Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group Z dr 0 r 0 2 V λ ( r, r 0 ) V λ ( r ) = 200 AV18 V(r) [MeV] 3 LO N 100 − 1 − 1 − 1 − 1 − 1 λ = 20 fm λ = 4 fm λ = 3 fm λ = 2 fm λ = 1.5 fm 0 − 100 0 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 r [fm] r [fm] r [fm] r [fm] r [fm]
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • elimination of coupling between low- and high momentum components, simplified many-body calculations • observables unaffected by resolution change (for exact calculations) • residual resolution dependences can be used as tool to test calculations Not the full story: RG transformation also changes three-body (and higher-body) interactions.
Ground state energies of nuclei based on consistently evolved 3NF interactions NN only NN 3N-induced NN 3N-full NN (N 3 LO) (a) (b) (c) -23 4 He -24 + 3NF (N 2 LO, 500 MeV) 20 MeV E gs [MeV] -25 oxygen isotopes -26 -27 -28 exp. -29 . -130 (d) (e) (f) -22 obtained in large many-body spaces 6 Li -24 20 MeV E gs [MeV] -26 -140 Energy (MeV) -28 -30 -150 -32 exp. NN only NN 3N-induced NN 3N-full . -34 (a) (b) (c) -60 2 4 6 8 10 12 14 2 4 6 8 10 12 14 2 4 6 8 10 12 14 -160 N max N max N max MR-IM-SRG 12 C -70 20 MeV E gs [MeV] IT-NCSM -80 -170 SCGF -90 Lattice EFT exp. @ -100 CC AME 2012 -180 . (d) (e) (f) -100 16 O 16 18 20 22 24 26 28 20 MeV E gs [MeV] -120 Mass Number A exp. -140 -160 -180 . 2 4 6 8 10 12 14 2 4 6 8 10 12 2 4 6 8 10 12 N max N max N max Roth, Langhammer, Calci, Binder, Navratil, PRL 107, 072501 (2011) • very promising results for light nuclei, issues for heavier nuclei • remarkable agreement of different MB calculations for a given Hamiltonian • calculations are based on NN (N 3 LO) and 3NF (N 2 LO) forces • need to quantify theoretical uncertainties
Calculations and measurements of neutron-rich nuclei 18 a 2.0 [ S 2n (theo) – S 2n (exp)] (MeV) 16 22 S 2n (MeV) 1.0 14 0.0 12 18 AME2003 –1.0 TITAN 10 NN+3N (MBPT) S 2n (MeV) –2.0 NN+3N (emp) 14 30 31 32 33 34 8 Neutron number, N 3 28 29 30 31 32 10 TITAN+ (3) (MeV) 2 AME2003 Experiment ISOLTRAP 6 ∆ n NN+3N (MBPT) 1 CC (ref. 5) KB3G GXPF1A 0 2 28 29 30 31 32 28 30 32 34 36 38 Neutron Number N Neutron number, N Gallant et al. Wienholtz et al. PRL 109, 032506 (2012) Nature 498, 346 (2013) • high precision mass measurements at TITAN showed that 52 Ca is 1.74 MeV more bound compared to atomic mass evaluation • neutron separation energies agree well with MBPT calculations based on NN+3NF chiral interactions • need to quantify theoretical uncertainties
Ground state energies of medium-mass and heavy nuclei -6 NN + 3N-induced (a) -7 -8 exp -9 � N 3 LO -10 � N 2 LO opt E / A [MeV] 0.5 (b) -0.5 NN + 3N-full (c) -7 -8 exp . -9 � Λ 3 N = 400 MeV / c -10 � Λ 3 N = 350 MeV / c 0.5 (d) -0.5 16 O 36 Ca 48 Ca 54 Ca 56 Ni 62 Ni 68 Ni 88 Sr 100 Sn 108 Sn 116 Sn 120 Sn 24 O 40 Ca 48 Ni 78 Ni 114 Sn 118 Sn 132 Sn 52 Ca 60 Ni 66 Ni 90 Zr 106 Sn Binder, Calci, Langhammer, Roth Phys. Lett B736, 119 (2014) • significant overbinding of heavy nuclei • need to quantify and reduce theoretical uncertainties
Ground state energies of medium-mass and heavy nuclei -6 NN + 3N-induced (a) -7 -8 exp -9 � N 3 LO -10 � N 2 LO opt • EFT power counting? E / A [MeV] 0.5 (b) -0.5 • missing NN and/or many-body contributions? NN + 3N-full (c) -7 • optimized fitting procedures? -8 exp . -9 � Λ 3 N = 400 MeV / c -10 � Λ 3 N = 350 MeV / c 0.5 (d) -0.5 16 O 36 Ca 48 Ca 54 Ca 56 Ni 62 Ni 68 Ni 88 Sr 100 Sn 108 Sn 116 Sn 120 Sn 24 O 40 Ca 48 Ni 78 Ni 114 Sn 118 Sn 132 Sn 52 Ca 60 Ni 66 Ni 90 Zr 106 Sn Binder, Calci, Langhammer, Roth Phys. Lett B736, 119 (2014) • significant overbinding of heavy nuclei • need to quantify and reduce theoretical uncertainties
Equation of state: Many-body perturbation theory central quantity of interest: energy per particle E/N H ( λ ) = T + V NN ( λ ) + V 3N ( λ ) + ... kinetic energy E = Hartree-Fock + + V NN V 3N V NN V NN V 3N V 3N V 3N 2nd-order + + + + + V 3N V NN V 3N V NN V 3N 3rd-order and beyond + . . . • “hard” interactions require non-perturbative summation of diagrams • with low-momentum interactions much more perturbative • inclusion of 3N interaction contributions crucial!
