Electromagnetic Interactions in Nuclei 2018, 811 October 2018 1 - - PowerPoint PPT Presentation

Collaborators:

• A. M. Shirokov, L. D. Blokhintsev, SINP MSU
• A. I. Mazur, PNU, Khabarovsk
• I. J. Shin, Y. Kim, Institute for Basic Sciences, Republic of Korea
• J. P. Vary, Iowa State University, USA
• G. Papadimitriou, Lawrence Livermore National Laboratory, USA
• R. Roth, Technische Universität, Darmstadt, Germany

Electromagnetic Interactions in Nuclei – 2018, 8–11 October 2018

1

Motivation

×

There are ab initio approaches for describing bound states of light nuclei (with A > 4 nucleons)

×

Green Function Monte Carlo (GFMC)

×

No-Core Shell Model (NCSM)

×

Coupled Cluster Method (CCM) …

×

Ab initio methods for description continuum spectrum states in light nuclear systems are developed no so well as bound states methods

×

We propose Single State HORSE method for description continuum spectrum states in light nuclear systems. This method based on ab initio calculations in NCSM

2

No-Core Shell Model (NCSM)

×

Modern version of Nuclear Shell Model

• B. R. Barrett, P. Navratil, J. P. Vary Progr. Part. Nucl. Phys. 69, 131 (2013).

× ×

Wave function is expanded to multiparticle oscillator basis

×

Parameters:

×

Nmax – max number of excitation quanta

×

ħΩ – parameter of oscillator function

HA = 1 A X

i<j

(pi − pj)2 2m + X

i<j

V NN

ij

+ X

i<j

V NNN

ijk

+ ...

0ħΩ +2ħΩ + + + … +4ħΩ … + + … + … + NmaxħΩ

6Li

3

HORSE (J-matrix)

×

Our method based on Harmonic Oscillator Representation of Scattering

• Equations. HORSE is reliable method for describing twoparticle scattering

×

– oscillator functions – oscillator quanta

×

Approximation

• f potential (short range)

×

Kinetic energy matrix stay full

×

Phase shifts

H`u`(E, r) = Eu`(E, r) R`

N(r)

tan δ`(E) = − SN`(E) − GNN(E)T `

N,N+2SN+2,`(E)

CN`(E) − GNN(E)T `

N,N+2CN+2,`(E) (N`)/2

X

n0=0

H`

NN 0hN 0`|⌫i = E⌫hN`|⌫i

CN`(E) = s ⇡n! Γ(n + ` + 3

2)

(−1)n Γ(−` + 1

2)

✓2E ~Ω ◆− `

2

exp ✓ − E ~Ω ◆

1F1

✓ −n − ` − 1 2, −` + 1 2; 2E ~Ω ◆ SN`(E) = s ⇡n! Γ(n + ` + 3

2)

✓2E ~Ω ◆ `+1

2

exp ✓ − E ~Ω ◆ L

`+ 1

2

n

✓2E ~Ω ◆

GNN 0(E) =

N 1

X

ν=0

h⌫|N`i hN 0`|⌫i Eν E

u`(E, r) =

∞

X

n=0

aN`(E) RN`(r) N = 2n + `

˜ V `

NN 0 =

⇢ V `

NN 0,

N, N 0 ≤ N = Nmax + Nmin 0, N, N 0 > N = Nmax + Nmin Q-space P-space

4

Single State HORSE

× ×

In P-space we may use various calculations with oscillator basis, for example, ab initio No-Core Shell Model calculations

×

Phase shift requires ALL eigenstates, number of them increase rapidly

×

E = Eν:

×

By varying Nmax and ħΩ we obtain Eν and δl in some interval

×

Parametrization of phase shifts

×

Search of S-matrix poles, which associated with bound, resonant states

tan δ`(E⌫) = − SNmax+Nmin+2,`(E⌫) CNmax+Nmin+2,`(E⌫)

tan δ`(E) = − SN`(E) − GNN(E)T `

N,N+2SN+2,`(E)

