1. Scales of critically stable few-body halo system Tobias Frederico Instituto Tecnológico de Aeronáutica São José dos Campos - Brazil  Marcelo T. Yamashita – Itapeva /Unesp  Lauro Tomio – IFT/Unesp/São Paulo  Antonio Delfino – UFF/Niterói  Sadhan K. Adhikari - IFT/Unesp/São Paulo  Collaborators FB18, Santos, Aug.21-26, 2006

2. FB18, Aug. 2006 Nuclear and Atomic weakly bound three-body halo systems How to study weakly bound three-body systems? Thomas-Efimov effect Scaling limit & limit cycle, scaling functions and correlations between observables General classification scheme: n-n-c or A-A-B Threshold conditions for an excited N+1 Efimov state Three-bosons: analytic structure & Efimov state trajectory Root mean square radii Four boson systems: new scale? Summary and perspectives OUTLINE

3. FB18, Aug. 2006 Two-neutron halo nucleus First observation 6 He T. Bjerge, Nature 138, 400 (1936) 11 Li colliding with some targets growth of the cross section Tanihata et al., Phys. Rev. Lett. 55, 2676 (1985) T. Kobayashi et al. Phys. Lett. B 232, 51 (1989) neutrons

4. FB18, Aug. 2006 Nuclear weakly bound three-body halo systems core n n core-neutron-neutron halo nuclei 11 Li 14 Be 20 C Binding energy ~ MeV or < MeV R nn (Exp) ~ 6 - 8 fm ( 11 Li) F. M. Marqués et al. Phys. Rev. C 64, 061301 (2001) M. Petrascu et al. Nucl. Phys. A 738, 503 (2004)

5. FB18, Aug. 2006 Atomic weakly bound three-body systems A B B A-B-B weakly bound molecules A-B-B weakly bound molecules ultra-low binding ~ mK or < mK 133 Cs 3 (trapped ultracold gas near a Feshbach resonance) 4 He 3 4 He 2 – 7 Li 4 He 2 – 6 Li 4 He 2 – 23 Na R 4 He- 4 He ~ 10 A o dimer R 4 He- 4 He ~ 50 A o

6. FB18, Aug. 2006 How to study weakly bound three-body systems? Use a realistic interaction and calculate the Hamiltonian eigenstates.... What details of the interaction are important for the results? Large systems are peculiar: size >> interaction range!....and the eigenfunction of the Hamiltonian satisfies a free Schrödinger equation almost everywhere for nonzero interparticle distances! asymptotic wf behaviour & universality Zero-range interaction

7. FB18, Aug. 2006 How to study weakly bound three-body systems? Charateristic phenomena: Thomas collapse (1935) and Efimov effect (1970) r o  0 |a|  ??? infinitely many weakly bound states |a|/r o  Thomas-Efimov effect! 8 8

8. FB18, Aug. 2006 How to study weakly bound three-body systems? Thomas-Efimov effect Skorniakov and Ter-Martirosian equations (1956) Thomas collapse:  8 Efimov effect:  0 Adhikari,TF,Goldman, PRL74 (1995) 487 = / = / 3 3 = /

9. FB18, Aug. 2006 Scaling limit: Frederico et al PRA60 (1999)R9 Yamashita et al PRA66(2003)052702 Scaling limit & limit cycle Limit cycle: Mohr et al Ann.Phys. 321 (2006)225 Efimov 1970 Scaling function

10. FB18, Aug. 2006 Scaling functions: Correlation between observables Correlation between S-wave observables Phillips plot: triton B.E. X doublet scattering length 2nd order neutron-deuteron polarization observables X triton B.E. Trapped atomic trimer B.E. X recombination rate

11. FB18, Aug. 2006 Three-boson wave function: Weakly bound system wave function & contact interaction + (1  2) + (1  3) q1q1 R1R1 (1) (2) (3)

12. FB18, Aug. 2006 General classification scheme: n-n-c or A-A-B BORROMEAN TANGO SAMBA ALL-BOUND bound state virtual state Yamashita, Tomio and T. F. Nucl. Phys. A 735, 40 (2004)

13. FB18, Aug. 2006 General classification scheme: n-n-c or A-A-B Scales: Energy of the bound/virtual nn system Energy of the bound/virtual nc system Energy of the Nth state of the nnc system A = mass of the core

14. FB18, Aug. 2006 Amorim,TF,Tomio PRC56(1997)2378 Borromean Samba Tango All-bound Halo-nuclei: Threshold for an excited N+1 Efimov state K nn =(B nn ) 1/2 K nc =(B nc ) 1/2 nn virtual nn bound nc virtual nc bound

15. FB18, Aug. 2006 Weakly bound molecules: Threshold for an excited N+1 Efimov state Delfino,TF,Tomio JCP 113 (2000) 7874 All-bound Tango Samba Borromean K aa =(B aa ) 1/2 K ab =(B ab ) 1/2

