⚛️ nuclear theory

Charge dependent nucleon-nucleon potentials in covariant chiral effective field theory

This paper derives charge-dependent nucleon-nucleon potentials up to next-to-next-to leading order within covariant chiral effective field theory by incorporating isospin-breaking contributions, yielding phase shifts that agree well with experimental data and providing a robust framework for future nuclear studies.

Original authors: Yang Xiao, Jun-Xu Lu, Chun-Yan Song, Li-Sheng Geng

Published 2026-07-24

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CC BY 4.0

Original authors: Yang Xiao, Jun-Xu Lu, Chun-Yan Song, Li-Sheng Geng

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic Lego set. At the very bottom of the stack are the tiniest bricks: protons and neutrons, the building blocks of every atom in your body and every star in the sky. But here's the mystery: these bricks don't just sit there; they are constantly talking to each other, pushing and pulling with invisible forces. This conversation is called the "nucleon-nucleon interaction." For decades, scientists have been trying to write the perfect dictionary for this conversation to understand how stars burn and how the core of a planet holds together.

However, there's a tricky twist in this story. In the world of these tiny bricks, there are two types of protons and neutrons that are almost identical twins, but not quite. One is the proton (which has a positive electric charge, like a tiny battery), and the other is the neutron (which is neutral, like a ghost). Usually, physicists treat them as perfect mirror images, assuming they behave exactly the same way when they hug or fight. But in reality, because the proton has that electric charge, it feels a little extra "static cling" from other protons that the neutron doesn't. This tiny difference, called "charge dependence," is like a slight stutter in the twins' conversation. If you ignore it, your dictionary is incomplete, and your predictions about how the universe is built might be slightly off. The big question is: how do we write a dictionary that accounts for these tiny, electric stutters without losing the main story?

This is where the team of researchers from Beihang University and their collaborators steps in. They decided to rewrite the rules of this cosmic conversation using a special, high-tech lens called "covariant chiral effective field theory." Think of this lens as a super-advanced camera that doesn't just take a picture of the bricks; it captures them moving at high speeds, respecting the rules of Einstein's relativity (where time and space are flexible). In the past, scientists had to simplify the camera to make the math work, often ignoring the "relativistic" effects or treating the electric charge as an afterthought.

In this new study, the authors built a complete, charge-dependent dictionary for the proton-proton and neutron-proton conversations, going all the way up to a level of detail they call "next-to-next-to leading order" (NNLO). This is like upgrading from a basic sketch to a 4K movie with surround sound. They didn't just guess; they calculated the forces using the full, complex math of relativity, including the fact that pions (the particles that carry the strong force between bricks) have slightly different weights depending on whether they are charged or neutral. They also added the "Coulomb force," which is the repulsive push between two positively charged protons, treating it with the same high-speed precision as the other forces.

The result? When they ran their calculations, the "movie" they created matched the real-world data almost perfectly. They compared their new, high-definition predictions against a massive database of experimental measurements known as "PWA93," which is like the gold standard of nuclear physics. For the most part, their new relativistic dictionary fit the data beautifully, especially at lower energies. It turns out that by keeping the full relativistic picture and carefully adding the charge-dependent details, they could explain the subtle differences between how protons and neutrons interact without needing to break the laws of physics.

The authors found that while the "charge-breaking" effects (the electric stutters) are small, they are crucial for getting the details right. They discovered that simply adding a few extra "contact terms"—which are like short-range, invisible handshakes between the particles—was enough to fix the discrepancies. Interestingly, they noted that while their method is a huge step forward for understanding the structure of the universe, it is still a work in progress. They didn't claim to have solved every mystery of the nucleus, but they did prove that their new, relativistic framework is a solid foundation. It suggests that in the future, we can use this same high-speed, charge-aware lens to study even more complex systems, like the dense matter inside neutron stars or the behavior of exotic nuclear matter, with much greater confidence than before.

Technical Summary: Charge Dependent Nucleon-Nucleon Potentials in Covariant Chiral Effective Field Theory

Problem Statement
Understanding nucleon-nucleon (NN) interactions is fundamental to nuclear physics, particularly for addressing nuclear structure and reaction problems. While high-precision NN potentials based on chiral effective field theory ( $\chi$ EFT) have successfully described neutron-proton (np) and proton-proton (pp) scattering data, a complete description requires accounting for charge dependence. This dependence arises from isospin-breaking contributions in both electromagnetic and strong interactions, leading to subtle but notable differences between pp and np systems. Although conventional non-relativistic $\chi$ EFT has incorporated these effects at higher orders, extending these interactions to a covariant framework presents specific challenges. Previous covariant chiral nuclear force studies successfully reproduced empirical saturation properties and np scattering data but had not systematically considered isospin-breaking effects. This omission is significant because proton-proton systems are subject to additional electromagnetic forces and unique charge-dependent effects that influence nuclear dynamics.

