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

TL;DR

This paper uses a nonlocal chiral SU(3) effective theory to calculate flavor asymmetries in the proton, aligning well with experimental data and predicting specific asymmetry behaviors for strange and light antiquarks.

Contribution

It introduces a novel nonlocal chiral effective theory approach to compute flavor asymmetries in the proton, constrained by baryon production data.

Findings

The $ar{d}-ar{u}$ asymmetry matches Fermilab E866 results.

The $s-ar{s}$ asymmetry is positive at $x>0$ with specific magnitude.

No sign change in asymmetry at large $x$ observed.

Abstract

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 $n$, $\Delta^{++}$, $\Lambda$ and $\Sigma^{*+}$ baryons in $pp$ collisions, we compute the shape of the light antiquark asymmetry $\bar{d}-\bar{u}$ in the proton and the strange asymmetry $s-\bar{s}$ in the nucleon sea. With these constraints, the magnitude of the $\bar{d}-\bar{u}$ 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 $\langle \bar d-\bar u \rangle \approx 0.09-0.17$. The $s-\bar s$ 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 $\langle x (s-\bar s) \rangle \approx (0.9-2.5) \times 10^{-3}$.

Figures

Tables

Table 1. Table 1: Moments Q ϕ ( n − 1 ) superscript subscript 𝑄 italic-ϕ 𝑛 1 Q_{\phi}^{(n-1)} of the quark distributions q ( = u , d , s ) q(=u,d,s) in the pseudoscalar mesons π + superscript 𝜋 \pi^{+} , K + superscript 𝐾 K^{+} and K 0 superscript 𝐾 0 K^{0} . The moments are normalized such that a ( 1 ) = 2 superscript 𝑎 1 2 a^{(1)}=2 .

$\varphi$	$U_{\varphi }^{\left(n-1\right)}$	$D_{\varphi }^{\left(n-1\right)}$	$S_{\varphi }^{\left(n-1\right)}$
${\pi }^{+}$	$\frac{1}{2}{a}^{\left(n\right)}$	$-\frac{1}{2}{a}^{\left(n\right)}$	0
$K^{+}$	$\frac{1}{2}{a}^{\left(n\right)}$	0	$-\frac{1}{2}{a}^{\left(n\right)}$
$K^0$	0	$\frac{1}{2}{a}^{\left(n\right)}$	$-\frac{1}{2}{a}^{\left(n\right)}$

Table 2. Table 2: Moments Q B ( n − 1 ) superscript subscript 𝑄 𝐵 𝑛 1 Q_{B}^{(n-1)} of the unpolarized quark distributions for q = u , d 𝑞 𝑢 𝑑 q=u,d or s 𝑠 s for octet baryons  B 𝐵 B . The spin-dependent moments Δ ​ Q B ( n − 1 ) Δ superscript subscript 𝑄 𝐵 𝑛 1 \Delta Q_{B}^{(n-1)} can be obtained from the entries here by the replacements { α ( n ) → α ¯ ( n ) , β ( n ) → β ¯ ( n ) , σ ( n ) → σ ¯ ( n ) } formulae-sequence → superscript 𝛼 𝑛 superscript ¯ 𝛼 𝑛 formulae-sequence → superscript 𝛽 𝑛 superscript ¯ 𝛽 𝑛 → superscript 𝜎 𝑛 superscript ¯ 𝜎 𝑛 \{\alpha^{(n)}\to\bar{\alpha}^{(n)},\beta^{(n)}\to\bar{\beta}^{(n)},\sigma^{(n)}\to\bar{\sigma}^{(n)}\} .

$B$	$U_B^{\left(n-1\right)}$	$D_B^{\left(n-1\right)}$	$S_B^{\left(n-1\right)}$
$p$	$\frac{5}{6}{\alpha }^{\left(n\right)}+\frac{1}{3}{\beta }^{\left(n\right)}+{\sigma }^{\left(n\right)}$	$\frac{1}{6}{\alpha }^{\left(n\right)}+\frac{2}{3}{\beta }^{\left(n\right)}+{\sigma }^{\left(n\right)}$	${\sigma }^{\left(n\right)}$
$n$	$\frac{1}{6}{\alpha }^{\left(n\right)}+\frac{2}{3}{\beta }^{\left(n\right)}+{\sigma }^{\left(n\right)}$	$\frac{5}{6}{\alpha }^{\left(n\right)}+\frac{1}{3}{\beta }^{\left(n\right)}+{\sigma }^{\left(n\right)}$	${\sigma }^{\left(n\right)}$
${\Sigma }^{+}$	$\frac{5}{6}{\alpha }^{\left(n\right)}+\frac{1}{3}{\beta }^{\left(n\right)}+{\sigma }^{\left(n\right)}$	${\sigma }^{\left(n\right)}$	$\frac{1}{6}{\alpha }^{\left(n\right)}+\frac{2}{3}{\beta }^{\left(n\right)}+{\sigma }^{\left(n\right)}$
${\Sigma }^0$	$\frac{5}{12}{\alpha }^{\left(n\right)}+\frac{1}{6}{\beta }^{\left(n\right)}+{\sigma }^{\left(n\right)}$	$\frac{5}{12}{\alpha }^{\left(n\right)}+\frac{1}{6}{\beta }^{\left(n\right)}+{\sigma }^{\left(n\right)}$	$\frac{1}{6}{\alpha }^{\left(n\right)}+\frac{2}{3}{\beta }^{\left(n\right)}+{\sigma }^{\left(n\right)}$
${\Sigma }^{-}$	${\sigma }^{\left(n\right)}$	$\frac{5}{6}{\alpha }^{\left(n\right)}+\frac{1}{3}{\beta }^{\left(n\right)}+{\sigma }^{\left(n\right)}$	$\frac{1}{6}{\alpha }^{\left(n\right)}+\frac{2}{3}{\beta }^{\left(n\right)}+{\sigma }^{\left(n\right)}$
$\Lambda$	$\frac{1}{4}{\alpha }^{\left(n\right)}+\frac{1}{2}{\beta }^{\left(n\right)}+{\sigma }^{\left(n\right)}$	$\frac{1}{4}{\alpha }^{\left(n\right)}+\frac{1}{2}{\beta }^{\left(n\right)}+{\sigma }^{\left(n\right)}$	$\frac{1}{2}{\alpha }^{\left(n\right)}+{\sigma }^{\left(n\right)}$
$\Lambda {\Sigma }^0$	$\frac{\sqrt{3}}{12}\left[{\alpha }^{\left(n\right)}-2{\beta }^{\left(n\right)}\right]$	$-\frac{\sqrt{3}}{12}\left[{\alpha }^{\left(n\right)}-2{\beta }^{\left(n\right)}\right]$	0
${\Xi }^0$	$\frac{1}{6}{\alpha }^{\left(n\right)}+\frac{2}{3}{\beta }^{\left(n\right)}+{\sigma }^{\left(n\right)}$	${\sigma }^{\left(n\right)}$	$\frac{5}{6}{\alpha }^{\left(n\right)}+\frac{1}{3}{\beta }^{\left(n\right)}+{\sigma }^{\left(n\right)}$
${\Xi }^{-}$	${\sigma }^{\left(n\right)}$	$\frac{1}{6}{\alpha }^{\left(n\right)}+\frac{2}{3}{\beta }^{\left(n\right)}+{\sigma }^{\left(n\right)}$	$\frac{5}{6}{\alpha }^{\left(n\right)}+\frac{1}{3}{\beta }^{\left(n\right)}+{\sigma }^{\left(n\right)}$

Table 3. Table 3: Moments Q T ( n − 1 ) superscript subscript 𝑄 𝑇 𝑛 1 Q_{T}^{(n-1)} of the unpolarized quark distributions for q = u , d 𝑞 𝑢 𝑑 q=u,d or s 𝑠 s for decuplet baryons T 𝑇 T . The results for the spin-dependent moments Δ ​ Q T ( n − 1 ) Δ superscript subscript 𝑄 𝑇 𝑛 1 \Delta Q_{T}^{(n-1)} , can be obtained by the replacements { θ ( n ) → θ ¯ ( n ) , ρ ( n ) → ρ ¯ ( n ) } formulae-sequence → superscript 𝜃 𝑛 superscript ¯ 𝜃 𝑛 → superscript 𝜌 𝑛 superscript ¯ 𝜌 𝑛 \{\theta^{(n)}\to\bar{\theta}^{(n)},\rho^{(n)}\to\bar{\rho}^{(n)}\} .

$T$	$U_T^{\left(n-1\right)}$	$D_T^{\left(n-1\right)}$	$S_T^{\left(n-1\right)}$
${\Delta }^{++}$	${\theta }^{\left(n\right)}+{\rho }^{\left(n\right)}$	${\rho }^{\left(n\right)}$	${\rho }^{\left(n\right)}$
${\Delta }^{+}$	$\frac{2}{3}{\theta }^{\left(n\right)}+{\rho }^{\left(n\right)}$	$\frac{1}{3}{\theta }^{\left(n\right)}+{\rho }^{\left(n\right)}$	${\rho }^{\left(n\right)}$
${\Delta }^0$	$\frac{1}{3}{\theta }^{\left(n\right)}+{\rho }^{\left(n\right)}$	$\frac{2}{3}{\theta }^{\left(n\right)}+{\rho }^{\left(n\right)}$	${\rho }^{\left(n\right)}$
${\Delta }^{-}$	${\rho }^{\left(n\right)}$	${\theta }^{\left(n\right)}+{\rho }^{\left(n\right)}$	${\rho }^{\left(n\right)}$
${\Sigma }^{\ast +}$	$\frac{2}{3}{\theta }^{\left(n\right)}+{\rho }^{\left(n\right)}$	${\rho }^{\left(n\right)}$	$\frac{1}{3}{\theta }^{\left(n\right)}+{\rho }^{\left(n\right)}$
${\Sigma }^{\ast 0}$	$\frac{1}{3}{\theta }^{\left(n\right)}+{\rho }^{\left(n\right)}$	$\frac{1}{3}{\theta }^{\left(n\right)}+{\rho }^{\left(n\right)}$	$\frac{1}{3}{\theta }^{\left(n\right)}+{\rho }^{\left(n\right)}$
${\Sigma }^{\ast -}$	${\rho }^{\left(n\right)}$	$\frac{2}{3}{\theta }^{\left(n\right)}+{\rho }^{\left(n\right)}$	$\frac{1}{3}{\theta }^{\left(n\right)}+{\rho }^{\left(n\right)}$
${\Xi }^{\ast 0}$	$\frac{1}{3}{\theta }^{\left(n\right)}+{\rho }^{\left(n\right)}$	${\rho }^{\left(n\right)}$	$\frac{2}{3}{\theta }^{\left(n\right)}+{\rho }^{\left(n\right)}$
${\Xi }^{\ast -}$	${\rho }^{\left(n\right)}$	$\frac{1}{3}{\theta }^{\left(n\right)}+{\rho }^{\left(n\right)}$	$\frac{2}{3}{\theta }^{\left(n\right)}+{\rho }^{\left(n\right)}$
${\Omega }^{-}$	${\rho }^{\left(n\right)}$	${\rho }^{\left(n\right)}$	${\theta }^{\left(n\right)}+{\rho }^{\left(n\right)}$

Table 4. Table 4: Moments of the unpolarized u 𝑢 u , d 𝑑 d and s 𝑠 s quark distributions in octet baryons B 𝐵 B arising from the B ​ B ​ ϕ ​ ϕ 𝐵 𝐵 italic-ϕ italic-ϕ BB\phi\phi tadpole vertex, as in Fig.  2 (j). The moments U ( tad ) ​ B ​ K 0 ​ K ¯ 0 ( n − 1 ) superscript subscript 𝑈 tad 𝐵 superscript 𝐾 0 superscript ¯ 𝐾 0 𝑛 1 U_{{\rm(tad)}BK^{0}\overline{K}^{0}}^{(n-1)} , D ( tad ) ​ B ​ K + ​ K − ( n − 1 ) superscript subscript 𝐷 tad 𝐵 superscript 𝐾 superscript 𝐾 𝑛 1 D_{{\rm(tad)}BK^{+}K^{-}}^{(n-1)} and S ( tad ) ​ B ​ π + ​ π − ( n − 1 ) superscript subscript 𝑆 tad 𝐵 superscript 𝜋 superscript 𝜋 𝑛 1 S_{{\rm(tad)}B\pi^{+}\pi^{-}}^{(n-1)} are zero for all baryons B 𝐵 B , and are not listed in the table.

