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Title: Skyrme N2LO functionals: first results on finite nuclei

Author: D. Davesne, P. Becker, A. Pastore, J. Navarro

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Skyrme N2LO

functionals: first results

on finite nuclei

Skyrme N2LO functionals: first results on

finite nuclei

D. Davesne, P. Becker,

A. Pastore, J. Navarro

Introduction

Infinite matter

calculations

D. Davesne, P. Becker, A. Pastore, J. Navarro

Application to

astrophysics LYVA1

Application to spherical

nuclei

Orsay, October 2017

First results

Conclusion and

perspectives

n

N2LO/N3LO extensions : physical motivation

n

Results in infinite matter

n

Extension of Gogny interaction

n

Application in astrophysics

n

Application to finite nuclei: first results

n

Conclusion

Linear response

formalism

N2LO/N3LO extensions : physical motivation

Skyrme N2LO

functionals: first results

on finite nuclei

D. Davesne, P. Becker,

A. Pastore, J. Navarro

Introduction

Infinite matter

calculations

n Construction of new effective interactions necessary!

Application to

astrophysics LYVA1

n Instabilities experienced with popular interactions (Skyrme,

Application to spherical

nuclei

Gogny)

n Initial idea (Skyrme) : expansion in powers of momentum (k2 )

→ systematic expansion up to kn ... which n???

n N2LO : n = 2 ; N3LO : n = 3 ; ...

2 2

n Gogny: e−r /µ , M3Y : e−µr /µr, ... : SAME kind of expansion

[F. Raimondi et al., Phys.Rev. C84 (2011) 064303]

First results

Conclusion and

perspectives

Linear response

formalism

N2LO/N3LO extensions : physical motivation

Skyrme N2LO

functionals: first results

on finite nuclei

D. Davesne, P. Becker,

A. Pastore, J. Navarro

Finite-range interaction D1S: infinite sum of partial waves.

Introduction

80

E/A [MeV]

60

Infinite matter

calculations

D1S

S-wave

S+P-wave

S+P+D-wave

S+P+D+F-wave

Application to

astrophysics LYVA1

Application to spherical

nuclei

First results

40

Conclusion and

perspectives

Linear response

formalism

20

0

-20

0

0.1

0.2

0.3

0.4

0.5

-3

ρ [fm ]

Only S, P, D and F (` < 4) waves necessary → N3LO good enough

Skyrme pseudo-potential N2LO/N3LO

Skyrme N2LO

functionals: first results

on finite nuclei

D. Davesne, P. Becker,

A. Pastore, J. Navarro

Introduction

1

V(r1 , r2 ) = t0 (1 + x0 Pσ ) + t3 (1 + x3 Pσ )ρα (R)

6

Skyrme (N1LO)

i

h 0

0

1

2

2

+ t1 (1 + x1 Pσ ) k + k + t2 (1 + x2 Pσ ) k · k

2

h

i

1

+ t1(4) (1 + x1(4) Pσ ) (k2 + k0 2 )2 + 4(k0 · k)2

4

Skyrme N2LO

(4)

(4)

0

2

02

+ t2 (1 + x2 Pσ )(k · k)(k + k )

i

0

h 0

1

+ t1(6) 1 + x1(6) Pσ (k 2 + k2 ) (k 2 + k2 )2 + 12(k0 · k)2

2

Skyrme N3LO

h

i

0

0

02

(6)

(6)

2 2

2

+ t2 1 + x2 Pσ (k · k) 3(k + k ) + 4(k · k)

n D and F partial waves included

n Gauge invariance

n Also includes:

n spin-orbit term W0

n tensor terms

Infinite matter

calculations

Application to

astrophysics LYVA1

Application to spherical

nuclei

First results

Conclusion and

perspectives

Linear response

formalism

Infinite matter: (S , T ) channels N2LO

Skyrme N2LO

functionals: first results

on finite nuclei

D. Davesne, P. Becker,

A. Pastore, J. Navarro

n Used as a preliminary test before dealing with finite nuclei

n First step: (S , T ) channels

Introduction

n Results compared to BHF calculations from Baldo and al.

Infinite matter

calculations

Application to

astrophysics LYVA1

15

10

5

0

-5

-10

-15

0

BHF

SLy5

NLO

N2LO

-5

-5

-10

-10

-15

-15

-20

-20

-25

-25

-30

-30

0

0.1

0.2

ρ [fm-3]

0.3

0

0.1

0.2

ρ [fm-3]

0.3

Application to spherical

nuclei

First results

(S=1,T=1)

15

10

5

0

-5

-10

-15

0

(S=0,T=1)

(S=1,T=0)

(S=0,T=0)

(1997)

Conclusion and

perspectives

Linear response

formalism

Infinite matter: (S , T ) channels N3LO

Skyrme N2LO

functionals: first results

on finite nuclei

40

35

30

25

20

15

10

5

0

0

-5

-10

-15

-20

-25

-30

-35

-40

-45

BHF

N1LO

N2LO

N3LO

0

0.2

0.4

0.6

0.8

0

0.2

ρ [fm-3]

n Agreement up to ρ = 0.8 fm−3

n Exploration of a new parameter space

0.4

0.6

ρ [fm-3]

0.8

Introduction

Infinite matter

calculations

(S=1,T=1)

40

35

30

25

20

15

10

5

0

0

-5

-10

-15

-20

-25

-30

-35

-40

-45

Application to

astrophysics LYVA1

Application to spherical

nuclei

First results

Conclusion and

perspectives

(S=0,T=1)

(S=1,T=0)

(S=0,T=0)

D. Davesne, P. Becker,

A. Pastore, J. Navarro

Linear response

formalism

Infinite matter: (S , T ) channels M3Y

Skyrme N2LO

functionals: first results

on finite nuclei

D. Davesne, P. Becker,

A. Pastore, J. Navarro

Introduction

30

Infinite matter

calculations

20

20

10

10

0

0

0

0

(S=1,T=1)

40

30

-10

-10

-20

-20

-30

-30

-40

0

0.2

0.4

0.6

0

0.2

-3

ρ [fm ]

0.4

0.6

-40

0.8

n M3Y takes into account nuclei and (S , T ) channels: both are not

incompatibles

Application to

astrophysics LYVA1

Application to spherical

nuclei

(S=0,T=1)

(S=1,T=0)

(S=0,T=0)

M3Y

40

First results

Conclusion and

perspectives

Linear response

formalism

Infinite matter: (S , T ) channels Gogny

Skyrme N2LO

functionals: first results

on finite nuclei

D. Davesne, P. Becker,

A. Pastore, J. Navarro

Not possible...

Introduction

20

3 Gaussians

D1M-fit

D1M

30

20

10

10

(S=1,T=1)

(S=0,T=0)

30

Infinite matter

calculations

40

40

-10

-20

-20

-30

-30

-40

0

0.2

0.4

0.6

... except with a third gaussian

0

0.2

-3

ρ [fm ]

0.4

0.6

-40

0.8

(S=0,T=1)

(S=1,T=0)

0

-10

Application to spherical

nuclei

First results

Conclusion and

perspectives

0

0

0

Application to

astrophysics LYVA1

Linear response

formalism

Determination of the three ranges

Skyrme N2LO

functionals: first results

on finite nuclei

D. Davesne, P. Becker,

A. Pastore, J. Navarro

Physical meaning of a range :

n Yukawa potential: related to masses (770, 490, 140 MeV)

n Gaussian potential??? → definition via the self-energy

Introduction

Infinite matter

calculations

Application to

astrophysics LYVA1

Application to spherical

nuclei

First results

Conclusion and

perspectives

0.25

0.2

Example:

RF/H Yukawa

mρ = 770 MeV

→ µ−1

Y =0.256 fm

RF/H Gogny

RF/H

0.15

0.1

0.05

0

0

0.5

1

1.5

2

2.5

3

→ R(µ−1

Y )=0.228 =R(µG )

→ µG = 0.475 fm

x

→ ranges: µ1 = 0.475 fm, µ2 = 0.746 fm , µ3 = 1.964 fm

Linear response

formalism

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