aa r X i v : . [ nu c l - t h ] J u l Is there a bound n? Avraham Gal a, ∗ , Humberto Garcilazo b a Racah Institute of Physics, The Hebrew University, 91904 Jerusalem, Israel b Escuela Superior de F´ısica y Matem´aticasInstituto Polit´ecnico Nacional, Edificio 9, 07738 M´exico D.F., Mexico
Abstract
The HypHI Collaboration at GSI argued recently for a n (Λ nn ) bound statefrom the observation of its two-body t + π − weak-decay mode. We deriveconstraints from several hypernuclear systems, in particular from the A = 4hypernuclei with full consideration of Λ N ↔ Σ N coupling, to rule out abound n. Keywords:
Faddeev equations, hypernuclei
PACS:
1. Introduction
The lightest established Λ hypernucleus is known since the early days ofhypernuclear physics to be H T =0 , in which the Λ hyperon is weakly boundto the T = 0 deuteron core, with ground-state (g.s.) separation energy B Λ ( H)=0.13 ± J P =
12 + . There is no evidence fora bound spin-flip partner with J P =
32 + . For a brief review on related resultsdeduced from past emulsion studies of light hypernuclei, see Ref. [1].As for H T =1 , given the very weak binding of the Λ hyperon in the T = 0g.s., and that the T = 1 N N system is unbound, it is unlikely to be particlestable against decay to Λ + p + n . Similarly, assuming charge independence,Λ nn is not expected to be particle stable. As early as 1959 just six yearsfollowing the discovery of the first Λ hypernucleus, it was concluded by Downsand Dalitz upon performing variational calculations of both T = 0 , N N ∗ Corresponding author: Avraham Gal, [email protected]
Preprint submitted to Physics Letters B July 2, 2018 ystems that the isotriplet ( n, H T =1 , He) hypernuclei do not form boundstates [2]. This issue was revisited in Refs. [3, 4, 5] using various versions ofNijmegen hyperon-nucleon (
Y N ) potentials within Λ
N N
Faddeev equationsfor states with total orbital angular momentum L = 0 and all possible valuesof total angular momentum J and isospin T . Again, no Λ nn bound state wasfound in any of these studies as long as H T =0 ( J P =
12 + ) was only slightlybound. Similar conclusions were reached in Refs. [6, 7, 8] based on chiralconstituent quark model
Y N interactions, and in Ref. [9] based on recentlyconstructed NLO chiral EFT
Y N interactions [10]. Note that Λ N ↔ Σ N coupling was fully implemented in the more recent n studies [4, 5, 6, 7, 8, 9].A more general discussion of stability vs. instability for n in the context ofneutral hypernuclei with strangeness − − n has been made recently by the HypHICollaboration [12] observing a signal in the t + π − invariant mass distribu-tion following the bombardment of a fixed graphite target by Li projectilesat 2 A GeV in the GSI laboratory. The binding energy of the conjecturedweakly decaying n is 0.5 ± ± σ =5.4 ± N N bound-state calculations that n cannot be particle stable.However, possible connections to other hypernuclear systems, in particularthe A = 4 bound isodoublet hypernuclei ( H, He), need to be explored.The present work addresses this issue by establishing connections that makeit clear why a bound n cannot be accommodated into hypernuclear physics.Assuming charge-symmetric Λ N interactions, V Λ p = V Λ n , we demonstratesome unacceptable implications of a bound n to Λ p scattering in Sect. 2, andto H T =0 in Sect. 3. Consequences of A = 4 hypernuclear spectroscopy withfull consideration of charge-symmetric Λ N ↔ Σ N couplings are derived for n in Sect. 4 by applying methods that differ from those used in the combinedanalysis of A = 3 and A = 4 hypernuclei by Hiyama et al. [5], reaffirming that n is unbound. Our results are discussed and summarized in Sect. 5, withadditional remarks made on the possible role of charge-symmetry breaking(CSB) and Λ N N interaction three-body effects, concluding that a bound ninterpretation of the t + π − signal in the HypHI experiment is outside thescope of present-day hypernuclear physics.2 . n vs. Λ p scattering To make a straightforward connection between the low-energy Λ N scat-tering parameters and the three-body Λ N N system we follow the method ofRef. [3] in solving
Y N N
Faddeev equations with two-body
Y N input pair-wise separable interactions constructed directly from given low-energy
Y N scattering parameters. For simplicity we neglect in this section the spin de-pendence of the low-energy Λ N scattering parameters, setting a s = a t forthe scattering length and r s = r t with values r =2.5 or 3.5 fm for the effectiverange, spanning thereby a range of values commensurate with most theoret-ical models and also with the analysis of measured Λ p cross sections at lowenergies [13]. By using Yamaguchi form factors within rank-one separableinteractions, we then compute critical values of scattering length a requiredto bind successively the T = 0 and T = 1 Λ N N systems, with results shownin Table 1.