16. FB18, Aug. 2006 Bound 3-body state -E 2 -E 3 Virtual 3-body state Three-body cut Two-body cut Three-bosons: analytic structure & Efimov state trajectory -E 3 (N) Three-body cut Bound 2-body state x x x -E 2 Virtual 3-body state x -E 3 (N+1) x

17. FB18, Aug. 2006 Efimov state trajectory: 2-body bound Efimov state trajectory: 2-body bound

18. FB18, Aug. 2006 -E 3 (N) 3-body Resonance Three-body cut x x Three-bosons: analytic structure & Efimov state trajectory Bound 3-body state -E 3 Three-body cut Virtual 2-body state x x 3-body Resonance x -E 3 (N+1)

19. FB18, Aug. 2006 Efimov state trajectory: 2-body virtual S-wave three-boson resonance Evidence of continuum resonances in recombination of ultracold Cs atoms

20. FB18, Aug. 2006 Evidence of continuum resonances in ultracold cesium gas M.T. Yamashita, “Triatomic states in ultracold gases” Parallel session R6-16, Friday

21. FB18, Aug. 2006 Threshold for an excited N+1 Efimov state Threshold for an excited N+1 Efimov state Arora, Mazumdar, Bhasin, PRC69(2004)061301(R) Mazuumdar, Rao, Bhasin, PRL97(2006)062503 Resonance in n+ 19 C

22. FB18, Aug. 2006 Root mean square radii CM A B B

23. FB18, Aug. 2006 Root mean square radii Scaling functions for the radii  = A or B + two-body bound state - two-body virtual state

24. FB18, Aug. 2006 Root mean square radii Yamashita, Tomio and T. F. Nucl. Phys. A 735, 40 (2004) Core Exp:

25. FB18, Aug. 2006 Root mean square radii nA bound nA virtual nA bound nA virtual

26. FB18, Aug. 2006 Root mean square radii BORROMEAN TANGO SAMBA ALL-BOUND bound state virtual state For a fixed E 3 > > >

27. FB18, Aug. 2006 Neutron-neutron correlation function Radii are experimentally extracted from correlation function R. Hanbury-Brown and R. Q. Twiss (HBT) - NATURE 177, 27 (1956) 178, 1046 (1956) 178, 1447 (1956) First used in astrophysics Nuclear Physics

28. FB18, Aug. 2006 pApA qAqA A n n' One-body density Breakup amplitude including the FSI between the neutrons  is the three-body wave function Neutron-neutron correlation function

29. FB18, Aug. 2006 F. M. Marqués et al. Phys. Rev. C 64, 061301 (2001) F. M. Marqués et al. Phys. Lett. B 476, 219 (2000) E 3 = 1.337 MeV E nA = 0.2 MeV E nn = 0.143 MeV asymptotic region ? x1.425 Neutron-neutron correlation function M. T. Yamashita, T. Frederico and L. Tomio Phys. Rev. C 72, 011601(R) (2005)

30. FB18, Aug. 2006 F. M. Marqués et al. Phys. Rev. C 64, 061301 (2001) M. Petrascu et al. Nucl. Phys. A 738, 503 (2004) E 3 = 0.29 MeV E nA = 0.05 MeV Enn = 0.143 MeV E 3 = 0.37 MeV E nA = 0.8 MeV E 3 = 0.37 MeV E nA = 0.05 MeV x2.5 Neutron-neutron correlation function

31. FB18, Aug. 2006 F. M. Marqués et al. Phys. Rev. C 64, 061301 (2001) E 3 = 0.973 MeV E nA = 4 MeV Enn = 0.143 MeV E 3 = 0.973 MeV E nA = 0 x1.12 Neutron-neutron correlation function

32. FB18, Aug. 2006 Results for different radii of the molecular system ABB Radii for weakly bound molecules Yamashita, Marques de Carvalho, Tomio, T. F., Phys. Rev. A 68, 012506 (2003)

33. FB18, Aug. 2006 Ground First excited Symbols from P. Barletta and A. Kievsky Phys. Rev. A 64, 042514 (2001) squares - Ground state circles - First excited state Weakly bound molecules

34. FB18, Aug. 2006 Four-boson system: new scale? no new scale new scale

35. FB18, Aug. 2006 Four-boson system: a new scale?

36. FB18, Aug. 2006 Four-boson system: a new scale? Tjon line: MeV

37. FB18, Aug. 2006 Summary and perspectives Zero-range model: classification of weakly-bound systems threshold conditions for excited states and resonances ( evidence of the trajectory of resonance in ultra-cold atoms) 6 He, 11 Li, 14 Be, 20 C 4 He- 4 He-( 4 He, 6 Li, 7 Li, 23 Na) Neutron-neutron correlation function Scattering, breakup of halo nuclei and weakly bound molecules: universal properties Weakly bound & large systems: few scales regime Exploration of the different possibilities of threshold conditions for resonances Evidence for a four-boson scale Four-boson excited states, resonances & scattering Flexibility: Next: