Determination of Light and Strange Quark Masses from Full Lattice QCD

We compute the light and strange quark masses m_l = (m_u+m_d)/2 and m_s, respectively, in full lattice QCD with N_f=2 flavors of light dynamical quarks. The renormalization constants, which convert bare quark masses into renormalized quark masses, are computed nonperturbatively, including the effect of quark-line disconnected diagrams. We obtain m_l=4.7(2)(3) MeV and m_s=119(5)(8) MeV in the MSbar scheme at the scale 2 GeV.

-QCDSF-UKQCD Collaboration -We compute the light and strange quark masses m ℓ = (mu + m d )/2 and ms, respectively, in full lattice QCD with N f = 2 flavors of light dynamical quarks. The renormalization constants, which convert bare quark masses into renormalized quark masses, are computed nonperturbatively, including the effect of quark-line disconnected diagrams. We obtain m M S ℓ (2 GeV) = 4.7 (2) (3) MeV and m M S s (2 GeV) = 119 (5)(8) MeV. PACS numbers: 12.15.Ff,12.38.Gc,14.65.Bt The light and strange quark masses are among the least well known parameters of the Standard Model. The reason is that quarks are confined, so that the masses must be determined indirectly through their influence on hadronic observables. This requires nonperturbative techniques. One such technique is lattice QCD.
The quark masses obtained directly in lattice calculations are bare quark masses at the cut-off scale a −1 , where a denotes the lattice spacing. For the lattice numbers to be useful for phenomenology, it is necessary to convert the bare quark masses to renormalized masses in some standard renormalization scheme. Because lattice perturbation theory converges badly, and the expansion coefficients are generally known to one loop order only, this ought to be done nonperturbatively. In full QCD a one-loop perturbative renormalization of the mass operator is totally inadequate even, as it does not account for the disconnected (flavor singlet) contribution shown in Fig. 1, which turns out to be comparable with the connected contribution at present lattice spacings.
In quenched QCD, in which the effect of sea quarks is neglected (and hence quark-line disconnected fermion loops are absent), several groups [1] have carried out an entirely nonperturbative calculation of the light and strange quark masses. Remarkably consistent results have been found. Previous calculations in full QCD, both with N f = 2 [2] and N f = 3 [3] flavors of sea quarks, employ perturbative renormalization to compute the relation between the bare and renormalized quark masses. These authors, except perhaps Eicker et al., find rather small values for the strange quark mass, which lie substantially below the central value quoted by the Particle Data Group [4].   In this Letter we shall present a first fully nonperturbative calculation of the light and strange quark masses in full QCD, including the effect of flavor singlet renormalization factors. We consider nonperturbatively O(a) improved Wilson fermions with N f = 2 flavors of degenerate dynamical quarks and the Wilson gauge field action. The calculation is done in two steps. We simulate dynamical gauge field configurations at four different values of the coupling, β, and at three different sea quark masses each. The latter are specified by the hopping parameter κ sea . The actual parameters, as well as the corresponding lattice spacings and pseudoscalar mass values, are shown in Fig. 2. We use the force parameter r 0 to set the scale. On these configurations we then perform a partially quenched calculation of the pseudoscalar mass, allowing for different sea and valence quark masses, from which we derive the physical quark masses. In Table 1 we list the hopping parameters of valence (κ val ) and sea quarks considered in this calculation.
The bare sea and valence quark masses are given by am sea = 1/(2κ sea )−1/(2κ c sea ) and am val = 1/(2κ val )−1/(2κ c sea ), respectively. We consider the case of degenerate valence quarks only. In Fig. 3 we show the partially quenched pseudoscalar mass m P S (κ sea , κ val ). The critical hopping parameter κ c sea is found by keeping β fixed and varying κ sea until m P S (κ sea , κ sea ) = 0. Similarly, we introduce a critical hopping parameter of the valence quarks, κ c val , which is found by varying κ val until m P S (κ sea , κ val ) = 0, while keeping β, κ sea fixed. Our calculation requires a precise determination of κ c sea and κ c val . We perform a global fit of the form to all 100 data points, where the parameters are taken to be second order polynomials in β. The fit gave χ 2 /dof = 0.67. The resulting values of κ c sea and κ c val are given in Table 1. The force parameter r 0 /a was computed from the static potential.
