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

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