Precise predictions for $\Lambda_b \to \Lambda_c$ semileptonic decays

We calculate the $\Lambda_b \to \Lambda_c \ell \nu$ form factors and decay rates for all possible $b\to c \ell\bar\nu$ four-Fermi interactions beyond the Standard Model, including nonzero charged lepton masses and terms up to order $\alpha_s\, \Lambda_\text{QCD}/m_{c,b}$ and $\Lambda_\text{QCD}^2/m_c^2$ in the heavy quark effective theory. At this order, we obtain model independent predictions for semileptonic $\Lambda_b \to \Lambda_c$ decays in terms of only two unknown sub-subleading Isgur-Wise functions, which can be determined from fitting LHCb and lattice QCD data. We thus obtain model independent results for $\Lambda_b\to \Lambda_c\ell\bar\nu$ decays, including predictions for the ratio $R(\Lambda_c) = {\cal B}(\Lambda_b\to \Lambda_c \tau\bar\nu) / {\cal B}(\Lambda_b\to \Lambda_c \mu\bar\nu)$ in the presence of new physics, that are more precise than prior results in the literature, and systematically improvable with better data on the decays with $\mu$ (or $e$) in the final state. We also explore tests of factorization in $\Lambda_b \to \Lambda_c\pi$ decays, and emphasize the importance of measuring at LHCb the double differential rate $d^2\Gamma(\Lambda_b\to\Lambda_c\ell\bar\nu) / (d q^2\, d\cos\theta)$, in addition to the $q^2$ spectrum.


I. INTRODUCTION
In a recent paper [1], it was shown that LHCb data for the semileptonic Λ b → Λ c µν decays [2] combined with lattice QCD calculations [3], provide sensitivity for the first time to sub-subleading O(Λ 2 QCD /m 2 c ) terms in the heavy quark effective theory (HQET) expansion [4,5] of the Λ b → Λ c semileptonic decay form factors, independent of |V cb |. The O(Λ 2 QCD /m 2 c ) corrections were found to have their expected characteristic size, suggesting that the expansion in Λ QCD /m c for baryon form factors is well-behaved up to Λ 2 QCD /m 2 c terms. The same framework also resulted in a new standard model (SM) prediction for the ratio which is significantly more precise than prior results [3,[6][7][8][9][10][11].
The ratio in Eq. (1) is of particular interest in light of the persistent hints of deviations from the SM, in the ratios at approximately the 4σ level, once the measurements for the D and D * final states are combined [12]. The Λ b → Λ c µν decays involve the same underlying b → cτ ν new physics (NP) operators as B → D ( * ) τν, but the HQET expansion for the ground-state baryon form factors is simpler than for mesons. The "brown muck" surrounding the heavy quark is in a spin and isospin zero ground state. A consequence of this is a simpler expansion of the form factors, in which the O(Λ QCD /m c,b , α s Λ QCD /m c,b ) subleading contributions are determined by the leading order Isgur-Wise function, reducing the number of free parameters in the form factor fits, and thereby providing sensitivity to O(Λ 2 QCD /m 2 c ) terms. The spread in the uncertainties quoted for theoretical predictions for R(D * ) in the SM are largely due to different estimates of O(Λ 2 QCD /m 2 c ) effects [13][14][15]. The very same hadronic matrix elements are also crucial to resolve tensions between inclusive and exclusive determinations of |V cb | [13][14][15][16][17][18][19][20][21]. The abundant sample of Λ b baryons produced at the LHC may therefore provide a complementary and theoretically cleaner laboratory to study the behavior of the heavy quark expansion, identify possible NP effects, and extract |V cb |.
In this paper, we expand and generalize the study of Ref. [1] beyond the SM, to include all b → cτν four-Fermi operators, including those containing right-handed (sterile) neutrinos.
We compute the relevant form factors including O(Λ 2 QCD /m 2 c ) terms, and compare the fit results of Ref. [1] to the lattice QCD determinations of not only the three vector and three axial vector SM form factors, but also the four NP tensor current form factors. We further emphasize the importance of measuring at LHCb the double differential rate d 2 Γ(Λ b → Λ c ν)/(dq 2 d cos θ) in addition to the q 2 spectrum, and also explore tests of factorization in Λ b → Λ c π decay.
In full QCD, the form factors of the SM currents were instead traditionally defined as [27], Our notation for the form factors follows Ref. [28]; the notation of Ref. [27] corresponds to an exchange of upper and lowercase symbols, F i ↔ f i and G i ↔ g i , in Eqs. (4) and (6). The relations between the form factors in Eqs. (4) and (6) are given in the Appendix A.

