From nuclear forces to neutron stars Achim Schwenk Indiana University Collaborators: S.K. Bogner, G.E. Brown, B. Friman, R.J. Furnstahl, T.T.S. Kuo, A. Nogga and A.P. Zuker Supported by DOE and NSF Prologue Supernova explosions fascinated mankind over a millenium Chaco Canyon, NM QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Supernova July 4, 1054 Understanding stellar collapse involves understanding nuclear physics Supernova SN1987a QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Outline 1) Introduction and motivation 2) Unifying nuclear forces 3) From light nuclei to nuclear matter 4) Superfluidity in neutron stars and neutron star cooling 5) Summary 1) Introduction and motivation Density Functional Theory A>100 Shell model, coupled cluster method A<100 Exact methods A≤12 GFMC, NCSM Effective (or soft) NN interactions Lattice QCD Nucleon-nucleon interaction (Effective theory of QCD) QCD Lagrangian adopted from Richter @ INPC2004 Crab pulsar Supernova July 4, 1054 nuclear structure: forces in nuclei, from 6He to neutron-rich nuclei QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. many-body aspects overlap with various fields nuclear structure: astrophysics: forces in nuclei, from 6He to neutron-rich nuclei stellar collapse, neutron stars QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. many-body aspects overlap with various fields nuclear structure: astrophysics: forces in nuclei, from 6He to neutron-rich nuclei stellar collapse, neutron stars QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. many-body aspects overlap with various fields cold Fermi gases: from Cooper pairs to condensed molecules, universal properties (from low-density neutron matter to 6Li and 40K) O’Hara et al., Science 298 (2002) 2179. nuclear structure: astrophysics: forces in nuclei, from 6He to neutron-rich nuclei stellar collapse, neutron stars QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. many-body aspects overlap with various fields QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. cold Fermi gases: high-density QCD: quark matter phases for realistic densities with strange quarks from Cooper pairs to condensed molecules, universal properties (from low-density neutron matter to 6Li and 40K) O’Hara et al., Science 298 (2002) 2179. Comparing nuclear to atomic forces Neutrons and protons are composite objects with interactions similar to atom-atom interactions. Differences: nuclear forces have strong spin dependences (spin-orbit and tensor forces) Comparing nuclear to atomic forces Neutrons and protons are composite objects with interactions similar to atom-atom interactions. Differences: nuclear forces have strong spin dependences (spin-orbit and tensor forces) For scattering: convenient to work in momentum space k k’ NN interactions well-constrained by NN scattering Many different NN potentials fit to data below k k CD Bonn, Machleidt PR C63 (2001) 024001 NN interactions well-constrained by NN scattering Many different NN potentials fit to data below k k Details not resolved for momenta larger than or distances CD Bonn, Machleidt PR C63 (2001) 024001 Models of nuclear forces and Effective Field Theory Argonne v18 = CD Bonn = Systematic EFT = + (1 exchange)2 + Woods-Saxon cores + one-boson exchange phenomenology (, , ,… ) Models of nuclear forces and Effective Field Theory Argonne v18 = CD Bonn = Systematic EFT = + (1 exchange)2 + Woods-Saxon cores + one-boson exchange phenomenology (, , ,… ) Comparing nuclear forces to the nucleon radius Compare scale to proton charge radius (hadronic r≈0.5 fm) Comparing nuclear forces to the nucleon radius Compare scale to proton charge radius (hadronic r≈0.5 fm) neutron Overlap of tails at average NN distance in nuclear matter (r≈1.8 fm) Large density overlap at attraction minimum or closer, quarks resolved Three-nucleon interactions when third particle present Evidence for NNN=3N interactions All exact calculations with only NN forces miss A=3,4 nucleon binding energies Exact GFMC results with only NN force miss A≤10 spectra Nogga et al., PRL 85 (2000) 944. Pieper, Wiringa, Ann. Rev. Nucl. Part. Sci. 51 (2001) 53. Evidence for NNN=3N interactions All exact calculations with only NN forces miss A=3,4 nucleon binding energies Exact GFMC results with only NN force miss A≤10 