OOn the lifetime of the hypertriton
F. Hildenbrand ∗ and H.-W. Hammer
1, 2, † Technische Universit¨at Darmstadt, Department of Physics, 64289 Darmstadt, Germany ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum f¨urSchwerionenforschung GmbH, 64291 Darmstadt, Germany (Dated: July 21, 2020)
Abstract
We calculate the lifetime of the hypertriton as function of the Λ separation energy B Λ in an effective fieldtheory with Λ and deuteron degrees of freedom. We also consider the impact of new measurements of theweak decay parameter of the Λ. While the sensitivity of the total width to B Λ is small, the partial widthsfor decays into individual final states and the experimentally measured ratio R = Γ He / (Γ He + Γ pd ) showa strong dependence. For the standard value B Λ = (0 . ± .
05) MeV, we find R = 0 . ± .
09, which isin good agreement with past experimental studies and theoretical calculations. For the recent STAR value B Λ = (0 . ± . ± .
11) MeV, we obtain R = 0 . ± . Keywords: effective field theory, hypernuclei, hypertriton, lifetime ∗ [email protected] † [email protected] a r X i v : . [ nu c l - t h ] J u l . INTRODUCTION The addition of hyperons to nuclear bound states extends the nuclear chart into a third di-mension. These so-called hypernuclei offer a unique playground for testing our understandinglow-energy Quantum Chromodynamics in nuclei beyond the u and d quark sector. A particularlyattractive feature of hypernuclei is that hyperons probe the nuclear interior without being affectedby the Pauli principle. There is a vigorous experimental and theoretical program in hypernuclearphysics that dates back as far as the 1950s [1].Here, we focus on the simplest hypernucleus, the hypertriton. The newest results on the life-time and binding energy of the hypertriton have created the so-called hypertriton puzzle. Thehypertriton consists of a neutron, a proton, and a Λ particle. Its structure has been studied usinghypernuclear interaction models as well as effective field theories (See, e.g., Refs. [1–6]). Further-more, first lattice QCD calculations of light hypernuclei have become available for unphysical pionmasses [7].Since the Λ separation energy of the hypertriton, B Λ , is small compared to the binding energyof the deuteron, B d ≈ . d bound state at low resolution. Themost frequently cited value for this separation energy is B Λ = (0 . ± .
05) MeV [8], resulting ina large separation of the Λ from the deuteron of about 10 fm [5]. However, recent results of theSTAR collaboration indicate that B Λ may be a factor three larger [9]. For a discussion of possibleimplications of the larger value for other hypernuclei, see Ref. [6].While the nucleus is stable against a breakup by strong interactions, the Λ is unstable againstweak decay with an energy release of about ∆ − M π ≈
38 MeV with ∆ = M Λ − m the baryon massdifference. An overview of the most relevant thresholds is given in Fig. 1. E .
22 MeV B Λ = 0 . − M π = 37 .
84 MeV5 .
50 MeV6 .
26 MeV H+ π He+ π d + p + π H Λ d + Λ p + n + Λ Figure 1. Most relevant thresholds for the hypertriton decay relative to the Λ d threshold. All energies aregiven in MeV; the figure is not up to scale. Experimentally, the hypertriton lifetime presents a puzzle. Old emulsion experiments give a verybroad range of values ranging from 100 ps up to 280 ps [15–20]. Newer heavy ion experiments,tend to lie significantly below the free Λ lifetime of about 260 ps [10–13]. However, recent resultsfrom ALICE yield a lifetime closer to the free Λ value [14]. An overview of experimental resultsfor the hypertriton lifetime from old emulsion efforts to the newest heavy-ion experiments is givenin Fig. 2.Theoretical investigations of the hypertriton started at the same time as the first experi-ments [22, 23]. Because B Λ is small compared to the deuteron binding energy, the decay of aquasifree Λ particle provides an intuitive picture of the hypertriton decay and one expects that thelifetime is driven by the free Λ width with small binding corrections. In the 1990s Congleton cal-culated the mesonic decays of the hypertriton in a Λ d picture within the closure approximation [2].2
965 1970 19750100200300400 τ H y p [ p s ] Figure 2. Compilation of lifetime measurements for the hypertriton. Blue squares show results obtained inaccelerators by different collaborations [10–14]. Earlier results from emulsion experiments are depicted byblack circles [15–20]. The red line is the PDG value for the free Λ lifetime, τ Λ , as reference[21]. Assuming a most likely pion momentum, he obtained a lifetime τ H about 10% shorter than thefree Λ lifetime τ Λ . This calculation also hinted that the details of the hypertriton wave func-tion do not seem to be important. Later complete three-body Faddeev calculations using realistichyperon-nucleon potentials confirmed this result [24]. Newer approaches combine the assets of bothcalculations, finding the impact of pionic final state interactions to be about 6% of Γ Λ [25]. RecentlyP´erez-Obiol et al. calculated the channel H (cid:55)→ π − + He based on NCSM wave functions for Heand the hypertriton [26]. Using the experimental branching ratio Γ He / (Γ He + Γ pd ) [16, 18, 20, 27]and varying B Λ by changing the short-distance cutoff in the NCSM, they found that all recent ex-perimental measurements of B Λ and τ Λ are internally consistent within their uncertainties. Thenon-mesonic decays are suppressed compared to the mesonic ones [28, 29].In this work, we address the hypertriton lifetime puzzle in a pionless effective field theory (EFT)approach with Λ d degrees of freedom. The pionless EFT framework provides a controlled, model-independent description of weakly-bound nuclei based on an expansion in the ratio of short- andlong-distance scales (see Refs. [30–34] for reviews). Since leptonic decays are strongly suppressed,we focus on the π -mesonic decays of the hypertriton into nucleon-deuteron and trinucleon finalstates. The choice of Λ d degrees of freedom is well motivated by the separation of scales between B Λ and the deuteron binding energy, as well as explicit three-body calculations in pionless EFTwith Λ pn degrees of freedom [3, 5]. Our approach has the advantage that B Λ enters as a free-parameter in the EFT and can be varied without changing other observables. In particular, weinvestigate the properties of the hypertriton decay for Λ separation energies in the range 0 ≤ B Λ ≤ α − [35, 36] correcting the previous value by 15%. This quantity encodes information on the relativecontributions of parity conserving and violating parts of the interaction. Preliminary results of ourwork were presented in Ref. [37].The structure of the paper is as follows. We start with an overview of the formalism and ourprocedure to fix the low-energy constants in Sec. II. After that, we discuss the calculation of the twomost prominent channels for mesonic decays, a weak decay of the bound Λ followed by the breakup into a nucleon and a deuteron in Sec. III and a weak decay of the bound Λ with a trinucleonin the final state in Sec. IV. We follow up with a discussion of our results for the dependence of3he lifetime on B Λ and α − in Sec. V. We then conclude with a summary and outlook in Sec. VI.A few calculational details are given in the Appendix. II. FORMALISMA. Preliminaries
Since the Λ separation energy of the hypertriton, B Λ ≈ .
13 MeV, is small compared to thebinding energy of the deuteron, B d ≈ . γ Λ3 ∼ (cid:113)(cid:0) mB − γ d (cid:1) / ≈ . γ d with γ d = 45 .
68 MeV the deuteron binding momentum and m the nucleon mass. Taking the deuteron breakup momemtum 1 . γ d as the high-energy scale, weestimate the expansion parameter of our effective field theory as γ Λ3 / (1 . γ d ) ≈ /
4. The Λ d picturefor the low-energy structure of the hypertriton is , e.g., supported by the work of Congleton [2]and our recent investigation of the hypertriton structure and matter radii [5], where a three-bodyframework with pn Λ and a two-body framework with Λ d degrees of freedom were compared. Sincethe deuteron is stable, the lifetime of the hypertriton is determined by the decay of a quasifreeΛ inside the hypertriton with small binding corrections. As discussed above, some measurementsfind the lifetime of the hypertriton to be about 30% shorter than the lifetime of the free Λ. Thepionless EFT description of the hypertriton in the Λ d picture provides an appropriate startingpoint to resolve this question.The main decay channels of the hypertriton are driven by the hadronic decay of the Λ: H (cid:55)→ π − + He , H (cid:55)→ π + H , H (cid:55)→ π − + d + p, H (cid:55)→ π + d + n, H (cid:55)→ π − + p + n + p, H (cid:55)→ π + p + n + n. (1)In the first line of Eq. (1), no breakup of the three-body nucleus takes place. Going down from topto bottom, more and more subsystems are broken up. The deuteron breakup processes have onlya small available phase space and are suppressed compared to the other ones. The correspondingpartial widths are a factor 100 smaller than the other hadronic decay channels [24]. The deuteronbreakup processes are therefore unlikely to resolve the lifetime puzzle. Moreover, the non-mesonicdecay branch of the hypertriton due to the reaction Λ N → N N is small and makes up only 1 . π − in the final state) and theneutral channels ( π in the final state) are connected via the empirical ∆ I = rule, setting theratio of the channels in Eq. (1) line for line approximately equal to 2.In the following, we describe the hypertriton in leading order pionless EFT with Λ d degrees offreedom. The typical momentum of the deuteron and the Λ in the hypertriton, γ Λ3 ≈
14 MeV, issmall compared to the pion mass and the deuteron binding momentum [5]. The pion in the outgoingstate is included with relativistic kinematics due to the large energy of M Λ − m − M π ≈ . M π released at the weak vertex. 4 . Fixing the weak interaction We use the free Λ decay to fix the weak interaction vertex. The non-leptonic decay matrixelement can be written as [38] M Λ (cid:55)→ pπ − = i √ G F M π ¯ u (cid:0) p (cid:48) (cid:1) (cid:104) ˜ A π + ˜ B π γ (cid:105) u ( p ) , (2)where ˜ A π is the parity violating (PV) amplitude while ˜ B π is parity conserving (PC). (Note thatthe pion has negative parity.) The factor √ pπ − channel. Therespective factor for the nπ channel is −
