Efimov Physics and Connections to Nuclear Physics
EEfimov Physics and
Connections to NuclearPhysics
A. Kievsky, L. Girlanda, M. Gattobigio, M.Viviani Istituto Nazionale di Fisica Nucleare, Largo Pontecorvo 3, 56100 Pisa, Italy Dipartimento di Matematica e Fisica ”E. De Giorgi”, Universit`a del Salento,I-73100 Lecce, Italy Universit´e Cˆote d’Azur, CNRS, Institut de Physique de Nice, 1361 route desLucioles, 06560 Valbonne, FranceAnnual Review of Particle and NuclearScience 2021. AA:1–27https://doi.org/10.1146/((please addarticle doi))Copyright © Keywords efimov physics, universal properties, gaussian characterization,few-body systems, discrete scale invariance, unitary limit
Abstract
Physical systems characterized by a shallow two-body bound or virtualstate are governed at large distances by a continuous-scale invariance,which is broken to a discrete one when three or more particles come intoplay. This symmetry induces a universal behavior for different systems,independent of the details of the underlying interaction, rooted in thesmallness of the ratio (cid:96)/a B (cid:28)
1, where the length a B is associatedto the binding energy of the two-body system E = ¯ h /ma B and (cid:96) is the natural length given by the interaction range. Efimov physicsrefers to this universal behavior, which is often hidden by the on-set ofsystem-specific non-universal effects. In this work we identify universalproperties by providing an explicit link of physical systems to theirunitary limit, in which a B → ∞ , and show that nuclear systems belongto this class of universality. a r X i v : . [ nu c l - t h ] F e b ontents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Universal characterization of two-body systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1. The characteristic potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2. Trajectories in the universal window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3. Correlations inside the universal window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83. The three-body universal window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1. The three-boson system: bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2. The three-boson system: scattering states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3. The three nucleon system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124. Characterization of the unitary window for more than three particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.1. The N -boson systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2. Collapse of finite-range interactions onto the zero-range model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3. The A ≤ universal window. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175. Implications of Efimov physics in determining the nuclear EFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.1. The nuclear physical point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2. The excited + state of He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.3. The saturation point of nuclear matter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1. Introduction
Studying a particular physical system we could wonder about the interactions that governthe underlying dynamics. Usually, the particular characteristics of those interactions arerevealed by the properties of those systems. The case of residual interactions is especiallyinteresting, as exemplified by the nuclear interaction, a residual interaction of QuantumChromodynamics (QCD), or by molecular structures built under residual effects of QuantumElectrodynamics (QED). We can imagine situations in which the residual interaction placesthe system in a particular energy region where the characteristics of the interaction becomeunimportant (we may consider this as a fine tuning). Similar situations could occur whena system is subject to a suitable external field. Following these ideas, and focusing ona non-relativistic theory, we can design a short-range tunable potential describing a two-particle system with mass m and refer to the unitary window as the region in the space ofthe potential parameters such that the scattering length a reaches a value close to infinity.When a is large the two-body system has a shallow (real or virtual) bound state whosebinding energy is governed by the scattering length, E ≈ ¯ h /ma . Its shallow character isdefined with respect to the typical energy of the system, ¯ h /m(cid:96) , where the typical lengthof the system (cid:96) could be for example the potential range. The limit (cid:96)/a → (cid:96)/a (cid:28) aboratories using external fields, like for trapped cold atoms with Feshbach resonances (1)or can be naturally produced. There are a few natural systems located inside this window,one is the dimer of two helium atoms. In fact the He molecule has an extremely lowbinding energy, E ≈ h /mr vdW ≈ . r vdW = 5 . a , a denoting, here and in the following, the Bohr radius. Nuclear physics is another example;the deuteron binding energy is E = 2 . h /m(cid:96) ≈
20 MeV, with the interaction length given in this case by the inverse ofthe pion mass m π , (cid:96) ∼ /m π ≈ . π -mesons. The mass of the pion m π is different from zero because of the soft explicit breakingterm introduced by the masses of the up and down quarks, but is still much lower than thetypical hadronic masses. Another interesting limit in QCD can be reached realizing that themass of the pion is close to a critical value at which the nucleon-nucleon scattering lengthsdiverge (19, 20). The S (singlet) a s and S (triplet) a t scattering lengths are functions ofthe up and down quark masses, or equivalently of m π which is related to the quark massesby the Gell-Mann-Oakes-Renner relation (21). It has been shown that for m π ≈
200 MeVboth scattering lengths diverge (22, 23). At the physical point, m π ≈
138 MeV, the valuesof the two scattering lengths are a s ≈ − . a t ≈ . (cid:96) ≈ . Q ∼ /a ofthe system and the underlying high momentum scale ∼ /(cid:96) (18, 24, 25, 26). This conditionis well fulfilled in both systems, atomic helium and nuclear physics. In the latter case thisapproach is known as pionless-EFT (24, 27, 28, 29). Using such an EFT, if the power-counting is correct (24, 27, 30, 31, 32), one can systematically improve the prediction ofthe observables. For instance, at low energies with E = ¯ h k /m , the s -wave phase shift δ determined by the effective range expansion (ERE) (33) k cot δ = − a + 12 r e k , a , whereas the finite-range nature of theinteraction, represented by the effective range r e , constitutes the next-to-the-leading orderterm (NLO). Inside the unitary window, there is an energy pole close to the two-particlethreshold relating the scattering and bound state properties. The extension of the ERE tothe negative energy pole results in 1 a B = 1 a + 12 r e a B , E = ¯ h /ma B defines the energy length a B . It could be positive (bound state) or ••
138 MeV, the valuesof the two scattering lengths are a s ≈ − . a t ≈ . (cid:96) ≈ . Q ∼ /a ofthe system and the underlying high momentum scale ∼ /(cid:96) (18, 24, 25, 26). This conditionis well fulfilled in both systems, atomic helium and nuclear physics. In the latter case thisapproach is known as pionless-EFT (24, 27, 28, 29). Using such an EFT, if the power-counting is correct (24, 27, 30, 31, 32), one can systematically improve the prediction ofthe observables. For instance, at low energies with E = ¯ h k /m , the s -wave phase shift δ determined by the effective range expansion (ERE) (33) k cot δ = − a + 12 r e k , a , whereas the finite-range nature of theinteraction, represented by the effective range r e , constitutes the next-to-the-leading orderterm (NLO). Inside the unitary window, there is an energy pole close to the two-particlethreshold relating the scattering and bound state properties. The extension of the ERE tothe negative energy pole results in 1 a B = 1 a + 12 r e a B , E = ¯ h /ma B defines the energy length a B . It could be positive (bound state) or •• Efimov Physics 3 egative (virtual state). Moreover, | a B | (cid:29) r e ∼ (cid:96) and a ∼ a B , so the ratio r e /a B representsa small parameter. In this energy region the two-body system is dominated by a continuousscale invariance (CSI) governed by the control parameter a B with violations of the order of r e /a B .The most remarkable property of systems at the unitary limit shows up at three-bodylevel through the Efimov effect (34, 35). The CSI in the two-body system is dynamicallybroken at the level of three bodies into a discrete scale invariance (DSI). When the strengthof the two-body interaction is such that there is a bound state at zero-energy, an infinitetower of geometrically distributed energy states appears in the three-body system withthe energy threshold E = 0 as an accumulating point. The energy ratio of successivelevels E n +13 /E n = e − π/s is a universal constant, with s depending on the mass ratio ofthe constituents; for three equal bosons s (cid:39) . e − π/s (cid:39) (1 / . . Theanomalous breaking of the symmetry gives rise to an emergent scale at the three-body levelwhich is usually referred to as the three-body parameter κ ∗ , giving the binding energy¯ h κ ∗ /m of a reference state belonging to the tower of states at the unitary point.This effect, predicted by V. Efimov around 50 years ago, was observed 35 years afterits prediction by the group of R. Grimm (36). An enormous amount of work, experimentalas well as theoretical, has been, and still is, dedicated to study this phenomenon. Anintroduction to this sector of research can be found in the following reviews and referencestherein (37, 38, 39, 40, 41, 42, 43, 44). The physics associated with the Efimov effect iscalled Efimov physics and the energy region in which the consequences of this effect canbe observed is called universal window, unitary window or Efimov window. Observationof universal behavior of systems belonging to this window allows to understand better theuniversal dynamics as has been shown recently in the analysis of the three- and four-neutronsystems. The evidence of a low-energy tetraneutron, observed in two experiments (45, 46),has been attributed to the universal long-range tail of that system (47, 48). Furthermore,arguments based on the separation of scales have been recently exploited to describe halonuclei, a sector of physics in which universal properties are expected to be observed, seeRefs. (49, 50, 51, 52) and references therein.In order to study universal behaviour in few-boson and few-fermion systems, theSchr¨odinger equation has been solved using two different variational methods. For sys-tem with three and four particles we have used the Hyperspherical Harmonic (HH) (53, 54)method and its unsymmetrized version (55, 56). For heavier systems we have implementeda version of the stochastic variational method (SVM) (57) using correlated-Gaussian func-tions as basis set.
2. Universal characterization of two-body systems
The dynamics of two-body systems inside the universal window are highly independent ofthe details of their mutual interaction. The systems satisfy an approximate CSI, exactlyverified in the case of a zero-range interaction. Though the zero-range case was used manytimes as a first approximation to describe systems inside the universal window, we proceeddifferently starting our description from the effective range expansion, Eq.(2), relating thethree parameters that determine the low-energy dynamics of the system. It can be cast inthe following compact form r e a = 2 r B a B here we have introduced the length r B = a − a B which, together with the energy length a B ,completely determines the S -matrix of systems having one bound state (58, 59). In the caseof a zero-range interaction, a = a B and r B = 0. To study the dynamics of the systems insidethe universal window, we make use of a two-parameter short-range potential and considerthis potential as a minimal low-energy representation of the two-particle interaction fixedby two low-energy data, a B (or a ) and r B (or r e ). In the following, to characterize the universal window we make use of a Gaussian potential: V ( r ) = V e − r /r , r is the interparticle distance, while the strength V and range r are parametersuseful to explore the low-energy dynamics associated with the existence of one (bound orvirtual) state close to threshold. For bound states, the wave function is obtained by solvingthe s -wave Schr¨odinger equation (cid:18) ∂ ∂z − mr V ¯ h e − z − r a B (cid:19) φ B ( z ) = 0 5.where z = r/r and φ B ( z ) is the reduced wave function. At zero energy, r /a B = 0, andat large separation values φ ( z → ∞ ) → − zr /a , from which the scattering length a isextracted. The zero-energy wave function, φ , also determines the effective range r e . Thesmall value of the ratio r e /a B can be used to characterize the unitary window and, limitingthe discussion to the case of one bound state, the energy values of a generic Gaussianpotential inside the window can be organized in the single curves shown in Fig. 1, panels(a) and (b). In panel (a) r e /a B is given as a function of r e /a . Real systems can be placedon the figure using the corresponding values of a , a B and r e . We analyze the dimer ofhelium atoms and the two-nucleon system. In the case of the dimer, experimental data arenot available for all those quantities, so we use values obtained with one of the most widelyused helium-helium interactions, the LM2M2 potential (60). For the purpose of the presentdiscussion, results obtained with this potential are considered equivalent to experimentaldata. For the two-nucleon system we use the experimental values or, equivalently, theresults of a realistic interaction, the AV18 potential (61), to determine the np and nn lowenergy parameters in states J π = 1 + and 0 + . Using the values given in Table 1, the fourcases shown in the figure by solid circles are on top of the Gaussian curve.In panel (b) of Fig. 1 the plot is reformulated in terms of the Gaussian range r , in sucha way that real systems are mapped on the Gaussian curve through their ratio a/a B . Theirpositions on the curve identify the characteristic range r indicated in the figure by thedashed lines. With this range, and the proper strength, a Gaussian potential reproducessimultaneously a and a B . The characteristic ranges for the deuteron, helium dimer, np and nn virtual sates are given in Table 1. The panel (b) of Fig. 1 defines a Gaussian characterization of the unitary window. Theposition of real systems on the Gaussian curve identifies the characteristic ranges. Theassociated Gaussian potentials can be considered as a low-energy representation of the two-body interaction of the systems. Through the variation of the Gaussian strength a system ••
The dynamics of two-body systems inside the universal window are highly independent ofthe details of their mutual interaction. The systems satisfy an approximate CSI, exactlyverified in the case of a zero-range interaction. Though the zero-range case was used manytimes as a first approximation to describe systems inside the universal window, we proceeddifferently starting our description from the effective range expansion, Eq.(2), relating thethree parameters that determine the low-energy dynamics of the system. It can be cast inthe following compact form r e a = 2 r B a B here we have introduced the length r B = a − a B which, together with the energy length a B ,completely determines the S -matrix of systems having one bound state (58, 59). In the caseof a zero-range interaction, a = a B and r B = 0. To study the dynamics of the systems insidethe universal window, we make use of a two-parameter short-range potential and considerthis potential as a minimal low-energy representation of the two-particle interaction fixedby two low-energy data, a B (or a ) and r B (or r e ). In the following, to characterize the universal window we make use of a Gaussian potential: V ( r ) = V e − r /r , r is the interparticle distance, while the strength V and range r are parametersuseful to explore the low-energy dynamics associated with the existence of one (bound orvirtual) state close to threshold. For bound states, the wave function is obtained by solvingthe s -wave Schr¨odinger equation (cid:18) ∂ ∂z − mr V ¯ h e − z − r a B (cid:19) φ B ( z ) = 0 5.where z = r/r and φ B ( z ) is the reduced wave function. At zero energy, r /a B = 0, andat large separation values φ ( z → ∞ ) → − zr /a , from which the scattering length a isextracted. The zero-energy wave function, φ , also determines the effective range r e . Thesmall value of the ratio r e /a B can be used to characterize the unitary window and, limitingthe discussion to the case of one bound state, the energy values of a generic Gaussianpotential inside the window can be organized in the single curves shown in Fig. 1, panels(a) and (b). In panel (a) r e /a B is given as a function of r e /a . Real systems can be placedon the figure using the corresponding values of a , a B and r e . We analyze the dimer ofhelium atoms and the two-nucleon system. In the case of the dimer, experimental data arenot available for all those quantities, so we use values obtained with one of the most widelyused helium-helium interactions, the LM2M2 potential (60). For the purpose of the presentdiscussion, results obtained with this potential are considered equivalent to experimentaldata. For the two-nucleon system we use the experimental values or, equivalently, theresults of a realistic interaction, the AV18 potential (61), to determine the np and nn lowenergy parameters in states J π = 1 + and 0 + . Using the values given in Table 1, the fourcases shown in the figure by solid circles are on top of the Gaussian curve.In panel (b) of Fig. 1 the plot is reformulated in terms of the Gaussian range r , in sucha way that real systems are mapped on the Gaussian curve through their ratio a/a B . Theirpositions on the curve identify the characteristic range r indicated in the figure by thedashed lines. With this range, and the proper strength, a Gaussian potential reproducessimultaneously a and a B . The characteristic ranges for the deuteron, helium dimer, np and nn virtual sates are given in Table 1. The panel (b) of Fig. 1 defines a Gaussian characterization of the unitary window. Theposition of real systems on the Gaussian curve identifies the characteristic ranges. Theassociated Gaussian potentials can be considered as a low-energy representation of the two-body interaction of the systems. Through the variation of the Gaussian strength a system •• Efimov Physics 5 r e /a -0.4-0.200.20.40.6 r e / a B Gaussiana=a B -0.4 -0.2 0 0.2 0.4 r /a -0.4-0.200.20.40.6 r / a B Gaussiana=a B -0.2 0 0.2 0.4 r /a -0.200.20.4 r / a B Gaussian λ V LM2M2 -1 -0.5 0 0.5 1 ε= B /a B f sc (a B /2) C a2
The inverse of the energy length as a function of the inverse of the scattering length, both in units of r e (panel (a)) and r (panel (b)), for a Gaussian potential. The position of selected real systems are indicated by the solid circles. Panel (c):position of the helium dimer (red circle), modified helium dimers (brown triangles) and the two-nucleon systems (blue,orange and green circles) on the Gaussian curve. Panel (d): collapse of the observables on the scaling function.Themodified helium dimers are shown as triangles. can (ideally) be moved along the unitary window. Physical systems exist at their physicalpoints, so the interaction has to be modified to move them from that point. At present,this can be done in the case of the residual interaction between atoms by applying magneticfields to change their electronic structure. The difficult technical implementations of thisprocedure could limit the knowledge of the new interaction allowing only to trace a fewparameters of it. We refer for example to the sector of trapped cold atoms in which theapplied magnetic field is related to changes in the two-body scattering length. When thisparameter is allowed to take large values (and eventually diverges) the system moves insidethe universal window. Since the window is characterized essentially by two parameters, a B and a , the lack of knowledge of the complete interaction is not important: the low-energy properties of the system inside the window are determined by them. Accordinglythe characterization of the universal window by the Gaussian potential could be of interest.To analyze possible trajectories along the unitary window we use as example the LM2M2 e dimer E [mK] a B [ a ] C a [ a − / ] (cid:112) (cid:104) r (cid:105) [ a ] a [ a ] r e [ a ] r [ a ]LM2M2 -1.3035 182.221 0.108985 67.015 189.415 13.845 10.03 λ = 1 .
