Identification of a universal three-body s-wave resonance in ^10He

Domain: biology
Published: 2026-03-31
Editorial outcome: major_revision

Abstract
Universality in physics describes the emergence of common features in vastly
different systems. One fascinating example is the Efimov phenomenon where
three-body resonantly interacting systems display universal properties. Efimov
states have been observed in ultra-cold-atom systems, but their manifestation
in nuclei remains elusive due to the long-range repulsive Coulomb interaction
and the stringent requirement for at least two s-wave resonances in its subsys-
tems. Recent theories propose another universality in three-body halos with only
one s-wave resonance. Here, we report the identification of a precursor of this
phenomenon in a neutron-rich 10He nucleus. With higher statistics and better
sensitivities than previous experiments, we identify two low-lying 0+ states of
10He at 1 MeV and 2 MeV above its two-neutron decay threshold, and deter-
mine an s-wave scattering length of 3.5 fm between 8He and neutron. It is
revealed that the lower energy state, the ground state of 10He, is a three-body
state with only s-wave interactions among its subsystems. This state manifests
as a resonance structure, which is a direct consequence of a universal long-range
three-body repulsion. Our work sheds new light on quantum halos with finite life-
times, providing a path toward their unified understanding across various scales
and fields.

Introduction
In few-body quantum systems with short-range interactions and large s-wave scatter-
ing lengths (a ) relative to the interaction range, universal low-energy phenomena can
emerge [1, 2]. One of the most prominent examples is the universality found by Efimov
that there are infinite series of three-body bound states with universal energy and spa-
tial structures when the inter-particle scattering length a is infinitely large [3–5]. The
phenomena in atomic and molecular systems governed by the electromagnetic inter-
actions have been extensively investigated with cold atoms (see ref. [6] and references
therein). Alternatively, in the world of nuclei dominated by the strong interaction,
nuclei around the neutron drip line are also expected to provide excellent opportunities
to search for such phenomena (see ref. [5] and references therein).

A key aspect underlying the few-body universality is the emergence of a long-range
three-body force induced by multiple s-wave scatterings [7]. For the Efimov state,
an attractive three-body force is created from the two-body subsystems which are
marginally bound or unbound. Such a halo state can mediate attractive force between
the other particles up to a substantially large distance, forming the Efimov bound
states. It has also been predicted that the Efimov’s universal three-body effect can
extend to resonant states [8] and virtual states [9]. Recently, it has been shown that
universal geometric properties for three-body nuclear halo states may also appear when
the s-wave interactions in part of the two-body subsystems are not resonant [10, 11].
Distinct from the Efimov states, the universal halo states originate from a long-range
repulsion and have not been observed experimentally in any atomic or nuclear systems
to date. In the present work, we examine the low-energy properties and the related
three-body effects in an unbound system in nuclear physics: the helium-10 (10He)
nucleus, consisting of a 8He core and two valence neutrons.

The nature of the 10He ground state has been the focus of keen debate since
its discovery in RIKEN in 1994 [12]. Up to now, two contradictory results exist for
the ground state energy of 10He. The 8He-n-n relative energy (E ) spectra from
the 11Li(-p) [12–14] and 14Be(-2p2n) [15] knockout reactions show a prominent low-
lying peak at 1.5(1) MeV (the weighted average of 1.2(3) MeV [12], 1.7(4) MeV [13],
1.54(11) MeV [14], and 1.60(25) MeV [15]) with a width Γ of 1-2 MeV. The 11Li(d,
3He) reaction shows a ground-state peak at 1.4(3) MeV (Γ 1.5 MeV) [16]. However,
the 3H(8He,p)10He two-neutron (2n) transfer reaction reported a higher ground-state
energy at 2.1(2) MeV (Γ 2 MeV) [17]. Currently, the most advanced theoretical
work on 10He is given in Refs. [18, 19]. On the one hand, it has been discussed that
the 10He ground state could be at 2 MeV, as observed by the 2n transfer reaction,
and has a [0p ]2 structure, while the initial-state effects due to the halo nature of
11Li or 14Be may lower the energy of 10He to 1.5 MeV as observed in knockout
reactions. Calculations with such effects qualitatively agree with the 10He invariant-
mass spectrum from 11Li(-p) [19] but similar calculations assuming a direct α-particle
removal from 14Be cannot explain the spectrum from 14Be(-α or -2p2n) [20]. On the
other hand, the 8He-n-n three-body model calculations predicted that 10He could have
two low-lying 0+ states with [1s ]2 and [0p ]2 valence neutron structures [18, 21],
that are built on the 1/2+ (8He ν1s ) or 1/2 (8He ν0p ) states of
9He. The observed 10He ground-state energy variation thus could be caused by two
overlapping 0+ states populated with different strengths in knockout and transfer
reactions [18, 22]. The [1s ]2 component, that is characterized as a three-body s-
wave resonance built from the s-wave virtual states in the n-n and 8He-n subsystems,
has not been identified in any experiments so far.

Here, we show the energy spectrum and three-body correlations of 10He measured
using the invariant-mass technique with higher statistics and better sensitivities than
previous experiments. It is concluded that the 10He ground state is a three-body s-
wave resonance, where both the n-n and 8He-n subsystems have zero relative angular
momenta. Such a state shows features that arise from the long-range repulsion and
represents an unbound analogue of the universal halo states predicted by Hongo and
Son [10].

Results
The two-neutron unbound 10He was produced through quasi-free knockout (p, 2p)
reaction at 250 MeV/nucleon, using an intense 11Li beam bombarding on the 150-
mm thick liquid hydrogen target of the MINOS device [23]. The 8He fragment and two
neutrons from the 10He decay were detected by the SAMURAI spectrometer [24] and
the neutron detector array NEBULA [25], respectively. The same setup was used in
Refs. [26–30]. Since 8He has no bound excited states, we do not require γ-ray detection
in coincidence.

The three-body relative energy E , which is also the energy of 10He above the
8He + 2n threshold, was reconstructed by the momenta of 8He and the two neutrons.
The typical energy resolution at E = 1 MeV is about σ = 0.15 MeV (see Methods).
To reject the cross-talk background mainly caused by neutron scattering within the
NEBULA detectors, we applied the procedures detailed in Ref. [31] (see Methods).
The overall 2n detection efficiency as a function of E is shown in the upper panel of
Fig. 1, which demonstrates sufficient sensitivity down to the decay threshold.

The lower panel of Fig. 1 shows the relative-energy spectrum after the efficiency
correction, which has much smaller statistical uncertainties compared to two pre-
vious experiments at 61 MeV/nucleon at RIKEN [12] and at 280 MeV/nucleon
at Gesellschaft fu¨r Schwerionenforschung (GSI) [14]. The present spectrum of 10He
shows a low-energy peak at E 1.5 MeV, consistent with the previous results in
Refs [12, 14]. In Refs. [12, 14], the peak at 1.5 MeV has been interpreted as the
ground state of 10He. In addition to this low-energy peak, the current data clearly
show a shoulder structure below 1 MeV, that was unresolved in the spectrum mea-
sured previously at RIKEN but was missing in the GSI result. The difference at small
relative energies has been reported before in the study of 11Li [32] and 6He [33], which
was attributed to the enhanced experimental sensitivity at low relative energies in the
current setup. The low-lying shoulder indicates that the peak at 1.5 MeV is not
the ground state of 10He. Above 2 MeV, the current data, with much better quality,
exhibit strength extending to higher excitation energies.

Here, we compare the data to theoretical calculations that combine the coupled-
channel 9Li + n + n three-body model of 11Li and the quasi-free knockout (p, pN)
reaction model. The three-body model reproduces well the ground-state properties of
the 11Li nucleus, including the 2n separation energy, matter and charge radii. The
dominant 2n configurations with a 9Li core are 47% [0p ]2 and a large s-wave mix-
ing of 44% [1s ]2 that generates the neutron halo structure and strong dineutron
correlations. The neutron configurations are in reasonable agreement with the recent
results from the 11Li(p, pn) reaction [27]. In the calculation, a 0p -orbital proton
is removed from the 9Li core, resulting in the population of the 8He ground state
that has a [0s ]2[0p ]4 neutron shell structure. The p-wave proton removal has
been confirmed by comparing the experimental momentum distributions of 10He to
the quasi-free knockout calculations (see Methods). The final 10He states were com-
puted with the complex-scaled solutions of Lippmann-Schwinger equation (CSLS)
method [34, 35], which is a suitable approach to describe three-body scattering states
including virtual states, and to investigate their decay mechanisms. We adopted the
8He-n folding potential and the realistic Argonne v8 force for the n-n potential [36].
Considering the uncertainty of the 8He-n interaction (see for instance a compilation
of resonances in 9He in ref. [37]), we allow variations in the parameters governing this
interaction. See Methods for details.

The calculated E distributions are shown in Fig. 1, which have been convo-
luted with the experimental resolution detailed in Methods. The calculation results
for the 8He-n subsystem, with the tuned 8He-n interaction to fit the current 10He
spectrum below 2 MeV, produced an s-wave scattering length between 8He and neu-
tron (a ) of 3.5 fm, in reasonable agreement with the results from some knockout
reactions ( 3.17(66) fm [14], 3 fm < a < 0 fm [38], 2.1(6) fm or 6.8(14) fm [39])
and the weak 8He-n s-wave attraction from the (p,p) resonance elastic scattering [40],
but not with the strong one (a 10 fm) reported in Ref. [41] and Ref. [42]. A
recent analysis of the (p, p) resonance elastic scattering data with more advanced
treatment of the isospin symmetry also suggests a small negative a [43]. In addi-
tion, the current calculations produce a broad p-wave resonance for 9He at 0.76 MeV
(Γ = 0.95 MeV), in reasonable agreement with the extracted 8He-n relative energy
spectrum from the present experiment [44].

To pin down the origin of the low-energy peak, we decomposed the E spectrum
into contributions from different decay channels in the final scattering states. It is
shown in Fig. 1 that the low-lying shoulder structure is dominated by the 8He +
[1s ]2 component, peaked at 0.96 MeV, while the peak at around 1.5 MeV in the
spectrum is due to the mixture of the 8He + [1s ]2 and 8He + [0p ]2
components. The 8He + [0p ]2 component gives a broad 10He resonance peaked
at 1.9 MeV, consistent with the 10He 0+ state reported by the 2n transfer reaction [17],
but with a smaller decay width (Γ 1.6 MeV). This finding resolves the long-standing
puzzle about the different 10He ground-state energies obtained from knockout and
2n transfer reactions. Our work suggests that the ground state of 10He has a nearly
pure [s]2 configuration. The 10He, which is traditionally considered as a doubly magic
nucleus, thus appears as a “double-intruder” nucleus with the last two neutrons mainly
in the 1s shell, in line with the intruder s-wave dominated ground state of 9He. The
intruder dominant configuration could be understood in the shell-model framework
due to the melting of the N=8 shell closure for the neutron-rich isotopes below Z=4.
In addition, the cross-section ratio for populating the excited 0+ state relative to the
ground state was extracted to be about 0.36. This value is in contrast to the estimation
of a simple direct-reaction model presented in Ref. [22], where a pure [sd]2-dominant
configuration in the ground state of 10He was expected to yield a ratio greater than
1.5.

Our theoretical calculations reproduce the data below E = 2 MeV, nevertheless,
they underestimate the strength at higher excitation energies. The reason might be
due to higher-order transitions or higher-lying states that are not accounted for in the
calculations. Indeed, existence of 2+ state with resonance energies around 4-6 MeV [14–
16, 45], 1 state near 5.5 MeV [17] and 3 state around 7.9 MeV [45] have been
suggested. Incorporation of these states into the fitting improves the agreement with
the E spectrum in the higher energy region, but it has minimal impact on the fit
results below 2 MeV. In our calculations, the spectrum’s shape is highly sensitive to
the 8He-n interaction, and the observed low-energy structure is reproduced only with
a weakly attractive 8He-n s-wave interaction. Futhermore, deactivating the final-state
interactions (FSI) results in a shift of the excitation strength towards higher energies,
coinciding with the disappearance of the [0p ]2 resonance and a broadening of the
peak from the [1s ]2 neutrons. See Methods for details. This indicates that the 10He
structure observed in the spectrum below 2 MeV primarily arises from the FSI rather
than the initial-state effects.

To understand the nature of the 10He low-lying states, we further investigated
their three-body correlations in the momentum space. Here, we choose to use the two
variables in the Jacobi-T and Jacobi-Y coordinates. As shown in Fig. 2, the Jacobi
momenta are defined as

m k m k 2m k m (k +k )
n 2 n 1 n c c 1 2
k = − ,k = −
x y
2m 2m +m
n n c
in the Jacobi-T system, and

m k m k (m +m )k m (k +k )
n c c 1 n c 2 n 1 c
k = − ,k = −
x y
m +m 2m +m
n c n c
in the Jacobi-Y system, where m and m are the masses of neutron and 8He, k , k
n c 1 2
and k are the momenta of neutron 1, neutron 2 and 8He in the 10He center-of-mass
c
frame. The three-body correlations are described by two parameters: ε=E /E and
x rel
cos(θ), where E is the relative energy of the two-body subsystem and θ is the angle
x
between k and k .
x y

The obtained correlations are shown in Fig. 2 in the energy region below E =
1 MeV. Fig. 3 shows those after projection to ε and cos(θ) for “T” and “Y” systems.
The cos(θ) spectrum in the Jacobi-T system in Fig. 3(b) and the ε spectrum in the
Jacobi-Y system in Fig. 3(c) show almost a featureless “phase-volume” distributions
that are consistent with the pure [1s ]2 configuration, in which the energy distri-
butions are proportional to ε (1 ε) [46]. This further confirms our assignment
× −
of [1s ]2 configuration for the 10He low-lying state below 1 MeV.