$B$	$U_{\left(\mathrm{tad}\right)B\varphi \varphi }^{\left(n-1\right)}$		$D_{\left(\mathrm{tad}\right)B\varphi \varphi }^{\left(n-1\right)}$		$S_{\left(\mathrm{tad}\right)B\varphi \varphi }^{\left(n-1\right)}$
	${\pi }^{+}{\pi }^{-}$	$K^{+}{K}^{-}$	${\pi }^{+}{\pi }^{-}$	$K^0\overline{K^0}$	$K^0\overline{K^0}$	$K^{+}{K}^{-}$
$p$	$-\frac{1}{3}{\alpha }^{\left(n\right)}+\frac{1}{6}{\beta }^{\left(n\right)}$	$-\frac{5}{12}{\alpha }^{\left(n\right)}-\frac{1}{6}{\beta }^{\left(n\right)}$	$\frac{1}{3}{\alpha }^{\left(n\right)}-\frac{1}{6}{\beta }^{\left(n\right)}$	$-\frac{1}{12}{\alpha }^{\left(n\right)}-\frac{1}{3}{\beta }^{\left(n\right)}$	$\frac{1}{12}{\alpha }^{\left(n\right)}+\frac{1}{3}{\beta }^{\left(n\right)}$	$\frac{5}{12}{\alpha }^{\left(n\right)}+\frac{1}{6}{\beta }^{\left(n\right)}$
$n$	$\frac{1}{3}{\alpha }^{\left(n\right)}-\frac{1}{6}{\beta }^{\left(n\right)}$	$-\frac{1}{12}{\alpha }^{\left(n\right)}-\frac{1}{3}{\beta }^{\left(n\right)}$	$-\frac{1}{3}{\alpha }^{\left(n\right)}+\frac{1}{6}{\beta }^{\left(n\right)}$	$-\frac{5}{12}{\alpha }^{\left(n\right)}-\frac{1}{6}{\beta }^{\left(n\right)}$	$\frac{5}{12}{\alpha }^{\left(n\right)}+\frac{1}{6}{\beta }^{\left(n\right)}$	$\frac{1}{12}{\alpha }^{\left(n\right)}+\frac{1}{3}{\beta }^{\left(n\right)}$
${\Sigma }^{+}$	$-\frac{5}{12}{\alpha }^{\left(n\right)}-\frac{1}{6}{\beta }^{\left(n\right)}$	$-\frac{1}{3}{\alpha }^{\left(n\right)}+\frac{1}{6}{\beta }^{\left(n\right)}$	$\frac{5}{12}{\alpha }^{\left(n\right)}+\frac{1}{6}{\beta }^{\left(n\right)}$	$\frac{1}{12}{\alpha }^{\left(n\right)}+\frac{1}{3}{\beta }^{\left(n\right)}$	$-\frac{1}{12}{\alpha }^{\left(n\right)}-\frac{1}{3}{\beta }^{\left(n\right)}$	$\frac{1}{3}{\alpha }^{\left(n\right)}-\frac{1}{6}{\beta }^{\left(n\right)}$
${\Sigma }^0$	0	$-\frac{1}{8}{\alpha }^{\left(n\right)}+\frac{1}{4}{\beta }^{\left(n\right)}$	0	$-\frac{1}{8}{\alpha }^{\left(n\right)}+\frac{1}{4}{\beta }^{\left(n\right)}$	$\frac{1}{8}{\alpha }^{\left(n\right)}-\frac{1}{4}{\beta }^{\left(n\right)}$	$\frac{1}{8}{\alpha }^{\left(n\right)}-\frac{1}{4}{\beta }^{\left(n\right)}$
${\Sigma }^{-}$	$\frac{5}{12}{\alpha }^{\left(n\right)}+\frac{1}{6}{\beta }^{\left(n\right)}$	$\frac{1}{12}{\alpha }^{\left(n\right)}+\frac{1}{3}{\beta }^{\left(n\right)}$	$-\frac{5}{12}{\alpha }^{\left(n\right)}-\frac{1}{6}{\beta }^{\left(n\right)}$	$-\frac{1}{3}{\alpha }^{\left(n\right)}+\frac{1}{6}{\beta }^{\left(n\right)}$	$\frac{1}{3}{\alpha }^{\left(n\right)}-\frac{1}{6}{\beta }^{\left(n\right)}$	$-\frac{1}{12}{\alpha }^{\left(n\right)}-\frac{1}{3}{\beta }^{\left(n\right)}$
$\Lambda$	0	$\frac{1}{8}{\alpha }^{\left(n\right)}-\frac{1}{4}{\beta }^{\left(n\right)}$	0	$\frac{1}{8}{\alpha }^{\left(n\right)}-\frac{1}{4}{\beta }^{\left(n\right)}$	$-\frac{1}{8}{\alpha }^{\left(n\right)}+\frac{1}{4}{\beta }^{\left(n\right)}$	$-\frac{1}{8}{\alpha }^{\left(n\right)}+\frac{1}{4}{\beta }^{\left(n\right)}$
${\Xi }^0$	$-\frac{1}{12}{\alpha }^{\left(n\right)}-\frac{1}{3}{\beta }^{\left(n\right)}$	$\frac{1}{3}{\alpha }^{\left(n\right)}-\frac{1}{6}{\beta }^{\left(n\right)}$	$\frac{1}{12}{\alpha }^{\left(n\right)}+\frac{1}{3}{\beta }^{\left(n\right)}$	$\frac{5}{12}{\alpha }^{\left(n\right)}+\frac{1}{6}{\beta }^{\left(n\right)}$	$-\frac{5}{12}{\alpha }^{\left(n\right)}-\frac{1}{6}{\beta }^{\left(n\right)}$	$-\frac{1}{3}{\alpha }^{\left(n\right)}+\frac{1}{6}{\beta }^{\left(n\right)}$
${\Xi }^{-}$	$\frac{1}{12}{\alpha }^{\left(n\right)}+\frac{1}{3}{\beta }^{\left(n\right)}$	$\frac{5}{12}{\alpha }^{\left(n\right)}+\frac{1}{6}{\beta }^{\left(n\right)}$	$-\frac{1}{12}{\alpha }^{\left(n\right)}-\frac{1}{3}{\beta }^{\left(n\right)}$	$\frac{1}{3}{\alpha }^{\left(n\right)}-\frac{1}{6}{\beta }^{\left(n\right)}$	$-\frac{1}{3}{\alpha }^{\left(n\right)}+\frac{1}{6}{\beta }^{\left(n\right)}$	$-\frac{5}{12}{\alpha }^{\left(n\right)}-\frac{1}{6}{\beta }^{\left(n\right)}$

Table 5. Table 5: Moments of the unpolarized u 𝑢 u , d 𝑑 d and s 𝑠 s quark distributions from the Kroll-Ruderman vertex for transitions from a proton initial state to octet and decuplet baryon intermediate states, as in Figs.  2 (e) and (h), respectively.

$B\varphi$	$U_{\left(\mathrm{KR}\right)B}^{\left(n-1\right)}$	$D_{\left(\mathrm{KR}\right)B}^{\left(n-1\right)}$	$S_{\left(\mathrm{KR}\right)B}^{\left(n-1\right)}$
$n{\pi }^{+}$	$-\frac{\sqrt{2}}{3}{\overline{\alpha }}^{\left(n\right)}+\frac{\sqrt{2}}{6}{\overline{\beta }}^{\left(n\right)}$	$\frac{\sqrt{2}}{3}{\overline{\alpha }}^{\left(n\right)}-\frac{\sqrt{2}}{6}{\overline{\beta }}^{\left(n\right)}$	0
${\Sigma }^0{K}^{+}$	$\frac{1}{12}{\overline{\alpha }}^{\left(n\right)}+\frac{1}{3}{\overline{\beta }}^{\left(n\right)}$	0	$-\frac{1}{12}{\overline{\alpha }}^{\left(n\right)}-\frac{1}{3}{\overline{\beta }}^{\left(n\right)}$
${\Sigma }^{+}{K}^0$	0	$\frac{\sqrt{2}}{12}{\overline{\alpha }}^{\left(n\right)}+\frac{\sqrt{2}}{3}{\overline{\beta }}^{\left(n\right)}$	$-\frac{1}{12}{\overline{\alpha }}^{\left(n\right)}-\frac{1}{3}{\overline{\beta }}^{\left(n\right)}$
${\Lambda }^0{K}^{+}$	$\frac{\sqrt{3}}{4}{\overline{\alpha }}^{\left(n\right)}$	0	$-\frac{\sqrt{3}}{4}{\overline{\alpha }}^{\left(n\right)}$
$T\varphi$	$U_{\left(\mathrm{KR}\right)T}^{\left(n-1\right)}$	$D_{\left(\mathrm{KR}\right)T}^{\left(n-1\right)}$	$S_{\left(\mathrm{KR}\right)T}^{\left(n-1\right)}$
${\Delta }^0{\pi }^{+}$	$\frac{1}{\sqrt{6}}{\overline{\omega }}^{\left(n\right)}$	$-\frac{1}{\sqrt{6}}{\overline{\omega }}^{\left(n\right)}$	0
${\Delta }^{++}{\pi }^{-}$	$\frac{1}{\sqrt{2}}{\overline{\omega }}^{\left(n\right)}$	$-\frac{1}{\sqrt{2}}{\overline{\omega }}^{\left(n\right)}$	0
${\Sigma }^{\ast 0}{K}^{+}$	$\frac{1}{2\sqrt{3}}{\overline{\omega }}^{\left(n\right)}$	0	$-\frac{1}{2\sqrt{3}}{\overline{\omega }}^{\left(n\right)}$
${\Sigma }^{\ast +}{K}^0$	0	$-\frac{1}{\sqrt{6}}{\overline{\omega }}^{\left(n\right)}$	$\frac{1}{\sqrt{6}}{\overline{\omega }}^{\left(n\right)}$

Table 6. Table 6: Moments Δ ​ Q B ​ T ( n − 1 ) Δ subscript superscript 𝑄 𝑛 1 𝐵 𝑇 \Delta Q^{(n-1)}_{BT} of the polarized u 𝑢 u , d 𝑑 d and s 𝑠 s quark distributions from the axial octet-decuplet transition.

