  We present a quenched lattice calculation of $f_V^\perp/f_V$, the coupling of
vector mesons to the tensor current normalized by the vector meson decay
constant. The chirally improved lattice Dirac operator, which allows us to
reach small quark masses, is used. We put emphasis on analyzing the quark mass
dependence of $f_V^\perp/f_V$ and find only a rather weak dependence. Our
results at the $\rho$ and $\phi$ masses agree well with QCD sum rule
calculations and those from previous lattice studies.
