# Final State Interaction Effects on theB+→J/ψρ+ Decay

Authors: Behnam Mohammadi; Hossein Mehraban
Journal: Advances in High Energy Physics
 (2012)

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## Abstract

The exclusive decay ofB+→J/ψρ+ is studied in the framework of the QCD factorization (QCDF) method and final state interaction (FSI). A direct B+→J/ψρ+ decay is only occurred via a tree and a penguin based on the quark diagram analysis. The result that is found by using the QCDF method is less than the experimental result, so, the role of FSI is considered. The intermediate states D+D̅0, D+*D̅0*, D+*D̅0, and D+D̅0* via the exchange of D- and D-* are contributed to the B+→J/ψρ+ decay. The above intermediate states is calculated by using the QCDF method. In the FSI effects the results of our calculations depend on “η” as the phenomenological parameter. The range of this parameter are selected from 1 to 2. For the exchanged particles D- and D-*, it is found that if η=1.58~1.83 is selected the numbers of the branching ratio are placed in the experimental range. The experimental branching ratio of B+→J/ψρ+ decay is (5±0.8)×10-5, and our prediction number is (1.42±0.36)×10-5 in the absence of FSI effects, and it becomes (4.2~5.8)×10-5 when FSI contributions are taken into account.


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## Body

## 1. Introduction

The importance of FSI in weak nonleptonicB meson decays is investigated by using a relativistic chiral unitary approach based on coupled channels [1–3]. The chiral Lagrangian approach is proved to be reliable for evaluating hadronic processes, but there are too many free parameters which are determined by fitting data, so that its applications are much constrained. Therefore, we have tried to look for some simplified models which can give rise to reasonable estimation of FSI [4, 5]. The FSI can be considered as a rescattering process of some intermediate two-body states with one particle exchange in the t-channel and computed via the absorptive part of the hadronic loop level (HLL) diagrams. The calculation with the single-meson-exchange scenario is obviously much simpler and straightforward. Moreover, some theoretical uncertainties are included in an off-shell form factor which modifies the effective vertices. Since the particle exchanged in the t-channel is off shell and since final state particles are hard, form factors or cutoffs must be introduced to the strong vertices to render the calculation meaningful in perturbation theory. If the intermediate two body mesons are hard enough, so that the perturbative calculation can make sense and work perfectly well, but the FSI can be modelled as the soft rescattering of the intermediate mesons. When one or two intermediate meson can reach a low-energy region where they are not sufficiently hard, one can be convinced that at this region the perturbative QCD approach fails or cannot result in reasonable values. If the intermediate mesons are soft, one can conjecture that at this region the nonperturbative QCD would dominate, and it could be attributed into the FSI effects. Because all FSI processes are concerning nonperturbative QCD [6], we have to rely on phenomenological models to analyze the FSI effects in certain reactions. In fact, after weak decays of heavy mesons, the particles produced can rescatter into other particle states through nonperturbative strong interaction. In the FSI the decay B+→J/ψρ+ is a very interesting mode [7]. We calculated the B+→J/ψρ+ decay according to QCDF method and selected the leading order Wilson coefficients at the scale mb and obtained the BR(B+→J/ψρ+)=(1.42±0.36)×10-5. The FSI can give sizable corrections, and we can utilize it [8]. Rescattering amplitude can be derived by calculating the absorptive part of triangle diagrams. In this case, intermediate states are D+D¯0, D+*D¯0*, D+*D¯0, and D+D¯0*. Then, we calculated the B+→J/ψρ+ decay according to HLL method. Taken FSI corrections into account the branching ratio of B+→J/ψρ+ becomes (4.2~5.8)×10-5 and the experimental result of this decay is (5±0.8)×10-5 [9].This paper is organized as follows. We present the calculation of QCDF forB+→J/ψρ+ decay in Section 2. In Section 3, we calculate the amplitudes of the intermediate states of B+→D+(*)D¯0(*) decays. Then, we present the calculation of HLL for B+→J/ψρ+ decay in Section 4. In Section 5, we give the numerical results, and in the last section, we have a short conclusion.


