Charmed and ϕ meson decay constants from 2+1-flavor lattice QCD
Ying Chen, Wei-Feng Chiu, Ming Gong, Zhaofeng Liu, Yunheng Ma
CCharmed and φ meson decay constants from 2+1-flavorlattice QCD Ying Chen , ∗ , Wei-Feng Chiu , Ming Gong , , Zhaofeng Liu , † , Yunheng Ma , ( χ QCD Collaboration) Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China
Abstract
On a lattice with 2+1-flavor dynamical domain-wall fermions at the physicalpion mass, we calculate the decay constants of D ( ∗ ) s , D ( ∗ ) and φ . The lattice sizeis 48 ×
96, which corresponds to a spatial extension of ∼ . a ≈ .
114 fm. For the valence light, strange and charm quarks, we useoverlap fermions at several mass points close to their physical values. Our resultsat the physical point are f D = 213(5) MeV, f D s = 249(7) MeV, f D ∗ = 234(6) MeV, f D ∗ s = 274(7) MeV, and f φ = 241(9) MeV. The couplings of D ∗ and D ∗ s to thetensor current ( f TV ) can be derived, respectively, from the ratios f TD ∗ /f D ∗ = 0 . f TD ∗ s /f D ∗ s = 0 . f D ∗ /f D = 1 . f D ∗ s /f D s = 1 . f D s /f D = 1 . f D ∗ s /f D ∗ = 1 . Meson decay constants are important nonperturbative quantities for the study of mesonleptonic decays, and their results from lattice Quantum Chromodynamics (QCD) havereceived much attention. The pseudoscalar meson decay constants ( f P ) can be neatlyused to determine the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, if com-bined with experiment measurements of the corresponding leptonic decays. The newestlattice QCD average of f P can be found in the review by Flavor Lattice Averaging Group(FLAG) [1]In principle vector meson decay constants f V can also be used to determine CKM ma-trix elements although experimental measurements of leptonic decays of vector mesonsare much harder than those of pseudoscalar mesons due to small branching ratios. With ∗ [email protected] † [email protected] a r X i v : . [ h e p - l a t ] O c t ncreasing statistics the leptonic decay of D ∗ s may be expected to be measured by BES-III or Belle II in the near future for the first time for a vector meson [2]. Then thecomparison of f D ∗ s from experiment and theoretical calculation can be used to study thelow energy properties of QCD.Furthermore, decay constants of heavy-light vector mesons can be used to test theaccuracy of heavy quark effective theory (HQET). Neglecting terms of O (1 /m Q ), where m Q is the heavy quark mass, one has f V /f P = 1 − α s ( m Q ) / (3 π ) [3] from the leadingorder QCD calculation, which implies that the ratio f V /f P approaches one since thestrong coupling constant α s ( m Q ) vanishes in the infinite heavy quark mass limit. Wecan obtain the corrections from the higher order terms in charmed mesons through theratio f V /f P from lattice QCD calculations. Also, the ratios f V /f P for charmed mesonsare input parameters for QCD factorization studies of charmed nonleptonic B mesondecays [4, 5]. Another important quantity f TV is the coupling of a vector meson to thetensor current. The nonperturbative determination of the ratio f TV /f V is important inlight cone QCD sum rule (LCSR) calculations of form factors in B to vector mesonsemileptonic decays (see discussions in [6, 7, 8]).In this paper, we present a lattice calculation of D ( ∗ ) s , D ( ∗ ) and φ meson decay con-stants in a lattice setup with chiral fermions, which are usually expected to be importantwhen light flavors are involved since chiral symmetry is a fundamental property of QCD.We use overlap fermions for valence quarks and carry out the calculation on 2+1-flavordomain wall fermion gauge configurations generated by the RBC-UKQCD Collabora-tions. The lattice size is big enough ( ∼ . f D ∗ s in literatures so far. Two of them were performed on 2-flavorgauge ensembles [9, 10]. The other two were performed on 2+1-flavor ensembles [2] and2+1+1-flavor ensembles [11], respectively. An unexpected large quenching effect of thestrange quark was observed in f D ∗ s and f D ∗ s /f D s from the 2-flavor result [9] (confirmedin [12] but with a reduced effect). While the 2-flavor result from [10] shows a much lesspronounced effect. In this study we give an independent 2+1-flavor calculation for f D ∗ s to compare with the aforementioned calculations.The rest of this paper is organized as follows. In Sec. 2 we give our framework of thecalculation, including the definitions of the decay constants and the lattice setup. Sec. 3presents the details of the analyses, the numerical results and discussions. Finally, wesummarize in Sec. 4. The decay constant f P of a pseudoscalar meson P is defined through (cid:104) | ¯ ψ ( x ) γ µ γ ψ ( x ) | P ( p ) (cid:105) = ip µ f P e − ipx , (1)2ith p µ being the momentum of the meson. By using the partially conserved axialvector current (PCAC) relation, f P can be obtained from the matrix element of thepseudoscalar density ( m + m ) (cid:104) | ¯ ψ (0) γ ψ (0) | P ( p ) (cid:105) = m P f P , (2)where m , are quark masses and m P is the pseudoscalar meson mass. For overlapfermions, the quark mass and pseudoscalar density ¯ ψ γ ψ renormalization constantscancel each other ( Z P = Z − m ) due to chiral symmetry. This makes f P obtained fromEq.