Universal aspects of light halo nuclei

Progress in Particle and Nuclear Physics 67 (2012) 939–994

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Progress in Particle and Nuclear Physics journal homepage: www.elsevier.com/locate/ppnp

Review

Universal aspects of light halo nuclei T. Frederico a,∗ , A. Delfino b , Lauro Tomio b,c , M.T. Yamashita c a

Departamento de Física, Instituto Tecnológico de Aeronáutica, 12.228-900, São José dos Campos, SP, Brazil

b

Instituto de Física, Universidade Federal Fluminense, 24210-346, Niterói, RJ, Brazil

c

Instituto de Física Teórica, Universidade Estadual Paulista, 01140-070, São Paulo, Brazil

article

info

Keywords: Halo nuclei Few-body systems Binding energies and masses Faddeev equation

abstract The theoretical status on universal aspects of weakly-bound neutron-rich light nuclei are reviewed, considering few-body approaches. We focus the review on the low-energy properties of light halo nuclei that can be treated within two- and three-body approaches (with one- and two-neutron halos), which are dominated by s-wave two-body interactions. The representative works studying the large two-neutron halos in light exotic nuclei with short-range interactions show that the general properties associated with the halo neutrons are model independent and obey scaling laws, which are functions of the low-energy observables of the neutron–neutron and neutron–core subsystems, with one additional scale that represents the physics of the three-body system at short-ranges. The scaling laws for the s-wave two-neutron halos are identified with limit-cycles in a renormalized zero-range three-body model. The necessary basic concepts for interpreting the physics of large halos, and also to treat the zero-range interaction in few-body systems, are given. © 2012 Elsevier B.V. All rights reserved.

Contents 1.

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Introduction............................................................................................................................................................................................. 940 1.1. Generalities on halo-nuclei investigations................................................................................................................................ 940 1.2. Few-body long-range correlations and weakly-bound halo nuclei ........................................................................................ 942 1.3. Universal few-body phenomena ............................................................................................................................................... 944 1.4. Low-energy scales and correlations between observables ...................................................................................................... 945 1.5. Renormalization and limit cycles .............................................................................................................................................. 947 1.6. Three-body halo-nuclei structure.............................................................................................................................................. 947 1.7. Outline of the review .................................................................................................................................................................. 948 Basic concepts in few-body problems ................................................................................................................................................... 949 2.1. Two-body problem ..................................................................................................................................................................... 949 2.1.1. Scattering, bound, virtual and resonant states .......................................................................................................... 949 2.1.2. Trajectory of S-matrix poles ....................................................................................................................................... 952 2.1.3. Two-body n–core halo system.................................................................................................................................... 953 2.2. Three-body problem ................................................................................................................................................................... 954 2.2.1. Thomas–Efimov Effect................................................................................................................................................. 954 2.2.2. Three-boson virtual state ............................................................................................................................................ 955 2.2.3. Bound, virtual and resonant Efimov states ................................................................................................................ 956 2.2.4. Scaling functions and limit cycles .............................................................................................................................. 957 2.2.5. Scaling of Efimov trimer energies............................................................................................................................... 958

Corresponding author. Tel.: +55 12 39475939. E-mail address: [email protected] (T. Frederico).

0146-6410/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ppnp.2012.06.001

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T. Frederico et al. / Progress in Particle and Nuclear Physics 67 (2012) 939–994

2.3. Illustration: Born–Oppenheimer three-particle model............................................................................................................ 959 Regularization, renormalization and EFT methods............................................................................................................................... 962 3.1. Subtracted T -matrix equations.................................................................................................................................................. 963 3.1.1. Renormalized Hamiltonian ......................................................................................................................................... 964 3.1.2. Renormalization group invariance ............................................................................................................................. 964 3.2. Examples: single-channel s- and p-waves ................................................................................................................................ 965 3.2.1. s-wave contact interaction.......................................................................................................................................... 965 3.2.2. p-wave contact interaction ......................................................................................................................................... 966 3.3. Subtracted Faddeev equations and renormalized Hamiltonian .............................................................................................. 967 3.4. Effective field theory and halo nuclei ........................................................................................................................................ 969 3.5. Beyond three-particles ............................................................................................................................................................... 971 Universality in three-body halo systems............................................................................................................................................... 971 4.1. Neutron–neutron–core systems ................................................................................................................................................ 972 4.1.1. Classification scheme: Borromean, Tango, Samba and all-bound systems ............................................................. 972 4.1.2. Short-range model for two-neutron halo .................................................................................................................. 972 4.1.3. Renormalized zero-range model ................................................................................................................................ 974 4.2. Halo Efimov states ...................................................................................................................................................................... 975 4.2.1. Critical conditions for binding excited halos ............................................................................................................. 975 4.2.2. Bound and virtual halo states in 20 C........................................................................................................................... 977 4.2.3. n-19 C scattering close to an Efimov state ................................................................................................................... 978 4.3. Two-neutron halo structure, wave-functions and mean-square distances............................................................................ 978 4.4. Two-neutron halos in 11 Li, 14 Be, 20 C and 22 C ............................................................................................................................ 980 4.4.1. Classification and sizes ................................................................................................................................................ 980 4.4.2. Mean separation distances in halo nuclei .................................................................................................................. 982 22 4.4.3. C — the largest halo nucleus .................................................................................................................................... 984 4.5. Neutron–neutron correlation functions .................................................................................................................................... 985 4.6. Halos with more than two-neutrons ......................................................................................................................................... 986 Summary ................................................................................................................................................................................................. 987 Acknowledgments .................................................................................................................................................................................. 990 References................................................................................................................................................................................................ 990

1. Introduction 1.1. Generalities on halo-nuclei investigations The structure of atomic nuclei at the neutron drip line, which defines the boundary region of an isotope chart where no additional neutrons can be added, revealed an unexpected exotic type of nucleus, discovered at Lawrence Berkeley National Laboratory in 1985, as reported by Tanihata et al. [1]. The experiments presented evidences that the most neutron-rich unstable isotope of lithium, the 11 Li, has a large unusual size similar to a heavy nucleus like gold, with 197 nucleons. This observation has challenged the naive belief that the radius of a nucleus depends only on the total number of nucleons (the mass number A) in a compact configuration, where all the protons and neutrons are distributed uniformly over the nuclear volume. It was found that this lithium nucleus has two extremely weakly-bound neutrons in an extended and diluted distribution around the 9 Li core forming a neutron halo, a term suggested by Hansen and Jonson [2]. In this situation the halo neutrons have a large probability of being located at distances very far (in a relative sense) from the core, in the classically forbidden region, being a prototype of a three-body light exotic nucleus at the neutron drip line. Experimental data [3,4] for 11 Li have shown that the matter radius, the root-mean-square (r.m.s.) radius deduced from the matter distribution, is about 1 fm larger than the charge radii. As reported in [3], where proton–nucleus elastic scattering was used for probing the nuclear matter distribution, the respective values for charge and matter radius are 2.55 (12) fm and 3.62 (19) fm, leading to a halo radius of about 6.54 (38) fm. The result for the matter radius is consistent with other experiments reported in [3] (even though a bit larger), where such radius is deduced from the total interaction cross-sections. Therefore, under the label of neutron-rich halo nuclei one is considering all kind of nuclei having a structure similar to 11 Li, with one or more neutrons (the halo) far apart from a core, which defines the rest of the nucleus. Another remarkable example of a neutron drip-line nucleus, which can be classified as a two-neutron halo system and a core, is the carbon-22 (22 C) [5]. The experiment reported in [5] provided a matter radius of 5.4 ± 0.9 fm, which was extracted via a finite-range Glauber analysis under an optical-limit approximation of the reaction cross section, for 22 C on a liquid hydrogen target, whereas an estimate for the r.m.s. distance of the halo neutron to the center-of-mass (c.m.) system (rn ) is of the order of 15 ± 4 fm [6]. This value gives to 22 C the status of the largest neutron halo discovered so far. In this review, our aim is to consider light halo-nuclei systems, where the maximum number of nucleons ranges up to the carbon isotopes, with no more than two neutrons in the halo. As observed by Cottle and Kemper [7], in their analysis of recent experiments on the structure of larger nuclei along the drip line [8], the nuclear behavior changes in a remarkable way by considering nuclei with charge larger than the carbon one, as verified in case of nitrogen-23 and oxygen-24, which have the same number of neutrons as 22 C, but are considerably more tightly bound than the later one [9,10]. In case of

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Fig. 1. Illustrative section of the nuclear chart with isotopes of light halo nuclei with charge number (z) less than 7 and neutron number (N) limited to 16. In particular, we are pointing out the halo-nuclei that, within three-body n–n–core configurations, can be considered as Borromean systems. Source: Adapted from Ref. [12].

Fig. 2. The Borromean rings and the three-body Borromean system. The particular name came from the use of three connected circles in the coat of arms of the aristocratic Borromeo family in Italy.

weakly-bound three-body systems with two-neutron halos such as 22 C, the core can be treated as a point-like structureless particle, when considering that most of its microscopic structure (with many neutrons and protons close together) are not affecting the general universal properties of the three-body bound system. The effect of the microscopic structure of the core in the halo properties are parameterized by the values of few number of observables, as the two- and three-body energies, which can be considered as inputs in effective models, as the renormalized zero range (RZR) approach proposed in [11]. In Fig. 1 we represent a section of the nuclear chart with isotopes of light halo nuclei that we are particularly concerned in this review, where halo effects have been shown to be more pronounced. After the experimental discovery of the two-neutron halo in 11 Li, it has been reported several other nuclear systems with neutron halos and far from the stability line. The investigations on these nuclei, before the experiments reported in [1], can be traced by following the 1979 review of Hansen [13], and one should realize that the first known halo-nucleus was reported by Bjerge in 1936 [14]. It is the 6 He, which is the longest-lived isotope of the α -particle, with a two-neutron halo. For a more recent account on the properties of 6 He, see Ref. [15]. We should note that both halo nuclei, 6 He and 11 Li, within three-body descriptions, n–n–core, have all the two-body subsystems (n–n and n–core) unbound, in what is known as Borromean configurations. A bound system with three particles is known as Borromean when all the subsystems are unbound, resembling the case of the three topological circles that are linked together in such a way that the binding is lost if one of the rings is removed (see illustration in Fig. 2). Nowadays many experimental facilities are used to study nuclei along the drip line in different countries and research groups (see e.g. [16]), with beams of exotic nuclei produced by different methods as described in Ref. [17]. The experimental status on exotic nuclei can be traced through recent reviews, such as Refs. [18,19]. When approaching the limits of nuclear stability, there are also relevant new decay modes to be considered. A review of such decay modes occurring close to the drip line can be found in Ref. [19]. The conditions for the occurrence of few-neutron halos in nuclei are reached when such neutrons are weakly bound to the core in the outermost orbits. The possibility of establishing dynamically long-range few-body correlations among the outermost neutrons and the core is at the basis of halo formation. Although, it looks reasonable to find large halos for nuclei near the drip line, when neutrons do not bind anymore, most of the experimental discoveries of large halos have been confined to relatively light nuclei. The existence of giant multi-neutron halos, not only with light but also with heavy cores, are still under investigation. A recent account on that can be found in Refs. [7,19].

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A motivation to study halo-nuclei systems in laboratories relies in the fact that these weakly-bound unstable nuclei, which decay naturally, have large asymmetries in the number of protons and neutrons. They are extreme cases of bound nuclei and define the neutron or proton drip line, challenging standard shell-model descriptions while privileging longrange few-body correlations [20]. Modern ab initio no-core shell model calculations of light-nuclei (see e.g. [21]) are also confronted by the large configurations attained by the neutrons of drip-line nuclei, while for proton-rich nuclei computations are available [22]. The study of light exotic unstable nuclei opens the possibility to go beyond the ones found in earth and look into what is created in a supernova explosion, or the nuclear mechanisms within extreme conditions allowing other cosmic events like novae and X-ray bursts. The majority of elements on earth are produced in such explosive stellar environments, involving unstable nuclei [23]. Light unstable halo-nuclei offer the opportunity to explore the manybody dynamics and nuclear interaction, that bind nucleons together to form the large variety of nuclei in our universe, in the classically forbidden region: a window to truly quantum mechanical effects. Exotic nuclei with neutron excess also challenge the common understanding of traditional nuclear physics, such as the single-particle description and the associated shell structure. Recent observations reported in Refs. [24,25] suggest that large neutron excess can change the sequence of single-particle states by altering the mean-field potential experienced by a single-particle in the nucleus. Studies of nuclei far from the β -stability region have begun to indicate that the familiar shell gaps do not persist in exotic systems. The halo structure of neutron-rich nuclei have considerable influence on the corresponding fusion cross section, presenting static and dynamical effects [26]. Static effects are associated with the increase of the fusion cross section in the whole energy interval due to the large matter extension of these exotic nuclei. The dynamical effects are associated with the strong coupling between the breakup and other reaction channels, which produce repulsive real dynamic polarization potentials that lead to the so called breakup threshold anomaly in the effective interaction near the Coulomb barrier energies and hinders the complete fusion cross section at energies above the Coulomb barrier [27,28]. For reactions with radioactive beams of exotic nuclei, see also Ref. [29]. The relevance of exotic unstable nuclei, through their unusual quantum properties, which can enlarge our understanding on quantum many-body problems and on the nuclear interaction, can further be appreciated by the vigorous experimental and theoretical research activities involving halo nuclei, which followed the discovery of 11 Li [1]. This can be traced back by the huge bibliography included in the already mentioned recent reviews on this subject, see e.g. [19,29], as well as in several other reviews devoted to experimental and theoretical aspects of halo nuclei, which are collected in Refs. [30–41]. In particular, for an introduction to halo-nuclei systems, accessible to graduate nuclear physics students, we can point out a recent book (see Chapter 13 of Ref. [12]) and the lecture notes of Al-Khalili [38], which are concerned with some universal characteristics of weakly-bound three-body systems that are relevant for model descriptions of neutron-rich halo nuclei. By considering several aspects derived from universal properties of neutron halos near the drip line, which are dominated by the large extension of the tail of the wave function outside the potential range, i.e., independent on the particular potential model, we can also mention a few interesting reviews, quoted in Refs. [42–47]. Finally, in the context of nuclear astrophysics, we have also a recent report [48], stressing the relevance of understanding the nuclear mechanism in the production of unstable nuclei. 1.2. Few-body long-range correlations and weakly-bound halo nuclei

• Two neutron halo-nuclei within a zero-range three-body model? Universal aspects of the physics of complex systems can only be separated out if one is able to simplify the description with a reduced number of parameters directly related to physical scales. This procedure is quite general in science when one attempts to find general laws governing complex systems. Light halo nuclei, such as 6 He, 11 Li, 14 Be, 20 C, and 22 C, fall within this category and can be treated as three-body n–n–core systems. The factorization of the halo wave function is justified when the core is much smaller than the size of the total system. In the case that two halo neutrons are weakly-bound the corresponding matter distribution heals to distances far from the core, leading to a halo radius that can be more than twice the corresponding charge radius, as found in case of 11 Li. Exchange effects are unimportant as halo neutrons are unlikely to be found within the core region, e.g., for 11 Li an estimate from the size of 9 Li and the halo radius gives a probability less than few percents. The cores, even in such light halo nuclei, are obviously systems with their own complex microscopic structure, considering the individual spin and isospin degrees of freedom of the nucleons and their interactions. However, all such complexity can, in many cases, be parameterized by few observables associated with the halo, such as the binding energy of the halo neutrons in the light exotic nuclei, which can be used as inputs in a simplified three-body model, together with the two-body observables, e.g. the neutron–neutron and neutron–core scattering lengths for s-wave halos. The neutron–neutron system has an s-wave virtual state, which can be present in the neutron–core subsystem of a Borromean nucleus, as for example in the case of neutron-9 Li and neutron-12 Be systems. For exotic nucleus where halo neutrons are weakly bound to a nuclear core, like in the 11 Li example, the absolute values of the scattering lengths are much larger than the nucleon–nucleon interaction range and the core size. In this respect, a zero-range model is quite appropriate to describe the system, because is free of extra potential-model parameters, and the input is given by low-energy observables, which fixes the unknown parameters in the corresponding necessary renormalization procedure [11]. The antisymmetrization of the two-neutron

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halo wave function is taken care by the antisymmetry of the spin singlet state, while in this model the configuration space part is symmetric under the exchange between the halo neutrons. Note that ultimately the finite range of the nucleon–nucleon interaction together with Pauli blocking effects provide a finite binding to nuclear systems in a nonrelativistic framework. In halo nuclei models with a structureless core, the short range physics, corresponding to residual core polarization and Pauli-blocking effects, the details of the neutron–core and nucleon–nucleon interactions are supplied to the model by fixing the separation energy of the halo neutrons and scattering lengths. Once universal aspects are recognized in a complex system, one can go deeper in the study of its microscopic structure. This is the case, for example, of the successful quantum atomic model, where the nuclear structure (in terms of neutrons and protons) should be considered only as corrections to a more general atomic theory. Another case, closer to our subject, is the few-nucleon interaction, where the main (universal) aspects are obtained within simplified approaches where only light meson exchanges are considered. The short-range part of the nucleon–nucleon interaction is simply parameterized, and considered in more detail only in a later stage, with multiple meson exchanges or within quantum-chromodynamical (QCD) inspired models.

• Point-like description of the core and polarization effects A relevant aspect to be discussed is the possibility of core polarization effects in halo nuclei, which could limit the effectiveness of a three-body description [49,50]. The importance of core polarization in halo nuclei was considered long time ago in a few works. In particular the work by Kuo et al. [51] demonstrated that core polarization is dramatically suppressed in halo-nuclei. From another side, a comment by Csótó [52] pointed out that, although core polarization is suppressed in neutron halos, it plays an essential role in their binding mechanism. Indeed, within a three-body renormalized zero-range approach, this problem was also discussed in Ref. [11], where it was clarified that the model results are not sensible to effects that influence the ground state energy of the system and/or the energies of the sub-systems, considering that such observables are given as inputs in such a three-body model. Therefore, even if the core polarization was to be of some residual importance, its effect would be taken into account through the energy of the ground state. In the light of experimental evidences, it is natural to start by looking for universal aspects that can be extracted from a model where the core is treated as a pointlike particle. The next step should check for possible corrections due to this simplified picture, where effects due to the core structure can be quantified. However, such corrections will no longer be of universal nature, but specific of the core nucleus that one is considering. The search for universal behavior of light halonuclei properties is clearly supported by experimental data, such that can be considered with quite well predictive power, as it will be shown in Section 4. In particular, the difference between the charge radius of 11 Li and 9 Li is well reproduced by the model, indicating that the movement of the core against the center-of-mass, without a noticeable change, is the source for the increase of the charge radius of 11 Li. Another aspect of neutron rich asymmetric light-nuclei at the drip line, which supports the factorization of the wave function in terms of a core and halo neutrons, is the occurrence of low-energy p-wave resonances, as discussed by Hussein [53] in the context of 11 Li. Two types of vibrations are to be considered when discussing the collective dipole excitation of such a loosely bound system: the usual isovector proton versus neutron vibration in the core, with the halo neutrons taken as mere spectators, and the oscillation of the whole core nucleus against the halo neutrons, which is known as Pygmy Dipole Resonance (PDR). The PDR, also known as soft Giant Dipole Resonance (GDR), is due to a dipole strength located close to the particle-emission threshold. As shown in Ref. [53], these PDR could enhance considerably the fusion ∗ probability of neutron-rich nuclei. The ratio between the excitation energy of the PDR (EPDR ) and the excitation energy of ∗ ∗ ∗ = the usual GDR (EGDR ≈ 80A−1/9 MeV) was predicted by Suzuki et al. [54] as EPDR /EGDR

 Nc ) [ ZN((NZ − ], where Nc refers to +N c )

the number of neutrons in the core, with N and Z being the total number of neutrons and protons of the whole nucleus. For the evidences of PDR, we can select the following high energy resolution experiments, reported in Refs. [55–60]. The PDR mode has been explained as being generated by oscillations of the weakly-bound neutrons with respect to the isospin symmetric core [61–63]. In the case of a Borromean nuclei as 11 Li, the PDR is a low-energy continuum state manifested in dipole transitions and is associated with the dynamics of the large two-neutron halo [64]. It should be understood as a manifestation of a universal three-body phenomenon in the continuum. The appearance of a low-energy resonant state is favored by the centrifugal barrier acting against the attraction generated by a long-range three-body correlation within the neutron–neutron–core system, when the subsystems have scattering lengths much larger than the size of the core and the nucleon–nucleon interaction range, as it happens for example, in the case of 11 Li.

• General aspects of halo nuclei within three-body models In this review we are concerned with the general properties of light exotic neutron-rich nuclei, which can be described within three-body models. The structure of these nuclei contains an inert core (c) surrounded by two weakly-bound neutrons (n–n) in a singlet spin state [11]. Three-body long-range correlations, characteristic of light unstable two-neutron halo nuclei, are quantified by the tail of the neutron–neutron–core (n–n–c) wave function, extending far beyond the radii of usual stable nuclei, which are of the 1

order of 1.25A 3 fm, where A is the number of nucleons. Examples are the low-mass exotic nuclei, such as 6 He, 11 Li, 14 Be, 20 C, 22 C etc., described as n–n–c systems (see, e.g., Refs. [6,11,34,65–75]). For an approach on the Borromean halo nuclei, such as 11 Li and 6 He, where the two-body n–core subsystems are unbound, see Ref. [66], which is based on a density-dependent

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contact interaction between the valence neutrons. Similar as other approaches, a good description of the ground-state properties of weakly-bound halo neutrons demands that the contact interaction is adjusted to produce a realistic scattering length and effective range. Three-body quantum mechanical effects are expected to play a dominant role on the long-wavelength properties of the halo neutrons, as their probability to be found at distances very far from the core in the classically forbidden region should be large. In the case of weakly-bound neutrons to the core, i.e., separation energies about one MeV or so, the wave function heals to distances far beyond the core radius and the nucleon–nucleon (NN) interaction range. It is worthwhile to observe that in this region of large distances, characterizing the halo, the three-body wave function is an eigenstate of the free Hamiltonian with attributes appearing in the form of universal scaling laws, independent on the detailed form of the short-range interaction. The short-range forces acting between the particles in an n–n–c description of halo nuclei have effective ranges (r0 ) of the order of one fermi, with the size of the core radius and the scattering lengths ann and anc generally larger than the effective ranges (for a recent review on nuclear forces see e.g. [45]). Furthermore, the neutron–neutron–core description of halo nuclei with a structureless core and two-body s-wave interactions, gives a halo wave function symmetric in configuration space for neutrons in the singlet spin state. In this case, a possible configuration for the three-body halo wave function is such that the halo neutrons are located relatively far from the core with the neutron–core system in an s-wave state. In addition, the core neutrons are localized within the compact core with a small overlap with the halo neutrons. In a three-boson system with two-body energies close to zero, the state of maximum symmetry with vanishing total orbital angular momentum resembles the situation described for the configuration space part of the halo wave function. Under such circumstances, and considering r0 /|a| ≪ 1 and large halo sizes, the low-energy properties of the halo three-body system in a state with zero total orbital angular momentum are driven by the Thomas collapse [76] and the associated Efimov effect [77], where the detailed form of the short-ranged interaction is less important. The pertinence of these effects, found for three-boson systems, in the study of two-neutron halo physics, stems from two facts: (i) large halo size, relatively to the radius of a compact core, and (ii) a wave function with halo neutrons in s-states. Particularly, the three-body binding energy may also have a strong increase by using a separable two-body interaction. It is indeed the case when the separable two-body interaction supports a two-body continuum bound state (pole of the S-matrix on the real axis of the complex momentum-plane) [78–80]. Within the next subsection, a retrospective is included on the relevance of Thomas collapse to nuclear physics, as well as its corresponding association with the three-body Efimov effect. 1.3. Universal few-body phenomena

• Thomas collapse and Efimov effect The best manifestation of the universality in quantum few-body physics can be observed in the case of weakly-bound three-body systems, by considering the collapse of the three-body ground-state energy, when the range of the two-body interaction goes to zero (with finite two-body binding), an effect discovered by Thomas [76] in 1935. This effect, nowadays known as Thomas collapse, is quite relevant in the understanding of the range of the nuclear forces, as recognized by Bethe and Bacher in their 1936 review on the nuclear physics status [81]. The collapse of the three-body system has been found in the context of nuclear physics, within a nonrelativistic quantum mechanical approach to the triton structure with shortrange NN interaction. Thomas pointed out that the triton binding has no lower bound in the limiting case in which the range of the two-body interaction shrinks to zero, once given the two-nucleon energies. Furthermore, he observed that the range of the nuclear force could not be below 10−13 cm from the observed value of the triton binding energy. Therefore, as pointed out by Bethe and Bacher [81], the collapse of the three-nucleon binding when the range of the forces goes to zero is clearly a proof of the finite range of the NN interaction, such that it is also referred as Thomas’s theorem by Hall [82]. The relevance of Thomas results to the study of nuclear forces is also emphasized in some works by Feshbach and collaborators [83,84]. Many other subsequent works, in the context of three-nucleon systems, have discussed the collapse of the three-body ground-state for zero-range two-body interactions. Here we can mention a few of them [78,85–92]. Particularly, Gibson and Stephenson [92] pointed out that the collapse could be understood by a qualitative argument presented by Wigner [93]. According to Wigner, the kinetic (T ) and potential (V ) energies in the deuteron and triton, when ignoring short-range repulsion and the problem of saturation, can be estimated as follows. For the deuteron, which has an almost vanishing binding energy in the nuclear scale, one has that 2T + V ≈ 0, while the triton binding energy is estimated as 3T + 2V ≈ −T , which has no lower bound when the triton has a vanishing size. A counterpart of the Thomas collapse is the Efimov effect [77], which can be verified in the limit of infinite scattering lengths with fixed force ranges. It corresponds to the appearance of an infinite sequence of weakly-bound three-body states condensing at the scattering threshold, with energies geometrically scaled, when the two-body scattering lengths diverge and the two-body systems are bound exactly at the dissociation edge. The three-boson states are formed by an attractive long-ranged ρ −2 potential (where ρ is the hyper-radius for the system), which acts on the lowest hyper-angular state. This potential is modified at the short range by the interaction which heals up to distances of the order of the scattering length [77]. The condensation of levels was originally predicted to occur in three-boson systems in s-wave states of maximum symmetry, a property that can be extended to asymmetric mass systems as the halo nuclei (see e.g. Ref. [39]). Indeed, it