## 2.B+→J/ψρ+ Decay in QCD Factorization

A detailed discussion of the QCDF approach can be found in [10, 11]. 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 J/ψρ+. In this section, we obtain the amplitude of B+→J/ψρ+ decay using QCDF method. In factorization approach, there are color-suppressed tree and allowed penguin diagrams to B+→J/ψρ+ decay. We adopt leading order Wilson coefficients at the scale mb for QCDF approach. The diagrams describing this decay are shown in Figure 1. According to the QCDF, the amplitude of B+→J/ψρ+ decay is given by
(2.1)AQCDF(B+⟶Jψρ+)=-iGF2fJ/ψmJ/ψ{2VBρ(mJ/ψ2)mB+mρ(ϵJ/ψ⋅ϵρ)(mB+mρ)A1Bρ(mJ/ψ2)-(ϵJ/ψ⋅pB)(ϵρ⋅pB)2A2Bρ(mJ/ψ2)mB+mρ-iɛμναβϵJ/ψμϵρνpBαp1β×2VBρ(mJ/ψ2)mB+mρ}×{VcbVcd*a2-VtbVtd*[a3+rχJ/ψ(a5+a7+a9)]},
where
(2.2)rχJ/ψ=2mJ/ψmbfJ/ψ⊥fJ/ψ.
The effective coefficients ai, which are specific to the factorization approach, and defined as
(2.3)ai=cieff+1Ncci+1eff(i=odd),ai=cieff+1Ncci-1eff(i=even),
where the quantities of cieff are effective Wilson coefficients at the renormalization scale μ for the b¯→d¯ transition. In the above amplitude the determination of a2 in the b→c current-current transitions has received a lot of attention, the quantities of a3, a5 and a7, a9 are the QCD-penguin and electroweak-penguin coefficients, respectively. Numerical values of ai(i=1,…,10) for representative value of the phenomenological parameter Nc are displayed in Section 5.Figure 1
The Feynman diagrams contributing to theB+→J/ψρ+ decay.


## 3. Amplitudes of Intermediate State

To estimate theB+→D+(*)D¯0(*) decays amplitudes of intermediate states [12], the QCDF method is again used. For B+→D+D¯0 decay, Feynman diagrams are shown in Figure 2, and the amplitude comes
(3.1)A(B+⟶D+D¯0)=iGF2fD{(b3+12b3,EW)F0BD(mB2-mD2)×{[a4+a10+rχD(a6+a8)](a1+a2)VcbVcd*+[a4+a10+rχD(a6+a8)]λp}+fBfD(b3+12b3,EW)λp},
where
(3.2)λp=∑p=u,c‍VpbVpd*,b3=CFNc2[c3A1i+c5(A3i+A3f)+Ncc6A3f],b3,EW=CFNc2[c9A1i+c7(A3i+A3f)+Ncc8A3f],A1i≈2παs[9(XA-4+π23)+rχD+rχD-XA2],A3i=0,A3f≈6παs(rχD++rχD-)(2XA2-XA),XA=(1+ρeiϕ)ln⁡mBΛQCD.
There are also some similar diagrams such as B+→D+*D¯0* and B+→D+*D¯0(D+D¯0*) decays, which have polarization vectors. In their amplitude, just contribution of a1 is considered, which has the dominant contribution. So the amplitude of B+→D+*D¯0* decay, read as
(3.3)A(B+⟶D+*D¯0*)=-iGF2fD*mD*VcbVcd*×{(ϵ1*⋅ϵ2*)(mB+mD*)A1BD*(mD*2)-(ϵ1⋅pB)(ϵ2⋅pB)2A2BD*(mD*2)mB+mD*},
where A1BD* and A2BD* are form factors for B→D* [13], and the amplitude of B+→D+*D¯0 is given by
(3.4)A(B+⟶D+*D¯0)=2GFmD*VcbVcd*(ϵD*⋅pD)[fD*F1BD(mD*2)+fDA0BD*(mD2)].Figure 2
B
+
→
D
+
D
¯
0 decay diagrams.


