using smoothed-step or sigmoid functions. These functions contain essential
singularities which make an analytic continuation into the complex energy
plane questionable. The pole lying near the ${D\bar{D}}$ threshold is not
present in the amplitude from which the higher two poles are extracted. A
coupled-channel analysis is performed above ${D_{s}\bar{D}_{s}}$ threshold,
but only a single parameterization is considered that restricts the possible
pole content – it can support two poles decoupled from each other, but cannot
straightforwardly describe a single resonance coupled to both the ${D\bar{D}}$
and ${D_{s}\bar{D}_{s}}$ channels.

The current paper presents a more complete study of the scattering system,
considering all possible channels, constrained by an order of magnitude more
energy levels, and using a variety of analytically well-behaved amplitudes. We
come to a completely different conclusion about the number of poles present,
and while the pion mass is different, it would be very surprising if _two_
additional poles move into the studied energy region under a modest change in
the light quark mass.

Figure 23: Comparison of the amplitudes extracted in the region near
${D\bar{D}}$-threshold region from energy levels in this work (with
$m_{\pi}\approx 391$ MeV), and other lattice calculations, Lang et al [27]
($m_{\pi}\approx 156$ MeV and 266 MeV) and Prelovsek et al [28]
($m_{\pi}\approx 280$ MeV). Presented as $k\cot\delta$ which has an effective
range expansion (upper panel) and $S$-wave elastic phase-shift $\delta$ (lower
panel).

###  VII.3 Interpretation and comparisons to other theoretical work

Our finding of a single relatively narrow resonance in each of $J^{PC}=0^{++}$
and $2^{++}$ can be compared to previous model-based predictions of the state
content of the energy region around the $D\bar{D}$ and $D_{s}\bar{D}_{s}$
thresholds.

Before the observation of the XYZ candidates, a leading picture of the
observed states above 3 GeV was in terms of charm-anticharm bound states
formed of heavy quarks moving in a static potential. Our results appear to
agree with the state counting of this picture, with the single $\chi_{c0}$ and
$\chi_{c2}$ resonances corresponding to the $2P$ radial excitations. The large
overlap of energy-levels near to the masses of these states with
$c\bar{c}$-like operators also supports a dominant role for charmonium-like
wavefunction components in these states. The fact that our scalar state is
slightly heavier than the tensor state323232albeit the effect is of limited
statistical significance. is in opposition to the prediction of short-distance
spin-orbit effects in this picture. This might indicate that the physics of
coupling to the open-charm decay channels is stronger than the relativistic
corrections to the interquark potential.

Potential models also predict a single nearby $2^{-+}$ state and a single
nearby $3^{++}$ state, as we found, and our determined masses lie within the
spread of predictions made by various model implementations.

In order to include some of the physics of hadron strong decay to pairs of
lighter hadrons, potential models are sometimes augmented with an application
of first-order perturbation theory in which an operator that produces a quark-
antiquark pair is introduced. The form of this operator is an assumption of
the approach, and a choice, known as the “${}^{3}\\!P_{0}$ model”, which
successfully describes some experimental meson decays, has the pair produced
with vacuum quantum numbers. The relative strengths of open-charm decays
predicted by the ${}^{3}\\!P_{0}$ model [71, 72] are not consistently
reflected in our extracted pole couplings. The large coupling to
${D^{\ast}\bar{D}^{\ast}}$ of the tensor resonance relative to the scalar
resonance _is_ a feature of the ${}^{3}\\!P_{0}$ model, its spin-recoupling
factors giving the amplitudes a ratio of 2. However, these factors do not work
universally, as the ratio of couplings for
$\chi_{c0}\to{D^{\ast}\bar{D}^{\ast}}$ and $\chi_{c0}\to{D\bar{D}}$ is
predicted to take value $1/\sqrt{3}$, which is in poor agreement with our
extracted couplings. The smallness of the tensor resonance coupling to
$D_{s}\bar{D}_{s}$ _may_ have an explanation in the $D$-wave threshold factor,
but a very similar reduction would be expected for ${D\bar{D}^{\ast}}$ decays
which is not seen, and the ${}^{3}\\!P_{0}$ model does not provide any
compensating factor here.

The observation of XYZ candidate states spurred theoretical consideration of
possible state constructions going beyond just $c\bar{c}$. In particular, the
$X(3872)$ at ${D\bar{D}^{\ast}}$ threshold and the $Z_{c}(3900)$ nearby have
been interpreted as providing evidence for strong long-distance meson-meson
interactions in $S$-wave, potentially strong enough to induce binding of
molecular-like meson-meson configurations.

Heavy quark spin symmetry applied to the charm quarks suggests similar strong
effects in the ${D^{\ast}\bar{D}^{\ast}}$ $S$-wave, and potentially
$0^{++},1^{+-},2^{++}$ partners of the $1^{++}$ $X(3872)$ [73, 74]. These
states may be bound relative to ${D^{\ast}\bar{D}^{\ast}}$, but because they
lie above $D\bar{D}$ and ${D\bar{D}^{\ast}}$ thresholds, they may manifest as
resonances.333333The longest-range process of one-pion exchange is not present
in elastic $D\bar{D}$ scattering, and hence a bound-state in $D\bar{D}$ must
be generated by some other process. It is suggested that these molecular
states appear _in addition_ to the $c\bar{c}$ states discussed above (or for
the physical eigenstates to be admixtures). The scalar and tensor resonances
found in the current calculation do have numerically significant couplings to
the kinematically closed ${D^{\ast}\bar{D}^{\ast}}$ channel, which may imply
they have significant ${D^{\ast}\bar{D}^{\ast}}$ components. However, the
state counting suggests that ${D^{\ast}\bar{D}^{\ast}}$ $S$-wave interactions
are not strong enough to generate additional states (at a pion mass of 391
MeV).

An approach to explaining at least some of the XYZs that does not directly
connect them to meson-meson thresholds is the suggestion that they contain
significant _compact tetraquark_ components. While the dynamics assumed in
models to get these states to bind varies [75, 76], inevitably such pictures
lead to many states beyond those expected in a $c\bar{c}$ only picture.
Tetraquark states are often proposed to lie within a few tens of MeV of meson-
meson thresholds with the same quark content and thus unambiguously
demonstrating such components is challenging [77]. The results presented in
this paper do not seem to support additional states of tetraquark origin, but
a natural criticism would be that the calculation did not include operators
resembling compact tetraquark configurations.343434Our meson-meson operators
have a spatial structure that is not compact, rather each meson samples the
entire volume of the lattice. An earlier calculation [50] performed on the
smallest volume lattice used here did include a basis of compact tetraquark
operators as well as meson-meson operators. This calculation found no
difference in the extracted finite volume spectrum when the tetraquark
operators were removed, suggesting that tetraquark components may not be
important.

###  VII.4 Experimental comparisons

The experimental status of the channels studied in this paper is at present
unclear. Peaks are seen in several processes but often $J^{PC}$ quantum
numbers are not known. Nor is it known how peaks in different final states
relate to each other.

It is not possible to directly compare the present work to experiment due to
the larger light quark mass, the known discretization effects illustrated by
the incorrect $J/\psi-\eta_{c}$ hyperfine splitting, leading to expected
differences of a few tens of MeV, and the other systematic uncertainties
discussed above. Nevertheless, we can present some discussion assuming
plausible extrapolations to the physical light and strange quark masses.

It has been observed in several studies that typically resonance properties
vary smoothly with changes in quark mass [78, 58, 79, 80, 81, 45, 52,
60].353535Although there are notable exceptions, for example Ref. [82] It has
proven to be reasonable in many cases to perform extrapolations based upon the
idea that the reduced couplings (pole couplings with the angular momentum
barrier divided out) are constant with changing quark mass. Predictions have
been made using this approach for $f_{2}$ resonances [42, 26], the $b_{1}$
resonance [25], $\rho_{J},\omega_{J}$ resonances [83], and a hybrid $\pi_{1}$
[26]. Typically these extrapolations assume that we know the physical mass of
the resonance from experiment.

For light quark resonances decaying to final states featuring a pion, there
can be a large increase in phase-space with reduction of light quark mass, and
a corresponding rapid growth in the decay width of resonances. However, in the
current case, consulting Table 4, we see that even though the light-quark
masses are high, because the charm-quark mass is much larger than the light-
quark mass, the differences with respect to experiment of the stable hadron
masses remain relatively small, and hence we do not expect particular large
changes in the resonance properties.

For the case of the single extracted scalar resonance, we might propose two
possible extrapolations: (a) if the resonance mass stays where it is (or
decreases slightly), there would be only a modest change in the $D\bar{D}$ and
$D_{s}\bar{D}_{s}$ phase-spaces, and the state would remain an isolated
relatively narrow resonance with decays to $D\bar{D}$, to $D_{s}\bar{D}_{s}$
(if this is still an open channel) and possibly to $J/\psi\,\omega$. (b) If
the resonance mass moves up slightly, getting close to or even above the
$D^{*}\bar{D}^{*}$ threshold, the large coupling to that channel in $S$-wave
_could_ generate a large total width for the state. In either case there will
be just a single scalar resonance.

Compared to this result there would appear to be a surfeit of experimental
scalar candidate states, as discussed in the introduction. What is not clear
is whether the features observed experimentally in differing production
processes and final states could actually be due to just a single resonance
appearing in the coupled-channel system. Production processes must share the
same pole singularities as the scattering $t$-matrix, but the real-energy axis
lineshape can be sculpted by polynomial energy-dependence from the production
factor,363636or even more rapid energy dependence if the Born-term has a
nearby singularity. which, if not accounted for, could lead to slightly
differing resonance masses and widths in different processes. It remains to be
seen if sufficiently rigorous coupled-channel analysis could resolve the
experimental $\chi_{c0}(3930),\chi_{c0}(3960)$ peaks, the broad
$\chi_{c0}(3860)$ enhancement, and possibly the $X(3915)$ as ultimately being
due to amplitudes featuring only a single scalar resonance pole.

For the tensor resonance, similar observations can be made. However, the
$D$-wave nature of the open decay modes means that the state will rapidly
become narrow if its mass decreases, while the large $S$-wave coupling to
$D^{*}\bar{D}^{*}$ might make it broad should it go up in mass. The results
are consistent with there being a single $\chi_{c2}(3930)$ resonance coupled
to ${D\bar{D}}$ [5, 6, 84], and the current experimental data is not
inconsistent with this having at most a small coupling to $D_{s}\bar{D}_{s}$
[10].

