[PPT] - Pairing in high-density neutron matter Arnau Rios Huguet Lecturer PowerPoint Presentation

SLIDE 1

Arnau Rios Huguet

Lecturer in Nuclear Theory Department of Physics University of Surrey

Pairing in high-density neutron matter

Nuclear Astrophysics Milan 18 September 2017

SLIDE 2

Summary

1. Neutron star motivation
2. Infinite matter BCS
3. Beyond-BCS with SCGF methods

2

SLIDE 3

Ho, Elshamouty, Heinke, Potekhin PRC 91 015806 (2015)

Cooling of CasA

3

Ho, et al., PRC 91 015806 (2015)

Cas A data

Page, et al., PRL 106 081101 (2011)

Ingredients

(a) Mass of pulsar (b) EoS (determines radius) (c) Internal composition (d) Pairing gaps (1S0 & 3PF2 channels) (e) Atmosphere composition

SLIDE 4

Complications

4

NN interaction is not unique ...but phase-shift equivalent!

30 60 90

δ [deg]

Nij CDBonn Av18 N3LO Nij93

50 100 150 200 250

E [MeV]

60 120 180

δ [deg]

1S0 3S1

S. Aoki, et al. Comput. Sci. Dis. 1 015009 (2008)

SLIDE 5

Complications

4

NN interaction is not unique

Non-uniqueness of nucleon forces ✘

...but phase-shift equivalent!

30 60 90

δ [deg]

Nij CDBonn Av18 N3LO Nij93

50 100 150 200 250

E [MeV]

60 120 180

δ [deg]

1S0 3S1

S. Aoki, et al. Comput. Sci. Dis. 1 015009 (2008)

SLIDE 6

Complications

4

NN interaction is not unique

Carlson et al., Phys. Rev. C 68 025802 (2003)

Strong short-range correlations

Non-uniqueness of nucleon forces ✘
Short-range core needs many-body treatment ✘
S. Aoki, et al. Comput. Sci. Dis. 1 015009 (2008)

SLIDE 7

Complications

4

NN interaction is not unique

Non-uniqueness of nucleon forces ✘
Short-range core needs many-body treatment ✘
Three-body forces needed for saturation ✘

Saturation point of nuclear matter

Li, Lombardo, Schulze et al. PRC 74 047304 (2006)

S. Aoki, et al. Comput. Sci. Dis. 1 015009 (2008)

SLIDE 8

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)

NN forces from EFTs of QCD

Chiral perturbation theory

π and N as dof
Systematic expansion
2N at N3LO - LECs from πN, NN
3N at N2LO - 2 more LECs
(Often further renormalized)

5

O ˆQ Λ ˙ Λ „ 1 GeV

ci ci π ci

SLIDE 9

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)

NN forces from EFTs of QCD

Chiral perturbation theory

π and N as dof
Systematic expansion
2N at N3LO - LECs from πN, NN
3N at N2LO - 2 more LECs
(Often further renormalized)

5

O ˆQ Λ ˙ Λ „ 1 GeV

ci ci π ci

SLIDE 10

In-medium interaction Ladder self-energy Dyson equation pp & hh Pauli blocking Spectral function

One-body properties Momentum distribution Thermodynamics & EoS Transport

Ramos, Polls & Dickhoff, NPA 503 1 (1989) Alm et al., PRC 53 2181 (1996) Dewulf et al., PRL 90 152501 (2003) Frick & Muther, PRC 68 034310 (2003) Rios, PhD Thesis, U. Barcelona (2007) Soma & Bozek, PRC 78 054003 (2008) Rios & Soma PRL 108 012501 (2012)

SCGF Ladder approximation

6

Self-consistent resummation
Energy and momentum integral
@Finite T (Matsubara)

−200 −100 100 200 200 400 600 −30 −25 −20 −15 −10 −5

Ω−2µ [MeV] P [MeV] Re T(Ω+; P,q=0) [MeV fm3]

