††thanks<EMAIL_ADDRESS>
# Improved action for contact effective field theory

L. Contessi  Université Paris-Saclay, CNRS-IN2P3, IJCLab, 91405 Orsay, France
M. Schäfer<EMAIL_ADDRESS>Nuclear Physics Institute of the Czech
Academy of Sciences, Řež 25068, Czech Republic  U. van Kolck  Université
Paris-Saclay, CNRS-IN2P3, IJCLab, 91405 Orsay, France  Department of Physics,
University of Arizona, Tucson, Arizona 85721, USA

###### Abstract

We present an improved action for renormalizable effective field theories
(EFTs) of systems near the two-body unitarity limit. The ordering of EFT
interactions is constrained, but not entirely fixed, by the renormalization
group. The remaining freedom can be used to improve the theory’s convergence,
to simplify its applications, and to connect it to phenomenological models. We
exemplify the method on a contact theory applied to systems of up to five 4He
atoms. We solve the EFT at LO including a subleading interaction that accounts
for part of the two-body effective range. We show that the effects of such
fake range can be compensated in perturbation theory at NLO, as long as the
fake range is smaller or comparable to the experimental effective range. These
results open the possibility of using similar improved actions for other many-
body systems.

##  I Introduction

Systems near the two-body unitarity limit — such as 4He atoms, where the two-
body scattering length is about 20 times larger than the range of the
interaction — can be described systematically with effective field theories
(EFTs) [1]. At leading order (LO), in the EFT expansion for bosons, two- and
three-body contact interactions reproduce Efimov physics [2]. At next-to-
leading order (NLO), the interaction range appears in the form of a two-
derivative two-body contact interaction [3] and renormalization requires a
four-body force [4]. Experiment and potential-model results for 4He can be
approximated in a controlled way, with NLO improving [4] on an already good LO
description of small [5] and large [6] clusters.

EFTs incorporate universality in a few LO parameters, and corrections to it at
subleading orders. It can also be applied to fermions such as nucleons, and
generalized to smaller distances by incorporating the longest-range
interactions such as pion exchange in nuclear physics [1], the Van der Waals
interaction between neutral toms [7], and the induced-dipole interaction
between a charged particle and a neutral atom [8]. EFTs make no specific
assumptions about the dynamics at distances much smaller than those of
interest. They are particularly useful when the short-range dynamics is either
unknown or hard to solve — as is the case for Quantum Chromodynamics (QCD) in
nuclear physics, where EFTs are now the mainstream [1].

However, current applications of EFT to many-body systems are not without
problems. Most serious is the instability of multi-state fermion systems at
LO. In contrast to experiment, the 6- [9], 16- [10, 11] and 40- [11] nucleon
ground states are not stable at LO, even when pion interactions are included
explicitly [12]. Even though it works well for up to four nucleons [13], an
expansion around the unitarity limit is unlikely to produce stable states at
LO [14, 15].

Stability in multi-component fermion system can be a small effect, in the
sense that the stable state is much closer to the nearest threshold than to
the threshold for break up into individual constituents. For example, the 16O
ground state is stable by only $\sim 15\%$ of its binding energy. There is
indication that unstable states might exist not too far from threshold [16]
and that they could be moved below threshold at higher orders in the EFT.
Unfortunately, it is unclear how that can be accomplished in perturbation
theory starting from a wavefunction of an unstable state.

So far, the solution to this problem has been to ignore the deficiencies of LO
and resum higher-order interactions into the exact solution of the many-body
Schrödinger equation with truncated potentials [17]. However, this approach
produces unrenormalized amplitudes [18]: observables depend on the chosen,
arbitrary regularization procedure. The problem stems from the increasingly
singular nature of higher-order EFT interactions, which poses significant
complications not seen for regular potentials. The simplest example is the
resummation of range corrections, which can produce stable states [11] but
whose renormalization violates [19] the Wigner bound [20]. This represents a
significant drawback since regulator dependence detracts from a clear
hierarchy of interaction orders: results depend on the choice of parameters
with arbitrarily many derivatives encoded in the specific regulator used. The
supposed higher-order contributions are no longer small nor amenable to
perturbation theory. Connection with the underlying theory is murked and a
priori error control is lost [18]. An alternative scheme for effective range
corrections, which might offer a chance to retain good renormalization
properties even when the range corrections are partially resummed, has been
proposed in Ref. [21].

The root of an EFT’s model independence is the renormalization group (RG),
which requires that sufficiently many interactions be present at each order to
remove essential dependence on the arbitrary regulator. At any order, the
remaining cutoff dependence must be small, in the sense that it can be reduced
arbitrarily with increasing momentum cutoff. Once this is achieved, cutoff
errors are no larger than uncertainties arising from the truncation of the
Hamiltonian, as long as the cutoff is taken beyond the breakdown scale of the
EFT. Small contributions can always be resummed at a given order, as long as
RG invariance is preserved. This in fact is implicitly done when using data
(which contain all orders in the EFT) to determine interaction strengths. One
possibility to improve the efficiency of EFT in many-body systems is to
determine interaction strengths from the corresponding data rather than few-
body data. This approach is phenomenologically fruitful for unrenormalized
interactions that include pions [22], but it precludes a systematic study of
the possible breakdown of the EFT as density increases with particle number.

We follow a different approach, inspired by the improved actions of lattice
quantum field theory [23]. There, irrelevant interactions are added to
relevant interactions to speed up convergence with a decreasing lattice
spacing. Here we include at LO a subleading interaction that simplifies
numerical calculations and thus may bypass the LO instability problem. The
clue to what interaction to include is offered by existing calculations (for
example, Ref. [10]), where stability is lost only at high momentum cutoffs.
Since a finite momentum cutoff amounts to a finite minimum distance, we modify
the LO interaction to account for a fake interaction range. We vary the fake
range within the limits of the LO error, as estimated by standard power
counting for contact EFTs. We show explicitly that at NLO its effects can be
removed in perturbation theory. The fake range is thus not a physical
parameter and power counting is preserved.

Operationally, our approach brings the LO of the EFT close to models employed
abundantly in nuclear and atomic physics, where a two-body potential of finite
range is used, sometimes in conjunction with three-body potentials — whose
importance is justified by EFT renormalization only when the two-body
potential has zero range [24, 25]. However, in these models typically the
potential, which is treated exactly, has its range fitted to data — for one of
many examples see Ref. [26]. In the EFT the physical two-body effective range
remains a perturbative, NLO effect. Introduction of the fake range at LO is
merely a way to account for small effects while preserving RG invariance and
the hierarchy of interactions. The fake range slightly changes the position of
shallow poles which already exist at LO, and enables perturbation theory with
respect to this new location.

We demonstrate our method using a contact EFT for resonant bosonic systems,
which we present in Sec. II. Up to four particles, this EFT is formally
identical to Pionless EFT [1] in the limit of Wigner’s SU(4) symmetry, but it
is simpler to benchmark against other theoretical approaches. Our results are
given in Sec. III, where we take atomic 4He as a specific example because it
exhibits a clearer separation of scales than the nuclear problem. Details are
relegated to Apps. A and B. The implications of this method to the logic
behind EFT calculations and its relation to phenomenological approaches are
discussed in Sec. IV.

##  II Theory

Contact EFT is a tool for describing nonrelativistic quantum systems where the
typical many-body momentum is small compared to the inverse of the interaction
range $R$ [1]. In this theory, interparticle interactions are expanded in
contact operators and derivatives, which require regularization and
renormalization. We work in coordinate space where the position of particle
$i$ is denoted by $\vec{r}_{i}$ and relative positions, by $\vec{r}_{ij}$. We
consider a cutoff regularization with the momentum cutoff $\Lambda$, which
replaces a Dirac delta function by a smeared function
$\delta_{\Lambda}(\vec{r}_{ij})$.

When the two-body scattering length $a_{2}$ dominates over other scales, such
as the two-body effective range $r_{2}$, $|a_{2}|\gg|r_{2}|$, a subset of
interactions needs to be treated nonperturbatively. The LO potential to be
iterated to all orders is [1]

${\mathcal{V}}^{(0)}=V^{(0)}_{\text{2B}}+V^{(0)}_{\text{3B}},$ (1)

in terms of two- and three-body interactions

$\displaystyle V^{(0)}_{\text{2B}}(\\{\vec{r}_{i}\\};\Lambda)$
$\displaystyle=C_{0}^{(0)}(\Lambda)\sum_{i<j}\delta_{\Lambda}(\vec{r}_{ij}),$
(2) $\displaystyle V^{(0)}_{\text{3B}}(\\{\vec{r}_{i}\\};\Lambda)$
$\displaystyle=D_{0}^{(0)}(\Lambda)\sum_{i<j<k}\sum_{cyc}\delta_{\Lambda}(\vec{r}_{ij})\delta_{\Lambda}(\vec{r}_{ik}).$
(3)

The two-body interaction $V^{(0)}_{\text{2B}}$ guarantees the presence of a
shallow two-body state [3], while the three-body interaction
$V^{(0)}_{\text{3B}}$ makes the three-body system well defined and introduces
[24, 25] a momentum scale which can be taken as

$Q_{3}\equiv\sqrt{\frac{2mB_{3}}{3}},$ (4)

where $B_{3}$ is the binding energy of the three-body ground state. The LO
low-energy constants (LECs) $C_{0}^{(0)}(\Lambda)$ and $D_{0}^{(0)}(\Lambda)$
are chosen to reproduce a two-body datum like the scattering length $a_{2}$
(large or infinite) and a three-body datum, such as $B_{3}$. All other
observables — for example, the ground-state binding energy $B_{N}$ of the
$N$-body system — can be predicted. The potential must depend on the cutoff so
that observables do not.

