$\Xi$-nuclear constraints from $\Xi^-$ emulsion capture events

All five KEK and J-PARC two-body $\Xi^-$+$^A$Z $\to$ $^{A'}_{\Lambda}$Z'+$^{A''}_{\Lambda}$Z'' capture events in light emulsion nuclei, including KISO and IBUKI in $^{14}$N, are consistent with Coulomb-assisted $1p_{\Xi^-}$ nuclear states. The underlying $\Xi$-nuclear potential is strongly attractive, with nuclear-matter depth $V_{\Xi}$ larger than 20 MeV. The recent $^{14}$N capture events KINKA and IRRAWADDY assigned by J-PARC E07 to $1s_{\Xi^-}$ nuclear states, and implying considerably shallower $V_{\Xi}$, have also another interpretation as $1p_{\Xi^0}$ nuclear states.


Experiment Event
where µ is the Ξ − -nucleus reduced mass and the complex strength parameters b 0 and b 1 are effective, generally density dependent ΞN isoscalar and isovector c.m. scattering amplitudes respectively. The density ρ = ρ n + ρ p is a nuclear density distribution normalized to the number of nucleons A and ρ exc = ρ n − ρ p is a neutron-excess density with ρ n = (N/Z)ρ p , implying that ρ exc = 0 for the N = Z emulsion nuclei 12 C and 14 N considered here. A finite-size Coulomb potential V c , including vacuumpolarization terms is added. For densities we used mostly harmonic-oscillator (HO) densities [16] where the r.m.s. radius of ρ p was set equal to that of the nuclear charge density [17]. Folding reasonably chosen ΞN interaction ranges other than corresponding to the proton charge radius, or using Modified Harmonic Oscillator densities, or replacing HO densities by realistic three-parameter Fermi density distributions, made little difference: all the calculated binding energies changed by a small fraction, about 0.03 MeV, of the uncertainty imposed by the ±0. 15 MeV experimental uncertainty of the 0.82 MeV 1p Ξ − binding energy in 12 C listed in Table 1. This holds also for adding a ρ exc 0 term induced by considering realistic differences of neutron and proton r.m.s. radii. Accepting the binding energy interval B 1p Ξ − =0.82±0.15 MeV for the two KEK E176 events listed in Table 1, a Ξ-nuclear potential strength of Re b 0 = 0.32 ± 0.01 fm follows for a fixed value Im b 0 = 0.01 fm. The sensitivity to variations of Im b 0 is minimal: choosing Im b 0 = 0.04 fm [7] instead of 0.01 fm increases Re b 0 by 0.01 fm to 0.33 ± 0.01 fm. The value Re b 0 = 0.32 ± 0.01 fm implies in the limit A → ∞ and ρ(r) → ρ 0 =0.17 fm −3 a depth value V Ξ = 24.3 ± 0.8 MeV in nuclear matter, compatible with that derived from AGS-E906 in Ref. [6] and in agreement with the range of values 21-24 MeV extracted from old emulsion events [18].
So far we have discussed a density independent t-matrix element b 0 in V opt , Eq. (1), to fit the Ξ − capture events in 12 C from Table 1. To explore how robust the deduced Ξ potential-depth value V Ξ = 24.3 ± 0.8 MeV is, we introduce the next to leading-order density dependence of V opt , replacing Re b 0 in Eq. (1) by where k F is the Fermi momentum corresponding to nuclear density ρ and b lab 0 = (1 + m Ξ − m N )b 0 is the lab transformed form of the c.m. scattering amplitude b 0 . Eq. (2) accounts for Pauli exclusion correlations in ΞN in-medium multiple scatterings [19,20]. Shorter-range correlations, disregarded here, were shown in Ref. [21] to contribute less than ∼30% of the long-range Pauli correlation term. Applying Eq. (2) in the present context, B 1p Ξ − ( 12 C)=0.82 MeV is refitted by Re b 0 =0.527 fm. The nuclear-matter Ξ-nuclear potential depth V Ξ decreases from 24.3±0.8 to 21.9±0.7 MeV, a decrease of merely 10%.

