njpNJOPFMNew Journal of PhysicsNJPNew J. Phys.1367-2630IOP Publishingnjpaa3da110.1088/1367-2630/18/9/093051aa3da1NJP-105470.R1PaperEffects of scalar leptoquark on semileptonic Λ_{b} decaysEffects of scalar leptoquark on semileptonic Λ_{b} decaysSahooSuchismitaMohantaRukmani1School of Physics, University of Hyderabad, Hyderabad -500046, Indiarmsp@uohyd.ernet.in
S Sahoo and R Mohanta
Author to whom any correspondence should be addressed.
We study the scalar leptoquark effects on the rare semileptonic decays of Λ_{b} baryon, governed by the quark level transition b→sl+l−. We estimate the branching ratios, forward–backward asymmetries, lepton polarisation parameters and the lepton flavour non-universality effects in these decay channels. We find significant deviations from the corresponding standard model predictions in some of the observables due to leptoquark effects. We also investigate the lepton flavour violating decays Λb→Λli−lj+, the branching ratios of which are found to be (10−10−10−9).
semileptonic Λ_{b} decaysleptoquark modelLFV decays13.30.Ce14.80.SvScience and Engineering Research Board http://dx.doi.org/10.13039/501100001843SB/S2/HEP-017/2013
The study of the rare B meson decays involving flavour changing neutral current (FCNC) transitions is very crucial, as they provide sensitive probe to look for new physics (NP) beyond the standard model (SM). These decays are highly suppressed in the SM due to Glashow–Iliopoulos–Maiani mechanism and occur only through one-loop level penguin and box diagrams. Recently, several anomalies have been observed in the rare semileptonic B decays mediated through the FCNC b→s transitions. The most prominent ones are the observation of 3.7σ deviation in the angular observable P5′ [1–3] of B→K*μ+μ− mode and the violation of lepton universality in the B→Kl+l− decays at the level of 2.6σ [4] by the LHCb experiment. In addition, LHCb has also observed significant discrepancy in the decay rates of the B→K*l+l− processes [5, 6]. Also the decay rate of the Bs→ϕμ+μ− process [7] has 3.3σ deviation form its SM value in the low q^{2} region. Furthermore, the observed discrepancy in the ratio of branching fractions of exclusive B→K(*)l+l− decay and the inclusive decays into dimuon over dielectron in the full q^{2} range [8] provide strong evidence of the presence of lepton non-universality.
The anomalies observed in b→sl+l− processes at LHCb [1, 2, 4–7] have attracted a lot of attention in recent times. The implications of these observations have been extensively studied both in the context of various NP models and in model independent ways [9–13]. These deviations which are at the level of (2–3)σ are not statistically significant enough to provide an unambiguous signal of NP. On the other hand they are also not small enough to be ignored completely and need to be scrutinized meticulously as many different ways as possible. If indeed they really evince the smoking gun signal of some kind of NP, such effects must also show up in other decay channels involving b→s transitions, such as the corresponding Λ_{b} transitions. Therefore, the study of the rare Λ_{b} decays is of utmost importance to obtain an unambiguous signal of NP. Including the baryonic decay mode Λb→Λ(→pπ−)μ+μ− in the Bayesian analysis of ∣ΔB∣=∣ΔS∣=1 transitions, a fit of the Wilson coefficients C9,10,, C′9,10 has been performed in [14], and it has been shown that, the shift to C_{9} prefers to be opposite to the one found in mesonic case. To be more specific, the shift in C_{9} in baryonic decay is found to be Δ9=C9−C9SM=1.6−0.9+0.7 , as compared to the mesonic case where its value is Δ9=−1.09−0.20+0.22 [11]. Whereas the corresponding shifts in C_{10} are in the same direction, i.e., Δ10=0.7−0.8+0.5 for the baryonic case Δ10=0.56−0.24+0.25 for mesonic case. As pointed out in [14], the observed discrepancy in the shift of C_{9} might arise from our incomplete understanding of the hadronic matrix elements of the two-point correlators of 1,⋯,6;8 with the quark electromagnetic current, which effectively shift the Wilson coefficients C_{7} and C_{9}. This could also be due to the large experimental uncertainties for the Λb→Λ(→pπ)μ+μ− observables. However, if this persists with improved statistics, this would constitute a breakdown of the universal structure of the transversity amplitudes at low recoil, as predicted by the operator product expansion.
The important distinction between the Λ_{b} baryon and B meson decays is the spin of the Λ_{b} baryon. Therefore, the number of degrees of freedom involved in the bound state of baryon is more, hence the systematic study of Λb→Λγ and Λb→Λμ+μ− are relatively less explored in comparison to their mesonic counter parts. Also the experimental data on various Λ_{b} decay channels are rather limited. Recently LHCb has reported the branching ratio of Λb→Λμ+μ− [15], which is found to be lower than its SM prediction. This decay process has been extensively studied in the literature both in the SM and in various beyond the SM scenarios [16–23]. To supplement these studies, in this paper we would like to analyze the rare baryonic decay processes Λb→Λl+l−, where l=e,μ,τ in the scalar leptoquark model. In recent times, the scalar leptoquark model has been received a lot of attention, as it can successfully explain most of the observed anomalies associated with the b→sll transitions. Leptoquarks are colour-triplet bosonic particles which can couple to a quark and lepton pair at the same time. The existence of leptoquark has been proposed in many extensions of the SM, such as grand unification model [24, 25], Pati–Salam model [26], extended technicolour model [27] and the composite models [28]. The leptoquark states can be classified as vectors (spin-1) or scalars (spin-0). They can be characterised by their Fermion no. F=3B+L, where B and L are the baryon no. and lepton no. respectively. Scalar leptoquarks may exist at TeV scale, and can give observable signatures in various low energy processes [33]. The phenomenology of scalar leptoquarks has been studied extensively in the literature [29–37]. In this paper, we would like to study the rare baryonic decay processes Λb→Λl+l− in the scalar leptoquark model. In particular, we estimate the decay rates, forward–backward (AFB) and lepton polarisation asymmetries in these modes. Furthermore, we explore the possibility of lepton non-universality parameter in Λ_{b} decays and also the lepton flavour violating (LFV) decays mediated via the scalar leptoquarks.