Subleading orders improve perturbatively on LO. At NLO there is a correction
due to the two-body effective range $r_{2}$ [3], which requires for
renormalization also a contact four-body force, introducing a four-body scale.
[4]. The NLO (non-local) potential therefore reads

$\mathcal{V}^{(1)}=V^{(1)}_{\text{2B}}+V^{(1)}_{\text{3B}}+V^{(1)}_{\text{4B}}$
(5)

with

$\displaystyle V_{\text{2B}}^{(1)}(\\{\vec{r}_{i}\\};\Lambda)$
$\displaystyle=$ $\displaystyle
C_{0}^{(1)}(\Lambda)\sum_{i<j}\delta_{\Lambda}(\vec{r}_{ij})$ $\displaystyle+$
$\displaystyle
C_{2}^{(1)}(\Lambda)\sum_{i<j}\left(\delta_{\Lambda}(\vec{r}_{ij})\overrightarrow{\nabla}^{2}_{ij}+\overleftarrow{\nabla}^{2}_{ij}\delta_{\Lambda}(\vec{r}_{ij})\right),$
$\displaystyle V_{\text{3B}}^{(1)}(\\{\vec{r}_{i}\\};\Lambda)$
$\displaystyle=$ $\displaystyle
D_{0}^{(1)}(\Lambda)\sum_{i<j<k}\sum_{cyc}\delta_{\Lambda}(\vec{r}_{ij})\,\delta_{\Lambda}(\vec{r}_{ik}),$
(7) $\displaystyle V_{\text{4B}}^{(1)}(\\{\vec{r}_{i}\\};\Lambda)$
$\displaystyle=$ $\displaystyle
E_{0}^{(1)}(\Lambda)\sum_{i<j<k<l}\,\prod_{[a\neq
b]\,\in\,\\{i,j,k,l\\}}\delta_{\Lambda}(\vec{r}_{ab}).$ (8)

The LEC $C_{2}^{(1)}(\Lambda)$ accounts for the effective range, while
$C_{0}^{(1)}(\Lambda)$ keeps $a_{2}$ at the desired value. The LEC
$D_{0}^{(1)}(\Lambda)$ can likewise be used to ensure $B_{3}$ remains
unchanged. Finally, $E_{0}^{(1)}(\Lambda)$ is fixed by a four-body datum, for
example $B_{4}$. All other observables are predicted with improved accuracy.

If $|a_{2}|Q_{3}\sim 1$, the finite value of the scattering length must be
accounted for at LO through a LEC whose running with $\Lambda$ is parametrized
by $a_{2}$, $C_{0}^{(0)}(\Lambda)\equiv C_{0}(a_{2},\Lambda)$. When, instead,
$|a_{2}|Q_{3}\gg 1$, effects induced by the finiteness of the scattering
length are subleading and can be introduced in an additional expansion in
powers of $(a_{2}Q_{3})^{-1}$ [13]. In this case,
$C_{0}^{(0)}(\Lambda)=C_{0}(a_{2}=\infty,\Lambda)$ is a fixed function of
$\Lambda$ and two-body unitarity is exact at LO. Moreover, more-body systems
exhibit discrete scale invariance [24, 25]. Depending on its magnitude, the
finite value of the scattering length could enter through
$C_{0}^{(1)}(\Lambda)$. However, in this case we can still include the finite
scattering length at LO without destroying renormalizability. The choice of
the expansion point $a_{2}$ is to a certain extent arbitrary and any large
value of the scattering length can be used as long as the perturbative
correction to the physical $a_{2}$ value remains small. This preserves the
model independence of the EFT, since different choices of expansion point are
equivalent up to the truncation error of each order. In fact, one frequently
fits at LO not $a_{2}$ but the binding energy of the two-body system, with the
difference being mostly corrected at NLO. The choice is usually guided by
practical considerations, for example whether one is interested predominantly
in bound-state properties or scattering. The formulation of contact EFT which
considers the finite scattering length at LO and a perturbative effective-
range correction together with a four-body scale at NLO has been successfully
applied to few-body systems, both atomic ${}^{4}\text{He}$ [4] and nuclear
[27, 28].

Accounting at LO for a finite scattering length even when close to the
unitarity limit is an example of a more general procedure, where a subset of
higher-order interactions $\Delta V$ are resummed into leading order,

$\tilde{\mathcal{V}}^{(0)}=V^{(0)}_{\text{2B}}+V^{(0)}_{\text{3B}}+\Delta V.$
(9)

This partial resummation of subleading orders only changes the parameters used
as the starting point to expand the theory. It would be entirely trivial were
we not seeking a systematic expansion with order-by-order renormalization and
corrections amenable to perturbation theory. As such, we are restricted to a
$\Delta V$ which does not change LO results by more than the expected LO
error, and which can be compensated at higher orders by an ${\cal O}(1)$
relative shift in their LECs. This ensures that the new EFT expansion is
equivalent to the standard one within the order-by-order truncation error.
Since the potential is not observable, we cannot in general check this
property without examining the convergence of physical properties such as
binding energies.

In the case of near unitarity, inclusion of the scattering length amounts to

$\Delta
V(\\{\vec{r}_{i}\\};\Lambda)=\left(C_{0}(a_{2},\Lambda)-C_{0}^{(0)}(\Lambda)\right)\sum_{i<j}\delta_{\Lambda}(\vec{r}_{ij}).$
(10)

Accounting for $a_{2}$ at LO can be compensated by changes in the higher-order
LECs such as $C_{0}^{(1)}(\Lambda)$. One might consider, similarly, to expand
the theory around a small effective-range value $r_{2}>0$. However, this
cannot be done with a $\Delta V$ in the form of a two-derivative contact
interaction because of the Wigner bound [20].

This issue can be circumvented by taking for $\Delta V$ an interaction with a
small but finite range $\tilde{R}$, which implicitly resums parts of not only
the two-body effective range but also the higher parameters in the effective-
range expansion. The most natural — but by no means unique — way of redefining
the expansion point is to take $\Delta V$ to have the same form as the
regularized LO interaction,

$\Delta
V(\\{\vec{r}_{i}\\};\Lambda)=\sum_{i<j}\left(\tilde{C}\,\delta_{\tilde{R}^{-1}}(\vec{r}_{ij})-C_{0}^{(0)}(\Lambda)\,\delta_{\Lambda}(\vec{r}_{ij})\right).$
(11)

The parameter $\tilde{C}$ can be fitted to the scattering length $a_{2}$. If
the range $\tilde{R}$ is sufficiently small, results from the entire potential
in Eq. (11) will not deviate from those of Eq. (2) more than the contributions
from subleading corrections. Since the main effect of the potential (11) is to
induce a fake effective range $\tilde{r}_{2}\sim\tilde{R}$, at NLO we fit the
$C_{0}^{(1)}$ and $C_{2}^{(1)}$ in the $V_{\text{2B}}^{(1)}$ of Eq. (II) to
the correct $a_{2}$ and $r_{2}$. The procedure is repeated at higher orders.

A similar improvement can be performed with the three-body force,

$\displaystyle\Delta V(\\{\vec{r}_{i}\\};\Lambda)$ $\displaystyle=$
$\displaystyle\sum_{i<j}\left(\tilde{C}\,\delta_{\tilde{R}^{-1}}(\vec{r}_{ij})-C_{0}^{(0)}(\Lambda)\,\delta_{\Lambda}(\vec{r}_{ij})\right)$
(12) $\displaystyle+$
$\displaystyle\sum_{i<j<k}\,\sum_{cyc}\left(\tilde{D}\,\delta_{\tilde{R}^{-1}}(\vec{r}_{ij})\,\delta_{\tilde{R}^{-1}}(\vec{r}_{ik})\right.$
$\displaystyle\left.\qquad-
D_{0}^{(0)}(\Lambda)\,\delta_{\Lambda}(\vec{r}_{ij})\,\delta_{\Lambda}(\vec{r}_{ik})\right).$

Here $\tilde{D}$ still fixes $B_{3}$, while the additional effects of
$\tilde{R}$ are minimized at next-to-next-to-leading (N2LO), when the two-
derivative contact three-body force appears [29]. Again, we must take
$\tilde{R}$ small enough to preserve EFT convergence.

