A convolution formalism for defining spatial densities of hadrons

TL;DR

This paper clarifies the concept of spatial densities of hadrons, showing how they relate to wave packets and internal structure, and introduces a convolution formalism to understand different density definitions.

Contribution

It introduces a convolution formalism that relates physical hadron densities to internal structure and wave packet effects, clarifying their interpretation.

Findings

Light front densities have a convolution structure.

Instant form densities approximate convolution for broad wave packets.

Relativistic corrections to the convolution formula are infinite.

Abstract

We clarify the meaning of spatial densities of hadrons. A physical density is given by the expectation value of a local operator for a physical state, and depends on both internal structure and the hadron's wave packet. In some particular cases, the physical density can be written as a convolution between a density function that depends on internal structure but not wave packet, and a smearing function that depends on wave packet but not internal structure. We show that the light front densities often encountered in the literature have this property but that instant form densities do not. For hadrons prepared in broad wave packets, physical instant form densities approximately obey such a convolution relation, with Breit frame densities as the apparent internal densities. However, there is an infinite series of relativistic corrections to this convolution formula.