# Has J-PARC E07 observed a $\Xi^{-}_{1s}$ nuclear state?

E. Friedman  A. Gal  Racah Institute of Physics, The Hebrew University, 91904
Jerusalem, Israel

###### Abstract

The Strangeness ${\cal S}=-2$ J-PARC E07 emulsion experiment published
recently two new $\Xi^{-}$–14N capture events, IRRAWADDY and IBUKI, identified
by observing weak-decay sequences of pairs of single-$\Lambda$ hypernuclei and
interpreted as $\Xi^{-}_{1s}$ and $\Xi^{-}_{1p}$ nuclear bound states,
respectively, with binding energies $B_{\Xi^{-}}^{1s}$=6.27$\pm$0.27 MeV and
$B_{\Xi^{-}}^{1p}$=1.27$\pm$0.21 MeV. $\Xi^{-}$ capture events in emulsion
play a major role in determining the $\Xi$-nuclear and the $\Lambda\Lambda$
potential strengths. Here we question the assignment of a $\Xi^{-}_{1s}$
nuclear bound state to IRRAWADDY and offer an alternative assignment as a
near-threshold $\Xi^{0}_{1p}$–14C nuclear bound state, slightly admixed with a
$\Xi^{-}_{1p}$–14N nuclear bound state component corresponding to IBUKI. We
also question using IBUKI’s $B_{\Xi^{-}}^{1p}$ value as is to determine the
$\Xi$-nuclear potential depth. Altogether, $\Xi^{-}$ capture events in 12C and
16O should prove less ambiguous than in 14N.

###### keywords:

$\Xi$ nuclear bound states.

##  1 Introduction

Experimental input to deciphering the Strangeness ${\cal S}=-2$ sector of
hypernuclear physics comes mostly from forming $\Xi^{-}$ hyperons by a
$(K^{-},K^{+})$ reaction on nuclear targets and stopping them in light-nuclei
emulsion [1]. Data collected in the last 30 years by KEK emulsion experiments
E176 [2] and E373 [3, 4] have advanced our understanding of $\Lambda\Lambda$
and $\Xi N$ nuclear dynamics [5]. New J-PARC E07 $\Xi^{-}$ capture data, some
of which were presented recently [6, 7, 8], may advance it further.

Because of the large momentum transfer in the $(K^{-},K^{+})$ nuclear reaction
which converts protons to $\Xi^{-}$ hyperons, only very few of the produced
high-energy $\Xi^{-}$ hyperons slow down sufficiently in the surrounding
nuclear emulsion to undergo Auger process, form high-$n$ atomic level, and
cascade down radiatively. In light-nuclei emulsion, strong-interaction
$\Xi^{-}p\to\Lambda\Lambda$ capture takes over radiative cascade beginning at
a $3D$ atomic state bound by 126, 175, 231 keV in C, N, O, respectively,
although the $3D$ strong-interaction shift is still small, less than 1 keV
[9]. Capture events are recorded by observing $\Lambda$ hyperons or
$\Lambda$-hypernuclear decay products. Interestingly, several observed decay
sequences of twin (pairs of) single-$\Lambda$ hypernuclei were found to follow
capture from a lower $\Xi^{-}$ orbit, namely Coulomb-assisted $1p_{\Xi^{-}}$
nuclear states in 12C and 14N [2, 4, 7].

Table 1: Reported two-body $\Xi^{-}$ capture events in 14N, $\Xi^{-}$+14N $\to$ ${}^{A^{\prime}}_{~{}\Lambda}$Z’+${}^{A^{\prime\prime}}_{~{}\Lambda}$Z”, to twin $\Lambda$ hypernuclei, some in ground states, some in specific excited states marked by asterisk. Of these, only IBUKI is uniquely assigned. Fitted $\Xi^{-}$ binding energies $B_{\Xi^{-}}^{1p}$ are listed. Experiment | Event |  ${}^{A^{\prime}}_{\Lambda}$Z’+${}^{A^{\prime\prime}}_{\Lambda}$Z” |  $B_{\Xi^{-}}^{1p}$ (MeV)
---|---|---|---
KEK E176 [2] | 14-03-35 |  ${}^{3}_{\Lambda}$H+${}^{12}_{~{}\Lambda}$B | 1.18$\pm$0.22
KEK E373 [4] | KISO |  ${}^{5}_{\Lambda}$He+${}^{10}_{~{}\Lambda}$Be∗ | 1.03$\pm$0.18
J-PARC E07 [7] | IBUKI |  ${}^{5}_{\Lambda}$He+${}^{10}_{~{}\Lambda}$Be | 1.27$\pm$0.21