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was in Ref. [67] where 20 C was suggested as a candidate to have an Efimov excited state, within an n–n–c configuration. Few years later, an analysis was performed in Ref. [11] for the critical conditions in which a two-neutron halo system could present an excited (bound or virtual) state. This analysis, together with the experimental available data, indicated that 20 C is on the verge of having such a state. For the recognition on the connection between Thomas collapse and the Efimov effect one can go back to one of the original works published in 1971 by Efimov [94]. A few years later, the common nature of both effects was also established in Ref. [95]. In Ref. [96] it was demonstrated that the equivalence between both effects occurs in three dimensions due to a divergence arising from essentially the same singular structure of the kernel of the scattering integral equation, whereas such divergence is absent in two dimensions.1 More recently, the Thomas collapse and the Efimov effect were also considered as being the two sides of the same coin, as one can find in Ref. [98] as well as in Ref. [99]. The equivalence of both effects are recognized by the validity of the same limit r0 /|a| → 0. However, such equivalence is questioned when considering real physical systems, because the range of the interaction r0 cannot be made exactly zero. As commented in Ref. [100], this fact prevents the observation of Thomas collapse, whereas there is no similar obstacle to observing the Efimov effect. The route towards the collapse of the three-body ground-state, when the interaction range vanishes, is closely related to the Efimov effect. In both extreme cases, three-body bound states are formed in the lowest scattering threshold and departs from it by forming an infinite sequence of geometrically separated levels as the interaction range goes to zero in respect to the scattering length. Technically, a scale transformation relates the collapsed and Efimov states, and they are obtained as solutions of the same dimensionless non-relativistic quantum-mechanical three-body eigenvalue equation [96]. Indeed, the triton binding energy misses a lower bound in the zero-range three-body equations proposed by Skorniakov and Ter-Martirosian [101], back in 1956, as a prototype of a model of the strong short-range interaction. The zero-range three-body equations have a scale invariance for |a|−1 = 0, while the kernel is strong enough to collapse the s-wave ground-state of maximum symmetry [102] (see e.g. [44]). The solution of the zero-range three-body model demanded the introduction of one physical three-body input [102] or an ultraviolet momentum cutoff [103]. The conclusion is such that the scale invariance has to be broken to give a definite size to the three-body system, which avoids the collapse and gives a physical scale to the s-wave three-body observables. One three-body scale correlates all low-energy s-wave three-body observables. In the configuration space, the three-body wave function of the zero-range model at large distances matches the one of a weakly bound state, which extends far away from the interaction range and, therefore, exhibits a universal structure. The zero-range wave function has a universal shape dictated by the free Hamiltonian and a spectator function solution of the Skorniakov and Ter-Martirosian equations (see e.g. [104]). The three-boson levels appear in a geometrical sequence due to the scale invariance (in the limit a → ±∞ or r0 → 0) and at all length scales. These levels correspond to Efimov states and gives the route of the Thomas collapse of three-body systems. The binding energies of two consecutive s-wave three-boson excited states obey the fixed ratio of ∼1/515, and their mean-square radii of these states scales by a factor of 22.7. The concepts of universality and scale invariance summarize our understanding of Efimov physics and Thomas collapse. They are useful in quite general situations, as universal properties of Brunnian configurations, which correspond to N-body Borromeans (with all subsystems unbound), as has been discussed in the case of four- and five-body systems [105,106]. A very nice description of the Efimov effect can be found in a solvable three-body model by using a Born–Oppenheimer approximation [107]. By a simple rescaling this model shows a correlation between three- and the two-body energies [108]. The sequence of weakly-bound s-wave three-body states requires one scale to be defined in the limit of a → ±∞. It can be chosen as the energy of one of the states in the sequence. The recognition of the importance of the concept of scales and model independence in the description of the few-body physics started together with the first calculations in three-nucleon systems and then extended to n–n–c halo-nuclei (see e.g. [34]). 1.4. Low-energy scales and correlations between observables The notion of scales in few-body systems allows to separate the long-wavelength physics from the short-range dominated one (see e.g. [39,44,96,109–111]), as minimum inputs are necessary to establish the binding and structure of large fewneutron halos. The source of the low-energy scales in nuclear physics is ultimately ΛQCD and quark masses [112] in Quantum Chromodynamics. Recent progress in lattice methods applied to Quantum Chromodynamics are promising to give a predictive calculational framework to nuclear physics [113]. The meaning of ‘‘scales’’ in few-body physics is a group of any physical quantities – for example, the binding energies of the system – which governs the behavior of the observables of this system (see e.g. [114]). Weakly-bound halo systems are mainly described by the tail of the wave function and do not depend on the particular shape of the interparticle potential [39,115]. The term ‘‘universality’’ is usually associated with the strict correlation between the low-energy properties of the

1 This was further discussed in Ref. [97], when considering three identical bosons at low energies in two dimensions, for short-range separable potentials whose range is much smaller than the size of the two-particle bound state. It was also shown that three-particle scattering observables will not depend on the potential shape, provided that the low-energy two-particle binding and scattering length are held fixed.

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system with the input scales, which does not depend on details of the interaction. Large few-body systems are in the ideal situation where the universal regime can be studied, in principle, by using any simple short-ranged potential, such as the square well potential [116]. Low-energy few-body halo systems approach the universal regime and can have their properties investigated by the use of contact s-wave potentials (see e.g. [11]), that allows to access the general basic characteristics of such systems as well the role of the relevant physical scales, which are reduced only to scattering lengths and the energy of one three-body level [114]. The introduction of the low-energy parameters of the two-body system, as effective ranges as well as higher partial waves with contact interaction terms can be done by using the systematic expansion provided by effective-field theory (EFT) [43,44]. The expansion parameter in EFT is chosen as the ratio between the low and high momentum scales of the system [109]. For one- and two-neutron halo systems, the low-momentum scale is associated with the size of the halo and the high-momentum scale with the nuclear force range. The application of the framework of EFT to halo-nuclei started with the analysis of the low-energy p-wave resonance in 5 He [117], as a neutron–core halo interacting with contact potentials. Several applications of EFT to light halo-nuclei with two and three-body structure were subsequently performed (see e.g. [47]). The example of the neutron–neutron–proton system showing low-energy physics of halo states and model independent effects is instructive. In this case only few low-energy parameters fix, to large extend, the long-wave length properties of the system without strong dependence on the details of the short-range part of the nuclear interaction. The nucleons in the s-wave spin doublet state can occupy a state of maximum symmetry, and tuning the range of the interaction to zero or the energies of the deuteron and the n–n virtual state to zero, the Thomas–Efimov effect shows up. Then, the minimum input to determine the low-energy properties of the doublet spin state of the trinucleon system in the zero-range limit is, besides the singlet and triplet NN scattering lengths, one s-wave three-body observable in this channel. Furthermore, the trinucleon virtual state [118–121] is indeed one Efimov excited state that moves into the second energy sheet through the two-body cut as the deuteron binding energy is decreased [122–124]. The effect of a typical three-nucleon scale was clearly seen in the Phillips plot, i.e., the model independent correlation between the doublet neutron–deuteron scattering length and triton binding energy [125,126], as an expression of the offshell sensitivity of the three-nucleon system [127,128]. For scattering energies, it is known the correlation between the elastic neutron–deuteron doublet phase-shift and the triton binding energy [129–131]. Correlations between low-energy observables in three-nucleon systems have been approached qualitatively giving further insights to realistic calculations. For example, the analysis of the Phillips plot in Refs. [132,133], as well as range corrections [134]. Two-body models with long-range effective interactions of 1/r 2 type inspired by the adiabatic approximation [107] and by the hyper-radial potential derived by Efimov, were applied to the three-nucleon problem to address Coulomb effects on bound and scattering states. Within such qualitative treatment, it was studied the correlation among scattering length and binding energy for the spin-doublet neutron–deuteron and the proton–deuteron systems [135]. The two-body potential models were parameterized to reproduce the correct binding energies of 3 H and 3 He and the correct s-wave spin-doublet neutron–deuteron scattering length 2 and and predicted the low energy nucleon–deuteron phase shifts, as well as the proton–deuteron scattering length to be 2 apd = 0.15 ± 0.1 fm. The dependence on the short-range part of the nuclear interaction, or off-shell sensitivity [136,137], is also evidenced by some other well-known correlations among other low-energy nuclear observables: Coester line [138,139] and Tjon line [140]. These are indications that few set of parameters are dominant in the corresponding dynamics. More general low-energy few-body properties and correlations were discussed in several works and reviews [39,44,141]. In [142] a manifestation of the Thomas collapse was discussed for the Tabakin-type [85] separable potentials, where the potential parameters allow two-boson s-wave poles close to a resonance, or when the pole is on the real momentum axis (continuum bound state). For a variational proof of the relation between the Efimov effect with the Thomas collapse, see also Ref. [143]. For a more recent discussion on the role of few-body scales in nuclear matter, the Ref. [144] is presenting some evidences for the scaling between light nuclei binding energies and the triton. In the case of absorptive short range potential, the three-boson system was also studied in a simplified model in Ref. [145]. By reducing the range to zero, it was demonstrated that the ratio between the width and the three-boson binding energy is proportional to the range. In general, a strong theoretical signature of neutron halo formation in nuclei is given by a dependence of the halo properties on the short-range physics through few scales. Historically, the view of light-nuclei as halo systems started with the deuteron, triton, 6 He and a family of other nuclei, with one, two, three and four valence neutrons in addition to a core. The Pauli principle, makes the properties of the halo neutrons sensitive up to a three-body scale. Therefore, several model independent features seen in the trinucleon systems, expressed by correlation between long-wavelength observables from the dependence on a three-body scale, generalize to asymmetric mass situations describing two-neutron halo systems. These universal aspects of light halo-nuclei will be thoroughly discussed in the forthcoming sections. We have to mention that correlations between few-body observables are quite relevant for cold atom physics with tunable interactions, where experiments [146–150], claim to have observed manifestations of the Efimov effect. Noteworthy, the cold-atom counterpart of several correlations between low-energy observables found in few-nucleon systems, can be experimentally investigated considering the actual possibilities of obtaining large scattering lengths by tuning the two-atom interaction through Feshbach resonance techniques [151,152].

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1.5. Renormalization and limit cycles The concept of a scale is commonly associated with the renormalization of a quantum field theory, which has properties known at a certain energy. A change in the physical scale makes the parameters of the theory run driven by the solution of renormalization group equations [153] such that the theory predictions remains unchanged. Generally, quantum Hamiltonian systems with ultraviolet divergences requiring renormalization may present renormalization group limit cycles [154]. A limit cycle in physics [153] refers to a model independent way to look for a manifestation of a hidden scale. One example is a correlation between physical quantities which is geometrically rescaled and replicates itself when a scale is multiplied by a given factor. After the scaling factor is removed all curves converge to a single function, i.e., the limit cycle. Quantum threeboson systems [154–156], in the limit of a zero-range interaction or large two-body scattering lengths, have limit cycles. They correspond to correlations between s-wave three-boson observables, described by scaling functions, as demonstrated in the particular case of the correlation between the energies of successive Efimov excited states for finite dimer energy [114]. The introduction of a scale is necessary for the regularization and renormalization of the three-body model with Diracdelta two-body potentials [157], (for a view of this problem in the context of effective field theories see e.g. [43,44]). The need for a new parameter beyond a to determine the properties of the low-energy s-wave state of a three-particle system in the state of maximum symmetry is a consequence of the Thomas–Efimov effect, which happens when the range of the interaction goes to zero in respect to a. The new parameter carries the three-body short-range physics that is required by the regularization of the three-body integral equations in the ultraviolet momentum region. Then, the step to renormalize the model is done by regularization of the kernel of the three-body equations, which can be done by introducing a subtraction at an energy scale, identified by −µ23 (or, momentum scale µ3 ). The regularization of the momentum integration is done by a subtracted form of the free three-body resolvent G0 (E ) − G0 (−µ23 ) in substitution to the free three-body Green’s function G0 (E ). The parameter −µ23 is an energy subtraction point, which can be identified with a three-body physical observable in the renormalization procedure [158,159]. The subtraction technique used to regularize the three-body zero-range equations is given by a generalization of the subtraction procedure applied in the two-body scattering case, where a renormalization group (RG) analysis was implemented for the two-body scattering equations. [157,160,161]. The extension of the subtraction technique to an arbitrary number of subtractions, with a non-relativistic derivation of the corresponding Callan–Symanzik (CS) equation [162–164] for the driven term, is given in Ref. [165]. This RG procedure was also applied to the NN scattering with effective interactions [166,167]. The subtracted equations can be derived from a renormalized Hamiltonian as shown in [168] (for other regularization and renormalization methods see e.g. [169,170] for the three-body problem and [171] in general). The subtracted form of the three-body integral equations for the Faddeev components of the transition matrix can be derived from a renormalized Hamiltonian, which contains two- and three-body interactions. This will be further discussed in Section 4. In this way, the method developed in Ref. [155] with a three-body potential that has a strength running to a limit-cycle can be in relation to the subtraction method [157]. The renormalized two-body Hamiltonian for a contact plus a Yukawa potential has been diagonalized and results for bound states exactly matches with the pole of the renormalized T -matrix (see [168]). Beyond the three-body system the four-boson problem in s-wave with the zero-range two-body interaction requires another regularization parameter that introduces a new scale. That scale is associated with four-body limit cycles that build a scaling function correlating four-body observables [172,173]. At the nuclear physics level the sensitivity on the number of short range scales stops at the three-body level as the nuclear interaction is dominated by two-body forces. This effect was verified by Tjon [140]. The physics of neutron halos is also expected to be sensitive only to short-range effects carried by a three-body scale. 1.6. Three-body halo-nuclei structure In experiments leading to breakup of 11 Li, as well as other neutron-rich nuclei, such as 14 Be or 20 C, the nuclear matter distributions near the surface can be obtained from elastic scattering or interaction cross section. The separation energy of the last two neutrons of 11 Li is known to be S2n = 0.25 ± 0.08 MeV, while 10 Li is unbound [174,175]. A precise recent measurement gives S2n = 369.15(65) keV, as reported in [176]. The binding energy between the two loosely bound neutrons is larger than the corresponding binding energies between each of them and the 9 Li core. As argued by Hansen and Jonson in Ref. [2], the strength of the neutron pairing is responsible for these differences in the separation energies. In view of that, a simple cluster-like structure for the light neutron-rich nuclei, can explain both the widths of the momentum distributions as well as the total cross sections. In case of 11 Li, for example, the suggested model should consider it as a 9 Li core with two neutrons. For a detailed account on the structure and reactions of the first discovered neutron-rich nuclei, considering in particular 11 Li, see e.g. Refs. [32,34,67] and references therein. Following the success in describing 11 Li as two neutrons and a core, and considering the discovery of several other neutron-rich nuclei, such as 14 Be, 6 He, 20 C, 22 C, many works treated such exotic nuclei within three-body models.

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In particular, considering that 11 Li, 14 Be, 20 C, 22 C, are weakly-bound systems, with extended wave-functions where details of the two-body interaction are not so relevant, the renormalized zero-range model considered in Ref. [11] was shown to be appropriate for description of light-halo nuclei. Therefore, three-body properties are set by low-energy parameters (scales), as the binding energy of the halo to the core, which is assumed to be structureless, and the corresponding scattering lengths of the two-body subsystems. The neutrons in the halo and the core are free to propagate everywhere except when they are superposed in the zerorange three-body model. The main physical simplification is the following: if three particles (1, 2, 3) interact, the third particle, commonly called spectator, receives only asymptotic information in relation to the pair (1, 2). This information is completely contained in the on-the-energy-shell two-body T -matrix defined in a three-body Hilbert space, which is not enough to shape the physics of the system that still needs one short-range scale. In this context, the effective long-ranged potential of the type 1/ρ 2 (where ρ is the hyperradius) derived by Efimov determines the lowest hyperangular component of the wave function of the two-neutron halo nuclei weakly-bound states, corresponding to orbital angular momentum zero, as explained in Ref. [39]. The 1/ρ 2 potential just brings the scale transformation of the free three-body Hamiltonian to the hyperradial differential equation. The potential acts in the region between the nucleon–core interaction range and the absolute value of the scattering length. Here, it is appropriate to remind the reader on the Landau and Lifshitz [177] well-known treatment of the inverse square radius potential. If the attractive dimensionless strength exceeds 1/4, the ground state collapses, corresponding to the Thomas collapse of the three-body bound state. The collapse is accompanied by an infinite number of states geometrically separated. The spectrum is stabilized by a short-ranged boundary condition, which corresponds to the introduction of a three-body short-range scale. Studies has been performed in Ref. [11] (see also [74,178]), in a three-body model of two-neutron halo systems to define the conditions for the appearance of one excited state with total orbital angular momentum zero close to the Efimov limit (infinite scattering length). These conditions define a universal two-dimensional parametric space, where the axis are given by the energies of the subsystems in units of the three-body ground-state energy. Several known exotic neutron-rich nuclei were studied in a perspective analysis, looking for three-body states with Efimov characteristics. Among the available data at that time, it was found that 20 C (described as n–n-18 C) was the strongest candidate to have an excited Efimov state. Actually, a better candidate was found with the recently discovered 22 C, in a model where 20 C is the core. Universal aspects of halo nuclei have been studied in several other theoretical works, such as Refs. [6,11,68–73] and the search for Efimov states in molecular systems Ref. [179] and references therein. The universal physics in the n–n-18 C [71,72], also shapes the low-energy properties of the scattering of a neutron and 19 C [180], presenting analytical features (virtual state and a pole in the effective range expansion) like the ones found in the doublet neutron–deuteron s-wave amplitude [118–121,129,130].

1.7. Outline of the review The main proposal of the present review is to give an account on the actual stage of studies related to universal aspects of light halo nuclei. As most of the neutron-rich nuclei have an apparent structure of a core with two-neutron halo, in a weakly-bound three-body system, it became natural to extend universal aspects verified for three-identical particles to the case with two different kind of particles. In such a case, we have to consider weakly-bound three-body systems where we can identify a core surrounded by two neutrons. Therefore, in the next sections, we intend to give an overall picture on the main aspects related to physical scales and universality, applied to light exotic nuclei close to the neutron drip line, following several previous studies related to this subject [11,74,75,178]. In Section 2, we revise some general concepts on scattering and bound-state properties of two and three-body systems, in order to fix notations and analytical structure of basic formalisms. The few-body physical scales are considered in the Section 2.2, within a renormalized treatment applied to three-body weakly-bound systems. Also in this subsection we revise some peculiar observed properties of three-body systems, long-time known in the nuclear physics context, such as the Thomas collapse and the Efimov effect. This Section 2 is concluded with an illustrative example of three-body system with two species of particles. Regularization and renormalization procedures, which are quite relevant in the studies of weaklybound three-body systems, within zero-range and EFT approaches, are considered in Section 3. Details and technical aspects related to regularization and renormalization approaches, which have been considered in the studies of three-body systems, are presented in this section. Universal physical aspects emerge from the studies considered in this section, such as the correlation between observables. The studies and formalism more directly concerned with neutron-rich nuclei are detailed in Section 4. In this section we revise studies related to the classification of halo nuclei in view of their two-body interactions (Section 4.1), critical conditions and scattering properties of halo nuclei, exemplified with the 20 C case (Section 4.2), sizes and mean separation distances between particles (Sections 4.3 and 4.4), and the neutron–neutron correlation functions (Section 4.5). The section is conclude with the Section 4.6, where we present some perspectives on theoretical approaches to halo nuclei with more than two neutrons. The Section 5 conclude the review with a summary and some more general perspectives.

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2. Basic concepts in few-body problems In this section we start by revising some text-book basic notions of scattering theory, for short-range central potential models, applied to two-body problem, in order to define clearly our notations, terminology and some frequent concepts appearing in few-body nuclear physics, related to the pole-nature of the singularities of the scattering matrix, which are directly associated to bound-states, virtual-states or resonant-states. Next, the formalism is developed from shortrange interactions to the most simple case of renormalized zero-range interactions, which has been shown to be a convenient framework to consider universal aspects of low-energy weakly-bound systems. Within a renormalized zerorange interaction, one can introduce the relevant physical scales directly in the renormalization procedure, such that the main observed effects cannot be claimed to be originated from a particular two-body parametrization. In this way, by considering a RZR model, one can first verify the universal characteristics of the problem in a physical region near the scaling limit, which is defined as the limit where the relation between observables does not depend on a particular form of the interaction. Deviation of the scaling limit, brought by range effects or some particular characteristics of the interactions due to the kind of particles one is leading (charged, bosonic or fermionic), can be treated systematically in a potential model. Following the formalism developed for the case of two-body interaction, in the next subsection we introduce the actual basic knowledge related to the case of three identical boson systems. Most of the notions we are going to describe in case of the three-body physics, such as the Thomas collapse, Efimov effect, Phillips line and Tjon line, were first studied in nuclear physics, at a time where the investigations were concentrated in the studies of the range of the nuclear forces. More recently, the interest in such effects have increased, considering that experiments in cold-atom laboratories, where the two-atom interaction can be altered, can be used to verify and test the extension of such few-body effects. 2.1. Two-body problem 2.1.1. Scattering, bound, virtual and resonant states The relevant aspects we are interested in, when considering a particular two-body interaction within a more complex few-body system, are related to the singularity structure of the corresponding two-body scattering matrix, from where one can obtain the full structure of poles, leading to bound, virtual and resonant two-body states. It is well known, for example, that we can have a bound state for the three-body system even in the case that all the two-body subsystems are unbound. Therefore, it is quite relevant to have a more general picture of the two-body scattering formalism, in particular for a defined short-ranged two-body interaction. The following concepts and notation given in this subsection will be useful for discussing prominent physical aspects of weakly-bound neutron-rich halo nuclei, which are dominated by model independent properties and amenable to be described by renormalized contact interactions. As usual for the application of basic non-relativistic scattering formalism in quantum mechanics, we assume that the two-body interactions decrease faster than the inverse-square of the distance from the interaction center, r −2 . We imply the two-body potential is in general local in coordinate space. However, considering that low-energy weakly-bound systems are not too much sensible to details of the interaction, we can take advantage of this fact by treating the two-body subsystem, within a more general three or four body system, with a potential in a separable form in momentum space. In this case the corresponding solution for the two-body scattering equation is simplified, leading to a closed form for the case the potential is described by just one separable term (rank-one separable two-body potential). In the scattering formalism, considering a fixed energy E = k2 (in units that h¯ = 1 and 2µ = 1, with µ the two-body reduced mass), which in general is a real positive quantity, the main relevant defined quantities, to be obtained for a given interaction, are the two closely related S- and T -matrices. In the momentum-space description of these two matrices, their on-shell energy elements, S (k) ≡ ⟨k|S |k⟩ and t (k2 ) ≡ ⟨k|t (k2 )|k⟩, are associated to direct scattering observables, as the differential and total cross-section (respectively, dσ /dΩ and σ ), or indirect ones, as the phase-shifts δℓ ≡ δℓ (k) of the partial ℓ-wave decomposition of the wave-function [181]. For a central potential, the on-shell partial-wave matrix element of the S-matrix carries the phase-shift of each outgoing partial-wave state of angular momentum ℓ: Sℓ (k) = e2iδℓ .

(1)

The S-matrix allows for a whole mapping of energies in the complex k-momentum plane, not only for the case of real quantities of k, where the energies are positive and given by E = k2 . When k is located on the imaginary axis, we have bound or virtual states (respectively, k = iκB or k = −iκV , for κB,V real positive), with the corresponding energies given by E = −|E | = −κB2 or E = −|E | = −κv2 . The resonant energy poles are obtained from nonzero real and imaginary parts of k, with Re(k) > 0 and Im(k) < 0 and close to the real energy axis. They are given by E = Eres − iΓ /2, where the resonant energy is defined by Eres ≡ Re(E ) = [Re(k)]2 − [Im(k)]2 and the decay width by Γ ≡ −2 Im(E ) = −4 [Re(k) Im(k)]. For real k, the Eq. (1) relates the usual outgoing solution above threshold, at kinetic energy k2 + iε . The time-reversed solution at k2 − iε leads Eq. (1) to have a reversed phase factor exp(−2iδℓ (k)) causing a discontinuity of Sℓ (k). Below the threshold, the energy becomes negative, suggesting that pure imaginary values of k would well describe this situation, since E = k2 . This dispersion relation causes a square-root cut in the whole complex energy-plane. The so-called physical energy sheet (first Riemann sheet) corresponds to the upper complex k-plane whereas the lower complex k-plane corresponds to the unphysical energy sheet (second sheet). To link both energy sheets, an analytical continuation is needed. For one of

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Fig. 3. Schematic representation of the S-matrix pole trajectories for a square-well potential with strength |λ| = V0 R20 , in the complex k-plane, considering s- and p-waves. The trajectories are indicated by the arrows, as one increases the values of λ.

the first detailed discussion and interpretation of the singularities of the scattering amplitude occurring in the unphysical Riemann sheet, see Ref. [182]. For a large class of well behaved central potentials, the analytical continuation of Sℓ (k) is a meromorphic function, satisfying the relation [181] Sℓ (k)Sℓ (−k) = Sℓ (k).Sℓ∗ (k∗ ) = 1.

(2)

From symmetric properties, poles of Sℓ (k) in the upper complex k plane can occur only in the positive imaginary axis and should be interpreted as bound states. In the lower part of the complex k plane, poles can occur either on the negative imaginary axis, interpreted as virtual states (negative energies, but non-normalizable eigenfunctions) or in pairs of conjugate complex numbers (defined as resonances when Re(k) > 0, with Im(k) < 0 and close to the real axis). Such poles can produce visible effects on the total scattering cross-section,

σ =

π  (2l + 1) |1 − Sℓ (k)|2 . 2

k

(3)

ℓ

In the two-body scattering problem, only few interactions allow for analytical solutions of the S-matrix. It is the case of a spherically symmetric rectangular potential studied long time ago by Nussenzveig [183]. Since the S-matrix poles of this particular case is very representative for other central short-ranged interactions, a schematic illustration of the corresponding trajectories under strength variation is presented in Fig. 3, where the dimensionless strength of the squarewell potential of depth V0 and range R0 is given by |λ| = V0 R20 with λ = −|λ|. With λ close to zero, for ℓ = 0, an infinite number of poles occur at kR0 = ±nπ − i ∞ (n = 0, 1, 2, . . .). As |λ| starts increasing from zero, the poles move upwards in the complex k-plane, with the pole with n = 0, on the negative imaginary axis (virtual state), reaching the origin for |λ| = (π /2)2 , at the point where a bound state starts to appear. Therefore, for |λ| ≥ (π /2)2 the pole with n = 0 is a bound-state pole. The other complex poles that occur in pairs, starting at kR0 = ±nπ (n ≥ 1) for λ = 0, move upwards in direction to the imaginary axis (decreasing the real part). As they meet the negative imaginary axis, a pair of virtual states emerges at the same point. By further increasing |λ|, the two poles of the pair move in opposite directions along the imaginary axis, with the one moving upwards favoring the emergence of an excited bound state with Im(k) > 0 (for details on the pole trajectories, see Ref. [183]). For ℓ ̸= 0, we have a similar behavior as in the case of ℓ = 0. However, in these cases, by increasing |λ|, the pair of complex poles meet at the origin, such that, by further increasing |λ|, we have a pair of bound and virtual states, as schematically represented in Fig. 3. Essentially, given a spherically symmetric central two-body interaction V (r ), all information on the scattering, bound and virtual states, for a given partial-wave ℓ, can be obtained from the elements of the T -matrix, given by the solution of the partial-wave Lippmann–Schwinger (LS) equation, tℓ (p, p′ ; k2 ) = Vℓ (p, p′ ) +

2

π



∞

p′′2 dp′′ Vℓ (p, p′′ )tℓ (p′′ , p′ ; k2 ) k2 − p′′2 + iε

0

,

(4)

where p and p′ are the absolute values of the momentum coordinates and Vℓ (p, p′ ) is the Fourier transform of V (r ), which is given by Vℓ (p, p′ ) =

∞



jℓ (pr )V (r )jℓ (p′ r )r 2 dr ,

(5)

0

where jℓ (p) is a spherical Bessel function for the variable p. The phase-shift δℓ ≡ δℓ (k), which is directly related to Sℓ (k) by Eq. (1), can be obtained from the on-shell T -matrix element tℓ (k) ≡ tℓ (k, k; k2 ), from where we can also verify a relation between Sℓ (k) and tℓ (k): tℓ (k) = −

eiδℓ sin(δℓ ) k

=−

1 k cot δℓ − ik

−→ Sℓ (k) = 1 − 2iktℓ (k).

