We investigate the B0→J/ψηK0 and B+→J/ψηK+ decay by using the Dalitz plot analysis. As we know there are tree, penguin, emission, and emission-annihilation diagrams for these decay modes in the factorization approach. The transition matrix element is factorized into a B→ηK form factor multiplied by J/ψ decay constant and also a B→K form factor multiplied by J/ψη decay constant. According to QCD factorization approach and using the Dalitz plot analysis, we calculate the branching ratios of the B0→J/ψηK0 and B+→J/ψηK+ three-body decay in view of the η-η' mixing and obtain the value of the (9.22-1.47+2.67)×10-5, while the experimental results of them are (8±4)×10-5 and (10.8±3.3)×10-5, respectively. In this research we also analyze the B(Bs)→ηcπK* decay which is similar to the previous decay, but there is no experimental data for the last decay. Since for calculations of the B(Bs)→ηcπK* decay we use assumptions of the B→J/ψηK decay, we hope that if this decay will be measured by the LHCb in the future, the experimental results will be in agreement with our calculations.
1. Introduction
The three-body decay of B0(+)→J/ψηK0(+) was originally measured by the BABAR [1] and BELLE [2] collaborations and later on tabulated by the Particle Data Group [3]. A long time ago, the D→Kππ decay was studied with the Dalitz plot analysis [4]. According to this technique, we can find many articles such as [5–9] which do not assume that the mass of the K-meson is heavy in front of the pi-meson one. So the momentum of the K-meson is sizable against the momentum of the pi-mesons. In these kinds of decay, when three particles are light, given that the theoretical momentums of the output particles are not directly calculable, momentums and form factors are written in terms of the s=(pB-p3)2 and t=(pB-p1)2 [10, 11] and the Dalitz plot analysis should be used for calculation of the decay rate integral from smin, tmin to smax, tmax. In our selected decay with the experimental values of BR(B0→J/ψηK0)=(8±4)×10-5 and BR(B+→J/ψηK+)=(10.8±3.3)×10-5 [3], we obtain BR(B→J/ψηK)=(9.22-1.47+2.67)×10-5 by using the Dalitz plot analysis.
Note that, to implement the η-η′ mixing, we will use the two-mixing-angle formalism proposed in [12, 13], in which one has
(1)i|η〉=cosθ8|η8〉-sinθ0|η0〉,|η′〉=sinθ8|η8〉+cosθ0|η0〉,
where η8 and η0 are, respectively, the flavor SU(3)-octet and SU(3)-singlet components. In the quark basis they are given by
(2)|η8〉=16|uu-+dd--2ss-〉,|η0〉=13|uu-+dd-+ss-〉.
The relations for the pseudoscalar decay constants in this mixing formalism involving the axial-vector currents Aμ8 and Aμ0 are
(3)〈0|Aμ8|η(p)〉=ifη8pμ,〈0|Aμ8|η′(p)〉=ifη′8pμ,〈0|Aμ0|η(p)〉=ifη0pμ,〈0|Aμ0|η′(p)〉=ifη′0pμ.
In order to get four unknown parameters (θ8, θ0, f8, and f0) allowed values one has to use as constraints the experimental decay widths of (η,η′)→γγ [3]:
(4)Γ(η⟶γγ)=(0.511±0.0026)keV,Γ(η′⟶γγ)=(4.338±0.1592)keV.
On the other hand [14, 15]
(5)Γ(η⟶γγ)=α2mη396π3(cθ0/f8-22sθ8/f0cθ0cθ8+sθ0sθ8)2,Γ(η′⟶γγ)=α2mη′396π3(sθ0/f8+22cθ8/f0cθ0cθ8+sθ0sθ8)2.
If fη′c is that large, the radiative decay J/ψ→η′γ may be dominated by a contribution where the cc- pair runs from the J/ψ to the η′ meson instead of being annihilated. On that supposition the width of that process can be calculated along the same lines as that one for the J/ψ→ηγ decay. The ratio of the two decay widths reads [3]
(6)RJ/ψ=Γ(J/ψ→η′γ)Γ(J/ψ→ηγ)=(mη′2(f8sinθ8+2f0cosθ0)mη2(f8cosθ8-2f0sinθ0))2=4.674±0.289.
