Towards resolving the $_\Lambda^3$H lifetime puzzle

Recent $_\Lambda^3$H lifetime measurements in relativistic heavy ion collision experiments have yielded values shorter by (30$\pm$8)% than the free $\Lambda$ lifetime $\tau_\Lambda$, thereby questioning the naive expectation that $\tau({_\Lambda^3}{\rm H})\approx\tau_\Lambda$ for a weakly bound $\Lambda$ hyperon. Here we apply the closure approximation introduced by Dalitz and coworkers to evaluate the $_\Lambda^3$H lifetime, using $_\Lambda^3$H wavefunctions generated by solving three-body Faddeev equations. Our result, disregarding pion final-state interaction (FSI), is $\tau({_\Lambda^3}{\rm H})$=(0.90$\pm$0.01)$\tau_\Lambda$. In contrast to previous works, pion FSI is found attractive, reducing further $\tau({_\Lambda^3}{\rm H})$ to $\tau({_\Lambda^3}{\rm H})$=(0.81$\pm$0.02)$\tau_\Lambda$. We also evaluate for the first time $\tau({_\Lambda^3}{\rm n})$, finding it considerably longer than $\tau_\Lambda$, contrary to the shorter lifetime values suggested by the GSI HypHI experiment for this controversial hypernucleus.


Introduction
3 Λ H, a pnΛ state with spin-parity J P = 1 2 + and isospin I = 0 in which the Λ hyperon is bound to a deuteron core by merely B Λ ( 3 Λ H)=0.13±0.05 MeV, presents in the absence of two-body ΛN bound states the lightest bound and one of the most fundamental Λ hypernuclear systems [1]. Its spin-parity 1 2 + assignment follows from the measured branching ratio of the two-body decay 3 Λ H → 3 He + π − induced by the free-Λ weak decay Λ → p + π − [2]. There is no experimental indication, nor theoretical compelling reason, for a bound J P = 3 2 + spin-flip excited state, and there is even less of a good reason to assume that an excited I = 1 state lies below the pnΛ threshold.  [3], HypHI [4], ALICE [5], STAR [6] and ALICE [7] in chronological order. Shown by horizontal lines are the free-Λ lifetime (solid), the world average of measured 3 Λ H lifetimes (dashed), and results of three calculations (dot-dashed), see text. We thank Benjamin Dönigus for providing this figure.
Given the loose binding of the Λ hyperon in 3 Λ H, it is of considerable theoretical interest to study the extent to which nuclear medium modification mechanisms impact the free Λ lifetime in such a diffuse environment. An updated compilation of measured lifetime values, τ exp ( 3 Λ H), is presented in Fig. 1. Shown also is the world-average value which is shorter by about 30% than the free-Λ lifetime τ Λ = 263 ± 2 ps. In sharp contrast with the large scatter of old bubble chamber and nuclear emulsion measurements from the 1960s and 1970s, the five recent measurements of τ ( 3 Λ H) in relativistic heavy ion experiments marked in the figure give values persistently shorter by (30±8)% than τ Λ . Also shown in the figure are τ calc ( 3 Λ H) values from three calculations that pass our judgement, two of which [8,9] claim 3 Λ H lifetimes that are by merely few percent shorter than τ Λ whereas the third one [10], approximating a genuine three-body 3 Λ H wavefunction by a Λd cluster wavefunction, obtained a lifetime shorter by as much as 12%. A lesson gained from the latter detailed calculation is that as long as the binding energy of 3 Λ H is reproduced, the lifetime calculation is rather insensitive to the fine de-tails of the particular ΛN interaction model chosen. The main uncertainty in these lifetime calculations arises in fact from the imprecisely known value of B Λ ( 3 Λ H). Few comments on the calculations in Refs. [8,9] are in order: (i) Rayet [11]. However, the RD decay rate expressions miss a recoil phase-space factor which, if not omitted inadvertently in print, would bring down their calculated 3 Λ H lifetime to 85% of τ Λ . As for other calculations that claimed 3 Λ H lifetimes much shorter than τ Λ , we were unable to reproduce the conceptually similar calculation of Ref. [12], nor to make sense out of a 3 Λ H decay rate calculation based on a nonmesonic ΛN → NN weak interaction hamiltonian [13]. (ii) Kamada et al. [9] used a 3 Λ H wavefunction obtained by solving threebody Faddeev equations with NN and Y N Nijmegen soft-core potentials to evaluate all three π − decay channels: 3 He + π − , d+p+π − and p+p+n+π − . The π 0 decay channels were related to the corresponding π − channels in a ratio 1:2 following the ∆I = 1 2 rule. This calculation resulted in 3 Λ H lifetime shorter by 1% than τ Λ , which upon adding an estimated 1.7% nonmesonic decay branch [14] gave a 3 Λ H lifetime 97% of τ Λ . In the present work we evaluate the closure-approximation 3 Λ H exchange matrix element using 3 Λ H wavefunctions obtained by solving three-body Faddeev equations, thereby combining elements of the two calculations reviewed briefly above [8,9] in order to assess critically the so called 'hypertriton lifetime puzzle'. Furthermore, we study semi-quantitatively final-state interaction (FSI) effects arising from the outgoing low energy pion. Such effects have not been treated satisfactorily to the best of our knowledge and have a potential of resolving the 3 Λ H lifetime puzzle. Finally, as a by-product of our considerations we evaluate for the first time the lifetime of 3 Λ n assuming that it is bound. The particle stability of 3 Λ n was conjectured by the GSI HypHI Collaboration having observed a 3 H+π − decay track [15]. However, it is unanimously opposed in recent theoretical works [16,17,18]. 3 Λ H and 3 Λ n The Λ weak decay rate considered here, Γ Λ ≈ Γ π − Λ + Γ π 0 Λ , accounts for the nonleptonic, mesonic decay channels pπ − (63.9%) and nπ 0 (35.8%), where each of these partial rates consists of parity-violating s-wave (88.3%) and parity-conserving p-wave (11.7%) terms, summing up to

