Energy-momentum tensor form factor D(t) of proton and neutron

TL;DR

This work shows that the D(t) form factor of charged hadrons acquires a universal QED-induced divergence D(t) ∝ 1/√{-t} as t → 0, while neutral hadrons like the neutron have a finite D(0). By constructing a neutron version of the Białynicki-Birula classical proton model and calibrating it to lattice QCD data, the authors demonstrate that proton and neutron D(t) are practically indistinguishable over a broad range of momentum transfers, and that electromagnetic effects on the proton’s D(t) become visible only at exceedingly small |t|. A regularization scheme is developed to make the proton D-term finite and comparable to the neutron’s, which aligns with lattice results and supports the practical use of a regularized D-term in phenomenology. The study also yields insights into EMT radii and proton–neutron size differences, showing the neutron is slightly smaller and that EMT radii provide a meaningful, electromagnetic-corrected notion of neutron size. Overall, the findings imply that, for current experiments and analyses, proton and neutron D(t) will look essentially identical, with QED effects effectively unreachable experimentally in the near term.

Abstract

The energy-momentum tensor (EMT) form factor $D(t)$ is finite and negative in hadronic models and lattice QCD when only strong forces are included. However, when electromagnetic forces are considered, the $D(t)$ of charged hadrons undergoes a dramatic change: at small $t$, it changes sign and diverges like $1/\sqrt{-t}$ as shown for the proton in the classical model by Białynicki-Birula based on residual nuclear forces which can be understood as a mean field approach. We construct an analogous neutron model and show that this framework accurately explains the electromagnetic proton-neutron mass difference. We demonstrate that, after appropriately rescaling the residual nuclear forces, the model can reproduce lattice data on the nucleon $D(t)$ up to $(-t)\lesssim 1\,$GeV$^2$ as well as QED effects. Based on this realistic model description, we show that the proton and neutron $D(t)$ form factors are practically indistinguishable down to $(-t) \approx 10^{-4}\rm GeV^2$ far below what can currently be accessed experimentally. We conclude that in the foreseeable future the $D(t)$ form factors of proton and neutron will practically look the same in experiments and phenomenology.

Figures (4)

• Figure 1: Dust distribution $\rho(r)$, scalar potential $g_{\rm s}\phi(r)$, vector potential $g_{\rm v} V_0(r)$, and Coulomb potential $eA_0(r)$ as functions of $r$ in the neutron (this work) and proton Varma:2020crx. The inserts illustrate that the fields are more strongly localized in the neutron.
• Figure 2: EMT densities $T_{00}(r)$, $s(r)$, $p(r)$ and normal force per unit area $\frac{2}{3}s(r)+p(r)$ as functions of $r$ for neutron (this work) and proton Varma:2020crx. The inserts show the outer regions $r\gtrsim 2\,\rm fm$ where the QED effects are manifest.
• Figure 3: Form factors $A(t)$ (a) and $D(t)$ (b) of neutron and proton vs. $(-t)$ in the classical model. (c) The same as panel (b) but with logarithmic $t$-scale to display the region $(-t)\ll 0.1\,\rm GeV^2$ where neutron and proton $D(t)$ exhibit different properties.
• Figure 4: (a) $D(t)$ of neutron and proton in the model with strong interaction parameters rescaled according to Eq. (\ref{['Eq:resc']}) by $\lambda_m = 0.85$ which yields the correct proton charge radius and $\lambda_g=3.2\pm 0.4$ which yields the displayed uncertainty band and a good description of the lattice data from Ref. Hackett:2023rif at $(-t)\lesssim 1\,{\rm GeV}^2$ that are shown in the figure for comparison. (b) The same as panel (a) but on a logarithmic $(-t)$-scale illustrating that the difference between $D(t)$ of neutron and proton becomes noticeable only for $(-t)\lesssim 10^{-4}\,\rm GeV^2$. (c) $D(t)$ of proton $(-t)\lesssim 10^{-6}\,\rm GeV^2$ in comparison to the QED prediction.