$B \to \pi \ell \nu$ and $B_s \to K \ell \nu$ form factors and $|V_{ub}|$ from 2+1-flavor lattice QCD with domain-wall light quarks and relativistic heavy quarks

4.3 MB·English

by  J. M. Flynn

FERMILAB-PUB-15-019-T
B → π(cid:96)ν and B → K(cid:96)ν form factors and |V | from 2+1-flavor lattice QCD with
s ub
domain-wall light quarks and relativistic heavy quarks
J. M. Flynn,1 T. Izubuchi,2,3 T. Kawanai,2,3,∗ C. Lehner,3 A. Soni,3 R. S. Van de Water,4 and O. Witzel5,†
(RBC and UKQCD Collaborations)
1School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK
2RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA
3Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA
4Theoretical Physics Department, Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
5Center for Computational Science, Boston University, Boston, MA 02215, USA
5 (Dated: April 28, 2015)
1
0 We calculate the form factors for B →π(cid:96)ν and Bs →K(cid:96)ν decay in dynamical lattice Quantum
2 Chromodynamics (QCD) using domain-wall light quarks and relativistic b quarks. We use the
(2+1)-flavor gauge-field ensembles generated by the RBC and UKQCD collaborations with the
r
p domain-wall fermion action and Iwasaki gauge action. For the b quarks we use the anisotropic
A cloveractionwitharelativisticheavy-quarkinterpretation. Weanalyzedataattwolatticespacings
of a ≈ 0.11,0.086 fm with unitary pion masses as light as M ≈ 290 MeV. We simultaneously
π
6 extrapolate our numerical results to the physical light-quark masses and to the continuum and
2 interpolate in the pion/kaon energy using SU(2) “hard-pion” chiral perturbation theory for heavy-
light meson form factors. We provide complete systematic error budgets for the vector and scalar

] formfactorsf (q2)andf (q2)forbothB →π(cid:96)ν andB →K(cid:96)ν atthreemomentathatspantheq2
t + 0 s
a range accessible in our numerical simulations. Next we extrapolate these results to q2 =0 using a
l model-independentz-parametrizationbasedonanalyticityandunitarity. Wepresentourfinalresults
-
p for f (q2) and f (q2) as the coefficients of the series in z and the matrix of correlations between
+ 0
e them; this provides a parametrization of the form factors valid over the entire allowed kinematic
h range. Our results agree with other three-flavor lattice-QCD determinations using staggered light
[ quarks,andhavecomparableprecision,therebyprovidingimportantindependentcross-checks. Both

B →π(cid:96)ν and B →K(cid:96)ν decays enable determinations of the Cabibbo-Kobayashi-Maskawa matrix
3 s
v element|Vub|. Toillustratethis,weperformacombinedz-fitofournumericalB →π(cid:96)ν form-factor
3 data with the experimental measurements of the branching fraction from BaBar and Belle leaving
7 the relative normalization as a free parameter; we obtain |Vub|=3.61(32)×10−3, where the error
3 includesstatisticalandallsystematicuncertainties. Thesameapproachcanbeappliedtothedecay
5 Bs → K(cid:96)ν to provide an alternative determination of |Vub| once the process has been measured
0 experimentally. Finally, in anticipation of future experimental measurements, we make predictions
. for B →π(cid:96)ν and B →K(cid:96)ν differential branching fractions and forward-backward asymmetries in
1 s
the Standard Model.
0
5
PACSnumbers: 11.15.Ha12.38.Gc13.20.He14.40.Nd
1
:
v
i I. INTRODUCTION probes for heavy new particles that may enter virtual
X loops. Decaysinvolvingτ leptonsareespeciallysensitive
r Semileptonic B-meson decays play an important role to charged Higgs bosons that arise in many new-physics
a
in the search for new physics in the quark-flavor sec- models (see e.g. Ref. [1] and references therein).
tor. Tree-level decays that occur via charged W-boson The decays B →π(cid:96)ν and Bs →K(cid:96)ν probe the quark-
exchange are used to obtain the Cabibbo-Kobayashi- flavor-changingtransitionb→u. IntheStandardModel,
Maskawa (CKM) matrix elements |Vub| and |Vcb|, while thedifferentialdecayratefortheseprocessesintheB(s)-
flavor-changing neutral-current decays provide sensitive meson rest frame is given by
∗ Present address: Forschungszentrum Ju¨lich, Institute for Ad-
† Presentaddress: HiggsCentreforTheoreticalPhysics,Schoolof
vancedSimulation,Ju¨lichSupercomputingCentre,52425Ju¨lich,
Physics & Astronomy, The University of Edinburgh, EH9 3FD,
Germany
UK
2
dΓ(B →P(cid:96)ν) G2|V |2 (q2−m2)2(cid:112)E2 −M2(cid:20)(cid:18) m2(cid:19)
(s) = F ub (cid:96) P P 1+ (cid:96) M2 (E2 −M2)|f (q2)|2
dq2 24π3 q4M2 2q2 B(s) P P +
B(s)
3m2 (cid:21)
+ (cid:96)(M2 −M2)2|f (q2)|2 , (1)
8q2 B(s) P 0
whereP denotesthelightpseudoscalarpionorkaonand nonperturbatively in Ref. [18]. We renormalize the lat-
q ≡(p −p )isthemomentumtransferredtotheoutgo- tice heavy-light vector current using the mostly nonper-
B P
ing charged-lepton-neutrino pair. The vector and scalar turbative method introduced in Ref. [19], in which we
form factors f (q2) and f (q2) parametrize the hadronic compute the bulk of the matching factor nonperturba-
+ 0
contributions to the electroweak decay and must be cal- tively [14, 16], with a small correction, that is close to
culated nonperturbatively, such as with lattice QCD. unity, evaluated in lattice perturbation theory [20, 21].
