#  7H ground state as a 3H+4n resonance

Emiko Hiyama  Department of Physics, Tohoku University, Sendai, 980-8578,
Japan and
RIKEN Nishina Center, 2-1 Hirosawa, Wako 351-0106, Japan  Rimantas Lazauskas
IPHC, CNRS/IN2P3, Université de Strasbourg, 67037 Strasbourg, France  Jaume
Carbonell  Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France

###### Abstract

We have investigated the possible existence of a 7H resonant state, considered
as a five-body system consisting of a 3H core with four valence neutrons. To
this aim, an effective n-3H potential is constructed in order to reproduce the
low energy elastic neutron scattering on 3H phase shifts and the 5H resonant
ground state in terms of 3H-n-n system. The variational Gaussian Expansion
Method is used to solve the 5-body Schrödinger equation, while the resonant
state parameters were estimated by means of the stabilization method. We have
not found any sign of a narrow low energy resonance in the vicinity of 3H+4n
threshold. However, we have identified a very broad structure at $E_{R}\approx
9$ MeV above this threshold, which corresponds to the 7H Jπ=1/2+ ground state.
In the vicinity of this state, we have also identified a broad structure
corresponding to the ground state of 6H isotope with quantum numbers
$J^{\pi}=2^{-}$.

###### keywords:

4H, 5H, 6H and 7H , Gaussian Expansion Method , Stabilization method , Few-
Nucleon problem , ab initio calculations

††journal: Physics Letters B

##  1 Introduction

Aside from the well-known stable isotopes - deuteron 2H (d) and tritium 3H (t)
- the H isotopic chain extends beyond these two familiar nuclei with well-
defined resonant states in 4H – clearly seen in the low energy n-t cross
section– and 5H system [1, 2, 3, 4]. The possibility for the simplest nucleus,
a single proton, to form an isotopic state as exotic as 7H constitutes an
exciting challenge both from experimental as well as from theoretical points
of view.

The first indication of a resonant 7H state came in 2003 from RIKEN [5] study
of p(8He,2p)7H reaction. These authors conclude the existence of a 7H
resonance near the t+4n threshold, although they were not able to determine
the resonance parameters.

This was immediately followed by a series of experiments in GANIL using a 8He
beam of $\approx$ 15 A MeV. The first one (E465S) [6, 7] was based on the
transfer reaction d(8He,3He)7H. They measured a clear structure in the missing
mass spectrum, which was attributed to the presence of a resonant state with
$E_{R}$=1.56 $\pm$ 0.27 above the t+4n threshold and $\Gamma$=1.74$\pm$ 0.72
MeV.

In the next GANIL experiment (E406S) [8, 9], the 12C(8He,13N)7H transfer
reaction was studied by employing a 8He beam of 15.4 A MeV and a 12C gas
target. Regardless of the small statistical evidence (7 events), these authors
suggested the existence of a very narrow resonant 7H state at
$E_{R}$=0.57+0.42-0.21 MeV and an estimated width $\Gamma$=0.09+0.94-0.06 MeV.
The same result has been recently published in [10].

GANIL results were not fully affirmed in a 2010 RIKEN experiment [11].
Nevertheless, a well-pronounced peak was observed and associated with a 7H
state, resulting in a peculiar structure at $\sim$2 MeV above the t+4n
threshold.

A recent experiment [12] at the Flerov Laboratory of Nuclear Reactions (JINR,
Dubna) employed a 8He beam of 26 A MeV to investigate the 2H(8He,3He)7H
reaction and suggested the existence of a 7H ground state at E=2.0(5) MeV as
well as the first excited state at E∗=6.5(5) MeV with $\Gamma$=2.0(5) MeV.

We should also mention the unsuccessful attempt of Ref. [13] to observe a
long-living quasi stable 7H nucleus in the reaction 2H(8He,7H)3He. The authors
of a former work estimated the 7H lifetime to be less than 1 ns and a lower
limit of 50-100 keV for the 7H energy above the t + 4n breakup threshold.

From the theoretical side, little is known despite the growing interest in
exotic nuclear systems. One needs to clarify the tendency – for or against the
stability – of the H isotopic chain when moving from p+3n to p+4n and to p+6n.
Our previous ab-initio calculation on 4H and 5H [14] shows a gain of stability
(i.e. proximity to the E-real axis) in the computed S-matrix poles when moving
from 4H to 5H ground state (see Fig 4 of this reference). If the results of
the GANIL second experiment [8, 9] would be confirmed, they will indicate a
further increase when moving to p+6n.

However, only a few attempts have been made in this direction. In Ref. [15]
the energy of 7H was roughly estimated to E(7H)$\sim$1.4 MeV (above t+4n) by
using a purely S-wave Volkov potential [16] and an exponential extrapolation
of energies calculated up to the Hyperspherical quantum number K= Kmin+6.

