Λ b → Λ ∗ (1520) ℓ + ℓ − form factors from lattice QCD
ΛΛ b → Λ ∗ (1520) (cid:96) + (cid:96) − form factors from lattice QCD Stefan Meinel and Gumaro Rendon Department of Physics, University of Arizona, Tucson, AZ 85721, USA Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA (Dated: September 19, 2020)We present the first lattice QCD determination of the Λ b → Λ ∗ (1520) vector, axial vector, andtensor form factors that are relevant for the rare decays Λ b → Λ ∗ (1520) (cid:96) + (cid:96) − . The lattice calculationis performed in the Λ ∗ (1520) rest frame with nonzero Λ b momenta, and is limited to the high- q region. An interpolating field with covariant derivatives is used to obtain good overlap with theΛ ∗ (1520). The analysis treats the Λ ∗ (1520) as a stable particle, which is expected to be a reasonableapproximation for this narrow resonance. A domain-wall action is used for the light and strangequarks, while the b quark is implemented with an anisotropic clover action with coefficients tunedto produce the correct B s kinetic mass, rest mass, and hyperfine splitting. We use three differentensembles 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. Wegive Standard-Model predictions for the Λ b → Λ ∗ (1520) (cid:96) + (cid:96) − differential branching fraction andangular observables in the high- q region. I. INTRODUCTION
Decays of b -hadrons that proceed through the flavor-changing neutral current transition b → s(cid:96) + (cid:96) − play an im-portant role in searching for physics beyond the Standard Model [1]. Global analyses of the increasingly preciseexperimental data point to lepton-flavor-nonuniversal shifts in one or more of the Wilson coefficients with respect totheir Standard-Model values [2, 3]. These deviations, along with further hints for violation of lepton-flavor universalityin b → cτ ¯ ν decays, have led to significant activity in constructing models of new fundamental physics, as reviewed forexample in Ref. [4].When searching for new physics in weak decays, it is important to consider multiple decay modes involving differentspecies of hadrons. Different decay modes may be sensitive to different combinations of operators in the effectiveHamiltonian, and will also differ in their experimental and theoretical systematic uncertainties. The benefits ofΛ b baryon decays in constraining ∆ B = ∆ S = 1 Wilson coefficients have been discussed by several authors [5–19]. Experimental data are available for the differential branching fraction and angular observables of Λ b → Λ( → pπ − ) µ + µ − [20–23], as well as the branching fraction of Λ b → Λ γ [24]. In Ref. [18], an analysis of b → sµ + µ − Wilsoncoefficients using all 33 independent angular observables of Λ b → Λ( → pπ − ) µ + µ − decays [23] and using Λ b → Λ formfactors from lattice QCD [25] was reported. Within the present uncertainties, the results are consistent both with theanomalies seen in B meson decays and with the Standard Model [18].Going beyond the lightest Λ baryon in the final state, the LHCb Collaboration has also reported first measurementsof Λ b → pK − (cid:96) + (cid:96) − decays, including CP asymmetries [26] and the muon-versus-electron ratio R pK − [27]. TheΛ b → pK − µ + µ − CP asymmetries were measured in the kinematic region with m pK − < q = m (cid:96) + (cid:96) − / ∈ [0 . , . ∪ [8 . , ∪ [12 . ,
15] GeV [26] to avoid large contributions from the φ , J/ψ , and ψ (cid:48) resonances; the ratio R pK − was measured for m pK − < q ∈ [0 . , .
0] GeV [27].The pK − -invariant-mass distribution of Λ b → pK − (cid:96) + (cid:96) − for q away from the φ , J/ψ , and ψ (cid:48) resonances is expectedto be similar to the distribution with q on-resonance. This pK − -invariant-mass distribution has been observed inΛ b → pK − J/ψ ( → (cid:96) + (cid:96) − ) [28]. As can be seen in Fig. 3 of Ref. [28], a large number of Λ ∗ baryon resonances contributeto this decay in overlapping mass regions. However, one resonance produces a narrow peak that clearly stands outabove the other contributions: the Λ ∗ (1520), which has a width of 15 . ± . J P = − . Thus, it may be feasible for LHCb to measure the Λ b → Λ ∗ (1520)( → pK − ) (cid:96) + (cid:96) − decay rate andangular observables for q in the nonresonant (rare-decay) region.The phenomenology of Λ b → Λ ∗ (1520)( → pK − ) (cid:96) + (cid:96) − was discussed in Refs. [17, 19], where the expressions forthe complete angular distribution were given (for unpolarized Λ b ), approximate relations among the Λ b → Λ ∗ (1520)form factors based on effective field theories were obtained, and numerical studies of the differential decay rate andangular observables were performed using form factors from a quark model [30]. The prospects for measurementsof Λ b → Λ ∗ (1520)( → pK − ) (cid:96) + (cid:96) − angular observables at LHCb were recently studied in Ref. [31]. Earlier workhad also considered the decay mode Λ b → Λ ∗ (1520)( → pK − ) γ , primarily as a probe of the photon polarization in b → sγ [10, 11]; the formalism for an amplitude analysis of Λ b → pK − γ was recently discussed also in Ref. [32].The authors of Ref. [10] pointed out that this mode may be easier to reconstruct in hadron-collider experiments thanΛ b → Λ( → pπ − ) γ , since the Λ has a long lifetime of cτ ≈ . a r X i v : . [ h e p - l a t ] S e p vertex locator without leaving any trace.To make predictions for the Λ b → Λ ∗ (1520)( → pK − ) (cid:96) + (cid:96) − 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 tensorcurrents 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 resultsin Ref. [34]). The lattice calculation of
12 + → − form factors is substantially more challenging than the calculationof
12 + →
12 + form factors, even when neglecting the strong decay of the − baryon in the analysis, as we do here.Correlation functions for negative-parity baryons have more statistical noise than correlation functions for the lightestpositive-parity baryons. Furthermore, at nonzero momenta, the irreducible representations of the lattice symmetrygroups mix positive and negative parities and also mix J = and J = . 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 theground state, the mixing with other J P values does not cause difficulties in isolating it). This has the effect that ourcalculation is limited to a relatively small kinematic region near q .This paper is organized as follows. Our definition of the Λ b → Λ ∗ (1520) form factors is presented in Sec. II. The lat-tice actions and parameters are given in Sec. III. Section IV explains our choices of the baryon interpolating fields andcontains numerical results for the hadron masses. The three-point functions and our method for extracting the indi-vidual form factors are described in Sec. V. We perform simple chiral, continuum, and kinematic extrapolations of theform 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 givenin Sec. VIII. Appendix A contains relations between our form factor definition and other definitions that have beenused 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 = − . It has a mass of 1519 . ± . . ± . N ¯ K , Σ π , or Λ ππ [29]. In this work, we treat the Λ ∗ (1520) asif it is a stable single-particle state. We expect this to be a reasonable approximation, given the relatively small widthand 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 (cid:104) Λ ∗ ( p (cid:48) , s (cid:48) ) | ¯ s Γ b | Λ b ( p , s ) (cid:105) for Γ ∈ { γ µ , γ µ γ , iσ µν q ν , iσ µν q ν γ } with q = p − p (cid:48) . These matrix elements are described by fourteen independent form factors that