02 -4.0905 102.864 0.149498 38.979 110.022 13.396 9.99 λ = 1 .
05 -11.137 62.3388 0.200802 24.678 69.4483 12.792 9.94 λ = 1 .
10 -30.358 37.7585 0.277525 16.024 44.7923 11.937 9.88 λ = 1 .
149 -57.981 27.3217 0.349857 12.362 34.2868 11.248 9.86
NN E [MeV] a B [fm] C a [fm − / ] (cid:112) (cid:104) r (cid:105) [fm] a [fm] r e [fm] r [fm] np (1 + ) -2.2245 4.318 0.885 1.967 5.419 1.753 1.559 np (0 + ) -0.066 -25.05 - - -23.74 2.77 1.83 nn (0 + ) -0.102 -20.19 - - -18.90 2.75 1.795He dimer E [mK] a B [ a ] C a [ a − / ] (cid:112) (cid:104) r (cid:105) [ a ] a [ a ] r e [ a ] r [ a ] λ = 0 . λ = 0 . Table 1 Low energy parameters of the helium dimer, calculated with the LM2M2interaction, and those of the np + and + states, and the nn + state calculated withthe AV18 interaction. For the + states the energy values are those of the virtualstate. Parameters of modified helium dimers, as explained in the text, are also shown.The values of the asymptotic normalization constant C a and of the mean square radiusare also reported. See the main text for more details. interaction of two helium atoms and define V λ = λV LM M . λ = 1 refers to the original potential whereas for slightly bigger and lower valuesof λ the system moves along the window. The different λ values generate fictitious heliumdimers mimicking possible modifications of the original potential. For selected cases of λ the corresponding low energy quantities are given in Table 1 and shown in Fig. 1, panel (c),as solid triangles. The two-nucleon systems are shown on the curve too and, for the givenvalues of λ , the position of these modified helium dimers travel along the curve coincidingin specific cases with the nuclear systems. For each case the range of the Gaussian potentialthat reproduces the values of a B and a is given in the last column of Table 1. We noticethat the Gaussian range of the modified dimers varies very little along the window showingthat it is possible to define a characteristic Gaussian range associated to the helium dimer.The above analysis is useful to characterize the universal behavior in terms of theposition of a system inside the universal window. Similar locations inside the universalwindow imply similar dynamical properties. This can be put in evidence using the wavefunction to calculate several observables, such as the mean square radius (cid:104) r (cid:105) = r (cid:90) ∞ dz z φ B ( z ) = a (cid:16) r B a (cid:17) + o ( (cid:16) r B a (cid:17) )) (cid:39) a B e r B /a B , φ B ( z > r B /r ) → C a e − zr /a B and directly related to the residue of the S -matrix at the momentum pole k = i/a B C a (cid:39) a B − r e /a B = 2 a B e r B /a B . ••
NN E [MeV] a B [fm] C a [fm − / ] (cid:112) (cid:104) r (cid:105) [fm] a [fm] r e [fm] r [fm] np (1 + ) -2.2245 4.318 0.885 1.967 5.419 1.753 1.559 np (0 + ) -0.066 -25.05 - - -23.74 2.77 1.83 nn (0 + ) -0.102 -20.19 - - -18.90 2.75 1.795He dimer E [mK] a B [ a ] C a [ a − / ] (cid:112) (cid:104) r (cid:105) [ a ] a [ a ] r e [ a ] r [ a ] λ = 0 . λ = 0 . Table 1 Low energy parameters of the helium dimer, calculated with the LM2M2interaction, and those of the np + and + states, and the nn + state calculated withthe AV18 interaction. For the + states the energy values are those of the virtualstate. Parameters of modified helium dimers, as explained in the text, are also shown.The values of the asymptotic normalization constant C a and of the mean square radiusare also reported. See the main text for more details. interaction of two helium atoms and define V λ = λV LM M . λ = 1 refers to the original potential whereas for slightly bigger and lower valuesof λ the system moves along the window. The different λ values generate fictitious heliumdimers mimicking possible modifications of the original potential. For selected cases of λ the corresponding low energy quantities are given in Table 1 and shown in Fig. 1, panel (c),as solid triangles. The two-nucleon systems are shown on the curve too and, for the givenvalues of λ , the position of these modified helium dimers travel along the curve coincidingin specific cases with the nuclear systems. For each case the range of the Gaussian potentialthat reproduces the values of a B and a is given in the last column of Table 1. We noticethat the Gaussian range of the modified dimers varies very little along the window showingthat it is possible to define a characteristic Gaussian range associated to the helium dimer.The above analysis is useful to characterize the universal behavior in terms of theposition of a system inside the universal window. Similar locations inside the universalwindow imply similar dynamical properties. This can be put in evidence using the wavefunction to calculate several observables, such as the mean square radius (cid:104) r (cid:105) = r (cid:90) ∞ dz z φ B ( z ) = a (cid:16) r B a (cid:17) + o ( (cid:16) r B a (cid:17) )) (cid:39) a B e r B /a B , φ B ( z > r B /r ) → C a e − zr /a B and directly related to the residue of the S -matrix at the momentum pole k = i/a B C a (cid:39) a B − r e /a B = 2 a B e r B /a B . •• Efimov Physics 7 alid up to third order, the scaling function f sc = 11 − r e /a B = e r B /a B . a B withcorrections given by the small parameter r e /a B = 2 r B /a or (cid:15) = 2 r B /a B . In the case of (cid:104) r (cid:105) and C a this is encoded in the scaling function f sc as explicitly shown in Fig. 1, panel(d). The values given in Table 1, properly divided by the indicated factors, are located inthe figure and result on top of the scaling function at the corresponding value of (cid:15) . Thecollapse on the curve is well verified for very different systems, in particular close to theunitary limit. This analysis puts in evidence the CSI and the universal characteristic of thewindow. Moreover, it shows that the dynamics is determined by the small parameter (cid:15) ascontinuously emerging from the unitary point ( (cid:15) = 0). When the interaction between two particles is strongly repulsive at short distances the two-body system is, as a consequence, highly correlated. For bound systems the probability tobe inside the repulsive core is very small. Accordingly, the wave function in that region isalmost zero and increases rapidly towards the attractive region. Therefore the total energyresults from a big cancellation between the kinetic and potential energy. Systems such asthe helium dimer or the deuteron are examples of this kind of correlation. It is interestingto analyze the description of these systems in terms of the low-energy parameters. Outsidethe interaction region the s -wave reduced wave function of the system is ψ B ( r → ∞ ) = C a e − r/a B P e to be in that region is defined as P e = C a (cid:90) ∞ r B e − r/a B dr = C a a B e − r B /a B = 11 − r e /a B e − r B /a B = e − r B /a B = 1 f sc , r B as the lower limit for two particles tobe considered outside the interaction region. Accordingly P e , the probability to be outsidethe interaction region, is the inverse of the scaling function. For weakly bound systemsthis quantity is governed by the ratio 2 r B /a B , therefore we consider the systems inside theunitary window as strongly correlated.