Other improvements could be made, but the two just mentioned are most natural
since they modify interactions already present at LO. To exemplify how the
freedom in the organization of the EFT impacts its results, we consider the
two different redefinitions of the LO interaction in Eqs. (11) and (12),
respectively

$\displaystyle\tilde{\mathcal{V}}_{I}^{(0)}$ $\displaystyle=$
$\displaystyle\tilde{C}\sum_{i<j}\delta_{\tilde{R}^{-1}}(\vec{r}_{ij})$ (13)
$\displaystyle+$ $\displaystyle
D_{0}^{(0)}(\Lambda)\sum_{i<j<k}\sum_{cyc}\delta_{\Lambda}(\vec{r}_{ij})\,\delta_{\Lambda}(\vec{r}_{ik}),$
$\displaystyle\tilde{\mathcal{V}}_{II}^{(0)}$ $\displaystyle=$
$\displaystyle\tilde{C}\sum_{i<j}\delta_{\tilde{R}^{-1}}(\vec{r}_{ij})$ (14)
$\displaystyle+$
$\displaystyle\tilde{D}\sum_{i<j<k}\sum_{cyc}\delta_{\tilde{R}^{-1}}(\vec{r}_{ij})\,\delta_{\tilde{R}^{-1}}(\vec{r}_{ik}).$

The redefined LO interaction is iterated to all orders. In the case of
$\tilde{\mathcal{V}}_{II}^{(0)}$, results are finite and the cutoff $\Lambda$
enters only in NLO calculations. For $\tilde{\mathcal{V}}_{I}^{(0)}$ the
cutoff $\Lambda$ is needed already at LO because of the three-body force. The
NLO interaction, Eq. (5), is included in first-order perturbation theory. The
form of the NLO interaction is not modified by the auxiliary interaction
$\Delta V$. However, the cutoff dependence of the NLO LECs change.

The fact that $\Delta V$ is a subset of higher-order interactions
automatically guarantees that a sufficiently small $\tilde{R}$ exists where
the above procedure can be carried out. What the maximum $\tilde{R}$ is a
question which we address in the following.

##  III Atomic 4He results

We analyze $\tilde{\mathcal{V}}_{I}^{(0)}$ and
$\tilde{\mathcal{V}}_{II}^{(0)}$ in systems ranging from two to five bosons.
We focus on the behaviour of LO and NLO with respect to the cutoff $\Lambda$
and the range $\tilde{R}$ of the auxiliary interaction. For convenience, we
display the $\Lambda$ and $\tilde{R}^{-1}$ dependence in units of the three-
body scale $Q_{3}$, Eq. (4). To be definite, we employ a Gaussian smearing of
the Dirac delta function,

$\delta_{\Lambda}(\vec{r})=\left(\frac{\Lambda}{\sqrt{\pi}}\right)^{3}~{}\text{exp}\left(-\dfrac{\Lambda^{2}}{4}\vec{r}^{\,2}\right).$
(15)

The large-cutoff limit of an observable $\mathcal{O}$ can be obtained from a
fit

$\mathcal{O}(\Lambda)=\mathcal{O}(\infty)+\frac{a}{\Lambda}+\frac{b}{\Lambda^{2}}$
(16)

with fitting parameters $a$, $b$. The $\Lambda^{-1}$ term accounts for
contributions that will be removed at next order once the cutoff exceeds the
theory’s breakdown scale, while the $\Lambda^{-2}$ term provides further
stability to the fits for moderate cutoff values.

We test our theory in systems made out of ${}^{4}\text{He}$ atoms, whose
inverse mass is $(\hbar c)^{2}/m=12.119~{}{\rm K\cdot\AA^{2}}$. As in previous
EFT work [5, 4], we constrain our theory with properties calculated [30, 31,
32, 33, 34] with either LM2M2 [35] or PCKLJS [36]
${}^{4}\text{He}$-${}^{4}\text{He}$ potential models: The values listed in
Table 1 are used here in lieu of experimental data, since only a few
experimental values [37, 38, 39] — also listed in Table 1 — are known.
Experimental data could replace potential-model input once more of them are
available.

|  LM2M2 |  PCKLJS |  Exp.
---|---|---|---
$a_{2}$ [Å] | 100.23 | 90.42(92) |
$r_{2}$ [Å] | 7.326 | 7.27 |
$B_{2}$ [mK] | 1.3094 | 1.6154 |  $1.3^{+0.25}_{-0.19}$; 1.76(15)
$B_{3}$ [mK] | 126.50 | 131.84 |
$B_{3}^{*}$ [mK] | 2.2779 | 2.6502 |
$B_{3}^{*}-B_{2}$ [mK] | 0.9685 | 1.0348 | 0.98(2)
$B_{4}$ [mK] | 559.22 | 573.90 |
$B_{5}$ [mK] | 1306.7 | - |
Table 1: Values for the two-body scattering parameters (scattering length
$a_{2}$ and effective range $r_{2}$) and $N$-body binding energies (ground
states $B_{N}$ and first-excited state $B_{3}^{*}$) of atomic 4He systems from
two phenomenological potentials LM2M2 [35] and PCKLJS [36], as extracted from
Refs. [30, 31, 32, 33, 34]. Known experimental values [37, 38, 39] are given
for comparison.

The structure of these atomic clusters is tied to the three-body scale
$Q_{3}\simeq 0.08$ Å-1. Since $|a_{2}|Q_{3}\simeq 8$, we expect an expansion
around the unitarity limit to converge. Nevertheless, for simplicity — and to
compare with previous calculations [5, 4] — we improve our LO to reproduce the
${}^{4}\text{He}$-${}^{4}\text{He}$ scattering length $a_{2}$, in addition to
the ground-state binding energy of the ${}^{4}\text{He}$ trimer, $B_{3}$. At
two-body level, the LO scattering fit is done using the Numerov algorithm. The
three-, four-, and five-body systems are solved with the stochastic
variational method (SVM) [40]. We predict the binding energies of the dimer,
$B_{2}$, trimer first-excited state, $B_{3}^{*}$, and of the tetramer and
pentamer ground states, $B_{4}$ and $B_{5}$. At NLO, we employ the distorted-
wave Born approximation. We fit the effective range $r_{2}$ and the ground-
state binding energy of the tetramer, $B_{4}$, and check for improvements in
$B_{2}$, $B_{3}^{*}$ and $B_{5}$.

Since we do not include the Van der Waals interaction explicitly, we would
expect the expansion parameter to be ${\cal O}(r_{\rm vdW}Q)$, where $r_{\rm
vdW}\sim 5$Å $\sim r_{2}$ and $Q$ is the typical particle momentum in the
bound state. For dimer properties, where $Q\sim a_{2}^{-1}$, the errors of the
expansion are expected to be $\sim r_{2}/a_{2}\simeq 10\%$ at LO and $\sim
r_{2}^{2}/a_{2}^{2}\simeq 1\%$ at NLO. The errors for more-body states is
likely larger but harder to estimate. Taking $Q\sim Q_{3}$, the only scale in
the unitarity limit, the errors would be $\sim 40\%$ at LO and $\sim 15\%$ at
NLO. These estimates include the power-counting argument that (i) the four-
body force enters at NLO and thus should produce LO errors comparable to those
of $r_{2}$; and (ii) the error at NLO should be given by the second Born
approximation for NLO interactions as well as the first Born approximation for
N2LO interactions, such as a three-body momentum-dependent force.
Nevertheless, previous studies of Contact EFT for 4He atomic systems [5, 4]
suggest that these errors are overestimated, perhaps because $Q_{3}$ is an
overestimate of the particle binding momentum.

In this section we show results of the theory fitted to LM2M2 quantities using
a derivative NLO term, Eq. (8). The same calculations were performed also for
a local, $r^{2}$ version defined in App. A. In App. B we summarize our
numerical results calculated with derivative and $r^{2}$ NLO interactions for
both LM2M2 and PCKLJS input. Our main conclusions are independent of the
specific input and form adopted for the NLO interaction.

###  III.1 Leading order

When the LO interaction $\mathcal{V}^{(0)}$, Eq. (1), is fitted to the
scattering length $a_{2}$, the resulting dimer binding energy $B_{2}$ depends
on the momentum cutoff $\Lambda$ as shown in Fig. 1. In the zero-range limit,
$B_{2}(\Lambda\rightarrow\infty)=1/ma_{2}^{2}$ and the ratio to the energy
obtained directly from the LM2M2 potential, $B_{2}^{\rm LM2M2}$ (see Table 1),
is $B_{2}(\Lambda\rightarrow\infty)/B_{2}^{\rm LM2M2}\simeq 0.93$. As
expected, results vary by less than 10% once the cutoff exceeds the expected
breakdown scale, $\Lambda/Q_{3}\hskip
1.99997pt\raisebox{2.15277pt}{$>$}\raisebox{-3.00003pt}{$\sim$}\hskip
1.99997pt(r_{\rm vdW}Q_{3})^{-1}\sim 3$.

Figure 1: Calculated dimer binding energy normalized to the dimer binding
energy obtained directly from the LM2M2 potential [35], $B_{2}/B_{2}^{\rm
LM2M2}$, as function of the cutoff $\Lambda$ in units of the three-body scale
$Q_{3}$. Results from the LO and NLO potentials, respectively
$\mathcal{V}^{(0)}$ and
$\mathcal{V}^{(0+1)}\equiv\mathcal{V}^{(0)}+\mathcal{V}^{(1)}$, are shown by
triangles and circles connected with dotted lines. At the two-body level the
improved potentials of type $I$ and $II$ are equivalent. The (blue) hatched
and shaded bands show the LO and NLO values from the improved potentials
$\tilde{\mathcal{V}}^{(0)}_{I}$/$\tilde{\mathcal{V}}^{(0)}_{II}$ and
$\tilde{\mathcal{V}}^{(0+1)}_{I}$/$\tilde{\mathcal{V}}^{(0+1)}_{II}$,
respectively, when the fake range is in the range
$6\leq(Q_{3}\tilde{R})^{-1}\leq 24$.