(6)

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Therefore, Sℓ (k) and tℓ (k) have common poles. In order to look for the poles of the two-body S-matrix we can use the principal value (P ) prescription. For the kind of integrals we are considering, let us define J (k). This prescription for an arbitrary regular function F in the integrand, leads to

J (k) ≡



2

π

0

∞

q2 dq k2

−

q2

+ iε

F (q) =

2

π

∞

 P

q2 dq k2

0

− q2

F (q) − ikF (k).

(7)

Considering the above prescription, the Eq. (4) for the T -matrix is given by 2

tℓ (p, p′ ; k2 ) = Vℓ (p, p′ ) +

π

∞

 P

p′′2 dp′′ Vℓ (p, p′′ )tℓ (p′′ , p′ ; k2 ) k2 − p′′2

0

− ikVℓ (p, k)tℓ (k, p′ ; k2 ).

(8)

The T -matrix is usually expressed in terms of the K -matrix, which is defined as Kℓ (p, p′ ; k2 ) = Vℓ (p, p′ ) +

2

π

 P

p′′2 dp′′ Vℓ (p, p′′ )Kℓ (p′′ , p′ ; k2 ) k2 − p′′2

.

(9)

The K -matrix has no fixed-point singularity and can be written in a subtracted form, dropping the principal value prescription, by Kℓ (p, p ; k ) = Vℓ (p, p ) + ′

′

2

2

dp′′



π

k2 − p′′

[p′′2 Vℓ (p, p′′ )Kℓ (p′′ , p′ ; k2 ) − k2 Vℓ (p′ , k)Kℓ (k, p′ ; k2 )].

(10)

In contrast to the T -matrix elements given by Eq. (4), the K -matrix elements are real for any real positive energy. Therefore, above the threshold at k2 = 0 no analytic cut exists. The on-shell T - and S-matrices can be related to K (k) ≡ K (k, k; k2 ) by tℓ (k) =

Kℓ (k) 1 + ikKℓ (k)

,

Sℓ (k) =

1 − ikKℓ (k) 1 + ikKℓ (k)

.

(11)

From Eqs. (6) and (11), the corresponding phase-shifts for the partial-wave ℓ can be obtained from the on-shell K -matrix, as Kℓ (k) = −

tan δℓ

. (12) k The Eq. (4), also written as (8), is defined in the first energy sheet and has a square-root unitarity cut along the real positive energy axis. It makes this equation not single valued in the whole energy plane, such that an analytical continuation through the cut from the first to the second energy sheet is needed. A way to perform this step is to deform the path of the p′′ integration [184] in Eq. (4), as schematically shown in left frame of Fig. 4. Therefore, the poles in the half-lower k-plane, corresponding to the second energy Riemann sheet, can be accessed by choosing the k complex values with negative imaginary parts [185,186]. In practice, as all the relevant scattering integrals we are considering have the simple structure J (k), given by the identity in the left-hand-side (lhs) of Eq. (7), we can follow the prescription which is presented in Ref. [122] to perform the analytical extension of such integrals to the second Riemann sheet of the complex energy plane. In this complex energy plane, the integral J (k) has a square-root branch point at zero energy and a branch cut along the positive real axis,2 if the q-integration path is along√the positive real axis as shown in Fig. 4(a), where the symbol × represents the pole of the denominator of J (k) at q = k2 + iε . In Fig. 4(b) we represent the analytical continuation in energy beyond the unitary cut by a contour deformation of the q-integration. The pole of J (k) moves across the real q axis into the lower half q plane (allowed because the integrand is analytic in the lower half q plane), where the complex k value corresponding to the analytically continued complex energy in the unphysical sheet is indicated by the × in Fig. 4(b). Therefore, to obtain the result of J (k) in the second complex energy sheet we need to change the contour to this second plane, as shown in Fig. 4(c) where the circled × represents the contribution of the residue at the pole of the integrand of J (k). With the above consideration, in the second sheet of the complex energy plane, J (k) has to be replaced by J II (k) = J (k) − 2ikF (k),

(13)

with J (k) given in (7). The second term in the right-hand-side (rhs) of the above equation is the contribution of the residue at the pole. Resonances and virtual states appear as poles in the T -matrix elements and, therefore, correspond to zeros of the denominator of the T -matrix extension. The case of bound and virtual states in the first and second complex energy sheet is represented in the right frame of Fig. 4. The analytical extension of the T -matrix to the second energy sheet is given by tℓII (p, p′ ; k2 ) = Vℓ (p, p′ ) +

2

π

∞

 0

p′′2 dp′′ Vℓ (p, p′′ )tℓII (p′′ , p′ ; k2 ) k2 − p′′2 − iε

− 2ikVℓ (p, k)tℓII (k, p′ ; k2 ),

(14)

2 In the complex analysis, a branch point of a multi-valued function is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point [See p. 46 of Mark J. Ablowitz and Athanassios S. Fokas (2003), Complex Variables: Introduction and Applications, Cambridge Texts in Applied Mathematics (2nd ed.), Cambridge University Press].

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T. Frederico et al. / Progress in Particle and Nuclear Physics 67 (2012) 939–994

a

b c Fig. 4. In the left frame, we represent the analytic continuation of the integral J (k) given in (7) by deforming the contour of momentum q integration, to pick up the pole contribution into the negative half-momentum-plane, which corresponds to the second energy Riemann sheet. In the right frame, we represent in the complex energy plane the particular case that a bound-state pole (in the first energy sheet) goes to a virtual-state pole (in the second energy sheet).

which can also be expressed in terms of the K -matrix equation (9). Instead of the K -matrix approach, one could also consider the Γ -matrix formalism, developed in Refs. [187–189], which follows de Kowalski–Sasakawa class of formalisms [190–192] to compute scattering observables. These formalisms keep all the advantages of the K -matrix approach, with no fixed-point singularity in the kernel of the integral equation. An extra advantage, of particular interest in this kind of formalisms, relies in their possibility to improve the iterative numerical convergence of the corresponding integral equations. In order to contextualize the relevance of the basic two-body formalism included in this section, we should note that this review is mostly concerned with halo nuclei where the two body properties are quite relevant. Therefore, the knowledge of what drives the physics of the core plus a nucleon is necessary. In many situations we will find situations where the neutron–core (n–c) system is unbound and the two-body information should be found by solving the analytical extension of the corresponding LS equation to reach virtual or resonant states. As an example, we can consider the case of 11 Li, composed by a 9 Li core plus two neutrons. In this case, the subsystem n–c, given by the 10 Li, is unbound, and can be described by the analytical extension of the corresponding LS equation, considering s-wave. On the other hand, virtual states in p-waves can also be found for three-body non-Borromean halo nuclei, such as 20 C with energy of about 1.6 times the corresponding neutron–core binding energy [193]. This may suggest further parameterizations of such non-Borromean three-body system, considering n–c as a virtual state, which applies to the examples of 11 Li and 14 Be. It may be possible that instead of a virtual p-wave three-body state in the non-Borromean situation, the Borromean halo nuclei presents a low-energy p-wave three-body resonance, or a Pygmy Dipole resonance. It was already shown [194] that a Borromean s-wave three-body system has low-energy resonances which appears when the barely bound s-wave subsystems becomes unbound. Furthermore, in the p-wave case the centrifugal barrier favors the formation of the continuum resonant state of the Borromean nuclei. Therefore, the appearance of a Pygmy dipole resonance in Borromean configurations dominated by s-wave interactions can be classified also as a universal phenomena of two-neutron halo-nuclei [64]. 2.1.2. Trajectory of S-matrix poles A particular case of two-body interaction that allows simple analytic solution for the LS equation is given in a separable form. There is a family of such interactions, with one or more separable terms, constructed in the past to facilitate the solution of the three-nucleon problem [195]. Let us consider, in s-wave, the case with one separable term, V = λ|g ⟩⟨g |, with a Yamaguchi form factor given by g (p) =

N 

αi

p2 + βi2 i =1

.

(15)

In this case, the potential and the corresponding two-body T -matrix are respectively given, in momentum space, by [181] V (p, p′ ) = λg (p)g (p′ ),

t (p, p′ , k2 ) = τ (k2 ) V (p, p′ ),

(16)

where

τ −1 (k2 ) = 1 −

2

π

V (p, p)

∞



p2 dp 0

k2

−

p2

+ iε

=1−λ

αi αj . ( k + i β ) ( k + iβj ) (βi + βj ) i i,j=1 N 

(17)

The interesting poles of the above separable T -matrix are the dynamical ones coming from τ (k2 ), since those coming from g (p)’s are given by the potential itself and are known as ‘static’ poles without  physical meaning. For the case with N = 1 in (15), considering α1 ≡ 1 and β1 = β , the poles are located at k = −iβ ± i

|λ|

2β

, considering attractive interaction (λ = −|λ|).

The only one bound-state, which is supported by this interaction, occurs for |λ| > 2β 3 , such that, for |λ| < 2β 3 , we have

T. Frederico et al. / Progress in Particle and Nuclear Physics 67 (2012) 939–994

953

Fig. 5. Schematic S-matrix pole trajectories for the case we have a rank-1 separable interaction, as shown in Eq. (16), with the form factor g (p) composed by two terms (N = 2), one positive with the other negative. The arrows indicate the direction for increasing values of the strength |λ|.

two virtual poles. When |λ| = 2β 3 , one of the poles is at k = 0, becoming a bound state, with the other, a virtual state pole, located at k = −i(2β). Next, considering the N = 2 case, in Fig. 5 we present the schematic T -matrix pole trajectory. In this case, we have a rank-1 separable potential with the form factor g (p) composed by two terms with opposite signals (α1 > 0 and α2 < 0). The attractive and repulsive parts of the final interaction are obtained from the combination of the two form factors of this rank-1 separable potential. As the strength of the interaction increases, the tendency is that the complex poles move to the real axis creating a continuum bound state (CBS). Such a pole is not related to an outgoing propagation wave, but instead to a localized function of momentum, with the corresponding Fourier transform decreasing exponentially as in the case of a bound-state wave function. This explains why this kind of pole is named CBS. If any two-body interaction allows for such a pole, beyond that of the usual two-body energy, the three-body system treated with this interaction brings the three-body ground-state for a value far away from the two-body energy scale [78,89]. For the N = 1 case, the range of the interaction is unambiguously related to the parameter β1 . However, for N = 2, the range of the two-body interaction is not so clear. Nevertheless, the observed collapsing behavior of the three-body system, in which one considers a two-body sub-system supporting a CBS, has been associated to the Thomas effect in Refs. [79,142]. Before we conclude this subsection, let us present the very useful low-energy effective range expression for ℓ = 0. In the neighborhood of the scattering threshold (k2 = 0), the T -matrix pole (6) may be expressed by two independent shape parameters, 1 + r0 k2 + O (k4 ), (18) a 2 where a is the two-body scattering length and r0 is the effective range of the two-body interaction. The structure of t0 (k), as a function of k2 , is given by Eq. (6), which has a cut along the real axis from k2 = 0 to k2 = +∞. Bound (virtual) states have positive (negative) values of a. In terms of T -matrix poles, occurs the following. Suppose we have a nearly zero negative imaginary pole (virtual state) for an effective attractive interaction. As we slightly increase the two-body strength, the pole approaches the real axis and the value of a is large and negative. If we keep increasing the strength, this negative imaginary pole reaches k2 = 0 and, at this point, we have a discontinuity in a, from −∞ to +∞. By still increasing the strength, a finite bound-state takes place with decreasing values of a. We anticipate here that both, r0 and a will have a crucial role in the three-boson system when they reach extreme values. For a zero-range interaction (r0 = 0), the T -matrix pole is just (1/a + ik)−1 , such that the bound-state energy will be −1/a2 . For r > r0 , the bound-state wave function is given by exp(−r /a)/r. As one should notice, what matters for the occurrence of the three-boson Thomas–Efimov effect is the divergence of the ratio a/r0 . k cot δ0 = −

1

2.1.3. Two-body n–core halo system The notion of halo nuclei can be introduced even in the case of a two-body model where a nucleon is bound by a core, as for example in the case of a neutron–proton system (deuteron), or n–10 Be. Let us assume a very simple interaction to treat this kind of system, with a spherical square well with depth V0 and range R0 , as considered before. The relevant question here is what fraction of the time the valence nucleon spends out of the range R0 . In order to reply this question, we should normalize the total wave function to find out the probabilities, Pint and Pout , for the valence nucleon to be inside and outside, respectively, the range  R0 . By considering the necessary boundary constraint that k2 /k1 + cot k1 R0 = 0, with 

k1 ≡ 2µ(V0 − B2 )/ h¯ 2 and k2 ≡ 2µ(B2 )/ h¯ 2 , where B2 is the two-body binding energy and µ the reduced two-body mass, we obtain the following expressions for the probabilities:

 Pint =

k1 R0 k1 R0 − tan(k1 R0 )

 Pout =



k1 k1 R0 − tan(k1 R0 )

1−



sin(2k1 R0 ) 2k1 R0

sin2 (k1 R0 ) k2



.



,

(19)

(20)

These normalized probabilities are represented in the panel (a) of Fig. 6, as functions of the two-body binding energy. As we observe, there is a large room in the nuclear binding energies to incorporate the interpretation of a halo, with a very weaklybound two-body system, where the valence nucleons are far apart from the core. The same information can be extracted

954

T. Frederico et al. / Progress in Particle and Nuclear Physics 67 (2012) 939–994

b

a

Fig. 6. Normalized probabilities (a) and root-mean-square radius (b) as functions of the binding energy, for a square-well model with fixed R0 = 2 fm and 2µ = 938 MeV.

from the calculation of the r.m.s. radius 2

⟨r ⟩ =

R20



k1 R0



k1 R0 − tan(k1 R0 )

 ⟨r 2 ⟩, where 1 3

+ [1 + 2k2 R0 + 2(k2 R0 )2 ]

−

cos(2k1 R0 ) 2(k1 R0 )2

sin2 (k1 R0 )



2(k2 R0 )3

−

[2(k1 R0 )2 − 1] sin(2k1 R0 ) 4(k1 R0 )3

.

(21)

In the panel (b) of Fig. 6 we show how the root-mean-square radius behaves as a function of the binding energy for a fixed R0 . Both panels shown in Fig. 6 indicate that loosely bound systems are basically governed by Pout > Pint , having very large sizes and breaking the usual common sense that the nuclear radius should increase as the cubic root of the number of nucleons. 2.2. Three-body problem 2.2.1. Thomas–Efimov Effect By following our general retrospective introduced in Section 1, on the Thomas collapse [76], and Efimov effect [77], in this section we will consider both effects in a unified approach applied to three identical bosons in s-wave. In such a case, the corresponding three-body homogeneous equations can be described by the so-called Skorniakov Ter-Martirosian equation using dimensionless quantities. Amado and Noble [196], have shown how the Efimov effect appears through the analysis of eigenvalue spectrum of the Faddeev kernel in momentum space for three-identical particles in the limit of large scattering lengths, and that the number of bound states is roughly (1/π ) ln(Λ|a|), where Λ is a momentum cutoff. They extended the proof to the case of two identical particles and a third one, in order to show that two light or two heavy three-particle systems present Efimov effect. Later on, it was derived [107] in a Born–Oppenheimer approximation the presence of a long range effective potential of the type 1/r 2 between the heavy particles when the heavy–light system has a zero binding energy, which induces the condensation of three-body levels at the threshold. In subsequent studies, more details of the Efimov effect in asymmetric mass systems were explored (see e.g. [11,39,44,107]). In the case of three identical particles, the long-range behavior of an effective two-body potential was also studied in Refs. [197,198], considering the relative s-wave state and non-zero twobody binding, where deviations from the 1/r 2 behavior is discussed. Based in the work of Amado and Noble, in a unified momentum space description of Efimov and Thomas effects, it has been claimed that these two apparently different effects due to Efimov and Thomas are related to the same singularity structure of the kernel of the Faddeev equation [96], and in appropriate units, the presence of one of these effects implies the presence of the other. The relation between Thomas and Efimov effects for three identical bosons can be seen through the Skorniakov and Ter-Martirosian equation for the bound-state and zero range potential [101]. The equation presented in a regularized and subtracted form in Ref. [158] was derived from the homogeneous subtracted equations for the Faddeev components of the three-body T -matrix [157], and it is written for s-wave as: f (y) =

−2/π  √ ± ϵ2 − ϵ3B + 34 y2

 0

∞

dxx2



+1

 dz

−1

1

ϵ3B + y2 + x2 + xyz

−

1 1 + y2 + x2 + xyz



f (x),

(22)

where the dimensionless quantities are ϵ2 = −E2 /µ3 and ϵ3B = −E3B /µ3 , with µ3 being the regularization parameter as defined in 1.5. The two-body input is the scattering amplitude with a bound or a virtual two-body state, respectively

T. Frederico et al. / Progress in Particle and Nuclear Physics 67 (2012) 939–994

955

√ distinguished by the sign of the scattering length given by the plus and minus sign in the ϵ2 . The units are such that the boson mass and h¯ are equal to one. The close relation between the three-body Thomas collapse and the Efimov effect [96] in s-wave states, is seen when the two-body energy in units of the regulator energy ϵ2 goes to zero in Eq. (22). For fixed µ3 this happens for E2 → 0, i.e., the Efimov limit, while for fixed E2 one has that µ3 → ∞, i.e., the intermediate three-body states explores the ultraviolet momentum region of the Skorniakov and Ter-Martirosian leading to the three-body collapse. In a situation where the strength of the short-potential is tuned to produce large scattering length |a| → ∞ in respect to the range, the wave function 1 heals in dimensionless units of distances ≫ µ− 3 , where it is essentially an eigenstate of the free Hamiltonian. This is the origin of the universal three-body physics, which means independence on the details of the short-ranged potential, besides the dependence on few parameters or scales. Furthermore, for non-zero angular momentum states, the kernel is compact for zero-range equation and there is no necessity of a regulator. In this simple model the physics of nonzero angular momentum states of three-identical boson is completely determined by E2 , which sets the scale of the problem. This has been explored in two-neutron halo situations, where virtual states are formed in higher partial waves [193]. The dimensionless form of Eq. (22) for µ3 → ∞ has a scale invariance property, which admits two solutions in momentum space with oscillatory behaviors [102], in correspondence to the wave function obtained by the Efimov longrange attractive potential, V (ρ) = −(s20 + 1/4)/ρ 2 , which acts in the lowest hyper-angular state. Here, ρ is the hyperradius, with ρ 2 given by the sum of the two squared relative Jacobi coordinates for the system, and s0 is a parameter given by s0 ≃ 1.00624 [77]. This hyperradius potential is modified at short distances by a short-range force, such that generates an extended wave-function that can be of the order of |a|. The hyper-radial wave-function has an oscillatory behavior like 1

Ψ (ρ) ∼ ρ ±is0 + 2 , where the combination of independent solutions describes the short-range physics (see also [39]), which is introduced as a boundary condition. In the framework of the effective field theory given by Ref. [44] a phase appears in the observables (a three-body parameter), which encodes the short-range physics for the long-wavelength properties of the s-wave three-particle system. It is illustrative to sketch the form of the three-boson wave function in coordinate space for the Skornyakov and Ter-Martirosian model (see e.g [104]). For the sake of notation, the particles are labeled by 1, 2 and 3. It is a solution of the free-Schrödinger equation for three-particles interacting with zero-ranged interactions, except when the particles overlap:

(E3 − H0 )Ψ = 0,

H0 =

3 4

⃗2i + p⃗2jk , q

(23)

⃗2i and p⃗2jk , with (i, j, k) being cyclic permutations where the kinetic energy operator is given in terms of the Jacobi momenta q ⃗ij ). The of 1, 2 and 3. The corresponding momentum- and configuration-space conjugate coordinates are (⃗ qi , ⃗ ri ) and (⃗ pij , R ⃗i , with R⃗ij ≡ R⃗i − R⃗j , and r⃗k = R⃗k − 1 (R⃗i + R⃗j ). vector position of the particle i in the center-of-mass frame is given by R 2 The solution of the free-Schrödinger equation in 3D is: Ψ =



where κij =

d3 q1



eı(κ23 R23 ) iq⃗1 ·⃗r1 1 e f (|⃗ q1 |µ− 3 ) + cyclic permutations, R23

(24)

3 2 q 4 k

⃗ − E3 from the solution of the free Schrödinger equation. The spectator functions are the weights of the

eigenstates of the kinetic energy operator with energy E3 . The irregular two-body Hankel function gives the tail of the relative two-body wave function for the three-body bound state and builds the eigenstate of the free three-particle Hamiltonian (see the appendix of [131]). Note that, the arguments are such that they are asymptotically decaying functions of the relative distance. One should also observe that, the three-particle eigenstate of a three-body Hamiltonian with a short-range interaction ⃗ij | ≫ R0 , where R0 represents the range, attaining a universal form. The dynamics is retained by should reduces to Ψ for |R the spectator functions, which carry the relevant low-energy scales of the three-particle system that are contained in the two-particle subsystems effective range parameters. 2.2.2. Three-boson virtual state The three-body system can present a virtual state, as already studied for the triton [120–122]. It was found that the triton virtual state is an Efimov virtual state, which becomes bound when two-nucleon subsystems are bound at the threshold. By considering bound the deuteron and the virtual neutron–neutron spin singlet state with an energy of −41.47 × 10−8 MeV they obtained four excited bound states. They observed that the triton virtual state first moves away from the lowest scattering branch-point, then stops and turns around and eventually becomes a bound state when the strength of the spin triplet NN interaction is increased. The other bound trinucleon states of this system become virtual and move into the left-hand cut with the increase of the strength of the triplet NN potential and do not become resonances. This behavior is repeated in n–n–c halo systems when the neutron can bind to the core, as in the case of 20 C [72] and the Efimov states do not become a resonance. That is different in the case of Borromean systems where an excited bound state turns into a continuum resonance when the interaction strength is increased [194]. The virtual state energy is obtained by the solution of the analytic continuation of Eq. (22) to the second Riemann energy sheet. The extension to the unphysical sheet of energy of the zero-range model [104] is done by passing through the elastic

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T. Frederico et al. / Progress in Particle and Nuclear Physics 67 (2012) 939–994

scattering cut, with branch point defined by the two-body bound energy. It follows the same analytical procedure developed in Ref. [199] to study three-body force effects in the triton virtual state energy and it is done as in the two-body case (14), where one sweeps over a pole and picks up the contribution of the residue. In the following, the steps given in [158] to derive the virtual state zero-range equation for the trimer, using [104,199] given as a generalization of (14) are reviewed. It starts by defining h(y) ≡ (ϵ2 − ϵ3 − 34 y2 )f (y) (ϵ3 > 0), and then Eq. (22) (still in the first energy sheet) is rewritten as h(y) = −

2



π

√ ϵ2 +

 ϵ3 +

4



1 + y2 + x2 + xyz

∞ 2

y2



+1

 dz

dxx

1

ϵ3 + y2 + x2 + xyz

−1

0



1

−

3

h(x) . ϵ2 − ϵ3 − 43 x2 + iδ

(25)

The corresponding analytic extension to the second Riemann energy sheet is made by adding a contribution from the twobody (boson-dimer) elastic scattering cut, that pick-up the contribution of the residue over a pole (see e.g. Eq. (14)), as follows: hV (y) = −

−

−

−



2

π

√ ϵ2 +

 ϵ3V +

3 4





π

√ ϵ2 +



1+

y2

+

 dz

−1

1

ϵ3V + y2 + x2 + xyz

1

1



3

4

ϵ3V +

dxx2

dz −1

0

hV (x)



x2

+1

dxx

ϵ3V + y2

1



− hV (x) ϵ2 − ϵ3V − 43 x2 + iδ ϵ2 − ϵ3V − 34 x2 − iδ   +1  ∞

1 + y2 + x2 + xyz 2

2

0



1

∞

y2

ϵ2 − ϵ3V − 43 x2 − iδ

+ xyz

1

y2

+ x2 + xyz

,

(26)

where the three-boson virtual state energy is real and denoted by −ϵ3V . The first integration can be  done analytically for 3 2 δ → 0. By replacing ϵ2 − ϵ3V ≡ 4 κ (note that for the three-body virtual state κ is defined as κ ≡ −i 43 (ϵ3V − ϵ2 ) ≡ −iκV ) in the above equation: hV (y) =

8 3

 κV 

√ ϵ2 +

+1



3



ϵ3V + y2



4

1

1



hV (−iκV ) ϵ3V + y2 − κV2 − iκV yz 1 + y2 − κV2 − iκV yz     +1  ∞ 1 2 √ 3 2 2 dxx dz − ϵ2 + ϵ3V + y 2 + x2 + xyz π 4 ϵ + y 3V 0 −1  1 hV (x) − . 1 + y2 + x2 + xyz ϵ2 − ϵ3V − 43 x2

×

dz

−

−1

(27)

The energy of the three-boson virtual state should be outside of the one-particle exchange cut from the first term in the right-hand side of Eq. (27), and given by

ϵ3cut + y2 + x2 + xyz = 0, with −1 ≤ z ≤ +1, y = x = −iκcut and ϵ3cut =

(28) 3 4

κ

2 cut

+ ϵ2 , which defines the cut between ϵ ≤ ϵ3cut ≤ 4ϵ2 . Therefore, 4 3 2

the three-boson virtual state energy should be found in the range ϵ2 ≤ ϵ3V ≤ 43 ϵ2 . The numerical solution of Eq. (27) given in reference [158] shows that, when ϵ2 is decreased, either meaning a decrease of the range of the two-body interaction or of the two-boson bound state energy, the bound state has precisely the same fate as previously found for finite short-range potentials [122]: the virtual Efimov state moves towards the cut and turns into a bound state. In the following more discussions on this point will be provided. 2.2.3. Bound, virtual and resonant Efimov states The movement of Efimov states as the two-body energy is varied, seems irrelevant in nuclear physics because it is not possible to tune the interaction in the nucleus, as it happens in the case of ultracold atomic traps by using a Feshbach resonance. However, the discovery of light exotic halo nuclei revealed some promising candidates to present an excited

T. Frederico et al. / Progress in Particle and Nuclear Physics 67 (2012) 939–994

a

957

b

Fig. 7. Schematic representation of the analytical cut structure for a three-boson system, showing how the three-body poles of the S-matrix move in the complex energy plane, as the dimensionless two-body energy ϵ2 is increased. The sketch in frame (a) represents the bound Nth state, followed by the next excited (N + 1)th state, which turns into a virtual state in the second energy-sheet plane (an S-matrix pole on the negative-real axis of the second Riemann energy sheet), crossing the two-body branch point. The unbound two-body system case (no two-body cut — only three-body cut) is sketched in frame (b), where the excited three-body bound state becomes a continuum resonance (S-matrix pole located in the positive real and negative imaginary parts of the second Riemann sheet of the complex plane).

