  We investigate the effects of possible new four-Fermi operators on $e^+$
$e^-$ annihilation experiments. They represent a class of new physics, which
has the potential to change the parameters of the $Z$ boson and at the same
time interferes with their determination from the data. We show that in the
presence of such operators the $Z$ parameters obtained from lineshape fits can
change significantly. Another important property of these operators is that
they spoil the factorization of the expression for the left-right asymmetry,
$A_{LR}$, into initial and final state couplings. Factorization and subsequent
cancellation of the final state coupling occurs in the Standard Model (after
correcting for photonic amplitudes) and is crucial for the interpretation of
$A_{LR}$ as a measurement of the effective weak mixing angle. Four-Fermi
operators may thus provide an explanation for the high value of the
polarization asymmetry as observed at SLC. However, the data from lower energy
$e^+$ $e^-$ annihilation severely constrains this class of operators and
virtually closes this possible loophole on how new physics might explain the
SLC/LEP discrepancy. We point out that if the surplus of observed $b$-pairs at
LEP is real and caused by these operators, there may be a significant effect on
the forward-backward asymmetry into $b$-quarks. It is this quantity which
presently gives the most precise determination of the weak mixing angle. We
present compact analytical expressions for the treatment of initial state
radiation in the presence of new contact operators.
