# Investigation of the Rare ExclusiveBc*→Dsνν- Decays in the Framework of the QCD Sum Rules

Authors: V. Bashiry
Journal: Advances in High Energy Physics
 (2014)

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## Abstract

ExclusiveBc*→Dsνν- decay is studied in the framework of the three-point QCD sum rules approach. The two gluon condensate contributions to the correlation function are calculated and the form factors of this transition are found. The decay width and total branching ratio for this decay are also calculated.


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## Body

## 1. Introduction

The standard model (SM) Higgs boson which is one of the most important components of the SM has been discovered by the ATLAS [1] and CMS [2] collaborations. Nowadays, we aim to find out the new physics beyond the SM. Heavy mesons with the different flavors like Bc and Bc* mesons can provide a good testing benchmark not only for the predictions of the SM but also for searching the new physics beyond SM. The LHCb experiment has aimed to test the SM predictions and discover the possible new physics signals. In this regard, a lot of the experimental data are released by the LHCb experiment [3].The dominant decay mode ofBc* is Bc*→Bcγ [4]. Rare Bc*→Dsνν¯ proceeds FCNC transitions. This decay is roughly of the same order as that of the Bc*→ηclν¯l [5]. In the SM framework, the rare Bc*→Dsνν¯ decay is dominated by the Z-penguin and box diagrams involving top quark exchanges. The theoretical uncertainties related to the renormalization scale dependence of running quark mass can be essentially neglected after the inclusion of next-to-leading order corrections [6]. This decay is theoretically very clean process in comparison with the semileptonic decays like the Bc*→Dsl+l- decay and is also sensitive to the new physics beyond the SM [7]. Moreover, this decay is complementary to the Bc*→Dsl+l- decay. Note that the direct calculation of physical observables such as form factors suffers from sizable uncertainties. These can be greatly reduced through a combined analysis of the rare Bc*→Dsνν¯ and Bc*→Dsl+l- [8] decays.These decays have not yet been measured by the LHCb. There are no theoretical studies relevant to the form factors and branching ratios ofBc*→Dsνν¯ decay. The form factors of these decays can be evaluated with the different approaches. Some of them are the light front, the constituent quark models [9], and the QCD sum rules. In this study the three-point QCD sum rules approach is used in the calculation of form factors. It is worth mentioning that the QCD sum rules have widely been utilized in calculation of the form factors (some of them can be found in [10–17]).The paper has 3 sections. In Section2, the effective Hamiltonian and the three-point QCD sum rules approach are presented for completeness. In Section 3, the numerical values of form factors are given and the sensitivity of the branching ratio is studied and conclusion is presented.


