for the Hadron Spectrum Collaboration

# Near-threshold states in coupled $DD^{\ast}-D^{\ast}D^{\ast}$ scattering
from lattice QCD

Travis Whyte<EMAIL_ADDRESS>Department of Applied Mathematics and Theoretical
Physics, Center for Mathematical Sciences,
University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK  School of
Mathematics, Trinity College, Dublin 2, Ireland  David J. Wilson
<EMAIL_ADDRESS>Department of Applied Mathematics and Theoretical
Physics, Center for Mathematical Sciences,
University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK  Christopher
E. Thomas<EMAIL_ADDRESS>Department of Applied Mathematics and
Theoretical Physics, Center for Mathematical Sciences,
University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

###### Abstract

The first determination of doubly-charmed isospin-0 coupled-channel
$DD^{\ast}-D^{\ast}D^{\ast}$ scattering amplitudes from lattice QCD is
presented. The finite-volume spectrum is computed for three lattice volumes
with a light-quark mass corresponding to $m_{\pi}\approx 391$ MeV and is used
to extract the scattering amplitudes in $J^{P}=1^{+}$ via the Lüscher
quantization condition. By analytically continuing the scattering amplitudes
to complex energies, a $T_{cc}$ pole corresponding to a virtual bound state is
found below $DD^{\ast}$ threshold. We also find a second pole,
$T_{cc}^{\prime}$, corresponding to a resonance pole below the kinematically
closed $D^{\ast}D^{\ast}$ channel, to which it has a strong coupling. A non-
zero coupling is robustly found between the $S$-wave $DD^{\ast}$ and
$D^{\ast}D^{\ast}$ channels producing a clear cusp in the $DD^{\ast}$
amplitude at the $D^{\ast}D^{\ast}$ threshold energy. This suggests that the
experimental $T_{cc}^{\prime}$ should be observable in $DD^{\ast}$ and
$D^{\ast}D^{\ast}$ final states at ongoing experiments.

##  I Introduction

The wealth of exotic hadron states discovered recently in experimentation has
driven theoretical efforts to understand their nature. In particular, the
recent observation of the doubly-charmed, manifestly exotic $T_{cc}^{+}(3875)$
at the LHCb experiment [1, 2] close in energy to ${DD^{\ast}}$ threshold
motivates further study. Having two-charm quarks and being charged, this state
goes beyond expectations from conventional quark model $q\bar{q}$ spectra,
with a minimal $cc\bar{u}\bar{d}$ quark content content [1]. LHCb investigates
several charged and neutral ${DD^{\ast}}$ combinations, finding only one
enhancement and implying a preference for isospin $I=0$.

Several aspects of the $T_{cc}$ make it particularly appealing for theoretical
studies. The exotic nature limits the possible decay modes, it arises close to
${DD^{\ast}}$ threshold and it is very narrow. This apparent simplicity should
help in determining the relevant degrees of freedom. However, the $T_{cc}$ and
the $D^{\ast}$ through which it decays are unstable resonances and undergo
decays to lighter hadrons. The unstable nature of the $D^{\ast}$ leads to a
three-body $DD\pi$ final state. Furthermore, two channels open close together,
due to the different charge combinations – the energy splitting of the
$D^{0}D^{\ast+}$ and $D^{+}D^{\ast 0}$ thresholds is of the order of the
observed width and so could be significant.

The intrinsic spin of the vector $D^{\ast}$ can combine with the $D$-meson in
many ways. Non-zero angular momenta are suppressed close to threshold and so
in the theoretical modeling of this system, it has been widely assumed that
$S$-wave is dominant, forming $J^{P}=1^{+}$. However, higher angular momenta
should also be investigated.

The LHCb ${DD^{\ast}}$ data spans a 30 MeV window up to 3900 MeV [1, 2]. A
little higher in energy the ${D^{\ast}{D}^{\ast}}$ channel opens, which may
bring additional features. A variety of theoretical models find a state with
quantum numbers $J^{P}=1^{+}$ below ${D^{\ast}{D}^{\ast}}$ threshold [3, 4, 5,
6, 7, 8, 9]. Some have associated this state as a heavy partner of the
$T_{cc}$ seen near ${DD^{\ast}}$ threshold. A prediction of a partner to the
$T_{cc}$ based on fits to the LHCb data and Heavy Quark Spin Symmetry [10] was
given in [11]. Subsequent predictions from chiral effective field theory,
including three body effects but assuming ${DD^{\ast}}$ and
${D^{\ast}{D}^{\ast}}$ are decoupled, appear in [12]. Its properties and decay
modes were studied in [13, 14].

Quantum Chromodynamics (QCD) is the theory of the strong interactions and it
is thus desirable to know what QCD predicts for the $T_{cc}$. A
nonperturbative approach to QCD making only controlled and systematically
improvable approximations is _lattice QCD_ , where the fundamental equations
of interacting quarks and gluons are discretized on a finite spacetime
lattice. The QCD path integral is sampled on the lattice using Monte Carlo
techniques, which allows for the extraction of correlation functions and the
calculation of the spectrum of the eigenstates of QCD on the lattice. Due to
the finite spatial volume of the lattice, momentum is quantized and only a
discrete spectrum is obtained.

The discrete spectra can then be used to determine infinite volume scattering
amplitudes that are continuous functions of energy using the Lüscher finite
volume formalism [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27].

The scattering amplitudes contain singularities, which are a model independent
way of understanding the spectroscopic content in terms of bound states,
virtual bound states, and resonances. The approach is reviewed in Ref. [28].

Lattice calculations of resonances typically use degenerate light-quark masses
that produce a pion that is too massive. When considering charm quarks, this
also results in a $D^{\ast}$ that is stable against decay to $D\pi$ for all
but the very smallest light-quark masses. This enables a simple treatment of
${DD^{\ast}}$ scattering using well-established two-hadron methods which have
demonstrated success in charmed [29, 30, 31, 32, 33] and charmonium systems
[34, 35, 36, 37, 38].

Investigations of ${DD^{\ast}}$ on the lattice have been made over a range of
light quark masses. Ref. [39] found that the ground state energy obtained
utilizing a basis of meson-meson and tetraquark operators in $J^{P}=1^{+}$
lies below the ${DD^{\ast}}$ threshold using three lattice spacings and
$m_{\pi}$ ranging from 153 to 689 MeV. This is suggestive of the presence of
attractive interactions, however the existence of a pole was not verified. In
Ref. [40], the HAL QCD method was used to extract $S$-wave phase shifts with
pions at nearly physical mass, locating a virtual bound state 45 keV below
${DD^{\ast}}$ threshold. Ref. [41] examined elastic ${DD^{\ast}}$ scattering
in $S$-wave for $I=0,1$, but did not verify the presence of a pole. Elastic
${DD^{\ast}}$ $S$-wave scattering was investigated in [42] for $m_{\pi}\approx
280$ MeV, and a pole in the scattering amplitudes corresponding to a virtual
bound state was found $9_{-7}^{+4}$ MeV below the ${DD^{\ast}}$ threshold. Its
quark mass dependence was examined in [43].

In this article, the first calculation of coupled-channel isospin-0
$J^{P}=1^{+}$ ${DD^{\ast}}-{D^{\ast}{D}^{\ast}}$ scattering from lattice QCD
is presented, finding a virtual bound-state below ${DD^{\ast}}$ threshold and
a _resonance_ just below ${D^{\ast}{D}^{\ast}}$ threshold coupled to both
channels. In Section II, the relevant partial wave configurations and
symmetries for ${DD^{\ast}}$ and ${D^{\ast}{D}^{\ast}}$ are discussed. In
Section III, the techniques used to extract the finite volume spectra are
reviewed, including operator construction and variational analysis of the
computed correlation functions. The finite volume spectra obtained from both
systems overall at rest and with non-zero total momentum for three lattice
volumes is also presented. In Section IV, the resulting scattering amplitudes
extracted from the finite volume spectra using the Lüscher formalism are
presented, including the scattering amplitudes of higher partial waves.
Section V contains an investigation of the singularities of the amplitudes. An
interpretation is given in Section VI and we summarize our results in Section
VII.

##  II Partial waves

We first consider the possible partial waves and their relevant symmetries for
${DD^{\ast}}$ and ${D^{\ast}{D}^{\ast}}$ with exact isospin symmetry. These
are classified by their total spin parity $J^{P}$, but for any given $J^{P}$
different orbital angular momenta may contribute due to the presence of the
vector $D^{\ast}$. It is convenient to use a spectroscopic notation
${\>\\!}^{{2S+1}\\!}{\ell}_{J}$ with total spin $S$ and orbital angular
momentum $\ell$. Partial waves grow at threshold like $k_{cm}^{2\ell}$ for cm
momentum $k_{cm}$ in the absence of any enhancement.

In $J^{P}=1^{+}$, the leading ${DD^{\ast}}$ contribution is in $S$-wave, but
it may also mix dynamically with $D$-wave to form a coupled
${\>\\!}^{{3}\\!}{S}_{1}-{\>\\!}^{{3}\\!}{D}_{1}$ system. $P$-waves produce
$J^{P}=0^{-},1^{-},2^{-}$ combinations, and further $D$-wave combinations
arise in other $J^{P}$, such as $3^{+}$. The relevant combinations up to
$F$-wave are summarized in Table 1.

Due to the presence of identical bosons in ${D^{\ast}{D}^{\ast}}$ states and
the requirement that the total wavefunction be symmetric under the interchange
of the two $D^{\ast}$, many ${D^{\ast}{D}^{\ast}}$ combinations are forbidden.
In $I=0$ the flavor wavefunction is antisymmetric, so the spin$\times$space
wavefunction is also required to be antisymmetric. This leads to the nonzero
partial waves given in Table 2, with $S=1$ for $\ell$-even and $S=0,2$ for
$\ell$-odd.

Table 1:  The allowed partial wave configurations up to $\ell=3$ for ${DD^{\ast}}$ in $I=0$. $\ell$ | ${\>\\!}^{{2S+1}\\!}{\ell}_{J}$ | $J^{P}$
---|---|---
0 | ${\>\\!}^{{3}\\!}{S}_{1}$ | $1^{+}$
1 | ${\>\\!}^{{3}\\!}{P}_{0,1,2}$ | $\\{0,1,2\\}^{-}$
2 | ${\>\\!}^{{3}\\!}{D}_{1,2,3}$ | $\\{1,2,3\\}^{+}$
3 | ${\>\\!}^{{3}\\!}{F}_{2,3,4}$ | $\\{2,3,4\\}^{-}$
Table 2:  The allowed ${D^{\ast}{D}^{\ast}}$ $I=0$ partial wave configurations up to $\ell=3$ and the symmetry (symmetric “S” or antisymmetric “A”) for the spin, space, and flavor wavefunctions which result in an overall wavefunction that is symmetric under the interchange of the two $D^{\ast}$. $\ell$ | ${\>\\!}^{{2S+1}\\!}{\ell}_{J}$ | $J^{P}$ | Spin | Space | Flavor | Total
---|---|---|---|---|---|---
0 | ${\>\\!}^{{3}\\!}{S}_{1}$ | $1^{+}$ | A | S | A | S
1 | ${\>\\!}^{{1}\\!}{P}_{1}$ | $1^{-}$ | S | A | A | S
${\>\\!}^{{5}\\!}{P}_{1,2,3}$ | $\\{1,2,3\\}^{-}$
2 | ${\>\\!}^{{3}\\!}{D}_{1,2,3}$ | $\\{1,2,3\\}^{+}$ | A | S | A | S
3 | ${\>\\!}^{{1}\\!}{F}_{3}$ | $3^{-}$ | S | A | A | S
${\>\\!}^{{5}\\!}{F}_{1,...,5}$ | $\\{1,2,3,4,5\\}^{-}$

##  III Determination of Lattice Energies

###  III.1 Lattice Setup

Lattice calculations were performed on anisotropic lattices with three volumes
having the same quark mass and lattice spacing. The volumes and the number of
configurations per volume are given in Table 3. In order to improve energy
resolution, the temporal lattice spacing $a_{t}$ is chosen to be finer than
the spatial one $a_{s}$ by a factor $\xi=a_{s}/a_{t}\approx 3.5$. Gauge
configurations were generated using a tree level improved Symanzik action in
the gauge sector and a Wilson-Clover action with $2+1$ flavors of dynamical
quarks in the fermion sector, where the strange quark has been tuned to
approximately its physical value [44, 45]. The mass of the $\Omega$ baryon is
used to set the scale when calculating quantities in physical units, which is
$a_{t}^{-1}=m^{\Omega}_{phys}/a_{t}m_{\Omega}=5667$ MeV [46], giving a pion
mass of $m_{\pi}\approx 391$ MeV. This corresponds to a spatial lattice
spacing $a_{s}\approx 0.12$ fm. The quenched charm quark also uses the Wilson-
Clover action and its mass is tuned to approximately reproduce the physical
mass of the $\eta_{c}$ meson [47]. This results in stable $D$ and $D^{\ast}$
mesons with masses approximately $1886$ MeV and $2010$ MeV respectively. In
the scattering analyses, we use the anisotropy determined from the dispersion
relation of the pion as the central value, which was found to be
$\xi=\xi_{\pi}=3.444(6)$ [48]. However, to account for the systematic
uncertainty from obtaining different anisotropies from the relativistic
dispersion relations for the $\pi$, $D$ and $D^{\ast}$ mesons across the three
lattice volumes, we utilize a value of $\xi=3.444(50)$ in this analysis. This
uncertainty spans the central values of the anisotropies determined from the
$\pi$, $D$ and $D^{\ast}$ mesons across the three lattice volumes and the
uncertainity is propagated through to the cm-frame energies obtained from the
lattice energies. Consequently, this is also propagated through into the
scattering amplitudes. See [33, 38] for further details of the dispersion
relations.

$(L/a_{s})^{3}\times(T/a_{t})$ | $N_{\mathrm{cfgs}}$ | $N_{\mathrm{tsrcs}}$ | $N_{\mathrm{vecs}}$
---|---|---|---
$16^{3}\times 128$ | 478 | 8 | 64
$20^{3}\times 256$ | 288 | 3-4 | 128
$24^{3}\times 128$ | 553 | 3-4 | 160
Table 3: A summary of the details for each ensemble used in this work.
$(L/a_{s})^{3}\times(T/a_{t})$ denotes the volume of the lattice with $L$ the
spatial extent and $T$ the temporal extent of the lattice. $N_{\mathrm{cfgs}}$
denotes the number of gauge-field configurations used. $N_{\mathrm{tsrcs}}$
denotes the number of time sources used per gauge-field configuration.
$N_{\mathrm{vecs}}$ denotes the number of vectors used in the distillation
basis.

| $a_{t}m$
---|---
$\pi$ | 0.06906(13)
$D$ | 0.33281(9)
$D^{\ast}$ | 0.35464(14)
$D^{\ast}_{0}$ | 0.40170(18)

| $a_{t}m$
---|---
$DD^{\ast}$ | 0.68745(17)
$D^{\ast}D^{\ast}$ | 0.70928(20)
$DD^{\ast}_{0}$ | 0.73381(20)
$DD\pi$ | 0.73468(18)

Table 4: (Left) The relevant stable masses for the $\pi$ [48], $D$, $D^{\ast}$
[38] and $D_{0}^{\ast}$ [31] and (right) kinematic thresholds for this study.

