Mapping the Nucleus: A Lattice QCD Approach to Structure Functions

Author: Denis Avetisyan

New calculations using lattice quantum chromodynamics and four-point correlation functions provide a direct path to understanding the internal structure of protons and neutrons.

The analysis details five distinct Wick contractions-<span class="katex-eq" data-katex-display="false">C_1C_{\hat{1}}, C_2C_{\hat{2}}, S_1S_{\hat{1}}</span> among others-that contribute to the four-point function in equation (9) when applied to baryons, with the specific mathematical form of these contractions dependent on the quark flavor of the involved currents. — The analysis details five distinct Wick contractions- $C_1C_{\hat{1}}, C_2C_{\hat{2}}, S_1S_{\hat{1}}$ among others-that contribute to the four-point function in equation (9) when applied to baryons, with the specific mathematical form of these contractions dependent on the quark flavor of the involved currents.

This review details a lattice QCD calculation of the hadronic tensor, extracting structure functions relevant to deep inelastic scattering and neutrino-nucleon interactions from current-current correlation functions in Euclidean space.

Determining non-perturbative hadronic contributions remains a central challenge in understanding lepton-hadron scattering processes. This work, ‘The hadronic tensor from four-point functions on the lattice’, presents a lattice QCD calculation of the hadronic tensor-a key object in describing these interactions-directly from the correlation of two quark-bilinear currents. By employing stochastic sources and analyzing four-point functions on a $223~\mathrm{MeV}$ pion mass ensemble with $0.085~\mathrm{fm}$ lattice spacing, we extend previous calculations to a broader range of momentum transfers relevant for extracting structure functions. Will these improved calculations provide new insights into the parton distribution functions and resonant contributions within hadrons?

Unveiling the Nucleon: A Journey into Hadronic Structure

The nucleon, comprising protons and neutrons, serves as the building block of atomic nuclei and, consequently, all visible matter; therefore, a complete understanding of its internal structure is paramount to advancing nuclear physics. This isn’t simply a matter of identifying constituent quarks, but also characterizing the complex interplay of the strong force – governed by the exchange of gluons – which binds them together. Investigations into the nucleon’s structure reveal a surprising dynamism, where quarks aren’t static particles but a fluctuating ‘sea’ of quarks and gluons, alongside virtual quark-antiquark pairs. Mapping this intricate landscape is crucial not only for accurately predicting the nucleon’s properties, like its mass and spin, but also for unraveling the fundamental principles governing the strong interaction itself, a force that dictates the behavior of matter at its most basic level. The quest to probe the nucleon’s structure represents a central challenge in modern physics, demanding innovative experimental techniques and theoretical frameworks.

Predicting the properties of hadrons, such as the proton and neutron collectively known as nucleons, presents a significant challenge for physicists due to the intricacies of the strong nuclear force. Unlike electromagnetism, which diminishes with distance, the strong force remains constant, binding quarks and gluons within hadrons with immense strength. This constant force creates a complex many-body problem, rendering perturbative calculations-standard tools in other areas of physics-unreliable. Traditional methods, reliant on simplifying assumptions about quark interactions, often fail to accurately reproduce experimentally observed hadron masses, magnetic moments, and decay patterns. The strong force also gives rise to phenomena like confinement, preventing the isolation of individual quarks, and dynamical symmetry breaking, which affects hadron masses in a non-trivial way. Consequently, alternative approaches, including lattice quantum chromodynamics and effective field theories, are crucial for gaining a deeper understanding of hadronic structure and overcoming the limitations of conventional calculations.

Deep Inelastic Scattering (DIS) functions as a powerful experimental technique for investigating the nucleon’s internal structure by bombarding it with high-energy leptons, such as electrons. However, interpreting the resulting scattering patterns isn’t straightforward; the strong force, governing interactions within the nucleon, complicates the analysis. Simply observing scattered leptons isn’t enough-theoretical frameworks, like Quantum Chromodynamics (QCD), are crucial for disentangling the complex dynamics at play. These frameworks allow physicists to relate the scattering cross-section to the distribution of quarks and gluons-the fundamental constituents-within the nucleon. The process involves calculating $F_2(x)$ , a structure function representing the probability of finding a quark carrying a fraction $x$ of the nucleon’s momentum. Accurate theoretical predictions, often achieved through perturbative QCD and increasingly sophisticated computational methods, are therefore essential to validate experimental results and refine understanding of hadronic structure.

