Size and Distance Scales in the Nucleon and in Dense Baryonic Matter

Abstract: This memorial tribute to Mannque Rho follows a line of thoughts and ideas that he continuously inspired and shaped over many decades: from the two-scales picture of low-energy nucleon structure to dense and cold baryonic matter as it is realized in the cores of neutron stars. Early groundbreaking concepts are recalled and updated by recent advanced analyses of the core' andcloud' sizes of the nucleon. Implications for dense nuclear matter are then discussed and confronted with empirical information from Bayesian inference analyses of neutron star observables.

• The paper demonstrates a common compact core radius (Rs = 0.50 ±0.01 fm) amidst mesonic clouds (Rs = 0.8–0.9 fm) using three independent form factors of the nucleon—isoscalar electric, axial, and mass.
• The core-plus-cloud structure allows baryonic cores to remain separated even at maximum densities, indicating evidence of quark deconfinement inside dense cores.
• In addressing neutron star compositions, the analysis found evidence for positive conformality across neutron stars and detailed equation-of-state constraints

Overview

This paper by Wolfram Weise, written as a memorial tribute to Mannque Rho, synthesizes two research threads that Rho shaped over his career: the two-scales ("core plus cloud") picture of the nucleon, and the physics of dense cold baryonic matter in neutron star cores. The central quantitative claim is that three independent form factors of the nucleon—isoscalar electric, axial, and mass (gravitational)—all point to a common compact core radius near 0.5 fm, surrounded by mesonic clouds that extend the charge radii to 0.8–0.9 fm. The paper then argues that this scale separation has direct consequences for dense matter: baryonic cores remain separated even at the maximum densities reachable in neutron stars, so quark deconfinement is not expected inside them.

The half-fermi core from spectral analyses

The analysis rests on unsubtracted dispersion relations for each form factor $G(q^2)$, whose mean-squared radii receive contributions split at a delineation scale $t_c \sim 1$ GeV$^2$ into low-mass (mesonic cloud) and high-mass (core) spectral regions.

Isoscalar electric form factor. The empirical inputs are $\sqrt{\langle r_S^2\rangle} = 0.78$ fm and $\sqrt{\langle r_V^2\rangle} = 0.90$ fm, built from the proton charge radius ($0.840 \pm 0.003 \pm 0.002$ fm) and neutron mean-squared radius. A minimal vector-meson-dominance estimate already gives $$\sqrt{\langle r_S^2\rangle_{\rm core}} \simeq 0.48 \pm 0.01$$ fm after subtracting the $\omega$ pole contribution $6/m_\omega^2$. The refined spectral analysis using precision fits to spacelike and timelike data yields a consistent core radius of $0.50 \pm 0.01$ fm.

A strong internal consistency check comes from the isovector channel: since the $t_c \sim 1$0 and $t_c \sim 1$1 valence cores should cancel in perfect isospin symmetry, the extracted isovector core radius should vanish—and it does, $t_c \sim 1$2 fm$t_c \sim 1$3. This cancellation supports the choice of delineation scale and indirectly confirms the core-plus-cloud decomposition; the entire isovector radius arises from the interacting two-pion cloud governed by the $t_c \sim 1$4 meson.

Axial form factor. Using the empirical axial radius ($t_c \sim 1$5 fm$t_c \sim 1$6 from dipole fits) and modeling the broad $t_c \sim 1$7 spectral function with its self-energy constrained by $t_c \sim 1$8 data, the axial core radius comes out as $t_c \sim 1$9 fm. The caveat here is that the uncertainty grows to roughly 25% if the more conservative "unbiased" fit to neutrino-deuteron scattering and muon capture data is used instead.

Mass form factor. The mass (gravitational) form factor, probed through threshold $^2$0 photoproduction where the amplitude couples to the QCD trace anomaly, carries most of the nucleon mass in its gluonic component $^2$1, with sigma terms contributing less than 10%. GlueX measurements give a mass radius $^2$2 fm. After correcting for the small $^2$3 and $^2$4 cloud pieces—whose sizes depend on the still-unsettled pion-nucleon sigma term ($^2$5–$^2$6 MeV across lattice and phenomenological determinations)—the gluonic core radius is $^2$7 fm, with an admittedly conservative error estimate that may be underestimated due to dipole-fit model dependence.