state and, in some cases there is large standard deviations in the energy of the two-body binding energies, which could make a given halo nucleus (approached by a core and two-neutrons) to be bound, virtual or resonant. However, in the original approach, Efimov considered only weakly-bound identical three-body systems for large scattering lengths. Nowadays the meaning of an Efimov state has been extended to interpret general properties of a shallow three-body state for |a| → ∞. The terminology includes asymmetric mass three particle systems, and mixing of particles obeying different statistics, such as mixture of bosons and fermions. Extension of this low-energy concept refers to virtual and resonant three-body states, which can be studied by considering the physical poles of the Faddeev components of the three-body transition matrix, by solving appropriate homogeneous equations for bound, virtual and resonant states. An excited shallow three-body bound state can turn into a virtual or a resonant state, by decreasing |ϵ2 | for the bound or virtual two-body state, respectively and in both cases the absolute value of the two-body scattering length increases. The trajectory of the three-body shallow state is sketched in Fig. 7 for two different situations: (i) the two-body subsystem has one bound state [frame (a)] and (ii) the subsystem has one virtual state [frame (b)]. On the negative part of the real axis it (N ) (N +1) is drawn two poles corresponding to the three-body bound states with energies ϵ3 and ϵ3 , where the upper indexes indicate the Nth and N + 1th states. The two possible situations for bound two-body or virtual states, are sketched in Fig. 7 when ϵ2 increases either by the decrease of the three-body short-range and or by the increase of the two-body binding energy |E2 |:

• The frame (a) of Fig. 7 shows the trajectory of an excited Efimov state for a two-body bound state. The shallow three-body bound state moves towards the two-body scattering cut and enters the second Riemann energy sheet becoming a virtual state. In this situation, the three-body bound state energy is swallowed by the two-body scattering cut, with the branch point at −ϵ2 becoming a three-body virtual state in the second Riemann sheet [122,158]. This effect was discovered in the context of the trinucleon system. (It remains even in the case that a three-body force is included [199]). In the complex momentum plane, the pole moves along the imaginary axis, from positive to negative values. • The frame (b) shows the movement of an excited Borromean Efimov state when the two-body systems has a virtual state. In this case the shallow bound state hits the three-body continuum branch point and migrates through the cut to the unphysical energy sheet becoming a continuum resonance [194] (see also [200,201]). We observe that in the case of a short-range potential model calculation the shallow three-body states can be moved either by introducing a three-body force, by changing the two-body binding through the strength of the short-ranged two-body interaction, or by changing the range of the potential. In the case of the zero-range model, the continuum three-boson resonance was calculated in Ref. [194] by applying a contour deformation in Eq. (22) when the two-body system has a virtual state. The momenta are rotated into the complex quantities: x(y) → x−iθ (ye−iθ ), with 0 < θ < π /4, which exposes the second Riemann energy sheet. For a large enough θ the solution of Eq. (22) in the complex plane is found for tan(2θ ) > −Im(ϵ3 )/Re(ϵ3 ). The method given in Ref. [202], shows how to calculate few-body resonances using harmonic traps in configuration space. 2.2.4. Scaling functions and limit cycles Any long-wavelength s-wave observable of three interacting particles and in 3D are sensitive to the details of the shortranged potential through one short-range parameter that can be rewritten, for example, in terms of the binding energy of the shallowest three-body state. Indeed, the solution of the regularized zero-range model for three-bosons depends on the regularization scale parameter µ3 as shown, for example, by Eqs. (22) and (27) of the preceding subsections. Then, s-wave observables are correlated to the regularization parameter that can be eliminated by fixing, for example, the binding energy

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of the shallowest three-body state, then a general universal correlation between the observable O3 , which represents either a scattering amplitude at an energy E or an excited three-body energy is written according to Ref. [114] as:

  O3 (E , E3 , {± |E2 |}) = |E3 |η F3

 

E E3

, ±

E2 E3

 ,

(29)

where the signs ± distinguish between a two-body bound or virtual state. The dimension of the observable gives the exponent η. For some short reviews with discussion on this approach, see also Refs. [115,203]. The correlation function (29) is found in the scaling limit where µ3 → ∞ when the function has gone over an infinite number of cycles. Bound states emerge from the lowest scattering threshold as µ3 grows and F3 as a function of the shallowest three-body bound state repeats itself. When ϵ2 goes to zero (unitary limit) the ratio between the energies of consecutive Efimov states is e2π/s with s = s0 ≃ 1.00624 [77]. In the limit µ3 → ∞ – scaling limit – the function F3 approaches a limit cycle. The scaling limit ensures that the wave function of the system is completely dominated by the effective Efimov long-range potential. In practice few cycles are enough to determine the correlation function, because ϵ3 ≪ 1 [158]. For asymmetric mass systems in the unitary limit the geometrical factor s depends on the mass ratios of the constituents [39,44]. The advantage of presenting correlation functions relies on the model independence of the approach. This means a regulator independence when considering a zero-range model, which allows comparison with any other short-range potential calculations. The Phillips plot [125,126] is historically the first suggestion of eliminating the model dependence by presenting results in terms of a correlation between the neutron–deuteron scattering length and triton binding energy. In the zero-range limit it is given by a scaling function as discussed in [132,133]. The suggestion of using dimensionless ratios in the arguments of the scaling function comes from [11,114], and is understood, e.g., by inspecting the subtracted form of the bound state equation for three-bosons (22) where only dimensionless quantities appear. The physical picture underlying the scaling functions or the correlations between s-wave observables for short-ranged potentials can be illustrated by the zero-range model when the ultraviolet regulator µ3 → ∞. The three-particle wave function for relative distances of the order of ∼(µ3 )−1 or less for low-energy scattering states with |E /µ23 | ≪ 1 or shallow bound states with |E3 /µ23 | ≪ 1 are essentially the same, parameterized by one scale. When the relative distances are larger then ∼(µ3 )−1 , only virtual two-body collisions happen, carrying the physical information fixed by scattering lengths. Then, the correlation between asymptotic observables of the scattering wave function and bound states, given by the scaling form of Eq. (29), seems to be possible not only for zero-range models but in general for short-range potential models.

2.2.5. Scaling of Efimov trimer energies The universal s-wave low-energy three-particle properties can be obtained with a zero-range interaction by calculating the functional form of the scaling function (29) for a particular observable using the subtracted form the Skorniakov and Ter-Martirosian equation [158] and the three-body transition matrix [160]. Other regularization schemes are possible, like momentum cutoff [103,114] or using a three-body force [155]. These different regularization schemes should provide identical form for the correlation between observables in the limit where the regularization parameter is let to infinity (see e.g. [114]). In this case the scaling limit is achieved, with scaling functions being the limit cycle of the correlation plots (see also [204]). Noteworthy to remark that, the study of low-energy properties of few-body systems, presenting results in the form of correlation between observables, should be preferable. This occurs because the dependence on a specific regulator parameter, which is particular for each scheme and not universal, is washed out. The general form of the scaling functions (29) for a three-boson observable O , when used for the particular case of the correlation between the energies of two consecutive trimers, N and N + 1 with the excited state labeled by the N + 1 is given by the limit cycle of the function: ( N +1 )

E3

(N )

E3

= lim

N →∞

ϵ3(N +1) ϵ3(N )

  =F

±

E2 (N )

E3

 ,

(30)

which is built by solving Eq. (22) in the limit of µ3 → ∞ [158], or when the ultraviolet regulator is set to infinity [114], and the limit N → ∞ is taken. The geometrically separated Efimov trimer states fix the value of F (0) = e−π /s0 = (22.7)−1 . The generalization of F for n–n–c systems is given in [11,74,178], which can be applied either to check for the possibility of bound two-neutron halos or molecular systems [179]. The universal function F represents the tower of trimer states close to the Efimov limit (see Fig. 8) as shown in [114]. The figure sketches the sequence of weakly bound trimer states as a function of the scattering length. By increasing the positive scattering length, states pumps out from the boson-dimer threshold (dots). At the Efimov limit, the infinite sequence of states condensing at zero energy are indicated by the full circles, continuing the variation of the scattering length to negative values the Borromean trimers dives into the three-body continuum.

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0

a<0

959

1/a

a>0

Fig. 8. Schematic representation of the trajectory of weakly-bound Efimov levels, as the inverse of the two-body scattering length a changes, from unbound (a < 0) to bound (a > 0) two-body cases.

The trimer excited state can be bound, virtual or resonant states and it is represented in a single plot [203]. The full sequence of states in Fig. 8 is resumed in the scaling plot of Fig. 9. In order to have sizable values for comparison, the twobody energy was subtracted from the excited three-body state, only in the cases that two-body is bound. The squares in Fig. 8 corresponds to the transition from (I) to (II) and  the inverted triangle from (III) to (IV). The plot from [114] is presented in the form of



(E3(N +1) − E2 )/E3(N ) for bound two-body systems; and in the form of

(E3(N +1) )/E3(N ) in the other case. In Ref. [194] the trimer continuum resonant states were calculated for a virtual dimer and

shown in the figure. The results from realistic short-range interactions for the helium trimer are given in the figure and confirm the model independence of the scaling function. Deviations due to the finite range can be seen in the right part of the plot, when the two-body binding increases and the scattering length decreases. As |a|/r0 decreases a deviation to the right of the curve is expected as the binding of an excited state is favored when the range increases as Fig. 9 shows. Indeed, this qualitative description of range corrections at the threshold are substantiated in Ref. [205]. In region (I) of Fig. 9, a trimer virtual state appears at the branch point 43 ϵ2 of the one-particle exchange cut in the second energy sheet and by further decreasing ϵ2 it becomes bound in region (II) [frame (a) of Fig. 7]. When the scattering length turns to be negative in region (III) of Borromean binding, the excited state approaches the three-body scattering threshold crit and turns to a continuum resonance in region (IV) [frame (b) of Fig. 7]. At the critical value of ϵ2B = 0.144ϵ3(N ) the virtual (N )

state becomes bound above an Nth trimer state [114,158], which corresponds to E2crit = 0.144E3 regions (I) and (II). At the critical value of the virtual two-boson binding energy, ϵ (N )

crit 2V

in the transition between

= 8.82 × 10−4 ϵ3(N ) [194], which

crit corresponds to E2V = 8.82 × 10−4 E3 the excited state turns into a continuum resonance. The elastic boson-dimer s-wave scattering length tends towards −∞ when the trimer virtual state approaches the scattering threshold and then turns to +∞ when the bound state is formed in the transition from region (I) to (II) in Fig. 9. The same analytical behavior appears in three-body systems with different particles, when at least one of the two-body subsystem is bound, as in the specific case of the low-energy s-wave neutron-19 C scattering, where such dependence was explored in Ref. [180]. The trimer continuum resonance appears only in the Borromean region (IV) and it was observed as a peak in the three-atom recombination measured near a Feshbach resonance, for negative scattering lengths, in a cold gas of cesium atoms [147]. The experimental shift in the position of the peak with temperature was shown to be in agreement with the dependence of the resonance energy with scattering length [211] (see also [212,213]). A theoretical description of weakly-bound atomic trimers, and the three-body recombination in ultracold systems, considering the experimental observations, was reported in Refs. [214,215]. Another experimental evidence for the scaling plot of Fig. 9 comes from [216] where it was measured the ratio between the scattering lengths corresponding to the trimer crossing the scattering threshold for a < 0 and a > 0 value of ∗ 7 |a− 2 |/|a2 | = 10.4(5) (14) in a cold gas of trapped Li in the state |F = 1, mF = 1⟩. The value obtained from the scaling curve of Fig. 9 in the zero-range model of |a− |/|a∗ | = 12.8 assuming that the reference trimer energy is unaltered from positive to negative scattering lengths is in agreement with the experimental ratio.

2.3. Illustration: Born–Oppenheimer three-particle model The Born–Oppenheimer three-particle model developed by Fonseca, Redish and Shanley in Ref. [107] is an analytically solvable model where the Efimov effect is clearly demonstrated, in a simple way allowing for a better intuitive understanding of the physical mechanism leading to this effect. Other demonstrations of this effect, in general requires more sophisticated

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N

(N +1)

/E3(N ) , as a function of the ratio between the two-body and three-body energies, E2 and E3 , respectively. The vertical lines separate the regions (I)–(IV). In region (I) the N + 1 trimer state is virtual. In region (II) the N + 1 trimer state is bound and in (III) it is a Borromean system. In region (IV) the N + 1 trimer state is a continuum resonance. Calculations with finite range potentials for 4 He3 from Ref. [206] given by the empty boxes (s-wave, N = 0), empty circles ((s + d)-waves, N = 0), and crossed dot (s-wave, N = 1). Other calculations for N = 0 and s-wave are from Refs. [207,208] (triangles), from Ref. [209] (crossed boxes) and from Ref. [210] (empty diamonds).

Fig. 9. Scaling function for the ratio of consecutive trimer energies, E3 (N )

Source: Extracted from Ref. [203].

Fig. 10. Schematic illustration of the three-body system considered in the FRS model with corresponding coordinates.

mathematical formalisms based on three-particle dynamical equations. The claim of Ref. [96] that the Efimov effect is closely related to the Thomas collapse of the three-body ground state, such that one of the effects implies the presence of the other, suggests that the Born–Oppenheimer three-particle model of Ref. [107] should also exhibit the Thomas effect under an appropriate limit of zero-range light–heavy interaction. In the present work we shall demonstrate the Thomas effect in the three-particle system consisting of two heavy and one light particles in case when the light–heavy interaction is of zero range. We study the effective heavy–heavy interaction in the Born–Oppenheimer model and find that this effective interaction, has a r −2 short range behavior responsible for the Thomas effect in the limit of zero range light–heavy interaction, which leads to an infinite number of s-wave bound states of infinite binding. The same heavy–heavy interaction has a r −2 long-range behavior responsible for the Efimov effect in the limit when the light–heavy system has just a zero-energy bound state, which leads to an infinite number of three-particle bound states, which accumulate at zero binding. This model has been recently revisited in Ref. [217]. In general we find that the effective heavy–heavy interaction in the Born–Oppenheimer model has a r −2 form in the range b > r > |a| when b is the range of the light–heavy interaction and a is the light–heavy scattering length. In the limit of zero-range light–heavy interaction b → 0, the effective heavy–heavy r −2 interaction extends to r = 0 and gives rise to collapse of the Thomas states at the origin. In the limit of infinite light–heavy scattering length (zero light–heavy binding) a → ∞, the effective heavy–heavy r −2 interaction extends to r = ∞ and leads to an infinite number of zero energy states. In the very special case with b = 0 and a = ∞, the effective heavy–heavy r −2 interaction extends from r = 0 to r = ∞ and one has simultaneously infinite number of Thomas and Efimov states. Our presentation here brings evidence to the claim that the existence of one effect implies the other. The three particle Born–Oppenheimer model is very simple and appropriate to start this subject. As our analysis will be based on the elegant analytical model of Fonseca et al. [107], hereafter FRS model, we present a brief resume of the same. In this model, we have two heavy particles of mass M and a light one of mass m, see Fig. 10, with the Hamiltonian H =−

1

µ

2 R

∇ −

1

ν

 2 r

∇ + V1 ⃗r −

⃗ R 2



 + V2 ⃗r +

⃗ R 2

 + V3 (R⃗),

(31)

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with

µ=

µ ¯ 2

,

2µ ¯

ν=

2µ ¯ +1

,

M

µ ¯ =

(32)

m

and 1

⃗ = ⃗r1 − ⃗r2 , R

⃗r = ⃗r3 − (⃗r1 + ⃗r2 ),

(33)

2

where ⃗ r1 and ⃗ r2 are coordinates of the two heavy particles and ⃗ r3 is that of the light particle. The use of the Born–Oppenheimer ansatz

Ψ (⃗r , R⃗) = ψ(⃗r , R⃗ )Φ (R⃗)

(34)

yields the following two equations

    1 1 ⃗ ⃗ ⃗ ⃗ − ∇ + V1 r − R + V2 r + R ψ(⃗r , R⃗) = E (R)ψ(⃗r , R⃗), ν 2 2   1 − ∇R2 + V3 (R) + E (R) Φ (R⃗) = E Φ (R⃗). µ 

1

2 r

(35)

(36)

⃗) of Eq. (35) serves as the effective potential for the movement of the heavy particles, given by Eq. (36). The eigenvalue E (R The light–heavy interactions, V1 and V2 , are chosen to be separable s-wave potentials V = λ|g ⟩ ⟨g |,

(37)

with Yamaguchi form factors

⟨p|g ⟩ =

1 p2

+ β2

.

(38)

The symmetric solution of Eq. (35) in the limit of very large heavy to light mass ratio, where µ ¯ = M /m is shown to be given analytically by

(β + 2κ0 + ξ )ξ = (β + κ0 )2

2β

e−(κ0 +ξ )R − e−β R

(β − κ0 − ξ )2

R



−

 β + κ0 + ξ e−β R , β − κ0 − ξ

(39)

where

κ 2 = −ν E (R),

κ02 = −ν E0 ,

and ξ ≡ κ − κ0 ,

(40)

⃗) the heavy–heavy effective interaction. The parameter β of the with E0 being the light–heavy binding energy and E (R Yamaguchi interaction is an inverse range parameter, which is defined by β −1 = b, where b is the range of the light–heavy interaction. In the limit of small light–heavy binding energy, we have κ0−1 = a > 0, where a is the scattering length of the light–heavy system. The parameter ξ was introduced as the difference between κ and κ0 for convenience in studying the Efimov effect. The Eq. (39) was derived in Ref. [107] for large R, which clearly shows how the Efimov effect appears due to ⃗). Let us now consider, in general, the behavior of E (R⃗), to explore the resulting effect in the the long-range behavior of E (R ⃗). For that, we start by considering three regions in the R-space, defined in the opposite case of a short-range behavior of E (R following way: (i) short range, with 0 < R < b; (ii) intermediate range, with b < R < a and the assumption of small light–heavy bound-state energy; (iii) long range, with a < R. For our actual purpose, by taking κ0 ≈ 0 and b ≈ 0 (such that β ≫ κ0 and β ≫ ξ ), the interest is in the intermediate and short-range solutions of Eq. (39). In the intermediate range, β R ≫ 1 and κ0 R ≪ 1, such that the Eq. (39) reduces to 1 −ξ R e , ⇒ ξ R ≡ A ≈ 0.567143. R By using Eq. (40), we obtain

ξ≈

κ = κ0 +

A

⇒ κ 2 = κ02 + 2κ0

A

(41)

A2

. (42) R R R2 In the Thomas limit, as b → 0 and a ≫ b for a ≫ R > b, the last term in the rhs of Eq. (42) dominates, such that one obtains E (R) = −

κ2 A2 = − 2. ν νR

+

(43)

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The effective interaction given by Eq. (43) is valid for small R values in the range a ≫ R > b, leading to the eventual collapse of the system when b → 0. In the Efimov limit, as κ0 → 0, we have a ≫ b, and the last term (rhs) of Eq. (42) again dominates (for a > R ≫ b) leading to Eq. (43). Now, with the effective interaction (43) valid for large values of R in the range a > R > b, as a → ∞, on obtains the infinite number of zero energy Efimov states. The FRS model also provides an interesting scaling property regarding to the two-body binding energy B2 of the subsystem and the binding energy B3 of the three-particle system [218]. Let us consider the effective interaction model of FRS in which R is the distance between the center-of-mass of the subsystem and the third particle: V (R) = −

1

ν



2 e−R/a a

R

+

e−2R/a R2



.

(44)

The Schrödinger equation to describe the three particle system may be given by an effective two-body model, where the subsystem threshold is taken into account,

  l(l + 1) 1 d2 + + V (R) ψ = (B2 − B3 )ψ. − 2 2 2m dR

(45)

2mR

Let us assume that B2 is small and given by B2 ≈ 1/a2 . Now, notice that V (R) = B2 V (R′ ) if R′ = may be rewritten as

    l(l + 1) B3 1 d2 ′ ′ + + V ( R ) ψ = 1 − ψ ′. − ′2 ′2 2m dR

2mR

B2

√ B2 R. Therefore, Eq. (45)

(46)

This implies that, within the Born–Oppenheimer approximation, a scaling should be observed between the three- and twoparticle binding energies, if such systems are bound. Interestingly, this statement should also be valid for ℓ ̸= 0. 3. Regularization, renormalization and EFT methods The description of large halos with one or two-neutrons by short-range contact interactions, where the long-wavelength physics is parameterized directly by observables, seems very attractive as the details of the short-range force are washed out. However, contact potentials produce ultraviolet divergences in the scattering equation as the intermediate state propagation explores states with arbitrarily large virtuality, which are not prevented to contribute significantly to the scattering. Therefore, the use of contact interactions to study the properties of the neutron halo demands a scheme to regularize and renormalize the scattering equation. A method to solve T -matrix equations with singular interactions was proposed in Ref. [165], which introduces a finite number of subtractions as a generalization of the one-subtracted equation enough to renormalize the nucleon–nucleon onepion-exchange potential plus a Dirac-delta term [161]. The multiple subtractive renormalization method of the scattering equation with contact potentials allows to get solutions of the two-body problem, for any finite momentum power in the matrix elements of the singular interaction, like the one below acting in s-wave states given by

⟨p|V |q⟩ =

1 

λij p2i q2j ,

(λij = λ∗ji )

(47)

i ,j = 0

where λij are the bare strengths. The unknown short range physics related to the divergent part of the potential are replaced by the renormalized strengths of the interaction, which are known from the scattering amplitude at some reference energy. The renormalized quantum mechanical theory of singular interactions should not depend on the arbitrary subtraction point [160], which means that it can be moved without affecting the physics (see e.g. [219,220]). As shown in Ref. [165], the driving term of the n-subtracted scattering equation changes as the subtraction point moves due to the requirement that the physics of the theory remains unchanged. The driving term of the subtracted scattering equation is a solution of a non-relativistic counterpart of a Callan–Symanzik equation [162–164], i.e., a first-order differential equation, which is a consequence of the invariance of the T -matrix under dislocations in the arbitrary subtraction point. The renormalization group (RG) equation matches the quantum mechanical theory at scales µ and µ + dµ, without changing its physical content [221,222]. Renormalization group analysis by boundary conditions in potential scattering has been discussed in [223]. The program of effective field theory applied to the nuclear force (see e.g. [45,109,141]) allows for a systematic expansion of the interaction and contact terms consistent with the symmetries of the problem. The fundamental point in the EFT expansion is the possibility to identify two well separated scales, one associated to low-momentum and another one to high-momentum. Large nucleon halos naturally leads to a low-momentum scale in correspondence to the size and a highmomentum scale to the force range. The application of EFT to halo-nuclei was suggested [117] to study the low energy 5 He p-wave resonance, where contact potentials were used. In the framework of EFT, renormalization and renormalization group techniques have also to be used to solve few-body problems (see e.g.[171]). The non-perturbative renormalization approach developed in Ref. [165], using successive subtractions in the kernel of the scattering equation to allow a finite solution for the T -matrix calculated with potentials including contact terms, has

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963

an associated fixed-point (renormalized) Hamiltonian. The general concept of fixed-point Hamiltonians can be found in [224–228]. The fixed-point Hamiltonian should have the property to be stationary in the parametric space of Hamiltonians, as a function of the subtraction point [228]. The subtraction point is the scale at which the quantum mechanical scattering amplitude is known [165]. In this context, the renormalization scale is given by an arbitrary subtraction point [229]. A sensible theory of singular interactions exists if and only if the subtraction point slides without affecting the physics of the renormalized theory [229]. This property is realized through the vanishing derivative of the renormalized Hamiltonian, in respect to the renormalization scale. This implies in the independence of the T -matrix in respect to the arbitrary subtraction scale, and in the renormalization group equations for the scattering amplitude. In one example, the discrete eigenvalues of a renormalized two-body Hamiltonian for a contact plus a Yukawa potential has been calculated and the eigenvalues matches the poles of the renormalized T -matrix in the negative energy region (see [168]). A point-like Dirac-delta interaction in the configuration space, which is given by a constant in momentum space, i.e., the first term of the bare interaction (47) with i = j = 0, has been used to study the s-wave three-particle problem of twoneutrons and a core. This is the case of the halo nuclei 11 Li, 14 Be, 20 C [11,70–72,74,75,230] and, more recently, 22 C [6]. The model needs an ultraviolet scale to regularize the kernel of the correspondent three-body equations. The need of a new parameter, beyond the two-particle scattering lengths to determine the properties of the low-energy s-wave state of n–n–c systems in the state of maximum symmetry, is a consequence of the Thomas–Efimov effect, extensively discussed in Section 2. The subtraction technique was applied to regularize the three-body zero-range equations with Dirac-delta interaction [157] and in particular to n–n–c systems [70]. The step to renormalize the model is done after the regularization of the kernel of the three-body equations by introducing a subtraction at an energy scale −µ23 , as exemplified in Section 2 for the threeboson problem. The subtraction parameter should be traded with a three-body physical observable in the renormalization procedure and let to infinity. Then, s-wave observables are determined by the correlation with a known physical quantity which achieves a limit cycle and the scaling function is determined (see Section 2). The subtraction method used to regularize and renormalize the three-body zero-range model is associated with a renormalized Hamiltonian (fixed point), which builds the subtracted form of the scattering equation resulting in a finite three-body T -matrix. The renormalized interaction can be separated in a two-body part and a three-body part which completely render the three-body T -matrix finite. The physical three-body scale is brought to the system by the three-body part of the renormalized interaction. The consequences of the renormalization group invariance of the three-body theory, like the Callan–Symanzik equation, and properties such as scale invariance and universality are discussed in this context. Furthermore, the method developed in Ref. [155], with a three-body potential that has a strength running to a limit-cycle, can be related to the subtraction method where the renormalized Hamiltonian interaction demands a three-body term. There is no room for a four-body scale in the neutron-halo systems, which is expected just to be sensitive up to the short-range effects carried by a three-body scale as the Pauli principle forbids configurations with three or more neutrons close together. However, in general, new limit cycles are expected in the four-boson problem in s-wave with the zerorange two-body interaction, which requires another regularization parameter and a scale. The existence of four-body limit cycles, which are manifested in a scaling function correlating four-body observables, was numerically demonstrated in Refs. [172,173]. In a broad sense, in nuclear physics the sensitivity on the number of short range scales stops at the three-body level, because the nuclear interaction is dominated by two-body forces, such that one can verify the strong correlation between the triton the and 4 He binding energy, as given by the Tjon line [140]. 3.1. Subtracted T -matrix equations A subtracted Lippmann–Schwinger equation allows to treat singular interactions of delta-type and higher derivatives, as e.g. the potential (47) and construct the RG equations for the nth order subtracted T -matrix. The scattering equation with n subtractions at an energy −µ2 is given by [165]: (+)

T (E ) = V (n) (E , −µ2 ) + (−1)n (E + µ2 )n V (n) (E , −µ2 )G0 (E )Gn0 (−µ2 )T (E )

(48) (+)

where the free Green’s function for the two-body system, with the appropriate boundary condition, is G0 (E ) = (E + iδ − H0 )−1 and the free Hamiltonian is H0 . The driven term is built recursively: V (n) (E , −µ2 ) = (1 + (−1)n (E + µ2 )n−1 V (n−1) (E , −µ2 )Gn0 (−µ2 ))−1 V (n−1) (E , −µ2 ).