Lattice QCD: A First-Principles Approach to Hadronic Calculations

Lattice Quantum Chromodynamics (QCD) offers a first-principles, non-perturbative approach to calculating properties of hadrons – composite particles made of quarks and gluons. Traditional perturbative QCD methods fail when the strong coupling constant becomes large, a regime common in low-energy hadronic physics. Lattice QCD circumvents this limitation by discretizing spacetime into a four-dimensional lattice, transforming the continuous QCD equations into a set of algebraic equations that can be solved numerically. This allows for the direct calculation of hadronic quantities, such as masses and decay constants, from the fundamental parameters of QCD – quark masses and the strong coupling constant – without relying on phenomenological models or fitting procedures. The accuracy of these calculations is limited by discretization errors, which can be systematically reduced by refining the lattice spacing and employing improved lattice actions.

Discretization of spacetime in Lattice QCD involves representing continuous space and time as a four-dimensional grid. This transforms the differential equations of QCD – which are analytically intractable for low-energy, non-perturbative regimes – into a set of algebraic equations that can be solved numerically. The computational cost scales rapidly with several factors, including the lattice spacing, volume, and the mass of the hadrons being calculated. Specifically, achieving accurate results necessitates lattices large enough to accommodate multiple hadrons and fine lattice spacing to reduce discretization errors, demanding high-performance computing resources and substantial computational time. Furthermore, the fermion formulation – in this case, Clover Fermions – adds to the computational complexity due to the need to account for quark propagation and interactions on the discrete spacetime lattice.

The central objective of our Lattice QCD calculations is the determination of the Hadronic Tensor $W^{\mu\nu}$ . This tensor functions as the fundamental link between theoretical predictions and experimentally measured quantities in Deep Inelastic Scattering (DIS). Specifically, the Hadronic Tensor encapsulates the probability amplitude for a virtual photon to fluctuate into a hadronic state, and its components directly contribute to the structure functions $F_2$ and $F_1$ observed in DIS experiments. By accurately calculating the Hadronic Tensor on the lattice, we can provide first-principles predictions for these structure functions, allowing for stringent tests of the Standard Model and searches for new physics.

Calculations within this Lattice QCD framework are performed on a discretized spacetime lattice with a spatial spacing of 0.085 femtometers (fm) and dimensions of 32³ x 128 in the spatial and temporal directions, respectively. The fermion action utilizes Clover Fermions, a formulation designed to improve the chiral properties of lattice quarks. These calculations are based on gauge configurations generated by the CLS Collaboration, providing a statistically significant and computationally efficient source of background gluon fields for the simulations. This lattice setup balances computational cost with the need to accurately represent the low-energy dynamics of Quantum Chromodynamics.

Computational Techniques: Extracting Precision from Four-Point Functions

The Hadronic Tensor, a fundamental quantity in quantum chromodynamics used to describe hadronic interactions, is calculated via the evaluation of four-point correlation functions. These functions, denoted as $\Gamma_{\mu\nu}$ , quantify the correlation between two hadronic currents, representing the creation and annihilation of hadrons. Specifically, the four-point function involves the vacuum expectation value of the time-ordered product of two currents at different spacetime points. The mathematical structure of this function directly relates to the amplitudes for various hadronic processes, allowing for the extraction of relevant physical observables when combined with appropriate kinematic factors and loop integrals.

Wick contractions are a fundamental technique in many-body quantum field theory used to evaluate vacuum expectation values of time-ordered products of operators. In the context of Lattice QCD calculations, these contractions systematically handle the pairing of creation and annihilation operators arising from the expansion of the four-point functions used to calculate hadronic observables. The $C_2$ contraction specifically addresses the connected contributions to the two-particle irreducible two-point function, representing the exchange of a single virtual particle between two operators. Utilizing the $C_2$ contraction reduces the computational complexity of the calculation by focusing on the most significant contributions while accurately representing the underlying physics of the hadronic interactions.