In comparison with the theoretical calculations, both the dσ/dε around 0 in “T”-ε,
shown in Fig. 3(a), and the dσ/dcos(θ) around 1 in “Y”-cos(θ), shown in Fig. 3(d),
are enhanced. This corresponds to the so-called “di-neutron” configuration that two
neutrons are emitted in opposite directions in momentum space, i.e. close in coordinate
space, with a very small relative energy. The reason for the stronger enhancements
than calculations is not fully understood. As mentioned before, a simple structure
model was assumed for the 8He core in the calculation. However, a more complex
structure is expected for the 8He core (see, for example, ref. [16]), which may lead to
a stronger “di-neutron” configuration than that considered in the model.

The above analyses demonstrate that the lowest-energy structure in 10He is a three-
body state with zero relative angular momenta (s-wave) among its subsystems. In a
two-body system, such a virtual state with orbital angular momentum L = 0, due to
the absence of centrifugal barrier, can only exist as a threshold peculiarity which can-
not be physically detected as a distinct peak. In contrast, the present case exhibits a
well-defined resonance structure, with positive energy and finite width, which is obvi-
ously different from the threshold peculiarity. Such an s-wave resonance peak appears
only in systems with 3 particles and can be attributed to a non-zero value of the
effective angular momentum K, which characterizes the long-range behaviour of the
three-body potential [2, 11, 18] (see Methods for the definition of K). Indeed, even for
a non-interacting three-body system, the hyperradial (i.e. three-body effective) poten-
tial is repulsive with K =3/2. More precisely, in our system, a large neutron-neutron
s-wave scattering length of a 18.6 fm [47] reduces the effective angular momen-
nn
tum to K = 1/2 (see Methods for a deduction of the effective angular momentum
based on the Faddeev approach). This is physically due to that a neutron occupy-
ing the halo virtual state can mediate an attractive interaction between the other
neutron and the 8He core up to a large distance of a , thereby effectively reduc-
nn
ing the strength of the repulsive hyperradial potential. Nevertheless, the hyperradial
potential retains its repulsive character in the 8He-n-n three-body system. This mech-
anism is similar to the emergence of long-range three-body attraction in the Efimov
effect [3, 7]. While K is imaginary in the Efimov effect, our system shows universal
long-range repulsion with K = 1/2 because of a small core-neutron s-wave scatter-
ing length of a 3.5 fm. Recently, Hongo and Son predicted that weakly bound
cn
Borromean 2n halo nuclei with small a and large a can exhibit a novel class
cn nn
of universal three-body bound states [10]. These states have universal energetic and
structural properties distinct from the Efimov states as a consequence of the universal
long-range repulsion [11]. Observation of the resonance structure for the three-body
s-wave “virtual” state in the 8He-n-n system can be consistently interpreted as aris-
ing from this long-range universal repulsion. Thus, the present work identifies the
ground state 10He as an unbound analogue of the universal 2n halo states [10, 11]. To
further unveil the nature of this novel universal quantum halo, it is pivotal to scruti-
nize their structural properties, with concurrent developments in low-energy nuclear
experiments and theoretical models on the three-body resonances.

In summary, we have reported on the low-lying states in 10He populated from the
11Li(p,2p) knockout reactions at 250 MeV/nucleon. Two low-lying 0+ states of 10He
were identified at about 1 MeV and 2 MeV, which have a [1s ]2
0+
and a [0p ]2
0+
neutron configuration, respectively, clarifying the long-standing puzzle about the dif-
ferent 10He ground-state energies obtained from knockout and 2n transfer reactions.
An s-wave scattering length of 3.5 fm between 8He and neutron was extracted
from the current 10He spectrum. The existence of a three-body s-wave resonance in
10He was conclusively elucidated through the high statistic invariant-mass spectrum
and the correlation analysis among the three decay particles. The three-body effect
analogous to the Efimov physics is presented as the origin for the three-body repulsive
barrier. The ground state of 10He represents an unbound analogue of the universal
three-body halo states. While these phenomena have been observed in a nuclear sys-
tem, we anticipate analogous behavior in other few-body systems, including exotic
hadrons and condensed matter systems, and particularly in ultracold atomic systems,
where tunable s-wave interactions between constituent particles enable systematic
exploration.

Methods
Production of the 11Li beam. The 11Li beam was produced at the Radioactive
Isotope Beam Factory operated by the RIKEN Nishina Center and the Center for
Nuclear Study of the University of Tokyo. Cocktail secondary beam was first produced
through fragmentation of a 345 MeV/nucleon 48Ca primary beam on a 9Be target.
The typical intensity of the primary beam is 400 particle nA. The secondary beam
was then purified and identified event by event using the BigRIPS two-stage fragment
separator [48]. The average energy of the 11Li beam is 246 MeV/nucleon, with a purity
of 70% and a typical intensity of 1 105 particles per second.

Invariant-mass method. The invariant mass M of 10He is expressed as

M = E P
i i
where E and P are the energy and momentum of the decay products, i.e. 8He and
i i
two neutrons in the current case. The relative energy E(E ) is defined as the difference
rel
between the invariant mass and the sum of the rest masses of the decay products.
The relative-energy resolutions have been obtained from GEANT4 simulations
considering the realistic experimental setup and the resolutions of the detectors. The
upper panel of Fig. 4 shows the resolutions as a function of the 8He-n-n three-body
relative energy. The energy resolution (1σ) can be well approximated by σ(E ) =
rel
0.0131 + 0.133 E 0.611 MeV. Some typical relative energy distributions fitted with
rel
symmetric Gaussian functions for fixed relative energies are shown in the lower panel
of Fig. 4. The distributions are slightly asymmetric below E = 0.5 MeV, which has
rel
been taken into account in the convolution of the theoretical distributions shown in
Fig. 1, although the effect is minor.

Cross-talk rejection. Neutrons are electrically neutral particles and are capable of
generating multiple detection hits within the neutron detector array mainly caused by
neutron scattering in the detector material. In the present work, to measure the energy
and momentum of two neutrons, rejection of such cross-talk background in the neutron
detector array NEBULA is required. In the data analysis, we applied 6 MeVee energy
threshold to the measured hits to reduce the γ background and applied the cross-talk
rejection procedures following the previous work presented in Ref. [31]. The parameters
for the cross-talk rejection conditions were obtained using the single-neutron events
taken from the experimental 11Li(p, pn)9Li + n reaction.

For the events of two neutrons detected in the same wall of NEBULA, the cross
talks due to a secondary recoil proton are identified using the following condition

dr dr dt dt
0 0
− + − <1,
where dr and dt are the distance and time difference between two hits, and parameters
of dr = 15.8 cm , R = 15.0 cm and dt = 0.32 ns and T = 1.8 ns were adopted.
0 0
In addition, the cross talks due to γ and scattered neutrons in larger dr and dt are
rejected through a two-dimensional selection applied to Q and β /β , where Q is
2 01 12 2
the energy deposit (unit in MeVee) of the secondary hit, β is the velocity (unit in
01
speed of light) between the reaction position and the primary hit, β is the velocity
12
between the primary and the secondary hits. The selection could be parameterized as
Q <108 β /β 60 and Q < 12 β /β +80.
2 01 12 2 01 12
For the events of two neutrons detected in the different wall of NEBULA, the cross
talks are mainly due to neutron scattering from the upstream wall to the downstream
one. Since the neutron’s velocity is reduced after the scattering, most of these cross
talks could be identified by a simple condition of β /β > 1. In the current anal-
01 12
ysis, we require that Q < 250 β /β 200, taking into account that Q , the
1 01 12 1
energy deposit of the primary hit, becomes larger when β is smaller in the neu-
12
tron scattering. Finally, some remaining γ cross talks are rejected by the condition of
1/β 1.
12
After applying the above procedures, we found that 96.9% of the cross talk is elim-
inated in the data, while 98.7% is eliminated in the simulation. We also confirmed that
varying the cross-talk rejection conditions reasonably yields consistent relative-energy
spectra, as long as the cross-talk rejection procedures are consistently incorporated
into the estimation of the 2n detection efficiency. This further demonstrates that the
above cross-talk rejection procedures are valid.

Momentum distributions of 10He. Parallel and perpendicular momentum dis-
tributions of 10He were extracted using the momenta of 8He and the outgoing two
neutrons. As shown in Fig. 5, the experimental distributions can be well reproduced
by the quasi-free knockout calculations that assume a p-wave proton removal from
11Li. The calculations were performed using the QTC (Quantum Transfer-to-the-
Continuum) model [49]. In the QTC calculations, the Paris Hamburg interaction [50]
was employed for the optical potentials, with densities of 10He from Variational Monte
Carlo calculations [51]. Reid93 interaction was adopted as the nucleon-nucleon inter-
action [52]. The single-particle wave function of the knocked-out proton was obtained
by solving the Schr¨odinger equation using a Woods-Saxon potential. The core radius
and the diffuseness of the Woods-Saxon potential were fixed at 1.25 fm and 0.65 fm,
respectively. The depth of the potential was tuned to produce the proton separation
energy of 15.76 MeV.

Theoretical calculations of the 10He energy spectra. The T-matrix for the
proton knockout from the 11Li is calculated as

(k ,k )= Ψ (k ,k ) Φ(8He) ψ˜ Vˆ Φ Φ(9Li) ψ ,
nn x y pp pp nn p
T ⊗ ⊗ | | ⊗ ⊗
where Ψ and Φ are the relative wave function of the two valence neutrons in the
nn nn
final scattering states and the initial state, respectively. Vˆ is the effective proton-
pp
proton interaction, which drives the knockout reaction. The wave functions of the
core nuclei, 8He and 9Li, are given by Φ(8He) and Φ(9Li), respectively. The scattering
wave function for the outgoing and knocked-out protons is given by ψ˜ and that for
pp
the initial incoming proton is given by ψ . The momenta, k and k , are the relative
p x y
momenta in a three-body Jacobi coordinate system.

In the present calculations, we assume that 8He in the final states is described by
an inert core whose configuration is given by [π0s ]2[ν0s ]2[ν0p ]4. In addition,
1/2 1/2 3/2
by considering the quasi-free knockout condition, we consider the two valence neutrons
of 11Li to remain as spectators through the reaction. Under these assumptions, we
calculate the cross section by factorizing it into the core and two-neutron parts as

d6σ
= Ψ (k ,k )Φ 2,
core nn x y nn dk dk KF |⟨ | ⟩|
x y
where is a kinematical factor and is an elemental cross section for the proton
core
K F
knockout from the 9Li core. Since the energy distributions of the knockout process is
dominated by the two-neutron part, we define a distribution function as

(k ,k )= Ψ (k ,k )Φ 2.
x y nn x y nn
W |⟨ | ⟩|
To calculate the distribution function , it is important to describe the relative
W
motion of the two valence neutrons not only for the 11Li ground state, Φ , but also for
nn
the final 8He + n + n three-body scattering states, Ψ . We here employ the coupled-
nn
channel 9Li + n + n three-body model [53] for the initial 11Li ground state. This
model well reproduces the observed quantities of the 11Li ground state, such as the
binding energy, charge and matter radii, and so on. For the final scattering states, we
use the complex-scaled solutions of the Lippmann-Schwinger equation (CSLS) [34, 35]
with the 8He + n + n orthogonality condition model (OCM) [21, 54]. The method
of CSLS enables us to describe three-body scattering states accurately by using the
complex-scaled Green’s function. The complex-scaled Green’s function is expanded
with the completeness relation of the complex-scaled eigenstates solved in the 8He +
n + n OCM. In the description of the final scattering states, we employ a realistic
Argonne v8 interaction for the n-n and a folding potential with the 8He core density
for the 8He-n. Due to a lack of the 8He-n scattering data, we tune the strength of the
8He-n potential to reproduce the observed 10He spectrum, while the functional form
of the 8He-n potential remains fixed as given in the folding procedure. The explicit
form of the folding potential is given in Ref. [54].

The calculated 8He-n-n relative energy spectra for different theoretical FSI sce-
narios are displayed in Fig. 6. The results demonstrate a strong dependence on the
treatment of FSI. The calculations, either omitting all FSI or with only the n-n FSI,
shift the excitation strength to higher energies resulting in a bump peaked below 1
MeV, which is inconsistent with the low-energy structure in the experimental spec-
trum. Furthermore, the spectrum’s shape is highly sensitive to the 8He-n interaction.
In Fig. 6, the calculated distributions, which correspond to 8He-n scattering lengths
of 3.9 fm a 2.7 fm, are also shown. The distributions become flatter when
cn
− ≤ ≤ −
the absolute value of a becomes smaller. The low-energy structure in the experi-
cn
mental 10He spectrum is reproduced only with a slightly negative a around 3.5
cn
fm. Besides, from the folding potential with depth fitted to the experimental data
−
, the
8He-n effective range is estimated as about 2 fm.

