? looking for the pieces of the puzzle n Diego Lonardoni FRIB - - PowerPoint PPT Presentation

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? looking for the pieces of the puzzle n Diego Lonardoni FRIB - - PowerPoint PPT Presentation

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From hypernuclei to neutron stars: p ? looking for the pieces of the puzzle n Diego Lonardoni FRIB Theory Fellow In collaboration with: Stefano Gandolfi, LANL Alessandro Lovato, ANL Francesco Pederiva, Trento

slide-1
SLIDE 1

In collaboration with:

✓

Stefano Gandolfi, LANL

✓

Alessandro Lovato, ANL

✓

Francesco Pederiva, Trento

✓

Francesco Catalano, Uppsala

Diego Lonardoni

FRIB Theory Fellow

From hypernuclei to neutron stars: looking for the pieces of the puzzle

Bormio, January 28, 2016

?

Λ

p n

slide-2
SLIDE 2

2

Strangeness in neutron stars: the hyperon puzzle

> 2.0M

stiff

M R

12 km

E ρb

ρ0

NS

n p e µ

slide-3
SLIDE 3

2

Strangeness in neutron stars: the hyperon puzzle

< 2.0M > 2.0M

stiff

M R

12 km

E ρb

ρ0

soft

µY = µN ρth

Y ' 2 3ρ0

NS

Λ Σ Ξ n p e µ

slide-4
SLIDE 4

2

Strangeness in neutron stars: the hyperon puzzle

< 2.0M > 2.0M

stiff

M R

12 km

E ρb

ρ0

∼ 2M

  • bs:

soft

µY = µN ρth

Y ' 2 3ρ0

NS

Λ Σ Ξ n p e µ

slide-5
SLIDE 5

2

Strangeness in neutron stars: the hyperon puzzle

< 2.0M > 2.0M

stiff

M R

12 km

E ρb

ρ0

∼ 2M

  • bs:

soft

µY = µN ρth

Y ' 2 3ρ0

NS

Λ Σ Ξ n p e µ

?

Hyperon puzzle

✓ Indication for the appearance of hyperons

in NS core

✓ Apparent inconsistency between theoretical

calculations and observations

slide-6
SLIDE 6

2

Strangeness in neutron stars: the hyperon puzzle

< 2.0M > 2.0M

stiff

M R

12 km

E ρb

ρ0

∼ 2M

  • bs:

soft

µY = µN ρth

Y ' 2 3ρ0

NS

Λ Σ Ξ n p e µ

?

Hyperon puzzle

✓ Indication for the appearance of hyperons

in NS core

✓ Apparent inconsistency between theoretical

calculations and observations Quantum Monte Carlo YN interaction

slide-7
SLIDE 7

3

Strangeness in QMC calculations

Quantum Monte Carlo (Auxiliary Field Diffusion Monte Carlo)

τ → ∞

projection

τ = it/~

imaginary time

|ψ(τ)i = e−(H−E0)τ|ψ(0)i ∂ ∂τ |ψ(τ)i = (H E0)|ψ(τ)i c0|ϕ0i

slide-8
SLIDE 8

3

Strangeness in QMC calculations

Quantum Monte Carlo (Auxiliary Field Diffusion Monte Carlo)

τ → ∞

projection

τ = it/~

imaginary time

|ψ(τ)i = e−(H−E0)τ|ψ(0)i ∂ ∂τ |ψ(τ)i = (H E0)|ψ(τ)i c0|ϕ0i

✓

nucleon-nucleon phenomenological interaction: Argonne & Urbana

H = X

i

p2

i

2mN + X

i<j

vij + X

i<j<k

vijk NN

scattering deuteron nuclei nuclear matter 2B: 3B: + +

slide-9
SLIDE 9

4

Strangeness in QMC calculations

Quantum Monte Carlo (Auxiliary Field Diffusion Monte Carlo)

τ → ∞

projection

τ = it/~

imaginary time

|ψ(τ)i = e−(H−E0)τ|ψ(0)i ∂ ∂τ |ψ(τ)i = (H E0)|ψ(τ)i c0|ϕ0i

✓

nucleon-nucleon phenomenological interaction: Argonne & Urbana

✓

hyperon-nucleon phenomenological interaction: Argonne like

H = X

i

p2

i

2mN + X

i<j

vij + X

i<j<k

vijk + X

λ

p2

λ

2mΛ + X

λ,i

vλi+ X

λ,i<j

vλij

scattering CSB* 2B: 3B:

A = 4 Λp

+

slide-10
SLIDE 10

4

Strangeness in QMC calculations

Quantum Monte Carlo (Auxiliary Field Diffusion Monte Carlo)

τ → ∞

projection

τ = it/~

imaginary time

|ψ(τ)i = e−(H−E0)τ|ψ(0)i ∂ ∂τ |ψ(τ)i = (H E0)|ψ(τ)i c0|ϕ0i

✓

nucleon-nucleon phenomenological interaction: Argonne & Urbana

✓

hyperon-nucleon phenomenological interaction: Argonne like

H = X

i

p2

i

2mN + X

i<j

vij + X

i<j<k

vijk + X

λ

p2

λ

2mΛ + X

λ,i

vλi+ X

λ,i<j

vλij

scattering CSB* 2B: 3B:

A = 4 Λp

no unique fit +

slide-11
SLIDE 11

5

Strangeness in QMC calculations

Quantum Monte Carlo (Auxiliary Field Diffusion Monte Carlo)

τ → ∞

projection

τ = it/~

imaginary time

|ψ(τ)i = e−(H−E0)τ|ψ(0)i ∂ ∂τ |ψ(τ)i = (H E0)|ψ(τ)i c0|ϕ0i

✓

nucleon-nucleon phenomenological interaction: Argonne & Urbana

✓

hyperon-nucleon phenomenological interaction: Argonne like

H = X

i

p2

i

2mN + X

i<j

vij + X

i<j<k

vijk + X

λ

p2

λ

2mΛ + X

λ,i

vλi+ X

λ,i<j

vλij

3B: no unique fit use QMC to fit hyp. exp. data

BΛ =E(A−1Z)−E(A

ΛZ)

slide-12
SLIDE 12

6

Strangeness in nuclei

BΛ [MeV] A-2/3

s

emulsion (K-,π-) (π+,K+) (e,e’K+) 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 0.0 0.1 0.2 0.3 0.4 0.5

  • D. L., F. Pederiva, S. Gandolfi, Phys. Rev. C 89, 014314 (2014)
slide-13
SLIDE 13

7

Strangeness in nuclei

  • D. L., F. Pederiva, S. Gandolfi, Phys. Rev. C 89, 014314 (2014)

BΛ [MeV] A-2/3

s

emulsion (K-,π-) (π+,K+) (e,e’K+) ΛN 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 0.0 0.1 0.2 0.3 0.4 0.5

17 ΛO 16 ΛO 5 ΛHe 4 ΛH, 4 ΛHe

3 ΛH

slide-14
SLIDE 14

8

Strangeness in nuclei

3 ΛH

VMC param

91 ΛZr 5 ΛHe 49 ΛCa 4 ΛH, 4 ΛHe

3 ΛH

41 ΛCa 17 ΛO 16 ΛO

  • D. L., F. Pederiva, S. Gandolfi, Phys. Rev. C 89, 014314 (2014)

BΛ [MeV] A-2/3

s

emulsion (K-,π-) (π+,K+) (e,e’K+) ΛN ΛN + ΛNN (I) 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 0.0 0.1 0.2 0.3 0.4 0.5

  • A. A. Usmani, Phys. Rev. C 52,

1773-1777 (1995)

(I)

slide-15
SLIDE 15

8

Strangeness in nuclei

3 ΛH

VMC param

91 ΛZr 5 ΛHe 49 ΛCa 4 ΛH, 4 ΛHe

3 ΛH

41 ΛCa 17 ΛO 16 ΛO

  • D. L., F. Pederiva, S. Gandolfi, Phys. Rev. C 89, 014314 (2014)

BΛ [MeV] A-2/3

s

emulsion (K-,π-) (π+,K+) (e,e’K+) ΛN ΛN + ΛNN (I) 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 0.0 0.1 0.2 0.3 0.4 0.5

  • A. A. Usmani, Phys. Rev. C 52,

1773-1777 (1995)

(I)

slide-16
SLIDE 16

9

Strangeness in nuclei

  • D. L., F. Pederiva, S. Gandolfi, Phys. Rev. C 89, 014314 (2014)
  • F. Pederiva, F. Catalano, D. L., A. Lovato, S. Gandolfi, arXiv:1506.04042 (2015)

AFDMC

fit

VMC param

BΛ [MeV] A-2/3

s

emulsion (K-,π-) (π+,K+) (e,e’K+) ΛN ΛN + ΛNN (I) ΛN + ΛNN (II) 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 0.0 0.1 0.2 0.3 0.4 0.5

91 ΛZr 5 ΛHe 49 ΛCa 4 ΛH, 4 ΛHe

3 ΛH

41 ΛCa 17 ΛO 16 ΛO

unique fit

slide-17
SLIDE 17

9

Strangeness in nuclei

  • D. L., F. Pederiva, S. Gandolfi, Phys. Rev. C 89, 014314 (2014)
  • F. Pederiva, F. Catalano, D. L., A. Lovato, S. Gandolfi, arXiv:1506.04042 (2015)

AFDMC

fit

VMC param

BΛ [MeV] A-2/3

s

emulsion (K-,π-) (π+,K+) (e,e’K+) ΛN ΛN + ΛNN (I) ΛN + ΛNN (II) 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 0.0 0.1 0.2 0.3 0.4 0.5

91 ΛZr 5 ΛHe 49 ΛCa 4 ΛH, 4 ΛHe

3 ΛH

41 ΛCa 17 ΛO 16 ΛO

unique fit three-body repulsive hyperon-nucleon force

slide-18
SLIDE 18

10

Strangeness in nuclei

  • F. Pederiva, F. Catalano, D. L., A. Lovato, S. Gandolfi, arXiv:1506.04042 (2015)

BΛ [MeV] A-2/3

208 89 40 28 16 12 9 7 6 5 4 3 s p d f g

emulsion (K-,π-) (π+,K+) (e,e’K+) AFDMC 0.0 5.0 10.0 15.0 20.0 25.0 30.0 0.0 0.1 0.2 0.3 0.4 0.5

slide-19
SLIDE 19

11

Strangeness in nuclei

  • F. Pederiva, F. Catalano, D. L., A. Lovato, S. Gandolfi, arXiv:1506.04042 (2015)

BΛ [MeV] A-2/3

208 89 40 28 16 12 9 7 6 5 4 3 s p d f g

emulsion (K-,π-) (π+,K+) (e,e’K+) AFDMC 0.0 5.0 10.0 15.0 20.0 25.0 30.0 0.0 0.1 0.2 0.3 0.4 0.5

no re-fit !!

slide-20
SLIDE 20

11

Strangeness in nuclei

  • F. Pederiva, F. Catalano, D. L., A. Lovato, S. Gandolfi, arXiv:1506.04042 (2015)

BΛ [MeV] A-2/3

208 89 40 28 16 12 9 7 6 5 4 3 s p d f g

emulsion (K-,π-) (π+,K+) (e,e’K+) AFDMC 0.0 5.0 10.0 15.0 20.0 25.0 30.0 0.0 0.1 0.2 0.3 0.4 0.5

no re-fit !! good interaction

slide-21
SLIDE 21

12

Strangeness in nuclei

  • F. Pederiva, F. Catalano, D. L., A. Lovato, S. Gandolfi, arXiv:1506.04042 (2015)

BΛ [MeV] A-2/3

208 89 40 28 16 12 9 7 6 5 4 3 s p d f g

emulsion (K-,π-) (π+,K+) (e,e’K+) AFDMC 0.0 5.0 10.0 15.0 20.0 25.0 30.0 0.0 0.1 0.2 0.3 0.4 0.5

statistics ?? calibration ??

slide-22
SLIDE 22

13

Strangeness in nuclei

  • F. Pederiva, F. Catalano, D. L., A. Lovato, S. Gandolfi, arXiv:1506.04042 (2015)
  • T. Gogami, et al., arXiv:1511.04801 (2015), arXiv:1511.02472 (2015)

BΛ [MeV] A-2/3

208 89 40 28 16 12 9 7 6 5 4 3 s p d f g

emulsion (K-,π-) (π+,K+) (e,e’K+) AFDMC 0.0 5.0 10.0 15.0 20.0 25.0 30.0 0.0 0.1 0.2 0.3 0.4 0.5

statistics ?? calibration ??

