# JVS-MuSiC: Japanese multispeaker singing-voice corpus

###### Abstract

Thanks to developments in machine learning techniques, it has become possible
to synthesize high-quality singing voices of a single singer [1, 2]. An open
multispeaker singing-voice corpus would further accelerate the research in
singing-voice synthesis. However, conventional singing-voice corpora [3, 4]
only consist of the singing voices of a single singer. We designed a Japanese
multispeaker singing-voice corpus called “JVS-MuSiC” with the aim to analyze
and synthesize a variety of voices. The corpus consists of 100 singers’
recordings of the same song, Katatsumuri, which is a Japanese children’s song.
It also includes another song that is different for each singer. In this
paper, we describe the design of the corpus and experimental analyses using
JVS-MuSiC. We investigated the relationship between 1) the similarity of
singing voices and perceptual oneness of unison singing voices and between 2)
the similarity of singing voices and that of speech. The results suggest that
1) there is a positive and moderate correlation between singing-voice
similarity and the oneness of unison and that 2) the correlation between
singing-voice similarity and speech similarity is weak. This corpus is freely
available online.

Index Terms: voice corpus, Japanese, speech synthesis

##  1 Introduction

Thanks to developments in machine learning techniques, it has become possible
to synthesize high-quality singing voices of a single singer [1, 2]. An open
multispeaker singing-voice corpus would further accelerate the research in
singing-voice synthesis. However, conventional singing-voice corpora [3, 4]
only consist of the singing voices of a single singer. We designed a Japanese
multispeaker singing-voice corpus called “JVS-MuSiC” with the aim to analyze
and synthesize a variety of voices. The corpus consists of 100 singers’
recordings of the same song, Katatsumuri, which is a Japanese children’s song.
It also includes another song that is different for each singer. In this
paper, we describe the design of the corpus and experimental analyses using
JVS-MuSiC. We investigated the relationship between 1) the similarity of
singing voices and perceptual oneness of unison singing voices and between 2)
the similarity of singing voices and that of speech. The results suggest that
1) there is a positive and moderate correlation between singing-voice
similarity and the oneness of unison and that 2) the correlation between
singing-voice similarity and speech similarity is weak. This corpus is freely
available online.

##  2 Current Japanese singing-voice corpora

All Japanese singing-voice corpora consist of only a single singer’s voices.
The singing-voice corpus included in the demo of HTS [3] consists of 31
Japanese children’s songs sung by a single female singer. JSUT-song [4]
consists of 25 songs in HTS sung by a different female singer. There is also a
singing-voice database of synthesized singing voices called Tohoku Kiritan’s
singing-voice database [5], which consists of 50 songs sung by Tohoku Kiritan,
who is a female character of VOICEROID.

##  3 Design of JVS-MuSiC

###  3.1 Structures

The directory structures of the corpus are listed below. The singer name is
formatted as jvs[SPKR_ID], indicating the speaker ID with the range of 1
through 100.

The main purpose of JVS-MuSiC is to cover various singers’ singing voices to
enable the analysis and synthesis of the personality of singing voices. The
100 singers are the same as those of JVS corpus [6]; thus, it is also possible
to investigate the relationship between speech and singing voices. The
following sections describe how we designed the corpus. .1 . .2  jvs001. .3
song_common. .4  wav. .5  raw.wav. .5  modified.wav. .5  modified_grouped.wav.
.4  mpd. .5  modified.mpd. .5  modified_grouped.mpd. .3  song_unique. .4  wav.
.5  raw.wav. .2  jvs002. .2  …. .2  jvs100. .2  similarity. .3
similarity_{name of group}.csv. .2  oneness. .3  oneness_{name of group}.csv.
.2  singer_info.txt.

####  3.1.1 Raw voice (raw.wav)

Recorded wav files of Katatsumuri and singer-dependent songs are stored in
song_common/wav/ and song_unique/wav/, respectively. Each singer sang
Katatsumuri in his or her favorite key and tempo; thus, the key and tempo vary
among singers. The key and tempo are not completely consistent in each
recording because the singers did not sing along with an example recording or
a guide melody sound.

####  3.1.2 Modified voices

There are two versions of modified voices.

Figure 1: Scatter plot of key and tempo.

  * •

modified.wav
We determined the closest key and tempo for each singer and modified raw
voices using the singing-voice modification software Melodyne [7], as if the
singer had sung in the key and tempo accurately. Figure 1 shows the
distribution of the keys and tempos of modified voices. There is a four-beat
silence at the beginning of each file. The key and tempo labels of the
modified voices are stored in mpd_label.txt, which consists of speaker ID,
gender, tempo (BPM: beats per minute), and key number (how many semitones
higher than the lowest one).

  * •

modified_grouped.wav
For facilitating analysis, we created a grouped version of modified wav files
and divided the 100 singers into six groups (three for each gender) by key. We
used Melodyne to unify the key within the group and the tempo within all
groups (100 BPM). We chose Melodyne for this purpose because it produces
empirically less change in voice timbre than with a time stretching- and
resampling-based method. Table 1 lists the keys and numbers of singers for the
groups.

Table 1: Description of singer groups Group | Key | # of singers
---|---|---
Male-low |  B$\flat$ | 17
Male-middle |  D$\flat$ | 16
Male-high |  E$\flat$ | 16
Female-low | A | 17
Female-middle |  B$\flat$ | 17
Female-high |  D$\flat$ | 17

####  3.1.3 MPD file (*.mpd)

modified.mpd and modified_grouped.mpd are Melodyne project documents that were
used to create the corresponding modified voices. We used Melodyne 4 Assistant
[7] to modify singing voices. The editor was not a professional engineer, but
had some experience in music production including vocal editing as an amateur.
We now describe the modification steps.

  1. 1.

Determination of key and tempo

We determined the closest key and tempo for each singing voice by listening to
the voice and looking at the graphical user interface.

  2. 2.

Editing pitch

We used the correct pitch macro to modify the pitch of musical notes. We set
both pitch center and pitch drift to 100% then manually checked and modified
all notes considering the perceptual naturalness.

  3. 3.

Editing time

We used the time tool to manually modify the onset and offset of all musical
notes considering the perceptual naturalness.

####  3.1.4 Similarity and oneness matrices (*.csv)

These csv files are the similarity and oneness matrices of the experimental
analysis mentioned in Section 4.

###  3.2 Recording

We hired 100 native Japanese professional speakers; 49 males and 51 females.
Their voices were recorded in a recording studio, and the recording for each
speaker was done within one day. The recordings were controlled by a
professional sound director. The voices were originally sampled at 48 kHz and
downsampled to 24 kHz by the Speech Signal Processing Toolkit [8], and the
16-bit/sample RIFF WAV format was used. These settings are the same as those
in the JVS corpus [6]. The total duration of the 100 files of the common song
was 49 min and 23 sec, and that of singer-dependent songs was 88 min and 3
sec.

