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

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

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

Experimental values for ξ of atomic gases ξ Authors 0.39(15)Bourdel et al., PRL (2004) 0.51(4)Kinast et al., Science (2005) 0.46(5)Partridge et al., Science (2006) 0.46Stewart et al., PRL (2006)

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

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

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! AA

CD-Bonn ( S ) of different We tuned only m, as attraction in S mainly from σ -exchange. depends sensitively on m. m [MeV] a [fm] original tunned ∞(-12070) ∞ s σ σ 1 0 σ

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

Phase shifts δ for HCSW potentials: with where E is the scattering energy.

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

Condition for unitarity potential is PotentialsV c /MeV r c /fm V b /MeV r b /fm a s / x 10 fm r eff /fm HCSW HCSW HCSW c c b cs e Three different HCSW unitarity potentials 6

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:

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

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

Ring diagram unitary ratio given by different unitarity potentials

Diagonal matrix elements of V NN

The ring-diagram unitary ratio near the unitary limit

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

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

Checking for four unitarity potentials V k / fm V / fm Sum ξ CDBonn HCSW HCSW HCSW HCSW HCSW NN F 0 24

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

ξ Method Ref ,0.568Padé approximationBaker et al., PRC(1999) 0.326Galitskii resummationHeiselberg et al., PRA(2001) 0.7Ladder approximationBruun et al., PRA(2004) 0.455Diagrammatic theoryPerali et al, PRL(2004) 0.42Density functional theoryPapenbrock et al., PRA(2005) 0.401NSR extension with pairing fluctuations Hu et al., EPL(2006) ε expansionNishidaet al., PRL(2006) Variational formalismHaussmann et al., PRA(2007) ε expansionChen et al., PRA(2007) 0.44(1)Quantum Monte CarloCarlson et al., PRL(2003) 0.42(1)Quantum Monte CarloAstrakharchik et al., PRL (2004) 0.44Ring-diagram and MSHFThis work (Dong, et al., PRC 2010) Comparison of recent calculated values on ξ

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 and Δ is effective ‘well-depth’. − F 1 + F 1

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

Check at unitary limit

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 ! 0 0 free

Thanks to organizers R. Marotta and N. Itaco