this phase is via the direct CP asymmetry in the decays $B^\pm\to\pi^\pm K^0$, which is given by $$A_{\rm CP}(\pi^+ K^0) =
   \frac{\mbox{B}(B^+\to\pi^+ K^0)-\mbox{B}(B^-\to\pi^-\bar K^0)}
        {\mbox{B}(B^+\to\pi^+ K^0)+\mbox{B}(B^-\to\pi^-\bar K^0)} 
   = \sin2\varphi\sin\phi_\rho \,.$$ In the SM, $\sin2\varphi\approx 2\varepsilon_a\sin\gamma$ is of order a few percent, and with realistic values for $\phi_\rho$ of order $10^\circ$–$20^\circ$ [@fact] one expects a very small CP asymmetry. However, in New Physics scenarios with new CP-violating couplings $\sin 2\varphi$ may be significantly larger than in the SM. An experimental finding that $A_{\rm CP}(\pi^+ K^0)=O(10\%)$ would constitute strong evidence for the existence of such an effect.

The exact theoretical expression for $R_*$ is $$\begin{aligned}
\label{Rstar}
   R_*^{-1} &=& 1 + 2\bar\varepsilon_{3/2}\cos\varphi
    \Big[ a\cos(\phi+\phi_a) - \cos\gamma\cos\phi \Big] \nonumber\\
   &&\mbox{}- 2\bar\varepsilon_{3/2}\sin\varphi
    \Big[ b\cos(\phi+\phi_b-\phi_\rho)
    - \sin\gamma\cos(\phi-\phi_\rho) \Big] \nonumber\\
   &&\mbox{}+ \bar\varepsilon_{3/2}^2 \left( 1 
    - 2a\cos\gamma\cos\phi_a - 2b\sin\gamma\cos\phi_b 
    + a^2 + b^2 \right) .\end{aligned}$$ In the SM $b=0$, $a=\delta_{\rm EW}$, $\phi_a=0$, and $\varphi\approx 0$ to very good approximation. Therefore $$\begin{aligned}
\label{Rstbound}
   R_*^{-1} &=& 1 + 2\bar\varepsilon_{3/2}\,
    (\delta_{\rm EW}-\cos\gamma)\cos\phi 
    + \bar\varepsilon_{3/2}^2
    (1 - 2\delta_{\rm EW}\cos\gamma + \delta_{\rm EW}^2)
    \nonumber\\
   &\le& \left( 1 + \bar\varepsilon_{3/2}\,
    |\delta_{\rm EW}-\cos\gamma| \right)^2 
    + \bar\varepsilon_{3/2}^2\sin^2\!\gamma \,.\end{aligned}$$ In the second step we have used the fact that $|\cos\phi|\le 1$ to obtain an upper bound on 