Methodology
The authors construct charge-dependent NN interactions up to next-to-next-to leading order (NNLO) within a covariant chiral effective field theory framework. The methodology involves the following key steps:

Derivation from Relativistic Lagrangians: Unlike non-relativistic approaches where static limits are assumed, the potentials are derived directly from relativistic Lagrangians using Dirac spinors and covariant propagators. This ensures manifest Lorentz covariance and naturally incorporates kinematic relativistic effects, such as retardation in pion exchanges.
Isospin-Symmetric and Breaking Potentials:
- Isospin-Symmetric: The baseline potential includes contact terms (CT), one-pion exchange (OPE), and two-pion exchange (TPE) contributions up to NNLO.
- Isospin-Breaking: The authors systematically incorporate isospin-breaking contributions, specifically:
  - Strong Interaction: Pion mass splitting ( $M_{\pi^\pm} \neq M_{\pi^0}$ ) in the OPE term at NLO. Nucleon mass differences are analyzed but found to contribute only at higher orders ( $v=6$ ) in the covariant counting and are thus neglected.
  - Electromagnetic Interaction: The static one-photon exchange (Coulomb force) is included. Higher-order electromagnetic corrections (e.g., vacuum polarization, magnetic moments) are noted as higher-order in the chiral expansion and are not explicitly included in the current calculation.
  - Contact Terms: New charge-symmetry-breaking (CSB) contact terms are introduced for the pp channel at NNLO. These include scalar, vector, axial-vector, and tensor bilinears ( $C^{pp}_{S,V,AV,T}$ ).
Scattering Equation and Regularization: The relativistic two-body scattering equation (Thompson equation) is solved in the $LSJ$ basis for isospin $I=0$ and $I=1$ . To handle ultraviolet divergences, a non-local Gaussian-type cutoff function with $\Lambda = 700$ MeV is applied. The Coulomb interaction is consistently incorporated within the Thompson equation using relativistic kinematic factors and matched to asymptotic Coulomb states to extract phase shifts.
Parameter Determination: A total of 23 low-energy constants (LECs) are determined: 19 for isospin-conserving interactions and 4 for isospin-breaking interactions. These are fitted simultaneously to np and pp scattering phase shifts (up to $J \le 2$ ) for laboratory energies ranging from 1 to 200 MeV, using the PWA93 partial wave analysis as the benchmark.

Key Contributions

First Systematic Covariant Treatment: This work provides the first systematic derivation of charge-dependent covariant chiral NN potentials up to NNLO, bridging the gap between covariant nuclear forces and isospin-breaking phenomena.
Relativistic Consistency: The approach ensures that the isospin-breaking sector remains strictly consistent with the previously developed isospin-symmetric nuclear force, avoiding ad-hoc "relativization" of non-relativistic results.
Inclusion of Specific Mechanisms: The derivation explicitly includes pion mass splitting in OPE and charge-dependent contact terms, while rigorously analyzing the power counting of nucleon mass differences in the covariant framework.

Results

Phase Shift Agreement: The calculated np and pp phase shifts show satisfactory agreement with the PWA93 partial wave analysis.
- np Scattering: At NNLO, the results agree well with PWA93 data in the fitted energy region. While the NLO description of the $^1S_0$ phase shift is slightly worse due to the lack of isospin-breaking LECs at that order, the NNLO results improve significantly, particularly in the coupled $^3S_1$ – $^3D_1$ channel, due to subleading TPE contributions.
- pp Scattering: The NLO and NNLO results closely match PWA93 phase shifts at low energies. The inclusion of the four additional pp contact terms at NNLO significantly improves the description of the $^1S_0$ channel, which showed large discrepancies at NLO.
Theoretical Uncertainties: The results generally fall within the estimated theoretical uncertainties of the EFT framework, even at higher energies where the description is less precise, consistent with the expected behavior of EFT.
Impact of Short-Range Corrections: Diagnostic calculations indicate that the additional pp-channel short-range corrections (the new CSB contact terms) primarily affect the $^1S_0$ channel, with moderate effects on $^3P_0$ and negligible effects on $^3P_1$ .

Significance and Claims
The authors claim that this work represents a significant step toward a fully relativistic and systematically improvable description of nucleon-nucleon interactions. The primary value of the formulation lies in its structural integrity: by treating pp and np systems on an equal footing within a Lorentz-invariant framework, it provides a solid foundation for future studies. Specifically, the results facilitate:

A deeper understanding of isospin-breaking effects in nuclear forces.
Future investigations into isospin-breaking phenomena in nuclear systems, such as disentangling charge-symmetry and charge-independence breaking effects.
Studies of mirror energy differences, charge-exchange processes, and electroweak transitions.
Determination of the symmetry energy and the equation of state for isospin-asymmetric matter within the covariant chiral EFT framework.

The paper emphasizes that the satisfactory agreement with PWA93 data confirms that the systematic inclusion of isospin-breaking contact terms at NNLO successfully compensates for the inherent differences between pp and np sectors in this covariant scheme.

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