B. Form factors in HQET
The ground state baryons are singlets of heavy quark spin symmetry, because the light degrees of freedom, the "brown muck", are in the spin-0 state. Hence, the baryon masses can be written as where the ellipsis denote terms suppressed by more powers of Λ QCD /m Q . The parameter Λ Λ is the energy of the light degrees of freedom in the m Q → ∞ limit. The λ Λ 1 parameter is related to the heavy quark kinetic energy in the Λ baryon. We use m Λ b = 5.620 GeV, m Λc = 2.286 GeV [29], and employ the 1S short distance mass scheme [30][31][32] to eliminate the leading renormalon ambiguities in the definition of the quark masses andΛ Λ . Details of the 1S scheme treatment can be found in Ref. [13]. In particular, we treat m 1S b = (4.71 ± 0.05) GeV and δm bc = m b − m c = (3.40 ± 0.02) GeV as independent parameters [33].
For the expansions of the form factors parametrizing the BSM currents, we obtain, Similar to f 3 and g 3 , neither of the h 3 and h 4 form factors receive Λ 2 QCD /m 2 c corrections. The structure of h 1,2,3 is similar to g 1,2,3 , while h 4 is non-zero only at O(α s Λ QCD /m c,b ).

C. Differential decay rates and forward-backward asymmetry
In Appendix B, we collect explicit expressions for the Λ b → Λ c ν amplitudes for all NP operators, including contributions from massless right-handed sterile neutrinos [45,46].
Including the charged lepton mass dependence, and defining θ as the angle between the lepton and the Λ c momentum in the dilepton rest frame, 1 the SM double differential decay rate is where and The double differential rate in Eq. (14) can be at most a degree-two polynomial in cos θ, and it was written in Eq. (14) in the Legendre polynomial basis, so that only the zeroth order term in the first line contributes to the dΓ/dq 2 , after integration over d cos θ.
The single differential rate in the SM is correspondingly and the forward-backward asymmetry is given by Our result in Eq. (17) agrees with those in Refs. [3,47]. Including all possible NP current operators and a nonzero charged lepton mass, our result for dΓ/dw as derived from Appendix B agrees with the result for SM neutrinos in Eq. (2.51) of Ref. [48]. We see from Eqs. (14) or (18)  To gain more information than obtainable from Eq. (14), the distribution of the Λ c decay products would have to be studied. Such an analysis would be simplest for two-body decays, such as Λ c → Λ(pπ − )π + [7]. This channel loses an order of magnitude in statistics compared to the commonly used Λ c → pKπ reconstruction, however, a model independent description of this three-body decay amplitude is not currently available. With much higher statistics and using Λ c → Λπ + , the measurement of all Λ b → Λ c form factors would be similar to that for Λ c → Λeν [50][51][52], requiring measuring distributions in three angles (as If NP only modifies the (axial)vector interactions (see e.g. Refs. [7,9,53] for other cases), which may be the most plausible scenario, then Eqs. (14) - (18) are simply modified via the and, in particular, In the m l = 0 limit, i.e., in the Λ c µν and Λ c eν modes, the forward-backward asymmetry only receives further contributions from tensor-(pseudo)scalar interference, even in the presence of arbitrary NP. The relation in Eq. (20) is then valid in the light lepton modes, as long as NP does not simultaneously generate (pseudo)scalar and tensor operators.
. The red band shows our fit of the HQET predictions to these data [2] and to the LQCD form factors [3]. The blue curve shows the fit results, setting the order Λ 2 QCD /m 2 c terms to zero. The gray band shows the LQCD prediction. Right: Our prediction for dΓ(Λ b → Λ c τν)/dq 2 normalized to R(Λ c ) from the same fit, with and without including the Λ 2 QCD /m 2 c terms.

