# Emergence of 4H $J^{\pi}=1^{-}$ resonance in contact theories

Lorenzo Contessi  Université Paris-Saclay, CNRS-IN2P3, IJCLab, 91405 Orsay,
France  IRFU, CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette, France
Martin Schäfer  The Racah Institute of Physics, The Hebrew University,
Jerusalem 9190401, Israel  Johannes Kirscher  Department of Physics, SRM
University - AP, Amaravati 522502, Andhra Pradesh, India  Theoretical Physics
Division, School of Physics and Astronomy, The University of Manchester,
Manchester, M13 9PL, UK  Institute for Nuclear Studies, Department of Physics,
The George Washington University, Washington DC 20052, USA  Rimantas Lazauskas
IPHC, IN2P3-CNRS/Université de Strasbourg BP 28, F-67037 Strasbourg Cedex 2,
France  Jaume Carbonel  Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405
Orsay, France

###### Abstract

We obtain the $s$\- and $p$-wave low-energy scattering parameters for n3H
elastic scattering using the pionless effective field theory at leading order.
The 4H $J^{\pi}=1^{-}$ resonance emerges as a consequence of a contact
interaction and discrete scale invariance. Results are extracted with three
numerical techniques: trapping the system in a harmonic oscillator trap,
solving the Faddeev-Yakubovsky equations in configuration space, and using an
effective two-body cluster approach. The renormalization of the theory for the
relevant amplitudes is assessed in a cutoff-regulator range between
$1\,\text{fm}^{-1}$ and $10\,\text{fm}^{-1}$.

Most remarkably, we find a cutoff-stable/RG-invariant resonance in the 4H
$J^{\pi}=1^{-}$ system. This $p$-wave resonance is thus a universal
consequence of the unitary deuteron system and an ensuing discretely scale
invariant three-body system with a scale set by the triton binding energy. The
stabilization of a resonant state in a few-fermion system solely through pure
contact interactions has a significant consequence for the powercounting of
the pionless theory. Specifically, the existence of a resonance implies that
the known inability of the theory’s leading order to bind larger nuclei like
16-oxygen, can be cured by perturbative insertion of sub-leading orders.

##  I Introduction

Contact effective field theory is a very successful approach in describing
bosonic Bazak:2016wxm ; Carlson:2017txq ; Bazak:2018qnu ; Bedaque:1998kg ;
Bedaque:1998km  and few-nucleon Bedaque:1999ve ; Barnea:2013uqa  systems
maintaining a tight connection to the underlying interaction Hammer:2019poc ;
Barnea:2013uqa . It promises a simple and clear framework to study the nuclear
force. However, recent works suggest that the nuclear version, pionless
effective field theory EFT($\not{\pi}$) , fails to predict stable bound states
for $p$-shell nuclei111Fermionic systems with more particles than fermionic
internal states at the leading order (LO) Stetcu:2006ey ; Contessi:2017rww ;
Dawkins:2019vcr ; Schafer:2020ivj . All these LO studies indicate that
beyond-$s$-shell nuclear systems, experimentally observed as bound, clusterize
into mutually unbound $s$-shell subcomponents. For instance, one of the
lightest bound $p$-shell nuclei, ${}^{6}{\rm Li}$, breaks into a $\rm{}^{4}He$
in its ground state and a deuteron (2H). In other words, ${}^{6}{\rm Li}$ is
predicted to be unbound in the zero-range contact limit Schafer:2020ivj .
Nonetheless, it can be argued that this effect is only due to the large LO
truncation uncertainty, and that the non-clustered state still exists as a
virtual state or a resonance above the threshold. Only after the inclusion of
higher order corrections these poles may be brought to the bound state region.
However, to preserve EFT($\not{\pi}$)  renormalizability, this can only be
done in perturbation theory. Since perturbative insertions of higher order
terms cannot create any pole ex novo, the possible absence of such poles in
the contact limit and within convergence radius of the EFT($\not{\pi}$) at LO
would put in question any chance of retrieving the correct nuclear description
beyond $s$-shell. The question is thus simple: Does LO EFT($\not{\pi}$)
interaction support the existence of such poles?

Historically, resonances were studied in (A$\leq$4)-body bosonic systems close
to the unitary limit Deltuva:2012ig ; deltuva2020energies  confirming the
existence of a tower of Efimov resonances. A recent study Habashi:2020qgw
proved the possibility of creating two-body $s$-shell resonances in the
contact limit with negative effective range. In LO EFT($\not{\pi}$)
framework, a resonance have been also found in neutral hypernuclear system
$\Lambda$nn Schafer:2020rba  and $s$-wave virtual states have been confirmed
as exited states of triton Rupak:2019  and hypertriton Schafer:2020rba .
Finally, a recent study of 3n confirmed the absence of $p$-wave resonance or
virtual state in this system Dietz:2021haj . At the present knowledge of the
authors, the calculations of $p$-shell systems stops at this level and no
studies have been done for discrete invariant systems of equal mass. The major
goal of this work is to study the emergent resonance states in 4H, a problem
never addressed before in this framework and that can be understood starting
from universal behaviors. The existence of an eventual $p$-shell resonance
using EFT($\not{\pi}$) at LO would open the possibility of finding the same
structure also in heavier nuclei, with the chance of stabilizing many-nucleon
systems by perturbative sub-leading corrections.