Taken together, these three independent analyses converge on what the paper calls a "half-fermi rule" for the nucleon core. As supporting context, lattice QCD computations of the scalar glueball's gravitational form factor give an even smaller mass radius of $^2$8 fm, suggesting that injecting three valence quarks into a compact gluonic energy density could set the observed nucleon-core scale.

The implication is twofold: there is no single universal "size" of the nucleon—the differing empirical radii are accounted for by distinct mesonic cloud components—and the localization of nearly massless valence quarks within a volume far smaller than $^2$9 fm$\sqrt{\langle r_S^2\rangle} = 0.78$0 implies spontaneously broken chiral symmetry in the core's vicinity.

Bayesian inference of the neutron-star equation of state

The second part confronts dense-matter questions with data-driven EoS constraints. The method parametrizes the squared sound speed $\sqrt{\langle r_S^2\rangle} = 0.78$1 segment-wise, converts trial EoSs through the Tolman-Oppenheimer-Volkov equations, and applies Bayesian inference against a database including heavy pulsars (up to the $\sqrt{\langle r_S^2\rangle} = 0.78$2 PSR J0952-0607), NICER mass-radius measurements, and GW170817 tidal deformabilities. Notably, chiral EFT constraints are imposed as a likelihood rather than a prior, at densities $\sqrt{\langle r_S^2\rangle} = 0.78$3, to avoid biasing the posterior toward hadronic assumptions near the edge of ChEFT validity.

The key results are:

Quantity	Result
Sound speed	Median exceeds conformal limit $\sqrt{\langle r_S^2\rangle} = 0.78$4 at $\sqrt{\langle r_S^2\rangle} = 0.78$5–$\sqrt{\langle r_S^2\rangle} = 0.78$6
Evidence for $\sqrt{\langle r_S^2\rangle} = 0.78$7 in all stellar centers	Bayes factor well above $\sqrt{\langle r_S^2\rangle} = 0.78$8
Neutron star radii	$\sqrt{\langle r_S^2\rangle} = 0.78$9 km median, nearly mass-independent; 95% band 11–13 km
Central density, $\sqrt{\langle r_V^2\rangle} = 0.90$0	$\sqrt{\langle r_V^2\rangle} = 0.90$1
Central density, $\sqrt{\langle r_V^2\rangle} = 0.90$2	$\sqrt{\langle r_V^2\rangle} = 0.90$3
Maximum central density	$\sqrt{\langle r_V^2\rangle} = 0.90$4, i.e. $\sqrt{\langle r_V^2\rangle} = 0.90$5 GeV/fm$\sqrt{\langle r_V^2\rangle} = 0.90$6

The super-conformal sound speed implies a stiff EoS capable of supporting the heaviest observed stars, naturally explained by repulsive correlations in a fermionic many-body system. The trace anomaly measure $\sqrt{\langle r_V^2\rangle} = 0.90$7 remains positive throughout the empirically constrained range; a Bayes-factor analysis gives strong evidence for negative $\sqrt{\langle r_V^2\rangle} = 0.90$8 at $\sqrt{\langle r_V^2\rangle} = 0.90$9–$0.840 \pm 0.003 \pm 0.002$0, meaning conformality is not reached in any neutron star core.

Two consistency tests bolster confidence: machine-learning inference of $0.840 \pm 0.003 \pm 0.002$1 using polytrope networks reproduces the Bayesian bands, and the updated NICER radius for PSR J0437-4715 (shifted from $0.840 \pm 0.003 \pm 0.002$2 km to $0.840 \pm 0.003 \pm 0.002$3 km) leaves the inferred pressure bands essentially unchanged.

On phase transitions, the data-driven EoS excludes all examined strong first-order transition scenarios with broad Maxwell coexistence regions at the 95% credibility level; a Bayes factor above $0.840 \pm 0.003 \pm 0.002$4 excludes such transitions for stars below $0.840 \pm 0.003 \pm 0.002$5, rendering twin-star solutions highly improbable. The only first-order transition known in nuclear physics—the liquid-gas transition—disappears in neutron-rich matter anyway.

