# Impact of Scalar Leptoquarks on Heavy Baryonic Decays

Authors: K. Azizi; A. T. Olgun; Z. Tavukoğlu
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

We present a study on the impact of scalar leptoquarks on the semileptonic decays ofΛb, Σb, and Ξb. To this end, we calculate the differential branching ratio and lepton forward-backward asymmetry defining the processes Λb→Λl+l-, Σb→Σl+l-, and Ξb→Ξl+l-, with l being μ or τ, using the form factors calculated via light cone QCD in full theory. In calculations, the errors of form factors are taken into account. We compare the results obtained in leptoquark model with those of the standard model as well as the existing lattice QCD predictions and experimental data.


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## Body

## 1. Introduction

The physics of transitions based onb→sl+l- at quark level constitute one of the main directions of the research in high energy and particle physics both theoretically and experimentally as new physics effects can contribute to such decay channels. The flavor changing neutral current (FCNC) transitions of Λb→Λl+l-, Σb→Σl+l-, and Ξb→Ξl+l- are among important baryonic decay channels that can be used as sensitive probes to indirectly search for new physics contributions. Particularly, the rare Λb→Λl+l- decay channel has been in the focus of much attention in recent years both theoretically and experimentally. The first measurement on the Λb→Λμ+μ- process has been reported by the CDF Collaboration [1] with 24 signal events and a statistical significance of 5.8 Gaussian standard deviations. Using the pp¯ collisions data samples corresponding to 6.8fb-1 and s=1.96 TeV collected by the CDF II detector, the differential branching ratio for the Λb→Λμ+μ- decay channel has been measured to be dBr(Λb0→Λμ+μ-)/dq2=[1.73±0.42(stat)±0.55(syst)]×10-6 [1]. The differential branching fraction of Λb0→Λμ+μ- decay channel has also been measured as dBr(Λb0→Λμ+μ-)/dq2=(1.18-0.08+0.09±0.03±0.27)×10-7GeV2/c4 at 15GeV2/c4≤q2≤20GeV2/c4 region by the LHCb Collaboration [2]. The LHCb Collaboration has also measured the lepton forward-backward asymmetries associated with this transition as AFBμ=-0.05±0.09(stat)±0.03(syst) at 15GeV2/c4≤q2≤20GeV2/c4 region [2]. The order of branching ratio in Λb→Λe+e- and Λb→Λτ+τ- as well as in Σb→Σl+l- and Ξb→Ξl+l- (for all leptons) indicates that these channels are all accessible at LHC (for details see [3–7]). We hope that with the RUN II data at the center of mass energy 13 TeV it will be possible to measure different physical quantities related to these FCNC loop level rare transitions in near future.The LHC RUN II may provide opportunities to search for various new physics scenarios. One of the important new physics models that has been proposed to overcome the problems of some inconsistencies between the SM predictions and experimental data is the leptoquark (LQ) model. Hereafter, by LQ model we mean a minimal renormalizable scalar leptoquark model which will be explained in some details in next section. As an example for the LHC constraints and prospects for scalar leptoquarks explaining theB¯→D(∗)τν¯ anomaly see [8]. LQs are hypothetical color triplet bosons that couple to leptons and quarks [9]. LQs carry both baryon (B) and lepton (L) quantum numbers with color and electric charges. The spin number of a leptoquark state can be 0 or 1, corresponding to a scalar leptoquark or vector leptoquark. If the leptoquarks violate both the baryon and lepton numbers, they are generally considered to be heavy particles at the level of O(1015) GeV in order to prevent the proton decay. For more detailed information about leptoquark models and the recent experimental and theoretical progress, see [10–30].In the light of progress about LQs, we calculate the differential branching ratio and lepton forward-backward asymmetry corresponding toΛb→Λl+l-, Σb→Σl+l-, and Ξb→Ξl+l- processes in a scalar LQ model. In the calculations, we use the form factors as the main inputs calculated from the light cone QCD sum rules in full theory. We also encounter the errors of the form factors to the calculations. We compare the regions swept by the LQ model with those of the SM and search for deviations of the LQ model predictions with those of the SM. We also compare the results with the available lattice predictions and experimental data.The outline of this article is as follows. In next section, we present the effective Hamiltonian responsible for the transitions under consideration both in the SM and LQ models. In Section3, we present the transition amplitude and matrix elements defining the above transitions. In Section 4, we calculate the differential decay rate and the lepton forward-backward asymmetry in the baryonic Λb→Λl+l-, Σb→Σl+l-, and Ξb→Ξl+l- channels and numerically analyze the results obtained. We compare the LQ predictions with those of the SM and existing lattice results and experimental data also in this section.


