for the Hadron Spectrum Collaboration

# Axial-vector $D_{1}$ hadrons in $D^{\ast}\pi$ scattering from QCD

Nicolas Lang<EMAIL_ADDRESS>School of Mathematics, Trinity
College, Dublin 2, Ireland  David J. Wilson<EMAIL_ADDRESS>DAMTP,
University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road,
Cambridge, CB3 0WA, UK

###### Abstract

We present $I=1/2$ $D^{\ast}\pi$ scattering amplitudes from lattice QCD and
determine two low-lying $J^{P}=1^{+}$ axial-vector $D_{1}$ states and a
$J^{P}=2^{+}$ tensor $D_{2}^{\ast}$. Computing finite-volume spectra at a
light-quark mass corresponding to $m_{\pi}=391$ MeV, for the first time, we
are able to constrain coupled $J^{P}=1^{+}$ $D^{\ast}\pi$ amplitudes with
${}^{2S+1}\ell_{J}\,=\,^{3}S_{1}$ and ${}^{3}\\!D_{1}$ as well as coupled
$J^{P}=2^{+}$ $D\pi\\{^{1}\\!D_{2}\\}$ and $D^{\ast}\pi\\{^{3}\\!D_{2}\\}$
amplitudes via Lüscher’s quantization condition. Analyzing the scattering
amplitudes for poles we find a near-threshold bound state, producing a broad
feature in $D^{\ast}\pi\\{^{3}\\!S_{1}\\}$. A narrow bump occurs in
$D^{\ast}\pi\\{^{3}\\!D_{1}\\}$ due to a $D_{1}$ resonance. A single resonance
is found in $J^{P}=2^{+}$ coupled to $D\pi$ and $D^{\ast}\pi$. A relatively
low mass and large coupling is found for the lightest $D_{1}$, suggestive of a
state that will evolve into a broad resonance as the light quark mass is
reduced. An earlier calculation of the scalar $D_{0}^{\ast}$ using the same
light-quark mass enables comparisons to the heavy-quark limit.

_Introduction_ — Since their discovery, the lightest scalar $D$-meson
excitations have raised questions about their mass ordering, widths and
composition. Heavy-quark spin symmetry suggests the same questions apply to
the axial vectors. At the same time, the $D$-meson resonances are exemplary
for several charmed states Choi  _et al._ (2003); Aaij  _et al._ (2020a, b,
c), some manifestly exotic, arising in coupled-channel systems and close to
thresholds. They may therefore serve as a place to obtain a more general
understanding of QCD dynamics among charmed hadrons.

In experiment, there are four low-lying positive-parity $D$-mesons Zyla  _et
al._ (2020): a scalar $D_{0}^{\ast}(2300)$, two axial-vectors $D_{1}(2430)$
and $D_{1}(2420)$, and a tensor $D_{2}^{\ast}(2460)$. The scalar and
$D_{1}(2430)$ are very broad and are thought to couple strongly to their
respective $D\pi$ and $D^{\ast}\pi$ decay modes. The $D_{1}(2420)$ and the
tensor $D_{2}^{\ast}$ are relatively narrow, the $D_{2}^{\ast}$ decaying into
both $D\pi$ and $D^{\ast}\pi$.

A range of theoretical approaches have been applied to understand these
states. Quark potential models Godfrey and Isgur (1985) provide a useful
qualitative starting point leading to four states arising from the $\ell=1$
singlet ${}^{1}\\!P_{1}$ and triplet ${}^{3}\\!P_{J}$ combinations. Charge-
conjugation is not a good symmetry so $\>\\!{}^{1}\\!P_{1}$ and
$\>\\!{}^{3}\\!P_{1}$ mix. This produces one $J^{P}=0^{+}$, two $1^{+}$ and a
$2^{+}$ state at similar masses Godfrey and Isgur (1985). Approaches
considering the coupling to decay channels Godfrey and Kokoski (1991); Bardeen
and Hill (1994); van Beveren and Rupp (2004); Kolomeitsev and Lutz (2004);
Hofmann and Lutz (2004); Burns (2014) are necessary when strong $S$-wave decay
modes are present. Recent studies have shown that the scalar $D_{0}^{\ast}$
pole may be far below its currently reported value Bardeen and Hill (1994);
van Beveren and Rupp (2003); Bardeen  _et al._ (2003); Albaladejo  _et al._
(2017); Du  _et al._ (2021); Gayer  _et al._ (2021), and the same could be
true of the broad $D_{1}$.

One particularly useful theoretical perspective is obtained by considering the
behaviour when the charm quark becomes infinitely heavy with respect to both
the light quarks and the scale of QCD interactions De Rujula  _et al._ (1976);
Rosner (1986); Godfrey and Kokoski (1991); Isgur and Wise (1991); Lu  _et al._
(1992); Bardeen  _et al._ (2003); Godfrey (2005); Colangelo  _et al._ (2012).
In this limit the spin of the heavy quark is conserved, and the $D$-meson
states can be characterized by the vector sum of the orbital angular momentum
and the light-quark spin. For the quark-model $\ell=1$ states two doublets
arise. One doublet contains the $D_{0}^{\ast}$ and one of the $D_{1}$ mesons,
and in the infinitely heavy-quark limit they decay exclusively via $S$-wave
interactions. The other contains a $D_{1}$ and the $D_{2}^{\ast}$ decaying
entirely by $D$-wave interactions.

Recent advances have enabled computations of the properties of hadron
resonances using _lattice_ QCD. Evidence for highly excited $D$-mesons has
been obtained, including patterns of states beyond the quark model with
apparent gluonic content Liu  _et al._ (2012); Moir  _et al._ (2013); Cheung
_et al._ (2016). These methods are able to determine the scattering amplitudes
and their spectroscopic content to compare with experiment. Relevant for the
axial-vector $D_{1}$ mesons, the scattering of hadrons with spin in coupled
$\>\\!{}^{3}\\!S_{1}$–$\>\\!{}^{3}\\!D_{1}$ amplitudes was first studied in a
weakly-interacting system Woss  _et al._ (2018) and later in the context of
the $b_{1}$ resonance Woss  _et al._ (2019). The scalar charmed resonances
have been investigated in both the charm-light Mohler  _et al._ (2013a); Moir
_et al._ (2016); Gayer  _et al._ (2021) and charm-strange Liu  _et al._
(2013); Mohler  _et al._ (2013b); Lang  _et al._ (2014); Bali  _et al._
(2017); Alexandrou  _et al._ (2020); Cheung  _et al._ (2021) flavors. A large
coupling to the strong-decay channels was found in both cases. The axial-
vectors were also studied in Refs. Mohler  _et al._ (2013a); Lang  _et al._
(2014); Bali  _et al._ (2017).

In this article, we determine the $D_{1}$ masses and couplings to $I=1/2$
$D^{\ast}\pi$ scattering from QCD. We also compute the $D_{2}^{\ast}$ and its
couplings to both $D^{\ast}\pi$ and $D\pi$ decay modes. With the
$D_{0}^{\ast}$ determined from the same lattices Moir  _et al._ (2016); Gayer
_et al._ (2021), we compare these states to experiment and probe the
predictions from the heavy-quark limit.

Figure 1: The black and grey points show the finite volume spectra obtained
from irreps up to $|\vec{P}|=1$ that contain $J^{P}\in\\{1^{+},2^{+}\\}$. Only
black points are used in the fit. The thin solid curves show the spectrum
anticipated in the absence of interactions between $D^{\ast}\pi$ (purple),
$D\pi$ (red) and $D^{\ast}\eta$ (green). Dotted lines represent kinematic
meson-meson thresholds. The $D\pi$ threshold lies below the displayed energy
range. The light yellow bands correspond to solutions of the determinant
condition Eq. 1 using the described parameterization. Additional irreps are
shown in the supplemental .1.

_Computing finite-volume spectra_ — Lattice QCD is a numerical approach
through which first-principles predictions of strongly-coupled QCD can be
obtained by Monte Carlo sampling the QCD path integral. Working in a
discretized finite euclidean volume $L^{3}\times T$ with spatial and temporal
lattice spacings $a_{s}$ and $a_{t}$, correlation functions can be computed
that determine the QCD spectrum in that volume.

The rotational symmetry of the finite cubic spatial boundary differs from an
infinite volume. At rest, we compute spectra within irreducible
representations (irreps) of the finite cubic group, rather than the continuous
orthogonal group. This results in a linear combination of multiple partial-
waves of definite $J^{P}$ within each irrep. Amplitudes grow at threshold like
$k^{2\ell}$ which commonly suppresses all but the lowest few partial waves at
low energies. We also consider non-zero overall momentum
$\vec{P}=\frac{2\pi}{L}(i,j,k)=[ijk]$, where the symmetry is further reduced;
notably, partial waves with both parities are present in the corresponding
irreps. The partial waves contributing to the irreps used in this calculation
are described in Ref. Woss  _et al._ (2018). Irreps are labeled by
$[ijk]\Lambda^{(P)}$.

