Proton-Neutron Pairing in N=Z Nuclei within the Quark-Meson-Coupling Energy Density Functional

This study demonstrates that incorporating isovector and isoscalar proton-neutron pairing correlations within the Quark-Meson-Coupling energy density functional and the quartet condensation model significantly improves the theoretical agreement with experimental binding energies for even-even N=Z nuclei across the mass range A=16 to A=120.

The Gist

The Big Picture: The Nuclear Dance Floor

Imagine an atomic nucleus not as a solid rock, but as a crowded dance floor inside a tiny ballroom. The dancers are protons (positively charged) and neutrons (neutral).

In most dance halls, the protons stick to their own group, and the neutrons stick to theirs. They hold hands with their own kind to stay stable. This is called "like-particle pairing."

But in a special type of nucleus where the number of protons equals the number of neutrons (called N=Z nuclei), something interesting happens. The protons and neutrons can also pair up with each other. It's like a mixed-gender dance where a proton and a neutron hold hands.

This paper asks a big question: Does this mixed dancing (proton-neutron pairing) make the nucleus hold together better? And if so, how much better?

The Problem: The "Missing" Energy

For a long time, scientists have used computer models to predict how heavy these nuclei are (their "binding energy"). Think of binding energy as the "glue" holding the dance floor together.

When scientists ran their old models, they found a problem: The glue wasn't strong enough. The models predicted the nuclei were lighter (less tightly bound) than what experiments actually showed. It was like predicting a bridge would hold 100 tons, but it actually held 105 tons. The models were missing about 2 tons of "glue."

Scientists suspected the missing glue was the proton-neutron dancing, but their old tools (called HFB or BCS models) were a bit clumsy at counting these specific dances. They often forgot to count the dancers exactly, or they treated the dance floor as perfectly round when it was actually squashed.

The New Tool: The "Quartet" Dance

The authors of this paper decided to try a new approach using two main ingredients:

1. The QMC Model (The Dance Floor Design):
   Instead of treating protons and neutrons as simple, solid marbles, this model treats them as bags of smaller particles called quarks. Imagine the dancers aren't just people, but people wearing special suits that change shape depending on how crowded the room is. This "Quark-Meson Coupling" (QMC) model gives a very accurate blueprint of the dance floor itself.

2. The QCM Model (The Quartet Condensation):
   This is the star of the show. Instead of just counting pairs (two dancers), this model counts quartets (groups of four: two protons and two neutrons).

   • The Analogy: Imagine the dance floor isn't just couples; it's groups of four friends doing a synchronized routine. The QCM model is a super-precise way of counting these groups. Crucially, it respects the rules of the universe perfectly: it counts every single dancer exactly (conserving particle number) and keeps the groups balanced (conserving isospin).

What They Did

The team took 100 different nuclei (from light ones like Oxygen to heavy ones like Tin) and ran a simulation:

1. They built the dance floor using the QMC blueprint.
2. They let the protons and neutrons dance, allowing them to pair up with their own kind and with each other, using the QCM method.
3. They calculated the total energy (how tight the glue is).

The Results: The Missing Glue Found!

The results were exciting:

• The Glue is Stronger: When they included the proton-neutron dancing, the calculated binding energy went up. It matched the real-world experimental data almost perfectly. The "missing 2 tons" of glue was found!
• Two Types of Dancing: There are two ways protons and neutrons can dance together:
  • Isovector (T=1): Like a standard waltz. This was the main contributor to the extra glue.
  • Isoscalar (T=0): A more exotic, "deuteron-like" dance. The paper checked if this exotic dance ever took over the whole floor (creating a "condensate").
• The Verdict on the Exotic Dance: The authors found that while the exotic dance happens, it doesn't usually take over the whole floor. The standard waltz (isovector) and the exotic dance (isoscalar) usually happen at the same time, competing for space. They don't see a sudden switch where the exotic dance becomes the only thing happening, which contradicts some older theories.

Why This Matters

1. Better Maps: This study gives us a much better map of how atomic nuclei hold together. This is crucial for understanding how stars explode (supernovae) and how heavy elements are created in the universe.
2. The "Quark" Advantage: By realizing that protons and neutrons are made of quarks that squish and change in the crowd, the model (QMC) was more accurate than previous models that treated them as rigid marbles.
3. The "Quartet" Trick: The method used (QCM) is a powerful new tool. It shows that if you want to understand these special nuclei, you have to count the groups of four perfectly, not just pairs.

