We analyze the baryonic semilepton decay Λb→Λμ+μ- in the scalar leptoquark models with X(3,2,7/6) and X(3,2,1/6) states, respectively. We also discuss the effects of these two NP models on some physical observables. For some measured observables, like the differential decay width, the longitudinal polarization of the dilepton system, the lepton-side forward-backward asymmetry, and the baryon-side forward-backward asymmetry, we find that the prediction values of SM are consistent with the current data in most q2 ranges, where the prediction values of these two NP models can also keep consistent with the current data with 1σ. However, in some q2 ranges, the prediction values of SM are difficult to meet the current data, but the contributions of these two NP models can meet them or keep close to them. For the double-lepton polarization asymmetries, PLT, PTL, PNN, and PTT are sensitive to the scalar leptoquark model X(3,2,7/6) but not to X(3,2,1/6). However, PLN, PNL, PTN, and PNT are not sensitive to these two NP models.

National Science Foundation11225523U13321031. Introduction

The current data have hinted at several anomalies in B decays induced by the flavor-changing neutral current (FCNC) processes b→sl+l-, which have been recognized as very important probes of the Standard Model (SM) and new physics (NP). For the baryonic semilepton decays, experimentally, Λb→Λμ+μ- decay has been observed by the CDF collaboration [1] and measured by the LHCb collaboration at CERN [2]. Theoretically, studies on the Λb→Λμ+μ- decay have been investigated in the SM and beyond the SM [3–16]. Their results showed that some observables of these processes are sensitive to the contributions of NP.

The leptoquark models have many kinds of states, not only vector ones, but also scalar ones. In regard to different decay processes, the different leptoquark states may produce different effects. For b→sμ+μ- processes, model independent constrains on leptoquarks are obtained in [17], where scalar leptoquark states with X=3,2,7/6 and X=3,2,1/6 have visible effects on the b→sμ+μ- processes of B meson decays. For Λb→Λμ+μ- decay, their quark level transitions are also b→sμ+μ-; therefore, in this paper, we try to examine the effects of scalar leptoquark models on some observables of Λb→Λμ+μ- decay, such as the differential decay width, the longitudinal polarization of the dilepton system, the lepton-side forward-backward asymmetry, and the baryon-side forward-backward asymmetry and double-lepton polarization asymmetries.

The outline of this paper is as follows. In the next section we present the SM theoretical framework for Λb→Λμ+μ- transitions. In Section 3, we introduce the employed scalar leptoquark models; the transition form factors are given in Section 4. In Section 5, we present the physical observables and numerical analyses. Finally, we will have a concluding section.

At quark level, the rare decay Λb→Λl+l- is governed by the b→sl+l- transition; its effective Hamiltonian in the SM can be written as(1)Heff=GFαemVtbVts∗22πC9effs¯γμ1-γ5bl¯γμl+C10s¯γμ1-γ5bl¯γμγ5l-2mbC7eff1q2s¯iσμνqν1+γ5bl¯γμl,where GF is the Fermi constant, αem=e2/4π is the fine-structure constant, and Vqq′ denote the CKM matrix elements.

Following [18], the effective Wilson coefficients in the high q2 region are given by(2)C7effq2=C7-13C3+43C4+20C5+803C6-αs4πC1-6C2F1,c7q2+C8F87q2,C9effq2=C9+43C3+649C5+6427C6+h0,q2-12C3-23C4-8C5-323C6+hmb,q2-72C3-23C4-38C5-323C6+hmc,q243C1+C2+6C3+60C5-αs4πC1F1,c9q2+C2F2,c9q2+C8F89q2,where the explicit expressions of these functions F8(7,9)(q2), h(mq,q2), F1,c(7,9)(q2), and F2,c(7,9)(q2) can be found in [19–21]. However, in the low q2 region, nonfactorizable hadronic effects are expected to have the sizeable corrections; these have not been calculated for the baryonic decay [21, 22]. According to [18], we use the effective Wilson coefficients C7eff(q2) and C9eff(q2) in (2) both in the low q2 region and in the high q2 region by increasing the 5% uncertainty.

3. Scalar Leptoquark Models

Here we consider two kinds of the minimal renormalizable scalar leptoquark models [17], containing one single additional representation of SU3×SU2×U1 where baryon number violation cannot be allowed in perturbation theory. There are only two such models which are represented as X(3,2,7/6) and X(3,2,1/6) under the SU(3)×SU(2)×U(1) gauge group.