The paper is organised as follows. In section 2 we present the effective Hamiltonian responsible for the b→sl+l− processes and the decay parameters for the semileptonic Λb→Λl+l− decays in the SM. The NP contribution due to the exchange of scalar leptoquark has been presented in section 3 and the constraints on the leptoquark parameter space has been obtained by using the measured branching ratios of the rare decays Bs→l+l−. In section 4, we present the numerical analysis for the branching ratios and other physical observables such as the forward–backward asymmetry, lepton polarisation asymmetry and the lepton non-universality by using the constrained leptoquark couplings. We compute the branching ratios of the LFV Λb→Λli−lj+ decays in sections 5 and 6 contains the summary and conclusion.
In this section, we will discuss the SM contributions to the branching ratios and other physical observables of the Λb→Λl+l−, l=e,μ,τ processes. The effective Hamiltonian describing the decay process Λb→Λl+l− involves the quark level transition b→sl+l− and is given by [38]eff=−4GF2VtbVts*∑i=16Ci(μ)Oi+C7e16π2(s¯σμν(msPL+mbPR)b)Fμν+C9effα4π(s¯γμPLb)l¯γμl+C10α4π(s¯γμPLb)l¯γμγ5l,where V_{qq′} are the CKM matrix elements, G_{F} denotes the Fermi constant, α is the fine-structure constant, C_{i}'s are the Wilson coefficients evaluated at the renormalized scale μ=mb [39] and PL,PR=(1∓γ5)/2 are the chiral operators. The sum over i includes the current–current operators i=1,2 and the QCD-penguin operators i=3,4,5,6.
In addition to the short distance contributions these processes also receive additional contributions arising from the long distance effects due to the real cc¯ resonant states of J/ψ,ψ′, i.e., Λb→ΛJ/ψ(ψ′)→Λl+l−. These resonance contributions can be included by modifying the Wilson coefficient C_{9}. Thus, the modified coefficient (C_{9}^{eff}) contains a perturbative part and a resonance part which can be written asC9eff=C9SM+Y(s)+C9res,where C_{9}^{SM} is the SM Wilson coefficient evaluated at the b quark mass scale [39], the perturbative part Y(s) receives contributions coming from one-loop matrix elements of the four quark operators [40] and the long distance resonance effect is given by [41]C9res=3πα2(3C1+C2+3C3+C4+3C5+C6)∑Vi=ψ(1S),⋯,ψ(6S)κVimViΓ(Vi→l+l−)mVi2−s−imViΓVi.Here the phenomenological parameter κ is taken to be 1.65 and 2.36 [42] for the lowest resonances J/ψ and ψ′ respectively in order to reproduce the correct branching ratio of (B→J/ψK*→K*l+l−)=(B→J/ψK*)(J/ψ→l+l−).
The matrix elements of the hadronic currents in (1) between initial Λ_{b} and the final Λ baryon can be parameterised in terms of various form factors which are presented in appendix A. Thus, using these matrix elements, the transition amplitude for the Λb→Λl+l− processes can be written as [16, 18](Λb→Λl+l−)=GFα2πVtbVts*[l¯γμl{u¯Λ(γμ(A1PR+B1PL)+iσμνqν(A2PR+B2PL))uΛb}+l¯γμγ5l{u¯Λ(γμ(D1PR+E1PL)+iσμνqν(D2PR+E2PL)+qμ(D3PR+E3PL))uΛb}],where the parameters A_{i}, B_{i}, D_{j} and E_{j} with i=1,2, j=1,2,3 are defined asAi=C9efffi−gi2−2mbq2C7fiT+giT2,Bi=C9efffi+gi2−2mbq2C7fiT−giT2,Dj=C10fj−gj2,Ej=C10fj+gj2.Using the transition amplitude (4), the double differential decay rate is given byd2Γdsˆdz=GF2α2212π5∣VtbVts*∣2mΛbvlλ1/2(1,r,sˆ)(sˆ,z),where(sˆ,z)=0(sˆ)+z1(sˆ)+z22(sˆ),sˆ=s/mΛb2 and z=pˆB·pˆl+ is the angle between the momenta of Λ_{b} and l+ in the dilepton invariant mass frame. The complete expressions for 0(sˆ), 1(sˆ) and 2(sˆ) are given in appendix B. Here vl=1−(4ml2/q2) and λ(1,r,sˆ)=(1−r)2−2sˆ(1+r)+sˆ2 is the triangle function with r=mΛ/mΛb. The physical allowed range for s ≡ q^{2} is4ml2≤s≤(mΛb−mΛ)2.Another interesting observable is the zero-crossing of the forward–backward asymmetry, wherein the position of the zero value of the forward–backward asymmetry parameter (A_{FB}) is very useful to look for the NP signal. The normalised forward–backward asymmetry is defined asAFB(sˆ)=∫01d2Γdsˆdzdz−∫−10d2Γdsˆdzdz∫01d2Γdsˆdzdz+∫−10d2Γdsˆdzdz,which can be simplified toAFB(sˆ)=1(sˆ)0(sˆ)+2(sˆ)/3.The polarisation asymmetries P_{i} (i=L,N,T) are defined asPi(sˆ)=dΓdsˆ(ηˆ=eˆi)−dΓdsˆ(ηˆ=−eˆi)dΓdsˆ(ηˆ=eˆi)+dΓdsˆ(ηˆ=−eˆi),where eˆi's are the unit vectors along the longitudinal, normal and transverse components of the l+ polarisation and ηˆ is a unit vector, used to write the l+ four-spin vector (s+), along the l+ spin in its rest frame ass+0=p⃗+·ηˆml,s⃗+=ηˆ+s+0El++mlp⃗+.Thus, the observables P_{L}, P_{T} and P_{N} correspond to longitudinal, transverse and normal polarisation asymmetries respectively. The observables P_{L} and P_{T} are P-odd, T-even, while P_{N} is P-even, T-odd and CP-odd. The explicit expressions for forward–backward asymmetry and all the polarisation parameters are taken from [16, 18, 19].
