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Lattice measurement of the couplings g_infinity and g_Bstar_B_pi

We present the results of a quenched lattice QCD study of the coupling g_hat, in the static heavy quark limit. After combining this with our previous results obtained by using propagating heavy quarks with a mass around the physical charm quark, we are able to interpolate to the b-quark sector. Our results are g_hat_infinity = 0.48 +/- 0.03 +/- 0.11, g_hat_b = 0.58 +/- 0.06 +/- 0.10, and g_Bstar_ B_p = 47 +/- 5 +/- 8.


Introduction
The coupling of the vector and pseudoscalar heavy-light mesons to the pion, g H * Hπ , and the related g Q defined through 1 have been extensively studied in the literature, both for phenomenological and theoretical reasons (a list of results can be found in refs. [1][2][3][4][5]). g Q is particularly important in the studies of the q 2 -dependence of the form factor F + (q 2 ) which encodes the non-perturbative QCD dynamics of D → π and B → π semileptonic decays. To better illustrate this, we consider the dispersion formula for F B→π + (q 2 ) : where m B * and f B * are the mass and the decay constant of the B * -meson. Near the B * -pole, F + can be approximated by the first term of the r.h.s of eq. (2), which is why g B * Bπ is so important. Inversely, from the q 2 -dependence deduced from the lattice data for F B→π + , one can extract indirectly the value of g B * Bπ (see e.g. [6]). Moreover the need for a more accurate estimate of g Q became recently more evident since the guidance of the chiral extrapolations by the predictions of the heavy meson chiral perturbation theory necessarily involvesĝ Q [7]. In the effective theory based on the combination of the heavy quark and chiral symmetries [8,9] one of the essential parameters is the coupling g ∞ which appears as the infinite quark mass limit of g Q : While the B * → Bπ decay is forbidden by the lack of phase space and therefore the coupling g B * Bπ cannot be measured in experiment, the coupling g D * Dπ has already been measured. CLEO reported recently [10]: i .e g exp.
where we used eq. (1) to obtain g c from g D * Dπ . Since the pion emerging from this decay is soft, the experimental measurement of Γ(D * + ) is difficult. The experimental confirmation of the CLEO results would be highly welcome.
Regarding the theoretical estimates the situation is still quite unsatisfactory. Typical values for g Q , as obtained by using the QCD sum rules (QCDSR) are low, around 0.3 [1], in disagreement with g exp.

c
. Such a disagreement with the experimental value was not, however, observed with similar couplings, such as g ρωπ , g N N π , or g Σ * c Λcπ . To overcome that problem, in ref. [11] it has been proposed to include the first radial excitations in the hadronic part of the sum rules, below the duality threshold. As a result the coupling g c becomes much larger ( g c ∼ 0.7), while the similar modification of the sum rule leaves the decay constants (f D and f D * ) unchanged.
Predictions of various quark models are in the range 0.3 g ∞ 0.8 [12]. In particular, in the model with the Dirac equation a value of g ∞ around 0.6 was found before the experimental result became available [2]. On the lattice, a quenched calculation by UKQCD [13] led to g ∞ = 0.42(4) (9). This exploratory study was carried out in the static heavy quark limit where the difficulties to isolate the ground state are well known. In addition it was performed on a coarse lattice, with a low statistics and with only two light quark masses from which it was extrapolated to the chiral limit. Finally, the indirect measurements using the lattice data for the form factors F + and F 0 [6] have extracted rather small values [6].
In our recent paper [4], we presented the results of the quenched lattice simulation in which we used the propagating heavy quarks with the mass around the physical charm quark. We obtained, g c = 0.67 ± 0.08 +0.04 −0.06 , and observed no apparent dependence on the heavy quark mass. Actually the 3 values of g Q measured directly in the neighbourhood of the c-quark mass were compatible within our error bars and shew only a very small negative slope (see section 4.3 of ref. [4]). This situation allowed a safe interpolation to the c-quark but the determination of g ∞ relied heavily on the hypothesis that the observed behaviour was valid over the whole mass-range. In order to check the validity of this asumption we have performed a lattice quenched computation of g In the static heavy quark limit, the outcome of which is presented in this paper. Our result is where the errors are statistical and systematic, respectively. Using this and the results we presented in ref. [4], we were able to interpolate to the b-quark. We get g b = 0.58 ± 0.06 ± 0.10 , (7) g B * Bπ = 47 ± 5 ± 8 .
By comparing eqs. (6) and (7), our results suggest that the 1/m B corrections to g ∞ are of the order of 20 − 30%.
The paper is organized as follows: after this introduction we explain in Sec. 1 the relation between g and the matrix elements that we compute on the lattice; in Secs. 2 and 3 we present the details of our lattice study and show our main results; Sec. 4 is devoted to the discussion of systematic errors; in Sec. 5 we interpolate to the physical B-meson mass; finally we summarize our findings in Sec. 6.

