Moments of parton distributions functions of the pion from lattice QCD using gradient flow

Abstract

We present a nonperturbative determination of the pion valence parton distribution function (PDF) moment ratios $\left\langle x^{n-1} \right\rangle / \left\langle x \right\rangle$ up to $n=6$, using the gradient flow in lattice QCD. As a testing ground, we employ SU($3$) isosymmetric gauge configurations generated by the OpenLat initiative with a pseudoscalar mass of $m_π\simeq 411~\text{MeV}$. Our analysis uses four lattice spacings and a nonperturbatively improved action, enabling full control over the continuum extrapolation, and the limit of vanishing flow time, $t\to0$. The flowed ratios exhibit O($a^2$) scaling across the ensembles, and the continuum-extrapolated results, matched to the $\overline {\text{MS}}$ scheme at $μ= 2$ GeV using next-to-next-to-leading order matching coefficients, show only mild residual flow-time dependence. The resulting ratios, computed with a relatively small number of configurations, are consistent with phenomenological expectations for the pion's valence distribution, with statistical uncertainties that are competitive with modern global fits. These findings demonstrate that the gradient flow provides an efficient and systematically improvable method to access partonic quantities from first principles. Future extensions of this work will target lighter pion masses toward the physical point, and applications to nucleon structure such as the proton PDFs and the gluon and sea-quark distributions.

Figures (12)

• Figure 1: Schematic representation of the optimized workflow for the contractions necessary to compute the three-point functions. This is repeated at each $\tau_s$ and $t$, while the arguments of the propagators and other various objects defined in the text are left implicit. We start with the forward propagator $S$ and act on it with the forward covariant derivative $\overrightarrow{D}_{\mu_n}$. For temporal $\mu_n$, we separately save $\tilde{S}_{\mu_n} = \{\mathcal{P}^1_{\mu_n} S, \mathcal{P}^0_{\mu_n} S\}$, while for spatial $\mu_n$, we save the linear combination $\tilde{S}_{\mu_n} = [\mathcal{P}^1_{\mu_n}+\mathcal{P}^0_{\mu_n}]S$. This object is then contracted with the gamma matrices and sequential propagator to construct $\mathcal{C}_{\mu_1\mu_2}$ for $n=2$. Two contractions are necessary for temporal $\mu_2$, and one for spatial. We then proceed to use these intermediate correlators to construct the full three-point function $C^{uu}_{\mu_1\mu_2}$ as defined in Eq. \ref{['eq:Cefficient']}. Starting with $\tilde{S}_{\mu_n}$, we then repeat the procedure with the next layer of covariant derivative, $\overrightarrow{D}_{\mu_{n-1}}$ and so forth, until reaching the maximum $n$ desired, with the $\mu_i$ iterations of the intermediate $\mu_1\mu_2$, $\mu_1\mu_2\mu_3$, etc. contractions ordered depth first.
• Figure 2: Examples of three-point function ratios $C_n/C_2$ for all $n$ and $\tau_s$ on ensemble a077, shown at five different flow times. For each case, we plot the corresponding effective estimator for the ground-state matrix element ratio, as described in Method 1. We compare three averaging intervals, $\delta \tau_{{\mathcal{O}}}=a$ (red band), $2a$ (green band), and $3a$ (lilac band).
• Figure 3: Examples of three-point function ratios for the a064 ensemble, shown as in Fig. 2. The green band corresponds to the effective estimator (Method 1) with $\delta\tau_{{\mathcal{O}}}=2a$. The brown band represents the AIC-weighted average of plateau fits over several regions, as described in Method 2.
• Figure 4: Examples of excited-state fit results (Method 3) for ensemble a077 and five different flow times. For each flow time, the fit includes three states and is performed simultaneously to the corresponding two-point function together with all three-point functions $C_n(t; \tau_s,\tau_{{\mathcal{O}}})$ for all $n$ and both sink times, $\tau_s$. From these fits we obtain the ground-state matrix elements for all $n$, and divide those with $n>2$ by the $n=2$ matrix element. The results are shown here as the pink band, plotted against the corresponding three-point function ratios ($\tau_s = 40a$ with blue points and $\tau_s = 42a$ with orange points). For comparison, the effective estimator (Method 1) with $\delta\tau_{{\mathcal{O}}}=2a$ is also included (green band).
• Figure 5: Investigation of the continuum extrapolation for all flow times of $n=3$ (top panel) and $n=4$ (bottom panel). Four different fits are considered, as described in the text, and each is shown only when its reduced $\chi^2_\nu < 1.5$. The shaded bands indicate the corresponding fit type, while the black data points mark the continuum-limit results obtained with the fit function adopted for the final analysis. These are shown for those values of $t/t_0$ where the continuum extrapolation is considered robust, according to the criteria discussed in the text.