First determination of the $CP$ content of D → π + π – π + π – and updated determination of the $CP$ contents of D → π + π – π 0 and D → K + K – π 0
Quantum-correlated ψ(3770)→DD¯ decays collected by the CLEO-c experiment are used to perform a first measurement of $F$ $^{4π}_{+}$, the fractional CP-even content of the self-conjugate decay D → π + π – π + π – , obtaining a value of 0.737 ± 0.028. An important input to the measurement comes from the use of D → $K$ $^{0}_{S}$π + π – and D → $K$ $^{0}_{L}$π + π – decays to tag the signal mode. This same technique is applied to the channels D → π + π – π 0 and D → K + K – π 0 , yielding $F$ $^{πππ0}_{+}$ =1.014 ± 0.045 ± 0.022 and $F$ $^{KKπ0}_{+}$ = 0.734 ± 0.106 ± 0.054, where the first uncertainty is statistical and the second systematic. These measurements are consistent with those of an earlier analysis, based on CP-eigenstate tags, and can be combined to give values of $F$ $^{πππ0}_{+}$ = 0.973 ± 0.017 and $F$ $^{KKπ0}_{+}$ = 0.732 ± 0.055. The results will enable the three modes to be included in a model-independent manner in measurements of the unitarity triangle angle γ using B ∓ → DK ∓ decays, and in time-dependent studies of CP violation and mixing in the D 0 D¯ 0 system.