Deeply Bound Pionic States in Sn at RIBF N. Ikeno (Nara Women’s Univ. M1) J. Yamagata-Sekihara (IFIC, Valencia Univ.) H. Nagahiro (Nara Women’s Univ.) D. Jido (YITP, Kyoto) S. Hirenzaki (Nara Women’s Univ.)

1. Introduction Pionic Atoms  X-ray spectroscopy ex) C. Batty, E. Friedman, and A. Gal, Phys. Rep. 287(1997)385  (d, 3 He) Reaction H. Toki, T. Yamazaki, Phys. Lett. B213(1988)129 H. Toki, S. Hirenzaki, T. Yamazaki, R. S. Hayano, Nucl. Phys. A501(1989)653 – 208 Pb Target (2p) – 206 Pb Target (1s, 2p) – 116,120, 124 Sn Target (1s) S. Hirenzaki et al. PRC44(91)2472 K. Itahashi et al. PRC62(00) S. Hirenzaki et al. PRC55(97)2719 H. Geissel et al. PRL88(02) Y. Umemoto et al. PRC62(00) K. Suzuki et al. PRL92(04)  Deeply bound pionic states exist  Strong correlation in potential parameter (Seki-Masutani(SM) Relation)  Isovector parameter b 1 of V opt was deduced (related to f  ) X-ray -- d 3 He --

1. Introduction Precision Spectroscopy of Pionic RIBF  122 Sn (d, 3 He) reaction  High Energy resolution  keV (about twice ~ three times better than GSI exp)  Better determination of V opt  Simultaneous 1s and 2s states Small experimental error (reduce systematic error) Small uncertainties in  n  Unique potential parameter sets (beyond SM relation) ?  Information on pion wave function    (wave function renormalization) inside nucleus ?  (Ref. Kolomeitsev, Kaiser, Weise, PRL 90(2003)092501,  D. Jido, T. Hatsuda, T. Kunihro, PLB670(2008)109 ) RIBF-054 K. Itahashi et al. RIBF-027 K. Itahashi et al. d 3 He -- 122 Sn Our Interest We expect to improve these points in the pre-exp.

2. Formulation  Klein Gordon equation  Optical Potential M. Ericson and T. E. O Ericson, Ann. Of. Phys, 36(1966)496 M. Krell and T. E. O Ericson, J. of Comp. Phys. 3. (1968)202 R. Seki, K. Masutani, Phys. Rev. C27(1983)2799 Kinematical factor       : S-wave  interaction : p-wave  interaction  : Lorentz-Lorenz correction parameter Parameters obtained by Nuclear density distribution

2. Formulation Seki-Masutani Relation R. Seki, K. Masutani, Phys. Rev. C27(1983)2799 b0b0 Re B 0 Contour plot of B.E. SM 208 Pb 1s state H. Toki, S. Hirenzaki, T. Yamazaki, R. S. Hayano, Nucl. Phys. 501(1989)653 Binding energies are almost unchanged along the line of SM relation. -> Pionic atom properties are determined by potential strength at  e ~   All potentials which satisfies the SM relation between potential parameters reproduce the experimental data.

2. Formulation Seki-Masutani Relation T. Yamazaki, S. Hirenzaki PLB557(03)20  Peak positions of the overlapping density are almost same for all states.  The effective nuclear density  e is almost same,  e ~    for all states. -> consistent with the expectation from the contour plot ee We calculate Sn case!! ->We expect to obtain new information on * pion at different  e in Sn-pion system * pion wave function behavior inside nucleus  - density : Overlapping density   Nuclear density Overlapping R ov peak

Principal quantum number n 3. Numerical Results for  -Sn systems 3.1 Effective Nuclear Density  Where does pion probe?  e [1/fm 3 ] 1s 2s2s 2s2s R ov peak [fm] 2s state probes inner part of the nucleus than 1s state. Pionic states which have larger n slightly probe inner part of the nucleus. Pion in all atomic states probes only narrow region of the density.

 Dependence of pionic state on b 0,ReB 0 3. Numerical Results for  -Sn systems 3.2 Contour plot B.E. [keV]  [keV] Re B 0 [m  -4 ] b 0 [m  -1 ] B.E.(1s)-B.E.(2s) 2s 1s Results of B.E. : Almost unchanged along the SM line Results of  : Slopes of the contour lines are somewhat shifted from that of SM line  (1s)-  (2s) 100keV 25keV 100keV 50keV

 Dependence of pionic state on b 0,ReB 0 3. Numerical Results for  -Sn systems 3.2 Contour plot B.E. [keV]  [keV] Re B 0 [m  -4 ] 2s B.E.(1s)-B.E.(2s) 1s  (1s)-  (2s) 40keV 9keV 8keV b 0 [m  -1 ] 80keV 18keV Previous experimental error ( 119 Sn target case) B.E. : 18keV  : 80keV

 Dependence of pionic state on b 0,ReB 0 3. Numerical Results for  -Sn systems 3.2 Contour plot BE [keV] Re B 0 [m  -4 ] Expected experimental error B.E. of 1s, 2s : 9keV B.E.(1s)-B.E.(2s) : 8keV (Experimental proposal)  : 40keV (Half of error at GSI) One could expect to fix parameters, if the error of widths can be further reduced b 0 [m  -1 ] 2s B.E.(1s)-B.E.(2s) 1s  [keV] b 0 [m  -1 ]  (1s)-  (2s) 40keV 9keV 8keV

Im B 0 [m  -4 ]  Dependence of pionic state on b 1,ImB 0 3. Numerical Results for  -Sn systems 3.2 Contour plot B.E. [keV] b1 -> ambiguity in  n Terashima et al. PRC77(2008) From the precise  n, we can expect to determine b 1 b 1 [m  -1 ]  [keV] b 1 [m  -1 ] 1s 2s  (1s)-  (2s) 40keV 9keV 8keV B.E.(1s)-B.E.(2s)

 We study behaviors of wave function at different points on SM line: in the {b 0,ReB 0 }plane 1s 2s The wave functions are closely related to the formation cross sections. Changes in wave function -> Changes of cross Section The wave functions are closely related to the formation cross sections. Changes in wave function -> Changes of cross Section b 0 [m  -1 ] Re B 0 [m  -4 ] 3. Numerical Results for  -Sn systems 3.3 Wave Function

Inside the nucleus, the different behaviors of the wave functions appear. But, because of distortion factor F(b), it seems to be difficult to observe them. 1s 2s H. Nagahiro et al. Nucl. Phys. A761(2005)92 Neutron wave function Different behaviors !! Distortion Factor

4. Summary  We have studied Pionic States in Sn The Effective nuclear density including non-yrast Various contour plots to guide exp. (We study the dependence of pionic states on the optical potential parameters.) * b 0 -ReB 0 correlation (SM) – strong * b 1 parameter dependence The behavior of pion wave function => info. on wave function renormalization?  We found that The accurate data lead to better determination of Vopt To fix one point on SM line is difficult To obtain the pion wave function behaviors inside nucleus is also difficult  We expect to obtain the better results for b 1  Future Formation Spectra of 1s and 2s data