Equation of state of symmetric nuclear matter, nuclear saturation ¯ l S � “Very soft potentials must be excluded because they do not give saturation; they give too much binding and too high density. In particular, a substantial tensor force is required.” Hans Bethe (1971)
Fitting the 3NF LECs at low resolution scales 200 0 AV18 3 LO (500 MeV) V low k NN from N V(r) [MeV] 3 LO N 100 − 1 3NF fit to E 3H and r 4He Λ 3NF = 2.0 fm − 1 − 1 − 5 λ = 20 fm λ = 1.5 fm Energy/nucleon [MeV] 0 − 10 − 100 0 1 2 3 4 4 1 2 3 4 r [fm] r [fm] − 15 ¯ l S 3rd order pp+hh − 20 NN only − 1 NN only Λ = 1.8 fm − 1 NN only Λ = 2.8 fm − 25 − 30 0.8 1.0 1.2 1.4 1.6 − 1 ] KH, Bogner, Furnstahl, Nogga, k F [fm � PRC(R) 83, 031301 (2011) “Very soft potentials must be intermediate (c D ) and short-range excluded because they do not (c E ) 3NF couplings fitted to few-body give saturation; systems at different resolution scales: they give too much binding and too high density. In particular, a r 4 He = 1 . 464 fm E 3 H = − 8 . 482 MeV substantial tensor force is required.” Hans Bethe (1971) c E term c D term c 1 , c 3 , c 4 terms
Fitting the 3NF LECs at low resolution scales 200 0 AV18 3 LO (500 MeV) V low k NN from N V(r) [MeV] 3 LO N 100 − 1 3NF fit to E 3H and r 4He Λ 3NF = 2.0 fm − 1 − 1 − 5 λ = 20 fm λ = 1.5 fm Energy/nucleon [MeV] 0 NN + 3N − 10 − 100 0 1 2 3 4 4 1 2 3 4 r [fm] r [fm] − 15 ¯ l S 3rd order pp+hh − 20 NN only − 1 Λ = 1.8 fm − 1 Λ = 2.8 fm − 1 NN only − 25 Λ = 1.8 fm − 1 NN only Λ = 2.8 fm − 30 0.8 1.0 1.2 1.4 1.6 − 1 ] KH, Bogner, Furnstahl, Nogga, k F [fm � PRC(R) 83, 031301 (2011) “Very soft potentials must be excluded because they do not give saturation; Reproduction of saturation point they give too much binding and too high density. In particular, a without readjusting parameters! substantial tensor force is required.” Hans Bethe (1971)
Fitting the 3NF LECs at low resolution scales 200 0 AV18 3 LO (500 MeV) V low k NN from N V(r) [MeV] 3 LO N 100 − 1 3NF fit to E 3H and r 4He Λ 3NF = 2.0 fm − 1 − 1 − 5 λ = 20 fm λ = 1.5 fm Energy/nucleon [MeV] 0 NN + 3N − 10 − 100 0 1 2 3 4 4 1 2 3 4 r [fm] r [fm] − 15 ¯ l S 3rd order pp+hh − 20 NN only − 1 Λ = 1.8 fm − 1 Λ = 2.8 fm − 1 NN only − 25 Λ = 1.8 fm − 1 NN only Λ = 2.8 fm − 30 0.8 1.0 1.2 1.4 1.6 − 1 ] KH, Bogner, Furnstahl, Nogga, k F [fm � PRC(R) 83, 031301 (2011) “Very soft potentials must be excluded because they do not give saturation; they give too much binding and too high density. In particular, a substantial tensor force is required.” Drischler, KH, Schwenk, Hans Bethe (1971) in preparation