Experimentally, 4H resonances are clearly seen in n3H elastic cross sections
PhysRev.119.1981 ; SEAGRAVE1972250 , $\pi^{-}$ absorption experiments
Sennhauser:1981tr ; Gornov:1991fg , and transfer reactions Miljanic:1986zz ;
Sidorchuk:2003fwa . However, the extracted resonance parameters differ
substantially, depending on the used experimental protocol and the underlying
hypothesis in the data analysis. For instance, results obtained by the same
$\pi^{-}$+9Be$\,\to\,$d+t+4H experiment, performed by the same authors, yields
for the lowest state ($E_{R}=3.0\pm 0.2$ MeV, $\Gamma=4.7\pm 1$ MeV)
Gornov:1991fg , ($E_{R}=2.0\pm 0.2$ MeV, $\Gamma=1.2\pm 0.2$ MeV) Gurov:2005sp
, and ($E_{R}=1.6\pm 0.1$ MeV, $\gamma^{2}=0.4\pm 0.1$ MeV) Gurov2005p , where
$\gamma$ is the reduced width (see the original paper), depending whether one
assumes the experimental signal to be created respectively by one, two, or
three resonant states. Independently, the $R$-matrix analysis of n3H cross
section Tilley:1992zz  concluded about a superposition of several $p$-waves
resonant states with quantum numbers $J^{\pi}$=$0^{-}$,$1^{-}_{I}$,
$1^{-}_{II}$,$2^{-}$ Lazauskas:2004uq . The groundstate parameters
($E_{R}$=3.2 MeV, $\Gamma$=5.4 MeV) are hardly compatible with the previous
many-state analysis. The discrepancies among different experimental approaches
further increase when including the transfer reaction experiments.

From a numerical point of view, computing the $S$-matrix pole of a resonance
is not an easy task. Microscopic calculations (in comparison with cluster
calculations) for nuclear few-body systems are therefore not abundant and up
to now have been performed for a limited amount of particles ($A<6$)
Li:2019pmg ; Lazauskas:2019cxj . This limit is often circumvented by changing
the calculation degrees of freedom and treating the many-body systems as a
few-body problem between clusters as done in Ref. Arai:2003ek ; deDiego:2007rd
introducing approximations in the solution. Resonances in 4H have been studied
in the past, for example, in Ref. deDiego:2007rd  using an effective two-body
n3H potential adjusted to reproduce the p3He experimental phase shifts. In
this reference, the resonance energy were calculated either directly computing
the $S$-matrix pole or by performing a $R$-matrix analysis of the phase
shifts, showing a sensible difference among the two methods (the $1^{-}_{I}$
resonance is located at energy $E_{R}=1.2$, $\Gamma=3.5$ MeV, and $E_{R}=3.6$,
$\Gamma=5.3$ MeV, respectively). This result further enhances the difference
between the position of the $S$-matrix pole and the resonance position
extracted from the cross section, making difficult any kind of comparison
between different calculations and experiments.

Other calculations have been done in Refs. Arai:2003ek ; Horiuchi:2013iw ;
Lazauskas:2019cxj ; Li:2021ado  using the resonating group method (RGM) with
complex scaling, Spin-dipole strength functions, Faddeev-Yakubovsky equations
(FYE) in configuration space, and No-core Gamov Shell Model. The results
differs from method to method varying in a range $E_{R}=\\{0.9-3.6\\}$ MeV and
$\Gamma=\\{1.0-5.3\\}$ MeV. This variation appears to be a consequence of the
different numerical techniques used more than on the interaction employed. In
fact, studying the position of the resonance with different realistic
interactions but the same theoretical scheme Lazauskas:2019cxj ; Li:2021ado
leads to comparable results.

In this letter, we study the emergence of a 3+1 $J^{\pi}=1^{-}$ resonance in
4H using LO EFT($\not{\pi}$) , thus demonstrating the possibility to create
n3H $p$-wave states with purely $s$-wave contact interaction. In the region of
resonance projectile neutron momentum is $\sim 0.5~{}m_{\pi}$, binding momenta
of nucleons within 3H target are $\sim 0.4~{}m_{\pi}$ (for a combined distance
from the complete disintegration threshold of $0.8~{}m_{\pi}$): thus 4H
resonant states are at the limit of convergence of EFT($\not{\pi}$) ,
nevertheless still observable in a LO calculation. The absence of any spin-
orbit or tensor term in the LO interaction causes degeneracy between
$J^{\pi}=0,^{-},1^{-}_{I},1^{-}_{II},2^{-}$ states which would be splitted
including higher orders. As 4H can still be treated as both a four-body and
two-cluster (triton-neutron) system, we choose to employ both the
methodologies starting from the same nuclear Hamiltonian.

