Hypernuclear constraints on the existence and lifetime of a deeply bound H dibaryon

We study to what extent the unique observation of ΛΛ hypernuclei by their weak decay into known Λ hypernuclei, with lifetimes of order 10 − 10 s, rules out the existence of a deeply bound doubly-strange ( S = − 2) H dibaryon. Treating 6 ΛΛ He (the Nagara emulsion event) in a realistic Λ − Λ − 4 He three-body model, we find that the 6 ΛΛ He → H + 4 He strong-interaction lifetime increases beyond 10 − 10 s for m H < m Λ + m n , about 176 MeV below the ΛΛ threshold, so that such a deeply bound H is not in conflict with hypernuclear data. Constrained by Λ hypernuclear ∆ S =1 nonmesonic weak-interaction decay rates, we evaluate the ∆ S =2 H → nn weak-decay lifetime of H in the mass range 2 m n ≲ m H < m Λ + m n . The resulting H lifetime is of order 10 4 s, many orders of magnitude shorter than required to qualify for a dark-matter candidate. A lower-mass absolutely stable H , m H ≲ 2 m n , is likely to be ruled out by established limits of nuclear stability such as for 16 O.


Introduction
The deuteron, with mass only 2.2 MeV below the sum of masses of its proton and neutron constituents, is the only particle-stable six-quark (hexaquark) dibaryon known so far.Here stabilty is regarded with respect to the lifetime of the proton, many orders of magnitude longer than the lifetime of the Universe (13.8 billion years [1]).Extending the very-light ud quark sector by the light strange quark s, Lattice-QCD (LQCD) calculations suggest two strong-interaction stable hexaquarks.Both are J π =0 + near-threshold s-wave dibaryons with zero spin and isospin: (i) a maximally strange S=−6 ssssss hexaquark classified as ΩΩ dibaryon member of the SU(3) flavor 28 f multiplet, and (ii) a strangeness S=−2 uuddss hexaquark, a 1 f H dibaryon, which is the subject of the present study.Whereas the LQCD calculation of ΩΩ reached m π values close to the physical pion mass [2], H dibaryon LQCD calculations have been limited to values of m π ∼ 400 MeV and higher (NPLQCD [3], HALQCD [4]) while following SU(3) f symmetry, where A very recent calculation of this type [5] finds the H dibaryon bound just by 4.6±1.3MeV with respect to the ΛΛ threshold.However, chiral extrapolation to physical quark masses values and thereby also to m π ≈ 0 [6] suggests that the H dibaryon becomes unbound by 13±14 MeV.Thus, a slightly bound 1 f H dibaryon is likely to become unbound with respect to the ΛΛ threshold in the SU(3) f -broken physical world, lying possibly a few MeV below the N Ξ threshold [7,8].
The H dibaryon was predicted in 1977 by Jaffe [9] to lie about 80 MeV below the 2m Λ =2231 MeV ΛΛ threshold.Dedicated experimental searches, beginning as soon as 1978 with a pp → K + K − X reaction [10] at BNL, have failed to observe a S=−2 dibaryon signal over a wide range of dibaryon masses below 2m Λ [11,12,13], notably BABAR's recent search at SLAC looking for a Υ(2S, 3S) → ΛΛ X decay [13].Furthermore, a simple argument questioning the existence of a strong-interaction stable S=−2 dibaryon was put forward by several authors, notably Dalitz et al. [14].It relates to the few established ΛΛ hypernuclei [15,16], foremost the lightest known one 6 ΛΛ He (the Nagara emulsion event) where a ΛΛ pair is bound to 4 He by 6.91±0.17MeV [17] exceeding twice the separation energy of a single Λ in 5 Λ He by merely ∆B ΛΛ ( 6 ΛΛ He)=0.67±0.17MeV.If H existed deeper than about 7 MeV below the ΛΛ threshold, 6  ΛΛ He could decay strongly, considerably faster than the ∆S=1 weak-interaction decay by which it has been observed and uniquely identified [17]: Further arguments questioning an H bound by less than about 7 MeV were put forward by Gal [18].