## 2. Sum Rules for theBc*→Dsνν¯ Transition Form Factors

The FCNCb→sνν¯ decay is described within the framework of the SM at the quark level by the effective Hamiltonian [18, 19]:
(1)Heff=GFα22πsin2θWVtbVts*X(x)b¯γμ(1-γ5)sν¯γμ(1-γ5)ν,
where GF is the Fermi constant, θW is the Weinberg angle, α is the fine structure coupling constant, and
(2)X(x)=X0(x)+αs4πX1(x).TheX0(x) is
(3)X0=x8[x+2x-1+3(x-2)(x-1)2ln⁡x],
where x=mt2/mW2. The explicit form of X1(x) is given in [18–20]. Note that X1(x) gives about 3% contribution to the X0(x) term [21].The Wilson coefficients (in our caseX0(x) and X1(x)) can be calculated in any gauge and they are gauge independent and the results should be gauge invariant. The Wilson coefficients are calculated in Rξ gauge. It is worth mentioning that local operators in the considered problem have anomalous dimensions. We have checked that taking into account anomalous dimensions can change numerical results at most 10%.The matrix element of the exclusiveBc*→Dsνν¯ decays is found by inserting initial meson state Bc* and final meson state Ds in (1):
(4)M=GFα22πsin2θWVtbVts*X(x)×〈Ds(pD)|s¯γμ(1-γ5)b|Bc*(pB,ɛ)〉ν¯γμ(1-γ5)ν,
where ɛ is the polarization vector of Bc* meson, pB is the momentum of the Bc*, and pD is the momentum of Ds meson. The matrix element of (4) is written in terms of the form factors as follows:
(5)〈Ds(pD)|s¯γμ(1-γ5)b|Bc*(pB,ɛ)〉=AV(q2)mBc*εμναβε*νpBαpDβ-iA0(q2)mBc*εμ*-iA+(q2)mBc*(ε*pD)Pμ-iA-(q2)mBc*(ε*pD)qμ.
Here, Lorentz invariant and parity conservation are considered. Also, Ai(q2), where i=V,0,+,- are the dimensionless transition form factors. Pμ=(pB+pD)μ and qμ=(pB-pD)μ is the transfer momentum or the momentum of the Z boson.The matrix element in terms of the form factors is as(6)M=GFα22πsin2θWVtbVts*X(x)×[iA1(q2)mBc*εμναβε*νpBαpDβ-iA0(q2)mBc*εμ*-iA+(q2)mBc*(ε*pD)Pμ-iA-(q2)mBc*(ε*pD)qμ]×ν¯γμ(1-γ5)ν,
where A1=-iAV.We try to calculate the aforementioned form factors by means of the QCD sum rules. The QCD sum rules begin with the following correlation functions:(7)ΠμνV-AV(pB2,pD2,q2)=i2∫‍d4xd4ye-ipBxeipDy×〈0|T[JDs(y)JμV-AV(0)JνBc*(x)]|0〉,
where the interpolating currents are JDs(y)=c¯γ5s and JνBc*(x)=b¯γνc the Ds and the Bc* meson states, respectively. JμV-AV=s¯γμ(1-γ5)b consists of the vector (V) and axial vector (AV) transition currents. After inserting the two complete sets of the Bc* and Ds meson, the correlation functions in (7) are written as follows:
(8)ΠμνV-AV(pB2,pD2,q2)=-(〈0|JDs|Ds(pD)〉〈Ds(pD)|JμV-AV|Bc*(pB,ɛ)〉×〈Bc*(pB,ɛ)|JνBc*|0〉)×((pD2-mDs2)(pB2-mBc*2))-1+⋯,
where “⋯” shows the contributions come from higher states and continuum of the currents with the same quantum numbers.The〈0|JDs|Ds(pD)〉 and 〈Bc*(pB,ɛ)|JνBc*|0〉 matrix elements are defined as follows:
(9)〈0|JDs|Ds(pD)〉=-ifDsmDs2ms+mc,〈Bc*(pB,ɛ)|JνBc*|0〉=fBc*mBc*εν,
where fBc and fDs are the leptonic decay constants of Bc* and Ds mesons, respectively. Using these equations and calculating the summation over the polarization of the vector meson Bc*, (8) is as follows:
(10)ΠμνV-AV(pB2,pD2,q2)=-fDsmDs2(mc+ms)fBc*mBc*(pD2-mDs2)(pB2-mBc*2)×[A0(q2)mBc*gμν+A+(q2)mBc*PμpBν+A-(q2)mBc*qμpBν+iA1(q2)mBc*εμναβpBαpDβ]+excitedstates.
This correlation function is calculated in terms of the quarks and gluons parameters by means of the operator product expansion (OPE) as
(11)ΠμνV-AV(pB2,pD2,q2)=Π0V-AVmBc*gμν+Π+V-AVmBc*PμpBν+Π-V-AVmBc*qμpBν+iΠ1V-AVmBc*εμναβpBαpDβ.
Each Πi with i=0,+,- and 1 contains the perturbative and nonperturbative parts as in the following:
(12)Πi=Πipert+Πinonpert.
