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Dipion invariant mass spectrum in $X(3872) \to J/ψππ$

It is pointed out that dipion invariant mass spectrum in $X (3872) \to ππJ/ψ$ is a useful probe for the $J^{PC}$ quantum number of the X(3872), complementary to the angular distributions. If X(3872) is a $^1 P_1$ state, the dipion has a peak at low $m_{ππ}$ region, which is not in accord with the preliminary Belle data. If X(3872) is a $^3 D_2$ or $^3 D_3$ state, the dipion spectrum shows a peak at high $m_{ππ}$ region, which is broader than the $ρ$ resonance that might come from the decay of a molecular state: $(D \bar{D^*}) (I=1) \to J/ψρ\to J/ψπ^+ π^-$. Better measurement of $m_{ππ}$ spectrum will shed light on the nature of X(3872).