The form factors for B -> pi l nu semileptonic decay from 2+1 flavors of domain-wall fermions
TThe form factors for B → π l ν semileptonic decayfrom + flavors of domain-wall fermions Taichi Kawanai ∗ Theoretical Research Division, Nishina Center, RIKEN, Wako 351-0198, JapanPhysics Department, Brookhaven National Laboratory, Upton, NY 11973, USARIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USAE-mail: [email protected]
Ruth S. Van de Water
Theoretical Physics Department, Fermi National Accelerator Laboratory, Batavia, IL 60510,USAE-mail: [email protected]
Oliver Witzel
Center for Computational Science, Boston University, 3 Cummington Mall, Boston, MA 02215,USAE-mail: [email protected]
We present a calculation of the B → π l ν form factors with domain-wall light quarks and rela-tivistic b -quarks on the lattice. We work with the 2 + z -parameterizationand impose the kinematic constraint f + ( ) = f ( ) at zero momentum transfer. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] N ov he B → π l ν form factors with domain-wall fermions Taichi Kawanai
1. Introduction
The precise determination of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element | V ub | provides a strong test of the standard model. The lattice calculation of the hadronic form factor f + plays an essential role in the determination of | V ub | from B → π l ν exclusive semileptonic de-cay. When combined with the hadronic form factor f + ( q ) , the value of | V ub | can be obtained byexperimental measurements of the differential decay rate via d Γ ( B → π l ν ) dq = G F | V ub | π m B (cid:2) ( m B + m π − q ) − m B m π (cid:3) / | f + ( q ) | , (1.1)where the momentum transfer q µ ≡ p µ B − p µπ and we neglect the mass of the outgoing lepton. Theform factors f + and f parameterize the hadronic element of the b → u vector current V µ ≡ i ¯ u γ µ b : (cid:104) π | V µ | B (cid:105) = f + ( q ) (cid:18) p µ B + p µπ − m B − m π q q µ (cid:19) + f ( q ) m B − m π q q µ . (1.2)Nonperturbative lattice-QCD provides a first-principles method for computing the B → π formfactors with controlled uncertainties.In these proceedings, we report on a lattice-QCD calculation of the B → π form factor withdomain-wall light quarks using the 2 + + f + ( q ) done by the HPQCD [1] and FNAL/MILC Collaborations [2]. Bothgroups use the MILC gauge configurations. Our calculation will provide an important independentcheck on existing calculations that use staggered light quarks.
2. Methodology
In practice, we calculate the form factors f (cid:107) and f ⊥ , which are more convenient for latticesimulations: (cid:104) π | V µ | B (cid:105) = (cid:112) m B [ v µ f (cid:107) ( E π ) + p µ ⊥ f ⊥ ( E π )] , (2.1)where v µ = p µ B / m B and p µ ⊥ = p µπ − ( p π · v ) v µ . In the B -meson rest frame, these form factors aredirectly proportional to the hadronic matrix elements of the temporal and spatial vector current: f (cid:107) = (cid:104) π | V | B (cid:105)√ m B , f ⊥ = (cid:104) π | V i | B (cid:105)√ m B p i π , (2.2)and the desired form factors f + and f can be obtained by the following relations: f + ( q ) = √ m B [ f (cid:107) ( E π ) + ( m B − E π ) f ⊥ ( E π )] (2.3) f ( q ) = √ m B m B − m π (cid:2) ( m B − E π ) f (cid:107) ( E π ) + ( E π − m π ) f ⊥ ( E π ) (cid:3) . (2.4)We employ the mostly nonperturbative method of Ref. [4] to match the lattice amplitude to thecontinuum matrix element: (cid:104) π | V µ | B (cid:105) = Z blV µ (cid:104) π | V µ | B (cid:105) with Z blV µ = ρ blV µ (cid:113) Z bbV Z llV . (2.5)2 he B → π l ν form factors with domain-wall fermions Taichi Kawanai
Table 1:
Heavy-light current renormalization factors used in our analysis. The values of ρ blV µ are obtained inthe chiral limit [3]. a ≈ .
11 fm a ≈ .
086 fm am l = . am l = . am l = . am l = . am l = . Z llV Z bbV ρ blV ρ blV i Table 2:
Lattice simulation parameters. a [fm] L × T am l am s M π [MeV] ≈ .
11 24 ×
64 0.005 0.040 329 1636 1 ≈ .
11 24 ×
64 0.010 0.040 422 1419 1 ≈ .
086 32 ×
64 0.004 0.030 289 628 2 ≈ .
086 32 ×
64 0.006 0.030 345 889 2 ≈ .
