Dimension-Six Terms in the Standard Model Lagrangian

TL;DR

This work rederives a complete, non-redundant basis for dimension-5 and dimension-6 operators built from Standard Model fields, enforcing SM gauge invariance and using equations of motion to eliminate redundant terms. Under baryon-number conservation, they find 59 independent dimension-6 operators, partitioned into 15 bosonic, 19 single-fermionic-current, and 25 four-fermion operators, with 4 additional four-fermion operators if $B$ is not conserved. The analysis both corrects and refines the classic Buchmüller–Wyler list, noting the sole dimension-5 operator $Q_{\nu\nu}$ and identifying the missing $Q_{lequ}^{(3)}$ among BW’s candidates, while detailing the complete operator basis and its CP, Lorentz, and gauge-symmetry structure. The results provide a practical and widely used SMEFT basis, clarifying which operators are most relevant for phenomenology, especially in weakly coupled UV completions that generate a subset at tree level.

Abstract

When the Standard Model is considered as an effective low-energy theory, higher dimensional interaction terms appear in the Lagrangian. Dimension-six terms have been enumerated in the classical article by Buchmueller and Wyler [3]. Although redundance of some of those operators has been already noted in the literature, no updated complete list has been published to date. Here we perform their classification once again from the outset. Assuming baryon number conservation, we find 15 + 19 + 25 = 59 independent operators (barring flavour structure and Hermitian conjugations), as compared to 16 + 35 + 29 = 80 in Ref.[3]. The three summed numbers refer to operators containing 0, 2 and 4 fermion fields. If the assumption of baryon number conservation is relaxed, 4 new operators arise in the four-fermion sector.