$\Lambda_c \to \Lambda^*(1520)$ form factors from lattice QCD and improved analysis of the $\Lambda_b \to \Lambda^*(1520)$ and $\Lambda_b \to \Lambda_c^*(2595,2625)$ form factors

We present the first lattice-QCD calculation of the form factors governing the charm-baryon semileptonic decays $\Lambda_c \to \Lambda^*(1520)\ell^+\nu_\ell$. As in our previous calculation of the $\Lambda_b \to \Lambda^*(1520)$ form factors, we work in the $\Lambda^*(1520)$ rest frame, but here we use four different heavy-baryon momenta instead of just two. Because of the lower mass of the $\Lambda_c$, the moderately-sized momenta used here are sufficient to determine the form factors in the full kinematic range of the semileptonic decay. We also update the analysis of our lattice results for the $\Lambda_b \to \Lambda^*(1520)$ and $\Lambda_b \to \Lambda_c^*(2595,2625)$ form factors by imposing exact relations among the different form factors at zero recoil that follow from rotational symmetry. Imposing these relations ensures the correct behavior of the angular observables near the endpoint.

To avoid mixing with unwanted lighter states on the lattice, we found it necessary to set the spatial momentum of the Λ * (1520) baryon to zero, and determine the q 2 -dependence of the form factors by varying the spatial momentum p of the initial-state Λ Q baryon instead [13]. The four-momentum transfer squared is then equal to q 2 = m 2 In the case Q = b, the large mass of the Λ b has the effect that very large values of p are needed to appreciably move q 2 away from q 2 max = (m Λ Q − m Λ * ) 2 . In Ref. [13], we performed the calculation for the two values p = (0, 0, 2) 2π L and p = (0, 0, 3) 2π L , where 2π L ≈ 0.47 GeV for the spatial lattice size L ≈ 2.7 fm, corresponding to q 2 /q 2 max ≈ 0.986 and q 2 /q 2 max ≈ 0.969, respectively. The situation is much more favorable for Q = c, because q 2 max is much smaller and because the energy E Λc increases more rapidly with p. Here we use the four different values p = (0, 0, 1) 2π L , p = (0, 1, 1) 2π L , p = (1, 1, 1) 2π L , and p = (0, 0, 2) 2π L , and these values are in fact sufficient to determine the shapes of the form factors in the full kinematic range relevant for the semileptonic decays Λ c → Λ * (1520) + ν , using only small extrapolations/interpolations. Consequently we are able to make Standard-Model predictions also for the fully integrated decay rates. These predictions and their implications are presented in an accompanying Letter [40].
We use helicity-based definitions of the 1 2 + → 1 2 − and 1 2 + → 3 2 − form factors [13,14]. It is known that helicity arXiv:2107.13140v2 [hep-lat] 26 Mar 2022 amplitudes, and hence helicity form factors, satisfy certain exact relations at the kinematic endpoint q 2 = q 2 max that follow from rotational symmetry [41,42]. For the 1 2 + → 1 2 + form factors, such relations were found by relating the helicity-based and non-helicity-based ("Weinberg") form factors in Refs. [5,9] and were already incorporated in the parametrizations used to fit the lattice results. When fitting our lattice QCD results for Λ b → Λ * (1520) and Λ b → Λ * c (2595, 2625) in Refs. [13,14], we did not impose any endpoint relations. Since then, we have found such relations (presented in Sec. II) also for the 1 2 + → 1 2 − and 1 2 + → 3 2 − cases by matching the helicity and non-helicity form factors, and they were proven rigorously in Ref. [43]. Our analysis of the Λ c → Λ * (1520) form factors (Sec. III) imposes these endpoint relations at q 2 max , as well as further exact relations at q 2 = 0. Given that the values of angular observables near q 2 max may be affected significantly by any small deviations from these relations, here we also provide updated fits of the lattice-QCD results for Λ b → Λ * (1520) (Sec. IV) and Λ b → Λ * c (2595, 2625) (Sec. V) in which we impose the constraints at q 2 max . We also present the correspondingly updated Standard-Model predictions for As already mentioned, our lattice results for Λ b → Λ * (1520) and Λ b → Λ * c (2595, 2625) are limited to small kinematic regions near q 2 max , so the relations at q 2 = 0 are not applicable. For Λ c → Λ * (1520), however, our lattice results cover nearly the full kinematic range and we impose the endpoint relations at both q 2 = q 2 max and q 2 = 0 in the following.