We use lattices with $2+1$ dynamical flavors of quark, where the strange quark
is approximately physical and the light quark produces a pion with
$m_{\pi}\approx 391$ MeV. The ratio of the spatial and temporal lattice
spacings on this anisotropic lattice is $a_{s}/a_{t}\approx 3.5$ Edwards  _et
al._ (2008); Lin  _et al._ (2009). We use three volumes with
$(L/a_{s})^{3}\times T/a_{t}=\\{16^{3},20^{3},24^{3}\\}\times 128$. The scale
is estimated using the $\Omega$ baryon leading to $a_{t}^{-1}=5667$ MeV
Edwards  _et al._ (2013). This corresponds to $a_{s}\approx
0.12\>\mathrm{fm}$, resulting in physical volumes between
$(2\>\mathrm{fm})^{3}$ and $(3\>\mathrm{fm})^{3}$. The distillation approach
is used, which both enhances signals from the required low energy modes, and
allows all of the Wick contractions specified by QCD to be computed
efficiently Peardon  _et al._ (2009). The charm-quark uses the same action as
the light and strange quarks Liu  _et al._ (2012). Crucial to this study, the
vector $D^{\ast}$ is stable at this light-quark mass Moir  _et al._ (2013).

The variational method is used to extract spectra from correlation functions
Michael (1985); Luscher and Wolff (1990); Blossier  _et al._ (2009). Special
care is needed in choosing operators with good overlap onto all the states
present within the investigated energy range. In this work, we use a large
basis of approximately-local $q\bar{q}$\- and meson-meson-like operators. The
latter are constructed from pairs of mesons obtained variationally from large
bases of $q\bar{q}$-like operators to reduce excited state contributions Dudek
_et al._ (2010); Thomas  _et al._ (2012); Dudek  _et al._ (2012). The
correlation functions form a matrix that is diagonalized using a generalized
eigenvalue approach. The time dependence of the eigenvalues yields the finite
volume spectrum $\\{E_{\mathfrak{n}}\\}$. These spectra expose the underlying
scattering amplitude through Lüscher’s finite volume quantization condition,
and extensions thereof Luscher (1991); Rummukainen and Gottlieb (1995);
Bedaque (2004); Kim  _et al._ (2005); Fu (2012); Leskovec and Prelovsek
(2012); Gockeler  _et al._ (2012); He  _et al._ (2005); Bernard  _et al._
(2011); Doring  _et al._ (2012); Hansen and Sharpe (2012); Briceno and Davoudi
(2013); Guo  _et al._ (2013); Briceno (2014) – these methods are reviewed in
Ref. Briceno  _et al._ (2018).

In Fig. 1 we present the finite volume spectra computed in irreps with
$\vec{P}=[000]$ and $[001]$ with contributions from $J^{P}=1^{+}$ and $2^{+}$.
In $[000]T_{1}^{+}$ and $[001]A_{2}$, we observe 3 energy levels below
$a_{t}E=0.45$ ($\approx 2550$ MeV) where only a single energy level would have
been expected based on the non-interacting spectrum. This indicates non-
trivial interactions in $J^{P}=1^{+}$. The $[000]E^{+}$ and $[000]T_{2}^{+}$
irreps also contain an extra energy level, suggestive of significant
$J^{P}=2^{+}$ interactions. Further irreps and a list of the operators used
are given in supplemental .1 and .2.

_Scattering amplitude determinations_ — The extension of Lüscher’s determinant
condition for the scattering of hadrons with arbitrary spin can be written
Briceno (2014),

$\displaystyle\det\Bigl{[}\bm{1}+i\bm{\rho}(E)\cdot\bm{t}(E)\cdot\bigl{(}\bm{1}+i\bm{\mathcal{M}}(E,L)\bigr{)}\Bigr{]}=0,$
(1)

where $E$ is the $\mathsf{cm}$ energy, $L$ is the spatial extent, $\bm{t}$ is
the infinite volume scattering $t$-matrix and $\bm{\mathcal{M}}$ is a matrix
of known functions of energy, which is dense in partial waves and dependent on
the irrep. $\bm{\rho}$ is a diagonal matrix of phase-space factors. Since
$\bm{t}$ has multiple unknowns at each value of energy, it is necessary to
introduce a parameterization. The $J^{P}=1^{+}$ wave can be parameterized
using a symmetric $2\times 2$ $K$-matrix, indexed by the $\>\\!{}^{3}\\!S_{1}$
and $\>\\!{}^{3}\\!D_{1}$ channel labels. A threshold factor is included to
promote the natural threshold behaviour of each $t$-matrix element,

$\displaystyle
t^{-1}_{ij}=(2k_{i})^{-\ell_{i}}\>K_{ij}^{-1}\>(2k_{j})^{-\ell_{j}}+I_{ij}\;,$
(2)

where $k_{i}$ are the $\mathsf{cm}$ momenta, $K_{ij}$ are the $K$-matrix
elements and $I_{ij}$ is a diagonal matrix. To respect unitarity,
$\mathrm{Im}I_{ij}=-\rho_{i}\delta_{ij}$, where $\rho_{i}=2k_{i}/E$ are the
phase space factors. The real part is either set to zero or a Chew-Mandelstam
phase space is used Wilson  _et al._ (2015a), where a logarithm is generated
from the known imaginary part.

Figure 2:  Scattering amplitudes for $J^{P}\in\\{0^{+}$, $1^{+}$, $2^{+}\\}$.
The outer uncertainty bands are obtained by varying the hadron masses and
anisotropy entering eq. 1 within their uncertainties. The energy levels from
irreps with a contribution from the respective $J^{P}$ are shown in black
below the horizontal axis. The $0^{+}$ amplitude determination is from Refs.
Moir  _et al._ (2016); Gayer  _et al._ (2021).

One useful form of $\bm{K}$ is

$\displaystyle
K_{ij}=\sum_{p=1}^{2}\>\frac{g_{p,i}g_{p,j}}{m_{p}^{2}-s}+\gamma_{ij}\;.$ (3)

The $K$-matrix pole mass $m_{p}$ and couplings $g_{p,i}$ are free parameters
that can efficiently produce resonances in a $t$-matrix. The $\gamma_{ij}$
form a real symmetric matrix of constants. The free parameters are determined
in a $\chi^{2}$ minimization as defined in Eq. 8 of Ref. Wilson  _et al._
(2015a), to find an amplitude that best describes the finite volume spectra
obtained in the lattice calculation. We choose to search for the solutions of
Eq. 1 using the eigenvalue decomposition method outlined in Ref. Woss  _et
al._ (2020), which is ideally suited to problems with multiple channels and
partial waves.

We determine the $J^{P}=(1,2)^{+}$ and $(0,1,2)^{-}$ partial waves (everything
up to $\ell=2$) simultaneously from the irreps in Fig. 1, plus
$[011]A_{2},B_{1},B_{2}$, $[111]A_{2},E_{2}$ and $[002]A_{2}$, resulting in 94
energy levels ($J^{P}=0^{+}$ does not contribute to any of these irreps). Each
wave is parameterized using a version of Eq. 3 with various parameters fixed
to zero. For example, for the $1^{+}$ wave we use the poles and the
$\gamma_{\>\\!{}^{3}\\!S_{1},\>\\!^{3}\\!S_{1}}$ element, and fix the other
$\gamma_{ij}$ to zero. We use a single pole for the $2^{+}$ amplitudes, and
only constants $\gamma$ for the $J^{-}$ waves. Further details of the
parameterization and the parameter values resulting from the
$\chi^{2}$-minimization are given in the supplemental .3. A
$\chi^{2}/N_{\text{dof}}=\tfrac{95.0}{94-15}=1.20$ is obtained. The spectra
from using these amplitudes in Eq. 1 are shown as orange curves in Fig. 1. The
amplitudes are plotted in Fig. 2.

The parameterization given in Eq. 3 is one of many reasonable choices. In
order to reduce possible bias from a specific choice, we vary the form. We
obtain 21 different parameterizations that describe the spectra, summarized in
supplemental .4. These are used when computing pole positions, couplings and
their uncertainties.

_Poles & Interpretation_ — Scattering amplitude poles describe the
spectroscopic content consisting of both unstable resonances and stable bound
states. The amplitude is analytically continued to complex-$s$. There is a
square-root branch cut beginning at each threshold leading to sheets that can
be labelled by the sign of the imaginary part of the momentum in each channel
$i$, $\mathrm{Im}k_{i}$. Close to a pole, the $t$-matrix is dominated by a
term $t_{ij}\sim\frac{c_{i}c_{j}}{s_{0}-s}$, where $s_{0}$ is the pole
position and $c_{i}$ are the channel couplings.

Figure 3: The $t$-matrix poles and couplings determined in this study. The top
and middle shows pole positions; the bottom shows couplings. The colored error
ellipses are the results from individual parameterizations and are used to
determine the quoted envelope over all parameterizations shown in black.

