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Charm and Strange Quark Masses and fDs from Overlap Fermions Yi-Bo Yang University of Kentucky

Ying Chen Chinese Academy of Sciences, China

Andrei Alexandru George Washington University

Shao-Jing Dong University of Kentucky, [email protected]

Terrence Draper University of Kentucky, [email protected]

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Repository Citation Yang, Yi-Bo; Chen, Ying; Alexandru, Andrei; Dong, Shao-Jing; Draper, Terrence; Gong, Ming; Lee, Frank X.; Li, Anyi; Liu, Keh-Fei;

Liu, Zhaofeng; and Lujan, Michael, "Charm and Strange Quark Masses and fDs from Overlap Fermions" (2015). Physics and Astronomy Faculty Publications. 390. https://uknowledge.uky.edu/physastron_facpub/390

This Article is brought to you for free and open access by the Physics and Astronomy at UKnowledge. It has been accepted for inclusion in Physics and Astronomy Faculty Publications by an authorized administrator of UKnowledge. For more information, please contact [email protected]. Authors Yi-Bo Yang, Ying Chen, Andrei Alexandru, Shao-Jing Dong, Terrence Draper, Ming Gong, Frank X. Lee, Anyi Li, Keh-Fei Liu, Zhaofeng Liu, and Michael Lujan

Charm and Strange Quark Masses and fDs from Overlap Fermions

Notes/Citation Information Published in Physical Review D, v. 92, no. 3, article 034517, p. 1-19.

© 2015 American Physical Society

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Digital Object Identifier (DOI) https://doi.org/10.1103/PhysRevD.92.034517

This article is available at UKnowledge: https://uknowledge.uky.edu/physastron_facpub/390 PHYSICAL REVIEW D 92, 034517 (2015) f Charm and strange quark masses and Ds from overlap fermions

Yi-Bo Yang,1,2 Ying Chen,1 Andrei Alexandru,3 Shao-Jing Dong,2 Terrence Draper,2 Ming Gong,1,2 Frank X. Lee,3 Anyi Li,4 Keh-Fei Liu,2 Zhaofeng Liu,1 and Michael Lujan3 1Institute of High Energy Physics and Theoretical Physics Center for Science Facilities, Chinese Academy of Sciences, Beijing 100049, China 2Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506, USA 3Department of Physics, The George Washington University, Washington, DC 20052, USA 4Institute for Nuclear Theory, University of Washington, Seattle, Washington 98195, USA (Received 20 October 2014; published 26 August 2015) We use overlap fermions as valence quarks to calculate meson masses in a wide quark mass range on the 2 þ 1-flavor domain-wall fermion gauge configurations generated by the RBC and UKQCD Collaborations. The well-defined quark masses in the overlap fermion formalism and the clear valence quark mass dependence of meson masses observed from the calculation facilitate a direct derivation of physical current quark masses through a global fit to the lattice data, which incorporates Oða2Þ and 4 4 Oðmca Þ corrections, chiral extrapolation, and quark mass interpolation. Using the physical masses of Ds, Ds and J=ψ as inputs, Sommer’s scale parameter r0 and the masses of charm quark and strange MS quark in the MS scheme are determined to be r0 ¼ 0.465ð4Þð9Þ fm, mc ð2 GeVÞ¼1.118ð6Þð24Þ GeV MS MS (or mc ðmcÞ¼1.304ð5Þð20Þ GeV), and ms ð2 GeVÞ¼0.101ð3Þð6Þ GeV, respectively. Furthermore, we observe that the mass difference of the vector meson and the pseudoscalar meson with the same valence quark content is proportional to the reciprocal of the square root of the valence quark masses. The hyperfine − splitting of charmonium, MJ=ψ Mηc , is determined to be 119(2)(7) MeV, which is in good agreement with 254 2 4 the experimental value. We also predict the decay constant of Ds to be fDs ¼ ð Þð Þ MeV. The masses of charmonium P-wave states χc0, χc1 and hc are also in good agreement with experiments.

DOI: 10.1103/PhysRevD.92.034517 PACS numbers: 11.15.Ha, 12.38.Aw, 12.38.Gc, 14.40.Pq

I. INTRODUCTION setup is a mixed action formalism where we use the overlap fermions as valence quarks and carry out the calculation on A large endeavor has been devoted by lattice QCD to the domain-wall gauge configurations generated by the determine the quark masses which are of great importance for precision tests of the standard model of particle physics RBC and UKQCD Collaborations. Both the domain-wall [1–14]. In the lattice QCD formulation, quark masses are fermion (DWF) and the overlap fermion are chiral fer- dimensionless bare parameters and their renormalized mions; as such, they do not have OðaÞ errors for the valence values at a certain scale should be determined through quark masses, and the additive renormalization for them is 10−9 physical inputs. For the light u; d quarks and the strange also negligible ( ) due to the overlap fermion imple- quark, their masses are usually set by the physical pion and mentation. It is also shown that the nonperturbative renormalization via chiral Ward identities or the regulari- kaon masses as well as the decay constants fπ and fK, where the chiral extrapolation is carried out through chiral zation independent/momentum subtraction (RI/MOM) perturbation theory [1,3,4]. For heavy quarks, the bare quark scheme can be implemented relatively easily. We have masses are first set in the vicinity of the physical region and explored this strategy and found that it is feasible for the physical point can be interpolated or extrapolated valence masses reaching even the charm quark region on through the quark mass dependence observed empirically the set of DWF configurations that we work on. from the simulation. In the above procedures, nonperturba- The RBC and UKQCD Collaborations have simulated tive quark mass renormalization is usually required to match 2 þ 1 flavor full QCD with dynamical domain-wall fer- the bare quark mass to the renormalized one at a fixed scale. mion (DWF) on several lattices in the last decade with pion For the heavy quark, the HPQCD collaboration [6] proposed masses as low as ∼300 MeV and volumes large enough for a promising scheme to obtain their masses from current- mesons (mπL>4) [3,10,12]. It turns out that the fermions current correlators of heavy quarkonium, which is free of the in this formalism with a finite fifth dimension Ls satisfy the quark mass renormalization [8]. Ginsparg-Wilson relation reasonably well and the chiral In this work we propose a global-fit strategy to determine symmetry breaking effects can be absorbed in the small the strange and charm quark masses which incorporates residual masses. As for the overlap fermion, its multimass simultaneously the Oða2Þ correction, the chiral extrapola- algorithm permits calculation of multiple quark propaga- tion, and the strange/charm quark interpolation. The lattice tors covering the range from very light quarks to the charm

1550-7998=2015=92(3)=034517(19) 034517-1 © 2015 American Physical Society YI-BO YANG et al. PHYSICAL REVIEW D 92, 034517 (2015) quark. This makes it possible to study the properties of results on quark masses and fDs are given in Sec. III, where charmonium and charmed mesons using the same fermion a thorough discussion of the statistical and systematic formulation for the charm and light quarks. Having errors is also presented. The summary and the conclusions multiple masses helps in determining the functional forms are presented in Sec. IV. for the quark mass dependence of the observables. In practice, we calculate the masses of charmonia and charm- II. NUMERICAL DETAILS strange mesons with the charm and strange quark mass 2 1 varying in a range, through which a clear observation of the Our calculation is carried out on the þ flavor domain valence quark mass dependence of meson masses can be wall fermion configurations generated by the RBC/ obtained. Similar calculations are carried out on six UKQCD Collaborations [17]. We use two lattice setups, 3 243 64 β 2 13 configuration ensembles and the results are treated as a namely, the L × T ¼ × lattice at ¼ . and the 323 64 β 2 25 β 2 13 total data set for the global fit as mentioned above. It should × lattice with ¼ . . For the ¼ . lattice, the ðsÞ be noted that the quark masses in the global fit are matched mass parameter of the strange sea quark is set to ms a ¼ to the renormalized quark masses at 2 GeV in the minimal- 0.04 and that of the degenerate u=d sea quark takes the subtraction scheme (MS scheme) by the quark mass ðsÞ 0 005 0 01 values of ml a ¼ . ; . , and 0.02, which give three renormalization constant Zm calculated in Ref. [15].In different gauge ensembles. However, it is found that the order to convert the quantities on the lattice to the values in ðsÞ physical strange quark mass parameter is actually ms a ¼ physical units, we take the following prescription. First, the 0.0348 [17] as determined by the physical Ω baryon mass, ratio of the Sommer scale parameter r0 to the lattice spacing this discrepancy has been corrected by the corresponding a, namely r0=a, is measured precisely from each gauge ðsÞ reweighting factors. Similarly, the ms a of the β ¼ 2.25 ensemble. Subsequently, r0=a’s in the chiral limit are used to replace the explicit finite-a dependence. Instead of lattice is set to 0.03 in generating the three gauge ensembles ðsÞ determining the exact value of r0 by a specific physical with ml a taking values 0.004, 0.006, and 0.008, but the ðsÞ quantity, we treat it as an unknown parameter and deter- physical ms a for β ¼ 2.25 is found to be 0.0273 [17]. mine it along with the quark masses through the global fit Since the physical values of the sea quark masses are not with physical inputs. the same on the two sets of configuration with different β, One of our major observations is that the masses of the we shall assess its systematic error by introducing a linear pseudoscalar and the vector mesons have clear contribu- ðsÞ ms dependent term in the global fitting formula and tions from the term proportional to the reciprocal of the observing the effects when the sea strange mass is shifted square-root of the valence quark masses, as predicted by a to the physical values determined by global fitting itself. On study based on a potential model of the quarkonium where the other hand, the explicit chiral symmetry breaking of the this kind of contribution is attributed to the scaling behavior domain wall fermions gives rise to the residual mass mresa of the spin-spin contact interaction of the valence quarks for the sea quarks which has been studied by RBC and [16]. This is also in quantitative agreement with the feature UKQCD [17]. These corrections to the light sea quark of the meson spectrum from experiments. After incorpo- masses are taken into account for the chiral limit. The rating this kind of mass dependence to the global fit, the parameters of the six gauge ensembles involved in this 1 experimental value of the hyperfine splitting of the S work are listed in Table I, and the numbers of configura- ψ η charmonium, the mass difference of J= and c, can be tions we used are listed in Table II. well reproduced after the charm quark mass, the strange We use the overlap fermion action for the valence quarks quark mass, and r0 at the physical point are determined by to perform a mixed-action study in this work. The massless using J=ψ, Ds and Ds masses as input. We also extract the overlap fermion operator Dov is defined as decay constant fDs of the Ds meson both from the partially 1 γ ϵ ρ conserved axial current relation and the direct definition of Dov ¼ þ 5 ðHWð ÞÞ; ð1Þ fDs along with the renormalization constant ZA of the axial vector current. The two derivations give consistent results where ϵðHWðρÞÞ is the sign function of the which are also in agreement with the experimental value Hermitian matrix HWðρÞ¼γ5DWðρÞ with DWðρÞ the within errors. The masses of charmonium P-wave states χ 0, χ 1 and h are also predicted and they are in good TABLE I. The parameters for the RBC/UKQCD configurations c c c ðsÞ ðsÞ [17]. ms a and ml a are the mass parameters of the strange sea agreement with experiments. ðsÞ This work is organized as follows. We give a detailed quark and the light sea quark, respectively. mres a is the residual description of our numerical study in Sec. II, where we mass of the domain wall sea quarks. focus on the derivation of r0=a and its chiral extrapolation, β 3 ðsÞ ðsÞ ðsÞ L × Tms aml amres a the quark mass renormalization, and the investigation of the 243 64 valence quark mass dependence of mesons, particularly the 2.13 × 0.04 0.005 0.01 0.02 0.00315(4) 2.25 323 × 64 0.03 0.004 0.006 0.008 0.00067(1) hyperfine splitting. The global fit details and the major

