Few-body resonances from finite-volume calculations Sebastian K - PowerPoint PPT Presentation

Few-body resonances from finite-volume calculations Sebastian K¨ onig FRIB TA Workshop “Connecting bound state calculations with scattering and reactions” NSCL, Michigan State University June 19, 2018 P. Klos, SK, J. Lynn, H.-W. Hammer, and A. Schwenk, arXiv:1805.02029 [nucl-th] Few-body resonances from finite-volume calculations – p. 1

Motivation terra incognita at the doorstep. . . ? ? ? bound dineutron state not excluded by pionless EFT Hammer + SK, PLB 736 208 (2014) recent indications for a three-neutron resonance state. . . Gandolfi et al. , PRL 118 232501 (2017) . . . although excluded by previous theoretical work Offermann + Gl¨ ockle, NPA 318 , 138 (1979); Lazauskas + Carbonell, PRC 71 044004 (2005) possible evidence for tetraneutron resonance Kisamori et al. , PRL 116 052501 (2016) Few-body resonances from finite-volume calculations – p. 2

4 Γ (MeV) 3 2 1 0 0 2 4 6 8 E R (MeV) Short (recent) history of tetraneutron states 1 2002: experimental claim of bound tetraneutron Marques et al. , PRC 65 044006 2 2003: several studies indicate unbound four-neutron system Bertulani et al. . JPG 29 2431; Timofeyuk, JPG 29 L9; Pieper, PRL 90 252501 3 2005: observable tetraneutron resonance excluded Lazauskas PRC 72 034003 Few-body resonances from finite-volume calculations – p. 3

Short (recent) history of tetraneutron states 1 2002: experimental claim of bound tetraneutron Marques et al. , PRC 65 044006 2 2003: several studies indicate unbound four-neutron system Bertulani et al. . JPG 29 2431; Timofeyuk, JPG 29 L9; Pieper, PRL 90 252501 3 2005: observable tetraneutron resonance excluded Lazauskas PRC 72 034003 4 2016: RIKEN experiment: possible tetraneutron resonance E R = (0 . 83 ± 0 . 65 stat. ± 1 . 25 syst. ) MeV , Γ � 2 . 6 MeV Kisamori et al. , PRL 116 052501 following this: several new theoretical investigations 5 complex scaling → need unphys. T = 3 / 2 3N force or strong rescaling Hiyama et al. , PRC 93 044004 (2016),; Deltuva, PLB 782 238 (2018) 4 incompatible predictions: 3 Γ (MeV) Fossez et al. , PRL 119 032501 (2017) 2 Shirokov et al. PRL 117 182502(2016) 1 Gandolfi et al. , PRL 118 232501 (2017) 0 0 2 4 6 8 E R (MeV) Few-body resonances from finite-volume calculations – p. 3

Short (recent) history of tetraneutron states 1 2002: experimental claim of bound tetraneutron Marques et al. , PRC 65 044006 2 2003: several studies indicate unbound four-neutron system Bertulani et al. . JPG 29 2431; Timofeyuk, JPG 29 L9; Pieper, PRL 90 252501 3 2005: observable tetraneutron resonance excluded Lazauskas PRC 72 034003 4 2016: RIKEN experiment: possible tetraneutron resonance E R = (0 . 83 ± 0 . 65 stat. ± 1 . 25 syst. ) MeV , Γ � 2 . 6 MeV Kisamori et al. , PRL 116 052501 following this: several new theoretical investigations 5 complex scaling → need unphys. T = 3 / 2 3N force or strong rescaling Hiyama et al. , PRC 93 044004 (2016),; Deltuva, PLB 782 238 (2018) 4 incompatible predictions: 3 Γ (MeV) Fossez et al. , PRL 119 032501 (2017) 2 Shirokov et al. PRL 117 182502(2016) 3 . 0 4 neutrons 2 . 0 1 3 neutrons 1 . 0 Gandolfi et al. , PRL 118 232501 (2017) 0 . 0 0 0 2 4 6 8 − 1 . 0 E (MeV) R WS = 4 . 5 fm E R (MeV) − 2 . 0 2 . 0 − 3 . 0 4 neutrons LO indications for three-neutron resonance. . . 0 . 0 NLO − 4 . 0 N 2 LO − 2 . 0 R WS = 6 . 0 fm . . . lower in energy than tetraneutron state − 5 . 0 − 4 . 0 R WS = 6 . 0 fm R WS = 7 . 5 fm − 6 . 0 − 6 . 0 − 3 . 0 − 2 . 0 − 1 . 0 Gandolfi et al. , PRL 118 232501 (2017) − 7 . 0 − 4 . 0 − 3 . 5 − 3 . 0 − 2 . 5 − 2 . 0 − 1 . 5 − 1 . 0 − 0 . 5 0 . 0 V 0 (MeV) Few-body resonances from finite-volume calculations – p. 3

How to tackle resonances? V ( r ) Resonances 2.0 1.5 metastable states 1.0 decay width ↔ lifetime 0.5 5 r 1 2 3 4 1 Look for jump by π in scattering phase shift: � simple � possibly ambiguous (background), need 2-cluster system δ ( E } 150 100 50 3.0E 0.5 1.0 1.5 2.0 2.5 Few-body resonances from finite-volume calculations – p. 4

How to tackle resonances? V ( r ) Resonances 2.0 1.5 metastable states 1.0 decay width ↔ lifetime 0.5 5 r 1 2 3 4 1 Look for jump by π in scattering phase shift: � simple � possibly ambiguous (background), need 2-cluster system δ ( E } 150 100 ↔ 50 3.0E 0.5 1.0 1.5 2.0 2.5 2 Find complex poles in S-matrix: e.g., Gl¨ ockle, PRC 18 564 (1978); Borasoy et al. , PRC 74 055201 (2006); . . . � direct, clear signature � technically challenging, needs analytic pot. Few-body resonances from finite-volume calculations – p. 4

