Probing new physics in semileptonic $\Sigma_b$ and $\Omega_b$ decays

Recently, several hints of lepton non-universality have been observed in the semileptonic B meson decays in terms of both in the neutral current ($b\to s l \bar{l}$) and charged current ($b\to c l \bar{\nu_{l}}$) transitions. Motivated by these inspiring results, we perform the analysis of the baryon decays ${\Sigma_b}\to \Sigma_c l \bar{\nu_{l}}$ and $\Omega_b\to \Omega_c l \bar{\nu_{l}} (l=e,\mu,\tau)$ which are mediated by $ b \to c l \bar{\nu_{l}}$ transitions at the quark level, to scrutinize the nature of new physics (NP) in the model independent method. We first use the experimental measurements of $\mathcal{B}( B\to D^{(*)} l\bar{\nu_{l}})$, ${\rm R}_{D^{(*)}}$ and ${\rm R}_{J/\psi}$ to constrain the NP coupling parameters in a variety of scenarios. Using the constrained NP coupling parameters, we report numerical results on various observables related to the processes ${\Sigma_b}\to \Sigma_c l \bar{\nu_{l}}$ and $\Omega_b\to \Omega_c l \bar{\nu_{l}}$, such as the branching ratios, the ratio of branching fractions, the lepton side forward-backward asymmetries, the hadron and lepton longitudinal polarization asymmetries and the convexity parameter. We also provide the $q^2$ dependency of these observables and we hope that the corresponding numerical results in this work will be testified by future experiments.


Introduction
Though the Standard Model (SM) is considered as the most fundamental and successful theory which describe almost all the phenomena of the particle physics, there are still some open issues 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 possible hints of NP have been observed in the B meson decay processes [1][2][3][4]. Even though the SM gauge 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 meson decays. The most basic experimental measurements which substantiate these anomalies are the ratio of the branching ratios R D ( * ) for b → clν l decay processes. The ratio which is defined as R D ( * ) = B(B→D ( * ) τντ ) B(B→D ( * ) ν ) with = e, µ has been measured first by the BaBar [5]. Besides Belle and LHCb also reported their results [6][7][8][9][10]. The experimental measurement results for these anomalies show that there is large deviations with their corresponding SM predictions. Very recently, the Belle Collaborations announced the latest measurements of R D ( * ) [11] R Belle D = 0.307 ± 0.037 ± 0.016, R Belle D * = 0.283 ± 0.017 ± 0.014, (1) which are in agreement with their SM predictions about within 0.2σ and 1.1σ, respectively, and their combination agrees with the SM predictions within 1.2σ. Although the tension between the latest measurement results and their SM predictions is obviously reduced, there is still 3.08σ corresponding SM predictions on combining all measurements in the global average fields. The latest averaged results reported by Heavy Flavor Averaging Group (HFAG) are [12] R avg D = 0.340 ± 0.027 ± 0.013, comparing with the SM predictions of R D ( * ) [12] R SM D = 0.299 ± 0.003, R SM D * = 0.258 ± 0.005.
One can see that above averaged experimental measurement results deviate from their SM predictions at 1.4σ and 2.5σ level, respectively.
Apart 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/ψτν τ ) B(B c → J/ψlν l ) = 0.71 ± 0.17 ± 0.18, which central value prediction of the SM is in the range 0.25∼0.28 and the experimental result has about 2σ tension with its SM prediction [14,15]. The uncertainties arise from the choice of the approach for the B c → J/ψ from factors [15][16][17][18].
It is also important and interesting to investigate the semileptonic baryon decays Σ b → Σ c lν l and Ω b → Ω c lν l which are mediated by the b → clν l transition at the quark level. Studying these 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 Σc(Ωc) which have a similar formalism to R D ( * ) . We will explore the NP effects on various observables for the Σ b → Σ c lν l and Ω b → Ω c lν l decays in the model independent effective field theory formalism.
