\begin{document}$ B_{c}(1P)^{+}\to B_{c}^{(*)+}\pi^{0} $\end{document}, which may provide additional information for the determination of the properties of the first orbital excitation states of \begin{document}$ B_{c}(1P)^{+} $\end{document}. By assuming a dual relation between the U(1) anomaly soft-gluon coupling for \begin{document}$ B_{c}(1P)^{+}\to B_{c}^{(*)+}\pi^{0} $\end{document} and the intermediate meson loop transitions, we can quantify the isospin-violating decay effects for these four P-wave states. We find that the partial decay width of \begin{document}$ B_{c0}^{*+}\to B_{c}^{+}\pi^{0} $\end{document} is about three orders of magnitude larger than that for \begin{document}$ B_{c2}^{*+}\to B_{c}^{+}\pi^{0} $\end{document}. It implies that \begin{document}$ B_{c0}^{*+} $\end{document} can be established in the \begin{document}$ B_{c}^{+}\pi^{0} $\end{document} decay channel as a single state. Meanwhile, the two axial-vector states \begin{document}$ B_{c1}^{+}/B_{c1}'^{+} $\end{document} can be possibly identified in \begin{document}$ B_{c1}^{+}/B_{c1}'^{+}\to B_{c}^{*+}\pi^{0} $\end{document} with comparable strengths. Although these isospin-violating decays turn out to be small, the theoretical predictions should be useful for guiding future experimental efforts."> Predictions for the isospin-violating decays of <inline-formula><tex-math id="M1">\begin{document}${{\boldsymbol B}_{\boldsymbol c}({\boldsymbol{1P}})^{\bf +}\bf\to{\boldsymbol B}_{\boldsymbol c}^{({\bf *})+}\boldsymbol\pi^{\bf 0}}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/8d291803-23bd-4018-a831-598c33c0148f/CPC-2025-0566_M1.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/8d291803-23bd-4018-a831-598c33c0148f/CPC-2025-0566_M1.png"/></alternatives></inline-formula> -
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