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using chiral perturbation theory』 2018.9.11-12 ELPH研究会 C023 『原子核中におけるハドロンの性質とカイラル対称性の役割』 in P-wave pion-nucleus interaction 『Effects of wavefunction renormalization P波成分に対する波動関数くりこみの影響 カイラル摂動論を用いたπ中間子-原子核相互作用の Tokyo Metropolitan University K. Aoki � Tokyo Institute of Technology D. Jido � Tottori University N. Ikeno � Nara Women ’ s University S. Hirenzaki

������ �������� ����� ����� ������ �������� ����� ����� π-nucleus experimental data 2 pion-nucleus elastic scattering K. Suzuki et al. PRL92, 072302 (2004) πN scattering length Friedman, et al. PRL93, 122302(2004) linear density approximation is not valid. Fits to pionic atom π-nucleus system: In-medium changes of π - N interaction π-nucleus S-wave optical potential [ linear density approximation ] ̶> ・mean free path in nuclear medium is as large as 5-10 fm ★low-energy π-nucleus interaction expected from πN interaction ・single step πN scattering should be dominant � � 1 + ω π 2 ω π V opt = − 4 π [ b 0 ( ρ p + ρ n ) + b 1 ( ρ p − ρ n )] M N b fit 0 , b fit b free , b free 1 0 1 ρ e ff ∼ 0 . 6 ρ 0 R = b free R = b free 1 1 = 0 . 68 = 0 . 78 ± 0 . 05 b fit b fit 1 1 b free ρ → Xb free ρ

NG boson ーNG bosons are written as their energy (momentum) expansion kinetic energy term correction to renormalization Wavefunction ★Optical potential in terms of wavefunction renormalization. ・self-energy Study in-medium pion properties by the effects of wavefunction renormalization ーWe expect that NG boson wavefunction renormalization is large ーNG boson-nucleon interaction has strong energy dependence Deeply bound pionic atom ーChiral perturbation theory (low-energy QCD effective theory) For NG bosons, such as pion…… wavefunction renormalization is large. 2. When self-energy (optical potential) has strong energy-dependence, 1. One of the higher order corrections beyond linear density approximation. ★Wavefunction renormalization ・energy ・momentum 3 Wavefunction renormalization � � � � ∂ Π � � � 1 + ∂ Π ∂ω 2 + ∂ Π 2 m π V opt = Π − ∇ ∇ Π ∂ω 2 ∂ q 2 ω q

4 Our study ①Effects of wavefunction renormalization in P-wave pion-nucleus interaction. ②Correction to kinetic energy term correction to kinetic energy term

�� �� �� �� �� �� �� �� �� �� �� �� � ������� ������� � ������ �� �� �� �� �� �� � ������� ������� � ������ 5 Kolomeitsev, Kaiser, Weise, PRL90, 092501 (2003) Jido, Hatsuda, Kunihiro, PRD63, 011901 (2001); PLB670, 109 (2008) ★P-wave term S-wave term correction to kinetic energy term ①Effects of wavefunction renormalization in P-wave pion-nucleus interaction. Our study ②Correction to kinetic energy term Partial wave expansion up to the P-wave Z = 1 + ∂ Π p � ′ Π = Π S + Π P ⃗ p cm · ⃗ cm � � ∂ω 2 = 1 + ∂ Π S ∂ω 2 + ∂ Π P p � ′ � ∂ω 2 ⃗ p cm · ⃗ cm p � ′ = 1 + z S + z P ⃗ p cm · ⃗ cm � � 1 + ∂ Π � � � � p � ′ p � ′ Π = 1 + z S + z P ⃗ p cm · ⃗ Π S + Π P ⃗ p cm · ⃗ cm cm ∂ω 2 p � ′ = (1 + z S ) Π S + [(1 + z S ) Π P + z P Π S ] ⃗ cm + · · · p cm · ⃗ �

� � � � � � 6 correction to ★P-wave term S-wave term kinetic energy term S-wave term Our study ★P-wave term coordinate space momentum space ②Correction to kinetic energy term ①Effects of wavefunction renormalization in P-wave pion-nucleus interaction. � � � � � 1 + ∂ Π � � � � p � ′ p � ′ Π = 1 + z S + z P ⃗ Π S + Π P ⃗ p cm · ⃗ p cm · ⃗ cm cm ∂ω 2 p � ′ = (1 + z S ) Π S + [(1 + z S ) Π P + z P Π S ] ⃗ p cm · ⃗ cm + · · · � � 1 + ∂ Π Π ( r ) = (1 + z S ) Π S + ∇ · [ − (1 + z S ) Π P − z P Π S ] ∇ ∂ω 2 = s ( r ) + ∇ · p ( r ) ∇

� � � ★Klein-Gordon equation ★In case of <0 ̶> instability ★kinetic term should be >0 kinetic energy term kinetic energy term correction to correction to kinetic energy term 7 T. E. O. Ericson and F. Myhrer, Phys. Lett. 74B(1978)163 ①Effects of wavefunction renormalization in P-wave pion-nucleus interaction. ②Correction to kinetic energy term Our study ω 2 − m 2 π + ∇ 2 − 2 m π V opt ( r ) � � φ ( r ) = 0 � � 1 − p ( r ) + ∂ Π ∂ω 2 + ∂ Π ∇ 2 φ ( r ) + [ ω 2 − m 2 π − s ( r ) + · · · ] φ ( r ) = 0 ∂ q 2

