Calculating the Isgur-Wise Function on the Lattice

arXiv:hep-lat/9212035v1 24 Dec 19921Calculating the Isgur-Wise Function on the LatticeClaude W. Bernard,a Yue Shenb ∗and Amarjit SonicaPhysics Department, Washington University, St. Louis, MO 63130, USAbPhysics Department, Boston University, Boston, MA 02215, USAcPhysics Department, Brookhaven National Laboratory, Upton, NY 11973, USAWe calculate the Isgur-Wise function by measuring the heavy-heavy meson transition matrix element on thelattice. The standard Wilson action is used for both the heavy and light quarks.

Our first numerical results arepresented.1. IntroductionBecause of the new flavor and spin symme-tries in the heavy quark effective field theory(HQEFT), the heavy-heavy meson decay matrixelement can be simplified and different decay pro-cesses can be related to each other.

For example,in the case of B →D decay the number of un-known form factors can be reduced from two toone, and we have [1]< Dv′|¯cγνb|Bv >= √mBmDCcbξ(v′·v)(v+v′)ν , (1)where vν, v′ν are the four-velocity and mB, mD arethe B and D meson mass, respectively. The con-stant Ccb comes from integrating the full QCDcontribution from the heavy quark mass scaledown to a renormalization scale µ ≪mDCcb =αs(mD)αs(mB)6/(33−2N) αs(mB)αs(µ)a(v·v′), (2)where a(v ·v′) is a slowly varying function of v ·v′and vanishes at v = v′ [1].

The Isgur-Wise func-tion ξ(v′·v) represents the interactions of the lightdegrees of freedom in the heavy meson system andcan thus be calculated only by nonperturbativemethods.On the lattice the heavy meson system can bestudied in two different approaches.One is tokeep the heavy quark dynamical by using thestandard Wilson action.This may require ex-trapolation to the physical heavy meson mass of∗Speaker at the Lattice ’92 conferenceinterest. The alternative is to integrate out theheavy quark first and derive an effective action in-cluding only the light degrees of freedom and thenperform numerical simulation using this effectiveaction.

Here we stay with the first approach.Using the flavor symmetry of HQEFT theIsgur-Wise function relevant to the B →D decayof Eq. (1) can be obtained also from the D →Delastic scattering matrix element [1]< Dv′|¯cγνc|Dv >= mDCcc(µ)ξ(v·v′)(v+v′)ν , (3)whereCcc(µ) =αs(mc)αs(µ)a(v·v′).

(4)This of course requires that the D meson be suf-ficiently heavy for the onset of the heavy quarklimit (HQL). Conventionally a B →D (here weuse B and D as generic names for heavy pseu-doscalar mesons, they do not necessarily repre-sent the physical B and D mesons) transition ma-trix can be parametrized as< Dp′|Vν|Bp >= f+(q2)(p′+p)ν+f−(q2)(p−p′)ν , (5)where q2 = (p′ −p)2 is the momentum transferbetween the initial and final states and Vν is avector current.It is easy to show that in theelastic scattering case one has f−(q2) = 0.

Usingthe relation vν = pν/m, Eq. (5) becomes< Dv′|Vν|Dv >= mDf+(q2)(v′ + v)ν .

(6)

2Comparing this with Eq. (3) one finds the simplerelation between f+ and ξf+ = Cccξ .

(7)The lattice calculation method for f+ has beenwell established [3,4], and thus the result can beeasily used to obtain ξ.2. Considerations in the Lattice Calcula-tionAt β = 6.0 the inverse lattice spacing a−1 ≈2.0GeV .

The HQL becomes valid when the heavyquark Q in a heavy meson has a mass mQ ≫ΛQCD ≈0.2GeV . The heaviest mass we can takeis order of one or less in lattice units, beyondwhich one would expect large lattice-spacing ar-tifacts.

At β = 6.0 this corresponds to a physi-cal mass in the range of D meson. This can beobtained by setting the hopping parameter forthe heavy quark Q to [3] κQ = 0.118.

For thelight quark we take the hopping parameter κq =0.152−0.155 and extrapolate to κq,cr = 0.157 [3].How far can v ·v′ change on the lattice? In ourcalculations we always have either the initial orthe final particle at rest.

