Universality in few-body Systems: from few-atoms to few-nucleons

Some aspects of universal properties are discussed in few-atom as well as in few- nucleon systems. When the two-body scattering length a is large the three-body system presents some peculiar characteristics such as the Efimov effect. In the present contribution, the zerorange theory deduced by V. Efimov is compared to results obtained using potential models. Accordingly range corrections are introduced. These corrections appear as a shift in the variable K*a with K* the three-body parameter. The three-boson spectrum, atom-dimer scattering and the recombination rate at threshold are analyzed using the finite-range theory proposed. In the case of nuclear systems, n – d scattering is discussed using the universal formula derived by Efimov. Furthermore, the spectrum up to N = 6 is discussed using the central Volkov potential.


Introduction
The scattering of two particles at very low energy E shows universal behavior encoded on the effective range function in that regime where k 2 = E/(h 2 /m), δ is the phase shift and r s is the effective range of the interaction. In fact, systems interacting differently but having equal values of the scattering length a and effective range r s share the same low energy behavior. Two different limits can be studied that produce interesting effects in the three-body system. The first one is the scaling limit: the range, r 0 , of the potential goes to zero. In this limit the two-body system presents a continuous scale invariance (CSI): the low energy observables depend on a single parameter, the two-body scattering length a. For example the s-wave phase shift reads k cot δ = −1/a and, for positive values of a, a two-body bound state appears (called dimer) with binding energy E 2 =h 2 /ma 2 .
In this context a appears as a control parameter. As it has been demonstrated by Thomas in 1935 [1], the three-body system collapses in this limit (this effect is known as the Thomas collapse). Using a variational argument Thomas showed that the three-body ground state energy scales ash 2 /mr 2 0 producing a system unbound from below as r 0 → 0. Remarkably, this collapse is independent of the magnitude of the two-body binding energy.
The second limit, which is subject of intense investigations, is the unitary limit. In this limit |a| >> r 0 and, in the case of positive and large values of a, a shallow dimer appears with binding energy E 2 ≈h 2 /ma 2 . As pointed out by V. Efimov in 1970 and subsequent papers [2,3], when a >> r 0 the three-boson system presents a new class of universal behavior. The spectrum of three interacting bosons results from a geometric series: E n+1 /E n → e −2π/s 0 with s 0 ≈ 1.00624 a universal constant. What is observed, introducing the three-body hyperradial variable ρ, is that a very extended hyperradial potential is generated which, as ρ → ∞, goes toh 2 /ma 2 . In the limit a → ∞ the hyperradial potential goes to zero as 1/ρ 2 supporting an infinite number of bound states. The short range physics can be implemented as a boundary condition at short distances (of the order of the interaction range r 0 ) and, as n increases, the universal spectrum results. More specifically when a >> r 0 a continuous scale invariance appears in the two-body system while, in the three-body system, it is broken to a discrete scale invariance (DSI).
The DSI imposes strong constrains on the observables. In addition to the geometric spectrum already mentioned in the case of three-boson bound states, the atom-dimer scattering length a AD has the following functional form deduced by Efimov [4] where d 1 , d 2 , d 3 are universal constants and κ * is the three-body parameter that fixes the energy scale at the unitary limit (see Ref. [5] for details). For collisions below the dimer breakup threshold the DSI imposes the following universal form for the effective range function with δ AD the atom-dimer phase-shift and c 1 , c 2 , φ universal functions depending on the product ka, where k 2 = (4/3)E/(h 2 /m), being E the center of mass energy of the process. As k → 0, ka cot δ AD → −1/a AD and at k = 0 the constants d 1 , d 2 , d 3 and c 1 (0), c 2 (0), φ(0) are related by simple trigonometric equalities. A parametrization of the functions c 1 , c 2 , φ can be found in Ref. [5]. The universal character of the effective range function can be used to evaluate a very different system: low energy nucleon-deuteron scattering [6]. The effective range function corresponding to nucleon-deuteron scattering at low energies, in the J = 1/2 + state, does not show the expected linear behavior but it presents a pole structure. First observations of this particular behavior have been done in Refs. [7,8,9] whereas in Ref. [10] explicit calculations of the effective range function have been done using nucleon-nucleon potentials. The pole structure has been related to a virtual state and, from the calculations, it was possible to extract the pole energy fitting the effective range formula with the form suggested by Delves [11]. Recently it was shown that the pole structure of the effective range function can be quantitatively related to the universal form given by equation (3) and, using the parametrization determined in the atomic three-helium system, that equation can be used to describe nucleon-deuteron scattering as well. In particular, using the universal function φ, the pole energy can be used to extract the three-body parameter κ * . In this way, the universal behavior imposed by the DSI is analyzed in systems with natural lengths that differ of several order of magnitude.