## 2. The Effective Hamiltonian and Wilson Coefficients

At the quark level the effective Hamiltonian, defining the above-mentionedb→sl+l- based transitions, in terms of Wilson coefficients and different operators in SM is generally defined as [32, 33](1)HSMeff=GFαemVtbVts∗22πC9effs-γμ1-γ5bl-γμl+C9′effs-γμ1+γ5bl-γμl+C10s-γμ1-γ5bl-γμγ5l+C10′s-γμ1+γ5bl-γμγ5l-2mbC7eff1q2s-iσμνqν1+γ5bl-γμl-2mbC7′eff1q2s-iσμνqν1-γ5bl-γμl,where GF is the Fermi weak coupling constant, αem is the fine structure constant at Z mass scale, Vtb and Vts∗ are elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, C9(′)eff, C10(′), and C7(′)eff are the SM Wilson coefficients, and q2 is the transferred momentum squared. Here the superscript “eff” refers to the shifts in the corresponding coefficients due to the effects of four-quark operators at large q2. The primed coefficients are ignored since the Hamiltonian does not receive any contribution from the corresponding operators in the SM. We collect the explicit expressions of the Wilson coefficients C9eff, C10, and C7eff in Appendix A.Considering the additional contributions arising from the exchange of scalar leptoquarks, the effective Hamiltonian is modified. The modified Hamiltonian in LQ model is obtained from (1) by the replacements C9eff→C9eff,tot, C9′eff→C9′eff,tot, C10→C10tot, and C10′→C10′tot. Here, C9eff,tot, C9′eff,tot, C10tot, and C10′tot, with the superscript “tot” being referring to “total,” are new Wilson coefficients. These coefficients contain contributions from both the SM and LQ models. Note that the Wilson coefficients C7eff and C7′eff remain unchanged compared to the SM. The new Wilson coefficients are given as (for details see, for instance, [20, 22–25]) (2)C9eff,tot=C9eff+C9LQ,C9′eff,tot=C9′eff+C9′LQ,C10tot=C10+C10LQ,C10′tot=C10′+C10′LQ,where the coefficients C9LQ and C10LQ receive contributions from the exchange of the scalar leptoquarks X(7/6)=(3,2,7/6) but the primed Wilson coefficients C9′LQ and C10′LQ pick up contributions from the exchange of the scalar leptoquarks X(1/6)=(3,2,1/6). Here we should remark that we consider the effects of the above two scalar leptoquarks on the Wilson coefficients since this representation does not allow for proton decay at tree-level. We do not consider the effects of the vector leptoquarks on the processes under consideration. Hence, in the present study we consider the minimal renormalizable scalar leptoquark models including one single additional representation of SU(3)×SU(2)×U(1) which guarantees that the proton does not decay. This requisite can only be satisfied by the models that have the representation of X(7/6)=(3,2,7/6) and X(1/6)=(3,2,1/6) scalar leptoquarks under the above gauge group (for details see, for instance, [18, 22]).Thus the coefficientsC9LQ and C10LQ are obtained as [20, 22–25](3)C9LQ=C10LQ=-π22GFαemVtbVts∗λe23λe22∗MY2,and the primed Wilson coefficients C9′LQ and C10′LQ are found as [20, 22–25](4)C9′LQ=-C10′LQ=π22GFαemVtbVts∗λs22λb32∗MV2,where Y and V are the two components of doublet LQ, X=(Vα,Yα), with MY and MV being representing the masses of the components of the scalar leptoquarks (for details on the LQ interaction Lagrangian and corresponding notations see [34]). It is assumed that each individual leptoquark contribution to the branching ratio does not exceed the experimental result. Here(5)0≤λe23λe22∗MY2=λs22λb32∗MV2≤5×10-9GeV-2,obtained via the fitting of the model parameters to the Bs→μ+μ- data [34]. In (5) we assumed that the contributions of the two components Y and V are equal.


