Tokyo Metropolitan University K.Aoki and D.Jido 2016.12.01,02 - PowerPoint PPT Presentation

Tokyo Metropolitan University K.Aoki and D.Jido 2016.12.01,02 ELPH 研究会 C015 マルチフレーバーで探るエキゾチックハドロンとハドロン多体系の物理

1. INTRODUCTION

Nucleus-hadron systems & par4al restora4on of chiral symmetry ★ Purpose of study. To prove spontaneous breaking of chiral symmetry from experimental fact. Prove reducKon of magnitude of quark condensate by restoring broken chiral symmetry . Chiral symmetry is parKally restored in finite density nuclear medium. We implant hadron into nucleus (hadron-nucleus system). Hadrons change their properKes in nuclear medium. ー In-medium hadron-nucleon interacKons vs. In-vacuum hadron-nucleon interacKons. ー reducKon of magnitude of quark condensate.

present status of par4al restora4on of chiral symmetry ●Deeply bound pionic atoms. Friedman et al. PRL93, ●π meson-nucleus elasKc scaTering. 122302 (2004) ＋ theoreKcal consideraKon K.Suzuki et al. PRL92, 072302 (2004) chiral symmetry is restored about 30%. wavefuncKon renormalizaKon ● One of the higher order correcKon beyond tρ approximaKon ●In-medium self-energy has strong energy-dependence wavefuncKon renormalizaKon has large contribuKon For NG boson low-energy QCD effecKve theory (chiral perturbaKon theory) NG bosons are wriTen as their energy expansion. ️ NG boson-nucleon interacKon has strong energy-dependence. wavefuncKon renormalizaKon is important as in-medium effects Jido,Hatsuda,Kunihiro, Kolomeitsev,Kaiser,Weise, Jido,Hatsuda,Kunihiro, PRD63,011901(2001) PRL90 092501(2003) PLB670,109 (2008)

connecKon of wavefuncKon renormalizaKon & Jido,Hatsuda,Kunihiro, parKal restoraKon of chiral symmetry PRD63,011901(2001) V(π, σ) low energy QCD effecKve theory [chiral perturbaKon theory] consistent with original chiral π symmetry ●angular direcKon: NG boson (pion) σ ●radial direcKon: σ mode m σ ●angular variable = dimensionless ●pion field has energy dimension ●F has energy dimension  � i ~ ⇡ · ~ ⌧ = pion decay constant U = exp =quark condensate. F renormalizaKon of parKal restoraKon of F is change. pion field. chiral symmetry.

Further studies. Does parKal restoraKon of chiral symmetry systemaKcally occur in other than pion-nucleus systems? OUR STUDY K + meson – nucleus system. ️ K + N amplitude. ・ elementary process of K + -nucleus interacKons. ️ K + self-energy in nuclear medium. ・ Taking into account all possible medium effect. -nucleon Fermi moKon -mass modificaKon -vertex correcKon - ️ wavefuncKon renormalizaKon Our goal ・ EvaluaKng reducKon of magnitude of quark condensate from in-medium K + N interacKon

Further studies. Does parKal restoraKon of chiral symmetry systemaKcally occur in other than pion-nucleus systems? OUR STUDY K + meson – nucleus system. ️ ✔ ️ K + N amplitude. ・ elementary process of K + -nucleus interacKons. ️ K + self-energy in nuclear medium. ・ Taking into account all possible medium effect. -nucleon Fermi moKon -mass modificaKon ・ large contribu4on as one of the -vertex correcKon in-medium effect. - ️ ✔ ️ wavefuncKon renormalizaKon ・ rela4on to par4al restora4on of chiral symmetry. Our goal ・ EvaluaKng reducKon of magnitude of quark condensate from in-medium K + N interacKon

2. Overview of K + - nucleus system.

K + - nucleon interac4on ★ K + N interacKon is repulsive and relaKvely weak. ー mean free path in nuclear medium is as large as 5 fm. ー comparable to typical nucleus size. Single step K + N interacKon should be dominant. [MulKple scaTering could be negligible.] K + is considered to be clean hadronic prove. ★ K + N interacKon has S= +1. ー There is no strong resonance. In contrast to S=−1 K bar N interacKon. ー Strong aTracKve interacKon and quasi bound state Λ(1405) ●Only K + exists in nuclear medium. ●It is relaKvely easy to consider in-medium effect to K + .

K + - nucleus interac4on. ★ ExpectaKon from K + N interacKon. Single step K + N interacKon should be dominant. [MulKple scaTering could be negligible.] σ ( K + A ) ' A σ ( K + N ) p lab < 800MeV / c ★ Experimental data…total cross secKon per nucleon. linear density approximaKon σ ( K + 12 C ) / 12 is not valid. > 1 1.2 at p lab =450 MeV/c σ ( K + d ) / 2 D.V.Bugg, et al Phys.Rev. 168(1968) 1466. ★ OpKcal potenKal… linear density (low density) approximaKon is not valid. 2 ω K + V opt = T ( K + N ) ρ