Using a variant of the Volkov potential, one could also estimate the energy of
7H in [5]. Since the experimental 8He binding energy is 31.4 MeV, we can
expect the binding energy of seven-body 7H to be -5.4 MeV, which is about 3
MeV above the t+4n threshold.

A more realistic estimation was realized in [17] based on the same Volkov
interaction. In order to solve the 7-body problem, these authors combined the
Antisymmetrized Molecular Dynamics (AMD) approach with generator coordinate
and stochastic variational methods. They found a state at E(7H)$\sim$ 7 MeV
above the t+4n threshold. All the calculations were performed within the bound
state approximation, failing to estimate the width of a such high energy
state. Some years later, these authors [18] realized a more advanced AMD
calculation, which suggested the presence of a state at E(7H) $\sim$ 4 MeV
above the t+4n threshold, again without giving its decay width.

Finally, it is worth noticing the recent work [19] in which 7H was considered
as a state formed from a rigid 3H core and four valence neutrons. The unbound
4-7H isotopes have been computed by using No-core Gamow Shell Model (NCGSM)
techniques with different phenomenological NN interactions. These authors
concluded that 7H should be a very narrow resonant state, even the sharpest
neutron resonance observed in the light nuclei chart. The value of $E_{R}$,
the real part of the complex energy ground state, varied from 2-3 MeV
depending on the NN model, and in all cases $\Gamma\approx$ 0.1 MeV. This is a
quite surprising result, since the predicted resonance is found to be well
above the $t+4n$ threshold, above the resonant 4H+3n and 5H+2n decay channels
and, nevertheless, its width is predicted to be much smaller than those of 4H
and 5H resonant states.

Recently, an experimental group at RIBF (NP1512-SAMURAI134) have performed a
study aiming to observe the 7H resonant states. The analysis of the collected
data is still in progress. Considering this situation, it is timely to study
the structure of 7H.

Contrary to our previous ab-initio works on 3n [20], 4n [21, 22, 23, 24], 4H
and 5H [14] we are not able to consider the 7H system as a genuine 7-N
problem. Our present work is thus an attempt to compute the 7H ground state
properties in the 5-body cluster approximation (t-n-n-n-n), which anyway
should be the dominant decay channel for a low energy resonant state. To this
aim, we have first constructed a n-t local interaction that we have adjusted
in order to reproduce the n-t phase shifts. We have also adjusted a simple
t-n-n three-body force to reproduce the ground state properties of 5H, now
described as a t-n-n system. It should be noted, however, that the
experimentally determined energy and decay widths of 5H represent only an
effective response to a particular reaction mechanism leading to the short-
living formation and instantaneous decay of a 5H state. Its value is thus
strongly dependent of the particular reaction mechanism. Therefore, the t-n-n
three-body force used in our model was adjusted in order to reproduce the 5H
resonance parameters computed in our previous ab initio calculation [14].

The paper is organized as follows. Section 2 contains the formal aspects of
the problem. These are essentially the t-n potential (2.1), the Gaussian
Expansion Method (GEM) used to solve the five-body Schrödinger equation for
the t-n-n-n-n system (2.2) and the Stabilization Method (SM) used to estimate
the complex energies of resonant states (2.3). Our numerical results for 6H
and 7H resonant state are given in Section 3 and Section 4 contains some
concluding remarks.

##  2 Formalism

An ab initio 7-body calculations of a resonant state is beyond the existing
technical capabilities. Since the ground state of 7H decays essentially into
the five-body 3H+4n channel, we have decided to describe its ground state as a
5-body system, based on a solid tritium core interacting with four valence
nucleons.

At the current stage, our 5-body code based on Gaussian expansion method is
not able to handle tensor interactions, and therefore our t-n interaction
model had to omit such terms. On the one hand, it is well known that the
tensor force is crucial in order to describe quantitatively the few-nucleon
data. On the other hand, light nuclei fulfill quite well the criteria of
Effective Field Theory [25, 26, 27, 28], according to which the main features
he weakly bound nuclei rely on very few parameters of the NN interaction. Our
previous studies of the lightest hydrogen isotopes [14] clearly confirm this
point. Indeed, by considering the phenomenological MT I-III NN potential,
which is limited to S-wave, we were able to qualitatively describe the n-t
scattering phase shifts and the position of 5H resonant state. We have
therefore decided to build our n-t effective interaction by adjusting its
parameters to reproduce the t-n phase shifts calculated by solving the ab-
initio four-nucleon scattering problem with MT I-III.

