Electroweak oblique parameters as a probe of the trilinear Higgs boson self-interaction

We calculate the two-loop contributions from a modified trilinear Higgs self-interaction, $\kappa_\lambda \lambda_{\rm SM} v h^3$, to the electroweak oblique parameters $S$ and $T$. Using the current bounds on $S$ and $T$ from electroweak measurements, we find the 95% C.L. constraint on the modified trilinear coupling to be $-14.0 \leq \kappa_\lambda \leq 17.4$. The largest effects on $S$ and $T$ arise from two insertions of the modified trilinear coupling that result in $T/ S \simeq -3/2$; remarkably, this is nearly parallel to the axis of the tightest experimental constraint in the $S$-$T$ plane. No contributions to $S$ and $T$ arise from a modified Higgs quartic coupling at two-loop order. These calculations utilized a gauge-invariant parameterization of the trilinear Higgs coupling in terms of higher dimensional operators $(H^\dagger H)^{n}$ with $n \ge 3$. Interestingly, the bounds on $\kappa_\lambda$ that we obtain are comparable to constraints from di-Higgs production at the LHC as well as recent bounds from single Higgs production at the LHC.