The bare-loop diagram given in Figure 1(a) is the contribution of the perturbative part. The nonperturbative part consists of the two gluon condensates diagrams {see Figures 2(a)–2(f)}. Hence, contributions of the light quark condensates {diagrams shown in Figures 1(b), 1(c), and 1(d)} vanish by applying the double Borel transformations [16].The bare-loop and light quarks condensates contributions toBc*→Dsl+l- transitions.
(a)
(b)
(c)
(d)Gluon condensate contributions toBc*→Dsν+ν- transitions.
(a)
(b)
(c)
(d)
(e)
(f)The following double dispersion integrals are the contributions of the bare-loop diagrams in the correlation function:(13)Πiper=-1(2π)2∫‍du∫‍dsρi(s,u,q2)(s-pB2)(u-pD2)+subtractionterms.
One of the basic methods to solve the Feynman Integrals in order to calculate the spectral densities ρi(s,u,q2) is Cutkosky rules where the quark propagators are replaced by Dirac Delta Functions: 1/(p2-m2)→-2πiδ(p2-m2), which indicates that all quarks are on-shell.Three delta functions appear as a result of the applying Cutkosky rules. These delta functions have to vanish at the same time. Therefore, we get the following inequality from the arguments of the delta functions:(14)-1≤2su+(s+u-q2)(mb2-s-mc2)+(mc2-ms2)2sλ1/2(mb2,s,mc2)λ1/2(s,u,q2)≤+1,
where λ(a,b,c)=a2+b2+c2-2ac-2bc-2ab.Following the standard calculations, the spectral densities are evaluated as(15)ρ1V-AV=NcI0(s,u,q2){C1(mb-mc)-(C2+1)mc+C2ms},ρ0V-AV=Nc2I0(s,u,q2)×{-2mc3+2msmc2-[(C1+C2+1)(-q2+s+u)+2C1s+2C2u]mc+mb[2mc2-2msmc+2C2u+C1(-q2+s+u)]+ms[2C1s+C2(-q2+s+u)]},ρ+V-AV=Nc2I0(s,u,q2)ρ+V-AV=×{C1(mb-2C2mc-mc+2C2ms)iiiiiiiiiiiiiiiiii-(2C2+1)(C2mc+mc-C2ms)},ρ-V-AV=Nc2I0(s,u,q2)ρ-V-AV=×{(2C2-1)(C2mc+mc-C2ms)iiiiiiiiiiiiiiiiiii+C1(mb-2C2mc-mc+2C2ms)},
where
(16)I0(s,u,q2)=14λ1/2(s,u,q2),C1=mc2(s-u-q2)+u(2mb2-s+u-q2)-ms2(s+u-q2)λ(s,u,q2)C2=s(2ms2+s-u-q2)-mb2(s+u-q2)-mc2(s-u+q2)λ(s,u,q2)Nc=3.
Now, it is aimed to calculate the nonperturbative part of (12) which consists of the gluon condensates diagrams shown in Figure 2. The gluon condensate contributions are calculated in Fock-Schwinger gauge because in this gauge the gluon field is expressed in terms of gluon field strength tensor directly. The following type of the integrals has to be calculated in order to get the results of the gluon condensate diagrams [15, 22]:
(17)I0[a,b,c]=∫‍d4k(2π)41[k2-mb2]a[(pB+k)2-mc2]b[(pD+k)2-ms2]c,Iμ[a,b,c]=∫‍d4k(2π)4kμ[k2-mb2]a[(pB+k)2-mc2]b[(pD+k)2-ms2]c,Iμν[a,b,c]=∫‍d4k(2π)4kμkν[k2-mb2]a[(pB+k)2-mc2]b[(pD+k)2-ms2]c,
where k is the momentum of the spectator quark c. The generic solutions for these integrals can be seen in [22, 23]. A part of our results for the contributions of the gluon condensate diagrams following similar methods shown in [22, 23] is given in the Appendix.The Borel transformations are applied for both phenomenological and QCD side {see (11)} in order to suppress the contributions of higher states and continuum. The QCD sum rules for the form factors (AV, A0, A+, and A-) are obtained by equalizing the Borel transformed forms of the physical side. The result is in the following formula:
(18)Ai(q2)=(ms+mc)emBc*2/M12emDs2/M22fBc*mBc*fDsmDs2×[1(2π)2∫umin⁡u0du∫smin⁡s0dsρiV-AV(s,u,q2)e-s/M12-u/M22iiiiiiiiiiii+1(2π)2∫umin⁡u0dui124π2CAi〈αsπG2〉].
Note that the contributions of the gluon condensates (CAi) are already considered in the numerical analysis. However, each of these explicit expressions is extremely long; it is found unnecessary to show all of them in this study. Therefore, one of these expressions (CAV) is shown as a sample in Appendix. The s0 and u0 are the continuum thresholds in s and u channels, respectively. Also smin⁡=(mb+mc)2 and umin⁡=(ms+mc)2.