spectra Nogga et al., PRL 85 (2000) 944. Pieper, Wiringa, Ann. Rev. Nucl. Part. Sci. 51 (2001) 53. Evidence for NNN=3N interactions All exact calculations with only NN forces miss A=3,4 nucleon binding energies Exact GFMC results with only NN force miss A≤10 spectra Nogga et al., PRL 85 (2000) 944. Pieper, Wiringa, Ann. Rev. Nucl. Part. Sci. 51 (2001) 53. Wish-list Is it possible to reduce the model dependence of nuclear forces? Cores strongly constrain the use of nuclear forces, with potential in harmonic oscillator basis: <0|V|n> Are nuclear interactions without cores possible? Connection to nuclei complicated How are 3N forces tractable? Can we estimate the error due to an imperfect description of 3N, 4N,… scattering? Reliable extrapolations to extremes: neutron/proton-rich nuclei, astrophysical densities and temperatures,… 2) Unifying nuclear forces Differences in NN interactions due to model assumptions at high momenta (short distances) n scattering amplitude = p = long-range exchange + + Low-momentum strength of VNN depends on how many intermediate states included Low-energy universality: Long-wavelength description should be independent of high-momentum details +… +… Integrating out high-momentum modes In momentum space, scattering amplitude (T matrix) given by k in matrix form VNN(k’,k) Integrating out high-momentum modes leads to effective interaction Vlow k (which reproduces the low-momentum T matrix) k’ Changes of effective interaction with cutoff are given by Renormalization Group (RG) eqn. Vlow k k’ k Starting from any NN interaction: Solutions to the RG eqn. evolve to a “universal” interaction Vlow k for cutoffs below 1S 0 Vlow k Bogner, Kuo, AS, Phys. Rep. 386 (2003) 1. Starting from any NN interaction: Solutions to the RG eqn. evolve to a “universal” interaction Vlow k for cutoffs below 1S 0 Vlow k Bogner, Kuo, AS, Phys. Rep. 386 (2003) 1. Main effect: constant shift from VNN (removes model-dependent cores) Collapse in all partial waves due to same long-distance physics and phase shift equivalence small differences related to spread in phase shift fit (Idaho misses 3F2) Shell-model matrix elements 1S 0 Low-momentum interactions are tractable in a shell-model basis <0|V|n> 3S 1 Vlow k similar to successful G matrix pseudopotential (without drawbacks of G) Plot of all two-body Vlow k vs. G matrix elements in MeV AS, Zuker, nucl-th/0501038. Simple phenomenology obtained by filling lowest oscillator orbits AS, Zuker, nucl-th/0501038. spin-orbit splittings 41Ca 17O exp.: 0f5/2-0f7/2 ≈ 6.0 MeV exp.: 0d3/2-0d5/2 ≈ 5.1 MeV spectra in Ca region Low-lying levels depend weakly on cutoff 3) From light nuclei to nuclear matter By construction: For Vlow k all NN observables are cutoffindependent and reproduced with 2 ≈ 1 Nucleons are composite objects: All NN potentials have corresponding 3N, 4N,… forces If one omits the many-body forces, calculations of lowenergy 3N, 4N,… observables will be cutoff-dependent By varying the cutoff, we can obtain an estimate for the effects of omitted 3N, 4N,… forces Nogga, Bogner, AS, PR C70 (2004) 061002(R). Cutoff dependence due to omitted 3N forces A=3,4 binding energies Exact Faddeev Vlow k only Cutoff dependence shows that three-body forces are inevitable Cutoff dependence is almost linear (explains Tjon line), with 3N contributions larger for core potentials Low-momentum 3N forces with only pion exchanges organized by chiral Effective Field Theory, leading-order 3N force van Kolck, PR C49 (1994) 2932; Epelbaum et al., PR C66 (2002) 064001. operator structure of any 3N force collapses at low energies to long (2π) c-terms intermediate (π) short-range D-term E-term Motivation of adjusting chiral 3N force to Vlow k: Low-energy universality indicates Vlow k effectively parameterizes a high-order chiral EFT interaction Two couplings fit to 3H and 4He Linear dependences in fits, consistent with perturbative 3N contributions 3N forces become perturbative for cutoffs nonperturbative at larger cutoffs, with expectation values beyond expected (Q/Λ)3 ~ (mπ/Λ)3 ≈ 0.1 of NN contribution NN scattering in free-space vs. nuclear matter Two sources of nonperturbative behavior: 1. Iterated interactions at low momenta are nonperturbative due to near bound states 2. Cores scatter strongly to high-momentum states (overwhelms 1.) For interactions with cores: second-order contributions larger than VNN n p NN scattering in free-space vs. nuclear matter Two sources of nonperturbative behavior: 1. Iterated interactions at low momenta are nonperturbative due to near bound states 2. Cores scatter strongly to high-momentum states (overwhelms 1.) For interactions with cores: second-order contributions larger than VNN n p P1>kF even with Pauli blocking P2>kF NN scattering in free-space vs. nuclear matter Two sources of nonperturbative behavior: 1. Iterated interactions at low momenta are nonperturbative due to near bound states 2. Cores scatter strongly to high-momentum states (overwhelms 1.) For low-momentum interactions: free-space nonperturbative due to near bound states n p NN scattering in free-space vs. nuclear matter Two sources of nonperturbative behavior: 1. Iterated interactions at low momenta are nonperturbative due to near bound states 2. Cores scatter strongly to high-momentum states (overwhelms 1.) For low-momentum interactions: free-space nonperturbative due to near bound states n p P1>kF small in matter due to occupied Fermi sea P2>kF NN scattering in free-space vs. nuclear matter Two sources of nonperturbative behavior: 1. Iterated interactions at low momenta are nonperturbative due to near bound states 2. Cores scatter strongly to high-momentum states (overwhelms 1.) For low-momentum interactions: leads to remarkable Hartree-Fock neutron matter equation of state P1>kF P2>kF n p Vision for nuclear forces Lattice QCD determination of low-energy parameters Chiral EFT interactions with consistent many-body forces and operators RG evolution to lower cutoffs for many-body applications 4) Superfluidity in neutron stars BCS theory: any attractive interaction renders Fermi surface unstable to pairing Pairing in nuclei: excitation energies show gap for N,Z even nuclei Pairing in neutron stars: some angular momentum carried by superfluid vortices trapped to neutron star crust vortices unpin suddenly, leading to glitches Δ ~ 1 MeV, Tc ~ 1010 K change of Crab pulsar period Neutron star interior From fastest pulsar: R < 75 km, > 0 /2 Typical: R ~ 10 km, M ~ 1.4 Msun Atmosphere (~10 cm) Envelope (~10 m blanket) from Horowitz, IU Crust (~1 km n-rich pasta superfluid neutrons, e ) Outer Core (superfluid n, supercond. protons, e, ) from Page, UNAM “Inner” Core ( ? ) Benchmark BCS gaps in neutron matter uses free-space NN interaction for pairing and free spectrum low-density: 1S0 pairing higher-density: 3P2 pairing Baldo et al., PR C58 (1998) 1921. phase-shift equivalent NN interactions give identical BCS gaps Benchmark BCS gaps in neutron matter uses free-space NN interaction for pairing and free spectrum low-density: 1S0 pairing higher-density: 3P2 pairing np gaps nn gaps Baldo et al., PR C58 (1998) 1921. phase-shift equivalent NN interactions give identical BCS gaps resolves charge-dependence of NN force: Beyond BCS theory: induced interactions Simple case: dilute gases (perturbative ) Gorkov et al., JETP 13 (1961) 1018, Heiselberg et al., PRL 85 (2000) 2418. screening/vertex corr. benchmark + induced interactions Induced interactions lead to a significant reduction of the gap [universal reduction for 2 spin states] RG resummation of particle-hole channels analogous to resummation of high-momentum modes in Vlow k Shankar, RMP 66 (1994) 129. Cutoff around the Fermi surface defines an effective theory for low-momentum particles and holes RG resummation of particle-hole channels analogous to resummation of high-momentum modes in Vlow k Shankar, RMP 66 (1994) 129. Cutoff around the Fermi surface defines an effective theory for low-momentum particles and holes Changes of the effective interaction with cutoff are included through RG eqn. Condensed matter results from the RG approach Zanchi, Schulz, PR B61 (2000) 13609, Halboth, Metzner, PRL 85 (2000) 5162, Salmhofer, Honerkamp, PTP 105 (2001) 1, Binz et al., Ann. Phys. 12 (2003) 704. 