Abstract
We study the dynamical generation of resonances in isospin singlet channels with mixing between two- and four-quark states. To this end we generalise a Bethe-Salpeter approach to four-quark states employed previously \cite{Heupel:2012ua} to accommodate for mixing diagrams. The $q\bar{q}q\bar{q}$ and $q\bar{q}$ components of the Bethe-Salpeter wave function (with light quarks $q\in\{u,d\}$) 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 $\pi\pi$ 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 $\sigma/f_0(500)$ that emerged in the past two decades from dispersive approaches \cite{Pelaez:2015qba}.
Highlights
There is perhaps no other state in the low-energy spectrum of QCD that has been puzzled over so intensely over the past decades than the isoscalar-scalar meson, the f0ð500Þ or σ meson
We study the dynamical generation of resonances in isospin singlet channels with mixing between twoand four-quark states
At least for the isospin-singlet cases like the χc1ð3872Þ, substantial admixtures of cc components seem possible [26]. We address this question on the example of the isospin-singlet 0þþ meson with light quark content using the functional approach via Dyson-Schwinger (DSE) and Bethe-Salpeter equations (BSE)
Summary
There is perhaps no other state in the low-energy spectrum of QCD that has been puzzled over so intensely over the past decades than the isoscalar-scalar meson, the f0ð500Þ or σ meson. The notion that the multiplet of light scalar mesons is incompatible with a conventional qqpicture goes back some way [6]: by assuming a dominant four-quark structure, interesting properties like inverted mass hierarchies and decay patterns are naturally explained This picture is supported by effective theory studies and large-Nc arguments (see, e.g., [5,7,8,9,10,11,12] and references therein) as well as lattice calculations [13,14,15,16] and functional methods [17,18].
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