for the Hadron Spectrum Collaboration

# Coupled-channel $J^{--}$ meson resonances from lattice QCD

Jozef J. Dudek<EMAIL_ADDRESS>Department of Physics, College of William and
Mary, Williamsburg, VA 23187, USA  Thomas Jefferson National Accelerator
Facility, 12000 Jefferson Avenue, Newport News, VA 23606, USA  Christopher T.
Johnson  Department of Physics, College of William and Mary, Williamsburg, VA
23187, USA

###### Abstract

We extend an earlier calculation within lattice QCD of excited light meson
resonances with $J^{PC}=1^{--},2^{--},3^{--}$ at the SU(3) flavor point in the
_singlet_ representation, by considering the _octet_ representation. In this
case the resonances appear in coupled-channel amplitudes, which we determine,
establishing the relative strength of pseudoscalar-pseudoscalar to
pseudoscalar-vector decays. Combining the new octet results with the prior
results for the singlet, we perform a plausible extrapolation to the physical
quark mass, and compare to experimental
$\rho^{\star}_{J},K^{\star}_{J},\omega^{\star}_{J}$ and $\phi^{\star}_{J}$
resonances.

††preprint: JLAB-THY-24-4079

##  I Introduction

Beyond the well-established lightest vector mesons, the $\rho,\omega,\phi$ and
$K^{*}$, the spectrum of $J^{PC}=J^{--}$ hadron resonances is quite poorly
understood, with even the number of such states below 2 GeV being unclear
experimentally. The “OZI-rule”, an empirical observation that in many cases
processes described by diagrams in which quark lines are completely
disconnected are heavily suppressed relative to those where all lines are
connected, is often taken as exact, and based upon it, assignments of flavor
structure for resonances made. However, this “rule” is not a demonstrated
feature of QCD in all channels, and its use may be introducing a systematic
error into our understanding of the hadron spectrum. Establishing the validity
of rules like this one is a major aim of modern hadron spectroscopy Shepherd
_et al._ (2016). An ongoing mystery is why there are no experimental candidate
states for light $2^{--}$ resonances outside the strange sector. These states
cannot decay into pairs of pseudoscalars, but can decay into pseudoscalar-
vector final states which are quite well explored experimentally.

An initial exploration of these excited $J^{--}$ meson resonances in first-
principles QCD appeared in Ref. Johnson and Dudek (2021). Lattice gauge-field
configurations having three flavors of dynamical quarks all set equal to the
physical strange quark mass were used to compute correlation functions from
which the discrete finite-volume spectrum in five different spatial volumes
were extracted. These energy levels, corresponding to the _singlet_
($\mathbf{1}$) representation of exact SU(3) flavor symmetry were used to
constrain elastic pseudoscalar-vector scattering amplitudes through the
Lüscher finite-volume formalism Luscher (1986a, b) (reviewed in Ref. Briceno
_et al._ (2018a)). The resulting amplitudes were found to contain pole
singularities that were identified as a narrow $3^{--}$ resonance, a broad
$2^{--}$ resonance, and two overlapping $1^{--}$ resonances. A crude but
plausible extrapolation to the physical quark masses was performed, and
properties of excited $\rho_{J}^{\star}$, $\omega_{J}^{\star}$, and
$\phi_{J}^{\star}$ resonances predicted, _assuming_ an exact implementation of
the “OZI-rule”.

This paper presents an extension of the calculation reported on in Ref.
Johnson and Dudek (2021) to consider, in addition, the independent _octet_
($\mathbf{8}$) representation of SU(3) flavor. $J^{P}=1^{-},3^{-}$ octet
states have pseudoscalar-pseudoscalar decays in addition to pseudoscalar-
vector decays, meaning that _coupled-channel_ amplitudes are required to
describe the relevant scattering. From the pole singularities of the
determined amplitudes we will be able to extract the relative strength of
these decay modes.

The “OZI-rule” assumptions assumed in the extrapolations of Ref. Johnson and
Dudek (2021) will be tested, to a certain extent, in this calculation by
comparing octet resonance decays to certain final states with those for
singlet resonances. Implications for amplitude extraction in a finite-volume
of the near degeneracy of the lightest stable octet and singlet vector mesons,
$\omega^{\scriptscriptstyle\mathbf{8}}$,
$\omega^{\scriptscriptstyle\mathbf{1}}$, will be explored, establishing that
the current level of statistical precision on the finite-volume energy
spectrum only allows us to determine with some precision the total decay rate
of octet resonances into
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$
plus
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$,
but not their relative strength.

Extrapolations comparable to those in Ref. Johnson and Dudek (2021) will be
presented, relaxing somewhat assumptions about the “OZI-rule”, with
$\rho_{J}^{\star}$ and $K^{\star}_{J}$ resonances following directly from the
new octet results, while $\omega_{J}^{\star}$ and $\phi_{J}^{\star}$
resonances will be predicted using a combination of the new octet and the
previous singlet results.

Novel features of the current calculation will be presented in this document,
while relevant background information can be found in Ref. Johnson and Dudek
(2021).

##  II Finite-Volume Spectrum

For the current calculation we use the same anisotropic lattices as Ref.
Johnson and Dudek (2021) which have three degenerate dynamical quark flavors,
tuned to approximately the physical strange quark mass, leading to a lightest
pseudoscalar mass near 700 MeV Edwards  _et al._ (2008); Lin  _et al._ (2009).
_Distillation_ Peardon  _et al._ (2009) is used to compute matrices of two-
point correlation functions across five lattice volumes presented in Table 1.
Discussion of the stable mesons on these lattices and their dispersion
relations can be found in Ref. Johnson and Dudek (2021), and we reproduce
their masses here in Table 2. These same lattices were also used to establish
the presence of an exotic $J^{PC}=1^{-+}$ resonance in Ref. Woss  _et al._
(2021).

$L/a_{s}$ | $14$ | $16$ | $18$ | $20$ | $24$
---|---|---|---|---|---
$N_{\mathrm{cfgs}}$ | $397$ | $490$ | $358$ | $477$ | $499$
$N_{\mathrm{vecs}}$ | $48$ | $64$ | $96$ | $128$ | $160$
$N_{\mathrm{tsrcs}}$ | $16$ | $4$ | $4$ | $4$ | $1$
Table 1:  Number of distillation vectors ($N_{\text{vecs}}$), gauge configurations ($N_{\text{cfgs}}$), and time-sources ($N_{\text{tsrcs}}$) used in computation of two-point correlation functions on each lattice volume. $\eta^{\scriptscriptstyle\mathbf{8}}(0^{-+})$ | 0.1478(1) | $\eta^{\scriptscriptstyle\mathbf{1}}(0^{-+})$ | 0.2017(11)
---|---|---|---
$\omega^{\scriptscriptstyle\mathbf{8}}(1^{--})$ | 0.2154(2) | $\omega^{\scriptscriptstyle\mathbf{1}}(1^{--})$ | 0.2174(3)
|  | $f_{0}^{\scriptscriptstyle\mathbf{1}}(0^{++})$ | 0.2007(18)
$f_{1}^{\scriptscriptstyle\mathbf{8}}(1^{++})$ | 0.3203(6) | $f_{1}^{\scriptscriptstyle\mathbf{1}}(1^{++})$ | 0.3364(14)
$h_{1}^{\scriptscriptstyle\mathbf{8}}(1^{+-})$ | 0.3272(6) | $h_{1}^{\scriptscriptstyle\mathbf{1}}(1^{+-})$ | 0.3288(17)
Table 2: Relevant stable hadron masses in temporal lattice units, $a_{t}m$.

The breaking of rotational symmetry on a cubic spatial lattice with a periodic
cubic spatial boundary leads to irreducible representations, _irreps_ , of the
reduced symmetry group which contain the subductions of states of definite
angular momentum Johnson (1982); Moore and Fleming (2006); Thomas  _et al._
(2012), as given in Table 3 111Note that Table 3 corrects a typographical
error in the corresponding table of Ref. Johnson and Dudek (2021) which
incorrectly indicated the presence of $2^{--}$ in $[111]\,A_{1}$. .

$[000]\,T_{1}^{-}$ | $1^{--}$ |  | $3^{--}$ |  |
---|---|---|---|---|---
$[000]\,E^{-}$ |  | $2^{--}$ |  |  |
$[000]\,T_{2}^{-}$ |  | $2^{--}$ | $3^{--}$ |  |
$[000]\,A_{2}^{-}$ |  |  | $3^{--}$ |  |
$[100]\,A_{1}$ | $1^{--}$ |  | $3^{--}$ | $\mathit{0^{+-}}$ | $\mathit{2^{+-}}$
$[100]\,B_{1}$ |  | $2^{--}$ | $3^{--}$ |  | $\mathit{2^{+-}}$
$[100]\,B_{2}$ |  | $2^{--}$ | $3^{--}$ |  | $\mathit{2^{+-}}$
$[111]\,A_{1}$ | $1^{--}$ |  | $\big{(}3^{--}\big{)}^{2}$ | $\mathit{0^{+-}}$ | $\mathit{2^{+-}}$
Table 3: Subductions of lowest $J^{PC}$ into cubic irreps with $C=-$;
superscripts indicate multiple embeddings.

The possible meson-meson channels contributing at low energies to scattering
in the relevant $J^{PC}$ are presented in Table 4, and the corresponding
finite-volume non-interacting energies (computed as described in Ref. Johnson
and Dudek (2021)) are used to select the basis of operators by including all
meson-meson operators Thomas  _et al._ (2012) having a non-interacting energy
below a certain cut-off (typically $a_{t}E_{\mathsf{cm}}\sim 0.46$). In
addition to these, a large set of $q\bar{q}$-like fermion bilinear operators
are included Dudek  _et al._ (2010).