CN`(E) − GNN(E)T `

N,N+2CN+2,`(E)

GNN 0(E) =

N 1

X

ν=0

h⌫|N`i hN 0`|⌫i Eν E

(N`)/2

X

n0=0

H`

NN 0hN 0`|⌫i = E⌫hN`|⌫i,

n = 0, ..., (N `)/2 N 0 = 2n0 + `

5

Parametrization & S-matrix poles search

×

Scattering amplitude

×

Effective-range function

×

Padé-approximation

×

S-matrix S = e2iδ has the same poles as f(E)

×

f(E) have poles where

×

From theory of functions of complex variables:

×

Bound, false pole ; virtual pole Resonance pole

f(E) = k2` K(E) − ik2`+1 K(E) = k2`+1 cot δ F(E) ≡ K(E) − ik2`+1 = 0 Υ = 1 2πi I

C

F0(E) F(E) dE Ep = 1 2πi I

C

E F0(E) F(E) dE number of zeroes zero’s position (S-matrix pole) Ep = Er + iΓ 2 Ep = −Eb k = √2µE ~

6

K(E) = w(n) + w(n)

1 E + w(n) 2 E2 + · · ·

1 + w(d)

1 E + w(d) 2 E2 + · · ·

Ep = |Ev|

7

×

Another way: Represent S-matrix as

× ×

S-matrix poles types and contributions to phase shift: Background Pole

S(k) = Θ(k)Sp(k) δ = φ + δp

Bound and false Virtual Resonance pole

• kr

kr Re(k)

• ikb(f)
• ir
• ikv

ikv ir ikb(f) Im(k)

k-plane

Sb(f) = kb(f) − ik kb(f) + ik Sv = kv + ik kv − ik Sr = (k − κ∗

r)(k + κr)

(k − κr)(k + κ∗

r), κr = kr − iγr

δb(E) = π − arctan s E |Eb| δf(E) = − arctan s E |Ef| δv(E) = arctan r E Ev

δr(E) = − arctan a √ E E − b2 Er = b2 − a2/2, Γ = a p 4b2 − a2

Parametrization & S-matrix poles search

NN interactions

×

JISP16 (J-Matrix Inverse Scattering Potential)

×

• A. M. Shirokov, J. P. Vary, A. I. Mazur, T. A. Weber
• Phys. Lett. B 644, 33 (2007)

×

Phenomenological potential obtained from np-phase shifts and deutron bound energy

×

Phase equivalent transformations for precise description of 4He and 16O

×

NN-interactions from chiral effective field theory: Idaho N3LO

×

• D. R. Entem, R. Machleidt
• Phys. Rev. C 68, 041001(R) (2003);

×

• E. Epelbaum, H. Krebs, U.-G. Meissner
• Phys. Rev. Lett. 115, 122301 (2015).

×

Daejeon16

×

• A. M. Shirokov, I. J. Shin, Y. Kim, M. Sosonkina, P. Maris, J. P. Vary Phys.
• Lett. B 761, 87 (2016)

×

Chiral interactions + phase equivalent transformations

8

nα-scattering

×

No-Core Shell Model calculations:

×

NN-interactions: JISP16 and Daejeon16

×

lowest eigenenergies E0 of 3/2-, 1/2- and 1/2+ states of 5He nuclei in bases with Nmax ≤ 18, 10 ≤ ħΩ ≤ 40 MeV

×

ground state energy of 4He nuclei in the same bases

×

Energy is calculated regarding the channel threshold

×

3/2-, 1/2- scattering states

×

1/2+ scattering state

E0(Nmax, ~Ω) = E

5He, 3 2 −,( 1 2 −)

(Nmax, ~Ω) − E

4He, gs

(Nmax, ~Ω) E0(Nmax, ~Ω) = E

5He, 1 2 +

(Nmax, ~Ω) − E

4He, gs

(Nmax − 1, ~Ω)

