Lattice QCD calculation of the two-photon exchange contribution to the muonic-hydrogen Lamb shift

We develop a method for lattice QCD calculation of the two-photon exchange contribution to the muonic-hydrogen Lamb shift. To demonstrate its feasibility, we present the first lattice calculation with a gauge ensemble at $m_{\pi }=142$ MeV. By adopting the infinite-volume reconstruction method along with an optimized subtraction scheme, we obtain $\Delta E_{$\text{TPE}$} = -28.9(4.9)~\mu$\text{eV}$ + 93.72~\mu$\text{eV}$/$\text{fm}$^2 \cdot\langle r_p^2 \rangle$,or$\Delta E_{$\text{TPE}$} = 37.4(4.9)~\mu$eV, which is consistent with the previous theoretical results in a range of 20-50 $\mu$eV.