##  VIII Summary

Figure 24:  Results from $J^{PC}=0^{++}$ (left column) and $J^{PC}=2^{++}$
(right column). Top panels: scattering $t$-matrix elements plotted as
$\rho_{i}\rho_{j}|t_{ij}|^{2}$. Middle: dots show the energy levels used to
constrain scattering amplitudes, with further energy levels at lower energies
that mostly constrain the $\eta_{c}\eta$ amplitudes. Bottom panels: resonance
pole positions on the “proximal” sheet, the closest unphysical sheet to the
real energies at which $s$-channel scattering occurs. The most significant
partial widths are also indicated, as determined from pole residues using Eq.
(30).

We have presented an investigation of the $\chi_{c0}$ and $\chi_{c2}$ channels
above ${D\bar{D}}$ threshold where resonant effects are observed in
experiment. Working in the approximation where charm-quark annihilation is
forbidden, for the first time we have been able to consider all of the
necessary channels up to $\psi\phi$ threshold. A summary of our key findings
is presented in Figure 24.

Working at $m_{\pi}\approx 391$ MeV, we find a quite simple picture with a
single resonance in both $J^{PC}=0^{++}$ and $2^{++}$ strongly coupled to
open-charm decay modes. Both resonances are found just below
${D^{\ast}\bar{D}^{\ast}}$ threshold around 4000 MeV with relatively narrow
widths of around 60 MeV, and both have a significant coupling to the
kinematically closed ${D^{\ast}\bar{D}^{\ast}}$ channel in $S$-wave. A key
difference between the resonances is that the ${D_{s}\bar{D}_{s}}$ coupling is
very small for the tensor resonance, but for the scalar state it is of
approximately equal strength to the coupling to ${D\bar{D}}$.

As a by-product of this work, in order to determine “background” partial waves
that appear in our lattice QCD calculation, we have found an $\eta_{c2}$ bound
state, and a $\chi_{c3}$ resonance, both of which have coupling to the
${D\bar{D}^{\ast}}$ channel. Exotic $J^{PC}=1^{-+}$ and $3^{-+}$ amplitudes
were found to be small below 4100 MeV.

Our results are in disagreement with other theoretical work reporting bound or
near-threshold states in ${D\bar{D}}$ in $S$-wave [15, 85, 86, 76, 87, 8, 9,
88], including a prior lattice QCD calculation [28].

The methods used in this paper may be applied to other sectors featuring
scattering of hadrons containing charm quarks. Particularly attractive targets
are the near-threshold vector-pseudoscalar enhancements, $X/\chi_{c1}(3872)$,
$T_{cc}(3875)^{+}$, and ${Z}_{c}/{T}^{b}_{\psi 1}(3900)^{+}$, whose
interaction dynamics are likely related to the states observed in the current
paper. A more complete calculation in a robust lattice QCD framework of these
systems will aid in understanding the inner workings of QCD at these energies.

###### Acknowledgements.

We thank our colleagues within the Hadron Spectrum Collaboration
(www.hadspec.org), in particular Raul Briceño, Andrew Jackura and Arkaitz
Rodas, and also acknowledge useful discussions with Igor Danilkin, Feng-Kun
Guo, Christoph Hanhart, Sasa Prelovsek, Steve Sharpe and Adam Szczepaniak. DJW
acknowledges support from a Royal Society University Research Fellowship. DJW
& CET acknowledge support from the U.K. Science and Technology Facilities
Council (STFC) [grant number ST/T000694/1]. JJD acknowledges support from the
U.S. Department of Energy contract DE-SC0018416 at William & Mary, and JJD &
RGE from contract DE-AC05-06OR23177, under which Jefferson Science Associates,
LLC, manages and operates Jefferson Lab. The software codes Chroma [89], QUDA
[90, 91], QUDA-MG [92], QPhiX [93], MG_PROTO [94], QOPQDP [95, 96], and
Redstar [97] were used. Some software codes used in this project were
developed with 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 used
the Cambridge Service for Data Driven Discovery (CSD3), part of which is
operated by the University of Cambridge Research Computing Service
(www.csd3.cam.ac.uk) on behalf of the STFC DiRAC HPC Facility
(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. Other components were provided by Dell EMC and
Intel using Tier-2 funding from the Engineering and Physical Sciences Research
Council (capital grant EP/P020259/1). This work also used the earlier DiRAC
Data Analytic system at the University of Cambridge. This equipment was funded
by BIS National E-infrastructure capital grant (ST/K001590/1), STFC capital
grants ST/H008861/1 and ST/H00887X/1, and STFC DiRAC Operations grant
ST/K00333X/1. DiRAC is part of the National E-Infrastructure. This work also
used clusters at Jefferson Laboratory under the USQCD Initiative and the LQCD
ARRA project. Propagators and gauge configurations used in this project were
generated using DiRAC facilities, at Jefferson Lab, and on the Wilkes GPU
cluster at the University of Cambridge High Performance Computing Service,
provided by Dell Inc., NVIDIA and Mellanox, and part funded by STFC with
industrial sponsorship from Rolls Royce and Mitsubishi Heavy Industries. Also
used was an award of computer time provided by the U.S. Department of Energy
INCITE program and supported in part under an ALCC award, and resources at:
the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science
User Facility supported under Contract DE-AC05-00OR22725; the National Energy
Research Scientific Computing Center (NERSC), a U.S. Department of Energy
Office of Science User Facility located at Lawrence Berkeley National
Laboratory, operated under Contract No. DE-AC02-05CH11231; the Texas Advanced
Computing Center (TACC) at The University of Texas at Austin; the Extreme
Science and Engineering Discovery Environment (XSEDE), which is supported by
National Science Foundation Grant No. ACI-1548562; and part of the Blue Waters
sustained-petascale computing project, which is supported by the National
Science Foundation (awards OCI-0725070 and ACI-1238993) and the state of
Illinois. Blue Waters is a joint effort of the University of Illinois at
Urbana-Champaign and its National Center for Supercomputing Applications.

##  Appendix A Dispersion relations and additional systematic uncertainty

In this appendix we give further details of the dispersion relations, stable
hadron masses and the additional systematic uncertainty included in the
spectrum. Figure 25 shows the result of determining stable hadron masses from
dispersion relation fits to each lattice volume individually.

Figure 26 shows the deviations of the computed energies from the dispersion
relation fitted to all three volumes simultaneously. The size of these
residuals, particularly for the $\chi_{cJ}$ states, motivates our addition of
a systematic error as described in the main text. Without this additional
systematic, the $\chi^{2}/N_{\mathrm{dof}}$ values for descriptions of
computed finite-volume spectra using scattering amplitudes in many cases are
large. By adding this systematic error, which may reflect discretization
effects or some other unaccounted-for systematic, we make the uncertainty on
the amplitude descriptions more accurately reflect the uncertainty in the
calculation. Additional details in the context of $DK$ scattering are given in
Ref. [45].

In Table 5 we provide details of the dispersion relation fits for stable
charmed and charmonium mesons using Eq. 2 that are shown in Fig. 1 and Table
2.

hadron |  | $a_{t}m$ |  | $\xi$ | $\chi^{2}/N_{\mathrm{dof}}$
---|---|---|---|---|---
$D$ |  | 0.33281(9) |  | 3.466(4) | $\frac{7.79}{7-2}=1.56$
$D_{s}$ |  | 0.34424(11) |  | 3.457(4) | $\frac{9.76}{7-2}=1.95$
$D^{\ast}$ |  | 0.35464(14) |  | 3.477(7) | $\frac{28.6}{16-2}=2.04$
$D_{s}^{\ast}$ |  | 0.36566(14) |  | 3.483(7) | $\frac{28.5}{16-2}=2.04$
$\eta_{c}$ |  | 0.52312(4) |  | 3.491(2) | $\frac{50.5}{7-2}=10.1^{\dagger}$
$\psi$ |  | 0.53715(5) |  | 3.491(2) | $\frac{128}{16-2}=9.17^{\ddagger}$
$h_{c}$ |  | 0.61662(26) |  | 3.450(22) | $\frac{31.4}{16-2}=2.24$
$\chi_{c0}$ |  | 0.60422(25) |  | 3.478(25) | $\frac{2.69}{7-2}=0.54$
$\chi_{c1}$ |  | 0.61488(46) |  | 3.462(26) | $\frac{20.9}{16-2}=1.49$
$\chi_{c2}$ |  | 0.62110(28) |  | 3.454(25) | $\frac{23.8}{27-2}=0.95$
$\eta_{c}^{\prime}$ |  | 0.64160(55) |  | 3.647(115) | $\frac{8.58}{7-2}=1.72$
$\psi^{\prime}$ |  | 0.64566(110) |  | 3.484(141) | $\frac{7.99}{10-2}=1.00^{\sharp}$
Table 5: Results of dispersion relation fits on the $L/a_{s}=24$ lattice with
$|\vec{n}|^{2}\leq 6$. Notes: †the largest individual deviation here is
1.95$\sigma$, and most are below 1$\sigma$, suggesting that correlations are
playing a significant role. ‡a single level, in $[002]A_{1}$, contributes most
of the tension here, having a $4\sigma$ deviation – removing this level
results in $\chi^{2}/N_{\mathrm{dof}}=\frac{27.7}{15-2}=2.13$ and compatible
values of $a_{t}m=0.53720(5)$ and $\xi=3.489(2)$. ♯in this case only momenta,
$|\vec{n}|^{2}\leq 4$ were used. Figure 25: Masses of stable hadrons obtained
from dispersion relation fits to each lattice volume individually,
$a_{t}m_{L}$, using Eq. 2. Fits include points with momentum
$|\vec{n}|^{\,2}\leq{3,4,6}$ for $L/a_{s}={16,20,24}$ respectively. Figure 26:
Residuals upon fitting stable hadron energies to the dispersion relation given
in Eq. 2 with independent values of $m$, $\xi$ for each hadron. Grey bands
indicate our selection of modest additional systematic uncertainties added to
energy levels for $J=0,1$ (left) and $J=2,3$ (right).

##  Appendix B Including the $\chi_{c0}(1P)$ bound-state in amplitudes

In this appendix we consider the two-channel rest-frame analysis of Section
V.1, but additionally include those energy levels lying far below
${\eta_{c}\eta}$ threshold which we have identified as being due to the stable
$\chi_{c0}(1P)$ state. The purpose is to show that the amplitude behavior
above threshold is unchanged upon this inclusion.

