The few scales of nuclei and nuclear matter
A. Delfino a, T. Frederico b, V. S. Timóteo c, and Lauro Tomio d
aInstituto de F́ısica, Universidade Federal Fluminense, 24210-900 Niterói, RJ,
Brasil
bDepartamento de F́ısica, Instituto Tecnológico de Aeronáutica, CTA, 12228-900,
São José dos Campos, Brasil
cCentro Superior de Educação Tecnológica, Universidade Estadual de Campinas,
13484-370, Limeira, SP, Brasil
dInstituto de F́ısica Teórica, Universidade Estadual Paulista, 01405-900, São
Paulo, Brasil
Abstract
The well known correlations of low-energy three and four-nucleon observables with a
typical three-nucleon scale (e.g. the Tjon line) is extended to light nuclei and nuclear
matter. Evidence for the scaling between light nuclei binding energies and the tri-
ton one are pointed out. We show that the saturation energy and density of nuclear
matter are correlated to the triton binding. From the available systematic nuclear
matter calculations, we verify the existence of bands representing these correlations.
PACS 21.45.+v, 21.65.+f, 21.30.Fe
Key words: Scaling, nonrelativistic few-body systems, nonrelativistic nuclear
matter
Two-nucleon interactions are typically constructed to fit scattering data and
deuteron properties. When such interactions are used to calculate three-nucleon
observables, the results exhibit some discrepancies [1]. Basically, they are ex-
plained as originated from different strengths of the two-nucleon tensor force
and short-range repulsions, provided that all realistic two-nucleon interactions
have the correct one-pion exchange tail. In four-nucleon bound state (4He )
calculations the discrepancies still remain. But, at least, they are correlated,
as seen in the binding energies of 4He (Bα) and triton (Bt), which lie on a
very narrow band [2], obtained when the short-range repulsion of the nucleon-
nucleon interaction is varied while two-nucleon informations (deuteron and
scattering) are kept fixed. This correlation is known as Tjon line [2]. Bα and
Bt follows an almost straight line in the range of about 1-2 MeV of variation
Preprint submitted to Elsevier 4 November 2018
http://arxiv.org/abs/0704.0481v1
of the triton binding energy around the experimental value. As the long-range
two-nucleon scales we have the deuteron binding energy (Bd) and the singlet
virtual-state energy (Bv).
Two-body short-ranged interactions, supporting very low two-body binding
energy and/or large scattering lengths, when used to calculate three-body sys-
tems, approach what we call the universal Thomas-Efimov limit [3]. By trying
to find the range r0 of the two-nucleon force, Thomas [4] showed that when
r0 → 0, while the two-body binding energy B2 is kept fixed, the three-body
binding energy goes to infinity (Thomas collapse). Much latter, Efimov [5]
showed that, in the limit B2 = 0(r0 6= 0) the number of three-body bound
states is infinite with an accumulation point at the common two- and three-
body threshold. Note that both the Thomas and Efimov effects are claimed to
be model independent, since they are due to a dynamically generated effective
three-body potential acting at distances outside the range of the two-body
potential. These apparently different effects are related to the same scaling
mechanism, as shown in Ref. [3]. In other words, the Thomas effect appears
when r0 is much smaller than the size of the two-body system (which is of the
order of the scattering length |a|), while the Efimov effect arises for |a| >> r0.
Therefore, what matters for both effects is the same condition: |a| >> r0 or
the ratio |a|/r0 >> 1. In terms of the two-body energies this is translated
m|B2|/h̄
r0 << 1 (m the boson mass). One would expect that the
Thomas-Efimov effect is manifested in weakly-bound quantum few-body sys-
tems which are much larger in size than the corresponding two-body effective
range. Notice that a zero binding energy for a free two-body system is not
known in nature. But, nowadays it was shown that, for trapped ultracold
gases of certain atomic species, it is possible to adjust the two-body scattering
length at very large values, using Feshbach resonance techniques, by tuning the
external magnetic field [6]. In this case, it is expected that the Thomas-Efimov
effect can be manifested [7].
The deuteron and triton may be viewed as low energy systems with large size
scales in which the range of the potential is smaller than the corresponding
healing distances of the wave functions, leading the nucleons to have a high
probability to be outside of the interaction range. Then, the low-energy prop-
erties of these systems can be studied with models that minimally includes the
physics of the Thomas-Efimov effect, as in the case of a few-body model with
renormalized pairwise s-wave zero-range force [8]. This approach shows that
all the low-energy properties of the three-body system are well defined in the
model, once one three-body scale and the two-body low-energy observables are
given. As a consequence, correlations between two three-body s−wave observ-
ables are expected to appear in model calculations with short-ranged interac-
tions. Along this line, some previous works (see [9] and references therein) have
studied weakly-bound halo states in exotic nuclei as well as possible Efimov
states for He-He-Alkali molecules.
The scaling of three-nucleon observables with the triton binding energy corre-
sponds to universal behaviors found when a three-body scale is varied. For ex-
ample, the Phillips plot [10] of the neutron-deuteron doublet scattering length,
as a function of the triton energy is nowadays one of the universal scalings
found in the three-nucleon system [11,12]. In general, it is observed for nu-
clear and molecular weakly bound three-body systems [13,14,15] the scaling
of observables with the three-body binding energy.
