Measurement of the Quasi-Two-Body B Decays

Abstract

We study the contributions of the 𝐵→𝜓(3770)𝐾[𝜓(3770)→𝐷𝐷], 𝐵→𝐾^∗(1410)𝜋[𝐾^∗(1410)→𝐾𝜋], and 𝐵→𝑋(3872)𝐾[𝑋(3872)→𝐽/𝜓𝛾, 𝜓(2𝑆)𝛾, 𝐷𝐷𝜋, 𝐽/𝜓𝜔, 𝐽/𝜓𝜋𝜋, and 𝐷𝐷^∗𝜋] quasi-two-body decays. There are no existing previous measurements of the three-body branching fractions for three final states of the 𝑋(3872)→𝐽/𝜓𝛾, 𝜓(2𝑆)𝛾, and 𝐷𝐷𝜋, but several quasi-two-body modes that can decay to this final state have been seen.

1. Introduction

The three-body mesondecays are generally dominated by intermediate vector and scalar resonances; this means that there are a resonance state and a pseudoscalar meson which they proceed by quasi-two-body decays [1–4]. In fact, analysis of three-body 𝐵 decays using the Dalitz plot technique leads us to the many quasi-two 𝐵 decays [1]. The study of the 𝐵→𝜓(3770)𝐾, 𝐵→𝐾^∗(1410)𝜋, and 𝐵→𝑋(3872)𝐾 via quasi-two-body decays was considered. These include 𝐵⁺→𝜓(3770)𝐾⁺ and 𝐵⁰→𝜓(3770)𝐾⁰, observed in the 𝜓(3770)→𝐷⁰𝐷⁰ channel and also seen in 𝜓(3770)→𝐷⁺𝐷⁻; 𝐵⁰→𝐾^∗(1410)⁺𝜋⁻, seen in 𝐾^∗(1410)⁺→𝐾⁰𝜋⁺; 𝐵⁺→𝑋(3872)𝐾⁺ and 𝐵⁰→𝑋(3872)𝐾⁰, seen in 𝑋(3872)→𝐽/𝜓𝜋⁺𝜋⁻, 𝐽/𝜓𝜔 and 𝐷^∗0𝐷⁰𝐾⁰ channel [5]. The decays 𝐵⁺→𝑋(3872)𝐾⁺ and 𝐵⁰→𝑋(3872)𝐾⁰ have also been observed with 𝑋(3872)→𝐽/𝜓𝛾, 𝜓(2𝑆)𝛾, and 𝐷⁰𝐷⁰𝜋⁰ [5]. In this paper, we do not perform a Dalitz plot analysis but instead use information on intermediate modes including narrow resonances by studying the two-body invariant mass distributions, because there are no existing previous measurement branching fractions for some of the three-body decays such as 𝑋(3872)→𝐽/𝜓𝛾, 𝜓(2𝑆)𝛾, and 𝐷⁰𝐷⁰𝜋⁰, but quasi-two-body modes that can decay to these final states have been seen. Hence, we present the results of a search for the three-body decay including short-lived intermediate two-body modes that can decay to these final states.

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 𝐵⁰→𝐾^∗(1410)⁺𝜋⁻ as an illustration. Under the factorization approach, its decay amplitude consists of three distinct factorizable terms: (i) the current-current process through the tree 𝑏→𝑢 transition, (ii) the transition process induced by 𝑏→𝑠 penguins, and (iii) the annihilation process. Note that weak-annihilation contributions are too small, so we ignore them in our calculations.

2. 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 3-body decay rate obeys the factorization relation [3] Br𝐵⟶𝑅𝑀⟶𝑀₁𝑀₂𝑀=Br(𝐵⟶𝑅𝑀)×Br𝑅⟶𝑀₁𝑀₂, (1) with 𝑅 being a vector meson resonance and 𝑀, 𝑀₁, and 𝑀₂ 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 𝑅+𝑀 state which seems to play an important role as indicated by experiments. We have to calculate the branching ratios of the Br(𝐵→𝑅𝑀) by using the Feynman quark diagrams and using the experimental information for the Br(𝑅→𝑀₁𝑀₂) decays as follows [5]: Br𝜓(3770)⟶𝐷⁰𝐷⁰=()𝜓()52±5%,Br3770⟶𝐷⁺𝐷⁻=()𝐾41±4%,Br^∗(1410)⁺⟶𝐾⁰𝜋⁺=()𝑋()⟶𝐽6.6±1.3%,Br3872𝜓𝜋⁺𝜋⁻𝑋()⟶𝐽>2.6%,Br3872𝑋()⟶𝐽𝜓𝜔>1.9%,Br3872𝜓𝛾>9×10⁻³,Br𝑋(3872)⟶𝐷⁰𝐷⁰𝜋⁰>3.2×10⁻³,Br𝑋(3872)⟶𝐷^∗0𝐷⁰>5×10⁻³,Br𝑋(3872)⟶𝜓(2𝑆)𝛾>3.0%. (2) 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.

