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{\sl Introduction}.
When a shallow state appears in a two-body system a continuous scale invariance
can be observed; many observables scale with the energy of this state.
Refering to $s$-waves, the system is characterized by the following $S$-matrix
\begin{equation}
        S(k)=\frac{k+i/a_B}{k-i/a_B} \frac{k+i/r_B}{k-i/r_B}
	\label{smatrix}
\end{equation}
where $a_B$ is the energy length of the two-body state,
$E_2=-\hbar^2/m\,a_B^2$. This state could be a bound state ($a_B>0$) or a virtual state ($a_B<0$).
The second pole, $i/r_B$, sometimes called redundant pole, assures the correct limit
$S(k\rightarrow\infty)=1$. It is given in terms of the positive length
$r_B=a-a_B$ where $a$ is the scattering length. The above $S$-matrix is equivalent
to the second order effective range expansion of the $s$-wave phase-shift
\begin{equation}
        k\cot\delta=-\frac{1}{a} + \frac{1}{2}{r_e}{k^2}
	\label{ERE}
\end{equation}
where $r_e$ is the effective range.
Extending the above equation to the complex plane, $k=i\kappa$, and recalling that the pole
of the $S$-matrix
is defined by the relation $k\cot\delta=ik$, the pole equation is obtained
\begin{equation}
        \kappa=\frac{1}{a} + \frac{1}{2}{r_e}{\kappa^2}
	\label{poleeq}
\end{equation}
with the two solutions
\begin{eqnarray}
	&\kappa_1=\frac{1}{a_B}=\frac{1}{r_e}(1-\sqrt{1-2r_e/a}) \\
	&\kappa_2=\frac{1}{r_B}=\frac{1}{r_e}(1+\sqrt{1-2r_e/a})
\end{eqnarray}
It should be noted that Eq.(\ref{ERE}) gives the low-energy dependence of the phase-shift
whereas the validity of Eq.(\ref{poleeq}) up to second order defines the shallow
characteristic of the state and it is represented by the $S$-matrix of Eq.(\ref{smatrix}).
In the fourties V. Bargmann studied the
relation between the phase-shifts and the scattering potential~\cite{bargmann}.
In particular the above $S$-matrix is exactly obtained by the Eckart potential~\cite{eckart}
\begin{equation}
        V_{Eck}(\beta,R,r)=-2\frac{\hbar^2}{mR^2}\frac{\beta e^{-r/R}}{(1+\beta e^{-r/R})^2}
\end{equation}
with a simple relation between the low energy parameters and the potential parameters
\begin{equation}
                \left\{
        \begin{array}{l}
                a=4R\frac{\beta}{\beta-1}  \\
                \\
                a_B=2R\frac{\beta+1}{\beta-1}
        \end{array}
                 \right. \hspace{1cm}\left\{
        \begin{array}{l}
                r_e=2R\frac{\beta+1}{\beta} \\
                \\
                r_B=2R
        \end{array}
                 \right.
                 \label{eckartp}
\end{equation}

The above relations make evident the continuous scale invariance: the scattering length,
the energy length and the effective range are all proportional to $r_B$.
This length represents the resolution scale at which the system can be observed.
In fact the specific form of the $S$-matrix given in Eq.(\ref{smatrix}),
or equivalently the effective range expansion
up to second order, fixes the capability of resolving the short-range physics. The picture that
emerges from this simple description is the following: when the scattering parameters, $a$ and
$r_e$, take values in the region $r_e/a<0.5$ the $S$-matrix is well represented by the form
given in Eq.(\ref{smatrix}) implying the presence of a shallow state.
This region defines the universal window; physical systems inside this window have properties
largely independent of the interaction details.

In Fig.1 a selected number of two-body systems are shown. They are
located inside the window by the values of the corresponding low energy parameters
taken from experiments or given by the respective interaction potentials.
In the figure the deuteron and the
singlet $^1S_0$ $np$ and $nn$ systems appear inside the universal window as well as the
atomic $^4$He and $^3$He dimers. The position of these systems
have been obtained calculating the low-energy parameters from realistic potentials widely used
in the literature, the Argone $v_{18}$ $NN$ interaction (AV18) and
the LM2M2 He-He interaction~\cite{AV18,LM2M2}.
The results, indicated by the solid circles, are well described by the two-pole $S$-matrix.
From the plot,
it is possible to see how far or how close a system is with respect to the unitarity point,
defined by $1/a_B=1/a=0$.


