Short- and long-range correlations in nuclear pairing phenomena

Short- and long-range correlations in nuclear pairing phenomena Ding, Rios, et al., Phys Rev C 94 2582 (26) Arnau Rios Huguet Lecturer in Nuclear Theory Department of Physics University of Surrey ECT Superfluids Workshop, 23 March 27

Summary. Neutron star motivation 2. Reminder of nuclear BCS 3. Beyond-BCS with SCGF methods 4. Dineutrons 2

Neutron star modeling Input # EoS Observable # Mass-Radius relation Input #2 Pairing gap Observable #2 Cooling curve superfluid Input #3 Crust-core Observable #3 Glitching Frequency crust spindown Time 3

Neutron star modeling Observable # Mass-Radius relation Input Consistent many-body theory Observable #2 Cooling curve superfluid Observable #3 Glitching Frequency crust spindown Time 3

Cooling of Car A Cas A data Ho, et al., PRC 9 586 (25) Page, et al., PRL 6 8 (2) Ingredients (a) Mass of pulsar (b) EoS (determines radius) (c) Internal composition (d) Pairing gaps ( S & 3 PF2 channels) (e) Atmosphere composition Ho, Elshamouty, Heinke, Potekhin PRC 9 586 (25) 4

Bayesian inversion is coming Beloin, Han, Steiner & Page, arxiv:62.4289 Ingredients (a) Mass of pulsar (b) EoS (determines radius) (c) Internal composition (d) Pairing gaps ( S & 3 PF2 channels) < Bayesian inversion from data (e) Atmosphere composition 5

Summary. Neutron star motivation 2. Reminder of nuclear BCS 3. Beyond-BCS with SCGF methods 4. Dineutrons 6

Bardeen-Cooper-Schrieffer pairing 3. S Neutron matter BCS gaps 3 PF2 Pairing gap, F [MeV] 2.5 2..5..5...5..5 2. 2.5 3. 3.5 Fermi momentum, k F [fm - ] BCS equation..5..5 2. 2.5 3. 3.5 Fermi momentum, k F [fm - ] CDBONN AV8 N3LO Dean & Hjorth-Jensen, Rev. Mod. Phys. 75 67 (23) + Single-particle spectrum choice: Angular gap dependence: " k = k2 2m + U(k) µ k 2 = X L k 2 L Maurizio, Holt & Finelli, Phys. Rev. C 9, 443 (24) 7

Bardeen-Cooper-Schrieffer pairing 3. S Neutron matter BCS gaps 3 PF2 Pairing gap, F [MeV] 2.5 2..5..5...5..5 2. 2.5 3. 3.5 Fermi momentum, k F [fm - ] BCS equation..5..5 2. 2.5 3. 3.5 Fermi momentum, k F [fm - ] CDBONN AV8 N3LO Dean & Hjorth-Jensen, Rev. Mod. Phys. 75 67 (23) + Single-particle spectrum choice: Angular gap dependence: " k = k2 2m + U(k) µ k 2 = X L k 2 L L k 2 Maurizio, Holt & Finelli, Phys. Rev. C 9, 443 (24) 7

Triplet pairing: phase shift equivalence Momentum, q 2 [fm - ] Momentum, q 2 [fm - ] Momentum, q 2 [fm - ] 8 6 4 2 8 6 4 2 8 6 4 2 3 P2 CDBonn 3 P2 Av8 3 P2 N3LO 2 4 6 8 Momentum, q [fm - ].2. -. -.2.2. -. -.2.2. -. -.2 δ [deg] δ [deg] δ [deg] 2 2 - - -2-3 NN phase shifts 3 P2 3 F2 Nij par SAID Nij data CDBonn Av8 N3LO ε 2...5 2. 2.5 3. 3.5-4.5.5 2 2.5 q bin [fm - ] Pairing gap, [MeV].. Neutron matter BCS gaps 3 PF2 CDBonn Av8 N3LO Fermi momentum, k F [fm - ] Srinivias & Ramanan, PRC 94 6433 (26) 8

NN forces from EFTs of QCD c i c i π O ˆQ GeV Chiral perturbation theory π and N as dof Systematic expansion 2N at N 3 LO - LECs from πn, NN 3N at N 2 LO - 2 more LECs (Often further renormalized) c i Weinberg, Phys. Lett. B 25 288 (99), NPB 363 3 (99) Entem & Machleidt, PRC 68, 4(R) (23) Tews, Schwenk et al., PRL, 3254 (23) Epelbaum, Frebs & Meissner, PRL 5, 223 (25) 9