0.54(4) MeV

slide-23
SLIDE 23

13

Strangeness in nuclei

  • F. Pederiva, F. Catalano, D. L., A. Lovato, S. Gandolfi, arXiv:1506.04042 (2015)
  • T. Gogami, et al., arXiv:1511.04801 (2015), arXiv:1511.02472 (2015)

BΛ [MeV] A-2/3

208 89 40 28 16 12 9 7 6 5 4 3 s p d f g

emulsion (K-,π-) (π+,K+) (e,e’K+) AFDMC 0.0 5.0 10.0 15.0 20.0 25.0 30.0 0.0 0.1 0.2 0.3 0.4 0.5

statistics ?? calibration ?? need of reliable data

0.54(4) MeV

slide-24
SLIDE 24

14

Strangeness in neutron stars

2.45(1)M 0.66(2)M M [M0] R [km] PNM N N + NN (I) 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 10 11 12 13 14 15 16 PSR J1614-2230 PSR J0348+0432 1.36(5)M

  • D. L., A. Lovato, S. Gandolfi, F. Pederiva, Phys. Rev. Lett. 114, 092301 (2015)

ρth

Λ = 0.24(1) fm−3

ρth

Λ = 0.34(1) fm−3

slide-25
SLIDE 25

15

Strangeness in neutron stars

2.45(1)M M [M0] R [km] PNM N N + NN (I) N + NN (II) 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 10 11 12 13 14 15 16 PSR J1614-2230 PSR J0348+0432 1.36(5)M 2.09(1)M

  • D. L., A. Lovato, S. Gandolfi, F. Pederiva, Phys. Rev. Lett. 114, 092301 (2015)

0.66(2)M ρth

Λ = 0.24(1) fm−3

ρth

Λ = 0.34(1) fm−3

ρth

Λ > 0.56 fm−3

no hyperon formation for

ΛNN (II) up to ρ ∼ 3.5ρ0

slide-26
SLIDE 26

15

Strangeness in neutron stars

2.45(1)M M [M0] R [km] PNM N N + NN (I) N + NN (II) 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 10 11 12 13 14 15 16 PSR J1614-2230 PSR J0348+0432 1.36(5)M 2.09(1)M

  • D. L., A. Lovato, S. Gandolfi, F. Pederiva, Phys. Rev. Lett. 114, 092301 (2015)

0.66(2)M

strong dependence upon the details of the three-body hyperon-nucleon force

ρth

Λ = 0.24(1) fm−3

ρth

Λ = 0.34(1) fm−3

ρth

Λ > 0.56 fm−3

no hyperon formation for

ΛNN (II) up to ρ ∼ 3.5ρ0

slide-27
SLIDE 27

16

Strangeness in nuclei

3-body interaction fit on symmetric hypernuclei singlet or triplet nucleon isospin state force: no dependence on

ΛNN

slide-28
SLIDE 28

16

Strangeness in nuclei

3-body interaction fit on symmetric hypernuclei

τi · τj = −3 PT =0 + PT =1 −3 PT =0 + CT PT =1

singlet or triplet nucleon isospin state force: no dependence on

ΛNN

control parameter: strength and sign of the nucleon isospin triplet channel isospin projectors sensitivity study: light- & medium-heavy hypernuclei

slide-29
SLIDE 29

17

Strangeness in nuclei

  • F. Pederiva, F. Catalano, D. L., A. Lovato, S. Gandolfi, arXiv:1506.04042 (2015)

BΛ [MeV] A-2/3

208 89 40 28 16 12 9 7 6 5 4 3 s

experiments CT = 1.0 CT = 1.5 CT = 0.0 CT = -1.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 0.0 0.1 0.2 0.3 0.4 0.5

slide-30
SLIDE 30

17

Strangeness in nuclei

  • F. Pederiva, F. Catalano, D. L., A. Lovato, S. Gandolfi, arXiv:1506.04042 (2015)

BΛ [MeV] A-2/3

208 89 40 28 16 12 9 7 6 5 4 3 s

experiments CT = 1.0 CT = 1.5 CT = 0.0 CT = -1.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 0.0 0.1 0.2 0.3 0.4 0.5

need of medium-heavy neutron-rich hypernuclei

slide-31
SLIDE 31

18

Conclusions

✓ The observation of massive neutron stars reopened the debate about the

presence of hyperons in the inner core

  • no general agreement among theoretical calculations
  • hyperon puzzle not yet solved: new hints?

✓ We developed a quantum Monte Carlo algorithm to study finite and infinite

hypernuclear systems:

  • a repulsive three-body ΛNN force is needed to reproduce the

experimental Λ separation energies for light- and medium-heavy hypernuclei

  • the predicted neutron star equation of state and maximum mass

strongly depend upon the details of the three-body ΛNN force

✓ Need of more constraints on hypernuclear interactions before drawing

conclusions on the role played by hyperons in neutron stars

  • accurate experimental investigation: medium-heavy

neutron-rich hypernuclei

  • accurate theoretical investigation

?

slide-32
SLIDE 32

19

Thank you!!

slide-33
SLIDE 33

20

Backup: the hyperon puzzle

X-ray/optical binaries Double– neutron star binaries White dwarf– neutron star binaries Main sequence– neutron star binaries Black widow pulsar Hulse–Taylor binary In M15 Double pulsar In NGC 6544 In NGC 6539 In Ter 5 2.95 ms pulsar In 47 Tuc In NGC 1851 In M5 In NGC 6440 In NGC 6441 In NGC 6752 4U1700-37 (32) Vela X-1 (33) Cyg X-2 (34) 4U 1538-52 (33) SMC X-1 (33) LMC X-4 (33) Cen X-3 (33) Her X-1 (33) XTE J2123-058 (35) 2S 0921-630 (36) 4U 1822-371 (37) EXO 1722-363 (38) B1957+20 (39) IGR J18027-2016 (40) J1829+2456 (42) J1829+2456 comp. (42) J1811-1736 (43) J1811-1736 comp. (43) J1906+0746 (44) J1906+0746 comp. (44) J1518+4904 (27) J1518+4904 comp. (27) B1534+12 (45) B1534+12 comp. (45) B1913+16 (46) B1913+16 comp. (46) B2127+11C (47) B2127+11C comp. (47) J0737-3039A (48) J0737-3039B (48) J1756-2251 (49) J1756-2251 comp. (49) J1807-2500B (29) J1807-2500B comp. ? (29) B2303+46 (31) J1012+5307 (50) J1713+0747 (51) B1802-07 (31) B1855+09 (52) J0621+1002 (53) J0751+1807 (53) J0437-4715 (54) J1141-6545 (55) J1748-2446I (56) J1748-2446J (56) J1909-3744 (57) J0024-7204H (56) B1802-2124 (58) J051-4002A (56) B1516+02B (59) J1748-2021B (60) J1750-37A (60) J1738+0333 (61) B1911-5958A (62) J1614-2230 (63) J2043+1711 (64) J1910+1256 (28) J2106+1948 (28) J1853+1303 (28) J1045-4509 (31) J1804-2718 (31) J2019+2425 (65) J0045-7319 (31) J1903+0327 (66) 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Neutron star mass (M◉)

  • J. M. Lattimer, Annu. Rev. Nucl. Part. Sci. 2012. 62:485–515

2010:

Mmax = 1.97(4)M

  • P. B. Demorest et al.

Nature 467, 1081 (2010)

2013:

Mmax = 2.01(4)M

  • J. Antoniadis et al.

Science 340, 1233232 (2013)

< 2010:

Mmax = 1.67(2)M

  • D. J. Champion et al.

Science 320, 1309 (2008)

slide-34
SLIDE 34

21

Backup: the hyperon puzzle

< 2.0M > 2.0M

stiff

M R

12 km

E ρb

ρ0

∼ 2M

  • bs:

✓ Approximated theoretical

many-body techniques

✓ Interactions poorly known

?

Hyperon puzzle soft

µY = µN ρth

Y ' 2 3ρ0

binding energies:

nuc : ∼ 3340 Λ hyp : ∼ 41 ΛΛ hyp : ∼ 5 Σ hyp : ∼ (1)

scattering data:

NN : ∼ 4300 Y N : ∼ 52

NS

Λ Σ Ξ n p e µ

QMC YN interaction

slide-35
SLIDE 35

22

Backup: the hyperon puzzle

R ∼ 12 km M ∼ 1.4 M

NS

  • P. Haensel, A. Y. Potekhin, D. G. Yakovlev

Neutron Stars 1, Springer 2007 ρ0 = 0.16 fm−3

ρ

ρ0 = 0.16 fm−3

?

Λ Σ Ξ πc Kc qp

940 MeV

1116 MeV

1200 MeV 1300 MeV

p n

3

Q = −1 : µb− = µn + µe Q = 0 : µb0 = µn Q = +1 : µb+ = µn − µe npe

slide-36
SLIDE 36

23

Backup: the hyperon puzzle

En

F

N

stiff

E ρb

ρ0

PNM

M R

12 km

TOV 1:1

> 2.0M

R ∼ 12 km M ∼ 1.4 M

NS

?

Λ Σ Ξ πc Kc qp n

n n n n

n

n n n n

n

n

n

n n

npe

slide-37
SLIDE 37

24

Backup: the hyperon puzzle

∼ 2 ÷ 3ρ0

soft stiff

ρth

Λ

E ρb

ρ0

µΛ = µn

HNM

> 2.0M M R

12 km

< 2.0M ∼ 2M

  • bs:

TOV 1:1

R ∼ 12 km M ∼ 1.4 M

NS

?

Λ Σ Ξ πc Kc qp En

F

N ρth

Λ

H n

n n n n

n

n n n n

n

n

n

n n

Λ

npe

slide-38
SLIDE 38

25

Backup: the hyperon puzzle

∼ 2 ÷ 3ρ0

soft stiff

ρth

Λ

E ρb

ρ0

µΛ = µn

HNM

> 2.0M M R

12 km

< 2.0M ∼ 2M

  • bs:

TOV 1:1

?

✓ Theoretical indication for hyperons in

NS core: softening of the EOS

✓ Magnitude of the softening: strongly

model dependent

✓ Observation of massive NS: stiff EOS

Hyperon puzzle

✓ Interactions poorly known

Problems

✓ Non trivial many-body problem: very

dense system, strong interactions QMC HN interaction

slide-39
SLIDE 39

26

Backup: the hyperon puzzle

TOV

  • J. Haidenbauer et al.,
  • Nucl. Phys. A 915

(2013) 24–58

lack of experimental data !!

∼ 2M

  • bs:

< 2.0M

binding energies: scattering data:

NN : ∼ 4300 nuc : ∼ 3340 Λ hyp : ∼ 41 ΛΛ hyp : ∼ 5 Σ hyp : ∼ (1) HN : ∼ 52

Proceedings of

The IX International Conference on Hypernuclear and Strange Particle Physics

HYP 2006

October 10-14, 2006 Mainz, Germany edited by

  • J. Pochodzalla and Th. Walcher

N Z |S|

1968 1968 1972

slide-40
SLIDE 40

27

Backup: terrestrial experiments

Green’s function Monte Carlo (GFMC)

A ≤ 12

nuclei

  • 100
  • 95
  • 90
  • 85
  • 80
  • 75
  • 70
  • 65
  • 60
  • 55
  • 50
  • 45
  • 40
  • 35
  • 30
  • 25
  • 20
  • 15
  • 10
  • 5

Energy (MeV)

α+n α+2n α+d

6He+n

α+t

6He+2n 7Li+n

2α

8Li+n

2α+n

8He+2n 9Be+n 6Li+α

3α

AV18 UIX IL7 Exp

2.2 1+ 2.2 1+ 2.2 1+ 2.2 1+ 2.2 1+ 2.2 1+ 2.2 1+ 2.2 1+

1+

2H

7.6 1/2+ 8.5 1/2+ 8.5 1/2+ 8.5 1/2+ 7.6 1/2+ 8.5 1/2+ 8.5 1/2+ 8.5 1/2+

1/2+

3H

24.1 0+ 28.4 0+ 28.4 0+ 28.3 0+ 24.1 0+ 28.4 0+ 28.4 0+ 28.3 0+

0+

4He

22.5 3/2− 21.9 1/2− 26.9 3/2− 25.8 1/2− 27.5 3/2− 26.2 1/2− 22.5 3/2− 21.9 1/2− 26.9 3/2− 25.8 1/2− 27.5 3/2− 26.2 1/2−

3/2− 1/2−

5He

23.8 0+ 21.9 2+ 20.4 2+ 19.6 1+ 19.0 0+ 28.0 0+ 26.1 2+ 23.9 1+ 29.2 0+ 26.9 2+ 29.3 0+ 27.5 2+ 23.7 2+ 23.7 0+ 23.8 0+ 21.9 2+ 20.4 2+ 19.6 1+ 19.0 0+ 28.0 0+ 26.1 2+ 23.9 1+ 29.2 0+ 26.9 2+ 29.3 0+ 27.5 2+ 23.7 2+ 23.7 0+

