Impact of neutron-proton pairing on the nucleon high-momentum distribution in symmetric nuclear matter

This study investigates the impact of neutron-proton pairing on the high-momentum tail of nucleon momentum distributions in symmetric nuclear matter, revealing that pairing contributes approximately 6% to the short-range correlation-induced high-momentum fraction and that this effect scales with the squared relative pairing gap normalized by the kinetic energy.

The Gist

The Big Picture: A Dance Floor of Particles

Imagine a crowded dance floor inside a nuclear reactor. This floor is packed with tiny dancers: protons (the positive ones) and neutrons (the neutral ones). In a perfect, calm world, these dancers would follow a strict rule: everyone stays in their assigned spot on the floor, and no one moves too fast. This is what physicists call a "normal" state.

However, the real world is messy. The dancers bump into each other, push, pull, and sometimes form tight couples. This paper investigates two specific types of "chaos" on this dance floor:

1. The "Bump" (Short-Range Correlations): Sometimes, two dancers accidentally crash into each other with huge force. They bounce off each other so hard that they shoot off the dance floor at incredible speeds. This creates a "tail" of very fast dancers.
2. The "Waltz" (Neutron-Proton Pairing): Sometimes, a neutron and a proton decide to hold hands and dance a slow, romantic waltz. Because they are holding hands, they move differently than the solo dancers.

The Question: The scientists wanted to know: Does the "Waltz" (pairing) change how many dancers end up flying off the floor at high speeds (the high-momentum tail)?

The Tools: A High-Tech Dance Simulator

To figure this out, the researchers built a super-complex computer simulation. Think of it like a video game engine for nuclear physics.

• The "Bump" Engine: They used a method called EBHF (Extended Brueckner-Hartree-Fock). Imagine this as a camera that tracks every time two dancers crash into each other. It knows that these crashes create the "high-speed tail" of particles.
• The "Waltz" Engine: They added a layer called BCS theory. This tracks the couples holding hands.
• The Challenge: Usually, these two engines don't play well together. If you try to simulate the "bumps" and the "waltzes" at the same time, the math gets messy and breaks. The authors developed a special way to combine these two engines so they could see how the "Waltz" affects the "Bumps."

The Discovery: A Small but Important Ripple

Here is what they found:

1. The "Bumps" are the Boss: The main reason particles fly off at high speeds is the violent crashes (Short-Range Correlations). This is the dominant force.
2. The "Waltz" Adds a Little Extra: When the neutron and proton hold hands (pairing), it actually makes slightly more particles fly off at high speeds than if they were just dancing alone.
3. The Magnitude: The "Waltz" adds about 6% to the number of high-speed particles.
   • Analogy: Imagine a stadium full of people. If 100 people are running at full speed because of a stampede (the "Bumps"), the "Waltz" is like a group of 6 extra people who decide to sprint because they are holding hands. It's a small number compared to the stampede, but it's definitely there and measurable.

The "Secret Sauce": Why It Happens

The paper explains why this happens using a concept called the Pairing Gap.

Think of the "Pairing Gap" as the strength of the hug between the neutron and proton.

• The stronger the hug (the larger the gap), the more the dance floor changes.
• The researchers found a simple rule: The amount of extra high-speed particles is directly related to the square of the hug strength.
• If the hug is twice as strong, the effect on the high-speed particles is four times stronger.

They also discovered that to get an accurate count, you have to account for a specific "correction factor" (called the $M_3$ term in the paper). Without this correction, the simulation underestimates how strong the "hug" is, and you miss the extra high-speed particles.

Why Should We Care?

You might ask, "Why does a 6% increase in fast particles matter?"

1. Neutron Stars: These are the densest objects in the universe, made almost entirely of neutrons and protons. Understanding how they move and interact helps us understand how neutron stars cool down, how they spin, and even how they create gravitational waves (ripples in space-time).
2. The "Tensor Force": The "hug" between neutrons and protons is caused by a mysterious force called the tensor force. This paper helps us understand how this force works over long distances, not just when particles crash into each other.
3. Better Models: By proving that pairing adds 6% to the high-speed tail, scientists can now build better models of nuclear matter. It's like realizing your map of a city was missing a small but important bridge; now that you've found it, your navigation is much more accurate.

The Bottom Line

The paper concludes that while the violent crashes between particles are the main reason for high-speed movement in nuclear matter, the gentle "waltz" of neutron-proton pairs plays a significant supporting role. It adds a measurable 6% to the high-speed crowd.

This confirms that to truly understand the nucleus (and the stars made of them), we can't just look at the crashes; we have to watch the couples dancing, too.

Technical Summary

1. Problem Statement

The nucleon momentum distribution in nuclear matter is a fundamental observable that reveals the nature of nucleon-nucleon (NN) correlations.

• Short-Range Correlations (SRCs): Strong repulsive cores and tensor forces in NN interactions deplete the Fermi sea and generate a High-Momentum Tail (HMT) ($k > k_F$). Experimental data suggests SRCs account for roughly 20% of nucleons forming correlated pairs, with neutron-proton (np) pairs dominating due to the tensor force.
• The Gap in Understanding: While SRCs are known to drive the HMT, the specific contribution of neutron-proton pairing (isoscalar pairing) to this high-momentum tail remains ambiguous. Pairing arises from the long-range attractive part of the tensor force. Theoretical frameworks often treat SRCs and pairing separately or struggle to incorporate them self-consistently because standard Brueckner-Hartree-Fock (BHF) approaches can develop spurious singularities when pairing is present.
• Core Question: To what extent does np pairing modify the HMT in symmetric nuclear matter, and how does this effect scale with density and self-energy corrections?