Given an experimental measurement of the branching We also improve the lattice heavy-light current through
fraction and a theoretical calculation of the form fac- O(α a).
s
tor(s), these decays enable a determination of the CKM Weanalyzedataonfivesea-quarkensembleswithuni-
matrixelement|V |. (ThecontributionfromfBπ(q2)in tary pions as light as ≈ 290 MeV and two lattice spac-
ub 0
Eq. (1) can be neglected for light leptons (cid:96) = e,µ given ings of a ≈ 0.11 and 0.086 fm. We simultaneously
the current experimental and theoretical precision.) To extrapolate our numerical results to the physical light-
date, both the BaBar and Belle experiments have mea- quark masses and to the continuum and interpolate in
sured B(B → π(cid:96)ν) [2–5], and the experimental uncer- the pion/kaon energy using SU(2) “hard-pion” chiral
taintywillcontinuetoimprovewiththecollectionofdata perturbation theory (χPT) for heavy-light meson form
at Belle II. The decay B →K(cid:96)ν has not yet been mea- factors [22, 23], which applies when the pion/kaon en-
s
sured, but we anticipate a result from LHCb in the next ergy is large compared to its rest mass. For B → π(cid:96)ν
few years. (B → K(cid:96)ν), we directly simulate in the momentum re-
s
The CKM matrix element |Vub| places a constraint on gion qm2ax > q2 ∼> 19.0 GeV2 (qm2ax > q2 ∼> 17.6 GeV2).
the apex of the CKM unitarity triangle [6–8]. Its value, Both statistical errors and discretization errors increase
however, is under scrutiny because of the long-standing at lower q2, which corresponds to larger pion/kaon en-
∼ 3σ disagreement between |V | obtained from exclu- ergies. To extend our results beyond the momenta ac-
ub
sive B → π(cid:96)ν decay and |V | obtained from inclusive cessible in our simulations, we extrapolate our results
ub
B →X (cid:96)ν decays, where X denotes all charmless final to q2 = 0 using a model-independent z-parametrization
u u
states with up quarks [6–11]. The value of |V | can also based on analyticity and unitarity [24, 25]. Our results
ub
in principle be obtained from leptonic B → τν decay, can be combined with current and future experimental
but the current determination from this process lies in measurements of the experimentally measured B →π(cid:96)ν
between those from exclusive and inclusive semileptonic and B → K(cid:96)ν branching fractions to obtain the CKM
s
decays, and is not as precise [11]. Further, B → τν is matrix element |V |.
ub
sensitivetocharged-Higgsbosonexchange,andtherefore There are two earlier published (2+1)-flavor calcu-
does not provide a clean Standard-Model determination lations of the B → π(cid:96)ν semileptonic form factor in
of |V |. Thus the decay B → K(cid:96)ν, once measured the literature by the HPQCD and Fermilab/MILC col-
ub s
experimentally, will provide an important new determi- laborations [26, 27]; updates of these works are in
nation of |V |. progress [28, 29]. In addition, HPQCD recently ob-
ub
In this paper we present a new calculation of the tained the first results for the B → K(cid:96)ν form factor
s
semileptonic form factors for B → π(cid:96)ν and B → K(cid:96)ν in Ref. [30]. Both groups use the MILC collaboration’s
s
in (2+1)-flavor lattice QCD. Preliminary results were asqtad-improvedstaggeredgauge-fieldensembles[31,32],
presented in Refs. [12, 13]. This is the second in a se- so their results are somewhat correlated. The differ-
ries of B-meson matrix-element calculations that uses ences between the two sets of calculations lie in the
the same lattice actions and ensembles, and our anal- choices of light valence- and b-quark actions. For the
ysis follows a similar approach to our earlier work on b quarks, HPQCD uses the NRQCD action [33] while
B-meson decay constants [14]. We use the gauge-field Fermilab/MILC uses a relativistic formulation similar to
ensembles generated by the RBC and UKQCD collabo- ours. Specifically,theyusetheFermilabinterpretationof
rations withthe domain-wallfermion action and Iwasaki theisotropiccloveraction[34]withthetadpole-improved
gluon action which include the effects of dynamical u,d, tree-level value of the clover coefficient c . The more
SW
andsquarks[15,16]. Forthebottomquarks,weusethe recent HPQCD calculation uses the HISQ action for the
Columbia version of the relativistic heavy-quark (RHQ) light valence quarks to reduce taste-breaking discretiza-
action introduced by Christ, Li, and Lin in Ref. [17], tioneffects,whileintheotherworkasqtadvalencequarks
with the parameters of the action that were obtained are used.