1+ 0+ 2+ 2+ 0+

6He

26.9 1+ 24.1 3+ 22.8 2+ 22.0 1+ 18.8 1+ 31.3 1+ 28.5 3+ 27.3 2+ 26.2 1+ 31.8 1+ 29.5 3+ 27.6 2+ 26.4 1+ 32.0 1+ 29.8 3+ 27.7 2+ 26.3 1+ 26.9 1+ 24.1 3+ 22.8 2+ 22.0 1+ 18.8 1+ 31.3 1+ 28.5 3+ 27.3 2+ 26.2 1+ 31.8 1+ 29.5 3+ 27.6 2+ 26.4 1+ 32.0 1+ 29.8 3+ 27.7 2+ 26.3 1+

1+ 1+ 3+ 2+ 1+

6Li

21.9 3/2− 20.7 1/2− 19.4 5/2− 26.3 3/2− 25.2 1/2− 23.9 5/2− 28.6 3/2− 26.7 1/2− 25.3 5/2− 28.8 3/2− 25.9 5/2− 25.9 (5/2)− 21.9 3/2− 20.7 1/2− 19.4 5/2− 26.3 3/2− 25.2 1/2− 23.9 5/2− 28.6 3/2− 26.7 1/2− 25.3 5/2− 28.8 3/2− 25.9 5/2− 25.9 (5/2)−

1/2− 3/2− 5/2− (5/2)−

7He

32.0 3/2− 32.2 1/2− 26.8 7/2− 26.4 5/2− 24.5 5/2− 23.0 7/2− 24.4 3/2− 23.8 1/2− 37.5 3/2− 37.6 1/2− 32.2 7/2− 31.1 5/2− 29.7 5/2− 28.1 7/2− 29.1 3/2− 27.0 5/2− 24.4 5/2− 39.1 3/2− 39.0 1/2− 34.9 7/2− 32.5 5/2− 31.3 5/2− 29.0 7/2− 39.2 3/2− 38.8 1/2− 34.6 7/2− 32.6 5/2− 31.8 5/2− 29.7 7/2− 30.5 3/2− 30.1 1/2− 32.0 3/2− 32.2 1/2− 26.8 7/2− 26.4 5/2− 24.5 5/2− 23.0 7/2− 24.4 3/2− 23.8 1/2− 37.5 3/2− 37.6 1/2− 32.2 7/2− 31.1 5/2− 29.7 5/2− 28.1 7/2− 29.1 3/2− 27.0 5/2− 24.4 5/2− 39.1 3/2− 39.0 1/2− 34.9 7/2− 32.5 5/2− 31.3 5/2− 29.0 7/2− 39.2 3/2− 38.8 1/2− 34.6 7/2− 32.6 5/2− 31.8 5/2− 29.7 7/2− 30.5 3/2− 30.1 1/2−

5/2− 5/2− 3/2− 1/2− 7/2− 5/2− 5/2− 7/2− 3/2− 1/2−

7Li

23.0 0+ 20.4 2+ 18.8 1+ 27.7 0+ 25.0 2+ 23.3 1+ 31.1 0+ 26.4 2+ 31.4 0+ 28.3 2+ 23.0 0+ 20.4 2+ 18.8 1+ 27.7 0+ 25.0 2+ 23.3 1+ 31.1 0+ 26.4 2+ 31.4 0+ 28.3 2+

1+ 0+ 2+

8He

32.7 2+ 32.1 1+ 31.5 0+ 30.1 3+ 29.7 2+ 31.0 1+ 27.1 4+ 26.3 3+ 29.1 1+ 28.7 2+ 38.8 2+ 37.8 1+ 36.9 0+ 35.4 3+ 35.2 1+ 32.3 4+ 41.2 2+ 40.5 1+ 38.1 0+ 38.2 3+ 36.9 2+ 37.8 1+ 34.8 4+ 35.9 2+ 41.3 2+ 40.3 1+ 35.9 0+ 39.0 3+ 38.1 1+ 34.8 4+ 35.2 3+ 35.9 1+ 32.7 2+ 32.1 1+ 31.5 0+ 30.1 3+ 29.7 2+ 31.0 1+ 27.1 4+ 26.3 3+ 29.1 1+ 28.7 2+ 38.8 2+ 37.8 1+ 36.9 0+ 35.4 3+ 35.2 1+ 32.3 4+ 41.2 2+ 40.5 1+ 38.1 0+ 38.2 3+ 36.9 2+ 37.8 1+ 34.8 4+ 35.9 2+ 41.3 2+ 40.3 1+ 35.9 0+ 39.0 3+ 38.1 1+ 34.8 4+ 35.2 3+ 35.9 1+

2+ 2+ 2+ 1+ 0+ 3+ 1+ 4+ 3+ 1+

8Li

46.3 0+ 43.7 2+ 36.2 4+ 31.0 2+ 30.8 1+ 29.7 3+ 29.1 2+ 55.2 0+ 52.1 2+ 44.2 4+ 36.4 2+ 37.0 1+ 35.2 3+ 56.5 0+ 53.5 2+ 45.3 4+ 39.9 2+ 38.5 1+ 36.8 3+ 35.9 4+ 35.4 0+ 36.6 1+ 56.5 0+ 53.4 2+ 45.1 4+ 39.6 2+ 38.3 1+ 37.3 3+ 36.6 4+ 36.3 0+ 36.4 2+ 46.3 0+ 43.7 2+ 36.2 4+ 31.0 2+ 30.8 1+ 29.7 3+ 29.1 2+ 55.2 0+ 52.1 2+ 44.2 4+ 36.4 2+ 37.0 1+ 35.2 3+ 56.5 0+ 53.5 2+ 45.3 4+ 39.9 2+ 38.5 1+ 36.8 3+ 35.9 4+ 35.4 0+ 36.6 1+ 56.5 0+ 53.4 2+ 45.1 4+ 39.6 2+ 38.3 1+ 37.3 3+ 36.6 4+ 36.3 0+ 36.4 2+

1+ 0+ 2+ 4+ 2+ 1+ 3+ 4+ 0+ 2+

8Be

33.7 3/2− 34.0 1/2− 32.1 5/2− 29.7 7/2− 40.9 3/2− 39.4 1/2− 37.9 5/2− 35.2 7/2− 37.3 3/2− 45.5 3/2− 43.4 1/2− 40.1 5/2− 45.3 3/2− 42.7 1/2− 41.0 5/2− 38.9 7/2− 33.7 3/2− 34.0 1/2− 32.1 5/2− 29.7 7/2− 40.9 3/2− 39.4 1/2− 37.9 5/2− 35.2 7/2− 37.3 3/2− 45.5 3/2− 43.4 1/2− 40.1 5/2− 45.3 3/2− 42.7 1/2− 41.0 5/2− 38.9 7/2−

3/2− 3/2− 1/2− 5/2− 7/2−

9Li

45.7 3/2− 44.7 1/2+ 45.5 5/2− 45.0 1/2− 42.8 5/2+ 40.3 7/2− 36.7 9/2− 55.1 3/2− 51.3 5/2− 50.9 1/2− 57.4 3/2− 55.9 1/2+ 55.1 5/2− 55.4 1/2− 54.4 5/2+ 53.2 3/2+ 51.0 7/2− 53.4 3/2− 46.5 7/2− 41.2 5/2+ 39.3 7/2+ 58.2 3/2− 56.5 1/2+ 55.7 5/2− 55.4 1/2− 55.1 5/2+ 53.5 3/2+ 51.8 7/2− 52.6 3/2− 46.9 7/2− 41.5 5/2+ 40.7 7/2+ 45.7 3/2− 44.7 1/2+ 45.5 5/2− 45.0 1/2− 42.8 5/2+ 40.3 7/2− 36.7 9/2− 55.1 3/2− 51.3 5/2− 50.9 1/2− 57.4 3/2− 55.9 1/2+ 55.1 5/2− 55.4 1/2− 54.4 5/2+ 53.2 3/2+ 51.0 7/2− 53.4 3/2− 46.5 7/2− 41.2 5/2+ 39.3 7/2+ 58.2 3/2− 56.5 1/2+ 55.7 5/2− 55.4 1/2− 55.1 5/2+ 53.5 3/2+ 51.8 7/2− 52.6 3/2− 46.9 7/2− 41.5 5/2+ 40.7 7/2+

9/2− 3/2− 1/2+ 5/2− 1/2− 5/2+ 3/2+ 7/2− 3/2− 7/2− 5/2+ 7/2+

9Be

19.8 0+ 26.6 0+ 30.3 0+ 19.8 0+ 26.6 0+ 30.3 0+

0+

10He

50.0 0+ 47.2 2+ 45.0 1− 47.0 2+ 43.0 3+ 59.5 0+ 56.0 2+ 55.8 2+ 64.3 0+ 60.5 2+ 58.8 2+ 57.3 1+ 65.0 0+ 61.6 2+ 59.0 1− 59.0 2+ 58.8 0+ 57.4 3,2+ 50.0 0+ 47.2 2+ 45.0 1− 47.0 2+ 43.0 3+ 59.5 0+ 56.0 2+ 55.8 2+ 64.3 0+ 60.5 2+ 58.8 2+ 57.3 1+ 65.0 0+ 61.6 2+ 59.0 1− 59.0 2+ 58.8 0+ 57.4 3,2+

3+ 1+ 0+ 2+ 1− 2+ 0+ 3,2+

10Be

48.6 3+ 51.6 1+ 47.2 2+ 43.5 2− 45.0 4+ 49.2 1+ 47.1 3+ 46.1 1+ 46.6 2+ 59.0 3+ 60.3 1+ 64.7 3+ 63.4 1+ 61.3 2+ 58.3 4+ 62.3 1+ 58.5 3+ 59.2 2+ 55.6 3+ 64.8 3+ 64.0 1+ 61.2 2+ 59.6 2− 58.7 4+ 62.6 1+ 60.0 3+ 59.6 1+ 58.8 2+ 56.1 3+ 48.6 3+ 51.6 1+ 47.2 2+ 43.5 2− 45.0 4+ 49.2 1+ 47.1 3+ 46.1 1+ 46.6 2+ 59.0 3+ 60.3 1+ 64.7 3+ 63.4 1+ 61.3 2+ 58.3 4+ 62.3 1+ 58.5 3+ 59.2 2+ 55.6 3+ 64.8 3+ 64.0 1+ 61.2 2+ 59.6 2− 58.7 4+ 62.6 1+ 60.0 3+ 59.6 1+ 58.8 2+ 56.1 3+

3+ 1+ 2+ 2− 4+ 1+ 3+ 1+ 2+ 3+

10B

48.6 3+ 51.6 1+ 47.2 2+ 43.5 2− 45.0 4+ 49.2 1+ 47.1 3+ 46.1 1+ 46.6 2+ 59.0 3+ 60.3 1+ 64.7 3+ 63.4 1+ 61.3 2+ 58.3 4+ 62.3 1+ 58.5 3+ 59.2 2+ 55.6 3+ 64.8 3+ 64.0 1+ 61.2 2+ 59.6 2− 58.7 4+ 62.6 1+ 60.0 3+ 59.6 1+ 58.8 2+ 56.1 3+ 48.6 3+ 51.6 1+ 47.2 2+ 43.5 2− 45.0 4+ 49.2 1+ 47.1 3+ 46.1 1+ 46.6 2+ 59.0 3+ 60.3 1+ 64.7 3+ 63.4 1+ 61.3 2+ 58.3 4+ 62.3 1+ 58.5 3+ 59.2 2+ 55.6 3+ 64.8 3+ 64.0 1+ 61.2 2+ 59.6 2− 58.7 4+ 62.6 1+ 60.0 3+ 59.6 1+ 58.8 2+ 56.1 3+