In the $J^{P}=1^{+}$ amplitudes, we find a bound-state close to threshold,
strongly coupled to the $\>\\!{}^{3}\\!S_{1}$ amplitude, that produces a broad
enhancement over the energy region where the amplitudes are constrained. The
$\>\\!{}^{3}\\!D_{1}$ amplitude has a coupling to this pole consistent with
zero, as shown in the lower left panel of Fig. 3. Due to the proximity of the
bound state pole to threshold, the $k_{i}^{\ell_{i}}$ factor severely dampens
any $\>\\!{}^{3}\\!D_{1}$ coupling. The narrow peak in the
$\>\\!{}^{3}\\!D_{1}$ amplitude is produced by a resonance pole dominantly
coupled to the $\>\\!{}^{3}\\!D_{1}$ amplitude, with only a small
$\>\\!{}^{3}\\!S_{1}$ coupling. We find a small but non-zero width in all but
two of the parameterizations. Both are rejected due to a larger $\chi^{2}$
than the rest of the amplitudes; the more subtle case is described further in
supplemental .5.

An additional pole is found at the upper edge of the fitting range close to
where the $\>\\!{}^{3}\\!S_{1}$ amplitude touches zero, as seen in Fig. 2.
When a zero occurs on the physical sheet, it is usually accompanied by a pole
on an unphysical sheet at a similar energy. Since this pole is so close to
both the upper limit of the energy range and the $D^{\ast}\eta$ an
$D_{s}^{\ast}\bar{K}$ channel openings, higher energy levels are required to
be certain of its presence. In one of the rejected amplitudes, this pole does
not arise. A similar feature occurs in $D\pi\\{\>\\!^{1}\\!S_{0}\\}$ elastic
scattering that either disappears or moves to higher energies once the
coupled-channel $D\eta$ and $D_{s}\bar{K}$ amplitudes are introduced Moir  _et
al._ (2016). We thus exclude this pole from the remaining discussion.

One pole is found in $J^{P}=2^{+}$, coupled to both
$D\pi\\{\>\\!^{1}\\!D_{2}\\}$ and $D^{\ast}\pi\\{\>\\!^{3}\\!D_{2}\\}$. No
other nearby poles are found.

In summary, poles are found at

$\sqrt{s_{D_{0}^{\ast}}}$ | = $2276(2)\,\mathrm{MeV}\quad(+)$ (from Refs. Moir  _et al._ (2016); Gayer  _et al._ (2021))
---|---
$\sqrt{s_{D_{1}}}$ | = $2397(2)\,\mathrm{MeV}\quad(+)$
$\sqrt{s_{D_{1}^{\prime}}}$ | = $\left(2475(3)-\tfrac{i}{2}5(3)\right)\,\mathrm{MeV}\;\;\quad(-)$
$\sqrt{s_{D_{2}^{\ast}}}$ | = $\left(2524(2)-\tfrac{i}{2}10(4)\right)\,\mathrm{MeV}\quad(-,-)\,.$

The signs in curved brackets indicate the sign of $\mathrm{Im}k_{i}$, and thus
the sheet where the pole is located. The quoted values correspond to the
envelope of uncertainties of all accepted parameterizations as well as mass
and anisotropy variations. The couplings are (in MeV):

$D_{0}^{\ast}$ |  $\;|c_{D\pi\\{\>\\!^{1}\\!S_{0}\\}}|$ $=$ 760(164) | (from Refs. Moir  _et al._ (2016); Gayer  _et al._ (2021))
---|---|---
$D_{1}$ |  $\;|c_{D^{\ast}\pi\\{\>\\!^{3}\\!S_{1}\\}}|$ $=$ 1007(123) | $|c_{D^{\ast}\pi\\{\>\\!^{3}\\!D_{1}\\}}|=2(3)$
$D_{1}^{\prime}$ |  $\;|c_{D^{\ast}\pi\\{\>\\!^{3}\\!S_{1}\\}}|$ $=$ 32(33) | $|c_{D^{\ast}\pi\\{\>\\!^{3}\\!D_{1}\\}}|=239(77)$
$D_{2}^{\ast}$ |  $\;|c_{D\pi\\{\>\\!^{1}\\!D_{2}\\}}|$ $=$ 234(11) |  $|c_{D^{\ast}\pi\\{\>\\!^{3}\\!D_{2}\\}}|=137(126)$ .

Comparing to experiment, the larger-than-physical light-quark mass must be
accounted for. $D$-meson masses larger than those found in experiment are
expected. The $D_{2}^{\ast}$ is found some 80 MeV above the experimental state
and the narrow $D_{1}^{\prime}$ is 52 MeV above experiment. The bound-state
pole that produces the broad feature in $S$-wave however is 15 MeV _below_ the
$D_{1}(2430)$ found at 2412(9) MeV Aaij  _et al._ (2020d); Zyla  _et al._
(2020). Considering the similarity of $D\pi\\{\>\\!^{1}\\!S_{0}\\}$ and
$D^{\ast}\pi\\{\>\\!^{3}\\!S_{1}\\}$ along with the light-quark mass
dependence of the $D_{0}^{\ast}$ determined in Ref. Gayer  _et al._ (2021),
the broad $D_{1}(2430)$ could in reality be produced by a pole at a much lower
mass. Another difference that arises as the light-quark mass is reduced, is
that $D\pi\pi$ opens and may introduce more mixing between channels. Recent
developments of three-body formalism will be essential in understanding these
processes Hansen and Sharpe (2015); Briceño  _et al._ (2019); Hansen and
Sharpe (2019); Blanton and Sharpe (2020, 2021a, 2021b); Blanton  _et al._
(2022); Mai and Döring (2017); Polejaeva and Rusetsky (2012); Hammer  _et al._
(2017a, b).

Crudely extrapolating the couplings to lighter pion masses requires a
kinematic factor, leading to
$|c^{\mathrm{ex.}}|=|c||k^{\mathrm{ex.}}(m_{r}^{\mathrm{ex.}})/k(m_{r})|^{\ell}$
Wilson  _et al._ (2015b); Woss  _et al._ (2021). The $D_{1}^{\prime}$
couplings, fixing $m_{D_{1}^{\prime}}^{\mathrm{ex.}}=2422$ MeV Zyla  _et al._
(2020), produce a consistent width to that observed experimentally,
$\Gamma_{D_{1}^{\prime}}=48\pm 30$ MeV. The $D_{2}^{\ast}$ couplings, fixing
$m_{D_{2}^{\ast}}^{\mathrm{ex.}}=2461$ MeV Zyla  _et al._ (2020), correspond
to $\Gamma_{D_{2}^{\ast}}=24\pm 13$ MeV, a little narrower than that observed
in experiment. The large coupling found for the bound $D_{1}$ suggests this
will evolve into a broad resonance as the light quark mass is reduced.

$D^{\ast}K$ in $I=0$ is related to $D^{\ast}\pi$ in $I=1/2$ by SU(3) flavor
symmetry Albaladejo  _et al._ (2017); Cheung  _et al._ (2021) and has
similarities. The experimental axial-vector $D_{s1}$ hadron masses follow a
similar pattern to that observed here, with one bound state and one narrow
resonance. The bound-state is significantly more-bound, and the resonance is
narrower. Evidence of a bound $D_{s1}$ was also found in other lattice studies
Lang  _et al._ (2014); Bali  _et al._ (2017).

_Heavy-quark limit comparisons_ — The heavy quark limit (as described in Refs.
De Rujula  _et al._ (1976); Rosner (1986); Godfrey and Kokoski (1991); Isgur
and Wise (1991); Lu  _et al._ (1992); Bardeen  _et al._ (2003); Godfrey
(2005)) captures many of the features observed in these states. Notably, the
prediction of decoupling between the two $1^{+}$ states is upheld in our
results. We could not rule out a $\>\\!{}^{3}\\!S_{1}$ component in the narrow
$D_{1}^{\prime}$ resonance; many of the parameterizations favor a small but
non-zero coupling. The $D\pi\\{\>\\!^{1}\\!S_{0}\\}$ and
$D^{\ast}\pi\\{\>\\!^{3}\\!S_{1}\\}$ amplitudes are remarkably similar in
terms of both singularities and energy dependence, producing near-threshold
bound states strongly coupled to the relevant nearby channel. Binding energies
are found to be $2(1)$ MeV and $6(2)$ MeV respectively. Heavy-quark symmetry
can also be used to relate the couplings of the narrow $D_{1}^{\prime}$ and
$D_{2}^{\ast}$. The results found here are not inconsistent with expectations
from the heavy-quark limit; a proper test needs a more precise determination
of the couplings, in particular $c_{D^{\ast}\pi\\{\>\\!^{3}\\!D_{2}\\}}$.