Uncertainties at the BCS + HF level Hard R=.9 fm R=. fm R=. fm R=.2 fm Soft Drischler, Kruger, Hebler, Schwenk, PRC 95 2432 (27)

Uncertainties at the BCS + HF level Hard R=.9 fm R=. fm Srinivias & Ramanan, PRC 94 6433 (26) R=. fm R=.2 fm Soft Drischler, Kruger, Hebler, Schwenk, PRC 95 2432 (27)

Summary. Neutron star motivation 2. Reminder of nuclear BCS 3. Beyond-BCS with SCGF methods 4. Dineutrons

Why short-range correlations? Deuteron momentum distribution n d (k) [GeV 3 ] 3 2 - -2-3 -4 Av8 CDBONN N3LO Av4 θ=8 o, <Q 2 >=2.7 θ=22 o, <Q 2 >=3.8 θ=26 o, <Q 2 >=4.8 θ=32 o, <Q 2 >=6.4 2 3 4 5 6 7 8 9 k [MeV] Fomin et al., PRL 8 9252 (22) Inclusive quasi-elastic e - scattering vs NN potential theory 2

Why short-range correlations? Deuteron momentum distribution n d (k) [GeV 3 ] 3 2 - -2-3 -4 Tensor Av8 CDBONN N3LO Av4 θ=8 o, <Q 2 >=2.7 θ=22 o, <Q 2 >=3.8 θ=26 o, <Q 2 >=4.8 θ=32 o, <Q 2 >=6.4 2 3 4 5 6 7 8 9 k [MeV] Fomin et al., PRL 8 9252 (22) Inclusive quasi-elastic e - scattering vs NN potential theory 2

Why short-range correlations? Deuteron momentum distribution n d (k) [GeV 3 ] 3 2 - -2-3 -4 Tensor 2 3 4 5 6 7 8 9 k [MeV] Av8 CDBONN N3LO Av4 θ=8 o, <Q 2 >=2.7 θ=22 o, <Q 2 >=3.8 θ=26 o, <Q 2 >=4.8 θ=32 o, <Q 2 >=6.4 SRC Fomin et al., PRL 8 9252 (22) Inclusive quasi-elastic e - scattering vs NN potential theory 2

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 Re T(Ω + ; P,q=) [MeV fm3] 5 5 2 25 3 6 One-body properties Momentum distribution Thermodynamics & EoS Transport Spectral function 4 P [MeV] 2 2 Ramos, Polls & Dickhoff, NPA 53 (989) Alm et al., PRC 53 28 (996) Dewulf et al., PRL 9 525 (23) Frick & Muther, PRC 68 343 (23) Rios, PhD Thesis, U. Barcelona (27) Soma & Bozek, PRC 78 543 (28) Rios & Soma PRL 8 25 (22) 3 Ω 2µ [MeV] 2

Advantages All kinds of NN interactions 3N interactions Short-range & tensor correlations Density & isospin dependence Access to off-shell spectral function Thermodynamically consistent Limitations Non-relativistic Missing diagrams? No nuclear surface 4

Quantitative SRC Av8 - k= (a) N3LO+SRG+3NF k= (d) - Energy weighted sum rules A(k,ω) [MeV - ] A(k,ω) [MeV - ] A(k,ω) [MeV - ] -2-3 -4-5 -6 - k=k F (b) -2-3 -4-5 -6 - -2-3 -4-5 -6 k=2k F (c) -75-5-25 25 5 75 Energy, ω µ [MeV] k=k F (e) k=2k F (f) -2-3 -4-5 -6 - -2-3 -4-5 -6 - -2-3 -4-5 -75-5-25 25 5 75-6 Energy, ω µ [MeV] m () k = m () k = m (2) k = Z Z Z sf variance 2 k = 2 k = Z + d! 2 A k(!) = d! 2!A k(!) = ~2 k 2 2m + k h i 2 + Z + d! 2!2 A k (!) = d! 2 [! m () k 2 k m() k ]2 A k (!) d! Im k(!) Rios, Carbone, Polls, arxiv:72.37 5

Quantitative SRC m (2) k moment [MeV2 ] 5x 4 4x 4 3x 4 2x 4 x 4 N3LO+SRG+3NF Sum rule [m k () ] 2 σ k 2 Av8 (d) Energy weighted sum rules m () k = m () k = m (2) k = Z Z Z d! 2 A k(!) = d! 2!A k(!) = ~2 k 2 2m + k h i 2 + d! 2!2 A k (!) = m () k 2 k m (2) k moment [MeV2 ] 3x 5 2x 5 x 5 (f) 2 4 6 Momentum, k [MeV] sf variance 2 k = 2 k = Z + Z + d! 2 [! m() k ]2 A k (!) d! Im k(!) Rios, Carbone, Polls, arxiv:72.37 5

Momentum distribution Single-particle occupation Momentum distribution, n(k).8.6.4.2 npkq A a : k a k.5.5 2 Momentum, k/k F E ρ=.6 fm -3, T= MeV Av8 CD-Bonn N3LO FFG ª d 3 k npkq p2 q3 ρ=.6 fm -3, T= MeV Neutron matter.5.5 2 2.5 3 Momentum, k/k F - -2-3 Momentum distribution, n(k) SNM: -3% depletion at low k, population at high k Dependence on NN interaction under control PNM: 4-5% depletion at low k 6

T= extrapolations Spectral function, A(k,ω) - -2-3 -4-5 -6 k= Av8 k F =.33 fm - k=k F T= MeV T=4 MeV T=5 MeV T=6 MeV T=8 MeV T=9 MeV T= MeV T= MeV T=2 MeV T=4 MeV T=5 MeV T=6 MeV T=8 MeV T=2 MeV k=2k F -7-5 -3-3 ω µ [MeV] -5-3 - 3 ω µ [MeV] -5-3 - 3 ω µ [MeV] 2 26 Temperature, T [MeV] 5 5 24 22 2 8 6 4 E/A [MeV]... Density, ρ [fm -3 ] 5 5 2 T [MeV] 2 7

Available data Av8 2 Temperature, T [MeV] 5 5... Density, ρ [fm -3 ] Self-energy, spectral function & thermodynamics 8

Neutron matter Pressure [MeV fm -3 ] 2 - -2 EoS for neutron matter: SRG..8.6.24.32.4 Density, ρ [fm -3 ] Tolman-Oppenheimer-Volkov dp G pm`4 pr 3 qp ` pq dr c 2 rpr 2Gm{c 2 q dm dr 4 c 2 r2 Error band from fits in ChPT c, c 3 parameters Finite temperature & higher densities available 23 4 5 7 Mass, M (Solar Masses) 2.8.6.4.2.8.6.4.2 6 8 2 4 6 Radius, R (km) Carbone, Polls & Rios, PRC 9 54322 (24) A. Carbone, Drischler, Hebeler, Schwenk, PRC 94 5437 (26) GR P < Causality 9

Neutron matter Pressure [MeV fm -3 ] 2 - -2 EoS for neutron matter: SRG..8.6.24.32.4 Density, ρ [fm -3 ] 23 4 5 7 Tolman-Oppenheimer-Volkov dp G pm`4 pr 3 qp ` pq dr c 2 rpr 2Gm{c 2 q dm dr 4 c 2 r2 Error band from fits in ChPT c, c 3 parameters Finite temperature & higher densities available Hebeler, Lattimer, Pethick, Schwenk ApJ 773 (23) Carbone, Polls & Rios, PRC 9 54322 (24) A. Carbone, Drischler, Hebeler, Schwenk, PRC 94 5437 (26) 9

Beyond BCS : SRC BCS is lowest order in Gorkov Green s function expansion T-matrix can be extended to paired systems Bozek, Phys. Rev. C 62 5436 (2); Muether & Dickhoff, Phys. Rev. C 72 5433 (25) 2

Beyond BCS : SRC Normal state BCS is lowest order in Gorkov Green s function expansion T-matrix can be extended to paired systems Bozek, Phys. Rev. C 62 5436 (2); Muether & Dickhoff, Phys. Rev. C 72 5433 (25) 2

Beyond BCS : SRC Normal state Superfluid Δ(kF) BCS is lowest order in Gorkov Green s function expansion T-matrix can be extended to paired systems Bozek, Phys. Rev. C 62 5436 (2); Muether & Dickhoff, Phys. Rev. C 72 5433 (25) 2