Thusv · v′ = ED/mDif v′ = (0, 0, 0, 1)E′D/mDif v = (0, 0, 0, 1),(8)where ED =pm2D + ⃗p2 .For a spatial lattice size L = 24, we have in-jected momenta⃗p = 2πL (1, 0, 0) , 2πL (1, 1, 0) . (9)Since mD ≈1.0 in lattice unit, one gets v · v′ ≈1.034 and 1.066 respectively.

To get larger valuesfor v · v′ one needs to inject larger lattice mo-menta which would in turn introduce large sta-tistical noise in the matrix element calculations.Thus in practice v ·v′ can not be much more than∼1.1.However, this apparent restriction in the rangeof v·v′ in the lattice calculations has little physicalconsequence as the validity of HQL requires thatthe momentum transfer between the initial andfinal light degrees of freedom, ∼Λ2QCD(v ·v′ −1),be ≪Λ2QCD[1]. This in turn means that v · v′should be close to one.Removing the lattice artifacts.

One of the ma-jor concerns in this calculation is the size of thelattice artifacts. Since the heavy meson mass isnear one in lattice units, the lattice artifacts couldbe significant.Ultimately the lattice artifactscan be brought under control either by compar-ing data at different β values or using the latticeimproved actions.

However, for simulations at agiven β value there are several ways to check thesize of the lattice artifacts.Eq. (5) and consequently Eq.

(6) requireLorentz invariance to hold. In Euclidean spacethe Lorentz transformation becomes a four di-mentional Euclidean rotation.The lattice the-ory, however, does not have exact Euclidean rota-tional invariance for finite lattice spacing a. Thusf−is not exactly zero.

The amplitude of f−(orf−/f+) gives a measure for the violation of theEuclidean invariance on lattice.We can also estimate the size of the lattice ar-tifacts by checking the simulation results againstknown continuum matrix element values at somespecial points. For example, when both the ini-tial and final D mesons are at rest, v = v′ =(0, 0, 0, 1), the continuum matrix element of ¯cγ4cis known because of the quark flavor current con-servation [1]< D|¯cγ4c|D >= 2mD .

(10)At this so-called “recoil-point” we have ξ(1) = 1.Both Eqs. (10) and (6) will have O(a) correctionson lattice.

These lattice artifacts may come fromdifferent origins. Part of the O(a) effect can beapproximately included by using a normalizationfactor< ψ(x) ¯ψ(0) >cont= 2κu0em < ψ(x) ¯ψ(0) >latt ,(11)where [2]em = 1 + 1u0 12κ −12κcr,(12)with u0 the “tadpole improvement” factor.Another correction comes from the use of (non-conserved) local vector current Vν = ¯cγνc [6].This effect can be corrected by introducing a

3Table 1The Isgur-Wise function. The heavy quark hopping parameter κQ and the lattice sizes are shown in thetable..118, 163 × 39.118, 243 × 39.135, 243 × 39v · v′1.0567(8)1.1103(15)1.0259(5)1.0512(10)1.0543(12)1.1059(23)ξ1.00(4)0.99(11)0.974(20)0.940(50)0.952(20)0.893(40)rescaling factor ZlocVin the vector current.

In per-turbation theory ZlocVis calculated to be [7]ZlocV= 1 −27.5 g216π2 . (13)In general the size of lattice artifacts will bemomentum dependent so the O(a) correction willbe different for Eq.

(10) and Eq. (6).

However,since we are using only small momentum injec-tions we may assume that the leading O(a) cor-rection does not depend on p, p′ and the Lorentzindex ν. We will use Eq.

(10) as the normaliza-tion condition for matrix element < Dv′|Vν|Dv >.This way both exp(m) and ZlocVfactors are takenout automatically (together with any other mul-tiplicative O(a) factors).3. Numerical resultsWe use the standard Wilson action for quarksin the quenched limit.Both heavy and lightquarks are treated as dynamical.

We use data atβ = 6.0 on 163 × 39 and 243 × 39 lattices. Thereare 19 configurations on the 163×39 lattice and 8configurations on the 243 × 39 lattice.

The tech-niques for measuring the two-point function andthe three-point matrix elements are standard [3].The fittings are done for time slices from t=10to t=15. To reduce the statistical error, we haveused symmetry properties of the Green functionsand averaged over ±t and ±⃗p directions.Comparing the measured f0 value at q2 = 0(f0(0) = f+(q2 = 0)) to the known continuumvalue f0(0) = 1, we observed that the lattice ar-tifacts are typically 20% −40% at κQ = 0.118and less than 10% at κQ = 0.135.