034517-2 … CHARM AND STRANGE QUARK MASSES AND fDs PHYSICAL REVIEW D 92, 034517 (2015) 0.45 TABLE II. The number of configurations of the six ensembles aV(r/a=2.828) used in this work. 0.44 0.43 243 64 323 64 × × 0.42 ðsÞ ml a 0.005 0.01 0.02 0.004 0.006 0.008 0.41 ncfg 99 107 100 53 55 50 0.4 0.39 aV(r/a=2.828) 0.38 usual Wilson-Dirac operator with a negative mass param- 0.37 eter −ρ. Thus the effective massive fermion propagator −1 0.36 Dc ðmaÞ can be defined through Dov as [18,19] 0 5 10 15 20 t/a 1 ρD D−1 ma ≡ ; with D ov 0.34 c ð Þ 0 c ¼ 1 − 2 aV(r/a=2.828) Dcð Þþma Dov= 0.33 ð2Þ 0.32 where ma is the bare mass of the fermion. From the γ ρ γ 0.31 Ginsparg-Wilson relation f 5;Dovg¼ aDov 5Dov, one a=2.828) r/ γ5 0 0 ( can check the relation f ;Dcð Þg ¼ [20], which implies V 0.3 that the mass term ma in Eq. (2) acts the same way as an a additive term to the chirally-invariant Dirac operator in the 0.29 continuum Dirac operator and thus there is no additive 0.28 mass renormalization. On the other hand, in order for the 0 5 10 15 20 chiral fermion to exist, ρ should take a value in the range t/a 0 < ρ < 2, so we take the optimal value ρ ¼ 1.5 which gives the smallest ðmaÞ2 error in the hyperfine splitting and FIG. 1 (color online). The plateau of heavy quark potential in −1 coarse/fine lattice ensembles with lightest sea quark masses. The the fastest production of Dc ðmaÞ [21]. Through the 243 64 ðsÞ 0 005 −1 upper panel is for the × lattice with ml a ¼ . , the multimass algorithm, quark propagators Dc ðmaÞ for lower panel for the 323 × 64 lattice with mðsÞa ¼ 0.004. two dozen different valence quark masses ma have been l calculated in the same inversion, such that we can calculate the physical quantities at each valence quark mass and Practically in each gauge ensemble, VðrÞ can be derived obtain clear observation of the quark mass dependence of from the precise measurement of Wilson loops Wðr; tÞ with these quantities. different spatial and temporal extensions ðr; tÞ as

hWðr; tÞi ∼ e−VðrÞt: ð4Þ A. The ratio of the Sommer scale and the lattice spacing The unique dimensionful parameter in the lattice for- Figure 1 shows the effective plateaus of VðrÞ at r=a ¼ mulation of QCD is the lattice spacing a, which is usually 2.828 with respect to t=a. One can see the measurements determined through a sophisticated scheme. Although are very precise and the plateaus last long enough (from 8 dimensionful quantities, such as fπ;fK, and hadron masses to 15 approximatively) for a precise determination of r0=a. have been used to determine the lattice spacing, the lattice VðrÞ is usually parametrized in the Cornell potential results need to be extrapolated to the continuum limit and form, physical pion mass in order for the experimental values to ec be used as inputs for such a determination. In contrast to the VðrÞ¼V0 − þ σr; ð5Þ hadronic quantities which have explicit dependence of r quark masses, the Sommer parameter, r0 (or r1), which where σ is the string tension. Considering the lattice is relatively easy to calculate, has been used to set the scale. spacing a explicitly, the potential one measures on the Still, it needs to be determined precisely at the chiral and lattice is actually continuum limits. r0 is defined by the relation [22], − ec σ 2 dVðrÞ aVðr=aÞ¼aV0 þð a Þr=a; ð6Þ r2 ¼ 1.65; ð3Þ r=a dr r r0 ¼ 2 and the aV0, ec, and σa can be obtained from a correlated where VðrÞ is the static potential in the heavy quark limit minimal-χ2 curve fitting to hVðr; tÞi’s through Eq. (6).For (r1 is defined similarly with 1.65 replaced by 1 [23]). each gauge ensemble, one can find the ratio

034517-3 YI-BO YANG et al. PHYSICAL REVIEW D 92, 034517 (2015)

TABLE III. r0=a’s and the sea quark masses renormalized in 6 r0/a1(mqa1) r0/a1(0) MS scheme at 2 GeV for the six ensembles (EN1, EN2 and EN3 r /a (m a ) at each of β) we are using. The residual masses of the light 0 2 q 2 5.5 r0/a2(0) domain wall sea quark have been included in the sea quark masses. The extrapolated values at the physical point (3.408 /a

0 5 (48) MeV) [1,7–11] are also listed. See the text for more details. r

EN1 EN2 EN3 Physical point 4.5 β 2 13 R ¼ . ml a 0.03653 0.02075 0.01286 r0=a 3.906(3) 3.994(3) 4.052(3) 4.114(10) 4 β 2 25 R ¼ . ml a 0.01323 0.01018 0.00713 0 0.01 0.02 0.03 0.04 0.05 r0=a 5.421(5) 5.438(6) 5.459(6) 5.494(3) (s),R mq a

R rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FIG. 2 (color online). Renormalized sea quark mass mq a dependence of r0=a. Square points for β ¼ 2.13 with lattice r0 1.65 − ec spacing a1, and circular points for β ¼ 2.25 with lattice ¼ σ 2 ð7Þ a a spacing a2. using Eq. (3). Table III lists the calculated r0=a’s for the six ensembles we are using. Note that the sea quark masses R as the extrapolated value of a linear fit in ml a for each (both light and strange) are bare quark masses of the lattice, i.e. domain wall fermion, the physical ones should include the residual masses (0.00315(4) and 0.00067(1) for the two r0 r0 ðmRa; aÞ¼ ðmphys;aÞþfðlÞðaÞðmR − mphysÞa: lattices respectively [17]) and the mass renormalization a l a l l l factor (1.578(2) and 1.527(6) correspondingly [17]) in the ð11Þ MS scheme at 2 GeV, i.e. Without the information of the lattice spacing, we have to ðsÞ;R ðsÞ ðsÞ ðsÞ ml a ¼ Zm ðml þ mresÞa ð8Þ do the fit with extrapolating r0=a to the chiral limit to produce a value of r0 and also the lattice spacings of the The r0 dependence on the lattice spacing a and the sea ensembles at β ¼ 2.13=2.25, then we can extrapolate r0=a 2 phys 3 408 48 quark mass up to Oða Þ can be expressed as [12,24] to ml ¼ . ð Þ MeV coming from the lattice aver-  X  age and iterate the fit until the r0=a is converged. The 0 a 2i extrapolated values CðaÞ are also listed in Table III. r0ða; m ;m Þ¼r 1 þ c a l s 0 i ðlÞ 6 08 44 β i Through such a fit, we get f ðaÞ¼ . ð Þ for ¼ ðlÞ 2 phys 2.13 and f ðaÞ¼6.19ð36Þ for β ¼ 2.25, which are þðcðlÞ þ dðlÞa ÞðmR − m Þ l l independent of the lattice spacing within errors. This ðsÞ ðsÞ 2 R phys ðlÞ þðc þ d a Þðms − ms Þ: ð9Þ implies that the coefficient c in Eq. (9) is very small and is consistent with zero. The strange sea quark mass has Note that the sea quark masses ml and ms should take the been tuned to be close to the physical point, so we ignore renormalized mass values at an energy scale in Eq. (9) in the strange sea quark mass dependence and treat the effect order for the c; d coefficients in the equation to be free of due to deviation from the physical point as a source of the the a-dependence. For the ensembles with the same β, systematic uncertainty. the behavior of r0=a with respect to the light sea quark In view of the above discussion for the subsequent global ðsÞ fits, we shall use the following fitting form mass ml a is shown in Fig. 2, where the square points are for the coarse lattice β ¼ 2.13 (243 × 64 lattices), and 0 1 a 2 l 2 R − phys the circular points are for the fine lattice β ¼ 2.25 r0ða; ml;msÞ¼r0ð þ c1a Þþd a ðml ml Þ: ð12Þ (323 × 64 lattices). In this work, we do not determine r0 (or the lattice B. Quark mass renormalization spacing) before the fit of the value of interest like the In lattice QCD, the bare quark masses are input param- hadron masses and decay constants. Instead, we will use eters in lattice units, say, mqa. However in the global fit three hadronic quantities to obtain the r0 (and also m and s including the continuum extrapolation to be discussed later, mc) with mqa has to be converted to the renormalized current quark R r0 mass m ðμÞ at a fixed scale μ and a fixed scheme (usually CðaÞ ≡ ðmphys;aÞð10Þ q a l MS) which appears uniformly in the global fitting formulas

