1. Study of sigma meson structure in chiral models Masayasu Harada (Nagoya Univ.) at Crossover 2012 (Nagoya University, July 12, 2012) Based on ・ M.H., H.Hoshino and Y.L.Ma, Phys. Rev. D85, 114027 (2012)

2. 1. Introduction

3. Clue to understand the chiral symmetry breaking “σ” particle (“QCD Higgs” particle) ・・・ Quantum fluctuation of the condensate PDG 2009 A candidate ・・・ f 0 (600) : lightest I=0 scalar meson However, f 0 (600) may not be a qq bar meson !

4. Scalar meson puzzle mass (MeV) Contradiction ?

5. Scalar meson puzzle mass (MeV) Consistent ?

6. ◎ Standard qq bar quark model assignment What is f 0 (600) ? 2 quark ( qq bar ) state “σ” particle 4 quark ( qqq bar q bar ) state Exotic hadron

7. Outline 1.Introduction 2.Linear sigma model for light quark sector including  meson  meson in D 1 → D  decay 4.Summary

8. 2. Linear sigma model for light quark sector including  meson

9. 2 and 4 quark states in linear sigma model 3×3 matrix fields ＆ (Linear Sigma Model): Scalar Pseudo scalar ScalarPseudo scalar ２ quark field ～ Different transformations under U(1) A : These transform in the same way under SU(3) L ×SU(3) R : ４ quark field ～ SU （３） R × SU （３） L ： U（1）A：U（1）A： Amir H. Fariborz, Renata Jora, and Joseph Schechter, PRD 72, 034001 (2005)

10. U(1) A Symmetry ? ◎ Anomaly is suppressed in the large Nc QCD Current is conserved. U(1) A is spontaneously broken by the quark condensate. ◎ Definition of the spontaneously broken charge Light-front axial charge is well-defined. S. Weinberg, Phys. Rev. 177 (1969) 2604.

11. When the U(1) A symmetry exists, 2-quark state and 4-quark state do not mix with each other. But, U(1) A symmetry is broken by anomaly explicitly by spontaneous chiral symmetry breaking ⇒ mixing between 2-quark state and 4-quark state mixing Lightest Heaviest 2 nd 3 rd

12. An effective Lagrangian Linear sigma model including 2-nonet fields includes SU(3)×SU(3) invariant terms which can include U(1)A variant - anomaly effect - terms. includes SU(3)×SU(3) variant terms. : SU(3) L ×SU(3) R invariant, U(1) A invariant. : SU(3) L ×SU(3) R invariant, U(1) A breaking (anomaly). : Explicit SU(3) L ×SU(3) R ×U(1) A breaking terms. (effects of current quark masses) constrained by anomaly matching with QCD

13. Phenomenological Analysis using a special form of the potential Note : A 2 /A 1 = m d /m u ; A 3 /A 1 = m s /m u A.H. Fariborz, R. Jora, and J. Schechter, PRD 79, 074014 (2009) generate mixing between 2-quark and 4-quark states

14. Phenomenological Study of f 0 (600) Inputs : fixed values m  = 137 MeV ; F  = 92.6 MeV m[ a 0 (980) ] = 987.4 MeV, m[ a 0 (1450) ] = 1474 MeV; A 2 /A 1 = m d /m u = 1 variable values 1200 MeV < m[  (1300) ] < 1400 MeV (exp: 1300 ± 100 MeV) 20 < A 3 /A 1 = m s /m u < 30 masses of I=0 scalar mesons Mass hierarchy is reasonably reproduced. Note that the light sigma meson appears automatically in the present model.

15. Quark Contents of f 0 (600) percentage of components fafa fbfb fcfc fdfd f a = (uu bar + dd bar )/√2 ; f b = ss bar ; f c = (usu bar s bar + dsd bar s bar )/√2 ; f d = udu bar d bar ex: for m[  (1300) ] = 1215 MeV, fa : fb : fc : fd = 0.36 : 0.04 : 0.36 : 0.24 → The sigma meson is about 40% 2-quark and 60% 4 quark. The lightest sigma is roughly about half 2-quark and half 4-quark state.

16. 3.  meson in D 1 → D  decay M.H., H.Hoshino and Y.L.Ma, Phys. Rev. D85, 114027 (2012)

17. “chiral doubling” excited states ground states heavy quark symmetry (heavy quark partner ) chiral symmetry (chiral partner) M D(0+,1+) – M D(0-,1-) ~ 0.43 GeV Chiral doubling seems to work. M.A.Nowak, M.Rho and I.Zahed, PRD48, 4370 (1993)

18. This scalar meson is made of 2-quarks. (Not a mass eigenstate) U(1) A of D mesons ◎ Assume D(0-, 1-) and D(0+, 1+) ~ c q bar ⇒ U(1) A eigenstates : D L ~ D(0-,1-) – D(0+,1+) → D L e -i D R ~ D(0-,1-) + D(0+,1+) → D R e +i ⇒ Only 2-quark light-mesons can couple to D mesons. note : M(2-quark) → M e +2i ; M’(4-quark) → M’ e -4i

19. D 1 → D  decay We determine g  and m  by fitting to  scattering.

20. 5. Summary ◎ We construct an effective model including the light scalar masons and the D mesons. U(1) A symmetry plays an important role: Heavy meson can couple only to 2 quark mesons. ◎ We study an effect of the sigma meson to D 1 → D  decay Our result indicates that we can get some clue to understand the composition of the sigma meson from future experiment.