Arnau Rios HuguetLecturer in Nuclear Theory

Department of PhysicsUniversity of Surrey

Pairing in high-density neutron matter

Nuclear Astrophysics Milan 18 September 2017

Summary

1. Neutron star motivation

2. Infinite matter BCS

3. Beyond-BCS with SCGF methods

2

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

Complications

4

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

0

30

60

90

δ [

deg

]

Nij

CDBonnAv18N3LONij93

0 50 100 150 200 250E [MeV]

0

60

120

180

δ [

deg

]

1S

0

3S

1

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

Complications

4

NN interaction is not unique

•Non-uniqueness of nucleon forces

...but phase-shift equivalent!

0

30

60

90

δ [

deg

]

Nij

CDBonnAv18N3LONij93

0 50 100 150 200 250E [MeV]

0

60

120

180

δ [

deg

]

1S

0

3S

1

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

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)

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)

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

OˆQ

˙

„ 1 GeV

ci

ci

π

ci

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

OˆQ

˙

„ 1 GeV

ci

ci

π

ci

In-medium interaction Ladder self-energy

Dyson equationpp & hh Pauli blocking

Spectral function

One-body properties Momentum distributionThermodynamics & 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 0 100 2000

200

400

600−30

−25

−20

−15

−10

−5

0

Ω−2µ [MeV]P [MeV]

Re

T(Ω

+; P,q

=0) [

MeV

fm3]

T-matrix at T=5 MeV

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

Spect

ral f

unct

ion, A

(k,ω

)

ω−µ [MeV]

k=0

-500 -300 -100 100 300ω−µ [MeV]

k=kF

-500 -300 -100 100 300ω−µ [MeV]

k=2kF

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

22.5 23

23.5 24

24.5 25

25.5 26

26.5 27

27.5

Av18 r=0.16

µ [

Me

V]

dataextrap

14

16

18

20

22

24

26

28

E/A

[M

eV

]44

46

48

50

52

54

56

58

60

0 5 10 15 20

K/A

[M

eV

]

T [MeV]0 5 10 15 20

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

n(k

=0

)

T [MeV]

22.5 23

23.5 24

24.5 25

25.5 26

26.5 27

27.5

Av18 r=0.16

µ [

Me

V]

dataextrap

14

16

18

20

22

24

26

28

E/A

[M

eV

]

44

46

48

50

52

54

56

58

60

0 5 10 15 20

K/A

[M

eV

]

T [MeV]0 5 10 15 20

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

n(k

=0

)

T [MeV]

Single-particle occupationMomentum 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 “Aa:kak

E

ªd3k

p2q3npkq “

Neutron matter

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

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 MeVT=18 MeVT=16 MeVT=14 MeVT=12 MeV

T=10 MeVT=8 MeVT=6 MeVT=4 MeVT=0 MeV

0.6

0.7

0.8

0.9

1

1.1

1.2

m*/m

Λ=500 MeVΛ=450 MeVΛ=414 MeV

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 200 400 600 800 1000

m*k/m

Momentum, k [MeV]0 200 400 600 800 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)

Available data

10

0

5

10

15

20

0.001 0.01 0.1 1

Te

mp

era

ture

, T

[M

eV

]

Density, ρ [fm-3]

Self-energy, spectral function & thermodynamics

Av18

In preparation, will be publicly available

Neutron matter

11

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

0

0.5

1

1.5

2

2.5

3

8 10 12 14 16

Mass,M(SolarMasses)

Radius, R (km)

N3LO+3BFSLy

0

0.5

1

1.5

2

2.5

3

8 10 12 14 16

GR

P <∞

Causality

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

0.5

1

1.5

2

2.5

3

8 10 12 14 16

Mass,M(SolarMasses)

Radius, R (km)

N3LO+3BFSLy

0

0.5

1

1.5

2

2.5

3

8 10 12 14 16

GR

P <∞

Causality

Summary

1. Neutron star motivation

2. Infinite matter BCS

3. Beyond-BCS with SCGF methods

12

Bardeen-Cooper-Schrieffer pairing

•Single-particle spectrum choice:

•Angular gap dependence:13

|k|2 =X

L

|Lk |2

"k =k2

2m+ U(k) µ

+

BCS equation

Bardeen-Cooper-Schrieffer pairing

•Single-particle spectrum choice:

•Angular gap dependence:13

|k|2 =X

L

|Lk |2

"k =k2

2m+ U(k) µ

+

BCS equation

Bardeen-Cooper-Schrieffer pairing

•Single-particle spectrum choice:

•Angular gap dependence:13

|k|2 =X

L

|Lk |2

"k =k2

2m+ U(k) µ

|k|2 =X

L

|Lk |2 |L

k |2

+

BCS equation

Bardeen-Cooper-Schrieffer pairing

•Single-particle spectrum choice:

•Angular gap dependence:13

|k|2 =X

L

|Lk |2

"k =k2

2m+ U(k) µ

|k|2 =X

L

|Lk |2 |L

k |2

+

BCS equation

3PF2 pairing: phase shift equivalence

14

3P2 CDBonn

0

2

4

6

8

Mo

me

ntu

m,

q2 [

fm-1

]

-0.2

-0.1

0

0.1

0.2

3F2 CDBonn

-0.05

0.05

0.15

0.25

0.35

0.45

0.55

0.65

3P2 Av18

0

2

4

6

8

Mo

me

ntu

m,

q2 [

fm-1

]

-0.2

-0.1

0

0.1

0.2

3F2 Av18

-0.05

0.05

0.15

0.25

0.35

0.45

0.55

0.65

3P2 N3LO

0 2 4 6 8Momentum, q1 [fm-1]

0

2

4

6

8

Mo

me

ntu

m,

q2 [

fm-1

]