##  4 Experimental analysis

###  4.1 Experimental conditions

We evaluated the inter-speaker similarity and oneness of unison for many pairs
of singers. We used grouped files because the key should be unified when
annotating similarity and must be unified when producing unison voices and
because too much pitch shifting may cause artifacts and changes in voice
timbre. For annotating perceptual similarity scores, we used 9.6-sec samples,
which were made by concatenating two singers’ voice samples of the same
musical phrase of 4.8 sec (eight beats). We followed Saito et al.’s study
[9]in which each listener scored the perceptual similarity for each pair of
speakers from $-3$ (completely different) to $+3$ (very similar). For the
evaluation of the oneness of unison, we used 9.6-sec (16 beats) samples of two
singers’ unison singing-voice samples. The unison voices were obtained by
mixing two singers’ voices. We balanced the volume of the two voices by
equalizing the mean squared amplitude when mixing. The listeners scored the
separateness of unison (i.e., how much the two singers’ voices were heard
separately, not as a united one) from 1 (heard as one) to 5 (heard
separately). The oneness of unison was obtained by inverting the sign of
separateness of unison. A final score for each speaker pair was obtained by
averaging listeners’ scores. For analysis, we normalized the measured values
into the range (0, 1), used the crowdsourcing platform “Lancers” [10], and
gathered ten participants for each pair of singers.

###  4.2 Correlation between similarity and oneness of unison

Figure 2 is the scatter plot of average similarity and oneness of unison for
all 784 ($=4{\binom{17}{2}}+2{\binom{16}{2}}$) pairs. The correlation
coefficient is 0.45 and $p$-value is $1.4\times 10^{-39}$. Table 2 shows the
group-wise results, which suggest that, to some extent, a pair of singers with
similar voices produces a united unison voice, which is often considered to
sound beautiful.

Figure 2: Scatter plot of average similarity and oneness of unison. Correlation coefficient is 0.45 and $p$-value is $1.4\times 10^{-39}$. Table 2: Correlation coefficients ($r$) and $p$-values for all groups Group | $r$ |  $p$-value
---|---|---
Male-low | $0.28$ | $8.2\times 10^{-4}$
Male-middle | $0.58$ | $<10^{-10}$
Male-high | $0.53$ | $3.6\times 10^{-10}$
Female-low | $0.30$ | $3.9\times 10^{-4}$
Female-middle | $0.36$ | $1.9\times 10^{-5}$
Female-high | $0.51$ | $2.0\times 10^{-10}$
All | $0.45$ | $<10^{-10}$

###  4.3 Correlation between singing-voice similarity and speech similarity

Figure 3: Scatter plot of average similarity of singing voice and speech.
Correlation coefficient is 0.17 and $p$-value is $1.9\times 10^{-6}$.

Figure 3 is the scatter plot of the average similarity of singing voice and
similarity of speech [6] for the 784 pairs. The correlation coefficient is
0.17 and $p$-value is $1.9\times 10^{-6}$. Although part of this low
correlation may be due to the change in voice timbre caused by pitch shifting,
this result suggests that a person’s singing voice and speech are
fundamentally different or that listeners perceive singing voices and speech
in different manners.

##  5 Conclusion

We introduced a Japanese multispeaker singing-voice corpus called JVS-MuSiC.
This corpus was designed for multispeaker singing-voice analysis and synthesis
and can be used for research in unison singing voices. We analyzed the
relationship between the similarity of singer pairs and unison singing voices.
The experimental results suggest that there is a positive and moderate
correlation between the similarity and oneness of unison. We also compared
singing-voice similarity to speech similarity and found that the correlation
between them was weak.

The similarity and oneness matrices are licensed with CC BY-SA 4.0. The audio
data and MPD files may be used for

  * •

Research by academic institutions

  * •

Non-commercial research, including research conducted within commercial
organizations

  * •

Personal use, including blog posts.

Our project page at
https://sites.google.com/site/shinnosuketakamichi/research-topics/jvs_music
describes the terms for commercial use.

Acknowledgments: Part of this work was supported by the SECOM Science and
Technology Foundation and the GAP foundation program of the University of
Tokyo.

## References

  * [1] M. Nishimura, K. Hashimoto, K. Oura, Y. Nankaku, and K. Tokuda,  “Singing voice synthesis based on deep neural networks,”  in Proc. INTERSPEECH, San Francisco, U.S.A., Sep. 2016, pp. 2478–2482.
  * [2] M. Blaauw and J. Bonada,  “A neural parametric singing synthesizer modeling timbre and expression from natural songs,”  Applied Sciences, vol. 7, no. 12, Dec. 2017.
  * [3] “HMM-based speech synthesis system (HTS),” http://hts.sp.nitech.ac.jp/.
  * [4] “JSUT-song,” https://sites.google.com/site/shinnosuketakamichi/publication/jsut-song.
  * [5] “Kiritan singing database,” https://zunko.jp/kiridev/login.php.
  * [6] S. Takamichi, K. Mitsui, Y. Saito, T. Koriyama, N. Tanji, and H. Saruwatari,  “JVS corpus: free Japanese multi-speaker voice corpus,”  arXiv, vol. abs/1908.06248, 2019.
  * [7] “Celemony — what is melodyne?,” https://www.celemony.com/en/melodyne/what-is-melodyne.
  * [8] “Speech signal processing toolkit (SPTK) http://sp-tk.sourceforge.net/,”  .
  * [9] Y. Saito, S. Takamichi, and H. Saruwatari,  “DNN-based speaker embedding using subjective inter-speaker similarity for multi-speaker modeling in speech synthesis,”  in Proc. SSW10. Vienna, Austria, Sep. 2019.
  * [10] “Lancers,” https://www.lancers.jp/.