$1^{--}$ | ${\color[rgb]{192,39,45}\eta^{\mathbf{8}}\eta^{\mathbf{8}}\\{\prescript{1\\!}{}{P}_{1}\\}},{\color[rgb]{47,122,121}\eta^{\mathbf{8}}\omega^{\mathbf{8}}\\{\prescript{3\\!}{}{P}_{1}\\}},{\color[rgb]{65,125,10}\eta^{\mathbf{8}}\omega^{\mathbf{1}}\\{\prescript{3\\!}{}{P}_{1}\\}}$
---|---
${\color[rgb]{0.6,0.6,0.6}f_{0}^{\mathbf{1}}\omega^{\mathbf{8}}\\{\prescript{3\\!}{}{S}_{1},\prescript{3\\!}{}{D}_{1}\\},\eta^{\mathbf{1}}\omega^{\mathbf{8}}\\{\prescript{3\\!}{}{P}_{1}\\},\omega^{\mathbf{8}}\omega^{\mathbf{8}}\\{\prescript{3\\!}{}{P}_{1},\prescript{5\\!}{}{P}_{1}\\}}$
$2^{--}$ | ${\color[rgb]{47,122,121}\eta^{\mathbf{8}}\omega^{\mathbf{8}}\\{\prescript{3\\!}{}{P}_{2},\prescript{3\\!}{}{F}_{2}\\}},{\color[rgb]{65,125,10}\eta^{\mathbf{8}}\omega^{\mathbf{1}}\\{\prescript{3\\!}{}{P}_{2},\prescript{3\\!}{}{F}_{2}\\}}$
${\color[rgb]{0.6,0.6,0.6}f_{0}^{\mathbf{1}}\omega^{\mathbf{8}}\\{\prescript{3\\!}{}{D}_{2}\\},\eta^{\mathbf{1}}\omega^{\mathbf{8}}\\{\prescript{3\\!}{}{P}_{2},\prescript{3\\!}{}{F}_{2}\\},\omega^{\mathbf{8}}\omega^{\mathbf{8}}\\{\prescript{5\\!}{}{P}_{2}\\}}$
$3^{--}$ | ${\color[rgb]{192,39,45}\eta^{\mathbf{8}}\eta^{\mathbf{8}}\\{\prescript{1\\!}{}{F}_{3}\\}},{\color[rgb]{47,122,121}\eta^{\mathbf{8}}\omega^{\mathbf{8}}\\{\prescript{3\\!}{}{F}_{3}\\}},{\color[rgb]{65,125,10}\eta^{\mathbf{8}}\omega^{\mathbf{1}}\\{\prescript{3\\!}{}{F}_{3}\\}}$
${\color[rgb]{0.6,0.6,0.6}f_{0}^{\mathbf{1}}\omega^{\mathbf{8}}\\{\prescript{3\\!}{}{D}_{3}\\},\eta^{\mathbf{1}}\omega^{\mathbf{8}}\\{\prescript{3\\!}{}{F}_{3}\\},\omega^{\mathbf{8}}\omega^{\mathbf{8}}\\{\prescript{5\\!}{}{P}_{3},\prescript{1\\!}{}{F}_{3},\prescript{5\\!}{}{F}_{3}\\}}$
Table 4: Meson-meson scattering partial-waves for each $J^{P}$– only waves
with $\ell\leq 3$ shown. Channels in gray excluded from the coupled-channel
amplitudes used to describe scattering in our selected energy region.

The computed matrices of correlation functions were analysed variationally as
in previous _hadspec_ papers by solving generalized eigenvalue problems. A
slight novelty in the current calculation is the use of a ‘model averaging’
procedure which weights various eigenvalue timeslice fitting-windows and
number of exponentials with a probability associated with a version of the
Akaike Information Criterion Jay and Neil (2021); Radhakrishnan  _et al._
(2022). In practice this leads to modestly more conservative error estimates
for some energy levels. The extracted discrete spectra across eight irreps are
presented in Figure 1 222In addition, spectra for irreps having leading
contributions from the (exotic) positive parity partial-waves listed in Table
3 were also computed, with no significant departures from non-interacting
energies observed in the energy region of interest. Based on this we neglect
scattering in these partial-waves in the remainder of the analysis. .

Figure 1: Finite-volume spectra in eight irreps across five lattice volumes.
Colored markers on vertical axis indicate kinematic thresholds for various
meson-meson channels, while the colored curves show the corresponding non-
interacting energies. Degeneracy of non-interacting levels is indicated by
$\\{n\\}$ for $n>1$. Points with errorbars represent energies extracted from
variational analysis of two-point correlator matrices, with some points
displaced horizontally for clarity. Levels with dominant overlap onto
decoupled channels $f_{0}^{\mathbf{1}}\omega^{\scriptscriptstyle\mathbf{8}}$
(brown),
$\eta^{\scriptscriptstyle\mathbf{1}}\omega^{\scriptscriptstyle\mathbf{8}}$
(sand) are indicated (only the lowest-lying such levels are shown) – these, as
well as all levels shown by gray points are not used to determine coupled-
channel amplitudes.

We observe that we are able to determine large numbers of excited states,
including many near-degenerate states in some energy regions having a high
density of energy levels, making good use of the orthogonality properties of
the generalized eigenvalue approach. As well as the energies, the variational
analysis also yields operator overlap factors for each state, and these
indicate that some states, colored brown or sand in Figure 1, overlap _only_
onto $f_{0}^{\mathbf{1}}\omega^{\scriptscriptstyle\mathbf{8}}$ and
$\eta^{\scriptscriptstyle\mathbf{1}}\omega^{\scriptscriptstyle\mathbf{8}}$
operators, suggesting that these channels are completely decoupled. We will
exclude these levels from further analysis. The operator overlaps show some
evidence of coupling to the
$\omega^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$
channel, but only at the highest energies, above the energy where we
anticipate resonances will lie. In order to avoid needing to consider
$\omega^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$
scattering, we will exclude all levels lying near to or above the lowest
$\omega^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$
non-interacting energy level in each irrep.

The $[000]\,A_{2}^{-}$ spectrum shows a clear ‘extra’ level, not lying close
to any non-interacting energy, near $a_{t}E_{\mathsf{cm}}\sim 0.42$ on the
smaller volumes, which is a likely indication of an isolated $3^{--}$
resonance. In the $[000]\,E^{-}$ spectrum, as well as a consistently present
level compatible with lying _between_ the lowest
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$,
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$
non-interacting energies, we observe what might be an avoided-level crossing
behavior around $a_{t}E_{\mathsf{cm}}\sim 0.42$. This would have a plausible
explanation as a $2^{--}$ resonance coupled to
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$
and/or
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$. In
$[000]\,T_{2}^{-}$, two ‘extra’ levels are clearly visible in the
$L/a_{s}=14,20$ spectra, compatible with the $2^{--}$ and $3^{--}$ resonances
proposed for the previous two irreps both being present here. Similar evidence
for these two resonances is present in the $[100]\,B_{1,2}$ spectra.

The $[000]\,T_{1}^{-}$, $[100]\,A_{1}$ and $[111]\,A_{1}$ spectra into which
$J^{PC}=1^{--}$ is subduced feature the stable
$\omega^{\scriptscriptstyle\mathbf{8}}$ as an isolated and essentially volume-
independent level which is not plotted in Figure 1. This bound-state lies well
below the
$\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}$
threshold and is not expected to have a significant impact above threshold,
and to the extent that it does, it should be reasonably modeled by a slowly
varying ‘background’ energy dependence in scattering amplitudes.

The $[000]\,T_{1}^{-}$ spectrum features several plausible avoided level
crossings relative to
$\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}$ non-
interacting energies near to $a_{t}E_{\mathsf{cm}}\sim 0.39$, which could be
explained in terms of a low-lying $1^{--}$ resonance. Two ‘extra’ levels
visible near $a_{t}E_{\mathsf{cm}}\sim 0.42$ might have an interpretation as
being due to the $3^{--}$ resonance inferred from the $[000]\,A_{2}^{-}$ irrep
spectra, supplemented by a second $1^{--}$ resonance. Patterns in the
$[100]\,A_{1}$ and $[111]\,A_{1}$ spectra would appear to be compatible with
this proposed resonance content.

To go beyond these qualitative statements, we must analyse these spectra in
terms of coupled-channel scattering amplitudes using the finite-volume
quantization condition. We will make use of all levels presented as black
squares in Figure 1, amounting to 154 levels in the irreps having leading
$2^{--}$ and $3^{--}$ subductions, and 119 levels in the irreps with leading
$1^{--}$.

##  III Scattering Amplitudes

The coupled-channel partial-wave $t$-matrix describing scattering in infinite
volume is related to the discrete spectrum in a periodic $L\times L\times L$
box by the finite-volume quantization condition,

$0=\det\Big{[}\bm{1}+i\bm{\rho}(E_{\mathsf{cm}})\,\bm{t}(E_{\mathsf{cm}})\,\big{(}\bm{1}+i\bm{\mathcal{M}}(E_{\mathsf{cm}},L)\big{)}\Big{]}\,,$
(1)

where the matrix $\bm{\mathcal{M}}$ contains known functions which encode the
kinematics of the finite cubic volume. Ref. Briceno  _et al._ (2018a) reviews
the origin of this relation, and its contemporary application to energy levels
obtained within lattice QCD calculations.

Eq. 1 only has solutions for $t$-matrices which satisfy the constraint of
multichannel unitarity, and parameterizations which achieve this are easily
constructed using a $K$-matrix Guo  _et al._ (2013),

$\big{[}\bm{t}^{-1}(s)\big{]}_{ij}=\frac{1}{(2k)^{\ell_{i}}}\big{[}\bm{K}^{-1}(s)\big{]}_{ij}\frac{1}{(2k)^{\ell_{j}}}+\bm{I}_{ij}(s)\,,$
(2)

where the correct threshold behavior for a channel with orbital angular
momentum $\ell$ is imposed. The matrix $\bm{K}(s)$ must be real and symmetric
in the space of coupled-channels, while the diagonal matrix $\bm{I}(s)$ has an
imaginary part given by the phase-space for each channel $i$, $-\rho_{i}(s)$.
$\bm{I}(s)$ can optionally be given a real part, corresponding to dispersively
improving $-\rho_{i}(s)$, in which case it is referred to as the “Chew-
Mandelstam phase-space”.

The likely presence of narrow resonances in the spectrum inspires the use of
explicit poles in $\bm{K}(s)$, as they provide an efficient parameterization
in such cases. As well as some number of poles, we may also consider including
polynomial behavior in $s$, i.e.

$K_{ij}(s)=\sum_{p}\frac{g^{(p)}_{i}\,g^{(p)}_{j}}{(m^{(p)})^{2}-s}+\gamma^{(0)}_{ij}+\gamma^{(1)}_{ij}s+\ldots$
(3)

For any given parameterization, and selection of parameter values, Eq. 1 can
be solved to yield the corresponding finite-volume spectrum for any desired
$L$ value Woss  _et al._ (2020), and this spectrum then compared to the
computed lattice QCD spectra. Minimization of a correlated $\chi^{2}$
quantifying the comparison will provide best fit values for the amplitude
parameters Wilson  _et al._ (2015a). The resulting $t$-matrix, with determined
parameter values (and correlated uncertainties), can then be analysed for its
pole content in the complex-energy plane,

$t_{ij}(s\sim s_{0})\sim\frac{c_{i}\,c_{j}}{s_{0}-s}\,,$ (4)

where the pole location is typically interpreted in terms of the mass and
width of a resonance, $\sqrt{s_{0}}=m_{R}\pm\tfrac{i}{2}\Gamma_{R}$, and the
pole residue is factorized in terms of the resonance’s complex-valued channel
couplings, $c_{i}$.