Total decay rate of
Here, ω π (q) and E N (q) are center of mass (cm) energies of the decay pion and the recoil nucleon, respectively, and q → q Λ ≈ 102 MeV/c in the free-space Λ → Nπ weak decay. The 2:1 approximate ratio of π − :π 0 decay rates, the so called ∆I = 1 2 rule in nonleptonic weak decays, implies that the final πN system is approximately in a well-defined I = 1 2 isospin state.

3 Λ H For 3
Λ H ground state (g.s.) weak decay, approximating the outgoing pion momentum by a mean valueq and using closure in the evaluation of the summed mesonic decay rate, one obtains In this equation we have omitted terms of order 0.5% of Γ(q) that correct for the use ofq in the two-body 3 Λ H→ π + 3 Z rate expressions [10]. We note that applying the ∆I = 1 2 rule to the isospin I = 0 decaying 3 Λ H g.s. , here too as in the free Λ decay, the ratio of π − :π 0 decay rates is approximately 2:1. The quantity η(q) in Eq. (2) is an exchange integral ensuring that the summation on final nuclear states is limited to totally antisymmetric states: Here χ( r Λ ; r N 2 , r N 3 ) is the real normalized spatial wavefunction of 3 Λ H, symmetric in the nucleon coordinates 2 and 3. This wavefunction, in abbreviated notation χ(1; 2, 3), is associated with a single spin-isospin term which is antisymmetric in the nucleon labels, such that s Λ = 1 2 couples to s 1 + s 2 = 1 to give S tot = 1 2 for the ground state and S tot = 3 2 for the spin-flip excited state (if bound), and t Λ = 0 couples trivially with t 1 + t 2 = 0. Eq. (2) already accounts for this spin-isospin algebra in 3 Λ H. For completeness we also list the total decay rate expression for 3 Λ H if its g.s. spin-parity were J P = 3 2 + : Since 0 < η(q) < 1, the dominant s-wave term here is weaker than in the free Λ decay, Eq. (1), implying that the 3 Λ H lifetime would have been longer than the free Λ lifetime, had its g.s. spin-parity been 3 2 + .

