We study the patterns of flavour violation in renormalisable extensions
of the Standard Model (SM) that contain vector-like quarks (VLQs)
in a single complex representation of either the SM gauge group

Article funded by SCOAP3

g_Z \approx 0.75$, such that for example $r' \approx 3$ can be reached with $g' X \approx 1.1$, still within the perturbative regime. The couplings of $T_d$ and $T_u$ differ just by a sign and factors 1/2. In distinction to $Z$-contributions in $\GSM$-models, both $Z$- and $Z'$-contributions in $\GSMUpr(\Phi)$ models decouple with large $\tan\beta$, see $K^{ij}$ in eq.~\eqref{eq:def:Gij-Kij}. \begin{table} \renewcommand{\arraystretch}{1.3} \centering{ \resizebox{\textwidth}{!}{ \begin{tabular}{|c|c|c|c|c|c|} \hline Model & $q$ & $\Delta^{q_i q_j}_L(Z^\prime)$ & $\Delta^{q_i q_j}_R(Z^\prime)$ & $\Delta^{q_i q_j}_L(Z)$ & $\Delta^{q_i q_j}_R(Z)$ \\ \hline \multicolumn{6}{|c|}{$\GSMUpr(S)$} \\ \hline $D$ & $d$ & $0$ & $(G^{ij})^*$ & $0$ & $0$ \\ \cline{2-6} \multirow{2}{*}{$Q_{V}$} & $d$ & $G^{ij}$ & $0$ & $0$ & $0$ \\ & $u$ & $V_{im}\, G^{mn}\, (V^{\dag})_{nj}$ & $0$ & $0$ & $0$ \\ \hline \multicolumn{6}{|c|}{$\GSMUpr(\Phi)$} \\ \hline $D$ & $d$ & $-r' K^{ij}$ & $0$ & $\left[1 - r' \xi_{ZZ'} \right] K^{ij}$ & $0$ \\ \cline{2-6} $Q_d$ & $d$ & $0$ & $-r' (K^{ij})^*$ & $0$ & $\left[1 - r' \xi_{ZZ'} \right](K^{ij})^*$ \\ \cline{2-6} \multirow{2}{*}{$T_d$} & $d$ & $-r' {K^{ij}}/{2}$ & $0$ & $ \left[1 - r' \xi_{ZZ'} \right] {K^{ij}}/{2}$ & $0$ \\ & $u$ & $-r' \, V_{im} K^{mn} (V^{\dag})_{nj}$ & $0$ & $ \left[1 - r' \xi_{ZZ'} \right] V_{im} K^{mn} (V^{\dag})_{nj}$ & $0$ \\ \cline{2-6} \multirow{2}{*}{$T_u$} & $d$ & $r' K^{ij}$ & $0$ & $-\left[1 - r' \xi_{ZZ'} \right] K^{ij} $ & $0$ \\ & $u$ & $r' V_{im} K^{mn}(V^{\dag})_{nj}/2$ & $0$ & $-\left[1 - r' \xi_{ZZ'} \right] V_{im} K^{mn}(V^{\dag})_{nj}/2$ & $0$ \\ \hline \end{tabular} } } \renewcommand{\arraystretch}{1.0} ]]>

dvv} and appendix~\ref{app:d->dll}, respectively. All Wilson coefficients in this section are formally at $\mu_{\rm EW}$, but since the corresponding operators are conserved currents under QCD, the RG evolution to the scale $\mu_b$ is trivial in all cases.\footnote{The usual mixing of $Q_9$ operators with current-current operators $Q_{1,2}$ present in the SM and affecting $C_9$ coefficient is fully negligible here because NP contributions to $C_{1,2}$ are tiny in all models.} The $V=Z,Z'$ contributions modify the Wilson coefficients and one-loop functions \begin{align} \label{eq:X_LR} C_{L\,(R)}^{ij,\nu} & = - \sum_{V} \frac{X_{L\,(R)}^{ij,\nu}(V)}{s_W^2}, & X_{L\,(R)}^{ij,\nu} (V) & = \frac{\Delta_L^{\nu\bar\nu}(V)}{g^2_{\rm SM} M_{V}^2} \frac{\Delta_{L\,(R)}^{ij}(V)}{\lambda_{ij}^{(t)}}, \end{align} which enter the expressions for $d_j\to d_i \bar\nu\nu$ decays like $\kpn$, $\klpn$ and also $B\to K^{(*)}\nu\bar\nu$ with more details in appendix~\ref{app:d->dvv}. The Wilson coefficients of the operators entering the $d_j\to d_i \ell\bar\ell$ transitions receive the following contributions \begin{align} \label{eq:DF1:bsll-C9} C_{9\,(9')}^{ij,\ell} & = - \sum_V \frac{\big[\Delta_R^{\ell\bar\ell}(V) + \Delta_L^{\ell\bar\ell}(V)\big]} {s_W^2 g_{\text{SM}}^2 M_{V}^2} \frac{\Delta_{L\,(R)}^{ij}(V)} {\lambda_{ij}^{(t)}} , \\ \label{eq:DF1:bsll-C10} C_{10\,(10')}^{ij,\ell} & = - \sum_V \frac{\big[\Delta_R^{\ell\bar\ell}(V) - \Delta_L^{\ell\bar\ell}(V)\big]} {s_W^2 g_{\text{SM}}^2 M_{V}^2} \frac{\Delta_{L\,(R)}^{ij}(V)}{\lambda_{ij}^{(t)}}, \end{align} where the leptonic $Z$ couplings are taken to be the ones of the SM except for $\GSMUpr(\Phi)$ models, where $Z-Z'$ mixing is included following~\eqref{eq:Z-lepton-coupl}. There are no $Z'$ contributions to $C_{10\,(10')}$, as the lepton couplings are vectorial, see~\eqref{eq:Zpr-lepton-coupl}. The purely leptonic decay $K_L \to \mu\bar\mu$ is described by ($\bar{s} \to \bar{d}$) \begin{align} \label{eq:YAK} Y_{\rm A}(K) & = Y_L^{\rm SM} + \frac{\left[\Delta_R^{\mu\bar\mu}(Z) - \Delta_L^{\mu\bar\mu}(Z) \right]} {g_\text{SM}^2 M_{Z}^2} \left[\frac{\Delta_L^{sd}(Z) - \Delta_R^{sd}(Z)}{\lambda_{sd}^{(t)}}\right] , \end{align} with $Y_L^{\rm SM} = 0.942$~\cite{Bobeth-ml-2013tba}. ]]>

