Valence parton distribution of the pion from lattice QCD: Approaching the continuum limit

We present a high-statistics lattice QCD determination of the valence parton distribution function (PDF) of the pion, with a mass of 300 MeV, using two very fine lattice spacings of $\mathrm{a}=0.06\mathrm{fm}$ and 0.04 fm. We reconstruct the $x$-dependent PDF, as well as infer the first few even moments of the PDF using leading-twist 1-loop perturbative matching framework. Our analyses use both RI-MOM and ratio-based schemes to renormalize the equal-time bilocal quark-bilinear matrix elements of pions boosted up to 2.4 GeV momenta. We use various model-independent and model-dependent analyses to infer the large-$x$ behavior of the valence PDF. We also present technical studies on lattice spacing and higher-twist corrections present in the boosted pion matrix elements.

Figure 1

The effective mass $E_{\mathrm{eff}}$ is shown as a function of source-sink separation $t_s$ for the $a=0.04\mathrm{fm}$ lattice. The filled and open symbols are obtained from SS and SP correlators respectively.

Figure 2

The dependence of the fitted values of energy levels on the fit range $[t_{\min },32a]$ is shown. The top-left and top-right panels show this dependence for the first excited level $E_1$ as obtained from two-state fits to SP correlator at $P_z=1.45\mathrm{GeV}$ and $P_z=1.94\mathrm{GeV}$ respectively. Similar results using SS correlator are shown in the two middle panels. The results for $E_1$ and $E_2$ obtained using three-state fits to the SS correlator (with prior on $E_0$ and $E_1$) are shown in the two bottom panels.

Figure 3

The energy of the ground state ($E_0$) and the first excited state ($E_1$) are shown as a function of $P_z$. The results from $a=0.04\mathrm{fm}$ are shown as filled symbols and those from $a=0.06\mathrm{fm}$ as the open symbols. The results shown in the plot for $E_0$ were obtained from an unconstrained two-state fit, while $E_1$ were obtained by fixing $E_0$ to its dispersion values.

Figure 4

The source-sink $t_s$ and operator insertion $\tau$ dependence of the ratio $R(t_s,\tau ;z,P_z)$, at fixed $z$ and $P_z=2\pi n_z/L$ are shown for the lattice spacing $a=0.06\mathrm{fm}$. The top-rows are for $n_z=0$, middles ones for $n_z=3$ and bottom ones for $n_z=5$. The left panels are for $z=0$, middle panels for $z=6a$ and right ones for $z=12a$ respectively. Each plot has left and right sub-panels. In the left sub-panels, the $t_s-\tau /2$ dependence is shown at $t_s=8a$ (red squares), $10a$ (green circles), and $12a$ (blue triangles). The corresponding colored bands are the $1-\sigma$ error bands from Fit(2,3) (see text). On the right subpanels, the extrapolation (grey band) to $t_s\to \infty$ is shown as a function of $a/t_s$ at fixed $\tau =t_s/2$.

Figure 5

The source-sink $t_s$ and operator insertion $\tau$ dependence of the ratio $R(t_s,\tau ;z,P_z)$, at fixed $z$ and $P_z=2\pi n_z/L$ are shown for the lattice spacing $a=0.04\mathrm{fm}$. The top-rows are for $n_z=0$, middles ones for $n_z=3$ and bottom ones for $n_z=5$. The left panels are for $z=0$, middle panels for $z=9a$ and right ones for $z=18a$ respectively. Each plot has left and right sub-panels. In the left sub-panels, the $t_s-\tau /2$ dependence is shown at $t_s=9a$ (red squares), $12a$ (green circles), $15a$ (blue triangles) and $18a$ (pink inverted-triangles). The corresponding colored bands are the $1-\sigma$ error bands from Fit(2,3) (see text). On the right sub-panels, the extrapolation (grey band) to $t_s\to \infty$ is shown as a function of $a/t_s$ at fixed $\tau =t_s/2$.

