and with maximally twisted Wilson fermions
We present a lattice QCD calculation of the heavy-light decay constants and performed with maximally twisted Wilson fermions, at four values of the lattice spacing. The decay constants have been also computed in the static limit and the results are used to interpolate the observables between the charm and the infinite-mass sectors, thus obtaining the value of the decay constants at the physical quark mass. Our preliminary results are , , . They are in good agreement with those obtained with a novel approach, recently proposed by our Collaboration (ETMC), based on the use of suitable ratios having an exactly known static limit.
The study of B-physics plays a fundamental role within flavour physics both in accurately testing the Standard Model and in the search of New Physics effects. To this aim it is crucial to have theoretical uncertainties under control, in particular those of the hadronic parameters computed on the lattice.
With the available computer power it is not possible to simulate quark masses in the range of the physical mass keeping, at the same time, finite volume and discretisation effects under control. In order to circumvent these problems, many different methods have been proposed so far (see ref.  for an up to date collection of results).
The approach that we have adopted and that we discuss below consists in using lattice QCD data with the heavy quark mass ranging from the charm region up to of the physical quark mass, together with the information coming from the static limit point. In order to deal with the simulated light quark mass and finite lattice spacing, a careful extrapolation to the chiral and continuum limits has been performed. An alternative method, based on the introduction of suitable ratios having an exactly known static limit, has been recently proposed and investigated by our Collaboration (ETMC) .
2 Heavy-light decay constant in the static limit of HQET
In order to extract the decay constant using maximally twisted lattice QCD, we need to evaluate the matrix element of the static-light local current. At maximal twist the pseudoscalar current in the physical basis, in terms of the twisted basis used in the numerical simulations (light quark fields ), is given by
where and are the pseudoscalar and scalar densities which renormalise with the and appropriate to the static-light framework.
We define and where is the lattice ground state. At maximal twist, the amplitude we need to compute is . The (bare) matrix elements and have been measured from an analysis following the static HQET spectrum study with twisted-mass fermions . The ETMC ensembles and [6, 7] have so far been considered (i.e. two lattice spacings). Here we concentrate on the lightest heavy-light meson state, the pseudoscalar meson which we call here the meson (or with a strange valence quark). We take the value of for the strange quark from the ETMC studies of the strange-light mesons [8, 9] which used the same gauge configurations as used here, namely at and at . We measure the correlation of operators at source and sink with a large choice of operators: local and smeared; parity conserving and non-conserving. We then make a simultaneous fit to a sub-matrix (typically ) in a given Euclidean time interval. We chose this -interval to have similar physical extent at different lattice spacings. We find that the non-local operators have weaker coupling to excited states, as expected. Such non-local operators can give a good determination of the energy levels but to extract the required matrix element (related to ) we need to include local operators in the fit. At we use a state fit with range but with the correlations that have local operators (at sink and/or source) we restrict to range to . This choice gives acceptable values of using correlated fits. We then make uncorrelated fits to determine the required energies and matrix elements with statistical errors determined by bootstrap. At the appropriate range is found to be for smeared correlators and for local ones. We have checked by making many different fits that the fit parameters are stable, within the statistical error assigned. For the correlations of mesons, we make similar fits but find that the minimum value has to be increased by 1 unit to preserve an acceptable (correlated) .
Then one computes and in order to get the matrix element renormalised in HQET at a specific scale . We have chosen to renormalise it in the scheme at and for this preliminary account of our work the renormalisation is done perturbatively at 1 loop order. is a continuum-like scheme defined within dimensional regularisation, while the regulator of our bare quantities is the inverse lattice spacing. So one needs a matching between both regularisations. It can be written as
where the renormalisation scheme and scale of the coupling constant is not specified at this level of perturbation theory. Expressions of are complicated and not illuminating, essentially due to the HYP-smeared static action and the improved part of the gluon propagator . Thus we have simply collected the numerical values of and in Table 1 for a boosted coupling (where and is the average plaquette value).
It turns out that the systematic error introduced by a poor determination of the ratio is minimal, especially on the ratio of the and decay constants. We thus present in fig. 1 the bare matrix element, which depends on the ratio only.