We also mea-sured the ratio f−/f+. We find the violation ofEuclidean invariance is typically 5% −15% forκQ = 0.118 and 3% −10% for κQ = 0.135.

Notethat the reduction of the lattice artifacts whenκQ is changed from 0.118 to 0.135 agrees withour intuitive expectations. One may try to usethe factors ZlocVand em to remove part of theO(a) effects.

At β = 6.0 we get from perturba-tive calculation Eq. (13) that ZlocV≈0.7 (usingthe shifted effective gauge coupling ˜g2 ≈1.7 assuggested in ref [8]).

The factor em is about 2 forκQ = .118 and 1.5 for κQ = .135. Including boththese factors the corrected f0(0) becomes 1 withinerrors.

This is in agreement with other observa-tions [5] that ZlocVand em factors seem to accountfor the largest part of the lattice artifacts.According to the discussions in Section 2, wedefine ξ(v · v′) = f+/f0(0) and list the resultsin Table 1.We emphasize that this definitionremoves all momentum-independent O(a) effectssimultaneously, including both em and ZlocVfac-tors. Note that in Eq.

(7) there is a factor Cccin the connection between f+ and ξ. This factorcomes from integrating out the QCD effects fromthe heavy quark scale down to a light scale µ. Forthe lattice calculation, however, µ is taken to beO(1/a).

Thus in our case µ ∼mD ∼mc. Alsov·v′ is very close to one.

So for practical purposeswe can set Ccc = 1 according to Eq. (4).We plot our results for the Isgur-Wise functionin Fig.

1. For comparison we also plotted the the-oretical bounds on the Isgur-Wise function.

Thetop and bottom curves are the upper and lowerbounds derived in ref. [9].They are obtainedusing the dispersion relation for the two-pointfunctions, with the requirements of unitarity andcausality and with some assumptions on the ana-lytic properties of the form factors.

The curve inthe middle is an upper bound on the Isgur-Wisefunction derived from current-algebraic sum rules[10]. This is a tighter upper bound.

Our data ob-tained on the 243×39 lattice appear to be, withinerrors, inside of the upper bound of Bjorken [10]and the lower bound of de Rafael and Taron[9].The data from the 163 × 39 lattice is consistent

4Table 2Comparison of the lattice calculation for the slope at v · v′ = 1 with various model calculations.Lattice[11][12][13][14][10,9]1.0(8)1.6(4)1.4(6)1.05(20)0.65(15)0.25 < ρ2 < 1.42with the Bjorken upper bound within rather largeerrors.Figure 1.The Isgur-Wise function is plottedagainst v · v′.The open circles represent dataon the 163 × 39 lattice.The solid circles andsquares represent data on the 243 × 39 lattice.The heavy quark hopping parameter is set atκQ = 0.118 (open and solid circles) and at κ =0.135 (squares) while the light quark hoppingparameters are extrapolated to the chiral limitκ = κc ≈0.157.Close to v · v′ = 1 the Isgur-Wise function canbe parametrized asξ(v · v′) = 1 −ρ2(v · v′ −1) + O((v · v′ −1)2) . (14)If we calculate the slope using the data point clos-est to the v · v′ = 1 axis, we get ρ2 = 1.0(8).

InTable 2 we list our result along with ρ2 valuesestimated by other authors.Note that the lattice meson mass we used isin the range of physical D meson.Thus theO(1/mQ) correction may be quite significant. Fora reliable calculation of the Isgur-Wise functionwe need to repeat this calculation for several dif-ferent masses and then extrapolate to the infinitemass limit.

For a check on the residual O(a) ef-fects we plan to repeat our calculation at β = 6.3.AcknowledgementC.B. was partially supported by the DOE un-der grant number DE2FG02-91ER40628.Y.S.was supported in part under DOE contractDE-FG02-91ER40676 and NSF contract PHY-9057173, and by funds from the Texas NationalResearch Laboratory Commission under grantRGFY92B6.

A.S. was supported in part by theDOE under grant number DE-AC0276CH00016.The computing for this project was done at theNational Energy Research Supercomputer Cen-ter in part under the “Grand Challenge” programand at the San Diego Supercomputer Center.REFERENCES1.Forreviews,seeM.Wise,CALT-68-1721, published in the Proceedings of theLake Louise Winter Institute, 1991 p.222;H. Georgi, preprint HUPT-91-A039, 1991.2.A. S. Kronfeld and P. B. Mackenzie, privatecommunications; also see G. P. Lepage, Nucl.Phys.

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