(2) free of renormalization.The vector meson decay constant f V is given by the matrix element of the vectorcurrent between the vacuum and vector meson V as (cid:104) | ¯ ψ (0) γ µ ψ (0) | V ( p, λ ) (cid:105) = m V f V (cid:15) µ ( p, λ ) , (3)where (cid:15) µ ( p, λ ) is the polarization vector of meson V ( p, λ ) with helicity λ . We use thelocal vector current on the lattice to compute the above matrix element for convenience.The price to pay is the need of a calculation of the finite renormalization constant for thelocal current, which was obtained nonperturbatively in Ref. [13] for our lattice setup.Besides f V , vector mesons have another decay constant f TV which is defined throughthe following matrix element of the tensor current (cid:104) | ¯ ψ (0) σ µν ψ (0) | V ( p, λ ) (cid:105) = if TV ( (cid:15) µ ( p, λ ) p ν − (cid:15) ν ( p, λ ) p µ ) . (4)Here in the tensor current σ µν = ( i/ γ µ , γ ν ]. Since the tensor current has a nonzeroanomalous dimension, we will give values of f TV in the commonly used MS scheme andat a scale µ = 2 GeV. The matching factor from the lattice to the continuum MS schemefor the tensor current was presented in Ref. [13]. Our calculation is carried out on the gauge configurations of N f = 2 + 1 domain wallfermions generated by the RBC-UKQCD Collaborations [14]. We use the gauge ensemblenamed as 48I with lattice size L × T = 48 ×
96 and pion mass m (sea) π = 139 . a − = 1 . La ∼ . mix , is 0 . [16], which is verysmall reflecting a small partial quenching effect. The multi-mass algorithm of overlapfermions [17] permits calculations of multiple quark propagators with a reasonable cost.We calculate propagators with a range of masses from the light to charm quark on 45configurations. The valence quark masses am (val) q ( q = l, s, c ) in lattice units are given inTable 1. The deflation algorithm is adopted to accelerate the inversion by projecting out3able 1: Parameters of gauge configurations used in this work. am (val) q ( q = l, s, c ) arethe valence quark mass parameters in lattice units and the corresponding pion masses(in MeV) are from Ref. [15]. The physical charm quark mass am phy c is estimated to bearound 0.73 (see below). L × T × a − (GeV) 1.730(4) N conf am (val) l m π / MeV 114(2), 135(2), 149(2), 208(2) am (val) s am (val) c am (val) l (as listed in Table 1) for the light valence quarksfor chiral interpolation. The corresponding pion masses range from 114 MeV to 208MeV [15]. Two strange quark mass parameters are used to extrapolate to the physicalstrange quark mass point. The bare charm quark masses that we use are around 0 .
72 inlattice units, which are not small. Although for chiral lattice fermions the discretizationerror due to the heavy quark mass starts at O (( am c ) ), it could still be large. Thus, weshall try to estimate the finite lattice spacing effects in our results for D -mesons. The matrix elements in Eq. (1), (3) and (4), from which the decay constants are defined,can be derived directly from the related two-point functions with the currents being thesink operators. Since the mesons involved in this study are all the ground state hadrons,in order for the matrix elements to be determined precisely, it is desired that the two-point functions are dominated by the contribution from the ground states. In this work,we adopt the Coulomb wall-source technique. That is to say, we perform the Coulombgauge fixing to the gauge configurations firstly, and then calculate the two-point functionsusing the following wall-source operators which are obviously gauge dependent, O ( W )Γ ( t ) = (cid:88) (cid:126)y,(cid:126)z ¯ ψ f ( (cid:126)y, t )Γ ψ f ( (cid:126)z, t ) , (5)where ψ f = u, d, s, ... and Γ = γ for pseudoscalar mesons and Γ = γ i ( i = 1 , , P -wave scattering states in the vectorchannels. 4or the sink operators, we use spatially extended operators O Γ ( (cid:126)x, t ; (cid:126)r ) by splitting thequark and anti-quark field with spatial displacement (cid:126)r , namely, O Γ ( (cid:126)x, t ; (cid:126)r ) ≡ ¯ ψ f ( (cid:126)x + (cid:126)r, t )Γ ψ f ( (cid:126)x, t ). The operators with the same spatial separation r ≡ | (cid:126)r | are averagedto guarantee the correct quantum number, and also to increase the statistics as a by-product. Thus, the two-point functions we calculate are C P ( r, t ) = 1 N r (cid:88) (cid:126)x, | (cid:126)r | = r (cid:104) | O γ ( (cid:126)x, t ; (cid:126)r ) O ( W ) † γ (0) | (cid:105) , (6) C V ( r, t ) = 13 N r (cid:88) (cid:126)x,i, | (cid:126)r | = r (cid:104) | O γ i ( (cid:126)x, t ; (cid:126)r ) O ( W ) † γ i (0) | (cid:105) , (7)and C T ( r = 0 , t ) = 13 (cid:88) (cid:126)x,i (cid:104) | O σ i ( (cid:126)x, t ) O ( W ) † γ i (0) | (cid:105) , (8)where N r is the number of O Γ ( (cid:126)x, t ; (cid:126)r )’s with the same | (cid:126)r | = r . The two-point func-tions C ( r, t ) with different r can be calculated simultaneously without expensive extrainversions. After the insertion of the intermediate states, the spectral expression of atwo-point function reads C ( r, t ) = (cid:88) n, | (cid:126)r | = r m n N r (cid:104) | O Γ ( (cid:126) , (cid:126)r ) | n (cid:105)(cid:104) n | O ( W ) † | (cid:105) e − m n t ≡ (cid:88) n Φ n ( r ) e − m n t , (9)where Φ n ( r ) is proportional to the Bethe-Salpeter amplitude N r (cid:80) | (cid:126)r | = r (cid:104) | O Γ ( (cid:126) , (cid:126)r ) | n (cid:105) forthe n -th state. Since the r dependences of Φ n ( r ) are different for different states in eachchannel, a proper linear combination of several C ( r, t )’s with different r may give anoptimal two-point function C ( ω, t ) ≡ (cid:80) ω i ω i C ( r i , t ) which is dominated by the groundstate.Obviously, the parameterization of Eq. (9) shows that the spectral weight Φ n ( r =0) is proportional to the matrix element that defines the decay constant of a specificmeson state. However, in order to get the decay constant, we need to remove the factor (cid:104) n | O ( W ) † Γ | (cid:105) , which is the matrix element of the wall-source operator O ( W ) † betweenthe vacuum and the meson state and can be derived from the wall-to-wall correlationfunction C W ( t ) = (cid:104) | O ( W ) ( t ) O ( W ) † (0) | (cid:105) . (10) To extract the meson masses, we apply two fitting strategies. One strategy is applyingcorrelated simultaneous fittings to the correlation functions with different r ’s using one5for vector mesons) or two (for pseudoscalar mesons) mass terms. The function formused in the simultaneous fits is C ( r, t ) = (cid:88) n =0 Φ n ( r ) (cid:104) e − m n t + e − m n ( T − t ) (cid:105) , (11)where T = 96 and the second term in the brackets on the right hand side comes fromthe propagation of the correlator in the negative time direction. Φ n ( r ) and m n are fittedwith the minimum χ method. We vary the number of mass terms to two or threeand check the stability of the fitting results. Within statistical uncertainties the fittedground state mass m does not depend on the number of mass terms. The upper limitof the fitting range [ t min , t max ] is chosen by the following criteria. For the pseudoscalarchannel t max is fixed to the maximum value where the relative errors of correlatorssatisfy δC/C ≤ t max is chosen by requiring δC/C ≤ χ / dof ≤ . t min to give our finalresults. The uncertainties are obtained from Jackknife analyses to take into account thecorrelations among the data as we repeat the fitting for each Jackknife ensemble.In the left panel of Fig. 1 we show the fitted ground state mass M D in lattice unitsas a function of t min . Here we finally choose the fitting range [11, 18] for the D meson. a M D t/am c =0.7200m l =0.0024 r=6.32ar=0 Figure 1: M D in lattice units as a function of t min (left panel). M D (the band in the rightgraph) from fitting range [11, 18] is compared with the corresponding effective massesfrom varies correlators (right panel).In the right panel of Fig. 1 the obtained ground state mass M D (the band in the graph)is compared with the corresponding effective masses M eff = log( C ( r, t ) /C ( r, t + 1)) fromvarious correlators with different r . The data points in magenta squares are from thecorrelator with r = 6 . a ( (cid:126)r = (2 , ,
0) and permutations averaged). The ones in bluetriangles are from the local sink correlator with r = 0. The ones in black circles are theeffective masses from a combination of two correlators C ( ω, t ) = C ( r = 1 , t ) + ωC ( r, t ) , (12)6able 2: The masses of D -mesons with am c = 0 .
72 extracted from two fitting strategies.The first errors are statistical from Jackknife analyses. The second errors are systematicerrors from variations in the center values as we vary t min . The two strategies giveconsistent results. am q am D am D ∗ ω and use various C ( r, t ) to make the effective massplateau from C ( ω, t ) appear as early as possible. This leads to our second fitting strategy.Different states with a same quantum number contribute differently to the correlators C Γ ( r, t ). And these contributions vary as r varies. Thus, it is possible to find a large r such that the contribution of the lowest excited state to ωC ( r, t ) cancels that to C ( r =1 , t ) and C ( ω, t ) is dominated by the ground state.In the right panel of Fig. 1, the black circles show a mass plateau which starts muchearlier than that from the correlator C ( r = 6 . a, t ) or C ( r = 0 , t ). Therefore, we canfit the combined correlator C ( ω, t ) easily with a single exponential term. We check thatthis fitting gives stable and consistent ground state mass as we vary the parameter ω .We also confirm that the results from the above two fitting strategies are in consistency.The fitting results of am D and am D ∗ from the two strategies with am c = 0 .
