Dirac masses and mixings in the (geo)SM(EFT) and beyond

Abstract

<jats:title>A<jats:sc>bstract</jats:sc> </jats:title><jats:p>We report a set of exact formulae for computing Dirac masses, mixings, and CP-violation parameter(s) from 3×3 Yukawa matrices <jats:italic>Y</jats:italic> valid when <jats:italic>YY</jats:italic><jats:sup>†</jats:sup> → <jats:italic>U</jats:italic><jats:sup>†</jats:sup><jats:italic>YY</jats:italic><jats:sup>†</jats:sup><jats:italic>U</jats:italic> under global <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathrm{U}{(3)}_{Q_L} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>U</mml:mi> <mml:msub> <mml:mfenced> <mml:mn>3</mml:mn> </mml:mfenced> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> flavour symmetry transformations <jats:italic>U</jats:italic>. 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \sqrt{2\left\langle {H}^{\dagger }H\right\rangle } $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mrow> <mml:mn>2</mml:mn> <mml:mfenced> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>†</mml:mo> </mml:msup> <mml:mi>H</mml:mi> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:msqrt> </mml:math></jats:alternatives></jats:inline-formula><jats:italic>/</jats:italic>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.</jats:p>