Testing Minimal Flavor Violation in Leptoquark Models of the $R_{K^{(*)}}$ Anomaly

The $R_{K^{(*)}}$ anomaly can be explained by tree level exchange of leptoquarks. We study the consequences of subjecting these models to the principle of minimal flavor violation (MFV). We consider MFV in the linear regime, and take the charged lepton Yukawa matrix to be the only spurion that violates lepton flavor universality. We find that a combination of constraints from a variety of processes -- $b\to s\mu\mu$, $b\to s\tau\tau$, $b\to s\nu\nu$, $b\bar b\to\tau\tau$ and $b\to c\tau\nu$ -- excludes MFV in these models.