We note that previous theoretical calculations of Ref. [18], as shown in Fig. 6(c)
of Ref. [18], which utilized 8He-n Gaussian potentials tuned to induce a p-wave reso-
nance of 9He around 2 MeV and an a around +2.9 fm, also reproduce the observed
cn
low-energy structure in the 10He spectrum reasonably well. In this case, the partial
decomposition in Ref. [18] leads to a consistent conclusion with us that the 10He spec-
trum below 1 MeV is dominated by the configuration of two neutrons in the s-wave
state. We also note that the universal halo states predicted by Hongo and Son [10]
only require a small absolute value of the core-neutron scattering length. Even if a is
cn
slightly positive, the 10He resonance can still be interpreted as an unbound analogue
of the universal Hongo-Son state.

We would like to comment on the different treatment of the Pauli principle between
our work and that presented in Ref. [18]. In the 8He + n + n OCM, the Pauli blocking
in the 8He-n subsystem is taken into account by using the projection operator, the so-
called pseudopotential, which excludes the Pauli forbidden states (occupied states by
neutrons in the 8He core) from the 8He-n relative motion microscopically. This treat-
ment of the Pauli blocking is important to investigate the knockout reaction, especially
for the s-waves in the 8He-n subsystems, because the internal wave function is cru-
cial to determine the distribution function quantitatively. The analysis in Ref. [18]
has been performed using the core repulsion potential to exclude the Pauli forbidden
states, resulting in a different internal wave function of 10He from the microscopic cal-
culations as shown in Ref. [55]. The Pauli repulsive core method is an approximate
method to exclude the Pauli forbidden states from inter-cluster motions but cannot
correctly describe nodal behaviors of inter-cluster motions. By contrast, the OCM is a
semimicroscopic approach in which the Pauli forbidden states are excluded accurately
via projection operators.

Deduction of the effective angular momentum for a three-body system.
Let us consider a three-body halo state consisting of a heavy core of mass M and two
identical light particles, each with mass m . Using the Faddeev approach, the wave
0
function of this state with zero internal relative angular momenta has the following
hyper-spherical adiabatic expansion [11]:

3
Ψ= ∞ F n (R) Φ(i)(α ;R) ,
R2 n i
n=1 i=1
where R is the hyperradius that gives the global size of the state and α is the hyper-
i
angle that describes the shape of the state. Here, the summation over i = 1,2,3
corresponds to three different Jacobi coordinate pairs. Φ(i)(α ;R) is the hyper-
n i
angular wave functions given by [2, 11]

sin s (R)(π α )E
Φ(i)(α ;R)= n 2 − i ,
n i sin2α i R
where s (R) are the solutions of
n

π 2 sin(s γ) sin(s π) R π sin(s π) R 2sin(s γ ) 2
cos(s )+
n
+
n2
cos(s )+
n2
=2
n ′
,
n n
− 2 s sin(2γ) s a − 2 s a s sin(2γ )
n n 2 3 n 12 n ′
with γ = arcsin 1 and γ = π γ. A = M/m represents the mass ratio between
1+A ′ 4 −L2 0
the core and the light particle.

Under the adiabatic approximation [2], which is valid for the current 8He-n-n three-
body system in the region of a R a , the hyperradial functions F (R) in
nn cn n
| | ≫ ≫ | |
Eq. (8) are the solutions of the hyperradial equation:

d2 2m
A dR2 + ℏ2 [E − V n (R)] √RF n (R)=0,
where m is a normalization mass that can be taken to be the particle’s mass m ,
0
and V (R) is the hyperradial potential defined as
n

ℏ 2 s2 1/4
V (R)= n− .
n 2m R2
In analogue to the definition of the two-body centrifugal potential, the effective
angular momentum K maybe defined by

ℏ 2 K(K+1)
V (R)= ,
n 2m R2
which results in K =s 1.
n − 2
Assume that the scattering length between the two identical particles is a , and
12
the scattering length between the core and the particle is a or a . Firstly, let us
23 13
consider a non-interacting system: a = a = a = 0. To satisfy Eq. (10), we need
12 13 23
sin(s π) = 0 to eliminate the divergent terms in the left-hand side of Eq. (10). This
n2
leads to s =2,4,6 and K = 3,7,11 .
n ··· 2 2 2 ···
Secondly, we consider another case that the particle-particle scattering length a
12
| |
is very large and the core-particle scattering length is much smaller than the size of
the state a = a << R. This condition is similar to the present case of 10He.
13 23 | | | |
Under such condition, the leading order term of the left-hand side of Eq. (10) becomes