2d Hubbard model: U t Eliminate modes according to phase diagram Leading instabilities: spin-density wave, d-wave SC t‘ Nuclear physics results With induced interactions: max. S-wave gap ∆ ≈ 0.8 MeV AS, Friman, Brown, NPA 713 (2003) 191. previous calculations including induced effects miss BCS gaps Nuclear physics results With induced interactions: max. S-wave gap ∆ ≈ 0.8 MeV AS, Friman, Brown, NPA 713 (2003) 191. For P-wave pairing: spin-orbit, tensor forces crucial first perturbative assessment of induced interactions AS, Friman, PRL 92 (2004) 082501. previous calculations including induced effects miss BCS gaps Note: 2nd order / Vlow k < 50% Nuclear physics results With induced interactions: max. S-wave gap ∆ ≈ 0.8 MeV AS, Friman, Brown, NPA 713 (2003) 191. For P-wave pairing: spin-orbit, tensor forces crucial first perturbative assessment of induced interactions AS, Friman, PRL 92 (2004) 082501. previous calculations including induced effects miss BCS gaps P-wave gaps < 10 keV possible Impact on neutron star cooling 951 years later from http://chandra.harvard.edu Neutron star structure Atmosphere (~10 cm) emission of photons, surface temperature from spectral fits Envelope (~10 m) controls heat leak from interior, temperature gradient Tinterior = 108 K ~ Tsurface = 106 K from Horowitz, IU Crust (~1 km n-rich pasta superfluid neutrons, e ) relevant for early cooling Outer Core (superfluid n, from Page, UNAM supercond. protons, e, ) Cooling of neutron stars for 105 yrs dominated by neutrino emission from interior nucleons In superfluid matter: neutrino emission suppressed due to gap, except for enhancement just below superfluid Tc surface temperatures correspond to typical Tint ~ 108 K ~ 10 keV enhancement from S-wave pairing early Isolated neutron star data from Yakovlev, Pethick, Ann. Rev. Astron. Astrophys. 42 (2004) 169. Constraints on P-wave gaps from neutron star cooling With P-wave gaps ∆ > 30 keV (e.g., free-space BCS gaps): Enhancement at intermediate stages and neutron stars cool too fast ∆ ~ 50 keV ∆ ~ 120 keV Yakovlev, Pethick, ARAA 42 (2004) 169. ∆ ~ 100 keV Blaschke et al., A&A 424 (2004) 979; Grigorian et al., astro-ph/0501678. Consistency with data requires P-wave gaps ∆ < 30 keV Possibility: core neutrons only superfluid after 105 yrs Implications of weak P-wave pairing minimal neutron star cooling Page et al., astro-ph/0403657; Page, astro-ph/0405196. most sensitive to P-wave gap, less severe dependence to size of gap Magnetized H atmosphere fits: 1) RX J0822-4247 (in Puppis A) 2) 1E 1207.4-5209 (in PKS 1209-52) 3) PSR 0538+2817 4) RX J0002+6246 (in CTB 1) 5) PSR 1706-44 6) PSR 0933-45 (in Vela) Blackbody atmosphere fits: 7) PSR 1055-52 P-wave gap ∆ < 10 keV 8) PSR 0656+14 9) PSR 0633+1748 "Geminga" 10) RX J1856.5-3754 11) RX J0720.4-3125 Upper limits: a b c & d: A) CXO J232327.8+584842 (in Cas A) An X-Ray Search for Compact Central Sources in Supernova Remnants Kaplan et al, ApJS 153, 269 (2004) B) PSR J0205+6449 (in 3C58) C) PSR J1124-5916 (in G292.0+1.8) from Page @ MINS 2005. D) RX J0007.0+7302 (in CTA 1) Implications of weak P-wave pairing minimal neutron star cooling Page et al., astro-ph/0403657; Page, astro-ph/0405196. most sensitive to P-wave gap, less severe dependence to size of gap Magnetized H atmosphere fits: 1) RX J0822-4247 (in Puppis A) 2) 1E 1207.4-5209 (in PKS 1209-52) 3) PSR 0538+2817 4) RX J0002+6246 (in CTB 1) 5) PSR 1706-44 6) PSR 0933-45 (in Vela) Blackbody atmosphere fits: 7) PSR 1055-52 P-wave gap ∆ < 10 keV 8) PSR 0656+14 9) PSR 0633+1748 "Geminga" 10) RX J1856.5-3754 11) RX J0720.4-3125 Upper limits: a b c & d: A) CXO J232327.8+584842 (in Cas A) An X-Ray Search for Compact Central Sources in Supernova Remnants Kaplan et al, ApJS 153, 269 (2004) B) PSR J0205+6449 (in 3C58) C) PSR J1124-5916 (in G292.0+1.8) Candidates for fast cooling, exotic cores D) RX J0007.0+7302 (in CTA 1) 5) Summary 1. RG removes model dependences from nuclear forces, results in “universal” low-momentum interaction 2. Low-momentum 3N forces are perturbative in this regime 3. Promising spin-orbit splittings, low-lying spectra and neutron matter nuclear matter: Bogner, AS, Furnstahl, Nogga, in prep. 4. Cutoff variation estimates theoretical errors needed for RIA 5. RG powerful for many-body applications: 2d Hubbard model, electronic excitations in atoms, pairing in neutron stars,… 6. Weak P-wave pairing in neutron star cores likely P-wave gaps important to identify rapid cooling neutron stars 7. Selected outlook: Neutrino interactions in supernovae (induced spin-orbit and tensor effects on neutrino rates?) Comparing neutron matter to cold atoms