It is necessary to study these decay modes both theoretically and experimentally to test the LFUV. There will be several difficulties to measure the branching ratio B(Σ b → Σ c lν l ) because Σ b decay strongly and their branching ratios will be very small [35]. Nevertheless it is feasible to measure B(Ω b → Ω c lν l ) as Ω b decays predominantly weakly and the branching ratio is significantly large. So it is worth to study these decay processes because they can provide very comprehensive information about possible NP.
It will draw very interesting results to investigate the implications of R D ( * ) on the processes The authors of Refs. [36][37][38][39][40][41][42][43] give the total decay rate Γ(in units of 10 10 s −1 ) from 1.44 to 4.3 for Σ b → Σ c eν e and from 1.29 to 5.4 for Ω b → Ω c eν e . It is worthwhile to note that the complexity of the baryon structures and the lack of precise predictions of various 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 NP scenarios. Using the NP coupling parameters constrained from the latest experimental limits from B(B → D ( * ) lν l ), R D ( * ) and R J/ψ , we investigate the NP effects of these anomalies on the differential branching fraction dB/dq 2 , the ratios of branching fractions R Ωc(Σc) (q 2 ), the lepton side forward-backward asymmetries A F B (q 2 ), the longitudinal polarizations P Σc(Ωc) L (q 2 ) of the daughter baryons Σ c (Ω c ), the longitudinal polarizations P l L (q 2 ) of the lepton l and the convexity parameter C l F (q 2 ). Note that there is different between our study and the Ref. [44], in which Ω b → Ω c lν l and Σ b → Σ c lν l have also been investigated in a model independent way. In our work the NP coupling parameters are assumed to be complex and we consider the constraints on the NP coupling parameters from the experimental limits of B(B → D ( * ) lν 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 describing the b → clν l transitions as well as the form factors, the helicity amplitudes and some observables of the processes Ω b → Ω c lν l and Σ b → Σ c lν l . Sec. 3 is devoted to the numerical results and discussions for the predictions within the SM and various NP scenarios. Our conclusions are given in Sec. 4.

Theory framework
The most general effective Lagrangian including both the SM and the NP contribution for B 1 → B 2 lν l decay processes, where B 1 = Σ b (Ω b ), B 2 = Σ c (Ω c ), mediated by the quark level transition b → clν l is given by [45,46] 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 5 )/2 as the projection operators. Here we note that the NP coupling parameters V L,R , S L,R , T L characterizing the NP contributions coming from the new vector, scalar and tensor interactions are associated with left handed neutrino and these NP coupling parameters are all zero in the SM. In our work we focus on a study of the vector and scalar type interactions, excepting the tensor interaction, and we assume that the NP coupling parameters V L,R and S L,R are complex.

Form factors and helicity amplitudes
The hadronic matrix elements of vector and axial vector currents for the decays B 1 → B 2 lν l are parametrized in terms of various hadronic form factors as follows: where σ µν = i 2 (γ µ γ ν − γ ν γ µ ), q µ = (p 1 − p 2 ) µ is the four momentum transfer. λ 1 and λ 2 are the helicities of the parent baryon B 1 and daughter baryon B 2 , respectively. Here B 1 represents the bottomed baryon Σ b or Ω b and B 2 represents the charmed baryon Σ c or Ω c . Using the equation of motion, we can obtain the hadronic matrix elements of the scalar and pseudo-scalar currents between these two baryons. The expressions for them can be written where m b and m c are the respective masses of b and c quarks calculated at the renormalization When both baryons are heavy, it is also convenient to parametrize the matrix element in the heavy quark limit, these matrix elements can be parametrized in terms of four velocities v µ and v µ as follows and M B 2 are the masses of the B 1 and B 2 baryons, respectively. The relationship of these two sets of form factors are related via [47] In our numerical analysis, we follow Ref. [38] and use the form factor inputs obtained in the framework of the relativistic quark model. In the heavy quark limit, the form factors can be expressed in terms of the Isgur-Wise function ζ 1 (w) as follows [38,41] and the values of ζ 1 (w) in the whole kinematic range, pertinent for our analysis, were mainly obtained from Ref. [38].
