Decay Constants of Pseudoscalar $D$-mesons in Lattice QCD with Domain-Wall Fermion

We present the first study of the masses and decay constants of the pseudoscalar $ D $ mesons in two flavors lattice QCD with domain-wall fermion. The gauge ensembles are generated on the $24^3 \times 48 $ lattice with the extent $ N_s = 16 $ in the fifth dimension, and the plaquette gauge action at $ \beta = 6.10 $, for three sea-quark masses with corresponding pion masses in the range $260-475$ MeV. We compute the point-to-point quark propagators, and measure the time-correlation functions of the pseudoscalar and vector mesons. The inverse lattice spacing is determined by the Wilson flow, while the strange and the charm quark masses by the masses of the vector mesons $ \phi(1020) $ and $ J/\psi(3097) $ respectively. Using heavy meson chiral perturbation theory (HMChPT) to extrapolate to the physical pion mass, we obtain $ f_D = 202.3(2.2)(2.6) $ MeV and $ f_{D_s} = 258.7(1.1)(2.9) $ MeV.

In the Standard Model (SM), the quark and antiquark of a charged pseudoscalar meson P (with quark content Qq) can decay into a charged lepton and its associated neutrino through a virtual W boson. This is the purely leptonic decay of the charged pseudoscalar meson. To the lowest order, the purely leptonic decay width can be written as where G F is the Fermi coupling constant, V Qq is the Cabibbo-Kobayashi-Maskawa (CKM) matrix element, m l is the mass of the lepton, M P is the mass of the charged pseudoscalar meson, and f P is the decay constant of the charged pseudocalar meson, which is defined by the matrix element of the axial vector current between the QCD vacuum and the one-particle state of the charged pseudoscalar meson, According to (1), experimental measurement of the leptonic decay width gives a determination of the product |V Qq |f P . If the value of f P can be obtained from another experimental measurement, then the value of |V Qq | can be determined, which is crucial for testing the SM via the unitarity of the CKM matrix, as a constraint for any new physics beyond the SM.
On the other hand, if the value of f P is unavailable from other experiments, then it must be determined theoretically from the first principles of QCD, before the value of |V Qq | can be fixed.
Theoretically, lattice QCD is a viable framework to tackle QCD nonperturbatively from the first principles of QCD, by discretizing the continuum space-time on a 4-dimensional space-time lattice [1], and computing physical observables by Monte Carlo simulation [2].
However, in practice, any lattice QCD calculation suffers from the discretization and finite volume errors, plus the systematic error due to the unphysically heavy u/d quark masses (with M π > 140 MeV). Moreover, since all quarks in QCD are excitations of Dirac fermion fields, it is vital to preserve all salient features of the Dirac fermion field on the lattice, in particular, the chiral symmetry of the massless Dirac fermion field. It is nontrivial to formulate Dirac fermion field with exact chiral symmetry at finite lattice spacing. This is realized through the domain-wall fermion (DWF) on the 5-dimensional lattice [3] and the overlap-Dirac fermion on the 4-dimensional lattice [4].
In June 2005, we determined the masses and decay constants of pseudoscalar mesons D and D s in quenched lattice QCD with exact chiral symmetry [5], before the CLEO Collaboration announced their high-statistics measurement of f D in July 2005. Our theoretical predictions of f D and f Ds turned out in good agreement with the experimental values from the CLEO Collaboration [6,7]. In 2007, we extended our study to the B-mesons [8], and determined the masses and decay constants of B s and B c , as well as the lowest-lying spectra of heavy mesons with quark contents bb, cb, sb, and cc.
To remove the systematic error due to the quenched approximation, it is necessary to simulate lattice QCD with dynamical quarks. For lattice QCD with exact chiral symmetry, the challenge is how to perform the hybrid Monte Carlo (HMC) simulation [9] such that the chiral symmetry is preserved at a high precision and all topological sectors are sampled ergodically.
