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Low-momentum interactions for nuclei

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. Introduction and motivation

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Low-momentum interactions for nuclei

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

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

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

  4. 1) Introduction and motivation Nuclear forces and the quantum many-body problem Choice of nuclearforce 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. 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 basisChange of resolution scale corresponds to changing thecutoff in nuclear forces. [This freedom is lost, if one uses thecutoff 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. 2) Low-momentum nucleon-nucleon interaction Many different NN interactions, fit to scattering data below (with χ2/dof ≈ 1) Details not resolved for relative momenta larger than or for distances Strong high-momentum components, model dependence S-wave < k | V | k > S-wave Argonne

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

  8. RG evolution also very useful for χEFT interactions - choose cutoff range in χEFT to include maximum known long-distance physics Λχ ~ 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 Vlow 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 byiteration, 1+2 = Lee-Suzuki transformation) [small changes, Q(ω=0) good for Elab below ~150 MeV] Both steps equivalent to RG equation Bogner et al., nucl-th/0111042. [Hermitize Vlow k or symmetrized RG equation (very small changes)]

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

  11. Collapse of off-shell matrix elements as well N2LON3LO ……

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

  13. Vlow k is much softer, without strong core at short-distance, see relative HO matrix elements < 0 | Vlow k | n > convergence will not require basis states up to ~ 50 shells

  14. Poor convergencein SM calculations (due to high-mom.components ~ 1 GeVin NN potentials) Unsatisfactorystarting-energy dependence(due to loss of BH self-consistency in2-body subsystem) Vlow k will lead to improvements NCSM from P. Navratil, talk @ INT-03-2003 coupled cluster 4He from T. Papenbrock, talk @ MANSC 2004

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

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

  17. 3) Cutoff dependence as tool to assess missing forcesPerturbative low-momentum 3N interactions All low-energyNN observables unchanged andcutoff-indep. Nijmegen partialwave analysis▲ CD Bonn○ Vlow k

  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. Cutoff dependence due to missing three-body forcesalong Tjon line Nogga, Bogner, AS, nucl-th/0405016. ▪ NN+3N forces Faddeev, Vlow k only + NN potentials Results for reasonable cutoffs seem closer to experimentRenormalization: three-body forces inevitable!

  20. 3H3Hebinding energies “bare” exp Faddeev, Vlow k only Cutoff dep. of low-energy 3N observables due to missingthree-body forces (see π EFT)A=3 details /

  21. Adjust low-momentum three-nucleon interaction to removecutoff dependence of A=3,4 binding energies Use leading-order effective field theory 3N force given byvan 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 Vlow k and χ EFT similar. long (2π) intermediate (π) short-range chiral counting: (Q/Λ)3 ~ (mπ/Λ)3rel. to 2N force due to Δ or high-mom. Nintermediate states(both effects inseparable at low energies) c-terms D-term E-termfixed by πN scattering(or NN PWA)

  22. Constraint on D- and E-termcouplings from fit to 3Hlinear dependence consistent withperturbativeE(3H) = E(Vlow k+2π 3NF) + cD < D-term > + cE < E-term > Second constraint from fit to 4He [E-term fixed by left Fig.]η=1: 4He fitted exactly η=1.01: deviation from exp. ≈ 600 keVnon-linearities at larger cutoffs

  23. 2 couplings for D- and E-terms fitted to 3H and 4HeWe find all 3N parts perturbative for cutoffs - <k2> ≈ (A-1) m <T> ≈ mπ2 < Λ2- 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 4He non-linear (a,b) and approximate solution (*) 20% is beyond(Q/Λ)3 ~ (mπ/Λ)3 <

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

  25. Pairing in neutron matterAS, Friman, Brown, NP A713 (2003) 191, AS, Friman, PRL 92 (2004) 082501. suppression due to polarization effects (spin-/LS fluct.) 1S0 pairing 3P2 pairing Vlow k Vlow k

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

  27. 5) Summary and priorities + Model-independent low-momentum interaction Vlow 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 Vlow k + 3NF, isospin dependence of 3N force sufficient? 10B? * Perturbative nuclear matter? Bogner, Furnstahl, AS, in prep. * Derivation of Vlow k from phase shifts + pion exchange Bogner, Birse, AS, in prep. * Calculations of SM effective interactions from Vlow k + 3N force in regions where two-body G matrix fails

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