(49)

The form of the n-subtracted T -matrix equation is constructed by performing successive subtractions of the scattering equation in −µ2 , that for convenience is chosen to be negative. For a regular potential (48), it is exactly equivalent to the (+) Lippmann–Schwinger equation T (E ) = V + VG0 (E )T (E ), provided that V (0) (E , −µ2 ) = V . For singular interactions, such as the one given by Eq. (47), the higher-order singularities of the two-body potential are (n) introduced in the driving term of the n-subtracted T -matrix through Vsing (−µ2 ). The finite renormalized strengths of the (n)

interaction, Vsing (−µ2 ), are determined by physical observables. The driving term reads: (n)

V (n) (E , −µ2 ) = (1 + (−1)n (E + µ2 )n−1 V (n−1) (E , −µ2 )Gn0 (−µ2 ))−1 V (n−1) (E , −µ2 ) + Vsing (−µ2 ).

(50)

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The choice of n is made by requiring the minimal number of subtractions which gives a finite T -matrix by solving of Eq. (48). By counting the momentum powers in the loop integrals from the iteration of the T -matrix equation, one can determine n for different singular terms in potentials like (47). The transition matrix has a parametric dependence on µ, which can be moved without changing the T -matrix, as long as the driven term runs with µ, as a solution of a RG equation [165]. Applications to NN scattering [166,167] with potentials at next-to-next-to-leading-order (N2 LO) in chiral effective theory [45], shows the practical use of the subtractive renormalization method. The construction of the driven term by the recursive formula (49) is discussed in detail in [166,167] up to d-waves. In particular, the one-pion exchange plus a contact interaction in the 3 S1 –3 D1 coupled waves has been renormalized with one subtraction [161] and the problem presents a limit cycle [231]. Further implementations of subtractive renormalization to treat the scattering amplitude in chiral effective theory is found in [232,233]. Applications of contact interactions to model one-neutron halo were performed for s-waves [11] and in p-waves [117], and in Section 3.2 the method of subtracted equations for the scattering amplitudes with contact potentials is reviewed for both the cases. 3.1.1. Renormalized Hamiltonian The renormalized Hamiltonian (fixed point) should have the property to be stationary in the parametric space of Hamiltonians, as a function of the subtraction point [228]. At the arbitrary subtraction point the T -matrix is known as input to fix the Hamiltonian. A sensible theory of singular interactions exists if and only if the subtraction point can move without affecting the physics of the renormalized theory [229]. This property is realized through the vanishing derivative of the renormalized Hamiltonian, in respect to the renormalization scale, which in our context is the subtraction point. This implies in the independence of the T -matrix in respect to the arbitrary subtraction scale, and in the renormalization group equations for the scattering amplitude. The renormalized Hamiltonian is build starting from the subtracted T -matrix equation (48), which is composed by the free Hamiltonian (H0 ) and the renormalized potential (VR ) [168]: HR = H0 + VR .

(51)

The Hamiltonian HR and the potential VR should be ‘fixed-point’ operators, as they do not depend on the subtraction point and from that the transition matrix. The T -matrix is a solution of the scattering equation corresponding to the fixed-point Hamiltonian: (+)

T (E ) = VR + VR G0 (E )T (E ),

(52)

which is exactly equivalent to the subtracted form (48). The renormalized potential comes by comparing (48) with (52): (+)

VR = [1 + V (n) G0 (E ) (1 − (−1)n (µ2 + E )n Gn0 (−µ2 ))]−1 V (n) .

(53)

The above fixed-point interaction is not well defined for singular interactions. Nevertheless, the corresponding T -matrix is finite, from the equivalence of the scattering equation with the n-subtracted T -matrix equation. Essentially, in addition to the renormalization subtraction procedure that was used in Ref. [165], it is satisfactory to have at hand the renormalized fixed-point interaction (53), with the purpose to relate to other approaches that make use of Hamiltonian diagonalization methods to solve problems with singular potentials. The operator VR is itself singular for contact interactions, we could also employ an ultraviolet momentum cutoff (Λ), defining a regularized interaction, to obtain the regularized T -matrix equation. By performing the limit Λ → ∞, the results for observables converge to the ones obtained through the direct use of the renormalized interaction. In particular, it implies that the eigenvalues of a renormalized Hamiltonian are stable in the limit Λ → ∞ (see e.g. [168]). 3.1.2. Renormalization group invariance The subtraction point in the renormalized interaction (53) is arbitrary and can be moved without affecting the physics of the model. The invariance under dislocations in µ requires that the driving term V (n) of subtracted equation should be changed by a definite prescription. The coefficients that appear in the driving term V (n) have to evolve according to the renormalization group method. The physical condition to derive the prescription to modify V (n) without altering the predictions of the theory, corresponds to the independence of the renormalized potential VR on the subtraction point, which reads:

∂ HR ∂ VR = 0 and = 0. 2 ∂µ ∂µ2

(54)

This states that the renormalized Hamiltonian does not depend on µ; it is a fixed-point Hamiltonian in this respect and ∂ T (E ) therefore the T -matrix, which is a solution of the scattering equation (52), and therefore ∂µ2 = 0. The renormalization group equation, satisfied by the driving term V (n) of the subtracted scattering equation is derived from the vanishing derivative of the renormalized potential in respect to µ [168], and it results in: (+) 2 ∂ V (n) (n) ∂ Gn (E ; −µ ) (n) = − V V , 2 2 ∂µ ∂µ

(55)

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965

which was also obtained in [165] by using the independence of the solution of the subtracted scattering equation on µ. The boundary condition of (55) is given by V (n) = V (n) (−µ2 ; E ) at some reference scale µ. Notice that, the subtracted potential V (n) obtained by solving the differential equation (55) is equal to the T -matrix for E = −µ2 and in particular, Eq. (55) can be written in terms of the T -matrix for n = 1 as:

  T (E ) dE d

E =−µ2

= −T (−µ2 )G20 (−µ2 )T (−µ2 ).

(56)

The coefficients in the renormalized potential run with the subtraction point in order to keep invariant the renormalized potential. Refs. [165,168] provide one example for s-wave interaction and in [167] for p-wave. Both cases can be useful in one-neutron halo physics. 3.2. Examples: single-channel s- and p-waves 3.2.1. s-wave contact interaction The neutron–core structure of 10 Li, 13 Be, 19 C, 21 C are examples of s-wave systems, which present either a virtual state or a bound state as in the case of 19 C. The low-energy s-wave phase-shift and T -matrix for the neutron–core scattering from contact interactions as (47) should be described by few parameters, namely the renormalized strengths at a given subtraction point. The transition matrix obtained from the bare interaction (47), using subtracted equations, demands as inputs the renormalized strengths of the interaction, λRij , which should come from the s-wave T -matrix at an reference energy −µ2 , as depicted below:

⟨p|T (−µ2 )|q⟩ = λR00 + λR10 (p2 + q2 ) + λR11 p2 q2 ,

(57)

where for simplicity we assume λR10 real. Notice that, the two-body problem with the bare interaction (47) is completely soluble having the values of the renormalized strengths at the reference energy −µ2 , used as input in construction of the driving term of the nth subtracted scattering equation. In this example, n = 3 is enough to render all the necessary integrals in subtracted scattering equation finite. The inhomogeneous term of (48) is built recursively according to [165],

⟨p|V (1) (−µ2 )|q⟩ = λR00 ,

1 −1 ⟨p|V (2) (−µ2 ; k2 )|q⟩ = [λ− R00 + I0 ] ,

⟨p|V (3) (−µ2 ; k2 )|q⟩ = λR00 + λR10 (p2 + q2 ) + λR11 p2 q2 , where λR00 ≡ [λR00 + I0 + I1 ] defined by: −1

Ii ≡ Ii (k2 , µ2 ) ≡

2



π

−1

(58)

. The units are such that h¯ and the mass are equal to one. Ii are functions of E = k and µ, 2

∞

dqq2

0

(µ2 + k2 )1+i (µ2 + k2 )1+i = 2 2 2 + i (2µ)1+2i (µ + q )

(i = 0, 1).

(59)

The solution of the s-wave n = 3 subtracted scattering equation, taking into account the driving term (58) gives the phaseshift k cot δ = −µ −

1

− [(λR00 J0 − k2 ) (2λR10 + λR11 k2 ) + λR11 J1 λR00 + (λ2R10 − λR00 λR11 k2 ) (λR00 J02 − J0 k2 + J1 )] × [λR00 (λR00 + (2λR10 + λR11 k2 )k2 + (λ2R10 − λR00 λR11 k2 ) (J0 k2 − J1 ))]−1 ,

(60)

where Ji ≡ Ji ( k , µ ) ≡ 2

2

2

π

∞



dqq 0

2+2i



µ2 + k2 µ2 + q2

3

 =

k2 + µ2 2µ

3

(3µ2 )i (i = 0, 1).

(61)

The values of the renormalized strengths are found by fitting phase-shifts. In case only the constant term of Eq. (47) is nonzero, the interaction is the Dirac-delta one and the scattering length a = [µ + (λR00 )−1 ]−1 is enough to fix the strength. Thus, the phase-shift becomes independent of µ. However, when λRij are nonzero, there is no such a simple composition strengths with µ. From the expression (60) for k cot δ , the non-trivial dependence on the scale µ in the low energy observables is shown. Besides the scattering length to fit the strengths, the effective range and the phase-shift at some specified higher energy can be used. In order to do that, the scale µ has to be chosen at some value, or better interpret µ as part of the physical information input. Notice that the subtraction point can move without modifying the physical content of the parametrization. Then, the RG equation (54) evolves the driving term from the reference µ to the new subtraction point. The evolution of the RG equation for the driving term in a pionless effective field theory for the 1 S0 state, with the potential (47) has been studied in [234].

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The renormalized two-body interaction is derived by introducing V (3) (−µ2 , k2 ) from Eq. (58) into (53). The results show the same structure of the bare interaction, but with the strengths given by functions of the energy and subtraction point:

⟨p|VR |q⟩ =

1 

Λij (k2 )p2i q2j (Λij = Λji ),

(62)

i ,j = 0

where Λij , that will not depend on the subtraction point, are given by:

Λ00 (k2 ) = Λ11 (k2 ) = Λ10 (k2 ) =

λR00 − (λR00 K1 + λR10 K2 )Λ10

;

(63)

λR11 − (λR10 K0 + λR11 K1 )Λ10 ; 1 + λR11 K2 + λR10 K1

(64)

1 + λR00 K0 + λR10 K1

λR10 + (λ2R10 − λR00 λR11 )K1 1 + D + (λ2R10 − λR00 λR11 ) (K12 − K0 K2 )

,

(65)

with D ≡ λR00 K0 + λR10 K1 + λR11 K2 + λR10 K1 and Ki=0,1,2 ≡ Ki (k , µ ) ≡ 2

2

2

∞



π

0

dqq2i+2



k2 − q2



1−

µ2 + k2 µ2 + q2

3 

.

(66)

The integrals Ki are divergent in order to exactly match and cancel the infinities of the Lippmann–Schwinger equation, which is obtained by this renormalizing procedure. The invariance under changes in the subtraction point, means that the derivatives ∂ Λij /∂µ2 vanish. These set of conditions gives the first order evolution equation for the renormalized strengths, equivalent to Eq. (54), with boundary conditions at the reference value of the subtraction point −µ2 and at each scattering energy k2 , considered as a parameter. 3.2.2. p-wave contact interaction The neutron-4 He has no bound states, with the 5 He being one example of single channel p-wave low-energy resonance, which appears in the J = 3/2 state (see e.g. [117]). The 6 He Borromean weakly-bound nucleus has in addition the 1 S0 two-neutron subsystem, which is a low-energy virtual state. The low-energy p-wave phase-shift and T -matrix for the neutron–core scattering can be derived either using effective field theory approach [117] as well as with the subtractive renormalization method. The simplest p-wave contact potential has the projected matrix element given by

⟨p|V |q⟩ = V (p, q) = λ1 p q.

(67)

In what follows, the problem is solved for two identical particles in a p-wave state with the subtracted scattering equation as presented in [167]. By counting the momentum powers in the loop-integral, which carries the virtual state propagation of the system, the minimal number of subtractions n = 2 is determined, to render finite the scattering amplitude for the bare potential (67). It means that at least two constants are required for the problem. If one works with a cutoff regularization indeed only fixing the scattering volume is not enough to render the scattering amplitude cutoff independent. The regularization of the p-wave Lippmann–Schwinger equation for the potential (67) with a momentum cutoff Λ gives the p-wave projected scattering amplitude as:

⟨p|T (k2 )|q⟩ = T (p, q; k2 ) = pq



1

λ1

−

2

Λ



π

dq′ q′4

0

1 k2 − q′2 + iϵ

 −1

,

(68)

with λ1 determined by the scattering volume α : 1

λ1

=−

2 3π

Λ3 +

1

α

.

(69)

By replacing λ1 in Eq. (68), one finds a subtracted form as: T (p, q; k2 ) = p · q



1

α

−

2

π

k2

Λ

 0

dq′ q′2

1 k2 − q′2 + iϵ

−1

.

(70)

By inspecting the integration, it remains a linear cutoff dependence. Therefore, the renormalization procedure using only one subtraction, for this p-wave problem is cutoff dependent, requiring an extra subtraction. The fitting of only the scattering volume is equivalent to the use of a n = 1 subtracted scattering equation, where the dependence on the cutoff is linear. Then, one more parameter is required to renormalize Eq. (70) in agreement with Refs. [117,235]. To renormalize the p-wave contact interaction besides the scattering volume the next term in the generalized effective range expansion for the k3 cot δ is necessary.

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967

The subtracted form of the scattering equation has a parameter µ that instead of the cutoff Λ regularizes the ultraviolet divergence, while allowing a straightforward RG treatment. The finite T -matrix solution of the n = 2 subtracted Eq. (48) was found in [167] as: T (p, q; k ) = pq



1

2

q′4 (k2 + µ2 )2

∞



 −1

dq − , (71) λR1 (k2 , µ2 ) π 0 (µ2 + q′2 )2 (k2 − q′2 + iϵ) where the renormalized strength λR1 can be fitted with the scattering volume. Another parameter, µ, intrinsic to this 2

′

method, still remains free. However, as an advantage of the subtraction method the driving term can be evolved by the RG equation (55) to any subtraction point. Just to provide an explicit application of the evolution by RG equation, Eq. (55) after partial wave projection reads: 2 ∂λR1 (k2 , µ2 ) = 2[λR1 (k2 , µ2 )]2 ∂µ2 π

∞



dq q4 0

(k2 + µ2 ) 3 (k2 + µ2 ) = [λ1 (k2 , µ2 )]2 , 2 2 3 (q + µ ) 4 µ

(72)

and the solution is 1

1

=

λR1 (k2 , µ2 )

λR1 (k2 , µ20 )

3

1

2

2

− k2 (µ − µ0 ) − (µ3 − µ30 ),

(73)

and the renormalized strength picks a dependence on k2 from the evolution. Choosing the reference scale at zero energy, the renormalized strength is the scattering volume. After solving the RG equation, the evolved potential is VR (p, q) = λR1 p q. The Eq. (71) gives the scattering amplitude and the phase-shift: k3 cot δ = −

1

λR1 (k2 , µ2 )

−

µ 2

(3k2 + µ2 ).

(74)

1 2 The evolution equation (73) shows that λ− R1 should be a linear function of k , which demands two constants. It means that 2 (74) has a momentum dependence containing a constant plus a k term, i.e., only two independent quantities appear: the scattering volume and an effective momentum. In fact, k3 cot δ (74) is the one obtained in the context of EFT [117]. The poles of the scattering amplitude, which appear in the complex k momentum-plane, are the cubic roots of [117]



1

µ3



3µ



1

1



3

− κ 2 − iκ 3 = 0 , (75) µ3 λ R 1 2 2 where the dimensionless momentum κ = k/µ is introduced. One pole is located on the positive imaginary axis, with the −

λR 1

+

2

−

2

k2 − ik3 = −

+

other two symmetrically located in the lower part of the complex momentum plane. 3.3. Subtracted Faddeev equations and renormalized Hamiltonian The subtraction technique used to define the two-body scattering amplitude in the previous subsections can be generalized to three-boson systems. The need for a new parameter beyond a is demanded by the Thomas–Efimov effect, i.e., collapse [76] of the three particle ground state, when the two-body interaction range r0 goes to zero, or the Efimov phenomenon as |a|/r0 → ∞ [96]. The singularity of the three-particle integral equation in the state of maximum symmetry associated with the Thomas collapse is avoided at the expense of regularizing the kernel of the three-body equation. This can done by a subtraction at an energy scale −µ23 . The Thomas collapse appears as µ3 → ∞. Taking in addition the twobody scattering length, this procedure determines the low-energy three-boson properties. The kernel of the three-particles integral equation is regularized by subtraction as G0 (E ) − G0 (−µ23 ) in substitution to the free three-body Green’s function G0 (E ) [157]. The presentation of observables in the s-wave state in a form of correlation, that achieves a limit cycle for µ3 → ∞, is preferred instead of the dependence on µ3 . As thoroughly explained in the case of the two-body problem, the advantage of using the subtracted approach relies on its explicit renormalization group invariance. Subtracted equations were applied to renormalize the bound and scattering three-body equations with zero-range potentials. Examples in the bound case are two-neutron halo nuclei 11 Li, 14 Be, 20 C [70–72,230] and 22 C [6]. In the scattering region, the neutron–deuteron scattering problem has been solved with a different form of subtracted equations [169]. The neutron-19 C s-wave elastic scattering [180] used the subtracted T -matrix equations exposed below. The subtracted Faddeev equations were given in [160] for the zero-range potential. At the subtraction point −µ23 the three-body T -matrix is the sum of the two-body T -matrices for all two-body subsystems for the zero-range potential as discussed in detail in the previous subsections 3 : T (−µ23 ) =

 i=1,2,3

t(ij)

 −µ23 −

q2k 2 mij,k



,

3 We omit the label R to indicate that the T -matrix is the result of a renormalized interaction to simplify the notation used in [203].

(76)

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where particles are labeled by i, j, k, and given cyclically as [(ijk) = (123), (231), (312)]. The argument of the two-body T -matrix is the energy of the two-particle subsystem in the center-of-mass of the three-body system. The Jacobi relative momentum canonically conjugated to the relative coordinate of the particle k to the center-of-mass of the pair (ij) is qk , and the corresponding reduced mass is mij,k ≡ (mi + mj )mk /(mi + mj + mk ) with mi the mass of the particle i = 1, 2, 3. The matrix elements in momentum space of the renormalized two-body transition operator for the zero-range interaction are given by

⟨⃗p′ |t(ij) (E )|⃗p⟩ =

1

4π

2m ij

 , (± 2mij |E(ij) | + i 2mij E )

(77)



where mij ≡ mi mj /(mi + mj ) is the reduced mass of the two-particle subsystem (ij), with E(ij) being the bound (+) or virtual (−) energy state. The subtracted integral equation for the three-body T -matrix is formally given by (48) with n = 1 [160], which is enough to avoid the Thomas collapse4 : (+)

T (E ) = T (−µ2(3) ) + T (−µ2(3) ) (G0 (E ) − G0 (−µ23 ))T (E )

=





t(ij)

−µ23 −

(ij)

q2k



2 [1 + (G(+) 0 (E ) − G0 (−µ3 ))T (E )].

2mk(ij)

(78)

Each term in the above sum is a Faddeev component of the three-body T -matrix. By writing the three-body T -matrix as a sum of the three components, the Faddeev equations are derived from (78) and are given by: Tk (E ) = t(ij)

q2k

 E−



2mij,k

2 [1 + (G(+) 0 (E ) − G0 (−µ3 )) (Ti (E ) + Tj (E ))].

(79)

The bound-state equation comes from (79) considering the pole of the T -matrix at the corresponding energy. One example obtained in this form is the regularized Skorniakov and Ter-Martirosian equation for the three-boson bound state (see Eq. (22)). The subtracted three-body scattering equation (79) leads to the Skorniakov and Ter-Martirosian original model when naively one let µ3 → ∞. However, the scale invariance of the zero-range three-body T -matrix equation in the ultraviolet momentum region is broken by the introduction of the finite scale µ3 , which stops the Thomas collapse in the s-wave state of maximum symmetry. The only inputs in the Faddeev integral equations are given by ϵ(ij) = E(ij) /µ23 . Therefore, the observables are determined by dimensionless parameters. As µ3 → ∞, an infinite tower of three-body states appear. The correlations between threebody observables tend to achieve a limit cycle, where the dependence on µ3 does not matter anymore and the theory in this sense is fully renormalized.

• Renormalized three-body Hamiltonian

(3)

The renormalized three-body interaction Hamiltonian, HRI , is a solution of (53) with n = 1, where V (1) is identified with (76) at the subtraction point −µ23 . An integral equation can be written as: (3B)

HRI =



t(ij)

 −µ23 −

(ij)

q2k



2mk(ij)

(3B) (1 − G0 (−µ23 )HR I ).

(80)

It is decomposed in three-terms in straight analogy to the Faddeev decomposition of the T -matrix: (3B)

HRI =

3 

(3B)

HRI (k) ,

(81)

k=1

where



(3B)

HRI (k) = t(ij)

−µ23 −

q2k



2mk(ij)

(3B) (1 − G0 (−µ23 )HR I ).

(82)

To obtain the set of equations which have the Faddeev components of the renormalized Hamiltonian as solution, we make use of the following form of the renormalized potential of the (ij) subsystem, which is given by Eq. (53) with n = 1: (2B)



VR(ij) = 1 + t(ij)



−µ23 −

q2k 2mij,k



G0 (−µ23 )

−1 t(ij)

 −µ23 −

4 Note that for n = 1 the driving term satisfies: V (1) (E , −µ2 ) = T (−µ2 ).

q2k 2mij,k



,

(83)

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969

where the two-body subsystem is immersed in the three-body Hilbert space. The argument of t(ij) can be chosen arbitrarily, as long as the two-body T -matrix is a solution of the one-subtracted equation, or it is found by evolving the appropriate RG equation. By algebraic manipulation of Eq. (82), one gets: (3B)

(2B)

(3B)

(3B)

HRI (k) = VR(ij) [1 − G0 (−µ2(3) ) (HRI (i) + HRI (j) )].

(84)

It is recognized from (84) that the renormalized three-body interaction can be written in a way where the two-body renormalized potential is separated out: (3B)



HR I =

(ij)

(2B)

(3B)

VR(ij) + VR .

(85)

The components of the three-body renormalized potential are finally defined as: (3B)

VR

3 

=

(3B)

VR(k) ,

(86)

k=1

with each component given the integral equation: (3B)

(2B)

(3B)

(3B)

VR(k) = −VR(ij) G0 (−µ23 ) (HRI (i) + HRI (j) ),

(87)

and the solution of Eq. (87) is:



(3)

VR(1)



 (3)   V  R(2)  =



(3) VR(3)

   VR(2()23)  (2)  1  −1  VR(31)  , 2 ˆ 1 + KR G0 (−µ3 ) (2)

(88)

VR(12)

where the three-body kernel Kˆ R is given by:



0



(2)

Kˆ R =  VR(31) (2) VR(12)

(2)

VR(23) 0 (2) VR(12)

VR(23)

(2)



(2)



VR(31)  ·

(89)

0

The three-body renormalized potential (86) that carries subtraction point is the counterpart in the subtraction method of the EFT three-body potential [155] necessary to renormalize the zero-range Skorniakov and Ter-Martirosian equations. As in the case of the renormalization of the two-body potential, the ill-defined integrals of the three-body formalism also requires cutoffs, if one wants to diagonalize the corresponding three-body Hamiltonian. After the diagonalization is performed, the cutoff can be removed. The subtraction and the EFT methods to treat the three-body problem should agree in respect to the limit-cycles which defines the scaling functions expressing correlations between three-body observables. 3.4. Effective field theory and halo nuclei Recent reviews on effective field theory program describe in detail its application to halo and light nuclei (see e.g. [47,111,236]). Here, we aim to discuss the relation of the EFT method with the renormalization techniques exposed in the preceding subsection. In Section 4, we review calculations of halo nuclei that are also consistent with EFT approach at leading order as well. The method of effective field theory used to investigate the nuclear force problem keeping pion exchanges and chiral constraints (see e.g. [45,109,141]) aims for a systematic expansion of the potential, while contact terms consistent with the symmetries of the problem are introduced to manage the short-range physics effects in the low-energy nuclear properties. The success of this program relies on the short-range property of the nuclear interaction, which allows to identify two well separated scales: a low-momentum and a high-momentum scales. For few-nucleon problems the nucleon–nucleon scattering lengths and the pion range are identified with these scales. The unknown short-range part of the interaction is expanded by contact terms, which can be encoded by an effective non-relativistic Lagrangian with local terms carrying powers of derivatives of the fermion fields. The two-particle interaction originated by using the field-theoretic language without pions is equivalent to the introduction of separable potentials in an effective quantum mechanics framework (see e.g. [237]). As an example we have the separable s-wave contact potential given by Eq. (47). Although, the matrix elements of the potential has a finite number of momentum powers, the expansion of k cot δ , Eq. (60), has an arbitrary power of k2 . That happens in part because the RG evolution introduces a parametric dependence on the energy for the four-term separable potential of (47). The correspondent renormalized Hamiltonian (62) depends on the energy with any power of k2 . Of course one can control the

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effective range expansion, but even at the level of the example, four independent parameters can be introduced. As one goes forward in introducing separable terms equivalent to higher field derivatives of the effective Lagrangian, what we have described becomes even more complex. The leading order (LO) effective separable potential in s-wave is a Dirac-delta and only the scattering length enters in the scattering amplitude. At the next-to-leading order (NLO), we have terms with higher powers of k2 (see Eq. (60)). Fortunately, large halo nuclei have a clear separation between low and high momentum scales, in correspondence to the size and the force range. Thus, it is reasonable to expect that the difference between LO and NLO results is of perturbative origin. 1 2 In the example, it means that when k− low is of the order of the halo size, the terms with dependence on powers of k or higher in the effective range expansion are small compared to klow , i.e., the contribution of the effective range is small compared to the halo size. Within the effective field theory method, the effective potential is valid up to a momentum, khigh , at which the errors become of order one. By using the contact interactions in the effective quantum mechanics approach [237], the contact term in LO is smeared out with a Gaussian form-factor, ⟨p|V |q⟩ = λ00 g (p)g (q) + · · · where the dots indicate higher order momentum dependent p2

interactions, and g (p) = exp(− Λ2 ). In this scheme renormalization means cutoff independence of the low-energy observables. It is achieved going to the limit of Λ → ∞, and looking for correlation between observables in halos with three-body structure as we have already discussed. The three-nucleon system has been first addressed within the EFT program (see e.g. [141]), after the recognition that a three-body potential [155] was necessary to renormalize the zero-range Skorniakov and Ter-Martirosian equations within such framework. The essential physics is a limit-cycle in the strength of the three-body potential as the cutoff increases, by keeping the low-energy s-wave three-body physics unchanged. The correspondence with the scaling function approach to treat the zero-range three-boson model [114], is evidenced when the limit-cycle of a correlation between s-wave observables is achieved. Formally, in working with subtracted Faddeev equations, the renormalized interaction contains a three-body potential, as derived in the previous section (see Eq. (86)). The application of EFT to halo-nuclei was suggested in Ref. [117] to study the low energy 5 He p-wave resonance, where contact potentials were used. It was later on extended to three-body halos as 11 Li, 14 Be and 20 C [74], with LO results for the critical conditions to bind an excited state for asymmetric mass systems in agreement with previous calculations done with s-wave contact interactions in Ref. [11].