## 4. Final State Interaction ofB+→J/ψρ+ Decay

In our approach, we consider only leading contributions in charmless FSI, when the FSI for decay is calculated, two-body intermediate states, such as,D+D¯0, D+*D¯0*, D+*D¯0, and D+D¯0* via the exchange of D- and D-* are contributed to the B+→J/ψρ+ decay. The quark model for B+→D+(*)D¯0(*)→J/ψρ+ diagram is shown in Figure 3. This diagram just shows tree level (a1 contribution), while there are other contributions that were calculated in the last section. The diagrams which determine the HLL effects on the rate of B+→J/ψρ+ decay are shown in Figure 4. At the HLL, the charge exchange reaction between the J/ψ and ρ can proceed either by exchange of a D or D* meson. In order to calculate the various Feynman diagrams for the reactions D+D¯→J/ψ+ρ. And so forth (were calculated in the nonrelativistic quark model [14]) we need the construct the effective three mesons vertices, such as, ψDD and ρDD. We begin with an SU(4) symmetry so as to include charm, and we introduce pseudoscalar (P=φaλa) and vector (Vμ=vaμλa) meson matrices, where φa and vaμ are pseudoscalar and vector multiplets and the λs are SU(4) generators. The free meson lagrangian is read [15]
(4.1)£0=Tr⁡(∂μP†∂μP)-12Tr⁡(∂μV†,ν-∂νV†,μ)(∂μVν-∂νVμ).
Interaction Lagrangians are generated by replacing the space-time derivative with a gauge covariant one ∂μf→Dμf=∂μf+[Aμ,f], where Aμ=-ig/2Vμ. The vector mesons are recognizable as playing roles of gauge bosons. As is usual in effective field theory strategies, to keep gauge consistency, we must collect terms of order g2. The effective lagrangian is read
(4.2)£int⁡=igTr⁡(PVμ∂μP-∂μPVμP)+12g2Tr⁡(PVμVμP-PVμPVμ)+igTr⁡(∂μVν[Vμ,Vν]+[Vμ,Vν]∂μVν)+g2Tr⁡(VμVν[Vμ,Vν]).
The order g terms deliver three-point vertices and the order g2 terms are responsible for so-called contact terms or four-point couplings which are necessary for a gauge invariant theory. The interactions of interest for this study are the following [5]:
