$\Lambda_b \to \Lambda_c^*(2595,2625)\ell^-\bar{\nu}$ form factors from lattice QCD

We present the first lattice-QCD determination of the form factors describing the semileptonic decays $\Lambda_b \to \Lambda_c^*(2595)\ell^-\bar{\nu}$ and $\Lambda_b \to \Lambda_c^*(2625)\ell^-\bar{\nu}$, where the $\Lambda_c^*(2595)$ and $\Lambda_c^*(2625)$ are the lightest charm baryons with $J^P=\frac12^-$ and $J^P=\frac32^-$, respectively. These decay modes provide new opportunities to test lepton flavor universality and also play an important role in global analyses of the strong interactions in $b\to c$ semileptonic decays. We determine the full set of vector, axial vector, and tensor form factors for both decays, but only in a small kinematic region near the zero-recoil point. The lattice calculation uses three different ensembles of gauge-field configurations with $2+1$ flavors of domain-wall fermions, and we perform extrapolations of the form factors to the continuum limit and physical pion mass. We present Standard-Model predictions for the differential decay rates and angular observables. In the kinematic region considered, the differential decay rate for the $\frac12^-$ final state is found to be approximately 2.5 times larger than the rate for the $\frac32^-$ final state. We also test the compatibility of our form-factor results with zero-recoil sum rules.

We normalize the baryon states as In the equations throughout this paper, Minkowski-space gamma matrices and the metric (g µν ) = diag(1, −1, −1, −1) are used, except where indicated otherwise.We introduce the notation and where q = p − p .We use a helicity basis for all form factors.For the J P = 1 2 − final state, our definition follows the one introduced previously for J P = 1 2 + final states [40] except for the changes resulting from the opposite parity [note the γ 5 in Eq. ( 7)]: For the J P = 3 2 − final state, we use the definition introduced by us in Ref. [39], which reads Only the vector and axial-vector form factors are needed to describe Λ b → Λ * c − ν decays in the Standard Model, but we also compute the tensor form factors. Above, σ µν = i 2 (γ µ γ ν − γ ν γ µ ).Note that the overall sign of the form factors for each decay mode depends on the phase conventions of the states.This means that also the relative overall sign between the two different final states is left undetermined.Relations between our form-factor definitions and alternative definitions used in the literature are given in Appendix A.

III. LATTICE ACTIONS AND PARAMETERS
The lattice actions and parameters used in this work are the same as in our calculation of Λ b → Λ * (1520) form factors [39], except that here the valence strange quark is replaced by a valence charm quark.For the latter, we employ the same form of action and analogous tuning conditions as for the bottom quark [41], i.e., an anisotropic clover action with bare parameters am E,B tuned to obtain the correct D s meson kinetic mass, rest mass, and hyperfine splitting (our notation for the bare parameters follows Ref. [42], while Ref. [41] uses . The values of these parameters are given in Table I.The gauge-field ensembles with 2 + 1 flavors of domain-wall fermions were generated by the RBC and UKQCD Collaborations [43,44].For the up and down valence quarks, we reuse the domain-wall propagators computed for Ref. [39].Our computation utilizes all-mode averaging [45,46], in which unbiased estimates with small statistical uncertainties are obtained at reduced cost by combining "exact" and "sloppy" samples.I. Parameters of the lattice actions, lattice spacings, and numbers of exact (ex) and sloppy (sl) samples computed for the correlation functions.The light-quark and gluon actions and the determination of the lattice spacings are described in Refs.[43,44].The form of the heavy-quark action is given in Ref. [41], where m0 = mQ, ζ = ν, cP = cE = cB.