###  2.1 The n-t potential

For numerical convenience, we have constructed a n-t potential in terms of
Gaussian regulators

$V^{SL}_{nt}(r)=\delta_{L0}\mid\phi_{0}>\lambda_{\infty}<\phi_{0}\mid\;+\;\sum_{i=1}^{3}\left[V_{i}^{c}\;+\;(-)^{L}V_{i}^{P}\;+\;{\hat{S}^{2}\over
2}V_{i}^{S}\;+\;(-)^{L}{\hat{S}^{2}\over 2}V_{i}^{SP}\right]\;e^{-a_{i}r^{2}}$
(1)

where $V_{i}^{c}$, $V_{i}^{P}$, $V_{i}^{S}$ and $V_{i}^{SP}$ are adjustable
parameters, and

$<\vec{r}\mid\phi_{0}>\equiv\phi_{0}(r)=e^{-a_{0}r^{2}}$

is an infinitely repulsive S-wave contribution accounting for the Pauli
forbidden state. In principle, $\lambda_{\infty}$ should be infinite, however
we found that value $\lambda_{\infty}$=105 MeV is sufficient to ensure an
accuracy of 100 keV for the calculated 7H binding energies. Potential (1) is
spin and angular momentum dependent but it has neither spin-orbit nor tensor
force. In spirit, it is rather similar to the MT I-III interaction used in out
previous studies of 4H and 5H systems [14], and will in particular provide
degenerate $J^{\pi}=0,1,2^{-}$ triplet P-wave resonant states states in 4H
(see Table I from [14]).

i |  | 1 | 2 | 3
---|---|---|---|---
$\alpha_{i}$ | fm-2 | 0.04458 | 0.271 | 0.1700
$V^{C}_{i}$ | MeV | 2.4975 | 96.245 | -57.19
$V^{P}_{i}$ | MeV | 2.4975 | -96.245 | 57.19
$V^{S}_{i}$ | MeV | -1.00725 | -13.755 | 0
$V^{SP}_{i}$ | MeV | -1.00725 | 13.755 | 0
Table 1: Parameters of the Vnt effective potential (1).

The corresponding parameters, adjusted to reproduce the experimental n-t phase
shifts, are provided in Table 1 with $a_{0}$=0.1979 068 fm-2. The results for
the n-t phase shifts obtained with this potential (and with ${\hbar^{2}\over
2\mu_{nt}}$=27.647 MeV fm-2) are plotted in Figure 1.

Figure 1: The S- and P-wave n-t phase shifts (solid lines for spin singlet
states and dashed lines for triplet ones) computed with the two-body potential
(1) are compared with the results obtained by solving four-nucleon problem
with MT I-III potential (cercles and squares)

As expected, this potential does not support any bound state, however it has
two broad P-wave resonant states: one in spin-singlet (S=0) and one in spin-
triplet (S=1) channels. The corresponding complex energies of these resonant
states are given in central column of Table 2. They are in qualitative
agreement with the values (right column) obtained by solving the four-nucleon
problem with the MT I-III potential and the Jost function/boundary condition
method to determine the S-matrix pole positions.

If the agreement between the n-t two-body model and four-body calculations is
not perfect, is essentially due to the fact that the n-t phase shifts
displayed in Figure 1 are not perfectly reproduced by our two-body potential
(1). Indeed we had to make some compromises when trying to fit simultaneously
all the partial wave using the simple operator form (1). Moreover, when
performing this fit we kept in mind to simultaneously reproduce within our
two-body model the resonant state of 5H and an effort was made to minimize the
effect of t-n-n three-body force. On the other hand, the four-nucleon
calculations by realistic models predict a $J^{\pi}=2^{-}$ resonant state at
slightly higher energy (E$\approx$1.2-2i) than MT I-III, what justifies the
slight discrepancies with our 2-body model prediction

| Vnt (1) | pnnn [14]
---|---|---
$L=1^{-},S=0$ | 1.28-2.61 i | 0.88(5)-2.20(5) i
$L=1^{-},S=1$ | 1.33-1.84 i | 1.08(3)-2.03(3) i
Table 2: Positions of 4H resonant states (central column) calculated with the
effective 2-body n-t potential (1) and (right column) by solving the 4N
problem with the MT I-III interaction. The parameters of (1) were adjusted to
the n-t scattering phase shifts displayed in Fig. (1).

When describing the 5H and 7H systems, the n-t interaction has been completed
with a neutron-neutron potential. We have taken the Minnesota model [29] which
consists in a superposition of two Gaussians with different ranges and
provides the neutron-neutron low energy parameters $a_{nn}$=-16.90 fm and
$r_{nn}$=2.88 fm.

As it was found in previous studies [30, 31], we were not able to reproduce
the experimental 5H ground state (Jπ=1/2+) resonance parameters when using the
local $V_{nt}$ and $V_{nn}$ interactions, although we were explicitly trying
to favour the appearance of 5H resonant state when adjusting parameters of
$V_{nt}$. We have thus adjusted the position of the 5H ground state by adding
a t-n-n 3-body force, having the same form as in Refs. [30, 31]:

Figure 2: 5H as a t-n-n three-body system.