Arguments of this kind, questioning the existence of a strong-interaction stable H dibaryon, were challenged 20 years ago by Farrar [19] who suggested that the H dibaryon may be a long-lived compact object with as small radius as 0.2 fm and as small mass as 1.5±0.2GeV, in which case it becomes absolutely stable, without disrupting the observed stability of nuclei.Once sufficiently abundant, relic H dibaryons would qualify as a cold Dark Matter (DM) candidate.Farrar's present estimate for the mass m H of such a compact dibaryon, often termed Sexaquark in these works, is between 1850 and 2050 MeV [20].Here, m H ≲ 1850 MeV is disfavored by the stability of oxygen [21], whereas the mass value 2050 MeV stems from the threshold value (m n + m Λ ) = 2055 MeV, above which the ∆S=1 strangeness changing weak decay H → nΛ would make H definitely short lived with respect to a lifetime of cosmological origin expected for a DM candidate.Following Farrar's conjecture of a deeply-bound compact S=−2 dibaryon, we present a realistic calculation of 6  ΛΛ He lifetime owing to the two-body strong-interaction decay reaction Eq. ( 2).Treating 6 ΛΛ He in a Λ − Λ − 4 He three-body model, it is found that the 6 ΛΛ He → H + 4 He strong-interaction lifetime is correlated strongly with m H , increasing upon decreasing m H such that it exceeds the hypernuclear ∆S=1 weak-decay lifetime scale of order 10 −10 s for m H < (m Λ + m n ).Therefore, hypernuclear physics by itself does not rule out an H-like dibaryon in this mass range.
Constrained by Λ hypernuclear ∆S=1 nonmesonic weak-interaction decay rates within leading-order (LO) effective field theory (EFT) approach, we present a realistic calculation of the ∆S=2 weak decay H → nn for H mass satisfying 2m n ≲ m H < (m n + m Λ ).The resulting H lifetimes are of order 10 5 s, in rough agreement with Donoghue, Golowich, Holstein [22] who followed a completely different high-energy physics methodology.Our calculated H lifetimes are 10 orders of magnitude shorter than the order of 10 8 yr reached in the 2004 Farrar-Zaharijas (FZ) calculation [23].Hence, a deeply bound H dibaryon would be far from qualifying for a DM candidate.

Spatial part
Here we follow the simple ansatz for the H dibaryon six-quark (6q) fully symmetric spatial wavefunction Ψ H given by FZ [23]: where N 6 is a normalization constant and ν is related to the H 'size' as detailed below.To transform this 6q Ψ H to a two-baryon form where each baryon B a and B b is described as a 3q cluster, we define relative coordinates ρ, λ and center-of-mass (c.m.) coordinates R: plus a total cm coordinate R = 1 2 (R a + R b ) = 1 6 6 i r i .Using these ρ and λ intrinsic quark coordinates, plus a relative coordinate r = (R b −R a ) between the baryonic 3q clusters B a and B b , Eq. (4) assumes the form where provides normalized 3q baryonic spatial wavefunction for baryon B j , j = a, b, and provides a normalized spatial wavefunction in the relative coordinate ⃗ r of the dibaryon B a B b .Note that all three components of H in Eq. ( 7) share the same root-mean-square (r.m.s.) radius value: (correcting an error: < r 2 H >= ν −1 in Ref. [23]), so that the radial extent of H is about 75% of the radial extent of each of its three components B a , B b , B a B b .We also note that a 3q baryon wavefunction, similar to Ψ H for 6q, implies a r.m.s.radius squared of slightly smaller according to Eq. ( 10) than when embedded within the H dibaryon.
