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Relativistic Chiral Mean Field Model for Finite Nuclei

Relativistic Chiral Mean Field Model for Finite Nuclei. Hiroshi Toki (RCNP/Osaka) in collaboration with Yoko Ogawa (RCNP/Osaka) Setsuo Tamenaga (RCNP/Osaka) Akihiro Haga (RCNP/Osaka). Pions in nuclei. Pion was introduced by Yukawa for nuclear force in 1934.

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Relativistic Chiral Mean Field Model for Finite Nuclei

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  1. Relativistic Chiral Mean Field Model for Finite Nuclei Hiroshi Toki (RCNP/Osaka) in collaboration with Yoko Ogawa (RCNP/Osaka) Setsuo Tamenaga (RCNP/Osaka) Akihiro Haga (RCNP/Osaka) Sep.18 2006 Erice

  2. Pions in nuclei • Pion was introduced by Yukawa for nuclear force in 1934. • Pions were isovector-peudoscalar bosons. (1940) Pauli, Dancoff, Oppenheimer, Schwinger • After shell model of Meyer-Jansen, pions were treated implicitly in nuclear physics • Now many physicists study pion for nuclei • Relativistic mean field model with projection • Renormalization of chiral symmetric model Sep.18 2006 Erice

  3. Ab initio calculation of light nuclei Pion 70 ~ 80 % C. Pieper and R. B. Wiringa, Annu. Rev. Nucl. Part. Sci.51(2001), nucl-th/0103005 Sep.18 2006 Erice

  4. Resolution Now and Then Y. Fujita et al., EPJ A 13 (’02) 411. H. Fujita et al., Dr. Th. & PRC Sep.18 2006 Erice

  5. Experiments High resolution GT (pionic) excitations High resolution (30keV) H. Fujita et al (RCNP) 2003 Tamii for (p, p’) Sep.18 2006 Erice

  6. Chiral sigma model Y. Ogawa et al. PTP (2004) Pion is the Goldstone boson of chiral symmetry Linear Sigma Model Lagrangian Polar coordinate Weinberg transformation Sep.18 2006 Erice

  7. where M = gsfp M* = M + gsj mp2 = m2 + l fp ms2 = m2 +3 l fp ~ ~ mw = gwfp mw* =mw + gwj Non-linear sigma model Lagrangian r = fp + j Sep.18 2006 Erice

  8. p w s, (0-) Relativistic chiral Mean Field Theory Parity mixed self-consistent mean field + Single particle state with parity mixing Intrinsic state (parity mixed state !!) H. Toki, S. Sugimoto, and K. Ikeda, Prog. Theor. Phys. 108(2002)903 Sep.18 2006 Erice

  9. Dirac equation Klein-Gordon equations Mean Field Equation Surface pion condensation Surface pion field Sep.18 2006 Erice

  10. Numerical results 9.2 40Ca 56Ni 9.0 N=20 N=28 8.8 8.6 8.4 Experiment 8.2 8.0 7.8 20 30 40 50 60 70 80 90 N=Z A (Mass number) Sep.18 2006 Erice

  11. No pion Pion Mean Field 56Ni Magic effect Parity mixed Pion produces spin-orbit splitting!! Sep.18 2006 Erice

  12. Gamow-Teller transition in Ni56 Sep.18 2006 Erice

  13. Symmetry projected RMF with pion The intrinsic state is obtained in RMF and it is parity-mixed and charge-mixed. Sep.18 2006 Erice

  14. Parity projection Kaiser, Fritsch Weise, NPA697 (2002) 0- 0- Finelli, Kaiser Vretener, Weise NPA770(2006) Sep.18 2006 Erice

