1. Systémes à petit nombre de corps (Few-body systems) IPHC Strasbourg, France Rimantas Lazauskas, IPHC Strasbourg, France

2. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France Few-body physics is… Goal: Goal: Provide (numerically) exact solutions for QM systems with N  2 Means: Means: General formalism for multidisciplinary applications Motivation: Motivation: Systems with N>2 posses qualitative peculiarities, which even in principle can not be described as a one-particle system No-adjustments!! Raw interaction (Hamiltonian) Complete description of QM systems: Bound states Resonant state Scattering process EM & EW reactions

3. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France In classical mechanics N>2 systems are non-integrable (chaotic) unlike quantum N>2 systems Thomas effect: N>2 boson system driven by the contact interactions collapse independently of C interactions collapse independently of C (Thomas, L. H. Phys. Rev. 47 (1935) 903.) Efimov states: binding energy of N=3 system can decrease when two- particle interaction is made more attractive. In addition, infinitely many weakly bound three-boson states appear in the limit B 2 Efimov states: binding energy of N=3 system can decrease when two- particle interaction is made more attractive. In addition, infinitely many weakly bound three-boson states appear in the limit B 2  (Efimov, V. Phys. Lett. B. 33, (1970) 563., seen exp: T. Kraemer et al., Nature 440 (2006) 315)

4. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France Bound states Few-body physics is … Wave function of finite size (bound to the box) Variational method, multiple ways to discretize wave function and solve the Schrödinger eq. Scattering Wave functions extend to infinity.

5. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France Schrödinger eq. is not enough (can not provide an unique solution) Faddeev-Yakubovski equations Faddeev-Yakubovski equations … The Formalism

6. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France x 13 x 23 y 23 y 13 y 12 x 12 1 2 3 1 2 3 1 2 3 4 y 4 1 2 3 1 2 3 z y z xx  12 66 The Formalism 3-body Faddeev eq.4-body Faddeev-Yakubovski eq.

7. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France Numerical solution Radial parts of wave function F  (x,y,z) are developed in the basis of piecewise splines, converting differential equations into linear algebra problem: (A-E·B)b=c Solution is searched by decomposing FY components in tripolar harmonic basis: Boundary conditions: 4 y 4 1 2 3 1 2 3 z y z x x

8. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France Interest Fundamental quantum mechanics problem Application: Application: any non-relativistic quantum mechanical system In particular: Nuclear physics: Nuclear physics: to test nuclear interaction models « In the past quarter century physicists have devoted a huge amount of experimentation and mental labor to this problem –probably more man-hours than have been given to any other scientific question in the history of mankind. » What Holds the Nucleus Together? by Hans A. Bethe Scientific American, September 1953 Cold atom systems …

9. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France e-e- Landau, Herring p+p+ e-e- p+p+  20 nodes in E=0 p + -H singlet wave function, indicating presence of 20 vibrational bound states (in agreement with variational calculations) 1 a=-29.3 (a. u.)  2 nodes in p + -H wave function at E=0, what does it mean? 3 a= 750 +/- 5(a. u.) p + -H scattering

10. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France p + -H scattering H 2 + 2p  u excited state Big scattering lengths are due to the existence of nearthreshold negative energy pole (bound state) Position of this pole can be extracted by the analytical continuation of effective range formula: E=(-1.125±0.03)*10 -9 (a. u.) E v =-1.08505*10 -9 (a. u.) k Im k Re

11. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France p + -H scattering H 2 + 2p  u excited state - + - + - + - + - + + 3.285*10 -12 (a.u.) Casimir-Polder (retardation) effect Radiative corrections (Lamb shift) J.C, R.L, D.Delande,L.Hilico, S. Kiliç: physics/0207007 (2002) Relativistic motion of the electron (Dirac Ham.) - + - + 4.73*10 -13 (a.u.) ~10 -15 (a.u.)

12. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France He atoms are chemically the most inert ( 4 He) 2 – the weakest bound molecule in its ground state excited ( 4 He) 3 excited state is state is Efimov state ( 4 He) n have nearthreshold excited and resonant states He molecules P. Bruhl et al. Phys. Rev. Lett 95 (2005) 063002 Superfluidity of He at low T

13. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France Direct bound state calculations are possible Trimer excited state – Efimov state B (mK) (mK) x rms (Å)P Dimer He 2 1.303599.4235.541.0 He 3 (g.s.)126.37165810.950.3301 He 3 (e.s.)2.268122.1104.30.7611 y x a 0 =115.6 Å Resolved ambiguity in He-(He) 2 scattering length a 0 =115.6 Å He trimers

14. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France 4 He tetramer ground state B=558 mK He 4 L=0 ground state binding energy B=558 mK and its properties are calculated by solving FY equations. Tripolar harmonic basis contained amplitudes up to max(l x,l y,l z )  8, which ensures three-digit accuracy. B (mK) (mK) x  rms (Å) He 4 557.741078.40

15. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France He-He 3 elastic scattering a 0 =103.7 Å We predict a resonant He-He 3 scattering length a 0 =103.7 Å k Im k Re x y z 0-energy scattering wave function contains two nodes in He-He 3 separation direction, indicating existence of He 4 L=0 excited state close to He 3 threshold.

16. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France 4 He tetramer excited state a 0 (Å)  E He (mK) He-He 2 100.21.3035 He-He 2 115.60.965 He-He 3 103.71.09 E (mK) (mK) r  rms (Å) He 4 560.053888.40 He* 4 127.5190034.4 Using FY equations we can calculate rotational states with the same ease as L=0 ones Possible existence of weakly bound He 3 and He 4 rotational states was explored: by enhancing He-He interaction analyzing low energy L>0 scattering wave functions do not have rotational states He 3 and He 4 do not have rotational states

17. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France Multineutron Controversial results in GANIL F.M. Marqués et al: Phys. Rev. C 65 (2002) 044006 et arxiv:nucl-ex/0504009 14 Be n n nn n p p n n p p n n n n n p p n n p p n n 10 Be + n nn n  E. Rich et al: proceedings Exxon conference 2004 n nn n p p n n n n p p n + n nn n  p n p + 8 He 6 Li 2H2H2H2H

18. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France Quest for multineutrons n n nn n p p n n p p n n n n n p p n n p p n n + n nn n  Recent experiment at GANIL suggested possible existence of a bound 4 neutron state (tetraneutron) 14 Be 10 Be F.M. Marqués et al: Phys. Rev. C 65 (2002) 044006. Interaction nn not well defined in P-waves, but still <20%… nn not bound, a nn =-18.59 fm (it needs only E nn  0.110 MeV) If they were bosons… E( 3 n)  0.94 MeV, E( 4 n)  9.0MeV,… But how to bind (nn)* ? nn S-waves untouchables!!! Fermions sometimes consolidate… 3 He atomcules

19. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France Quest for multineutrons Effect of 3BF? … less than 1%!!! Enhancment of 3BF? Higher partial waves (P,D,F..)? … should be too strong, Too strong charge dependency breaking,   3.5 2 n becomes resonant in P waves E=5.2-2.5i MeV Violating nuclear properties: inversing order of nuclear states 4 H becomes strongly bound B>40 MeV!!! 1S01S0 P,D,F,… Violating nuclear properties B triton  211 MeV Abnormal nuclear matter density … should be too strong,

20. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France Basics of 2-body resonances Even if not bound, they still should be somethere… Re(k) Im(k) Resonance 1 S 0 : a nn =-18.59 fm (virtual state E nn  0.110 MeV)

21. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France Basics of 2-body resonances Even if not bound, they still should be somethere… Resonance is complex eigenvalue of the Hamiltonian: their eigenfunctions are divergent!!!

22. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France Multineutron resonances Even if not bound, they still should be somethere… Resonance is complex eigenvalue of the Hamiltonian: their eigenfunctions are divergent!!! But what is a resonance in multiparticle system without any bound states But what is a resonance in multiparticle system without any bound states n nn n p p n n n n p p n + n nn n  p n p + 8 Be 6 Li 2H2H2H2H D. Beaumel et al., (IPN Orsay)

23. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France 3 n and 4 n resonances are explored using realistic NN interaction Reid93 in conjunction with additional attractive force: All the resonance trajectories end up in III-rd energy quadrant, with |E img |>6 MeV Multineutron resonances

24. Peculiarities of some simple few-body systems 15 Février, 2009 Rimantas Lazauskas, IPHC Strasbourg, France Rigorous description of very cold p-H scattering E v =-1.08*10 -9 (a. u.)Prediction of H 2 + 2p  u excited state, with proton separation energy only: E v =-1.08*10 -9 (a. u.) a 0 =103.7 Å. We predict, for the first time, 4 He -( 4 He) 3 scattering length a 0 =103.7 Å. B=1.09 mK. Resonant value of this length, indicates existence of weakly bound ( 4 He) 4 excited state. We predict its binding energy (relative to He 3 ground state) B=1.09 mK. Non-existence of ( 4 He) 3 and ( 4 He) 4 rotational states have been shown. We have tested if some ‘brave’ experimental claims on the possible existence of bound or resonant tetraneutron can be supported with theoretical background We have demonstrated that theoretical nuclear interaction models exclude existence of both bound or physically observable resonant pure neutron systems with A≤4. Summary I have presented Faddeev-Yakubovski equations, which is rigorous and powerful method in describing QM few particle systems This method is very general and can be applied to study very different physical systems.