Table 1: Values of the spin-independent Λ N scattering length a required to bind T = 0 and T = 1 Λ N N states as indicated, for two representative values of thespin-independent effective range r , and calculated values of the Λ p total cross sec-tion at p Λ =145 MeV/c. The measured value at the lowest momentum bin avail-able is σ totΛ p ( p Λ =145 ±
25 MeV/c)=180 ±
22 mb [13]. Calculated values of B Λ ( H T =0 )are listed in the last column for Λ N interactions that just bind n, in contrast to B expΛ ( H)=0.13 ± B T =0Λ =0 B T =0Λ =0.13 MeV B T =1Λ =0 ( n just bound) r a σ totΛ p a σ totΛ p a σ totΛ p B T =0Λ (fm) (fm) (mb) (fm) (mb) (fm) (mb) (MeV)2.5 − − − − − − N scattering lengths are seen to be requiredto bind n, and the low-energy Λ p cross sections thereby implied exceedsubstantially the measured cross sections as shown by the Λ N cross sectionsevaluated at the lowest momentum bin reported in Ref. [13]. Of the three B Λ values tested in the table, only B T =0Λ =0.13 MeV is consistent with thereported Λ p cross sections, including their uncertainties. In the last columnof the table we also listed the Λ separation energies in H that result once n has just been brought to bind. These calculated values are much too bigto be reconciled with B expΛ ( H)=0.13 ± . n vs. H The n vs. H discussion in this section is limited to using s -wave Λ N effective interactions, providing a straightforward extension of earlier stud-ies [2, 3]. Effects of possibly substantial Λ N ↔ Σ N coupling, as generatedby strong one-pion exchange in Nijmegen meson-exchange potentials [14] andin recent chirally based potentials [10], are discussed in Sect. 4. Table 2: Λ separation energies B Λ ( H T =0 ) (in MeV) calculated for both J P =
12 + ,
32 + ,using Λ N separable interactions based on the low-energy parameters Eq. (1) with V t multiplied by a factor x up to values allowing n to become bound, as indicated byfollowing the values of its Fredholm determinant (FD) at E = 0. x n FD( E = 0) B Λ [ H T =0 (
12 + )] B Λ [ H T =0 (
32 + )]1.00 0.55 0.096 unbound1.10 0.47 0.147 0.1241.20 0.39 0.211 0.4481.30 0.31 0.288 0.9861.40 0.21 0.381 1.7041.50 0.12 0.488 2.5981.60 +0.015 0.612 3.6591.61 +0.004 0.625 3.7721.62 − n and H using simpleYamaguchi separable s -wave interactions fitted to prescribed input valuesof singlet and triplet scattering lengths a and effective ranges r , therebyrelaxing the spin-independence assumption of the preceding section. Of thefour Nijmegen interaction models A,B,C,D studied there, only C reproducesthe observed binding energy of H, binding also the
32 + spin-flip excitedstate just 11 keV above the
12 + g.s. To get rid of this excited state, we haveslightly changed the input parameters of model C. In this model, denotedC’, the input Λ N low-energy parameters are (in fm): a s = − . , r s = 3 . , a t = − . , r t = 3 . . (1)The H T =0 ( J P =
12 + ,
32 + ) separation energies obtained by solving the appro-priate Λ
N N
Faddeev equations are listed in Table 2. The row marked x = 14orresponds to using Λ N interaction based on the low-energy parametersEq. (1), and subsequent rows correspond to multiplying the Λ N triplet in-teraction V t by x > n ( H T =1 ).Inspection of Table 2 shows that while the Λ separation energies increaseupon varying x , a by-product of this increase is that H T =0 (
32 + ) quicklyovertakes H T =0 (
12 + ) becoming H g.s. This is understood by observing thatthe weights with which V t and the singlet interaction V s enter a simple foldingexpression for the Λ–core interaction are given by J P = 12 + : ( T + 12 ) V t + ( 32 − T ) V s , J P = 32 + : 2 V t , (2)so that V t is the only Λ N -interaction component affecting H T =0 (
32 + ) besidesbeing more effective in binding n than binding H T =0 (
12 + ). Subsequently,beginning with x = 1 . n becomes bound as indicated by the corre-sponding Fredholm determinant at E = 0 going through zero. Note that the(2 J +1)-averaged B T =0Λ ( H) is then ≈ n is therefore in strong disagreement with thebinding energy B expΛ ( H)= 0.13 ± H g . s . and withits spin-parity J P =
