  A search is presented for heavy vector-like quarks (VLQs) that couple only to
light quarks in proton-proton collisions at $\sqrt{s} =$ 8 TeV at the LHC. The
data were collected by the CMS experiment during 2012 and correspond to an
integrated luminosity of 19.7 fb$^{-1}$. Both single and pair production of
VLQs are considered. The single-production search is performed for down-type
VLQs (electric charge of magnitude 1/3), while the pair-production search is
sensitive to up-type (charge of magnitude 2/3) and down-type VLQs. Final states
with at least one muon or one electron are considered. No significant excess
over standard model expectations is observed, and lower limits on the mass of
VLQs are derived. The lower limits range from 400 to 1800 GeV, depending on the
single-production cross section and the VLQ branching fractions B to W, Z, and
Higgs bosons. When considering pair production alone, VLQs with masses below
845 GeV are excluded for B(W) = 1.0, and below 685 GeV for B(W) = 0.5, B(Z) =
B(H) = 0.25. The results are more stringent than those previously obtained for
single and pair production of VLQs coupled to light quarks.