3
Our form-factor calculation with domain-wall light A. Form factors
quarks and RHQ b quarks has the advantage that dis-
cretization errors from the light quarks and gluons are TheB →π(cid:96)νandB →K(cid:96)νsemileptonicformfactors
s
simpler, such that the SU(2) heavy-light meson χPT parametrize the hadronic matrix element of the b → u
expressions are continuum-like. Further, as compared vector current Vµ ≡uγµb:
to the Fermilab/MILC calculation, we tune the coeffi-
cient of the clover term in the b-quark action nonper- (cid:32) M2 −M2 (cid:33)
(cid:104)P|Vµ|B (cid:105)=f (q2) pµ +pµ − B(s) Pqµ
turbatively and improve the heavy-light vector current (s) + B(s) P q2
through O(α a), whereas Fermilab/MILC only improve
s
it through O(a). Thus, for similar values of the lattice M2 −M2
+f (q2) B(s) Pqµ, (2)
spacing, discretization errors from the heavy-quark ac- 0 q2
tion and current are smaller in our calculation. Our new
results for the B → π(cid:96)ν and B → K(cid:96)ν form factors where f (q2) and f (q2) are the vector and scalar form
s + 0
therefore enable important independent determinations factors, respectively. It is convenient in lattice simula-
of the CKM matrix element |V |. tions to instead calculate the form factors f (E ) and
ub (cid:107) P
f (E ), which are defined by
This paper is organized as follows. Section II provides ⊥ P
anoverviewofthelatticecalculation. Firstwedefinethe (cid:104)P|Vµ|B (cid:105)=(cid:113)2M (cid:2)vµf (E )+pµf (E )(cid:3) , (3)
needed matrix elements and form factors in Sec. IIA. (s) B(s) (cid:107) P ⊥ ⊥ P
Next we present the lattice actions and parameters in
where E is the outgoing light pseudoscalar meson en-
Sec.IIB.Then, inSec.IICwedescribetherenormaliza- P
ergy, vµ ≡ pµ /M is the B -meson velocity, and
tion and improvement of the heavy-light vector current B(s) B(s) (s)
operator. Section III presents the numerical analysis. pµ⊥ ≡ pµP − (pP · v)vµ. In the B(s)-meson rest frame,
First, in Secs. IIIA and IIIB we fit lattice two-point whichwewilluseforoursimulations, f(cid:107) andf⊥ arepro-
and three-point correlators to extract the needed me- portional to the hadronic matrix elements of the tempo-
son masses and matrix elements, respectively. Then, in ral and spatial vector currents:
Sec.IIICweextrapolateournumericaldatatothephysi-
(cid:104)P|V0|B (cid:105)
callight-quarkmassesandcontinuum,andinterpolatein f (E )= (s) , (4)
the pion/kaon energy, using SU(2) hard-pion χPT. Sec- (cid:107) P (cid:112)2MB(s)
tion IV provides complete error budgets for f+(q2) and (cid:104)P|Vi|B(s)(cid:105) 1
f0(q2)atthreemomentumvaluesthatspantherangeac- f⊥(EP)= (cid:112)2M pi . (5)
cessible in our numerical simulations; for clarity, we dis- B(s) P
cuss each source of systematic uncertainty in a separate Thevectorandscalarformfactorscanbeeasilyobtained
subsection. In Section V we extrapolate our form-factor from f and f via
(cid:107) ⊥
datatoq2 =0usingamodel-independentz parametriza-
tion. We present our results for f+(q2) and f0(q2) as the f (q2)= 1 (cid:104)f (E )+(cid:0)M −E (cid:1)f (E )(cid:105),
coefficients of the series in z and the matrix of correla- + (cid:112)2M (cid:107) P B(s) P ⊥ P
B(s)
tions between them; this provides a model-independent
(6)
parametrization of the form factors valid over the entire
(cid:112)
allowed kinematic range. We illustrate the phenomeno- f (q2)= 2MB(s) (cid:104)(cid:0)M −E (cid:1)f (E )
logical utility of our form-factor results in Sec. VI. First, 0 M2 −M2 B(s) P (cid:107) P
iincaSleBc. V→IAπ,(cid:96)νwefopremrf-ofarmctoarcdoamtabinweidthz-tfihteoefxopuerrinmuemnetar-l +(cid:0)EPB2(s−) MP2(cid:1)Pf⊥(EP)(cid:105). (7)
measurements of the branching fraction from BaBar and
Belle to determine |V |. Next, in Sec. VIB, we make
ub
predictions for Standard-Model observables for the de- B. Actions and parameters
cay processes B → π(cid:96)ν and B → K(cid:96)ν with (cid:96) = µ,τ in
s
anticipation of future experimental measurements. Sec- We use the (2 + 1)-flavor domain-wall fermion and
tion VII concludes with a comparison of our results with Iwasakigauge-fieldensemblesgeneratedbytheRBCand
otherlatticedeterminations,andwithanoutlookforthe UKQCD collaborations [15, 16]. We perform measure-
future. ments at five different light sea-quark masses m and at
l
two lattice spacings of a ≈ 0.11 fm (a−1 ≈ 1.729 GeV)
anda≈0.086fm(a−1 ≈2.281GeV).Thelightsea-quark
masses m correspond to pion masses of 289 MeV (cid:46)
l
M (cid:46)422 MeV. The up and down sea-quark masses are
II. LATTICE CALCULATION π
degenerate and the strange sea-quark mass m is tuned
h
within 10% of its physical value. The spatial volumes
Here we present the setup of our numerical lattice cal- are approximately (2.6 fm)3, such that M L ≥ 4. We
π
culation. summarize the simulation parameters in Table I.
4
TABLE I. Lattice simulation parameters [15, 16]. The columns list the lattice volume, approximate lattice spacing, light (m)
l
andstrange(m )sea-quarkmasses,residualchiralsymmetrybreakingparameterm ,physicalu/d-ands-quarkmass,unitary
h res
pionmass,numberofconfigurationsanalyzedandnumberofsources. Thetildesoveram andam denotethatthesevalues
(cid:101)u/d (cid:101)s
include the residual quark mass.
(cid:0)L(cid:1)3×(cid:0)T(cid:1) ≈a(fm) a−1 [GeV] am am am am am M [MeV] # configs. # time sources
a a l h res (cid:101)u/d (cid:101)s π
243×64 0.11 1.729(25) 0.005 0.040 0.003152 0.00136(4) 0.0379(11) 329 1636 1
243×64 0.11 1.729(25) 0.010 0.040 0.003152 0.00136(4) 0.0379(11) 422 1419 1
323×64 0.086 2.281(28) 0.004 0.030 0.0006664 0.00102(5) 0.0280(7) 289 628 2
323×64 0.086 2.281(28) 0.006 0.030 0.0006664 0.00102(5) 0.0280(7) 345 889 2
323×64 0.086 2.281(28) 0.008 0.030 0.0006664 0.00102(5) 0.0280(7) 394 544 2
In the valence sector we use for the light quarks the product:
domain-wall action [35, 36] and generate propagators
withperiodicboundaryconditionsinspaceandtimeand
(cid:113)
withthesamedomain-wallheight(M5 =1.8)andextent Zbl =ρbl ZbbZll. (9)
of the fifth dimension (L = 16) as in the sea sector. Vµ Vµ V V
s
We generate both unitary light valence-quark propaga-
torswiththesamemassasthelightseaquarksandprop-
Most of the heavy-light current renormalization comes
agators with a mass close to the physical strange quark.
