Spin and momentum fraction carried by partons in the nucleon

Technical Summary: Spin and Momentum Fraction Carried by Partons in the Nucleon

Problem Statement
The fundamental origin of the proton's momentum and spin remains a central question in nucleon and particle physics. Since the European Muon Collaboration (EMC) experiment in the late 1980s revealed that valence quarks account for only a small fraction of the proton's spin, the "proton spin puzzle" has driven decades of theoretical and experimental effort. While phenomenological analyses via deep inelastic scattering (DIS) and global parton distribution function (PDF) fits provide constraints, a first-principles determination of how momentum and angular momentum are distributed among quarks (flavors $u, d, s, c$) and gluons within Quantum Chromodynamics (QCD) has been challenging. Previous lattice QCD studies often relied on unphysical quark masses requiring chiral extrapolations or lacked a continuum limit, introducing systematic uncertainties.

Methodology
This work presents a first-principles calculation of the proton's momentum and angular momentum decomposition using lattice QCD. The study employs four $N_f = 2+1+1$ twisted mass clover-improved fermion ensembles containing up, down, strange, and charm quarks. Crucially, the quark masses are tuned to their physical values, and the ensembles feature similar physical volumes but different lattice spacings ($a \approx 0.049$ to $0.079$ fm). This setup allows for a direct continuum extrapolation at the physical pion mass point without chiral extrapolation.

The calculation focuses on the matrix elements of the QCD energy-momentum tensor (EMT), $T^{\mu\nu}$, decomposed into quark ($T^{\mu\nu}_q$) and gluon ($T^{\mu\nu}_g$) contributions. The authors compute nucleon two- and three-point functions, utilizing smearing techniques (stout smearing for gauge links) to enhance the ground-state overlap and improve signal-to-noise ratios. Both connected and disconnected contributions are evaluated using improved techniques.

Renormalization is performed non-perturbatively in the RI'-MOM scheme, followed by a perturbative conversion to the $\overline{\text{MS}}$ scheme at $\bar{\mu} = 2$ GeV. A significant methodological improvement over previous studies involves the full non-perturbative determination of the mixing matrix between the flavor-singlet quark EMT and the gluon EMT. Additionally, the authors implement improved renormalization conditions to suppress rotational breaking effects and mixing with gauge non-invariant operators.

The gravitational form factors (GFFs), $A_{20}(Q^2)$ and $B_{20}(Q^2)$, are extracted by solving an over-constrained system of linear equations. The momentum fraction $\langle x \rangle$ is obtained from $A_{20}(0)$, while the total angular momentum $J$ is derived from $J = \frac{1}{2}[A_{20}(0) + B_{20}(0)]$. The orbital angular momentum $L$ is subsequently calculated as $L = J - \frac{1}{2}\Delta\Sigma$, where the intrinsic spin $\frac{1}{2}\Delta\Sigma$ is taken from updated axial charge calculations. Continuum limits are established using both constant and linear fits in $a^2$, weighted by the Akaike Information Criterion (AIC).

Key Contributions

1. First Complete Continuum Limit Decomposition: This is the first lattice QCD study to provide a complete decomposition of the proton spin and momentum into all quark flavors ($u, d, s, c$) and gluons using only physical quark masses and a continuum extrapolation.
2. Non-Perturbative Mixing: The study fully determines the mixing between the flavor-singlet quark operator and the gluon operator non-perturbatively, a critical step for accurate singlet and gluon extraction.
3. Systematic Control: By utilizing four lattice spacings and physical pion masses, the work controls discretization effects and eliminates uncertainties associated with chiral extrapolations.
4. Intrinsic and Orbital Separation: The paper provides a simultaneous determination of intrinsic quark spin and orbital angular momentum contributions for all parton species.

Results
The authors report the following results in the $\overline{\text{MS}}$ scheme at 2 GeV:

• Momentum Sum Rule: The total momentum fraction is $\langle x \rangle_N = 0.995(60)(29)$, satisfying the momentum sum rule $\langle x \rangle_q + \langle x \rangle_g = 1$ within uncertainties. The gluon carries approximately 37.2% of the momentum, while the total quark contribution is 62.3%.
• Spin Sum Rule: The total angular momentum is $J_N = 0.507(43)(65)$, consistent with the proton spin of $1/2$.
• Flavor Decomposition:
  • Momentum: The up quark carries the largest fraction (35.7%), followed by the down quark (19.2%). Strange and charm quarks contribute ~4.9% and ~2.5%, respectively.
  • Spin: The up quark provides a significant positive contribution to the spin ($J_u \approx 0.239$), while the down quark contribution is small and positive ($J_d \approx 0.025$) due to a cancellation between a negative intrinsic spin and a large positive orbital angular momentum. The strange quark contribution is small and negative, while the charm contribution is consistent with zero.
  • Gluon: The gluon contributes approximately 41.9% to the total spin.
• Orbital Angular Momentum: The orbital angular momentum is found to be significant, with the down quark providing a large positive contribution ($L_d \approx 0.228$) that compensates for its negative intrinsic spin.

Significance and Claims
The paper claims to resolve the long-standing proton spin puzzle by demonstrating, from first principles, that valence quarks account for only about half of the proton's spin and momentum. The remaining contributions arise from sea quarks and gluons, with the gluon contribution to both sums being approximately 40%. The results confirm the EMC finding that valence quarks are not the sole carriers of proton spin.

The study validates the momentum and spin sum rules directly from QCD without relying on phenomenological assumptions. The results for momentum fractions and intrinsic quark spins are in agreement with global PDF analyses and other lattice QCD calculations, though this work achieves higher precision for the strange quark contribution (finding a non-zero negative value) and provides the first continuum-limit results for the orbital angular momentum decomposition. The detection of a small, non-zero charm quark contribution to the momentum fraction is also highlighted as an indication of intrinsic charm effects in the nucleon.

Ultimately, this work provides the first quantitatively complete parton decomposition of the proton's momentum and angular momentum directly from QCD, serving as a benchmark for future experimental programs such as the Electron-Ion Collider.