Determining the sign of the $Z$-penguin amplitude

We point out that the precision measurements of the pseudo observables $R_b^0$, ${\mathcal{A}}_b$, and $A_{\mathrm{FB}}^{0,b}$ performed at the CERN LEP and the Stanford Linear Collider suggest that in models with minimal-flavor violation the sign of the $Z$-penguin amplitude is identical to the one present in the standard model. We determine the allowed range for the nonstandard contribution to the Inami-Lim function $C$ and show, by analyzing possible scenarios with positive and negative interference of standard model and new physics contributions, that the derived bound holds in each given case. Finally, we derive lower and upper limits for the branching ratios of $K^{+}\to {\pi }^{+}\nu \overline{\nu }$, $K_L\to {\pi }^0\nu \overline{\nu }$, $K_L\to {\mu }^{+}{\mu }^{-}$, $\overline{B}\to X_{d,s}\nu \overline{\nu }$, and $B_{d,s}\to {\mu }^{+}{\mu }^{-}$ within constrained minimal-flavor violation, making use of the wealth of available data collected at the $Z$ pole.

Figure 1

Relative deviations ${\delta }_n$ for low (upper panel) and high values (lower panel) of $M$. The solid, dashed, and dotted curves correspond to $n=1$, 2, and 3, respectively. In obtaining the numerical values we have set $M_Z=91\mathrm{GeV}$ and $m_t=165\mathrm{GeV}$. See text for details.

Figure 2

Examples of one-loop vertex diagrams that result in a nonuniversal correction to the $Z\to d_j{\overline{d}}_i$ transition in assorted NP scenarios with CMFV. See text for details.

Figure 3

Relative difference $\delta C_{\mathrm{NP}}$ in the THDM, the MSSM, the mUED, and the LHT model as a function of $M_H^{\pm }$, $M_{{\tilde{\chi }}_1}^{\pm }$, $1/R$, and $x_L$. The allowed parameter regions after applying experimental and theoretical constraints are indicated by the colored (grayish) bands and points. See text for details.

Figure 4

The upper (lower) panel displays the constraints on $\Delta C_7^{\mathrm{eff}}$ and $\Delta C$ within CMFV scanning $\Delta B^{l^{+}{l}^{-}}$ in the range $\pm 0.1$ (set to zero) that follow from a combination of the $Z\to b\overline{b}$ POs with flavor observables. The colors encode the frequentist $1-\mathrm{CL}$ level and the corresponding 68% and 95% probability regions as indicated by the bars on the right side of the panels. See text for details.

Figure 5

The upper (lower) panel displays future constraints on $\Delta C_7^{\mathrm{eff}}$ and $\Delta C$ within CMFV scanning $\Delta B^{\nu \overline{\nu }}$ in the range $\pm 0.4$ (set to zero) that are based on flavor observables only. The colors encode the frequentist $1-\mathrm{CL}$ level and the corresponding 68% and 95% probability regions as indicated by the bars on the right side of the panels. See text for details.