1. The Fermi constant is taken as G F = 1 . × − GeV − [21]. In the following, we will only calculate the width for the charged pion channeland obtain the width for the corresponding neutral pion channel by applying isospin symmetryand the ∆ I = rule. (See Sec. V for more details.) Due to the small binding momentum of thehypertriton, it is sufficient to treat the baryons non-relativistically. The non-relativistic reductionof the decay matrix element is W k ≡ M reducedΛ (cid:55)→ pπ − = i √ G F M π (cid:18) A π + B π M Λ + m σ · k (cid:19) , (3)with k the momentum of the pion and σ the usual Pauli spin-matrices (see also Refs. [24, 28]).Note that we have redefined the amplitudes for the PC ( B π ) and PV ( A π ) part to absorb somenormalization factors of the matrix element.It is now straightforward to calculate free width of the Λ, according to the diagram given inFig. 3. The Λ is assumed to be at rest, while the momentum of the outgoing pion is denoted k and the one of the nucleon is p . We obtain: Figure 3. Vertex for the decay of the free Λ (dashed line) into a nucleon (solid line) and a pion (wiggly line).The weak decay operator W k is indicated by the black box. Γ p Λ = (cid:90) d k (2 π ) d p (2 π ) ω k (2 π ) δ (3) ( p + k ) δ (cid:18) ∆ − ω k − p m (cid:19) (cid:88) spins |W k | , (4)with ω k = √ M π + k the relativistic energy of the pion and ∆ = M Λ − m the baryon massdifference, which is released at the weak vertex W k . The δ functions fix the momentum of theoutgoing pion to be ¯ k = √ (cid:113) − (cid:112) m ( m + 2∆ m + M π ) + m + ∆ m. (5)5he resulting width is then given byΓ p Λ = G F M π π m ¯ km + ω ¯ k (cid:32) A π + (cid:18) B π M Λ + m (cid:19) ¯ k (cid:33) . (6)The associated lifetime 1 / Γ Λ = τ Λ = (263 ±
2) ps is experimentally established very well [21]. Weuse this observable together with the polarization of the Λ P Λ = A π B π M Λ + m ¯ kA π + (cid:16) B π M Λ + m (cid:17) ¯ k = α − , (7)which determines the Λ decay parameter α − to fix the weak interaction strength. Up to 2018 thewidely accepted value was α − = 0 . ± .
013 [21], but new results from the BESIII Collaborationsuggest a significantly higher value α BESIII − = 0 . ± . ± .
004 [35]. Also an independentestimation from kaon-photo production suggests a value of α KP − = 0 . ± . ± .
005 [36] closeto the results of BESIII. The results for the PV and PC amplitudes A π and B π as determined byEqs. (6), (7) and the experimental Λ lifetime are depicted in Fig. 4. The different experimentalresults for α − are marked explicitly. − − − . . . B π A π Figure 4. Weak Λ-decay amplitudes A π and B π determined by Eqs. (6), (7) as a parameter plot of theΛ-decay parameter in the range − ≤ α − ≤
1. The free Λ lifetime is fixed to the experimental value.Different experimental results for α − are marked as indicated in the legend. C. Hypertriton as two-body system
The typical momentum scales of the deuteron and the Λ in the hypertriton are small comparedto the rest masses (see, e.g., Refs. [3, 5]), so they can be treated non-relativistically. Hence singleparticle propagators are given by iS d, Λ ,N ( p , p ) = ip − p M i + i(cid:15) , (8)6ith M i the respective particle masses of the deuteron and the Λ and M N ≡ m the nucleon mass. + + + ... Figure 5. Diagrams contributing to the hypertriton propagator in the effective field theory with deuteron(double line) and the Λ (dashed line) degrees of freedom.
The full propagator of the interacting Λ d system in the dimer picture (cf. [39]) is depicted inFig. 5. Evaluating the geometric series, we obtain the full ”dimer” propagator: iD H ( p , p ) = 2 πµ Λ d g − i − γ Λ d + (cid:114) − µ Λ d (cid:16) p − p M Λ + M d + i(cid:15) (cid:17) , (9)which has a pole at the Λ separation energy of the hypertriton, B Λ . The residue of the pole isthe wave function renormalization Z H ( B Λ ) = π ( µ Λ d g ) √ µ Λ d B Λ . For convenience, we will use thereduced wave function renormalization ¯ Z H = g Z H , where the coupling constant g has beendivided out, in the following sections.We now go on to calculate the weak decay of the hypertriton in the Λ d picture. III. ND CHANNELS
The main contribution to the hypertriton lifetime for small Λ separation energy B Λ is expectedto come from the nucleon-deuteron channels, since in the limit of vanishing B Λ , all other channelsclose. To be precise, we expect Γ (cid:0) H (cid:55)→ π − / + N + d (cid:1) (cid:55)→ B Λ (cid:55)→
0, because theoutgoing states do not correspond to those of a free Λ decay plus a spectator deuteron. Thereforewe need to retrieve the free Λ width in the limit B Λ (cid:55)→ N d channels . At leading order,diagrams with and without a final state interaction between the decay nucleon and the deuteroncontribute, see also Fig. 6. We neglect pionic final state interactions, since the pions are Goldstonebosons which interact weakly. Furthermore, all pionic scattering lengths, measured in pionic atomsor calculated in HB χ PT, are smaller than few percent of the inverse pion mass [40–45] and phaseshifts are still small at the relevant energies [46–48]. Recent calculations indicate that they maychange the result by up to 6 −