dqq}. Since the flavour-diagonal $Z$ couplings are given by the SM ones to the order we are working in, no dependence on $q$ arises. The non-vanishing contributions to the $|\Delta F| = 1$ Wilson coefficients are conveniently rewritten as NP contributions to the Inami-Lim $Z$-penguin function $C$ (see appendices~\ref{app:d->dvv} and~\ref{app:d->dll})\footnote{Note that whereas the SM contribution to the function $C$ is gauge dependent this shift is gauge independent.} \begin{align} C_{L(R)}^{ij} & = - \frac{g_Z}{2 g_{\rm SM}^2 M_Z^2} \frac{\Delta_{L(R)}^{ij}(Z)}{\lambda_{ij}^{(u)}}. \end{align} It contributes at the scale $\mu_{\rm EW}$ to the Wilson coefficients of the QCD- and EW-penguin operators~\cite{Buras-ml-2015jaq}, \begin{align} \label{eq:ddqq-Z-contr} C_{3(5')}^{ij} & = \frac{\alpha}{6\pi}\frac{C_{L(R)}^{ij}}{s_W^2}, & C_{7(9')}^{ij} & = \frac{\alpha}{6\pi} 4 C_{L(R)}^{ij}, & C_{9(7')}^{ij} & = - \frac{\alpha}{6\pi} \frac{c_W^2}{s_W^2} 4 C_{L(R)}^{ij}\,. & \end{align} The RG evolution induces also non-vanishing contributions for the remaining QCD- and EW-penguin operators at lower scales relevant for Kaon and $B$-meson decays. Here we are mainly interested in CP violation in the Kaon sector, especially $\epe$. \looseness=-1 It is known from various analyses of $\epe$, see~\cite{Buras-ml-2015jaq} and references therein, that NP has to generate contributions to the Wilson coefficients of $O_8 \propto (V-A) \otimes (V+A)$ or $O'_{8} \propto (V+A) \otimes (V-A)$ operators at the low energy scale in order to be able to modify significantly the SM predictions. This requires the presence of both LH flavour-violating couplings and RH flavour-diagonal couplings of $Z$ or $Z^\prime$ in the case of $O_8$, or RH flavour-violating couplings and LH flavour-diagonal couplings in the case of $O_{8'}$. But in the models considered quark couplings of the $Z^\prime$ are either LH \emph{or} RH, hence such contributions can only be generated as a higher-order effect. Given that $(V-A)$ and $(V+A)$ flavour-diagonal $Z$ couplings to SM quarks are always present, tree-level $Z$ exchanges fully dominate. NP contributions to $O_{9,10} \propto (V-A) \otimes (V-A)$ or $O'_{9,10} \propto(V+A) \otimes (V+A)$ operators are negligible due to their suppressed hadronic matrix elements relative to the ones of $O_8$ and $O'_8$. This can be clearly seen in the semi-numeric expression~\eqref{eq:epe-seminum} for $\epe$, where the coefficients of $C^{(\prime)}_7$, which mixes into $C^{(\prime)}_8$, is largely enhanced w.r.t.\ all others. Whether $O_8$ or $O'_8$ is generated depends on whether a given model has $(V-A)$ or $(V+A)$ flavour-violating couplings: \begin{itemize} \item Within the $\GSM$- and $\GSMUpr(\Phi)$-models, the pattern of NP contributions to $\epe$ is as follows \begin{equation} \begin{aligned} \mbox{singlets} & : & D & & & \to & (O_8)\,, \\ \label{QVQd} \mbox{doublets} & : & Q_V, & & Q_d & \to & (O'_8)\,, \\ \mbox{triplets} & : & T_d, & & T_u & \to & (O_8)\,. \end{aligned} \end{equation} \item In $\GSMUpr(S)$-models $\epe$ remains SM-like, which could become problematic as we discuss briefly below. \end{itemize} Tree-level $Z$ contributions to $\epe$ have been recently considered in detail in ref.~\cite{Buras-ml-2015jaq}, where explicit expressions for the relevant hadronic matrix elements $\langle Q_8(m_c)\rangle_2$ and $\langle Q^\prime_8(m_c)\rangle_2$ can be found. Whereas these matrix elements differ only by sign from each other, their Wilson coefficients differ also in magnitude, the one of $Q^\prime_8$ being larger by a factor of $c_W^2/s_W^2=3.33$. This can also be seen in eq.