The semileptonic Bs(B)→K2⁎(a2,f2)lν, l=τ,μ, transitions are investigated in the frame work of the three-point QCD sum rules. Considering the quark condensate contributions, the relevant form factors of these transitions are estimated. The branching ratios of these channel modes are also calculated at different values of the continuum thresholds of the tensor mesons and compared with the obtained data for other approaches.
Isfahan University of Technology Research Council1. Introduction
Investigation of the B meson decays into tensor mesons is useful in several aspects such as CP asymmetries, isospin symmetries, and the longitudinal and transverse polarization fractions. A large isospin violation has already been experimentally detected in B→ωK2∗(1430) mode [1]. Also, the decay mode B→ϕK2∗(1430) is mainly dominated by the longitudinal polarization [2, 3], in contrast with B→ϕK∗, where the transverse polarization is comparable with the longitudinal one [4]. Therefore, nonleptonic and semileptonic decays of B meson can play an important role in the study of the particle physics.
In the flavor SU(3) symmetry, the light p-wave tensor mesons with JP=2+ containing isovector mesons a2(1320), isodoublet states K2∗(1430), and two isosinglet mesons f2(1270) and f2′(1525) are building the ground state nonet which has been experimentally established [5, 6]. The quark content qq¯ for the isovector and isodoublet tensor resonances is obvious. The isoscalar tensor states, f2(1270) and f2′(1525), have mixing wave functions where mixing angle should be small [7, 8]. Therefore, f2(1270) is primarily a (uu¯+dd¯)/2 state, while f2′(1525) is dominantly ss¯ [9].
As a nonperturbative method, the QCD sum rules is a well established technique in the hadron physics since it is based on the fundamental QCD Lagrangian [10]. The semileptonic decays of B to the light mesons involving π, K(K∗,K0∗), and a1 have been studied via the three-point QCD sum rules (3PSR), for instance, B→πlν [11], B→Kl+l-, B→K∗l+l-[12–14], Bs→K0∗lν [15], Bs→(K0∗,f0)l+l- [16], and B→a1l+l- [17]. The determination of the form factor value T1(0)=0.35±0.05 relevant for the B→K∗γ and B→K∗l+l- [14, 18] decays allowed prediction of the ratio Γ(B→K∗γ)/Γ(b→sγ)=0.17±0.05, which agrees with the experimental measurements [19–21]. The obtained results of the decay B→πlν [11] and simulations on the lattice [22–24] are in a reasonable agreement.
In this work, we investigate B(Bs)→K2∗(a2,f2)lν decays within the 3PSR method. For analysis of these decays, the form factors and their branching ratio values are calculated. So far, the form factors of the semileptonic decays B(Bs)→K2∗(a2,f2)lν have been studied via different approaches such as the LCSR [25], the perturbative QCD (PQCD) [5], the large energy effective theory (LEET) [26–28], and the ISGW II model [29]. A comparison of our results for the form factor values in q2=0 and branching ratio data with predictions obtained from other approaches, especially the LCSR, is also made.
The plan of the present paper is as follows: the 3PSR approach for calculation of the relevant form factors of B(Bs)→K2∗(a2,f2)lν decays is presented in Section 2. In the final section, the value of the form factors in q2=0 and the branching ratio of the considered decays are reported. For a better analysis, the form factors and differential branching ratios related to these semileptonic decays are plotted with respect to the momentum transfer squared q2.
2. Theoretical Framework
In order to study B(Bs)→K2∗(a2,f2)lν decays, we focus on the exclusive decay Bs→K2∗ via the 3PSR. The Bs→K2∗lν decay governed by the tree level b→u transition (see Figure 1). In the framework of the 3PSR, the first step is appropriate definition of correlation function. In this work, the correlation function should be taken as(1)Παβμp2,p′2,q2=i∬eip′x-py0∣TjαβK2∗xjμ0jBsy∣0d4xd4y,where p and p′ are four-momentum of the initial and final mesons, respectively. q2 is the squared momentum transfer and T is the time ordering operator. jμ=u¯γμ(1-γ5)b is the transition current. jBs and jαβK2∗ are also the interpolating currents of Bs and the tensor meson K2∗, respectively. With considering all quantum numbers, their interpolating currents can be written as follows [33]:(2)jBsy=b¯yγ5sy,jαβK2∗x=i2s¯xγαD↔βxux+s¯xγβD↔αxux,where D↔μ(x) is the four-derivative vector with respect to x acting at the same time on the left and right. It is given as (3)D↔μx=12D→μx-D←μx,D→μx=∂→μx-ig2λaAμax,D←μx=∂←μx+ig2λaAμax,where λa and Aμa(x) are the Gell-Mann matrices and the external gluon fields, respectively. It should be noted that the second current in (2) interpolates a spin 2 particle for massless quarks. In the general case, to describe a spin 2 state one has to use a current such that the trace of jαβK2∗ vanishes.