This approach to determine scattering amplitudes from lattice QCD spectra is
now well established, having been applied successfully to a range of coupled-
channel scattering systems Dudek  _et al._ (2014); Wilson  _et al._ (2015a,
b); Moir  _et al._ (2016); Dudek  _et al._ (2016); Briceno  _et al._ (2018b);
Woss  _et al._ (2019); Prelovsek  _et al._ (2021); Lang and Wilson (2022);
Bulava  _et al._ (2024); Wilson  _et al._ (2024a, b). Ultimately, its ability
to constrain non-zero scattering amplitudes relies upon the departure of
computed lattice QCD energy levels from non-interacting energy levels in a
finite-volume. This leads to a particular challenge in the current
calculation, where the near degeneracy of the stable
$\omega^{\scriptscriptstyle\mathbf{8}}$,
$\omega^{\scriptscriptstyle\mathbf{1}}$ mesons in systems where there is the
presence of both
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$ and
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$
channels, leads to ambiguous scattering amplitude descriptions of the spectra.
In order to illustrate the issue, we will explore it within a simple toy-
model.

###  III.1 Separation of
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$,
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$
channels.

The near degeneracy of the stable $\omega^{\scriptscriptstyle\mathbf{8}}$ and
$\omega^{\scriptscriptstyle\mathbf{1}}$ mesons ensures that
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$,
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$
non-interacting levels always lie very close to each other in all volumes, as
indicated by the blue and green curves in Figure 1. One important effect of
this is the limited ability of the finite-volume quantization condition, Eq.
1, to distinguish between scattering in these two channels. Here we will
propose simple toy models of scattering in which the ratio of couplings of a
resonance to these channels is a parameter, and explore the sensitivity of the
finite-volume spectrum to this ratio.

Figure 2: Finite-volume spectra corresponding to simple toy models of
resonance scattering having fixed coupling combination
$g_{\eta^{\mathbf{8}}\omega^{\mathbf{8}}}^{\,2}+g_{\eta^{\mathbf{8}}\omega^{\mathbf{1}}}^{\,2}$,
but varying ratio,
$r=g_{\eta^{\mathbf{8}}\omega^{\mathbf{1}}}/g_{\eta^{\mathbf{8}}\omega^{\mathbf{8}}}=0.1,0.632,1.0,10.0$,
as described in the text. Gray bands indicate the toy model resonance masses
and their widths.

We begin by examining the $[000]\,A_{2}^{-}$ spectrum using a simple amplitude
model of a single $3^{--}$ resonance coupled to
$\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}$,
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$ and
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$.
The amplitude is constructed using a single $K$-matrix pole fixed in location
at $a_{t}m=0.4227$, with a fixed $F$-wave coupling to
$\eta^{\mathbf{8}}\eta^{\mathbf{8}}$ having a value inspired by fits reported
later in this paper. Four values for the $K$-matrix $F$-wave pole couplings to
$\eta^{\mathbf{8}}\omega^{\mathbf{8}},\eta^{\mathbf{8}}\omega^{\mathbf{1}}$,
are considered, where the ratio of the couplings, ${r\equiv
g_{\eta^{\mathbf{8}}\omega^{\mathbf{1}}}/g_{\eta^{\mathbf{8}}\omega^{\mathbf{8}}}}=0.1,0.632,1.0,10.0$
is varied, while keeping
$g_{\eta^{\mathbf{8}}\omega^{\mathbf{8}}}^{\,2}+g_{\eta^{\mathbf{8}}\omega^{\mathbf{1}}}^{\,2}$
constant. The value ${r=\sqrt{2/5}\approx 0.632}$ is that proposed in Ref.
Johnson and Dudek (2021) to implement exactly the OZI-rule.

The corresponding finite-volume spectra in the $[000]\,A_{2}^{-}$ irrep,
obtained by solving the finite-volume quantization condition for this
amplitude, are presented in the leftmost panel of Figure 2 along with the
lattice QCD computed spectra from Figure 1. We emphasize that these colored
points are not a fit to the computed data, but the amplitude parameter values
are selected to make the comparison to the lattice QCD points meaningful. We
can see that only those energy levels lying close to
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$,
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$
non-interacting energies show any visible dependence upon $r$, and that even
in those cases, the level of variation over two orders of magnitude change in
$r$ is not significantly larger than the statistical uncertainty on the
computed energy levels.

The middle panel of Figure 2, for $[000]\,E^{-}$, shows a similar amplitude
construction for a $2^{--}$ resonance coupled to
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$ and
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$
with $P$-wave couplings whose ratio is varied as above, with the sum of
squares again kept constant 333Modest $F$-wave couplings, required to get the
correct number of energy levels in the spectrum, are kept fixed.. Once again
we see an insensitivity to the ratio of couplings.

The $[000]\,T_{1}^{-}$ irrep shown in the right panel of Figure 2 is described
by an amplitude with _two_ $K$-matrix poles, each coupled to
$\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}$,
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$ and
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$ in
$P$-wave 444The $3^{--}$ amplitude of the leftmost panel is also included for
this irrep.. The same value of channel coupling ratio is used for the two
poles, and again we see very little sensitivity in the spectrum to the value
of $r$.

Moving-frame irrep spectra were also explored, and these showed no additional
sensitivity to the value of $r$. In summary, it appears to be unlikely that
fits to the lattice QCD computed spectra, having the current level of
statistical precision, will be capable of constraining the ratio of couplings
to the
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}},\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$
channels. Explicit amplitude fits to our computed spectra presented later in
this paper will confirm these expectations, finding that while the sum of
squares of the couplings is rather precisely determined (as is the
corresponding combination of $t$-matrix pole couplings), their ratio is poorly
determined.

###  III.2 $2^{--}$, $3^{--}$ amplitudes.

We will proceed to describe 154 levels spread over five irreps and five
volumes, as presented in the upper panels of Figure 1, in terms of coupled-
channel amplitudes describing $2^{--}$ and $3^{--}$ partial-waves. The set of
channels needed are those presented in color in Table 4.

The simplest amplitudes that prove capable of describing the spectra feature
just a single pole in each of the $2^{--}$ and $3^{--}$ $K$-matrices, and no
additional polynomial, with use of the Chew-Mandelstam phase-space subtracted
at the pole location. Given the discussion in the previous section, in both
$J^{PC}$ cases we fix the ratio of
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}},\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$
$K$-matrix pole couplings to $r=0.632$, in $P$-wave for $2^{--}$, and $F$-wave
for $3^{--}$ 555Both $F$-wave channel couplings in $2^{--}$ are allowed to
freely float.. The following parameter values give amplitudes that describe
the 154 levels with a quite reasonable $\chi^{2}$,

$m(3^{--})=$ | $0.4227(9)\,a_{t}^{-1}$ | $\begin{bmatrix}[r]1&-0.4&0.2\\\ &1&0.1\\\ &&1\\\ \end{bmatrix}$
---|---|---
$g_{\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}};F}(3^{--})=$ | $1.76(8)\,a_{t}^{2}$
$g_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}};F}(3^{--})=$ | $-2.02(29)\,a_{t}^{2}$
$m(2^{--})=$ | $0.4209(10)\,a_{t}^{-1}$ | $\begin{bmatrix}[r]1&0.1&-0.3&0.1\\\ &1&0.1&0.2\\\ &&1&-0.8\\\ &&&1\end{bmatrix}$
$g_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}};P}(2^{--})=$ | $0.369(14)$
$g_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}};F}(2^{--})=$ | $-1.3(6)\,a_{t}^{2}$
$g_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}};F}(2^{--})=$ | $-1.9(7)\,a_{t}^{2}$
$\chi^{2}/N_{\text{dof}}=\frac{213.8}{154-7}=1.45$ ,
(5)

where the only large parameter (anti-)correlation is one we could have
anticipated, between the floating $F$-wave
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$ and
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$
couplings 666Not explicitly shown is the fact that no large correlations were
found between the $2^{--}$ and $3^{--}$ amplitude parameters..

The obtained amplitudes are shown777In order to be able to better resolve
weaker amplitudes, we are choosing to plot $\rho|t|$ in this paper, rather
than $\rho^{2}|t|^{2}$ as we have in previous _hadspec_ papers. in the
leftmost panels of Figures 3, 4, where we see clear narrow peaks in both
$3^{--}$ and $2^{--}$. While the relative height of the peaks in $t$-matrix
elements including either
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$ or
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$
will be sensitive to the fixed value of $r$, and will change if we change the
value of $r$, these plots indicate the level of statistical precision that can
be obtained from the spectra presented in Figure 1.

Figure 3: $3^{--}$ coupled-channel scattering amplitudes with coupling ratio
$g_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}}/g_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}$
fixed to value 0.632. Left panel: single $K$-matrix pole (see Eq. 5). Right
panel: $K$-matrix pole plus constants.  Figure 4: $2^{--}$ coupled-channel
scattering amplitudes with $P$-wave coupling ratio
$g_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}};P}/g_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}};P}$
fixed to value 0.632. Left panel: single $K$-matrix pole (see Eq. 5). Right
panel: $K$-matrix pole plus constants.

The $3^{--}$ amplitude of Eq. 5 is found to house a $t$-matrix pole on the
proximal sheet above
$\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}},\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}},\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$
thresholds, located at

$a_{t}\sqrt{s_{0}}=0.4225(9)\pm\tfrac{i}{2}\,0.0046(4)\,,$

with channel couplings,

$\displaystyle
a_{t}c_{\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}\\{\prescript{1\\!}{}{F}_{3}\\}}$
$\displaystyle=0.049(2)\,e^{\mp i\pi 0.023(2)}\,,$ $\displaystyle
a_{t}c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}\\{\prescript{3\\!}{}{F}_{3}\\}}$
$\displaystyle=0.020(3)\,e^{\pm i\pi 0.967(3)}\,,$ $\displaystyle
a_{t}c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}\\{\prescript{3\\!}{}{F}_{3}\\}}$
$\displaystyle=0.012(2)\,e^{\pm i\pi 0.966(3)}\,,$

which we observe to be close to being real-valued. The ratio of the magnitudes
of $\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$,
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$
pole couplings,

$\left|\frac{c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}\\{\prescript{3\\!}{}{F}_{3}\\}}}{c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}\\{\prescript{3\\!}{}{F}_{3}\\}}}\right|=0.6012(5)\,,$

clearly inherits a value close to the fixed $r=0.632$ of the $K$-matrix pole
couplings.