The Thomas-collapse of the three-body energy in systems of maximum wave-
function symmetry implies the existence of a three nucleon scale (identified
with the triton binding energy) governing the short-range behavior of the
wave-function. Four nucleons can also form a state of maximum symmetry,
allowing in principle the collapse of such configuration, independently of the
three-nucleon collapse [8]. This is under discussion and it is suggested in [16]
that the four-body scale is not independent of the three-body one. However,
in their work it was introduced a three-body force to stabilize the shallowest
three-body state, against the variation of the cut-off. The three-body interac-
tion can be attractive or repulsive and their conclusion lies on the repulsive
sector. We note that the attractive part indicates a possible independent be-
havior of the four body ground-state energy from the three-body one. Certainly
this point merits further discussions and so far, we think, it is still open the
possibility of a four-body scale. Anyway, as the nucleon-nucleon interaction
is strongly repulsive at short range and therefore the probability of four nu-
cleons to be simultaneously in a volume ∼ r30 is quite small, presumably the
four-nucleon scale itself has much less opportunity to be evidenced in realistic
nuclear models. Indeed, this is indicated by the existence of the Tjon line. Due
to that, as we will see later, the four-nucleon binding energy is eliminated in
favor of the triton binding energy. In this respect it is worthwhile to note that
Platter et al. extended the effective field theory framework applied to four-
bosons [16] to calculate the 4He binding energy by controlling the triton energy
through a repulsive effective three-nucleon force. Within their approach [17]
the Tjon line is reproduced.
In a nuclear scenario dominated by an interaction with a range smaller than
the nucleon-nucleon scattering lengths, and considering the triton and 4He
nuclear sizes yet larger than the force range, the picture of nuclei would be of
a many-body system with the wave-function being an eigenfunction of the free
Hamiltonian almost everywhere. The Pauli principle allows only up to four nu-
cleons at the same position, forbidding certain particular configurations with
overlap of more particles. If more than four particles are allowed to overlap,
it would imply that the asymptotic information from the interaction of the
cluster would go beyond of those already fixed by the low-energy observables
of two, three and four nucleons. By some unknown reason the parameters
of Quantum Chromodynamics are close to this limit. It was conjectured in
Ref. [18] that a small change in the light quark masses away from their physi-
cal values could put the deuteron and the singlet virtual state at zero binding
energy, and therefore the above idealized picture of the nuclear systems could
not be far from reality. It is quite amazing thinking that nuclear wave func-
tions could heal much beyond the interaction range. Therefore, the details of
the long wavelength structure of nuclei are given by the free Hamiltonian and
by few-nucleon scales, which determine the wave function at short distances.
The universal behavior of the scaling functions are due to that.
If one wonders about the neutron matter within a non-relativistic quantum
framework, in the limit of a zero-range force, we could say that the only scale
in this case is the neutron-neutron scattering length. Therefore, the binding
energy of neutron droplets will be strongly correlated to that quantity, which
is the only physical scale in this situation allowed by the Pauli principle. This
discussion has been performed in the context of three neutron systems [19].
Moreover, it was concluded in Ref. [20] that stable tetra-neutron droplets
would imply a major change in the neutron-neutron scattering length.
Another example of the dominance of only two-body scale appears in three-
boson systems in two dimensions, where the Thomas-Efimov effect is ab-
sent [3]. In this case, only two-body low energy scales are enough to define the
many-body properties in the limit of a zero-range interaction. The low-energy
properties of a many-body system of spin-zero particles in two dimensions
will be sensitive only to the two-boson binding energy. Even in the case where
bosons are trapped, since the essential singularity of the point-like configura-
tion is not affected by the confining force as the harmonic one.
For the sake of generality, we start with the observables Bd, Bv, Bt and Bα
as the scales determining the asymptotic properties of nuclei [9]. Then, in the
limit of a zero-range interaction, we write the binding energy of a nucleus with
mass number A and isospin projection Iz, considering isospin breaking effects,
B(A, Iz) = A Bt B (βv, βd, βα, A, Iz) , (1)
where βa = Ba/Bt with a = v, d and α.
According to the Tjon line, βα remains approximately constant for a variety
of two-nucleon potentials and the parametrization of the numerical results,
given in MeV, for several two-nucleon potentials is
Bα = 4.72 (Bt − 2.48) , (2)
which for B
t = 8.48 MeV gives B
α = 28.32 MeV. Using (2) in (1),
R(A, Iz) = B(A, Iz)/A = Bt R (Bt, A, Iz) , (3)
where in the scaling function R(A, Iz) the values of Bd and Bv are fixed to
the experimental values. The dependence of Bα with Bt for realistic nucleon-
nucleon potentials is given by Eq. (2).