A. SM form factor fits
The methods used to fit dΓ(Λ b → Λ c µν)/dq 2 measured by LHCb [2] and lattice QCD (LQCD) calculation of the (axial)vector form factors [3] were described in Ref. [1], and are only briefly recapitulated here. LHCb measured the q 2 spectrum in 7 bins, normalized to unity [2], reducing the effective degrees of freedom in the spectrum from 7 to 6. This measurement is shown as the data points in the left plot in Fig. 1. Our fits to the LHCb data use the measured and predicted partial rates in each bin. This procedure differs slightly from the fits performed by LHCb [2], which used the square root of dN corr /dw evaluated at the midpoint in the seven unfolded w bins. The right plot in Fig. 1 shows our prediction for The lattice QCD results [3] for the six (axial)vector form factors are published as fits to the BCL parametrization [54], using either 11 or 17 parameters. We derive predictions for f 1,2,3 and g 1,2,3 using the 17 parameter result at three q 2 values, q 2 = 1 GeV 2 , q 2 max /2, q 2 max − 1 GeV 2 for a total of eighteen form factor values, constructing a covariance matrix from their correlation structure. The values of q 2 are chosen to sample both ends and the middle of the  slightly differs from the prescription in Ref. [3], which used the maximal differences of the form factor values between the two parametrizations, and cannot preserve the correlation structure between the form factor values. The 18 form factor values used in our fits are shown as data points in Fig. 2. The LQCD predictions, following the prescription of Ref. [3], are shown as gray bands. The uncertainties are in good agreement. Similarly, the gray band in Fig. 1 (left plot) shows the LQCD prediction for the normalized spectrum, using the BCL parametrization.
In our fits, m 1S b and δm bc are constrained using Gaussian uncertainties. The leading order Isgur-Wise function is fitted to quadratic order in w − 1 Alternative expansions using the conformal parameters z or z * [47,[54][55][56] instead of w yield nearly identical fits. Therefore, we do not explore the differences in the unitarity bounds between meson and baryon form factors [57]. Fits with ζ linear in either w, z, or z * are poor, while adding more q 2 values to our sampling indicates no preference for the inclusion of higher order terms in w −1. In the fitsb 1,2 are assumed to be constants, which is appropriate at the current level of sensitivity. With better experimental and lattice constraints in the future, the sensitivity to lifting these assumptions should be tested.
Fit results combining the LHCb and LQCD results are shown in Table I, and in Fig. 2 by red bands. To test the importance of the Λ 2 QCD /m 2 c terms, we also perform a fit with the order Λ 2 QCD /m 2 c terms, parametrized byb 1,2 , set to zero. These fits are shown in Fig. 2 as blue bands, and the corresponding fit values are provided in Table I. This is a much poorer fit, changing χ 2 /ndf from 7.2/20 to 18.8/22. We do not include explicitly an uncertainty for neglected higher order terms in Eqs. (12) and (13). Four form factors, f 3 , g 3 , h 3 , and h 4 receive no Λ 2 QCD /m 2 c corrections, so the agreement of f 3 and g 3 with the LQCD results in the plots in the bottom row in Fig. 2 indicates that these higher order corrections are probably small. The order ε c ε b corrections to f 3 and g 3 are given by two new functions of w, b 5 and b 6 [27], while the ε 3 c corrections to f 3 and g 3 also vanish. Thus, including such corrections, b 5 and b 6 would simply accommodate the 0.5 σ − 1 σ differences between the LQCD results and our fit for f 3 and g 3 . The impact of this is small, for example, setting f 3 = 0 does not perceptibly change the SM prediction for R(Λ c ) compared to Eq. (1), while setting g 3 = 0 changes the SM prediction from R(Λ c ) = 0.324 ± 0.004 in Eq. 1 by about 1σ, to 0.320 ± 0.003. In Fig. 3 show our fit results for ratios of form factors (red bands) and the LQCD predictions (gray bands). The top plot shows f 1 /g 1 , which HQET predicts to be O(1), whereas the four ratios f 2 /f 1 and g 2 /g 1 (second row) and f 3 /f 1 and g 3 /g 1 (third row) are predicted to be O(ε c,b , α s ). The ratio, f 1 /g 1 (= f ⊥ /g ⊥ ), is determined by Eq. (12) as so the enhancement of f 1 relative to g 1 is a model independent prediction of HQET, as seen in the top plot in Fig. 3.