from the flavor-conserving factors Zbb and Zll. The re-
On the coarser ensembles we choose am = 0.0343 and V V
s mainingfactorρbl isexpectedtobeclosetounitybecause
on the finer ensembles am =0.0272. V
s mostoftheradiativecorrections,includingcontributions
For the bottom quarks, we use the Columbia version
from tadpole graphs, cancel [37].
of the relativistic heavy quark (RHQ) action [17] to con-
trol heavy-quark discretization errors introduced by the Bothflavor-conservingrenormalizationfactorsZVbband
large lattice b-quark mass. We use the anisotropic O(a) ZVll were computed nonperturbatively in previous works.
improved Wilson-clover action with the following three We computed ZVbb for our earlier calculation of B-meson
parameters: the bare-quark mass m a, clover coefficient decay constants from the matrix element of the b → b
0
cP, and anisotropy parameter ζ. In this work we use the vector current between two Bs mesons [14]. We can also
RHQ parameterstuned nonperturbatively in Ref.[18] to take advantage of the fact that for domain-wall fermions
reproduce the experimentally measured Bs-meson mass ZVll = ZAll up to corrections of O(amres) and use the de-
and hyperfine splitting; we list their values in Table II. terminationofZAll fromRef.[16]. Theflavoroff-diagonal
renormalizationfactorρbl iscalculatedatO(α )inmean-
We reduce autocorrelations between our lattices by V s
field improved lattice perturbation theory [38] and eval-
shiftingthegaugefieldsbyarandom4-vectorbeforecre-
ating the sources for the valence-quark propagators used uated at the MS coupling αMS(µ=a−1). Our perturba-
s
in the 2-point and 3-point correlation functions. This tive computation extends the work of Ref. [39] to bilin-
random4-vectorshiftisequivalenttoplacingthesources ears with one relativistic heavy quark in the Columbia
at random positions in spacetime but simplifies the sub- formulation and one domain-wall light quark. For αMS,
s
sequent analysis. On the finer ensembles, we double the weuseEq.(167)ofRef.[39],whichdoesnottakeintoac-
statistics by using two sources per configuration sepa- count sea-quark effects. Because sea-quark effects enter
rated by half the lattice temporal extent. at two loops, however, and the rest of the computation
is performed at one loop, the error introduced by setting
N =0 is of the same size as other remaining truncation
f
errors. Furtherdetailsoftheperturbativecalculationwill
C. Operator renormalization and improvement
be provided in a forthcoming publication [21]. Table III
shows the renormalization factors used in this work.
Tomatchthelatticeamplitudestothecontinuumma-
trix elements, we multiply by the heavy-light renormal- We reduce discretization errors in the heavy-light vec-
ization factor ZVblµ: ttohrecmurarternixtbeyleimmepnrtovoinfgthitethtrreoeu-glehvOel(αhesaav).y-Wligehctomvepcutoter
current
(cid:104)P|V |B (cid:105)=Zbl (cid:104)P|V |B (cid:105), (8)
µ (s) Vµ µ (s)
where Vµ and Vµ are the continuum and lattice current V0(x)=q(x)γµQ(x), (10)
operators, respectively. Following Ref. [19] we calculate µ
therenormalizationfactorZbl usingamostlynonpertur-
Vµ
bative method in which we express Zbl as the following plusmatrixelementsoftheseadditionalsingle-derivative
Vµ
5
TABLE II. Tuned RHQ parameters on the 243 and 323 ensembles [18]. The errors listed for m a, c , and ζ are from left
0 P
to right: statistics, heavy-quark discretization errors, the lattice scale uncertainty, and the uncertainty in the experimental
measurement of the B -meson hyperfine splitting, respectively.
s
m a c ζ
0 P
a≈0.11 fm 8.45(6)(13)(50)(7) 5.8(1)(4)(4)(2) 3.10(7)(11)(9)(0)
a≈0.086 fm 3.99(3)(6)(18)(3) 3.57(7)(22)(19)(14) 1.93(4)(7)(3)(0)
TABLE III. Operator renormalization factors and improvement coefficients. The flavor-conserving Z factors were obtained
nonperturbatively [14, 16]. The ρ factors and improvement coefficients cn were computed at one loop in mean-field improved
i
lattice perturbation theory and are evaluated at αMS(a−1) [21].
s
Zll Zbb αMS(a−1) ρ ρ c3 c4 c1 c2 c3 c4
V V s V0 Vi t t s s s s
a≈0.11 fm 0.71689(51) 10.039(25) 0.23 1.02658 0.99723 0.0558 -0.0099 -0.00079 0.0018 0.0485 -0.0033
a≈0.086 fm 0.74469(13) 5.256(8) 0.22 1.01661 0.99398 0.0547 -0.0095 -0.0012 0.00047 0.0480 -0.0020
operators III. ANALYSIS
→−
V1(x)=q(x)2D Q(x), (11)
µ µ Herewepresentourdeterminationsoftheformfactors
←−
Vµ2(x)=q(x)2DµQ→−(x), (12) fva+l(uqe2s)oafnqd2 ∼>f0(1q92.)0fGoreVB2 (→q2π∼>(cid:96)ν17(.6BsGe→V2K) a(cid:96)νcc)esastibllaergine
Vµ3(x)=q(x)2γµγiDiQ(x), (13) our numerical simulations.
←− Our analysis proceeds in three steps: First, in
V4(x)=q(x)2γ γ D Q(x), (14)
µ µ i i Sec. IIIA, we fit the pion, kaon, and B -meson 2-point
(s)
where the covariant derivatives are defined by correlation functions to obtain the ground-state meson
masses. The results for these meson masses then enter
→− 1 our 3-point correlator fits in Sec. IIIB to obtain the lat-
D Q(x)= (U (x)Q(x+µˆ)
µ 2 µ tice form factors f (E ) and f (E ) at fixed values of
(cid:107) P ⊥ P
−Uµ†(x−µˆ)Q(x−µˆ)), (15) the pion/kaon energy EP. In Sec. IIIC, we interpolate
the renormalized values for f (E ) and f (E ) in en-
←− 1 (cid:107) P ⊥ P
q(x)D = (q(x+µˆ)U†(x) ergy, and extrapolate to the physical light-quark masses
µ 2 µ
andthecontinuumlimit,usingSU(2)hard-pionχPTfor-
−q(x−µˆ)U (x−µˆ)). (16)
µ mulatedforheavy-lightmesons. Toavoidpossiblebiases
due to analysis choices, we use the same fit functions in
The temporal and spatial O(a)-improved vector-current
the correlator and chiral fits for both processes B →π(cid:96)ν
operators are given by the following sums:
and B → K(cid:96)ν, and fitting ranges that are as close as
s
Vimp(x)=V0(x)+c3V3(x)+c4V4(x), (17) possible.
