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CP violation in the two-Higgs-doublet model: an example

In a general two-scalar-doublet model without fermions, there is a unique source of CP violation, $ J_1 $, in the gauge interactions of the scalars. It arises in the mixing of the three neutral physical scalars $ X_1 $, $ X_2 $ and $ X_3 $. CP violation may be observed via different decay rates for $ X_1 \rightarrow H^+ W^- $ and $ X_1 \rightarrow H^- W^+ $ (or, alternatively, for $ H^+ \rightarrow X_1 W^+ $ and $ H^- \rightarrow X_1 W^- $ --- depending on which decays are kinematically allowed). I compute the part of those CP-violating decay-rate differences which is proportional to $ J_1 $. The CP-invariant final-state-interaction phase is provided by the absorptive parts of the one-loop diagrams. I check the gauge invariance of the whole calculation.