B. Tensor form factors
LQCD results [48] for the tensor form factors are available, and may be compared to HQET predictions from our fits to the (axial)vector form factors, via Eqs. (13). 2 The correspondence between the four form factors used in this paper for the tensor current, h 1 , h 2 , h 3 , h 4 , defined in Eq. (5), and those used in the LQCD calculation [48], h + , h ⊥ , h + , h ⊥ , are given in Appendix A. In the former basis, only one form factor, h 1 , is nonzero in the heavy quark limit, while the four form factors of the LQCD basis are equal to one another in this limit. Note in particular that h 1 = h + .   Fig. 7 in Appendix A. In this basis the uncertainties are not strongly q 2 dependent.) Unlike the fits in Sec. III A, the LQCD results for the tensor form factors are not an input to our fits, so there is no free parameter in these comparisons. Figure 7 shows that the order ε c terms, which are fully determined by HQET in Eq. (13), combined with the definitions in Eq. (A6), account for the near equality of h ⊥ and h + , the slight enhancement of h ⊥ , and the substantial enhancement of h + . The top left plot in Fig. 4 shows a tension between our fit and the LQCD determination of h 1 = h + , visible in all plots in Fig. 7. In addition, the LQCD result for h 1 prefers a slightly smaller curvature than our prediction.
This is similar to what is seen for f 1 and g 1 in the top row of Fig. 2: The LQCD results prefer a smaller curvature at small q 2 . This is related to the observation that LQCD rate in Fig. 1 falls more quickly at small q 2 than the LHCb measurement.

C. R(Λ c ) predictions with new physics
LHCb expects that the precision of the measurement of R(Λ c ) can compete with that of R(D ( * ) ) in the future [60]. For the SM prediction we obtained [1] R(Λ c ) = 0.324 ± 0.004 .
Our form factor fit, combined with the expressions for the NP rates in Appendix B and the HQET predictions in Eqs. (13), allows for precision computation of R(Λ c ) for arbitrary NP contributions (see e.g. Refs [7,9,53]  In Fig. 6 we compare the variation in R(Λ c )/R(Λ c ) SM with the corresponding ratios for D ( * ) , as a function of each NP coupling, assuming they are real. An error band, corresponding to the uncertainties in the fit of Ref. [1], is also shown. In some cases the errors are imperceptible. We see that the NP sensitivity of R(Λ c ) is typically between the R(D * ) and R(D) variations.

IV. FACTORIZATION AND Λ
The LHCb measurement of the dΓ(Λ b → Λ + c µ −ν )/dq 2 spectrum [2] is normalized to unity, and the LCQD results for the Λ b → Λ c form factors are also independent of |V cb |. Thus, our fit is sensitive to hadronic parameters, but it cannot be combined with the present LHCb data to extract |V cb |. One may, however, use the LHCb measurement of dΓ(Λ b → Λ + c µ −ν )/dq 2 to test factorization in Λ b → Λ c π, or to extract |V cb | assuming factorization (see also Ref. [61]).
For B → D ( * ) π decays, it has long been known that the ratios B(B − → D 0 π − )/B(B 0 → D + π − ) 1.9 and B(B − → D * 0 π − )/B(B 0 → D * + π − ) 1.8 [29] deviate substantially from unity, the prediction in the heavy quark limit. This implies that O(Λ QCD /m c ) contributions to the amplitudes enter at the 30% level, and deviations from factorization in the heavy quark limit are substantial.
At leading order in the heavy quark expansion, the Λ b → Λ c π matrix element factorizes such that the nonleptonic rate is related to the semileptonic rate at q 2 = m 2 π via where f π = 131 MeV is the pion decay constant, and C 1,2 are the usual Wilson coefficients in the effective Hamiltonian, satisfying (C 1 + C 2 /3) |V ud | 1. (Uncertainties in this linear combination, f π , and τ Λ b are neglected.) In Eq. (24), we write the Λ c eν final state to emphasize that the semileptonic rate has to be evaluated neglecting lepton masses. In Λ b → Λ c µν decay, measured by LHCb, the impact of m µ = 0 is substantial at q 2 = m 2 π . Combining the factorization relation in Eq. (24), our fit for the form factors, and |V cb | = 4.22 ± 0.08 × 10 −2 [29] predicts B(Λ b → Λ c π) = (3.6 ± 0.3) × 10 −3 , where this uncertainty is from the fit and |V cb |. By comparison, the measured nonleptonic branching ratio [29] is 3 Conversely, assuming factorization, one could use Eqs. (25) in Eq. (24) to extract |V cb | =