0 0 t 0 t 0
Wepropagatestatisticalerrorsthroughouttheanalysis
Vimp(x)=V0(x)+c1V1(x)+c2V2(x)
i i s i s i via a single-elimination jackknife procedure. We avoid a
+c3sVi3(x)+c4sVi4(x). (18) directdependenceonthelatticescalebycarryingoutour
analysis in units of the B -meson mass. The B -meson
We calculate the coefficients cn and cn at one loop s s
t s mass plays a special role because we tuned parameters
in mean-field improved lattice perturbation theory [21];
of the b-quark action to match the experimental value.
the values of the coefficients evaluated at αMS(a−1) are
s Thus we can obtain the form factors in physical units
shown in Table III.
after the chiral-continuum extrapolation by multiplying
by Mexp. to the appropriate power. With this approach,
Bs
the uncertainty onthe lattice scale enters onlyindirectly
via the values of the RHQ parameters.
WeusetheChromasoftwarelibraryforlatticeQCDto
compute our numerical data for the lattice 2-point and
3-point correlation functions [40].
6
TABLE IV. Time ranges used in two-point and three-point fits to determine the lattice meson masses and form factors. For
the three-point fits, we use the same range for all operators and momenta.
[t , t ]
min max
2-point fits 3-point fits
M M M M fBπ fBπ fBsK fBsK
π K B Bs (cid:107) ⊥ (cid:107) ⊥
a≈0.11 fm [12,23] [12,23] [7,30] [10,30] [6,10] [6,10] [6,10] [6,10]
a≈0.086 fm [16,30] [16,30] [9,30] [13,30] [8,13] [8,13] [8,13] [8,13]
A. Two-point correlator fits fine ensemble with am = 0.004. To minimize bias, we
l
use the same fit range for all ensembles with the same
To obtain the lattice B → P amplitude, we first lattice spacing (although different for light and heavy-
(s)
calculate the following two-point correlation functions: lightmesons); thesefitrangesaregiveninTableIV.The
resultingpion/kaonandB -mesonmassesonallensem-
CP(t,p(cid:126)P)=(cid:88)eip(cid:126)P·(cid:126)x(cid:104)OP†((cid:126)x,t)OP((cid:126)0,0)(cid:105), (19) bles are given in Tables XV(s)and XVI, respectively.
(cid:126)x
CB(s)(t)=(cid:88)(cid:104)OB†(s)((cid:126)x,t)O˜B(s)((cid:126)0,0)(cid:105), (20) shoInuldthseatciosnfytintuhuemdislipmerits,iotnhereplaitoinonanEdP2k=aonMeP2ne+rgp(cid:126)ie2Ps
(cid:126)x and the amplitudes of the two-point functions Z =
P
C˜B(s)(t)=(cid:88)(cid:104)O˜B†(s)((cid:126)x,t)O˜B(s)((cid:126)0,0)(cid:105), (21) |W(cid:104)0e|OobPt|aπi(cid:105)n| tshheoualmdpbleituinddeespfernomdenctororfeltahteedmpolmateenatuumfitsp(cid:126)Pto.
(cid:126)x
where O = qγ q and O = Qγ q are interpolating
P 5 B(s) 5
operators for the light pseudoscalar and B -meson, re- (cid:114)
(s) 2E C (t,p(cid:126) )
spectively. Both pions and kaons are simulated with a Z (t)= P P P (24)
pointsourceandpointsink,whereasb-quarkpropagators P e−EPt
are generated with a gauge-invariant Gaussian smeared
source [41, 42] to reduce excited state contamination.
usingthesamefitrangesasforthemasses. Figure2com-
We employ the same smearing parameters optimized in
pares the measured pion and kaon energies and ampli-
Ref. [18] and denote a smeared source in Eqs. (19)-(21)
tudes with continuum expectations on the a≈0.086 fm,
with a tilde above the operator.
am = 0.004 ensemble. The measured kaon energies
We obtain the pion or kaon energy and B -meson l
(s) and amplitudes agree remarkably well with the predic-
mass from the exponential decay of the correlators in
tions from the continuum dispersion relation, to within
Eqs. (19) and (20). The correlator in Eq. (21) is used to
5% even at the largest momentum ap(cid:126) = 2π(2,0,0)/L.
normalizetheB →P three-pointfunction. Weworkin K
(s) Although the pion data is not precise enough to draw
the B -meson rest frame such that only pions or kaons
(s) strong quantitative conclusions, the measured energies
carry nonzero momentum. In our analysis we use data
and amplitudes still agree with continuum expectations
withdiscretelatticepionmomentathrough2π(1,1,1)/L
withinthelargestatisticaluncertaintiesforallmomenta.
and kaon momenta through 2π(2,0,0)/L. We average
Dispersion-relation plots for the other ensembles show
theresultsforallequivalentmomenta,i.e. withdifferent
similar behavior.