The $2^{--}$ amplitude of Eq. 5 is found to house a $t$-matrix pole on the
proximal sheet above both
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}},\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$
thresholds, located at

$a_{t}\sqrt{s_{0}}=0.4202(10)\pm\tfrac{i}{2}\,0.0104(8)\,,$

with channel couplings,

$\displaystyle
a_{t}c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}\\{\prescript{3\\!}{}{P}_{2}\\}}$
$\displaystyle=0.078(3)\,e^{\mp i\pi 0.04(3)}\,,$ $\displaystyle
a_{t}c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}\\{\prescript{3\\!}{}{P}_{2}\\}}$
$\displaystyle=0.048(2)\,e^{\mp i\pi 0.04(3)}\,,$ $\displaystyle
a_{t}c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}\\{\prescript{3\\!}{}{F}_{2}\\}}$
$\displaystyle=0.012(6)\,e^{\pm i\pi 0.930(5)}\,,$ $\displaystyle
a_{t}c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}\\{\prescript{3\\!}{}{F}_{2}\\}}$
$\displaystyle=0.017(6)\,e^{\pm i\pi 0.928(5)}\,,$

which are all observed to be close to real-valued. As we might expect, the
$P$-wave couplings are significantly larger than the $F$-wave couplings, which
are rather imprecisely determined. The ratio of $P$-wave couplings,

$\displaystyle\left|\frac{c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}\\{\prescript{3\\!}{}{P}_{2}\\}}}{c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}\\{\prescript{3\\!}{}{P}_{2}\\}}}\right|$
$\displaystyle=0.6212(2)\,,$

again inherits the fixed $r=0.632$ of the $K$-matrix pole coupling, while the
$F$-wave coupling ratio,

$\displaystyle\left|\frac{c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}\\{\prescript{3\\!}{}{F}_{2}\\}}}{c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}\\{\prescript{3\\!}{}{F}_{2}\\}}}\right|$
$\displaystyle=1.5\pm 1.1\,,$

is rather imprecise.

Use of a pure single-pole form for the $K$-matrix forces a factorization of
the $t$-matrix for _all_ real energy values, rendering it rank one, and this
is not physically well motivated in general. A straightforward way to loosen
this assumption is to add to the $K$-matrix pole a symmetric matrix of real
constants. If the spectrum is described allowing such constants in the
$3^{--}$ $K$-matrix, a $\chi^{2}/N_{\mathrm{dof}}=193.2/(154-13)=1.37$ is
obtained, which is seen to be slightly lower than that in Eq. 5. The
corresponding amplitude is shown in the right panel of Figure 3, where we
observe that the strong peak feature is qualitatively unchanged.

An analogous addition of a matrix of constants to the $2^{--}$ $K$-matrix,
yields a ${\chi^{2}/N_{\mathrm{dof}}=177.4/(154-17)=1.29}$, which is again a
modest improvement over Eq. 5. The amplitude, presented in the right panel of
Figure 4, shows the same peak structure, with some mild adjustment of the
relative strength of $F$-wave versus $P$-wave.

Figure 5: Resonance pole in $3^{--}$ $t$-matrix over a range of
parameterizations, including variation of fixed value of $r$. Top panel: pole
location in lower half-plane of proximal sheet. Numerical value quoted
accounts for parameterization variation. Middle panel: Magnitude of pole
couplings to
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$,
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$.
Blue quarter-circle shows a conservative estimate of
$\left(|c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}}|^{2}+|c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}|^{2}\right)^{1/2}$.
Bottom panel: Magnitude of pole coupling to
$\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}$.
Figure 6: Resonance pole in $2^{--}$ $t$-matrix over a range of
parameterizations, including variation of fixed value of the ratio of
$K$-matrix $P$-wave pole couplings to
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$,
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$.
Top panel: pole location in lower half-plane of proximal sheet. Numerical
value quoted accounts for parameterization variation. Middle panel: Magnitude
of pole couplings to
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$,
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$ in
$P$-wave. Blue quarter-circle shows a conservative estimate of
$\left(|c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}}|^{2}+|c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}|^{2}\right)^{1/2}$.
Bottom panel: Magnitude of pole couplings to
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$,
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$ in
$F$-wave.

In order to assess sensitivity to the particular amplitude parameterization
form, and to the fixed value of $r$, we attempted to describe the finite-
volume spectra using a range of choices, including allowing constants to be
added to the $K$-matrix poles, replacing such constants with a matrix
proportional to $s$, and using the simple phase-space, rather than the Chew-
Mandelstam form. For the $3^{--}$ case, the results of this variation for the
resonance pole and its couplings are presented in Figure 5. The points are
color-coded according to the $\chi^{2}/N_{\mathrm{dof}}$ with which they
describe the spectrum. The top panel shows that the lattice QCD spectrum
requires there to be a $3^{--}$ resonance whose pole position is very well
determined, irrespective of the details of the parameterization, or the value
of $r$ selected. The middle panel shows, as anticipated, that while the
couplings to
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$ and
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$ are
not individually well determined, the value of the combination
${\big{(}|c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}}|^{2}+|c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}|^{2}\big{)}^{1/2}}$
(indicated by the blue quarter-circle) _can_ be established. The plot
indicates that, on the basis of $\chi^{2}$, there is perhaps a slight
preference for a ratio of couplings less than unity in the $3^{--}$ case. The
bottom panel shows that the pole coupling to
$\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}$ is
rather well determined, regardless of the details of the parameterization, or
the fixed value of $r$.

Variations of the $2^{--}$ amplitude lead to resonance pole variations shown
in Figure 6, where again we observe that the lattice QCD spectrum demands
there is a pole on the proximal sheet at a location which is quite precisely
determined. The $P$-wave couplings to
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$ and
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$ are
again not individually determined, but the sum of their squares is. The ratio
of $F$-wave $K$-matrix pole couplings is not fixed in our procedure, and we
see that description of the spectrum suggests a ratio of such couplings that
may be larger than unity, albeit with the couplings being not very precisely
determined.

###  III.3 $1^{--}$ amplitudes

As discussed in Section II, the spectra in the lower panels of Figure 1
suggest the presence of _two_ $1^{--}$ resonances. We will begin an
investigation of this by presenting, as an example, one successful scattering
amplitude description of all 119 energy levels across the $[000]\,T_{1}^{-}$,
$[100]\,A_{1}$, and $[111]\,A_{1}$ irreps.

Since the $3^{--}$ partial wave also subduces into these irreps, we must
include it in the finite-volume quantization condition – we choose to fix the
$3^{--}$ amplitude to one obtained in the previous section888A single pole
$K$-matrix with $r=0.632$ – sensitivity to this particular amplitude choice
was found to be weak., and vary only parameters in the $1^{--}$ amplitude
parameterization.

The $1^{--}$ amplitude we will use as illustration features two $K$-matrix
poles, where the pole coupling ratios,
$r^{(0)}=g^{(0)}_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}}/g^{(0)}_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}$
and
$r^{(1)}=g^{(1)}_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}}/g^{(1)}_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}$
are each fixed to value 0.632. These poles are supplemented with a complete
symmetric matrix of unfixed constants, and use of the Chew-Mandelstam phase-
space subtracted at $s=(m^{(0)})^{2}$, the location of the lower-mass
$K$-matrix pole. A best-fit description of the finite-volume spectra is
obtained with parameter values,

$m^{(0)}=$ | $0.3891(13)\,a_{t}^{-1}$ | $\begin{bmatrix}[r]1&0.2&-0.4\\\ &1&0.2\\\ &&1\\\ \end{bmatrix}$
---|---|---
$g^{(0)}_{\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}}=$ | $0.227(13)$
$g^{(0)}_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}=$ | $0.882(58)$
$m^{(1)}=$ | $0.4198(11)\,a_{t}^{-1}$ | $\begin{bmatrix}[r]1&-0.2&-0.1\\\ &1&0.1\\\ &&1\\\ \end{bmatrix}$
$g^{(1)}_{\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}}=$ | $-0.194(20)$
$g^{(1)}_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}=$ | $0.173(38)$
$\gamma(\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}|\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}})=$ | $(-0.49\pm 0.47)\,a_{t}^{2}$ | $\begin{bmatrix}[r]1&-0.1&0.0&0.0&-0.1&-0.1\\\ &1&0.1&0.4&0.4&-0.3\\\ &&1&0.1&0.2&0.3\\\ &&&1&0.9&-0.4\\\ &&&&1&-0.1\\\ &&&&&1\\\ \end{bmatrix}$
$\gamma(\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}|\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}})=$ | $(2.2\pm 1.8)\,a_{t}^{2}$
$\gamma(\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}|\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}})=$ | $(-1.7\pm 1.3)\,a_{t}^{2}$
$\gamma(\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}|\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}})=$ | $(9.1\pm 3.5)\,a_{t}^{2}$
$\gamma(\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}|\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}})=$ | $(3.9\pm 2.4)\,a_{t}^{2}$
$\gamma(\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}|\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}})=$ | $(9.2\pm 3.8)\,a_{t}^{2}$
$\chi^{2}/N_{\text{dof}}=\frac{175.4}{119-12}=1.64$ ,
(6)

where the only significant parameter correlations not shown above are between
$g^{(0)}_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}$
and
$\gamma(\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}|\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}})$
and
$\gamma(\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}|\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}})$.

The $t$-matrix elements corresponding to the above description are shown in
Figure 7, along with the $t$-matrix poles on the proximal sheet above
$\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}},\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}},\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$
thresholds. As was also found to be the case in the $SU(3)$ flavor _singlet_
case in Ref. Johnson and Dudek (2021), the lighter resonance has a larger
total width than the narrow heavier resonance. Notably, while the singlet case
is a single-channel process, and hence tightly constrained by elastic
unitarity, such that interference between the two resonances was forced to
manifest as a sharp dip, the octet case is a system of coupled channels, where
overlapping resonances can present in one of several ways. In fact we see that
depending upon the element of the $t$-matrix considered, the two resonances
can appear as two peaks, as a peak and a dip (in
$t(\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}|\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}})$),
or as a peak and a ‘shoulder’ (in
$t(\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}|\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}})$).