3+ 1+ 2+ 2− 4+ 1+ 3+ 1+ 2+

72.8 0+ 93.3 0+ 82.9 0+ 92.2 0+ 84.5 0+ 72.8 0+ 93.3 0+ 82.9 0+ 92.2 0+ 84.5 0+

0+ 0+

12C

α+n α+2n α+d

6He+n

α+t

6He+2n 7Li+n

2α

8Li+n

2α+n

8He+2n 9Be+n 6Li+α

3α

Argonne v18 without & with Vijk GFMC Calculations

23 August 2012

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B.S. Pudliner, V.R. Pandharipande, J. Carlson, S.C. Pieper & R.B. Wiringa, Quantum Monte Carlo calculations of nuclei with A ≤ 7, Phys. Rev. C 56, 1720-1750 (1997).
  • L. Lapikas, J. Wesseling & R.B. Wiringa, Nuclear structure studies with the 7Li(e,e′p) reaction, Phys. Rev. Lett. 82, 4404-4407 (1999).
R.B. Wiringa, S.C. Pieper, J. Carlson & V.R. Pandharipande, Quantum Monte Carlo calculations of A = 8 nuclei, Phys. Rev. C 62, 014001-1:23 (2000). K.M. Nollett, R.B. Wiringa & R. Schiavilla, Six-body calculation of the alpha-deuteron radiative capture cross section, Phys. Rev. C 63, 024003-1:13 (2001). S.C. Pieper, V.R. Pandharipande, R.B. Wiringa & J. Carlson, Realistic models of pion-exchange three-nucleon interactions, Phys. Rev. C 64, 014001-1:21 (2001). S.C. Pieper, R.B. Wiringa & J. Carlson, Quantum Monte Carlo calculations of excited states in A=6,−,8 nuclei, Phys. Rev. C 70, 054325-1:11 (2004). S.C. Pieper, K. Varga & R.B. Wiringa, Quantum Monte Carlo calculations of A = 9,10 nuclei, Phys. Rev. C 66, 044310-1:14 (2002). R.B. Wiringa & S.C. Pieper, Evolution of Nuclear Spectra with Nuclear Forces, Phys. Rev. Lett. 89, 182501-1:4 (2002). S.C. Pieper & R.B. Wiringa, Quantum Monte Carlo Calculations of Light Nuclei, Annu. Rev. Nucl. Part. Sci. 51, 53-90 (2001). S.C. Pieper, Can Modern Nuclear Hamiltonians Tolerate a Bound Tetraneutron?, Phys. Rev. Lett. 90, 252501-1:4 (2003). S.C. Pieper, Quantum Monte Carlo calculations of light nuclei, Nucl. Phys. 751, 516c-532c (2005). A.H. Wuosmaa et al., Neutron Spectroscopic Factors in 9Li from 2H(8Li,p)9Li, Phys. Rev. Lett. 94, 082502-1:4 (2005).
  • R. Schiavilla, R.B. Wiringa, S.C. Pieper & J. Carlson, Tensor Forces and the Ground-State Structure of Nuclei, Phys. Rev. Lett. 98, 132501-1:4 (2007).
K.M. Nollett, S.C. Pieper, R.B. Wiringa, J. Carlson & G.M. Hale, QMC Calculations of Neutron-α Scattering, Phys. Rev. Lett. 99, 022502-1:4 (2007). L.E. Marcucci, M. Pervin et al., QMC calcs of magnetic moments & M1 transitions in A≤7 nuclei including meson-exchange currents, Phys. Rev. C 78, 065501-1:10 (2008). We use the 132,000-processor IBM Blue Gene/P of the Argonne Leadership Computing Facility under a DOE INCITE award and computers at Argonne’s Mathematics and Computer Science Division and Laboratory Computing Resource Center

The main figure compares computed and experimental energies of nuclear states. The computations were made using just the Argonne v18 (AV18) NN potential and AV18 plus the Urbana-IX or the Illinois-2 NNN potentials.

  • 1
1
  • 1
1 femtometer MJ = 0 femtometer
  • 1
1
  • 1
1 MJ = 1 Surfaces of density = 0.24 fm-3 in polarized deuteron states. The distinctive structures are induced by the strong tensor potentials which result from the pion-exchange component of the nucleon-nucleon interaction. 1 2 3 4 1 2 3 4 5 6 7 Ec.m. (MeV) σJL (b) n + 4He scattering − AV18+IL2 1 2 + 1 2
  • 3
2
  • R-Matrix (data)
Pole location States that are above particle emission thres- hold should really be computed as scattering
  • states. This figure shows
the first few partial waves for n+4He scatter- ing; i.e. the low-lying resonant states of the unbound nucleus 5He. The results are shown as partial-wave cross sections. The solid black curves represent the experimental
  • data. The J = 3/2-
resonance is well reproduced, both in location and width. 0.5 1 1.5 2 2.5 3 0.000 0.005 0.010 0.015 0.020 r12 (fm) ρpp(r12) (fm-3) 4,6,8He - AV18 + IL2 - GFMC proton-proton distributions 4He 6He 8He The proton-proton two-body density in 4,6,8He is an indica- tor of the size of the alpha core of these
  • nuclei. Many calcula-
tions assume this core is not modified by the additional neutrons. Our A-nucleon calculations show a small, but signi- ficant, suppression of the peak density and increase in the rms pp
  • radius. This implies ~80
and ~350 keV excitations
  • f the alpha cores of
6,8He, respectively.
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Energy (MeV) 1S0 Vijk IL2 Exp 0+

2n

1/2+

3H

0+

4n

0+

4He

1/2+

5H

3/2−

5He

0+

6n

0+

6He

1+

6Li

Argonne v18 With Illinois-2 + modifications

There is an experimental claim of a bound tetra- neutron (4n). Our Ham- iltonian predicts at most a (likely very broad) reson- ance at +2 MeV. The figure shows attempts to produce a 4n with nega- tive energy by changing the Hamiltonian. Modi- fication of the 1S0 poten- tial gives a bound dineutron and signifi- cantly overbinds other
  • nuclei. Adding a T=3/2
NNN potential doesn’t effect 2n or 4He but very much overbinds heavier systems; in fact 6n becomes the most stable A=6 system! We conclude that a bound 4n is very unlikely.
  • 100
100 200 300 10-11 10-10 10-9 10-8 10-7 10-6 pm [MeV/c] ρ(pm) [(MeV/c)-3] 7Li(e,e′p)6He(Jπ) - Argonne v18 + Urbana IX - VMC Wave Functions 6He(0+) × 10 6He(2+) The shape and magnitude
  • f (e,e′p) differential
cross sections are given by the overlap of the wave functions of the target and residual nuclear states. This normalization is usually expressed as a spectro- scopic factor. In this figure overlaps of 7- and 6-nucleon VMC wave functions have been used to directly compute the
  • verlap with no adjust-
ment to fit the data. The resulting spectro- scopic factors are 0.41 for the 0+ and 0.19 for the 2+ states; consider- ably smaller than conven- tional (Cohen-Kurath) shell-model values of 0.59 and 0.40. This reduction is due to the strong short-range and tensor correlations.
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Energy (MeV) IL2
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N+8O Exp. 1/2+ 3/2− 1/2− 5/2+ 3/2+ 1n 0+ 2n 3/2− 1/2− 3n 0+ 4n 1/2− 3/2− 5n 0+ 6n 1/2− 3/2− 7n 0+ 8n 1/2+ 5/2+ 3/2+ 9n 0+ 10n 5/2+ 1/2+ 11n 0+ 12n 1/2+ 5/2+ 13n 0+ 14n
  • Ext. well: V=-35.5, R=3, a=1.1

with Argonne v18 + Illinois-2

Neutron drops are systems of interacting nucleons bound in an artificial external well which may be thought of as representing the protons of a real nucleus. By including
  • nly neutrons in the calcula-
tion, the isospin degree of freedom is suppressed and a larger system can be studied than for real nuclei (so far 14n which is much easier than 12C, our biggest nucleus). This figure shows results for a well chosen to mimic the protons in oxygen. The Nn energies are compared with experimental N+8O values; the Nn energies have been shifted to match 8n to
  • 16O. One can clearly see
the effects of nn pairing. .05 .1 .5 1 5 10 1 5 10 50 100 500 1000 Ec.m. (MeV) S Factor Astrophysical S factor for α-d capture E2 component E1 component Total Kiener (Coulomb dissociation) Mohr (direct capture) Robertson (direct capture) 0.67 0.69 0.71 0.73 500 1000 1500 6Li was produced in the big bang by the alpha- deuteron capture reaction at energies of 20 to 200 keV, for which good data does not exist. This is a VMC computation using 6-nucleon wave functions
  • f the rate, expressed as
the astrophysical S
  • factor. The low-energy
Kiener data are indirect and were extracted in a model-dependent way from 6Li dissociation on 208Pb; no theoretical calculation gives a constant S at low energy.
  • 4
  • 2
2 4
  • 2
2 4
  • 5
  • 3
  • 1
1 3 5 r = (x2+y2)1/2 (fm) z (fm) Constant density contours in 8Be(0+) .1 .05 .02 .01 .005 .002 .001
  • Lab. Frame
.3 .25 .2 .15 .1 .05 .02 .01 .005 .002 .001 Body-fixed. Frame The density of 8Be(0+) in the laboratory frame is spherically symmetric (left panel). The one- body structure of the 8Be(0+,2+,4+) states consists of an alpha particle and four p-shell
  • nucleons. But strong
NN correlations shape the p-shell nucleons into a second alpha, which can be seen in the body-fixed frame (right panel). 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
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τ (MeV-1) E(τ) (MeV) 10B(gs, 1+, 2+, 4+) - AV18 + Illinois-4 - 〈H〉 10B(4+) 10B(2+) 10B(1+) 10B(3+) GFMC acts on a trial wave function (ΨT) with exp[-(H-E0)τ] where τ is the imaginary time. This filters excited- state contamination out
  • f ΨT. The figure shows
energies from GFMC propagation of 10B states as functions of τ, starting from the VMC values at τ=0. Solid and dashed lines show the averages and statistical errors used in the main figure. The large and rapid change for small τ indicates that small admixtures of highly excited (~1 GeV) states are being removed.
slide-41
SLIDE 41

28

Green’s function Monte Carlo (GFMC)

A ≤ 12

nuclei

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Energy (MeV)

α+n α+2n α+d

6He+n

α+t

6He+2n 7Li+n

2α

8Li+n

2α+n

8He+2n 9Be+n 6Li+α

3α

AV18 UIX IL7 Exp

2.2 1+ 2.2 1+ 2.2 1+ 2.2 1+ 2.2 1+ 2.2 1+ 2.2 1+ 2.2 1+

1+

2H

7.6 1/2+ 8.5 1/2+ 8.5 1/2+ 8.5 1/2+ 7.6 1/2+ 8.5 1/2+ 8.5 1/2+ 8.5 1/2+

1/2+

3H

24.1 0+ 28.4 0+ 28.4 0+ 28.3 0+ 24.1 0+ 28.4 0+ 28.4 0+ 28.3 0+

0+

4He

22.5 3/2− 21.9 1/2− 26.9 3/2− 25.8 1/2− 27.5 3/2− 26.2 1/2− 22.5 3/2− 21.9 1/2− 26.9 3/2− 25.8 1/2− 27.5 3/2− 26.2 1/2−

3/2− 1/2−

5He

23.8 0+ 21.9 2+ 20.4 2+ 19.6 1+ 19.0 0+ 28.0 0+ 26.1 2+ 23.9 1+ 29.2 0+ 26.9 2+ 29.3 0+ 27.5 2+ 23.7 2+ 23.7 0+ 23.8 0+ 21.9 2+ 20.4 2+ 19.6 1+ 19.0 0+ 28.0 0+ 26.1 2+ 23.9 1+ 29.2 0+ 26.9 2+ 29.3 0+ 27.5 2+ 23.7 2+ 23.7 0+

1+ 0+ 2+ 2+ 0+

6He

26.9 1+ 24.1 3+ 22.8 2+ 22.0 1+ 18.8 1+ 31.3 1+ 28.5 3+ 27.3 2+ 26.2 1+ 31.8 1+ 29.5 3+ 27.6 2+ 26.4 1+ 32.0 1+ 29.8 3+ 27.7 2+ 26.3 1+ 26.9 1+ 24.1 3+ 22.8 2+ 22.0 1+ 18.8 1+ 31.3 1+ 28.5 3+ 27.3 2+ 26.2 1+ 31.8 1+ 29.5 3+ 27.6 2+ 26.4 1+ 32.0 1+ 29.8 3+ 27.7 2+ 26.3 1+

1+ 1+ 3+ 2+ 1+

6Li

21.9 3/2− 20.7 1/2− 19.4 5/2− 26.3 3/2− 25.2 1/2− 23.9 5/2− 28.6 3/2− 26.7 1/2− 25.3 5/2− 28.8 3/2− 25.9 5/2− 25.9 (5/2)− 21.9 3/2− 20.7 1/2− 19.4 5/2− 26.3 3/2− 25.2 1/2− 23.9 5/2− 28.6 3/2− 26.7 1/2− 25.3 5/2− 28.8 3/2− 25.9 5/2− 25.9 (5/2)−

1/2− 3/2− 5/2− (5/2)−

7He

32.0 3/2− 32.2 1/2− 26.8 7/2− 26.4 5/2− 24.5 5/2− 23.0 7/2− 24.4 3/2− 23.8 1/2− 37.5 3/2− 37.6 1/2− 32.2 7/2− 31.1 5/2− 29.7 5/2− 28.1 7/2− 29.1 3/2− 27.0 5/2− 24.4 5/2− 39.1 3/2− 39.0 1/2− 34.9 7/2− 32.5 5/2− 31.3 5/2− 29.0 7/2− 39.2 3/2− 38.8 1/2− 34.6 7/2− 32.6 5/2− 31.8 5/2− 29.7 7/2− 30.5 3/2− 30.1 1/2− 32.0 3/2− 32.2 1/2− 26.8 7/2− 26.4 5/2− 24.5 5/2− 23.0 7/2− 24.4 3/2− 23.8 1/2− 37.5 3/2− 37.6 1/2− 32.2 7/2− 31.1 5/2− 29.7 5/2− 28.1 7/2− 29.1 3/2− 27.0 5/2− 24.4 5/2− 39.1 3/2− 39.0 1/2− 34.9 7/2− 32.5 5/2− 31.3 5/2− 29.0 7/2− 39.2 3/2− 38.8 1/2− 34.6 7/2− 32.6 5/2− 31.8 5/2− 29.7 7/2− 30.5 3/2− 30.1 1/2−