72 aresummarized in Table 2 for comparison. The second strategy gives smaller statisticaluncertainties since the mass plateau from the combined correlator appears earlier andthus data points with less errors are used in fittings. Similar advantages of the secondstrategy are observed in the analyses of other meson masses, therefore we adopt strategyII to obtain meson masses in the following.The results of the pion and kaon masses are shown in Table 3. The pion mass andthe combination m ss ≡ m K − m π are used to fix the physical up (degenerate withthe down quark) and strange quark mass respectively. From Table 3 we can see that2 m K − m π is independent of the pion mass (or equivalently the up/down quark mass)within the statistical uncertainties. This is exactly what we expect from the lowest-orderanalysis of chiral perturbation theory and it is the reason why we use this combination.The results of the meson masses and decay constants will be interpolated/extrapolatedto the physical point where ( a m π ) phys = 0 . a m ss (phys) ≡ a (2 m K − m π ) phys = 0 . m phys π = 139 . m phys K = 493 . φ and K ∗ at our valence quark masses. Fromthe data we see that the mass of K ∗ barely depends on the light quark mass with7able 3: Masses of pion and kaon with statistical uncertainties from Jackknife analyses. am s am q am K am π a (2 m K − m π )0.0580 0.0017 0.2608(24) 0.0659(12) 0.1317(25)0.0024 0.2621(20) 0.0780(12) 0.1313(21)0.0030 0.2631(19) 0.0861(12) 0.1310(20)0.0060 0.2689(20) 0.1202(12) 0.1302(22)0.0650 0.0017 0.2755(22) 0.0659(12) 0.1475(24)0.0024 0.2769(22) 0.0780(12) 0.1473(24)0.0030 0.2780(21) 0.0861(12) 0.1472(23)0.0060 0.2833(18) 0.1202(12) 0.1461(21)Table 4: Masses and decay constants of φ and K ∗ with the statistical uncertainties. Thefitting range of correlators for φ is t ∈ [11 , K ∗ is t ∈ [8 , am s am φ af bare φ am q am K ∗ m K ∗ at the physical point, we use thefollowing interpolation/extrapolation form m K ∗ ( m π , m ss ) = m phys K ∗ + b ∆ m π + b ∆ m ss , (13)where ∆ m π = m π − m π (phys) and ∆ m ss = m ss − m ss (phys). This is the Taylor expansionaround the physical u/d and strange quark masses and we keep only the lowest order,i.e., the linear terms since our quark masses are close to their physical values. Then weobtain m phys K ∗ = 895(10) MeV , (14)where the error includes the statistical/fit uncertainty and the uncertainty of the latticespacing. The parameter b from the fitting is consistent with zero within uncertainty asexpected from the raw data.For the mass of φ we do a linear extrapolation to the physical point a m ss (phys) =0 . a m ss at each of the two strange quark masses we use the average of the four values inthe last column of Table 3. This extrapolation gives m phys φ = 1 . m phys K ∗ and m phys φ are in good agreement withtheir experiment values. This means that the finite lattice spacing effects in the studyof light hadrons are smaller than our current statistical uncertainties.Vector mesons can decay to two pseudoscalar mesons through P -wave. On ourlattice the minimal nonzero momentum is 226 MeV, which is not small. The thresholdsof P -wave decays for φ , D ∗ and D ∗ s mesons are not open on our lattice. But K ∗ candecay to Kπ on our lattice. We observed mass plateaus for the K ∗ meson but not forthe scattering states of Kπ , which we believe are suppressed by the usage of Coulombgauge wall source when calculating the 2-point functions [15]. The agreement of m phys K ∗ and m phys φ (from our interpolation/extrapolation) with their experimental values tells usthat it is safe to ignore the threshold effects at our current precision.The masses of D s and D ∗ s mesons are listed in Table. 5. We use the experimentalvalue of D s (together with m ss (phys) in the above) to set the physical charm (andstrange) quark mass. With our lattice spacing we have ( am D s ) phys = 1 . m D s = 1968 . m D ∗ s to the physicalstrange and charm quark mass point: m D ∗ s ( m ss , m D s ) = m phys D ∗ s + b ∆ m ss + b ∆ m D s , (16)where ∆ m D s = m D s − ( m D s ) phys and b is another free parameter. From this we obtain m phys D ∗ s = 2 . , (17)9able 5: Masses and decay constants of D s and D ∗ s with statistical uncertainties. Thefitting range of correlators for D s is t ∈ [17 , D ∗ s is t ∈ [12 , f bare D ∗ s /f D s is collected in the last column. am c am s am D s af D s am D ∗ s af bare D ∗ s f bare D ∗ s /f D s m D ∗ s to the physical point by using Eq.(16). am D ∗ s is plotted as a function of a ∆ m ss (left panel) or a ∆ m D s (right panel). Theoctagon is the result at the physical strange and charm quark mass point.10able 6: Masses and decay constants of D and D ∗ with statistical uncertainties. Thefitting range of correlators for D is t ∈ [11 , D ∗ is t ∈ [10 , f bare D ∗ /f D is collected in the last column. am c am l am D af D am D ∗ af bare D ∗ f bare D ∗ /f D . m D ∗ s on the strange quark mass is relatively small. Therefore, the slopeof the straight lines in the left plot of Fig. 2 is small. This is also the reason why thetwo lines in the right plot are very close to each other. The dependence on the charmquark mass is apparent. From the position of the physical point in the left plot we canread the physical charm quark mass is around am c = 0 . D and D ∗ mesons are listed in Table. 