$V_{tnn}(\rho)=-V_{0}\;e^{-{\rho^{2}\over b_{3}^{2}}}$ (2)

where

$\rho^{2}=\sum_{i<j}{m_{i}m_{j}\over mM}\;(\vec{r}_{j}-\vec{r}_{i})^{2}\qquad
M=\sum_{i}m_{i}$ (3)

and $m$ and arbitrary mass. Setting $m=m_{n}$ one has

$\rho^{2}={m_{n}\over M}r_{nn}^{2}+{m_{t}\over M}r_{nt}^{2}+{m_{t}\over
M}r_{nt}^{2}\qquad M=2m_{n}+m_{t}$

in which $r_{nn},r_{nt},r_{nt}$ denote the corresponding relative distances,
as shown in figure (2) 111Notice that in terms of Jacobi coordinates
$\displaystyle\vec{x}_{3}$ $\displaystyle=$ $\displaystyle\sqrt{{1\over
m}{2m_{1}m_{2}\over(m_{1}+m_{2})}}\;(r_{2}-r_{1})$ $\displaystyle\vec{y}_{3}$
$\displaystyle=$ $\displaystyle\sqrt{2m_{3}(m_{1}+m_{2})\over
mM}\;\left(r_{3}-{m_{1}r_{1}+m_{2}r_{2}\over m_{1}+m_{2}}\right)$ on has
$\rho_{J}^{2}=x_{3}^{2}+y_{3}^{2}=2\rho^{2}$

###  2.2 Solution of the 3- and 5-body problems with Gaussian expansion method

Figure 3: Faddeev components and corresponding Jacobi coordinates of the t-n-n
sytem. Figure 4: Different topologies for the 5-body components and
corresponding Jacobi coordinates of the t-n-n-n-n system.

We aim to solve the Schrödinger equation for 5H and 7H within the framework of
t-n-n and t-n-n-n-n 3- and 5-body cluster models respectively, by means of the
Gaussian Expansion Method (GEM) [32, 33]. It consists in first splitting the
total wave function $\Psi$ in terms of 3 (and 9) components, which correspond
to the schematic decomposition of the 3- (and 5-body) systems in different
clusters. These components are depicted in Figures 3 and 4, together with the
corresponding set of Jacobi coordinates on which they are naturally expressed.

The total Hamiltonian and the Schrödinger equation are given by

$\displaystyle H=K+\sum_{a,b}V_{ab}\;+\sum_{abc}V_{abc}$ (4)
$\displaystyle(H-E)\,\Psi^{JM}_{TT_{z}}(^{5,7}{\rm H})=0$ (5)

where $K$ is the kinetic-energy operator, $V_{ab}$ is the interaction between
the constituent particle $a$ and $b$ and $V_{abc}$ the three-body force.

The total wavefunctions of 5H and 7H are expressed as sum of amplitudes
corresponding to the rearrangement channels shown in Figs 3 and 4. In the $LS$
coupling scheme they read, respectively:

$\displaystyle\Psi^{JM}_{TT_{z}}(^{5}{\rm H})\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\sum_{c=1}^{3}\;\sum_{n,N}\;\sum_{l,L}\;\sum_{S,\Sigma,I}C^{(c)}_{nlNLS\Sigma
I}$ (6) $\displaystyle\times$ $\displaystyle{\cal
A}_{N}\Big{[}\Phi(t)\big{[}\chi_{\frac{1}{2}}(t)\big{[}\chi_{\frac{1}{2}}(n_{1})\chi_{\frac{1}{2}}(n_{2})\big{]}_{S}\big{]}_{\Sigma}\times\big{[}\phi^{(c)}_{nl}({\bf
r}_{c})\psi^{(c)}_{NL}({\bf
R}_{c})\big{]}_{I}\Big{]}_{JM}\;\Big{[}\eta_{\frac{1}{2}}(t)\big{[}\eta_{\frac{1}{2}}(n_{1})\eta_{\frac{1}{2}}(n_{2})\big{]}_{t}\Big{]}_{T,T_{z}}$
(7) $\displaystyle\Psi^{JM}_{TT_{z}}(^{7}{\rm H})\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\sum_{c=1}^{9}\;\sum_{n,N}\;\sum_{l,L,\lambda,\alpha,\beta}\;\sum_{S,\Sigma,I,K}\;C^{(c)}_{nlNL\lambda\alpha\beta
S\Sigma IK}\;\;{\cal A}_{N}$ $\displaystyle\times$
$\displaystyle\Big{[}\Phi(t)\big{[}\chi_{\frac{1}{2}}(t)\big{[}\big{[}\big{[}\chi_{\frac{1}{2}}(n_{1})\chi_{\frac{1}{2}}(n_{2})\big{]}_{s}\chi_{\frac{1}{2}}(n_{3})\big{]}_{\Sigma}\chi_{\frac{1}{2}}(n_{4})\big{]}_{S_{4}}\big{]}_{S}\big{[}\big{[}\big{[}\phi^{(c)}_{nl}({\bf
r}_{c})\psi^{(c)}_{NL}({\bf
R}_{c})\big{]}_{I}\xi^{(c)}_{\nu\lambda}(\mbox{\boldmath$\rho$}_{c})\big{]}_{I_{4}}\zeta^{(c)}_{\alpha\beta}({\bf
s}_{c})\big{]}_{K}\Big{]}_{JM}\>$ $\displaystyle\times$
$\displaystyle\big{[}\eta_{\frac{1}{2}}(t)\big{[}\big{[}\big{[}\eta_{\frac{1}{2}}(n_{1})\eta_{\frac{1}{2}}(n_{2})\big{]}_{t}\eta_{\frac{1}{2}}(n_{3})\big{]}_{\tau}\eta_{\frac{1}{2}}(n_{4})\big{]}_{T_{4}}\big{]}_{T,T_{z}}\;,$
(9)

