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2nd August 2023

Universality of Three Identical Bosons
with Large, Negative Effective Range

Harald W. Grießhammera,111Email<EMAIL_ADDRESS>
_and_

Ubirajara van Kolckb,c,222Email<EMAIL_ADDRESS>
a Institute for Nuclear Studies, Department of Physics,
The George Washington University, Washington DC 20052, USA

b Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France

c Department of Physics, University of Arizona, Tucson AZ 85721, USA

“Resummed-Range Effective Field Theory” is a consistent nonrelativistic
effective field theory of contact interactions with large scattering length
$a$ and an effective range $r_{0}$ large in magnitude but negative. Its
leading order is non-perturbative, and its observables are universal in the
sense that they depend only on the dimensionless ratio $\xi:=2r_{0}/a$ once
the overall distance scale is set by $|r_{0}|$. In the two-body sector, the
relative position of the two shallow $S$-wave poles in the complex plane is
determined by $\xi$. We investigate three identical bosons at leading order
for a two-body system with one bound and one virtual state ($\xi\leq 0$), or
with two virtual states ($0\leq\xi<1$). Such conditions might, for example, be
found in systems of heavy mesons. We find that no three-body interaction is
needed to renormalise (and stabilise) Resummed-Range EFT at LO. A well-defined
ground state exists for $0.366\ldots\leq\xi\leq-8.72\ldots$. Three-body
excitations appear for even smaller ranges of $\xi$ around the “quasi-
unitarity point” $\xi=0$ ($|r_{0}|\ll|a|\to\infty$) and obey discrete scaling
relations. We explore in detail the ground state and the lowest three
excitations and parametrise their trajectories as function of $\xi$ and of the
binding momentum $\kappa_{2}^{-}$ of the shallowest $2\mathrm{B}$ state from
where three-body and two-body binding energies are identical to zero three-
body binding. As $|r_{0}|\ll|a|$ becomes perturbative, this version turns into
the “Short-Range EFT” which needs a stabilising three-body interaction and
exhibits Efimov’s Discrete Scale Invariance. By interpreting that EFT as a
low-energy version of Resummed-Range EFT, we match spectra to determine
Efimov’s scale-breaking parameter $\Lambda_{\ast}$ in a renormalisation scheme
with a “hard” cutoff. Finally, we compare phase shifts for scattering a boson
on the two-boson bound state with that of the equivalent Efimov system.

##  1 Introduction

The three-boson system displays a remarkable universality when the two-boson
subsystem is near the unitarity limit, where the inverse scattering length
$a^{-1}$, the effective range $r_{0}$ and other effective-range-expansion
parameters are small on the scale set by the relative momentum. At that point,
the two-body ($2\mathrm{B}$) system is scale invariant, as reflected in a
zero-energy pole of the scattering amplitude, and the three-boson system has
Discrete Scale Invariance in the form of a geometric tower of so-called Efimov
states [1, 2, 3, 4]. As $|a^{-1}|$ increases, these states display the
remarkable twin properties that they are bound (dubbed Borromean binding) even
when the $2\mathrm{B}$ subsystems are unbound ($a<0$), and that they become
unstable as attraction increases ($a>0$). The Efimov effect also occurs in
multi-component fermions. For example, light nuclei are perturbatively close
to unitarity [5, 6]. Efimov states have attracted considerable attention from
theory and experiment alike [7, 8].

The Efimov effect takes place in a non-relativistic system where the
$2\mathrm{B}$ scattering length $|a|\gg R$ is much larger than the interaction
range $R$, while the other effective-range parameters, and especially $r_{0}$,
are comparable to $R$. In this article, we consider the universal properties
of the three-boson system when the $2\mathrm{B}$ effective range is large as
well, $|r_{0}|\gg R$. While the Efimov effect appears when the depth of an
attractive $2\mathrm{B}$ potential is precisely adjusted, the situation
envisioned here arises when the potential has attractive and repulsive parts
that nearly balance each other. The consequence is that the $2\mathrm{B}$
system has two low-energy $S$-wave poles, whose positions on the complex
relative-momentum plane depend on the relative magnitudes and signs of $a$ and
$r_{0}$, encapsulated in the ratio

$\xi:=\frac{2r_{0}}{a}\;\;.$ (1.1)

This is in contrast to the single (bound or virtual) pole of the near-
unitarity case, and this changes the structure of more-body systems
dramatically.

While two-body systems with effective ranges which are large in magnitude and
negative appear to be an exception, they do occur most likely in scattering
systems with a resonance just above zero momentum, and hence in a
$k\cot\delta$ with large negative curvature (i.e. effective range). There are
indications of their existence, for example, in the $D^{*}_{s}(2317)$
described as bound state of the $DK$-$D_{s}\eta$ coupled-channel system [9].
Some of the $\Xi\mathrm{N}$ and $\Xi\Xi$ two-fermion systems may show large
$\xi<0$ as well [10, 11, 12].

The consequences of shallow poles can be captured by an effective field theory
(EFT) designed for relative momenta $k\ll R^{-1}$; see e.g. ref. [13] for a
recent review. It contains all possible contact interactions, with a finite
number of them contributing non-perturbatively at leading order (LO), and the
remainder entering perturbatively at higher orders. The existence of a single
shallow pole near unitarity requires a single, non-derivative $2\mathrm{B}$
contact interaction at LO. The three-boson system is not, however, well
defined [14], unless the non-derivative three-body ($3\mathrm{B}$) contact
interaction is also present at LO to ensure consistency with the
renormalisation group [15, 16]. This $3\mathrm{B}$ interaction breaks
continuous scale invariance down to a discrete subgroup, generating the Efimov
effect. The related properties of more-body systems continue to be studied
extensively. We will refer to this version as “Short-Range EFT”, which in
Nuclear Physics takes the form of “Pionless EFT” and “Halo/Cluster EFT”,
depending on the elementary degrees of freedom being, respectively, just
nucleons or including also tight nucleon clusters. In Atomic Physics, it is
sometimes also referred to as “Contact EFT”.

To accommodate a second shallow two-body pole, the two-derivative
$2\mathrm{B}$ contact interaction must be included non-perturbatively at LO
[17, 18]. We call this the “Resummed-Range EFT”. Renormalisability then
constrains [19, 20] the effective range to be negative,

$r_{0}<0\;\;,$ (1.2)

so that the Wigner bound [21] is satisfied [22, 23]. Consequently, this EFT
admits five combinations of poles of the scattering amplitude on the complex
relative-momentum plane:

  1. (i)

the resonance case ($\xi>1$), with two poles on the lower half-plane which are
symmetric with respect to the imaginary axis;

  2. (ii)

a virtual double-pole ($\xi=1$), namely a double pole on the negative
imaginary axis;

  3. (iii)

two virtual states ($0<\xi<1$), that is, two single poles on the negative
imaginary axis;

  4. (iv)

one virtual and one quasi-bound state ($\xi=0$), i.e. a single pole on the
negative imaginary axis and another at the origin (“quasi-unitarity”); and

  5. (v)

one bound and one virtual state (positive/negative imaginary axis, resp.;
$\xi<0$).

In all cases, the poles are symmetric with respect to the point
$-\mathrm{i}/r_{0}$ on the imaginary axis. Typically, one scans through these
pole configurations in this sequence as attraction increases [17]. They can
also be described by replacing the two-derivative contact interaction by a
dimeron field [24, 18], which moreover accounts more naturally for two poles
nearly symmetrical with respect to the origin. However, the constraint
$r_{0}<0$ must be imposed by hand in the dimeron formulation.

We will describe later that $3\mathrm{B}$ bound states exist only for cases
(iii) to (v), and only for a range of $\xi$ around zero. Of particular
interest is case (v), where a third boson can scatter on the two-boson bound
state. This is also the situation in Weinberg’s discussion of “elementary”
particles [25] and widely employed [26] in the context of new exotic hadronic
states since the discovery of the $X(3872)$ [27]. The $2\mathrm{B}$ sector was
analysed from the EFT perspective in ref. [18]. We look here for $3\mathrm{B}$
bound states which emerge as poles of the homogeneous version of the Faddeev
equation [28, 29].

We also discuss Borromean systems which emerge for cases (iii) and (iv)
($0\leq\xi<1$), namely bound $3\mathrm{B}$ systems whose $2\mathrm{B}$
subsystems are unbound into two virtual states. While there is then no
scattering on a $2\mathrm{B}$ state, our equations remain valid but a formal
study of $3\mathrm{B}$ scattering is not presented here.

In Resummed-Range EFT, the faster fall-off of the $2\mathrm{B}$ amplitude with
momentum suggests that no $3\mathrm{B}$ interaction is necessary to
renormalise the $3\mathrm{B}$ problem at LO, and indeed our results converge
as the cutoff regulator increases. The $3\mathrm{B}$ spectrum and scattering
are thus predicted uniquely, up to corrections from higher-order effective-
range parameters. At LO, the magnitude $|r_{0}|$ of the $2\mathrm{B}$
effective range fixes the overall scale of the spectrum, and the relative
positions of the various levels is universal in the sense that it depends
solely on $\xi$. In the “quasi-unitarity” limit $|\xi|\ll 1$, we expect an
approximately geometric tower of states truncated from below at binding
momentum $\kappa_{3}\sim|r_{0}^{-1}|$. Away from this limit, more significant
effects of $r_{0}$ are expected. As $|\xi|$ increases, we will confirm that
the spectrum differs from Efimov’s, with levels disappearing at different
points either at the $2\mathrm{B}$ threshold (for $\xi<0$) or at vanishing
$3\mathrm{B}$ binding (for $\xi>0$). We will in particular address the
question whether the $3\mathrm{B}$ ground-state energy vanishes before the
critical value $\xi=1$ of case (ii) is reached, namely the point at which the
two $2\mathrm{B}$ virtual states coalesce as a precursor to the appearance of
a resonance. We will also see that as either $|\xi|$ or the relative momentum
$k$ increases, scattering phase shifts deviate more dramatically from
Efimov’s.

Frequently, the consequences of Efimov universality are obtained from finite-
range potentials whose effective-range effects are made small—see e.g. the
highly precise calculations of ref. [30]. Some of our results could no doubt
be extracted from calculations with specific potentials with small shape
parameters. However, typically Efimov universality calculations are done with
potentials (such as Gaußian) where the $2\mathrm{B}$ effective range is
positive. Here, we isolate instead the effects of a _negative_ effective
range, imposed by renormalisability of the EFT at $2\mathrm{B}$ level. This
avoids $2\mathrm{B}$ poles in the upper complex-momentum half-plane which are
difficult to interpret: those not on the imaginary axis [31, 32], and those on
the imaginary axis that are not associated with bound states (“redundant
poles”) [33, 34, 35]. Nevertheless it has been suggested [36] that the
effective ranges in the two-nucleon system, which are positive and somewhat
large, should be accounted for at LO through the use of dibaryon fields.
Unfortunately, problems then appear in the three-nucleon system [37], unless
the dibaryon is replaced by a nonlocal potential of unclear origin [38]. In
contradistinction, we find no pathologies when $r_{0}<0$, and its Resummed-
Range EFT appears well-defined.

This article is organised as follows. In sect. 2, we present the formalism for
the solution of the Faddeev equation at LO. Section 3 starts with an overview
of our findings for bound states (sect. 3.1) and then presents results with a
discussion of the lowest four bound-state trajectories as function of $\xi$
and of the binding momentum $\kappa_{2}^{-}$ of the shallowest $2\mathrm{B}$
state (sect. 3.2), followed by a comparison of these trajectories (sect. 3.3),
their parametrisation (sect. 3.4), and finally an interpretation of the Efimov
effect in Short-Range EFT as a low-energy version of the Resummed-Range EFT
which allows determination of the Efimov parameter (sect. 3.5). A presentation
of the highlights in scattering (sect. 4) is followed by the customary summary
and outlook in sect. 5.

##  2 Formalism

We first formulate Resummed-Range (Pionless) EFT at LO in momentum space. We
highlight the main features of the $2\mathrm{B}$ scattering amplitude which
are needed as input in the integral equation for the scattering amplitude of a
third boson on the $2\mathrm{B}$ bound state and for the $3\mathrm{B}$ bound-
state equation. We also briefly review the $3\mathrm{B}$ system in Short-Range
EFT and its Efimov physics, for ease of comparison of our results in the next
section.

###  2.1 The Two-Boson System

The LO amplitude of the Resummed-Range EFT arises from the non-perturbative
solution of the $2\mathrm{B}$ system with two contact interactions: one
without derivatives, the other with two [17, 18]. Corrections are
systematically accounted for through higher-derivative interactions and
included in perturbation theory [17, 18]. They generate higher effective-range
parameters, starting with the shape parameter at next-to-leading order, as
well as kinematically relativistic and other effects. This EFT can
alternatively be formulated with a dimer field [24, 18]. We focus on the
universal aspects at LO as the input into the $3\mathrm{B}$ problem.

In this EFT, the LO on-shell amplitude of a system of two identical bosons of
mass $M$ and relative momentum $k$ is

$\mathcal{A}_{2}(k)=\frac{4\pi}{M}\;\frac{1}{\frac{1}{a}-\frac{r_{0}}{2}\,k^{2}+\mathrm{i}\,k}\;\;,$
(2.1)

where $a$ is the scattering length and $r_{0}\leq 0$ the effective range. It
is convenient to choose one of these two dimensionful parameters to set the
overall scale so that systems with the same dimensionless ratio (1.1) are
self-similar. Since $r_{0}\leq 0$ is taken as non-positive throughout, we
render momenta dimensionless by measuring them in units of its magnitude,
$|r_{0}^{-1}|$,

$K:=k\,|r_{0}|\;\;.$ (2.2)

This helps track some signs and makes sure that there are no confusions with
positive momenta in scattering processes. The dimensionless version of the
above amplitude reads then

$A_{2}(K):=-\frac{M}{4\pi|r_{0}|}\mathcal{A}_{2}(k)=\frac{1}{\frac{\xi-K^{2}}{2}-\mathrm{i}K}\;\;.$
(2.3)

The $S$-wave scattering phase shift $\delta(K)$ at momentum $K\geq 0$ is
obtained from

$K\cot\delta(K)=A_{2}^{-1}(K)+\mathrm{i}K\;\;.$ (2.4)

This amplitude has two poles, which for $\xi\leq 1$ ($r_{0}\leq 0\leq a$ or
$a\leq 2r_{0}\leq 0$) are on the imaginary axis of the complex-$K$ plane at

$K_{2}^{\pm}=:\mathrm{i}\,\kappa_{2}^{\pm}\;\mbox{ with
}\;\kappa_{2}^{\pm}:=-\left[1\pm\sqrt{1-\xi}\right]\;\;.$ (2.5)

Following the definition of ref. [17], a real $\kappa_{2}$ represents thus a
pole in $K$ on the imaginary axis, and a positive one represents a bound
state. The deeper pole $\kappa_{2}^{+}$ is on the negative imaginary axis,
corresponding to a virtual state with a positive residue for $\mathrm{i}$
times the $S$ matrix. In contrast, the shallow pole $\kappa_{2}^{-}$ lies for
$\xi<0$ (namely $r_{0}\leq 0\leq a$) on the positive imaginary axis with a
positive residue, representing a bound state, or for $\xi>0$ (namely $a\leq
2r_{0}\leq 0$) on the negative imaginary axis, representing a virtual state.
Unwrapping the rescaling by $|r_{0}|$, the shallow state has a binding energy

$B_{2}=\frac{(\kappa_{2}^{-})^{2}}{M\,r_{0}^{2}}>0\;\;,$ (2.6)

which for $\xi\ll 1$ turns into the familiar effective-range expression
$B_{2}(\xi\ll 1)\simeq 1/(Ma^{2})$. In the “quasi-unitarity” limit $\xi\to 0$
($|a|\to\infty$), $\kappa_{2}^{-}\to 0$ becomes quasi-bound, i.e. it is no
longer a pole of the $S$ matrix. We call this “quasi-unitarity” since the
other pole, $\kappa_{2}^{+}\to-2$ still sets a scale in the $2\mathrm{B}$
system, in contrast to full unitarity which has no $2\mathrm{B}$ scale.

In the many-body problem, the $2\mathrm{B}$ contribution enters through the
amplitude $\mathcal{A}_{2}$ once the motion with respect to the many-body
centre-of-mass frame is taken into account. Denoting the three-momentum and
energy of the $2\mathrm{B}$ state by $\vec{p}\mkern 2.0mu\vphantom{p}$ and
$p_{0}$, respectively, one replaces $k$ by
$\mathrm{i}\sqrt{-Mp_{0}+\vec{p}\mkern 2.0mu\vphantom{p}^{2}/4}$ in eq. (2.1).
For scattering on the bound state, the $2\mathrm{B}$ wave-function
renormalisation is

$\mathcal{Z}_{2}:=\frac{M}{4\pi}\left[\frac{\partial}{\partial
p^{0}}\mathcal{A}_{2}^{-1}\left(\mathrm{i}\sqrt{-Mp^{0}+\frac{\vec{p}\mkern
2.0mu\vphantom{p}^{2}}{4}}\right)\right]_{Mp^{0}=-B_{2}}^{-1}=\frac{2}{M|r_{0}|}\left(1-\frac{1}{\sqrt{1-\xi}}\right)=\frac{2}{M|r_{0}|}\;\frac{1}{1+\frac{1}{\kappa_{2}^{-}}}\;\;.$
(2.7)

In the limit $\xi\to 0$, one finds $\mathcal{Z}_{2}\to 2/(Ma)$.

###  2.2 The Three-Boson System

We consider now the scattering of a boson on the two-boson bound state at
total centre-of-mass energy $E$ and relative momentum $k$ at LO in this
theory. The corresponding amplitude is dominated by the $S$ wave. It can
sustain $3\mathrm{B}$ bound states.

The Faddeev equation [28, 29] for the $S$-wave amplitude with off-shell
momentum $p$ is

$\displaystyle t(ME;k,p)=$
$\displaystyle\;\frac{4\pi}{kp}\ln\frac{p^{2}+k^{2}+pk-ME}{p^{2}+k^{2}-pk-ME}$
(2.8)
$\displaystyle-\frac{2}{\pi}\int\limits_{0}^{\infty}\mathrm{d}\;\\!\\!q\;\frac{q}{p}\;\ln\frac{p^{2}+q^{2}+pq-
ME}{p^{2}+q^{2}-pq-
ME}\;\frac{t(ME;k,q)}{\frac{1}{a}-\sqrt{\frac{3q^{2}}{4}-ME}-\frac{r_{0}}{2}(ME-\frac{3q^{2}}{4})}\;\;,$

where the correct cut structures are made obvious by replacing the real $ME$
by $ME+\mathrm{i}\epsilon$. After rescaling to dimensionless momenta as before
and including for the energy

$\mu^{2}:=ME\,r_{0}^{2}\;\;,$ (2.9)

the amplitude at on-shell momentum $K$ and off-shell momentum $Q$ becomes

$\displaystyle T(\mu^{2};K,P):=$ $\displaystyle\;\frac{t(E;k,p)}{r_{0}^{2}}$
$\displaystyle=$
$\displaystyle\;\frac{4\pi}{KP}\ln\frac{P^{2}+K^{2}+PK-\mu^{2}}{P^{2}+K^{2}-PK-\mu^{2}}+\frac{4}{\pi}\int\limits_{0}^{\infty}\mathrm{d}\;\\!\\!Q\;\mathcal{K}(\mu^{2};P,Q)\;T(\mu^{2};K,Q)\;\;,$
(2.10)

where we introduced the kernel

$\mathcal{K}(\mu^{2};P,Q):=\frac{Q}{P}\;\ln\frac{P^{2}+Q^{2}+PQ-\mu^{2}}{P^{2}+Q^{2}-PQ-\mu^{2}}\;\frac{1}{\xi+(3Q^{2}-4\mu^{2})/4+\sqrt{3Q^{2}-4\mu^{2}}}\;\;.$
(2.11)

In scattering ($K>0$),
$\mu^{2}=3K^{2}/4-(\kappa_{2}^{-})^{2}+\mathrm{i}\epsilon$. Accounting for the
$2\mathrm{B}$ wave-function renormalisation of eq. (2.7), the relation to the
scattering phase shift is

$K\cot\delta_{3}(K)=\;\frac{3\pi}{2}\left[\left(1-\frac{1}{\sqrt{1-\xi}}\right)T(\mu^{2};K,K)\right]^{-1}-\mathrm{i}\,K\;\;.$
(2.12)

In addition, there can be bound-state solutions. Let us denote the
dimensionless energy of the $j$th three-body bound-state by
$\mu^{2}=-(\kappa^{(j)}_{3})^{2}<0$, and the corresponding “binding momentum”
as $\kappa^{(j)}_{3}=\sqrt{|\mu^{2}|}\in\mathbbm{R}_{0}^{+}$ for a bound state
pole at $\mathrm{i}\,\kappa_{3}$. By construction and in complete analogy to
the $2\mathrm{B}$ case, this is a pole which lies on the positive imaginary
axis in the complex relative-momentum plane. Bound states are then the
nontrivial solutions of the homogeneous equation with $K=0$,

$\Gamma^{(j)}(P):=T(\mu^{2}=-(\kappa^{(j)}_{3})^{2};0,P)\propto\int\limits_{0}^{\infty}\mathrm{d}\;\\!\\!Q\;\mathcal{K}(-(\kappa^{(j)}_{3})^{2};P,Q)\;\Gamma^{(j)}(Q)\;\;.$
(2.13)

We choose the proportionality constant for ease of plotting results such that
$\Gamma^{(j)}(0)=1$. This vertex function characterises the bound-state
residue, and hence its wave function. While the two-body system is bound only
for $\xi<0$, we will find that $3\mathrm{B}$ bound states exist also for
$\xi=0$, when the shallowest $2\mathrm{B}$ state is quasi-bound, and for
$\xi>0$, when it is virtual.

###  2.3 The Efimov Version: Short-Range EFT

It is instructive to compare $3\mathrm{B}$ properties in Resummed-Range EFT
with those in the more widely known “Efimov version”, namely the Short-Range
EFT in which the effective range is considered to be a perturbation [13]. In
this EFT, leading order is independent of the effective range, and the
$2\mathrm{B}$ system’s amplitude is given by eq. (2.1) with $r_{0}=0$ and a
single pole at binding momentum $1/a$. The three-boson integral equation is
then found by replacing in eq. (2.10) the $2\mathrm{B}$ propagator,

$\frac{1}{\xi+(3Q^{2}-4\mu^{2})/4+\sqrt{3Q^{2}-4\mu^{2}}}\;\to\;\frac{1}{-2\gamma+\sqrt{3Q^{2}-4\mu^{2}}}\;\;,$
(2.14)

where $\gamma\propto 1/a$ is the (rescaled) magnitude of the binding momentum
of the $2\mathrm{B}$ system (pole at $\mathrm{i}\gamma$). We choose to again
rescale each with $|r_{0}|$ to render them dimensionless. The resulting
$3\mathrm{B}$ equation is in agreement with, for example, ref. [39].

It is well-explored that for renormalisation this more ultraviolet-sensitive
propagator needs a (dimensionless and properly normalised) cutoff-dependent
$3\mathrm{B}$ interaction $H(\Lambda)$ to be added to the one-body “exchange
potential” [15, 16]. The logarithm in eqs. (2.10) and (2.11) is thus replaced
by

$\ln\frac{P^{2}+Q^{2}+PQ-\mu^{2}}{P^{2}+Q^{2}-PQ-\mu^{2}}\;\to\;\ln\frac{P^{2}+Q^{2}+PQ-\mu^{2}}{P^{2}+Q^{2}-PQ-\mu^{2}}+\frac{2PQ}{\Lambda^{2}}\,H(\Lambda)\;\;.$
(2.15)

Like $P,Q$, the cutoff $\Lambda$ is given in units of $|r_{0}^{-1}|$. The
running of $H(\Lambda)$ with the hard-momentum cutoff $\Lambda$ in the
$3\mathrm{B}$ integral equation is for $\Lambda\gg\gamma$ approximately given
by [15, 16]

$H(\Lambda)\approx-A\;\frac{\sin[s_{0}\ln\frac{\Lambda}{\Lambda_{\ast}}-\mathrm{arccot}s_{0}]}{\sin[s_{0}\ln\frac{\Lambda}{\Lambda_{\ast}}+\mathrm{arccot}s_{0}]}\;\;.$
(2.16)

with $A\approx 1$ and $s_{0}=1.0062378\dots$ the real solution to the
transcendental equation

$8s_{0}\;\sinh\frac{\pi s_{0}}{6}=\sqrt{3}\;\cosh\frac{\pi s_{0}}{2}\;\;.$
(2.17)

Here, $\Lambda_{\ast}$ is a scale fixed by some datum, breaking the Continuous
Scale Invariance for $\gamma=0$ down to a discrete one by dimensional
transmutation. Like the cutoff $\Lambda$, it is measured in units of
$|r_{0}^{-1}|$. Its precise value depends on the chosen renormalisation
scheme, which in our case employs a “hard” cutoff regularisation. Its
consequence is the Efimov effect [1, 2, 3, 4]: the $3\mathrm{B}$ spectrum is
geometric, and the binding-momentum ratio between adjacent states is
$e^{-\pi/s_{0}}$. Discrete Scale Invariance is exact at leading order in
Short-Range EFT even away from unitarity for $\gamma\to e^{-\pi/s_{0}}\gamma$
— it is broken only if $\gamma$ remains fixed, as in Nuclear Physics. The
consequence is that the “Efimov plot” of the $3\mathrm{B}$ binding momentum as
function of the $2\mathrm{B}$ binding momentum $\gamma$ remains self-similar
under radial dilations by multiples of $e^{-\pi/s_{0}}$; see e.g. ref. [7]. In
particular, the ratio between points where the $3\mathrm{B}$ states disappear
— either $\mu=0$ for $\gamma<0$ or $\sqrt{|\mu^{2}|}=\gamma>0$ — should also
be $e^{-\pi/s_{0}}$.

As $\xi\to 0$ in the Resummed-Range EFT we consider here, one $2\mathrm{B}$
pole becomes very shallow, $\kappa_{2}^{-}\to\gamma\to 0$, while the other
corresponds to momenta comparable to the inverse of the effective range,
$\kappa_{2}^{+}\to-2$. For $|\xi|\ll 1$, a régime of momenta
$|K|\ll|\kappa_{2}^{+}|$ exists in which $\kappa_{2}^{+}$ only enters as a
high-momentum effect. In that case, the $2\mathrm{B}$ amplitude (2.3),

$A_{2}(K)=\frac{1}{-\kappa_{2}^{-}-\mathrm{i}K}\left[1+{\cal
O}\left(\frac{|K|,\kappa_{2}^{-}}{|\kappa_{2}^{+}|}\right)\right]\;\;,$ (2.18)

has a single pole $\mathrm{i}\kappa_{2}^{-}$. Thus, in the quasi-unitarity
limit $\xi\to 0$ and at sufficiently small momenta, the spectrum and phase
shifts should be equally well described by Short-Range EFT with a
$3\mathrm{B}$ interaction $H(\Lambda)$ with parameter $\Lambda_{\ast}$.
Sufficiently far up in the spectrum, the binding momenta
$\kappa_{3\mathrm{QU}}^{(j)}:=\kappa_{3}^{(j)}(\xi=0)$ must obey

$\lim_{j\to\infty}\frac{\kappa_{3\mathrm{QU}}^{(j)}}{\kappa_{3\mathrm{QU}}^{(j+1)}}=e^{\pi/s_{0}}=22.6944\ldots\;\;.$
(2.19)

Away from quasi-unitarity, results depend on $\xi$ in principle for all
states. Again, these effects should be small in the evolution of highly
excited states, for which Short-Range EFT holds. For these, one should thus
reproduce consequences of Discrete Scale Invariance even away from unitarity.
The values of $\xi$ and $\kappa_{2}^{-}$ where the $j$th $3\mathrm{B}$ state
has zero binding, $\xi_{\mathrm{ZB}}^{(j)}:=\xi(\kappa_{3}^{(j)}=0)$ and
$\kappa_{2\mathrm{ZB}}^{-\>(j)}:=\kappa_{2}^{-}(\kappa_{3}^{(j)}=0)$, are
expected to obey

$\lim_{j\to\infty}\frac{\xi_{\mathrm{ZB}}^{(j)}}{\xi_{\mathrm{ZB}}^{(j+1)}}=\lim_{j\to\infty}\frac{\kappa_{2\mathrm{ZB}}^{-\>(j)}}{\kappa_{2\mathrm{ZB}}^{-\>(j+1)}}=e^{\pi/s_{0}}=22.6944\ldots\;\;.$
(2.20)

Likewise, for the values of $\xi$ and $\kappa_{2}^{-}$ where the binding
energy of the $j$th $3\mathrm{B}$ state and the shallow $2\mathrm{B}$ state
coincide, $\xi_{\rm
thr}^{(j)}:=\xi(\kappa_{3}=\kappa_{2}^{-}=:\kappa_{\mathrm{thr}})$,

$\lim_{j\to\infty}\frac{\xi_{\mathrm{thr}}^{(j)}}{\xi_{\mathrm{thr}}^{(j+1)}}=\lim_{j\to\infty}\frac{\kappa_{\mathrm{thr}}^{(j)}}{\kappa_{\mathrm{thr}}^{(j+1)}}=e^{\pi/s_{0}}=22.6944\ldots\;\;.$
(2.21)

Since eq. (2.10) converges for large momenta, in Resummed-Range EFT we expect
the existence of a $3\mathrm{B}$ ground state, in contradistinction to the
Efimov effect, where an infinite tower of states of arbitrarily large binding
energy exists due to perfect Discrete Scale Invariance. This provides a
“natural” cutoff to the tower of Efimov states at $\xi=0$. The $2\mathrm{B}$
effective range $r_{0}$ (and therefore the existence of the second pole at
$\mathrm{i}\,\kappa_{2}^{+}$) also fixes the position of the tower, which in
Short-Range EFT is determined by $\Lambda_{\ast}$. Resummed-Range EFT can thus
be seen as an underlying theory to Short-Range EFT where one can _predict_
$\Lambda_{\ast}$, rescaled by the remaining dimensionful parameter $r_{0}$.
The deepest bound states should of course show the largest deviation from the
Efimov behaviour. As a rough estimation, one might expect departures to scale
with $\sim{\rm max}(\kappa_{2}^{-},\kappa_{3})/|\kappa_{2}^{+}|$. Likewise,
scattering at momenta $K\hskip
1.99997pt\raisebox{2.15277pt}{$>$}\raisebox{-3.00003pt}{$\sim$}\hskip
1.99997pt|\kappa_{2}^{+}|$ differs from the one in Short-Range EFT.