~\eqref{eq:ddqq-Z-contr}, remembering that the Wilson coefficients of $Q_8$ and $Q^\prime_8$ at $\mu=m_c$ are directly related to the Wilson coefficients of $Q_7$ and $Q^\prime_7$ at $\mu_{\rm EW}$, respectively. Finally let us mention that the top-Yukawa generated 1stLLA contributions to $|\Delta F|=1$ operators in $\GSM$ models discussed in section~\ref{sec:GSM-RGE} induce operators with the same chiral structure as already present from the $Z$-exchange due to $\psi^2 H^2 D$ operators. In particular the $\psi^2 H^2 D$ Wilson coefficients generate $\psi^4$ Wilson coefficients via the mixing given in~\eqref{eq:ADM:qq1}--\eqref{eq:RGE:ud1} \begin{align} \Wc[(1,3)]{Hq} & \to \Wc[(1,3)]{qq},\; \Wc[(1)]{qu} \\ \Wc{Hd} & \to \Wc[(1)]{qd},\; \Wc[(1)]{ud} \end{align} where ${\cal O}^{(1,3)}_{qq} \sim (V-A) \otimes (V-A)$, ${\cal O}^{(1)}_{qu,qd} \sim (V-A) \otimes (V+A)$ and ${\cal O}^{(1)}_{ud} \sim (V+A) \otimes (V+A)$. Given their additional suppression w.r.t.\ existing contributions we do not consider these contributions further. The status of $\epe$ in the SM can be summarized as follows. The RBC-UKQCD lattice collaboration calculating hadronic matrix elements of all operators, but not including isospin-breaking effects, finds~\cite{Blum-ml-2015ywa, Bai-ml-2015nea} \begin{align} \label{eq:epe:LATTICE} (\epe)_\text{SM} & = (1.38 \pm 6.90) \times 10^{-4}\qquad {\rm (RBC-UKQCD)}. \end{align} Using the hadronic matrix elements of QCD- and EW-penguin $(V-A)\otimes (V+A)$ operators from RBC-UKQCD lattice collaboration~\cite{Blum-ml-2015ywa, Bai-ml-2015nea} but extracting the matrix elements of $(V-A)\otimes (V-A)$-penguin operators from the CP-conserving $K\to\pi\pi$ amplitudes and including isospin breaking effects, one finds~\cite{Buras-ml-2015yba} \begin{align} \label{eq:epe:LBGJJ} (\epe)_\text{SM} & = (1.9 \pm 4.5) \times 10^{-4}\qquad {\rm (BGJJ)}\,. \end{align} This result differs by $2.9\,\sigma$ from the experimental world average from the NA48~\cite{Batley-ml-2002gn} and KTeV~\cite{AlaviHarati-ml-2002ye, Abouzaid-ml-2010ny} collaborations, \begin{align} \label{eq:epe:EXP} (\epe)_\text{exp} & = (16.6 \pm 2.3) \times 10^{-4} , \end{align} suggesting that models providing enhancement of $\epe$ are favoured. A new analysis in ref.~\cite{Kitahara-ml-2016nld} confirms these findings \begin{align} \label{KNT} (\epe)_\text{SM} & = (1.1 \pm 5.1) \times 10^{-4}\qquad {\rm (KNT)}\,. \end{align} These results are supported by upper bounds on the matrix elements of the dominant penguin operators from the large-$N_c$ dual-QCD approach~\cite{Buras-ml-2015xba, Buras-ml-2016fys}, which allows to derive an upper bound on $\epe$~\cite{Buras-ml-2015yba}, \begin{align} \label{eq:epe:BoundBGJJ} (\epe)_\text{SM} \le (8.6 \pm 3.2) \times 10^{-4} , \end{align} still $2\,\sigma$ below the experimental data. In particular it has been demonstrated in ref.~\cite{Buras-ml-2016fys} that final state interactions are much less relevant for $\epe$ than previously claimed in refs.~\cite{Antonelli-ml-1995gw, Bertolini-ml-1995tp, Frere-ml-1991db, Pallante-ml-1999qf, Pallante-ml-2000hk, Buchler-ml-2001np, Buchler-ml-2001nm, Pallante-ml-2001he}. These findings diminish significantly hopes that improved lattice QCD calculations will be able to bring the SM prediction for $\epe$ to agree with the experimental data in~\eqref{eq:epe:EXP}, motivating additionally to search for NP models capable of alleviating this tension. In fact it has been demonstrated that in general models with flavour-changing $Z$ and $Z^\prime$ exchanges~\cite{Buras-ml-2015yca, Buras-ml-2015jaq}, in the Littlest Higgs model with $T$-parity~\cite{Blanke-ml-2015wba}, 331 models~\cite{Buras-ml-2015kwd, Buras-ml-2016dxz} and supersymmetric models~\cite{Tanimoto-ml-2016yfy, Kitahara-ml-2016otd, Endo-ml-2016aws} agreement with the data for $\epe$ can be obtained, with interesting implications for other flavour observables. We will see in section~\ref{sec:numerics} that also in VLQ models large NP contributions to $\epe$ are possible, such that agreement with the data in~\eqref{eq:epe:EXP} can be obtained with a significant impact not only on rare $K$ decays but also $B$ decays. ]]>