− sin(sn π 2 sn )cos(sn π 2 ) a R 23 . Since the right-hand side of Eq. (10) is of the order unity (recall
that 0<γ <π/4 and s =0), to satisfy Eq. (10), we need the leading order term of the
′ n
left-hand side vanish, i.e.: cos(s π) = 0 or sin(s π) = 0. This leads to s =1,2,3
n2 n2 n
···
and therefore K = 1,3,5 . The wave function of the low-energy three-body state
2 2 2 ···
is dominated by the channel n=1 at large R. In comparison with the non-interacting
three-body system, the effective angular momentum K is thus reduced from 3/2 to
1/2. The hyperradial potentials for K = 3/2 and K = 1/2 are shown in Fig. 7, where
we also show the Efimov attractive three-body potential for comparison.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Referee report for the paper\nFirst identification of a universal three-body s-wave resonance in 10 He\nby Y.L. Sun et al\nThe new highly accurate data on 10He populated in the proton knockout reaction\nfrom 11Li halo nucleus is reported in this work. The Authors report on the\nlow-energy enhancement in the invariant mass spectrum of 10He, which was not\nidentified in the previous studies due to insufficient resolution or/and statistics.\nThese data are interpreted as providing evidence to existence of s-wave\nresonance in 10He, built on three attractive s-wave pairwise interactions.\nSuch a resonance is discussed as representative of the \"universal\" long-range\ndynamics of Efimov type. The search for these class of phenomenon was\nwas so far unsuccessful in nuclear systems, therefore such a discovery\ncould be a real \"gem\" in the studies of exotic phenomena near the nuclear\ndriplines.\nIn my analysis of situation I fully rely on the reliability of the data.\nThese are provided by one of leading groups in the field with broad expertise\nin analogous studies. So, in the following analysis of the proposed\ninterpretation complete reliability of the data is implied.\nThere are important questions about interpretation of the data. It can be\nfound in the literature that alternative interpretations are possible.\nThese interpretations are for a long time \"on the market\" and should\nnot be neglected without significant explanation.\nIt should be noted that alternative interpretations are also not trivial\nand are related to questionable types of the long-range dynamics in\nnuclear systems, which are also of significant interest.\nI find the experimental observations of this paper and various types of\nassociated physical phenomena sufficiently interesting to be published in Nature.\nHowever, major revision of the manuscript in aspects concerning the\ndata interpretation is due (see the discussion below).\nI would strongly recommend this paper for publication in Nature if my\ncriticism concerning interpretation is fully taken into account.\n=========================================================================\nPoint 1. Scattering length in the 9He=8He+n.\nBroad theoretical exploratory studies of the 10He spectra sensitivity to the\ns-wave 8He-n interaction have been performed in Ref. [18] by Grigorenko and Zhukov.\n-------------------------------------------------------------------------\nSince further we rely on the results of Refs. [18, 19], it is necessary\nto comment to their description in this work:\n\"Currently, the most advanced theoretical work on 10He is given in\nRefs. [18, 19]. On the one hand, it has been discussed that the\n10He ground state could be at ∼ 2 MeV, as observed by the 2n\ntransfer reaction, and has a [0p_1/2]^2_{0+} structure, while\nthe initial state effects due to the halo nature of 11Li or\n14Be may lower the energy of 10He to ∼1.5 MeV as observed in\nknockout reactions. Calculations with such effects qualitatively\nagree with the 10He invariant-mass spectrum from 11Li(-p) [19]\nbut cannot explain the spectrum from 14Be(-2p2n) [20].\"\n(1) The low energy strength in the 10He spectrum has mixed character\n[0s_1/2]^2_{0+} + [0p_1/2]^2_{0+} with around 50% mixing. Important\nspecial attention was paid in [18] to evolution of these components\nin various conditions.\n(2) The statement that the results if [19] \"cannot explain the\nspectrum from 14Be(-2p2n)\" is not relevant, since the calculations\nin [19] were about alpha knockout, while there were no clear identification\nof the reaction channel in [20] and \"14Be(-2p2n)\" (not alpha removal!) is\njust one of reasonable assumptions of this work.\n---------------------------------------------------------------------------\nLet's have a look\nin Fig. 5f of this work. The solid curve shows the spectrum which is very\nconsistent with the new experimental data. In Fig. 6c the decomposition\nof exactly this spectrum is shown demonstrating energy split between [s^2]_0 and\n[p^2]_0 components at about 1 MeV, which is, again, very consistent with\ntheoretical results shown in Fig. 1 of this work. The complete three-body\ncorrelations for this case also can be found in Fig. 13c of Ref. [18]. These\nare calculated not exactly for the excitation energy range, used in Fig. 2\nof the Refereed paper, but they looks very similar to the recent data.\nSo, the agreement of this specific theoretical prediction with the data seem\nto be very good. However, this specific calculation corresponds to quite a\ndifferent scattering length 2.9 fm in the 8He+n, which is not ATTRACTIVE,\nbut REPULSIVE. This is not in accord about the basic message of this work\nabout universal Efimov-type phenomenon.\nLet's have a look on the situation in a different way. The energy of the\np_{1/2} resonant state used in the calculations of this work is mentioned\nto be 0.76 MeV. The results for such p_{1/2} position can be found in\nFig. 5d of [18]. They should be between dotted and dash-dotted curve,\nproducing peak between 30 and 200 keV. So, there seem to be strong\ncontradiction between theoretical approaches of this work and of Ref.\n[18]. According to [18] the universal s-wave structure built on the\ns-wave attractions in all three subsystems should have been seen as\nextreme low-energy (100 keV? 200 keV?) structure in this reaction.\nI would mention one qualitatively suspicious moment in the provided theoretical\ncalculations. Based on the energy of p_{1/2} state 0.76 MeV, the trivial\nindependent particle approximation should provide the energy ∼1.5 MeV\nfor the [p_{1/2}^2]_0 configuration. The peak energy for this configuration in\nFig. 1 of this work is ∼1.9 MeV. This means that the paring interaction\nis effectively repulsive for the [p_{1/2}^2]_0 configuration. This is quite\nunexpected result, which I see as strange for the three-body model and which\nit is not within the systematics of effective pairing energies for the\np_{1/2} subshell, which can be traced for the other members of N=8\nisotone. In Ref. [18] these effective pairing energies vary from 0.5 to 1.8 MeV,\nwhich is smaller then the typical values in the p_{3/2} subshell, but\nnot negative at least.\nThe strong theoretical controversy is quite intriguing, considering the\nnice agreement between the three-body calculations of this group [33] and\n[Grigorenko et al. PRC 102, 014611 (2020)] for another quite complicated\nperipheral phenomenon: soft dipole excitation in the continuum of 6He system.\nIt is not impossible that quite old calculations of Ref. [18] are deficient\nin some way. However, unless this deficiency is demonstrated they can not\nbe just disregarded: these are published and well documented results, while\nfor the calculations of this work not much details are available. The Authors\nshould either find out what is wrong with calculations of [18] or rely on\nthem as alternative interpretation of the data.\n---------------------------------------------------------------------------\nSome separate note concerning the 8He+n scattering length issue.\nI would like to pay attention of the Authors to results of paper\n[Khirk et al., Phys. Rev. C 111, L051601 (2025)].\nIn this work the data of Uberseder et al. [39] were reinterpreted with more advanced\ntreatment of isospin symmetry. The scattering length in the 8He+n channel\ndeduced in this work was weakly attractive as -2.7 fm, with no need for s_{1/2}\nresonance. This is quite close to the deduced in this work value −3.45 fm.\nHowever, the p_{1/2} state should be at about 2.3 MeV (original\nanalysis of Uberseder et al. do not allow any resonances with l>0 and\nE<3 MeV at all).\n---------------------------------------------------------------------------\n=========================================================================\nPoint 2. Reaction mechanism issue.\nThe structures promoted in this paper as \"two low-lying 0+ states of 10He\nat ∼1 MeV and ∼2 MeV above its two-neutron decay threshold\" according to\ncalculations of [18] are strongly sensitive to the reaction mechanism.\nLet's compare solid curves Fig. 5 c and f in [18]. One is well consistent\nwith the data of the refereed work the other is shifted to higher energy\nto more than 1 MeV. The same if we compare Fig. 6 b and c in [18]. For\none the [s^2]_0 and [p^2]_0 components are well energy split (as in the\ncalculations of this work), but for the\nother they more or less overlaps, so we can hardly speculate about\nseparate [s^2]_0 and [p^2]_0 states.\nThe results of the refereed work are based on some realistic reaction model\nwhere the 11Li WF is present. So, all these effects should be there, they\nare just not discussed. These could be easily checked if I am right\nby the Authors of this work in two reasonably simple ways:\n(1) By performing calculations without 10He FSI - just plane waves.\nTypically people try to check, whether the signal they see is not provided\njust by a plane wave, without any dynamics.\n(2) By changing the reaction mechanism by using not a halo nucleus 11Li,\nbut a compact system. For example 14C(-4p). Not very realistic reaction\nfor practical realization, but theoretically we may play games. If the\nstrength function is strongly qualitatively sensitive to the reaction\nmechanism, then you can hardly speculate about resonance states. Resonant\nstates is something independent on reaction mechanism by definition.\nAlso, if your population cross section strongly depends on the reaction\nmechanism then what is your \"universality\" about? This is not about that the\nstory is not interesting and important, but about carefully choosing terms\nfor the discussion.\nI am a bit amused that this topic is total missing in the manuscript,\nsince the Authors has recently studies soft dipole excitation in 6He [33],\nwhere they do not attempt to interpret the E1 strength function in terms\nof resonance position and width.\n=========================================================================\nPoint 3. Possibility of 1- state population.\nThe excitations of 10He with J>0 are only briefly discussed in this work\nas possibly contributing the high-energy part of the strength function.\nThis is not substantiated with any calculations. However, it was shown\nin [19] that due to extreme halo extent of the 11Li WF the specific mechanism of\nproton knockout may lead to population of 1- as the lowest energy\ncomponent of the 10He spectrum. It should be noted that mechanism of\nsuch a knockout is not just s-wave removal\nand do not lead to typical s-wave momentum distributions\nof the spectator in the outgoing channel (see Fig. 2 of [19]).\nThe population of the 1- configuration was predicted on the level up to\n25% to the 0+ population rate. However, the low-energy region, where\na specific enhancement was observed in the 11Li(-p) reaction, the\n1- may provide dominating contribution.\n-------------------------------------------------------------------------\nPlease check the following:\nText:\n\"As shown in Fig.4, the distributions can be well reproduced by the quasi\nfree knockout calculations that assume a p-wave proton removal from 11Li.\"\nCaption Fig.4:\n\"s-wave (red solid line) or p-wave (blue solid line)\"\nThere is agreement with the red solid line in the figure, which is in\ncontradiction with the text.\n-------------------------------------------------------------------------\n=========================================================================\nMinor things\nI find the presentation of data of [Korsheninnikov et al] misleading in Fig. 1.\nThe original paper contains the data on CD_2 and C targets, which are not in\nfull agreement. The conclusions of [Korsheninnikov et al] are based on subtracted,\nspectrum which has large error bars, while the authors of this work arbitrarily\nhave chosen one of initial spectra with comfortably small error bars.\nI sincerely believe that considering all the problematic issues discussed above\nthe deduced scattering length precision \"−3.45(20) fm\" is a strong overestimation\nof the achievable precision. Something like \"∼ -3.5 fm\" will be more realistic,\nsince criterion of the error determination is unclear.\nThe Authors use two Figures (2 and 3) to show the complete three-body correlation\npatterns and their projections. I think it could be more instructive to demonstrate\nevolution of the correlation patterns with energy. According to the theoretical\ninterpretation there are energy ranges where the [s^2]_0 component is totally\ndominant and where the [p^2]_0 contribution is important. It is a bit strange\nnot to try to observe the difference in the correlations.\nThe same is somehow true about Fig. 4. When gated on the excitations of 10He\nthese spectra may (or may not) demonstrate some differences related to\npeculiarities of reaction mechanisms populating [s^2]_0 and [p^2]_0 configurations\n(if there are any).\nI find the topic \"Deduction of the effective angular momentum for a\nthree-body system\". On one hand, the relevance of this derivation to the topic\nof the paper is doubtful. On the other hand, this derivation is valid within the\nspecific adiabatic framework. The conclusion of this derivation is trivial:\nbetween \"no s-wave interaction\" and \"Efimov case\" there will be a continuum\nof effective interactions as the pairwise scattering lengths (all together\nsimultaneously or in some separate manner) tend from zero to minus infinity.\nDo not think this deserve publication in the Nature.\n===========================================================================\nTo finalize the discussion. All the theoretical points above are quite complicated.\nI doubt that within the limited volume this particular publication the\nmentioned controversies could be well resolved.\nHowever, these theoretical discussions also do not devaluate the potential importance of\nthe current experimental observation and its interest to a broad scientific community.\nMy proposal is to reformulate the paper from modality \"solidly understood in a unique way\"\n(which I doubt the Authors can actually prove) to \"there are different possible\nintriguing interpretations of the data\". Possible alternative interpretations should\nbe clearly discussed in the paper. They are also not trivial and may be of significant\nscientific interest by themselves as ideas existing in parallel with ideology of\nEfimov-type phenomena. The title of the paper should be also changed to a broader\ncontext and statements about \"resonant states\" replaced everywhere with something\nneutral like \"peaks in the population cross section\".", "reviewer_2": "Report on \"First identification of a universal three-body s-wave resonance in $^{10}$He\"\nThe subject of this paper is the experimental identification of an Efimov state (resonance) in the $^{10}$He system.\nAlthough Efimov states were first observed in cold atom systems and have long\nbeen discussed in the context of atomic He$_{3}$, they were originally predicted\nwithin the realm of nuclear physics by Efimov. I believe it is extremely\nimportant to identify nuclear systems that can be described by this universal\nmodel, further highlighting the universality of Efimov physics. For this reason,\nI think the present paper, especially given its experimental nature, deserves\npublication.\nI have only a few remarks, which do not pertain to the experimental techniques\nthat are well explained.\n1. A simple remark is on line 6, where the emergence of an infinite tower of\nthree-body states is related, strictly speaking, to an infinite (and not just a\nsufficiently large) scattering length.\n2. The explanation of the resonance is mostly based on the shell model\ninterpretation of the state. It would be useful for a reader from another\ncommunity (thanks to the universality of Efimov physics) to at least provide a\nsketch of what is happening and why the state appears as a double-intruder\nnucleus (lines 45-..., 90-..., and 122).\n3. The explanation of the state rests heavily on the shell model. However, there\nare other ways to view it that rely on both the scattering length, which has\nbeen extracted (as mentioned again in line 199), and on the effective range\n(Kievsky A., Gattobigio M., Girlanda L., Viviani M., *Efimov Physics and\nConnections to Nuclear Physics.*, Annu. Rev. Nucl. Part. Sci **71**, 465,\n2021). It would be important to try to extract the effective range to allow\nfor other interpretations of the same state.