Measurement of the Dipion Mass Spectrum in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3872</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>→</mml:mo><mml:mi>J</mml:mi><mml:mo>/</mml:mo><mml:mi>ψ</mml:mi><mml:msup><mml:mi>π</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup></mml:math>Decays

We measure the dipion mass spectrum in $X(3872)\ensuremath{\rightarrow}J/\ensuremath{\psi}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decays using $360\text{ }\text{ }{\mathrm{pb}}^{\ensuremath{-}1}$ of $\overline{p}p$ collisions at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$ collected with the CDF II detector. The spectrum is fit with predictions for odd $C$-parity ($^{3}S_{1}$, $^{1}P_{1}$, and $^{3}D_{J}$) charmonia decaying to $J/\ensuremath{\psi}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$, as well as even $C$-parity states in which the pions are from ${\ensuremath{\rho}}^{0}$ decay. The latter case also encompasses exotic interpretations, such as a ${D}^{0}{\overline{D}}^{*0}$ molecule. Only the $^{3}S_{1}$ and $J/\ensuremath{\psi}\ensuremath{\rho}$ hypotheses are compatible with our data. Since $^{3}S_{1}$ is untenable on other grounds, decay via $J/\ensuremath{\psi}\ensuremath{\rho}$ is favored, which implies $C=+1$ for the $X(3872)$. Models for different $J/\ensuremath{\psi}\ensuremath{-}\ensuremath{\rho}$ angular momenta $L$ are considered. Flexibility in the models, especially the introduction of $\ensuremath{\rho}\ensuremath{-}\ensuremath{\omega}$ interference, enables good descriptions of our data for both $L=0$ and 1.

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