086 32 ×
64 0.008 0.030 394 544 2The flavor-conserving renormalization factors Z bbV and Z llV are computed nonperturbatively on thelattice and the factor ρ blV µ is computed at one loop in mean-field improved lattice perturbation the-ory [5]. Most of the heavy-light current renormalization factor comes from Z bbV and Z llV , such that ρ blV µ is expected to be close to unity [6]. In this study, the factor Z bbV is calculated using the charge-normalization condition Z bbV (cid:104) B s | V bb , | B s (cid:105) = m B where V bb , is the b → b lattice vector current.We use the values of Z llV obtained by the RBC/UKQCD collaborations by exploiting the fact that Z A = Z V for domain-wall fermions [7]. The renormalization factors used in this analysis are sum-marized in Table 1.We improve the b → u vector current through O ( α S a ) . At this order we need only computeone additional matrix element with a single-derivative operator. We calculate the improvementcoefficient at 1-loop in mean-field improved lattice perturbation theory.
3. Computational setup
We use the 2 + T / m a , the clover3 he B → π l ν form factors with domain-wall fermions Taichi Kawanai a - / f ^ ( a E p ) c /d.o.f. = 0.90, p-value = 53% preliminary m l / m s = 0.005/0.04 coarse m l / m s = 0.01/0.04 coarse m l / m s = 0.004/0.03 fine m l / m s = 0.006/0.03 fine m l / m s = 0.008/0.03 finechiral-continuum f ^ a / f || ( a E p ) c /d.o.f. = 1.41, p-value = 13% preliminary m l / m s = 0.005/0.04 coarse m l / m s = 0.01/0.04 coarse m l / m s = 0.004/0.03 fine m l / m s = 0.006/0.03 fine m l / m s = 0.008/0.03 finechiral-continuum f || Figure 1:
The B → π form factors f ⊥ (left) and f (cid:107) (right), shown in units of the lattice spacing on the 24 ensembles. The black curves with gray error bands show the chiral-continuum extrapolated f (cid:107) and f ⊥ withstatistical errors. coefficient c P , and the anisotropy parameter ξ ) [10, 12]. Here we employ values determined non-perturbatively in Ref. [13].
4. Analysis f lat (cid:107) and f lat ⊥ The lattice form factors f lat (cid:107) and f lat ⊥ are obtained from the following ratios of correlationfunctions at large source-sink separation: R B → π , µ ( E π , t , t snk ) = C B → π , µ ( E π , t , t snk ) (cid:113) C π ( E π , t ) C B ( t snk − t ) (cid:114) E π e − E π t e − m B ( t snk − t ) . (4.1)After multiplying the results for R B → π , and R B → π , i by Z blV : f ⊥ ( E π ) = Z blV i lim t , t snk → ∞ p i R B → π , i ( E π , t , t snk ) (4.2) f (cid:107) ( E π ) = Z blV lim t , t snk → ∞ R B → π , ( E π , t , t snk ) , (4.3)we obtain the renormalized form factors f (cid:107) and f ⊥ as a function of pion energy on each ensemble,shown in Fig. 1. We use the form factor data through momentum (cid:126) p = π ( , , ) / L in this analysis. In order to extrapolate simultaneously the form factors to the physical light-quark mass andthe continuum, we employ NLO SU ( ) hard-pion chiral perturbation theory [14] suitably modifiedto incorporate leading discretization errors from the domain-wall and Iwasaki actions. Thus the fitfunctions depend on the pion mass m ll , pion-energy E π and squared lattice spacing a : f ⊥ ( m ll , E π , a ) = c ( ) ⊥ E π + ∆ (cid:16) + δ f ⊥ + c ( ) ⊥ m ll + c ( ) ⊥ E π + c ( ) ⊥ E π + c ( ) ⊥ a (cid:17) (4.4) f (cid:107) ( m ll , E π , a ) = c ( ) (cid:107) (cid:16) + δ f (cid:107) + c ( ) (cid:107) m ll + c ( ) (cid:107) E π + c ( ) (cid:107) E π + c ( ) (cid:107) a (cid:17) , (4.5)4 he B → π l ν form factors with domain-wall fermions Taichi Kawanai f + / q [GeV ] f + f preliminary m l / m s = 0.005/0.04 coarse m l / m s = 0.01/0.04 coarse m l / m s = 0.004/0.03 fine m l / m s = 0.006/0.03 fine m l / m s = 0.008/0.03 finechiral-continuum f +/0 Figure 2:
The B → π form factors f + and f The black curves with gray error bands show the chiral-continuum extrapolated form factors with statistical errors with the four evenly-spaced synthetic data pointsused in the q extrapolation overlaid. Errors shown are statistical only. where the quantity ∆ is the mass difference m B ∗ − m B , fixed to the experimental value [15], andensures the proper location of the pole at the B ∗ mass in the physical form factor f + . The function δ f contains logarithmic functions of the pion mass: ( π f π ) δ f (cid:107) / ⊥ = − g I ( m ll ) − I ( m ll ) + I ( m ll ) + ( m ll − m ll ) ∂ I ( m ll ) ∂ m ll , (4.6)where I ( m ll ) = m ll log ( m ll / Λ ) and f π = . g , we use g B ∗ B π = .