spatial directions but the same total |p(cid:126) |. We effectively
P
double our statistics by folding the two-point correlators The kaon data, for which both the energies and am-
atthetemporalmidpointofthelattice,therebyaveraging plitudes are statistically well resolved, provides an ac-
forward- and backward-propagating states. curate measure of momentum-dependent discretization
At sufficiently large lattice times, the ground-state effects, while the pion data provides only a rough cross-
masses and energies can be determined from simple two- check. On all ensembles, the measured pion and kaon
pointcorrelatorratios. Wedefinethelightpseudoscalar- energiesbothagreewithinstatisticalerrorswiththepre-
meson effective energy and B -meson effective mass as dictions from the continuum dispersion relation, and the
(s)
(cid:20) (cid:21) measured pion and kaon amplitudes agree with the zero-
C (t,p(cid:126) )+C (t+2,p(cid:126) )
E (t,p(cid:126) )=cosh−1 P P P P , (22) momentum result. Thus, in our determinations of the
P P C (t+1,p(cid:126) )
P P latticeformfactorsf andf inthenextsection, weuse
(cid:107) ⊥
(cid:34) (cid:35)
C (t)+C (t+2) pion and kaon energies calculated from the continuum
MB(s)(t)=cosh−1 B(s)C (tB+(s)1) . (23) dispersion relation (rather than the measured values) to
B(s) reduce the statistical uncertainties. Although we do not
We perform correlated, constant-in-time, fits to these use the amplitudes obtained from Eq. (24) in our subse-
expressions, choosing fit ranges without visible excited- quent form-factor determinations, the observed momen-
statecontaminationthatleadtoacceptablepvalues. Fig- tumindependenceofZ providesfurthersupportforthis
P
ure 1 shows example meson-mass determinations on our strategy.
7
0.15
2.36
aMp = 0.12611(51), p = 37% aMB = 2.3203(14) , p = 11%
0.14 2.34
(t) 0.13 (t)B2.32
Mp M
a a
0.12 2.30
2.28
0.11
0 5 10 15 20 25 30 0 5 10 15 20 25 30
time slice time slice
2.38
0.25 aMp = 0.23249(42), p = 17% aMBs = 2.3509(11) , p = 7%
2.36
(t)K 0.24 (t)Bs2.34
M M
a 0.23 a
2.32
0.22
2.30
0 5 10 15 20 25 30 0 5 10 15 20 25 30
time slice time slice
FIG. 1. Effective masses of the pion (upper left), kaon (bottom left), B meson (upper right) and B meson (bottom right) on
s
thea≈0.086fmensemblewitham =0.004. Shadedbandsshowthecorrelatedfitresultswithjackknifestatisticalerrorsover
l
the fit ranges used. All results are shown in lattice units.
1.3 1.3
1.2 (a→pπ)2 1.2 (a→pπ)2
2+p )π 1.1 (0)Zπ 1.1
22E / (m 01..90 () / pππ 01..90
Z
0.8 0.8
0.7 0.7
0 0.05 0.1 0.15 0 0.05 0.1 0.15
(a→pπ)2 (a→pπ)2
1.2 1.2
(afip )2 (afip )2
K K
222)+p / (mEK 011...901 (0)) / (ZpZKKK011...901
0.8 0.8
0 0.05 0.1 0.15 0 0.05 0.1 0.15
(afip )2 (afip )2
K K
FIG.2. Comparisonofpion(top)andkaon(bottom)energies(left)andamplitudes(right)withcontinuum-limitexpectations
on the a ≈ 0.086 fm ensemble with am = 0.004. The dashed lines show a power-counting estimate of the leading O((ap(cid:126))2)
l
momentum-dependent discretization errors.
8
0.4
0R3,0
−1/2 Bs 0.3 tsnk = 24
M t = 26
snk
t = 28
snk
0.2 tsnk = 30
t = 32
snk
FIG. 3. Three-point correlation function used to obtain the
B →P form factors. The single and double lines correspond 0 5 10 15 20 25 30
tolight-andb-quarkpropagators,respectively. ForB →π(cid:96)ν, time slice
the spectator-quark mass (ml(cid:48)) is the same as the light sea-
quarkmass,whileforBs →K(cid:96)ν,thespectator-quarkmassis FIG. 4. Unimproved three-point ratio R30,0 for p(cid:126)π = 0 with
closetothephysicalms. Thelightdaughter-quarkmass(ml) several source-sink separations tsnk on the a ≈ 0.086 fm en-
isalwaysequaltotothelightsea-quarkmass. Blackandgrey semble with am =0.004.
l
circles denote local and smeared operators, respectively.
with
B. Three-point correlator fits
R (t,t ,p(cid:126) )
3,µ snk P
wtioeTncosa:elcxutlraatcettthhee dfoelsliorwedinBg(tsh)r→eePpohinadtrcoonrircelaamtiponlitufudnecs-, = (cid:113)CPC(t3i,m,µp(cid:126)p()t,C˜tBsn(ks),(p(cid:126)tP) −t)(cid:114)e−EPte2−EMPB(tsnk−t),
2 P 2 snk
(28)
Cimp(t,t ,p(cid:126) )
3,µ snk P whereweusethecontinuumdispersionrelationandmea-
=(cid:88)eip(cid:126)P·(cid:126)y(cid:104)O˜† ((cid:126)x,t )Vimp((cid:126)y,t)O ((cid:126)0,0)(cid:105), (25) suredlightpseudoscalar-mesonmassM toconstructthe
B(s) snk µ P P
(cid:126)x,(cid:126)y energyEP. Todeterminetheoptimalsource-sinksepara-
tionforC (t,t ,p(cid:126) ),wecarriedoutadedicatedstudy.