Extracting g ∞
The coupling g H * Hπ (with H ∈ {D, B}) is defined by the matrix element where q = p − p ′ and ǫ λ (p) is the polarization of the vector meson. To determine g H * Hπ , we can consider the matrix element of the divergence of the light-light axial current, A µ = qγ µ γ 5 q. In the limit where the pion is soft, we can write where the dots stand for terms suppressed when q 2 is small and close to the m 2 π -pole. To bring out g H * Hπ from (9) we express the l.h.s in terms of the form factors, A 0,1,2 , evaluated on the lattice. The standard parametrization reads Note that when q = 0, the soft pion limit is verified, since Moreover, if we choose q = 0 and p = p ′ = 0, eq. (10) simplifies to As discussed in our previous paper [4], when q 2 = 0 the form factors A 0,1,2 are related to one another because the q 2 -poles in eq. (10) are unphysical and therefore their residues have to cancel. This leads to the relation When taking the divergence of the axial current in (10), only the term in A 0 remains on the r.h.s. so that by using eq. (9), we arrive at Finally, inserting the definition (1) leads to the expression for g Q Owing to the heavy quark symmetry, the vector and the pseudoscalar heavy-light mesons are degenerate in the static limit and eq. (15) simply becomes Thus, the value of g ∞ (in the static heavy quark limit) is simply given by the form factor A 1 at q = 0.
2 Strategy for the calculation of g ∞ on the lattice To determine the form factor A 1 (0) from the lattice evaluation of the matrix element H |A µ | H * , we use eq. (11) where H and H * now refer to the pseudoscalar and vector mesons containing an infinitely heavy quark. We consider the two-and three-point correlation functions, C 2 and C 3 µν , with local (L) and smeared (S) interpolating fields for mesons consisting of a static heavy and a propagating light quarks.
In the case in which both the source and the sink are smeared, the two-point functions are defined as where P S and V S µ are the smeared interpolating fields of the pseudoscalar and vector mesons. C SL 2 (t x ) is obtained by simply replacing P S → P L =qγ 5 h, V S → V L =qγ µ h, where h stands for the static heavy quark field. The equality between the pseudoscalar and vector meson correlation functions in eq. (17) is a consequence of the heavy quark symmetry. When the excited states are decoupled and the ground state isolated (at large enough t x ), the two-point function becomes where E H is the binding energy of the static heavy-light meson, and The three-point Green functions with smeared interpolating fields are defined in the following way, where the vector and pseudoscalar mesons are inserted at the origin and at a fixed time t y , respectively. The insertion time t x of the (local) axial current (A µ =qγ µ γ 5 q) varies in the range 0 − t y . The matrix element we are considering, H |A µ | H * , is then extracted from the following ratio: The time dependence cancels in the ratio, so that a plateau in time of R SS (t x ) directly leads to the value of A 1 (0), which for m q → 0 becomes the desired coupling g ∞ . In what follows we will use the notation in which A 1 (0) ≡ g ∞ , even though this is strictly true only for m q → 0. In our simulation we choose to work with the usual (improved) Wilson light quarks and with the static heavy quark by using the Eichten-Hill action [15]. In terms of quark propagators and the Wilson line, the three-point correlation function (20) can be written as where S q (x; y; U) = q(x)q(y) is the propagator of the light quark q. In our study the two light quarks are degenerate, m u = m d ≡ m q . The brackets . . . U indicate the average over the background gauge field configurations (U), whereas P x y is the ordered product of links in the time direction (Wilson line), witht being the unit vector in the time direction.
To improve the statistical quality of the signal we implement the recent proposal of ref. [17] and replace the simple link variables in the Wilson line by the so called "fat links", defined as In order to isolate the ground state from the correlators (22) at the smallest time separations, it is necessary to devise an efficient smearing procedure. We use the proposal of ref. [14] and replace the quark fields q(x) by The wave function φ(r) is chosen in such a way that the overlap with the ground state is increased. We take φ(r) = e −r/R b , where R b is a parameter which is fixed by requiring that the smearing be optimal. Note that it is not necessary to normalize the wave function since the normalisation factors cancel in the ratio (21). The smearing also includes the so called fuzzing: the gauge links U F are fuzzed by N F iterations of the following procedure where P is used to project the fuzzed fields onto SU (3), and C is a real parameter.