Equation (3) generalizes the concept of the Tjon line to nuclei. Recent calcu-
lations using the AV18 nucleon-nucleon potential plus three-body forces [21]
show that there is a systematic improvement of the binding energy results
for He, Li, Be and B isotopes simultaneously with the triton binding energy,
when models are tuned to fit Bt. It is important to note that these AV18
calculations have at least two three-body parameters that are fitted to Bt and
nuclear matter saturation properties. Consequently, one could argue that such
calculations cannot provide evidence for one-parameter correlation. The fit-
ting to nuclear matter calculation presumably is not that important for light
nuclei in view of the dominance of the triton binding (or three-body correla-
tions) in the four-nucleon bound state as given by the Tjon line. Therefore,
it is reasonable to think that three-body correlations are quite important for
light-nuclei, since that even for the alpha particle where the nucleons are in a
very compact configuration this occurs. In our opinion, the fitting of nuclear
matter saturation properties has more to do with the approximations done
in nuclear matter calculations. The three-body potential should be somewhat
tuned, which is, probably, not so important for light nuclei once the triton
binding attains its physical value.
For nuclear matter properties calculation using a variety of two-nucleon poten-
tials, in which the tensor strength was varied but the deuteron binding energy
was kept fixed, it was shown that these interactions cannot quantitatively ac-
count for nuclear saturation [22,23]. Coester et al. [22] observed that, in an
energy versus density plot, the saturation points of nuclear matter obtained
by employing different realistic potentials are located along a band (Coester
band). Also, in a relativistic framework it was observed such strong correla-
tion [24]. The displayed nuclear matter binding energy (BA/A ≡ B(A,0)/A)
versus saturation density [ ρo = (2/3)k
2, with kF the Fermi momentum]
results are within a narrow band [22]. The observation given in Ref. [22] have
been studied by many other authors that have used nuclear matter binding
energies and saturation densities from different two-nucleon interactions. The
main argumentation, as also in the case of three-nucleon calculations, is that
this effect comes from different strengths of the two-nucleon tensor force and
short range repulsion, which changes the triton binding energy, while keeping
fixed the low-energy two-body scales.
Basically, nuclear matter saturates due to the composed repulsive and attrac-
tive short-range two-nucleon potential. Since, it may also be seen as a typical
low-energy problem, it is natural to question whether any connection exists
30 40 50 60 70
 [ MeV ]
Fig. 1. Infinite nuclear matter binding energy as a function of EF extracted from
Ref. [25] (solid circles and squares). The squares includes the single particle contri-
bution in the continuum. The full triangle is given by the empirical values.
between the proper few-body scales, Bd , Bv and Bt with those of the many-
body problem, like the BA/A and the Fermi energy EF = h̄
2k2F/(2mN) . For
light nuclei there is strong evidences of scaling between Bd , Bv and Bt as
expressed by Eq. (3).
Here we are arguing that the scales of nuclear matter, BA/A and EF , are
determined by Bd , Bv and Bt . Therefore, we suppose that going to the
infinite isospin symmetrical nuclear matter, A → ∞ and Iz = 0, the limit
B (βv, βd, βα, A, Iz = 0)
=Bt G (βv, βd, βα) , (4)
is well defined and expresses the correlation between the binding energy of the
nucleon in nuclear matter with the few-nucleon scales. The Fermi energy
EF = Bt EF (βv, βd, βα) , (5)
will be correlated as well to the few-nucleon binding energies.
The aim of this work is to study the possible correlation of the nuclear matter
binding energy per nucleon with Bd , Bv and Bt , in order to improve our
understanding of the general and important scaling of observables. Our inves-
tigation is based on Eqs. (4) and (5), motivated by our previous discussion
that leads to Eq. (1) and in the several works that recognize the role played by
the low-energy few-body scales [8,9,13,12,14,15], in defining the observables of
few-nucleon systems.
In the present framework, the universal scaling functions connect the proper
-9 -8.5 -8 -7.5 -7 -6.5 -6
 [MeV]
Fig. 2. BA/A as a function of Bt extracted from Ref. [25] (solid circles and squares).
The squares includes the single particle contribution in the continuum. The full
triangle is given by the empirical values.
scales of the few-body system with those of the many-body system, as given by
Eqs. (4) and (5). Different potentials, which describe the deuteron and the two-
nucleon scattering properties give different values of Bt , Bα BA/A and EF .
As we have done in deriving Eq. (3) for a class of changes in the short-range
part of the nuclear force that keeps the deuteron and low energy scattering
properties unchanged, and taking into account that for these variations of
the potential the 4He and triton binding energies are strongly correlated as
given by the Tjon line, one can rewrite Eqs. (4) and (5) in order to get a one
parameter scaling:
=Bt G (Bt) (6)
for fixed Bd and Bv, where the only true dependence in the class of potential
variations is dominated by Bt. The analogous expression for the Fermi energy
EF = Bt EF (Bt) , (7)
where EF scale with Bt.
In the perspective of the one parameter functions of Eqs. (6) and (7), it is
clear that one could express Et as a function of EF and immediately get
= C(EF ), (8)
-9 -8.5 -8 -7.5 -7 -6.5 -6
 [MeV]
Fig. 3. EF as a function of Bt extracted from Ref. [25] (solid circles and squares).
The squares includes the single particle contribution in the continuum. The full
triangle is given by the empirical values.
the correlation implied by the Coester band.
In order to enlighten our discussion we bring a variety of nuclear matter bind-
ing energies BA/A at the corresponding saturation density, represented by
the Fermi momenta kF , calculated from different two-nucleon potentials. In
Fig. 1, we present the well known Coester band in which the results for BA/A
and EF are showed. The two distinct bands represent the nuclear matter cal-
culations with and without the single-particle continuum contributions. The
empirical values are BA/A = 16 MeV and EF = 37.8 MeV from [27].