Schematic picture of the spectator mechanism for the Bs→K2∗lν decay.
The correlation function is a complex function of which the imaginary part comprises the computations of the phenomenology and real part comprises the computations of the theoretical part (QCD). By linking these two parts via the dispersion relation, the physical quantities are calculated. In the phenomenological part of the QCD sum rules approach, the correlation function in (1) is calculated by inserting two complete sets of intermediate states with the same quantum numbers as Bs and K2∗. After performing four integrals over x and y, it will be(4)Παβμ=-0∣jαβK2∗∣K2∗p′K2∗p′∣jμ∣BspBsp∣jBs∣0p2-mBs2p′2-mK2∗2+higher states.In (4), the vacuum to initial and final meson state matrix elements is defined as(5)0∣jαβK2∗∣K2∗p′,ε=fK2∗mK2∗2εαβ,0∣jBs∣Bsp=-ifBsmBs2mb+ms,where fK2∗ and fBs are the leptonic decay constants of K2∗ and Bs mesons, respectively. εαβ is polarization tensor of K2∗. The transition current gives a contribution to these matrix elements and it can be parametrized in terms of some form factors using the Lorentz invariance and parity conservation. The correspondence between a vector meson and a tensor meson allows us to get these parametrizations in a comparative way (for more information see [5]). The parametrization of B→T form factors is analogous to the B→V case except that ε is replaced by εT, as follows:(6)cVK2∗p′,ε∣u¯γμ1-γ5b∣Bsp=-iεTμ∗mBs+mK2∗A1q2+ip+p′μεT∗·qA2q2mBs+mK2∗+iqμεT∗·q2mK2∗q2A3q2-A0q2+ϵμνρσεT∗νpρp′σ2Vq2mBs+mK2∗(7)with A3q2=mBs+mK2∗2mK2∗A1q2-mBs-mK2∗2mK2∗A2q2,A00=A30,where q=p-p′, P=p+p′, and εTμ∗=pλ/mBsεμλ. The factor cV accounts for the flavor content of particles: cV=2 for a2, f2 and cV=1 for K2∗ [34]. Inserting (5) and (6) in (4) and performing summation over the polarization tensor as (8)εμνεαβ=12TμαTνβ+12TμβTνα-13TμνTαβ,where Tμν=-gμν+pμ′pν′/mK2∗2, the final representation of the physical side is obtained as(9)Παβμ=fBsmBsmb+msfK2∗mK2∗2p2-mBs2p′2-mK2∗2V′q2iϵβμρσpαpρp′σ+A0′q2pαpβpμ′+A1′q2gβμpα+A2′q2pαpβpμ+higher states.For simplicity in calculations, the following redefinitions have been used in (9): (10)V′q2=Vq2mBs+mK2∗,A0′q2=-mK2∗A3q2-A0q2q2,A1′q2=-mBs+mK2∗2A1q2,A2′q2=A2q22mBs+mK2∗.Now, the QCD part of the correlation function is calculated by expanding it in terms of the OPE at large negative value of q2 as follows:(11)Παβμ=Cαβμ0I+Cαβμ30∣Ψ¯Ψ∣0+Cαβμ40∣GρνaGaρν∣0+Cαβμ50∣Ψ¯σρνTaGaρνΨ∣0+⋯,where Cαβμ(i) are the Wilson coefficients, I is the unit operator, Ψ¯ is the local fermion field operator, and Gρνa is the gluon strength tensor. In (11), the first term is contribution of the perturbative and the other terms are contribution of the nonperturbative part.