3,µ snk P
where the improved lattice temporal and spatial lattice We computed the unimproved ratio R0 for several val-
3,µ
vector currents Vimp are defined in Eqs. (17) and (18). ues of the source-sink separation on one a ≈ 0.086 fm
µ
As shown in Fig. 3, we fix the location of the pion or and one a ≈ 0.11 fm ensemble, choosing those with the
kaon at the temporal origin and the location of the B lightest sea-quark mass because they are most sensitive
(s)
meson at time t , and vary the location of the current to excited-state contamination. Figure 4 shows the ra-
snk
operator over all time slices in between. In our calcula- tio R0 for B → π with p(cid:126) = 0 for several source-sink
3,0 π
tions, the mass of the light daughter quark (l) is always separations on the a ≈ 0.086 fm ensemble. All plateaus
equal to the light sea-quark mass. For B → π decay, overlap within statistical uncertainties in the region far
the spectator-quark mass (l(cid:48)) also equals the light sea- from both the source and the sink. The results for the
quark mass. For B → K decay, the spectator-quark ratios R0 and R0 at nonzero momenta and on the
s 3,0 3,i
mass is close to that of the physical strange quark. We a≈0.11 fm ensemble look similar. Because the statisti-
useaGaussian-smearedsequentialsourceforthebquark cal errors increase with larger source-sink separation, we
intheB mesontoreduceexcited-statecontamination. chose t = 26 (20) on the a ≈ 0.086 fm (a ≈ 0.11 fm)
(s) snk
We insert discrete nonzero momentum at the local cur- ensembles. This corresponds to approximately the same
rent operator through p(cid:126) =2π(1,1,1)/L for B →π and physical distance for the two lattice spacings.
π
p(cid:126) = 2π(2,0,0)/L for B → K (recall that the B Figure 5 shows the O(α a)-improved ratios R and
K s (s) s 3,0
meson is at rest). To improve statistics, we compute R /pi for different momenta on the a ≈ 0.086 fm en-
3,i P
the three-point correlators with both positive and nega- semble with am = 0.004. Results for other ensem-
l
tivesource-sinkseparations(±t );wealsoaverageover bles look similar. We perform correlated, constant-in-
snk
equivalent spatial momenta. time, fits to these ratios using fit ranges without visible
Thelatticeformfactorsflat andflat areobtainedfrom excited-statecontaminationthatleadtoacceptablepval-
(cid:107) ⊥
thefollowingratiosofcorrelationfunctionsfarawayfrom ues. To minimize bias, we use the same fit range for all
both the pion/kaon source and the B -meson sink: momenta and ensembles with the same lattice spacing;
(s)
these fit ranges are given in Table IV. The complete fit
flat(p(cid:126) )= lim R (t,t ,p(cid:126) ), (26) resultsforthethree-pointratiosaregiveninTablesXVII
(cid:107) P 0(cid:28)t(cid:28)tsnk 3,0 snk P and XVIII.
1 Finally, we obtain the renormalized B → P(cid:96)ν form
flat(p(cid:126) )= lim R (t,t ,p(cid:126) ), (27) (s)
⊥ P 0(cid:28)t(cid:28)tsnk piπ 3,i snk P factors f(cid:107) and f⊥ in the continuum after multiplying by
9
2.0 0.4
π
/Bπp3,i 1.0 BπR3,0 0.2
1/2 MRBs -1/2M Bs →p = 2π(0,0,0)/L, p = 28%
π
0.0 →→→ppp πππ === 222πππ(((111,,,110,,,100)))///LLL,,, ppp === 885950%%% 0.0 →→→ppp πππ === 222πππ(((111,,,110,,,100)))///LLL,,, ppp === 6 1936%%%
0 5 10 15 20 0 5 10 15 20
time slice time slice
2.0 0.4
K
p
/BKs3,i 1.0 BKRs3,0 0.2
1/2 MRBs →p = 2π(1,0,0)/L, p = 72% -1/2M Bs →→pp K == 22ππ((10,,00,,00))//LL,, pp == 2384%%
K K
→p = 2π(1,1,0)/L, p = 33% →p = 2π(1,1,0)/L, p = 10%
0.0 →p K = 2π(1,1,1)/L, p = 14% 0.0 →p K = 2π(1,1,1)/L, p = 11%
K K
→p = 2π(2,0,0)/L, p = 49% →p = 2π(2,0,0)/L, p = 32%
K K
0 5 10 15 20 0 5 10 15 20
time slice time slice
FIG.5. O(α a)-improvedratiosR /pi (left)andR (right)witht =26onthea≈0.086fmensemblewitham =0.004.
s 3,i P 3,0 snk l
Plots for B →πlν are on the top and B →Klν are on the bottom. Fit ranges and fit results with jackknife statistical errors
s
are shown as horizontal bands.
the heavy-light renormalization factors Zbl given in Ta- of the strange-quark mass, as well as on the value of
Vµ
ble III: theb-quarkmassforB-mesonformfactors. “Hard-pion”
χPT, which was introduced by Flynn and Sachrajda for
f(cid:107)(p(cid:126)P)=ZVbl0f(cid:107)lat(p(cid:126)P), (29) thelight-pseudoscalar-mesondecayK →π(cid:96)ν inRef.[43]
f (p(cid:126) )=Zblflat(p(cid:126) ). (30) and later extended to heavy-light-meson decays by Bi-
⊥ P Vi ⊥ P jnens and Jemos in Ref. [23], applies in the kinematic
regime where the pion or kaon energy is large compared
toitsrestmass. Almostallofourlatticesimulationdata
C. Chiral-continuum extrapolation
is in this hard-pion (or kaon) regime. We can obtain the
expressionsfortheB →π(cid:96)ν andB →K(cid:96)ν formfactors
We extrapolate the renormalized lattice form factors s
in hard-pion/kaon χPT by taking the limit of the con-
to the physical light-quark mass, and interpolate in the
tinuumexpressionsfromRef.[22]asM /E →0,where
pion or kaon energy using next-to-leading order (NLO) π P
P =π,K denotes the final-state pseudoscalar meson.