Figure 7: $1^{--}$ coupled-channel scattering amplitudes of Eq. 6, together
with $t$-matrix pole content on the proximal sheet above
$\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}},\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}},\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$
thresholds.

In Figure 8 we present factorized residue couplings at the two $t$-matrix
poles, where the heavier, narrower pole ($\mathsf{b}$) is observed to have
couplings which are nearly real, while the lighter, broader pole
($\mathsf{a}$) has couplings which have a nonzero phase. The two poles are
found to have comparable magnitude of coupling to
$\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}$,
while $\mathsf{a}$ has significantly larger couplings to
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}},\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$
than $\mathsf{b}$. We remind the reader than the relative size of couplings to
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$ and
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$ for
each resonance, and hence the magnitude of different $t$-matrix elements in
Figure 7, is at this stage essentially determined by our choices of fixed
$r^{(0)},r^{(1)}$. Nevertheless, it should be clear that the finite-volume
spectrum computed in lattice QCD leads to statistically quite precise
amplitudes within this model.

Figure 8: Complex-valued $t$-matrix pole couplings for $1^{--}$ scattering
amplitudes of Eq. 6, with poles labelled as in Figure 7.  Figure 9: Finite-
volume spectra corresponding to amplitude in Eq. 6 (orange curves) compared to
computed lattice QCD spectra (black and gray points). Purple and blue
horizontal bands indicate the masses and widths of the $1^{--}$ and $3^{--}$
resonances present in the amplitudes.

The quality of the description of the finite-volume spectra can be seen in
Figure 9, where the orange curves correspond to the spectrum obtained by
solving the finite-volume quantization condition for the amplitude of Eq. 6
(and the fixed $3^{--}$ amplitude). This amplitude is observed to give a
faithful description of the lattice QCD computed spectrum used to constrain
it, and in fact appears to quite well reproduce the pattern of many levels (in
gray) that were not included in the $\chi^{2}$ minimization999Our ability to
make use of densely-packed energy levels in explicit fits is somewhat limited
by our current implementation of algorithms to find solutions of the
quantization condition and match them to computed lattice QCD energy levels.
This is done many hundreds or thousands of times during a $\chi^{2}$
minimization, and hence must be extremely reliable. Further improvements
beyond those described in Ref. Woss  _et al._ (2020) are under current
investigation..

In Section III.1 we presented an argument which indicates that we do not
expect the ratios of couplings to
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$ and
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$ to
be well determined using the constraint supplied by our finite-volume spectra,
but nevertheless, we can _attempt_ to describe the finite-volume spectra
allowing the couplings ratios to float. As an illustration, consider a fit to
the 46 energy levels in the $[000]\,T_{1}^{-}$ irrep using an amplitude built
from a $1^{--}$ $K$-matrix featuring just two poles and no constant terms,
with $r^{(1)}$ (the
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}},\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$
coupling ratio for the higher mass pole) fixed to 0.632, but $r^{(0)}$ allowed
to float in the fit. Such a fit yields a quite reasonable
$\chi^{2}/N_{\mathrm{dof}}=62.5/(46-7)=1.60$, with parameter values,

$m^{(0)}=$ | $0.3890(13)\,a_{t}^{-1}$ | $\begin{bmatrix}[r]1&0.2&0.1&-0.1\\\ &1&0.1&0.1\\\ &&1&{\color[rgb]{192,39,45}\bm{-1}}\\\ &&&1\end{bmatrix}$
---|---|---
$g^{(0)}_{\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}}=$ | $0.181(12)$
$g^{(0)}_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}=$ | $0.227(161)$
$r^{(0)}=$ | $3.17\pm 2.43$
$m^{(1)}=$ | $0.4230(16)\,a_{t}^{-1}$ | $\begin{bmatrix}[r]1&-0.1&0.0\\\ &1&0.4\\\ &&1\\\ \end{bmatrix}$
$g^{(1)}_{\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}}=$ | $-0.207(25)$
$g^{(1)}_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}=$ | $0.120(58)$
(7)

where none of the (not shown) parameter correlations between the two poles is
larger than 0.3. We observe that while most parameters are determined with
good statistical precision, there are very large errors on
$g^{(0)}_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}$
and $r^{(0)}$, but that these parameters are 100% anti-correlated, such that
if this anti-correlation is accounted for,

$\displaystyle\left(\big{(}g^{(0)}_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}\big{)}^{2}+\big{(}g^{(0)}_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}}\big{)}^{2}\right)^{1/2}$
$\displaystyle\quad\quad=\left|g^{(0)}_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}\right|\left(1+\big{(}r^{(0)}\big{)}^{2}\right)^{1/2}=0.761(20)\,,$

which is rather precisely determined. This anti-correlation between $r$ and
$g_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}$
proves to be a general feature of attempts to allow the ratio of couplings to
float for either the lower or higher pole. This observation, along with the
fairly precise determination of the sum of squares of the
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}},\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$
couplings is exactly what was indicated by the toy model analysis presented in
Section III.1. As such, as was the case for the $2^{--},3^{--}$ presented in
Section III.2, we will proceed by considering amplitude variations in which
the ratios $r^{(0)},r^{(1)}$ are fixed to various values, allowing other
parameters to float.

Figure 10: Resonance poles in $1^{--}$ $t$-matrix over a range of
parameterizations on the proximal sheet above
$\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}},\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}},\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$
thresholds. Top panel: pole location in lower half-plane of proximal sheet.
Numerical value quoted accounts for parameterization variation. Middle panels:
Magnitude of $P$-wave pole couplings to
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$,
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$.
Blue quarter-circles show a conservative estimate of
$\left(|c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}}|^{2}+|c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}|^{2}\right)^{1/2}$.
Bottom panels: Magnitude of pole coupling to
$\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}$.
Green points have low $\chi^{2}$, orange points somewhat larger $\chi^{2}$ and
red points larger still. Thinnest lines correspond to fitting only
$[000]\,T_{1}^{-}$, medium thickness to $[000]\,T_{1}^{-},[100]\,A_{1}$, and
thickest to all 119 levels in $[000]\,T_{1}^{-},[100]\,A_{1}$ and
$[111]\,A_{1}$. Lighter color points are for an amplitude having two
$K$-matrix poles but no constants.

A large number of variations of amplitude parameterization, and subsets of
fitted energy levels were considered, with particular focus on amplitudes with
two $K$-matrix poles and no free constants, and amplitudes that also include a
matrix of constants. In each case we used fixed $r^{(0)},r^{(1)}$ values
(which may differ from each other) including $0.1,0.3,0.632,1.0,3.0$ and
$10.0$. The resulting amplitudes feature $t$-matrix poles whose properties are
summarized in Figure 10.

We observe that the pole locations show relatively little sensitivity to the
choice of amplitude parameterization or $r^{(0)},r^{(1)}$ value, particularly
if only the lowest obtained $\chi^{2}$ values are retained. The pole residue
couplings to the
$\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}$
channel are similarly insensitive, and are seen to be of similar magnitude for
the two poles. The lighter pole, $\mathsf{a}$, again shows the property that
the ratio of pseudoscalar-vector couplings is not well determined, while the
sum of their squares is – based upon the lowest $\chi^{2}$ values obtained,
there might be a slight preference for the ratio being less than unity. For
the heavier pole, $\mathsf{b}$, the pseudoscalar-vector couplings are smaller
in magnitude, and there appears to be no constraint on their ratio.

##  IV Resonance Interpretation

Having established the resonance content of the octet
$J^{PC}=1^{--},2^{--},3^{--}$ partial waves at the $SU(3)$ flavor point, we
proceed to a discussion of their properties, followed by some speculation as
to the properties of the corresponding resonances at the physical light quark
mass. In order to put the resonance properties in physical units, we must set
the scale of the lattice. The temporal lattice spacing is obtained by
comparing the $\Omega$ baryon mass computed on these lattices to the
experimentally measured value,
${a_{t}^{-1}=m_{\Omega}^{\mathrm{phys}}/(a_{t}m_{\Omega}^{\mathrm{lat}})=4655\,\mathrm{MeV}}$
with a statistical uncertainty that is irrelevant compared to the statistical
errors on the computed quantities.

A summary of the $t$-matrix pole properties of resonances101010We give the
properties of the pole on the proximal sheet above all relevant thresholds.,
with uncertainties that include the variation over parameterization choice is
given below:

$\displaystyle\bm{3^{--},\,\omega^{\scriptscriptstyle\mathbf{8}}_{3}}$
$\displaystyle\sqrt{s_{0}}=1968(6)\pm\tfrac{i}{2}23(4)\,\mathrm{MeV}$
$\displaystyle\big{|}c_{\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}}\big{|}=223(14)\,\mathrm{MeV}$
$\displaystyle\left(\big{|}c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}\big{|}^{2}+\big{|}c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}}\big{|}^{2}\right)^{1/2}=140(28)\,\mathrm{MeV}$
$\displaystyle\Gamma_{\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}}=18(2)\,\mathrm{MeV},\;\;\Gamma_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}+\Gamma_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}}=5(2)\,\mathrm{MeV}$

$\displaystyle\bm{2^{--},\,\omega^{\scriptscriptstyle\mathbf{8}}_{2}}$
$\displaystyle\sqrt{s_{0}}=1966(6)\pm\tfrac{i}{2}51(7)\,\mathrm{MeV}$
$\displaystyle\left(\big{|}c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}\\{\prescript{3\\!}{}{P}_{2}\\}}\big{|}^{2}+\big{|}c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}\\{\prescript{3\\!}{}{P}_{2}\\}}\big{|}^{2}\right)^{1/2}=386(51)\,\mathrm{MeV}$
$\displaystyle\left(\big{|}c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}\\{\prescript{3\\!}{}{F}_{2}\\}}\big{|}^{2}+\big{|}c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}\\{\prescript{3\\!}{}{F}_{2}\\}}\big{|}^{2}\right)^{1/2}=233(61)\,\mathrm{MeV}$

$\displaystyle\bm{1^{--},\,\omega^{\scriptscriptstyle\mathbf{8}}_{\mathsf{a}}}$
$\displaystyle\sqrt{s_{0}}=1786(8)\pm\tfrac{i}{2}87(11)\,\mathrm{MeV}$
$\displaystyle\big{|}c_{\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}}\big{|}=233(19)\,\mathrm{MeV}$
$\displaystyle\left(\big{|}c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}\big{|}^{2}+\big{|}c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}}\big{|}^{2}\right)^{1/2}=587(28)\,\mathrm{MeV}$
$\displaystyle\Gamma_{\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}}=19(3)\,\mathrm{MeV},\;\;\Gamma_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}+\Gamma_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}}=60(6)\,\mathrm{MeV}$

$\displaystyle\bm{1^{--},\,\omega^{\scriptscriptstyle\mathbf{8}}_{\mathsf{b}}}$
$\displaystyle\sqrt{s_{0}}=1957(8)\pm\tfrac{i}{2}35(8)\,\mathrm{MeV}$
$\displaystyle\big{|}c_{\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}}\big{|}=270(33)\,\mathrm{MeV}$
$\displaystyle\left(\big{|}c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}\big{|}^{2}+\big{|}c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}}\big{|}^{2}\right)^{1/2}=177(51)\,\mathrm{MeV}$
$\displaystyle\Gamma_{\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}}=26(6)\,\mathrm{MeV},\;\;\Gamma_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}+\Gamma_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}}=8(5)\,\mathrm{MeV}$

In this summary we have made use of the PDG-advocated approach to define
partial-widths in terms of pole couplings Workman and Others (2022)
($\Gamma_{i}=\tfrac{\rho_{i}}{m_{R}}|c_{i}|^{2}$). We observe in each case
that the sum of partial widths is in reasonable agreement with the total width
defined as twice the imaginary part of the pole position, as one might expect
for relatively narrow resonances.

