Unitarity potentials and neutron matter at unitary limit

1. Unitarity potentials and neutron matter at unitary limit T.T.S. Kuo (Stony Brook) Collaborators: H. Dong (Stony Brook), R. Machleidt (Idaho)

2. AA • Atom-Atom interaction V for trapped cold fermionic gases can be experimentally tuned by external magnetic field, giving many-body problems with tunable interactions: • By tuning V to Feshbach resonance, scattering length . At this limit (unitary limit), interesting physics observed. AA

3. gas in BEC • Near the Feshbach resonance ( ) • At , the equation of state (EOS) • Above often known as ‘Bertsch challenge problems’ gas in BCS this is BCS-BEC cross-over has an ‘universal’ form: with ξ=0.44 for ‘all’ gases. depends only on

4. Experimental values for ξof atomic gases +0.05 -0.12

5. Neutron matter is a two-species fermionic system, should have same unitary-limit properties as cold fermi gas, and neutron-neutron fm, it is rather long. • We study the EOS of neutron matter at and near the unitary limit, using different unitarity potentials If is universal, then results should be independent of the potentials as long as their

6. How to obtain unitarity potentials with ? tuned CDBonn meson-exchange potential tuned square-well ‘box’ potentials • How to calculate ground state energy ? ring-diagram and model-space HF methods • Results and discussions

7. AA • Atom-Atom interactions V can be experimentally varied by tuning external magnetic field. • How to vary the NN interaction ? Can we tune experimentally? May use Brown-Rho scaling to tune , namely slightly changing its meson masses. • Ask Machleidt to help!

8. 1 0 • CD-Bonn ( S ) of different • We tuned only m , as attraction in S mainly from σ-exchange. depends sensitively on m . s σ 1 σ 0 σ

9. We have also used hard-core square-well (HCSW) potentials • Their scattering length ( ) and effective range ( ) can be obtained analytically. We can have many HSCW unitarity potentials

10. Phase shiftsδfor HCSW potentials: with where E is the scattering energy.

11. From phase shift δ, the scattering length is where The effective range also analytically given. with

12. Condition for unitarity potential is c c b e c s Three different HCSW unitarity potentials 6

13. Ground state energy shift • Above is quasi-boson RPA By summing the pphh ring diagrams to all orders, the transition amplitudes Y are given by the RPA equations:

14. Model-space approach: • Space (k > Λ) integrated out: renormalized to has strong short range repulsion is smooth and energy independent • Space (k ≤ Λ) use to calculate all-order sum of ring diagrams Note we need of specific scattering length including

15. of specific scattering length Starting from a bare CD-Bonn potential of scattering length a, given by • obtained from solving the above T-matrix equivalence equations using the iteration method of Lee-Suzuki-Andreozzi

16. Ring diagram unitary ratio given by different unitarity potentials

17. Diagonal matrix elements of V NN

18. The ring-diagram unitary ratio near the unitary limit

19. F When choosing Λ= k , ring-diagram method becomes a Model-Space HF method, and E /A given by simple integral Here means Λ= k . 0 F

20. MSHF has simple relation between ξand : is highly accurately simulated by momentum expansions: where V , V and V are constants. Then 4 2 0 Above is a strong sum-rule and scaling constraint for at the unitary limit.

21. Checking for four unitarity potentials -1 2 4 0 F NN

22. k F low-k F Comparison of V from four different unitarity potentials (Λ= k =1.2 fm)

23. Comparison of recent calculated values on ξ

24. MSHF single-particle (s.p.) potential is with k =(k - k )/2, k =(k + k )/2. • The MSHF s.p. spectrum is which can be well approximated by m* is effective mass andΔis effective ‘well-depth’ . + − 1 F 1 F

25. At the unitary limit, m* and Δof MSHF should obey the linear constraint We have checked this constraint.

26. Check at unitary limit

27. Summary and outlook: • Our results have provided strong ‘numerical’ evidences that the ratio ξ= E / E is a universal constant, independent of the interacting potentials as long as they have . • However, it will be still challenging to prove this universality analytically ! free 0 0

28. Thanks to organizers R. Marotta and N. Itaco