{\sl The two-body scale}.
The low-energy parameters used to place the systems on the plot can be used
to determine the Eckart parameters, $\beta$, $R$ constructing in this way a low
energy representation of the interaction. Once the potential is determined, it
is possible to vary its strength $\beta$, maintaining its range $R=r_B/2$ constant
in such a way that the system is moved from its
physical point along the curve, eventually arriving at the unitarity point ($\beta=1$).
This is a well defined path that continuously link the physical point to the unitarity point.
It is defined in terms of the two-body scale, or resolution scale, $r_B=2R$ which remains
constant along the path. In the deuteron case, $r_B=1.19\,$fm
whereas for the $^4$He dimer it results $r_B=7.2\, a_0$. Examples of trajectories
inside the universal window are given in Ref.~\cite{kievsky2021}.

\begin{figure}
	\includegraphics[width=\textwidth]{eckart2.pdf}
	\caption{The two poles of the $S$-matrix, $\kappa_{i}$ with $i=1,2$,
	are shown as a function of the inverse
	of the scattering length, both in units of the effective range $r_e$. Physical systems
	are located on the curves as explained on the text.}
\end{figure}

Though the Eckart potential can be directly parametrized from the values of the
low-energy parametres, other potentials can be used to represent the
universal window. The two-parameter gaussian potential
\begin{equation}
	V_G(\beta_G,R_G,r)=-\frac{\hbar^2}{mR_G^2}\beta_G e^{-(r/R_G)^2}
\end{equation}
is widely used in this context and low-energy parametrizations of the nuclear and atomic
potentials have been produced many times in the literature~\cite{kievsky2017,gatto2019}.
However in
the gaussian case there is no analytical expression between the scattering and potential
parameters, they have to be calculated numerically. At the unitarity point the relation
is $r_e/R_G=1.43522$ and $\beta_G=2.684$~\cite{raquel}. Using the deuteron values for the binding
energy, $E_d=2.224\,$MeV, and triplet scattering length, ${}^3a=5.419\,$fm, to determine the
gaussian parameters $\beta_G=3.898,R_G=1.56\,$fm, it is possible to define a path to the unitarity
point by varying $\beta_G$ with $R_G$ constant.
Accordingly at the unitarity point the effective range should be
$r_e=1.43522R_G$, resulting for the deuteron in that limit $r_e=2.24\,$fm.

The gaussian path is a good representation of a link between the physical point and the
unitarity point when the interaction potential is multiplied by a global factor.
This can be verified by scaling the AV18 in the spin-isospin channel $S,T=1,0$
making the deuteron binding energy almost zero. This is achieved by multiplying the potential
in that channel
by the factor $\lambda=0.8$. After this operation the deuteron binding energy is almost zero, the
triplet scattering length diverges and the effective range is $r_e=2.35\,$ fm, which is
slightly larger than the values predicted by the gaussian path. However,
this very small difference indicates
that multiplying the interaction by a global factor, the physical system moves from its physical
point towards the unitarity point almost following the gaussian path. We recall that the gaussian
path is defined by varying the gaussian strength maintaining constant its range determined by the value
of the low-energy parameters at the physical point. In the case of the helium dimer the gaussian
exponent at the physical point is $R_G=10.03\,a_0$ predicting
an effective range at unitarity of $14.4\,a_0$. Scaling the LM2M2 potential by a factor
$\lambda=0.974$, the dimer has zero energy and the effective range is $14.5\,a_0$,
extremely close to the predicted value. In summary, we have introduced a two-body scale,
defined in terms of the gaussian (or Eckart) range, that identifies a path
linking the physical point to the unitary point. We have shown that a physical system,
almost follows this path when its interaction is scaled by a global factor.

{\sl The nuclear three-body parameter}.
To make contact with the studies of Refs.~\cite{gatto2019,deltuva2020}
we use a gaussian potential to extend the analysis of the universal window
to the three-body system. Fig.2 shows the binding momentum of three, equal mass, bosons (upper panel)
and three, equal mass, spin-isospin 1/2 fermions (lower panel), calculated using a
generic gaussian potential with range $R_G$, as a function of the inverse of the energy length $a_B$,
both
in units of the gaussian range $R_G$. In the case of three fermions  we consider the
$s$-wave channels with spin-isospin $S,T=0,1$ and $1,0$. Accordingly the gaussian
interaction potential is
\begin{equation}
	V_G(r)=-\sum_{\mu=0,1}\frac{\hbar^2}{m(R_G^\mu)^2}\beta_G^\mu\, e^{-(r/R_G^\mu)^2}{\cal P}_\mu
\end{equation}
with $\mu=0,1$ the singlet and triplet channels respectively and ${\cal P}_\mu$ a projector on
those channels. We choose the potential to have the same ranges in both channels, $R_G=R_G^0=R_G^1$
and the corresponding strengths, $\beta_G^\mu$, are related to give the ratio
of the single and triplet scattering lengths ${}^1a/{}^3a=-4.38$, which is the ratio of the
nuclear singlet and triplet scattering lengths~\cite{gatto2019}.