Another interesting observable is the lepton universality violation (LUV) parameter, which has been recently observed by the LHCb collaboration in B+→K+l+l− process and has 2.6σ deviation from its SM predicted value [6]. Analogously, we define the parameter (RΛ) as the ratio of branching fractions of Λb→Λl+l− into dimuon over dielectron asRΛ=Br(Λb→Λμ+μ−)Br(Λb→Λe+e−).
NP contribution due to scalar leptoquark exchange
In this section we will consider the effect of scalar leptoquarks to the Λb→Λl+l− decay processes. The exchange of leptoquarks will contribute additional operators to the SM effective Hamiltonian and thus, the various observables may deviate significantly from their corresponding SM values. The scalar leptoquark multiplets with representations X(3,2,7/6) and X(3,2,1/6) under the SM gauge group SU(3)C×SU(2)L×U(1)Y conserve baryon and lepton numbers and do not allow proton decay. These baryon and lepton number conserving scalar leptoquarks can have sizable Yukawa couplings and could be light enough to be accessible in accelerator searches. Thus, they could potentially contribute to the b→sl+l− transitions and one can constrain the underlying couplings from experimental data on Bs→l+l− processes as well as from Bs−B¯s mixing.
The interaction Lagrangian of the scalar leptoquarks X=(3,2,7/6) with the SM bilinear fermions is given as [33, 34]=−λuiju¯RiXTϵLLj−λeije¯RiX†QLj+h.c.,where i,j are the generation indices, X is the leptoquark doublet, Q_{L} (L_{L}) denotes the left handed quark (lepton) doublet, the right handed up-type quark (charged lepton) singlet is represented by u_{R} (e_{R}) and ϵ=iσ2 is a 2 × 2 matrix. The multiplets defined above are represented asX=VαYα,QL=uLdL,andLL=νLeL.Now expanding the SU(2) indices, the interaction Lagrangian (14) takes the form=−λuiju¯αRi(VαeLj−YανLj)−λeije¯Ri(Vα†uαLj+Yα†dαLj)+h.c..Thus, from equation (16) one can obtain the interaction Hamiltonian for b→sli+li− processes after performing the Fierz transformation asLQ=λei3λei2*8MY2[s¯γμ(1−γ5)b][l¯iγμ(1+γ5)li]=λei3λei2*4MY2(O9+O10).Comparing (17) with the corresponding SM effective Hamiltonian (1), one can obtain the new Wilson coefficients asC9NP=C10NP=−π22GFαVtbVts*λei3λei2*MY2.Similarly, the interaction Lagrangian due to the exchange of the scalar leptoquark X=(3,2,1/6) is=−λdijd¯αRi(VαeLj−YανLj)+h.c.,which contributes to the primed Wilson coefficients (C9,10′) corresponding to the semileptonic electroweak penguin operators 9,10′ (i.e., the right-handed counter parts of the SM operators 9,10) and are given asC9′NP=−C10′NP=π22GFαVtbVts*λs2iλb3i*MV2.Thus, from the above equations (18) and (20), one can find that there are four additional Wilson coefficients C9,10(′)NP, which will contribute to the b→sl+l− processes due to the scalar leptoquark exchange. Thus, the modified parameters (5) in the amplitude (4), becomeAi=C9′NPfi+gi2+(C9eff+C9NP)fi−gi2−2mbq2C7SMfiT+giT2,Bi=(C9eff+C9NP)fi+gi2−2mbq2C7SMfiT−giT2+C9′NPfi−gi2,Dj=C10′NPfj+gj2+(C10SM+C10NP)fj−gj2,Ej=(C10SM+C10NP)fj+gj2+C10′NPfj−gj2.Next, we have to find out the constraints on the leptoquark couplings to see how various observables behave in the LQ model. The detailed calculation of the constraint on the new leptoquark parameter space has been presented in [29–31], therefore here we will simply quote the main results. We constrain the leptoquark coupling by comparing the theoretical [44] and experimental [45–47] branching ratios of Bs→l+l− processes and the Bs−B¯s mixing data [8]. For completeness, here we briefly outline the procedure for obtaining the constraints from Bs→μ+μ− process and Bs−B¯s mixing, however, the technical details can be found in [29–31].