The DSI imposes constrains also on the form of the S-matrix for energies above the dimer threshold. The following peculiar form for the recombination rate at threshold [12,13] has been deduced: and using the large value of the factor e 2πs 0 ≈ 515, the above equation can be approximated by where γ = 1.16 [5]. As it has been shown recently [14], this equation can be used to describe very different systems as recombination of three atoms or three nucleons. The above discussion shows that the study of universality in few-body systems is of interest in several fields of research, ranging from cold-atoms to nuclear physics. In atomic physics, where the Efimov effect has been observed for the first time [15], discrepancies arise between the theoretical prediction and the experimental determination of the ratio between a * and a − [16,17], that means between the scattering lengths at which the Efimov state disappears in the atomdimer and in the three-atom continuum, respectively. The solution to this puzzle is probably hidden in finite-range corrections to the universal formulas (see Ref. [18] and references therein for a recent account to the problem). A recent study has discussed these corrections from a new perspective [19].
In nuclear physics these studies can be applied to describe halo nuclei where a cluster description is justified. We can mention the observation of universal aspects in the scattering of a neutron on a neutron-halo nucleus having a large scattering length, as for example the n− 19 C system (see Refs. [20,21] and Ref. [22] for a recent review). Also the light nuclei spectrum shows some universal characteristic as it has been discussed in Ref. [23].

The three-boson spectrum
The spectrum of the three-boson system (called trimer) having a large two-body scattering length a has been derived by Efimov in the zero-range limit [4]. The Efimov's binding energy equation is where E n 3 /E 2 = tan 2 ξ and the function Δ(ξ) is an universal function defined in the range −π ≤ ξ ≤ −π/4 (a parametrization of Δ(ξ) in this range can be found in [5]). The twobody binding energy is defined in the zero-range limit as E 2 =h 2 /ma 2 and the three-body parameter κ * is determined by the energy E * u of the n * level at the unitary limit. The scattering length a appears as a control parameter, fixing its value the above equation has to be solved simultaneously with the condition E n 3 /(h 2 /ma 2 ) = tan 2 ξ to find the the binding energies E n 3 at each value of the angle ξ. In this context the three-body parameter appears as a scale parameter. The universal character of the above equation can be seen using dimensionless quantities as with n = E n 3 /E * u . This form of the equation explicitly shows DSI. At fixed values of the angle ξ the ratio between two different levels is constant and it results In the case of two consecutive levels the ratio E n+1 3 /E n 3 = e −2π/s 0 ≈ 1/515.03 is obtained. The behavior of the energy for two consecutive levels, n = 0, 1, as given in equation (7), is shown in figure 1. In order to see the two levels in a single plot the quantity ( n ) 1/4 is plotted as function of √ x with x = κ * a. In the figure the following points are indicated: a n − and a * n are the values of the two-body scattering length at which the trimer disappears into the three-body continuum or into the atom-dimer continuum, respectively.
The Efimov equations are exact in the zero-range limit. They have been derived imposing a boundary condition in the short-range region encoded in the three-body parameter κ * . The same problem has been studied using Effective Field Theories (EFT) [24]. In this approach the short-range physics enters through low energy constants, fitted to reproduce selected data. One particular form to apply this theory was suggested by Lepage [25]. The interaction between two The dimensionless quantity ( n ) 1/4 as a function of (x) 1/2 for n = 0, 1. The particular values of the two-body scattering length a n − and a n * are indicated. The long-dashed line indicates the atom-dimer threshold.
