The X(3872) tetraquarks in B and B_s decays

We discuss how the latest data on X(3872) in B and B_s decays speak about its tetraquark nature. The established decay pattern, including the up to date observations by CMS, are explained by the mixing of two quasi-degenerate, unresolvable, neutral states. The same mechanism also explains isospin violations in X decays and strongly suggests that the lurking charged partners are required to have very small branching fractions in J/psi rho^pm, well below the current experimental limits. In addition, a new prediction on the decay into J/psi omega final states is attained. The newest experimental observations are found to give thrust to the simplest tetraquark picture and call for a definitive, in-depth study of final states with charged rho mesons.

We discuss how the latest data on X(3872) in B and Bs decays speak about its tetraquark nature. The established decay pattern, including the up to date observations by CMS, are explained by the mixing of two quasi-degenerate, unresolvable, neutral states. The same mechanism also explains isospin violations in X decays and strongly suggests that the lurking charged partners are required to have very small branching fractions in J/ψ ρ ± , well below the current experimental limits. In addition, a new prediction on the decay into J/ψ ω final states is attained. The newest experimental observations are found to give thrust to the simplest tetraquark picture and call for a definitive, in-depth study of final states with charged ρ mesons.
Comparing to other similar decays, the following pattern is observed We will show how this pattern clearly emerges from the simplest decay diagram in Fig. 1 and a compact tetraquark picture of the X(3872). In addition, the pattern in (1) and (2), combined with our previous analysis [2] of the branching fractions of X(3872) → J/ψ + 2π/3π, allows to determine uniquely the mixing and couplings of the two . From these results we derive two new predictions 1. The branching ratio of the decays of B mesons into J/ψ + 3π 2. A definite range for the production of the charged tetraquark X ± in B decays to be compared with the present limit R − 2π < 1 [3]. These predictions can be tested experimentally and, if supported, would provide a decisive clarification on the nature of the X(3872). The compact tetraquark model was developed in [2,4,5]. It proposes that X(3872) belongs to a complex of four-quark bound states: X u , X d and X ± = [cu][cd], [cd] [cū]. These states are expected to be very close in mass.
In a different line, proximity to the DD * threshold of X(3872) and the notable lack of evidence of additional states nearby are the motivations of the alternative, molecular models of X(3872) and other exotic hadrons, which we do not consider in this letter (exotic hadrons are reviewed in [6][7][8][9][10][11][12]).

arXiv:2005.08764v1 [hep-ph] 18 May 2020
In a first estimate, Ref. [4] gave a X d − X u separation close to 2(m d − m u ) ∼ 7 MeV. However, a second state close to X(3872) has not been observed, and upper bounds have been given for the branching ratios of B meson decays into X ± [14,15]. Building on the analysis of isospin breaking hadron masses [16,17], which takes into account the effect of the electromagnetic interactions, it was suggested [2] that X u and X d are much closer in mass than expected, so as to be two unresolved lines inside the J/ψ π + π − peak. This quasi-degeneracy is reached assuming a separation of scales between the diquark size and the size of the whole diquark-antidiquark composite state. This conjecture was further addressed in [13].
Another result obtained in [2] was that X u −X d mixing, estimated from the branching ratios of X(3872) → J/ψ+2π or 3π, would push the branching ratio for the production of X ± in B meson decays well below the experimental limits quoted in Reffs. [14,15], thus calling for more refined searches.
Assuming a tetraquark X(3872), one has to create a light quark pair from the sea in the blob of Fig. 1, so that the overall weak decay is b + u, d, s B + ,B 0 ,Bs −→c + cs + (dd or uū) sea + u, d, s The decays B → X K are then described by two amplitudes: A 1 , where thes forms the Kaon with the spectator u or d quark, and A 2 , where it forms the Kaon with a d or u quark from the sea. In terms of the unmixed states With near degeneracy of X u,d , even a small qq annihilation amplitude inside the tetraquark could produce sizeable mixing. We consider the mass eigenstates in the isospin basis, namely (we can take cos φ > 0, so that −π/4 < φ < +π/4). It is straightforward 1 to compute the rate for B going to X(3872), the sum of two unresolved, almost degenerate lines, followed by decay into J/ψ + 2π/3π, as function of the mixing angle φ and of the ratio of the isospin zero and isospin one amplitudes, 2A 1 + A 2 , A 2 , respectively. The result [2] is reported in the two panels of Fig. 2 Let us now turn to the results (1) and (2). From Eqs. (5) to (7), and recalling that one easily finds the ratio of the B + to B 0 rates in (2). The result is where z = A few observations are in order 1. We have summed over the rates of X 1 and X 2 , as required by the hypothesis [2] that the two neutral states are both within the J/ψρ width.
Using the experimental branching ratios [3] and adding errors in quadrature, we find The corresponding region in φ, z space is reported in Fig. 2 For B 0 s decay, only the spectator quark can lead to the φ meson in the final state. The decay is described by one amplitude, A 3 , with the same role as Assuming A 3 = A 1 and neglecting A 2 we find R s0 2π (Solution 1) = 1.35 R s0 2π (Solution 2) = 0.08 (15) The pattern found by CMS selects uniquely Solution 1. This fact has a simple interpretation. In Solution 1, A 2 is very small and the mixing is such that the contribution of X u dominates in B 0 decay. Thus, to a good approximation, meson formation in B 0 decay is dominated by the spectator quark as in B s decay. One may wonder if A 3 = A 1 is compatible with the different spin of the final mesons in B s and B 0 decays. The near experimental equality of the branching ratios B(B 0 → K * 0 X(3872) → K * 0 J/ψ π + π − ) = (4.0 ± 1.5) × 10 −6 B(B 0 → K 0 X(3872) → K 0 J/ψ π + π − ) = (4.3 ± 1.3) × 10 −6 (16) is a control case, same flavor and different meson spin, that reassures that this is the case. Using the parameters of Solution 1, one obtains the two predictions in (3) and (4). We conclude that the new results by CMS mark an advancement in the understanding of the X(3872) problem and call for a few more steps to do on the experimental side which would allow to safely decide among existing interpretations.