1. Low-momentum interactions for nuclei Achim Schwenk Indiana University Nuclear Forces and the Quantum Many-Body Problem Institute for Nuclear Theory, Oct. 7, 2004 with Scott Bogner, Tom Kuo and Andreas Nogga supported by DOE and NSF

2. 1)Introduction and motivation 2)Low-momentum nucleon-nucleon interaction Bogner, Kuo, AS, Phys. Rep. 386 (2003) 1. 3)Cutoff dependence as a tool to assess missing forces Perturbative low-momentum 3N interactions Nogga, Bogner, AS, nucl-th/0405016. 4)Selected applications: perturbative 4 He, 16 O and 40 Ca, perturbative nuclear matter? Coraggio et al., PR C68 (2003) 034320 and nucl-th/0407003. Bogner, Furnstahl, AS, in prep. pairing in neutron matter AS, Friman, Brown, NP A713 (2003) 191, AS, Friman, PRL 92 (2004) 082501. 5)Summary and priorities Outline

3. 1) Introduction and motivation Nuclear forces and the quantum many-body problem Choice of nuclear force starting point: If system is probed at low energies, short-distance details are not resolved. Improvement of many-body methods: Bloch-Horowitz, NCSM, CCM, DFT + effective actions, RG techniques,…

4. Nuclear forces and the quantum many-body problem Choice of nuclear force starting point: Use low-momentum degrees- of-freedom and replace short- dist. structure by something simpler w/o distorting low- energy observables. Infinite number of low-energy potentials (diff. resolutions), use this freedom to pick a convenient one. 1) Introduction and motivation Improvement of many-body methods: Bloch-Horowitz, NCSM, CCM, DFT + effective actions, RG techniques,…

5. What depends on this resolution? - strength of 3N force relative to NN interaction - strength of spin-orbit splitting obtained from NN force - size of exchange-correlations, Hartree-Fock with NN interaction bound or unbound - convergence properties in harmonic oscillator basis Change of resolution scale corresponds to changing the cutoff in nuclear forces. [This freedom is lost, if one uses the cutoff as a fit parameter (or cannot vary it substantially).] Observables are independent of the cutoff, but strength of NN, 3N, 4N,… interactions depend on it! Explore…

6. Many different NN interactions, fit to scattering data below (with χ 2 /dof ≈ 1) Details not resolved for relative momenta larger than or for distances Strong high-momentum components, model dependence 2) Low-momentum nucleon-nucleon interaction S-wave S-wave Argonne

7. Separation of low-momentum physics + renormalization Integrate out high-momentum modes and require that the effective potential V low k reproduces the low-momentum scattering amplitude calculated from potential model V NN Cutoff Λ is boundary of unresolved physics [NB: cutoff only on potential] V low k sums high-momentum modes (according to RG eqn) V low k = V NN

8. RG evolution also very useful for χEFT interactions - choose cutoff range in χEFT to include maximum known long-distance physics Λ χ ~ 500-700 MeV for N3LO - run cutoff down lower for application to nuclear structure (e.g., to Λ ≈ 400 MeV) - observables (phase shifts, …) preserved - higher-order operators induced by RG comp. fitting a χEFT truncation at lower Λ (less accurate)

9. Details on V low k construction: 1. Resummation of high-momentum modes in energy-dep. effective interaction (BH equation) [largest effect] 2. Converting energy to momentum dependence through equations of motion; below to second order (to all orders by iteration, 1+2 = Lee-Suzuki transformation) [small changes, Q(ω=0) good for E lab below ~150 MeV] Both steps equivalent to RG equation Bogner et al., nucl-th/0111042. [Hermitize V low k or symmetrized RG equation (very small changes)]

10. Exact RG evolution of all NN models below leads to model-independent low-mom. interaction V low k (all channels) V low k

11. Collapse of off-shell matrix elements as well N 2 LO N 3 LO ……

12. Collapse due to same long-distance (π) physics + phase shift equivalence

13. V low k is much softer, without strong core at short-distance, see relative HO matrix elements convergence will not require basis states up to ~ 50 shells

14. Poor convergence in SM calculations (due to high-mom. components ~ 1 GeV in NN potentials) Unsatisfactory starting-energy dependence (due to loss of BH self-consistency in 2-body subsystem) V low k will lead to improvements from P. Navratil, talk @ INT-03-2003 from T. Papenbrock, talk @ MANSC 2004 NCSM coupled cluster 4 He