3
Λ n For 3 Λ n(I = 1, J P = 1 2 + ) weak decay, it is necessary to distinguish between decays induced by Λ → p + π − and those induced by Λ → n + π 0 . In the first case the spectator neutrons are 'frozen' to their s shell in both initial and final state, without having to recouple spins or consider exchange integrals for the final proton. This means that the 3 Λ n → (pnn) + π − weak decay rate will be given in the closure approximation essentially by the Λ → p + π − free-space weak-decay rate. In the other case of Λ → n + π 0 induced decays, production of a low-momentum neutron is suppressed by the Pauli principle on account of the two neutrons already there in the initial 3 Λ n state. To a good approxiamtion, perhaps to just a few percent, this 3 Λ n weak decay branch may be disregarded. Our best estimate for the 3 Λ n weak decay rate is then given by where the coefficient 0.641 is the free-space Λ → p + π − fraction of the total Λ → N + π weak decay rate. Evaluating the ratio Γ( 3 Λ n)/Γ Λ for the choicē q = q Λ one obtains where the factor 1.114 follows from the difference between E 3N (q Λ ) and E N (q Λ ) in the recoil phase-space factors. Our predicted 3 Λ n lifetime is then but likely not shorter than 350 ps upon assigning 5% contribution from the π 0 decay branch. This lifetime is way longer than the 181 +30 −24 ±25 ps or 190 +47 −35 ±36 ps lifetimes deduced from the ndπ − and tπ − alleged decay modes of 3 Λ n [15,19]. Note that adding a potentially unobserved proton could perhaps reconcile these deduced lifetimes with τ ( 4 Λ H)=194 +24 −26 ps [20].

Three-body 3 Λ H wavefunctions
To have as simple input as possible to a genuine three-body description of the weakly bound 3 Λ H we constructed baryon-baryon s-wave separable interactions of Yamaguchi forms by fitting to the corresponding low-energy scattering parameters. In particular, the binding energy of the deuteron, limited to a 3 S 1 channel, is reproduced while the 1 S 0 nn system is unbound. For the ΛN interaction we follow Ref. [18] by choosing values (in fm) close to those used in Nijmegen models: Solving a set of three-body Faddeev equations, we get B Λ ( 3 Λ H)=0.094 MeV. Scaling up slightly the ΛN 1 S 0 interaction we also generated binding energies values of B Λ ( 3 Λ H)=0.13 and 0.17 MeV. In these calculations the Faddeev integral equations were solved using a momentum-space Gauss mesh of 32 points: so that half the integration points satisfy q < 1 fm −1 , thereby taking good care of the small q (large r) region which is of utmost importance for the diffuse 3 Λ H. The results prove numerically stable already upon using 20 Gauss mesh points. Finally, we note that to bind 3 Λ n in the present Faddeev equations formalism one could scale up, e.g., the ΛN 3 S 1 interaction by as much as 1.62 [18]. However, given the rough estimate in Eq. (7) for the 3 Λ n weak decay rate, we will not pursue further this issue.

Evaluation of the 3 Λ H exchange integral η(q)
To calculate the total decay rate, Eq. (2), it is useful to evaluate the exchange integral η(q) of Eq. (3) with 3 Λ H momentum-space wavefunctions. Denoting byχ( p Λ ; p N 2 , p N 3 ) the normalized Fourier transform of χ( r Λ ; r N 2 , r N 3 ), the expression for η(q) assumes the form Factoring out the overall cm dependence, the wavefunctionχ may be replaced by Ψ( p 1 , q 1 ), derived by solving the appropriate Faddeev equations in momentum space in terms of two Jacobi coordinates: the latter is the relative momentum of 1 (the Λ) with respect to the cm of nucleons 2 and 3. Integration in Eq. (10) is limited then to these two vector variables p 1 and q 1 . To evaluate η(q) of Eq. (10), using Ψ( p 1 , q 1 ) instead ofχ( p Λ ; p N 2 , p N 3 ), one needs to implement in Ψ * the consequences of transforming the variables p Λ and p N 2 into p N 2 + q and p Λ − q, respectively, inχ * . In the total cm system one may identify p Λ with q 1 , p N 2 with p 1 − 1 2 q 1 and p N 3 with − p 1 − 1 2 q 1 , so that p Λ → p N 2 + q and p N 2 → p Λ − q give rise to respectively. Eq. (10) is thereby transformed to To evaluate this form of the exchange integral η(q) one needs to express the Faddeev three-body wavefunction Ψ as a function of the two variables p 1 and q 1 . This requires careful attention since the Faddeev decomposition T = T 1 +T 2 +T 3 of the total T matrix into three partial T j matrices coupled to each other by the Faddeev equations T j = t j (1+G 0 k =j T k ) implies a similar decomposition of the bound-state wavefunction ψ into three components where G 0 is the three-body free Green's function and φ is a three-body plane wave. The natural momentum variables for each Ψ j component are p j and q j , so we need to switch in Ψ 2 and Ψ 3 from their respective momentum bases to p 1 and q 1 . This naturally involves integration on the angle between p j and q j , j = 1, so that the redressed Ψ 2 and Ψ 3 necessarily develop ℓ > 0 partial waves in addition to their dominant s-wave component. We have omitted such unwanted ℓ = 0 partial waves. The error incurred in this approximation may be estimated by evaluating the normalization integral of Ψ in two ways, first with each component Ψ j in its natural coupling scheme, and then with all three components expressed in terms of the p 1 , q 1 variables, thus giving rise to a normalization integral smaller by about 2.5%.

