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In this work, the two-photon-exchange (TPE) effect in $ep\rightarrow en\pi^+$at small $-t$ is discussed. In the previous work, the TPE contribution with one$\pi$ intermediate state is estimated numerically within a hadronic model underthe pion-dominance approximation. Here we extend the discussion to include one$\rho$ intermediate state. The TPE contribution can be described by one scalarfunction in the limit $m_e\rightarrow 0$, the dispersion relation (DR)satisfied by this scalar function is analysed. The analytic expressions for theimaginary parts of the TPE contributions from one $\pi$ or one $\rho$intermediate state are given within the hadronic model. Combining theseanalytic expressions and the DR, the corresponding real parts of the TPEcontributions can be estimated easily at any available region. This can helpthe further experimental analysis to include the TPE contributions in aconvenient way. The numeric results show that the TPE correction with one$\rho$ intermediate state is much smaller than that with one $\pi$ intermediatestate in the current energy region. These results suggest that the TPEcontribution with an elastic state is the main TPE contribution in$ep\rightarrow en\pi^+$ at small $-t$.

Two-Photon-Exchange effect in $ep\rightarrow en\pi^+$ at small $-t$ with the hadronic model and dispersion relation approach