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Improved Heavy Quark Effective Theory Currents

arXiv:hep-ph/9203221v1 26 Mar 1992hep-ph@xxx/9203221UCLA/92/TEP/9MCGILL/92–11Improved Heavy Quark Effective Theory CurrentsOscar F. Hern´andezDepartment of PhysicsMcGill UniversityErnest Rutherford Physics BuildingMontr´eal, Qu´e., Canada H3A 2T8Brian R. HillDepartment of PhysicsUniversity of CaliforniaLos Angeles, CA 90024AbstractIt is hoped that the accuracy of a variety of lattice calculations will be improved byperturbatively eliminating effects proportional to the lattice spacing. In this paper,we apply this improvement program to the heavy quark effective theory currentswhich cause a heavy quark to decay to a light quark, and renormalize the resultingoperators to order αS.

We find a small decrease in the amount that the operatorneeds to be renormalized, relative to the unimproved case.3/92

1. IntroductionIt is hoped that the accuracy of a variety of lattice calculations will be improved bythe elimination of effects proportional to powers of the lattice spacing, a [1][2].

This improvement program can be implemented order by order in a and thestrong coupling g2. Calculationally, it is relatively easy to perform the leading orderin either of these expansions.

For matrix elements of vector currents, both types ofleading order corrections are thought to be in the 20 to 30% range [3], so it is atthe same time important to include them, and reasonable to stop at this order.Much of the application of the improvement program to hadronic matrixelements has been to the order a improvement of the Wilson fermion action andoperators [4]–[6].Alternative methods for measuring matrix elements involvingquarks that are heavy relative to the QCD scale have been proposed [7][8]. Theaction and operators in this method can be thought of as discretizations of theheavy quark effective field theory action and operators [8][9].

In this paper, westudy the application of the improvement program to this action and the currentswhich cause a heavy quark to decay to a light quark.The outline of the paper is as follows. In the next section, we discuss improve-ment of the heavy quark action, and in section three, we discuss improvement of theaforementioned currents.

As is generally the case, the improved lattice currents needto be related to their continuum counterparts. This is done perturbatively in g2where the scale is set by the lattice spacing.

Although quite different in motivation,this perturbative calculation is similar to the renormalization of other discretizationsof the heavy quark effective theory currents performed in reference [10].Thatcalculation and the present one rely on the framework and results of reference [11],which are summarized in section four before the new perturbative results arepresented. In section five we give our conclusions.2.

Improvement of the Heavy Quark ActionIn this section, we argue that under two criteria of improvement, there is no need tomodify the heavy quark action. The arguments are tree level, but this is adequatefor the leading order improvement discussed in the introduction.1

Roughly, the on-shell improvement condition formulated by L¨uscher andWeisz [2] is that spectral quantities such as the location of single-particle polesshould not be corrected by terms proportional to powers of the lattice spacing.Examining the momentum space propagator for the heavy quark action in thecontinuum [9],1p0 + iǫ,(2.1)and comparing it with its lattice counterpart [11],1−i(eip0a −1)/a + iǫ,(2.2)we see that although the propagators differ at order a, the pole is located at p0 = 0 toall orders in a. We immediately conclude that the action has no need of improvementunder the on-shell improvement condition, to all orders in a at tree level.A second argument yielding this conclusion is to compare the heavy quarkpropagator in position space in an external gauge field on the lattice to the samething in the continuum.† In the continuum, the propagator from x to y is [12],−iδ3(y −x)θ(y0 −x0)P exp −igZ y0x0dt A0(t, x).

(2.3)The discretization of the action with a nearest neighbor, one-sided time derivativefollows from this propagator [11].On the lattice, the heavy quark propagatorfrom m to n in a background field is [7][8],−ia3 δnmθ(n0 −m0)U0(n −ˆ0)†U0(n −2ˆ0)† · · · U0(m)†. (2.4)In this expression, δnm is the three-dimensional Kronecker δ–function, θ(n0 −m0)is 1 if n0 ≥m0 and 0 otherwise, and U0(m) is the lattice gauge link in the timedirection at site m.If the correspondence between continuum and lattice gauge fields is [2],U0(n) = P exp igaZ 10 dt A0((n0 + 1 −t)a, na),(2.5)then the lattice propagator perfectly reproduces its continuum counterpart.Ifone modifies the heavy quark action (by choosing any other discretization of the† We thank Estia Eichten for discussions leading to this argument.2

time derivative) the position space lattice propagator (2.4) is multiplied by a c-number factor depending only on n0−m0 but is otherwise unchanged (as long asthe discretization does not include spatial links). With the correspondence (2.5),there is clearly no closer approximation to the continuum propagator (2.3) than thepropagator (2.4) obtained from the action currently in use.3.

