Low-virtuality photon transitions $\gamma^*\to f\bar f$ and the photon-to-jet conversion function

The calculation of electroweak corrections to processes with jets in the final state involves contributions of low-virtuality photons leading to jets in the final state via the singular splitting $\gamma^* \to q\bar q$. These singularities can be absorbed into a photon-to-jet "fragmentation function", better called "conversion function", since the physical final state is any hadronic activity rather than an identified hadron. Using unitarity and a dispersion relation, we relate this $\gamma^* \to q\bar q$ conversion contribution to an integral over the imaginary part of the hadronic vacuum polarization and thus to the experimentally known quantity $\Delta\alpha^{(5)}_{\mathrm{had}}(M^2_{\rm Z})$. Therefore no unknown non-perturbative contribution remains that has to be taken from experiment. We also describe practical procedures following subtraction and phase-space-slicing approaches for isolating and cancelling the $\gamma^* \to q\bar q$ singularities against the photon-to-jet conversion function. The production of Z+jet at the LHC is considered as an example, where the photon-to-jet conversion is part of a correction of the order $\alpha^2/\alpha_{\rm s}$ relative to the leading-order cross section.