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Supercurrent dephasing by electron-electron interactions

We demonstrate that in sufficiently long diffusive superconducting-normal-superconducting ($SNS$) junctions dc Josephson current is exponentially suppressed by electron-electron interactions down to zero temperature. This suppression is caused by the effect of {\it Cooper pair dephasing} which occurs in the normal metal and defines a new fundamental length scale $L_φ$ in the problem. Provided the temperature length exceeds $L_φ$ this dephasing length can be conveniently extracted from equilibrium measurements of the Josephson current.