To describe the scattering system including the deep bound-state, a $K$-matrix
with a pole term and a matrix of constants is used, with a Chew-Mandelstam
phase-space subtracted at the $K$-matrix pole location. Describing 13 energy
levels results in amplitude parameters,

$a_{t}m=$ | $(0.60402\pm 0.00037\pm 0.00004)$ | $\begin{bmatrix}[r]1.00&0.52&0.02&0.53&0.08&-0.09\\\ &1.00&0.45&0.91&0.54&0.37\\\ &&1.00&0.19&0.99&0.98\\\ &&&1.00&0.28&0.10\\\ &&&&1.00&0.96\\\ &&&&&1.00\end{bmatrix}$
---|---|---
$a_{t}g_{\eta_{c}\eta}=$ | $(0.23\pm 0.09\pm 0.02)$
$a_{t}g_{{D\bar{D}}}=$ | $(0.23\pm 0.63\pm 0.15)$
$\gamma_{\eta_{c}\eta\to\eta_{c}\eta}=$ | $(0.97\pm 0.53\pm 0.08)$
$\gamma_{\eta_{c}\eta\to{D\bar{D}}}=$ | $(0.05\pm 1.54\pm 0.31)$
$\gamma_{{D\bar{D}}\to{D\bar{D}}}=$ | $(1.11\pm 3.24\pm 0.07)$
|  $\chi^{2}/N_{\mathrm{dof}}=\frac{8.19}{13-6}=1.17$ ,

(38)

and the resulting amplitude is presented in Fig. 27, which is comparable to
Fig. 7.

We performed a limited exploration of the space of possible parameterization
variations, finding amplitudes with $g_{{D\bar{D}}}=0$ that described the data
as well as the above description. We were not able to find amplitudes with a
pole term but with $g_{\eta_{c}\eta}$ set to zero that described the data
well. Including moving-frame energy levels do not change these conclusions.

Figure 27: As Figure 7, but including levels well below $\eta_{c}\eta$
threshold due to the stable $\chi_{c0}(1P)$ state. Amplitude as given by Eq.
38.

##  Appendix C $J^{PC}=0^{++}$ amplitude determinations with 3 and 5 channels

In this appendix we investigate scalar amplitudes determined using highest
energies that lie between the two channel region in sections V.1 & V.7 and the
7 channel region in sections V.2 & V.8. The aim is to more closely inspect the
${D_{s}\bar{D}_{s}}$ near-threshold region, and to show that inclusion of the
${D^{\ast}\bar{D}^{\ast}}$ channel is not essential to extract the resonance
pole (although the extra levels and wider energy coverage are helpful).

###  C.1 $J^{PC}=0^{++}$ below $\psi\omega$ threshold

We determine scattering amplitudes in the region where ${\eta_{c}\eta}$,
${D\bar{D}}$ and ${D_{s}\bar{D}_{s}}$ are active, below the ${\psi\omega}$
threshold at $a_{t}E_{\mathsf{cm}}=0.690$. In this case we opt to neglect the
${\eta_{c}\eta^{\prime}}$ channel which is assumed to be decoupled, and
exclude those levels dominated by overlap with operators resembling
${\eta_{c}\eta^{\prime}}$. One motivation for considering only these levels is
that the description in this energy region appears to have some tension in the
larger energy region $S$-wave amplitude determinations, for example see the
left panel of Fig. 17.

We make use of rest-frame irreps ($L/a_{s}=16,20,24$) and moving-frame irreps
($L/a_{s}=20,24$) with energies up to $a_{t}E_{\mathsf{cm}}=0.690$. This
includes 5 levels dominated by overlap with ${D_{s}\bar{D}_{s}}$ operators
very close to ${D_{s}\bar{D}_{s}}$ threshold from $[000]A_{1}^{+}$ and
$[002]A_{1}$. The $[001]A_{1}$ and $[111]A_{1}$ irreps are included but they
do not have any levels dominated by ${D_{s}\bar{D}_{s}}$ operators in this
energy region.

An example amplitude determined from these energies is given by a $K$-matrix
of constants $K_{ij}=\gamma_{ij}$. In this case, we subtract the Chew-
Mandelstam function at the threshold of each channel. The resulting parameter
values are,

$\gamma_{{\eta_{c}\eta}\to{\eta_{c}\eta}}=$ | $(1.39\pm 0.58\pm 0.40)$ | $\begin{bmatrix}[r]1.00&0.23&0.70&0.10&0.92&0.07\\\ &1.00&0.32&-0.28&0.26&-0.55\\\ &&1.00&0.18&0.91&0.13\\\ &&&1.00&0.14&0.84\\\ &&&&1.00&0.14\\\ &&&&&1.00\end{bmatrix}$
---|---|---
$\gamma_{{D\bar{D}}\to{D\bar{D}}}=$ | $(0.13\pm 0.35\pm 0.22)$
$\gamma_{{D_{s}\bar{D}_{s}}\to{D_{s}\bar{D}_{s}}}=$ | $(4.87\pm 3.84\pm 3.28)$
$\gamma_{{\eta_{c}\eta}\to{D\bar{D}}}=$ | $(0.57\pm 0.32\pm 0.08)$
$\gamma_{{\eta_{c}\eta}\to{D_{s}\bar{D}_{s}}}=$ | $(3.18\pm 1.35\pm 0.96)$
$\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}}=$ | $(-0.87\pm 1.07\pm 0.45)$
|  $\chi^{2}/N_{\mathrm{dof}}=\frac{53.0}{60-6-8}=1.15$ ,
(39)

where the central value of the
$\gamma_{{D_{s}\bar{D}_{s}}\to{D_{s}\bar{D}_{s}}}$ parameter is numerically
larger than in the other amplitudes in this work, although with a large
uncertainty. The description of the finite-volume energy levels is improved in
the region of ${D_{s}\bar{D}_{s}}$ threshold, as can be seen in Fig. 28. The
amplitudes are plotted in Fig. 29.

Figure 28: As Figure 7, except solutions from the three-channel amplitudes in
Eq. 39 determined from $[000]\,A_{1}^{+}$ and moving frames $A_{1}$ irreps are
shown as the orange curves. Figure 29: Three-channel $J^{PC}=0^{++}$
scattering amplitudes in Eq. 39 determined from $[000]A_{1}^{+}$ and moving
frame $A_{1}$ irreps.

With this selection of levels, stronger features can be seen at
${D_{s}\bar{D}_{s}}$ threshold than when considering a wider energy range.
Amplitudes determined from this limited selection of energies feature poles on
unphysical sheets in the region of $a_{t}\sqrt{s}\approx 0.68$, which may be
at complex locations in coupled-channel cases, or on the real axis below
${D_{s}\bar{D}_{s}}$ threshold in decoupled cases. In the case presented in
Eq. 39, there is a pole on a “hidden” sheet, although within uncertainties the
imaginary part is consistent with zero, as summarized in Table 6.

$\mathrm{sign}\>\mathrm{Im}$ |
---|---
$(k_{{\eta_{c}\eta}},k_{{D\bar{D}}},k_{{D_{s}\bar{D}_{s}}})$ | $a_{t}\sqrt{s_{0}}$
$(+,-,-)$ | $(0.675\pm 0.022)\pm\frac{i}{2}(0.027\pm 0.028)$
$(+,+,-)$ | $(0.681\pm 0.012)\pm\frac{i}{2}(0.029\pm 0.045)$
Table 6: Poles in three channel ${\eta_{c}\eta}-{D\bar{D}}-{D_{s}\bar{D}_{s}}$
amplitudes, Eq. 39.

The large uncertainties on the parameters and the plotted amplitudes indicate
that the constraint on ${D_{s}\bar{D}_{s}}$ is not particularly strong. We
have highlighted this because it is the most extreme amplitude behavior near
${D_{s}\bar{D}_{s}}$ threshold that the finite-volume spectra can tolerate.
However, no poles have been found _close to_ ${D_{s}\bar{D}_{s}}$ threshold in
this work, in contrast to Ref. [28] which finds one within a few MeV. The
energy shifts seen in the current calculation are smaller and the
corresponding interactions are weak. Adding further levels at higher energies
pushes the solution towards a smaller ${D_{s}\bar{D}_{s}}$ amplitude just
above threshold, the pole then moves away, and the coupling weakens. The poles
in Table 6 thus appear to be a property of describing energy levels in too
small a region, and are not a good overall reflection of the findings in this
work.

###  C.2 Determining $J^{PC}=0^{++}$ up to ${D^{\ast}\bar{D}^{\ast}}$
threshold only

The $0^{++}$ resonance identified in sections V.2 & V.8 lies below
${D^{\ast}\bar{D}^{\ast}}$ threshold and so, in principle, we should be able
to extract it without considering the ${D^{\ast}\bar{D}^{\ast}}$ and
$\psi\phi$ channels, although the strong overlap with
${D^{\ast}\bar{D}^{\ast}}$ operators of several states below threshold (as
seen in Fig 3) suggest there is merit in including the kinematically closed
${D^{\ast}\bar{D}^{\ast}}$ channel.

In this section we consider a coupled
${\eta_{c}\eta}-{D\bar{D}}-{\eta_{c}\eta^{\prime}}-{D_{s}\bar{D}_{s}}-{\psi\omega}$
scattering system below ${D^{\ast}\bar{D}^{\ast}}$ threshold, taking the
opportunity to test some of the properties of the amplitudes at lower
energies. As with the main analysis, we begin by using only at-rest energies
before making use of both rest-frame and moving-frame energies together.

####  C.2.1 Rest-frame energies only

Working below $a_{t}E_{\mathsf{cm}}=0.709$, there are 30 levels over the three
volumes (including three $\eta_{c}\eta^{\prime}$ levels) from
$[000]A_{1}^{+}$. Following the methods outlined in the main text, we make use
of a $K$-matrix with a pole and constants,
$K_{ij}=\frac{g_{i}g_{j}}{m^{2}-s}+\gamma_{ij}$, and a Chew-Mandelstam phase
space subtracted at the $K$-matrix pole. One representative result is,

$a_{t}m=$ | $(0.70398\pm 0.00128\pm 0.00035)$ | $\begin{bmatrix}[r]1.00&-0.10&-0.57&-0.27&-0.16&-0.10\\\ &1.00&0.14&-0.07&0.28&0.01\\\ &&1.00&0.09&0.01&0.00\\\ &&&1.00&-0.14&0.17\\\ &&&&1.00&-0.03\\\ &&&&&1.00\end{bmatrix}$
---|---|---
$a_{t}g_{{D\bar{D}}}=$ | $(0.0867\pm 0.0155\pm 0.00143)$
$a_{t}g_{{D_{s}\bar{D}_{s}}}=$ | $(0.1281\pm 0.0222\pm 0.00393)$
$\gamma_{\eta_{c}\eta\to\eta_{c}\eta}=$ | $(0.07\pm 0.098\pm 0.048)$
$\gamma_{{D\bar{D}}\to{\psi\omega}}=$ | $(2.04\pm 0.620\pm 0.208)$
$\gamma_{\eta_{c}\eta^{\prime}\to\eta_{c}\eta^{\prime}}=$ | $(3.17\pm 1.49\pm 0.66)$
|  $\chi^{2}/N_{\mathrm{dof}}=\frac{41.9}{30-6}=1.75$ ,
(40)

where as previously all parameters not listed are set equal to zero. This
amplitude is plotted in the top panel of Fig. 30.