are functions of q only.Possible definitions of these form factors were given, for example, in Refs. [17, 30, 33–36]. Here we use a helicity-baseddefinition. We first presented such a definition in Ref. [34]; the choice used here differs from that in Ref. [34] only bya q -dependent rescaling to avoid divergences in the form factors at the endpoint q = ( m Λ b − m Λ ∗ ) . We use thestandard relativistic normalization of states, (cid:104) Λ b ( k , r ) | Λ b ( p , s ) (cid:105) = δ rs E Λ b (2 π ) δ ( k − p ) , (1) (cid:104) Λ ∗ ( k (cid:48) , r (cid:48) ) | Λ ∗ ( p (cid:48) , s (cid:48) ) (cid:105) = δ r (cid:48) s (cid:48) E Λ ∗ (2 π ) δ ( k (cid:48) − p (cid:48) ) , (2)and introduce Dirac and Rarita-Schwinger spinors satisfying (cid:88) s u ( m Λ b , p , s )¯ u ( m Λ b , p , s ) = m Λ b + /p, (3) (cid:88) s (cid:48) u µ ( m Λ ∗ , p (cid:48) , s (cid:48) )¯ u ν ( m Λ ∗ , p (cid:48) , s (cid:48) ) = − ( m Λ ∗ + /p (cid:48) ) (cid:18) g µν − γ µ γ ν − m ∗ p (cid:48) µ p (cid:48) ν − m Λ ∗ ( γ µ p (cid:48) ν − γ ν p (cid:48) µ ) (cid:19) . (4)We introduce the notation (cid:104) Λ ∗ ( p (cid:48) , s (cid:48) ) | ¯ s Γ b | Λ b ( p , s ) (cid:105) = ¯ u λ ( m Λ ∗ , p (cid:48) , s (cid:48) ) G λ [Γ] u ( m Λ b , p , s ) , (5)and s ± = ( m Λ b ± m Λ ∗ ) − q . (6)The form factors f , f + , f ⊥ , f ⊥ (cid:48) , g , g + , g ⊥ , g ⊥ (cid:48) , h + , h ⊥ , h ⊥ (cid:48) , (cid:101) h + , (cid:101) h ⊥ , and (cid:101) h ⊥ (cid:48) are defined via G λ [ γ µ ] = f m Λ ∗ s + ( m Λ b − m Λ ∗ ) p λ q µ q + f + m Λ ∗ s − ( m Λ b + m Λ ∗ ) p λ ( q ( p µ + p (cid:48) µ ) − ( m b − m ∗ ) q µ ) q s + + f ⊥ m Λ ∗ s − (cid:18) p λ γ µ − p λ ( m Λ b p (cid:48) µ + m Λ ∗ p µ ) s + (cid:19) + f ⊥ (cid:48) m Λ ∗ s − (cid:18) p λ γ µ − p λ p (cid:48) µ m Λ ∗ + 2 p λ ( m Λ b p (cid:48) µ + m Λ ∗ p µ ) s + + s − g λµ m Λ ∗ (cid:19) , (7) G λ [ γ µ γ ] = − g γ m Λ ∗ s − ( m Λ b + m Λ ∗ ) p λ q µ q − g + γ m Λ ∗ s + ( m Λ b − m Λ ∗ ) p λ ( q ( p µ + p (cid:48) µ ) − ( m b − m ∗ ) q µ ) q s − − g ⊥ γ m Λ ∗ s + (cid:18) p λ γ µ − p λ ( m Λ b p (cid:48) µ − m Λ ∗ p µ ) s − (cid:19) − g ⊥ (cid:48) γ m Λ ∗ s + (cid:18) p λ γ µ + 2 p λ p (cid:48) µ m Λ ∗ + 2 p λ ( m Λ b p (cid:48) µ − m Λ ∗ p µ ) s − − s + g λµ m Λ ∗ (cid:19) , (8) G λ [ iσ µν q ν ] = − h + m Λ ∗ s − p λ ( q ( p µ + p (cid:48) µ ) − ( m b − m ∗ ) q µ ) s + − h ⊥ m Λ ∗ s − ( m Λ b + m Λ ∗ ) (cid:18) p λ γ µ − p λ ( m Λ b p (cid:48) µ + m Λ ∗ p µ ) s + (cid:19) − h ⊥ (cid:48) m Λ ∗ s − ( m Λ b + m Λ ∗ ) (cid:18) p λ γ µ − p λ p (cid:48) µ m Λ ∗ + 2 p λ ( m Λ b p (cid:48) µ + m Λ ∗ p µ ) s + + s − g λµ m Λ ∗ (cid:19) , (9) G λ [ iσ µν q ν γ ] = − (cid:101) h + γ m Λ ∗ s + p λ ( q ( p µ + p (cid:48) µ ) − ( m b − m ∗ ) q µ ) s − − (cid:101) h ⊥ γ m Λ ∗ s + ( m Λ b − m Λ ∗ ) (cid:18) p λ γ µ − p λ ( m Λ b p (cid:48) µ − m Λ ∗ p µ ) s − (cid:19) − (cid:101) h ⊥ (cid:48) γ m Λ ∗ s + ( m Λ b − m Λ ∗ ) (cid:18) p λ γ µ + 2 p λ p (cid:48) µ m Λ ∗ + 2 p λ ( m Λ b p (cid:48) µ − m Λ ∗ p µ ) s − − s + g λµ m Λ ∗ (cid:19) . (10)The requirement that physical matrix elements are non-singular for q → q = ( m Λ b − m Λ ∗ ) imposes certainrequirements on the behavior of the form factors in this limit [17]. More information on this behavior can be obtainedfrom heavy-quark effective theory [36] if the strange quark is treated as a heavy quark. For our definition, we expectall form factors to be finite and nonzero at q = q . Relations between our form factors and other definitions usedin 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 UKQCDcollaborations [37, 38]. These ensembles include the effects of 2+1 flavors of sea quarks, implemented with a domain-wall action [39–41]; the gauge action used is the Iwasaki action [42]. The main parameters of the ensembles andvalence-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 equalto the sea light-quark masses, and valence strange-quark masses tuned to the physical values, which are slightly lowerthan the sea strange-quark masses. For the b -quark propagators, we use the anisotropic clover action discussed inRef. [43], but with parameters am ( b ) Q , ξ ( b ) , c ( b ) E,B newly tuned by us to obtain the correct B s kinetic mass, rest mass,and hyperfine splitting. Label N s × N t β a [fm] am u,d am (sea) s am (val) s am ( b ) Q ξ ( b ) c ( b ) E,B N ex N sl C01 24 ×
64 2 .
13 0 . .
01 0 .
04 0 . . . . ×
64 2 .
13 0 . .
005 0 .
04 0 . . . . ×
64 2 .
25 0 . .
004 0 .
03 0 . . . . a , were determined in Ref. [38]. The bottom-quark is implemented with the action described in Ref. [43], but with parameters am ( b ) Q , ξ ( b ) , c ( b ) E,B newly tuned by us to obtain the correct B s kinetic mass, rest mass, and hyperfine splitting. The last twocolumns give the numbers of exact (ex) and sloppy (sl) samples used for the calculation of the correlation functions withall-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 thefollowing sections), as well as 32 “sloppy” samples with reduced conjugate-gradient iteration count in the computationof the light and strange quark propagators. For the light quarks, we also used deflation based on the lowest 400eigenvectors 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 chosenoverall 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 andneglects QED, which means that we have exact isospin symmetry, and the Λ b and Λ ∗ (1520) both have I = 0. Thecontinuous space-time symmetries on the other hand are reduced to discrete symmetries by the cubic lattice. Atzero momentum, the relevant symmetry group is O , the double cover of the cubic group [46], and we still have thefull parity symmetry. At zero momentum, the continuum J P = ± and J P = ± irreps subduce identically to the G g/u and H g/u irreps; the next-higher values of J that appear in these irreps are J = and J = , respectively. Inthis case we can therefore safely construct the interpolating fields for both the Λ b and the Λ ∗ (1520) using continuumsymmetries. At nonzero momenta, we no longer have parity symmetry, and the relevant symmetry groups are LittleGroups of O [47–49]. An interpolating field that would have J P = − in the continuum then also couples to J P =
32 + , and in some cases even J P =
12 + (for example, for momentum direction (0 , , J = also contains J = ), which would make isolating the Λ ∗ (1520) extremely difficult. For this reason, we performthe lattice calculation in the Λ ∗ (1520) rest frame, giving nonzero momentum to the Λ b instead. Since the Λ b is thelightest baryon with quark content udb , any contributions from mixing with opposite parity and higher J only appearas 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( O Λ b ) γ = 12 (cid:15) abc ( Cγ ) αβ (cid:16) (cid:101) d aα (cid:101) u bβ (cid:101) b cγ − (cid:101) u aα (cid:101) d bβ (cid:101) b cγ (cid:17) = (cid:15) abc ( Cγ ) αβ (cid:101) d aα (cid:101) u bβ (cid:101) b cγ , (11)where (cid:101) q denotes a smeared quark field. We use gauge-covariant Gaussian smearing of the form (cid:101) q = (cid:18) σ N Gauss (cid:101) ∆ (cid:19) N Gauss q, (12)where (cid:101) ∆ q ( x ) = 1 a (cid:88) j =1 (cid:104) (cid:101) U j ( x ) q ( x + a (cid:98) j ) − q ( x ) + (cid:101) U † j ( x − a (cid:98) j ) q ( x − a (cid:98) j ) (cid:105) , (13)and the gauge links (cid:101) U are APE-smeared (in the case of the up, down, and strange quarks) or Stout-smeared (inthe case of the bottom quark). The values used for the smearing parameters are given in Table II. We average over Up, down, and strange quarks Bottom quarks N Gauss σ Gauss /a N
APE α APE N Gauss σ Gauss /a N
Stout ρ Stout
Coarse 30 4 .
350 25 2 . .
000 10 0 . .
728 25 2 . .