# Bound isoscalar axial-vector $bc\bar{u}\bar{d}$ tetraquark $T_{bc}$ from
lattice QCD using two-meson and diquark-antidiquark variational basis

M. Padmanath<EMAIL_ADDRESS>The Institute of Mathematical Sciences, a
CI of Homi Bhabha National Institute, Chennai, 600113, India  Archana
Radhakrishnan<EMAIL_ADDRESS>Department of Theoretical
Physics, Tata Institute of Fundamental Research,
Homi Bhabha Road, Mumbai 400005, India  Nilmani Mathur
<EMAIL_ADDRESS>Department of Theoretical Physics, Tata Institute
of Fundamental Research,
Homi Bhabha Road, Mumbai 400005, India

###### Abstract

We report a lattice QCD study of the heavy-light meson-meson interactions with
an explicitly exotic flavor content $bc\bar{u}\bar{d}$, isospin $I\\!=\\!0$,
and axialvector $J^{P}=1^{+}$ quantum numbers in search of possible tetraquark
bound states. The calculation is performed at four values of lattice spacing,
ranging $\sim$0.058 to $\sim$0.12 fm, and at five different values of valence
light quark mass $m_{u/d}$, corresponding to pseudoscalar meson mass $M_{ps}$
of about 0.5, 0.6, 0.7, 1.0, and 3.0 GeV. The energy eigenvalues in the
finite-volume are determined through a variational procedure applied to
correlation matrices built out of two-meson interpolating operators as well as
diquark-antidiquark operators. The continuum limit estimates for
$D\bar{B}^{*}$ elastic $S$-wave scattering amplitude are extracted from the
lowest finite-volume eigenenergies, corresponding to the ground states, using
amplitude parametrizations supplemented by a lattice spacing dependence. Light
quark mass $m_{u/d}$ dependence of the $D\bar{B}^{*}$ scattering length
($a_{0}$) suggests that at the physical pion mass
$a_{0}^{phys}=+0.57(^{+4}_{-5})(17)$ fm, which clearly points to an attractive
interaction between the $D$ and $\bar{B}^{*}$ mesons that is strong enough to
host a real bound state $T_{bc}$, with a binding energy of
$-43(_{-7}^{+6})(_{-24}^{+14})$ MeV with respect to the $D\bar{B}^{*}$
threshold. We also find that the strength of the binding decreases with
increasing $m_{u/d}$ and the system becomes unbound at a critical light quark
mass $m^{*}_{u/d}$ corresponding to $M^{*}_{ps}=2.73(21)(19)$ GeV.

††preprint: IMSc/23/05, TIFR/TH/23-14

The discovery of a doubly charmed tetraquark111We follow the nomenclature that
a “tetraquark” refers to any bound state or resonance with dominant four-quark
Fock component, whether it is a compact four-quark object or a two-meson
molecule or a mixture of both., $T_{cc}$, marks an important milestone [1] in
spectroscopy of hadrons. Phenomenologically, doubly heavy tetraquarks in the
heavy quark limit are long hypothesized to form deeply bound states [2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12] with binding energy $\mathcal{O}$(100 MeV) with
respect to the elastic strong decay threshold. While doubly bottom tetraquarks
are suitable candidates for such deeply bound states, as predicted by multiple
lattice QCD calculations [13, 14, 15, 16, 17, 18], $T_{cc}$ is found to be
$360$ keV below the lowest two-meson threshold ($D^{0}D^{*+}$). A handful of
recent experimental developments involving multiple heavy quark production
such as the recent discoveries of $\Xi_{cc}$ [19], $T_{cc}$ [1], reports of
tri-$J/\psi$ [20], associated $J/\psi\Upsilon$ [21], and di-$\Upsilon$ [22]
productions, and recent proposals of inclusive search strategies [23, 24]
augment promising prospects for the doubly heavy hadron sector in the near
future. In light of these advancements, a doubly heavy tetraquark with a
bottom and a charm quark with a valence quark configuration $T_{bc}\equiv
bc\bar{u}\bar{d}$ is going to be one of the most sought-after hadron in this
decade [25]. In this work, using lattice QCD calculations, we show a clear
evidence of an attractive interaction between the $D$ and $\bar{B}^{*}$ mesons
that is strong enough to host a real bound state $T_{bc}$. This finding will
further boost the search for such bottom-charm tetraquarks.

The phenomenological picture on deeply bound doubly heavy tetraquarks is based
on a compact heavy diquark-light antidiquark interpretation [14, 26], whereas
the shallow binding energy of $T_{cc}$ could possibly be a reflection of its
dominant noncompact molecular nature [8, 27]. Bottom-charm tetraquarks form an
intermediate platform, where there could be complicated interplay between
these pictures. A collective and refined knowledge of the low energy spectra
in all these three doubly heavy systems ($T_{bb}$, $T_{bc}$ and $T_{cc}$)
could culminate in a deeper understanding of strong interaction dynamics
across a wide quark mass regime spanning from charm to the bottom quarks. The
isoscalar bottom-charm tetraquarks with quantum numbers [$I(J^{P})=0(1^{+})$]
have been investigated previously both using lattice [28, 29, 30] and
nonlattice methodologies [5, 6, 8, 9, 31, 32, 11, 12, 26, 33, 34, 35, 36, 37,
38, 39, 40]. The predictions from nonlattice approaches are quite scattered
from being unbound to deeply bound, whereas the difference in conclusions from
the three existing lattice QCD investigations[28, 29, 30] call for more
detailed efforts in this regard.

In this work, we perform a lattice QCD simulation of coupled $D\bar{B}^{*}$
and $\bar{B}D^{*}$ two-meson channels222We work in the isosymmetric limit with
no QED effects and $m_{u}=m_{d}$. Hence we choose to call the degenerate
($D^{+}B^{-},~{}D^{0}\bar{B}^{0}$) threshold as $D\bar{B}$, and equivalently
for others like $D\bar{B}^{*}$, $\bar{B}D^{*}$ and $D^{*}\bar{B}^{*}$. that
are the relevant lowest two strong decay thresholds, in the order of
increasing energies, $E_{D\bar{B}^{*}}=M_{\bar{B}^{*}}+M_{D}$ and
$E_{\bar{B}D^{*}}=M_{\bar{B}}+M_{D^{*}}$, where $M_{h}$ is the mass of the
hadron $h$. The extracted finite-volume ground state energies are utilized to
constrain the continuum extrapolated elastic $D\bar{B}^{*}$ scattering
amplitudes following the Lüscher’s finite-volume prescription [41, 42]. The
light quark mass $m_{u/d}$ dependence of the extracted amplitudes suggests a
binding energy of $-43(^{+6}_{-7})(^{+14}_{-24})$ MeV for the
$bc\bar{u}\bar{d}$ tetraquark pole with respect to $E_{D\bar{B}^{*}}$ at the
physical point $m_{u/d}^{phys}$.

Label |  Symbol |  $a~{}[fm]$ |  $N_{s}^{3}\times N_{t}$ |  $M_{ps}^{sea}$
---|---|---|---|---
$S_{1}$ |  |  0.1207(11) |  $24^{3}\times 64$ |  305
$S_{2}$ |  |  0.0888(8) |  $32^{3}\times 96$ |  312
$S_{3}$ |  |  0.0582(4) |  $48^{3}\times 144$ |  319
$L_{1}$ |  |  0.1189(9) |  $40^{3}\times 64$ |  217
Table 1: Relevant details of the lattice QCD ensembles used. The lattice
spacing estimates are measured using the $r_{1}$ parameter [43]. $L_{1}$
refers to large spatial volume, and $S_{1},~{}S_{2}$, and $S_{3}$ refer to
small spatial volumes.