3. The three-body universal window
In this section we discuss the three-body universal window for three equal bosons and threeequal fermions with 1 / nside the universal window (36, 62, 63, 64, 65, 66, 67). On the other side it would be offundamental importance to understand correlations between low-energy properties and thespecific location of a system inside the window. In particular, in the case of nucleons, thesecorrelations will be taken as signatures of universal behavior. In the case of a zero-range interaction the three-body system turns out to be unbound frombelow (Thomas collapse (68)). Its spectrum, deduced by V. Efimov, is given by the Efimovradial law (34, 35): E ( n )3 E = tan θ E ( n )3 + E = e − n − n ∗ ) π/s e ∆( θ ) /s E ∗ . θ , the binding energy of level n , E ( n )3 , is determined simul-taneously by the two-body binding energy, which in the zero-range limit ( a = a B ) is E = ¯ h /ma , and by the binding energy of level n ∗ at the unitary limit, E ∗ = ¯ h κ ∗ /m ,defining the three-body parameter κ ∗ . The function ∆( θ ) is a universal function, the samefor all levels, governing the values of the three-body binding energy inside the window.With the above definition, ∆( − π/
2) = 1, and parametrizations of the universal functionexist (37, 44, 69) for θ varying in the range [ − π, − π/ θ = − π/ E = 0 and thespectrum shows the Efimov effect: a geometrical tower of states with constant energy ratios E ( n )3 /E ( n +1)3 = e π/s where, in the case of three-equal bosons, the universal number is s = 1 . . . . . The zero-range spectrum of Eq.(13) verifies a DSI. It results invariantwhen the scattering length a is scaled by the factor e mπ/s , with m an integer number,maintaining invariant the three-body parameter κ ∗ and the angle θ .The zero-range model can be extended to consider the finite-range character of theinteraction. In this case the Thomas collapse is not present any more and the three-bodyspectrum can be written as E ( n )3 + E = e ∆ ( n )3 ( θ ) /s E ( n ) ∗ n = 0 , , . . . indicates the energy levels and ∆ ( n )3 is the n -level function defined as∆ ( n )3 ( θ ) = s log E ( n )3 + E E ( n ) ∗ . E = ¯ h /ma B and E ( n ) ∗ = ¯ h [ κ ( n ) ∗ ] /m is the energy of level n at the unitary limit, definingthe three-body parameter of each level, κ ( n ) ∗ . When finite-range potentials are used tocompute the n -level function the following behavior is verified (70)∆ ( n )3 ( θ ) → ∆( θ ) n > E ( n ) ∗ E ( n +1) ∗ → e π/s n > . n = 0), show range effects.Starting from n = 2 the energy spectrum closely tends to the zero-range spectrum of ••
2) = 1, and parametrizations of the universal functionexist (37, 44, 69) for θ varying in the range [ − π, − π/ θ = − π/ E = 0 and thespectrum shows the Efimov effect: a geometrical tower of states with constant energy ratios E ( n )3 /E ( n +1)3 = e π/s where, in the case of three-equal bosons, the universal number is s = 1 . . . . . The zero-range spectrum of Eq.(13) verifies a DSI. It results invariantwhen the scattering length a is scaled by the factor e mπ/s , with m an integer number,maintaining invariant the three-body parameter κ ∗ and the angle θ .The zero-range model can be extended to consider the finite-range character of theinteraction. In this case the Thomas collapse is not present any more and the three-bodyspectrum can be written as E ( n )3 + E = e ∆ ( n )3 ( θ ) /s E ( n ) ∗ n = 0 , , . . . indicates the energy levels and ∆ ( n )3 is the n -level function defined as∆ ( n )3 ( θ ) = s log E ( n )3 + E E ( n ) ∗ . E = ¯ h /ma B and E ( n ) ∗ = ¯ h [ κ ( n ) ∗ ] /m is the energy of level n at the unitary limit, definingthe three-body parameter of each level, κ ( n ) ∗ . When finite-range potentials are used tocompute the n -level function the following behavior is verified (70)∆ ( n )3 ( θ ) → ∆( θ ) n > E ( n ) ∗ E ( n +1) ∗ → e π/s n > . n = 0), show range effects.Starting from n = 2 the energy spectrum closely tends to the zero-range spectrum of •• Efimov Physics 9 q.(13). The practical use of Eq.(13) and Eq.(14) depends on the knowledge of the universalor level functions ∆( θ ) or ∆ ( n )3 ( θ ) respectively. In the first case it is possible to solvethe Skorniakov-Ter-Martirosian (STM) equations (71) for different values of the two-bodyscattering length a to cover the region of interest given by − π < θ < − π/ n -level function along the unitary windowis related to the knowledge of the interaction in that region. In general the interaction isknown at one point, the physical point, and to explore the unitary window some assumptionsare needed. Many times scaled potentials have been used to slightly increase or reduce theirstrength as a way to explore the universal window, here we use the Gaussian potential ofEq.(4) as the reference interaction to characterize the universal window.The results for a Gaussian potential of range r with variable strength can be summa-rized in the following equations (72, 73, 74, 75) a B κ ( n )3 = tan θ r κ ( n )3 = γ ( n )3 e ∆ ( n )3 ( θ ) / s sin θ , γ ( n )3 = r κ ( n ) ∗ and E ( n )3 = ¯ h [ κ ( n )3 ] /m . The level functions ∆ ( n )3 are computed solvingthe Schr¨odinger equation with a Gaussian potential with variable strength whereas thepure numbers r κ ( n ) ∗ = γ ( n )3 , define the three-body parameter of each level at θ = − π/
2. Itshould be noticed that ∆ ( n )3 and γ ( n )3 are the same for all Gaussian potentials.In Fig. 2 the first three levels of the Gaussian potential are shown (solid lines) in a[( r /a B ) / , − [ r κ ( n )3 ] / ] plot. The powers 1 / / r /a B = 0 axis indicates the first γ n values whereas the points where the bound states disappear into the three-body continuum, E ( n )3 = 0, are the corresponding values of the scattering length a ( n ) − shown as solid diamonds,in units of the Gaussian range. Using those values the almost model independent quantitiescan be extracted κ (0) ∗ a (0) − = − .
14 20. κ (1) ∗ a (1) − = − .
57 21. κ (2) ∗ a (2) − = − . . κ (0) ∗ a (0) − ≈ − . n = 1 , a − κ ∗ = 1 .
507 (76).In Fig. 2 the two levels of the helium trimer using the LM2M2 interaction, E (0)3 =126 . E (1)3 = 2 .
27 mK are shown as solid squares. Noticing that E = 1 .
303 mK,the position of these data on the plot are fixed through the angle θ defined as E ( n )3 /E =tan θ . The axis value of r /a B = 0 . r (0)0 = 11 . a with which a Gaussian potentialreproduces the dimer and ground state trimer energies. From that value, the three-body
10 Kievsky et al. arameters of the helium trimer, ground and excited states, can be estimated (77) E (0) ∗ = ¯ h m (cid:20) γ r (0)0 (cid:21) = 83 . E (1) ∗ = ¯ h m (cid:20) γ r (0)0 (cid:21) = 0 .
157 mK 24.in complete agreement with the predictions given in literature (78, 79). Moreover at thethree-atom continuum the characteristic range predicts the value a (0) − = − . a in agree-ment with the helium values at that point, see Ref. (78). Using the scaled van der Waalslength of helium, ˜ r vdW = λ / r vdW , the Gaussian trajectory predicts a (0) − / ˜ r vdW ≈ − .
6, inclose agreement with the universal value observed in van der Waals species, see Ref. (44)and references therein.
Considering three equal, spin 0, atoms as representative of the three-boson system, theGaussian characterization of the universal window can be applied to study the atom-dimerscattering length a AD . In the zero-range limit its expression, derived by Efimov (80), is a AD /a B = d + d tan[ s ln( κ ∗ a B ) + d ] , -0.4 -0.2 0 0.2 0.4 0.6 0.8 (r /a B ) -1.5-1-0.50 -[r κ ( n ) ] / -r κ (0)* = -0.4883-r κ (1)* = -0.02125-r κ (2)* = -0.0009362a (0)- = -4.37 r a (1)- = -74.0 r a (2)- = -1613 r Figure 2
The dimensionless quantity, − [ r κ ( n )3 ] / , for n = 0 , ,
3, are shown as a function of ( r /a B ) / fora Gaussian potential (solid lines). The dashed lines are the results from the zero-range model.Notable values at θ = − π/ − π are shown as solid circles and diamonds respectively. Thesolid squares represent the two levels of the helium trimer on the Gaussian n = 0 , ••
3, are shown as a function of ( r /a B ) / fora Gaussian potential (solid lines). The dashed lines are the results from the zero-range model.Notable values at θ = − π/ − π are shown as solid circles and diamonds respectively. Thesolid squares represent the two levels of the helium trimer on the Gaussian n = 0 , •• Efimov Physics 11 here d , d and d are universal numbers and κ ∗ is the three-body parameter belonging toone of the three-body energy branches. The log-periodic functional form of the observableis a consequence of the constraints imposed by the DSI. As a B → ∞ , the ratio a AD /a B forms different branches with asymptotes located at values of a B at which the three-bodylevels disappear into the atom-dimer continuum. In the case of finite-range interactions weuse the parametrization proposed in Ref. (72) a AD /a B = d + d tan[ s ln( κ ( n ) ∗ r ( a B /r ) + Γ ( n )3 ) + d ] , κ ( n ) ∗ r = γ ( n )3 , is used as the driving term and we have introducedthe finite-range three-body parameter Γ ( n )3 , as discussed in Refs. (72, 73), to absorb finite-range corrections.We analyze the behavior of a AD inside the unitary window using a Gaussian potential.Following Ref. (77), we show in Fig. 3 (left panel) two branches of the function a AD /a B (violet solid line) using γ (1)3 = 0 . d = 1 . d = − . d = − .