569 calculated as a part of this RHQ project [16]. Fig. 1 shows the resulting chiral-continuumextrapolation of f ⊥ and f (cid:107) . q to zero recoil We must extrapolate the lattice data to lower q (larger E π ) to reach the kinematic regionwhere experimental measurements are most precise. Using chiral-continuum extrapolated latticedata in the range of simulated pion energies, we first generate four synthetic data points of the formfactors f + and f used in the q extrapolation to the full kinematic range, as shown in Fig. 2.In this study, we employ the model-independent z -expansion fit to extrapolate to low momen-tum transfer [17–20]. As a first step, we consider mapping the variable q on to a new variable z defined as z = (cid:112) t + − q − √ t + − t (cid:112) t + − q + √ t + − t (4.7)where t ± = ( m B ± m π ) . This transformation maps the semileptonic region 0 < q < t − onto smallvalues of z between 0 . < z < .
22 when we choose t = . t + . The B → π form factors areanalytic in the semileptonic region except at the location of the B ∗ pole, so the form factors f + and f can be expressed as convergent power series: f ( q ) = P ( q ) φ ( q , t ) ∞ ∑ k = a ( k ) ( t ) z ( q , t ) k , (4.8)where the function P ( q ) is the Blaschke factor that contains subthreshold poles, and the outerfunction φ ( q , t ) is an arbitrary analytic function. Unitarity constrains the sum of the squares of5 he B → π l ν form factors with domain-wall fermions Taichi Kawanai f + / q [GeV ] c /d.o.f. = 0.41, p-value = 66% q preliminary f + f mixed z-fit 0.000.050.100.150.20 -0.3 -0.2 -0.1 0 0.1 0.2 P f f + / z c /d.o.f. = 0.41, p-value = 66% preliminary f + f mixed z-fit Figure 3:
Extrapolation in q of the factors f + and f using the model-independent BGL z -parametrization,and imposing the kinematic constraint f + ( q = ) = f ( q = ) . The left plot shows f + / vs. q , whilethe right plot shows P φ f + / vs. z . The black curves with error bands show the parametrization of the formfactors over the full kinematic range. Errors are statistical only. the coefficients because the decay process of B → π l ν semileptonic decay is related to the scatteringprocess l ν → B π by crossing symmetry. When the outer function is chosen as in Ref. [18], the sumof the squares of the coefficients is bounded by unity: ∑ Nk = ( a ( k ) ) ≤ N . Therefore only asmall number of terms is needed to accurately describe the shape of the form factors over the fullkinematic range.Fig. 3 shows fits of the form factors f + and f using the z -parametrization in Boyd, Grinstein,and Lebed (BGL) [18]. In the fits, we includes terms up to z for f + and z for f , and the kinematicconstraint f + ( q = ) = f ( q = ) is imposed. The resulting slope and curvature for the B → π l ν vector form factor f + are a ( )+ / a ( )+ = − . ± .
63 (4.9) a ( )+ / a ( )+ = − . ± . , (4.10)where the errors are from statistics only. Our preliminary values of the slope and curvature areconsistent with the independent lattice determination from the Fermilab Lattice and MILC Collab-orations [2].
5. Outlook
We are currently estimating the systematic uncertainties in the form factors f + and f . Weexpect the dominant source of error to be from the chiral-continuum extrapolation, and that ourtotal error will be competitive with that of Ref. [2].Once we have a complete error budget, we will extrapolate our results from the simulatedpion energies to the full q range using the z -parametrization. Currently we are using the z -parametrization of Boyd, Grinstein, and Lebed, but we will also consider the alternative parametriza-tion of Bourrely, Caprini, and Lellouch [20]. We will perform the z -fit to the lattice data alone toprovide a model-independent parametrization of our result valid over the full kinematic range. We6 he B → π l ν form factors with domain-wall fermions Taichi Kawanai will also perform a simultaneous z -fit of our data and experimental measurements of the B → π differential branching fraction to obtain | V ub | .Our results will provide an important independent check on existing calculations, all of whichuse staggered light quarks.
6. Acknowledgments
The authors wish to thank our collaborators in the RBC and UKQCD Collaborations for help-ful discussions. Computations for this work were mainly performed on resources provided by theUSQCD Collaboration, funded by the Office of Science of the U.S. Department of Energy, as wellas computers at BNL and Columbia University. T. Kawanai was partially supported by JSPS Strate-gic Young Researcher Overseas Visits Program for Accelerating Brain Circulation (No. R2411).O.W. acknowledges support at Boston University by the U.S. DOE grant DE-SC0008814. BNLis operated by Brookhaven Science Associates, LLC under Contract No. DE-AC02-98CH10886with the U.S. Department of Energy. Fermilab is operated by Fermi Research Alliance, LLC, underContract No. DE-AC02-07CH11359 with the U.S. Department of Energy.
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