Nonlocal effective field theory and its applications

TL;DR

The article surveys nonlocal effective field theory as a natural framework for incorporating hadron finite size and nonpointlike interactions. It details nonlocal chiral EFT for nucleon GPDs and gravitational form factors, nonlocal QED with a solid-quantization formalism to address lepton $g-2$ anomalies, and extensions to curved spacetime yielding a nonlocal energy-momentum tensor and gravitational form factors. Across these sections, gauge invariance is preserved via gauge links, with results expressed as convolution integrals or form-factor decompositions that connect to lattice QCD and experimental data. The work demonstrates that nonlocality regularizes loops, yields finite results without new particles, and offers testable predictions for GPDs, sea-quark asymmetries, and gravitational couplings, while highlighting parameter sensitivities and the need for further phenomenological constraints. Overall, nonlocal EFT provides a cohesive toolkit for probing hadron structure and electroweak/gravitational couplings beyond point-like approximations.

Abstract

We review recent applications of nonlocal effective field theory, focusing in particular on nonlocal chiral effective theory and nonlocal quantum electrodynamics (QED), as well as an extension of nonlocal effective theory to curved spacetime. For the chiral effective theory, we discuss the calculation of generalized parton distributions (GPDs) of the nucleon at nonzero skewness, along with the corresponding gravitational (or mechanical) form factors, within the convolution framework. In the QED application, we extend the nonlocal formulation to construct the most general nonlocal QED interaction, in which both the propagator and fundamental QED vertex are modified due to the nonlocal Lagrangian, while preserving the Ward-Green-Takahashi identities. For consistency with the modified propagator, a solid quantization is proposed, and the nonlocal QED is applied to explain the lepton $g-2$ anomalies without the introduction of new particles or interactions. Finally, with an extension of the chiral effective action to curved spacetime, we investigate the nonlocal energy-momentum tensor and gravitational form factors of the nucleon with a nonlocal pion-nucleon interaction.

Figures (15)

• Figure S1: One-loop diagrams for the proton to pseudoscalar meson (dashed lines) and octet baryon (solid lines) or decuplet baryon (double solid lines) splitting functions up to the fourth chiral order: (a)--(c) octet baryon rainbow diagrams, (d)--(g) octet baryon Kroll-Ruderman diagrams, (h)--(j) tadpole diagrams, (k)--(l) bubble diagrams, (m)--(o) decuplet baryon rainbow diagrams, (p)--(q) octet-decuplet transition rainbow diagrams, (r)--(u) decuplet baryon Kroll-Ruderman diagrams. The crossed circles ($\otimes$) represent the interaction with external vector field from the minimal substitution, filled circles ($\bullet$) denote additional gauge link interaction with the external field, black squares ($\blacksquare$) represent the magnetic interaction in Eq. (\ref{['lomag']}), and gray squares (${\textcolor{gray}{\blacksquare}}$) denote the interaction in Eq. (\ref{['adm']}).
• Figure S2: Electric and magnetic GPDs for light antiquarks: (a)$xH^{\bar{u}}$, (b)$xE^{\bar{u}}$, (c)$xH^{\bar{d}}$, and (d)$xE^{\bar{d}}$, versus parton momentum fraction $x$ and four-momentum transfer squared $-t$, for cutoff mass $\Lambda=1$ GeV at a scale $Q=1$ GeV.
• Figure S3: Light antiquark flavor asymmetry for the (a) electric $xH^{\bar{d}-\bar{u}}$ and (b) magnetic $xE^{\bar{d}-\bar{u}}$ GPDs versus parton momentum fraction $x$ and four-momentum transfer squared $-t$, for regulator parameter $\Lambda = 1$ GeV.
• Figure S4: Light antiquark asymmetries for the electric $xH^{\bar{u}-\bar{d}}$ (red bands) and magnetic $xE^{\bar{u}-\bar{d}}$ (blue bands) GPDs versus parton momentum fraction $x$ at four-momentum transfer squared of $t=0$ [ (a), (b)] and $t=-0.25$ GeV$^2$ [ (c), (d)], for cutoff parameter $\Lambda = 1.0(1)$ GeV. The asymmetries are shown at the scale $Q=1$ GeV, except for the electric asymmetry at $t=0$, which is compared with the $x(\bar{d}-\bar{u})$ PDF asymmetry from the JAM global QCD analysis Cocuzza:2021cbi (yellow band) at the scale $Q=m_c$.
• Figure S5: Electric and magnetic GPDs for the strange and antistrange quarks: (a)$xH^s$, (b)$xE^s$, (c)$xH^{\bar{s}}$, and (d)$xE^{\bar{s}}$ versus the parton momentum fraction $x$ and four-momentum transfer squared $-t$, for $\Lambda=1$ GeV, at the scale $Q=1$ GeV.