038 and Γ (1)3 = 0 . r /a B = 0 . . n = 1 level (lower green diamond) andon the n = 2 level (lower orange diamond) are shown too. These points correspond to thecrossing of a straight line passing through the origin, defined by the angle E (1)3 /E = tan θ ,with the n = 1 , n = 1 level, the value of the axis r /a B = 0 . a AD /a B = 1 .
19 (higher green diamond). Therefore the Gaussiancharacterization of the unitary window predicts the atom-dimer scattering length to be a AD = 1 . a B . Using the LM2M2 value, a B = 182 . a , the value a AD = 217 a isobtained which has to be compared to the LM2M2 value for this quantity of 218 . a (81).This demonstrates the capability of the Gaussian characterization of the universal windowto take into account accurately finite-range effects. Accordingly, within an EFT we considerthis result at the NLO level, in the sense that it includes range corrections.The DSI allows to map the excited state of the trimer on a higher branch as it isgiven by the lower orange diamond on the left panel of Fig. 3, corresponding to the values r /a B = 0 . a AD /a B = 1 .
17 (higher orange diamond). The prediction is now a AD ≈ a . Within an EFT we consider this result as corresponding to the LO of theEFT, as the n = 2 or higher branches have almost negligible finite-range effects. Thissimple analysis shows the strong correlation existing between low-energy observables insidethe unitary window. Moreover it shows how the different branches can be used to estimatefinite-range effects. Recent studies of the three-boson continuum can be found in (82) The two-nucleon system in states J π = 0 + and 1 + belongs to the universal window. The 0 + state is an s -wave state whereas the 1 + has a dominant s -wave component at low energies,in the case of the deuteron it is about 95%. The lightest nuclei, H, H, He and He havelarge probabilities to be in L = 0 and therefore we expect to observe universal properties.Important questions to be clarified are the lack of excited states in the three- and four-
12 Kievsky et al. ucleon systems. Moreover the doublet neutron-deuteron scattering length, a nd ≈ .
65 fmhas a very small value compared to the triplet neutron-proton scattering length a np ≈ . a s and a t . Among differentpossibilities we choose to maintain the ratio a s /a t close to the experimental value, a s /a t = − .
38, in our exploration of the unitary window (83). Therefore the change in one value fixesthe value of the other. To characterize the universal window we construct a spin-dependentGaussian potential with different strengths and ranges in the spin-isospin channels
S, T =0 , , V ( r ) = V e − r /r P + V e − r /r P (r /a B ) -2-1012 -[r κ ( n ) ] / a AD /a B r /a B -2-1012 -(r κ ) / a nd /a B2 a nd k p Figure 3
Left panel: The a AD /a B function (violet solid line) inside the ( r /a B ) / , − [ r κ ( n )3 ] / diagram, as explained in the text.Right panel: The a nd /a B function (violet solid line) and the pole momentun k p (multiplied by a nd ) inside the( r /a B ) , − [ r κ ( n )3 ] / diagram, as explained in the text. ••
Left panel: The a AD /a B function (violet solid line) inside the ( r /a B ) / , − [ r κ ( n )3 ] / diagram, as explained in the text.Right panel: The a nd /a B function (violet solid line) and the pole momentun k p (multiplied by a nd ) inside the( r /a B ) , − [ r κ ( n )3 ] / diagram, as explained in the text. •• Efimov Physics 13 here P projects onto the S, T = 0 , P onto the S, T = 1 , J π = 1 / + state considering r = r , for which choice, at the unitary limit, the spectrum coincides with the boson case.The Gaussian strengths are varied to examine the plane ( r /a B , − r κ ( n )3 ), with E ( n )3 =¯ h [ κ ( n )3 ] /m being the binding energy of level n and E = ¯ h /ma B the two-body bindingenergy of the triplet state. In Fig. 3, right panel, we show the ground state, n = 0 (blue line)and first excited state, n = 1 (green line) of the J = 1 / + three-nucleon system whereas thered line is the ground state of the 1 + two-nucleon system. The H nucleus is mapped on theGaussian ground state curve as a blue circle at coordinates verifying κ (0)3 a B = tan θ = 1 . .
48 MeV withthe deuteron binding energy of 2 .
224 MeV. At that point r /a B = 0 .
457 from which thecharacteristic Gaussian range r (0)0 = 1 .
97 fm can be estimated and used to assign a valueof the three-nucleon system at unitarity through the quantity κ (0) ∗ r = 0 . E (0) ∗ ≈ .
55 MeV in good agreement with previous estimates (22, 84).Furthermore, we also show in the right panel of Fig. 3 the doublet neutron-deuteronscattering length, a nd , calculated with the Gaussian interaction, as the violet curve (in unitsof the energy length a B ). It corresponds to a fit of the numerical results using the formgiven by Eq.(26), see Ref. (77). Two branches are shown, with the dashed vertical line, theasymptote at r /a B = 0 . a nd diverges and the firstexcited state disappears into the 1+2 continuum. Using the characteristic range r (0)0 = 1 . a B = 18 . .
12 MeV and a scattering length around 20 fm, very far fromthe corresponding physical values. This simple analysis explains the one level structureof H in terms of its position inside the Gaussian characterization of the unitary window.The correlation between the ground state and the doublet scattering length can be studiedlooking at the value of a nd /a B for r /a B = 0 . a nd /a B = 0 .
08, indicated as the upper blue solidcircle in the figure. Using the deuteron energy length a B = 4 .
32 fm, the resulting doubletscattering length is a nd ≈ . .
65 fm, however this analysis explains the very low value of this quantity if compared tothe value of the np triplet scattering length. We observe the very delicate region in which a nd is located, where slightly different values of r /a B could produce large variations of a nd , including a change of sign. The Gaussian characterization maps a nd in the correct(positive) region clarifying the strong correlation between this quantity and the H energy, aproperty observed already many years ago (85). It is possible to use the higher branch of the a nd /a B curve to determine the size of finite-range corrections. The triton point is locatedon the n = 1 level (lower green circle) at r /a B = 0 .
015 corresponding to a nd /a B = 0 . n = 0 level. As for the boson case,these two estimates can be considered in the EFT as corresponding to the NLO and LOrespectively. This simple analysis explains some peculiarities of the nuclear system strictlycorrelated to its location inside the universal window.Finally we discuss the evolution of the three-nucleon virtual state after the n = 1 levelcrosses the 1 + 2 continuum. Following Refs. (77, 86) the S -matrix energy pole, E P = − h k p / m is determined from the s -wave low energy phase-shifts calculated using theGaussian potential of Eq.(27). The behavior of the a nd k p function is shown in Fig. 3 (rightpanel) as a cyan solid line fitting the numerical calculations (cyan diamonds). This functioncrosses the physical point at r /a B = 0 .
457 from which the triton virtual state, E p =
14 Kievsky et al. . − .
25 0 .
00 0 .
25 0 . r /a B − − − − − r κ ( m ) N E (0)3 E (0)4 E (0)5 E (0)6 E (1)4 E (1)5 E (1)6 Figure 4 N ≤ a B , both in units or r . The squares represent the groundstates of the He N clusters calculated with the realistic TTY potential (91). .
48 MeV can be extracted. The Gaussian characterization explains this value in agreementto experimental determinations and theoretical investigations (87, 88, 59, 89, 49, 90).
