Heavy flavour precision physics from Nf = 2 + 1 + 1 lattice simulations

We present precision lattice calculations of the pseudoscalar decay constants of the charmed sector as well as determinations of the bottom quark mass and its ratio to the charm quark mass. We employ Nf=2+1+1 dynamical quark gauge configurations generated by the European Twisted Mass Collaboration, using data at three values of the lattice spacing and pion masses as low as 210 MeV. Strange and charm sea quark masses are close to their physical values.


Introduction
Physical processes in the heavy quark sector offer the possibility to get some of the more stringest tests of the Standard Model and to search for possible footprints of New Physics dynamics, by directly challenging the unitarity constraints of the CKM matrix.
Lattice QCD has already entered the precision era as the accuracy of numerical computations is becoming comparable to that of experiments. For some of the relevant hadronic quantities in Flavour Physics the goal of per cent precision has been achieved. State-of-theart lattice calculations involve O(a)-improved fermionic actions with N f = 2, 2+1 and 2+1+1 dynamical flavours with the smallest simulated pion masses being today at the physical point or slightly higher and employing three or more values of the lattice spacing. For a review with a critical evaluation of lattice results and averages, see [1]. First computations with four non-degenerate quark flavours including electromagnetic effects have also been presented recently [2]. * Corresponding author, dimopoulos@roma2.infn.it Direct computations by many lattice collaborations have shown that the cutoff effects in the D-sector are small and under control. Moreover, considerable progress has been recently made in flavour physics at the b mass, with the help of both effective theories approaches and thanks to the implementation of some innovative methods. All these progresses have allowed the determination of a number of B-physics parameters (e.g. m b , f B and f B s ) with controlled systematic uncertainties.
Lattice methods are an invaluable tool to obtain direct determinations of hadronic quantities relevant for the computation of many of the so called golden plated processes such as decay constants, form factors and bag parameters. For instance, the width of the D and D s leptonic decays is given, to lowest order, by (1) Thus lattice computations of the quantities f D and f Ds gives access to the determination of the CKM matrix elements, |V cd | and |V cs |, respectively, as in Eq. (1) all the arXiv:1411.0484v1 [hep-lat] 3 Nov 2014 rest is known experimentally. On the experimental side also the accuracy of the measurements of the D [3,4,5] and D s [6,7] leptonic width has progressively improved during the years.
Lattice QCD provides a first principles' way to compute quark masses. This is possible since quark masses enter as parameters in the QCD Lagrangian and their values can be extracted by matching hadron masses calculated on the lattice with their experimental values. The accuracy of quark mass estimates depends on the conversion from the lattice regularisation to continuum renormalisation schemes. Quark mass ratios instead can be computed in a fully non-perturbative way and are free of renormalisation scheme ambiguities. We notice, here, that the knowledge of the b-quark mass value and to less extent that of the c-quark mass plays an important rôle in the study of the Higgs decay to bb and cc [8].
The European Twisted Mass Collaboration (ETMC) has undertaken an extensive program of heavy quark physics calculations on the lattice using two and four dynamical flavours. Here we present the results of the computation of the D (s) pseudoscalar meson decay constants (in the isospin symmetric limit) and the b to c quark mass ratio obtained using gauge configurations with N f = 2 + 1 + 1 dynamical quarks. The main (preliminary) results in these proceedings are f D = 208.7(5.2) MeV, f Ds = 247.5(4.1) MeV, (2) f Ds f D = 1.186 (21), For completeness we remind the recent ETMC determinations of the c-quark mass and the charm to strange quark mass ratio published in [9]:

Lattice setup
ETMC has produced gauge configurations with N f = 2 + 1 + 1 dynamical quarks [11] employing the Iwasaki gluon action [12] and the Wilson Twisted Mass fermionic action for the sea quarks [13]. Automatic O(a)-improvement is guaranteed both for the light and heavier quarks by tuning at maximal twist whilst the drawback of the mixing between the strange and charm sectors [14] is avoided in the valence by using the Osterwalder-Seiler fermions [15]. We have data ensembles at three values of the lattice spacing in the range [0.06, 0.09] fm. Simulated pion masses lie in the interval [210,440] MeV. Thanks to the properties of the twisted mass action light quarks in the sea and all types of quarks in the valence enjoy multiplicative mass renormalisation, Z m = 1/Z P , which is computed nonperturbatively using the RI -MOM scheme [9]. Moreover owing to PCAC, at maximal twisted angle no normalisation constant is needed in the computation of the decay constants. In Ref. [9] we have presented our computation for the quark masses of the In this work the computation of the decay constants in the charmed region as well as the determination of the b-quark mass are performed using (Gaussian) smearing meson operators [16,17] combined with APE smeared links [18] in order to reduce both the coupling of the interpolating field with the excited states and the gauge noise of the links involved in the smeared fields. (For an alternative preliminary analysis of the charmed decay constants that use local point correlators see Ref. [19].) A summary of the most important details of our simulations is given in Table 1 [9]. We denote with aµ , aµ s and aµ c − aµ h , the light, strange-like, charm-like and somewhat heavier bare quark masses, respectively, entering in the valence sector computations.