• Recent applications of EFT: Coulomb problems Neutron halos of light-nuclei are suitable to be treated in momentum space and/or in configuration space representations. In momentum space, while the effective contact potentials can be easily introduced in the problem, the Coulomb interaction has to be treated carefully. It has to be considered in effective theory descriptions of weakly-bound or low-energy resonances involving few clusters and/or a proton, like e.g. 8 Be or α –α low-energy scattering (see e.g. [238–240]). We have not discussed so far Coulomb problems with contact interactions, and it is useful to have a short view in what are the developments following some aspects of the problem in the context of Halo Effective Field Theory. The application of halo EFT to the 8 Be or α –α low-energy scattering has to deal with a subtle interplay between the strong and Coulomb interaction in respect to the power counting for the α –α system as shown in [238–240]. The approach is based on an expansion around the region where no Coulomb force is manifested, with the 8 Be ground state at the threshold. A narrow resonance at an energy of about 0.1 MeV is verified in their calculations, with a good description of existing low-energy α –α phase-shifts and known effective-range parameters. However, a considerable amount of fine tuning of the parameters was necessary. In Ref. [241] also low-lying narrow resonances for contact interactions plus Coulomb repulsion were addressed, and it was shown that the two-body scattering amplitude at leading order is the sum of a Coulombmodified background term and a Breit–Wigner amplitude with parameters renormalized by Coulomb interactions. A similar separation and dressing of the T -matrix parameters for a Dirac-delta plus attractive Coulomb potentials using a subtracted scattering equation was found in [242], when studying the bound state problem in a different context. It was suggested in [240] that in systems with three α ’s, the Hoyle state could be originated from an Efimov excited state that due to the Coulomb repulsion approaches the threshold for the α -8 Be continuum. In an ab-initio calculation of the low-lying states of 12 C, the 0+ 2 resonance, i.e., the Hoyle state, has been found by solving the twelve nucleon problem with lattice effective field theory methods (see e.g. [243]), just near the α -8 Be threshold at NNLO O (Q 3 ) [244]. The relation with the cluster picture having an Efimov state approaching the threshold as the Coulomb potential is turned on, as suggested in [240], has still to be explored relating EFT lattice results with a EFT cluster description. A coupled-channel effective field theory [245] that address the problem of describing low-energy two-body scattering for systems with two open channels with different thresholds was extended to systems with Coulomb interaction [246]. The approach was applied to the p + 7 Li and n + 7 Be coupled channels which couple close to the threshold of a n + 7 Be to a J P = 2− state of 8 Be. At NLO the parametrization of the p + 7 Li phase-shift and the 7 Be(n, p)7 Li cross section data was successful using four parameters. The application of the EFT program to three-nucleons with charge has been also addressed. The proton–proton–neutron system was studied in EFT without pions for bound and scattering problems including electromagnetic interactions. The 3 He problem was studied in [247], where Coulomb potential between the protons was included in the set of integral equations through the full off-shell Coulomb T -matrix. It was considered the contact s-wave two-nucleon interactions with renormalized strengths set to the scattering lengths and a three-body contact interaction. The integral equations were solved

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by introducing a sharp momentum cutoff to avoid the Thomas collapse, and therefore it is necessary a three-body scale. The strength of the contact three-body potential showed the characteristic limit-cycle behavior when the 3 He binding energy was fixed, as well as in the case of the 3 H (see also [141]). The proton–deuteron scattering was investigated in [248] and the low-energy elastic p–d s-wave phase-shifts computed. A NNLO calculation of phase-shift in the quartet spin channel exhibited a good agreement with available data down to close to zero-energy. Note that, the Pauli principle in the quartet state does not allow the Thomas collapse and therefore only nucleon–nucleon parameters are enough to determine the observables. Differently, in the doublet channel the calculation were done in NLO, taking into account a LO three-body force fitted to the keeping fixed the scattering length. The phase-shifts were calculated with an special integration mesh to allow to account for the Coulomb potential, subleading three-body forces effects were expect to be small (see [248]). 3.5. Beyond three-particles The subtraction technique used to define the two- and three-body T -matrices can be generalized to four-particle systems. The addition of a fourth particle to the three-body system with zero-range two-body interactions requires another regularization parameter, due to new terms in the coupled integral equations, not directly identified with the three-body kernel [249]. Within the EFT framework approaches considered so far it is believed that the four-boson physics of contact interactions is fixed by the properties of the two- and three-boson systems [237,250]. This belief is confronted with the outcome of the subtractive regularization and renormalization approach considered in Ref. [249] for the ground-state, and more recently verified when considering excited four-body states in Refs. [172,173]. As shown in these works, the approach allows one to introduce in an independent way the three- and four-body scaling parameters in order to test their possible independent behavior. Therefore, within the picture of EFT, near unitarity (infinite scattering length) the tower of all tetramers attached to each Efimov trimer, depends only on the trimer energy [250,251]. The debate had given a new dimension when it was found by Ref. [172] a limit-cycle associated with the scaling function associated with the correlation between two consecutive tetramer energies at the unitary limit for a fixed trimer energy. It was shown that previous four-boson calculations, indeed had the effect of the four-body scale consistent with the new scaling function. The introduction of the new regularization parameter generalizes the subtraction method applied to two- and three-body equations. A new regularizing parameter (‘‘four-body scale’’), µ24 , is introduced in the Faddeev–Yakubovski (FY) formalism. The new parameter appears in the integrands associated to the presence of the fourth particle, in order to allow the complete regularization of all the momentum integrals. Technically, among the eighteen FY components, only the ones corresponding to the partitions (ijk + l, jki + l and kij + l) will fully describe the three-body subsystem (ijk), where the three-body scale brought by the parameter µ3 enters in the subtracted form of the free Green’s function. The parameter µ24 enters in the subtracted form of the Green’s functions which appear in the remaining fifteen components (see [172,173,249]). Therefore, in the three components associated to the threebody subsystem, the regularization is done by (3)

G0 (E ) −→ G0 (E ) ≡ G0 (E ) − G0 (−µ23 ),

(90)

with the new scale appearing in the regularization of the other fifteen components: (4)

G0 (E ) −→ G0 (E ) ≡ G0 (E ) − G0 (−µ24 ).

(91)

The physical picture underlying the regularization procedure of the subtracted FY equations can be described as follows. The three-body scale parameterizes the short-range physics in the virtual propagation of the interacting three-boson subsystem within the four-body system. The four-body scale parameterizes the short-range physics beyond the three-body one, explored in the four-boson virtual-state propagation between two different fully interacting three-body or disjoint twobody clusters. As shown in [249], the increase of µ4 /µ3 leads to the collapse of the four-boson ground state. We should also note that the freedom in exploring the dependence on the regularization parameter µ4 is in principle possible. However, it is unlikely to be relevant in the physics of neutron halos. The possibility will be a mixture of protons and neutrons in the halo, apart of the inclusion of the Coulomb potential. Cold-atom physics can be an alternative. The physical conditions in cold-atom traps that allow to search for a short-range four-boson independent scale, when approaching the Feshbach resonance, are discussed in Ref. [172]. 4. Universality in three-body halo systems The different correlations between low-energy observables of two-neutron halo nuclei, assuming neutron–neutron and neutron–core on-shell parameters as inputs, are universal, namely independent on the detailed form of the short-range interactions. The low-energy s-wave observables of the two-neutron halo nuclei, where the halo neutrons occupy s-wave states, depend on one more quantity, the two-neutron separation energy. Large two-neutron halos are in the ideal situation where the universal regime can be studied, in principle, by using any simple short-ranged potential. In particular, a contact s-wave potential is appropriate in this case, as it allows to make predictions with a minimum set of physical parameters.

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The reason for the insensitivity on the detailed form of the interparticle potentials relies on the large size of typical weaklybound neutron halos. Therefore, a large probability exists for such neutrons to be located outside the potential range in a region where the corresponding wave function is an eigenstate of the free Hamiltonian, or in the classically forbidden region. Let us remark that the concept of universality worked out along this review does not mean that the same set of parameters of a given interaction will describe the two neutron halo nuclei of different systems, as for example the cases of 11 Li, 14 Be and heavy carbon isotopes. In each case, the short-range s-wave n–core and n–n interactions should be fitted to the corresponding scattering lengths. In the following, the limiting case of a short-range interaction, namely the zero-range potential, is used. This kind of potential is not affected by the nuclear medium, as it was designed to describe particles (neutrons) living far away from the core, having little overlap with it. In this section, the review on n–n–core universal behavior starts with the presentation of a general classification scheme, which is based on the kind of two-particle interactions, where four possibilities exist. Evidently, in view of the fact that the n–n interaction is known as a virtual one, for real halo-nuclei systems only two cases are of interest, according to the interaction of the core with each halo neutron, which can be an unbound state (10 Li) as it happens with 11 Li, or a bound state (19 C) as in case of 20 C. Within the renormalized zero-range model, the possibilities of halo Efimov excited states are quantified in terms of the input parameters, by reviewing the critical conditions for binding excited halos, shown as a curve in a parametric space. Bound and virtual excited halo states in 20 C are shown to be theoretically possible by changing the 19 C binding. The effect of an Efimov state close to the continuum is shown by recalling the low-energy n-19 C scattering studies. The two-neutron halo structure, wave-functions and radii, are discussed for 11 Li, 14 Be, 20 C and 22 C. The classification of halo systems and the corresponding sizes show systematic trends, with results being presented for the mean separation distances of the neutrons and the core. Particular attention is devoted to recent studies on 22 C, which has the largest twoneutron halo found so far, with a size of about 15 fm. The n–n correlation functions for 11 Li and 14 Be are also reviewed. We close the section with a discussion on halos with more than two-neutrons. 4.1. Neutron–neutron–core systems 4.1.1. Classification scheme: Borromean, Tango, Samba and all-bound systems Apart from intrinsic particle properties, such as spin, isospin or charge, the possible four situations of a general n–n–c system, regarding to the two-body interactions (bound or virtual), is shown schematically in Fig. 11. Within the four possibilities, we are also considering the case where the particle n is not restricted only to the case where it is identified with the neutron, such that one can also consider situations in which we have the n–n–c system identified with some other nuclear or molecular three-particle interactions. The four configurations, schematically represented in Fig. 11 are (see [70]): (i) (ii) (iii) (iv)

all-bound, when both subsystems n–n and n–c are bound; Tango [252], when the subsystems n–n and n–c are, respectively, bound and virtual. Samba [70], when the subsystems n–n and n–c are, respectively, virtual and bound. Borromean [39], when both subsystems n–n and n–c are unbound, with virtual-state energies.

Note that in two-neutron halo systems the configurations (i) and (ii) are of course hypothetical and are discussed with the sake of being complete. In general, when an excited N + 1 state crosses the lowest scattering threshold, it becomes a virtual state if at least one pair has a bound state, i.e., when considering the configurations (i)–(iii). In the Borromean case (iv), the excited state turns into a continuum resonance [194]. The two possibilities, (iii) and (iv), in the above classification, are realized in the three-body configuration case of twoneutron halo nuclei, as, for example, the cases of 20 C, which is a Samba configuration, or 6 He and 11 Li, which are in Borromean configurations. The exotic carbon isotope, 22 C, is possibly in a Borromean configuration, if the subsystem n-20 C is not bound [5,6,73]. It was suggested in [6], that by changing the yet unknown negative n-20 C scattering length an excited s-wave three-body excited state of 22 C could be formed from a resonance that migrates from the continuum. The case of a Samba configuration (iii), as 20 C, which is built by two halo neutrons with 18 C, where in 19 C the halo s-wave neutron is bound, when the neutron separation energy in 19 C is decreased with fixed two-neutron separation energy of 20 C a virtual state of 20 C cross the threshold and turns to an excited bound state [71,72]. The elastic neutron-19 C cross-section presents a peak when the nearby pole in the s-wave amplitude from the bound or virtual 20 C state are about to hit the scattering threshold [180]. It turns out that the s-wave phase-shift has also a pole given by zero in the s-wave amplitude, that moves as the low-energy parameters of the neutron–neutron-19 C system are modified, showing rapid change in the cross-section when the phase-shift pole is in the physical region [180] (see also [253]). In Section 4.2 this point will be discussed further. 4.1.2. Short-range model for two-neutron halo The idealized picture of the neutron–neutron–core (n–n–c) halo nuclei corresponds to a system, which is large compared to the interaction range and core radius, with constituents interacting through the short-range nuclear force. To keep the generality of the following discussion and before going to the limit of contact forces, where the core is considered structureless and interaction range vanishes, the case of a short-range one-term (rank-1) s-wave separable two-body potential is described below (see [68,237]).

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Fig. 11. Classification scheme for a three-body system composed by two particle species, with two identical ones, according to the nature of their two-body interactions.

The particles labeled by n are identical fermions that interact in s-wave, in a spin antisymmetric state. In this case, the wave-function in configuration space is symmetric under exchange of the fermions, allowing formally to treat them as spinless bosons. The n–n system at low energies is dominated by the s-wave channel, with a large scattering length, corresponding to a virtual state energy of −143 keV. The neutron–core potential is assumed to be dominated by the s-wave channel as well. A virtual state close to threshold appears for n-9 Li, n-12 Be and n-20 C. For the last case, it is not known the virtual state energy of 21 C. The n-18 C system has a weakly-bound state. This simple picture is deficient in the case of 6 He, where 5 He is p-wave resonance. The model Hamiltonian for n–n–c system, with a pairwise potential, is written as H = H0 + Vnn + Vnc + Vn′ c

(92)

and in a simplified description where the spin of the core is neglected, the neutrons n′ and n are in a singlet spin state with a configuration space wave function symmetric under permutation. The kinetic energy operator, written in terms of the Jacobi relative momenta eliminates the center-of-mass motion, and it is given by H0 =

⃗2n q 2µn,nc

+

⃗2c p 2µnc

=

⃗2c q 2µc ,nn

+

⃗2c p 2µnn

=

⃗2n′ q 2µn,nc

+

⃗2n′ p 2µnc

,

(93)

where, in terms of the rest frame momenta, ⃗ ki (i = α, β, γ ), we have the following transformation:

 ⃗γ = ⃗kγ q

⃗γ = µαβ and p

⃗kα mα

−

⃗kβ mβ

 ,

(94)

⃗γ and the relative momentum of the pair the relative momentum of the particle γ to the center-of-mass of the pair αβ is q αβ is p⃗γ , with (α, β, γ ) = cyclic permutations of the particles (n, n′ , c ), which have masses mn and mc . The reduced masses are given by µαβ ≡ mα mβ /(mα + mβ ) and µγ ,αβ ≡ mγ (mα + mβ )/(mα + mβ + mγ ). The two-body potential in the form of a rank-1 separable interaction given in operator form is Vαβ = λαβ |χαβ ⟩⟨χαβ |, where λαβ is the strength, and the form-factor ⟨⃗ pγ |χαβ ⟩ ≡ gαβ (pγ ) depends only on the modulus of the relative momentum of the pair for a s-wave interaction. Note that, the contact interaction is defined by a constant form factor g (pγ ) = 1. With the s-wave zero-range interaction, all the corresponding components of the two-body T -matrix, for an arbitrary energy E, are given in a simple form, ⟨⃗ p′γ |Tαβ (E )|⃗ pγ ⟩ = ταβ (E ), depending only on the bound or virtual s-wave energies of the pair αβ and corresponding masses. The example of the contact s-wave interaction for equal mass particles, where the LO term is the Dirac-delta, was presented in detail in Section 3.2.1, where derivative terms were also considered. The homogeneous equations for the Faddeev components of the bound-state three-body wave function for the rank-1  potential is written in terms of the spectator functions fγ (qγ ) = ⟨⃗ qγ | ⟨χαβ |Ψ ⟩, where the projection on s-wave states is considered. The n–n–c system has two independent spectator functions denoted by fn (qn ) and fc (qc ). The coupled Faddeev equations for the bound state can be written in terms of the spectator functions and in the notation of [74] (see also [68]), they are given by: fn (q) =

fc (q) =

1



⃗; A))Gn0 (P (⃗k, q⃗; A), k; B3 )tn (k; B3 )fn (k) d3 k [gnc (P (⃗ q, ⃗ k; A))gnc (P (⃗ k, q 8π + gnc (P (⃗k, p⃗; 1/A))gnn (P (⃗k, q⃗; 1))Gc0 (P (⃗k, q⃗; 1), k; B3 )tc (k; B3 )fc (k)], 1 4π



⃗; 1/A))Gn0 (P (⃗k, q⃗; 1/A), k; B3 ) tn (k; B3 ) fn (k), d3 k gnn (P (⃗ q, ⃗ k; 1))gnc (P (⃗ k, q

(95) (96) n ,c

where tn and tc correspond to n–c and n–n pairs, respectively. The form-factors gnn (p) and gnc (p), and the resolvent G0 , depend on the absolute values of the shifted momenta, given by

 

P (⃗q, ⃗k; A) ≡ ⃗k +

  . 1 + A ⃗ q

(97)

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In the above expressions, B3 = |E3 | is the three-body bound-state energy and A (= mc /mn ) is given by the number of nucleons in the core. The resolvents Gn0 and Gc0 , for the spectator neutron and core, respectively, are given by



Gn0 (p, q; B3 ) = B3 + Gc0

A+1 2A



(p, q; B3 ) = B3 + p + 2

p2 +

A+2 4A

A+2 2(A + 1) 2

 −1

q2

−1

, (98)

.

q

The two-body T -matrices are solutions of the Lippmann–Schwinger equation for the n–n and n–c interactions. By describing both subsystems with the rank-1 separable potential, one should start by determining the strengths λnc and λnn , such that the corresponding n–c (bound or virtual state) and n–n (virtual state) are reproduced. As discussed in Section 2, these kind of two-body state energies should appear as simple poles located on the imaginary axis of the momentum-space formulation of the two-body T -matrix. By considering only one form-factor, i.e., gnn (p) = gnc ≡ g (p) = exp(−p2 /Λ2 ) as considered in Ref. [74], the following expressions are obtained for tn and tc : tn (p; B3 ) = −anc

A+1



πA 

e

2 2 a2 nc Λ

 √



2

erfc



|anc |Λ  √ 

2 2 −2 ann 2 2 e ann Λ erfc tc (p; B3 ) = π |ann |Λ

− anc Bn (p; B3 ) e 

− ann Bc (p; B3 ) e

2Bn (p;B3 ) Λ2

2Bc (p;B3 ) Λ2

√ erfc

2Bn (p; B3 )

Λ

√

2Bc (p; B3 )

erfc

−1

Λ

−1

, (99)

,

where

Bn (p; B3 ) ≡

2A

A+1



A+2

B3 +

2(A + 1)

p

2



,

A+2

Bc (p; B3 ) ≡ B3 +

4A

p2 .

(100)

In the above equations, erfc(x) is the usual defined complementary error function, which converges to 1 when x ≪ √ 1. For a short-range potential the cutoff parameter is large compared to all other momentum scales, i.e., Λ ≫ 1/|a| and B . The effective range expansion is fulfilled for this separable potential, and in the limit of Λ → ∞, only the scattering length survives and the effective range goes to zero, which leads the zero-range potential. The zero-range model has to be regularized and renormalized. In the next, we use the framework reviewed in Section 3, where the regularization is done by introducing, as a convenient regulator, an energy subtraction parameter in the propagator of the integral equations in momentum space. 4.1.3. Renormalized zero-range model The zero-range model provides model independent predictions for the three-body properties when the scaling functions are properly formulated. The bound state equation, is the homogeneous form of (79). The subtracted zero-range equations are solved for a given subtraction point µ, chosen as the momentum units, then the binding energy, ϵ , for the n–n–c system is a function of the two masses, (mn , mc ), and of the two-body binding energies, (ϵnn , ϵnc ), which corresponds to bound or virtual states. Note that an unknown scale is set as our units, and the dependence on the subtraction point is due to the Thomas collapse. The bound-state coupled integral equations for the spectator functions reads [70]:

fn (q) =



A+1

3/2

2A

1

π

  |ϵ| +

 ×

2(A + 1)

1

|ϵ| + q2 +

A+1 2 k 2A

 −

fc (q) =

q2 (A + 2)

|ϵ| + 

− + kqy

1 (A+1) 2 q 2A

+

(A+1) 2 k 2A

−1 ∓

√

ϵnc 

∞



k2 dk

− + kqy 1+ A  −1 

A+1 2 k 2A

dy −1

0



1 1 + q2 +

1



+ kqy

fc (k)



1 (A+1) 2 q 2A

+

(A+1) 2 k 2A

+

kqy

 fn (k) .

∞ (A + 2) 2 √ |ϵ| + q ∓ ϵnn k2 dk π 2A 0   1  1 1 × dy − fn (k). |ϵ| + (A2+A1) q2 + k2 + kqy 1 + (A2+A1) q2 + k2 + kqy −1

2

(101)

A

(102)

In all the formalism considered in this section, we keep the usual units such that h¯ = 1 and mn = 1. In front of the energy square-root, the minus (−) sign refers to a bound state subsystem and the plus sign (+) to a virtual state subsystem.

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975

0.5 0.4 0.3

mc mn =

{

18C

1 9 18 100

Enc / E3

0.2

20C

0.1 22

0

C

-0.1 -0.2 -0.8

-0.6

-0.4 -0.

-0.2 -0.

0

0. 2

0.4

Enn / E3 Fig. 12. Critical threshold for the appearance of an N + 1 excited Efimov state in an n–n–c system. The two-body energies, Enn and Enc , given in terms of a reference three-body bound state E3 , can be bound (positive square-root values) √ √ or virtual (negative square-root values). An increasing number of excited three-body states should emerge as one decreases the ratios Enn /E3 and Enc /E3 . The exact Efimov limit, with infinite number of excited three-body bound states, is given by Enn = Enc = 0. The source of the experimental data of the exotic carbon nuclei 18 C and 20 C is indicated in Refs. [11,74], with the two neutron separation energy for 22 C quoted in the reference by Tanaka et al. [5].

As described in previous subsection, four types of n–n–c systems are possible according to the two-body interactions: Borromean, configuration [39], when all the two-body subsystems are unbound; tango configuration [252], when we have two unbound and one bound subsystems; Samba configuration [70], when just one of the two-body subsystems is unbound; and all-bound configuration, when there is no unbound subsystems. The spectator functions, fn and fc , are different for the ground and excited states. As verified in Refs. [11,70,159] the scaling functions approaches the limit-cycle fast, and for many purposes one can use calculations done for the ground state. The properties of excited bound states can be extracted from the scaling functions written in terms of dimensionless ratios of the two and three-body quantities. 4.2. Halo Efimov states 4.2.1. Critical conditions for binding excited halos The condition to bind an N + 1 excited Efimov state in a general three-body system with two-identical particles, depends on two dimensionless parameters given by the two-body energies (bound or virtual), Enn and Enc , in units of a reference threebody bound state energy E3N , when the interaction range goes to zero. The labels n–n and n–c were originally considered in the nuclear physics context, to label neutron–neutron and neutron–core interactions [11] (see also [178]). However, when deriving the critical conditions in Ref. [11], it was already clear the possibilities to extend the approach to more general three-body systems, such that the n–n system was not restricted only to the realistic well-known unbound neutron–neutron interaction. Of course that, by considering extensions to other three-body systems, a more appropriate notation could be α –α –β , with α and β referring to different particles. This notation is indeed applied to molecular systems composed by lithium, sodium and helium atoms in the search for weakly bound molecules that could have excited Efimov state [179]. As our focus in the present section is the exotic halo-nuclei systems, for convenience we keep the n–n–c notation. The critical boundary, which separates the regions where an excited (N appears, is presented in Fig. 12. It  + 1) Efimov  (N )

(N )

is conveniently shown in a two-dimensional parametric space defined by Enn /E3 and Enc /E3 , with numerical results considered for mass ratios, A ≡ mc /mn , varying from A = 1 to A = 100. The boundary curves shown in this figure, first derived in Ref. [11], were updated in Ref. [74] within a different approach, by using effective field theory. More recently, range corrections were also calculated in Ref. [75]. The mathematical structure for the kernel of the  integral equations  for systems of three distinct particles does not prevent (N )

(N )

the Thomas collapse or the Efimov effect when Enn /E3 and Enc /E3 go towards zero. Therefore, the central point of Fig. 12 corresponds to  the situation  where an infinity set of geometrically separated states appears. When the point defined (N )

(N )

by the coordinates ( Enn /E3 , Enc /E3 ) approaches the boundary, the excited states above N (the reference one), cross the lowest scattering threshold and, at the boundary, only one state (N + 1) survives at zero binding energy. The light exotic nuclei are characterized by a neutron halo, with radius larger than the core size, in which the last neutrons are weakly bound to the core [1,32–34]. The structure of these halo nuclei can be described as a weakly-bound few-body system [32–34]. The possibility of Efimov states to be present in light exotic nuclei, as in 11 Li, 12 Be and 20 C was pointed out,

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T. Frederico et al. / Progress in Particle and Nuclear Physics 67 (2012) 939–994

Fig. 13. Three-dimensional schematic plot, combining the results of the boundary region to exist at least one Efimov state in an n–n–c system, given in Fig. 12, with the part of the scaling plot shown in Fig. 9 corresponding to bound identical three-boson systems. In the defined z-axis, E2 is fixed to zero for the cases that the two-body system is in a virtual state. Source: Extracted from Ref. [46].

originally, in Refs. [67,254,255]. These nuclei are characterized by the small values of the one or two neutrons separation energies. We describe them by an inert core of mass mc and two weakly-bound neutrons. Thus, we have two possible pairwise interactions, one for neutron–core and another for neutron–neutron system. The positions of the carbon isotopes with mass numbers 18, 20 and 22, considered as two-halo systems, are represented in the boundary plot given in Fig. 12, as examples, in view of their specific characteristics with the corresponding available experimental data. By considering the available data of other well-known halo-nuclei, such as 11 Li and 12 Be, it was found no room for an excited Efimov state within this three-body approach, such that their corresponding point will be outside the limiting region represented in Fig. 12. The position of 12 Be in this representation can be seen in Fig. 6 of Ref. [46]. In the cases that we have shown, one should also note that the isotope 18 C is excluded to have an excited halo state. The situation is more favorable (to have an excited Efimov state) in the other two represented cases, 20 C and 22 C, as they are close to the critical boundary. For 22 C, it may also be the case that the two-neutron halo presents a continuum resonance [6]. This halo nuclei system will be discussed in the Section 4.4.3 of this review, where results of calculations for the mean-square-distance of a halo neutron with respect to the center-of-mass of 22 C are presented. The model assumption, as considered in Ref. [6], is that n-20 C is unbound, with a virtual state energy near zero. The region where we have bound excited three-body systems can also be represented in a three-dimensional plot, as given in Ref. [46] and  shown in Fig. 13. The figure is a schematic representation combining Figs. 9 and 12, where the surface (N )

(x ≡

(N )

Enn /E3 , y ≡ Enc /E3 ) defines the limiting region in order to have at least one excited bound Efimov state. Outside this surface region (for larger x and/or y) the three-body state will be virtual or resonant, as we have discussed. In the next following section, using the zero-range two-body interactions, we revise some studies on the sizes of weakly-bound systems, in the case we have two identical particles, such as exotic halo nuclei n–n–c, or mixing of two kind of atoms (α –α –β ). See Ref. [256], discussed in the context of cold-atom systems, for some universal dynamical aspects of binary mixtures. In the zero-range limit and leaving out the core spin, there are two possibilities for each n–n and n–core two-body subsystems, as they can be bound (+) or virtual (−). By considering γ = n or c, we introduce the following definitions:

Knγ

    Enγ  ≡ ± (N ) , E 

where

3



|Enγ | = 

1

2mnγ anγ

.

