Effect of sigma meson on theD1(2430)→Dππdecay

Abstract

We study the effect of sigma meson on the ${D}_{1}(2430)\ensuremath{\rightarrow}D\ensuremath{\pi}\ensuremath{\pi}$ decay by constructing an effective Lagrangian preserving the chiral symmetry and the heavy quark symmetry. The sigma meson is included through a linear sigma model, in which both the $q\overline{q}$ and $qq\overline{q}\overline{q}$ states are incorporated respecting their different $U(1{)}_{A}$ transformation properties. We first fit the sigma meson mass and $\ensuremath{\sigma}\mathrm{\text{\ensuremath{-}}}\ensuremath{\pi}\mathrm{\text{\ensuremath{-}}}\ensuremath{\pi}$ coupling constant to the $I=0$, $S$-wave $\ensuremath{\pi}\mathrm{\text{\ensuremath{-}}}\ensuremath{\pi}$ scattering data. Then, we show how the differential decay width $d\ensuremath{\Gamma}({D}_{1}\ensuremath{\rightarrow}D(\ensuremath{\pi}\ensuremath{\pi}{)}_{I=0,L=0})/d{m}_{\ensuremath{\pi}\ensuremath{\pi}}$ depends on the quark structure of the sigma meson. We find that our study, combing with the future data, can give a clue to understand the sigma meson structure.