We consider a scalar field, such as the Higgs boson H, coupled to gluons via the effective operator H tr G_{mu nu} G^{mu nu} induced by a heavy-quark loop. We treat H as the real part of a complex field phi which couples to the self-dual part of the gluon field-strength, via the operator phi tr G_{SD mu nu} G_{SD}^{mu nu}, whereas the conjugate field phi^dagger couples to the anti-self-dual part. There are three infinite sequences of amplitudes coupling phi to quarks and gluons that vanish at tree level, and hence are finite at one loop, in the QCD coupling. Using on-shell recursion relations, we find compact expressions for these three sequences of amplitudes and discuss their analytic properties.
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