Lifetime of the hypertriton

Conflicting values of the hypertriton lifetime $\tau({}_\Lambda^3\mathrm{H})$ were derived in relativistic heavy ion (RHI) collision experiments over the last decade. A very recent ALICE Collaboration measurement is the only experiment where the reported $\tau({}_\Lambda^3\mathrm{H})$ comes sufficiently close to the free-$\Lambda$ lifetime $\tau_\Lambda$,as expected naively for a very weakly bound $\Lambda$ in ${}_\Lambda^3\mathrm{H}$. We revisited theoretically this ${}_\Lambda^3\mathrm{H}$ lifetime puzzle, using ${}_\Lambda^3\mathrm{H}$ and ${}^3\mathrm{He}$ wave functions computed within the abinitio no-core shell model employing interactions derived from chiral effective field theory to calculate the two-body decay rate $\Gamma({}_\Lambda^3\mathrm{H}\to{}^3\mathrm{He}+\pi^-)$. We found significant but opposing contributions arising from $\Sigma NN$ admixtures in ${}_\Lambda^3\mathrm{H}$ and from $\pi^- -{}^3\mathrm{He}$ final-state interaction. To derive $\tau({}_\Lambda^3\mathrm{H})$, we evaluated the inclusive $\pi^-$ decay rate $\Gamma_{\pi^-}({}_\Lambda^3\mathrm{H})$ by using the measured branching ratio $\Gamma({}_\Lambda^3\mathrm{H}\to{}^3\mathrm{He}+\pi^-)/\Gamma_{\pi^-}({}_\Lambda^3\mathrm{H})$ and added the $\pi^0$ contributions through the $\Delta I = \frac{1}{2}$ rule. The resulting $\tau({}_\Lambda^3\mathrm{H})$ varies strongly with the rather poorly known $\Lambda$ separation energy $E_{\mathrm{sep}}({}_\Lambda^3\mathrm{H})$ and it is thus possible to associate each one of the distinct RHI $\tau({}_\Lambda^3\mathrm{H})$ measurements with its own underlying value of $E_{\mathrm{sep}}({}_\Lambda^3\mathrm{H})$.

It can be approximated as a bound state of a Λ hyperon and a deuteron ( 2 H) with tiny Λ separation energy of E sep ( 3 Λ H) = 0.148(40) MeV [2]. The lifetime of such a loosely bound system is expected to be comparable to the lifetime of the free Λ, τ Λ = 263(2) ps [3], which is almost completely (99.7%) governed by its nonleptonic Λ → N+π weak decay mode. Yet, the world average of measured 3 Λ H lifetime, τ( 3 Λ H) = 223 +12 −11 ps [4], is by ∼ 20% shorter than τ Λ . This so-called 'hypertriton lifetime puzzle' has been strengthened recently by the ALICE Collaboration's τ( 3 Λ H) value [5] which is consistent with τ Λ and in tension with previous STAR [6] and HypHI [7] Collaborations measurements. The latest STAR Collaboration's measurement [8] does not seem to be conclusive since their reported τ( 3 Λ H) value is consistent within experimental uncertainties with τ Λ but its central value is 20% shorter. The measured 3 Λ H lifetimes from recent RHI experiments together with latest microscopic calculations are summarized in table 1. ≈ τ Λ Hildenbrand, Hammer [12] 2 Method

