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Lattice QCD Calculation of Axial-Vector Generalized Parton Distribution Moments and Their Implications for Nucleon Spin Structure


Core Concepts
This research paper presents a novel lattice QCD calculation of axial-vector generalized parton distribution (GPD) moments, providing insights into the quark helicity, orbital angular momentum, and spin-orbit correlations within a nucleon.
Abstract
  • Bibliographic Information: Bhattacharya, S., Cichy, K., Constantinou, M., Gao, X., Metz, A., Miller, J., Mukherjee, S., Petreczky, P., Steffens, F., & Zhao, Y. (2024). Moments of Axial-Vector GPD from Lattice QCD: Quark Helicity, Orbital Angular Momentum, and Spin-Orbit Correlation. arXiv:2410.03539v1 [hep-lat].

  • Research Objective: This study aims to calculate the Mellin moments of the twist-2 axial-vector generalized parton distribution (GPD) eH(x, ξ, t) at zero skewness (ξ) using lattice QCD, and to analyze these moments to gain insights into the spin structure of the nucleon.

  • Methodology: The research employs a lattice QCD calculation based on an Nf = 2 + 1 + 1 twisted mass fermions ensemble with clover improvement. The analysis utilizes the short-distance factorization framework on ratio-scheme renormalized quasi-GPD matrix elements. Both iso-vector and iso-scalar cases are considered, utilizing next-to-leading-order perturbative matching.

  • Key Findings: The study successfully determines the Mellin moments of eH up to the fifth order for the first time. The results indicate a clear signal and expected t-dependence for these moments. The analysis of these moments provides insights into the quark helicity and orbital angular momentum contributions to the nucleon spin, as well as the spin-orbit correlations of the quarks.

  • Main Conclusions: The calculated moments of the axial-vector GPD offer valuable information about the nucleon spin structure. The study demonstrates the feasibility of extracting higher moments using the employed lattice QCD approach, paving the way for further investigations into the three-dimensional structure of hadrons.

  • Significance: This research significantly contributes to the understanding of nucleon spin dynamics by providing valuable data on axial-vector GPD moments. The findings have implications for interpreting experimental results from facilities like the Electron-Ion Collider and enhancing theoretical models of nucleon structure.

  • Limitations and Future Research: The study acknowledges limitations such as neglecting disconnected contributions and gluon mixing in the iso-scalar case. Future research could address these limitations and explore the impact of these contributions on the extracted moments. Additionally, extending the analysis to non-zero skewness would provide a more complete picture of the GPD and nucleon spin structure.

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Stats
The lattice QCD calculations are based on an Nf = 2 + 1 + 1 twisted mass fermions ensemble with a lattice spacing of a = 0.093 fm and a pion mass of mπ = 260 MeV. The study considers both the iso-vector and iso-scalar cases, utilizing next-to-leading-order perturbative matching. The analysis focuses on the zero-skewness axial-vector GPD eH(x, 0, t) over a range of momentum transfer values (t). The study extracts Mellin moments of eH up to the fifth order.
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Deeper Inquiries

How would the inclusion of disconnected contributions and gluon mixing in the iso-scalar case affect the calculated moments and the interpretation of nucleon spin structure?

Answer: Including disconnected contributions and gluon mixing in the iso-scalar case would significantly impact the calculated moments and the interpretation of nucleon spin structure. Here's how: Disconnected Contributions: These contributions arise from quark lines that are not directly connected to the three-point function operator. While computationally demanding, they are crucial for a complete picture of the nucleon's structure, especially for the iso-scalar case. Neglecting them could lead to an underestimation of the total quark contributions to the nucleon spin. Gluon Mixing: The iso-scalar axial-vector current is not a conserved current, leading to mixing with the gluon operator under renormalization. This mixing becomes increasingly important at higher orders in perturbation theory. Ignoring it can lead to an inaccurate determination of the iso-scalar moments, particularly at large momentum transfers. Impact on Interpretation: Quark Helicity: Disconnected contributions could shift the value of the iso-scalar quark helicity ΔΣ = Δu + Δd. A more accurate determination of ΔΣ would refine our understanding of the flavor decomposition of the nucleon spin. Orbital Angular Momentum: Gluon mixing can significantly affect the extraction of the iso-scalar angular momentum Jq. This, in turn, would impact the determination of the quark orbital angular momentum Lq, which is crucial for understanding the nucleon spin puzzle. Spin-Orbit Correlations: A more precise determination of both the quark helicity and orbital angular momentum, after accounting for disconnected contributions and gluon mixing, would lead to a more reliable calculation of the spin-orbit correlations Cq. This would provide a more complete picture of the interplay between quark spin and orbital motion within the nucleon. In summary, incorporating disconnected contributions and gluon mixing in the iso-scalar case is essential for a precise understanding of the nucleon spin structure. These contributions are expected to have a non-negligible impact on the calculated moments, particularly at higher orders and large momentum transfers, leading to a more accurate and nuanced picture of the nucleon's internal dynamics.