###  III.2 Details of the Calculation

In order to extract the finite volume spectra from correlation matrices, we
follow the well-established variational method of the generalized eigenvalue
problem (GEVP) [49, 50, 51] and summarize our implementation [52] here. Given
a basis of operators $\\{\mathcal{O}_{i}\\}$ that interpolate the quantum
numbers of interest, we solve the GEVP
$C_{ij}(t)v^{\mathfrak{n}}_{j}(t)=\lambda_{\mathfrak{n}}(t,t_{0})C_{ij}(t_{0})v^{\mathfrak{n}}_{j}(t)$
where

$C_{ij}(t)=\langle 0|\mathcal{O}_{i}(t)\mathcal{O}_{j}^{\dagger}(0)|0\rangle$
(1)

is the matrix of correlation functions and $t_{0}$ is an appropriately chosen
reference timeslice. The eigenvalue, or “principal correlator”,
$\lambda_{\mathfrak{n}}(t,t_{0})$, is used to obtain the energy
$E_{\mathfrak{n}}$ of the $\mathfrak{n}^{th}$ eigenstate
$|\mathfrak{n}\rangle$ through a fit to the form
$\lambda_{\mathfrak{n}}(t,t_{0})=(1-A_{\mathfrak{n}})e^{-E_{\mathfrak{n}}(t-t_{0})}+A_{\mathfrak{n}}e^{-E_{\mathfrak{n}}^{\prime}(t-t_{0})}$,
where the second exponential is used to allow for contamination from excited
states. The eigenvectors, $v^{\mathfrak{n}}_{j}(t)$, give the operator-state
overlaps

$Z_{i}^{\mathfrak{n}}=\langle\mathfrak{n}|\mathcal{O}_{i}^{\dagger}(0)|0\rangle=\sqrt{2E_{\mathfrak{n}}}e^{(E_{\mathfrak{n}}t_{0})/2}(v^{\mathfrak{n}}_{j})^{*}C_{ji}(t_{0})\,,$
(2)

and are used to construct variationally optimized operators
$\Omega_{\mathfrak{n}}^{\dagger}=\sum_{i}v^{\mathfrak{n}}_{i}\mathcal{O}_{i}^{\dagger}$,
the optimal linear combination of the basis of operators to interpolate state
$\mathfrak{n}$.

Because the finite spatial volume and discretization of the lattice break the
rotational symmetry of the the infinite volume continuum, angular momentum $J$
is no longer a good quantum number, and mesons at rest with definite continuum
$J$ are labelled by the finite cubic group irreducible representation (irrep)
$\Lambda$, which we refer to as _subduction_ [53, 52]. Additionally, the
rotational symmetry is further broken for mesons in flight, and so such states
are labelled by an irrep of the little group LG$(\vec{P})$ [54, 55]. The
subductions for pseduoscalar-vector and vector-vector scattering relevant to
$D^{(\ast)}D^{\ast}$ can be found in Appendix A.

The basis of interpolating operators $\\{\mathcal{O}_{i}\\}$ must be chosen to
be sufficiently large and with suitable structures such that all levels in the
energy region of interest can be extracted reliably.

We employ meson-meson-like operators [48] of the form

$\begin{split}\mathcal{O}^{\Lambda,\mu\,{\dagger}}_{D^{(\ast)}D^{\ast}}(\vec{P})&=\sum_{\begin{subarray}{c}\mu_{1},\mu_{2}\\\
\vec{p_{1}},\vec{p_{2}}\end{subarray}}\mathcal{C}(\vec{P}\Lambda\mu;\vec{p_{1}}\Lambda_{1}\mu_{1};\vec{p_{2}}\Lambda_{2}\mu_{2})\\\
&\times\Omega_{D^{(\ast)}}^{\Lambda_{1}\mu_{1}\,{\dagger}}(\vec{p_{1}})\Omega_{D^{\ast}}^{\Lambda_{2}\mu_{2}\,{\dagger}}(\vec{p_{2}})\,,\end{split}$
(3)

with definite overall momentum $\vec{P}=\vec{p}_{1}+\vec{p}_{2}$, transforming
in an irrep $\Lambda$ and row $\mu$ of the appropriate symmetry group. Here
the single meson operators,
$\Omega^{\Lambda_{i}\mu_{i}\,{\dagger}}_{D^{(\ast)}}(\vec{p}_{i})$, are
variationally optimised fermion bilinears
$\bar{\psi}_{\ell}\bm{\Gamma}\psi_{c}$ which interpolate the $D^{(\ast)}$ with
momentum $\vec{p}_{i}$ in irrep $\Lambda_{i}$, and $\bm{\Gamma}$ is a product
of $\gamma$-matrices and gauge covariant derivatives [52, 55]. Up to three
derivatives are used for mesons with $\vec{p}_{i}=0$ and up to two derivatives
for mesons with $\vec{p}_{i}\neq 0$, giving a large basis of operators in each
irrep. The lattice Clebsch-Gordan coefficients, $\mathcal{C}$, couple the
single meson operators to produce a $D^{(\ast)}D^{\ast}$ operator projected to
$\Lambda=\Lambda_{1}\otimes\Lambda_{2}$ with overall momentum $\vec{P}$, and
all momenta related to $\vec{p}_{1}$ and $\vec{p}_{2}$ by allowed lattice
rotations are summed over subject to the constraint that
$\vec{P}=\vec{p}_{1}+\vec{p}_{2}$. Isospin indices and the sum over isospin
components to give overall isospin-0 have been suppressed in Eq. (3). More
details are given in Ref. [48] and pseudoscalar-vector scattering was first
considered in this framework in Ref. [56].

At the light-quark masses chosen, the first inelastic three-hadron channel is
$DD\pi$, given in Table 4. This leads to a large window in energy where only
${DD^{\ast}}$ and ${D^{\ast}{D}^{\ast}}$ are kinematically open. Included in
the basis are all two hadron operators which in the absence of interactions
correspond to a two hadron cm-frame energy, i.e. non-interacting energy, below
the $DD\pi$ threshold. Also included are a limited amount of additional
operators whose non interacting energies lie above the $DD\pi$ threshold to
ensure a robust basis. In [57], it was shown that the inclusion of local
compact $qq\bar{c}\bar{c}$ operators had no significant effect on the
extracted finite volume spectra of ${DD^{\ast}}$ and
${D^{\ast}{D}^{\ast}}$-like states in $I=0$, $J^{P}=1^{+}$ on the $16^{3}$
lattice used here, so they are not included in the basis. The complete bases
of operators used to interpolate the $D^{(\ast)}D^{\ast}$ states, both for
zero and nonzero total momentum, can be found in Appendix B.

Distillation vectors [58] are utilized in order to obtain the necessary two-
point functions to form the correlation matrix of Eq. (1). The distillation
basis is formed using the lowest eigenmodes of the 3D gauge covariant
Laplacian restricted to a time-slice $t$ of the lattice, which projects out
the high frequency modes of the gauge fields. This projection creates smeared
quark fields that are used in the interpolating operators, allowing reliable
extraction of many states in the finite volume spectrum, such as those
resembling mesons with various lattice-permitted momenta. Additionally, the
use of distillation vectors allows for the required Wick contractions to be
efficiently computed. The number of distillation vectors and the number of
time sources are given in Table 3.

###  III.3 Finite Volume Spectra

Figure 1: The extracted finite-volume spectra for the $[000]\Lambda^{P}$
irreps. Points in black are used in the subsequent scattering analysis, while
points in grey are not used. Solid curved lines denote the non interacting
energies with the number of degenerate levels labelled in braces for
${DD^{\ast}}$ (green) and ${D^{\ast}{D}^{\ast}}$ (blue) when the degeneracy is
larger than one. Colored dashed lines denote relevant kinematic thresholds.
The $D_{0}^{*}D$ threshold ($a_{t}E_{cm}=0.73381$) has been ommitted due to
its proximity to the $DD\pi$ threshold ($a_{t}E_{cm}=0.73468$).

Figure 2: As Figure 1, but for the $[n_{x}n_{y}n_{z}]A_{2}$ irreps.
Overlapping points have been displaced horizontally for clarity.

Here the finite volume spectra in the $I=0$ doubly-charmed sector are
presented. The irreps are labelled $[n_{x}n_{y}n_{z}]\Lambda^{(P)}$, where
$[n_{x}n_{y}n_{z}]=\frac{2\pi}{L}(n_{x},n_{y},n_{z})=\vec{P}$ is the quantized
momentum, while $\Lambda$ labels the irrep with parity $P$. For $\vec{P}=0$,
we calculate the spectrum for the $T_{1}^{+},A_{1}^{-},E^{-},E^{+}$ and
$A_{2}^{+}$ irreps, using the basis of operators given in Table 11 of Appendix
B. Also calculated is the spectra for some irreps with $\vec{P}\neq 0$, namely
$[001],[002],[011]$ and $[111]A_{2}$, using the basis of operators given in
Table 12 of Appendix B. The use of the $A_{2}$ irreps allows us to constrain
the $J^{P}=1^{+}$ scattering amplitudes in the full coupled-channel region,
where the $[000]T^{+}_{1}$ irrep would offer only a small amount of
constraint. The additional rest frame irreps $A_{1}^{-},E^{-},E^{+}$ and
$A_{2}^{+}$ allow us to constrain other partial waves that subduce into
$[000]T_{1}^{+}$ and the $A_{2}$ irreps. In Tables 7, 8, 9 and 10 in Appendix
A we display the relevant partial wave subductions into these irreps for
pseudoscalar-vector [56] and identical vector-vector meson scattering.

Figures 1 and 2 show the extracted finite volume spectra for the irreps with
$\vec{P}=0$ and $\vec{P}\neq 0$, respectively. The relevant kinematic
thresholds and non interacting energies, the expected cm-frame energies in the
absence of any interaction, are labeled with their degeneracy. Points in black
are those used in the subsequent scattering analysis, while those in grey are
not used. In the $[000]T_{1}^{+}$ and $[002]A_{2}$ irreps, the ground state
energies display significant shifts down from the ${DD^{\ast}}$ threshold,
which is similar to previous lattice studies in $I=0$. In addition, the energy
levels near the ${D^{\ast}{D}^{\ast}}$ threshold have significant shifts
downwards, consistent with the spectrum obtained in [57] on a single lattice
volume. A similar picture is present in the $[n_{x}n_{y}n_{z}]A_{2}$ irreps.
While the degeneracies of the non interacting levels are large in these
irreps, we observe that some energy levels near ${D^{\ast}{D}^{\ast}}$
threshold have significant shifts down away from the non interacting levels.

In Figure 3, we show the principal correlators with their corresponding fits,
and the operator-state overlaps
$Z_{i}^{\mathfrak{n}}=\langle\mathfrak{n}|\mathcal{O}_{i}(0)|0\rangle$ for the
first eight states in the $[000]T_{1}^{+}$ irrep, which are representative of
the states extracted for the other irreps on each lattice volume. The
normalization is fixed for each operator $i$ by requiring the largest observed
magnitude of $Z_{i}^{\mathfrak{n}}$ over all $\mathfrak{n}$ to be 1.

As discussed in Ref. [38], there is a tension between the $D$ and $D^{\ast}$
meson masses on the three volumes due to some unaccounted for uncertainty. We
allow for this by adding an additional systematic uncertainty to the finite-
volume energy levels using

$a_{t}\delta E_{\mathrm{latt}}\rightarrow a_{t}\sqrt{\delta
E_{\mathrm{latt}}^{2}+\delta E^{2}_{\mathrm{syst}}},$ (4)

where $a_{t}\delta E_{\mathrm{latt}}$ is the uncertainty of the energy level
and we set $a_{t}\delta E_{\mathrm{syst}}=5\times 10^{-4}$. The addition of
this systematic error relieves the tension in the observed masses obtained
from the relativistic dispersion fits of the $D$ and $D^{\ast}$ which may have
been attributed to discretization effects or other additional sources of
unknown systematic error. This added systematic error is propagated through to
the uncertainty in the final scattering amplitudes. Further details can be
found in [38] in the context of charmonium scattering and [33] in the context
of $DK$ scattering.

Figure 3:  The principal correlators, plotted as
$\lambda_{\mathfrak{n}}(t,t_{0})e^{E_{\mathfrak{n}}(t-t_{0})}$ with
$t_{0}=12a_{t}$, for the $[000]T_{1}^{+}$ irrep on the lattice of spatial
extent $L/a_{s}=24$. The result of the fit to the principle correlators is
displayed in red, with the $\chi^{2}/N_{\mathrm{dof}}$ and extracted
$E_{\mathfrak{n}}$ from the fit. Points shown in gray were not used in the
fit. Insets show histograms for the operator-state overlaps (normalized as
explained in the text) for ${DD^{\ast}}$ (green) and ${D^{\ast}{D}^{\ast}}$
(blue) operators ordered by increasing non interacting energy.

##  IV Scattering Analysis

In order to connect the finite volume energies with the infinite volume
scattering amplitudes, we utilize the Lüscher quantization condition [15],
extended to the coupled-channel scattering of hadrons of arbitrary mass and
spin [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]

$\mathrm{det}[{\bf 1}+i{\pmb{\rho}}\cdot{\bf t}\cdot({\bf 1}+i\pmb{\cal
M})]=0$ (5)

where $\pmb{\rho}(E_{cm})$ is a diagonal matrix of the phase space factors
$2k/E_{cm}$ with $k$ the scattering momentum in the cm-frame, ${\bf
t}(E_{cm})$ is the infinite volume scattering t-matrix, and $\pmb{{\cal
M}}(E_{cm},L)$ is a known matrix dependent on the volume and irrep and
$E_{cm}$ is the energy in the cm-frame [27]. The solutions of Eq. (5),
$\\{E_{\mathfrak{n}}\\}$, correspond to the finite volume energies for an
appropriately parameterized ${\bf t}(E_{cm})$, which is related to the unitary
scattering $S$-matrix via ${\bf S}={\bf 1}+2i\pmb{\rho}^{1/2}\cdot{\bf
t}\cdot\pmb{\rho}^{1/2}$.

In coupled-channel scattering with multiple partial waves, the matrices in Eq.
(5) are in the space of all hadron-hadron scattering channels and partial
waves. There are then multiple elements of the $t$-matrix that are unknown at
any value of $E_{cm}$ and must be determined. A convienient parameterization
for coupled-channel scattering is the $K$-matrix parameterization

$[t^{-1}(s)]_{\ell
SJa,\ell^{\prime}S^{\prime}Jb}=\frac{1}{(2k_{a})^{\ell}}[K^{-1}(s)]_{\ell
SJa,\ell^{\prime}S^{\prime}Jb}\frac{1}{(2k_{b})^{\ell^{\prime}}}+\delta_{JJ^{\prime}}\delta_{\ell\ell^{\prime}}\delta_{SS^{\prime}}I_{ab}(s)$
(6)

where the indices run over orbital angular momentum $\ell$, spin $S$, angular
momentum $J$ (the $t$-matrix is block diagonal in $J$), and hadron scattering
channel $a,b$, with $s=E_{cm}^{2}$ and where the cm-frame momenta
$k_{a,b}^{\ell}$ factors promote the expected behavior of partial wave
amplitudes close to threshold. Unitarity of the $S$-matrix is ensured if the
symmetric matrix $[K^{-1}(s)]_{\ell SJa,\ell^{\prime}S^{\prime}Jb}$ is real
for real energies and the diagonal matrix $I_{ab}$ satisfies
$\textrm{Im}I_{ab}=-\rho_{a}\delta_{ab}$. The real part of $I_{ab}$ is either
chosen to be zero (simple phase space), or a Chew-Mandelstam phase space is
used, defined in terms of a dispersive integral as given in App. B of Ref.
[59]. There is freedom in how to parameterise the elements of
$[K^{-1}(s)]_{\ell SJa,\ell^{\prime}S^{\prime}Jb}$ and here we employ
polynominals in $s=E_{cm}^{2}$,

$K(s)_{\ell SJa,\ell^{\prime}S^{\prime}Jb}=\sum_{n}\gamma_{\ell
SJa,\ell^{\prime}S^{\prime}Jb}^{(n)}s^{n},$ (7)

where $\gamma_{\ell SJa,\ell^{\prime}S^{\prime}Jb}^{(n)}$ are real parameters
to be determined. The parameters are determined using a correlated
${\chi}^{2}$ minimization of the difference between the finite volume energies
computed on the lattice and the finite volume energies following from
solutions of Eq. (5) [59, 60].