Conditions and results of the simulation
The main results presented in this work are obtained from the simulation performed on a 24 3 ×28 lattice with periodic boundary conditions, at β = 6.2. We collect 160 independent SU(3) gauge configurations in the quenched approximation. The Wilson fermion action is non-perturbatively O(a) improved with c SW = 1.614 [18]. The light quark propagators are computed with the following Wilson hopping parameters: corresponding to quark masses around the physical strange quark. We will present our results using our preferred set of parameters, t y = 13 a, R max = 5 a, R b = 3.0 a, N F = 5 and C = 4. The motivation for this choice of parameters, the effect of the smearing (25) and of the modification of the static heavy quark action (24), will be discussed in section 4, where we will also quantify the systematic effects involved in our calculation.

Study of the two-point functions
Light-light operator: The axial current appearing in eq. (20) consists of two light quarks. Even though we use the improved Wilson quarks, for our purpose there is no need to improve the bare axial current since the form factor A 1 is insensitive to the presence of the O(a) improvement term. On the other hand, the improvement of the renormalization constant is of course necessary. We use where the quark mass aρ is obtained from the axial Ward identity, ∂ µ A µ = 2ρP (P =qγ 5 q).
The non-perturbatively evaluated renormalization constants are: Z [19,20] andb A = 1.19 (6) [19] at β = 6.2. We have checked that the inverse lattice spacing obtained from the pion decay constant is consistent with the one we reported in our previous paper. Since the temporal extension of the lattice used in our previous paper is much larger than the present one we will use, whenever needed, a −1 (f π ) = 2.71(12) GeV from ref. [4].
Static heavy-light correlators: The binding energy E H and the constant Z are obtained from the two-point correlation function as indicated in eq. (18). To identify the plateau in time where the ground state is properly isolated, we study the effective binding energy, The superscript "SS" refers to the situation in which both interpolating fields are smeared. Note also that the projection 1+γ 0 2 of the static quark onto the positive energy states reduces the contamination of the signal coming from the opposite side of the periodic lattice. Therefore, the time reversal symmetry over the time dimension of the lattice is not preserved in the heavy quark limit and the two-point functions are no longer symmetric 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Figure 1: Signals for the effective binding energies obtained by using eq. (29) in case of Smeared-Smeared ("SS") and Smeared-Local ("SL") sources. In both cases the light quark mass corresponds to κ q = 0.1344, whereas for the static quark we use the "fat" links as discussed in the text (see eq. (24)]).
with respect to t = T /2. We illustrate in fig. 1 the signals for the effective binding energies of the C SS 2 and C SL 2 correlators. By fitting to the form (18) in the time interval in which the plateau is observed, we extract E H and Z. We have also considered a fit with two exponentials, but did not observe any noticeable effect. The double exponential fit becomes necessary only for when performing simulations without the "fattening" indicated in eq. (24) because in that case the plateaus of E eff (t) are much shorter.
We first calculate the B-meson decay constant, f static Bs . Rather than providing an accurate value for this quantity (a precision determination of f static Bs , has already been made in ref. [17]), our intention is only to test the validity of our computation. Z L is not obtained directly from the local-local correlators because the corresponding correlators are much less stable and statistically noisier than the smeared-local ones. For this reason Z L is extracted from In table 1, we present the values of E SS H , E SL H , Z S , and Z L for a static heavy quark and light quarks with masses corresponding to the κ q i 's specified in eq. (27). The decay constant f static Bs is obtained from Z L using m Bs = 5.37 GeV, i.e. the physical B s -meson mass,   (3), the static heavy-light axial current renormalization constant computed non-perturbatively in ref. [16]. Notice that this value of Z stat A is obtained by using the Eichten-Hill static heavy quark action, i.e. without including the effects of the fattening (24). In ref. [17], however, it has been argued, on the basis of the observed behaviour of the step scaling function, that the effect on Z stat A of the modification of the static quark action is only very small. Therefore the value we are using is certainly sufficient to achieve our present goal which, we recall, is to check the compatibility of our results with previous ones.
In fig. 2, we provide an illustration of the signal (plateau) of the ratio R z (t), and plot the corresponding f static Bq , as obtained by using eq. (31). From the linear interpolation to the strange quark mass (corresponding to m 2 ss = 2m 2 K − m 2 π ), we obtain f Bs , which we quote as This value agrees quite well with the one, f static Bs = 225 ± 10 MeV, recently obtained also at β = 6.2, in ref. [17].