Although derived for a specific spatially symmetric wavefunction, Eq. ( 4), the relationships noted above between various < r 2 > values hold for any spatially symmetric form chosen for H. Establishing physically one such 'size' value determines necessarily all other 'size' values.However, the choice of a specific 'size' value is constrained by the choice of H binding energy value, as demonstrated in Table 1 for B a = B b = Λ.In this particular case, the Schroedinger equation in the relative coordinate r ΛΛ was solved for assumed binding energy values B ΛΛ , using attractive Gaussian ΛΛ potential of the form is a strength parameter fitted to given values of B ΛΛ and is a zero-range Dirac δ (3) (r) function in the limit λ → ∞, smeared over distance of √ < r 2 > λ = √ 6/λ (0.612 fm for λ = 4 fm −1 chosen here).As expected, once B ΛΛ increases beyond a nuclear physics scale of roughly 20 MeV or so, < r 2 ΛΛ > decreases below 1 fm down to ≈ 0.5 fm.The corresponding H r.m.s.radius values are even smaller: < r 2 H >/ < r 2 ΛΛ > = ( √ 5/3) ≈ 0.745 according to Eqs. (10,11).Taking a shorter-range Gaussian potential, say with λ = 5 fm −1 , has a relatively small effect on < r 2 ΛΛ > which decreases between 3% to 12% as B ΛΛ increases from 5 to 1000 MeV.In passing we comment that the constraint imposed on < r 2 H > by assuming a definite value of B ΛΛ , or vice versa, was overlooked in Ref. [23].

Spin-Flavor-Color part
To complete the discussion of the H dibaryon wavefunction we note that the fully symmetric spatial 6q wavefunction Ψ H , Eq. ( 4), needs to be supplemented by a singlet 1 S total spin S=0 component, represented by a 6q SU(2) Young tableau and by singlet 1 F total flavor (F ) and 1 C total color (C) components, each represented by its own 6q SU(3) Young tableau Each of these S, F and C tableaux accommodates five components.In spin space, only one component corresponds to S a (uds) = S b (uds) = 1 2 and S a (ud) = S b (ud) = 0 implied by a ΛΛ dibaryon component, and in flavor space, again, only one component corresponds to 8 a (uds) and 8 b (uds) with isospin I a (ud) = I b (ud) = 0 for a ΛΛ dibaryon component.In color space, too, only one component corresponds to colorless 1 a (uds) and 1 b (uds) baryons.Hence, up to a phase, we assign in each of these three spaces a coefficient of fractional parentage 1/5 to Ψ H of Eq. (7).Finally, having chosen the ΛΛ component over the ΣΣ and N Ξ components of H, see Eq. ( 1), involves a Clebsch-Gordan coefficient of magnitude 1/8 which together with the former coefficients amounts to supplementing the spatially symmetric ΛΛ wavefunction ψ ΛΛ , Eq. ( 7) for B a = B b = Λ, by a flavor-color-spin factor 1/1000: ψΛΛ = 1/1000 × ψ ΛΛ .
Representing the H dibaryon spatially by the fairly small-size ψΛΛ wavefunction rather than by the normal-size normalized ψ ΛΛ means that its initialand final-state interactions with 'normal' baryonic matter are negligible, in agreement with arguments reviewed in Ref. [20].Accordingly, no final-state interaction between H and 4 He is introduced in the strong-interaction decay 6 ΛΛ He → H + 4 He studied in Sect. 4 below.

