Two-photon-exchange effects in the unpolarized $\mu p$ scattering within the hadronic model

In this work, the two-photon-exchange (TPE) effects in the unpolarized $\mu p$ scattering are discussed within the hadronic model where the intermediate states $N,\mathrm{\Delta }$ and $\sigma$ are considered. The contribution from the $N$ intermediate is close to the results given by Ref. \cite{Tomalak2014} at the small $Q$ and there is a sizeable difference when $Q>0.25$GeV (where $Q^2$ is the four momentum transfer). The contributions from the $\mathrm{\Delta }$ and the $\sigma$ intermediate states are much smaller than that from the $N$ intermediate at the small $Q$. In the kinematic region with $k_i\subseteq [0.01,0.3]$GeVand$Q \leq0.4$GeV(where$k_i$ is the three momentum of initial muon at Lab frame), a naive expression for the TPE contributions is given, which can be used directly for other analysis.