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8-1-2015
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.
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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