The J-PARC light-nuclei emulsion experiment E07 reported two $\Xi^{-}$ capture
events in 14N, uniquely identified by their subsequent single-$\Lambda$
hypernuclear decay sequences, and with sufficiently small binding-energy
uncertainty (say $\lesssim 0.3$ MeV) to become potentially informative. One
event, named IBUKI, was assigned as a $\Xi^{-}_{1p}$–14N bound state, with
binding energy $B_{\Xi^{-}}^{1p}$ consistent with those found in KEK
experiments E176 and E373 for $\Xi^{-}$–14N capture events listed in Table 1.
All three listed $B_{\Xi^{-}}^{1p}$ values are slightly above 1 MeV,
significantly higher than the purely-Coulomb $2P$ atomic binding energy value
of about 0.4 MeV in 14N, thereby corresponding to a Coulomb-assisted
$1p_{\Xi^{-}}$ nuclear state in 14N that evolves from a $2P$ atomic state.
Unfortunately, as argued in our recent work [10], it is just the highest of
five $\Xi^{-}_{1p}$–14Ng.s. configuration states with energy spread of $\sim
1.3$ MeV, which makes IBUKI unfit to determine accurately the $\Xi$-nuclear
potential strength.

Figure 1: $\Xi^{-}_{1s}$ and $\Xi^{-}_{1p}$ nuclear states in 14N, assigned in
KEK-E373 and J-PARC E07 emulsion experiments by interpreting $\Xi^{-}$ capture
events that lead to observed twin-$\Lambda$ hypernuclear decays. Figure
provided by Dr. K. Nakazawa, based on recent results from Refs. [4, 7, 8].

The other event, named IRRAWADDY [8], was assigned as a $\Xi^{-}_{1s}$ state.
Its accurately determined binding energy of 6.27$\pm$0.27 MeV is not confirmed
reliably by any event reported so far. Figure 1 focuses on IBUKI and IRRAWADDY
along with two other similar $\Xi^{-}$–14N states, KISO [4] and KINKA [8],
respectively, deduced by revisiting KEK-E373 capture events. Whereas assigning
the upper of the two KISO candidates in Fig. 1 to a $\Xi^{-}_{1p}$ state is
compatible with the same IBUKI assignment discussed above, each of the two
energy intervals defining the KINKA candidate is too broad to constrain
meaningfully the assignment of IRRAWADDY as a $\Xi^{-}_{1s}$ bound state in
14N.

Given the relatively small experimental uncertainty in the reported
IRRAWADDY’s binding energy, it is of prime interest and importance to study it
in some detail. We note that $2P\to 1S$ radiative decay rates are of order 1%
of $3D\to 2P$ radiative decay rates [11, 12] suggesting that $\Xi^{-}$ capture
from a nuclear $\Xi^{-}_{1s}$–14N state is suppressed to this order relative
to capture from a nuclear $\Xi^{-}_{1p}$–14N state. Assigning a
$\Xi^{-}_{1s}$–14N bound state to IRRAWADDY is therefore questionable. Here we
explore a different assignment, as follows.