The best-fit values of the (η-η′) mixing parameters yield
(7)θ8=(-22.2±1.8)∘,θ0=(-8.7±2.1)∘,f8=(167.296±7.842)MeV,f0=(154.23±5.228)MeV,
which are used to calculate the decay rates in which η and/or η′ are involved. In the B meson decay into states with an η, in the case of B→J/ψηK, since the η meson is described by the uu-, dd-, and ss- combination, there are two different color-suppressed internal W-emission Feynman diagrams; the B0→J/ψηK0 decay mode contains ss- and dd- pairs while the B+→J/ψηK+ decay mode includes ss- and uu- pairs of the η components. In addition, there is a penguin diagram where the dd- and uu- pairs are considered for both decay modes. The diagrams in which J/ψ is emitted via three-gluon exchange are called “hairpin” diagrams so that uu- pairs of the η components are used. Before giving the matrix elements for the B→J/ψηK decay, we discuss the parametrization of the decay constants and form factors which appear in the factorized form of the hadronic matrix elements. In the tree and penguin levels, the K and η mesons are placed in the form factors and the J/ψ meson is placed in the decay constant [16] in which the vector meson’s decay constant, such as J/ψ, is expressed in terms of the matrix element 〈J/ψ|c-γμc|0〉=fJ/ψmJ/ψϵJ/ψ [17]. We also have a B→K form factor multiplied by J/ψη decay constant in the emission diagram with an emission-annihilation level.
The present analysis contains nonfactorizable effects, whereas the hadronic physics governing the B→M1 transition and the formation of the emission particle M2 is genuinely nonperturbative; nonfactorizable interactions connecting the two systems (hard-scattering kernels) are dominated by hard gluon exchange. The hard-scattering kernels are calculable in perturbation theory, which starts at tree level and, at higher order in αs, contains nonfactorizable corrections from hard gluon exchange.
The B0(+)→J/ψηK0(+) decay channels can also receive contributions through intermediate resonances K2*(1430) and K3*(1780), namely, B→K*(→Kη)J/ψ. Therefore, in order to get a reliable estimation on the branching fraction, it is important to have an estimate of the resonant contributions.
2. Amplitudes of the B0→J/ψηK0 and B+→J/ψηK+ Decay 2.1. The Dalitz Plot Analysis2.1.1. Nonresonant Background
In the factorization approach, the Feynman diagrams for three-body B0→J/ψηK0 and B+→J/ψηK+ decay are shown in Figure 1; the η meson is produced from three uu-, dd-, and ss- components; according to Figure 1 to draw the Feynman diagrams of the B0→J/ψηK0 decay, two dd- and ss- components are used for tree level and just dd- component is considered for penguin contribution. These topics are shown in (a), (c), and (e) panels. For B+→J/ψηK+ decay, the uu- and ss- components can be used for tree and uu- pairs for penguin contributions. The panels of (b), (d), and (f) show the mentioned content. Panels (g)–(k) show the emission and emission-annihilation diagrams in which J/ψ is emitted via three-gluon exchange, which are the so-called hairpin diagrams; as we can see, both decay modes have the same amplitudes. Under the factorization approach, the B+→J/ψηK+ and B0→J/ψηK0 decay amplitudes consist of three distinct factorizable terms: (i) the tree and penguin processes, 〈B→ηK〉×〈0→J/ψ〉, (ii) the J/ψ meson emission process, 〈B→K〉×〈0→J/ψη〉, and (iii) the emission-annihilation process, 〈B→0〉×〈0→J/ψηK〉, where 〈A→B〉 denotes an A→B transition matrix element. Here 〈B→ηK〉 denotes two-meson transition matrix element. The leading nonfactorizable diagrams in Figure 2 should be taken into account. To this, we employ the QCD factorization framework, which incorporates important theoretical aspects of QCD like color transparency, heavy quark limit, and hard scattering and allows us to calculate nonfactorizable contributions systematically.
Quark diagrams illustrating the processes B0→J/ψηK0 and B+→J/ψηK+ decay. Panels (a)–(d) show the tree, (e)-(f) show the penguin, (g)-(h) show the emission, and (i)–(k) show the emission-annihilation transitions. The diagrams in which J/ψ is emitted via gluon exchange are called “hairpin” diagrams.