The limited set of energy levels provides relatively little constraint for the
$\psi\omega$ channel, with only three levels dominated by $\psi\omega$
operators being present very close to $\psi\omega$ threshold. However, in this
case, the $\psi\omega$ amplitude has an interesting shape, it produces a dip
around the pole position and has a relatively sharp rise from threshold,
resulting in a shallow peak slightly _below_ the resonance mass. The
uncertainties are large and this feature does not survive the addition of more
energies, but it does show how small strength features in very weakly coupled
channels do not always resemble the dominant resonance.

Adding more free parameters can produce a lower $\chi^{2}/N_{\mathrm{dof}}$,
in particular when allowing non-zero $\gamma_{{D\bar{D}}\to{D\bar{D}}}$ and
$\gamma_{{D_{s}\bar{D}_{s}}\to{D_{s}\bar{D}_{s}}}$. However these also result
large uncertainties on the determined amplitudes. Freedom in the
$\gamma_{{D\bar{D}}\to{\psi\omega}}$ parameter can be interchanged with
$\gamma_{{\psi\omega}\to{\psi\omega}}$ or $g_{\psi\omega}$ with little effect
on the amplitudes within uncertainties, and only small changes in the
$\chi^{2}/N_{\mathrm{dof}}$.

Figure 30: A selection of scattering amplitudes determined below
$a_{t}E_{\mathsf{cm}}=0.709$ (just below ${D^{\ast}\bar{D}^{\ast}}$ threshold)
in $J^{PC}=0^{++}$. Peaks are seen across all parameterizations in
${D\bar{D}}$ that correspond to a resonance pole consistent with that given in
the main text, strongly coupled to ${D\bar{D}}$ and ${D_{s}\bar{D}_{s}}$. The
top panel shows amplitudes obtained from 30 rest frame energies. The middle
panel shows amplitudes obtained from 75 energies including moving frames. The
bottom panel highlights a slightly different amplitude found with hadron
masses set to 1$\sigma$ below their best-fit values.

####  C.2.2 Including moving-frame energies

Using rest-frame energies up to $a_{t}E_{\mathsf{cm}}=0.709$ and the same
selection of moving-frame energies up to $a_{t}E_{\mathsf{cm}}=0.690$ that
were previously used in section C.1, results in 75 levels to constrain the
$S$-wave amplitudes. We fix the $J^{PC}=2^{++}$ result to the reference
amplitude given in Eq. 15, and for simplicity we fix all other contributing
partial-waves to zero. We also fix the strength of the decoupled
${\eta_{c}\eta^{\prime}}$ channel via
$\gamma_{{\eta_{c}\eta^{\prime}}\to{\eta_{c}\eta^{\prime}}}=3$, consistent
with the determination in Eqs. 28 and 40. An example amplitude determined from
these levels is,

$a_{t}m=$ | $(0.7037\pm 0.0013\pm 0.0007)$ | $\begin{bmatrix}[r]1.00&0.31&-0.46&-0.22&0.08&0.47&-0.06\\\ &1.00&-0.06&-0.05&-0.32&0.42&-0.01\\\ &&1.00&0.21&-0.11&0.06&0.00\\\ &&&1.00&0.14&0.10&0.09\\\ &&&&1.00&0.04&0.05\\\ &&&&&1.00&0.03\\\ &&&&&&1.00\end{bmatrix}$
---|---|---
$a_{t}g_{{D\bar{D}}}=$ | $(0.081\pm 0.016\pm 0.001)$
$a_{t}g_{{D_{s}\bar{D}_{s}}}=$ | $(0.133\pm 0.022\pm 0.009)$
$\gamma_{\eta_{c}\eta\to\eta_{c}\eta}=$ | $(0.02\pm 0.08\pm 0.05)$
$\gamma_{{D\bar{D}}\to{D\bar{D}}}=$ | $(-0.53\pm 0.21\pm 0.11)$
$\gamma_{{D_{s}\bar{D}_{s}}\to{D_{s}\bar{D}_{s}}}=$ | $(0.16\pm 1.02\pm 0.34)$
$\gamma_{{\psi\omega}\to{\psi\omega}}=$ | $(1.02\pm 2.01\pm 0.26)$
$\gamma_{\eta_{c}\eta^{\prime}\to\eta_{c}\eta^{\prime}}=$ |  $3$ (fixed) |
|  $\chi^{2}/N_{\mathrm{dof}}=\frac{89.9}{75-7-11}=1.58$ .

In this case, the second uncertainties are obtained by varying the scattering
hadron masses to their upper values in Table 2 ($m_{i}\to m_{i}+\delta
m_{i}$), but only half a sigma in the negative direction ($m_{i}\to
m_{i}-\delta m_{i}/2$). When varying the masses to their lower values
($m_{i}\to m_{i}-\delta m_{i}$), we find qualitatively slightly different
solutions which are discussed below. This amplitude is plotted in the central
panel of Fig. 30.

In this reduced energy region below ${D^{\ast}\bar{D}^{\ast}}$ threshold, when
performing amplitude determinations at the extreme lower end of the ranges of
the hadron masses ($m_{i}\to m_{i}-\delta m_{i}$ in Table 2), we observe a
second class of solution for the ${D_{s}\bar{D}_{s}}$ amplitude, shown in the
bottom panel of Fig. 30. We see that this solution has a significantly
stronger turn-on of the $D_{s}\bar{D}_{s}\to D_{s}\bar{D}_{s}$ amplitude at
threshold, and an atypical behavior at higher energies where a strong peak
appears in $D\bar{D}\to D\bar{D}$.

The parameters corresponding to this solution using the rather extreme mass
values $m_{i}-\delta m_{i}$ are,

$a_{t}m=$ | $(0.7069\pm 0.0010)$ | $\begin{bmatrix}[r]1.00&-0.17&-0.62&-0.32&0.00&-0.04&-0.12\\\ &1.00&-0.14&-0.08&-0.71&0.08&0.00\\\ &&1.00&0.14&0.26&-0.18&0.07\\\ &&&1.00&0.21&0.27&0.11\\\ &&&&1.00&0.13&0.11\\\ &&&&&1.00&0.09\\\ &&&&&&1.00\end{bmatrix}$
---|---|---
$a_{t}g_{{D\bar{D}}}=$ | $(0.102\pm 0.019)$
$a_{t}g_{{D_{s}\bar{D}_{s}}}=$ | $(0.065\pm 0.057)$
$\gamma_{\eta_{c}\eta\to\eta_{c}\eta}=$ | $(-0.04\pm 0.10)$
$\gamma_{{D\bar{D}}\to{D\bar{D}}}=$ | $(-0.39\pm 0.23)$
$\gamma_{{D_{s}\bar{D}_{s}}\to{D_{s}\bar{D}_{s}}}=$ | $(\;\;2.87\pm 1.02)$
$\gamma_{{\psi\omega}\to{\psi\omega}}=$ | $(-0.22\pm 1.19)$
$\gamma_{{\eta_{c}\eta^{\prime}}\to{\eta_{c}\eta^{\prime}}}=$ | 3 (fixed) |
|  $\chi^{2}/N_{\mathrm{dof}}=\frac{91.9}{75-7-11}=1.61$ ,

and this amplitude has a virtual bound-state pole strongly coupled to the
${D_{s}\bar{D}_{s}}$ channel at ${a_{t}\sqrt{s_{0}}=0.6656\pm 0.0157}$ with
$a_{t}|c_{{D_{s}\bar{D}_{s}}}|=0.413\pm 0.056$. We highlight this result in
part because it is resembles somewhat the solution found in Ref. [28], but
only qualitatively, as the pole is roughly 125 MeV below ${D_{s}\bar{D}_{s}}$
threshold while Ref. [28] reports a pole within a few MeV of the
${D_{s}\bar{D}_{s}}$ threshold.

We consider this solution to be disfavored as it only appears when an extreme
choice is made for all scattering hadron masses, but even here a scalar
resonance appears, in reasonable agreement with our other determinations, with
a pole position $a_{t}\sqrt{s_{0}}=0.7068(9)\pm\frac{i}{2}0.0059(26)$ on the
proximal sheet. Large couplings to ${D\bar{D}}$ and ${D_{s}\bar{D}_{s}}$ and
small couplings to the ${\eta_{c}\eta}$ and ${\psi\omega}$ channels are found.

It should also be noted that while the central values look quite different,
the uncertainty bands are largely consistent across sections C.2.1, C.2.2, and
also the main results given above, as can be seen in Fig. 30.

##  Appendix D A $J^{PC}=2^{++}$ toy-model study

The purpose of this appendix is to show that the couplings to the
kinematically closed ${D^{\ast}\bar{D}^{\ast}}$ channel can be determined
reliably from the volume-dependence of energy levels. We illustrate the
sensitivity using a simplified two-channel system with a resonance coupling to
an open $D$-wave channel (which we call ${D\bar{D}}$) and a closed $S$-wave
channel (${D^{\ast}\bar{D}^{\ast}}$). For an approximately fixed resonance
mass and width, we show that the spectra are sensitive to the value of the
coupling to ${D^{\ast}\bar{D}^{\ast}}$. This toy-model also contains a further
example of the $K$-matrix pole “coupling-ratio phenomenon” described in
Section V.5.

We utilize a two-channel version of the Flatté amplitude Eq. 14, where the
lower channel is ${D\bar{D}}{\\{{\>\\!}^{{1}\\!}{D}_{2}\\}}$ (with $D$-wave
suppression close to threshold) and the higher channel is
${D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{S}_{2}\\}}$ (an $S$-wave channel
that can open rapidly). The pole parameter is set to $a_{t}m=0.7$, however we
subtract the real correction from
$-ig_{{D^{\ast}\bar{D}^{\ast}}}^{2}\rho_{{D^{\ast}\bar{D}^{\ast}}}(m^{2})$, as
described after Eq. 14, so that the pole parameter $m$ retains its meaning. We
initially fix $a_{t}g_{{D^{\ast}\bar{D}^{\ast}}}=1.6$ which is a
representative value giving
${D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{S}_{2}\\}}$ amplitudes similar
to those found throughout this work. The $g_{{D\bar{D}}}$ coupling is then
chosen so that a $t$-matrix pole width
$a_{t}\Gamma=-2\,\mathrm{Im}\,a_{t}\sqrt{s_{0}}=0.0116$ is obtained
(corresponding to $\approx$ 66 MeV). We then reduce
$g_{{D^{\ast}\bar{D}^{\ast}}}$ and adjust $g_{{D\bar{D}}}$ in order to
maintain an approximately constant $t$-matrix pole position
$a_{t}\sqrt{s_{0}}=0.697\pm\frac{i}{2}0.0116$.