## 3. Transition Amplitude and Matrix Elements

Generally, the amplitude of the transition responsible for theΛb→Λl+l-, Σb→Σl+l-, and Ξb→Ξl+l- baryonic decays is provided with sandwiching the effective Hamiltonian between the initial and final baryonic states,(6)MBQ→Bl+l-=BpHeffBQp+q,s,where B represents Λ, Σ, and Ξ baryons and Q corresponds to b quark. To get the transition amplitude, we need to consider the following transition matrix elements parametrized in terms of twelve form factors in full QCD, that is, without any expansion in the heavy quark mass or large hadron energies:(7)Bps-γμ1-γ5bBQp+q,s=u-Bpγμf1q2+iσμνqνf2q2+qμf3q2-γμγ5g1q2-iσμνγ5qνg2q2-qμγ5g3q2uBQp+q,s,Bps-γμ1+γ5bBQp+q,s=u-Bpγμf1q2+iσμνqνf2q2+qμf3q2+γμγ5g1q2+iσμνγ5qνg2q2+qμγ5g3q2uBQp+q,s,Bps-iσμνqν1+γ5bBQp+q,s=u-Bpγμf1Tq2+iσμνqνf2Tq2+qμf3Tq2+γμγ5g1Tq2+iσμνγ5qνg2Tq2+qμγ5g3Tq2uBQp+q,s,Bps-iσμνqν1-γ5bBQp+q,s=u-Bpγμf1Tq2+iσμνqνf2Tq2+qμf3Tq2-γμγ5g1Tq2-iσμνγ5qνg2Tq2-qμγ5g3Tq2uBQp+q,s,where uBQ and uB represent spinors of the initial and final states, respectively. fi(T) and gi(T) (i running from 1 to 3) are transition form factors. The values of these form factors corresponding to Λb→Λl+l-, Σb→Σl+l-, and Ξb→Ξl+l- transitions and calculated via light cone sum rules in full theory are taken from [3, 4] and [5], respectively (for form factors of Λb channel calculated with different phenomenological models see also, for instance, [35–37]). These form factors are also available in lattice QCD in Λ channel [31].Using the above transition matrix elements in terms of form factors, we get the amplitude of the transitionsΛb→Λl+l-, Σb→Σl+l-, and Ξb→Ξl+l- in the SM and LQ as (8)MSMBQ→Bl+l-=GFαemVtbVts∗22πu-BpγμA1SMR+B1SML+iσμνqνA2SMR+B2SML+qμA3SMR+B3SMLuBQp+q,sl-γμl+u-BpγμD1SMR+E1SML+iσμνqνD2SMR+E2SML+qμD3SMR+E3SMLuBQp+q,sl-γμγ5l,MtotBQ→Bl+l-=GFαemVtbVts∗22πu-BpγμA1totR+B1totL+iσμνqνA2totR+B2totL+qμA3totR+B3totLuBQp+q,sl-γμl+u-BpγμD1totR+E1totL+iσμνqνD2totR+E2totL+qμD3totR+E3totLuBQp+q,sl-γμγ5l,where R=(1+γ5)/2 and L=(1-γ5)/2 and the calligraphic coefficients are collected in Appendix B.


## 4. Physical Observables

In this section we would like to calculate some physical observables such as the differential decay width, the differential branching ratio, and the lepton forward-backward asymmetry for the considered decay channels.


