\begin{document}$ a_1(1260) $\end{document}"> \begin{document}$ \gamma p \to a_1(1260)^+ n $\end{document} and \begin{document}$ \gamma p \to \pi^+\pi^+\pi^- n $\end{document} reactions via the \begin{document}$ \pi $\end{document}-exchange mechanism. For the \begin{document}$ \gamma p \to \pi^+\pi^+\pi^- n $\end{document} reaction, we perform a calculation of the differential and total cross-sections by considering the contributions of the \begin{document}$ a_1(1260) $\end{document} intermediate resonance decaying into \begin{document}$ \rho \pi $\end{document} and then into \begin{document}$ \pi^+\pi^+\pi^- $\end{document}. Besides, the non-resonance process is also considered. With a lower mass of \begin{document}$ a_1(1260) $\end{document}, the experimental data for the invariant \begin{document}$ \pi^+\pi^+\pi^- $\end{document} mass distributions can be fairly well reproduced. For the \begin{document}$ \gamma p \to a_1(1260)^+ n $\end{document} reaction, with the model parameters, the total cross-section is of the order of 10 μb at the photon beam energy \begin{document}$ E_{\gamma} $\end{document}~2.5 GeV. It is expected that the model calculations in this work could be tested by future experiments."> Study of the <inline-formula><tex-math id="M1">\begin{document}$ {{a_1}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/391aa38e-7482-4d50-a973-80cef1cc3ec7/CPC-2018-0451_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/391aa38e-7482-4d50-a973-80cef1cc3ec7/CPC-2018-0451_M1.png"/></alternatives></inline-formula>(1260) resonance in the <inline-formula><tex-math id="M2">\begin{document}$ {{\gamma p \to \pi^+\pi^+\pi^- n}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/391aa38e-7482-4d50-a973-80cef1cc3ec7/CPC-2018-0451_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/391aa38e-7482-4d50-a973-80cef1cc3ec7/CPC-2018-0451_M2.png"/></alternatives></inline-formula> reaction -
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