We begin with a brief description of the interaction theory (LO
EFT($\not{\pi}$) ) before introducing the three numerical techniques which
were used to reach our main conclusion. The results leading to that thesis are
presented in the third part prior to a summary.

##  II Theory and numerical methods

The theory used in this work is the EFT($\not{\pi}$) vanKolck:1999mw
truncated at LO. The lowest order consists of contact two-body and a contact
three-body interactions with corresponding low energy constants (LECs) fitted
to reproduce physical observables. To further simplify the theory we truncate
it around the SU(4) symmetry Konig:2016utl . Therefore, nucleon-nucleon spin-
singlet and spin-triplet interactions are identical and only one contact term
appears in the two-body sector. This adds further uncertainty to the theory of
order $(a_{0}m_{\pi})^{-1}\sim 30\%$ on the top of the usual EFT($\not{\pi}$)
LO truncation error. The contact interactions are renormalized by introducing
a Gaussian regulator and a momentum cutoff $\Lambda$. This yields the
following two- and three-body potentials:

$V(\textbf{r})=C_{0}\sum_{i<j}e^{-\frac{r_{ij}^{2}\Lambda^{2}}{4}},$ (1)
$W(\textbf{r})=D_{0}\sum_{i<j<k}\sum_{cyc}e^{-\frac{(r_{ij}^{2}+r_{ik}^{2})\Lambda^{2}}{4}},\\\
$ (2)

where the cutoff dependent LECs ($C_{0}$ and $D_{0}$) are fitted on two-body
observables and triton binding energy ($B_{3}=8.48$ MeV) and $cyc$ denotes the
cyclic sum. In order to assess the uncertainty introduced by the adoption of
SU(4) symmetry, we use two different parametrizations for the two-body LEC
$C_{0}$: one at unitarity (with $|a_{0}|>10^{5}$ fm and referred to as
“unitary”) and one fitting the deuteron binding energy $B_{2}=2.22$ MeV
(referred as “nuclear”). We assume a degenerate nucleon mass $m=938.858$ MeV.

To extract the effective range parameters and resonance energy we employ three
different methods. On one hand, we use two four-body microscopic techniques:
Faddeev-Yakubovsky equations (FYE) in configuration space Lazauskas:2004hq ;
Lazauskas:2019hil  and the stochastic variational method (SVM) Suzuki:1998bn
with harmonic oscillator trap. On the other hand, we use a potential folding
technique to obtain an effective two-body problem representing a neutron
scattering on a solid triton core.

Employing the above interaction we extract the triton-neutron scattering
parameters ($a_{L}$ and $r_{L}$) for $L=0$ and $L=1$ trough the effective
range expansion (ERE) around the triton-neutron threshold:

$k^{2L+1}\text{cot}(\delta_{L})=-\frac{1}{a_{L}}+\frac{1}{2}r_{L}k^{2}+...\,,$
(3)

where $\delta_{L}$ is the triton-neutron phase-shift at angular momentum $L$
and $k$ is the relative momentum of the fragments in the center-of-mass frame.
Resonances will, therefore, emerge as poles of the system’s $T$-matrix

$T=\frac{2\pi}{\mu}\frac{1}{\text{cot}(\delta)-i}=\frac{2\pi}{\mu}\frac{k^{2L+1}}{-\frac{1}{a_{L}}+\frac{1}{2}r_{L}k^{2}+\dots-
ik^{2L+1}}$ (4)

where $\mu\approx 3/4m$ is the reduce mass of the n3H system and can be
extracted either by fitting phase shifts to the ERE formula or directly
through solving the Schrödinger equation for complex energies. From a
theoretical point of view, extracting the resonance energy with the two
methods is not equivalent.

Solving the Schrödinger equation for complex energies is affected by the
EFT($\not{\pi}$) LO truncation error ($\xi_{\mbox{EFT($\not{\pi}$) }}\approx
k_{\rm res}/m_{\pi}\sim 80\%$), while extracting the resonance position using
the ERE of the triton-neutron phase shift and using RGM formalism (see below)
is equivalent to the halo-EFT expanded around the n3H treshold. In the latter
case, a new scale, the breaking momentum of 3H, is introduced at
$K_{d-n}\simeq 94$ MeV/c. Since the scattering parameters are extracted using
EFT($\not{\pi}$) at LO we expect the two methods to agree inside the
truncation error $\xi_{Halo}\approx k^{*}_{res}/K_{d-n}\simeq 70\%$ using only
the scattering volume and $\xi_{Halo}^{2}$ if the effective range $r_{1}$ is
also included in the ERE.