4. Characterization of the unitary window for more than three particles
The Gaussian characterization of the universal window can be extended to describe systemscomposed by more than three particles. The DSI, which emerges in the three-body sectorand gives rise to the Efimov spectrum, strongly constrains the
N >
A > A nucleons, the spatial-symmetric wave function is dominant only up to four particles, and deviations from thebosonic-Efimov scenario appear for the A > N -boson systems The unitary window for N bosons can be characterized using the Gaussian potential ofEq. (4). Tuning the strength of the potential the ground- and excited-state energies, E (0) N and E (1) N , are calculated as a function of the two-body scattering length a , or equivalentlythe energy length a B . The results of these calculations are presented in Fig. 4 for N =4 , ,
6. Results for E (0) N , up to N = 70, can be found in Ref. (79). A striking featureof the N = 4 , , N ,there are two bound states, one deep and one shallow, below each N − N -body system. Studies in the four-body systemexists (92, 93, 94, 95, 96, 97). The existence of the twin-level structure is not restrictedto a number of particles N ≤
6; for the Gaussian potential, the pattern is maintained ••
6; for the Gaussian potential, the pattern is maintained •• Efimov Physics 15 p to N = 12 (73, 98). For a number of particles N >
12 a third level appears as oneconsequence of the finite range character of the force. The DSI smears out allowing for atransition between universal and non-universal behavior as the study of the unitary windowis extended to consider deep bound states (79). Limiting the discussion to the two-levelstructure, Eq.(19) is extended for
N > a B κ ( m ) N = tan θ r κ ( m ) N = γ ( m ) N e ∆ ( m ) N ( θ ) / s sin θ m = 0 , m = 0 being the N -body ground state and m = 1 the excited state close to the( N − γ ( m ) N = r κ ( m ) ∗ ,N , determining the energies at theunitary limit, E ( m ) ∗ ,N , are characteristic of every Gaussian potential and their values, up to N = 6, are given in Table 2. The energy of the level m is E ( m ) N = ¯ h [ κ ( m ) N ] /m and ∆ Nm ( θ )is the Gaussian level function for N bosons in the states m = 0 , ( m ) N ( θ ) = s log E ( m ) N + E E ( m ) ∗ ,N . N = 4 case, ∆ ( m )4 ( θ ) is explicitly given in Ref. (70) where it is compared to the zero-range four-body universal function. To put in evidence the DSI character of the N -bosonsystem, many efforts have been done to determine universal ratios between the N -bodybound state energies at the unitary point in the limit of zero-range interaction. Precisenumbers exist for N = 4 (92) whereas estimates exist for higher systems (38, 99, 100,101, 102). The Gaussian ratios γ (0) N /γ (1) N and γ ( m ) N +1 /γ ( m ) N can be inferred from the values inTable 2.As illustration of the effectiveness of the Gaussian characterization we analyze He N clusters, largely studied with realistic helium-helium interactions (103, 91). We map thesesystems on the Gaussian curves of Fig. 4 (solid squares) using the energies calculated inRef. (91). The position on the Gaussian curve fixes the ground state characteristic radius r (0) N for each N -body system; it can be used to predict the energy of the ground and excitedstate of the clusters at the unitary limit. The corresponding results are given in Table 2.For the sake of comparison, the results of He-He potential HFD-HE2 (104), re-scaled atthe unitary limit as discussed in Ref. (79), are shown in the last column. A remarkableagreement, better than 2%, is obtained.In general, the knowledge of the N = 2 − E (0) N /N , of the homogeneoussystem. A strict correlation between the low-energy dynamics of the few-body system andthe many-body system, induced from the position of the system inside the universal window,exists (79, 105, 106, 107, 108, 109, 110). We make one more step in the study of universal behavior of real systems located insidethe unitary window showing that the curves describing the N -body energies as a functionof the energy length for different number of particles are actually the same curve. Thisis a manifestation of the strong constraints imposed by the DSI and controlled by the
16 Kievsky et al. γ (0) N γ (1) N r (0) N E (0) ∗ ,N E (1) ∗ ,N E (0) ∗ ,N (HFD-HE2)4 1.1847 0.512 11.85 a a a Table 2 Gaussian pure numbers r κ ( m ) ∗ ,N = γ ( m ) N , m = 0 , at unitarity, the characteristicrange and the energies at the unitary limit for bosonic helium (first rows) and fournucleons (last row). In the boson case, the results using the re-scaled HFD-HE2potential at the unitary limit are shown in the last column. three-body parameter (106). Following Refs. (72, 111, 74) the Efimov radial law, extendedin Eqs.(19) and (29) to describe finite-range interactions, can be related to the three-bodyuniversal function by the introduction of the N -body finite-range parameter Γ ( n ) N at differentbranches. In the specific case of N = 3, Eq.(19) is modified by explicitly relating the finite-range spectrum to the zero-range universal function ∆( θ ), as follows κ ( n )3 a B = tan θ , κ ( n ) ∗ a B + Γ ( n )3 = e − ∆( θ ) / s cos θ , N -body system, ground and excited state, has its finite-rangeparameter Γ (0) N and Γ (1) N , so that Eq.(29) is modified as κ ( m ) N a B = tan θ , κ ( m ) ∗ ,N a B + Γ ( m ) N = e − ∆( θ ) / s cos θ , m = 0 and the excited m = 1 states tothe three-body universal function. In Fig. 5 we see that the finite-range parameter Γ ( m ) N ,which encodes the finite-range corrections, can be used to make both, the ground andexcited states of the few-body systems, collapsing on the three-body universal curve givenby the Efimov radial law in Eq.(13). This is a clear sign that these systems belong to thesame universality class and that their spectra are constrained by a DSI governed by thethree-body parameter κ (0) ∗ . A ≤ universal window We have already discussed the universal character of the three-nucleon low energy spectrumshowing the existence of strong correlations between observables related to the position ofthe two- and three-nucleon systems inside the universal window. These properties suggestthe possibility of describing nuclear physics as continuously linked to the unitary limit (20,113, 114, 115). Here we show the A = 4 , L = 0, nuclear spectrum along the nuclearcut a s /a t = − .
38 using the Gaussian two-channel potential of Eq. (27). First of all, notconsidering the Coulomb interaction, the A = 2 − H and He, are degenerate, moreover there is aninfinite tower of excited states at unitarity as in the boson case. As the value of r /a B increases the excited states disappear one by one and the last one, indicated in the figure as H ∗ , disappears at r /a B = 0 .
101 resulting in the observed one level structure of H. Thefour body spectrum has similar behaviour to the bosonic case: it has a two-level structure,a deep state corresponding to He, and one excited state, He ∗ , close to the three-body ••
101 resulting in the observed one level structure of H. Thefour body spectrum has similar behaviour to the bosonic case: it has a two-level structure,a deep state corresponding to He, and one excited state, He ∗ , close to the three-body •• Efimov Physics 17 − . − . − . . . . E ( ) N / E / ( κ (0) ∗ ,N a B + Γ (0) N ) N = 3 N = 4 N = 5 N = 6 − . − . − . . . . E ( ) N / E / ( κ (1) ∗ ,N a B + Γ (1) N ) N = 3 N = 4 N = 5 N = 6 Figure 5
The N boson ground (left panel) and first excited (right panel) binding energies, E (0) N and E (1) N , inunits of E as a function of the inverse of the energy length a B , in units of the N -body parameter κ ( m ) ∗ ,N shifted by the finite-range parameter Γ ( m ) N . Different symbols represent different potentialmodels as described in Ref. (74). Adapted from Ref.(74) with permission. threshold. To be noticed that this state, which is a resonance, results bound withoutconsidering the Coulomb interaction (84). The H and He nuclei can be mapped on theGaussian curves through the angles defined by the corresponding energy ratios, E /E and E /E . They are indicated in Fig. 6, left panel, as a green solid square ( H) and as a redsolid square ( He). In the He case it should be taken E = 29 . r (0) = 2 .
078 fm, from which the binding energies at unitarity, E (0) ∗ , and E (1) ∗ , , can be deduced. They are reported on Table 2.The spectrum of the A = 6 systems along the nuclear cut is reported in the right panelof Fig. 6. There are two different A = 6 states discriminated by their spin-isospin quantumnumbers: the He with S = 0 , T = 1, and the Li with S = 1 , T = 0. Interestingly, neitherare present at the unitary limit, being above the corresponding thresholds He and He + d , respectively. As r /a B moves toward positive values they emerge from their thresholds,first Li at r /a B = 0 .
07, and then He at r /a B = 0 .