The correlation of BA/A with Bt expressed by (6) is plotted in Fig. 2, for
the same set of results given in Fig. 1. The two-nucleon potentials present
different values for the triton binding energy Bt , while the two-nucleon low-
energy observables are fixed. We observe that the scaling function EtG(Et)
is quite linear in the interval of about 2 MeV including the triton binding
energy. We note that the ratio BA/A depends strongly on Bt. We understand
this fact as a reminiscent manifestation of the three-body scale in the nuclear
matter results obtained with only two-body correlations. In Fig. 3, we show the
correlation between the Fermi energy and the triton binding energy. In general,
we observe that the increase in the three-body scale leads to the increase of the
Fermi energy, which is reasonable in view of the scaling function (7). However,
we observe that the empirical values disagrees with the general trend of the
correlation, a problem that could already be anticipated by looking at the
Coester band in Fig. 1.
The inclusion of three-body correlations, which carries the dynamics that sta-
bilizes the Thomas collapse, presumably has a repulsive effect diminishing the
saturation density and somewhat the nuclear matter binding. Due the short-
range repulsion of the nucleon-nucleon interaction, the nuclear matter tends
to saturate at large densities if only two-body correlations are considered and
the empirical binding is achieved. The dynamically generated three-body sta-
bilization mechanism carried only through the three-body correlations should
appear in the nucleon-nucleon interaction range, however such repulsive con-
tribution is absent if only two-body correlations are considered in the evalu-
ation of nuclear matter properties. Therefore, we suspect that the inclusion
of three-body correlations in nuclear matter calculations will bring the corre-
lation curve of the BA/A with Bt in Fig. 2 toward the empirical values; and
also in Fig. 3, where the saturation densities would be possibly found at lower
values for a given triton binding energy.
Recent sophisticated many-body calculations [26], where the triton binding en-
ergy and nuclear matter saturation density were adjusted through two three-
body parameters make subtle the clear appreciation of the real role of the
three-body correlations, once the fit of these realistic forces mixes differences
that come from the interaction and from the many-body approximations in nu-
clear matter calculations. It is beyond our work the discussion of the quite in-
volved many-body approximations needed to perform such calculations. Nev-
ertheless, our conjecture is obviously based on qualitative arguments, which
implies that just one three-body scale parameter is relevant for a systematic
description of light nuclei and nuclear matter.
The one parameter dependence in Eqs. (6) and (7) suggests to plot the dimen-
sionless quantity BA/(ABt) as a function of the ratio EF/Bt, which should look
as an almost linear correlation. We display in Fig. 4 the values for BA/(A Bt)
versus EF/Bt . As we could anticipate, the results show a clear linear cor-
relation. We are tempted to say that, if the correlation is extrapolated and
assuming that the binding and saturation densities somewhat decreases when
three-body correlations are considered, it looks to be possible that the empir-
ical values would be consistent with the correlation band.
In summary, we suggest for the first time a possible scaling of nuclei asymp-
totic properties, in particular the nuclear binding energies with the triton
binding energy, substantiated by recent realistic calculations of light nuclei.
This observation generalizes to the many-nucleon context the correlations be-
tween observables found in the three and four-nucleon systems. Beyond that,
we found that the original correlation between the nuclear matter binding en-
ergy per nucleon with the Fermi momentum described by the Coester band
can now be seen as robustly represented by the scaling of nuclear matter prop-
erties with the triton binding energy. The values of Bt carry different aspects
of the used two- and three-nucleon potentials. To verify the extension of our
conjecture, we propose that one could control the strength of the three-body
force without a many-body parameter in order produce different triton binding
4 5 6 7 8 9
Fig. 4. Infinite nuclear matter binding energy as a function of EF , both in units
of the triton binding energy. The calculation results are extracted from Ref. [25]
(solid circles and squares). The squares includes the single particle contribution in
the continuum. The full triangle represents the empirical values.
energies and nuclear matter properties. We emphasizing that the nuclear mat-
ter results should be obtained within the same approximations for all models.
In this way, we expect that the plotted results in figures 2 and 3 should lie in
a very narrow band.
Our discussion may turn in a simple way to systematize results of possible
forthcoming realistic calculations for many-nucleon systems. Consistent with
our conclusions, it was argued in Ref. [18] that QCD implies that nuclear
physics is close to the Thomas-Efimov limit. In this reference, it was conjec-
tured that a small change in the light quark masses away from their physical
values could be enough to move the deuteron and the singlet virtual state to
zero binding energies. Therefore, nuclear physics could be dominated by long-
range universal effective forces making the scalings not only a subtlety but
also an evident reality in ab-initio non-relativistic nuclear model calculations.
Acknowledgments. This work was partially supported by Fundação de Amparo
à Pesquisa do Estado de São Paulo and Conselho Nacional de Desenvolvimento
Cient́ıfico e Tecnológico. V. S. T. would like to thank FAEPEX/UNICAMP
for partial financial support.
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	References

  The well-known correlations of low-energy three and four-nucleon observables
with a typical three-nucleon scale (e.g., the Tjon line) is extended to light
nuclei and nuclear matter. Evidence for the scaling between light nuclei
binding energies and the triton one are pointed out. We argue that the
saturation energy and density of nuclear matter are correlated to the triton
binding energy. The available systematic nuclear matter calculations indicate a
possible band structure representing these correlations.