###  II.1 Faddeev-Yakubovsky-equation formalism

The Faddeev-Yakubovsky formalism decomposes the few-body Schrödinger/Lippman-
Schwinger equation into a set of equations in order to organize the
overlapping spectra and divergences of the few-body scattering problem.
Details about this complex and on our particular numerical realization can be
found in Refs. Lazauskas:2004hq ; Lazauskas:2019hil . The appropriate boundary
conditions on the formalism which yield the n3H scattering lengths (volumes)
relevant for this work,

$\Psi(t,\boldsymbol{y})=\lim_{k\rightarrow
0}\,\left(\psi_{t}\,\left[j_{L}(ky)/k^{L}+a_{L}(k)\,n_{L}(ky)k^{L+1}\right]\,Y_{L}(\hat{\boldsymbol{y}})\right)\quad,$
(5)

fix the solutions of the differential FYE equations in the far asymptote where
the radial distance $\left|\boldsymbol{y}\right|$ between the neutron and 3H
is larger than the support of $\hat{V}$ and $\hat{W}$. Here, $j_{L}(z)$ and
$n_{L}(z)$ are spherical Bessel functions of the first and second kind,
respectively, and $\psi_{t}$ is the 3H ground state wave function whose
determination precedes n3H scattering calculations. ERE parameters are
extracted in the limit $k\to 0~{}$MeV/c. In this limit, both terms in the
bracket remain finite functions of a distance $|y_{n-^{3}{\rm H}}|$.

Resonant states are obtained by imposing the canonical boundary conditions on
the FYE, namely a purely outgoing wave in the region where the neutron
projectile is no longer interacting with any target 3H nucleons:

$\Psi(t,\boldsymbol{y})=\psi_{t}\,\left[h_{L}^{+}(k_{\rm res}y)+F_{L}(k_{\rm
res})h_{L}^{-}(k_{res}y)\right]\,Y_{L}(\hat{\boldsymbol{y}})\quad.$ (6)

Now, $h_{L}^{-}(z)$ and $h_{L}^{+}(z)$ are Spherical Bessel functions of the
third kind, and the solution corresponds to a resonant state for those complex
momenta $k_{\rm res}$ at which the function $F_{L}(k_{\rm res})$ vanishes,
i.e. , where the wave function $\Psi(t,\boldsymbol{y})$ represents a purely
outgoing n3H waves.

###  II.2 Harmonic oscillator trap

The second microscopic approach involves the introduction of a harmonic
oscillator (HO) trapping potential

$V_{\rm
HO}(\textbf{r})=\frac{m}{2A}\omega^{2}\sum_{i<j}(\textbf{r}_{i}-\textbf{r}_{j})^{2}$
(7)

with oscillator frequency $\omega$ into $A=4$-body 4H system.

Assuming that the range of an interaction is much smaller than the typical HO
trap range $R<<b_{\rm HO}=\sqrt{2/(m\omega)}$, $\hbar=1$, one can match
asymptotic n3H part of a trapped 4H wavefunction to the trapped solution of
effective n3H two-body system with no nuclear interaction considered. The n3H
phasehifts $\delta_{L}$ at relative momentum $k$ are then extracted using
generalized Busch formula for an arbitrary orbital momentum $L$ Suzuki:2009

$(-)^{L+1}\left(\sqrt{4\mu\omega}\right)^{2L+1}\frac{\Gamma\left(3/4+L/2-\epsilon^{n}_{\omega}/2\omega\right)}{\Gamma\left(1/4-L/2-\epsilon^{n}_{\omega}/2\omega\right)}=k^{2L+1}~{}{\rm
cot}\left(\delta_{L}\right),$ (8)

where $k=\sqrt{2\mu\epsilon^{n}_{\omega}}$,
$\Gamma(x)=\int_{0}^{\infty}z^{x-1}e^{-z}dz$, and
$\epsilon^{n}_{\omega}=E_{\omega}^{n}(^{4}{\rm H})-E_{\omega}(^{3}{\rm H})$ is
an energy of the $n$-th 4H excited state in a trap with respect to the n3H
threshold. Here, bound state energies $E_{\omega}(^{3}{\rm H})$,
$E_{\omega}^{n}(^{4}{\rm H})$ are calculated using Stochastic Variational
Method Suzuki:1998bn . Both $s$-wave ($L=0$) and $p$-wave ($L=1$) n3H phase
shifts are extracted applying HO trap lengths $40~{}{\rm fm}\leq b_{\rm
HO}\leq 60$ fm.