_Summary_ — The dynamically-coupled ${}^{2S+1}\ell_{J}=\>\\!^{3}\\!S_{1}$ and
$\>\\!{}^{3}\\!D_{1}$ $D^{\ast}\pi$ scattering amplitudes in $I=1/2$ have been
computed from QCD for the first time. Working at $m_{\pi}=391$ MeV, the
$D^{\ast}\pi\\{\>\\!^{3}\\!S_{1}\\}$ amplitude is dominated by a pole just
below threshold, whose influence extends over a broad energy region. This pole
is found below the experimental state despite the larger pion mass, analogous
to the scalar sector. There is a narrow resonance coupled dominantly to the
$D^{\ast}\pi\\{\>\\!^{3}\\!D_{1}\\}$ amplitude. The
$D^{\ast}\pi\\{\>\\!^{3}\\!S_{1}\\}\to D^{\ast}\pi\\{\>\\!^{3}\\!D_{1}\\}$
amplitude is consistent with zero in the constrained energy region.

The importance of understanding $S$-wave interactions between pairs of hadrons
in QCD cannot be understated. The strength of interactions observed here
between a vector and a pseudoscalar not only extends our understanding of
$D$-meson decays; it may point toward a resolution of some of the many other
puzzles found with interacting charmed hadrons.

###### Acknowledgements.

We thank our colleagues within the Hadron Spectrum Collaboration
(www.hadspec.org). DJW acknowledges support from a Royal Society University
Research Fellowship. DJW acknowledges support from the U.K. Science and
Technology Facilities Council (STFC) [grant number ST/T000694/1]. The software
codes Chroma Edwards and Joo (2005), QUDA Clark  _et al._ (2010); Babich  _et
al._ (2010a), QPhiX Joó  _et al._ (2013), and QOPQDP Osborn  _et al._ (2010);
Babich  _et al._ (2010b) were used to compute the propagators required for
this project. 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 clusters at
Jefferson Laboratory under the USQCD Initiative and the LQCD ARRA project, and
the authors acknowledge support from the U.S. Department of Energy, Office of
Science, Office of Advanced Scientific Computing Research and Office of
Nuclear Physics, Scientific Discovery through Advanced Computing (SciDAC)
program, and the U.S. Department of Energy Exascale Computing Project. This
research was supported in part under an ALCC award, and used resources of the
Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory,
which is supported by the Office of Science of the U.S. Department of Energy
under Contract No. DE-AC05-00OR22725. This research is also 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. This
work is also part of the PRAC “Lattice QCD on Blue Waters”. This research used
resources of the National Energy Research Scientific Computing Center (NERSC),
a DOE Office of Science User Facility supported by the Office of Science of
the U.S. Department of Energy under Contract No. DEAC02-05CH11231. The authors
acknowledge the Texas Advanced Computing Center (TACC) at The University of
Texas at Austin for providing HPC resources that have contributed to the
research results reported within this paper. Gauge configurations were
generated using resources awarded from the U.S. Department of Energy INCITE
program at Oak Ridge National Lab, NERSC, the NSF Teragrid at the Texas
Advanced Computer Center and the Pittsburgh Supercomputer Center, as well as
at Jefferson Lab.

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## Supplemental

###  .1 Energy levels from other irreps

In Fig. 4 we present the remaining irreps used to constrain the amplitudes in
addition to those presented in the main text.

Figure 4: Energy levels and finite-volume solutions of the determinant
condition from remaining irreps used in the fit, that are not shown in figure
1.

###  .2 Operator tables

In tables 1 and 2 we summarize the operators used to compute the correlation
matrices from which the spectra are determined. Further details of the
$\bar{q}\mathbf{\Gamma}q$ operators are given in Ref. Dudek  _et al._ (2010).
The construction of two-body operators is described in Refs. Thomas  _et al._
(2012); Dudek  _et al._ (2012). Operators labeled $f_{0}$, $\rho$ and $D_{0}$
resemble three-body states as is described in Refs. Woss  _et al._ (2019).
These are only found to contribute far above the energy region considered in
this study.