Results for the neutron matter equation of state 20 (1) neutron matter is a unique E NN+3N,eff E NN+3N,eff Energy/nucleon [MeV] 3N < 2.5 fm -1 system for chiral EFT: 2.0 < 15 only long-range 3NF 10 contribute in leading order = 1.8 fm -1 = 2.0 fm -1 5 = 2.4 fm -1 = 2.8 fm -1 0 0 0. 05 0. 10 0.15 0 0. 05 0. 10 0. 15 c E term c D term c 1 , c 3 , c 4 terms [fm -3 ] [fm -3 ] KH and Schwenk PRC 82, 014314 (2010) pure neutron matter NN only, EM 2.5 E NN+3N,eff +c 3 +c 1 uncertainties 20 NN only, EGM E NN+3N,eff +c 3 uncertainty KH and Schwenk PRC 82, 014314 (2010) Energy/nucleon [MeV] P [10 33 dyne / cm 2 ] 2.0 (1) + E NN (2) E NN 15 1.5 3N 10 1.0 neutron star 0.5 5 matter 0 KH, Lattimer, Pethick, Schwenk, 0 0.2 0.4 0.6 0.8 1.0 0 PRL 105, 161102 (2010) 0 0.05 0.10 0.15 [ 0 ] [fm -3 ]
First application to isospin asymmetric nuclear matter • uncertainty bands determined by set of 7 Hamitonians n p Drischler, KH, Schwenk, x = in preparation n p + n n
First complete calculations of neutron matter at N 3 LO 2 LO 3N EM 500 MeV + N 20 3 LO 3N + 4N EM 500 MeV + N 2 LO 3N EM 500 MeV, RG evolved + N SCGF (Carbone et al.) CC (Hagen et al.) 15 MBPT (Corragio et al.) E/N [MeV] 10 5 0 0 0.05 0.1 0.15 Tews, Krueger, KH, Schwenk, KH, Holt, Menendez, Schwenk, n [fm -3 ] PRL 110, 032504 (2013) in press • bands include uncertainties from many-body calculations and NN, 3NF and 4NF • good agreement with other methods • significant contributions from 3NF at N3LO
Symmetry energy and neutron skin constraints 0.3 !"#$ 0.2 S(208Pb) (fm) "%&! 0.1 0.0 -50 0 50 100 150 200 slope of neutron EOS Brown, Piekarewicz, PRL 85, 5296 (2000) 1303.4662 PRC 85, 041302 (2012) KH, Lattimer, Pethick, Schwenk, ApJ 773,11 (2013) neutron skin constraint from neutron matter results: S v = ∂ 2 E/N � � � ∂ 2 x r skin [ 208 Pb] = 0 . 14 − 0 . 2 fm � ρ = ρ 0 ,x =1 / 2 ∂ 3 E/N � L = 3 � KH, Lattimer, Pethick, Schwenk, PRL 105, 161102 (2010) � ∂ρ∂ 2 x 8 � ρ = ρ 0 ,x =1 / 2 • neutron matter give tightest constraints • in agreement with all other constraints
Symmetry energy and neutron skin constraints ab intio coupled cluster calculations of neutron skin and dipole polarizability of 48 Ca 3.5 R p (fmD 3.4 3.3 3.2 A B C Hagen et al., 0.15 0.18 0.21 3.4 3.5 3.6 2.0 2.4 2.8 α D (fm n D R skin (fmD R n (fmD
Constraints on the nuclear equation of state (EOS) A Massive Pulsar in a A two-solar-mass neutron star measured using Compact Relativistic Binary Shapiro delay P. B. Demorest 1 , T. Pennucci 2 , S. M. Ransom 1 , M. S. E. Roberts 3 & J. W. T. Hessels 4,5 30 a 20 Timing residual ( μ s) 10 0 –10 –20 –30 –40 –40 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 b Orbital phase (turns) Demorest et al., Nature 467, 1081 (2010) Antoniadis et al., Science 340, 448 (2013) New constraints from recent observations: M max = 1 . 65 M � → 1 . 97 ± 0 . 04 M � → 2 . 01 ± 0 . 04 M � Calculation of neutron star properties require EOS up to high densities. Strategy: Use observations to constrain the high-density part of the nuclear EOS.