##  3 Results for Bound States

In this section, we present numerical solutions of the homogeneous equation
(2.13) for bound states. Because of the bijection between $\xi$ and
$\kappa_{2}^{-}$ and because we will compare to the results of Short-Range EFT
at the same $2\mathrm{B}$ binding momentum $\gamma=\kappa_{2}^{-}$, we report
results interchangeably as function of $\kappa_{2}^{-}=\sqrt{1-\xi}-1$ (eq.
(2.5)) and of $\xi=-\kappa_{2}^{-}(\kappa_{2}^{-}+2)$.

Practically, the bound state is found by discretising the kernel on a quasi-
logarithmic mesh in $(P,Q)\in\mathbbm{R}_{0}^{+}$ which weighs low momenta
stronger than higher ones, and scanning for nontrivial solutions at discrete
$\mu^{2}=-\kappa_{3}^{2}$ of

$\det[1-\mathcal{K}(\mu^{2};P,Q)]=0\;\;.$ (3.1)

The vertex function $\Gamma^{(j)}(Q)$ is the eigenvector to the kernel with
zero eigenvalue. Unless otherwise specified, we use $n=500$ points at a cutoff
$\Lambda=45.2548\dots|r_{0}^{-1}|$. There, residual cutoff-dependence is
negligible, and results are quite stable against $n$ or different mappings
between the grid and $P,Q$. We consider the four lowest bound states,
extrapolate to shallower states, and compare with Discrete Scale Invariance of
Short-Range EFT, sect. 2.3.

###  3.1 Bound States: Overview

A qualitative illustration of the trajectories as $\kappa_{2}^{-}$ ($\xi$) is
varied is offered in fig. 1, where the $3\mathrm{B}$ binding momentum
$\kappa_{3}$ is sketched as function of $\kappa_{2}^{-}$ (increasing to the
right) or $\xi$ (decreasing to the right). The top of the figure shows
qualitatively the relative position of the poles of the $3\mathrm{B}$ and
$2\mathrm{B}$ bound states in the complex-momentum plane.

Figure 1: (Colour on-line) Qualitative trajectory of a $3\mathrm{B}$ bound
state as the $2\mathrm{B}$ binding momentum increases — i.e. $\xi$ decreases —
to the right. Bottom: the four regions and three points of interest; see text.
The solid line indicates the $3\mathrm{B}$ binding momentum, while the green
dashed lines in the sub-threshold (unstable) and unbound regions are
speculative. In the gray region, the $3\mathrm{B}$ system is unstable as its
binding is smaller than in the $2\mathrm{B}$ subsystem. The “polar”
coordinates $(\theta,\rho(\theta))$ of eq. (3.6) are indicated in green for
one point. Top: relative positions of the $2\mathrm{B}$ pole $\kappa_{2}^{-}$
(blue dot) and the $3\mathrm{B}$ pole $\kappa_{3}$ (red cross).

The figure leads to a natural classification of the trajectory of each state
into four regions, separated by three points of interest whose positions are
different for different states:

  1. (1)

Sub-Threshold (Unstable) Region: For $\kappa_{3}(\xi)<\kappa_{2}^{-}(\xi)$,
the $3\mathrm{B}$ system is less bound than the $2\mathrm{B}$ one, and thus
unstable to the breakup into $1+2$ subsystems. This happens for values
$\xi<\xi_{\mathrm{thr}}<0$ which are negative and sufficiently large in
magnitude (i.e. $\kappa_{2}^{-}>\kappa_{\mathrm{thr}}>0$ sufficiently large —
the $2\mathrm{B}$ system in case (v) of the Introduction).

  2. (2)

Threshold Point: At a some $\xi_{\mathrm{thr}}<0$, the $3\mathrm{B}$ and
$2\mathrm{B}$ systems have identical binding momenta
$\kappa_{\mathrm{thr}}:=\kappa_{3}(\xi_{\mathrm{thr}})\stackrel{{\scriptstyle!}}{{=}}\kappa_{2}^{-}(\xi_{\mathrm{thr}})\in\mathbbm{I}^{+}$,
and hence identical binding energies (still case (v) of the Introduction). The
$3\mathrm{B}$ state emerges from the continuum.

  3. (3)

All-Bound Region: For larger values $0>\xi>\xi_{\mathrm{thr}}$,
$\kappa_{3}>\kappa_{2}^{-}>0$, so that both the $3\mathrm{B}$ and all
$2\mathrm{B}$ sub-systems are bound (for $2\mathrm{B}$: case (v) of the
Introduction).

  4. (4)

Quasi-Unitarity Point: At $\xi=0$, the binding momentum of the shallowest
$2\mathrm{B}$ bound state vanishes, $\kappa_{2}^{-}(\xi=0)=0$, corresponding
to an infinite scattering length (case (iv) of the Introduction). This is the
point at which comparison with the Short-Range EFT of sect. 2.3 is most
fruitful. The second $2\mathrm{B}$ state continues to be virtual and has
binding momentum $\kappa_{2}^{+}(\xi=0)=-2$ in units of $|r_{0}^{-1}|$,
providing a $2\mathrm{B}$ scale for the $3\mathrm{B}$ system’s binding
momentum $\kappa_{3\mathrm{QU}}:=\kappa_{3}(\xi=0)\in\mathbbm{I}^{+}$.

  5. (5)

Borromean Region: As $\xi>0$ turns positive, the $3\mathrm{B}$ system
continues to be bound, $\kappa_{3}(\xi_{\mathrm{ZB}}>\xi>0)>0$, despite the
fact that all $2\mathrm{B}$ sub-systems become unbound and now contain two
virtual states, $\kappa_{2}^{\pm}(\xi)<0$ (case (iii) of the Introduction).

  6. (6)

Zero-Binding Point: The binding momentum (and energy) of the $3\mathrm{B}$
system becomes zero at some $\xi_{\mathrm{ZB}}>0$:
$\kappa_{3}(\xi_{\mathrm{ZB}})=0$. The $2\mathrm{B}$ system is already
unbound, $\kappa_{2\mathrm{ZB}}^{-\>}:=\kappa_{2}^{-}(\xi_{\mathrm{ZB}})$ with
$\kappa_{2\mathrm{ZB}}^{-\>}<0$ (case (iii) of the Introduction).

  7. (7)

Unbound Region: For $\xi>\xi_{\mathrm{ZB}}>0$ (namely
$\kappa_{2}^{-}(\xi)<\kappa_{2\mathrm{ZB}}^{-\>}<0$), the pole of the
$3\mathrm{B}$ system is no longer on the positive imaginary axis. Both three-
and two-body (cases (i-iii) of the Introduction) systems are unbound.

Finally, we note that the $2\mathrm{B}$ binding momentum has a zero real part,
$\mathrm{Re}[\mathrm{i}\,\kappa_{2}^{\pm}]=0$, except in the last region, and
only for $\xi>1$. Likewise, the three-body state is bound in all but the last
region, and therefore the real part of its pole vanishes,
$\mathrm{Re}[K_{3}^{\mathrm{pole}}]=\mathrm{Im}[\kappa_{3}]=0$, while its
imaginary part is non-negative,
$\mathrm{Im}[K_{3}^{\mathrm{pole}}]=\mathrm{Re}[\kappa_{3}]\geq 0$.

The fate of the $3\mathrm{B}$ trajectory in the sub-threshold and unbound
regions will be discussed in future publications [40]. Close to the threshold,
$\xi\simeq\xi_{\rm thr}$, the three-body state is a large “halo” system where
one boson is on average far from the two-body bound state. Therefore, the
system reduces to a “core” whose constituents are relatively tightly bound,
surrounded by another particle which is only loosely bound, and both interact
predominantly in an $S$ wave. In such an $S$-wave two-body (core-particle)
system without long-range interactions, one expects a single pole in the
vicinity of the threshold. When the interaction strength is changed smoothly,
continuity dictates that it goes from a bound state (on the positive imaginary
axis in momentum) to a virtual state (on the negative imaginary axis in
momentum, namely a negative-energy pole on the second Riemann sheet in
energy), or vice-versa. This is the case e.g. for an Efimov state [41], and we
expect the same here. On the other hand, near $\xi_{\mathrm{thr}}$, no
$2\mathrm{B}$ bound state exists, only $2\mathrm{B}$ virtual states. Like in
the sub-threshold region, the $3\mathrm{B}$ state continues on the imaginary
axis as an unbound state. This might be a virtual state, or it might collide
with another $3\mathrm{B}$ pole to acquire a real momentum component,
mirroring the behaviour of $2\mathrm{B}$ poles interpreted as resonance,
namely two poles at $\mathrm{Re}[\kappa_{3}]<0$,
$\pm\mathrm{Im}[\kappa_{3}]\neq 0$. We cannot exclude the possibility of the
collision happening at the origin. In fact, this might be the case in Short-
Range EFT, as refs. [42, 43] report a non-vanishing resonance width already
just past the zero-binding point. Whatever the exact mechanism, one does not
expect a $3\mathrm{B}$ bound state to be resurrected for
$\xi>\xi_{\mathrm{ZB}}$ or $\xi<\xi_{\mathrm{thr}}$. Indeed, a scan for
$3\mathrm{B}$ bound states in these cases was unsuccessful.

###  3.2 The Lowest Three-Body States

Figure 2: (Colour on-line) Vertex functions of the first four states as
functions of the momentum in units of $|r_{0}^{-1}|$, on a logarithmic scale
in $Q$. Ten values of $\xi$ are shown in linear spacing between
$\xi_{\mathrm{thr}}$ and $\xi_{\mathrm{ZB}}$ of the respective state, plus at
$\xi_{\mathrm{thr}}$ and $\xi=0$. The zero-binding state at
$\xi_{\mathrm{ZB}}$ has support only at $Q\to 0$ due to its infinite extend in
coordinate space and is thus not shown. The legend lists values as
$\kappa_{2}^{-}\,(\xi)$:$\;\kappa_{3}$, with binding momenta given in
$|r_{0}^{-1}|$.

A lowest-energy (ground) three-body bound state exists only for
$0.367\ldots\geq\xi\geq-8.726\ldots$ (namely
$-0.204\ldots\leq\kappa_{2}^{-}\leq 2.119\ldots$). Outside this range, a scan
for binding momenta $\kappa_{3}$ in the range $10^{2}\gtrsim\xi\gtrsim-10^{2}$
all the way to the cutoff $\Lambda$ and varying $\Lambda$ to several hundred
$|r_{0}^{-1}|$ found no $3\mathrm{B}$ bound states. This is strong indication
that no $3\mathrm{B}$ bound states exist when the $2\mathrm{B}$ pole is either
a double virtual pole or a resonance ($\xi\geq 1$,
$\mathrm{Re}[\mathrm{i}\,\kappa_{2}^{-}]\neq 0$). A number of excited bound
states do exist, depending on $\xi$. In this subsection, we present results
for the ground state, which we label $j=0$, and the first three excited
states, $j=1,2,3$. As to be detailed in sub-sect. 3.5, the trajectory of the
$3$rd excitation is near-indistinguishable from the one in Short-Range EFT
with the same $3\mathrm{B}$ binding momentum at quasi-unitarity
($\kappa_{2}^{-}=\xi=0$) and can thus be taken to represent the trajectory of
an Efimov state.