$BT$	$\Delta U_{BT}^{\left(n-1\right)}$	$\Delta D_{BT}^{\left(n-1\right)}$	$\Delta S_{BT}^{\left(n-1\right)}$
$n{\Delta }^0$	$-\frac{1}{\sqrt{3}}{\overline{\omega }}^{\left(n\right)}$	$\frac{1}{\sqrt{3}}{\overline{\omega }}^{\left(n\right)}$	0
$p{\Delta }^{+}$	$-\frac{1}{\sqrt{3}}{\overline{\omega }}^{\left(n\right)}$	$\frac{1}{\sqrt{3}}{\overline{\omega }}^{\left(n\right)}$	0
${\Sigma }^0{\Sigma }^{\ast 0}$	$-\frac{1}{2\sqrt{3}}{\overline{\omega }}^{\left(n\right)}$	$-\frac{1}{2\sqrt{3}}{\overline{\omega }}^{\left(n\right)}$	$\frac{1}{\sqrt{3}}{\overline{\omega }}^{\left(n\right)}$
${\Sigma }^{+}{\Sigma }^{\ast +}$	$\frac{1}{\sqrt{3}}{\overline{\omega }}^{\left(n\right)}$	0	$-\frac{1}{\sqrt{3}}{\overline{\omega }}^{\left(n\right)}$
${\Sigma }^{-}{\Sigma }^{\ast -}$	0	$-\frac{1}{\sqrt{3}}{\overline{\omega }}^{\left(n\right)}$	$\frac{1}{\sqrt{3}}{\overline{\omega }}^{\left(n\right)}$
$\Lambda {\Sigma }^{\ast 0}$	$\frac{1}{2}{\overline{\omega }}^{\left(n\right)}$	$-\frac{1}{2}{\overline{\omega }}^{\left(n\right)}$	0

Table 7. Table 7: Contributions to the integral ⟨ d ¯ − u ¯ ⟩ ≡ ∫ 0 1 𝑑 x ​ ( d ¯ − u ¯ ) delimited-⟨⟩ ¯ 𝑑 ¯ 𝑢 superscript subscript 0 1 differential-d 𝑥 ¯ 𝑑 ¯ 𝑢 \langle\bar{d}-\bar{u}\rangle\equiv\int_{0}^{1}dx(\bar{d}-\bar{u}) from the π ​ N 𝜋 𝑁 \pi N rainbow, π ​ Δ 𝜋 Δ \pi\Delta rainbow and π 𝜋 \pi bubble diagrams in Fig.  1 , for the best fit parameters Λ π ​ N = 1.0 ​ ( 1 ) subscript Λ 𝜋 𝑁 1.0 1 \Lambda_{\pi N}=1.0(1)  GeV and Λ π ​ Δ = 0.9 ​ ( 1 ) subscript Λ 𝜋 Δ 0.9 1 \Lambda_{\pi\Delta}=0.9(1)  GeV. The contributions from the various terms in Eqs. ( 45 )–( 48 ) are listed individually, as are the combined contributions from x > 0 𝑥 0 x>0 and x = 0 𝑥 0 x=0 , and the local and nonlocal terms, to the total. Note that some numbers do not sum to the totals because of rounding.

diagram		$⟨\overline{d}-\overline{u}⟩$
$\pi N$ (rbw)	$f_N^{\left(\mathrm{on}\right)}$	${0.152}_{-0.030}^{+0.032}$
	$f_{\pi }^{\left(\delta \right)}$	$-\left({0.079}_{-0.018}^{+0.020}\right)$
	$\delta f_{\pi }^{\left(\delta \right)}$	${0.044}_{-0.009}^{+0.010}$
total $\pi N$		${0.116}_{-0.022}^{+0.022}$
$\pi \Delta$ (rbw)	$f_{\Delta }^{\left(\mathrm{on}\right)}$	$-\left({0.044}_{-0.012}^{+0.012}\right)$
	$f_{\Delta }^{\left(\mathrm{on}\mathrm{end}\right)}$	$-\left({0.009}_{-0.003}^{+0.004}\right)$
	$f_{\Delta }^{\left(\delta \right)}$	${0.002}_{-0.001}^{+0.001}$
	$f_{\pi }^{\left(\delta \right)}$	${0.039}_{-0.010}^{+0.010}$
	$\delta f_{\pi }^{\left(\delta \right)}$	$-\left({0.022}_{-0.005}^{+0.005}\right)$
total $\pi \Delta$		$-\left({0.033}_{-0.010}^{+0.010}\right)$
$\pi$ (bub)	$f_{\pi }^{\left(\delta \right)}$	${0.099}_{-0.022}^{+0.025}$
	$\delta f_{\pi }^{\left(\delta \right)}$	$-\left({0.054}_{-0.012}^{+0.013}\right)$
total $\pi$ bubble		${0.044}_{-0.010}^{+0.012}$
total
$x>0$		${0.099}_{-0.046}^{+0.047}$
$x=0$		${0.028}_{-0.007}^{+0.008}$
local		${0.159}_{-0.039}^{+0.041}$
nonlocal		$-\left({0.032}_{-0.008}^{+0.008}\right)$

Table 8. Table 8: Contributions from octet Y = Λ , Σ 0 , Σ + 𝑌 Λ superscript Σ 0 superscript Σ Y=\Lambda,\Sigma^{0},\Sigma^{+} and decuplet Y ∗ = Σ ∗ 0 , Σ ∗ + superscript 𝑌 superscript Σ absent 0 superscript Σ absent Y^{*}=\Sigma^{*0},\Sigma^{*+} hyperons to the average number (in units of 10 − 2 superscript 10 2 10^{-2} ) and momentum carried (in units of 10 − 3 superscript 10 3 10^{-3} ) by s 𝑠 s and s ¯ ¯ 𝑠 \bar{s} quarks in the nucleon from diagrams in Figs.  1 and 2 , for dipole regulator mass parameters Λ K ​ Y = 1.1 ​ ( 1 ) subscript Λ 𝐾 𝑌 1.1 1 \Lambda_{KY}=1.1(1)  GeV and Λ K ​ Y ∗ = 0.8 ​ ( 1 ) subscript Λ 𝐾 superscript 𝑌 0.8 1 \Lambda_{KY^{*}}=0.8(1)  GeV. Note that some of the numbers do not sum to the totals because of rounding.

		$⟨\overline{s}⟩$	$⟨x\overline{s}⟩$			$⟨s⟩$	$⟨xs⟩$
		$(\times {10}^{-2})$	$(\times {10}^{-3})$			$(\times {10}^{-2})$	$(\times {10}^{-3})$
$KY\left(\mathrm{rbw}\right)$	$f_Y^{\left(\mathrm{on}\right)}$	${1.39}_{-0.54}^{+0.69}$	${1.33}_{-0.56}^{+0.74}$	$YK\left(\mathrm{rbw}\right)$	$f_Y^{\left(\mathrm{on}\right)}$	${1.39}_{-0.54}^{+0.69}$	${1.67}_{-0.63}^{+0.78}$
	$f_K^{\left(\delta \right)}$	$-\left({1.66}_{-0.63}^{+0.79}\right)$	0		$f_Y^{\left(\mathrm{off}\right)}$	$-\left({4.01}_{-1.42}^{+1.68}\right)$	$-\left({5.35}_{-1.83}^{+2.12}\right)$
	$\delta f_K^{\left(\delta \right)}$	${1.12}_{-0.41}^{+0.50}$	0		$\delta f_Y^{\left(\mathrm{off}\right)}$	${2.70}_{-0.92}^{+1.07}$	${3.12}_{-1.02}^{+1.13}$
					$f_K^{\left(\delta \right)}$	${1.66}_{-0.63}^{+0.79}$	${2.82}_{-1.07}^{+1.35}$
				$YK\left(\mathrm{KR}\right)$	$f_Y^{\left(\mathrm{off}\right)}$	${4.01}_{-1.42}^{+1.68}$	${6.29}_{-2.15}^{+2.50}$
					$f_K^{\left(\delta \right)}$	$-\left({3.31}_{-1.26}^{+1.58}\right)$	$-\left({6.66}_{-2.53}^{+3.18}\right)$
				$YK\left(\delta \mathrm{KR}\right)$	$\delta f_Y^{\left(\mathrm{off}\right)}$	$-\left({2.70}_{-0.92}^{+1.07}\right)$	$-\left({3.68}_{-1.20}^{+1.33}\right)$
					$\delta f_K^{\left(\delta \right)}$	${1.12}_{-0.41}^{+0.50}$	${2.24}_{-0.82}^{+1.01}$
total octet		${0.85}_{-0.32}^{+0.40}$	${1.33}_{-0.56}^{+0.74}$	total octet		${0.85}_{-0.32}^{+0.40}$	${0.46}_{-0.14}^{+0.14}$
$K\left(\mathrm{bub}\right)$	$f_K^{\left(\delta \right)}$	${4.85}_{-1.84}^{+2.32}$	0	$K\left(\mathrm{tad}\right)$	$f_K^{\left(\delta \right)}$	${4.85}_{-1.84}^{+2.32}$	${7.87}_{-2.98}^{+3.76}$
	$\delta f_K^{\left(\delta \right)}$	$-\left({3.27}_{-1.20}^{+1.47}\right)$	0	$K\left(\delta \mathrm{tad}\right)$	$\delta f_K^{\left(\delta \right)}$	$-\left({3.27}_{-1.20}^{+1.47}\right)$	$-\left({5.30}_{-1.94}^{+2.38}\right)$
total bubble		${1.59}_{-0.64}^{+0.85}$	0	total tadpole		${1.59}_{-0.64}^{+0.85}$	${2.57}_{-1.04}^{+1.38}$
$K{Y}^{\ast }\left(\mathrm{rbw}\right)$	$f_{Y^{\ast }}^{\left(\mathrm{on}\right)}$	${0.09}_{-0.07}^{+0.13}$	${0.06}_{-0.04}^{+0.09}$	$Y^{\ast }K\left(\mathrm{rbw}\right)$	$f_{Y^{\ast }}^{\left(\mathrm{on}\right)}$	${0.09}_{-0.07}^{+0.13}$	${0.10}_{-0.08}^{+0.14}$
	$f_{Y^{\ast }}^{\left(\mathrm{on}\mathrm{end}\right)}$	${0.04}_{-0.03}^{+0.07}$	${0.03}_{-0.03}^{+0.06}$		$f_{Y^{\ast }}^{\left(\mathrm{on}\mathrm{end}\right)}$	${0.04}_{-0.03}^{+0.07}$	${0.04}_{-0.03}^{+0.07}$
	$f_{Y^{\ast }}^{\left(\delta \right)}$	$-\left({0.01}_{-0.01}^{+0.01}\right)$	0		$f_{Y^{\ast }}^{\left(\mathrm{off}\right)}$	$-\left({0.59}_{-0.42}^{+0.72}\right)$	$-\left({0.75}_{-0.52}^{+0.89}\right)$
	$f_K^{\left(\delta \right)}$	$-\left({0.15}_{-0.11}^{+0.20}\right)$	0		$f_{Y^{\ast }}^{\left(\mathrm{off}\mathrm{end}\right)}$	${0.17}_{-0.12}^{+0.23}$	${0.21}_{-0.15}^{+0.29}$
	$\delta f_K^{\left(\delta \right)}$	${0.11}_{-0.08}^{+0.14}$	0		$\delta f_{Y^{\ast }}^{\left(\mathrm{off}\right)}$	${0.34}_{-0.24}^{+0.45}$	${0.38}_{-0.27}^{+0.47}$
					$f_K^{\left(\delta \right)}$	${0.18}_{-0.13}^{+0.24}$	${0.26}_{-0.19}^{+0.34}$
					$f_{Y^{\ast }}^{\left(\delta \right)}$	${0.01}_{-0.01}^{+0.01}$	${0.01}_{-0.01}^{+0.02}$
					$\delta f_{Y^{\ast }}^{\left(\delta \right)}$	$-\left({0.07}_{-0.05}^{+0.11}\right)$	$-\left({0.10}_{-0.07}^{+0.16}\right)$
				$Y^{\ast }K\left(\mathrm{KR}\right)$	$f_{Y^{\ast }}^{\left(\mathrm{off}\right)}$	${0.59}_{-0.42}^{+0.72}$	${1.02}_{-0.71}^{+1.21}$
					$f_{Y^{\ast }}^{\left(\mathrm{off}\mathrm{end}\right)}$	$-\left({0.17}_{-0.12}^{+0.23}\right)$	$-\left({0.29}_{-0.21}^{+0.39}\right)$
					$f_K^{\left(\delta \right)}$	$-\left({0.34}_{-0.24}^{+0.44}\right)$	$-\left({0.65}_{-0.47}^{+0.85}\right)$
					$f_{Y^{\ast }}^{\left(\delta \right)}$	$-\left({0.02}_{-0.01}^{+0.03}\right)$	$-\left({0.03}_{-0.02}^{+0.05}\right)$
					$\delta f_{Y^{\ast }}^{\left(\delta \right)}$	${0.05}_{-0.03}^{+0.08}$	${0.09}_{-0.07}^{+0.15}$
				$Y^{\ast }K\left(\delta \mathrm{KR}\right)$	$\delta f_{Y^{\ast }}^{\left(\mathrm{off}\right)}$	$-\left({0.34}_{-0.24}^{+0.45}\right)$	$-\left({0.51}_{-0.36}^{+0.63}\right)$
					$\delta f_K^{\left(\delta \right)}$	${0.11}_{-0.08}^{+0.14}$	${0.22}_{-0.15}^{+0.27}$
					$\delta f_{Y^{\ast }}^{\left(\delta \right)}$	${0.02}_{-0.02}^{+0.04}$	${0.05}_{-0.03}^{+0.07}$
total decuplet		${0.08}_{-0.06}^{+0.12}$	${0.09}_{-0.07}^{+0.15}$	total decuplet		${0.08}_{-0.06}^{+0.12}$	${0.04}_{-0.03}^{+0.04}$
total		${2.51}_{-1.02}^{+1.36}$	${1.42}_{-0.62}^{+0.89}$	total		${2.51}_{-1.02}^{+1.36}$	${3.08}_{-1.20}^{+1.55}$
non $\delta$-function		${1.52}_{-0.64}^{+0.88}$	${1.42}_{-0.62}^{+0.89}$	non $\delta$-function		${1.52}_{-0.64}^{+0.88}$	${2.25}_{-0.92}^{+1.21}$
$\delta$-function		${0.99}_{-0.50}^{+0.60}$	0	$\delta$-function		${0.99}_{-0.50}^{+0.60}$	${0.82}_{-0.63}^{+0.67}$
local		${4.55}_{-1.77}^{+2.23}$	${1.42}_{-0.62}^{+0.89}$	local		${4.55}_{-1.77}^{+2.23}$	${6.58}_{-2.60}^{+3.14}$
nonlocal		$-\left({2.04}_{-0.92}^{+1.04}\right)$	0	nonlocal		$-\left({2.04}_{-0.92}^{+1.04}\right)$	$-\left({3.50}_{-1.46}^{+1.66}\right)$