During 2011-2012, we demonstrated that it is feasible to perform large-scale dynamical QCD simulations with the optimal domain-wall fermion (ODWF) [10], which not only preserves the chiral symmetry to a good precision, but also samples all topological sectors ergodically. To recap, we perform HMC simulations of two flavors QCD on the 16 3 × 32 lattice (with lattice spacing a ∼ 0.1 fm), for eight sea-quark masses corresponding to the pion masses in the range 228-565 MeV. Our results of the topological susceptibility [12], as well as the mass and decay constant of the pseudoscalar meson [13], are all in good agreement with the sea-quark mass dependence predicted by the next-to-leading order (NLO) chiral perturbation theory (ChPT). This asserts that the nonperturbative chiral dynamics of the sea-quarks are well under control in our HMC simulations. In this paper, we perform HMC simulations of two flavors QCD with ODWF on the 24 3 × 48 lattice (with lattice spacing a ∼ 0.062 fm), with the purpose of studying the charm physics in lattice QCD with exact chiral symmetry.
In general, the 5-dimensional lattice Dirac operator of ODWF can be written as [14] [ where ρ s = cω s +d, σ s = cω s −d, and c, d are constants. The indices x and x ′ denote the sites on the 4-dimensional space-time lattice, and s and s ′ the indices in the fifth dimension, while the lattice spacing a and the Dirac and color indices have been suppressed. The weights {ω s , s = 1, · · · , N s } along the fifth dimension are fixed according to the formula derived in [10] such that the maximal chiral symmetry is attained. Here D w is the standard Wilson Dirac operator plus a negative parameter −m 0 (0 < m 0 < 2), where U µ (x) denotes the link variable pointing from x to x+μ. The operator L is independent of the gauge field, and it can be written as where N s is the number of sites in the fifth dimension, m ≡ rm q , m q is the bare quark mass, Including the action of Pauli-Villars fields (with bare mass m P V = 1/r), the partition function of ODWF in a gauge background can be written as which can be integrated successively to obtain the fermion determinant of the effective 4dimensional Dirac operator [10] where Here S opt (H) = HR Z (H), where R Z (H) is the Zolotarev optimal rational approximation of For HMC simulation of lattice QCD with ODWF, it is crucial to perform the even-odd preconditioning on the ODWF operator (3) such that the condition number of the conjugate gradient is reduced and the memory consumption is halved. Now, separating the even and the odd sites on the 4-dimensional space-time lattice, (3) can be written as We further rewrite it in a more symmetric form by defining and Then Eq. (8) becomes where the Schur decomposition has been used in the last equality, with the Schur complement Since det D = det S −1 1 · det C · det S −1 2 , and S 1 and S 2 do not depend on the gauge field, we can just use C for the Monte Carlo simulation. After including the Pauli-Villars fields (with m P V = 1/r), the pseudofermion action for two-flavors QCD (in the isospin symmetry limit m u = m d ) can be written as where φ and φ † are complex scalar fields carrying the same quantum numbers (color, spin) of the quark fields. Including the gluon fields, the partition function for 2 flavors QCD can be written as where S g [U] is the lattice action for the gluon field. Here we use the plaquette gauge action Further details of our HMC simulations of two flavors QCD can be found in Ref. [15] and a forthcoming long paper.  (4) 347(7)(6) C 0.020 501 3.044(13)(11) 0.000028 (3) 474(6)(5) We perform the hybrid Monte Carlo simulation of two flavors QCD on the 24 3 × 48 lattice with the plaquette gauge action at β = 6/g 2 = 6.10, for three sea-quark masses  Table I. In Fig. 1, we plot the histogram of the topological charge (Q t ) distribution for these three ensembles. Evidently, the probability distribution of Q t for each ensemble behaves like a Gaussian, and it becomes more sharply peaked around Q t = 0 as the sea-quark mass gets smaller. Here the topological charge  projecting the zero modes and the low-lying eigenmodes of the overlap Dirac operator for each gauge configuration, with the same procedures as outlined in Ref. [12], and we will report our results in a forthcoming long paper.
To determine the lattice scale, we use the Wilson flow [16] with the condition [17] {t 2 E(t) } t=t 0 = 0.3, to obtain √ t 0 /a for each gauge ensemble. Our procedures are as follows. First, we compute the Wilson flow for each gauge ensemble of the 2-flavors QCD on the 16 3 × 32 lattice at β = 5.95 [13], and obtain the value of √ t 0 /a. By linear extrapolation, we obtain We compute the valence quark propagator with the point source at the origin, and with parameters exactly the same as those of the sea-quarks (N s = 16 and λ min /λ max = 0.05/6.2).