9% of the free Λ width [25, 26]. However, this is beyond the leadingorder accuracy of 25%. The final state interaction between the outgoing nucleon and deuteron isdescribed by the scattering amplitude for a shallow bound state with binding momentum γ Nd : A ( E ) = 2 πµ Nd (cid:104) − γ Nd + (cid:112) − µ Nd E − i(cid:15) (cid:105) − , (10)We tune γ Nd such that the correct He binding energy B He with respect to the dp threshold isreproduced. Utilizing the momentum δ function, the width Γ Nd is given byΓ pd =2 π (cid:90) (cid:90) d p (2 π ) d k (2 π ) ω k (cid:88) spins (cid:12)(cid:12) M (cid:0) H (cid:55)→ πpd (cid:1)(cid:12)(cid:12) × δ (cid:32) ∆ − B Λ − ω k − p M d − ( k + p ) m (cid:33) , (11) If the deuteron breakup is also included, decays into three nucleons contribute at threshold as well. Hyp + T Hyp A Figure 6. Decay of the hypertriton into nucleon-deuteron channels. The amplitude A depicts the final stateinteraction between the decay nucleon and the deuteron from Eq. (10). with k the outgoing pion momentum and p the deuteron momentum. The invariant matrix elementis the sum of the diagrams in Fig. 6: M (cid:0) H (cid:55)→ πN d (cid:1) = M Nd + M FSI Nd . It can be most easily seenthat the limit B Λ (cid:55)→ M pd is then given by M pd ( k , p ) = (cid:113) ¯ Z H ( B Λ ) S Λ (cid:18) − B Λ − p M d , − p (cid:19) W k , (12)which is directly related to the normalization of the hypertriton wave function. Therefore theexpression given in Eq. (11) contains a so-called Dirac series in the limit B Λ (cid:55)→ p Λ .Including now final state interactions and moving away from the limit B Λ (cid:55)→ M FSI pd reads S M FSI pd = i (cid:90) d q (2 π ) S Λ ( q , q ) S d ( − B Λ − q , − q ) S p (∆ + q − ω k , q − k ) (cid:124) (cid:123)(cid:122) (cid:125) = I q ( k,B Λ ) × A (cid:18) ∆ − B Λ − ω k − k M (cid:19) (cid:113) ¯ Z H ( B Λ ) W k . (13)The energy shift in the amplitude A is due to the boost of the nucleon-deuteron system in thehypertriton decay. M denotes here the total mass of the pd system.Now we proceed to the evaluation of the integral I q ( k, B Λ ). Due to the energy release at theweak vertex, the nucleon propagator S N has up to two poles in the q loop momentum integrationdepending on the angle between the outgoing pion momentum k and q . We end up with thefollowing expression I q ( k, B Λ ) = 2 mµ dΛ k (2 π ) (cid:90) d q q ln (cid:34) mµ dΛ B Λ + mq − µ Nd qk + µ Nd m k mµ dΛ B Λ + mq + 2 µ Nd qk + µ Nd m k (cid:35) q + ¯ q q − ¯ q (14)with ¯ q = 1 m (cid:112) µ Nd ( − m ( B Λ + ω k − ∆) + k ( µ Nd − m )) , which can be evaluated utilizing the principal value method.The evaluation of the phase space restricts the allowed momenta since the energy delta functionin Eq. (11) depends on the angle between k and p . Evaluating the angular integration between k and p leaves two Heaviside step functions Θ behind, restricting the area of integration. The phase8pace reads ρ ( k, p ) = mkpω k (cid:2) Θ (cid:0) φ + ( k, p ) (cid:1) − Θ (cid:0) φ − ( k, p ) (cid:1)(cid:3) with φ ± ( k, p ) = k m ± kpm + p µ Nd + 2 ( B Λ + ω k − ∆) (15)so that Γ pd = 1(2 π ) (cid:90) (cid:90) d p d k ρ ( k, p ) 12 (cid:88) spins (cid:12)(cid:12) M (cid:0) H (cid:55)→ πpd (cid:1)(cid:12)(cid:12) . (16)For more details see App. A. We emphasize that the phase space integrals are evaluated exactlyand no closure approximation is assumed. IV. HELIUM/TRITON CHANNEL
The second contribution to the hypertriton decay in our theory comes from decays into trinu-cleon final states, i.e., He and H. As before we calculate the decay into He and a charged pionand infer the neutral channel using the ∆ I = 1 / T Hyp He Figure 7. Decay of the hypertriton into He and a charged pion. A similar diagram with an outgoing tritonexists in the neutral decay channel. Γ He phase space looks similar to the free oneΓ He = (cid:90) (cid:90) d p (2 π ) d k (2 π ) ω k (cid:88) spins |M He | (2 π ) δ (3) ( p + k ) δ (cid:18) ∆ − ω k − p M He (cid:19) (17)with ∆ = M H Λ − M He and p is now the momentum of the outgoing He nucleus. ¯ Z He is the He wave function renormalization, constructed in a similar way to the hypertriton one. In fact wecan reuse the calculation for the phase space from the free Λ width together with the loop analysisdone before for the
N d case. We obtainΓ He = G F M π π ¯ kM He M He + ω ¯ k ¯ Z H ( B Λ ) ¯ Z He ( B He ) (cid:32) A π + 19 (cid:18) B π M Λ + m (cid:19) ¯ k (cid:33) (cid:12)(cid:12) I q (cid:0) ¯ k, B Λ (cid:1)(cid:12)(cid:12) . (18)Using relativistic kinematics, the momentum of the outgoing pion is fixed to¯ k = (cid:114)(cid:16) M H + M He − M π (cid:17) − M H M He M H . (19)9 . RESULTSA. Partial decay width and dependence on α − α − A π B π .
642 1 . − . .
721 1 . − . .