To compute the portion of the perturbative part (Figure 1), using the Feynman rules for the bare loop, we obtain(12)Cαβμ0=-i4∬eip′x-pyTrSsx-yγαD↔βxSu-xγμ1-γ5Sbyγ5+Trα⟷βd4xd4y;taking the partial derivative with respect to x of the quark free propagators and performing the Fourier transformation and using the Cutkosky rules, that is, 1/p2-m2→-2iπδ(p2-m2), imaginary part of Cαβμ(0) is calculated as(13)ImCαβμ0=18π∫δk2-ms2δp+k2-mb2δp′+k2-mu22k+p′βTrk+msγαp′+k+muγμ1-γ5p+k+mbγ5+α⟷βd4k,where k is four-momentum of the spectator quark s. To solve the integral in (13), we will have to deal with the integrals such as I0, Iα, Iαβ, and Iαβμ with respect to k. For example, Iαβμ can be as (14)Iαβμs,s′,q2=∫kαkβkμδk2-ms2δp+k2-mb2δp′+k2-mu2d4k,where s=p2 and s′=p′2. I0, Iα, Iαβ, and Iαβμ can be taken as an appropriate tensor structure as follows:(15)I0=14λs,s′,q2,Iα=B1pα+B2pα′,Iαβ=D1gαβ+D2pαpβ+D3pαpβ′+pα′pβ+D4pα′pβ′,Iαβμ=E1gαβpμ+gαμpβ+gβμpα+E2gαβpμ′+gαμpβ′+gβμpα′+E3pαpβpμ+E4pα′pβpμ+pαpβ′pμ+pαpβpμ′+E5pα′pβ′pμ+pα′pβpμ′+pαpβ′pμ′+E6pα′pβ′pμ′.The quantities λ(s,s′,q2), Bl(l=1,2), Dj(j=1,…,4), and Er(r=1,…,6) are indicated in Appendix. Using the relations in (15), Im[Cαβμ(0)] can be calculated for each structure corresponding to (9) as follows:(16)ImCαβμ0=ρViϵβμρσpαpρp′σ+ρ0pαpβpμ′+ρ1gβμpα+ρ2pαpβpμ,where the spectral densities ρi(i=V,0,1,2) are found as (17)ρVs,s′,q2=24B1λB1ms-mb+B2ms-mu+msI0,ρ0s,s′,q2=12D2ms-mb+D3ms-mu+2B1ms-2E4mb-ms,ρ1s,s′,q2=3B12ms2mb+mu-ms-ms2mbmu+u+Δms-mu+Δ′ms-mb+6D1ms-mu-24E1mb-ms,ρ2s,s′,q2=24D2ms+E3ms-mb.Using the dispersion relation, the perturbative part contribution of the correlation function can be calculated as follows:(18)Ci0=∬ρis,s′,q2s-p2s′-p′2ds′ds.
For calculation of the nonperturbative contributions (condensate terms), we consider the condensate terms of dimensions 3, 4, and 5 related to the contributions of the quark-quark, gluon-gluon, and quark-gluon condensate, respectively. They are more important than the other terms in the OPE. In the 3PSR, when the light quark is a spectator, the gluon-gluon condensate contributions can be easily ignored [35]. On the other hand, the quark condensate contributions of the light quark, which is a nonspectator, are zero after applying the double Borel transformation with respect to both variables p2 and p′2, because only one variable appears in the denominator. Therefore, only two important diagrams of dimensions 3, 4, and 5 remain from the nonperturbative part contributions. The diagrams of these contributions corresponding to Cαβμ(3) and Cαβμ(5) are depicted in Figure 2. After some calculations, the nonperturbative part of the correlation function is obtained as follows:(19)CV3+CV5=-2κp2-mb22p′2-mu2,C03+C05=-4κp2-mb22p′2-mu2,C13+C15=κp2-mb2p′2-mu2+κmb+mu2-q2p2-mb22p′2-mu2,C23+C25=-4κp2-mb22p′2-mu2,where κ=(ms2-m02/2)/160∣ss¯∣0, m02=(0.8±0.2)GeV2 [35], and 0∣s¯s∣0=0.8±0.20∣u¯u∣0, 0∣u¯u∣0=0∣d¯d∣0=-(0.240±0.010GeV)3; that is, we choose the value of the condensates at a fixed renormalization scale of about 1 GeV [36, 37].
The diagrams of the effective contributions of the condensate terms.