### 4.1. The Differential Decay Width

Using the decay amplitudes and transition matrix elements in terms of form factors, we find the differential decay rate defining the transitions under consideration in the LQ model as(9)d2Γtotds^dzz,s^=GF2αem2mBQ16384π5VtbVts∗2vλ1,r,s^T0tots^+T1tots^z+T2tots^z2,where v=1-4ml2/q2 is the lepton velocity, λ=λ(1,r,s^)=(1-r-s^)2-4rs^ is the usual triangle function, s^=q2/mBQ2, r=mB2/mBQ2, and z=cos⁡θ with θ being the angle between momenta of the lepton l+ and BQ in the center of mass of leptons. The calligraphic T0tot(s^), T1tot(s^), and T2tot(s^) functions are given in Appendix B.


### 4.2. The Differential Branching Ratio

Using the expression of the differential decay width, in this subsection, we numerically analyze the differential branching ratio in terms ofq2 for the decay channels under consideration. For this aim, we present the values of some input parameters and the quark masses in MS¯ scheme used in the numerical analysis in Tables 1 and 2 [9]. Using the numerical values in these tables and the expressions presented in Appendix A, we find the values/intervals C7eff=-0.295,  C9eff=[1.573,6.625],  C10=-4.260,  C9eff,tot=[2.793,4.394],  C9′eff,tot=[0,1.586],  C10tot=[-5.846,-4.260], and C10′tot=[-1.586,0] for the corresponding Wilson coefficients. Since C9eff(tot) depend on q2, the above intervals for these coefficients denote the maximum and minimum values obtained varying q2 in the physical region, that is, [0–20]GeV2. In the case of coefficients with label “tot” the above intervals are obtained considering the intervals for related parameters in (5). Note that we will use directly the expressions of the Wilson coefficients in the numerical analyses instead of the above-mentioned values/intervals. We shall remark that the above-mentioned values/intervals for C7eff, C9eff, and C10 are consistent with the ones obtained in [38–43] for Wilson coefficients using the global fits to b→sl+l- data. We would also like to compare the intervals for four Wilson coefficients C9eff,tot, C9′eff,tot, C10tot, and C10′tot, which are relevant to the LQ model with the values extracted in [44] from experimental data on observables of Λb→Λμ+μ- in a (9,10,9′,10′) scenario assuming uncorrelated independent contributions to these coefficients. In [44] the values C9=6.0-0.8+0.8, C9′=0.5-1.8+1.3, C10=-1.3-1.1+1.3, and C10′=2.3-1.3+0.8 are obtained. The comparison of the intervals obtained in the present study with those of [44] shows that our prediction on the range of C9′ exactly remains inside the interval obtained in [44]. For other coefficients although the values obtained in these works are comparable in some regions, we overall see considerable differences between the predictions of two studies. The difference in C9 can be attributed to the fact that in [44] the authors use the data only in the interval q2=[15–20]GeV2 to extract its value.Table 1
The values of some input parameters used in our analysis [9].
Some input parameters
Values
m
Λ
b
5.6195 GeV
m
Λ
1.11568 GeV
τ
Λ
b
1.451
×
1
0
-
12
s
m
Σ
b
5.807 GeV
m
Σ
1.192 GeV
τ
Σ
b
1.391
×
1
0
-
12
s
m
Ξ
b
5.791 GeV
m
Ξ
1.314 GeV
τ
Ξ
b
1.464
×
1
0
-
12
s
m
W
80.385 GeV
G
F
1.166
×
1
0
-
5
G
e
V
-
2
α
e
m
1/137
|
V
t
b
V
t
s
∗
|
0.040Table 2
The values of quark masses inMS¯ scheme [9].
Quarks
Masses inMS¯ scheme
m
c
(
1.275
±
0.025
)
G
e
V
m
b
(
4.18
±
0.03
)
G
e
V
m
t
16
0
-
4.3
+
4.8
G
e
VAs we previously said, we use the values of form factors calculated via light cone QCD sum rules in full theory and available for all channels under consideration from [3–5]. These form factors are also available in lattice QCD in Λ channel [31]. The differential branching ratios of decay channels under consideration on q2, in the SM and LQ models, at μ and τ lepton channels are plotted in Figures 1–6. Note that, in these figures, the form factors are encountered with their uncertainties in both models. The bands in LQ model are due to both the constrained regions of some parameters presented in (5) and errors of form factors. In these figures, we show the charmonia veto regions by the vertical shaded bands. We do not present the results for e channel in the figures, because the predictions of μ channel are very close to those of the e channel. In Figure 1, we also show the experimental data provided by LHCb [2] and lattice predictions [31]. From these figures it is clear that(i)
the bands of differential branching ratios in terms ofq2 obtained in SM for all baryonic processes at both lepton channels remain inside the bands of the LQ model. The LQ model bands are wider and somewhere show considerable discrepancies from the SM predictions for all channels roughly at whole physical regions of q2;(ii)
the SM band for the differential branching fraction inΛb→Λμ+μ- channel roughly coincides with all the lattice predictions borrowed from [31]. This band also defines all the experimental data provided by the LHCb Collaboration except that in the interval 18GeV2/c4≤q2≤20GeV2/c4, which can not be described by the SM. This datum coincides with the LQ band. As is also seen from this figure the lattice QCD predictions on the differential branching fraction in Λb→Λμ+μ- channel show considerable discrepancies with the experimental data in the interval 15GeV2/c4≤q2≤20GeV2/c4.Figure 1
The dependence of the differential branching ratio onq2 for the Λb→Λμ+μ- transition in the SM and LQ models. The experimental data are taken from the LHCb Collaboration [2]. The lattice predictions are borrowed from [31]. The vertical shaded bands indicate the charmonia veto regions.Figure 2
The dependence of the differential branching ratio onq2 for the Λb→Λτ+τ- transition in the SM and LQ models. The vertical shaded band indicates the charmonia veto region.Figure 3
The dependence of the differential branching ratio onq2 for the Σb→Σμ+μ- transition in the SM and LQ models. The vertical shaded bands indicate the charmonia veto regions.Figure 4
The dependence of the differential branching ratio onq2 for the Σb→Στ+τ- transition in the SM and LQ models. The vertical shaded band indicates the charmonia veto region.Figure 5
The dependence of the differential branching ratio onq2 for the Ξb→Ξμ+μ- transition in the SM and LQ models. The vertical shaded bands indicate the charmonia veto regions.Figure 6
The dependence of the differential branching ratio onq2 for the Ξb→Ξτ+τ- transition in the SM and LQ models. The vertical shaded band indicates the charmonia veto region.


