We present a detailed analysis of the rare exclusive Higgs-boson decays into a single vector meson and a photon and investigate the possibility of using these processes to probe the light-quark Yukawa couplings. We work with an effective Lagrangian with modified Higgs couplings to account for possible new-physics effects in a model-independent way. The h->Vgamma{} decay rate is governed by the destructive interference of two amplitudes, one of which involves the Higgs coupling to the quark anti-quark pair inside the vector meson. We derive this amplitude at next-to-leading order in alpha_s using QCD factorization, including the resummation of large logarithmic corrections and accounting for the effects of flavor mixing. The high factorization scale mu~m_h ensures that our results are rather insensitive to the hadronic parameters characterizing the light-cone distribution amplitude of the vector meson. The second amplitude arises from the loop-induced effective hgammagamma* and hgamma Z* couplings, where the off-shell gauge boson converts into the vector meson. We devise a strategy to eliminate theoretical uncertainties related to this amplitude to almost arbitrary precision. This opens up the possibility to probe for O(1) modifications of the c- and b-quark Yukawa couplings and O(30) modifications of the s-quark Yukawa coupling in the high-luminosity LHC run. In particular, we show that measurements of the ratios Br(h->Upsilon(nS)gamma)/Br(h->gammagamma) and Br(h->bb)/Br(h->gammagamma) can provide complementary information on the real and imaginary parts of the b-quark Yukawa coupling. More accurate measurements would be possible at a future 100 TeV proton-proton collider.
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