9

3/2- state of nα-scattering: JISP16

Er , MeV Γ, MeV Ξ, keV SS-HORSE Nmax ≤ 18 0.89 0.99 70 SS-HORSE Nmax ≤ 4 0.89 1.01 106 From experiment 0.80 0.65

10 20 30 40 h [MeV] 5 10 15 E [MeV] Nmax= 2 4 6 8 10 12 14 16 18

n, 3/2

• JISP16

Manual selection 5 10 15 Ecm [MeV] 30 60 90 120 1 [degrees] Exp. Nmax= 2 4 6 8 10 12 14 16 18

n, 3/2

• JISP16

5 10 15 Ecm [MeV] 30 60 90 120 1 [degrees] Exp. Fit exp. Nmax= 2 4 6 8 10 12 14 16 18 SS-HORSE

n, 3/2

• JISP16

Manual selection

• Phys. Rev. C 94, 064320 (2016)

tan δ`(E⌫) = − SN+2,`(E⌫) CN+2,`(E⌫) δ1 = − arctan a √ E E − b2 − a b2 √ E + d √ E 3

10

1/2- state of nα-scattering: JISP16

Er , MeV Γ, MeV Ξ, keV SS-HORSE Nmax ≤ 18 1.856 5.456 85 SS-HORSE Nmax ≤ 6 1.834 5.511 193 From experiment 2.07 5.57

• Phys. Rev. C 94, 064320 (2016)

tan δ`(E⌫) = − SN+2,`(E⌫) CN+2,`(E⌫) δ1 = − arctan a √ E E − b2 − a b2 √ E + d √ E 3

11 10 20 30 40 h [MeV] 5 10 15 20 25 E [MeV] Nmax= 2 4 6 8 10 12 14 16 18

n, 1/2

• JISP16

Manual selection 5 10 15 20 25 Ecm [MeV] 30 60 1 [degrees] Exp. Nmax= 2 4 6 8 10 12 14 16 18

n, 1/2

• JISP16

5 10 15 20 25 Ecm [MeV] 30 60 1 [degrees] Exp. Fit exp. Nmax= 4 6 8 10 12 14 16 18 SS-HORSE

n, 1/2

• JISP16

Manual selection

1/2+ state of nα-scattering: JISP16

Er , MeV Γ, MeV Ξ, keV SS-HORSE Nmax ≤ 17 non-resonant 120 SS-HORSE Nmax ≤ 7 non-resonant 168

• Phys. Rev. C 94, 064320 (2016)

tan δ`(E⌫) = − SN+2,`(E⌫) CN+2,`(E⌫)

12

δ0 = π − arctan s E |Eb| + c √ E + √ E 3 + f √ E 5

10 20 30 40 h [MeV] 10 20 30 40 E [MeV] Nmax= 3 5 7 9 11 13 15 17

n, 1/2

+

JISP16

Manual selection 10 20 30 40 Ecm [MeV] 60 90 120 150 180 0 [degrees] Exp. Nmax= 3 5 7 9 11 13 15 17

n, 1/2

+ 10 20 30 40 Ecm [MeV] 60 90 120 150 180 0 [degrees] Exp. Fit exp. Nmax= 5 7 9 11 13 15 17 SS-HORSE

n, 1/2

+

JISP16

Manual selection

nα-scattering: Daejeon16

Er , MeV Γ, MeV Ξ, keV 3/2- Single State HORSE Daejeon16 0.68 0.52 22 JISP16 0.89 0.99 70 From experiment 0.80 0.65 1/2- Single State HORSE Daejeon16 2.45 5.07 48 JISP16 1.856 5.456 85 From experiment 2.07 5.57 1/2+ Single State HORSE Daejeon16 non-resonant 119 JISP16 non-resonant 120

arXiv: 1808.03394

13

5 10 15 20 E [MeV] 30 60 90 120 150 180 [degrees] Experiment Nmax= 12 (11) 14 (13) 16 (15) 18 (17) Daejeon16 JISP16

n Daejeon16 1/2

• 3/2
• 1/2

++20

• w0 + w1E + w2E2 =

= −k2l+1 CN+2,l(E) SN+2,l(E)