667 10 0 . α APE is defined as in Eq. (8) of Ref. [51], and we apply N APE such sweeps. The Stout smearing is definedin Ref. [52]. “forward” and “backward” two-point functions given by C (2 , Λ b , fw) αβ ( p , t ) = (cid:88) y e − i p · ( y − x ) (cid:68) ( O Λ b ) α ( x + t, y ) ( O Λ b ) β ( x , x ) (cid:69) , (14) C (2 , Λ b , bw) αβ ( p , t ) = (cid:88) y e − i p · ( x − y ) (cid:68) ( O Λ b ) α ( x , x ) ( O Λ b ) β ( x − t, y ) (cid:69) . (15)The Λ b masses obtained from single-exponential fits in the time region of ground-state dominance are given in thelast column of Table III.Even at zero momentum, constructing an interpolating field with a good overlap to the Λ ∗ (1520) proved to benontrivial. In a first, unsuccessful attempt, we tried the form( O Λ ∗ ) (old) jγ = (cid:15) abc ( Cγ j ) αβ (cid:16) − γ (cid:17) γδ (cid:16)(cid:101) u aα (cid:101) s bβ (cid:101) d cδ − (cid:101) d aα (cid:101) s bβ (cid:101) u cδ (cid:17) , (16)which can be projected to the H u irrep by contracting the index j (which runs over the spatial directions) with P kj (3 / = g kj − γ k γ j . (17)Even though the resulting interpolating field has the correct values for all exactly conserved quantum numbers, itis found to have poor overlap with the Λ ∗ (1520) and much greater overlap with higher-mass J P = − states. Theeffective mass for the two-point function computed with O (old)Λ ∗ on the C005 ensemble is shown with the red circles inFig. 1, and shows a “false plateau” at higher mass before the signal is swamped by noise. A previous lattice QCDstudy 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 / ] has an internal structure corresponding to total quarkspin S = 3 /
2, total quark orbital angular momentum L = 0, and flavor- SU (3) octet, while quark models suggest thatthe Λ ∗ (1520) dominantly has an L = 1, S = 1 /
2, and flavor- SU (3)-singlet structure [54]. To obtain L = 1, a suitablespatial structure of the interpolating field is needed, which can be achieved using covariant derivatives [55]. For themain calculations in this work we use the form( O Λ ∗ ) jγ = (cid:15) abc ( Cγ ) αβ (cid:16) γ (cid:17) γδ (cid:104)(cid:101) s aα (cid:101) d bβ ( (cid:101) ∇ j (cid:101) u ) cγ − (cid:101) s aα (cid:101) u bβ ( (cid:101) ∇ j (cid:101) d ) cγ + (cid:101) u aα ( (cid:101) ∇ j (cid:101) d ) bβ (cid:101) s cγ − (cid:101) d aα ( (cid:101) ∇ j (cid:101) u ) bβ (cid:101) s cγ (cid:105) , (18)which has L = 1, S = 1 /
2, and is a flavor- SU (3) singlet. The covariant derivatives, which are defined as (cid:101) ∇ j (cid:101) q ( x ) = 12 a (cid:104) (cid:101) U j ( x ) (cid:101) q ( x + a (cid:98) j ) − (cid:101) U † j ( x − a (cid:98) j ) (cid:101) q ( x − a (cid:98) j ) (cid:105) , (19)change the parity, so the projector (1 + γ ) / O (old)Λ ∗ ,we project the two-point functions C (2 , Λ ∗ , fw) jkαβ ( t ) = (cid:88) y (cid:68) ( O Λ ∗ ) jα ( x + t, y ) ( O Λ ∗ ) kβ ( x , x ) (cid:69) , (20) C (2 , Λ ∗ , bw) jkαβ ( t ) = (cid:88) y (cid:68) ( O Λ ∗ ) jα ( x , x ) ( O Λ ∗ ) kβ ( x − t, y ) (cid:69) (21) We use the Minkowski-space metric tensor ( g µν ) = diag(1 , − , − , −
1) and Minkowski-space gamma matrices throughout this paper,except where indicated with a subscript “E.” t/a . . . . . . a E e ff O (old)Λ ∗ O Λ ∗ FIG. 1. The effective masses computed for the two-point functions with the old and new Λ ∗ interpolating fields, on the C005ensemble. The horizontal lines indicate the time ranges used and energies obtained from single-exponential fits.Label m π [GeV] m K [GeV] m N [GeV] m Λ [GeV] m Σ [GeV] m Λ ∗ [GeV] m Λ b [GeV]C01 0 . . . . . . . . . . . . . . . . . . . . . to the H u irrep with P kj (3 / . 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 propagatorswith derivative sources only for the light quarks. The effective mass for C (2 , Λ ∗ ) computed on the C005 ensembleis shown with the green squares in Fig. 1, and shows a plateau at a significantly lower mass, which we identify (inthe single-hadron/narrow-width approximation) with the Λ ∗ (1520) resonance. The Λ ∗ (1520) masses obtained fromsingle-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 givenin the same table. For the three ensembles we have, the mass differences m Λ ∗ − m Σ − m π are found to be in the rangefrom approximately 80 to 150 MeV (physical value: 192 MeV), while m Λ ∗ − m N − m K ranges from approximately − ∗ (1520) in the narrow-width approximation. A proper finite-volume scattering analysis with L¨uscher’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 inthe 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 (cid:104) | O Λ b | Λ b ( p , s ) (cid:105) = ( Z (1)Λ b + Z (2)Λ b γ ) u ( m Λ b , p , s ) , (22)and (cid:104) | ( O Λ ∗ ) j | Λ ∗ ( , s (cid:48) ) (cid:105) = Z Λ ∗ γ u j ( m Λ ∗ , , s (cid:48) ) . (23)As everywhere in this paper, | Λ ∗ ( , s (cid:48) ) (cid:105) denotes the lowest-energy 3 / − state. For the Λ b at nonzero momentum,it is necessary to have the two separate coefficients Z (1)Λ b and Z (2)Λ b that may also depend on p , because the spatial-only smearing of the quark fields breaks hypercubic symmetry (and because the lattice itself also breaks the Lorentzsymmetry). The spectral decomposition of C (2 , Λ b ) ( p , t ) then reads C (2 , Λ b ) ( p , t ) = 12 v ( Z (1)Λ b + Z (2)Λ b γ )(1 + /v )( Z (1)Λ b + Z (2)Λ b γ ) e − E Λ b t + (excited-state contributions) (24)with v µ = p µ /m Λ b , while the spectral decomposition of C (2 , Λ ∗ ) ( t ) after projection with P (3 / becomes P jl (3 / C (2 , Λ ∗ ) lk ( t ) = − Z ∗ (1 + γ ) (cid:18) g jk − γ j γ k (cid:19) e − m Λ ∗ t + (excited-state contributions) . (25)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 C (3 , fw) j γ δ ( p , Γ , t, t (cid:48) ) = (cid:88) y , z e − i p · ( y − z ) (cid:10) ( O Λ ∗ ) jγ ( x , x ) J Γ ( x − t + t (cid:48) , y ) ( O Λ b ) δ ( x − t, z ) (cid:11) , (26) C (3 , bw) j δ γ ( p , Γ , t, t − t (cid:48) ) = (cid:88) y , z e − i p · ( z − y ) (cid:68) ( O Λ b ) δ ( x + t, z ) J † Γ ( x + t (cid:48) , y ) ( O Λ ∗ ) jγ ( x , x ) (cid:69) , (27)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 (cid:48) is the current-insertion time. To match the currents to the continuum MS scheme, we employ the mostlynonperturbative method described in Refs. [57, 58]. Specifically, we use J Γ = ρ Γ (cid:113) Z ( ss ) V Z ( bb ) V [¯ s Γ b + a d ¯ s Γ γ E · ∇ b ] , (28)where Z ( ss ) V and Z ( bb ) V 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 closeto 1 and are computed using one-loop lattice perturbation theory [59], and the term with coefficient d removes O ( a )discretization errors at tree level. In Eq. (28), γ E denotes the three Euclidean spatial gamma matrices, γ j E = − iγ j .The values of Z ( ss ) V , Z ( bb ) V , and d are given in Table IV. For the residual matching factors ρ Γ of the vector and axial-vector currents, we use the one-loop values given in Table III of Ref. [60]. These matching factors were computed forslightly different values of the parameters in the b -quark action [43], but are not expected to depend strongly on theseparameters. For the residual matching factors of the tensor currents, one-loop results were not available and we setthem to the tree-level values equal to unity. Following Ref. [25], we estimate the resulting systematic uncertainty inthe tensor form factors at scale µ = m b to be equal to 2 times the maximum value of | ρ γ µ − | , | ρ γ µ γ − | , whichis 0 . O in the weak Hamiltonian to the Λ b → Λ ∗ (1520) (cid:96) + (cid:96) − differential decay rate at high q are relatively small, so the larger systematic uncertainty in the tensor form factorsis unproblematic.Both the forward and backward three-point functions are computed using light and strange quark propagators withsources (Gaussian-smeared, with and without derivatives) located at ( x , x ). Given the more complicated interpolatingfield for the Λ ∗ (compared to that for the Λ in Ref. [25]), here we apply the sequential-source method for the b -quarkpropagators through the weak current, and not through the Λ b interpolating field as was done in Ref. [25]. Thismethod fixes t (cid:48) rather than t , but we only computed the three-point functions for t = 2 t (cid:48) , t = 2 t (cid:48) + a , and t = 2 t (cid:48) − 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 away from q , as shown inFig. 