<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>σ</mml:mi></mml:math>-meson: Four-quark versus two-quark components and decay width in a Bethe-Salpeter approach
Nico Santowsky, Gernot Eichmann, Christian S. Fischer, Paul C. Wallbott, Richard Williams
Abstract
We study the dynamical generation of resonances in isospin singlet channels with mixing between two- and four-quark states. To this end, we generalize a Bethe-Salpeter approach to four-quark states employed previously to accommodate for mixing diagrams. The $q\overline{q}q\overline{q}$ and $q\overline{q}$ components of the Bethe-Salpeter wave function (with light quarks $q\ensuremath{\in}{u,d}$) are determined consistently in a symmetry-preserving truncation of the underlying Dyson-Schwinger equations. As a prominent example, we deal with the isospin-singlet ${0}^{++}$ meson with light quark content. We find that the $\ensuremath{\pi}\ensuremath{\pi}$ contribution of the four-quark component is mainly responsible for the low (real part of the) mass of the resulting state. We also study the analytic structure in the complex momentum plane and find a branch cut at the two-pion threshold and a singularity in the second Riemann sheet indicating a considerable decay width. Our findings are in excellent qualitative agreement with the general picture for the $\ensuremath{\sigma}/{f}_{0}(500)$ that emerged in the past two decades from dispersive approaches.