Calculation of nucleon strange quark content with dynamical overlap quarks
We calculate the nucleon strange quark content directly from disconnected three-point functions. Numerical simulations are carried out in two-flavor QCD using the overlap quark action with up and down quark masses down to a fifth of the physical strange quark mass. To improve the statistical accuracy, we calculate the nucleon two-point functions with the low-mode averaging technique, whereas the all-to-all quark propagator is used for the disconnected quark loop. We obtain the parameter, which is the ratio of the strange and light quark contents, at the physical point. This is in a good agreement with our earlier calculation from the nucleon spectrum through the Feynman-Hellmann theorem.
Calculation of nucleon strange quark content with dynamical overlap quarks
JLQCD collaboration: K. Takeda††thanks: Speaker. , S. Aoki, S. Hashimoto, T. Kaneko, T. Onogi, N. Yamada
Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki, 305-8571, Japan
Riken BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA
High Energy Accelerator Research Organization (KEK), Ibaraki 305-0801, Japan
School of High Energy Accelerator Science, The Graduate University for Advanced Studies
(Sokendai), Ibaraki 305-0801, Japan
Department of Physics, Osaka University Toyonaka, Osaka 560-0043, Japan
The nucleon strange quark content is an important parameter to determine the cross section of the scattering of dark matter candidates from the nucleon [1, 2]. It can not be measured directly by experiments, and only lattice QCD can provide a model-independent and nonperturbative determination. A precise lattice calculation is, however, very challenging, because only disconnected diagrams contribute to the strange quark content and they are computationally very expensive to calculate with the conventional method. In addition, the scalar operator has a vacuum expectation value (VEV), which diverges towards the continuum limit. We need to subtract the VEV contribution and this induces a substantial uncertainty in the strange quark content.
In our previous study , we avoid the above mentioned difficulties by calculating the strange quark content from the quark mass dependence of the nucleon mass through the Feynman-Hellmann theorem
We refer to this method as the spectrum method in the following. This method is, however, not applicable to other interesting matrix elements, such as the strange quark spin fraction of the nucleon. In this article, therefore, we attempt a direct determination of strange quark content from nucleon matrix element including a disconnected diagram. To this end, we employ the overlap quark action, which has exact chiral symmetry, and improved measurement methods, such as the low-mode averaging (LMA) technique [4, 5] and the use of the all-to-all quark propagator .
2 Simulation details
Gauge ensembles of two-flavor QCD are generated on a lattice using the Iwasaki gauge action and the overlap quark action. We set the gauge coupling at which the lattice spacing determined from the Sommer scale fm is fm. Our simulation is accelerated by introducing a topology fixing term into our lattice action , and we simulate only the trivial topological sector in this study. We take four values of bare up and down quark masses and 0.050, which cover a range of the pion mass MeV. Statistics are 100 independent configurations at each quark mass. We refer readers to  for further details on our configuration generation. In our measurement, we take two values of the valence strange quark mass and 0.100, which are close to the physical mass determined from our analysis of the meson spectrum .
The strange quark content can be extracted from nucleon two- and three-point functions
where is the strange scalar operator, represents the temporal coordinate of the nucleon source operator, and () is the temporal separation between the nucleon source and sink (quark loop). We calculate and with two choices of the projector corresponding to the forward and backward propagation of the nucleon. We then take the average over the two choices of with appropriately chosen temporal separations and . The averaged correlators, which are denoted by and in the following, show reduced statistical fluctuation.
For further improvement of the statistical accuracy, we employ the low-mode averaging(LMA) technique [4, 5] to calculate and . In this method, the quark propagator is expanded in terms of the eigenmodes of the Dirac operator . We calculate the contribution of 100 low-modes exactly as
The remaining contribution from the higher modes is taken from by that of the conventional point-to-all propagator. With this decomposition of the quark propagator, is divided into eight contributions
It is expected that dominates at large temporal separation . The statistical accuracy of can be remarkably improved by averaging over the location of the nucleon source operator. We also improve the statistical accuracy of other contributions (, …, ) by using point-to-all propagators averaged over 4 or 8 different source locations. The nucleon piece of the disconnected correlator is calculated in the same way.