Improved Heavy-Light CurrentsSince the lattice heavy quark action is not in need of improvement, we review thesituation for the light quark action. Expressions for improved lattice heavy-lightcurrents quickly follow from this discussion.The improved Wilson fermion action proposed by Sheikholeslami and Wohlert [6]is obtained from an improved action with next-to-nearest neighbor couplings [4] bya change of variables.

The resulting action is the Wilson action plus an additionalpiece,∆SI = −ia4 ar4Xnµνq(n)σµνgPµν(n)q(n). (3.1)Pµν(n) is the sum of plaquettes defined in reference [5] and goes to Fµν(n) inthe continuum limit, while σµν = [γµ, γν]/2.

The action has the advantage thatit only contains nearest neighbor couplings. One can most easily obtain improvedoperators for use with this action by starting with local bilinears and making thereplacement [3][6],q(n) →q(n) −r2XµγµhUµ(x)q(n + µ) −Uµ(n −ˆµ)†q(n −ˆµ)i.

(3.2)A similar replacement is made for q(n).Actually, a two-parameter family of transformations, all of which yield opera-tors improved to order a, can be obtained by using the equations of motion for thelight and heavy quark. The effect of applying the equation of motion to the lightquark field has been discussed in reference [5] and the effect of applying the equationof motion to the heavy quark field has been studied under the guise of temporallysplit operators [10].

In either case, the operator renormalization is changed by anamount which comes from the self-energy. We will not consider this generalizationfurther.3

The most general heavy-light bilinear in the full continuum theory is,J(x) = b(x)Γq(x). (3.3)Here Γ is any Dirac matrix, and q is the light quark field.

In the heavy quarkeffective theory, the corresponding operator (3.3) is [9],b†(x)(1 0)Γq(x). (3.4)In a Dirac basis, the two-by-four matrix preceding Γ takes the form (1 0) andprojects onto the upper two rows of Γ.

Applying the above recipe to the localoperator b†(n)(1 0)Γq(n) we are led to consider,b†(n)(1 0)Γq(n) −r2b†(n)(1 0)ΓXµγµhUµ(x)q(n + µ) −Uµ(n −ˆµ)†q(n −ˆµ)i. (3.5)The first term in this expression is the usual local current used to determinehadronic matrix elements.

The calculations necessary to renormalize this part of thecurrent were mostly done in previous papers [11][13], however it receives additionalcontributions coming from the new term in the action (3.1). The remainder of thecurrent, which is naively order a, will also affect the renormalization of the currentwhen it appears in loop diagrams with loop momenta of order a−1.

We now turnto the perturbative renormalization of the improved heavy-light current with theimproved Wilson action of Sheikholeslami and Wohlert.4. Renormalization of the Improved CurrentThe lattice renormalization of the heavy-light current (3.5) gives the ratio of thelattice operator to its counterpart in the continuum theory.We can divide thediagrams that contribute to this ratio into two parts.

Those that give heavy andlight quark wave function renormalization, and the 1PI vertex correction diagrams.The difference between wave function renormalization of the heavy quark onthe lattice and in the continuum is g2/(12π2) times a constant e = 4.53 computedin references [11] and [13].† The corresponding light quark wave function renormal-ization constant was calculated in [5]. The results for the self energy-graphs which† Here we are using the reduced value of e which is appropriate if one fits corre-lation functions containing the propagator (2.4) to Ae−B(n0−m0)a [10][11][13].4

rd1d2∆Σ11.005.46−7.22−9.210.755.76−7.23−8.620.506.30−7.00−7.800.257.37−5.72−6.730.008.790.00−6.04Table 1. Previously Computed r-dependent Quantities [5][11][13].we will need are given in terms of ∆Σ1, defined in reference [5] (when consultingtheir expressions for ∆Σ1, note that we have taken F0001 = 1.31 [14]).We now turn to the vertex correction diagrams of figures 1 and 2.Thetechniques we use to evaluate these diagrams have been discussed in detail in anumber of references [10][11], and will not be reviewed here.At zero external momentum, 1(a) vanishes.

As noted below equation (3.5),the operator has two pieces: one which is the local unimproved operator, and anadditional piece coming from improvement.Thus the contribution of 1(b) canbe split up into two parts which we will call the “unimproved” and “improved”contributions here and in the following paragraph. The “unimproved” contributionis [11][13],g212π2 (d1 + Gd2).

