Pairing in high-density neutron matter Arnau Rios Huguet Lecturer in Nuclear Theory Department of Physics University of Surrey Nuclear Astrophysics Milan 18 September 2017
Summary 1. Neutron star motivation 2. Infinite matter BCS 3. Beyond-BCS with SCGF methods 2
Cooling of CasA Cas A data Ho, et al. , PRC 91 015806 (2015) Page, et al. , PRL 106 081101 (2011) Ingredients (a) Mass of pulsar (b) EoS (determines radius) (c) Internal composition (d) Pairing gaps ( 1 S 0 & 3 PF 2 channels) Ho, Elshamouty, Heinke, Potekhin (e) Atmosphere composition PRC 91 015806 (2015) 3
Complications NN interaction is not unique ...but phase-shift equivalent! 90 1 S 0 60 δ [deg] 30 0 180 3 S 1 Nij CDBonn 120 δ [deg] Av18 N3LO Nij93 60 0 S. Aoki, et al. Comput. Sci. Dis. 1 015009 (2008) 0 50 100 150 200 250 E [MeV] 4
Complications NN interaction is not unique ...but phase-shift equivalent! 90 1 S 0 60 δ [deg] 30 0 180 3 S 1 Nij CDBonn 120 δ [deg] Av18 N3LO Nij93 60 0 S. Aoki, et al. Comput. Sci. Dis. 1 015009 (2008) 0 50 100 150 200 250 E [MeV] • Non-uniqueness of nucleon forces ✘ 4
Complications NN interaction is not unique Strong short-range correlations Carlson et al., Phys. Rev. C 68 025802 (2003) S. Aoki, et al. Comput. Sci. Dis. 1 015009 (2008) • Non-uniqueness of nucleon forces ✘ • Short-range core needs many-body treatment ✘ 4
Complications NN interaction is not unique Saturation point of nuclear matter Li, Lombardo, Schulze et al. PRC 74 047304 (2006) S. Aoki, et al. Comput. Sci. Dis. 1 015009 (2008) • Non-uniqueness of nucleon forces ✘ • Short-range core needs many-body treatment ✘ • Three-body forces needed for saturation ✘ 4
NN forces from EFTs of QCD Chiral perturbation theory π c i • π and N as dof • Systematic expansion • 2N at N 3 LO - LECs from π N, NN c i • 3N at N 2 LO - 2 more LECs ˆ Q ˙ • (Often further renormalized) O Λ Λ „ 1 GeV c i Weinberg, Phys. Lett. B 251 288 (1990), NPB 363 3 (1991) Entem & Machleidt, PRC 68 , 041001(R) (2003) Tews, Schwenk et al. , PRL 110 , 032504 (2013) Epelbaum, Frebs & Meissner, PRL 115 , 122301 (2015) 5
NN forces from EFTs of QCD Chiral perturbation theory π c i • π and N as dof • Systematic expansion • 2N at N 3 LO - LECs from π N, NN c i • 3N at N 2 LO - 2 more LECs ˆ Q ˙ • (Often further renormalized) O Λ Λ „ 1 GeV c i Weinberg, Phys. Lett. B 251 288 (1990), NPB 363 3 (1991) Entem & Machleidt, PRC 68 , 041001(R) (2003) Tews, Schwenk et al. , PRL 110 , 032504 (2013) Epelbaum, Frebs & Meissner, PRL 115 , 122301 (2015) 5
SCGF Ladder approximation In-medium interaction Ladder self-energy • Self-consistent resummation • Energy and momentum integral • @Finite T (Matsubara) pp & hh Pauli blocking Dyson equation T-matrix at T=5 MeV 0 Re T( Ω + ; P,q=0) [MeV fm3] − 5 − 10 − 15 − 20 − 25 − 30 600 400 Spectral function 200 200 100 0 P [MeV] 0 − 100 − 200 Ω − 2 µ [MeV] One-body properties Ramos, Polls & Dickhoff, NPA 503 1 (1989) Momentum distribution Alm et al. , PRC 53 2181 (1996) Thermodynamics & EoS Dewulf et al. , PRL 90 152501 (2003) Transport Frick & Muther, PRC 68 034310 (2003) Rios, PhD Thesis, U. Barcelona (2007) Soma & Bozek, PRC 78 054003 (2008) Rios & Soma PRL 108 012501 (2012) 6
T=0 extrapolations k=k F k=2k F k=0 10 0 T=0 MeV Av18 T=4 MeV 10 -1 k F =1.33 fm -1 T=5 MeV Spectral function, A(k, ω ) T=6 MeV T=8 MeV 10 -2 T=9 MeV T=10 MeV T=11 MeV 10 -3 T=12 MeV T=14 MeV T=15 MeV 10 -4 T=16 MeV T=18 MeV T=20 MeV 10 -5 10 -6 10 -7 -500 -300 -100 100 300 -500 -300 -100 100 300 -500 -300 -100 100 300 ω− µ [MeV] ω− µ [MeV] ω− µ [MeV] 27.5 28 data 27 26 extrap 26.5 24 26 E/A [MeV] µ [MeV] 25.5 22 25 20 24.5 24 18 23.5 16 23 44 0.89 22.5 14 0 5 10 15 20 0 5 10 15 20 7 T [MeV] T [MeV]