Equations

$$Q^{(n-1)}$$

$$⟨N(p)|{\mathcal{O}}_q^{{\mu }_1\cdots {\mu }_n}|N(p)⟩=2Q^{(n-1)}{p}^{{\mu }_1}\cdots p^{{\mu }_n},$$

$${\mathcal{O}}_q^{{\mu }_1\cdots {\mu }_n}=i^{n-1}\bar{q}{\gamma }^{\{{\mu }_1}{\overleftrightarrow{D}}^{{\mu }_2}\cdots {\overleftrightarrow{D}}^{{\mu }_n\}}q,$$

$${\mathcal{O}}_q^{{\mu }_1\cdots {\mu }_n}=\sum _j{c}_{q/j}^{(n)}{\mathcal{O}}_j^{{\mu }_1\cdots {\mu }_n},$$

$$c_{q/j}^{(n)}$$

$$⟨N(p)|{\mathcal{O}}_j^{{\mu }_1\cdots {\mu }_n}|N(p)⟩=2f_j^{(n)}{p}^{\{{\mu }_1}\cdots p^{{\mu }_n\}},$$

$$f_j^{(n)}$$

$$q(x)$$

$$\begin{split}\bar{q}(x)&=\sum\limits_{B,T,\phi}\big{[}\big{(}f^{\rm(rbw)}_{\phi B}+f^{\rm(rbw)}_{\phi T}+f^{\rm(bub)}_{\phi}\big{)}\otimes\bar{q}_{\phi}\big{]}(x),\end{split}$$

$$\begin{split}q(x)=&\ Z_{2}\,q^{(0)}(x)\ +\ \sum\limits_{B,T,\phi}\bigg{\{}\big{[}\big{(}f^{\rm(rbw)}_{\phi B}+f^{\rm(rbw)}_{\phi T}+f^{\rm(bub)}_{\phi}\big{)}\otimes q_{\phi}\big{]}(x)\\ &+\ \big{[}{\bar{f}}^{\rm(rbw)}_{B\phi}\otimes q_{B}\big{]}(x)+\ \big{[}{\bar{f}}^{\rm(KR)}_{B}\otimes q^{\rm(KR)}_{B}\big{]}(x)+\ \big{[}\delta{\bar{f}}^{\rm(KR)}_{B}\otimes q^{(\delta)}_{B}\big{]}(x)\\ &+\ \big{[}{\bar{f}}^{\rm(rbw)}_{T\phi}\otimes q_{T}\big{]}(x)+\ \big{[}{\bar{f}}^{\rm(KR)}_{T}\otimes q^{\rm(KR)}_{T}\big{]}(x)+\ \big{[}\delta{\bar{f}}^{\rm(KR)}_{T}\otimes q^{(\delta)}_{T}\big{]}(x)\\ &+\ \big{[}{\bar{f}}^{\rm(tad)}_{\phi}\otimes q^{\rm(tad)}_{\phi}\big{]}(x)+\ \big{[}\delta{\bar{f}}^{\rm(tad)}_{\phi}\otimes q^{(\delta)}_{\phi}\big{]}(x)\bigg{\}},\end{split}$$

$$\begin{split}\mathcal{O}^{\mu_{1}\cdots\mu_{n}}_{q}&=a^{(n)}i^{n}\frac{f_{\phi}^{2}}{4}\Big{\{}\mathrm{Tr}\left[U^{\dagger}\lambda^{q}\partial_{\mu_{1}}\cdots\partial_{\mu_{n}}U\right]+\mathrm{Tr}\left[U\lambda^{q}\partial_{\mu_{1}}\cdots\partial_{\mu_{n}}U^{\dagger}\right]\Big{\}}\\ &+\Big{[}\alpha^{(n)}(\overline{\cal B}\gamma^{\mu_{1}}{\cal B}\lambda^{q}_{+})+\beta^{(n)}(\overline{\cal B}\gamma^{\mu_{1}}\lambda^{q}_{+}{\cal B})+\sigma^{(n)}(\overline{\cal B}\gamma^{\mu_{1}}{\cal B})\,\mathrm{Tr}[\lambda^{q}_{+}]\Big{]}\,p^{\mu_{2}}\cdots p^{\mu_{n}}\\ &+\Big{[}\bar{\alpha}^{(n)}(\overline{\cal B}\gamma^{\mu_{1}}\gamma_{5}{\cal B}\lambda^{q}_{-})+\bar{\beta}^{(n)}(\overline{\cal B}\gamma^{\mu_{1}}\gamma_{5}\lambda^{q}_{-}{\cal B})+\bar{\sigma}^{(n)}(\overline{\cal B}\gamma^{\mu_{1}}\gamma_{5}{\cal B})\,\mathrm{Tr}[\lambda^{q}_{-}]\Big{]}\,p^{\mu_{2}}\cdots p^{\mu_{n}}\\ &+\Big{[}\theta^{(n)}(\overline{\cal T}_{\alpha}\gamma^{\alpha\beta\mu_{1}}\lambda^{q}_{+}{\cal T}_{\beta})+\rho^{(n)}(\overline{\cal T}_{\alpha}\gamma^{\alpha\beta\mu_{1}}{\cal T}_{\beta})\,{\rm Tr}[\lambda^{q}_{+}]\Big{]}\,p^{\mu_{2}}\cdots p^{\mu_{n}}\\ &+\Big{[}\bar{\theta}^{(n)}(\overline{\cal T}_{\alpha}\gamma^{\mu_{1}}\gamma_{5}\lambda^{q}_{-}{\cal T}^{\alpha})+\bar{\rho}^{(n)}(\overline{\cal T}_{\alpha}\gamma^{\mu_{1}}\gamma_{5}{\cal T}^{\alpha}\,{\rm Tr}[\lambda^{q}_{-}]\Big{]}\,p^{\mu_{2}}\cdots p^{\mu_{n}}\\ &-\sqrt{\frac{3}{2}}\,\bar{\omega}^{(n)}\Big{[}(\overline{\cal B}\Theta^{\mu_{1}\mu}\lambda^{q}_{-}{\cal T}_{\mu})+(\overline{\cal T}_{\mu}\Theta^{\mu\mu_{1}}\lambda^{q}_{-}{\cal B})\Big{]}\,p^{\mu_{2}}\cdots p^{\mu_{n}}\\ &+\mathrm{permutations}-\mathrm{Tr},\end{split}$$

$$U=\exp \left(i\frac{\sqrt{2}\varphi }{f_{\varphi }}\right),$$

$$\lambda_{\pm}^{q}=\frac{1}{2}\big{(}u\lambda^{q}u^{\dagger}\pm u^{\dagger}\lambda^{q}u\big{)}\,,$$

$$\Theta^{\mu\nu}=g^{\mu\nu}-\big{(}Z+\tfrac{1}{2}\big{)}\gamma^{\mu}\gamma^{\nu}.$$