Study of the three-point functions
We observe that, for t x not too close to 0 or t y , the three-point function, C SS 3 ii (0, t x , t y ), introduced in eq. (20), does not depend on t x . The reason for this is that, in the static limit, the ground states of the vector and pseudo-scalar mesons are degenerate. Therefore, by writing C SS 3 ii (0, t x , t y ) as where k and l label the excitations, one sees that in the diagonal contribution (k = l) the exponential time dependence writes which is independent of t x (t y is kept fixed). The same is true for the ratio (21) in which the denominator is composed of two correlators C 2 which are degenerate in this case so that the exponential t x dependence explicitly cancels out, and one ends up with In other words, the plateau for the ratio R SS (t x ) is equivalent to a plateau of C SS 3 ii (t x ). A remarkable feature is that C 3 already develops a plateau at very a small time t x . The physical interpretation could be that in the static limit, the non-diagonal matrix elements of the axial current simply vanish, which implies that the t x -dependence in the l.h.s. of eq. (34) vanishes as well when k = l. Note that this is the case in a quark model framework where the axial current reduces to a light quark spin operator. Since the spatial wave functions of the ground state and the excited states are orthogonal, the spin operator does verify eq. (36) 2 . As shown in eq. (21), the value of the bare coupling g ∞ , is obtained from the fit of R SS (t x ) to a constant. We illustrate the signal for R SS (t x ) in fig. 3. The corresponding results are listed in table 2, where we also give the final results for g ∞ = Z I A g (0) ∞ , for our set of light quarks (27). We also include the values of g ∞ when the light quark is either the strange or the u/d quark. From the linear extrapolation to the chiral limit 3 , shown in fig. 4, we finally obtain, The results presented in this section have been obtained after having fixed t y = 13a < T /2. This choice was made after we performed several simulations with t y /a ∈ [8, 13] and studied R SS (t x )Z 2 S for each t y . We have checked that the result for R SS (t x )Z 2 S does not change as soon as t y gets larger than 10a. To separate the sources as far as possible we have chosen to fix t y at 13 a.

Smearing, Fattening and Systematic uncertainties
As we already mentioned, to improve the statistical quality of the correlation functions computed with the static heavy quark action, we used the "fattening" procedure [17]. In addition, we use the smearing of the heavy-light interpolating fields [14], which helps isolating the lowest bound state at lower time separations. In this section we explain how the parameters of the smearing procedure were chosen (25) and show the benefits of using the "fat link" static heavy action instead of the standard Eichten-Hill one. After that discussion, we enumerate the sources of systematic uncertainties and estimate the related errors.

Smearing
The smearing procedure involves four parameters: R max , R b , N F and C (see eqs. (25)(26)). The value of R max represents half the size of the hypercube where the smearing wave function lives and it is set to the maximal value allowed by the memory limitations. In our case, R max = 5 a. The parameter R b , which appears in the wave function φ(r) = e −r/R b , is not fixed by any physical requirement other than improving the shape of the plateau for R Z . The smeared-local correlators are much more sensitive to the variation of R b than the smeared-smeared ones. In fig.5 we show the shape of R Z (t) for R b ∈ [1.0 a , 4.0 a]. We see that if R b ≥ 3.0 a, the value of R Z (t) on the plateau, and thus also the corresponding f static Bq , remains unchanged. The reason is that, as the wave function φ(r) lives in a typical volume of radius R max = 5 a, the shape of the normalized wave function term [(r + 1/2) 2 φ(r)] in eq. (25) does not vary much when changing R b in 3a − 5a. Of course, all the physical quantities are built in such a way that, at the end, the effect of the normalization of φ(r) (i.e. the unwanted effect of the smearing) cancels. On the basis of these observations, we fix R b = 3a. 4 The remaining two parameters, C and N F , enter in the fuzzing procedure (26). As far as C is concerned, it turns out that its value does not make any significant effect on our result. We varied its value in the range C ∈ [0.8 , 4.0], and the shape of R Z (t) does not change at all (of course, within our present statistical accuracy). On the other hand, the number of fuzzing iterations, N F , is chosen to be N F = 5. The effect of each iteration is to fuzz the spatial links by the gauge fields in the immediate neighborhood. By several iterations we can restore the effect of the wave function inside the volume of radius R b , where φ(r) is supposed to live. As N F iterations cover a volume of typical radius (N F + 1) 1/2 a, we should require N F ∼ 8 to fill a volume of radius R b = 3.0 a. It turns out, however, that for N F ≥ 6, the signal for R Z begins to increase sharply: a large number of fuzzing iterations destroy the quantum fluctuations as it was observed in the cooling procedures used to study instantons [21]. We checked that R Z , and thus f static B , remain stable for N F ∈ [3,5], and for the final simulation we chose N F = 5.