12 + . n vs. H Λ N ↔ Σ N coupling cannot be ignored in quantitative calculations of Λhypernuclear binding energies. One-pion exchange induces a strong couplingin the Y N S − D channel which dominates the effective V t contribution in 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 S channel, in contrast, Λ N ↔ Σ N coupling is weak. Here weemploy G -matrix 0 s N s Y effective interactions devised by Akaishi et al. [16]from the Nijmegen soft-core interaction model NSC97 and used in bindingenergy calculations of the A = 4 , ≈ +g . s . –1 +exc spin-doubletlevels in the isodoublet hypernuclei H– He which cannot be reconciled withtheory without substantial Λ N ↔ Σ N contribution. These 0 s N s Y effec-tive interactions were extended by Millener to the p shell and tested there5uccessfully in a comprehensive analysis of hypernuclear γ -ray measurements[17]. For a recent application to neutron-rich hypernuclei, see Ref. [18]. The0 s N s Y Λ N ↔ Σ N effective interaction V ΛΣ assumes a spin-dependent cen-tral interaction form V ΛΣ = ( ¯ V ΛΣ + ∆ ΛΣ ~s N · ~s Y ) p / ~t N · ~t ΛΣ , (3)where ~t ΛΣ converts a Λ to Σ in isospace, with matrix elements¯ V ΛΣ = 2 . .
35) MeV , ∆ ΛΣ = 5 . .
76) MeV (4)derived from the Nijmegen model version NSC97e (NSC97f) as given inRef. [18] (Ref. [19]). As for the diagonal 0 s N s Y interactions, we will con-strain the spin-dependent Λ N interaction ∆ ΛΛ matrix elements by fitting,together with ¯ V ΛΣ and ∆ ΛΣ , to the excitation spectrum of the A = 4 hyper-nuclei. Finally, the detailed properties of the Σ N interaction hardly matterin view of the large energy denominators of order M Σ − M Λ ≈
80 MeV withwhich they appear. The binding-energy contribution arising from V ΛΣ is thengiven to a good approximation schematically by |h V ΛΣ i| /
80 (in MeV).The nonvanishing matrix elements of the spin-independent term in Eq. (3)are given in closed form by h ( J N T, s Σ t Σ ) J T | V ΛΣ (∆ ΛΣ = 0) | ( J N T, s Λ t Λ ) J T i = p T ( T + 1) / V ΛΣ , (5)where s Σ = s Λ = , t Σ = 1, t Λ = 0. This term is diagonal in the nuclearcore, specified here by its total angular momentum J N and isospin T , withmatrix elements that resemble the Fermi matrix elements in β decay of thecore nucleus. Similarly, matrix elements of the spin-spin term in Eq. (3)involve the SU(4) generator P j ~s Nj ~t Nj for the core, connecting core stateswith large Gamow-Teller transition matrix elements. A complete listing ofthese Λ N ↔ Σ N Fermi and Gamow-Teller matrix elements together withcorresponding Λ N spin-spin matrix elements for the A = 3 , V ΛΣ are listed in the last two rows, including two-body as well as three-body terms.The last two columns of the table list matrix elements and binding-energycontributions for the A = 4 states, marked here by H. Fermi and Gamow-Teller contributions are added coherently because both ¯ V ΛΣ and ∆ ΛΣ connectto the same and only spin-isospin SU(4) 0 s N s Σ intermediate state available.6 able 3: Nonvanishing Λ N 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 threerows for H( T, J P ) and H( T, J P ) 0 s Λ states. Estimates of the total ΛΣ contributionsto binding energies, using the NSC97e parameter values (4), are given in MeV in the lasttwo rows. Note: ∆ ΛΛ is positive for binding-energy contributions. H(0,
12 + ) H(0,
32 + ) H(1,
12 + ) H( ,0 + ) H( ,1 + )Λ N ( × ∆ ΛΛ ) 1 − / − × ¯ V ΛΣ ) – – 2 p / × ∆ ΛΣ ) √ / − − ( | F | + | GT | ) 0.243 – 0.373 – – | (F + GT) | – – – 0.574 0.036The Λ N ↔ Σ N transition matrix elements are seen to provide about half ofthe observed 1.1 MeV 0 +g . s . –1 +exc splitting in the A = 4 hypernuclei, the restmust then be assigned to the Λ N spin-spin matrix element ∆ ΛΛ . For the A = 3 states, marked here by H, Fermi and Gamow-Teller contributionsare added incoherently owing to different intermediate states involved inthese transitions, with binding-energy contributions obtained upon assumingimplicitly same-size nucleon and hyperon wavefunctions as for A = 4. Since H(0,
12 + ) is weakly bound, the actual A = 3 contributions are expected to besomewhat suppressed, with matrix-element suppression factor η estimated tobe about η ≈ N spin-spin and ΛΣtransition binding-energy negative contributions to H(0,