のとき 積分を実行すると次のような方程式が得られる のフィット 異なるポテンシャルを用いた 表 表 なる と 結合の強さがある程度まで大きくなるとクォークの質量は の解を持つ E.Friedman,A.Gal Table 1 raKo of K + - nucleus opKcal potenKal. Phys.Rept. 452(2007) 89 t ρ t ρ P LAB [MeV/c] Re b 0 [fm] Im b 0 [fm] X ≡ | t free ρ | | arg( t free ρ ) | [deg] V OP T experiment 488 -0.203(26) 0.172(7) 1.14 0.33 t ρ expectaKon -0.178 0.153 t free ρ 531 -0.196(39) 0.202(9) 1.17 1.27 t ρ -0.172 0.170 t free ρ 656 -0.220(50) 0.262(12) 1.27 2.31 t ρ -0.165 0.213 t free ρ 714 -0.242(53) 0.285(15) 1.34 5.15 t ρ -0.161 0.228 t free ρ tρ : Fits to K + -nucleus t free ρ: In-vacuum K + N interacKon (expectaKon). scaTering data (experiment). ・ |t free ρ| is increased by 14 – 34 % by in-medium effects. ・ ImX is small compared to ReX. | t ρ | = X | t free ρ | in-medium effects

3. Method

Outline ️ ✔ ️ K + N amplitude. ・ elementary process of K + -nucleus interacKons. ️ K + self-energy in nuclear medium. ・ Taking into account all possible medium effect. -nucleon Fermi moKon -mass modificaKon -vertex correcKon - ️ ✔ ️ wavefuncKon renormalizaKon

K + N amplitude(1) K + p → K + p K + n → K + n Using the 3-flavor chiral perturbaKon theory up to NLO. Weinberg-Tomozawa Born (crossed) NLO f K D = 0 . 80 , F = 0 . 46 χ 2 fits to the exparimental data. I=1 (2 parameters) & I=0 (2 parameters)

K + N amplitude(2) ●Tree-level calculaKon…Amplitude is real. We want to calculate the imaginary part. ●We want to reproduce more wide energy range data. ★ UnitalizaKon Summing up higher order diagrams to saKsfy unitarity of amplitude. = + + + ・・・ New parameter is appearing. SubtracKon constant a .

K + N amplitude(3) χ 2 fits to the experimental data. I=1 DifferenKal cross secKon P LAB = 726MeV / c (many data) I=0 Total cross secKon P LAB = 350 ∼ 800MeV / c (less data) Isospin0 Isospin1 common parameter ・・・ subtracKon constant a

Outline ️ ✔ ️ K + N amplitude. ・ elementary process of K + -nucleus interacKons. ️ K + self-energy in nuclear medium. ・ Taking into account all possible medium effect. -nucleon Fermi moKon -mass modificaKon -vertex correcKon - ️ ✔ ️ wavefuncKon renormalizaKon

wavefuncKon renormalizaKon ●K + self-energy describe interacKon with nuclear medium. Σ = T ( K + N ) ρ ●WavefuncKon renormalizaKon factor. � Z ' 1 + ∂ Σ � � ∂ E 2 � E 2 = ω 2 K + 2 ω K + V opt = ZT ( K + N ) ρ

4. Results

I=1 TOTAL CROSS SECTION 16 14 TOTAL CROSS SECTION [mb] 12 10 8 6 ChPT Bugg 1968 4 Bowen 1970 Adams 1971 Bowen 1973 2 Carroll 1973 Cameron 1974 0 0 100 200 300 400 500 600 700 800 P LAB [MeV/c] ●DifferenKal cross secKon at P LAB =726 MeV/c. b I=1 =6.18×10 -4 d I=1 =3.00×10 -4 [MeV -1 ] ●χ2/N =0.68 a =-1.224

I=0 TOTAL CROSS SECTION 25 ChPT Bowen 1970 Bowen 1973 TOTAL CROSS SECTION [mb] Carroll 1973 20 15 10 5 0 0 100 200 300 400 500 600 700 800 P LAB [MeV/c] ●Total cross secKon P lab = 350 ー 800 MeV/c b I=0 =4.13×10 -4 d I=0 =-9.80×10 -4 [MeV -1 ] ●χ2/N =6.85 a=-1.224

のフィット 積分を実行すると次のような方程式が得られる 異なるポテンシャルを用いた 表 表 なる と 結合の強さがある程度まで大きくなるとクォークの質量は の解を持つ のとき � Z ' 1 + ∂ Σ Table 2 wavefuncKon renormalizaKon factor Z . � � ∂ E 2 � E 2 = ω 2 [evaluated at saturaKon density] K + Weinberg-Tomozawa term evaluated at threshold: |Z| = 1.08 t ρ t ρ P LAB [MeV/c] | Z | | arg(Z) | [deg] X ≡ | t free ρ | | arg( t free ρ ) | [deg] 488.0 1.11 3.87 1.14 0.33 531.0 1.09 1.68 1.17 1.27 656.0 1.04 0.04 1.27 2.31 714.0 1.03 0.11 1.34 5.15 [our calculaKon] [literature values] | t ρ | = | Z || t free ρ | | t ρ | = X | t free ρ | ・ ImZ is small compared to ReZ. ・ At low-energy |t free ρ| is increased by about 10 % only by wavefuncKon renormalizaKon.

5. Summary & Outlook

Summary ● We have constructed K + N elasKc scaTering amplitude based on chiral perturbaKon theory. ● We have carried out unitalizaKon. ● To determine low-energy constants and subtracKon constant , we have carried out χ 2 fit. ●We have calculated wavefuncKon renormalizaKon as one of the in-medium effect. ●Only wavefuncKon renormalizaKon, |t free ρ| is increased by 3- 11 %.

Outlook Whether wavefuncKon renormalizaKon is large compared to other in-medium effects or not. more precise calculaKon of K + self-energy. Taking into account other in-medium effects. ー Fermi moKon ー mass modificaKon ー vertex correcKon 3-flavor in-medium chiral perturbaKon theory.