6. The following ansatz is usedto interpolate/extrapolate our numerical results to the physical quark mass point: m D ( ∗ ) ( m π , m D s ) = m phys D ( ∗ ) + b ∆ m π + b ∆ m ss + b ∆ m D s . (18)Here the term b ∆ m ss appears because our lattice results m D s are not calculated at thephysical strange quark mass and m phys D s is used to set the physical charm quark mass.We get m phys D = 1 . m phys D ∗ = 2 . D mesonmass agrees with the PDG2018 value m D ± = 1 . σ . However our D ∗ meson mass is heavier than the PDG2018 value m D ∗± = 2 . Z A (= Z V ) Z T /Z A (2 GeV) Z T (2 GeV)1.1025(16) 1.055(31) 1.163(34) Before we go into the data analyses for the meson decay constants, we present first therenormalization constants (RCs) for the local vector current and the tensor current. TheRCs of quark bilinear operators for our lattice setup (overlap fermions on domain-wallfermion configurations) were calculated nonperturbatively in Refs. [13, 19]. For the 48Iensemble used in this work we employed both the RI/MOM and the RI/SMOM schemesto calculate those constants nonperturbatively [13]. The matching factors to the MSscheme for the local axial vector current Z A and for the tensor current (at scale 2 GeV)are listed in Table. 7. Because we use chiral fermions, we have Z V = Z A which was alsoconfirmed numerically in Ref. [13]. f P and f V To obtain decay constants f P and f V we perform simultaneous fits to the wall-to-point( C P/V ( r = 0 , t )) and wall-to-wall ( C W ( t )) correlators for a given meson M . Thesefittings are with two exponentials and the ground state mass is constrained within 10 σ to its fitted result from the above strategy II as we determined the meson masses.After removing the matrix element of the source operator (cid:104) | O ( W )Γ | M (cid:105) from the spectralweight of C P/V ( r = 0 , t ), we obtain (cid:104) | O Γ | M (cid:105) and then the decay constants f P/V byusing Eqs.(2,3) and using the fitted meson mass. This fitting and calculation processis repeated for each Jackknife sample to get the statistical uncertainty of f P/V . For f V obtained from the local vector current we need to multiply it with Z V (= Z A ) asdiscussed in Section 3.2.1. In the following we use a superscript “bare” to indicate decayconstants obtained directly from the local vector current.The bare decay constants f V in lattice units for the φ meson at our two strangequark masses are given in the third column of Table 4. The two center values are almostthe same and our statistical uncertainty is big ( ∼ af bare φ . If we do a constant fit to the two numbers,then we obtain ( af bare φ ) phys = 0 . f phys φ = 241(9) MeV after multiplying itwith 1 /a and Z V . If we do a linear extrapolation to the physical strange quark masspoint a m ss (phys) = 0 . f phys φ = 243 . .
3) MeV. We choosethe value with a larger error from the constant fit as our result at the physical point.Therefore, we give f phys φ = 241(9)(2) MeV (20)12s our final result, where the second error comes from the difference between the constantfit and the linear extrapolation and is treated as a systematic error.We use f D s to estimate our discretization error due to the large charm quark masssince we cannot extrapolate to the continuum limit with only one lattice spacing. Thedecay constant in lattice units af D s for all charm and strange quark masses are given inTable 5. One can use the function form given in Eq. (16) (replacing m D ∗ s with f D s ) toextrapolate/interpolate our lattice results in Table 5 to the physical charm and strangequark mass point. What we find is af phys D s = 0 . f phys D s = 249(5) MeV . (21)The difference in the center values of f phys D s calculated in this work and in our previouswork (254(2)(4) MeV) [20] is 5 MeV or 2%. Since our previous result was obtained inthe continuum limit, we treat this 2% difference as an estimate of the discretization errorand assign it to all our decay constants for the charmed mesons in this work.The vector meson decay constant af bare D ∗ s from our lattice data is given in the sixthcolumn of Table 5. Again we use the function form Eq. (16) (replacing m D ∗ s with f D ∗ s ) toextrapolate/interpolate our lattice results to the physical charm and strange quark masspoint. The fitting is shown on the left panel of Fig. 3. Compared with the case of am D ∗ s the quark mass dependence of af bare D ∗ s is hard to see with the relatively big statisticalerrors.Figure 3: The interpolation/extrapolation of f D ∗ s to the physical point by using functionform Eq.(16) (left panel). The right panel shows the interpolation/extrapolation of f D ∗ by using function form Eq.