particles is replaced by an effective potential parametrized in such a way that some relevant data are reproduced. For example, a two-parameter potential (as a gaussian) can be constructed to reproduce the scattering length and the effective range as given in equation (1). This effective interaction will describe the low energy dynamics of the system with good accuracy. The level of accuracy is given by the energy scale. Equation (1) is accurate at energies E << E 0 , with E 0 =h 2 /mr 2 0 , and errors in that formula appear at order of (E/E 0 ) 2 . In the three-body system the leading order (LO) prescription of the EFT is that a three-body counter term has to be considered. At this order the EFT approach is based in a Hamiltonian with two coupling constants, one for the two body sector, related with the control parameter a, and one for the three-body sector, related to the scale parameter κ * . The Efimov equations can be derived in this approach as has been shown if Ref. [26].
Following the above discussion, universal aspects of a three-boson system can be studied using a potential model. Following Refs. [6,14,27] the three-helium system is taken as a reference system. At the two-body level, one of the most commonly used He-He potentials, i.e., the LM2M2 interaction [28], is taken as the reference interaction. In particular, in order to explore the (a −1 , κ) plane (κ = sign(E)[|E|/(h 2 /m)] 1/2 and E the energy level), we modify this potential as:  Examples of this strategy exist in the literature [29,30]. For λ ≈ 0.9743 the interaction is close to the unitary limit (a → ∞) and for λ = 1 the values predicted by the LM2M2 are recovered. The scattering length is a = 189.41 a 0 , the two-body energy E 2 = −1.303 mK and the effective range r s = 13.845 a 0 , with the mass parameterh 2 /m = 43.281307 (a 0 ) 2 K. Noticing the r 0 ≈ r s , the two-helium system has the r 0 /a << 1 allowing a description based on the Efimov universal theory, having in mind that the range corrections have to be considered as well. As an effective potential we define an attractive two-body gaussian (TBG) potential [27,31] with range r 0 = 10 a 0 and strength V 0 fixed to reproduce the values of a given by V λ (r). For example with the strength V 0 = −1.2343566 K, corresponding to λ = 1, the LM2M2 low-energy data are closely reproduced, E 2 = −1.303 mK, a = 189.42 a 0 , and r s = 13.80 a 0 . In figure 2 the two interactions are compared, the differences at short distanced are evident.
When the TBG potential is used in the three-atom system it produces a ground state binding energy appreciable deeper than that one calculated using the V λ (r) interaction. For example, at λ = 1 the LM2M2 helium-trimer ground-state binding energy is E 0 3 = 126.4 mK, whereas the one obtained using the TBG potential is 151.32 mK. Following the LO EFT prescription an hypercentral three-body (H3B) (repulsive) interaction can be introduced The ratio between the excited state of the trimer and the dimer binding energy as a function of κ * a calculated for the TBG potential (red circles) and for the TBG+H3B potential (green squares). The dashed line represents the zero-range theory of equation (6) whereas the solid line has been obtained using equation (13).
with the strength W 0 tuned to reproduce the trimer E 0 3 obtained with V λ (r) for the different values of λ. In the above equation ρ 2 123 = 2 3 (r 2 12 + r 2 23 + r 2 31 ) is the hyperradius of three identical particles and ρ 0 gives the range of the three-body force that can be fixed as ρ 0 = r 0 .
Varying λ from the unitary limit to λ = 1.1 a set of values for the ground state binding energy E 0 3 and the first excited state E 1 3 using the TBG and TBG+H3B potentials can be obtained in a broad range of a. The results can be compared to the predictions given by the Efimov's binding energy equation. Fixing n * = 1, the three-body parameter κ * is determined by calculating E 1 3 at the unitary limit; we obtain κ * = 2.119 × 10 −3 a −1 0 and κ * = 1.899 × 10 −3 a −1 0 for the TBG and TBG+H3B, respectively.
As it has been shown in Refs. [6,14,19], in order to be in accord with the numerical results obtained solving a finite-range potential, the universal relation equation (6) must be modified in the following way κ * a + Γ n = e (n−n * )π/s 0 exp [−Δ(ξ)/2s 0 ]/ cos ξ .