Lattice setup: We use four lattice QCD ensembles (see Table 1 for relevant
details) with $N_{f}=2+1+1$ dynamical Highly Improved Staggered Quark (HISQ)
fields generated by the MILC collaboration [43]. The charm and strange quark
masses in the sea are tuned to their respective physical values, whereas the
dynamical light quark masses correspond to sea pion masses as listed in Table
1. We utilize a partially quenched setup on these configurations with valence
quark fields up to the charm quark masses realized using an overlap fermion
action as in Refs. [44, 45]. We employ a nonrelativistic QCD (NRQCD)
Hamiltonian [46] for the bottom quark. Following the Fermilab prescription
[47], the bare charm [48, 49] and bottom [50] quark masses on each ensemble
are tuned using the kinetic mass of spin averaged $1S$ quarkonia
$\\{a\overline{M}_{kin}^{\bar{Q}Q}={3\over 4}aM_{kin}(V)+{1\over
4}aM_{kin}(PS)\\}$ determined on the respective ensembles. The bare strange
quark mass is set by equating the lattice estimate for the fictitious
pseudoscalar $\bar{s}s$ meson mass to 688.5 MeV [51].

For the valence $m_{u/d}$, we investigate five different cases: three
unphysical quark masses [corresponding to approximate pseudoscalar meson
masses $M_{ps}\sim$0.5, 0.6, and 1.0 GeV], the strange quark mass
[$M_{ps}\sim$0.7 GeV] and the charm quark mass [$M_{ps}\sim$3.0 GeV]. We
evaluate the finite-volume spectrum for all these five quark masses on all
four ensembles, investigate the scattering of $D$ and $\bar{B}^{*}$ mesons in
all five cases and then extract the $m_{u/d}$ (otherwise $M_{ps}$) dependence
of the scattering parameters.

Interpolators and measurements: The finite-volume spectrum is determined from
Euclidean two-point correlation functions $\mathcal{C}_{ij}(t)$, between
interpolating operators $\mathcal{O}_{i,j}(\mathbf{x},t)$ with desired quantum
numbers, given by

$\mathcal{C}_{ij}(t)=\sum_{\mathbf{x}}\left<\mathbb{O}_{j}^{\dagger}(0)\mathcal{O}_{i}(\mathbf{x},t)\right>\approx\sum_{n}\mathbb{Z}_{j}^{n\dagger}Z_{i}^{n}e^{-E^{n}t}.$
(1)

Here $E^{n}$ is the energy of the $n^{th}$ state and
$Z_{i}^{n}=\bra{0}\mathcal{O}_{i}\ket{n}$ is the operator-state overlap
between the sink operator $\mathcal{O}_{i}$ and state $n$. We use $\mathbb{O}$
and $\mathbb{Z}$ to represent the source operator and overlaps to distinguish
them from that for the sink as we follow a wall-source to point-sink
construction in our $\mathcal{C}_{ij}$ evaluations. This is well-established
procedure in ground state energy determination, despite the non-Hermitian
setup in Eq. (1) (see Refs. [14, 28, 52, 16, 29, 53] for details). We use the
following set of linearly independent, yet Fierz related [54], operators,

$\displaystyle\mathcal{O}_{1}(x)$ $\displaystyle=$
$\displaystyle[\bar{u}(x)\gamma_{i}b(x)][\bar{d}(x)\gamma_{5}c(x)]$
$\displaystyle-[\bar{d}(x)\gamma_{i}b(x)][\bar{u}(x)\gamma_{5}c(x)]$
$\displaystyle\mathcal{O}_{2}(x)$ $\displaystyle=$
$\displaystyle[\bar{u}(x)\gamma_{5}b(x)][\bar{d}(x)\gamma_{i}c(x)]$ (2)
$\displaystyle-[\bar{d}(x)\gamma_{5}b(x)][\bar{u}(x)\gamma_{i}c(x)]$
$\displaystyle\mathcal{O}_{3}(x)$ $\displaystyle=$
$\displaystyle(\bar{u}(x)^{T}\Gamma_{5}\bar{d}(x)-\bar{d}(x)^{T}\Gamma_{5}\bar{u}(x))(b(x)\Gamma_{i}c(x)).$

$\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ are two-meson operators of the type
$D\bar{B}^{*}$ and $\bar{B}D^{*}$, respectively. $\mathcal{O}_{3}$ is a
diquark-antidiquark type operator. Here $\Gamma_{k}=C\gamma_{k}$ with
$C=i\gamma_{y}\gamma_{t}$ being the charge conjugation matrix and the diquarks
(antidiquarks) in the color antitriplet (triplet) representations. Other high
lying two-meson ($D^{*}\bar{B}^{*}$) and three-meson ($D\bar{B}\pi$)
interpolators are ignored in this analysis as they are sufficiently high in
energy to have any effects on the extracted ground states. Bilocal two-meson
interpolators with nonzero internal meson momenta are also not considered,
which would be an important step ahead [55]. We also compute two-point
correlation functions for $\bar{B}$, $\bar{B}^{*}$, $D$, and $D^{*}$ mesons,
using standard local quark bilinear interpolators ($\bar{q}~{}\Gamma~{}Q$)
with spin structures $\Gamma\sim\gamma_{5}$ and $\gamma_{i}$ for pseudoscalar
and vector quantum numbers, respectively.

Analysis: The correlation matrices $\mathcal{C}$ evaluated for the basis in
Eq. (2) are analyzed following a variational procedure [56] by solving the
generalized eigenvalue problem (GEVP),
$\mathcal{C}(t)v^{n}(t)=\lambda^{n}(t)\mathcal{C}(t_{0})v^{n}(t)$. The
eigenvalues in the large time limit represent the time evolution the low lying
eigenenergies $\mathcal{E}^{n}$ as $\lim_{t\to\infty}\lambda^{n}(t)\sim
A_{n}e^{-\mathcal{E}^{n}t}$. The corresponding eigenvectors $v^{n}(t)$ are
related to the operator-state-overlaps $Z_{i}^{n}$.