Nonfactorizable diagrams for B0→J/ψηK0 and B+→J/ψηK+ decay.
The matrix elements of the B→J/ψηK decay amplitude are given by
(8)〈JψηK|Heff|B〉∝〈Jψ|(cc-)V-A|0〉〈ηK|(sb-)V-A|B〉+〈Jψη|(su-)V-A|0〉〈K|(ub-)V-A|B〉+〈JψηK|(su-)V-A|0〉〈0|(ub-)V-A|B〉.
For the current-induced process, the two-meson transition matrix element 〈ηK|(sb-)V-A|B〉 has the general expression as [10]
(9)〈η(p1)K(p2)|(sb-)V-A|B〉=ir(pB-p1-p2)μ+iω+(p2+p1)μ+iω-(p2-p1)μ,
and the decay constant is defined as [17]
(10)〈0|Vμ|Jψ(ϵ3,p3)〉=fJ/ψmJ/ψϵ3.
The direct three-body decay of mesons in general receives two distinct contributions: one from the point-like weak transition and the other from the pole diagrams that involve three-point or four-point strong vertices [6]. So the r, ω+, and ω- form factors are computed from point-like and pole diagrams; we also need the strong coupling of B*B(*)η, B*B(*)K, and BBηK vertices. These form factors are given by [10]
(11)r=fB2fηfK-fBfηfKpB·(p2-p1)(pB-p1-p2)2-mB2+2gfBs*fηfKmBmBs*(pB-p1)·p1(pB-p1)2-mBs*2-4g2fBfηfKmBmBs*(pB-p1-p2)2-mB2×p1·p2-p1·(pB-p1)p2·(pB-p1)/mBs*2(pB-p1)2-mBs*2,ω+=-gfηfKfBs*mBs*mBmBs*(pB-p1)2-mBs*2[1-(pB-p1)·p1mBs*2]+fB2fηfK,ω-=gfηfKfBs*mBs*mBmBs*(pB-p1)2-mBs*2[1+(pB-p1)·p1mBs*2].
The other two-body matrix element can be related to the η and J/ψ matrix element of the weak interaction current
(12)〈η(p1)Jψ(p3)|(uu-)V-A|0〉=(p3-p1)μFJ/ψη(q2),
where FJ/ψη(q2) is the J/ψ to η transition form factor and needs to be determined from experiment [2]. This transition occurs by two gluons where both of the gluons are off shell or by two-photon decay widths of the η meson in the J/ψ→ηγ decay [19]. The contribution of gluonic wave function to the transition form factor Fg*g*-η(Q12,Q22) has been tested in [20, 21], and the B→K form factor is defined as follows [17]:
(13)〈K(p2)|(ub-)V-A|B〉=((pB+p2)μ-mB2-m22q2qμ)F1(q2)+mB2-m22q2qμF0(q2),
where qμ=(pB-p2)μ. The emission-annihilation matrix element is assumed to be [22]
(14)〈η(p1)K(p2)Jψ(p3)|(su-)V-A|0〉=2ifK(p2μ-pB·p2pB2-p32pBμ)FJ/ψηK(q2),
and the form factor FJ/ψηK(q2) is parameterized as
(15)FJ/ψηK(q2)=11-q2/Λχ2,
where Λχ=830 Mev is the chiral-symmetry breaking scale. Then the matrix elements read
(16)〈η(p1)K(p2)Jψ(ϵ3,p3)|Heff|B〉∝ifJ/ψmJ/ψ(ϵ3·p3r+(ϵ3·p2+ϵ3·p1)ω++(ϵ3·p2-ϵ3·p1)ω-(ϵ3·p2+ϵ3·p1))+((F1BK(q2)-F0BK(q2))mB2-m22(p1+p3)2(p3·pB-p1·pB+p3·p2-p1·p2)F1BK(q2)-mB2-m22(p1+p3)2(m32-m12)(F1BK(q2)-F0BK(q2)))×FJ/ψη-2fB(p2·pB)fK(1-mB2mB2-m32)FJ/ψηK,
where under the Lorentz condition ϵ3·p3=0. The J/ψ meson polarization vectors become
(17)ϵ3(λ=0)=(|p→3|,0,0,p30)m3,ϵ3(λ=±1)=∓(0,1,±i,0)2.