Figure 3: (Colour on-line) Binding momenta $\kappa_{3}$ of the $3\mathrm{B}$ system in $|r_{0}^{-1}|$ as function of the $2\mathrm{B}$ binding momentum $\kappa_{2}^{-}$ (bottom scale, in $|r_{0}^{-1}|$) and of $\xi$ (top scale), for the ground state (black) and the $1$st (green), $2$nd (red) and $3$rd (blue) excitations. A logarithmic scale is used on both axes, with positive and negative values on the $\kappa_{2}^{-}$ ($\xi$) axis stitched together by a gray line. In the gray region, a $3\mathrm{B}$ state is unstable because it has smaller binding energy than the $2\mathrm{B}$ state, $\kappa_{3}<\kappa_{2}^{-}$, i.e. it disappears into the $2\mathrm{B}$ continuum. The right axis shows the colour-coded value of the $3\mathrm{B}$ binding momentum of the corresponding state at the quasi-unitarity limit ($\xi=0$, dotted horizontal lines) and at threshold ($\xi_{\mathrm{thr}}/\kappa_{\mathrm{thr}}$, solid horizontal and vertical lines). The top axis also shows the colour-coded values $\xi_{\mathrm{thr}}$ at threshold and at the zero-momentum state, $\xi_{\mathrm{ZB}}:=\xi(\kappa_{3}=0)$ (dashed vertical lines); the bottom axis contains the corresponding values of $\kappa_{\mathrm{thr}}$ and $\kappa_{2\mathrm{ZB}}^{-\>}$. Arrows are colour-coded with the corresponding state to mark the points of zero binding ($\kappa_{3}=0$, vertical arrows); the threshold ($\xi_{\mathrm{thr}}/\kappa_{\mathrm{thr}}$, horizontal arrows), and of quasi-unitarity ($\xi=0$, horizontal arrows). Per subsect. 3.5, the trajectory of the $3$rd excitation in this and all subsequent figures is within line thickness indistinguishable from the one in Short-Range EFT with the same $3\mathrm{B}$ binding momentum at quasi-unitarity ($\kappa_{2}^{-}=\xi=0$).  | zero binding | quasi-unitarity | threshold
---|---|---|---
state |  $\xi_{\mathrm{ZB}}:=\xi(\kappa_{3}=0)$ |  |  |
($\kappa_{2\mathrm{ZB}}^{-\>}:=\kappa_{2}^{-}(\xi_{\mathrm{ZB}})$) | $\kappa_{3\mathrm{QU}}:=\kappa_{3}(\xi=0)$ | $\xi_{\mathrm{thr}}:=\xi(\kappa_{2}=\kappa_{3})$ | $\kappa_{\mathrm{thr}}:=\kappa_{3}(\xi_{\mathrm{thr}})$ |
ground ($j=0$) |  $3.6689(1)\cdot 10^{-1}$ |  |  |
($-2.04318(6)\cdot 10^{-1}$) | $2.35412(3)\cdot 10^{-1}$ | $-8.7258(1)$ | $2.11862(2)$ |
$1$st excited ($j=1$) |  $1.3855(1)\cdot 10^{-2}$ |  |  |
($-6.9517(5)\cdot 10^{-3}$) | $1.03030(5)\cdot 10^{-2}$ | $-2.6580(1)\cdot 10^{-1}$ | $1.25108(1)\cdot 10^{-1}$ |
$2$nd excited ($j=2$) |  $6.0279(2)\cdot 10^{-4}$ |  |  |
($-3.0144(1)\cdot 10^{-4}$) | $4.53987(1)\cdot 10^{-4}$ | $-1.2680(3)\cdot 10^{-2}$ | $6.320(1)\cdot 10^{-3}$ |
$3$rd excited ($j=3$) |  $2.6538(3)\cdot 10^{-5}$ |  |  |
($-1.3269(2)\cdot 10^{-5}$) | $2.00039(5)\cdot 10^{-5}$ | $-5.632(2)\cdot 10^{-4}$ | $2.810(3)\cdot 10^{-4}$ |
$j\to\infty$ ($\kappa_{2}^{-}\to-\frac{\xi}{2}$) |  $0.3102(1)\;\mathrm{e}^{-j\frac{\pi}{s_{0}}}$ |  |  |
($\rule[-6.88889pt]{0.0pt}{3.44444pt}\kappa_{2}^{-}=-0.1551(1)\;\mathrm{e}^{-j\frac{\pi}{s_{0}}}$) | $0.23381(8)\;\mathrm{e}^{-j\frac{\pi}{s_{0}}}$ | $-6.62(4)\;\mathrm{e}^{-j\frac{\pi}{s_{0}}}$ | $3.31(2)\;\mathrm{e}^{-j\frac{\pi}{s_{0}}}$ |
$\frac{\mbox{ground}}{\mbox{$1$st excited}}$ | $26.4808(2)$ | $22.849(1)$ | $32.828(2)$ | $16.934(2)$
(rel. dev. to $\mathrm{e}^{\frac{\pi}{s_{0}}})$ |  ($1.668(1)\cdot 10^{-1}$) |  ($6.81(3)\cdot 10^{-3}$) |  ($4.4653(1)\cdot 10^{-1}$) |  ($-2.537(1)\cdot 10^{-1}$)
$\frac{\mbox{$1$st excited}}{\mbox{$2$nd excited}}$ | $22.9847(2)$ | $22.695(1)$ | $20.963(5)$ | $19.795(3)$
(rel. dev. to $\mathrm{e}^{\frac{\pi}{s_{0}}})$ |  ($1.279(1)\cdot 10^{-2}$) |  ($2.7(4)\cdot 10^{-5}$) |  ($-7.6529(2)\cdot 10^{-2}$) |  ($-1.277(2)\cdot 10^{-1}$)
$\frac{\mbox{$2$nd excited}}{\mbox{$3$rd excited}}$ | $22.7142(6)$ | $22.694(5)$ | $22.556(3)$ | $22.488(5)$
(rel. dev. to $\mathrm{e}^{\frac{\pi}{s_{0}}})$ |  ($8.65(5)\cdot 10^{-4}$) |  ($0(4)\cdot 10^{-6}$) |  ($-7.9(5)\cdot 10^{-3}$) |  ($-9.0(1)\cdot 10^{-3}$)
ratio $j\to\infty:\;\mathrm{e}^{\frac{\pi}{s_{0}}}$ | $22.6944\dots$ | $22.6944\dots$ | $22.6944\dots$ | $22.6944\dots$
Table 1: Characteristics of the point of zero binding energy
$\xi_{\mathrm{ZB}}:=\xi(\kappa_{3}=0)$, of the quasi-unitarity limit
($\kappa_{2}^{-}=\xi=0$) with binding momentum
$\kappa_{3\mathrm{QU}}:=\kappa_{3}(\xi=0)$, and of the threshold point
$\kappa_{\mathrm{thr}}(\xi_{\mathrm{thr}}):=\kappa_{3}(\xi_{\mathrm{thr}})\stackrel{{\scriptstyle!}}{{=}}\kappa_{2}^{-}(\xi_{\mathrm{thr}})$
where the binding energies of the $2\mathrm{B}$ and $3\mathrm{B}$ system are
identical. Values are given for the ground and first three excited states,
plus extrapolated values for the Efimov-like states. The lower part lists the
ratios of the properties between adjacent states and in parenthesis their
relative deviations
$\displaystyle\mathrm{e}^{-\pi/s_{0}}\;\frac{\mbox{\footnotesize property of
$(j-1)$st state}}{\mbox{\footnotesize property of $j$th state}}-1$ from the
scaling factor $\mathrm{e}^{\pi/s_{0}}\approx 22.6944\ldots$. Binding momenta
are quoted in $|r_{0}^{-1}|$, and their real parts are zero. Uncertainties
combine extrapolation and numerics.

The vertex functions $\Gamma^{(j)}(Q)$ shown in fig. 2 for a number of $\xi$
values display the expected behaviour: with decreasing binding energy, the
typical momentum is shifted to lower $Q$, making the state wider; and the
vertex function of the $j$th excitation has $j$ nodes, just as the operator
$1-\mathcal{K}(\mu^{2};P,Q)$ of eq. (2.11) also has $j$ negative eigenvalues.

The respective binding momenta are given in fig. 3 as function of $\xi$ or
$\kappa_{2}^{-}$, aligning with the qualitative behaviour of fig. 1. The
number of bound $3\mathrm{B}$ states increases as $|\xi|\to 0$: none above the
$2\mathrm{B}$ state at $\xi=-10$, but one at $\xi=-1$, two at $\xi=-0.1$,
three at $\xi=-0.01$, and four at $\xi=-0.0001$. Likewise, states disappear
one by one as $\xi$ increases from zero.

Table 1 lists the values at the points of interest for the first four states,
and their ratios between different states. Given are the two thresholds
$\xi_{\mathrm{ZB}}$ and $\xi_{\mathrm{thr}}$ with their respective
$2\mathrm{B}$ binding momenta $\kappa_{2\mathrm{ZB}}^{-\>}$ and
$\kappa_{\mathrm{thr}}$, as well as the $3\mathrm{B}$ binding momentum at
quasi-unitarity, $\kappa_{3\mathrm{QU}}$. These results are inferred from
computations at $13$ cutoffs, equidistant in $\ln\Lambda$ between
$16\,|r_{0}^{-1}|$ and $128\,|r_{0}^{-1}|$, with grids of $500$, $1000$ and
$1500$ points, and independent variations of the mappings between the grid and
the momentum variables $P,Q$. The uncertainties are from the $n\to\infty$ and
$\Lambda\to\infty$ extrapolation and numerics combined. Determining the ratios
both independently and using the extracted values of $\xi$ and $\kappa_{3}$
leads to central values and errors which are consistent with each other. The
reported values and uncertainties of all quantities were determined to be
consistent using the network of individual results and from stability
considerations.

###  3.3 Comparison between Trajectories

We first focus on comparisons between the trajectories of the ground and first
three excited states; subsect. 3.5 will compare them to the results of Short-
Range EFT. Perhaps the most striking feature of table 1 and fig. 3 is the
apparent self-similarity of the different trajectories
$\kappa_{3}^{(j)}(\xi)$, indicating the approximate invariance under a
discrete scaling transformation.

For a quantitative comparison, we first rescale the trajectory of the $j$th
state,
$(\kappa_{2}^{-},\kappa_{3})\;\to\;\mathrm{e}^{j\pi/s_{0}}(\kappa_{2}^{-},\kappa_{3})$,
by the Efimov ratio expected for $j\to\infty$ from the discussion in sect.
2.3. The result is shown on the top of fig. 4 on logarithmic scales, and at
its bottom using linear scales. The ratios of the points at threshold, quasi-
unitarity and zero binding are listed in table 1, together with uncertainties.
There are indeed a few noticeable differences: The ground state has the
“shortest” rescaled trajectory, and length grows with each excitation. While
the rescaled zero-binding point and $3\mathrm{B}$ binding momentum at quasi-
unitarity remain largely in place as $j$ is increased, subsequent excitations
stretch the threshold position by more than half of the ground-state value.
The logarithmic scale on the top of fig. 4 makes one intuitively underestimate
the significant effect, which is better captured on the linear scale at the
bottom. Asymptotically for $j\to\infty$, the trajectory of
$\kappa_{3}^{-\,{(j)}}$ stretches between zero binding at
$\kappa_{2\mathrm{ZB}}^{-\>(j\to\infty)}=-0.155\dots\;\mathrm{e}^{-j\pi/s_{0}}$
and a threshold at
$\kappa_{\mathrm{thr}}^{(j\to\infty)}=3.3\dots\;\mathrm{e}^{-j\pi/s_{0}}$, or
minus-twice those values in $\xi$ since $\kappa_{2}^{-}(\xi)\to-\xi/2$ for
$|\xi|\to 0$. Similarly, quasi-unitarity is reached for
$\kappa_{3\mathrm{QU}}^{(j\to\infty)}=0.2338\dots\;\mathrm{e}^{-j\pi/s_{0}}$.

Figure 4: (Colour on-line) The rescaled trajectories of the binding momentum
$\mathrm{e}^{j\pi/s_{0}}\,\kappa_{3}$ of the $j$th $3\mathrm{B}$ state as
function of the binding momentum $\mathrm{e}^{j\pi/s_{0}}\,\kappa_{2}^{-}$ of
the shallowest $2\mathrm{B}$ state, both in units of $|r_{0}^{-1}|$. Top:
_logarithmic_ scales on both axes, with $\kappa_{2}^{-}>0$ and $<0$ stitched
together, and the top and right axes indicating the universal values at
threshold, quasi-unitarity and zero binding for $j\to\infty$ as provided in
table 1. Centre: _linear_ scales on both axes, rendering the different
threshold and zero-binding points of the states more prominent. Bottom:
difference between the $j$th and $3$rd state. Colour-coding,
vertical/horizontal lines, arrows and shaded area as in fig. 3.

The main differences among trajectories are found where the state goes unbound
or unstable, and for the ground state. To highlight them, we re-plot in fig. 4
the same data on linear scales as differences
$\mathrm{e}^{j\pi/s_{0}}\,\kappa_{3}^{(j)}-e^{3\pi/s_{0}}\,\kappa_{3}^{(0)}$
between the rescaled binding momentum of the $j$th state and the shallowest
state investigated, $j=3$, as function of
$\mathrm{e}^{j\pi/s_{0}}\,\kappa_{2}^{-}$. This plot shows again that the
differences are large towards the threshold points, and non-negligible towards
zero binding. However, little difference occurs around quasi-unitarity, where
the three curves very nearly intersect at one point for
$\kappa_{2}^{-}=\xi=0$. Deviations are most striking for the ground state,
visible for the first excitation, and barely noticeable for the second one.
Remarkably, though, they are not very large even for the ground state. Note
that for $\kappa_{2}^{-}\simeq\kappa_{2\mathrm{ZB}}^{-\>}$
($\xi\simeq\xi_{\mathrm{ZB}}$), differences are artificially enhanced by the
different points where each state goes unbound, since
$e^{3\pi/s_{0}}\,\kappa_{3}^{(0)}\to 0$ before
$\mathrm{e}^{j\pi/s_{0}}\,\kappa_{3}^{(j)}$ does.

Figure 5: (Colour on-line) The universal function
$\mathcal{R}^{(j)}(x_{\kappa}):=\kappa_{3}^{(j)}/\kappa_{\mathrm{thr}}^{(j)}$
of the binding momentum of the $j$th state as function of the rescaling
variable $x_{\kappa}:=\kappa_{2}^{-}/\kappa_{\mathrm{thr}}^{(j)}$ of the
respective state, both on a logarithmic scale. The inverse of the ratio for
$\kappa_{3\mathrm{QU}}$ at quasi-unitarity ($\kappa_{2}^{-}=\xi=0$) and for
$\kappa_{2\mathrm{ZB}}^{-\>}$ at zero binding ($\kappa_{3}=0$) are given on
the top and left, as provided in the second and last column of table 4,
respectively. Colour-coding, vertical/horizontal lines, arrows and
$\kappa_{2}^{-}>0$ and $<0$ stitched together as in fig. 3.

Another way to characterise differences and commonalities of trajectories is
to rescale the $2\mathrm{B}$ and $3\mathrm{B}$ binding momenta by their common
value $\kappa_{\mathrm{thr}}^{(j)}$ at threshold, defining

$\mathcal{R}^{(j)}\left(x_{\kappa}:=\frac{\kappa_{2}^{-}}{\kappa_{\mathrm{thr}}^{(j)}}\right):=\frac{\kappa_{3}^{(j)}(\kappa_{2}^{-})}{\kappa_{\mathrm{thr}}^{(j)}}\;\;.$
(3.2)

The rescaled trajectories are shown in fig. 5 as function of $x_{\kappa}$.
Their characteristic values at zero binding and quasi-unitarity are given in
table 4, where these are shown for the first four states within the same
trajectory and extrapolated for $j\to\infty$, which were derived as described
for table 1. The function $\mathcal{R}^{(j)}$ is not only universal for any
given state, but it also appears to approach a common, universal function for
shallower states. While differences are clearly seen for the ground state and
first-excited state, already for the second-excited state values are very
close to those for the third-excited state. A similar plot results if the
rescaled trajectories are displayed as function of the rescaled variable
$x_{\xi}:=\xi/\xi_{\mathrm{thr}}$, although some differences arise from the
nontrivial mapping between $\xi$ and $\kappa_{2}^{-}$, which only approaches a
simple rescaling $x_{\kappa}\to x_{\xi}$ from $\xi\to-2\kappa_{2}^{-}$ as
$\xi\to 0$.

###  3.4 Parametrising Trajectories

These figures in combination highlight two striking features which are
confirmed by a closer numerical analysis. First, the trajectories as function
of $x_{\kappa}$ are extremely similar towards threshold, $x_{\kappa}=1$. A fit

$\mathcal{R}(x_{\kappa}\to
1)=x_{\kappa}+0.098(1)\;(1-x_{\kappa})^{2}+\mathcal{O}((1-x_{\kappa})^{3})$
(3.3)

is right between the $j=0,1$ trajectories which are bit higher, and the
$j=2,3$ trajectories. The error includes the uncertainties of the individual
fit and is stable against including more fit terms. The relative residuals
$\frac{\mathrm{data}(x_{\kappa})-\mathrm{param}(x_{\kappa})}{\mathrm{data}(x_{\kappa})}$
increases from $0$ by construction at threshold to $\pm 0.4\%$ at
$x_{\kappa}=0.5$ and $\pm 1.5\%$ at $x_{\kappa}=0.3$. This nicely quantifies
the extremely slow convergence between the binding momenta of the
$3\mathrm{B}$ and $2\mathrm{B}$ systems towards threshold.