750 1 . − . A π and B π for different α − and τ Λ = 263 . In our calculation, we use the free Λ lifetime, τ Λ = 263 . α − tofix the values of the weak couplings A π and B π in Eq. (3). The corresponding couplings for differentinput values of α − discussed in subsection II B are given in Table I. The remaining momentumintegrals in in the expressions for the widths, Eqs. (16) and (18), are evaluated numerically, ex-ploiting the correlation between charged and uncharged decay channels from the ∆ I = 1 / N d final state interaction in the hypertriton decay can be visualizedby plotting the differential rate dΓ pd d k , where k is the final pion momentum for fixed B Λ . Theresult for B Λ = 0 .
13 MeV is depicted in the left panel of Fig. 8. For small pion momenta the k [MeV] d Γ p d d k π (cid:2) M e V − s − (cid:3) / α − = − . α − = − . α − = − .
80 81 8233 . · . . . . . . . . . k [MeV] d Γ P C / P V p d d k π / d Γ p d d k π Figure 8. Left panel: logarithmic plot of the differential rate dΓ pd d k for different values of α − indicated in thelegend at fixed Λ separation energy, B Λ = 0 .
13 MeV. Results including (excluding) final state interactionsare shown by solid (dashed) lines, respectively. The inset shows the small dependence on α − . Right panel:relative contribution of the parity conserving (dashed lines) and parity violating part (solid lines) to the fulldifferential rate. N d final state interactions (solid lines) reduce the differential width by an order of magnitudecompared to the calculation without final state interactions (dashed lines). The new larger Λ10ecay parameter α − shifts the partial widths slightly upwards as shown in the inset of Fig. 8, butthe overall sensitivity is small. It is instructive to consider the parity conserving and violatingparts separately. Indeed, the 17% change in the decay parameter shifts the contribution of theparity conserving part moderately, as indicated in the right panel of Fig. 8. The parity violatingpart gives a smaller contribution over the full range of pion momenta k but shows roughly theopposite behavior. Hence, although the relative contribution of the parity violating term and theparity conserving term change moderately, their sum only changes slightly as seen in the left panelof Fig. 8. This behavior is expected from the scaling behavior of Eqs. (16), (18) with A π and B π .A similar trend is reflected in the partial widths discussed below. B. Width results and comparison with theory and experiment
The results for the different partial widths are summarized in Fig. 9. The two prominentexperimental values for the Λ separation energy, B Λ = (0 . ± .
05) MeV [8] and B Λ = (0 . ± . ± .
11) MeV [9] are indicated by the shaded light (green) and dark (blue) rectangular areas,respectively. The calculated partial widths and ratios are given by the shaded bands explained inthe legend. The bands cover the parameter space − . ≤ α − ≤ − . B Λ the N d channel dominates, since the allowed phase space for the decay intoa bound state is smaller and for B Λ (cid:55)→ B Λ increases, the decay into a trinucleon bound state becomes more and more dominant. Note thatfor B Λ (cid:55)→ ∆ both partial decay widths go to zero as expected and the hypertriton becomes stableagainst the weak decay, since the energy release at the weak vertex would be below the Λ separationenergy.While the full hypertriton width Γ H Λ does only moderately depend on B Λ , and the correlationappears small, the partial widths show a strong dependence. As a consequence, the experimen-tally measured ratio of the partial width into He divided by the partial width into He and pd ,Γ He / (Γ He + Γ pd ), is also very sensitive to B Λ . Hence this quantity appears to be better suitedto determine B Λ indirectly than the total width [2].The partial widths for the Λ separation energies B Λ = 0 .
13 MeV and B Λ = 0 .
41 MeV anddifferent values of are listed in Table II. All partial widths have an uncertainty of 25% from higherorders in the EFT expansion. Standard error propagation leads to an absolute uncertainty of 0.09in the ratio R = Γ He / (Γ He + Γ pd ) given in the second last line of Table II. Our results with α − Observable B Λ = 0 .
13 MeV B Λ = 0 .
41 MeV α − .
642 0 .
721 0 .
750 0 .
642 0 .
721 0 . pd + Γ nd ) / Γ Λ .
629 0 .
636 0 .
640 0 .
438 0 .
446 0 . He + Γ H ) / Γ Λ .
387 0 .
371 0 .
364 0 .
574 0 .
550 0 . H / Γ Λ .
016 1 .
007 1 .
003 1 .
012 0 .
994 0 . He / (Γ He + Γ pd ) 0 .
362 0 .
368 0 .
365 0 .
563 0 .
551 0 . τ H [ps] 259 . . . . . . α − . The results assume theempirical isospin rule . The widths are given as a fraction of the Λ free width corresponding to τ Λ = 263 . compare very well with the result obtained by Ref. [24]. Note that the peak of the differential11 . . . . . . . . . . . . B Λ [MeV] Γ i / Γ Λ Γ Λ Γ nFSINd Γ FSINd Γ He Γ H Γ He / (Γ He + Γ pd ) Figure 9. Partial decay widths Γ i in units of the free Λ width Γ Λ as a function of the Λ separation energy B Λ . The ratio Γ He / (Γ He + Γ pd ) is also shown. The bands show the uncertainty from the Λ weak decayparameter α − . The experimental values B Λ = (0 . ± .
05) MeV[8] and B Λ = (0 . ± . ± .
