Leptonic decay constants for D-mesons from 3-flavour CLS ensembles

We report on the status of an ongoing effort by the RQCD and ALPHA Collaborations, aimed at determining leptonic decay constants of charmed mesons. Our analysis is based on large-volume ensembles generated within the CLS effort, employing N_f=2+1 non-perturbatively O(a) improved Wilson quarks, tree-level Symanzik-improved gauge action and open boundary conditions. The ensembles cover lattice spacings from a ~ 0.09 fm to a ~ 0.05 fm, with pion masses varied from 420 to 200 MeV. To extrapolate to the physical masses, we follow both the 2m_l+m_s=const. and the m_s=const. lines in parameter space.


Introduction and computational setup
The pseudoscalar decay constants f D and f D s encode the QCD contributions in leptonic decays of Dand D s -mesons, respectively. Theory input on the decay constants in conjunction with experimental data allows the CKM matrix elements V cd and V cs to be tightly constrained. Apart from being an important test of the Standard Model, hints of new physics may be discovered in the charm sector. For an overview of lattice QCD results, see Ref. [1]. An update on last year's report [2] and our ongoing effort on the computation of f D (s) is presented in the following.
We utilize ensembles generated within the Coordinated Lattice Simulations (CLS) effort, with N f = 2 + 1 non-perturbatively O(a) improved Wilson-Sheikholeslami-Wohlert (clover) fermions and tree-level improved Lüscher-Weisz gauge action, employing the openQCD [3] open-source software package. Open boundary conditions in temporal direction are used in order to avoid topological freezing, making the use of very fine lattice spacings of a ≈ 0.0854 − 0.039 fm (β = 3.4 − 3.85) feasible [4,5]. We also employ twisted-mass reweighting for light quarks [6], to prevent instabilities resulting from accidental near-zero eigenmodes of the Dirac operator, and rational approximation for the strange quark with appropriate reweighting. Expectation values of physical observables are therefore obtained by The twisted-mass (W 0 ) and rational approximation reweighting (W 1 ) factors are defined in Ref. [7] (Eqs. (3.2) and (3.5), respectively).
Speaker, e-mail: k.eckert@uni-muenster.de Apart from additional ensembles for β = 3.4 and β = 3.55, our analysis now also includes ensembles for the lattice spacings with β = 3.46, β = 3.7 and β = 3.85. We follow two lines in the light and strange quark mass plane: (i) The average lattice quark mass (m = (2m l + m s ) /3) is kept fixed such that the sum of the renormalized quark masses is constant up to O(a) effects (ensembles available for all β values). (ii) The renormalized strange quark mass is kept constant, again up to O(a) effects (ensembles available for β = 3.4 and β = 3.55). For details on how an almost constant renormalized strange quark mass was achieved, see Ref. [8]. We only mention that the vector Ward identity masses are defined as where κ crit is the critical hopping parameter value at which the axial Ward identity (i.e., PCAC) quark mass in the symmetric limit, m l = m s , vanishes.
The two lines are referred to as m = const. line (first proposed in [9]) andm s = const. line, respectively, and exemplarily visualized in Fig. 1 for the ensembles with β = 3.4. Note that the figure displays the PCAC strange quark massm s , for which the tuning to constant values was actually done but which corresponds to constantm s up to very small corrections. Having two lines in the quark mass plane available enables us to tightly constrain the chiral extrapolation, by enforcing extrapolations along both lines to intersect at the physical point. For further details concerning the computational setup see Refs. [2,7,8,10].   Table 1. Details of the ensembles analyzed so far for the two trajectories to the physical point, keeping m fixed to the value at the symmetric point (m = m sym ) and keeping the renormalized strange quark mass (m s ) approximately equal to the physical value (m (phys) s ). The light and strange quark hopping parameters are denoted κ l and κ s , respectively. The lattice volumes N t × N 3 s , the pion (m π ) and kaon (m K ) masses and the statistics given by the number of molecular dynamics units (N MD ) are also indicated.

Observables
The pseudoscalar decay constants f D and f D s are the low-energy QCD contributions to the leptonic decays of D-and D s -mesons. They are defined as the matrix elements where A qc µ = qγ µ γ 5 c is the axial vector current for quark flavours q = l(ight), s and |D q (p) is a pseudoscalar meson state at momentum p with quantum numbers corresponding to the D-(for q = l) or D s -meson (for q = s). We make use of the pseudoscalar operator P qc = qγ 5 c to remove O(a) discretization effects from the axial operator, obtaining the improved axial current with the standard notation for lattice forward and backward derivatives. Renormalization then proceeds via A Here, m qc and m denote the bare vector Ward identity quark mass combinations The improvement coefficients c A and Z A have been calculated non-perturbatively in Refs [12,13]. b A has been determined non-perturbatively in Ref [14] and in a preliminary analysis the same authors find b A to be consistent with zero. Since m includes only the light and strange quark, the mass dependent corrections in Eq. (5) are dominated by the charmed mass term m qc . Therefore we neglect the term proportional tob A m in our analysis.
In order to extract the matrix elements of Eq. (3), we evaluate the two-point functions at zero momentum, with y 0 being the timeslice of the source insertion and x 0 that of the sink. Starting from the spectral decomposition of the two-point functions it can be shown [10] that for large time separations x 0 − y 0 the correlators behave as where the bare decay constant f bare qc corresponds to 0|A qc,I 4 |D q /m D q , while A(y 0 ) encodes the matrix element 0 |P qc | D q plus possible boundary contaminations at the source position.