Eigenenergy extraction proceeds via fitting the eigenvalue correlators,
$\lambda_{n}(t)$, or the ratios
$R^{n}(t)=\lambda^{n}(t)/\mathcal{C}_{m_{1}}(t)\mathcal{C}_{m_{2}}(t)$, with
the expected asymptotic exponential behaviour. Here, $\mathcal{C}_{m_{i}}$ is
the two-point correlation function for the meson $m_{i}$. $R^{n}(t)$ is
empirically known to efficiently mitigate correlated noise between the product
of two single hadron correlators and the interacting correlator for the two-
hadron system [57]. Note that the automatic cancellation of the additive quark
mass offset, inherent to NRQCD formulation, is an added advantage in using
$R^{n}(t)$ for the fits. The systematics associated with the chosen time
interval for fitting are assessed by varying the lower boundary of the time
interval, $t_{min}$, with a fixed upper boundary, $t_{max}$, chosen
considering the noise level. In Figure 1, we present a representative plot
showing this $t_{min}$ dependence of the energy splittings
($\Delta\mathcal{E}^{n}$) determined from the fits to $\lambda^{n}(t)$ and
$R^{n}(t)$, respectively. The energy differences are evaluated from
$\lambda^{n}(t)$ using the relation
$\Delta\mathcal{E}^{n}=\mathcal{E}^{n}-M_{m_{1}}-M_{m_{2}}$, whereas the fits
to $R^{n}(t)$ directly yield the respective estimates. We choose the optimal
$t_{min}$ values where the two different procedures found to agree
asymptotically in time. We also perform additional checks considering an
alternative quark smearing with different smearing widths to affirm our energy
estimates, see Appendix A. Our final results are based on fitting the ratio
correlators $R^{n}(t)$.

Figure 1: $t_{min}$ dependence of the $\Delta\mathcal{E}^{0}$ fit estimates
determined from the fits to $\lambda^{0}$ and $R^{0}(t)$ for the case
$M_{ps}\sim 700$ MeV in the finest ensemble. Here the superscript 0 refers to
the lowest eigenenergy.  Figure 2: The GEVP eigenenergies in finite-volume for
isoscalar axialvector $bc\bar{u}\bar{d}$ channel on the $L_{1}$ ensemble. Five
panels show the results obtained at various pseudoscalar meson masses
($M_{ps}$) 0.5, 0.6, 0.7, 1.0, and 3.0, respectively.

Finite-volume eigenenergies: In Figure 1, we present the finite-volume GEVP
eigenenergies, in lattice units, for the isoscalar axialvector
$bc{\bar{u}}{\bar{d}}$ channel. The results shown are for the $L_{1}$ ensemble
at the five different $m_{u/d}$ values corresponding to $M_{ps}\sim$ 0.5, 0.6,
0.7, 1.0, and 3.0 GeV. Note that these estimates include the additive offsets
related to the NRQCD-based bottom quark dynamics. The non-interacting two-
meson energy levels corresponding to $D\bar{B}^{*}$ and $\bar{B}D^{*}$
thresholds are indicated as dotted horizontal line segments and those related
to $\bar{B}^{*}D^{*}$ threshold by dashed lines for each $M_{ps}$. The lowest
eigenenergy or the ground state energy is dominated by the $Z_{1}^{0}$ factor
corresponding to $\mathcal{O}_{1}$, that is related to the $D\bar{B}^{*}$
threshold and is determined unambiguously by the operator $\mathcal{O}_{1}$,
see Ref. [58] for details. The most important observation is a clear trend for
negative energy shifts in the ground state energies, which can be observed in
all the cases, indicating a possible attractive interaction between the $D$
and $\bar{B}^{*}$ mesons [59]. A similar pattern of low lying eigenenergies
and ground state negative energyshifts are also observed in other ensembles,
see details in Ref. [58]. We expect that for our choice of interpolating
operators and the accessible values of $t$, $\mathcal{E}^{0}$ will be an
accurate estimate of $E^{0}$, whereas our setup is unable to accurately
estimate excited-state energies. This means the excited eigenenergies
presented in Figure 1 may not correspond to the higher lying elastic
excitations of the $D\bar{B}^{*}$ channel. The location of lowest two non-
interacting finite-volume levels related to the $D\bar{B}^{*}$ channel along
with the ground state eigenenergies are presented in Appendix B. Hence we
focus only on the ground state energies ($\mathcal{E}^{0}\sim E^{0}$) for the
rest of the analysis.

Figure 3: The ground state energies in units of $E_{D\bar{B}^{*}}$ on all
ensembles (see Table 1 for color-symbol conventions) for all $M_{ps}$ values
(different vertical panels).

In Figure 3, we present the ground state energy estimates, in units of
$E_{D\bar{B}^{*}}$, at various $M_{ps}$ and for all the ensembles. These
estimates are evaluated as $E^{n}=\Delta E^{n}+M_{D}+\tilde{M}_{\bar{B}^{*}}$,
where $\Delta E^{n}$ is the estimate from fit to $R^{n}(t)$,
$\tilde{M}_{\bar{B}^{*}}=M_{\bar{B}^{*}}-0.5\overline{M}^{\bar{b}b}_{lat}$
$+0.5\overline{M}^{\bar{b}b}_{phys}$ accounts for the NRQCD additive offset,
and $\overline{M}^{\bar{b}b}_{lat}(\overline{M}^{\bar{b}b}_{phys})$ refers to
the spin averaged mass of the $1S$ bottomonium measured on the lattice
(experiments). The eigenenergies clearly show a trend of decreasing energy
spitting, hence decreasing interaction strength, with increasing $M_{ps}$.
Another interesting feature to note here is the nonzero lattice spacing ($a$)
dependence of the ground state energies on similar volume ensembles ($S_{1}$,
$S_{2}$, $S_{3}$), which we account for through an $a$ dependence in the
parametrized amplitude as discussed below.

In Figure 3, we also indicate the branch point location of the left hand cut
(lhc) arising out of an off-shell pion exchange process for different $M_{ps}$
by horizontal dashed lines. Recent developments point to the importance of lhc
effects on virtual subthreshold poles related to the $T_{cc}$ tetraquark [60].
Such effects on bound states are the subject of future studies where one could
successfully solve the relevant three particle integral equations. This is
beyond the scope of this work and we ignore such effects in our analysis.

$D\bar{B}^{*}$ scattering amplitude: Assuming these energy splittings in
ground states are purely described by elastic scattering in the $D\bar{B}^{*}$
system, we utilize them to constrain the associated $S$-wave scattering
amplitude following Lüscher’s finite-volume prescription [41, 42]. For the low
energy scattering of $D$ and $\bar{B}^{*}$ mesons, where other multi-particle
thresholds are sufficiently high [61], in the $S$-wave leading to the total
angular momentum and parity $J^{P}=1^{+}$, the scattering phase shifts
$\delta_{l=0}(k)$ are related to the finite-volume energy spectrum through
$kcot[\delta_{0}(k)]=2Z_{00}[1;(\frac{kL}{2\pi})^{2})]/(L\sqrt{\pi})$. Here,
$k$($E_{cm}=\sqrt{s}$) is the momentum (energy) in the center of momentum
frame such that
$4sk^{2}=(s-(M_{D}+M_{\bar{B}^{*}})^{2})(s-(M_{D}-M_{\bar{B}^{*}})^{2})$. We
follow the procedure outlined in Appendix B of Ref. [62] to constrain the
amplitude. A sub-threshold pole in the $S$-wave scattering amplitude
$t=({\mathrm{cot}}\delta_{0}-i)^{-1}$ occurs when
$k{\mathrm{cot}}\delta_{0}=\pm\sqrt{-k^{2}}$ for scattering in $S$-wave.