The few scales of nuclei and nuclear matter
A. Delfino a, T. Frederico b, V. S. Timóteo c, and Lauro Tomio d
aInstituto de F́ısica, Universidade Federal Fluminense, 24210-900 Niterói, RJ,
Brasil
bDepartamento de F́ısica, Instituto Tecnológico de Aeronáutica, CTA, 12228-900,
São José dos Campos, Brasil
cCentro Superior de Educação Tecnológica, Universidade Estadual de Campinas,
13484-370, Limeira, SP, Brasil
dInstituto de F́ısica Teórica, Universidade Estadual Paulista, 01405-900, São
Paulo, Brasil
Abstract
The well known correlations of low-energy three and four-nucleon observables with a
typical three-nucleon scale (e.g. the Tjon line) is extended to light nuclei and nuclear
matter. Evidence for the scaling between light nuclei binding energies and the tri-
ton one are pointed out. We show that the saturation energy and density of nuclear
matter are correlated to the triton binding. From the available systematic nuclear
matter calculations, we verify the existence of bands representing these correlations.
PACS 21.45.+v, 21.65.+f, 21.30.Fe
Key words: Scaling, nonrelativistic few-body systems, nonrelativistic nuclear
matter
Two-nucleon interactions are typically constructed to fit scattering data and
deuteron properties. When such interactions are used to calculate three-nucleon
observables, the results exhibit some discrepancies [1]. Basically, they are ex-
plained as originated from different strengths of the two-nucleon tensor force
and short-range repulsions, provided that all realistic two-nucleon interactions
have the correct one-pion exchange tail. In four-nucleon bound state (4He )
calculations the discrepancies still remain. But, at least, they are correlated,
as seen in the binding energies of 4He (Bα) and triton (Bt), which lie on a
very narrow band [2], obtained when the short-range repulsion of the nucleon-
nucleon interaction is varied while two-nucleon informations (deuteron and
scattering) are kept fixed. This correlation is known as Tjon line [2]. Bα and
Bt follows an almost straight line in the range of about 1-2 MeV of variation
Preprint submitted to Elsevier 4 November 2018
http://arxiv.org/abs/0704.0481v1
of the triton binding energy around the experimental value. As the long-range
two-nucleon scales we have the deuteron binding energy (Bd) and the singlet
virtual-state energy (Bv).
Two-body short-ranged interactions, supporting very low two-body binding
energy and/or large scattering lengths, when used to calculate three-body sys-
tems, approach what we call the universal Thomas-Efimov limit [3]. By trying
to find the range r0 of the two-nucleon force, Thomas [4] showed that when
r0 → 0, while the two-body binding energy B2 is kept fixed, the three-body
binding energy goes to infinity (Thomas collapse). Much latter, Efimov [5]
showed that, in the limit B2 = 0(r0 6= 0) the number of three-body bound
states is infinite with an accumulation point at the common two- and three-
body threshold. Note that both the Thomas and Efimov effects are claimed to
be model independent, since they are due to a dynamically generated effective
three-body potential acting at distances outside the range of the two-body
potential. These apparently different effects are related to the same scaling
mechanism, as shown in Ref. [3]. In other words, the Thomas effect appears
when r0 is much smaller than the size of the two-body system (which is of the
order of the scattering length |a|), while the Efimov effect arises for |a| >> r0.
Therefore, what matters for both effects is the same condition: |a| >> r0 or
the ratio |a|/r0 >> 1. In terms of the two-body energies this is translated
m|B2|/h̄
r0 << 1 (m the boson mass). One would expect that the
Thomas-Efimov effect is manifested in weakly-bound quantum few-body sys-
tems which are much larger in size than the corresponding two-body effective
range. Notice that a zero binding energy for a free two-body system is not
known in nature. But, nowadays it was shown that, for trapped ultracold
gases of certain atomic species, it is possible to adjust the two-body scattering
length at very large values, using Feshbach resonance techniques, by tuning the
external magnetic field [6]. In this case, it is expected that the Thomas-Efimov
effect can be manifested [7].
The deuteron and triton may be viewed as low energy systems with large size
scales in which the range of the potential is smaller than the corresponding
healing distances of the wave functions, leading the nucleons to have a high
probability to be outside of the interaction range. Then, the low-energy prop-
erties of these systems can be studied with models that minimally includes the
physics of the Thomas-Efimov effect, as in the case of a few-body model with
renormalized pairwise s-wave zero-range force [8]. This approach shows that
all the low-energy properties of the three-body system are well defined in the
model, once one three-body scale and the two-body low-energy observables are
given. As a consequence, correlations between two three-body s−wave observ-
ables are expected to appear in model calculations with short-ranged interac-
tions. Along this line, some previous works (see [9] and references therein) have
studied weakly-bound halo states in exotic nuclei as well as possible Efimov
states for He-He-Alkali molecules.