### 4.3. The Lepton Forward-Backward Asymmetry

In this subsection, we present the results of the lepton forward-backward asymmetry (AFB) which is one of useful observables to search for NP effects. This quantity is defined as(10)AFBs^=∫01d2Γ/ds^dzz,s^dz-∫-10d2Γ/ds^dzz,s^dz∫01d2Γ/ds^dzz,s^dz+∫-10d2Γ/ds^dzz,s^dz.In order to see how predictions of LQ scenario deviate from those of the SM, we plot the dependence of the lepton forward-backward asymmetry onq2 for the channels under discussion in Figures 7–12. In Figure 7, we also present the measured values of the leptonic forward-backward-asymmetries by the LHCb Collaboration [2] as well as the lattice QCD predictions [31] in the Λb→Λμ+μ- decay channel. From these figures, we read that(i)
in all decay channels the LQ model predictions demonstrate considerable discrepancies from the SM predictions;(ii)
the SM band on the lepton forward-backward asymmetry inΛb→Λμ+μ- channel coincides with the existing lattice QCD predictions borrowed from [31];(iii)
ignoring from the small intersection of the SM narrow bands with errors of the experimental data at very low and high values ofq2, the LQ model, against the SM, can describe all data available in Λb→Λμ+μ- channel. The lattice QCD predictions in this channel also show sizable differences with the experimental data.Figure 7
The dependence of theAFB on q2 for the Λb→Λμ+μ- transition in the SM and LQ models. The experimental data are taken from the LHCb Collaboration [2]. The lattice predictions are borrowed from [31]. The vertical shaded bands indicate the charmonia veto regions.Figure 8
The dependence ofAFB on q2 for the Λb→Λτ+τ- transition in the SM and LQ models. The vertical shaded band indicates the charmonia veto region.Figure 9
The dependence ofAFB on q2 for the Σb→Σμ+μ- transition in the SM and LQ models. The vertical shaded bands indicate the charmonia veto regions.Figure 10
The dependence ofAFB on q2 for the Σb→Στ+τ- transition in the SM and LQ models. The vertical shaded band indicates the charmonia veto region.Figure 11
The dependence ofAFB on q2 for the Ξb→Ξμ+μ- transition in the SM and LQ models. The vertical shaded bands indicate the charmonia veto regions.Figure 12
The dependence ofAFB on q2 for the Ξb→Ξτ+τ- transition in the SM and LQ models. The vertical shaded band indicates the charmonia veto region.


