Extraordinary Nature of the Nucleon Scalar Charge and Its Densities as a Signal of Nontrivial Vacuum Structure of QCD

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 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 approach 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 QCD.