Closely related are the results of the redefined LO potentials
$\tilde{\mathcal{V}}^{(0)}_{I}$, Eq. (13), and
$\tilde{\mathcal{V}}^{(0)}_{II}$, Eq. (14), which are equivalent at the two-
body level. Since their two-body parts depend solely on the size of the fake
range $\tilde{R}$, no $\Lambda$ dependence is introduced. In Fig. 1 the
resulting dimer energies from $\tilde{\mathcal{V}}^{(0)}_{I}$ or
$\tilde{\mathcal{V}}^{(0)}_{II}$ would show as horizontal lines that intersect
the calculated $\mathcal{V}^{(0)}(\Lambda)$ energies where the inverse fake
range reaches the cutoff,
$\tilde{\mathcal{V}}^{(0)}_{I}(\tilde{R}^{-1}=\Lambda)=\tilde{\mathcal{V}}^{(0)}_{II}(\tilde{R}^{-1}=\Lambda)=\mathcal{V}^{(0)}(\Lambda)$.
For clarity, we represent the results for $6\leq(Q_{3}\tilde{R})^{-1}\leq 24$
as a horizontal band. The band thus matches the residual cutoff dependence of
$\mathcal{V}^{(0)}$. For decreasing fake range, $B_{2}/B_{2}^{\rm LM2M2}$
attains the $B_{2}(\Lambda\rightarrow\infty)/B_{2}^{\rm LM2M2}$ value. As the
fake range increases, the ratio stays well within the expected $\sim 10\%$
truncation error and reaches $1$ for $\tilde{R}_{c}^{-1}\simeq 4.8\,Q_{3}$,
where the physical effective range is roughly reproduced already at the LO.
Allowing even larger fake ranges leads to a deterioration of our results —
this region will be addressed in Sec. III.3. The values of $B_{2}/B_{2}^{\rm
LM2M2}$ at various fake ranges can be found in Table 2 of App. B.

In Fig. 2, we present the LO predictions for the $B_{3}/B_{3}^{*}$ (left
panel) and $B_{4}/B_{3}$ (right panel) binding energy ratios as a function of
the increasing cutoff $\Lambda$. Predictions for $B_{5}/B_{3}$ are given in
Fig. 3. The results are for the improved LO potentials
$\tilde{\mathcal{V}}^{(0)}_{I}$ and $\tilde{\mathcal{V}}^{(0)}_{II}$ with a
few representative values of the inverse fake range in the range
$6\leq(Q_{3}\tilde{R})^{-1}\leq 24$. We compare them to the standard LO
potential $\mathcal{V}^{(0)}$.

|
---|---
Figure 2: Binding energies for the excited trimer $B_{3}^{*}$ (left panel) and
ground tetramer $B_{4}$ (right panel) at LO, normalized to the ground-trimer
binding energy $B_{3}$, as function of the cutoff $\Lambda$ in units of the
three-body scale $Q_{3}$. Results for the LO potential $\mathcal{V}^{(0)}$ are
shown by triangles connected with a dotted line. Results for the improved LO
potential $\tilde{\mathcal{V}}^{(0)}_{I}$ are given in red with different
symbols for different inverse fake-range values: $\tilde{R}^{-1}=6\,Q_{3}$
(diamonds connected by a dash-dotted line), $\tilde{R}^{-1}=12\,Q_{3}$
(squares, dashed line), and $\tilde{R}^{-1}=24\,Q_{3}$ (circles, solid line).
The blue shaded band shows the LO values from $\tilde{\mathcal{V}}^{(0)}_{II}$
when the fake range is in the range $6\leq(Q_{3}\tilde{R})^{-1}\leq 24$. The
(green) horizontal dashed lines mark the result of the full LM2M2 potential
[34]. Figure 3: Same as Fig. 2 but for the ground-pentamer binding energy
$B_{5}$. The LM2M2 result is from Ref. [30]

The ratios obtained with $\mathcal{V}^{(0)}$ converge with $\Lambda$, and
using the extrapolation function (16) we obtain
$B_{3}/B_{3}^{*}(\Lambda\to\infty)=59.089$,
$B_{4}(\Lambda\to\infty)/B_{3}=4.150(6)$, and
$B_{5}(\Lambda\to\infty)/B_{3}=9.27(5)$, which are consistent with Refs. [5,
4]. Here, the number in the parenthesis represents the numerical uncertainty,
if non-negligible, which should be further inflated by the expected LO
theoretical error.

Because of the three-body force, the calculated
$\tilde{\mathcal{V}}^{(0)}_{I}$ energies depend on $\Lambda$. While
$\mathcal{V}^{(0)}$ results show the expected residual cutoff dependence
$\propto\Lambda^{-1}$, $\tilde{\mathcal{V}}^{(0)}_{I}$ energies appear, based
on the limited amount of points available, to be compatible with a
$\Lambda^{-2}$ behavior at large $\Lambda$. This would imply that the
$\Lambda^{-1}$ dependence of LO results — which with an unimproved action is
compensated by NLO corrections — is removed by the fake range. This is not
unexpected for two- and three-body systems, where improvement will include
errors of order $\tilde{R}$ instead. Since the four-body force is driven by
the two-body range, we cannot conclude that the absence of $\Lambda^{-1}$
dependence in higher-body systems implies that the four-body force is not
strictly required for NLO renormalizability. This conclusion would contradict
evidence from previous work [4, 27].

For $6\leq(Q_{3}\tilde{R})^{-1}\leq 24$ the calculated
$\tilde{\mathcal{V}}^{(0)}_{I}$ binding energies converge at sufficiently
large $\Lambda$ to values that are within the LO error of the full LM2M2
result. We observe that all $\tilde{\mathcal{V}}^{(0)}_{I}$ results get closer
to the LM2M2 values the larger $\Lambda$ and the fake range $\tilde{R}$ are.
Cutoff-extrapolated values for various fake ranges are given in Table 3 of
App. B.

On the other hand, when the cutoff is smaller than the inverse fake range but
larger than the supposed breakdown scale of the theory, we see an enhancement
of the cutoff dependence in the results. This phenomenon is best illustrated
by the behavior of $\tilde{\mathcal{V}}^{(0)}_{I}(\tilde{R}^{-1}=24Q_{3})$ in
Fig. 3, where the calculated pentamer $B_{5}/B_{3}$ ratios at
$\Lambda=6\,Q_{3}$ and $12\,Q_{3}$ are very different from the same quantities
with the unimproved action. For $\Lambda$ larger than the breakdown scale of
the theory, the convergence pattern sets in only for $\Lambda\hskip
1.99997pt\raisebox{2.15277pt}{$>$}\raisebox{-3.00003pt}{$\sim$}\hskip
1.99997pt\tilde{R}^{-1}$. Since $\tilde{R}^{-1}$ functions as a two-body
cutoff, this effect is in agreement with what was found in Ref. [41].

In contrast to $\tilde{\mathcal{V}}^{(0)}_{I}$, the improved
$\tilde{\mathcal{V}}^{(0)}_{II}$ interaction depends solely on the fake range.
As for the dimer, we depict the $\Lambda$-independent
$\tilde{\mathcal{V}}^{(0)}_{II}$ results as straight horizontal bands which
represent the range $6\leq(Q_{3}\tilde{R})^{-1}\leq 24$. Again, the
corresponding bands match the residual cutoff dependence of the
$\mathcal{V}^{(0)}(\Lambda)$ results. Tables 4 and 5 of App. B give the energy
ratios at various fake ranges. Note that $\mathcal{V}^{(0)}$ results converge
from below, but to a larger value than the exact result only for the trimer
excited state. Thus, only in this case does $\mathcal{V}^{(0)}_{II}$ give
results closer to exact than the asymptotic $\mathcal{V}^{(0)}$ value. In any
case, the width of the $\mathcal{V}^{(0)}_{II}$ band is not larger than the
difference between $\mathcal{V}^{(0)}$ and exact results.

Partially accounting for the physical two-body effective range through the
fake range ($\tilde{\mathcal{V}}^{(0)}_{I}$) brings LO results closer to the
“exact” results. Incorporating this fake range in the three-body force as well
($\mathcal{V}^{(0)}_{II}$) does not always accomplish the same, perhaps
reflecting the fact that the range of the three-body force is not fixed by
that of the two-body force. On the other hand, the different ranges
$\tilde{R}$ and $\Lambda^{-1}$ of, respectively, two- and three-body
interactions in $\tilde{\mathcal{V}}^{(0)}_{I}$ should be selected with
caution, as it is often required that $\Lambda\hskip
1.99997pt\raisebox{2.15277pt}{$>$}\raisebox{-3.00003pt}{$\sim$}\hskip
1.99997pt\tilde{R}^{-1}$. Moreover, $\tilde{\mathcal{V}}^{(0)}_{II}$ decreases
significantly the number of few-body calculations to be performed at this
order since it does not require independent variation of both ranges.

In either case, the range of results is comparable to the difference between
standard LO ($\mathcal{V}^{(0)}$) and exact model results. We examine next the
extent to which fake range effects can be removed perturbatively at NLO.

