Parton distributions from nonlocal chiral SU(3) effective theory: Flavor asymmetries

Using recently derived results for one-loop hadronic splitting functions from a nonlocal implementation of chiral effective theory, we study the contributions from pseudoscalar meson loops to flavor asymmetries in the proton. Constraining the parameters of the regulating functions by inclusive production of $n$, ${\mathrm{\Delta }}^{++}$, $\mathrm{\Lambda }$, and ${\mathrm{\Sigma }}^{*+}$ baryons in $pp$ collisions, we compute the shape of the light antiquark asymmetry $\overline{d}-\overline{u}$ in the proton and the strange asymmetry $s-\overline{s}$ in the nucleon sea. With these constraints, the magnitude of the $\overline{d}-\overline{u}$ asymmetry is found to be compatible with that extracted from the Fermilab E866 Drell-Yan measurement, with no indication of a sign change at large values of $x$, and an integrated value in the range $⟨\overline{d}-\overline{u}⟩\approx 0.09–0.17$. The $s-\overline{s}$ asymmetry is predicted to be positive at $x>0$, with compensating negative contributions at $x=0$, and an integrated $x$-weighted moment in the range $⟨x(s-\overline{s})⟩\approx (0.9-2.5)\times {10}^{-3}$.

Figure 1

One-meson loop diagrams contributing to quark and antiquark PDFs in the nucleon, representing (a) the rainbow diagram with octet baryon (solid lines) intermediate state; (b) the rainbow diagram with decuplet baryon (double solid lines) intermediate state; and (c) the meson (dashed lines) bubble diagram. The symbol “$\otimes$” represents an operator insertion.

Figure 2

Contributions to quark PDFs in the nucleon from baryon coupling diagrams, representing (a) coupling to the bare nucleon; (b),(c) contributions from wave function renormalization with octet and decuplet baryon intermediate states; (d) the rainbow diagram with octet baryon; (e),(f) Kroll-Ruderman and gauge link (filled circle) Kroll-Ruderman diagrams with octet baryon; (g) the rainbow diagram with decuplet baryon; (h),(i) Kroll-Ruderman diagrams with decuplet baryon; and (j),(k) meson tadpole and gauge link tadpole diagrams.

Figure 3

Differential inclusive hadron production cross section $\sigma (y,k_{\perp }^2)$ vs $\overline{y}$ for (a) $pp\to nX$ at $k_{\perp }^2=0$ [38]; (b) $pp\to nX$ integrated over $k_{\perp }^2$ [39]; (c) $pp\to {\mathrm{\Delta }}^{++}X$ integrated over $k_{\perp }^2$ [40], compared with the fitted nonlocal pion exchange contributions for ${\mathrm{\Lambda }}_{\pi N}=1.0(1)\mathrm{GeV}$ and ${\mathrm{\Lambda }}_{\pi \mathrm{\Delta }}=0.9(1)\mathrm{GeV}$ (solid red lines and pink $1\sigma$ uncertainty bands) and with Pauli-Villars regularization (dashed red lines) for ${\mathrm{\Lambda }}_{\pi N}^{(\mathrm{PV})}=0.3\mathrm{GeV}$ and ${\mathrm{\Lambda }}_{\pi \mathrm{\Delta }}^{(\mathrm{PV})}=0.64\mathrm{GeV}$.

Figure 4

The flavor asymmetry of the proton $x(\overline{d}-\overline{u})$ vs $x$ from the lowest order pion exchange (solid red curve and pink band), with cutoff parameters ${\mathrm{\Lambda }}_{\pi N}=1.0(1)\mathrm{GeV}$ and ${\mathrm{\Lambda }}_{\pi \mathrm{\Delta }}=0.9(1)\mathrm{GeV}$, including nucleon on-shell (dashed blue line), $\mathrm{\Delta }$ on-shell (dashed green line), and $\mathrm{\Delta }$ end-point (dotted green line) contributions, and compared with the asymmetry extracted from the Fermilab E615 Drell-Yan experiment [4].

Figure 5

Contributions to the $⟨\overline{d}-\overline{u}⟩$ moment vs the dipole cutoff parameter $\mathrm{\Lambda }$ ($={\mathrm{\Lambda }}_{\pi N}$ or ${\mathrm{\Lambda }}_{\pi \mathrm{\Delta }}$) from (a) the $\pi N$ rainbow diagram [Fig. 1], including on-shell (solid red line), and local (dotted blue line) and nonlocal (dot-dashed green line) $\delta$-function terms; (b) the pion bubble [Fig. 1], including local (dotted blue line) and nonlocal (dot-dashed green line) $\delta$-function pieces; (c) the $\pi \mathrm{\Delta }$ rainbow [Fig. 1], including on-shell (solid red line), end-point (dashed red line), local (dotted blue line) and nonlocal (dot-dashed green line) $\delta$-function, and local decuplet $\delta$-function (dotted black line) contributions.

Figure 6

Differential inclusive hadron production cross section $\sigma (y,k_{\perp }^2)$ vs $\overline{y}$ for (a) $pp\to \mathrm{\Lambda }X$ at $k_{\perp }=0.075\mathrm{GeV}$ [39]; (b) $pp\to \mathrm{\Lambda }X$ integrated over $k_{\perp }^2$ [41]; and (c) $pp\to {\mathrm{\Sigma }}^{*+}X$ integrated over $k_{\perp }^2$ [42], compared with the fitted nonlocal kaon exchange contributions for dipole regulator parameters ${\mathrm{\Lambda }}_{K\mathrm{\Lambda }}=1.1(1)\mathrm{GeV}$ and ${\mathrm{\Lambda }}_{K{\mathrm{\Sigma }}^{*}}=0.8(1)\mathrm{GeV}$ (solid red lines and pink $1\sigma$ uncertainty bands).

Figure 7

Kaon loop contributions to (a) antistrange PDF $x\overline{s}$ from the octet and decuplet rainbow diagrams [Figs. 1 and 1]; (b) strange quark PDF $xs$ from the octet rainbow [Fig. 2], Kroll-Ruderman [Figs. 2 and 2], and tadpole [Figs. 2 and 2] diagrams; (c) strange PDF $xs$ from the decuplet rainbow [Fig. 2] and Kroll-Ruderman [Figs. 2 and 2] diagrams; (d) strange asymmetry $x(s-\overline{s})$, showing the local and nonlocal (gauge) octet and decuplet contributions, along with the total asymmetry. The PDFs are computed with the best fit regulator parameters ${\mathrm{\Lambda }}_{K\mathrm{\Lambda }}={\mathrm{\Lambda }}_{K\mathrm{\Sigma }}=1.1\mathrm{GeV}$ and ${\mathrm{\Lambda }}_{K{\mathrm{\Sigma }}^{*}}=0.8\mathrm{GeV}$.

Figure 8

Contributions to the $⟨x(s-\overline{s})⟩$ moment vs the dipole cutoff parameter $\mathrm{\Lambda }$ ($={\mathrm{\Lambda }}_{KY}$, for $Y=\mathrm{\Lambda }$ or $\mathrm{\Sigma }$ hyperons, or ${\mathrm{\Lambda }}_{K{\mathrm{\Sigma }}^{*}}$) from (a) the $KY$ octet rainbow [Figs. 1 and 2] and Kroll-Ruderman [Figs. 2–2] diagrams; (b) the $K$ bubble [Fig. 1] and tadpole [Figs. 2 and 2] diagrams; (c) the $K{\mathrm{\Sigma }}^{*}$ decuplet rainbow [Figs. 1 and 2] and Kroll-Ruderman [Fig. 2 and 2] diagrams.