Effective theory analysis for vector-like quark model

We study a model with a down-type SU(2) singlet vector-like quark (VLQ) as a minimal extension of the standard model (SM). In this model, flavor-changing neutral currents (FCNCs) arise at tree level and the unitarity of the $3\times 3$ Cabibbo-Kobayashi-Maskawa (CKM) matrix does not hold. In this paper, we constrain the FCNC coupling from $b\to s$ transitions, especially $B_s\to {\mu }^{+}{\mu }^{-}$ and $\overline{B}\to X_s\gamma$ processes. In order to analyze these processes we derive an effective Lagrangian that is valid below the electroweak symmetry breaking scale. For this purpose, we first integrate out the VLQ field and derive an effective theory by matching Wilson coefficients up to one-loop level. Using the effective theory, we construct the effective Lagrangian for $b\to s{\gamma }^{(\ast )}$. It includes the effects of the SM quarks and the violation of CKM unitarity. We show the constraints on the magnitude of the FCNC coupling and its phase by taking account of the current experimental data on $\Delta M_{B_s}$, $Br[B_s\to {\mu }^{+}{\mu }^{-}]$, $Br[\overline{B}\to X_s\gamma ]$, and CKM matrix elements, as well as theoretical uncertainties. We find that the constraint from $Br[B_s\to {\mu }^{+}{\mu }^{-}]$ is more stringent than that from $Br[\overline{B}\to X_s\gamma$]. We also obtain a bound for the mass of the VLQ and the strength of the Yukawa couplings related to the FCNC coupling of the $b\to s$ transition. Using the CKM elements that satisfy the above constraints, we show how the unitarity is violated on the complex plane.