Master formula for one-loop renormalization of bosonic SMEFT operators

Understand

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.