  We investigate quarkonium mass spectra in external constant magnetic fields
by using QCD sum rules. We first discuss a general framework of QCD sum rules
necessary for properly extracting meson spectra from current correlators
computed in the presence of strong magnetic fields, that is, a consistent
treatment of mixing effects caused in the mesonic degrees of freedom. We then
implement operator product expansions for pseudoscalar and vector heavy-quark
current correlators by taking into account external constant magnetic fields as
operators, and obtain mass shifts of the lowest-lying bound states $\eta_c$ and
$J/\psi$ in the static limit with their vanishing spatial momenta. Comparing
results from QCD sum rules with those from hadronic effective theories, we find
that the dominant origin of mass shifts comes from a mixing between $\eta_c$
and $J/\psi$ with a longitudinal spin polarization, accompanied by other
subdominant effects such as mixing with higher excited states and continua.