5/2− 5/2− 3/2− 1/2− 7/2− 5/2− 5/2− 7/2− 3/2− 1/2−

7Li

23.0 0+ 20.4 2+ 18.8 1+ 27.7 0+ 25.0 2+ 23.3 1+ 31.1 0+ 26.4 2+ 31.4 0+ 28.3 2+ 23.0 0+ 20.4 2+ 18.8 1+ 27.7 0+ 25.0 2+ 23.3 1+ 31.1 0+ 26.4 2+ 31.4 0+ 28.3 2+

1+ 0+ 2+

8He

32.7 2+ 32.1 1+ 31.5 0+ 30.1 3+ 29.7 2+ 31.0 1+ 27.1 4+ 26.3 3+ 29.1 1+ 28.7 2+ 38.8 2+ 37.8 1+ 36.9 0+ 35.4 3+ 35.2 1+ 32.3 4+ 41.2 2+ 40.5 1+ 38.1 0+ 38.2 3+ 36.9 2+ 37.8 1+ 34.8 4+ 35.9 2+ 41.3 2+ 40.3 1+ 35.9 0+ 39.0 3+ 38.1 1+ 34.8 4+ 35.2 3+ 35.9 1+ 32.7 2+ 32.1 1+ 31.5 0+ 30.1 3+ 29.7 2+ 31.0 1+ 27.1 4+ 26.3 3+ 29.1 1+ 28.7 2+ 38.8 2+ 37.8 1+ 36.9 0+ 35.4 3+ 35.2 1+ 32.3 4+ 41.2 2+ 40.5 1+ 38.1 0+ 38.2 3+ 36.9 2+ 37.8 1+ 34.8 4+ 35.9 2+ 41.3 2+ 40.3 1+ 35.9 0+ 39.0 3+ 38.1 1+ 34.8 4+ 35.2 3+ 35.9 1+

2+ 2+ 2+ 1+ 0+ 3+ 1+ 4+ 3+ 1+

8Li

46.3 0+ 43.7 2+ 36.2 4+ 31.0 2+ 30.8 1+ 29.7 3+ 29.1 2+ 55.2 0+ 52.1 2+ 44.2 4+ 36.4 2+ 37.0 1+ 35.2 3+ 56.5 0+ 53.5 2+ 45.3 4+ 39.9 2+ 38.5 1+ 36.8 3+ 35.9 4+ 35.4 0+ 36.6 1+ 56.5 0+ 53.4 2+ 45.1 4+ 39.6 2+ 38.3 1+ 37.3 3+ 36.6 4+ 36.3 0+ 36.4 2+ 46.3 0+ 43.7 2+ 36.2 4+ 31.0 2+ 30.8 1+ 29.7 3+ 29.1 2+ 55.2 0+ 52.1 2+ 44.2 4+ 36.4 2+ 37.0 1+ 35.2 3+ 56.5 0+ 53.5 2+ 45.3 4+ 39.9 2+ 38.5 1+ 36.8 3+ 35.9 4+ 35.4 0+ 36.6 1+ 56.5 0+ 53.4 2+ 45.1 4+ 39.6 2+ 38.3 1+ 37.3 3+ 36.6 4+ 36.3 0+ 36.4 2+

1+ 0+ 2+ 4+ 2+ 1+ 3+ 4+ 0+ 2+

8Be

33.7 3/2− 34.0 1/2− 32.1 5/2− 29.7 7/2− 40.9 3/2− 39.4 1/2− 37.9 5/2− 35.2 7/2− 37.3 3/2− 45.5 3/2− 43.4 1/2− 40.1 5/2− 45.3 3/2− 42.7 1/2− 41.0 5/2− 38.9 7/2− 33.7 3/2− 34.0 1/2− 32.1 5/2− 29.7 7/2− 40.9 3/2− 39.4 1/2− 37.9 5/2− 35.2 7/2− 37.3 3/2− 45.5 3/2− 43.4 1/2− 40.1 5/2− 45.3 3/2− 42.7 1/2− 41.0 5/2− 38.9 7/2−

3/2− 3/2− 1/2− 5/2− 7/2−

9Li

45.7 3/2− 44.7 1/2+ 45.5 5/2− 45.0 1/2− 42.8 5/2+ 40.3 7/2− 36.7 9/2− 55.1 3/2− 51.3 5/2− 50.9 1/2− 57.4 3/2− 55.9 1/2+ 55.1 5/2− 55.4 1/2− 54.4 5/2+ 53.2 3/2+ 51.0 7/2− 53.4 3/2− 46.5 7/2− 41.2 5/2+ 39.3 7/2+ 58.2 3/2− 56.5 1/2+ 55.7 5/2− 55.4 1/2− 55.1 5/2+ 53.5 3/2+ 51.8 7/2− 52.6 3/2− 46.9 7/2− 41.5 5/2+ 40.7 7/2+ 45.7 3/2− 44.7 1/2+ 45.5 5/2− 45.0 1/2− 42.8 5/2+ 40.3 7/2− 36.7 9/2− 55.1 3/2− 51.3 5/2− 50.9 1/2− 57.4 3/2− 55.9 1/2+ 55.1 5/2− 55.4 1/2− 54.4 5/2+ 53.2 3/2+ 51.0 7/2− 53.4 3/2− 46.5 7/2− 41.2 5/2+ 39.3 7/2+ 58.2 3/2− 56.5 1/2+ 55.7 5/2− 55.4 1/2− 55.1 5/2+ 53.5 3/2+ 51.8 7/2− 52.6 3/2− 46.9 7/2− 41.5 5/2+ 40.7 7/2+

9/2− 3/2− 1/2+ 5/2− 1/2− 5/2+ 3/2+ 7/2− 3/2− 7/2− 5/2+ 7/2+

9Be

19.8 0+ 26.6 0+ 30.3 0+ 19.8 0+ 26.6 0+ 30.3 0+

0+

10He

50.0 0+ 47.2 2+ 45.0 1− 47.0 2+ 43.0 3+ 59.5 0+ 56.0 2+ 55.8 2+ 64.3 0+ 60.5 2+ 58.8 2+ 57.3 1+ 65.0 0+ 61.6 2+ 59.0 1− 59.0 2+ 58.8 0+ 57.4 3,2+ 50.0 0+ 47.2 2+ 45.0 1− 47.0 2+ 43.0 3+ 59.5 0+ 56.0 2+ 55.8 2+ 64.3 0+ 60.5 2+ 58.8 2+ 57.3 1+ 65.0 0+ 61.6 2+ 59.0 1− 59.0 2+ 58.8 0+ 57.4 3,2+

3+ 1+ 0+ 2+ 1− 2+ 0+ 3,2+

10Be

48.6 3+ 51.6 1+ 47.2 2+ 43.5 2− 45.0 4+ 49.2 1+ 47.1 3+ 46.1 1+ 46.6 2+ 59.0 3+ 60.3 1+ 64.7 3+ 63.4 1+ 61.3 2+ 58.3 4+ 62.3 1+ 58.5 3+ 59.2 2+ 55.6 3+ 64.8 3+ 64.0 1+ 61.2 2+ 59.6 2− 58.7 4+ 62.6 1+ 60.0 3+ 59.6 1+ 58.8 2+ 56.1 3+ 48.6 3+ 51.6 1+ 47.2 2+ 43.5 2− 45.0 4+ 49.2 1+ 47.1 3+ 46.1 1+ 46.6 2+ 59.0 3+ 60.3 1+ 64.7 3+ 63.4 1+ 61.3 2+ 58.3 4+ 62.3 1+ 58.5 3+ 59.2 2+ 55.6 3+ 64.8 3+ 64.0 1+ 61.2 2+ 59.6 2− 58.7 4+ 62.6 1+ 60.0 3+ 59.6 1+ 58.8 2+ 56.1 3+

3+ 1+ 2+ 2− 4+ 1+ 3+ 1+ 2+ 3+

10B

48.6 3+ 51.6 1+ 47.2 2+ 43.5 2− 45.0 4+ 49.2 1+ 47.1 3+ 46.1 1+ 46.6 2+ 59.0 3+ 60.3 1+ 64.7 3+ 63.4 1+ 61.3 2+ 58.3 4+ 62.3 1+ 58.5 3+ 59.2 2+ 55.6 3+ 64.8 3+ 64.0 1+ 61.2 2+ 59.6 2− 58.7 4+ 62.6 1+ 60.0 3+ 59.6 1+ 58.8 2+ 56.1 3+ 48.6 3+ 51.6 1+ 47.2 2+ 43.5 2− 45.0 4+ 49.2 1+ 47.1 3+ 46.1 1+ 46.6 2+ 59.0 3+ 60.3 1+ 64.7 3+ 63.4 1+ 61.3 2+ 58.3 4+ 62.3 1+ 58.5 3+ 59.2 2+ 55.6 3+ 64.8 3+ 64.0 1+ 61.2 2+ 59.6 2− 58.7 4+ 62.6 1+ 60.0 3+ 59.6 1+ 58.8 2+ 56.1 3+

3+ 1+ 2+ 2− 4+ 1+ 3+ 1+ 2+

72.8 0+ 93.3 0+ 82.9 0+ 92.2 0+ 84.5 0+ 72.8 0+ 93.3 0+ 82.9 0+ 92.2 0+ 84.5 0+

0+ 0+

12C

α+n α+2n α+d

6He+n

α+t

6He+2n 7Li+n

2α

8Li+n

2α+n

8He+2n 9Be+n 6Li+α

3α

Argonne v18 without & with Vijk GFMC Calculations

23 August 2012

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B.S. Pudliner, V.R. Pandharipande, J. Carlson, S.C. Pieper & R.B. Wiringa, Quantum Monte Carlo calculations of nuclei with A ≤ 7, Phys. Rev. C 56, 1720-1750 (1997).
  • L. Lapikas, J. Wesseling & R.B. Wiringa, Nuclear structure studies with the 7Li(e,e′p) reaction, Phys. Rev. Lett. 82, 4404-4407 (1999).
R.B. Wiringa, S.C. Pieper, J. Carlson & V.R. Pandharipande, Quantum Monte Carlo calculations of A = 8 nuclei, Phys. Rev. C 62, 014001-1:23 (2000). K.M. Nollett, R.B. Wiringa & R. Schiavilla, Six-body calculation of the alpha-deuteron radiative capture cross section, Phys. Rev. C 63, 024003-1:13 (2001). S.C. Pieper, V.R. Pandharipande, R.B. Wiringa & J. Carlson, Realistic models of pion-exchange three-nucleon interactions, Phys. Rev. C 64, 014001-1:21 (2001). S.C. Pieper, R.B. Wiringa & J. Carlson, Quantum Monte Carlo calculations of excited states in A=6,−,8 nuclei, Phys. Rev. C 70, 054325-1:11 (2004). S.C. Pieper, K. Varga & R.B. Wiringa, Quantum Monte Carlo calculations of A = 9,10 nuclei, Phys. Rev. C 66, 044310-1:14 (2002). R.B. Wiringa & S.C. Pieper, Evolution of Nuclear Spectra with Nuclear Forces, Phys. Rev. Lett. 89, 182501-1:4 (2002). S.C. Pieper & R.B. Wiringa, Quantum Monte Carlo Calculations of Light Nuclei, Annu. Rev. Nucl. Part. Sci. 51, 53-90 (2001). S.C. Pieper, Can Modern Nuclear Hamiltonians Tolerate a Bound Tetraneutron?, Phys. Rev. Lett. 90, 252501-1:4 (2003). S.C. Pieper, Quantum Monte Carlo calculations of light nuclei, Nucl. Phys. 751, 516c-532c (2005). A.H. Wuosmaa et al., Neutron Spectroscopic Factors in 9Li from 2H(8Li,p)9Li, Phys. Rev. Lett. 94, 082502-1:4 (2005).
  • R. Schiavilla, R.B. Wiringa, S.C. Pieper & J. Carlson, Tensor Forces and the Ground-State Structure of Nuclei, Phys. Rev. Lett. 98, 132501-1:4 (2007).
K.M. Nollett, S.C. Pieper, R.B. Wiringa, J. Carlson & G.M. Hale, QMC Calculations of Neutron-α Scattering, Phys. Rev. Lett. 99, 022502-1:4 (2007). L.E. Marcucci, M. Pervin et al., QMC calcs of magnetic moments & M1 transitions in A≤7 nuclei including meson-exchange currents, Phys. Rev. C 78, 065501-1:10 (2008). We use the 132,000-processor IBM Blue Gene/P of the Argonne Leadership Computing Facility under a DOE INCITE award and computers at Argonne’s Mathematics and Computer Science Division and Laboratory Computing Resource Center

The main figure compares computed and experimental energies of nuclear states. The computations were made using just the Argonne v18 (AV18) NN potential and AV18 plus the Urbana-IX or the Illinois-2 NNN potentials.

