cannot be particle stable. However, possible connections to other hypernuclear systems, in particular the A = 4 bound isodoublet hypernuclei ( 4 Λ H, 4 Λ He), need to be explored. The present work addresses this issue by establishing connections that make it clear why a bound 3 Λ n cannot be accommodated into hypernuclear physics. Assuming charge-symmetric ΛN interactions, V Λp = V Λn , we demonstrate some unacceptable implications of a bound 3 Λ n to Λp scattering in Sect. 2, and to 3 Λ H T =0 in Sect. 3. Consequences of A = 4 hypernuclear spectroscopy with full consideration of charge-symmetric ΛN ↔ ΣN couplings are derived for 3 Λ n in Sect. 4 by applying methods that differ from those used in the combined analysis of A = 3 and A = 4 hypernuclei by Hiyama et al. [5], reaffirming that 3 Λ n is unbound. Our results are discussed and summarized in Sect. 5, with additional remarks made on the possible role of charge-symmetry breaking (CSB) and ΛNN interaction three-body effects, concluding that a bound 3 Λ n interpretation of the t + π − signal in the HypHI experiment is outside the scope of present-day hypernuclear physics.

3 Λ n vs. Λp scattering
To make a straightforward connection between the low-energy ΛN scattering parameters and the three-body ΛNN system we follow the method of Ref. [3] in solving Y NN Faddeev equations with two-body Y N input pairwise separable interactions constructed directly from given low-energy Y N scattering parameters. For simplicity we neglect in this section the spin dependence of the low-energy ΛN scattering parameters, setting a s = a t for the scattering length and r s = r t with values r=2.5 or 3.5 fm for the effective range, spanning thereby a range of values commensurate with most theoretical models and also with the analysis of measured Λp cross sections at low energies [13]. By using Yamaguchi form factors within rank-one separable interactions, we then compute critical values of scattering length a required to bind successively the T = 0 and T = 1 ΛNN systems, with results shown in Table 1. Table 1: Values of the spin-independent ΛN scattering length a required to bind T = 0 and T = 1 ΛN N states as indicated, for two representative values of the spin-independent effective range r, and calculated values of the Λp total cross section at p Λ =145 MeV/c. The measured value at the lowest momentum bin available is σ tot Λp (p Λ =145±25 MeV/c)=180±22 mb [13]. Calculated values of B Λ ( 3 Λ H T =0 ) are listed in the last column for ΛN interactions that just bind 3 Λ n, in contrast to Exceptionally large values of ΛN scattering lengths are seen to be required to bind 3 Λ n, and the low-energy Λp cross sections thereby implied exceed substantially the measured cross sections as shown by the ΛN cross sections evaluated at the lowest momentum bin reported in Ref. [13].
Λ n vs. 3 Λ H discussion in this section is limited to using s-wave ΛN effective interactions, providing a straightforward extension of earlier studies [2,3]. Effects of possibly substantial ΛN ↔ ΣN coupling, as generated by strong one-pion exchange in Nijmegen meson-exchange potentials [14] and in recent chirally based potentials [10], are discussed in Sect. 4.
using ΛN separable interactions based on the low-energy parameters Eq. (1) with V t multiplied by a factor x up to values allowing 3 Λ n to become bound, as indicated by following the values of its Fredholm determinant (FD) at E = 0. Following Ref. [3] we solve Faddeev equations for 3 Λ n and 3 Λ H using simple Yamaguchi separable s-wave interactions fitted to prescribed input values of singlet and triplet scattering lengths a and effective ranges r, thereby relaxing the spin-independence assumption of the preceding section. Of the four Nijmegen interaction models A,B,C,D studied there, only C reproduces the observed binding energy of 3 Λ H, binding also the 3 2 + spin-flip excited state just 11 keV above the 1 2 + g.s. To get rid of this excited state, we have slightly changed the input parameters of model C. In this model, denoted C', the input ΛN low-energy parameters are (in fm): The 3 Λ H T =0 (J P = 1 2 + , 3 2 + ) separation energies obtained by solving the appropriate ΛNN Faddeev equations are listed in Table 2. The row marked x = 1 corresponds to using ΛN interaction based on the low-energy parameters Eq. (1), and subsequent rows correspond to multiplying the ΛN triplet in- . Inspection of Table 2 shows that while the Λ separation energies increase upon varying x, a by-product of this increase is that 3 s. This is understood by observing that the weights with which V t and the singlet interaction V s enter a simple folding expression for the Λ-core interaction are given by being more effective in binding 3 Λ n than binding 3 Λ H T =0 ( 1 2 + ). Subsequently, beginning with x = 1.614, 3 Λ n becomes bound as indicated by the corresponding Fredholm determinant at E = 0 going through zero. Note that the MeV, in rough agreement with the spin-independent analysis of the previous section (cf. first row in Table 1). Similar results are obtained when replacing the parameters (1) of model C' by those of model C, used in Ref. [3], and repeating the procedure summarized in Table 2. A bound 3 Λ n is therefore in strong disagreement with the binding energy B exp Λ ( 3 Λ H)= 0.13±0.05 MeV determined for 3 Λ H g.s. and with its spin-parity J P = 1 2 + .