A. Lattice parameters and extraction of the form factors
Our Λ c → Λ * (1520) lattice calculation closely follows the one for Λ b → Λ * (1520) [13], and uses gauge-field configurations generated by the RBC and UKQCD Collaborations [49,50] with 2 + 1 flavors of domain-wall fermions. For the u, d, s valence quarks, we use the same Shamir domain-wall action with the same N 5 = 16 and aM 5 = 1.8 as used for the u, d, s sea quarks [49,50]. The valence u, d masses are set equal to the sea u, d masses, while the valence s masses are set equal to the physical values as determined with sub-MeV precision in Ref. [50]. There, m π , m K , and m Ω were used to determine the light and strange quark masses and the lattice spacing. The main parameters of the three ensembles and of the quark propagators computed thereon are given in Table I. For the valence charm quark, we use an anisotropic clover action with the mass am B tuned nonperturbatively such that the D s -meson rest mass, kinetic mass, and hyperfine splitting match the experimental values [39]. These observables calculated on each ensemble are found to agree with experiment within 0.4%, 1.0%, and 1.4% (or better) precision, respectively. The c → s currents are renormalized using the mostly nonperturbative method described in Refs. [51,52]. That is, the renormalized currents are written as  I. Parameters of the three data sets used to determine the Λc → Λ * (1520) form factors. The ensemble generation is described in Ref. [49] and the lattice spacings were determined in Ref. [50]. Above, L = Nsa is the spatial lattice size and we provide the values of the momentum unit, 2π/L, for convenience. The parameters am B are the mass, anisotropy parameter, and chromoelectric/chromomagnetic clover coefficients in the anisotropic clover action used for the charm quark [14]. We use all-mode averaging [53,54] with 32 sloppy and 1 exact sample per gauge configuration; Nex and N sl are the total numbers of exact and sloppy samples, respectively.
where Z (ss) V and Z (cc) V are the matching factors of the temporal components of the s → s and c → c vector currents, determined nonperturbatively using charge conservation, ρ Γ are the residual matching factors that are numerically close to 1 and are computed using one-loop lattice perturbation theory, and the term with coefficient d (c) 1 removes O(a) discretization errors at tree level. The values of these parameters are given in Table II. The residual matching factors for the vector and axial-vector currents were computed by C. Lehner at one loop in mean-field-improved lattice perturbation theory, originally for Ref. [10]. The perturbative calculation was performed for a slightly different tuning of the charm-action parameters, and we therefore assign a larger systematic uncertainty to the residual matching factors, as discussed in Sec. III B. Here we also determine the Λ c → Λ * (1520) tensor form factors for completeness, even though they are not needed to describe Λ c → Λ * (1520) + ν in the Standard-Model. One-loop results are not available for the tensor-current residual matching factors and we set them equal to 1 ± 0.05 (this estimate should be viewed as corresponding to a renormalization scale µ = m c ). As in Refs. [13,14], in our estimates of systematic uncertainties we will also account for the incomplete (tree-level only) O(a) improvement of the currents.
The extraction of the form factors from the three-point and two-point correlation functions was performed as in Ref. [13] by computing the quantities where X ∈ {V, A, T V, T A} and λ ∈ {0, +, ⊥, ⊥ }. Above, R is a ratio of products of three-point and two-point functions that becomes equal to f 2 ⊥ at large t, and is illustrated in Fig. 1. The objects S    Table III.

B. Chiral and continuum extrapolations
The final step in determining the physical Λ c → Λ * (1520) form factors is to fit suitable parametrizations describing the dependence on the momentum transfer, the lattice spacing, and the light-quark mass (or, equivalently m 2 π ) to the form factor data points shown in Table III. Because we will impose the constraints discussed in Sec. II B, which relate different form factors, we perform global, fully correlated fits to all form factors: one "nominal" fit, and one "higher-order" fit that will be used to estimate systematic uncertainties.