Generalization for scattering with repulsive Coulomb interaction

×

Auxiliary short-range potential

×

Optimal d is classic turning point

• f highest oscillator state

×

Phase shifts δSh of scattering with VSh and phase shifts δ of scattering with VNucl + VCoul linked

(through matching condition for wave function at r = d) jl, nl, Fl, Gl – Bessel, Neumann, Coulomb regular and irregular functions

×

Single State HORSE:

r V

Sh(r)

Coulomb Nuclear

r = d

14

V Sh = ⇢ V Nucl + V Coul, r ≤ d 0, r > d tan δ` = − Wd(j`, F`) + Wd(n`, F`) tan δSh

`

Wd(j`, G`) + Wd(n`, G`) tan δSh

`

Wd(φ, χ) = ✓dφ dr χ − φ dχ dr ◆

• r=d

tan δ`

• E⌫
• = − SN+2,`
• E⌫
• Wd(n`, F`) + CN+2,`(E⌫) Wd(j`, F`)

SN+2,`

• E⌫
• Wd(n`, G`) + CN+2,`
• E⌫
• Wd(j`, G`)

Generalization for scattering with repulsive Coulomb interaction

×

Approximation through Coulomb modified

×

scattering amplitude

×

effective-range function

• J. Hamilton, I. Øverbö and B. Tromborg, Nucl. Phys. B 60, 443 (1973)

×

Search of poles of f(E)

c`⌘ =

`

Y

n=1

(1 + ⌘2/n2)−1 (` > 0), c0⌘ = 1 η = Z1Z2α r µc2 2E H(η) = Γ0(iη) Γ(iη) + 1 2iη − ln(iη)

15

f(E) = k2` K(E) − 2ηk2`+1H(η)(c`⌘)−1 K(E) = k2`+1(c`⌘)−1 ⇢ 2πη exp (2πη) − 1 [cot δ` − i] + 2ηH(η)

• ,

pα-scattering

×

No-Core Shell Model calculations:

×

NN-interactions: JISP16 and Daejeon16

×

lowest eigenenergies E0 of 3/2-, 1/2- and 1/2+ states of 5Li nuclei in bases with Nmax ≤ 18, 10 ≤ ħΩ ≤ 40 MeV

×

ground state energy of 4He nuclei in the same bases

×

Energy is calculated regarding the channel threshold

×

3/2-, 1/2- scattering states

×

1/2+ scattering state

×

16

E0(Nmax, ~Ω) = E

5Li, 3 2 −,( 1 2 −)

(Nmax, ~Ω) − E

4He, gs

(Nmax, ~Ω) E0(Nmax, ~Ω) = E

5Li, 1 2 +

(Nmax, ~Ω) − E

4He, gs

(Nmax − 1, ~Ω) w0 + w1E + w2E2 = −k2l+1(clη)−1 ⇢ 2πη exp (2πη) − 1 × SN+2,l(E) Wb(nl, Gl) + CN+2,l(E) Wb(jl, Gl) SN+2,l(E) Wb(nl, Fl) + CN+2,l(E) Wb(jl, Fl) + i

• −2ηH(η)
• .

pα-scattering

17

Er , MeV Γ, MeV Ξ, keV 3/2- Single State HORSE JISP16 1.84 1.80 43 Daejeon16 1.52 1.05 24 From experiment 1.69 1.23 1/2- Single State HORSE JISP16 3.54 6.04 63 Daejeon16 3.21 5.63 50 From experiment 3.18 6.60 1/2+ Single State HORSE JISP16 non-resonant 105 Daejeon16 non-resonant 99

5 10 15 20 E [MeV] 30 60 90 120 150 180 [degrees] Experiment Nmax= 4 6 8 10 (9) 12 (11) 14 (13) 16 (15) 18 (17) SS-HORSE

p JISP16 1/2

++20

• 1/2
• 3/2
• 5

10 15 20 E [MeV] 30 60 90 120 150 180 [degrees] Experiment Nmax= 12 (11) 14 (13) 16 (15) 18 (17) Daejeon16 JISP16

p Daejeon16 1/2

• 3/2
• 1/2

++20

• arXiv: 1808.03394

n–6He scattering

×

No-Core Shell Model calculations:

×

NN-interaction: Daejeon16

×

lowest eigenenergies E0 of 3/2-, 1/2-, 5/2-, 3/2+, 1/2+ and 5/2+ states

• f 7He nuclei in bases with Nmax ≤ 17, 10 ≤ ħΩ ≤ 40 MeV

×

ground state energy of 6He nuclei in the same bases

×

Energy is calculated regarding the channel threshold

×

for waves with negative parity

×

for waves with positive parity

18

E0(Nmax, ~Ω) = E

7He

(Nmax, ~Ω) − E

6He, gs

(Nmax, ~Ω) E0(Nmax, ~Ω) = E

7He

(Nmax, ~Ω) − E

6He, gs

(Nmax − 1, ~Ω)

3/2- state of n–6He scattering: Daejeon16

19

2 4 6 8 E [MeV] 30 60 90 120 150 [degrees] Nmax= 2 4 6 8 10 12 14 16 n-

6He, 3/2

• , Daejeon16, Variant1

2 4 6 E [MeV] 30 60 90 120 150 [degrees] 12 14 16 SS-HORSE n-

6He, 3/2

• , Daejeon16, Variant1

Er , MeV Γ, MeV Ξ, keV SS-HORSE Nmax ≤ 16

0.28 0.13 32

From experiment

0.44 0.16 w(n) + w(n)

1 E + w(n) 2 E2

1 + w(d)

1 E + w(d) 2 E2

= = −k2l+1 CN+2,l(E) SN+2,l(E)

10 20 30 40 50 h [MeV] 1 2 3 4 E [MeV] Nmax= 2 4 6 8 10 12 14 16 n-

6He, 3/2

• , Daejeon16

1/2- state of n–6He scattering: Daejeon16

20

Er , MeV Γ, MeV Ξ, keV SS-HORSE Nmax ≤ 16

2.8 4.3 570

From experiment

1.2 1.0

5 10 15 E [MeV] 15 30 45 [degrees] Nmax= 2 4 6 8 10 12 14 16 n-

6He, 1/2

• , Daejeon16, Variant1

5 10 15 E [MeV] 15 30 45 [degrees] 12 14 16 SS-HORSE n-

6He, 1/2

• , Daejeon16, Variant1

w0 + w1E + w2E2 = = −k2l+1 CN+2,l(E) SN+2,l(E)

10 15 20 25 30 35 40 h [MeV] 5 10 15 E [MeV] Nmax= 2 4 6 8 10 12 14 16 n-

6He, 1/2

• , Daejeon16

5/2- state of n–6He scattering: Daejeon16

21

Er , MeV Γ, MeV Ξ, keV SS-HORSE Nmax ≤ 16

3.7 1.4 651

From experiment

3.3 2.2 w(n) + w(n)

1 E

1 + w(d)

1 E + w(d) 2 E2 + w(d) 3 E3 =

= −k2l+1 CN+2,l(E) SN+2,l(E)

5 10 15 E [MeV] 30 60 90 120 [degrees] Nmax= 2 4 6 8 10 12 14 16 n-

6He, 5/2

• , Daejeon16, Variant1

5 10 15 E [MeV] 30 60 90 120 [degrees] 12 14 16 SS-HORSE n-

6He, 5/2

• , Daejeon16, Variant1

10 20 30 40 50 h [MeV] 5 10 15 E [MeV] Nmax= 2 4 6 8 10 12 14 16 n-

6He, 5/2

• , Daejeon16

1/2+ state of n–6He scattering: Daejeon16

22

Er , MeV Γ, MeV Ξ, keV SS-HORSE Nmax ≤ 17

non-resonant 885

From experiment

non-resonant

5 10 15 E [MeV] 60 90 120 150 180 [degrees] Nmax= 3 5 7 9 11 13 15 17 n-

6He, 1/2 +, Daejeon16, Variant1

5 10 15 E [MeV] 60 90 120 150 180 [degrees] 13 15 17 SS-HORSE n-

6He, 1/2 +, Daejeon16, Variant1

w0 + w1E + w2E2 = = −k2l+1 CN+2,l(E) SN+2,l(E)