11) MeV[9] are indicated by the shaded light (green) and dark (blue) rectangular areas, respectively. The EFTuncertainties are discussed in the main text. . decay width is slightly shifted due to the different particle thresholds. Considering only the phasespace it seems reasonable that the width is decreasing for larger B Λ since the available phasespace gets smaller. The result obtained by Congleton [2] is in agreement with the ratio R , whichwas measured before [16, 18, 20, 27]. However, the total width is about 13% higher. Althoughthe decay constant changes by up to 17% compared to the old value α − , the impact on thedecay rates is much smaller for small binding energies B Λ . While the change of the partial decaywidth is in the order of a few percent, the total width changes barely at all. We note that theCoulomb interaction is not included explicitly in this calculation, which might shift the lifetime inthe charged channel. However, part of the Coulomb interaction is included implicitly due to thetuning of γ Nd to reproduce the correct trinucleon binding energy (see Eq. (10)). Our calculationsupports the picture that for small B Λ the lifetime of the hypertriton is mainly determined by thefree Λ lifetime with some small corrections.The results of this work compare differently to the recent heavy ion collision experiments. Ourresults for low binding energy B Λ lie within the error bars of the value close to the free Λ width [14],while other measurements tend to lie lower [10–13]. Despite giving values for the lifetime within12 large range 60 −
400 ps (see also Fig. 2), older emulsion experiments give relatively consistentexperimental values for the branching ratio R = Γ He / (Γ He + Γ pd ) ranging from R = 0 . ± .
07 to0 . ± .
07 [16, 18, 20, 27]. Both values are in agreement with our value R | B Λ =0 .
13 MeV = 0 . ± . B Λ = 0 .
13 MeV, while the ratio R | B Λ =0 .
41 MeV = 0 . ± .
09 comes out much larger, seealso Table II. Further on, this value is larger than the value of R STAR = 0 . ± . ± . R values, our calculationthus favors smaller binding energies up to B Λ = 0 .
20 MeV. Taking into account the uncertaintyin our calculation and the experimental errors for R , however, the recent STAR result B Λ =0 . ± . ± .
11 MeV [9] cannot be excluded.
C. Effects of isospin splitting
A discussed above, we have explicitly calculated the charged pion channels and estimated theneutral pion channels by applying the empirical ∆ I = 1 / M π − = 139 .
57 MeV and neglected the Coulomb repulsion betweenthe deuteron and the proton. To estimate the accuracy of this approximation, we also calculatedthe neutral channels explicitly using the neutral pion mass and the triton binding energy as input.The latter leads to a change in the final state trinucleon binding momentum γ Nd in Eq. (10) ofabout 10%. This change, however, is absorbed completely by kinematic changes and differences inthe masses. Overall, we obtain a shift by 1% upwards for the sum of the channels decaying into adeuteron, while the the width for decay into the trinucleon bound states goes down by about 2%.Hence the correction to the total width is negligibly small ( < . R moves up by4%, resulting in R = 0 .
38. This shift is significantly smaller than the estimated uncertainty of ourleading order calculation.
VI. SUMMARY AND OUTLOOK
In this work, we have investigated the dependence of the hypertriton lifetime on the Λ separationenergy with an pionless EFT with deuteron, nucleon, and Λ degrees of freedom. The validity ofsuch a picture for the low-energy structure of the hypertriton was justified in a recent investigationof the hypertriton structure and matter radii [5], where a three-body framework with pn Λ and atwo-body framework with Λ d degrees of freedom were compared in the context of pionless EFT.The EFT framework allows us to vary the Λ separation energy while keeping all other low-energyconstants constant. The uncertainty in the partial widths from higher-order contributions in theΛ d picture is estimated to be of order 25%. It can be reduced by going beyond the leading orderin the EFT expansion.We focus on the dominant hadronic decay channels with πN d and π -trinucleon final states.These channels make up 97.4% of the total width of the hypertriton [24] and thus provide the keyto understanding the hypertriton lifetime puzzle. We explicitly calculate the decay channels withneutral pions in the final state, evaluating all phase space intergrals exactly. The ∆ I = 1 / I = 1 / d picture in the closure approximation, differs by10% which is well within the EFT uncertainties. We also investigate the impact of recent changes13n the weak decay parameter α − , correcting the previous value by 15% [35, 36]. While there aremoderate changes in the parity conserving and parity violating contributions, the change in thetotal rate is small.For the commonly accepted value of the Λ separation energy, B Λ = (0 . ± .
05) MeV [8], wefind the hypertriton width Γ H = (1 . ... . Λ , depending on the input value for α − , to beclose to the free Λ width. Varying B Λ between zero and 2 MeV, the width increases first andthen decreases, reaching 90% of the free Λ width at B Λ = 2 MeV. This increase is due to theopening of additional decay channels for the hypertriton compared to the free Λ decay, but thewidth eventually has to go down due to the decreasing phase space as B Λ increases, eventuallyvanishing as B Λ approaches ∆ − M π . For physically reasonable values of B Λ , the lifetime ofthe hypertriton is not very sensitive to B Λ . However the partial widths and the experimentallymeasured branching ratio R = Γ He / (Γ He + Γ pd ) depend strongly on the Λ separation energy.Our result of R | B Λ =0 .
13 MeV = 0 . ± .
09 is consistent with the experimental measurements of R [13, 16, 18, 20, 27], which favor small Λ separation energies, B Λ ≤ .
