Probing new physics in semileptonic Σ b and Ω b decays
Jin-Huan Sheng, Jie Zhu, Xiao-Nan Li, Quan-Yi Hu, Ru-Min Wang
PProbing new physics in semileptonic Σ b and Ω b decays Jin-Huan Sheng ∗ , Jie Zhu , Xiao-Nan Li , Quan-Yi Hu and Ru-Min Wang † School of Physics and Electrical Engineering , Anyang Normal University, Anyang, Henan 455000, China . College of Physics and Communication Electronic , Jiangxi Normal University, Nanchang, Jiangxi 330022, China . Abstract
Recently, several hints of lepton non-universality have been observed in the semilep-tonic B meson decays in terms of both in the neutral current ( b → sl ¯ l ) and charged current( b → cl ¯ ν l ) transitions. Motivated by these inspiring results, we perform the analysis ofthe baryon decays Σ b → Σ c l ¯ ν l and Ω b → Ω c l ¯ ν l ( l = e, µ, τ ) which are mediated by b → cl ¯ ν l transitions at the quark level, to scrutinize the nature of new physics (NP) in the model in-dependent method. We first use the experimental measurements of B ( B → D ( ∗ ) l ¯ ν l ), R D ( ∗ ) and R J/ψ to constrain the NP coupling parameters in a variety of scenarios. Using theconstrained NP coupling parameters, we report numerical results on various observablesrelated to the processes Σ b → Σ c l ¯ ν l and Ω b → Ω c l ¯ ν l , such as the branching ratios, theratio of branching fractions, the lepton side forward-backward asymmetries, the hadronand lepton longitudinal polarization asymmetries and the convexity parameter. We alsoprovide the q dependency of these observables and we hope that the corresponding nu-merical results in this work will be testified by future experiments. ∗ [email protected] † [email protected] a r X i v : . [ h e p - ph ] S e p Introduction
Though the Standard Model (SM) is considered as the most fundamental and successful theorywhich describe almost all the phenomena of the particle physics, there are still some openissues that are not discussed in the SM, like matter-antimatter asymmetry, dark matter, etc.Although there is no direct evidence for NP beyond the SM has been found, some possiblehints of NP have been observed in the B meson decay processes [1–4]. Even though the SMgauge interactions are lepton flavor universal, the hints of lepton flavor universal violation(LFUV) have also been observed in several anomalies relative to the semileptonic B mesondecays. The most basic experimental measurements which substantiate these anomalies arethe ratio of the branching ratios R D ( ∗ ) for b → cl ¯ ν l decay processes. The ratio which is definedas R D ( ∗ ) = B ( B → D ( ∗ ) τ ¯ ν τ ) B ( B → D ( ∗ ) (cid:96) ¯ ν (cid:96) ) with (cid:96) = e, µ has been measured first by the BaBar [5]. Besides Belleand LHCb also reported their results [6–10]. The experimental measurement results for theseanomalies show that there is large deviations with their corresponding SM predictions. Veryrecently, the Belle Collaborations announced the latest measurements of R D ( ∗ ) [11]R Belle D = 0 . ± . ± . , R Belle D ∗ = 0 . ± . ± . , (1)which are in agreement with their SM predictions about within 0 . σ and 1 . σ , respectively, andtheir combination agrees with the SM predictions within 1 . σ . Although the tension betweenthe latest measurement results and their SM predictions is obviously reduced, there is still 3 . σ corresponding SM predictions on combining all measurements in the global average fields. Thelatest averaged results reported by Heavy Flavor Averaging Group (HFAG) are [12]R avg D = 0 . ± . ± . , R avg D ∗ = 0 . ± . ± . , (2)comparing with the SM predictions of R D ( ∗ ) [12]R SM D = 0 . ± . , R SM D ∗ = 0 . ± . . (3)One can see that above averaged experimental measurement results deviate from their SMpredictions at 1 . σ and 2 . σ level, respectively.2part from R D and R D ∗ measurements, the ratio R J/ψ has also been measured by LHCb [13]R
J/ψ = B ( B c → J/ψτ ¯ ν τ ) B ( B c → J/ψl ¯ ν l ) = 0 . ± . ± . , (4)which central value prediction of the SM is in the range 0.25 ∼ σ tension with its SM prediction [14, 15]. The uncertainties arise from the choice ofthe approach for the B c → J/ψ from factors [15–18].These deviations between the experimental measurements and their SM predictions areperhaps from the uncertainties of hadronic transition form factors. This may imply the leptonflavor universality is violated, which is the hint of the existence of NP. Many works have beendone based on model independent framework [19–25] or specific NP models by introducingnew particles such as leptoquarks [26–28], SUSY particles [29, 30], charged Higgses [31–33], ornew vector bosons [34].It is also important and interesting to investigate the semileptonic baryon decays Σ b → Σ c l ¯ ν l and Ω b → Ω c l ¯ ν l which are mediated by the b → cl ¯ ν l transition at the quark level. Studyingthese processes not only can provide an independent determination of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element | V cb | , but also can confirm the LFUV in R Σ c (Ω c ) which havea similar formalism to R D ( ∗ ) . We will explore the NP effects on various observables for theΣ b → Σ c l ¯ ν l and Ω b → Ω c l ¯ ν l decays in the model independent effective field theory formalism.It is necessary to study these decay modes both theoretically and experimentally to test theLFUV. There will be several difficulties to measure the branching ratio B (Σ b → Σ c l ¯ ν l ) becauseΣ b decay strongly and their branching ratios will be very small [35]. Nevertheless it is feasibleto measure B (Ω b → Ω c l ¯ ν l ) as Ω b decays predominantly weakly and the branching ratio issignificantly large. So it is worth to study these decay processes because they can provide verycomprehensive information about possible NP.It will draw very interesting results to investigate the implications of R D ( ∗ ) on the processesΩ b → Ω c l ¯ ν l and Σ b → Σ c l ¯ ν l . The authors of Refs. [36–43] give the total decay rate Γ(in units of10 s − ) from 1 .
