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Related content Modes of Internal Motion of Three-Body Systems Containing a Lighter Particle* Duan Yi-wu, Qiu Guo-chun and Bao Cheng-guang - Two-neutron halo state of 15B around 3.48 MeV by a three-body model Dong Bai, Zhongzhou Ren and Tiekuang Dong - Bound 0+ Excited States of Three-Body Systems with Short-Range Two-Body Interactions Lin Qi-Hu and Ren Zhong-Zhou - This content was downloaded from IP address 186.217.236.61 on 23/08/2019 at 17:51 https://doi.org/10.1088/1742-6596/915/1/012002 http://iopscience.iop.org/article/10.1088/0253-6102/14/3/353 http://iopscience.iop.org/article/10.1088/0253-6102/14/3/353 http://iopscience.iop.org/article/10.1088/1674-1137/42/6/064103 http://iopscience.iop.org/article/10.1088/1674-1137/42/6/064103 http://iopscience.iop.org/article/10.1088/0256-307X/30/5/052102 http://iopscience.iop.org/article/10.1088/0256-307X/30/5/052102 http://iopscience.iop.org/article/10.1088/0256-307X/30/5/052102 http://iopscience.iop.org/article/10.1088/0256-307X/30/5/052102 http://iopscience.iop.org/article/10.1088/0256-307X/30/5/052102 https://oasc-eu1.247realmedia.com/5c/iopscience.iop.org/244505455/Middle/IOPP/IOPs-Mid-JPCS-pdf/IOPs-Mid-JPCS-pdf.jpg/1? 1 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1234567890 ISNDMAFBP IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 915 (2017) 012002 doi :10.1088/1742-6596/915/1/012002 Scaling behavior of scattering observables for three-body systems near the unitary limit Lauro Tomio1,2, M A Shalchi1, M T Yamashita1, M R Hadizadeh3,4 and T Frederico2 1Instituto de F́ısica Teórica, Univ. Estadual Paulista (UNESP), 01140-070, São Paulo, SP, Brazil (E-mail: tomio@ift.unesp.br) 2Instituto Tecnológico de Aeronáutica, DCTA,12228-900, São José dos Campos, Brazil 3Institute for Nuclear and Particle Physics and Department of Physics and Astronomy, Ohio University, Athens, OH 45701, USA 4College of Science and Engineering, Central State University, Wilberforce, OH 45384, USA Abstract. This contribution reports recent investigations on low-energy scaling properties of three-body systems, by considering elastic s−wave collisions of a particle in a bound-state formed by the remaining two-body system. First, some previous results for the case of the halo nucleus 20C will be revised, for the neutron−19C scattering properties near the critical condition for the occurrence of an excited bound state in 20C, within a neutron−neutron−18C configuration. In our approach, we consider the Faddeev formalism with renormalized zero-range two-body interactions. Next, by considering the actual possibilities for verification of low-energy scaling properties in cold-atom laboratories, the approach is extended to atomic strongly-mass- imbalanced three-body systems, with two identical heavy particles and a light one. In this case, we consider that the heavy particle is being scattered by the light-heavy weakly-bound dimer. Our preliminary results for scattering observable are evidencing the universal scaling features. 1. Introduction First shown by Efimov [1] for the case that we have three-identical particles in the unitary limit (more specifically, when the absolute value of the two-body scattering length |a| becomes infinity), a discrete geometrical scaling emerges in the corresponding bound-state energy spectrum, which has an infinite number of three-body levels. Considered as a quantum mechanics pathology in the beginning [2], the Efimov effect was demonstrated in [3] to arise from essentially the same singularity structure of the kernel of the scattering equation, which is also responsible for the Thomas collapse [4] of the three-body ground state when the two- body range r0 is reduced to zero. As shown in Ref. [3], the trace of the kernel will diverge as ln(|a|/r0), allowing the Efimov and Thomas effects to be described in a unified way. The Efimov effect has been discussed occasionally over the years in the nuclear and atomic-molecular physics context [5–13] as related to some universal aspects of three-body quantum systems. The experimental observation of this effect in different ultracold laboratories [14–18], which was possible in view of the quantum-mechanical control of the two-body interactions provided by Feshbach resonance mechanisms [19], attracted a lot of new theoretical and experimental investigations on low-energy quantum few-body systems. Also, the existence of the long-standing prediction of an excited Efimov state in the helium trimer [9] was reported recently in Ref. [20]. As a summary of the research activity on this topic, in atomic and nuclear physics, we should http://creativecommons.org/licenses/by/3.0 2 1234567890 ISNDMAFBP IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 