At present, there are several measurements of $B$ decays that exhibit discrepancies with the predictions of the SM, and suggest the presence of new physics (NP) in $b to s mu^+ mu^-$ transitions. Many NP models have been proposed as explanations. These involve the tree-level exchange of a leptoquark (LQ) or a flavor-changing $Z$ boson. In this paper we examine whether it is possible to distinguish the various models via CP-violating effects in $B to K^{(*)} mu^+ mu^-$. Using fits to the data, we find the following results. Of all possible LQ models, only three can explain the data, and these are all equivalent as far as $b to s mu^+ mu^-$ processes are concerned. In this single LQ model, the weak phase of the coupling can be large, leading to some sizeable CP asymmetries in $B to K^{(*)} mu^+ mu^-$. There is a spectrum of $Z$ models; the key parameter is $g_L^{mumu}$, which describes the strength of the $Z$ coupling to $mu^+mu^-$. If $g_L^{mumu}$ is small (large), the constraints from $B^0_s$-${bar B}^0_s$ mixing are stringent (weak), leading to a small (large) value of the NP weak phase, and corresponding small (large) CP asymmetries. We therefore find that the measurement of CP-violating asymmetries in $B to K^{(*)} mu^+ mu^-$ can indeed distinguish among NP $b to s mu^+ mu^-$ models
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