$[000]T_{1}^{+}$ | $[000]E^{+}$ | $[000]T_{2}^{+}$ | $[001]A_{2}$ | $[001]E_{2}$ | $[001]B_{1}$ | $[001]B_{2}$
---|---|---|---|---|---|---
$D_{[000]}\rho_{[000]}$ (1) |  $D_{[100]}\pi_{[100]}$ (1) |  $D_{[110]}\pi_{[110]}$ (1) |  $D_{[100]}\rho_{[000]}$ (1) |  $D_{[100]}\pi_{[110]}$ (1) |  $D_{[100]}\pi_{[110]}$ (1) |  $D_{[111]}\pi_{[110]}$ (1)
$D_{[100]}\rho_{[100]}$ (2) |  $D_{[110]}\pi_{[110]}$ (1) |  ${D^{*}}_{[100]}\pi_{[100]}$ (1) |  $D_{[000]}{f_{0}}_{[100]}$ (1) |  $D_{[110]}\pi_{[100]}$ (1) |  $D_{[110]}\pi_{[100]}$ (1) |  $D_{[110]}{f_{0}}_{[100]}$ (1)
$D_{[100]}{f_{0}}_{[100]}$ (1) |  $D_{[200]}\pi_{[200]}$ (1) |  ${D^{*}}_{[000]}\rho_{[000]}$ (1) |  $D_{[100]}{f_{0}}_{[000]}$ (1) |  $D_{[111]}\pi_{[110]}$ (1) |  $D_{[100]}\eta_{[110]}$ (1) |  ${D^{*}}_{[100]}\pi_{[110]}$ (2)
${D^{*}}_{[000]}\pi_{[000]}$ (1) |  $D_{[100]}\eta_{[100]}$ (1) |  $\bar{q}\mathbf{\Gamma}q$ (29) |  ${D^{*}}_{[000]}\pi_{[100]}$ (1) |  $D_{[110]}\eta_{[100]}$ (1) |  $D_{[110]}\eta_{[100]}$ (1) |  ${D^{*}}_{[110]}\pi_{[100]}$ (2)
${D^{*}}_{[100]}\pi_{[100]}$ (2) |  $D_{[110]}\eta_{[110]}$ (1) |  |  ${D^{*}}_{[100]}\pi_{[000]}$ (1) |  $D_{[000]}\rho_{[100]}$ (1) |  ${D_{s}}_{[100]}\bar{K}_{[110]}$ (1) |  $\bar{q}\mathbf{\Gamma}q$ (20)
${D^{*}}_{[110]}\pi_{[110]}$ (3) |  ${D_{s}}_{[100]}\bar{K}_{[100]}$ (1) |  |  ${D^{*}}_{[110]}\pi_{[100]}$ (2) |  $D_{[100]}\rho_{[000]}$ (1) |  ${D_{s}}_{[110]}\bar{K}_{[100]}$ (1) |
${D^{*}}_{[000]}\eta_{[000]}$ (1) |  ${D_{s}}_{[110]}\bar{K}_{[110]}$ (1) |  |  ${D^{*}}_{[100]}\eta_{[000]}$ (1) |  ${D_{s}}_{[110]}\bar{K}_{[100]}$ (1) |  $\bar{q}\mathbf{\Gamma}q$ (12) |
${D^{*}}_{[100]}\eta_{[100]}$ (2) |  $\bar{q}\mathbf{\Gamma}q$ (4) |  |  ${D_{s}^{*}}_{[100]}\bar{K}_{[000]}$ (1) |  ${D^{*}}_{[000]}\pi_{[100]}$ (1) |  |
${D^{*}}_{[000]}\rho_{[000]}$ (1) |  |  |  ${D_{0}}_{[100]}\pi_{[000]}$ (1) |  ${D^{*}}_{[100]}\pi_{[000]}$ (1) |  |
${D_{s}^{*}}_{[000]}\bar{K}_{[000]}$ (1) |  |  |  $\bar{q}\mathbf{\Gamma}q$ (32) |  ${D^{*}}_{[110]}\pi_{[100]}$ (3) |  |
${D_{s}^{*}}_{[100]}\bar{K}_{[100]}$ (2) |  |  |  |  ${D^{*}}_{[000]}\eta_{[100]}$ (1) |  |
${D_{0}}_{[100]}\pi_{[100]}$ (1) |  |  |  |  ${D^{*}}_{[100]}\eta_{[000]}$ (1) |  |
$\bar{q}\mathbf{\Gamma}q$ (44) |  |  |  |  ${D^{*}}_{[100]}{f_{0}}_{[000]}$ (1) |  |
|  |  |  |  ${D_{s}^{*}}_{[100]}\bar{K}_{[000]}$ (1) |  |
|  |  |  |  $\bar{q}\mathbf{\Gamma}q$ (44) |  |
Table 1: Operators included in the computation of the spectra shown in figure 1. For meson-meson operators, the numbers in parentheses indicate degeneracies in the non-interacting case. For the $\bar{q}q$-like operators (last line of each column), the number in parentheses gives the number of different $\Gamma$ constructions used. $[000]T_{1}^{-}$ | $[011]A_{2}$ | $[011]B_{1}$ | $[011]B_{2}$ | $[111]A_{2}$ | $[111]E_{2}$ | $[002]A_{2}$
---|---|---|---|---|---|---
$D_{[100]}\pi_{[100]}$ (1) |  $D_{[110]}\pi_{[110]}$ (1) |  $D_{[100]}\pi_{[100]}$ (1) |  $D_{[110]}\pi_{[110]}$ (1) |  $D_{[111]}\rho_{[000]}$ (1) |  $D_{[100]}\pi_{[110]}$ (1) |  $D_{[100]}\rho_{[100]}$ (1)
$D_{[100]}\eta_{[100]}$ (1) |  $D_{[110]}\rho_{[000]}$ (1) |  $D_{[110]}\pi_{[110]}$ (1) |  $D_{[111]}\pi_{[100]}$ (1) |  $D_{[111]}{f_{0}}_{[000]}$ (1) |  $D_{[110]}\pi_{[100]}$ (1) |  $D_{[100]}{f_{0}}_{[100]}$ (1)
${D^{*}}_{[100]}\pi_{[100]}$ (1) |  $D_{[110]}{f_{0}}_{[000]}$ (1) |  $D_{[210]}\pi_{[100]}$ (1) |  $D_{[110]}\rho_{[000]}$ (1) |  ${D^{*}}_{[110]}\pi_{[100]}$ (2) |  $D_{[211]}\pi_{[100]}$ (1) |  $D_{[200]}{f_{0}}_{[000]}$ (1)
$\bar{q}\mathbf{\Gamma}q$ (20) |  ${D^{*}}_{[100]}\pi_{[100]}$ (2) |  $D_{[100]}\eta_{[100]}$ (1) |  $D_{[100]}{f_{0}}_{[100]}$ (1) |  ${D^{*}}_{[111]}\pi_{[000]}$ (1) |  $D_{[100]}\eta_{[110]}$ (1) |  ${D^{*}}_{[100]}\pi_{[100]}$ (1)
|  ${D^{*}}_{[110]}\pi_{[000]}$ (1) |  $D_{[110]}\rho_{[000]}$ (1) |  ${D^{*}}_{[000]}\pi_{[110]}$ (1) |  ${D^{*}}_{[111]}\eta_{[000]}$ (1) |  $D_{[110]}\eta_{[100]}$ (1) |  ${D^{*}}_{[200]}\pi_{[000]}$ (1)
|  ${D^{*}}_{[111]}\pi_{[100]}$ (2) |  ${D_{s}}_{[100]}\bar{K}_{[100]}$ (1) |  ${D^{*}}_{[100]}\pi_{[100]}$ (2) |  ${D_{s}^{*}}_{[111]}\bar{K}_{[000]}$ (1) |  $D_{[111]}\rho_{[000]}$ (1) |  ${D^{*}}_{[210]}\pi_{[100]}$ (2)
|  ${D^{*}}_{[110]}\eta_{[000]}$ (1) |  ${D^{*}}_{[000]}\pi_{[110]}$ (1) |  ${D^{*}}_{[110]}\pi_{[000]}$ (1) |  ${D_{0}}_{[111]}\pi_{[000]}$ (1) |  $D_{[110]}{f_{0}}_{[100]}$ (1) |  ${D^{*}}_{[100]}\eta_{[100]}$ (1)
|  ${D_{s}^{*}}_{[110]}\bar{K}_{[000]}$ (1) |  ${D^{*}}_{[100]}\pi_{[100]}$ (1) |  ${D^{*}}_{[111]}\pi_{[100]}$ (1) |  $\bar{q}\mathbf{\Gamma}q$ (36) |  ${D_{s}}_{[100]}\bar{K}_{[110]}$ (1) |  ${D^{*}}_{[200]}\eta_{[000]}$ (1)
|  ${D_{0}}_{[110]}\pi_{[000]}$ (1) |  ${D^{*}}_{[110]}\pi_{[000]}$ (1) |  ${D^{*}}_{[110]}\eta_{[000]}$ (1) |  |  ${D_{s}}_{[110]}\bar{K}_{[100]}$ (1) |  ${D_{s}^{*}}_{[200]}\bar{K}_{[000]}$ (1)
|  $\bar{q}\mathbf{\Gamma}q$ (52) |  ${D^{*}}_{[111]}\pi_{[100]}$ (2) |  ${D_{s}^{*}}_{[110]}\bar{K}_{[000]}$ (1) |  |  ${D^{*}}_{[100]}\pi_{[110]}$ (3) |  $\bar{q}\mathbf{\Gamma}q$ (32)
|  |  ${D^{*}}_{[100]}\eta_{[100]}$ (1) |  $\bar{q}\mathbf{\Gamma}q$ (52) |  |  ${D^{*}}_{[110]}\pi_{[100]}$ (3) |
|  |  ${D^{*}}_{[110]}\eta_{[000]}$ (1) |  |  |  ${D^{*}}_{[111]}\pi_{[000]}$ (1) |
|  |  ${D^{*}}_{[110]}{f_{0}}_{[000]}$ (1) |  |  |  ${D^{*}}_{[111]}\eta_{[000]}$ (1) |
|  |  ${D_{s}^{*}}_{[110]}\bar{K}_{[000]}$ (1) |  |  |  ${D_{s}^{*}}_{[111]}\bar{K}_{[000]}$ (1) |
|  |  $\bar{q}\mathbf{\Gamma}q$ (44) |  |  |  $\bar{q}\mathbf{\Gamma}q$ (60) |
Table 2: Same as table 1 but for the spectra shown in figure 4.

###  .3 Reference parameterization

The reference parameterization consists of partial wave amplitudes in
$J^{P}=1^{+}$, $2^{+}$, $0^{-}$, $1^{-}$ and $2^{-}$. Both $D\pi$ and
$D^{\ast}\pi$ contribute in various combinations of total spin $J$, angular
momentum $\ell$ and spin $S$, identified below as ${}^{2S+1}\ell_{J}$
(spectroscopic notation). Although the focus of this study is _elastic_
$D^{\ast}\pi$ scattering, each partial wave $J^{P}$ can receive contributions
from multiple ${}^{2S+1}\ell_{J}$ combinations which can couple. For
$J^{P}=1^{+}$, we have

$\displaystyle\bm{t}{(J^{P}=1^{+})}=\left(\begin{matrix}t(D^{\ast}\pi\\{\>\\!^{3}\\!S_{1}\\}\to
D^{\ast}\pi\\{\>\\!^{3}\\!S_{1}\\})&t(D^{\ast}\pi\\{\>\\!^{3}\\!S_{1}\\}\to
D^{\ast}\pi\\{\>\\!^{3}\\!D_{1}\\})\\\ t(D^{\ast}\pi\\{\>\\!^{3}\\!D_{1}\\}\to
D^{\ast}\pi\\{\>\\!^{3}\\!S_{1}\\})&t(D^{\ast}\pi\\{\>\\!^{3}\\!D_{1}\\}\to
D^{\ast}\pi\\{\>\\!^{3}\\!D_{1}\\})\end{matrix}\right)\,.$ (4)

For $J^{P}=2^{+}$, only one $D^{\ast}\pi$ combination arises, however $D\pi$
also contributes,

$\displaystyle\bm{t}{(J^{P}=2^{+})}=\left(\begin{matrix}t(D\pi\\{\>\\!^{1}\\!D_{2}\\}\to
D\pi\\{\>\\!^{1}\\!D_{2}\\})&t(D\pi\\{\>\\!^{1}\\!D_{2}\\}\to
D^{\ast}\pi\\{\>\\!^{3}\\!D_{2}\\})\\\ t(D^{\ast}\pi\\{\>\\!^{3}\\!D_{2}\\}\to
D\pi\\{\>\\!^{1}\\!D_{2}\\})&t(D^{\ast}\pi\\{\>\\!^{3}\\!D_{2}\\}\to
D^{\ast}\pi\\{\>\\!^{3}\\!D_{2}\\})\end{matrix}\right)\,.$ (5)

For $J^{P}=0^{-}$ and $2^{-}$, we consider just a single
$t(D^{\ast}\pi\\{^{3}P_{J}\\}\to D^{\ast}\pi\\{^{3}P_{J}\\})$ combination. In
$J^{P}=1^{-}$, there are two possible $\ell=1$ combinations,

$\displaystyle\bm{t}{(J^{P}=1^{-})}=\left(\begin{matrix}t(D\pi\\{\>\\!^{1}\\!P_{1}\\}\to
D\pi\\{\>\\!^{1}\\!P_{1}\\})&t(D\pi\\{\>\\!^{1}\\!P_{1}\\}\to
D^{\ast}\pi\\{\>\\!^{3}\\!P_{1}\\})\\\ t(D^{\ast}\pi\\{\>\\!^{3}\\!P_{1}\\}\to
D\pi\\{\>\\!^{1}\\!P_{1}\\})&t(D^{\ast}\pi\\{\>\\!^{3}\\!P_{1}\\}\to
D^{\ast}\pi\\{\>\\!^{3}\\!P_{1}\\})\end{matrix}\right)\,.$ (6)

We parameterize each $J^{P}$ using a $K$-matrix coupling the relevant hadron-
hadron ${}^{2S+1}\ell_{J}$ combinations. For the $J^{P}=1^{+}$ partial wave,
we use

$\displaystyle
K_{ij}=\frac{g_{0,i}g_{0,j}}{m_{0}^{2}-s}+\frac{g_{1,i}g_{1,j}}{m_{1}^{2}-s}+\gamma_{ij}$
(7)

where $i,j$ refer to the ${}^{2S+1}\ell_{J}$ hadron-hadron combinations. In
this study ${}^{3}\ell_{J}$ is always $D^{\ast}\pi$ and ${}^{1}\ell_{J}$ is
always $D\pi$. Any parameter not given a value below, such as many of the
$\gamma_{ij}$, is implicitly fixed to zero. Where we specify the channel label
$i$ explicitly using spectroscopic notation, we write $g_{p}(i)$ instead of
$g_{p,i}$ and $\gamma(i\to j)$ instead of $\gamma_{ij}$. Superscripts in
parentheses are used for coefficients of $s$ and indicate the order of the
term.