$$\begin{split}\mathcal{O}^{\mu_{1}\cdots\mu_{n}}_{\Delta q}&=\Big{[}\bar{\alpha}^{(n)}(\overline{\cal B}\gamma^{\mu_{1}}\gamma_{5}{\cal B}\lambda^{q}_{+})+\bar{\beta}^{(n)}(\overline{\cal B}\gamma^{\mu_{1}}\gamma_{5}\lambda^{q}_{+}{\cal B})+\bar{\sigma}^{(n)}(\overline{\cal B}\gamma^{\mu_{1}}\gamma_{5}{\cal B})\,\mathrm{Tr}[\lambda^{q}_{+}]\Big{]}\,p^{\mu_{2}}\cdots p^{\mu_{n}}\\ &+\Big{[}\alpha^{(n)}(\overline{\cal B}\gamma^{\mu_{1}}{\cal B}\lambda^{q}_{-})+\beta^{(n)}(\overline{\cal B}\gamma^{\mu_{1}}\lambda^{q}_{-}{\cal B})+\sigma^{(n)}(\overline{\cal B}\gamma^{\mu_{1}}{\cal B})\,\mathrm{Tr}[\lambda^{q}_{-}]\Big{]}\,p^{\mu_{2}}\cdots p^{\mu_{n}}\\ &+\Big{[}\bar{\theta}^{(n)}(\overline{\cal T}_{\alpha}\gamma^{\mu_{1}}\gamma_{5}\lambda^{q}_{+}{\cal T}^{\alpha})+\bar{\rho}^{(n)}(\overline{\cal T}_{\alpha}\gamma^{\mu_{1}}\gamma_{5}{\cal T}^{\alpha}){\rm Tr}[\lambda^{q}_{+}]\Big{]}\,p^{\mu_{2}}\cdots p^{\mu_{n}}\\ &+\Big{[}\theta^{(n)}(\overline{\cal T}_{\alpha}\gamma^{\alpha\beta\mu_{1}}\lambda^{q}_{-}{\cal T}_{\beta})+\rho^{(n)}(\overline{\cal T}_{\alpha}\gamma^{\alpha\beta\mu_{1}}{\cal T}_{\beta}){\rm Tr}[\lambda^{q}_{-}]\Big{]}\,p^{\mu_{2}}\cdots p^{\mu_{n}}\\ &-\sqrt{\frac{3}{2}}\,\bar{\omega}^{(n)}\Big{[}(\overline{\cal B}\Theta^{\mu_{1}\mu}\lambda^{q}_{+}{\cal T}_{\mu})+(\overline{\cal T}_{\mu}\Theta^{\mu\mu_{1}}\lambda^{q}_{+}{\cal B})\Big{]}\,p^{\mu_{2}}\cdots p^{\mu_{n}}\\ &+\mathrm{permutations}-\mathrm{Tr}.\end{split}$$

$$(\overline{\mathcal{B}}\mathcal{B})$$

$$(\overline{\mathcal{B}}\mathcal{B}A)$$

$$(\overline{\mathcal{B}}A\mathcal{B})$$

$$({\overline{\mathcal{T}}}_{\alpha }{\mathcal{T}}_{\beta })$$

$$({\overline{\mathcal{T}}}_{\alpha }A{\mathcal{T}}_{\beta })$$

$$({\overline{\mathcal{T}}}_{\alpha }A\mathcal{B})$$

$$\begin{array}{rl}{\mathcal{O}}_q^{{\mu }_1\cdots {\mu }_n}& =Q_{\varphi }^{(n-1)}{\mathcal{O}}_{\varphi }^{{\mu }_1\cdots {\mu }_n}+Q_B^{(n-1)}{\mathcal{O}}_{B{B}^{\prime }}^{{\mu }_1\cdots {\mu }_n}+Q_T^{(n-1)}{\mathcal{O}}_{T{T}^{\prime }}^{{\mu }_1\cdots {\mu }_n}\\ & +Q_{(\mathrm{tad})B\varphi \varphi }^{(n-1)}{\mathcal{O}}_{B\varphi \varphi }^{{\mu }_1\cdots {\mu }_n}+Q_{(\mathrm{KR})B}^{(n-1)}{\mathcal{O}}_{B{B}^{\prime }\varphi }^{{\mu }_1\cdots {\mu }_n}+Q_{(\mathrm{KR})T}^{(n-1)}{\mathcal{O}}_{BT\varphi }^{{\mu }_1\cdots {\mu }_n}.\end{array}$$

$${\mathcal{O}}_{\varphi }^{{\mu }_1\cdots {\mu }_n}$$

$${\mathcal{O}}_{B{B}^{\prime }}^{{\mu }_1\cdots {\mu }_n}$$

$${\mathcal{O}}_{T{T}^{\prime }}^{{\mu }_1\cdots {\mu }_n}$$

$${\mathcal{O}}_{B\varphi \varphi }^{{\mu }_1\cdots {\mu }_n}$$

$${\mathcal{O}}_{B{B}^{\prime }\varphi }^{{\mu }_1\cdots {\mu }_n}$$

$${\mathcal{O}}_{BT\varphi }^{{\mu }_1\cdots {\mu }_n}$$

$$Q_{\varphi }^{(n-1)}$$

$$Q_B^{(n-1)}$$

$$Q_T^{(n-1)}$$

$$Q_{(\mathrm{tad})B\varphi \varphi }^{(n-1)}$$

$$Q_{(\mathrm{KR})B}^{(n-1)}$$

$$Q_{(\mathrm{KR})T}^{(n-1)}$$

$$a^{(1)}=2.$$

$$U_{{\pi }^{+}}^{(n-1)}=-D_{{\pi }^{+}}^{(n-1)}=U_{K^{+}}^{(n-1)}=-S_{K^{+}}^{(n-1)}=D_{K^0}^{(n-1)}=-S_{K^0}^{(n-1)}=\frac{1}{2}{a}^{(n)}.$$

$${\alpha }^{(n)}$$

Full text

††preprint: JLAB-THY-19-2996, ADP-19-15/T1095

I Introduction

It is well known that a complete characterization of nucleon substructure must go beyond three valence quarks. One of the great challenges of modern hadron physics is to unravel the precise role of hidden flavors in the structure of the nucleon. The observation of the $\bar{d}-\bar{u}$ flavor asymmetry in the light quark sea of the proton Arneodo:1994sh ; Ackerstaff:1998sr ; Baldit ; Towell:2001nh , following its prediction by Thomas a decade earlier Thomas:1983fh on the basis of chiral symmetry breaking Thomas:1982kv ; Thomas:1981vc , has been one of the seminal results in hadronic physics over the past two decades. It has led to a major reevaluation of our understanding of the role of the non-valence components of the nucleon and their origin in QCD Geesaman:2018ixo .

The role that strange quarks, in particular, play in the nucleon has also been the focus of attention in hadronic physics for many years. Early polarized deep-inelastic scattering (DIS) experiments suggested that a surprisingly large fraction of the proton’s spin might be carried by strange quarks Ashman , in contrast to the naive quark model expectations Ellis . One of the guiding principles for understanding the nonperturbative features of strange quarks and antiquarks in the nucleon sea has been chiral symmetry breaking in QCD. While the generation of $s\bar{s}$ pairs through perturbative gluon radiation typically produces symmetric $s$ and $\bar{s}$ distributions (at least up to two loop corrections Catani04 ), any significant difference between the momentum dependence of the $s$ and $\bar{s}$ distributions would be a clear signal of nonperturbative effects Signal87 ; Malheiro97 ; Sufian18 .

In the previous paper nonlocal-I , we presented the proton $\to$ pseudoscalar meson ($\phi$) $+$ baryon splitting functions for the intermediate octet ($B$) and decuplet ($T$) baryon configurations in nonlocal chiral effective theory He1 ; He2 . From the calculated splitting functions, the parton distribution functions (PDFs) of the nucleon are obtained as convolutions of these with PDFs of the intermediate state mesons and baryons Chen02 ; XGWangPLB ; XGWangPRD . Here we apply the results from nonlocal-I to compute, for the first time within the nonlocal theory, sea quark PDF asymmetries in the proton, including the light antiquark flavor asymmetry $\bar{d}-\bar{u}$ and the strange quark asymmetry $s-\bar{s}$. Using SU(3) relations for the intermediate state hadron PDFs, the only parameters in the calculation of the asymmetries are the mass parameters appearing in the ultraviolet regulator functions. These will be determined by fitting cross section data from inclusive baryon production in high energy $pp$ scattering, using the same splitting functions that appear in the PDF asymmetries.

We begin in Sec. II by summarizing the convolution formulas for the quark and antiquark PDFs in terms of the fluctuations of the nucleon into its meson-baryon light-cone components. The calculation of the PDFs of the intermediate state baryons and mesons in the chiral theory is discussed in detail in Sec. III. Numerical results for the sea quark asymmetries are presented in Sec. IV, where we compare the results for $\bar{d}-\bar{u}$ with those extracted from Drell-Yan and semi-inclusive DIS measurements, and compare predictions for $s-\bar{s}$ asymmetries with some recent PDF parametrizations. Finally, in Sec. V we summarize our results and discuss future measurements which could further constrain the PDF asymmetries experimentally.

II Convolution formulas

Using the crossing symmetry properties of the spin-averaged PDFs, $q(-x)=-\bar{q}(x)$, the $n$-th Mellin moment ($n\geq 1$) of the distribution for a given flavor $q$ ($q=u,d,s,\ldots$) is defined by

[TABLE]

In the operator product expansion, the moments $Q^{(n-1)}$ are related to matrix elements of local twist-two, spin-$n$ operators $\mathcal{O}_{q}^{\mu_{1}\cdots\mu_{n}}$ between nucleon states with momentum $p$,

[TABLE]

where the operators are given by

[TABLE]

with \overleftrightarrow{D}=\frac{1}{2}\big{(}\overrightarrow{D}-\overleftarrow{D}\big{)}, and the braces $\{\,\cdots\}$ denote symmetrization of Lorentz indices. The effective theory allows the quark operators ${\cal O}_{q}$ to be matched to hadronic operators ${\cal O}_{j}$ with the same quantum numbers Chen02 ,

[TABLE]

where the coefficients $c^{(n)}_{q/j}$ are the $n$-th moments of the PDF $q_{j}(x)$ in the hadronic configuration $j$,

[TABLE]

The nucleon matrix elements of the hadronic operators ${\cal O}_{j}^{\mu_{1}\cdots\mu_{n}}$ are given in terms of moments of the splitting functions $f_{j}(y)$,

[TABLE]

where

[TABLE]

with $y$ the light-cone momentum fraction of the nucleon carried by the hadronic state $j$. The operator relation in Eq. (4) then gives rise to the convolution formula for the PDFs Chen02 ; XGWangPRD ,

[TABLE]

where $q_{j}^{v}\equiv q_{j}-\bar{q}_{j}$ is the valence distribution for the quark flavor $q$ in the hadron $j$. The complete set of splitting functions $f_{j}(y)$ for octet and decuplet baryons is given in Ref. nonlocal-I .

In the present analysis we work under the basic assumption that the bare baryon states are composed of three valence quarks plus quark-antiquark pairs that are generated perturbatively through gluon radiation. Such contributions will effectively cancel in any differences of PDFs, such as $\bar{d}-\bar{u}$ or $s-\bar{s}$. We therefore focus only on the nonperturbative contributions to sea quark PDFs which arise from pseudoscalar meson loops. In this approximation antiquark distributions arise only from diagrams involving direct coupling to mesons, as in the meson rainbow and bubble diagrams in Fig. 1. The meson loop contribution to the antiquark PDFs in the nucleon can then be written as

[TABLE]

where $f^{\rm(rbw)}_{\phi B}$ and $f^{\rm(rbw)}_{\phi T}$ represent splitting functions from the rainbow diagrams with octet and decuplet baryons in Fig. 1(a) and (b), respectively, $f^{\rm(bub)}_{\phi}$ is the splitting function for the meson bubble diagram in Fig. 1(c), and ${\bar{q}}_{\phi}(x)$ is the antiquark PDF in the meson.