Neutron star radius constraints incorporation of beta-equilibrium: neutron matter neutron star matter parametrize piecewise high-density extensions of EOS: • use polytropic ansatz p ∼ ρ Γ • range of parameters Γ 1 , ρ 12 , Γ 2 , ρ 23 , Γ 3 limited by physics 37 crust EOS (BPS) 3 neutron star matter 36 with c i uncertainties 2 log 10 P [dyne / cm 2 ] 35 crust 1 34 33 32 31 13.0 13.5 14.0 1 12 23 [g / cm 3 ] log 10 KH, Lattimer, Pethick, Schwenk, ApJ 773, 11 (2013) KH, Lattimer, Pethick, Schwenk, PRL 105, 161102 (2010)
Constraints on the nuclear equation of state use the constraints: recent NS observations M max > 1 . 97 M � causality � v s ( ρ ) = dP/d ε < c KH, Lattimer, Pethick, Schwenk, ApJ 773,11 (2013) constraints lead to significant reduction of EOS uncertainty band
Constraints on the nuclear equation of state use the constraints: fictitious NS mass M max > 2 . 4 M � causality � v s ( ρ ) = dP/d ε < c KH, Lattimer, Pethick, Schwenk, ApJ 773,11 (2013) increased systematically reduces width of band M max
Constraints on neutron star radii WFF1 PCL2 3 3 WFF2 SQM1 36 36 WFF3 SQM2 AP4 SQM3 AP3 PS 2.5 2.5 causality MS1 MS3 log 10 P [dyne / cm 2 ] Mass [M sun ] Mass [M sun ] GM3 2 2 35 35 ENG PAL GS1 1.5 1.5 GS2 34 34 1 1 0.5 0.5 33 33 0 0 8 8 10 10 12 12 14 14 16 16 14.2 14.2 14.4 14.4 14.6 14.6 14.8 14.8 15.0 15.0 15.2 15.2 15.4 15.4 [g / cm 3 ] log 10 Radius [km] Radius [km] KH, Lattimer, Pethick, Schwenk, ApJ 773, 11 (2013) see also KH, Lattimer, Pethick, Schwenk, PRL 105, 161102 (2010) • low-density part of EOS sets scale for allowed high-density extensions • current radius prediction for typical neutron star: 1 . 4 M � 9 . 7 − 13 . 9 km
Constraints on neutron star radii WFF1 PCL2 Large 3 3 WFF2 SQM1 observatory 36 36 WFF3 SQM2 for X-ray AP4 SQM3 AP3 PS 2.5 2.5 timing causality MS1 MS3 log 10 P [dyne / cm 2 ] Mass [M sun ] Mass [M sun ] GM3 2 2 35 35 ENG PAL GS1 1.5 1.5 GS2 34 34 1 1 0.5 0.5 33 33 0 0 8 8 10 10 12 12 14 14 16 16 14.2 14.2 14.4 14.4 14.6 14.6 14.8 14.8 15.0 15.0 15.2 15.2 15.4 15.4 [g / cm 3 ] log 10 Radius [km] Radius [km] KH, Lattimer, Pethick, Schwenk, ApJ 773, 11 (2013) see also KH, Lattimer, Pethick, Schwenk, PRL 105, 161102 (2010) • low-density part of EOS sets scale for allowed high-density extensions • current radius prediction for typical neutron star: 1 . 4 M � 9 . 7 − 13 . 9 km • new observatories could significantly improve constraints
Representative set of EOS 3 1000 2.5 causality Mass [M sun ] P [MeV / fm 3 ] 100 2 1.5 10 1 0.5 1 0 8 10 12 14 16 100 1000 [MeV / fm 3 ] Radius [km] KH, Lattimer, Pethick, Schwenk, ApJ 773, 11 (2013) • constructed 3 representative EOS compatible with uncertainty bands for astrophysical applications: soft, intermediate and stiff • allows to probe impact of current theoretical EOS uncertainties on astrophysical observables
Gravitational wave signals from neutron star binary mergers 3.6 3.4 3.2 3 f peak [kHz] 2.8 − 22 2.6 x 10 h + at 50 Mpc 2 2.4 − 21 0 f peak − 2 log(h + (f)f 1/2 /Hz 1/2 ) 2.2 − 22 0 5 10 15 20 t [ms] 2 − 23 eosUU 1.8 − 24 10 11 12 13 14 15 16 R 1.6 [km] Shen − 25 Bauswein and Janka, PRL 108, 011101 (2012), 0 1 2 3 4 5 Bauswein, Janka, KH, Schwenk, PRD 86, 063001 f [kHz] • simulations of NS binary mergers show strong correlation between between of the GW spectrum and the radius of a NS f peak • measuring is key step for constraining EOS systematically at large f peak ρ
Future directions, open problems • develop the most advanced chiral Hamiltonians to enable controlled microscopic calculations of matter and light as well as medium-mass nuclei • improve EOS constraints at high densities (LOFT, GW waves, ?), explore limits of chiral EFT interactions • extend EOS calculations to finite temperature • calculate response functions and neutrino interactions in matter • benchmarks between different many-body frameworks based on a set of Hamiltonians • derivation of systematic uncertainty estimates by performing order-by-order calculations in chiral expansion
In collaboration with: C. Drischler, T. Krüger, R. Roth, R. Furnstahl, S. More A. Schwenk, I. Tews S. Bogner E. Epelbaum, H. Krebs A. Gezerlis, A. Nogga J. Lattimer C. Pethick J. Golak, R. Skibinski international collaborator in computing support: JUROPA
Thank you!