Figure 1: Left panel : $s$-wave scattering length $a_{0}$ and effective range
$r_{0}$ calculated using microscopic techniques, SVM with Busch Formula and
solving the Faddeev-Yakubovsky equation (red), and n3H folding using RGM
(blue) for the ”nuclear” case. Right panel : the same as in the left panel but
for $p$-wave scattering parameters - scattering volume $a_{1}$ and $p$-wave
effective range $r_{1}$. For each scattering parameter the corresponding
dashed line represents an extrapolation to the contact limit
($\Lambda\rightarrow\infty$) using the function in eq. 11. The extrapolated
values together with extrapolation uncertainties are shown by the shaded
bands.

###  II.3 Resonating Group Method

For neutron-projectile momenta much smaller than any excitation scale of the
triton, $k\ll k_{d-n}$, the description of n-3H scattering with a halo EFT
Hammer:2022lhx  becomes appropriate. The resonating-group expansion of the
four-body wave function resembles the accompanying transition from nuclear
(“microscopic”) degrees of freedom to a fundamental trimer (3H) and an atom
(neutron) field. Halo EFTs must account for exchange effects between particles
within their elemental fields, e.g. , the Pauli repulsion between the
projectile neutron and one of the neutrons inside the triton, via
renormalization constraints, i.e. , through the introduction of, possibly
redundant, (n-3H)-interaction vertices. The so-called resonating-group method
expands the wave function in a double product of fragment-internal ($\phi$)
and relative ($\chi$) states:

$\Psi=\sum_{\alpha_{j}}\prod_{m=1}^{\alpha_{j}}\phi^{j}_{m}\prod_{n=1}^{\alpha_{j}-1}\chi^{j}_{n}\quad.$
(9)

With a properly chosen interaction, this method is formally identical to a
halo EFT. The compound fields of the halo EFTs are identified with cluster
states $\phi^{j}$ of a given partition $\alpha_{j}$ of the $A$ nucleons, and
the effective vertices of the EFT are expressed in a set of integral kernels.
Deriving the latter is tantamount to matching the halo EFT with a microscopic,
few-body description, and we achieve this matching analytically.

In short, we utilize the technique with the same boundary conditions as in the
FY approach eq. 5 but find solutions to the four-body Schrödinger equation in
a subspace in which only variations of the relative function between the
triton-core $\psi_{t}$ and the neutron are admitted. Therefore (see
wildermuth1977unified  for a pedagogical exposition), the four-body problem
reduces to a two-body for the relative motion between the triton and neutron.
With a Gaussian expansion of the core, the Gaussian regulated EFT potentials
translate analytically into an effective core-atom interaction with a local
and a non-local, energy-dependent component which fully accounts for the
identity and associated permutation properties of the neutron projectile and
its core copy:

$\left(\hat{T}_{\rm rel}-E+\hat{U}^{(2)}_{\rm
eff}(\boldsymbol{y})\right)\chi(\boldsymbol{y})+\int\hat{U}^{(3)}_{\rm
eff}(\boldsymbol{y},\boldsymbol{y}^{\prime},E)\chi(\boldsymbol{y}^{\prime})d\boldsymbol{y}^{\prime}=0\quad.$
(10)

No additional renormalization of the effective triton-neutron potentials
222Written in the so-called parameter representation,
$\boldsymbol{y}^{\prime}$ is an additional expansion label. As such,
permutation operators have no effect on it. Hence, one does not expect a
symmetric form for the kernel $\hat{U}^{(3)}_{\rm
eff}(\boldsymbol{y},\boldsymbol{y}^{\prime},E)$ in which the core-projectile
exchange has been already considered (see fig. (2)). is required, and we
obtain their dependence on $C_{0}$ and $D_{0}$ (see eqs. 1 and 2) by
analytically integrating out fast, i.e. , core-internal degrees of freedom. In
effect, eq. 10 represents a halo EFT with two degrees of freedom: the extended
but inert triton and a neutron, with an interaction parametrized by the
fundamental, nuclear theory. A necessary matching constraint between the two
descriptions is the wave function of the three-nucleon state, i.e. , the core.
To satisfy this constraint, we feed the single-particle densities of the SVM
calculation to the RGM integration over the fast degrees of freedom, thereby
obtaining $\hat{U}^{(2)/(3)}_{\rm eff}$ as functions of the expansion
coefficients of these densities in a set of $\leq 8$ Gaussian functions.

We then solve the integro-differential eq. 10, which is projected into partial
waves, to obtain the relative $s$\- and $p$-wave functions via a standard
finite-difference method. The amplitude $a_{L}(k)$ is subsequently found with
boundary conditions identical to eq. 5, albeit, for real momenta, only.