Figure 11: Spectrum of octet and singlet resonances at the $SU(3)$ flavor
point. A solid line indicate the state’s pole mass while the bands show the
pole width (uncertainties are not shown). Kinematic thresholds for
$\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}$,
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$,
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$,
and
$\omega^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$
indicated by the dashed lines.

Figure 11 compares the spectrum of determined octet resonances with the
previously determined spectrum of singlet resonances taken from Ref. Johnson
and Dudek (2021). The only significant mass difference is seen to be between
the $3^{--}$ states, where the singlet is somewhat heavier than the octet. The
similarity in width of the lighter octet and singlet $1^{--}$ states should be
viewed as a coincidence given that the octet state is allowed to decay into
$\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}$, a
mode forbidden to the singlet, and similarly for the width of the $3^{--}$
states.

We can perform a limited investigation of realization of the OZI-rule in this
data by using the common decay mode of both singlet and octet resonances to
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$.
Our inability to determine separately the
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$ and
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$
decay rates (discussed above) restricts what we can achieve here, but we can
at least ask if the expectations of “pure OZI” as outlined in Ref. Johnson and
Dudek (2021) are compatible with the couplings obtained made in this paper. In
the “pure OZI” limit, the ratio,

$\left|\frac{g^{\mathbf{8}}}{g^{\mathbf{1}}}\right|\equiv\frac{|c(\omega_{J}^{\mathbf{8}}\to\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}})|}{|c(\omega_{J}^{\mathbf{1}}\to\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}})|}=\frac{\sqrt{5}}{4}\approx
0.56\,,$ (8)

where the notation on the left matches with that in Ref. Johnson and Dudek
(2021). Using the OZI-rule value of
${c(\omega_{J}^{\mathbf{8}}\to\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}})/c(\omega_{J}^{\mathbf{8}}\to\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}})=0.632}$,
we obtain the following ratios for the resonances shown in Figure 11:

$\displaystyle\frac{|c(\omega_{\mathsf{a}}^{\mathbf{8}}\to\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}})|}{|c(\omega_{\mathsf{a}}^{\mathbf{1}}\to\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}})|}$
$\displaystyle=0.76(6)\,,$
$\displaystyle\frac{|c(\omega_{\mathsf{b}}^{\mathbf{8}}\to\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}})|}{|c(\omega_{\mathsf{b}}^{\mathbf{1}}\to\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}})|}$
$\displaystyle=0.62(30)\,,$
$\displaystyle\frac{|c(\omega_{2}^{\mathbf{8}}\to\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}\\{\prescript{3\\!}{}{P}_{2}\\})|}{|c(\omega_{2}^{\mathbf{1}}\to\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}\\{\prescript{3\\!}{}{P}_{2}\\})|}$
$\displaystyle=0.40(5)\,,$
$\displaystyle\frac{|c(\omega_{2}^{\mathbf{8}}\to\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}\\{\prescript{3\\!}{}{F}_{2}\\})|}{|c(\omega_{2}^{\mathbf{1}}\to\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}\\{\prescript{3\\!}{}{F}_{2}\\})|}$
$\displaystyle=0.72(21)\,,$
$\displaystyle\frac{|c(\omega_{3}^{\mathbf{8}}\to\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}})|}{|c(\omega_{3}^{\mathbf{1}}\to\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}})|}$
$\displaystyle=0.40(19)\,,$

and none of these are observed to be in significant tension with the value
expected from the OZI-rule.

In summary the only hint of any significant departure from the OZI-rule at the
$SU(3)$ flavor point, and hence of a role for disconnected diagrams is the
mass of the narrow singlet $3^{--}$ resonance being somewhat larger than the
mass of the corresponding octet resonance. Since higher angular momentum
states (a $3^{--}$ resonance decays in $F$-waves) have wavefunctions with more
rapid spatial variation, we might expect them to be more sensitive to spatial
discretization, so we defer from drawing any strong conclusions about this
mass difference until calculations like this one can be performed on lattices
with more than one lattice spacing.

We now proceed to consider an attempt to infer some properties of $J^{--}$
resonances at the physical light quark mass based upon the empirical
observation from several prior lattice QCD calculations that resonance pole
couplings, when scaled by a factor $k^{\ell}$ reflecting the angular momentum
barrier in the decay, appear to be largely independent of the quark mass Woss
_et al._ (2021).

###  IV.1 Estimating $J^{--}$ resonance properties at the physical $u,d$ quark
mass

Upon breaking the $SU(3)$ flavor symmetry by lowering the (still degenerate)
$u,d$ quark masses below the strange quark mass, the degenerate isospin–1
parts of the octet ($\rho^{\star}$) separate from the degenerate isospin–1/2
parts ($K^{\star}$), and the previously independent isospin–0 parts of the
octet and singlet can admix to form two states, which might be denoted
$\omega^{\star}$, $\phi^{\star}$. The use of $\omega$ and $\phi$ as symbols
reflects a ‘lore’, based upon assuming the OZI-rule, that the octet and
singlet will strongly admix to produce dominantly hidden light-quark
$u\bar{u}+d\bar{d}$ ($\omega$), and hidden strange quark $s\bar{s}$ ($\phi$)
states.

In Ref. Johnson and Dudek (2021), partial decay widths into pseudoscalar-
vector final states of the $\rho^{\star}$, $\omega^{\star}$ and $\phi^{\star}$
resonances were extrapolated from the _singlet_ couplings, assuming a perfect
implementation of the OZI-rule. The calculation reported on in this paper
which gives us also the _octet_ couplings to pseudoscalar-vector and
pseudoscalar-pseudoscalar final states allows us to perform a more complete
study.

The physical decay momentum is required to evaluate the scaled couplings, and
we currently lack a lattice QCD estimate of the resonance masses at the
physical point, so we resort to using experimental masses where available,
taken from the PDG averages Workman and Others (2022).

A number of factors limit the predictive power of our approach. Firstly, the
already discussed lack of constraint on the ratio of couplings to
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$ and
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$,
which we will assume lies in a region around the OZI-rule value. Secondly,
$K^{\star}$ states are not eigenstates of charge-conjugation (or $G$-parity)
and hence a $J^{P}$ state can in general be an admixture of $J^{P-}$ and
$J^{P+}$ basis states, the latter of which we have not computed in this
study111111Since $1^{-+}$ and $3^{-+}$ are exotic in the sense of being
inaccessible to a $q\bar{q}$ pair, and because lightest hybrid mesons with
these quantum numbers lie at much heavier mass Dudek  _et al._ (2009, 2010);
Dudek (2011); Dudek  _et al._ (2013), we expect them to be negligible
components in the $K^{*}$, $K^{*}_{3}$ resonances.. Finally, the degree of
admixture of octet and singlet basis states to form the $\omega^{\star}$ and
$\phi^{\star}$ eigenstates is completely unknown to us using only a
calculation at the $SU(3)$ flavor point, where octet and singlet are
independent representations. In this last case, we will indicate the
sensitivity of our results to the value of the octet-singlet mixing angle.

As in Ref. Johnson and Dudek (2021), we make use of decomposition of elements
of the $SU(3)$ representations into states labelled by isospin and strangeness
de Swart (1963); Woss  _et al._ (2021). For example, for pseudoscalar-
pseudoscalar decays of the octet,

$\displaystyle\big{|}\mathbf{8}_{2}$
$\displaystyle;S=0,I=1,I_{z}=+1\big{\rangle}$
$\displaystyle=\sqrt{\tfrac{1}{6}}\left(K^{+}\overline{K}^{0}-\overline{K}^{0}K^{+}\right)+\sqrt{\tfrac{1}{3}}\left(\pi^{+}\pi^{0}-\pi^{0}\pi^{+}\right)$
$\displaystyle\big{|}\mathbf{8}_{2}$
$\displaystyle;S=1,I=\tfrac{1}{2},I_{z}=+\tfrac{1}{2}\big{\rangle}$
$\displaystyle=\sqrt{\tfrac{1}{12}}\left(K^{+}\pi^{0}-\pi^{0}K^{+}\right)-\sqrt{\tfrac{1}{6}}\left(K^{0}\pi^{+}-\pi^{+}K^{0}\right)+\tfrac{1}{2}\left(K^{+}\eta_{8}-\eta_{8}K^{+}\right)$
$\displaystyle\big{|}\mathbf{8}_{2}$
$\displaystyle;S=0,I=0,I_{z}=0\big{\rangle}$
$\displaystyle=\tfrac{1}{2}\left(K^{+}K^{-}-K^{-}K^{+}-K^{0}\overline{K}^{0}+\overline{K}^{0}K^{0}\right)\,,$

which will be used to evaluate decays of the $\rho^{\star}$, $K^{\star}$
and $\omega^{\star}/\phi^{\star}$ resonances respectively.