###  III.2 Next-to-leading order

At NLO, we study binding energies obtained with the improved potentials
$\tilde{\mathcal{V}}_{I}^{(0+1)}\equiv\tilde{\mathcal{V}}^{(0)}_{I}+\mathcal{V}^{(1)}$
and
$\tilde{\mathcal{V}}_{II}^{(0+1)}\equiv\tilde{\mathcal{V}}^{(0)}_{II}+\mathcal{V}^{(1)}$
and how they compare with the standard potential
$\mathcal{V}^{(0+1)}\equiv\mathcal{V}^{(0)}+\mathcal{V}^{(1)}$.

The LM2M2 effective range $r_{2}$, given in Table 1, is fitted at NLO. The
cutoff convergence of $B_{2}$, obtained from $\mathcal{V}^{(0+1)}$, is seen in
Fig. 1 to rapidly reach the exact result within 1%, as expected. As a
consequence, the width of the horizontal band for the improved potential at
NLO,
$\tilde{\mathcal{V}}^{(0+1)}_{I}(\tilde{R}^{-1}=\Lambda)=\tilde{\mathcal{V}}^{(0+1)}_{II}(\tilde{R}^{-1}=\Lambda)=\mathcal{V}^{(0+1)}(\Lambda)$
with $6\leq(Q_{3}\tilde{R})^{-1}\leq 24$, is narrow and much smaller than the
change in standard results from LO to NLO. As before, as long as the fake
range remains smaller than the physical effective range, improved-potential
results preserve the accuracy of the unimproved results.

The ratio $B_{4}/B_{3}$ is used to constrain the NLO four-body interaction,
thus reproducing the LM2M2 line on the right panel of Fig. 2. The calculated
$B_{3}/B_{3}^{*}$ and $B_{5}/B_{3}$ binding-energy ratios are shown as a
function of the momentum cutoff $\Lambda$ in Fig. 4. As a benchmark, we
extrapolate our $\mathcal{V}^{(0+1)}$ trimer and pentamer results for
$\Lambda\rightarrow\infty$ using Eq. (16). We obtain
$B_{3}/B_{3}^{*}(\Lambda\to\infty)=55.803$ and
$B_{5}(\Lambda\to\infty)/B_{3}=10.17(8)$. The extrapolated pentamer ratio
agrees with the earlier result of Ref. [4], which was obtained from the same
$\mathcal{V}^{(0+1)}$ potential renormalized to LM2M2 quantities. For the
trimer, the ratio obtained here slightly differs from Ref. [4], which employed
PCKLJS quantities as reference. Even better agreement is found using the
PCKLJS setting, as can be seen in Table 4 of App. B. As noted in Ref. [4], the
change of only $\sim 10\%$ compared to LO suggests the expansion parameter is
smaller than expected, with asymptotic NLO values differing from exact ones by
at most a few percent.

|
---|---
Figure 4: Binding energies for the excited trimer $B_{3}^{*}$ (left panel) and
ground pentamer $B_{5}$ (right panel) up to NLO, normalized to the ground-
trimer binding energy $B_{3}$, as function of the cutoff $\Lambda$ in units of
the three-body scale $Q_{3}$. Results for the LO and NLO potentials
$\mathcal{V}^{(0)}$ and $\mathcal{V}^{(0+1)}$ are shown by, respectively empty
triangles connected with a dotted line and empty circles connected with a
solid line. Results for the improved NLO potentials
$\tilde{\mathcal{V}}^{(0+1)}_{I}$ and $\tilde{\mathcal{V}}^{(0+1)}_{II}$ at
$\tilde{R}^{-1}=12\,Q_{3}$ are given by, respectively, (red) half-filled
circles connected by a (red) solid line and (blue) solid circles connected by
a (blue) solid line. The (red) hatched area and the (blue) shaded band
indicate the spread of $\mathcal{V}^{(0+1)}_{I}$ and
$\mathcal{V}^{(0+1)}_{II}$ results, respectively, for fake-range values in the
range $6\leq(Q_{3}\tilde{R})^{-1}\leq 24$. (For the pentamer, we show
$\mathcal{V}^{(0+1)}_{I}$ results only for one fake-range value, as obtaining
accurate results for the entire range is numerically challenging.) The (green)
horizontal dashed lines mark the result of the full LM2M2 potential [34, 30].

We have found our $\tilde{\mathcal{V}}^{(0+1)}_{I}$ NLO calculations to be
rather sensitive to the accuracy in the description of the LO wavefunction.
The sensitivity is especially pronounced when the LO two- and three-body
potentials have very different ranges $\tilde{R}$ and $\Lambda^{-1}$. As a
result, large cancellations occur in various potential channels, which in turn
lead to relatively slow convergence in our few-body calculations. While for
trimer and tetramer we were able to obtain accurate wavefunctions, we
encountered challenges with the pentamer ground state. Consequently, for
$\tilde{\mathcal{V}}^{(0+1)}_{I}$ we depict in Fig. 4 the $B_{3}/B_{3}^{*}$
ratio for a range of fake-range values, $6\leq(Q_{3}\tilde{R})^{-1}\leq 24$,
but $B_{5}/B_{3}$ only for $\tilde{R}^{-1}=12\,Q_{3}$. For clarity, pentamer
results for smaller $\tilde{R}^{-1}$ are not displayed in the figure, but the
corresponding $\Lambda\rightarrow\infty$ values are given in App. B. Results
for larger $\tilde{R}^{-1}$ are at the moment beyond the reach of our few-body
method.

The resulting $\tilde{\mathcal{V}}^{(0+1)}_{I}$ values for the
$B_{3}/B_{3}^{*}$ and $B_{5}/B_{3}$ ratios follow the pattern observed when
using solely the LO $\tilde{\mathcal{V}}^{(0)}_{I}$ part: rather fast
stabilization once the momentum cutoff $\Lambda$ is large enough to suppress
regulator-induced, and thus arbitrary, effects such as a finite range
$\propto\Lambda^{-1}$ in the three-body force. Comparison with Fig. 2 shows
that the perturbative NLO correction $\tilde{\mathcal{V}}^{(1)}$ shifts the
$B_{3}/B_{3}^{*}$ and $B_{5}/B_{3}$ ratios closer to the LM2M2 values. At the
same time, the residual $\Lambda$ dependence decreases with respect to the LO
results.

The $\tilde{\mathcal{V}}^{(0+1)}_{II}$ type of improved EFT potential
introduces $\Lambda$ dependence solely through the perturbative inclusion of
$\mathcal{V}^{(1)}$ at NLO. The variation of our
$\tilde{\mathcal{V}}^{(0+1)}_{II}$ results in the range
$6\leq(Q_{3}\tilde{R})^{-1}\leq 24$ is shown for the trimer as well as the
pentamer in Fig. 4. For any given $\tilde{R}^{-1}$ in this interval, the
resulting $B_{3}/B_{3}^{*}$ and $B_{5}/B_{3}$ ratios stabilize with the
increasing cutoff. The corresponding quantities, extrapolated to
$\Lambda\rightarrow\infty$, are listed in Tables 4 and 5 of App. B. There is a
considerably milder residual cutoff dependence over the larger cutoff region
than in the $\tilde{\mathcal{V}}^{(0+1)}_{I}$ case. This is likely linked to
the same fake ranges being employed in both two- and three-body potentials of
the LO $\tilde{\mathcal{V}}^{(0)}_{II}$ interaction.

Computationally, there is a significant difference between the
$\tilde{\mathcal{V}}^{(0+1)}_{I}$ and $\tilde{\mathcal{V}}^{(0+1)}_{II}$ few-
body calculations, since for $\tilde{\mathcal{V}}^{(0)}_{II}$ the Schrödinger
equation needs to be solved only once for each fake range. This is especially
beneficial once many $\tilde{R}$ values need to be considered. Furthermore, in
the $\tilde{\mathcal{V}}^{(0+1)}_{II}$ case we do not observe severe numerical
problems. This is likely related to the same $\tilde{R}$ values for the two-
and three-body LO potentials.

Despite the computational differences, the NLO results are qualitatively
similar for both types of improvements we consider here. For all trimer NLO
results we notice relatively fast convergence with both increasing $\Lambda$
value and EFT order. This can be understood by the shallow nature of the
trimer excited state where the corresponding small typical momentum induces
only a minor contribution from the further subleading corrections. The
inclusion of the NLO terms yields a $B_{3}/B_{3}^{*}$ ratio which is already
close to the “exact” value calculated directly with the LM2M2 potential. Even
for the pentamer the cutoff dependence is mild once the cutoff is somewhat
higher than for the trimer, probably as a consequence of a higher
characteristic binding momentum. NLO results also improve significantly if
compared with the exact result.

We observe that the trimer $\tilde{\mathcal{V}}^{(0+1)}_{II}$ band is
relatively wide even at large cutoff values. For this improved potential, the
$\tilde{R}^{-1}=12\,Q_{3}$ results tend to differ the most from both the
standard EFT potential and the “exact” LM2M2 value. Excluding the
$B_{3}/B_{3}^{*}$ ratios calculated for fake ranges close to
$\tilde{R}^{-1}=12\,Q_{3}$, the $\tilde{\mathcal{V}}^{(0+1)}_{II}$ band
shrinks significantly, making it more similar at large $\Lambda$ to the band
calculated with $\tilde{\mathcal{V}}^{(0+1)}_{I}$. In
$\tilde{\mathcal{V}}^{(0+1)}_{I}$ the three-body fake range is absent.
Therefore, we attribute this large width primarily to a still substantial
influence of the subleading three-body contributions induced around
$\tilde{R}^{-1}=12\,Q_{3}$ by accidental cancellations. The same statements
hold for the pentamer, but the effect seems to be quenched by the presence of
the four-body force.

