large they are. The corresponding bound on $|X_R|$ is $$\label{phiarb}
   |X_R| \le 
   \sqrt{1 - 2\delta_{\rm EW}\cos\gamma + \delta_{\rm EW}^2}
   \le 1+\delta_{\rm EW} \,.$$ Note that the extremal value is the same as in the SM, i.e., isospin-conserving New Physics effects cannot lead to a value of $|X_R|$ exceeding $1+\delta_{\rm EW}$. In the left-hand plot in Figure \[fig:case12\] we show the upper bound on $X_R$ versus $\gamma$ in the SM, and for New Physics scenarios with different values of $\varphi$. The three choices of $\varphi$ shown correspond to $|\rho|\approx 0.27$, 0.58 and 1. The gray curve shows the upper bound obtained by varying $\varphi$. We observe that isospin-conserving New Physics can enhance the value of $X_R$ relative to the SM, but only by a moderate amount.

Next we consider New Physics effects that violate isospin symmetry, but we first restrict ourselves to the important subclass of models which do not contain significant new CP-violating phases. Then $\varphi$ and $b$ still vanish, and we obtain $$\begin{aligned}
\label{Rsta}
   R_*^{-1} &=& 1 + 2\bar\varepsilon_{3/2} \Big[
    a\cos(\phi+\phi_a) - \cos\gamma\cos\phi \Big] 
    + \bar\varepsilon_{3/2}^2 
    (1 - 2a\cos\phi_a\cos\gamma + a^2) \nonumber\\
   &\le& \left[ 1 + \bar\varepsilon_{3/2} \left(
    |a|+|\cos\gamma| \right) \right]^2
    + \bar\varepsilon_{3/2}^2\sin^2\!\gamma \,.\end{aligned}$$ Again, a lower bound can 