The slopes around quasi-unitarity are very similar. Fitting

$\mathcal{R}^{(j)}(x_{\kappa}\to
0)=\frac{\kappa_{3\mathrm{QU}}^{(j)}}{\kappa_{\mathrm{thr}}^{(j)}}+a^{(j)}\;x_{\kappa}+\mathcal{O}(x_{\kappa}^{2})$
(3.4)

with the constants at quasi-unitarity from table 4 gives noticeable deviations
from the straight line only for $|x_{\kappa}|>0.005$. The slope increases
slightly from the ground state ($a^{(0)}=0.8289(3)$) via the first excitation
($a^{(1)}=1.035(1)$) to $a^{(j=2\to\infty)}=1.057(1)$. These fits are again
quite stable against varying the data range or including higher-order terms.

The approach to zero binding appears quite uniform for all states, too. An
expansion

$\mathcal{R}^{(j)}\left(x_{\kappa}\to\frac{\kappa_{2\mathrm{ZB}}^{-\>{(j)}}}{\kappa_{\mathrm{thr}}^{(j)}}\right)=\left(x_{\kappa}-\frac{\kappa_{2\mathrm{ZB}}^{-\>{(j)}}}{\kappa_{\mathrm{thr}}^{(j)}}\right)^{\alpha{(j)}}$
(3.5)

leads to approximately the same exponent $\alpha(j)\approx 0.58(3)$ for each
state, but numerical stability is more elusive. The exponent grows as the fit
range is increased, while the residuals remain comparable.

Finally, we parametrise the whole trajectory $\mathcal{R}^{(j)}(x_{\kappa})$
of each state, not just the behaviour at the three “critical” points. We are
inspired by Efimov’s universal function $\Delta(\xi)$ [2, 3, 4] to transform
the curve from Cartesian coordinates in the
$(x_{\kappa},\mathcal{R}^{(j)}(x_{\kappa}))$ plane to polar ones,

$\begin{split}x_{\kappa}&=\sqrt{2}\;\rho^{(j)}(\theta)\;\cos(\theta+\frac{\pi}{4})\;\;,\\\
\mathcal{R}^{(j)}(x_{\kappa})&=\sqrt{2}\;\rho^{(j)}(\theta)\;\sin(\theta+\frac{\pi}{4})\;\;,\end{split}$
(3.6)

as shown in fig. 1. We choose $\theta$ to measure the angle formed by the
threshold point, the origin, and any particular point on the trajectory
(chosen in the mathematically positive sense and continuous between the first
and second quadrant),

$\theta:=\arctan\frac{\mathcal{R}^{(j)}(x_{\kappa})}{x_{\kappa}}\;-\;\frac{\pi}{4}\;\;.$
(3.7)

From our choice of normalisation and using $\mathcal{R}(x_{\kappa}\to 1)\to
1$, the radius

$\rho^{(j)}(\theta):=\frac{1}{\sqrt{2}}\;\sqrt{x_{\kappa}^{2}(\theta)+(\mathcal{R}^{(j)}(\theta))^{2}}$
(3.8)

is $1$ at threshold. The coordinates of the points of interest and regions are
then

$\begin{array}[]{lll}\mbox{Threshold
Point:}&\displaystyle\theta_{\mathrm{thr}}=0&\mbox{ with
}\displaystyle\rho^{(j)}_{\mathrm{thr}}=1\\\ \mbox{All-Bound
Region:}&\displaystyle\theta\in]0;\frac{\pi}{4}[&\\\ \mbox{Quasi-Unitarity
Point:}&\displaystyle\theta_{\mathrm{QU}}=\frac{\pi}{4}&\mbox{ with
}\displaystyle\rho^{(j)}_{\mathrm{QU}}=\frac{\kappa_{3\mathrm{QU}}^{(j)}}{\sqrt{2}\,\kappa_{\mathrm{thr}}^{(j)}}\\\
\mbox{Borromean
Region:}&\displaystyle\theta\in]\frac{\pi}{4};\frac{3\pi}{4}[&\\\\[4.30554pt]
\mbox{Zero Binding
Point:}&\displaystyle\theta_{\mathrm{ZB}}=\frac{3\pi}{4}&\mbox{ with
}\displaystyle\rho^{(j)}_{\mathrm{ZB}}=\left|\frac{\kappa_{2\mathrm{ZB}}^{-\>{(j)}}}{\sqrt{2}\,\kappa_{\mathrm{thr}}^{(j)}}\right|\end{array}$
(3.9)

Concretely, we are inspired by an idea for $\Delta(\xi)$ by Gattobigio et al.
[46] to parametrise

$\rho^{(j)}_{\mathrm{param}}(\theta)=\exp\left(\sum\limits_{n=1}^{7}\;c_{n}^{(j)}\;\theta^{n/2}\right)\;\;.$
(3.10)

The powers of square-roots are essential to capture the behaviour towards the
threshold, $\theta\to 0$. Gattobigio et al. use a slightly different
definition of the angle and have one more parameter since the scale of their
function is not fixed at one point. Here, $\rho^{(j)}(0)=1$ for all states by
normalisation, and the scale can be restored by multiplying with the threshold
value, $\kappa_{\mathrm{thr}}^{(j)}$. We find a more economical
parametrisation with an exponential function of powers. Without an exhaustive
search, we did not find more efficient combinations of powers of roots, but a
(non-exponentiated) polynomial of $x^{n/4}$ comes close.

Our determination of the coefficients differ somewhat from Gattobigio et al.
[46], who optimise the fit constrained by a universal number of significant
figures. We first perform an unconstrained fit for all coefficients $c_{n}$,
$n=1,\dots,7$. The resulting $c_{1}$ to $c_{4}$ are then rounded to $5$
significant figures. With these coefficients fixed, we then re-fit the other
ones, $c_{5}$ to $c_{7}$, and truncate that result at $6$ significant figures.
Finally, we compare $\rho_{\mathrm{param}}$ of the final fit, with all
coefficients truncated as prescribed, to that of the first fit, in which none
of the coefficients were rounded beyond machine precision. The difference
between first and final fit function is never bigger than $\pm 1.5\cdot
10^{-5}$ for each state. The magnitude of the relative deviation between the
two is $\lesssim\pm 0.2\%$. At quasi-unitarity, truncated and free version are
near-identical for all states. The estimated variance (root of the sum-squared
of all residuals) is $\\{0.00116;0.00039;0.00033;0.0015\\}$, ordered from the
ground state to the highest excitation. This measure worsen by $\lesssim
0.02\%$ from the un-truncated to the truncated parametrisation ($0.17\%$ for
the $2$nd excitation). The adjusted coefficient of determination $R^{2}$
worsens by $\lesssim 0.6\%$ ($0.3\%$ for the $2$nd excitation), and the
Bayesian Information Criterion (in the Gaußian approximation) by $\lesssim
0.02\%$ ($0.15\%$ for the $2$nd excitation). In both the first and final
parametrisations, the residuals are very similar for all states. Variants of
this procedure, like different numbers of significant figures on different
parameter combinations or iterative fits, lead to results of at best similar
quality for the same overall number of significant figures — and worse quality
for smaller numbers.

state | $c_{1}^{(j)}$ | $c_{2}^{(j)}$ | $c_{3}^{(j)}$ | $c_{4}^{(j)}$ | $c_{5}^{(j)}$ | $c_{6}^{(j)}$ | $c_{7}^{(j)}$ | $w_{\mathrm{pred}}^{(j)}$
---|---|---|---|---|---|---|---|---
ground ($j=0$) | $-4.7545$ | $4.3710$ | $-7.4404$ | $11.714$ | $-11.0015$ | $5.54726$ | $-1.12595$ | $0.00009$
$1$st excited ($j=1)$ | $-4.2764$ | $1.0734$ | $-1.6005$ | $5.4777$ | $-6.58104$ | $3.64577$ | $-0.768214$ | $0.00003$
$2$nd excited ($j=2)$ | $-4.6823$ | $1.4352$ | $-1.2221$ | $4.1199$ | $-5.08001$ | $2.87318$ | $-0.613720$ | $0.00003$
$3$rd excited ($j=3)$ | $-4.7003$ | $1.1952$ | $0.014514$ | $1.5657$ | $-2.44535$ | $1.53708$ | $-0.348975$ | $0.00011$
Table 2: The coefficients of the parametrisation $\rho_{\mathrm{param}}^{(j)}$
of the $j$th state in eq. (3.10), in $\mathrm{rad}^{-n/2}$, with $5$ and $6$
significant figures, respectively; see text. In the last column,
$w^{(j)}_{\mathrm{pred}}(\theta)$ estimates the $68\%$ confidence interval
$\rho_{\mathrm{param}}^{(j)}(\theta)\pm w_{\mathrm{pred}}^{(j)}$ for a
prediction at a single, new $\theta$.

Table 2 contains the parameter values for the ground and first three excited
states from a least-square fit to $208$ points per trajectory, including
higher densities of values exactly at and very close to threshold, quasi-
unitarity and zero binding. We do not quote parameter uncertainties since all
entries of the correlation matrix are bigger than $0.66$ in magnitude,
indicating that they are highly (anti-)correlated. A parametrisation with
fewer coefficients is therefore likely to exist. The last column in the table
contains the half-width of the parametrisation’s $68\%$ confidence prediction
interval $\rho_{\mathrm{param}}^{(j)}(\theta)\pm w_{\mathrm{pred}}^{(j)}$ for
the likely position of a new datum at one single $\theta$. This includes the
variation on both the parameters and the data scatter (variation of “future”
individual value). Within uncertainties, it can be treated as angle-
independent. The half-width of the $95\%$ interval is well-approximated by
$2w_{\mathrm{pred}}^{(j)}$.

Figure 6: (Colour on-line) The data and fitted function of trajectories of the
ground and first three excited states in polar form, eq. (3.6) in a linear-log
plot. For ease of comparison with the other figures, the angle $\theta$
increases to the left, i.e. the threshold is on the right. Therefore, slopes
have the opposite sign to what the figure seems to imply at first glance.
Values at zero binding are indicated on the left, those at quasi-unitarity on
the right, each in the colour of the corresponding state. As detailed in
subsect. 3.5, the blue solid line of the $3$rd excitation is indistinguishable
from the result using Short-Range EFT with the same $3\mathrm{B}$ binding
momentum at quasi-unitarity ($\kappa_{2}^{-}=\xi=0$).

Figure 6 shows the result, together with the values of $\rho^{(j)}$ at points
of interest. As fig. 7 also demonstrates, the residuals increase towards the
threshold to $\leq 0.001$ in magnitude. While $\rho$ is big there, the
relative residuals are with $\pm 0.1\%$ actually smaller than in the other
regions. Otherwise, residuals are never bigger than $0.00005$ in magnitude.
Relative residuals can reach $0.5\%$ in magnitude in the Borromean region
around $105^{\circ}$. The $68\%$ confidence interval
$\rho_{\mathrm{param}}^{(j)}(\theta)\pm w_{\mathrm{conf}}^{(j)}$ of the
uncertainty of the mean shows substantial dependence on $\theta$. It is
minimal around quasi-unitarity and maximal at zero binding; see fig. 7. That
reflects the high number of high-quality points around the former, and the
notorious difficulty to adequately map the approach $\kappa_{3}\to 0$ for the
latter. On top of that, it must by construction vanish at $\theta=0$, but the
rapid fall bears yet again witness to the difficulty close to threshold to
numerically uniquely disentangle a $3\mathrm{B}$ energy very close to the
$2\mathrm{B}$ energy. Remember that this confidence interval captures only the
variation in parameter estimates, while the prediction interval
$w_{\mathrm{pred}}$ of table 2 in addition accounts for the variation induced
by new data and is thus considerably less $\theta$ dependent and, as expected,
larger than $w_{\mathrm{conf}}$.

Figure 7: (Colour on-line) Quality-of-fit assessment for the trajectory of the
ground and first three excited states in polar form. The threshold is on the
right; $\theta$ increases to the left. Top: residuals (rescaled by $10^{4}$).
Centre: relative residuals. Bottom: function $w^{(j)}_{\mathrm{conf}}(\theta)$
(rescaled by $10^{4}$) parametrising the $68\%$ (solid lines) and $95\%$
(dashed) confidence interval of the uncertainty around the mean,
$\rho_{\mathrm{param}}^{(j)}(\theta)\pm w_{\mathrm{conf}}^{(j)}(\theta)$.

The fit coefficients of table 2 approach roughly universal values with
increasing excitation and appear all of similar, “natural” size. This
indicates at best limited fine-tuning, despite a rather strong correlation
matrix which masks a common limit for the coefficients as $j\to\infty$. Other
reasonable parametrisations with the same number of parameters and similar
rounding prescriptions show qualitatively similar behaviour but more
pronounced fine-tuning and correlations. Clearly, the detailed numbers for the
coefficients, for the uncertainty estimates and for the fit criteria depend
significantly on the number and positions of the angles $\theta_{i}$ at which
“data” for the fit are produced. For example, a fit with very many data close
to threshold usually leads to distorted parametrisations which are poor at
quasi-unitarity and zero binding. Overall, we believe that the distribution of
points shown in the residuals of fig. 7 is an adequate compromise for the
present investigation.

point | property |  ground ($j=0$) |  $1$st excited |  $2$nd excited |  $3$rd excited
---|---|---|---|---|---
threshold: | $\rho^{(j)}(0)$ |  $1$ (normalisation)
|  slope $(\rho^{(j)}_{\mathrm{param)}})^{\prime}(\theta\to 0)$ $[\mathrm{rad}^{-1}]$ |  $\displaystyle\frac{c_{1}^{(j)}}{2\;\sqrt{\theta}}\to\infty$
quasi-unitarity: | $\rho^{(j)}_{\mathrm{param}}(\frac{\pi}{4})$ | $0.078571(7)$ | $0.058244(2)$ | $0.050779(2)$ | $0.050143(8)$
|  from table 1 | $0.078571(1)$ | $0.058232(3)$ | $0.050794(8)$ | $0.05034(5)$
| relative residual | $-0.00(1)\%$ | $-0.02(1)\%$ | $0.03(2)\%$ | $0.4(1)\%$
|  slope $(\rho^{(j)}_{\mathrm{param}})^{\prime}(\frac{\pi}{4})$ $[\mathrm{rad}^{-1}]$ | $-0.06557(7)$ | $-0.06042(1)$ | $-0.053653(1)$ | $-0.052971(3)$
|  from $x_{\kappa}^{(j)}\to 0$ fit | $-0.0651(1)$ | $-0.0603(1)$ | $-0.0537(1)$ | $-0.0532(1)$
| relative residual | $-0.7(2)\%$ | $-0.2(2)\%$ | $-0.1(2)\%$ | $-0.4(2)\%$
zero binding: | $\rho^{(j)}_{\mathrm{param}}(\frac{3\pi}{4})$ | $0.06895(5)$ | $0.03953(2)$ | $0.03393(1)$ | $0.03359(7)$
|  from table 1 | $0.068193(2)$ | $0.039290(3)$ | $0.033726(5)$ | $0.03339(4)$
| relative residual | $-1.1(1)\%$ | $-0.62(5)\%$ | $-0.62(5)\%$ | $-0.60(2)\%$
|  slope $(\rho^{(j)}_{\mathrm{param)}})^{\prime}(\frac{3\pi}{4})$ $[\mathrm{rad}^{-1}]$ | $0.0155(4)$ | $0.00439(2)$ | $0.00427(2)$ | $0.0052(5)$
Table 3: Values from the parametrisation $\rho_{\mathrm{param}}^{(j)}$, eq.
(3.10) and of derivatives at points of interest, compared to those derived
from table 1. The relative residual is
$1-\rho_{\mathrm{param}}^{(j)}/\rho_{\mathrm{table}}$.