(18). The quark mass dependence is hard to see with therelatively big statistical errors. The octagons show the results at the physical strangeand charm quark mass point.From the extrapolation/interpolation we get ( af bare D ∗ s ) phys = 0 . /a = 1 . Z V = 1 . f phys D ∗ s = 274(5) MeV.Here the uncertainty includes the errors from the statistics, extrapolation/interpolation,13attice spacing and Z V . If we assign a 2% discretization error, then we finally get f phys D ∗ s = 274(5)(5) MeV . (22)At each quark mass combination we find the ratio f bare D ∗ s /f D s as given in the lastcolumn of Table 5. The statistical error is from Jackknife by using the Jackknife estimatesof f D ∗ s and f D s . Then the ratio is extrapolated/interpolated to the physical quark masspoint by using the function form in Eq.(16) (replacing m D ∗ s with the ratio). What we findis ( f bare D ∗ s /f D s ) phys = 0 . Z V and assigning a 2% discretizationerror, we obtain ( f D ∗ s /f D s ) phys = 1 . . (23)The decay constants f D and f D ∗ and the ratio f D ∗ /f D from our lattice data areshown in Table 6. Similarly to the above analyses for f D s and f D ∗ s , we get f phys D = 213(2)(4) MeV , f phys D ∗ = 234(3)(5) MeV , (24)( f D ∗ /f D ) phys = 1 . . (25)Here the first error comes from statistics and the interpolation/extrapolation to thephysical quark mass point by using Eq.(18) with the replacement of m D ( ∗ ) by the decayconstants or their ratio. For f D ∗ the error of Z V is also included in the first error. Thesecond error is the 2% systematic uncertainty due to the finite lattice spacing. As anexample, the interpolation of f D ∗ to the physical pion mass is shown in the right panelof Fig. 3. Since our four light quark masses are distributed around and close to thephysical point (the same is also true for our charm quark masses), the uncertainty of f D ∗ at the physical point is smaller than those of the lattice data.Now we turn to the ratios f D s /f D and f D ∗ s /f D ∗ which reflect the size of SU(3) flavorsymmetry breaking. These ratios can be calculated in two ways. One is using our finalresults for f D ( ∗ )( s ) at the physical quark mass point. By doing this we get 1 . f D s /f D = 1 . f D ∗ s /f D ∗ = 1 . f D s /f D = 1 . f D ∗ s /f D ∗ = 1 . , (26)which we take as our final results for the two ratios. They tell us that SU(3) flavorsymmetry breaking effects are of size ∼ f D s /f D agrees with theresult from the RBC-UKQCD Collaborations in Ref. [21], which uses unitary latticesetups with eight gauge ensembles including the 48I used in this work.14 .2.3 f TV /f V Because of the bad signal-to-noise ratio in C T ( r = 0 , t ) we do not directly determine thedecay constant f TV but calculate the ratio f TV /f V from the ratio of two-point functions f TV f V = lim t →∞ C T ( r = 0 , t ) C V ( r = 0 , t ) ≡ lim t →∞ R ( t ) . (27)The cancelation of statistical fluctuations from the numerator C T ( r = 0 , t ) and thedenominator C V ( r = 0 , t ) leads to a better signal for the ratio R ( t ) since both two-pointfunctions are calculated on the same gauge ensemble and thus are correlated. At thelarge time limit the contributions from the higher states to the two-point functions aresuppressed by their heavier masses. Then from Eq.(3) and Eq.(4) one can derive thatthe ratio approaches f TV /f V for the ground state since the other factors in the numeratorand the denominator cancel out. Fig. 4 shows the ratio R ( t ) for D ∗ and D ∗ s in the leftand right panel respectively. The uncertainties δR ( t ) of the ratio shown in the figureare from Jackknife analyses. As we can see, this ratio approaches a plateau at large t . n (cid:217) (cid:145) > (cid:30) I (cid:134) ¥ (cid:254) f (cid:27) (cid:148) f P C ~ Œ f T K * /f K * t/am s =0.0650m l =0.0030R=0.6950 ± 0.0007 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 5 10 15 20 25 f T f /f f t/am s =0.0650R=0.7163 ± 0.0008 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 6 8 10 12 14 16 18 20 R ( t ) t/am c =0.7200m l =0.0030 D* f T D s * /f D s * t/am c =0.7200m s =0.650R=0.8673 ± 0.0002 ª , (cid:10) §(cid:142) (cid:159) (cid:254) : (cid:254) § K ∗ , φ , D ∗ , D ∗ s (cid:27) f TV /f V ( v ƒ ›(cid:18) z ~ Œ ) " (cid:133) • (cid:130) (cid:143) v k v (cid:10) ı (cid:27) (cid:159) (cid:254) : § • (cid:130) (cid:144) · (cid:137) (cid:10) (cid:152) (cid:135) (cid:130) £ § (cid:13) (cid:154) c [ / (cid:138) (cid:226) ˆ (cid:14) (cid:135) (cid:216) (cid:137) (cid:209) (cid:27) œ “ ( ~ X [85] [86]) ? [ (cid:220) " (cid:138) (cid:130) (cid:27) [ (cid:220) fi † v (cid:10) — (cid:10) § (cid:13) (cid:141) ı (cid:27) º Œ ‹ ƒ (cid:26) [ (cid:220) C (cid:26) ( J " , (cid:9) § • (cid:130) (cid:27) (cid:218) O (cid:216) (cid:11) A ’ d § (cid:142) ‰ I (cid:216) O (cid:145) (cid:27) (cid:216) (cid:11) (cid:140) (cid:216) (cid:8) " m π § m ss · ˆ (cid:14) (cid:135) (cid:216) (cid:27) + (cid:222) (cid:145) § (cid:13) • (cid:130) (cid:192) J m D s (cid:144) · ˇ (cid:143) D s (cid:27) (cid:252) : ’ Ø … Œ (cid:27) & (cid:210) (cid:154) ~ — " m ss (cid:218) m D s K ' O · m π (cid:218) m ss (cid:27) … Œ § ⁄ – • (cid:130) k Ø z (cid:135) m ( val ) s (cid:218) m ( val ) c (cid:130) [ (cid:220) (cid:26)(cid:20) m ss ( ms ; m π ) = m phy ) π (cid:218) m D s ( mc ; m phy ) ss ) " (cid:129) (cid:0) • (cid:130) r m π § m ss (cid:218) m D s (cid:23) ‰ (cid:143)(cid:212) n (cid:27): § (cid:26)(cid:20) (cid:129) “ ( J " (cid:143) (cid:210) · ‘ , • (cid:130) (cid:242) K ( ∗ ) , φ (cid:218) D ( ∗ )( s ) (cid:27) (cid:159) (cid:254) (cid:218) P C ~ Œ U (cid:236) e “ ? [ (cid:220) (cid:181) A ( m u/d , m s , m c ) = A ( phy ) + b ∆ m π ( m u/d ) + b ∆ m ss ( m s ) + b ∆ m D s ( m c ) , (3.25) (cid:217) ¥ A (cid:210) · • (cid:130) (cid:135) ˜ (cid:18) (cid:27) (cid:159) (cid:254) ‰ P C ~ Œ § (cid:13) ∆ m (cid:147) L (cid:10) • (cid:130) (cid:192) (cid:18) (cid:27) º (cid:236) (cid:134) (cid:212) n (cid:138) (cid:27) (cid:11) m − m ( phy ) " [ (cid:220) (cid:27) L § (cid:140) – º (cid:236) ª " • (cid:130) u y § (cid:131) ’ u (cid:159) (cid:254) § P n (cid:217) (cid:145) > (cid:30) I (cid:134) ¥ (cid:254) f (cid:27) (cid:148) f P C ~ Œ f T K * /f K * t/am s =0.0650m l =0.0030R=0.6950 ± 0.0007 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 5 10 15 20 25 f T f /f f t/am s =0.0650R=0.7163 ± 0.0008 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 6 8 10 12 14 16 18 20 f T D * /f D * t/am c =0.7200m l =0.0030R=0.8633 ± 0.0002 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 6 8 10 12 14 16 18 20 R ( t ) t/am c =0.7200m s =0.650 D s * ª , (cid:10) §(cid:142) (cid:159) (cid:254) : (cid:254) § K ∗ , φ , D ∗ , D ∗ s (cid:27) f TV /f V ( v ƒ ›(cid:18) z ~ Œ ) " (cid:133) • (cid:130) (cid:143) v k v (cid:10) ı (cid:27) (cid:159) (cid:254) : § • (cid:130) (cid:144) · (cid:137) (cid:10) (cid:152) (cid:135) (cid:130) £ § (cid:154) c [ / (cid:138) (cid:226) ˆ (cid:14) (cid:135) (cid:216) (cid:137) (cid:209) (cid:27) œ “ ( ~ X [85] [86]) ? [ (cid:220) " (cid:138) (cid:130) (cid:27) [ (cid:220) fi † v (cid:10) — (cid:10) § (cid:13) (cid:141) ı (cid:27) º Œ ‹ ƒ (cid:26) [ (cid:220) C (cid:26) ( J " , (cid:9) § • (cid:130) (cid:27) (cid:218) O (cid:216) (cid:11) A ’ d § (cid:142) ‰ I (cid:216) O (cid:145) (cid:27) (cid:216) (cid:11) (cid:140) (cid:216) (cid:8) " m π § m ss · ˆ (cid:14) (cid:135) (cid:216) (cid:27) + (cid:222) (cid:145) § • (cid:130) (cid:192) J m D s (cid:144) · ˇ (cid:143) D s (cid:27) (cid:252) : ’ Ø … Œ (cid:27) & (cid:210) (cid:154) ~ — " m ss (cid:218) m D s K ' O · m π (cid:218) m ss (cid:27) … Œ § ⁄ – • (cid:130) k Ø z (cid:135) m ( val ) s (cid:218) m ( val ) c (cid:130) [ (cid:220) (cid:26)(cid:20) m ss ( ms ; m π ) = m phy ) π (cid:218) m D s ( mc ; m phy ) ss ) " (cid:129) (cid:0) • (cid:130) r m π § m ss (cid:218) m D s (cid:23) ‰ (cid:143)(cid:212) n (cid:27): § (cid:26)(cid:20) (cid:129) “ ( J " (cid:143) (cid:210) · ‘ , • (cid:130) (cid:242) K ( ∗ ) , φ (cid:218) D ( ∗ )( s ) (cid:27) (cid:159) (cid:254) (cid:218) P C ~ Œ U (cid:236) e “ ? [ (cid:220) (cid:181) A ( m u/d , m s , m c ) = A ( phy ) + b ∆ m π ( m u/d ) + b ∆ m ss ( m s ) + b ∆ m D s ( m c ) , (3.25) (cid:217) ¥ A (cid:210) · • (cid:130) (cid:135) ˜ (cid:18) (cid:27) (cid:159) (cid:254) ‰ P C ~ Œ § ∆ m (cid:147) L (cid:10) • (cid:130) (cid:192) (cid:18) (cid:27) º (cid:236) (cid:134) (cid:212) n (cid:138) (cid:27) (cid:11) m − m ( phy ) " [ (cid:220) (cid:27) L § (cid:140) – º (cid:236) ª " • (cid:130) u y § (cid:131) ’ u (cid:159) (cid:254) § P Figure 4: Ratio of two-point functions R ( t ) for D ∗ (left panel) and D ∗ s (right panel).We do constant fits to R ( t ) in the range [ t min , t max ] to get f TV /f V , where t max is fixedto the maximum value of t with δR/R ≤ t min is varied to check the stability ofthe fitting results. The variation ranges of t min are indicated by the red lines in Fig. 4.We make sure all the fittings give consistent results. In this way we get the bare valueof f TV /f V at each quark mass point. As an example, the numerical results of this ratiofor D ∗ s are presented in Table 8.Then we use Eq.(16) and Eq.(18) to interpolate/extrapolate our raw data to thephysical quark mass point for f TD ∗ s /f D ∗ s and f TD ∗ /f D ∗ respectively. After multiplying theresults with the renormalization factor Z T /Z A (2 GeV) = 1 . f TD ∗ s /f D ∗ s ) phys = 0 . f TD ∗ /f D ∗ ) phys = 0 . f TD ∗ s /f D ∗ s at various valence quark masses. am c am s ( f TD ∗ s /f D ∗ s ) bare D ( ∗ )( s ) and φ in units of MeV. f TV /f V is given in the MSscheme at the scale 2 GeV. D s D ∗ s D D ∗ φf P/V /MeV 249(7) 274(7) 213(5) 234(6) 241(9) f TV /f V - 0.92(4) - 0.91(4) -at the scale 2 GeV. Here the first uncertainty includes the errors from statistics andinterpolation/extrapolation and the error of Z T /Z A (2 GeV), and is dominated by theerror of the renormalization factor. The second uncertainty is from the finite latticespacing effect. We calculated the decay constants f P , f V and f TV /f V of the charmed and light mesonsincluding D ( ∗ )( s ) and φ by using 2+1-flavor domain wall fermion gauge configurations at onelattice spacing. The valence overlap fermion has 4, 2 and 4 mass values respectively forthe light, strange and charm quarks. We use the