Figure 6

An example for summation method, $\mathrm{Sum}({\tau }_0)$, that uses only insertion points $\tau >{\tau }_0=2a$ are shown as a function of $t_s/a$ at fixed $z$ and $P_z$. The left panel is at $a=0.04\mathrm{fm}$ with $z=16a$ and $n_z=2$. The right panel is at $a=0.06\mathrm{fm}$ with $z=12a$ and $n_z=2$. In each panel, the red circles are the lattice data. In addition, there are three different curves—the red one is the straight line summation fit to the data, the blue one is the corrected summation fit SumExp that includes $\exp [-(E_1-E_0)t_s]$ correction, and the green one is the expected summation curve from the two-state fit Fit(2,3).

Figure 7

The bare matrix elements $h^B(z,P_z)$ from excited state extrapolations are shown as a function of $z/a$ for the $a=0.06\mathrm{fm}$ ensemble. The results from $n_z=0$, 2, 4 and 5 are shown in the top-left, top-right, bottom-left, and bottom-right are shown. The top part of each panel shows the $z/a$ dependence using two-state extrapolation Fit(2,3) using $t_s/a=8$, 10 and 12. The bottom part of each panel shows the deviation, $\mathrm{\Delta }$, of the different extrapolation methods Fit(2,2), Fit(3,1), Sum(2), SumExp(3) from the method Fit(2,3). The scatter of these differences from 0 (shown by dashed line) characterizes the robustness of the extrapolation.

Figure 8

The bare matrix elements $h^B(z,P_z)$ from excited state extrapolations are shown as a function of $z/a$ for the $a=0.04\mathrm{fm}$ ensemble. The results from $n_z=0$, 2, 4 and 5 are shown in the top-left, top-right, bottom-left, and bottom-right are shown. The top part of each panel shows the $z/a$ dependence using two-state extrapolation Fit(2,3) using $t_s/a=9$, 12, 15 and 18. The bottom part of each panel shows the deviation, $\mathrm{\Delta }$, of the different extrapolation methods as explained in Fig. 7. Deviations of results from Sum(2) from 2- and 3-state fit results are seen. But we find that corrected sum SumExp(2) is consistent with the fit results. This is a result of the observation in Fig. 6.

Figure 9

The result for the local bare matrix element $h^B(z=0,P_z)$ is shown as a function of $P_z$. The red and blue open symbols are the estimates of the bare matrix elements on $a=0.06\mathrm{fm}$ and $a=0.04\mathrm{fm}$ lattices. The estimated value of $h^B(z=0,P_z=0)$ for $a=0.04\mathrm{fm}$ after correcting for wrap-around effect (see text) is shown as the filled blue square. The red and the blue dashed curves are the modeled lattice spacing effects using an ansatz, $h^B(z=0,P_z)=h^B(z=0,P_z=0)+b(P_{z}a{)}^2$ for $a=0.06\mathrm{fm}$ and 0.04 fm respectively, with fixed $b=-0.0813$ in both cases.

Figure 10

Comparison of renormalized matrix elements at fixed $P_z=1.45\mathrm{GeV}$ in the ratio scheme with generalized nonzero values of the reference momentum $P_z^0$. The different colored symbols correspond to different $P_z^0=0$, 0.48 and 0.97 GeV. The effect of changing $P_z^0$ is significant, but it does not cause a big difference in signal-to-noise ratio at smaller $z$ that we are interested in. It is the expectation that matrix elements with $P_z,P_z^0>\{{\mathrm{\Lambda }}_{\mathrm{QCD}},m_{\pi }\}$ suffer from lesser higher-twist contamination.