20 MeV. The result for R at the recent STAR value B Λ = 0 . ± . ± .
11 MeV [9], R | B Λ =0 .
41 MeV = 0 . ± .
09, comesout significantly higher. Moreover, this value is significantly larger than the value of R STAR =0 . ± . ± .
08 reported by the STAR collaboration [13]. Taking into account the experimentalerrors and the uncertainty from higher orders in our calculation, we can not exclude the STARresult B Λ = 0 . ± . ± .
11 MeV [9] but there is some tension.An investigation similar in spirit to ours was carried out by P´erez-Obiol et al. [26]. Theycalculated the width for decay into a charged pion and He using NCSM wave functions for thehypertriton and the helion, including final state interactions. Using the ∆ I = 1 / R as input, they determined the full hypertriton width. Varying the Λseparation energy by adjusting the ultraviolet cutoff in the NCSM calculation, they calculatedthe width for different values of B Λ , although the calculations were not fully converged for allconsidered values of B Λ . Their calculation suggests that the STAR values for B Λ and R are fullyconsistent with each other. The slight tension between our calculation and Ref. [26] requires furtherstudy, especially regarding the different dynamical inputs and strategies in the calculations.In the EFT calulation, this requires the inclusion of higher orders. The first correction wouldcome from the Λ d effective range which can be taken from Ref. [5]. In order to calculate thecontribution from the deuteron breakup channel a four-body calculation of the hypertriton decaywith pn Λ π degrees of freedom is required. According to Refs. [25, 26] pionic final state interactionscould affect the width at the 10% level which would also be relevant at next-to-leading order. Hereit might be easier to return to a theory with a fundamental deuteron to reduce complexity. ACKNOWLEDGMENTS
We thank M. G¨obel and W. Elkamhawy for useful discussions. This work was funded bythe Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Projektnummer279384907 - SFB 1245 and the Federal Ministry of Education and Research (BMBF) under con-tracts 05P15RDFN1 and 05P18RDFN1. 14 ppendix A: Calculation details
In order to evaluate the loop integral given in Eq. (13), we perform the q integration with themeans of standard contour integration resulting in an integral containing two factors I q ( k, B Λ ) = (cid:90) d q (2 π ) (cid:20) − B Λ − q µ dΛ (cid:21) − (cid:34) ∆ − B Λ − ω k − q m d − ( q − k ) m (cid:35) − . (A1)Due to the positive energy ∆ and the dependence on q · k the second term has a complex polestructure with up to two poles, which can in principle fall on top of each other, depending on theangle between the loop momentum q and the external momentum of the pion k . In contrast, thefirst term is always negative, and therefore never develops a pole. Hence it is adroit to shift theangular dependence to the first term, leading to I q ( k, B Λ ) = (cid:90) d q (2 π ) (cid:20) − B Λ − q µ dΛ − µ Nd q · k mµ dΛ − µ Nd m µ dΛ k (cid:21) − × (cid:20) ∆ − B Λ − ω k − q µ Nd + µk m − k m (cid:21) − . (A2)The angular integration can now be done independently of the second propagator and one obtainsEq. (14). [1] A. Gal, E. V. Hungerford, and D. J. Millener, Rev. Mod. Phys. , 035004 (2016), arXiv:1605.00557[nucl-th].[2] J. G. Congleton, J. Phys. G18 , 339 (1992).[3] H. W. Hammer, Nucl. Phys.
A705 , 173 (2002), arXiv:nucl-th/0110031 [nucl-th].[4] R. Wirth, D. Gazda, P. Navr´atil, A. Calci, J. Langhammer, and R. Roth, Phys. Rev. Lett. , 192502(2014), arXiv:1403.3067 [nucl-th].[5] F. Hildenbrand and H. W. Hammer, Phys. Rev.
C100 , 034002 (2019), arXiv:1904.05818 [nucl-th].[6] H. Le, J. Haidenbauer, U.-G. Meißner, and A. Nogga, Phys. Lett. B , 135189 (2020),arXiv:1909.02882 [nucl-th].[7] S. R. Beane, E. Chang, S. D. Cohen, W. Detmold, H. W. Lin, T. C. Luu, K. Orginos, A. Parreno, M. J.Savage, and A. Walker-Loud (NPLQCD), Phys. Rev.
D87 , 034506 (2013), arXiv:1206.5219 [hep-lat].[8] M. Juric et al. , Nucl. Phys.
B52 , 1 (1973).[9] J. Adam et al. (STAR), Nature Phys. (2020), 10.1038/s41567-020-0799-7, arXiv:1904.10520 [hep-ex].[10] B. I. Abelev et al. (STAR), Science , 58 (2010), arXiv:1003.2030 [nucl-ex].[11] C. Rappold et al. , Nucl. Phys.
A913 , 170 (2013), arXiv:1305.4871 [nucl-ex].[12] J. Adam et al. (ALICE), Phys. Lett.
B754 , 360 (2016), arXiv:1506.08453 [nucl-ex].[13] L. Adamczyk et al. (STAR), Phys. Rev.
C97 , 054909 (2018), arXiv:1710.00436 [nucl-ex].[14] S. Acharya et al. (ALICE), Phys. Lett.