Could the observed t-dependence of the higher moments provide insights into the spatial distribution of quark orbital angular momentum within the nucleon?

Answer: Yes, the observed t-dependence of the higher moments of Generalized Parton Distributions (GPDs) can indeed provide valuable insights into the spatial distribution of quark orbital angular momentum (OAM) within the nucleon. Here's how: Fourier Transform and Impact Parameter Space: The t-dependence of GPDs is directly related to the transverse spatial distribution of partons within the nucleon through a Fourier transform. By analyzing the t-dependence of the GPD moments, we can gain information about the distribution of quarks in the transverse plane as a function of their longitudinal momentum fraction x. Higher Moments and OAM: Higher moments of GPDs, particularly the second moment, are sensitive to the quark OAM. The t-dependence of these higher moments can therefore reveal how the quark OAM is distributed in the transverse plane. For instance, a steeper t-dependence at higher moments might suggest a more localized distribution of quark OAM near the nucleon's center. Model-Independent Access: Lattice QCD calculations of GPD moments offer a model-independent way to access the spatial distribution of quark OAM. By studying the t-dependence of these moments, we can go beyond one-dimensional parton distribution functions (PDFs) and obtain a three-dimensional picture of the nucleon's internal structure. Connecting t-dependence to OAM distribution: The precise connection between the t-dependence of higher moments and the spatial distribution of quark OAM is complex and requires a careful analysis. However, some general insights can be obtained: Slope of t-dependence: A steeper slope at higher moments suggests a more compact spatial distribution of the corresponding quark OAM. Nodes in t-dependence: The presence of nodes (points where the moment changes sign) in the t-dependence can indicate transitions between regions of positive and negative OAM contributions within the transverse plane. In conclusion, the t-dependence of higher GPD moments provides a valuable window into the spatial distribution of quark OAM within the nucleon. By combining lattice QCD calculations with theoretical insights, we can use this information to construct a more complete and nuanced picture of the nucleon's three-dimensional structure and the role of quark OAM in shaping it.

What are the potential implications of these findings for understanding the behavior of matter in extreme conditions, such as those found in neutron stars?

Answer: The findings from these Lattice QCD calculations of GPD moments, particularly those related to quark angular momentum, have significant potential implications for understanding the behavior of matter under extreme conditions, such as those found in neutron stars. Here's why: Equation of State: The equation of state (EOS) of dense nuclear matter is crucial for modeling neutron stars. It describes the pressure and energy density of matter as a function of density. The internal structure of nucleons, including the contribution of quark angular momentum, can influence the EOS, especially at high densities where nucleons are expected to overlap significantly. Neutron Star Structure: The structure and properties of neutron stars, such as their mass-radius relationship and moments of inertia, are sensitive to the EOS. A better understanding of the nucleon spin structure, including the role of quark OAM, can lead to more accurate predictions of neutron star observables and constrain their internal composition. Cooling Mechanisms: Neutron stars cool down over time through various processes, including neutrino emission. The efficiency of these cooling mechanisms can be affected by the properties of dense nuclear matter, including the spin structure of nucleons. A detailed understanding of quark angular momentum contributions could refine models of neutron star cooling. Phase Transitions: At extremely high densities, nuclear matter might undergo phase transitions to exotic states, such as quark matter. The properties of these phases, and the conditions under which they form, could be influenced by the internal structure of nucleons, including the spatial distribution of quark angular momentum. Connecting Nucleon Structure to Neutron Stars: The connection between nucleon structure and neutron star properties is complex and requires sophisticated theoretical modeling. However, the insights gained from lattice QCD calculations of GPD moments can provide valuable input for: Effective Field Theory Approaches: These approaches use the properties of nucleons as building blocks to describe nuclear matter at high densities. Information about quark angular momentum can be incorporated into these models to improve their predictive power. First-Principles Simulations: While computationally challenging, efforts are underway to simulate dense nuclear matter directly from QCD. Lattice QCD calculations of nucleon structure can provide benchmarks and guidance for these simulations. In conclusion, a deeper understanding of nucleon spin structure, including the spatial distribution of quark angular momentum, is crucial for unraveling the mysteries of matter under extreme conditions. The findings from these lattice QCD calculations can contribute to more accurate models of neutron stars, shedding light on their formation, evolution, and ultimate fate.
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