Figure 11

Quantifying the residual systematic effects after significant statistical error reduction by the process of normalizing $z=0$ matrix element to 1 [see Eq. (17) and Eq. (19)]. The plot compares the different normalization types, $\mathcal{M}$ and ${\mathcal{M}}_{\mathrm{add}}$, and the matrix element without any imposed normalization ${\mathcal{M}}_0$. In the above plot, we show results for the $a=0.04\mathrm{fm}$ ensemble.

Figure 12

Comparison of renormalized matrix elements at two lattice spacings $a=0.06\mathrm{fm}$ (red circles) and 0.04 fm (blue squares) plotted as a function of $\nu =z{P}_z$. The top panel uses RI-MOM renormalization scheme. The real and imaginary parts of the renormalized RI-MOM matrix element are shown as closed and open symbols. The bottom panel uses ratio scheme with $n_z^0=1$. The variation of data by $\pm 2\%$ is shown as the red and blue bands.

Figure 13

The phase (top) and the magnitude (bottom) of the $P_z=0$ matrix element in RI-MOM scheme with $(P_z^R,P_{\perp }^R)=(1.29,2.98)\mathrm{GeV}$ are shown. The expectations from the 1-loop leading twist results are shown as the red bands. In the bottom panel, the actual lattice data is shown using filled black circles. The absolute part clearly suffers from a leading $z^2$ correction shown as the blue straight line. The twist-2 target mass correction is shown as the green curve for comparison. The lattice data after subtracting the $z^2$ correction term is shown using open circles.

Figure 14

The effect of higher-twist term $k{z}^2$ on the pion matrix element at fixed small $P_z=0.43\mathrm{GeV}$ is discussed. The left panel compares the usual ratio, $\mathcal{M}(z,P_z,P_z^0=0)$, and the ratio ${\mathcal{M}}^{\prime }$ derived from RI-MOM matrix element after the subtraction of $k{z}^2$ [see Eq. (33)]. The right panel compares the actual RI-MOM matrix element and the RI-MOM matrix element with $k{z}^2$ subtraction with the result expected by applying the conversion factor $Z_{\text{ratio}\to \mathrm{RI}}$ to the ratio $\mathcal{M}$ [see Eq. (34)].

Figure 15

Plot of the ratio, $\mathcal{M}(z,P_z,P_z^0)$, shown as a function of $z{P}_z$ using $a=0.06\mathrm{fm}$ data with $P_z^0=0.43\mathrm{GeV}$. The points of same colored symbols have the same value of $z$, and hence the $z{P}_z$ dependence comes from the variation in $P_z$. The corresponding colored bands are the fits of Eq. (30) to data at each fixed $z$. Inset: The data points are at fixed $z=a$. The blue curve is the expectation for the $z{P}_z$ dependence at $z=a$ based on the moments obtained at $z=5a$ without correcting for $(P_{z}a{)}^2$ lattice terms. The red curve is obtained after accounting for such lattice corrections.

Figure 16

The $z$ dependence of $⟨x^2⟩$ obtained by fitting rational polynomial functions in $z{P}_z$ to matrix elements $\mathcal{M}(z,P_z,P_z^0)$ at different fixed values of $z$, with $P_z^0=2\pi n_z^0/L$. The left panel shows results as obtained with $n_z^0=1$ and the right panel for $n_z^0=2$. In each case, the red and blue points correspond to $a=0.06\mathrm{fm}$ and 0.04 fm lattice spacings. The curves are fits to a functional form $⟨x^2⟩+2r(z/a{)}^{-2}$ to capture possible $(a{P}_z{)}^2$ corrections (refer text).

Figure 17

$⟨x^2⟩$ from combined fits of the rational polynomial function in $z{P}_z$ to $\mathcal{M}(z,z{P}_z,P_z^0)$ for the data $z\in [z_{\min },z_{\max }]$. The dependence of $⟨x^2⟩$ on $z_{\max }$ is shown. For each $z_{\max }$, values from three different $z_{\min }$ are shown. The results for $a=0.06\mathrm{fm}$ and 0.04 fm are shown in the left and right panels. The grey bands are the estimates—the inner band includes only statistical error, and outer one includes both statistical and systematic error (see text).