dvv}. \end{itemize} Considering next $\GSM$ and $\GSMUpr(\Phi)$ models in which tree-level $Z$ contributions to $\Delta F=1$ processes dominate, the most notable feature comes from the tree-level decoupling of the VLQs depicted in figure~\ref{fig:tree-decoupl-2}, which implies a relationship between the flavour-changing $Z$ and $Z'$ couplings in these models, again owing to the equality of~\eqref{eq:Yuk:H} and~\eqref{eq:Yuk:Phi} upon $H \leftrightarrow \Phi$. Below the scale $\mu_M$ in both models a $\psi^2 \varphi^2 D$ operator is generated, with the same Wilson coefficient, where $\varphi = H,\Phi$ in $\GSM$ and $\GSMUpr(\Phi)$ models, respectively. The covariant derivative is the same in both models, up to the additional $\UonePr$ part in $\GSMUpr(\Phi)$ models. Upon spontaneous symmetry breaking at the scale $\mu_{\rm EW}$, this operator becomes $\propto v^2$ in $\GSM$ models and $\propto v_1^2 = c^2_\beta\, v^2$ in $\GSMUpr(\Phi)$ models. Consequently, in $\GSMUpr(\Phi)$ models all $Z$ and $Z'$ couplings $\propto c^2_\beta \Delta^{ij}$ are suppressed by $c^2_\beta = (1 + \tan^2\beta)^{-1}$ w.r.t.\ $Z$ couplings $\propto \Delta^{ij}$ in $\GSM$ models, see~\eqref{eq:def:Gij-Kij},~\eqref{eq:Dij} and table~\ref{tab:GSMP}. Note that the additional modifications from $Z-Z'$ mixing in $\GSMUpr(\Phi)$ models do not affect the dependence on the $\lambda_i^{\rm VLQ}$. The suppression by $c^2_\beta$ can be only softened by going to very small $\tan\beta$. In order to guarantee perturbativity of the top-quark Yukawa coupling $0.3 \lesssim \tan \beta$~\cite{Branco-ml-2011iw}. In appendix~\ref{app:scalar:S+H+Phi} we discuss further constraints on $\tan\beta$ in $\GSMUpr(\Phi)$ models from the measured $Z$ mass and partial widths to leptons, which for $M_Z < M_{Z'}$ allow at most $2 \lesssim \tan\beta$, i.e.\ $c^2_\beta \lesssim 0.2$. Depending on the choice of $g'$ and $v_S$, this bound becomes even stronger. Therefore, VLQ effects in $|\Delta F|=1$ FCNC processes are generically suppressed in $\GSMUpr(\Phi)$ models w.r.t.\ $\GSM$ models. As an example one might consider the Wilson coefficient $C_9^{ij}$ given in~\eqref{eq:DF1:bsll-C9}, governing $d_j \to d_i \ell\bar\ell$. The suppression factor in $\GSMUpr(\Phi)$ versus $\GSM$ models is \begin{align} \frac{(C_9^{ij})_{\GSMUpr(\Phi)}}{(C_9^{ij})_{\GSM}} & = c^2_\beta \left[ 1 - r' \xi_{ZZ'} - \frac{g'}{g_Z} \frac{4 Q'_\ell}{(1 - 4 s_W^2)} \xi_{ZZ'} - \frac{g'}{g_Z} \frac{4 Q'_\ell}{(1 - 4 s_W^2)}\frac{M_Z^2}{M_{Z'}^2}\right] . \end{align} \looseness=-1 The mixing angle $\xi_{ZZ'} \sim M_Z^2/M_{Z'}^2$ is small in most of the parameter space, such that $(1 - 4 s_W^2)^{-1} \sim 10$ is overcompensated. The comparison of the first three terms with the last one in the brackets also shows the relative size of the $Z'$ to $Z$ contribution in $\GSMUpr(\Phi)$ models, which is also suppressed by $M_Z^2/M_{Z'}^2$. Consequently VLQ contributions to semileptonic $|\Delta F|=1$ FCNC decays are in most cases suppressed in $\GSMUpr(\Phi)$ w.r.t.