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Two-Photon-Exchange effects in $ep\rightarrow enπ^+$ and their corrections to separated cross sections $σ_{\textrm{L,T,LT,TT}}$ at small $-t$

In this work, the two-photon-exchange (TPE) effects in $ep\rightarrow enπ^+$ at small $-t$ are discussed within a hadronic model. Under the pion dominance approximation the TPE contribution to the amplitude can be described by a scalar function in the limit $m_e\rightarrow 0$. The TPE contributions to the amplitude and the unpolarized differential cross section are both estimated when only the elastic intermediate state is considered. We find that the TPE corrections to the unpolarized differential cross section are about from $-4\%$ to $-20\%$ at $Q^2=1$--$ 1.6$ GeV$^2$. After considering the TPE corrections to the experimental data sets of unpolarized differential cross section, we analyze the TPE corrections to the separated cross sections $σ_{\textrm{L,T,LT,TT}}$. We find that the TPE corrections (at $Q^2=1$--$1.6$ GeV$^2$) to $σ_{\textrm{L}}$ are about from $-10\%$ to $-30\%$, to $σ_{\textrm{T}}$ are about $20\%$, and to $σ_{\textrm{LT,TT}}$ are much larger. By these analysis, we conclude that the TPE contributions in $ep\rightarrow enπ^+$ at small $-t$ are important to extract the separated cross sections $σ_{\textrm{L,T,LT,TT}}$ and the electromagnetic magnetic form factor of $π^+$ in the experimental analysis.