SU(2) chiral perturbation theory for heavy-light mesons
(HMχPT) in the “hard-pion” limit. In the SU(2) the-
ory, the strange-quark mass is integrated out, and only The NLO SU(2) χPT full-QCD expressions for the
thelight-quarks’degrees-of-freedomareincluded. There- B → π(cid:96)ν and B → K(cid:96)ν form factors in the hard-
s
fore the chiral logarithms for B → π(cid:96)ν (B → K(cid:96)ν) pion/kaon limit are functions of the pion mass M , pion
s π
depend on the pion mass and the pion (kaon) energy. orkaonenergyE ,andlatticespacinga. Theyhavetwo
P
The SU(2) low-energy constants depend upon the value general forms:
10
(cid:20) (cid:18) δf M2 E E2 a2 (cid:19)(cid:21)
fB(s)P (M ,E ,a2)=c(1) 1+ (cid:107) +c(2) π +c(3) P +c(4) P +c(5) (31)
no pole π P np (4πf)2 np Λ2 np Λ np Λ2 np Λ2a4
32
1 (cid:20) (cid:18) δf M2 E E2 a2 (cid:19)(cid:21)
fB(s)P(M ,E ,a2)= c(1) 1+ ⊥ +c(2) π +c(3) P +c(4) P +c(5) , (32)
pole π P E +∆ p (4πf)2 p Λ2 p Λ p Λ2 p Λ2a4
P 32
one with a pole at E = −∆ = M − M and
P B∗ B(s) TABLE V. Constants used in the chiral and continuum ex-
one without. Here the B∗ resonance corresponds to
trapolations of the B → π(cid:96)ν and B → K(cid:96)ν form fac-
a state with flavor bu and quantum numbers JP = s
tors [10, 16, 51].
0+ for f and 1− for f . The experimentally mea-
(cid:107) ⊥
sured vector-meson mass is M = 5.3252(4) GeV [10]. a ≈0.11 fm ≈0.086 fm
B∗
The scalar B∗ meson has not been observed experimen- a−1 1.729 GeV 2.281 GeV
aµ 2.348 1.826
tally, but its value has been estimated theoretically us-
f 130.4 MeV
ing heavy-quark and chiral-symmetry arguments to be π
g 0.57
M (0+) = 5.63(4) GeV [44], while the 0+-0− split- b
B∗ Λ 1 GeV
χ
ting has been estimated in (2+1)-flavor lattice QCD to
be M (0+) − M ∼ 400 MeV [45]. In our chiral-
B∗ B
continuum extrapolations we include the effects of res-
using Λ =500 MeV,1 we estimate these to be about
onances below the Bπ and B K production thresholds, QCD
s 5% on the 323 ensembles. The remaining discretization
i.e. q2 < (M +M )2. For B → π(cid:96)ν, the B∗ meson
B(s) P errors–light-quarkandgluondiscretizationerrorsinthe
lies below the Bπ production threshold, so we include a
heavy-light current, and heavy-quark discretization er-
pole in the fit for fBπ taking ∆Bπ = 45.78 MeV from
⊥ ⊥ rors from both the action and current – are expected
experiment [10]. The predicted value of M (0+) is well
B∗ from power counting to be much smaller. In Secs. IVE
aboveM +M , however, sowedonotincludeapolein
B π and IVF, we estimate their sizes to be below 2%. We
thefitoffBπ. ForB →K,bothM andM (0+)are
(cid:107) s B∗ B∗ therefore expect light-quark and gluon discretization er-
belowM +M ,soweincludeapoleinthefitsforboth
Bs K rors from the action to dominate the scaling behavior of
f⊥BsK andf(cid:107)BsK,taking∆B⊥sK =−41.6MeVfromexper- theformfactors,suchthatincludingana2 terminthefit
iment [10] and taking ∆BsK =263 MeV from the model will largely remove these contributions. We will add the
(cid:107)
estimate in Ref. [44]. The precise value of M (0+) has remaining subdominant discretization errors a posteriori
B∗
to the systematic error budget after the chiral fit.
little impact on the fit because the pole location is so far
In addition to the pion masses and pion/kaon ener-
outside the semileptonic region, but we vary its value by
gies,severalparametersentertheexpressionsinEqs.(31)
agenerousamountwhenestimatingthechiral-continuum
and(32). Forcompleteness, wecompilethevaluesofthe
extrapolation error in Sec. IVA.
fixedparametersinourchiralfitsinTableV.Weusethe
Theone-loopchirallogarithmsarethesameforf and
(cid:107) lattice spacings and low-energy constant µ obtained in
f , but differ for B →π(cid:96)ν and B →K(cid:96)ν:
⊥ s Ref. [16] from the RBC/UKQCD analysis of light pseu-
δfBπ =−34(cid:0)3gb2+1(cid:1)Mπ2log(cid:18)MΛ2π2(cid:19) (33) dthoescPalDarGmveasloune mofasfsπes=an1d30d.4e(c2a)yMcoenVsta[1n0t]s,. anWdetauksee
Λ = 1 GeV for the scale in the chiral logarithms. We
χ
3 (cid:18)M2(cid:19) use the B∗Bπ coupling constant g = 0.57(8) obtained
δfBsK =− M2log π , (34) b
4 π Λ2 in our companion analysis also using the RBC/UKQCD
domain-wall+Iwasaki ensembles and the RHQ action for
where g is the B∗Bπ coupling constant. At tree level, the b-quarks [51].
b
the mass of a pion composed of two domain-wall quarks We perform correlated chiral-continuum fits to the
is given in terms of the light-quark mass by data calculated on all five sea-quark ensembles listed in
Table I using the full-QCD NLO SU(2) hard-pion/kaon
HMχPT expressions. For B → π(cid:96)ν, we include dis-
M2 =2µ(m +m ), (35)
π l res crete lattice momenta up to p(cid:126) = 2π(1,1,1)/L, which
π
where µ is a leading-order low-energy constant.
We include a term proportional to a2 in the chiral fit
functions Eqs. (31) and (32) to account for the domi-
1 Recentthree-andfour-flavorlattice-QCDcalculationstypically
nant lattice-spacing dependence. To make the a2 ana-
givevaluesforΛ intherangeofabout300–400MeV[46–50].
MS
lytic term dimensionless with an expected coefficient of The2013FlavorLatticeAveragingGroup(FLAG)reviewquotes
O(1) in χPT, we normalize it using the lattice spacing the range Λ(3) =339(17) MeV for three active flavors [11]. To
onthefiner323 ensemblesa32. Discretizationerrorsfrom beconservatMivSe,wetakeaslightlylargervalueΛQCD=500MeV
the domain-wall and Iwasaki actions are of O(aΛ )2; forthepower-countingestimatesthroughoutthiswork.
QCD