Using samples of 2.25*10^8 J/psi events and 1.06*10^8 psi events collected with the BES III detector, we study the f0(980) to a0(980) and a0(980) to f0(980) transitions in the processes J/psi --> phi f0(980) --> phi a0(980) and chi_c1 --> pi^0 a0(980) --> pi^0 f0(980), respectively. Evidence for f0(980) --> a0(980) is found with a significance of 3.4 sigma, while in the case of a0(980) --> f0(980) transition, the significance is 1.9 sigma. Measurements and upper limits of both branching ratios and mixing intensities are determined.
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