2. Methodology

The authors employ a unified theoretical framework combining the Extended Brueckner–Hartree–Fock (EBHF) approach with off-shell BCS theory to self-consistently treat both SRCs and pairing.

• Self-Energy Calculation (EBHF):
  • The nucleon self-energy $\Sigma(k, \omega)$ is calculated up to the third order in the hole-line expansion to account for SRCs.
  • $M_1$ (First Order): The standard BHF term (ladder diagrams).
  • $M_2$ (Second Order): The rearrangement term, accounting for particle-hole excitations.
  • $M_3$ (Third Order): A renormalization term crucial for describing the partial occupation of hole states below the Fermi surface induced by NN correlations. This term is essential for a reliable description of np pairing.
• Pairing Formalism (Off-shell BCS):
  • The gap equation is solved using off-shell propagators, incorporating the full self-energy $\Sigma(k, \omega)$.
  • The calculation uses the realistic Argonne V18 (Av18) two-body interaction.
  • The framework solves the gap equation self-consistently with the density constraint to determine the pairing gap $\Delta(k)$ and chemical potential $\mu$.
• Spectral Functions and Momentum Distributions:
  • The spectral function $A(k, E)$ is derived from the Green's function, including the pairing gap.
  • The momentum distribution $n(k)$ is obtained by integrating the spectral function over energy.
  • The study compares the BCS state (with pairing) against the Normal state (without pairing) to isolate the specific contribution of pairing to the HMT.

3. Key Contributions

• Unified Framework: The paper successfully integrates SRCs (via EBHF ladder resummations) and np pairing (via off-shell BCS) in a single self-consistent calculation, avoiding the singularities that plague standard approaches.
• Role of Third-Order Terms ($M_3$): The authors demonstrate that the third-order renormalization term ($M_3$) is not negligible. It acts with a sign opposite to the first-order term ($M_1$), significantly enhancing the pairing gap and correcting the spectral function near the Fermi surface.
• Quantification of Pairing Contribution: The study provides a quantitative measure of how much np pairing contributes to the high-momentum tail relative to SRCs.
• Scaling Law Identification: The authors identify a qualitative measure for the pairing effect on the HMT: the squared relative pairing gap, $\tilde{\Delta}_F^2 / E^{*2}_{kF}$, where $\tilde{\Delta}_F$ is the effective pairing gap and $E^*_{kF}$ is the kinetic energy at the Fermi surface using the effective mass.

4. Key Results

• Spectral Function Modifications:
  • In the BCS state, the single-peaked spectral function of the normal state splits into two peaks (Cooper pair partners) near the Fermi energy.
  • Pairing enhances the depletion below the Fermi surface and increases occupation above it.
  • The inclusion of $M_3$ leads to a more pronounced impact on the spectral function compared to calculations using only $M_1$ and $M_2$.
• High-Momentum Tail (HMT) Enhancement:
  • The momentum distribution in the BCS state is slightly higher than in the normal state even at high momenta ($k \gg k_F$).
  • HMT Ratio ($N_{BCS}/N_{normal}$): Defined as the ratio of high-momentum nucleons in the BCS state to the normal state.
    • The ratio peaks at approximately 1.06 around a density of $\rho \approx 0.052 \text{ fm}^{-3}$.
    • This indicates that the contribution of np pairing to the HMT is approximately 6% of the contribution arising from SRCs.
  • The effect is density-dependent, following the evolution of the pairing gap.
• Dependence on Self-Energy Order:
  • Calculations including $M_3$ yield a larger pairing gap and a more significant HMT enhancement than those excluding it.
  • The $M_1 + M_2$ calculation actually yields a smaller gap than $M_1$ alone because $M_2$ introduces a large imaginary part (stronger correlations) that quenches the gap, whereas $M_3$ counteracts this and restores/enhances the gap.
• Correlation with Relative Pairing Gap:
  • The density dependence of the HMT ratio closely tracks the squared relative pairing gap ($\tilde{\Delta}_F^2 / E^{*2}_{kF}$). This suggests that the ratio serves as a robust qualitative indicator of the pairing effect on the momentum distribution.

5. Significance and Outlook

• Physical Insight: The results confirm that while SRCs are the primary driver of the HMT, the long-range tensor force associated with np pairing generates a non-negligible, observable enhancement (approx. 6%) to the high-momentum tail.
• Astrophysical Implications: Since the HMT influences quasiparticle strength near the Fermi surface, these findings have implications for neutron star transport properties, cooling rates, and glitch phenomena.
• Future Directions:
  • The current study uses the Av18 potential, which has a strong tensor force. The authors suggest that using chiral effective field theory (EFT) interactions (which have weaker tensor components) might alter the interplay between SRCs and pairing.
  • Future work should incorporate polarization corrections to the pairing interaction, which were omitted in this study due to complexity but are necessary for a complete understanding.

In conclusion, this paper establishes that np pairing is a significant, albeit secondary, contributor to the high-momentum tail in nuclear matter, and its effect can be quantitatively linked to the squared relative pairing gap within a self-consistent EBHF-BCS framework.