###  IV.1 ${DD^{\ast}}$ Scattering Below ${D^{\ast}{D}^{\ast}}$ Threshold

We first consider scattering where only the ${DD^{\ast}}$ channel is
kinematically open, which we refer to as elastic scattering despite the
inclusion of multiple partial waves. As is common when particles with non-zero
spin are present, different combinations with the same $J^{P}$ can dynamically
couple, such as ${\>\\!}^{{3}\\!}{S}_{1}$ and ${\>\\!}^{{3}\\!}{D}_{1}$ in
$J^{P}=1^{+}$. Ignoring the contribution of higher partial waves, Eq. (5)
reduces to a single equation for $S$-wave scattering and the $t$-matrix can be
written in terms of a single real function, the scattering phase shift
$\delta_{0}(E_{cm})$, via $t=\frac{1}{\rho}e^{i\delta_{0}}\sin(\delta_{0})$.
In principle, the S-wave phase shift could be extracted using the
$[000]T_{1}^{+}$ irrep alone. However, many volumes would be required to have
a set of energy levels that finely samples the energy dependence. We will
therefore also consider irreps with overall non-zero momentum, but these have
contributions from additional partial waves which must be constrained.

The possible partial wave configurations for ${DD^{\ast}}$ are given in Table
1 up to $\ell=3$ and the irreps to which they contribute are given in Tables 7
and 9 in Appendix A, which show partial wave contributions up to $\ell=4$. As
we wish to extract the $J^{P}=1^{+}$ scattering amplitudes, we utilize the
$[000]T_{1}^{+}$ irrep, where the ${\>\\!}^{{3}\\!}{S}_{1}$ and
${\>\\!}^{{3}\\!}{D}_{1,3}$ partial waves contribute. To increase the amount
of constraint compared with using $[000]T_{1}^{+}$ alone, we additionally
utilize the $[001]$, $[002]$, $[011]$ and $[111]$ $A_{2}$ irreps, which also
have contributions from ${\>\\!}^{{3}\\!}{S}_{1}$ and
${\>\\!}^{{3}\\!}{D}_{1}$. However, these irreps with overall non zero
momentum receive contributions from many other partial waves which must be
constrained to extract the $J^{P}=1^{+}$ scattering amplitude, notably the
${\>\\!}^{{3}\\!}{P}_{0,2}$, ${\>\\!}^{{3}\\!}{D}_{2,3}$ and
${\>\\!}^{{3}\\!}{F}_{2,3,4}$ partial waves, which we refer to as _background
waves_ 111In general, we refer to all partial waves not in $J^{P}=1^{+}$ as
background waves.. To this end, we also utilize the $[000]A_{1}^{-}$ and
$[000]E^{-}$ irreps, with lowest partial wave contribution
${\>\\!}^{{3}\\!}{P}_{0}$ and ${\>\\!}^{{3}\\!}{P}_{2}$, respectively.

In total, we use use 36 energy levels below ${D^{\ast}{D}^{\ast}}$ threshold
up to $a_{t}E_{{cm}}=0.705$.222The energy levels in $[000]A_{2}^{+}$ and
$[000]E^{+}$, with lowest partial wave contribution ${\>\\!}^{{3}\\!}{D}_{3}$
and ${\>\\!}^{{3}\\!}{D}_{2}$, respectively, lie above the energy region
considered here, and as such are not utilized. This is far enough below
${D^{\ast}{D}^{\ast}}$ threshold to ensure that no energy levels which have
dominant overlap with ${D^{\ast}{D}^{\ast}}$ operators are included. We
explicitly parameterize the partial waves ${\>\\!}^{{3}\\!}{S}_{1}$,
${\>\\!}^{{3}\\!}{P}_{0,2}$ and ${\>\\!}^{{3}\\!}{D}_{1,2,3}$. In this energy
region, we assume that partial waves with $\ell\geq 3$ are negligible as they
are expected to be highly suppressed in the absence of nearby singularities
due to threshold momentum factors. ${\>\\!}^{{3}\\!}{P}_{1}$ is not
considered, as this partial wave does not contribute to any of the irreps
used.

As mentioned previously, the ${\>\\!}^{{3}\\!}{S}_{1}$ and
${\>\\!}^{{3}\\!}{D}_{1}$ partial waves can dynamically mix in $J^{P}=1^{+}$
which contributes an off-diagonal matrix element to Eq. (7). The scattering
amplitude describing such a coupling could be expected to have a magnitude
similar to P-wave based on the threshold momentum factors of Eq. (6) and as
such could be significant. However, we fix this and all other off-diagonal
elements to zero for simplicity, which keeps Eq. (7) diagonal. This constraint
will be relaxed in the full coupled-channel calculation of the subsequent
section.

Utilizing a $K$-matrix parameterization of the form of Eq. (7), we find that a
reasonable description can be obtained with a constant and linear term in $s$
for ${\>\\!}^{{3}\\!}{S}_{1}$, and constant elements for others,

$\gamma^{(0)}_{{\>\\!}^{{3}\\!}{S}_{1}}=$ | $(8.89\pm 0.81\pm 1.16)$ | $\begin{bmatrix}[r]1.00&-0.97&0.60&0.55&0.35&0.25&0.13\\\ &1.00&-0.54&-0.49&-0.33&-0.24&-0.09\\\ &&1.00&0.74&0.44&0.17&0.29\\\ &&&1.00&0.49&0.22&0.17\\\ &&&&1.00&0.28&-0.62\\\ &&&&&1.00&-0.30\\\ &&&&&&1.00\end{bmatrix}$
---|---|---
$\gamma^{(1)}_{{\>\\!}^{{3}\\!}{S}_{1}}=$ | $(-241\pm 35\pm 70)\cdot a_{t}^{2}$
$\gamma^{(0)}_{{\>\\!}^{{3}\\!}{P}_{0}}=$ | $(181\pm 15\pm 24)\cdot a_{t}^{2}$
$\gamma^{(0)}_{{\>\\!}^{{3}\\!}{P}_{2}}=$ | $(76\pm 9\pm 17)\cdot a_{t}^{2}$
$\gamma^{(0)}_{{\>\\!}^{{3}\\!}{D}_{1}}=$ | $(1218\pm 390\pm 719)\cdot a_{t}^{4}$
$\gamma^{(0)}_{{\>\\!}^{{3}\\!}{D}_{2}}=$ | $(-1408\pm 761\pm 288)\cdot a_{t}^{4}$
$\gamma^{(0)}_{{\>\\!}^{{3}\\!}{D}_{3}}=$ | $(-410\pm 248\pm 414)\cdot a_{t}^{4}$
|  $\chi^{2}/N_{\mathrm{dof}}=\frac{39.7}{36-7}=1.37$ ,
(8)

where we have suppressed the index on the hadronic scattering channel since
only ${DD^{\ast}}$ scattering is present. For brevity, we have labeled the
matrix elements $\gamma_{\ell SJa,\ell^{\prime}S^{\prime}Jb}^{(n)}$ in Eq. (7)
as $\gamma_{{\>\\!}^{{2S+1}\\!}{\ell}_{J}}^{(n)}$. In this parameterization
and all others that follow, any parameter not listed is implicitly fixed to
zero. The first errors are statistical while the second are systematic errors
arising from the envelope of parameter values obtained by performing separate
$\chi^{2}$-minimizations using the 1$\sigma$ deviations from the central
values of the $D$ and $D^{\ast}$ mesons masses and anisotropy. In the square
brackets we display the correlation between the individual parameters.

Figure 4: (Top) ${DD^{\ast}}$ elastic scattering amplitudes in the
${\>\\!}^{{3}\\!}{S}_{1}$, ${\>\\!}^{{3}\\!}{P}_{0,2}$ and
${\>\\!}^{{3}\\!}{D}_{1,2,3}$ partial waves for the parameterization given by
Eq. (8). Inner shaded bands reflect the statistical uncertainty on the
parameterization and outer bands represent the systematic uncertainty obtained
by varying the mass of the $D$ and $D^{\ast}$ mesons and the anisotropy as
described in the text. (Bottom) The energy levels used to constrain the
scattering amplitudes.

Fig. 4 shows the scattering amplitudes from the $K$-matrix parameterization of
Eq. (8). We observe a rapid turn on of the $S$-wave amplitude at ${DD^{\ast}}$
threshold, suggestive of a pole in the $t$-matrix close to threshold. The
$P$-wave amplitudes are significantly non-zero. However, we observe that
${\>\\!}^{{3}\\!}{P}_{0}$ and ${\>\\!}^{{3}\\!}{P}_{2}$ differ which can
perhaps be anticipated from the larger downward shifts for the lowest levels
in $[000]A_{1}^{-}$ compared with $[000]E^{-}$. This will be further discussed
in Section IV.2. The $D$-wave amplitudes are found to be negligible.

The $S$-wave $t$-matrix (in the absence of higher partial waves) can be recast
to the form $t=\frac{E_{{cm}}}{2}\frac{1}{k\cot\delta_{0}-ik}$, which makes it
obvious that a pole singularity will manifest through an intersection of
$k\cot\delta_{0}$ with $ik$ at an energy below threshold. An intersection of
$k\cot\delta_{0}$ with $-ik$ corresponds to a pole on the real axis below
threshold on the physical sheet, i.e. a bound state. In contrast, an
intersection of $k\cot\delta_{0}$ with $ik$ corresponds to a pole below
threshold on the unphysical sheet, i.e. a virtual bound state.

Fig. 5 (top) shows a plot of $k\cot\delta_{0}$ as a function of $k^{2}$ for
the extracted $S$-wave phase shift (bottom). We observe a clear intersection
of the $k\cot\delta_{0}$ curve with $ik$, corresponding to a virtual bound
state. The position of the virtual bound state using this elastic analysis
will be discussed in Section V.1. In the next section we will expand upon the
determination of the scattering amplitudes and the virtual bound state using a
coupled-channel analysis.

Figure 5: (Top) The $S$-wave amplitude from Eq. (8), shown as
$(a_{t}k)\cot\delta_{0}$ (green), along with $\pm a_{t}|k|$ (blue) for
$k^{2}<0$. The inner shaded band reflects the statistical uncertainty while
the outer band represents the systematic uncertainty obtained by varying the
mass of the $D$ and $D^{\ast}$ mesons and the anisotropy as described in the
text. (Bottom) The corresponding phase shift, $\delta_{0}$. Discrete points
are determined from individual energy levels using the determinant condition
Eq. (5). Those with large statistical error due to their proximity to non
interacting levels have been plotted in grey to lessen their visual impact.

###  IV.2 Coupled-channel ${DD^{\ast}}$, ${D^{\ast}{D}^{\ast}}$ scattering

Building upon the parameterization of elastic ${DD^{\ast}}$ scattering, we now
consider higher energies where both the ${DD^{\ast}}$ and
${D^{\ast}{D}^{\ast}}$ channels are kinematically accessible and so we
consider coupled-channel ${DD^{\ast}}$-${D^{\ast}{D}^{\ast}}$ scattering. In
this higher-energy region, partial waves that are suppressed close to
threshold have more phase space and can become important. In order to have
sufficient constraint, 109 energy levels are used from both the rest-frame
irreps and those with non-zero overall momentum, up to energies of
$a_{t}E_{cm}=0.73$. This is far enough below the $DD\pi$ and $D_{0}^{\ast}D$
thresholds to ensure that no three-body effects or contributions from
$D_{0}^{\ast}D$ scattering arise.

On each volume large correlations are observed between the energy levels. This
has been seen before on these lattices in calculations with charm quarks, as
described in Refs. [33, 38]. It is possible that the correlations between
energy levels are not reliably determined given the number of gauge
configurations used ($\approx 500$ in the case of the $24^{3}$ lattice) and
the number of independent numbers in the correlation matrix (which grows like
$N_{\mathrm{levels}}^{2}$). As found previously, some method for taming these
large correlations is necessary in the coupled-channel case, which is not
required in the elastic analysis as the number of required energy levels to
provide constraint for the scattering amplitudes is significantly smaller
there.

In this study, we use “singular value limits” in the $\chi^{2}$ minimization.
Here the replacement $\lambda_{i}\rightarrow\mathrm{max}(\lambda_{i},\sigma)$
for the eigenvalues, $\lambda_{i}$, of the correlation matrix is made, where
$\sigma=\tau\lambda_{1}$, $\tau$ is the singular value limit and $\lambda_{1}$
is the largest eigenvalue of the correlation matrix. This has the effect of
lowering the correlations between finite volume energies, but increases the
uncertainties [61]. An alternative is to use “singular value cuts”, wherein
eigenvalues below $\sigma$ are set to zero, rather than a limit, as was used
in Refs. [33, 38]. In practical terms, the approaches are similar, however for
a given value of the cutoff $\tau$, “limiting” is found to be less
conservative than “cutting”. We choose a value of $\tau=10^{-2}$. For further
discussion of the use of singular value limits, see Appendix D.

In this coupled-channel analysis, we seek to determine the $J^{P}=1^{+}$
scattering amplitudes of both ${DD^{\ast}}$ and ${D^{\ast}{D}^{\ast}}$. These
amplitudes can be simultaneously extracted by using energy levels around and
above ${D^{\ast}{D}^{\ast}}$ threshold in the same irreps utilized in the
elastic analysis, as energy levels dominated by ${D^{\ast}{D}^{\ast}}$
operators begin to appear just below ${D^{\ast}{D}^{\ast}}$ threshold (see
Figure 3). The high degeneracy of non-interacting levels in the upper part of
the spectra for the irreps utilized mirrors the large number of partial waves
that are present, and these must be accounted for when extracting the
$J^{P}=1^{+}$ ${DD^{\ast}}$ and ${D^{\ast}{D}^{\ast}}$ scattering amplitudes.

In the elastic analysis, we considered all partial waves for ${DD^{\ast}}$
with $\ell\leq 2$ that contribute to the irreps utilized. These partial waves
are also considered in this coupled-channel analysis. As we now seek to
constrain the scattering amplitudes up to $a_{t}E_{cm}=0.73$, additional
energy levels in the $[000]A_{1}^{-}$ and $[000]E^{-}$ irreps are available,
which we can use to further constrain the
${DD^{\ast}}\\{{\>\\!}^{{3}\\!}{P}_{0,2}\\}$ amplitudes. Additionally, the
energy levels in the $[000]A_{2}^{+}$ and $[000]E^{+}$ irreps can be used to
further constrain the ${\>\\!}^{{3}\\!}{D}_{2,3}$ partial waves. We also
choose to include ${\>\\!}^{{3}\\!}{F}_{2}$, which contributes to the
$[000]E^{-}$ and all $A_{2}$ irreps with non zero momentum, to account for the
large degeneracy of non-interacting ${DD^{\ast}}$ levels in the upper part of
the spectrum.

We must also determine ${D^{\ast}{D}^{\ast}}$ amplitudes in these irreps.
There are fewer ${D^{\ast}{D}^{\ast}}$ combinations due to Bose symmetry as
discussed in Section II; see Table 2 and the irreps to which they contribute
in Tables 8 and 10. In contrast to the ${DD^{\ast}}$ partial waves, we do not
include all ${D^{\ast}{D}^{\ast}}$ partial waves with $\ell\leq 2$. Of the
allowed ${D^{\ast}{D}^{\ast}}$ partial waves, we consider
${\>\\!}^{{3}\\!}{S}_{1}$, ${\>\\!}^{{3}\\!}{D}_{1}$,
${\>\\!}^{{5}\\!}{P}_{2}$ and ${\>\\!}^{{5}\\!}{F}_{2}$. Similarly to
${DD^{\ast}}$, the ${D^{\ast}{D}^{\ast}}$ ${\>\\!}^{{3}\\!}{S}_{1}$ and
${\>\\!}^{{3}\\!}{D}_{1}$ partial waves contribute to $[000]T_{1}^{+}$ and all
non-zero momentum $A_{2}$ irreps, while the ${\>\\!}^{{5}\\!}{P}_{2}$ and
${\>\\!}^{{5}\\!}{F}_{2}$ partial waves contribute to $[000]E^{-}$ and all
non-zero momentum $A_{2}$ irreps. The choice to include
${\>\\!}^{{5}\\!}{P}_{2}$ is motivated by the magnitude of the scattering
amplitude of ${DD^{\ast}}$ in ${\>\\!}^{{3}\\!}{P}_{2}$, which was observed to
be significant. A similar effect could be expected of the
${D^{\ast}{D}^{\ast}}$ ${\>\\!}^{{5}\\!}{P}_{2}$ partial wave. Additionally, a
fourth amplitude is required to satisfy the quadruple degeneracy in the upper
part of the spectrum in the $[111]A_{2}$ irrep (see Figure 2). In Appendix C,
we show there is a freedom in the choice of which partial wave to parameterize
for, which we choose to be the partial wave ${\>\\!}^{{5}\\!}{F}_{2}$.

