The hadronic tensor from four-point functions on the lattice

This paper presents preliminary results from a lattice QCD study using stochastic sources and a clover fermion ensemble to calculate four-point functions, thereby extending the computation of the hadronic tensor to a broader range of momentum transfers essential for understanding neutrino-nucleon scattering structure functions.

The Gist

The Big Picture: Taking a "Slow-Motion" Photo of a Proton

Imagine you want to understand how a proton (a tiny particle inside an atom) behaves when it gets hit by a neutrino (a ghost-like particle that rarely interacts with anything). To do this, physicists need to calculate something called the Hadronic Tensor.

Think of the Hadronic Tensor as a complex blueprint or a recipe card. It contains all the secret instructions on how the proton reacts when struck. If you have this recipe, you can predict exactly what happens in experiments at giant particle colliders.

The problem? This recipe is written in a language called "Minkowski space" (the real world of time and space), but the supercomputers physicists use (called Lattices) only speak "Euclidean space" (a mathematical version where time is treated like a fourth dimension of space). It's like trying to read a book written in French using a dictionary that only translates to Japanese. You can't read it directly; you have to do a difficult translation.

The Challenge: The "Inverse Problem"

In this paper, the authors are trying to solve a puzzle known as the Inverse Problem.

• The Analogy: Imagine you have a cake (the proton) and you bake it. You can easily measure the ingredients you put in (the currents). But now, imagine you only have the smell of the cake wafting through the kitchen over time (the data on the lattice). Your job is to look at that smell and figure out exactly what the cake looked like and what ingredients were used.
• The Difficulty: Usually, the "smell" (the data) fades away very quickly. If you only have a few seconds of data, it's hard to guess the whole recipe. The authors realized that to get a good recipe, they need to smell the cake for a longer time and from different angles.

What They Did: The "Stochastic Flashlight"

To get better data, the team used a new technique involving stochastic sources.

• The Analogy: Imagine trying to take a photo of a dark room.
  • Old way: You shine a flashlight in one spot, take a picture, move the flashlight, take another picture. This is slow and you might miss things.
  • Their new way: They used a "stochastic source," which is like turning on a strobe light that flashes randomly everywhere in the room at once. This allows them to capture the entire room (all possible momentum transfers) in a single snapshot.

By using this "strobe light" method, they could calculate the interactions of the proton with a much wider range of energy levels than ever before. This is crucial because the "Deep Inelastic Scattering" (the high-energy crash we care about) happens at very specific, high-energy levels.

The Setup: A Digital Sandbox

They built a digital universe (a lattice) to run their simulation:

• The Grid: A 32x32x32x128 grid of points.
• The Particles: They simulated protons made of "up" and "down" quarks.
• The Conditions: They set the "pion mass" (a related particle) to a specific value to make the math work, even though it's slightly heavier than in the real world. They are working on refining this later.

The Results: Seeing the Signal

They ran the simulation and looked at the data. Here is what they found:

1. Stability: They checked if their "cake" was stable. They looked at the data at different times and found it was flat and consistent. This means their simulation wasn't being messed up by "noise" (excited states).
2. The Decay: When they looked at how the signal changed over time (the $\tau$ dependence), they saw it fading away, just like a smell dissipating.
   • The Catch: The signal fades very fast, especially when the energy is high. It's like trying to hear a whisper in a hurricane; after a short distance, the whisper is gone.
3. The Momentum: They tested different "momentum transfers" (how hard the neutrino hits the proton). They found that for the highest energies, the signal disappears so quickly that it's hard to get a clear picture with their current setup.

The Conclusion: "We're Just Getting Started"

The authors are very honest about where they stand:

• The Good News: They successfully built the "strobe light" method and got the first-ever preliminary results for this specific type of calculation. They proved the method works.
• The Bad News: Because the signal fades so fast, they can't yet solve the "Inverse Problem" (translate the smell back into the recipe) with perfect accuracy. They are currently stuck with the proton sitting still (zero momentum).
• The Next Step: To fix this, they need to:
  1. Move the proton: Simulate protons that are actually moving (non-zero momentum). This opens up a "window" where the signal stays visible longer.
  2. Refine the grid: Use a finer grid (smaller pixels) to catch the signal before it fades.
  3. Add more ingredients: Include more complex interactions (like disconnected quark loops) to get the full picture.

