Two-nucleon systems at $m_{\pi}\approx292$ MeV from lattice QCD

Using lattice QCD with $N_f=2+1$ ensembles at a pion mass of approximately 292 MeV, this study determines finite-volume energies of two-nucleon systems in the $^3S_1$ and $^1S_0$ channels and extracts scattering amplitudes via Lüscher's method and the Non-Perturbative Hamiltonian framework, revealing that both the deuteron and di-neutron channels exhibit virtual state poles with binding energies of $6^{+5}_{-3}$ MeV and $11^{+6}_{-5}$ MeV, respectively.

The Gist

Imagine the universe as a giant, invisible Lego set. The smallest bricks in this set are particles called quarks, and when three of them snap together, they form nucleons (protons and neutrons), which are the building blocks of everything we see, including the sun and your own body.

Physicists want to understand exactly how these nucleons stick together to form atomic nuclei. The "instruction manual" for how they interact is a complex theory called Quantum Chromodynamics (QCD). However, calculating these interactions on a computer is incredibly difficult because the math is messy and the signals are faint.

This paper is like a team of master builders using a supercomputer to simulate a tiny, controlled version of this Lego world to see how two nucleons behave when they get close to each other.

Here is a breakdown of what they did and found, using simple analogies:

1. The Simulation Setup: A Bigger, Heavier Lego World

Usually, scientists try to simulate the real world exactly as it is. But in this study, the researchers decided to change the "weight" of the Lego bricks.

• The Change: They simulated a world where the particles (pions) that help glue nucleons together are about three times heavier than they are in our real universe.
• Why? It's like trying to learn how to juggle by starting with heavy bowling balls instead of light tennis balls. It's harder, but it helps them test their tools and see if their methods work before trying the real thing.
• The Tools: They used three different-sized "rooms" (computer grids) to hold these particles. To get a clear picture, they used a special technique called distillation. Think of this as using a high-definition camera lens that filters out the static noise, allowing them to see the particles clearly without the "blur" that usually ruins these calculations.

2. The Experiment: Two Nucleons Dancing

The team watched how two nucleons behaved in two specific "dance styles" (scientific channels):

• The "Deuteron" Dance (3S1): This is the pair that usually sticks together to form the nucleus of a hydrogen atom (deuterium).
• The "Di-neutron" Dance (1S0): This is a pair of neutrons trying to stick together.

They watched these pairs in two ways:

1. Sitting Still: The pair was at rest in the center of the room.
2. Moving: The pair was zooming across the room.

3. The Big Question: Do They Stick?

In our real world, the deuteron pair sticks together tightly (it's a bound state), while the di-neutron pair usually flies apart.

The researchers wanted to know: In this "heavy particle" world, do they still stick?

To answer this, they used two different mathematical "rulers" to measure the interaction:

• Ruler A (Lüscher's Method): This is a standard tool that looks at the energy levels of the particles in the box to figure out how they scatter.
• Ruler B (NPHF): This is a newer, alternative tool that tries to account for the "long-range" forces (like a long elastic band) that might be pulling the particles.

4. The Discovery: The "Virtual" Ghost

Here is the surprising result: In this heavy-particle world, neither pair actually stuck together to form a permanent bond.

Instead, both pairs exhibited what physicists call a "virtual state."

The Analogy:
Imagine two people trying to hug.

• A Bound State is like a firm, permanent hug. They are locked together.
• A Resonance is like a high-five that happens very quickly and then they bounce apart.
• A Virtual State (what they found here) is like two people leaning in for a hug, getting very close, feeling a strong pull, but just barely missing the embrace before being pushed apart by the momentum. They are "almost" stuck, but not quite.

The paper found that in this specific simulation:

• The "Deuteron" pair was "almost" stuck, with a "binding energy" (how close they were to sticking) of about 6 MeV.
• The "Di-neutron" pair was also "almost" stuck, with a binding energy of about 11 MeV.

5. Checking the "Long Elastic Band"

The researchers were worried that their "Ruler A" might be missing a subtle force (the long-range pull of the pion) that could change the result. So, they used "Ruler B" (NPHF) to check.

The Result: Both rulers agreed. Even when they accounted for the long-range forces, the particles were still just "virtual states." They were attracted to each other, but not strongly enough to form a permanent bond in this heavy-particle world.

Summary

The paper concludes that at this specific, heavier mass for the particles, the universe is a place where nucleons are almost friends, but not quite. They lean in close and feel a strong pull, but they don't lock arms to form a stable nucleus.

This doesn't mean our real universe is like this (in our real world, the deuteron does stick). Instead, this study proves that the computer tools the scientists are using are working correctly. It shows that by changing the "weight" of the particles, they can see how the nature of nuclear forces changes, helping them understand the rules of the universe better when they eventually simulate the real, physical world.