Another difference to the microscopic formulations is the smaller breakdown
momentum $K_{d-n}$. Compared to $K_{d-n}$, the resonance momentum remains
relatively large even when calculated with respect the n3H threshold ($k_{\rm
res}^{halo}$), as pertinent for halo-EFT formulation. Therefore, the naïve
estimate for the theoretical uncertainty of the halo EFT is larger than the
expected EFT($\not{\pi}$)  uncertainty: $k_{\rm res}^{halo}/K_{d-n}>k_{\rm
res}/m_{\pi}$. However, the renormalizability of the theory, i.e. , the cutoff
independence inside the truncation error of the theory for the resonance-
related observables, and the sheer existence of the pole will be still be
accessible in the RGM formulation.

##  III Results and discussion

Figure 2: Dependence of the effective RGM $L=1$ “nuclear” potential
$L(L+1)y^{-2}+\hat{U}_{\rm eff}^{(2)}(y)+\hat{U}_{\rm eff}^{(3)}(y,y)$ on the
distance, $y$ between the neutron and the center of mass of the triton on the
diagonal $y^{\prime}=y$ (left panel, $1.5\,{\rm fm}^{-1}\leq\Lambda\leq
10\,{\rm fm}^{-1}$). The topography of the non-local part $\hat{U}_{\rm
eff}^{(3)}(y,y^{\prime})$ (right panel, $\Lambda=10$ fm-1) exhibits an off-
diagonal attractive pocket (second quadrant of the contour plot) which does
not vanishes in the zero-range limit. The potentials were obtained at $E=0$
with a five-dimensional Gaussian calibration of the core wave function to SVM
ground states.

We begin discussing EFT($\not{\pi}$) LO predictions in the n3H scattering
system with our results for the effective-range parameters $a_{L}$ and
$r_{L}$. Their cutoff-$\Lambda$ dependence is shown in fig. (1), where we
compare values obtained with the four-body formulations (red circles) with the
RGM extractions. We find the difference between the former two, i.e. , HO trap
and FYE, $\simeq 0.1-1\%$ for $a_{0}$ and $\simeq 1-10\%$ for $a_{1}$. The HO
trap method uncertainty in the effective range $r_{0}$ and $r_{1}$ predictions
are within $0.1$ fm and $0.1$ fm-1, respectively.

The observed behavior for all parameters resembles the LO-EFT($\not{\pi}$)
-characteristic $1/\Lambda$ falloff. For $\Lambda\geq 4~{}{\rm fm^{-1}}$, we
thus fit the function

$O(\Lambda)=O(\infty)+\frac{\alpha}{\Lambda}$ (11)

in order to extrapolate the contact/zero-range limit
($\Lambda\rightarrow\infty$). We find $a_{0}(\infty)=3.06(1)$ fm,
$r_{0}(\infty)=1.59(7)$ fm and $a_{1}(\infty)=-9.8(1.6)$ fm3,
$r_{1}(\infty)=1.3(4)$ fm-1, with an uncertainty set by extrapolation errors.
For the “unitary” $NN$ case, $a_{0}(\infty)=2.27(1)$ fm,
$r_{0}(\infty)=0.47(3)$ fm, and $a_{1}(\infty)=-5.1(1)$ fm3,
$r_{1}(\infty)=1.8(1)$ fm-1. With the sign convention of eq. 3, $a_{0}>0$
relates to an overall repulsive $s$-wave interaction, as intuitively expected
for Pauli forbidden states. Consistently, we find no bound-state solutions in
the $L=0^{+}$ n3H channel for any of the considered $\Lambda$s. In contrast,
$a_{1}<0$ for all $\Lambda$ indicates the presence of attractive $p$-wave n3H
interaction. With decreasing cutoff, $|a_{1}|$, and the strength of the
interaction, increases until a bound state emerges at $\Lambda<0.75\,{\rm
fm}^{-1}$.

The character of an attractive odd partial wave and a repulsive even partial
wave interaction is found in the nuclear and unitary scenarios. Scattering
length and volume decrease when removing the two-body scale in the unitary
limit. Compared with a three-body scale set by
$R_{3}=1/\sqrt{-2mE_{3}/\hbar^{2}}\simeq 6$ fm, i.e. , the only scale present
in the unitary system, the scattering parameters are natural:
$|a_{0}(\infty)|\simeq 0.4\,R_{3}$, $|r_{0}(\infty)|\simeq 0.1\,R_{3}$,
$|a_{1}(\infty)|^{1/3}\simeq 0.3\,R_{3}$, and $|r_{1}(\infty)|^{-1}\simeq
0.1\,R_{3}$.

Our four-body and RGM results for $a_{0}$ match sign and magnitude of the
calculations and experiments available in literature (within EFT($\not{\pi}$)
uncertainty). This qualitative agreement holds in the “nuclear” and the
“unitary” cases. Specifically, we compare our results to the coherent
scattering length
$a_{0}^{c}=\nicefrac{{1}}{{4}}a^{S=0}_{0}+\nicefrac{{3}}{{4}}a^{S=1}_{0}$, and
use $a_{0}^{c}({\rm theo})\approx 3.6$ fm Lazauskas:2019hil  and
$a_{0}^{c}({\rm exp})=3.5\,-\,3.9$ fm Tilley:1992zz  as theoretical and
experimental benchmarks.

