  In high-energy leptonic collisions, such as at a multi-TeV muon collider, the
collinear splittings of the electroweak (EW) gauge bosons and leptons are the
dominant phenomena, and the scattering processes should thus be formulated in
terms of the EW parton distribution functions (EW PDFs). We complete this
formalism in the Standard Model to include the QCD sector and evaluate the
quark and gluon PDFs inside a lepton at the double-log accuracy. The splittings
of the photon and subsequently the quarks and gluons control the quark/gluon
PDFs below the EW scale. The massive gauge bosons lead to substantial
contributions at high scales. The jet production cross section can reach the
order of a few nb (50 pb) in $e^+e^-$ ($\mu^+\mu^-$) collisions, at the TeV
c.m. energies with a moderate acceptance cut, that governs the overall event
shape up to about $p_T^j \sim 60$ GeV.