The helicity amplitudes can be defined by [47][48][49][50][51] where λ 2 and λ W denote the respective helicities of the daughter baryon and W − of f −shell , In the rest frame of the parent baryon B 1 , the vector and axial vector hadronic helicity amplitudes in the terms of the various form factors and NP coupling parameters are given by [44,[47][48][49][50][51] So the total left-handed helicity amplitude is Similarly, the scalar and pseudoscalar helicity amplitudes associated with the form factors and NP coupling parameters G S and G P can be written as The results of above helicity amplitudes in SM can be obtained by setting V L,R = 0 and S L,R = 0.

The observables for
After including the NP contributions, the differential decay distribution for Σ b → Σ c lν l and Ω b → Ω c lν l in term of q 2 , θ l and helicity amplitudes can be written as [47,49] where the θ l is the angle between the directions of the parent baryon B 1 and final lepton l three momentum vector in the dilepton rest frame.
After integrating over the cos θ l of Eq. (10), we can obtain the normalized differential decay with Besides the differential decay rate, other interesting observables are also investigated and they can be written as follows: * The total differential branching fraction * The lepton side forward-backward asymmetries parameter * The convexity parameter * The longitudinal polarization asymmetries parameter of daughter baryons Ω c (Σ c ) where dΓ λ 2 =±1/2 /dq 2 are the individual helicity dependent differential decay rates, whose detailed expressions are given in Ref. [50].
* The longitudinal polarization asymmetries parameter of the charged lepton where dΓ λ l =±1/2 /dq 2 are differential decay rates for positive and negative helicity of lepton and their detailed expressions are also given in Ref. [50].
* The ratios of the branching fractions Note that integrating the numerator and denominator over q 2 separately before taking the ratio, we can get the average values of all the observables such as A l FB , C l F , P l L , P Ωc(Σc) L and R Ωc(Σc) .

Numerical analysis and discussion
In this section, we will give our results within SM and various NP scenarios in a model independent way. We present the constrained NP coupling parameter space and give the numerical results of the observables displayed in Eqs. The constraints about these NP coupling parameters obtained from various B meson decay processes have been also discussed in Refs. [1,[49][50][51][56][57][58]. The NP coupling parameters are assumed complex or real in these references and corresponding experimental data which     for both Ω b → Ω c lν l and Σ b → Σ c lν l . All the convexity parameters C l F are negative and C τ F is much larger than C l F (l = e, µ). The ratio of branching ratio R Ωc is slightly larger than R Σc .
The behaviors of each observable as a function of q 2 for the processes Ω b → Ω c lν l and Σ b → Σ c lν l are similar to each other. So we only take Ω b → Ω c lν l decays as an example to illustrate in detail and the same goes in the following text. The SM predictions for the q 2 dependency of different observables in the reasonable kinematic range for Ω b → Ω c lν l are displayed in Fig. 3. In this figure, we compare the distributions of the each observable and the red dot dash line, blue and green line represents the e, µ and τ mode, respectively. The q 2 dependency of dΓ/dq 2 , A l F B , C l F and P l L are distinct for three generation leptons. But we can find that the variation tendency of dΓ/dq 2 , A l F B , C l F and P l L for e and µ modes is almost same except in small q 2 region. The total differential decay rate for e is maximum at q 2 min and minimum at q 2 max , however, the result for τ is maximum when q 2 ≈ 8GeV 2 and approaches zero at q 2 min and q 2 max . For µ mode, dΓ/dq 2 changes to zero quickly when q 2 = m 2 µ due to the effect of µ mass. All the A l F B approach to zero at q 2 max . The A e F B is