General minimal flavor violation

A model independent study of the minimal flavor violation (MFV) framework is presented, where the only sources of flavor breaking at low energy are the up and down Yukawa matrices. Two limits are identified for the Yukawa coupling expansion: linear MFV, where it is truncated at the leading terms, and nonlinear MFV, where such a truncation is not possible due to large third generation Yukawa couplings. These are then resummed to all orders using nonlinear $\sigma$-model techniques familiar from models of collective breaking. Generically, flavor-diagonal $CP$ violating (CPV) sources in the UV can induce $O(1)$ CPV in processes involving third generation quarks. Because of a residual $U(2)$ symmetry, the extra CPV in $B_d-{\overline{B}}_d$ mixing is bounded by CPV in $B_s-{\overline{B}}_s$ mixing. If operators with right-handed light quarks are subdominant, the extra CPV is equal in the two systems, and is negligible in processes involving only the first two generations. We find large enhancements in the up-type sector, both in CPV in $D-\overline{D}$ mixing and in top flavor violation.