32 + ) with respect to H(0,
12 + ) g.s., it is safe to conclude that H(0,
32 + ) is unbound.Focusing on discussion of H(1,
12 + ), particularly relative to H(0,
12 + ) g.s.,we first go to the SU(4) limit of nuclear-core dynamics in which the dineutronbecomes bound and degenerate with the deuteron, and where the differencein Λ separation energies of H(1,
12 + ) and H(0,
12 + ) according to Table 3 isgiven (in MeV) by δB Λ ≡ B T =1Λ ( 12 + ) − B T =0Λ ( 12 + ) = η (0 . − . − η ∆ ΛΛ . (6)To maximize this energy difference we neglect the Λ N spin-spin contribution,thereby letting ∆ ΛΛ →
0, and compensate by doubling the ΛΣ contributionin order to keep E (1 + ) − E (0 + ) ≈ . H intact. For η = 1, expected7o be a fair approximation in this SU(4) limit, we obtain δB maxΛ =0.26 MeV,and so by charge independence the Λ separation energy in this hypotheti-cally bound n with respect to the bound dineutron core is 0.39 ± nn Faddeev equations we fit a Λ N spin-independent Yamaguchi separableinteraction that reproduces B Λ ( n)=0.39 MeV, with B ( n)=2.23 MeV as inthe deuteron. For a chosen value of 2.5 fm for the Λ N effective range, thisrequires a Λ N scattering length of − nn interaction we usedYamaguchi separable potential determined by the N N T = 0 low-energy pa-rameters a s =5.4 fm, r s =1.75 fm, resulting in B ( n)=2.23 MeV which equalsthe deuteron binding energy in this SU(4) limit. We then perform a seriesof Λ nn Faddeev calculations keeping the Λ N interaction as is, but breakingSU(4) progressively by varying the nn interaction to reach a s = − r s =2.88 fm as appropriate in the real world to the unbound dineutron. Thisis documented in Table 4. Table 4: Binding energy B ( n) (in MeV) of two neutrons in a separable Yamaguchi po-tential specified by scattering length a s and effective range r s (both in fm) in the S channel, and Λ separation energy B Λ ( n) (in MeV) obtained by solving Λ nn Faddeevequations with a separable Yamaguchi Λ N spin-independent interaction specified by scat-tering length a = − .
804 fm and effective range r = 2 . B ( n) approx values areobtained using Eq. (7). a s r s B ( n) B ( n) approx B Λ ( n)5.4 1.75 2.23 2.24 0.395.4 2.25 2.79 2.87 0.275.4 2.881 4.98 – 0.166.0 2.881 2.86 3.20 0.117.0 2.881 1.64 1.68 0.069.0 2.881 0.80 0.80 0.0113.0 2.881 0.32 0.32 0.00317.612 2.881 0.16 0.16 – − B ( n) and the n binding energy B ( n)= B ( n)+ B Λ ( n) upon varying the N N low-energy scattering parameters from values given by the T = 0 pn T = 1 nn interaction. Thisis done in two stages. First, increasing the effective range while keeping thescattering length fixed, B ( n) increases whereas B Λ ( n) steadily decreases. In the second stage, while keeping the effective range fixed at its final empiri-cal nn value, the scattering length is varied by increasing it and then crossingfrom a large positive value associated with a loosely bound dineutron to theempirical large negative value of a nn associated with a virtual dineutron.During this stage, B ( n) too decreases steadily until n is no longer bound.With B Λ ( n) ≪ B ( n) holding over the full range of variation exhibitedin Table 4, it is clear that the behavior of B ( n) follows closely that of B ( n). For fairly small values of B ( n), say B ( n) . B ( n) is quiteaccurately reproduced by the effective-range expansion approximation B ( n) approx = ~ M n r s (cid:18) − r − r s a s (cid:19) , (7)as shown by comparing the exact and approximate values of B ( n) listed inthe table.It is worth noting in Table 4 that the dissociation of n occurs while thedineutron is still bound, although quite weakly. The final result of no nbound state, for a virtual dineutron and Λ N low-energy scattering parame-ters listed in the caption to Table 4, should come at no surprise given that aconsiderably larger-size Λ N scattering length was found to be required in theFaddeev calculations listed in Table 1 to bind n. Although a specific valueof 2.5 fm for the Λ N effective range was used in our actual demonstration,similar results are obtained for other reasonable choices of the Λ N effectiverange.