positive while A τ F B is negative and great increasing with q 2 over the all q 2 region. Besides, A µ F B changes to -0.4 quickly when q 2 = m 2 µ and there is a zero-crossing point, which lies in the low q 2 region. All the C l F are negative in the whole q 2 region and at the large q 2 limit C l F are zero. At the low q 2 range C e F is around -1.5 when q 2 = q 2 min , and C µ F ≈ −1.4 when q 2 ≈ 0.4GeV 2 , while C µ F changes to zero quickly when q 2 = m 2 µ due to the effect of the lepton mass. This behavior indicates that the cosθ distribution in q 2 ∈ [0.4, 11.23] is strongly parabolic. On the contrary, the C τ F is small in the whole ranges, which implies a straight-line behavior of the cosθ distribution. The P Ωc L are zero for three modes at q 2 max . The results of P Ωc L for e and µ modes completely coincide and it is around 0.6 at q 2 = q 2 min = m 2 l . The P e L is -1 over the all q 2 region and it is similar to µ mode except for low q 2 region. When q 2 = m 2 µ , the P µ L changes to 0.4 quickly. While for the τ mode, the behavior is quite different and P τ L take only positive values for entire q 2 values. The R Ωc show an almost positive slope over the whole q 2 region and R Ωc is around 0 when q 2 = q 2 min . Because the R Ωc is ratios of the differential branching fraction with the heavier τ in the final state to the differential branching fraction with the lighter lepton in the final state, the result of 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 order to avoid repetition, we only display the q 2 dependency of each observable for decay Ω b → Ω c τν τ and the results are displayed in Fig. 4. In the figure we report the q 2 dependency of the observables dΓ/dq 2 , A τ F B (q 2 ), C τ F (q 2 ), P τ L (q 2 ), P Ωc L (q 2 ) and R Ωc (q 2 ) for Ω b → Ω c τν τ transition including the contribution of only one NP vector or scalar type coupling parameter, and we incorporate both SM and NP result. In the Fig. 4, the band for the input parameters (form factors and V cb ) and different NP coupling parameters restricted by the relative experimental values of the processes B → D ( * ) lν l and B c → J/ψlν l are represented with that different colors.   Figure 3: The SM predictions for the q 2 dependent observables dΓ/dq 2 , A l F B (q 2 ), C l F (q 2 ), P Ωc L (q 2 ), P l L (q 2 ) and R Ωc (q 2 ) relative to the decays Ω b → Ω c lν l ( = e, µ, τ ). The red dot dash line, blue and green 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 only come 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 coupling appears in the H V λ 2 ,λ W and H A λ 2 ,λ W only. From Eq. (11), it is clear to find that the dΓ/dq 2 depends on (1 + V L ) 2 only. Using the constrained range of V L which are displayed in the Fig. 1, one can see that the deviation from the SM prediction due to the V L coupling is observed only in the total differential decay rate dΓ/dq 2 and the observable is proportional to (1 + V L ) 2 . The dΓ/dq 2 is largely enhanced in the whole q 2 region. Moreover, the factor (1+V L ) 2 appears both in the numerator and denominator of the expressions which describe other observables simultaneously. So the NP dependency cancels in the ratios and we do not see any deviation from the SM prediction for other observables. * Similar to V L , the NP coupling parameter V R is also included in the vector and the axial-vector helicity amplitudes. In this case, the dΓ/dq 2 depends on both (1 + V R ) 2 and (1 − V R ) 2 . Hence, there is no cancellation of NP effects in the ratios and there is deviation in each observable from the SM prediction. The deviation of dΓ/dq 2 from their SM prediction is not so significant, while, it is very significant for other observables. The effects of the V R coupling are rather significant on the observables P τ L (q 2 ), P Ωc L (q 2 ) and R Ωc (q 2 ), especially in largest q 2 region for P τ L (q 2 ) and R Ωc (q 2 ) and lowest q 2 region for P Ωc L (q 2 ).