15. Comparison of V low k to G matrix elements up to N=4 BUT: V low k is a bare interaction!

16. Comments: 1.Renormalization of high-momentum modes in free- space easier (before going to a many-body system) 2.Soft interaction avoids need for G matrix resummation (which was introduced because of strong high- momentum components in nuclear forces) 3.V low k is energy-independent, no starting-energy dep. 4.Cutoff is not a parameter, no “magic” value, use cutoff as a tool…

17. All low-energy NN observables unchanged and cutoff-indep. Nijmegen partial wave analysis ▲ CD Bonn ○ V low k 3) Cutoff dependence as tool to assess missing forces Perturbative low-momentum 3N interactions

18. All NN potentials have a cutoff (“P-space of QCD”) and therefore have corresponding 3N, 4N, … forces. If one omits the many-body forces, calculations of low-energy 3N, 4N, … observables will be cutoff dependent. By varying the cutoff, one can assess the effects of the omitted 3N, 4N, … forces. Nogga, Bogner, AS, nucl-th/0405016.

19. Potential model dependence in A=3,4 systems (Tjon line) Nogga, Kamada, Glöckle, PRL 85 (2000) 944. + NN potentials ▪ NN+3N forces Cutoff dependence due to missing three-body forces along Tjon line Nogga, Bogner, AS, nucl-th/0405016. Faddeev, V low k only Results for reasonable cutoffs seem closer to experiment Renormalization: three-body forces inevitable!

20. Cutoff dep. of low-energy 3N observables due to missing three-body forces (see π EFT) A=3 details exp “bare” 3 H 3 He binding energies Faddeev, V low k only /

21. Adjust low-momentum three-nucleon interaction to remove cutoff dependence of A=3,4 binding energies Use leading-order effective field theory 3N force given by van Kolck, PR C49 (1994) 2932; Epelbaum et al., PR C66 (2002) 064001. Motivation: At low energies, all phenomenological 3N forces (from ω, ρ,… exchange, high-mom. N, Δ,… intermed. states) collapse to this form; cutoffs in V low k and χ EFT similar. due to Δ or high-mom. N intermediate states (both effects inseparable at low energies) long (2π) intermediate (π) short-range chiral counting: (Q/Λ) 3 ~ (m π /Λ) 3 rel. to 2N force c-terms D-term E-term fixed by πN scattering (or NN PWA)

22. Constraint on D- and E-term couplings from fit to 3 H linear dependence consistent with perturbative E( 3 H) = E(V low k +2π 3NF) + c D + c E Second constraint from fit to 4 He [E-term fixed by left Fig.] η=1: 4 He fitted exactly η=1.01: deviation from exp. ≈ 600 keV non-linearities at larger cutoffs

23. 2 couplings for D- and E-terms fitted to 3 H and 4 He We find all 3N parts perturbative for cutoffs - ≈ (A-1) m ≈ m π 2 < Λ 2 - c- and E-terms increase and cancel - 3N force parts increase by factor ~ 5 from A=3 to A=4 - Larger cutoffs: contributions become nonperturbative, fit to 4 He non-linear (a,b) and approximate solution (*) < 20% is beyond (Q/Λ) 3 ~ (m π /Λ) 3

24. 5) Selected applications Perturbative calculations for 4 He, 16 O and 40 C Coraggio et al., PR C68 (2003) 034320 nucl-th/0407003. Results for V low k V low k from N3LO exact 7.30 V low k binds nuclei on HF level (contrary to all other microscopic NN interactions) V low k from (all for Λ=2.1 fm -1 )

25. Pairing in neutron matter AS, Friman, Brown, NP A713 (2003) 191, AS, Friman, PRL 92 (2004) 082501. suppression due to polarization effects (spin-/LS fluct.) V low k 1 S 0 pairing 3 P 2 pairing

26. No strong core simpler many-body starting point Neutron matter EoS AS, Friman, Brown, NP A713 (2003) 191. BHF Bao et al. NP A575 (1994) 707. Simple V low k Hartree-Fock FHNC Akmal et al. PR C58 (1998) 1804. nuclear matter from low-mom. NN + 3N int. Bogner, Furnstahl, AS, in prep.

27. 5) Summary and priorities + Model-independent low-momentum interaction V low k + Cutoff independence as a tool, not fit parameter + Three-body forces required by renormalization, perturbative low-momentum 3N interaction * Cutoff dependence and convergence properties of NCSM and CCM results obtained from V low k + 3NF, isospin dependence of 3N force sufficient? 10 B? * Perturbative nuclear matter? Bogner, Furnstahl, AS, in prep. * Derivation of V low k from phase shifts + pion exchange Bogner, Birse, AS, in prep. * Calculations of SM effective interactions from V low k + 3N force in regions where two-body G matrix fails