## 4.1. The Differential Decay Width

Using the decay amplitudes and transition matrix elements in terms of form factors, we find the differential decay rate defining the transitions under consideration in the LQ model as(9)d2Γtotds^dzz,s^=GF2αem2mBQ16384π5VtbVts∗2vλ1,r,s^T0tots^+T1tots^z+T2tots^z2,where v=1-4ml2/q2 is the lepton velocity, λ=λ(1,r,s^)=(1-r-s^)2-4rs^ is the usual triangle function, s^=q2/mBQ2, r=mB2/mBQ2, and z=cos⁡θ with θ being the angle between momenta of the lepton l+ and BQ in the center of mass of leptons. The calligraphic T0tot(s^), T1tot(s^), and T2tot(s^) functions are given in Appendix B.


## 4.2. The Differential Branching Ratio

Using the expression of the differential decay width, in this subsection, we numerically analyze the differential branching ratio in terms ofq2 for the decay channels under consideration. For this aim, we present the values of some input parameters and the quark masses in MS¯ scheme used in the numerical analysis in Tables 1 and 2 [9]. Using the numerical values in these tables and the expressions presented in Appendix A, we find the values/intervals C7eff=-0.295,  C9eff=[1.573,6.625],  C10=-4.260,  C9eff,tot=[2.793,4.394],  C9′eff,tot=[0,1.586],  C10tot=[-5.846,-4.260], and C10′tot=[-1.586,0] for the corresponding Wilson coefficients. Since C9eff(tot) depend on q2, the above intervals for these coefficients denote the maximum and minimum values obtained varying q2 in the physical region, that is, [0–20]GeV2. In the case of coefficients with label “tot” the above intervals are obtained considering the intervals for related parameters in (5). Note that we will use directly the expressions of the Wilson coefficients in the numerical analyses instead of the above-mentioned values/intervals. We shall remark that the above-mentioned values/intervals for C7eff, C9eff, and C10 are consistent with the ones obtained in [38–43] for Wilson coefficients using the global fits to b→sl+l- data. We would also like to compare the intervals for four Wilson coefficients C9eff,tot, C9′eff,tot, C10tot, and C10′tot, which are relevant to the LQ model with the values extracted in [44] from experimental data on observables of Λb→Λμ+μ- in a (9,10,9′,10′) scenario assuming uncorrelated independent contributions to these coefficients. In [44] the values C9=6.0-0.8+0.8, C9′=0.5-1.8+1.3, C10=-1.3-1.1+1.3, and C10′=2.3-1.3+0.8 are obtained. The comparison of the intervals obtained in the present study with those of [44] shows that our prediction on the range of C9′ exactly remains inside the interval obtained in [44]. For other coefficients although the values obtained in these works are comparable in some regions, we overall see considerable differences between the predictions of two studies. The difference in C9 can be attributed to the fact that in [44] the authors use the data only in the interval q2=[15–20]GeV2 to extract its value.Table 1
The values of some input parameters used in our analysis [9].
Some input parameters
Values
m
Λ
b
5.6195 GeV
m
Λ
1.11568 GeV
τ
Λ
b
1.451
×
1
0
-
12
s
m
Σ
b
5.807 GeV
m
Σ
1.192 GeV
τ
Σ
b
1.391
×
1
0
-
12
s
m
Ξ
b
5.791 GeV
m
Ξ
1.314 GeV
τ
Ξ
b
1.464
×
1
0
-
12
s
m
W
80.385 GeV
G
F
1.166
×
1
0
-
5
G
e
V
-
2
α
e
m
1/137
|
V
t
b
V
t
s
∗