In the leptoquark model the branching ratio for the Bs→μ+μ− mode can be given asBr(Bs→μ+μ−)=GF216π3τBsα2fBs2MBsmμ2∣VtbVts*∣2∣C10SM+C10NP−C10′NP∣21−4mμ2MBs2=BrSM1+C10NP−C10′NPC10SM2≡BrSM∣1+reiϕNP∣2,where BrSM is the SM branching ratio and the parameters r and ϕNP are defined asreiϕNP=C10NP−C10′NPC10SM.Now comparing the SM theoretical prediction of Br(Bs→μ+μ−) [44]Br(Bs→μ+μ−)∣SM=(3.65±0.23)×10−9,with the corresponding experimental value [45–47]Br(Bs→μ+μ−)=(2.9±0.7)×10−9,and assuming that each individual leptoquark contribution to the branching ratio does not exceed the experimental result, one can obtain the bound on the NP parameters r and ϕNP. The allowed parameter space in r−ϕNP plane which is compatible with the 1σ range of the experimental data is0≤r≤0.35,withπ/2≤ϕNP≤3π/2.These bounds can be translated to obtain the bounds for the leptoquark couplings as0≤∣λμ23λμ22*∣MY2=∣λs22λb32*∣MV2≤5×10−9GeV−2forπ/2≤ϕNP≤3π/2.Similarly, one can obtain the upper bound on the product of various combination of leptoquark couplings from Bs→l+l− processes which are presented in table 1. Using the bounds on leptoquark couplings one can obtain the constraints on new Wilson coefficients using the equations (18) and (20).
Constraints on the leptoquark couplings obtained from various leptonic Bs→l+l− decays [29], where M_{S} denotes the mass of the scalar LQ.
In this subsection, we will discuss the constraint on leptoquark couplings from the Bs−B¯s mixing, which in the SM, proceeds through the box diagram with internal top quark and W boson exchange. The effective Hamiltonian describing the ΔB=2 transition is given as [48]eff=GF216π2∣VtbVts*∣2MW2S0(xt)ηB(s¯b)V−A(s¯b)V−A,where ηB is the QCD correction factor and S0(xt) is the loop function given in [48].
Thus, the Bs−B¯s mixing amplitude in the SM, can be written asM12SM=12MBs〈B¯s∣eff∣Bs〉=GF212π2MW2∣VtbVts*∣2ηBBˆsfBs2MBsS0(xt).The corresponding mass difference can be computed from the mixing amplitude through ΔMs=2∣M12∣. Now using the particle masses from [8], ηB=0.551, the Bag parameter BˆBs=1.320±0.017±0.030 and the decay constant fBs=225.6±1.1±5.4 from [49], the value of ΔMs in the SM, is found asΔMsSM=(17.426±1.057)ps−1,which is in good agreement with the experimental result [8]ΔMs=17.761±0.022ps−1.For X(3,2,7/6) LQ, the mixing amplitude receives additional contribution from leptoquark and charged lepton in the box diagram whereas for X(3,2,1/6) both charged lepton and neutrino will contribute to the mixing amplitude. The effective Hamiltonian due to the leptoquark X(3,2,7/6) is given byeff=∑i=e,μ,τ(λbiλsi*)2128π21MS2Imi2MS2(b¯γμPLs)(b¯γμPLs),and for X(3,2,1/6) leptoquark the corresponding effective Hamiltonian becomeseff=∑i=e,μ,τ(λbi*λsi)2128π21MS2Imi2MS2+1MS2(b¯γμPRs)(b¯γμPRs).where the loop function I(x) is given asI(x)=1−x2+2xlogx(1−x)2.Thus, the contribution to the mixing amplitude due to the exchange of scalar leptoquark is given byM12LQ=(λ32*λ22)2192π2MS2ηBBˆBsfBs2MBs,forX(3,2,1/6),M12LQ=(λ32λ22*)2384π2MS2ηBBˆBsfBs2MBs,forX(3,2,7/6).Including both the SM and leptoquark contributions the total mass difference is given asΔMs=ΔMsSM1+c16GF2∣VtbVts*∣2mW2S0(xt)(λ32λ22*)2MS2,where the constant c = 1 for X(3,2,1/6) and 1/2 for X(3,2,7/6). Now varying the mass difference (ΔMs/ΔMsSM) within its 1σ allowed range [8], the constraint on ∣λ32λ22/MS∣ is found to be [31]0≤λ32λ22MS≤7.5×10−5GeV−1,forX(3,2,7/6),0≤λ32λ22MS≤5.0×10−5GeV−1,forX(3,2,1/6).In order to relate this results with the bounds obtained Bs→μμ process, we scale the couplings obtained from Bs−B¯s mass difference for a benchmark leptoquark mass of 1 TeV and the bounds in equation (37) is translated as0≤∣λ32λ22MS2∣≤7.5×10−8GeV−2,forX(3,2,7/6),0≤∣λ32λ22MS2∣≤5.0×10−8GeV−2,forX(3,2,1/6),which are reasonably higher than those of obtained from Bs→μμ process. Hence in our analysis, we will use the bounds (26) as discussed in the previous subsection.
