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 Buchmüller 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, 5 new operators arise in the four-fermion sector.

LHCb has reported deviations from the Standard Model in $b\to s{\mu }^{+}{\mu }^{-}$ transitions for which a new neutral gauge boson is a prime candidate for an explanation. As this gauge boson has to couple in a flavor nonuniversal way to muons and electrons in order to explain $R_K$, it is interesting to examine the possibility that also lepton flavor is violated, especially in the light of the CMS excess in $h\to {\tau }^{\pm }{\mu }^{\mp }$. In this article, we investigate the perspectives to discover the lepton-flavor violating modes $B\to K^{(*)}{\tau }^{\pm }{\mu }^{\mp }$, $B_s\to {\tau }^{\pm }{\mu }^{\mp }$ and $B\to K^{(*)}{\mu }^{\pm }{e}^{\mp }$, $B_s\to {\mu }^{\pm }{e}^{\mp }$. For this purpose we consider a simplified model in which new-physics effects originate from an additional neutral gauge boson ($Z^{\prime }$) with generic couplings to quarks and leptons. The constraints from $\tau \to 3\mu$, $\tau \to \mu \nu \overline{\nu }$, $\mu \to e\gamma$, $g_{\mu }-2$, semileptonic $b\to s{\mu }^{+}{\mu }^{-}$ decays, $B\to K^{(*)}\nu \overline{\nu }$ and $B_s$–${\overline{B}}_s$ mixing are examined. From these decays, we determine upper bounds on the decay rates of lepton-flavor violating $B$ decays. $\mathrm{Br}(B\to K\nu \overline{\nu })$ limits the branching ratios of lepton-flavor violating $B$ decays to be smaller than $8\times {10}^{-5}(2\times {10}^{-5})$ for vectorial (left-handed) lepton couplings. However, much stronger bounds can be obtained by a combined analysis of $B_s$–${\overline{B}}_s$, $\tau \to 3\mu$, $\tau \to \mu \nu \overline{\nu }$ and other rare decays. The bounds depend on the amount of fine-tuning among the contributions to $B_s$–${\overline{B}}_s$ mixing. Allowing for a fine-tuning at the percent level we find upper bounds of the order of ${10}^{-6}$ for branching ratios into $\tau \mu$ final states, while $B_s\to {\mu }^{\pm }{e}^{\mp }$ is strongly suppressed and only $B\to K^{(*)}{\mu }^{\pm }{e}^{\mp }$ can be experimentally accessible (with a branching ratio of order ${10}^{-7}$).

We assume that New Physics effects are parametrized within the Standard Model Effective Field Theory (SMEFT) written in a complete basis of gauge invariant operators up to dimension 6, commonly referred to as “Warsaw basis”. We discuss all steps necessary to obtain a consistent transition to the spontaneously broken theory and several other important aspects, including the BRST-invariance of the SMEFT action for linear R_ξ-gauges. The final theory is expressed in a basis characterized by SM-like propagators for all physical and unphysical fields. The effect of the non-renormalizable operators appears explicitly in triple or higher multiplicity vertices. In this mass basis we derive the complete set of Feynman rules, without resorting to any simplifying assumptions such as baryon-, lepton-number or CP conservation. As it turns out, for most SMEFT vertices the expressions are reasonably short, with a noticeable exception of those involving 4, 5 and 6 gluons. We have also supplemented our set of Feynman rules, given in an appendix here, with a publicly available Mathematica code working with the FeynRules package and producing output which can be integrated with other symbolic algebra or numerical codes for automatic SMEFT amplitude calculations.

This paper contains the complete set of Feynman rules for the minimal supersymmetric extension of the standard model. Propagators and vertices are computed in the 't Hooft-Feynman gauge, convenient for perturbative calculations beyond the tree level.