034517-4 … CHARM AND STRANGE QUARK MASSES AND fDs PHYSICAL REVIEW D 92, 034517 (2015) for different lattice spacings. This requires the renormal- C. The quark mass list β ization constant Zm of the quark mass for each to be Since we are using the overlap fermion action for valence settled beforehand. quarks, we can take advantage of the multimass algorithm ψˆ When we use the chirally regulated field ¼ with little computation overhead to calculate the valence 1 − 1 ψ ð 2 DovÞ in the definition of the interpolation fields quark propagators for dozens of different quark masses ma and currents for the overlap fermion, the renormalization in the same inversion. Subsequently, multiple quantities can constants of scalar (ZS), pseudoscalar (ZP), vector (ZV), be calculated at these valence quark masses, such that their and axial vector (ZA) currents obey the relations ZS ¼ ZP quark mass dependence can be clearly observed. Because and ZV ¼ ZA due to chiral symmetry. In addition, Zm can we have not determined the concrete values of lattice −1 be derived from ZS by the relation Zm ¼ ZS . In Ref. [15], spacings yet, we first estimate the meson masses in the the RI-MOM scheme is adopted to do the nonperturbative strange and the charm quark mass regions using the renormalization on the lattice to obtain the renormalization approximate values a−1 ∼ 1.75 GeV for β ¼ 2.13 and constants under that scheme; those values are then con- a−1 ∼ 2.30 GeV for β ¼ 2.25 as determined by RBC verted from the RI-MOM scheme to the MS scheme using and UKQCD [17] where both the sea and valence quarks ratios from continuum perturbation theory. The relations are domain-wall fermions. We obtain the dimensionless between Z’s mentioned above are verified, and the renorm- masses of the pseudoscalar and vector mesons for different alization constants obtained are listed in Table IV. Besides valence quark masses through the relevant two-point the statistical error, systematic errors including those from functions which are calculated with the Z3 grid source the scheme matching and the running of quark masses to increase statistics [21]. The physical strange quark mass in the MS scheme are also considered in Ref. [15]. The is estimated to be around msa ¼ 0.06 for β ¼ 2.13 and systematic error from the running quark mass in the MS msa ¼ 0.04 for β ¼ 2.25; thus we choose the msa region to scheme is negligibly small, while the one from scheme be msa ∈ ½0.0576; 0.077 and msa ∈ ½0.039; 0.047 for the matching is at four loops, and has a size of about 1.4%. The two lattices, respectively. We cover a wider range for MS 2 the charm quark mass, i.e. [0.29, 0.75] and [0.38, 0.57] errors of ZS ð GeVÞ quoted above include both the statistical and systematic ones. A systematic discussion for the two lattices to study the charmonium and charm- on the renormalization of overlap fermion on domain wall strange mesons. The concrete strange and charm quark fermion sea is given in Ref. [15]. mass parameters in this work are listed in Table V. With the above prescriptions, we can replace the renor- It should be noted that for β ¼ 2.13 at physical charm malized quark masses and a by the bare quark mass quark mass, discretization artifacts prevent us from com- ’ parameters m a, r0, Z ð2 GeV;aÞ and CðaÞ defined in puting charmonium states masses using point-source q m ≥ 0 7 Eq. (10) as propagators. For amc . , the quark propagators receive an unphysical contribution related to the locality radius of the overlap operator. This can be seen from a heavy quark R CðaÞ mq ð2 GeVÞ¼Zmð2 GeV;aÞðmqaÞ : ð13Þ r0 expansion of the propagator:   1 1 2 In this way r0 enters into the global fitting formula to be D D ≈ 1 − þ þ : discussed in Sec. II D as a new parameter and can be D þ m m m m fitted simultaneously with other parameters in the fitting formulas. The off-diagonal (nonlocal) elements of the operator D − The discretization errors of the renormalization constant decay exponentially like e r=rð0Þ [25]. For large masses the 2 Zm could induce an extra ma term on the quark mass quark propagator will be dominated by the first term in the dependence of a given meson mass. Note that the lattice expansion, D=m, and at large distances the decay rate will 1 spacings used in Ref. [15] are slightly different from those be set by =rð0Þ rather than m. This regime should set in which will be obtained in this paper, and it could be around the point where the quark mass is comparable with considered as a source of the ma2 dependence.

TABLE V. The bare mass parameters for valence strange and TABLE IV. The renormalization constants of the overlap charm quarks in this study. fermion on domain wall fermion sea. Both the statistical error β 2 13 and the systematic errors from scheme matching and running of ¼ . msa 0.0576 0.063 0.067 0.071 0.077 quark masses in the MS scheme are considered. mca 0.29 0.33 0.35 0.38 0.40 0.42 0.45 0.48 0.50 0.53 0.55 0.58 0.60 0.61 ZS Zm 0.63 0.65 0.67 0.68 0.70 0.73 0.75

β ¼ 2.13 1.127(9)(19) 0.887(7)(15) β ¼ 2.25 msa 0.039 0.041 0.043 0.047 β ¼ 2.25 1.056(6)(24) 0.947(6)(20) mca 0.38 0.46 0.48 0.50 0.57

034517-5 YI-BO YANG et al. PHYSICAL REVIEW D 92, 034517 (2015) 2 Mass of pseudoscalar c-s 0.6, pt 1.9 0.6, wall 2.4 0.7, pt 1.8 0.7, wall 2.2 ) 2 1.7 V a 1.8 M 1.6 (Ge 2464-005 PS 1.6 2464-010 1.5 M 1.4 2464-020 3264-004 1.4 1.2 3264-006 3264-008 1 1.3 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0 5 10 15 20 25 30 R R t m c+m s(GeV) FIG. 3 (color online). Effective mass plot of the pseudoscalar Mass of vector c-s meson mass (Ma) at fixed quark masses (ma). The plateau in the ma ¼ 0.7 case with point source is not reliable. The one with 2.4 coulomb gauge fixed wall source is better, while the plateau still 2.2 drops down for the t>25 region. This plot is based on the result 2 from the gauge ensemble at β ¼ 2.13 with the light sea quark ðseaÞ 0 05 1.8 mass ml a ¼ . . (GeV) 2464-005 V 1.6 2464-010 M 1.4 2464-020 1 β 2 13 3264-004 =rð0Þ.For ¼ . the locality radius for the quark 1.2 3264-006 3264-008 bilinear state is about 1.5a. In Fig. 3 we plot the effective 1 mass for the pseudoscalar cc¯ state using both point sources 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 R R and wall-sources. For am ¼ 0.7 both propagators show m c+m s(GeV) signs of this unphysical state at large times, but for the wall- source propagator the effect of the unphysical state is FIG. 4 (color online). The masses of the pseudoscalar and weaker and the effective mass forms a plateau whereas for vector cs¯ mesons are plotted with respect to the renormalized R R mc þ ms (tentatively taking r0 ¼ 0.46) for the six RBC/UKQCD the point source it never plateaus. We use the wall-source R R propagators to extract the masses for charmonium states. gauge ensembles, where the linear behaviors in mc þ ms are clearly seen. The horizontal lines in the plot are the physical value We have used two-term fitting to account for the effect of of D in the upper panel and D in the lower panel. the excited state. But for safety, we have to use the coulomb s s wall source propagator to construct the charmonium correlators in the three ensembles with β ¼ 2.13 since converted into values in the physical unit using this scale the physical mca is around 0.7. Note that we continue to parameter. Note that we are focusing on the behavior at the use the point source propagator for the cs¯ system to obtain moment, instead of the precise values of the masses here. It the decay constant of Ds. It should be safe since the is interesting that the Ds and Ds masses are almost R R unphysical mode is much heavier than MðDsÞ or MðDsÞ.In completely linear in mc þ ms for both lattices. The light the case of the β ¼ 2.25 ensembles, the correlators based sea quark mass dependence is very weak for Ds masses in on the point source propagators are used and the standard the upper panel of Fig. 4 but sizable for Ds masses in the interpolation of the physical charm quark is applied, since lower panel. On the other hand, the slopes with respect to R R the physical mca is around 0.5 and does not suffer from the mc þ ms are approximately the same for Ds and Ds, while problem of the unphysical state. they still slightly depend on the lattice spacing. The red horizontal lines in the figure show the physical Ds and Ds D. The quark mass dependence of meson masses masses, and the intersection regions with the data indicate where the physical m and m should be. Figure 5 is similar For each of the six gauge ensembles, we calculate the c s η J=ψ masses of the pseudoscalar and the vector mesons of the cs¯ to Fig. 4, but for c and , where one can see the similar η J=ψ and cc¯, with the strange and charm quark taking all the feature of the charm quark mass dependence of c and possible values in Table V. Figure 4 shows the quark mass masses. Based on the above observations, we assume tentatively dependence of cs¯ mesons, where the upper panel is for the dominance by linear dependence of meson masses on the pseudoscalar (Ds) and the lower panel is for the vector quark masses, (Ds). The abscissa is the sum of the renormalized strange and charm quark masses at 2 GeV in the MS scheme, which ð0Þ M ðm ;m ;mÞ¼A0 þ A1m þ A2m þ A3m þ; are converted through Eq. (13) by tentatively taking c s l c s l r0 ¼ 0.46 fm, for example. Meson masses are also ð14Þ

034517-6 CHARM AND STRANGE QUARK MASSES AND fD … PHYSICAL REVIEW D 92, 034517 (2015) - s Mass of pseudoscalar cc form is not well known but can be investigated empirically 3.2 from the lattice observations such as that of Eq. (14). The 3 polynomial with respect to m a in the parentheses takes 2.8 c into account the lattice artifacts of the lattice quark actions. 2.6 Since we use chiral fermion actions both for the sea quarks

(GeV) 2.4 2464-005 (domain wall fermions) and the valence quarks (overlap PS 2.2 2464-010

M fermions), chiral symmetry guarantees that they are auto- 2 2464-020 matically improved to Oða2Þ and higher order artifacts due 1.8 3264-004 3264-006 to the heavy quark are even powers of mca [26]. In the 1.6 3264-008 4 4 ensembles with β ¼ 2.13, even the effect of the mca term 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 could be important since the m a of the physical charm R c m c(GeV) quark is around 0.7. It motivates us to use a large number of - quark mass parameter values in those ensembles to deter- Mass of vector cc mine this effect precisely. There should be also similar 3.2 terms for ml and ms, but they are much smaller in 3 comparison with that of mc and can be neglected. Also 2.8 2

