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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

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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/ )Selected applications: perturbative 4 He, 16 O and 40 Ca, perturbative nuclear matter? Coraggio et al., PR C68 (2003) and nucl-th/ Bogner, Furnstahl, AS, in prep. pairing in neutron matter AS, Friman, Brown, NP A713 (2003) 191, AS, Friman, PRL 92 (2004) )Summary and priorities Outline

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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,…

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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,…

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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…

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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

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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

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RG evolution also very useful for χEFT interactions - choose cutoff range in χEFT to include maximum known long-distance physics Λ χ ~ 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)

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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/ [Hermitize V low k or symmetrized RG equation (very small changes)]

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

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

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

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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

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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, INT from T. Papenbrock, MANSC 2004 NCSM coupled cluster 4 He

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

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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…

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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

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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/

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

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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 /

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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) 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)

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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

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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

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5) Selected applications Perturbative calculations for 4 He, 16 O and 40 C Coraggio et al., PR C68 (2003) nucl-th/ 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 )

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

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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) nuclear matter from low-mom. NN + 3N int. Bogner, Furnstahl, AS, in prep.

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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

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