Abstract
The isospin-breaking decay $X(3872)\to(D^*\bar D+ \bar D^*D) \to\pi^0D\bar D\to\pi^0\pi^+\pi^-$ is discussed. In its amplitude there is a triangle logarithmic singularity, due to which the dominant contribution to $BR(X(3872)\to\pi^0\pi^+\pi^-)$ comes from the production of the $\pi^+\pi^-$ system in a narrow interval of the invariant mass $m_{\pi^+\pi^-}$ near the value of $2m_{D^0} \approx 3.73$ GeV. The analysis shows that $BR(X(3872)\to\pi^0 \pi^+\pi^-)$ can be expected at the level of $10^{-3}$--$10^{-4}$. This estimate includes, in particular, the assumption that the $S$-wave inelastic scattering length $|\alpha''_{D^0\bar D^0 \to\pi^+\pi^-}|\approx1/(2m_{D^{*+}})\approx0.25\ \mbox{GeV}^{-1}$.
Highlights
The state Xð3872Þ [or χc1ð3872Þ [1]] was first observed in 2003 by the Belle Collaboration in the process B → KðXð3872Þ → πþπ−J=ψÞ [2]. It was observed in many other experiments in other processes and decay channels [1]
The Xð3872Þ is a narrow resonance in non-ðDÃ0D 0 þ D Ã0D0Þ decay channels, ΓX < 1.2 MeV [3], and its mass coincides practically with the DÃ0D 0 threshold [1]
In addition to decay into πþπ−J=ψ [2,6,7], the Xð3872Þ decays into ωJ=ψ [8,9,10,11], DÃ0D 0 þ c:c. [12,13], γJ=ψ [14,15,16], γψð2SÞ [14,15,16], and π0χc1ð1PÞ [17,18]
Summary
The state Xð3872Þ [or χc1ð3872Þ [1]] was first observed in 2003 by the Belle Collaboration in the process B → KðXð3872Þ → πþπ−J=ψÞ [2]. The search for Xð3872Þ in decay channels that do not contain charmed particles or charmonium states [i.e., in channels other than DÃ0D 0 þ c:c., D0D 0π0, πþπ−J=ψ, ωJ=ψ, γJ=ψ, γψð2SÞ, πþπ−ηcð1SÞ, πþπ−χc1ð1PÞ, and π0χc1ð1PÞ] is of great interest [1,25,26,27,28,29,30,31,32,33,34,35]. The branching ratio of the decay Xð3872Þ → π0πþπ− can be expected at the level of 10−3–10−4 due to the transition mechanism Xð3872Þ → ðDÃDþ D ÃDÞ → π0DD → π0πþπ− In this case, the main contribution to BRðXð3872Þ → π0πþπ−Þ comes from the production of πþπ− pairs in a narrow interval of the invariant mass mπþπ− near the value of 2mD0 ≈ 3.73 GeV. As for the nature of X(3872), our calculations implicitly imply for this state the conventional ccnature, i.e., that it is a compact charmonium state similar to the states χc1ð1PÞ, ψð2SÞ, ψð3770Þ, and so on, and to describe its decays one can use the effective phenomenological Lagrangian approach [25,26,27,28,29,30]
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