In summary: This paper is a "proof of concept." The team built a powerful new tool to look inside the proton, took some blurry but promising first photos, and is now promising to build a better camera and a faster computer to get the crystal-clear images needed to understand the universe's fundamental building blocks.

Technical Summary

1. Problem Statement

The hadronic tensor ($W_{\mu\nu}$) is a fundamental non-perturbative object required to calculate cross-sections for lepton-hadron interactions, such as neutrino-nucleon scattering and Deep Inelastic Scattering (DIS). While traditionally parameterized by structure functions ($F_1, F_2$) which encode information on Parton Distribution Functions (PDFs) and hadron resonances, calculating these directly from first principles is challenging.

• The Challenge: Lattice QCD operates in Euclidean spacetime, whereas the physical hadronic tensor is defined in Minkowski spacetime via a commutator of currents involving a time integral. Direct calculation on the lattice is impossible because the time integration cannot be performed directly.
• The Inverse Problem: The lattice calculation yields a Euclidean version of the tensor ($W^E$) related to the physical tensor via a Laplace transform. Reconstructing the physical structure functions from $W^E$ constitutes an ill-posed inverse problem.
• Kinematic Limitations: Previous lattice studies focused on the resonance region. Extracting DIS structure functions requires access to a much broader kinematic range (high momentum transfer $Q^2$ and specific Bjorken-$x$). Furthermore, for a nucleon at rest ($\vec{p}=0$), the integration range in the inverse problem barely covers the DIS region unless extremely large momentum transfers are used, where lattice artifacts become significant.

2. Methodology

The authors propose a direct calculation of the hadronic tensor using four-point correlation functions on the lattice, employing stochastic sources to access a broad kinematic region.

A. Theoretical Framework

• Euclidean Formulation: Instead of the commutator, the authors calculate the time-ordered product of two currents separated by Euclidean time $\tau$. This quantity, $W^E_{\mu\nu}(\vec{p}, \vec{q}, \tau)$, is related to the Minkowski tensor via a Laplace transform.
• Structure Function Decomposition: In Euclidean space, the tensor is decomposed into four structure functions ($A, B, C, D$) rather than the two physical ones ($F_1, F_2$). The authors impose the Ward identity ($q^\mu W_{\mu\nu} = 0$) as additional constraints to relate the Euclidean functions to the physical ones.
• Inverse Problem Strategy: The physical structure functions $F_1$ and $F_2$ are obtained by inverting the Laplace transform relation. The authors note that for $\vec{p}=0$, the "deep inelastic window" is narrow, making reconstruction difficult. They plan to address this by calculating for non-zero nucleon momentum ($\vec{p} \neq 0$) in future work.

B. Lattice Setup and Simulation

• Ensemble: The study uses the CLS ensemble S100 with $N_f = 2+1$ flavors of O($a$)-improved Sheikholeslami-Wohlert (clover) fermions.
  • Pion mass: $m_\pi = 223$ MeV.
  • Lattice spacing: $a = 0.085$ fm.
  • Volume: $32^3 \times 128$.
• Correlation Functions: The core calculation involves the four-point function:
  $$C^{4pt}_{jj'}(\vec{p}, \vec{z}; t, \tau_1, \tau_2) = \langle \Gamma \mathcal{O}(\vec{p}, t_0+t) J_j(\vec{x}+\vec{z}, t_0+\tau_1) J_{j'}(\vec{z}, t_0+\tau_2) \mathcal{O}(\vec{p}, t_0) \rangle$$
• Wick Contractions: For the nucleon, five types of contractions exist. The authors restrict the current analysis to connected contributions only.
  • For the electromagnetic current in a proton, the $C_1$ (disconnected-like connected) contributions largely cancel due to charge factors ($4/9 - 2/9 - 2/9 \approx 0$).
  • Therefore, the analysis focuses on $C_2$-type contractions (quark lines connecting source to sink via both currents).
• Computational Techniques:
  • Stochastic Sources: To evaluate the matrix element for all spatial vectors $\vec{z}$ (necessary for the Fourier transform to momentum space), the authors use time-local stochastic sources ($N_{stoch}=96$ per timeslice).
  • Source-Sink Separation: Calculations are performed at $t \in \{8a, 10a, 12a\}$ to control excited state contamination.
  • Smearing: Gaussian smearing is applied to nucleon source and sink operators.

