Master formula for one-loop renormalization of bosonic SMEFT operators

Using background-field method and super-heat-kernel expansion, we derive a master formula for the one-loop UV divergences of the bosonic dimension-6 operators in Standard Model Effective Field Theory (SMEFT). This approach reduces the calculation of all the UV divergences to algebraic manipulations. Using this formula we corroborate results in the literature for the one-loop anomalous dimension matrix of SMEFT obtained via diagrammatic methods, considering contributions from the operators $X^3, \phi^6, \phi^4 D^2, X^2 \phi^2$ of the Warsaw basis. The formula is derived in a general way and can be applied to other quantum field theories as well.