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Nucleon strange quark content in 2+1-flavor QCD

We calculate the strange quark content of the nucleon directly from its disconnected three-point function in N_f=2+1 QCD. Chiral symmetry is crucial to avoid a possibly large contamination due to operator mixing, and is exactly preserved by employing the overlap quark action. We also use the all-to-all quark propagator and the low-mode averaging technique in order to accurately calculate the relevant nucleon correlator. Our preliminary result extrapolated to the physical point is f_{T_s} = 0.013(12)(16). This is in good agreement with our previous estimate in N_f=2 QCD as well as those from our indirect calculations using the Feynman-Hellmann theorem.


Introduction
The strange quark content of the nucleon N|ss|N is a fundamental quantity on the nucleon structure. It represents the effect of strange quark on the nucleon mass M N where m s is the strange quark mass. This parameter is also relevant to direct experimental searches for the dark matter, as one of its candidates, namely neutralino, may interact with the nucleon most strongly through the strange quark content. Therefore, N|ss|N is crucial to assess the sensitivity of the experiments [1]. We calculate N|ss|N from lattice QCD simulations in this study. The relevant nucleon threepoint function is purely composed of a disconnected diagram, whose computational cost is prohibitively high with the conventional simulation methods. We overcome this difficulty by using the methods of the low-mode averaging [2,3] and the all-to-all propagator [4].
Another advantage of our study is that chiral symmetry is exactly preserved by using the overlap quark action. We point out in this article that explicit symmetry breaking with conventional lattice fermions induces operator mixing leading to a possibly large contamination in N|ss|N .

Simulation setup
We simulate N f =2 + 1 QCD with the Iwasaki gauge and the overlap quark actions. The lattice spacing is determined as a = 0.112(1) fm using the Ω baryon mass as an input. We simulate two values of degenerate up and down quark masses m ud = 0.035 and 0.050 on a N 3 s ×N t = 16 3 × 48 lattice. The strange quark mass is set to m s = 0.080 and 0.100, which are close to the physical mass m s,phys = 0.081 fixed from M K . We also push our simulations to lighter masses m ud = 0.015 and 0.025 on a larger lattice 24 3 × 48 but with a single value of m s = 0.080. The four values of m ud cover a range of M π ∼ 290 − 520 MeV with a condition M π L 4 satisfied. We have accumulated about 50 independent gauge configurations at each combination of m ud and m s .
We calculate the nucleon three-point function where N = ε abc (u T a Cγ 5 d b )u c is the nucleon interpolating operator, andss is the strange scalar operator on the lattice. We denote the temporal separation between the nucleon source and sink (scalar operator) by ∆t (∆t s ). This correlator is calculated with four choices of the source location, namely t src = 0, 12, 24 and 36 with y = 0, and two choices of the projection operator Γ = Γ ± = (1 ± γ 4 )/2, which correspond to the forward and backward propagating nucleons. We then take the average C 3pt (∆t, ∆t s ) = 1 to reduce its statistical fluctuation. We also calculate the nucleon two-point function C 2pt (∆t) in a similar way.
We calculate the disconnected strange quark loop fromss and its vacuum expectation value ss using the all-to-all propagator. It is expected that low-lying modes of the Dirac operator D contribute dominantly to low-energy dynamics of QCD, and we calculate this contribution exactly where the number of eigenmodes N e is 160 (240) on our smaller (larger) lattice. The contribution of the remaining high-modes is estimated stochastically. We generate a single complex Z 2 noise vector η(x) for each configuration, and divide it into , which have non-zero elements only for a single combination of color and spinor indices on two consecutive time-slices. The high-mode contribution is then estimated as We note that, in this report, C 3pt on 24 3 × 48 is calculated without the high-mode contribution in the quark loop, our measurement of which is currently underway. In our study in N f = 2 QCD [5,6], we observed that the result for N|ss|N is well dominated by the low-modes and does not change significantly by ignoring the high-mode contribution. We can improve the statistical accuracy of the nucleon correlators by using LMA. Let us divide C 2pt into two contributions C 2pt,low and C 2pt,high : C 2pt,low is the two-point function constructed only from the low-mode part of the quark propagator (D −1 ) low , and C 2pt,high is the remaining contribution. We can average over all possible source points (y,t src ) for C 2pt,low . The piece representing the nucleon propagation in C 3pt can be calculated in a similar way. In this study, we average over all t for t src but 16 spatial locations for y at each t src .
In our previous study in N f = 2 QCD, we observed that the smearing of the nucleon source and sink operators N is crucial to obtain a clear signal of C 3pt at reasonably small ∆t. We therefore construct N from the quark field with the Gaussian smearing where the parameters are set to ω = 20 and N = 400. This smeared operator is not gauge invariant but we fix the gauge to the Coulomb gauge. We use q smr for both C 3pt and C 2pt .