\n4. The grand-angular momentum K is discussed in the Methods section, but I'm not\nsure that the introduction of K in line 171 without a previous explanation is\ncorrect.\nTo summarize, the paper deserves publication as it sheds a nuclear\nlight on Efimov physics, which originates from Nuclear Physics.", "reviewer_3": "Summary\nThe manuscript describes measurements of 10He produced through (p, 2p) reactions of a ∼250 MeV/u 11Li beam with a\nliquid hydrogen target. Analysis of the three-body relative energy and Jacobi correlation spectra utilized a coupled-channel\nthree-body model of 11Li combined with the quasi-free knockout reaction model, and complex-scaled solutions of the\nLippmann-Schwinger equation were used to compute final states in 10He. The low-energy structure in the three-body\nrelative energy spectrum is explained by two 0+ states with 8He cores, one with a [1s1/2]2 neutron component and another\nwith a [0p1/2]2 neutron component. By tuning the 8He-n interaction to fit the 10He spectrum below 2 MeV, the authors\nextracted an s-wave scattering length between 8He and the neutron. The Jacobi-T normalized relative energy and Jacobi-Y\nangle 8He-n-n correlations gated on relative energies < 1 MeV show enhancements that are qualitatively reproduced by the\nthree-body calculations. The authors conclude that the observation of the two 0+ states resolves the long-standing difference\nbetween 10He ground state energies measured from knockout and 2n transfer. Furthermore, the 10He ground state is a\nthree-body s-wave resonance that exemplifies a novel class of universal three-body bound states.\nRevisions\nThe paper is well-constructed, concise, and it provides new information suggesting a resolution to a long-standing question\nabout the structure of 10He. The following revisions would strengthen the paper and should be satisfactorily addressed:\n1. How is the 1-sigma energy resolution dependence as a function of relative energy determined? Does this come from a\nprevious measurement? A Monte Carlo simulation of the setup? A brief explanation or reference in Section 2 is necessary to\nsupport the stated relationship. An addition to the top panel in Fig. 1 to show the resolution alongside the efficiency or an\nillustration and accompanying explanation in the methods section would help further convince the reader of the setup’s\nsensitivity.\n2. Including additional information in the Methods section about the cross-talk rejection procedures would benefit the reader.\nFor example, reference [31] includes empirically derived values for the same-wall cross-talk rejection filters based on the\n7Li(p,n)7Be test case. Are similar values used here?\n3. Some clarification of the analysis procedure is needed to help the reader understand the comparison between data and\ncalculation shown in Fig. 1.\na. How does the output from the three-body 11Li model + quasi-free knockout model inform the calculation of the final 10He\nstates? Do the reaction calculations provide cross-sections that scale the contributions of the 10He states in the relative\nenergy spectrum?\nb. What parameters for the 8He-n interaction are varied and how? Including an additional figure in the Methods section that\nshows how these parameters impact the agreement with data would help reinforce the assertion from lines 131 – 133: “the\nspectrum’s shape is highly sensitive to the 8He-n interaction, and the observed low-energy structure is reproduced only with\na weakly attractive 8He-n s-wave interaction.”\n4. The legend and caption for Fig. 1 should be revised to clarify what is indicated by the green dashed curve labeled ‘others.’\nIt’s not clear if this is referring to the discussion in lines 124 – 131 regarding higher-lying states.\n5. Does the Argonne v8’ force include/account for the large neutron-neutron s-wave scattering length ann ∼ 19 fm referenced\nin line 174?\n6. Is it possible to extract cross-sections for populating the two 0+ states? Or a ratio of these cross-sections? This could be\nuseful for comparing to previous predictions e.g. reference [22].\n7. The [2s1/2]2 should be [1s1/2]2 to be consistent with notation elsewhere (e.g. the legend of Fig. 1 and line 196).\n8. In lines 173-174 and in the Methods section, the 10He ground state is shown to have an effective angular momentum\nK=1/2 which should be associated with a repulsive effective three-body potential indicated by the black curve in Fig. 5. The\nnext line seems contradictory: “This is physically due to that a neutron occupying the halo virtual state can mediate an\nattractive interaction between the other neutron and the 8He core up to large distance …” And line 180 emphasizes again\nthe 10He system shows long-range repulsion. The explanation of how the attractive interaction arises needs to be clarified.\n9. In the methods subsection “deduction of the effective angular momentum for a three-body system,” additional context is\nneeded around line 304 to explain or justify why the right hand side of Eq.(6) is O(1) so that the leading order term of the left-\nhand side must vanish."}, "author_response": "Reply to comments of Reviewer #1\n-----------------------------------------------------------------------------------------------------------\nThe new highly accurate data on 10He populated in the proton knockout reaction from 11Li halo\nnucleus is reported in this work. The Authors report on the low-energy enhancement in the\ninvariant mass spectrum of 10He, which was not identified in the previous studies due to\ninsufficient resolution or/and statistics. These data are interpreted as providing evidence to\nexistence of s-wave resonance in 10He, built on three attractive s-wave pairwise interactions.\nSuch a resonance is discussed as representative of the \"universal\" long-range dynamics of\nEfimov type. The search for these class of phenomenon was was so far unsuccessful in nuclear\nsystems, therefore such a discovery could be a real \"gem\" in the studies of exotic phenomena\nnear the nuclear driplines.\nIn my analysis of situation I fully rely on the reliability of the data. These are provided by one of\nleading groups in the field with broad expertise in analogous studies. So, in the following\nanalysis of the proposed interpretation complete reliability of the data is implied.\nThere are important questions about interpretation of the data. It can be found in the literature\nthat alternative interpretations are possible. These interpretations are for a long time \"on the\nmarket\" and should not be neglected without significant explanation. It should be noted that\nalternative interpretations are also not trivial and are related to questionable types of the\nlong-range dynamics in nuclear systems, which are also of significant interest.\nI find the experimental observations of this paper and various types of associated physical\nphenomena sufficiently interesting to be published in Nature. However, major revision of the\nmanuscript in aspects concerning the data interpretation is due (see the discussion below). I\nwould strongly recommend this paper for publication in Nature if my\ncriticism concerning interpretation is fully taken into account.\nReply: Thank you to the reviewer for the comments, especially for directing us deeper to the\npioneer theoretical work of Ref. [18] regarding the interpretation of the experimental data.\nFirstly, we would like to clarify the differences between Ref. [18] and the theoretical framework\nof the present work. One of the differences is in the treatment of the Pauli blocking in the 8He-n\ninteraction. The treatment of the Pauli blocking can affect both the 10He structure and the\ninitial-state 11Li wavefunction. Ref. [18] employs a local repulsive potential to take the effect of\noccupied 0s orbit into account. In our model, the Pauli forbidden states are excluded by using\n1/2\nthe so-called Pauli projection method with pseudopotential. A comparison between different\nmethods for the three-body states could be found in Ref. [I.J. Thompson et al., Phys. Rev. C 61\n(2000) 024318]. The differences are small for bound and resonant states, but larger for\ncontinuum states. Larger differences are expected for halo nuclei where the s-wave channel plays\na significant role, such as 11Li.\n[18] adopts Gaussian potentials for different partial waves for the 8He-n subsystem. The\nevolution of s-wave and p-wave components were checked by tuning the strengths of the s-wave\nand p-wave Gaussian potentials separately. In our theoretical model, we take the folding\npotential for the 8He-n subsystem. This does not allow us to tune the s-wave and p-wave\ninteractions separately. However, we have indeed checked the evolution of the s&p components\nby tuning the potential depth of the folding potential. Considering the uncertainty of the 8He-n\ninteraction, this is the only parameter we have tuned to reproduce the experimental data.\nSecondly, we would like to emphasize that our main conclusion that the observed 10He ground\nstate is a universal 3-body s-wave resonance related to the prediction by Hongo and Son (Ref.\n[10]) doesn’t change depending on the details of nuclear-structure interpretation or the sign of\nneutron-core scattering length. Since the Hongo-Son state only requires a small “absolute value”\nof the core-neutron scattering length, even if the neutron-core scattering length is slightly\npositive, the 10He resonance can still be interpreted as an unbound analogue of the universal\nHongo-son state; The sign of the scattering length only plays a very minor role as shown in Ref\n[10]. Furthermore, in Ref. [18], it is also stated that “The state with the [s ]2 structure should\n1/2\nbe characterized by domination of the component with the lowest possible value of the\ngeneralized angular momentum K = 0. However, the centrifugal barrier L(L + 1)/2Mρ2 is\nnot zero even in the channel with K = 0, as it depends on the “effective angular momentum”\nL = K + 3/2. This means that the low-energy three-body virtual state may exist in the form\nof a real resonance peak, not a threshold peculiarity as the two-body virtual state.” Our\nclaim on the universal properties of the 3-body s-wave state is consistent with the above\nstatement in Ref. [18]. We note that the definition of “effective angular momentum” in Ref. [18]\nbased on the hyperspherical harmonics model is slightly different to our definition based on the\nFaddeev approach. We take the induced long-range interaction into account. In our definition, the\neffective angular momentum for the three-body s-wave state is reduced from non-interacting\nvalue 3/2 (lowest value given in Ref. [18]) to 1/2 due to the mediated attraction originating from\nthe hyper-angular motion.\n==========================================================\nPoint 1. Scattering length in the 9He=8He+n.\nBroad theoretical exploratory studies of the 10He spectra sensitivity to the s-wave 8He-n\ninteraction have been performed in Ref. [18] by Grigorenko and Zhukov.\n-------------------------------------------------------------------------\nSince further we rely on the results of Refs. [18, 19], it is necessary to comment to their\ndescription in this work:\n\"Currently, the most advanced theoretical work on 10He is given in Refs. [18, 19]. On the one\nhand, it has been discussed that the 10He ground state could be at ∼ 2 MeV, as observed by the\n2\n2n transfer reaction, and has a [0p_1/2]^2_{0+} structure, while\nthe initial state effects due to the halo nature of 11Li or 14Be may lower the energy of 10He to ∼1.5 MeV as observed in\nknockout reactions. Calculations with such effects qualitatively\nagree with the 10He invariant-mass spectrum from 11Li(-p) [19] but cannot explain the spectrum\nfrom 14Be(-2p2n) [20].\"\n(1) The low energy strength in the 10He spectrum has mixed character [0s_1/2]^2_{0+} +\n[0p_1/2]^2_{0+} with around 50% mixing. Important special attention was paid in [18] to\nevolution of these components in various conditions.\nReply: We agree with the reviewer that the low-energy region of the 10He spectrum has mixed\ncharacter [1s ]2+ [0p ]2. But in our case, it is not 50% mixing. As shown in Fig. 1 of our\n1/2 1/2\nmanuscript, the percentages of the [1s ]2 and [0p ]2 components are 72% and 26%,\n1/2 1/2\nrespectively, which in our calculations mainly depend on the initial state wave function of 11Li,\nthe final 10He wave function (governed by the 8He-n and n-n interactions), and also the phase\nspace of the final 10He states as a function of the excitation energy. As mentioned before, in our\ntheoretical model, we take the folding potential for the 8He-n subsystem. This does not allow us\nto tune the s-wave and p-wave interactions separately as done in Ref. [18]. However, we have\nindeed checked the evolution of the s&p components by tuning the potential depth of the folding\npotential. Considering the uncertainty of the 8He-n interaction, this is the only parameter we have\ntuned to reproduce the experimental data.\n(2) The statement that the results if [19] \"cannot explain the spectrum from 14Be(-2p2n)\" is not\nrelevant, since the calculations in [19] were about alpha knockout, while there were no clear\nidentification of the reaction channel in [20] and \"14Be(-2p2n)\" (not alpha removal!) is just one\nof reasonable assumptions of this work.\nReply: We agree with the reviewer that the calculations in Ref. [19] assumed a direct α removal\nfrom 14Be, while 10He data in Ref. [20] could be populated by either 2p2n removal or α removal\nfrom 14Be. Nevertheless, we think the discussions in Ref. [19] and Ref. [20] show an important\nstep to understand the 10He spectrum from 14Be. We would like to keep this sentence and modify\nit accordingly as below:\n“Calculations with such effects qualitatively agree with the 10He invariant-mass spectrum from\n11Li(-p) [19] but similar calculations assuming a direct α-particle removal from 14Be cannot\nexplain the spectrum from 14Be(-α or -2p2n) [20].”\n---------------------------------------------------------------------------\nLet's have a look in Fig. 5f of this work. The solid curve shows the spectrum which is very\nconsistent with the new experimental data. In Fig. 6c the decomposition of exactly this spectrum\nis shown demonstrating energy split between [s^2]_0 and [p^2]_0 components at about 1 MeV,\nwhich is, again, very consistent with theoretical results shown in Fig. 1 of this work. The\ncomplete three-body correlations for this case also can be found in Fig. 13c of Ref. [18]. These\n3\nare calculated not exactly for the excitation energy range, used in Fig. 2 of the Refereed paper,\nbut they looks very similar to the recent data. So, the agreement of this specific theoretical\nprediction with the data seem to be very good. However, this specific calculation corresponds to\nquite a different scattering length 2.9 fm in the 8He+n, which is not ATTRACTIVE, but\nREPULSIVE. This is not in accord about the basic message of this work about universal\nEfimov-type phenomenon.\nLet's have a look on the situation in a different way. The energy of the p_{1/2} resonant state\nused in the calculations of this work is mentioned to be 0.76 MeV. The results for such p_{1/2}\nposition can be found in Fig. 5d of [18]. They should be between dotted and dash-dotted curve,\nproducing peak between 30 and 200 keV. So, there seem to be strong\ncontradiction between theoretical approaches of this work and of Ref. [18]. According to [18] the\nuniversal s-wave structure built on the s-wave attractions in all three subsystems should have been seen as\nextreme low-energy (100 keV? 200 keV?) structure in this reaction.\nReply: Since the reviewer thinks of the solid curve in Fig. 5f or Fig. 6c in Ref. [18] looks very\nconsistent with the experimental data, we have compared this curve with the data. The results are\nshown in the figure below. Note that we have convoluted the curve with our experimental\nresolution, and to be consistent with our analysis, the amplitude of the curve has been tuned to\nbetter reproduce the data below 2 MeV. We found that this calculation reproduces the shape of\nthe data reasonably well, although the distribution below 1 MeV is slightly overestimated\ncompared to our theoretical calculations. This calculation corresponds to a 8He-n s-wave\nscattering length of +2.9 fm and a p-wave resonance of 9He at 2 MeV.\nWe would like to note that a different scattering length of +2.9 fm does not change the basic\nmessage of the current work. Although the position of the p-wave resonance of 9He is different\nwith ours, it is consistent with our interpretation that the distribution below 1 MeV is dominated\nby the configuration of two neutrons in the s-wave state. 1) In the Faddeev approach, this state,\nwith a positive peak position and a finite width, could exist due to the universal repulsive\nhyper-radial potential in the three-body system. Small positive scattering length does not change\n4\nthis conclusion. 2) Since the Hongo-Son state only requires a small absolute value of the\ncore-neutron scattering length, this state is still an unbound analogue of the universal Hongo-Son\nstate. We have added this information in “Theoretical calculations of the 10He energy spectra” in\nthe Method section.