Valence parton distribution function of pion from fine lattice

We present a lattice QCD study of the valence parton distribution inside the pion within the framework of Large Momentum Effective Theory. We use a mixed action approach with 1-HYP smeared valence Wilson clover quarks on $2+1$ flavor HISQ sea with the valence quark mass tuned to 300 MeV pion mass. We use $48^3\times 64$ lattice at a fine lattice spacing $a=0.06\mathrm{fm}$ for this computation. We renormalize the quasi parton distribution function matrix element in the nonperturbative regularization independent momentum subtraction (RI-MOM) scheme. As a byproduct, we test the validity of a 1-loop matching procedure by comparing the RI-MOM renormalized quasi parton distribution function matrix element with off-shell quark external states as computed in the continuum 1-loop perturbation theory with the lattice results at $a=0.04$ and 0.06 fm. By applying the RI-MOM to $\overline{\mathrm{MS}}$ one-loop matching, implemented through a fit to phenomenologically motivated parton distribution functions, we obtain the valence parton distribution function of pion.

Figure 1

The effective masses $E_{\mathrm{eff}}$ from the pion two-point functions with the boosted Coulomb gauge Gaussian source and sink for different momenta as a function of the source-sink separation $t$. The horizontal lines are the energy levels from the continuum dispersion relation with $m_{\pi }=300\mathrm{MeV}$.

Figure 2

The systematical dependence of the ground state $E_{\pi }$ and the first excited state $E_1$ on the fit range $[t_{\min },t_{\max }]$ is shown. In the left panels, we show such a dependence for the pion SP correlator at two different $P_z$. For each $t_{\min }$, data from $t_{\max }=24a$ and $32a$ are shown. The black solid line is the value of $E_{\pi }$ expected from the continuum dispersion relation. The red patterned band is our best estimate of $E_1$ using the SP correlator. In the right panels, the fit systematics of $E_1$ for the SS correlator is shown. The red band is the prior used for $E_1$ from the SP correlator (same as the one in the left panels). The different symbols are the various fit strategies.

Figure 3

The energies of the ground state and the first two excited states as functions of $P_z$. The red, blue, and black symbols correspond to $E_{\pi }$, $E_1$, and $E_2$, respectively. For each color, the different symbols correspond to different fitting methods (2-state and 3-state fit with or without prior on the ground state) and the types of source-sink (SP or SS). The lines show the expected dispersion relations for the pion and its first excited state. The arrows are the PDG values of $\pi (1300)$ and $\pi (1800)$ which are shifted to account for $m_{\pi }=300\mathrm{MeV}$.

Figure 4

The ratio of the three-point function to the two-point function, $R(t,\tau ;z,P_z,\mathrm{\Gamma })$ is shown as function of $\tau -t/2$ for ${\gamma }_t$ (top row) and ${\gamma }_z$ (bottom row) for $z/a=0$, 4 and 8 (from left to right) and $P_z=1.29\mathrm{GeV}$. The central values of the two-state fits of the lattice results for different source-sink separations are shown as the curves. The horizontal band corresponds to the extrapolated result for the infinite source-sink separation.

Figure 5

The $t$ dependence of $R_{\mathrm{sum}}(t;z=0,{\gamma }_z)$ for $P_z=1.29\mathrm{GeV}$ with ${\tau }_o=1$ (red) and ${\tau }_o=2$ (blue). The solid lines are the straight line fit $h_{{\gamma }_z}^B(z,P_z)t+\text{const}$ to the data. The dashed lines are the expected curve for $R_{\mathrm{sum}}(t;z=0,{\gamma }_z)$ using the fit(2,2) best fit parameters.

Figure 6

The ratio of the matrix elements for ${\gamma }_z$ to ${\gamma }_t$ as a function of $P_z$. The curve shows the expected result, $P_z/E_{\pi }(P_z)$.

Figure 7

The bare matrix elements $h_{{\gamma }_t}^B(z,P_z)$ (left) and $h_{{\gamma }_z}^B(z,P_z)$ (right) as a function of quark-antiquark separation $z$. The panels in the top row show results for $P_z=1.29\mathrm{GeV}$, while the panels in the bottom row show the results for $P_z=1.72\mathrm{GeV}$. The different symbols are from various methods of $t\to \infty$ extrapolation.

