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). ( p i − p j ) 2 H A = 1 X X V NN X V NNN + + + ... × ij ijk A 2 m i<j i<j i<j Wave function is expanded to multiparticle oscillator basis × 6 Li 0 ħ Ω +2 ħ Ω + + + … +4 ħ Ω … + + … + … + N max ħ Ω Parameters: × N max – max number of excitation quanta × ħ Ω – parameter of oscillator function × 3
HORSE ( J -matrix) Our method based on Harmonic Oscillator Representation of Scattering × Equations. HORSE is reliable method for describing twoparticle scattering ∞ X H ` u ` ( E, r ) = Eu ` ( E, r ) u ` ( E, r ) = a N ` ( E ) R N ` ( r ) × R ` – oscillator functions N ( r ) n =0 – oscillator quanta N = 2 n + ` N, N 0 ≤ N = N max + N min Approximation P -space ⇢ V ` NN 0 , × V ` ˜ NN 0 = N, N 0 > N = N max + N min Q -space of potential (short range) 0 , Kinetic energy matrix stay full × S N ` ( E ) − G NN ( E ) T ` N , N +2 S N +2 , ` ( E ) tan δ ` ( E ) = − Phase shifts × C N ` ( E ) − G NN ( E ) T ` N , N +2 C N +2 , ` ( E ) ◆ ` +1 s ⇡ n ! ✓ 2 E ✓ ◆ ✓ 2 E ◆ − E 2 ` + 1 S N ` ( E ) = exp L 2 n Γ ( n + ` + 3 ~ Ω ~ Ω ~ Ω 2 ) ◆ − ` s ( − 1) n ⇡ n ! ✓ 2 E ✓ ◆ ✓ − n − ` − 1 2 , − ` + 1 2; 2 E ◆ − E 2 C N ` ( E ) = exp 1 F 1 Γ ( n + ` + 3 Γ ( − ` + 1 2 ) 2 ) ~ Ω ~ Ω ~ Ω ( N � ` ) / 2 N � 1 h ⌫ | N ` i h N 0 ` | ⌫ i X X H ` NN 0 h N 0 ` | ⌫ i = E ⌫ h N ` | ⌫ i G NN 0 ( E ) = � E ν � E 4 n 0 =0 ν =0
Single State HORSE S N ` ( E ) − G NN ( E ) T ` N , N +2 S N +2 , ` ( E ) × tan δ ` ( E ) = − C N ` ( E ) − G NN ( E ) T ` N , N +2 C N +2 , ` ( E ) N � 1 h ⌫ | N ` i h N 0 ` | ⌫ i X G NN 0 ( E ) = � E ν � E ( N � ` ) / 2 ν =0 N 0 = 2 n 0 + ` X H ` NN 0 h N 0 ` | ⌫ i = E ⌫ h N ` | ⌫ i , n = 0 , ..., ( N � ` ) / 2 n 0 =0 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 × tan δ ` ( E ⌫ ) = − S N max + N min +2 , ` ( E ⌫ ) E = E ν : × C N max + N min +2 , ` ( E ⌫ ) By varying N max and ħ Ω we obtain E ν and δ l in some interval × Parametrization of phase shifts × Search of S -matrix poles, which associated with bound, resonant states × 5
Parametrization & S -matrix poles search k 2 ` √ 2 µE Scattering amplitude f ( E ) = × k = K ( E ) − ik 2 ` +1 ~ K ( E ) = k 2 ` +1 cot δ Effective-range function × 2 E 2 + · · · K ( E ) = w ( n ) + w ( n ) 1 E + w ( n ) Padé-approximation 0 × 1 + w ( d ) 1 E + w ( d ) 2 E 2 + · · · S -matrix S = e 2 i δ has the same poles as f(E) × F ( E ) ≡ K ( E ) − ik 2 ` +1 = 0 f(E) have poles where × From theory of functions of complex variables: × 1 F 0 ( E ) 1 E F 0 ( E ) I I Υ = E p = F ( E ) dE F ( E ) dE 2 π i 2 π i C C number of zeroes zero’s position ( S -matrix pole) Bound, false pole ; virtual pole E p = | E v | × E p = − E b E p = E r + i Γ Resonance pole 6 2
Parametrization & S -matrix poles search Another way: × Im(k) k-plane Represent S -matrix as ik b(f) i � r S ( k ) = Θ ( k ) S p ( k ) ik v Background Pole -k r k r Re(k) - ik v - i � r δ = φ + δ p - ik b(f) × S-matrix poles types and contributions to phase shift: × Bound and false Virtual Resonance pole S b ( f ) = k b ( f ) − ik S r = ( k − κ ∗ r )( k + κ r ) S v = k v + ik r ) , κ r = k r − i γ r k b ( f ) + ik ( k − κ r )( k + κ ∗ k v − ik √ s δ r ( E ) = − arctan a E E δ b ( E ) = π − arctan | E b | r E E − b 2 δ v ( E ) = arctan s E v E δ f ( E ) = − arctan p E r = b 2 − a 2 / 2 , Γ = a 4 b 2 − a 2 | E f | 7