T-matrix at T=5 MeV

SLIDE 11

T=0 extrapolations

7 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

500
300
100

100 300

Av18 kF=1.33 fm-1

Spectral function, A(k,ω) ω−µ [MeV]

k=0

500
300
100

100 300

ω−µ [MeV]

k=kF

500
300
100

100 300

ω−µ [MeV]

k=2kF

T=20 MeV T=18 MeV T=16 MeV T=15 MeV T=14 MeV T=12 MeV T=11 MeV T=10 MeV T=9 MeV T=8 MeV T=6 MeV T=5 MeV T=4 MeV T=0 MeV

22.5 23 23.5 24 24.5 25 25.5 26 26.5 27 27.5 µ [MeV] data extrap 14 16 18 20 22 24 26 28 E/A [MeV] 44 5 10 15 20 T [MeV] 5 10 15 20 0.89 T [MeV]

SLIDE 12

Single-particle occupation

Momentum distribution

8

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)

npkq “ A a:

kak

E

ν ª d3k p2πq3 npkq “ ρ

Neutron matter

Coraggio, Holt, et al. PRC 89 044321 (2014)

SLIDE 13

Effective masses

9

Neutron matter

0.6 0.7 0.8 0.9 1 1.1 1.2

ρ=0.16 fm-3

m*/m

T=20 MeV T=18 MeV T=16 MeV T=14 MeV T=12 MeV T=10 MeV T=8 MeV T=6 MeV T=4 MeV T=0 MeV

0.6 0.7 0.8 0.9 1 1.1 1.2 m*/m

Λ=500 MeV Λ=450 MeV Λ=414 MeV

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 200 400 600 800 1000 m*

k/m

Momentum, k [MeV] 200 400 600 800 1000 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 m*

w/m

Momentum, k [MeV]

In preparation, will be publicly available Coraggio, Holt, et al. PRC 89 044321 (2014)

SLIDE 14

Available data

10

5 10 15 20 0.001 0.01 0.1 1 Temperature, T [MeV] Density, ρ [fm-3]

Self-energy, spectral function & thermodynamics Av18

In preparation, will be publicly available

SLIDE 15

Neutron matter

11

Mass-Radius relation from SCGF calculations
Cut-off variation (N3LO) and/or SRG evolution

0.5 1 1.5 2 2.5 3 8 10 12 14 16 Mass, M (Solar Masses) Radius, R (km) N3LO+3BF SLy 0.5 1 1.5 2 2.5 3 8 10 12 14 16 GR P < ∞ Causality

SLIDE 16

Neutron matter

11

Mass-Radius relation from SCGF calculations
Cut-off variation (N3LO) and/or SRG evolution

Hebeler, Lattimer, Pethick, Schwenk ApJ 773 11 (2013)

0.5 1 1.5 2 2.5 3 8 10 12 14 16 Mass, M (Solar Masses) Radius, R (km) N3LO+3BF SLy 0.5 1 1.5 2 2.5 3 8 10 12 14 16 GR P < ∞ Causality

SLIDE 17

Summary

1. Neutron star motivation
2. Infinite matter BCS
3. Beyond-BCS with SCGF methods

12

SLIDE 18

Bardeen-Cooper-Schrieffer pairing

Single-particle spectrum choice:
Angular gap dependence:

13

|∆k|2 = X

L

|∆L

k |2

εk = k2 2m + U(k) − µ

+ BCS equation

SLIDE 19

Bardeen-Cooper-Schrieffer pairing

Single-particle spectrum choice:
Angular gap dependence:

13

|∆k|2 = X

L

|∆L

k |2

εk = k2 2m + U(k) − µ

+ BCS equation

SLIDE 20

Bardeen-Cooper-Schrieffer pairing

Single-particle spectrum choice:
Angular gap dependence:

13

|∆k|2 = X

L

|∆L

k |2

εk = k2 2m + U(k) − µ

|∆k|2 = X

L

|∆L

k |2 ≈ |∆L k |2

+ BCS equation

SLIDE 21

Bardeen-Cooper-Schrieffer pairing

Single-particle spectrum choice:
Angular gap dependence:

13

|∆k|2 = X

L

|∆L

k |2

εk = k2 2m + U(k) − µ

|∆k|2 = X

L

|∆L

k |2 ≈ |∆L k |2

+ BCS equation

SLIDE 22

3PF2 pairing: phase shift equivalence

14

3P2 CDBonn

2 4 6 8 Momentum, q2 [fm-1]

0.2
0.1

0.1 0.2

3P2 Av18

2 4 6 8 Momentum, q2 [fm-1]

0.2
0.1

0.1 0.2

3P2 N3LO

2 4 6 8 Momentum, q1 [fm-1] 2 4 6 8 Momentum, q2 [fm-1]

0.2
0.1

0.1 0.2

10 20

δ [deg] NN phase shifts

1

1 2

δ [deg]

Nij par SAID Nij data CDBonn Av18 N3LO

0.5 1 1.5 2 2.5

qbin [fm

1]
4
3
2
1

δ [deg]

3P2 3F2

ε2

Neutron matter BCS gaps

Srinivias & Ramanan, PRC 94 064303 (2016)

SLIDE 23

3BF effect: estimate

15

0.5 1 1.5 2 2.5 3

1S0

Pairing gap, ∆ [MeV]

BCS NN SRC NN BCS NN+3NF SRC NN+3NF

0.5 1 1.5 2 2.5 3

(a)

0.01 0.1 1 0.0 0.5 1.0 1.5 2.0 2.5

3PF2

Pairing gap, ∆ [MeV] Fermi momentum, kF [fm-1]

0.01 0.1 1 0.0 0.5 1.0 1.5 2.0 2.5

(b)

Singlet gap: 3NF reduce closure
Triplet gap: 3NF increase gap

⇒NN forces

SLIDE 24

Drischler, Kruger, Hebler, Schwenk, PRC 95 024302 (2017)

Uncertainties at the BCS + HF level

16

R=0.9 fm R=1.0 fm R=1.1 fm R=1.2 fm “Hard” “Soft”

SLIDE 25

Drischler, Kruger, Hebler, Schwenk, PRC 95 024302 (2017)

Uncertainties at the BCS + HF level

16

R=0.9 fm R=1.0 fm R=1.1 fm R=1.2 fm “Hard” “Soft”

Srinivias & Ramanan, PRC 94 064303 (2016)

SLIDE 26

Summary

1. Neutron star motivation
2. Infinite matter BCS
3. Beyond-BCS with SCGF methods

17

SLIDE 27

Bozek, Phys. Rev. C 62 054316 (2000); Muether & Dickhoff, Phys. Rev. C 72 054313 (2005)

Beyond BCS 101: SRC

BCS is lowest order in Gorkov Green’s function expansion
T-matrix can be extended to paired systems

18

Normal state

SLIDE 28

Bozek, Phys. Rev. C 62 054316 (2000); Muether & Dickhoff, Phys. Rev. C 72 054313 (2005)

Beyond BCS 101: SRC

BCS is lowest order in Gorkov Green’s function expansion
T-matrix can be extended to paired systems

18

Normal state Superfluid Δ(kF)

SLIDE 29

Bozek, Phys. Rev. C 62 054316 (2000); Muether & Dickhoff, Phys. Rev. C 72 054313 (2005)

Beyond BCS 101: SRC

BCS is lowest order in Gorkov Green’s function expansion
T-matrix can be extended to paired systems

18

Normal state Superfluid Δ(kF)

?

SLIDE 30

+ BCS+SRC gap equation

∆L

k =

X

L0

Z

k0

hk|V LL0

nn |k0i

2 p ¯ χ2

k0 + |∆k0|2 ∆L0 k0

BCS gap equation + Gorkov gap equation

∆L

k =

X

L0

Z

k0

hk|V LL0

nn |k0i

2ξk0 ∆L0

k0

1 2ξk = Z

ω

Z

ω0

1 − f(ω) − f(ω0) ω + ω0 A(k, ω)As(k, ω0)

1 2¯ χk = Z

ω

Z

ω0

1 − f(ω) − f(ω0) ω + ω0 A(k, ω)As(k, ω0)

+ BCS+Z-factor equation

εk = k2 2m + U(k)