The scaling of three-nucleon observables with the triton binding energy corre-
sponds to universal behaviors found when a three-body scale is varied. For ex-
ample, the Phillips plot [10] of the neutron-deuteron doublet scattering length,
as a function of the triton energy is nowadays one of the universal scalings
found in the three-nucleon system [11,12]. In general, it is observed for nu-
clear and molecular weakly bound three-body systems [13,14,15] the scaling
of observables with the three-body binding energy.
The Thomas-collapse of the three-body energy in systems of maximum wave-
function symmetry implies the existence of a three nucleon scale (identified
with the triton binding energy) governing the short-range behavior of the
wave-function. Four nucleons can also form a state of maximum symmetry,
allowing in principle the collapse of such configuration, independently of the
three-nucleon collapse [8]. This is under discussion and it is suggested in [16]
that the four-body scale is not independent of the three-body one. However,
in their work it was introduced a three-body force to stabilize the shallowest
three-body state, against the variation of the cut-off. The three-body interac-
tion can be attractive or repulsive and their conclusion lies on the repulsive
sector. We note that the attractive part indicates a possible independent be-
havior of the four body ground-state energy from the three-body one. Certainly
this point merits further discussions and so far, we think, it is still open the
possibility of a four-body scale. Anyway, as the nucleon-nucleon interaction
is strongly repulsive at short range and therefore the probability of four nu-
cleons to be simultaneously in a volume ∼ r30 is quite small, presumably the
four-nucleon scale itself has much less opportunity to be evidenced in realistic
nuclear models. Indeed, this is indicated by the existence of the Tjon line. Due
to that, as we will see later, the four-nucleon binding energy is eliminated in
favor of the triton binding energy. In this respect it is worthwhile to note that
Platter et al. extended the effective field theory framework applied to four-
bosons [16] to calculate the 4He binding energy by controlling the triton energy
through a repulsive effective three-nucleon force. Within their approach [17]
the Tjon line is reproduced.
In a nuclear scenario dominated by an interaction with a range smaller than
the nucleon-nucleon scattering lengths, and considering the triton and 4He
nuclear sizes yet larger than the force range, the picture of nuclei would be of
a many-body system with the wave-function being an eigenfunction of the free
Hamiltonian almost everywhere. The Pauli principle allows only up to four nu-
cleons at the same position, forbidding certain particular configurations with
overlap of more particles. If more than four particles are allowed to overlap,
it would imply that the asymptotic information from the interaction of the
cluster would go beyond of those already fixed by the low-energy observables
of two, three and four nucleons. By some unknown reason the parameters
of Quantum Chromodynamics are close to this limit. It was conjectured in
Ref. [18] that a small change in the light quark masses away from their physi-
cal values could put the deuteron and the singlet virtual state at zero binding
energy, and therefore the above idealized picture of the nuclear systems could
not be far from reality. It is quite amazing thinking that nuclear wave func-
tions could heal much beyond the interaction range. Therefore, the details of
the long wavelength structure of nuclei are given by the free Hamiltonian and
by few-nucleon scales, which determine the wave function at short distances.
The universal behavior of the scaling functions are due to that.
If one wonders about the neutron matter within a non-relativistic quantum
framework, in the limit of a zero-range force, we could say that the only scale
in this case is the neutron-neutron scattering length. Therefore, the binding
energy of neutron droplets will be strongly correlated to that quantity, which
is the only physical scale in this situation allowed by the Pauli principle. This
discussion has been performed in the context of three neutron systems [19].
Moreover, it was concluded in Ref. [20] that stable tetra-neutron droplets
would imply a major change in the neutron-neutron scattering length.
Another example of the dominance of only two-body scale appears in three-
boson systems in two dimensions, where the Thomas-Efimov effect is ab-
sent [3]. In this case, only two-body low energy scales are enough to define the
many-body properties in the limit of a zero-range interaction. The low-energy
properties of a many-body system of spin-zero particles in two dimensions
will be sensitive only to the two-boson binding energy. Even in the case where
bosons are trapped, since the essential singularity of the point-like configura-
tion is not affected by the confining force as the harmonic one.
For the sake of generality, we start with the observables Bd, Bv, Bt and Bα
as the scales determining the asymptotic properties of nuclei [9]. Then, in the
limit of a zero-range interaction, we write the binding energy of a nucleus with
mass number A and isospin projection Iz, considering isospin breaking effects,
B(A, Iz) = A Bt B (βv, βd, βα, A, Iz) , (1)
where βa = Ba/Bt with a = v, d and α.
According to the Tjon line, βα remains approximately constant for a variety
of two-nucleon potentials and the parametrization of the numerical results,
given in MeV, for several two-nucleon potentials is
Bα = 4.72 (Bt − 2.48) , (2)
which for B
t = 8.48 MeV gives B
α = 28.32 MeV. Using (2) in (1),
R(A, Iz) = B(A, Iz)/A = Bt R (Bt, A, Iz) , (3)
where in the scaling function R(A, Iz) the values of Bd and Bv are fixed to
the experimental values. The dependence of Bα with Bt for realistic nucleon-
nucleon potentials is given by Eq. (2).