|
0.040Table 2
The values of quark masses inMS¯ scheme [9].
Quarks
Masses inMS¯ scheme
m
c
(
1.275
±
0.025
)
G
e
V
m
b
(
4.18
±
0.03
)
G
e
V
m
t
16
0
-
4.3
+
4.8
G
e
VAs we previously said, we use the values of form factors calculated via light cone QCD sum rules in full theory and available for all channels under consideration from [3–5]. These form factors are also available in lattice QCD in Λ channel [31]. The differential branching ratios of decay channels under consideration on q2, in the SM and LQ models, at μ and τ lepton channels are plotted in Figures 1–6. Note that, in these figures, the form factors are encountered with their uncertainties in both models. The bands in LQ model are due to both the constrained regions of some parameters presented in (5) and errors of form factors. In these figures, we show the charmonia veto regions by the vertical shaded bands. We do not present the results for e channel in the figures, because the predictions of μ channel are very close to those of the e channel. In Figure 1, we also show the experimental data provided by LHCb [2] and lattice predictions [31]. From these figures it is clear that(i)
the bands of differential branching ratios in terms ofq2 obtained in SM for all baryonic processes at both lepton channels remain inside the bands of the LQ model. The LQ model bands are wider and somewhere show considerable discrepancies from the SM predictions for all channels roughly at whole physical regions of q2;(ii)
the SM band for the differential branching fraction inΛb→Λμ+μ- channel roughly coincides with all the lattice predictions borrowed from [31]. This band also defines all the experimental data provided by the LHCb Collaboration except that in the interval 18GeV2/c4≤q2≤20GeV2/c4, which can not be described by the SM. This datum coincides with the LQ band. As is also seen from this figure the lattice QCD predictions on the differential branching fraction in Λb→Λμ+μ- channel show considerable discrepancies with the experimental data in the interval 15GeV2/c4≤q2≤20GeV2/c4.Figure 1
The dependence of the differential branching ratio onq2 for the Λb→Λμ+μ- transition in the SM and LQ models. The experimental data are taken from the LHCb Collaboration [2]. The lattice predictions are borrowed from [31]. The vertical shaded bands indicate the charmonia veto regions.Figure 2
The dependence of the differential branching ratio onq2 for the Λb→Λτ+τ- transition in the SM and LQ models. The vertical shaded band indicates the charmonia veto region.Figure 3
The dependence of the differential branching ratio onq2 for the Σb→Σμ+μ- transition in the SM and LQ models. The vertical shaded bands indicate the charmonia veto regions.Figure 4
The dependence of the differential branching ratio onq2 for the Σb→Στ+τ- transition in the SM and LQ models. The vertical shaded band indicates the charmonia veto region.Figure 5
The dependence of the differential branching ratio onq2 for the Ξb→Ξμ+μ- transition in the SM and LQ models. The vertical shaded bands indicate the charmonia veto regions.Figure 6
The dependence of the differential branching ratio onq2 for the Ξb→Ξτ+τ- transition in the SM and LQ models. The vertical shaded band indicates the charmonia veto region.


