Lattice QCD calculation of form factors for Λ b →Λ(1520) ℓ + ℓ − decays
LLattice QCD calculation of form factors for Λ b → Λ ( ) (cid:96) + (cid:96) − decays Stefan Meinel ∗ Department of Physics, University of Arizona, Tucson, AZ 85721, USARIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USAE-mail: [email protected]
Gumaro Rendon
Department of Physics, University of Arizona, Tucson, AZ 85721, USAE-mail: [email protected]
Experimental results for mesonic b → s µ + µ − decays show a pattern of deviations from Standard-Model predictions, which could be due to new fundamental physics or due to an insufficientunderstanding of hadronic effects. Additional information on the b → s µ + µ − transition can beobtained from Λ b decays. This was recently done using the process Λ b → Λ µ + µ − , where the Λ is the lightest strange baryon. A further interesting channel is Λ b → p + K − µ + µ − , where the p + K − final state receives contributions from multiple higher-mass Λ resonances. The narrowestand most prominent of these is the Λ ( ) , which has J P = − . Here we present an ongoinglattice QCD calculation of the relevant Λ b → Λ ( ) form factors. We discuss the choice ofinterpolating field for the Λ ( ) , and explain our method for extracting the fourteen Λ b → Λ ( ) helicity form factors from correlation functions that are computed in the Λ ( ) restframe. We present preliminary numerical results at a pion mass of 340 MeV and a lattice spacingof 0 .
11 fm. This calculation uses a domain-wall action for the u , d , and s quarks and a relativisticheavy-quark action for the b quark, and is based on gauge-field configurations generated by theRBC and UKQCD Collaborations. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] M a r attice QCD calculation of form factors for Λ b → Λ ( ) (cid:96) + (cid:96) − decays Stefan Meinel
1. Introduction
Flavor-changing neutral-current decays of bottom hadrons play an important role in the searchfor physics beyond the Standard Model. The effective Hamiltonian describing b → s (cid:96) + (cid:96) − decaysat low energies [1] contains the operators O ( (cid:48) ) = m b e ¯ s σ µν P R ( L ) b F µν , O ( (cid:48) ) = ¯ s γ µ P L ( R ) b ¯ (cid:96) γ µ (cid:96), O ( (cid:48) ) = ¯ s γ µ P L ( R ) b ¯ (cid:96) γ µ γ (cid:96), (1.1)as well as four-quark and gluonic operators. The Wilson coefficients C i of these operators encodethe short-distance physics and can be computed perturbatively in the Standard Model and in variousnew-physics scenarios. The values of C i can also be constrained by fitting the decay rates andangular distributions measured in experiments, provided that the relevant hadronic matrix elementsare known. Global analyses of experimental data for mesonic b → s µ + µ − decays, which use acombination of several theoretical methods [including lattice QCD for O ( (cid:48) ) , O ( (cid:48) ) , and O ( (cid:48) ) ],yield best-fit values for C that are approximately 25% below the Standard-Model prediction (see,e.g., Refs. [2, 3, 4, 5]). However, the results for C also depend on nonlocal matrix elementsinvolving the four-quark operators O and O , which are enhanced by charmonium resonances,and