Intrinsic strangeness and charm of the nucleon using improved staggered fermions

We calculate the intrinsic strangeness of the nucleon, $⟨N|\overline{s}s|N⟩-⟨0|\overline{s}s|0⟩$, using the MILC library of improved staggered gauge configurations using the Asqtad and HISQ actions. Additionally, we present a preliminary calculation of the intrinsic charm of the nucleon using the HISQ action with dynamical charm. The calculation is done with a method which incorporates features of both commonly used methods, the direct evaluation of the three-point function and the application of the Feynman-Hellman theorem. We present an improvement on this method that further reduces the statistical error, and check the result from this hybrid method against the other two methods and find that they are consistent. The values for $⟨N|\overline{s}s|N⟩$ and $⟨N|\overline{c}c|N⟩$ found here, together with perturbative results for heavy quarks, show that dark matter scattering through Higgs-like exchange receives roughly equal contributions from all heavy quark flavors.

Figure 1

Feynman diagram of an incoming neutralino interacting with a sea strange quark loop in the proton, mediated by a Higgs boson. The overall interaction amplitude depends on the intrinsic strangeness of the nucleon which must be computed on the lattice.

Figure 2

A history of calculations of the nucleon strangeness. The “natural scale”, given by the perturbative QCD calculation [23] as discussed in 2a, is shown as a vertical blue dashed line [3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26].

Figure 3

A schematic illustration of the strangeness of the nucleon: the presence of the valence quarks creates a “bubble” in the vacuum chiral condensate.

Figure 4

A schematic depiction of the perturbative approach to $\frac{\partial M_N}{\partial m_c}$ taken by Shifman and Kryjevski at leading order, inspired by a similar presentation by Kryjevski in Ref. [23]. Changing the mass of a heavy quark changes the scale at which it freezes out and thus affects the running of the coupling constant, affecting all low-mass scales equally.

Figure 5

The correlation between nucleon propagators of length $5a$, $10a$, and $15a$ with the strange quark condensate as a function of the distance from the propagator source on one of the MILC $a=0.12\mathrm{fm}$ ensembles. The source and sink location of the propagators are marked as green vertical lines.

Figure 6

Results for $⟨N|\overline{q}q|N⟩$ using the improved hybrid method averaged over the $a\approx 0.12\mathrm{fm}$ ensembles on which the needed measurements are available in the lattice regularization. Each graph shows the values of $⟨N|\overline{s}s|N⟩$ (red circles) and $⟨N|\overline{u}u|N⟩$ (blue crosses) at varying pad sizes. Negative pad sizes, with significant systematic bias, are included to make the trend of this bias plain. The result using the unimproved method (i.e. the limit of large pad size) on the newly-computed condensate and propagator measurements is shown as a starburst; the result using the unimproved method on the old data is shown as a fancy square for comparison. Note that the new data use different random time sources for the nucleon correlator. The uncertainty in the difference between adjacent values is shown as a horizontal carat centered on the left point; if the right point lies within the bracketed range then the difference between the two is less than one standard deviation. A horizontal band corresponding to the width of the systematic error estimate from the use of the improved method is shown.

Figure 7

$⟨N|\overline{u}u|N⟩$ (blue crosses) and $⟨N|\overline{s}s|N⟩$ (red circles) vs. $T_{\min }$ using the (improved) hybrid method. The three panels show the average values on the 0.12 fm, 0.09 fm, and 0.06 fm ensembles, respectively; note the difference in the y-axis scale between the plots. The uncertainty in the difference between adjacent values (obtained by jackknife) is shown as a horizontal carat centered on the left point. The size of the 5% systematic error estimate from excited state pollution is shown with a green bar.

Figure 8

$⟨N|\overline{s}s|N⟩$ on the Asqtad ensembles using data from the improved hybrid method where it is available, in the $\overline{\mathrm{MS}}$ (2 GeV) regularization scheme, adjusted to the correct value of $m_s$, along with the linear chiral and continuum fit. Data from the 0.12, 0.09, and 0.06 fm ensembles are shown by red octagons, green diamonds, and blue squares, respectively. The black line is the chiral fit in the continuum limit; the chiral fit at $a\approx 0.12\mathrm{fm}$ is shown by the red dashed line. Symbol area is proportional to the number of lattices in the gauge ensemble. The black point marked by a cross is the evaluation at the physical point, along with the combined statistical and continuum-extrapolation error.