Not accounted for are the ${D^{\ast}{D}^{\ast}}$ partial waves
${\>\\!}^{{3}\\!}{D}_{2,3}$, ${\>\\!}^{{5}\\!}{P}_{3}$, and
${\>\\!}^{{5}\\!}{F}_{3,4,5}$. In Appendix C, we justify neglecting the
partial waves, and show they do not have an effect on the $J^{P}=1^{+}$
scattering amplitudes. The ${D^{\ast}{D}^{\ast}}$ ${\>\\!}^{{1}\\!}{P}_{1}$,
${\>\\!}^{{5}\\!}{P}_{1}$ and ${\>\\!}^{{5}\\!}{F}_{1}$ partial waves are not
considered as these do not contribute to the irreps utilized.

Analogously to the elastic case, we utilize a $K$-matrix parameterization of
the form of Eq. (7) to determine simultaneously the $J^{P}=1^{+}$ amplitude
and the smaller $J^{P}=0^{-},2^{-},2^{+}$ and $3^{+}$ background waves.
Performing this amplitude determination, we find a good description of the
spectra using a constant plus linear term in $s$ for the ${DD^{\ast}}$ and
${D^{\ast}{D}^{\ast}}$ $S$-wave amplitudes and constants for all
others,333Correlations are present between the parameters in all $J^{P}$.
However, for brevity we have chosen to show only the larger diagonal blocks.
Parameter correlations between different $J^{P}$ are typically much smaller
than those within a given $J^{P}$.

$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}\rightarrow{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}=$ | $(6.68\pm 0.53\pm 0.31)$ | $\begin{bmatrix}[r]1.00&-0.80&0.33&-0.12&0.25&0.07&0.10\\\ &1.00&0.19&0.23&-0.27&-0.00&-0.04\\\ &&1.00&0.27&-0.04&0.06&0.09\\\ &&&1.00&-0.95&0.04&0.06\\\ &&&&1.00&-0.03&-0.02\\\ &&&&&1.00&0.03\\\ &&&&&&1.00\end{bmatrix}$
---|---|---
$\gamma^{(1)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}\rightarrow{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}=$ | $(-59\pm 13\pm 21)\cdot a_{t}^{2}$
$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}=$ | $(4.11\pm 0.51\pm 0.94)$
$\gamma^{(0)}_{{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}=$ | $(12.7\pm 2.5\pm 2.8)$
$\gamma^{(1)}_{{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}=$ | $(-122\pm 53\pm 71)\cdot a_{t}^{2}$
$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{1}\\}\rightarrow{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{1}\\}}=$ | $(149\pm 63\pm 42)\cdot a_{t}^{4}$
$\gamma^{(0)}_{{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{1}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{1}\\}}=$ | $(-504\pm 178\pm 1095)\cdot a_{t}^{4}$
$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{P}_{0}\\}\rightarrow{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{P}_{0}\\}}=$ | $(66\pm 5\pm 14)\cdot a_{t}^{2}$ | $\begin{bmatrix}[r]1.00&-0.01&0.14&0.30&0.08\\\ &1.00&0.14&-0.73&0.05\\\ &&1.00&-0.02&-0.18\\\ &&&1.00&-0.00\\\ &&&&1.00\end{bmatrix}$
---|---|---
$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{P}_{2}\\}\rightarrow{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{P}_{2}\\}}=$ | $(29\pm 3.9\pm 3.9)\cdot a_{t}^{2}$
$\gamma^{(0)}_{{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{5}\\!}{P}_{2}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{5}\\!}{P}_{2}\\}}=$ | $(83\pm 11\pm 4)\cdot a_{t}^{2}$
$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{F}_{2}\\}\rightarrow{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{F}_{2}\\}}=$ | $(-126\pm 1095\pm 3181)\cdot a_{t}^{6}$
$\gamma^{(0)}_{{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{5}\\!}{F}_{2}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{5}\\!}{F}_{2}\\}}=$ | $(-25297\pm 15400\pm 15133)\cdot a_{t}^{6}$
$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{2}\\}\rightarrow{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{2}\\}}=$ | $(82\pm 64\pm 93)\cdot a_{t}^{4}$ | $\begin{bmatrix}[r]1.00&0.07\\\ &1.00\end{bmatrix}$
---|---|---
$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{3}\\}\rightarrow{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}=$ | $(-76\pm 38\pm 83)\cdot a_{t}^{4}$
|  $\chi^{2}/N_{\mathrm{dof}}=\frac{121.3}{109-14}=1.28$ ,
(9)

where the first and second uncertainties are as described in Section III.1. We
refer to Eq. (9) as the reference parameterization. Very little deviation is
observed in the parameters describing the $J^{P}=0^{-},2^{-},2^{+}$ and
$3^{+}$ background waves from this determination to when they are determined
independently from rest-frame irreps alone. Of particular importance in the
reference parameterization is the nonzero value of the matrix element
$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}$.
This matrix element allows the ${DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ and
${D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ channels to mix and is
critical to obtaining a parameterization suitable for replicating the finite
volume spectra, which will be demonstrated in Section V.3.

Figures 7 and 8 display the finite volume spectra obtained from Eq. (5) using
the reference parameterization of Eq. (9) for the at rest and in flight
irreps, respectively. The lattice energies are well replicated across all
irreps. An avoided level crossing occurs in the $[000]T_{1}^{+}$ irrep just
below ${D^{\ast}{D}^{\ast}}$ threshold, indicating some coupling between the
channels.

In order to reduce bias caused by the use of a single parameterization, we
also consider 13 additional parameterizations listed in Table 5, some of which
include the effects of $S$ and $D$-wave mixing for both ${DD^{\ast}}$ and
${D^{\ast}{D}^{\ast}}$. Out of the 14 parameterization variations, two were
found to give $\chi^{2}/N_{\mathrm{dof}}>1.5$ and so were not used in the
subsequent analysis.

Table 5: Variations of the $K$-matrix parameterization used in coupled-channel scattering. The entries of the table display the order of the polynomial in $s$ utilized. $-$ denotes that the parameter was fixed to 0 for that parameterization. The reference parameterization Eq. (9) is dentoted in bold with $\ast$. ${\dagger}$ denotes a parameterization that is _not_ used in the subsequent analysis. $\ddagger$ denotes parameterizations whose $D^{\ast}D^{\ast}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ $\rightarrow$ $D^{\ast}D^{\ast}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ $t$-matrix elements are statistically consistent with zero. All background waves used a polynomial of order 0 and a Chew-Mandelstam phase space, so they are not included in the table. Phase Space |  $DD^{\ast}$$\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ |  $DD^{\ast}$$\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ |  $D^{\ast}D^{\ast}$$\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ |  $DD^{\ast}$$\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ |  $D^{\ast}D^{\ast}$$\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ |  $DD^{\ast}$$\\{{\>\\!}^{{3}\\!}{D}_{1}\\}$ |  $D^{\ast}D^{\ast}$$\\{{\>\\!}^{{3}\\!}{D}_{1}\\}$ | $\chi^{2}/N_{\mathrm{dof}}$
---|---|---|---|---|---|---|---|---
$\rightarrow$$DD^{\ast}$$\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ |  $\rightarrow$$D^{\ast}D^{\ast}$$\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ |  $\rightarrow$$D^{\ast}D^{\ast}$$\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ |  $\rightarrow$$DD^{\ast}$$\\{{\>\\!}^{{3}\\!}{D}_{1}\\}$ |  $\rightarrow$$D^{\ast}D^{\ast}$$\\{{\>\\!}^{{3}\\!}{D}_{1}\\}$ |  $\rightarrow$$DD^{\ast}$$\\{{\>\\!}^{{3}\\!}{D}_{1}\\}$ |  $\rightarrow$$D^{\ast}D^{\ast}$$\\{{\>\\!}^{{3}\\!}{D}_{1}\\}$
Chew-Man. | 0 | 0 | 1 | $-$ | $-$ | 0 | 0 | 1.45
1 | 0 | 1 | $\bm{-}$ | $\bm{-}$ | 0 | 0 | $\bm{1.28^{*}}$
0 | 1 | 1 | $-$ | $-$ | 0 | 0 | 1.36
1 | 1 | 0 | $-$ | $-$ | 0 | 0 | $1.30^{\ddagger}$
1 | 1 | 1 | $-$ | $-$ | 0 | 0 | 1.29
1 | 0 | 1 | 0 | 0 | 0 | 0 | 1.23
1 | 0 | 1 | 0 | 0 | $-$ | $-$ | 1.31
Simple | 0 | 0 | 1 | $-$ | $-$ | 0 | 0 | $1.63^{{\dagger}}$
1 | 0 | 1 | $-$ | $-$ | 0 | 0 | 1.30
0 | 1 | 1 | $-$ | $-$ | 0 | 0 | $1.53^{{\dagger}}$
1 | 1 | 0 | $-$ | $-$ | 0 | 0 | $1.33^{\ddagger}$
1 | 1 | 1 | $-$ | $-$ | 0 | 0 | 1.31
| 1 | 0 | 1 | 0 | 0 | 0 | 0 | 1.25
| 1 | 0 | 1 | 0 | 0 | $-$ | $-$ | 1.32

Figure 6: (Left, top to bottom) Scattering amplitudes for
${\>\\!}^{{3}\\!}{S}_{1}$, ${\>\\!}^{{3}\\!}{D}_{1}$,
${\>\\!}^{{3}\\!}{P}_{0,2}$, ${\>\\!}^{{3}\\!}{D}_{2,3}$ and
${\>\\!}^{{3}\\!}{F}_{2}$ for ${DD^{\ast}}$ and ${\>\\!}^{{3}\\!}{S}_{1}$,
${\>\\!}^{{3}\\!}{D}_{1}$, ${\>\\!}^{{5}\\!}{P}_{2}$ and
${\>\\!}^{{5}\\!}{F}_{2}$ for ${D^{\ast}{D}^{\ast}}$ arising from the
reference parameterization given by Eq. (9). Inner bands denote the
statistical uncertainty while outer bands denote the systematic uncertainty
obtained from varying the $D$ and $D^{\ast}$ masses and the anisotropy.
(Right, top to bottom) Scattering amplitudes arising from all
parameterizations given in Table 5 with $\chi^{2}/N_{\mathrm{dof}}<1.5$. The
shaded band denotes the systematic uncertainty for the reference
parameterization. (Bottom) The energy levels used in the scattering analysis.

Figure 6 (left) shows the amplitudes arising from the reference
parameterization of Eq. (9) and for all parameterization variations (right)
given in Table 5. In all parameterizations, we observe a rapid opening of the
${DD^{\ast}}$ $S$-wave amplitude at threshold and a pronounced cusp-like
feature peaked at ${D^{\ast}{D}^{\ast}}$ threshold. The diagonal
${D^{\ast}{D}^{\ast}}\rightarrow{D^{\ast}{D}^{\ast}}$ and off-diagonal
${DD^{\ast}}\rightarrow{D^{\ast}{D}^{\ast}}$ amplitudes both display a rapid
opening at ${D^{\ast}{D}^{\ast}}$ threshold. These three features of the
amplitudes suggest the presence of a pole close to the ${D^{\ast}{D}^{\ast}}$
threshold, which would account for the cusp. We also observe that the
${\>\\!}^{{3}\\!}{D}_{1}$ waves and the coupling between
${\>\\!}^{{3}\\!}{S}_{1}$ and ${\>\\!}^{{3}\\!}{D}_{1}$ waves in both hadron-
hadron channels are consistent with zero. The background waves in
$J^{P}=0^{-},2^{-},2^{+}$ and $3^{+}$ are well constrained in the region near
${D^{\ast}{D}^{\ast}}$ threshold with background waves having a partial wave
configuration of $\ell\geq 2$ being statistically consistent with zero. As in
the elastic case, we observe that the ${DD^{\ast}}$ amplitudes in the
${\>\\!}^{{3}\\!}{P}_{0}$ and ${\>\\!}^{{3}\\!}{P}_{2}$ channels are
significantly nonzero, with a large difference in magnitude at higher
energies.

Figure 7: As Fig. 1, but with the finite volume spectra obtained from Eq. (5)
using the reference parameterization of Eq. (9) given by yellow points.

Figure 8: As Figure 7, but for the $[n_{x}n_{y}n_{z}]A_{2}$ irreps.
Overlapping points have been displaced horizontally for clarity.

To summarize our key findings of these various amplitude variations, as can be
seen in Figs. 4 and 6 for both the elastic and coupled-channel cases, we find
a strong $S$-wave interaction, moderate $P$-wave interactions and only weak
interactions in $D$-wave and higher. The coupled-channel effects are
significant between ${DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ and
${D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ with a pronounced cusp
clearly visible. There is no appreciable coupling found between
${\>\\!}^{{3}\\!}{S}_{1}$ and ${\>\\!}^{{3}\\!}{D}_{1}$ in both hadron-hadron
channels. The ${DD^{\ast}}\\{{\>\\!}^{{3}\\!}{P}_{0}\\}$ is the next most
significant amplitude - it shows no sharp features but is large across both
elastic and coupled-channel scattering amplitudes. We now turn to
investigating the singularities in the scattering amplitudes, from which
masses, widths, and channel couplings of any states present can be obtained.

##  V Scattering amplitude singularities

Unstable hadrons can appear as resonating peaks in the scattering of lighter,
stable hadrons. The least model-dependent way to describe these, that can
unify bound states and resonances in multiple channels, is as the complex
poles of a scattering amplitude. The real and imaginary parts of the pole
position then correspond to the mass and width of the state, and the residue
factorizes into the channel couplings. Sharp features in scattering amplitudes
are often found to be due to a nearby pole.

Experiment usually relies on production and decay processes, where an initial
state excited by a high-energy interaction decays into the hadrons of
interest. This is not simple $s$-channel scattering as has been computed here.
However due to unitarity, the scattering $t$-matrix multiplies the production
amplitude and thus the same singularities are present.

The scattering amplitudes used in this work are analytic functions of
$s=E_{{cm}}^{2}$, except for branch cuts at each hadron-hadron threshold.
These are thus relatively easy to search for complex poles, and such a
procedure is well-defined. The amplitudes used do not include the full
complexity that a complete description of these processes requires, such as
the left-hand cuts arising below the ${DD^{\ast}}$ and ${D^{\ast}{D}^{\ast}}$
thresholds. We will return to this issue later.

###  V.1 Poles in ${DD^{\ast}}$ S-wave scattering

We begin by considering the elastic ${DD^{\ast}}$ scattering region below
${D^{\ast}{D}^{\ast}}$ threshold. In our parameterization for elastic
${DD^{\ast}}$ scattering Eq. (8), we found a good description of the finite
volume energy levels with no ${\>\\!}^{{3}\\!}{S}_{1}-{\>\\!}^{{3}\\!}{D}_{1}$
coupling. In the limit that the ${\>\\!}^{{3}\\!}{D}_{1}$ wave vanishes, then
the existence of a pole in the elastic scattering $S$-wave $t$-matrix can be
elucidated through the intersection of $a_{t}k\cot\delta_{0}$ with the curve
$ia_{t}k$ at energies below threshold. If the $D$-wave amplitude is non-
negligible in magnitude, the corrections to the $S$-wave phase shift from the
determinant condition could significantly alter the intersection of
$a_{t}k\cot\delta_{0}$. However, the $D$-wave amplitude was found to be small
in Section IV.1 and such corrections are negligible to the $S$-wave phase
shift and the intersection of $a_{t}k\cot\delta_{0}$ with $ia_{t}k$. This
intersection can be seen in Fig. 5 and occurs at

$\mathrm{Im}(a_{t}k)=-0.0326(25)$ (10)

at an energy of

$(a_{t}E_{{cm}})=a_{t}\sqrt{s_{0}}=0.6844(5),$ (11)

which corresponds to a virtual bound state.