915 (2017) 012002 doi :10.1088/1742-6596/915/1/012002 mention as relevant some previous reviews, as Refs. [21–24]. In particular, for more recent ones, we can mention Ref. [25] when considering universal aspects in light halo nuclei; and Refs. [26,27] for the case of ultracold quantum gases. For a simplified pedagogical description of the Efimov effect, we can suggest Ref. [28], where this effect is shown to be a manifestation of a well-known quantum mechanics anomaly [29]. The investigations of Efimov related effects in the scattering region is quite appealing in view of recent interest in three-body systems α − α − β with non-identical masses, mα �= mβ . In nuclear physics, the actual interest goes to weakly-bound halo-nuclei systems, with two neutrons (n) in the halo and a core (c), where one can consider the scattering of a neutron by the n− c weakly-bound sub-system. With increasing possibilities to be studied experimentally, we have ultra-cold atomic systems where one heavy atom is colliding with low-energy to a weakly-bound dimer formed by the remaining two particles. Before examining the scattering region, it is first relevant to remind that in the unitary limit, it was demonstrated that two levels of the three-body Efimov spectrum are related by an exponential scaling factor given by exp (2π/s0), where s0 is a constant that varies according to the mass-ratio mα/mβ [23]. In the case of an identical-mass system, we have the maximum energy-ratio predicted to be ∼ 515, such that it will be quite difficult to be experimentally verified in laboratory, due to the large splitting of the energy levels. However, one can easily identify from the mass-dependence of the discrete scaling factor that optimal situations, better than the equal-mass cases detected for example in [16], can occur for mass-imbalanced systems with mα >> mβ , when the level splitting could be more easily experimentally identified. For instance, in case of mα = 100mβ , the ratio between consecutive levels of the bound-state energy spectrum is given by exp (2π/s0) ∼ 4.7. Therefore, the actual experimental investigation are more promising in this direction, by considering the atom-molecule collision with two kind of atomic species [18, 30, 31]. By following this line of investigation in cold-atom experiments, we have the Heidelberg group studying the extreme mass-imbalance atomic mixtures composed of Caesium and Lithium atomic species [27]. With the actual possibilities to manipulate Lithium(Li)-Caesium(Cs) collisions [32], as well as considering that LiCs molecules can be generated in ultracold experiments [33], we understand that more favorable conditions are accessible by now to probe the rich Efimov physics in cold- atom laboratories by investigating the low-energy collision of a Cs atom in a weakly-bound LiCs molecule. In a more general perspective, by also including other possible atomic species, we can consider the scattering of an atomic specie α in a weakly-bound α−β system, formα >> mβ . For that, in our following theoretical approach, we assume that the two-body interactions are such that there is no interaction between the two heavy particles. Further, by controlling the strength of the α− β binding in laboratory through Feshbach resonance mechanism, we understand that appropriate conditions can be created for atom-molecule collisions at very low energies. Within our approach to identify the scaling behavior of scattering observables for three-body systems near the unitary limit, let us recall some recent discussion related to the k cot δR0 pole behavior in the weakly bound three-body nuclear physics case of carbon-20 (20C), within the three-body neutron-neutron−18C model, where the subsystem neutron−18C is bound. In this case, k is the on-energy shell momentum (incoming and final), with δR0 being the real part of the s−wave phase shift. This case, with two light particles and a heavy one, was shown to resemble the well-known equal-mass neutron-deuteron case, which was studied about forty years ago [34–36]. In contrast with these studies, when considering the present experimental possibilities in cold-atom laboratories, we can verify that the limit of two-heavy and one light particle is quite more interesting to be explored near the unitary limit. As in this limit, for the case mα >> mβ , we can identify several bound-state poles for the three-body spectrum with levels close together, in correspondence, a sequence of poles is expected to appear in the scattering observable k cot δR0 for low-energy collisions of the heavy particle in the α− β bound 3 1234567890 ISNDMAFBP IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 915 (2017) 012002 doi :10.1088/1742-6596/915/1/012002 sub-system, which can be identified by the corresponding minima in the cross sections. In the next, we have two sections reporting results for scattering observables in mass- asymmetric cases. Section 2 is concentrated in presenting some results and analysis related do the halo-nucleus 20C, with section 3 having its focus in a more general mass-imbalanced atomic system with two heavy and one light particle. Section 4 gives some concluding remarks. 