(4.3)£DDρ=igDDρ(Dρμ∂μD¯-∂μDρμD¯),£D*Dρ=-gD*Dρϵμναβ(D∂μρν∂αD¯β*+∂μDν*∂αρβD¯),£D*D*ρ=igD*D*ρ{∂μDν*ρμD¯*ν-Dν*ρμ∂μD¯*ν+(D*ν∂μρν-∂μDν*ρν)D¯*μ+D*μ(ρν∂μD¯ν*-∂μρνD¯*ν)},£ψDD=igψDDψμ(∂μDD¯-D∂μD¯),£ψD*D=-gψD*Dϵμναβ∂μψν(∂αDβ*D¯+D∂αD¯β*),£ψD*D*=-igψD*D*{ψμ(∂μD*νD¯ν*-D*ν∂μD¯ν*)+(∂μψνD*ν-ψν∂μD*ν)D¯*μ+D*μ(ψν∂μD¯ν*-∂μψνD¯*ν)},£ψρDD=-gρDDgψDDψμDρμD¯,£ψρD*D*=gψD*D*gρD*D*(2D*νρμD¯ν*-D*νρνD¯*μ-D*μρνD¯ν*).
Here ϵ0123=+1 and we define the charm meson iso-doublets as
(4.4)D¯T=(D¯0,D-),D=(D0,D+),D¯*T=(D¯*0,D*-),D*=(D*0,D*+).
In this approach there are several coupling constants in (4.3). Methods for constraining them will be discussed below. The O(g2) terms carry one power of coupling constant for each three-point vertices from which the contact term collapses. The coupling constants gψDD, gρDD, gψD*D*, and gρD*D* can be derived in the standard framework of the vector meson dominance model (VDM) [16–18]. By using VDM to the coupling of γDD we have
(4.5)γρgρDD=παem,γJ/ψgψDD=43παem,
where
(4.6)γρ=3Γρ→e+e-mραem,γJ/ψ=3ΓJ/ψ→e+e-mJ/ψαem.
The same method can be used for the γD*D* coupling. Vector dominance gives
(4.7)gρD*D*=gρDD,gψD*D*=gψDD.
For the ρD*D and ψD*D couplings the VDM can be applied to the radiative decays of D* into D, that is, D*→Dγ. By using the same technique we have
(4.8)γρgρD*D=13παem(gV±+2gV0),γJ/ψgψDD=43παemgV±=43παemgV0,
where gV±,0 are calculated in the model prediction based on relativistic potential model that we follow [19]. With the above preparation we can write out the decay amplitude involving HLL contributions with the following formula:
(4.9)A(B(pB)⟶M(p1)M(p2)⟶M(p3)M(p4))=12∫‍d3p1→2E1(2π)3d3p2→2E2(2π)3×(2π)4δ4(pB-p1-p2)M(B⟶M1M2)×G(M1M2⟶M3M4),
for which both intermediate mesons (M1,M2) are pseudoscalar. And
(4.10)A(B(pB)⟶M(p1)M(p2)⟶M(p3)M(p4))aaB(pB)=-iGF22∫‍d3p1→2E1(2π)3d3p2→2E2(2π)3aaB(pB)=×(2π)4δ4(pB-p1-p2)fD*mD*VcbVcd*aaB(pB)=×[2A2BM1(m22)mB+m1(ϵ1*⋅ϵ2*)(mB+m1)A1BM1(m22)-(ϵ1*⋅pB)(ϵ2*⋅pB)2A2BM1(m22)mB+m1]aaB(pB)=×G(M1M2⟶M3M4),
in which both meson are vector. Also G(M1M2→M3M4) involves hadronic vertices factor, which is defined as
(4.11)〈D(p3)ρ(ɛ2,p2)|i£|D(p1)〉=-igDDρϵ2⋅(p1+p3),〈D*(ϵ3,p3)ρ(ϵ2,p2)|i£|D(p1)〉=-i2gD*Dρϵμναβɛ2μϵ3*νp1αp2β,〈D*(ϵ3,p3)ρ(ϵ2,p2)|i£|D*(ϵ1,p1)〉=-iϵ1βϵ2νϵ3λ[2p1νgαβ-(p1+p2)αgνβ+2p2βgαν],〈D(p3)ψ(ϵ2,p2)|i£|D(p1)〉=-igψDDϵ2⋅(p1+p3),〈D*(ϵ3,p3)ψ(ϵ2,p2)|i£|D(p1)〉=-igψD*Dϵμναβϵ2μϵ3*νp1αp2β,〈D*(ϵ3,p3)ψ(ϵ2,p2)|i£|D*(ϵ1,p1)〉=-iϵ1βϵ2νϵ3λ[2p1νgαβ-(p1+p2)αgνβ+2p2βgαν].