Equation (3) generalizes the concept of the Tjon line to nuclei. Recent calcu-
lations using the AV18 nucleon-nucleon potential plus three-body forces [21]
show that there is a systematic improvement of the binding energy results
for He, Li, Be and B isotopes simultaneously with the triton binding energy,
when models are tuned to fit Bt. It is important to note that these AV18
calculations have at least two three-body parameters that are fitted to Bt and
nuclear matter saturation properties. Consequently, one could argue that such
calculations cannot provide evidence for one-parameter correlation. The fit-
ting to nuclear matter calculation presumably is not that important for light
nuclei in view of the dominance of the triton binding (or three-body correla-
tions) in the four-nucleon bound state as given by the Tjon line. Therefore,
it is reasonable to think that three-body correlations are quite important for
light-nuclei, since that even for the alpha particle where the nucleons are in a
very compact configuration this occurs. In our opinion, the fitting of nuclear
matter saturation properties has more to do with the approximations done
in nuclear matter calculations. The three-body potential should be somewhat
tuned, which is, probably, not so important for light nuclei once the triton
binding attains its physical value.
For nuclear matter properties calculation using a variety of two-nucleon poten-
tials, in which the tensor strength was varied but the deuteron binding energy
was kept fixed, it was shown that these interactions cannot quantitatively ac-
count for nuclear saturation [22,23]. Coester et al. [22] observed that, in an
energy versus density plot, the saturation points of nuclear matter obtained
by employing different realistic potentials are located along a band (Coester
band). Also, in a relativistic framework it was observed such strong correla-
tion [24]. The displayed nuclear matter binding energy (BA/A ≡ B(A,0)/A)
versus saturation density [ ρo = (2/3)k
2, with kF the Fermi momentum]
results are within a narrow band [22]. The observation given in Ref. [22] have
been studied by many other authors that have used nuclear matter binding
energies and saturation densities from different two-nucleon interactions. The
main argumentation, as also in the case of three-nucleon calculations, is that
this effect comes from different strengths of the two-nucleon tensor force and
short range repulsion, which changes the triton binding energy, while keeping
fixed the low-energy two-body scales.
Basically, nuclear matter saturates due to the composed repulsive and attrac-
tive short-range two-nucleon potential. Since, it may also be seen as a typical
low-energy problem, it is natural to question whether any connection exists
30 40 50 60 70
 [ MeV ]
Fig. 1. Infinite nuclear matter binding energy as a function of EF extracted from
Ref. [25] (solid circles and squares). The squares includes the single particle contri-
bution in the continuum. The full triangle is given by the empirical values.
between the proper few-body scales, Bd , Bv and Bt with those of the many-
body problem, like the BA/A and the Fermi energy EF = h̄
2k2F/(2mN) . For
light nuclei there is strong evidences of scaling between Bd , Bv and Bt as
expressed by Eq. (3).
Here we are arguing that the scales of nuclear matter, BA/A and EF , are
determined by Bd , Bv and Bt . Therefore, we suppose that going to the
infinite isospin symmetrical nuclear matter, A → ∞ and Iz = 0, the limit
B (βv, βd, βα, A, Iz = 0)
=Bt G (βv, βd, βα) , (4)
is well defined and expresses the correlation between the binding energy of the
nucleon in nuclear matter with the few-nucleon scales. The Fermi energy
EF = Bt EF (βv, βd, βα) , (5)
will be correlated as well to the few-nucleon binding energies.
The aim of this work is to study the possible correlation of the nuclear matter
binding energy per nucleon with Bd , Bv and Bt , in order to improve our
understanding of the general and important scaling of observables. Our inves-
tigation is based on Eqs. (4) and (5), motivated by our previous discussion
that leads to Eq. (1) and in the several works that recognize the role played by
the low-energy few-body scales [8,9,13,12,14,15], in defining the observables of
few-nucleon systems.
In the present framework, the universal scaling functions connect the proper
-9 -8.5 -8 -7.5 -7 -6.5 -6
 [MeV]
Fig. 2. BA/A as a function of Bt extracted from Ref. [25] (solid circles and squares).
The squares includes the single particle contribution in the continuum. The full
triangle is given by the empirical values.
scales of the few-body system with those of the many-body system, as given by
Eqs. (4) and (5). Different potentials, which describe the deuteron and the two-
nucleon scattering properties give different values of Bt , Bα BA/A and EF .
As we have done in deriving Eq. (3) for a class of changes in the short-range
part of the nuclear force that keeps the deuteron and low energy scattering
properties unchanged, and taking into account that for these variations of
the potential the 4He and triton binding energies are strongly correlated as
given by the Tjon line, one can rewrite Eqs. (4) and (5) in order to get a one
parameter scaling:
=Bt G (Bt) (6)
for fixed Bd and Bv, where the only true dependence in the class of potential
variations is dominated by Bt. The analogous expression for the Fermi energy
EF = Bt EF (Bt) , (7)
where EF scale with Bt.
In the perspective of the one parameter functions of Eqs. (6) and (7), it is
clear that one could express Et as a function of EF and immediately get
= C(EF ), (8)
-9 -8.5 -8 -7.5 -7 -6.5 -6
 [MeV]
Fig. 3. EF as a function of Bt extracted from Ref. [25] (solid circles and squares).
The squares includes the single particle contribution in the continuum. The full
triangle is given by the empirical values.
the correlation implied by the Coester band.