Figure 1 

Quark diagrams illustration of the processes of the 𝐵+(0)→𝜓𝐾+(0) and 𝑋𝐾+(0) decays.

Figure 2 

Quark diagrams illustration of the process of the 𝐵0→𝐾∗+𝜋− decay.

A detailed discussion of the QCD factorization (QCDF) approach can be found in [6–9]. Factorization is a property of the heavy-quark limit, in which we assume that the 𝑏 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 two-body final states 𝜓𝐾⁺⁽⁰⁾, 𝑋𝐾⁺⁽⁰⁾, and 𝐾^∗+𝜋⁻. In this section, we obtain the amplitude of 𝐵⁺⁽⁰⁾→𝜓𝐾⁺⁽⁰⁾,  𝐵⁺⁽⁰⁾→𝑋𝐾⁺⁽⁰⁾, and 𝐵⁰→𝐾^∗+𝜋⁻ decays by using the QCDF method. We adopt leading order Wilson coefficients at the scale 𝑚_𝑏 for QCDF approach. According to the QCDF, the amplitudes of the 𝐵⁺⁽⁰⁾→𝜓𝐾⁺⁽⁰⁾, 𝐵⁺⁽⁰⁾→𝑋𝐾⁺⁽⁰⁾, and 𝐵⁰→𝐾^∗+𝜋⁻ decays are given by 𝐴𝐵⁺⁽⁰⁾⟶𝜓(3770)𝐾⁺⁽⁰⁾=√2𝐺_𝐹𝑚_𝜓𝑓_𝜓𝜖_𝜓⋅𝑝_𝐵𝐹₁^𝐵𝐾𝑚²_𝜓×𝑎₂𝑉_𝑐𝑏𝑉^∗_𝑐𝑠+𝑎₃𝜆_𝑝,𝐴𝐵⁺⁽⁰⁾⟶𝑋(3872)𝐾⁺⁽⁰⁾=√2𝐺_𝐹𝑚_𝑋𝑓_𝑋𝜖_𝑋⋅𝑝_𝐵𝐹₁^𝐵𝐾𝑚²_𝑋×𝑎₂𝑉_𝑐𝑏𝑉^∗_𝑐𝑠+𝑎₃𝜆_𝑝,𝐴𝐵⁰⟶𝐾^∗(1410)⁺𝜋⁻=√2𝐺_𝐹𝑚_𝐾^∗𝑓_𝐾∗𝜖_𝐾^∗⋅𝑝_𝐵𝐹₁^𝐵𝜋𝑚²_𝐾^∗×𝑎₁+𝑎₂𝑉_𝑢𝑏𝑉^∗_𝑢𝑠+𝑎₄𝜆_𝑝, (3)(4)(5) where 𝜖_𝑉⋅𝑝_𝐵=𝑚_𝐵𝑚_𝑉𝑝_𝑉,𝑉=𝜓,𝑋,𝐾^∗,𝜆_𝑝=_{𝑝=𝑢,𝑐}𝑉_𝑝𝑏𝑉^∗_𝑝𝑠,𝐹₁^𝐵𝑃𝑞²=𝐹(0)1−𝑎_𝐹𝑞²/𝑚²_𝐵+𝑏_𝐹𝑞²/𝑚²_𝐵²,(𝑃=𝐾,𝜋), (6) where |𝑝_𝑉|=(𝑝⁰_𝑉)²−(𝑚_𝑉)² is the absolute value of the 3-momentum of the vector meson in the 𝐵 rest frame.

