We present a study of Bs0 decays to the CP-odd final state J/psi f0(980) with J/psi -> mu+ mu- and f0(980) -> pi+ pi-. Using ppbar collision data with an integrated luminosity of 3.8/fb collected by the CDF II detector at the Tevatron we measure a Bs0 lifetime of tau(Bs0 -> J/psi f0(980)) = 1.70 -0.11+0.12(stat) +-0.03(syst) ps. This is the first measurement of the Bs0 lifetime in a decay to a CP eigenstate and corresponds in the standard model to the lifetime of the heavy Bs0 eigenstate. We also measure the product of branching fractions of Bs0 -> J/psi f0(980) and f0(980) -> pi+ pi- relative to the product of branching fractions of Bs0 -> J/psi phi and phi -> K+ K- to be R_f0/phi = 0.257 +_0.020(stat) +-0.014(syst), which is the most precise determination of this quantity to date.
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