  • 1
1
  • 1
1 femtometer MJ = 0 femtometer
  • 1
1
  • 1
1 MJ = 1 Surfaces of density = 0.24 fm-3 in polarized deuteron states. The distinctive structures are induced by the strong tensor potentials which result from the pion-exchange component of the nucleon-nucleon interaction. 1 2 3 4 1 2 3 4 5 6 7 Ec.m. (MeV) σJL (b) n + 4He scattering − AV18+IL2 1 2 + 1 2
  • 3
2
  • R-Matrix (data)
Pole location States that are above particle emission thres- hold should really be computed as scattering
  • states. This figure shows
the first few partial waves for n+4He scatter- ing; i.e. the low-lying resonant states of the unbound nucleus 5He. The results are shown as partial-wave cross sections. The solid black curves represent the experimental
  • data. The J = 3/2-
resonance is well reproduced, both in location and width. 0.5 1 1.5 2 2.5 3 0.000 0.005 0.010 0.015 0.020 r12 (fm) ρpp(r12) (fm-3) 4,6,8He - AV18 + IL2 - GFMC proton-proton distributions 4He 6He 8He The proton-proton two-body density in 4,6,8He is an indica- tor of the size of the alpha core of these
  • nuclei. Many calcula-
tions assume this core is not modified by the additional neutrons. Our A-nucleon calculations show a small, but signi- ficant, suppression of the peak density and increase in the rms pp
  • radius. This implies ~80
and ~350 keV excitations
  • f the alpha cores of
6,8He, respectively.
  • 120
  • 100
  • 80
  • 60
  • 40
  • 20
Energy (MeV) 1S0 Vijk IL2 Exp 0+

2n

1/2+

3H

0+

4n

0+

4He

1/2+

5H

3/2−

5He

0+

6n

0+

6He

1+

6Li

Argonne v18 With Illinois-2 + modifications

There is an experimental claim of a bound tetra- neutron (4n). Our Ham- iltonian predicts at most a (likely very broad) reson- ance at +2 MeV. The figure shows attempts to produce a 4n with nega- tive energy by changing the Hamiltonian. Modi- fication of the 1S0 poten- tial gives a bound dineutron and signifi- cantly overbinds other
  • nuclei. Adding a T=3/2
NNN potential doesn’t effect 2n or 4He but very much overbinds heavier systems; in fact 6n becomes the most stable A=6 system! We conclude that a bound 4n is very unlikely.
  • 100
100 200 300 10-11 10-10 10-9 10-8 10-7 10-6 pm [MeV/c] ρ(pm) [(MeV/c)-3] 7Li(e,e′p)6He(Jπ) - Argonne v18 + Urbana IX - VMC Wave Functions 6He(0+) × 10 6He(2+) The shape and magnitude
  • f (e,e′p) differential
cross sections are given by the overlap of the wave functions of the target and residual nuclear states. This normalization is usually expressed as a spectro- scopic factor. In this figure overlaps of 7- and 6-nucleon VMC wave functions have been used to directly compute the
  • verlap with no adjust-
ment to fit the data. The resulting spectro- scopic factors are 0.41 for the 0+ and 0.19 for the 2+ states; consider- ably smaller than conven- tional (Cohen-Kurath) shell-model values of 0.59 and 0.40. This reduction is due to the strong short-range and tensor correlations.
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  • 20
  • 10
Energy (MeV) IL2
  • 19
N+8O Exp. 1/2+ 3/2− 1/2− 5/2+ 3/2+ 1n 0+ 2n 3/2− 1/2− 3n 0+ 4n 1/2− 3/2− 5n 0+ 6n 1/2− 3/2− 7n 0+ 8n 1/2+ 5/2+ 3/2+ 9n 0+ 10n 5/2+ 1/2+ 11n 0+ 12n 1/2+ 5/2+ 13n 0+ 14n
  • Ext. well: V=-35.5, R=3, a=1.1

with Argonne v18 + Illinois-2

Neutron drops are systems of interacting nucleons bound in an artificial external well which may be thought of as representing the protons of a real nucleus. By including
  • nly neutrons in the calcula-
tion, the isospin degree of freedom is suppressed and a larger system can be studied than for real nuclei (so far 14n which is much easier than 12C, our biggest nucleus). This figure shows results for a well chosen to mimic the protons in oxygen. The Nn energies are compared with experimental N+8O values; the Nn energies have been shifted to match 8n to
  • 16O. One can clearly see
the effects of nn pairing. .05 .1 .5 1 5 10 1 5 10 50 100 500 1000 Ec.m. (MeV) S Factor Astrophysical S factor for α-d capture E2 component E1 component Total Kiener (Coulomb dissociation) Mohr (direct capture) Robertson (direct capture) 0.67 0.69 0.71 0.73 500 1000 1500 6Li was produced in the big bang by the alpha- deuteron capture reaction at energies of 20 to 200 keV, for which good data does not exist. This is a VMC computation using 6-nucleon wave functions
  • f the rate, expressed as
the astrophysical S
  • factor. The low-energy
Kiener data are indirect and were extracted in a model-dependent way from 6Li dissociation on 208Pb; no theoretical calculation gives a constant S at low energy.
  • 4
  • 2
2 4
  • 2
2 4
  • 5
  • 3
  • 1
1 3 5 r = (x2+y2)1/2 (fm) z (fm) Constant density contours in 8Be(0+) .1 .05 .02 .01 .005 .002 .001
  • Lab. Frame
.3 .25 .2 .15 .1 .05 .02 .01 .005 .002 .001 Body-fixed. Frame The density of 8Be(0+) in the laboratory frame is spherically symmetric (left panel). The one- body structure of the 8Be(0+,2+,4+) states consists of an alpha particle and four p-shell
  • nucleons. But strong
NN correlations shape the p-shell nucleons into a second alpha, which can be seen in the body-fixed frame (right panel). 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
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τ (MeV-1) E(τ) (MeV) 10B(gs, 1+, 2+, 4+) - AV18 + Illinois-4 - 〈H〉 10B(4+) 10B(2+) 10B(1+) 10B(3+) GFMC acts on a trial wave function (ΨT) with exp[-(H-E0)τ] where τ is the imaginary time. This filters excited- state contamination out
  • f ΨT. The figure shows
energies from GFMC propagation of 10B states as functions of τ, starting from the VMC values at τ=0. Solid and dashed lines show the averages and statistical errors used in the main figure. The large and rapid change for small τ indicates that small admixtures of highly excited (~1 GeV) states are being removed.

AV18 UIX IL7 Exp

The main figure compares computed and experimental energies of nuclear states. The computations were made using just the Argonne v18 (AV18) NN potential and AV18 plus the Urbana-IX or the Illinois-2 NNN potentials.

9Be+n 6Li+α

50.0 0+ 47.2 2+ 45.0 1− 47.0 2+ 43.0 3+ 59.5 0+ 56.0 2+ 55.8 2+ 64.3 0+ 60.5 2+ 58.8 2+ 57.3 1+ 65.0 0+ 61.6 2+ 59.0 1− 59.0 2+ 58.8 0+ 57.4 3,2+ 50.0 0+ 47.2 2+ 45.0 1− 47.0 2+ 43.0 3+ 59.5 0+ 56.0 2+ 55.8 2+ 64.3 0+ 60.5 2+ 58.8 2+ 57.3 1+ 65.0 0+ 61.6 2+ 59.0 1− 59.0 2+ 58.8 0+ 57.4 3,2+

3+ 1+ 0+ 2+ 1− 2+ 0+ 3,2+

10Be

48.6 3+ 51.6 1+ 47.2 2+ 43.5 2− 45.0 4+ 49.2 1+ 47.1 3+ 46.1 1+ 46.6 2+ 59.0 3+ 60.3 1+ 64.7 3+ 63.4 1+ 61.3 2+ 58.3 4+ 62.3 1+ 58.5 3+ 59.2 2+ 55.6 3+ 64.8 3+ 64.0 1+ 61.2 2+ 59.6 2− 58.7 4+ 62.6 1+ 60.0 3+ 59.6 1+ 58.8 2+ 56.1 3+ 48.6 3+ 51.6 1+ 47.2 2+ 43.5 2− 45.0 4+ 49.2 1+ 47.1 3+ 46.1 1+ 46.6 2+ 59.0 3+ 60.3 1+ 64.7 3+ 63.4 1+ 61.3 2+ 58.3 4+ 62.3 1+ 58.5 3+ 59.2 2+ 55.6 3+ 64.8 3+ 64.0 1+ 61.2 2+ 59.6 2− 58.7 4+ 62.6 1+ 60.0 3+ 59.6 1+ 58.8 2+ 56.1 3+

3+ 1+ 2+ 2− 4+ 1+ 3+ 1+ 2+ 3+

10B

48.6 3+ 51.6 1+ 47.2 2+ 43.5 2− 45.0 4+ 49.2 1+ 47.1 3+ 46.1 1+ 46.6 2+ 59.0 3+ 60.3 1+ 64.7 3+ 63.4 1+ 61.3 2+ 58.3 4+ 62.3 1+ 58.5 3+ 59.2 2+ 55.6 3+ 64.8 3+ 64.0 1+ 61.2 2+ 59.6 2− 58.7 4+ 62.6 1+ 60.0 3+ 59.6 1+ 58.8 2+ 56.1 3+ 48.6 3+ 51.6 1+ 47.2 2+ 43.5 2− 45.0 4+ 49.2 1+ 47.1 3+ 46.1 1+ 46.6 2+ 59.0 3+ 60.3 1+ 64.7 3+ 63.4 1+ 61.3 2+ 58.3 4+ 62.3 1+ 58.5 3+ 59.2 2+ 55.6 3+ 64.8 3+ 64.0 1+ 61.2 2+ 59.6 2− 58.7 4+ 62.6 1+ 60.0 3+ 59.6 1+ 58.8 2+ 56.1 3+

3+ 1+ 2+ 2− 4+ 1+ 3+ 1+ 2+

9Be+n 6Li+α

Backup: terrestrial experiments

slide-42
SLIDE 42

29

?

PNM

E [MeV] ρb [fm-3] AV8’ AV8’ + UIX AV8’ + IL7 5 10 15 20 25 0.04 0.08 0.12 0.16 0.20 0.24

2.45(1)M 1.74(1)M

  • P. Maris, J. P. Vary, S. Gandolfi, J. Carlson, S. C. Pieper, Phys. Rev. C 87, 054318 (2013)

3BF NNN

light nuclei: not enough to constraint the interaction

Backup: terrestrial experiments

slide-43
SLIDE 43

30

TOV

  • J. Haidenbauer et al.,
  • Nucl. Phys. A 915

(2013) 24–58

lack of experimental data !!

∼ 2M

  • bs:

< 2.0M

binding energies: scattering data:

NN : ∼ 4300 nuc : ∼ 3340 Λ hyp : ∼ 41 ΛΛ hyp : ∼ 5 Σ hyp : ∼ (1) HN : ∼ 52

Proceedings of

The IX International Conference on Hypernuclear and Strange Particle Physics

HYP 2006

October 10-14, 2006 Mainz, Germany edited by

  • J. Pochodzalla and Th. Walcher

N Z |S|

1968 1968 1972

52 ΛV

  • S. N. Nakamura, Hypernuclear workshop, JLab, May 2014

updated from: O. Hashimoto, H. Tamura, Prog. Part. Nucl. Phys. 57, 564 (2006)

? ? ?

new experimental proposals

Backup: terrestrial experiments

slide-44
SLIDE 44

31

AZ

  • e, e0K+A

Λ[Z − 1] AZ

  • K−, π0A

Λ[Z − 1] AZ

  • π−, K0A

Λ[Z − 1] AZ

  • K−, π−A

ΛZ AZ

  • π+, K+A

ΛZ

⇢ ⇢ n n K− π− Λ u u ¯ u ¯ u d d d d s s n K+ π+ Λ u u u u d d d ¯ d s ¯ s n ⇢ ⇢ n n ⇢ n      K+ Λ e e0 p u u u u d d s ¯ s γ∗ ⇢ ⇢ n n K− Λ u u ¯ u ¯ u d d s s

π0

u u p Λ u u d d s ¯ s ⇢ ⇢ n n K0 π− p d d ¯ u u

AZ

  • π−, K+A+1

Λ [Z − 2] AZ

  • K−, π+A+1

Λ [Z − 2]

✓ Charge conserving reactions ✓ Single charge exchange reactions (SCX) ✓ Double charge exchange reactions (DCX)

Backup: terrestrial experiments

slide-45
SLIDE 45

32

  • H. Hotchi et al., Phys. Rev. C 64, 044302 (2001)

89 ΛY

MHY = q (Eπ + MA − EK)2 − (p2

π + p2 K − 2pπ pK cos θ)

BΛ = MA−1 + MΛ − MHY ✓ dσ dΩ ◆ = A ρx · NA · 1 Nbeam · fbeam · NK εexp · dΩ ¯ σ2−14 = Z θ=14

θ=2

✓ dσ dΩ ◆ dΩ , Z θ=14

θ=2 dΩ 89Y

  • π+, K+89

Λ Y

SKS spectrometer KEK 12-GeV Proton Synchrotron Japan

Backup: terrestrial experiments

slide-46
SLIDE 46

33

Backup: hyperon-nucleon interaction

hyperon-nucleon interaction ?