2. At the same time, discretization errors are expected to grow with p , and the number of b -quark sequentialpropagators that need to be computed is proportional to the number of choices for p . In this first lattice study of Z ( bb ) V Z ( ss ) V d ( b )1 Coarse 9 . . . . . . Z ( bb ) V using the charge-conservation condition from ratios of B s two-point and three-point functions. The values of Z ( ss ) V are taken from Ref. [38]. The O ( a )-improvement coefficients d ( b )1 were computed at tree level in mean-field-improved perturbation theory. t/a Coarse 4 , , ..., , , ..., . . . . . . . | p | [GeV] q [ G e V ] ← p = (0 , , πL FIG. 2. The value of the four-momentum transfer squared as a function of the Λ b momentum in the Λ ∗ rest frame. The verticaldashed line indicates the largest momentum we use in this calculation. the Λ b → Λ ∗ form factors, we therefore used only two different choices: p = (0 , , πL and p = (0 , , πL . 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 / , the spectral decomposition of the forward three-point function reads P jl (3 / C (3 , fw) l ( p , Γ , t, t (cid:48) ) = − v Z Λ ∗ γ (cid:18) g jλ − γ j γ λ − γ j g λ (cid:19) G λ [Γ] 1 + /v Z (1)Λ b + Z (2)Λ b γ ) e − m Λ ∗ ( t − t (cid:48) ) e − E Λ b t (cid:48) + (excited-state contributions) , (29)while the decomposition of the backward three-point function is given by the Dirac adjoint. Here, G λ [Γ] are, up tosmall 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. Thesecond 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 contri-bution, we form the ratios R jkµν ( p , t, t (cid:48) ) X = Tr (cid:104) P jl (3 / C (3 , fw) l ( p , Γ µX , t, t (cid:48) ) (1 + /v ) C (3 , bw) m ( p , Γ νX , t, t − t (cid:48) ) P mk (3 / (cid:105) Tr (cid:104) P lm (3 / C (2 , Λ ∗ ) lm ( t ) (cid:105) Tr (cid:104) (1 + /v ) C (2 , Λ b ) ( p , t ) (cid:105) , (30)where X ∈ { V, A, T V, T A } and Γ µV = γ µ , Γ µA = γ µ γ , Γ µT V = iσ µν q ν , Γ µT A = iσ µν γ q ν , and the traces are over theDirac indices. To isolate the individual helicity form factors, we then contract with the timelike, longitudinal, andtransverse polarization vectors (cid:15) (0) = ( q , q ) , (cid:15) (+) = ( | q | , ( q / | q | ) q ) , (cid:15) ( ⊥ , j ) = ( 0 , e j × q ) , (31)and define R X ( p , t, t (cid:48) ) = g jk (cid:15) (0) µ (cid:15) (0) ν R jkµν ( p , t, t (cid:48) ) X , (32) R X + ( p , t, t (cid:48) ) = g jk (cid:15) (+) µ (cid:15) (+) ν R jkµν ( p , t, t (cid:48) ) X , (33) R X ⊥ ( p , t, t (cid:48) ) = p j p k (cid:15) ( ⊥ ,l ) µ (cid:15) ( ⊥ ,l ) ν R jkµν ( p , t, t (cid:48) ) X , (34) R X ⊥ (cid:48) ( p , t, t (cid:48) ) = (cid:20) (cid:15) ( ⊥ ,m ) j (cid:15) ( ⊥ ,m ) k − p j p k (cid:21) (cid:15) ( ⊥ ,l ) µ (cid:15) ( ⊥ ,l ) ν R jkµν ( p , t, t (cid:48) ) X . (35)Repeated Latin indices are summed only over the spatial directions, while repeated Greek indices are summed overall four spacetime directions. The above quantities are equal to the squares of the individual form factors timescertain combinations of the hadron masses and energies. For a given value of t , the excited-state contamination willbe minimal for t (cid:48) = t/
2. Using this choice and removing the kinematic factors, we evaluate R V ( p , t ) = 48 E Λ b ( E Λ b − m Λ b )( m Λ b − m Λ ∗ ) R V ( p , t, t/ f + (excited-state contributions) , (36) R V + ( p , t ) = 48 E Λ b ( E Λ b + m Λ b )( m Λ b + m Λ ∗ ) R V + ( p , t, t/ f + (excited-state contributions) , (37) R V ⊥ ( p , t ) = − E Λ b ( E Λ b − m Λ b ) ( E Λ b + m Λ b ) R V ⊥ ( p , t, t/ f ⊥ + (excited-state contributions) , (38) R V ⊥ (cid:48) ( p , t ) = − E Λ b ( E Λ b − m Λ b ) ( E Λ b + m Λ b ) R V ⊥ (cid:48) ( p , t, t/ f ⊥ (cid:48) + (excited-state contributions) , (39) R A ( p , t ) = 48 E Λ b ( E Λ b + m Λ b )( m Λ b + m Λ ∗ ) R A ( p , t, t/ g + (excited-state contributions) , (40) R A + ( p , t ) = 48 E Λ b ( E Λ b − m Λ b )( m Λ b − m Λ ∗ ) R A + ( p , t, t/ g + (excited-state contributions) , (41) R A ⊥ ( p , t ) = − E Λ b ( E Λ b + m Λ b ) ( E Λ b − m Λ b ) R A ⊥ ( p , t, t/ g ⊥ + (excited-state contributions) , (42) R A ⊥ (cid:48) ( p , t ) = − E Λ b ( E Λ b + m Λ b ) ( E Λ b − m Λ b ) R A ⊥ (cid:48) ( p , t, t/ g ⊥ (cid:48) + (excited-state contributions) , (43) R T V + ( p , t ) = 48 E Λ b ( E Λ b + m Λ b ) q R T V + ( p , t, t/ h + (excited-state contributions) , (44) R T V ⊥ ( p , t ) = − E Λ b ( E Λ b + m Λ b ) ( E Λ b − m Λ b ) ( m Λ b + m Λ ∗ ) R T V ⊥ ( p , t, t/ h ⊥ + (excited-state contributions) , (45)0 R T V ⊥ (cid:48) ( p , t ) = − E Λ b ( E Λ b + m Λ b ) ( E Λ b − m Λ b ) ( m Λ b + m Λ ∗ ) R T V ⊥ (cid:48) ( p , t, t/ h ⊥ (cid:48) + (excited-state contributions) , (46) R T A + ( p , t ) = 48 E Λ b ( E Λ b − m Λ b ) q R T A + ( p , t, t/ (cid:101) h + (excited-state contributions) , (47) R T A ⊥ ( p , t ) = − E Λ b ( E Λ b − m Λ b ) ( E Λ b + m Λ b ) ( m Λ b − m Λ ∗ ) R T A ⊥ ( p , t, t/ (cid:101) h ⊥ + (excited-state contributions) , (48) R T A ⊥ (cid:48) ( p , t ) = − E Λ b ( E Λ b − m Λ b ) ( E Λ b + m Λ b ) ( m Λ b − m Λ ∗ ) R T A ⊥ (cid:48) ( p , t, t/ (cid:101) h ⊥ (cid:48) + (excited-state contributions) . (49)Since t (cid:48) and t must both be integer multiples of the lattice spacing, here we imply an average over the two valuesof t (cid:48) closest to t/ t/a . The excited-state contributions in the above quantities will decay exponentially as afunction 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: S V,T Vλ ( p , t, t (cid:48) ) = Tr (cid:104) M ( λ ) µj P jl (3 / C (3 , fw) l ( p , Γ µV,T V , t, t (cid:48) ) (1 + /v )2 (cid:105) , (50) S A,T Aλ ( p , t, t (cid:48) ) = Tr (cid:104) γ M ( λ ) µj P jl (3 / C (3 , fw) l ( p , Γ µA,T A , t, t (cid:48) ) (1 + /v )2 (cid:105) , (51)where λ ∈ { , + , ⊥ , ⊥ (cid:48) } and M (0) µj = (cid:15) (0) µ (cid:15) (0) j , (52) M (+) µj = (cid:15) (+) µ (cid:15) (0) j , (53) M ( ⊥ ) µj = (cid:88) i =1 (cid:15) ( ⊥ ,i ) µ (cid:15) ( ⊥ ,i ) j , (54) M ( ⊥ ) ij = i(cid:15) (0) l γ l γ (cid:15) (0) m (cid:15) mij , M ( ⊥ )0 j = 0 , (55) M ( ⊥ ) µj = − M ( ⊥ ) µj + M ( ⊥ ) µj , (56) M ( ⊥ (cid:48) ) µj = M ( ⊥ ) µj + M ( ⊥ ) µj , (57)with the polarization vectors as defined in Eq. (31). As before, repeated Latin indices are summed only over thespatial directions. To improve the signals, we use the average of the forward three-point function and the Diracadjoint of the backward three-point function instead of just C (3 , fw) . We can isolate the form factors, up to commonoverlap factors and exponentials, in the following way: S V ( p , t, t (cid:48) ) = 3 E Λ b m Λ b ( E Λ b − m Λ b )( E Λ b + m Λ b )( m Λ b − m Λ ∗ ) S V ( p , t, t (cid:48) )= f Z Λ ∗ ( Z (1)Λ b m Λ b + Z (2)Λ b E Λ b ) e − m Λ ∗ ( t − t (cid:48) ) e − E Λ b t (cid:48) + (excited-state contributions) , (58)1 S V + ( p , t, t (cid:48) ) = 3 E Λ b m Λ b ( E Λ b − m Λ b ) / ( E Λ b + m Λ b ) / ( m Λ b + m Λ ∗ ) S V + ( p , t, t (cid:48) )= f + Z Λ ∗ ( Z (1)Λ b m Λ b + Z (2)Λ b E Λ b ) e − m Λ ∗ ( t − t (cid:48) ) e − E Λ b t (cid:48) + (excited-state contributions) , (59) S V ⊥ ( p , t, t (cid:48) ) = 3 E Λ b m Λ b E Λ b − m Λ b )( E Λ b + m Λ b ) S V ⊥ ( p , t, t (cid:48) )= f ⊥ Z Λ ∗ ( Z (1)Λ b m Λ b + Z (2)Λ b E Λ b ) e − m Λ ∗ ( t − t (cid:48) ) e − E Λ b t (cid:48) + (excited-state contributions) , (60) S V ⊥ (cid:48) ( p , t, t (cid:48) ) = E Λ b m Λ b E Λ b − m Λ b )( E Λ b + m Λ b ) S V ⊥ (cid:48) ( p , t, t (cid:48) )= f ⊥ (cid:48) Z Λ ∗ ( Z (1)Λ b m Λ b + Z (2)Λ b E Λ b ) e − m Λ ∗ ( t − t (cid:48) ) e − E Λ b t (cid:48) + (excited-state contributions) , (61) S A ( p , t, t (cid:48) ) = 3 E Λ b m Λ b ( E Λ b + m Λ b )( E Λ b − m Λ b )( m Λ b + m Λ ∗ ) S A ( p , t, t (cid:48) )= g Z Λ ∗ ( Z (1)Λ b m Λ b + Z (2)Λ b E Λ b ) e − m Λ ∗ ( t − t (cid:48) ) e − E Λ b t (cid:48) + (excited-state contributions) , (62) S A + ( p , t, t (cid:48) ) = 3 E Λ b m Λ b ( E Λ b − m Λ b ) / ( E Λ b + m Λ b ) / ( m Λ b − m Λ ∗ ) S A + ( p , t, t (cid:48) )= g + Z Λ ∗ ( Z (1)Λ b m Λ b + Z (2)Λ b E Λ b ) e − m Λ ∗ ( t − t (cid:48) ) e − E Λ b t (cid:48) + (excited-state contributions) , (63) S A ⊥ ( p , t, t (cid:48) ) = − E Λ b m Λ b E Λ b − m Λ b ) ( E Λ b + m Λ b ) S A ⊥ ( p , t, t (cid:48) )= g ⊥ Z Λ ∗ ( Z (1)Λ b m Λ b + Z (2)Λ b E Λ b ) e − m Λ ∗ ( t − t (cid:48) ) e − E Λ b t (cid:48) + (excited-state contributions) , (64) S A ⊥ (cid:48) ( p , t, t (cid:48) ) = − E Λ b m Λ b E Λ b − m Λ b ) ( E Λ b + m Λ b ) S A ⊥ (cid:48) ( p , t, t (cid:48) )= g ⊥ (cid:48) Z Λ ∗ ( Z (1)Λ b m Λ b + Z (2)Λ b E Λ b ) e − m Λ ∗ ( t − t (cid:48) ) e − E Λ b t (cid:48) + (excited-state contributions) , (65) S T V + ( p , t, t (cid:48) ) = − E Λ b m Λ b ( E Λ b − m Λ b ) / ( E Λ b + m Λ b ) / q S T V + ( p , t, t (cid:48) )= h + Z Λ ∗ ( Z (1)Λ b m Λ b + Z (2)Λ b E Λ b ) e − m Λ ∗ ( t − t (cid:48) ) e − E Λ b t (cid:48) + (excited-state contributions) , (66) S T V ⊥ ( p , t, t (cid:48) ) = − E Λ b m Λ b E Λ b − m Λ b )( E Λ b + m Λ b ) ( m Λ b + m Λ ∗ ) S T V ⊥ ( p , t, t (cid:48) )= h ⊥ Z Λ ∗ ( Z (1)Λ b m Λ b + Z (2)Λ b E Λ b ) e − m Λ ∗ ( t − t (cid:48) ) e − E Λ b t (cid:48) + (excited-state contributions) , (67)2 S T V ⊥ (cid:48) ( p , t, t (cid:48) ) = − E Λ b m Λ b E Λ b − m Λ b )( E Λ b + m Λ b ) ( m Λ b + m Λ ∗ ) S T V ⊥ (cid:48) ( p , t, t (cid:48) )= h ⊥ (cid:48) Z Λ ∗ ( Z (1)Λ b m Λ b + Z (2)Λ b E Λ b ) e − m Λ ∗ ( t − t (cid:48) ) e − E Λ b t (cid:48) + (excited-state contributions) , (68) S T A + ( p , t, t (cid:48) ) = 3 E Λ b m Λ b ( E Λ b + m Λ b ) / ( E Λ b − m Λ b ) / q S T A + ( p , t, t (cid:48) )= (cid:101) h + Z Λ ∗ ( Z (1)Λ b m Λ b + Z (2)Λ b E Λ b ) e − m Λ ∗ ( t − t (cid:48) ) e − E Λ b t (cid:48) + (excited-state contributions) , (69) S T A ⊥ ( p , t, t (cid:48) ) = − E Λ b m Λ b E Λ b + m Λ b )( E Λ b − m Λ b ) ( m Λ b − m Λ ∗ ) S T A ⊥ ( p , t, t (cid:48) )= (cid:101) h ⊥ Z Λ ∗ ( Z (1)Λ b m Λ b + Z (2)Λ b E Λ b ) e − m Λ ∗ ( t − t (cid:48) ) e − E Λ b t (cid:48) + (excited-state contributions) , (70) S T A ⊥ (cid:48) ( p , t, t (cid:48) ) = − E Λ b m Λ b E Λ b + m Λ b )( E Λ b − m Λ b ) ( m Λ b − m Λ ∗ ) S T A ⊥ (cid:48) ( p , t, t (cid:48) )= (cid:101) h ⊥ (cid:48) Z Λ ∗ ( Z (1)Λ b m Λ b + Z (2)Λ b E Λ b ) e − m Λ ∗ ( t − t (cid:48) ) e − E Λ b t (cid:48) + (excited-state contributions) . (71)The excited-state contributions decay exponentially faster than the ground-state contributions. The unwanted factorsof Z Λ ∗ ( Z (1)Λ b m Λ b + Z (2)Λ b E Λ b ) e − m Λ ∗ ( t − t (cid:48) ) e − E Λ b t (cid:48) 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 magnitudeof a single reference form factor as in Sec. V A, and multiplying with ratios of the projected three-point functions S Xλ ( p , t, t (cid:48) ). We choose f ⊥ (cid:48) to be the reference form factor because the results for the corresponding R V ⊥ (cid:48) show goodplateaus and reasonably small statistical uncertainties (see the third plot from the left in the top row of Fig. 3). Weagain set t (cid:48) = t/
2, and define the functions F Xλ ( p , t ) = S Xλ ( p , t, t/ S V ⊥ (cid:48) ( p , t, t/ (cid:113) R V ⊥ (cid:48) ( p ) , (72)where R V ⊥ (cid:48) ( p ) denotes the result of a constant fit to R V ⊥ (cid:48) ( p , t ) in the region of ground-state saturation. The functions F Xλ ( p , t ) are equal to the individual helicity form factors up to excited-state contamination that decays exponentiallywith t . We perform constant fits to F Xλ ( p , t ) in the plateau regions, requiring good quality-of-fit and stability undervariations of the starting time. Plots of F Xλ ( p , t ) and the associated fits for one ensemble and one momentum areshown in Fig. 3. All fit results are listed in Table VI. The uncertainties were computed using statistical bootstrap.3 t/a . . . F V + t/a . . F V ⊥ t/a . . . . R V ⊥ t/a F V t/a F A + t/a F A ⊥ t/a − . − . . F A ⊥ t/a . . . F A t/a − . − . . . F TV + t/a − . − . . F TV ⊥ t/a . . F TV ⊥ t/a − − F TA + t/a − − F TA ⊥ t/a − . . F TA ⊥ FIG. 3. Numerical results for the quantities F Xλ ( p , t ), defined in Eq. (72), as a function of the source-sink separation, for p = (0 , , πL and for the F004 ensemble. Also shown is R V ⊥ (cid:48) ( p , t ), which is used to extract the square of the reference formfactor f ⊥ (cid:48) . The horizontal lines indicate the ranges and extracted values of constant fits. Form factor | p | / (2 π/L ) C01 C005 F004 f . . . . . . f + . . . . . . f ⊥ . − . − . . . . f ⊥ (cid:48) . . . . . . g . . . . . . g + . . . . . . g ⊥ . . . . . . g ⊥ (cid:48) − . − . − . − . − . − . h + . . . − . . . h ⊥ − . − . − . − . − . − . h ⊥ (cid:48) . . . . . . (cid:101) h + − . − . − . − . − . − . (cid:101) h ⊥ − . − . − . − . − . − . (cid:101) h ⊥ (cid:48) − . − . − . − . − . − . F Xλ ( p , t ) in the plateau regions, for each ensemble andfor the two different Λ b momenta. 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 thekinematics, the light-quark mass (or, equivalently, m π ), and the lattice spacing to the results given in Table VI. Giventhat we have data for only two different momenta that correspond to values of q near the kinematic endpoint, wedescribe the kinematic dependence of each form factor by a linear function of the dimensionless variable w ( q ) = v · v (cid:48) = m b + m ∗ − q m Λ b m Λ ∗ . (73)We expect this description to be accurate only in the high- q region. To allow for dependence on the light-quark massand lattice spacing, we use the model f ( q ) = F f (cid:2) C f ( m π − m π, phys ) / (4 πf π ) + D f a Λ (cid:3) + A f (cid:104) (cid:101) C f ( m π − m π, phys ) / (4 πf π ) + (cid:101) D f a Λ (cid:105) ( w − , (74)with independent fit parameters F f , A f , C f , D f , (cid:101) C f , and (cid:101) 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 f ( q ) = F f + A f ( w − , (75)which only depend on the parameters F f and A f . The model (74) can be thought of as expansions of both thezero-recoil form factors F f and the slopes A f in terms of the light-quark mass and the square of the lattice spacing.The limited number of data points made it necessary to constrain the size of the coefficients C f , D f , (cid:101) C f , and (cid:101) D f tobe not unnaturally large. To this end, we introduced Gaussian priors for C f , D f , (cid:101) C f , and (cid:101) D f with central valuesequal to 0 and widths equal to 10.Our results for the physical-limit parameters F f and A f are given in Table VII. The full 28 ×
28 covariance matrixof the parameters for all fourteen form factors is available as an ancillary file. The form factors in the physical limitare plotted as the solid magenta curves with 1 σ uncertainty bands in Figs. 4 and 5. The dashed-dotted, dashed, anddotted 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 welldescribed by the model.Given that we only have three ensembles of gauge configurations and two momentum values, it is difficult to obtaindetailed data-based estimates of the systematic uncertainties that remain after extrapolation to m π = m π, phys and f F f A f f . − . . f + . . f ⊥ − . . f ⊥ (cid:48) . − . g . . g + . − . . g ⊥ . − . . g ⊥ (cid:48) − . . h + . − . h ⊥ − . − . h ⊥ (cid:48) . − . (cid:101) h + − . . . (cid:101) h ⊥ − . . . (cid:101) h ⊥ (cid:48) − . . q region, are given by f = F f + A f ( w − w = v · v (cid:48) = ( m b + m ∗ − q ) / (2 m Λ b m Λ ∗ ). The 28 × .