Since the nucleon correlators and damp rapidly as increases, it is essential to reduce the contamination from excited states at small . In this study, we employ the Gaussian smearing
for both of the source and sink operators. The parameters and are chosen so that the effective mass of shows a good plateau. For comparison, we repeat our measurement with the local sink operator. In this additional measurement, we test the local and an exponential source operator . The parameter is chosen so that the distribution of the smeared quark is close to that of the Gaussian smearing (2).
To calculate the disconnected quark loop in , we construct the all-to-all quark propagator as proposed in . The low-mode contribution is the same as in (2) and the contribution from the high-modes is estimated by employing the noise method with the dilution technique . We prepare a single noise vector for each configuration and it is split into vectors () which have non-zero elements only for single color and spinor indices and two consecutive time-slices. The high-mode contribution is then given by
where is the solution of the linear equation
and is the projection operator to the subspace spanned by the low-modes.
We also tested the all-to-all quark propagator to calculate the high-mode contributions in (2). It turned out, however, that these contributions have large statistical error due to the insufficient number of the noise samples. We therefore use and calculated with the LMA in the following analysis.
3 Matrix element at simulated quark masses
We extract the unrenormalized matrix element from the ratio
Figure 1 shows dependence of at our heaviest quark mass with a fixed . We observe a clear plateau between the nucleon source and sink with the Gaussian smeared operator. On the other hand, the plateau is unclear if the local operator is used for the source and/or sink. It is therefore crucial for a reliable determination of from to reduce contamination from the excited states by appropriately smearing the nucleon operators.
The situation is similar at two smaller quark masses and 0.025. As shown in Fig. 2, however, we do not observe a clear signal even with the smeared source and sink at our smallest quark mass . To observe a clear plateau of at such small , we may need more statistics as well as a larger lattice to suppress finite volume corrections, which are possibly sizable at at . We leave such a calculation for future studies, and omit data at this in the following analysis.
In this report, is determined by the following simple two-step fits. First, we carry out a constant fit to in terms of . The fit result, which we denote by , is plotted as a function of in Fig. 3. We then extract by a constant fit to at . As seen in Fig. 3, do not show significant dependence with this range of . We therefore expect that extraction of ground state signal is well under control.
4 Strange quark content at physical point
As seen in Fig. 3, the fit result for does not have significant dependence at each . This leads us to interpolate to the physical strange quark mass using a linear form in terms of . Fit results as well as fitted data at are plotted as a function of in Fig. 4.
At next-to-leading order of heavy baryon chiral perturbation theory (HBChPT)  the nucleon mass can be written as , where are functions of the low-energy constants (LECs) in HBChPT. We note that the contributions from decuplet baryons are neglected. The Feynman-Hellmann theorem (1) then implies that the dependence of comes from the terms from and loops. By using the leading order relation and , we obtain
where and depend on the LECs and .
We extrapolate at by the linear form (4) with treated as fitting parameters. This chiral extrapolation is plotted in Fig. 4. We obtain at the physical point, where the error is statistical only. This is converted to the phenomenologically relevant parameters
As shown in Fig. 5, we observe a good agreement with our estimate from the spectrum method . The same figure also shows that previous studies with the Wilson-type fermions [12, 13, 14] led to rather large values for the strange quark content . It is argued in [3, 15] that the explicit chiral symmetry breaking induces a mixing between the scalar operators of sea and valence quarks and leads to a substantial uncertainty in the strange quark content.
In this article, we report on our calculation of the strange quark content directly from the nucleon matrix element. We determine and with an accuracy of . The key points leading to this accuracy are the use of the improved measurement techniques, namely the LMA and the all-to-all quark propagator, as well as the appropriately smeared operator both for nucleon source and sink. It is an interesting subject in the future to extend this study to other matrix elements containing disconnected diagram such as the quark spin fraction of the nucleon.
We observe a good agreement with our previous estimate from the spectrum method. Chiral symmetry preserved by the overlap action plays a crucial role in avoiding the unwanted operator mixing for the Wilson-type actions. For more precise determination, we need to extend our calculation to QCD and larger volumes. Our preliminary estimate with the spectrum method is reported at this conference . A direct determination from nucleon disconnected functions in QCD is also in progress.
Numerical simulations are performed on Hitachi SR11000 and IBM System Blue Gene Solution at High Energy Accelerator Research Organization (KEK) under a support of its Large Scale Simulation Program (No. 09-05). This work is supported in part by the Grant-in-Aid of the Ministry of Education (No. 19540286, 20105001, 20105002, 20105003, 20340047, 21674002 and 21684013).
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