Little attention if any was given to mixing between $\Xi^{-}$ states in 14N
and $\Xi^{0}$ states in 14C, induced by the $\Xi^{-}p\leftrightarrow\Xi^{0}n$
strong-interaction charge exchange. Nuclear ground-state mass differences for
$Z\to(Z-1)$ isobars in the main light-nuclei emulsion elements 12C, 14N and
16O are listed in Table 2. Note that the nuclear mass difference $M(^{14}{\rm
C})-M(^{14}{\rm N})$ is smaller than the $\Xi$ hyperon mass difference
$m_{\Xi^{-}}-m_{\Xi^{0}}=6.85\pm 0.21$ MeV [14]. This means that $\Xi^{0}$–14C
bound states, if and when exist, lie always below the $\Xi^{-}$–14N threshold.
To be more precise, the $\Xi^{0}$–14C threshold for a virtual
14N($\Xi^{-},\Xi^{0}$)14C charge-exchange reaction is at $\Xi^{-}$–14N energy
of

$-(6.85\pm 0.21)+0.67=-(6.18\pm 0.21)~{}{\rm MeV}.$ (1)

Remarkably, the $\Xi^{0}$–14C threshold energy in $\Xi^{-}$–14N, Eq. (1), is
within the reported IRRAWADDY bound-state energy interval $-$(6.27$\pm$0.27)
MeV.

Table 2: Ground-state nuclear mass differences (in MeV) for $Z\to Z-1$ isobars of the main light-nuclei emulsion elements 12C, 14N and 16O [13]. $M(^{12}$B)–$M(^{12}$C) |  $M(^{14}$C)–$M(^{14}$N) |  $M(^{16}$N)–$M(^{16}$O)
---|---|---
13.88 | 0.67 | 10.93

The aim of the present note is to show that this remarkable coincidence may
not be accidental, and that IRRAWADDY is likely to stand for a near-threshold
$\Xi^{0}_{1p}$–14C bound state that has nothing to do with a
$\Xi^{-}_{1s}$–14N bound state suggested by E07. To do so, we first discuss in
Sect. 2 conditions for a near-threshold $\Xi^{0}_{1p}$–14C bound state using a
$\Xi$-nuclear potential constrained by our recent work [10] on
$\Xi^{-}_{1p}$–12C capture events. We then evaluate in Sect. 3 the mixing
induced by the $\Xi N$ strong interaction between a $\Xi^{-}_{1p}$–14N bound
state identified with J-PARC E07 IBUKI and a $\Xi^{0}_{1p}$–14C bound state
lying about 5 MeV below IBUKU, within the J-PARC E07 experimental uncertainty
of IRRAWADDY. Provided such mixing is not prohibitively small, $E1$ radiative
deexcitation of the $\Xi^{-}_{3D}$–14N atomic state equally well to both
$\Xi^{-}_{1p}$–14N and $\Xi^{0}_{1p}$–14C nuclear states is possible. And
indeed we find it to be the case, thereby motivating the title of this note:
has J-PARC E07 observed a $\Xi^{-}_{1s}$ nuclear state? A brief Discussion and
Summary Sect. 4. ends the paper.

##  2 $\Xi^{0}_{1p}$–14C binding

Following the optical model methodology introduced in our recent work on low-
energy $\Xi^{-}$–nuclear interactions [10], the strong-interaction optical
potential used here is of the form

$V_{\rm
opt}(r)=-\frac{2\pi}{\mu}(1+\frac{A-1}{A}\frac{\mu}{m_{N}})[b_{0}(\rho)\,\rho(r)+b_{1}\,\rho_{\rm
exc}(r)].$ (2)

Here $\mu$ is the $\Xi^{-}$-nuclear reduced mass, and $b_{0}$ and $b_{1}$ are
effective $\Xi N$ c.m. scattering amplitudes related to the isoscalar $V_{0}$
and isovector $V_{\tau}$ components, respectively, of the two-body $s$-wave
$\Xi N$ interaction $V_{\Xi N}$,