Figure 18

$⟨x^4⟩$ from combined fits of the rational polynomial function in $z{P}_z$ to $\mathcal{M}(z,z{P}_z,P_z^0)$ for the data $z\in [z_{\min },z_{\max }]$. The description of the points are the same in Fig. 17.

Figure 19

$⟨x^6⟩$ from combined fits of the rational polynomial function in $z{P}_z$ to $\mathcal{M}(z,z{P}_z,P_z^0)$ for the data $z\in [z_{\min },z_{\max }]$ using priors $⟨x^2{⟩}_v$ and $⟨x^4{⟩}_v$. The dependence of $⟨x^6⟩$ on $z_{\max }$ is shown. At each $z_{\max }$, values from $z_{\min }=a,2a$ and $3a$ are shown.

Figure 20

The plot shows the model boosted pion matrix element $\mathcal{M}(z,P_z,P_z^0)$ at $P_z=1.29\mathrm{GeV}$, $P_z^0=0.43\mathrm{GeV}$ as a function of $z$, constructed based on the JAM valence PDF [79] (solid curves) and ASV result [35] (dashed curves) at $\mu =3.2\mathrm{GeV}$. The red, green and blue curves are the matrix elements constructed using Eq. (30) using values of ${\alpha }_s(\mu )$, ${\alpha }_s(2\mu )$ and ${\alpha }_s(\mu /2)$ respectively. The lattice data for $\mathcal{M}(z,P_z,P_z^0)$ from $a=0.06\mathrm{fm}$ lattice are also shown.

Figure 21

Renormalized pion matrix elements in various schemes are shown as a function of $z{P}_z$ along with the best fits using PDFs $f_v^{\pi }(x)$ of the form $x^{\alpha }(1-x{)}^{\beta }(1+s\sqrt{x}+tx)$. The top and bottom rows show the results for $a=0.06\mathrm{fm}$ and $a=0.04\mathrm{fm}$ respectively. The leftmost panels are using the ratio renormalization scheme with the reference momentum $P_z^0=2\pi /(La)$, which is $P_z^0=0.43\mathrm{GeV}$ for $a=0.06\mathrm{fm}$ and $P_z^0=0.48\mathrm{GeV}$ for $a=0.04\mathrm{fm}$. The middle panels use $P_z^0=4\pi /(La)$, which is $P_z^0=0.86\mathrm{GeV}$ for $a=0.06\mathrm{fm}$ and $P_z^0=0.97\mathrm{GeV}$ for $a=0.04\mathrm{fm}$. The rightmost panels are in the RI-MOM scheme with the renormalization momentum $(P_z^R,P_{\perp }^R)=(1.29,2.98)\mathrm{GeV}$ for $a=0.06\mathrm{fm}$ case and (1.93,3.34) GeV for $a=0.04\mathrm{fm}$ case. Here, the real and imaginary parts are shown. In each panel, the different colored symbols are the actual lattice data at different pion momentum $P_z$. The corresponding similarly colored bands are the results of the combined fit over the fixed range $z\in [2a,0.5]\mathrm{fm}$ from different $P_z>P_z^0$ in the ratio scheme, and $n_z=3$, 4, 5 in the case of RI-MOM scheme.

Figure 22

The valence PDF of pion $f_v^{\pi }(x)$ at $\mu =3.2\mathrm{GeV}$. The top and bottom panels are for $a=0.06\mathrm{fm}$ and $a=0.04\mathrm{fm}$ respectively. The left panels show $f_v^{\pi }(x)$ and the right panel re-plot the same data as $x{f}_v^{\pi }(x)$. The ansatz $f_v^{\pi }(x)=\mathcal{N}{x}^{\alpha }(1-x{)}^{\beta }(1+s\sqrt{x}+tx)$ was used for this reconstruction of valence PDF via their ability to describe pion matrix elements in real space in different ratio schemes involving ranges of quark-antiquark separation $z\in [2a,0.5]\mathrm{fm}$. In each panel, such results for $f_v^{\pi }(x)$ based on ratio schemes with reference momenta $P_z^0=2\pi n_z^0/L$ with $n_z^0=1$, 2 are shown as different colored bands. For comparison, the JAM result at the same $\mu$ is shown as the black band.