3
Λ n vs. 4 Λ H ΛN ↔ ΣN coupling cannot be ignored in quantitative calculations of Λ hypernuclear binding energies. One-pion exchange induces a strong coupling in the Y N 3 S 1 − 3 D 1 channel which dominates the effective V t contribution in 3 Λ H three-body calculations, independently of whether using NSC97-related Y N interactions as in Refs. [4,5] or NLO chiral Y N interactions in Ref. [15]. In the Y N 1 S 0 channel, in contrast, ΛN ↔ ΣN coupling is weak. Here we employ G-matrix 0s N 0s Y effective interactions devised by Akaishi et al. [16] from the Nijmegen soft-core interaction model NSC97 and used in binding energy calculations of the A = 4, 5 Λ hypernuclei. Of particular significance in the present context is the ≈1.1 MeV splitting of the 0 + g.s. -1 + exc spin-doublet levels in the isodoublet hypernuclei 4 Λ H-4 Λ He which cannot be reconciled with theory without substantial ΛN ↔ ΣN contribution. These 0s N 0s Y effective interactions were extended by Millener to the p shell and tested there successfully in a comprehensive analysis of hypernuclear γ-ray measurements [17]. For a recent application to neutron-rich hypernuclei, see Ref. [18]. The 0s N 0s Y ΛN ↔ ΣN effective interaction V ΛΣ assumes a spin-dependent central interaction form where t ΛΣ converts a Λ to Σ in isospace, with matrix elements derived from the Nijmegen model version NSC97e (NSC97f) as given in Ref. [18] (Ref. [19]). As for the diagonal 0s N 0s Y interactions, we will constrain the spin-dependent ΛN interaction ∆ ΛΛ matrix elements by fitting, together withV ΛΣ and ∆ where s Σ = s Λ = 1 2 , t Σ = 1, t Λ = 0. This term is diagonal in the nuclear core, specified here by its total angular momentum J N and isospin T , with matrix elements that resemble the Fermi matrix elements in β decay of the core nucleus. Similarly, matrix elements of the spin-spin term in Eq. (3) involve the SU(4) generator j s N j t N j for the core, connecting core states with large Gamow-Teller transition matrix elements. A complete listing of these ΛN ↔ ΣN Fermi and Gamow-Teller matrix elements together with corresponding ΛN spin-spin matrix elements for the A = 3, 4 Λ hypernuclei is given in the first three rows of Table 3, and the resulting binding-energy contributions arising from V ΛΣ are listed in the last two rows, including twobody as well as three-body terms.
The last two columns of the table list matrix elements and binding-energy contributions for the A = 4 states, marked here by 4 Λ H. Fermi and Gamow-Teller contributions are added coherently because bothV ΛΣ and ∆ ΛΣ connect to the same and only spin-isospin SU(4) 0s N 0s Σ intermediate state available. Table 3: Nonvanishing ΛN spin-spin matrix elements as well as Fermi (F) and Gamow-Teller (GT) nonvanishing matrix elements of V ΛΣ , Eq. (3), are listed in the first three rows for 3 Λ H(T, J P ) and 4 Λ H(T, J P ) 0s Λ states. Estimates of the total ΛΣ contributions to binding energies, using the NSC97e parameter values (4), are given in MeV in the last two rows. Note: ∆ ΛΛ is positive for binding-energy contributions.
