Ab initio charge form factors and radii of light isoscalar nuclei: Role of the two-body charge density

Using the Jacobi-coordinate No-Core Shell Model with chiral interactions, this study demonstrates that including two-nucleon charge density operators is essential for accurately predicting the charge form factors and radii of light isoscalar nuclei like $^6$Li and $^8$Be, thereby resolving the long-standing issue of charge radius underestimation in *ab initio* calculations.

The Gist

Imagine an atomic nucleus not as a solid marble, but as a bustling, chaotic dance floor filled with tiny dancers (protons and neutrons). Scientists have long tried to map out exactly how these dancers are arranged and how much space they take up. This paper is like a high-tech attempt to draw that map using a new, ultra-precise set of rules.

Here is the breakdown of what the researchers did and found, explained simply:

The Goal: Measuring the "Size" of the Dance Floor

In physics, the "size" of a nucleus is measured by its charge radius. Think of this as the average distance between the center of the dance floor and the outer edge of the dancers. Scientists also look at form factors, which are like a "fingerprint" of the nucleus. If you shine a light (an electron beam) at the nucleus, the way the light bounces off tells you about the shape and arrangement of the dancers inside.

For a long time, scientists have been using a sophisticated rulebook called Chiral Effective Field Theory (think of it as the "Laws of Physics for Tiny Dancers") to predict these sizes. However, there was a problem: their predictions were consistently too small. The calculated nuclei were always a bit too tight and compact compared to what experiments actually showed.

The New Ingredient: The "Team Dance"

The researchers realized they were missing a crucial piece of the puzzle.

• The Old Way (One-Body): They previously calculated the charge by looking at each dancer individually. "This proton has a charge, that neutron has a charge," and they just added them up.
• The New Way (Two-Body): The paper argues that you can't just look at dancers in isolation. Sometimes, two dancers interact so closely that they create a new effect together. It's like a "team dance" where the space they occupy together is different than the sum of their individual spaces.

The authors added these "two-body charge density" effects into their calculations. Think of this as realizing that when two dancers hold hands and spin, they create a "charge cloud" that isn't just the sum of their individual charges.

The Experiment: Testing on Small Groups

To test this idea, they focused on two light nuclei: Lithium-6 and Beryllium-8. These are like small dance troupes (6 and 8 dancers, respectively).

They used a powerful computer method called the Jacobi-coordinate No-Core Shell Model. Imagine this as a super-accurate simulation that tracks every single dancer's movement without ignoring anyone. They fed their new "team dance" rules into this simulation.

The Results: Finally Getting the Size Right

The results were a big success:

1. The Shape Matched: When they included the "team dance" (two-body) effects, the predicted "fingerprint" (form factor) of the nucleus matched the experimental data much better, especially when the "light" hit the nucleus at sharper angles (higher momentum).
2. The Size Fixed: The most important finding was about the size. The old calculations (looking only at individual dancers) underestimated the size of the nucleus. By adding the "team dance" effects, the predicted size grew slightly, bringing it into perfect alignment with real-world measurements.

The Takeaway

The paper concludes that to accurately understand the size of an atomic nucleus, you cannot just count the individual protons and neutrons. You must account for how they interact in pairs.

The Analogy:
Imagine trying to measure the size of a crowd of people.

• Old Method: You measure the width of one person and multiply by the number of people. This gives you a number that is too small because it ignores the space people need to stand next to each other.
• New Method: You realize that when people stand in groups, they create a "personal bubble" that expands the total area. By accounting for these group bubbles (the two-body effects), your measurement of the crowd's total size becomes accurate.

The authors state that this "two-body" correction is essential. It solves a long-standing mystery where physics theories kept predicting nuclei that were slightly too small, finally bridging the gap between theory and reality.

Technical Summary

Technical Summary: Ab Initio Charge Form Factors and Radii of Light Isoscalar Nuclei

Problem Statement
Electromagnetic probes provide a direct link between experimental cross-sections and nuclear structure properties, particularly charge form factors (FFs) and radii. While high-precision experimental data for light nuclei have become available, theoretical models based on modern chiral Effective Field Theory ($\chi$EFT) have historically struggled to accurately predict nuclear charge radii, often underestimating them. A critical gap exists in understanding the role of two-body charge density operators, particularly in nuclei heavier than the deuteron and $A \le 4$ systems. Previous studies established the importance of two-body currents for the deuteron and light nuclei, but their specific impact on the charge FFs and radii of $p$-shell nuclei like $^6$Li and $^8$Be remained unclear.

