A phenomenological analysis on isospin-violating decay of X(3872)

In a molecular scenario, we investigate the isospin-breaking hidden charm decay processes of X(3872), i.e., $$X(3872) \rightarrow \pi ^+ \pi ^- J/\psi $$ X ( 3872 ) → π + π - J / ψ , $$X(3872) \rightarrow \pi ^+ \pi ^- \pi ^0 J/\psi $$ X ( 3872 ) → π + π - π 0 J / ψ , and $$X(3872)\rightarrow \pi ^0\chi _{cJ}$$ X ( 3872 ) → π 0 χ cJ . We assume that the source of the strong isospin violation comes from the different coupling strengths of X(3872) to its charged components $$D^{*+} D^-$$ D ∗ + D - and neutral components $$D^{*0 } {\bar{D}}^0$$ D ∗ 0 D ¯ 0 as well as the interference between the charged meson loops and neutral meson loops. The former effect could fix our parameters by using the measurement of the ratio $$\Gamma [X(3872) \rightarrow \pi ^+ \pi ^- \pi ^0 J/\psi ]/\Gamma [X(3872) \rightarrow \pi ^+ \pi ^- J/\psi ]$$ Γ [ X ( 3872 ) → π + π - π 0 J / ψ ] / Γ [ X ( 3872 ) → π + π - J / ψ ] . With the determined parameter range, we find that the estimated ratio $$\Gamma [X(3872) \rightarrow \pi ^0 \chi _{c1}/\Gamma [X(3872) \rightarrow \pi ^+ \pi ^- J/\psi ]$$ Γ [ X ( 3872 ) → π 0 χ c 1 / Γ [ X ( 3872 ) → π + π - J / ψ ] is well consistent with the experimental measurement from the BESIII collaboration. Moreover, the partial width ratio of $$\pi ^0 \chi _{cJ}$$ π 0 χ cJ for $$J=0,1,2$$ J = 0 , 1 , 2 is estimated to be $$1.77{-}1.65:1:1.09{-}1.43$$ 1.77 - 1.65 : 1 : 1.09 - 1.43 , which could be tested by further precise measurements of BESIII and Belle II.