Elastic and resonance structures of the nucleon from hadronic tensor in lattice QCD: implications for neutrino-nucleon scattering and hadron physics

This paper presents a lattice QCD calculation of the Euclidean hadronic tensor to extract nucleon elastic form factors and identify resonance structures, including the Roper resonance, thereby establishing a foundational framework for computing inclusive neutrino-nucleon scattering cross sections.

The Gist

The Big Picture: Listening to the Nucleon's "Echo"

Imagine a proton (a nucleon) not as a solid marble, but as a complex, vibrating drum. When you hit a drum, it doesn't just make one sound; it produces a fundamental tone (the "elastic" sound) and a whole bunch of higher-pitched overtones or "ringing" sounds (the "resonance" structures).

For decades, physicists have tried to understand exactly what these vibrations look like inside the proton. This is crucial because when neutrinos (ghostly particles that rarely interact with matter) smash into protons, they create these vibrations. To predict what happens in massive neutrino experiments like DUNE, scientists need a perfect map of these vibrations.

This paper is a major step toward creating that map using Lattice QCD, which is essentially a super-computer simulation of the universe's strongest force (the strong nuclear force) on a grid.

The New Tool: The "Hadronic Tensor"

Traditionally, to study a proton, physicists would hit it once with a probe (like a photon) and measure the result. This is like tapping a drum once and listening to the single note.

In this paper, the researchers used a new, more complex method called the Hadronic Tensor.

• The Analogy: Instead of tapping the drum once, imagine tapping it twice in rapid succession. The first tap excites the drum, and the second tap listens to how the drum is still vibrating from the first tap.
• The Result: By analyzing the relationship between these two "taps" (mathematically represented as a four-point function), the researchers can see not just the main note, but the entire "spectrum" of sounds the drum makes. This allows them to see the proton's internal structure, including its "ringing" states (resonances), all in one go.

What They Did: Two Main Tasks

The team performed two main tasks with this new method:

1. Checking the Main Note (Elastic Scattering)
First, they wanted to make sure their new "double-tap" method worked correctly. They calculated the proton's basic electric shape (the Sachs electric form factor) using this new method.

• The Result: They compared their new "double-tap" results with the old, trusted "single-tap" method. The numbers matched perfectly. This proved that their new, more complex tool is reliable and accurate.

2. Listening to the Ringing (Resonance Structures)
Next, they looked at what happens after the main note fades. They looked for the "overtones"—the excited states of the proton.

• The Discovery: Using a sophisticated mathematical technique called Bayesian Reconstruction (think of it as a high-tech audio equalizer that tries to reconstruct a song from a blurry recording), they found a distinct "bump" or structure in the data.
• The Location: This bump appeared at an energy level about 0.5 to 0.7 GeV higher than the proton's normal mass.
• The Identity: They interpret this bump as a mix of several things:
  • The Roper Resonance (a well-known excited state of the proton, often called N(1440)).
  • Other similar heavy particles.
  • Multi-particle states (like a proton temporarily turning into a proton plus a pion).

The Challenge: A Blurry Photo

The authors are very honest about the limitations.

• The Analogy: Imagine trying to take a photo of a fast-moving race car at night. You get a picture, but it's a bit blurry. You can clearly see a car is there, and you can tell it's moving fast, but you can't clearly distinguish if it's a Ferrari or a Lamborghini, or if there are two cars overlapping.
• The Reality: The computer simulation is powerful, but the "blur" (statistical noise) is still too high to separate the individual "ringing" states perfectly. They can see the group of excited states, but they can't yet isolate the Roper resonance from the others with 100% precision.

The Comparison: Theory vs. Reality

To see if their "blurry photo" made sense, they compared their results to real-world data from the CLAS experiment (a real particle accelerator).

• They calculated a specific property called the Longitudinal Helicity Amplitude (a measure of how the proton spins and responds to the hit).
• The Outcome: Their theoretical numbers were within a factor of three of the real experimental data. Given that their simulation used a "heavy" version of the pion (a particle inside the proton) and a small grid, this is a very promising first step. It suggests the method is on the right track.

