SMEFT ATLAS of $\Delta F=2$ Transitions

Abstract

We present a model-independent anatomy of the $\Delta F=2$ transitions $K^0-{\bar{K}}^0$, $B_{s,d}-{\bar{B}}_{s,d}$ and $D^0-{\bar{D}}^0$ in the context of the Standard Model Effective Field Theory (SMEFT). We present two master formulae for the mixing amplitude $[M_{12}{]}_{\text{BSM}}$. One in terms of the Wilson coefficients (WCs) of the Low-Energy Effective Theory (LEFT) operators evaluated at the electroweak scale ${\mu }_{\text{ew}}$ and one in terms of the WCs of the SMEFT operators evaluated at the BSM scale $\Lambda$. The coefficients $P_a^{ij}$ entering these formulae contain all the information below the scales ${\mu }_{\text{ew}}$ and $\Lambda$, respectively. Renormalization group effects from the top-quark Yukawa coupling play the most important role. The collection of the individual contributions of the SMEFT operators to $[M_{12}{]}_{\text{BSM}}$ can be considered as the SMEFT ATLAS of $\Delta F=2$ transitions and constitutes a travel guide to such transitions far beyond the scales explored by the LHC. We emphasize that this ATLAS depends on whether the down-basis or the up-basis for SMEFT operators is considered. We illustrate this technology with tree-level exchanges of heavy gauge bosons ($Z^{\prime }$, $G^{\prime }$) and corresponding heavy scalars.

Comments: 76 pages, 13 figures