Contributions to quark PDFs can in principle come from both meson coupling and baryon coupling diagrams. The latter are illustrated in Fig. 2, and involve the bare nucleon coupling [Fig. 2(a)], wave function renormalization [Fig. 2(b) and (c), with octet and decuplet baryon intermediate states, respectively], baryon rainbow [Fig. 2(d) and (g)], Kroll-Ruderman [Fig. 2(e) and (h)], and meson tadpole [Fig. 2(j)] diagrams, along with gauge link dependent Kroll-Ruderman [Fig. 2(f) and (i)] and tadpole [Fig. 2(k)] diagrams. Within the valence approximation, all of these diagrams will contribute to the $u$ and $d$ quarks in the nucleon. However, for the strange quark the bare coupling and wave function renormalization diagrams do not contribute. The total nonperturbative contribution from meson loops to the quark PDF in the nucleon can then be written

[TABLE]

where $q^{(0)}$ is the quark PDF in the bare nucleon, and the wave function renormalization $Z_{2}$ arises from the summation over the diagrams in Figs. 2(a)–(c) Ji:2013bca . Following Ref. XGWangPRD , we will work in terms of the same momentum fraction $y$ for all meson and baryon coupling diagrams in Figs. 1 and 2. Using the same definition of the convolution integral as in Eq. (8), it will be convenient therefore to define for each of the splitting functions in Eq. (10) involving the coupling to baryons the shorthand notation ${\bar{f}}_{j}(y)\equiv f_{j}(1-y)$ [see Sec. IV.2 below]. Explicit expressions for the splitting functions $f^{\rm(rbw)}_{B\phi}$, $f^{\rm(KR)}_{B}$, $\delta f^{\rm(KR)}_{B}$, $f^{\rm(rbw)}_{T\phi}$, $f^{\rm(KR)}_{T}$, $\delta f^{\rm(KR)}_{T}$, $f^{\rm(tad)}_{\phi}$ and $\delta f^{\rm(tad)}_{\phi}$, which represent the diagrams in Figs. 2(d)–(k), respectively, are given in Ref. nonlocal-I . The corresponding quark PDFs for the intermediate state octet and decuplet baryons are discussed in the next section.

III Bare baryon and meson PDFs

To calculate the contributions to the quark and antiquark distributions in the proton in the convolution formulas (9) and (10) requires the proton $\to$ baryon $+$ meson splitting functions and the PDFs of the baryons and mesons to which the current couples. The full set of splitting functions was presented in our previous paper, Ref. nonlocal-I . In this section we derive the (valence) PDFs of the bare baryon and meson intermediate states using the same chiral SU(3) EFT framework that was used to compute the splitting functions.

III.1 Operators and moments

In the effective theory the quark level operators are matched to a sum of hadronic level operators whose matrix elements [see Eq. (4)] are given by the moments of the splitting functions, as in Eq. (6). Identifying all possible contributions from octet and decuplet baryon intermediate states that transform as vectors, the most general expression for the quark vector operator $\mathcal{O}^{\mu_{1}\cdots\mu_{n}}_{q}$ is given by XGWangPRD ; Shanahan:2013xw

[TABLE]

where “Tr” denotes traces over Lorentz indices. In the first term of Eq. (11), the operator $U$ represents pseudoscalar meson fields $\phi$,

[TABLE]

where $f_{\phi}$ is the pseudoscalar decay constant, and the coefficients $a^{(n)}$ are related to moments of quark and antiquark PDFs in the pseudoscalar mesons. The flavor operators $\lambda_{\pm}^{q}$ are defined by

[TABLE]

where $\lambda^{q}={\rm diag}(\delta_{qu},\delta_{qd},\delta_{qs})$ are diagonal $3\times 3$ quark flavor matrices.

In the remaining terms of Eq. (11), the operators $\cal B$ and ${\cal T}_{\alpha}$ represent octet and decuplet baryon fields, respectively, and we define the Dirac tensors $\gamma^{\alpha\beta\rho}=\frac{1}{2}\{\gamma^{\mu\nu},\gamma^{\alpha}\}$ and $\gamma^{\mu\nu}=\frac{1}{2}[\gamma^{\mu},\gamma^{\nu}]$. The coefficients $\{\alpha^{(n)},\beta^{(n)},\sigma^{(n)}\}$ and $\{\bar{\alpha}^{(n)},\bar{\beta}^{(n)},\bar{\sigma}^{(n)}\}$ are related to moments of the spin-averaged and spin-dependent PDFs in octet baryons, while $\{\theta^{(n)},\rho^{(n)}\}$ and $\{\bar{\theta}^{(n)},\bar{\rho}^{(n)}\}$ are related to moments of spin-averaged and spin-dependent PDFs in decuplet baryons, respectively. The coefficients $\bar{\omega}^{(n)}$ are given in terms of moments of spin-dependent octet–decuplet transition PDFs, where the octet–decuplet transition tensor operator $\Theta^{\mu\nu}$ is defined as

[TABLE]

Here $Z$ is the decuplet off-shell parameter, and since physical quantities do not depend on $Z$, it is convenient to choose $Z=1/2$ to simplify the form of the spin-3/2 propagator Hacker:2005fh ; Nath:1971wp .

For the Kroll-Ruderman diagrams in Fig. 2(e), (f), (h) and (i), the presence of the pseudoscalar field at the vertex introduces hadronic axial vector operators, whose contribution to the quark axial vector operator can in general be written as

[TABLE]

From the transformation properties of the operators $\mathcal{O}^{\mu_{1}\cdots\mu_{n}}_{q}$ and $\mathcal{O}^{\mu_{1}\cdots\mu_{n}}_{\Delta q}$ under parity Moiseeva13 , the sets of coefficients $\{\alpha^{(n)},\beta^{(n)},\sigma^{(n)},\theta^{(n)},\rho^{(n)}\}$ and $\{\bar{\alpha}^{(n)},\bar{\beta}^{(n)},\bar{\sigma}^{(n)},\bar{\theta}^{(n)},\bar{\rho}^{(n)}\}$ in (15) are the same as those in the spin-averaged operators in (11).

The operators $\cal B$ and $\cal T_{\alpha}$ appearing in Eqs. (11) and (15) can be written in terms of the SU(3) baryon octet fields $B^{ij}$ (which include $N$, $\Lambda$, $\Sigma$ and $\Xi$ fields) and decuplet baryon fields $T_{\alpha}^{ijk}$ (which include $\Delta$, $\Sigma^{*}$, $\Xi^{*}$ and $\Omega$ fields) using the relations Labrenz:1996jy ; XGWangPRD

[TABLE]

and

[TABLE]

where $\varepsilon^{klm}$ is the antisymmetric tensor. Applying the relations (16) and (17), the vector operator $\mathcal{O}^{\mu_{1}\cdots\mu_{n}}_{q}$ in Eq. (11) can then be more intuitively rearranged in the form

[TABLE]

The individual vector hadronic operators in (18) are given by

[TABLE]

and correspond to the insertions in the diagrams of Figs. 1, 2(d), 2(g), 2(j), 2(e), and 2(h), respectively. The coefficients $Q_{j}^{(n-1)}$ of each of the operators are defined in terms of Mellin moments of the corresponding parton distributions in the intermediate mesons and baryons, as in Eq. (5),

[TABLE]

where the PDFs correspond to those appearing in the convolution expressions in Eqs. (9) and (10). Each of the moments $Q^{(n-1)}_{j}$ can be expressed in terms of the coefficients $\{a^{(n)};\alpha^{(n)},\beta^{(n)},\sigma^{(n)};\theta^{(n)},\rho^{(n)}\}$ appearing in Eq. (11), as discussed below.

In particular, for the meson PDFs, the contributions to the $U_{\phi}^{(n-1)}$, $D_{\phi}^{(n-1)}$ and $S_{\phi}^{(n-1)}$ moments are listed in Table 1 for the $\phi=\pi^{+},K^{+}$ and $K^{0}$ mesons. Conservation of valence quark number fixes the normalization of the $n=1$ moment of the meson distribution, such that

[TABLE]

Note that in the SU(3) symmetric limit, the $u$-quark moments in $\pi^{+}$ and $K^{+}$ are equivalent, as are the $s$-quark moments in $K^{+}$ and $K^{0}$, while the $d$-quark moments in $\pi^{+}$ and $K^{0}$ have equal magnitude but opposite sign,

[TABLE]

The results for other charge states ($\pi^{-}$, $\pi^{0}$, $K^{-}$ and $\overline{K}^{0}$) are obtained from those in Table 1 using charge symmetry. Unlike in baryons, the sea quark distributions in mesons are flavor symmetric. In the simplest valence quark models the sea quark distributions in pions and kaons are zero.

For the moments of the quark PDFs in the intermediate state baryons, the contributions from the $u$, $d$ and $s$ flavors to the octet baryon moments $Q_{B}^{(n-1)}$ are given in terms of combinations of $\{\alpha^{(n)},\beta^{(n)},\sigma^{(n)}\}$ and listed in Table 2 for baryons $B=p,n,\Sigma^{\pm,0},\Lambda,\Xi^{-,0}$ as well as for the $\Lambda$-$\Sigma^{0}$ interference. Solving for the coefficients, one can write these as linear combinations of the individual $u$, $d$ and $s$ quark moments in the proton,

[TABLE]

Assuming the strangeness in the intermediate state nucleon to be zero (or equivalently, that the $u$ content of $\Sigma^{-}$, for example, vanishes), one finds for the lowest ($n=1$) moments,

[TABLE]

For the quark PDFs in the decuplet baryon intermediate states $T$, the moments $Q_{T}^{(n-1)}$ for the individual $u$, $d$ and $s$ flavors are given in terms of combinations of $\{\theta^{(n)},\rho^{(n)}\}$, and are listed in Table 3 for $T=\Delta,\Sigma^{*},\Xi^{*}$ and $\Omega^{-}$. Solving for the coefficients $\theta^{(n)}$ and $\rho^{(n)}$ in terms of the moments in the $\Delta^{+}$ baryon, one has

[TABLE]

Again, assuming zero strangeness in the $\Delta^{+}$, the $n=1$ moments are given by

[TABLE]

For the moments of the distributions generated by the tadpole diagrams in Fig. 2(j), in Table 4 we list the contributions $Q^{(n-1)}_{{\rm(tad)}B\phi\phi}$ for the $u$, $d$ and $s$ flavors in each octet baryon $B$. Note that the combinations involving $K^{0}\overline{K}^{0}$ do not contribute to the $u$-quark moments, those involving $K^{+}K^{-}$ do not contribute to the $d$-quark moments, and the contributions from $\pi^{+}\pi^{-}$ to the $s$-quark moments are also zero.

Finally, to complete the set of the contributions to the unpolarized PDFs, in Table 5 we list the moments $Q_{{\rm(KR)}B}^{(n-1)}$ and $Q_{{\rm(KR)}T}^{(n-1)}$ of the Kroll-Ruderman induced quark distributions from Fig. 2(e) and (h), for the transitions from a proton initial state to intermediate states including octet $B$ and decuplet $T$ baryons, respectively. (Similar results can be derived for other octet or decuplet baryon initial states, but are not listed here to avoid unnecessary detail.) Note that, unlike for all other contributions from the diagrams in Fig. 2, the moments $Q_{{\rm(KR)}B}^{(n-1)}$ and $Q_{{\rm(KR)}T}^{(n-1)}$ are given in terms of the coefficients $\bar{\alpha}^{(n)}$, $\bar{\beta}^{(n)}$ and $\bar{\omega}^{(n)}$, which are related to moments of the spin-dependent parton distributions.

