Decay $X(3872)\to\pi^0\pi^+\pi^-$ and $S$-wave $D^0\bar D^0 \to\pi^+\pi^-$ scattering length

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 $|\alpha''_{D^0\bar D^0 \to\pi^+\pi^-}|\approx1/(2m_{D^{*+}})\approx0.25\ \mbox{GeV}^{-1}$.

As for the nature of X(3872), our calculations implicitly imply for him the conventional cc nature.
If the virtual invariant mass squared of the X(3872) resonance s 1 falls in the range then, in the range of the invariant mass squared of the π + π − system s 2 = m 2 Figure 2: The diagram of the decay X(3872) → (D * 0D0 + D * 0 D 0 ) → π 0 D 0D0 → π 0 π + π − . In the X(3872) mass region, all intermediate particles in the triangle loop can be near or directly on the mass shell. As a consequence, a logarithmic singularity in the imaginary part of the amplitude emerges in the hypothetical case of the stable D * 0 meson. The 4momenta of corresponding particles are denoted as p1, p2, and p3; p 2 1 = s1 is the squared invariant mass of the X(3872) resonance or of the final π 0 π + π − system; p 2 2 = s2 = m 2 π + π − is the squared invariant mass of the final π + π − system; and the imaginary part of the amplitude of the diagram in Fig. 2 contains the triangle logarithmic singularity. [37][38][39][40]. Below, we see that this singularity leads to the resonancelike enhancement in the π + π − mass spectrum at √ s 2 = m π + π − ≈ 2m D 0 ≈ 3.73 GeV, i.e., near the D 0D0 threshold.
The decay X(3872) → π 0 π + π − can also be produced via the charged intermediate states, Fig.  3). From the isotopic symmetry for the coupling con- Figure 3: The diagram of the decay X(3872) → π 0 π + π − corresponding to the charged intermediate state contributions, stants, it follows that the contributions of the diagrams in Figs. 2 and 3 exactly compensate each other and the isospin breaking decay X(3872) → π 0 π + π − is absent, if m D * + = m D * 0 and m D + = m D 0 . However, the D * 0D0 and D * + D − thresholds in the variable √ s 1 differ by 8.23 MeV (m D * 0 + mD0 = 3.87168 GeV, m D * + + m D − = 3.87991 GeV) and the D 0D0 and D + D − thresholds in the variable √ s 2 differ by 9.644 MeV (2m D 0 = 3.72966 GeV, 2m D ± = 3.73930 GeV). Therefore, in the region of the variables √ s 1 and √ s 2 that is significant for the decay X(3872) → π 0 π + π − (i.e., for √ s 1 ≈ m X ≈ m D * 0 + mD0, where m X is the nominal mass of the X(3872) equal to 3.87169 GeV [1], and √ s 2 ≈ 2mD0 ≈ 3.73 GeV), the contributions from the neutral (see Fig. 2) and charged (see Fig. 3) intermediate states weakly compensate each other and the contribution of the diagram in Fig. 2 dominates. We write the differential probability for the decay of the virtual state X(3872) to π 0 π + π − in the form where D X (s 1 ) is the inverse propagator of the X(3872) resonance [25,27,28] that takes into account the couplings of X(3872) with the D * D +D * D decay channels and as well as with all non-(D * D +D * D)decay channels; and dΓ(X → π 0 π + π − ; s 1 , s 2 )/d √ s 2 is the X → π 0 π + π − differential decay width in the variable √ s 2 = m π + π − caused by the sum of the diagrams in Figs. 2 and 3.
The X(3872) resonance propagator constructed in Refs. [25,27,28] has good analytical and unitary properties (as for the case of scalar mesons [41,42]). The inverse propagator D X (s 1 ) has the form [25,27,28] where Γ non = Σ i Γ i is the total width of the X(3872) decay to all non-(D * D +D * D) channels which in the narrow region of the X(3872) peak ( where ρ ab (s 1 ) = s 1 − m and g A is the coupling constant of X with the D * 0D0 channel. At m where ρ ab (s 1 ) = m The sum of the probabilities of the X(3872) decay to all modes satisfies the unitarity [25,27,28] BR(X → (D * 0D0 + c.c.)) +BR(X → (D * + D − + c.c)) + Σ i BR(X → i) = 1. (11) The coupling of the X(3872) with the D * 0D0 system was introduced in Refs. [25][26][27][28] by means of the Lagrangian and the range of possible values of the coupling constant g 2 A /(16π) was determined from the analysis of the experimental data [3,6,8,9,13,15].