(103)

In the above, mnγ and anγ are, respectively, the reduced mass and scattering length of the particle pair nγ . In the scaling limit, the consecutive (N + 1)th and Nth three-body energy levels are related by a scaling function, which can be written as: (N +1)

E3

(N )

E3

= F (Knn , Knc ; A),

(104)

where A ≡ mc /mn . The extreme conditions for the existence of the (N + 1)th excited state on the top of the Nth state, are described by the following critical conditions:

F (Knn , Knc ; A) = 0,

(105)

T. Frederico et al. / Progress in Particle and Nuclear Physics 67 (2012) 939–994

977

when both (n–n) and (n–c) energies are virtual state levels. In the other cases, where at least one subsystem is bound, let us say the n–c system, we have the critical scaling function given by: Enc

F (Knn , Knc ; A) =

E3

.

(106)

The critical curves are the boundaries, which limits the possibility of presence of excited states, in the parametric space defined by the n–n–core ground-state energy, and by the scattering lengths of the two-body subsystems. In Fig. 12, the points in the parametric plane given by the energy ratios defined by Knc and Knn , which are inside the critical boundary regions, present at least one Efimov excited state. The calculations performed in Refs. [67,254,255], with short-ranged local potentials, which presented at least one excited Efimov state, have parameters completely consistent with the scaling limit, once they are placed in the interior of the region delimited by the critical boundary. These results have also generalized the conclusions given in Ref. [257] on the critical conditions for the existence of bound-state in three-body systems. 4.2.2. Bound and virtual halo states in 20 C A recent topic that appeared in the nuclear physics context is related to the trajectory of Efimov bound states for a threebody n–n–c system consisting of a core and two neutrons. The specific system under investigation was the 20 C, with the 18 C as the core. The question is whether the trajectory of the bound states poles could be associated with a mass asymmetry of the system. The bound state energies are given by solving the Eqs. (101) and (102). The analytical extension of these coupled equations to the second Riemann sheet is made by a contour passing through the cut defined by the neutron-18 C energy, εnc . Before starting this analytical extension, let us consider the Eqs. (101) and (102) with the following definitions: Ki=1,2 (q, k; E ) ≡ Gi (q, k; E ) − Gi (q, k; −µ2 ) Gi (q, k; E ) =

1



dy −1

(107)

1

E−

A+1 q2 A+Ai−1

−

− Akqy i−1 −1

A+1 2 k 2A

,

(108)

  −2  A+2 2 τnn (q; E ) ≡ |εnn | + q −E , π 4A  −1  3/2   1 A+1 (A + 2) 2 q −E . τnc (q; E ) ≡ |εnc | − π 2A 2(A + 1)

(109)

(110)

Next, we redefine fc and fn , as fc (q; E ) ≡ hc (q; E ),

fn (q; E ) ≡

hn (q; E ) q2

− k2i + iϵ

,

(111)

where, by considering E3v a virtual three-body state energy, we have

 ki = −iκv =

2(A + 1)

A+2

(E3v − εnc ).

(112)

For convenience, instead to use the above definition for τnc , we define Tnc , as



−1 Tnc (q; E ) ≡ π

A+1

 32 

2A

 |εnc | +



A+2



q2

A+1

2

 −E .

(113)

Then, we have the analytical extension of the Eqs. (101) and (102) in the second Riemann sheet, as: hn (q; E3v ) =

2(A + 1)

A+ 2



Tnc (q; E3v ) π κv K2 (q, −iκv ; E3v )hn (−iκv ; E3v )+

∞

dkk2

+



K1 (q, k; E3v )hc (k; E3v ) +

0

K2 (q, k; E3v ) hn (k; E3v ) k2 + κv2 ∞

  hc (q; E3v ) = τnn (q; E3v ) π κv K1 (−iκv , q; E3v )hn (−iκv ; E3v ) +



,

K1 (k, q; E3v ) hn (k; E3v )

(114)



. (115) k2 + κv2 The virtual states are delimited by the analytic cut of the elastic scattering amplitude in the complex plane, corresponding to the second term in the rhs of Eq. (114). In this case, the cut is given by the zero of the denominator of G2 (q, k; E3v ) [shown 2 in Eq. (108)], where −1 < y < 1 and q = k = −iκcut . With |E3v | = |εnc | + 2(AA++21) κcut , we obtain the analytic branch points, with the cut given by 2

dkk

0

2(A + 1)

A+2

|εnc | < |E3v | <

2(A + 1)

A

|εnc |.

(116)

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T. Frederico et al. / Progress in Particle and Nuclear Physics 67 (2012) 939–994

Fig. 14. Three-body n–n-18 C results for the first excited state, with respect to the threshold (|E20 C − E19 C |) for varying the neutron separation energy in 19 C (Sn ≡ |E19 C |). Three-body bound (virtual) states occur when |E19 C | is approximately smaller (larger) than 170 keV. Source: Adapted from Ref. [72].

For n–n-18 C (A = 18), a virtual state energy can be found in the energy interval between the threshold of the elastic scattering and the starting of the above cut (116):

|εnc | < |E3v | < 1.9 |εnc |.

(117) 20

19

20

In Fig. 14, we display the trajectory of the C energy as the C energy is changed, whereas the C ground state energy is S2n = 3.5 ± 0.2 MeV, as given in Ref. [258]. We note that around 170 keV we have the bound–virtual state transition. When |E19 C | is smaller than 170 keV the system is bound; for larger values the system becomes unbound, going to a virtual state. The current adopted value for 19 C, as given in the AME2003 compilation [258], is Sn = 576.8 ± 93.7 keV, suggesting that, besides the ground state, 20 C has a virtual state with energy of about −100 keV below the neutron-19 C elastic scattering threshold, as one could check by inspecting the figure. In the inset, we show the region where the transition occurs. We can clearly see that no different behavior is found from that obtained for identical masses. This could be anticipated, considering that the mass asymmetry does not change the analytical form of the kernel. 4.2.3. n-19 C scattering close to an Efimov state The increase of the elastic scattering cross section, for three-body energies close to the two-body threshold, is verified when |E19 C | approaches the pole of the virtual/excited state, as presented in Fig. 15. A natural consequence, on the investigation of the role that the difference of masses can play, is to study the n–19 C low-energy scattering near the conditions for the appearance of an Efimov state. In this case it was verified that it presents the same qualitative behavior as the neutron–deuteron scattering. In this situation the s-wave phase-shift (δ0R ) presents a zero, or a pole in k cot δ0R , when the system has an Efimov excited or virtual state. In this case we have to adopt a parametrization of k cot δ0R similar to that one proposed by Van Oers and Seagrave to fit low energy data for the n–d s-wave doublet state just below the elastic threshold, which is given by k cot δ0R =

1 −a − + β EK + γ EK2 n −19 C

1 − EK /E0

,

(118)

where an-19 C is the n-19 C scattering length, with β and γ being the effective range parameters to be adjusted. E0 is the position of the pole with respect to the threshold for elastic scattering and EK is the center-of-mass kinetic energy. This function is presented in Fig. 16. 4.3. Two-neutron halo structure, wave-functions and mean-square distances One of the main concerns in nuclear physics experiments is to understand the nuclear mechanism in the production of unstable nuclei. From breakup reaction experiments with halo nuclei, as described in many works and reviews, such as in Refs. [34,65–67]), one can extract relevant information on the structure of such nuclei, their sizes, binding properties, and

T. Frederico et al. / Progress in Particle and Nuclear Physics 67 (2012) 939–994

0

30

60

90

120

150

979

180

Fig. 15. n-19 C elastic cross sections (in barns) versus the CM kinetic energies, for different 19 C bound energies. In the left-hand-side frame we show results for two cases that generate three-body energies close to the threshold: E19 C = −150 keV (main figure) and −180 keV (inset), producing respectively three-body bound and virtual states. In the rhs we have results for E19 C = −500 keV. The dashed lines correspond to the approximate formula for the cross-sections including only the pole of the bound or virtual state in the s-wave scattering amplitudes. Source: Extracted from Ref. [72].

100

R

(1-E/E0) kcot δ0 (fm)

10 1 0.1 0.01 0.001

0

400

800 1200 E (ke V)

1600

2000

Fig. 16. (1 − E /E0 )k cot δ0R as a function of the center-of-mass kinetic energy. From bottom to top the curves were obtained by considering the following 19 C energies: 200, 400, 600, 800 and 850 keV. Source: Extracted from Ref. [180].

correlations between the components. Such information will be quite helpful, and necessary, for an appropriate theoretical model descriptions, which will help us to improve our understanding on the nuclear structure. Therefore, in order to explore the rich physics of light unstable nuclei, within three-body descriptions with a two-neutron halo (n–n–c), one should first consider the structure of the two-body subsystem n–c. Taking into account that the n–n interaction (virtual state) is already well-known, one should study the one-neutron halo nuclei (n–c) properties, for a valid model description of the n–n–c system. In the next, we will present the basic renormalized zero-range formalism, for the wave-functions and radii of such weakly-bound systems, in a three-body model with two neutrons and a core. The formalism will be further used in Section 4.4, in the classification of the halo-nuclei through their sizes and mean-separation between the components. The n–n–c wave function for the zero-range interaction is a solution of the free Schrödinger equation, except when the particles overlap. In momentum space, the wave function is built from spectator functions, fn (⃗ qn ) and fc (⃗ qc ), which are solutions of the Faddeev coupled equations solutions of the set of coupled equations (101)–(102). We remind that, the ⃗n and for the particle c relative Jacobi momentum of the particle n with respect to the center-of-mass of n–c is given by q ⃗c . with respect to n–n by q In the next, we revise the basic formalism detailed in Refs. [70,159], used to calculate the different average separation distances in weakly-bound ααβ states, in nuclear or atomic systems, respectively. The correspondent notation, in this case, is such that the (n, n, c ) particle labels are replaced by (α, α, β). For a short review on the structure of exotic three-body systems, see also Ref. [259].

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T. Frederico et al. / Progress in Particle and Nuclear Physics 67 (2012) 939–994

The n–n–c wave function eigenstate of the Schrödinger equation for the Dirac-delta interaction are given in terms of the spectator functions with α = n and β = c. In momentum space they are given by5 :

 

 

 

 

z − 2  + fα ⃗ z + 2 fβ (|⃗ y|) + fα ⃗ ⃗ y

⃗ y

  , Ψαα (⃗y, ⃗z ) =  +2 2 +2 2 ⃗ + ⃗z 2 1 + A4A ⃗ + ⃗z 2 ϵ + A4A y y      ⃗  Ay z − A+1  + fα (|⃗ fβ ⃗ z+ y|) + fα ⃗  Ψαβ (⃗y, ⃗z ) =  +1 2 +1 2 ⃗z + 2(AA++21) y⃗2 1 + A2A ⃗z + ϵ + A2A

⃗ y A+1

(119)

  

A+2 y2 2(A+1)

⃗

,

⃗ is the relative momentum of the spectator particle to the pair in units of where ⃗ z is the relative momentum of the pair and y µ3 . Note that the sub-indexes of Ψ in Eq. (119) just denote the pair of Jacobi relative momenta used to evaluate the wavefunction. For αγ with γ = α or β , one has the relative momentum between α and γ and the relative momentum of the third particle to the center-of-mass of the system αγ . The mean-square separation distances are obtained from the derivative of the Fourier transform of the respective matter density with respect to the momentum transfer squared. The Fourier transform of the one and two-body densities defines ⃗. For the root-meanthe respective form factors, Fβ (q2 ) and Fαγ (q2 ), as a function of the dimensionless momentum transfer q square (r.m.s.) distance of the particle γ (=α or β ) to the center-of-mass system, we have 

2

⟨rγ ⟩ = −6 1 −

2

mγ

dFγ (q2 )  dq2

2mα + mβ

  

,

(120)

q 2 =0

where Fα (q2 ) = Fβ (q2 ) =

 

d3 yd3 z Ψαβ d3 yd3 z Ψαα

 ⃗+ y 

⃗ q 2

   ⃗ q , ⃗z Ψαβ y⃗ − , ⃗z 2

   ⃗ q ⃗ + , ⃗z Ψαα y⃗ − , ⃗z . y ⃗ q

2

(121)

2

And, for the mean-square distance between the particles α and γ , we have 2 ⟨rαγ ⟩ = −6

dFαγ (q2 ) 

  

dq2

,

(122)

q2 =0

where Fαγ (q ) = 2



d yd z Ψαγ 3

3



⃗, ⃗z + y

⃗ q 2



Ψαγ



⃗, ⃗z − y

⃗ q 2



.

(123)

A qualitative analysis of the regularized Faddeev coupled equations for the zero-range interaction, as given in Eqs. (101) and (102), can evidence what happens to the sizes of the three-body halo nucleus when we keep the three-body energy fixed and change the nature of the systems going from the All-Bound situation to the Borromean. The two-body bound or √ √ virtual states are distinguished by the ± signs in front of ϵnc and ϵnn , + for a virtual state and − to bound. Thus, for the All-Bound case we have only negative signs where for the Borromean case all signs are positive. So, the sequence of the configurations according to the effective attraction of the kernel is given by: All-Bound < Samba < Tango < Borromean. These differences are directly reflected in the sizes of these systems: for a same ϵ the constituents of the system should be closer for a less attractive kernel in order to form a bound state, thus the sequence of their sizes is exactly the opposite of the sequence of the attraction of the kernel. This simple interpretation has been supported by numerical calculations. 4.4. Two-neutron halos in 11 Li, 14 Be, 20 C and 22 C 4.4.1. Classification and sizes The mean-square separation distances of a generic three-body system, as derived in the previous section, can also be described by scaling functions, which are convenient to investigate the scaling properties of such weakly-bound systems as the halo nuclei. The scaling functions for the separation distances between two particles (n−γ , where γ ≡ n, c) and between

5 The regularization term is included in the wave function in [70,159].

T. Frederico et al. / Progress in Particle and Nuclear Physics 67 (2012) 939–994

981

b

r2

r2

E3

E3

a

K

K

K

K

2 ⟩|E | [frame (a)] and 2 ⟩|E | [frame (b)] for an n–n–c system, with m = m and K 2 = 0.1 as functions of Fig. 17. Dimensionless products ⟨rnc ⟨rnn 3 3 c n nn Knc /|Knn | [the Knγ , with γ = (n, c ), are defined by Eq. (103)]. Different possibilities for the particle interactions are considered, as indicated inside the frames. The transition between the configurations occurs when Knc = 0 (vertical line). Source: Extracted from Ref. [70].





a

b

2 ⟩|E | and (b) 2 ⟩|E |, for m /m = 200 and E /E = K 2 = 0.1 as functions of K /|K | [K Fig. 18. Dimensionless products (a) ⟨rnc ⟨rnn 3 3 c n nn 3 nc nn nn and Knc are nn defined by Eq. (103)]. In (a) the dashed line represents a subsystem n–n bound and the dot-dashed an n–n virtual. In (b) the solid and dotted lines represent, respectively, a bound and a virtual n–n pair. The vertical lines passing through Enc = 0 define, respectively, on its left and right sides a virtual and a bound n–c pair. Source: Extracted from Ref. [70].





each particle to the center-of-mass of the three-body system can be generally written by the following dimensionless expressions:





  rn2γ

⟨

⟩|E3 | = Rnγ

±  

r2

CM

⟨ γ ⟩|E3 | = Rγ

±

Enn E3

Enn E3

 ,± 

,±

Enc E3

Enc E3

 ,A ,

(124)

 ,A .

(125)

Within this topic we are reporting studies where the three particle system can be considered in a more broad picture, such that the two identical particles could be considered as bound or virtual (instead of the more restricted nuclear case, where the n–n interaction is virtual). Therefore, the present conclusions on the classification and sizes of weakly-bound three-body systems can be applied to atomic and molecular systems. In Fig. 17, we present numerical results of calculations for the n–c and n–n root-mean-square radius as functions of √ Knc /|Knn | for fixed Knn = ± 0.1 and A = 1, corresponding to bound (+) or virtual (−) subsystems [see Eq. (103) for the definition of Knγ ]. The configurations for which the n–n pair is virtual (dot-dashed line) are smaller than the ones that have the n–n pair bound (dashed line). The mean-square distance between the two identical particles [frame (b) of Fig. 17]  2 ⟩|E | when the configuration type is modified for a fixed three-body exhibits the same qualitative behavior as found for ⟨rnc 3 energy. These conclusions are still valid for a heavy c particle with A = 200 or a light one with A = 0.1. In Fig. 18 we show results for the mean square distances in dimensionless quantity, considering the same fixed energy ratio Enn /E3 = 0.1 as in Fig. 17, but with a quite different and large mass ratio, mc /mn = 200. In the frame (a) of Fig. 18, the  r.m.s. of n–c is shown for an n–n subsystem bound (dashed line) and virtual (dot-dashed). We can see the increase of 2 ⟩|E | when passing from a Tango to an All-Bound system. In the dot-dashed the increase of the sizes happens when ⟨rnc 3



2 ⟩|E |. In this case, n–n is passing from a Borromean to Samba system. In the frame (b) of Fig. 18 we show results for ⟨rnn 3 virtual in the dotted line and bound in the solid one. The increase of the sizes follows the same characteristic behavior in both the frames, such that one can verify that the Tango and All-Bound systems are always larger than systems with the Borromean and Samba configurations, when considering fixed three-body energies.

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Fig. 19. Classification scheme for an n–n–c system. The solid and dashed lines represent, respectively, two-body bound and virtual states.

By examining the results presented in Figs. 17 and 18, respectively for the cases with A = 1 and A = 200, it is important to note that the behavior remains for different mass ratios. The distance between the solid and dotted curves, when the subsystem n–n changes from virtual to bound, decreases with A. For A tending to zero this difference becomes smaller and smaller as the long range interaction between n–n is dominated by the lighter particle c, as already demonstrated in an adiabatic approximation of the three-body system. We have illustrated that from the nature of the two-body subsystems (bound or virtual), when considering two particle species in the three-body system, one can classify the different configurations as shown in Fig. 19. These configurations present a systematical increase in their sizes, for a fixed three-body binding energy, when the system changes following the sequence: Borromean, Tango, Samba, and All-bound. This curious effect happens because the interaction of the pair is less attractive in a virtual state and to keep the same binding the three-body system has naturally to shrink, as shown schematically in the figure. In the case where the two-body energies are zero the scaling functions given by Eqs. (124) and (125) depend only on the mass ratio of the particles (A = mc /mn ). Thus, they are reduced to



⟨rn2γ ⟩|E3 | = Rn′ γ (A) and



CM

⟨rγ2 ⟩|E3 | = R′ γ (A).

(126)

These scaling functions, for the mean-square distances in terms of A, are presented in Fig. 20, considering ground and excited states, in the limit such that Enn = Enc = 0. One can observe in frame (a) of Fig. 20, that the results almost saturate above 2 A ≈ 3 to the values found in the limit of A = ∞. The calculations for A = ∞ give for ⟨rnn ⟩ the values of 0.69/E3 for N = 0 and 0.61/E3 for N = 1. Therefore, for ground states, the root-mean-square distance between two n in the three-body  2 ⟩ are 0.45/E3 system can be estimated by 0.83 h¯ 2 /(E3 mn ), in the limit of zero pairwise binding energies. Our results for ⟨rnc

2 ⟩E is achieved faster by increasing A than for 2 ⟩E , for N = 0 and 0.40/E3 for N = 1. The saturation value for ⟨rnn ⟨rnc 3 3 which depends on the difference in the masses of the pair of particles. The mean-square distance of one of the particles γ (= n or c) to the center-of-mass of the system can be obtained from





⟨rγ2 ⟩E3 as a function of A. One sees that, for the infinitely heavy c  2 ⟩, while ⟨rc2 ⟩E3 = 0 as the heavy particle should rest in the center-ofparticle, the results for ⟨rn2 ⟩ are the same of the ⟨rnc the results shown in frame (b) of Fig. 20, where we plot mass of the system in this limit.

4.4.2. Mean separation distances in halo nuclei The formalism presented in Section 4.4.1 was given for a generic three-body system, and now we will report on how it applies to light exotic nuclei approached by a core and two neutrons dominated by s-wave interactions. The scaling function (126) can be tested by comparing its result with the experimental data of 6 He, 11 Li and 14 Be. Evidently, this kind of comparison is only illustrative as the subsystems 5 He, 10 Li and 13 Be do not have a binding energy equal to zero. It is important to note that the observables of these weakly-bound systems are strongly dominated by the tail of the wave-function and even in the case of 6 He, in which the interaction n-5 He is repulsive in s-wave and attractive in p-wave, we can see a good reasonable agreement with our zero-range calculation in s-wave, as shown in Fig. 21. However, we remark that the case of 6 He, cannot be considered realistic in the model, because 5 He was treated as an s-wave state. In the Fig. 21 we can see that the r.m.s. radii between two particles and from the particles to the center-of-mass of the three-body system saturate √ very fast as we increase A, but they diverge when A → 0. The average momentum of particle A|E3 |, which obviously tends to zero when A → 0 making that the dimensionless quantities ccan be estimated as ∼ 

⟨rn2γ ⟩|E3 | and

⟨rγ2 ⟩|E3 | increase infinitely as a consequence of the increase of the size of the system. The results tend to √

finite values after multiplying them by A. As expected the distances of c to the center-of-mass tends to zero and n to the center-of-mass tends to the n–c distance as A goes to infinity.

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a 0.8

0.6

b

0.6

0.4

0.2

0.0

 ⟨rn2γ ⟩ [frame (a)] and ⟨rγ2 ⟩E3 [frame (b)] in units of 1/E3 (E3 is the three-body energy), given as functions of the mass ratio mc /mn , in the limit Enn = Enc = 0. In (a) we have the r.m.s. distance between the particles n and γ ≡ (n, c ), whereas in (b) we have the r.m.s. distance of particle γ to the center-of-mass of the three-body system. The ground-state (N = 0) is represented by solid lines (γ = n) and dot-dashed lines (γ = c); with the excited state (N = 1) represented by dashed lines (γ = n) and dotted lines (γ = c).

Fig. 20. From the above results one can extract the root-mean-square (r.m.s.) distances



Source: Extracted from Ref. [70].

b

a

  ⟨rn2γ ⟩|E3 | [panel (a)] and ⟨rγ2 ⟩|E3 | [panel (b)] (γ = n, c ) as functions of mc /mn for zero two-body energies. The  2 ⟩, obtained from [260,261] (see also Table 1). In frame (b), the experimental available data for 11 Li experimental data shown in frame (a) are only for ⟨rnn were extracted from Ref. [4] (dot-dashed line, where γ = c) and Ref. [3] (solid line, where γ = n). The plots were adapted from [70]. However, in (b), we

Fig. 21. Dimensionless products

present updated experimental results, in complete agreement with model results.

The experimental values of the charge radius of 9 Li and 11 Li are given in [4] as 2.217(35) and 2.467(37) fm, respectively, 

such that

2 9 2 11 ( Li)⟩ = 1.08(11) fm. A neutron halo radius of 6.54(38) fm was obtained from the extracted ⟨rch ( Li)⟩ − ⟨rch

matter radius in the experiment performed by [3]. Together with S2n = 369.15(65) keV, reported in  [176] for 11 Li, the 9 11 experimental value of the root-mean-square distance of Li in respect to the center-of-mass of Li ( ⟨rc2 ⟩) in units of

√

h¯ / mn S2n , is 0.10(1) and the halo radius ( ⟨rn2 ⟩) in such units is 0.617(36), these values should be compared with the theoretical results extracted from Fig. 21, of 0.10 and 0.61, respectively. The agreement with the experimental supports the model assumptions.



The calculations presented in Fig. 21 assume subsystems with zero binding energies, while the neutron–neutron virtual state energy is −143 keV, which produces small corrections to the results shown, if the 10 Li virtual state energy is close to the scattering threshold (c.f. Table 1).