The finite-range nature of the interaction has been taken into account by the substitution h 2 /ma 2 → E 2 in the first row. The main modification however is the introduction of the shift Γ n . In figure 3 we collect our numerical results for the ratio E 1 3 /E 2 as a function of κ * a for the TBG potential (circles) and for the TBG+H3B potential (squares). The dashed line corresponds to the zero-range theory given in equation (6), whereas the solid line, which goes on top of the numerical results, has been obtained using equation (13) with Γ 1 ≈ 4 × 10 −2 .  (2) whereas the solid and dotted line are the finite-range theory of equation (14). In the latter a slightly different parametrization of the constants d 1 , d 2 and d 3 is used.

Atom-dimer scattering
The effective potential models, TBG and TBG+H3B, described in the previous section can be used to describe atom-dimer scattering and the recombination rate K 3 at threshold. To this end the dimer-atom scattering length a AD and the s-wave atom-dimer phase δ can be calculated. The calculations have been done using the hyperspherical harmonic (HH) method as described in Refs. [31,32]. The results for the atom-dimer scattering length are collected in figure 4 where the ratio a AD /a B is given in terms of the product κ * a (the energy scattering length a B is defined from the dimer binding energy as E 2 =h 2 /ma 2 B ). It can be observed that the calculated points, given as full circles (TBG potential) and full squares (TBG+H3B potential), are on a curve shifted with respect to the dashed line representing the zero-range theory of equation (2) with the parametrization given in Ref. [5]. We can interpret again the shift as produced by the finite-range character of the calculations. Accordingly equation (2) can be modified to describe finite-range interactions as Using the values Γ * = Γ 1 , the solid line is obtained in figure 4. Furthermore, modifying slightly the values of the universal constants as d 1 = 1.531, d 2 = −2.141 and d 3 = 1.100, from those given in Ref. [5], a better description of the numerical results is obtained as it is shown by the dotted line in the figure.
The calculations have been extended to describe atom-dimer scattering at energies below the dimer breakup threshold for different values of a. As it has been done for the zero-range theory of the binding energies and atom-dimer scattering length, equation (3) has to be modified to consider finite-range interactions. Following Ref. [6], the effective range function can be written as The results are collected in figure 5 (full squares) at different values of κ * a = κ * a + Γ * . In the figure different patterns can be observed. For the smallest values of κ * a the behavior is almost linear in all the energy range. Starting at values κ * a ≈ 0.4 a curvature appears close to zero energy, pointing out to an emergent pole structure that becomes evident at larger values of κ * a. Specifically, the pole appears when a AD changes sign (see figure 4) or when the argument of the cotangent function in equation (15) becomes zero (or nπ). The shadow plot in the first row of figure 5 corresponds to the case λ = 1 and describes 4 He-4 He 2 scattering (full triangles). The shadow plot in the second row corresponds to nucleon-deuteron scattering as discussed below. The solid curves are obtained using the finite-range theory of equation (15) with the parametrization of Ref. [5]. We can observe a noticeable agreement along the whole range of values.
The effective range function in the case of neutron-deuteron scattering has a peculiar form [10]. Moreover Efimov claimed that this process can be described with the universal formula of equation (3) [4]. In order to verify this property, equation (15) can be applied to quantitatively describe n − d scattering at low energies. To this end the results of Ref. [10], in which n − d scattering has been described using a spin dependent central potential, can be used. In that reference the value of a nd = 0.71 fm has been obtained for the n − d scattering length, and the effective range function has been parametrized as with E p = −160 keV and r s ≈ −127 fm. It should be noticed that this particular parametrization of the effective range function can be simple related to equation (15) in the low energy limit.