backup slides
Contributions of many-body forces at N 3 LO in neutron matter NN 3N 4N • first calculations of N 3 LO 3NF and 4NF contributions to EOS of neutron matter • found large contributions in Hartree Fock appr., comparable to size of N 2 LO contributions Tews, Krüger, KH, Schwenk PRL 110, 032504 (2013) Two-pion-exchange 3N Two-pion − one-pion-exchange 3N Pion-ring 3N Two-pion-exchange − contact 3N 4 4 2 2 E/N [MeV] 0 0 -2 -2 EM 500 MeV -4 EGM 450/700 MeV -4 EGM 450/500 MeV 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0.2 n [fm -3 ] n [fm -3 ] n [fm -3 ] n [fm -3 ]
Contributions of many-body forces at N 3 LO in neutron matter NN 3N 4N • first calculations of N 3 LO 3NF and 4NF contributions to EOS of neutron matter • found large contributions in Hartree Fock appr., comparable to size of N 2 LO contributions • 4NF contributions small Tews, Krüger, KH, Schwenk PRL 110, 032504 (2013) a e f Two-pion-exchange 3N Two-pion − one-pion-exchange 3N Pion-ring 3N Two-pion-exchange − contact 3N Three-pion-exchange 4N V Three-pion-exchange 4N V Pion-pion-interaction 4N V pRelativistic-corrections 3Np 0.4 0.4 EGM 450/500 MeV 4 4 EGM 450/700 MeV 0.3 0.3 EM 500 MeV 0.2 0.2 2 2 E/N [MeV] E/N [MeV] 0.1 0.1 0 0 0 0 -0.1 -0.1 -2 -2 -0.2 -0.2 EM 500 MeV -0.3 -0.3 -4 EGM 450/700 MeV -4 EGM 450/500 MeV -0.4 -0.4 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0.2 n [fm -3 ] n [fm -3 ] n [fm -3 ] n [fm -3 ] n [fm -3 ] n [fm -3 ] n [fm -3 ] n [fm -3 ]
N 3 LO contributions in nuclear matter (Hartree Fock) Two-pion-exchange 3N Two-pion- -one-pion-exchange 3N Pion-ring 3N Two-pion-exchange- -contact 3N 4 4 2 2 0 0 E/N [MeV] -2 -2 -4 -4 -6 -6 EM 500 MeV1 1 EGM 450/700 MeV -8 -8 EGM 450/500 MeV1 1 -10 -10 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15 n [fm -3 ] n [fm -3 ] n [fm -3 ] n [fm -3 ] Three-pion-exchange 4N V a Three-pion-exchange 4N V c Three-pion-exchange 4N V e Relativistic-corrections 3N 0.4 0.4 0.2 0.2 E/N [MeV] 0 0 -0.2 -0.2 EGM 450/500 MeV1 -0.4 -0.4 1 EM 500 MeV EGM 450/700 MeV1 1 -0.6 -0.6 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0.2 Pion-pion-interaction 4N V f n [fm -3 ] Two-pion-exchange-contact 4N V k n [fm -3 ] n [fm -3 ] -contact 4N V l n [fm -3 ] -two-contact 4N V n Two-pion-exchange- Pion-exchange- EM 500 MeV11 EGM 500 MeV11 EM 500 MeV11 0.5 0.5 EGM 450/700 MeV EGM 450/700 MeV EGM 450/700 MeV 0.4 EGM 450/500 MeV1 EGM 450/500 MeV1 EGM 450/500 MeV1 0.4 1 1 1 E/N [MeV] 0.3 0.3 0.2 0.2 0.1 0.1 0 0 -0.1 -0.1 Krüger, Tews, KH, Schwenk -0.2 -0.2 PRC88, 025802 (2013) 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0.2 n [fm -3 ] n [fm -3 ] n [fm -3 ] n [fm -3 ]
N 3 LO contributions in nuclear matter (Hartree Fock) Two-pion-exchange 3N Two-pion- -one-pion-exchange 3N Pion-ring 3N Two-pion-exchange- -contact 3N 4 4 2 2 0 0 E/N [MeV] -2 -2 -4 -4 -6 -6 EM 500 MeV1 1 EGM 450/700 MeV -8 -8 EGM 450/500 MeV1 1 -10 -10 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15 n [fm -3 ] n [fm -3 ] n [fm -3 ] n [fm -3 ] Three-pion-exchange 4N V a Three-pion-exchange 4N V c Three-pion-exchange 4N V e Relativistic-corrections 3N Conclusions/Indications: 0.4 0.4 • N 3 LO 3N contributions significant 0.2 0.2 E/N [MeV] • N 3 LO 4N contributions small 0 0 -0.2 -0.2 EGM 450/500 MeV1 -0.4 -0.4 1 EM 500 MeV EGM 450/700 MeV1 1 -0.6 -0.6 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0.2 Pion-pion-interaction 4N V f n [fm -3 ] Two-pion-exchange-contact 4N V k n [fm -3 ] n [fm -3 ] -contact 4N V l n [fm -3 ] -two-contact 4N V n Two-pion-exchange- Pion-exchange- EM 500 MeV11 EGM 500 MeV11 EM 500 MeV11 0.5 0.5 EGM 450/700 MeV EGM 450/700 MeV EGM 450/700 MeV 0.4 EGM 450/500 MeV1 EGM 450/500 MeV1 EGM 450/500 MeV1 0.4 1 1 1 E/N [MeV] 0.3 0.3 0.2 0.2 0.1 0.1 0 0 -0.1 -0.1 Krüger, Tews, KH, Schwenk -0.2 -0.2 PRC88, 025802 (2013) 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0.2 n [fm -3 ] n [fm -3 ] n [fm -3 ] n [fm -3 ]
Chiral 3N forces at subleading order (N 3 LO) Goal Calculate matrix elements of 3NF in a momentum partial-wave decomposed form, which is suitable for all these few- and many-body frameworks.