The interaction Lagrangian for the scalar leptoquark X(3,2,7/6) couplings to the fermion bilinear can be written as(3)L=-λuiju¯RiXTϵLLj-λeije¯RiX†QLj+h.c.,where i,j=1,2,3 are the generation indices, the couplings λ are in general complex parameters, uR(eR) is the right-hand up-type quark (charged lepton) singlet, X is the scalar leptoquark doublet, ϵ=iσ2 is a 2×2 matrix, and QL(LL) is the left-hand quark (lepton) doublet.

After performing Fierz transformation, the contribution to the interaction Hamiltonian for b→sμ+μ- is(4)H=λμ32λμ22∗8M7/62s¯γμ1-γ5bμ¯γμ1+γ5μ=λμ32λμ22∗4M7/62O9+O10,which can be written in the style of the SM effective Hamiltonian as(5)H=-GFα2πVtbVts∗C9NPO9+C10NPO10.Then we obtain the new Wilson coefficients(6)C9NP=C10NP=-π22GFVtbVts∗αλμ32λμ22∗M7/62.

The interaction Lagrangian for the scalar leptoquark X=(3,2,1/6) couplings to the fermion bilinear can be written as(7)L=-λdijd¯RiXTϵLLj+h.c.

After performing Fierz transformation, the contribution to the interaction Hamiltonian for the b→sμ+μ- is(8)H=λs22λb32∗8M1/62s¯γμ1+γ5bμ¯γμ1-γ5μ=λs22λb32∗4M1/62O9′-O10′,where O9′ and O10′ are six-dimensional operators obtained from O9 and O10 by the replacement PL↔PR. Writing in the style of the SM effective Hamiltonian, we obtain the new Wilson coefficients:(9)C9′NP=-C10′NP=π22GFVtbVts∗αλs22λb32∗M1/62.

In [23], comparing the bounds on NP coupling parameters obtained from Bs→μ+μ-, B¯d0→Xsμ+μ-, and Bs-B¯s mixing, respectively, the authors obtain the following results:(10)0≤λμ32λμ22∗M7/62=λs22λb32∗M1/62=λs32λb22∗MS2≤5×10-9GeV-2,for π2≤ϕNP≤3π2,where the bounds will be used in the process of our calculations.

4. Transition Form Factors

For Λb→Λμ+μ- decay, these form factors have been calculated in the framework of QCD light-cone sum rules (LCSR) in the low q2 region [22] and lattice QCD in the high q2 region [18], respectively. All of them use the helicity-based definition of the form factors [24]:(11)Λp′,s′s¯γμbΛbp,s=u¯Λp′,s′f0q2mΛb-mΛqμq2+f+q2mΛb+mΛs+pμ+p′μ-mΛb2-mΛ2qμq2+f⊥q2γμ-2mΛs+pμ-2mΛbs+p′μuΛbp,s,Λp′,s′s¯γμγ5bΛbp,s=-u¯Λp′,s′γ5g0q2mΛb+mΛqμq2+g+q2mΛb-mΛs-pμ+p′μ-mΛb2-mΛ2qμq2+g⊥q2γμ+2mΛs-pμ-2mΛbs-p′μuΛbp,s,Λp′,s′s¯iσμνqνbΛbp,s=-u¯Λp′,s′h+q2q2s+pμ+p′μ-mΛb2-mΛ2qμq2+h⊥q2mΛb+mΛγμ-2mΛs+pμ-2mΛbs+p′μuΛbp,s,Λp′,s′s¯iσμνqνγ5bΛbp,s=-u¯Λp′,s′γ5h~+q2q2s-pμ+p′μ-mΛb2-mΛ2qμq2+h~⊥q2mΛb-mΛγμ+2mΛs-pμ-2mΛbs-p′μuΛbp,s,where q=p-p′ and s±=(mΛb±mΛ)2-q2. The fit functions of helicity-based form factors can be found in equations (133)–(135) of [22] and equation (49) of [18].

5. Physical Observables and Numerical Analyses5.1. Some Measured Observables

According to [25], the Λb polarization at the LHC has been measured to be small and compatible with zero, and polarization effects will average out for the symmetric ATLAS and CMS detectors, so we consider the initial baryon Λb as unpolarized. The fourfold differential rate of the Λb→Λ(→a(1/2+),b(0-))l+l- can be written as [26](12)d4Γdq2dcosθldcosθΛdϕ=38πK1sssin2θl+K1cccos2θl+K1ccosθl+K2sssin2θl+K2cccos2θl+K2ccosθlcosθΛ+K3scsinθlcosθl+K3ssinθlsinθΛsinϕ+K4scsinθlcosθl+K4ssinθlsinθΛsinϕ,where the angles θl and θΛ denote the polar directions of l- and a(1/2+), respectively. ϕ is the azimuthal angle between the l+l- and a(1/2+)b(0-) decay planes, and the explicit expressions of the coefficients Ki can be found in [26].

