The Higgs-Graviton Couplings: from Amplitudes to the Action

In this paper we study the coupling of scalar (Higgs) particles ($\varphi$) with gravitons ($h$) and their possible effects. The general form of the 3-point interaction $\varphi (p)h(1)h(2)$ 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 $\varphi R^2$, $\varphi R^{\mu \nu }{R}_{\mu \nu }$ and $\phi R^{\mu\nu\rho\sigma} R_{\mu\nu\rho\sigma}$,aswellastheterm$\epsilon_{\mu\nu \alpha\beta} \phi_5 R^{\mu\nu}_{\rho\sigma} R^{\alpha\beta\rho\sigma}$forthecaseofapseudo-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 $\varphi \to h{h}^{\ast }(\to XX)$, with an off-shell graviton ($h^{\ast }$) 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.