Chiral 3N forces at subleading order (N 3 LO) Goal Calculate matrix elements of 3NF in a momentum partial-wave decomposed form, which is suitable for all these few- and many-body frameworks. Challenge Due to the large number of matrix elements, the traditional way of computing matrix elements requires extreme amounts of computer resources.
Chiral 3N forces at subleading order (N 3 LO) V 123 @ fm 4 D V 123 @ fm 4 D V 123 @ fm 4 D Goal o o o o o o o o o o o oo o o o o o oo o o o o oo o o o o o o o o o o o o o o o o o o o o 8 q @ fm - 1 D o o - 0.003 o 8 q @ fm - 1 D o o o o 2 4 6 o o 2 4 6 o - 0.001 o o - 0.0001 - 0.004 Calculate matrix elements of 3NF in a momentum partial-wave decomposed o o o o o - 0.0002 - 0.002 - 0.005 o o o o - 0.0003 o - 0.003 o o - 0.006 form, which is suitable for all these few- and many-body frameworks. o o - 0.0004 o o o o oo - 0.004 o - 0.007 oo o o o - 0.0005 8 q @ fm - 1 D o o o o - 0.005 o o 2 4 6 V 123 @ fm 4 D V 123 @ fm 4 D V 123 @ fm 4 D Challenge o o o o o 0.002 o o o o o o o o o oo o o 0.0010 0.0010 o o o o o o o o o o 0.001 o o oo 0.0005 o o o o o o 0.0005 o Due to the large number of matrix elements, the traditional way of o o 8 q @ fm - 1 D o o o o o o o o o o 8 q @ fm - 1 D o 2 4 6 o 2 4 6 o o o 8 q @ fm - 1 D o o o o o o o - 0.001 o o 2 4 6 - 0.0005 computing matrix elements requires extreme amounts of computer resources. o o o oo o o o o o o o o o o o o o - 0.002 o o o - 0.0005 o o o o - 0.0010 Strategy Development of a general framework, which allows to decompose efficiently arbitrary local 3N interactions. • perfect agreement with results based on traditional approach • speedup factors of >1000 • very general , can also be applied to pion-full EFT, N 4 LO terms, currents...
Incorporation in different many-body frameworks Faddeev, Hyperspherical harmonics Faddeev-Yakubovski Bacca (TRIUMF), Barnea (Hebrew U.) Nogga (Juelich), Witala (Kracow) no-core shell model coupled cluster method Roth (TU Darmstadt), Binder, Hagen, Papenbrock Navratil (TRIUMF), Vary (Iowa) (Oak Ridge) Self-consistent valence shell model !" #$ ! Greens function Holt (TRIUMF), Menendez, Simonis, Schwenk (TU Darmstadt) Barbieri (Surrey), Duguet, Soma (CEA) V NN Many-body In-medium SRG V 3N V NN V NN perturbation theory Bogner (MSU), Hergert (OSU), V 3N V 3N V 3N Holt (TRIUMF) V NN V 3N V 3N Required inputs: 1. consistent NN and 3N forces at N 3 LO in partial-wave-decomposed form 2. softened forces for judging approximations and pushing to heavier nuclei
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • generate unitary transformation which decouples low- and high momenta H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution successively in small steps: d λ = [ η λ , H λ ] • generator can be chosen and tailored to different applications η λ • observables are preserved due to unitarity of transformation
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group Z dr 0 r 0 2 V λ ( r, r 0 ) V λ ( r ) = 200 AV18 V(r) [MeV] 3 LO N 100 − 1 − 1 − 1 − 1 − 1 λ = 20 fm λ = 4 fm λ = 3 fm λ = 2 fm λ = 1.5 fm 0 − 100 0 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 r [fm] r [fm] r [fm] r [fm] r [fm]
Systematic decoupling of high-momentum physics: The Similarity Renormalization Group • elimination of coupling between low- and high momentum components, simplified many-body calculations • observables unaffected by resolution change (for exact calculations) • residual resolution dependences can be used as tool to test calculations Not the full story: RG transformation also changes three-body (and higher-body) interactions.
