  The Standard Model prediction for $\mu$-$e$ scattering at
Next-to-Next-to-Leading Order (NNLO) contains non-perturbative QCD
contributions given by diagrams with a hadronic vacuum polarization insertion
in the photon propagator. By taking advantage of the hyperspherical integration
method, we show that the subset of hadronic NNLO corrections where the vacuum
polarization appears inside a loop, the irreducible diagrams, can be calculated
employing the hadronic vacuum polarization in the space-like region, without
making use of the $R$ ratio and time-like data. We present the analytic
expressions of the kernels necessary to evaluate numerically the two types of
irreducible diagrams: the two-loop vertex and box corrections. As a cross
check, we evaluate these corrections numerically and we compare them with the
results given by the traditional dispersive approach and with analytic two-loop
vertex results in QED.