Charmed decay constants
We use two point correlation functions with pseudoscalar interpolating operators, P(x) = q 1 (x)γ 5 q 2 (x), that in periodic lattice have the typical form: We take the Wilson parameters of the two valence quarks of the pseudoscalar meson to be opposite in order to guarantee that the cutoff effects on the pseudoscalar mass are O(a 2 µ) [21,22,23]. We then consider two cases, using smeared source only and source and sink both smeared. As for ξ PP , this is given by ξ PP = 0|P L |ps ps|P S |0 in the first case and ξ PP = 0|P S |ps ps|P S |0 in the second one, where L and S indicate local and smeared operators. By combining the two kinds of correlators it is easy to obtain the matrix element of the local operator g ps = 0|P L |ps which serves for computing the pseudoscalar decay constant (via PCAC) given by: where µ 1,2 are the masses of the valence quarks that form the pseudoscalar meson with mass M ps . The use of sinh M ps (rather than M ps ) in Eq. (8) is beneficial for reducing the discretisation errors. For the computation of f Ds we tune, via well controlled interpolations, one of the valence quark masses to the value of the strange mass and the other to the value of the charm mass, both taken from Ref. [9]. In this way, for each value of the sea light quark mass and of the three lattice spacings, we get estimates for the decay constant f cs . Then a simultaneous extrapolation to the physical value of the u/d quark mass and to the continuum limit can be performed in order to obtain f Ds . In the present analysis we consider the quantity (  fitting data corresponding to light pseudoscalar masses with M < 350 MeV. Finite volume systematics are estimated by fitting data corresponding to L > 2.6 fm. Discretisation systematic errors have been estimated by fitting data either from the two finest lattice spacings or from the two coarsest ones, and also by estimating the difference of our results from the finest lattice to the continuum limit. Moreover, we have also included the propagated error due to the m s,c uncertainties as well as the systematic effect of the quark mass renormalisation constant (RC) computed in two ways that differ by O(a 2 ) effects. Our central value is the weighted average over the results from all the analyses described above. Our (preliminary) result for f Ds reads f Ds = 247.5 (3.0) stat+ f it (2.7) syst [4.1] MeV, (9) where we report in square brackets the total error (∼ 1.6%) that is the sum in quadrature of the statistical and systematic uncertainties. For the full error budget see Table 2.
In Fig. 2 we compare our result with those computed in other lattice studies and with the PDG estimate based on experimental results and unitarity assumptions. Some tension between the PDG estimate and the most precise lattice results is still present.
In order to determine the SU(3) symmetry breaking ratio f Ds / f D we measure on our data sets the double   For the results of other lattice studies we refer to (from top to bottom) [24,25,26,27,28,29,30]). For the PDG result see [31]. expected from the large cancellation between the SU(2) chiral logarithms [32,33]. The quantity R f in the continuum limit and at the physical pion mass point multiplied with ( f K / f π ) (taken from Ref. [20]) will provide the result for f Ds / f D . We try the following fit ansätze: where ξ = (2B 0 µ )/(4π f 0 ) 2 with B 0 and f 0 determined in Ref. [9]. We have applied finite size corrections using Ref. [34]. Among the available estimates for the (D * Dπ) coupling we have usedĝ = 0.61(7) that in our case leads to the most conservative estimate for the chiral fit systematic uncertainty. The chiral and continuum limit extrapolation is shown in Fig. 3. Moreover we have performed an analysis similar to the one for f Ds in order to estimate our systematic uncertainties. The full error budget is given in Table 2 [14], (12) f Ds / f D = 1.186 (9) stat+ f it (19) syst [21], (13) and each one of the total errors (in square brackets) is the sum in quadrature of the statistical error and the systematic one. We combine the results from Eqs. (9) and (13) to get our (preliminary) result for the decay constant of the Dmeson, namely f D = f Ds /( f Ds / f D ), which reads: where also in this case the total error written in square brackets (∼ 2.5%) is the sum in quadrature of the statistical and systematic uncertainties. For the complete error budget see Table 2. In Figs. 4 and 5 we present a world comparison between lattice results for f Ds / f D and f D , respectively. In both figures the PDG estimate is also included. For some recent non-lattice estimates of the charmed decay constants, see Refs. [35,36,37,38].