For the latter, recall that spin-dependent PDFs are related to matrix elements of the axial vector operators $\mathcal{O}^{\mu_{1}\cdots\mu_{n}}_{\Delta q}$ in Eq. (15), which, using the relations (16) and (17), can be expanded in terms of hadronic axial vector operators with coefficients given by moments $\Delta Q^{(n-1)}_{j}$ of the spin-dependent distributions. In analogy to the expansion in Eq. (18), we therefore expand the axial vector operators as

[TABLE]

where only the operators relevant for the calculation of unpolarized PDFs are listed. (The remaining terms not listed in Eq. (27) will be relevant for the calculation of spin-dependent PDFs in the proton XDWang_spin .) More explicitly, the axial vector hadronic operators in (27) are given by

[TABLE]

with the corresponding moments $\Delta Q_{j}^{(n-1)}$ of the spin-dependent PDFs defined by

[TABLE]

For simplicity, in Eq. (29) we restrict ourselves to the diagonal octet ($B=B^{\prime}$) and diagonal decuplet ($T=T^{\prime}$) cases, with respective spin-dependent PDFs $\Delta q_{B}(x)$ and $\Delta q_{T}(x)$, and the octet–decuplet transition distribution, $\Delta q_{BT}(x)$. In particular, the moments $\Delta Q_{B}^{(n-1)}$ of the spin-dependent PDFs in octet baryons can be obtained from the entries in Table 2 by substituting $\{\alpha^{(n)}\to\bar{\alpha}^{(n)},\beta^{(n)}\to\bar{\beta}^{(n)},\sigma^{(n)}\to\bar{\sigma}^{(n)}\}$, while the moments $\Delta Q_{T}^{(n-1)}$ of the spin-dependent PDFs in decuplet baryons are obtained from Table 3 with the replacements $\{\theta^{(n)}\to\bar{\theta}^{(n)},\rho^{(n)}\to\bar{\rho}^{(n)}\}$. For the octet-decuplet axial transition distribution, the moments, $\Delta Q^{(n-1)}_{BT}$, are given in terms of the coefficient $\bar{\omega}$ in Eq. (15) and are listed in Table 6. Solving for the octet coefficients in terms of the moments of the spin-dependent proton PDFs in the proton, one has, in analogy with Eq. (23), the relations

[TABLE]

Similarly, for the decuplet case, the coefficients $\bar{\theta}^{(n)}$ and $\bar{\rho}^{(n)}$ can be written in terms of the moments of the spin-dependent PDFs of quarks in the $\Delta^{+}$ bayron,

[TABLE]

The moments of the octet–decuplet transition operators can be related to the moments of the octet baryon operators via the SU(3) relation

[TABLE]

for all $n$. For the $n=1$ octet baryon moments, in particular, the coefficients are given in terms of axial vector charges $F$ and $D$,

[TABLE]

In terms of moments of the spin-dependent proton PDFs, for the octet–decuplet transition vertex, $\bar{\omega}^{(1)}$ is given by the SU(3) symmetry relation Shanahan:2013xw ,

[TABLE]

which also reproduces the relation ${\cal C}=-2D$ between the meson–octet–decuplet baryon coupling ${\cal C}$ and the meson-octet coupling $D$ Jenkins:1991ts . Note that through Eq. (32) the quark distributions in the Kroll-Ruderman diagrams with decuplet baryon intermediate states in Fig. 2(h) are related to the spin-dependent distributions of quarks in proton.

This completes the discussion of the moments of the PDFs of the various mesons and baryons appearing in the intermediate states in the diagrams of Fig. 2. From these, in the next section we derive relations for the $x$ dependence of the PDFs themselves.

III.2 SU(3) relations for baryon and meson PDFs

In the previous section we derived relations between the coefficients of the various operators in $\mathcal{O}^{\mu_{1}\cdots\mu_{n}}_{q}$ and $\mathcal{O}^{\mu_{1}\cdots\mu_{n}}_{\Delta q}$ and the $n$-th Mellin moments of the quark distributions in Eqs. (22)–(26) and Eqs. (30)–(34). Since these relations are valid for all moments $n$, one can derive from them explicit expressions for the $x$ dependence of the PDFs.

For the valence distributions in the pion and kaon, from Eq. (22) and Table 1 one has

[TABLE]

for all values of $x$. For the PDFs in the baryons, to simplify notations we shall label the bare distributions in the proton without an explicit baryon subscript, $q(x)\equiv q_{p}(x)$, and those in the $\Delta^{+}$ baryon by $q_{\Delta}(x)\equiv q_{\Delta^{+}}(x)$. Starting with the quark distributions in the SU(3) octet baryons, from Table 2 the individual $u$-, $d$- and $s$-quark flavor PDFs can be written in terms of the proton PDFs as

[TABLE]

For the quark distributions in the SU(3) decuplet baryons, from Table 3 the $u$-, $d$- and $s$-quark PDFs can be written in terms of the PDFs in the $\Delta^{+}$ as

[TABLE]

In our actual numerical calculations, for simplicity we approximate $q_{\Delta}(x)\approx q(x)$, and assume valence quark dominance for the bare states, so that $s(x)\approx s_{\Delta}(x)\approx 0$.

For the PDFs arising from the tadpole diagrams in Fig. 2(j), from Table 4 the $u$-, $d$- and $s$-quark distributions can be written as

[TABLE]

The distributions associated with the tadpole gauge link diagrams in Fig. 2(g) turn out to be the same as those for the regular tadpole diagrams,

[TABLE]

Turning now to the Kroll-Ruderman diagrams in Fig. 2(e) and 2(h), for a proton initial state the corresponding PDFs are expressed in terms of spin-dependent PDFs in the proton, $\Delta q(x)\equiv\Delta q_{p}(x)$. From Table 5, for the octet baryon intermediate states the $u$-, $d$- and $s$-quark distributions are given by

[TABLE]

Similarly, for the decuplet baryon intermediate states the individual quark flavor Kroll-Ruderman distributions are given by

[TABLE]

The PDFs associated with the KR gauge link diagrams in Figs. 2(f) and 2(i) are the same as those for the regular KR diagrams,

[TABLE]

With this set of distributions in the SU(3) octet and decuplet baryons and mesons, and the proton $\to$ meson $+$ baryon splitting functions from Ref. nonlocal-I , we can finally proceed with the computation of the meson loop contributions to the quark and antiquark PDFs in the proton, as in Eqs. (9) and (10). In the following section we focus on the calculation of specific PDF asymmetries in the proton numerically.

IV Sea quark asymmetries in the proton

To illustrate the calculation of the contributions to PDFs from pseudoscalar meson loops within the nonlocal chiral effective theory framework, we consider the examples of the flavor asymmetry in the light antiquark sea in the proton, $\bar{d}-\bar{u}$, and the strange–antistrange asymmetry in the nucleon, $s-\bar{s}$. In both quantities perturbatively generated contributions from gluon radiation effectively cancel, at least up to next-to-next-to-leading order corrections in $\alpha_{s}$ Catani04 , so that observation of large asymmetries may be indicative of nonperturbative effects Signal87 ; Malheiro97 ; Sufian18 .

For the numerical calculation of the meson–bayron splitting functions, earlier work used various regularization prescriptions, including sharp transverse momentum cutoffs, Pauli-Villars regularization, as well as phenomenological vertex form factors Holtmann96 ; MST98 ; Burkardt13 ; Salamu15 ; XGWangPLB ; XGWangPRD . At times the prescriptions have been imposed in rather ad hoc ways, without necessarily ensuring that the relevant symmetries, such as Lorentz, chiral, and local gauge symmetries, are necessarily respected. In the present work we for the first time perform the calculation within nonlocal regularization, which is consistent with all of the above symmetry requirements. An advantage of the nonlocal method is that only a single parameter, $\Lambda$, is needed to regulate all of the on-shell, off-shell and $\delta$ functions associated with each of the diagrams in Figs. 1 and 2.

Following Ref. nonlocal-I , in the present analysis we adopt a dipole shape in the meson virtuality $k^{2}$ for the regulator functions for the one-loop contributions, parametrized by a cutoff parameter $\Lambda$,

[TABLE]

where $D_{\Lambda}=k^{2}-\Lambda^{2}+i\varepsilon$. The cutoff $\Lambda$ can be determined by fitting the calculated meson-exchange cross section to differential cross sections data for inclusive baryon production in high-energy $pp$ scattering, $pp\to BX$, for different species of baryon $B$. Summing over the particles $X$ in the final state, the differential inclusive baryon production cross sections can be written as

[TABLE]

where $E$ is the incident proton energy and $\bar{y}\equiv 1-y$ is the longitudinal momentum fraction of the incident proton carried by the produced baryon $B$. In Eq. (44) we have used the fact that for spin-averaged scattering the differential cross section is independent of the azimuthal angle. Available data exist on inclusive neutron and $\Delta^{++}$ production Flauger:1976ju ; Blobel:1978yj ; Barish:1975uv , as well as on $\Lambda$ and $\Sigma^{*+}$ production Blobel:1978yj ; Jaeger:1974pk ; Bockmann:1977sc in the hyperon sector. In principle, the cutoffs may depend on the baryon $B$, although within the SU(3) symmetry framework we do not expect large variations among the different $\Lambda$ values.

Once the cutoffs are determined and the one-loop splitting functions are fixed, these can then be convoluted with the various meson and baryon PDFs in Eqs. (9) and (10) to compute the contributions to the PDFs in the proton. In the numerical calculations the input PDFs of the pion and kaon are taken from Aicher et al. Aicher:2010cb . The spin-averaged PDFs of the proton are from Ref. Martin:1998sq , while the spin-dependent PDFs are taken from Ref. Leader:2010rb . Since the valence pion and proton PDFs are reasonably well determined, at least compared with the sea quark distributions, using other pion Barry:2018ort ; Owens84 ; SMRS92 ; GRV92 ; Wijesooriya05 or proton JMO13 ; ForteWatt13 parametrizations will not lead to significant differences.

IV.1 $\bar{d}-\bar{u}$ asymmetry

Turning to the light antiquark asymmetry in the proton sea, within the chiral effective theory framework the primary source of the asymmetry is the meson rainbow and bubble diagrams in Fig. 1. In this approximation the $\bar{d}-\bar{u}$ difference does not depend directly on the structure of the baryon coupling diagrams in Fig. 2, but only on the splitting functions and the substructure of the pion. More specifically, from Eq. (9) one can write the contribution to the $\bar{d}-\bar{u}$ difference in the proton as the convolution

[TABLE]

where the first (octet rainbow) term in the brackets is from Fig. 1(a), the second and third (decuplet rainbow) terms correspond to Fig. 1(b), and the fourth (bubble) term is from Fig. 1(c). Using the notations of Ref. nonlocal-I , the splitting functions in Eq. (45) for the rainbow and bubble diagrams can be expressed in terms of octet and decuplet basis functions. In particular, for the $\pi N$ configuration the function $f^{\rm(rbw)}_{\pi^{+}n}$ is given by a sum of nucleon on-shell and $\delta$-function contributions,

[TABLE]

where $D$ and $F$ are the SU(3) flavor coefficients, and $f=93$ MeV is the pseudoscalar meson decay constant. Explicit forms for the basis functions are given in Ref. nonlocal-I for the dipole regulator $\widetilde{F}(k)$ in Eq. (43). The on-shell function $f^{\rm(on)}_{N}$ is nonzero for $y>0$, while the local $f^{(\delta)}_{\pi}$ and nonlocal $\delta f^{(\delta)}_{\pi}$ functions are proportional to $\delta(y)$ nonlocal-I , and therefore contribute to the $\bar{d}-\bar{u}$ asymmetry only at $x=0$ Burkardt13 ; Salamu15 . In the pointlike limit, in which the form factor cutoff $\Lambda\to\infty$, the nonlocal function $\delta f^{(\delta)}_{\pi}$ vanishes; however, at finite $\Lambda$ values it remains nonzero.