1p Ξ − states in 14 N
Applying Eqs. (1,2) to 14 26 MeV, Pauli correlations included. This binding energy is considerably higher than the value B Ξ − =1.15±0.20 MeV obtained from the three events assigned in Table 1 to Ξ − capture in 14 N. To resolve this apparent dicrepancy, we note that the calculated B 1p Ξ − ( 14 N) corresponds to a (2Λ + 1)-average of binding energies for a triplet of states Λ π = (0 − , 1 − , 2 − ) obtained by coupling a 1p Ξ − state to J π ( 14 N g.s. )=1 + , as shown in Fig. 1. (3). The (2Λ + 1)-averaged energy −1.96 MeV was calculated using the same Pauli-corrected optical potential parameter b 0 that yields a 12 C g.s. + 1p Ξ − state at −0.82 MeV, corresponding to the Ξ − capture events in 12 C listed in Table 1. The energy splittings marked in Fig. 1 follow from a shell-model quadrupole-quadrupole spin independent residual interaction V ΞN , where F (2) is the corresponding Slater integral. A representative value of F (2) ΞN = −3 MeV is used here, smaller than the value F (2) ΛN = −3.7 MeV established empirically for p-shell Λ hypernuclei [22], in accordance with a ΞN strong interaction somewhat weaker than the ΛN strong interaction. A single 3 D 1 14 N g.s. component providing a good approximation to the full intermediate-coupling g.s. wavefunction [23] was assumed in the present evaluation. Fig. 1 shows a triplet of 14 N g.s. + 1p Ξ − levels, spread over more than 1 MeV. The least bound of these states, Λ π = 0 − , is shifted upward by 0.84 MeV from the (2Λ + 1) averaged position at −1.96 ± 0.26 MeV to E(0 − ) = −1.12 ± 0.26 MeV. This is consistent with the averaged position E = −1.15 ± 0.20 MeV of the three Ξ − 14 N g.s. capture events listed in Table 1. The Λ π = 0 − state assumes spin-parity J π = 1 2 − when Pauli-spin s Ξ − = 1 2 is introduced, but its position is unaffected by spin dependent ΞN residual interactions in leading order. We are not aware of any good reason why capture has not been seen from the other two states with Λ π = 1 − , 2 − . This may change when more events are collected at the next stage of the ongoing J-PARC E07 emulsion experiment.

1s Ξ − states in 1N?
In addition to the Ξ − 1p -14 N capture events listed as KISO and IBUKI in Table 1, the J-PARC lightnuclei emulsion experiment E07 reported also two other events KINKA and IRRAWADDY, assigned as Ξ − 1s -14 N states, see Fig. 2. We note that 2P → 1S radiative decay rates are of order 1% of 3D → 2P radiative decay rates [13,14] suggesting that Ξ − capture from a nuclear Ξ − 1s -14 N state is suppressed to this order relative to capture from a nuclear Ξ − 1p -14 N state. Assigning a Ξ − 1s -14 N bound state to IRRAWADDY, and by default also KINKA which-given its large uncertainty-is not inconsistent with IRRAWADDY, is therefore questionable.   It has been conjectured by us recently [24] that IRRAWADDY is a near-threshold Ξ 0 1p -14 C bound state that has nothing to do with a Ξ − 1s -14 N bound state suggested by E07. The mixing induced by the ΞN strong interaction between a Ξ − 1p -14 N bound state identified with IBUKI and a Ξ 0 1p -14 C bound state lying about 5 MeV below IBUKU, within the J-PARC E07 experimental uncertainty of IRRAWADDY, is sufficiently strong to make the E1 radiative deexcitation of the Ξ − 3D -14 N atomic state populate equally well both Ξ − 1p -14 N and Ξ 0 1p -14 C nuclear states.