the approximations used for these matrix elements need further scrutiny.In addition to the commonly studied B and B s decays, the b → s (cid:96) + (cid:96) − couplings can also beprobed in decays of Λ b baryons (see Table 1 for a comparison of the most important semilep-tonic decay modes). Recently, the authors of Ref. [15] included, for the first time, the LHCb re-sults for the differential branching fraction and three angular observables of the decay Λ b → Λ ( → p + π − ) µ + µ − [16] in an analysis of the Wilson coefficients C , (cid:48) , , (cid:48) . From a theoretical point ofview [14, 17], this decay combines the best aspects of B → K (cid:96) + (cid:96) − (having only a single QCD-stable hadron in the final state, which simplifies the lattice QCD calculation of the form factors)and B → K ∗ ( → K π ) (cid:96) + (cid:96) − (providing a large number of observables that give full sensitivity to allDirac structures in the effective Hamiltonian). The fits performed in Ref. [15] prefer a positive shiftin C , contrary to previous fits of only mesonic decays. This behavior could hint at large duality vi-olations in the high- q operator product expansion that is used to approximate the nonlocal matrixelements of O and O . Unfortunately, the statistical uncertainties in the Λ b → Λ ( → p + π − ) µ + µ − data [16] are still quite large. One experimental challenge with this decay is that the hadron in thefinal state, the lightest Λ baryon, is electrically neutral and long-lived. It is therefore worth explor-ing decays proceeding through unstable Λ ∗ resonances, which can immediately decay into chargedProbes all Final hadron Charged hadrons from LQCDDirac structures QCD-stable b -decay vertex Refs. B + → K + (cid:96) + (cid:96) − × (cid:88) (cid:88) [6, 7, 8, 9] B → K ∗ ( → K + π − ) (cid:96) + (cid:96) − (cid:88) × (cid:88) [10, 11, 12] B s → φ ( → K + K − ) (cid:96) + (cid:96) − (cid:88) × (cid:88) [10, 11, 12] Λ b → Λ ( → p + π − ) (cid:96) + (cid:96) − (cid:88) (cid:88) × [13, 14, 15] Λ b → Λ ∗ ( → p + K − ) (cid:96) + (cid:96) − (cid:88) × (cid:88) This work
Table 1:
Comparison of exclusive b → s (cid:96) + (cid:96) − decay channels. attice QCD calculation of form factors for Λ b → Λ ( ) (cid:96) + (cid:96) − decays Stefan Meinel particles such as p + K − and produce tracks in the particle detectors that originate from the b -decayvertex.The p + K − -invariant-mass distribution in Λ b → p + K − µ + µ − decays is expected to be similarto that in Λ b → p + K − J / ψ . As can be seen in Fig. 3 of Ref. [18], a large number of Λ ∗ resonancescontribute to this decay in overlapping mass regions. However, one resonance produces a narrowpeak that clearly stands out above the other contributions: the Λ ( ) , which is the lightest reso-nance with J P = − . The Λ ( ) has a width of 15 . ± . pK , Σ π , Λ ππ , and, less importantly, Σ ππ . Given the small width, a naive analysis inwhich the Λ ( ) is treated as if it were QCD-stable is expected to be quite accurate, and istherefore justified in a first lattice QCD calculation of Λ b → Λ ( ) form factors. When workingin the Λ ( ) rest frame, the lowest energy level in the finite