Consider the decay of B meson into three particles of masses m1, m2, and m3. Denote their 4 momenta by pB, p1, p2, and p3, respectively. Energy-momentum conservation is expressed by
(18)pB=p1+p2+p3.
Define the following invariants:
(19)s12=(p1+p2)2=(pB-p3)2,s13=(p1+p3)2=(pB-p2)2,s23=(p2+p3)2=(pB-p1)2.
The three invariants s12, s13, and s23 are not independent; it follows from their definitions together with 4-momentum conservation that
(20)s12+s13+s23=mB2+m12+m22+m32.
We take s12=s and s23=t, so we have s13=mB2+m12+m22+m32-s-t. In the center of mass of η(p1) and K(p2), according to Figure 3, we find
(21)|p→1|=|p→2|=12s-4m12,p10=p20=12s,|p→3|=p33=12s(mB2-m32-s)2-4sm32,p30=12s(mB2-m32-s),|ϵ→3|=12m3s(mB2-m32-s),ϵ30=12m3s(mB2-m32-s)2-4sm32,
and the cosine of the helicity angle θ between the direction of p→2 and that of p→3 reads
(22)cosθ=14|p→2||p→3|(mB2+m32+2m22-s-2t).
Definition of helicity angle θ, for the decay B→J/ψηK.
With these definitions, we obtain multiplying of the 4-momentum conservation as
(23)ϵ3·(p2+p1)=2p10ϵ30,ϵ3·(p2-p1)=2|ϵ→3||p→1|cosθ.
In this framework, nonfactorizable contributions (hard-scattering and vertex corrections) to B→KηJ/ψ can be obtained by calculating the diagrams in Figure 2. Each of the diagrams in Figure 2 contains a leading-power contribution relevant to power-suppressed terms, which are not factorized in general. An important class of such power-suppressed effects is related to certain higher-twist meson distribution amplitudes. Fortunately, it turns out that ratios of the different hard-scattering contribution have very small uncertainties so we just put the vertex corrections in the calculation of the Wilson coefficients.
Now we can derive the nonresonant amplitude with η-η′ mixing angle θp as
(24)MNR(B⟶η(p1)K(p2)Jψ(p3))=iGF22(16cosθp-13sinθp)2×[mB2-m22mB2+m12+m22+m32-s-t(a2Vcb*Vcs+a3λp)×(1-4m12sω+(mB2-m32-s)2-4sm32+ω-(mB2-m32-s)1-4m12scosθ(mB2-m32-s)2-4sm32)fJ/ψ-2ia2Vub*Vus(mB2-m22mB2+m12+m22+m32-s-t(t-s)F1BK-mB2-m22mB2+m12+m22+m32-s-t×(F1BK-F0BK)mB2-m22mB2+m12+m22+m32-s-t)FJ/ψη+2i(a1Vub*Vus+a4λp)fBfK(s+t-m12-m32)×(1-mB2mB2-m32)FJ/ψηK].
The vertex corrections to the B→J/ψηK decay, denoted as fI in QCDF, have been calculated in the NDR scheme and can be adopted directly. Their effects can be combined into the Wilson coefficients associated with the factorizable contributions [18]:
(25)a1=c1+c23+αs4πCF3c2(-18+12lnmbμ+fI),a2=c2+c13+αs4πCF3c1(-18+12lnmbμ+fI),a3=c3+c43+αs4πCF3c4(-18+12lnmbμ+fI),a4=c4+c33+αs4πCF3c3(-18+12lnmbμ+fI),λp=∑p=u,cVpb*Vps,
where
(26)fI=26fJ/ψ∫dxϕJ/ψL(x)[2r2(1-x)1-r2x3(1-2x)1-xln(x)-3πifdfdfdfdfdffffiggg+3ln(1-r2)+2r2(1-x)1-r2x],
and ϕJ/ψL(x) is the J/ψ meson asymptotic distribution amplitude which is given by [23]
(27)ϕJ/ψL(x)=9.58fJ/ψ26x(1-x)[x(1-x)1-2.8x(1-x)]0.7.