The dispersive part of the rescattering amplitude can be obtained from the absorptive part via the dispersion relation [6, 20]:
(4.12)DisM(mB2)=1π∫s∞‍AbsM(s′)s′-mB2ds′,
where s′ is the square of the momentum carried by the exchanged particle and s is the threshold of intermediate states, in this case s~mB2. Unlike the absorptive part, the dispersive contribution suffers from the large uncertainties arising from the complicated integrations. So the amplitudes of the mode B+→D+(p1)D¯0(p2)→J/ψ(p3,ϵ3)ρ+(p4,ϵ4), where D- and D-* mesons are exchanged, respectively, are given by
(4.13)Abs(4a)=∫-11‍|p1→|d(cos⁡θ)16πmBA(B+⟶D+D¯0)(-i)gψDD(2ϵ3⋅p1)×(-i)gDDρ(2ϵ4⋅p2)F2(q2,mD2)q2-mD2=-∫-11‍|p1→|d(cos⁡θ)4πmBgψDDgDDρA(B+⟶D+D¯0)F2(q2,mD2)q2-mD2H1,
where θ is the angel between p→1 and p→3, q is the momentum of the exchange D meson, and
(4.14)H1=ϵ3μp1μϵ4νp2ν=(p10-|p1→|)(p20-|p2→|),q2=m12+m32-2E1E3|p1→||p3→|cos⁡θ,
and F(q2,mD2) is the form factor defined to take care of the off-shell of the exchange particles, which introduced as [6, 21]
(4.15)F(q2,mD2)=(Λ2-mD2Λ2-q2)n.
The form factor (i.e., n=1) normalized to unity at q2=mD2. mD and q are the physical parameters of the exchange particle and Λ is a phenomenological parameter. It is obvious that for q2→0, F(q2,mD2) becomes a number. If Λ≫mi then F(q2,mD2) turns to be unity, whereas, as q2→∞ the form factor approaches to zero and the distance becomes small and the hadron interaction is no longer valid. Since Λ should not be far from the mD and q, we choose
(4.16)Λ=mD+ηΛQCD,
where η is the phenomenological parameter that its value in the form factor is expected to be of the order of unity and can be determined from the measured rates, and
(4.17)Abs(4b)=∫-11‍|p1→|d(cos⁡θ)16πmBA(B+⟶D+D¯0)(-i)2gD*Dρϵμναβϵ4μp2αp4β×(-i)gψD*Dϵρσληϵ3ρp1λp3η(gσν-qσqνmD*2)F2(q2,mD*2)q2-mD*2=-∫-11‍|p1→|d(cos⁡θ)16πmBgψD*DgD*DρA(B+⟶D+D¯0)F2(q2,mD*2)q2-mD*2H2,
where
(4.18)H2=-2[(p1⋅p4)(p2⋅p3)-(p1⋅p2)(p3⋅p4)]+p30[p40(p1⋅p2)-(p20-|p2→|)(p1⋅p4)]+(p10-|p1→|)[(p20-|p2→|)(p3⋅p4)-p40(p2⋅p3)].
