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We investigate the discovery potential of the $a_1(1260)$ photoproduction inthe $\gamma p \to a_1(1260)^+ n$ and $\gamma p \to \pi^+\pi^+\pi^- n$ reactionsvia the $\pi$-exchange mechanism. For the $\gamma p \to \pi^+\pi^+\pi^- n$reaction, we perform a calculation for the differential and total crosssections by including the contributions from the $a_1(1260)$ intermediateresonance decaying into $\rho \pi$ then into $\pi^+\pi^+\pi^-$. Besides, thenon-resonance process is also considered. With a lower mass of $a_1(1260)$, weget a fairly good description of the experimental data for the invariant$\pi^+\pi^+\pi^-$ mass distributions. For the $\gamma p \to a_1(1260)^+ n$reaction, with our model parameters, the total cross section is of the order of10 $\mu$b at photon beam energy $E_{\gamma} \sim$ 2.5 GeV. It is expected thatour model calculations could be tested by future experiments.

Study of $a_1(1260)$ in the $\gamma p \to \pi^+\pi^+\pi^- n$ reaction