3. Numerical Results

To proceed with the numerical calculations, we need to specify the input parameters. For the CKM matrix elements, we use 𝑉_𝑢𝑏=0.00351^+0.00015_−0.00014, 𝑉_𝑢𝑠=0.22534±0.00065, 𝑉_𝑐𝑏=0.0412^+0.001_−0.0005, and 𝑉_𝑐𝑠=0.97344±0.00016 [5]. For 𝐵→𝐾 and 𝐵→𝜋 form factors, a good parametrization for the 𝑞² dependence can be given in terms of three parameters (see (6)). We fix for 𝐵→𝐾 transition 𝐹(0)=0.374, 𝑎_𝐹=1.42, 𝑏_𝐹=0.434 [10] and for 𝐵→𝜋 transition 𝐹(0)=0.25, 𝑎_𝐹=1.73, 𝑏_𝐹=0.95 [11], namely, 𝐹^𝐵𝐾(𝑚²_𝜓)=0.962, 𝐹^𝐵𝐾(𝑚²_𝑋)=1.283, and 𝐹^𝐵𝜋(𝑚²_𝐾^∗₍₁₄₁₀₎)=0.284. The meson masses and decay constants needed in our calculations are (in units of Mev) 𝑚_𝐵=5279.25±0.17, 𝑚_𝜓=3772.15±0.33,  𝑚_𝑋=3871.68±0.17,  𝑚_𝐾^∗₍₁₄₁₀₎=1414±15 [5]; 𝑓_𝜓=384±14 [12]; 𝑓_𝑋=446±16 [13]. The Wilson coefficients 𝑐_𝑖 have been calculated in different schemes. In this paper we will use consistently the naive dimensional regularization (NDR) scheme. The values of 𝑐_𝑖 at the scale 𝜇=𝑚_𝑏 at the leading order (LO) and next to leading order (NLO) are shown in Table 1 [9, 14]. Numerical values of effective coefficients 𝑎_𝑖 for 𝑏→𝑠 transition at 𝑁_𝑐=3 are shown in Table 2. According to Table 2, it is not much difference between effective coefficients 𝑎_𝑖 at the LO and NLO; therefore, we use them at the LO: 𝑎₁=1.03,  𝑎₂=0.11,  𝑎₃=0.002, and 𝑎₄=−0.03 [9, 14].

Table 1 

Wilson coefficients 𝑐_𝑖 in the NDR scheme at the leading order and next to leading order.

Table 2 

Numerical values of effective coefficients 𝑎_𝑖 for 𝑏→𝑠 transition at the leading and next to leading order and 𝑁_𝑐=3.

Now we are able to calculate the branching ratios of the 𝐵⁺⁽⁰⁾→𝜓(3770)𝐾⁺⁽⁰⁾,  𝐵⁺⁽⁰⁾→𝑋(3872)𝐾⁺⁽⁰⁾ and 𝐵⁰→𝐾^∗(1410)⁺𝜋⁻ decays by using (3)–(5) as follows Br𝐵⁺⁽⁰⁾⟶𝜓(3770)𝐾⁺⁽⁰⁾=2.95^+1.10_−0.86×10⁻⁴,Br𝐵⁺⁽⁰⁾⟶𝑋(3872)𝐾⁺⁽⁰⁾=3.12^+1.00_−0.85×10⁻⁴,Br𝐵⁰⟶𝐾^∗(1410)⁺𝜋⁻=2.57^+0.58_−0.52×10⁻⁵. (7) The experimental result for Br(𝐵→𝜓(3770)𝐾) which turns out to be (4.9±1.3)×10⁻⁴ [5] is in very good agreement with our prediction. As we know the branching ratio of 𝐵→𝑋(3872)𝐾 decay has already been estimated: (a) in [15] they have used QCD sum rules to calculate the branching ratio and predicted (0.38±0.06)×10⁻⁶, (b) in [16] they have analyzed the decay 𝐵→𝑋𝐾 and the decays of 𝐵 into 𝐷⁰𝐷^0∗𝐾 and 𝐷^0∗𝐷⁰𝐾 near the threshold for the charm mesons by separating the decay amplitudes into short-distance factors and long-distance factors, and they have predicted for branching ratio 2.9×10⁻⁵, while the experimental result of this decay is Br(𝐵→𝜓(3770)𝐾)<3.2×10⁻⁴ [17]. By using (1), (2), and (7) the branching ratios of quasi-two-body decays are summarized in Table 3. Note that the experimental BRs for 𝑋(3872) decay modes (see (2)) only have lower bounds, so the theory predictions involving these modes in Table 3 only have lower bounds.

Table 3 

Branching ratios of quasi-two-body decays obtained from the studies of three-body decays based on the factorization approach.

4. Conclusion

In this research we have calculated the branching ratios of the 𝐵→𝜓(3770)𝐾[𝜓(3770)→𝐷𝐷], 𝐵→𝐾^∗(1410)𝜋[𝐾^∗(1410)→𝐾𝜋], and 𝐵→𝑋(3872)𝐾[𝑋(3872)→𝐽/𝜓𝛾, 𝜓(2𝑆)𝛾, 𝐷𝐷𝜋, 𝐽/𝜓𝜔, 𝐽/𝜓𝜋𝜋 and 𝐷𝐷^∗𝜋] 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 𝐵→𝐽/𝜓𝛾𝐾,  𝜓(2𝑆)𝛾𝐾, and 𝐷𝐷𝜋𝐾, but quasi-two-body modes that can decay to these final states have been seen.