The differential decay width is(13)dΓdq2=2K1ss+K1cc.

The longitudinal polarization of the dilepton system is(14)FL=2K1ss-K1cc2K1ss+K1cc.

The lepton-side forward-backward asymmetry is(15)AFBl=32K1c2K1ss+K1cc.

The baryon-side forward-backward asymmetry is(16)AFBΛ=122K2ss+K2cc2K1ss+K1cc.

In the process of numerical analyses, we consider the theoretical uncertainties of all input parameters. For the form factors, we use the results of QCD light-cone sum rules (LCSR) in the low q2 region [22] and lattice QCD in the high q2 region [18]. Comparing to the current data which have been measured by LHCb collaboration [27], we plot the dependence of four observables mentioned above on the full physical region except the intermediate region of q2 in Figure 1.

The dependence of dΓ/dq2, FL, AFBμ, and AFBΛ on q2, respectively.

From Figure 1, we obtain the following results:

For the differential decay width dΓ/dq2, its prediction values of SM are consistent with the current data in the ranges of 0.1<q2<1GeV2/c2 and 15<q2<16GeV2/c2. When we consider the effects of these two NP models, the theoretical predictions are still consistent with the experimental results with 1σ in these ranges. However, in the remaining ranges, its prediction values of SM and these two NP models are difficult to meet the current data. But in the large q2 region, the prediction values of the scalar leptoquark X(3,2,7/6) are more closer to the current data.

For the longitudinal polarization FL of the dilepton system, its prediction values of SM and these two NP models are consistent with the current data both in the low q2 region and in the high q2 region, respectively. In the low q2 region, the prediction value of the scalar leptoquark X(3,2,7/6) enhances that of SM, but the opposite result happens in the scalar leptoquark X(3,2,1/6). There are not obvious difference results between the SM and these two NP models in the high q2 region.

For the lepton-side forward-backward asymmetry AFBμ, in the range of 0.1<q2<1GeV2/c2, its prediction value of SM is consistent with the current data with 1σ, but the result of the scalar leptoquark X(3,2,7/6) is more closer to the central value of the current data than that of SM. In the high q2 region, its prediction value of SM is lower than the current data. But the result of the scalar leptoquark X(3,2,7/6) can meet the current data in the range of 15<q2<16GeV2/c2.

For the baryon-side forward-backward asymmetry AFBΛ, except in the range of 16<q2<18GeV2/c2, the current data in the remaining ranges can be met both in the SM and in these two NP models, respectively. When we consider the NP effects, this observable shows strong dependence on the scalar leptoquark X(3,2,1/6). However, there are not obvious difference results between the SM and scalar leptoquark X(3,2,7/6).

5.2. Double-Lepton Polarization Asymmetries

The definition of the double-lepton polarization asymmetry can be written as(17)Pij=dΓs→i-,s→j+/ds^-dΓ-s→i-,s→j+/ds^-dΓs→i-,-s→j+/ds^-dΓ-s→i-,-s→j+/ds^dΓs→i-,s→j+/ds^+dΓ-s→i-,s→j+/ds^+dΓs→i-,-s→j+/ds^+dΓ-s→i-,-s→j+/ds^,where s→i(j)-(+) is the orthogonal unit vector in the rest frame of the leptons; its explicit explanation and nine double-lepton polarization asymmetries are presented in [28].

In [28], the form factors are defined as follows:(18)Λps¯γμ1-γ5bΛbp+q=u¯Λpγμf1q2+iσμνqνf2q2+qμf3q2-γμγ5g1q2-iσμνγ5qνg2q2-qμγ5g3q2uΛbp+q,Λps¯iσμνqν1+γ5bΛbp+q=u¯Λpγμf1Tq2+iσμνqνf2Tq2+qμf3Tq2+γμγ5g1Tq2+iσμνγ5qνg2Tq2+qμγ5g3Tq2uΛbp+q,where uΛb and uΛ are spinors of Λb and Λ baryons, respectively.