The couplings to the hadron channels are more straightforwardly determined
from the $t$-matrix. At energies close to the position of a pole, the
scattering $t$-matrix can be written

$t_{lSJa,l^{\prime}S^{\prime}Jb}\sim\frac{c_{lSJa}c_{l^{\prime}S^{\prime}Jb}}{s_{0}-s}$
(12)

where the residue is factorized into the couplings $c_{lSJa}$ and
$c_{l^{\prime}S^{\prime}Jb}$ that quantify the strength of the coupling to the
decay channel. Performing this factorization for the $t$-matrix parameterized
with the reference parameterization of Eq. (8), we find the coupling of the
virtual bound state to be

$a_{t}c^{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}=0.2571(69).$ (13)

We expand upon the position and coupling of this pole in the subsequent
subsection using the amplitudes determined in coupled-channel analysis of
Section IV.2.

###  V.2 Poles in coupled-channel ${DD^{\ast}}-{D^{\ast}{D}^{\ast}}$ S-wave
scattering

In coupled-channel ${DD^{\ast}}$-${D^{\ast}{D}^{\ast}}$ scattering, the
complex $s$-plane is divided into 4 sheets due to the branch cut present at
each threshold. The sheets are labelled by the sign of the imaginary part of
the cm-frame momentum for each hadron channel using the notation
$(\mathrm{sign}[\mathrm{Im}(k_{{DD^{\ast}}}]),\mathrm{sign}[\mathrm{Im}(k_{{D^{\ast}{D}^{\ast}}})])$.
The physical sheet, where physical scattering occurs, is defined by
$\mathrm{Im}(k_{{DD^{\ast}}}),\mathrm{Im}(k_{{D^{\ast}{D}^{\ast}}})>0$. We
refer to this as sheet I, or equivalently the $(+,+)$ sheet. Causality forbids
that poles off the real energy axis occur on the physical sheets. However,
they are allowed to occur on the unphysical sheets II $(-,+)$, III $(-,-)$ and
IV $(+,-)$ where they occur as complex-conjugate pairs in $s$. It is poles on
these unphysical sheets that correspond to resonances with
$\sqrt{s_{0}}=m\pm\frac{i}{2}\Gamma$, where $m$ is the mass of the resonance
and $\Gamma$ its width. A pole off the real energy axis on an unphysical sheet
can influence physical scattering if it is close to the physical sheet. In the
case of sheets II and III, it is the lower half plane that is in close
proximity, whereas for sheet IV it is the upper half. We will denote the
relevant sheets using the shorthand
$\mathrm{I}_{u},\mathrm{II}_{l},\mathrm{III}_{l}$ and $\mathrm{IV}_{u}$. Close
to the pole, the scattering $t$-matrix is of the form of Eq. (12).

Figure 9:  (Left) The positions of the poles in the complex
${D^{\ast}{D}^{\ast}}$ momentum plane for all parameterizations with a
$\chi^{2}/N_{\mathrm{dof}}<1.5$. Blue diamonds denote the position of the pole
in sheet II, green circles denote the position of the pole in sheet III and
purple squares the position of the pole in sheet IV. The positions of the
poles arising from the reference parameterization are given in black, where
the inner error bars are statistical and the outer error bars are the
systematic error arising from varying the mass and anisotropy as described in
Section IV.1. Physical scattering is denoted by the red line in sheet I and
grey contour lines are plotted for constant real (solid) and imaginary
(dashed) $\sqrt{s}$. (Right) The pole positions in the complex $\sqrt{s}$
plane. The insert shows the position of the sheet II and sheet III poles for
each parameterization, which have been displaced vertically for clarity.

For the case of coupled-channel ${DD^{\ast}}-{D^{\ast}{D}^{\ast}}$ scattering
in $J^{P}=1^{+}$, three relevant poles are found for all parameterizations
that adequately describe the finite volume spectra. Figure 9 shows the
position of the three poles in the complex $k_{{D^{\ast}{D}^{\ast}}}$ plane
(left) and their position in the complex $\sqrt{s}$-plane (right). A pole at
real $s$ on sheet $\mathrm{II}_{l}$ is found below ${DD^{\ast}}$ threshold,
corresponding to the virtual bound state discussed in the elastic analysis of
Section IV.1, along with a pole at real $s$ on sheet $\mathrm{III}_{l}$ below
the ${DD^{\ast}}$ threshold. It is shown that the sheet $\mathrm{III}_{l}$
pole and $\mathrm{II}_{l}$ poles are related for the reference
parameterization in the subsequent subsection. More interestingly, we observe
a pole in $\mathrm{IV}_{u}$ with a very narrow width that lies close to the
real axis and slightly below ${D^{\ast}{D}^{\ast}}$ threshold. In [62] it was
shown that complex poles in $\mathrm{II}_{l}$ and $\mathrm{IV}_{u}$ can
influence physical scattering, and induce an observable cusp in the amplitude.
In Section V.3, we will show that the $\mathrm{IV}_{u}$ pole is responsible
for the cusp in the ${DD^{\ast}}$ $S$-wave amplitude.

Figure 10:  The values of the coupling of the poles to
${DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ for the sheet II (left) and sheet
III (center) poles and the couplings of the sheet IV pole to
${DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ and
${D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ (right). The couplings
from the reference parameterization are given in black, where the inner error
bars are statistical and the outer error bars are the systematic error from
varying the mass and anisotropy as described in Section IV.1.

Taking a conservative envelope over the pole positions and couplings for all
reasonable parameterizations, we find for the three poles

$\begin{split}a_{t}\sqrt{s}_{\mathrm{II}}&=0.6765(55)\\\
a_{t}\sqrt{s}_{\mathrm{III}}&=0.6759(68)\\\
a_{t}\sqrt{s}_{\mathrm{IV}}&=0.7007(62)+\frac{i}{2}0.0020(22)\\\ \end{split}$
(14)

with couplings

$\begin{split}a_{t}c_{\mathrm{II}}^{{{DD^{\ast}}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}&=i0.36(7)\\\
a_{t}c_{\mathrm{III}}^{{{DD^{\ast}}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}&=i0.30(21)\\\
a_{t}c_{\mathrm{IV}}^{{{DD^{\ast}}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}&=-0.07(4)+i0.10(7)\\\
a_{t}c_{\mathrm{IV}}^{{{D^{\ast}{D}^{\ast}}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}&=0.00(4)+i0.33(8)\\\
\end{split}$ (15)

where the uncertainties are taken as the envelope of the individual
uncertainties for each parameterization, and the central values are taken from
the center of the envelope. The couplings of the poles for all
parameterizations with $\chi^{2}/N_{\mathrm{dof}}<1.5$ are given in Figure 10.
The couplings of the sheet II and sheet III poles to ${D^{\ast}{D}^{\ast}}$
$\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ are not reported due to the distance of these
poles to the ${D^{\ast}{D}^{\ast}}$ threshold. The sheet IV pole has nonzero
couplings to both channels with the real part of the coupling to the
${D^{\ast}{D}^{\ast}}$ channel being consistent with zero. In addition, the
width is consistent with zero. All three poles have a small coupling to
$D$-wave in both hadron-hadron channels, but due to the negligible $D$-wave
amplitudes in both hadron-hadron channels, we do not report them.

Though the value of the width of the sheet IV pole shows a statistical
consistency with zero using a conservative envelope over the pole positions
from each parameterization, we show in the subsequent section for the
reference parameterization that the coupling to both
${DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ and
${D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ is necessarily non zero,
as is the width.

###  V.3 Varying the ${DD^{\ast}}\rightarrow{D^{\ast}{D}^{\ast}}$ $S$-wave
Amplitude

We examine a variation of the reference parameterization Eq. (9) where the
coupling between the ${DD^{\ast}}$ and ${D^{\ast}{D}^{\ast}}$ is reduced to
zero. This is used to examine the sensitivity of the reference
parameterization to the observed coupling and also to observe the behavior of
the sheet IV pole as the ${D^{\ast}{D}^{\ast}}$ channel is decoupled from
${DD^{\ast}}$. We change the parameter
$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}$,
which couples the ${DD^{\ast}}$ and ${D^{\ast}{D}^{\ast}}$ channels in
$S$-wave, by decreasing it towards zero in constant intervals while keeping
all other parameters fixed. Table 6 displays the values of
$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}$
and the associated $\chi^{2}$ for the parameterization. We observe that as the
value of the coupling parameter is decreased towards zero, the overall
$\chi^{2}$ increases. This suggests that the determined spectra is
incompatible with zero coupling between the channels for this
parameterization.

Figure 11: The scattering amplitude (top) and trajectory of the sheet IV pole
in the complex ${D^{\ast}{D}^{\ast}}$ momentum (middle) and complex energy
(bottom) planes as the
$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}$
matrix element goes to zero. The transition of the amplitudes from color to
grey denotes a decrease in the value of the
$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}$
matrix element.

Figure 11 shows the amplitudes (top) and poles position (bottom) for each
value of the coupling parameter. We observe that as the coupling parameter
decreases, the near threshold cusp in the ${DD^{\ast}}$ $S$-wave amplitude
becomes less pronounced, eventually disappearing when the coupling is
completely turned off. This is accompanied by a rapid opening of the
${D^{\ast}{D}^{\ast}}$ $S$-wave amplitude. The behavior of the amplitude can
be explained by examining the position of the sheet IV pole. As the coupling
parameter is decreased, the sheet IV pole moves to the axis with purely
imaginary momentum in conjunction with the disappearance of the cusp. We
interpret this resonance as a virtual bound state in the limit that the two
hadron scattering channels are decoupled, which correspondingly enhances the
${D^{\ast}{D}^{\ast}}$ $S$-wave amplitude upon opening. However, due to the
open ${DD^{\ast}}$ channel and nonzero coupling, the pole is moved to a
complex energy, manifesting as a resonance which is responsible for the cusp
in the ${DD^{\ast}}$ $S$-wave amplitude.

Also examined is the behavior of the sheet II and sheet III poles as the
hadron-hadron channels are decoupled. Table 6 shows the momentum of the sheet
II and III poles in the ${D^{\ast}{D}^{\ast}}$ plane for the different values
of the coupling parameter. When the two channels are fully decoupled, the
sheet III pole becomes an exact “mirror” of the sheet II pole with opposite
sign of momentum. Due to this behavior and in conjunction with its distance
from the physical sheet, we do not prescribe it any physical significance.

Table 6: The $\chi^{2}$ of the fit and the positions of the sheet II and sheet III poles in the complex ${D^{\ast}{D}^{\ast}}$ momentum plane as the ${DD^{\ast}}$-${D^{\ast}{D}^{\ast}}$ coupling parameter is reduced to zero. The ${\>\\!}^{{3}\\!}{S}_{1}$ label on ${DD^{\ast}}$ and ${D^{\ast}{D}^{\ast}}$ has been suppressed for brevity in the parameter $\gamma$. |  | Sheet II | Sheet III
---|---|---|---
$\gamma^{(0)}_{{DD^{\ast}}\rightarrow{D^{\ast}{D}^{\ast}}}$ | $\chi^{2}$ | Im$(k_{{DD^{\ast}}})$ | Im$(k_{{D^{\ast}{D}^{\ast}}})$ | Im$(k_{{DD^{\ast}}})$ | Im$(k_{{D^{\ast}{D}^{\ast}}})$
4.11 | 121.26 | -0.0493(36) | 0.1003(18) | -0.0529(43) | -0.1021(22)
3.60 | 127.47 | -0.0480(35) | 0.0996(17) | -0.0507(39) | -0.1010(20)
3.08 | 140.04 | -0.0469(34) | 0.0991(16) | -0.0489(37) | -0.1001(18)
2.57 | 156.40 | -0.0460(32) | 0.0987(15) | -0.0473(33) | -0.0993(16)
2.06 | 174.20 | -0.0453(31) | 0.0984(14) | -0.0461(31) | -0.0988(15)
1.54 | 191.53 | -0.0448(31) | 0.0981(14) | -0.0452(31) | -0.0983(14)
1.02 | 206.91 | -0.0444(30) | 0.0980(14) | -0.0446(30) | -0.0980(14)
0 | 223.26 | -0.0441(32) | 0.0978(14) | -0.0441(32) | -0.0978(14)

To summarize our key findings on the coupling of the ${DD^{\ast}}$ and
${D^{\ast}{D}^{\ast}}$ channels, we find that the coupling of the
${DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ and
${D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ channels is necessarily
nonzero in the reference parameterization in order to describe the finite
volume spectra. The non zero width of the sheet IV pole is directly related to
the coupling between the two channels and is responsible for the cusp observed
in the ${DD^{\ast}}$ $S$-wave amplitude at the ${D^{\ast}{D}^{\ast}}$
threshold energy. In addition, the sheet III pole is demonstrated to be a
mirror pole of the sheet II pole.

###  V.4 Poles in other $J^{P}$

The $1^{+}$ amplitudes contain $S$-waves, which are large and have sharp
features. The majority of the other amplitudes determined are small and do not
have features one usually associates with nearby poles. The next largest
amplitude is ${\>\\!}^{{3}\\!}{P}_{0}$, which rises relatively quickly for a
$P$-wave and reaches a value $\rho^{2}|t|^{2}\approx 0.5$. We did not perform
very many amplitude variations, given that there is not much constraint based
on the energy levels present in the $[000]A^{-}_{1}$ and $A_{2}$ irreps.
However, the reference parameterization in $J^{P}=0^{-}$ contains a pole on
the second Riemann sheet at

$\displaystyle a_{t}\sqrt{s}_{\mathrm{II}}$ $\displaystyle=(0.7011\pm
0.0022)-\tfrac{i}{2}(0.062\pm 0.011)$ $\displaystyle
a_{t}|c_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{P}_{0}\\}}|$ $\displaystyle=0.2963\pm
0.0125$ (16)

where the quoted uncertainities are statistical. We stress that this is just a
single parameterisation and so no firm conclusions should be drawn. With such
a small amount of constraint, higher order amplitudes can easily introduce
complex poles on the physical sheet444This occurs in all amplitudes attempted
with a $K$-matrix with a linear polynomial in $s$.. Given that relatively few
energy levels are available, we do not investigate this further. However, it
is possible this resonance pole is a feature in $J^{P}=0^{-}$
${DD^{\ast}}\\{{\>\\!}^{{3}\\!}{P}_{0}\\}$ scattering. The width is large,
more than 300 MeV in physical units.

Other more distant poles can be found in some of the other background partial
waves. We do not consider any of these important as they are far away from
where we constrain the scattering amplitudes and have a negligible effect on
physical scattering.

##  VI Interpretation

In all parameterizations capable of describing the finite volume spectra, we
find a virtual bound state below ${DD^{\ast}}$ threshold which we identify
with the $T_{cc}$, accompanied by a mirror pole on sheet III. In addition, we
find a nearby complex resonance pole on sheet IV below ${D^{\ast}{D}^{\ast}}$
threshold, which we denote $T_{cc}^{\prime}$, an as-yet unobserved state. This
is responsible for the cusp in the ${DD^{\ast}}$ $S$-wave amplitude, and rapid
rise in the ${D^{\ast}{D}^{\ast}}$ $S$-wave amplitude.

In order to present results in physical units, the lattice scale has been set
using the $\Omega$ baryon mass as described in Section III.1. The virtual
bound state’s position and coupling are found to be

$\begin{split}\sqrt{s}&=3834(31)\,\mathrm{MeV}\\\
|c^{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}|&=2040(400)\,\mathrm{MeV,}\\\
\end{split}$ (17)

where the uncertainties are taken as the envelope of the uncertainties for the
pole position and coupling found from each parameterization, and the central
values are taken to be the center of the envelope. The position of the virtual
bound state is approximately $62$ MeV below ${DD^{\ast}}$ threshold.