44 to 4 . b → Σ c e ¯ ν e and from 1.29 to 5.4 for Ω b → Ω c e ¯ ν e . It is worthwhileto note that the complexity of the baryon structures and the lack of precise predictions ofvarious form factors may lead to the variations in the prediction of the total decay rate Γ.In this paper we will give the predictions of various observables within SM and different NPscenarios. Using the NP coupling parameters constrained from the latest experimental limits3rom B ( B → D ( ∗ ) l ¯ ν l ), R D ( ∗ ) and R J/ψ , we investigate the NP effects of these anomalies on thedifferential branching fraction d B /dq , the ratios of branching fractions R Ω c (Σ c ) ( q ), the leptonside forward-backward asymmetries A F B ( q ), the longitudinal polarizations P Σ c (Ω c ) L ( q ) of thedaughter baryons Σ c (Ω c ), the longitudinal polarizations P lL ( q ) of the lepton l and the convexityparameter C lF ( q ). Note that there is different between our study and the Ref. [44], in whichΩ b → Ω c l ¯ ν l and Σ b → Σ c l ¯ ν l have also been investigated in a model independent way. In ourwork the NP coupling parameters are assumed to be complex and we consider the constraints onthe NP coupling parameters from the experimental limits of B ( B → D ( ∗ ) l ¯ ν l ), R J/ψ and R D ( ∗ ) .However, NP coupling parameters are set to real and only R D ( ∗ ) is considered in Ref. [44].Our paper is organized as follows. In Sec.2 we briefly introduce the effective theory de-scribing the b → cl ¯ ν l transitions as well as the form factors, the helicity amplitudes and someobservables of the processes Ω b → Ω c l ¯ ν l and Σ b → Σ c l ¯ ν l . Sec. 3 is devoted to the numeri-cal results and discussions for the predictions within the SM and various NP scenarios. Ourconclusions are given in Sec. 4. The most general effective Lagrangian including both the SM and the NP contribution for B → B l ¯ ν l decay processes, where B = Σ b (Ω b ), B = Σ c (Ω c ), mediated by the quark leveltransition b → cl ¯ ν l is given by [45, 46] L eff = − G F √ V cb (cid:40) (1 + V L ) ¯ l L γ µ ν L ¯ q L γ µ b L + V R ¯ l L γ µ ν L ¯ q R γ µ b R + S L ¯ l R ν L ¯ q R b L + S R ¯ l R ν L ¯ q L b R + T L ¯ l R σ µν ν L ¯ q R σ µν b L (cid:41) + h . c . , (5) where G F is the Fermi constant, V cb is the CKM matrix elements and ( q, b, l, ν ) L,R = P L,R ( q, b, l, ν )are the chiral quark (lepton) fields with P L,R = (1 ∓ γ ) / V L,R , S L,R , T L characterizing the NP contributionscoming from the new vector, scalar and tensor interactions are associated with left handedneutrino and these NP coupling parameters are all zero in the SM. In our work we focus ona study of the vector and scalar type interactions, excepting the tensor interaction, and weassume that the NP coupling parameters V L,R and S L,R are complex.4 .1 Form factors and helicity amplitudes
The hadronic matrix elements of vector and axial vector currents for the decays B → B l ¯ ν l are parametrized in terms of various hadronic form factors as follows: M Vµ = (cid:104) B , λ | ¯ cγ µ b | B , λ (cid:105) = ¯ u ( p , λ ) (cid:2) f ( q ) γ µ + if ( q ) σ µν q ν + f ( q ) q µ (cid:3) × u ( p , λ ) ,M Aµ = (cid:104) B , λ | ¯ cγ µ γ b | B , λ (cid:105) = ¯ u ( p , λ ) (cid:2) g ( q ) γ µ + ig ( q ) σ µν q ν + g ( q ) q µ (cid:3) γ × u ( p , λ ) , where σ µν = i ( γ µ γ ν − γ ν γ µ ), q µ = ( p − p ) µ is the four momentum transfer. λ and λ are thehelicities of the parent baryon B and daughter baryon B , respectively. Here B represents thebottomed baryon Σ b or Ω b and B represents the charmed baryon Σ c or Ω c . Using the equationof motion, we can obtain the hadronic matrix elements of the scalar and pseudo-scalar currentsbetween these two baryons. The expressions for them can be written (cid:104) B , λ | ¯ cb | B , λ (cid:105) = ¯ u ( p , λ ) × (cid:20) f ( q ) qm b − m c + f ( q ) q m b − m c (cid:21) × u ( p , λ ) , (cid:104) B , λ | ¯ cγ b | B , λ (cid:105) = ¯ u ( p , λ ) × (cid:20) − g ( q ) qm b + m c − g ( q ) q m b + m c (cid:21) γ × u ( p , λ ) , where m b and m c are the respective masses of b and c quarks calculated at the renormalizationscale µ = m b .When both baryons are heavy, it is also convenient