Beyond BCS : SRC Normal state? Superfluid Δ(kF) BCS is lowest order in Gorkov Green s function expansion T-matrix can be extended to paired systems Bozek, Phys. Rev. C 62 5436 (2); Muether & Dickhoff, Phys. Rev. C 72 5433 (25) 2

BCS gap equation + BCS+SRC gap equation L k = X Z L k hk V LL nn k i 2 p 2k L + k 2 2 k = Z! Z k +! f(!) f(! )! +! A(k,!)A s (k,! ) Gorkov gap equation L k = X Z L k hk V LL nn 2 k 2 k = k i Z! Z L k +! f(!) f(! )! +! A(k,!)A s (k,! ) Muether & Dickhoff, Phys. Rev. C 72 5433 (25) 2

Beyond BCS : SRC 2 k = Z! Z Full off-shell! f(!) f(! )! +! A(k,!)A(k,! ) Quasi-particle BCS 2 k = 2 " k µ Denominator, χ(k) [MeV] 8 6 4 2.5.5 2 Momentum, k/k F T= MeV T=4 MeV T=5 MeV T=6 MeV T=8 MeV T=9 MeV T= MeV T= MeV T=2 MeV T=4 MeV T=5 MeV T=6 MeV T=8 MeV T=2 MeV qp denominator, χ qp (k) [MeV] 8 6 4 2 Av8 k F =.33 fm -.5.5 2 Momentum, k/k F 22

Beyond BCS : 3 SD Symmetric matter 3 SD pairing gap Pairing gap, [MeV] 4 2 8 6 4 2 BCS CDBonn BCS Av8 SRC CDBonn SRC Av8..5..5 2. 2.5 Fermi momentum, k F [fm - ] Massive gaps 3 SD channel but No evidence of strong np nuclear pairing Short-range correlations deplete gap 3BF effect? Short-range effects? Muether & Dickhoff, PRC 72 5433 (25) Maurizio, Holt & Finelli, PRC 9, 443 (24) 23

Beyond BCS : SRC Neutron matter CDBonn Av8 N3LO 3 S S S Pairing gap, [MeV] 2.5 2.5.5 S BCS SRC..5..5 Fermi momentum, k F [fm - ].5..5 Fermi momentum, k F [fm - ].5..5 Fermi momentum, k F [fm - ].9.8 3 PF2 BCS SRC CDBonn Av8 N3LO Pairing gap, [MeV].7.6.5.4.3.2 3 PF2 3 PF2 3 PF2. BCS+SRC..5 2. 2.5 3. Fermi momentum, k F [fm - ] L k = X Z L k hk V LL nn k i 2 p 2k L + k 2..5 2. 2.5 3. Fermi momentum, k F [fm - ] k + 2 k = Z! Z..5 2. 2.5 3. Fermi momentum, k F [fm - ]! f(!) f(! )! +! A(k,!)A s (k,! ) 24

Beyond BCS 2: LRC? ph recoupled G-matrix Effective Landau parameters V pair = h V i = 4 X 2,2 X ( ) S (2S + )h2 G ph ST 2 ph i A h2 G ST 2 i A (22 ) S,T ST (q) = ST (q) ST (q) F ST Bare NN potential only is not the only possible interaction Diagram (a): nuclear interaction Diagram (b): in-medium interaction, density and spin fluctuations Diagram (c): included by Landau parameters Cao, Lombardo & Schuck, Phys Rev C 74 643 (26) 25

Landau parameters Landau parameters.6 F.4 G F.2 /(+F )+G /(+G ).8.6.4.2.5.5 2 2.5 3 Fermi momentum, k F [fm - ] Diagonal V [MeVfm -3 ] Diagonal V [MeVfm -3 ] 3 S 2 - -2 Bare (a) -3 3 3 2 P2 - -2-3 -4 (b) -5 2 3 4 5 Momentum, k [fm - ] Effective V S= LRC V S= LRC (c) (d).8.6.4.2 -.2 -.4 -.6 2 3 4 5 -.8 Momentum, k [fm - ] 26