For the $T_{cc}^{\prime}$ resonance pole on sheet IV, the pole position and
couplings are found to be

$\begin{split}\sqrt{s}&=3971(35)+\frac{i}{2}11(13)\,\mathrm{MeV}\\\
|c^{{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}|&=1870(450)\,\mathrm{MeV}\\\
|c^{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}|&=692(195)\,\mathrm{MeV},\\\
\end{split}$ (18)

which is approximately $49$ MeV below ${D^{\ast}{D}^{\ast}}$ threshold,
coupled most strongly to the kinematically closed ${D^{\ast}{D}^{\ast}}$
channel. We also examined the coupling of the virtual bound state and the
resonance to both hadron-hadron channels in $D$-wave, and found that the
coupling of both poles is negligible. The results are summarized in Fig. 12,
where we plot the $J^{P}=1^{+}$ scattering amplitudes, the energy levels used
to constrain them, and the positions of the virtual bound state and the
resonance in the complex energy plane.

Figure 12: Left: The extracted $J^{P}=1^{+}$ amplitudes for each
parameterization with $\chi^{2}/N_{\mathrm{dof}}<1.5$ (top) and the energy
levels used to constrain them (bottom). The shaded band denotes the systematic
uncertainty for the reference parameterization. Right: The positions of the
sheet II virtual bound state and the sheet IV resonance in the complex energy
plane (top) with the extracted energy levels (bottom).

###  VI.1 Systematic Effects and Left-Cut Physics

Working at a heavier than physical light-quark mass, that produces a pion with
$m_{\pi}\approx 391$ MeV, a direct comparison with experiment is not possible.
Since the experimentally-observed $T_{cc}$ is so close to threshold, fine
details like the isospin splitting of the $D^{0}D^{\ast+}$ and $D^{+}D^{\ast
0}$ thresholds and effects due to the unstable vector $D^{\ast}$ (which is
stable in this calculation) may be significant in determining the properties
of the $T_{cc}$.

With the scale setting used, the charm quarks produce an $\eta_{c}$ with
$m_{\eta_{c}}\approx 2965$ MeV, slightly lower than the experimental value.
The presence of the heavy charm quark results in a relative insensitivity to
the light-quark mass and so the $D$ and $D^{\ast}$ masses in this work are not
that far from those in nature (the $D$ in this calculation has a mass 1886
MeV, and the $D^{\ast}$ mass is 2010 MeV).

In addition, the charm quarks used in this work produce a $J/\psi-\eta_{c}$
hyperfine splitting $\approx 33$ MeV lower than its experimental value [47,
38]. The hyperfine splitting of the $D^{\ast}$ and the $D$ is found to be 124
MeV, around 17 MeV lower than the observed hyperfine splittings (the charged
and neutral cases differ slightly) [63, 38]. We thus expect systematic
differences of a few 10s of MeV to be present. Notably, the ${DD^{\ast}}$ and
${D^{\ast}{D}^{\ast}}$ thresholds are slightly closer to each other in this
calculation than in nature.

The effects of left-cut physics arising from meson exchange in the scattering
$t,u$-channels have not been accounted for in this calculation. They are also
not currently included in the most general finite volume formalism applicable
to coupled-channel scattering. The finite volume energy levels should include
the necessary physics, but this has not been included in translating the
lattice QCD energy levels into scattering information. At the masses used in
this calculation, a $u$-channel cut due to $\pi$ exchange555This is the
equivalent of the “short nucleon cut” cut seen in $N\pi$ scattering ends at
$a_{t}E_{lhc}=0.68432$, roughly 18 MeV below threshold, and only a little
below the lowest energy level extracted ($a_{t}E=0.68474(46)$ in
$[000]T_{1}^{+}$ on the $L/a_{s}=16$ lattice). The virtual bound state pole we
have identified would lie on this cut. The effect occurs again a similar
distance below ${D^{\ast}{D}^{\ast}}$ threshold and here we have many energy
levels overlapping this cut from the elastic ${DD^{\ast}}$ region. Some
modification is likely required to account for these cuts; at the very least,
small imaginary parts must be introduced in the scattering amplitudes.
However, as the amplitudes are consistent with the finite volume spectrum, it
is possible that the effects of the left-cut physics are not that large.

Efforts are underway to add the necessary terms to the two-body Lüscher
formalism to include these effects [64, 65]. A perhaps more general
alternative appears to be to continue the three-body formalism a short way
below threshold to describe this system [66].

Lippmann-Schwinger and effective field theory methods applied to the
${DD^{\ast}}$ system at larger-than-physical pion masses, where the $D^{\ast}$
is stable, can be found in Refs. [67, 68, 43]. There it is shown that the
presence of the left-cut below ${DD^{\ast}}$ threshold arising from one pion
exchange affects both the nature of the pole and its position. Similar
complications can be expected below ${D^{\ast}{D}^{\ast}}$ threshold, and so
extra work is needed to understand how the structures observed in this study
are affected.

###  VI.2 Comparison to Previous Work and Phenomenology

Other works have studied this channel. Considering only elastic scattering,
Ref. [42] working at a light-quark mass corresponding to $m_{\pi}=280$ MeV,
finds a virtual bound state 9.9 MeV below ${DD^{\ast}}$ threshold in $S$-wave.
Using the HAL QCD method with almost-physical pions, Ref. [40] finds a virtual
bound state much closer to threshold. While these results point toward a
ground-state $T_{cc}$ that gets closer to threshold with decreasing light-
quark mass, further work is needed to understand the structures in the complex
plane before firm conclusions can be made.

Phenomenological studies have previously suggested that strong effects may be
seen at ${D^{\ast}{D}^{\ast}}$ threshold [3, 4, 5, 6, 7, 8, 9]. One source of
motivation for this is the infinitely heavy quark limit where the $D$ and
$D^{\ast}$ become degenerate, and the interactions of
${DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ and
${D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ are identical [11]. Away
from this limit, the degeneracy is broken and the relation no longer holds.
However, given the large charm quark mass relative to the other relevant
energy scales, corrections which should be proportional to $1/m_{c}$ could be
modest. Similarities are found between ${DD^{\ast}}$ and
${D^{\ast}{D}^{\ast}}$ close to threshold in ${\>\\!}^{{3}\\!}{S}_{1}$ in this
calculation. Phenomenological approaches do not usually quantify the mixing
between ${DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$ and
${D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}$; it is often assumed to be
zero. Here we have found it to be non-zero and significant.

In experiment, the $T_{cc}^{+}(3875)$ and $X/\chi_{c1}(3872)$ are seen close
to thresholds involving a vector $D^{\ast}$-meson. These produce apparent
excesses in $DD$ or $D\bar{D}$ final states respectively, due to the vector
$D^{\ast}$ decaying to a $D$ and a low-momentum pion, or photon, that then
escape detection. This is explained in Ref. [2], see for example Fig. 7
therein, where peaks are visible close to the pseudoscalar-pseudoscalar
thresholds. With the $T_{cc}^{\prime}$ in mind, the present work suggests it
should be visible in both ${DD^{\ast}}$ and ${D^{\ast}{D}^{\ast}}$ final
states. One may then anticipate that a contribution from $T_{cc}^{\prime}$
might already be present in the data and could possibly be seen in $DD$ and/or
$DD\pi$ mass distributions. There are no other clear peaks visible in Ref.
[2], but it is possible that they may be revealed by future analyses with
improved data.

In this work we also computed several partial waves that were expected to be
suppressed relative to the ${\>\\!}^{{3}\\!}{S}_{1}$ amplitudes. In
particular, we found ${DD^{\ast}}\\{{\>\\!}^{{3}\\!}{P}_{0}\\}$ to be
relatively large. Large, statistically-significant downward shifts are seen in
$[000]A_{1}^{-}$ where this partial wave appears in relative isolation. The
resulting amplitudes describing the levels were large enough to support
resonance poles in some cases, although we also found descriptions with very
different poles, and left-cut effects are also present [68]. Nevertheless, the
interactions in ${DD^{\ast}}\\{{\>\\!}^{{3}\\!}{P}_{0}\\}$ appear quite
strongly attractive.

The magnitude of the $P$-wave amplitudes in $J^{P}=0^{-}$ and $2^{-}$ are
significantly different. An argument for the observed asymmetry comes from the
one pion exchange model. In this model, the potential is directly proportional
to the spin matrix element in the tensor interaction [69, 70]. The magnitude
of this matrix element is directly proportional to the Racah coefficient,
which for $J^{P}=0^{-}$ and $2^{-}$ is $1/3$ and $1/30$, respectively. Such a
difference in magnitude for the spin matrix element leads to a weaker
potential for $J^{P}=2^{-}$ than for $0^{-}$, leading to a smaller scattering
amplitude as is observed in this calculation.

##  VII Summary

In summary, we have presented the first coupled-channel ${DD^{\ast}}$,
${D^{\ast}{D}^{\ast}}$ scattering calculation from lattice QCD in $S$-wave for
isospin-0. Using 109 energy levels, we determined the $J^{P}=1^{+}$ scattering
amplitude by constraining background waves in the ${DD^{\ast}}$ and
${D^{\ast}{D}^{\ast}}$ channels for 12 parameterization variations that
adequately describe the finite volume spectrum. In all successful variations,
we observe a pole in the $S$-wave amplitudes below ${DD^{\ast}}$ threshold
corresponding to a virtual bound state at approximately $\sqrt{s}\approx
3834(31)$ MeV, commonly identified as the $T_{cc}$. A pole corresponding to a
resonance $T_{cc}^{\prime}$ is also found just below ${D^{\ast}{D}^{\ast}}$
threshold at $\sqrt{s}\approx 3971(35)$ MeV with a width of $\Gamma\approx
11(13)$ MeV, which predominantly couples to the kinematically closed
${D^{\ast}{D}^{\ast}}$ channel. The presence of the nearby resonance pole
below ${D^{\ast}{D}^{\ast}}$ threshold enhances the ${DD^{\ast}}$ $S$-wave
amplitude, resulting in clear cusp at ${D^{\ast}{D}^{\ast}}$ threshold. As
ongoing experiments are currently investigating the isospin-0 $J^{P}=1^{+}$
charm sector, it should be possible to find the experimental $T_{cc}^{\prime}$
in both ${DD^{\ast}}$ and ${D^{\ast}{D}^{\ast}}$ final states.

###### Acknowledgements.

We thank our colleagues within the Hadron Spectrum Collaboration
(www.hadspec.org), particularly Jozef Dudek, Raul Briceño and Andrew Jackura
for useful discussions. TW & DJW acknowledges support from a Royal Society
University Research Fellowship. TW, DJW & CET acknowledge support from the
U.K. Science and Technology Facilities Council (STFC) [grant numbers
ST/T000694/1, ST/X000664/1]. TW also acknowledges support from Science
Foundation Ireland [grant number 21/FFP-P/10186] The software codes Chroma
[71], QUDA [72, 73], QUDA-MG [74], QPhiX [75], MG_PROTO [76], QOPQDP [77, 78],
and Redstar [79] 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.

## Data access statement

Reasonable requests for data, such as energy levels and correlations, can be
directed to the authors and will be considered in accordance with the Hadron
Spectrum Collaboration’s policy on sharing data.

##  Appendix A Subduction Tables

In this Appendix, we list the pseudoscalar-vector and identical vector-vector
partial waves that contribute to the finite volume irreducible
representations. Note that for identical vector-vector scattering, certain
partial waves are not allowed by Bose symmetry (see Table 2) and these are
ommitted from the table.