) included in Eq. (15) is the explicit artifact in terms of a 2.6 which comes from the lattice gauge action and other 2.4 (GeV sources of a-dependence. V 2.2 2464-005

M 2464-010 However, this does not complete the investigation of the 2 2464-020 valence quark mass dependence of the meson masses. For 1.8 3264-004 3264-006 example, if we apply the functional form above in the 1.6 3264-008 2 correlated fit of the mass of Ds, the χ =d:o:f: is 4.6, much 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 larger than unity. R m c(GeV) The reason is simple. Given the linear behavior described above, it would be expected that the mass difference of the FIG. 5 (color online). The quark mass dependence of the vector and the pseudoscalar mesons with the same flavor ¯ masses of the pseudoscalar and vector cc mesons is illustrated content is also proportional to the valence quark mass. But in the plots for the six RBC/UKQCD gauge ensembles, where the R the experimental results give a different story: for example, linear behaviors in mc (tentatively taking r0 ¼ 0.46) are clearly Mρ − Mπ ∼ 630 MeV, MK − MK ∼ 400 MeV, MD − seen. The horizon lines in the plot are the physical value ηc in the M ∼ 140 M − M ∼ 140 M ψ − Mη ∼ upper panel and J=Ψ in the lower panel. D MeV, Ds Ds MeV, J= c 117 MeV, etc. do not have a linear dependence in the sum of their constituent quark masses. where the coefficients Ai can be different for different This motivates us to explore the subtle aspect of the mesons, but are independent of the lattice spacing and quark mass dependence of the hyperfine splitting with a quark masses, since m ;m ;m here are the current closer view. In the following section, we will see that c s l 2 quark masses in the continuum QCD Lagrangian, which including this effect reduces the χ =d:o:f: from (4.6) can be defined at an energy scale through a renormal- mentioned above to ∼1.0. ization scheme and are independent of the lattice spacing a. E. Hyperfine splitting Although Figs. 4 and 5 suggest that the a-dependence is mild, it is incorporated with the usual generic formula of In the constituent quark potential model, the vector 13 the charm quark mass dependence of the physical observ- meson and the pseudoscalar mason are depicted as S1 11 ables for the charmonium and charm-light mesons on the and S0 states, respectively, and their mass difference (the Δ lattice. It is expressed as [26] hyperfine splitting) HFS comes from the spin-spin contact interaction of the valence quark and antiquark. A prelimi- Δ nary study on the behavior of HFS with respect to the Mðmc;ms;ml;aÞ quark mass m on the lattice has been performed in 2 4 q ffiffiffiffiffiffi ¼ Mðm ;m ;mÞ × ð1 þ B1ðm aÞ þ B2ðm aÞ Δ ∝ 1 p c s l c c Refs. [21,27], where one finds that HFS = mq 6 2 4 þ OððmcaÞ ÞÞ þ C1a þ Oða Þ: ð15Þ describes the data surprisingly well for mq ranging from the charm quark mass region down to almost the chiral With the help of chiral perturbation theory, Mðmc;ms;mlÞ region. (See Fig. 4 and Fig. 5 in Ref. [21].) For heavy in Eq. (15) could be the theoretical function for the quantity quarkonium, this behavior can be understood qualitatively M which is better known for light quarks. But for as follows. In the quark potential model, the perturbative charmonium and charm-strange mesons, the functional spin-spin interaction gives

034517-7 YI-BO YANG et al. PHYSICAL REVIEW D 92, 034517 (2015)

16πα hs1 · s2i 1 − hs1 · s2i 0 from the gauge ensembles at β ¼ 2.13=2.25 with the Δ ¼ s S¼ S¼ jΨð0Þj2; ð16Þ HFS 9 2 lightest sea quark mass is plotted versus mR mR in mQ q1 þ q2 Fig. 6, where one can see that such a combination is where m is the mass of the heavy quark, s1 2 are the spin consistent with a constant within one sigma in each Q ; χ2 1 10 operators of the heavy quark and antiquark, and Ψð0Þ is ensemble, with q=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid:o:f: ¼ . from the correlated fit the vector meson wave function at the origin. In view of the 1 R R with only the = mq1 þ mq2 term for all the data points fact that charmonium and bottomonium have almost the in the six ensembles. The constants in the two ensem- same 2S − 1S and 1P − 1S mass splittings (N. B. this bles in the Fig. 6 are different which could be due to an equal spacing rule extends to light mesons as well, albeit Oða2Þ or Oðm Þ effect. This suggests the following qualitatively) it is argued [16] that the size of the heavy l functional form quarkonium should scale as

R 1 A4 þ A5ml 2 ∝ Δ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 B0a : 19 rQQ¯ pffiffiffiffiffiffiffi ð17Þ HFS ¼ ð þ Þ ð Þ m R R δ Q mq1 þ mq2 þ m in the framework of the nonrelativistic Schrödinger The parameter δm is included since if δm is zero, the equation. This prediction is checked against the leptonic hyperfine splitting will diverge in the chiral limit. With decay widths and the fine and hyperfine splittings [16] of δm ¼ 0,theχ2=d:o:f: is 0.87 which is better than the charmonium and upsilon and it holds quite well. Since 2 −3 2 former fit without any Oða Þ or Oðm Þ effect, and the Ψð0Þ scales as ðr ¯ Þ = , one finds from Eqs. (16) and (17) l QQ χ2=d:o:f: is almost the same when we set δm ∼ 0.07. that Δ−2 R R Figure 7 shows the HFS versus mq1 þ mq2 with the 1 experimental data points from the review of the Particle Δ ∝ HFS pffiffiffiffiffiffiffi : ð18Þ Data Group in 2014 [28]. The data of the charm-strange m Q and charm-charm system in the ensemble with lightest sea quark mass at β ¼ 2.25 and the correlated fit of Even though the above argument is for heavy quar- those data with δm ¼ 0.068 (the reason we choose this konium, it is interesting to see how far down in quark value will be discussed in Sec. III A) are also plotted on mass it is applicable with a slight modification. In the Fig. 7. The fit we obtained could explain the splittings present study, we also check the quark mass dependence Mρ − Mπ, M − M within 10% level, while the B Δ K K of HFS for the charm-strange systems. Forqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi clarity of meson and bottomonium cases are beyond the scope of Δ R R illustration, the combined quantity HFS mq1 þ mq2 this form.

0.3 10000 PDG.live ) s-c, β=2.25 3/2 0.25 c-c, β=2.25 M * -M 1000 B (s) B(s) ) (GeV

) 0.2 -2 2 q M ψ-Mη J/ c +m

0.15 (GeV 1 Mψ -Mη 100 b b q -2

0.1 HFS

Δ M *-M M * -M K K D (s) D(s) *sqrt(m 10 0.05 Mρ-Mπ HFS s-c, β=2.13 Δ s-c, β=2.25 0 1 0.6 0.8 1 1.2 0.01 0.03 0.1 0.3 1 3 10 mq +mq (GeV) m +m (GeV) 1 2 q1 q2

FIG. 6 (color online). The quark mass dependence of the Δ−2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FIG. 7 (color online). The mq1 þ mq2 dependence of the HFS, Δ − R R vs the physical quarks mass coming from PDG values [28] hyperfine splittings HFS ¼ mV mPS times mq1 þ mq2 for renormalized in MS scheme at 2 GeV. From left to right, the HFS cs¯ systems from the gauge ensemble with the lightest sea quark for Mρ − Mπ, MK − MK, MD − MD , MJ=ψ − Mη , MB − mass at β ¼ 2.13=2.25. One can see that such a combination is ðsÞ ðsÞ c ðsÞ M MΨ − Mη consistent with a constant within one sigma for each ensemble, BðsÞ and b b are plotted for comparison. The solid line in indicated by a solid line in the plot, obtained from an independent the plot is based on the correlated fit of the simulation data points constant fit for each of the two ensembles. with δm ¼ 0.068 GeV.

034517-8 … CHARM AND STRANGE QUARK MASSES AND fDs PHYSICAL REVIEW D 92, 034517 (2015) Finally, the global fit formula for the meson system is In view of the observation from Fig. 6 and Fig. 7 that the hyperfine splitting is primarily determined by the square R R R Mðmc ;ms ;ml ;aÞ root term, one expects that the parameters A0, A1, A2 and A3 of the corresponding pseudoscalar and vector meson R R R ¼ A0 þ A1mc þ A2ms þ A3ml masses to be the same within errors. It is true that one could do the polynomial expansion 1 R around the physical point, but then the fit would need to be þðA4 þ A5m Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l R R iterated until the initial value for the center point of the mc þ ms þ δm polynomial expansion converges to the true physical point. 1 2 R 2 R 4 2 × ð þ B0a þ B1ðmc aÞ þ B2ðmc aÞ ÞþC1a When we do this, we obtain consistent results compared to ð20Þ those of our preferred square-root fit, but the apparent simplicity of an interpolation is illusory. With the square where δm ≈ 70 MeV is a constant parameter, the terms root term, we can skip the iteration, and, as a byproduct, R 2 have a possibly useful phenomenological form. A5ml and B0a are introduced for the light sea quark mass and lattice spacing dependence of Δ . Note that A2 is set HFS III. THE GLOBAL FIT AND RESULTS to zero for the charm quark-antiquark system, and A1 is expected to be close to 1 (or 2) for the meson masses of Actually the meson masses measured from lattice QCD R cs¯ ðcc¯ Þ system. We keep the mc a correction to the fourth simulations are dimensionless values. Since we will be R4 4 order (mc a ∼ 0.25 for the physical charm quark mass at determining the lattice spacing in a global fit, the fit the ensembles at β ¼ 2.13, and just 0.0625 for the case at formula in Eq. (20) cannot be used directly. Instead, we β ¼ 2.25), which turns out to be enough (and necessary for shall multiply the lattice spacing a to both sides of Eq. (20) the charmonium case) in the practical study. and modify the expression to

1 1 1 0 0 R 0 R 0 R 0 0 R Ma ¼ A0 þ A1ðm aÞþA2ðm aÞþA3ðm aÞþðA4 þ A5CðaÞðm aÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c s l l 3=2 R R CðaÞ CðaÞ ðm a þ m a þ δmr0=CðaÞÞ c s 1 1 0 0 R 2 0 R 4 0 × 1 þ B0 þ B1ðm aÞ þ B2ðm aÞ þ C1 ; ð21Þ CðaÞ2 c c C3ðaÞ