IV. TWO-POINT FUNCTIONS AND HADRON MASSES
We now move to the discussion of the baryon interpolating fields, two-point functions, and results for the masses.For the Λ b , everything is identical to Ref. [39].The Λ * c (2625) has the same isospin and spin-parity quantum numbers as the Λ * (1520) (I = 0, J P = 3 2 − ), but with a charm quark instead of a strange quark.We therefore use the interpolating field which differs from Eq. ( 18) of Ref. [39] only by the replacement s → c.As before, this form will work only at zero momentum.The tilde indicates gauge-covariant Gaussian smearing of the quark fields with the parameters given in Table II.The field (18) actually has nonzero overlap with both the Λ * c (2595) and the Λ * c (2625), and we can isolate the J = 1 2 and J = 3 2 components 1 using the projectors The zero-momentum Λ * c two-point functions are defined like those for the Λ * in Ref. [39], and after applying the above projectors their spectral decomposition reads At this point the reader may wonder why we did not analyze the Λ * (1405) with J P = 1 2 − in Ref. [39], despite being able to project to J P = 1 2 − with the available data.The reason is that we do not trust the single-hadron/narrow-width approximation for the Λ * (1405), which has a larger decay width than the Λ * (1520) and likely a two-pole structure [48].
The masses extracted from single-exponential fits to our results for P jl (1/2) C (2,Λ * c ) and P jl (3/2) C (2,Λ * c ) in the plateau regions are given in Table III, along with the masses of potential decay products.The latter are not used in our determination of the form factors but are included to assess whether the Λ * c baryons are stable under the strong interactions for our quark masses.We find that both m Λ * c,1/2 and m Λ * c,3/2 are lower than all of the following: becomes consistent with zero for the F004 ensemble within the statistical uncertainties.The results are of course affected by the finite volume to some degree, but it appears likely that both the Λ * c,1/2 and the Λ * c,3/2 are stable hadrons at least on the C01 and C005 ensembles, where the energies are well below all thresholds.
1 At zero momentum, the continuum J P = 1 2 − and J P = 3 2 − irreducible representations subduce identically to the G u 1 and H u irreducible representations of the double-cover of the cubic group [47]; the next-higher values of J P that subduce to the same cubic irreps are Hadron masses in GeV.We did not compute Σc two-point functions in this work and the Σc masses were estimated by adding the Σc − Λc mass differences computed in Ref. [42] on the same ensembles with a slightly different tuning of the charm-quark action to the Λc masses computed here.
We also performed simple chiral-continuum extrapolations of m Λ * c,1/2 and m Λ * c,3/2 of the form with fit parameters m , c J , d J , and constants f π = 132 MeV, Λ = 300 MeV.These fits yield m MeV, m = 2742(43) MeV.To estimate systematic uncertainties associated with the choice of fit model, we additionally performed higher-order fits of the form with Gaussian priors h J,HO = 0 ± 10 and g J,HO = 0 ± 10, and computed the systematic uncertainties using where m, σ m denote the central value and uncertainty obtained using the parameter values and covariance matrix of the nominal fit and m HO , σ 2 m,HO denote the central value and uncertainty obtained using the parameter values and covariance matrix of the higher-order fit.In this way we finally obtain which are consistent with the experimental values of m Λ * c,1/2 = 2592.25 (28) MeV, m Λ * c,3/2 = 2628.11(19) MeV [24].Plots of the extrapolations are shown in Fig. 1.Note that we do not use the chiral-continuum extrapolations of the baryon masses in our determination of the form factors; we use the lattice baryon masses when computing the form factors on each ensemble, and then extrapolate only the form factors themselves.The mass extrapolations merely provide a test of our methodology.Finally, in Table III [24] are also shown.