Momentum distribution Single-particle occupation ª d 3 k A E a : p 2 π q 3 n p k q “ ρ n p k q “ k a k ν Neutron matter �� � �� � � ����� �� �� ��� ��� ������� ���� ��� �������� ������������� ���� ������ �������� ��� ��� �� �� �� �� ��� ��� ��� ��� �� �� �� �� ��� ��� �� �� �� �� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��������� ��� � ��������� ��� � • Dependence on NN interaction under control • PNM: 4-5% depletion at low k • N3LO+3NF = N3LO 2NF + N2LO 3NF @ Λ =414-500 MeV (cutoff variation only) Coraggio, Holt, et al. PRC 89 044321 (2014) 8
Effective masses ρ = 0.16 fm -3 Neutron matter 1.2 1.2 1.1 1.1 1 1 m * /m m * /m 0.9 0.9 T=20 MeV T=10 MeV Λ =500 MeV 0.8 0.8 T=18 MeV T=8 MeV Λ =450 MeV T=16 MeV T=6 MeV Λ =414 MeV 0.7 0.7 T=14 MeV T=4 MeV T=12 MeV T=0 MeV 0.6 0.6 1.4 1.4 1.3 1.3 1.2 1.2 1.1 1.1 w /m k /m 1 1 m * m * 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0 200 400 600 800 1000 0 200 400 600 800 1000 Momentum, k [MeV] Momentum, k [MeV] Coraggio, Holt, et al. PRC 89 044321 (2014) In preparation, will be publicly available 9
Available data Av18 20 Temperature, T [MeV] 15 10 5 0 0.001 0.01 0.1 1 Density, ρ [fm -3 ] Self-energy, spectral function & thermodynamics In preparation, will be publicly available 10
Neutron matter GR 3 3 P < ∞ N3LO+3BF SLy 2.5 2.5 Mass, M (Solar Masses) Causality 2 2 1.5 1.5 1 1 0.5 0.5 0 0 8 8 10 10 12 12 14 14 16 16 Radius, R (km) • Mass-Radius relation from SCGF calculations • Cut-off variation (N3LO) and/or SRG evolution 11
Neutron matter GR 3 3 P < ∞ N3LO+3BF SLy 2.5 2.5 Mass, M (Solar Masses) Causality 2 2 1.5 1.5 1 1 0.5 0.5 0 0 8 8 10 10 12 12 14 14 16 16 Radius, R (km) Hebeler, Lattimer, Pethick, Schwenk ApJ 773 11 (2013) • Mass-Radius relation from SCGF calculations • Cut-off variation (N3LO) and/or SRG evolution 11
Summary 1. Neutron star motivation 2. Infinite matter BCS 3. Beyond-BCS with SCGF methods 12
Bardeen-Cooper-Schrieffer pairing � � � � � ������ � �� � ���� � � ��� ���� � ���� ��� ��� ��� � ���� ��� ������� ���� � ����� � ���� ��� � � ��� ��� ��� ��� � � ��� ��� � � ���� ���� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ����� ��������� � � ��� �� � ����� ��������� � � ��� �� � BCS equation + ε k = k 2 • Single-particle spectrum choice: 2 m + U ( k ) − µ • Angular gap dependence: X | ∆ L | ∆ k | 2 = k | 2 13 L
Bardeen-Cooper-Schrieffer pairing � � � � � ������ � �� � ���� � � ��� ���� � ���� ��� ��� ��� � ���� ��� ������� ���� � ����� � ���� ��� � � ��� ��� ��� ��� � � ��� ��� � � ���� ���� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ����� ��������� � � ��� �� � ����� ��������� � � ��� �� � BCS equation + ε k = k 2 • Single-particle spectrum choice: 2 m + U ( k ) − µ • Angular gap dependence: X | ∆ L | ∆ k | 2 = k | 2 13 L
Bardeen-Cooper-Schrieffer pairing � � � � � ������ � �� � ���� � � ��� ���� � ���� ��� ��� ��� � ���� ��� ������� ���� � ����� � ���� ��� � � ��� ��� ��� ��� � � ��� ��� � � ���� ���� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ����� ��������� � � ��� �� � ����� ��������� � � ��� �� � BCS equation + ε k = k 2 • Single-particle spectrum choice: 2 m + U ( k ) − µ • Angular gap dependence: X X | ∆ L | ∆ L k | 2 ≈ | ∆ L | ∆ k | 2 = | ∆ k | 2 = k | 2 k | 2 13 L L
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