B797 , 134905 (2019), arXiv:1907.06906 [nucl-ex].[15] M. M. Block, R. Gessaroli, J. Kopelman, S. Ratti, M. Schneeberger, L. Grimellini, T. Kikuchi, L. Lend-inara, L. Monari, W. Becker, and E. Harth, (1964), 10.5170/CERN-1964-001.63.[16] G. Keyes, M. Derrick, T. Fields, L. G. Hyman, J. G. Fetkovich, J. McKenzie, B. Riley, and I. T. Wang,Phys. Rev. Lett. , 819 (1968).[17] R. E. Phillips and J. Schneps, Phys. Rev. , 1307 (1969).[18] G. Keyes, M. Derrick, T. Fields, L. G. Hyman, J. G. Fetkovich, J. Mckenzie, B. Riley, and I. T. Wang,Phys. Rev. D1 , 66 (1970).[19] G. Bohm et al. , Nucl. Phys. B16 , 46 (1970), [Erratum: Nucl. Phys.B16,523(1970)].[20] G. Keyes, J. Sacton, J. H. Wickens, and M. M. Block, Nucl. Phys.
B67 , 269 (1973).[21] M. Tanabashi et al. (Particle Data Group), Phys. Rev.
D98 , 030001 (2018).
22] M. Rayet and R. H. Dalitz, (1966), [Nuovo Cim.A46,786(1966)].[23] B. Ram and W. Williams, Nucl. Phys.
B28 , 566 (1971).[24] H. Kamada, J. Golak, K. Miyagawa, H. Witala, and W. Gloeckle, Phys. Rev.
C57 , 1595 (1998),arXiv:nucl-th/9709035 [nucl-th].[25] A. Gal and H. Garcilazo, Phys. Lett.
B791 , 48 (2019), arXiv:1811.03842 [nucl-th].[26] A. P´erez-Obiol, D. Gazda, E. Friedman, and A. Gal, (2020), arXiv:2006.16718 [nucl-th].[27] M. Block, in
Sienna International Conference on Elementary Particles , edited by G. Bernandini andG. Puppi (Societa de Fisicia, 1963) p. 62.[28] J. Golak, K. Miyagawa, H. Kamada, H. Witala, W. Gloeckle, A. Parreno, A. Ramos, and C. Bennhold,Phys. Rev.
C55 , 2196 (1997), [Erratum: Phys. Rev.C56,2892(1997)], arXiv:nucl-th/9612065 [nucl-th].[29] A. P´erez-Obiol, A. Nogga, and D. R. Entem,
Proceedings, 12th International Conference on Hyper-nuclear and Strange Particle Physics (HYP 2015): Sendai, Japan, September 7-12, 2015 , JPS Conf.Proc. , 022002 (2017).[30] S. R. Beane, P. F. Bedaque, W. C. Haxton, D. R. Phillips, and M. J. Savage, “At the frontier ofparticle physics—handbook of qcd,” (World Scientific, 2001) Chap. From hadrons to nuclei: crossingthe border, pp. 133–271.[31] P. F. Bedaque and U. van Kolck, Ann. Rev. Nucl. Part. Sci. , 339 (2002).[32] E. Epelbaum, H.-W. Hammer, and U.-G. Meissner, Rev. Mod. Phys. , 1773 (2009).[33] H. W. Hammer, C. Ji, and D. Phillips, J. Phys. G , 103002 (2017), arXiv:1702.08605 [nucl-th].[34] H.-W. Hammer, S. K¨onig, and U. van Kolck, Rev. Mod. Phys. , 025004 (2020), arXiv:1906.12122[nucl-th].[35] M. Ablikim et al. (BESIII), Nature Phys. , 631 (2019), arXiv:1808.08917 [hep-ex].[36] D. G. Ireland, M. D¨oring, D. I. Glazier, J. Haidenbauer, M. Mai, R. Murray-Smith, and D. R¨onchen,Phys. Rev. Lett. , 182301 (2019), arXiv:1904.07616 [nucl-ex].[37] H.-W. Hammer, Talk at workshop ”Origin of nuclear clusters in hadronic collisions”, CERN, May 19-20,2020, https://indico.cern.ch/event/893621/timetable/ .[38] B. R. Holstein, Weak interactions in nuclei (1985).[39] E. Braaten and H. W. Hammer, Phys. Rept. , 259 (2006), arXiv:cond-mat/0410417 [cond-mat].[40] V. Bernard, N. Kaiser, and U. G. Meissner, Phys. Rev. C , 2185 (1995), arXiv:hep-ph/9506204.[41] N. Fettes, U.-G. Meissner, and S. Steininger, Nucl. Phys. A , 199 (1998), arXiv:hep-ph/9803266.[42] P. Hauser et al. , Phys. Rev. C , 1869 (1998).[43] U.-G. Meissner, U. Raha, and A. Rusetsky, Eur. Phys. J. C , 213 (2005), [Erratum: Eur.Phys.J.C45, 545 (2006)], arXiv:nucl-th/0501073.[44] I. Schwanner, G. Backenstoss, W. Kowald, L. Tauscher, H. Weyer, D. Gotta, and H. Ullrich, Nucl.Phys. A , 253 (1984).[45] S. Beane, V. Bernard, E. Epelbaum, U.-G. Meissner, and D. R. Phillips, Nucl. Phys. A , 399(2003), arXiv:hep-ph/0206219.[46] L. D. Roper, R. M. Wright, and B. T. Feld, Phys. Rev. , B190 (1965).[47] D. Brayshaw and E. Ferreira, Phys. Lett. B , 139 (1977).[48] J. Arvieux and A. Rinat, Nucl. Phys. A , 205 (1980)., 205 (1980).