V. THREE-POINT FUNCTIONS AND FORM FACTORS
As in Ref. [39], we compute forward and backward three-point functions where p is the Λ b momentum, Γ is the Dirac matrix in the b → c weak current, t is the source-sink separation, and t is the current-insertion time.With both the b and c quarks implemented using anisotropic clover actions, the current now includes O(a)-improvement terms for both quarks: Here, γ E = (γ j E ) = (−iγ j ) are the Euclidean spatial gamma matrices, and − → ∇ are the gauge-covariant symmetric lattice derivatives.The overall matching factors in the current are written as ρ Γ Z (cc) V Z (bb) V [49,50], where Z (QQ) V are the matching factors for the flavor-conserving temporal vector currents Qγ 0 Q.We determined the values of Z (QQ) V nonperturbatively using the charge-conservation condition for three-point functions with D s and B s meson interpolating fields; the results are given in Table IV.With this choice, the residual matching factors ρ Γ are equal to 1 at tree level and can be computed in perturbation theory without introducing large uncertainties.For the vector and axial-vector currents, we use the one-loop results given in Table III of Ref. [5].Here we use more accurately tuned parameters in the b-and c-quark actions, but we expect the resulting change in the matching factors to be negligible.For the tensor currents, one-loop results are not presently available so we set ρ σµν = 1 and estimate the resulting systematic uncertainty at µ = m b to be 4.04% as in Ref. [10].The values of the O(a)-improvement coefficients for all currents are also computed at tree level and are given in Table IV.
We generated data for the same two choices of Λ b momenta as in Ref. [39], p = (0, 0, 2) 2π L and p = (0, 0, 3) 2π L , and for slightly larger source-sink separations: t/a = 6...14 at the coarse lattice spacing and t/a = 8...16 at the fine Coarse lattice (C01, C005) 9.0631(84) 1.35761 (16)   , determined using charge-conservation from ratios of zero-momentum Bs and Ds two-point and three-point functions, as well as the values of the O(a)-improvement coefficients, computed at tree level in mean-field-improved perturbation theory.
lattice spacing.Here we project the Λ * c field in the three-point functions to both J = 1 2 and J = 3 2 , and the spectral decompositions read where contain the form factors as explained in Sec.II.In the following, we introduce a label X ∈ {V, A, T V, T A} denoting the type of weak current, such that the matrix Γ in Eq. ( 32) is equal to We also introduce a label λ ∈ {0, +, ⊥, ⊥ } for the different helicities.As in Ref. [39], we compute the quantities where (p, t, t/2) are linear projections of the three-point functions proportional to the form factor with current X and helicity λ.In this way, the relative signs of the form factors are preserved, and F (J P )X λ (p, t) becomes equal to the form factor of interest at large t, which is then extracted from a constant fit.The choice of reference form factor (X ref , λ ref ) is arbitrary in principle, and we select it based on the signal-to-noise ratio and quality of the ground-state plateau.
The equations for J P = 3 2 − were given in Ref. [39] and we do not repeat them here.For − )X λ (p, t) is similar to that used previously for J P = 1 2 + in Refs.[5,51].We define where for any four-vector n, and e j denotes the three-dimensional unit vector in direction j.Above, repeated Greek indices are summed over from 0 to 3, while Latin indices are summed only over the spatial directions.The ratios − )X λ (p, t, t ) are equal to kinematic factors depending on the baryon energies times the squares of individual helicity form factors, up to excited-state contamination that decays exponentially for t and t − t both large.We then set t = t/2 [or average over (t + a)/2 and (t − a)/2 in the case of odd t/a] and divide out the kinematic factors to obtain R The linear projections of the three-point functions are constructed using where with the polarization vectors To improve the signals, we use the average of the forward three-point function and the Dirac adjoint of the backward three-point function instead of just C (3,fw) .We then divide out appropriate kinematic factors to obtain such that the unwanted factors of (t−t ) e −EΛ b t cancel in Eq. ( 36) at large t.
− )X λ (p, t) and our constant fits thereof are shown in Fig. 2. For J P = 3 2 − , we use X ref = V , λ ref =⊥ as in Ref. [39].Sample results for F (p, t) and our constant fits thereof are shown in Fig. 3.The values of the form factors obtained from the constant fits are listed in Tables V and VI.The fits were done individually for each form factor and take into account the correlations between the data at different t.The values of χ 2 /d.o.f.range between approximately 0.5 and 1.0, where typically d.o.f.= 4.The correlations between the results for different form factors and different momenta on a given ensemble were evaluated using bootstrap resampling.t/a 0.0 0.5 , defined in Eq. ( 36), as a function of the source-sink separation, for p = (0, 0, 2) 2π L and for the F004 ensemble.Also shown is R , defined in Eq. ( 36), as a function of the source-sink separation, for p = (0, 0, 2) 2π L and for the F004 ensemble.Also shown is R

