We investigate new physics models that can increase the lifetime differences in the $B_q$--$bar{B}_q$ systems ($q = d,s$) above their standard model values. If both $B_q$ as well as $bar{B}_q$ can decay to a final state through flavour dependent new physics interactions, the so-called Grossman bound may be evaded. As examples, we consider the scalar leptoquark model and $lambda$-type R-parity violating supersymmetry. We find that models with a scalar leptoquark can enhance $DeltaGamma_s/Gamma_s$ all the way up to its experimental upper bound and $DeltaGamma_d/Gamma_d$ to as much as $sim 2.5%$, at the same time allowing the CP violating phase $beta_s$ to vary between $- 45^circ$ and $20^circ$. R-parity violating supersymmetry models cannot enhance the lifetime differences significantly, but can enhance the value of $beta_s$ up to $sim pm 20^circ$. This may bring the values of $DeltaGamma_q/Gamma_q$ as well as $beta_s$ within the measurement capabilities of $B$ factories and LHCb. We also obtain bounds on combinations of these new physics couplings, and predict enhanced branching ratios of $B_{s/d} to tau^+ tau^-$.
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