Nucleon unpolarized second Mellin moments using lattice QCD ensembles with physical quark masses and in the continuum limit

Technical Summary: Nucleon Unpolarized Second Mellin Moments using Lattice QCD Ensembles with Physical Quark Masses and in the Continuum Limit

Problem and Motivation
Understanding the internal structure of the nucleon, specifically how quark and gluon constituents generate its mass, momentum, and spin, remains a central challenge in Quantum Chromodynamics (QCD). While the proton is composed of quarks and gluons, the precise decomposition of its momentum and angular momentum among these degrees of freedom requires a rigorous theoretical framework. The matrix elements of the QCD energy-momentum tensor (EMT) provide this framework, encoding the mechanical and dynamical structure of hadrons. At zero momentum transfer, these matrix elements yield the momentum fractions ($\langle x \rangle$) and, via Ji's sum rule, the total angular momentum ($J$) carried by quarks and gluons.

Previous lattice QCD calculations have made significant progress but often relied on simulations at unphysical pion masses, necessitating chiral extrapolations that introduce systematic uncertainties. Furthermore, the determination of the gluon contribution and the mixing between the quark singlet and gluon operators under renormalization have remained challenging due to signal-to-noise issues and the complexity of non-perturbative renormalization. A complete, first-principles determination of the flavor decomposition of nucleon momentum and spin, including the continuum limit at the physical pion mass, has been a long-standing goal.

Methodology
This work presents a lattice QCD determination of nucleon matrix elements of the quark and gluon energy-momentum tensors using four gauge ensembles generated with $N_f=2+1+1$ twisted-mass clover-improved fermions. The up, down, strange, and charm quark masses are tuned to approximately their physical values. The four ensembles (labeled B64, C80, D96, and E112) feature similar physical volumes but distinct lattice spacings ($a \approx 0.080, 0.068, 0.057,$ and $0.049$ fm), enabling a direct extrapolation to the continuum limit at the physical pion mass point.

The calculation involves:

1. Correlation Functions: Computation of nucleon two- and three-point functions to extract bare generalized form factors (GFFs), $A_{20}(Q^2)$ and $B_{20}(Q^2)$. Both connected and disconnected quark contributions, as well as gluon contributions, are evaluated.
2. Excited State Control: Analysis of effective masses and ratios of three- to two-point functions using two-state and three-state fits, as well as the summation method, to isolate ground-state matrix elements.
3. Renormalization: Non-perturbative determination of renormalization functions in the RI'-MOM scheme, including the full mixing matrix between the quark singlet and the gluon operator. This mixing is treated non-perturbatively, and the results are converted to the $\overline{\text{MS}}$ scheme at a scale of $\mu = 2$ GeV.
4. Continuum Extrapolation: Linear and constant extrapolations in $a^2$ are performed for all observables. The final results are obtained using an Akaike Information Criterion (AIC) model averaging of these extrapolations.
5. Systematic Uncertainties: Systematic errors arising from excited-state contamination, $Q^2$-dependence fits (dipole vs. constant), and the dependence on stout smearing steps for the gluon operator are quantified and added in quadrature to the statistical errors.

Key Contributions and Results
The paper provides the first continuum extrapolation of nucleon momentum and angular momentum decomposition directly at the physical pion mass, eliminating systematic errors associated with chiral extrapolation.

• Momentum Fractions: The total nucleon momentum fraction is found to be $\langle x \rangle_N = 0.995(60)(24)(1)(16)$, satisfying the momentum sum rule within uncertainties. The gluon contribution is approximately 40% ($\langle x \rangle_g \approx 0.372$), while the sea quark contribution is about 10%. The up-quark carries roughly twice the momentum of the down-quark ($\langle x \rangle_u \approx 0.357$, $\langle x \rangle_d \approx 0.192$).
• Angular Momentum: The total angular momentum is determined to be $J_N = 0.507(43)(63)(17)(0)$, consistent with the proton spin of $1/2$. The gluon contribution to the spin is approximately 40%, and the sea quark contribution is roughly 10%.
• Flavor Decomposition: The study provides a complete flavor decomposition for the form factor $B_{20}(0)$ and the orbital angular momentum $L_q$. The results show that $B_{20}(0)$ for up and down quarks have opposite signs and comparable magnitudes, leading to an increase in the up-quark angular momentum and a decrease in the down-quark angular momentum.
• Orbital Angular Momentum: Using intrinsic quark spin values from previous analyses, the orbital angular momentum $L_q$ is derived. The up-quark orbital angular momentum is found to be negative, reducing its total contribution to the spin, while the down-quark orbital angular momentum is positive, largely compensating for its negative intrinsic spin.

Significance
The authors claim that this work provides a comprehensive, first-principles determination of the quark and gluon decomposition of the nucleon's momentum and spin. By performing the continuum extrapolation at the physical pion mass and treating the quark-gluon mixing non-perturbatively, the study resolves systematic uncertainties that have plagued previous calculations. The results confirm the momentum and spin sum rules from first principles, offering a quantitative understanding of the nucleon's internal structure that is directly relevant for global parton distribution function analyses and precision studies at facilities such as the Electron-Ion Collider. The paper emphasizes that this full flavor decomposition, including a complete budget of systematic uncertainties, allows for a reliable comparison with phenomenological analyses.