Abstract
We report a set of exact formulae for computing Dirac masses, mixings, and CP-violation parameter(s) from 3×3 Yukawa matrices Y valid when YY† → U†YY†U under global mathrm{U}{(3)}_{Q_L} flavour symmetry transformations U. The results apply to the Standard Model Effective Field Theory (SMEFT) and its ‘geometric’ realization (geoSMEFT). We thereby complete, in the Dirac flavour sector, the catalogue of geoSMEFT parameters derived at all orders in the sqrt{2leftlangle {H}^{dagger }Hrightrangle } /Λ expansion. The formalism is basis-independent, and can be applied to models with decoupled ultraviolet flavour dynamics, as well as to models whose infrared dynamics are not minimally flavour violating. We highlight these points with explicit examples and, as a further demonstration of the formalism’s utility, we derive expressions for the renormalization group flow of quark masses, mixings, and CP-violation at all mass dimension and perturbative loop orders in the (geo)SM(EFT) and beyond.
Highlights
(or even explain) this distinct flavour structure
8Note that we do not present these models in the context of a matching calculation to theSMEFT, but rather as example UV constructions with BSM field and symmetry content remaining in the spectrum — our goal is to show that our model-independent formulae from above can be used to study the Dirac flavour sector of specific models in the literature, and as a result we only give relevant details to that end below
The practical implementation of these latter formulae requires knowledge of the RGE of the 11 flavour invariants that exclusively compose them, which has been given at one- and two-loop perturbative orders for quark flavour invariants in minimally flavour-violating (MFV) theories [23]
Summary
Our goal is to extend the definition of all-orders parameters in the geoSMEFT to the fermion mass and mixing sector, i.e. those quantities defined in the fermion mass-eigenstate basis, where the different generations of a given fermion family (e.g. u, c, t) can be physically distinguished by their distinct masses. Our goal is to derive expressions for the Yukawa eigenvalues yi and CKM mixing parameters θij and δ, in terms of Yrp alone, such that these Lagrangian parameters are defined at all orders in the vT /Λ expansion characterizing the geoSMEFT. To this end we first consider the two-generation case, where one can expand the flavour indices of Y, finding. When three fermion generations are present, obtaining analogous expressions for yi, θij, and δ becomes intractable if approached with standard diagonalization techniques as above, regardless of concerns over the basis-dependence of the results
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