For diagrams (4c) and (4d) in Figure 4 the absorptive part of the amplitude corresponding to the process of B+→D+*(p1,ϵ1)D¯0*(p2,ϵ2)→J/ψ(p3,ϵ3)ρ+(p4,ϵ4), where D- and D-* mesons are exchanged, respectively, is read as
(4.19)Abs(4c)=-iGF22∫‍d3p1→2E1(2π)3d3p2→2E2(2π)3(2π)4δ4(pB-p1-p2)×(-i)gψD*Dϵμναβϵ3μp3νϵ1αp1β(-i2)gD*Dρϵρσληϵ4ρp4σϵ2λp2ηF2(q2,mD2)q2-mD2×fD*mD*VcbVcd*{(ϵ1*⋅ϵ2*)(mB+mD*)A1BD*(mD*2)-(ϵ1⋅pB)(ϵ2⋅pB)2A2BD*(mD*2)mB+mD*}=iGF16πmBfD*mD*gψD*DgD*DρVcbVcd*×∫-11‍|p1→|d(cos⁡θ)F2(q2,mD2)q2-mD2×{(mB+mD*)A1BD*(mD*2)H3-2A2BD*(mD*2)mB+mD*H3′},
where
(4.20)H3=ϵμναβϵρσληϵ3μp3νp1βϵ4ρp4σp2η(-gχα+p1αp1χmD*2)(-gλχ+p2λp2χmD*2)=-2[(p1⋅p2)(p3⋅p4)-(p1⋅p4)(p2⋅p3)]+p40[(p10-|p1→|)(p2⋅p3)-p30(p1⋅p2)]+(p20-|p2→|)[p30(p1⋅p4)-(p10-|p1→|)(p3⋅p4)],H3′=ϵμναβϵρσληϵ3μp3νp1βϵ4ρp4σp2η(-pBα+(p1⋅pB)p1αmD*2)(-pBλ+(p2⋅pB)p2λmD*2)=mB2{(p10-|p1→|)[p40(p2⋅p3)-(p20-|p2→|)(p3⋅p4)]-2[(p1⋅p2)(p3⋅p4)-(p1⋅p4)(p2⋅p3)]+p30[(p20-|p2→|)(p1⋅p4)-p40(p1⋅p2)]+[p3⋅p4-p30p40,p10(p20-|p2→|)+p20(p10-|p1→|)-p1⋅p2]+3p40[p30(p1⋅p2)-p10(p2⋅p3)]+[p1⋅p4-p40(p10-|p1→|)][p2⋅p3-p30(p20-|p2→|)]-2p20[p30(p1⋅p4)-p10(p3⋅p4)]+p20p30[p10p40-p1⋅p4](p10-|p1→|)},Abs(4d)=-iGF22∫‍d3p1→2E1(2π)3d3p2→2E2(2π)3(2π)4δ4(pB-p1-p2)×(-i)gψD*D*ϵ1μϵ3ρ[2p1ρgμβ-(p1+p3)βgμρ+2p3μgβρ](-gαβ+qαqβmD*2)×(-i)gD*D*ρϵ2νϵ4λ[2p2λgαν-(p2+p4)αgνλ+2p4νgαλ]F2(q2,mD*2)q2-mD*2×fD*mD*VcbVcd*{(ϵ1*⋅ϵ2*)(mB+mD*)A1BD*(mD*2)-(ϵ1⋅pB)(ϵ2⋅pB)2A2BD*(mD*2)mB+mD*}=iGF162πmBfD*mD*VcbVcd*∫-11‍|p1→|d(cos⁡θ)F2(q2,mD*2)q2-mD*2×{(mB+mD*)A1BD*(mD*2)H4-2A2BD*(mD*2)mB+mD*H4′},
where
(4.21)H4=(-gαβ+qαqβmD*2)(-gμχ+p1μp1χmD*2)(-gχν+p2νp2χmD*2)ϵ3ρϵ4λ×(2p1ρgμβ-(p1+p3)βgμρ+2p3μgβρ)(2p2λgαν-(p2+p4)αgνλ+2p4νgαλ),H4′=(-gαβ+qαqβmD*2)(-pBμ+(p1⋅pB)p1μmD*2)(-pBν+(p2⋅pB)p2νmD*2)ϵ3ρϵ4λ×(2p1ρgμβ-(p1+p3)βgμρ+2p3μgβρ)(2p2λgαν-(p2+p4)αgνλ+2p4νgαλ).
For diagrams (4e) and (4f) in Figure 4 the absorptive part of the amplitude corresponding to the process of B+→D+*(p1,ϵ1)D¯0(p2)→J/ψ(p3,ϵ3)ρ+(p4,ϵ4), where D- and D-* mesons are exchanged, respectively, becomes
(4.22)Abs(4e)=GF2∫‍d3p1→2E1(2π)3d3p2→2E2(2π)3(2π)4δ4(pB-p1-p2)×VcbVcd*mD*(-i)gψD*Dϵμναβϵ3μϵ1νp1αp3β(-i)gDDρ(ϵ4⋅q)(ϵ1⋅p2)×[fD*F1BD(mD*2)+fDA0BD*(mD2)]F2(q2,mD2)q2-mD2=-GF42πmB(fD*F1BD(mD*2)+fDA0BD*(mD2))VcbVcd*mD*gψD*DgDDρ×∫-11‍|p1→|d(cos⁡θ)F2(q2,mD2)q2-mD2H5,