Evaluating four-point functions in Lattice QCD calculations is computationally intensive; therefore, time-local stochastic sources are implemented to improve efficiency. This technique involves introducing random noise, represented by the stochastic sources, at each time slice. The current analysis utilizes 96 stochastic sources per timeslice, which provides a statistically significant sampling of possible noise configurations. By averaging over these numerous source configurations, the computational cost associated with direct evaluation of the four-point function is substantially reduced, allowing for a more feasible calculation of hadronic properties.

Lattice Quantum Chromodynamics (LQCD) calculations are performed in Euclidean spacetime, a mathematical construct differing from the Minkowski spacetime of physical experiments. To bridge this gap and extract physically observable quantities, a Laplace Transform is applied to the calculated correlation functions. This transformation analytically continues the results from Euclidean momentum space to Minkowski momentum space, allowing for direct comparison with experimental data. The Laplace Transform effectively reverses the Wick rotation $t \rightarrow -i\tau$ , converting Euclidean time τ to Minkowski time $t$ and enabling the calculation of quantities like structure functions in a physically relevant spacetime.

This analysis utilizes a dataset comprising 890 lattice QCD configurations. This number of configurations represents a preliminary assessment of the computational techniques employed; while sufficient for initial validation and methodological refinement, it is not yet at the scale required for definitive, statistically robust physical results. Further calculations with a significantly larger ensemble of configurations will be necessary to reduce statistical uncertainties and provide a comprehensive assessment of systematic effects, ultimately allowing for precise determination of the target hadronic observables.

The quantity <span class="katex-eq" data-katex-display="false">W^{\mathrm{E},u}_{44}</span> depends on <span class="katex-eq" data-katex-display="false">\bar{\tau}</span> for a fixed <span class="katex-eq" data-katex-display="false">\vec{q}=(2,3,4)2\pi/L</span>, while its derivative, <span class="katex-eq" data-katex-display="false">W^{\mathrm{E},d}_{44}</span>, and the electromagnetic contribution <span class="katex-eq" data-katex-display="false">W^{\mathrm{E},\mathrm{em}}_{44}</span> each exhibit a strong dependence on τ, modulated by the source-sink separation distance. — The quantity $W^{\mathrm{E},u}_{44}$ depends on $\bar{\tau}$ for a fixed $\vec{q}=(2,3,4)2\pi/L$ , while its derivative, $W^{\mathrm{E},d}_{44}$ , and the electromagnetic contribution $W^{\mathrm{E},\mathrm{em}}_{44}$ each exhibit a strong dependence on τ, modulated by the source-sink separation distance.

Unveiling Nucleon Structure: Towards Precise Parton Distribution Functions

The Hadronic Tensor, a fundamental quantity in nuclear physics, serves as the bridge between the internal structure of nucleons and their observable responses to external probes, such as electrons or neutrinos. By carefully analyzing this tensor, physicists can extract the Structure Functions, $F_1$ and $F_2$ , which quantify how momentum is distributed among the nucleon’s constituent quarks and gluons. These functions essentially map the nucleon’s internal architecture, revealing the probabilities of finding a particular quark or gluon carrying a specific fraction of the nucleon’s total momentum. Consequently, a precise determination of the Hadronic Tensor allows for a detailed characterization of the nucleon’s response to high-energy interactions and provides crucial insights into the strong force that binds these particles together.

Structure Functions, $F_1$ and $F_2$ , serve as a crucial window into the nucleon’s internal architecture by revealing how momentum is distributed amongst its constituent particles – quarks and gluons. These functions don’t depict static images, but rather probability distributions, detailing the likelihood of finding a particular constituent carrying a specific fraction of the nucleon’s total momentum. This momentum distribution is formally described by Parton Distribution Functions (PDFs), which are fundamental quantities in quantum chromodynamics. Understanding PDFs is critical for interpreting high-energy scattering experiments, as they allow physicists to predict the outcome of collisions involving nucleons and to probe the strong force that binds these constituents together. Essentially, the Structure Functions provide the experimental access point, while the PDFs represent the theoretical framework for deciphering the nucleon’s dynamic internal landscape.