2. Universal properties in Neutron−19C scattering The low-energy properties of the elastic s−wave scattering for the n−19C near the critical condition for the occurrence of an excited Efimov state in n−n−18C was studied in Refs. [37,38], by considering renormalized zero-range as well as finite-range interactions. By fixing the two- neutrons separation energy in 20C with available experimental data, it is studied the scaling of the real (δR0 ) and imaginary parts of the s−wave phase-shift with the variation of the n−18C binding energy. Universal characteristics are identified for the pole-position of k cot(δR0 ) and effective-range parameters. It was verified that the excited state of 20C goes to a virtual state when increasing the n−18C binding energy, resembling the neutron-deuteron behavior in the triton. The basic formalism for a three-body halo nucleus formed by a core and two neutrons interacting via two-body separable potential is presented in Refs. [37,38]. The n−n interaction was considered fixed, such that it was obtained the usual virtual-state energy, Enn = −143 keV. For the n− c subsystem (Enc = E19C), the bound-state energies are varied within a range given by experimental available data, ranging from −160±110 keV [39] to −530±130 keV [40]. The n − n − c three-body ground-state binding energy is fixed at E20C = − 3.5 MeV. The units for this case are such that � = 1, with momentum variables in fm−1 and the unit conversion given by � 2/mn = 41.47 MeV fm2. The core-mass number is defined as A = mc/mn, with μnc = Amn/(A+1) being the reduced mass for the n− c system, such that we have A = 18 and μnc = (18/19)mn in the specific system we are considering. The zero-range two-body interaction considered for the s−wave elastic n − (nc) scattering formalism in Ref. [37] was extended to include finite-range two-body interactions, with separable form, given by Vij(p, p ′) = λij ( 1 p2 + β2 ij )( 1 p′2 + β2 ij ) , (1) where ij = nn or nc, respectively, for the n − n or n − c two-body subsystems. λij and βij refer to the strength and range rij of the respective two-body interaction. For negative bound or virtual two-body energies Eij , the corresponding relations for the strengths and ranges are λij = −2πμij βij(βij ± κij) , rij = 1 βij + 2βij (βij ± κij)2 , (2) where κij = √−2μijEij , with (+) for bound and (−) for virtual states. By following the formalism given in Ref. [38] for the on-shell scattering amplitude hn(k;E), where E is the three-body energy E ≡ E(ki), we obtain: hn(k;E) = eiδ0 sin δ0 k = 1 k cot δ0 − ik , (3) with the on-shell relative momentum given by k ≡ ki ≡ |�ki| = |�kf | = √ 2μn(nc) (E − Enc) and the reduced mass is μn(nc) = mn(A+ 1)/(A+ 2). 4 1234567890 ISNDMAFBP IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 915 (2017) 012002 doi :10.1088/1742-6596/915/1/012002 The validity of the picture found with the renormalized zero-range model in Ref. [37] is verified with finite-range interactions in [38] compatible with the nuclear force. The analysis follow a suggestion given in Ref. [34] for a parameterization of the effective-range expansion of k cot δR0 , where δ R 0 is the real part of the corresponding s−wave phase shifts. With E0 being the pole position for the energy, we have k cot δR0 = −a−1 + b E + c E2 1− E/E0 , (4) where a [n − (nc) scattering length], b and c are the adjustable parameters. When considering short-ranged interactions, such that we have fixed the two-neutron separation energy in 20C (3.5 MeV [39]) and the n − n virtual state energy (Enn = −143 keV), as the 19C binding energy is varied from 0.2 to 0.8 MeV, the corresponding parameters are given in the following Table 1, where rnn and rnc are the corresponding ranges. Table 1. One-neutron separation energy in 19C, obtained by the given low-range parameters of the separable potential. The values of βnc were obtained by fitting the two-neutron separation energy in 20C (3.5 MeV [39]), with the n − n interactions fixed by the virtual state energy, Enn = −143 keV. In this case, we fix βnn =24.5 fm−1 (rnn =0.1228). |E19C|(keV) 200 400 600 800 830 βnc(fm −1) 18.970 17.036 15.592 14.395 14.234 rnc(fm) 0.157 0.174 0.190 0.205 0.207 The parameters for the low-range potential shown in Table 1 provide results consistent with the ones