Measurement of the Quasi-Two-Body B Decays

We study the contributions of the $B\rightarrow \psi(3770)K[\psi(3770)\rightarrow D\bar{D}]$, $B\rightarrow K^*(1410)\pi[K^*(1410)\rightarrow K\pi]$ and $B\rightarrow X(3872)K[X(3872)\rightarrow J/\psi\gamma, \psi(2S)\gamma, D\bar{D}\pi, J/\psi\omega, J/\psi\pi\pi$ and $D\bar{D}^*\pi]$ quasi-two-body decays. There are no existing previous measurement of the three-body branching fractions for three final states of the $X(3872)\rightarrow J/\psi\gamma$, $\psi(2S)\gamma$ and $D\bar{D}\pi$ but several quasi-two-body modes that can decay to this final state have been seen.

In general factorization approach, to obtain the amplitudes of the two-body decays, the Feynman quark diagrams should be plotted, and quasi-two-body decays of the heavy mesons can be also expressed in terms of some quark-graph amplitudes.For example we take  0 →  * (1410) +  − as an illustration.Under the factorization approach, its decay amplitude consists of three distinct factorizable terms: (i) the current-current process through the tree  →  transition, (ii) the transition process induced by  →  penguins, and (iii) the annihilation process.Note that weak-annihilation contributions are too small, so we ignore them in our calculations.

Quasi-Two-Body Decay Amplitudes
It is known that in the narrow width approximation, in the models we use to obtain the amplitudes of the decays, the 3body decay rate obeys the factorization relation [3] Br with  being a vector meson resonance and ,  1 , and  2 are pseudoscalar and vector final state mesons.The intermediate vector meson contributions to three-body decays are identified through the vector current, and their effects are described in terms of the Breit-Wigner formalism.The Breit-Wigner resonant term associated to quasi-two-body  +   state which seems to play an important role as indicated by experiments.We have to calculate the branching ratios of the Br( → ) by using the Feynman quark diagrams and using the experimental information for the Br( →  1  2 ) decays as follows [5]: Br ( (3872) →  (2) ) > 3.0%. ( We calculate the branching ratios of the intermediate states two-body decays.Feynman diagrams related to these decays are shown in Figures 1 and 2. A detailed discussion of the QCD factorization (QCDF) approach can be found in [6][7][8][9].Factorization is a property of the heavy-quark limit, in which we assume that the  quark mass is parametrically large.The QCDF formalism allows us to compute systematically the matrix elements of the effective weak Hamiltonian in the heavy-quark limit for certain twobody final states  +(0) ,  +(0) , and  * +  − .In this section, we obtain the amplitude of  +(0) →  +(0) ,  +(0) →  +(0) , and  0 →  * +  − decays by using the QCDF method.We adopt leading order Wilson coefficients at the scale   for QCDF approach.According to the QCDF, the amplitudes of the  +(0) →  +(0) ,  +(0) →  +(0) , and  0 →  * +  − decays are given by ( 0 →  * (1410) where is the absolute value of the 3momentum of the vector meson in the  rest frame.

Conclusion
In this research we have calculated the branching ratios of the  → (3770)[(3770) → ],  →  * (1410) [ * (1410) → ], and  → (3872)[(3872) → /, (2), , /, / and  * ] decays in the framework of the quasi-two-body method.We have also measured the branching ratios of two-body decays including the short-lived intermediate mesons by using the QCDF method.Our calculation results are shown in Table 3.There are no existing previous measurement branching fractions for some of the three-body decays such as  → /, (2), and , but quasi-two-body modes that can decay to these final states have been seen.

Table 1 :
Wilson coefficients   in the NDR scheme at the leading order and next to leading order.

Table 2 :
Numerical values of effective coefficients   for  →  transition at the leading and next to leading order and   = 3.