Figure 8

The RI-MOM renormalization constants using $\cancel{p}$-projection at lattice spacing $a=0.06\mathrm{fm}$ at renormalization scale $p_z=1.29\mathrm{GeV}$, $p_{\perp }=1.49\mathrm{GeV}$ are shown. In the top-left panel, the real and imaginary parts of the renormalization constant for ${\gamma }_t$ qPDF operator are shown as a function of quark-antiquark separation $z$ in physical units. On the top-right panel, the self-energy divergent part $e^{-c|z|}$ of the Wilson line is removed from renormalization constant for ${\gamma }_t$ qPDF. Similarly, in the bottom-left panel, the diagonal part $Z_{{\gamma }_z{\gamma }_z}$ and the mixing term $Z_{{\gamma }_{z}1}$ are shown, and the corresponding values after the removal of self energy divergence are shown in the bottom-right panel.

Figure 9

The $(pa{)}^2$ dependence of the renormalization factors $Z_q$ (black squares), $Z_{{\gamma }_t{\gamma }_t}(z=0)$ (red triangles), and $Z_V=Z_q{Z}_{{\gamma }_t{\gamma }_t}(0)$ (blue diamonds) are shown for the $a=0.06\mathrm{fm}$ ensemble.

Figure 10

The behavior of $\zeta (z)$ (symbols) with the quark-antiquark separation $z$, is compared with the expectation from 1-loop perturbation theory (bands) when $p_{\perp }^R$ is slightly away from $p_{\perp }$. The red symbol and bands are the real part of $\zeta$ while the blue ones correspond to the imaginary part. The bands enclosed by solid curves corresponds to the momenta $(p_z,p_{\perp })$ for the 0.04 fm data, and similarly the band enclosed by the dashed curves corresponds to $(p_z,p_{\perp })$ of the 0.06 fm data (see text). On the left and right panels, the comparisons are made for ${\zeta }_{{\gamma }_t}$ and ${\zeta }_{{\gamma }_z}$ respectively. In each of the panels, the data from two different lattice spacings are also shown ($a=0.04\mathrm{fm}$ as filled circles and $a=0.06\mathrm{fm}$ as open circles). For the data shown, $p_z=p_z^R=1.92\mathrm{GeV}$. The transverse momentum of the quark $p_{\perp }=1.58\mathrm{GeV}$, and the transverse renormalization momentum is chosen to be 1.5 times $p_{\perp }$.

Figure 11

The renormalization flow of ${\zeta }_{{\gamma }_t}$ (left panel) and ${\zeta }_{{\gamma }_z}$ (right panel) with the transverse renormalization scale, $p_{\perp }^R$ are shown at fixed $z=0.12\mathrm{fm}$. The data from two different lattice spacings, $a=0.04\mathrm{fm}$ (filled circles) and $a=0.06\mathrm{fm}$ (open circles), shown at fixed $z=0.12\mathrm{fm}$ and $p_z=p_z^R=1.92\mathrm{GeV}$. The transverse momenta of the quark at $a=0.04$ and 0.06 fm are $p_{\perp }=1.67$ and 1.48 GeV, respectively, and they are chosen to be roughly equal for this comparison. The real part of $\zeta$ is shown in red while the imaginary part is shown in blue. The band enclosed by the solid red (blue) curves corresponds to the 1-loop result for real (imaginary) parts of $\zeta$ at $(p_z,p_{\perp })$ for the 0.04 fm data, and similarly the band enclosed by the dashed curves corresponds to $(p_z,p_{\perp })$ of the 0.06 fm data.

Figure 12

The dependence of $\zeta$ on the longitudinal momentum $p_z^R$. ${\zeta }_{{\gamma }_z}$ is shown as a function of $z$ for a specific choice of $p_z$ and $p_{\perp }=p_{\perp }^R$. The uncertainty bands for the real and imaginary parts for the leading-order expectation are shown using bands enclosed by solid lines. The symbols are the lattice data determined at lattice spacing $a=0.04\mathrm{fm}$.

Figure 13

The real and imaginary parts of the lattice data for $q_{{\gamma }_t}(z,p)$ are compared with the purely oscillatory model (dashed curves) at short-distances, and the damped oscillatory ansatz (solid curves) at larger $|z|$. The data correspond to quark momentum $(p_z,p_{\perp })=(1.29,1.49)\mathrm{GeV}$.

Figure 14

The effective frequency of oscillations $\omega$ (top) and effective screening mass $m_{\mathrm{scr}}$ (bottom) as extracted from $q_{{\gamma }_t}(z,p)$ are shown. The various values of $p=(p_z,p_{\perp })$ for the data are tabulated in the plots. The values of $\omega$ are normalized with respect to $p_z$. A 1-HYP smeared Wilson line was used in all the cases shown in the plots, and the corresponding self-energy $ca=0.1586$ was subtracted to obtain $m_{\mathrm{scr}}$.

