The Higgs-Graviton Couplings: from Amplitudes to the Action

In this paper we study the coupling of scalar (Higgs) particles ($phi$) with gravitons ($h$) and their possible effects. The general form of the 3-point interaction $phi(p) h(1)h(2)$ can be derived using the scaling behavior of the spinor variables under the little group; the resulting vertices exhibit such simplicity, that some simplifications should be hidden in the expressions obtained from the extended scalar action. To investigate this, we study an extended Einstein-Hilbert action that besides the minimal coupling, it also includes terms of the form $phi R^2$, $phi R^{mu u} R_{mu u}$ and $phi R^{mu urhosigma} R_{mu urhosigma}$, as well as the term $epsilon_{mu u alphabeta} phi_5 R^{mu u}_{rhosigma} R^{alphabetarhosigma}$ for the case of a pseudo-scalar ($phi_5$). The resulting vertices satisfy KLT-type relations, i.e., they can be written as the square of the coupling of the Higgs with gluons. We find that the amplitude for the Higgs decay into a pair of gravitons (on-shell) only receives a contribution coming from the square of the Riemann tensor. Similar results are obtained for the 3-body decay $phi to h h^* (to XX)$, with an off-shell graviton ($h^*$) that goes into the final state $XX$. One could expect that these quadratic terms can produce new loop effects, however we find that the new contribution from this non-minimal coupling to the graviton self-energy, also vanishes for on-shell gravitons.