The ΛN ↔ ΣN transition matrix elements are seen to provide about half of the observed 1.1 MeV 0 + g.s. -1 + exc splitting in the A = 4 hypernuclei, the rest must then be assigned to the ΛN spin-spin matrix element ∆ ΛΛ . For the A = 3 states, marked here by 3 Λ H, Fermi and Gamow-Teller contributions are added incoherently owing to different intermediate states involved in these transitions, with binding-energy contributions obtained upon assuming implicitly same-size nucleon and hyperon wavefunctions as for A = 4. Since  Table 3 is given (in MeV) by To maximize this energy difference we neglect the ΛN spin-spin contribution, thereby letting ∆ ΛΛ → 0, and compensate by doubling the ΛΣ contribution in order to keep E(1 + ) − E(0 + ) ≈ 1.1 MeV in 4 Λ H intact. For η = 1, expected to be a fair approximation in this SU(4) limit, we obtain δB max Λ =0.26 MeV, and so by charge independence the Λ separation energy in this hypothetically bound 3 Λ n with respect to the bound dineutron core is 0.39±0.05 MeV. Precisely the same result is obtained if Nijmegen model NSC97f ΛΣ matrix elements from (4), in parentheses there, are used instead. Next, by solving Λnn Faddeev equations we fit a ΛN spin-independent Yamaguchi separable interaction that reproduces B Λ ( 3 Λ n)=0.39 MeV, with B( 2 n)=2.23 MeV as in the deuteron. For a chosen value of 2.5 fm for the ΛN effective range, this requires a ΛN scattering length of −1.804 fm. For nn interaction we used Yamaguchi separable potential determined by the NN T = 0 low-energy parameters a s =5.4 fm, r s =1.75 fm, resulting in B( 2 n)=2.23 MeV which equals the deuteron binding energy in this SU(4) limit. We then perform a series of Λnn Faddeev calculations keeping the ΛN interaction as is, but breaking SU(4) progressively by varying the nn interaction to reach a s =−17.6 fm and r s =2.88 fm as appropriate in the real world to the unbound dineutron. This is documented in Table 4. Table 4: Binding energy B( 2 n) (in MeV) of two neutrons in a separable Yamaguchi potential specified by scattering length a s and effective range r s (both in fm) in the 1 S 0 channel, and Λ separation energy B Λ ( 3 Λ n) (in MeV) obtained by solving Λnn Faddeev equations with a separable Yamaguchi ΛN spin-independent interaction specified by scattering length a = −1.804 fm and effective range r = 2.5 fm. The B( 2 n) approx values are obtained using Eq. (7). The table demonstrates the behavior of the dineutron binding energy B( 2 n) and the 3 Λ n binding energy B( 3 Λ n)=B( 2 n)+B Λ ( 3 Λ n) upon varying the NN low-energy scattering parameters from values given by the T = 0 pn interaction down to the empirical values for the T = 1 nn interaction. This is done in two stages. First, increasing the effective range while keeping the scattering length fixed, B( 2 n) increases whereas B Λ ( 3 Λ n) steadily decreases. 1 In the second stage, while keeping the effective range fixed at its final empirical nn value, the scattering length is varied by increasing it and then crossing from a large positive value associated with a loosely bound dineutron to the empirical large negative value of a nn associated with a virtual dineutron. During this stage, B( 2 n) too decreases steadily until 3 Λ n is no longer bound. With B Λ ( 3 Λ n)≪ B( 2 n) holding over the full range of variation exhibited in Table 4, it is clear that the behavior of B( 3 Λ n) follows closely that of B( 2 n). For fairly small values of B( 2 n), say B( 2 n) 3 MeV, B( 2 n) is quite accurately reproduced by the effective-range expansion approximation as shown by comparing the exact and approximate values of B( 2 n) listed in the table.
It is worth noting in Table 4 that the dissociation of 3 Λ n occurs while the dineutron is still bound, although quite weakly. The final result of no 3 Λ n bound state, for a virtual dineutron and ΛN low-energy scattering parameters listed in the caption to Table 4, should come at no surprise given that a considerably larger-size ΛN scattering length was found to be required in the Faddeev calculations listed in Table 1 to bind 3 Λ n. Although a specific value of 2.5 fm for the ΛN effective range was used in our actual demonstration, similar results are obtained for other reasonable choices of the ΛN effective range.

