Physics models the coefficient $C_{8g}$ can be enhanced with respect to its SM value by an order of magnitude [@glue1; @George; @Alex; @CGGi]. Using the QCD factorization approach of [@fact], we find that $$a_{8g} = \frac{2\alpha_s}{3\pi}\,(1+\chi)(-\lambda_t)
   \approx 4\times 10^{-3} \,.$$ Note that the magnitude of the left-hand side in (\[rhobound\]) is bounded by unity. This implies a nontrivial upper bound on the possible CP-violating New Physics contributions to the penguin operators, given by $$\label{rhonumeric}
   \left|\,\mbox{Im}\Big[ 
   (\bar c_5^d - \bar c_4^d - \chi\,\bar c_2^d)
   + \frac13\,(\bar c_6^d - \bar c_3^d - \chi\,\bar c_1^d)
   - a_{8g}\,C_{8g} \Big] \right| < 
   \frac{4|\lambda_u|}{3\bar\varepsilon_{3/2}} \,.$$ Using $|V_{ub}/V_{cb}|=0.085\pm 0.015$, corresponding to $|\lambda_u|=(7.5\pm 1.3)\times 10^{-4}$, and taking for $\bar\varepsilon_{3/2}$ the value in (\[epsval\]), we find that the right-hand side of this bound is less than $7.4\times 10^{-3}$ at 90% confidence level. For comparison, we note that in the SM the magnitude of the combination of penguin coefficients entering above is about $3\times 10^{-3}$. However, due to the smallness of the weak phase of $\lambda_t$ the imaginary part of this combination is much smaller.

Model-independent bounds on $R_*$ in the presence of New Physics
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The most important observable in the exploration of New Physics effects in $B\to\pi K$ 