$V_{\Xi
N}=V_{0}+V_{\sigma}{\vec{\sigma}}_{\Xi}\cdot{\vec{\sigma}}_{N}+V_{\tau}{\vec{\tau}}_{\Xi}\cdot{\vec{\tau}}_{N}+V_{\sigma\tau}{\vec{\sigma}}_{\Xi}\cdot{\vec{\sigma}}_{N}\,{\vec{\tau}}_{\Xi}\cdot{\vec{\tau}}_{N}\,,$
(3)

with $V$s functions of $r_{\Xi N}$. The density $\rho=\rho_{n}+\rho_{p}$ in
Eq. (2) is a nuclear density distribution normalized to $A$, and $\rho_{\rm
exc}=\rho_{n}-\rho_{p}$ is a neutron-excess density with
$\rho_{n}=(N/Z)\rho_{p}$, implying that $\rho_{\rm exc}=0$ for the $N=Z$
emulsion nuclei 12C, 14N and 16O. The r.m.s radius of $\rho_{p}$ was set equal
to that of known nuclear charge densities, as discussed in our earlier work
[10]. Finally, the density-dependent $b_{0}(\rho)$ is given by

$b_{0}(\rho)=\frac{{\rm Re}\,b_{0}}{1+\frac{3k_{F}}{2\pi}{\rm Re}\,b_{0}^{\rm
lab}}+{\rm
Im}\,b_{0},\,\,\,\,\,\,\,\,\,\,k_{F}=(3{\pi}^{2}\rho/2)^{\frac{1}{3}},$ (4)

with Fermi momentum $k_{F}$ and $b_{0}^{\rm
lab}=(1+\frac{m_{\Xi^{-}}}{m_{N}})b_{0}$. Eq. (4) accounts for Pauli
correlations in $\Xi N$ in-medium multiple scatterings [15].

Using $b_{0}=(0.527+{\rm i}0.010)$ fm from fitting $B^{1p}_{\Xi^{-}}=0.82$ MeV
in 12C [10], together with $b_{1}=0$, failed to produce a $\Xi^{0}_{1p}$–14C
bound state. This agrees with $\Xi^{-}_{1p}$–12C being a Coulomb-assisted
bound state, unbound without the $\Xi^{-}$-nuclear Coulomb potential $V_{C}$.
However, a likely source of additional attraction for $\Xi^{0}_{1p}$–14C, with
$N\neq Z$, is provided by the $\Xi N$ strong-interaction $V_{\tau}$ isovector
component in terms of $b_{1}\neq 0$. Indeed, varying $b_{1}$ between 0.360 to
0.420 fm produces $\Xi^{0}_{1p}$–14C binding energy of 10.6 to 178 keV,
comfortably within IRRAWADDY’s binding energy uncertainty. These $b_{1}$
values, relative to $b_{0}$, are of the same sign and order of magnitude
expected from the HAL-QCD lattice derivation of $V_{\Xi N}$ [16] and from the
chiral EFT calculation at NLO of $\Xi$ in nuclear matter [17]. The
$\Xi^{0}_{1p}$ radial wavefunctions generated in this way reach a maximum
value between 2.54 to 2.58 fm. Interestingly, the $\Xi^{-}_{1p}$–14N Coulomb-
assisted bound state wavefunction generated for the same value of $b_{0}$, but
$b_{1}=0$ as appropriate to the $T=0$ 14N, reaches its maximum at 2.62 fm,
overlapping to better than 0.98 with any of the $\Xi^{0}_{1p}$–14C
wavefunctions in the range considered above.

##  3 $\Xi^{0}_{1p}$–14C $\leftrightarrow$ $\Xi^{-}_{1p}$–14N mixing

To evaluate the $\Xi^{-}_{1p}$–14N to $\Xi^{0}_{1p}$–14C mixing amplitude we
use realistic shell-model wavefunctions for 14Ng.s. and 14Cg.s. [18]:

$|{{}^{14}{\rm
C}}(J^{\pi}=0^{+};T=1)\rangle=0.7729\,{{}^{1}S}+0.6346\,{{}^{3}P}\,,$ (5)
$|{{}^{14}{\rm
N}}(J^{\pi}=1^{+};T=0)\rangle=0.1139\,{{}^{3}S}+0.2405\,{{}^{1}P}+0.9639\,{{}^{3}D}\,,$
(6)