Distance scales in dense matter and the two-scales scenario

Combining the two threads, the paper estimates average baryon-baryon distances via a hexagonal-lattice packing argument with excluded-volume corrections calibrated to $0.840 \pm 0.003 \pm 0.002$6 fm. Even at $0.840 \pm 0.003 \pm 0.002$7, distances remain slightly above 1 fm—larger than the 0.5 fm core diameter scale but comparable to the cloud extension. Since the mesonic cloud size grows with density as $0.840 \pm 0.003 \pm 0.002$8 decreases while the gluonic core stays stable (a NJL-type quark-diquark model shows less than 10% core growth up to saturation density and under 15% at $0.840 \pm 0.003 \pm 0.002$9), the scale separation $$\sqrt{\langle r_S^2\rangle_{\rm core}} \simeq 0.48 \pm 0.01$$0 widens with compression.

The resulting scenario proceeds through three regimes: isolated baryons with two-body exchange below $$\sqrt{\langle r_S^2\rangle_{\rm core}} \simeq 0.48 \pm 0.01$$1; delocalizing, percolating meson clouds generating many-body forces at $$\sqrt{\langle r_S^2\rangle_{\rm core}} \simeq 0.48 \pm 0.01$$2–$$\sqrt{\langle r_S^2\rangle_{\rm core}} \simeq 0.48 \pm 0.01$$3; and only beyond $$\sqrt{\langle r_S^2\rangle_{\rm core}} \simeq 0.48 \pm 0.01$$4—beyond any neutron-star interior—touching and overlap of valence-quark cores, which must additionally overcome short-range NN repulsion before deconfinement. This directly contradicts early large-bag-model expectations of deconfinement near $$\sqrt{\langle r_S^2\rangle_{\rm core}} \simeq 0.48 \pm 0.01$$5, and reinforces Rho's Cheshire Cat viewpoint that no sharp, physically observable confinement boundary exists. Any eventual deconfinement at higher density would proceed, per prior work cited here, as a continuous crossover rather than a phase transition, accompanied by chiral symmetry restoration.

Limitations and open questions

Several caveats bear on the conclusions. The core-radius extraction depends on the delineation scale $$\sqrt{\langle r_S^2\rangle_{\rm core}} \simeq 0.48 \pm 0.01$$6 separating cloud and core spectral regions; the vanishing isovector core radius is offered as evidence that the choice is not grossly wrong, but the dependence is acknowledged as potentially model-based. The mass-radius result carries larger uncertainties (of order 10%) tied to dipole-fit assumptions in the GlueX extraction and to the inconsistent sigma-term determinations ($$\sqrt{\langle r_S^2\rangle_{\rm core}} \simeq 0.48 \pm 0.01$$7 ranging from about 44 to 56 MeV). The axial core radius relies on the $$\sqrt{\langle r_S^2\rangle_{\rm core}} \simeq 0.48 \pm 0.01$$8 pole approximation and inherits sizable experimental ambiguity if unbiased fits are used. On the astrophysical side, the EoS is empirically constrained only up to roughly $$\sqrt{\langle r_S^2\rangle_{\rm core}} \simeq 0.48 \pm 0.01$$9; extrapolations beyond carry no data support, and the inference cannot resolve which baryonic or quark degrees of freedom contribute to $\omega$0—only that their interactions must be strongly repulsive. The dense-matter scenario itself is qualitative: it assumes core stability from one model calculation and infers cloud expansion from chiral counting rules, without a first-principles treatment of percolating mesonic fields at $\omega$1–$\omega$2.

Conclusion

The paper consolidates modern form-factor analyses into a quantitatively supported two-scales picture: a common half-fermi core carrying baryon number and most of the nucleon mass via the gluonic trace anomaly, embedded in current-specific mesonic clouds that generate the observed spread of radii. When combined with Bayesian inference of neutron-star observables—which yield stiff EoSs, super-conformal sound speeds, central densities capped near $\omega$3, and no evidence for strong first-order transitions—it follows that baryonic cores in even the heaviest neutron stars never overlap, placing deconfinement beyond anything realized in cold stellar matter. The open question the framework leaves is a quantitative description of the intermediate regime where percolating meson clouds generate many-body forces while cores remain distinct.