6 ΛΛ He wavefunction 3.1. Three-body approximation
Given the tight binding of 4 He, we treat the six-body 6 ΛΛ He as a threebody ΛΛα system with spatial coordinates r α , r Λ 1 , r Λ 2 .Starting from the two relative Λα vector coordinates r Λ 1 α = r Λ 1 − r α and r Λ 2 α = r Λ 2 − r α , we transform to their relative and c.m. coordinates A reasonable simple approximation of the Pionless-EFT (/ πEFT) 6 ΛΛ He wavefunction calculated in Ref. [24] is then to use a factorized ansatz: where the wavefunctions ϕ ΛΛ and Φ ΛΛ are chosen as Gaussians constrained by requiring that ϕ ΛΛ reproduces the r.m.s.radius of the coordinate r ΛΛ in the 6-body 6 ΛΛ He / πEFT calculation [24] as discussed below.Note that the r.m.s.radius value of the c.m. Gaussian Φ ΛΛ is half that of the Gaussian ϕ ΛΛ .Finally, the 4 He core wavefunction ϕ α within 6  ΛΛ He is approximated by a freespace 4 He wavefunction identical with that for 4 He in the 6 ΛΛ He → H + 4 He strong-interaction decay.
Studying the / πEFT 5 Λ He five-body calculation [25] we note that B exp Λ ( 5 Λ He) is nearly reproduced by choosing Eq. ( 14) for ΛN contact terms, with cutoff values λ = 1.25 fm −1 or λ = 1.50 fm −1 for ΛN scattering length versions Alexander(B) and χEFT(NLO19).Going over to the / πEFT 6 ΛΛ He six-body calculation [24], the ΛΛ r.m.s.distance computed for these cutoff values is < r 2 ΛΛ > = 3.65 ± 0.10 fm, which we adopt for the r.m.s.radius of the Gaussian ϕ ΛΛ in Eq. (19).Note that ϕ ΛΛ appears as a bound-state wavefunction in spite of the ΛΛ interaction being much too weak to form a bound state; it is the 4 He nuclear core that stabilizes the two Λs in 6  ΛΛ He.We also note that since this value refers to a weakly 'bound' ΛΛ pair in 6  ΛΛ He, it is considerably larger than < r 2 ΛΛ > values listed in Table 1 for a tightly bound H dibaryon.

Short-range behavior
Eq. ( 19) provides a simple wavefunction for two loosely bound Λ hyperons held together by 4 He, disregarding the short-range repulsive component of the ΛΛ interaction which is manifest in LQCD calculations [26].To account for the short-range repulsion effect on the 6 ΛΛ He → H + 4 He decay rate, we modify ϕ ΛΛ in Eq. ( 19) by introducing a short-range correlation (SRC) factor [1 − j 0 (κr)], where j 0 is a spherical Bessel function of order zero: Choosing κ = 2.534 fm −1 , corresponding to 500 MeV/c in momentum space, nearly reproduces the ΛΛ G-matrix calculation in Ref. [27] (Fig. 2 there and related text).We therefore replace Φ 6 ΛΛ He , Eq. ( 19), by for use as initial 6 ΛΛ He wavefunction in the 6 ΛΛ He → H + 4 He decay rate calculation reported below.

6
ΛΛ He → H + 4 He decay rate We assume that the strong-interaction decay 6 ΛΛ He → H + 4 He of a loosely 'bound' ΛΛ pair in 6  ΛΛ He into a ΛΛ pair constituent of a tightly bound 1 F H dibaryon, flying off 4 He with momentum k H in their c.m. system, is triggered by the ΛΛ strong interaction V ΛΛ extracted near threshold.The spatial dependence of the decay matrix element is given by < Ψ f |V ΛΛ |Ψ i >, where Ψ i stands for the initial 6  ΛΛ He wavefunction, Eq. ( 21), and where ψ ΛΛ , Eq. ( 9), is renormalized by the flavor-color-spin factor 1/1000, see Eq. ( 17), thereby accounting for the elimination of ΣΣ and N Ξ components of H.Note that in agreement with the overall attraction of the BB interaction in the 1 F channel [26] no SRC factor was introduced in Eq. ( 22) for Ψ f .We note that the calculations reported below disregard the slight difference between the inner 3q structure of each Λ hyperon in the H dibaryon to that in 6 ΛΛ He.For a 3q baryon size of about 0.5 fm [28], this neglect is well justified in the range of B ΛΛ values considered here.
