  The B_s to mu+ mu- decay plays an outstanding role in tests of the Standard
Model and physics beyond it. The LHCb collaboration has recently reported the
first evidence for this decay at the 3.5 sigma level, with a branching ratio in
the ballpark of the Standard Model prediction. Thanks to the recently
established sizable decay width difference of the B_s system, another
observable, A^mumu_DeltaGamma, is available, which can be extracted from the
time-dependent untagged B_s to mu+ mu- rate. If tagging information is
available, a CP-violating asymmetry, S_mumu, can also be determined. These two
observables exhibit sensitivity to New Physics that is complementary to the
branching ratio. We define and analyse scenarios in which these quantities
allow us to discriminate between model-independent effective operators and
their CP-violating phases. In this context we classify a selection of popular
New Physics models into the considered scenarios. Furthermore, we consider
specific models with tree-level FCNCs mediated by a heavy neutral gauge boson,
pseudoscalar or scalar, finding striking differences in the predictions of
these scenarios for the observables considered and the correlations among them.
We update the Standard Model prediction for the time-integrated branching ratio
taking the subtle decay width difference effects into account. We find (3.56
+/- 0.18) x 10^-9, and discuss the error budget.