Computation of m b and m b /m c
We perform the determination of the b-quark mass employing the ratio method described in detail in Refs. [30,39,40]. We present here a variant of this  Figure 4: We compare the available continuum limit determinations for f Ds / f D from lattice studies that use N f = 2, 2+1 and 2+1+1 dynamical flavours. "ETMC '14" result refers to the present work. For the results of the other lattice studies we refer to (from top to bottom) [24,26,28,29,30]). For the PDG result see [31].  [24,25,26,27,28,29,30]). For the PDG result see [31]. method and we build the quantity Q h = M hs /(M h ) γ , where M hs and M h are the heavy-strange and heavylight pseudoscalar masses, respectively. The parameter γ is a free one and may take values at will in the interval [0, 1). By HQET arguments we know that for the asymptotic behaviour we get: where µ pole h is the heavy quark pole mass. We then consider a sequence of heavy quark masses expressed in the MS-scheme at the scale of 2 GeV such that any two successive masses have a common and fixed ratio i.e. µ (n) = λµ (n−1) , n = 2, 3, . . . . The next step is to construct at each value of the sea quark mass and the lattice spacing the following ratios: with n = 2, 3, . . . and we have used the relation µ pole h = ρ(µ h , µ * ) µ h (µ * ) between the MS renormalised quark mass (at the scale of 2 GeV) and the pole quark mass. The ρ's are known perturbatively up to N 3 LO. For each pair of heavy quark masses we then carry out a simultaneous chiral and continuum fit of the quantity defined in Eq. (16) to obtain y Q (µ h ) ≡ y Q (µ h , λ; µ u/d , µ s , a = 0). By construction this quantity involves (double) ratios of pseudoscalar meson masses at successive values of the heavy quark mass, so we expect that discretisation errors will be under control. In fact this is the case even for the largest values of the heavy quark mass used in this work, see Fig. 6. Since we have taken into account the matching of QCD onto HQET concerning the evaluation of a heavy-light pseudoscalar mass, M hs/ , our ratio y Q (µ h ) has been defined in such a way that the following ansatz is sufficient to describe the µ h -dependence 1 in which the constraint lim µ h →∞ y Q (µ h ) = 1 has already been incorporated. This fit is reported in Fig. 7. Finally, we compute the b-quark mass via the chain equation  [31], respectively. Notice that the quantity Q h (µ (1) h ) for any value of µ (1) h around the charm quark mass is safely computed in the continuum limit and at the physical pion mass. For instance, using quark mass RC of the M2-type (see [9]) and setting as input µ (1) h = 1.148 GeV and γ = 0.75 we find (λ, K) = (1.1588, 10). Thus, the b-quark mass in the MS-scheme at the scale of 2 GeV is given by µ b = λ K µ (1) h . Our preliminary result for the bquark mass is given by the average over two estimates obtained using either M1 or M2-type quark mass RCs while their half difference is taken as an additional systematic error. This reads (14) syst [16] GeV, (19) where the total error (in brackets) is the sum in quadrature of the statistical and the systematic ones. For a complete error budget we refer to  verified that for a large range of values of γ ∈ [0, 1) we get fully compatible final results 2 for m b . The freedom of choosing γ allows for better control of systematic uncertainties stemming from discretisation effects and the fit ansatz Eq. (17). Finally, the ratio method offers the advantage of determining the ratio m b /m c in a simple and fully nonperturbative way. By setting µ (1) h = µ c we repeat the above ratio method analysis and we find m b /m c = 4.40(6) stat+ f it (5) syst [8] (20) A complete error budget is also reported in Table 3. In Figs. 8 and 9 we present a comparison between lattice results for m b and m b /m c , respectively. For non-lattice estimate of m b see [41].   For the results of other lattice studies we refer to (from top to bottom) [42,43,44,30]. For the PDG value see [31].  Figure 9: Comparison between the two available continuum limit determinations for m b /m c obtained from fully non-perturbative studies. For the "HPQCD" result see [43].