(4.1)We have defined the c-number G by GΓ = γ0Γγ0. The analytical expressions ford1 and d2 can be found in reference [11].

We tabulate the constant ∆Σ1 and theconstants d1 and d2 in Table 1 for several values of the Wilson mass parameter r.Errors in Table 1 and 2 are at most O(1) in the last decimal place.The “improved” contribution from diagram 1(b) can be combined with thecontribution from the other three diagrams in figure 1. In order to compactly quotethe analytic expressions for this result, we use the notation of references [15] and [16]for the following commonly occurring combinations:∆1 =Xµ sin2 lµ2 ,∆4 =Xµ sin2 lµ,∆2 = ∆4 + 4r2∆21,∆5 =Xµ sin2 lµ2 sin2 lµ.

(4.2)5

r∆d1∆d2∆d′1∆d′21.00−6.64−5.843.436.160.75−5.30−3.752.534.940.50−3.61−1.871.553.660.25−1.52−0.450.552.150.000.000.000.000.00Table 2. Changes to d1 and d2 as a function of r.Additional combinations, ∆(3)1 , ∆(3)2 , and ∆(3)4 , are the same as above except thesums on µ run only from 1 to 3.Given these definitions the total “improved”contribution from figure 1 to the vertex renormalization is,g212π2 (∆d1 + G∆d2),(4.3)where,∆d1 = −3r216Z d4lπ24 −∆1∆2,∆d2 = −r32Z d3lπ∆(3)1∆(3)2.

(4.4)We now turn to the contributions depicted in figure 2 which are due to theadditional term in the action (3.1). In these figures, the insertion of this term isdenoted with a cross.

A tadpole diagram which vanishes has not been depicted.The result for the two diagrams depicted is,g212π2 (∆d′1 + G∆d′2),(4.5)where,∆d′1 = r24Z d4lπ2∆4(3 −∆1) + ∆54∆1∆2,∆d′2 = r2Z d3lπ∆(3)4 (1 + r2∆(3)1 )4∆(3)1 ∆(3)2. (4.6)Combining the various results in this section, we find that the ratio of thelattice to continuum operators is,1 +g212π2(d1 + ∆d1 + ∆d′1) + (d2 + ∆d2 + ∆d′2)G + 12e −12∆Σ1 −1.

(4.7)The dependance on µa has been eliminated by setting µ = 1/a.6

5. ConclusionsWe illustrate the use of the results of the previous section for the case of mostinterest, the current which determines the B meson decay constant, fB.

In thatcase, for reasonable values of the input parameters, the ratio of the continuumeffective theory to full theory bilinears is numerically 0.98 [9][13].We need tomultiply this by the ratio given in eq. (4.7).

For Γ = γ0γ5, the constant G (whichappears in (4.7)) is −1. We take g2 = 1.8 which is appropriate for effects arisingfrom the scale π/a with 1/a = 2 GeV as argued in [17].

Taking values from tables 1and 2 with r = 1.00, equation (4.7) gives 1.23 and the product of the two ratios is1.20 (as compared to 1.28 in the unimproved case). To obtain the physical value offB, one divides the lattice results for the improved current by this number.The operator we have renormalized is corrected both to order g2 and to order a.As noted in the introduction both of these corrections are thought to be in the 20to 30% range.

Analytically, the next perturbative corrections are proportional tog2a, g4, or a2 times powers of g2 ln a [3], and there is numerical evidence that thesefurther corrections are at the few per cent level for currents made from two Wilsonfermion fields [3]. Thus it is hoped that the improved currents renormalized herewill lead to a considerably more precise lattice determination of fB and other heavyquark matrix elements.Note Added.

While preparing this manuscript we learned of unpublished workon this subject [18][19] referenced in a paper on lattice measurements of severalquantities using an improved action [20]. Martinelli and Rossi [18] argue that itis unnecessary to change the heavy quark action up to order a2.

This is consistentwith our conclusion in section two that the heavy quark action is not in needof improvement at any order in a. The result of Borrelli and Pittori [19] cited inreference [20] is consistent with our factor above if one takes g2 = 1.0 rather than 1.8.AcknowledgementsOFH was supported in part by the National Science and Engineering ResearchCouncil of Canada, and les Fonds FCAR du Qu´ebec.

BRH was supported in part bythe Department of Energy under Contract No. DE–AT03–88ER 40383 Mod A006–Task C.7

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Figure CaptionsFig. 1:The vertex correction diagrams resulting from improving the light quarkoperatorFig.

2:The vertex correction diagrams resulting from improving the action9

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