Hypernuclei in halo/cluster EFT Shung-Ichi Ando Sunmoon University, - PowerPoint PPT Presentation

Hypernuclei in halo/cluster EFT Shung-Ichi Ando Sunmoon University, Asan, Republic of Korea arXiv:1512.07674 31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 1

Outline • Singular potentials: Limit cycle and Efimov states in three-body systems 4 ΛΛ H as ΛΛ d system in halo EFT • 6 • ΛΛ He as ΛΛ α system in cluster EFT • Summary 31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 2

Limit cycle • Three-body systems in unitary (asymptotic) limit • If an interaction is singular, the system exhibits cyclic singularities, so called limit cycle. • It is necessary to introduce a counter term for renormalization. • Efimov-like bound states Infinitely many three-body bound states (whose energies B ( n ) ) appear, for three-boson case, e − 2 π/s 0 � n − n ∗ � B ( n ) = κ 2 ∗ /m , where s 0 ≃ 1 . 00624 and e π/s 0 ≃ 22 . 7 . 31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 3

Halo/Cluster EFT • Effective Field Theories (EFTs) • Model independent approach • Separation scale • Counting rules • Parameters should be fixed by experiments • For the study of three-body systems the unitary limit can be chosen as a first approximation. 31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 4

4 ΛΛ H at LO • ΛΛ d • 3 Λ H, B Λ = 0 . 13 MeV • d , B 2 = 2 . 22 MeV • S -waves are considered at LO. • S = 0 : no limit cycle, one parameter γ Λ d , and we 4 find no bound state for ΛΛ H and a 0 = 16 . 0 ± 3 . 0 fm for Λ - 3 Λ H scattering. 4 • S = 1 : ΛΛ H shows a limit cycle, three parameters, a ΛΛ , γ Λ d , g 1 (Λ c ) , and the three-body interaction is fixed by using the results of the potential models. 31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 5

Two-body part: ΛΛ in 1 S 0 state • Dressed dibaryon propagator = + + + ... • Renormalized dressed dibaryon propagator 4 π 1 D s ( p 0 , � p ) = . m Λ y 2 � 1 − m Λ p 0 + 1 p 2 − iǫ s a ΛΛ − 4 � a ΛΛ = − 1 . 2 ± 0 . 6 fm , from 12 C( K − , K + ΛΛ X ) reaction [Gasparyan et al. , PRC85(2012)015204]. 31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 6

Two-body part: Λ d in 3 Λ H channel • Dressed 3 Λ H propagator = + + + ... • Renormalized dressed 3 Λ H propagator 2 π 1 D t ( p 0 , � p ) = � , µ Λ d y 2 � � t 1 p 2 γ Λ d − − 2 µ Λ d p 0 − 2( m Λ + m d ) � with � γ Λ d = 2 µ Λ d B Λ . 31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 7

Three-body part: S = 1 channel = + + + + = + K ( a ) ( p, k ; E ) − g 1 (Λ c ) a ( p, k ; E ) = Λ 2 c � Λ c � � � � − 1 K ( a ) ( p, l ; E ) − g 1 (Λ c ) 1 dll 2 l 2 ,� D t E − l a ( l, k ; E ) 2 π 2 Λ 2 2 m Λ 0 c � Λ c − 1 � 1 � l 2 ,� dll 2 K ( b 1) ( p, l ; E ) D s E − l b ( l, k ; E ) , 2 π 2 2 m d 0 b ( p, k ; E ) = K ( b 2) ( p, k ; E ) � Λ c − 1 � 1 � l 2 ,� dll 2 K ( b 2) ( p, l ; E ) D t E − l a ( l, k ; E ) , 2 π 2 2 m Λ 0 31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 8

where   p 2 + l 2 + 2 µ Λ d  − 2 µ Λ d E m d y 2 m d t  , K ( a ) ( p, l ; E ) = ln p 2 + l 2 − 2 µ Λ d 6 pl − 2 µ Λ d E m d � p 2 + 2 µ Λ d l 2 + pl − m Λ E m Λ � � 2 m Λ y s y t K ( b 1) ( p, l ; E ) = ln , − p 2 + m Λ 2 µ Λ d l 2 − pl − m Λ E 3 2 pl 2 µ Λ d p 2 + l 2 + pl − m Λ E m Λ � � � 2 m Λ y s y t K ( b 2) ( p, l ; E ) = ln . − 2 µ Λ d p 2 + l 2 − pl − m Λ E m Λ 3 2 pl 31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 9

