Λb → Λ∗(1520)`+`− form factors from lattice QCD

We present the first lattice QCD determination of the Λb → Λ∗(1520) vector, axial vector, and tensor form factors that are relevant for the rare decays Λb → Λ∗(1520)`+`−. The lattice calculation is performed in the Λ∗(1520) rest frame with nonzero Λb momenta, and is limited to the high-q 2 region. An interpolating field with covariant derivatives is used to obtain good overlap with the Λ∗(1520). The analysis treats the Λ∗(1520) as a stable particle, which is expected to be a reasonable approximation for this narrow resonance. A domain-wall action is used for the light and strange quarks, while the b quark is implemented with an anisotropic clover action with coefficients tuned to produce the correct Bs kinetic mass, rest mass, and hyperfine splitting. We use three different ensembles of lattice gauge-field configurations generated by the RBC and UKQCD collaborations, and perform extrapolations of the form factors to the continuum limit and physical pion mass. We give Standard-Model predictions for the Λb → Λ∗(1520)`+`− differential branching fraction and angular observables in the high-q region.

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To make predictions for the Λ b → Λ * (1520)(→ pK − ) + − decay observables in the Standard Model and beyond, the Λ b → Λ * (1520) form factors corresponding to the matrix elements of the b → s vector, axial vector, and tensor currents are required. These form factors have previously been studied in a quark model [30,33]. In the following, we present the first lattice-QCD determination of the Λ b → Λ * (1520) form factors (we reported preliminary results in Ref. [34]). The lattice calculation of 1 2 + → 3 2 − form factors is substantially more challenging than the calculation of 1 2 + → 1 2 + form factors, even when neglecting the strong decay of the 3 2 − baryon in the analysis, as we do here.
Correlation functions for negative-parity baryons have more statistical noise than correlation functions for the lightest positive-parity baryons. Furthermore, at nonzero momenta, the irreducible representations of the lattice symmetry groups mix positive and negative parities and also mix J = 1 2 and J = 3 2 . To avoid having to deal with this mixing, we perform our calculation in the Λ * (1520) rest frame and give the Λ b nonzero momentum (since the Λ b is the ground state, the mixing with other J P values does not cause difficulties in isolating it). This has the effect that our calculation is limited to a relatively small kinematic region near q 2 max . This paper is organized as follows. Our definition of the Λ b → Λ * (1520) form factors is presented in Sec. II. The lattice actions and parameters are given in Sec. III. Section IV explains our choices of the baryon interpolating fields and contains numerical results for the hadron masses. The three-point functions and our method for extracting the individual form factors are described in Sec. V. We perform simple chiral, continuum, and kinematic extrapolations of the form factors as discussed in Sec. VI. We then use the extrapolated form factors to calculate the Λ b → Λ * (1520)µ + µ − differential decay rate and angular observables in the Standard Model, presented in Sec. VII. Conclusions are given in Sec. VIII. Appendix A contains relations between our form factor definition and other definitions that have been used in the literature.

II. DEFINITIONS OF THE FORM FACTORS
The Λ * (1520) is the lightest of the strange baryon resonances with I = 0 and J P = 3 2 − . It has a mass of 1519.5±1.0 MeV, a width of 15.6 ± 1.0 MeV, and decays mainly into NK, Σπ, or Λππ [29]. In this work, we treat the Λ * (1520) as if it is a stable single-particle state. We expect this to be a reasonable approximation, given the relatively small width and given the other sources of uncertainty in our calculation. In the following, we denote the Λ * (1520) as simply Λ * .
We are interested in the matrix elements Λ * (p , s )|sΓb |Λ b (p, s) for Γ ∈ {γ µ , γ µ γ 5 , iσ µν q ν , iσ µν q ν γ 5 } with q = p − p . These matrix elements are described by fourteen independent form factors that are functions of q 2 only. Possible definitions of these form factors were given, for example, in Refs. [17,30,[33][34][35][36]. Here we use a helicity-based definition. We first presented such a definition in Ref. [34]; the choice used here differs from that in Ref. [34] only by a q 2 -dependent rescaling to avoid divergences in the form factors at the endpoint q 2 max = (m Λ b − m Λ * ) 2 . We use the standard relativistic normalization of states, Λ * (k , r )|Λ * (p , s ) = δ r s 2E Λ * (2π) 3 δ 3 (k − p ), (2) and introduce Dirac and Rarita-Schwinger spinors satisfying We introduce the notation and The form factors The requirement that physical matrix elements are non-singular for q 2 → q 2 max = (m Λ b − m Λ * ) 2 imposes certain requirements on the behavior of the form factors in this limit [17]. More information on this behavior can be obtained from heavy-quark effective theory [36] if the strange quark is treated as a heavy quark. For our definition, we expect all form factors to be finite and nonzero at q 2 = q 2 max . Relations between our form factors and other definitions used in the literature are given in Appendix A.