Beyond BCS 2: results S Neutron matter CDBonn Av8 N3LO 3 S S S Pairing gap, [MeV] 2.5 2.5.5 S BCS SRC SRC+LRC..5..5 Fermi momentum, k F [fm - ].5..5 Fermi momentum, k F [fm - ].5..5 Fermi momentum, k F [fm - ] Pairing gap, [MeV].9.8.7.6.5.4.3.2 3 PF2 BCS SRC SRC+LRC 3 PF2 CDBonn 3 PF2 Av8 3 PF2 N3LO...5 2. 2.5 3. Fermi momentum, k F [fm - ]..5 2. 2.5 3. Fermi momentum, k F [fm - ]..5 2. 2.5 3. Fermi momentum, k F [fm - ] LRC S ( 3 PF2) produces (anti-)screening 27

Summary Pairing gap, [MeV]. 3. 2.5 2..5..5 N3LO (c) S Pairing gap, [MeV]... N3LO 3 PF2 (c) BCS BCS+LRC SRC SRC+LRC...5..5 Fermi momentum, k F [fm - ]...5 2. 2.5 3. Fermi momentum, k F [fm - ] JST pk F k q 2 pk F k 2 q 2 L pk F q pk F k q 2 ` k pk F k 2 q 2 ` k 3 Δ k k k2 k3 CDBonn 8.8.5.39.45.8 Singlet Av8 4.7.4..44.78 N3LO 5.85.46.48.42 CDBonn.4.3.56 2.8 Triplet Av8.7..35 2.8.5 N3LO.6..69 2.79.53 28

Ladder approximation with 3BF Two-body interaction Effective interactions 2 In-medium T-matrix = + In-medium T-matrix = + Self-energy Self-energy Dyson equation 2 Carbone et al., PRC 88 54326 (23) A. Carbone, PhD Thesis 29

3BF effect: estimate Pairing gap, [MeV] 2.5 2.5.5 3 S (a) BCS NN SRC NN BCS NN+3NF SRC NN+3NF spectrum 2 NN forces Pairing gap, [MeV]. 3 PF2 (b) Singlet gap: 3NF reduce closure Triplet gap: 3NF increase gap Model dependence to be explored SRG dependence for systematics...5..5 2. 2.5 Fermi momentum, k F [fm - ] 3

BCS+HF level Singlet gaps with 3NF Triplet gaps with 3NF Drischler, Kruger, Hebler, Schwenk, PRC 95 2432 (27) 3

Beyond BCS pairing: overview S 3 PF2 Pairing gap, F [MeV] 3. 2.5 2..5..5 BCS N3LO BCS CDBONN BCS AV8 N3LO CDBONN AV8 CCDK GIPSF CLS MSH SFB SCLBL BCS N3LO BCS CDBONN BCS AV8 N3LO CDBONN AV8 SYHHP EEHOr EEHO AO.....5..5 2. 2.5 3. 3.5 Fermi momentum, k F [fm - ]..5..5 2. 2.5 3. 3.5. Fermi momentum, k F [fm - ] Effect is robust: independent of NN potential 3NF effect not included in SRC, BCS indicates small Singlet results very close to Cao+Lombardo+Schuck Triplet results are significantly different 32

Summary. Neutron star motivation 2. Reminder of nuclear BCS 3. Beyond-BCS with SCGF methods 4. Dineutrons 33

Dineutrons BHF G-matrix.2 bnn [ MeV ].4.6.8 AV8 Paris N3LO 2N N3LO 2N +N2LO 3N Poles kf=.6 fm - <r> [ fm ] 2 8 AV8 Paris N3LO 2N N3LO 2N +N2LO 3N r 2 Ψ 2 [ fm - ].2. Deuteron (free space) AV8 Paris N3LO 2N N3LO 2N +N2LO 3N 6.2.4.6.8 k F [ fm - ] 2 3 r [ fm ] Isaule, Arellano, Rios, in preparation 34

Collaborators +A. Carbone +D. Ding, W. H. Dickhoff +A. Polls UNIVERSITAT DE BARCELONA U B +C. Barbieri +V. Somà +H. Arellano, F. Isaule 35

Conclusions Ab initio nuclear theory to treat correlations Talk to us if you need quantitative predictions! Different NN forces provide robust predictions Challenges ahead Full self-consistent Gorkov Consistent treatment of LRC Pairing in isospin asymmetric matter a.rios@surrey.ac.uk @riosarnau 36