2.1.2. Resonant Contributions
According to Figure 1, the decay channels of B→J/ψηK can also receive contributions through intermediate resonances K2*(1430) and K3*(1780). Resonant effects are described in terms of the usual Breit-Wigner formalism
(28)〈ηK|(s-b)V-A|B〉R=∑igKi*→KηmKi*2-(pη+pK)2-imKi*ΓKi*→Kη×∑polϵKi*·(pK-pη)〈Ki*|(s-b)V-A|B〉,
where
(29)〈Ki*(p1,ϵ1)|Vμ-Aμ|B(pB)〉=-i(ϵKi*-εKi*·qq2qμ)(mB+mKi*)A1BKi*+i((pB+pKi*)μ-mB2-mKi*2q2qμ)(ϵKi*·q)×A2BKi*mB+mKi*,gKi*→Kη=12πmKi*2ΓKi*→Kη2pc3,
where Ki* denote K2*(1430), K3*(1780), and pc is the c.m. momentum. In determining the coupling of Ki*→Kη, we have used the partial widths ΓK*0(1430)→K0η=0.164 Mev, ΓK*+(1430)→K+η=0.148 Mev, and ΓK3*(1780)→Kη=47.70 Mev measured by PDG [3]. Then the decay amplitude through resonance intermediate reads
(30)MR(B⟶JψηK)=-iGF26mJ/ψfJ/ψ(a2Vcb*Vcs+a3λp)∑i(2ϵ→Ki*·p→K)×[A2BKi*mB+mKi*(ϵKi*·ϵJ/ψ)(mB+mKi*)A1BKi*-(ϵKi*·pJ/ψ)×(ϵJ/ψ·(pB+pKi*))A2BKi*mB+mKi*]×gKi*→KηmKi*2-s-imKi*ΓKi*→Kη.
Finally by using the full amplitude, the decay rate of B→J/ψηK is then given by [22]
(31)Γ(B⟶JψηK)=1(2π)332mB3×∫sminsmax∫tmintmax|MNR(B⟶JψηK)dsdsffffdsddsdsdsdf+MR(B⟶JψηK)|2dsdt,
where
(32)tmin,max(s)=m12+m32-12s(λ(s,mB,m3)(mB2-s-m32)(s-m22+m12)∓λ(s,mB2,m32)λ(s,m12,m22)),smin=(m1+m2)2,smax=(mB-m3)2,
where λ(x,y,z)=x2+y2+z2-2(xy+xz+yz).
3. Amplitudes of the B→ηcπK* and Bs→ηcπK* Decay
The Feynman diagrams for three-body decay of B→ηcπK* and Bs→ηcπK*, in the factorization approach, are shown in Figure 4 and types of these decay modes can be obtained from the following options.
For choices of q-=s- and
q1=q1′=d, decay mode becomes B0→ηcπ0K*0,
q1=d,q1′=u, decay mode becomes B0→ηcπ-K*+,
q1=q1′=u, decay mode becomes B+→ηcπ0K*+,
q1=u,q1′=d, decay mode becomes B+→ηcπ+K*0.
For selection of q-=d- and
q1=s,q1′=u, decay mode becomes Bs0→ηcπ+K*-,
q1=s,q1′=d, decay mode becomes Bs0→ηcπ0K-*0.
For this decay, according to Figure 2, the decay constant is defined as
(33)〈0|Aμ|ηc(p)〉=ifηcpηc.
Then the nonresonant amplitude can be obtained by
(34)M(B⟶π(p1)K*(p2)ηc(p3))=-GF22[(1-mB2mB2-m32)fηc(a2Vcb*Vcs+a3λp)×(GF22|Aμ|2rm32+ω+(mB2-s-m32)+ω-(2t+s-mB2-2m22-m32)GF22)+2a4λpfBfK*(s+t-m12-m32)×(1-mB2mB2-m32)FηcπK].
In the Bs decay modes we useVcd instead of Vcs (also within the λp). Note that when the final states contain π0 meson, the decay amplitudes multiplied by 1/2. In addition, several intermediate resonant states involving K1(1270), K1(1400), K*(1410), K2*(1430), K*(1680), and K3*(1820) resonances are used in the calculations [3].
Quark diagrams illustrating the processes B(Bs)→ηcπK* decay.
4. Numerical Results
The theoretical input parameters used in our analysis, together with their respective ranges of uncertainty, are summarized below.