To describe the amplitudes of the D * → Dπ 0 decays, we use the expression where ǫ D * is the polarization four-vector of the D * meson, p π 0 and p D are the four-momenta of π 0 and D, respectively; g D * + D + π 0 = −gD * 0D0 π 0 . The effective vertex of the X(3872) → (D * D + D * D) → π 0 DD → π 0 π + π − transition corresponding to the sum of the diagrams in Figs. 2 and 3, in which the π + π − system is produced in the S wave, can be written as where ǫ X is the is the polarization four-vector of the X(3872), the amplitudes F 0 (s 1 , s 2 ) and F + (s 1 , s 2 ) describe the contributions from the neutral and charged intermediate D * D states, respectively, and We assume the S wave amplitudes of the processes D 0D0 → π + π − and D +D− → π + π − (entering in the amplitudes of the diagrams in Figs. 2 and 3) to be equal and approximate them in the region of the DD thresholds by an s 2 -independent constant g D 0D0 π + π − . Taking into account Eqs. (12), (13), and (14), the amplitude F 0 (s 1 , s 2 ) can be written in the form The four-vector under the integral sign we transform as follows This shows that after reducing the numerator and denominator in Eq. (16) by the factor ((k − p 3 ) 2 − m 2 D 0 ), the divergent part of the integral is proportional to p 1µ (i.e., the four-moment of the X(3872) resonance) and does not contribute to F 0 (s 1 , s 2 ) because (ǫ X , p 1 ) = 0. The analysis also shows that the term proportional to k µ (m 2 (17) gives a negligible contribution to the integral in Eq. (16) in the √ s 1 and √ s 2 region under consideration. Thus we get The amplitude F + (s 1 , s 2 ) is obtained from Eq. (18) by replacing the masses of neutral D * and D mesons by the masses of their charged partners. For the numerical calculation of the amplitudes F 0 (s 1 , s 2 ) and F + (s 1 , s 2 ), we use the method developed in Ref. [43]. Using Eq. (14) we express the differential width dΓ(X → π 0 π + π − ; s 1 , s 2 )/d √ s 2 in terms of the invariant amplitude G Xπ 0 π + π − (s 1 , s 2 ): where p(s 1 , s 2 ) = The width of the decay X → π 0 π + π − as a function of s 1 has the form Γ(X → π 0 π + π − ; s 1 ) and the probability of this decay is given by the expression Equations (22) and (23) indicate the kinematically allowable limits of integration. In fact, the main contributions in Eqs. (22) and (23) are concentrated in much smaller intervals. We now estimate the coupling constants g D * 0 D 0 π 0 and g D 0D0 π + π − .
For the total decay width of the D * 0 meson, only its upper limit is known so far: Γ D * 0 < 2.1 MeV [1]. On the other hand, the total decay width of the D * + meson and the branching ratio of the D * + → (Dπ) + decay are well known [1]: Γ D * + ≈ 83.6 keV, BR(D * + → (Dπ) + ) ≈ 98.4%. Assuming the isotopic symmetry for the coupling constants g D * Dπ , we have where p Dπ denotes the momentum of the final D or π meson in the D * rest frame. From here we find the decay width Γ D * 0 →D 0 π 0 ≈ 36 keV and the coupling constant Using also the value of BR(D * 0 → D 0 π 0 ) ≈ 64.7% [1], we get an estimate for the total decay width of the D * 0 meson: Γ D * 0 ≈ 55.6 keV. Here we note in passing the following. As the examples [44][45][46][47][48][49] show, the instability of the vector mesons in the intermediate states (i.e., the finiteness of their total widths) is important to take into account when estimating the contributions of logarithmic triangle singularities. In this case, Γ D * 0 is small. Nevertheless, its accounting in the D * 0 propagator (by replacing m 2 D * 0 → m 2 D * 0 − im D * 0 Γ D * 0 ) noticeably smoothes the logarithmic singularity in the amplitude of the diagram in Fig. 2 and the computed probability of the X(3872) → π 0 π + π − decay is reduced by approximately 30% as compared to that for Γ D * 0 = 0.
It is not yet clear whether the mass of the X(3872) state lies slightly above or slightly below the D * 0D0 threshold. The ±0.17 MeV uncertainty that the Particle Data Group [1] indicates allows for both possibilities. Tables II and III show the estimates for BR(X → π 0 π + π − ) at the same values of g 2 A /(16π) and Γ non as in Tab. I but for m X = 3.87169 ± 0.00017 GeV.
The present work is partially supported by the program No. II.15.1 of fundamental scientific researches of the Siberian Branch of the Russian Academy of Sciences, the project No. 0314-2019-0021.