Probing nucleon strange and charm distributions with lattice QCD

We present the first lattice-QCD calculation of the unpolarized strange and charm parton distribution functions using large-momentum effective theory. We use a lattice ensemble with ($2+1+1$) flavors of highly improved staggered quarks generated by the MILC Collaboration, with lattice spacing $a\approx 0.12\mathrm{fm}$ and $M_{\pi }\approx 310\mathrm{MeV}$, and clover valence fermions with two valence pion masses: 310 and 690 MeV. We use momentum-smeared sources to improve the signal up to nucleon boost momentum $P_z=2.15\mathrm{GeV}$, and determine nonperturbative renormalization factors in regularization independent momentum subtraction scheme. We compare our lattice results with the matrix elements obtained from matching the parton distribution functions (PDFs) from CT18NNLO and NNPDF3.1NNLO global fits. Our data support the assumptions of strange-antistrange and charm-anticharm symmetry that are commonly used in global PDF fits.

Figure 1

Illustration of the three-point correlator involving a strange-quark long-link operator, which forms a disconnected diagram. The dashed line indicates the spatial displacement of the Wilson link with the choice of operator $\mathrm{\Gamma }$. The gray blobs show the nucleon source and sink, separated by $t_{\mathrm{sep}}$ in Euclidean time direction. Sea-quark and gluon interactions, although present in the lattice configurations, are omitted from this schematic diagram.

Figure 2

Example ratio plot for real (top) and imaginary (bottom) strange matrix elements from $n_z=4$, $z=2$, $t_{\mathrm{sep}}=6$ as function of insertion time $t$, centered by half of $t_{\mathrm{sep}}$. Different data points indicate $N_{\mathrm{LP}}\in \{1000,2000,3000,4000,8000\}$ noise sources used in the calculation (slightly shifted in $t$ to make the data points visible). The error is effectively reduced by using larger $N_{\mathrm{LP}}$.

Figure 3

Comparison of the imaginary part of the strange quasi-PDF renormalized matrix elements obtained from different fit strategies at $n_z=3$.

Figure 4

Extrapolation to physical pion mass for the imaginary part of strange (top) and charm (bottom) MEs at $n_z=4$. The extrapolated results are very close to the $M_{\pi }\approx 310\mathrm{MeV}$ results. The charm distribution is much smaller, thus noisier with larger relative errors. The band indicates the extrapolated matrix elements at the physical pion mass.

Figure 5

The real (top) and imaginary (bottom) parts of the strange quasi-PDF matrix elements in coordinate space from our calculations at physical pion mass with $P_z\in [0.43,2.15]\mathrm{GeV}$, along with those from CT18 and NNPDF NNLO in RI/MOM renormalized scale of 2.3 GeV. The CT18 analysis assumes $s(x)=\overline{s}(x)$, so their results are exactly zero after matching and Fourier transformation. Our real matrix elements at $P_z>1\mathrm{GeV}$ are consistent with zero, supporting strange-antistrange symmetry, while our imaginary ones are smaller than global-fit results.

Figure 6

The real (top) and imaginary (bottom) parts of the charm quasi-PDF matrix elements in coordinate space derived from global-fit PDFs compared with our renormalized nucleon quasi-PDF MEs at $P_z\in [0.43,2.15]\mathrm{GeV}$. The real part is consistent with zero, while the imaginary part is within the bounds of NNPDF3.1, but smaller than CT18 results.

Figure 7

The dispersion relation for the nucleon. The speed of light $c$ from the linear fit $E_0^2=c^2{P}_z^2+c^4{M}_N^2$ is slightly smaller than but consistent with 1.

Figure 8

Comparison of the imaginary parts of the renormalized nucleon strange-PDF matrix elements as functions of $z$ using different momentum-smearing parameters and symmetrization methods. The results are obtained at $M_{\pi }\approx 310\mathrm{MeV}$ and nucleon boost momentum $P_z=0.87\mathrm{GeV}$.

Figure 9

Ratio plots for the imaginary part of strange quasi-PDF at $n_z=2$, $z=4$ (left) and the real part of charm quasi-PDF at $n_z=2$, $z=2$ (right). The ratios are plotted as data points with error bars and the fitted results are plotted in colored bands. The gray band is the ground-state matrix elements obtained from the fit.

Figure 10

RI/MOM-renormalized strange (left) and charm (right) nucleon quasi-PDF matrix element as a function of $z{P}_z$ at $P_z\in [0.44,2.15]$ at $M_{\pi }\approx 310\mathrm{MeV}$ (top) and $M_{\pi }\approx 690\mathrm{MeV}$ (bottom). The real matrix elements are consistent with zero within 2 sigma for $P_z>1\mathrm{GeV}$; suggesting quark-antiquark symmetry.

Figure 11

The $x$-dependent strange (left) and charm (right) PDF from a naive truncated inverse Fourier transformation and inverse matching for $P_z=\{1.29,1.72,2.15\}\mathrm{GeV}$ at extrapolated physical pion mass.

Figure 12

Full matching results for the strange quasi-PDF in momentum space (left) and coordinate space (right) including both quark and gluon contributions. We can see that the fraction of gluon contribution (blue curve) is negative and negligible at very small $z{P}_z$, and goes up to 20% at the largest $z{P}_z$ we cover.

Figure 13

Full matching results for the charm quasi-PDF in momentum space (left) and coordinate space (right) including both quark and gluon contributions. The gluon contribution (blue curve) for charm is much more significant, comparable to the charm contribution (red curve) and drag the total contribution downwards.