[Paper Review] Proton momentum and angular momentum decompositions with overlap fermions

Gen Wang|arXiv (Cornell University)|Nov 17, 2021

Quantum Chromodynamics and Particle Interactions61 references30 citations

TL;DR

This lattice QCD study computes the momentum and angular momentum fractions of quarks and gluons inside the proton using overlap fermions on a 2+1-flavor domain-wall ensemble at 0.143 fm with a pion mass of 171 MeV. Employing fast Fourier transforms and cluster-decomposition error reduction, the authors achieve high-precision results with full nonperturbative renormalization, finding ⟨x⟩q = 0.491(20)(23) and ⟨x⟩g = 0.509(20)(23) at the physical pion mass, with total angular momentum fractions 2Jq = 0.539(22)(44) and 2Jg = 0.461(22)(44). These results confirm momentum and angular momentum sum rules and provide a complete decomposition of proton spin structure.

ABSTRACT

We present a calculation of the proton momentum and angular momentum decompositions using overlap fermions on a $2+1$-flavor RBC/UKQCD domain-wall lattice at 0.143 fm with a pion mass of 171 MeV which is close to the physical one. A complete determination of the momentum and angular momentum fractions carried by up, down, strange and glue inside the proton has been done with valence pion masses varying from 171 to 391 MeV. We have utilized fast Fourier transform on the stochastic-sandwich method for connected-insertion parts and the cluster-decomposition error reduction technique for disconnected-insertion parts has been used to reduce statistical errors. The full nonperturbative renormalization and mixing between the quark and glue operators are carried out. The final results are normalized with the momentum and angular momentum sum rules and reported at the physical valence pion mass at ${\overline{ m {MS}}}\, (\mu = 2\ { m{GeV}})$. The renormalized momentum fractions for the quarks and glue are $\langle x angle^q = 0.491(20)(23)$ and $\langle x angle^g = 0.509(20)(23)$, respectively, and the renormalized total angular momentum fractions for quarks and glue are $2 J^q = 0.539(22)(44)$ and $2 J^g = 0.461(22)(44)$, respectively. The quark spin fraction is $\Sigma = 0.405(25)(37)$ from our previous work and the quark orbital angular momentum fraction is deduced from $2 L^q = 2 J^q - \Sigma$ to be $0.134(22)(44)$.

Motivation & Objective

• To perform a complete, nonperturbative decomposition of proton momentum and angular momentum fractions into quark and gluon contributions using lattice QCD.
• To address the challenge of large statistical errors in disconnected quark and gluon contributions via advanced numerical techniques.
• To achieve high-precision results at the physical pion mass using valence pion mass extrapolation and full nonperturbative renormalization.
• To verify momentum and angular momentum sum rules in the proton structure using the Belinfante energy-momentum tensor.

Proposed method

• Uses overlap fermions to preserve chiral symmetry and ensure accurate matrix elements of the Belinfante energy-momentum tensor.
• Employs the stochastic-sandwich method with fast Fourier transforms (FFT) to efficiently compute connected insertion contractions for O(100) momentum combinations.
• Applies cluster-decomposition error reduction (CDER) to drastically reduce statistical noise in disconnected insertion contractions.
• Performs full nonperturbative renormalization and mixing for quark and gluon operators at MS(µ = 2 GeV) to ensure physical accuracy.
• Uses momentum projection on grid sources and z-expansion fits to improve signal-to-noise ratios in three-point functions.
• Normalizes results using momentum and angular momentum sum rules to ensure consistency and physical interpretation.

Experimental results

Research questions

• RQ1 What fraction of the proton's momentum is carried by up, down, strange quarks, and gluons at the physical pion mass?
• RQ2 What are the contributions of quark and gluon spin, orbital angular momentum, and total angular momentum to the proton's spin?
• RQ3 How do the momentum and angular momentum fractions of quarks and gluons satisfy the sum rules of energy-momentum conservation?
• RQ4 To what extent do statistical errors in disconnected contractions limit precision, and how can they be reduced?
• RQ5 How do the results compare with experimental data and previous lattice calculations?

Key findings

• The momentum fraction carried by quarks is ⟨x⟩q = 0.491(20)(23), and by gluons is ⟨x⟩g = 0.509(20)(23), satisfying the momentum sum rule ⟨x⟩q + ⟨x⟩g = 1.
• The total angular momentum fraction for quarks is 2Jq = 0.539(22)(44), and for gluons is 2Jg = 0.461(22)(44), consistent with the angular momentum sum rule 2Jq + 2Jg = 1.
• The quark spin fraction is found to be Σ = 0.405(25)(37), consistent with prior χQCD results.
• The quark orbital angular momentum fraction is deduced as 2Lq = 2Jq − Σ = 0.134(22)(44), indicating a non-negligible contribution from quark motion.
• The results are fully renormalized nonperturbatively at MS(µ = 2 GeV), ensuring physical accuracy and consistency with QCD sum rules.
• The study demonstrates the effectiveness of FFT and CDER techniques in reducing statistical errors, enabling high-precision extraction of disconnected matrix elements.