These $K$-matrix parameters are inserted into a $t$-matrix that includes
momenta raised to the power required by the near-threshold behaviour,
$k_{i}^{\ell}$, appropriate for each element,

$\displaystyle
t_{ij}^{-1}=\frac{1}{(2k_{i})^{\ell_{i}}}K^{-1}_{ij}\frac{1}{(2k_{j})^{\ell_{j}}}+I_{ij}$
(8)

where $I_{ij}$ is a Chew-Mandelstam phase space as described in appendix B of
Ref. Wilson  _et al._ (2015a).

The $K$-matrices for the remaining amplitudes are similar. For $J^{P}=2^{+}$
we use just a single pole,

$\displaystyle K_{ij}=\frac{g_{2,i}g_{2,j}}{m_{2}^{2}-s}+\gamma_{ij}$ (9)

where the channels $i,j$ are $D\pi(^{1}D_{2})$ and $D^{\ast}\pi(^{3}D_{2})$.

The negative parity waves are parameterized by constants $K_{ij}=\gamma_{ij}$.
The corresponding plots of $\rho_{i}\rho_{j}|t_{ij}|^{2}$ can be seen in
figure 5. The $0^{-}$ and $2^{-}$ waves receive contributions from only
$D^{\ast}\pi$. The $1^{-}$ wave has contributions from both
$D\pi\\{^{1}P_{1}\\}$ and $D^{\ast}\pi\\{^{3}P_{1}\\}$. The $1^{-}$ amplitude
contains the $D^{\ast}$ bound state far below threshold. We exclude this level
from all the amplitude determinations, as there is not a large obvious effect
on the spectrum above threshold. However, the parameterization is capable of
describing an effect in the physically-allowed $D\pi$ and $D^{\ast}\pi$
scattering region above threshold. Further details of the $D^{\ast}$ bound-
state on these lattices can be found in Ref. Moir  _et al._ (2016).

The parameter values and their correlations are:

$m_{0}=$ | $(0.42294\pm 0.00017\pm 0.00011)\,\cdot\,a_{t}^{-1}$ | $\left.\rule{0.0pt}{36.98866pt}\right\\}$ $J^{P}=1^{+}$
---|---|---
$m_{1}=$ | $(0.43691\pm 0.00028\pm 0.00006)\,\cdot\,a_{t}^{-1}$
$g_{0}({\>\\!{}^{3}\\!S_{1}})=$ | $(0.528\pm 0.022\pm 0.022)\,\cdot\,a_{t}^{-1}$
$g_{0}({\>\\!{}^{3}\\!D_{1}})=$ | $(0.390\pm 0.850\pm 0.089)\,\cdot\,a_{t}$
$g_{1}({\>\\!{}^{3}\\!S_{1}})=$ | $(0.014\pm 0.012\pm 0.002)\,\cdot\,a_{t}^{-1}$
$g_{1}({\>\\!{}^{3}\\!D_{1}})=$ | $(-6.40\pm 1.36\pm 0.21)\,\cdot\,a_{t}$
$\gamma(\>\\!^{3}\\!S_{1}\to\>\\!^{3}\\!S_{1})=$ | $(10.3\pm 1.2\pm 0.6)$
$m_{2}=$ | $(0.44546\pm 0.00029\pm 0.00006)\,\cdot\,a_{t}^{-1}$ | $\left.\rule{0.0pt}{19.91684pt}\right\\}$ $J^{P}=2^{+}$
$g_{2}(\>\\!^{1}\\!D_{2})=$ | $(1.730\pm 0.065\pm 0.013)\,\cdot\,a_{t}$
$g_{2}(\>\\!^{3}\\!D_{2})=$ | $(3.02\pm 0.92\pm 0.09)\,\cdot\,a_{t}$
$\gamma(\>\\!^{1}\\!D_{2}\to\>\\!^{1}\\!D_{2})=$ | $(222\pm 172\pm 225)\,\cdot\,a_{t}^{4}$
$\gamma(\>\\!^{3}\\!P_{0}\to\>\\!^{3}\\!P_{0})=$ | $(-62\pm 54\pm 24)\,\cdot\,a_{t}^{2}$ |  $\;\>\left.\right\\}$ $J^{P}=0^{-}$
$\gamma(\>\\!^{1}\\!P_{1}\to\>\\!^{1}\\!P_{1})=$ | $(15.3\pm 3.3\pm 10.2)\,\cdot\,a_{t}^{2}$ | $\left.\rule{0.0pt}{8.5359pt}\right\\}$ $J^{P}=1^{-}$
$\gamma(\>\\!^{3}\\!P_{1}\to\>\\!^{3}\\!P_{1})=$ | $(-107\pm 16\pm 18)\,\cdot\,a_{t}^{2}$
$\gamma(\>\\!^{3}\\!P_{2}\to\>\\!^{3}\\!P_{2})=$ | $(86\pm 21\pm 18)\,\cdot\,a_{t}^{2}$ |  $\;\>\left.\right\\}$ $J^{P}=2^{-}$

$\setcounter{MaxMatrixCols}{15}\begin{bmatrix}[r]1.00&0.38&\text{-}0.22&\text{-}0.05&\text{-}0.10&\text{-}0.00&\text{-}0.16&0.28&\text{-}0.02&\text{-}0.04&\text{-}0.13&\text{-}0.10&\text{-}0.32&\text{-}0.15&\text{-}0.12\\\
&1.00&0.01&\text{-}0.02&0.16&0.32&\text{-}0.02&0.34&0.11&\text{-}0.17&\text{-}0.11&0.12&\text{-}0.23&\text{-}0.04&\text{-}0.31\\\
&&1.00&\text{-}0.44&\text{-}0.03&0.40&0.96&0.03&0.03&0.28&0.18&0.36&0.10&\text{-}0.24&\text{-}0.24\\\
&&&1.00&0.52&\text{-}0.61&\text{-}0.54&\text{-}0.01&0.01&\text{-}0.34&\text{-}0.05&\text{-}0.06&0.01&0.41&0.26\\\
&&&&1.00&\text{-}0.06&\text{-}0.05&\text{-}0.03&\text{-}0.02&\text{-}0.07&0.08&0.27&0.03&0.12&0.01\\\
&&&&&1.00&0.40&0.01&0.08&\text{-}0.03&\text{-}0.05&0.39&0.01&\text{-}0.11&\text{-}0.71\\\
&&&&&&1.00&\text{-}0.00&\text{-}0.00&0.39&0.23&0.32&0.11&\text{-}0.32&\text{-}0.21\\\
&&&&&&&1.00&0.11&\text{-}0.03&0.00&\text{-}0.02&\text{-}0.18&\text{-}0.09&\text{-}0.10\\\
&&&&&&&&1.00&\text{-}0.17&\text{-}0.17&0.04&0.08&\text{-}0.03&\text{-}0.10\\\
&&&&&&&&&1.00&0.39&\text{-}0.16&0.01&\text{-}0.19&0.40\\\
&&&&&&&&&&1.00&0.00&\text{-}0.16&\text{-}0.06&0.24\\\
&&&&&&&&&&&1.00&0.06&0.04&\text{-}0.48\\\ &&&&&&&&&&&&1.00&0.14&0.11\\\
&&&&&&&&&&&&&1.00&\text{-}0.04\\\ &&&&&&&&&&&&&&1.00\end{bmatrix}$
---

$\displaystyle\chi^{2}/N_{\mathrm{dof}}=\frac{95.0}{94-15}=1.20\,.$ (10)

The first uncertainty on the parameter values is the from the $\chi^{2}$
minimum. The second uncertainty is the largest deviation found between the
central values of the parameters after varying the scattering hadron masses
and anisotropy within their uncertainties.

Figure 5: As figure 2 but for $J^{P}\in\\{0^{-}$, $1^{-}$, $2^{-}\\}$.