lattice volume can be identifiedwith the resonance in the narrow-width approximation; in the rest frame, the pK , Σ π , Λ ππ , and Σ ππ scattering-like states will appear at higher energies due to the nonzero back-to-back momentarequired for a coupling to J P = − .In the following, we will use the notation Λ ∗ to refer to the Λ ( ) . The Λ b → Λ ∗ matrixelements of the b → s vector, axial vector, and tensor currents [as needed for O ( (cid:48) ) , O ( (cid:48) ) , and O ( (cid:48) ) ] are described by 14 form factors [20]. Following the approach of Ref. [21], we havederived a new helicity-based definition of the Λ b → Λ ∗ form factors. The decomposition for thevector current reads (cid:104) Λ ∗ ( p (cid:48) , s (cid:48) ) | ¯ s γ µ b | Λ b ( p , s ) (cid:105) = ¯ u λ ( p (cid:48) , s (cid:48) ) (cid:34) f ( m Λ b − m Λ ∗ ) p λ q µ m Λ b q + f + ( m Λ b + m Λ ∗ ) p λ ( q ( p µ + p (cid:48) µ ) − ( m Λ b − m Λ ∗ ) q µ ) m Λ b q s + + f ⊥ (cid:32) p λ γ µ m Λ b − p λ ( m Λ b p (cid:48) µ + m Λ ∗ p µ ) m Λ b s + (cid:33) + f ⊥ (cid:48) (cid:32) p λ γ µ m Λ b − p λ p (cid:48) µ m Λ b m Λ ∗ + p λ ( m Λ b p (cid:48) µ + m Λ ∗ p µ ) m Λ b s + + s − g λ µ m Λ b m Λ ∗ (cid:33)(cid:35) u ( p , s ) , (1.2)where q = p − p (cid:48) , s ± = ( m Λ b ± m Λ ∗ ) − q , and the form factors f , f + , f ⊥ , f ⊥ (cid:48) are functions of q . Above, ¯ u λ ( p (cid:48) , s (cid:48) ) is the Rarita-Schwinger spinor for the Λ ∗ . Similar relations are obtainedfor the currents ¯ s γ µ γ b (form factors g , g + , g ⊥ , g ⊥ (cid:48) ), ¯ s i σ µν q ν b (form factors h + , h ⊥ , h ⊥ (cid:48) ), and¯ s i σ µν γ q ν b (form factors (cid:101) h + , (cid:101) h ⊥ , (cid:101) h ⊥ (cid:48) ).
2. Interpolating field for the Λ ( ) We work in the Λ ( ) rest frame to allow an exact projection to the J P = − quantumnumbers, and also for the reasons discussed in Sec. 1. In a first (unsuccessful) attempt at calculatingthe form factors, we used the interpolating field Λ ( old ) j γ = ε abc ( C γ j ) αβ (cid:16) ˜ u a α ˜ s b β ˜ d c γ − ˜ d a α ˜ s b β ˜ u c γ (cid:17) , (2.1)which has isospin 0 as required, and which we projected to J P = − by contracting with P k j = (cid:0) g k j − γ k γ j (cid:1) − γ (above, the tilde on the quark fields denotes gauge-covariant Gaussian smearing).2 attice QCD calculation of form factors for Λ b → Λ ( ) (cid:96) + (cid:96) − decays Stefan Meinel N s × N t β a m ( sea ) u , d a m ( sea ) s a m ( val ) u , d a m ( val ) s a [fm]24 ×
64 2.13 0.005 0.04 0.005 0.0323 0.1106(3) m π [MeV] m K [MeV] m N [MeV] m Λ [MeV] m Σ [MeV]340(1) 550(2) 1168(5) 1272(5) 1320(6) Table 2:
Lattice parameters and preliminary results for selected hadron masses. Details on the lattice actionsand ensemble generation can be found in Ref. [22]. We use all-mode-averaging (AMA) [23] with 1 exactand 32 sloppy measurements per configuration. t / a . . . . . . a E e ff J P = − Λ ( old ) Λ ( new ) Figure 1:
Effective-energy plot for the two-point functions computed with the interpolating fields definedin Eqs. (2.1) and (2.2). The preliminary results shown here are from 311 configurations (with AMA). Theenergies obtained from the fits are 1878 ( ) MeV and 1740 ( ) MeV.
