Toward an estimate of the amplitude $X(3872)\to\pi^0\chi_{c1}(1P)$

The well-known model of the triangle diagrams with $D^*\bar DD^*$ and $\bar D^*D\bar D^*$ mesons in the loops is compared with the modern data on the amplitude of the $X(3872)\to\pi^0\chi_{c1}(1P)$ decay. Considering the $X(3872)$ object as a $\chi_{c1}(2P)$ charmonium state, we introduce a parameter $\xi$ characterizing the scale of the isotopic symmetry violation in this decay and find a lower limit of $\xi\simeq0.0916$. The model incorporates the only fitted parameter associated with the form factor. We analyze in detail the influence of the form factor on the amplitude $X(3872)\to\pi^0 \chi_{c1}(1P)$ and on the parameter $\xi$. As the suppression of the amplitude by the form factor increases, $\xi$ increases. Due to the fact that the $X(3872)$ resonance is located practically at the threshold of the $D^0\bar D^{*0}$ channel, the amplitude of $X(3872) \to\pi^0\chi_{c1}(1P)$ turns out to be proportional to $\sqrt{m_d -m_u}$. Using the estimating values for the coupling constants $g_{XD\bar D^*}$, $g_{\chi_{ c1}D\bar D^*}$, and $g_{D^{*0}D^{*0}\pi^0}$, we show that the model of the triangle loop diagrams is in reasonable agreement with the available data. Apart from the difference in the masses of neutral and charged charmed mesons, any additional exotic sources of isospin violation in $X(3872)\to\pi^0\chi_{c1}(1P)$ (such as a significant difference between the coupling constants $g_{XD^0\bar D^{*0}}$ and $g_{XD^+ D^{*-}}$) are not required to interpret the data. This indirectly confirms the isotopic neutrality of the $X(3872)$, which is naturally realized for the $c\bar c$ state $\chi_{c1}(2P)$.