We parametrize the elastic $D\bar{B}^{*}$ scattering amplitude in terms of the
scattering length $a_{0}$ in an effective range expansion near the threshold,
supplemented by a lattice spacing dependence. This is required to incorporate
the cut-off effects observed in the ground state energy estimates. We find
that a linear functional form given by
$k{\mathrm{cot}}\delta_{0}=A^{[0]}+aA^{[1]}$, where $A^{[0]}=-1/a_{0}$,
accommodate the $a$ dependence of the $k{\mathrm{cot}}\delta_{0}$ estimates.
We present the fit results for $A^{[0]}=-1/a_{0}$ in Figure 4 (circle symbols)
as a function of $M_{ps}$ involved. Alternative fitting choices with a leading
quadratic dependence or using only data from non-charm $M_{ps}$ are consistent
with results in Figure 4, see details in Supplemental material [58].

The sign of $A^{[0]}=-1/a_{0}$ determines the fate of the near-threshold pole,
if there exists one. A negative (positive) value of $A^{[0]}$($a_{0}$)
indicates that the interaction potential is strong enough to form a real bound
state[63]. After considering all possible systematics, we find that for all
the non-charm light quark masses, $A^{[0]}$ is negative, which indicates an
attractive interaction strong enough to host a real bound state. On the
contrary, at the charm point, despite the unambiguous negative energy shifts
in the ground states, the attraction is weak to host any real bound state as
suggested by the positive value of $k{\mathrm{cot}}\delta_{0}$ in the
continuum limit. This observation goes in line with the phenomenological
expectation for doubly heavy four quark ($QQ^{\prime}l_{1}l_{2}$) systems with
$m_{l_{1}}=m_{l_{2}}$ that the binding increases with increased relative
heaviness of the heavy quarks with respect to its light quark content[14, 26,
64].

Now we investigate the light quark mass ($m_{u/d}$) or $M_{ps}$ dependence of
the fitted parameters. To this end, we consider three different
parametrizations: a linear dependence
($f_{l}(M_{ps})=\alpha_{c}+\alpha_{l}M_{ps}$) to probe the heavy light quark
mass case, a leading $M_{ps}^{2}$ dependence
($f_{s}(M_{ps})=\beta_{c}+\beta_{s}M_{ps}^{2}$) to assess the chiral
behaviour, and a quadratic dependence
($f_{q}(M_{ps})=\theta_{c}+\theta_{l}M_{ps}+\theta_{s}M_{ps}^{2}$) to quantify
the associated systematics. In Figure 4, we show the fit results for this
$M_{ps}$ dependence in colored bands. The two stars represent $A^{[0]}$ at the
physical $M_{ps}$ (equivalently the physical scattering length $a_{0}^{phys}$)
and the critical $M_{ps}$ at which $A^{[0]}$ changes its sign or above which
the system becomes unbound. It is indeed desired to have more points in the
intermediate mass regime between the charm and the strange

Figure 4: Continuum extrapolated $k{\mathrm{cot}}\delta_{0}$ or
$A^{[0]}=-1/a_{0}$ estimates of the $D\bar{B}^{*}$ system as a function of
$M_{ps}^{2}$ in units of $E_{D\bar{B}^{*}}$. The dotted vertical line close to
the $y$-axis indicates $M_{ps}=M_{\pi}^{phys}$. The two star symbols represent
the amplitude at $M_{ps}=M_{\pi}^{phys}$ and the critical $M_{ps}=M^{*}_{ps}$
above which the system becomes unbound.

quark masses to further constrain the dependence. Yet, our fits demonstrate
near independence in the fit forms as can be observed from the consistency
between the error bands from different fit forms.

Based on the fit form $f_{s}(M_{ps})$ in the chiral regime, we find that the
scattering length of the $D\bar{B}^{*}$ system at the physical light quark
mass ($m_{u/d}^{phys}$), corresponding to $M_{ps}=M_{\pi}^{phys}$, to be

$a_{0}^{phys}=0.57(^{+4}_{-5})(17)\mbox{~{}fm}.$ (3)

The asymmetric errors indicate the statistical uncertainties, whereas the
second parenthesis quotes the systematic uncertainties with the most dominant
contribution arising from the chiral extrapolation fit forms. The positive
value of the scattering length at $M_{ps}=M_{\pi}^{phys}$, at the level of
3$\sigma$ uncertainty, is an unambiguous evidence for the strength of the
$D\bar{B}^{*}$ interaction potential to host a real $bc\bar{u}\bar{d}$
tetraquark bound state $T_{bc}$ with binding energy

$\delta m_{T_{bc}}=-43(^{+6}_{-7})(^{+14}_{-24})\mbox{~{}MeV},$ (4)

with respect to $E_{D\bar{B}^{*}}$. The first parenthesis indicates the
statistical errors and the second one quantifies various systematic
uncertainties added in quadrature. The pseudoscalar meson mass, corresponding
to the critical light quark mass, where $a_{0}$ diverges, is found to be
$M^{*}_{ps}=2.73(21)(19)\mbox{~{}GeV}$. This critical point also signifies
that QCD dynamics within such exotic systems is such that at a heavy light
quark mass the system of quarks perhaps reaches the unitary gas limit, as
indicated by the divergent scattering length [65]. For $M_{ps}\geq
M^{*}_{ps}$, the $T_{bc}$ system remains unbound.

Systematic uncertainties: Our lattice setup together with the bare bottom and
charm quark mass tuning procedure has been demonstrated to reproduce the $1S$
hyperfine splittings in quarkonia with uncertainties less than 6 MeV [53, 50].
We observe the effects of such a mistuning of either of the heavy quark mass
on the energy splittings we extract are very small compared to the statistical
errors. Additionally, our strategy of evaluating the energy differences and
working with mass ratios has also been shown to significantly mitigate the
systematic uncertainties related to heavy quark masses [52, 53]. This is
observed to be the case in this study as well, leading to transparent signals
for the ground state energy as shown in Figures 1, 1, and 3. Our fitting
procedure involves careful and conservative determination of statistical
errors, and uncertainties related to the excited-state-contamination and fit-
window errors. Additional checks using alternative quark smearing procedures
also agree with our energy estimates, see Appendix A. The amplitude
determination and followed extrapolations are performed with results from
varying the fit-windows to evaluate the uncertainties propagated to our final
results. The uncertainties related to the fit forms used in chiral
extrapolations are observed to be the most dominant, as is evident from Figure
4. We assume the partially quenched setup involving ensembles with different
sea pion masses, we utilize, have negligible effects on the energy splittings
we extract for the explicitly exotic $T_{bc}$ tetraquark, similar to what was
observed for heavy hadrons in Refs. [66, 67]. Uncertainty related to scale
setting is also found to be negligible in comparison to the statistical
uncertainties in the energy splittings.