These gaussian potentials are used to calculate the three-body spectrum.
The curves in Fig.2 show the solutions obtained using a generic
gaussian interaction independent of the
range used for the calculation. In particular, at the unitary point, the ground state binding
energy, $E_*^{(0)}$, depends on the gaussian range $R_G$ by the relation $R_G \kappa_*^{(0)}=0.4883$
and results
\begin{equation}
	E_*^{(0)}=\frac{\hbar^2}{m} [\kappa_*^{(0)}]^2=\frac{\hbar^2}{m}
	\frac{0.4883^2}{ [R_G^{(0)}]^2}
\end{equation}

\begin{figure}[h]
	\includegraphics[width=\textwidth]{threeb0.pdf}\\
	\includegraphics[width=\textwidth]{threef1.pdf}
	\caption{The binding momentum of three equal mass bosons (upper panel)
	and of three equal mass sipn-isospin 1/2 fermions (lower panel),
	calculated using a generic gaussian interaction with range $R_G$
	as a function of the two-body energy length, both in units of
	$R_G$. Notable points are given, as indicated on the text. The
	groubd state $^4$He trimer as well as the triton, $^3$H, are located
	on the corresponding curve.}
\end{figure}

On the figure the dimer and the first three trimer levels, for bosons, and the
deuteron and the first two three-nucleon levels are shown. The notable points
$R_G\kappa_*^{(n)}$ at the unitarity point are explicitely given and are the same
for boson and fermions. In fact, in that limit, the boson and fermion gaussian
potentials are the same. It should be noticed that the ratio $\kappa_*^{(n)}/\kappa_*^{(n+1)}$
tends to the zero-range factor of 22.7. For bosons, the scattering length values at which the
three-boson system dissociates in three atoms, $a_-^{(n)}$, are given in units of the gaussian range,
and also in this case, their ratio tends to the universal factor of 22.7.
Another interesting issue is how the excited states disappear into the 1+2 continuum. For bosons, the
ground and first excited states are bound along the window whereas the second excited state
disappears into the continuum as shown by the orange curve. In the case of fermions already the
first excited state disappears into the continuum at the point indicated by the vertical dashed line.
The $^4$He trimer and $^3$H ground states are shown as (blue) solid squares.
Their positions on the $x$-axis allow to determine the corresponding values of $R_G$.
By construction, this particular range is the range of a gaussian potential that, for a given
value of its strength, reproduces simultaneously the dimer and trimer energies in the case of
bosons, and the deuteron and $^3$H energies in the case of nucleons.
For the $^4$He trimer it results $R_G/a_B=0.061$ and, considering that the LM2M2 potential
predicts a dimer of $1.303\,$mK, the energy length is $a_B=182.2\,a_0$, hence the corresponding
gaussian range is $R_G=11.15\,a_0$.
In the $^3$H case it results $R_G/a_B=0.457$, the deuteron energy length is $4.318\,$fm and,
therefore $R_G=1.97\,$fm. Using the pure number at the unitarity limit,
$R_G\kappa_*^{(0)}=0.4883$, valid for both systems,
we can predict the Helium and nuclear three-body pameters, defined as the binding momentum
$\kappa_*^{(0)}$ at the unitarity point.
\begin{eqnarray}
	&{\rm \,\, Helium:\,\,\,}\kappa_*^{(0)}= 0.0438\, a_0^{-1}  \\
	&{\rm \,\, nuclear:\,\,\,}\kappa_*^{(0)}= 0.248\, {\rm fm}^{-1}
\end{eqnarray}
The three-body parameters can be used to predict the ground state binding energies at the unitarity limit.
In the case of the Helium trimer it results $E_*^{(0)}\approx 83\,$mK. In the
three-nucleon case it results $E_*^{(0)}\approx 2.55\,$MeV. The last step in this analysis is
to calculate
the binding energy at unitarity using the scaled LM2M2 and AV18 potentials. For the latter we
include the Urbana IX three-body force since the two-body potential alone is not sufficient to
describe the $^3$H binding energy. Making the calculation of both three-body systems at
unitarity we obtain $E_*^{(0)}=84.0\,$mK for the Helium trimer and
$E_*^{(0)}=2.65\,$MeV for the triton. Looking at these results, from one side we can conclude
that the gaussian characterization of the universal window introduces a link between the physical
and the unitarity points. Secondly, using the scaled potentials it would be possible to study
the physical systems in different locations inside the universal window. In particular it
would be possible to study the nuclear levels at the unitarity point to see the dominance of
the scale invariance.

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