Numerical analysis
After having the detailed knowledge about the SM observables and the bound on the new leptoquark couplings, we now proceed for numerical analysis. We have taken the particle masses and the life time of Λ_{b} baryon from [8]. The q^{2} dependence of form factors derived in the light cone sum rule (LCSR) approach can be parameterised asfi(q2)=fi(0)1−a(q2/mΛb2)+b(q2/mΛb2)2,where the values of the parameters fi(0), a and b and are listed in table 2 [21]. The other form factors are related to these two and the HQET form factors (F1,2) through [21]f2T=g2T=f1=g1=F1+mΛmΛbF2,f2=g2=f3=g3=F2mΛb,f1T=g1T=F2mΛbq2.In the lattice QCD formalism, the Λb→Λ helicity form factors, i.e., f+,⊥,0, g+,⊥,0, h+,⊥ and h˜+,⊥ in the physical limit can have the simple form [23]f(q2)=11−q2/(mpolef)2[a0f+a1fz(q2)+a2f[z(q2)]2],where the values and uncertainties of the parameters a_{0}^{f}, a_{1}^{f} and a_{2}^{f} from the higher-order fit are given in table 5 of [23]. These helicity form factors are related to the form factors fi(T) and gi(T) used in this work as follows:f+=f1−q2mΛb+mΛf2,f⊥=f1−(mΛb+mΛ)f2,f0=f1+q2mΛb−mΛf3,g+=g1+q2mΛb−mΛg2,g⊥=g1+(mΛb−mΛ)g2,g0=g1−q2mΛb+mΛg3,h+=f2T−mΛb+mΛq2f1T,h⊥=f2T−f1TmΛb+mΛ,h˜+=g2T+mΛb−mΛq2g1T,h˜⊥=g2T+g1TmΛb−mΛ.In our analysis, we have taken the form factors computed in the LCSR approach for low q^{2} region (as these are not so well-behaved in the high q^{2} regime), and for high q^{2} theory we have used the lattice QCD calculations of Λb→Λ form factors [23]. The values of the Wilson coefficients used in our analysis are evaluated at the renormalisation scale μ≈mb=4.8GeV. In the LQ model, the NP contributions to the branching ratios and forward–backward asymmetry parameters are encoded in the new Wilson coefficients. By using the above input parameters and the values of the new Wilson coefficients, we show in figure 1, the q^{2} variation of branching ratio of Λb→Λe+e− (top left panel), Λb→Λμ+μ− (top right panel) and Λb→Λτ+τ− (bottom panel) processes for the full kinematically accessible physical region. In these plots, we have shown the contributions arising from the exchange of X=(3,2,7/6) leptoquark. The SM contributions are represented by blue lines and the grey bands denote the theoretical uncertainties arising due to the uncertainties associated with the CKM matrix elements and the hadronic form factors. The green bands represent the leptoquark contributions to the branching ratios. The bin-wise experimental results for Λb→Λμ+μ− process [15] are shown by black data points. There is slight deviation in the decay distribution between the predicted and observed data. The corresponding results coming from the exchange of the X=(3,2,1/6) LQ are shown in figure 2. From these figures, one can see that the branching ratios of Λb→Λe+e− and Λb→Λτ+τ− decay processes deviate significantly from their SM predictions, whereas the NP effects on Λb→Λμ+μ− branching ratio is not so prominent. In table 3, we present the integrated values of branching ratio for all the above processes, where we have used the veto windows as (8GeV2<ml+l−2<11GeV2) and (12.5GeV2<ml+l−2<15GeV2) [15], to eliminate the backgrounds coming from the dominant resonances Λb→ΛJ/ψ(ψ′) with J/ψ(ψ′)→l+l−. The predicted branching ratio for Λb→Λμ+μ− is almost consistent with the observed data Br(Λb→Λμ+μ−)=(1.08±0.28)×10−6 [8]. Also, as seen from table 3, the experimental result can be accommodated in the leptoquark model. Within the SM, the forward backward asymmetry parameters in the B→Kl+l− decay processes are identically zero since they only involve scalar and tensor types of currents, whereas B→Kl+l− processes are described by only vector-type interactions. However, for semileptonic Λb→Λl+l− decay processes, the forward backward asymmetry depends on two combinations of the Wilson coefficients Re(C7effC10*) and Re(C9effC10*) [16] and thus, can have negative values in the SM. The contribution due to the new Wilson coefficients (C9,10NP(′)) may enhance the rate of asymmetries and can shift the zero position of these asymmetries. In figure 3, the variation of forward–backward asymmetry for Λb→Λμ+μ− (left panel), Λb→Λτ+τ− (right panel) modes are depicted with respect to q^{2} both in the SM and in the X=(3,2,7/6) LQ model including the LD contributions and the corresponding integrated values are presented in table 3. Similarly the variation of forward–backward asymmetries for X=(3,2,1/6) LQ exchange are shown in figure 4. We found no significant deviation of the zero position of A_{FB} from its SM value due to the leptoquark contributions in Λb→Λμ+μ− process. However, there is certain discrepancy between the observed and predicted results in the high q^{2} regime. The forward–backward asymmetry for Λb→Λτ+τ− process however, has significant deviation from the SM in both the X=(3,2,7/6) and X=(3,2,1/6) leptoquark model. Besides the branching ratios and forward–backward asymmetry parameters of Λb→Λl+l− processes, the NP effects can also be observed in the lepton polarisation asymmetries. In the left panel of figure 5, the distribution of the longitudinal (top), transverse (middle) and normal (bottom) polarisation components for Λb→Λμ+μ− process are shown both in the SM and in the X=(3,2,7/6) LQ model, and the corresponding plots for Λb→Λτ+τ− process are presented in the right panel. The integrated values of all the three polarisations in the full physical phase space have been presented in table 3. In figure 6, we have shown the variation of the different polarisation parameters for Λb→Λμ+μ− process in the X=(3,2,1/6) leptoquark model. It is found from table 3, that the transverse and normal polarisation values are very small in the SM and even the leptoquark model does not give any significant deviation.