where
(4.23)H5=ϵμναβϵ3μp1αp3β(ϵ4⋅p2)(-p2ν+(p1⋅p2)p1νmD*2),Abs(4f)=GF2∫‍d3p1→2E1(2π)3d3p2→2E2(2π)3(2π)4δ4(pB-p1-p2)×VcbVcd*mD*(-i)gψD*D*ϵ1μϵ3νϵα[2p1νgμβ-(p1+p3)βgμν+2p3μgβν]×(i)2gD*Dρϵρσληϵ4ρϵσp2λp4η[fD*F1BD(mD*2)+fDA0BD*(mD2)]F2(q2,mD*2)q2-mD*2=GF8πmB(fD*F1BD(mD*2)+fDA0BD*(mD2))VcbVcd*mD*gψD*D*gD*Dρ×∫-11‍|p1→|d(cos⁡θ)F2(q2,mD*2)q2-mD*2H6,
where
(4.24)H6=ϵρσληϵ4ρϵ3νp2λp4η[2p1νgμβ-(p1+p3)βgμν+2p3μgβν]×(-gασ+qαqσmD*2)(-p2μ+(p1⋅p2)p1μmD*2).
For diagrams (4g) and (4h) in Figure 4 the absorptive part of the amplitude corresponding to the process of B+→D+(p1)D¯0*(p2,ϵ2)→J/ψ(p3,ϵ3)ρ+(p4,ϵ4), where D- and D-* mesons are exchanged, respectively, is given
(4.25)Abs(4g)=GF2∫‍d3p1→2E1(2π)3d3p2→2E2(2π)3(2π)4δ4(pB-p1-p2)×VcbVcd*mD*(i2)gD*Dρϵμναβϵ4μϵ2νp2αp4β(-i)gψDD(ϵ3⋅q)(ϵ2⋅p1)×[fD*F1BD(mD*2)+fDA0BD*(mD2)]F2(q2,mD2)q2-mD2=GF4πmB(fD*F1BD(mD*2)+fDA0BD*(mD2))VcbVcd*mD*gψDDgD*Dρ×∫-11‍|p1→|d(cos⁡θ)F2(q2,mD2)q2-mD2H7,
where
(4.26)H7=ϵμναβϵ4μp2αp4β(ϵ3⋅p1)(-p1ν+(p1⋅p2)p2νmD*2),Abs(4h)=GF2∫‍d3p1→2E1(2π)3d3p2→2E2(2π)3(2π)4δ4(pB-p1-p2)×VcbVcd*mD*(-i)gD*D*ρϵ2μϵ4νϵβ[2p2μgαν-(p2+p4)αgμν+2p4νgαμ]×(-i)gψD*Dϵρσληϵ3ρϵσp1λp3η[fD*F1BD(mD*2)+fDA0BD*(mD2)]F2(q2,mD*2)q2-mD*2=GF82πmB(fD*F1BD(mD*2)+fDA0BD*(mD2))VcbVcd*mD*gψD*DgD*D*ρ×∫-11‍|p1→|d(cos⁡θ)F2(q2,mD*2)q2-mD*2H8,
where
(4.27)H8=ϵρσληϵ3ρϵ4μp1λp3η[2p2μgαν-(p2+p4)αgμν+2p4νgαμ]×(-gβσ+qβqσmD*2)(-p1ν+(p1⋅p2)p2νmD*2).
As the bridge between the dispersive part of FSI amplitude and the absorptive part, the dispersion relation is
(4.28)Dis4(mB2)=1π∫s∞‍Abs(4a)(s′)+Abs(4b)(s′)+Abs(4c)(s′)⋯Abs(4h)(s′)s′-mB2ds′.
According to the FSI, the amplitude of B+→J/ψρ+ decay is given by
(4.29)AFSI(B+⟶D+(*)D¯0(*)⟶Jψρ+)=Abs(4a)+Abs(4b)+Abs(4c)+Abs(4d)+Abs(4e)+Abs(4f)+Abs(4g)+Abs(4h)+Dis4(mB2).
Concerning the QCDF amplitude and FSI correction, then the total decay amplitude turns to
(4.30)A(B+⟶Jψρ+)=AQCDF+iAFSI.Figure 3
Quark level diagram forB+→D+(*)D¯0(*)→J/ψρ+.HLL diagrams for long-distancet-channel contribution to B+→J/ψρ+.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)