III. LATTICE ACTIONS AND PARAMETERS
Our calculation utilizes three different ensembles of gauge-field configurations generated by the RBC and UKQCD collaborations [37,38]. These ensembles include the effects of 2+1 flavors of sea quarks, implemented with a domainwall action [39][40][41]; the gauge action used is the Iwasaki action [42]. The main parameters of the ensembles and valence-quark actions are listed in Table I; see Table III for the resulting hadron masses. To compute the u, d, and s-quark propagators, we use the same domain-wall action as for the sea-quarks, with valence light-quark masses equal to the sea light-quark masses, and valence strange-quark masses tuned to the physical values, which are slightly lower than the sea strange-quark masses. For the b-quark propagators, we use the anisotropic clover action discussed in Ref. [43], but with parameters am E,B newly tuned by us to obtain the correct B s kinetic mass, rest mass, and hyperfine splitting.  [38]. The bottom-quark is implemented with the action described in Ref. [43], but with parameters am E,B newly tuned by us to obtain the correct Bs kinetic mass, rest mass, and hyperfine splitting. The last two columns give the numbers of exact (ex) and sloppy (sl) samples used for the calculation of the correlation functions with all-mode averaging [44,45].
Our calculation employs all-mode averaging [44,45] to reduce the cost for the light and strange quark propagators. On each gauge-configuration, we computed one exact sample for the relevant correlation functions (discussed in the following sections), as well as 32 "sloppy" samples with reduced conjugate-gradient iteration count in the computation of the light and strange quark propagators. For the light quarks, we also used deflation based on the lowest 400 eigenvectors to reduce the cost and improve the accuracy of the propagators. On a given gauge-field configuration, the different samples correspond to different source locations on a four-dimensional grid, with a randomly chosen overall offset.