The variation of branching ratio of Λb→Λe+e− (left panel), Λb→Λμ+μ− (right panel) and Λb→Λτ+τ− (bottom panel) with respect to low and high q^{2} including the LD contributions, both in the SM and in the X=(3,2,7/6) leptoquark model. In each plot, the green band represents the leptoquark contribution and the blue solid line is for the SM. The grey band represents the theoretical uncertainty arises due to the input parameters in the SM. The black data points in Λb→Λμ+μ− process represent the bin-wise experimental data.
Same as figure 1 for X=(3,2,1/6) LQ exchange.
The forward–backward asymmetry variation of Λb→Λμ+μ− (left panel) and Λb→Λτ+τ− (right panel) with respect to q^{2} for X=(3,2,7/6) LQ exchange. The black data points in Λb→Λμ+μ− process represent the bin-wise experimental data.
Same as figure 3 for X=(3,2,1/6) LQ exchange.
The plots in the left panel represent the longitudinal (top), transverse (middle) and normal (bottom) polarisations for Λb→Λμ+μ− precess with respect to q2/mΛb2 in the X=(3,2,7/6) LQ model. The corresponding plots for Λb→Λτ+τ− mode are shown in the right panel.
The polarisation plots of Λb→Λμ+μ− process for X=(3,2,1/6) LQ exchange.
Numerical values of the form factor f1(0), f2(0) and the parameters involved in the double fit for Λb→Λ transition.
Parameter
LCSR (twist-3) [21]
f1(0)
0.14−0.01+0.02
a
2.91−0.07+0.1
b
2.26−0.08+0.13
f2(0)(10−2GeV−1)
−0.47−0.06+0.06
a
3.4−0.05+0.06
b
2.98−0.08+0.09
The predicted integrated values of the branching ratio, forward–backward asymmetry, lepton polarisation asymmetry and the lepton non-universality with respect to their respective q^{2} range for the Λb→Λμ(τ)+μ(τ)− processes in the SM and the LQ model.
Observables
SM prediction
Values in Y=7/6 LQ model
Values in Y=1/6 LQ model
Br(Λb→Λe+e−)
(1.168±0.134)×10−6
(1.168−1.91)×10−6
(1.168−2.13)×10−6
Br(Λb→Λμ+μ−)
(1.165±0.132)×10−6
(1.165−1.37)×10−6
(1.165−1.52)×10−6
〈AFBμ〉
−0.567
−0.567→−0.446
−0.567→−0.54
〈PLμ〉
0.34
0.3−0.34
0.24−0.34
〈PTμ〉
−4.5×10−4
−(4.5→2.87)×10−4
−(0.45→3.26)×10−3
〈PNμ〉
−0.0192
−0.0192→−0.013
−0.0192→−0.012
Br(Λb→Λτ+τ−)
(2.13±0.215)×10−7
(2.13−4.38)×10−7
(2.13−8.32)×10−7
〈AFBτ〉
−0.38
−0.38→3.2×10−3
−0.38→7.68×10−2
〈PLτ〉
0.075
0.047−0.075
6.3×10−3−0.075
〈PTτ〉
−2.3×10−3
−(7→2.3)×10−3
(−0.23→2.0)×10−2
〈PNτ〉
−0.05
−0.05→8.1×10−3
−0.05→0.0316
〈RΛbμe〉
0.997
0.67−0.997
0.68−0.997
〈RΛbμe〉[q2∈(1,6)]
0.998
0.71−0.998
0.74−0.998
Analogous to the lepton flavour non-universality parameter R_{K}, i.e., the ratio of branching fractions of B→Kμ+μ− over B→Ke+e−, we would like to see whether it is possible to observe lepton non-universality in the Λ_{b} decays. We have define these parameters as e.g., RΛμe=Br(Λb→Λμ+μ−)/Br(Λb→Λe+e−). In figure 7, we show the variation of lepton nonuniversality parameter RΛμe (top-right panel), RΛτe (bottom-left panel) and RΛτμ (bottom-right panel) in their respective q^{2} region. Also, we show the low-q^{2} behaviour of RΛμe (top-left panel), in the range 1≤q2≤6GeV2. These results are for X=(3,2,7/6) leptoquark. Similarly the lepton nonuniversality plot for X=(3,2,1/6) leptoquark exchange is shown in figure 8. The integrated values of the lepton non-universality parameter in both SM and LQ model are presented in table 3. We found that there is significant violation of lepton universality in Λ_{b} decays, though there is no experimental evidence so far. The violation of lepton universality is more pronounced for the processes having τ as final particle. However, as the reconstruction of tau events are extremely difficult, this observable may not be sensitive enough to be observed in near future. As seen from the top-left panel of figures 7 and 8, the parameter RΛμe is very promising for the Belle II experiment, as the LHCb, being a hadronic machine works better in muon mode than electron.
The variation of lepton universality violation RΛμe (top-right panel), RΛτe (bottom-left panel) and RΛτμ (bottom-right panel) with respect to q^{2} for X=(3,2,7/6) LQ exchange. Here RΛμe (top-left panel) shows the non-universality in the low q2∈[1,6] region.
In this section, we will compute the branching ratios of LFV Λ_{b} decays mediating through the exchange of scalar leptoquarks. The LFV decay processes are extremely rare in the SM as they are either two-loop suppressed with tiny neutrino masses in one of the loop or proceed through box diagram (which is also highly suppressed due to tiny neutrino mass). However, they can occur at tree level in the LQ model and are expected to have significantly large branching fractions. The observation of neutrino oscillation has provided unambiguous evidence for lepton flavour violation in the neutral lepton sector which in turn provides motivation to explore other LFV transitions such as li→ljγ, li−→lj−lk+lk−, B→li±lj∓ etc. Though there is no direct experimental evidence for such processes, but there exists experimental upper bounds on some of these modes. The LFV decays in the B meson and in the charged lepton sector have been widely investigated in the literature [29, 36, 50]. Therefore, it is interesting to see whether LFV decays could be observed in Λ_{b} decays also.