In Fig. 31 we show the amplitudes and the finite volume spectra resulting from
this procedure. Below $a_{t}E_{\mathsf{cm}}=0.7$, on the lower half of the
resonance hump, we see almost no variation as these parameters are changed.
Similarly, the finite volume spectra in this energy region show little
sensitivity.

On the other hand _above_ $a_{t}E_{\mathsf{cm}}=0.7$, significant differences
are observed. An avoided level crossing occurs in every irrep around the
position of the lowest ${D^{\ast}\bar{D}^{\ast}}$ non-interacting energy with
departures proportional to the size of $g_{{D^{\ast}\bar{D}^{\ast}}}$. These
deviations are significantly larger than typical uncertainties in the computed
spectrum and so it is plausible that the coupling to this channel can be well-
determined.

We also observe the beginning of the onset of the coupling-ratio phenomenon in
this toy model given that there is only a relatively small difference between
the amplitudes with $a_{t}g_{{D^{\ast}\bar{D}^{\ast}}}=0.8$ and
$a_{t}g_{{D^{\ast}\bar{D}^{\ast}}}=1.6$, which correspond to very similar
coupling ratios, $g_{{D^{\ast}\bar{D}^{\ast}}}/g_{{D\bar{D}}}=1.05,0.096$
respectively.

Figure 31: Toy model amplitudes consisting of
${D\bar{D}}{\\{{\>\\!}^{{1}\\!}{D}_{2}\\}}$ and
${D^{\ast}\bar{D}^{\ast}}\\{{\>\\!}^{{5}\\!}{S}_{2}\\}$ as described in the
text. Upper panel: Scattering amplitudes plotted as
$\rho_{i}\rho_{j}|t_{ij}|^{2}$ for each of the parameters given in the top
right. Circles on the horizontal axis indicate threshold energies. Lower
panel: The solid green and blue curves show non-interacting energies
corresponding to ${D\bar{D}}$ and ${D^{\ast}\bar{D}^{\ast}}$ respectively.
Degeneracies are not indicated since only a single level is expected for each
hadron-hadron pair when only a single combination is present in the Lüscher
determinant condition Eq. 5. The dashed horizontal lines indicate kinematic
thresholds. The grey band and horizontal solid grey line indicate the mass and
width of the resonance pole. The dotted horizontal line indicates both the
position of the mass parameter $a_{t}m$, and the centre of the peak seen in
the ${D\bar{D}}{\\{{\>\\!}^{{1}\\!}{D}_{2}\\}}$ amplitudes. The red and grey
spectrum of curves show the finite volume spectra obtained from the Lüscher
determinant condition Eq. 5 corresponding to the same-colored amplitudes from
the upper panel.

##  Appendix E Summary of amplitude parameterizations

In this appendix we summarise the parameterizations of the $J^{PC}=3^{++}$,
$2^{++}$ and $0^{++}$ amplitudes.

###  E.1 Summary of $J^{PC}=3^{++}$ parameterizations

In Table 7 we summarize 10 $J^{PC}=3^{++}$ parameterization variations,
describing 16 energy levels in the $[000]A_{2}^{+}$ irrep as introduced in
section V.3. All eigenvalues of the data correlation matrix were above the
cutoff of $\Lambda=0.02$. The amplitudes are shown in Fig. 10.

amplitude parameters | $\chi^{2}/N_{\mathrm{dof}}$ | other details
---|---|---
$m$, $g_{{D\bar{D}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}}$,
$\gamma_{\psi\omega\to\psi\omega{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}}$,
$\gamma_{\psi\omega\to\psi\omega{\\{{\>\\!}^{{5}\\!}{D}_{3}\\}}}$,
$\gamma_{\psi\phi\to\psi\phi{\\{{\>\\!}^{{5}\\!}{D}_{3}\\}}}$
$\quad\&\,g_{{D_{s}\bar{D}_{s}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}},g_{{D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{D}_{3}\\}}},\gamma_{{D\bar{D}^{\ast}}\to{D\bar{D}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}},\gamma_{{D_{s}\bar{D}_{s}^{\ast}}\to{D_{s}\bar{D}_{s}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}},\gamma_{{D^{\ast}\bar{D}^{\ast}}\to{D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{D}_{3}\\}}}$ | $\frac{8.34}{16-10}=1.39$ | reference amp
$\quad\&\,\gamma_{{D_{s}\bar{D}_{s}^{\ast}}\to{D_{s}\bar{D}_{s}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}},\gamma_{{D^{\ast}\bar{D}^{\ast}}\to{D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{D}_{3}\\}}}$ | $\frac{8.47}{16-7}=0.94$ |
$\quad\&\,\gamma_{{D_{s}\bar{D}_{s}^{\ast}}\to{D_{s}\bar{D}_{s}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}},\gamma_{{D^{\ast}\bar{D}^{\ast}}\to{D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{D}_{3}\\}}}$ | $\frac{8.45}{16-7}=0.94$ | simple phase space
$\quad\&\,\gamma_{{D\bar{D}^{\ast}}\to{D\bar{D}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}},\gamma_{{D_{s}\bar{D}_{s}^{\ast}}\to{D_{s}\bar{D}_{s}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}},\gamma_{{D^{\ast}\bar{D}^{\ast}}\to{D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{D}_{3}\\}}}$ | $\frac{8.34}{16-8}=1.04$ |
$\quad\&\,g_{{D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{D}_{3}\\}}},\gamma_{{D\bar{D}^{\ast}}\to{D\bar{D}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}},\gamma_{{D_{s}\bar{D}_{s}^{\ast}}\to{D_{s}\bar{D}_{s}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}},\gamma_{{D^{\ast}\bar{D}^{\ast}}\to{D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{D}_{3}\\}}}$ | $\frac{8.50}{16-8}=1.06$ |
$\quad\&\,g_{{D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{D}_{3}\\}}},\gamma_{{D\bar{D}^{\ast}}\to{D\bar{D}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}},\gamma_{{D_{s}\bar{D}_{s}^{\ast}}\to{D_{s}\bar{D}_{s}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}},\gamma_{{D^{\ast}\bar{D}^{\ast}}\to{D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{D}_{3}\\}}}$ | $\frac{8.48}{16-8}=1.06$ | simple phase space
$\quad\&\,g_{{D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{D}_{3}\\}}},\gamma_{{D\bar{D}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}\to{D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{D}_{3}\\}}},\gamma_{{D_{s}\bar{D}_{s}^{\ast}}\to{D_{s}\bar{D}_{s}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}}$ | $\frac{4.65}{16-8}=0.58$ |
$\quad\&\,g_{{D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{D}_{3}\\}}},\gamma_{{D\bar{D}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}\to{D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{D}_{3}\\}}},\gamma_{{D_{s}\bar{D}_{s}^{\ast}}\to{D_{s}\bar{D}_{s}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}},\gamma_{{D^{\ast}\bar{D}^{\ast}}\to{D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{D}_{3}\\}}}$ | $\frac{4.33}{16-9}=0.62$ |
$\quad\&\,g_{{D_{s}\bar{D}_{s}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}},\gamma_{{D_{s}\bar{D}_{s}^{\ast}}\to{D_{s}\bar{D}_{s}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}},\gamma_{{D^{\ast}\bar{D}^{\ast}}\to{D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{D}_{3}\\}}}$ | $\frac{8.47}{16-8}=1.06$ |
$\quad\&\,g_{{D_{s}\bar{D}_{s}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}},g_{{D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{D}_{3}\\}}},\gamma_{{D_{s}\bar{D}_{s}^{\ast}}\to{D_{s}\bar{D}_{s}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}},\gamma_{{D^{\ast}\bar{D}^{\ast}}\to{D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{D}_{3}\\}}}$ | $\frac{8.47}{16-9}=1.21$ |
Table 7: $J^{PC}=3^{++}$ parameterization variations. Parameters not listed
are fixed to zero. A $K$-matrix constant parameter $\gamma$ is diagonal in
${\>\\!}^{{2S+1}\\!}{\ell}_{J}$ unless otherwise stated. A Chew-Mandelstam
phase space with a pole subtraction point is used unless otherwise stated.

###  E.2 Summary of $J^{PC}=2^{++}$ parameterizations

In Table 8 we summarize 24 $J^{PC}=2^{++}$ parameterization variations
obtained including energies from moving frame irreps. Many more
parameterization forms were attempted but only those with good $\chi^{2}$
minima are retained. Four additional parmeterizations were obtained with the
same free parameters as the reference amplitude, two with the scattering
hadron masses set to their central values $\pm 1\sigma$, as given in Table 2,
and two with the anisotropy set to its upper and lower values $3.444\pm 0.006$
as determined from the pion. By default we apply a cutoff on data correlation
eigenvalues of $\Lambda=0.02$ as mentioned in the text, with further details
given in Appendix F.