\ $\GSM$ models. However, there are exceptions related to the fact that with the parametric suppression of the $Z$ and $Z^\prime$ couplings, the values of Yukawa couplings are weaker constrained by $\Delta F=1$ transitions than in $\GSM$ models and the constraints on Yukawas are governed this time by $\Delta F=2$ processes. A detailed numerical analysis in the next section then shows that the allowed NP effects in $\Delta M_K$ are in fact significantly larger than in $\GSM$ models. For a given flavour-changing transition the correlations between different $|\Delta F|=1$ observables depend on whether $Z^{(\prime)}$ have LH or RH flavour-violating quark couplings and the size of the corresponding leptonic $Z^{(\prime)}$ couplings. A summary is given in table~\ref{tab:DNA:bsll-WC}, where in addition to $\GSM$ and $\GSMUpr(\Phi)$ models we include $\GSMUpr(S)$ models discussed already above. The generically small NP contributions in $C_9^{(\prime) ij,\ell}$ compared to $C_{10}^{(\prime)ij, \ell}$ and $C_{L(R)}^{ij,\nu}$ in $\GSM$ models are due to the smallness of leptonic vector $Z$ couplings relative to the axial-vector ones. The additional generic suppression of NP effects in $\GSMUpr(\Phi)$ w.r.t.\ $\GSM$ is due to the aforementioned suppression by $c^2_\beta$. We observe that in $\GSM$ models significant NP effects in $\kpn$, $\klpn$, $B_{s,d}\to \mu\bar\mu$, $B\to K^{(*)} \mu\bar\mu$ and $B\to K^{(*)}\nu\bar\nu$ are possible, but the LHCb anomalies in angular observables in $B\to K^* \mu\bar\mu$ cannot be explained in these models because the vector coupling of $Z$ to muons is suppressed by $(1 - 4 s_W^2) \sim 0.1$ w.r.t.\ the axial-vector coupling of the $Z$. LFU of $Z$ couplings precludes also the explanation of the violation of this universality in $R_K$, hinted at by LHCb data. \begin{table} \addtolength{\arraycolsep}{4pt} \renewcommand{\arraystretch}{1.5} \centering \begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|} \hline & \multicolumn{5}{|c|}{$\GSM$} & \multicolumn{2}{c|}{$\GSMUpr(S)$} & \multicolumn{4}{c|}{$\GSMUpr(\Phi)$} \\ & $D$ & $Q_V$ & $Q_d$ & $T_d$ & $T_u$ & $D$ & $Q_V$ & $D$ & $Q_d$ & $T_d$ & $T_u$ \\ \hline $C_9^{ij,\ell}$ & \red & --- & --- & \red & \red & --- & \green & \sred & --- & \sred & \sred \\ \hline $C_9^{\prime\, ij,\ell}$ & --- & \red & \red & --- & --- & \green & --- & --- & \sred & --- & --- \\ \hline $C_{10}^{ij,\ell}$ & \green & --- & --- & \green & \green & --- & --- & \sgreen & --- & \sgreen & \sgreen \\ \hline $C_{10}^{\prime\, ij,\ell}$ & --- & \green & \green & --- & --- & --- & --- & --- & \sgreen & --- & --- \\ \hline $C_L^{ij,\nu}$ & \green & --- & --- & \green & \green & --- & \green & \sgreen & --- & \sgreen & \sgreen \\ \hline $C_R^{ij,\nu}$ & --- & \green & \green & --- & --- & \green & --- & --- & \sgreen & --- & --- \\ \hline \end{tabular} \renewcommand{\arraystretch}{1.0} \setlength{\tabcolsep}{2pt} ]]>

svv:obs} {\cal R}_{B\to K^{(*)} \nu\bar\nu} & = \frac{Br(B\to K^{(*)} \nu\bar\nu)}{Br(B\to K^{(*)} \nu\bar\nu)_{\rm SM}}, & {\cal R}_{F_L} & = \frac{F_L(B\to K^* \nu\bar\nu)}{F_L(B\to K^* \nu\bar\nu)_{\rm SM}} \end{align} via~\cite{Buras-ml-2014fpa} \begin{align} {\cal R}_{B\to K \nu\bar\nu} & = (1-2\eta)\epsilon^2, & {\cal R}_{B\to K^* \nu\bar\nu} & = (1+\kappa_\eta\eta)\epsilon^2, & {\cal R}_{F_L} & = \frac{1+2\eta}{1+\kappa_\eta \eta} , \end{align} where $\kappa_\eta$ is form-factor dependent and given in ref.~\cite{Buras-ml-2014fpa}. The Belle~II experiment is expected to measure these branching ratios with 30\% uncertainty~\cite{Aushev-ml-2010bq} if they are of the size as predicted in the SM. In RH scenarios large VLQ effects are excluded due to the strong complementarity of the $|\Delta F|=1$ constraints from $Br(B_s\to \mu\bar\mu)$ and $Br(B^+\to K^+ \mu\bar\mu)$ as mentioned above. $\epsilon$ has to be larger than one in these cases. The VLQ effects for $M_{\rm VLQ} = 1$\,TeV can lead to a rather large suppression in LH scenarios for $\epsilon$ while $\eta = 0$, leading to maximally correlated ${\cal R}_{B\to K^{(*)} \nu\bar\nu}$. The suppression is smaller for $M_{\rm VLQ} = 10$\,TeV, whereas ${\cal R}_{F_L} = 1$. The correlation plot is shown in figure~\ref{fig:GSM-crr-bs-2}. It will be challenging to distinguish the small deviations from SM predictions in RH scenarios; however, large (suppression) effects are possible and LH and RH scenarios are well distinguishable. A measurement of $\epsilon$ significantly larger than one would challenge all $\GSM$ scenarios with a single VLQ representation. \end{itemize} \clearpage \begin{figure} \centering \includegraphics[width=0.46\textwidth]{pdf/v2_epsetabsnunu_GSM_QV.pdf} { ]]>