where the ${}^{2S+1}L$ spin-orbital basis wavefunctions refer to the two-hole
$(1p_{N})^{-2}$ configuration in the nuclear $p$-shell. A check on the $S$ and
$P$ amplitudes is provided by the near vanishing of the weak-interaction
Gamow-Teller (GT) matrix element $\langle{{}^{14}{\rm
N}}(1^{+};0)|{\sum\vec{\sigma}_{i}\tau^{+}_{i}}|{{}^{14}{\rm
C}}(0^{+};1)\rangle$ responsible for the extremely long lifetime of
14C${}_{\rm g.s.}(0^{+};1)$ in its $\beta$-decay to 14N${}_{\rm
g.s.}(1^{+};0)$. The near vanishing of this GT matrix element occurs owing to
a strong cancelation between the ($\Delta L$=0, $\Delta S$=1)
${}^{3}S_{1}\leftrightarrow{{}^{1}{S_{0}}}$ and
${}^{1}P_{1}\leftrightarrow{{}^{3}{P_{0}}}$ GT transitions, persisting also
for the same ($\Delta L$=0, $\Delta S$=1) transitions induced by the strong-
interaction $V_{\sigma\tau}$ in mixing $\Xi^{-}_{1s}$–14N$(1^{+};0)$ with
$\Xi^{0}_{1s}$–14C$(0^{+};1)$. However, the $\Xi^{-}_{1p}$–14N$(1^{+};0)$ to
$\Xi^{0}_{1p}$–14C$(0^{+};1)$ mixing is governed by a normal-strength $\Delta
L$=2 matrix element that couples the dominant ${}^{3}D_{1}$ amplitude in
14N${}_{\rm g.s.}(1^{+};0)$ to the ${}^{1}S_{0}$ and ${}^{3}P_{0}$ amplitudes
in 14C${}_{\rm g.s.}(0^{+};1)$.

Table 3: Coefficients of $F^{(2)}_{\Xi N}(V_{\sigma\tau})$ in mixing matrix elements between two $J^{\pi}={\frac{1}{2}}^{-}$ states: $\Xi^{0}_{1p}$–14C$(0^{+};1)$ and the $\Xi^{-}_{1p}$–14N$(1^{+};0)$ IBUKI state. Entries are labeled by ${}^{2S+1}L$ components of $A=14$ wavefunctions given by Eqs. (5) and (6). ${}^{2S+1}L$ |  ${}^{3}S$ |  ${}^{1}P$ |  ${}^{3}D$
---|---|---|---
${}^{1}S$ | – | – | $\frac{4{\sqrt{2}}}{5{\sqrt{5}}}$
${}^{3}P$ | – | $\frac{2{\sqrt{2}}}{5{\sqrt{3}}}$ | $\frac{6}{5{\sqrt{5}}}$

The $\Xi^{-}_{1p}$–14N$(1^{+};0)$ configuration consists of five states: two
$J^{\pi}={\frac{1}{2}}^{-}$ states, two $J^{\pi}={\frac{3}{2}}^{-}$ states and
one $J^{\pi}={\frac{5}{2}}^{-}$ state. The IBUKI state, judging by its energy,
is a $J^{\pi}={\frac{1}{2}}^{-}$ state specified by coupling
$s_{\Xi^{-}}=\frac{1}{2}$ Pauli spin to ${\cal L}^{\pi}=0^{-}$, where
$\vec{\cal L}={\vec{J}}_{N}+{\vec{\ell}}_{\Xi}$, see Fig. 2 in Ref. [10].
Things are simpler for the $\Xi^{0}_{1p}$–14C$(0^{+};1)$ configuration where a
single $J^{\pi}={\frac{1}{2}}^{-}$ state is obtained by coupling
$s_{\Xi^{0}}=\frac{1}{2}$ Pauli spin to ${\cal L}^{\pi}=1^{-}$. Mixing matrix
elements of
$V_{\sigma\tau}{\vec{\sigma}}_{\Xi}\cdot{\vec{\sigma}}_{N}\,{\vec{\tau}}_{\Xi}\cdot{\vec{\tau}}_{N}$
between these $J^{\pi}={\frac{1}{2}}^{-}$ states are given in Table 3 for
14C$(0^{+};1)$ and 14N$(1^{+};0)$ ground-state wavefunctions given in Eqs. (5)
and (6) by their $LS$ basis expansion. The numbers listed in the table need to
be multiplied by the corresponding Slater integral $F^{(2)}_{\Xi
N}(V_{\sigma\tau})$.