The Bottom Line

Think of the nucleus as a complex puzzle. For years, scientists were missing a few pieces (the proton-neutron pairing). This paper didn't just find the missing pieces; it also built a better picture frame (the QMC model) and a better way to fit the pieces together (the QCM quartet method).

The result? The puzzle is now complete, and the picture of how these atomic nuclei hold together is clearer and more accurate than ever before.

Technical Summary

1. Problem Statement

The role of proton-neutron ($pn$) pairing correlations in nuclei with equal numbers of neutrons and protons ($N=Z$) remains a subject of intense theoretical and experimental debate. Key issues include:

• Excess Binding: $N=Z$ nuclei often exhibit higher binding energies than neighboring nuclei, a phenomenon not fully explained by standard mean-field theories.
• Isovector vs. Isoscalar: While isovector ($T=1$) pairing is well-established, the existence and dominance of isoscalar ($T=0$, deuteron-like) pairing in the ground states of heavy $N=Z$ nuclei ($A > 100$) are controversial.
• Limitations of HFB: Standard Hartree-Fock-Bogoliubov (HFB) calculations often fail to exactly conserve particle number, angular momentum, and isospin. Furthermore, predictions of $T=0$ dominance in heavy nuclei are highly sensitive to deformation and may be artifacts of symmetry breaking in HFB.
• Model Dependence: Previous studies using Skyrme or Gogny functionals often neglect the quark substructure of nucleons, which can lead to density-dependent effects missed in point-particle approximations.

The authors aim to investigate these issues by extending the Quark-Meson Coupling (QMC) Energy Density Functional (EDF) to include explicit proton-neutron pairing correlations treated within the Quartet Condensation Model (QCM).

2. Methodology

A. Theoretical Framework

The study combines two distinct models:

1. Quark-Meson Coupling (QMC) EDF:
   • Unlike traditional EDFs (Skyrme/Gogny) that treat nucleons as point particles, QMC models nucleons as clusters of confined quarks interacting via scalar ($\sigma$), vector ($\omega$), and isovector ($\rho$) meson fields.
   • This approach naturally generates density-dependent interactions and spin-orbit/tensor terms without ad-hoc parameters.
   • The mean field is generated using the QMC$\pi$-III functional, which includes pion-exchange Fock terms and full spin-orbit interactions.
2. Quartet Condensation Model (QCM):
   • Instead of the BCS or HFB approximations, the ground state is described as a condensate of quartets (two protons + two neutrons) coupled to total isospin $T=0$.
   • Key Advantage: The QCM wavefunction exactly conserves particle number, total angular momentum projection ($J_z=0$), and total isospin ($T=0$), avoiding the symmetry-breaking issues inherent in HFB.
   • The ground state is approximated as $|QCM\rangle = [Q^\dagger_{T=1} + (\Delta^\dagger_0)^2]^{n_q} |0\rangle$, where $Q^\dagger$ represents isovector quartets and $\Delta^\dagger_0$ represents isoscalar pairs.

B. Pairing Interactions

• Isovector ($T=1$): A zero-range pairing force with a density-dependent term derived consistently from the QMC framework. The strength is scaled by factors ($s$) fitted to odd-even mass differences in semi-magic nuclei ($^{16}$O, $^{40}$Ca, $^{100}$Sn regions).
• Isoscalar ($T=0$): Assumed to be proportional to the isovector force ($V^{T=0} = w V^{T=1}$) with a scaling factor $w=1.6$, guided by Gamow-Teller strength calculations.
• Matrix Elements: The authors derive explicit matrix elements for zero-range interactions in axially deformed bases, noting that off-diagonal isoscalar matrix elements can be positive, which suppresses isoscalar correlations relative to isovector ones.

C. Computational Scheme

• Self-Consistency: The calculation iterates between the QMC mean field (generating single-particle states) and the QCM (determining occupation probabilities and pairing fields) until convergence.
• Scope: Calculations cover even-even $N=Z$ nuclei in the mass range $16 \le A \le 120$, including sd-shell, pfg-shell, and nuclei above $^{100}$Sn.
• Deformation: Axial deformation ($\beta_2$) is treated self-consistently, allowing for the investigation of shape coexistence.