Technical Summary

Technical Summary: Two-nucleon systems at $m_\pi \approx 292$ MeV from lattice QCD

Problem Statement
Understanding low-energy nuclear phenomena through Quantum Chromodynamics (QCD) remains a central objective in nuclear physics. While lattice QCD has achieved percent-level accuracy for single-nucleon properties, the study of multi-nucleon systems faces significant challenges, primarily the signal-to-noise problem and controversies regarding the existence of bound states in two-nucleon systems. Previous studies at heavy pion masses ($m_\pi \gtrsim 700$ MeV) suggested deeply bound deuteron and di-neutron states, while studies at intermediate masses ($m_\pi \sim 300-500$ MeV) using asymmetric correlators (local hexaquark operators at the source and momentum-space operators at the sink) also supported bound states. However, recent investigations employing variational techniques, distillation, and boosted frames have raised tensions with these earlier findings, suggesting that previous identifications of bound dinucleons might have been artifacts of asymmetric correlators or misidentifications of the spectrum. Furthermore, standard scattering analysis methods, such as Lüscher's finite-volume formalism, often neglect the effects of the left-hand cut, which becomes significant at lower pion masses where the cut ($k^2 = -m_\pi^2/4$) approaches the scattering threshold.

Methodology
This study investigates nucleon-nucleon interactions in the $^3S_1$ (deuteron) and $^1S_0$ (di-neutron) channels at a pion mass of approximately $m_\pi \approx 292$ MeV. The calculations utilize three $N_f = 2+1$ ensembles (C24P29, C32P29, C48P29) generated by the CLQCD collaboration, featuring a fixed lattice spacing $a = 0.10530(18)$ fm but varying spatial volumes ($L/a = 24, 32, 48$).

Key methodological features include:

1. Distillation Smearing: The distillation quark smearing method is employed to compute quark propagators. This allows for the construction of momentum-space interpolating operators at both the source and sink, ensuring symmetric correlators and avoiding potential spectral determination issues associated with asymmetric correlators. Local hexaquark operators are explicitly avoided.
2. Finite-Volume Spectroscopy: Energy levels are extracted in both the rest frame and a moving frame ($P = \frac{2\pi}{L}(0,0,1)$) using the variational method with a Generalized Eigenvalue Problem (GEVP). The analysis focuses on specific lattice irreducible representations (irreps) corresponding to the $^3S_1$ and $^1S_0$ channels.
3. Scattering Analysis (Lüscher Method): Scattering amplitudes are extracted using the generalized Lüscher finite-volume method. The scattering amplitude is parameterized via a K-matrix approach ($T^{-1} = K^{-1} - i\rho$) with a polynomial expansion in the Mandelstam variable $s$. The analysis is restricted to energy levels above the left-hand cut to ensure the validity of the standard Lüscher formula.
4. Scattering Analysis (NPHF): To investigate the potential impact of the left-hand cut arising from one-pion exchange, the Non-Perturbative Hamiltonian Framework (NPHF) is employed as an alternative. This method parameterizes the dinucleon interaction potential (including short-range contact terms and one-pion exchange) and fits the Hamiltonian eigenvalues to the lattice spectra. This approach explicitly includes the left-hand cut effects.

Key Results

• Virtual States: Both analysis methods indicate that at $m_\pi \approx 292$ MeV, neither the $^3S_1$ nor the $^1S_0$ channel forms a bound state. Instead, both channels exhibit a virtual state pole.
  • In the $^3S_1$ channel, a virtual state pole is found with a binding energy of $6^{+5}_{-3}$ MeV.
  • In the $^1S_0$ channel, a virtual state pole is found with a binding energy of $11^{+6}_{-5}$ MeV.
• Consistency of Methods: The results obtained from the Lüscher method (K-matrix parameterization) and the NPHF method are consistent within uncertainties. The NPHF analysis confirms that the inclusion of the one-pion exchange potential (long-range interaction) does not significantly alter the pole positions compared to the short-range only scheme, suggesting that the long-range force is negligible for s-wave scattering at this pion mass.
• Pole Locations: The poles are located on the unphysical Riemann sheet close to the real energy axis below the threshold, characteristic of virtual states. The physical sheet poles found in the K-matrix analysis are located near or below the left-hand cut and are deemed unphysical for the current analysis.

Significance and Claims
The paper claims to bridge the gap between previous studies at heavy pion masses and the physical point. By employing symmetric correlators via distillation and analyzing the system at a relatively low pion mass ($292$ MeV), the study provides evidence that the two-nucleon systems in the $^3S_1$ and $^1S_0$ channels are characterized by virtual states rather than bound states at this mass scale.

The authors emphasize that the use of symmetric correlators avoids the ambiguities associated with asymmetric correlators used in earlier works. Furthermore, the consistency between the Lüscher method and the NPHF method suggests that, at this specific pion mass, the effects of the left-hand cut and long-range one-pion exchange are not dominant enough to change the qualitative nature of the interaction (virtual vs. bound). The paper concludes that while these results clarify the situation at $m_\pi \approx 292$ MeV, the formation of a bound deuteron at the physical point likely requires a controlled study including the $^3S_1$–$^3D_1$ coupled-channel mixing and explicit treatment of the physical pion mass, which remains a priority for future lattice QCD investigations.