The Higgs-Graviton Couplings: from Amplitudes to the Action
In this paper we study the coupling of scalar (Higgs) particles ($ϕ$) with gravitons ($h$) and their possible effects. The general form of the 3-point interaction $ϕ(p) h(1)h(2)$ 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 $ϕR^2$, $ϕR^{μν} R_{μν}$ and $ϕR^{μνρσ} R_{μνρσ}$, as well as the term $ε_{μναβ} ϕ_5 R^{μν}_{ρσ} R^{αβρσ}$ for the case of a pseudo-scalar ($ϕ_5$). 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 $ϕ\to h h^* (\to XX)$, 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.