For the $\pi\Delta$ contributions to the asymmetry in Eq. (45), the splitting function for the rainbow diagram in Fig. 1(b) includes several regular and $\delta$-function terms,

[TABLE]

where $\overline{M}=M+M_{\Delta}$, and ${\cal C}$ is the meson-octet-decuplet baryon coupling, which is related to the $\pi N\Delta$ coupling constant $g_{\pi N\Delta}$ by ${\cal C}=\sqrt{2}f\,g_{\pi N\Delta}$ Salamu15 ; nonlocal-I . As for the $\pi N$ case, the on-shell function for the $\Delta$ intermediate state, $f^{\rm(on)}_{\Delta}$, is nonzero for $y>0$, with a shape that is qualitatively similar to $f^{\rm(on)}_{N}$, but peaking at smaller $y$ because of the positive $\Delta$–nucleon mass difference nonlocal-I ; Holtmann96 ; MST98 . The on-shell end-point function, $f^{\rm(on\,end)}_{\Delta}$, also has a similar shape for finite $\Lambda$, but in the $\Lambda\to\infty$ limit is associated with an end-point singularity that gives a $\delta$-function at $y=1$. The off-shell components of the $\Delta$ propagator induce several terms that are proportional to $\delta$-functions at $y=0$. The functions $f^{(\delta)}_{\pi}$ and $\delta f^{(\delta)}_{\pi}$ are equivalent to those in Eq. (46), while $f^{(\delta)}_{\Delta}$ is a new function that appears only for the decuplet intermediate state Salamu15 .

Finally, the bubble diagram contribution to the $\bar{d}-\bar{u}$ asymmetry, $f^{\rm(bub)}_{\pi}$, in Fig. 1(c) is given by the same combination of basis $\delta$-function contributions as for the rainbow diagrams,

[TABLE]

Although this term gives a nonzero PDF only at $x=0$, since it contributes to the integral of $\bar{d}-\bar{u}$, it will indirectly affect the normalization for $x>0$. On the other hand, experimental cross sections are in practice available only for $x>0$, so that the $\delta$-function pieces are generally difficult to constrain directly, especially in regularization schemes that use different regulator parameters for the $\delta$-function and $y>0$ contributions Salamu15 . The advantage of the nonlocal approach employed here is that by consistently introducing a vertex form factor in coordinate space in the nonlocal Lagrangian nonlocal-I , the same regulator function then appears in all splitting functions derived from the fundamental interaction, which in our case is parametrized through the single cutoff $\Lambda$. Even if experimental data constrain only contributions at $x>0$, such as from the on-shell functions, once determined these can then be used to compute other contributions, including those at $x=0$.

Following Refs. Holtmann96 ; XGWangPLB ; XGWangPRD , we can constrain the parameter $\Lambda$ for the octet intermediate states by comparing the one-pion exchange contribution with the differential cross section for the inclusive charge-exchange process $pp\to nX$ at $y>0$,

[TABLE]

where $s$ is the invaraiant mass squared of the reaction. The function $\hat{f}_{N}^{(\rm on)}(y,k_{\bot}^{2})$ in Eq. (49) is the unintegrated on-shell nucleon splitting function, which is related to the corresponding integrated splitting function $f_{N}^{\rm(on)}(y)$ in Eq. (46) by (see also Eq. (63) in Ref. nonlocal-I )

[TABLE]

The cross section $\sigma_{\rm tot}^{\pi^{+}p}(ys)$ in Eq. (49) is the total $\pi^{+}p$ scattering cross section evaluated at the center of mass energy $ys$. In the numerical calculations, we use the (approximately energy independent) empirical value $\sigma_{\rm tot}^{\pi^{+}p}=23.8(1)$ mb Hufner:1992cu . For the SU(3) couplings we take $D=0.85$ and $F=0.41$, which gives a triplet axial charge $g_{A}=(2\bar{\alpha}^{(1)}-\bar{\beta}^{(1)})/3=D+F=1.26$ and an octet axial charge $g_{8}=\bar{\alpha}^{(1)}+\bar{\beta}^{(1)}=3F-D=0.38$.

The results for the differential neutron production cross section are shown in Fig. 3 versus $\bar{y}$. The experimental data are typically presented as a function of the ratio $2p_{L}/\sqrt{s}$, where $p_{L}$ is the longitudinal momentum of the produced baryon in the center of mass frame; at high energies, however, this is equivalent to $\bar{y}$. In Fig. 3(a) we compare our results with the neutron production data from the ISR at CERN at energies $\sqrt{s}$ between $\approx 31$ and 63 GeV for $0^{\circ}$ neutron production angles, or $k_{\bot}^{2}=0$ Flauger:1976ju . Data from the hydrogen bubble chamber experiment at the CERN proton synchrotron at $\sqrt{s}\approx 5$ and 7 GeV Blobel:1978yj are shown in Fig. 3(b) for the $k_{\bot}$-integrated neutron cross section. Because the pion-exchange processes is dominant only at large $\bar{y}$ McKenney16 , with contributions from background processes such as the exchange of heavier mesons MT93 ; Holtmann96 ; Kopeliovich12 becoming more important at lower $\bar{y}$, we include data only in the region $\bar{y}>0.7$. Corrections from rescattering and absorption are also known to play a role in inclusive hadron production, and are estimated to be around 20% at high values of $\bar{y}$ D'Alesio00 ; Kopeliovich12 ; H1_01 . A good description of the single and double differential neutron data can be achieved with a cutoff parameter $\Lambda_{\pi N}=1.0(1)$ GeV. A marginally larger value is found if fitting only the double differential data, and slightly smaller value for just the $k_{\bot}$-integrated cross section, but consistent within the uncertainties.

For the inclusive production of decuplet baryons, the invariant differential cross section for an inclusive $\Delta^{++}$ in the final state can be written for $y>0$ as

[TABLE]

where $\sigma_{\rm tot}^{p\pi^{-}}$ is the total $\pi^{-}p$ scattering cross section. In our numerical calculations we assume this to be charge independent, so that $\sigma_{\rm tot}^{\pi^{-}p}\approx\sigma_{\rm tot}^{\pi^{+}p}$, and for the coupling constant ${\cal C}$ we take the SU(6) symmetric value ${\cal C}=-2D=-1.72$. The functions $\hat{f}_{\Delta}^{(\rm on)}(y,k_{\bot}^{2})$ and $\hat{f}_{\Delta}^{(\rm on\,end)}(y,k_{\bot}^{2})$ in (51) are the unintegrated decuplet on-shell and on-shell end-point splitting functions, which are related to the corresponding integrated splitting functions (see Eqs. (86)–(88) in nonlocal-I ) by the identities

[TABLE]

respectively. The $k_{\bot}^{2}$-integrated $\Delta^{++}$ cross section is shown in Fig. 3(c) compared with hydrogen bubble chamber data taken at Fermilab for $\sqrt{s}\approx 20$ GeV Barish:1975uv . A good fit to the data is obtained with a value of the decuplet cutoff of $\Lambda_{\pi\Delta}=0.9(1)$ GeV, which is slightly smaller than that for the neutron production cross sections.

To examine the model dependence of the analysis, for comparison we also fitted the hadron production cross sections in Figs. 3(a)–(c) using instead the Pauli-Villars regularization for the local effective theory XGWangPRD ; XGWangPLB . The explicit forms of the Pauli-Villars regularized octet on-shell splitting functions can be found in XGWangPRD ; XGWangPLB . The result for the sum of the decuplet on-shell and on-shell end-point functions is as in Eq. (96) of XGWangPRD , with the integral regularized by a factor $(1+4D_{\pi\Delta}/D_{\Lambda\Delta})$, where $D_{\pi\Delta}$ and $D_{\Lambda\Delta}$ are momentum dependent functions given in Eq. (86) of XGWangPRD . The results for the best fit Pauli-Villars mass parameters $\Lambda_{\pi N}^{\rm(PV)}=0.30$ GeV and $\Lambda_{\pi\Delta}^{\rm(PV)}=0.64$ GeV are illustrated by the dashed curves in Fig. 3, and are similar to those for the nonlocal calculation. While there is some difference in the shape of the calculated $k_{\perp}$-integrated neutron production cross section in Fig. 3(b) at smaller values of $\bar{y}$, in the region where the data provide constraints the Pauli-Villars results lie within the uncertainty bands of the nonlocal curves.

Using the values of $\Lambda_{\pi N}$ and $\Lambda_{\pi\Delta}$ for our nonlocal calculation constrained by the $pp$ cross sections in Fig. 3, we next evaluate the flavor asymmetry $\bar{d}-\bar{u}$ from the convolution of the splitting functions and the pion PDF in Eq. (45). The results for $x(\bar{d}-\bar{u})$ are shown in Fig. 4, and compared with the asymmetry extracted from the E866 Drell-Yan lepton-pair production data from Fermilab Towell:2001nh . At nonzero $x$ values only the on-shell nucleon and $\Delta$ and end-point $\Delta$ terms contribute to the asymmetry, each of which is indicated in Fig. 4. The positive nucleon on-shell term makes the largest contribution, which is partially cancelled by the negative $\Delta$ contributions. For the values of the cutoffs used here, the end-point term is relatively small compared with the on-shell $\Delta$ component.

Although the $\delta$-function contributions to the flavor asymmetry are not directly visible in Fig. 4, their effect can be seen in the lowest moment of the asymmetry,

[TABLE]

The contributions from the individual on-shell, end-point and $\delta$-function components of the $\pi N$ and $\pi\Delta$ rainbow and the $\pi$ bubble diagrams to the moment are shown in Fig. 5 versus the dipole cutoff parameter $\Lambda$ ($=\Lambda_{\pi N}$ or $\Lambda_{\pi\Delta}$), for the approximate ranges of values found in the fits in Fig. 3. For the best fit values $\Lambda_{\pi N}=1.0(1)$ GeV and $\Lambda_{\pi\Delta}=0.9(1)$ GeV, the contributions from the individual terms in Eqs. (45)–(48) are listed in Table IV.1, along with the combined contributions from the $x>0$ and $x=0$ terms, and the local and nonlocal terms, to the total integrated result. The nucleon on-shell term is the most important component, with a contribution that is within $\approx 20\%$ of the total integrated value $\langle\bar{d}-\bar{u}\rangle=0.127^{+0.044}_{-0.042}$, where the errors here reflect the uncertainties on the cutoff parameters. The on-shell and end-point $\pi\Delta$ terms yield overall negative contributions, with magnitude $\approx 30\%$ of the on-shell $\pi N$. The various $\delta$-function terms from all three diagrams in Fig. 1 cancel to a considerable degree, with the $x=0$ contribution making up $\approx 20\%$ of the total. Furthermore, the breakdown into the local and nonlocal pieces shows that the latter is negative with magnitude $\approx 20\%$ of the local.