In cases of resonance decays to final states containing the $\eta$, we will
make use of the empirical observation that the $\eta$ is very close to being
an $SU(3)$ flavor octet state121212See also Ref. Dudek  _et al._ (2013) for
determinations of the mixing on lattices related to those used in this paper.,
and set $\eta=\eta_{8}$. In decays to final states containing the
$\omega(782)$ or the $\phi(1020)$, we will treat these as ideally flavor mixed
so that,

$\displaystyle\omega=\tfrac{1}{\sqrt{2}}\left(u\bar{u}+d\bar{d}\right)$
$\displaystyle=\sqrt{\tfrac{1}{3}}\,\omega_{8}+\sqrt{\tfrac{2}{3}}\,\omega_{1}$
$\displaystyle\phi=s\bar{s}$
$\displaystyle=-\sqrt{\tfrac{2}{3}}\,\omega_{8}+\sqrt{\tfrac{1}{3}}\,\omega_{1}\,,$

which naturally introduces sensitivity to the ratio of couplings to
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$ and
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$.

We will compute partial widths for pseudoscalar-pseudoscalar and pseudoscalar-
vector final states, and then sum these to estimate the total hadronic width,
but since we cannot predict partial widths into channels which are
kinematically closed at the $SU(3)$ flavor point (as we have no constraint on
their couplings), generally we will predict lower limits on the total width of
each resonance. In the $1^{--}$ sector, we will assume that the two states
present at the $SU(3)$ point evolve into two states (in each flavor channel)
at the physical point without mixing with each other. We will not propagate
the statistical errors on the $SU(3)$ flavor point couplings to avoid the
appearance of any kind of quantified uncertainty on these (potentially crude)
model-dependent predictions. We will present only a limited phenomenological
comparison with experimental data, reserving a more complete discussion for a
future publication.

####  IV.1.1 $\rho^{\star}_{J}$ resonances

The isovector resonances depend only upon the octet couplings presented in the
current paper. We scale them as described above, e.g.

$g(\rho^{\star}_{J}\to\pi\pi)=\sqrt{\frac{2}{3}}\,\big{|}c(\omega^{\mathbf{8}}_{J}\to\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}})\big{|}\,\left|\frac{k_{\mathrm{phys}}}{k_{\mathrm{lat}}}\right|^{\ell}\,,$

where for pseudoscalar-pseudoscalar decays, $\ell=J$, and where we are
considering the coupling to include the sum over possible final state charge
configurations. Using this coupling, the corresponding partial-width can be
obtained,

$\Gamma(\rho^{\star}_{J}\to\pi\pi)=\frac{\rho_{\mathrm{phys}}}{m_{\mathrm{phys}}}g(\rho^{\star}_{J}\to\pi\pi)^{2}\,.$

For resonance decays to pseudoscalar-vector final states, our lack of
constraint on the
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$,
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$
coupling ratio is expressed in relationships like,

$\displaystyle g(\rho^{\star}_{J}\to\pi\omega)$
$\displaystyle=\sqrt{\frac{1}{15}}\frac{1+\sqrt{10}\,r}{\sqrt{1+r^{2}}}\,\tilde{c}\,,$
$\displaystyle g(\rho^{\star}_{J}\to\pi\phi)$
$\displaystyle=-\sqrt{\frac{2}{15}}\frac{1-\sqrt{5/2}\,r}{\sqrt{1+r^{2}}}\,\tilde{c}\,,$

where $\tilde{c}$ is the well-constrained combination
$\left(|c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}|^{2}+|c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}}|^{2}\right)^{1/2}$,
while $r$ is the poorly constrained ratio,
$|c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}}|/|c_{\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}}|$.
Notice that in the limit $r\to\sqrt{2/5}\approx 0.632$, the decay
$\rho^{\star}_{J}\to\pi\phi$ has zero coupling, in line with the OZI-rule
forbidding production of the completely disconnected $\phi=s\bar{s}$ in this
decay.

$\bm{\rho_{3}(1690)}$ – With $r=0.632$, the $F$-wave decays of this state
extrapolate to

$\pi\pi$ | $K\overline{K}$ | $\pi\omega$ | $K\overline{K}^{*}$ | $\eta\rho$ | $\pi\phi$ | total
---|---|---|---|---|---|---
53 | 7 | 20 | 2 | 1 | 0 | 83 MeV .

The $\pi\omega$ decay is the only relevant mode to have significant
sensitivity to the value of $r$, becoming close to zero for small value of
$r$, and reaching a maximum near 35 MeV for $r\sim 3$. As such the predicted
total width is relatively insensitive to $r$, not exceeding 100 MeV, which is
somewhat lower than the PDG average width, although the PDG reports
considerable variation in width between analysis of different final states.
The significant experimental branch into $\pi\pi\pi\pi$ (excluding
$\pi\omega$) would not be captured in our extrapolation, so an under estimate
of width may be appropriate.

$\bm{\rho_{2}(1690)}$ – Given the absence of an experimental candidate for
this state, we set the mass equal to the PDG mass of the $\rho_{3}$ considered
above. The extrapolated partial-widths (summing $P$-wave and $F$-wave
contributions) for $r=0.632$ are

$\pi\omega$ | $K\overline{K}^{*}$ | $\eta\rho$ | $\pi\phi$ | total
---|---|---|---|---
108 | 24 | 13 | 0 | 145 MeV .

Again, the $\pi\omega$ mode shows sensitivity to the chosen value of $r$, and
a non-negligible branch into $\pi\phi$ is generated away from the OZI-rule
point, but this partial-width does not get above 10 MeV for $r\lesssim 3$.
Because only pseudoscalar-vector decays are present here, the total width is
more sensitive to the value of $r$, being as low as 80 MeV for small $r$ or as
large as 200 MeV for $r\sim 3$. The possibility of a kinematically open
$S$-wave decay into $\pi a_{2}$ (which will dominantly populate a
$\pi\pi\pi\pi$ final state), could lead to a much larger total width for this
state, and might explain why it has not yet been seen experimentally.

$\bm{\rho(1450)}$ – Using the PDG average mass of 1465 MeV, for $r=0.632$ we
predict

$\pi\pi$ | $K\overline{K}$ | $\pi\omega$ | $K\overline{K}^{*}$ | $\eta\rho$ | $\pi\phi$ | total
---|---|---|---|---|---|---
39 | 8 | 230 | 20 | 18 | 0 | 314 MeV .

The rather large total width of this state is in line with the PDG summary
which suggests a total width near 400 MeV. The width is dominated by the
$\pi\omega$ mode, which is sensitive to the choice of $r$, with significantly
smaller total width at smaller values of $r$, and a total width near 450 MeV
for larger values of $r$.

$\bm{\rho(1700)}$ – Using the PDG average mass of 1720 MeV, for $r=0.632$ we
predict

$\pi\pi$ | $K\overline{K}$ | $\pi\omega$ | $K\overline{K}^{*}$ | $\eta\rho$ | $\pi\phi$ | total
---|---|---|---|---|---|---
41 | 12 | 12 | 5 | 2 | 0 | 72 MeV .

This prediction of a relatively narrow resonance is in poor agreement with
experiment, where a much broader state is required, although a relatively
small rate into $\pi\omega$ was suggested in the analysis of experimental data
presented in Ref. Clegg and Donnachie (1994). The large mass of this state
might suggest that there are additional decay modes kinematically accessible
at the physical point not accounted for in our calculation, and these could be
significant contributions to the total width.

####  IV.1.2 $K^{\star}_{J}$ resonances

Our knowledge of the excited kaon resonances comes mostly from the LASS
experiment, which published results on $K\pi,K\pi\pi,K\eta\ldots$ final states
using a low-energy kaon beam in the early 1980s. One particularly relevant
result of theirs is the claim that data in the $K\omega$ final state is best
described in terms of _two_ $J^{P}=2^{-}$ states Aston  _et al._ (1993) –
within a $q\bar{q}$ quark model these would correspond to admixtures of the
$\prescript{3\\!}{}{D}_{2}(2^{--}),\prescript{1\\!}{}{D}_{2}(2^{-+})$ basis
states.

Our kaon resonance predictions depend only on the octet couplings computed in
this paper.

$\bm{K^{*}_{3}(1780)}$ – With $r=0.632$, the $F$-wave decays of this state
extrapolate to

$K\pi$ | $K\eta$ | $K\rho$ | $K^{*}\pi$ | $K^{*}\eta$ | $K\omega$ | $K\phi$ | total
---|---|---|---|---|---|---|---
30 | 13 | 9 | 14 | 0 | 3 | 1 | 70 MeV .

This small total width is well below the PDG width of 161 MeV, and variation
of the value of $r$ cannot resolve the discrepancy. There remains the
possibility of relatively large $P$-wave vector-vector decays whose strength
we have not estimated.

$\bm{K_{2}(1779)}$ – Since we have no constraint on possible mixing between a
$2^{--}$ basis state and a $2^{-+}$ basis state, we will proceed assuming that
one $2^{-}$ kaon resonance is dominantly the $2^{--}$ state we’ve computed.
With ${r=0.632}$ we have

$K\rho$ | $K^{*}\pi$ | $K^{*}\eta$ | $K\omega$ | $K\phi$ | total
---|---|---|---|---|---
55 | 74 | 3 | 17 | 9 | 158 MeV .

Variation of $r$ changes this total width only by about 20% up or down. The
lighter of the two $2^{-}$ states in the LASS analysis Aston  _et al._ (1993)
has a total width of roughly this size.

$\bm{K^{*}(1410)}$ – With $r=0.632$ this suprisingly low-mass state (it is
lighter than the $\rho(1450)$ despite needing to contain a strange quark) has
partial widths

$K\pi$ | $K\eta$ | $K\rho$ | $K^{*}\pi$ | $K^{*}\eta$ | $K\omega$ | $K\phi$ | total
---|---|---|---|---|---|---|---
19 | 9 | 41 | 95 | 0 | 12 | 0 | 176 MeV .

This predicted total width is only slightly below the PDG average for this
state, and the LASS results have this state seen in $K^{*}\pi$ but not in
$K\rho$, somewhat supporting the hierarchy of partial widths coming from
broken $SU(3)$ flavor symmetry. The total width of this state grows somewhat
for smaller values of $r$.

$\bm{K^{*}(1680)}$ – Using the PDG average mass of 1718 MeV, and $r=0.632$ we
have

$K\pi$ | $K\eta$ | $K\rho$ | $K^{*}\pi$ | $K^{*}\eta$ | $K\omega$ | $K\phi$ | total
---|---|---|---|---|---|---|---
24 | 16 | 5 | 7 | 0 | 2 | 1 | 56 MeV .