✓ -EFT (NLO)

χ

  • J. Haidenbauer, S. Petschauer, N. Kaiser,
  • U. -G. Meißner, A. Nogga, W. Weise,
  • Nucl. Phys. A 915 (2013) 24–58

✓ one boson exchange model

Nijmegen & Jülich

  • Th. A. Rijken, M. M. Nagels, Y. Yamamoto,

Few-Body Syst. (2013) 54, 801

  • J. Haidenbauer, Ulf-G. Meißner,
  • Phys. Rev. C 72, 044005 (2005)

✓ effective - mean field models

  • E. Hiyama,Y. Yamamoto,
  • Prog. Theor. Phys. (2012) 128 (1) 105

H.-J. Schulze, E. Hiyama

  • Phys. Rev. C 90, 047301 (2014)
  • cluster approach
  • Skyrme-Hartree-Fock
  • A. A. Usmani, F. C. Khanna, J. Phys. G: Nucl.
  • Part. Phys. 35 (2008) 025105

✓ phenom. pion exchange model

Argonne-Urbana like good for QMC

slide-47
SLIDE 47

34

✓ 2-body interaction: AV18 & Usmani

Λ Λ Σ N N π π

Note: vertex

ΛπΣ 2π exchange

forbidden

ΛπΛ vertex A = 4

CSB

ΛN NN 8 > > < > > : vij = X

p=1,18

vp(rij) O p

ij

O p=1,8

ij

= n 1, σij, Sij, Lij · Sij

  • ⊗

n 1, τij

  • 8

> > < > > : vλi = X

p=1,4

vp(rλi) O p

λi

O p=1,4

λi

= n 1, σλi

  • ⊗

n 1, τ z

i

  • NN

scattering deuteron scattering

Λp

Backup: hyperon-nucleon interaction

slide-48
SLIDE 48

35

N N N N N N ∆ π π π π N N N N N N N N π π N N N ∆

NNN vijk = AP

2π O2π,P ijk

+ AS

2π O2π,S ijk

+ AR OR

ijk

nuclei nuclear matter

✓ 3-body interaction: Urbana IX & Usmani

VMC calc. no unique fit

Λ Λ π π N N N N Λ Λ Σ π π N N N N Σ π π N Λ Λ N N

ΛNN vλij = CP O2π,P

λij

+ CS O2π,S

λij

+ WD OR

λij

Backup: hyperon-nucleon interaction

slide-49
SLIDE 49

36

vCSB

λi

= Cτ T 2

π (rλi) τ z i

vλij = v2π,P

λij

+ v2π,S

λij

+ vD

λij

vλi = v0(rλi) + 1 4vσT 2

π(rλi) σλ · σi

✓ 2-body interaction ✓ 3-body interaction

charge symmetric charge symmetry breaking (spin independent)

  • A. R. Bodmer, Q. N. Usmani, Phys.Rev.C 31, 1400 (1985)

use QMC to fit on

  • hyp. exp. data

8 > > > > > < > > > > > : v2π,P

λij

= −CP 6 n Xiλ , Xλj

  • τi · τj

v2π,S

λij

= CS Z (rλi) Z (rλj) σi · ˆ riλ σj · ˆ rjλ τi · τj vD

λij = WD T 2 π (rλi) T 2 π (rλj)

 1 + 1 6σλ·(σi + σj)

  • Backup: hyperon-nucleon interaction
slide-50
SLIDE 50

37

v0(r) = vc(r) − ¯ v T 2

π(r)

vc(r) = Wc ⇣ 1 + e

r−¯ r a

⌘−1 ¯ v = (vs + 3vt)/4 vσ = vs − vt Yπ(r) = e−µπr µπr ξY (r) Tπ(r) =  1 + 3 µπr + 3 (µπr)2 e−µπr µπr ξT (r) µπ = mπ ~ = 1 ~ mπ0 + 2 mπ± 3 ξY (r) = ξ1/2

T

(r) = 1 − e−cr2 Zπ(r) = µπr 3 h Yπ(r) − Tπ(r) i Xλi = Yπ(rλi) σλ · σi + Tπ(rλi) Sλi Sλi = 3 (σλ · ˆ rλi) (σi · ˆ rλi) − σλ · σi

Constant Value Unit Wc 2137 MeV ¯ r 0.5 fm a 0.2 fm vs 6.33, 6.28 MeV vt 6.09, 6.04 MeV ¯ v 6.15(5) MeV vσ 0.24 MeV c 2.0 fm−2 Cτ

  • 0.050(5)

MeV CP 0.5 ÷ 2.5 MeV CS ' 1.5 MeV WD 0.002 ÷ 0.058 MeV

Backup: hyperon-nucleon interaction

slide-51
SLIDE 51

38

Backup: strangeness in QMC calculations

ground state

τ → ∞ E = hψ|H|ψi hψ|ψi τ → ∞ E0

projection

τ = it/~

imaginary time

✓ diffusion Monte Carlo

|ψ(0)i = |ψT i =

∞

X

n=0

cn|ϕni |ψ(τ)i = e−(H−E0)τ|ψ(0)i ∂ ∂τ |ψ(τ)i = (H E0)|ψ(τ)i c0|ϕ0i =

∞

X

n=0

e−(En−E0)τcn|ϕni

slide-52
SLIDE 52

39

Backup: strangeness in QMC calculations

✓ diffusion Monte Carlo

  • r, s

w

initial walkers

τ = Mdτ dτ ⌧ 1

imaginary time evolution: propagator final walkers

hSR|ψ(τ + dτ)i = Z dR0dS0hSR|e(HE0)dτ|R0S0ihS0R0|ψ(τ)i

  • r∗, s∗

w

slide-53
SLIDE 53

40

Backup: strangeness in QMC calculations

diffusion in coordinate space branching of configurations

✓ diffusion Monte Carlo

  • r0, s0

w

  • r, s

w

initial walkers

τ = Mdτ dτ ⌧ 1

imaginary time evolution: propagator final walkers

hSR|ψ(τ + dτ)i = Z dR0dS0hSR|e(HE0)dτ|R0S0ihS0R0|ψ(τ)i

propagator:

H = T + V (r) + V (s)

problem !!

∞ ← τ ∞ ← M

slide-54
SLIDE 54

41

Backup: strangeness in QMC calculations

✓ auxiliary field

GFMC: A ≤ 12 components many body

|Si : 2A A! (A Z)!Z!

components single particle

|Si = O

i

|Sii : 4A

AFDMC: A ∼ 90 rotation over spin-isospin configurations auxiliary field Idea: Hubbard-Stratonovich transformation

e− 1

2 γdτO2 =

1 √ 2π Z dx e− x2

2 +√−γdτxO

P ∼ e− 1

2 γdτO2

e− 1

2 γdτO2 O

i

|Sii 6= O

i

| ˜ Sii

slide-55
SLIDE 55

42

Backup: strangeness in QMC calculations

V (x) ψ(x)

dτ dτ

diffusion (DMC) rotation (AF) branching

dτ

r

r0

s s0 s00

√ dτ

dτ

ψ0(x) V (x)

dτ

V (x) ψ(x)

✓ auxiliary field diffusion Monte Carlo

slide-56
SLIDE 56

43

Backup: strangeness in QMC calculations

✓ auxiliary field diffusion Monte Carlo

  • imaginary time projection
  • stochastic method

error estimate: exact ground state

σ ∼ 1/ √ N

✓ extended wavefunction: nucleons + hyperons ✓ new propagation:

  • hyperon diffusion
  • nucleon & hyperon spinor rotations

H = X

i

p2

i

2mN + X

i<j

vij + X

i<j<k

vijk + X

λ

p2

λ

2mΛ + X

λ,i

vλi + X

λ,i<j

vλij

non-strange strange

slide-57
SLIDE 57

44

Backup: strangeness in QMC calculations

fit on exp. values

BΛ = E A−1Z

  • − E

A

ΛZ

  • > 0

✓ auxiliary field diffusion Monte Carlo

  • imaginary time projection
  • stochastic method

error estimate: exact ground state

σ ∼ 1/ √ N H = X

i

p2

i

2mN + X

i<j

vij + X

i<j<k

vijk + X

λ

p2

λ

2mΛ + X

λ,i

vλi + X

λ,i<j

vλij

non-strange strange

p

p n n

Λ

p

p n

n

ex: BΛ

5

ΛHe

  • = E

4He

  • − E

5

ΛHe

slide-58
SLIDE 58

45

Backup: strangeness in QMC calculations

s.p. orbitals plane waves

ψT (R, S) = Y

λi

f ΛN

c

(rλi) ψN

T (RN, SN) ψΛ T (RΛ, SΛ)

Φ(R, S) = A " Nκ Y

i=1

ϕ

✏ (ri, si)

# = det n ϕ

✏ (ri, si)

  • ψκ

T (Rκ, Sκ) =

Y

i<j

f κκ

c (rij) Φκ(Rκ, Sκ)

κ = N, Λ

n

sλ = ✓ uλ vλ ◆

λ

= uλ|Λ "iλ + vλ|Λ #iλ si =     ai bi ci di    

i

= ai|p "ii + bi|p #ii + ci|n "ii + di|n #ii

slide-59
SLIDE 59

46

Backup: strangeness in QMC calculations

V SD

NN + V SD ΛN = 1

2

3NN

X

n=1

λ[σ]

n

⇣ O[σ]

n

⌘2 + 1 2

3NN

X

n=1 3

X

α=1

λ[στ]

n

⇣ O[στ]

nα

⌘2 + 1 2

NN

X

n=1 3

X

α=1

λ[τ]

n

⇣ O[τ]

nα

⌘2 + 1 2

NΛ

X

n=1 3

X

α=1

λ[σΛ]

n

⇣ O[σΛ]

nα

⌘2 + 1 2

NNNΛ

X

n=1 3

X

α=1

B[σ]

n

⇣ O[σΛN]

nα

⌘2 +

NN

X

i=1

B[τ]

i

τ z

i

A[σ]

iα,jβ

A[στ]

iα,jβ

A[τ]

ij

C[σ]

λµ

direct calculation diagonalization:

λn eigenvalues

eigenvectors

ψn On = σn ψn

slide-60
SLIDE 60

47

Backup: strangeness in QMC calculations

computing time

  • 5000 configurations, 3 time steps: nucleus & hypernucleus
  • 10 nodes @ Edison (NERSC)
  • 2 socket 12-core Intel "Ivy Bridge" processor @ 2.4 GHz

240 processors

1024 2048 3072 4096 5120 6144 7168 8192 # nodes 0.05 0.1 0.15 0.2 0.25 1/t (sec

  • 1)

16 MPI ranks per node

AFDMC scaling @ Mira (ANL)

32,768 configurations, 25 steps, 28 nucleons in a periodic box, ρ=0.16 fm

  • 3

~128000 processors

  • S. Gandolfi, unpublished

system CPU time BΛ error

41 ΛCa - 40Ca

∼ 30 k hrs ∼ 0.75 MeV

49 ΛCa - 48Ca

∼ 55 k hrs ∼ 0.75 MeV

91 ΛZr - 90Zr

∼ 350 k hrs ∼ 0.75 MeV

209 ΛPb - 208Pb

∼ 4.2 M hrs ∼ 0.75 MeV AFDMC ∼ A3 σ ∼ 1/ √ N

calculation accessible

BΛ in all waves, A ± 1

slide-61
SLIDE 61

48

BΛ [MeV] A-2/3

208 89 40 28 16 12 9 7 6 5 4 3 s p d f g

emulsion (K-,π-) (π+,K+) (e,e’K+) AFDMC 0.0 5.0 10.0 15.0 20.0 25.0 30.0 0.0 0.1 0.2 0.3 0.4 0.5

  • D. L., A. Lovato, S. Gandolfi, F. Pederiva, arXiv:1508.04722 (2015)
  • 4

3

  • no bound

3 Λn

⇣

1 2 +⌘

System BΛ Bexp

Λ

∆BΛ ∆Bexp

Λ 3 ΛH

⇣

1 2 +⌘

0.23(9) 0.13(5)

4 ΛH

  • 0+

1.95(9) 2.04(4) 0.42(11) 0.35(6)