The 6 ΛΛ He → H + 4 He decay rate, or equivalently the corresponding strong-interaction width of 6 ΛΛ He, is given by [29]: where µ Hα is the H − 4 He reduced mass and < Ψ f |V ΛΛ |Ψ i > is a product of two factors, as follows.
The other factor in < Ψ f |V ΛΛ |Ψ i > is an overlap matrix element between the initial ΛΛ − α Gaussian wavefunction Φ ΛΛ (R ΛΛ ) in 6  ΛΛ He and the final outgoing H − α plane-wave exp (ik H • R H ). Note that R H has nothing to do with the relatively small size of H.In the following we identify R H with the corresponding argument R ΛΛ in Eq. ( 19), both defined relative to 4 He and denoted below simply by R. The square of this overlap matrix element times 4π from d kH in Eq. ( 23) is given by where a Φ = 2 < R 2 ΛΛ > /3 = 1.49±0.04fm.As shown in Table 2, I(k H ; a Φ ) varies strongly with k H over 18 decades as B ΛΛ is varied from 100 MeV (76 MeV above m Λ + m n ) to 400 MeV (47 MeV below 2m n ).This is caused by the increased oscillations of exp(ik H •R) vs. the smoothly varying Φ ΛΛ (R) in Eq. (25).And on top of that, the ±0.04 fm uncertainty of a Φ makes I(k H ; a Φ ) uncertain by a factor of 4 larger/smaller values than for the mean value a Φ = 1.49fm at B ΛΛ = 176 MeV corresponding to the m Λ + m n threshold, increasing to a factor about 20 at B ΛΛ = 400 MeV.
The final values of 6 ΛΛ He → H + 4 He decay rate Γ listed in Table 2 account for the two factors in < Ψ f |V ΛΛ |Ψ i > considered above.Notably, Γ decreases over 17 decades, reflecting the strong variation of I(k H ; a Φ ) with B ΛΛ , and the 6  ΛΛ He strong-interaction lifetime τ = ℏ/Γ increases as strongly over this range of B ΛΛ values.In particular, for H binding energy B ΛΛ = 176 MeV (m Λ + m n threshold), and given the a Φ uncertainty cited above, τ lies in the interval [1.1 × 10 −9 s, 1.7 × 10 −8 s], exceeding by far the weak-interaction lifetime scale set by the free-Λ lifetime τ Λ = 2.6 × 10 −10 s, so that the robust observation of ΛΛ hypernuclei by their weak-interaction decay modes does not rule out the existence of a deeply bound H dibaryon with mass below Figure 1: ∆S = 1 Λ → n weak-interaction diagrams in free space (a) or in Λ hypernuclei (b), see Dalitz [31], and ∆S = 2 ΛΛ → nn weak-interaction diagram (c), all involving emission (a) or exchange (b,c) of a π 0 meson.Weak-interaction and strong-interaction coupling constants g w and g s , respectively, are denoted by circles.

ΛΛ nonmesonic weak decays
Having realized that a deeply bound H dibaryon lying below m Λ + m n is not in conflict with the weak-decay lifetime scale τ Λ ∼ 10 −10 s of all observed ΛΛ hypernuclei, we now estimate the leading ∆S = 2 weak-interaction decay rate of H, that of the H → nn two-body decay.H is represented here, as above, by its deeply bound ΛΛ component.Although ∆S = 2 ΛΛ → nn transitions are not constrained directly by experiment, they are related to ∆S = 1 Λn → nn transitions which are constrained by ample lifetime data in Λ hypernuclei [15].These ∆S = 1 transitions, including the pion-exchange transition depicted in Fig. 1(b), proceed in Λ hypernuclei with a total rate comparable to the Λ → nπ 0 free-space decay rate associated with Fig. 1(a).The weak-interaction coupling constant g w extracted from the free-space Λ lifetime, and proved to be relevant in the Λn → nn nonmesonic decay of Λ hypernuclei, could then be used as shown in Fig. 1(c) to estimate the strength of the ∆S = 2 ΛΛ → nn weak-decay transition.