The next step is to apply the Borel transformations with respect to p2(p2→M12) and p′2(p′2→M22) on the phenomenological as well as the perturbative and nonperturbative parts of the correlation functions and equate these two representations of the correlations. The following sum rules for the form factors are derived:(20)V′q2=mb+msemBs2/M12emK2∗2/M22fBsmBsfK2∗mK2∗2-12π2∫ms2s0′∫sLs0ρVs,s′,q2e-s/M12e-s′/M22+B~CV3+CV5ds′ds,An′q2=mb+msemBs2/M12emK2∗2/M22fBsmBsfK2∗mK2∗2-12π2∫ms2s0′∫sLs0ρns,s′,q2e-s/M12e-s′/M22+B~Cn3+Cn5ds′ds,where n=0,…,2 and s0 and s0′ are the continuum thresholds in the initial and final channels, respectively. The lower limit in the integration over s is sL=mb2+mb2/mb2-q2s′. Also, B~ transformation is defined as follows:(21)B~1p2-mb2mp′2-mu2n=-1m+nΓnΓme-mb2/M12e-mu2/M22M12m-1M22n-1,where M12 and M22 are Borel mass parameters.
In (20), to subtract the contributions of the higher states and the continuum, the quark-hadron duality assumption is also used; that is, it is assumed that(22)ρhigherstatess,s′=ρs,s′θs-s0θs′-s0′.
We would like to provide the same results for B→a2lν and B→f2lν decays. With a little bit of change in the above expressions such as s↔d(u) and mK2∗↔ma2(mf2), we can easily find similar results in (20) for the form factors of the new transitions.
3. Numerical Analysis
In this section, we numerically analyze the sum rules for the form factors V(q2), A0(q2), A1(q2), and A2(q2) as well as branching ratio values of the transitions B(Bs)→T, where T can be one of the tensor mesons K2∗,a2, or f2. The values of the meson masses and leptonic decay constants are chosen as presented in Table 1. Also, mb = 4.820 GeV, ms = 0.150 GeV [38], mτ = 1.776 GeV, and mμ = 0.105 GeV [30].
The values of the meson masses [30] and decay constants [31, 32] in GeV.
Meson
Bs
B
K2∗
a2
f2
Mass
5.366
5.279
1.425
1.318
1.275
Decay constant
0.222±0.012
0.186±0.014
0.118±0.005
0.107±0.006
0.102±0.006
From the 3PSR, it is clear that the form factors also contain the continuum thresholds s0 and s0′ and the Borel parameters M12 and M22 as the main input. These are not physical quantities; hence the form factors should be independent of these parameters. The continuum thresholds, s0 and s0′, are not completely arbitrary, but these are in correlation with the energy of the first exiting state with the same quantum numbers as the considered interpolating currents. The value of the continuum threshold s0B(Bs)=35GeV2 [39] is calculated from the 3PSR. The values of the continuum threshold s0′ for the tensor mesons K2∗, a2, and f2 are taken to be s0K2∗=3.13GeV2, s0a2=2.70GeV2, and s0f2=2.53GeV2, respectively [9]. In this work, the variations of s0T(T=a2,K2∗,f2) are considered to be ±0.2. In these regions, the dependence of the form factors on the continuum threshold values is very small. For instance, we have shown the variations of the form factor A1Bs→K2∗(q2) for different values of s0K2∗ in Figure 3. As can be seen, these plots are very close to each other.
The form factor of A1Bs→K2∗ on q2 for different values of s0K2∗.
We search for the intervals of the Borel parameters so that our results are almost insensitive to their variations. One more condition for the intervals of these parameters is the fact that the aforementioned intervals must suppress the higher states, continuum, and contributions of the highest-order operators. In other words, the sum rules for the form factors must converge. As a result, we get 8GeV2≤M12≤12GeV2 and 4GeV2≤M22≤8GeV2. To show how the form factors depend on the Borel mass parameters, as examples, we depict the variations of the form factors V, A0, A1, and A2 for Bs→K2∗lν at q2=0 with respect to the variations of the M12 and M22 parameters in their working regions in Figure 4. From these figures, it is revealed that the form factors weakly depend on these parameters in their working regions.
The form factor of Bs→K2∗ on M12 and M22.
In the Borel transform scheme, the ratio of the nonperturbative to perturbative part of the form factor VBs→K2∗ is about Vnon-per(0)/Vper(0)≃13%. This value confirms that the higher order corrections are small, constituting a few percent, and can easily be neglected. Our calculation shows that the same suppression is observed for all other form factors.