\n“We note that previous theoretical calculations of Ref. [18], as shown in Fig. 6(c) of Ref. [18],\nwhich utilized 8He-n Gaussian potentials tuned to induce a p-wave resonance of 9He around 2\nMeV and an a around +2.9 fm, also reproduce the observed low-energy structure in the 10He\ncn\nspectrum reasonably well. In this case, the partial decomposition in Ref. [18] leads to a\nconsistent conclusion with us that the 10He spectrum below 1 MeV is dominated by the\nconfiguration of two neutrons in the s-wave state. We also note that the universal halo states\npredicted by Hongo and Son [10] only require a small absolute value of the core-neutron\nscattering length. Even if a is slightly positive, the 10He resonance can still be interpreted as an\ncn\nunbound analogue of the universal Hongo-Son state.”\nOn the other hand, according to Ref. [18], Fig. 5d corresponds to a p-wave resonance of 9He\naround 0.74 MeV (indeed close to our value of 0.76 MeV.), and various s-wave scattering lengths\nof -10/-5/-1.4/1.8/2.9 fm (-5 fm is close to our value of ∼-3.5 fm). All curves in Fig. 5d shows\nnarrow peaks below 1 MeV that apparently cannot reproduce the experimental data. This is\nindeed in contradiction with our calculation results. It is indeed valuable to investigate the\nreasons between the calculations of Ref. [18] and ours. We have tried our best to investigate the\nreasons based on the information given in Ref. [18].\n1) We found that the initial state wave function of 11Li (details given in N.B. Shulgina et al.,\nNuclear Physics A 852 (2009) 175-199) adopted in Ref. [18] and ours are rather consistent. Both\nwave functions consider a large mixture of the s-wave and the p-wave configurations, and\nreproduce the charge-radius, matter radius of 11Li and 9Li.\n2) As mentioned in the previous reply, one reason may be related to the different 8He-n potential\nused in our model and in Ref. [18]. Ref. [18] adopted Gaussian form potentials. In our model, we\nemploy a folding potential with the 8He core density for the 8He-n interaction. The explicit form\nof the potential is given in Ref. [S. Aoyama et al., Phys. Rev. C 55, 393 2379–2384 (1997).]. We\nhave added this information in “Theoretical calculations of the 10He energy spectra” in the\nMethod section.\n3) Another reason may be related to the treatment of the Pauli blocking for the 8He-n interaction.\nThis will result in differences in the internal part of the 8He-n-n three-body wave functions. Ref.\n[18] includes a Pauli repulsive core in the 8He-n interaction. In our model, for the final scattering\nstates, we use the complex-scaled solutions of the Lippmann-Schwinger equation (CSLS) with\nthe 8He + n + n orthogonality condition model (OCM). The Pauli forbidden states are excluded\nby using the so-called Pauli projection method with pseudopotential. A comparison between\ndifferent Pauli blocking methods for the three-body states could be found in Ref. [I.J. Thompson\n5\net al., Phys. Rev. C 61 (2000) 024318]. The differences between “Pauli repulsive core method”\nand “Pauli projection method” are small for bound and resonant states, but larger for continuum\nstates. Larger differences are expected for halo nuclei where the s-wave channel plays a\nsignificant role, such as 11Li. As far as we know, Pauli repulsive core method is an approximate\nmethod to exclude the Pauli forbidden states from the inter-cluster motion but cannot correctly\ndescribe nodal behaviors of the inter-cluster motion, while the OCM is a semimicroscopic\napproach in which the Pauli forbidden states are excluded accurately by using the projection\noperator. We have added this information in “Theoretical calculations of the 10He energy spectra”\nin the Method section.\n“We would like to comment on the different treatment of the Pauli principle between our work\nand that presented in Ref. [18]. In the 8He + n + n OCM, the Pauli blocking in the 8He-n\nsubsystem is taken into account by using the projection operator, the so-called pseudo potential,\nwhich excludes the Pauli forbidden states (occupied states by neutrons in the 8He core) from the\n8He-n relative motion microscopically. This treatment of the Pauli blocking is important to\ninvestigate the knockout reaction, especially for the s-waves in the 8He-n subsystems, because\nthe internal wave function is crucial to determine the distribution function W quantitatively. The\nanalysis in Ref. [18] has been performed using the core repulsion potential to exclude the Pauli\nforbidden states, resulting in a different internal wave function of 10He from the microscopic\ncalculations as shown in Ref. [55]. The Pauli repulsive core method is an approximate method to\nexclude the Pauli forbidden states from inter-cluster motions but cannot correctly describe nodal\nbehaviors of inter-cluster motions. By contrast, the OCM is a semimicroscopic approach in\nwhich the Pauli forbidden states are excluded accurately via projection operators.”\nI would mention one qualitatively suspicious moment in the provided theoretical calculations.\nBased on the energy of p_{1/2} state 0.76 MeV, the trivial independent particle approximation\nshould provide the energy ∼1.5 MeV for the [p_{1/2}^2]_0 configuration. The peak energy for\nthis configuration in Fig. 1 of this work is ∼1.9 MeV. This means that the paring interaction\nis effectively repulsive for the [p_{1/2}^2]_0 configuration. This is quite unexpected result, which\nI see as strange for the three-body model and which it is not within the systematics of effective\npairing energies for the p_{1/2} subshell, which can be traced for the other members of N=8\nisotone. In Ref. [18] these effective pairing energies vary from 0.5 to 1.8 MeV, which is smaller\nthen the typical values in the p_{3/2} subshell, but not negative at least.\nThe strong theoretical controversy is quite intriguing, considering the nice agreement between\nthe three-body calculations of this group [33] and [Grigorenko et al. PRC 102, 014611 (2020)]\nfor another quite complicated peripheral phenomenon: soft dipole excitation in the continuum of\n6He system.\nReply: In the present calculation, the energy of the p state of 9He is 0.76 MeV with a decay\n1/2\nwidth of 0.95 MeV, and the 0 + resonance of 10He is located at 1.4 MeV in the complex-energy\n2\nplane, with its decay width of 1.0 MeV. Since the [0p ]2 decay channel has a small phase space\n1/2\nin the low-energy region, the cross section decaying into this channel is suppressed in the\n6\nlow-energy region, and then the peak is shifted to slightly higher energy (∼1.9 MeV in Fig. 1)\nthan the expected one. We also note that the 0 + resonance has a small s-wave mixing and such\n2\nan s-wave component also decays into the [1s ]2 channel shown as the red dashed line in Fig. 1.\n1/2\nIt is not impossible that quite old calculations of Ref. [18] are deficient in some way. However,\nunless this deficiency is demonstrated they can not be just disregarded: these are published and\nwell documented results, while for the calculations of this work not much details are available.\nThe Authors should either find out what is wrong with calculations of [18] or rely on them as\nalternative interpretation of the data.\nReply: In Ref. [18], the detailed analysis has been performed and, we think, the calculations are\nquite accurate within the model the authors employed. However, they use the Pauli repulsion\ncore for the 8He-n potential, and the wave function obtained with the repulsion core is much\ndifferent from that obtained in the microscopic calculations, such as RGM and OCM, especially\nin the internal region. To discuss the initial state effect, namely the effect of the “11Li source”,\nquantitatively, an accurate description of the internal wave function would be crucial. We think\nthat the analysis in Ref. [18] is thought-provoking and is correct in a qualitative way, but it is not\nsufficient in a quantitative way. In the Method section of the revised manuscript, we have\nexplained this point in detail. We think more detailed investigations, such as performing\ncalculations with the same treatment of Pauli blocking, will be beyond the scope of the current\nwork. As an alternative interpretation of the data, it is now explicitly mentioned in the Method\nsection that the calculations of Ref. [18] with a scattering length of +2.9 fm reproduce the data\nreasonably well. However, this does not change the main conclusion of this work.\n---------------------------------------------------------------------------\nSome separate note concerning the 8He+n scattering length issue. I would like to pay attention of\nthe Authors to results of paper [Khirk et al., Phys. Rev. C 111, L051601 (2025)].\nIn this work the data of Uberseder et al. [39] were reinterpreted with more advanced treatment\nof isospin symmetry. The scattering length in the 8He+n channel deduced in this work was\nweakly attractive as -2.7 fm, with no need for s_{1/2} resonance. This is quite close to the\ndeduced in this work value −3.45 fm. However, the p_{1/2} state should be at about 2.3 MeV\n(original analysis of Uberseder et al. do not allow any resonances with l>0 and E<3 MeV at all).\nReply: We thank the reviewer for pointing out the recent work by Khirk et al. But after going\nthrough the paper [Khirk et al., Phys. Rev. C 111, L051601 (2025)], we do not find the value of\n-2.7 fm mentioned by the reviewer. We have modified our manuscript to include the conclusion\nof this work on the 8He-n s-wave interaction.\n“The calculation results for the 8He-n subsystem, with the tuned 8He-n interaction to fit the\ncurrent 10He spectrum below 2 MeV, produced an s-wave scattering length between 8He and\nneutron (a ) of −3.45(20) ∼3.5 fm, in reasonable agreement with the results from some knockout\ncn\nreactions(−3.17(66) fm [14], −3 fm < a < 0 fm [37], −2.1(6) fm or −6.8(14) fm [38]) and the\ncn\nweak 8He-n s-wave attraction from the (p, p) resonance elastic scattering [39], but not with the\n7\nstrong one (a ≤ −10 fm) reported in Ref. [40] and Ref. [41]. A recent analysis of the (p, p)\ncn\nresonance elastic scattering data with more advanced treatment of the isospin symmetry also\nsuggests a small negative a [Khirk et al…]. …”\ncn\n---------------------------------------------------------------------------\nPoint 2. Reaction mechanism issue.\nThe structures promoted in this paper as \"two low-lying 0+ states of 10He at ∼1 MeV and ∼2\nMeV above its two-neutron decay threshold\" according to calculations of [18] are strongly\nsensitive to the reaction mechanism. Let's compare solid curves Fig. 5 c and f in [18]. One is well\nconsistent with the data of the refereed work the other is shifted to higher energy to more than 1\nMeV. The same if we compare Fig. 6 b and c in [18]. For one the [s^2]_0 and [p^2]_0\ncomponents are well energy split (as in the calculations of this work), but for the other they more\nor less overlaps, so we can hardly speculate about separate [s^2]_0 and [p^2]_0 states.\nThe results of the refereed work are based on some realistic reaction model where the 11Li WF is\npresent. So, all these effects should be there, they are just not discussed. These could be easily\nchecked if I am right by the Authors of this work in two reasonably simple ways:\n(1) By performing calculations without 10He FSI - just plane waves.\nTypically people try to check, whether the signal they see is not provided just by a plane wave,\nwithout any dynamics.\nReply: We have indeed performed calculations without any FSI and with only the n-n FSI. As\nshown in the figure below, deactivating the FSI results in a shift of the excitation strength\ntowards higher energies, consistent with the disappearance of the [0p ]2 resonance and a\n1/2\nbroadening of the peak from the [1s ]2 neutrons. Calculations without any FSI or with only the\n1/2\nn-n FSI produce a bump below 1 MeV, and cannot reproduce the experimental E distribution.\nrel\nWe therefore conclude that it is important to include the 8He-n FSI for the observed 10He\nstructure. We have also added this information in “Theoretical calculations of the 10He energy\nspectra” in the Methods section.\n8\n(2) By changing the reaction mechanism by using not a halo nucleus 11Li, but a compact system.\nFor example 14C(-4p). Not very realistic reaction for practical realization, but theoretically we\nmay play games. If the strength function is strongly qualitatively sensitive to the reaction\nmechanism, then you can hardly speculate about resonance states. Resonant\nstates is something independent on reaction mechanism by definition.\nAlso, if your population cross section strongly depends on the reaction mechanism then what is\nyour \"universality\" about? This is not about that the story is not interesting and important, but\nabout carefully choosing terms for the discussion.\nI am a bit amused that this topic is total missing in the manuscript, since the Authors has recently\nstudies soft dipole excitation in 6He [33], where they do not attempt to interpret the E1 strength\nfunction in terms of resonance position and width.\nReply: We are not sure about the terminology of “reaction mechanism” mentioned by the\nreviewer. It seems to us it means the “initial-state effect”. If so, in our opinion, the population\ncross section is always strongly dependent on the reaction mechanism. Regarding the position\nand width of a resonance, they might also depend on the reaction mechanism. For a narrow\nresonance, its position and width should be independent of the reaction mechanism. On the other\nhand, for a broad resonance, the position and width may show dependence on the initial state, as\ndemonstrated by the solid curves in Fig. 5 c and f in Ref. [18].\nIn the present analysis, we assume that the proton knockout does not disturb the configuration of\nthe valence neutrons, so that the decaying neutrons are spectators through the reaction. Under\nthis assumption, the \"sources\" of [1s ]2 and [0p ]2 configurations are generated by the equal\n1/2 1/2\namount to each other just after the knockout process. We also note that the source effect in the\npresent calculation has a smaller contribution than that discussed in Ref. [18]. Details of our\ncalculations have been added in the Method section.\nThe “universality” in our work does not refer to the energy position and width of the 10He states\nis universal, instead it refers to that we found the 10He ground state populated from 11Li(p, 2p)\nreaction has a universal property, i.e. it is a three-body s-wave state governed by a universal\nrepulsion. Considering the small absolute value of the core-neutron scattering length, we propose\nthat this state is an unbound analogue of the universal Hong-Son halo state. We note again that\nthe above two aspects (universal repulsion; candidates of universal halo) only require the\nabsolute value of 8He-n scattering length is small, and the sign of the 8He-n scattering length\ndoes not matter.\nWe think discussion of the population of 10He from other reaction channels like 14C(-4p) is\nbeyond the scope of the current work. To the best of our knowledge, no reliable theoretical\nframework currently exists that can treat four-proton removal. But it is indeed interesting to\n9\nperform such an experiment to see whether the same 10He states (or similar peak structure) could\nbe populated.\nFinally, the soft dipole excitation of 6He [33] is another story. In Ref. [33], 6He is populated via\nCoulomb breakup. It has been reported for a long time that the Coulomb breakup reaction of halo\nnuclei populates continuum states instead of resonances. Indeed, the soft dipole excitation of 6He\nfrom the Pb target was found to be dominated by the 1- continuum state. Therefore, there are no\ndiscussions in Ref. [33] about resonance and its position/width.\n===============================================================\nPoint 3. Possibility of 1- state population.\nThe excitations of 10He with J>0 are only briefly discussed in this work as possibly contributing\nthe high-energy part of the strength function. This is not substantiated with any calculations.\nHowever, it was shown in [19] that due to extreme halo extent of the 11Li WF the specific\nmechanism of proton knockout may lead to population of 1- as the lowest energy component of\nthe 10He spectrum. It should be noted that mechanism of such a knockout is not just s-wave\nremoval and do not lead to typical s-wave momentum distributions of the spectator in the\noutgoing channel (see Fig. 2 of [19]).\nThe population of the 1- configuration was predicted on the level up to 25% to the 0+ population\nrate. However, the low-energy region, where a specific enhancement was observed in the\n11Li(-p) reaction, the 1- may provide dominating contribution.\nReply: In our calculations, in the same manner as Ref. [19], we assume a p -orbitalproton is\n3/2\nremoved from the 9Li core populating a 8He core in the ground state. The neutrons of 11Li thus\nremain as spectrators, i.e. two neutrons always stay in the same orbitals and are coupled to 0+\nstates. Therefore, the populated 10He final state from 11Li could be only in positive parity states.