Effect of "fattening" the static heavy quark propagator
In this subsection we show that replacing the link variables in the Eichten-Hill action by the "fat" ones (see eq. (24)), does make the plateau for the effective binding energies much larger, including a visible reduction of the statistical noise. In fact, after replacing U t → U fat t in the Wilson line, the plateau gets extended to t ∼ 12 a − 13 a, which is actually the upper limit (T /2 = 14 a). From the comparison of the effective energy, eq. (29), obtained by using the "fat" Wilson line and of the standard one (i.e. without fattening), we can clearly see the improvement in the signal. This is illustrated in fig. 6, where in the case of "fat" Wilson line, the statistics is half what it was in the other case. This improvement is still more striking in the case where one uses the so-called Hyp-fattening (cf. refs. [22,23]) instead of the standard one (see ref. [24]). We see that not only the plateau appears much clearer, but the signal remains longer in time. We also see that the two values of the effective binding energies are different, as expected, since the heavy quark lattice action was modified. In table 3, we compare the results obtained in both cases, i.e. with and without using the fattening procedure. Before closing this subsection, we note also that, as far as the stability of the signal is concerned, the effect of replacing the standard Eichten-Hill action by the fat-link one is much more rewarding in the case of the heavy-light decay constant than for our g ∞ . This is so because the operator whose matrix element is related to f static B includes a static quark, whereas the one related to g ∞ does not. The fat link action however helps improving the statistical quality of the signal for g ∞ . We note that, besides this improvement, the use of fattened links leads to a dramatic decrease of the value of f static  Table 3: Values of f static Bs and g ∞ as computed from the usual Eichten-Hill static quark action and the new procedure with "fat" links. Parameters of both simulations are kept identical. In particular, to determine g ∞ we have taken in this case t y = 10 a.
had been found with the Eichten-Hill action. The fact that the plain EH formalism faces problems when trying to study f B has been known for some time. In refs. [25,27] it was shown that the smearing led to systematically lower values 5 . We show here that a similar effect results from the use of fattened links. In ref. [24] we show that the Hyp-fattening confirms fully the present results.

Systematic errors
• Smearing: We combine in quadrature the effects of the variation of all the parameters of the smearing procedure on the resulting value for g ∞ (see discussion in sec. 4.1). We estimate that systematics to be ±15%.

• Discretization errors:
In our study we implemented the full O(a) improvement of the Wilson QCD action and the axial current. As we discussed in the text, the improvement of the bare axial current does not influence the value of g ∞ . As for the renormalization constant, we used the non-perturbatively determined value, including the coefficient b A , which ensures the elimination of the artifacts of O(aρ).
Our main result is obtained from the simulation at β = 6.2. In order to study the O(a) effects, we also performed the simulations at β = 6.1 and β = 6.0, keeping the physical volume approximately the same and rescaling the smearing parameters. In fig. 7, we show the chiral behavior of g ∞ , as computed in all three simulations. We see that both the results accessed directly from our simulations and those obtained after the linear extrapolation agree within the error bars. To give the reader a more quantitative insight into this effect, we collect in table 4 the results for g ∞ and f static Bs from all three simulations.
Since in this work we do not deal with propagating heavy quarks (which are usually the dominant source of discretization errors in the heavy-light meson observables), and since we do not observe any noticeable discretization error on g ∞ from our  simulations at three different lattice spacings, we will not account for any extra discretization error in our overall systematic uncertainty 6 .
• Finite volume: In this work we did not carry out a detailed study of the finite volume effects. Instead, we will rely on our previous study with the propagating heavy quark where the finite volume effects are estimated to be ±6% [4]. This uncertainty will be added in our overall systematic error estimate.
• Chiral extrapolations: Since we have only three values for the bare light-quark masses, we cannot make a detailed study of the effect of the chiral extrapolation like the one we did in [4]. The values of g ∞ that are directly accessed from our simulations follow a very smooth linear behavior when the value of the light quark mass is changed. The result (37) is obtained from the linear extrapolation. To that we will add an error of ±15%, as estimated in our previous paper [4]. This error is considered symmetric for the reasons explained in detail in ref. [28]: the chiral logs in the full (unquenched) theory have a tendency to lower the value of the coupling g ∞ , whereas those in the quenched theory do the opposite (increase the value g ∞ ).  Table 4: Values of f static Bs and g ∞ from three simulations at β = 6.0, 6.1 and 6.2. Consistent results are found when comparing these studies.
After combining the above sources of systematic uncertainty in quadrature, we end up with 22% of error to the value given in eq. (37), i.e. g ∞ = 0.48 ± 0.03 ± 0.11 . (38)