-0.2

-0.1

0

0.1

0.2

3F2 N3LO

0 2 4 6 8Momentum, q1 [fm-1]

-0.05

0.05

0.15

0.25

0.35

0.45

0.55

0.65

0

10

20

δ [

deg

]

NN phase shifts

-1

0

1

2δ [

deg

]

Nij par

SAIDNij data

CDBonnAv18N3LO

0 0.5 1 1.5 2 2.5

qbin

[fm-1

]

-4

-3

-2

-1

0

δ [

deg

]

3P

2

3F

2

ε2

Neutron matter BCS gaps

Srinivias & Ramanan, PRC 94 064303 (2016)

3BF effect: estimate

15

0

0.5

1

1.5

2

2.5

31S0

Pairin

g g

ap,

∆ [M

eV

]

BCS NNSRC NNBCS NN+3NFSRC NN+3NF

0

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

Pa

irin

g g

ap,

∆ [

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

1

2

GII

⇒NN forces

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”

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)

Summary

1. Neutron star motivation

2. Infinite matter BCS

3. Beyond-BCS with SCGF methods

17

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

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)

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)

?

+

BCS+SRC gap equationL

k = X

L0

Z

k0

hk|V LL0

nn |k0i2p2k0 + |k0 |2

L0

k0

BCS gap equation

+

Gorkov gap equation

Lk =

X

L0

Z

k0

hk|V LL0

nn |k0i2k0

L0

k0

1

2k=

Z

!

Z

!0

1 f(!) f(!0)

! + !0 A(k,!)As(k,!0)

1

2k=

Z

!

Z

!0

1 f(!) f(!0)

! + !0 A(k,!)As(k,!0)

+BCS+Z-factor equation

"k =k2

2m+ U(k) µ+

+"k =

k2

2m+ U(k) µ+

Zk’Zk

Beyond BCS 101: SRC

20

Full off-shell Quasi-particle ⇔ BCS1

2k=

1

2|"k µ|1

2k=

Z

!

Z

!0

1 f(!) f(!0)

! + !0 A(k,!)A(k,!0)

0

20

40

60

80

100

0 0.5 1 1.5 2

De

no

min

ato

r, χ

(k)

[Me

V]

Momentum, k/kF

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

0

20

40

60

80

100

0 0.5 1 1.5 2

Av18 kF=1.33 fm-1

qp

de

no

min

ato

r, χ

qp(k

) [M

eV

]

Momentum, k/kF

Neutron matter

21

Beyond BCS 101: SRC

0.5

1

1.5

2

2.5

3

0.0 0.5 1.0 1.5

Pa

irin

g g

ap

, ∆

[M

eV

]

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

Pa

irin

g g

ap

, ∆

[M

eV

]

Fermi momentum, kF [fm-1]

CDBonn3PF2 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

2k=

Z

!

Z

!0

1 f(!) f(!0)

! + !0 A(k,!)As(k,!0)L

k = X

L0

Z

k0

hk|V LL0

nn |k0i2p2k0 + |k0 |2

L0

k0

1S01S0

1S0

3PF23PF2 3PF2

Neutron matter

22

Beyond BCS 101: SRC

+

BCS+

SRC

1

2k=

Z

!

Z

!0

1 f(!) f(!0)

! + !0 A(k,!)As(k,!0)L

k = X

L0

Z

k0

hk|V LL0

nn |k0i2p2k0 + |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

Pairinggap,Δ[MeV]

N3LOΛ=414 MeVΛ=450 MeVΛ=500 MeVCDBonnAv18

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+NNNCDBonnAv18

0.0 0.5 1.0 1.5

(c)

0.0 0.5 1.0 1.5

1S0

SRC

N3LO+NNNCDBonnAv18

0.0 0.5 1.0 1.5

(d)

10-2

10-1

100

1.0 1.5 2.0

3PF2

Pairinggap,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)

•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

•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

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

?

𝒱pair=

ph recoupled G-matrix Effective Landau

parameters

ST (q) =0ST (q)

1 0ST (q) FST

h11|V|11i = 1

4

X

2,20

X

S,T

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

020iAh201|GphST |21

0iA(220)

Landau parameters

25

-5

-4

-3

-2

-1

0

1

2

3

0 1 2 3 4 5

3P2

(b)Dia

gonal V

[M

eV

fm-3

]

Momentum, k [fm-1]

0 1 2 3 4 5-0.8

-0.6

-0.4

-0.2

0

VS=1LRC

(d)

Momentum, k [fm-1]

-30

-20

-10

0

10

20

301S0

(a)Dia

gonal V

[M

eV

fm-3

]

0

0.2

0.4

0.6

0.8

VS=0LRC

(c)

0.5 1 1.5 2 2.5 3Fermi momentum, k

F [fm

-1]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6 F0

G0

F0/(1+F

0)+G

0/(1+G

0)

Lan

dau

par

amet

ers

“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

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

Pairin

g g

ap,

∆ [M

eV

]

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

Pairin

g g

ap,

∆ [M

eV

]

Fermi momentum, kF [fm-1]

CDBonn3PF2 BCS

SRCSRC+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

1S01S0

1S0

3PF23PF2 3PF2

3BF effect: uncertainty estimate

27

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

1

2

GII

⇒spectrum

⇒NN forces

3BF effect: uncertainty estimate

27

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

1

2

GII

⇒spectrum

⇒NN forces

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

Superfluid in the core

28

Collaborators

29

+A. Carbone

+D. Ding, W. H. Dickhoff

+A. Polls

+C. Barbieri

+V. Somà

+H. Arellano, F. Isaule

UNIVERSITAT DE BARCELONAU

B

•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

a.rios@surrey.ac.uk@riosarnau

Postdoc position opening imminently