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Gluon Gravitational $ D$-Form Factor: The $σ$-Meson as a Dilaton Confronted with Lattice Data II

Roy Stegeman, Roman Zwicky

TL;DR

The paper tests the dilaton effective theory for QCD by analyzing gluon gravitational D-form factors $D^H(q^2)$ of light and strange hadrons via lattice QCD at $m_\pi \approx 450$ MeV and $m_\pi \approx 170$ MeV. In the dilaton EFT, $D^H(q^2)$ contains a sigma-pole term $r_{\sigma}^H/(q^2 - m_\sigma^2)$ (with specific forms for bosons, e.g. $D^\pi(q^2) = (r_{\sigma}^\pi q^2)/(q^2 - m_\sigma^2) - 1$) plus a background, and the residues $r_{\sigma}^H$ follow universal soft-breaking patterns; representative values such as $r_{\sigma}^\pi = 2/3$, $r_{\sigma}^N = 0.91\ \text{GeV}^2$, $r_{\sigma}^\rho = 0.35\ \text{GeV}^2$, and $r_{\sigma}^\Delta = 1.77\ \text{GeV}^2$ are extracted, with gluon- fractions $z_g$ around $0.64$ at $\mu=2$ GeV guiding the separation into gluon components. Fits to the lattice data with $m_\sigma(450\text{ MeV}) = 850(50)$ MeV and $m_\sigma(170\text{ MeV}) = 550(50)$ MeV yield good agreement (\chi^2/ d.o.f. ~ 1–2), reinforcing the picture that the $\sigma$ acts as a (pseudo) dilaton near an infrared fixed point, while heavy hadrons show deviations due to explicit scale-breaking.

Abstract

We investigate the gluon gravitational form factors of the $π$, $N$, $ρ$, and $Δ$ using lattice QCD data at $m_π\approx 450 \text{MeV}$ and $m_π\approx 170 \text{MeV}$. We base the analysis on fits to a simple $σ/f_0(500)$-meson pole, supplemented by a polynomial background term. The fitted residues agree with predictions from dilaton effective theory, in which the $σ$-meson acts as the dilaton, the pseudo Goldstone boson of spontaneously broken scale symmetry. We derive new dilaton-based predictions for the $ρ$- and $Δ$-gravitational form factors, and comment on the $η_{c}$- and $η_b$-form factors in the context of the dilaton interpretation. These results reinforce our earlier findings, based on lattice total (quark and gluon) gravitational form factors, and provide further evidence that QCD dynamics may be governed by an infrared fixed point.

Gluon Gravitational $ D$-Form Factor: The $σ$-Meson as a Dilaton Confronted with Lattice Data II

TL;DR

The paper tests the dilaton effective theory for QCD by analyzing gluon gravitational D-form factors of light and strange hadrons via lattice QCD at MeV and MeV. In the dilaton EFT, contains a sigma-pole term (with specific forms for bosons, e.g. ) plus a background, and the residues follow universal soft-breaking patterns; representative values such as , , , and are extracted, with gluon- fractions around at GeV guiding the separation into gluon components. Fits to the lattice data with MeV and MeV yield good agreement (\chi^2/ d.o.f. ~ 1–2), reinforcing the picture that the acts as a (pseudo) dilaton near an infrared fixed point, while heavy hadrons show deviations due to explicit scale-breaking.

Abstract

We investigate the gluon gravitational form factors of the , , , and using lattice QCD data at and . We base the analysis on fits to a simple -meson pole, supplemented by a polynomial background term. The fitted residues agree with predictions from dilaton effective theory, in which the -meson acts as the dilaton, the pseudo Goldstone boson of spontaneously broken scale symmetry. We derive new dilaton-based predictions for the - and -gravitational form factors, and comment on the - and -form factors in the context of the dilaton interpretation. These results reinforce our earlier findings, based on lattice total (quark and gluon) gravitational form factors, and provide further evidence that QCD dynamics may be governed by an infrared fixed point.

Paper Structure

This paper contains 17 sections, 48 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Gravitational Form Factors in Dilaton Effective Theory
  3. Spin-$1$ and spin-$\frac{3}{2}$ dilaton couplings with soft breaking
  4. Explicit gravitational form factors for all relevant spins
  5. The gluon-fraction of the $\sigma$-meson decay constant
  6. Dilaton Residues of the $\pi$, $N$, $\rho$ and $\Delta$ at $m_\pi \approx 450 \,\hbox{MeV}$
  7. The $\pi$-meson residue
  8. The $N$-baryon residue
  9. The $\rho$-meson residue
  10. The $\Delta$-baryon residue
  11. Fits to the Lattice Data
  12. Fit results and interpretation
  13. The dilaton interpretation of the $D(0) < 0$ hypothesis
  14. Heavy Hadron $D$-Form Factors are Different
  15. Conclusions and Discussion
  16. ...and 2 more sections

Figures (4)

  • Figure 1: Illustration of the dilaton Goldberger-Treiman mechanism.
  • Figure 2: Fits using the ansatz \ref{['eq:Bansatz']} for the baryons and \ref{['eq:Mansatz']} for the mesons, compared to the MIT lattice QCD data Pefkou:2021fni (first four) and Hackett:2023nkrHackett:2023rif (last two). Corresponding fit-parameters are given in Tab. \ref{['tab:450']}. The dark green line is the best-fit result and the light green band denotes the $68\%$-confidence interval. See footnote \ref{['foot:conv']} for our conventions compared to Pefkou:2021fni, explaining the sign-difference in the $\rho$-plot.
  • Figure 3: The gluon-residue fits for the $m_\pi \approx 450 \,\hbox{MeV}$ lattice data (first four) and the $m_\pi \approx 170 \,\hbox{MeV}$ lattice data (remaining two), compared against the dilaton effective theory predictions, see Tab. \ref{['tab:450']}. To gauge the agreement it is worthwhile to consider the $\chi^2$/d.o.f. in that table.
  • Figure 4: Ratio of gluon- to quark-part of the nucleon Hackett:2023nkr (left) and the pion Hackett:2023rif (right) form factors of the MIT-data at $m_\pi \approx 170\,\hbox{MeV}$, for $\mu = 2 \,\hbox{GeV}$ in the $\overline{\text{MS}}$-scheme. We note that $A$ and $J$ are more or less constant and roughly equal whereas the $D$-form factor raises in the gluon-part towards zero momentum transfer.