IV. TWO-POINT FUNCTIONS AND HADRON MASSES
We now proceed to the discussion of the baryon interpolating fields. Our lattice calculation uses m u = m d and neglects QED, which means that we have exact isospin symmetry, and the Λ b and Λ * (1520) both have I = 0. The continuous space-time symmetries on the other hand are reduced to discrete symmetries by the cubic lattice. At zero momentum, the relevant symmetry group is 2 O, the double cover of the cubic group [46], and we still have the full parity symmetry. At zero momentum, the continuum J P = 1 2 ± and J P = 3 2 ± irreps subduce identically to the G g/u 1 and H g/u irreps; the next-higher values of J that appear in these irreps are J = 7 2 and J = 5 2 , respectively. In this case we can therefore safely construct the interpolating fields for both the Λ b and the Λ * (1520) using continuum symmetries. At nonzero momenta, we no longer have parity symmetry, and the relevant symmetry groups are Little Groups of 2 O [47 -49]. An interpolating field that would have J P = 3 2 − in the continuum then also couples to J P = 3 2 + , and in some cases even J P = 1 2 + (for example, for momentum direction (0, 1, 1), the only irrep containing J = 3 2 also contains J = 1 2 ), which would make isolating the Λ * (1520) extremely difficult. For this reason, we perform the lattice calculation in the Λ * (1520) rest frame, giving nonzero momentum to the Λ b instead. Since the Λ b is the lightest baryon with quark content udb, any contributions from mixing with opposite parity and higher J only appear as excited-state contamination, which will be suppressed exponentially for large Euclidean time separations.
We take the interpolating field for the Λ b in position space to be where q denotes a smeared quark field. We use gauge-covariant Gaussian smearing of the form where and the gauge links U are APE-smeared (in the case of the up, down, and strange quarks) or Stout-smeared (in the case of the bottom quark). The values used for the smearing parameters are given in Table II. We average over  [50] with parameter αAPE is defined as in Eq. (8) of Ref. [51], and we apply NAPE such sweeps. The Stout smearing is defined in Ref. [52].
"forward" and "backward" two-point functions given by The Λ b masses obtained from single-exponential fits in the time region of ground-state dominance are given in the last column of Table III. Even at zero momentum, constructing an interpolating field with a good overlap to the Λ * (1520) proved to be nontrivial. In a first, unsuccessful attempt, we tried the form which can be projected to the H u irrep by contracting the index j (which runs over the spatial directions) with 1 Even though the resulting interpolating field has the correct values for all exactly conserved quantum numbers, it is found to have poor overlap with the Λ * (1520) and much greater overlap with higher-mass J P = 3 2 − states. The effective mass for the two-point function computed with O (old) Λ * on the C005 ensemble is shown with the red circles in Fig. 1, and shows a "false plateau" at higher mass before the signal is swamped by noise. A previous lattice QCD study of Λ * -baryon spectroscopy using interpolating fields similar to Eq. (16) also did not find a Λ * (1520)-like state [53]. The problem is that O (old) Λ * [after projection with P kj (3/2) ] has an internal structure corresponding to total quark spin S = 3/2, total quark orbital angular momentum L = 0, and flavor-SU (3) octet, while quark models suggest that the Λ * (1520) dominantly has an L = 1, S = 1/2, and flavor-SU (3)-singlet structure [54]. To obtain L = 1, a suitable spatial structure of the interpolating field is needed, which can be achieved using covariant derivatives [55]. For the main calculations in this work we use the form which has L = 1, S = 1/2, and is a flavor-SU (3) singlet. The covariant derivatives, which are defined as change the parity, so the projector (1 + γ 0 )/2 is used to obtain negative overall parity. As we did previously for O (old) Λ * , we project the two-point functions Label to the H u irrep with P kj (3/2) . In Eq. (18), we eliminated covariant derivatives acting on the strange-quark fields using "integration by parts," which is possible only at zero momentum. In this way, the calculation requires propagators with derivative sources only for the light quarks. The effective mass for C (2,Λ * ) computed on the C005 ensemble is shown with the green squares in Fig. 1, and shows a plateau at a significantly lower mass, which we identify (in the single-hadron/narrow-width approximation) with the Λ * (1520) resonance. The Λ * (1520) masses obtained from single-exponential fits in the plateau regions for all ensembles are given in the second-to-last column of Table III. We also computed the pion, kaon, nucleon, Lambda, and Sigma two-point functions and obtained the masses given in the same table. For the three ensembles we have, the mass differences m Λ * − m Σ − m π are found to be in the range from approximately 80 to 150 MeV (physical value: 192 MeV), while m Λ * − m N − m K ranges from approximately −20 to +100 MeV (physical value: 89 MeV). These results support our identification of the extracted energy level with the Λ * (1520) in the narrow-width approximation. A proper finite-volume scattering analysis with Lüscher's method [56] is beyond the scope of this work. Here we just note that the lowest noninteracting N -K and Σ-π scattering states in the H u irrep must have nonzero back-to-back momenta and their energies are well above m Λ * for our lattice volumes (this is another benefit of working in the Λ * rest frame).
For later reference, we also define overlap factors of the interpolating fields with the baryon states of interest as and As everywhere in this paper, |Λ * (0, s ) denotes the lowest-energy 3/2 − state. For the Λ b at nonzero momentum, it is necessary to have the two separate coefficients Z Λ b that may also depend on p, because the spatialonly smearing of the quark fields breaks hypercubic symmetry (and because the lattice itself also breaks the Lorentz symmetry). The spectral decomposition of C (2, Λ b ) (p, t) then reads with v µ = p µ /m Λ b , while the spectral decomposition of C (2,Λ * ) (t) after projection with P (3/2) becomes The excited-state contributions decay exponentially faster with t than the ground-state contributions shown here.

