Lattice HQET with exponentially improved statistical precision

We introduce an alternative discretization for static quarks on the lattice retaining the O(a) improvement properties of the Eichten-Hill action. In this formulation, statistical fluctuations are reduced by a factor which grows exponentially with Euclidean time, x_0. For the first time, B-meson correlation functions are computed with good statistical precision in the static approximation for x_0>1 fm. At lattice spacings a \approx 0.1 fm, a \approx 0.08 fm and a \approx 0.07 fm the B_s-meson decay constant is determined in static and quenched approximations. A correction due to the finite mass of the b-quark is estimated by combining these static results with a recent determination of F_Ds.

1. B-physics matrix elements such as the B-meson decay constant F B are obtained from lattice correlation functions at large Euclidean time. Considerable interest lies in the treatment of the b-quark in the leading order of HQET, the static approximation [1,2]: in this framework non-perturbative renormalization can be performed, the continuum limit exists and also 1/m b corrections can in principle be taken into account [3][4][5][6].
Progress along this line has been hampered by large statistical errors in the static approximation. In particular it has been observed [7] that the errors of a B-meson correlation function roughly grow as where E stat is the ground state energy of a B-meson in the static approximation with the Eichten-Hill action 1 , for the static quark [2]. Eq. (1) is problematic because the requirement R NS ≪ 1 is satisfied only for x 0 of the order of ∆ −1 and this time interval shrinks rapidly to zero in the continuum limit a → 0 where E stat ∼ e 1 ×g 2 0 /a with some number e 1 . In the attempt to eliminate the discretization errors by reducing the lattice spacing, a, one is then limited more and more by unwanted contaminations by higher energy states and it has been very difficult to compute matrix elements in the static approximation [1,[8][9][10][11]. Since the exponent in eq. (1) is dominated by a divergent term, it is plausible that one may reduce it by changing the discretization. Here we will demonstrate that this is indeed possible while remaining with roughly the same discretization errors.
In [3] it has been shown that energy differences computed with the action eq. (2) are O(a)-improved if the relativistic sector (light quarks and gluons) is O(a)-improved. Furthermore, apart from the usual mass dependent factor, 1 + b stat A am q , the static axial current, is on-shell O(a)-improved after adding only one correction term, 1 For a more precise definition of the theory and for any unexplained notation we refer to [3].
We want to retain these properties of the theory. They are guaranteed if the lattice Lagrangian is invariant under the following symmetry transformations (we do not list the usual ones such as parity and cubic invariance) [3].
i) Heavy quark spin symmetry: ii) Local conservation of heavy quark flavor number: Keeping these symmetries intact, there is little freedom to modify the action. We may, however, alter the way the gauge fields enter the discretized covariant derivative, D 0 . To this end we choose with W (x, 0) a generalized gauge parallel transporter with the gauge transformation properties of U(x, 0). In particular we take W (x, 0) to be a function of the link variables in the neighborhood of x, which is invariant under spatial cubic rotations and does have the correct classical continuum limit such that D 0 = ∂ 0 +A 0 +O(a 2 ). This is enough to ensure that the universality class as well as O(a)-improvement are unchanged in comparison to eq. (3). Since we expect the size of remaining higher order lattice artifacts to be moderate if one keeps the action rather local, we here consider only choices where W (x, 0) is a function of gauge fields in the immediate neighborhood of x, x + a0. We choose where and where the so-called HYP-link, V HYP (x, 0), is a function of the gauge links located within a hypercube [12,13]. In the latter case we take the parameters α 1 = 0.75, α 2 = 0.6, α 3 = 0.3 [12]. The choices (9) -(11) will be motivated further in [14]. It is worth pointing out that a covariant derivative of the general type used above has first been introduced in [15]. In this reference it was considered for the Kogut-Susskind action for relativistic quarks and with a different motivation.
2. Next we have to study the scaling behaviour of observables computed with the actions S S h , S A h , S HYP h which are obtained by inserting W S , W A , W HYP into eqs. (8) and (2). In [14] this scaling behaviour is analyzed in depth for various observables and various choices for the static action in perturbation theory and non-perturbatively. Here we will present only one example. The necessity of such an investigation can be underlined by the following consideration.
The static potential can be seen as an energy for a static quark with action S h and an antiquark with the corresponding Sh [16]. Hence, the static force is one indicator for the scaling behavior of these actions. In [13], rather large a 2 -effects have been seen in the short-distance force for S HYP h and S HYP h .
One may therefore worry about large a-effects, in particular in correlation functions of the static-light axial current, where static and light quarks propagate also close to each other. With the new actions, A stat 0 is O(a)-improved once [14] c stat c stat is set in eq. (5). The improvement coefficient b stat A is set to its tree-level value b stat A = 1/2 in this work.