## 5. Numerical Results

Numerical values of effective coefficientsai for b¯→d¯ transition at Nc=3 are given by [22]
(5.1)a1=1.05,a2=0.053,a3=0.0048,a4=-0.0461-0.0124i,a5=-0.0045,a6=-0.0597-0.0124i,a7=0.00003-0.00018i,a8=0.00045-0.00006i,a9=-0.0095-0.00018i,a10=-0.0014-0.00006i.
The Cabibbo-Kobayashi-Maskawa (CKM) matrix is a 3×3 unitary matrix as [9]
(5.2)V=(VudVusVubVcdVcsVcbVtdVtsVtb).
The elements of the CKM matrix can be parameterized by three mixing angles A,λ,ρ and a CP-violating phase η(5.3)Vud=1-λ22,Vus=λ,Vub=Aλ3(ρ-iη),Vcd=-λ,Vcs=1-λ22,Vcb=Aλ2,Vtd=Aλ3(1-ρ-iη),Vts=-Aλ2,Vtb=1.
The results for the Wolfenstein parameters are
(5.4)λ=0.2257±0.001,A=0.814±0.02,ρ¯=0.135±0.023,η¯=0.349±0.016.
We use the central values of the Wolfenstein parameters and obtain
(5.5)Vud=0.9745,Vus=0.2257,Vub=0.0035,Vcd=0.2257,Vcs=0.9745,Vcb=0.0415,Vtd=0.0087,Vts=0.0415,Vtb=1.
The meson masses and decay constants needed in our calculations are taken as (in units of Mev)
(5.6)mB+=5278±0.3,mD±=1869±0.2,mD¯0=1864.84±0.17,mD±*=2010.27±0.17,mD¯0*=2006.97±0.19,mJ/ψ=3096.916±0.011,mb=4.20±0.12,mρ=775.49±0.34,fB+=176±42,fJ/ψ=0.416,fJ/ψ⊥=0.405,fD±=fD¯0=222.6±19.5,fD±*=fD¯0*=230±20,
The Borel mass squares M12 and M22 and continuum thresholds s0 and s0′ are auxiliary parameters, hence the physical quantities should be independent of them. The parameters s0 and s0′ are determined from the conditions that guarantee the sum rules for form factors to have the best stability in the allowed M12 and M22 region. The working regions for M12 and M22 as well as the values for continuum thresholds are determined in [13]. They choose the values s0=35±5GeV2, s0′=7±1GeV2, M12=17.0±2.5GeV2, M22=7±1GeV2 from those working values for auxiliary parameters. The values of the form factors are given by
(5.7)F0BD(mD2)=0.86±0.22,F1BD(mD*2)=0.95±0.23,A0BD*(mD2)=2.50±0.60,A1BD*(mD*2)=1.09±0.26,A2BD*(mD*2)=1.82±0.46[13],A1Bρ=A2Bρ=0.26,VBρ=0.33[10].
Nonuniversal annihilation phases are [23]
(5.8)ϕ=-55(PP),ϕ=-20∘(PV),ϕ=-70∘(VP).
The signs of these phases are not predicted. As a result in [23], their predictions for the signs of CP asymmetries must be taken with caution. We use other input parameters as follows:
(5.9)ΛQCD=0.225Gev,CF=43,N=3,GF=1.166×10-5,αs=0.2244,ρ=0.5,rχD±=1,gDDρ=gD*D*ρ=2.52,gψDD=gψD*D*=7.71,gD*Dρ=2.82,gψD*D=8.64[5].
By using the input parameters and according to the QCDF method of B+→J/ψρ+ decay in Section 2, we get
(5.10)BR(B+⟶Jψρ+)=(1.42±0.36)×10-5.
We note that our estimate of branching ratio of B+→J/ψρ+ decay according to QCDF method seems less than the experimental result. Before calculating the B+→J/ψρ+ decay amplitude via FSI, we have to compute the intermediate state amplitude, for the B+→D+D¯0 decay amplitude we get
(5.11)A(B+⟶D+D¯0)=(2.31±0.27)×10-7.
Now, using the above amplitude and according to the FSI, we are able to calculate the branching ratio of B+→J/ψρ+ decay with different values of η which are shown in Table 1 and Figure 5.Table 1
The branching ratio ofB+→J/ψρ+ decay with η=1~2 and experimental data (in units of 10-5).
η
1
1.2
1.4
1.6
1.8
2
EXP [9]
BR
1.8
8
-
0.35
+
0.37
2.3
3
-
0.35
+
0.37
3.0
5
-
0.35
+
0.36
4.0
0
-
0.35
+
0.36
5.3
9
-
0.35
+
0.37
7.3
2
-
0.35
+
0.36
5
-
0.80
+
0.80Figure 5
The variation of the branching ratio ofB+→J/ψρ+ with η=1~2.2.


## 6. Conclusion

In this work, we have calculated the contribution of thet-channel FSI, that is, inelastic rescattering processes to the branching ratio of B+→J/ψρ+ decay. For evaluating the FSI effects, we have only considered the absorptive part of the HLL because both hadrons which produced via the weak interaction are on their mass shells.We have calculated the branching ratio ofB+→J/ψρ+ decay by using FSI. The experimental result of this decay is BR(B+→J/ψρ+)=(5±0.8)×10-5 and according to QCDF method and assuming the FSI corrections into account, our results of the branching ratio are (1.42±0.36)×10-5 and (4.2~5.8)×10-5, respectively.There exist some phenomenological parameters in our calculations on FSI, such as,η in (4.16). We have introduced the phenomenological parameter η that its value in the form factor is expected to be of order unity and can be determined from the measured rates. For a given exchanged particle, we have used η=1~2, and the branching ratios are 1.88~7.32. We have considered that the factor of η has important role to obtain of branching ratio. Note that [6] for the exchange particles D* and D they have fixed η=2.2 and according to Figure 5, we have found that if η=1.58~1.83 is selected (which is in good agreement with the reference [6] selection), the numbers of the branching ratio of the B+→J/ψρ+ decay are placed in the experimental range.



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*Source: 203692-2012-07-19.xml*