V. THREE-POINT FUNCTIONS AND FORM FACTORS
To determine the form factors, we compute forward and backward three-point functions where p is the momentum of the Λ b , Γ is the Dirac matrix in the b → s current J Γ , t is the source-sink separation, and t is the current-insertion time. To match the currents to the continuum MS scheme, we employ the mostly nonperturbative method described in Refs. [57,58]. Specifically, we use where Z are the matching factors of the temporal components of the s → s and b → b vector currents, determined nonperturbatively using charge conservation, ρ Γ are residual matching factors that are numerically close to 1 and are computed using one-loop lattice perturbation theory [59], and the term with coefficient d 1 removes O(a) discretization errors at tree level. In Eq. (28), γ E denotes the three Euclidean spatial gamma matrices, V , and d 1 are given in Table IV. For the residual matching factors ρ Γ of the vector and axialvector currents, we use the one-loop values given in Table III of Ref. [60]. These matching factors were computed for slightly different values of the parameters in the b-quark action [43], but are not expected to depend strongly on these parameters. For the residual matching factors of the tensor currents, one-loop results were not available and we set them to the tree-level values equal to unity. Following Ref. [25], we estimate the resulting systematic uncertainty in the tensor form factors at scale µ = m b to be equal to 2 times the maximum value of |ρ γ µ − 1|, |ρ γ µ γ5 − 1|, which is 0.05316. Note that the contributions from the operator O 7 in the weak Hamiltonian to the Λ b → Λ * (1520) + − differential decay rate at high q 2 are relatively small, so the larger systematic uncertainty in the tensor form factors is unproblematic.
Both the forward and backward three-point functions are computed using light and strange quark propagators with sources (Gaussian-smeared, with and without derivatives) located at (x 0 , x). Given the more complicated interpolating field for the Λ * (compared to that for the Λ in Ref. [25]), here we apply the sequential-source method for the b-quark propagators through the weak current, and not through the Λ b interpolating field as was done in Ref. [25]. This method fixes t rather than t, but we only computed the three-point functions for t = 2t , t = 2t + a, and t = 2t − a. We generated data for nine different separations on the coarse lattices and ten different separations on the fine lattices, as shown in Table V.
Due to the large mass of the Λ b , large values of p are needed to appreciably move q 2 away from q 2 max , as shown in Fig. 2. At the same time, discretization errors are expected to grow with p, and the number of b-quark sequential propagators that need to be computed is proportional to the number of choices for p. In this first lattice study of were computed at tree level in mean-field-improved perturbation theory. the Λ b → Λ * form factors, we therefore used only two different choices: p = (0, 0, 2) 2π L and p = (0, 0, 3) 2π L . Here, L = N s a are the spatial lattice extents, which are approximately 2.7 fm for all three ensembles.
After projection with P (3/2) , the spectral decomposition of the forward three-point function reads while the decomposition of the backward three-point function is given by the Dirac adjoint. Here, G λ [Γ] are, up to small lattice-discretization and finite-volume effects, the linear combinations of form factors defined in Eqs. (7)- (10).
To extract the form factors, we utilize two different types of combinations of correlation functions. The first type (Sec. V A) allows us to extract the absolute magnitudes of individual form factors, but not their relative signs. The second type (Sec. V B) allows us to extract ratios of different form factors in which the sign information is preserved.

A. Extracting the squares of individual form factors
To remove the unwanted overlap factors and cancel the exponential time-dependence for the ground-state contribution, we form the ratios where X ∈ {V, A, T V, T A} and Γ µ V = γ µ , Γ µ A = γ µ γ 5 , Γ µ T V = iσ µν q ν , Γ µ T A = iσ µν γ 5 q ν , and the traces are over the Dirac indices. To isolate the individual helicity form factors, we then contract with the timelike, longitudinal, and transverse polarization vectors and define Repeated Latin indices are summed only over the spatial directions, while repeated Greek indices are summed over all four spacetime directions. The above quantities are equal to the squares of the individual form factors times certain combinations of the hadron masses and energies. For a given value of t, the excited-state contamination will be minimal for t = t/2. Using this choice and removing the kinematic factors, we evaluate Since t and t must both be integer multiples of the lattice spacing, here we imply an average over the two values of t closest to t/2 for odd t/a. The excited-state contributions in the above quantities will decay exponentially as a function of the source-sink separation t.