V. CONCLUSIONS
Fitting the LHCb measurement of the normalized q 2 spectrum for Λ b → Λ c µν decay [2], and the six (axial)vector form factors calculated in lattice QCD [3], one can test HQET relations and the applicability of power counting. In Ref. [1] we found that the Λ 2 QCD /m 2 . We observed some tension between our results based on HQET and those in Ref. [48], at a magnitude greater than the Λ 2 QCD /m 2 c corrections (see the top left figure for the h 1 = h + form factor in Fig. 4).
The small uncertainties in our fit to the (axial)vector form factors, combined with HQET predictions for the form factors at O(Λ 2 QCD /m 2 c ) allowed us to derive precise predictions for R(Λ c ) for arbitrary NP. We studied the NP impacts on R(Λ c ), including their correlations with R(D ( * ) ). The NP sensitivity of R(Λ c ) typically falls between those of R(D * ) and R(D). We also explored tests of factorization in Λ b → Λ c π decay. Factorization in the heavy quark limit, combined with |V cb | measurements and our fit to the semileptonic form factors, implies a mildly lower nonleptonic rate than is measured, consistent with corrections to the factorization relations arising at O(Λ QCD /m c ). 4 Regarding the behavior of the heavy quark expansion, the decay constants also satisfy the HQET scaling better than was thought in the 1990s. The N f = 2 + 1 + 1 FLAG [68] averages, f B = (186 ± 4) MeV and f D = (212 ± 1.5) MeV, yield f B /f D 0.88, which is not inconsistent with the leading order HQET relation [69,70] LHCb measurements of the double differential rate d 2 Γ(Λ b → Λ c ν)/(dq 2 d cos θ), in addition to the q 2 spectrum, will provide the most differential information measurable in the massless lepton channels (µ and e), if the details of the Λ c decay are ignored. Besides the q 2 spectrum and the (q 2 dependent) forward-backward asymmetry, this double differential distribution involves a third function of q 2 , which can help constrain form factors and test heavy quark symmetry. If the absolute normalization and the double differential rate of semileptonic Λ b → Λ c decays can be measured, it will provide a fully complementary path to extract |V cb |, explore the b → cτ ν anomalies, and test HQET. We look forward to these developments.
follow the definitions in Ref. [72], Finally, the translation between the h 1,2,3,4 tensor form factors used in this paper, defined in Eq. (5), and those defined in Eq. (2.14) in Ref. [48] are and in the opposite direction, In the heavy quark limit, the tensor form factors calculated in LQCD and shown in Fig. 2 of Ref. [48] satisfy h

Appendix B: Amplitudes
In this appendix we collect explicit expressions for the Λ b → Λ c ν amplitudes, including mass terms and right-handed sterile neutrino contributions. These amplitudes correspond to those used in the Hammer code [71].
As in Ref. [73], we write explicit expressions for theb →c amplitudes rather than b → c, defining the basis of NP operators to be The lower index of β denotes the ν chirality and the lower index if α is that of the c quark. Operators for the CP conjugate b → c processes follow by Hermitian conjugation.
(The correspondence between the α, β coefficients and the basis typically chosen for b → c operators can be found in Ref. [25].) The Λ b → Λ c ν process has four external spins: s b = ±, s c = 1, 2, s = 1, 2 and s ν = ±. (We label the Λ c and spin by 1 and 2, to match the conventions of Ref. [73] for massive spinors on internal lines.) Helicity angles and momenta are similarly defined with respect to theb →c process.
Definitions for the conjugate process follow by replacing all particles with their antiparticles.
The single physical polar helicity angle, θ , defines the orientation of the lepton momenta in their center of mass reference frame, with respect to −p Λ b , as shown in Fig. 4 of Ref. [25].
If subsequent Λ c → ΛY decays are included coherently, one further defines φ and φ Λ as twist angles of the -ν and Λ-Y decay planes, with the combination φ − φ Λ becoming a physical phase. Our phase conventions match the spinor conventions of Ref. [73] for not only τ but also Λ c decay amplitudes. This amounts to requiring the inclusion in the τ and/or Λ c decay amplitudes of an additional spinor phase function, h s (s ν ) and h sc (s b ), defined with respect to s ν and s b , such that h 1 (−) = 1 = h 2 (+), h 1 (+) = e iφ and h 2 (−) = e −iφ . Under these conventions, the Λ b → Λ c ν amplitudes themselves are independent of φ − φ Λ .
For compact expression of the amplitudes, it is convenient to define along with The Λ b → Λ c ν amplitudes obey the conjugation relation As bsc s sν w, in which the exchange √ w 2 − 1 → − √ w 2 − 1 implies also w − ↔ w + . One then need only write the s b = − amplitudes, with the s b = + amplitudes following via Eq. (B4). Further r Λ (q 2 − ρ ) × A, the explicit amplitudes are The total differential rate for Λ b → Λ c ν is obtained from these expressions via s b ,sc,s ,sν |A s b ,sc,s ,sν | 2 dw sin θ dθ .