to parametrize the matrix element inthe heavy quark limit, these matrix elements can be parametrized in terms of four velocities v µ and v (cid:48) µ as follows M Vµ = (cid:104) B , λ | ¯ cγ µ b | B , λ (cid:105) = ¯ u ( p , λ ) (cid:2) F ( w ) γ µ + F ( w ) v µ + F ( w ) v (cid:48) µ (cid:3) u ( p , λ ) ,M Aµ = (cid:104) B , λ | ¯ cγ µ γ b | B , λ (cid:105) = ¯ u ( p , λ ) (cid:2) G ( w ) γ µ + G ( w ) v µ + G ( w ) v (cid:48) µ (cid:3) γ u ( p , λ ) , where w = v.v (cid:48) = (cid:0) M B + M B − q (cid:1) / M B M B , M B and M B are the masses of the B and B baryons, respectively. The relationship of these two sets of form factors are related via [47] f ( q ) = F ( q ) + ( m B + m B ) (cid:20) F ( q )2 m B + F ( q )2 m B (cid:21) ,f ( q ) = F ( q )2 m B + F ( q )2 m B ,f ( q ) = F ( q )2 m B − F ( q )2 m B ,g ( q ) = G ( q ) − ( m B − m B ) (cid:20) G ( q )2 m B + G ( q )2 m B (cid:21) ,g ( q ) = G ( q )2 m B + G ( q )2 m B ,g ( q ) = G ( q )2 m B − G ( q )2 m B . (6)In our numerical analysis, we follow Ref. [38] and use the form factor inputs obtained inthe framework of the relativistic quark model. In the heavy quark limit, the form factors canbe expressed in terms of the Isgur-Wise function ζ ( w ) as follows [38, 41] F ( w ) = G ( w ) = − ζ ( w ) ,F ( w ) = F ( w ) = 23 2 w + 1 ζ ( w ) ,G ( w ) = G ( w ) = 0 , (7)and the values of ζ ( w ) in the whole kinematic range, pertinent for our analysis, were mainlyobtained from Ref. [38].The helicity amplitudes can be defined by [47–51] H V/Aλ ,λ W = M V/Aµ ( λ ) ε † µ ( λ W ) , (8)where λ and λ W denote the respective helicities of the daughter baryon and W − off − shell , In therest frame of the parent baryon B , the vector and axial vector hadronic helicity amplitudes inthe terms of the various form factors and NP coupling parameters are given by [44, 47–51] H V = (1 + V L + V R ) (cid:112) Q − (cid:112) q (cid:104) ( M B + M B ) f ( q ) − q f ( q ) (cid:105) , A = (1 + V L − V R ) (cid:112) Q + (cid:112) q (cid:104) ( M B − M B ) g ( q ) + q g ( q ) (cid:105) ,H V + = (1 + V L + V R ) (cid:112) Q − (cid:104) − f ( q ) + ( M B + M B ) f ( q ) (cid:105) ,H A + = (1 + V L − V R ) (cid:112) Q + (cid:104) − g ( q ) − ( M B − M B ) g ( q ) (cid:105) ,H V t = (1 + V L + V R ) (cid:112) Q + (cid:112) q (cid:104) ( M B − M B ) f ( q ) + q f ( q ) (cid:105) ,H A t = (1 + V L − V R ) (cid:112) Q − (cid:112) q (cid:104) ( M B + M B ) g ( q ) − q g ( q ) (cid:105) , where Q ± = ( M B ± M B ) − q and f i , g i ( i = 1 , ,
3) are the various form factors. Either fromparity or from explicit calculation, it is clear to find that H V − λ − λ W = H Vλ λ W and H A − λ − λ W = − H Aλ λ W . So the total left-handed helicity amplitude is H λ λ W = H Vλ λ W − H Aλ λ W (9)Similarly, the scalar and pseudoscalar helicity amplitudes associated with the form factorsand NP coupling parameters G S and G P can be written as H SPλ = H Sλ − H Pλ ,H S = ( S L + S R ) (cid:112) Q + m b − m q (cid:104) ( M B − M B ) f ( q ) + q f ( q ) (cid:105) ,H P = ( S L − S R ) (cid:112) Q − m b + m q (cid:104) ( M B + M B ) g ( q ) − q g ( q ) (cid:105) , one can see that H S − λ − λ W = H Sλ λ W and H P − λ − λ W = − H Pλ λ W . The results of above helicityamplitudes in SM can be obtained by setting V L,R = 0 and S L,R = 0. Σ b → Σ c l ¯ ν l and Ω b → Ω c l ¯ ν l After including the NP contributions, the differential decay distribution for Σ b → Σ c l ¯ ν l andΩ b → Ω c l ¯ ν l in term of q , θ l and helicity amplitudes can be written as [47, 49] d Γ( B → B l ¯ ν l ) dq dcosθ l = N (cid:18) − m l q (cid:19) (cid:20) A + m l q A +2 A + 4 m l (cid:112) q A (cid:21) , (10) where N = G F | V cb | q (cid:113) λ ( M B , M B , q )2 π M B ,λ ( a, b, c ) = a + b + c − ab + bc + ca ) , = 2 sin θ l (cid:0) H , + H − , (cid:1) + (1 − cos θ l ) H , + +(1 + cos θ l ) H − , − ,A = 2 cos θ l (cid:0) H , + H − , (cid:1) + sin θ l (cid:0) H , + + H − , − (cid:1) + 2 (cid:0) H ,t + H − ,t (cid:1) − θ l (cid:0) H , H ,t + H − , H − ,t (cid:1) ,A = H SP , + H SP − , ,A = − cos θ l (cid:0) H , H SP , + H − , H SP − , (cid:1) + (cid:0) H ,t H SP , + H − ,t H SP − , (cid:1) , the θ l is the angle between the directions