Figure 2: A schematic plot of electromagnetic $E1$ transitions deexciting the
$\Xi^{-}_{3D}$–14Ng.s. atomic state to lower $\Xi^{-}$–14Ng.s. nuclear states
and to a $\Xi^{0}_{1p}$–14Cg.s. nuclear state from which $\Xi$ hyperon capture
may occur. The vertical $E1$ sequence (in red) feeds successively the
$\Xi^{-}_{1p}$–14Ng.s. IBUKI state and the $\Xi^{-}_{1s}$–14Ng.s. IRRAWADDY
state according to the common interpretation. The diagonal $E1$ transition (in
green) feeds the $\Xi^{0}_{1p}$–14Cg.s. state marked at the ${}^{14}{\rm
C}+\Xi^{0}$ threshold of $-$6.18 MeV in ${}^{14}{\rm N}+\Xi^{-}$. This state
is discussed in the main text as providing an alternative interpretation of
IRRAWADDY.

Using Eqs. (5) and (6) for the nuclear $LS$ basis amplitudes and the $LS$
matrix elements listed in Table 3, one gets the mixing matrix element between
the $J^{\pi}={\frac{1}{2}}^{-}$ relevant $\Xi^{0}_{1p}$–14Cg.s. and
$\Xi^{-}_{1p}$–14Ng.s. states in terms of $F^{(2)}_{\Xi N}(V_{\sigma\tau})$:

$\langle\Xi^{0}_{1p}-^{14}{\rm C}_{\rm
g.s.}|V_{\sigma\tau}{\vec{\sigma}}_{\Xi}\cdot{\vec{\sigma}}_{N}\,{\vec{\tau}}_{\Xi}\cdot{\vec{\tau}}_{N}|\Xi^{-}_{1p}-^{14}{\rm
N}_{\rm g.s.}\rangle=(0.705+0.050)F^{(2)}_{\Xi N}(V_{\sigma\tau}).$ (7)

Here the two matrix elements involving the dominant ${}^{3}D$ component in
14N$(1^{+};0)$ contribute coherently to give most of the mixing,
0.705$F^{(2)}$, and the contribution of the $P$-states mixing, 0.050$F^{(2)}$,
is small on this scale. For a conservative estimate we take $F^{(2)}_{\Xi
N}(V_{\sigma\tau})\approx-0.6$ MeV, which is 20% of the value of $F^{(2)}_{\Xi
N}(V_{0})$ used in Ref [10] to reproduce IBUKI, thereby getting
$\approx\,-$0.453 MeV for the overall mixing matrix element. The resulting
amplitude of IBUKI’s admixture into $\Xi^{0}_{1p}$–14Cg.s., now identified
with IRRAWADDY, is obtained upon dividing it by $\Delta E_{1p}\approx 5$ MeV
for the energy difference between the two admixed $1p_{\Xi}$ nuclear states:
$-0.453/5=-0.0906$, and 0.00821 upon squaring it. The electromagnetic (e.m.)
$E1$ transition rate from $3D_{\Xi^{-}}$ to $1p_{\Xi}$ states through their
$\Xi^{-}_{1p}$–14Ng.s. components goes like $(\Delta E)^{3}$, with a large
factor of $(6.1/1.1)^{3}\approx 171$ in favor of populating the lower,
$\Xi^{0}_{1p}$–14Cg.s. state. Altogether one gets a ratio of
0.00821$\times$171=1.40 between the two $E1$ rates, making them roughly equal
to each other. A schematic plot of these two scenarios is given by Fig. 2.