Figure 23

Valence PDF of pion at $\mu =3.2\mathrm{GeV}$ extracted from RI-MOM renormalized matrix elements. The left and the right panels show data for $a=0.06\mathrm{fm}$ and $a=0.04\mathrm{fm}$. The red and blue bands are the PDFs that best describes the RI-MOM data at two different RI-MOM scales $(P_z^R,P_{\perp }^R)$. For comparison, the PDF as extracted using ratio scheme with $P_z^0=2\pi n_z^0/(La)$ with $n_z^0=1$ is shown as the green band. The JAM estimate [79] of $f_v^{\pi }(x)$ is shown as the black band.

Figure 24

Comparison of PDF as extracted using a simple two-parameter $x^{\alpha }(1-x{)}^{\beta }$ ansatz and from a four-parameter $x^{\alpha }(1-x{)}^{\beta }(1+s\sqrt{x}+tx)$ ansatz.

Figure 25

The first four valence PDF moments $⟨x^n⟩$ as inferred from the best estimates of PDFs $f_v^{\pi }(x)$ that best describes the equal-time pion matrix elements in ratio and RI-MOM renormalization schemes. The top panels are for $a=0.06\mathrm{fm}$ and bottom ones for $a=0.04\mathrm{fm}$. The dependence of $⟨x^n⟩$ on all the variables in the lattice analysis is summarized in the above plot. The foremost variable is the range of quark-antiquark separations used in the fits $z\in [z_1,z_2]$. Such variations are bunched together as blocks separated by the dashed lines along the y-axis. The second variable factor is the renormalization scheme of the matrix elements: it could be RI-MOM scheme or ratio scheme at reference scale $P_z^0$. At fixed $z\in [z_1,z_2]$, four different renormalization points are shown: ratio scheme at $n_z^0=1$ (ratio-1), $n_z^0=2$ (ratio-2), RI-MOM scheme at two different scales $P^R$, denoted as RI-1 and RI-2. This scheme and scale variations are bunched together within the dotted lines. The tertiary variable is the fit ansatz: the results obtained using the ansatz $f_v^{\pi }(x)=\mathcal{N}{x}^{\alpha }(1-x{)}^{\beta }$ are shown in red and those using $f_v^{\pi }(x)=\mathcal{N}{x}^{\alpha }(1-x{)}^{\beta }(1+s\sqrt{x}+tx)$ are shown in blue.

Figure 26

The exponents $\alpha$ and $\beta$ inferred from the estimates of PDFs $f_v^{\pi }(x)=x^{\alpha }(1-x{)}^{\beta }(1+\dots )$ that best describes the ratio and RI-MOM real space data. The first two panels are for $a=0.06\mathrm{fm}$ and last two for $a=0.04\mathrm{fm}$. The dependence of the exponents on all the variables in the lattice analysis is summarized in the above plot. The notation is similar to Fig. 25.

Figure 27

The plot shows the effective large-$x$ exponent ${\beta }_{\mathrm{eff}}(n)$ as a function of $n$. The data points are obtained by using values of moments $⟨x^n{⟩}_v$ obtained from the model independent combined fits. The smaller error bar is only the statistical error and larger error bar is statistical plus systematic error. The red and blue points are the results using our $a=0.06\mathrm{fm}$ and 0.04 fm respectively. The red and blue bands are the expectation for the behavior of ${\beta }_{\mathrm{eff}}(n)$ as obtained from the model-dependent fits to the pion matrix elements. The green curve is the expectation using the JAM data [79] and the purple dashed curve is obtained using the ASV analysis [35].