experiment values of m π , m ss ≡ m K − m π and m D s to set the physical light, strange and charm quark masses. The masses of D , D ∗ ( s ) , φ and K ∗ at the physical point are found by interpolation/extrapolation usingthe lowest order of Taylor expansion (i.e., a linear interpolation/extrapolation) since ourvalence quark masses are close to their physical values.The masses m D , m D ∗ s , m φ and m K ∗ obtained from our lattice calculation are in goodagreement with their experiment measurements. The D ∗ mass we found is 1% higherthan its experiment value. The center value of f D s from this calculation is 2% away fromour previous lattice QCD calculation extrapolated to the continuum limit [20]. Thus,we estimate the discretization uncertainty in this work to be around 2%.The final results of this work for the decay constants are given in Eqs.(20,22-26,28).Quadratically adding together the statistical/fitting uncertainty and the systematic un-certainty, we get the decay constants in Table 9 and some of their ratios in Table 10.For the light vector meson φ the statistical error dominates the uncertainties. Whilefor the heavy mesons the discretization error and the error from Z T /Z A (when needed)16able 10: Ratios of decay constants for D ( ∗ )( s ) . f D ∗ /f D f D ∗ s /f D s f D s /f D f D ∗ s /f D ∗ f TD ∗ s /f D ∗ s and f TD ∗ /f D ∗ are the first lattice QCD calculations, which can be used as input parameters for LCSRcalculations of form factors in B to vector meson semileptonic decays.Our number f φ = 241(9) MeV is lower than the N f = 2 lattice simulation result inRef. [22], which gives f φ = 308(29) MeV. This may be due to the dynamical strangequark effects. Note our f φ is in good agreement with that in [23], which is also a2+1-flavor lattice calculation. The experimental value of f φ can be extracted fromΓ( φ → e + e − ) = 1 . φ → e + e − ) = 4 πα m φ f φ . (29)Inputting α em = 1 / .
036 and m φ = 1019 . f exp φ = 227(2)MeV. Our result agrees with the experiment value at 1 . σ .Our value for f D is 213(5) MeV, which agrees with other lattice QCD calculationswith 2-flavor [24], 2+1-flavor [25, 26, 27] and 2+1+1-flavor [28, 29] simulations. Com-bining the latest experimental average f D + | V cd | = 45 . .
05) MeV from PDG2018 [18]and our f D = 213(5) MeV, one gets | V cd | = 0 . . (30)Here the two errors are from the lattice calculation and experiment, respectively.In Fig. 5 we compare f D ∗ ( s ) and the ratio f D ∗ s /f D s from this work and other lat-tice QCD calculations [2, 9, 10, 11, 12]. The values from 2+1-flavor and 2+1+1-flavorsimulations are in consistency. There might be a tension between 2-flavor calculationsand the other calculations including the dynamical strange quark. This may reflect anunexpected large quenching effect from the strange quark. However the 2-flavor calcula-tion of f D ∗ s /f D s in [10] shows that this quenching effect is not so significant as that seenin [9]. The two calculations employ different lattice actions of the two-flavor theory. Thecomputation in [12] is performed on the same 2-flavor gauge ensembles as used in [9] andemploys the analysis method as used in [11]. It gives a f D ∗ s /f D s with a smaller strangequark quenching effect, and therefore is more in agreement with [10]. Thus, more latticeQCD calculations, especially those with two dynamical flavors, are certainly welcome toclarify this situation.The ratios of decay constants of charmed mesons in Table 10 show that the size ofheavy quark symmetry breaking is about 10%. While the size of SU(3) flavor symmetrybreaking is around 17%. 17igure 5: Comparisons of f D ∗ ( s ) (left panel) and f D ∗ s /f D s (right panel) from lattice QCDcalculations.To better control the systematic uncertainty from discretization effects in our work,we need to perform our calculation at more lattice spacings in the future. Also weneed to include the quark-line disconnected diagram for the φ meson two-point function.To accurately estimate the threshold effects of strong decays of vector mesons, furtherstudies on larger volumes are necessary. Acknowledgements
This work was supported by the National Key Research and Development Program ofChina (No. 2017YFB0203200). We thank RBC-UKQCD collaborations for sharing thedomain wall fermion configurations. This work was partially supported by the NationalNatural Science Foundation of China (NSFC) under Grant 11935017. This research usedresources of the Oak Ridge Leadership Computing Facility at the Oak Ridge NationalLaboratory, which is supported by the Office of Science of the U.S. Department of Energyunder Contract No. DE-AC05-00OR22725. This work used Stampede time under theExtreme Science and Engineering Discovery Environment (XSEDE), which is supportedby National Science Foundation Grant No. ACI-1053575. We also thank the NationalEnergy Research Scientific Computing Center (NERSC) for providing HPC resourcesthat have contributed to the research results reported within this paper.
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