$\cal{A}_{N}$ stands for antisymmetrization operator between two or four
neutrons. $\chi_{\frac{1}{2}}(t)$ and $\chi_{\frac{1}{2}}(n_{i})$ are the
triton and neutron spin functions respectively, and $\eta_{\frac{1}{2}}(t)$
and $\eta_{\frac{1}{2}}(n_{i})$ are triton and neutrons isospin functions
respectively. The total isospin (and its projection on z-axis) $T(T_{z})$ are
equal 3/2(-3/2) and 5/2(-5/2) for 5H and 7H nuclei respectively. In the GEM
the functional form of $\phi_{nlm}({\bf r})$, $\psi_{NLM}({\bf R})$,
$\xi^{(c)}_{\nu\lambda\mu}(\mbox{\boldmath$\rho$}_{c})$ and
$\zeta^{(c)}_{\kappa\alpha\beta}({\bf s}_{c})$ is:

$\displaystyle\phi_{nlm}({\bf r})$ $\displaystyle=$ $\displaystyle
r^{l}\,e^{-(r/r_{n})^{2}}Y_{lm}({\widehat{\bf r}})\;,$
$\displaystyle\psi_{NLM}({\bf R})$ $\displaystyle=$ $\displaystyle
R^{L}\,e^{-(R/R_{N})^{2}}Y_{LM}({\widehat{\bf R}})\;,$
$\displaystyle\xi_{\nu\lambda\mu}(\mbox{\boldmath$\rho$})$ $\displaystyle=$
$\displaystyle\rho^{\lambda}\,e^{-(\rho/\rho_{\nu})^{2}}Y_{\lambda\mu}({\widehat{\mbox{\boldmath$\rho$}}})\;,$
$\displaystyle\zeta_{\kappa\alpha\beta}({\bf s})$ $\displaystyle=$
$\displaystyle s^{\alpha}\,e^{-(s/s_{\beta})^{2}}Y_{\alpha\beta}({\widehat{\bf
s}})\>,$ (10)

where the Gaussian range parameters are chosen according to geometrical
progressions:

$\displaystyle r_{n}$ $\displaystyle=$ $\displaystyle
r_{1}a^{n-1}\qquad\enspace(n=1-n_{\rm max})\;,$ $\displaystyle R_{N}$
$\displaystyle=$ $\displaystyle R_{1}A^{N-1}\quad(N\\!=1-N_{\rm max})\;,$
$\displaystyle\rho_{\nu}$ $\displaystyle=$
$\displaystyle\rho_{1}\tilde{a}^{\nu-1}\qquad(\nu\\!=1-\nu_{\rm max})\;,$
$\displaystyle s_{\beta}$ $\displaystyle=$ $\displaystyle
s_{1}\tilde{A}^{\kappa-1}\qquad(\kappa=1-\kappa_{\rm max})\>.$ (11)

The eigenenergy $E$ in Eq. (5) and the coefficients $C^{(c)}$ are determined
by inserting respectively the expansions (7) and (9) in (5) and applying the
Rayleigh-Ritz variational method. This results into a generalized eigenvalue
equation $AC=\lambda BC$ of typical dimension d$\sim 60000$. It is worth
noticing that the different, not orthogonal, components of the total wave
function represented in Fig. 4, are all crucial in order to reach a converged
solution of the Schrödinger equation with limited numerical resources.

###  2.3 The stabilization method

To access the complex energy eigenvalues of a resonant state, we have used the
so-called stabilization (or real scaling) method [34, 35]. It consists in
computing the real eigenvalues of the Hamiltonian matrix representation and
studying their (approximate) ”stability” when varying a scaling parameter
(generically denoted $\alpha$) of the basis sate, for instance, the Gaussian
range parameters $\nu_{n}\to\alpha\nu_{n}$.

Figure 5: Example of stabilization graph as a function of the stabilization
parameter $\alpha$

The method is based in the observation, illustrated in Figure 5, that when
varying $\alpha$, the eigenvalue corresponding to the real part of a resonant
state ($E_{r}$) has an avoided crossing with the real eigenvalues
corresponding to the continuum ($E_{c}$). The analysis of this level repulsion
near the crossing point, defined by the intersection of the asymptotes to both
curves, provides an estimation of the resonance parameters ($E_{R},\Gamma$).
In Figure 5, the level crossing between a resonant state ($E_{r}$) and a
continuous state ($E_{c}$) moving down, is produced at $\alpha=\alpha_{c}$ and
defines the resonant energy $E_{R}$ (dashed horizontal line). This feature is
a signature of a resonance state of the Hamiltonian with real energy $E_{R}$
and a width given by [35]

$\Gamma=2\Delta E\;\frac{\sqrt{\mid S_{r}\mid\mid S_{c}\mid}}{|S_{r}-S_{c}|}$
(12)

where $\Delta E$ (in red) is the energy difference between the two curves at
the crossing, and $S_{r}$ and $S_{c}$ are respectively the slopes of the
resonant and continuous levels asymptotes at the crossing.