00 1 .
01 1 .
02 1 .
03 1 .
04 1 . f .
00 1 .
01 1 .
02 1 .
03 1 .
04 1 . . . . . . . . . g .
00 1 .
01 1 .
02 1 .
03 1 .
04 1 . . . . . . . . . . f + .
00 1 .
01 1 .
02 1 .
03 1 .
04 1 . g + .
00 1 .
01 1 .
02 1 .
03 1 .
04 1 . − . − . . . . . f ⊥ .
00 1 .
01 1 .
02 1 .
03 1 .
04 1 . g ⊥ .
00 1 .
01 1 .
02 1 .
03 1 .
04 1 . w . . . . . . . . f ⊥ .
00 1 .
01 1 .
02 1 .
03 1 .
04 1 . w − . − . − . − . − . − . − . . g ⊥ FIG. 4. Chiral and continuum extrapolations of the vector and axial vector form factors. The solid magenta curves with 1 σ statistical-uncertainty bands show the form factors in the physical limit. The dashed-dotted, dashed, and dotted curves showthe 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. .
00 1 .
01 1 .
02 1 .
03 1 .
04 1 . − . − . . . . . h + .
00 1 .
01 1 .
02 1 .
03 1 .
04 1 . − − − − − e h + .
00 1 .
01 1 .
02 1 .
03 1 .
04 1 . − . − . − . − . − . − . − . − . . h ⊥ .
00 1 .
01 1 .
02 1 .
03 1 .
04 1 . − − − − − e h ⊥ .
00 1 .
01 1 .
02 1 .
03 1 .
04 1 . w . . . . . . . h ⊥ .
00 1 .
01 1 .
02 1 .
03 1 .
04 1 . w − . − . − . − . . e h ⊥ FIG. 5. Like 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 valencelight-quark masses, the total systematic uncertainties in the Λ b → Λ(1115) form factors at high q were found to beapproximately 5%, plus the 5.3% matching uncertainty in the tensor form factors as discussed in Sec. V. We roughlyestimate the systematic uncertainties in the Λ b → Λ ∗ (1520) form factors to be 1.5 times larger, i.e. 7.5%, plus theextra 5.3% matching uncertainty for the tensor form factors which is unchanged here. This increased estimate alsoallows for larger heavy-quark discretization errors associated with the nonzero Λ b momenta used here. VII. Λ b → Λ ∗ (1520) (cid:96) + (cid:96) − OBSERVABLES
To calculate the Λ b → Λ ∗ (1520) (cid:96) + (cid:96) − observables, we employ the usual operator-product expansion that allows usto express the decay amplitude in terms of local hadronic matrix elements [61]. For the differential decay rate in the8Standard Model, we finddΓd q = G F α · π m b | V tb V ∗ ts | υ √ s + s − (cid:2) A (cid:0) m (cid:96) + q (cid:1) + A q υ + 6 A t m (cid:96) (cid:3) , (76)where υ = (cid:112) − m (cid:96) /q , and the quantities A , A , and A t are given by A = (cid:12)(cid:12)(cid:12)(cid:12) H (cid:18) − , , (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) H (cid:18) − , − , (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) H (cid:18) , , (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) H (cid:18) , − , − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) H (cid:18) , , − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) H (cid:18) , − , − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (77) A = (cid:12)(cid:12)(cid:12)(cid:12) H (cid:18) − , , (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) H (cid:18) − , − , (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) H (cid:18) , , (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) H (cid:18) , − , − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) H (cid:18) , , − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) H (cid:18) , − , − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (78) A t = (cid:12)(cid:12)(cid:12)(cid:12) H (cid:18) t, , (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) H (cid:18) t, − , − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . (79)Here, H and H are linear combinations of hadronic helicity amplitudes with the appropriate Wilson coefficients: H = − m b q C eff7 ( q ) ( H T + H T ) + C eff9 ( q ) ( H V − H A ) , (80) H = C ( H V − H A ) . (81)In terms of the form factors, the helicity amplitudes (in our sign conventions) for the vector, axial-vector, and tensorcurrents are equal to H V (cid:18) t, , (cid:19) = H V (cid:18) t, − , − (cid:19) = − f ( m Λ b − m Λ ∗ ) √ s − (cid:112) q , (82) H V (cid:18) , , (cid:19) = H V (cid:18) , − , − (cid:19) = − f + ( m Λ b + m Λ ∗ ) √ s + (cid:112) q , (83) H V (cid:18) , , − (cid:19) = − H V (cid:18) − , − , (cid:19) = − f ⊥ √ s + √ , (84) H V (cid:18) , − , − (cid:19) = H V (cid:18) − , , (cid:19) = f ⊥ (cid:48) √ s + , (85) H A (cid:18) t, , (cid:19) = − H A (cid:18) t, − , − (cid:19) = g ( m Λ b + m Λ ∗ ) √ s + (cid:112) q , (86) H A (cid:18) , , (cid:19) = − H A (cid:18) , − , − (cid:19) = g + ( m Λ b − m ∗ Λ ) √ s − (cid:112) q , (87) H A (cid:18) , , − (cid:19) = − H A (cid:18) − , − , (cid:19) = − g ⊥ √ s − √ , (88) H A (cid:18) , − , − (cid:19) = − H A (cid:18) − , , (cid:19) = − g ⊥ (cid:48) √ s − , (89)9and H T (cid:18) t, , (cid:19) = H T (cid:18) t, − , − (cid:19) = 0 , (90) H T (cid:18) , , (cid:19) = H T (cid:18) , − , − (cid:19) = − h + ( m Λ b + m Λ ∗ ) √ s + (cid:112) q , (91) H T (cid:18) , , − (cid:19) = H T (cid:18) − , − , (cid:19) = − h ⊥ √ s + √ , (92) H T (cid:18) , − , − (cid:19) = H T (cid:18) − , , (cid:19) = h ⊥ (cid:48) √ s + , (93) H T (cid:18) t, , (cid:19) = − H T (cid:18) t, − , − (cid:19) = 0 , (94) H T (cid:18) , , (cid:19) = − H T (cid:18) , − , − (cid:19) = (cid:101) h + ( m Λ b − m ∗ Λ ) √ s − (cid:112) q , (95) H T (cid:18) , , − (cid:19) = − H T (cid:18) − , − , (cid:19) = − (cid:101) h ⊥ √ s − √ , (96) H T (cid:18) , − , − (cid:19) = − H T (cid:18) − , , (cid:19) = − (cid:101) h ⊥ (cid:48) √ s − . (97)For the effective Wilson coefficients C eff7 ( q ) and C eff9 ( q ), we use the expressions given in Eqs. (65) and (66) ofRef. [25]. The Wilson coefficients C through C , the strong and electromagnetic couplings, and the b and c quarkmasses are also evaluated as in Ref. [25]. We take | V tb V ∗ ts | = 0 . ± . B / d q = τ Λ b dΓ / d q , the Λ b lifetime τ Λ b = (1 . ± . { f , f + , f ⊥ , f ⊥ (cid:48) } , { g , g + , g ⊥ , g ⊥ (cid:48) } , { h + , h ⊥ , h ⊥ (cid:48) } , { (cid:101) h + , (cid:101) h ⊥ , (cid:101) h ⊥ (cid:48) } , but uncorrelated across different groups. The additional 5 .