obtained in Ref. [37] when using renormalized zero-range interactions. In Fig. 1 we present the results obtained by considering short-range interactions, reproduced from Ref. [38]. From this Fig. 1, by a fitting procedure, one can obtain the values for the parameters of the effective-range expansion (4). The corresponding parameters are shown in Table 2. Table 2. Effective-range parameters, obtained by fitting Eq. (4) to Fig. 1, when considering different values of |E19C| (first column) with short-range potentials. |E19C| −1/a 104 b 108 c E0 d rnc (keV) (fm−1) (fm.keV)−1 (fm.keV2)−1 (keV) (fm−1) (fm) 200 0.006 5.579 5.717 1304 0.831 0.157 400 0.066 6.742 9.144 749.0 0.622 0.174 600 0.234 9.840 13.16 402.9 0.652 0.190 800 1.798 44.67 35.19 78.86 2.153 0.205 830 5.149 119.8 85.78 28.98 5.497 0.207 Finally, we observe that in analogy with the known behavior of the elastic neutron-deuteron doublet s−wave scattering amplitude, the results presented in [37,38] confirm that the real part of the elastic s−wave phase shift (δR0 ) reveals a zero when the n − n − c system is close to an excited Efimov state (bound or virtual). For more details on this investigation, see Ref. [38], where the zero-range results are being compared with finite-range ones, by using separable one-term two-body interactions with different parametrizations. In this reference, results for the s−wave absorption parameter are also presented. By considering more realistic potential models, a recent investigation by Deltuva [41] is confirming the main conclusions on the universal properties verified for the n−19C scattering in the models used in Refs. [37, 38]. 5 1234567890 ISNDMAFBP IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 915 (2017) 012002 doi :10.1088/1742-6596/915/1/012002 0 500 1000 1500 2000 E k (keV) 0.001 0.01 0.1 1 10 100 (1 -E k/E 0)k co tδ 0R ( fm -1 ) 200 keV 400 keV 600 keV 800 keV 830 keV Figure 1. The function (1 − EK/E0)k cot δ R 0 , where E ≡ EK and E0 is the pole position, is given in terms of EK ≡ k2/(2μn,nc), by considering one-term separable low-range potentials (large β values), shown in Table 1. For each case, we plot the results obtained for different values of the two-body binding energy |Enc|, as indicated inside the frame (adapted from Ref. [38]). 3. Scaling behavior in atom-dimer α− (α− β) scattering with mα � mβ As motivated in our introduction, large mass imbalanced systems can better be investigated in cold-atom experiments in view of possibilities to alter the two-body scattering length by tuning a Feshbach resonance with magnetic fields. Therefore, it is interesting to extend the approach considered in section 2 to the case of the collision of a heavy atom α on a dimer formed by α and a lighter atom β, in the case that mα � mβ . For this kind of system, one can also consider a Born-Oppenheimer (BO) approach in addition to the basic Faddeev three-body formalism that was used for the n − (n −18 C) system, and check for the properties of the s-wave elastic cross section, while keeping the essence of the Efimov physics. For that aim, we have examined the low-energy collision of a heavy particle α with a weakly- bound two-body α− β system. The heavy-heavy interaction is assumed zero as this should not affect the driving Efimov physics, which is originated by the heavy-light resonant interaction as already pointed out long ago by Fonseca and Shanley [7]. In this case the Efimov physics of the system is driven by the mass ratio and the ratio between the weakly-bound energy of the two- body system Bαβ and a three-body energy scale, which can be chosen as the three-body bound state energy [25]. The two-body bound-state energy can also be represented by the corresponding very-large positive two-body scattering length, which is given by a ≡ aαβ = √ (2μαβ/�2)/Bαβ , where μαβ is the reduced mass between α and β particles. In this case, for a very large mass ratios mα � mβ , namely for our purpose we choose mα/mβ = 20 we can show that the minimum of the s-wave elastic cross-section appears and more than one. These minima are associated with the poles of k cot δR0 , which are associated with the long-range Efimov potential, strengthen by the mass ratio effect that implies a smaller geometrical ratio between the Efimov states as larger is the mass asymmetry in the heavy-heavy- light system. Our results, which are verified for several small values of Bαβ , by using Faddeev approach as well as a BO approximation [42], are being exemplified in two panels of Fig. 2 for the cases that the mass-ratios is given by A = mβ/mα = 0.08 and 0.05, when considering Bαβ = 0.05B3, 6 1234567890 ISNDMAFBP IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 915 (2017) 012002 doi :10.1088/1742-6596/915/1/012002 