As was the case in the isovector channel, the small value of the $SU(3)$–limit
coupling to pseudoscalar-vector causes this resonance to be rather narrow,
much narrower than the rather broad experimental state. The total width is
largely independent of the value of $r$. The possibility of a large $S$-wave
$\pi K_{1}(1270)$ branch can be considered as an explanation of the
discrepancy.

####  IV.1.3 Isoscalar $\omega^{\star}_{J},\phi^{\star}_{J}$ resonances

Given our current lack of any constraint on the mixing of $SU(3)$ flavor octet
and singlet components to form the physical eigenstates away from the $SU(3)$
flavor point, we will present results as a function of the mixing angle,
$\theta$, which appears in,

$\displaystyle\omega^{\star}$
$\displaystyle=\cos\theta\,\omega_{8}-\sin\theta\,\omega_{1}\,,$
$\displaystyle\phi^{\star}$
$\displaystyle=\sin\theta\,\omega_{8}+\cos\theta\,\omega_{1}\,,$

where the ideal flavor mixing case $\omega^{\star}\sim u\bar{u}+d\bar{d}$,
${\phi^{\star}\sim s\bar{s}}$ corresponds to $\theta=-54.74^{\circ}$. The
sensitivity of our partial-width predictions to this mixing angle, and to the
value of $r$, is presented in Figure 12.

Figure 12: Partial widths (in MeV) of $\omega_{J}$(blue) and $\phi_{J}$(green)
resonances with masses given in the text, as a function of the octet-singlet
mixing angle $\theta$. In each case the darker curve corresponds to $r=0.632$
while the two lighter curves correspond to $r=0$ and $r=1$. Top to bottom
panels present $3^{--}$, $2^{--}$, $1^{--}_{\mathsf{a}}$,
$1^{--}_{\mathsf{b}}$.

$\bm{\omega_{3}(1670),\phi_{3}(1850)}$ – Using the PDG masses for these
states, with $r=0.632$ and the ideal flavor mixing angle (i.e. assuming
perfect OZI-rule) we obtain partial-widths

| $K\overline{K}$ | $\pi\rho$ | $K\overline{K}^{*}$ | $\eta\omega$ | $\eta\phi$ | total
---|---|---|---|---|---|---
$\omega_{3}$ | 6 | 61 | 2 | 1 | 0 | 70 MeV
$\phi_{3}$ | 29 | 0 | 20 | 0 | 3 | 51 MeV .

The sensitivity to $r$ and the mixing angle, $\theta$, is presented in the top
row of Figure 12, where the largest sensitivity is seen to be in the partial
widths of these two resonances to the $\pi\rho$ final-state.

$\bm{\omega_{2}(1667),\phi_{2}(1854)}$ – Given the lack of any experimental
candidate states in the $2^{--}$ isoscalar sector, we choose to reuse the
$3^{--}$ masses from above. With $r=0.632$ and the ideal flavor mixing angle
we obtain partial-widths (summed over $F$-wave and $P$-wave decays),

| $\pi\rho$ | $K\overline{K}^{*}$ | $\eta\omega$ | $\eta\phi$ | total
---|---|---|---|---|---
$\omega_{2}$ | 347 | 40 | 14 | 0 | 401 MeV
$\phi_{2}$ | 12 | 137 | 0 | 34 | 183 MeV .

Clearly this extrapolation predicts a very large width for the $\omega_{2}$
state, even before any possible $S$-wave decays are considered. The
sensitivity to $r$ and the mixing angle, $\theta$ is presented in the second
row of Figure 12, where it is clear that for any choice of mixing angle, at
least one of $\omega_{2},\phi_{2}$ must be broad.

$\bm{\omega(1410),\phi(1680)}$ – using PDG masses and with ${r=0.632}$ and the
ideal flavor mixing angle we obtain partial-widths

| $K\overline{K}$ | $\pi\rho$ | $K\overline{K}^{*}$ | $\eta\omega$ | $\eta\phi$ | total
---|---|---|---|---|---|---
$\omega$ | 7 | 468 | 2 | 6 | 0 | 482 MeV
$\phi$ | 25 | 10 | 191 | 0 | 37 | 263 MeV .

The sensitivity to $r$ and the mixing angle, $\theta$ is presented in the
third row of Figure 12, where it is clear that for any value of the mixing
angle, both light vector meson resonances should be broad. The $\omega(1410)$
resonance is very poorly determined experimentally, with the PDG average for
its total width being computed from completely incompatible extractions
varying from 100 MeV to 880 MeV. The PDG average of $e^{+}e^{-}$ results has
the $\phi(1680)$ with a width of only $150(50)$ MeV, well below our
prediction, but examination of the data entering this average includes results
like those in Ref. Aubert  _et al._ (2008), where $e^{+}e^{-}\to
K\overline{K}^{*},\eta\phi$ isoscalar cross-sections feature a single broad
bump described as a resonance with a width over 300 MeV.

$\bm{\omega(1670),\phi(1850)}$ – The mass used for the $\omega$ state
corresponds to the PDG average for a state claimed in multiple experiments,
while the $\phi$ mass is based on the $\phi_{3}$ mass, as there is currently
no second $\phi$ resonance candidate. Using $r=0.632$ and the ideal flavor
mixing angle we obtain partial-widths

| $K\overline{K}$ | $\pi\rho$ | $K\overline{K}^{*}$ | $\eta\omega$ | $\eta\phi$ | total
---|---|---|---|---|---|---
$\omega$ | 11 | 28 | 3 | 1 | 0 | 43 MeV
$\phi$ | 28 | 0 | 12 | 0 | 4 | 44 MeV .

The sensitivity to $r$ and the mixing angle, $\theta$ is presented in the
bottom row of Figure 12, where we see that the decays accounted for in this
analysis cannot generate large widths for these states. As was the case for
the $\omega(1410)$, the $\omega(1670)$ width varies considerably between
experiments, but it is certainly significantly larger than any of the values
presented in the bottom right panel of Figure 12.

##  V Summary

We have presented a first calculation within lattice QCD of the $1^{--}$,
$2^{--}$ and $3^{--}$ coupled-channel scattering amplitudes in the octet
representation of $SU(3)$ flavor, finding relatively narrow resonances in each
case. We demonstrated that the near degeneracy of the stable
$\omega^{\scriptscriptstyle\mathbf{8}}$ and
$\omega^{\scriptscriptstyle\mathbf{1}}$ mesons severely limits our ability to
determine independently the
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{8}}$ and
$\eta^{\scriptscriptstyle\mathbf{8}}\omega^{\scriptscriptstyle\mathbf{1}}$
couplings of these resonances, but that the sum of these two modes is well
determined, as is the coupling to
$\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}$ (when
allowed by symmetries). The obtained results are broadly compatible with the
accepted “OZI-rule” phenomenology in which disconnected quark-line diagrams
are suppressed.

The results of this paper, taken together with earlier results on the $SU(3)$
flavor _singlet_ channel, were used to perform a speculative extrapolation to
the physical light quark mass, in order to make predictions for the properties
of $\rho_{J}^{\star},K_{J}^{\star},\omega_{J}^{\star}$ and $\phi_{J}^{\star}$
resonances. In order to make more reliable QCD predictions without undue
model-dependence it will be necessary to perform lattice calculations with
lower values of the light quark mass, explicitly breaking the $SU(3)$ flavor
symmetry. This will lead to a number of complications such as the growth in
the number of scattering channels (e.g.
$\eta^{\scriptscriptstyle\mathbf{8}}\eta^{\scriptscriptstyle\mathbf{8}}\to\pi\pi,K\overline{K},\eta\eta$)
and the presence of three-meson scattering channels in the energy regions of
interest Hansen  _et al._ (2021).

###### Acknowledgements.

We thank our colleagues within the Hadron Spectrum Collaboration, in
particular D.J. Wilson for his assistance with the analysis code and C.E.
Thomas for comments on an earlier version of the manuscript. JJD and CTJ
acknowledge support from the U.S. Department of Energy contract DE-SC0018416
at William & Mary, and contract DE-AC05-06OR23177, under which Jefferson
Science Associates, LLC, manages and operates Jefferson Lab. This work
contributes to the goals of the U.S. Department of Energy _ExoHad_ Topical
Collaboration, Contract No. DE-SC0023598. The software codes Chroma Edwards
and Joo (2005) and QUDA Clark  _et al._ (2010); Babich  _et al._ (2010); Clark
_et al._ (2016) were used. The authors acknowledge support from the U.S.
Department of Energy, Office of Science, Office of Advanced Scientific
Computing Research and Office of Nuclear Physics, Scientific Discovery through
Advanced Computing (SciDAC) program. Also acknowledged is support from the
Exascale Computing Project (17-SC-20-SC), a collaborative effort of the U.S.
Department of Energy Office of Science and the National Nuclear Security
Administration. This work was performed using the Cambridge Service for Data
Driven Discovery (CSD3) operated by the University of Cambridge Research
Computing Service (www.hpc.cam.ac.uk), provided by Dell EMC and Intel using
Tier-2 funding from the Engineering and Physical Sciences Research Council
(capital grant EP/P020259/1), and DiRAC funding from STFC (www.dirac.ac.uk).
The DiRAC component of CSD3 was funded by BEIS capital funding via STFC
capital grants ST/P002307/1 and ST/R002452/1 and STFC operations grant
ST/R00689X/1. DiRAC is part of the National e-Infrastructure. This work was
also performed on clusters at Jefferson Lab under the USQCD Collaboration and
the LQCD ARRA Project. This research was supported in part under an ALCC
award, and used resources of the Oak Ridge Leadership Computing Facility at
the Oak Ridge National Laboratory, which is supported by the Office of Science
of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. This
research used resources of the National Energy Research Scientific Computing
Center (NERSC), a DOE Office of Science User Facility supported by the Office
of Science of the U.S. Department of Energy under Contract No. DE-
AC02-05CH11231. The authors acknowledge the Texas Advanced Computing Center
(TACC) at The University of Texas at Austin for providing HPC resources. Gauge
configurations were generated using resources awarded from the U.S. Department
of Energy INCITE program at the Oak Ridge Leadership Computing Facility, the
NERSC, the NSF Teragrid at the TACC and the Pittsburgh Supercomputer Center,
as well as at the Cambridge Service for Data Driven Discovery (CSD3) and
Jefferson Lab. This work was performed in part using computing facilities at
William & Mary which were provided by contributions from the National Science
Foundation (MRI grant PHY-1626177), and the Commonwealth of Virginia Equipment
Trust Fund.

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