Figure 3:  Left panel : Complex energy plane with calculated 4H $1^{-}$
resonance positions (red crosses) for multiple momentum cutoff $\Lambda$. The
black error bars represent the position of the $1_{I}^{-}$ and $1_{II}^{-}$
resonances calculated in Ref Lazauskas:2019cxj  for realistic interaction
model. Right panel: The absolute value, real, and imaginary part of the
complex resonance momentum $k_{res}$ as a function of increasing $\Lambda$.
The momentum is related to resonance energies via eq. 12. The resonance pole
was extracted directly solving the four-nucleon problem trough the Faddeev-
Yakubovsky equations.

The RGM predictions for the “nuclear” parametrization are $a_{0}(\infty)=5.71$
fm, $r_{0}(\infty)\simeq 3.15$ fm, $a_{1}(\infty)=-10.49$ fm3,
$r_{1}(\infty)=4.33$ fm-1, and $a_{0}(\infty)=3.3$ fm, $r_{0}(\infty)\simeq
1.8$ fm, $a_{1}(\infty)=-2.5$ fm3, $r_{1}(\infty)=10.1$ fm-1 for the “unitary”
case. These findings are consistent with the above in sign and thus with the
attractive/repulsive character but differ in absolute value, providing a more
repulsive interaction. This is consistent with the lack of excitations in the
considered triton core, which naively introduces a variational shift in the
system groundstate. Still RGM results retain the same functional dependence on
$\Lambda$ for all scattering parameters. Regarding renormalization-group
dependence, in particular, on the stability of pole varying the cutoff, both
theories are apparently equally useful for the amplitudes under consideration.

Although the effective n3H interaction is not an observable, we use the RGM to
extract its $p$-wave component to acquire a deeper insight into the interplay
between repulsive and attractive parts and their evolution with $\Lambda$.
This allows for a heuristic assessment of whether or not one should expect a
resonant-type solution. In fig. (2), we visualize this interaction for the
”nuclear” case in the zero-energy approximation. In the left panel, we display
the dependence of the sum of the angular momentum barrier, the local and non-
local (at $y^{\prime}=y$ ) parts of the coordinate space interaction, eq. 10,
as a function of the relative coordinate $y$ between the triton center of mass
and the neutron. For small cutoffs, the potentials form a positive energy
pocket at $y\simeq 0.5$ fm. The pocket vanishes in the contact limit
$\Lambda\to\infty$, where the potential is visibly incapable of forming quasi-
stationary states like resonances. While the pocket suggests the possibility
to form resonant states at small cutoffs, we find that its formation demands
the inclusion of the non-local interaction, i.e. , one must take into account
the exchange between projectile and core neutron. In the right panel, we
investigate the character of the non-local part with a contour plot at
$\Lambda=10$ fm-1 and zero energy. The dominance of a repulsive part (red)
prevails in the non-local potential for most points with the notable exception
of a weak attractive region $min(\hat{U}_{\rm eff}^{(3)})\approx$-0.35 MeV;
blue) far off the diagonal where $y^{\prime}>y$. The pocket, which does not
vanish in the zero contact limit, is apparently too shallow to support a bound
state, but it might suffice to form a broad resonance state.

With the Faddeev-Yakubovsky approach, we search for a pole corresponding to a
resonance in the $J^{\pi}=1^{-}$ channel, directly. The results of this search
are displayed in fig. (3). In the left panel, the cutoff dependence of the
pole position in the complex energy plane is shown (“nuclear” case). The
trajectory passes the n3H threshold, $B_{3}=8.484$ MeV (dashed line) and
stabilizes for $\Lambda\gtrsim 6~{}{\rm fm}^{-1}$ at $E_{\rm res}=-8.7-i3.1$
MeV. In order to study the pole convergence, we translate resonance energies
to complex n3H momenta $k_{\rm res}$:

$k_{\rm res}=\sqrt{2\mu\tilde{E}_{\rm res}},~{}~{}~{}~{}\tilde{E}_{\rm
res}=E_{\rm res}+B_{t}\quad.$ (12)

The $\Lambda$ dependence of ${\rm Re}(k_{\rm res})$, ${\rm Im}(k_{\rm res})$,
and $|k_{\rm res}|$ is shown in the right panel of fig. (3). The resonance
momentum stabilizes for $\Lambda\gtrsim m_{\pi}$, i.e. , the breakdown scale
of the theory. Its absolute value $|k_{\rm res}|$ does not exceed $0.35~{}{\rm
fm}^{-1}\simeq 0.5~{}m_{\pi}$.

