The $Z$ decay width in the SMEFT: $y_t$ and $\lambda$ corrections at one loop

Abstract

We calculate one loop $y_t$ and $\lambda$ dependent corrections to ${\bar{\Gamma }}_Z,{\bar{R}}_f^0$ and the partial $Z$ widths due to dimension six operators in the Standard Model Effective Field Theory (SMEFT), including finite terms. We assume $\mathrm{CP}$ symmetry and a $\mathrm{U}(3{)}^5$ symmetry in the UV matching onto the dimension six operators, dominantly broken by the Standard Model Yukawa matrices. Corrections to these observables are predicted using the input parameters $\{{\hat{\alpha }}_{ew},{\hat{M}}_Z,{\hat{G}}_F,{\hat{m}}_t,{\hat{m}}_h\}$ extracted with one loop corrections in the same limit. We show that at one loop the number of SMEFT parameters contributing to the precise LEPI pseudo-observables exceeds the number of measurements. As a result the SMEFT parameters contributing to LEP data are formally unbounded when the size of loop corrections are reached until other data is considered in a global analysis. The size of these loop effects is generically a correction of order $\sim \%$ to leading effects in the SMEFT, but we find multiple large numerical coefficients in our calculation at this order. We use a $\overline{\mathrm{MS}}$ scheme, modified for the SMEFT, for renormalization. Some subtleties involving novel evanescent scheme dependence present in this result are explained.

38 pages + refs, 11 figures, v3: Acknowledgement updated with grant numbers

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