0 0 0 0 0 3=2 0 1=2 with A0 ¼ A0r0;A1 ¼ A1;A2 ¼ A2;A3 ¼ A3;A4 ¼ A4r0 ;A5 ¼ A5r0 ; 0 2 0 0 0 3 B0 ¼ B0r0;B1 ¼ B1;B2 ¼ B2;C1 ¼ C1r0: ð22Þ

with mR fixed to the physical point (3.408(48) MeV) 0Ds l A0 0Ds R 0Ds R 0Ds R R M A mc A ms A m [1,7–11]. We have kept the Ma and m a combinations Ds ¼ þ 1 þ 2 þ 3 l q r0 as they are, since Ma is measured directly on the lattice and   D 0D R ’ A0 s A s 1 renormalized mq a s are used as the parameters of the sea þ 4 þ 5 mR pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ’ 3=2 1=2 l R R quark and valence quark actions. For the explicit a s which r r mc þ ms þ δm  0 0  are not accompanied by a mass term, we have replaced 0Δ 0Δ 1 A4 A5 R them with r0=CðaÞ from Eq. (10). Δ ¯ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi HFS;cs ¼ 3=2 þ 1=2 ml R R ; Now, the global fit can be performed for all the relevant r0 r0 mc þ ms þ δm quantities using the measured results from the six ensem- 0J=ψ A ψ ψ ψ 0 0J= mR 0J= mR 0J= R bles. It should be noted that the parameters to be fitted with MJ=ψ ¼ þ A1 c þ A2 s þ A3 ml 0 0 0 r0 this expression are A0 1 2 3 4 5, B0 1 2, and C1 defined in ; ; ; ; ; ; ;  ψ ψ  Eq. (22) for each physical quantity, and the universal A0J= A0J= 1 þ 4 þ 5 mR pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : parameter δm. For comparison, we have O(200) data points 3=2 1=2 l R R r0 r0 mc þ ms þ δm for each physical quantity in the cs¯ system (the total number of data points on all the six ensembles), and the ð23Þ corresponding number in the cc¯ system is O(50). Once we We see that they depend on the renormalized charm and have fitted the coefficients A0 for the three dimen- R R 0;1;2;3;4;5 strange quark masses mc and ms , and the scale parameter − sionless quantities MDs a; MDs a MDs a, and MJ=ψ a,we r0 in the continuum limit. From the physical values of 1 9685 Δ ¯ ≡ − 0 1438 can examine their dimensionful expressions, MDs ¼ . GeV, HFS;cs MDs MDs ¼ . GeV,

034517-9 YI-BO YANG et al. PHYSICAL REVIEW D 92, 034517 (2015)

Δ ¯ ≡ − “ ” TABLE VI. The fitting parameters [defined in Eq. (20)] for MDs , MDs and HFS;cs MDs MDs . We list the default fit (keeping every parameter, listed as the first line of each channel), the “optimal” case (dropping the parameters which are consistent with zero, listed as the second line of each quantity), and also the parameters obtained in the global fit combining all the S-wave quantities (the third 2 Δ ¯ χ line of the MDs and HFS;cs cases. MDs does not have this line since we do not use it in the global fit). The =d:o:f of first two cases are close, while the parameters from the optimal case have higher precision.

2 χ =d:o:f:A0 A1 A2 A3 A4 A5 B0 B1 B2 C1 −0 499 200 −0 14 8 MDs 1.04 1.343(140) 0.881(34) 0.791(25) 0.28(3) . ð Þ 0.3(2) . ð Þ 0.07(10) 0.09(10) 0.11(13) 1.07 1.200(54) 0.913(19) 0.824(16) 0.26(2) −0.338ð36Þ 0.061(6) −0.18ð2Þ 1.297(23) 0.891(8) 0.850(8) 0.30(2) −0.450ð16Þ 0.057(3) −0.18ð1Þ

−0 172 55 −0 07 33 −0 10 8 −0 004 19 −0 01 13 MDs 0.94 1.17(8) 0.904(34) 0.853(16) 0.64(31) . ð Þ . ð Þ . ð Þ 0.088(34) . ð Þ . ð Þ 0.95 1.14(4) 0.913(16) 0.840(14) 0.55(7) −0.137ð18Þ 0.070(6) −0.19ð2Þ Δ −0 025 18 −0 36 22 HFS;cs¯ 0.88 . ð Þ 0.02(3) 0.02(3) 0.5(5) 0.164(7) 0.23(15) 0.36(16) . ð Þ 0.30(20) 0.07(4) 0.91 0.158(7) 0.23(4) 0.35(12) 0.157(3) 0.29(6) 0.37(8)

and MJ=ψ ¼ 3.0969 GeV as inputs, we can determine we obtain from the latter one have higher precision. So R R compared with using M as input, replacing it with the mc ;ms and r0. Ds − Note that the quark masses here are the ones renormal- splitting MDs MDs gives more precise results for ized under given scheme at given scale, specifically the predictions of the charm/strange quark mass and r0. Δ ≡ MSð2 GeVÞ in our case. We ignore the tiny experimental For the same reason, we discuss the splitting HFS;cc¯ − uncertainties of these values. MJ=ψ Mηc instead of Mηc itself. The use of the J=ψ mass instead of the ηc mass as one Note that we did the correlated fit for each quantity input is based on two considerations. First, the exper- independently (turn on all the coefficients) and optimized imental ηc mass is not as precisely determined as that for the fit (turn off the negligible coefficients) to obtain the first J=ψ. Second, the omission of the cc¯ annihilation in the two rows of each quantity in Table VI, then did the fully calculation of charmonium masses necessarily introduces correlated global fit for all the S-wave quantities to obtain Δ ¯ systematic uncertainties. This kind of uncertainty is the third row of MDs and HFS;cs. The case of MDs does not expected to be smaller for J=ψ than for ηc [29]. have such a line since we do not use this quantity directly in We list in Table VI the fitting parameters [defined in the global fit. Due to the correlation between different Δ ¯ ≡ − Eq. (20)] for MDs, MDs and HFS;cs MDs MDs .We quantities, in Table VI, the parameters listed in the second “ ” Δ ¯ give both the default fit (keeping every parameter, line of MDs and HFS;cs are slightly different with that in the listed as the first line of each quantity) and the “optimal” third line which are used for the final results and in the case (dropping the parameters which are consistent with following discussion. zero, listed as the second line of each channel). The χ2=d:o:f of two cases are close, while the parameters A. Systematic errors from the optimal case have higher precision. We note that In Tables VII and VIII we list both statistical and the values of the coefficient A0 of the constant term and systematic errors. For the statistical error we use the the coefficients A1, A2, and A3 of the terms with linear jackknife error of the global fit. Since we apply a global quark-mass dependence on M obtained from the Ds fit for data in all of six ensembles (two lattice spacings with default fit are consistent with those obtained for the three sea masses each), the errors from the Oða2Þ and default fit of M within errors. 4 4 Ds Oðm a Þ corrections, and linear chiral extrapolation have Δ c So for the splitting HFS;cs¯ , these corresponding coef- been included in the statistical error. ficients should be, and are, consistent with zero. To obtain For the systematic errors, we consider those concerning results with higher precision we thus force these coeffi- r0, those of Z ðaÞ, the global parameter δm, continuum/ “ ” Δ m cients to be zero in our optimal fit for HFS;cs¯ . In the first chiral extrapolation, the correlated fit cutoff, a possible Δ two rows of HFS;cs¯ (the sixth and seventh rows from the electromagnetic effect, the effect from the missing charm top) in Table VI, we show that both the default fit with all sea, the one from the mixed action and the heavy quark data the parameters [defined in Eq. (20) as deduced from the points in the ensembles at β ¼ 2.13. combined parameters defined in Eq. (21) by using r0 to be (1) Since r0 is the scale we want to determine in determined in the following section] and the optimal the global fit, we need to consider only two fit excluding the constant term and linear-quark-mass-ffiffiffiffiffiffi systematic errors: one from the statistical error of 1 p 2 dependence terms [thus keeping only the = mq term CðaÞ¼r0ðaÞ=a, and the other from the nonzero a 2 and its Oða Þ corrections] are consistent, but the parameters dependence of r0ðaÞ.

034517-10 … CHARM AND STRANGE QUARK MASSES AND fDs PHYSICAL REVIEW D 92, 034517 (2015)