\nThe population of 10He depends on the initial state configuration. In Ref. [19], the 1- state are\ndominated by two components of hyperspherical decomposition: K = 1 (one neutron in p wave\nand one neutron in s wave) and K = 3 (one neutron in p wave and one neutron in d wave.). Our\ninitial state wave function of 11Li does not have such configurations, therefore, no 1- state is\npopulated in our calculations.\n-------------------------------------------------------------------------\nPlease check the following:\nText: \"As shown in Fig.4, the distributions can be well reproduced by the quasi free knockout\ncalculations that assume a p-wave proton removal from 11Li.\" Caption Fig.4: \"s-wave (red solid\nline) or p-wave (blue solid line)\"\nThere is agreement with the red solid line in the figure, which is in contradiction with the text.\nReply: Thank you to the reviewer for pointing out the typo in the text. We have modified the\ncaption of Fig. 4 to “s-wave (red blue solid line) or p-wave (blue red solid line)”\n==============================================================\nI find the presentation of data of [Korsheninnikov et al] misleading in Fig. 1. The original paper\ncontains the data on CD_2 and C targets, which are not in full agreement. The conclusions of\n[Korsheninnikov et al] are based on subtracted, spectrum which has large error bars, while the\nauthors of this work arbitrarily have chosen one of initial spectra with comfortably small error\nbars.\nReply: We did not arbitrarily choose the initial spectra of Korsheninnikov et al. We actually\nfollowed the previous work of [H. Johansson et al. Nuclear Physics A 354 842, 15 – 32 (2010).],\nwhich also compared the GSI data with the data on CD2 target.\nIndeed, in the paper of Korshenininikov et al., both the CD2 target data and deuteron target data\nare shown. It is mentioned explicitly in [Korsheninnikov et al] that the peak position from CD2\ntarget, C target and deuteron target are the same. We therefore do not think showing the data on\nCD2 target is problematic.\nI sincerely believe that considering all the problematic issues discussed above the deduced\nscattering length precision \"−3.45(20) fm\" is a strong overestimation of the achievable precision.\nSomething like \"∼ -3.5 fm\" will be more realistic, since criterion of the error determination is\nunclear.\nReply: The scattering length value of -3.45(20) fm was obtained by chi2 minimization for the\nexperimental spectrum below 2 MeV with tuning of the 8He-n interaction strength that produces\ndifferent s-wave scattering length in our calculation (See also “Theoretical calculations of the\n10He energy spectra” in the updated Methods section). As shown in the figure below (all\ndistributions with area normalized to unity), the distributions become flatter when the absolute\nvalues of 8He-n scattering length become smaller. The 10He spectrum below 2 MeV is well\nreproduced with an a value around -3.45 fm. Considering that -3.45(20) is a model dependent\ncn\nvalue and the model in Ref. [18] with a slightly positive scattering length +2.9 fm also\nreproduces the data reasonably well, we agree with the reviewer to use a value of ∼ -3.5 fm\nthroughout the text.\npatterns and their projections. I think it could be more instructive to demonstrate the evolution of the\ncorrelation patterns with energy. According to the theoretical interpretation there are energy\nranges where the [s^2]_0 component is totally dominant and where the [p^2]_0 contribution is\nimportant. It is a bit strange not to try to observe the difference in the correlations.\nThe same is somehow true about Fig. 4. When gated on the excitations of 10He these spectra\nmay (or may not) demonstrate some differences related to peculiarities of reaction mechanisms\npopulating [s^2]_0 and [p^2]_0 configurations (if there are any).\nReply: It is indeed very interesting to investigate the evolution of the correlation patterns and the\nmomentum distributions with the excitation energy of 10He. In the current work, we see that the\nmomentum distributions do not show significant dependence on the excitation energy, while the\ncorrelation patterns indeed show some dependencies. However, since the current work focuses\non the three-body s-wave state below 1 MeV, and the current calculations do not reproduce the\nE distribution above 2 MeV, such investigations will be done in the future.\nrel\nI find the topic \"Deduction of the effective angular momentum for a\nthree-body system\". On one hand, the relevance of this derivation to the topic\nof the paper is doubtful. On the other hand, this derivation is valid within the\nspecific adiabatic framework. The conclusion of this derivation is trivial:\nbetween \"no s-wave interaction\" and \"Efimov case\" there will be a continuum\nof effective interactions as the pairwise scattering lengths (all together\nsimultaneously or in some separate manner) tend from zero to minus infinity.\nDo not think this deserve publication in the Nature.\nReply: We respectfully disagree with the reviewer’s opinion. Our experimental data has clearly\ndemonstrated the existence of the s-wave three-body resonance peak, and therefore it is\nnecessary to explain at least qualitatively why an s-wave resonant state can emerge; Textbooks\non quantum mechanics often emphasize that higher angular-momentum two-body states can\nexhibit resonant behavior, while s-wave state does not show resonant state but rather becomes a\nvirtual state due to the absence of centrifugal repulsion. In contrast, our data reveals an s-wave\nresonance, which makes it essential to explain its physical origin. The effective angular\nmomentum discussion, although qualitative, offers a transparent physical picture: the s-wave\nthree-body resonance arises from the interplay between a short-range attraction and long-range\nuniversal repulsion. Regarding the adiabatic approximation, which the reviewer has expressed\ndoubts about, we would like to emphasize that it remains a well-established and reliable\napproximation as long as the condition |a | >> R >> |a | is satisfied (see Braaten Hammer\nnn He-n\nPhys. Rep. 428, 259 (2006) or Naidon SciPost Phys. 15, 123 (2023) ). We acknowledge that the\nthree-body resonant state may also be interpreted in an alternative framework (e.g., conventional\nnuclear structure theories). However, it is equally important to recognize recent advances in\nuniversal halo three-body studies, such as the work of Hongo and Son Phys. Rev. Lett. 2022. For\nthis reason, we believe it is scientifically fair to retain the effective angular momentum argument\n12\nand its connection to the Hongo–Son’s work, while also revising the manuscript to reflect the\nreviewer’s valid point that other interpretations deserve to be mentioned [see revisions made in\nresponse to the reviewer’s other comments].\nWe have added a short sentence to clarify the validity of the adiabatic approximation in the\npresent work:\n“Under the adiabatic approximation [2], which is valid for the current 8He-n-n three-body system\nin the region of a >> R >> a , the hyperradial functions F (R) in Eq. (5) are the solutions of\nnn cn n\nthe hyperradial equation:”\n==============================================================\nTo finalize the discussion. All the theoretical points above are quite complicated. I doubt that\nwithin the limited volume this particular publication the mentioned controversies could be well\nresolved. However, these theoretical discussions also do not devaluate the potential importance\nof the current experimental observation and its interest to a broad scientific community.\nMy proposal is to reformulate the paper from modality \"solidly understood in a unique way\"\n(which I doubt the Authors can actually prove) to \"there are different possible\nintriguing interpretations of the data\". Possible alternative interpretations should be clearly discussed in the\npaper. They are also not trivial and may be of significant scientific interest by themselves as\nideas existing in parallel with ideology of Efimov-type phenomena. The title of the paper should\nbe also changed to a broader context and statements about \"resonant states\" replaced everywhere\nwith something neutral like \"peaks in the population cross section\".\nReply: We thank the reviewer again for the valuable comments. There are indeed differences\nbetween our calculations and that in Ref. [18]. In the revised version of the manuscript, we have\nadded details of our calculations and also explain the reasons that may result in these differences.\nWe think more detailed investigations, such as performing calculations with the same potentials\nand/or with the same treatment of Pauli blocking, will be beyond the scope of the current work.\nAlthough there exist differences between our calculations and that in Ref. [18], we do not think\nthese differences will change the main conclusion of present work, i.e. the low-energy state\nbelow 1 MeV of 10He is a three-body s-wave state and it could be an unbound analogue of the\nuniversal halo state predicted recently by Hong and Son. The three-body s-wave state appears in\nthe experimental spectrum as a real resonance peak distinct from the two-body s-wave virtual\nstate. We therefore respectfully disagree with the reviewer’s opinion to change the title of the\npaper.\n-----------------------------------------------------------------------------------------------------------\n1) Comment:\nThe subject of this paper is the experimental identification of an Efimov state (resonance) in the\n10He system.\nAlthough Efimov states were first observed in cold atom systems and have long been discussed\nin the context of atomic He3, they were originally predicted within the realm of nuclear physics\nby Efimov. I believe it is extremely important to identify nuclear systems that can be described\nby this universal model, further highlighting the universality of Efimov physics. For this reason,\nI think the present paper, especially given its experimental nature, deserves publication.\nI have only a few remarks, which do not pertain to the experimental techniques that are well\nexplained.\n1. A simple remark is on line 6, where the emergence of an infinite tower of three-body states is\nrelated, strictly speaking, to an infinite (and not just a sufficiently large) scattering length.\nReply: We have modified the text to “…energy and spatial structures when the inter-particle\nscattering length a is sufficiently infinitely large.”\ns\n2. The explanation of the resonance is mostly based on the shell model interpretation of the state.\nIt would be useful for a reader from another community (thanks to the universality of Efimov\nphysics) to at least provide a sketch of what is happening and why the state appears as a\ndouble-intruder nucleus (lines 45-..., 90-..., and 122).\nReply: We have modified the text to explain the reason.\n“The 10He, which is traditionally considered as a doubly magic nucleus, thus appears as a\n“double-intruder” nucleus with the last two neutrons in the 1s shell, in line with the intruder\n1/2\ns-wave dominated ground state of 9He. The intruder dominant configuration could be understood\nin the shell-model framework due to the melting of the N = 8 shell closure for the neutron-rich\nisotopes below Z = 4.”\n3. The explanation of the state rests heavily on the shell model. However, there are other ways to\nview it that rely on both the scattering length, which has been extracted (as mentioned again in\nline 199), and on the effective range (Kievsky A., Gattobigio M., Girlanda L., Viviani M.,\n*Efimov Physics and Connections to Nuclear Physics.*, Annu. Rev. Nucl. Part. Sci **71**, 465,\n2021). It would be important to try to extract the effective range to allow for other interpretations\nof the same state.\nReply: We have tried to extract the effective range, but compared to the scattering length, the\nexperimental E distribution is not sensitive to the effective range, and the extracted value has\nrel\nlarge uncertainties. On the other hand, our explanation of the state as a candidate of the universal\nrecent work [https://arxiv.org/abs/2507.19171] that including effective range correction does not\nsignificantly change the universal property of the Hongo-Son state.\n4. The grand-angular momentum K is discussed in the Methods section, but I'm not sure that the\nintroduction of K in line 171 without a previous explanation is correct.\nReply: The effective angular momentum K, which characterizes the long-range behaviour the\nthree-body potential, comes naturally when the low-energy 3-body problem is treated using the\nhyper-spherical formalism. We have added an explanation of the grand-angular momentum K.\n“Such an s-wave resonance peak appears only in systems with ≥ 3 particles and can be attributed\nto a non-zero value of the effective angular momentum K, which characterizes the long-range\nbehaviour of the three-body potential [2, 11, 18] (see Methods for the definition of K).”\nTo summarize, the paper deserves publication as it sheds a nuclear light on Efimov physics,\nwhich originates from Nuclear Physics.\n-----------------------------------------------------------------------------------------------------------\nSummary\nThe manuscript describes measurements of 10He produced through (p, 2p) reactions of a ∼250\nMeV/u 11Li beam with a liquid hydrogen target. Analysis of the three-body relative energy and\nJacobi correlation spectra utilized a coupled-channel three-body model of 11Li combined with\nthe quasi-free knockout reaction model, and complex-scaled solutions of the\nLippmann-Schwinger equation were used to compute final states in 10He. The low-energy\nstructure in the three-body relative energy spectrum is explained by two 0+ states with 8He\ncores, one with a [1s1/2]2 neutron component and another with a [0p1/2]2 neutron component. By\ntuning the 8He-n interaction to fit the 10He spectrum below 2 MeV, the authors extracted an\ns-wave scattering length between 8He and the neutron. The Jacobi-T normalized relative energy\nand Jacobi-Y angle 8He-n-n correlations gated on relative energies < 1 MeV show enhancements\nthat are qualitatively reproduced by the three-body calculations. The authors conclude that the\nobservation of the two 0+ states resolves the long-standing difference between 10He ground state\nenergies measured from knockout and 2n transfer. Furthermore, the 10He ground state is a\nthree-body s-wave resonance that exemplifies a novel class of universal three-body bound states.\nRevisions\nThe paper is well-constructed, concise, and it provides new information suggesting a resolution\nto a long-standing question about the structure of 10He. The following revisions would\nstrengthen the paper and should be satisfactorily addressed:\n1. How is the 1-sigma energy resolution dependence as a function of relative energy determined?\nDoes this come from a previous measurement? A Monte Carlo simulation of the setup? A brief\nexplanation or reference in Section 2 is necessary to support the stated relationship. An addition\nto the top panel in Fig. 1 to show the resolution alongside the efficiency or an illustration and\naccompanying explanation in the methods section would help further convince the reader of the\nsetup’s sensitivity.\nReply: The energy resolution was obtained based on Monte Carlo simulation of the setup. As\nsuggested by the reviewer, we have added additional information in the Methods section about\nthe energy resolution:\n8He-n-n three-body relative energy. Each black point corresponds to the resolution for one fixed relative energy in\nthe simulation, and the black solid line is the fit result which has been given in the main text. The lower panel\nshows the simulated relative energy distributions for fixed relative energies of 0.1 MeV, 0.5 MeV, 1.0 MeV, 2.0\nMeV, 4.0 MeV and 6.0 MeV, respectively. The red solid lines are the fit results using Gaussian functions.\n“The relative-energy resolutions have been obtained from GEANT4 simulations considering the\nrealistic experimental setup and the resolutions of the detectors. The upper panel of Fig. 4 shows\nthe resolutions as a function of the 8He-n-n three-body relative energy. The energy resolution\n(1σ) can be well approximated by σ(E ) = 0.0131 + 0.133 E 0.611 MeV. Some typical relative\nrel rel\nenergy distributions fitted with symmetric Gaussian functions for fixed relative energies are\nshown in the lower panel of Fig. 4. The distributions are slightly asymmetric below E = 0.5\nrel\nMeV, which has been taken into account in the convolution of the theoretical distributions shown\nin Fig. 1, although the effect is minor.”\n2. Including additional information in the Methods section about the cross-talk rejection\nprocedures would benefit the reader.\nFor example, reference [31] includes empirically derived values for the same-wall cross-talk rejection filters based on the\n7Li(p,n)7Be test case. Are similar values used here?