+ +

εk = k2 2m +

+

Zk’ Zk

SLIDE 31

Beyond BCS 101: SRC

20

Full off-shell Quasi-particle ⇔ BCS

1 2¯ χk = 1 2|εk − µ|

1 2¯ χk = Z

ω

Z

ω0

1 − f(ω) − f(ω0) ω + ω0 A(k, ω)A(k, ω0) 20 40 60 80 100 0.5 1 1.5 2

Denominator, χ(k) [MeV] Momentum, k/kF

T=20 MeV T=18 MeV T=16 MeV T=15 MeV T=14 MeV T=12 MeV T=11 MeV T=10 MeV T=9 MeV T=8 MeV T=6 MeV T=5 MeV T=4 MeV T=0 MeV

20 40 60 80 100 0.5 1 1.5 2

Av18 kF=1.33 fm-1 qp denominator, χqp(k) [MeV] Momentum, k/kF

SLIDE 32

Neutron matter

21

Beyond BCS 101: SRC

0.5 1 1.5 2 2.5 3

0.0 0.5 1.0 1.5

Pairing gap, ∆ [MeV] Fermi momentum, kF [fm-1]

CDBonn

1S0 BCS

SRC

0.5 1.0 1.5

Fermi momentum, kF [fm-1]

Av18

0.5 1.0 1.5

Fermi momentum, kF [fm-1]

N3LO

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1.0 1.5 2.0 2.5 3.0

Pairing gap, ∆ [MeV] Fermi momentum, kF [fm-1]

CDBonn

3PF2 BCS

SRC 1.0 1.5 2.0 2.5 3.0

Fermi momentum, kF [fm-1]

Av18

1.0 1.5 2.0 2.5 3.0

Fermi momentum, kF [fm-1]

N3LO

+

BCS+SRC

1 2χk = Z

ω

Z

ω0

1 − f(ω) − f(ω0) ω + ω0 A(k, ω)As(k, ω0)

∆L

k =

X

L0

Z

k0

hk|V LL0

nn |k0i

2 p ¯ χ2

k0 + |∆k0|2 ∆L0 k0

1S0 1S0 1S0 3PF2 3PF2 3PF2

SLIDE 33

Neutron matter

22

Beyond BCS 101: SRC

+

BCS+SRC

1 2χk = Z

ω

Z

ω0

1 − f(ω) − f(ω0) ω + ω0 A(k, ω)As(k, ω0)

∆L

k =

X

L0

Z

k0

hk|V LL0

nn |k0i

2 p ¯ χ2

k0 + |∆k0|2 ∆L0 k0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5

1S0

BCS NN Pairing gap, Δ [MeV]

N3LO Λ=414 MeV Λ=450 MeV Λ=500 MeV CDBonn Av18

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5

(a)

0.0 0.5 1.0 1.5

1S0

BCS NN+NNN

N3LO+NNN

0.0 0.5 1.0 1.5

(b)

0.0 0.5 1.0 1.5

1S0

BCS+εk+Ζk

N3LO+NNN CDBonn Av18

0.0 0.5 1.0 1.5

(c)

0.0 0.5 1.0 1.5

1S0

SRC

N3LO+NNN CDBonn Av18

0.0 0.5 1.0 1.5

(d)

10-2 10-1 100 1.0 1.5 2.0

3PF2

Pairing gap, Δ [MeV] Fermi momentum, kF [fm-1] 10-2 10-1 100 1.0 1.5 2.0

(e)

1.0 1.5 2.0

3PF2

Fermi momentum, kF [fm-1] 1.0 1.5 2.0

(f)

1.0 1.5 2.0

3PF2

Fermi momentum, kF [fm-1] 1.0 1.5 2.0

(g)

1.0 1.5 2.0

3PF2

Fermi momentum, kF [fm-1] 1.0 1.5 2.0

(h)

SLIDE 34

Massive gaps 3SD1 channel but…
No evidence of strong np nuclear pairing
3NF do not alter picture significantly
Short-range correlations deplete gap

Beyond BCS 101: 3SD1

23

Symmetric matter

Muether & Dickhoff, PRC 72 054313 (2005) Rios, Polls & Dickhoff, arXiv:1707.04140 Maurizio, Holt & Finelli, PRC 90, 044003 (2014)