amplitude parameters | $\chi^{2}/N_{\mathrm{dof}}$ | other details
---|---|---
$m$, $g_{{D\bar{D}}}$, $g_{{D\bar{D}^{\ast}}}$, $g_{{D_{s}\bar{D}_{s}}}$,
$g_{{D^{\ast}\bar{D}^{\ast}}}$, $\gamma_{\eta_{c}\eta\to\eta_{c}\eta}$,
$\gamma_{{\psi\omega}\to{\psi\omega}}$, $\gamma_{\psi\phi\to\psi\phi}$
| $\frac{65.3}{86-7-23}=1.17$ |
$\quad\&\quad g_{{D\bar{D}}}$ (freed) | $\frac{65.1}{86-7-23}=1.16$ |  $g_{{D\bar{D}^{\ast}}}=-40$ (fixed)
$\quad\&\quad\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}}$ | $\frac{62.8}{86-8-23}=1.14$ | reference amp Eq. 15
$\quad\&\quad\gamma_{{D\bar{D}}\to{D\bar{D}}},\gamma_{{D\bar{D}^{\ast}}\to{D\bar{D}^{\ast}}},\gamma_{{D_{s}\bar{D}_{s}}\to{D_{s}\bar{D}_{s}}},\gamma_{{D^{\ast}\bar{D}^{\ast}}\to{D^{\ast}\bar{D}^{\ast}}}$ | $\frac{64.0}{86-11-23}=1.23$ |
$\quad\&\quad\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}},\gamma_{{D\bar{D}}\to{D\bar{D}^{\ast}}},\gamma_{{D\bar{D}^{\ast}}\to{D_{s}\bar{D}_{s}}},\gamma_{{D_{s}\bar{D}_{s}}\to{D_{s}\bar{D}_{s}}}$ | $\frac{56.9}{86-11-23}=1.09$ |
$\quad\&\quad\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}},\gamma_{{D\bar{D}^{\ast}}\to{D\bar{D}^{\ast}}},\gamma_{{D^{\ast}\bar{D}^{\ast}}\to{D^{\ast}\bar{D}^{\ast}}}$ | $\frac{65.3}{86-10-23}=1.23$ |
$\quad\&\quad\gamma_{{D\bar{D}^{\ast}}\to{D\bar{D}^{\ast}}},\gamma_{{D^{\ast}\bar{D}^{\ast}}\to{D^{\ast}\bar{D}^{\ast}}}$ | $\frac{65.3}{86-9-23}=1.21$ |
$\quad\&\quad\gamma_{{D\bar{D}^{\ast}}\to{D_{s}\bar{D}_{s}}}$ | $\frac{60.0}{86-8-23}=1.09$ |
$\quad\&\quad\gamma_{{D^{\ast}\bar{D}^{\ast}}\to{D^{\ast}\bar{D}^{\ast}}}$ | $\frac{65.3}{86-8-23}=1.19$ |
$\quad\&\quad\gamma_{{D\bar{D}^{\ast}}\to{\psi\omega}}$ | $\frac{65.2}{86-7-23}=1.16$ |  $\gamma_{{\psi\omega}\to{\psi\omega}}=0$ (fixed)
| $\frac{99.1}{86-7-12}=1.48$ | $\Lambda=0.01$
| $\frac{74.6}{86-7-19}=1.24$ | $\Lambda=0.016$
| $\frac{54.4}{86-7-28}=1.07$ | $\Lambda=0.024$
| $\frac{45.3}{86-7-35}=1.03$ | $\Lambda=0.032$
| $\frac{36.8}{86-7-40}=0.94$ | $\Lambda=0.040$
$\quad\&\quad\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}}$ | $\frac{63.3}{86-8}=0.81$ | uncorrelated
|  | simple phase space
$\quad\&\quad\gamma_{{\psi\omega}{\\{{\>\\!}^{{3}\\!}{D}_{2}\\}}\to{\psi\omega}{\\{{\>\\!}^{{3}\\!}{D}_{2}\\}}}$ | $\frac{66.9}{86-8-23}=1.22$ |
$\quad\&\quad\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}}+\gamma_{{D_{s}\bar{D}_{s}}\to{D^{\ast}\bar{D}^{\ast}}}$ | $\frac{70.9}{86-9-23}=1.31$ |
$\quad\&\quad\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}}+\gamma_{{D_{s}\bar{D}_{s}}\to{D^{\ast}\bar{D}^{\ast}}}$ | $\frac{68.1}{86-9-23}=1.26$ |  $g_{D\bar{D}}$ = 20
$\quad\&\quad\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}}+\quad\gamma_{{D\bar{D}^{\ast}}\to{D\bar{D}^{\ast}}}+\quad\gamma_{{D^{\ast}\bar{D}^{\ast}}\to{D^{\ast}\bar{D}^{\ast}}}$ | $\frac{69.4}{86-10-23}=1.31$ |
|  | simple phase space
| $\frac{106.6}{86-7-12}=1.59$ | $\Lambda=0.010$
| $\frac{78.8}{86-7-19}=1.31$ | $\Lambda=0.016$
| $\frac{69.5}{86-7-23}=1.24$ | $\Lambda=0.020$
| $\frac{57.0}{86-7-28}=1.12$ | $\Lambda=0.024$
Table 8: $J^{PC}=2^{++}$ parameterization variations. Parameters not listed
are fixed to zero. A Chew-Mandelstam phase space with a $K$-matrix pole
subtraction point is used unless otherwise stated. An cutoff on data
correlation eigenvalues of $\Lambda=0.02$ is used unless otherwise stated. We
fix $g_{{D\bar{D}}}=10\cdot a_{t}$ unless indicated otherwise.

###  E.3 Summary of $J^{PC}=0^{++}$ parameterizations

####  E.3.1 Coupled ${\eta_{c}\eta}$ and ${D\bar{D}}$ scattering below
${\eta_{c}\eta^{\prime}}$ and ${D_{s}\bar{D}_{s}}$ thresholds

This section gives further details on the amplitudes variations used in
Section V.7, where $K$-matrices are determined using energies from
$[000]A_{1}^{+}$, $[001]A_{1}$, $[111]A_{1}$ and $[002]A_{1}$, resulting in 43
levels. In two of the fits a reduced selection of energies is used, removing
$[002]A_{1}$ levels and resulting in 31 levels. A data correlation eigenvalue
cutoff of $\Lambda=0.02$ is used, resulting in 5 resets for the 43 level
selection, and 3 resets for the 31 level selection. Using two or three
constant parameter terms $\gamma_{ij}$ in the $K$-matrix results in 10
parameterisations in total, as summarized in Table 9.

$\gamma_{\eta_{c}\eta\to\eta_{c}\eta}$ | $\gamma_{\eta_{c}\eta\to{D\bar{D}}}$ | $\gamma_{{D\bar{D}}\to{D\bar{D}}}$ | $\qquad{\chi^{2}}/{N_{\mathrm{dof}}}\qquad$
---|---|---|---
with Chew-Mandelstam phase space
0.37(16) | -0.65(18) | 0.06(49) |  $\frac{28.4}{31-3-3}=1.14$ $(\ast)$
0.37(15) | -0.64(16) | 0.15(34) | $\frac{40.5}{43-3-5}=1.16$
– | -0.45(14) | 0.14(32) | $\frac{47.6}{43-2-5}=1.32$
0.40(12) | – | -0.39(24) | $\frac{48.6}{43-2-5}=1.35$
0.37(14) | -0.61(15) | – | $\frac{40.8}{43-2-5}=1.13$
with simple phase space
0.36(11) | -0.51(15) | -0.02(40) |  $\frac{45.0}{31-3-3}=1.80$ $(\ast)$
0.36(14) | -0.63(15) | 0.14(31) | $\frac{40.4}{43-3-5}=1.15$
– | -0.46(14) | 0.13(31) | $\frac{47.4}{43-2-5}=1.32$
0.40(12) | – | -0.39(25) | $\frac{48.5}{43-2-5}=1.35$
0.36(14) | -0.61(15) | – | $\frac{40.6}{43-2-5}=1.13$
Table 9: Parameterization variations for two-channel
${\eta_{c}\eta}-{D\bar{D}}$ $S$-wave scattering amplitudes including moving
frame energies. The first row in each block, indicated by $(\ast)$, uses only
31 levels, excluding levels from $[002]A_{1}$. ‘–’ indicates that a parameter
is fixed to zero. The number of degrees of freedom is taken to be
$N_{\mathrm{dof}}=N_{\mathrm{levels}}-N_{\mathrm{pars.}}-N_{\mathrm{reset}}$
using a data-correlation eigenvalue cutoff of $\Lambda=0.02$ as discussed in
Appendix F.

####  E.3.2 Coupled-channel scattering up to $\psi\phi$ threshold at rest and
$a_{t}E_{\mathsf{cm}}=0.69$ in moving frames

In Table 10 we summarize $J^{PC}=0^{++}$ parameterization variations working
up to $\psi\phi$ threshold while including moving-frame information. One
example with parameter values and correlations is given in Eq. 28. These
amplitudes are plotted in Fig. 18, and are used when determining $t$-matrix
poles in Section VI.1.

amplitude parameters | $\chi^{2}/N_{\mathrm{dof}}$ | other details
---|---|---
$m$, $g_{{D\bar{D}}}$, $g_{{D_{s}\bar{D}_{s}}}$,
$g_{{D^{\ast}\bar{D}^{\ast}}}$, $\gamma_{\eta_{c}\eta\to\eta_{c}\eta}$,
$\gamma_{\psi\phi\to\psi\phi}$
$\quad\quad\&\,g_{{\psi\omega}},\,\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}},\,\gamma_{\eta_{c}\eta^{\prime}\to\eta_{c}\eta^{\prime}},\,\gamma_{{\psi\omega}{\\{{\>\\!}^{{5}\\!}{D}_{4}\\}}\to{\psi\omega}{\\{{\>\\!}^{{5}\\!}{D}_{4}\\}}}$ | $\frac{91.0}{90-10-16}=1.42$ | reference amp
$\quad\quad\&\,\gamma_{{\psi\omega}\to{\psi\omega}}$ | $\frac{91.6}{90-7-16}=1.37$ |
$\quad\quad\&\,g_{{\psi\omega}},\,\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}}$ | $\frac{91.5}{90-8-16}=1.39$ |
$\quad\quad\&\,g_{{\psi\omega}},\,\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}},\,\gamma_{{\psi\omega}{\\{{\>\\!}^{{5}\\!}{D}_{4}\\}}\to{\psi\omega}{\\{{\>\\!}^{{5}\\!}{D}_{4}\\}}}$ | $\frac{91.2}{90-9-16}=1.40$ |
$\quad\quad\&\,\gamma_{{D\bar{D}}\to{D\bar{D}}},\,\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}},\,g_{{\psi\omega}}$ | $\frac{87.1}{90-9-16}=1.34$ |
$\quad\quad\&\,\gamma_{{D\bar{D}}\to{D\bar{D}}},\,\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}},\,\gamma_{{\psi\omega}\to{\psi\omega}},\,\gamma_{{\psi\omega}\to{D^{\ast}\bar{D}^{\ast}}}$ | $\frac{94.7}{90-10-16}=1.48$ |
$\quad\quad\&\,\gamma_{{D\bar{D}}\to{D\bar{D}}},\,\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}},\,\gamma_{{\psi\omega}\to{\psi\omega}}$ | $\frac{94.7}{90-9-16}=1.46$ |
$\quad\quad\&\,\gamma_{{D_{s}\bar{D}_{s}}\to{D_{s}\bar{D}_{s}}},\,\gamma_{{\psi\omega}\to{\psi\omega}}$ | $\frac{93.6}{90-9-16}=1.44$ |
$\quad\quad\&\,\gamma_{{D^{\ast}\bar{D}^{\ast}}\to{D^{\ast}\bar{D}^{\ast}}},\,\gamma_{{\psi\omega}\to{\psi\omega}}$ | $\frac{92.8}{90-9-16}=1.43$ |
$\quad\quad\&\,\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}},\,\gamma_{{\psi\omega}\to{\psi\omega}},\,\gamma_{\eta_{c}\eta^{\prime}\to\eta_{c}\eta^{\prime}}$ | $\frac{95.0}{90-9-16}=1.46$ |
$\quad\quad\&\,\gamma_{{\psi\omega}\to{\psi\omega}}$ | $\frac{92.9}{90-7-16}=1.39$ | simple phase space
$\quad\quad\&\,\gamma_{{\psi\omega}\to{\psi\omega}},\,\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}}$ | $\frac{90.6}{90-8-16}=1.37$ | simple phase space
$\quad\quad\&\,\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}},\,g_{{\psi\omega}}$ | $\frac{143.0}{90-8-6}=1.88$ |  $\Lambda=0.01$ ($\ast$)
$\quad\quad\&\,\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}},\,g_{{\psi\omega}}$ | $\frac{59.4}{90-8-35}=1.26$ |  $\Lambda=0.04$ ($\ast$)
$\quad\quad\&\,\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}},\,\gamma_{{\psi\omega}\to{\psi\omega}}$ | $\frac{143.4}{90-8-6}=1.89$ |  $\Lambda=0.01$ ($\ast$)
$\quad\quad\&\,\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}},\,\gamma_{{\psi\omega}\to{\psi\omega}}$ | $\frac{115.3}{90-8-11}=1.62$ | $\Lambda=0.016$
$\quad\quad\&\,\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}},\,\gamma_{{\psi\omega}\to{\psi\omega}}$ | $\frac{82.2}{90-8-20}=1.33$ | $\Lambda=0.024$
$\quad\quad\&\,\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}},\,\gamma_{{\psi\omega}\to{\psi\omega}}$ | $\frac{69.8}{90-8-28}=1.29$ | $\Lambda=0.032$
$\quad\quad\&\,\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}},\,\gamma_{{\psi\omega}\to{\psi\omega}}$ | $\frac{59.4}{90-8-35}=1.26$ |  $\Lambda=0.04$ ($\ast$)
$\quad\quad\&\,\gamma_{{D\bar{D}}\to{D_{s}\bar{D}_{s}}},\,\gamma_{{\psi\omega}\to{\psi\omega}}$ | $\frac{71.0}{90-8}=0.87$ | uncorrelated
Table 10: $J^{PC}=0^{++}$ parameterization variations. Parameters not listed
are fixed to zero. A Chew-Mandelstam phase space with a $K$-matrix pole
subtraction point is used unless otherwise stated. A data-correlation
eigenvalue cutoff of $\Lambda=0.02$ is used unless otherwise stated. If
$\gamma_{{\psi\omega}{\\{{\>\\!}^{{5}\\!}{D}_{4}\\}}\to{\psi\omega}{\\{{\>\\!}^{{5}\\!}{D}_{4}\\}}}$
is not listed, it is fixed to 300. If
$\gamma_{\eta_{c}\eta^{\prime}\to\eta_{c}\eta^{\prime}}$ is not listed, it is
fixed to 3. All meson-meson channels are ${\>\\!}^{{1}\\!}{S}_{0}$ unless
otherwise stated. Amplitudes marked ($\ast$) are provided for comparison with
Table 8, and they are not included on the plots or used in the analysis.