Table 3 contains the values of $\rho^{(j)}$ at the points of interest in the
parametrisation, as well as its values for the derivatives at quasi-unitarity
and zero binding. As the table shows, these match the values derived from
tables 1 and 4 within the quoted uncertainties to better than $0.03\%$ at
quasi-unitarity ($0.4\%$ for the $3$rd excitation). At zero binding, the
tension is about $0.6\%$ but still within uncertainties ($1\%$ for the ground
state). In addition, the table compares to the direct determination of the
slope $\mathcal{R}^{(j)}$ with respect to $x_{\kappa}$ at quasi-unitarity, eq.
(3.4), translated to the slope $(\rho^{(j)})^{\prime}(\theta)$ via

$\frac{\mathrm{d}\mathcal{R}^{(j)}(x_{\kappa})}{\mathrm{d}x_{\kappa}}=\frac{(\rho^{(j)})^{\prime}(\theta)\;\cos[\theta+\frac{\pi}{4}]+\rho(\theta)\;\sin[\theta+\frac{\pi}{4}]}{(\rho^{(j)})^{\prime}(\theta)\;\cos[\theta+\frac{\pi}{4}]-\rho(\theta)\;\sin[\theta+\frac{\pi}{4}]}\;\;,$
(3.11)

at $\theta_{\mathrm{QU}}=\frac{\pi}{4}$ with
$\rho_{\mathrm{QU}}=\kappa_{3\mathrm{QU}}^{(j)}/(\sqrt{2}\kappa_{\mathrm{thr}}^{(j)})$
from table 4. All results are in reasonable agreement.

All this makes us confident that the truncated parameters of table 2 provide
an adequate representation of the best fit. Accounting for all information,
including the confidence and prediction uncertainty intervals of the fit as
well as the comparison to the other extractions, an uncertainty of about
$0.5\%$ overall seems reasonable, and somewhat more towards the zero-binding
endpoints. This should in any case suffice to address physical systems, where
experimental uncertainties and higher-than-leading order corrections enter as
well.

###  3.5 Comparison to Short-Range EFT

The universality of Resummed-Range EFT as $j\to\infty$ can be understood from
the discussion in sect. 2.3. Still, it is instructive to consider in more
detail the transition (2.14) in the $2\mathrm{B}$ propagator from Resummed-
Range to Short-Range EFT. Since $P,Q,\mu$ are dimensionless and
$\mu^{2}=-\kappa_{3}^{2}<0$ for a bound $3\mathrm{B}$ state, one can
differentiate two regions in the propagator of the integral equation (2.11).
For $3Q^{2}+4\kappa_{3}^{2}\gtrsim 16$, the effective-range term
$(3Q^{2}+4\kappa_{3}^{2})/4$ cannot be neglected and may indeed dominate over
the Short-Range EFT term of the propagator, $\sqrt{3Q^{2}+4\kappa_{3}^{2}}$.
The integral converges quickly. The scale of the loop momentum is set by the
binding momentum, $Q\hskip
1.99997pt\raisebox{2.15277pt}{$<$}\raisebox{-3.00003pt}{$\sim$}\hskip
1.99997pt\kappa_{3}$. As a consequence, we expect that bound states can be
supported at $\kappa_{3}\lesssim 2$. Interestingly, this is not far from our
numerical finding for the ground state, $\kappa_{3}^{(0)}\lesssim 2.1$. In
contrast, for $3Q^{2}+4\kappa_{3}^{2}\ll 16$, the effective-range term becomes
a correction while the propagator reduces to

$\frac{1}{\xi+(3Q^{2}+4\kappa_{3}^{2})/4+\sqrt{3Q^{2}+4\kappa_{3}^{2}}}\to\frac{1}{\xi+\sqrt{3Q^{2}+4\kappa_{3}^{2}}}\to\frac{1}{\sqrt{3}\;Q}\;\;,$
(3.12)

and hence to the propagator of Short-Range EFT, for a shallow bound state when
$|\xi|,\kappa_{3}\ll 1$. Therefore, bound states are to be expected close to
quasi-unitarity in Resummed-Range EFT, and the previous subsections confirm
that with a tower of states near $\xi\approx 0$. In general, this sequence of
régimes implies that range effects should be noticeable for lower-lying states
in the spectrum. On top of that, we might expect the largest effect for the
largest binding momenta, namely where the $3Q^{2}+4\kappa_{3}^{2}$ term should
be largest. As we saw before, that is near the point where the state
disappears at the $2\mathrm{B}$ threshold.

At a given $\xi$, $\kappa_{2}^{+}(\xi)$ remains fixed while the binding
momenta of highly excited states $j\to\infty$ approach zero, and are therefore
much smaller than the inverse effective range. We thus expect an expansion in
powers of $|r_{0}|$,

$\kappa_{3}^{(j)}(\xi(\theta))=\kappa_{\mathrm{Ef}}^{(j)}(\theta)\left[1+b_{1}^{(j)}(\theta)\;\kappa_{\mathrm{Ef}}^{(j)}(\theta)+b_{2}^{(j)}(\theta)\;(\kappa_{\mathrm{Ef}}^{(j)}(\theta))^{2}+\ldots\right]\;\;,$
(3.13)

where $\kappa_{\mathrm{Ef}}^{(j)}(\theta)$ are the binding momenta for the
Efimov tower at finite scattering length but vanishing effective range, but in
units of $|r_{0}^{-1}|$, and $b_{i}^{(j)}(\theta)$ are functions which are
universal for systems with large, negative effective range. Discrete Scale
Invariance translates into equal spacing for the Efimov tower, keeping the
angle $\theta$ unaffected but $\rho\to\mathrm{e}^{-\pi/s_{0}}\rho$. Therefore,
it is sensible to use $\theta$ as independent variable in eq. (3.13). Thus,
the ratio between adjacent states of the various properties of the special
points (threshold, quasi-unitarity, and zero binding) should approach the
Efimov ratio $e^{\pi/s_{0}}$ as we go up the bound spectrum; see eqs. (2.19)
to (2.21). The proximity of our tower to Efimov’s can be seen in the lower
rows of table 1. Only the features of the ground state deviate by more than
$10\%$ of the Efimov ratio, and only at threshold and zero binding. The
deviations at these extreme points is ${\cal
O}(\kappa_{2}^{-}/\kappa_{2}^{+})$, as naively expected when
$\kappa_{3}^{(j)}=\kappa_{2}^{-}$ and $\kappa_{3}^{(j)}=0$, respectively. At
quasi-unitarity, on the other hand, deviations seem smaller, ${\cal
O}((\kappa_{3}^{(j)}/\kappa_{2}^{+})^{2})$.

| ratio threshold to zero binding |  ratio quasi-unitarity |
---|---|---|---
to zero binding |  ratio threshold |  |  |
to quasi-unitarity |  |  |  |
state | $\displaystyle\frac{\xi_{\mathrm{thr}}}{\xi_{\mathrm{ZB}}}$ | $\displaystyle\frac{\kappa_{\mathrm{thr}}}{\kappa_{2\mathrm{ZB}}^{-\>}}$ | $\displaystyle\frac{\kappa_{3\mathrm{QU}}}{\kappa_{2\mathrm{ZB}}^{-\>}}$ | $\displaystyle\frac{\kappa_{\mathrm{thr}}}{\kappa_{3\mathrm{QU}}}$
ground | $-23.783(1)$ | $-10.3692(3)$ | $-1.1526(1)$ | $8.9963(1)$
$1$st excited | $-19.185(3)$ | $-17.993(1)$ | $-1.482(1)$ | $12.1413(1)$
$2$nd excited | $-21.04(3)$ | $-20.97(1)$ | $-1.506(3)$ | $13.921(3)$
$3$rd excited | $-21.22(5)$ | $-21.2(3)$ | $-1.51(3)$ | $14.05(2)$
$j\to\infty$ |  $-21.3(2)$ | $-1.51(4)$ | $14.1(3)$
Efimov $j\to\infty$ |  $-21.3048\dots$ [44] with [7] |  |
$-21.306(1)$ [45] with [7] |  $-1.50763$ [44] |  |  |
$-1.5077(1)$ [45] |  $14.131\dots$ |  |  |
$=(0.0707645\dots)^{-1}$ [7] |  |  |  |
Table 4: Characteristics of the trajectories from threshold to zero
$3\mathrm{B}$ binding energy of the ground and first three excited state and
their extrapolated values for the Efimov-like states. Our ratios between
quasi-unitarity and zero-binding are inferred from the other ratios.
Uncertainties combine extrapolation and numerics. The last row lists the
ratios in leading-order Short-Range EFT, with references.

In addition, the universal trajectory $\mathcal{R}^{(j\to\infty)}$ should
become identical to the one of Short-Range EFT, whose characteristic values
are well-explored [44, 7, 45]. When these quantities are rescaled by the
threshold parameters, universal numbers also emerge within each trajectory of
the highly excited levels. Indeed, there is agreement of our values at the
$1\%$ level already for the second excitation, and excellent agreement for the
extrapolations $j\to\infty$. Table 4 shows that ratios between threshold and
zero binding, $\kappa_{\mathrm{thr}}^{(j)}/\kappa_{2\mathrm{ZB}}^{-\>}$
($\xi_{\mathrm{thr}}^{(j)}/\xi_{\mathrm{ZB}}$), approach $\simeq-21.3$ as
$j\to\infty$, compared with the literature value $\simeq-21.305$ [7, 44, 45].
Similarly, the ratios
$\kappa_{\mathrm{thr}}^{(j)}/\kappa_{3\mathrm{QU}}^{(j)}$ between threshold
and quasi- unitarity approach $\simeq 14.1$, while the literature value of
Short-Range EFT is $\simeq 14.131$ [7]. The two literature values [44, 45] for
the ratio between quasi-unitarity and zero binding are identical within the
error bar quoted in ref. [45], and our result agrees with both. This confirms
yet again that the EFT with non-perturbative effective-range corrections
includes Short-Range EFT for sufficiently high excitations. Thus, we recover
Discrete Scale Invariance far up in the bound spectrum.

However, there is an important difference to Resummed-Range EFT. In Short-
Range EFT, the position of the geometric tower of $3\mathrm{B}$ states is
fixed by the dimensionful scale parameter $\Lambda_{\ast}$ of the
$3\mathrm{B}$ interaction $H(\Lambda)$. In contrast, here the position of the
tower is a parameter-free prediction once the $2\mathrm{B}$ pole positions are
determined from, say, $2\mathrm{B}$ scattering. In other words, Resummed-Range
EFT can be seen as an underlying theory to Short-Range EFT, _predicting_
$\Lambda_{\ast}$ in units of $|r_{0}^{-1}|$. The reason is, of course, that
Short-Range EFT has no remaining natural scale once the $2\mathrm{B}$ binding
is zero, while the second pole at $\kappa_{2}^{+}=-2$ still provides one in
Resummed-Range EFT.

To illustrate this prediction, we focus on $\xi=0$. The binding momenta at
quasi-unitarity are a particular case of eq. (3.13) when $\theta=\pi/4$. The
proximity of the ratio of binding momenta at quasi-unitarity and unitarity in
table 1 suggests the expansion at $\theta=\pi/4$ converges even for the ground
state. We therefore computed $H(\Lambda)$ in Short-Range EFT at $\gamma=0$
such that the binding momenta $\kappa_{3}^{(j)}$ of the ground state or any of
its first three excitations is identical to the results of Resummed-Range EFT
at $\xi=0$ which are listed in table 1. We employed $156$ values of
$\Lambda|r_{0}|$ between $32$ and $16,384$, equidistant in $\ln\Lambda$. From
these, we determined $\Lambda_{\ast}|r_{0}|$ using eq. (2.16), setting it
close to $1$ considering that it is by eq. (2.16) only unique only up to
multiples of $\mathrm{e}^{\pi/s_{0}}$. First, we performed a three-parameter
fit to $\Lambda_{\ast}$, the amplitude $A$, and the frequency $s_{0}$. Since
we found that $s_{0}=1.00623(1)$ is statistically identical to the predicted
value of eq. (2.17), $s_{0}=1.006237\ldots$, we then used the predicted value
of $s_{0}$ to eliminate one parameter. The results for each state are reported
in table 5. They are sufficiently stable against varying numerics (grid size,
mapping, spacing) and cutoffs that we are confident in the extraction error
estimate. All fits have an adjusted coefficient of determination $R^{2}$ of
better than $1-10^{-8}$. The amplitude $A=0.87866(1)$ is well-identical within
error bars for all states, and agrees with the value of $A=0.879$ first
reported in ref. [47]. The renormalisation scale $\Lambda_{\ast}|r_{0}|$
decreases minimally from the ground state to the first excitation, even less
to the second one, and is stable for the third excitation and beyond. We see
no substantial quality-of-fit differences for different levels. That (in
regularisation with a “hard” cutoff)

$\Lambda_{\ast}^{(j\geq 2)}=0.610206(1)\;|r_{0}^{-1}|$ (3.14)

is near-identical for all levels can be traced to the small deviations from
the Efimov ratio at quasi-unitarity discussed in the preceding subsection; see
also table 1.

state |  scale parameter $\Lambda_{\ast}\;[|r_{0}^{-1}|]$ |  amplitude $A$
---|---|---
ground ($j=0$) | $0.614379(2)$ | $0.87866(2)$
$1$st excited ($j=1$) | $0.610223(1)$ | $0.87866(1)$
$2$nd excited ($j=2$) | $0.610206(1)$ | $0.87866(1)$
$3$rd excited ($j=3$) | $0.610206(1)$ | $0.87866(1)$
$j\to\infty$ | $0.610206(1)$ | $0.87866(1)$
Table 5: Scale parameter $\Lambda_{\ast}$ in $|r_{0}^{-1}|$ and amplitude $A$
of the $3\mathrm{B}$ interaction $H(\Lambda)$ at $1/a=0$ in Short-Range EFT
with “hard” cutoff regularisation, fixed to reproduce the results of Resummed-
Range EFT at quasi-unitarity ($\xi=0$). Uncertainties combine fit and
numerics.