U. Lombardo’s talk

SLIDE 35

Massive gaps 3SD1 channel but…
No evidence of strong np nuclear pairing
3NF do not alter picture significantly
Short-range correlations deplete gap

Beyond BCS 101: 3SD1

23

Symmetric matter

Muether & Dickhoff, PRC 72 054313 (2005) Rios, Polls & Dickhoff, arXiv:1707.04140 Maurizio, Holt & Finelli, PRC 90, 044003 (2014)

U. Lombardo’s talk

SLIDE 36

Cao, Lombardo & Schuck, Phys Rev C 74 064301 (2006)

Beyond BCS 201: LRC

24

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

?

✔

𝒲pair=

ph recoupled G-matrix Effective Landau parameters

ΛST (q) = Λ0

ST (q)

1 − Λ0

ST (q) × FST

h1¯ 1|V|1¯ 1i = 1 4 X

2,20

X

S,T

()S(2S + 1)h12|Gph

ST |1020iAh20¯

1|Gph

ST |2¯

10iAΛ(220)

SLIDE 37

Landau parameters

25

5
4
3
2
1

1 2 3 1 2 3 4 5

3P2

(b)

Diagonal V [MeVfm-3] Momentum, k [fm-1]

1 2 3 4 5-0.8

0.6
0.4
0.2

VS=1

LRC

(d)

Momentum, k [fm-1]

30
20
10

10 20 30 1S0

(a)

Diagonal V [MeVfm-3]

0.2 0.4 0.6 0.8

VS=0

LRC

(c)

0.5 1 1.5 2 2.5 3 Fermi momentum, kF [fm

1]

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

F0 G0 F0/(1+F0)+G0/(1+G0)

Landau parameters

“Bare” Effective

Effective Landau parameters
Not consistent (yet) with NN force
LRC in nuclei will be different
Small correction
Repulsive in singlet
Attractive in triple

SLIDE 38

Neutron matter

26

Beyond BCS 201: results 1S0

0.5 1 1.5 2 2.5 3

0.0 0.5 1.0 1.5

Pairing gap, ∆ [MeV] Fermi momentum, kF [fm-1]

CDBonn

1S0 BCS

SRC SRC+LRC

0.5 1.0 1.5

Fermi momentum, kF [fm-1]

Av18

0.5 1.0 1.5

Fermi momentum, kF [fm-1]

N3LO

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1.0 1.5 2.0 2.5 3.0

Pairing gap, ∆ [MeV] Fermi momentum, kF [fm-1]

CDBonn

3PF2 BCS

SRC SRC+LRC 1.0 1.5 2.0 2.5 3.0

Fermi momentum, kF [fm-1]

Av18

1.0 1.5 2.0 2.5 3.0

Fermi momentum, kF [fm-1]

N3LO

LRC 1S0 (3PF2) produces (anti-)screening

1S0 1S0 1S0 3PF2 3PF2 3PF2

SLIDE 39

3BF effect: uncertainty estimate

27

Singlet gap: 3NF reduce closure
Triplet gap: 3NF increase gap
LRC effect to be explored

1 2

⇒spectrum ⇒NN forces

SLIDE 40

3BF effect: uncertainty estimate

27

Singlet gap: 3NF reduce closure
Triplet gap: 3NF increase gap
LRC effect to be explored

1 2

⇒spectrum ⇒NN forces

SLIDE 41

Ho et al., Sci. Adv. 2015;1:e1500578 Ho et al., 0.7566/JPSCP.14.010805

Superfluid in the core

28

SLIDE 42

Collaborators

29

+A. Carbone

+D. Ding, W. H. Dickhoff +A. Polls

+C. Barbieri +V. Somà +H. Arellano, F. Isaule

UNIVERSITAT DE BARCELONA

U B

SLIDE 43

Ab initio nuclear theory to treat beyond-BCS

correlations

Different NN forces provide robust predictions
Challenges ahead
Full self-consistent Gorkov
Consistent treatment of LRC
Pairing in isospin asymmetric matter

Conclusions

30