4 ΛHe

  • 0+

2.37(9) 2.39(3)

Backup: strangeness in nuclei

slide-62
SLIDE 62

48

BΛ [MeV] A-2/3

208 89 40 28 16 12 9 7 6 5 4 3 s p d f g

emulsion (K-,π-) (π+,K+) (e,e’K+) AFDMC 0.0 5.0 10.0 15.0 20.0 25.0 30.0 0.0 0.1 0.2 0.3 0.4 0.5

  • D. L., A. Lovato, S. Gandolfi, F. Pederiva, arXiv:1508.04722 (2015)
  • 4

3

  • CSB interaction not compatible with

A = 7 7

ΛHe, 7 ΛLi, 7 ΛBe

  • systems
  • E. Hiyama et al., Phys. Rev. C 80 (2009) 054321

new experiments: @ MAMI, JLab

4 ΛH 4 ΛHe @ J-PARC

problem no bound

3 Λn

⇣

1 2 +⌘

System BΛ Bexp

Λ

∆BΛ ∆Bexp

Λ 3 ΛH

⇣

1 2 +⌘

0.23(9) 0.13(5)

4 ΛH

  • 0+

1.95(9) 2.04(4) 0.42(11) 0.35(6)

4 ΛHe

  • 0+

2.37(9) 2.39(3)

Backup: strangeness in nuclei

slide-63
SLIDE 63

49

Backup: strangeness in nuclei

  • D. L., A. Lovato, S. Gandolfi, F. Pederiva, arXiv:1508.04722 (2015)

BΛ [MeV] A-2/3

208 89 40 28 16 12 9 7 6 5 4 3 s p d f g

emulsion (K-,π-) (π+,K+) (e,e’K+) AFDMC 0.0 5.0 10.0 15.0 20.0 25.0 30.0 0.0 0.1 0.2 0.3 0.4 0.5

  • 4

3

  • 5

ΛHe

AFDMC NSHH NN Minn E

  • 37.69(8)

E

  • 37.77(10)

+ ΛN BΛ 6.95(9) BΛ 6.99(10) NN AV40 E

  • 39.46(12)

E

  • 39.54(10)

+ ΛN BΛ 6.70(16) BΛ 6.84(10)

preliminary

  • F. Ferrari Ruffino

benchmark: validation interactions + methods

slide-64
SLIDE 64

50

Backup: strangeness in nuclei

  • D. L., S. Gandolfi, F. Pederiva, Phys. Rev. C 87, 041303(R) (2013)

BΛ =E(A−1Z)−E(A

ΛZ)

NN potential

5 He 17 O

VN VN + VNN VN VN + VNN Argonne V4’ 7.1(1) 5.1(1) 43(1) 19(1) Argonne V6’ 6.3(1) 5.2(1) 34(1) 21(1) Minnesota 7.4(1) 5.2(1) 50(1) 17(2) Expt. 3.12(2) 13.0(4)

nuclear effects cancel at most Idea:

✓

slide-65
SLIDE 65

51

Backup: strangeness in nuclei

double hypernuclei

Λ

  • D. L., F. Pederiva, S. Gandolfi, Phys. Rev. C 89, 014314 (2014)

vλµ =

3

X

k=1

⇣ v(k) + v(k)

σ

σλ · σµ ⌘ e−µ(k)r2

λµ

  • E. Hiyama, et al., Phys. Rev. C 66, 024007 (2002)

System E BΛ(Λ) ∆BΛΛ

4He

  • 32.67(8)

— —

5 ΛHe

  • 35.89(12)

3.22(14) —

6 ΛΛHe

  • 40.6(3)

7.9(3) 1.5(4)

6 ΛΛHe

Expt. 7.25 ± 0.19+0.18

−0.11

1.01 ± 0.20+0.18

−0.11

slide-66
SLIDE 66

52

Backup: strangeness in nuclei

4He

  • S. Gandolfi, A. Lovato, J. Carlson, K. E. Schmidt, Phys. Rev. C 90, 061306(R) (2014)
  • F. Pederiva, F. Catalano, D. L., A. Lovato, S. Gandolfi, arXiv:1506.04042 (2015)

AV40 AV40+UIXc AV60 AV70 exp

4He

  • 32.83(5)
  • 26.63(3)
  • 27.09(3)
  • 25.7(2)
  • 28.295

16O

  • 180.1(4)
  • 119.9(2)
  • 115.6(3)
  • 90.6(4)
  • 127.619

40Ca

  • 597(3)
  • 382.9(6)
  • 322(2)
  • 209(1)
  • 342.051

48Ca

  • 645(3)
  • 414.2(6)

– –

  • 416.001

Hamiltonian AFDMC GFMC AV40

  • 32.83(5)
  • 32.88(6)

AV40+UIXc

  • 26.63(3)
  • 26.82(8)

AV60

  • 27.09(3)
  • 26.85(2)

AV70

  • 25.7(2)
  • 26.2(1)

N2LO (R0 = 1.0 fm)

  • 24.41(3)
  • 24.56(1)

N2LO (R0 = 1.2 fm)

  • 25.77(2)
  • 25.75(1)
slide-67
SLIDE 67

53

Backup: strangeness in nuclei

  • A. A. Usmani, S. C. Pieper, Q. N. Usmani,
  • Phys. Rev. C 51, 2347-2355 (1995)

[ fm-3 ] r [ fm ] Nucleons (x3)

16O 17 O 17 O

0.00 0.05 0.10 0.15 0.20 0.0 1.0 2.0 3.0 4.0

single particle densities and radii

p 16O

  • = 2.50(2) fm

p 17

ΛO

  • = 2.52(3) fm

Λ 17

ΛO

  • = 2.2(1) fm

rms : exp : 2.79 fm

unpublished

slide-68
SLIDE 68

54

Backup: strangeness in neutron stars

hyper-nuclear matter

n n

n n

n n

n

n

n n n n n

n

n n n

p

Λ

Λ Λ

p p Σ

n

n

Σ

slide-69
SLIDE 69

55

Backup: strangeness in neutron stars

EOS

     EHNM ≡ EHNM(ρb) EHNM ≡ EHNM(ρb) PHNM ≡ PHNM(ρb)

TOV

( M(R) Mmax

AFDMC calculations neutrons + lambdas

EHNM ≡ EHNM(ρb, xΛ)

lambda-neutron matter equilibrium condition: chemical potentials

µΛ(ρb, xΛ) = µn(ρb, xΛ)

PNM hyperon fraction energy per particle

n n

n n

n n

n n

n

n

n

n n n n

n

n n n n

n

Λ

Λ Λ

n n n

slide-70
SLIDE 70

56

Backup: strangeness in neutron stars

EHNM(ρb, xΛ) = h EPNM((1 − xΛ)ρb) + mn i (1 − xΛ) + h EF

Λ (xΛρb) + mΛ

i xΛ + f(ρb, xΛ)

neutrons + lambdas

( ρn = (1 − xΛ)ρb ρΛ = xΛρb    ρb = ρn + ρΛ xΛ = ρΛ ρb

Problem1: limitation in xΛ due to simulation box Problem2: finite size effects Problem3: fitting procedure cluster expansion

ρΛρn ρb ρΛρnρn ρb ρΛρnρnρn ρb

,

f(ρb, xΛ) ρΛρΛρn ρb

, ,

slide-71
SLIDE 71

57

Backup: strangeness in neutron stars

0.0 0.2 0.4 0.6 0.8 1.0

  • b

[ f m

  • 3

] 0.0 0.1 0.2 0.3 0.4 0.5 x 1.0 2.0 3.0 4.0 µ [ GeV ]

µn(ρb, xΛ) µΛ(ρb, xΛ)        µn(ρb, xΛ) = EPNM(ρn) + ρn ∂EPNM ∂ρn + mn + f(ρb, xΛ) + ρb ∂f ∂ρn µΛ(ρb, xΛ) = EF

Λ (ρΛ) + ρΛ

∂EF

Λ

∂ρΛ + mΛ + f(ρb, xΛ) + ρb ∂f ∂ρΛ µΛ(ρb, xΛ) = µn(ρb, xΛ)

equilibrium condition:

xΛ ≡ xΛ(ρb) ρth

Λ

@ xΛ → 0

slide-72
SLIDE 72

58

Backup: strangeness in neutron stars

E [MeV] b [fm-3] PNM N 20 40 60 80 100 120 140 0.0 0.1 0.2 0.3 0.4 0.5 0.6

particle fraction b [fm-3] n

  • 0.2

0.3 0.4 0.5 0.6 10-2 10-1 100 AV8’+UIX

  • D. L., A. Lovato, S. Gandolfi, F. Pederiva, Phys. Rev. Lett. 114, 092301 (2015)

ρth

Λ = 0.24(1) fm−3

slide-73
SLIDE 73

59

Backup: strangeness in neutron stars

ρth

Λ = 0.24(1) fm−3

ρth

Λ = 0.34(1) fm−3

E [MeV] b [fm-3] PNM N N + NN (I) 20 40 60 80 100 120 140 0.0 0.1 0.2 0.3 0.4 0.5 0.6

particle fraction b [fm-3] n

  • 0.2

0.3 0.4 0.5 0.6 10-2 10-1 100 AV8’+UIX

  • D. L., A. Lovato, S. Gandolfi, F. Pederiva, Phys. Rev. Lett. 114, 092301 (2015)
slide-74
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60

Backup: strangeness in neutron stars

0.66(2)M 2.09(1)M M [M0] c [fm-3] PNM N N + NN (I) N + NN (II) 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 0.0 0.2 0.4 0.6 0.8 1.0 PSR J1614-2230 PSR J0348+0432 2.45(1)M 1.36(5)M

  • D. L., A. Lovato, S. Gandolfi, F. Pederiva, Phys. Rev. Lett. 114, 092301 (2015)
slide-75
SLIDE 75

61

Backup: strangeness in neutron stars

  • Phys. Rev. Lett. 114, 092301 (2015)

M [M0] R [km] PNM N N + NN (I) N + NN (II) 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 10 11 12 13 14 15 16 PSR J1614-2230 PSR J0348+0432 PSR J0348+0432

NN+NNN

BB+NNN BB+BBB

NN

  • Phys. Rev. C 90, 045805 (2014)

AFDMC G-matrix

slide-76
SLIDE 76

62

Backup: strangeness in neutron stars

  • Phys. Rev. Lett. 114, 092301 (2015)

M [M0] R [km] PNM N N + NN (I) N + NN (II) 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 10 11 12 13 14 15 16 PSR J1614-2230 PSR J0348+0432 PSR J0348+0432

NN+NNN

BB+NNN BB+BBB

NN

  • Phys. Rev. C 90, 045805 (2014)

AFDMC G-matrix

BΛ [MeV] A-2/3 ΛN ΛN + ΛNN (I) ΛN + ΛNN (II)

  • 30.0
  • 25.0
  • 20.0
  • 15.0
  • 10.0
  • 5.0

0.0 0.0 0.1 0.2 0.3 0.4 0.5

  • Phys. Rev. C 90, 045805 (2014)
  • Phys. Rev. C 89, 014314 (2014), arXiv:1506.04042 (2015)

AFDMC G-matrix

slide-77
SLIDE 77

63

Backup: strangeness in neutron stars

  • Phys. Rev. Lett. 114, 092301 (2015)

BΛ [MeV] A-2/3 ΛN ΛN + ΛNN (I) ΛN + ΛNN (II)

  • 30.0
  • 25.0
  • 20.0
  • 15.0
  • 10.0
  • 5.0

0.0 0.0 0.1 0.2 0.3 0.4 0.5

  • Phys. Rev. C 90, 045805 (2014)
  • Phys. Rev. C 89, 014314 (2014), arXiv:1506.04042 (2015)

PSR J0348+0432

NN+NNN

BB+NNN BB+BBB

NN

  • Phys. Rev. C 90, 045805 (2014)

G-matrix

M [M0] c [fm-3] PNM N N + NN (I) N + NN (II) 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 0.0 0.2 0.4 0.6 0.8 1.0 PSR J1614-2230 PSR J0348+0432

AFDMC

slide-78
SLIDE 78

64

Backup: strangeness in nuclei

vλij = v2π,P

λij

+ v2π,S

λij

+ vD

λij

✓ 3-body interaction

−3 PT =0 + CT PT =1

isospin projectors fit on symmetric hypernuclei

8 > > > > > < > > > > > : v2π,P

λij

= −CP 6 n Xiλ , Xλj

  • τi · τj

v2π,S

λij

= CS Z (rλi) Z (rλj) σi · ˆ riλ σj · ˆ rjλ τi · τj vD

λij = WD T 2 π (rλi) T 2 π (rλj)

 1 + 1 6σλ·(σi + σj)

  • control parameter:

strength and sign of the nucleon isospin triplet channel sensitivity study: light- & medium-heavy hypernuclei

τi · τj = −3 PT =0 + PT =1