$\Lambda^{P}$ | $T^{+}_{1}$ | $A^{-}_{1}$ | $E^{-}$ | $E^{+}$ | $A^{+}_{2}$
---|---|---|---|---|---
$J^{P}({\>\\!}^{{2S+1}\\!}{\ell}_{J})~{}$ |  | $0^{-}\left({\>\\!}^{{3}\\!}{P}_{0}\right)$ |  |  |
$1^{+}\left(\begin{matrix}{\>\\!}^{{3}\\!}{S}_{1}\\\ {\>\\!}^{{3}\\!}{D}_{1}\end{matrix}\right)$ |  |  |  |
|  | $2^{-}\left(\begin{matrix}{\>\\!}^{{3}\\!}{P}_{2}\\\ {\>\\!}^{{3}\\!}{F}_{2}\end{matrix}\right)$ | $2^{+}\left({\>\\!}^{{3}\\!}{D}_{2}\right)$ |
$3^{+}\left(\begin{matrix}{\>\\!}^{{3}\\!}{D}_{3}\\\ {\>\\!}^{{3}\\!}{G}_{3}\end{matrix}\right)$ |  |  |  | $3^{+}\left(\begin{matrix}{\>\\!}^{{3}\\!}{D}_{3}\\\ {\>\\!}^{{3}\\!}{G}_{3}\end{matrix}\right)$
$4^{+}\left({\>\\!}^{{3}\\!}{G}_{4}\right)$ | $4^{-}\left(\begin{matrix}{\>\\!}^{{3}\\!}{F}_{4}\\\ {\>\\!}^{{3}\\!}{H}_{4}\end{matrix}\right)$ | $4^{-}\left(\begin{matrix}{\>\\!}^{{3}\\!}{F}_{4}\\\ {\>\\!}^{{3}\\!}{H}_{4}\end{matrix}\right)$ | $4^{+}\left({\>\\!}^{{3}\\!}{G}_{4}\right)$ |
Table 7: Subduction table for pseudoscalar-vector scattering overall at rest. $\Lambda^{P}$ | $T^{+}_{1}$ | $A^{-}_{1}$ | $E^{-}$ | $E^{+}$ | $A^{+}_{2}$
---|---|---|---|---|---
$J^{P}({\>\\!}^{{2S+1}\\!}{\ell}_{J})~{}$ |  |  |  |  |
$1^{+}\left(\begin{matrix}{\>\\!}^{{3}\\!}{S}_{1}\\\ {\>\\!}^{{3}\\!}{D}_{1}\end{matrix}\right)$ |  |  |  |
|  | $2^{-}\left(\begin{matrix}{\>\\!}^{{5}\\!}{P}_{2}\\\ {\>\\!}^{{5}\\!}{F}_{2}\end{matrix}\right)$ | $2^{+}\left({\>\\!}^{{3}\\!}{D}_{2}\right)$ |
$3^{+}\left(\begin{matrix}{\>\\!}^{{3}\\!}{D}_{3}\\\ {\>\\!}^{{3}\\!}{G}_{3}\end{matrix}\right)$ |  |  |  | $3^{+}\left(\begin{matrix}{\>\\!}^{{3}\\!}{D}_{3}\\\ {\>\\!}^{{3}\\!}{G}_{3}\end{matrix}\right)$
$4^{+}\left({\>\\!}^{{3}\\!}{G}_{4}\right)$ | $4^{-}\left(\begin{matrix}{\>\\!}^{{5}\\!}{F}_{4}\\\ {\>\\!}^{{5}\\!}{H}_{4}\end{matrix}\right)$ | $4^{-}\left(\begin{matrix}{\>\\!}^{{5}\\!}{F}_{4}\\\ {\>\\!}^{{5}\\!}{H}_{4}\end{matrix}\right)$ | $4^{+}\left({\>\\!}^{{3}\\!}{G}_{4}\right)$ |
Table 8: Subduction table for identical vector-vector scattering overall at rest. $\Lambda^{P}$ | $[00n]A_{2}$ | $[0nn]A_{2}$ | $[nnn]A_{2}$
---|---|---|---
$J^{P}({\>\\!}^{{2S+1}\\!}{\ell}_{J})~{}$ | $0^{-}\left({\>\\!}^{{3}\\!}{P}_{0}\right)$ | $0^{-}\left({\>\\!}^{{3}\\!}{P}_{0}\right)$ | $0^{-}\left({\>\\!}^{{3}\\!}{P}_{0}\right)$
$1^{+}\left(\begin{matrix}{\>\\!}^{{3}\\!}{S}_{1}\\\ {\>\\!}^{{3}\\!}{D}_{1}\end{matrix}\right)$ | $1^{+}\left(\begin{matrix}{\>\\!}^{{3}\\!}{S}_{1}\\\ {\>\\!}^{{3}\\!}{D}_{1}\end{matrix}\right)$ | $1^{+}\left(\begin{matrix}{\>\\!}^{{3}\\!}{S}_{1}\\\ {\>\\!}^{{3}\\!}{D}_{1}\end{matrix}\right)$
$\begin{matrix}\\\ 2^{-}\left(\begin{matrix}{\>\\!}^{{3}\\!}{P}_{2}\\\ {\>\\!}^{{3}\\!}{F}_{2}\end{matrix}\right)\end{matrix}$ | $\begin{matrix}2^{+}\left({\>\\!}^{{3}\\!}{D}_{2}\right)\\\ 2^{-}\left(\begin{matrix}{\>\\!}^{{3}\\!}{P}_{2}\\\ {\>\\!}^{{3}\\!}{F}_{2}\end{matrix}\right)_{[2]}\end{matrix}$ | $\begin{matrix}\\\ 2^{-}\left(\begin{matrix}{\>\\!}^{{3}\\!}{P}_{2}\\\ {\>\\!}^{{3}\\!}{F}_{2}\end{matrix}\right)\end{matrix}$
$\begin{matrix}3^{+}\left(\begin{matrix}{\>\\!}^{{3}\\!}{D}_{3}\\\ {\>\\!}^{{3}\\!}{G}_{3}\end{matrix}\right)\\\ \phantom{3^{-}\left({\>\\!}^{{3}\\!}{F}_{3}\right)}\end{matrix}$ | $\begin{matrix}3^{+}\left(\begin{matrix}{\>\\!}^{{3}\\!}{D}_{3}\\\ {\>\\!}^{{3}\\!}{G}_{3}\end{matrix}\right)_{[2]}\\\ 3^{-}\left({\>\\!}^{{3}\\!}{F}_{3}\right)\end{matrix}$ | $\begin{matrix}3^{+}\left(\begin{matrix}{\>\\!}^{{3}\\!}{D}_{3}\\\ {\>\\!}^{{3}\\!}{G}_{3}\end{matrix}\right)_{[2]}\\\ 3^{-}\left({\>\\!}^{{3}\\!}{F}_{3}\right)\end{matrix}$
$4^{-}\left(\begin{matrix}{\>\\!}^{{3}\\!}{F}_{4}\\\ {\>\\!}^{{3}\\!}{H}_{4}\end{matrix}\right)_{[2]}$ | $4^{-}\left(\begin{matrix}{\>\\!}^{{3}\\!}{F}_{4}\\\ {\>\\!}^{{3}\\!}{H}_{4}\end{matrix}\right)_{[3]}$ | $4^{-}\left(\begin{matrix}{\>\\!}^{{3}\\!}{F}_{4}\\\ {\>\\!}^{{3}\\!}{H}_{4}\end{matrix}\right)_{[2]}$
Table 9: Subduction table for pseudoscalar-vector at overall nonzero momentum. The numbers in square brackets indicate the number of times a given amplitude appears (number of embeddings). $\Lambda^{P}$ | $[00n]A_{2}$ | $[0nn]A_{2}$ | $[nnn]A_{2}$
---|---|---|---
$J^{P}({\>\\!}^{{2S+1}\\!}{\ell}_{J})~{}$ |  |  |
$1^{+}\left(\begin{matrix}{\>\\!}^{{3}\\!}{S}_{1}\\\ {\>\\!}^{{3}\\!}{D}_{1}\end{matrix}\right)$ | $1^{+}\left(\begin{matrix}{\>\\!}^{{3}\\!}{S}_{1}\\\ {\>\\!}^{{3}\\!}{D}_{1}\end{matrix}\right)$ | $1^{+}\left(\begin{matrix}{\>\\!}^{{3}\\!}{S}_{1}\\\ {\>\\!}^{{3}\\!}{D}_{1}\end{matrix}\right)$
$\begin{matrix}\\\ 2^{-}\left(\begin{matrix}{\>\\!}^{{5}\\!}{P}_{2}\\\ {\>\\!}^{{5}\\!}{F}_{2}\end{matrix}\right)\end{matrix}$ | $\begin{matrix}2^{+}\left({\>\\!}^{{3}\\!}{D}_{2}\right)\\\ 2^{-}\left(\begin{matrix}{\>\\!}^{{5}\\!}{P}_{2}\\\ {\>\\!}^{{5}\\!}{F}_{2}\end{matrix}\right)_{[2]}\end{matrix}$ | $\begin{matrix}\\\ 2^{-}\left(\begin{matrix}{\>\\!}^{{5}\\!}{P}_{2}\\\ {\>\\!}^{{5}\\!}{F}_{2}\end{matrix}\right)\end{matrix}$
$\begin{matrix}3^{+}\left(\begin{matrix}{\>\\!}^{{3}\\!}{D}_{3}\\\ {\>\\!}^{{3}\\!}{G}_{3}\end{matrix}\right)\\\ \phantom{3^{-}\left(\begin{matrix}{\>\\!}^{{5}\\!}{P}_{3}\\\ {\>\\!}^{{5}\\!}{F}_{3}\end{matrix}\right)}\end{matrix}$ | $\begin{matrix}3^{+}\left(\begin{matrix}{\>\\!}^{{3}\\!}{D}_{3}\\\ {\>\\!}^{{3}\\!}{G}_{3}\end{matrix}\right)_{[2]}\\\ 3^{-}\left(\begin{matrix}{\>\\!}^{{5}\\!}{P}_{3}\\\ {\>\\!}^{{5}\\!}{F}_{3}\end{matrix}\right)\end{matrix}$ | $\begin{matrix}3^{+}\left(\begin{matrix}{\>\\!}^{{3}\\!}{D}_{3}\\\ {\>\\!}^{{3}\\!}{G}_{3}\end{matrix}\right)_{[2]}\\\ 3^{-}\left(\begin{matrix}{\>\\!}^{{5}\\!}{P}_{3}\\\ {\>\\!}^{{5}\\!}{F}_{3}\end{matrix}\right)\end{matrix}$
$4^{-}\left(\begin{matrix}{\>\\!}^{{5}\\!}{F}_{4}\\\ {\>\\!}^{{5}\\!}{H}_{4}\end{matrix}\right)_{[2]}$ | $4^{-}\left(\begin{matrix}{\>\\!}^{{5}\\!}{F}_{4}\\\ {\>\\!}^{{5}\\!}{H}_{4}\end{matrix}\right)_{[3]}$ | $4^{-}\left(\begin{matrix}{\>\\!}^{{5}\\!}{F}_{4}\\\ {\>\\!}^{{5}\\!}{H}_{4}\end{matrix}\right)_{[2]}$
Table 10: Subduction table for identical vector-vector mesons at overall
nonzero momentum. The numbers in square brackets indicate the number of times
a given amplitude appears (number of embeddings).

##  Appendix B Operator Tables

In Tables 11 and 12 we list the ${DD^{\ast}}$ and ${D^{\ast}{D}^{\ast}}$
operators used in this study, with their relative momentum denoted by
$[n_{x}n_{y}n_{z}]$, including their multiplicity. Operators listed in grey
were included in the basis in the variational extraction of energy levels, but
energy levels associated with these operators were not used in the scattering
analysis.

$[000]T_{1}^{+}$ | $[000]A^{-}_{1}$
---|---
$L/a_{s}=16$ | $L/a_{s}=20$ | $L/a_{s}=24$ | $L/a_{s}=16$ | $L/a_{s}=20$ | $L/a_{s}=24$
$D[000]D^{\ast}[000]$ | $D[000]D^{\ast}[000]$ | $D[000]D^{\ast}[000]$ | $D[001]D^{\ast}[001]$ | $D[001]D^{\ast}[001]$ | $D[001]D^{\ast}[001]$
$D^{\ast}[000]D^{\ast}[000]$ | $D^{\ast}[000]D^{\ast}[000]$ | $D[001]D^{\ast}[001]\times 2$ | $D[011]D^{\ast}[011]$ | $D[011]D^{\ast}[011]$ | $D[011]D^{\ast}[011]$
$D[001]D^{\ast}[001]\times 2$ | $D[001]D^{\ast}[001]$ | $D^{\ast}[000]D^{\ast}[000]$ |  | $D^{\ast}[011]D^{\ast}[011]$ | $D[111]D^{\ast}[111]$
${\color[rgb]{.75,.75,.75}D^{\ast}[001]D^{\ast}[001]\times 2}$ | $D^{\ast}[001]D^{\ast}[001]\times 2$ | $D[011]D^{\ast}[011]\times 3$ |  | $D[111]D^{\ast}[111]$ | $D^{\ast}[011]D^{\ast}[011]$
${\color[rgb]{.75,.75,.75}D[011]D^{\ast}[011]\times 3}$ | $D[011]D^{\ast}[011]\times 3$ | $D^{\ast}[001]D^{\ast}[001]\times 2$ |  |  | $D[002]D^{\ast}[002]$
| $D^{\ast}[011]D^{\ast}[011]\times 3$ | $D[111]D^{\ast}[111]\times 2$ |  |  |
| $D[111]D^{\ast}[111]\times 2$ | $D^{\ast}[011]D^{\ast}[011]\times 3$ |  |  |
|  | $D[002]D^{\ast}[002]\times 2$ |  |  |
|  | $D^{\ast}[111]D^{\ast}[111]\times 2$ |  |  |
$[000]E^{-}$ | $[000]E^{+}$
$L/a_{s}=16$ | $L/a_{s}=20$ | $L/a_{s}=24$ | $L/a_{s}=16$ | $L/a_{s}=20$ | $L/a_{s}=24$
$D[001]D^{\ast}[001]$ | $D[001]D^{\ast}[001]$ | $D[001]D^{\ast}[001]$ | $D[011]D^{\ast}[011]$ | $D[011]D^{\ast}[011]$ | $D[011]D^{\ast}[011]$
$D[011]D^{\ast}[011]\times 2$ | $D^{\ast}[001]D^{\ast}[001]$ | $D[011]D^{\ast}[011]\times 2$ |  | $D[111]D^{\ast}[111]$ | $D[111]D^{\ast}[111]$
$D^{\ast}[001]D^{\ast}[001]$ | $D[011]D^{\ast}[011]\times 2$ | $D^{\ast}[001]D^{\ast}[001]$ |  | $D^{\ast}[011]D^{\ast}[011]$ | $D^{\ast}[011]D^{\ast}[011]$
| $D^{\ast}[011]D^{\ast}[011]\times 2$ | $D[111]D^{\ast}[111]$ |  |  | $D^{\ast}[111]D^{\ast}[111]$
|  | $D^{\ast}[011]D^{\ast}[011]\times 2$ |  |  |
|  | $D[002]D^{\ast}[002]$ |  |  |
|  | $D^{\ast}[111]D^{\ast}[111]$ |  |  |

$[000]A^{+}_{2}$
---
| $L/a_{s}=16$ | $L/a_{s}=20$ | $L/a_{s}=24$
---|---|---
$D[011]D^{\ast}[011]$ | $D[011]D^{\ast}[011]$ | $D[011]D^{\ast}[011]$
| $D^{\ast}[011]D^{\ast}[011]$ | $D^{\ast}[001]D^{\ast}[001]$
| $D[111]D^{\ast}[111]$ | $D[111]D^{\ast}[111]$
|  | $D^{\ast}[011]D^{\ast}[011]$
|  | $D^{\ast}[111]D^{\ast}[111]$

Table 11: Operators for the at rest irreps. $[001]A_{2}$ | $[002]A_{2}$
---|---
$L/a_{s}=20$ | $L/a_{s}=24$ | $L/a_{s}=20$ | $L/a_{s}=24$
$D[000]D^{\ast}[001]$ | $D[000]D^{\ast}[001]$ | $D[001]D^{\ast}[001]$ | $D[001]D^{\ast}[001]$
$D[001]D^{\ast}[000]$ | $D[001]D^{\ast}[000]$ | $D[000]D^{\ast}[002]$ | $D[000]D^{\ast}[002]$
$D^{\ast}[001]D^{\ast}[000]$ | $D[001]D^{\ast}[011]\times 2$ | $D^{\ast}[001]D^{\ast}[001]$ | $D[011]D^{\ast}[011]\times 2$
$D[001]D^{\ast}[011]\times 2$ | $D[011]D^{\ast}[001]\times 2$ | $D[002]D^{\ast}[000]$ | $D[002]D^{\ast}[000]$
$D[011]D^{\ast}[001]\times 2$ | $D^{\ast}[001]D^{\ast}[000]$ | $D[011]D^{\ast}[011]\times 2$ | $D^{\ast}[001]D^{\ast}[001]$
$D^{\ast}[011]D^{\ast}[001]\times 4$ | $D[001]D^{\ast}[002]$ | $D^{\ast}[002]D^{\ast}[000]$ | $D[001]D^{\ast}[012]\times 2$
$D[001]D^{\ast}[002]$ | $D[011]D^{\ast}[111]\times 2$ | $D[001]D^{\ast}[012]\times 2$ | $D[111]D^{\ast}[111]\times 2$
$D[011]D^{\ast}[111]\times 2$ | $D[111]D^{\ast}[011]\times 2$ | $D^{\ast}[011]D^{\ast}[011]\times 2$ | $D[012]D^{\ast}[001]\times 2$
| $D[002]D^{\ast}[001]$ | $D[012]D^{\ast}[001]\times 2$ | $D^{\ast}[002]D^{\ast}[000]$
| $D^{\ast}[011]D^{\ast}[001]\times 4$ | $D[111]D^{\ast}[111]\times 2$ | $D^{\ast}[011]D^{\ast}[011]\times 2$
| $D[011]D^{\ast}[012]\times 2$ |  | $D[011]D^{\ast}[112]\times 2$
|  |  | $D[112]D^{\ast}[011]\times 2$
$[011]A_{2}$ | $[111]A_{2}$
$L/a_{s}=20$ | $L/a_{s}=24$ | $L/a_{s}=20$ | $L/a_{s}=24$
$D[000]D^{\ast}[011]$ | $D[000]D^{\ast}[011]$ | $D[000]D^{\ast}[111]$ | $D[000]D^{\ast}[111]$
$D[001]D^{\ast}[001]\times 2$ | $D[001]D^{\ast}[001]\times 2$ | $D[001]D^{\ast}[011]\times 2$ | $D[001]D^{\ast}[011]\times 2$
$D[011]D^{\ast}[000]$ | $D[011]D^{\ast}[000]$ | $D[111]D^{\ast}[000]$ | $D[011]D^{\ast}[001]\times 2$
$D^{\ast}[011]D^{\ast}[000]\times 2$ | $D[001]D^{\ast}[111]\times 2$ | $D[011]D^{\ast}[001]\times 2$ | $D[111]D^{\ast}[000]$
$D^{\ast}[001]D^{\ast}[001]\times 2$ | $D[011]D^{\ast}[011]\times 3$ | $D^{\ast}[111]D^{\ast}[000]$ | $D^{\ast}[111]D^{\ast}[000]$
$D[001]D^{\ast}[111]\times 2$ | $D[111]D^{\ast}[001]\times 2$ | $D^{\ast}[011]D^{\ast}[001]\times 4$ | $D^{\ast}[011]D^{\ast}[001]\times 4$
$D[011]D^{\ast}[011]\times 3$ | $D^{\ast}[011]D^{\ast}[000]\times 2$ |  | $D[001]D^{\ast}[112]\times 2$
$D[111]D^{\ast}[001]\times 2$ | $D^{\ast}[001]D^{\ast}[001]\times 2$ |  | $D[011]D^{\ast}[012]\times 3$
| $D[001]D^{\ast}[012]\times 2$ |  | $D[111]D^{\ast}[002]\times 2$
| $D[011]D^{\ast}[002]\times 2$ |  | $D[112]D^{\ast}[001]\times 2$
| $D[012]D^{\ast}[001]\times 2$ |  | $D[002]D^{\ast}[111]\times 2$
| $D[002]D^{\ast}[011]\times 2$ |  | $D[012]D^{\ast}[011]\times 3$
| $D^{\ast}[111]D^{\ast}[001]\times 4$ |  |
| $D^{\ast}[011]D^{\ast}[011]\times 3$ |  |
Table 12: Operators for the in flight irreps.