Why This Matters (According to the Paper)

The paper emphasizes that this is the first major step toward calculating "inclusive" scattering.

• Inclusive means counting everything that happens, not just the clean, simple hits.
• Currently, models used to predict neutrino behavior often struggle with the messy middle ground between simple hits and total destruction (Deep Inelastic Scattering).
• By proving that the Hadronic Tensor method can capture both the clean hits and the messy "ringing" states, this work lays the foundation for a unified theory. In the future, this could help scientists build better models for neutrino experiments, helping them understand the universe's fundamental forces more accurately.

Summary

This paper is like a physicist successfully testing a new, high-tech microphone. They proved it can hear the main drumbeat clearly (matching old methods) and that it can also pick up the complex, messy ringing that follows. While the recording is still a bit fuzzy and they can't yet identify every single instrument in the band, they have successfully proven that this new microphone works and can hear the whole orchestra.

Technical Summary

1. Problem Statement

The paper addresses the challenge of determining the hadronic tensor ($W_{\mu\nu}$) from first principles using Lattice Quantum Chromodynamics (LQCD). The hadronic tensor is central to understanding inclusive lepton-nucleon scattering, which is crucial for:

• Neutrino Physics: Accurately modeling neutrino-nucleus ($\nu-A$) interactions for experiments like DUNE and Hyper-K. Current models rely on phenomenological inputs for quasi-elastic (QE), resonance (RES), and deep inelastic scattering (DIS) regions, particularly in the few-GeV energy range where systematic uncertainties dominate.
• Hadron Spectroscopy: Understanding the excitation spectrum of baryons (resonances) and the transition between elastic, resonance, and DIS regions within a unified framework.

Existing LQCD methods often focus on exclusive processes (e.g., 3-point functions for form factors) or specific kinematic regions. A major hurdle is the inverse problem: converting Euclidean-space lattice correlators (calculated at imaginary time) into Minkowski-space spectral functions (physical energy spectra) to extract resonance structures and form factors.

2. Methodology

The authors employ a novel approach combining the hadronic tensor formalism with advanced spectral reconstruction techniques:

• Formalism:

  • They calculate the Euclidean hadronic tensor ($W^E_{\mu\nu}$) using a four-point correlation function involving two current insertions ($J_\mu, J_\nu$) separated by Euclidean time $\tau$.
  • The tensor is related to the physical Minkowski tensor via an inverse Laplace transform, which is an ill-posed inverse problem due to limited and noisy lattice data.
  • The calculation focuses on connected insertions (ignoring disconnected diagrams, which are suppressed for electromagnetic currents) using the charge density operator ($J_4$).

• Numerical Setup:

  • Lattice: RBC/UKQCD $32I_{fine}$ domain wall fermion ensembles.
  • Parameters: Lattice spacing $a \approx 0.06$ fm, pion mass $m_\pi = 371$ MeV (unphysical, heavy pion), volume $32^3 \times 64$.
  • Configurations: 645 gauge configurations with 8 source locations.
  • Smearing: A fast fermion smearing scheme is used to enhance ground-state overlap and reduce computational cost.

• Extraction Techniques:

  1. Bayesian Reconstruction (BR): Used as a qualitative tool to reconstruct the spectral density $\rho(\omega)$ from the 4pt/2pt correlation ratio. This helps identify the presence and location of spectral peaks (elastic and excited states) without assuming a specific functional form.
  2. Multi-Exponential Fitting: Used for quantitative extraction. The ratio of 4-point to 2-point functions is fitted to a sum of exponentials:
     $$R(\tau) \propto W_0 e^{-\Delta E_0 \tau} + W_1 e^{-\Delta E_1 \tau} + \dots$$
     where $W_n$ are spectral weights related to form factors and $\Delta E_n$ are energy gaps.

