  We show using a model independent approach that the deviation in measured
B_d^0-\bar{B}_d^0 mixing phase caused by pollution from another amplitude,
within the Standard Model, is always less in magnitude, and has the same sign
as the weak phase of the polluting amplitude. The exception is to have large
destructive interference between the two amplitudes. We demonstrate that any
deviation larger than a few degrees is only possible if the observed decay rate
results from fine tuned cancellations between significantly larger quark level
amplitudes. These simple observations have very significant consequences for
signals of New Physics.