5. Discussion and conclusion
We have shown in this work that the Λ N interactions required to bind n are inconsistent with the measured Λ p scattering cross sections at lowenergies, with H g . s . binding energy, and with the 0 +g . s . –1 +exc excitation en-ergy of the A = 4 Λ hypernuclei. Although simple Λ N interactions wereused to simulate the more realistic NSC97 interactions, the consequences of A decrease of B Λ upon increasing one of the effective ranges in a few-body calculationwas noted and discussed by Gibson and Lehman [20]. n for Λ hypernuclear data are sufficiently strong thatthe use of more refined interactions is unlikely to modify any of the conclu-sions reached here. Of the three hypernuclear systems related here to n,we attach special significance to the A = 4 Λ hypernuclei where only the1.1 MeV 0 +g . s . –1 +exc excitation energy is involved in our model building. Thisexcitation energy is intimately connected to Λ N ↔ Σ N coupling effects inthe A = 4 hypernuclei [16] which have been further incorporated and testedsuccessfully to reproduce excitation spectra in p -shell hypernuclei [17]. Wejudiciously 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 satisfactorilyin few-body calculations that use theoretically derived Y N potentials, asstressed recently by Nogga [15]. This difficulty might be associated withmissing three-body Λ
N N interaction terms, other than those incorporatedhere by including Λ N ↔ Σ N coupling.Of the Λ N N interactions considered in past hypernuclear calculations,those arising from an intermediate Σ(1385) hyperon resonance [21] are inde-pendent of the spin of the Λ and thus would not affect the 0 +g . s . –1 +exc spin-flipexcitation upon which our considerations rest. The spin-isospin dependenceof the central component of this interaction is given by − ( ~τ · ~τ ~σ · ~σ ) whichassumes the same value +3 for both J P =
12 + states in the A = 3 hypernuclei.A dispersive Λ N N repulsive contribution with Λ spin dependence given by(1+ ~σ Λ · ~S ), where ~S = ( ~σ + ~σ ), was considered in VMC calculationsof light hypernuclei [22]. This gives 1( ) for the T = 1(0) , J P =
12 + A = 3states, namely more repulsion for n than for H g . s . . Another form of disper-sive Λ N N contribution suggested in Ref. [23] depends on spin and isospinthrough the factor − ~τ · ~τ ( ~σ · ~σ + ~σ Λ · ~S ) which assumes values +3( − T = 1(0), J P =
12 + states, repulsive for n while attractive for H g . s . .The latter two dispersive Λ N N interaction forms were found in Ref. [24] ca-pable 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 n relative to H g . s . . This brief survey of three-body Λ N N phenomenologyoffers, therefore, no plausible solution of the n puzzle.A comment on CSB effects in light Λ hypernuclei and whether or notCSB might resolve the n puzzle is in order before concluding the presentstudy. For the known T = isodoublet of A = 4 hypernuclear 0 +g . s . levels∆ B expΛ ( A = 4) ≡ B Λ ( He) − B Λ ( H)=0.35 ± Y N interaction models, see Table 9 in10ef. [15] where the recently constructed NLO chiral
Y N interactions [10] areshown to yield only ∆ B calcΛ ( A = 4) ≈
50 keV. This ∆ B Λ ( A = 4) arises largelyfrom kinetic energies depending on which charged Σ hyperon mass is used.The same CSB effect will result in smaller B Λ ( n) values relative to thosecalculated, as done here, using a charge symmetric calculation. Therefore,CSB contributions are also unlikely to resolve the n puzzle.How does one then explain the HypHI t + π − signal which is naturallyassigned to the two-body weak decay n → t + π − ? This problem is aggravatedby a similar one addressing a d + π − signal, also observed in the HypHIexperiment, the most straightforward assignment of which would be due tothe two-body weak decay of a bound Λ n system: n → d + π − . No plausiblesolution has been offered to these puzzles and more work on other possibleorigins of d + π − and t + π − signals is called for. Acknowledgements
We thank Nir Barnea, Daniel Gazda, Emiko Hiyama, Jiˇr´ı Mareˇs andJean-Marc Richard for useful discussions. A.G. acknowledges support bythe EU initiative FP7, HadronPhysics3, under the SPHERE and LEANNIScooperation programs. H.G. is supported in part by COFAA-IPN (M´exico).
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