\nReply: Yes, similar values have been used here for the same-wall cross-talk rejection filters. The\ncross talk rejection procedure of current work is similar to the rejection condition described in\nRef. [31]. We have now included “cross-talk rejection” in the Methods section regarding the\ndetailed parameters used in the cross-talk rejection filters:\n17\n“Cross-talk rejection. Neutrons are electrically neutral particles and are capable of generating\nmultiple detection hits within the neutron detector array mainly caused by neutron scattering. In\nthe present work, to measure the energy and momentum of two neutrons, rejection of such\ncross-talk background in the neutron detector array NEBULA is required. In the data analysis,\nwe applied 6 MeVee energy threshold to the measured hits to reduce the γ background and\napplied the cross-talk rejection procedures following the previous work presented in Ref. [31].\nThe parameters for the cross-talk rejection conditions were obtained using the single-neutron\nevents taken from the experimental 11Li(p, pn)9Li + n reaction.\nFor the events of two neutrons detected in the same wall of NEBULA, the cross talks due to a\nsecondary recoil proton are identified using the following condition\nwhere dr and dt are the distance and time difference between two hits, and parameters of dr =\n0\n15.8 cm , R = 15.0 cm and dt = 0.32 ns and T = 1.8 ns were adopted. In addition, the cross talks\n0\ndue to γ and scattered neutrons in larger dr and dt are rejected through a two-dimensional\nselection applied to Q and β /β , where Q is the energy deposit (unit in MeVee) of the\n2 01 12 2\nsecondary hit, β is the velocity (unit in speed of light) between the reaction position and the\n01\nprimary hit, β is the velocity between the primary and the secondary hits. The selection could\n12\nbe parameterized as Q < 108 × |β /β | - 60 and Q < -12 × | β /β | + 80. For the events of\n2 01 12 2 01 12\ntwo neutrons detected in the different wall of NEBULA, the cross talks are mainly due to\nneutron scattering from the upstream wall to the downstream one. Since the neutron’s velocity is\nreduced after the scattering, most of these cross talks could be identified by a simple condition\nof |β /β | > 1. In the current analysis, we require that Q < 250 × |β /β | − 200, taking into\n01 12 1 01 12\naccount that Q , the energy deposit of the primary hit, becomes larger when |β | is smaller in the\n1 12\nneutron scattering. Finally, some remaining γ cross talks are rejected by the condition of 1/β ∼\n12\n1. After applying the above procedures, we found that 96.9% of the cross talk is eliminated in\nthe data, while 98.7% is eliminated in the simulation. We also confirmed that varying the\ncross-talk rejection conditions reasonably yields consistent relative-energy spectra, as long as the\ncross-talk rejection procedures are consistently incorporated into the estimation of the 2n\ndetection efficiency. This further demonstrates that the above cross-talk rejection procedures are\nvalid.”\n3. Some clarification of the analysis procedure is needed to help the reader understand the\ncomparison between data and calculation shown in Fig. 1.\na. How does the output from the three-body 11Li model + quasi-free knockout model inform the\ncalculation of the final 10He states? Do the reaction calculations provide cross-sections that scale\nthe contributions of the 10He states in the relative energy spectrum?\nReply: The initial wave function (11Li-p w.f.) was computed from the three-body 11Li model +\nquasi-free knockout of a p-wave proton. The final 10He wave function (10He w.f.) was computed\n18\nwith the complex-scaled solutions of the Lippmann-Schwinger equation (CSLS) method. The\ncross section for the population of the three-body continuum can be obtained by calculating the T\nmatrix from the initial state (11Li-p w.f.) to the final state (10He w.f.).\nIn our calculations, the 10He final state wave function is expressed using three-body Jacobi\ncoordinates. The three-body relative energy can be easily deduced using the Jacobi momenta.\nThe cross section is thus computed naturally as a function of the relative energy. We have added\nadditional context about the details of the calculations in the Methods section to make the\nexplanation clearer. See “Theoretical calculations of the 10He energy spectra” in the Method\nsection.\nb. What parameters for the 8He-n interaction are varied and how? Including an additional figure\nin the Methods section that shows how these parameters impact the agreement with data would\nhelp reinforce the assertion from lines 131 – 133: “the spectrum’s shape is highly sensitive to the\n8He-n interaction, and the observed low-energy structure is reproduced only with a weakly\nattractive 8He-n s-wave interaction.”\nReply: In our calculation of the 10He wave function, we adopted the 8He-neutron folding\npotential given in Aoyama et al., PRC 55, 2379 (1997). Considering the uncertainty of the\n8He-neutron interaction, the depth of the potential is varied to better reproduce the experimental\ndata. This results in different 8He-neutrons scattering lengths. The figure below shows how the\nthree-body relative energy distributions (area normalized to unity) change when we tune\npotential depths corresponding to various s-wave scattering lengths. The distributions become\nflatter when the absolute values of the 8He-n scattering length become smaller. The low-energy\nstructure in the experimental 10He spectrum is reproduced only with a slightly negative a\ncn\naround −3.5 fm. We have added this information in “Theoretical calculations of the 10He energy\nspectra” in the Method section.\n19\n4. The legend and caption for Fig. 1 should be revised to clarify what is indicated by the green\ndashed curve labeled ‘others.’ It’s not clear if this is referring to the discussion in lines 124 – 131\nregarding higher-lying states.\nReply: ‘others’ means two neutrons in partial waves other than 1s and 0p , such as d wave\n1/2 1/2\netc. We have clarified the label ‘others’ in the caption of Fig. 1.\n“……that is decomposed to three components: 8He + [1s ]2 (red dashed line), 8He +\ng.s. 1/2 g.s.\n[0p ]2 (blue dashed line) and others (two neutrons in partial waves other than 1s and 0p ,\n1/2 1/2 1/2\ngreen dashed line).”\n5. Does the Argonne v8’ force include/account for the large neutron-neutron s-wave scattering\nlength ann ∼ 19 fm referenced in line 174?\nReply: Yes, Argonne v8’ force accounts for the large neutron-neutron s-wave scattering length.\nArgonne v8’ force [B. S. Pudliner et al., Phys. Rev. C 56, 1720–1750 (1997)] have been\ndetermined by fitting to NN scattering data in the range of 0-350 MeV. It was also fit to nn\nscattering length measured in experiments [-18.7(6) fm from O. Schori et al., Phys. Rev. C 35,\n2252-2257 (1987)] and the deuteron binding energy.\n6. Is it possible to extract cross-sections for populating the two 0+ states? Or a ratio of these\ncross-sections? This could be useful for comparing to previous predictions e.g. reference [22].\nReply: Extraction of the absolute cross-sections needs further corrections on the acceptance of\nthe two outgoing protons in the data analysis. Since the two-proton acceptance of the two 0+\nstates should be the same, we have extracted the ratio and compared it to the predictions in Ref.\n[22]. From Fig. 1, we extracted the percentages of the [1s ]2 and [0p ]2 components, which are\n1/2 1/2\n72% and 26%, respectively. Therefore, the ratio for populating the excited 0+ state relative to the\nground state is 0.26/0.72 = 0.36.\nIn Ref. [22], the 0+ states of 10He are described as a linear combination of the [p]2 and [sd]2 basis\nstates. The cross-section ratio for the excited 0+ state relative to the ground state (exc./g.s.) is\nestimated as a function of the assumed fraction of the [sd]2 component in 10He ground state, as\nshown in the figure below by the long-dashed line.\ntaken into account. We found the mixture in the 0+ states is not so strong, which probably comes\nfrom the unbound nature of 10He, and the ground state of 10He has a nearly pure [s]2 component,\nwhich means b2 ∼ 100% in the definition of Ref. [22]. According to the long-dash line in the\nabove figure, the (exc./g.s.) ratio will be larger than 1.5. This is in contrast to our value of 0.36.\nHowever, no details or references are given in Ref. [22] about the adopted ‘simple direct-reaction\nmodel’. We are not able to track the reason why the predicted ratio is different from ours.\nNevertheless, we have added this information in the main text:\n“This finding resolves the long-standing puzzle about the different 10He ground-state energies\nobtained from knockout and 2n transfer reactions. Our work suggests that the ground state of\n10He has a nearly pure [s]2 configuration. The 10He, which is traditionally considered as a doubly\nmagic nucleus, thus appears as a “double-intruder” nucleus with the last two neutrons mainly in\nthe 1s shell, in line with the intruder s- wave dominated ground state of 9He. The intruder\n1/2\ndominant configuration could be understood in the shell-model framework due to the melting of\nthe N = 8 shell closure for the neutron-rich isotopes below Z = 4. In addition, the cross-section\nratio for populating the excited 0+ state relative to the ground state was extracted to be about\n0.36. This value is in contrast to the estimation of a simple direct-reaction model presented in\nRef. [22], where a pure [sd]2-dominant configuration in the ground state of 10He was expected to\nyield a ratio greater than 1.5. ”\n7. The [2s ]2 should be [1s ]2 to be consistent with notation elsewhere (e.g. the legend of Fig. 1\n1/2 1/2\nand line 196).\nReply: The [2s ]2 in line 115 has been corrected to [1s ]2.We have also double-checked the\n1/2 1/2\nmain text to make sure all the notations are consistent.\n8. In lines 173-174 and in the Methods section, the 10He ground state is shown to have an\neffective angular momentum K=1/2 which should be associated with a repulsive effective\nthree-body potential indicated by the black curve in Fig. 5. The next line seems contradictory:\n“This is physically due to that a neutron occupying the halo virtual state can mediate an attractive\ninteraction between the other neutron and the 8He core up to large distance …” And line 180\nemphasizes again the 10He system shows long-range repulsion. The explanation of how the\nattractive interaction arises needs to be clarified.\nReply: The repulsive hyper-radial potential is reduced due to the induced three-body attraction,\nbut the hyper-radial potential still remains repulsive in the current 8He-n-n system. To make the\nexplanation cleaner, we have modified the text to:\n21\n“Indeed, even for a non-interacting three-body system, the hyper-radial (i.e. three-body effective)\npotential is repulsive with K = 3/2. More precisely, in our system, a large neutron-neutron s-wave\nscattering length of |a | ∼ 18.6 fm [45] reduces the effective angular momentum to K = 1/2 (see\nnn\nMethods for a deduction of the effective angular momentum based on the Faddeev approach).\nThis is physically due to that a neutron occupying the halo virtual state can mediate an attractive\ninteraction between the other neutron and the 8He core up to large distance of ∼|a |,\nnn\nthereby\neffectively reducing the strength of the repulsive hyper-radial potential. Nevertheless, the\nhyper-radial potential retains its repulsive character in the 8He–n–n three-body system. This\nmechanism is similar to the emergence of long range three-body attraction in the Efimov effect\n[3, 7].”\n9. In the methods subsection “deduction of the effective angular momentum for a three-body\nsystem,” additional context is needed around line 304 to explain or justify why the right hand\nside of Eq.(6) is O(1) so that the leading order term of the left-hand side must vanish.\nReply: It is easy for the reader to check that the right-hand side of Eq. (6) (Eq. (10) in the revised\nmanuscript) could be rewritten to . Since 0< γ' < π/4 and s ≠ 0 , the\nn\nright-hand side of Eq. (6) is basically [sin(x)/x]2, if we take x = s γ'. Since sin(x)/x is O(1), so\nn\ndoes the right-hand side of Eq. (6). We have modified the main text as below:\n“Since the right-hand side of Eq. (10) is of the order unity (recall that 0< γ' < π/4 and s ≠ 0),\nto\nn\nsatisfy Eq. (10), we need the leading order term of the left-hand side vanish, i.e.: ……”\n-----------------------------------------------------------------------------------------------------------\nReviewer #2 (Remarks to the Author):\n# Report on \"First identification of a universal three-body s-wave resonance in $^{10}$He\"\nThe authors have addressed most of my concerns. However, their response regarding the\neffective range, which I had previously inquired about, remains unconvincing.\nThe paper they cite solves the Schrödinger equation only for cases where both the scattering\nlength and the effective range are negative. This constraint (a negative effective range) is\nmandatory in their scheme, which involves summing to all orders.\nIn contrast, the spirit of the article by Kievsky et al. (Annu. Rev. Nucl. Part. Sci 71, 465,\n2021) is different. It has been shown that a positive effective range can also be handled.\nMoreover, the effective range is not a correction to the infinite scattering-length result, nor is it\npresented as such in the paper cited by the authors.\nI understand that extracting the effective range can be subject to large errors; still, I believe\nproviding it would be very important. Its sign, in particular, could be of great significance.\nThis remains my main concern. I can understand if the authors are not interested in analyzing\ntheir results further to provide this estimation, perhaps because they believe their result is solid\nregardless of the effective range. However, they rely on only one type of EFT, where the\neffective range is either treated as a perturbation or is required to be negative.\nDespite my concerns, the paper deserves publication as it brings the physics of universal states\nback into the focus of Nuclear Physics.\nReply: We appreciate further comments of reviewer #2 regarding the 8He-neutron effective\nrange. As detailed in the Method section, in our analysis, we have taken the folding potential\nas the 8He-n interaction. The depth of the potential has been tuned to reproduce the\nexperimental E distribution of 10He. From the folding potential with depth fitted to the\nrel\nexperimental data, the 8He-neutron effective range is estimated as about 2 fm. Since both the\n8He-n scattering length and 8He-n effective range are much smaller than the neutron-neutron\nscattering length, we expect that the leading order halo EFT (Hongo and Son, Phys. Rev. Lett\n128, 212501 (2022)) should be valid for the 8He-n-n system. We have added the estimated\neffective range value in the Method section.\nThe modified Method section now reads:\n“……Furthermore, the spectrum’s shape is highly sensitive to the 8He-n interaction. In Fig. 6,\nthe calculated distributions, which correspond to 8He-n scattering lengths of −3.9 fm ≤ a ≤\ncn\n−2.7 fm, are also shown. The distributions become flatter when the absolute value of a\ncn\nbecomes smaller. The low-energy structure in the experimental 10He spectrum is reproduced\nonly with a slightly negative a around −3.5 fm. Besides, from the folding potential with depth\ncn\nfitted to the experimental data, the 8He-n effective range is estimated as about 2 fm.”\n-----------------------------------------------------------------------------------------------------------\nReviewer #3 (Remarks to the Author):\nThe manuscript describes measurements of 10He produced through (p, 2p) reactions of a ∼250\nMeV/u 11Li beam with a liquid hydrogen target. Analysis of the three-body relative energy and\nJacobi correlation spectra utilized a coupled-channel three-body model of 11Li combined with\nthe quasi-free knockout reaction model, and complex-scaled solutions of the Lippmann-\nSchwinger equation were used to compute final states in 10He. The low-energy structure in the\nthree-body relative energy spectrum is explained by two 0+ states with 8He cores, one with a\n[1s1/2]2 neutron component and another with a [0p1/2]2 neutron component. By tuning the\n8He-n interaction to fit the 10He spectrum below 2 MeV, the authors extracted an s-wave\nscattering length between 8He and the neutron. The Jacobi-T normalized relative energy and\nJacobi-Y angle 8He-n-n correlations gated on relative energies < 1 MeV show enhancements\nthat are qualitatively reproduced by the three-body calculations. The authors conclude that the\nobservation of the two 0+ states resolves the long-standing difference between 10He ground\nstate energies measured from knockout and 2n transfer. Furthermore, the 10He ground state is\na three-body s-wave resonance that exemplifies a novel class of universal three-body bound\nstates.\nThis reviewer's comments have been sufficiently addressed by the author's revision, and\npublication is recommended. Many thanks to the authors for their thoughtful and thorough\nresponses to this reviewer's comments.\nReply: We sincerely thank the reviewer for evaluating our manuscript and recommending its\npublication.\n2"}]

Decision letter