Electroweak precision observables in a fourth generation model with general flavour structure

Abstract

We calculate the contributions to electroweak precision observables (EWPOs) due to a fourth generation of fermions with the most general (quark-) flavour structure (but assuming Dirac neutrinos and a trivial flavour structure in the lepton sector). The new-physics contributions to the EWPOs are calculated at one-loop order using automated tools (FeynArts/FormCalc). No further approximations are made in our calculation. We discuss the size of non-oblique contributions arising from Z–quark–anti-quark vertex corrections and the dependence of the EWPOs on all CKM mixing angles involving the fourth generation. We find that the electroweak precision observables are sensitive to two of the fourth-generation mixing angles and that the corresponding constraints on these angles are competitive with those obtained from flavour physics. For non-trivial 4×4 flavour structures, the non-oblique contributions lead to relative corrections of several permille and should be included in a global fit.

Appendix A: Parametrisation of the CKM matrix

The CKM matrix of the SM3 depends on three mixing angles θ ₁₂, θ ₁₃ and θ ₂₃ and one complex phase δ ₁₃. Its elements are given by

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where c _ij =cosθ _ij and s _ij =sinθ _ij . The SM4 CKM matrix is parametrised by three additional mixing angles θ ₁₄, θ ₂₄ and θ ₃₄ and two additional phases δ ₁₄ and δ ₂₄. In terms of these parameters, it is then written as

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with

$$ \begin{array}{@{}ll} \multicolumn{2}{@{}l}{V_{ud} = c_{12} c_{13} c_{14},\qquad V_{us} = c_{13} c_{14} s_{12},} \\ \noalign {\vspace {5pt}} \multicolumn{2}{@{}l}{V_{ub} = c_{14} s_{13}e^{-i\delta_{13}},\qquad V_{ub'} = s_{14}e^{-i\delta_{14}},} \\ \noalign {\vspace {5pt}} \multicolumn{2}{@{}l}{V_{cb'} = c_{14} s_{24}e^{-i\delta_{24}},\qquad V_{tb'} = c_{14} c_{24} s_{34},} \\ \noalign {\vspace {5pt}} \multicolumn{2}{@{}l}{V_{t'b'} = c_{14} c_{24} c_{34},} \\ \noalign {\vspace {5pt}} V_{cd} =& -c_{23} c_{24} s_{12}\\ \noalign {\vspace {5pt}} &{} + c_{12} \bigl(-c_{24} s_{13} s_{23} e^{i\delta_{13}} - c_{13} s_{14} s_{24}e^{i(\delta_{14}-\delta_{24})} \bigr), \\ \noalign {\vspace {5pt}} V_{cs} =& c_{12} c_{23} c_{24} \\ \noalign {\vspace {5pt}} & {}+ s_{12} \bigl(-c_{24} s_{13} s_{23} e^{i\delta_{13}} - c_{13} s_{14} s_{24}e^{i(\delta_{14}-\delta_{24})}\bigr), \\ \noalign {\vspace {5pt}} V_{cb} =& c_{13} c_{24} s_{23} -s_{13} s_{14} s_{24}e^{i(\delta_{14}-\delta_{13}-\delta_{24})}, \\ \noalign {\vspace {5pt}} V_{td} =& -s_{12} \bigl(-c_{34} s_{23} - c_{23} s_{24} s_{34} e^{i\delta_{24}} \bigr) \\ \noalign {\vspace {5pt}} &{} + c_{12} \bigl(-c_{13} c_{24} s_{14} s_{34} e^{i\delta_{14}}\\ \noalign {\vspace {5pt}} &{} -s_{13}e^{i\delta_{13}} \bigl(c_{23} c_{34} - s_{23} s_{24} s_{34} e^{i\delta_{24}} \bigr) \bigr), \\ \noalign {\vspace {5pt}} V_{ts} =& c_{12} \bigl(-c_{34} s_{23} -c_{23} s_{24} s_{34} e^{i\delta_{24}} \bigr) \\ \noalign {\vspace {5pt}} &{} + s_{12} \bigl(-c_{13} c_{24} s_{14} s_{34} e^{i\delta_{14}} \\ \noalign {\vspace {5pt}} &{}- s_{13}e^{i\delta_{13}} \bigl(c_{23} c_{34} - s_{23} s_{24} s_{34} e^{i\delta_{24}} \bigr) \bigr), \\ \noalign {\vspace {5pt}} V_{tb} =& -c_{24} s_{13} s_{14} s_{34} e^{i(\delta_{14}-\delta_{13})}\\ \noalign {\vspace {5pt}} &{} + c_{13} \bigl(c_{23} c_{34} - s_{23} s_{24} s_{34} e^{i\delta_{24}} \bigr), \\ \noalign {\vspace {5pt}} V_{t'd} =& -s_{12} \bigl(-c_{23} c_{34} s_{24}e^{i\delta_{24}} + s_{23} s_{34} \bigr) \\ \noalign {\vspace {5pt}} &{} + c_{12} \bigl(-c_{13} c_{24} c_{34} s_{14}e^{i\delta_{14}}\\ \noalign {\vspace {5pt}} &{}- s_{13}e^{i\delta_{13}} \bigl(-c_{34} s_{23} s_{24}e^{i\delta_{24}} - c_{23} s_{34} \bigr) \bigr), \\ \noalign {\vspace {5pt}} V_{t's} =& c_{12} \bigl(-c_{23} c_{34} s_{24}e^{i\delta_{24}} + s_{23} s_{34} \bigr) \\ \noalign {\vspace {5pt}} &{} + s_{12} \bigl(-c_{13} c_{24} c_{34} s_{14}e^{i\delta_{14}} \\ \noalign {\vspace {5pt}} &{}- s_{13}e^{i\delta_{13}} \bigl(-c_{34} s_{23} s_{24}e^{i\delta_{24}} -c_{23} s_{34} \bigr) \bigr), \\ \noalign {\vspace {5pt}} V_{t'b} =& -c_{24} c_{34} s_{13} s_{14} e^{i(\delta_{14}-\delta_{13})}\\ \noalign {\vspace {5pt}} &{} + c_{13} \bigl(-c_{34} s_{23} s_{24}e^{i\delta_{24}} - c_{23} s_{34} \bigr). \end{array} $$

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For θ ₁₄=θ ₂₄=θ ₃₄=δ ₁₄=δ ₂₄=0 the SM4 CKM matrix assumes block-diagonal form with the SM3 CKM matrix in the first 3×3 block:

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