Feasible Deviations from Unitarity with Vector-Like Quark Singlets

Abstract

We deduce pertinent relations between the elements of the CKM matrix, and find that not all of these are totally compatible with experiment and/or the assumption of the $3 \times 3$ unitarity. We identify complex phases in the CKM-elements which may signal deviations from unitary (DU). We focus on DUs induced by VLQ-singlets, and the possibility of having significant DUs of the first and second rows of the CKM matrix, together with DUs in its columns. We make a thorough analysis of models with the lowest amount of singlets and find a useful set of parametrizations crucial in coherently exploring the parameter-space of the proposed cases. We test the feasibility of each model, confronting them with the restrictions imposed by several important flavor observables. Special attention is given to the neutral kaon and $D^0$-meson sectors, particularly to the parameters $ε_K$ and $x_D$. We find that for the most elementary VLQ-singlet cases, the DUs in the second row must roughly accompany the DUs of the first row. However, in cases with more elaborate combinations of VLQ-singlets, the DUs in the second row may be very large, and even substantially exceed those of the first row. This is what happens in models with sufficient mingling of the two sectors, e.g. in a 2-up-1-down VLQ-singlet scenario. Until now, the analysis of this joining of the up and down sectors with VLQs has not been described in the literature in great detail.

Figures (6)

• Figure 1: Deviations from unitarity of the first column of the CKM matrix squared ($\delta^2_1$) vs deviations from unitarity of the first row of the CKM matrix squared ($\Delta^2_1$). The dark(light) green regions represents the allowed regions at $1 \sigma$ ($2 \sigma$) for the deviation from unitarity of the first row of the CKM matrix $\left(\Delta^\text{exp}_1\right)^2=(1.70\pm0.72)\times 10^{-3}$ParticleDataGroup:2024cfk, corresponding to deviation from unitarity for $\Delta^\text{exp}_1$ around $0.041$. The dark(light) blue regions represent the regions obtained from \ref{['eq:delta1Delta1']} at $1 \sigma$ ($2 \sigma$), considering the PDG values for $|V_{td}|$, $|V_{us}|$, $|V_{cd}|$, $|V_{ub}|$.
• Figure 2: Plot of $\sin \beta_k$ as a function of $\sin \xi_1$ as in \ref{['eq:generalized8']}. In gray, we present the 1$\sigma$ (darker) and 2$\sigma$ (lighter) regions for $\sin\beta_K$ in the SM. We use the PDG values for the $\gamma$ phase and the moduli of the CKM matrix elements without assuming unitarity. In red we present the 1$\sigma$ (darker) and 2$\sigma$ (lighter) regions for $\sin\beta_K$ allowing for deviations of unitarity in the first and second row. Here we use $|V_{14} ||V_{24}|=\left(\Delta^\text{exp}_1\right)^2=(1.70\pm0.72)\times 10^{-3}$. The dashed lines represent the central values for each case.
• Figure 3: Diagrams contributing to $K^0-\overline{K^0}$ mixing that include effects of mixing both with up- and down-type VLQ-singlets. Here, $\alpha,\beta$ represent up-type quarks. In green we highlight CKM couplings, while in blue(red) we highlight down(up)-type FCNC couplings.
• Figure 4: The $(1U,1D)$ VLQ-singlet model. On the left, we plot the deviations from unitarity of the second row of the CKM matrix ($\Delta_2$) vs deviations from unitarity of the first row of the CKM matrix ($\Delta_1$). The different regions (from lighter to darker) indicate the $68.3\%$ CL, $95.5\%$ CL and the $99.7\%$ CL regions.
• Figure 5: The $(2U,0D)$ VLQ-singlet model. On the left, we plot the deviations from unitarity of the second row of the CKM matrix ($\Delta_2$) vs deviations from unitarity of the first row of the CKM matrix ($\Delta_1$). On the right, the green points are a plot of $\sin \beta_K$vs deviations from unitarity of the first row of the CKM matrix ($\Delta_1$) and the black lines are the limits of the SM $1 \sigma$ region for $\sin \beta_K$ obtained assuming $3\times 3$ unitarity. On both figures, the different regions (from lighter to darker) indicate the $68.3\%$ CL, $95.5\%$ CL and the $99.7\%$ CL regions.