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Table 1 Results of the neutron–neutron root-mean-square radii in halo nuclei. The cores (c) are identified in the first column, with the absolute values of |E3 | in the second column. The |E3 | are given by the center values of the two-neutron separation energies S2n , as given in Ref. [270], except for Lithium. In case of Li, the data, with maximum (0.32 MeV) and center value (0.29 MeV), are from Ref. [271]. In the third column, several values are considered in our inputs for |Enc |, suggested in the literature. For bound two-body subsystem, |Enc | is equal to the one-neutron separation  energy Sn . The virtual (unbound) states are 2 ⟩ are obtained from Refs. [260,261]. indicated by (v), with the n–n virtual energy fixed to a well-known value: Enn = −143 keV. The experimental ⟨rnn Source: Extracted from [70]. Core (c)

E3 (MeV)

Enc (MeV)



4

He

0.973

0 0.3(v) 4.0(v) [262]

5.1 4.6 3.6

9

Li

0.32

0 0.8(v) [263]

9.2 5.9

9

Li

0.29

0 0.05(v) [264–267] 0.8(v) [263]

9.7 8.5 6.7

6.6 ± 1.5

9

Li

0.37

0 0.05(v) [264–267] 0.8(v) [263]

8.6 7.7 6.2

6.6 ± 1.5

2 ⟩ (fm) ⟨rnn



2 ⟩ ⟨rnn (fm) exp

5.9 ± 1.2 6.6 ± 1.5

12

Be

1.337

0 0.2(v) [268]

4.6 4.2

5.4 ± 1.0

18

C

3.50

0.16 [270] 0.53 [269]

3.0 4.4

– –

The successful reproduction of the 11 Li charge radius indicates that 9 Li core does not deform significantly. Of course this result should be in the future corroborated by calculations, within the model, of the shift of the magnetic moment and electric quadrupole of 11 Li in respect to 9 Li as measured in [272]. Note that even a simple three-body model with s-wave interaction provides angular momentum to the recoil core which are compensated by the angular momentum of the halo neutrons in respect to the center-of-mass, such that the total angular orbital momentum vanishes (c.f. the halo wave function (119)), leading to corrections to the magnetic moment and to the electric quadrupole of 11 Li in respect to 9 Li. We should remind that the effect of the internal dynamics of the core in our model is parameterized by the given two-neutron separation energy, which is considered as an input quantity, differently from ab-initio calculations. The mean separation distances between the neutrons in the two-neutron halo nuclei 11 Li, 14 Be and 20 C were estimated using as input of the calculation the separation energy of two neutrons, the neutron–neutron and neutron–core energies in [70]. As discussed in that work, despite the simplified description, i.e., using s-wave contact interaction with subtracted equations, the results are in reasonable consistency with the experimental data. In Table 1, the zero-range results for the radii are compared to the experimental values. We can see that the results  2 ⟩ for the 6 He indicates that the 5 He may have an energy very close to zero, or even be weakly bound, which of ⟨rnn does not agree with other data in the literature which suggest a much higher virtual state energy. In this specific case the discrepancy can be understood taking into account that the interaction for n-4 He is repulsive in the s-wave and attractive in p-wave, producing a larger virtual energy for the 5 He. In the case of 10 Li our approximation is much better because the s-wave interaction is attractive and the virtual state is close to the scattering threshold. Even assuming that the model only involves the s-wave interactions, it already take into account the higher partial waves through the input two-neutron separation energy. Range corrections, to these results has been added by Canham and Hammer [75]. (For experiments on the determination of effective matter radii of light neutron-rich nuclei see e.g. Ref. [273].) 4.4.3. 22 C — the largest halo nucleus The interesting case of the exotic nucleus 22 C was recently formulated as a two-neutron halo plus a 20 C core within the renormalized zero-range model [6]. This study allowed to put some constrain in the two neutron separation energy, S2n , as well as in the 21 C virtual state energy using the recently extracted matter radius of 5.4 ± 0.9 fm [5]. It gives an estimation of the mean-square distance of a halo neutron with respect to the center-of-mass of this nucleus as given by 6

 rn ≃

where rn ≡

22 2

  20 20 C 2 22 C 2 (RM ) − (RM ) ≈ 15 ± 4 fm,

(127)

22

  i i ⟨rn2 ⟩ and RMC ≡ ⟨(RMC )2 ⟩ (matter radius), with i = 20, 22, with this value, 22 C has the largest known neutron

halo along the neutron drip line. The estimated value for rn allows to get some information on S2n and on the

21

C virtual

6 Note that, this expression corrects a misprint in the corresponding one given in Ref. [6]. We also should observe that this approximation is not exactly the same used in Ref. [274].

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Fig. 22. In the three-body model n–n-20 C, the two halo-neutron separation energy, S2n , is shown as a function of the root-mean-square distance of a halo neutron, with respect to the 22 C center-of-mass. The solid curves represent results obtained by considering 21 C unbound, with two virtual-state energies Enc = 0 (upper curve) and −100  keV (lower curve). (This plot corrects results presented in Ref. [6].) The experimental point and corresponding error bars define a region with 11 fm ≤ ⟨rn2 ⟩ ≤ 19 fm [5]. The quoted 2n binding in 22 C is not well defined, being less than 1.36 MeV. In our plot we represent S2n from zero to 600 keV.

state energy using the scaling function (120), in suitable units for the nuclear system as shown in Fig. 22. The quoted value (exp) of S2n = 0.42 ± 0.94 MeV from [5] is also shown in the plot. This analysis indicates that S2n is expected to be found below ∼0.12 MeV (see also [274]), where the 22 C is approximated by a Borromean configuration with a point-like 20 C and two s-wave halo neutrons. √ The estimate of the root-mean-square distance of 20 C to the center-of-mass, in units of h¯ / m n S2n , is 0.049 (see Fig. 21).

Considering in addition that S2n < 0.12 MeV for 22 C, the charge radius of 22 C in respect to 20 C is

2 22 2 20 ⟨rch ( C)⟩ − ⟨rch ( C)⟩ >

0.9 fm, with the lower bound comparable to the value found for Li [6]. We remark that a virtual-state energy of 21 C close to zero, would make the 22 C, within Borromean nuclei configurations, the most promising candidate to present an excited bound Efimov state or a continuum three-body resonance, as shown in Fig. 12. It is noteworthy that the 22 C nucleus has an s-wave neutron halo with a large extension, even larger than the halo in 11 Li, suggesting that the estimate of the difference between the mean square charge radius of the two carbon isotopes could be even better in than in the case of 11 Li. Let us recall the discussion in [6], where it was pointed out that the three-body approximation considered for 22 C is justified because the mean distance of the halo neutrons in 22 C is much larger then the size of 20 C, and moreover the halo neutrons in 20 C are bound with about 3.5 MeV, one order of magnitude greater than S2n in 22 C. In the case of 22 C the halo neutrons have much larger probability to experience the long-range 1/r 2 potential derived by Efimov than in 20 C, as the corresponding wave function tail is extending far beyond the size of 20 C. The Efimov physics should be much more evident in the properties of 22 C ground state than in the corresponding properties of 20 C. A more microscopic treatment of 22 C and 20 C in 5-body model, the four neutrons out of 18 C, should be in a fully antisymmetric wave function due to the proposed separation of scales. We should remark that as the s-wave radial wave functions corresponding to the neutrons in the halo of 20 C and in 22 C have different sizes, in principle it is not forbidden that an antisymmetric wave function can be built. If all spectator neutron interactions are dominated by only s-waves, as in our model, the Pauli exclusion principle would make the halo neutrons in 22 C much less bound than in 20 C, which indeed seems to be the case. Therefore, we believe that the simple picture presented by our three-body model for 22 C does not exclude a three-body model for 20 C as having a two-neutron halo or an Efimov state for 20 C very near the scattering threshold [11]. We should also note that the recent experiment performed by Kobayashi et al. [275] concluded that a significant 2s21/2 valence neutron configurations are present in both 11

20 C and 22 C, by investigating one- and two-neutron removal reactions on a carbon target, which gives strong support to the above pictures of 20 C and 22 C.

4.5. Neutron–neutron correlation functions The r.m.s. radii of the halo nuclei can be measured indirectly by considering the dissociation of their constituents in the field of a heavy nucleus target. In the specific case of the two-neutron r.m.s. radius, it can be measured considering the neutron–neutron correlation function, Cnn , as a function of the relative momentum between the neutrons, pnn , and defined as

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Fig. 23. Two-neutron correlation for the halo of 14 Be (left side) and 11 Li (right side), as a function of the relative n–n momentum, pnn . Left side: the solid curve gives the model results for S2n = 1.337 MeV, and Enc = 0.2 MeV. When compared with data, the model result is multiplied by 1.425. Experimental data are from [260] (open triangles) and [261] (full circles). Right side: the model results (inset) are given for three cases. (i) S2n = 0.29 MeV and Enc = 0.05 MeV (solid line); (ii) S2n = 0.37 MeV and Enc = 0.8 MeV (dashed line); (iii) S2n = 0.37 MeV and Enc = 0.05 MeV (dotted line). In the main rms body of the figure, the solid curve (when rnn = 8.5 fm [70]) presents the corresponding curve of the inset multiplied by 2.5; the dot-dot-dashed curve, rms the model presented in [276] with rnn = 8.3 fm. The experimental data are from [260] (full circles) and [276] (empty circles). Source: Extracted from Ref. [230].

 3 ⃗nn )|2 d qnn |Φ (⃗ qnn , p C (⃗ pnn ) =  3 , d qnn ρ(⃗ qnc )ρ(⃗ qn′ c ) ⃗nn ⃗nn q q ⃗nc ≡ p⃗nn − ⃗n′ c ≡ −⃗pnn − q and q . 2

(128)

2

The one-body density is

ρ(⃗qnc ) =



    ⃗nc − q⃗n′ c 2 q −⃗qnc − q⃗n′ c ,  .

d3 qn′ c Φ

2

(129)

Φ ≡ Φ (⃗qnn , p⃗nn ) is the corresponding breakup amplitude of three-body wave function including the final state interaction ⃗nn is the relative momentum between the spectator particle c and the center-of-mass of the (FSI) between the neutrons. q ⃗nn the relative momentum between the neutrons. n–n subsystem; and p (−) ⃗(−) The final state interaction (FSI) is introduced directly in the inner product Φ ≡ ⟨⃗ qnn ; p pnn ⟩ refers nn |Ψ ⟩, where the ket |⃗ to the n–n scattered wave given by the Lippmann–Schwinger equation. Φ is given by Φ = Ψ (⃗qnn , p⃗nn ) −

1/(2π 2 ) √ − Enn + ipnn



d3 p

Ψ (⃗qnn , p⃗) , − p2 + iϵ

p2nn

(130)

where Ψ is the three-body wave function, as given in Ref. [70]. −Enn is the n–n virtual state energy taken as −143 keV. The results of our calculations for the n–n correlation function Cnn of the systems 14 Be and 11 Li are shown in Fig. 23. They are shown as functions of the relative n–n momentum, and compared with experimental results. We can see that our calculation agrees very well with the experimental points when the curves are multiplied by a normalization factor (given in the figure captions). We should also point out that the required limit of Cnn → 1 for pnn → ∞ is achieved but not from above, but from below, which possibly shows the effect of considering the FSI with a coherent source of neutrons. 4.6. Halos with more than two-neutrons The concepts and model discussed in this review to treat halo-nuclei in a three-body model can also be extended to the case of nuclei with three or more neutrons in the halo. By considering the Faddeev–Yakubovski formalism, within the simplified renormalized zero-range model, one can easily describe some static properties of halo nuclei where a four-body treatment may be appropriate. We could mention a few cases of halo nuclei that have their properties being intensively investigated, demanding for a simplified theoretical description. This is the case, for example, of the 12 Li [277], that could be treated in a model having the 9 Li as the core, with three-neutrons in the halo.

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By studying this kind of system in a model with more than three-particles, one may have a deeper insight to the significant physics of the shallow bound states, such as the possible occurrence of Efimov-like effect for more than threeparticles. This matter, which is of great theoretical relevance for the understanding of some basic principles of fewbody quantum mechanics with short-range potentials, and long-time ago discussed by several works [278–282], got a revival in recent years [172,237,249–251,283–287] in view of the experimental possibilities opened by the studies with trapped cold atoms [216,288–290] where the two-body interaction can be controlled. The halo-nuclei properties that are being investigated in nuclear physics experiments can give a complementary picture to these studies, with the support of appropriate few-body theoretical descriptions. Amado and Greenwood [278], in 1973, claimed to have demonstrated the non-existence of an analogous Efimov effect with more than three bosons. In recent model calculations, within the EFT approach [237,286,287] the authors have found two four-boson bound states below the three-body ground state energy. Another numerical investigation was reported in Ref. [250] on tetramers attached to each excited trimers. They concluded that each excited trimer (in the unitary limit) has, attached to it, two tetramer levels below, with their energies (trimer and tetramers) presumably in a fixed ratio [250]. By using the Faddeev–Yakubovski (FY) formalism [291–293] in the continuum with two-body separable potential near the unitary limit, it was also revealed in [251] two resonances attached to each trimer. The four-body problem with Dirac-delta potential discussed in Section 3.5 can be treated by the method of subtracted equations, where one can have the freedom to consider a new scale (subtraction point) [249] independent from the one introduced in the three-body subsystem. By considering this four-body scale, one is able to verify the occurrence of a new limit cycle, similar to the Efimov limit cycle that happens in the three-body case. The four-body limit cycle is revealed by a scaling function that relates the energies of successive tetramer states, obtained in a numerical calculation using the FY equations in the zero-range limit [172]. In view of the interpretation on the results that were obtained [250,251], suggesting that the energy of the trimer in the unitary limit completely determines the tetramer energies, the emergence of an independent scale in the four-body problem is still under debate. However, as it was shown in [172], the others available numerical results, near or at the unitary limit, are consistent with the derived tetramer scaling plot, where the emergence of new independent scale was verified. The disentanglement of three and four-body scales would imply in an independent movement of the energy levels of the three and four-body spectra, which will change the ratio between their binding energies. It is clear that this assumption, theoretically allowed, needs a deeper experimental investigation with trapped ultracold gases much closer to a Feshbach resonance. In this regard, indications of three-body short-range effects were found recently in cold atom experiments [294,295], suggesting that an induced three-body force in the open channel could have a sizable effect, as previously discussed in Ref. [249]. The recent results reported in Ref. [172] also suggest further numerical investigations within other approaches, in order to clarify the scope of the nonuniversal corrections and on the possible need of an independent new scale in the four-body problem, as pointed out in Ref. [296]. In what concerns the physics of the halo nuclei, with three or more neutrons and a core, the Pauli principle avoids the emergence of a short range scale beyond the three-body one. Therefore, it seems likely and fortunately that the properties of these halos will be dominated by the scattering lengths and energies of the neutron–neutron–core system, that will lead to interesting applications for nuclei like 12 Li, 21 C or other ones. 5. Summary The main purpose of the present review is to emphasize the utility of analyzing theoretical results through correlations between s-wave halo properties in three-body systems of two-neutrons and a core. For systems with large sizes the details of the interactions are far less important than the low-energy two-body parameters that they fit, apart a typical short-range three-body scale. The Thomas [76] and Efimov [77] effects drive the physics of such systems irrespectively to the details of the interaction form, that are resumed in few two-body low energy parameters and one three-body scale that represents the off-shell sensitivity for the low-energy three-body observables. The large and shallow halo states in s-waves are dominated by the Efimov physics [132,133], which gives them universal properties. In the limit of a zero-range force or infinite scattering length the properties of these systems are described by scaling functions [114] that depend only on dimensionless ratios of observables. These scaling functions gives a starting point for the study of the properties of these neutrons in halo and can be calculated through a zero-range model, which can be regularized and renormalized with the aid of a subtractive approach [157]. The method has also been applied to twobody problems with contact potentials [165], with the advantage of the formulation being renormalization group invariant, leading to a renormalization group equation for the evolution of the driving term of the subtracted scattering equations. Many of the suggestions of correlations among observables required to extract the physics of contact interactions had their origin in the nucleon–nucleon and three-nucleon problems. Since the first studies in few-nucleon systems, remarkable phenomena and model independence of the correlation among observables for nuclear systems with short-ranged forces were found [77,125,126,138,139]. At the basis of the dependence on a three-body short-range scale is the Thomas collapse in the limit of a zero-range force [76] and the closely related Efimov effect [77]. We discussed some main landmark papers that established the concepts underlying the low-energy correlations among few-body observables, providing a bridge with the modern language of renormalization and effective field theory. We do not attempt to make a review on EFT as there are many recent ones devoted to the subject, but in what concerns the halo physics

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we discuss the relation with many aspects, which the driven physics is already carried by contact forces. The usefulness of such an approach based on contact forces is just to allow the identification of the main scaling laws, signalizing the properties the halo should have. Several of the scaling laws, e.g., for binding, structure and reactions involving up to the three-body system, which composes the light halo-nuclei and reviewed here, have been inspired by the strong correlations among few-nucleon observables quite well-known in nuclear physics. These are, for instance, the Phillips plot [125,126], which shows a model independent correlation between values of the triton binding energy and the neutron–deuteron doublet scattering length for potentials with the same low-energy two-nucleon observables. Moreover, the Tjon line [140] exhibits the correlation between the binding energies of 4 He and 3 H nuclei. The physical reason for such correlations is the off-shell dependence for short-range potentials resumed in one parameter. It brings the ultraviolet physics down to low energies due to the Thomas collapse of the three-body ground state when the interaction range goes to zero. The Thomas collapse was recognized as a proof of the finite range of nuclear forces [81] and it is closely related to the Efimov effect [77] happening for infinite scattering lengths (see also [96]). Notice that the Tjon line is valid in the scope of nuclear physics, where the interaction is dominated by two-body forces. In more general four-boson systems this correlation can be broken, and a family of Tjon lines with different slopes appear [172,249]. We provided a short historical overview on the light halo nuclei concepts, addressing to the appearance of quantum few-body long-range correlations in weakly-bound halos states, that establish the physics of the neutron halo mostly in the classically forbidden region, outside the potential range. Under this circumstance, one realizes why universal few-body phenomena as the Efimov effect is important to the s-wave two-neutron halo nuclei. The observables of these tenuous states are dominated by few scales which determines the tail of the wave function, and creates strong correlation among the halo observables, considering the low-energy scattering parameters of the neutron–neutron and neutron–core subsystems. The scaling laws correlating observables in the neutron–neutron–core system, can be derived in the limit of a zerorange interaction, and therefore the model has an essential singularity, i.e., the Thomas collapse, which needs regularization and renormalization. One possibility is to use subtracted equations which are consistent with renormalization group invariance [165]. The correlations between observables are the limit cycle scaling functions achieved in the limit where the subtraction point is driven towards infinity [158]. These scaling functions depend solely on the dimensionless ratios of two and one three-body quantity. Both, can be chosen as the energies bound or virtual of the two-particle subsystems and the energy of the shallow neutron–neutron–core system. For s-wave observables one three-body scale is required, while for non-zero angular momentum all properties of the states are determined just by the n–n and n–core scattering lengths. Weakly-bound two-neutron halo nuclei close to the drip line present a three-body structure with wave functions healing far beyond the size of stable nuclei. Examples of such exotic nuclei are: 6 He, 11 Li, 14 Be, 20 C and 22 C. The neutron–core systems 10 Li, 13 Be, and 21 C have a virtual s-wave state close to the threshold, while 19 C is weakly bound, and 5 He is a low-energy p-wave resonance. With exception of 6 He, the other quoted exotic nuclei can be studied with s-wave interactions and fulfill the requirement of being shallow states with large halos. 20 C is on the border line for application of such models as it is quite compact. On the other hand 22 C is quite large. The study of two-neutron halo systems demands basic concepts in few-body physics, associated with the S-matrix poles in the complex plane, namely the bound state pole, the virtual state pole on the second energy sheet as well as the resonances. The three-body system presents for bosons in the state of maximum symmetry, the Thomas and Efimov effects when r0 /|a| → 0. When the dimer moves between a weakly-bound and virtual state, Efimov trimers emerges from the lowest scattering threshold, for a > 0 they turn or come from virtual states [122], while for a < 0 they turn or comes from a continuum resonance [194]. To calculate the virtual states, an extension to the second energy sheet of the zero-range trimer equation has to be performed [199]. In the case of a continuum resonance a complex rotation method in the momentum variables of the integral equation is performed to reveal the second energy sheet, where the resonance lies. A scaling function correlates the energies of two successive trimers, being the shallowest one be virtual, bound or resonant [114,158,194]. At the same time this scaling function determines the full tower of Efimov states and corresponds to a limit cycle of the scaling law. To give an illustrative discussion of scaling we presented the Born–Oppenheimer three-particle model for subsystem energies close to zero [107]. We review at length the regularization, renormalization of contact interactions by using subtracted T -matrix equations and the relation with EFT methods [44,109,141] applied to two and three-body systems. The subtracted T -matrix equations, which can be used with any number of subtractions as required by the ultraviolet behavior of the interaction, has an associated renormalized Hamiltonian and interaction, that when used together with standard scattering equation, automatically gives rise to a subtracted equation. The renormalized Hamiltonian, albeit singular, is renormalization group invariant as imposed by the physical constraint that the T -matrix should be independent of the arbitrary subtraction point, leading to renormalization group equations [168]. We presented two examples for s- and p-waves, the last case [167] matches the expression found within the methods of effective field theory and applied to 5 He [117]. It is worthwhile to note that, in the case of s-wave the example considers a contact potential with matrix elements containing up fourth power of momentum, and the resulting T -matrix while being independent on the subtraction point for running strengths, the effective range expansion has any power of the energy. Using the subtraction method, the Faddeev equations were written in a suitable form [157] to be used with a Dirac-delta interaction. Associated with that a renormalized Hamiltonian, was derived formally. The interaction contains the renormalized potentials for two and three particles. At this point, we relate the subtracted three-body approach to the EFT treatment of the contact interaction, which introduces in leading order a

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three-body potential [155] to remove cutoff dependence. We add a short review on the inclusion of Coulomb potentials within EFT of two and three-body systems. Beyond three-particles, the four-body problem with Dirac-delta potential can be treated by the method of subtracted equations [249], and a freedom comes of a new subtraction point, different from the one introduced in the three-body subsystem, which is associated with a four-body scale. A new limit cycle appears and a scaling function relates the energies of successive tetramer states [172], at odd of effective field theory [237] and the finding that for each Efimov trimer only two tetramers are possible [250], which suggested that the energy of the trimer in the unitary limit completely determines the tetramer energies. Despite that, these calculations are consistent with the tetramer scaling plot. The full description of the interwoven three and four-body cycles demands further precise calculations of the tetramer resonances. The situation simplifies in neutron halos, where the Pauli principle acts and the properties of these halos are determined by the scattering lengths and energy of the neutron–neutron–core system, having interesting applications for nuclei like 12 Li [277] or 21 C. The universal aspects in the three-body halo systems [39], classified as Borromean (all subsystems are unbound), tango (one subsystem is bound) [252], Samba [70] (two subsystems are bound) and all-bound systems. In two-neutron halo nuclei, Borromean (e.g. 6 He, 11 Li, 14 Be, 22 C) and Samba (e.g. 20 C) are the possible classifications, as the neutron–neutron system does not bind. For one term separable s-wave interactions the bound/virtual state equations simplifies considerably [68,237] and allows to retrieve the zero-range equations for unit form factors [11], and then a subtraction is introduced to coop with the renormalization framework and renormalization group equations for the three-body problem (see e.g. [70]). The halo Efimov states can be calculated and the critical conditions for binding excited halos define a boundary on the parametric space of the subsystem energies measured in units of the shallowest neutron–neutron–core bound state, which allows at least one excited state on the top of that reference one [11,74]. Towards the origin of the plot the Efimov condition is met. In these asymmetric systems, when at least one of the subsystems is bound, the bound excited Efimov state when crossing the scattering threshold becomes a virtual state. This is the case of Samba type nuclei as 20 C [71,72]. The n- 19 C s-wave scattering amplitude close to an Efimov state exhibits the characteristic virtual state pole [180], and has many of the features observed in the neutron–deuteron doublet s-wave scattering amplitude. One of these features is a pole in k cot δ [121] (see also [129]), implying that the s-wave cross-section vanishes at the pole if in the scattering region. In the case of Borromean type nuclei, as possibly 22 C the excited Efimov state turns to a continuum resonance, when the scattering lengths have their magnitude decreased [6]. An analysis of the structure and sizes of the weakly-bound two-neutron halo, reveals the limit cycles for the different average separation distances expressed by scaling functions. The estimate of the average separations of the constituents of two-neutron halos for 11 Li, 14 Be, 20 C and 22 C has been done based on the scattering lengths and two neutron separation energy. A convenient scaling function applied to the case of a Borromean 22 C [6] allows to correlate radii, energy of 21 C and two neutron separation energy, using a recent extraction of the radius [5] which gave a large two-neutron halo in 22 C healing beyond 10 fm’s, to constraint the poorly known energies of 21 C and 22 C. The two-neutron separation energy in 22 C is estimated to be below ∼0.4 MeV. Ref. [6] suggested that with the virtual-state energy of 21 C close to zero, the 22 C is the most promising Borromean candidate to present an excited bound Efimov state or a continuum three-body resonance. The neutron–neutron correlation function gives relevant information on the long wave-length structure of the twoneutron halo. The zero-range model provides consistent results with data for the Borromean nuclei 11 Li and 14 Be (see [230]), once the calculation is rescaled at zero relative momentum [230]. It was found theoretically that the asymptotic region is achieved at much larger relative momentum than the ones adopted to normalize the data and that is the reason for the necessity of rescaling. Along the review, finite range effects are discussed in several places, in respect to the corrections in the trimer scaling function [114,205], which provides the critical boundary to bind an excited Efimov state in asymmetric mass systems [74], with the effect on the mean separation distances between the constituent three particles of the halo nuclei [75]. It is worthwhile to mention, the new opportunities to gain complementary information on the universal properties of neutron halos by performing experiments with few-atom and molecular systems in ultracold laboratories. Previous theoretical developments made in the context of nuclear physics can now be tested in atomic and molecular physics. For example, the low-energy scaling functions can be studied with different kind of particles, from Bosons to Fermions. Apart of microscopic quantum effects, as the spin interactions, it has been noticed that many concepts and basic phenomena in few-body physics, when considering low-energy observables in s- waves, could be easily extended to few-atom systems [206,209,297]. In view of the recent developments of cold-atom laboratories, it was possible to access and confirm universal aspects of three-particle systems, namely the Efimov physics [290]. Tuning the atom–atom scattering lengths by using resonance Feshbach techniques was essential for these studies. As the scaling functions depends only on dimensionless quantities and appropriate for large systems, it is conceivable that cold atom traps could act as a quantum halo simulation device, giving in a controlled way complementary information on the structure of the halo, associating with the nuclear core with a heavy atom and the neutrons with two fermionic atoms in two polarization states. Certainly, cold atoms provide much richer situations, but for nuclear halo studies the simplest case is enough. The possibilities of simulating with cold atoms, nuclei far from the stability region, like the heavy isotopes of hydrogen, lithium, carbon and other weakly-bound systems in a controlled quantum device is within reach. Moreover, it could allow to access properties of large halos in rich fermionic environment at finite temperature, which have implications for astrophysics. The traditional few-body area is in a time of many challenges where accurate numerical techniques are already at our disposal to solve few-body problems, as one can verify from the collection of methods and benchmark calculations

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presented in [298,299]. By considering a three-dimensional approach, without partial-wave decomposition, one can trace the bibliography through Refs. [300–302]. In the near future, certainly, the knowledge and techniques used to treat nuclei far from the stability line in a few-body perspective will join to more complex situations, such as the case of the nuclear reaction 8 B + 58 Ni, within a three-body description, 7 Be + p + 58 Ni [303,304]. Other reactions, such as 6 He + 208 Pb, can be address to the four-body configuration n + n + 4 He + 208 Pb. This problem was treated recently within a four-body ContinuumDiscretized Coupled-Channels (CDCC) [305,306]. In all the above mentioned situations the bound and continuum three-body wave function will be necessary. From a simplified approach, the first three-body wave-function candidate would be that constructed from a renormalized zero-range model, once its long-range tail contains reliable information. One theoretical perspective in halo physics is to investigate models that essentially carry few scales and present a rich and universal phenomenology. The low-energy properties of the structure and reaction of few-neutron halo nuclei offer exciting possibilities to investigate the classically forbidden region, far from the interaction range, where the realm of quantum mechanics and non-locality are expressed by few-scales. In help to simplify the theoretical assumptions comes the Pauli principle that restricts neutron-halos to require information beyond the three-body scale. The accretion of neutrons forming a halo on the top of a light and compact nuclear core is dominated essentially by at most one three-body scale, the neutron–core and neutron–neutron low-energy parameters like the cases of 12 Li and 21 C as three neutrons and a core. Furthermore, numerical tools to solve few-body problems are evolving rapidly, allowing the investigation of such large systems, which certainly will give access to model descriptions of novel halo phenomena. Acknowledgments For discussions on some specific aspects of this review, we would like to thank Dr. Mohammadreza Hadizadeh, Dr. Raquel S. Marques de Carvalho, and Daneele S. Ventura. We also thank the Brazilian agencies Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for partial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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