Nuclear and hypernuclear wave functions
The initial-and final-state 3 Λ H and 3 He wave functions in equation (1) are computed within the ab initio no-core shell model (NCSM) approach [20,21], where nuclei and hypernuclei are described as systems of A nonrelativistic particles interacting through realistic nucleon-nucleon (NN), 3-nucleon (NNN) and hyperon-nucleon (YN) interactions. In this work we employed a version of NCSM formulated in translationally-invariant relative Jacobi-coordinate harmonic oscillator (HO) basis which is suitable for dealing with few-body systems [22,23]. The many-body wave function is cast as an expansion in a complete set of basis states where the HO states |NλJT are characterized by the HO frequency ω and the expansion is truncated by the maximum number N max of HO excitations above the lowest configuration allowed by Pauli principle. In equation (3), N is the total number of HO excitations of all particles and J π T are the total angular momentum, parity and isospin. The quantum number λ labels all additional quantum numbers and the sum over N is restricted by parity to an even or odd sequence. The energy eigenstates are obtained by solving the Schrödinger equation with the intrinsic Hamiltonian Here, the masses m i and single-particle momenta p i for i ≤ A − 1 correspond to the nucleons and for i = A to hyperons. The H CM is the free center-of-mass (CM) Hamiltonian and the mass term ∆M = i<A m i − M 0 with M 0 the reference mass of a hypernuclear system containing only nucleons and a Λ hyperon is introduced to account for the mass difference of coupled Λand Σ-hypernuclear states. For the nuclear V NN +V NNN interactions in equation (5) we employ the NNLO sim potentials [24] that are based on chiral effective field theory (χEFT) up to next-tonext-to-leading order (NNLO). The NNLO sim is a family of 42 different interactions where each potential is associated with one of seven different regulator cutoffs Λ NN = 450, 475, . . . , 575, 600 MeV and six different maximum scattering energies in the laboratory system T max Lab = 125, 158, . . . , 257, 290 MeV at which the experimental NN scattering cross sections data base used to constrain the respective interaction was truncated. For the purpose of this work precise wave functions of 3 He and the 'core nucleus' 2 H are required. Since certain low-energy properties of 2 H and 3 He were included in the pool of fit data, their energies are accurately described for all NNLO sim interactions, E2 H, 3 [24]. For NNLO sim , the NCSM ground-state (g.s.) energies of 2 H and 3 He exhibit a good convergence with the NCSM model-space truncation N max as demonstrated in figure 1 for ω = 14 MeV using the NNLO sim (Λ NN = 500 MeV, T max Lab = 290 MeV) NN+NNN interaction. The g.s. energies are converged within few keV already at N max ≈ 30 for a wide range of HO frequencies.
For the YN sector (V YN in equation (5)) we use the leading order (LO) coupled-channel Bonn-Jülich SU(3)-based χEFT model [25]. The potential consists of pseudoscalar π, K, η meson exchanges and baryon-baryon contact interaction terms. The interaction is regularized by a smooth momentum cutoff Λ YN ranging from 550 to 700 MeV. Unfortunately, calculations of 3 Λ H g.s. energy exhibit a slower convergence with the NCSM model-space truncation N max . This can be attributed to the very small binding energy of 3 Λ H and, consequently, the long tail of the 3 Λ H wave function in coordinate space. Nevertheless, well-converged results for the 3 Λ H g.s. energy can be obtained for N max ≈ 70 [26]. However, as shown in Sect. 3, it is not possible to achieve full convergence for the transition operator matrix element in equation (1) even in the largest feasible NCSM model spaces and it is necessary to extrapolate the finite-space results into infinite model space. We employ the infrared (IR) extrapolation scheme developed for nuclear NCSM in [27] and generalized recently to hypernuclear NCSM in [28]. In this scheme, the truncation of the HO basis in terms of N max and ω is translated into the associated infrared (L IR ) and ultraviolet (UV) (Λ UV ) scales and IR correction formulae can be systematically derived for observables to extrapolate to infinite model space, L IR → ∞. The LO correction for energies and the expected magnitude of subleading corrections σ IR [29] are where (E UV ∞ , a UV 0 , k UV ∞ ) are parameters determined from fit to the NCSM-calculated energies with weights proportional to the inverse of σ IR . Note that this procedure slightly differs and should improve the one employed in [1]. Additional ultraviolet (UV) corrections to equation (6) can be substantial unless Λ UV Λ NN , Λ YN . A large-enough Λ UV scale can be identified by performing calculations at a fixed Λ UV -by choosing appropriate (N max , ω) model-space parameters-and monitoring the UV dependence [29]. We find that 1000 Λ UV 1200 MeV is sufficient to achieve UV-convergence and to perform IR extrapolations. This is shown in figure 2 where the 3 Λ H g.s. energy E UV

Pion wave function
The pion wave function in equation (1) was generated from a standard optical potential constrained by data of pionic atoms across the periodic table [30,31]. In order to extrapolate from the near-threshold region relevant for pionic atoms to q π = 114.4 MeV in the π − − 3 He CM system, the potential parameters were adjusted using the πN [32], as well as π-nucleus elastic scattering amplitudes [33,34].

Results
In figure 3, we show the 2-body 3 Λ H π − decay rate Γ(  While the UV convergence for Λ UV 1000 MeV was not fully achieved, the calculations still give meaningful, sufficiently IRconverged results. The missing UV corrections to equation (6) depend only on short-range details of the employed interactions truncated by Λ UV . In table 2, we show the extrapolated Λ separation energies, two-body π − decay rates, and lifetime values for several HO basis UV scales, 800 ≤ Λ UV ≤ 1400 MeV. The τ( 3 Λ H) values were deduced from Γ( 3 Λ H→ 3 He + π − ) using the procedure detailed in Sect. 2.1, tacitly assuming that the branching ratio R 3 is independent of E sep ( 3 Λ H). The Λ separation energy rapidly decreases with reducing the UV scale from Λ UV = 1200 MeV to Λ UV = 800 MeV which lowers the two-body π − rate and enhances τ (

Summary
We performed a new microscopic calculation of the hypertriton π − two-body decay rate Γ( 3 Λ H→ 3 He+π − ) employing 3 Λ H and 3 He three-body wave functions generated by ab initio hypernuclear no-core shell model (Y-NCSM) using realistic YN and NN+NNN interactions derived from χEFT. Using the ∆I = 1 2 rule and the measured branching ratio R 3 to include the remaining π 0 and 3-plus 4-body 3 Λ H decay channels, we deduced the 3 Λ H lifetime τ( 3 Λ H). The main findings of this study are: (i) Replacing pionic plane wave (PW) by realistic π − − 3 He distorted wave (DW) enhances Γ( 3 Λ H→ 3 He+π − ) by ≈ 15%. (ii) The ΣNN admixtures in 3 Λ H reduce the purely ΛNN decay rate by ≈ 10% due to interference effects, despite their 0.5% contribution to the norm of the 3 Λ H wave function. (iii) The lifetime τ( 3 Λ H) varies strongly with the rather poorly known E sep ( 3 Λ H). It is then possible to associate each of the distinct RHI measured τ( 3 Λ H) values with its own underlying value of E sep ( 3 Λ H). Remarkably, the most recent ALICE Collaboration's [5] E sep ( 3 Λ H) = 72(63)(36) keV central value is almost the same as our lowest E Λ UV =800 sep = 69 keV and our lifetime τ Λ UV =800 ( 3 Λ H) = 234 ± 27 ps is then consistent with their reported τ( 3 Λ H) = 253 ± 11 ± 6 ps. Only future experiments expected at MAMI, JLab, J-PARC, and CERN will hopefully pin down E sep ( 3 Λ H) with a better precision than 50 keV and lead to a resolution of the 3 Λ H lifetime puzzle.