As in Refs. [13,14], we fit the shapes of the form factors using power series in the dimensionless variable (w − 1), where such that w = 1 corresponds to This expansion is expected to converge for |w − 1| smaller than |w s − 1|, where w s denotes the position of the branch point or pole that is closest to w = 1. Such singularities arise from on-shell intermediate states with four-momentum q produced by thesΓc weak current. The D-K twoparticle branch cut (in infinite volume) starts at q 2 = (m D + m K ) 2 which, for physical hadron masses, corresponds to |w − 1| ≈ 0.72. The exact isospin symmetry in our calculation forbids D s -π intermediate states. The three-particle D s -π-π branch cut starts at at q 2 = (m Ds + 2m π ) 2 corresponding to |w − 1| ≈ 0.64. In addition, single-particle intermediate states result in poles at q 2 equal to the masses of these particles squared. The experimentally observed masses [39] of the lightestcs mesons with the J P quantum numbers occurring in the different form factors are given in Table IV. The lowest-lying single-particle state is the pseudoscalar D s meson (which contributes a pole to the form factor g , which corresponds to |w − 1| ≤ 0.085. Thus, the series is expected to converge in the entire region of interest (using the lattice hadron masses instead of the experimental masses changes the numerical values slightly but does not affect this conclusion).
Because we now have data for four different Λ c momenta, we are able to go beyond the first order in the expansion in (w − 1); we find that second order is sufficient. Furthermore, we choose to factor out the lowest-lying poles from the single-particle states. While this is not necessary for convergence, it may make the convergence slightly more rapid. The pole masses are set to the physical values listed in Table IV. In the nominal fit, each form factor f is parametrized as where the factors describe the dependence on the pion mass and lattice spacing (we set f π = 132 MeV, Λ = 300 MeV). This functional form corresponds to the lowest nontrivial order in an expansion in the light-quark mass m u,d ∝ m 2 π and the lattice spacing (as discussed at the beginning of Sec. III A, the strange and charm quark masses are already tuned accurately to their physical values on each ensemble, requiring no extrapolation). The use of the chirally symmetric domain-wall action for the light and strange quarks, and of a nonperturbatively tuned relativistic-heavy-quark action [55] for the charm quark ensure the absence of O(a) discretization errors, except for the effects of the incomplete improvement of the c → s current. Systematic uncertainties from this incomplete improvement and from neglected higher-order or nonanalytic terms are estimated by varying the fit form, as discussed later in this section.
By construction, L f n = 1 in the physical limit a = 0, m π = m π,phys = 135 MeV, such that only the parameters a f n (along with the constant pole massses) are needed to describe the form factors in that limit. No priors are used for the parameters a f n , while Gaussian priors with central values 0 and widths 10 are used for the coefficients C f n and D f n , following Refs. [13,14]. To ensure that the physical-limit form factors satisfy the endpoint relations of Sec. II B, we eliminate the following a f 0 parameters using Eqs. (6)- (12), and the following a f 2 parameters using using Eqs. (13)- (16), Here, Λ Q = Λ c and Λ * q,3/2 = Λ * (1520), and To report the value of χ 2 /dof of the fit, we need to make a choice for the number of free parameters to be subtracted from the number of data points to obtain the number of degrees of freedom. If we count all parameters as free, the nominal fit has χ 2 /dof ≈ 1.22. However, the results for the coefficients C f 2 and D f 2 are all consistent with zero and their uncertainty is approximately equal to the prior width, suggesting that these parameters have little effect on the quality of the fit and should not be counted. With that choice, we find χ 2 /dof ≈ 0.80. The values of the physical-limit parameters are given in the first three columns of Table V, and the full covariance matrix is available as Supplemental Material [56]. Plots of the fits are shown in Figs. 3 and 4.