The current methodology offers a fundamentally new way to determine nucleon structure, providing calculations directly from first principles – an “ab-initio” approach – rather than relying on adjustable parameters or simplified assumptions. This is particularly valuable because it complements existing strategies for understanding the proton and neutron. While experimental measurements, like deep inelastic scattering, provide crucial data, they require interpretation through theoretical frameworks. Similarly, phenomenological models, built to describe observed data, often contain inherent biases. This ab-initio determination, rooted in the fundamental equations of quantum chromodynamics, serves as an independent check on both experimental results and existing models, ultimately refining the precision and reliability of Parton Distribution Functions – essential quantities describing the momentum distribution of quarks and gluons within the nucleon and, consequently, its interactions.

To rigorously assess the reliability of the calculated hadronic tensor and subsequent structure functions, the study employed a systematic variation in the lattice spacing through calculations performed with timeslice separations of 8a, 10a, and 12a. This approach allowed researchers to investigate potential discretization effects – systematic errors arising from the finite spacing between lattice points. By comparing results obtained from these different timeslice separations, any dependence on the lattice spacing could be identified and quantified, ensuring the final determination of parton distribution functions is robust and minimizes uncertainties stemming from numerical approximations. This careful control over systematic effects represents a crucial step towards obtaining accurate, ab initio insights into the nucleon’s internal structure.

The Euclidean structure functions <span class="katex-eq" data-katex-display="false">A_A(a)</span> and <span class="katex-eq" data-katex-display="false">B_B(b)</span> exhibit a dependence on <span class="katex-eq" data-katex-display="false">|\vec{q}|</span> for various values of τ at <span class="katex-eq" data-katex-display="false">t=10</span>a, demonstrating the behavior of electromagnetic currents within an unpolarized proton. — The Euclidean structure functions $A_A(a)$ and $B_B(b)$ exhibit a dependence on $|\vec{q}|$ for various values of τ at $t=10$ a, demonstrating the behavior of electromagnetic currents within an unpolarized proton.

The pursuit of extracting structure functions from lattice QCD, as detailed in this work, exemplifies a dedication to clarity and precision. It’s a process of revealing underlying order from complex interactions-a principle echoing Bertrand Russell’s observation, “The point of contact between the aesthetic and the logical is that both deal with structure.” This calculation of the hadronic tensor using four-point functions isn’t merely about obtaining numerical results; it’s about constructing a comprehensible framework for understanding deep inelastic scattering. The elegance lies in the method’s ability to translate the complexities of quantum chromodynamics into a form that yields observable, meaningful data, making the system durable and comprehensible.

The Path Forward

The calculation of hadronic tensors, as demonstrated, is not merely an exercise in computational physics; it is an attempt to reconcile the abstract beauty of quantum chromodynamics with the messy reality of nucleon structure. While the presented four-point function approach offers a direct route to structure functions, the limitations are, predictably, illuminating. Current statistics demand refinement, and the extension to higher momentum transfer-the truly interesting regime-remains a considerable challenge. A good interface is invisible to the user, yet felt; similarly, a robust theoretical calculation should yield physical results without shouting its internal struggles.

Future progress hinges not simply on increased computational power, though that will undoubtedly be necessary, but on elegant algorithmic improvements. One envisions, for example, a more efficient treatment of the Wick contractions, or perhaps a clever application of machine learning to accelerate the analysis of the lattice data. Every change should be justified by beauty and clarity. The pursuit of precision, therefore, must be tempered by a commitment to simplicity, lest the calculation become an opaque monument to its own complexity.

Ultimately, this work is a step towards a complete understanding of the nucleon’s internal landscape. The goal is not merely to reproduce existing experimental data, but to predict new phenomena, to reveal the hidden symmetries and dynamics that govern the strong interaction. It is a slow, painstaking process, but one guided by the conviction that the universe, at its deepest level, is fundamentally elegant.

Original article: https://arxiv.org/pdf/2602.22379.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

Unveiling the Nucleon: A Journey into Hadronic Structure

Lattice QCD: A First-Principles Approach to Hadronic Calculations

Computational Techniques: Extracting Precision from Four-Point Functions

Unveiling Nucleon Structure: Towards Precise Parton Distribution Functions

The Path Forward

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