Figure 15

Top panels: The RI-MOM renormalized qPDF matrix element in real space $h_{{\gamma }_t}^R(z,P_z,P^R)$ at pion momenta $P_z=0.82$, 1.29, and 1.72 GeV are shown at fixed RI-MOM renormalization scale $(P_z^R,P_{\perp }^R)=(1.29,2.98)\mathrm{GeV}$. The red and blue points are the real and imaginary parts of the actual data, respectively. The bands were obtained by fitting the two-parameter phenomenologically motivated real space qPDF matrix element to the data over a range $z\in [-z_{\max },z_{\max }]$—the solid band is for $z_{\max }=1.44\mathrm{fm}$ and patterned one for $z_{\max }=0.72\mathrm{fm}$. Bottom panels: The two parameter $u-d$ PDF $f_{u-d}(x)$ (dashed lines) at $\mu =3.2\mathrm{GeV}$, and the matched qPDFs ${\tilde{q}}_{u-d}(x)$ ($1-\sigma$ error bands) that describe the real space qPDF on the top panels are shown. To avoid clutter, only the central values of $f_{u-d}(x)$ are shown as dashed lines. The results from different $z_{\max }$ are shown in red and blue.

Figure 16

The top and bottom panels show our estimated pion valence PDF at $\mu =3.2\mathrm{GeV}$ using ${\gamma }_t$ qPDF at $P_z=1.29$ and 1.72 GeV, respectively. The results using multiple RI-MOM scales $(P_z^R,P_{\perp }^R)$ are shown using different colored error bands. On the left panels, the results for $f_v^{\pi }(x)$ are shown, while on the right panels the results for $x{f}_v^{\pi }(x)$ are shown. For all the cases shown, the fit range was held fixed at $z_{\max }=0.98\mathrm{fm}$. The solid line (with a small error band around it) is the JAM result [59] for pion valence PDF at the same $\mu$.

Figure 17

The $1-\sigma$ confidence region ellipse of the exponents $a$ and $b$ in the model PDF at $\mu =3.2\mathrm{GeV}$ that best describes the real space RI-MOM qPDF is shown. The solid lines and dashed lines correspond to $P_z=1.29\mathrm{GeV}$ and $P_z=1.72\mathrm{GeV}$. For each of these pion momenta, the different colored lines correspond to different RI-MOM scale $P^R$. The black point is the JAM value [59] for valence pion PDF. The black straight line is the line of constant first moment of valence PDF, $⟨x⟩=0.215$.

Figure 18

In the top and bottom panels, the real (red) and imaginary (blue) parts of the renormalized real space ${\gamma }_z$ qPDF matrix element are shown for pion momenta $P_z=1.29\mathrm{GeV}$ and 1.72 GeV, respectively. The data points are the actual lattice data. The bands are the expected matched ${\gamma }_z$ qPDF matrix element starting from our best estimate for valence pion PDF obtained using ${\gamma }_t$ qPDF analysis.

Figure 19

Effective masses for $P_z=0$ using Gaussian sources with 40 steps of Wuppertal smearings, 90 steps of Wuppertal smearings, and Coulomb gauge.

Figure 20

Effective masses for different values of $\zeta$ with 50 configurations; green, blue, and black points correspond to momentum 0.86, 1.29, and 1.72 GeV, respectively.

Figure 21

The ratio of the three-point function to the two-point function, $R(t,\tau ;z,P_z,{\gamma }_z)$ for $\mathrm{\Gamma }={\gamma }_z$ is shown as function of $\tau -t/2$ for $z/a=0$, 4, and 8 (from left to right), and $P_z=0$, 0.483, 0.86, and 1.72 GeV (top to bottom). The corresponding plots for $P_z=1.29\mathrm{GeV}$ are shown in Fig. 4 in the main text. The central values of the two-state fits to the lattice results for different source-sink separations are shown as the curves. The horizontal band corresponds to the extrapolated result for infinite source-sink separation.

Figure 22

The ratio of the three-point function to the two-point function, $R(t,\tau ;z,P_z,{\gamma }_t)$ for $\mathrm{\Gamma }={\gamma }_t$ is shown as function of $\tau -t/2$ for $z/a=0$, 4, and 8 (from left to right), and $P_z=0$, 0.483, 0.86, and 1.72 GeV (top to bottom). The corresponding plots for $P_z=1.29\mathrm{GeV}$ are shown in Fig. 4 in the main text. The central values of the two-state fits to the lattice results for different source-sink separations are shown as the curves. The horizontal band corresponds to the extrapolated result for infinite source-sink separation. The case of $P_z=0$, in the top-most panels, is special due to the presence of the effect of lattice periodicity, and, hence, the various symbols and curves for the top-most panels are explained in detail in the text of Appendix pp7.