Abstract
We calculate a set of one-loop corrections to $h\to b\bar b$ and $h\to \tau\bar \tau$ decays in the dimension-6 Standard Model effective field theory (SMEFT). In particular, working in the limit of vanishing gauge couplings, we calculate directly in the broken phase of the theory all large logarithmic corrections and in addition the finite corrections in the large-$m_t$ limit. Moreover, we give exact results for one-loop contributions from four-fermion operators. We obtain these corrections within an extension of the widely used on-shell renormalisation scheme appropriate for SMEFT calculations, and show explicitly how UV divergent bare amplitudes from a total of 21 different SMEFT operators are rendered finite within this scheme. As a by-product of the calculation, we also compute to one-loop order the logarithmically enhanced and finite large-$m_t$ corrections to muon decay in the limit of vanishing gauge couplings, which is necessary to implement the $G_F$ input parameter scheme within the SMEFT.
Highlights
The Standard Model effective field theory (SMEFT) approach is justified as long as the new physics scale ΛNP characteristic of the masses of as yet undiscovered particles is much larger than the electroweak scale, a scenario which seems quite likely given the absence of direct evidence for new particles in the Run-I data
We have calculated a set of one-loop corrections to h → bb and h → τ τdecay rates within the SMEFT
In this procedure counterterms related to wavefunction, mass, and electric charge renormalisation are determined from one-loop two-point functions directly in the broken phase of the theory as in the on-shell scheme used in SM calculations
Summary
We introduce the elements of the SMEFT which are necessary to describe the tree-level h → bb and h → τ τdecay amplitudes. The operators appearing in the Lagrangian are naturally defined in the unbroken phase of the gauge theory, where the vacuum expectation value of the Higgs field vanishes. A complete set of 59 gauge-invariant dimension-6 operators was first established in [31] (a refinement of the over-complete basis originally proposed in [32]), and is listed in table 1. The Wilson coefficients Ci of the dimension-6 operators implicitly contain two inverse powers of ΛNP, and are dimensionful. The labeling convention of the operators appearing in table 1 is applied to the corresponding Wilson coefficient. The Wilson coefficient of the operator QdH is CdH.
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