of the parent baryon B and final lepton l threemomentum vector in the dilepton rest frame.After integrating over the cos θ l of Eq. (10), we can obtain the normalized differential decayrate d Γ( B → B l ¯ ν l ) dq = 8 N (cid:18) − m l q (cid:19) [ B + m l q B + 32 B + 3 m l (cid:112) q B ] , (11)with B = H + H − + H + + H − − , B = H + H − + H + + H − − + 3 (cid:16) H t + H − t (cid:17) , B = (cid:16) H SP (cid:17) + (cid:16) H SP − (cid:17) , B = H t H SP + H − t H SP − . Besides the differential decay rate, other interesting observables are also investigated andthey can be written as follows:* The total differential branching fraction d B ( B → B l ¯ ν l ) dq = τ Ω b (Σ b ) d Γ( B → B l ¯ ν l ) dq . (12)* The lepton side forward-backward asymmetries parameter A l FB ( q ) = (cid:18) (cid:90) − d cos θ l d Γ dq d cos θ l (cid:90) d cos θ l d Γ dq d cos θ l (cid:19)(cid:46) d Γ dq . (13)* The convexity parameter C lF ( q ) = 1 d Γ /dq d d (cos θ l ) (cid:32) d Γ dq d cos θ l (cid:33) . (14)* The longitudinal polarization asymmetries parameter of daughter baryons Ω c (Σ c ) P Ω c (Σ c ) L ( q ) = dΓ λ =1 / / d q − dΓ λ = − / / d q dΓ / d q , (15)where dΓ λ = ± / /dq are the individual helicity dependent differential decay rates, whosedetailed expressions are given in Ref. [50].* The longitudinal polarization asymmetries parameter of the charged lepton P lL ( q ) = dΓ λ l =1 / / d q − dΓ λ l = − / / d q dΓ / d q , (16)where dΓ λ l = ± / /dq are differential decay rates for positive and negative helicity of leptonand their detailed expressions are also given in Ref. [50].* The ratios of the branching fractionsR Ω c (Σ c ) ( q ) = d B ( B → B τ ¯ ν τ ) /dq d B ( B → B (cid:96) ¯ ν (cid:96) ) /dq . (17)Note that integrating the numerator and denominator over q separately before taking the ratio,we can get the average values of all the observables such as (cid:104) A l FB (cid:105) , (cid:104) C lF (cid:105) , (cid:104) P lL (cid:105) , (cid:104) P Ω c (Σ c ) L (cid:105) and (cid:104) R Ω c (Σ c ) (cid:105) . In this section, we will give our results within SM and various NP scenarios in a model inde-pendent way. We present the constrained NP coupling parameter space and give the numericalresults of the observables displayed in Eqs. (12)-(17) for Ω b → Ω c l ¯ ν l and Σ b → Σ c l ¯ ν l transitionsincluding the contributions of different NP coupling parameters. In order to get the allowedNP coupling parameter space in various NP scenarios, we will impose the 2 σ constraint coming9rom the latest experimental values of the observables B ( B → D ( ∗ ) l ¯ ν l ), R D ( ∗ ) and R J/ψ . Thespecific expressions of these observables for B → D ( ∗ ) l ¯ ν l and B c → J/ψl ¯ ν l processes used inour work can easily be found in the Refs. [50–54].In our numerical computation about above various observables, except for the transitionform factors and the NP coupling parameters, the values of the other input parameters such asthe particle masses, decay constants, mean lives and some relevant experimental measurementdata of B ( B → D ( ∗ ) l ¯ ν l ) are mainly taken from the Particle Data Group (PDG) [55]. Therelevant experimental data about R D ( ∗ ) and R J/ψ used in this work are listed in Eqs. (2) and (4).Note that, in the model independent analysis, we assume that all the NP coupling parametersare complex and we consider only one NP coupling existing in Eq. (5) at one time and keep itinterference with the SM.Firstly, we obtain the constrained range of NP coupling parameters V L , V R , S L and S R byusing the recent experimental measurement results, and then examine the NP effects on theobservables which are displayed in Sec. 2 by using the constrained NP coupling parameters.The constrained the range of four NP coupling parameters V L , V R , S L and S R are shown inthe Fig. 1, and the results can be intuitively displayed by both real-imaginary and modulus-phases of the NP coupling parameters in the figure. There are few references that discussthe relationship between modulus and phases of the NP coupling parameters. The constrainedresults on the real, imaginary and modulus of the NP coupling parameters are listed in the