Spacelike and timelike kernels for the hadronic vacuum polarization contribution to the muon lifetime

The explicit expressions for the spacelike kernels $K_Π^{(2)}(Q^2)$ and $K_{D}^{(2)}(Q^2)$ for the hadronic vacuum polarization contribution to the muon lifetime $ΔΓ_{\text{had}}$ are derived. The obtained results enable one to evaluate $ΔΓ_{\text{had}}$ within such methods as lattice studies, MUonE project, and others.