* The effects of the scalar NP coupling S L come into the scalar and pseudoscalar helicity amplitudes H S λ 2 ,λ W and H P λ 2 ,λ W . One can see that it is different from V L and V R coupling scenarios. From Eq. (11) one can see that dΓ/dq 2 depends on S L and S 2 L in this case. So there is also no cancellation in the numerator and denominator of the expressions in other observables simultaneously. We can find that the deviation from their SM prediction is more pronounced than that with V L and V R NP coupling except A τ F B (q 2 ). The deviation from the SM prediction for dΓ/dq 2 is most prominent at q 2 ≈ 8.8GeV 2 . When consider the value of the S L NP coupling, there may or may not be a zero crossing in the P τ L (q 2 ), while there is no zero crossing for P τ L (q 2 ) in the SM prediction. Besides, the deviations from their SM prediction for P τ L (q 2 ) and R Ωc (q 2 ) are most prominent at largest q 2 region. There are some differences between our results and Ref. [44] for S L NP coupling scenario. In Ref. [44], there are two constraint results for S L NP coupling and they are S L ∈ [−0. * From last column in Fig. 4 considering the S R NP coupling, the change trend of each observable 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 dΓ/dq 2 depends on S R and S 2 R . The deviation from the SM prediction of dΓ/dq 2 may be less obvious than the S L scenario. However, it is larger for C τ F (q 2 ) compared to S L scenario. In the P τ L (q 2 ), the zero -crossing point may shift slightly towards a lower q 2 value than in the S L case.  Figure 4: The SM (gray) and NP predictions in the presence of V L (first column), V R (second column), S L (third column) and S R (fourth column) coupling for the q 2 dependency observables dΓ/dq 2 , A τ F B (q 2 ), C τ F (q 2 ), P τ L (q 2 ), P Ωc L (q 2 )and R Ωc (q 2 ) relative to the decay Ω b → Ω c τν τ .
Finally, we also explore the impact of these four combinations for vector and scalar type couplings 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 τν τ process. We find that the NP predictions of the same observables in these four combinations NP scenarios show a similar variation tendency to the increasing of q 2 and have similar deviations to their corresponding S L and S R predictions, except that the value of the corresponding longitudinal axis is different. In order to avoid repetition, we do not display the results of different combinations anymore. At the same time we find that similar conclusions can be also made for the Σ b → Σ c τν τ decay process. In this work we have used the helicity formalism to get various angular decay distribution and have performed a model independent analysis of baryonic Ω b → Ω c lν l and Σ b → Σ c lν l decay processes. In this work we considered the NP coupling parameters to be complex in our analysis. In order to constrain the various NP coupling parameters, we have assumed that only one NP coupling parameter is present one time. We have gotten strong bounds on the phases and strengths of the various NP coupling parameters from the latest experimental limits of B → D ( * ) lν l and B c → J/ψlν l . Using the constrained NP coupling parameters, we have estimated various observables of the Ω b → Ω c lν l and Σ b → Σ c lν l baryon decays in the SM and various NP scenarios in a model independent way. The numerical results have been presented for e, µ and τ mode respectively in SM. We also display the q 2 dependency of different observables for Ω b → Ω c τν τ process within the SM and various NP coupling scenarios.

Summary
The results show that dΓ/dq 2 including any kind of NP couplings are all enhanced largely and have significant deviations comparing to their SM predictions in whole q 2 region. In the V L scenario, the observables A τ F B (q 2 ), C τ F (q 2 ), P τ L (q 2 ), P Ωc(Σc) L (q 2 ) and R Ωc(Σc) (q 2 ) are the same as their corresponding SM predictions because the coefficient (1 + V L ) 2 appears in the numerator and the denominator of the expressions which describing these observables simultaneously. We noticed a profound deviation in all angular observables of the semileptonic baryonic b → cτν τ process due to the additional contribution of V R , S L and S R couplings to the SM. The deviations from their SM prediction of P τ L (q 2 ) and R Ωc (q 2 ) are most prominent at largest q 2 region. Till now there are only some experimental data about the non-leptonic decay of Ω b and Σ 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 → clν 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 hope the results of the observables discussed in this work can be tested at experimental facilities at BEPCII, LHCb and Belle II.