Summary: We have performed a lattice QCD simulation of coupled
$D\bar{B}^{*}$-$\bar{B}D^{*}$ scattering with explicitly exotic flavor
$bc\bar{u}\bar{d}$ and $I(J^{P})=0(1^{+})$. Following a rigorous extraction of
finite-volume eigenenergies and continuum extrapolated elastic $D\bar{B}^{*}$
scattering amplitudes for the five light quark masses studied, we determine
the light quark mass dependence of the elastic $D\bar{B}^{*}$ scattering
length $a_{0}$. We observe unambiguous negative energy shifts between the
interacting and non-interacting finite-volume energy levels. Our estimate for
$a_{0}^{phys}$ (Eq. (3)) is positive, indicating an attractive interaction
between the $D$ and $\bar{B}^{*}$ mesons, which is strong enough to host a
real bound state with binding energy $\delta
m_{T_{bc}}=-43(^{+6}_{-7})(^{+14}_{-24})$ MeV. We find that the strength of
interaction is such that this $bc\bar{u}\bar{d}$ tetraquark becomes unbound at
$M^{*}_{ps}$, which is close to the $\eta_{c}$ meson mass.

In this work, we make several important steps ahead to arrive at robust
inference on the nature of interaction between the $D$ and $\bar{B}^{*}$
mesons. Our main strategy has been to determine the signature of scattering
length in $D\bar{B}^{*}$ interactions at the physical pion mass
$a_{0}^{phys}$. Our results indicate that $a_{0}^{phys}$ is positive, which
suggests that attractive $D\bar{B}^{*}$ interactions are strong enough to host
a real bound state. Further theoretical investigations are desired to reduce
the uncertainties in the binding energy of $T_{bc}$ with respect to
$E_{D\bar{B}^{*}}$. Fully dynamical simulations on several more ensembles,
with different volumes and improvized fermion actions, high statistics studies
with lighter $m_{u/d}$, etc. are a few other improvisations that can further
constrain the relevant scattering amplitude. Additionally, future works
involving Hermitian correlation matrices at rest as well as in moving frames
and those using bilocal two-meson interpolators with nonzero relative meson
momenta aimed at reliable excited state extraction would be a few important
steps ahead [68, 62, 69, 55]. We hope that our observations and inferences in
this work will motivate more theoretical efforts and experimental searches for
such states.

###### Acknowledgements.

This work is supported by the Department of Atomic Energy, Government of
India, under Project Identification Number RTI 4002. M.P. gratefully
acknowledges support from the Department of Science and Technology, India,
SERB Start-up Research Grant No. SRG/2023/001235. We are thankful to the MILC
collaboration and in particular to S. Gottlieb for providing us with the HISQ
lattice ensembles. We thank Sara Collins for a careful reading of the
manuscript. We thank the authors of Ref. [70] for making the TwoHadronsInBox
package utilized in this work. We also thank Gunnar Bali, Parikshit Junnarkar,
Alexey Nefediev, Sayantan Sharma, Stephen R. Sharpe, and Tanishk Shrimal for
discussions. Computations were carried out on the Cray-XC30 of ILGTI, TIFR.
Amplitude analyses were performed on Nandadevi computing cluster at IMSc
Chennai. N. M. would also like to thank A. Salve and K. Ghadiali for
computational support.

Appendix A: Ground state energy plateau.- In this work we have utilized a
wall-source point-sink setup to construct the necessary two-point correlation
functions. The use of such an asymmetric setup implies the effective energies
$aE_{eff}=[ln(C(t)/C(t+\delta t))]/\delta t$ could approach their asymptotic
values as rising-from-below, due to the nonpositive definite nature of the
coefficients in a spectral decomposition, in contrast to a falling-from-above
behaviour in a symmetric setup. In Figure 5, we show the effective mass in
wall-source point-sink setup with the brown-circle ($R^{2}=0$) which rises
from below.

To avoid any ambiguity in selecting the plateau regions of effective masses of
such correlators, we also employ a wall-source box-sink setup [29], which
asymptotically approaches the symmetric limit. In the symmetric limit, the
effective masses are expected to follow a conventional falling-from-above
feature, modulo the statistical noise. To this end, we vary the smearing
radius $R$ to investigate the time dependence of effective mass plateaus in
the approach to the symmetric limit. In Figure 5, we present a comparison of
the effective energy (top) and effective energy splittings (bottom) determined
using different quark sink smearing procedures for the case of $M_{ps}\sim
700$ MeV on the finest ensemble. Clearly the rising-from-below behaviour is
gradually disappearing in the approach to the symmetric limit. It is also
evident that the results at the large time limit from point-sink and box-sink
are very much consistent with each other affirming our assessment on effective
mass plateau in choosing a fit range. Such a behavior of effective masses with
varying smearing radii was also observed in Ref. [29]. In the large time
limit, where the signal quality is still good, all of sink smearing cases
suggest consistent negative energy shifts. This is evident from the large time
behaviour of energy splittings presented in the bottom panel of Figure 5,
where the correlated statistical noise, not related to the excited state
contamination, is suppressed between the numerator and denominator in the
ratio correlators $R^{n}(t)$.

The agreement of energy splitting estimates from fits to $R^{n}(t)$ with those
evaluated from separate fits to the GEVP eigenvalue correlators
$\lambda^{n}(t)$ and the single-meson correlators
$\mathcal{C}_{D/\bar{B}^{*}}$ at large times (see Figure 1) already rules out
the usual concern of accidental partial cancellation of excited state
contaminations in $R^{n}(t)$. The consistency at large times between ground
state energy plateaus from different sink-smearing radii observed in top panel
of Figure 5 further affirms the reliable isolation of the ground state
plateau. Note also that the magnitude of such cancellations and the ground
state saturation times could be different in different lattice QCD ensembles.
All the ground state estimates for noncharm $M_{ps}$ values in our study are
determined from the time intervals approximately between 1.5(2) fm [$t_{min}$]
to 2.3(2) fm [$t_{max}$]. The consistent ground state saturation times across
different ensembles with different specifications further imply the
reliability of our ground state saturation, despite our asymmetric setup.

Figure 5: Comparison of effective energy (top) and effective energy splitting
(bottom) for the ground state as determined using three different smearing
radii applied on the quark fields at the sink timeslice. The legend indicates
the smearing radius squared in units of the lattice spacing [29]. The blue
horizontal band indicates the final fit estimate for the energy and energy
splitting. The results presented are for the case $M_{ps}\sim 700$ MeV on the
finest ensemble.