3. Key Contributions

1. First QMC+QCM Application: This is the first application of the QCM to the QMC EDF, bridging the gap between quark-level nuclear structure and many-body pairing correlations.
2. Exact Symmetry Conservation: By using QCM, the study provides results where particle number and isospin are exactly conserved, offering a benchmark against HFB predictions that rely on symmetry restoration (which is computationally expensive and often approximate).
3. Density-Dependent Pairing: The derivation of the pairing interaction directly from the QMC framework ensures consistency between the mean field and the pairing force, a feature often missing in phenomenological EDFs.

4. Results

A. Binding Energies

• Improvement over Mean Field: Standard QMC mean-field calculations (without pairing) underestimate binding energies for $N=Z$ nuclei. The inclusion of isovector pairing via QCM significantly improves agreement with experimental data, reducing residuals by $\sim 2$ MeV for sd-shell nuclei and $1-2$ MeV for pfg-shell nuclei.
• Role of Isoscalar Pairing: Switching on isoscalar pairing generally provides only a small additional contribution ($\sim 300-600$ keV) to the total binding energy for most nuclei.
• Double-Magic Nuclei: Unlike HFB/BCS where pairing vanishes in closed shells, QCM predicts significant pairing energies (several MeV) for double-magic nuclei ($^{16}$O, $^{40}$Ca, $^{56}$Ni, $^{100}$Sn). This suggests that mean-field parameters fitted to these nuclei in BCS frameworks may need re-evaluation when using beyond-mean-field methods to avoid double-counting.

B. Isovector vs. Isoscalar Competition

• Coexistence: Isovector and isoscalar correlations coexist in all studied nuclei, a feature guaranteed by the exact conservation of isospin in QCM.
• Suppression Mechanism: In several nuclei (e.g., $^{20}$Ne, $^{64}$Ge), the isoscalar $pn$ pairing energy exceeds the isovector $pn$ component. However, the total isovector pairing (including like-particle $nn$ and $pp$) remains dominant. The suppression of the isoscalar channel is attributed to the fluctuating signs of off-diagonal matrix elements in the $T=0$ channel.

C. Deformation and Shape Coexistence

• Shape Coexistence: The binding energy curves as a function of deformation ($\beta_2$) suggest shape coexistence in nuclei like $^{20}$Ne, $^{32}$S, $^{36}$Ar, $^{64}$Ge, and $^{68}$Se.
• Impact of Isoscalar Pairing: Isoscalar pairing has a minor effect on the equilibrium deformation for most nuclei, except for $^{20}$Ne, where it creates a second local minimum near zero deformation.

D. Heavy Nuclei ($A > 100$)

• No Dominant Isoscalar Phase: Contrary to some HFB predictions suggesting a transition to a dominant $T=0$ phase in heavy $N=Z$ nuclei (e.g., $^{108}$Xe), the self-consistent QMC+QCM calculations find no evidence of a clearly dominant isoscalar phase. The isoscalar contribution remains smaller than the total isovector pairing.
• Smooth Transitions: The variation of pairing energies with deformation is smooth, lacking the sudden phase transitions predicted by HFB, which the authors attribute to the lack of exact particle-number conservation in HFB.

5. Significance and Conclusions

• Validation of QCM: The study confirms that the QCM is a robust framework for describing $N=Z$ nuclei, providing accurate binding energies and correctly handling symmetry conservation.
• Refutation of HFB Artifacts: The results challenge the HFB prediction of a dominant isoscalar ground state in heavy $N=Z$ nuclei, suggesting that such predictions may be artifacts of symmetry breaking and the lack of particle-number projection.
• Implications for EDFs: The finding that pairing contributes significantly to the binding of double-magic nuclei implies that EDF parameters (like those in QMC, Skyrme, or Gogny) fitted using BCS/HFB assumptions may be inconsistent when applied to methods that go beyond mean-field theory. A re-fitting of parameters is likely necessary for self-consistent beyond-mean-field calculations.
• Future Outlook: The work establishes a foundation for extending these calculations to nuclei with small neutron excess, where $pn$ pairing effects are reduced but still non-negligible.

In summary, this paper demonstrates that while proton-neutron pairing is crucial for the binding of $N=Z$ nuclei, the isovector channel remains dominant even in heavy nuclei, and the inclusion of isoscalar correlations does not lead to a global phase transition to a deuteron-like condensate in the ground state.