Taking the derivatives of the quark propagator with respect to am sea and am val , we obtain for the renormalized sea and valence quark masses [5] where Z S m and Z N S m are singlet and nonsinglet renormalization constants of the mass operator. Partially quenched chiral perturbation theory to NLO predicts [6] From (4) follows that m R val vanishes where the partially quenched pseudoscalar mass vanishes, which happens at the value κ val = κ c val . If we insert this value into (3) we obtain the ratio In Fig. 4 we plot Z S m /Z N S m for all data sets. The effect of the quark-line disconnected diagram is found to be significant. The numbers depend only mildly on the sea quark mass.
It remains to determine Z N S m . We compute Z N S S = (Z N S m ) −1 nonperturbatively [7] in the RI-MOM scheme. The result is converted to the more popular M S and RGI schemes by a three-loop perturbative calculation [8]. In Fig. 5 we show the nonsinglet Z RGI S as a function of the renormalization scale µ. We find that the nonperturbative scale dependence of Z RI−MOM S is matched by the three-loop conversion factor for (r 0 µ) 2 20. We obtain Z RGI S from a fit to the plateau as indicated by the solid line. The result varies by a few percent only over our range of sea quark masses at any given β value. In the M S scheme at µ = 2 GeV we have [8] Z MS S (2 GeV) = 1.461 Z RGI S . At our smallest lattice spacing, a ≈ 0.07 fm, Z MS S ≈ 0.6, which is certainly beyond the range of one-loop perturbation theory, tadpole-improved or not.
Having unscrambled renormalized valence and sea quark masses, we are now able to fit our data by the partially quenched chiral formula (4) and determine the physical quark masses from it. In the process we replace all masses m by dimensionless quantities mr 0 . It turns out that the data are not sensitive to logarithmic variations in the renormalized valence quark mass (parameters C and E). We therefore have chosen to fit our partially quenched pseudoscalar masses by the formula In Fig. 6 we plot our data for our largest β value. The slope of the data depends only rather weakly on the renormalized sea quark mass. Perhaps most of the effect is washed out by having used r 0 to set the scale. The solid curve shows the result of the fit in the limit of vanishing sea quark mass. We find good scaling properties. The fit parameter A varies by less than 5% over our range of β values.
To fix the scale r 0 in physical units, we extrapolate recent dimensionless nucleon masses, m N r 0 , found by the CP-PACS, JLQCD and QCDSF-UKQCD collaborations jointly to the physical pion mass, following [9]. This gives the value r 0 = 0.467 fm, which we will use here. A similar result was quoted in [10]. The average mass of the up and down quarks, m ℓ = (m u +m d )/2, is found from extrapolating m P S to the physical π 0 mass, setting m R val = m R sea in (6). We obtain m MS ℓ (2 GeV) r 0 = 0.00981(19), 0.00987(17), 0.00986(18) and 0.01044(19) at β = 5.20, 5.25, 5.29 and 5.40, respectively. Similarly, the strange quark mass, m s , is obtained from the lattice value of m R val that brings m P S to the physical K 0 mass, while m R sea is kept fixed at the corresponding physical sea quark mass m R ℓ . Owing to the fact that the valence quarks are degenerate, we then have [6] This finally gives m MS s (2 GeV) r 0 = 0.2525(50), 0.2544(45), 0.2545(47) and 0.2671(48) at β = 5.20, 5.25, 5.29 and 5.40, respectively. To be able to extrapolate our results to the continuum limit, we need to know a/r 0 in the chiral limit. In Fig. 7 we show our data for r 0 /a together with a fit (fitting function: exponential of a polynomial in β, m sea ), and in Fig. 8 we show the light and strange quark masses as a function of the chirally extrapolated lattice spacing. Because our fermionic action is nonperturbatively O(a) improved, we expect the error due to the finite cut-off to be at most of O(a 2 ). A linear extrapolation in (a/r 0 ) 2 to the continuum limit is therefore appropriate. We estimate the systematic error on r 0 to be of the order of 7%. We then obtain m MS s (2 GeV) = 119 (5) where the first error is statistical and the second systematic. In particular, m s /m ℓ = 26 (1), in good agreement with leading order chiral perturbation theory.
To summarize, we have performed a lattice calculation of the light and strange quark masses in full QCD with N f = 2 flavors of light dynamical quarks. Our calculation differs from previous calculations in several respects. We use nonperturbatively O(a) improved Wilson fermions and perform simulations at four different couplings with 0.07 ≤ a ≤ 0.12 fm, which allows an extrapolation to the continuum limit. Furthermore, an entirely nonperturbative scheme of mass renormalization, both for sea and valence quark masses, is devised, including the effect of quark-line disconnected contributions. The identification of renormalized sea and valence quark masses greatly facilitates the extrapolation to the chiral limit. We believe that our method will be useful in other applications of partially quenched chiral perturbation theory as well.