B. Extracting ratios of form factors
To preserve the sign information, we define the following linear projections of three-point functions: where λ ∈ {0, +, ⊥, ⊥ } and with the polarization vectors as defined in Eq. (31). As before, repeated Latin indices are summed only over the spatial directions. 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 can isolate the form factors, up to common overlap factors and exponentials, in the following way: The excited-state contributions decay exponentially faster than the ground-state contributions. The unwanted factors of Z Λ * (Z t will cancel in ratios of the above quantities at large times.

C. Results for the form factors with relative signs preserved
The fourteen form factors with relative sign information preserved can now be obtained by extracting the magnitude of a single reference form factor as in Sec. V A, and multiplying with ratios of the projected three-point functions S X λ (p, t, t ). We choose f ⊥ to be the reference form factor because the results for the corresponding R V ⊥ show good plateaus and reasonably small statistical uncertainties (see the third plot from the left in the top row of Fig. 3). We again set t = t/2, and define the functions where R V ⊥ (p) denotes the result of a constant fit to R V ⊥ (p, t) in the region of ground-state saturation. The functions F X λ (p, t) are equal to the individual helicity form factors up to excited-state contamination that decays exponentially with t. We perform constant fits to F X λ (p, t) in the plateau regions, requiring good quality-of-fit and stability under variations of the starting time. Plots of F X λ (p, t) and the associated fits for one ensemble and one momentum are shown in Fig. 3. All fit results are listed in Table VI. The uncertainties were computed using statistical bootstrap.

VI. CHIRAL AND CONTINUUM EXTRAPOLATIONS OF THE FORM FACTORS
The final step in the analysis of the form factors is to fit suitable functions describing the dependence on the kinematics, the light-quark mass (or, equivalently, m 2 π ), and the lattice spacing to the results given in Table VI. Given that we have data for only two different momenta that correspond to values of q 2 near the kinematic endpoint, we describe the kinematic dependence of each form factor by a linear function of the dimensionless variable We expect this description to be accurate only in the high-q 2 region. To allow for dependence on the light-quark mass and lattice spacing, we use the model with independent fit parameters F f , A f , C f , D f , C f , and D f for each form factor f . Here, we introduced f π = 132 MeV and Λ = 300 MeV to make all parameters dimensionless. In the physical limit m π = m π,phys = 135 MeV, a = 0, the fit functions reduce to the form Our results for the physical-limit parameters F f and A f are given in Table VII. The full 28 × 28 covariance matrix of the parameters for all fourteen form factors is available as an ancillary file. The form factors in the physical limit are plotted as the solid magenta curves with 1σ uncertainty bands in Figs. 4 and 5. The dashed-dotted, dashed, and dotted curves show the fit models evaluated at the pion masses and lattice spacings of the individual data sets C01, C005, and F004, respectively, where the uncertainty bands are omitted for clarity. We see that the data are well described by the model.
Given that we only have three ensembles of gauge configurations and two momentum values, it is difficult to obtain detailed data-based estimates of the systematic uncertainties that remain after extrapolation to m π = m π,phys and  (24) 12.0(2.9) h ⊥ −3.01 (25) 12.2(2.8) h ⊥ −0.144 (24) 0.74 (37) TABLE VII. The fit parameters describing the form factors in the physical limit. The parametrizations, which are accurate only in the high-q 2 region, are given by . The 28 × 28 covariance matrix of all fit parameters is available as an ancillary file. The uncertainties given here are statistical only; see the main text for a discussion of systematic uncertainties.    Fig. 4, but for the tensor form factors. a = 0. In Ref. [25], which used the same lattice actions and lattice spacings but included additional lower valence light-quark masses, the total systematic uncertainties in the Λ b → Λ(1115) form factors at high q 2 were found to be approximately 5%, plus the 5.3% matching uncertainty in the tensor form factors as discussed in Sec. V. We roughly estimate the systematic uncertainties in the Λ b → Λ * (1520) form factors to be 1.5 times larger, i.e. 7.5%, plus the extra 5.3% matching uncertainty for the tensor form factors which is unchanged here. This increased estimate also allows for larger heavy-quark discretization errors associated with the nonzero Λ b momenta used here.
To calculate the Λ b → Λ * (1520) + − observables, we employ the usual operator-product expansion that allows us to express the decay amplitude in terms of local hadronic matrix elements [61]. For the differential decay rate in the FIG. 6. The Λ b → Λ * (1520) + − differential branching fraction in the high-q 2 region calculated in the Standard Model using our form factor results. Note that the factor of B(Λ * → pK − ) is not included here.
A 2. In the following, we use the convention that we do not include the factor of B Λ * = B(Λ * → pK − ) in the angular coefficients L i , which means that the integral of Eq. (100) over cos θ , cos θ Λ * , and φ is equal to dΓ/dq 2 for the primary decay Λ b → Λ * (1520) + − . As in Ref. [17], we define the CP-averaged, normalized angular observables as Our predictions for S 1c , S 1cc , S 1ss , S 2c , S 2cc , S 2ss , S 3ss , S 5s , and S 5sc are shown in Figs. 7 and 8. Two further combinations of interest are the fraction of longitudinally polarized dileptons and the lepton-side forward-backward asymmetry these are shown in Fig. 9. In the kinematic region considered here, our results for all angular observables are remarkably close to those predicted using quark-model form factors [30], shown in Refs. [17] and [19]. 16