Methodology
The authors perform ab initio calculations for the isoscalar nuclei $^6$Li and $^8$Be using the Jacobi-coordinate No-Core Shell Model (J-NCSM). The theoretical framework relies on:

• Interactions: Chiral semilocal momentum-space regularized (SMS) two-nucleon (NN) potentials at order $N^4LO^+$ and consistent three-nucleon (3N) potentials at $N^2LO$. The calculations utilize momentum cutoffs $\Lambda_N = 450$ and $500$ MeV.
• Operators: Consistently regularized one- and two-nucleon electromagnetic charge operators. The two-body charge density includes contact (short-range) and one-pion-exchange (1$\pi$) terms.
• Wave Functions: Nuclear many-body wave functions are obtained via J-NCSM. To ensure consistency with the Simulated Renormalization Group (SRG) evolved interactions, a unitary transformation is applied to the two-body densities to eliminate SRG artifacts.
• Parameter Determination: The low-energy coupling constants (LECs) for the two-body charge density are determined as follows:
  • LECs for the isospin-zero-to-isospin-zero channel are fixed using deuteron charge and quadrupole FFs (from previous work).
  • The remaining LEC for the isospin-one-to-isospin-one channel is fixed by fitting to the experimental charge radius of $^4$He ($1.67824(83)$ fm).
• Observables: Charge FFs are calculated using a transition-density formalism convoluted with one- and two-body density matrices. Charge radii are decomposed into contributions from the matter radius, proton/neutron intrinsic radii, Darwin-Foldy (DF), spin-orbit (SO), two-body contact, and one-pion-exchange terms.

Key Contributions and Results

1. Form Factors: The inclusion of two-nucleon (2N) charge density contributions significantly alters the predicted charge FFs for $^6$Li and $^8$Be, particularly at intermediate and large momentum transfers ($q$). The 2N contributions shift the diffraction minima to lower $q$ values, bringing the theoretical predictions into much better agreement with experimental data compared to calculations using only one-body (1N) operators.
2. Charge Radii:
   • $^4$He: The J-NCSM results converge with Faddeev-Yakubovsky (FY) calculations using bare interactions. The 2N contributions increase the charge radius by approximately $0.04$ fm, a value largely insensitive to the SRG flow parameter.
   • $^6$Li: The 2N contributions increase the charge radius by approximately $0.03$ fm. While the total radius calculation shows some dependence on the model space size ($N_{HO}$), the 2N contribution itself ($\Delta R_{Cont+1\pi}$) exhibits stable convergence. The final predicted radii, including tail corrections for the basis space, are consistent with experimental measurements.
   • $^8$Be: Despite being unbound, calculations show a similar trend where 2N contributions increase the radius by $\approx 0.035$ fm.
3. Magnitude of Effects: The study quantifies that the 2N contact and 1$\pi$ contributions to the charge radius are significantly larger than the Darwin-Foldy and spin-orbit terms.
4. System Size Dependence: The relative size of the 2N contributions stabilizes as the system size increases from $^4$He to $^6$Li and $^8$Be. The authors note that this contribution does not simply scale with the number of nucleon pairs or $\alpha$-clusters but appears driven by short-distance currents involving $s$-shell nucleons.

Significance
The paper claims that two-nucleon charge operators are essential for achieving accurate ab initio predictions of nuclear charge radii and form factors in light isoscalar nuclei. The results address the long-standing issue of the underestimation of nuclear charge radii in calculations based on modern chiral interactions. By demonstrating that consistently regularized 2N operators reduce the discrepancy between theory and experiment, the work validates the necessity of including these operators alongside nuclear forces within the same $\chi$EFT framework. The authors conclude that while the current study focuses on light isoscalar systems, the developed framework provides a pathway for extending these consistent calculations to heavier nuclei and resolving systematic errors in nuclear structure predictions.