★Lorentz-Lorentz correction ★potential parameters kinetic energy term correction to M. Ericson, T. E. O. Ericson, Ann. Phys. 36(66)496 the potential We compare real part of 8 ★kinematical factor ★linear density R. Seki and K. Masutani, PRC27(83)2799 ②Correction to kinetic energy term Our study ①Effects of wavefunction renormalization in P-wave pion-nucleus interaction. P-wave phenomenological potential (Ericson-Ericson type) 2 µV P ( r ) = 4 π ∇ · [ c ( r ) + � − 1 2 C 0 ρ 2 ( r )] L ( r ) ∇ � � c ( r ) = � − 1 c 0 [ ρ p ( r ) + ρ n ( r )] + c 1 [ ρ n ( r ) − ρ p ( r )] 1 b 0 = − 0 . 0283 m − 1 b 1 = − 0 . 12 m − 1 π π c 0 = 0 . 223 m − 3 c 1 = 0 . 25 m − 3 π π B 0 = 0 . 042 i m − 4 C 0 = 0 . 10 i m − 6 1 π π L ( r ) = λ = 1.0 3 πλ [ c ( r ) + � − 1 1 + 4 2 C 0 ρ 2 ( r )] � 1 = 1 + m π � 2 = 1 + m π 2 M N M N

9 Method ①Effects of wavefunction renormalization in P-wave pion-nucleus interaction. ②Correction to kinetic energy term ★How to construct self-energyΠ? STEP1 elementary process of π - -nucleus interactions ̶> πN elastic scattering amplitude STEP2 π - self-energy based onπ - N amplitude STEP3 Wavefunction renormalization Momentum derivative correction to kinetic energy term

χ 2 fitting STEP1: πN elastic scattering amplitude Our study Born term（u-channel） NLO term Weinberg-Tomozawa term 10 differential cross section at T π = 25.8 MeV NLO term Born term（s-channel） Weinberg-Tomozawa term Chiral perturbation theory [low-energy QCD effective theory] π − p → π − p f π g A c 1 , c 2 , c 3 , c 4 π + p → π + p g A f π c 1 , c 2 , c 3 , c 4

11 STEP2: π - self-energy in nuclear medium based on π - N amplitude Model of self-energy (optical potential) ̶> linear density approximation ★πN elastic scattering amplitude ★π - self-energy (optical potential) assuming isospin symmetry Momentum derivative Our study STEP3: Wavefunction renormalization T π + p T π − p T π + p = T π − n 2 m π V opt = Π = − T π − p ρ p − T π − n ρ n Z = 1 + ∂ Π ∂ q 2 = − ∂ T π − p ∂ Π ∂ q 2 ρ p − ∂ T π − n ∂ω 2 ρ n = 1 − ∂ T π − p ∂ω 2 ρ p − ∂ T π − n ∂ q 2 ∂ω 2 ρ n � � � � � �

12 Results

PLB 633 (2006) T pi = 25.8 MeV pion kinetic energy 13 H. Dens et al., Fig. 1 π − p → π − p differential cross section 2 T pi =25.8 MeV/c d � /d � [mb/sr] 1.5 1 0.5 0 0 30 60 90 120 150 180 � c.m. [deg] ����� c 1 = − 0 . 8 × 10 − 3 � c 2 = 2 . 8 × 10 − 3 � c 3 = − 4 . 1 × 10 − 3 � c 4 = 3 . 9 × 10 − 3 � [MeV − 1 ] χ 2 /N = 4 . 2

T pi = 19.9 MeV Fig. 2 T pi = 43.3 MeV T pi = 37.3 MeV T pi = 32.0 MeV π − p → π − p differential cross section 2 2 T pi =19.9 MeV/c T pi =32.0 MeV/c d � /d � [mb/sr] d � /d � [mb/sr] 1.5 1.5 1 1 0.5 0.5 0 0 0 30 60 90 120 150 180 0 30 60 90 120 150 180 � c.m. [deg] � c.m. [deg] 2 2 T pi =37.3 MeV/c T pi =43.3 MeV/c d � /d � [mb/sr] d � /d � [mb/sr] 1.5 1.5 1 1 0.5 0.5 0 0 0 30 60 90 120 150 180 0 30 60 90 120 150 180 � c.m. [deg] � c.m. [deg]

Fig. 3 differential cross section T pi = 25.8 MeV T pi = 19.9 MeV π + p → π + p 2 T pi =19.9 MeV/c d � /d � [mb/sr] 1.5 1 0.5 0 0 30 60 90 120 150 180 � c.m. [deg] 2 T pi =25.8 MeV/c d � /d � [mb/sr] 1.5 1 0.5 0 0 30 60 90 120 150 180 � c.m. [deg]

16 ★P-wave wavefunction renormalization is considerably small. threshold T π =0 Fig. 4 S-wave wavefunction renormalization for in-medium πN interaction. ★S-wave wavefunction renormalization gives 50% enhancement ∂ω = 1 − ∂ T π − p ∂ω 2 ρ p − ∂ T π − n 0.6 ∂ω 2 ρ n 121 Sn 0.5 p − ∂ T π − n 0.4 ∂ω 2 ρ n z S 0.3 Z = 1 + ∂ Π 0.2 = 1 − ∂ T π − p ∂ω 2 ∂ω 2 ρ p = 1 − ∂ T π − p ∂ω 2 ρ p − ∂ T π − n 0.1 ∂ω 2 ρ n 0 0 2 4 6 8 10 r [fm]

★P-wave derivative is considerably small. 17 threshold T π =0 Fig. 5 S-wave momentum derivative − ∂ T π − p ∂ q 2 ρ p − 0 121 Sn -0.1 -0.2 p − ∂ T π − n ρ n ∂ q 2 q -0.3 z S -0.4 − ∂ T π − p ∂ q 2 ρ p − ∂ T π − n ρ n ∂ q 2 -0.5 -0.6 0 2 4 6 8 10 r [fm]