VI. CHIRAL AND CONTINUUM EXTRAPOLATIONS OF THE FORM FACTORS
As in Ref. [39], we extrapolate the lattice results for the form factors to the continuum limit and the physical pion mass using the model with fit parameters for each form factor f , and using the kinematic variable where depending on the final state considered.In the physical limit m π = m π,phys , a = 0, the functions reduce to This parametrization corresponds to a Taylor expansion of the shape of the form factors around the endpoint w = 1, i.e. an expansion in powers of (w − 1); because we have lattice results for only two different kinematic points near w = 1.01 and w = 1.03, we work only to first order, and we expect the parametrization to become unreliable for large (w − 1).Our results for F f and A f from fits using Eq. ( 67) are given in the first two columns of Table VII, and the values and full covariance matrices (evaluated using bootstrap) are also provided as supplemental files.As can be seen in Figs. 4, 5, 6, and 7, the lattice data are well described by the model.The fits of the individual form factors have χ 2 /d.o.f. in the range from approximately 0.5 to 1.5, where we count F f , A f , C f , and D f as parameters that are primarily constrained by the data, such that d.o.f.= 6 − 4 = 2. Again following Ref.[39], to estimate systematic uncertainties associated with the chiral and continuum extrapolations, we also performed "higher-order" fits including additional terms describing the dependence on the lattice spacing and pion mass, No priors were used for the parameters F f , A f , F f HO , A f HO .The Gaussian priors for the parameters describing the lattice-spacing and pion-mass dependence were chosen as in Ref. [39] except for E f HO and E f HO .These coefficients describe the effects of the incomplete O(a) improvement of the weak currents in Eq. (32), and here we take the prior widths for E f HO and E f HO to be two times larger than in Ref. [39], based on the observation in Ref. [5] that these effects may be larger for a heavy-to-heavy current than for a heavy-to-light current.These widths allow for missing O(a) corrections as large as 10% at the coarse lattice spacing, motivated by the large b-quark momenta used here.In the higher-order fits, we also multiplied the data for each form factor with Gaussian random distributions of central value 1 and appropriate widths to incorporate estimates of systematic uncertainties associated with the residual matching factors ρ Γ (2% for the vector and axial vector currents, 4.04% for the tensor currents [10]) and systematic uncertainties associated with neglecting the down-up quark-mass difference and QED corrections [O((m d − m u )/Λ) ≈ 0.8% and O(α e.m. ) ≈ 0.7%].Furthermore, to include the scale-setting uncertainty, we also promoted the lattice spacings to fit parameters with Gaussian priors according to the values and uncertainties shown in Table I.All of our lattice calculations were performed with m π L > 4, and we therefore expect finite-volume effects to be negligible at least for the heavier pion masses where the Λ * c (2595) and Λ * c (2625) are well below strong-decay thresholds.However, we are unable to provide a quantitative estimate of finite-volume effects in the extrapolated form factors.
In the physical limit, the higher-order fits reduce to the same form as in Eq. ( 69) but with parameters F f HO and A f HO .Our results for these parameters are given in the last two columns in Table VII and also in supplemental files.For any observable O, we evaluate the form-factor systematic uncertainty using 2625) form factors at the physical pion mass and in the continuum limit.The nominal parameters F f and A f are used to evaluate the central values and statistical uncertainties, while the "higher-order" parameters F f HO and A f HO are used in combination with the nominal parameters to evaluate systematic uncertainties as explained in the main text.Files containing the parameter values and the covariance matrices are provided as supplemental material.
where O, σ O denote the central value and uncertainty calculated using {F f , A f } and their covariance matrix, and O HO , σ 2 O,HO denote the central value and uncertainty calculated using {F f HO , A f HO } and their covariance matrix.We find that the (vector and axial-vector) form-factor systematic uncertainties result in an approximately 12