svv:obs} are based on formulae given in ref.~\cite{Buras-ml-2014fpa}. These expressions account for the lepton-non-universal contribution of VLQ's w.r.t.\ the neutrino flavour in $\GSMUpr$ models. However, the particular structure of the gauged $\UonePr$~\eqref{eq:UonePr-lep-charges} leads to a cancellation of the numerically leading interference contributions of the SM and new physics~\cite{Altmannshofer-ml-2014cfa}. The $Br(\kpn)$ receives in the SM the numerically leading contribution from the ``top''-sector, when decoupling heavy degrees of freedom at $\mu_{\rm EW}$, which yields directly the local ${\cal O}_L^{sd,\nu}$ operator $(\nu = e, \mu, \tau)$. Further, a non-negligible ``charm''-sector arises from double-insertions of hadronic and semi-leptonic $|\Delta S| = 1$ operators when decoupling the charm quark at $\mu_c \sim m_c$, which is enhanced due to the strong CKM hierarchy $(\lambda_{sd}^{(t)} \propto \lambda^5) \; \ll \; (\lambda_{sd}^{(c)} \propto \lambda^2)$, where $\lambda = |V_{us}|$ is the Cabibbo angle. This is usually expressed in the effective Hamiltonian of the SM as~\cite{Buras-ml-2006gb} \begin{align} {\cal H}_{\rm eff} & = {\cal N} \sum_\nu \left[\lambda_{sd}^{(c)} X_{c}^\nu + \lambda_{sd}^{(t)} X_L^{\rm SM} \right] {\cal O}_L^{sd,\nu} , \end{align} with ${\cal N} = G_F \alpha_{e}/(2 \sqrt{2}\pi s_W^2)$, where $X_{c}^e = X_{c}^\mu \neq X_c^\tau$. The NP contributions in VLQ-models cannot compete with the SM contribution to the tree-level processes entering the ``charm''-sector, since they are suppressed by an additional factor $(M_W/M_{\rm VLQ})^2$. In consequence, NP contributes to the ``top''-sector only \begin{align} X_L^{\rm SM} \quad \to \quad X_{t}^\nu & = X_L^{\rm SM} + X_L^{sd,\nu} + X_R^{sd,\nu} \equiv X_L^{\rm SM} + X_{\rm NP}^\nu , \end{align} with $X_{L,R}^{sd,\nu}$ given in eq.~\eqref{eq:X_LR}, such that the top-sector becomes neutrino-flavour dependent. The experimental measurement averages over the three neutrino flavours, \begin{equation} \label{eq:Br-kpivv-1} Br(\kpn) \!=\! \frac{\kappa_+ (1 \!+\! \Delta_{\rm EM})}{\lambda^{10}} \frac{1}{3} \sum_{\nu} \left[ \mbox{Im}^2 \Big(\lambda_{sd}^{(t)} X_{t}^\nu \Big) \!+\! \mbox{Re}^2 \Big(\lambda_{sd}^{(c)} X_{c}^\nu \!+\! \lambda_{sd}^{(t)} X_{t}^\nu \Big)\right] , \end{equation} with the assumption that $\lambda_{sd}^{(c)} X_{c}^\nu$ is real. The NNLO QCD results of the functions $X_{c}^\nu$~\cite{Buras-ml-2006gb} together with long distance contributions~\cite{Isidori-ml-2005xm} are combined into \begin{align} P_c & = \frac{1}{\lambda^4} \left( \frac{2}{3} X_{c}^e + \frac{1}{3} X_{c}^\tau \right) = \left(\frac{0.2252}{\lambda}\right)^4 (0.404 \pm 0.024), \end{align} where $\lambda = 0.2252$ has been used in ref.~\cite{Buras-ml-2015qea}. The factor \begin{align} \kappa_+ & = r_{K^+} \frac{3 \alpha^2(M_Z) \lambda^8}{2 \pi^2 s_W^4} Br(K\to \pi e\bar{\nu}_e) = 0.5173(25) \times 10^{-10} \left[\frac{\lambda}{0.225}\right]^8 \end{align} contains the experimental value $Br(K\to \pi e\bar{\nu}_e)$ and the isospin correction $r_{K^+}$ and has been evaluated in ref.~\cite{Mescia-ml-2007kn} (table 2) including various corrections. Further $\Delta_{\rm EM} = -0.003$ for $E^\gamma_{\rm max} \approx 20$\,MeV~\cite{Mescia-ml-2007kn}. If one takes into account the different value of $s_W^2 = 0.231$ taken in ref.