In the higher-order fit, the data for each form factor f are fitted with where L f n,HO = 1 + C f n,HO We use Gaussian priors for the parameters C f n,HO , H f n,HO , D f n,HO , G f n,HO with central values equal to 0 and widths equal to 10. The terms with coefficients E f n,HO allow for effects resulting from the incomplete O(a) improvement (done at tree level only) of the heavy-light currents [13]. Because the largest momentum used here is only 2/3 times the one in Ref. [13], we reduce the prior widths of E f n,HO by the same factor, to 0.2 (with central values 0). This allows for the effect of the missing radiative corrections to the O(a) improvement to be as large as 3.3 percent at the coarse lattice spacing, which is substantially larger than observed in Ref. [5], where a numerical comparison between full one-loop and incomplete O(a) improvement was performed. We also incorporate the systematic uncertainties associated with the matching of the heavy-light currents by multiplying each form factor with Gaussian random distributions of central value 1 and width corresponding to the estimated uncertainty. The residual matching factors computed at one loop for the c → s vector and axial-vector currents are very close to their tree-level values of 1 (see Table II). We include a 1% matching uncertainty for the vector and axial-vector form factors, which would allow for two-loop corrections with coefficients of α 2 s that are about six times the size of the one-loop coefficients of α s . This estimate also allows for some small changes in the matching coefficients due to the slightly different tuning of the charm-quark-action parameters. For the tensor currents, the residual matching factors were set equal to their tree-level values because a one-loop calculation was not available. The procedure used in Ref. [13] to estimate the resulting systematic uncertainty would yield an unrealistically small value in the c → s case, and we instead include a 5% uncertainty (this estimate should be viewed as corresponding to a renormalization scale µ = m c ). Also recall that the tensor form factors are not needed to describe Λ c → Λ * (1520) + ν in the Standard-Model. In the higher-order fit, we furthermore include the estimated uncertainty from the missing isospin-breaking/QED corrections, also by multiplying with further Gaussian random distributions of central value 1 and width corresponding to the estimated uncertainty (0.9%), and we include the scale-setting uncertainty by promoting the lattice spacings to fit parameters, constrained to have the known values and uncertainties.
The parameters a f n,HO obtained from the higher-order fit are listed in the last three columns of Table V, and again their full covariance matrix is available as a supplemental file. As in Refs. [5,9,13,14], we evaluate the systematic form-factor uncertainty of any observable O through where O, σ O denote the central value and uncertainty obtained using the parameter values and covariance matrix of the nominal fit and O HO , σ 2 O,HO denote the central value and uncertainty obtained using the parameter values and covariance matrix of the higher-order fit. The systematic and statistical uncertainties are then added in quadrature to obtain the total uncertainties, which are shown as the darker bands in Figs. 3 and 4.
To discuss the uncertainties in a representative observable, we consider the Λ c → Λ * (1520)e + ν e total (i.e., integrated over the full q 2 range) decay rate. Using the nominal fit to compute the central value and statistical uncertainty and the higher-order fit to compute the total systematic uncertainty, we find Γ(Λ c → Λ * (1520)e + ν e ) |V cs | 2 = (0.00267 ± 0.00039 stat. ± 0.00018 syst. ) ps −1 . The relative uncertainties are 14.6% statistical and 6.7% systematic. 1 To assess the breakdown of systematic uncertainties into individual sources, we also performed five additional fits that differ from the nominal fit by including only subsets of the higher-order terms or only selected additional uncertainties: 1. To estimate the uncertainty associated with the continuum extrapolation, we add the terms with the coefficients E f n,HO and G f n,HO . 2. To estimate the uncertainty associated with the chiral extrapolation, we add the terms with the coefficients H f n,HO . 3. To estimate the uncertainty associated with the matching of the heavy-light currents, we multiply each form factor with Gaussian random distributions of central value 1 and width corresponding to the estimated uncertainty (1% for the vector and axial-vector form factors, 5% for the tensor form factors).
4. To estimate the uncertainty associated with the missing isospin-breaking/QED corrections, we multiply each form factor with Gaussian random distributions of central value 1 and width corresponding to the estimated uncertainty (0.9%).

5.
To estimate the uncertainty associated with the scale setting, we promote the lattice spacings to fit parameters, constrained to have the known values and uncertainties.