VII. COMPARISON WITH ZERO-RECOIL SUM RULES
At zero recoil (w = 1), approximate sum-rule bounds on the size of heavy-to-heavy form factors can be derived using the operator product expansion and heavy-quark effective theory [25,26,[52][53][54][55].In Ref. [25], it was found that the lattice results for the Λ b → Λ c form factors with the J P = 1 2 + final state (which constitute the "elastic" contribution to the sum rule) almost completely saturate the bounds derived through order 1/m 3 , apparently leaving very little room for "inelastic" contributions from other final states such as the Λ * c 's considered here.However, in the case of B-meson decays, the size of 1/m 4 and 1/m 5 corrections has been found to be approximately 33% of the size of the 1/m 2 and 1/m 3 terms [26,55].Allowing for effects of this size also for Λ b decays, the authors of Ref. [26] then obtained estimates of the size of the inelastic contributions, which are expected to be dominated by Λ b → Λ * c (2595) and Λ b → Λ * c (2625).When expressed in terms of our form-factor definitions using the relations given in Appendix A 3, Eqs.( 46), ( 48), (50), and (52) of Ref. [26] become G inel,1/2 = 1 3 g G inel,3/2 = 1 18 The zero-recoil sum-rule estimate obtained in Ref. [26] is G inel,1/2 + G inel,3/2 ≈ 0.040 +0.049 −0.052 .
(79) Thus, our result for the axial current falls within the range given in Ref. [26], while our result for the vector current is slightly above the upper limit.
where θ is the helicity angle of the charged lepton and A (J) , B (J) , C (J) are functions of q 2 only [26].The J = 1 2 , 3 2 superscript is used to distinguish the Λ * c (2595) and Λ * c (2625) final states.The equations for A (J) , B (J) , and C (J) in terms of the form factors are given in Ref. [26] (where A (J) = Γ ( ) 0 a (J) etc.) and can be converted to our conventions using the relations in Appendix A 3. The integral over cos θ yields the q 2 -differential decay rate dΓ (J) dq 2 = 2A (J) + 2 3 and we also consider two angular observables [26]: the forward-backward asymmetry  The Standard-Model predictions for dΓ (J) /dq 2 /|V cb | 2 and for the angular observables using our form-factor results are shown in Fig. 8.Note that at leading order in heavy-quark effective theory, the differential decay rate for the J = 1 2 final state would be a factor 2 smaller than the differential rate for J = 3 2 , and the lepton-side angular observables

.
t), which is used to extract the square of the reference form factor f The horizontal lines indicate the ranges and extracted values of constant fits.

FIG. 4 .
FIG. 4. Chiral and continuum extrapolations of the Λ b → Λ * c (2595) vector and axial vector form factors.The solid magenta curves show the form factors in the physical limit a = 0, mπ = 135 MeV, with inner light magenta bands indicating the statistical uncertainties and outer dark magenta bands indicating the total uncertainties.The dashed-dotted, dashed, and dotted curves show the fit functions evaluated at the lattice spacings and pion masses of the individual data sets C01, C005, and F004, respectively, with uncertainty bands omitted for clarity.

q 2 [GeV 2 ]FIG. 8 .F
FIG. 8. Λ b → Λ * c (2595) − ν (left) and Λ b → Λ * c (2625) − ν (right) observables in the high-q 2 region calculated in the Standard Model using our form-factor results.From top to bottom: the differential decay rate divided by |V cb | 2 , the forward-backward asymmetry, and the flat term.In each case, we show results for = µ and = τ (the results for = e would look the same as for = µ in this kinematic region).The bands indicate the total (statistical + systematic) uncertainties.

TABLE II .
[39]mters of the quark-field smearing used in the baryon interpolating fields.See Ref.[39]for explanations.
(34)lso list the hyperfine splittings m Λ * c,3/2 − m Λ * c,1/2 computed on each ensemble.Their relative uncertainties are too large to obtain a useful chiral-continuum extrapolation, but the results are consistent within < 2σ with the experimental value of 35.86(34)MeV on each ensemble.FIG. 1. Chiral and continuum extrapolations of our results for the Λ * c,1/2 and Λ * c,3/2 masses.The inner error bands are statistical only and the outer bands include estimates of the systematic uncertainties associated with these extrapolations.The experimental values from Ref.

TABLE IV .
The values of the nonperturbative matching factors Z

TABLE V .
Values of the Λ b → Λ * c,1/2 form factors for each ensemble and for the two different Λ b momenta.
t), which is used to extract the square of the reference form factor f

TABLE VI .
Values of the Λ b → Λ * c,3/2 form factors for each ensemble and for the two different Λ b momenta.