TABLE VII. The quark mass and Sommer scale r0 after chiral TABLE VIII. Charmonium spectrum results and fDs after and linear Oða2Þ extrapolation. The statistical error σðstatÞ, and chiral and linear Oða2Þ extrapolation, in unit of GeV. σ σ ∂r0 twelve systematic errors from r0 ðr0=aÞ and ð∂a2Þ, the mass Δ ¯ Mχ Mχ 1 M f renormalization (MR) σðMR=statÞ and σðMR=sysÞ, the strange HFS;cc c0 c hc Ds sea quark mass dependence σðSSQMDÞ, the parameter δm PDG [28] 0.1132(7) 3.4148(3) 3.5107(1) 3.5254(2) 0.258(6) σðδmÞ, the chiral and continuum extrapolation σðchiralÞ and The work 0.1188 3.439 3.524 3.518 0.2536 σðaÞ, the possible m − m effect σðu − dÞ, the cutoff of the u d σ correlated fit σðcutÞ, the electromagnetic effect σðEMÞ and the ðstatÞ 0.0021 0.037 0.043 0.011 0.0022 σ heavy quark artifact σðheavyÞ in the ensembles at β ¼ 2.13 are ðr0=aÞ 0.0002 0.001 0.003 0.004 0.0001 ∂r0 listed below the central values. σ 2 ð∂a Þ 0.0018 0.008 0.034 0.056 0.0016 2 σðMRÞ 0.0008 0.008 0.009 0.039 0.0007 χ =d:o:f r0(fm) m (GeV) m (GeV) c s σðSSQMDÞ 0.0027 0.008 0.009 0.005 0.0021 PDG [28] 1.09(3) 0.095(5) σðδmÞ 0.0007 0.001 0.002 0.004 0.0005 σ This work 1.05 0.465 1.118 0.101 ðchiralÞ 0.0041 0.001 0.018 0.012 0.0006 σ − σðstatÞ 0.004 0.006 0.003 ðu dÞ 0.0000 0.001 0.000 0.008 0.0002 σ σðr0=aÞ 0.002 0.001 0.000 ðaÞ 0.0008 0.003 0.004 0.003 0.0006 σ ∂r0 ðcutÞ 0.0005 0.000 0.001 0.000 0.0007 σð 2Þ 0.005 0.007 0.004 ∂a σðEMÞ 0.0008 0.001 0.003 0.004 0.0006 σðMR=statÞ 0.001 0.022 0.000 σðheavyÞ 0.0036 0.018 0.036 0.008 0.0025 σðMR=sysÞ – 0.003 0.000 σðall sysÞ 0.0068 0.023 0.052 0.070 0.0036 σðSSQMDÞ 0.006 0.004 0.002 σðallÞ 0.0069 0.044 0.066 0.071 0.0043 σðδmÞ 0.001 0.001 0.000 σðchiralÞ 0.004 0.006 0.003 σðu − dÞ 0.001 0.001 0.000 σðaÞ 0.002 0.002 0.001 (2) For the nonperturbative mass renormalization factor σ ðcutÞ 0.001 0.001 0.000 in the RI/MOM scheme, ZmðaÞ, two kinds of σðEMÞ 0.002 0.002 0.001 systematic errors are involved. σ ðheavyÞ 0.005 0.007 0.001 (a) One is from the statistical errors of Z ðaÞ.We σ m ðall sysÞ 0.009 0.024 0.006 follow the same procedure as the case of the σ all 0.010 0.025 0.007 ð Þ statistical errors of CðaÞ to estimate the resulting effect on the quantities of interest: namely, for each lattice spacing we redo the fit with the value (a) Our global fits use the central values for CðaÞ. of ZmðaÞ changed by 1σ, and then combine in The effect of the statistical errors of CðaÞ for quadrature the differences. For the strange and each value of a used in the fit is incorporated into charm quark masses, this is the largest of the four a systematic error as follows: For each lattice systematic errors we tabulate. A lot of this is due spacing, we repeat the global fit with the value of to the magnification of the statistical errors of 1σ CðaÞ changed by and calculate the resulting ZmðaÞ, which are independent at the two lattice difference for each quantity of interest, namely spacings, upon extrapolation in lattice spacing. r0, ms and mc, and then combine in quadrature This error will be marked with σðMR=statÞ. the differences for each lattice spacing. This (b) On the other hand, the systematic errors of the error will be marked with σðr0=aÞ. perturbative matching from RI/MOM to the MS (b) For simplicity, we constrain the fit parameter scheme, and the running of the mass renormal- r0ðaÞ at the physical point to be constant as a ization factor to the scale of 2 GeV in MS function of the lattice spacing in the global fits. scheme, are independent of lattice spacing and But in principle there could be an a-dependence, are totally the same for any simulation at any a with nonzero c1 in Eq. (12). In the work of RBC- lattice spacing. Furthermore, since the physical a −0 25 14 UKQCD [17], their fit gives c1 ¼ . ð Þ.For quantities like meson mass or decay constant are a χ2 such small c1, the of the fit is almost un- independent of renormalization scheme or en- a changed: repeating the fit with c1 ¼0.25 ergy scale, this systematic error will not con- changes the χ2=d:o:f: by 0.15%. For each quantity tribute to those quantities, only to the quark of interest, the change in its fit value is reported as masses. For the quark masses, the systematic a small systematic error in Table VII. Had this errors are independent of simulation and so are been larger, it would have been incorporated into a not magnified by linear extrapolation in lattice a statistical error instead, using c1 as a fit parameter, spacing. These, then, are expected to be very but this was determined to be unnecessary apos- small, which is what we see. This error will be ∂r0 marked with σðMR=sysÞ. teriori. This error will be marked with ∂a2.

034517-11 YI-BO YANG et al. PHYSICAL REVIEW D 92, 034517 (2015) 2.02 (3) Since the strange quark mass used in the domain- a=0.111 fm wall configurations (∼120 MeV for the β ¼ 2.13 a=0.083 fm 2 continuum ensembles and ∼110 MeV for the β ¼ 2.25 ensem- bles) are not equal to the physical strange quark 1.98

mass, a systematic error is induced. )(GeV) s In Ref. [17], the reweighting of the strange quark 1.96 mass is used to correct the values obtained from the M(D original samples. In view of the fact that the strange 1.94 sea quark mass has different values in the two sets of 1.92 the ensembles with different lattice spacing, it pro- 0 10 20 30 40 50 60 70 vides another way to estimate the systematic error m (MeV) from the mismatch of the strange sea quark mass. l We can add the strange sea quark mass dependence a=0.111 fm terms into the functional form in Eq. (20) with 3.14 a=0.083 fm coefficients A6 and A7, continuum 3.12 R R R Mðmc ;ms ;ml ;aÞ

)(GeV) 3.1 ψ R R R A mR ¼ A0 þ A1mc þ A2ms þ A3ml þ 6 s;sea M(J/ 3.08 1 R A mR pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3.06 þðA4 þ A5ml þ 7 s;seaÞ mR þ mR þ δm c s 0 10 20 30 40 50 60 70 1 2 R 2 R 4 × ð þ B0a þ B1ðmc aÞ þ B2ðmc aÞ Þ ml (MeV) 2 þ C1a : ð24Þ FIG. 8 (color online). The interpolated values of the MDs and MJ=ψ from the data points of the two charm quark mass values Since the form of the dependence of the lattice which bracket the physical one, for each ensemble with different spacing and the strange sea quark mass are different, β, versus the renormalized sea quark mass. The lattice spacing it is possible to distinguish them in the global fit. dependence of all the three quantities is manifest given the R precision of the data. At the same time, the sea quark dependence After we obtain the coefficients, the condition ms ¼ R of them is not negligible and trends similarly for each β. ms;sea is applied to predict the Sommer scale r0, the quark masses, and the other quantities. The χ2 of the fit with the strange sea quark mass extrapolation is plot in Fig. 8 the interpolated values for the MðDsÞ and almost the same as the one in the default case MðJ=ψÞ versus the renormalized sea quark mass, for without such an extrapolation (1.06 vs 1.07), and the each ensemble with different β. The error bands of the values of each quantity of interest in two ways of fit correlated fit and linear extrapolation of the sea quark are consistent within error. We use the resulting mass in the different lattice spacings, and the con- difference of each quality of interest as the estimate tinuum limit of them (the experimental inputs) are also of this systematic error. This error will be marked plotted in the same figure. Most of the interpolated with σðSSQMDÞ. values are consistent with the fit, and the few excep- (4) In Sec. II E, we induce a parameter δm ∼ tional ones reflect the statistical scatter from such a Oð70Þ MeV since it provides better understanding simple interpolation, compared to a global fit over a of the light vector-pseudoscalar meson mass large quark mass region. differences. In the correlated fit including all the Figure 8 shows the sea quark mass dependence of S-wave related quantities, the value minimizes the χ2 the interpolated values and that of the global fit, for the is δm ¼ 68 MeV. To estimate the systematic error quantities we used as the inputs such as MðDsÞ and by this global parameter, we repeat the fit with δm ¼ MðJ=ψÞ. It is obvious that the slope of the sea quark 68 14 MeV (20% uncertainty) which changes the mass dependence is ∼0.5 from Fig. 8 and not χ2=d:o:f: by 1% and the changes for the fit value of negligible, given the precision of the data points. each quantity of interest is reported as a systematic Since the low lying meson mass in the charmonium error in Table VII. This error will be marked system does not involve any valence light quark, there with σðδmÞ. is no chiral perturbation theory available here to R (5) After we obtain the quark mass mc;s and Sommer scale provide a reliable functional form of the sea quark r0, we can do the interpolation on the data points of the mass dependence. To estimate the systematic error 2 two neighboring charm/strange quark masses, and from the chiral extrapolation, we added the ml term

034517-12 … CHARM AND STRANGE QUARK MASSES AND fDs PHYSICAL REVIEW D 92, 034517 (2015) (following the twisted mass work [13]) into the (7) Figure. 8 also shows that the lattice spacing depend- functional form of the charmonium mass quantities ence (Oða2Þ) based on the functional form with 2 4 4 andrepeatedthefit,andtookthechangesforthefit Oða Þ correction is obvious. In addition, Oðmca Þ value of each quantity of interest as a systematic error. dependence is not negligible in the MðJ=ψÞ case. 3 On the other hand, a mK term involving valence With the ensembles at only two values of lattice strange quark and light sea quark could appear in the spacing, we cannot justify the systematic error of functional form of the Ds or Ds masses due to the such a lattice spacing dependence before we have the 3 chiral perturbative theory, like the mπ term in the K or ensembles at β > 2.25. But, if we change the func- K masses. Since the valence strange quark mass is tional form of the lattice spacing dependence in the much heavier than the light sea quark mass, the effect chiral limit into of this term will not deviate far from a linear 3 0 1 2 2 2 4 dependence of m . So we added the m term into M ðaÞ¼Að þ B1mca þÞþC0a þ Oða Þ l K   the functional form and took the change as a system- 2 ∼ 1 2 2 1 − C0a 4 atic error. Að þ B1mca þÞ= þ Oða Þ A For the decay constant of f , Ref. [30] shows a Ds ¯ 2 partially quenched form for its chiral behavior, ¯ 2 2 ¯ 2 C0 4 ∼ A 1 B1m a C0a a ð þ c þÞþ þ ¯ 2 A 2 msl 2 4 f ¼ c0 1 þ b1m log þ Oðmca ÞÞ ð28Þ Ds sl Λ2 QCD 2 ¯ ¯ 2 2 mss and solve A and C0 with the continuum limit at two b2 m − m þ ð ss llÞ log Λ2 lattice spacings, we find that the S-wave quantities QCD are just changed by about 1 MeV and the changes of the P-wave quantities are a few MeVs. For each of þ c1ms þ c2ml þ ð25Þ the quantities of interest, we combine all the changes of the input quantities and the change of that in which m is the mass of the valence pseudoscalar ss quantity itself in quadrature, and treat it as a possible ss¯ , m is the pion mass in sea, and m is the valence- ll sl estimate of the systematic error of the lattice spacing sea mixed kaon mass. We use the difference between dependence. This error will be marked with σðaÞ. this form and the trivial linear form as an estimate of (8) Due to the precision of the data, the correlated fit the systematic error from the chiral extrapolation. requires a cutoff for the small eigenvalue of the All the estimates of the error due to the chiral correlation matrix of the data points. The cutoff of extrapolation form will be marked with σðchiralÞ. the global fit is set to be 10−11, and we changed the (6) For the possible m − m effects, if the sea quark u d 10−111 mass dependence of a quantity from the u=d cutoff into , repeated the fit, and took the degenerated ensembles is changes for the fit value of each quantity of interest as the systematic error. This effect is very small in the 10−12 − 10−10 AðmlÞ¼A0 þ A3ml; ð26Þ cutoff region and the change of the χ2=d:o:f is just 0.2%. This error will be marked then we can rewrite it into with σðcutÞ. (9) As in Ref. [2], we can estimate the electromagnetic u − u AðmlÞ¼A0 þ A3mu þðA3 A3Þmd ð27Þ effect by modifying the mass of Ds by 1 MeV. In our case, we used MðDsÞ and MðDsÞ − MðDsÞ as the u with a reasonable assumption A3 ∈ ð0;A3Þ for the input, so we modified this two quantities by 1 MeV u=d nondegenerate case. Then the upper bound of independently, and combined the changes in quad- the nondegenerate effect happens at the boundary of rature to estimate this systematic error. This error u the range of A3, in extrapolating ml to mu (2.079 will be marked with σðEMÞ. (94) MeV) or md (4.73(12) MeV), not the average of (10) Our simulation is based on the 2 þ 1 flavor domain- them 3.408(48) MeV [1,7–11]. So the above esti- wall sea configuration which does not have any mate of the systematic error of the chiral extrapo- charm quark in the sea. Reference [29] shows that lation also includes the possible mu − md effects. without the disconnected charm diagram of the For all the quantities of interest, these effects are not correlation function, the hyperfine splitting will larger than the standard estimate 0.2% which comes decrease by a few MeV. But this affects only the − from ðmd muÞ=mp. mass of ηc which is not the input quantity. So we As seen in Fig. 8, the slopes are close to each think this effect will be negligible. other, so we can expect that the effect will not be (11) The mixed action will inevitably introduce partial σ − Δ large. This error will be marked with ðu dÞ. quenching. However, the low-energy constant mix