Discussion
Two Ξ-nuclear scenarios are listed in Table 2. In the first one, two KEK-E176 12 C events [8], with B 1p Ξ − = 0.82 ± 0.15 MeV, serve as input for setting up the strength of the Ξ-nuclear optical potential. Other Ξ − binding energies are then predicted, as listed in the first row of Table 2. We note that a value V Ξ 20 MeV implies a substantially stronger in-medium ΞN attraction than reported by some recent free-space model evaluations (HAL-QCD [25], EFT@NLO [26,27] and RMF [28]), all of which satisfy V Ξ 10 MeV. A notable exception is provided by versions ESC16*(A,B) of the latest Table 2. Ξ − -12 C and Ξ − -14 N binding energies in 1s and 1p states, B 1s Ξ − and B 1p Ξ − , plus Ξ nuclear potential depths V Ξ (ρ 0 ) at nuclear-matter density ρ 0 = 0.17 fm −3 calculated using a density-dependent optical potential Eqs. (1,2) with Re b 0 fitted to binding energies underlined for each input choice. All entries are in MeV.
9.82 0.82 11.78 1.96 21.9 J-PARC E07 [12] 4.94 0.31 6.27 0.50 13.8 Nijmegen extended-soft-core ΞN interaction model [29], in which values of V Ξ higher than 20 MeV are derived. However, these large values are reduced substantially by ΞNN three-body contributions within the same ESC16* model. Choosing instead the J-PARC E07 14 N IRRAWADDY event, with B 1s Ξ − = 6.27 ± 0.27 MeV [12] as input, gives rise to different predictions as listed in the second row of the table. The difference between the two sets of predictions is striking, particularly for the Ξ − 1s binding energies. This large difference is reflected also in the Ξ-nuclear potential depths at nuclear matter density, V Ξ (ρ 0 ), listed in the last column of the table. Equally interesting is the difference between the two sets with regard to Ξ − 1p bound states. In particular, the Ξ − 1p -12 C binding energy constrained by B 1s Ξ − ( 14 N)=6.27±0.27 MeV comes out in the second row exceedingly small, substantially disagreeing with that determined from the two KEK E176 capture events [8] underlined in the first row. As for the calculated Ξ − 1p -14 N binding energies, since J( 14 N) 0, see Fig. 1, these cannot be compared directly with IBUKI's binding energy of 1.27±0.21 MeV from Fig. 2.
Recent Skyrme-Hartree-Fock (SHF) calculations [30] presented global fits to comprehensive Ξnuclear data, including B 1s Ξ − =8.00 MeV for KINKA and B 1p Ξ − =1.13 MeV for the mean of KISO and IBUKI. Apart from small nonlocal potential terms and effective mass corrections, the SHF Ξ-nuclear mean-field potential V Ξ (ρ N ) consists of two terms: Large-scale SHF fits of the corresponding Ξ potential depths, and their sum, are listed in the first row of Table 3. Listed in the lower rows are Ξ potential depths obtained in the optical potential methodology [31] when fitting just the two Ξ − states in 14 N as used in the SHF calculations [30]. Good agreement is observed between the two methods, with V Ξ (ρ 0 ) ≈ 14 MeV, similar to the depth value listed in the second row of Table 2 using IRRAWADDY alone. We note that the Ξ − 1p state in 12 C (see Table 2) comes out unbound in the SHF calculations unless 12 C is made artificially deformed [30]. Table 3. Ξ-nuclear potential depths (in MeV) from a large-scale SHF fit [30] and from our V opt two-parameter fits to just B 1s The solution proposed here to the difficulty of interpreting IRRAWADDY as a Ξ − 1s bound state in 14 N is by pointing out that it could correspond to a Ξ 0 1p -14 C bound state, something that cannot occur kinematically in the other light-emulsion nuclei 12 C and 16 O. Given that in this nuclear mass range capture rates from 1s Ξ − states are estimated to be two orders of magnitude below capture rates from 1p Ξ − states [13,14], this Ξ 0 1p -14 C new assignment addresses satisfactorily the capture rate hierarchy.