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 scales μ=mb/2, μ=mb, and μ=2mb at the next to leading order (NLO) are shown in Table 1.
The Wilson coefficients ci in the NDR scheme at three different choices of the renormalization scale μ [18].
NLO
c1
c2
c3
c4
μ=mb/2
1.137
−0.295
0.021
−0.051
μ=mb
1.081
−0.190
0.014
−0.036
μ=2mb
1.045
−0.113
0.009
−0.025
There is a potentially quite large error that could come from the uncertainty in the parameter g available on the form factors. This parameter is determined from the D*→Dπ decay and we use [10]
(35)g=0.3±0.1.
For the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, we use the values of the Wolfenstein parameters and obtain
(36)|Vud|=0.97425±0.00022,|Vus|=0.2252±0.0009,|Vub|=0.00415±0.00049,|Vcd|=0.230±0.011,|Vcs|=1.006±0.023,|Vcb|=0.0409±0.0011,|Vtd|=0.0084±0.0006,|Vts|=0.0429±0.0026,|Vtb|=0.89±0.07.
The meson masses and decay constants needed in our calculations are taken as (in units of Mev) [3]
(37)mBs*=5415.4-2.1+2.4,mB0=5279.58±0.17,mB±=5279.25±0.17,mJ/ψ=3096.916±0.011,mηc=2981.0±1.1,mK*=891.66±0.26,mη=547.853±0.024,mK0=497.614±0.024,mK±=493.677±0.016,mπ0=134.9766±0.0006,mπ±=139.57018±0.00035,fB=176±42,fBs*=220,fJ/ψ=418±9,fηc=387±7,fη=131±7,fK=159.8±1.84,fK*=217±5,
and the form factors at zero momentum transfer are taken as [17]
(38)iF0BK=F1BK=0.35±0.05,A1BK*=0.35±0.07,A2BK*=0.34±0.06.
Using the parameters relevant to the B→J/ψηK and B(Bs)→ηcπK* decay, we calculate the branching ratios of this decay, which are shown in Table 2. Note that, as we mentioned before, both decay of B0→J/ψηK0 and decay of B+→J/ψηK+ have similar amplitudes. Since the masses of the B0 and B+ and also K0 and K+ mesons are very close to each other, both branching ratios are the same.
Branching ratios of B→J/ψηK (in units of 10-5) and B(Bs)→ηcπK* (in units of 10-4) decay by using the Dalitz plot analysis.
Mode
μ=mb/2
μ=mb
μ=2mb
Exp. [3]
B0→J/ψηK0
2.42-0.39+0.70
9.22-1.47+2.67
17.73-3.16+4.64
8±4
B+→J/ψηK+
2.42-0.39+0.70
9.22-1.47+2.67
17.73-3.16+4.64
10.8±3.30
B0→ηcπ-K*+, B+→ηcπ+K*0
3.62-0.41+0.71
13.82-1.56+2.70
26.00-2.93+5.08
—
B0→ηcπ0K*0,B+→ηcπ0K*+
1.81-0.21+0.36
6.91-0.78+1.35
13.00-1.47+2.54
—
Bs0→ηcπ+K*-
0.20-0.02+0.04
0.75-0.09+0.14
1.40-0.16+0.27
—
Bs0→ηcπ0K*0
0.10-0.01+0.02
0.38-0.05+0.07
0.70-0.08+0.14
—
5. Conclusion
In this work, we have calculated the branching ratios of the B0→J/ψηK0 and B+→J/ψηK+ decay by using the Dalitz plot analysis. In this calculation we have used factorizable terms, nonfacorizable effects, and η-η′ mixing. According to QCD factorization approach, we have obtained BR(B→J/ψηK)=(9.22-1.47+2.67)×10-5 while the experimental results of them are (8±4)×10-5 and (10.8±3.3)×10-5, respectively. The branching ratios obtained by applying the Dalitz plot analysis are compatible with the experimental results.
Moreover we have analyzed the B(Bs)→ηcπK* decay which is similar to the B→J/ψηK decay, but there are no experimental data for the last decay. Since for calculations of the B(Bs)→ηcπK* decay we have used assumptions of the B→J/ψηK decay, we hope that if this decay will be measured by the LHCb in the future, the experimental results will be in agreement with our calculations.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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