20 30 40 h [MeV] 5 10 15 E [MeV] Nmax= 3 5 7 9 11 13 15 17 n-

6He, 1/2 +, Daejeon16

3/2+ state of n–6He scattering: Daejeon16

23

Er , MeV Γ, MeV Ξ, keV SS-HORSE Nmax ≤ 17

4.0 4.4 131

From experiment

non-resonant

5 10 15 20 E [MeV]

• 30

30 [degrees] Nmax= 3 5 7 9 11 13 15 17 n-

6He, 3/2 +, Daejeon16, Variant1

5 10 15 E [MeV] 15 30 45 [degrees] Nmax= 15 17 SS-HORSE n-

6He, 3/2 +, Daejeon16, Variant1

w(n) + w(n)

1 E + w(n) 2 E2

1 + w(d)

1 E

= = −k2l+1 CN+2,l(E) SN+2,l(E)

5 10 15 20 25 30 35 40 h [MeV] 5 10 15 E [MeV] Nmax= 3 5 7 9 11 13 15 17 n-

6He, 3/2 +, Daejeon16

5/2+ state of n–6He scattering: Daejeon16

24

Er , MeV Γ, MeV Ξ, keV SS-HORSE Nmax ≤ 17

3.9 4.7 135

From experiment

non-resonant

5 10 15 E [MeV]

• 60
• 30

30 60 [degrees] Nmax= 3 5 7 9 11 13 15 17 n-

6He, 5/2 +, Daejeon16, Variant1

5 10 15 20 E [MeV] 30 [degrees] Nmax= 15 17 SS-HORSE n-

6He, 5/2 +, Daejeon16, Variant1

w(n) + w(n)

1 E + w(n) 2 E2

1 + w(d)

1 E

= = −k2l+1 CN+2,l(E) SN+2,l(E)

20 30 40 h [MeV] 5 10 15 E [MeV] Nmax= 3 5 7 9 11 13 15 17 n-

6He, 5/2 +, Daejeon16

Generalization for true-multiparticle scattering

×

Continuum spectrum states can be expanded to hyperspherical basis with hypermomentum K = Kmin, Kmin+ 2, …

×

In case of 4n → 4n (Kmin = 2) it is reasonable to use only one hyperspherical harmonic, other are suppressed by centrifugal barrier , ρ – hyperradius,

×

General theory of true-multiparticle scattering:

• S. A. Zaytsev, Yu. F. Smirnov, A. M. Shirokov,
• Theor. Math. Phys. 117, 1291 (1998)

×

Wave function of A-particle system N – oscillator quanta, – (3A - 3)-dimensional oscillator functions

×

One-channel case (one value of K) S-matrix S = e2iδ, – the same as in two-particle scattering can be calculated in No-Core Shell Model

25

L(L + 1)/ρ2 L = K + (3A − 6)/2 |Eβi = X

NKγ

hNKγ|Eβi |NKγi |NKγi tan δ(E) = − SNL(E) − GNN(E)T L

N,N+2SN+2,L(E)

CNL(E) − GNN(E)T L

N,N+2CN+2,L(E)

SNL(E), CNL(E) GNN

4n → 4n scattering with JISP16

×

Single-State HORSE: E = Eν:

× ×

No-Core Shell Model: lowest eigenvalue with 0+ was calculated in bases with Nmax ≤ 18, 1 ≤ ħΩ ≤ 40 MeV

×

P-space include all hyperspherical harmonics

26

tan δ(Eν) = − SN+2,L(Eν) CN+2,L(Eν)

N = Nmax + Nmin = Nmax + 2

• Phys. Rev. Lett. 117, 182502 (2016)

