Establishing the Primary HEFT as a Precision Benchmark for UV-HEFT Matching

We match the real Higgs triplet model (RHTM) onto HEFT under different parameter choices and power-counting schemes, thereby obtaining several representative HEFT formulations and clarifying their relations. We establish the primary HEFT (pHEFT) as a benchmark framework, demonstrating that alternative HEFT constructions can be systematically derived from it. A key advantage of the pHEFT construction is its parameter choice, which maintains linear relations between the UV Lagrangian parameters and squared heavy masses. By strictly employing the inverse squared heavy masses as the expansion parameters without imposing additional constraints, pHEFT preserves maximal ultraviolet (UV) information and ensures higher perturbative accuracy by avoiding the additional truncations inherent in more complex, non-linear formulations or extra constraints. Through the analysis of the $Z_2$-symmetric real singlet model and the 2HDM, we illustrate the criteria for identifying viable primary HEFT constructions in UV models with scalar extensions. Furthermore, for the first time, we derive the HEFT operators of the RHTM involving fermions.