The sum rules for the form factors are truncated at about 0≤q2≤11GeV2. The dependence of the form factors V, A0, A1, and A2 on q2 for B→T transitions is shown in Figure 5. However, it is necessary to obtain the behavior of the form factors with respect to q2 in the full physical region, 0≤q2≤(mB(Bs)-mT)2, in order to calculate the decay width of the B→T transitions. So, to extend our results, we look for a parametrization of the form factors in such a way that in the region 0≤q2≤(mB(Bs)-mT)2, this parametrization coincides with the sum rules predictions. Our numerical calculations show that the sufficient parametrization of the form factors with respect to q2 is as follows:(23)fq2=f01-aq2/mBBs2+bq2/mBBs22.The values of the parameters f(0), a, and b for the transition form factors of B→T are given in Table 2.
Parameter values appearing in the fit functions of the B→Tlν decays.
Form factor
f(0)
a
b
VBs→K2∗
0.13
2.19
0.83
A1Bs→K2∗
0.10
1.36
0.09
VB→a2
0.13
2.10
0.75
A1B→a2
0.11
1.45
0.23
VB→f2
0.12
2.01
0.60
A1B→f2
0.10
1.40
0.16
A0Bs→K2∗
0.23
3.77
4.21
A2Bs→K2∗
0.05
0.21
−2.99
A0B→a2
0.26
3.71
4.03
A2B→a2
0.09
0.63
0.46
A0B→f2
0.24
3.70
4.02
A2B→f2
0.09
0.46
0.29
The SR predictions for the form factors of the B(Bs)→Tlν transitions on q2.
In Table 3, our results for the form factors of B→Tlν decays in q2=0 are compared with those of other approaches such as the LCSR, the PQCD, the LEET, and the ISGW II model. Our results are in good agreement with those of the LCSR, PQCD, and LEET in all cases.
Comparison of the form factor values of B→Tlν decays in q2=0 in different approaches.
Form factor
This work
LCSR [25]
PQCD [5]
LEET [26–28]
ISGW II [29]
VBs→K2∗
0.13±0.03
0.15±0.02
0.18-0.04+0.05
—
—
A0Bs→K2∗
0.23±0.06
0.22±0.04
0.15-0.03+0.04
—
—
A1Bs→K2∗
0.10±0.02
0.12±0.02
0.11-0.02+0.03
—
—
A2Bs→K2∗
0.05±0.01
0.05±0.02
0.07-0.02+0.02
—
—
VB→a2
0.13±0.03
0.18±0.02
0.18-0.04+0.05
0.18±0.03
0.32
A0B→a2
0.26±0.07
0.21±0.04
0.18-0.04+0.06
0.14±0.02
0.20
A1B→a2
0.11±0.04
0.14±0.02
0.11-0.03+0.03
0.13±0.02
0.16
A2B→a2
0.09±0.02
0.09±0.02
0.06-0.01+0.02
0.13±0.02
0.14
VB→f2
0.12±0.04
0.18±0.02
0.12-0.03+0.03
0.18±0.02
0.32
A0B→f2
0.24±0.06
0.20±0.04
0.13-0.03+0.04
0.13±0.02
0.20
A1B→f2
0.10±0.02
0.14±0.02
0.08-0.02+0.02
0.12±0.02
0.16
A2B→f2
0.09±0.02
0.10±0.02
0.04-0.01+0.01
0.13±0.02
0.14
At the end of this section, we would like to present the differential decay widths of the process under consideration. Using the parametrization of these transitions in terms of the form factors, the differential decay width for B→Tlν transition is obtained as (24)dΓB→Tlνdq2=GFVub2λmB2,mT2,q2256mB3π3q21-ml2q22XL+X++X-,where ml represents the mess of the charged lepton. The other parameters are defined as (25)XL=19λmT2mB22q2+ml2h02q2+3λml2A02q2,X±=2q232q2+ml2λ8mT2mB2mB+mTA1q2∓λmB+mTVq22,h0q2=12mTmB2-mT2-q2mB+mTA1q2-λmB+mTA2q2.Integrating (24) over q2 in the whole physical region and using Vub=(3.89±0.44)×10-3 [30], the branching ratios of the B→Tlν are obtained. The differential branching ratios of the B→Tlν decays on q2 are shown in Figure 6. The branching ratio values of these decays are also obtained as presented in Table 4. Furthermore, this table contains the results estimated via the PQCD. Considering the uncertainties, our estimations for the branching ratio values of the B→Tlν decays are in consistent agreement with those of the PQCD.