The form factors fi(T) and gi(T) in (18) are related to the helicity form factors f+,⊥,0, g+,⊥,0, h+,⊥, and h~+,⊥ in (11) as follows:(19)f+=f1-q2mΛb+mΛf2,f⊥=f1-mΛb+mΛf2,f0=f1+q2mΛb-mΛf3,g+=g1+q2mΛb-mΛg2,g⊥=g1+mΛb-mΛg2,g0=g1-q2mΛb+mΛg3,h+=f2T-mΛb+mΛq2f1T,h⊥=f2T-f1TmΛb+mΛ,h~+=g2T+mΛb-mΛq2g1T,h~⊥=g2T+g1TmΛb-mΛ.

The amplitude for Λb→Λμ+μ- decay can be written in terms of twelve form factors in (18), and we find that(20)M=Gα82πVtbVts∗l¯γμ1-γ5lu¯ΛpA1-D1γμ1+γ5+B1+E1γμ1-γ5+iσμνqνA2-D21+γ5+B2-E21-γ5+qμA3-D31+γ5+B3-E31-γ5uΛbp+q+l¯γμ1+γ5lu¯ΛPA1+D1γμ1+γ5+B1+E1γμ1-γ5+iσμνqνA2+D21+γ5+B2+E21-γ5+qμA3+D31+γ5+B3+E31-γ5uΛbp+q, where(21)A1=-2mbq2C7efff1T+g1T+C9eff+C9NPf1-g1+C9′NPf1+g1,A2=A11⟶2,A3=A11⟶3,Bi=Aigi⟶-gi;giT⟶-giT,D1=C10eff+C10NPf1-g1+C10′NPf1+g1,D2=D11⟶2,D3=D11⟶3,Ei=Digi⟶-gi.Because the vector and axial vector currents are not renormalized, we neglect the anomalous dimensions of coefficients C9′NP and C10(′)NP [29].

We also plot the dependence of the double-lepton polarization asymmetries on the full physical region except the intermediate region of q2 in Figure 2 and find the following:

Double-lepton polarization asymmetries PLT, PTL, PNN, and PTT of this decay process are sensitive to the contribution of the scalar leptoquark X(3,2,7/6) but not to that of the scalar leptoquark X(3,2,1/6).

For double-lepton polarization asymmetry PLT of Λb→Λμ+μ- decay, the contribution of the scalar leptoquark X(3,2,7/6) can enhance its maximum value of SM prediction from 0.48 to 0.65 in the low q2 region. For this decay process, these effects of these two NP models on PTL which is not presented in this paper are similar to PLT, respectively.

For double-lepton polarization asymmetry PNN of Λb→Λμ+μ- decay, when we consider the contribution of the scalar leptoquark X(3,2,7/6), its value of SM prediction can be enhanced quite a lot both in the low q2 region and in the high q2 region. For this decay process, the contribution of the scalar leptoquark X(3,2,7/6) to PTT is similar to PNN, respectively.

For double-lepton polarization asymmetries PLN, PNL, PNT, and PTN, their values of SM prediction are almost zero, and the effects of these two NP models are not significant on them.

The dependence of double-lepton polarization asymmetries PLT, PNN, and PTT on q2, respectively.

6. Conclusions

We calculate the differential decay width, the longitudinal polarization of the dilepton system, the lepton-side forward-backward asymmetry, and the baryon-side forward-backward asymmetry and double-lepton polarization asymmetries of Λb→Λμ+μ- decay in the scalar leptoquark model X(3,2,7/6) and X(3,2,1/6), respectively. Using the constrained parameter spaces from Bs→μ+μ- and Bd→Xsμ+μ- decays, we depict the correlative figures between these observables and the momentum transfer q2, respectively. We find, for the differential decay width, the longitudinal polarization of the dilepton system, the lepton-side forward-backward asymmetry, and the baryon-side forward-backward asymmetry, which have been measured by LHCb collaboration; most of their current data can be met both in the SM and in these two NP models. However, some of their current data still cannot be met in the SM. When we consider the effects of these two NP models, like the lepton-side forward-backward asymmetry AFBμ in the range of 0.1<q2<1GeV2/c2, its current data with 1σ can be met. For the double-lepton polarization asymmetries, PLT, PTL, PNN, and PTT show strong dependence on the scalar leptoquark model X(3,2,7/6) but not on X(3,2,1/6), respectively. However, the prediction values of PLN, PNL, PTN, and PNT in the SM are almost zero, and they also show weak dependence on these two NP models.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is supported by the National Science Foundation under Contracts nos. 11225523 and U1332103.

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