How to tackle resonances? V ( r ) Resonances 2.0 1.5 metastable states 1.0 decay width ↔ lifetime 0.5 5 r 1 2 3 4 1 Look for jump by π in scattering phase shift: � simple � possibly ambiguous (background), need 2-cluster system δ ( E } 150 100 ↔ 50 3.0E 0.5 1.0 1.5 2.0 2.5 2 Find complex poles in S-matrix: e.g., Gl¨ ockle, PRC 18 564 (1978); Borasoy et al. , PRC 74 055201 (2006); . . . � direct, clear signature � technically challenging, needs analytic pot. 3 Put system into periodic box! Few-body resonances from finite-volume calculations – p. 4

Finite periodic boxes physical system enclosed in finite volume (box) typically used: periodic boundary conditions � volume-dependent energies Few-body resonances from finite-volume calculations – p. 5

Finite periodic boxes physical system enclosed in finite volume (box) typically used: periodic boundary conditions � volume-dependent energies L¨ uscher formalism Physical properties encoded in the L -dependent energy levels! infinite-volume S-matrix governs discrete finite-volume spectrum PBC natural for lattice calculations. . . . . . but can also be implemented with other methods Few-body resonances from finite-volume calculations – p. 5

General bound-state volume dependence volume dependence ↔ overlap of asymptotic wave functions L¨ uscher, Commun. Math. Phys. 104 177 (1986); . . . � κ A | N − A = 2 µ A | N − A ( B N − B A − B N − A ) Volume dependence of N -body bound state ∆ B N ( L ) ∝ ( κ A | N − A L ) 1 − d/ 2 K d/ 2 − 1 ( κ A | N − A L ) � � /L ( d − 1) / 2 as L → ∞ ∼ exp − κ A | N − A L ( L = box size, d no. of spatial dimensions, K n = Bessel function) SK and D. Lee, PLB 779 , 9 (2018) channel with smallest κ A | N − A determines asymptotic behavior Few-body resonances from finite-volume calculations – p. 6

Numerical results SK and D. Lee, PLB 779 , 9 (2018) 2.5 0 D = 1, a latt = 1/3, k = 2 D = 3, a latt = 1/2, k = 2 0 −5 N = 2 log(L ΔB) −2.5 N = 3 −10 N = 2 log(ΔB) N = 4 N = 3 −5 −15 N = 5 −7.5 −20 −10 −25 −30 −12.5 5 10 15 20 25 0 10 20 30 40 50 L L → straight lines ↔ excellent agreement with prediction ֒ N B N L min . . . L max κ fit κ 1 | N − 1 d = 1 , V 0 = − 1 . 0 , R = 1 . 0 2 0.356 20 . . . 48 0 . 59536(3) 0.59625 3 1.275 15 . . . 32 1 . 1062(14) 1.1070 4 2.859 12 . . . 24 1 . 539(3) 1.541 5 5.163 12 . . . 20 1 . 916(21) 1.920 d = 3 , V 0 = − 5 . 0 , R = 1 . 0 2 0.449 15 . . . 24 0 . 6694(2) 0.6700 3 2.916 4 . . . 14 1 . 798(3) 1.814 Few-body resonances from finite-volume calculations – p. 7

Finite-volume resonance signatures L¨ uscher formalism: phase shift ↔ box energy levels � Lp � 2 p cot δ 0 ( p ) = 1 � E ( L ) � πLS ( η ) , η = , p = p 2 π L¨ uscher, Nucl. Phys. B 354 531 (1991); . . . resonance contribution � avoided level crossing Wiese, Nucl. Phys. B (Proc. Suppl.) 9, 609 (1989); . . . Few-body resonances from finite-volume calculations – p. 8

Finite-volume resonance signatures L¨ uscher formalism: phase shift ↔ box energy levels � Lp � 2 p cot δ 0 ( p ) = 1 � E ( L ) � πLS ( η ) , η = , p = p 2 π L¨ uscher, Nucl. Phys. B 354 531 (1991); . . . resonance contribution � avoided level crossing Wiese, Nucl. Phys. B (Proc. Suppl.) 9, 609 (1989); . . . p 12 no interaction , δ ( p ) = 0 10 ֒ → free levels ∼ 1 /L 8 6 4 2 10 L 2 4 6 8 Few-body resonances from finite-volume calculations – p. 8

Finite-volume resonance signatures L¨ uscher formalism: phase shift ↔ box energy levels � Lp � 2 p cot δ 0 ( p ) = 1 � E ( L ) � πLS ( η ) , η = , p = p 2 π L¨ uscher, Nucl. Phys. B 354 531 (1991); . . . resonance contribution � avoided level crossing Wiese, Nucl. Phys. B (Proc. Suppl.) 9, 609 (1989); . . . p δ ( p } 12 p 1 2 3 4 5 6 10 - 0.1 - 0.2 8 - 0.3 6 - 0.4 4 2 10 L 2 4 6 8 Few-body resonances from finite-volume calculations – p. 8

Finite-volume resonance signatures L¨ uscher formalism: phase shift ↔ box energy levels � Lp � 2 p cot δ 0 ( p ) = 1 � E ( L ) � πLS ( η ) , η = , p = p 2 π L¨ uscher, Nucl. Phys. B 354 531 (1991); . . . resonance contribution � avoided level crossing Wiese, Nucl. Phys. B (Proc. Suppl.) 9, 609 (1989); . . . p δ ( p } 12 2.5 10 2.0 1.5 8 1.0 6 0.5 p 4 1 2 3 4 5 6 2 10 L 2 4 6 8 Few-body resonances from finite-volume calculations – p. 8