##  4 Discussion and summary

Table 4: $\Xi^{-}$–12C and $\Xi^{-}$–14N binding energies (MeV) in $1s$ and $1p$ states, plus $\Xi$ nuclear potential depths (MeV) at nuclear-matter density $\rho_{0}=0.17\,$fm-3, calculated using the density-dependent optical potential Eqs. (2,4) with Re$\,b_{0}$ fitted to binding energies underlined for each input choice. Input |  $\Xi^{-}_{1s}$–12C |  $\Xi^{-}_{1p}$–12C |  $\Xi^{-}_{1s}$–14N |  $\Xi^{-}_{1p}$–14N | $-V_{\Xi}(\rho_{0})$
---|---|---|---|---|---
KEK E176 [2] | 9.82 | $\underline{0.82}$ | 11.78 | 1.96 | 21.9
J-PARC E07 [8] | 4.94 | 0.25 | $\underline{6.27}$ | 0.50 | 13.8

Before summarizing it is appropriate to discuss briefly the role played by the
choice of $\Xi^{-}$ capture data input in predicting $\Xi$-nuclear bound
states and potential depths. In our recent work [10] two KEK-E176 12C events
[2], with $B_{\Xi^{-}}^{1p}=0.82\pm 0.14$ MeV, served as input for setting up
the strength of the $\Xi$-nuclear optical potential. Other $\Xi^{-}$ binding
energies are then predicted, as listed in the first row of Table 4. Choosing
alternatively the J-PARC E07 14N IRRAWADDY event [8] as input, with
$B_{\Xi^{-}}^{1s}=6.27\pm 0.27$ MeV, 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 $\Xi^{-}_{1s}$ binding energies,
which in both 12C and 14N come out about twice larger in the first row than in
the second one. This large difference is reflected also in the $\Xi$-nuclear
potential depths at nuclear matter density, $-V_{\Xi}(\rho_{0})$, listed in
the last column of the table. Equally interesting is the difference between
the two sets with regard to $\Xi^{-}_{1p}$ bound states. In particular, the
$\Xi^{-}_{1p}$–12C binding energy constrained by
$B_{\Xi^{-}}^{1s}(^{14}$N)=6.27$\pm$0.27 MeV comes out in the second row
exceedingly small, substantially disagreeing with that determined from the two
KEK E176 capture events [2] underlined in the first row. As for the calculated
$\Xi^{-}_{1p}$–14N binding energies, since $J({{}^{14}{\rm N}})\neq 0$ these
cannot be compared directly [10] with IBUKI’s binding energy of 1.27$\pm$0.27
MeV in 14N [7].

The solution offered here to the difficulty of interpreting IRRAWADDY as a
$\Xi^{-}_{1s}$ bound state in 14N is by pointing out that it could correspond
to a $\Xi^{0}_{1p}$–14C bound state, something that cannot happen
kinematically in the other light-emulsion nuclei 12C and 14N. We showed in the
present work that radiative $E1$ deexcitation of the $\Xi^{-}$–14N atom,
proceeding through the $\Xi^{-}_{3D}$ atomic state, leads at comparable rates
to two $\Xi_{1p}$-nuclear states coupled by $\Xi^{-}p\leftrightarrow\Xi^{0}n$
strong-interaction charge exchange. One state, of dominantly
$\Xi^{-}_{1p}$–14N structure is the one assigned commonly to IBUKI, and the
other one, of dominantly $\Xi^{0}_{1p}$–14C structure is assigned here to
IRRAWADDY. The latter assignment contrasts with viewing IRRAWADDY as a
$\Xi^{-}_{1s}$–14N state, an assignment motivated only by its binding energy
of a few MeV. Given that in this nuclear mass range capture rates from
$1s_{\Xi^{-}}$ states are estimated to be two orders of magnitude below
capture rates from $1p_{\Xi^{-}}$ states [11, 12], this $\Xi^{0}_{1p}$–14C new
assignment addresses satisfactorily the capture rate hierarchy. Establishing
$1s_{\Xi^{-}}$ states in light emulsion nuclei requires focusing on capture
events in 12C and 16O, where the $\Xi^{-}p\leftrightarrow\Xi^{0}n$ coupling is
ineffective.

## Acknowledgments

We thank Dr. K. Nakazawa for useful correspondence with one of us (A.G.)
following his talk at HYP2022. This work was supported by the European Union’s
Horizon 2020 research and innovation programme under grant agreement No.
824093.

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