Pion exchange is not the only contributor to the nonmesonic weak decay (NMWD) of Λ hypernuclei.Owing to the large momentum transfer in Fig. 1(b), pion exchange generates mostly a tensor 3 S 1 → 3 D 1 transition which is Pauli forbidden for nn, so shorter-range meson exchanges need to be considered.However, the next candidate of pseudoscalar meson-exchange, K meson-exchange, interferes destructively with pion exchange in the Λn → nn 1 S 0 → 1 S 0 parity-conserving (PC) transition of interest [32].It is useful then to follow an EFT approach initiated by Jun [33] and applied systematically by Parreño et al. [34,35] where Figs.1(b,c) are supplemented by Figs.2(a,b) respectively.The square vertices in these figures stand at leading-order (LO) for 1 S 0 → 1 S 0 low-energy constants (LECs) denoted schematically by g w g s and g 2 w , respectively.These vertices incorporate effects of heavier-meson (and thus shorter-range) exchange diagrams which are poorly known.Furthermore, we note that the smallness of the Λ intrinsic asymmetry parameter a Λ measured at KEK in the NMWD of 5  Λ He and 12 Λ C [36] suggests that the Λn → nn 1 S 0 → 3 P 0 parity-violating (PV) amplitude, disregarded here, is substantially smaller than the 1 S 0 → 1 S 0 PC amplitude considered below.Since the momentum p f ≈ 420 MeV/c of each of the final neutrons in Fig. 2(a) is much larger than the Fermi momentum of the initial neutron, the Λ hypernuclear decay rate induced by this diagram is well approximated by a quasi-free expression tested in studies of Σ hypernuclear widths [37,38], where ρ n = 0.084 fm −3 is the neutron density in nuclear matter and 1 4 stands for the fraction of initial-state neutrons satisfying S Λn = 0. To calculate the ∆S = 1 two-body reaction cross section σ Λn→nn at LO, we use a ∆S = 1 1 S 0 → 1 S 0 contact interaction C (λ) 1 δ λ (r), Eq. ( 14), with a LEC C (λ) 1 (Λn) determined by fitting the r.h.s. of Eq. ( 26) to Γ n = (0.35 ± 0.04)Γ Λ where Γ Λ = ℏ/τ Λ .We used a value Γ n /Γ p = 0.55 ± 0.10 from 12  Λ C NMWD measurements (see Table XIII of Ref. [15]) assuming that all ΛN NMWD modes sum up to Γ Λ .Note that lifetimes of heavier hypernuclei are shorter by ∼ 25% than τ Λ = 263 ps (τ hyp ≈ 210 ps [39,40]) owing most likely to ΛN N NMWD modes [15].