Strange quark content at simulated quark masses
We extract the strange quark content N|ss|N on the lattice from the ratio of C 3pt and C 2pt In order to identify a plateau of R(∆t, ∆t s ), it is helpful to consider the same ratio but approximated by taking only the low-mode contribution in the scalar loop. Figure 1 shows an example of the approximated ratio as a function of ∆t s . The significant change near ∆t s ∼ 0 and ∆t is due to a contamination from excited states. We therefore fit R(∆t, ∆t s ) to a constant form in ∆t s =[5, ∆t − 5], which is well separated from the nucleon source and sink. Note that this fit and the following analysis on 16 3 × 48 are carried out using R(∆t, ∆t s ) without the approximation. While we use the approximated ratio on 24 3 × 48 as mentioned in the previous section, we confirm in the analysis on 16 3 × 48 that the high-mode contribution to the quark loop has only small effect to N|ss|N .
The result of the constant fit in terms of ∆t s is denoted as R(∆t) and is plotted as a function of ∆t in Fig. 2. The stability of R(∆t) at ∆t ≥ 12 suggests that these data are well dominated by the ground state contribution. We extract N|ss|N by a constant fit to R(∆t) in this region.

Chiral extrapolation to the physical point
In Fig. 3 We note that this is in good agreement with our previous estimate in N f = 2 QCD [5,6] as well as our indirect determinations through the Feynman-Hellmann theorem in N f = 2 [8] and N f = 2 + 1 QCD [9].

Renormalization ofss
In (4.2), f T s is expressed with the bare quantities assuming that the operator m ss s is renormalization invariant as in the continuum limit. This is, however, not the case if chiral symmetry is explicitly broken by the lattice fermions of the choice. To illustrate this, let us consider the renormalization of the scalar operatorss in the flavor SU (3) symmetric limit for simplicity. Using the flavor triplet quark field ψ,ss can be written as a linear combination of the flavor singlet and octet scalar operators where λ 8 is a Gell-Mann matrix, and we put the superscript "phys" to the renormalized operators to distinguish them from bare lattice operators. The singlet and octet operators are renormalized as in renormalization schemes which respect chiral symmetry. The operator (ss) phys is thus expressed as (ss) phys = 1 3 This implies thatss can mix with the up and down quark operators unless Z 0 = Z 8 . The difference Z 0 −Z 8 comes from disconnected diagrams, such as that in Fig. 4, contributing only to Z 0 . These diagrams are in fact forbidden by chiral symmetry: the quark loop with the scalar operatorss =s R s L +s L s R has to flip the chirality, while such a flip does not occur with the quark-quark-gluon vertices in the diagrams. This implies Z 0 = Z 8 , which also holds at finite quark masses in mass independent renormalization schemes. The renormalization ofss thus reduces to a simple multiplicative renormalization (ss) phys = Z Ss s, Z S = Z 8 = Z 0 , (5.5) and the operator m ss s is renormalization invariant provided that chiral symmetry is exactly preserved as in our study. When chiral symmetry is explicitly broken by a lattice fermion formulation, (5.5) is modified as The second term represents the mixing with the light quark contents due to Z 0 = Z 8 . This may lead to a large contamination in (ss) phys , because the light quark contents induce a connected diagram, the magnitude of which is much larger than that of the disconnected one. Previous direct calculations with the Wilson-type fermions [10,11,12] in fact obtained rather large values f T s ∼ 0.3 -0.5 compared to our result (4.2) probably due to this contamination. The explicit symmetry breaking also induces the mixing of the flavor-singlet operatorψψ with lower dimensional operators, such as the third term in (5.6), through the renormalization (5.2). This contribution must be subtracted as a part of the vacuum expectation value ofss. Due to the cubic divergence, this results in a large cancellation, with which the calculation is potentially noisier.

Conclusion
We calculate the nucleon strange quark content in N f =2 + 1 QCD. Chiral symmetry is exactly preserved by employing the overlap quarks in order to avoid large contaminations from the operator mixing. We also use the LMA and the all-to-all propagator to accurately calculate the relevant nucleon correlators. Our result of f T s is in good agreement with our previous study in N f = 2 QCD as well as our indirect calculations for N f = 2 and 3 using the Feynman-Hellmann theorem: all our studies consistently favor small strange quark content f T s ≈ 0.02.