Heavy mass interpolation to the b-quark
We can now combine our static heavy quark result, g ∞ , with those obtained in our previous study in which propagating heavy quarks with masses around the physical charm quark were used [4], to interpolate to the b-quark sector. For that purpose we use the spin-averaged mass of the heavy-light mesons, i.e.
Motivated by the heavy quark symmetry, we fit our results to the linear and quadratic forms: These fits are shown in fig. 8, from which we read off the values at 1/m B = (5.314 GeV) −1 , i.e. g b : As our final estimate we will take the average of the two, and add the difference in the overall systematics. Thus we have which by means of eq. (1) gives

Summary and conclusions
In this paper we made a study of g ∞ , the coupling of the lowest-lying doublet of heavylight mesons to a charged pion, in the infinitely heavy quark limit. The static heavy quark action used in this work is the Eichten and Hill one with fat-links, which ensures a better statistical quality of the correlation functions computed on the lattice. Our result, in the quenched approximation, is where the errors are statistical and systematic respectively. The most significant sources of systematic uncertainty are those related to the chiral extrapolation and to the smearing procedures used to suppress the contribution of the higher excited states in the relevant correlation functions. That result is then combined with the ones reported in our previous work [4], in which we used propagating heavy quarks with masses close to the one of the physical charm quark. Interpolation to the b-quark sector leads to g b = 0.58 ± 0.06 ± 0.10 and g B * Bπ = 47 ± 5 ± 8 .
The quantities we have studied in this paper,ĝ ∞ and g B * B π , are very interesting because 1 they are of high phenomenological relevance, 2 they escape direct measurement, 3 their theoretical determinations are very widespread.
Still, as the attentive reader will have noticed, the errors remain larger large. Referring for instance to eq. (28) above, the statistical error is 6% and the systematic one 22%.
Concerning the statistical error a first factor is obviously the performance level of the available computing facilities. Since this is getting higher and higher rather fast and, at the same time, new computational techniques have been discovered which greatly improve the signal/noise ratio this source of uncertainty will probably be tamed very soon. Some of the systematic errors are no serious problems : most of the various renormalization and improvement constants we need have now been non perturbatively determined with an accuracy at the %-level. Note also that some of the (less well-known) heavy-light constants (e.g. Z static Bs ) simply drop out of our computation. Some other ones are more serious : The chiral extrapolation (which we have very roughly estimated to contribute by 15%) can be attacked on both the theoretical side (determination of the chiral logs) and the numerical one.
In this study we have used the static limit only as a limit point for our extrapolations from the finite-mass region. It is possible to go beyond this approach and to determine higher order 1 M Q coefficients, which would allow a kind of "educated fit" and to decresase the theoretical uncertainties. This is part of an ambitious program which has been undertaken by the Alpha-group 3 years ago and is still under progress [29][30][31][32].
There remains the question of the unquenching. Its possibility while using the fattening procedures has been established (see ref. [33]) at least in the staggered-fermions case and probably with Wilson fermions too. This is one of the tasks we should undertake in the future.
These are just a few hints regarding what will have to be done now but it is impossible at present to outline any really sensible quantitative perspective regarding the time evolution in the near future of the various uncertainty factors we have just mentionned. Still, pursuing in the direction of a "pure QCD" determination of g is clearly of primary importance.
Informatiques (Paris-Sud, Orsay) and purchased thanks to a funding from the Ministère de l'Education Nationale and the CNRS. This work was supported in part by the European Union Human Potential Program under contract HPRN-CT-2000-00/45, Hadrons/Lattice Phenomenology.