By comparing the decay rates calculated using each of these fits with that calculated using the nominal fit, we obtain the error budget shown in Table VI. While not quantified explicitly, we expect finite-volume errors to be much smaller than the statistical uncertainties, give that (i) all of our data sets have m π L > 4 and (ii) for a narrow non-S-wave resonance like the Λ * (1520), at zero total momentum the energy level in the finite volume caused by the resonance will be far below all multi-hadron scattering states with the same quantum numbers, and the narrow-width approximation is expected to be very accurate.  Our full Standard-Model predictions for the Λ c → Λ * (1520) + ν differential and integrated decay rates and angular observables, along with a comparison to the quark-model calculation of Ref. [57], are presented in an accompanying Letter [40]. Our new fits to the Λ b → Λ * (1520) lattice-QCD results differ from those in Ref. [13] in the following ways: (i), we perform simultaneous, fully correlated fits with a single χ 2 function to all form factors; (ii), we enforce the seven constraints at q 2 max [Eqs. (6)(7)(8)(9)(10)(11)(12)] by eliminating redundant parameters before the fit using Eqs. (24)- (30); and (iii), we include pole factors in the parametrizations, which now read f HO (q 2 ) = 1 The factors L f n and L f n,HO describe the lattice-spacing and pion-mass dependence and are identical to Ref. [13]; in the physical limit, L f n = L f n,HO = 1. The pole masses used are given in Table VII. We find that including the pole factors in the fits has negligible impact on the values of the form factors in the kinematic region 1 ≤ w ≤ 1.05, but there is no harm in doing so and it could potentially slightly improve the description farther away from this region. Counting a f 0 , a f 1 , and the two parameters in L f 0 as free parameters [13], the nominal fit has χ 2 /dof ≈ 0.75. Tables and plots of the fit results are shown in Appendix A. The uncertainties of some of the form factors are reduced noticeably compared to Ref. [13] as a result of the additional constraints at q 2 max . The impact of these improvements on the Standard-Model predictions for Λ b → Λ * (1520) + − is illustrated for the differential branching fraction in Fig. 5 and for two of the angular observables (defined in Refs. [13,18]) in Fig. 6. Updated plots of additional observables are given in Appendix A. The uncertainties of all the angular observables considered here now vanish at the endpoint q 2 = q 2 max , where these observables take on the exact values S 1c → 0, FIG. 5. The Λ b → Λ * (1520) + − differential branching fraction in the high-q 2 region calculated in the Standard Model. The blue solid curve is obtained using the improved form factor results with the exact endpoint constraints, while the gray dashed curve shows the previous results without these constraints from Ref. [13].
The uncertainties near the endpoint are also reduced substantially, as expected. Our previous predictions in Ref. [13] are mostly consistent with the new results within the (old) uncertainties, with deviations at the 2σ level seen in some angular observables at the endpoint, such as F L and A F B as shown in Fig. 6 67) and (70) of Ref. [14] (the poles in q 2 caused by B c bound states are very far away from the physical region, and we do not include them in our form-factor parametrizations.) In the limit of zero lattice spacing and physical pion mass, these functions reduce to (1) and (2) to eliminate the parameters F f Tables and plots of the fit results are shown in Appendix B. A comparison of the Λ b → Λ * c (2595)µ −ν and Λ b → Λ * c (2625)µ −ν observables computed with and without the endpoint constraints is shown in Fig. 7 (see Ref. [17] for the definitions). We see that imposing the endpoint constraints in the form-factor fits has substantially increased the precision near q 2 max , compared to Ref. [14]. The angular observable F H now become exactly equal to 1 at q 2 = q 2 max , with uncertainty vanishing toward that point. All of our previous predictions are consistent with the new, more precise results. The updated results for the tau-lepton final states are shown in Appendix B. Finally, note that in Ref. [14], we had evaluated the combinations of Λ b → Λ * c (2595) and Λ b → Λ * c (2625) form factors that appear in zero-recoil sum rules [17]. The updated results for these combinations are F inel,1/2 + F inel,3/2 = 0.0942 ± 0.0075 stat ± 0.0081 syst , (57) G inel,1/2 + G inel,3/2 = 0.0162 ± 0.0015 stat ± 0.0019 syst , which are consistent with the previous results and slightly more precise. As before, our result for the axial current falls within the range given in Ref. [17], while our result for the vector current is slightly above the upper limit.    [17], in the high-q 2 region calculated in the Standard Model. From top to bottom: the differential decay rate divided by |V cb | 2 , the forward-backward asymmetry, and the flat term. The blue solid curves are obtained using the improved form factor results with the exact endpoint constraints, while the gray dashed curves show the previous results without these constraints from Ref. [14]. The new results for the decays to tau leptons are given in Appendix B.