VIII. CONCLUSIONS
We have presented the first lattice-QCD calculation of the form factors describing the Λ b → Λ * (1520) matrix elements of the vector, axial vector, and tensor b → s currents. Similarly to the lattice calculation of B → K * (892) form factors in Ref. [63], this work treats the Λ * (1520) as a stable particle. Even in this approximation, our work required overcoming several challenges. The simplest choices of three-quark interpolating fields with I = 0 and J P = 3 2 − dominantly couple to higher-lying states; a previous lattice-QCD study of Λ-baryon spectroscopy [53] in fact was unable to identify the Λ * (1520) for this reason. Here we solved this problem by including gauge-covariant spatial derivatives in the interpolating field, at the expense of having to compute additional propagators with derivative sources. We also used all-mode averaging [44,45] to overcome the poor signal-to-noise ratios in the correlation functions involving the Λ * (1520). Traditionally, lattice-QCD calculations of heavy-to-light form factors have been performed in the rest frame of the heavy hadron, giving the final-state light hadron nonzero momentum. However, at nonzero momentum an interpolating field that would have J P = 3 2 − in the continuum then also couples to J P = 3 2 + , and in some cases even J P = 1 2 + , which would make isolating the Λ * (1520) extremely difficult. For this reason, we performed the lattice calculation in the Λ * (1520) rest frame, giving nonzero momentum to the Λ b instead. While this choice eliminates the problem of mixing with unwanted lighter states, it also limits the accessible q 2 range to be very close to q 2 max . We performed the calculation for two different Λ b momenta, |p| ≈ 0.935 GeV and |p| ≈ 1.402 GeV, corresponding to q 2 /q 2 max ≈ 0.986 and q 2 /q 2 max ≈ 0.969, respectively. This only allowed linear fits of the q 2 -dependence (or, equivalently, w-dependence), which yield the values of the form factors at q 2 max and their slopes. Using three different ensembles of gauge fields on lattices that all have approximately the same spatial volume, we performed extrapolations linear in a 2 and m 2 π , with independent coefficients for the slopes and intersects of the form factors, to the physical limit.
Looking ahead, lower values of q 2 could be reached using the moving-NRQCD action [64] for the b quark, which enables much higher Λ b momenta while keeping discretization errors under control, but requires a more complicated matching of the currents to continuum QCD. Furthermore, a more rigorous analysis of Λ b → Λ * (1520) form factors that treats the Λ * (1520) as a resonance in coupled-channel p-K, Σ-π scattering may be possible using the finite-volume formalism of Refs. [65,66], but this would still not include Λ-π-π three-particle contributions.
Using our form factor results, we have obtained Standard-Model predictions for the Λ b → Λ * (1520) + − differential branching fraction and several Λ b → Λ * (1520)(→ pK − ) + − angular observables at high q 2 . The uncertainty in the differential branching fraction in the region considered is approximately 20 percent, while some angular observables are more precise due to their reduced dependence on the form factors and benefits from correlations. We predict a somewhat (∼ 30%) lower dB/dq 2 than the quark model of Ref. [30]. Our results for the angular observables are also quite close to those computed using the quark-model form factors [17]. We look forward to future experimental results for Λ b → Λ * (1520) + − .