Tab. 1clearly. From Fig. 1 we can see that present experimental data give quite strong bounds onthe relevant coupling parameters, in particular, modulus and phase of V L is strongly restricted.The constrained range of V L and S L , V R and S L are shown in Fig. 2 (a1-a4) and (b1-b4),respectively. From Fig. 2 (a1-a4) we can see that the values of Re[ S L ] and Im[ S L ] are in smallrange compared with the values of Re[ V L ] and Im[ V L ]. From Fig. 2 (b1-b4), it is clear to find theresult of V R - S L presents an axial symmetric phenomenon, and the scattered points are mainlydistributed around the origin. Because the distribution relationship of V L - S R and V R - S R aresimilar to the Fig. 2, we do not show the relationship of V L - S R and V R - S R anymore.The constraints about these NP coupling parameters obtained from various B meson decayprocesses have been also discussed in Refs. [1, 49–51, 56–58]. The NP coupling parametersare assumed complex or real in these references and corresponding experimental data which10 a ) - - - - - - - - [ VL ] I m [ V L ] ( a ) - - - - - [ VR ] I m [ V R ] ( a ) - - - - [ SL ] I m [ S L ] ( a ) - - - - - [ SR ] I m [ S R ] ( b ) - - - | VL | ϕ ( r ad ) ( b ) - - - | VR | ϕ ( r ad ) ( b ) - - - | SL | ϕ ( r ad ) ( b ) - - - | SR | ϕ ( r ad ) Figure 1:
The bounds on both real-imaginary ( a - a ) and modulus-phase ( b - b ) parts of the complexcoupling parameters VL,VR, SL and SR coming from the relevant experimental constraints. Table 1:
The allowed ranges of V L , V R , S L and S R NP coupling coefficients .Decay mode NP coefficients Min value Max Value Max of | V i ( S i ) | ( i = L.R )(Re[ V L ] , Im[ V L ]) ( − . , − . . , . b → cl ¯ ν l (Re[ V R ] , Im[ V R ]) ( − . , − . . , . S L ] , Im[ S L ]) ( − . , − . . , . S R ] , Im[ S R ]) ( − . , − . . , . D ( ∗ ) and R J/ψ . But few references considerthe experimental data of B ( B → D ( ∗ ) l ¯ ν l ) which are considered in our work. In our analysis,we use the experimental data of R D ( ∗ ) , R J/ψ and B ( B → D ( ∗ ) l ¯ ν l ) to constrain the space ofthe corresponding NP coupling parameters. We get more severe bounds on the phases andstrengths of the NP coupling parameters and we also give the relationship between modulusand phase of four NP coupling parameters which are not discussed in many previous references.Employing the theoretical framework described in Sec. 2, the SM predictions are reportedfor processes Ω b → Ω c l ¯ ν l and Σ b → Σ c l ¯ ν l . In Tab. 2, we list the average values of Γ, (cid:104) P lL (cid:105) ,11 a ) - - - - - - - - [ VL ] R e [ S L ] ( a ) - - - - - - - - [ VL ] I m [ S L ] ( a ) - - - - - - [ VL ] R e [ S L ] ( a ) - - - - - - [ VL ] I m [ S L ] ( b ) - - - - - - - - [ VR ] R e [ S L ] ( b ) - - - - - - - - [ VR ] I m [ S L ] ( b ) - - - - - - [ VR ] R e [ S L ] ( b ) - - - - - - [ VR ] I m [ S L ] Figure 2:
The bounds on both real and imaginary parts of the complex coupling parameters VL andSL ( a − a ), VR and SL ( b − b ) coming from the relevant experimental results. (cid:104) P Ω c (Σ c ) L (cid:105) , (cid:104) A lF B (cid:105) , (cid:104) C lF (cid:105) and (cid:104) R Ω c (Σ c ) (cid:105) for e , µ and τ mode respectively. From Tab. 2, one can seethat the results for e mode and µ mode are close for Ω b → Ω c l ¯ ν l and Σ b → Σ c l ¯ ν l processes. Thetotal decay rates Γ (in units of 10 s − ) at l = e, µ are observed to be larger than the resultat l = τ , and same phenomenon arises in (cid:104) P Ω c (Σ c ) L (cid:105) and A lF B . The lepton polarization fractions P lL for the e and µ are negative, but one for the τ mode is positive. The forward-backwardasymmetries A lF B for e and µ mode are positive, but one of the τ mode is negative. The hadronpolarization fractions P Ω c (Σ c ) L are about 0.58 at l = e, µ , and the result is about 0.35 at l = τ for both Ω b → Ω c l ¯ ν l and Σ b → Σ c l ¯ ν l . All the convexity parameters (cid:104) C lF (cid:105) are negative and (cid:104) C τF (cid:105) is much larger than (cid:104) C lF (cid:105) ( l = e, µ ). The ratio of branching ratio (cid:104) R Ω c (cid:105) is slightly larger than (cid:104) R Σ c (cid:105) .The behaviors of