$S$-wave $K{K}^{*}$ interactions in a finite volume and the $f_1(1285)$

Lattice QCD simulations provide a promising way to disentangle different interpretations of hadronic resonances, which might be of particular relevance to understand the nature of the so-called $XYZ$ particles. Recent studies have shown that in addition to the well-established naive quark model picture, the axial-vector meson $f_1(1285)$ can also be understood as a dynamically generated state built upon the $K{K}^{*}$ interaction. In this work, we calculate the energy levels of the $K{K}^{*}$ system in the $f_1(1285)$ channel in finite volume using the chiral unitary approach. We propose to calculate the loop function in the dimensional regularization scheme, which is equivalent to the hybrid approach adopted in previous studies. We also study the inverse problem of extracting the bound state information from synthetic lattice QCD data and comment on the difference between our approach and the Lüscher method.

Figure 1

Energy levels of the $K{K}^{*}$ system with $\tilde{G}$ obtained from the dimensional regularization scheme in comparison with those obtained in the hybrid approach (left) and the Lüscher approach (right) with $q_{\max }=4000\mathrm{MeV}$. The lattice size $L$ is given in units of $1/m_{\pi }$, where $m_{\pi }$ is the physical pion mass.

Figure 2

Finite-volume corrections, $\delta G$, for the $K{K}^{*}$ system calculated in the dimensional regularization scheme and the cutoff scheme with a sharp cutoff, $q_{\max }$, at the energy $E=1250\mathrm{MeV}$ (left) and $E=1400\mathrm{MeV}$ (right) and with $L=2.5/m_{\pi }$.

Figure 3

Mass of the $f_1(1285)$ as a function of the lattice size $L$ at six combinations of light and strange quark masses, from lower to upper corresponding to Conf1 to Conf6 of Table 1, respectively.