Evaluation formula for the limit cycle • In the asymmetric limit, there is no scale in the integral equations. The scale invariance suggests that the power-law behavior for the amplitude a ( p ) ∼ p − 1+ s . • After Mellin transformations we have 1 = C 1 I 1 ( s ) + C 2 I 2 ( s ) I 3 ( s ) . • It has imaginary solutions for s , s = ± is 0 , s 0 = 0 . 4492 · · · , and thus e π/s 0 ≃ 1 . 09 × 10 3 . 31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 10

• Evaluation formula for the limit cycle (for the ΛΛ d system) 1 = C 1 I 1 ( s ) + C 2 I 2 ( s ) I 3 ( s ) , with √ � µ Λ(Λ d ) √ m Λ µ d (ΛΛ) µ Λ(Λ d ) 1 m d 2 C 1 = , C 2 = , 3 π 2 µ 3 / 2 6 π µ Λ d µ Λ d Λ d where µ d (ΛΛ) = 2 m Λ m d / (2 m Λ + m d ) , and sin[ s sin − 1 � 1 � 2 a ] 2 π I 1 ( s ) = , � π � s cos 2 s sin[ s cot − 1 � √ 4 b − 1 � ] 2 π 1 I 2 ( s ) = , � π b s/ 2 � s cos 2 s s b s/ 2 sin[ s cot − 1 � √ 4 b − 1 � ] 2 π I 3 ( s ) = , � π � cos 2 s and a = 2 µ Λ d m Λ and b = 2 µ Λ d . m d 31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 11

Numerical results: S = 1 channel • With g 1 (Λ c ) , ( B ΛΛ , a ΛΛ ) = (I) ( 0 . 2 MeV, − 0 . 5 fm), (II) ( 0 . 6 , − 1 . 5 ), (III) ( 1 . 0 , − 2 . 5 ). 20 (I) (II) 15 (III) 10 5 g 1 ( Λ c ) 0 -5 -10 -15 -20 10 1 10 2 10 3 10 4 10 5 10 6 Λ c (MeV) 31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 12

Numerical results: S = 1 channel 1.2 Λ c = 300 MeV 1.1 = 150 MeV 1 = 50 MeV (II), (III) 0.9 0.8 ΛΛ (MeV) 0.7 0.6 B 0.5 0.4 0.3 0.2 0.1 0.5 1 1.5 2 2.5 3 3.5 -a ΛΛ (fm) 31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 13

6 ΛΛ He at LO • ΛΛ α ( S = 0 ) • 5 Λ He, B Λ ≃ 3 MeV • First excited energy of α , B 1 ≃ 20 MeV • The limit cycle appears, three parameters, a ΛΛ , γ Λ α , g (Λ c ) , at LO, and the three-body interaction is fixed by using the Nagara event, B ΛΛ ≃ 6 . 93 MeV 31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 14

Numerical results: • With g (Λ c ) (Input: B ΛΛ = 6.93MeV) 10 a ΛΛ = -1.8 fm = -1.2 fm = -0.6 fm 5 g( Λ c ) 0 -5 -10 100 1000 10000 100000 Λ c (MeV) Λ n = Λ 0 exp( nπ/s 0 ) , s 0 ≃ 1 . 05 , exp( π/s 0 ) ≃ 19 . 9 . 31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 15

Numerical results: • Without g (Λ c ) 12 a ΛΛ = - 1.8 fm = - 1.2 fm 11 = - 0.6 fm 10 9 ΛΛ (MeV) 8 7 B 6 5 4 300 350 400 450 500 550 600 650 Λ c (MeV) 31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 16 • r c = Λ − 1 ≃ 0 . 35 to 0 . 5 fm.

Numerical results: • With g (Λ c ) (Input: B ΛΛ = 6.93MeV, a ΛΛ = − 0.5fm) 14 Λ c = 430 MeV 13 = 300 MeV = 170 MeV 12 Potential models 11 ΛΛ (MeV) 10 9 B 8 7 6 5 -3 -2.5 -2 -1.5 -1 -0.5 0 1/a ΛΛ (fm -1 ) [Filikhin and Gal, NPA707,491(2002)] 31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 17

Summary • Halo/cluster EFTs at LO for the light hypernuclei are constructed. • Those three-body systems described by means of EFTs at LO exhibit a limit cycle in the asymptotic limit which implies the formation of bound states. • For more conclusive results, we need to have the exp. data and include higher order corrections. • We have applied the present approach to the study of nn Λ system [SIA, Raha, Oh, PRC92(2015)024325] . 31st Reimei workshop, JAEA, Tokai, Japan, Jan. 18-20, 2016 – p. 18