Alternatively, the resonance position can also be found as pole of the
effective-range-expanded n3H $T$-matrix in eq. 4. With the ERE parameters,
$a_{1}$ and $r_{1}$, calculated with four-body methods, we find $k_{\rm
res}^{\rm halo}\simeq 0.41-0.23\,i$ fm-1. The value is close to the estimated
breakdown $K_{d-n}$ of the halo approach implying a relatively large
uncertainty. Within this uncertainty margin, $k_{\rm res}^{\rm halo}$ is
consistent with the pole obtained with the FY method: $k_{\rm res}\simeq
0.23-0.25\,i$ fm-1 for $\Lambda=10$ fm-1.

For reasons stated in the introduction regarding the model-dependence in data,
and the ensuing discrepancies in experimental values, we abstain from a
comparison of our predictions to data. To justify this with another example,
we note that the real part of the energy of a $S$-matrix pole is, in general,
found smaller than the one extracted with a $R$-matrix analysis. This behavior
is exemplified by the transfer reactions 2H(t,p)4H and 3H(t,d)4H
Sidorchuk:2004ntw . The $R$-matrix analysis of the results obtains for the
lowest 4H state $E_{R}=3.05\pm 0.19$ MeV and $\Gamma=4.18\pm 1.02$ MeV. The
corresponding $S$-matrix pole, extracted by the same authors, led to
$E_{R}=1.99\pm 0.37$ MeV and $\Gamma=2.85\pm 0.3$ MeV. Therefore, a fair
comparison can only be made with other theoretical $S$-matrix-pole
predictions.

For a quantitative comparison, we consider the pole location predicted in
Lazauskas:2019cxj ; Lazauskas:2019hil ; Arai:2003ek ; deDiego:2007rd :
$E_{R}=(0.9\,-\,1.23)$ MeV and $\Gamma=(3.5\,-\,5.8)$ MeV. The LO
EFT($\not{\pi}$) pole is closer to the real axis and narrower, $E_{R}=-0.22$
MeV and $\Gamma=1.6$ MeV. The size of the deviation appears to be consistent
with the effect induced by SU(4) symmetry breaking found in Lazauskas:2019cxj
(i.e. , the difference between two black error bars in fig. (3)). On the point
of view of EFT($\not{\pi}$) , the deviation is also consistent with the
expected truncation uncertainty.

Finally, for the “unitary” parametrization, the FY calculations find a wider
resonance compared with the “nuclear” case. We find the numerical tracing with
$\Lambda$ prohibitive and practical only for cutoffs of the order of $m_{\pi}$
and we were not able to determine a prediction for the zero-range limit
$\Lambda\to\infty$. For the considered cutoffs the resonance has an energy
$E_{R}=-8.156$ MeV and $\Gamma=2.4$ MeV (corresponding to an absolute
$k_{res}=0.58\,m_{\pi}$). The pole is still expected to be inside the
truncation error of the theory and the deviation with respect to the “nuclear”
one is in line with the expected $\approx 30\%$ error for the SU(4) theory
truncation.

##  IV Conclusions

In this work, we predict, model-independently, the existence of a stable
$J^{\pi}=1^{-}$ resonance in 4H with the zero-range theory of nuclear forces:
leading-order pionless effective field theory (EFT($\not{\pi}$) ). We reach
this result with three independent numerical techniques, two microscopic
methods, and an analytically-matched halo effective theory. This finding of a
stable $p$-wave resonance complements and generalizes studies on such states
with pure contact theories which were, as of now, limited to $s$-wave
channels. Our results agree qualitatively with each other and with earlier
simulations which address the existence of a 4H resonance with high-precision
interaction models.

Most remarkable is the emergence of $p$-wave poles from a contact $s$-wave
contact theory between two and three particles. For the usefulness of
EFT($\not{\pi}$)  beyond $s$-shell nuclei, this is good news. The absence of
stable bound states for nuclei larger than 4He at LO is thus no longer a
fundamental problem, when we conjecture that an existing $p$-wave resonance
might be moved on the stable region by the perturbative insertion of sub-
leading orders. This transmutation of $p$-wave resonances into bound states,
obviously, cannot occur in the absence of the former.

## acknowledgement

We thank H. W. Grießhammer, N. Walet, and U. van Kolck for helpful
discussions. MS was supported by the Pazy Foundation and the Israel Science
Foundation grant 1086/21, JK by STFC ST/P004423/1 and the US Department of
Energy under contract DE-SC0015393, and LC and JC in part by the National
Science Foundation under Grant No. NSF PHY-1748958. We were granted access to
the HPC resources of TGCC/IDRIS under the allocation 2022-A0110506006 made by
GENCI (Grand Equipement National de Calcul Intensif).

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