3% matching uncertainty inthe tensor form factors was assumed to be 100% correlated between all tensor form factors.Our prediction for the differential branching fraction in the high- q region is shown in Fig. 6. Here we have set m (cid:96) = 0, which, in this kinetic region, is a good approximation for both electrons and muons. We only show resultsabove q = 16 GeV because our lattice data only reach down to approximately 16 . , and our parametrizationof the q -dependence of the form factors is not expected to be reliable for lower q . In this kinematic region, ournumerical results for d B / d q are approximately 30% lower than those obtained using the quark-model form factorsof Ref. [30].In the narrow-width approximation for the Λ ∗ (1520) and for m (cid:96) = 0, the Λ b → Λ ∗ (1520)( → pK − ) (cid:96) + (cid:96) − four-folddifferential decay distribution in the Standard Model has the formd Γd q d cos θ (cid:96) d cos θ Λ ∗ d φ = 38 π (cid:104) cos θ Λ ∗ (cid:0) L c cos θ (cid:96) + L cc cos θ (cid:96) + L ss sin θ (cid:96) (cid:1) + sin θ Λ ∗ (cid:0) L c cos θ (cid:96) + L cc cos θ (cid:96) + L ss sin θ (cid:96) (cid:1) + sin θ Λ ∗ (cid:0) L ss sin θ (cid:96) cos φ + L ss sin θ (cid:96) sin φ cos φ (cid:1) + sin θ Λ ∗ cos θ Λ ∗ cos φ ( L s sin θ (cid:96) + L sc sin θ (cid:96) cos θ (cid:96) )+ sin θ Λ ∗ cos θ Λ ∗ sin φ ( L s sin θ (cid:96) + L sc sin θ (cid:96) cos θ (cid:96) ) (cid:105) , (100)where the angular coefficients L i are functions of q only [17]. The expressions for the L i in terms of form factors aregiven in Ref. [17], using a slightly different definition of the form factors that is related to ours as shown in Appendix0 . . . . . q [GeV ] × − d B / d q [GeV − ] FIG. 6. The Λ b → Λ ∗ (1520) (cid:96) + (cid:96) − differential branching fraction in the high- q region calculated in the Standard Model usingour 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 angularcoefficients L i , which means that the integral of Eq. (100) over cos θ (cid:96) , cos θ Λ ∗ , and φ is equal to d Γ /dq for the primarydecay Λ b → Λ ∗ (1520) (cid:96) + (cid:96) − . As in Ref. [17], we define the CP-averaged, normalized angular observables as S i = L i + L i d(Γ + Γ) / d q . (101)Our predictions for S c , S cc , S ss , S c , S cc , S ss , S ss , S s , and S sc are shown in Figs. 7 and 8. Two furthercombinations of interest are the fraction of longitudinally polarized dileptons F L = 1 − L cc + 2 L cc )3 dΓ / d q (102)and the lepton-side forward-backward asymmetry A (cid:96)F B = L c + 2 L c / d q ; (103)these are shown in Fig. 9. In the kinematic region considered here, our results for all angular observables are remarkablyclose to those predicted using quark-model form factors [30], shown in Refs. [17] and [19].1 . . . . . q [GeV ] − . . . . S c . . . . . q [GeV ] . . . . . . S cc . . . . . q [GeV ] . . . . . . S ss . . . . . q [GeV ] − . . . . S c . . . . . q [GeV ] . . . . . S cc . . . . . q [GeV ] . . . . . S ss FIG. 7. The Λ b → Λ ∗ (1520)( → pK − ) (cid:96) + (cid:96) − angular observables S c , S cc , S ss , S c , S cc , and S ss in the high- q regioncalculated in the Standard Model using our form factor results. . . . . . q [GeV ] − . − . − . − . . . S ss . . . . . q [GeV ] . . . . . . S s . . . . . q [GeV ] − . − . − . − . . S sc FIG. 8. The Λ b → Λ ∗ (1520)( → pK − ) (cid:96) + (cid:96) − angular observables S ss , S s , and S sc in the high- q region calculated in theStandard Model using our form factor results. . . . . . q [GeV ] . . . . . F L . . . . . q [GeV ] − . − . . . . . . A ‘F B FIG. 9. The Λ b → Λ ∗ (1520)( → pK − ) (cid:96) + (cid:96) − fraction of longitudinally polarized dileptons and the lepton-side forward-backwardasymmetry in the high- q region calculated in the Standard Model using our form factor results. VIII. CONCLUSIONS
We have presented the first lattice-QCD calculation of the form factors describing the Λ b → Λ ∗ (1520) matrixelements 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 workrequired overcoming several challenges. The simplest choices of three-quark interpolating fields with I = 0 and J P = − dominantly couple to higher-lying states; a previous lattice-QCD study of Λ-baryon spectroscopy [53] in factwas unable to identify the Λ ∗ (1520) for this reason. Here we solved this problem by including gauge-covariant spatialderivatives in the interpolating field, at the expense of having to compute additional propagators with derivativesources. We also used all-mode averaging [44, 45] to overcome the poor signal-to-noise ratios in the correlationfunctions involving the Λ ∗ (1520). Traditionally, lattice-QCD calculations of heavy-to-light form factors have beenperformed in the rest frame of the heavy hadron, giving the final-state light hadron nonzero momentum. However, atnonzero momentum an interpolating field that would have J P = − in the continuum then also couples to J P =
32 + ,and in some cases even J P =
12 + , which would make isolating the Λ ∗ (1520) extremely difficult. For this reason, weperformed the lattice calculation in the Λ ∗ (1520) rest frame, giving nonzero momentum to the Λ b instead. While thischoice eliminates the problem of mixing with unwanted lighter states, it also limits the accessible q range to be veryclose to q . We performed the calculation for two different Λ b momenta, | p | ≈ .
935 GeV and | p | ≈ .
402 GeV,corresponding to q /q ≈ .
986 and q /q ≈ . q -dependence(or, equivalently, w -dependence), which yield the values of the form factors at q and their slopes. Using threedifferent ensembles of gauge fields on lattices that all have approximately the same spatial volume, we performedextrapolations linear in a and m π , with independent coefficients for the slopes and intersects of the form factors, tothe physical limit.Looking ahead, lower values of q could be reached using the moving-NRQCD action [64] for the b quark, whichenables much higher Λ b momenta while keeping discretization errors under control, but requires a more complicatedmatching of the currents to continuum QCD. Furthermore, a more rigorous analysis of Λ b → Λ ∗ (1520) form factorsthat treats the Λ ∗ (1520) as a resonance in coupled-channel p - K , Σ- π scattering may be possible using the finite-volumeformalism 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) (cid:96) + (cid:96) − differentialbranching fraction and several Λ b → Λ ∗ (1520)( → pK − ) (cid:96) + (cid:96) − angular observables at high q . The uncertainty in thedifferential branching fraction in the region considered is approximately 20 percent, while some angular observablesare more precise due to their reduced dependence on the form factors and benefits from correlations. We predicta somewhat ( ∼ d B /dq than the quark model of Ref. [30]. Our results for the angular observables arealso quite close to those computed using the quark-model form factors [17]. We look forward to future experimentalresults for Λ b → Λ ∗ (1520) (cid:96) + (cid:96) − . ACKNOWLEDGMENTS
We thank Danny van Dyk and S´ebastien Descotes-Genon for discussions, and the RBC and UKQCD Collaborationsfor making their gauge field ensembles available. SM is supported by the U.S. Department of Energy, Office of Science,Office of High Energy Physics under Award Number DE-SC0009913. GR is supported by the U.S. Department ofEnergy, Office of Science, Office of Nuclear Physics, under Contract No. DE-SC0012704 (BNL). The computations forthis work were carried out on facilities at the National Energy Research Scientific Computing Center, a DOE Officeof Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No.DE-AC02-05CH1123, and on facilities of the Extreme Science and Engineering Discovery Environment (XSEDE) [67],which is supported by National Science Foundation grant number ACI-1548562.
