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^-, obtaining a value of 0.737±0.028. An important input to the measurement comes from the use of D→K_S⁰π+π- and D→K_L⁰π+π- decays to tag the signal mode. This same technique is applied to the channels D→π⁺π^-π⁰ and D→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^∓ decays, and in time-dependent studies of CP violation and mixing in the D⁰D⁰ system.