Applications of chiral 3N forces at N 3 LO Problem: Basis size for converged results of ab initio calculations including 3N forces grows rapidly with the number of particles. Calculations limited to light nuclei. Strategy: Use SRG transformations to decouple low- and high momentum states. Required basis size decreases drastically. First implementation of consistent SRG evolution of 3NF in a momentum basis: Hebeler PRC(R) 85, 021002 (2012)
Applications of chiral 3N forces at N 3 LO Problem: Transformation to HO basis: Basis size for converged results of ab initio calculations including -6 λ =infty 3N forces grows rapidly with the number of particles. -1 λ =3.0 fm -6.5 -1 λ =2.0 fm Calculations limited to light nuclei. -1 λ =1.8 fm -1 -7 λ =1.6 fm -1 E gs [MeV] λ =1.4 fm Strategy: -7.5 Use SRG transformations to decouple low- and high momentum states. -8 Required basis size decreases drastically. -8.5 First implementation of consistent 3 H (N2LO 450/500 MeV) -9 SRG evolution of 3NF in a momentum basis: 0 2 4 6 8 10 12 14 16 18 20 22 N max Hebeler PRC(R) 85, 021002 (2012)
First results for neutron matter equation of state based on consistently evolved 3N (N 2 LO) forces 20 EM 500 MeV -1 n=n s λ =2.8 fm EM 500 MeV 20 -1 λ =2.4 fm 18 Energy per neutron [MeV] -1 Energy per neutron [MeV] λ =2.0 fm 3N-full -1 λ =1.8 fm 16 15 14 3N-induced 10 12 NN-only 10 5 3N-full 8 3N-induced NN-only 0 6 0 0.05 0.1 0.15 0.2 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 − 3 ] − 1 ] λ [fm n [fm KH and Furnstahl, PRC 87, 031302(R) (2013) • 3NF contributions treated in Hartree-Fock approximation • no indications for unnaturally large 4N force contributions
3NF evolution in momentum basis: Current developments and applications • application to infinite systems ‣ equation of state (first applications to neutron matter) NN-only NN + 3N-ind. NN + 3N-full -100 -110 -120 exp. E [MeV] -130 -140 16 O -150 Ω = 20 MeV ‣ systematic study of induced many-body contributions -160 -170 . -120 -140 -160 exp. E [MeV] -180 24 O -200 Ω = 20 MeV -220 -240 NN-only NN + 3N-ind. NN + 3N-full . -250 2 4 6 8 10 12 14 2 4 6 8 10 12 14 2 4 6 8 10 12 14 -300 exp. • transformation of evolved interactions to oscillator basis -350 E [MeV] -400 -450 40 Ca -500 Ω = 20 MeV -550 -600 . -300 ‣ application to nuclei, complimentary to HO evolution exp. -400 E [MeV] -500 48 Ca -600 Ω = 20 MeV -700 (already implemented and tested) . -800 2 4 6 8 10 12 14 2 4 6 8 10 12 14 2 4 6 8 10 12 14 e max e max e max k 2 low k k ’ • study of various generators k ‣ different decoupling patterns (e.g. V low k ) Λ 2 k � 2 ‣ improved efficiency of evolution Λ 1 ‣ suppression of many-body forces Λ 0 • evolution of arbitrary operators ‣ needed for all observables ‣ study of correlations in nuclear systems factorization
RG evolution of 3N interactions in momentum space | pq α � 1 | pq α � 2 | pq α � 3 3 3 3 p 1 1 q q 1 p q p 2 2 2 • represent interaction in basis | pq α ⇥ i � | p i q i ; [( LS ) J ( ls i ) j ] J J z ( Tt i ) T T z ⇥ • explicit equations for NN and 3N flow equations 4 ] 4 ] 4 ] 4 ] s [fm s [fm s [fm s [fm 0.1 0.1 0.1 0.1 0.01 0.01 0.01 0.01 0.001 0.001 0.001 0.001 0.0001 0.0001 0.0001 0.0001 − 8.1 − 8.1 − 8.1 − 8.1 − 8.1 − 8.1 − 8.1 − 8.1 dV ij N α =42 N α =42 N α =42 N α =42 = [[ T ij , V ij ] , T ij + V ij ] , 550/600 MeV 550/600 MeV 550/600 MeV 550/600 MeV − 8.2 − 8.2 − 8.2 − 8.2 − 8.2 − 8.2 − 8.2 − 8.2 J 12 max =5 J 12 max =5 J 12 max =5 J 12 max =5 ds NN-only NN-only NN-only NN-only dV 123 E gs [MeV] E gs [MeV] E gs [MeV] E gs [MeV] NN + 3N-induced NN + 3N-induced NN + 3N-induced = [[ T 12 , V 12 ] , V 13 + V 23 + V 123 ] NN + 3N-full NN + 3N-full − 8.3 − 8.3 − 8.3 − 8.3 − 8.3 − 8.3 − 8.3 − 8.3 ds + [[ T 13 , V 13 ] , V 12 + V 23 + V 123 ] 450/500 MeV 600/500 MeV − 8.4 − 8.4 − 8.4 − 8.4 − 8.4 − 8.4 − 8.4 − 8.4 450/700 MeV + [[ T 23 , V 23 ] , V 12 + V 13 + V 123 ] 600/700 MeV + [[ T rel , V 123 ] , H s ] − 8.5 − 8.5 − 8.5 − 8.5 − 8.5 − 8.5 − 8.5 − 8.5 exp . 7 7 7 7 1.5 1.5 1.5 1.5 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 10 10 10 10 15 15 15 15 Bogner, Furnstahl, Perry PRC 75, 061001(R) (2007) − 1 ] − 1 ] − 1 ] − 1 ] λ [fm λ [fm λ [fm λ [fm Hebeler PRC(R) 85, 021002 (2012)