###  .4 Parameterization variations

In addition to the parameterization defined in section .3, we use the
following amplitude variations. These include terms linear in $s$ and pole
terms with numerators linear in $s$. All parameterizations use the $K$-matrix
formalism and each block of the $K$-matrix corresponding to a $J^{P}$
combination can be given by one master formula, in which various parameters
may be fixed to zero or floated in the fit. The complete list is given in
table 3. For the $J^{P}=1^{+}$ block of the $K$-matrix we use

$K_{ij}=\sum_{p\in\\{0,1\\}}\frac{\big{(}g_{p,i}+g^{(1)}_{p,i}s\big{)}\big{(}g_{p,j}+g^{(1)}_{p,j}s\big{)}}{m_{p}^{2}-s}+\gamma_{ij}+\gamma^{(1)}_{ij}s\;.$
(11)

Here $i$, $j$ represent the two mixing partial waves of $D^{\ast}\pi$,
$\\{\>\\!^{3}\\!S_{1}\\}$ and $\\{\>\\!^{3}\\!D_{1}\\}$. For the $J^{P}=2^{+}$
block we have

$K_{ij}=\frac{\big{(}g_{2,i}+g^{(1)}_{2,i}s\big{)}\big{(}g_{2,j}+g^{(1)}_{2,j}s\big{)}}{m_{2}^{2}-s}+\gamma_{ij}+\gamma^{(1)}_{ij}s\;.$
(12)

The two channels represented by $i$, $j$ are $D\pi\\{\>\\!^{1}\\!D_{2}\\}$ and
$D^{\ast}\pi\\{\>\\!^{3}\\!D_{2}\\}$. The masses $m_{0}$, $m_{1}$ and $m_{2}$
in all $J^{P}=1^{+}$ and $J^{P}=2^{+}$ parameterizations are always free fit
parameters. For the $J^{P}=0^{-}$ amplitude, corresponding to
$D^{\ast}\pi\\{\>\\!^{3}\\!P_{0}\rightarrow\>\\!^{3}\\!P_{0}\\}$, we attempt
fits of both a $K$-matrix including a pole term and one with a constant term
only. The general $K$-matrix reads

$\displaystyle
K=\frac{g^{2}_{3}(\>\\!^{3}\\!P_{0})}{m_{3}^{2}-s}+\gamma(\>\\!^{3}\\!P_{0}\rightarrow\>\\!^{3}\\!P_{0})\;.$
(13)

When the pole term is included, i.e. $g^{(3)}(\>\\!^{3}\\!P_{0})\neq 0$, the
mass is fixed to $m_{3}=0.4707\cdot a_{t}^{-1}$, based on observations from
the $[000]A_{1}^{-}$ irrep as shown in Ref. Moir  _et al._ (2013). There is
not sufficient constraint to allow $m_{3}$ to float. The $J^{P}=1^{-}$ and
$J^{P}=2^{-}$ amplitudes are well-described by a $K$-matrix with constant and
linear terms only. For $J^{P}=1^{-}$ we have

$K_{ij}=\gamma_{ij}\;,$ (14)

where $i$, $j$ represent combinations of $D\pi\\{\>\\!^{1}\\!P_{1}\\}$ and
$D^{\ast}\pi\\{\>\\!^{3}\\!P_{1}\\}$. For $J^{P}=2^{-}$ we have a single
channel only and the $K$-matrix reads

$K=\gamma(\>\\!^{3}\\!P_{2}\rightarrow\>\\!^{3}\\!P_{2})+\gamma^{(1)}(\>\\!^{3}\\!P_{2}\rightarrow\>\\!^{3}\\!P_{2})s\;.$
(15)

Most parameterizations use the Chew-Mandelstam prescription (indicated by CM
in table 3), as given in equation 8. For the subtraction point we typically
choose the mass parameter of the lowest pole in the respective channel.
Further details are given in appendix B of Ref. Wilson  _et al._ (2015a). For
some amplitudes, we subtract at threshold instead. We also fit amplitude
variations that use a simple phase space $\rho_{i}=2k_{i}/\sqrt{s}$, with a
$t$-matrix given by

$\displaystyle
t_{ij}^{-1}=\frac{1}{(2k_{i})^{\ell_{i}}}K^{-1}_{ij}\frac{1}{(2k_{j})^{\ell_{j}}}-i\rho_{i}\delta_{ij}\;.$
(16)

We only include amplitude determinations in table 3 which have an acceptable
fit quality and correspond to amplitudes with have no unphysical features such
as complex poles on the first (physical) Riemann sheet.