3. Key Contributions

1. Extension to Broad Kinematics: Unlike previous studies limited to resonance regions, this work targets the DIS region by utilizing stochastic sources to calculate four-point functions over a wide range of momentum transfers ($\vec{q}$).
2. Direct Four-Point Calculation: The paper demonstrates a direct evaluation of the two-current matrix element for an unpolarized nucleon, moving beyond the Feynman-Hellmann method used in earlier structure function studies.
3. Handling the Inverse Problem: The authors provide a detailed framework for relating Euclidean structure functions ($A, B, C, D$) to physical ones, explicitly addressing the constraints imposed by the Ward identity and the limitations of the $\vec{p}=0$ frame.
4. Preliminary Data Validation: The study validates the signal quality and excited state suppression, showing that the data is consistent with theoretical expectations (e.g., charge conservation at $\vec{q}=0$).

4. Results

The authors present preliminary results based on 890 configurations with the nucleon at rest ($\vec{p}=0$):

• Excited State Contamination: Analysis of the dependence on the average insertion time ($\bar{\tau}$) shows the data is flat across different source-sink separations ($8a, 10a, 12a$), indicating that excited state contamination is small and manageable.
• $\tau$-Dependence:
  • For $\vec{q}=0$, the signal reproduces the valence quark count for $\tau > 0$ and vanishes for $\tau < 0$ (pure sea contribution), consistent with charge conservation.
  • For $\vec{q} \neq 0$, the signal exhibits an exponential decay, characteristic of resonance contributions.
  • Signal quality improves with larger $|\vec{q}|$ due to rotational averaging within the $H_4$ symmetry group.
• Euclidean Structure Functions ($A$ and $B$):
  • The authors solved the over-determined system of equations to extract $A$ and $B$.
  • Observation: The signal decays rapidly with $\tau$. For $|\vec{q}| > 3$ GeV and $\tau > 0.3$ fm, the signal is consistent with zero.
  • This rapid decay highlights the difficulty of the inverse problem and suggests a need for finer lattice spacings to resolve the $\tau$-dependence better.

5. Significance and Future Work

• Significance: This work represents a critical step toward calculating DIS structure functions and PDFs directly from lattice QCD without relying on model-dependent assumptions. It establishes the feasibility of using four-point functions with stochastic sources to access high-momentum transfer regimes.
• Current Limitations: The restriction to $\vec{p}=0$ limits the kinematic window for the inverse problem, making the extraction of physical structure functions currently unfeasible. The analysis is also limited to connected contributions ($C_2$) and electromagnetic currents.
• Future Directions:
  1. Non-Zero Momentum: Extend simulations to $\vec{p} \neq 0$ to broaden the deep inelastic kinematic window.
  2. Full Wick Contractions: Include $C_1$ (connected) and $S_2$ (leading disconnected) contributions, which are essential for non-electromagnetic currents and flavor-separated PDFs.
  3. Axial-Vector Currents: Incorporate axial currents to enable calculations for neutrino-nucleon scattering.
  4. Finer Lattices: Simulate on finer lattices to improve the resolution of the $\tau$-dependence and mitigate discretization errors in the derivative operators.

In summary, this paper lays the methodological groundwork and provides initial validation for a direct lattice QCD approach to the hadronic tensor, paving the way for future ab-initio determinations of nucleon structure functions.