[Paper Review] Gravitational form factors of the proton from lattice QCD

Daniel C. Hackett, Dimitra A. Pefkou+1 more|arXiv (Cornell University)|Oct 12, 2023

Superconducting Materials and Applications109 references4 citations

TL;DR

This lattice QCD study presents the first flavor-decomposed determination of the proton's gravitational form factors (GFFs)—A(t), J(t), and D(t)—in the momentum transfer range 0 ≤ −t ≤ 2 GeV², separating contributions from up, down, strange quarks, and gluons. It reveals the first-principles constraints on individual parton contributions to the proton's mass, spin, and mechanical structure, including the D-term and mechanical radii, with results consistent with recent experimental extractions for gluon and valence quark GFFs.

ABSTRACT

The gravitational form factors (GFFs) of a hadron encode fundamental aspects of its structure, including its shape and size as defined from e.g., its energy density. This work presents a determination of the flavor decomposition of the GFFs of the proton from lattice QCD, in the kinematic region $0\leq -t\leq 2~ ext{GeV}^2$. The decomposition into up-, down-, strange-quark, and gluon contributions provides first-principles constraints on the role of each constituent in generating key proton structure observables, such as its mechanical radius, mass radius, and $D$-term.

Motivation & Objective

• To determine the flavor decomposition of the proton's gravitational form factors (GFFs) A(t), J(t), and D(t) from first principles using lattice QCD.
• To quantify the individual contributions of up, down, strange quarks, and gluons to the proton's mechanical properties, including mass radius, mechanical radius, and D-term.
• To provide constraints on the t-dependence of GFFs for each parton species, enabling a deeper understanding of the proton's internal structure.
• To compute energy and radial force densities and associated mechanical radii from the GFFs, allowing comparison of different size measures of the proton.

Proposed method

• Lattice QCD simulations are performed on a single Nf=2+1 flavor gauge ensemble with pion mass ≈170 MeV, lattice spacing ≈0.091 fm, and volume 48³×96.
• Matrix elements of the energy-momentum tensor (EMT) are computed via three-point functions, with bare matrix elements extracted from ratios of three- and two-point functions at large Euclidean times.
• The EMT is decomposed into quark and gluon contributions, with flavor-specific components (singlet, non-singlet) used to isolate up, down, and strange quark contributions.
• Renormalization is performed in the $ar{\mathrm{MS}}$ scheme at μ=2 GeV using irreducible representation-specific coefficients from prior work.
• Fits to the t-dependence of GFFs are performed using both the n-pole model (n=2, dipole) and z-expansion parametrization, with χ²/d.o.f. used to select optimal fits.
• The mechanical radii and energy/force densities are computed from the renormalized GFFs using analytical expressions derived from the EMT matrix elements.

Experimental results

Research questions

• RQ1 What are the individual contributions of up, down, strange quarks, and gluons to the proton's gravitational form factors A(t), J(t), and D(t)?
• RQ2 How do the t-dependences of the flavor-separated GFFs compare to experimental extractions and theoretical expectations?
• RQ3 What are the mechanical radii and energy density distributions of the proton, and how do they differ from mass radii?
• RQ4 To what extent do the gluon and quark contributions to the D-term and mechanical structure agree with recent experimental data?
• RQ5 How do discretization effects influence the extracted GFFs, and what is the consistency between different irreducible representations in the lattice calculation?

Key findings

• The gluon GFF Dg(t) and the valence quark GFF Dv1(t) are consistent with recent experimental extractions from 2018 and 2020.
• The t-dependence of the gluon contribution to A(t), Ag(t), is consistent with one of two experimental analyses but in tension with the other, including an updated analysis.
• The individual up-, down-, and strange-quark GFFs are computed for the first time from first-principles lattice QCD, providing new constraints on their mechanical roles.
• The mechanical radius of the proton, derived from the GFFs, is found to be 0.83(10) fm, differing from the mass radius due to distinct contributions of energy and momentum densities.
• The D-term contributions from individual flavor channels are quantified: Dg(0) = -2.57(84), Dq(0) = -1.30(49), Dv1(0) ≈ 0.009(23), and Dv2(0) = -0.77(15), with the gluon contribution being dominant and negative.
• The z-expansion and dipole model fits to the GFFs yield consistent results, with the dipole model (n=2) selected based on χ²/d.o.f., and the z-expansion with kmax=2 providing the best fit quality.