  15. Charge and parity projected RMF Sep.18 2006 Erice

  16. projection -2- Sep.18 2006 Erice

  17. Energy components and radius Y. Ogawa et al., PRC73 (2006) 34301 Sep.18 2006 Erice

  18. Parity projection Wave function Sep.18 2006 Erice

  19. Density distribution and form factor Sep.18 2006 Erice

  20. He4 and He5 Myo et al (2005) Sep.18 2006 Erice

  21. Phase shifts for various partial waves Sep.18 2006 Erice

  22. Higher partial waves Sep.18 2006 Erice

  23. Coleman-Weinberg mechanism forspontaneous chiral symmetry breakingin the massless chiral sigma model • We want to include the vacuum polarization • for the study of nuclei. • Nobody have succeeded to work out the renormalization • for chiral symmetric lagrangian • We take the Coleman-Weinberg mechanism for • this program. Sep.18 2006 Erice

  24. Chiral sigma model : sigma field before the chiral symmetry breaking : sigma field after the chiral symmetry breaking Chiral sigma model is renormalizable but ... Sep.18 2006 Erice

  25. Unstable potential J. Boguta, NPA501, 637 (1989) With nucleon loop Unnatural size of interactionR. J. Furnstahl, et al. NPA618, 446 (1997) Unstable Problems of chirally symmetric renormalization Stable Sep.18 2006 Erice

  26. Reasons for these problems 1: The number of counterterms and renormalization conditions T. D. Lee and M. Margulies, PRD11, 1591, 1975 T. Matsui and B. D. Serot, Ann. of Phys. 144, 107, 1982 2: Cancellation between nucleon and boson loops does not occur. A. D. Jackson et al, NPA407, 495, 1983 E. M. Nyman and M. Rho, PLB60, 134, 1976 The masses of sigma and pi mesons become tachyonic. Sep.18 2006 Erice

  27. New chirally symmetric renormalization(Coleman-Weinberg) Sep.18 2006 Erice

  28. Loop contributions in φ4 theory  Sep.18 2006 Erice

  29. Bare potential Nucleon loop Boson loop Spontaneous chiral symmetry breaking (Local minimum condition) Massless nucleon and boson loops The differences among boson and fermion loops are sign and coupling constants, but both of them have the same functional form. Sep.18 2006 Erice

  30. One-loop corrections as origin of SCSB Input Output Sep.18 2006 Erice

  31. After SCSB The stable effective potential Before SCSB Sep.18 2006 Erice

  32. Summary of renormalization • We construct the massless chiral model with nucleon and boson loops in the Coleman-Weinberg scheme. • We obtain a stable effective potential with Dirac sea in the chiral model for the first time. • SCSB is caused by the in-balance between nucleon and boson loops, and both nucleon and bosons become massive at the same time. • By introducing nucleon and boson loops to the RMF theory, naturalness restores in the massless chiral sigma model. • As future works, we would like to study the properties of finite nuclei, hadronic matter at finite temperature and at high density. Sep.18 2006 Erice

  33. Conclusion • We have developed the relativistic chiral mean field model for finite nuclei • Spin and charge projection is essential • Pion provides a half of spin-orbit splitting • We have succeeded to obtain renormalized chiral meson-baryon Lagrangian (vacuum effect) Sep.18 2006 Erice

  34. The renormalization scale Radiative corrections as origin of SSB Coleman & Weinberg redefine two renormalization conditions before the symmetry breaking in the massless ϕ4 theory in order to avoid a logarithmical singularity. S. R. Coleman and E. Weinberg, PRD 7, 1888 (1973) Sep.18 2006 Erice

  35. One-boson loop with chiral symmetry Sep.18 2006 Erice

  36. We take the limit especially for two cases ... Dependence on renormalization scale Sep.18 2006 Erice

  37. Break Preserve Renormalization and Symmetry 1: The restoration of chiral symmetry 2: All of one-loop corrections are perfectly cancelled. Symmetry (Chiral symmetry and symmetry between fermion and boson) Sep.18 2006 Erice

  38. Naturalness and naive dimensional analysis (NDA) H. Georgi, Adv. Nucl. Phys. 43, 209 (1993) If naturalness holds, a dimensionless coefficient κn is O(1). For example, we check the bare potential in massless φ4 theory. As the next case, we check the vacuum fluctuation from nucleon loop in the Walecka model. Sep.18 2006 Erice

  39. Estimation of naturalness Sep.18 2006 Erice

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