In a molecular scenario, we investigate the isospin-breaking hidden charm decay processes of X(3872), i.e., X(3872)→π+π-J/ψ\documentclass[12pt]{minimal}
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\begin{document}$$X(3872) \rightarrow \pi ^+ \pi ^- J/\psi $$\end{document}, X(3872)→π+π-π0J/ψ\documentclass[12pt]{minimal}
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\begin{document}$$X(3872) \rightarrow \pi ^+ \pi ^- \pi ^0 J/\psi $$\end{document}, and X(3872)→π0χcJ\documentclass[12pt]{minimal}
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\begin{document}$$X(3872)\rightarrow \pi ^0\chi _{cJ}$$\end{document}. 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-\documentclass[12pt]{minimal}
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\begin{document}$$D^{*+} D^-$$\end{document} and neutral components D∗0D¯0\documentclass[12pt]{minimal}
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\begin{document}$$D^{*0 } {\bar{D}}^0$$\end{document} 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 Γ[X(3872)→π+π-π0J/ψ]/Γ[X(3872)→π+π-J/ψ]\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma [X(3872) \rightarrow \pi ^+ \pi ^- \pi ^0 J/\psi ]/\Gamma [X(3872) \rightarrow \pi ^+ \pi ^- J/\psi ]$$\end{document}. With the determined parameter range, we find that the estimated ratio Γ[X(3872)→π0χc1/Γ[X(3872)→π+π-J/ψ]\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma [X(3872) \rightarrow \pi ^0 \chi _{c1}/\Gamma [X(3872) \rightarrow \pi ^+ \pi ^- J/\psi ]$$\end{document} is well consistent with the experimental measurement from the BESIII collaboration. Moreover, the partial width ratio of π0χcJ\documentclass[12pt]{minimal}
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\begin{document}$$\pi ^0 \chi _{cJ}$$\end{document} for J=0,1,2\documentclass[12pt]{minimal}
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\begin{document}$$J=0,1,2$$\end{document} is estimated to be 1.77-1.65:1:1.09-1.43\documentclass[12pt]{minimal}
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\begin{document}$$1.77{-}1.65:1:1.09{-}1.43$$\end{document}, which could be tested by further precise measurements of BESIII and Belle II.
