Dimension-8 SMEFT Matching Conditions for the Low-Energy Effective Field Theory

In particle physics, the modern view is to categorize things in terms of effective field theories (EFTs). Above the weak scale, we have the SMEFT, formed when the heavy new physics (NP) is integrated out, and for which the Standard Model (SM) is the leading part. Below $M_W$, we have the LEFT (low-energy EFT), formed when the heavy SM particles ($W^\pm$, $Z^0$, $H$, $t$) are also integrated out. In order to determine how low-energy measurements depend on the underlying NP, it is necessary to compute the matching conditions of LEFT operators to SMEFT operators. These matching conditions have been worked out for all LEFT operators up to dimension 6 in terms of SMEFT operators up to dimension 6 at the one-loop level. However, this is not sufficient for all low-energy observables. In this paper we present the momentum-independent matching conditions of all such LEFT operators to SMEFT operators up to dimension 8 at tree level.