It is interesting to compare the deviations we observe in the Efimov spectrum
or, equivalently, in $\Lambda_{\ast}$ for a “hard” cutoff regularisation, to
Short-Range EFT at subleading orders, where $r_{0}\neq 0$. In a specific
physical system, where other contributions beyond the effective range exist,
$\Lambda_{\star}$ would be fixed by the energy of a particular $3\mathrm{B}$
level, for example the ground state. The $2\mathrm{B}$ interaction
corresponding to the range is then included in perturbation theory — for
complete NLO and N2LO calculations, see refs. [48] and [49], respectively. The
coefficients $b_{1}^{(j)}(\theta)$ in eq. (3.13) are the matrix elements of
these perturbations of LO states. In first-order perturbation, a correction to
the $3\mathrm{B}$ interaction is necessary for renormalisation, and the
determination of $b_{1}^{(j)}(\theta)$ requires an additional datum, various
choices of which are described in ref. [48]. However, Discrete Scale
Invariance requires

$b_{1}^{(j)}(\theta)=b_{1}(\theta)$ (3.15)

to be the _same_ for all levels, as the LO states are the same apart from the
scaling factor. In other words, if $r_{0}$ were to scale — that is, made a
spurion field in the language of Particle Physics — as the inverse of a
momentum, shuffling $3\mathrm{B}$ levels, then the normalised
$\kappa_{\mathrm{Ef}}^{(j)}(\theta)$ would be invariant and
$b_{1}^{(j)}(\theta)$ cannot depend on the level $j$ [50]. If we determine the
NLO $3\mathrm{B}$ interaction by requiring one of the levels at unitarity to
remain unchanged, then

$b_{1}\left(\frac{\pi}{4}\right)\stackrel{{\scriptstyle!}}{{=}}0$ (3.16)

and all levels remain unchanged [51]. In a strict comparison of just
effective-range effects, we match Short-Range EFT at LO to Resummed-Range EFT
at LO by assuming the third (or a higher) excited state agree, thus
determining $\Lambda_{\star}$ as done above. That implies that the range
corrections are quadratic in $r_{0}$ at quasi-unitarity, which explains why
deviations from the Efimov spectrum are ${\cal
O}((\kappa_{3}^{(j)}/\kappa_{2}^{+})^{2})$. Such quadratic corrections appear
at N2LO in Short-Range EFT, i.e. the effective-range interaction enters in
second-order perturbation theory. Renormalisation requires a new, two-
derivative $3\mathrm{B}$ interaction [16, 52, 49], which breaks Discrete Scale
Invariance and results in coefficients $b_{2}^{(j)}(\theta)$ which are not
necessarily the same for all states. The new $3\mathrm{B}$ interaction
strength can in fact be determined by fitting a second $3\mathrm{B}$ level at
(quasi-)unitarity. Not surprisingly, binding energies in Resummed-Range EFT
become increasingly different from those at unitarity in Short-Range EFT as we
move towards the ground state of the quasi-unitarity spectrum. Approximating
$\kappa_{3\mathrm{QU}}^{(j)}$ by the
$\left(\kappa_{\mathrm{Ef}}^{(j)}(\pi/4)\right)^{2}$ term, we find from eq.
(3.13) and table 1 for example for the ground state $b_{2}^{(0)}(\pi/4)\approx
0.1$, which is small but not totally unnatural. With $b_{2}^{(j\geq
1)}(\pi/4)$ of similar magnitudes, the quadratic correction in eq. (3.13)
quickly disappears in the numerical noise. The argument above does not require
$b_{1}(\theta\neq\pi/4)$ to vanish, but even so corrections to the Efimov
tower quickly become very small as we go up the spectrum.

Indeed, we checked directly that the trajectory of the $3$rd excitation is
practically identical to the trajectory of an Efimov state whose $3\mathrm{B}$
interaction $H$ (2.16) is determined to reproduce the same binding energy at
(quasi-)unitarity, $\gamma=\xi=0$. The small differences of $<0.4\%$ we find
are within numerical and extrapolation uncertainties, and are dwarfed by the
thickness of the lines in the plots. Therefore, the blue lines labelled “$3$rd
excited ($j=3$)” in figs. 4 to 6 can safely be identified with the Efimov
trajectory. In particular, the difference between rescaled trajectories in
fig. 4 is indeed a difference between each trajectory and the Efimov
trajectory; and the parametrisation of the $3$rd excitation in table 2 is for
the Efimov trajectory, within the uncertainties induced by the strong
correlations between parameters.

As we go down the spectrum, the deviations from Discrete Scale Invariance
become more noticeable. One way to gauge the departure from the Efimov
spectrum is through

$r^{(j)}(\theta):=\frac{e^{j\pi/s_{0}}}{e^{3\pi/s_{0}}\kappa_{3}^{(3)}(\xi)}\left[\frac{e^{j\pi/s_{0}}\kappa_{3}^{(j)}(\xi)}{e^{3\pi/s_{0}}\kappa_{3}^{(3)}(\xi)}-1\right]=b_{1}(\theta)+b_{2}^{(j)}(\theta)\,\kappa_{3}^{(j)}(\xi(\theta))+\ldots\;\;,$
(3.17)

which vanishes for $j=3$. This ratio measures the relative deviation between
levels when all are rescaled to the ground state, in units of the binding
momentum of the $3$rd state, rescaled to the $j$th state. It is plotted in
fig. 8 for the various states we consider. Towards zero binding, $\theta\to
3\pi/4$, the binding momentum
$\kappa_{3}^{(j)}(\kappa_{2}^{-\,{(j)}}\searrow\kappa_{2\mathrm{ZB}}^{-\>{(j)}})\to
0$ of each state becomes increasingly small. That makes its subtraction from
the (rescaled) $3$rd excitation in eq. (3.17) numerically increasingly
unreliable. Therefore and in lieu of future studies of the details of
$r^{(j)}(\theta)$ [40], we refrain from showing results for
$\theta>7\pi/12=105^{\circ}$.

Following eq. (3.13), the linear correction in the effective range is
approximately given by this ratio for the second excited state,

$b_{1}(\theta)\simeq r^{(2)}(\theta)\;\;.$ (3.18)

It decreases near-monotonically from zero binding to threshold, passing as
expected through 0 at $\theta=\pi/4$. The first excited state has sizeable
corrections from the quadratic term,

$r^{(1)}(\theta)-b_{1}(\theta)\simeq
b_{2}^{(1)}(\theta)\;e^{2\pi/s_{0}}\;\kappa_{3}^{(3)}(\xi(\theta))\;\;,$
(3.19)

where $e^{2\pi/s_{0}}\;\kappa_{3}^{(3)}(\xi)$ reaches 0.15 at threshold, using
the numbers from table 1. As can be seen from fig. 8, the quadratic
coefficient $b_{2}^{(1)}(\theta)$ changes sign at $\theta\approx\pi/4$ and
always opposes the linear correction. While the expansion (3.13) is expected
to apply for the excited states, ground-state corrections can be large,

$r^{(0)}(\theta)-b_{1}(\theta)=b_{2}^{(0)}(\theta)\;e^{3\pi/s_{0}}\;\kappa_{3}^{(3)}(\xi(\theta))+\ldots\;\;,$
(3.20)

since $e^{3\pi/s_{0}}\;\kappa_{3}^{(3)}(\xi<0)\hskip
1.99997pt\raisebox{2.15277pt}{$>$}\raisebox{-3.00003pt}{$\sim$}\hskip
1.99997pt0.23$. In fact, we see from fig. 3 that this expansion should not
converge once $\kappa_{2}^{-}\gtrsim\sqrt{2}-1\approx 0.4$ ($\xi\hskip
1.99997pt\raisebox{2.15277pt}{$<$}\raisebox{-3.00003pt}{$\sim$}\hskip
1.99997pt-1$), where $\kappa_{3}^{(0)}\hskip
1.99997pt\raisebox{2.15277pt}{$>$}\raisebox{-3.00003pt}{$\sim$}\hskip
1.99997pt1$. The effects of terms quadratic and higher in $r_{0}$ have the
same qualitative behaviour as in the first excitation, but are larger.

Figure 8: (Colour on-line) The ratio $r^{(j)}(\theta)$, eq. (3.17), in the
region where the numerics is considered stable for all states.

Since $b_{1}(\theta)$ grows in magnitude away from $\theta=\pi/4$, we expect
the largest corrections at the two other points of particular interest, namely
zero-binding and threshold. This is indeed revealed by the larger deviations
between Resummed-Range and Short-Range EFTs there. These deviations are, as
one would expect, in the opposite direction to those found for (small)
$r_{0}>0$ in Short-Range EFT [51]. The deviations at these end points can be
connected through universal numbers. Equation (3.13) can be applied directly
at threshold ($\theta=0$), where $\kappa_{3}^{(j)}=\kappa_{\rm thr}^{(j)}$, by
setting $\kappa_{\mathrm{Ef}}^{(j)}(0)=\gamma_{\rm thr}^{(j)}$ to the
$2\mathrm{B}$ binding momentum where the $j$th $3\mathrm{B}$ excited state
becomes stable in LO Short-Range EFT. At zero binding we can introduce a
similar expansion,

$\kappa_{2{\rm ZB}}^{-(j)}=\gamma_{\rm
ZB}^{(j)}\left[1+b_{1}\left(\frac{3\pi}{4}\right)\;\gamma_{\rm
ZB}^{(j)}+\ldots\right]\;\;,$ (3.21)

where $\gamma_{\rm ZB}^{(j)}$ is the (negative) $2\mathrm{B}$ binding momentum
at which the $j$th $3\mathrm{B}$ excited state disappears in LO Short-Range
EFT. The shifts at the two end points relative to the unitarity limit are then

$\frac{\kappa_{2{\rm ZB}}^{-(j)}}{\gamma_{\rm
ZB}^{(j)}}-1=\frac{b_{1}(\frac{3\pi}{4})}{b_{1}(0)}\;\frac{\gamma_{\rm
ZB}^{(j)}}{\gamma_{\rm thr}^{(j)}}\left(\frac{\kappa_{\rm
thr}^{(j)}}{\gamma_{\rm thr}^{(j)}}-1\right)+\ldots\;\;.$ (3.22)

A similar relation was obtained in ref. [48].

After this first exploration of the bound spectrum, of its discrete scaling
properties, of the trajectories of each state as $\xi=2r_{0}/a$ is varied, and
of the emergence of the Efimov tower in Resummed-Range EFT, we now turn to
scattering.

##  4 Some Scattering Results

The differences in bound states compared to Short-Range EFT prompts us to look
at the scattering of one boson on the bound state of the other two which
exists for $\kappa_{2}^{-}>0$ ($\xi<0$). A detailed exploration of scattering
is left for a future publication [53]; here, we highlight some of its rich
structure and concentrate on the transition to Short-Range EFT.

The scattering equation eq. (2.10) is solved using the Hetherington-Schick
method on a contour in the fourth quadrant of the complex $P,Q$ plane, inside
the wedge $0\geq\mathrm{Im}[P,Q]\geq-2\kappa_{2}^{-}\;\mathrm{Re}[P,Q]/K$
where the amplitude is known to be analytic [54, 55, 56, 57, 58]. Its
advantage is that it readily applies to momenta $K$ above the breakup
threshold

$K_{\mathrm{thr}}:=\sqrt{\frac{4}{3}}\;\kappa_{2}^{-}$ (4.1)

where phase shifts and $K\cot\delta$ become complex. The underlying code has
already been used in refs. [59, 60, 52, 61, 62]. We use up to $n=1300$ points
with an exponential mapping which weighs more both small $P,Q$ and the region
around $K$. The results are stable against using different grids, mappings and
contours, although arcs of circles starting at $(P,Q)=0$ and ending at cutoff
$\Lambda$ appear to provide the most stability. We chose to show results for
one representative “high” cutoff of $\Lambda=45.2548\dots|r_{0}^{-1}|$;
results with half this value are nearly indistinguishable even at the high end
of the $K\leq 11.3137\dots$ we report on, that is, up to a quarter of the
cutoff. More details at low $K$ are available from plots of $K\cot\delta$
since phase shifts necessarily go to zero (modulo $\pi$) for $K\to 0$. In
particular, one recovers the effective-range expansion,

$K\cot\delta(K\to
0)=-\frac{1}{a_{3}}+\frac{r_{3}}{2}\;K^{2}+\mathcal{O}(K^{4})\;\;,$ (4.2)

where $a_{3}$ and $r_{3}$ are the scattering length and effective range for
$\mathrm{B}(\mathrm{B}\mathrm{B})$ scattering, measured yet again in units
$|r_{0}|$ of the effective range of the $2\mathrm{B}$ system.

To explore similarities and differences with Short-Range EFT, it is
instructive to consider more quantitatively the $\kappa_{2}^{-}\to 0$ limit of
the propagator of the integral equation (2.11), as we did for bound states but
now using $\mu^{2}=3K^{2}/4-(\kappa_{2}^{-})^{2}$. The integral equation is
dominated by momenta $Q$ for which its $2\mathrm{B}$ propagator (remember the
relation $\xi=-\kappa_{2}^{-}(\kappa_{2}^{-}+2)$)

$\frac{1}{\xi+\frac{3}{4}(Q^{2}-K^{2})+(\kappa_{2}^{-})^{2}+2\sqrt{\frac{3}{4}(Q^{2}-K^{2})+(\kappa_{2}^{-})^{2}}}$
(4.3)

is large, namely only for $Q\sim K$. It is this integrable singularity of the
$3\mathrm{B}$ amplitude at $Q=K$ on the real axis which makes the complex
contour of the Hetherington-Schick method so attractive. Similarly to the
bound-state case, we expect the range contribution to be relatively small in
the limit $\kappa_{2}^{-}\to 0$ when $3(Q^{2}-K^{2})\hskip
1.99997pt\raisebox{2.15277pt}{$<$}\raisebox{-3.00003pt}{$\sim$}\hskip
1.99997pt16$, which translates into $Q\sim K\hskip
1.99997pt\raisebox{2.15277pt}{$<$}\raisebox{-3.00003pt}{$\sim$}\hskip
1.99997pt2\sim|\kappa_{2}^{+}|$. As we obtained before, the deeper bound state
sets the momentum scale for the transition from Resummed-Range to Short-Range
EFT.

Figure 9 shows the results when the $2\mathrm{B}$ binding is around the
threshold point, namely where the $3\mathrm{B}$ ground state emerges from the
continuum ($\kappa_{2}^{-}\sim\kappa_{\mathrm{thr}}^{(0)}\approx 2$). In the
sub-threshold region close to the threshold,
$\kappa_{2}^{-}>\kappa_{\mathrm{thr}}$, the scattering length of the
$\mathrm{B}(\mathrm{B}\mathrm{B})$ system is negative, $K\cot\delta(K\to
0)>0$, and $\delta(K\to 0)\to 0$, as expected for a system without a
$3\mathrm{B}$ bound state. Conversely, in the all-bound region close to the
first threshold, the $\mathrm{B}(\mathrm{B}\mathrm{B})$ scattering length is
positive, $K\cot\delta(K\to 0)<0$, and $\delta(K\to 0)\to\pi$, indicating the
presence of a bound $3\mathrm{B}$ state. Both real and imaginary parts of
$K\cot\delta$ have monotonic behaviours in this momentum region, but the
results in Short-Range EFT diverge sharply from those of Resummed-Range EFT
for $K\gtrsim 2$, as expected.