~\cite{Mescia-ml-2007kn} compared to our value in table~\ref{tab:input}, then $\kappa_+ = 0.5150 \times 10^{-10} \, (\lambda/0.225)^8$. The sum~\eqref{eq:Br-kpivv-1} contains the SM contribution and further the interference of SM$\times$NP and NP$\times$NP. Besides $P_c$ at NNLO in the SM contribution, the NLO numerical values \begin{align} X_c^e & = 10.05 \times 10^{-4} , & X_c^\tau & = 6.64 \times 10^{-4} , \end{align} for $\mu_c = 1.3$\,GeV are used for the interference of SM$\times$NP. The branching fraction of $\klpn$ is obtained again by averaging over the three neutrino flavours \begin{align} Br(\klpn) & = \frac{\kappa_L }{\lambda^{10}} \frac{1}{3} \sum_{\nu} \mbox{Im}^2 \Big(\lambda_{sd}^{(t)} X_{t}^\nu \Big) , \end{align} with \begin{align} \kappa_L & = \kappa_+ \frac{r_{K_L}}{r_{K_+}} \frac{\tau_{K_L}}{\tau_{K_+}} = 2.231(13) \times 10^{-10} \left[\frac{\lambda}{0.225}\right]^8 . \end{align} The numerical value is from ref.~\cite{Mescia-ml-2007kn} (table 2) and it decreases to $\kappa_L = 2.221 \times 10^{-10} \, (\lambda/0.225)^8$ when rescaling with our value of $s_W^2$. ]]>

dll} The effective Lagrangian for $d_j\to d_i \ell\bar\ell$ ($i\neq j$) is adopted from ref.~\cite{Bobeth-ml-2007dw}, \begin{align} \label{eq:EFT:ddll} {\cal L}_{d\to d\ell\bar\ell} & = \frac{4 G_F}{\sqrt{2}} \frac{\alpha_e}{4\pi} \, \lambda^{(t)}_{ij} \sum_{a} \sum_{\ell} C_a^{ij, \ell} O_a^{ij, \ell} + \mbox{h.c.} , \end{align} were the sum over $a$ extends over the $|\Delta F| = 1$ operators \begin{align} \label{eq:operators:ddll} O_{9\,(9')}^{ij,\ell} & = [\bar{d}_i \gamma_\mu P_{L\,(R)} d_j][\bar\ell \gamma^\mu \ell] , & O_{10\,(10')}^{ij,\ell} & = [\bar{d}_i \gamma_\mu P_{L\,(R)} d_j][\bar\ell \gamma^\mu \gamma_5 \ell] , \end{align} whereas scalar $O_{\rm S,P(S',P')}^{\ell}$ and tensorial operators $O_{\rm T(T5)}^{\ell}$ are not generated in the context of VLQ models. In the SM the only non-zero Wilson coefficients, \begin{align} C_9^{ij,\ell} \big|_{\rm SM} & = \frac{1}{s_W^2} \left[(1 - 4 s_W^2) C - B - s_W^2 D \right] \equiv \frac{Y_0}{s_W^2} - 4 Z_0, \\ C_{10}^{ij,\ell} \big|_{\rm SM} & = \frac{1}{s_{W}^2} \left(B - C \right) \equiv -\frac{Y_0}{s_W^2}, \end{align} are lepton-flavour universal and also universal w.r.t.\ down-type quark transitions, as the CKM elements have been factored out. All other Wilson coefficients vanish at the scale $\mu_{\rm EW}$. The functions $B,C,D$ depend again on the ratio $x_t \equiv m_t^2/M_W^2$ of the top-quark and $W$-boson masses and give two gauge-independent combinations $Y_0(x_t) \equiv C(x_t) - B(x_t)$ and $Z_0(x_t) \equiv C(x_t) + D(x_t)/4$, that are given in the SM as \begin{align} \label{eq:Y-SM} Y_0(x) & = \frac{x}{8} \left(\frac{x - 4}{x - 1} + \frac{3 x \ln x}{(x - 1)^2} \right) , \\ \label{eq:Z-SM} Z_0(x) & = \frac{18 x^4 - 163 x^3 + 259 x^2 - 108 x}{144 (x-1)^3} + \frac{32 x^4 - 38 x^3 - 15 x^2 + 18 x}{72 (x-1)^4} \ln x - \frac{1}{9} \ln x . \end{align} In the predictions of $Br(B_{d,s}\to \mu\bar\mu)$ and the mass-eigenstate rate asymmetry $A_{\Delta \Gamma}(B_{d,s}\to \mu\bar\mu)$ we include for the SM contribution the NNLO QCD~\cite{Hermann-ml-2013kca} and NLO EW~\cite{Bobeth-ml-2013tba} corrections, whereas NP contributions are included at LO. The values of the decay constants $F_{B_{d,s}}$ are collected in table~\ref{tab:input}. The branching fractions $Br(B^+ \to (\pi^+,\, K^+) \mu\bar\mu)$ at high dilepton invariant mass $q^2$ are predicted within the framework outlined in refs.~\cite{Grinstein-ml-2004vb, Beylich-ml-2011aq, Bobeth-ml-2011nj}. We neglect contributions from QCD penguin operators, which have small Wilson coefficients and the NLO QCD corrections to matrix elements of the charged-current operators~\cite{Seidel-ml-2004jh, Greub-ml-2008cy}, but include the contributions $\sim V_{ub}^{} V_{ud(s)}^*$. The form factors and their uncertainties are adapted from lattice calculations~\cite{Lattice-ml-2015tia, Bailey-ml-2015nbd} for $B\to \pi$ and~\cite{Bailey-ml-2015dka} for $B\to K$ with a summary given in~\cite{Du-ml-2015tda}. We add additional relative uncertainties of 15\% for missing NLO QCD corrections and 10\% for possible duality violation~\cite{Beylich-ml-2011aq} in quadrature. The predictions for observables of $B\to K^*\mu\bar\mu$ are based on refs.