Parton distributions from nonlocal chiral SU(3) effective theory: Splitting functions

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Parton distributions from nonlocal chiral SU(3) effective theory: Splitting functions

Y. Salamu, Chueng-Ryong Ji, W. Melnitchouk, A. W. Thomas, and P. Wang

Phys. Rev. D 99, 014041 – Published 29 January 2019

Abstract

We present a new formulation of pseudoscalar meson loop corrections to nucleon parton distributions within a nonlocal covariant chiral effective field theory, including contributions from SU(3) octet and decuplet baryons. The nonlocal Lagrangian, constrained by requirements of local gauge invariance and Lorentz-invariant ultraviolet regularization, generates additional interactions associated with gauge links. We use these to compute the full set of $proton \to meson + baryon$ splitting functions, which in general contain on-shell and off-shell contributions, in addition to $δ$ -function terms at zero momentum, along with nonlocal contributions associated with the finite size of the proton. We illustrate the shapes of the various local and nonlocal functions numerically using a simple example of a dipole regulator.

Received 20 June 2018
Corrected 24 July 2019

DOI:https://doi.org/10.1103/PhysRevD.99.014041

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Deep inelastic scattering Effective field theory Quantum chromodynamics

Nuclear PhysicsParticles & Fields

Corrections

24 July 2019

Correction: A second affiliation has been inserted for the first author.

Authors & Affiliations

Y. Salamu^1,2, Chueng-Ryong Ji³, W. Melnitchouk⁴, A. W. Thomas⁵, and P. Wang^1,6

¹Institute of High Energy Physics, CAS, Beijing 100049, China
²School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
³North Carolina State University, Raleigh, North Carolina 27695, USA
⁴Jefferson Lab, Newport News, Virginia 23606, USA
⁵CoEPP and CSSM, Department of Physics, University of Adelaide, Adelaide SA 5005, Australia
⁶Theoretical Physics Center for Science Facilities, CAS, Beijing 100049, China

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Issue

Vol. 99, Iss. 1 — 1 January 2019

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Images

Figure 1
Diagrams representing the interaction of an external current (denoted by the crossed circles) with the proton involving SU(3) octet [(a)–(g)] and decuplet [(h)–(j)] states: (a) and (h) are for meson coupling rainbow diagrams; (b) and (i) are for octet and decuplet baryon coupling rainbow diagrams; (c) and (k) are for Kroll-Ruderman; (d) and (j) are for Kroll-Ruderman type diagrams generated by the gauge link (denoted by the filled circle); (e) is for meson tadpole; (f) is for meson bubble; and (g) is for meson tadpole diagram generated by the gauge link.
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Figure 2
Splitting functions versus meson momentum fraction $y$ for the proton dissociations into (a) $N + π$ , (b) $Λ + K$ , (c) $Δ + π$ , and (d) $Σ^{*} + K$ state, for the on-shell $f^{(on)}$ (red solid curves), off-shell $f^{(off)}$ (blue dashed), and nonlocal off-shell $δ f^{(off)}$ (black dotted) contributions. For the decuplet $Δ$ and $Σ^{*}$ states, additional contributions from on-shell endpoint $f^{(on end)}$ (red dot-dashed) and off-shell endpoint $f^{(off end)}$ (blue dot-dot-dashed) are included. All results correspond to the covariant dipole form factor in Eq. (56) with cutoff mass $Λ = 1 GeV$ .
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Figure 3
Normalized splitting functions $f_{i} (y) / ⟨ f_{i} ⟩$ for the (a) on-shell endpoint and (b) off-shell endpoint contributions for the $Δ + π$ intermediate state, for different values of the dipole cutoff mass $Λ$ (1 GeV to 1 TeV) and a fixed value of the constant $Ω_{0} = 100 {GeV}^{2}$ .
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Figure 4
Splitting functions versus $y$ for proton dissociations into various meson–baryon intermediate states as in Fig. 2, but for the total contributions to the meson-coupling rainbow diagrams in Fig. 1 and 1 (red solid curves), baryon-coupling rainbow diagrams in Fig. 1 and 1 (blue dashed), KR diagrams in Fig. 1 and 1 (green dot-dashed), and nonlocal KR diagrams in Fig. 1 and 1 (black dotted). Contributions from the tadpole and bubble diagrams in Fig. 1 at $y = 0$ are not shown here.
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Figure 5
Decomposition of the splitting function for the nucleon-coupling rainbow diagram in Fig. 1 for (a) the nonlocal chiral theory with dipole regulator, and (b) the local chiral theory with a symmetry preserving Pauli-Villars regulator. The value of the Pauli-Villars mass parameters $Λ$ is determined by normalizing to the momentum carried by the interacting nucleon, $⟨ y ⟩ = \int_{0}^{1} d y y f (y)$ , for the dipole regulator with $Λ = 1 GeV$ .
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Figure 6
Integrals of splitting functions $⟨ f ⟩$ versus $Λ$ , for (a) $N + π$ , (b) $Λ + K$ , (c) $Δ + π$ and (d) $Σ^{*} + K$ intermediates states, for the on-shell (red solid curves), off-shell (blue dashed), nonlocal off-shell (blue dotted), local $δ$ -function (green dot-dashed), and nonlocal $δ$ -function (green dotted) contributions. The decuplet states include additional contributions from on-shell endpoint (red dot-dot-dashed) and off-shell end point (blue dot-dot-dashed) terms. All results correspond to the covariant dipole form factor in Eq. (56).
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