A theoretical study has derived the explicit expressions for the spacelike kernels K_Π⁽²⁾(Q²) and K_D⁽²⁾(Q²) entering the hadronic vacuum polarization contribution ΔΓ_had to the muon decay rate. These quantities, which had not previously been written in closed form, were the missing piece needed to bring different calculation methods to bear on the final value of the muon decay rate.

Hadronic vacuum polarization is one of the main sources of uncertainty in the theoretical determination of the muon lifetime and, by extension, in the Fermi constant extracted from it. Its calculation requires non-perturbative input on the strong interaction at low energies, and the usual approaches — e⁺e⁻ to hadrons dispersion relations or higher-order corrections — cannot always be applied directly.

With the expressions obtained, ΔΓ_had can be evaluated using techniques that work in momentum space, such as lattice quantum chromodynamics studies or the MUonE project, which proposes measuring elastic muon-electron scattering to access the hadronic vacuum polarization in the spacelike region. Both routes now share a common analytical framework to convert their results into a contribution to the decay rate.

The advance is of a formal nature, but it opens the door to comparing the different determinations of ΔΓ_had in a consistent way and to reducing their systematic error. Cross-checks with lattice data and with future MUonE results will be the next step to consolidate the muon lifetime value and to test the Standard Model.