~\cite{Bobeth-ml-2008ij} and~\cite{Bobeth-ml-2010wg} for low- and high-$q^2$ regions, respectively. The corresponding results for $B\to K^*$ form factors in the two regions are from the LCSR calculation~\cite{Straub-ml-2015ica} and the lattice calculations~\cite{Horgan-ml-2013hoa, Horgan-ml-2015vla}. The measurement of $Br(K_L \to \mu\bar\mu)$ provides important constraints on its short-distance (SD) contributions, despite the dominating long-distance (LD) contributions inducing uncertainties that are not entirely under theoretical control. In particular there is the issue of the sign of the interference of the SD part $\chi_{\rm SD}$ of the decay amplitude of $K_L \to \mu\bar\mu$ with the LD parts. Allowing for both signs implies a conservative bound $|\chi_{\rm SD}| \leq 3.1$~\cite{Isidori-ml-2003ts}. Relying on predictions of this sign based on the quite general assumptions stated in~\cite{Isidori-ml-2003ts, DAmbrosio-ml-1996kjn, GomezDumm-ml-1998gw} one finds $-3.1 \leq \chi_{\rm SD}\leq 1.7$ which we employ in most of this work. Note, however, that a different sign is found\footnote{We thank G.~D'Ambrosio and J-M.~G{\'e}rard for the discussion on this point.} in~\cite{DAmbrosio-ml-1996kjn, Gerard-ml-2005yk}, implying $-1.7 \leq \chi_{\rm SD}\leq 3.1$. In light of this situation, we comment on the impact of the more conservative choice where appropriate, which includes both sign choices. ]]>

dqq} The effective Lagrangian for $d_j\to d_i q\bar{q}$ ($i\neq j$) is adopted from ref.~\cite{Buras-ml-1993dy}, where the definition of the operators can be found and here we restrict ourselves to $\bar{s}\to \bar{d}$, i.e.\ $ij=sd$. At the scale $\mu_{\rm EW}$ ($N_f = 5$) it reads \begin{equation} \label{eq:EFT:ddqq:Nf5} \begin{aligned} {\cal L}_{d\to dq\bar{q}} = - \frac{G_F}{\sqrt{2}} \, \lambda_{sd}^{(u)} \Bigg\{ & (1-\tau) \big[z_1 (O_1 - O_1^c) + z_2 (O_2 - O_2^c)\big] \\ & + \sum_{a=3}^{10} (\tau v_a + v_a^{\rm NP}) O_a + \sum_{a=3}^{10} v'_a O'_a \Bigg\} + \mbox{h.c.} , \end{aligned} \end{equation} where $O_{1,2}^{(c)}$ denote current-current operators. The sum over $a$ extends over the QCD- and EW-penguin operators and we included their chirality-flipped counterparts $O'_a = O_a [\gamma_5 \to -\gamma_5]$. Thereby we assume that VLQ contributions to other operators are strongly suppressed. The Wilson coefficients are denoted as $z_a$, $v_a^{(\rm NP)}$ and $v'_a$, taken at the scale $\mu_{\rm EW}$. For the SM-part, CKM unitarity was used, \begin{align} \tau & \equiv\lambda_{sd}^{(u)} \big/ \lambda_{sd}^{(t)} , \end{align} and we introduced a new physics contribution $v_a^{\rm NP}$ as shown above, which is related to the VLQ-contribution~\eqref{eq:ddqq-Z-contr} as \begin{align} v_a^{\rm NP} & = C_a^{sd} , & v'_a & = C_{a'}^{sd} . \end{align} The RG evolution at NLO in QCD and QED leads to the effective Hamiltonian at a scale $\mu\lesssim \mu_c \sim m_c$ ($N_f=3$) \begin{align} \label{eq:EFT:ddqq:Nf3} {\cal H}_{d\to dq\bar{q}} & = \frac{G_F}{\sqrt{2}} \, \lambda_{sd}^{(u)} \left\{ z_1 O_1 + z_2 O_2 + \sum_{a=3}^{10} [z_a + \tau y_a + v_a^{\rm NP}] O_a + \sum_{a=3}^{10} v'_a O'_a \right\} + \mbox{h.c.} , \end{align} after decoupling of $b$- and $c$-quarks at scales $\mu_{b,c}$~\cite{Buras-ml-1993dy}, where $y_a \equiv v_a - z_a$ and all Wilson coefficients are at the scale $\mu$. \begin{table} \renewcommand{\arraystretch}{1.3} \centering \begin{tabular}{|c|rrr|r|} \hline $a$ & $p^{(0)}_a$ & $p^{(6)}_a$ & $p^{(8)}_a$ & $P_a$ \\ \hline 3 & $7.45$ & $-3.40$ & $-3.50$ & $2.85$ \\ 5 & $1.70$ & $30.62$ & $-18.74$ & $4.91$ \\ 7 & $-102.02$ & $-1.32$ & $2040.38$ & $1447.91$ \\ 9 & $36.72$ & $4.42$ & $-21.28$ & $23.06$ \\ \hline \end{tabular} \renewcommand{\arraystretch}{1.0} ]]>