where B3 is an energy scale that we assume equal to the three-body ground-state energy. The novel feature emerging in these two panels is verified by the possibility that more zeros can be identified in the cross-section (or more poles in the k cot δR0 ), as the mass asymmetry increases. This observation is closely related to the dominance of the Efimov long-range potential, which becomes more visible as the mass asymmetry increases. In a semi-classical description, such zeros should come as a consequence of the interference between different classical trajectories that pass around the two sides of the bound target. Therefore, it is conceivable that more zeros of the s−wave cross-section should appear as the intensity of the Efimov long-range potential raises. As a final remark in this case, we should point that the ratio between the positions of the scattering momentum k(n) of the two-zeros shown in the upper plot of Fig. 2, for A = 0.05, is given by √ E(n)/E(n+1) = 6.5, when the two-body energy is Bαβ = 0.05B3. This result, when compared with the corresponding Efimov ratio (note that √ B (n) 3 /B (n+1) 3 ≈ 5 for Bαβ = 0, in this mass asymmetric case), strongly suggests the plausibility of the above semi-classical description. 0.01 0.1 √ ⎯⎯ E/B3 10 -2 10 0 10 2 10 4 σ ( a. u. ) 10 -2 10 0 10 2 10 4 A=0.05 A=0.08 Figure 2. Results obtained for the total s−wave cross-section σ (in arbitrary units) as a function of √ E/B3, by considering the mass-ratios A ≡ mβ/mα =0.05 and 0.08, with Bαβ/B3 = 0.05. As shown, by decreasing the mass ratio A (from lower to upper panels), we can already identify the emergence of the second minimum in σ. For A =0.05, the ratio between the scattering momenta where the two minima are occurring (with Bαβ/B3 = 0.05), which is ∼ 6.5, can already be compared with the ratio between the Efimov levels in the three-body bound-state spectrum (where √ B (n) 3 /B (n+1) 3 ≈ 5 for Bαβ = 0). 7 1234567890 ISNDMAFBP IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 915 (2017) 012002 doi :10.1088/1742-6596/915/1/012002 4. Concluding remarks In the introduction, our motivations to study mass-asymmetric systems in nuclear and atomic cases are presented. Following that, in section 2, we shortly review results already obtained on the low-energy properties of the elastic s−wave neutron−19C elastic scattering near the critical condition for the occurrence of an excited Efimov state. The analysis of that results was done in close analogy with the case of the s−wave neutron-deuteron doublet phase-shift, where it is well known that at low-energies the properties of the neutron-deuteron doublet state is dominated by the Efimov physics, and the parameters of the effective range expansion of the neutron-deuteron doublet s−wave phase-shift present universal scaling laws with the triton binding energy for fixed nucleon-nucleon scattering lengths and effective ranges. Next, by following the approach considered for the neutron−19C scattering, it was communicated some examples on large mass-asymmetric cases of interest in recent cold-atom investigations. The low-energy s-wave elastic cross-section results were obtained for the collision between a heavy atom α and a weakly-bound heavy-light (α − β) sub-system. These studies are being motivated by the actual experimental possibilities, along the lines as reviewed in Ref. [27]. We should also point out the relevance of an extension of the present investigation to hetero-nuclear ultracold quantum gases. As being verified, the scattering observable k cot δR0 presents poles, corresponding to the zeros of the s−wave cross section, which can be directly connected with the Efimov physics, as observed when approaching the unitary limit. Here we have presented a sample of results for the mass-imbalanced system, considering mβ/mα = 0.08 and 0.05 in the case that Bαβ = 0.05B3. As shown, a second pole in k cot δR0 can be found as we increase the imbalance between the masses of the components of the three-body system. A more detailed investigation is in progress. Acknowledgments This is a contribution in memory of Prof. S. A. Sofianos, a friend and active participant in the few-body community. For partial support, we thank the Brazilian agencies Coordenação de Aperfeiçoamento de Pessoal de Nı́vel Superior - CAPES [Proc. no. 88881.030363/2013-01(MTY and MAS) and a Senior Visitor Program at the Instituto Tecnológico de Aeronáutica (LT)], Conselho Nacional de Desenvolvimento Cient́ıfico e Tecnológico - CNPq [grants no. 302701/2013- 3(MTY), 306191/2014-8(LT), 308486/2015-3 (TF)], and Fundação de Amparo á Pesquisa do Estado de São Paulo - FAPESP [grant no. 2016/01816-2(MTY)]. 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