Matching the real Higgs triplet extension of Standard Model to HEFT

We make a first matching of the real Higgs triplet extension (RHTE) of the standard model to the Higgs effective field theory (HEFT), which is also known as electroweak chiral Lagrangian (EWChL). In the RHTE, the ratio $\xi$ of two VEVs separately from triplet and doublet is a tiny quantity, due to the constrained custodial symmetry violation. Based on the nonlinear representation~\cite{Song:2024kos} of RHTE and functional method, we get the HEFT in the power counting of $\xi$. By comparing its phenomenological results of $hh\to hh$, $W W \to h h$, $Z Z \to hh $ to SMEFT's, we show its convergence property is as good as SMEFT's. Besides we explicitly show that the HEFT is available in the phase transition region where SMEFT fails.