Most importantly, at large cutoff NLO results for the two types of improvement
differ only by a few percent. They are very close to the non-improved NLO
results, demonstrating that at the NLO level the choice of LO interaction was
irrelevant as far as observables are concerned.

###  III.3 Maximal fake range

So far, we have presented our results using improved LO and NLO potentials
with fake ranges carefully selected to be smaller than the range of the
$\rm{}^{4}He$-$\rm{}^{4}He$ interaction. Here, we will describe effects which
become increasingly dominant as the fake range increases and becomes
comparable to the size of the atomic interaction.

In Fig 5, we display the dependence of our $\tilde{\mathcal{V}}^{(0+1)}_{II}$
results for the trimer and pentamer on the inverse fake range
$\tilde{R}^{-1}$. We show the values for both
$B_{3}/B_{3}^{*}(\Lambda\to\infty)$ and $B_{5}(\Lambda\to\infty)/B_{3}$
obtained from the extrapolation with Eq. (16) at fixed $\tilde{R}^{-1}$. We
also display the spread of results for $\Lambda\geq 6Q_{3}$. For comparison we
show also the corresponding bands for the standard interactions
$\mathcal{V}^{(0)}$ and $\mathcal{V}^{(0+1)}$.

|
---|---
|
Figure 5: Binding energies for the excited trimer $B_{3}^{*}$ (left panel) and
ground pentamer $B_{5}$ (right panel) up to NLO, normalized to the ground-
trimer binding energy $B_{3}$, as function of the inverse fake range
$\tilde{R}^{-1}$ in units of the three-body scale $Q_{3}$. Results for the
improved NLO potential $\tilde{\mathcal{V}}^{(0+1)}_{II}$ for cutoff values
$\Lambda\geq 6\,Q_{3}$ are given by a (blue) shaded band; extrapolated values
for $\Lambda\rightarrow\infty$ at the given $\tilde{R}^{-1}$ are marked by
(blue) stars. For comparison, the variation of LO ($\mathcal{V}^{(0)}$) and
NLO ($\mathcal{V}^{(0+1)}$) results for cutoff $\Lambda\geq 6\,Q_{3}$ are
shown by light and dark (gray) hatched bands, respectively. The (green)
horizontal dashed lines mark the result of the full LM2M2 potential [34, 30].
The vertical dotted line indicates the fake range value,
$\tilde{R}_{c}^{-1}\simeq 4.8\,Q_{3}$, where the improved LO potential
$\tilde{\mathcal{V}}_{II}^{(0)}$ exactly reproduces the LM2M2 effective range.

The figure demonstrates that the extrapolated results become consistent with
the standard EFT formulation, $\mathcal{V}^{(0+1)}$, as both the fake range
and the cutoff approach the contact limit $\tilde{R}^{-1}\rightarrow\infty$
and $\Lambda\rightarrow\infty$, where
$\tilde{\mathcal{V}}^{(0+1)}_{II}=\mathcal{V}^{(0+1)}$. For quite a range of
fake-range values, the arbitrary choice of fake-range improvement at LO can be
compensated perturbatively at NLO. Both the positions and widths of the NLO
bands are similar.

As $\tilde{R}^{-1}$ decreases, there are two special $\tilde{R}^{-1}$ values
visible in Fig. 5. As we discussed in the previous subsection,
$\tilde{R}^{-1}=12\,Q_{3}$ represents the situation where our NLO results
differ the most from the “exact” LM2M2 result, likely due to the induced
subleading correction in the three-body force. If that is the case, then the
difference will be largely suppressed at N2LO, where the first three-body
subleading contribution enters. The second special value,
$\tilde{R}_{c}^{-1}=4.8\,Q_{3}$, represents the situation at which the LM2M2
physical effective range $r_{2}$ is reasonably well reproduced in our
calculations already at LO. In this case, NLO corrections besides the four-
body scale are absorbed in the LO fake range, and the variation of our few-
body NLO results with respect to the cutoff $\Lambda$ is minimal. Moreover, at
this small $\tilde{R}^{-1}$ the corresponding potentials become reasonably
soft, which greatly facilitates their treatment in few- and many-body
calculations. Consequently, it might be argued that this is the most
convenient point to fix the LO fake range.

On the other hand, it is also a delicate choice. Once
$\tilde{R}^{-1}<\tilde{R}_{c}^{-1}$ results begin to reveal increasing
fluctuations and quickly deteriorate. Continuing past this point, the
resummation procedure likely becomes nonperturbative, and the standard power
counting can no longer be sustained. This sets a hard limit for the LO
resummation in terms of a maximal fake range $\tilde{R}$ that is allowed in
order to maintain theory convergence. The deterioration of our results is more
dramatic for the pentamer. With the limitations discussed above, we find a
similar outcome when employing $\tilde{\mathcal{V}}^{(0)}_{I}$. Since the
“wall” gets steeper when going from trimer to pentamer, the choice
$\tilde{R}=\tilde{R}_{c}$ might become even more delicate for larger clusters.

From these results, we conclude that a theory with the modified LO range is
equivalent to the standard Contact EFT as long as $\tilde{R}\hskip
1.99997pt\raisebox{2.15277pt}{$<$}\raisebox{-3.00003pt}{$\sim$}\hskip
1.99997ptr_{2}$, effectively setting the boundary for possible theory
resummation.

##  IV Discussion and conclusions

We have described how we can improve the action of a renormalizable and
convergent power counting by taking advantage of the partial flexibility in
the definition of each order. Higher-order EFT interactions are increasingly
singular and demand perturbation theory for renormalization. Model
independence requires insensitivity to the regularization procedure and its
cutoff. Adding to LO subleading effects that preserve order-by-order
renormalizability can provide benefits such as improved convergence and easier
access to numerical results.

We based our improvement on the observation that sometimes finite values of
the cutoff $\Lambda$, which endows LO interactions with a range, give results
closer to exact results than the $\Lambda\to\infty$ limit. Thus, we used two
types of LO improvement which include an arbitrary fake range for the two-body
interaction, with one of them also including the same fake range in the three-
body force. We have illustrated this technique employing as benchmark a well-
known phenomenological potential for $\rm{}^{4}He$ atoms. Previous studies
[29, 5, 4] had already demonstrated that an EFT with contact interactions can
reproduce the binding energies of systems of up to six atoms systematically.
We replicated these results and used them for comparison. To test our approach
further, we conducted our study using also a second potential model as well as
two different forms for NLO interactions — non-local and local versions. We
found no significant differences in the results and conclusions among these
variations.

As could be expected, improved-LO results were almost always better than
unimproved results. More crucially, we found that NLO results were not
significantly affected by these modifications as long as the LO interaction
remained perturbatively close to the contact limit. In other words, for an
extensive range of values the arbitrary choice of fake range is compensated
with the perturbative corrections stipulated by the standard power counting.
In this range, the improved actions we considered can be used just as other
improvements that preserve renormalization-group invariance, such as the
choice of low-energy data used to determine interactions strengths. However,
we noticed a sudden change in behavior when the arbitrary fake range used at
LO increases beyond the physical effective range, suggesting that using this
method when they are equal should be done with care.

In spirit, our approach is similar to that of Ref. [42], where a finite cutoff
was kept at each order, with the hope that the cutoff range could be increased
order by order. For the case where we chose to improve both two- and three-
body potentials with the same fake range, the inverse of our fake range
functions as the cutoff at LO in Ref. [42]. However, our improvement is more
general as it does not substitute for a cutoff at LO when the three-body force
is not improved, nor for a cutoff at subleading orders.

The improvement is meant to minimize NLO contributions, and the possibility
arises that LO interactions can be chosen to effectively reduce the error to
N2LO. While the results presented above are promising, in order to achieve a
full elimination of NLO contributions $\mathcal{V}^{(0)}$ must be improved to
account for not only the two-body effective range but also the four-body scale
introduced through the four-body force. It is unlikely that this can be done
without including a four-body potential. For the sake of model independence
and renormalization, it would be essential to study N2LO contributions to make
sure NLO effects have indeed been entirely removed. If successful, this
technique could be iterated to resum more subleading aspects of the theory.
However, since the number of subleading interactions grows, it will be
increasingly difficult to incorporate them in an improved LO.

The more improvements are made, the closer one gets to phenomenological
models. Typically such models have a finite range for each component of the
potential. These ranges are either fitted to data or varied within narrow
intervals (for example, Ref. [43]). One might argue that a phenomenological
potential can be used as an auxiliary interaction, essentially transforming a
model into the lowest order of an improved EFT action. A phenomenological
potential may include contributions to several orders of a contact theory,
partially accounting for them. However, indiscriminate improvement destroys
order-by-order renormalization and obscures power counting. It is crucial to
check the renormalizability of the theory and model independence by varying
the auxiliary potential used and calculating at least one order of the power
counting more than the ones that have been resummed. Moreover, the calculation
of subleading contributions can be simplified by choosing the auxiliary
interaction to maximize the stability of numerical codes used in the wave-
function calculation. Our improvement is controlled, as it is constrained by
the small size of NLO corrections. Because power counting is preserved, the
type of interactions and the number of physical parameters at each order are
unchanged — for example, for systems sufficiently closed to unitarity, LO
physics is still determined by a single, three-body parameter [13]. One can
also still estimate the theoretical error of a calculation a priori from the
expected size of the first order not accounted for.

