# Measurement of the Quasi-Two-BodyB Decays

Authors: Hossein Mehraban; Behnam Mohammadi
Journal: Advances in High Energy Physics
 (2013)

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## Abstract

We study the contributions of theB→ψ(3770)K[ψ(3770)→DD̅], B→K*(1410)π[K*(1410)→Kπ], and B→X(3872)K[X(3872)→J/ψγ, ψ(2S)γ, DD-π, 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 DD̅π, but several quasi-two-body modes that can decay to this final state have been seen.


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## Body

## 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)→D0D-0 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 D0D-0π0 [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 D0D-0π0, 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 takeB0→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]
(1)Br(B⟶RM⟶M1M2M)=Br(B⟶RM)×Br(R⟶M1M2),
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]:
(2)Br(ψ(3770)⟶D0D-0)=(52±5)%,Br(ψ(3770)⟶D+D-)=(41±4)%,Br(K*(1410)+⟶K0π+)=(6.6±1.3)%,Br(X(3872)⟶Jψπ+π-)>2.6%,Br(X(3872)⟶Jψω)>1.9%,Br(X(3872)⟶Jψγ)>9×10-3,Br(X(3872)⟶D0D-0π0)>3.2×10-3,Br(X(3872)⟶D-*0D0)>5×10-3,Br(X(3872)⟶ψ(2S)γ)>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.Figure 1
Quark diagrams illustration of the processes of theB+(0)→ψK+(0) and XK+(0) decays.Figure 2
Quark diagrams illustration of the process of theB0→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 mb 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
(3)A(B+(0)⟶ψ(3770)K+(0))=2GFmψfψ(ϵψ·pB)F1BK(mψ2)×(a2VcbVcs*+a3λp),(4)A(B+(0)⟶X(3872)K+(0))=2GFmXfX(ϵX·pB)F1BK(mX2)×(a2VcbVcs*+a3λp),(5)A(B0⟶K*(1410)+π-)=2GFmK*fK*(ϵK*·pB)F1Bπ(mK*2)×[(a1+a2)VubVus*+a4λp],
where
(6)ϵV·pB=mBmV|p→V|,(V=ψ,X,K*),λp=∑p=u,c‍VpbVps*,F1BP(q2)=F(0)1-aF(q2/mB2)+bF(q2/(mB2))2,(P=K,π),
where |p→V|=(pV0)2-(mV)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 useVub=0.00351-0.00014+0.00015, Vus=0.22534±0.00065, Vcb=0.0412-0.0005+0.001, and Vcs=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, aF=1.42, bF=0.434 [10] and for B→π transition F(0)=0.25, aF=1.73, bF=0.95 [11], namely, FBK(mψ2)=0.962, FBK(mX2)=1.283, and FBπ(mK*(1410)2)=0.284. The meson masses and decay constants needed in our calculations are (in units of Mev) mB=5279.25±0.17, mψ=3772.15±0.33,  mX=3871.68±0.17,  mK*(1410)=1414±15 [5]; fψ=384±14 [12]; fX=446±16 [13]. The Wilson coefficients ci have been calculated in different schemes. In this paper we will use consistently the naive dimensional regularization (NDR) scheme. The values of ci at the scale μ=mb at the leading order (LO) and next to leading order (NLO) are shown in Table 1 [9, 14]. Numerical values of effective coefficients ai for b-→s- transition at Nc=3 are shown in Table 2. According to Table 2, it is not much difference between effective coefficients ai 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 coefficientsci in the NDR scheme at the leading order and next to leading order.
c
1
c
2
c
3
c
4
NLO
1.081
-
0.190
0.014
-
0.036
LO
1.117
-
0.268
0.012
-
0.027Table 2
Numerical values of effective coefficientsai for b-→s- transition at the leading and next to leading order and Nc=3.
a
1
a
2
a
3
a
4
NLO
1.02
0.17
0.003
-
0.02
LO
1.03
0.11
0.002
-
0.03Now we are able to calculate the branching ratios of theB+(0)→ψ(3770)K+(0),  B+(0)→X(3872)K+(0) and B0→K*(1410)+π- decays by using (3)–(5) as follows
(7)Br(B+(0)⟶ψ(3770)K+(0))=(2.95-0.86+1.10)×10-4,Br(B+(0)⟶X(3872)K+(0))=(3.12-0.85+1.00)×10-4,Br(B0⟶K*(1410)+π-)=(2.57-0.52+0.58)×10-5.
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 D0*D-0K 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.
Decay mode
Theory
Exp. [5]
Units
ψ
(
3770
)
K
+;
ψ
→
D
0
D
-
0
1.56
±
0.32
1.6
±
0.4
×10-4
ψ
→
D
+
D
-
12.28
±
2.61
9.4
±
3.5
×10-5
ψ
(
3770
)
K
0;
ψ
→
D
0
D
-
0
1.56
±
0.32
<1.23
×10-4
ψ
→
D
+
D
-
1.23
±
0.26
<1.88
×10-4
X
(
3872
)
K
+;
X
→
J
/
ψ
π
+
π
-
>9.17
8.6
±
0.8
×10-6
X
→
J
/
ψ
γ
>3.20
2.1
±
0.4
×10-6
X
→
ψ
(
2
S
)
γ
>10.59
4
±
4
×10-6
X
→
D
0
D
-
0
π
0
>0.016
1.0
±
0.4
×10-4
X
→
D
-
0
*
D
0
>0.20
8.5
±
2.6
×10-5
X
(
3872
)
K
0;
X
→
J
/
ψ
π
+
π
-
>9.17
4.3
±
1.3
×10-6
X
→
J
/
ψ
γ
>3.20
<2.41
×10-6
X
→
ψ
(
2
S
)
γ
>1.06
<6.61
×10-6
X
→
D
0
D
-
0
π
0
>0.016
1.7
±
0.8
×10-4
X
→
D
-
0
*
D
0
>0.20
1.2
±
0.4
×10-4
X
→
J
/
ψ
ω
>6.72
6.0
±
3.2
×10-6
K
*
(
1410
)
+
π
-;
K
*
→
K
0
π
+
2.53
±
0.21
<3.8
×10-6


## 4. Conclusion

In this research we have calculated the branching ratios of theB→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, DD-π, 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 DD-πK, but quasi-two-body modes that can decay to these final states have been seen.



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