  We present a study of the strong isospin-breaking (IB) effect, due in QCD to
the difference between $u$- and $d$-quark masses, in the leptonic decay
constants of charmed and beauty pseudoscalar and vector mesons using the method
of QCD sum rules. We apply the sum-rule analysis to the decay constants of
mesons containing one heavy quark and one light quark with the light mass in
the range from the average $u/d$ quark mass to the strange-quark mass. We then
analyse the dependence of the decay constants on the light quark mass and
extract with good accuracy the IB ratios of decay constants at leading order in
the mass difference $(m_d - m_u)$, obtaining: $(f_{D^+} - f_{D^0}) / f_{D} =
0.0047 (6)$, $(f_{D^{*+}} - f_{D^{*0}}) / f_{D^*} = 0.0068 (9)$, $(f_{B^0} -
f_{B^+}) / f_{B} = 0.0047 (6)$, and $(f_{B^{*0}} - f_{B^{*+}}) / f_{B^*} =
0.0045 (5)$, which yield: $f_{D^+} - f_{D^0} = 0.97 \pm 0.13$ MeV, $f_{D^{*+}}
- f_{D^{*0}} = 1.73 \pm 0.27$ MeV, $f_{B^0} - f_{B^+} = 0.90 \pm 0.13$ MeV,
$f_{B^{*0}} - f_{B^{*+}} = 0.81 \pm 0.11$ MeV. In the case of the $D$-meson our
finding is consistent with recent lattice QCD results, whereas it is much lower
in the case of the $B$-meson showing a tension of $\approx 3$ standard
deviations.