Comparison of the branching ratio values of B→Tlν decays with those of the PQCD (in units of 10-4).
This work
PQCD [5]
BrB→a2μν
0.82±0.25
1.16-0.57+0.81
BrBs→K2∗μν
0.65±0.20
0.73-0.33+0.48
BrB→f2μν
0.77±0.23
0.69-0.34+0.48
BrB→a2τν
0.51±0.17
0.41-0.20+0.29
BrBs→K2∗τν
0.35±0.11
0.25-0.12+0.17
BrB→f2τν
0.53±0.18
0.25-0.13+0.18
The differential branching ratios of the semileptonic B→Tlν decays on q2.
It should be noted that the uncertainties in the branching ratio values come from the form factors, the CKM parameter, and the meson and lepton masses which are about 30% of the central values.
In summary, we considered Bs(B)→K2∗(a2,f2)lν channels and computed the relevant form factors considering the contribution of the quark condensate corrections. Our results are in good agreement with those of the LCSR, PQCD, and LEET in all cases. We also evaluated the total decays widths and the branching ratios of these decays. Our branching ratio values of these decays are in consistent agreement with those of the PQCD.
Appendix
In this appendix, the explicit expressions of the coefficients λ(s,s′,q2), Bl(l=1,2), Dj(j=1,…,4), and Er(r=1,…,6) are given. (A.1)λs,s′,q2=s2+s′2+q22-2sq2-2s′q2-2ss′,B1=I0λs,s′,q22s′Δ-Δ′u,B2=I0λs,s′,q22sΔ′-Δu,D1=-I02λs,s′,q24ss′ms2-sΔ′2-s′Δ2-u2ms2+uΔΔ′,D2=-I0λ2s,s′,q28ss′2ms2-2ss′Δ′-6s′2Δ2-2u2s′ms2+6s′uΔΔ′-u2Δ′2,D3=I0λ2s,s′,q24ss′ums2+4ss′ΔΔ′-3suΔ′2-3uΔ2s′-u3m32+2u2ΔΔ′,D4=I0λ2s,s′,q2-6s′uΔΔ′+6s2Δ′2-8s2s′ms2+2u2sms2+u2Δ2+2ss′Δ2,E1=I02λ2s,s′,q28s′2ms2Δs-2s′ms2Δu2-4ums2Δ′ss′+u3ms2Δ′-2s′2Δ3+3s′uΔ2Δ′-2Δ′2Δss′-Δ′2Δu2+usΔ3,E2=I02λ2s,s′,q28s2ms2Δ′s′-2s2Δ′3-4ums2Δss′-2Δ2Δ′ss′+3usΔ′2Δ-2sms2Δ′u2+s′uΔ3+u3ms2Δ-Δ2Δ′u2,E3=-I0λ3s,s′,q248sms2Δs′3-24ss′2ums2Δ′-12ss′2Δ′2Δ+6suΔ′3s′-20s′3Δ3+30s′2uΔ2Δ′-12s′2ms2Δu2-12s′Δ′2Δu2+6s′u3ms2Δ′+u3Δ′3,E4=-I0λ3s,s′,q216s2ms2Δ′s′2-4s2Δ′3s′-12ss′2Δ2Δ′-24ss′2ums2Δ+3u3Δ′2Δ+18suΔ′2Δs′-4sΔ′3u2+10s′2uΔ3+6s′u3ms2Δ-12s′Δ2Δ′u2-2ms2Δ′u4+4ss′u2ms2Δ′,E5=-I0λ3s,s′,q216s2ms2Δs′2-24s2s′ums2Δ′-12s2s′Δ′2Δ+10us2Δ′3-4ss′2Δ3+4ss′u2ms2Δ+18suΔ2Δ′s′+6su3ms2Δ′-12sΔ2Δu2-4s′Δ3u2-2ms2Δu4+3u3Δ2Δ′,E6=-I0λ3s,s′,q248s3ms2Δ′s′-20s3Δ′3-12s2Δ2Δ′s′-24s2s′ums2Δ-12s2ms2Δ′u2+30us2Δ′2Δ+6suΔ3s′-12sΔ2Δ′u2+6su3ms2Δ+u3Δ3,Δ=s+ms2-mb2,Δ′=s′+ms2-mu2,u=s+s′-q2.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
Partial support of the Isfahan University of Technology Research Council is appreciated.
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