034517-13 YI-BO YANG et al. PHYSICAL REVIEW D 92, 034517 (2015)

for the overlap on RBC-UKQCD DWF configura- PDG tions, as calculated by the combined DWF and ETMC ’14 overlap proapgators, is very small [31]. As a result, BMW ’10 the mass of the mixed pion involving the light HPQCD ’10 ∼300 valence and sea which corresponds to MeV Laiho Van de Water ’11 ∼10 pion is only shifted by MeV on the ensembles MILC ’09 we used. RBC/KEK/Nagoya The present work does not involve valence light RBC/UKQCD ¯ quarks; we only calculate charmonium and cs this work mesons. The only relevant terms from the χPT to ¯ 3 80 90 100 110 120 130 140 150 the cs mesons are the nonanalytic terms mK in the Λ m 2GeV(MeV) Ds, Ds masses and the logðmK= Þ term in fDs .We s studied the effects of these terms and found that (a) contributions from these terms in chiral extrapola- tion are up to one sigma of the statistical error which are included in the systematic errors as σðchiralÞ in PDG Table VII and VIII. Since the mixed kaon mass is to be shifted by 5 MeV only, this gives a deviation of ALPHA ’13 ∼3% in m3 and 0.15% in log m =Λ . Such small K ð K Þ HPQCD ’10 changes on top of very small contributions from the χPT terms other than the linear dependence of ml are ETMC ’14 not possible to discern and thus they are neglected in the present work. this work The mixed action effect only appears in cases involving a valence-sea mixed meson such as ex- 1.2 1.25 1.3 1.35 1.4 1.45 1.5 2GeV ploring a scattering state with a two-valence-quark mc (GeV) interpolation field, which is not relevant to this work, (b) except for the decay constant of Ds [30]. MS (12) As shown in Fig. 3, the masses extracted for amc FIG. 9 (color online). The prediction of ms ð2 GeVÞ (upper MS above 0.7 could be suspicious, even when using the panel) and mc ðmcÞ (lower panel) from this work, compared to wall-source propagator. So a possible estimate of those of other works. this systematic error is to remove the data points above 0.7 from the analysis and determine how the MS result are affected. It turns out that this does not In Fig. 9(a), we plot our results of ms ð2 GeVÞ to change the charm/strange quark mass and Sommer compare with the 2 þ 1 flavor ones listed in lattice scale much, but affects other quantities by up to one averages, and another recent lattice calculation [13]. The sigma of the statistical error. error bar in the plot and the following ones include both Another issue related to the amc error is that we statistical and systematic errors. Since we determine the forced the coefficient B0 ¼ 0 for all the quantities strange quark mass by the cs¯ spectrum in which the strange except the hyperfine splittings which we observed quark mass only has a minor contribution, our result of ms this effect clearly in Fig. 6. So we can turn on this is not quite precise, but it is consistent with the exper- coefficient in the global fit, repeat it and take the imental data and the results of the other groups. Besides the difference as the estimate of this error. Contrary to statistical error, the systematic error from the a2 depend- the previous estimate, the change here affects the ence of the Sommer scale r0, and that from the chiral charm quark mass and the Sommer scale up to one extrapolation are as large as the statistical one, and sigma of the statistical error, and also slightly affects contribute substantially to the total uncertainty. the other quantities. Our prediction of the value of mMS 2 GeV is 1.118(6) The error combined by the above two estimates in c ð Þ MS quadrature will be marked with σðheavyÞ. (24) GeV. To obtain mc ðmcÞ, we applied the quark mass running in Refs. [32,33]. Note that the uncertainty of B. The charm and strange quark masses and mMSðm Þ, indicated by the black band in Fig. 10,isnot r c c Sommer scale parameter 0 just the rescaling of the error at 2 GeV with the running MS 2 MS 2 factor from 2 GeV to the one of mMSðm Þ. It means that the Our results for mc ð GeVÞ, ms ð GeVÞ and r0 are c c pffiffiffi 2 MS listed in Table VII. The χ =d:o:f: of the fully correlated fit error bar of mc ðmcÞ will be suppressed by ∼ 2 compared Δ Δ including MðDsÞ, HFS;cs¯ , MJ=ψ , HFS;cc¯ and fDs is 1.05. to the estimate from naively rescaling. We repeat the

034517-14 … CHARM AND STRANGE QUARK MASSES AND fDs PHYSICAL REVIEW D 92, 034517 (2015) 1.6 mc(2GeV) mc(mc) PDG

1.4 HPQCD ’06

PACS-CS ’08 ) (GeV) μ

( 1.2 FNL ’09 c m FNL-MILC ’12

1 this work

1 1.5 2 2.5 3 90 100 110 120 130 140 μ (GeV) Charmonium Hyperfine Splitting (MeV) μ FIG. 10 (color online). The running charm quark mass mcð Þ FIG. 11 (color online). The prediction of the hyperfine splitting μ versus the scale . Since the mass mcðmcÞ is fully correlated to of charmonium in this work, compared to these of other works that scale, the uncertainty of mcðmcÞ by the runningpffiffiffi from a given and experiment. Note the lattice results have not included 2 scale will be suppressed by approximately . the cc¯ annihilation diagram which is expected to lower HFS by 1–5 MeV [29,47,48].

MS running for the upper/lower band of mc ð2 GeVÞ, obtain the on-shell scales of them, and average the changes lattice result [46] shows that the dynamical simulation could actually get a value close to experiment. At the comparing to the one of the central value of mMS 2 GeV in c ð Þ same time, Refs. [29,47,48] show that without the dis- MS quadrature as the estimate of the error of mc ðmcÞ. The connected charm diagram of the correlation function, the MS value of mc ðmcÞ, 1.304(5)(20) GeV, is plotted in hyperfine splitting will increase by a few MeV. So the Fig. 9(b) to compare with those of ALPHA [14], correct lattice prediction of the hyperfine splitting should HPQCD [8] and ETMC [13]. Considering the fact that be slightly larger than the physical value for a dynamical HPQCD used Oð500Þ configurations per ensemble and simulation without the disconnected charm diagram. Our that only Oð50 − 100Þ per ensemble are used in this prediction of the value of the hyperfine splitting of work, the difference in precision between the results of charmonium, 119(2)(7) MeV, is plotted in Fig. 11 to HPQCD and this work reflects the different statistics to compare with experiment and other lattice results based a certain extent. on 2 þ 1 flavor configurations. Δ Note that the results of the strange/charm quark mass are Figure 12 shows the interpolated values of HFS;cc¯ , based based on the ensembles at only two lattice spacings and the on the data points of the neighboring two charm quark systematics of the finite lattice spacing effect are not fully masses which bracket the physical one, for each ensemble under control. We still need ensembles at least one more with different β, versus the renormalized sea quark mass. lattice spacing to access the full Oða4Þ errors.

0.18 C. Charmonium spectrum a=0.111 fm 0.17 a=0.083 fm Having determined the charm quark mass, we can 0.16 continuum predict the charmonium spectrum with the J=ψ mass used 0.15 )(GeV) as input. c 0.14 η 0.13 There is a long story regarding the mass of ηc in )-M( 0.12 experiment and lattice calculation. In experiment, Belle c ψ 0.11

[34] and BES [35] obtained a value smaller than 2980 MeV M( 0.1 about 10 years ago, while the BABAR result is around 0.09 – 2983 MeV [36,37]. At present, all of their results [38 40] 0.08 are consistent with each other in the range 2982–2986, and 0 10 20 30 40 50 60 70 ml (MeV) the PDG average of the ηc mass is 2983.7(7) [28]. In quenched lattice calculations, the hyperfine splitting Δ ψ η FIG. 12 (color online). The interpolated values of HFS;cc¯ on the result, namely the mass difference between J= and c, data points of the two charm quark masses which bracket is much smaller than the physical value, only around the physical one, for each ensemble with different β, versus 50–90 MeV, such as in Refs. [41–44]. Such a difference the renormalized sea quark mass. Note that the Oðm4a4Þ effect is is understood to be due to the effects of the shift of the large so that then the continuum limit based on our functional coupling constant in a quenched simulation [45]. A recent form is between the data at the two finite lattice spacings.