##  Appendix C Coupled-Channel Parameterization with ${D^{\ast}{D}^{\ast}}$
${\>\\!}^{{5}\\!}{P}_{3}$, ${\>\\!}^{{3}\\!}{D}_{2,3}$ and
${\>\\!}^{{5}\\!}{F}_{3}$ Partial Waves

The reference parameterization of Eq. (9) and the 13 parameterization
variations consider ${D^{\ast}{D}^{\ast}}$ in ${\>\\!}^{{3}\\!}{S}_{1}$,
${\>\\!}^{{3}\\!}{D}_{1}$, ${\>\\!}^{{5}\\!}{P}_{2}$ and
${\>\\!}^{{5}\\!}{F}_{2}$ partial waves. The ${\>\\!}^{{5}\\!}{F}_{2}$ partial
wave was introduced in part to enable the description of a set of four levels
in the $[111]A_{2}$ irrep that would be degenerate in the absence of meson-
meson interactions. Other ${D^{\ast}{D}^{\ast}}$ combinations such as
${\>\\!}^{{3}\\!}{D}_{3}$, ${\>\\!}^{{5}\\!}{P}_{3}$ and
${\>\\!}^{{5}\\!}{F}_{3}$ also contribute to the $[111]A_{2}$ irrep and so
could potentially be used instead of ${\>\\!}^{{5}\\!}{F}_{2}$, but the
${\>\\!}^{{3}\\!}{D}_{2}$ partial wave does not contribute to $[111]A_{2}$ and
so could not. To examine the effects of these partial waves, three separate
additional variations of the reference parameterization were considered
wherein ${\>\\!}^{{3}\\!}{D}_{3}$, ${\>\\!}^{{5}\\!}{P}_{3}$ and
${\>\\!}^{{5}\\!}{F}_{3}$ were in turn used in place of
${\>\\!}^{{5}\\!}{F}_{2}$.

${D^{\ast}{D}^{\ast}}$ PW | $\gamma^{(0)}_{{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{2S+1}\\!}{\ell}_{J}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{2S+1}\\!}{\ell}_{J}\\}}$ | $\chi^{2}/N_{\mathrm{dof}}$
---|---|---
${\>\\!}^{{3}\\!}{D}_{3}$ | $(37\pm 239)\cdot a_{t}^{4}$ | $361.06/(109-14)=3.80$
${\>\\!}^{{5}\\!}{P}_{3}$ | $(-9.49\pm 7.22)\cdot a_{t}$ | $121.54/(109-14)=1.28$
${\>\\!}^{{5}\\!}{F}_{3}$ | $(10800\pm 48300)\cdot a_{t}^{6}$ | $122.08/(109-14)=1.29$

Table 13: The values of the parameters describing the
${\>\\!}^{{3}\\!}{D}_{3}$, ${\>\\!}^{{5}\\!}{P}_{3}$ and
${\>\\!}^{{5}\\!}{F}_{3}$ ${D^{\ast}{D}^{\ast}}$ scattering amplitudes along
with the associated $\chi^{2}/N_{\mathrm{dof}}$ of the fit.

Table 13 displays the $\chi^{2}/N_{\mathrm{dof}}$ and the value of the
parameter describing the ${\>\\!}^{{3}\\!}{D}_{3}$, ${\>\\!}^{{5}\\!}{P}_{3}$
and ${\>\\!}^{{5}\\!}{F}_{3}$ ${D^{\ast}{D}^{\ast}}$ scattering amplitudes
from these three fits. A poor $\chi^{2}/N_{\mathrm{dof}}$ is obtained for the
parameterization using the ${\>\\!}^{{3}\\!}{D}_{3}$ partial wave because
${\>\\!}^{{3}\\!}{D}_{3}$ does not provide a solution to the determinant
condition for reasonable values of the parameter in the energy region of the
quadruple degeneracy in the $[111]A_{2}$ irrep. Additional partial waves are
required in order to saturate the observed approximate degeneracy. In
contrast, the use of ${\>\\!}^{{5}\\!}{P}_{3}$ or ${\>\\!}^{{5}\\!}{F}_{3}$
yields a $\chi^{2}/N_{\mathrm{dof}}$ similar to that of the reference
parameterization, but does not offer any further improvement. In addition, the
parameters describing the ${\>\\!}^{{5}\\!}{P}_{3}$ and
${\>\\!}^{{5}\\!}{F}_{3}$ scattering amplitudes are approximately consistent
with zero. This indicates that the finite volume spectrum is insensitive to
the choice between ${\>\\!}^{{5}\\!}{P}_{3}$, ${\>\\!}^{{5}\\!}{F}_{2}$ and
${\>\\!}^{{5}\\!}{F}_{3}$, and any of these three partial waves can be
employed.

We also considered a variation of the reference parameterization where the
effects of the ${D^{\ast}{D}^{\ast}}$ ${\>\\!}^{{5}\\!}{P}_{3}$,
${\>\\!}^{{3}\\!}{D}_{2,3}$ and ${\>\\!}^{{5}\\!}{F}_{2,3}$ partial waves were
included simultaneously, which accounts for all ${DD^{\ast}}$ and
${D^{\ast}{D}^{\ast}}$ partial waves with $\ell\leq 2$ that contribute to the
irreps utilized. Using the reference parameterisation with the addition of a
constant term for each of the ${\>\\!}^{{3}\\!}{D}_{2,3}$,
${\>\\!}^{{5}\\!}{P}_{3}$ and ${\>\\!}^{{5}\\!}{F}_{3}$ ${D^{\ast}{D}^{\ast}}$
amplitudes gives,

$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}\rightarrow{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}=$ | $(6.72\pm 0.56)$ | $\begin{bmatrix}[r]1.00&-0.80&0.35&-0.09&0.24&0.10&0.06\\\ &1.00&0.17&0.21&-0.26&-0.01&-0.03\\\ &&1.00&0.29&-0.05&0.07&0.03\\\ &&&1.00&-0.95&0.06&0.02\\\ &&&&1.00&-0.03&-0.01\\\ &&&&&1.00&0.02\\\ &&&&&&1.00\end{bmatrix}$
---|---|---
$\gamma^{(1)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}\rightarrow{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}=$ | $(-60\pm 13)\cdot a_{t}^{2}$
$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}=$ | $(4.16\pm 0.53)$
$\gamma^{(0)}_{{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}=$ | $(13.1\pm 2.6)$
$\gamma^{(1)}_{{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{S}_{1}\\}}=$ | $(-128\pm 54)\cdot a_{t}^{2}$
$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{1}\\}\rightarrow{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{1}\\}}=$ | $(151\pm 57)\cdot a_{t}^{4}$
$\gamma^{(0)}_{{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{1}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{1}\\}}=$ | $(-831\pm 267)\cdot a_{t}^{4}$

$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{P}_{0}\\}\rightarrow{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{P}_{0}\\}}=$ | $(66\pm 5)\cdot a_{t}^{2}$ | $\begin{bmatrix}[r]1.00&0.06&0.12&0.00&0.26&-0.02&0.00\\\ &1.00&0.13&0.00&-0.70&-0.03&0.00\\\ &&1.00&0.00&-0.03&-0.75&0.00\\\ &&&0.00&0.00&0.00&0.00\\\ &&&&1.00&0.00&0.00\\\ &&&&&1.00&0.00\\\ &&&&&&0.00\end{bmatrix}$
---|---|---
$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{P}_{2}\\}\rightarrow{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{P}_{2}\\}}=$ | $(30\pm 4)\cdot a_{t}^{2}$
$\gamma^{(0)}_{{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{5}\\!}{P}_{2}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{5}\\!}{P}_{2}\\}}=$ | $(77\pm 18)\cdot a_{t}^{2}$
$\gamma^{(0)}_{{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{5}\\!}{P}_{3}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{5}\\!}{P}_{3}\\}}=$ | $(-5.80\pm 21.0)\cdot a_{t}^{2}$
$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{F}_{2}\\}\rightarrow{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{F}_{2}\\}}=$ | $(-391\pm 998)\cdot a_{t}^{6}$
$\gamma^{(0)}_{{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{5}\\!}{F}_{2}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{5}\\!}{F}_{2}\\}}=$ | $(-1680\pm 56840)\cdot a_{t}^{6}$
$\gamma^{(0)}_{{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{5}\\!}{F}_{3}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{5}\\!}{F}_{3}\\}}=$ | $(18233\pm 70489)\cdot a_{t}^{6}$

$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{2}\\}\rightarrow{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{2}\\}}=$ | $(82\pm 64)\cdot a_{t}^{4}$ | $\begin{bmatrix}[r]1.00&0.05&-0.02&0.03\\\ &1.00&0.03&0.04\\\ &&1.00&-0.14\\\ &&&1.00\end{bmatrix}$
---|---|---
$\gamma^{(0)}_{{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{3}\\}\rightarrow{DD^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}=$ | $(-78\pm 31)\cdot a_{t}^{4}$
$\gamma^{(0)}_{{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{2}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{2}\\}}=$ | $(-3229\pm 1134)\cdot a_{t}^{4}$
$\gamma^{(0)}_{{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{3}\\}\rightarrow{D^{\ast}{D}^{\ast}}\\{{\>\\!}^{{3}\\!}{D}_{3}\\}}=$ | $(352\pm 262)\cdot a_{t}^{4}$
|  $\chi^{2}/N_{\mathrm{dof}}=\frac{116.4}{109-18}=1.28$ .

The $\chi^{2}/N_{\mathrm{dof}}$ is the same as in the reference
parameterization, suggesting that the inclusion of these additional partial
waves is not necessary to describe the finite volume spectrum. Furthermore,
the values of parameters describing the $J^{P}=1^{+}$ scattering amplitudes
display very little deviation between this parameterization and the reference
parameterization. The central values of these parameters in this
parameterization are well contained within the statistical uncertainty of the
parameters in the reference parameterization. It is also observed that the
statistical uncertainties of the parameters common to this and the reference
parameterization display very little deviation between the two
parameterizations. This indicates that these partial waves have little effect
on the scattering amplitudes in $J^{P}=1^{+}$ and can be neglected, justifying
the approach of Sec. IV.2.

##  Appendix D SVD Limits of the Data Correlation Matrix

As discussed in Section IV.2, we employ the use of singular value limits in
the correlated $\chi^{2}$ minimization used for the parameter determination.
It would be preferable to perform the $\chi^{2}$ minimization without any
manipulation of the underlying correlations of the data, however large data
correlations between energy levels on each of the three lattice volumes result
in an unreasonable large $\chi^{2}/N_{\mathrm{dof}}$.

Large correlations between data points can arise when the smallest eigenvalues
of the data correlation matrix are underestimated. This can occur when the
number of Monte Carlo samples is of the order of the square of the number of
data points, which is precisely the situation in this work. Across all irreps
and lattice volumes, we have utilized 109 energy levels. The largest
$L/a_{s}=24$ volume provided 56 energy levels to constrain the scattering
amplitudes while having $553$ gauge configurations (Monte Carlo samples). It
is then not possible to reliably determine the data covariance matrix as the
number of independent numbers in the correlation matrix is greater than the
number of Monte Carlo samples.

In singular value limits, the singular values (eigenvalues) of the symmetric
correlation matrix undergo the replacement
$\lambda_{i}\rightarrow\mathrm{max}(\lambda_{i},\sigma)$ for
$\sigma=\tau\lambda_{1}$, where $\tau$ is the singular value limit. This is in
contrast to singular value cuts, wherein the replacement
$\lambda_{i}\rightarrow\mathrm{max}(\lambda_{i},\infty)$ is made for all
$\lambda_{i}<\sigma=\tau\lambda_{1}$ meaning the corresponding modes are
removed from the data covariance matrix inverse.

In Appendix D of Ref. [61], singular value limits are shown to be equivalent
to adding an additional statistical uncertainty to the data (see Eq. D17
therein), as the replacement sets the eigenvalue to at least its true value.
As such, taking a singular value limit can be seen as a conservative approach.
Singular value cuts can be interpreted in the same way, however the
replacement sets the eigenvalue to $\infty$, thus making it the most extreme
and conservative case of singular value limits. This can be ameliorated by
modifying the interpretation of the $\chi^{2}/N_{\mathrm{dof}}$ as a measure
of the quality of the fit. When a singular value cut is used, it is to natural
reduce the number of degrees of freedom by the number of modes cut, such that
$\hat{N}_{\mathrm{dof}}=N_{\mathrm{data}}-N_{\mathrm{cut}}-N_{\mathrm{pars}}$,
as was done in Refs. [33, 38]. For singular value limits, we retain the usual
definition, $N_{\mathrm{dof}}=N_{\mathrm{data}}-N_{\mathrm{pars}}$.

SVD limit | Number of Replaced Modes | $\chi^{2}/N_{\mathrm{dof}}$
---|---|---
$\tau$ | (total: 6+47+56)
0 | $0+0+0$ | $759.10/(109-14)\quad=\quad 7.99$
0.003 | $0+26+33$ | $284.28/(109-14)\quad=\quad 2.99$
0.006 | $0+34+40$ | $177.24/(109-14)\quad=\quad 1.87$
$\\!\\!\\!{}^{{\dagger}}\bm{0.010}$ | $\bm{0+37+46}$ | $\bm{121.26/(109-14)=\;\;1.28}$
$0.015$ | $0+40+48$ | $\;\>88.39/(109-14)\quad=\quad 0.93$
$0.020$ | $0+42+49$ | $\;\>69.35/(109-14)\quad=\quad 0.73$
uncorrelated | - | $169.51/(109-14)\quad=\quad 1.78$

SVD cut |  Number of Modes Cut | $\chi^{2}/\hat{N}_{\mathrm{dof}}$
---|---|---
$\tau$ | (total: 6+47+56)
0.0016 | $0+21+23$ |  ${119.65}/(109-14-44)$ = 2.35
0.0020 | $0+22+27$ |  ${69.60}/(109-14-49)$ = 1.52
0.0024 | $0+25+29$ |  ${62.01}/(109-14-54)$ = 1.51
0.0032 | $0+28+33$ |  ${48.90}/(109-14-61)$ = 1.44

Table 14: The number of eigenmodes replaced, $\chi^{2}$ and
$\chi^{2}/N_{\mathrm{dof}}$ for each value of the singular value limit/cut
$\tau$. The number of modes replaced for the three lattice volumes is denoted
as $(16^{3}+20^{3}+24^{3})$. The value of $\tau$ used in the main text is
denoted with ${\dagger}$ in bold.

Performing a range of amplitude determinations with various choices of SVD
limits and cuts, we find a range of reasonable choices. Table 14 shows the
value of the $\chi^{2}$ for the reference parameterization (Eq. (9)) for fully
correlated ($\tau=0$) data, uncorrelated data and five values of $\tau$,
including the number of eigenvalues replaced for each lattice volume. In the
lower half of Table 14, we show the $\chi^{2}$ values obtained when using SVD
cuts. For any of the fits listed in Table 14 with a
$\chi^{2}/N_{\mathrm{dof}}\lesssim 2$, the central values of the determined
amplitudes and poles are largely unchanged within uncertainties. In general,
larger values of $\tau$ lead to broader uncertainties.

Fig. 13 shows the eigenvalues of the data correlation matrix for each lattice
volume, normalized by the largest eigenvalue $\lambda_{1}$. It is observed
that the $\chi^{2}$ for the fully correlated parameterization is large and
cannot be trusted as a valid parameterization. Using a singular value limit of
$\tau=0.003$ results in a decrease of the $\chi^{2}$, but not enough to have
confidence in the parameterization. A limit of $\tau=0.01$ results in a
reasonable $\chi^{2}/N_{\mathrm{dof}}$. The $\chi^{2}$ is lower than that of
the uncorrelated fit, which reflects the additional uncertainty introduced by
this approach.

Figure 13:  The eigenvalues of the correlation matrix on each lattice volume
normalized by the largest eigenvalue $\lambda_{1}$. The value of $\tau$ used
in the main analysis is denoted with ${\dagger}$ in bold.

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