We consider now a step scaling function, Σ stat A , which gives the change of the renormalized static axial current in a Schrödinger functional (SF) scheme [17], when the renormalization scale is changed from µ = 1/L to µ = 1/(2L). Its continuum limit is known for a few values of L [17]. This quantity is thus a good observable to search for a-effects. In Fig. 1 we show Σ stat A (3.48, a/L), where the first argument parameterizes L in terms of the SF-couplingḡ 2 (L) = 3.48. O(a)improvement is employed as in [17] but we consider the different actions for the static quark introduced above. All of them lead to Σ stat A (3.48, a/L) at finite a/L differing from the continuum limit by about the same amount. Supported also by further such studies [14], we conclude that within the set of actions studied none is particularly distinguished by its scaling behavior.
defined in the Schrödinger functional with T = 3L/2, L/a = 24, β = 6/g 2 0 = 6.2 and a vanishing background field [3]. Here, as a novelty compared to previous applications, a wave function ω(x) is introduced to construct an interpolating Bmeson field in terms of the boundary quark fields ζ l and ζ h . It may be exploited to reduce the contribution of excited B-meson states to the correlation function, but this does not concern us yet. At the moment we simply consider ω(x) = 1 and form the ratio R NS , eq. (1), for the different actions. From now on we set the light quark mass to the strange quark mass, taken from [18] following exactly [19] concerning the technical details. 2 Figure 2 shows that in all cases R NS grows exponentially with x 0 . For the Eichten-Hill action, also the effective coefficient ∆, describing the growth for x 0 = 1 fm − 2 fm, is roughly given by E stat − m π in agreement with eq. (1), while for the other actions this is not the case. Most importantly for the other actions, ∆ is reduced by a factor around 4, and with the statistics in our example a distance of x 0 ≈ 2 fm is reached with S HYP h if one requires R NS ≤ 2%. The actions S A h , S S h behave only slightly worse. 4. This reduction of statistical errors enables us to choose ω(x) such that a long and precise plateau is visible in the effective energy, as shown in Fig. 3. Neither position nor length of the plateau depend sensitively on the details of ω, as long as it is chosen such that the first excited state in the B-meson channel is canceled to a good approximation. For the figure as well as for the following, we have chosen ω ∈ {Ω 1 , Ω 2 } with where a 0 = 0.1863 r 0 and the (dimensionless) coefficients N i are chosen such that a 3 x ω 2 i (x) = 1. The B-meson decay constant is then obtained from the renormalization group invariant matrix element [22] of the static axial current, where The renormalization factor, Z RGI , relates the bare matrix element to the renormalization group invariant one [17]. Its regularization dependent part is computed exactly as in that reference, but for the new actions. In Table 1 we give results for Φ RGI (x 0 ) for three values of the lattice spacing and selected choices of T, T ′ , x 0 , highlighting what we selected for further analysis. These numbers do not change significantly if we vary the improvement coefficients c stat A and b stat A , which are known only in perturbation theory, by factors of two. We thus extrapolate our results quadratically in the lattice spacing and arrive at our estimate for the continuum limit r 3/2 0 Φ RGI = 1.78 (13) .
5. The result eq. (22) may be used to compute F Bs by taking account of the mass dependent function [17] using the 3-loop anomalous dimension [23] and the associated matching coefficient between HQET and QCD [24]. M b denotes the renormalization group invariant bquark mass [4,5]. With this we arrive at r 0 F stat Bs = 0.57(4). A correction due to the finite mass of the b-quark can be computed by connecting the static result eq. (22) and by a linear interpolation in the inverse meson mass. Here we have used recent computations of the D s -meson decay constant [25] and of the charm quark mass [19].
In this way we obtain Conservatively, we may attribute an additional ≈ 5 % uncertainty to the fit ansatz used, but our personal estimate is that this error is significantly smaller and it will soon be quantified [26]. One should remember that eq. (24) refers to the quenched approximation and as in [25] a 12 % scale ambiguity may be estimated from the slope of the linear interpolation.
6. An interesting point is that the potential in full QCD may be computed replacing the time-like links in the Wilson loop (or Polyakov loops) by the different W i introduced above. In particular the "HYP-link potential" [13] may be used. Depending on which W i is chosen, the static potentials differ from each other, but all of them approach the continuum limit with O(a 2 ) corrections if the action used for the dynamical fermions is O(a)-improved. This property follows from the considerations of [16] applied to the static actions introduced above, which satisfy all the necessary requirements. This virtue of e.g. the HYP-link potential was not obvious before. Using it, better precision can be reached and some signs of string breaking [27] may become visible.
7. To summarize, we have shown that a modification of the Eichten-Hill static ac-tion can be found which keeps lattice artifacts in heavy-light correlation functions moderate but reduces statistical errors to a level making the region x 0 > 1.5 fm accessible. Furthermore, the new action can be used without change for dynamical fermions and also to compute the static potential with dynamical fermions. As a demonstration of the usefulness of this reduction of statistical errors, we have computed F Bs in the quenched approximation, by joining the continuum limit of the static approximation estimated with the new action with the previously determined continuum limit of F Ds by means of a linear interpolation. This procedure can systematically be improved by computing 1) the mass dependence around m c , 2) the 1/m corrections to the static limit and 3) repeating the whole analysis with dynamical fermions. Work along these lines is in progress and a more detailed investigation of the properties of various static quark actions is in preparation.