The stabilization method is closely connected with the Complex Scaling Method
(CSM) [36] but lacks of a sound mathematical ground, providing only a rough
estimation of the resonance width. This method has been recently applied, in
conjunction with the Gaussian Expansion Method, to determine the resonance
positions of a doubly heavy tetraquark system [37].

##  3 Results

Our initial effort was to establish properties of the 5H ground state within
our model, i.e., by considering it as a t-n-n three-body system interacting
via the t-n potential of eq.(1) and Minnesota nn pairwise interaction [29] in
conjunction with the hyperradial 3-body force eq.(2). The solution was
obtained, as in Ref. [31], by combining GEM and CSM, which directly provides
the position of the S-matrix pole in the complex energy plate corresponding to
the resonant state.

Figure 6: Energy trajectory (E=$E_{R}$+i $E_{I}$) of 5H ground state in
complex plane, obtained by varying the strength $V_{0}$ of three-body force
(2) for three different values of $b_{3}$. The numbers along the curves denote
the values of $V_{0}$ (in MeV). The filled blue square corresponds to the ab-
initio result of the 5H ground state computed in Ref. [14]; the up-pointing
triangle in green corresponds to the experimental value from [2].

The complex energy values E=$E_{R}$+i $E_{I}$ ($E_{I}\equiv{\Gamma/2}$) of the
5H Jπ=1/2+ ground state as a function of the three-body force strength
$V_{0}$, defined in eq (2), are displayed in Figure 6 in the ($E_{R}$
,$E_{I}$) plane for several values of the range parameter $b_{3}$. The numbers
along the curves denote the $V_{0}$ values in MeV. The ab-initio value for
this state computed in [14] is indicated by a filled blue square ( ) and the
experimental data from [2] by a filled green up-pointing triangle
($\color[rgb]{0,1,0}{\bigtriangleup}\color[rgb]{0,0,0}$) with corresponding
error bars.

As one can see, the best result is obtained with $b_{3}$=8 fm and $V_{0}$=2.5
MeV for which one gets E(5H)=1.9 - 1.2 i MeV in full agreement with our
previous ab-initio theoretical result, obtained for the realistic NN
interaction models [14]. This value is quite close to, but different from, the
experimental one [2] as the later one reflects the footprint of the 5H
resonant state in a complex nuclear reaction, which strongly depends on the
reaction mechanism but also on the experimental protocol, while its relation
with the S-matrix singularity is only indirect.

Notice that reproducing the phenomenology and the ab-initio results requires a
relatively large value of the range parameter $b_{3}$ compared in comparison
with Refs. [30, 31]. This is related to the choice of our n-t potential, with
the particular functional form of eq. (1). The presence of strongly repulsive
S-wave channels, manifested by the $\phi_{o}$ term with parameter
$a_{0}\approx$ 0.20 fm-2, implies the exclusion of a large spatial region to
project out the symmetry forbidden states in 4H. Any attractive contribution,
aimed to account for the spin-dependent effects in the n-t S-waves, as well as
to describe the energy dependence of the n-t phase shifts, will require an
even larger interaction range to have a significant effect. Furthermore, any
attempt to reproduce the 5H resonance position also favored large interaction
ranges. This is an obvious consequence of simulating the Pauli repulsion
between the valence neutrons and those of the triton core. As the n-n
interaction is resonant in 1S0 state, dineutron correlations take place at
rather large distances, and this generates also a rather large exclusion
region.

Having fixed the parameters of the t-n and t-n-n interactions, we have
estimated the complex energy of 7H ground state by using the stabilization
method, briefly described in section (2.3). An illustrative example of results
provided by this approach in the case of 7H, is displayed in Figure 7. It
reveals the stabilization graph $E(\alpha)$ – that is the lower values of the
7H Hamiltonian spectra as a function of the stabilization parameter $\alpha$ –
corresponding to the 3-body parameters $b_{3}$=8 fm and $V_{0}$=3 MeV.

Figure 7: 7H stabilization graph: lower eigenvalues of the 7H hamiltonian as a
function of the stabilization parameter $\alpha$ corresponding to the three-
body force parameters $b_{3}$=8 fm and $V_{0}$=3 MeV in (2).