##  Appendix F Data covariance eigenvalue cutoff

Relatively large data correlations between the energy levels on each lattice
volume are found in this work. For small selections of energy levels, such as
those obtained using only rest-frame energies, this does not present a
problem, but for larger selections of energy levels, such as those using
moving-frames, inverting the data covariance for use in a correlated
$\chi^{2}$ produces an object of questionable validity. Given the use of
ensembles of typically $\sim 500$ gauge configurations, one should not expect
to be able to reliably determine all components of data covariance matrices of
increasingly high rank. This issue is most relevant for the amplitude
determinations in Sections V.8 and V.6.

In Fig. 32, we show the eigenvalues $\lambda_{i}$ of the data correlation
matrices, normalised to the largest eigenvalue, $\lambda_{1}$ for the two
largest sets of spectra relevant to scalar and tensor scattering, including
moving-frame energies. A steep dropoff in value is observed for the smallest
value eigenvalues, and we infer that this is associated with the these modes
being poorly determined. We choose to place a cut on the allowed values when
performing fits, removing the eigenvectors associated with the cut eigenvalues
from the matrix inverse. Our default choice is to retain only those modes with
$\Lambda=\lambda/\lambda_{1}>0.02$. We have explored a range of values of this
cut between $\Lambda=0.01$ and 0.04, and have reported the modest sensitivity
to this choice in earlier appendices.

We have also explored other approaches, such as artificially setting the
correlations to zero, “shrinkage” which interpolates between fully correlated
and uncorrelated [98, 99], and using “eigenvalue limits” rather than hard
cutoffs [55]. The outcomes are broadly similar as can be inferred from Figs.
14 & 18, which include amplitudes determined using several different values of
$\Lambda$ and one where the correlations are set to zero, as shown in Tables
10 and 8 respectively. Provided the smallest eigenmodes or the most extreme
correlations are tamed using one of these methods, the results are in good
agreement. We consider $\Lambda=0.02$ to be a conservative choice.

Figure 32: The eigenvalues, $\lambda_{i}$, of the data correlation matrix
normalized to the largest eigenvalue $\lambda_{1}$, ordered in decreasing
magnitude. We observe a steep falloff above $i\approx 30$ where the directions
in the eigenspace are unlikely to be reliably determined. The number of resets
on the plot indicates the number of modes that are discarded on the $20^{3}$
\+ $24^{3}$ volumes, with cuts of $\Lambda=0.04$ (light grey), 0.02 (dark
grey) and 0.01 (black). The “$0^{++}$” and “$2^{++}$” refer to the correlation
matrix used in the largest amplitude determinations of these $J^{PC}$ from
Sections V.8 and V.6 respectively.

##  Appendix G Additional scattering amplitude poles in $J^{PC}=2^{++}$

The $J^{PC}=2^{++}$ amplitudes determined in Section V.6 feature a single
narrow resonance pole that is systematically present across many
parameterizations, but in addition other poles can be present which vary in
location and which do not have obvious interpretations. We explore these in
this Appendix. In particular we investigate the origin of the closest of these
additional poles using simplified elastic and two-channel systems that capture
the main features of the amplitudes used in this work. We explore the
dependence of these poles on the $g_{D\bar{D}^{\ast}}$ parameter and propose
an alternative parameterization where the additional poles do not arise.
Ultimately we find that the narrow resonance pole on the proximal
$\scriptstyle{({D\bar{D}}[-],{D\bar{D}^{\ast}}[-],{D_{s}\bar{D}_{s}}[-],{D^{\ast}\bar{D}^{\ast}}[+])}$
sheet is the only nearby pole singularity necessary to describe the finite-
volume spectra.

Figure 33 shows the $t$-matrix poles found for a range of parameterizations,
where the nearby pole on the proximal sheet (in red) is observed to show very
little variation over parameterization. In section VI.1 we discuss “mirror”
poles in the context of the scalar amplitudes, and these poles on “hidden”
sheets are to be expected with a large number of Riemann sheets and several
decoupled hadron-hadron channels in each case. Many of these poles can be
ignored due to their distance on the Riemann surface from physical scattering.
In Figure 33 the green and blue points show such “mirror” poles, and we
observe that they show a greater scatter over parameterization variation than
the pole on the proximal sheet.

Figure 33: Poles of the $J^{PC}=2^{++}$ amplitudes plotted in the complex
$k_{{D^{\ast}\bar{D}^{\ast}}}$ and $\sqrt{s}$ planes. The resonance pole on
the $(-,-,-,+)$ sheet is shown in red. A second nearby pole observed on the
$(+,+,+,-)$ sheet is shown in green. Several other more distant poles are also
present as described in the text. In particular, there are virtual bound state
poles on several sheets below ${D\bar{D}}$ threshold. For each pole the
reference parameterization is highlighted in black.

Figure 33 also shows, in grey, poles found _on the physical sheet_. Such poles
indicate a breakdown of causality, but depending upon how close they are to
real scattering energies, the pathology may not be of any practical relevance.
The origin of these poles can be traced back to the presence of the
$k_{i}^{-\ell_{i}}$ barrier factor in Eq. 6 for $D$-wave channels. Such
barriers are necessary to promote the expected behavior of amplitudes at
threshold 373737Which matches the behavior of the Lüscher Zeta functions at
threshold., but unless some other part of the amplitude suppresses their
effect at higher-energies, they can give rise to unwanted energy dependence.

The amplitudes presented in Section V.5 feature a large contribution to the
denominator of the $t$-matrix from the ${D\bar{D}^{\ast}}$ channel,

$\displaystyle\sim
g^{2}_{{D\bar{D}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{2}\\}}}(2k_{{D\bar{D}^{\ast}}})^{4}\rho_{{D\bar{D}^{\ast}}}\,,$

where values of $g_{{D\bar{D}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{2}\\}}}$ are
found between $-30\;a_{t}$ and $-40\;a_{t}$. The dominance of this term over
others in the denominator offers an explanation of the presence of physical
sheet poles. A simple way to see this is by plotting the positions of the
poles of the amplitudes in the complex-$k_{{D\bar{D}^{\ast}}}$ plane, as is
done in Fig. 34. Using the reference parameterization in Eq. 15, the position
of the complex-conjugate pair (in $s$) of poles due to the resonance are
plotted in blue. The physical sheet poles are shown in red, and a virtual
bound state pole that also arises is shown in green.

These poles can be compared to those present in a simple toy-amplitude
featuring a single elastic ${D\bar{D}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{2}\\}}$
amplitude constructed from just a $K$-matrix pole,
$K=g^{2}_{{D\bar{D}^{\ast}}}/{(m_{0}^{2}-s)}$. In this case, the $t$-matrix
has a denominator
$D=m_{0}^{2}-s-ig_{D\bar{D}^{\ast}}^{2}(2k_{D\bar{D}^{\ast}})^{5}/\sqrt{s}$,
and with $g_{{D\bar{D}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{2}\\}}}=-40\>a_{t}$ and
$a_{t}m=0.71$; numerically the final term dominates the behavior. In this case
the denominator equals zero at five points shown in orange in Fig. 34, which
lie very close to the roots of $10^{-6}-i(a_{t}k_{{D\bar{D}^{\ast}}})^{5}$,
shown in grey. This “roots of unity”-like phenomenon is unavoidable with a
large coupling and the $D$-wave threshold factor in the denominator.383838It
is straightforward to observe similar solutions for $\ell>0$ in a very simple
amplitude such as a scattering length approximation,
$k^{2\ell+1}\cot\delta_{\ell}=1/a_{\ell}$.

