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Four-Fermi Operators in $e^+$ $e^-$ Annihilation Experiments and Uncertainties in $Z$ Boson Properties

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.