The improved-action method might also shed light on why additional
approximations usually performed in many-body calculations could still be
compatible with EFTs without necessarily compromising renormalizability. Such
approximations are exemplified by methods like many-body perturbation theory
[44] or approaches utilizing SRG transforms to soften interactions [45, 46].
In the former case, only part of the expected LO contributions is treated
exactly [47], while in the latter induced many-body correlations are typically
omitted. A possible justification for this procedure is a change in power
counting as the number of particles become large. (For an example in Chiral
EFT, see Ref. [48].) Or, perhaps, it can be interpreted that these
approximations amount to the inclusion of subleading interactions at LO, which
cancel the neglected effects.

We conclude that, even when subleading interactions can be accounted for
explicitly, it is beneficial to start from a LO that captures as many of the
qualitative aspects of the system as possible. For many-fermion systems, in
particular, an improved LO might be able to provide a stable state upon
perturbative corrections are much simpler to implement. We plan to tackle this
issue in a future publication.

## acknowledgement

We thank N. Barnea, B. Bazak, H. W. Grießhammer, A. Lovato, and F. Pederiva
for the useful exchanges and discussions. The work of M. Schäfer was supported
by the Czech Science Foundation GAČR grant 22-14497S. This material is based
upon work supported in part by the U.S. Department of Energy, Office of
Science, Office of Nuclear Physics, under award DE-FG02-04ER41338.

##  Appendix A Local NLO formulation

In addition to the non-local formulation of the NLO presented in Sec. II, Eq.
(II); we also employ a simplified, coordinate-space version,

$\displaystyle\tilde{V}_{\text{2B}}^{(1)}(\\{\vec{r}_{i}\\};\Lambda)$
$\displaystyle=$
$\displaystyle\tilde{C}_{0}^{(1)}(\Lambda)\sum_{i<j}\delta_{\Lambda}(\vec{r}_{ij})$
(17) $\displaystyle+$
$\displaystyle\tilde{C}_{2}^{(1)}(\Lambda)\sum_{i<j}r^{2}_{ij}\delta_{\Lambda}(\vec{r}_{ij}).$

These interactions differ from each other by derivatives of the wavefunction,
which, in the contact limit, affect only non-zero relative angular momentum of
the two particles. In this appendix, we derive the local NLO formulations
starting from the standard EFT operator of the NLO effective-range term. In
particular, we consider the NLO contribution to the energy of a pair of
particles using the Gaussian regulator (Eq. (15)).

First, note that one can write

$\bra{\psi^{(0)}}\left(p^{\prime
2}+p^{2}\right)\ket{\psi^{(0)}}=\bra{\psi^{(0)}}\left[\left(\vec{p}\,^{\prime}-\vec{p}\right)^{2}+2\vec{p}\,^{\prime}\cdot\vec{p}\right]\ket{\psi^{(0)}}.$
(18)

Where $p$ and $p^{\prime}$ are the incoing and outgoing momenta of the
particles in the center of mass frame and $\psi^{(0)}$ is the LO wavefunction
of the state on which the perturbation theory is applied. Next, if we denote
the Fourier transform of $\delta_{\Lambda}(\vec{r})$ by
$f_{\Lambda}(\vec{p})$, then

$f_{\Lambda}(\vec{p}\,^{\prime})f_{\Lambda}(\vec{p})=f_{\Lambda}(\vec{p}-\vec{p}\,^{\prime})\left[1+{\cal
O}\left(\frac{\vec{p}\,^{\prime}\cdot\vec{p}}{\Lambda^{2}}\right)\right].$
(19)

The $\vec{p}\,^{\prime}\cdot\vec{p}$ terms contribute only to $P$ and higher
waves. Thus, denoting $q=p-p^{\prime}$ and up to contributions to partial
waves higher than $S$, we can write

$\displaystyle\bra{\psi^{(0)}}\left(p^{\prime
2}+p^{2}\right)\ket{\psi^{(0)}}=\int\frac{d^{3}p\,^{\prime}}{(2\pi)^{3/2}}\int\frac{d^{3}p}{(2\pi)^{3/2}}\,\psi^{*(0)}(\vec{p}\,^{\prime})\,\psi^{(0)}(\vec{p})\,f_{\Lambda}(\vec{p}-\vec{p}\,^{\prime})\left(\vec{p}\,^{\prime}-\vec{p}\right)^{2}+\ldots$
$\displaystyle=\int\frac{d^{3}r\,^{\prime}}{(2\pi)^{3/2}}\,\psi^{*(0)}(\vec{r}\,^{\prime})\int\frac{d^{3}r}{(2\pi)^{3/2}}\,\psi^{(0)}(\vec{r})\int\frac{d^{3}p\,^{\prime}}{(2\pi)^{3/2}}\,e^{-i\vec{p}\,^{\prime}\cdot(\vec{r}\,^{\prime}-\vec{r})}\int\frac{d^{3}q}{(2\pi)^{3/2}}\,e^{i\vec{q}\cdot\vec{r}}\,\vec{q}^{\,2}\,f_{\Lambda}(\vec{q})+\ldots$
$\displaystyle=-\int\frac{d^{3}r^{\prime}}{(2\pi)^{3/2}}\psi^{*(0)}(\vec{r}\,^{\prime})\int\frac{d^{3}r}{(2\pi)^{3/2}}\psi^{(0)}(\vec{r})\,\delta(\vec{r}\,^{\prime}-\vec{r})\,\nabla^{2}\delta_{\Lambda}(\vec{r})+\ldots$
(20)
$\displaystyle=\frac{\Lambda^{2}}{4}\int\frac{d^{3}r}{(2\pi)^{3/2}}\,\psi^{*(0)}(\vec{r})\,\psi^{(0)}(\vec{r})\left(2-r^{2}\Lambda^{2}\right)\delta_{\Lambda}(\vec{r})+\ldots\;.$

The “$\ldots$” contain induced contributions to $P$ waves. We notice that
these contributions are suppressed by only one power of the breakdown scale,
while $P$-wave interactions are expected to be suppressed by three powers.
However, in $S$ or $D$-wave observables, at least two such interactions are
needed, which makes these interactions to induce errors at least one order
higher than we work at. From this analysis, we deduce that the results derived
from the derivative and local NLO interactions are equivalent within the
framework of this study and its outcomes. This conclusion is further
substantiated by the data provided in App. (B).

##  Appendix B Tables of results

In this appendix we list the numerical results obtained for the binding
energies of the systems considered in this work at various values of the fake
range $\tilde{R}$ for both $\tilde{\mathcal{V}}_{I}^{(0)}$ and
$\tilde{\mathcal{V}}_{II}^{(0)}$ interactions, Eqs. (13) and (14). For
comparison we also present results for $\tilde{R}=0$ obtained from the
“standard” $\mathcal{V}^{(0)}$, Eq. (1). For the NLO interaction, we consider
the two forms, two-derivative, Eq. (II), and $r^{2}$, Eq. (17). The binding
energies $B_{N}^{(*)}$, $N=2-5$, are given at LO and NLO with input parameters
fitted to two different sets of low-energy data predicted with the LM2M2 or
PCKLJS potential models, see Table 1. The results for LM2M2 are plotted in the
main text. All results are extrapolated in the regulator cutoff $\Lambda$
according to Eq. (16) and the extrapolation error is given. No EFT truncation
errors are shown.

At the two-body level, the interactions $\tilde{\mathcal{V}}_{I}^{(0)}$ and
$\tilde{\mathcal{V}}_{II}^{(0)}$ are equivalent. In Table 2 we give our
results for the two-body binding energy $B_{2}$ normalized to the respective
model binding energy $B_{2}^{\rm Model}$. Results are essentially
indistinguishable except perhaps for the local NLO potential at the largest
value of the fake range for PCKLJS, where it differs by about $3\%$ from the
direct model result and the NLO non-local interaction.

| $B_{2}/B_{2}^{\text{Model}}$
---|---
| LM2M2 | PCKLJS
$\tilde{R}^{-1}$ [$Q_{3}$] | LO | NLO $\nabla^{2}$ | NLO $r^{2}$ | LO | NLO $\nabla^{2}$ | NLO $r^{2}$
4.8 | 0.99 | 1.00 | 0.99 | 0.99 | 1.00 | 1.03
6.0 | 0.98 | 1.00 | 1.00 | 0.98 | 1.00 | 1.00
12 | 0.95 | 0.99 | 0.99 | 0.95 | 1.00 | 1.00
18 | 0.94 | 0.99 | 1.00 | 0.94 | 0.99 | 1.00
24 | 0.93 | 0.99 | 1.00 | 0.93 | 0.99 | 1.00
30 | 0.93 | 0.99 | 1.00 | 0.93 | 0.99 | 1.00
36 | 0.93 | 0.99 | 1.00 | 0.93 | 0.99 | 1.00
42 | 0.93 | 0.99 | 1.00 | 0.93 | 0.99 | 1.00
$\infty$ | 0.93 | 0.99 | 1.00 | 0.93 | 0.99 | 1.00
Table 2: Cutoff-extrapolated ($\Lambda\rightarrow\infty$) dimer binding energy
$B_{2}$ for several values of the inverse fake range $\tilde{R}^{-1}$, in
units of the three-body scale $Q_{3}$. The binding energy was calculated with
the LO potential $\tilde{\mathcal{V}}_{I}^{(0)}$ (equivalent at two-body level
to $\tilde{\mathcal{V}}_{II}^{(0)}$) and two versions of the corresponding NLO
interaction, $\nabla^{2}$ and $r^{2}$. The symbol “$\infty$” marks the result
for the “standard” $\mathcal{V}^{(0)}$. Calculated energies are given as
ratios with respect to the “exact” value $B_{2}^{\text{Model}}$ listed in
Table 1 for either LM2M2 or PCKLJS phenomenological models. Extrapolation
errors are negligible. No EFT truncation errors are shown.

