Extraordinary nature of the nucleon scalar charge and its densities as a signal of nontrivial vacuum structure of QCD

TL;DR

This paper explores the unusual behavior of the nucleon scalar charge and its densities, linking them to the nontrivial vacuum structure of QCD and suggesting a delta-function singularity in the twist-3 PDF as a signal of this complex vacuum.

Contribution

It introduces the connection between nucleon scalar densities and the QCD vacuum structure, proposing a novel interpretation of the twist-3 PDF $e(x)$ as having a delta-function singularity at zero.

Findings

Nucleon scalar density approaches a nonzero constant at infinity.

The twist-3 PDF $e(x)$ likely contains a delta-function at $x=0$.

The singularity indicates a nontrivial QCD vacuum structure.

Abstract

It is widely known that the nucleon scalar charge is proportional to the pion-nucleon sigma term as one of the important low energy observables of QCD. Especially interesting to us is the physics of the nucleon scalar charge densities. This comes from the fact that the corresponding operator has the same quantum number as the physical vacuum. It indicates unusual behavior of the the nucleon scalar density as a function of the distance $r$ from the nucleon center. Namely, it would not be reduced down to zero at the spatial infinity but rather approaches some nonzero constant corresponding to the vacuum quark condensate. Naturally, this unique nature of the nucleon scalar density in the position space also affects the corresponding density in the momentum space, i.e. the corresponding parton distribution function (PDF) as a function of the Bjorken variable $x$. This PDF is known as the chiral-odd twist-3 PDF $e (x)$. We argue that $e(x)$ is likely to have a delta-function type singularity at $x=0$, and that the appearance of this singularity can be interpreted as a signal of the nontrivial vacuum structure of the QCD.