Figure 9: (Colour on-line) Scattering of one boson from a bound two-boson
system with resummed effective range for parameter sets around the threshold
at which the $3\mathrm{B}$ ground state emerges from the continuum
($\kappa_{\mathrm{thr}}^{(0)}=2.11\ldots$,
$\xi_{\mathrm{thr}}^{(0)}=-8.72\ldots$). Left: $K\cot\delta$ at low momenta
$K$. Right: phase shifts for momenta $K$ up to $\Lambda/4$. Top: real parts.
Bottom: imaginary parts, nonzero only above the
$\mathrm{B}(\mathrm{B}\mathrm{B})\to\mathrm{B}\mathrm{B}\mathrm{B}$
disintegration threshold, whose position $K_{\mathrm{thr}}$ at a given
$\kappa_{2}^{-}$ is denoted by colour-coded crosses and vertical lines. Solid
lines: Resumed-Range EFT. Dashed: Short-Range EFT.

As $\kappa_{2}^{-}$ decreases, so does the break-up threshold, and non-
monotonicity appears in $K\cot\delta$ at smaller momenta — momenta well inside
the region where Short-Range EFT is expected to hold. In fig. 10, we address
binding momenta $\kappa_{2}^{-}\in\\{0.5;0.4\\}$ in the all-bound region
between the threshold for the bound ground state
($\kappa_{\mathrm{thr}}^{(0)}\approx 2.1$) and for the first excitation
($\kappa_{\mathrm{thr}}^{(1)}\approx 0.1$), as well as
$\kappa_{2}^{-}\in\\{\frac{1}{30};\frac{1}{40}\\}$ between that for the first
and second excitation ($\kappa_{\mathrm{thr}}^{(2)}\approx\frac{1}{160}$). In
both cases, discrepancies to Short-Range EFT become prominent at a somewhat
lower scale $K\approx 1$. It is particularly remarkable that both capture the
“hump” around $K\approx 0.25$ for
$\kappa_{2}^{-}\in\\{\frac{1}{30};\frac{1}{40}\\}$.

Figure 10: (Colour on-line) Real (top) and imaginary (bottom) part of
$K\cot\delta$ at low momenta $K$ for small $\kappa_{2}^{-}(\xi)$ but not close
to thresholds $\kappa_{\mathrm{thr}}^{(0)}\approx 2.1$,
$\kappa_{\mathrm{thr}}^{(1)}\approx 0.12$ or
$\kappa_{\mathrm{thr}}^{(2)}\approx 0.006$. Colour coding, dashing, crosses
and vertical lines as in fig. 9.

Finally, fig. 11 scans phase shifts and low-momentum $K\cot\delta$ for generic
values of $\kappa_{2}^{-}$ from well above $\kappa_{\mathrm{thr}}^{(0)}$ for
the ground state to before the emergence of the first excited state (top
figure), and from there to just above the second excitation (lower figure).
For large $\kappa_{2}^{-}$, phase shifts are small, $a_{3}<0$ is small, and
the amplitude could be treated with the Born approximation. That conforms with
the estimate that for $\kappa_{2}^{-}\gg K$, $Q^{2}-K^{2}$ can be treated
around $\delta(K\to 0)=0$ as perturbation in eq. (4.3). As the first threshold
$\kappa_{\mathrm{thr}}^{(0)}\approx 2.1$ is crossed, $K\cot\delta(K\to 0)$ and
the scattering length change sign, and $\delta(K\to 0)\to\pi$, as discussed
above. As the breakup threshold lowers, imaginary parts become much bigger.

As $\kappa_{2}^{-}$ approaches from above the threshold
$\kappa_{\mathrm{thr}}^{(1)}\simeq 0.125$ for the emergence of the first
$3\mathrm{B}$ excited state, interesting structures appear in the real and
imaginary parts of the phase shifts. They are all stable against variation in
the number of points, complex contour, exponential mapping and cutoff
variation. The “shoulder” in $\mathrm{Re}[\delta]$ for $\kappa_{2}^{-}=1$
around $K\approx 3$ becomes a prominent plateau as $\kappa_{2}^{-}\to 0.5$
decreases even further. Between $0.5$ and $0.4$, namely at about $0.43$ which
is close to the logarithmic mean $\kappa_{2}^{-}\approx 0.46$ of
$\kappa_{\mathrm{thr}}^{(0)}$ and $\kappa_{\mathrm{thr}}^{(1)}$, the plateau
turns into a valley: For smaller $\kappa_{2}^{-}$, $\mathrm{Re}[\delta]$ does
not approach $0$ as $K\to\infty$, but $\pi$. Remarkably, this elongated flat
structure from $K\approx 1$ up to about $4$ has $\mathrm{Re}[\delta]\simeq
80^{\circ}$. It is thus very close to saturating the unitarity bound
($\delta=90^{\circ}$), albeit all of the subsystems have nonzero binding
energies. The plateau (valley) in $\mathrm{Re}[\delta]$ is followed by a sharp
decrease (increase), which is correlated with a sharp peak in
$\mathrm{Im}[\delta]$. With decreasing $\kappa_{2}^{-}$, its height increases,
correlated with an increasing maximum slope and wanders to larger $K$. Large
$\mathrm{Im}[\delta]$ corresponds to a small elasticity parameter $\eta$, thus
we are witnessing a trough in $\eta$. This is a marker for a resonance in a
multi-channel system, as an $S$-matrix pole at complex energy $E_{\rm
R}-i\Gamma_{\rm R}/2$ leads to a minimum of $\eta$ at centre-of-mass energy
$E_{R}$ with width determined by $\Gamma_{\rm R}$ [63]. However, since the
structure cannot adequately be captured by a Breit-Wigner form with constant
background as suggested in ref. [64] and we do at present not locate a complex
pole directly, we leave it for future work to confirm the presence of
resonances, map their trajectory in the complex $K$ plane, and relate them
more directly to the emergence of excited states [53].

Figure 11: (Colour on-line) Evolution of $K\cot\delta$ at low momenta (left)
and of the phase shift $\delta$ up to $K=\Lambda/4$ (right) for
$\kappa_{2}^{-}\in[15;1]$ ($\xi\in[-255;-3])$ (top) and
$\kappa_{2}^{-}\in[1;0.02]$ ($\xi\in[-3;-0.\overline{04}])$ (bottom). The red
solid line ($\kappa_{2}^{-}=1$, $\xi=-3$) is shown on both plots for
orientation. Colour-coded vertical lines and crosses denote the respective
breakup threshold.

Around $\kappa_{2}^{-}\approx 0.2$ close to the emergence of the first
excitation from the continuum, the scattering length changes sign so that
again $\delta(K\to 0)\to 0$ for even smaller $\kappa_{2}^{-}$. At the same
time, the phase shifts drop dramatically from $\delta(0)=\pi$ to $0$ at very
small $K$. That implies that the effective range of
$\mathrm{B}(\mathrm{B}\mathrm{B})$ scattering becomes very large, as also seen
from the large curvature in $K\cot\delta(K\to 0)$. In fact,
$\mathrm{Re}[K\cot\delta]$ blows up just before threshold, indicating a zero
of the scattering amplitude. Note that the phase shift approaches the
asymptotic free-particle limit $\delta\to 0$ or $\pi$ much more rapidly for
large $K$ as $\kappa_{\mathrm{thr}}\searrow 0$ is decreased. Indeed, for
$Q^{2},K^{2}\gg Q,K\gtrsim\kappa_{2}^{-}$, eq. (4.3) reduces at leading order
to a quasi-Coulombic form $4/[3(Q^{2}-K^{2})]$ in momentum space. This
suggests that, like in electromagnetism, phase shifts rapidly become
perturbative as $K\to\infty$.

At long last, it is also worth noticing that while the $2\mathrm{B}$ effective
range is by definition always negative to provide a well-defined EFT, the
effective range $r_{3}$ of $\mathrm{B}(\mathrm{B}\mathrm{B})$ scattering, eq.
(4.2), is always positive. We did not find any $\xi$ for which it is negative.

##  5 Summary and Conclusions

We studied the three-boson bound states and scattering in an Effective Field
Theory with two shallow two-body $S$-wave poles [17, 24, 18] at leading order.
At this order, properties of few-body systems are determined by two
parameters, say the two-body effective range $r_{0}$ and the dimensionless
ratio $\xi=2r_{0}/a$, where $a$ is the $2\mathrm{B}$ scattering length. The
emerging Resummed-Range EFT is self-consistent and does not need a three-body
interaction at leading order to be renormalisable.

In this EFT, the effective range provides a lower bound for the spectrum, and
three-body bound states are found only in the range
$\xi_{\mathrm{thr}}\simeq-8.726<\xi<\xi_{\mathrm{ZB}}\simeq 0.367$ for the
ground state. We see no indications for bound three-body systems with values
$\xi$ outside that narrow window. In particular, this is still far away from
the point $\xi=1$ where the two virtual two-body states collide to form a
resonance.

At first glance, the spectrum appears quite close to a truncated Efimov
spectrum, but noticeable differences exist for the low-lying states; see e.g.
table 1 and fig. 4. In particular, the ground state’s threshold point (where
$3\mathrm{B}$ and $2\mathrm{B}$ states have the same binding energy) is with
$\kappa_{\mathrm{thr}}^{(0)}\approx 2.12\;|r_{0}^{-1}|$ about $30\%$ smaller
than in the Efimov case, $\kappa_{\mathrm{thr}}^{(j\to\infty)}\approx
3.31\;|r_{0}^{-1}|$, and the difference is still about $15\%$ for the first
excitation. Conversely, the zero binding point of the ground state is about
$30\%$ further away than that of an Efimov state, and that of the first
excitation, about $10\%$. Thus, the ground and lower excited states become
unstable against two-body decay considerably earlier than in the Efimov
version, while they survive considerably longer as bound states when the
Efimov state is already virtual. Overall, the parametrisation in sect. 3.4
shows that trajectories of the ground and first few excited states are
considerably shorter in $\kappa_{2}^{-}$ than that of a corresponding Efimov
state. Furthermore, the three-body ground-state momentum at quasi-unitarity
(vanishing shallowest two-body binding momentum) is only $\simeq
0.2\;|r_{0}^{-1}|$. It becomes comparable to or larger than $|r_{0}^{-1}|$
only for $\xi\hskip
1.99997pt\raisebox{2.15277pt}{$<$}\raisebox{-3.00003pt}{$\sim$}\hskip
1.99997pt-3$ ($\kappa_{2}^{-}\gtrsim|r_{0}^{-1}|$). Excited states, when they
exist, all have momenta $\hskip
1.99997pt\raisebox{2.15277pt}{$<$}\raisebox{-3.00003pt}{$\sim$}\hskip
1.99997pt0.1\;|r_{0}^{-1}|$.

Consequently, the ground and first excited state are close to threshold not
well described with Short-Range EFT [13] (with a single low-energy two-body
state). The higher spectrum is more amenable, if one is only interested in
accuracies on the level of a few per-cent. Even for the excited spectrum,
there is a remarkable difference between Resummed-Range and Short-Range EFT.
In the latter, the relative position of the levels is fixed by Discrete Scale
Invariance, while the absolute position requires a three-body parameter since
no two-body scale is left. In contradistinction, in Resummed-Range EFT, this
parameter is determined by the effective range, or, equivalently, by the
binding momentum $\kappa_{2}^{+}(\xi=0)=-2\;|r_{0}^{-1}|$ of the second,
“deep” bound two-body state even when $\kappa_{2}^{-}(\xi=0)=0$. This natural
two-body scale remains, in contrast to unitarity in Short-Range EFT which has
no two-body scale at all. This is the most striking difference between many-
body systems in the two versions.

A main finding of our investigation is therefore: If one interprets Short-
Range EFT as low-energy version of Resummed-Range EFT, then absolute position
of the Efimov tower is fixed, and the corresponding Efimov scale is determined
as $\Lambda_{\ast}=0.610206(1)\;|r_{0}^{-1}|$ (in a renormalisation scheme
with “hard” cutoff regularisation); see eq. (3.14). It will be very
interesting to put this prediction to the test in systems where Resummed-Range
EFT applies.

Larger differences with Short-Range EFT appear in scattering at momenta
comparable with $|r_{0}^{-1}|$. This leads to quite interesting structures in
the phase shifts whose predictions could also be explored in physical systems.
There are, for example, plateaus for certain ranges of $\xi\approx 0.5$ where
the real part of the phase shift is near-independent of the scattering
momentum $K$ and close to the unitarity limit $\delta\approx 90^{\circ}$
albeit none of the subsystems are scale invariant. These plateaus are followed
by sharp cliffs or walls, which are correlated with maxima of the imaginary
part of the phase shifts — a potential sign of a resonance. Remarkably, we did
not find a parameter range for which the three-body effective range is
negative, while the two-body one always must be to obey the Wigner bound.

Indeed, we expect that the size of deviations of Resummed-Range EFT from the
Short-Range EFT, and hence from Discrete Scale Invariance, grows larger in the
spectrum of more-boson systems. Its upper part should still display Discrete
Scale Invariance, for example the four-boson system should have two states
associated with each Efimov state [65]. But that is not necessarily the case
for the lower part, and in particular for the ground state. At unitarity in
Short-Range EFT, the ground-state binding energy per particle grows nearly
linearly with the number of bosons [66, 67], before tapering off according to
liquid-drop-type saturation [67]. It is likely that ground states will have
momenta comparable to or larger than $|r_{0}^{-1}|$. At unitarity, the four-
boson ground state, for example, is already more bound than the three-boson
ground state by a factor of about $4.6$ [65]. At quasi-unitarity, the factor
will be different but, if it is not very different, the four-boson ground-
state binding momentum should still be close to $|r_{0}^{-1}|$. Moreover, the
evolution of the binding energies per particle as function of particle number
will now depend on $\xi$, in such a way that $\xi\to 0$ should approach the
unitarity results of ref. [67]. Further work to clarify the effects of
$r_{0}<0$ and large $|a|$ on the spectrum of more-boson systems would be of
high interest.

Obvious further extensions include systems of three identical fermions; bosons
with different masses; more than three particles; the study of higher-than-
leading order effects, namely the inclusion of range parameters like the shape
parameter in perturbation; a more detailed study of scattering states
including when none of the two-body systems is bound; a study of the virtual
and resonance states of the three-body system, especially for $\xi$
($\kappa_{2}^{-}$) outside the range which allows for bound states (including
two-body resonances); and of course the confrontation of our predictions with
data of suitable systems with large but negative effective two-body effective
range and large-in-magnitude scattering length in Atomic, Nuclear and Particle
Physics.

## Acknowledgements

We are particularly indebted to Lorenzo Contessi, Alireza Dehghani, Hans-
Werner Hammer, Johannes Kirscher, Sebastian König, Manolo Pavón Valderrama,
and Daniel R. Phillips for discussions. Johann Haidenbauer and Christoph
Hanhart provided context on physical systems with potentially large and
negative effective range. Instrumental for this research were the warm
hospitality and financial support for HWG’s stays at IJCLab Orsay, UvK’s stay
at George Washington University, and for both stays at the Kavli Institute for
Theoretical Physics which is supported in part by the National Science
Foundation under Grant No. NSF PHY-1748958. This material is based upon work
supported in part by the U.S. Department of Energy, Office of Science, Office
of Nuclear Physics, under awards DE-SC0015393 (HWG) and DE-FG02-04ER41338
(UvK).

## Data Availability Statement

All data underlying this work are available in full upon request from the
authors.

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