We investigate new physics models that can increase the lifetime differences in the Bq-B̄q systems (q=d, s) above their standard model values. If both Bq as well as B̄q can decay to a final state through flavor dependent new physics interactions, the so-called Grossman bound may be evaded. As examples, we consider the scalar leptoquark model and λ′′ -type R-parity violating supersymmetry. We find that models with a scalar leptoquark can enhance ΔΓs/Γs all the way up to its experimental upper bound and ΔΓd/Γd to as much as ∼2.5%, at the same time allowing the CP violating phase βs to vary between -45° and 20°. R-parity violating supersymmetry models cannot enhance the lifetime differences significantly, but can enhance the value of βs up to ∼±20°. This may bring the values of ΔΓq/Γq as well as βs within the measurement capabilities of B factories and the B physics experiment at the Large Hadron Collider (LHCb). We also obtain bounds on combinations of these new physics couplings, and predict enhanced branching ratios of Bs/d→τ+τ-. © 2007 The American Physical Society.