Once the matrix element has been renormalised in the scheme at the scale a NLO running of perturbation theory  has been applied to evolve it to a scale . This is what is needed to perform a fit together with the relativistic data matched to HQET at the same scale (see next section).
where denotes the simulated pion masses, stands for the light decay constant in the chiral limit, while and are free fit parameters. The coupling has been fixed to [15, 16], and we have checked that a change of in the value of results in a shift in which is well below the statistical error. The chiral extrapolation of , and the ratio is shown in fig. 1. This figure also illustrates that we find consistent results at our two available lattice spacings within the relatively large errors. We do not have enough data to include explicit discretisation error terms in the fit formula. However it seems that cut-off effects are quite small. This is more evident for the ratio which is consistent with having no cutoff effects (see right plot of fig. 1).
3 Relativistic results and interpolation to the physical quark mass
We perform an interpolation of the heavy-light () decay constants from the charm region up to the bottom mass, by including data in the static limit calculated in the HQET as explained in the previous section. The lattice QCD data used in this analysis are at four values of the lattice spacing (corresponding to ), that is we have used the configuration ensembles denoted in [6, 7] as , , and , respectively. We have simulated for each ensemble 16 heavy quark masses in the range . Quark propagators with different valence masses are obtained using the so called multiple mass solver method . In fig. 2 we show for illustrative purpose the effective masses at and for few quark mass combinations.
The analysis is performed by studying the dependence of the decay constants, more precisely of the quantity , as a function of the meson masses, as in our recent analysis of the and decay constants .
In order to make use of the HQET scaling low we introduce for each simulated meson mass the HQET quantity that is finite in the static limit :
which has been obtained through the NLO matching from QCD to HQET and evolving at NLO to the renormalisation scale given by the experimental value of the meson mass. For () we first study the dependence on the light/strange quark mass at fixed heavy mass through the following functional forms
We note that the fit forms above follow from the HMChPT formulae [12, 13, 14], which we have already used in the static sector (see eq. (4)). A dependence of the coefficients on the lattice spacings is allowed, in order to account for discretisation effects. The extrapolation/interpolation to the physical light/strange quark mass is performed by replacing in eq. (6) , . This first step provides the values of the decay constants at the physical light/strange quark mass for every simulated lattice spacing and heavy quark mass, or equivalently the quantities .
The second step consists in studying the dependence of , included the available static points, on the heavy quark mass and on the lattice spacing, in order to interpolate to the quark mass and to extrapolate to the continuum limit. Several functional forms with different and discretisation terms have been tried, which can be written in a compact way as
where is a reference meson mass with the same simulated heavy quark mass as in the fitted quantity and the light quark mass is fixed to a similar value for all data. We have performed correlated fits by assuming the static results uncorrelated with the relativistic data.
The results for the decay constants and are finally obtained by replacing in eq. (7) , setting the lattice spacing equal to zero and performing the matching from HQET back to QCD at NLO.
The dependence of the decay constants on the meson mass is shown in fig. 3 where, for illustrative purpose, we also show curves corresponding to one of the various fits. The discretisation terms included in the shown fits are of , and for both and .
We observe that with our data it is not possible to determine the coefficients of more than three discretisation terms for each fit and that, in some cases, only two out of three parameters turn out to be different from zero. About twenty of these fits have a chi square per degree of freedom of order one or smaller and are considered in deriving our final result for and . The spread among these fits is included in the systematic uncertainty.
Our preliminary results for , and the ratio read222The results given in the present proceedings are based on a larger statistical sample w.r.t to the values presented at the Conference and cited in ref. .
where: i) the first error is of statistical plus fitting origin, ii) the second error, estimated through the spread of the results obtained with functional forms containing different discretisation terms, represents the residual uncertainty due to the continuum limit and to the mass interpolation, iii) the third error takes into account the effect of the systematic uncertainty on the static point.
where the uncertainty is the sum in quadrature of the statistical and systematic errors. The two sets of results are in very good agreement, thus providing further confidence on their robustness. We note that the results in eq. (9) are obtained from a subset of the data analysed in the present study. The inclusion of the full set of data is in program for a forthcoming publication.
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