Figure 9

$⟨N|\overline{s}s|N⟩$ on the Asqtad ensembles with the highest statistics, those with at least 1600 configurations, along with the ensemble with $m_l=m_s$ which has a large impact on the chiral extrapolation. Symbols are the same as Fig. 8.

Figure 10

The intrinsic strangeness of the nucleon on the MILC Asqtad ensembles using the direct method with a propagator length of $10a$, as a function of $\overline{s}s$ insertion time $T_1$. Vertical dashed lines represent the sink for the propagator, while the horizontal lines show the average using the improved hybrid method with the “conservative” padding choice.

Figure 11

The intrinsic strangeness of the nucleon on the MILC Asqtad ensembles using the direct method for multiple propagator lengths, as a function of $\overline{s}s$ insertion time $T_1$. Vertical dashed lines represent the source and sink for the various propagators, with the symbol used to indicate the result using that propagator length indicated above the sink. The horizontal lines show the average using the improved hybrid method on these ensembles with the “conservative” padding choice.

Figure 12

Results for the nucleon strangeness $⟨N|\overline{s}s|N⟩$ from the direct method. The results are averaged over four adjacent condensate measurements at locations centered on the value of $t_1$ shown, and averaged over two adjacent propagator lengths. The intrinsic strangeness of the nucleon on the MILC Asqtad ensembles using the direct method for multiple propagator lengths, as a function of $\overline{s}s$ insertion time $T_1$. The results are averaged over four adjacent condensate measurements at locations centered on the value of $t_1$ shown, and averaged over two adjacent propagator lengths. Dotted vertical lines indicate the source and sink locations, with the sink position shown equidistant between the sinks of the two adjacent propagators that are averaged. Symbols above these dotted lines indicate the plot symbols corresponding to that propagator length. The horizontal lines show the average using the improved hybrid method on these ensembles with the “conservative” padding choice.

Figure 13

Results for $M_N$ on the two ensembles with $\beta =7.10$, $a{m}_l=0.0093$, $a{m}_s=0.031$, $a{m}_{\mathrm{val}}=0.0062$ (black octagons) and $\beta =7.10$, $a{m}_l=0.0062$, $a{m}_s=0.0186$ (red squares), at varying values of $T_{\min }$, using a two-state fit method. Symbol size indicates fit quality. Absent data indicate fits that failed to converge or converged to nonsensical values.

Figure 14

The intrinsic strangeness of the nucleon on the MILC HISQ ensembles, using the hybrid method. The left pane shows a constant fit; the right pane shows a linear chiral fit, with the slope constrained using a Bayesian prior from the Asqtad fit. Ensembles with $a\approx (0.15,0.12,0.09,0.06)\mathrm{fm}$ are shown as violet fancy squares, red octagons, green diamonds, and blue squares, respectively. Symbol area is proportional to the number of gauge configurations in the ensemble. The fit at the continuum is shown as a black dotted line, while the fit evaluated at $a=0.12\mathrm{fm}$ is shown as a red dashed line. The fit evaluated at the continuum is shown as a black cross, and the quoted error includes the uncertainty in the continuum extrapolation with the prior as discussed in the text.

Figure 15

The intrinsic charm content of the nucleon on the MILC HISQ ensembles using the hybrid method. The left pane shows the result using the larger minimum propagator distances $t_{\min }$ used for the strangeness calculation; the right pane shows the result using the smaller minimum propagator distances discussed in the text. Ensembles with $a\approx (0.15,0.12,0.09,0.06)\mathrm{fm}$ are shown as violet fancy squares, red octagons, green diamonds, and blue squares, respectively. Symbol area is proportional to the number of gauge configurations in the ensemble. The fit at the continuum is shown as a black dotted line, while the fit evaluated at $a=0.12\mathrm{fm}$ is shown as a red dashed line. The fit evaluated at the continuum is shown as a black cross, and the quoted error includes the uncertainty in the continuum extrapolation with the prior as discussed in the text.