Analysis details
Elaborating on our earlier status report in [2], we describe our current analysis setup in this section. It features point-to-all propagators with Wuppertal smearing [15,16] and APE-smoothed links [17] for the pseudoscalar source and sink operator P qc . In order to gain statistics, source operators are inserted at three different positions. For β = 3.4, for example, the source positions are y 0 /a = 30, 47 and 65.
We exploit time-reversal symmetry of the correlators and average the forward propagating part of the C X (x 0 , 30) correlator with source position at y 0 /a = 30 with the backwards propagating part of the C X (x 0 , 65) correlator, while for the C X (x 0 , 47) correlator we average the forward and backward propagating parts. This yields four correlators, two for both the axial and the pseudoscalar case. In order to determine a region where a one-state fit to the ground state can be done safely, we first perform a double-exponential fit of the form C X (x 0 , y 0 ) = c X,1 e −m Dq (x 0 −y 0 ) + c X,2 e −M Dq (x 0 −y 0 ) ,  where the second term represents the first excited state with mass M D q . At first, this is done for the pseudoscalar and axial correlators separately. Since all sources are placed far away from the boundary such that boundary contaminations are expected to be negligible, A(y 0 ) = const. (see Eq. (9)) holds and the amplitudes can be enforced to be the same for different source positions (e.g., c P,1 has the same value for both types of source-position-averaged pseudoscalar correlators). We vary both the starting point and the end point of the fit interval in order to find the optimal fit range, indicated by a minimum of the obtained χ 2 -values. In addition, a careful visual inspection of the fit quality is performed. Then the point x min 0 , where the contributions from the exited states have sufficiently decayed, is given by the criterion where ∆C X denotes the statistical error of the correlator. Finally, x min 0 is taken as the start point for a simultaneous fit of all four correlators to a single-exponential form: The bare pseudoscalar decay constant f bare D qc (q = l for the D-meson and q = s for the D s -meson) is then given by the ratio For the end point x max 0 of the fit range we choose roughly the point, at which the ratio of the statistical error and mean of the correlator exceeds 3 %. Both the double-exponential and the singleexponential fits are illustrated in Fig. 3 for the example of the H105 ensemble.
Since our setup does not include a dynamical charm quark, κ charm has to be fixed for each ensemble. Based on a subset of statistics, we first estimate κ charm from the spin-flavour-averaged 1S mass combination M X = 6m D * + 2m D + 3m D * s + m D s /12 along the m = const. line and from the spin-averaged mass combination M X = 3m D * s + m D s /4 along them s = const. line. Afterwards, simulations with full statistics were performed at two values of κ charm slightly above and below the estimated value, allowing for an uncertainty in the lattice spacing a of ±2 %. A linear interpolation of M X in 1/κ charm to the physical point, given by the central value of a, then fixes the physical κ

Preliminary results
The results described in this section represent the status of the project at the time of the conference and are summarized in Figs. 4 and 5. Statistical error analyses are performed using bootstrap techniques, where the bin size is varied to perform an extrapolation of the error to infinite bin size, as well as adopting the so-called Γ-method, which extracts the statistical errors from the evaluation of autocorrelation functions [18]. Possible sources of systematic errors have still to be accounted for in the final analysis.  As the left panel of Fig. 4 shows, the majority of ensembles at our disposal has been included in the analysis so far. Ensembles for the m = const. line are now available for every lattice spacing (drawn in blue), while for them s = const. line ensembles have been included for β = 3.4 and β = 3.55 only (drawn in green). The physical point in the quark mass plane is indicated by a red triangle. Results for the pseudoscalar decay constant f D are presented in the right panel of Fig. 4, as a function of m 2 π . The left panel of Fig. 5 shows the corresponding results in the case of f D s , while in the right panel the ratio f D s / f D is presented along with the recent FLAG [1] averages for N f = 2 + 1 and N f = 2 + 1 + 1 at the physical point. Combined linear fits in the right panel do not yet represent a full chiral and continuum extrapolation, but have been included in order to demonstrate consistency with the FLAG  results. Along the m = const. line, the ratio of f D s / f D is expected to be one at the symmetric point m l = m s , which is also in agreement with the linear fits employed. This finding, in conjunction with the observed absence of large discretization and finite-size effects, leads us to conclude that these preliminary estimates of the D (s) -meson decay constants in three-flavour lattice QCD are intermediate promising results.

Outlook
In order to improve on the precision of our measurements, we will continue to increase statistics, as well as to perform a careful analysis of the statistical errors. Further steps still to be done include gaining a good control over systematic effects and performing combined chiral and continuum extrapolations. Furthermore, a cross check of the results via a second method of extracting the decay constants, utilizing the axial Ward identity (PCAC) quark mass [19], is being prepared. An investigation of the size of the effect from a slight mistuning of the quark mass trajectories [7,10] is also planned.