In the last figure a level repulsion manifests at $\alpha\approx$1.1 in the
energy range E= 7-10 MeV with $\Delta E\approx$ 3 MeV, indicated by a vertical
double arrow. This is a signature of a resonant state at $E_{R}\approx$ 8.8
MeV (denoted by an horizontal blue line) and an estimated width
$\Gamma=2E_{I}\approx$ 3.1 MeV given by eq. (12).

Following this approach, we have computed the 7H ground state energy as a
function of the t-n-n three-body force strength parameter $V_{0}$. The range
of $V_{tnn}$ was fixed to $b_{3}$=8 fm and we have varied its strength $V_{0}$
from $V_{0}=7$ MeV until it reaches the value $V_{0}=2.5$, closely reproducing
position of the 5H resonant state established in our previous ab-initio
calculations [14].

The final results are summarized in Figure 8. The complex energies of 7H are
indicated by black dots joined by black dashed line. For comparison we have
also included in the same figure (blue dots joined by blue dashed line) the
results of 5H, already described in Figure 6.

For $V_{0}$=7 MeV, the 7H appears to be bound with respect to the t+4n
threshold by E$\approx$-0.5 MeV. When decreasing $V_{0}$ this bound state
turns into a resonance. When the strength of t-n-n force is further reduced to
its realistic value $V_{0}$=2.5 – reproducing position of the 5H – the 7H
resonance is found at $E_{R}\approx$ 9.5 MeV with an estimated width
$\Gamma=2E_{I}\approx$ 3.5 MeV. If we increase the value $V_{0}=3$ MeV,
producing 5H ground state at the lowest point compatible with the experiment,
the 7H resonance is found at $E_{R}=8.5$ MeV and $\Gamma=3$ MeV. These finding
conflicts with a presence of a narrow 7H resonant state in the vicinity of
t+4n threshold. We still find a signature of a broad state at relatively high
energy sufficient to break triton core in single particles. This high energy
domain is beyond the range of validity of our model, which supposes existence
of a solid tritium core. One expects that the widths of 7H state should become
even larger in a more realistic model, where tritium core is allowed to break.

Figure 8: In black, the 7H ground state complex energy trajectory (E=$E_{R}$+i
$E_{I}$) as a function of the strength $V_{0}$ of three-body force (2).
Results ware obtained with fixed value $b_{3}$=8 fm, reproducing the ab-initio
5H ground state. The results for 5H (corresponding to Fig 6) are displayed in
blue for comparison, and the filled blue square corresponds to the ab-initio
result of the 5H ground state computed in Ref. [14].

As one can see our results are in no way compatible with the existence of a
narrow 7H resonant state as being claimed in some experimental works, mainly
[6, 7, 8, 11, 12]. Although all these experiments suffer from a very poor
statistics, there is a huge gap between these two views. The only possible
link between our results and the experimental claims is to identify our values
with what the authors of Ref. [12] claim to be the first excited state of 7H
with $E^{*}$=6.5(5) and $\Gamma$=2.0(5).

In relation with the previous theoretical works, our conclusions are in the
same line than those of Ref. [17] who obtain E(7H)$\sim$ 7 MeV. Our findings
are however in a sharp contrast with the recent work [19]. These authors
provide an energy of the ground 7H which varies from 2-3 MeV, depending on the
NN interaction considered, and in all cases a very narrow width $\Gamma\approx
0.1$ MeV (see Fig 4 of this reference, with a factor 10 enhancement in the 7H
widths).

It is worth emphasizing that our differences with the results of [19] come
already from the very input of both approaches. Indeed, in a former work [38],
the same authors using the same method found very narrow 4H resonant states
(see Fig 1), placing them at $E_{R}\approx$1.7- 0.9 i MeV , in agreement with
some R-matrix analysis [41] . However, it is well established that
experimental R-matrix results and the S-matrix pole positions may differ
significantly for broad resonant states, which is in particular true for 4H
case as stated in [40, 30, 14]. As a consequence of this original choice, [19]
obtained also an extremely narrow 5H resonance (see Fig 4), which is not
supported neither by theory nor by the experiment.

In our opinion, there seems to exist an intrinsic limitation in the GSM used
in [19], which makes this method to systematically underestimate the widths of
resonant states. It is due to the fact that GSM employs only single particle
basis, which are not well adapted to describe collective cluster channels and
thus limits the decay possibilities of a resonant states. In the same way, if
we would have restricted the variation of the basis parameters in the
stabilization method described in Section 2.3, for instance by just limiting
to few selected components of Figure 4, we would have obtained a significant
reduction of the estimated 7H width.

For the sake of completeness, we have also computed the resonant ground state
of 6H, considered as a t-n-n-n four-body system By using the same nn, t-n, and
t-n-n interactions than those used to obtain the 5H and 7H, in particular
$V_{0}$=2.5 and $b_{3}=$8 fm, we estimate for 6H ground state the resonance
parameters $E_{R}$=10 MeV and $\Gamma=$4 MeV. Both, energy and width, are
sligthly larger than those of 7H, though compatible.