Results for $B_{N\geq 3}^{(*)}$ differ slightly depending on whether one
employs $\tilde{\mathcal{V}}_{I}^{(0)}$ or $\tilde{\mathcal{V}}_{II}^{(0)}$.
In Tables 3, 4, and 5 one finds the binding energies of the excited trimer
$B_{3}^{*}$ and of the ground tetramer and pentamer, $B_{4}$ and $B_{5}$
respectively. They are normalized to the ground trimer energy $B_{3}$, which
is fitted to the respective potential-model result.

| $B_{3}/B_{3}^{*}$ | $B_{4}/B_{3}$ | $B_{5}/B_{3}$
---|---|---|---
$\tilde{R}^{-1}$ [$Q_{3}$] | LM2M2 | PCKLJS | LM2M2 | PCKLJS | LM2M2 | PCKLJS
4.8 | 55.656 | 49.688 | 4.36(0) | 4.30(0) | 10.1(1) | 9.9(3)
6.0 | 56.986 | 50.947 | 4.12(1) | 4.07(1) | 9.2(1) | 9.0(1)
12 | 58.068 | 52.100 | 3.98(5) | 3.93(5) | 8.6(6) | 8.5(5)
18 | 58.418 | 52.469 | 4.01(6) | 3.96(6) | 8.6(3) | 8.5(1)
24 | 58.661 | 52.728 | 4.04(4) | 4.00(4) | 8.63(1) | 8.5(3)
30 | 58.762 | 52.868 | 4.08(2) | 4.04(2) | 8.7(4)∗ | 8(1)∗
36 | 58.981 | 53.061 | 4.12(3) | 4.07(3) | 8.8(9)∗ | 9(2)∗
42 | 59.204 | 53.277 | 4.15(1) | 4.11(1) | 9(2)∗ | 8(3)∗
$\infty$ | 59.250 | 53.335 | 4.14(1) | 4.10(1) | 9.27(5) | 9.13(5)
Model | 55.482 | 49.751 | 4.42 | 4.35 | 10.33 | -
Table 3: Cutoff-extrapolated ($\Lambda\rightarrow\infty$) LO binding energies $B_{3}^{*}$ of the trimer excited state, $B_{4}$ of the tetramer ground state, and $B_{5}$ of the pentamer ground state for several values of the inverse fake range $\tilde{R}^{-1}$, in units of the three-body scale $Q_{3}$. Binding energies were calculated with the LO potential $\tilde{\mathcal{V}}_{I}^{(0)}$. They are given as ratios with respect to the “exact” trimer ground energy $B_{3}$ listed in Table 1 for either LM2M2 or PCKLJS phenomenological models, which is fitted in the EFT. Extrapolation errors, if sizeable, are given in parenthesis. The pentamer values marked with an asterisk are suspected to be less numerically stable than other results due to the large $\tilde{R}^{-1}$ employed; although this is reflected in the large extrapolation errors, they should be taken with caution. The last row gives the corresponding ratios calculated directly with LM2M2 and PCKLJS potential models [35, 36]. | $B_{3}/B_{3}^{*}$ | $B_{4}/B_{3}$
---|---|---
| LM2M2 | PCKLJS | LM2M2 | PCKLJS
$\tilde{R}^{-1}$ [$Q_{3}$] | LO | NLO $\nabla^{2}$ | NLO $r^{2}$ | LO | NLO $\nabla^{2}$ | NLO $r^{2}$ | LO | NLO | LO | NLO
4.8 | 55.024 | 55.121 | 55.024 | 49.176 | 49.232 | 49.241 | - |  Fitted | - |  Fitted
6.0 | 55.949 | 56.479 | 56.479 | 50.091 | 50.486 | 50.473 | 3.76 | 3.71
12 | 57.605 | 58.112 | 58.060 | 51.722 | 51.869 | 51.828 | 3.41 | 3.36
18 | 58.107 | 57.234 | 57.087 | 52.235 | 51.229 | 51.022 | 3.58 | 3.54
24 | 58.565 | 56.523 | 56.173 | 52.589 | 50.533 | 50.193 | 3.72 | 3.68
$\infty$ | 59.089 | 55.803 | 55.964 | 52.729 | 49.833 | 49.995 | 4.14(1) | 4.10(1)
Model | 55.533 | 49.747 | 4.42 | 4.35
Table 4:  Cutoff-extrapolated ($\Lambda\rightarrow\infty$) LO and NLO binding energies $B_{3}^{*}$ of the trimer excited state and $B_{4}$ of the tetramer ground state for several values of the inverse fake range $\tilde{R}^{-1}$, in units of the three-body scale $Q_{3}$. Binding energies were calculated with the LO potential $\tilde{\mathcal{V}}_{II}^{(0)}$ and two versions of the corresponding NLO interaction, $\nabla^{2}$ and $r^{2}$. They are given as ratios with respect to the “exact” trimer ground energy $B_{3}$ listed in Table 1 for either LM2M2 or PCKLJS phenomenological models, which is fitted in the EFT. Extrapolation errors, if sizeable, are given in parenthesis. The tetramer $B_{4}/B_{3}$ ratio has been fitted to adjust the strength of the NLO four-body force. The last row gives the corresponding ratios calculated directly with LM2M2 and PCKLJS potential models [35, 36]. | $B_{5}/B_{3}$
---|---
| LM2M2 | PCKLJS
$\tilde{R}^{-1}$ [$Q_{3}$] | LO | NLO $\nabla^{2}$ | NLO $r^{2}$ | LO | NLO $\nabla^{2}$ | NLO $r^{2}$
4.8 | 9.68 | 9.92 (0) | 10.25(0) | 9.50 | 9.74(0) | 9.85(0)
6.0 | 7.40 | 10.10(1) | 10.13(1) | 7.28 | 9.89(1) | 10.41(5)
12 | 6.56 | 10.29(5) | 10.60(5) | 6.44 | 10.09(5) | 9.88(6)
18 | 7.21 | 9.78(6) | 10.07(6) | 7.09 | 9.59(6) | 9.62(4)
24 | 7.70 | 9.64(4) | 9.82(4) | 7.57 | 9.44(4) | 9.66(2)
$\infty$ | 9.27(5) | 9.76(1) | 10.05(1) | 9.13(5) | 9.57(1) | 9.86(1)
Model | 10.33 | -
Table 5: Same as in Table 2 but for the pentamer binding energy $B_{5}$.

LO binding-energy ratios obtained with $\tilde{\mathcal{V}}_{I}^{(0)}$ are
reported in Table 3. We present a large range of fake ranges. For the smaller
values of $\tilde{R}$, pentamer results are probably less reliable than the
rest. For comparison, the corresponding ratios calculated directly with LM2M2
and PCKLJS potential models [35, 36] are also given. As it can be seen, the
trend for PCKLJS results is similar to LM2M2: results are quite close to the
“exact” results at the largest value of the fake range and end up about 10%
off for $\tilde{R}=0$.

Results obtained with $\tilde{\mathcal{V}}_{II}^{(0)}$ at LO and the two forms
of the NLO interaction — non-local and local — are reported in Tables 4 and 5.
LO binding-energy ratios follow the same trend as
$\tilde{\mathcal{V}}_{I}^{(0)}$, but are smaller, the difference increasing
with particle number and reaching over 10% for $B_{5}/B_{3}$. Agreement with
“exact” results slightly improves for the excited trimer but deteriorates for
the tetramer and pentamer. At NLO, the two types of potential give the same
results within a few percent, and binding energies are essentially independent
of the fake range. As expected, the inclusion of the two-body effective range
makes a very small effect for the excited trimer when the fake range is
numerically close to the experimental value of the effective range, but as the
fake range is removed the effect remains no larger than 10%. In the end,
excellent agreement with the “exact” result is obtained. The four-body force
is fitted to the tetramer ground-state energy and the combined NLO effects
increase the pentamer binding energy. For LM2M2, where an “exact” result
exists [35], NLO halves the pentamer discrepancy.

We see that the results, although different in detail, are qualitatively the
same for LM2M2 and PCKLJS, and for the two versions of the NLO potential.
Differences are well within the expected truncation errors discussed in the
main text. Results presented in the main text for LM2M2 for the non-local NLO
form are therefore representative of the theory.

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