5 10 15 20 25 30 E [MeV] 30 60 90 120 [degrees] Nmax= 2 4 6 8 10 12 14 16 18

4n, gs

10 20 30 40 h [MeV] 5 10 15 20 25 30 E [MeV] Nmax= 2 4 6 8 10 12 14 16 18

4n, gs

JISP16

4n → 4n scattering with JISP16

Er, MeV Γ, MeV Ef, keV Ξ, keV 10 ≤ Nmax ≤ 12

0.844 1.38 –54.9

17.5 10 ≤ Nmax ≤ 18

0.844 1.38 –54.9

43.8

27

10 20 30 40 h [MeV] 5 10 15 20 25 30 E [MeV] Nmax= 2 4 6 8 10 12 14 16 18

4n, gs JISP16

5 10 15 20 25 30 E [MeV] 30 60 90 120 [degrees] All bases Small bases

4n, gs JISP16

• Phys. Rev. Lett. 117, 182502 (2016)

4n → 4n scattering with Daejeon16

×

NCSM: Nmax ≤ 20,

1 ≤ ħΩ ≤ 40 MeV

× Used only E ≤ 7 MeV energies

28

Er, MeV Γ, MeV Ef, keV Ξ, keV

0.997 1.60 –63.4

47.9

5 10 15 h [MeV] 2 4 6 E [MeV] Nmax= 2 4 6 8 10 12 14 16 18 20

4n, gs

Daejeon16

5 10 15 20 25 30 E [MeV] 30 60 90 120 [degrees] Nmax= 2 4 6 8 10 12 14 16 18 20

4n, gs

Daejeon16

2 4 6 E [MeV] 30 60 90 120 [degrees] SS-HORSE

4n, gs

Daejeon16

4n → 4n scattering with SRG-evolved Idaho N3LO

29

×

NCSM: Nmax ≤ 20, 1 ≤ ħΩ ≤ 40 MeV with NN-interaction from chiral effective field theory with flow parameters Λ = 1.5 fm-1 and 2 fm-1

×

Used only E ≤ 7 MeV energies

Λ, fm-1 Er, MeV Γ, MeV Ef, keV Ξ, keV 1.5 0.783 1.15 –52.1 29.0 2.0 0.846 1.29 –54.5 31.7

5 10 15 h [MeV] 2 4 6 E [MeV] Nmax= 2 4 6 8 10 12 14 16 18 20

4n, gs

N3LO, SRG, =1.5

5 10 15 20 25 30 E [MeV] 30 60 90 120 [degrees] Nmax= 2 4 6 8 10 12 14 16 18 20

4n, gs

N3LO, SRG, =1.5

2 4 6 E [MeV] 30 60 90 120 [degrees] SS-HORSE

4n, gs

N3LO, SRG, =1.5

4n → 4n scattering with “bare” Idaho N3LO

30

×

NCSM: Nmax ≤ 20, 1 ≤ ħΩ ≤ 40 MeV with NN-interaction from chiral effective field theory

×

No resonance! But virtual pole appears

5 10 h [MeV] 2 4 6 E [MeV]

Nmax= 2 4 6 8 10 12 14 16 18 20

4n, gs N3LO

5 10 15 20 25 30 E [MeV] 30 60 90 [degrees]

Nmax= 2 4 6 8 10 12 14 16 18 20

4n, gs N3LO

5 10 E [MeV] 30 60 90 [degrees]

SS-HORSE

4n, gs N3LO

|Ev|, keV Ξ, keV

15.2

19.4

Conclusions

×

Method Single State HORSE for description of continuum spectrum states is proposed

×

Method gives reasonable result in nucleon-α scattering with realistic NN-interactions

×

Method applied for description n-6He scattering and resonances in 7He

×

Method applied for description four-neutron scattering, resonance state with energy Er = 0.7-1.1 MeV and Γ = 1.1-1.7 MeV is obtained with various realistic interactions; with Idaho N3LO bare NN-interaction there is no resonance

31

32

Thank you for attantion!