Results disregarding pion FSI
We evaluated numerically the 3 Λ H exchange integral η(q) of Eq. (13) for two values of the closure momentumq discussed in Ref. [8] and for several values of B Λ ( 3 Λ H) allowed by its experimental uncertainty. Our results are listed in Table 1, compared with those of Congleton [10] who considered the same range of B Λ ( 3 Λ H) values. As argued by RD [8], and followed up by Congleton [10], the appropriate choice for the 3 Λ H pionic decay closure momentumq is the empirical peak valueq = 96 MeV/c in the π − weak decay continuum spectrum. To study the sensitivity of η(q) to a small departure from this accepted value ofq, we also evaluated η(q) forq = 104 MeV/c, a value a bit larger than that for the free Λ decay which was used in calculations that preceded RD. The variation of η(q) withq over the momentum interval studied is quite weak. Forq = 96 MeV/c, our calculated value of η(q) is about 70% of Congleton's value. This apparent discrepancy can be shown to arise from his use of a Λd cluster model for 3 Λ H: specifically by (i) limiting the full Faddeev wavefunction Ψ, Eq. (14), to Ψ 1 which is the component most natural to represent a Λd cluster, and (ii) suppressing then in the three-body free Green's function G 0 the dependence on the Λ momentum, we obtain a value of η(q = 96 MeV/c)=0.238±0.038, in good agreement with the value listed for 'Λd cluster' in Table 1.
Using η(q = 96 MeV/c)=0.146±0.021 from Table 1 in Eq. (2) for 3 Λ H, and noting Eq. (1) for the free Λ decay, we obtain the following 3 Λ H mesonic decay rate: Adding a ≈1.7% nonmesonic weak decay branch [14], our Faddeev result is This interim result has to be supplemented by possible contributions from pion FSI, a subject dealt with in the next section.

