Insights into Meson and Baryon Structure using Continuum Schwinger Function Methods

TL;DR

The paper surveys how continuum Schwinger function methods reveal Emergent Hadron Mass in QCD by unifying the gluon mass scale, a process-independent effective charge, and dressed-quark masses into a single, nonperturbative picture of hadron structure. It presents parameter-free predictions for pion, kaon, and nucleon electromagnetic and gravitational form factors, including a gluon-driven mass scale $m_{hat} \approx 0.43$ GeV and a running coupling $\alpha_{hat}(k^2)$, yielding $M(k^2)$ and a light-quark constituent-like mass $M(0) \approx m_N/3$. Key results include a zero in the proton electric form factor $G_E^p(Q^2)$ near $Q^2 \sim 9$ GeV$^2$, a nonzero neutron electric form factor, and a positive-definite mass form factor ${\cal M}(Q^2)$ up to at least $Q^2=100$ GeV$^2$, all consistent with lattice-QCD and data-driven inferences. The framework connects QCD dynamics to measurable observables and outlines tests for future high-luminosity facilities (e.g., JLab, EIC) that can validate QCD as a fundamental, confining four-dimensional quantum field theory and illuminate the mechanism of Emergent Hadron Mass.

Abstract

The bulk of visible mass is supposed to emerge from nonperturbative dynamics within quantum chromodynamics (QCD). Following years of development and refinement, continuum and lattice Schwinger function methods have recently joined in revealing the three pillars that support this emergent hadron mass (EHM); namely, a nonzero gluon mass-scale, a process-independent effective charge, and dressed-quarks with running masses that take constituent-like values at infrared momenta. One may argue that EHM and confinement are inextricably linked; and theory is now working to expose their manifold expressions in hadron observables and highlight the types of measurements that can be made in order to validate the paradigm. This contribution sketches these ideas via the unified explanation of pion and proton electromagnetic and gravitational form factors.

Figures (6)

• Figure 1: Pion elastic electromagnetic form factor, $Q^2 F_\pi(Q^2)$. Legend. Purple curves -- CSM predictions from Ref. Yao:2024drm; dashed green curve -- CSM results obtained using perturbation theory integral representations Chang:2013niaGao:2017mmp; grey down-triangles -- lattice-regularised QCD Ding:2023fac. Data (gold) -- diamond Horn:2007ug; circles and squares Huber:2008id. Panel B only. Dot-dashed red curve -- monopole with mass fixed by empirical pion charge radius Dally:1982zkAmendolia:1986wjCui:2021aee: $r_\pi \approx 0.66\,$fm. Further: green up triangle -- estimated uncertainty of forthcoming JLab measurement at the highest accessible $Q^2$ point E12-19-006; and black asterisks -- anticipated uncertainty of EIC data, whose coverage should extend to $Q^2 \approx 35\,$GeV$^2$Aguilar:2019tebArrington:2021biu. The central magnitude of these points was chosen arbitrarily.
• Figure 2: Panel A. SPM prediction for $\mu_p G_E^p(Q^2)/ G_M^p(Q^2)$. The image also depicts modern parameter-free Faddeev equation predictions Yao:2024uej and the result obtained via a subjective phenomenological fit to the world's electron + nucleon scattering data Ye:2017gyb. Panel B. CSM predictions for $\mu_n G_E^n/G_M^n$: Fad-I -- dashed orange curve within like-coloured band; and Fad-II -- solid red curve within like-coloured band. (See Ref. Yao:2024uej for details.) Data: proton -- Refs. Jones:1999rzGayou:2001qdPunjabi:2005wqPuckett:2010acPuckett:2017flj; and neutron -- Refs. Madey:2003avRiordan:2010id.
• Figure 3: Nucleon gravitational form factors. Curves -- CSM predictions Yao:2024ixu. The bracketing bands mark the extent of $1\sigma$ SPM uncertainty. Each overall (species-summed) result is independent of resolving scale, $\zeta$; but, naturally, species decompositions evolve with $\zeta$. Points in each panel -- lattice-QCD results reproduced from Ref. Hackett:2023rif: black squares -- total form factor; red diamonds -- quark; blue circles -- glue component.
• Figure 4: Panel A. Sketch representing the relative sizes of the in-proton spacetime volumes occupied by quark and gluon mass-energy distributions at resolving scale $\zeta_2$. Panel B. Combined as required, the scale-dependent distributions in the upper panel yield the scale-invariant (observable) mass-energy distribution (purple 3/4-sphere) in this image. Plainly, the spacetime volume occupied by the proton mass-energy distribution is far smaller than that occupied by the charge distribution (grey 3/4-sphere). (Images based on information in Eqs. \ref{['RadOrder']}, \ref{['z2ratio']}.)
• Figure 5: Combined $A$, $J$, $D$ lattice-QCD ${\mathpzc g}/{\mathpzc q}$ results (grey points): grey line -- uncertainty weighted average of all lattice points; and grey band -- $1\sigma$ around the central value: $0.82(18)$. CSM prediction in Eq. \ref{['gonqconstant']} -- solid purple line and band. Resolving scale: $\zeta = \zeta_2$. Separated function comparisons are drawn in Ref. Yao:2024ixu.