Figure 28

Estimate of continuum extrapolation of $⟨x^2{⟩}_v$ and $⟨x^4{⟩}_v$ from combined fits to $a=0.04\mathrm{fm}$ and $a=0.06\mathrm{fm}$ data using the ansatz in Eq. (60). For the case shown, $n_z^0=1$, $[z_1,z_2]=[a,0.5\mathrm{fm}]$, and analyzed using 4-parameter PDF ansatz. The black circles are the data from the analysis at the two fixed $a$. The bands are the $a^2$ extrapolations using the combined fits.

Figure 29

Our determinations of valence PDF of pion, $f_v^{\pi }(x,\mu )$, at factorization scale $\mu =3.2\mathrm{GeV}$. Top panel: the PDF determination from $a=0.04\mathrm{fm}$ data (red) and $a=0.04\mathrm{fm}$ data (blue). Bottom panel: our estimate of PDF in the continuum limit (blue). In both top and bottom panels, the darker inner band includes only the statistical error. The lighter outer band includes both statistical error as well as the systematic errors. Our estimates are compared with the FNAL E-0615 estimate [45] (green symbols), ASV estimate [35] (green dashed line), JAM estimate [79] (black band) and xFitter analysis [84] (purple line). Insets: the same data are replotted as the traditional $x{f}_v^{\pi }(x)$ versus $x$.

Figure 30

The vector current renormalization factor $Z_V$ as function of $p_R$ in lattice units.

Figure 31

The quark field renormalization factor $Z_q$ as function of $p_R$ in lattice units.

Figure 32

The renormalization factor $Z_{{\gamma }_t{\gamma }_t}$ as function of $p_R$ in lattice units.

Figure 33

${\chi }^2/\mathrm{dof}$ for the 2-parameter (red) and 4-parameter (blue) fits is shown from various fit ranges and renormalization schemes. This plot accompanies Fig. 25 and Fig. 26. The description of the plot is similar to Fig. 25. The left panel is for $a=0.06\mathrm{fm}$ and the right one for $a=0.04\mathrm{fm}$.

Figure 34

The value of large-$x$ exponent, $\beta ({\mu }^2)$, for the valence PDF at the $\overline{\mathrm{MS}}$ scale ${\mu }^2$.

Figure 35

The plot shows different set of 2-parameter PDFs, $f(x,\alpha ,\beta )$, that have fixed $⟨x⟩=0.23$. The different colored curves correspond to different values of $\beta$ centered around $\beta =1$. The dashed line is the value $x^{*}=0.57$ which is the fixed point for the PDFs.

Figure 36

Plot shows the first three even moments obtained on $a=0.06\mathrm{fm}$ lattice in a model independent manner as described in Sec. 7. The description of the three plots is similar to that in Fig. 17, Fig. 18, and Fig. 19. In addition to the two values of $P_z^0\ne 0$, this plot includes $P_z^0=0$ reference scale for the ratio.

Figure 37

The plot is on the 4-parameter PDF ansatz fits to the $P_z^0=0$ ratio data on $a=0.06\mathrm{fm}$ lattice. The top panel shows the ratio data a function of $P_{z}z$ along with the fits. The bottom panel shows the PDF extracted from $P_z^0=0$ ratio with other ratio data at nonzero $P_z^0$ discussed in the main text. The description of the plots are similar to the ones in Fig. 21 and Fig. 22.

Figure 38

The valence PDF with $\beta$ being a free parameter (red) and $\beta$ being fixed at a value 2 (blue) in the ansatz $f_v^{\pi }(x)=\mathcal{N}{x}^{\alpha }(1-x{)}^{\beta }(1+s\sqrt{x}+tx)$. For comparison, the JAM18 [79] curve (black) and ASV [35] result (green) are also shown.