• Analysis Strategy:

  • Elastic Region: Extracted the nucleon Sachs electric form factor ($G_E$) and compared it with traditional 3-point function results.
  • Resonance Region: Identified structures above the nucleon mass. Due to limited resolution, the first excited state is treated as an effective state representing a mixture of $N(1440)$ (Roper), $N(1710)$, and negative parity states ($N(1535)$, etc.).
  • Kinematics: Calculated for various momentum transfers $\vec{q}$ to determine $Q^2$ dependence.

3. Key Contributions

• First Major Step in Inclusive LQCD: This work presents the first significant lattice QCD calculation of the hadronic tensor to determine inclusive $N \to X$ contributions, bridging the gap between elastic scattering and deep inelastic scattering.
• Validation of Formalism: Successfully demonstrated that the hadronic tensor formalism yields elastic form factors ($G_E$) consistent with the conventional 3-point function method, validating the approach for future high-precision studies.
• Resonance Structure Identification: Identified a broad resonance structure approximately 0.5 – 0.7 GeV above the nucleon mass in the Bayesian reconstruction. This structure is interpreted as a mixture of the Roper resonance ($N(1440)$) and other nearby states.
• Transition Form Factors: Extracted the transition electric form factor ($G^*_E$) and the longitudinal helicity amplitude ($S_{1/2}$) for the nucleon-to-resonance transition, comparing them with experimental CLAS data.

4. Key Results

• Elastic Form Factor ($G_E$):

  • The $G_E(Q^2)$ extracted from the 4-point hadronic tensor matches the results from the 3-point function calculation within uncertainties for $Q^2 \in [0, 1.74]$ GeV$^2$.
  • This confirms the feasibility of using the hadronic tensor approach for elastic properties.

• Resonance Structure:

  • Location: A distinct structure appears at an invariant mass of $\sim 1.7 - 1.9$ GeV (0.5–0.7 GeV above the nucleon).
  • Composition: Likely a mixture of $1/2^+$ states (Roper $N(1440)$, $N(1710)$) and $1/2^-$ states ($N(1535)$, $N(1650)$). The $\Delta(1232)$ resonance is not observed, likely due to the dominance of the charge current $J_4$ and the specific kinematic suppression of the $\Delta$ contribution in this channel.
  • Limitation: The current lattice statistics and volume prevent the resolution of individual resonances within this structure; they appear as a single broad peak.

• Transition Form Factors & Helicity Amplitudes:

  • The extracted transition form factor $G^*_E(Q^2)$ and longitudinal helicity amplitude $S_{1/2}(Q^2)$ for the effective excited state are compared to CLAS experimental data for the $N \to N(1440)$ transition.
  • Magnitude: The lattice results for $S_{1/2}$ are within a factor of 3 of the experimental data in the range $Q^2 \approx 0.08 - 0.48$ GeV$^2$.
  • Discrepancies: The differences are attributed to the unphysical pion mass (371 MeV), finite volume effects, and the fact that the lattice result represents a mixture of multiple resonances rather than a pure Roper state.

5. Significance and Implications

• Neutrino Oscillation Experiments: This work provides a pathway to calculate inclusive neutrino-nucleon cross-sections directly from QCD, reducing reliance on phenomenological models. This is critical for reducing systematic uncertainties in experiments like DUNE and Hyper-K, particularly in the resonance region where current models are uncertain.
• Unified Framework: The hadronic tensor formalism allows for the simultaneous study of elastic, resonance, and deep inelastic scattering regions within a single calculation, offering a more comprehensive view of nucleon structure.
• Future Directions: The paper establishes the methodology for future calculations with physical pion masses and larger volumes, which will be necessary to resolve individual resonances (like separating the Roper from $N(1710)$) and to apply finite-volume corrections (Lüscher formalism) to match infinite-volume Minkowski results.
• Methodological Advancement: The successful application of Bayesian Reconstruction and multi-exponential fitting to 4-point functions demonstrates a robust technique for solving the inverse problem in LQCD, applicable to other inclusive observables.

In conclusion, while the current results are semi-quantitative due to unphysical simulation parameters, this study proves the feasibility of using the hadronic tensor in LQCD to access inclusive scattering physics, marking a pivotal step toward precision neutrino physics and hadron spectroscopy.