In simple coupled
${D\bar{D}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{2}\\}}-{D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{S}_{2}\\}}$
systems, an additional term
$-ig_{D^{\ast}\bar{D}^{\ast}}^{2}(2k_{D^{\ast}\bar{D}^{\ast}})/\sqrt{s}$
arises in the denominator. Adding this results in a very close agreement with
the solutions obtained from the amplitudes determined from lattice QCD
energies, as shown by the pale blue-green points in Fig. 34. Fig. 34 shows
only sheets where $\mathrm{Im}\,k_{D^{\ast}\bar{D}^{\ast}}>0$, but poles are
also present on sheets with $\mathrm{Im}\,k_{D^{\ast}\bar{D}^{\ast}}<0$. In
this highly simplified two-channel system, an additional group of five poles
arises that are approximately the complex-conjugates in
$k_{{D\bar{D}^{\ast}}}$ of those in Fig. 34.

Figure 34: The “roots of unity”-like phenomena present in the $2^{++}$
amplitudes. The grey circles show the solutions of
$10^{-6}-i(a_{t}k_{{D\bar{D}^{\ast}}})^{5}=0$, an arbitrary but simple choice
whose zeros approximate the positions of the observed poles, and the large
dashed circle shows $|a_{t}k_{{D\bar{D}^{\ast}}}|=10^{-6/5}$. The orange
circles are the solutions of
$m_{0}^{2}-s-ig_{D\bar{D}^{\ast}}^{2}(2k_{D\bar{D}^{\ast}})^{5}/\sqrt{s}=0$.
The pale blue-green circles are the solutions of
${m_{0}^{2}-s-ig_{D\bar{D}^{\ast}}^{2}(2k_{D\bar{D}^{\ast}})^{5}/\sqrt{s}-ig_{D^{\ast}\bar{D}^{\ast}}^{2}(2k_{D^{\ast}\bar{D}^{\ast}})/\sqrt{s}=0}$
which closely mimics the observed behavior of the amplitudes determined above.
The points with error bars are the relevant poles of the reference
parameterization. The open circles on axes are hadron-hadron thresholds.

The sensitivity of the additional poles to the value of
$g_{{D\bar{D}^{\ast}}}$ can be explored. By default we have fixed
$g_{{D\bar{D}}}=10\,a_{t}$, but owing to the coupling-ratio phenomenon, there
is very little sensitivity to this choice. We may now consider in addition
fixing $g_{{D\bar{D}^{\ast}}}$ to a range of values, and redetermine the
remaining parameters by $\chi^{2}$ minimization for each choice. For
simplicity we do this using only the energies from the $[000]\,E^{+}$ and
$[000]\,T_{2}^{+}$ irreps. In Fig. 35, the result of this procedure is shown.
Four clusters of poles are plotted, the resonance pole on the proximal sheet
$(-,-,-,+)$ is shown below the real $\sqrt{s}$ axis, and three clusters of
spurious poles on the $(+,+,+,\pm)$ sheets are plotted above the axis. The
closest of these on the $(+,+,+,+)$ sheet is the equivalent of the physical
sheet pole shown in red in Fig. 34, and the nearby pole on the $(+,+,+,-)$
sheet is the “mirror” obtained by switching to the unphysical
${D^{\ast}\bar{D}^{\ast}}$ sheet. These closest two poles move away from the
constrained energy region as $g_{D\bar{D}^{\ast}}$ is reduced. The resonance
pole position on the proximal sheet is very well determined and relatively
insensitive to the precise value of $g_{D\bar{D}^{\ast}}$. The $\chi^{2}$ is
also shown and has a smooth dependence on $g_{D\bar{D}^{\ast}}$. We see that
the energy levels clearly do favor a large value of $g_{D\bar{D}^{\ast}}$ and
the proximity of the physical sheet pole can be associated with this.

Figure 35: Left: Pole positions in $\sqrt{s}$ from the $2^{++}$ amplitudes as
a function of $g_{D\bar{D}^{\ast}}$ colored according to the values shown in
the right figure. Four clusters of poles are shown. The tight cluster below
the real axis is the resonance pole on the proximal sheet. The other poles are
strongly dependent on the value of $g_{D^{\ast}\bar{D}^{\ast}}$. Right: the
$\chi^{2}$ value computed from the $[000]\,E^{+}$ and $[000]\,T_{2}^{+}$
irreps at the same values of $g_{D\bar{D}^{\ast}}$.

Since the narrow resonance we are interested in lies some way above
${D\bar{D}^{\ast}}$ threshold, we might anticipate that its properties would
not be overly sensitive to the barrier behavior at threshold. This can be
explored by crudely replacing the $k_{{D\bar{D}^{\ast}}}^{5}$ factor with a
lower power in the parameterization. This introduces a mismatch with the
behavior of the Lüscher Zeta functions, but in practice this only has a
significant effect for energies very close to or more severely _below_
threshold. By restricting to consideration of the $[000]\,E^{+}$ and
$[000]\,T_{2}^{+}$ spectra, which have no energy levels at or below threshold
that have significant overlap onto ${D\bar{D}^{\ast}}$-like operators, we
anticipate that we do not introduce a serious error into the analysis.

Using a similar parameterization to the reference parameterization determined
from $[000]\,E^{+}$ and $[000]\,T_{2}^{+}$ energies, as given in Eq. 13, we
artificially modify the $\ell_{i}$ for
${D\bar{D}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{2}\\}}$ terms in Eq. 6 to take
value 1 rather than 2. This leads to a term in the denominator with only three
powers of momentum rather than five. Describing 47 energy levels results in an
amplitude,

$a_{t}m=$ | $(0.7037\pm 0.0011\pm 0.0001)$ | $\begin{bmatrix}[r]1.00&-0.17&-0.18&-0.24&-0.01&-0.15&0.08&-0.07&-0.16&0.01\\\ &1.00&-0.17&-0.56&-0.00&0.11&0.27&-0.03&0.06&-0.10\\\ &&1.00&0.29&-0.02&0.03&-0.57&-0.06&-0.09&-0.06\\\ &&&1.00&-0.00&0.04&-0.32&0.07&0.10&-0.04\\\ &&&&1.00&-0.00&0.02&-0.02&0.01&0.00\\\ &&&&&1.00&0.01&0.02&0.01&-0.01\\\ &&&&&&1.00&0.08&0.10&0.09\\\ &&&&&&&1.00&0.08&0.23\\\ &&&&&&&&1.00&0.03\\\ &&&&&&&&&1.00\end{bmatrix}$
---|---|---
“$g_{{D\bar{D}^{\ast}}{\\{{\>\\!}^{{3}\\!}{D}_{2}\\}}}$” $=$ | $\mathit{(-4.39\pm 0.70\pm 0.17)}$
$g_{{D_{s}\bar{D}_{s}}{\\{{\>\\!}^{{1}\\!}{D}_{2}\\}}}=$ | $(-0.32\pm 3.49\pm 0.97)\cdot a_{t}$
$g_{{D^{\ast}\bar{D}^{\ast}}{\\{{\>\\!}^{{5}\\!}{S}_{2}\\}}}=$ | $(1.74\pm 0.22\pm 0.13)\cdot a_{t}^{-1}$
$g_{{\psi\omega}{\\{{\>\\!}^{{5}\\!}{S}_{2}\\}}}=$ | $(0.00\pm 0.22\pm 0.06)\cdot a_{t}^{-1}$
$\gamma_{\eta_{c}\eta{\\{{\>\\!}^{{1}\\!}{D}_{2}\\}},\eta_{c}\eta{\\{{\>\\!}^{{1}\\!}{D}_{2}\\}}}=$ | $(22.0\pm 23.9\pm 7.83)\cdot a_{t}^{4}$
$\gamma_{{D\bar{D}}{\\{{\>\\!}^{{1}\\!}{D}_{2}\\}},{D_{s}\bar{D}_{s}}{\\{{\>\\!}^{{1}\\!}{D}_{2}\\}}}=$ | $(163\pm 189\pm 44)\cdot a_{t}^{4}$
$\gamma_{{\psi\omega}{\\{{\>\\!}^{{5}\\!}{S}_{2}\\}},{\psi\omega}{\\{{\>\\!}^{{5}\\!}{S}_{2}\\}}}=$ | $(-0.88\pm 0.45\pm 0.05)$
$\gamma_{{\psi\omega}{\\{{\>\\!}^{{3}\\!}{D}_{2}\\}},{\psi\omega}{\\{{\>\\!}^{{3}\\!}{D}_{2}\\}}}=$ | $(561\pm 513\pm 132)\cdot a_{t}^{4}$
$\gamma_{\psi\phi{\\{{\>\\!}^{{5}\\!}{S}_{2}\\}},\psi\phi{\\{{\>\\!}^{{5}\\!}{S}_{2}\\}}}=$ | $(1.33\pm 0.78\pm 0.04)$
$g_{{D\bar{D}}{\\{{\>\\!}^{{1}\\!}{D}_{2}\\}}}=$ |  $10\cdot a_{t}$ (fixed) |
|  $\chi^{2}/N_{\mathrm{dof}}=\frac{49.1}{47-10}=1.33$ .
(41)

This amplitude and its resonance pole position are shown in Fig. 36, alongside
those of other $2^{++}$ amplitudes given in Eq. 13 and Eq. 15.

The nearby resonance pole on the proximal sheet appears at
$a_{t}\sqrt{s_{0}}=(0.7016\pm 0.0013)-\frac{i}{2}(0.013\pm 0.003)$, which, as
anticipated, is essentially the same location as when the correct $D$-wave
barrier behavior was present for ${D\bar{D}^{\ast}}$. On the other hand,
physical sheet poles are found in completely different locations for this
amplitude, with the nearest being at $a_{t}\sqrt{s_{0}}=(0.651\pm
0.010)+\frac{i}{2}(0.169\pm 0.052)$, which is very far from physical
scattering.

In summary we conclude that the physical sheet poles found in the amplitudes
presented in the main text are an artifact of the $D$-wave barrier factors for
the ${D\bar{D}^{\ast}}$channel, while the narrow resonance pole which is the
dominant feature of the $2^{++}$ amplitude is a robust result.

Figure 36: The amplitude and resonance pole position from the amplitude in Eq.
(41) using the modified threshold factors compared with other amplitudes
obtained in this work. In the left and upper panels we show the diagonal
${D\bar{D}}$, ${D\bar{D}^{\ast}}$ and ${D^{\ast}\bar{D}^{\ast}}$ amplitudes
compared with the amplitudes given in Eq. (13) using only rest-frame energies
(dotted curves) and Eq. (15) including also moving frame energies (pink
curves). In the lower right panel, a comparison of the resonance pole
positions are shown, including the value obtained considering all
parameterization variations in table 8.

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