Appendix A: Relations between different form factor definitions
In this appendix we provide the relations between two other definitions of Λ b → Λ ∗ (1520) form factors used in theliterature and our definition.4
1. Non-helicity-based definition
This definition is used in Refs. [30, 33]. For the vector and axial vector currents, it has the same structure as thedefinition of Λ b → Λ ∗ c (2625) form factors in Ref. [35]. In the notation of our Eq. (5), it is given by G λ [ γ µ ] = v λ ( F γ µ + F v µ + F v (cid:48) µ ) + F g λµ , (A1) G λ [ γ µ γ ] = v λ ( G γ µ + G v µ + G v (cid:48) µ ) γ + G g λµ γ , (A2) G λ [ iσ µν q ν ] = v λ (cid:0) F T γ µ + F T v µ + F T v (cid:48) µ (cid:1) + F T g λµ , (A3) G λ [ iσ µν q ν γ ] = v λ (cid:0) G T γ µ + G T v µ + G T v (cid:48) µ (cid:1) γ + G T g λµ γ . (A4)Note that only four of the six tensor form factors in this definition are independent. The relation to our definition is F = m Λ b m Λ ∗ s − ( f ⊥ + f ⊥ (cid:48) ) , (A5) F = m b m Λ ∗ q s + s − (cid:2) ( m Λ b − m Λ ∗ ) s − f − m Λ ∗ q ( f ⊥ − f ⊥ (cid:48) ) − ( m Λ b + m Λ ∗ ) (cid:0) m b − m ∗ − q (cid:1) f + (cid:3) , (A6) F = m Λ b m Λ ∗ q s + s − (cid:2) − m Λ ∗ ( m Λ b − m Λ ∗ ) s − f − m Λ b m Λ ∗ q f ⊥ + 2 q ( m Λ b m Λ ∗ − s + ) f ⊥ (cid:48) + m Λ ∗ ( m Λ b + m Λ ∗ ) (cid:0) m b − m ∗ + q (cid:1) f + (cid:3) , (A7) F = f ⊥ (cid:48) , (A8) G = m Λ b m Λ ∗ s + ( g ⊥ + g ⊥ (cid:48) ) , (A9) G = m b m Λ ∗ q s + s − (cid:2) − ( m Λ b + m Λ ∗ ) s + g − m Λ ∗ q ( g ⊥ − g ⊥ (cid:48) ) + ( m Λ b − m Λ ∗ ) (cid:0) m b − m ∗ − q (cid:1) g + (cid:3) , (A10) G = m Λ b m Λ ∗ q s + s − (cid:2) m Λ ∗ ( m Λ b + m Λ ∗ ) s + g + 2 m Λ b m Λ ∗ q g ⊥ − q ( m Λ b m Λ ∗ + s − ) g ⊥ (cid:48) − m Λ ∗ ( m Λ b − m Λ ∗ ) (cid:0) m b − m ∗ + q (cid:1) g + (cid:3) , (A11) G = g ⊥ (cid:48) , (A12) F T = − m Λ b m Λ ∗ ( m Λ b + m Λ ∗ ) s − ( h ⊥ + h ⊥ (cid:48) ) , (A13) F T = m b m Λ ∗ s + s − (cid:2) m Λ ∗ ( m Λ b + m Λ ∗ )( h ⊥ − h ⊥ (cid:48) ) + ( m b − m ∗ − q ) h + (cid:3) , (A14) F T = m Λ b m Λ ∗ s + s − (cid:2) m Λ b + m Λ ∗ )( m Λ b m Λ ∗ h ⊥ − ( m Λ b m Λ ∗ − s + ) h ⊥ (cid:48) ) − m Λ ∗ (cid:0) m b − m ∗ + q (cid:1) h + (cid:3) , (A15) F T = − ( m Λ b + m Λ ∗ ) h ⊥ (cid:48) , (A16) G T = m Λ b m Λ ∗ ( m Λ b − m Λ ∗ ) s + ( (cid:101) h ⊥ + (cid:101) h ⊥ (cid:48) ) , (A17) G T = m b m Λ ∗ s + s − (cid:2) − m Λ ∗ ( m Λ b − m Λ ∗ )( (cid:101) h ⊥ − (cid:101) h ⊥ (cid:48) ) + ( m b − m ∗ − q ) (cid:101) h + (cid:3) , (A18) G T = m Λ b m Λ ∗ s + s − (cid:2) m Λ b − m Λ ∗ )( m Λ b m Λ ∗ (cid:101) h ⊥ − ( m Λ ∗ m Λ b + s − ) (cid:101) h ⊥ (cid:48) ) − m Λ ∗ (cid:0) m b − m ∗ + q (cid:1) (cid:101) h + (cid:3) , (A19) G T = ( m Λ b − m Λ ∗ ) (cid:101) h ⊥ (cid:48) . (A20)5
2. Helicity-based definition used by Descotes-Genon and Novoa Brunet
Reference [17] uses a helicity-based definition that differs from ours only by simple kinematic factors: f Vt = m Λ ∗ s + f , (A21) f V = m Λ ∗ s − f + , (A22) f V ⊥ = m Λ ∗ s − f ⊥ , (A23) f Vg = f ⊥ (cid:48) , (A24) f At = m Λ ∗ s − g , (A25) f A = m Λ ∗ s + g + , (A26) f A ⊥ = m Λ ∗ s + g ⊥ , (A27) f Ag = − g ⊥ (cid:48) , (A28) f T = m Λ ∗ s − h + , (A29) f T ⊥ = m Λ ∗ s − h ⊥ , (A30) f Tg = ( m Λ b + m Λ ∗ ) h ⊥ (cid:48) , (A31) f T = m Λ ∗ s + (cid:101) h + , (A32) f T ⊥ = m Λ ∗ s + (cid:101) h ⊥ , (A33) f T g = − ( m Λ b − m Λ ∗ ) (cid:101) h ⊥ (cid:48) . (A34)Similarly, Ref. [36], which considers Λ b → Λ ∗ c , contains another helicity-based definition (for the vector and axial-vectorform factors only) that also differs from ours only by simple kinematic factors. [1] T. Blake, G. Lanfranchi, and D. M. Straub, “Rare B Decays as Tests of the Standard Model,” Prog. Part. Nucl. Phys. (2017) 50–91, arXiv:1606.00916 [hep-ph] .[2] M. Alguer´o, B. Capdevila, A. Crivellin, S. Descotes-Genon, P. Masjuan, J. Matias, and J. Virto, “Emerging patterns ofNew Physics with and without Lepton Flavour Universal contributions,” Eur. Phys. J. C79 no. 8, (2019) 714, arXiv:1903.09578 [hep-ph] .[3] J. Aebischer, W. Altmannshofer, D. Guadagnoli, M. Reboud, P. Stangl, and D. M. Straub, “ B -decay discrepancies afterMoriond 2019,” Eur. Phys. J. C no. 3, (2020) 252, arXiv:1903.10434 [hep-ph] .[4] D. Buttazzo, A. Greljo, G. Isidori, and D. Marzocca, “ B -physics anomalies: a guide to combined explanations,” JHEP (2017) 044, arXiv:1706.07808 [hep-ph] .[5] M. Gremm, F. Kruger, and L. M. Sehgal, “Angular distribution and polarization of photons in the inclusive decayΛ b → X s γ ,” Phys. Lett. B355 (1995) 579–583, arXiv:hep-ph/9505354 [hep-ph] .[6] T. Mannel and S. Recksiegel, “Flavor changing neutral current decays of heavy baryons: The Case Λ b → Λ γ ,” J. Phys. G24 (1998) 979–990, arXiv:hep-ph/9701399 [hep-ph] .[7] C.-S. Huang and H.-G. Yan, “Exclusive rare decays of heavy baryons to light baryons: Λ b → Λ γ and Λ b → Λ (cid:96) + (cid:96) − ,”Phys. Rev. D59 (1999) 114022, arXiv:hep-ph/9811303 [hep-ph] . [Erratum: Phys. Rev.D61,039901(2000)].[8] G. Hiller and A. Kagan, “Probing for new physics in polarized Λ b decays at the Z ,” Phys. Rev. D65 (2002) 074038, arXiv:hep-ph/0108074 [hep-ph] .[9] C.-H. Chen, C. Q. Geng, and J. N. Ng, “T violation in Λ b → Λ (cid:96) + (cid:96) − decays with polarized Λ,” Phys. Rev. D65 (2002)091502, arXiv:hep-ph/0202103 [hep-ph] .[10] F. Legger and T. Schietinger, “Photon helicity in Λ b → pKγ decays,” Phys. Lett. B645 (2007) 204–212, arXiv:hep-ph/0605245 [hep-ph] . [Erratum: Phys. Lett.B647,527(2007)].[11] G. Hiller, M. Knecht, F. Legger, and T. Schietinger, “Photon polarization from helicity suppression in radiative decays ofpolarized Λ b to spin-3/2 baryons,” Phys. Lett. B649 (2007) 152–158, arXiv:hep-ph/0702191 [hep-ph] . [12] P. B¨oer, T. Feldmann, and D. van Dyk, “Angular Analysis of the Decay Λ b → Λ( → Nπ ) (cid:96) + (cid:96) − ,” JHEP (2015) 155, arXiv:1410.2115 [hep-ph] .[13] S. Meinel and D. van Dyk, “Using Λ b → Λ µ + µ − data within a Bayesian analysis of | ∆ B | = | ∆ S | = 1 decays,” Phys. Rev. D94 no. 1, (2016) 013007, arXiv:1603.02974 [hep-ph] .[14] T. Blake and M. Kreps, “Angular distribution of polarised Λ b baryons decaying to Λ (cid:96) + (cid:96) − ,” JHEP (2017) 138, arXiv:1710.00746 [hep-ph] .[15] D. Das, “Model independent New Physics analysis in Λ b → Λ µ + µ − decay,” Eur. Phys. J. C78 no. 3, (2018) 230, arXiv:1802.09404 [hep-ph] .[16] H. Yan, “Angular distribution of the rare decay Λ b → Λ( → Nπ ) (cid:96) + (cid:96) − ,” arXiv:1911.11568 [hep-ph] .[17] S. Descotes-Genon and M. Novoa Brunet, “Angular analysis of the rare decay Λ b → Λ(1520)( → NK ) (cid:96) + (cid:96) − ,” JHEP (2019) 136, arXiv:1903.00448 [hep-ph] .[18] T. Blake, S. Meinel, and D. van Dyk, “Bayesian Analysis of b → sµ + µ − Wilson Coefficients using the Full AngularDistribution of Λ b → Λ( → p π − ) µ + µ − Decays,” Phys. Rev. D no. 3, (2020) 035023, arXiv:1912.05811 [hep-ph] .[19] D. Das and J. Das, “The Λ b → Λ ∗ (1520)( → N ¯ K ) (cid:96) + (cid:96) − decay at low-recoil in HQET,” JHEP (2020) 002, arXiv:2003.08366 [hep-ph] .[20] CDF
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