In order to enlighten our discussion we bring a variety of nuclear matter bind-
ing energies BA/A at the corresponding saturation density, represented by
the Fermi momenta kF , calculated from different two-nucleon potentials. In
Fig. 1, we present the well known Coester band in which the results for BA/A
and EF are showed. The two distinct bands represent the nuclear matter cal-
culations with and without the single-particle continuum contributions. The
empirical values are BA/A = 16 MeV and EF = 37.8 MeV from [27].
The correlation of BA/A with Bt expressed by (6) is plotted in Fig. 2, for
the same set of results given in Fig. 1. The two-nucleon potentials present
different values for the triton binding energy Bt , while the two-nucleon low-
energy observables are fixed. We observe that the scaling function EtG(Et)
is quite linear in the interval of about 2 MeV including the triton binding
energy. We note that the ratio BA/A depends strongly on Bt. We understand
this fact as a reminiscent manifestation of the three-body scale in the nuclear
matter results obtained with only two-body correlations. In Fig. 3, we show the
correlation between the Fermi energy and the triton binding energy. In general,
we observe that the increase in the three-body scale leads to the increase of the
Fermi energy, which is reasonable in view of the scaling function (7). However,
we observe that the empirical values disagrees with the general trend of the
correlation, a problem that could already be anticipated by looking at the
Coester band in Fig. 1.
The inclusion of three-body correlations, which carries the dynamics that sta-
bilizes the Thomas collapse, presumably has a repulsive effect diminishing the
saturation density and somewhat the nuclear matter binding. Due the short-
range repulsion of the nucleon-nucleon interaction, the nuclear matter tends
to saturate at large densities if only two-body correlations are considered and
the empirical binding is achieved. The dynamically generated three-body sta-
bilization mechanism carried only through the three-body correlations should
appear in the nucleon-nucleon interaction range, however such repulsive con-
tribution is absent if only two-body correlations are considered in the evalu-
ation of nuclear matter properties. Therefore, we suspect that the inclusion
of three-body correlations in nuclear matter calculations will bring the corre-
lation curve of the BA/A with Bt in Fig. 2 toward the empirical values; and
also in Fig. 3, where the saturation densities would be possibly found at lower
values for a given triton binding energy.
Recent sophisticated many-body calculations [26], where the triton binding en-
ergy and nuclear matter saturation density were adjusted through two three-
body parameters make subtle the clear appreciation of the real role of the
three-body correlations, once the fit of these realistic forces mixes differences
that come from the interaction and from the many-body approximations in nu-
clear matter calculations. It is beyond our work the discussion of the quite in-
volved many-body approximations needed to perform such calculations. Nev-
ertheless, our conjecture is obviously based on qualitative arguments, which
implies that just one three-body scale parameter is relevant for a systematic
description of light nuclei and nuclear matter.
The one parameter dependence in Eqs. (6) and (7) suggests to plot the dimen-
sionless quantity BA/(ABt) as a function of the ratio EF/Bt, which should look
as an almost linear correlation. We display in Fig. 4 the values for BA/(A Bt)
versus EF/Bt . As we could anticipate, the results show a clear linear cor-
relation. We are tempted to say that, if the correlation is extrapolated and
assuming that the binding and saturation densities somewhat decreases when
three-body correlations are considered, it looks to be possible that the empir-
ical values would be consistent with the correlation band.
In summary, we suggest for the first time a possible scaling of nuclei asymp-
totic properties, in particular the nuclear binding energies with the triton
binding energy, substantiated by recent realistic calculations of light nuclei.
This observation generalizes to the many-nucleon context the correlations be-
tween observables found in the three and four-nucleon systems. Beyond that,
we found that the original correlation between the nuclear matter binding en-
ergy per nucleon with the Fermi momentum described by the Coester band
can now be seen as robustly represented by the scaling of nuclear matter prop-
erties with the triton binding energy. The values of Bt carry different aspects
of the used two- and three-nucleon potentials. To verify the extension of our
conjecture, we propose that one could control the strength of the three-body
force without a many-body parameter in order produce different triton binding
4 5 6 7 8 9
Fig. 4. Infinite nuclear matter binding energy as a function of EF , both in units
of the triton binding energy. The calculation results are extracted from Ref. [25]
(solid circles and squares). The squares includes the single particle contribution in
the continuum. The full triangle represents the empirical values.
energies and nuclear matter properties. We emphasizing that the nuclear mat-
ter results should be obtained within the same approximations for all models.
In this way, we expect that the plotted results in figures 2 and 3 should lie in
a very narrow band.
Our discussion may turn in a simple way to systematize results of possible
forthcoming realistic calculations for many-nucleon systems. Consistent with
our conclusions, it was argued in Ref. [18] that QCD implies that nuclear
physics is close to the Thomas-Efimov limit. In this reference, it was conjec-
tured that a small change in the light quark masses away from their physical
values could be enough to move the deuteron and the singlet virtual state to
zero binding energies. Therefore, nuclear physics could be dominated by long-
range universal effective forces making the scalings not only a subtlety but
also an evident reality in ab-initio non-relativistic nuclear model calculations.
Acknowledgments. This work was partially supported by Fundação de Amparo
à Pesquisa do Estado de São Paulo and Conselho Nacional de Desenvolvimento
Cient́ıfico e Tecnológico. V. S. T. would like to thank FAEPEX/UNICAMP
for partial financial support.
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