With the interpolating field Λ ( old ) j γ , the numerical results for the ratios of three-point and two-pointfunctions used to extract the form factors were very noisy and did not show plateaus. We thennoticed that a previous lattice QCD study of Λ -baryon spectroscopy using interpolating fields sim-ilar to Eq. (2.1) in fact did not find a Λ ( ) -like state [24], while the calculation of Ref. [25],which included interpolating fields with covariant derivatives, did. This can be understood fromquark models, in which the Λ ( ) dominantly has an L = S = /
2, and flavor- SU ( ) singletstructure [26], very different from Eq. (2.1). We therefore now use the interpolating field Λ ( new ) j γ = ε abc ( C γ ) αβ (cid:104) ˜ s a α ˜ d b β ( ∇ j ˜ u ) c γ − ˜ s a α ˜ u b β ( ∇ j ˜ d ) c γ + ˜ u a α ( ∇ j ˜ d ) b β ˜ s c γ − ˜ d a α ( ∇ j ˜ u ) b β ˜ s c γ (cid:105) , (2.2)which matches the structure suggested by nonrelativistic quark models and has naturally negativeparity, so that it can be projected to J P = − by contracting with P k j = (cid:0) g k j − γ k γ j (cid:1) + γ (notethe plus sign). In Eq. (2.2), covariant derivatives acting on the strange quark have been eliminatedusing “integration by parts” (which is possible only at zero momentum). Numerical results for thetwo-point functions, with the lattice parameters given in Table 2, are shown in Fig. 1. The two-point function of the new interpolating field (2.2) shows a plateau at an energy close to m N + m K and m Σ + m π , as expected for the Λ ( ) , while the two-point function of the old interpolating3 attice QCD calculation of form factors for Λ b → Λ ( ) (cid:96) + (cid:96) − decays Stefan Meinel field (2.1) shows an apparent plateau at a significantly higher energy that is likely associated withone or more states that have a larger overlap with an S = / SU ( ) -octet structure.
3. Extracting the form factors from ratios of three-point and two-point functions
To determine the Λ b → Λ ( ) form factors, we compute three-point functions C ( , fw ) j γ δ ( p , Γ , t , t (cid:48) ) = ∑ y , z e − i p · ( y − z ) (cid:68) Λ ( new ) j γ ( x , x ) J Γ ( x − t + t (cid:48) , y ) Λ b δ ( x − t , z ) (cid:69) , (3.1)where J Γ = ρ Γ (cid:113) Z ( ss ) V Z ( bb ) V [ ¯ s Γ b + a d ¯ s Γ γγγ · ∇∇∇ b ] is the renormalized and O ( a ) -improved b → s current, Λ b δ = ε abc ( C γ ) αβ ˜ u a α ˜ d b β ˜ b c δ is the interpolating field for the Λ b , p is the momentum ofthe Λ b , and t is the source-sink separation. The bottom quark is implemented with the relativisticheavy-quark action of Ref. [27]. Using also the time-reversed backward three-point function andthe Λ ∗ and Λ b two-point functions, we form the ratios R jk µν ( p , t , t (cid:48) ) X = Tr (cid:104) P jl C ( , fw ) l ( p , Γ µ X , t , t (cid:48) ) ( / p + m Λ b ) C ( , bw ) m ( p , Γ ν X , t , t − t (cid:48) ) P mk (cid:105) Tr (cid:104) P lm C ( , Λ ∗ ) lm ( t ) (cid:105) Tr (cid:104) ( / p + m Λ b ) C ( , Λ b ) ( p , t ) (cid:105) , (3.2)where X ∈ { V , A , TV , TA } and Γ µ V = γ µ , Γ µ A = γ µ γ , Γ µ TV = i σ µν q ν , Γ µ TA = i σ µν γ q ν . We thencontract with the timelike, longitudinal, and transverse polarization vectors ε ( ) = ( q , q ) , ε (+) = ( | q | , ( q / | q | ) q ) , ε ( ⊥ , j ) = ( , e j × q ) (3.3)as follows: R X ( p , t , t (cid:48) ) = g jk ε ( ) µ ε ( ) ν R jk µν ( p , t , t (cid:48) ) X , (3.4) R X + ( p , t , t (cid:48) ) = g jk ε (+) µ ε (+) ν R jk µν ( p , t , t (cid:48) ) X , (3.5) R X ⊥ ( p , t , t (cid:48) ) = p j p k ε ( ⊥ , l ) µ ε ( ⊥ , l ) ν R jk µν ( p , t , t (cid:48) ) X , (3.6) R X ⊥ (cid:48) ( p , t , t (cid:48) ) = (cid:20) ε ( ⊥ , m ) j ε ( ⊥ , m ) k − p j p k (cid:21) ε ( ⊥ , l ) µ ε ( ⊥ , l ) ν R jk µν ( p , t , t (cid:48) ) X . (3.7)Up to excited-state contamination that is suppressed at large time separations, these quantities areequal to the squares of the individual helicity form factors times known kinematic factors. Forexample, in the case of the vector current we obtain the helicity form factors by computing R V ( p , t ) = (cid:115) E Λ b m Λ b R V ( p , t , t / )( E Λ b − m Λ b )( m Λ b − m Λ ) ( E Λ b + m Λ b ) = f + ( excited-state contribs. ) , (3.8) R V + ( p , t ) = (cid:115) E Λ b m Λ b R V + ( p , t , t / )( E Λ b + m Λ b )( m Λ b + m Λ ) ( E Λ b − m Λ b ) = f + + ( excited-state contribs. ) , (3.9) R V ⊥ ( p , t ) = (cid:115) − E Λ b m Λ b R V ⊥ ( p , t , t / )( E Λ b − m Λ b ) ( E Λ b + m Λ b ) = f ⊥ + ( excited-state contribs. ) , (3.10) R V ⊥ (cid:48) ( p , t ) = (cid:115) − E Λ b m Λ b R V ⊥ (cid:48) ( p , t , t / )( E Λ b − m Λ b ) ( E Λ b + m Λ b ) = f ⊥ (cid:48) + ( excited-state contribs. ) . (3.11)Preliminary numerical results for these quantities for all 14 helicity form factors at momentum p = ( , , ) π L are shown in Fig. 2. Reasonably good signals are obtained for most form factors.4 attice QCD calculation of form factors for Λ b → Λ ( ) (cid:96) + (cid:96) − decays Stefan Meinel . . . . . . . R V + . . . . . . . R V ⊥ . . . . . . R V ⊥ . . . . . . . R V . . . . . . R A + . . . . . . R A ⊥ . . . . . R A ⊥ . . . . . R A t / a . . . . . R TV + . . . . . R TV ⊥ . . . . . . R TV ⊥ t / a . . . . . . R TA + t / a . . . . . . R TA ⊥ t / a . . . . . R TA ⊥ Figure 2:
Preliminary results for the functions R Xi ( p , t ) , defined in Eqs. (3.8)-(3.11) for the vector currentand similarly for the other currents, at the Λ b -momentum p = ( , , ) π L . These functions become equalto the Λ b → Λ ( ) helicity form factors at the given momentum for large source-sink separation t . Theresults shown here are from 77 configurations (with AMA).
4. Next steps
The drawback of working in the Λ ( ) rest frame is that very large Λ b momenta are re-quired to appreciably move q away from q = ( m Λ b − m Λ ∗ ) , as illustrated in Fig. 3. Withthe relativistic heavy-quark action used so far, this introduces potentially large heavy-quark dis-cretization errors. We therefore plan to perform additional calculations in which the b quark isimplemented with moving NRQCD [28], which will allow us to reach much higher momenta. Wealso plan to substantially increase statistics and add two ensembles to study the lattice-spacing andlight-quark-mass dependence of the results. . . . . . . . | p | [ GeV ] q [ G e V ] Λ ( ) rest frame ← p = ( , , ) π L . . . . . . . | p | [ GeV ] q [ G e V ] Λ b rest frame Figure 3:
Value of q as a function of the Λ b momentum in the Λ ( ) rest frame (left), and as a functionof the Λ ( ) momentum in the Λ b rest frame (right). attice QCD calculation of form factors for Λ b → Λ ( ) (cid:96) + (cid:96) − decays Stefan Meinel
Acknowledgments:
This work is supported by National Science Foundation Grant Number PHY-1520996, and by the RHIC Physics Fellow Program of the RIKEN BNL Research Center. High-performance computing resources were provided by XSEDE (supported by National Science Foun-dation Grant Number OCI-1053575) and NERSC (supported by U.S. Department of Energy GrantNumber DE-AC02-05CH11231).
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