Strategy: Use a lower-resolution version low-pass filter 1 S 0 − 1 k = 2 fm 60 phase shift (degrees) AV18 phase shifts 40 20 low-pass filter 0 − 20 0 100 200 300 E lab (MeV)
Strategy: Use a lower-resolution version low-pass filter 1 S 0 − 1 1 S 0 − 1 k = 2 fm k = 2 fm 60 60 phase shift (degrees) phase shift (degrees) AV18 phase shifts AV18 phase shifts 40 40 20 20 low-pass filter 0 0 after low-pass filter − 20 − 20 0 100 200 300 0 100 200 300 E lab (MeV) E lab (MeV) • truncated interaction fails completely to reproduce original phase shifts • problem: low- and high momentum states are coupled by interaction!
First Quantum Monte Carlo based on chiral EFT interactions Problem: Current QMC frameworks can only applied to local Hamiltonians. Conventional interactions derived within chiral EFT are nonlocal . Strategy: Use freedom in the choice of operators and the type of regulator to construct local Hamiltonians up to N 2 LO: • regulate in coordinate space in relative distance: f ( r ) = 1 − e − ( r/R 0 ) 4 • use isospin dependent terms instead of non-local operators at NLO 70 50 0 30 3 P 0 3 P 2 LO LO 60 NLO -5 NLO 40 N 2 LO N 2 LO PWA PWA 50 Phase Shift [deg] Phase Shift [deg] -10 30 20 40 -15 20 30 -20 10 10 20 LO LO NLO NLO N 2 LO N 2 LO 1 S 0 3 P 1 -25 10 0 PWA PWA 0 0 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 Lab. Energy [MeV] Lab. Energy [MeV] Lab. Energy [MeV] Lab. Energy [MeV] Gezerlis, Tews, Epelbaum, Gandolfi, KH, Nogga, Schwenk, PRL 111, 032501 (2013)
First Quantum Monte Carlo based on chiral EFT interactions 15 perfect agreement for soft interactions, first direct validation 10 E/N [MeV] of perturbative calculations QMC (2010) 5 AFDMC N 2 LO 0.8 fm (2nd order) 0.8 fm (3rd order) 1.2 fm (2nd order) 1.2 fm (3rd order) Gezerlis, Tews, Epelbaum, Gandolfi, KH, Nogga, Schwenk 0 PRL 111, 032501 (2013) 0 0.05 0.1 0.15 n [fm -3 ] Greens Function Monte Carlo calculations for light nuclei based on chiral interactions currently in progress
Decoupling in 3NF matrix elements θ = π T = J = 1 2 12 Λ / ˜ Λ 550 / 600 MeV 450 / 500 MeV KH, PRC(R) 85, 021002 (2012) see also KH, Furnstahl, PRC(R) 87, 031302 (2013) ξ 2 = p 2 + 3 2 p hyperradius: hyperangle: 4 q 2 tan θ = √ 3 q same decoupling patterns like in NN interactions
Resolution dependence of nuclear forces Effective theory for NN, 3N, many-N interactions: QCD Λ � Λ chiral quarks+gluons/partons: Q � m π Λ chiral typical momenta in nuclei: Q ∼ m π chiral EFT: nucleons interacting via pion exchanges and short-range contact interactions Λ pionless n 9 Li large scattering length physics: Q � m π n pionless EFT: unitary regime, non-universal corrections
Problem: Traditional “hard” NN interactions − k � k � V V 3N k − k � k � | V | k ⇥ � • constructed to fit scattering data (long-wavelength information!) • “hard” NN interactions contain repulsive core at small relative distance • strong coupling between low and high-momentum components, hard to solve! Claim: Problems due to high resolution from interaction.
Wavelength and resolution size of resolvable structures depends on the wavelength
Wavelength and resolution size of resolvable structures depends on the wavelength
Wavelength and resolution size of resolvable structures depends on the wavelength
Wavelength and resolution size of resolvable structures depends on the wavelength
Wavelength and resolution size of resolvable structures depends on the wavelength
Wavelength and resolution size of resolvable structures depends on the wavelength
Recommend
More recommend