Evaluating σ Λn→nn at rest, Eq. ( 26) assumes the form where the initial, at-rest, ψ Λn and final ψ (kn) nn wavefunctions are given by with k n the neutron momentum release.Note that both repulsive short-range initial-state interaction (ISI) 1−exp(− 1 6 q 2 c r 2 ) and final-state interaction (FSI) 1 − j 0 (q c r), where q c = m ω = 3.97 fm −1 , start as 1  6 q 2 c r 2 at small r.For λ = 4 fm −1 , Eq. ( 27) yields C 1 (Λn).We assign a factor of two systematical uncertainty to this choice by varying g s between 1/2 and 2 about the chosen value g s = 1.Altogether, the H → nn decay rate is given then by where ψΛΛ (r) is defined in Eq. ( 22).H → nn decay rate values Γ H and their associated lifetime values τ H , as calculated using λ = 4 fm −1 in Eq. ( 29), are listed in Table 3.A weak dependence of τ H on the H mass (m H = 2m Λ − B ΛΛ ) is noted, in contrast to the strong dependence observed in Table 2 for the strong-interaction lifetime of 6 ΛΛ He caused by the rapid exponential decrease exp(−a 2 Φ k 2 H ) upon increasing k H in Eq. ( 25).The relatively large value a Φ ≈ 1.5 fm extracted from 6  ΛΛ He is replaced here by a considerably smaller value of less than 0.3 fm owing to the considerably smaller size parameters of the deeply bound H wavefunction ψ and of the contact term δ λ (r), resulting altogether in a weak k n dependence.The calculated H → nn lifetimes listed in Table 3 are then uniformly of order 10 5 s, less than 1 yr, many orders of magnitude shorter than cosmological time scales commensurate with the Universe age.Regarding the model dependence of the calculated lifetimes, τ H ∼ 10 5 s, we note that it depends weakly on the range parameter of the LECs C from Γ n has an opposite effect; In total, the resulting PW value of Γ H is a factor of 2 to 3 smaller and the PW value of τ H 2 to 3 larger than listed in Table 3.This model dependence is of the same scale as the factor of four systematical uncertainty noted earlier, arising from the choice of g s = 1 in relating C

Concluding remarks
In this work we considered hypernuclear constraints on the existence and lifetime of an hypothetical deeply-bound doubly-strange H dibaryon.Re-garding the existence of H, it was found by considering the unambiguously identified 6  ΛΛ He double-Λ hypernucleus [16] that its strong-interaction lifetime for decay to 4 He + H would increase substantially upon decreasing m H , exceeding for m H < m Λ + m n by far the 10 −10 s weak-interaction hypernuclear lifetime scale.Thus, the unique observation of double-Λ hypernuclei through weak decay to single-Λ hypernuclei does not rule out on its own the existence of a deeply-bound H dibaryon in the mass range 2m n to m Λ + m n , defying doubts raised by Dalitz et al. 35 years ago [14], but in agreement with the 20 years old claim by FZ [23].Special attention was given in our evaluation to constrain hadronic cluster sizes by their binding energies and, indeed, our own conclusion follows from respecting the large difference between the r.m.s.radius of the loosely bound ΛΛ pair in 6  ΛΛ He and that of the compact ΛΛ pair within the H dibaryon.And furthermore, the anticipated ΛΛ short-range repulsion was fully considered by incorporating the SRC factor (1 − j 0 (κr)), Eq. ( 20), into the ΛΛ wavefunction.
Provided an H-like dibaryon exists in the mass range 2m n to m Λ + m n , it was found that its ∆S = 2 decay lifetime τ (H → nn) would be quite long, of the order of 10 5 s, but many orders of magnitude shorter than cosmological lifetimes comparable to the age of the Universe that H would need to qualify for a DM candidate.To reach this conclusion we used a weak-interaction EFT approach [34] constrained by experimentally known, largely nonmesonic, hypernuclear lifetimes [15].Our conclusion is in stark disagreement with that reached by FZ [23] using outdated by now hard-core strong-interaction nuclear models and, furthermore, disregarding the constraint imposed on the H radius r H by its binding energy.
We have not considered in the present work the case for an H-like dibaryon with mass below the nn threshold, a scenario likely to be ruled out by the established stability of several key nuclei, notably 16 O.A straightforward calculation of the hypothetical two-body decay rate Γ( 16 O → H + 14 O) gives indeed a rate many orders of magnitude larger than the upper bound established for oxygen by Super-Kamiokande [21].A report of this calculation is in preparation.is part of a project funded by the EU Horizon 2020 Research & Innovation Programme under grant agreement 824093.

Table 2 :
6ΛΛ He → H +4He decay rate Γ, Eq. (23), and decay time ℏ/Γ for some representative values of H binding energy B ΛΛ and their associated k H and I(k H ; a Φ ) values for a Φ = 1.49fm, see text.B ΛΛ = 176 MeV corresponds to m H = m Λ + m n .