19. At the physical point, the lightnuclear spectrum (without considering the Coulomb interaction) consists in one level for H and He, which are degenerated, two levels for He and one level for He and for Li.The evolution of the excited He state considering the Coulomb interaction is discussed insection 5.2. The present analysis gives a simple explanation of the light nuclear spectrumas emerging continuously from the unitary limit.
5. Implications of Efimov physics in determining the nuclear EFT
The Gaussian characterization of physical systems in the Efimov window corresponds to aregularized version of the LO EFT description, where the Gaussian range is the inverse ofthe ultraviolet cutoff. In this formulation, finite range effects are implicitly contained inthe cutoff and disappear as the latter is removed, recovering a scale invariant description.Even in this limit, a scale has nevertheless to be introduced at the 3-body level in the formof a dimensionful 3-body parameter (37, 44): as a matter of fact, the short-distance two-body dynamics does not decouple in the 3-body sector and manifests itself as an additional3-body interaction in the LO EFT, designed to absorb all the cutoff dependence in the
18 Kievsky et al. . . . . r /a B ) / − . − . − . − . − . − . − . . − ( r κ N ) / d H ∗ H He ∗ He . . . . . r /a B − . − . − . − . − . − . − . − r κ N He He+ d He Li Figure 6
Left panel: Square root of the binding momenta κ , κ ( n )3 , κ ( m )4 for d , H, and He respectivelyalong the nuclear cut a s /a t = − . a B , both in unitsof the Gaussian range r . The position of H and He at the physical point are given as a greenand red solid squares respectively. Right panel: Binding momentum along the nuclear cut for A = 6 as a function of a B both in units of r . In both panels the Coulomb interaction has notbeen taken into account. Adapted from Ref.(84) with permission. zero range limit. By specifying the corresponding strength through a 3-body datum, thecontinuous scale invariance is broken to a discrete scale invariance.The sensitivity of the 3-body system to the short distance two-body dynamics dependssolely on the proximity to the unitary limit, and persists after the inclusion of finite rangeeffects (117). In particular, it also applies if the EFT is interpreted as a finite cutoff effectivetheory `a la Lepage (118, 119), where the renormalization is done implicitly, through thefitting of low-energy constants, and cutoff independence is only attained up to neglectedhigher orders. With a finite cutoff the Thomas collapse is avoided and there is a well defined3-body ground state as well as, close to unitarity, all the higher Efimov states, as exemplifiedin Fig. 2. In this perspective the cutoff is interpreted as a physical parameter related to theintrinsic scale of the theory, and therefore it is bound to assume values inside a given naturalrange. This constraint also identifies the spectrum of 3-body bound states as a functionof the 2-body scattering length, or alternatively of the 2-body binding energy. Variationsof the cutoff within the natural range induce drastic changes in the 3-body spectrum, themore so the closest the system is to the unitary limit, due to the existence of densely spacedEfimov states. Thus, close to the unitary limit, the extreme sensitivity to the cutoff alsoaffects the finite-cutoff theory, because of the strong correlations between the ground stateand the other bound states, reflecting the remaining DSI. The sensitivity also concerns thecontinuum states, as exemplified in Fig. 3, where one can verify that small changes in r produce a change of sign in the scattering length. The introduction of a LO 3-body forceallows to set correctly the 3-body ground state energy, bringing the lowest branch of theEfimov plot to the curve that follows the evolution of the physical state to the unitary limit.Stated differently, one can say that, without a 3-body force at LO, the scale of the 3-body ••
Left panel: Square root of the binding momenta κ , κ ( n )3 , κ ( m )4 for d , H, and He respectivelyalong the nuclear cut a s /a t = − . a B , both in unitsof the Gaussian range r . The position of H and He at the physical point are given as a greenand red solid squares respectively. Right panel: Binding momentum along the nuclear cut for A = 6 as a function of a B both in units of r . In both panels the Coulomb interaction has notbeen taken into account. Adapted from Ref.(84) with permission. zero range limit. By specifying the corresponding strength through a 3-body datum, thecontinuous scale invariance is broken to a discrete scale invariance.The sensitivity of the 3-body system to the short distance two-body dynamics dependssolely on the proximity to the unitary limit, and persists after the inclusion of finite rangeeffects (117). In particular, it also applies if the EFT is interpreted as a finite cutoff effectivetheory `a la Lepage (118, 119), where the renormalization is done implicitly, through thefitting of low-energy constants, and cutoff independence is only attained up to neglectedhigher orders. With a finite cutoff the Thomas collapse is avoided and there is a well defined3-body ground state as well as, close to unitarity, all the higher Efimov states, as exemplifiedin Fig. 2. In this perspective the cutoff is interpreted as a physical parameter related to theintrinsic scale of the theory, and therefore it is bound to assume values inside a given naturalrange. This constraint also identifies the spectrum of 3-body bound states as a functionof the 2-body scattering length, or alternatively of the 2-body binding energy. Variationsof the cutoff within the natural range induce drastic changes in the 3-body spectrum, themore so the closest the system is to the unitary limit, due to the existence of densely spacedEfimov states. Thus, close to the unitary limit, the extreme sensitivity to the cutoff alsoaffects the finite-cutoff theory, because of the strong correlations between the ground stateand the other bound states, reflecting the remaining DSI. The sensitivity also concerns thecontinuum states, as exemplified in Fig. 3, where one can verify that small changes in r produce a change of sign in the scattering length. The introduction of a LO 3-body forceallows to set correctly the 3-body ground state energy, bringing the lowest branch of theEfimov plot to the curve that follows the evolution of the physical state to the unitary limit.Stated differently, one can say that, without a 3-body force at LO, the scale of the 3-body •• Efimov Physics 19 round state is a cutoff effect, and as such it is affected by a sizeable uncertainty. Close to theunitary limit, this uncertainty would propagate to all the tower of Efimov states, resultingin a very poor description of the shallowest ones. Thus, the EFT would be totally unable todescribe those states which should, on the contrary, better fit in the domain of applicabilityof the EFT. The intrinsic length scale of the underlying interaction can be reconstructedby locating the systems on the universal curves through the value of the correspondingGaussian range r . For example, a N -body physical system, having energy E (0) N , is mappedon the Gaussian characterization of the universal window through the energy ratio E (0) N /E ,where E is the energy of the corresponding two-body system. Limiting the discussion toequal particles and a single two-body energy level, this procedure is unambiguous. Theposition of the system fixes the characteristic radius r (0) N with which a Gaussian potentialwith variable strength describes a path linking the physical point, determined by ( E , E (0) N ),to the unitary point, determined by E = 0 and E (0) ∗ ,N = [ γ (0) N ] ¯ h /m [ r (0) N ] . Consideringdifferent values of N of the same physical system, different characteristic ranges are obtainedas it is clear on Figs. 4,6. Though these different Gaussian potentials are useful to determinethe paths to the unitary limit, they define different potentials in each N -body sector, withdifferent ranges all having the same order of magnitude.By introducing a 3-body force at the LO all these different descriptions can be unifiedas deriving from a single underlying effective Lagrangian, comprising two- and three-bodycontact interactions (117). The great complexity of QCD interactions produces very disparate phenomena at variousscales. In the chiral limit, spontaneous chiral symmetry breaking takes place, leading to theemergence of long-range collective modes, the Goldstone bosons, represented by the pions.Thus the chiral limit defines a critical point. Since chiral symmetry is only approximate,the pions acquire a mass but they keep their Goldstone bosons’ character in that theirinteractions are weak at low energies. This enables in turn the perturbative approach tonuclear interactions known as the Chiral-EFT or ChEFT (13, 14, 15, 16, 17, 18, 120, 121,122). Within this approach, the 3-nucleon interaction is only a small perturbation, arisingat the third order of the perturbative scheme.Although not as directly linked to the QCD parameters as the chiral limit, anothercritical point can be identified in the parameter space, corresponding to the unitary limit.In this case the separation of scales is provided by the large scattering lengths, resultingin a different (pionless) EFT (24, 27, 28, 29). The two low-energy expansion schemes aredifferent. In particular, for the reasons already explained, in the pionless EFT the 3-nucleonforce is part of the LO description.The question of the actual importance of the 3-nucleon force depends on which one ofthe two critical points can be considered as closer to the physical point. Furthermore, whilethe chiral regime of very small quark masses is outside of the Efimov window, ruled by thebehaviour in the unitary limit, because the scattering lengths are natural in that limit (123),the unitary regime is met for values of the pion mass around 200 MeV (22, 23) where theChEFT should still apply. This means that the ChEFT treatment of the 3-nucleon forcecould have to be modified accordingly, by promoting it to the LO, as required by Efimovphysics (117). Indeed, although formally consistent, the ChEFT expansion scheme wouldfail in reproducing the universal correlations arising at the unitary limit, unless very high