As discussed earlier, these processes occur at tree level due to the exchange of scalar leptoquarks. In the leptoquark model the effective Hamiltonian for b→sli−lj+ LFV process is given as [29, 36]LQ=GLQ(s¯γμPLb)(l¯iγμ(1+γ5)lj),where the coefficient G_{LQ} isGLQ=λi3λj2*8MY2.Using the form factors given in the appendix A, the amplitude for the LFV Λb→Λli−lj+ decay is given by(Λb→Λli−lj+)=GLQ[(l¯iγμ(1+γ5)lj){u¯Λ(γμ(A1′PR+B1′PL))uΛb+u¯Λiσμνqν(A2′PR+B2′PL)uΛb+qμu¯Λ(A3′PR+B3′PL)uΛb}].The coefficients Ak′ and Bk′ in (45) are related to the form factors throughAk′=fk−gk2andBk′=fk+gk2,k=1,2,3.Now using this transition amplitude, the branching ratio for the Λb→Λli−lj+ process is given asd2Γdsˆdcosθ=∣GLQ∣226π3mΛb5λ1λ2sˆI(sˆ),whereI(sˆ)=I0(sˆ)+I1(sˆ)cosθ+I2(sˆ)cos2θ,withI0(sˆ)=14(∣A1′∣2+∣B1′∣2+mΛb2sˆ(∣A2′∣2+∣B2′∣2))[(1−r)2−sˆ2]−2rsˆ1−mi2+mj2q2(Re(A1′B1′*)+mΛb2sˆRe(A2′B2′*))−Re(A2′B2′*)r−(mi2−mj2)2mΛb2+sˆ(mi2+mj2)+2mΛbsˆ1−(mi2+mj2)2q2[(Re(A1′A2′*)+Re(B1′B2′*))r(1−t)−(Re(A1′B2′*)+Re(B1′A2′*))(t−r)]+(mi2+mj2)mΛb[(Re(A1′A3′*)+Re(B1′B3′*))r(1−t)+(Re(A1′B3′*)+Re(B1′A3′*))(t−r)]+sˆ(mi2+mj2)−(mi2−mj2)2mΛb2t2(∣A3′∣2+∣B3′∣2)−rRe(A3′B3′*),I1(sˆ)=λ1λ2sˆ−12sˆ(∣A1′∣2−∣B1′∣2)+(mj2−mi2)(1−t−sˆ2)(∣A2′∣2+∣B2′∣2)+12mj2−mi2mΛb[r(Re(A1′A2′*)+Re(B1′B2′*))−(Re(A1′B2′*)+Re(B1′A2′*))]+12mj2−mi2mΛb[r(Re(A1′A3′*)+Re(B1′B3′*))+(Re(A1′B3′*)+Re(B1′A3′*))]+sˆ2(mi2−mj2)(Re(A2′A3′*)+Re(B2′B3′*)),andI2(sˆ)=λ1λ2sˆ2−14(∣A1′∣2+∣B1′∣2−mΛb2sˆ(∣A2′∣2+∣B2′∣2)).Here, λ1=λ (as defined in section 3), λ2=mˆi4+mˆj4+sˆ2−2(mˆi2mˆj2+mˆi2sˆ+mˆj2sˆ), and t=(1+r−sˆ)/2. The full kinematically accessible physical range for these processes is given by(mi+mj)2≤q2≤(mΛb−mΛ)2.As there is no intermediate particle in the SM which can decay into two leptons of different flavours, so in comparison with the Λb→Λl+l− processes, LFV decays have no long distance QCD contributions and dominant charmonium resonance background. The required input values for numerical evaluation are taken from [8] and the values of the q^{2} dependent form factors are taken from LCSR approach [21]. To determine the values of various LQ couplings, which are involved in the LFV decays, we use the following assumptions. As we know that the expansion parameter of the CKM matrix in the Wolfenstein parametrization (λ), can be related to the down type quark masses as λ∼(md/ms)1/2 in the quark sector, while in the lepton sector one can have the same order for λ with the relation λ∼(mli/mlj)1/4. Hence, for other required leptoquark coupling, we assume that the coupling between different generation of quarks and leptons follow the simple scaling laws, i.e., λij=(mi/mj)1/4λii with j>i. Thus, using the values of the leptoquark coupling as given in table 1, one can obtain the bound on required LQ couplings involved in LFV decays. Using these values we plot the variation of branching ratio of LFV decays such as Λb→Λμ−e+ (top left panel), Λb→Λτ−e+ (top right panel) and Λb→Λτ−μ+ (lower panel) with respect to q^{2} in figure 9 and the predicted upper limits of the branching ratios are given in table 4. So far there is no experimental evidence on the LFV Λ_{b} decays. However, since the predicted branching ratios are (10−9), they can be searched at LHCb and exploration/observation of these modes would definitely shed some light in the leptoquark scenarios.
The variation of branching ratio of LFV Λb→Λμ−e+ (left panel), Λb→Λτ−e+ (right panel) and Λb→Λτ−μ+ (bottom panel) processes with respect to q^{2} in the X=(3,2,7/6) leptoquark model. Here the required leptoquark couplings are computed by using the scaling ansatz λij=(mi/mj)1/4λii.
