First determination of the CP content of D → π+π−π+π− and updated determination of the CP contents of D → π+π−π0 and D → K+K−π0

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−D→π+π−π+π−, obtaining a value of 0.737±0.0280.737±0.028. An important input to the measurement comes from the use of D→KS0π+π− and D→KL0π+π− decays to tag the signal mode. This same technique is applied to the channels D→π+π−π0D→π+π−π0 and D→K+K−π0D→K+K−π0, 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∓B∓→DK∓ decays, and in time-dependent studies of CP   violation and mixing in the D0D‾0 system.