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 4 He and 16 O × 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
n α -scattering No-Core Shell Model calculations: × NN -interactions: JISP16 and Daejeon16 × lowest eigenenergies E 0 of 3/2 - , 1/2 - and 1/2 + states of 5 He nuclei in × bases with N max ≤ 18, 10 ≤ ħ Ω ≤ 40 MeV ground state energy of 4 He nuclei in the same bases × Energy is calculated regarding the channel threshold × 3/2 - , 1/2 - × 5 He , 3 4 He, gs − , ( 1 − ) E 0 ( N max , ~ Ω ) = E ( N max , ~ Ω ) − E ( N max , ~ Ω ) 2 2 scattering states 0 0 1/2 + × 5 He , 1 + 4 He, gs E 0 ( N max , ~ Ω ) = E ( N max , ~ Ω ) − E ( N max − 1 , ~ Ω ) 2 scattering state 0 0 9
3/2 - state of n α -scattering: JISP16 15 N max = 2 - n � , 3/2 E r , MeV Γ , MeV Ξ , keV 4 6 JISP16 8 SS-HORSE 10 Manual selection 0.89 0.99 70 10 12 N max ≤ 18 14 E [MeV] 16 18 SS-HORSE 0.89 1.01 106 N max ≤ 4 5 From 0.80 0.65 experiment 0 0 10 20 30 40 Phys. Rev. C 94 , 064320 (2016) h � [MeV] √ tan δ ` ( E ⌫ ) = − S N +2 , ` ( E ⌫ ) δ 1 = − arctan a E − b 2 − a E √ � √ � 3 E + d E C N +2 , ` ( E ⌫ ) b 2 120 120 Exp. 90 90 Fit exp. - - � 1 [degrees] � 1 [degrees] n � , 3/2 Exp. n � , 3/2 N max = 2 N max = 2 4 JISP16 JISP16 60 4 60 6 6 8 Manual selection 8 10 10 12 12 30 14 30 14 16 16 18 18 SS-HORSE 0 0 10 0 5 10 15 0 5 10 15 E cm [MeV] E cm [MeV]
1/2 - state of n α -scattering: JISP16 N max = 2 25 - n � , 1/2 E r , MeV Γ , MeV Ξ , keV 4 6 JISP16 8 SS-HORSE 20 10 1.856 5.456 85 Manual selection 12 N max ≤ 18 14 E [MeV] 15 16 18 SS-HORSE 1.834 5.511 193 N max ≤ 6 10 From 2.07 5.57 5 experiment 0 0 10 20 30 40 Phys. Rev. C 94 , 064320 (2016) h � [MeV] √ tan δ ` ( E ⌫ ) = − S N +2 , ` ( E ⌫ ) δ 1 = − arctan a E − b 2 − a E √ � √ � 3 E + d E C N +2 , ` ( E ⌫ ) b 2 60 60 - n � , 1/2 JISP16 Exp. � 1 [degrees] � 1 [degrees] Fit exp. Exp. N max = 4 N max = 2 30 30 6 4 8 - 6 n � , 1/2 10 8 12 10 JISP16 14 12 16 14 18 Manual selection 16 SS-HORSE 18 0 0 11 0 10 20 0 5 10 15 20 25 5 15 25 E cm [MeV] E cm [MeV]
1/2 + state of n α -scattering: JISP16 40 N max = 3 + n � , 1/2 E r , MeV Γ , MeV Ξ , keV 5 7 JISP16 9 30 SS-HORSE 11 Manual selection non-resonant 120 13 N max ≤ 17 15 E [MeV] 17 20 SS-HORSE non-resonant 168 N max ≤ 7 10 0 0 10 20 30 40 Phys. Rev. C 94 , 064320 (2016) h � [MeV] s tan δ ` ( E ⌫ ) = − S N +2 , ` ( E ⌫ ) E � 3 + f √ � √ � √ � 5 δ 0 = π − arctan | E b | + c E + E E C N +2 , ` ( E ⌫ ) 180 180 Exp. Exp. Fit exp. N max = 3 + n � , 1/2 N max = 5 150 5 150 7 7 9 9 11 � 0 [degrees] 11 � 0 [degrees] 13 120 120 13 15 15 17 17 SS-HORSE 90 90 + n � , 1/2 JISP16 60 60 Manual selection 12 0 10 20 30 40 0 10 20 30 40 E cm [MeV] E cm [MeV]
n α -scattering: Daejeon16 Experiment n � Daejeon16 w 0 + w 1 E + w 2 E 2 = 180 N max = 12 (11) + +20 o 1/2 14 (13) 150 = − k 2 l +1 C N +2 ,l ( E ) 16 (15) - 3/2 18 (17) � [degrees] 120 S N +2 ,l ( E ) Daejeon16 JISP16 90 - 1/2 60 30 arXiv: 1808.03394 0 0 5 10 15 20 E [MeV] E r , MeV Γ , MeV Ξ , keV Single State Daejeon16 0.68 0.52 22 HORSE 3/2 - JISP16 0.89 0.99 70 From experiment 0.80 0.65 Single State Daejeon16 2.45 5.07 48 HORSE 1/2 - JISP16 1.856 5.456 85 From experiment 2.07 5.57 Single State Daejeon16 non-resonant 119 1/2 + HORSE JISP16 non-resonant 120 13
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