034517-15 YI-BO YANG et al. PHYSICAL REVIEW D 92, 034517 (2015)   4 4 X Note that the Oðm a Þ effect is large so that the continuum G ðtÞ¼ s¯ðxÞγ4γ5cðxÞc¯ð0Þγ4γ5sð0Þ limit based on our functional form is between the chiral A4A4 x~ extrapolation at the two finite lattice spacings. So the X  present result would have an additional systematic error ¯ γ ¯ 0 γ γ 0 GPA4 ðtÞ¼ sðxÞ 5cðxÞcð Þ 4 5sð Þ due to the functional form of the continuum extrapolation, x~ and then would be changed somewhat if we had ensembles X  β 2 25 ¯ γ γ ¯ 0 γ 0 at > . . GA4PðtÞ¼ sðxÞ 4 5cðxÞcð Þ 5sð Þ As mentioned in the beginning of Sec. III,wefit  x~  the hyperfine splitting instead of the mass of ηc,and X ¯ γ ¯ 0 γ 0 list it and its statistical and systematic uncertainty in GPPðtÞ¼ sðxÞ 5cðxÞcð Þ 5sð Þ ð32Þ Table VIII. x~ Since P-wave charmonium states are very noisy compared to the S-wave states, including them into to improve the precision of f . Combining the results of the global fit will make the result quite unstable. Ds these four correlation functions, we can get two kinds of Therefore we do not include them in the global fit. matrix elements h0js¯γμγ5cjD i and h0js¯γ5cjD i required in Rather, we just do the correlated fit for the data points s s Eq. (31) and Eq. (30), and then obtain f . The vector/ with different mass parameters, and use the quark mass Ds axial-vector renormalization factor required in Eq. (31) and Sommer scale r0 with their correlations as the is 1.111(6) for the β 2.13 lattice and 1.086(2) for the inputs. Table VIII also shows results for the mass of the ¼ β ¼ 2.25 lattice, as given in Ref. [15]. P-wave charmonium states which are in good agreement We found that the average of two estimate of f (254(2) with experiment. PS (4) MeV) obtained from Eq. (31) and (30) provides a Our prediction of f shown in Table VIII based on the Ds prediction consistent with those of the f obtained from global fit of the S-wave quantities will be discussed in the PS these two equations separately (253(2)(5) and 255(3)(4)), next section. χ2 while the =d:o:f of the averaged fPS is smaller (0.8 for the

D. Decay constant of Ds 0.27

For a pseudoscalar (PS) meson, its decay constant fPS is defined through the hadronic matrix element 0.26 0.25 ih0js¯γμγ5cjPSi¼f pμ; ð29Þ

PS (GeV) s

D 0.24 f with pμ the momentum of the PS meson. 0.23 f (m ,m phys) Using the Ward identity of the partially conserved axial Ds c s current (PCAC) [49], the decay constant fPS could be also 0.22 obtained by 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

mc(GeV) 0 ¯γ 2 ðmq1 þ mq2 Þh js 5cjPSi¼MPSfPS; ð30Þ 0.27 with MPS being the mass of the PS meson. 0.26 In a lattice simulation with local operators, the renormalization of the vector/axial-vector current is 0.25

not equal to unity. So the f obtained from Eq. (29) (GeV)

PS s

D 0.24 requires the axial-vector renormalization factor to get f the physical result: 0.23 f (m phys,m ) Ds c s 0 ψ¯ γ γ ψ bare ZAh j a 4 5 bjPSi ¼ MPfPS: ð31Þ 0.22 0.08 0.09 0.1 0.11 0.12 0.13 m (GeV) On the other hand, the pseudoscalar current and mass s renormalization involved in Eq. (30) are canceled FIG. 13 (color online). The charm quark mass dependence (Z Z ≡ 1). This makes the f from Eq. (30) free of PS m Ds (upper panel) and the strange quark mass dependence (lower the renormalization. panel) of fDs with the other quark mass close to the physical In this work, we construct four kinds of correlation point. This plot is based on the β ¼ 2.13 ensemble with lightest functions sea quark mass as an illustration.

034517-16 … CHARM AND STRANGE QUARK MASSES AND fDs PHYSICAL REVIEW D 92, 034517 (2015) is very close to the one obtained by HPQCD (0.4661 PDG (38) fm), and the one determined by RBC-UKQCD (0.48 FNAL+MILC ’14 (1)). With the r0 obtained here, the lattice spacing of the ALPHA ’13 β ¼ 2.13 and 2.25 ensembles are 0.112(3) and 0.084(2) fm, Light front respectively [or 1.75(4) and 2.33ð5Þ GeV−1, respectively]. Field correlators The strange/charm quark masses we obtain are QCD sum rules PQL MS 2 0 101 3 6 HPQCD ms ð GeVÞ¼ . ð Þð Þ GeV; this work MS mc ðmcÞ¼1.304ð5Þð20Þ GeV; 220 240 260 280 300 320 340 MS 2 1 118 6 24 f (MeV) mc ð GeVÞ¼ . ð Þð Þ GeV; Ds and; FIG. 14 (color online). The prediction of fD in this work, s mMS compared to those of other works. c ð2 GeVÞ¼11.1ð0.8Þ: ð34Þ MS ms averaged case vs. 1.2 for the two separated cases). The final Both the strange and charm masses are consistent with their result is listed in Table VIII. PDG averages [28] which includes many calculations from It is interesting to show the charm/strange quark mass the lattice simulation. dependence on the f in Fig. 13, in which the dependence Ds For the charmonium hyperfine splitting, our result of the charm quark mass is much stronger than that of the strange quark mass, when the other quark mass is fixed Δ ¯ ¼ 119ð2Þð7Þ MeV ð35Þ around the physical point. HFS;cc The uncertainty of our prediction of f mostly comes Ds is consistent the PDG average of 113.7(7) MeV [28]. from the statistical error (about 0.9%); both the systematic Considering the possible effect of the disconnected diagram error from the mismatch of strange sea quark mass and its (∼1 − 5 MeV) [29,47,48], our prediction could be smaller physical value, and the heavy quark artifact in the ensem- by one sigma, and thus even better in agreement. Besides β 2 13 2 bles at ¼ . are around 0.8%; that from the a the hyperfine splitting, we also checked the mass spectrum dependence of the Sommer scale r0 is around 0.6%. The 3 439 44 of the P-wave mesons, Mχc0 ¼ . ð Þ GeV, Mχc1 ¼ effects from the other systematic errors are smaller than 3.524ð66Þ GeV, and M ¼ 3.518ð71Þ GeV. The uncer- 0.3%. Note that the systematic error due to the finite lattice hc tainty of all of them are at the 2% level and the values are in spacing seems to be small based on the functional form of the continuum we used, but it is not fully under control 1.6 since the simulation is based on the ensembles at just two Exp (PDG) this work (M ,M * and M ψ as Input) lattice spacings. Ds Ds J/

The comparison with other results of fDs is illustrated 1.4 in Fig. 14. 0.095(5) 3.52541(16) IV. CONCLUSIONS 1.2 1.09(3) 0.1132(7) 3.51066(7) 0.2575(46) 3.41475(31) In this work, we used six ensembles of the 2 1 flavor þ Ratio gauge configurations with the domain wall sea quarks 1 from RBC-UKQCD Collaboration, which include two lattice spacings each with three different light sea quark 0.465(10) 0.101(7) 3.518(71) masses, to do the simulation for the spectrum of cs¯ and 0.8 1.118(25) 0.119(7) 3.524(66) 0.2536(43) 3.439(44) cc¯ . With the global fit scheme, we can determine the charm/strange quark masses and Sommer scale r0 using 0.6 input from three physical quantities, M , M − M , r (fm) m m M ψ-Mη Mχ Mχ M f Ds Ds Ds 0 c s J/ c c0 c1 hc Ds and MJ=ψ . Note that the results are based on the ensembles at only two lattice spacings and the system- FIG. 15 (color online). We list the ratio of our simulation results atics of the finite lattice spacing effect are not fully under to PDG averages. Note that the numbers of PDG averages are in control. italic type, and all the numbers are in unit of GeV, except r0. For Our prediction of the Sommer scale parameter r0, we list its value from HPQCD (0.4661(38) fm) and RBC- UKQCD (0.48(1) fm) for reference. Note that the values of the 0 465 4 9 renormalized charm and strange quark masses are those at 2 GeV r0 ¼ . ð Þð Þ fm ð33Þ in MS scheme.

034517-17 YI-BO YANG et al. PHYSICAL REVIEW D 92, 034517 (2015) agreement with experimental results 3.4148(3) GeV, ACKNOWLEDGMENTS 3.5107(1) GeV and 3.5254(2) GeV, within one sigma. We thank the RBC and UKQCD Collaborations for Another important prediction of this work is that of f . Ds making their DWF configurations available to us. This Our result work is supported in part by the National Science Foundation of China (NSFC) under Grants f 254 2 4 MeV 36 Ds ¼ ð Þð Þ ð Þ No. 11335001, No. 11075167, No. 11105153, No. 11405178 and also by the U.S. Department of is in agreement with experiment at 257.5(4.6) MeV, and Energy under Grants No. DE-FG05-84ER40154 and other lattice simulations and phenomenology calculations. No. DE-FG02-95ER-40907. Y. C. and Z. L. also acknowl- The ratio of our results for various quantities to their edge the support of NSFC under No. 11261130311 (CRC corresponding PDG averages [28] are plotted in Fig. 15,to 110 by DFG and NSFC). A. A. acknowledges the support provide a direct comparison of their consistency. of NSF CAREER through Grant No. PHY-1151648. Z. L. The calculation in this work is based on configurations at and M. G. are partially supported by the Youth Innovation two lattice spacings. We still need ensembles at least one Promotion Association of CAS. We thank the Center for more lattice spacing to access the full Oða4Þ errors, and computational Sciences of the University of Kentucky for lighter sea quark masses closer to those of the physical their support. This research used resources of the Oak ones, to confirm their systematic effects. Besides that, Ridge Leadership Computing Facility at the Oak Ridge reducing the systematic error from the strange sea quark National Laboratory, which is supported by the Office of being not at the physical point and including the discon- Science of the U.S. Department of Energy under Contract nected charm diagram, could result in better estimates. No. DE-AC05-00OR22725.

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