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, (2012)

1. Introduction

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 !

Scalar meson puzzle mass (MeV) Contradiction ?

Scalar meson puzzle mass (MeV) Consistent ?

◎ 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

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

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

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, (2005)

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.

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

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

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, (2009) generate mixing between 2-quark and 4-quark states

Phenomenological Study of f 0 (600) Inputs : fixed values m  = 137 MeV ; F  = 92.6 MeV m[ a 0 (980) ] = 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.

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.

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

“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)

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

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

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.