In summary, our theoretical analysis of the heavier well identified H-isotopes
4H, 5H, 6H and 7H can be described in Figure 9. As it is customary in the
experimental tables, it represents the resonance energies $E_{R}$ of their
ground states relative to the corresponding t+$x$n threshold, with an
overimposed square denoting the total width $\Gamma$ (all units are in MeV).

The results of 6H ($E_{R}$=10 MeV, $\Gamma$=4 MeV) and 7H ($E_{R}$=9.5 MeV,
$\Gamma$=3.5 MeV) are those obtained in the present work, within the t-n-n-n-n
cluster approximation and $V_{tnn}$ parameters $b_{3}$=8 fm and $V_{0}$=2.5
MeV. They are compatible with the experimental value $E_{gs}$=7.3$\pm$1 MeV,
$\Gamma_{gs}$=5.8 $\pm$ 2 MeV obtained by [41] in the 11B($\pi^{-}$,p4He)6H
reaction. They indicate that the big jump in $E_{R}$ already manifests in 6H.

The values for 4H and 5H correspond respectively to $2^{-}$ ($E_{R}$=1.2 MeV,
$\Gamma$=4 MeV) and $1/2^{+}$ ($E_{R}$=1.85 MeV, $\Gamma$=2.4 MeV), both
supposed to be the experimental ground states, as they are provided by the ab
initio calculations and N3LO interaction in our previous work [14] (see Table
I and II). For 5H, the energy fully coincides with the value of the t-n-n
cluster approximation displayed in Figs 6 (blue squared corresponding to
$b_{3}$=8 fm and $V_{0}$=2.5 MeV). For 4H, it is slightly different from the
$J^{\pi}$=0-,1-,2- degenerate value of the two-body ground state ($E_{R}$=1.3
MeV, $\Gamma$=3.8 MeV) provided by the n-t potential (1).

Figure 9: Predictions for the ground states of heavier H-isotopes. 7H ground
state ($E_{R}$=9.5 MeV and $\Gamma$=3.5 MeV) is obtained in the t-n-n-n-n
cluster approach with the parameters of the t-n-n interaction (b=8 fm
$V_{0}$=2.5 MeV) adjusted to reproduce the 5H ground state. 4H ($2^{-}$) and
5H values are those provided by ab-initio calculations with realistic NN
interactions and are taken from [14].

The tendency to stability observed between 4H and 5H, pointed out in our
previous work [14], is totally reversed by the heavier isotopes. Both, 6H and
7H ground states are at relatively high energies ($E_{R}\sim$ 9 MeV) and are
broad ($\Gamma\sim$ 4 MeV) resonant states, which suggest the end point of the
H isotopic chain rather than a stability region opened by the neutron richer
isotopes.

In view of the strong contradiction with previous experimental and theoretical
works, it would be highly desirable that new high statistic experiments and
independent calculations could bring some light into this disarming situation.

##  4 Conclusion

We have obtained the complex energy position of the 7H resonant ground state
considered as a 5-body t+n+n+n+n system.

To this aim, we first constructed a n-t local potential whose parameters were
adjusted to reproduce the low energy neutron scattering on triton phase
shifts. An additional 3-body force was needed to reproduce the 5H ground state
in the t+n+n approximation.

The solution of the 5-body Schrödinger equation has been obtained by means of
the variational Gaussian expansion approach and the resonance parameters have
been estimated by using the stabilization (or real scaling) method.

Our estimated parameters of 7H ground state are $E_{R}$=9.5 MeV and
$\Gamma$=3.5 MeV, that is quite a broad state when compared with the lighter H
isotopes (4H and 5H). Our findings are in sharp contrast with some
experimental values [5, 8, 11, 12] as well as with the theoretical results of
[19].

By using the same interaction parameters than for 7H, a similar calculation
was performed to describe the ground state of the 6H isotope, considered as a
3H+3n four-body system. The ground state was identified to have $J=2^{-}$ and
is situated at $E_{R}\approx$10 MeV and $\Gamma\approx$4 MeV.

Even within the limits of our calculation, we must exclude the presence of a
narrow resonant state in 7H. The comparison with 4H and 5H makes unlikely that
the H isotopic chain could be continued, at least an experimentally observed
state, beyond 7H.

## Aknowledgements

We are sincerely grateful to V. Lapoux and F. M. Marqués for enlightening
discussions concerning the experimental results. We were granted access to the
HPC resources of TGCC/IDRIS under the allocation A0110506006 by GENCI (Grand
Equipement National de Calcul Intensif). This work was supported by french
IN2P3 for a theory project ”Neutron-rich light unstable nuclei”, by the
japanese Grant-in-Aid for Scientific Research on Innovative Areas
(No.18H05407) and by the Pioneering research project ’RIKEN Evolution of
Matter in the Universe Program’. It was completed during the program Living
Near Unitarity at Kavli Institute for Theoretical Physics, University of Santa
Barbara (California). We thank the organizers and the staff members of this
Institute for their kind invitation and financial support, from the National
Science Foundation Grant No. NSF PHY-1748958.

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