The predicted upper limits of the branching ratios, which are obtained using the upper limits of the LQ couplings, of LFV Λb→Λli−lj+ processes, l=e,μ,τ in the X=(3,2,7/6) leptoquark model. Also the required leptoquark couplings are computed by using the scaling ansatz λij=(mi/mj)1/4λii.
Decay process
Predicted branching ratio
Λb→Λμ−e+
1.56 × 10^{−9}
Λb→Λτ−e+
3.2 × 10^{−10}
Λb→Λτ−μ+
4.6 × 10^{−9}
Conclusion
In this paper, we have studied the rare semileptonic Λb→Λl+l−, l=e,μ,τ baryonic decays in the scalar leptoquark model. The leptoquark parameter space has been constrained using the experimental limits on the branching ratios of the two body leptonic decays Bs→l+l−. We have computed the branching ratios, the forward–backward and lepton polarisation asymmetries (PL,T,N) using the new leptoquark couplings. We have shown explicitly the results for both the relevant X=(3,2,7/6) and X=(3,2,1/6) leptoquark models. The zero-position of the forward–backward asymmetry is found to be insensitive to the additional leptoquark effect. These models also give negligible contribution to the transverse polarisation asymmetry. In addition, we also estimated the LUV parameters in these decays analogous to R_{K} in B→Kl+l− process. The LFV Λ_{b} decays are also studied and the predicted upper limits on these branching ratios are found to be (10−10−10−9), which could be searched in the LHCb experiment.
Acknowledgments
We would like to thank Science and Engineering Research Board (SERB) for financial support through grant No. SB/S2/HEP-017/2013.
The transition form factors for Λb(P)→Λ(p′) decays can be parameterised as [16, 43]〈Λ(p′)∣s¯γμb∣Λb(P)〉=f1u¯ΛγμuΛb+f2u¯ΛiσμνqνuΛb+f3qμu¯ΛuΛb,〈Λ(p′)∣s¯γμγ5b∣Λb(P)〉=g1u¯Λγμγ5uΛb+g2u¯Λiσμνqνγ5uΛb+g3qμu¯Λγ5uΛb,〈Λ(p′)∣s¯iσμνb∣Λb(P)〉=fTu¯ΛiσμνuΛb+fTVu¯Λ(γμqν−γνqμ)uΛb+fTS(Pμqν−Pνqμ)u¯ΛuΛb,〈Λ(p′)∣s¯iσμνγ5b∣Λb(P)〉=gTu¯Λiσμνγ5uΛb+gTVu¯Λ(γμqν−γνqμ)γ5uΛb+gTS(Pμqν−Pνqμ)u¯Λγ5uΛb,and for dipole operators〈Λ(p′)∣s¯iσμνqνb∣Λb(P)〉=f1Tu¯ΛγμuΛb+f2Tu¯ΛiσμνqνuΛb+f3Tqμu¯ΛuΛb,〈Λ(p′)∣s¯iσμνqνγ5b∣Λb(P)〉=g1Tu¯Λγμγ5uΛb+g2Tu¯Λiσμνqνγ5uΛb+g3Tqμu¯Λγ5uΛb.with q=P−p′ andf2T=fT−fTSq2,f1T=[fTV+fTS(mΛ+mΛb)]q2,f1T=−q2(mΛb−mΛ)f3T,g2T=gT−gTSq2,g1T=[gTV+gTS(mΛ−mΛb)]q2,g1T=q2(mΛb+mΛ)g3T.
The complete expressions for 0,1,2(sˆ) functions required to calculate the double differential decay rate is given by [18]0(sˆ)=32ml2mΛb2sˆ(1+r−sˆ)(∣D3∣2+∣E3∣2)+64ml2mΛb3(1−r−sˆ)Re(D1*E3+D3E1*)+64mΛb2r(6ml2−sˆmΛb2)Re(D1*E1)+64ml2mΛb3r(2mΛbsˆRe(D3*E3)+(1−r+sˆ)Re(D1*D3+E1*E3))+32mΛb2(2ml2+mΛb2sˆ)((1−r+sˆ)mΛbrRe(A1*A2+B1*B2)−mΛb(1−r−sˆ)Re(A1*B2+A2*B1)−2r[Re(A1*B1)+mΛb2sˆRe(A2*B2)])+8mΛb2(4ml2(1+r−sˆ)+mΛb2[(1−r)2−sˆ2])(∣A1∣2+∣B1∣2)+8mΛb4(4ml2[λ+(1+r−sˆ)sˆ]+mΛb2sˆ[(1−r)2−sˆ2])(∣A2∣2+∣B2∣2)−8mΛb2(4ml2(1+r−sˆ)−mΛb2[(1−r)2−sˆ2])(∣D1∣2+∣E1∣2)+8mΛb5sˆvl2(−8mΛbsˆrRe(D2*E2)+4(1−r+sˆ)rRe(D1*D2+E1*E2)−4(1−r−sˆ)Re(D1*E2+D2*E1)+mΛb[(1−r)2−sˆ2][∣D2∣2+∣E2∣2]),1(sˆ)=−16mΛb4sˆvlλ{2Re(A1*D1)−2Re(B1*E1)+2mΛbRe(B1*D2−B2*D1+A2*E1−A1*E2)}+32mΛb5sˆvlλ{mΛb(1−r)Re(A2*D2−B2*E2)+rRe(A2*D1+A1*D2−B2*E1−B1*E2)},and2(sˆ)=8mΛb6vl2λsˆ((∣A2∣2+∣B2∣2+∣D2∣2+∣E2∣2)−8mΛb4vl2λ(∣A1∣2+∣B1∣2+∣D1∣2+∣E1∣2).
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