First, we solve the following linear system (with even-odd preconditioned CG), where B −1 x,s;x ′ ,s ′ = δ x,x ′ (P − δ s,s ′ + P + δ s+1,s ′ ) with periodic boundary conditions in the fifth dimension. Then the solution of (18) gives the valence quark propagator For each gauge ensemble, we measure the time-correlation functions for pseudoscalar (P ) and vector (V ) mesons, In general, the decay constant f P of a charged pseudoscalar meson P with quark content Qq is defined by (2). In lattice QCD with exact chiral symmetry, we can use the axial Ward idenity ∂ µ (qγ µ γ 5 Q) = (m q + m Q )qγ 5 Q, to obtain where the pseudoscalar mass M P a and the decay amplitude z ≡ | 0|qγ 5 Q|P ( 0) | can be obtained by fitting the pseudoscalar time-correlation function C P (t) to the formula where the excited states have been neglected.
To measure the chiral symmetry breaking due to finite N s , we compute the residual mass according to the formula [14] m res = tr(D c + m q ) −1 where (D c + m q ) −1 denotes the valence quark propagator with m q equal to the sea-quark mass, tr denotes the trace running over the color and Dirac indices, and the brackets · · · U denote averaging over an ensemble of gauge configurations. In Table I, we list the residual mass of each ensemble. We see that the residual mass is at most ∼ 1% of the bare quark mass, amounting to ∼ 0.17 MeV, which is expected to be much smaller than other systematic errors. In Table I Table I Table II, where the error denotes the combined statistical and systematic error.
Using the value of m c in Table II, Table II, we measure  TABLE II Table III, where the error denotes the combined statistical and systematic error.  For the decay constants f D and f Ds , we use HMChPT [20] to extrapolate to the physical M π = 140 MeV. In general, for the pseudoscalar meson with quark content (cq), the NLO formula for N f = 2 reads where ξ = M 2 π /(4πf ) 2 , ξ q = M 2 qq /(4πf ) 2 , M qq is the mass of the pseudoscalar meson with quark content qq, g c = 0.61(7) which is determined by the experimental measurement of the coupling g D * →Dπ [21], and κ and c 1 are low-energy constants. Forq equal tod, (22) reduces reliably. Nevertheless, we do not expect it to be much larger than the systematic errors due to the chiral extrapolation and the scaling violations, beacuse our lattice action is free from O(a) discretization errors. Now, assuming the discretization error to be ∼ 2 MeV, together with the systematic errors due to the scaling violations and the chiral extrapolation, we which is in good agreement with the experimental values [22,23], as well as the experimental average f D = 204.6 ± 5.0 MeV [24].
Next, we turn to the decay constant of D s meson. Fitting the data of ensembles (A)-(C) to (22) withq =s [see the upper curve in Fig. 2 (a)], we obtain κ = 0.2790 (12) and which is in good agreement with the experimental values [7,25,26], as well as the experimental average f Ds = 257.5 ± 4.6 MeV [24].
Since the ratio f Ds /f D is expected to have systematic error less than those of f D and f Ds , our results of f D and f Ds yields in good agreement with the experimental average f Ds /f D = 1.258 ± 0.038 [24].
To summarize, we perform the first study of the masses and decay constants of the results [27]. Since our calculation is done at one single lattice spacing, we cannot perform the extrapolation to the continuum limit. Nevertheless, we do not expect the combined systematic errors much larger than our estimates in (24) and (25), since the lattice spacing (a ∼ 0.062 fm) is sufficiently fine, and our lattice action is free from O(a) lattice artifacts.
Likewise, since our calculation is done on a single volume, the finite volume effect cannot be estimated reliably. However, it is believed that the finite volume error of physical observables involving heavy (charm/bottom) quarks is smaller than other systematic ones. We will address these issues with calculations on different volumes as well as several lattice spacings.