## 4.3. The Lepton Forward-Backward Asymmetry

In this subsection, we present the results of the lepton forward-backward asymmetry (AFB) which is one of useful observables to search for NP effects. This quantity is defined as(10)AFBs^=∫01d2Γ/ds^dzz,s^dz-∫-10d2Γ/ds^dzz,s^dz∫01d2Γ/ds^dzz,s^dz+∫-10d2Γ/ds^dzz,s^dz.In order to see how predictions of LQ scenario deviate from those of the SM, we plot the dependence of the lepton forward-backward asymmetry onq2 for the channels under discussion in Figures 7–12. In Figure 7, we also present the measured values of the leptonic forward-backward-asymmetries by the LHCb Collaboration [2] as well as the lattice QCD predictions [31] in the Λb→Λμ+μ- decay channel. From these figures, we read that(i)
in all decay channels the LQ model predictions demonstrate considerable discrepancies from the SM predictions;(ii)
the SM band on the lepton forward-backward asymmetry inΛb→Λμ+μ- channel coincides with the existing lattice QCD predictions borrowed from [31];(iii)
ignoring from the small intersection of the SM narrow bands with errors of the experimental data at very low and high values ofq2, the LQ model, against the SM, can describe all data available in Λb→Λμ+μ- channel. The lattice QCD predictions in this channel also show sizable differences with the experimental data.Figure 7
The dependence of theAFB on q2 for the Λb→Λμ+μ- transition in the SM and LQ models. The experimental data are taken from the LHCb Collaboration [2]. The lattice predictions are borrowed from [31]. The vertical shaded bands indicate the charmonia veto regions.Figure 8
The dependence ofAFB on q2 for the Λb→Λτ+τ- transition in the SM and LQ models. The vertical shaded band indicates the charmonia veto region.Figure 9
The dependence ofAFB on q2 for the Σb→Σμ+μ- transition in the SM and LQ models. The vertical shaded bands indicate the charmonia veto regions.Figure 10
The dependence ofAFB on q2 for the Σb→Στ+τ- transition in the SM and LQ models. The vertical shaded band indicates the charmonia veto region.Figure 11
The dependence ofAFB on q2 for the Ξb→Ξμ+μ- transition in the SM and LQ models. The vertical shaded bands indicate the charmonia veto regions.Figure 12
The dependence ofAFB on q2 for the Ξb→Ξτ+τ- transition in the SM and LQ models. The vertical shaded band indicates the charmonia veto region.


## 5. Conclusion

In the present work, we have performed a comprehensive analysis of the semileptonicΛb→Λl+l-, Σb→Σl+l-, and Ξb→Ξl+l- rare processes in the SM as well as the scalar leptoquark model. Using the parametrization of the matrix elements in terms of form factors calculated via light cone QCD sum rules in the full theory, we calculated the differential decay width and numerically analyzed the differential branching fraction and the lepton forward-backward asymmetry in terms of q2 in different heavy baryonic decay channels for both μ and τ leptons in both scenarios. We compared the predictions of the LQ model on the considered physical observables with those of the SM and the existing lattice QCD predictions as well as experimental data in Λb→Λμ+μ- channel. We observed that the predictions of the LQ model in all channels show considerable discrepancies with those of the SM on both the differential decay width and lepton forward-backward asymmetry. The SM results for both the observables considered in the present study are consistent with the existing predictions of lattice QCD. Except the interval 18GeV2/c4≤q2≤20GeV2/c4, the SM band describes the existing experimental data on the differential branching ratio in Λb→Λμ+μ- transition. The datum in 18GeV2/c4≤q2≤20GeV2/c4 coincides with the LQ model prediction.In the case of lepton forward-backward asymmetry, the SM, overall, can not describe the experimental data existing inΛb→Λμ+μ- channel, while the LQ model band coincides with the experimental data.More experimental data inΛb→Λτ+τ- as well as Σb→Σl+l- and Ξb→Ξl+l- with both leptons are needed to compare with the theoretical predictions. We hope that, with the RUN II data, it will be possible to measure different physical quantities related to such FCNC transitions at LHCb in near future. Comparison of the future experimental data with the theoretical predictions on different physical quantities in various decay channels can help us better explain some anomalies between the SM predictions and the experimental data. Any sizable discrepancy between the theoretical predictions on physical observables with the experimental data can be considered as an indication of new physics effects and may help us in the course of searching for the new particles like leptoquarks.Note Added. When preparing this work we noticed that a part of our work, namely, the Λb→Λl+l- channel has been investigated in [34, 45] within the same framework. In these studies the authors use the form factors, as the main inputs, calculated in heavy quark effective theory while we use the form factors calculated via light cone QCD sum rules in full theory.



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*Source: 7435876-2017-10-09.xml*