each observable as a function of q for the processes Ω b → Ω c l ¯ ν l andΣ b → Σ c l ¯ ν l are similar to each other. So we only take Ω b → Ω c l ¯ ν l decays as an exampleto illustrate in detail and the same goes in the following text. The SM predictions for the q dependency of different observables in the reasonable kinematic range for Ω b → Ω c l ¯ ν l aredisplayed in Fig. 3. In this figure, we compare the distributions of the each observable andthe red dot dash line, blue and green line represents the e , µ and τ mode, respectively. The12 dependency of d Γ /dq , A lF B , C lF and P lL are distinct for three generation leptons. But wecan find that the variation tendency of d Γ /dq , A lF B , C lF and P lL for e and µ modes is almostsame except in small q region. The total differential decay rate for e is maximum at q min andminimum at q max , however, the result for τ is maximum when q ≈ GeV and approaches zeroat q min and q max . For µ mode, d Γ /dq changes to zero quickly when q = m µ due to the effectof µ mass. All the A lF B approach to zero at q max . The A eF B is positive while A τF B is negativeand great increasing with q over the all q region. Besides, A µF B changes to -0.4 quickly when q = m µ and there is a zero-crossing point, which lies in the low q region. All the C lF arenegative in the whole q region and at the large q limit C lF are zero. At the low q range C eF is around -1.5 when q = q min , and C µF ≈ − . q ≈ . GeV , while C µF changes to zeroquickly when q = m µ due to the effect of the lepton mass. This behavior indicates that the cosθ distribution in q ∈ [0 . , .
23] is strongly parabolic. On the contrary, the C τF is small inthe whole ranges, which implies a straight-line behavior of the cosθ distribution. The P Ω c L arezero for three modes at q max . The results of P Ω c L for e and µ modes completely coincide and itis around 0.6 at q = q min = m l . The P eL is -1 over the all q region and it is similar to µ modeexcept for low q region. When q = m µ , the P µL changes to 0.4 quickly. While for the τ mode,the behavior is quite different and P τL take only positive values for entire q values. The R Ω c show an almost positive slope over the whole q region and R Ω c is around 0 when q = q min .Because the R Ω c is ratios of the differential branching fraction with the heavier τ in the finalstate to the differential branching fraction with the lighter lepton in the final state, the resultof this observable do not distinguish for the different leptons in the final state.Next, we proceed to investigate the effects of these four NP coupling parameters V L , V R , S L and S R on the above observables for various NP scenarios in a model independent way. In orderto avoid repetition, we only display the q dependency of each observable for decay Ω b → Ω c τ ¯ ν τ and the results are displayed in Fig. 4. In the figure we report the q dependency of theobservables d Γ /dq , A τF B ( q ), C τF ( q ), P τL ( q ), P Ω c L ( q ) and R Ω c ( q ) for Ω b → Ω c τ ¯ ν τ transitionincluding the contribution of only one NP vector or scalar type coupling parameter, and weincorporate both SM and NP result. In the Fig. 4, the band for the input parameters (formfactors and V cb ) and different NP coupling parameters restricted by the relative experimentalvalues of the processes B → D ( ∗ ) l ¯ ν l and B c → J/ψl ¯ ν l are represented with that different colors.13able 2: The SM central values for the decay rate Γ, the lepton polarization fraction (cid:104) P lL (cid:105) , thehadron polarization fraction (cid:104) P Σ c (Ω c ) L (cid:105) , the forward-backward asymmetry (cid:104) A lF B (cid:105) , the convexity factor (cid:104) C lF (cid:105) and the ratio of branching ratio (cid:104) R Σ c (Ω c ) (cid:105) for the e mode, µ mode and τ mode of Ω b → Ω c l ¯ ν l and Σ b → Σ c l ¯ ν l decays. Ω b → Ω c lν Σ b → Σ c lνe mode µ mode τ mode e mode µ mode τ modeΓ × s − (cid:104) P lL (cid:105) -1.123 -1.093 0.135 -1.135 -1.131 0.132 (cid:104) P Ω c (Σ c ) L (cid:105) (cid:104) A lF B (cid:105) (cid:104) C lF (cid:105) -1.170 -1.140 -0.135 -1.178 -1.148 -0.139 (cid:104) R Ω c (Σ c ) (cid:105) R Ω c = 0 .