## 3. Numerical Analysis

Having known the matrix element, that is, (6), the decay rate for Bc*→Dsνν¯ decay is evaluated as follows:
(19)dΓdq2=α2Gf2λ1/2(mBc*2,mDs2,q2)|VtbVts*|2vX2(x)3072π5mBc*3sin4θW×{|A0|2(mBc*4-2mBc*2(mDs2-5q2)+(mDs2-q2)2)λ2(mBc*2,mDs2,q2)mBc*4-2Re[A+A0*]×((mBc*6-(mDs2-q2)3-mBc*4(3mDs2+q2)+mBc*2(3mDs4-2mDs2q2-q4))×(mBc*2)-1)+2|A1|2q2λ(mBc*2,mDs2,q2)mBc*2+|A+|2λ2(mBc*2,mDs2,q2)mBc*4}.The expression for the decay rate shows that we need to know the input parameters shown in Table1, taken from [24].Table 1
The values of the input parameters [24].
|
V
t
b
|
0.7
7
-
0.24
+
0.18
|
V
t
s
|
(
40.6
±
2.7
)
×
1
0
-
3
τ
B
c
*
(
0.452
±
0.033
)
×
1
0
-
12
s
α
(
m
w
2
)
1
/
128
sin
⁡
2
⁡
θ
W
0.2315
m
t
173.07
±
0.52
±
0.72GeV
m
W
80.385
±
0.015GeV
m
B
c
*
6.2745
±
0.0018GeV
m
D
s
1968.30
±
0.11MeV
f
D
s
(
206.7
±
8.5
±
2.5
)MeV
m
b
(
4.18
±
0.03
)
GeV
m
c
(
μ
=
m
c
)
1.275
±
0.015
GeVMoreover, the values of the leptonic decay constantsfBC*=0.415±0.031 GeV [25] and the gluon condensate 〈(αs/π)G2〉=0.012 GeV4 [26] are necessary for the evaluation of the form factors. In addition, the form factors contain four auxiliary parameters: the Borel mass squares M12 and M22 and the continuum threshold s0 and u0. The form factors are assumed to be independent of or weakly dependent on these auxiliary parameters in the suitable chosen regions named as “working regions.”The contributions proportional to the highest power of1/M1,22 are supposed to be less than about 30% of the contributions proportional to the highest power of M1,22. The lower bound of the M12 and M22 can be determined by the above condition. In addition, the contributions of continuum must be less than that of the first resonance. This helps us to fix the upper bound of the M12 and M22. Therefore, we find the suitable region for the Borel mass parameters in the following intervals; 10GeV2≤M12≤25GeV2 and 4GeV2≤M22≤10GeV2.The numerical value of thes0 and u0 is supposed to be less than the mass squared of the first excited state meson with the same quantum numbers. In other words, the s0 and u0 are between mass squared of the ground sate meson and excited state meson with the same quantum numbers. The following regions for the s0 and u0 are chosen: (mBc*+0.3)2≤s0≤(mBc*+0.7)2 and (mDs+0.3)2≤u0≤(mDs+0.7)2.The form factors depend on theq2. The detail of the dependence is complicated. We fit them to the following function:
(20)F(q2)=a1-q2/mfit2+b(1-q2/mfit2)2.Thea, b, and mfit are given in Table 2.Table 2
Parameters appearing in the form factors of theBc*→Dsνν- decay in a four-parameter fit, for M12=15GeV2, M22=6 GeV2, s0=46 GeV2, and u0=6 GeV2.
m
fit
a
b
A
1
(
q
2
)
5.01
±
1.1
-
0.14
±
0.04
0.26
±
0.08
A
0
(
q
2
)
6.44
±
1.4
-
0.11
±
0.03
0.17
±
0.06
A
+
(
q
2
)
5.00
±
1.08
-
0.14
±
0.04
0.28
±
0.08
A
-
(
q
2
)
4.98
±
1.07
-
0.14
±
0.04
0.28
±
0.08The origin of the errors in Table2 is the variation of s0, u0, and M1,2 in the chosen intervals and the uncertainties of the input parameters.In order to evaluate the branching ratio of theBc*→Dsνν¯ decay, the mean life time of the Bc* meson is needed. For the time being there is no experimental data on the mean life time of this meson. We follow the theoretical methods like Bethe-Salpeter model [27] and potential model [28] and estimate that the mean life time of the Bc* meson is in the order of the mean life time of the Bc meson. We assume that the total life time τBc≈τBc=0.452×10-12 s [24]. Using the mean life time and the q2 dependence of the form factors given by (20) in the kinematical allowed region [0≤q2≤(mBc*-mDs)2] we study the branching ratios for Bc*→Dsνν¯ decay. Our results for three different values of the q2=(1,6,12) GeV2 are presented in Table 3. In addition, Figure 3 depicts the dependence of the branching ratio on q2 for full kinematical allowed region.Table 3
Values for the branching ratio of theBc*→Dsνν- decay at three different values of the dileptonic invariant mass.
q
2 (GeV2)
B
r
(
q
2
)
(
B
c
*
→
D
s
ν
ν
-)
1
1.83
×
1
0
-
10
6
9.68
×
1
0
-
10
12
3.99
×
1
0
-
9Figure 3
The dependence of the branching ratio onq2 for Bc*→Dsμ+μ- transitions.Finally, we calculate the integrated branching ratio forBc*→Dsνν¯ decay as follows:
(21)Br=∫0(mBc*-mDs)2‍Br(q2)dq2=(5.47±1.30)×10-8.To sum up, we investigated the branching ratio and decay rate of theBc*→Dsνν¯ decay. The form factors of this decay were found in the framework of the QCD sum rules. In addition, the contributions of the two gluon condensates diagrams to the correlations function were obtained.



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