Quark distribution inside a pion in many-flavor ($2+1$)-dimensional QCD using lattice computations: UV listens to IR

We study the changes in the short-distance quark structure of the Nambu-Goldstone boson when the long-distance symmetry-breaking scales are depleted controllably. We achieve this by studying the valence parton distribution function (PDF) of pion in $2+1$ dimensional two-color QCD, with the number $N$ of massless quarks as the tunable parameter that slides the theory from being strongly broken for $N=0$ to the conformal window for $N>4$, where the theory is gapped by the fixed finite volume. We perform our study nonperturbatively using lattice simulations with $N=0$, 2, 4, 8 flavors of nearly massless two-component Wilson-Dirac sea quarks and employ the leading-twist formalism (LaMET/SDF) to compute the PDF of pion at a fixed valence mass. We find that the relative variations in the first few PDF moments are only mild compared to the changes in decay constant, but the shape of the reconstructed $x$-dependent PDF itself dramatically changes from being broad in the scale-broken sector to being sharply peaked in the near-conformal region, best reflected in PDF shape observables such as the cumulants.

Figure 1

The spectral content of pion two-point functions in $N=0$, 2, 4, 8 quark flavors from left to right. Top: the effective mass from pion two-point function at different momenta (different colored symbols) shown as a function of source-sink separations $t_s/a$ in lattice units. The bands are the expected effective mass from two-state fits to the two-point function. Bottom: the dispersion relation between ground state (red circles) and the first excited state (purple triangle) energy on boosted momentum is shown. For comparison, the expected single particle dispersion in the continuum (blue dashed curve) and on the lattice (black curve) are shown. The ground state pion energy extracted from axial-vector ${\mathcal{A}}_1$ correlator are also shown at nonzero momenta.

Figure 2

The estimation of matrix element $h_{00}(z_1,P_1)$ by the two-state fits to the ratio, $R(t_s,\tau )$. Some sample fits to the operator insertion point, $\tau$, and the source-sink separation, $t_s$, dependence of $R$ are shown in the different panels; the top panels are at $z_1=3a$ and the bottom ones are $z_1=6a$, at a momentum of $n_1=3$. For each $z_1$, the panels from left to right are from different number of flavors.

Figure 3

The matrix elements at $N=0$, 2, 4, 8 (left to right) obtained by extrapolation are shown as a function of quark-antiquark separation $z_1/a$. The different colored symbols correspond to different fixed momenta $P_1$, which are shown in the legend in units of gauge coupling $g^2$.

Figure 4

The determination of pion decay constant $F_{\pi }$. The data points are the effective $F_{\pi }^{\mathrm{eff}}(t_s)$ as determined from the $⟨{\mathcal{A}}_3\pi ⟩(t_s)$ correlator, and the curves are fits to $F_{\pi }+A{e}^{-\delta m{t}_s}$.

Figure 5

The changes to the infrared scales when the number of massless quark flavor is changed. Top panel: the quark condensate as a function of decay constant, both implicitly depending on the number of quark flavors. Bottom panel: a similar plot showing how the mass gap between the pion and the axial vector changes as a function of decay constant.

Figure 6

A cross-check that at least an application of small number of Stout smearing to the Wilson line connecting quark-antiquark is harmless. The pion bilocal matrix element at $n_1=3$ momentum in $N=0$ theory with 2-stout and 6-stout smeared Wilson line insertions are compared and shown to be consistent once the ratio is taken.

Figure 7

The pion bilocal quark bilinear matrix elements, $\tilde{\mathcal{M}}(z_1,P_1)$ from $N=0$, 2, 4 and 8 flavor theories are shown in the four panels. In each panel, the matrix elements from $P_1=1.05,1.57,2.09g^2$ are put together and shown as a function of $z_1{P}_1$. The color and symbols differentiate the data at fixed $P_1$. The bands are the expectations for $\tilde{\mathcal{M}}(z_1,P_1)$ based on the fits to the leading twist expression in Eq. (30). The bands extend over points included in the fit.

Figure 8

The effect of reduction in $F_{\pi }^2$ on the bilocal matrix element (Ioffe-time distribution) $\mathcal{M}(\nu )$ is shown. The bands are inferred from the fits to lead-twist expression in Eq. (30) and taking its $z_1\to 0,P_1\to \infty$ limit at fixed $P_1{z}_1=\nu$. The matrix element is shown as a function of $\nu =P^{+}{\xi }^{-}$ (Ioffe time). The different colored bands are at different $F_{\pi }^2{g}^{-2}$.

Figure 9

Top: the correlation between decay constant and the valence PDF moments. The filled symbols are obtained from model-independent fits and the open ones from model-dependent PDF ansatz fits. For model-independent fits only the even moments are directly obtained, whereas the odd moments $⟨x{⟩}_v$ and $⟨x^3{⟩}_v$ were obtained indirectly by definition in Eq. (32). The curves are quadratic fits in order to interpolate the data. Bottom: the correlation between decay constant and the cumulants ${\kappa }_4$ and ${\kappa }_6$ of $u-d$ PDF.

Figure 10

Reconstruction of valence PDF, $f_v(x)$, by fits to two-parameter ansatz. The left panels show the fits (bands) to the bilocal matrix element $\tilde{\mathcal{M}}(z_1,P_1)$ (points) via leading-twist expression in Eq. (30). The middle panels show the inferred valence PDF, $f_v(x)$. The different colored bands correspond to different fit ranges $[0,z_1]$. The right panels show $x{f}_v(x)$. Top to bottom are $N=0$, 2, 4, 8 theories respectively.

Figure 11

Left: the reconstructed valence pion PDFs, $f_v(x)$, and Right: their corresponding momentum distribution, $x{f}_v(x)$ are shown as a function of pion decay constant that characterizes the vacuum of different $N=0$, 2, 4, 8 flavor theories.

Figure 12

The matrix element ${\mathcal{M}}^B(z_1,P_1=0)$ (before forming ratios) is shown as a function of quark-antiquark separation $g^2{z}_1$ in units of coupling. The different colored symbols are the data taken from $N=0$, 2, 4, 8 flavor theories. The black dashed line is the expectation based on target mass corrections from leading-twist trace terms. The blue dashed line is the behavior modeled by Eq. (a2) to take into account the screening behavior of the Wilson line.