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The on-shell matrix elements, or couplings $g_{H H^*(H_1)\gamma}$, describingthe $B(D)_q^* \to B(D)_q \gamma$ and $B_{1 q} \to B_q \gamma$ ($q = u,d,s$)radiative decays, are determined from light-cone sum rules at next-to-leadingorder for the first time. Two different interpolating operators are used forthe vector meson, providing additional robustness to our results. For the$D^*$-meson, where some rates are experimentally known, agreement is found. Thecouplings are of additional interest as they govern the lowest pole residue inthe $B(D) \to \gamma$ form factors which in turn are connected toQED-corrections in leptonic decays $B(D) \to \ell \bar \nu$. Since thecouplings and residues are related by the decay constants $f_{H^*(H_1)}$ and$f^T_{H^*(H_1)}$ respectively, we determine them at next-leading order as aby-product. The quantities $\{ f_{H^*}^T, f_{H_1},f_{H_1}^T\}$ have notpreviously been subjected to a QCD sum rule determination. All results arecompared with the existing experimental and theoretical literature.

Radiative Decays of Heavy-light Mesons and the $f_{H,H^*,H_1}^{(T)}$ Decay Constants