VI. CONCLUSIONS
In summary, here we have extended our lattice studies of heavy-baryon semileptonic decays to negative-parity baryons in two ways: (i), we performed the first calculation of the Λ c → Λ * (1520) form factors describing the charm-to-strange decays Λ c → Λ * (1520) + ν , and (ii), we improved our determinations of the Λ b → Λ * (1520) and Λ b → Λ * c (2595, 2625) form factors such that the required relations between different helicity form factors at the kinematic endpoint q 2 = q 2 max are satisfied exactly. In contrast to the Λ b decays, for Λ c → Λ * (1520) it is possible to determine the form factors in the full kinematic range occuring in the semileptonic decays using just moderately-sized initial-baryon momenta-this is a consequence of the much lower mass of the Λ c . This allows us to predict the total Λ c → Λ * (1520) + ν decay rates in the Standard Model with 15.9% uncertainty, of which 14.6% are statistical and 6.7% are systematic. As in our previous study of Λ b → Λ * (1520)µ + µ − , the estimate of systematic uncertainties does not include finite-volume effects / effects associated with the unstable nature of the Λ * (1520). While we believe these effects to be small in our case (for a narrow, non-S-wave resonance and at zero spatial momentum, the energy level caused by the resonance will be far below all scattering states for typical lattice sizes), only a new, more complicated and more expensive calculation using the proper multi-hadron formalism [59][60][61] would be able to fully control this issue. An experimental measurement of the Λ c → Λ * (1520) + ν branching fraction would of course provide a valuable check of our methodology, which is largely shared also with the Λ b -decay calculations.
When performing the combined chiral/continuum/kinematic extrapolations of the Λ c → Λ * (1520) form factors, we have enforced exact relations among the different helicity form factors in the physical limit at the kinematic endpoints q 2 = 0 and q 2 max . These relations ensure that angular observables approach exactly the values predicted by rotational symmetry at the endpoints, and also ensure that transforming the helicity form factors to a non-helicity basis (if desired) does not introduce singularities at the endpoints. In our previous analyses of the Λ b → Λ * (1520) and Λ b → Λ * c (2595, 2625) helicity form factors [13,14], we did not explicitly impose the endpoint relations when fitting the lattice results, resulting in them being satisfied only approximately (this statement refers only to the relations at q 2 = q 2 max , since the results for the Λ b decays are limited to the vicinity of that endpoint). In the present work, we have updated the fits to the Λ b → Λ * (1520) and Λ b → Λ * c (2595, 2625) form factors by imposing the endpoint relations at q 2 = q 2 max , and we have re-calculated the differential decay rates and angular observables of Λ b → Λ * (1520)(→ pK − )µ + µ − and Λ b → Λ * c (2595, 2625) −ν in the Standard Model. The predictions are now more precise, and the angular observables exactly approach the values predicted by rotational symmetry at q 2 = q 2 max . As already mentioned in Ref. [14] and further analyzed in Ref. [31], our lattice results for the Λ b → Λ * c (2595, 2625) form factors imply large higher-order corrections in heavy-quark effective theory near q 2 = q 2 max , in particular for the Λ * c (2595) final state with J P = 1 2 − . Our improved form-factor results are more precise but are consistent with the previous results and therefore do not alter this conclusion. As before, near q 2 max we find the Λ b → Λ * c (2595)µ −ν µ differential decay rate to be significantly larger than the Λ b → Λ * c (2625)µ −ν µ differential decay rate, whereas the total decay rates measured in experiment [62] have the opposite order. We therefore expect the differential decay rates to cross at some value of q 2 lower than covered by our lattice results. Such a crossing is in fact seen in the quark-model predictions of Ref. [44]. Also note that the authors of Ref. [28,29] suggested an exotic structure of the Λ * c (2595), possibly with two resonance poles of which only one is a heavy-quark symmetry partner of the Λ * c (2625). This warrants further investigation.     10. Updated Standard-Model predictions of the Λ b → Λ * (1520)(→ pK − ) + − observables in the high-q 2 region (continued in Fig. 11). See Refs. [13,18] 11. Updated Standard-Model predictions of the Λ b → Λ * (1520)(→ pK − ) + − observables in the high-q 2 region (continuation of Fig. 10). See Refs. [13,18] for the definitions.