Discussion and conclusion
We have shown in this work that the ΛN interactions required to bind 3 Λ n are inconsistent with the measured Λp scattering cross sections at low energies, with 3 Λ H g.s. binding energy, and with the 0 + g.s. -1 + exc excitation energy of the A = 4 Λ hypernuclei. Although simple ΛN interactions were used to simulate the more realistic NSC97 interactions, the consequences of accepting a bound 3 Λ n for Λ hypernuclear data are sufficiently strong that the use of more refined interactions is unlikely to modify any of the conclusions reached here. Of the three hypernuclear systems related here to 3 Λ n, we attach special significance to the A = 4 Λ hypernuclei where only the 1.1 MeV 0 + g.s. -1 + exc excitation energy is involved in our model building. This excitation energy is intimately connected to ΛN ↔ ΣN coupling effects in the A = 4 hypernuclei [16] which have been further incorporated and tested successfully to reproduce excitation spectra in p-shell hypernuclei [17]. We judiciously avoided relying on the absolute binding energy of the 0 + g.s. of the A = 4 Λ hypernuclei because it has not been yet reproduced satisfactorily in few-body calculations that use theoretically derived Y N potentials, as stressed recently by Nogga [15]. This difficulty might be associated with missing three-body ΛNN interaction terms, other than those incorporated here by including ΛN ↔ ΣN coupling.
Of the ΛNN interactions considered in past hypernuclear calculations, those arising from an intermediate Σ(1385) hyperon resonance [21] are independent of the spin of the Λ and thus would not affect the 0 + g.s. -1 + exc spin-flip excitation upon which our considerations rest. The spin-isospin dependence of the central component of this interaction is given by −( τ 1 · τ 2 σ 1 · σ 2 ) which assumes the same value +3 for both J P = 1 2 + states in the A = 3 hypernuclei. A dispersive ΛNN repulsive contribution with Λ spin dependence given by (1+ 1 3 σ Λ · S 12 ), where S 12 = 1 2 ( σ 1 + σ 2 ), was considered in VMC calculations of light hypernuclei [22]. This gives 1( 1 3 ) for the T = 1(0), J P = 1 2 + A = 3 states, namely more repulsion for 3 Λ n than for 3 Λ H g.s. . Another form of dispersive ΛNN contribution suggested in Ref. [23] depends on spin and isospin through the factor − τ 1 · τ 2 ( σ 1 · σ 2 + σ Λ · S 12 ) which assumes values +3(−3) for the T = 1(0), J P = 1 2 + states, repulsive for 3 Λ n while attractive for 3 Λ H g.s. . The latter two dispersive ΛNN interaction forms were found in Ref. [24] capable of accounting for a substantial fraction of the 0 + g.s. -1 + exc excitation in the A = 4 hypernuclei, but obviously neither of them would add attraction to 3 Λ n relative to 3 Λ H g.s. . This brief survey of three-body ΛNN phenomenology offers, therefore, no plausible solution of the 3 Λ n puzzle. A comment on CSB effects in light Λ hypernuclei and whether or not CSB might resolve the 3 Λ n puzzle is in order before concluding the present study. For the known T = 1 2 isodoublet of A = 4 hypernuclear 0 + g.s. levels ∆B exp Λ (A = 4)≡B Λ ( 4 Λ He)−B Λ ( 4 Λ H)=0.35±0.04 MeV [1] is exceptionally large and defies explanation in modern Y N interaction models, see Table 9 in Ref. [15] where the recently constructed NLO chiral Y N interactions [10] are shown to yield only ∆B calc Λ (A = 4) ≈ 50 keV. This ∆B Λ (A = 4) arises largely from kinetic energies depending on which charged Σ hyperon mass is used. The same CSB effect will result in smaller B Λ ( 3 Λ n) values relative to those calculated, as done here, using a charge symmetric calculation. Therefore, CSB contributions are also unlikely to resolve the 3 Λ n puzzle. How does one then explain the HypHI t + π − signal which is naturally assigned to the two-body weak decay 3 Λ n→ t+π − ? This problem is aggravated by a similar one addressing a d + π − signal, also observed in the HypHI experiment, the most straightforward assignment of which would be due to the two-body weak decay of a bound Λn system: 2 Λ n→ d + π − . No plausible solution has been offered to these puzzles and more work on other possible origins of d + π − and t + π − signals is called for.