$J^{P}$ | Parameterization | Free parameters (couplings and polynomial) | $N_{\text{pars}}$ | $\chi^{2}/N_{DoF}$
---|---|---|---|---
$1^{+}$ |  eq. 11 (CM, pole 0) | $\\{g_{0}(\>\\!^{3}\\!D_{1}),\;g_{1}(\>\\!^{3}\\!D_{1}),\;g_{0}(\>\\!^{3}\\!S_{1}),\;g_{1}(\>\\!^{3}\\!S_{1}),\;\gamma(\>\\!^{3}\\!S_{1}\to\>\\!^{3}\\!S_{1})\\}$ | 15 | 1.20
$\\{g_{0}^{(1)}(\>\\!^{3}\\!D_{1}),\;g_{1}(\>\\!^{3}\\!D_{1}),\;g_{0}(\>\\!^{3}\\!S_{1}),\;g_{1}(\>\\!^{3}\\!S_{1}),\;\gamma(\>\\!^{3}\\!S_{1}\to\>\\!^{3}\\!S_{1})\\}$ | 15 | 1.20
$\\{g_{1}(\>\\!^{3}\\!D_{1}),\;g_{0}(\>\\!^{3}\\!S_{1}),\;g_{1}(\>\\!^{3}\\!S_{1}),\;\gamma(\>\\!^{3}\\!S_{1}\to\>\\!^{3}\\!S_{1})\\}$ | 14 | 1.19
$\\{g_{0}(\>\\!^{3}\\!D_{1}),\;g_{1}^{(1)}(\>\\!^{3}\\!D_{1}),\;g_{0}(\>\\!^{3}\\!S_{1}),\;g_{1}(\>\\!^{3}\\!S_{1}),\;\gamma(\>\\!^{3}\\!S_{1}\to\>\\!^{3}\\!S_{1})\\}$ | 15 | 1.21
$\\{g_{0}(\>\\!^{3}\\!D_{1}),\;g_{0}(\>\\!^{3}\\!S_{1}),\;g_{1}(\>\\!^{3}\\!S_{1}),\;\gamma(\>\\!^{3}\\!S_{1}\to\>\\!^{3}\\!S_{1})\\}$ | 14 | _1.24_
$\\{g_{0}(\>\\!^{3}\\!D_{1}),\;g_{1}(\>\\!^{3}\\!D_{1}),\;g_{0}^{(1)}(\>\\!^{3}\\!S_{1}),\;g_{1}(\>\\!^{3}\\!S_{1}),\;\gamma(\>\\!^{3}\\!S_{1}\to\>\\!^{3}\\!S_{1})\\}$ | 15 | 1.20
$\\{g_{0}(\>\\!^{3}\\!D_{1}),\;g_{1}(\>\\!^{3}\\!D_{1}),\;g_{0}(\>\\!^{3}\\!S_{1}),\;\gamma(\>\\!^{3}\\!S_{1}\to\>\\!^{3}\\!S_{1})\\}$ | 14 | 1.20
$\\{g_{0}(\>\\!^{3}\\!D_{1}),\;g_{1}(\>\\!^{3}\\!D_{1}),\;g_{0}(\>\\!^{3}\\!S_{1}),\;g_{0}^{(1)}(\>\\!^{3}\\!S_{1}),\;g_{1}(\>\\!^{3}\\!S_{1}),\;\gamma(\>\\!^{3}\\!S_{1}\to\>\\!^{3}\\!S_{1})\\}$ | 16 | 1.22
$\\{g_{0}(\>\\!^{3}\\!D_{1}),\;g_{1}(\>\\!^{3}\\!D_{1}),\;g_{0}(\>\\!^{3}\\!S_{1}),\;g_{1}(\>\\!^{3}\\!S_{1}),\;\gamma(\>\\!^{3}\\!S_{1}\leftrightarrow\>\\!^{3}\\!D_{1}),\;\gamma(\>\\!^{3}\\!S_{1}\to\>\\!^{3}\\!S_{1})\\}$ | 16 | 1.16
$\\{g_{0}(\>\\!^{3}\\!D_{1}),\;g_{1}(\>\\!^{3}\\!D_{1}),\;g_{0}(\>\\!^{3}\\!S_{1}),\;g_{1}(\>\\!^{3}\\!S_{1}),\;\gamma(\>\\!^{3}\\!S_{1}\to\>\\!^{3}\\!S_{1}),\;\gamma^{(1)}(\>\\!^{3}\\!S_{1}\to\>\\!^{3}\\!S_{1})\\}$ | 16 | 1.21
$\\{g_{0}(\>\\!^{3}\\!D_{1}),\;g_{1}(\>\\!^{3}\\!D_{1}),\;g_{0}(\>\\!^{3}\\!S_{1}),\;g_{1}(\>\\!^{3}\\!S_{1})\\}$ | 14 | _1.35_
$2^{+}$ |  eq. 12 (CM, pole 2) | $\\{g_{2}(\>\\!^{1}\\!D_{2}),\;g_{2}(\>\\!^{3}\\!D_{2}),\;\gamma(\>\\!^{1}\\!D_{2}\to\>\\!^{1}\\!D_{2})\\}$ | 15 | 1.20
$\\{g_{2}^{(1)}(\>\\!^{1}\\!D_{2}),\;g_{2}(\>\\!^{3}\\!D_{2}),\;\gamma(\>\\!^{1}\\!D_{2}\to\>\\!^{1}\\!D_{2})\\}$ | 15 | 1.20
$\\{g_{2}(\>\\!^{1}\\!D_{2}),\;g_{2}^{(1)}(\>\\!^{3}\\!D_{2}),\;\gamma(\>\\!^{1}\\!D_{2}\to\>\\!^{1}\\!D_{2})\\}$ | 15 | 1.20
$\\{g_{2}(\>\\!^{1}\\!D_{2}),\;g_{2}(\>\\!^{3}\\!D_{2})\\}$ | 14 | 1.21
$\\{g_{2}(\>\\!^{1}\\!D_{2}),\;g_{2}(\>\\!^{3}\\!D_{2}),\;\gamma^{(1)}(\>\\!^{1}\\!D_{2}\to\>\\!^{1}\\!D_{2})\\}$ | 15 | 1.20
(CM, thresh.) | $\\{g_{2}(\>\\!^{1}\\!D_{2}),\;g_{2}(\>\\!^{3}\\!D_{2}),\;\gamma(\>\\!^{1}\\!D_{2}\to\>\\!^{1}\\!D_{2})\\}$ | 15 | 1.20
(w/o CM) | $\\{g_{2}(\>\\!^{1}\\!D_{2}),\;g_{2}(\>\\!^{3}\\!D_{2}),\;\gamma(\>\\!^{1}\\!D_{2}\to\>\\!^{1}\\!D_{2})\\}$ | 15 | 1.20
$0^{-}$ |  eq. 13 (CM, thresh.) | $\\{\gamma(\>\\!^{3}\\!P_{0}\to\>\\!^{3}\\!P_{0})\\}$ | 15 | 1.20
(CM, pole 3) | $\\{g_{3}(\>\\!^{3}\\!P_{0}),\;\gamma(\>\\!^{3}\\!P_{0}\to\>\\!^{3}\\!P_{0})\\}$ | 16 | 1.22
no $J^{P}=0^{-}$ | - | 14 | 1.20
$1^{-}$ |  eq. 14 (CM, thresh.) | $\\{\gamma(\>\\!^{1}\\!P_{1}\to\>\\!^{1}\\!P_{1}),\;\gamma(\>\\!^{3}\\!P_{1}\to\>\\!^{3}\\!P_{1})\\}$ | 15 | 1.20
$\\{\gamma(\>\\!^{1}\\!P_{1}\to\>\\!^{1}\\!P_{1}),\;\gamma(\>\\!^{3}\\!P_{1}\to\>\\!^{3}\\!P_{1}),\;\gamma(\>\\!^{1}\\!P_{1}\leftrightarrow\>\\!^{3}\\!P_{1})\\}$ | 16 | 1.20
(w/o CM) | $\\{\gamma(\>\\!^{1}\\!P_{1}\to\>\\!^{1}\\!P_{1}),\;\gamma(\>\\!^{3}\\!P_{1}\to\>\\!^{3}\\!P_{1})\\}$ | 15 | 1.20
$2^{-}$ |  eq. 15 (CM, thresh.) | $\\{\gamma(\>\\!^{3}\\!P_{2}\to\>\\!^{3}\\!P_{2})\\}$ | 15 | 1.20
$\\{\gamma(\>\\!^{3}\\!P_{2}\to\>\\!^{3}\\!P_{2}),\;\gamma^{(1)}(\>\\!^{3}\\!P_{2}\to\>\\!^{3}\\!P_{2})\\}$ | 16 | 1.19
$\\{\gamma^{(1)}(\>\\!^{3}\\!P_{2}\to\>\\!^{3}\\!P_{2})\\}$ | 15 | 1.21
Table 3: Amplitude variations and the corresponding fit results. The first row
of each block belonging to one combination of $J^{P}$ corresponds to the
reference parameterization described in section .3 (last column printed in
bold face). The free parameters belong to the respective referenced equation.
The phase space prescription is given in parentheses. For each variation in a
given $J^{P}$ block of the $K$-matrix, all other blocks correspond to the
reference parameterization. Couplings and polynomial coefficients not listed
under “Free parameters” are fixed to zero. Amplitudes, where the entry in the
last column is printed in italics, were removed from the calculation of the
final result due to a comparatively large $\chi^{2}$-minimum (see also section
.5).

###  .5 Parameterization with $g_{1}(\>\\!^{3}\\!D_{1})=0$

Figure 6: Upper panel: The $\chi^{2}$ value obtained after minimization as a
function of $g_{1}({\>\\!{}^{3}\\!D_{1}})$. The vertical grey band shows the
value and uncertainty on this parameter obtained from the determination of the
reference parameterization, $g_{1}({\>\\!{}^{3}\\!D_{1}})=(-6.40\pm 1.36)\cdot
a_{t}$. Lower panel: The width of the pole
$\Gamma=-2\;\mathrm{Im}\sqrt{s_{0}}$ as a function of
$g_{1}({\>\\!{}^{3}\\!D_{1}})$. The open circular markers indicate values of
$g_{1}({\>\\!{}^{3}\\!D_{1}})$ where a fit was performed, the solid red-black
line in both panels is for visual guidance only.

In one of the parameterization variations, the $K$-matrix pole-coupling
parameter to the $\>\\!{}^{3}\\!D_{1}$ amplitude for the pole with a higher
mass is fixed to $g_{1}(\>\\!^{3}\\!D_{1})=0$. This differs from the reference
parameterization in section .3 only by the change to this parameter. After
minimization, we find $\chi^{2}/N_{\mathrm{dof}}=\frac{99.24}{94-14}=1.24$,
slightly larger than the reference parameterization but not unreasonably
large. The imaginary part of the higher $J^{P}=1^{+}$ resonance pole found in
this amplitude around $a_{t}\mathrm{Re}\sqrt{s_{0}}\approx 0.437$ is much
smaller compared to the other parameterizations, which requires further
investigation.

This parameterization and the reference parameterization can be interpolated
by adjusting $g_{1}(\>\\!^{3}\\!D_{1})$ from the value in the reference
parameterization $g_{1}({\>\\!{}^{3}\\!D_{1}})=(-6.40\pm 1.36)\cdot a_{t}$ to
zero. We adjust $g_{1}({\>\\!{}^{3}\\!D_{1}})$ in small steps and determine
the remaining parameters using the $\chi^{2}$ minimization procedure.

In Fig. 6, the $\chi^{2}$ value and $\Gamma=-2\>\mathrm{Im}\sqrt{s_{0}}$ are
shown as a function of $g_{1}({\>\\!{}^{3}\\!D_{1}})$. As expected, we see a
clear minimum in the $\chi^{2}$ at the value of $g_{1}({\>\\!{}^{3}\\!D_{1}})$
obtained from the reference parameterization. There is a symmetry in
$g_{1}({\>\\!{}^{3}\\!D_{1}})\to-g_{1}({\>\\!{}^{3}\\!D_{1}})$. $K_{ij}$ is
invariant under certain sign changes, in particular when both
$g_{1}({\>\\!{}^{3}\\!S_{1}})$ and $g_{1}({\>\\!{}^{3}\\!D_{1}})$ change sign
simultaneously. The relative sign of $g_{1}({\>\\!{}^{3}\\!S_{1}})$ and
$g_{1}({\>\\!{}^{3}\\!D_{1}})$ is relevant, as are the signs of the $\gamma$
parameters. The imaginary part of the pole position determined from the minima
is approximately quadratic in $g_{1}({\>\\!{}^{3}\\!D_{1}})$.

We see that the spectra clearly favour a non-zero value, and that
$g_{1}({\>\\!{}^{3}\\!D_{1}})=0$ is not special. However, at the level of a
few standard deviations, we cannot exclude a vanishingly small value. This is
reflected in the quoted value of the width in the main text, $\Gamma=(5\pm 3)$
MeV.

A better determination of the width of this pole could be obtained by
utilizing larger volumes. This would provide additional non-interacting meson-
meson energies in the region of this state. The avoided level crossings
induced by the resonance would then improve the amplitude determination.