From the values of E p and a nd it is possible to determine the values of a and κ * using the universal function φ and equation (14). The values a = 4.075 fm and κ * a = 0.5779 are obtained. The negative energy E p of the pole implies that it appears in the negative region similar to what happens in figure 5 at intermediate κ * a values. With these numerical values the shadow panel of the second row in figure 5 shows a comparison between equation (15) and the n − d scattering results of Ref. [10] (full circles). A noticeable agreement is observed. Is should be stressed that the universal functions used in equation (15) are those obtained through the analysis of the atomic system. Finally the application of the finite-range theory is discussed in the case of the recombination rate at threshold corresponding, in the case of three-helium atoms, to the process 4 He+ 4 He+ 4 He → 4 He+ 4 He 2 . The coefficient can be defined as K 3 = α(ha 4 /m). The DSI imposes α to be a log-periodic function, as given in equation (4), whose simplified version is given in equation (5). In order to describe the numerical results obtained using finite-range potentials, the following modification in the definition of K 3 has to be introduced with the simplified form where, as in the effective range function, a has been replaced by a B and the shift Γ + has been introduced. The results obtained in Ref. [14] are shown in figure 6 as circles (TBG potential) The solid curves have been calculated using equation (15) for different values of κ * a.
and squares (TBG+H3B potential). The dashed line represents equation (4) using the value γ = 1.16 from Ref. [5]. From the figure it is evident that the calculated points organize in a curve shifted with respect to the universal curve. The solid line represents equation (17), with the same value of γ and Γ + ≈ 6 × 10 −2 . Again the finite-range theory obtained by modifying the zero-range equations with a shift in the variable κ * a describes the calculated points with good accuracy. Moreover the values of the shifts Γ 1 , Γ * and Γ + are of the same order.

Light nuclei spectrum
In this section the spectrum obtained for N ≤ 6 fermions is discussed using a central potential.
The intention here is to see if, in the case of fermions, some of the universal properties discussed  (4) whereas the solid line is the finite-range theory of equation (17).
before can be identified in the spectrum of light nuclei. To this end the following central potential, the Volkov potential, is used with V R = 144.86 MeV, R 1 = 0.82 fm, V A = −83.34 MeV, R 2 = 1.60 fm. The fermions are considered to have the same mass, chosen to be equal to the reference mass m, and corresponding toh 2 /m = 41.47 MeV fm 2 (nuclear case). With this parametrization of the potential, the twonucleon system has binding energy E 2 = 0.54592 MeV and scattering length a = 10.082 fm.
For N > 2 the use of the s-wave version of the potential produces a spectrum much closer to the experimental one. This is a direct consequence of the weakness of the nuclear interaction in p-waves. The results are obtained after diagonalizing the Hamiltonian matrix constructed using the HH basis without symmetrization as discussed in Ref. [23].
With the parametrization of the potential given above, the 3 H and 3 He binding energies result to be 8.431 MeV and 7.725 MeV, very close to the experimental values. In fact, this parametrization was chosen to correctly describe the three-nucleon binding energies and, for this reason, the deuteron binding energy results artificially low. As a consequence 3 H has an excited state close to the nucleon-deuteron threshold. Using a more realistic interaction this state should move above the threshold becoming the virtual state discussed in the previous section. In despite of its simplicity this model predicts the 4 He binding energy with an error below 5% and, more interesting, the almost correct position of the 0 + excited state, as is shown in figure 7. Without considering the spin-dependence of the force the 6 He ground state energy results too bound.

Conclusions
The zero-range theory for few-body systems having a large two-body scattering length a has been studied and discussed using potential models. Following the prescription of Lepage [25] an effective interaction has been constructed that reproduces the same low energy data: the two-body scattering a, acting as a control parameter, and the energy wave number κ * of one selected level of the trimer at the unitary limit, acting as a scale parameter. Using this model different observables have been calculated solving the Schrödinger equation: the three-boson energy spectrum, the atom-dimer scattering length, the atom-dimer effective range function and the recombination rate at threshold. The analysis of the numerical results allowed to judge the capability of the zero-range theory to describe finite-range systems. The conclusion was that a finite-range scale parameter Γ n has to be introduced to correct the original zero-range theory. This parameter appears as a finite-range scaling parameter.
Using the finite-range theory it was possible to describe very different systems as atom-dimer scattering and neutron-deuteron scattering using the same parametrization of the universal effective range function, equation (15), obtained studying atomic systems. As a final study, the spectrum of light nuclear systems has been analyzed using a central potential, the Volkov potential. Fixing only the 3 H binding energy, this simple model describes the binding energies of the ground and excited states in 4 He with an error below 5%. The 6 He results slightly over bind. Studies including the spin dependence of the interaction are at present underway.