370 R Σ c = 0 . × × × × q d / dq ( s - ) - - - q A F B l ( q ) - - - - q C F l ( q ) q P L Ω c ( q ) - - - q P L l ( q ) q R Ω c ( q ) Figure 3:
The SM predictions for the q dependent observables d Γ /dq , A lF B ( q ), C lF ( q ), P Ω c L ( q ), P lL ( q ) and R Ω c ( q ) relative to the decays Ω b → Ω c l ¯ ν l ( (cid:96) = e, µ, τ ). The red dot dash line, blue andgreen line represent the e , µ and τ mode, respectively. The SM and four NP scenarios are distinguished by gray (SM), red ( V L ), green ( V R ) , blue ( S L )and cyan ( S R ) colors, respectively. In the Fig. 4, we suppose that the NP contributions onlycome from one NP coupling and we find the following remarks:* When we only consider the effect of vector NP coupling V L , the effect of this NP couplingappears in the H Vλ ,λ W and H Aλ ,λ W only. From Eq. (11), it is clear to find that the d Γ /dq depends on (1 + V L ) only. Using the constrained range of V L which are displayed in theFig. 1, one can see that the deviation from the SM prediction due to the V L coupling isobserved only in the total differential decay rate d Γ /dq and the observable is proportionalto (1 + V L ) . The d Γ /dq is largely enhanced in the whole q region. Moreover, the factor(1+ V L ) appears both in the numerator and denominator of the expressions which describeother observables simultaneously. So the NP dependency cancels in the ratios and we donot see any deviation from the SM prediction for other observables.15 Similar to V L , the NP coupling parameter V R is also included in the vector and theaxial-vector helicity amplitudes. In this case, the d Γ /dq depends on both (1 + V R ) and (1 − V R ) . Hence, there is no cancellation of NP effects in the ratios and there isdeviation in each observable from the SM prediction. The deviation of d Γ /dq from theirSM prediction is not so significant, while, it is very significant for other observables. Theeffects of the V R coupling are rather significant on the observables P τL ( q ), P Ω c L ( q ) andR Ω c ( q ), especially in largest q region for P τL ( q ) and R Ω c ( q ) and lowest q region for P Ω c L ( q ).* The effects of the scalar NP coupling S L come into the scalar and pseudoscalar helicityamplitudes H Sλ ,λ W and H Pλ ,λ W . One can see that it is different from V L and V R couplingscenarios. From Eq. (11) one can see that d Γ /dq depends on S L and S L in this case. Sothere is also no cancellation in the numerator and denominator of the expressions in otherobservables simultaneously. We can find that the deviation from their SM prediction ismore pronounced than that with V L and V R NP coupling except A τF B ( q ). The deviationfrom the SM prediction for d Γ /dq is most prominent at q ≈ . GeV . When consider thevalue of the S L NP coupling, there may or may not be a zero crossing in the P τL ( q ), whilethere is no zero crossing for P τL ( q ) in the SM prediction. Besides, the deviations fromtheir SM prediction for P τL ( q ) and R Ω c ( q ) are most prominent at largest q region. Thereare some differences between our results and Ref. [44] for S L NP coupling scenario. InRef. [44], there are two constraint results for S L NP coupling and they are S L ∈ [ − . , . − . , − .
4] respectively. The authors use S L ∈ [ − . , − .
4] when consider the NPeffect of S L . If S L ∈ [ − . , .
1] in their analysis, their result are similar to our work forthis scenario.* From last column in Fig. 4 considering the S R NP coupling, the change trend of eachobservable are similar to the S L scenario. Because NP effects which come from the S R NP coupling are also encoded in the scalar and pseudoscalar helicity amplitudes only, the d Γ /dq depends on S R and S R . The deviation from the SM prediction of d Γ /dq maybe less obvious than the S L scenario. However, it is larger for C τF ( q ) compared to S L scenario. In the P τL ( q ), the zero -crossing point may shift slightly towards a lower q value than in the S L case. 16 MVL Only × × × × q d Γ / dq ( s - ) SMVR Only × × × × × × q d Γ / dq ( s - ) SMSL Only4 6 8 10 1205.0 × × × × q d Γ / dq ( s - ) SMSR Only4 6 8 10 1205.0 × × × × q d Γ / dq ( s - ) - - - - - - q A F B τ ( q ) SMVR Only4 6 8 10 12 - - - - - - q A F B τ ( q ) SMSL Only4 6 8 10 12 - - - - - - q A F B τ ( q ) SMSR Only - - - - - - q A F B τ ( q ) - - - - - - q C F τ ( q ) SMVR Only4 6 8 10 12 - - - - - - q C F τ ( q ) SMSL Only4 6 8 10 12 - - - - q C F τ ( q ) SMSR Only4 6 8 10 12 - - - - - - q C F τ ( q ) q P L τ ( q ) SMVR Only4 6 8 10 12 - - q P L τ ( q ) SMSL Only4 6 8 10 12 - q P L τ ( q ) SMSR Only4 6 8 10 12 - q P L τ ( q ) q P L Ω c ( q ) SMVR Only4 6 8 10 120.00.20.40.60.8 q P L Ω c ( q ) SMSL Only4 6 8 10 120.00.20.40.60.8 q P L Ω c ( q ) SMSR Only4 6 8 10 120.00.20.40.60.8 q P L Ω c ( q ) q R Ω c ( q ) SMVR Only4 6 8 10 120.00.51.01.52.0 q R Ω c ( q ) SMSL Only4 6 8 10 120.00.51.01.5 q R Ω c ( q ) SMSR Only4 6 8 10 120.00.51.01.5 q R Ω c ( q ) Figure 4:
The SM (gray) and NP predictions in the presence of V L (first column), V R (secondcolumn), S L (third column) and S R (fourth column) coupling for the q dependency observables d Γ /dq , A τF B ( q ), C τF ( q ), P τL ( q ), P Ω c L ( q )and R Ω c ( q ) relative to the decay Ω b → Ω c τ ¯ ν τ . Finally, we also explore the impact of these four combinations for vector and scalar typecouplings such as V L - S L , V L - S R , V R - S L , and V R - S R to above various observables for Ω b → Ω c τ ¯ ν τ q and have similar deviationsto their corresponding S L and S R predictions, except that the value of the correspondinglongitudinal axis is different. In order to avoid repetition, we do not display the results ofdifferent combinations anymore. At the same time we find that similar conclusions can be alsomade for the Σ b → Σ c τ ¯ ν τ decay process. Several anomalies R D ( ∗ ) and R J/ψ observed in the semileptonic B meson decays have indicatedthe hints of LFUV and attracted the attention of many researchers. Many works about baryondecays Λ b → Λ c ( p ) l ¯ ν l and Ξ b → Ξ c (Λ) l ¯ ν l have been done to investigate the NP effects of aboveanomalies on the precess b → c ( u ) l ¯ ν l . These baryon decays not only can provide an independentdetermination of the CKM matrix element | V cb | but also may be further confirmation of the hintsof LFUV that is helpful in exploring NP. At present, there exist few quantitative measurementfor the semileptonic decay of Ω b and Σ b due the complexity baryons structures and the lack ofprecise predictions of various form factors. It is indeed necessary to investigate the semileptonicbaryon decays Ω b → Ω c l ¯ ν l and Σ b → Σ c l ¯ ν l both theoretically and experimentally to test theLFUV.In this work we have used the helicity formalism to get various angular decay distributionand have performed a model independent analysis of baryonic Ω b → Ω c l ¯ ν l and Σ b → Σ c l ¯ ν l decay processes. In this work we considered the NP coupling parameters to be complex inour analysis. In order to constrain the various NP coupling parameters, we have assumedthat only one NP coupling parameter is present one time. We have gotten strong bounds onthe phases and strengths of the various NP coupling parameters from the latest experimentallimits of B → D ( ∗ ) l ¯ ν l and B c → J/ψl ¯ ν l . Using the constrained NP coupling parameters,we have estimated various observables of the Ω b → Ω c l ¯ ν l and Σ b → Σ c l ¯ ν l baryon decays inthe SM and various NP scenarios in a model independent way. The numerical results havebeen presented for e , µ and τ mode respectively in SM. We also display the q dependency ofdifferent observables for Ω b → Ω c τ ¯ ν τ process within the SM and various NP coupling scenarios.18he results show that d Γ /dq including any kind of NP couplings are all enhanced largely andhave significant deviations comparing to their SM predictions in whole q region. In the V L scenario, the observables A τF B ( q ), C τF ( q ), P τL ( q ), P Ω c (Σ c ) L ( q ) and R Ω c (Σ c ) ( q ) are the same astheir corresponding SM predictions because the coefficient (1 + V L ) appears in the numeratorand the denominator of the expressions which describing these observables simultaneously. Wenoticed a profound deviation in all angular observables of the semileptonic baryonic b → cτ ¯ ν τ process due to the additional contribution of V R , S L and S R couplings to the SM. The deviationsfrom their SM prediction of P τL ( q ) and R Ω c ( q ) are most prominent at largest q region.Till now there are only some experimental data about the non-leptonic decay of Ω b andΣ b , and there is poor quantitative measurement of the semileptonic decay rates of Ω b and Σ b ,Though there is no experimental measurement on these baryonic b → cl ¯ ν l decay processes,the study of this work is found to be very crucial in order to shed light on the nature of NP.In the near future, more data on Ω b will be obtained by the LHCb experiments and we hopethe results of the observables discussed in this work can be tested at experimental facilities atBEPCII, LHCb and Belle II. Acknowledgements
We would like to thank Yuan-Guo Xu for providing us some helpful discussion and constantencouragement on the manuscript. This work was supported by the National Natural ScienceFoundation of China (Contracts Nos. 11675137 and 11947083) and the Key Scientific ResearchProjects of Colleges and Universities in Henan Province (Contract No. 18A140029).
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