Two-Photon-Exchange effects in $eprightarrow enpi^+$ and their corrections to separated cross sections $sigma_{textrm{L,T,LT,TT}}$ at small $-t$

In this work, the two-photon-exchange (TPE) effects in $eprightarrow enpi^+$ at small $-t$ are discussed within a hadronic model. The TPE contributions to the amplitude and the unpolarized differential cross section are both estimated and we find that the TPE corrections to the unpolarized differential cross section are about $-4%sim-15%$ at $Q^2=1$GeV$^2sim1.6$GeV$^2$. After considering the TPE corrections to the experimental data sets of unpolarized differential cross section, we analyse the TPE corrections to the separated cross sections $sigma_{textrm{L,T,LT,TT}}$. We find that the TPE corrections (at $Q^2=1$GeV$^2sim1.6$GeV$^2$) to $sigma_{textrm{L}}$ are about $-10%sim -20%$, to $sigma_{textrm{T}}$ are about $20%$ and to $sigma_{textrm{LT,TT}}$ are much larger. By these analysis, we conclude that the TPE contributions in $eprightarrow enpi^+$ at small $-t$ are important to extract the separated cross sections $sigma_{textrm{L,T,LT,TT}}$ and the electromagnetic magnetic form factor of $pi^+$ in the experimental analysis.