20 Kievsky et al. rders in the low-energy expansion are reached, so as to include the needed 3-nucleon force.In order to study the impact of the explicit inclusion of the pion-range interactions onthe sensitivity to the short-distance dynamics which was discussed previously, we make useof the following lowest order Hamiltonian (117) H LO = T + (cid:88) i 03 MeV is the average pion mass, g A = 1 . 29 is the nucleon axial couplingconstant and F π = 2 f π = 184 . 80 MeV is the pion decay constant. The regularizationparameter β is used to smoothly relate the chiral LO Hamiltonian to the pionless ( β → ∞ )LO Hamiltonian. Moreover, H LO includes a three-body term of the form W ( i, j, k ) = W e − r ij /r e − r ik /r . r the three-body range, r ij = | r i − r j | , and the sum in Eq.(33) includes cyclicpermutations of the three particles. + state of He As a first application of the pionless LO Hamiltonian we study the evolution of the Heexcited state turning on adiabatically the Coulomb interaction by considering V EM = (cid:15)e /r .The parameters of the two-body potential are fixed to reproduce the np scattering lengthand effective range in channels S, T = 0 , , H. Turning on smoothly the Coulomb interaction varying (cid:15) from 0 to 1, the two-body potential does not changes whereas the strength of the three-bodyterm is modified, maintaining its range fixed, to reproduce the triton energy at each step.The results are shown in Fig. 7 where, for (cid:15) = 0, we observe one A = 3 state and the two A = 4 states (38, 84, 94, 96, 101, 124). As the value of the Coulomb interaction grows toits full value, (cid:15) = 1, the degeneracy between the H and He is removed and the values ofthe ground- and excited-state energies of He change. For (cid:15) ≈ . 75 the He excited statedisappears onto the H+p threshold; a polynomial fit gives the critical value at (cid:15) ∗ = 0 . (cid:15) = 1 the correct low-energy three- and four-nucleon spectrum is recovered (84). The application of the H LO , given in Eq.(33) to the case of A = 3 , •• 75 the He excited statedisappears onto the H+p threshold; a polynomial fit gives the critical value at (cid:15) ∗ = 0 . (cid:15) = 1 the correct low-energy three- and four-nucleon spectrum is recovered (84). The application of the H LO , given in Eq.(33) to the case of A = 3 , •• Efimov Physics 21 − . . . . . . − − He He ∗ H He . . . . . . (cid:15) . . . . . . E A ( M e V ) Figure 7 Evolution of the A = 3 , (cid:15) . The full Coulomb interaction corresponds to (cid:15) = 1.The four-body excited state disappears at the critical value (cid:15) ∗ = 0 . (cid:15) = 1 the experimentalenergies of H, He and He are reproduced within a 1% accuracy. Adapted from Ref.(84) withpermission. calculated using the Brueckner–Bethe–Goldstone (BBG) quantum many-body theory inthe Brueckner–Hartree–Fock (BHF) approximation (see e.g. (126, 127, 128) and referencestherein). In the calculations the three-nucleon force has been reduced to an effective, densitydependent two-body force, by averaging over the coordinates of the third nucleon (127).The energy per particle E/A of symmetric nuclear matter (SNM) is shown in Fig. 8 forvarious parametrizations of the two- and three-body forces. In each panel, for a fixed valueof the OPEP regulator β of the two-body force, the saturation curve (i.e. E/A as a functionof the nucleonic density ρ ) of SNM is shown using four different values of the three-nucleonforce range r , determined to describe the H binding energy. The empirical saturationpoint of SNM ( ρ = 0 . ± . 01 fm − , E/A | ρ = − . ± . r compatible with a correct description of He, with the best description obtained in thepionless case, β → ∞ . This is another example of strict correlations, in this case for thenuclear system, between the low-energy few-body properties and the many-body system,which are induced by the physics of the universal window. SUMMARY POINTS 1. The Gaussian characterization of the univeral window discussed in this article isbased on the simplest description of the one level two-body S -matrix(129, 130) S ( k ) = k + i/a B k − i/a B k + i/r B k − i/r B , r B (cid:28) a B , 22 Kievsky et al. E / A [ M e V ] r = 1.0 fmr = 1.2 fmr = 1.4 fmr = 1.6 fm r = 1.2 fmr = 1.3 fmr = 1.35 fmr = 1.4 fm r = 1.0 fmr = 1.2 fmr = 1.25 fmr = 1.4 fm ρ [fm -3 ]-20-1001020 E / A [ M e V ] r = 1.0 fmr = 1.1 fmr = 1.15 fmr = 1.2 fm ρ [fm -3 ] r = 1.0 fmr = 1.1 fmr = 1.15 fmr = 1.2 fm ρ [fm -3 ] r = 1.2 fmr = 1.3 fmr = 1.4 fmr = 1.5 fm β = ∞ β =10 fm β = 5 fm β = 2 fm β = 1.8 fm β = 1 fm Figure 8 Energy per particle of symmetric nuclear matter E/A as a function of the nucleonic density ρ for several combinations ofthe LO two- and three-body interactions. The yellow box denoted the empirical saturation point. Adapted from Ref.(125)with permission. we have selected a Gaussian potential to reproduce this behavior in the two-bodysector and used it to extend the description to larger systems.2. Two-body systems manifest a CSI, depicted in Fig. 1, which is broken to a DSI inthree-body systems. The adopted procedure allows to address the impact of finiterange corrections on three-body levels. Interestingly, only the first two levels areaffected in a significant way, with the higher levels tending rapidly to the zero rangelimit. The position of a physical system on the lowest level is controlled by a three-body datum. Then its spectrum and correlations with the low-energy scatteringstates are completely determined. Examples have been shown for two very differentsystems, the helium trimer and the three-nucleon system. Furthermore, the studyhas been extended to larger systems showing how they are still constrained by theDSI.3. We have highlighted a number of properties, for systems belonging to the universalwindow, that can be understood as consequences of their position inside the window.The present analysis suggests that the EFT describing nuclear interactions should •• E/A as a function of the nucleonic density ρ for several combinations ofthe LO two- and three-body interactions. The yellow box denoted the empirical saturation point. Adapted from Ref.(125)with permission. we have selected a Gaussian potential to reproduce this behavior in the two-bodysector and used it to extend the description to larger systems.2. Two-body systems manifest a CSI, depicted in Fig. 1, which is broken to a DSI inthree-body systems. The adopted procedure allows to address the impact of finiterange corrections on three-body levels. Interestingly, only the first two levels areaffected in a significant way, with the higher levels tending rapidly to the zero rangelimit. The position of a physical system on the lowest level is controlled by a three-body datum. Then its spectrum and correlations with the low-energy scatteringstates are completely determined. Examples have been shown for two very differentsystems, the helium trimer and the three-nucleon system. Furthermore, the studyhas been extended to larger systems showing how they are still constrained by theDSI.3. We have highlighted a number of properties, for systems belonging to the universalwindow, that can be understood as consequences of their position inside the window.The present analysis suggests that the EFT describing nuclear interactions should •• Efimov Physics 23 ncorporate a three-nucleon term at LO, independently if the pions are integratedout or not. This important consequence is based on the extreme sensitivity of thethree-nucleon system to the cutoff effects at LO. The inclusion of a three-nucleonforce at LO in the nuclear hamiltonian will have significant consequences in thedescription of nuclei using precise interactions derived from ChEFT (121, 122, 117,131, 132).4. As the Gaussian characterization is used to describe systems with larger numberof particles, system-specific non-universal behavior starts to emerge (79, 105). In-deed, the position of a system inside the universal window determines the two-bodyGaussian potential from the values of a B and a whereas the three-body bindingenergy determines the strength of the three-body potential. The use of this two-plus three-body potential to describe heavier systems introduces a dependence ona short range scale which is a non universal effect. This effect can be incorporatedtuning the range of the three-body interaction, a parameter that can be used toimprove the convergence of the EFT expansion, or by including higher orders.5. Efimov physics has substantial implications for the dynamical description of systemslocated inside the universal window. These systems are strongly constrained by an(approximate) scale invariance. A thorough analysis of its consequences in themany-body sector is an important task which is at present intensively pursued. 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