Pion FSI effects
The pion emitted in the 3 Λ H decay is dominantly s-wave pion, see Eq.
(1). Optical model fits of measured 1s pionic atom level shifts and widths across the periodic table [21,22] suggest that the underlying πN s-wave interaction term in nuclei at low energy is weakly repulsive and that the attractive πN p-wave term has negligible effect on 1s pionic states. The corresponding s-wave induced π − nuclear optical potential is given by 1 in terms of fitted real πN scattering lengths: isoscalar b 0 = −0.0325 fm and isovector b 1 = −0.126 fm [23]. With these negative signs one gets π − repulsion in the majority of stable nuclei, those with N ≥ Z. However, in the few Z > N available nuclear targets like hydrogen and helium this repulsion is reversed into attraction owing to the isovector term flipping sign under Z ↔ N. This is confirmed by the attractive 1s level shifts observed in the π − 1 H and π − 3 He atoms [24]. One therefore expects attractive FSI in the 3 He+π − decay channel of 3 Λ H, and repulsive FSI in the 3 H+π 0 decay channel where the πN isovector term associated with b 1 vanishes while the weakly repulsive isoscalar term associated with b 0 remains in effect. Altogether we have verified that the sum of these two FSI contributions to the 3 Λ H decay rate is nearly zero. Pion FSI in the context of 3 Λ H decay has been considered elsewhere only by RD [8], who indeed found it weakly repulsive and decreasing the 3 Λ H decay rate by 1.3%, and in Ref. [25] where the attraction in the 3 He+π − decay channel was overlooked.
The preceding argumentation on the role of pion FSI in the 3 Λ H decay is incomplete, if not misleading. The ∆I = 1 2 rule in the Λ → Nπ weak decay implies that the 3N + π final states are good (I = 1 2 , I z = − 1 2 ) isospin states which are coherent combinations of ppnπ − and pnnπ 0 configurations. For I = 1 2 , and in the Born approximation applied to the optical potential Eq. (17), the pion-nuclear scattering length is given by 3b 0 − 2b 1 which is considerably more attractive than the scattering length 3b 0 − b 1 valid for 3 He+π − alone. The difference between these two expressions arises from charge exchange transitions between the nearly degenerate charge states of (I = 1 2 , I z = − 1 2 ) good-isospin states, such as between 3 He+π − and 3 H+π 0 . To estimate the effect of pion FSI on the 3 Λ H lifetime we consider the twobody decay mode to the coherent state made of these two components. The 3 Λ H decay amplitude to this final state is given by a distorted-wave (DW) formfactor wherej 0 is a pion DW evolving via FSI from a pion plane-wave (PW) spherical Bessel function j 0 . The vectors r and ρ are Jacobi coordinates: r stands for the Λ → N 'active' baryon relative to the cm of the spectator nucleons, and ρ denotes the relative coordinate of the spectator nucleons. In Eq. (18), r N = 2 3 r stands for the coordinate of the 'active' baryon with respect to the cm of the 3N final system. The Φ α are properly normalized L = 0 initial and final A = 3 wavefunctions. For this first estimate we approximated each Φ α ( r, ρ) by a product ψ α (r)φ α (ρ), with ψ α (r) given by as generated by Yukawa separable potentials with strength and range parameters fitted to yield the respective N or Λ separation energies (wave numbers κ α ) and realistic r.m.s. radii. The values chosen for the parameters β α and κ α are given in units of fm −1 as follows: The formfactor (18) reduces then to where is the overlap integral of the two φs and is the same for both PW and DW pions. For this reason, the choice of the 'deuteron' wavefunctions φ α (ρ) is not discussed further here. For the pion DWj 0 we used a continuum wavefunction, also generated from a separable Yukawa potential: where f (q) = 3b 0 − 2b 1 = 0.180 + i0.0483 fm is a π-nuclear s-wave scattering amplitude with values of b 0 and b 1 taken from SAID [26] at 31 MeV pion kinetic energy, compared to 0.155 fm for the threshold values listed following Eq. (17). The variation of both b 0 and b 1 with energy in the SAID analysis is rather slow. In the actual calculation we dropped the small imaginary part of f (q) so as to account also for the few percent pion absorption contributions 3 Λ H → pnn to the 3 Λ H lifetime. For β π we used a representative value of 0.806 fm −1 corresponding to a r.m.s. radius of a separable Yukawa formfactor 1.754 fm (matter radius of 3 He).
Evaluating the formfactors F DW (q) and F PW (q), where the latter is obtained from the former by reducingj 0 to j 0 , we find a pion FSI enhancement factor |F DW (q)/F PW (q)| 2 = 1.097. Strictly speaking this ≈10% increase of the 3 Λ H decay rate applies to the two-body final states 3 He+π − and 3 H+π 0 . The cooresponding two-body decay modes take roughly 36.5% of the total decay rate in the present s-wave calculation, and a more quantitative calculation needs to be done for the remaining ≈63.5% pionic final states which are mostly p + d + π − and n + d + π 0 . A rough estimate gives a 1.074 enhancement factor for these decay modes, in which case the overall pion DW enhancement factor is 1.082. Details will be given elsewhere.

Conclusion
In this work we evaluated the mesonic decay rate of 3 Λ H by considering the closure-approximation decay-rate expression Eq. (2). The 3 Λ H exchange integral η(q), Eq. (3), which provides input to Eq. (2) was calculated within a fully three-body Faddeev equations model of 3 Λ H g.s. . Forq = 96 MeV/c, as suggested by RD [8], our calculated value of η(q) listed in Table 1 leads to a 9% increase of the 3 Λ H decay rate over the free Λ decay rate. This result supersedes Congleton's result [10] of 12% increase based on a Λd cluster model of 3 Λ H which gave a value of η(q) about 50% higher than our Faddeev equations model value. Adding a 1.7% nonmesonic decay rate contribution [14], we get a total of about 10% decrease of the 3 Λ H lifetime with respect to the free Λ lifetime which, by itself, does not resolve the 3 Λ H lifetime puzzle.
Furthermore, we discussed for the first time semi-quantitatively the pion FSI effect on the 3 Λ H lifetime, finding it in the s-wave approximation to shorten further the 3 Λ H lifetime by ≈6%. Altogether, our calculated 3 Λ H lifetime amounts to 84% of τ Λ , with uncertainty which we estimate as ≈2%, including also pion FSI p-wave contributions. More precise calculations are required to verify this result.
Last, as a by-product of our formulation of the A = 3 hypernuclear lifetime, we showed in simple terms that the lifetime of 3 Λ n, if bound, is considerably longer than τ Λ , in disagreement with the shorter lifetime with respect to τ Λ extracted from the HypHI events assigned to this hypernucleus. Pion FSI should be repulsive in this case, increasing the 3 Λ n lifetime by a few more percents.