Comment on "$\sigma$-meson: Four-quark versus two-quark components and decay width in a Bethe-Salpeter approach"

TL;DR

This paper critiques a recent covariant approach to tetraquark modeling, highlighting inconsistencies in the derived equations that challenge the validity of the proposed Bethe-Salpeter kernels.

Contribution

It identifies fundamental flaws in the equations used to describe tetraquarks, specifically their reducibility, which questions the approach's correctness.

Findings

The equations imply a reducible $q\bar{q}$ Bethe-Salpeter kernel.

The derived equations are inconsistent with established quantum field theory principles.

The critique suggests the need for revised formulations in tetraquark studies.

Abstract

In a recent paper by N. Santowsky et al. [Phys. Rev. D 102, 056014 (2020)], covariant coupled equations were derived to describe a tetraquark in terms of a mix of four-quark states $2q 2\bar{q}$ and two-quark states $q\bar{q}$. These equations were expressed in terms of vertices describing the disintegration of a tetraquark into identical two-meson states, into a diquark-antidiquark pair, and into a quark-antiquark pair. We show that these equations are inconsistent as they imply a $q\bar{q}$ Bethe-Salpeter kernel that is $q\bar{q}$-reducible.