Appendix B: Elastic $D\bar{B}^{*}$ excitations.- Gaining access to higher
lying elastic excitations in the $D\bar{B}^{*}$ channel is an important step
ahead towards constraining the energy dependence of the amplitude over a long
energy range. However, within the wall-smearing setup, all the nonzero
momentum excitations are significantly suppressed. This suppression is exact
in a free theory, and is empirically confirmed from the early plateauing and
from the quality of signals in the interacting theory. While this suppression
is advantageous in ground state energy determination (see Refs.[14, 28, 52,
16, 29, 53] for details), the suppressed coupling to the nonzero momentum
excitations implies that the access to higher two-meson elastic excitations
with nonzero relative meson momenta are restricted in the wall-smearing setup.
This implies other methodologies that facilitate the use of bilocal two-meson
interpolators with separately momentum projected mesons are necessary in
future studies [71, 72, 55]333While this letter was being reviewed, a
preprint, Ref. [73] appeared which utilizes bilocal two-meson interpolators in
their analysis, utilizing the methods in Ref. [72].. In this respect, it is
informative to know the location of the lowest non-interacting level with
nonzero relative meson momenta and whether it is close enough to influence the
ground state energies in any substantial way. Considering this, in Figure 6 we
present the ground state eigenenergies along with the $D\bar{B}^{*}$ threshold
and the next lowest elastic $D\bar{B}^{*}$ excitation with nonzero relative
meson momentum determined using the continuum dispersion relation that is
assumed in the finite-volume quantization condition [41, 42]. Clearly, the
location of this first non-interacting elastic excitation is sufficiently high
to have any nonnegligible effects on the extracted the ground state energies.

Figure 6: The ground state energy eigenvalues in the background of lowest two
non-interacting $D\bar{B}^{*}$ finite-volume levels units of
$E_{D\bar{B}^{*}}$ on all ensembles (see Table 1 for color-symbol conventions)
for all $M_{ps}$ values (different vertical panels).

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Supplemental material

##  I Low lying eigenenergies on all ensembles

Figure 1: Low-lying finite-volume eigenenergies for isoscalar axialvector
$bc\bar{u}\bar{d}$ channel on the four ensembles studied. The near degenerate
eigenlevels are slightly shifted horizontally for clarity.

In this section, we present the finite-volume GEVP eigenenergies of the
isoscalar axialvector $bc{\bar{u}}{\bar{d}}$ channel that we extract on all
four ensembles listed in Table I of the main article, at the five different
$m_{u/d}$ values corresponding to $M_{ps}\sim$ 0.5, 0.6, 0.7, 1.0, and 3.0
GeV. The eigenenergies shown in lattice units include the additive offsets
related to the NRQCD-based dynamics of heavy bottom quarks. The non-
interacting two-meson energy levels corresponding to $D\bar{B}^{*}$ and
$\bar{B}D^{*}$ thresholds are indicated as dotted horizontal line segments for
each lattice and each $M_{ps}$. The $D^{*}\bar{B}^{*}$ threshold in each case
is also shown in the figure by dashed lines. Note that the use of wall-
smearing setup restricts any direct access to the elastic two-meson
excitations with nonzero relative meson momenta. This means although a
reliable ground state extraction could be made, the excited eigenenergies may
not represent the real elastic excitations.

##  II Operator-state overlaps

Figure 2: Modulus of normalized sink operator-state overlaps
$|\tilde{Z}_{i}^{n}|$ for an eigenenergy indicated by $n={0,1,2}$ and an
operator represented by $\mathcal{O}_{i}$, where $i={1,2,3}$ on the $L_{1}$
ensemble. The errors in the normalized overlap factors are smaller than the
symbols and hence are suppressed.

In Figure 2, we present the modulus of normalized sink operator-state overlaps
$|\tilde{Z}_{i}^{n}|$, normalized such that its largest value for any given
operator $\mathcal{O}_{i}$ across all the eigenenergies $\\{n\\}$ is unity
[74, 75]. $\tilde{Z}_{i}^{n}$ quantifies the relative relevance of any given
operator across all the eigenenergies. The $|\tilde{Z}_{i}^{n}|$ values are
presented for all $M_{ps}$ cases on the $L_{1}$ ensemble. Each square marker
corresponds to the $|\tilde{Z}_{i}^{n}|$ for a given operator
$\mathcal{O}_{i}$ on to a given eigenenergy $n$. Each horizontal panel stands
for an $M_{ps}$ indicated on the right-hand side, whereas the vertical lines
in each horizontal panel part $|\tilde{Z}_{i}^{n}|$ for different operators
indicated on the top panel. The $x$-axis ticks refer to the three finite-
volume eigenenergies we have extracted. $\mathcal{O}_{1}$, the two-meson
operator related to $D\bar{B}^{*}$ threshold, can be seen to have the largest
overlap with the ground state and has significantly small overlaps with the
excited eigenenergies. $\mathcal{O}_{2}$, the two-meson operator related to
$\bar{B}D^{*}$ threshold, has the largest overlap with the first excited
eigenenergy and a very small overlap with the ground state. $\mathcal{O}_{2}$
also have nonnegligible overlap factors with the second excited eigenenergy
indicating $\bar{B}D^{*}$-type two-meson Fock component, which decreases with
increasing $M_{ps}$. On the other hand, $\mathcal{O}_{3}$, the diquark-
antidiquark type operator, have substantial overlap factors with all
eigenenergies at the two lightest $M_{ps}$ values, whereas with an increased
$M_{ps}$ its largest overlap is with the second excited eigenenergy. Note that
$\mathcal{O}_{3}$ is Fierz related to two-meson interpolators [54], and the
large $\tilde{Z}_{3}^{n}$ values of $\mathcal{O}_{3}$ for all $n$ could be
related to this underlying connection between two-meson and diquark-
antidiquark operators.

A summary from the above observations on overlap factors is as follows.
$\mathcal{O}_{1}$ predominantly determines the ground state, whereas it has
significantly small coupling with the excited eigenenergies. Similar patterns
of overlap factors are also observed for other ensembles, all of which
indicate that $\mathcal{O}_{1}$ predominantly determines the ground state. The
two excited eigenenergies have strong two-meson and diquark-antidiquark Fock
components in the two lightest $M_{ps}$ values. One could also evaluate and
investigate the normalized source operator-state overlaps from the left
eigenvectors of $\mathcal{C}$ in Eq. (1) in the main draft, which also leads
to the same conclusions.

##  III Operator basis dependence

Figure 3: Operator basis dependence of the low lying eigenenergies of the
$L_{1}$ ensemble and $M_{ps}\sim$700 MeV for all possible operator basis that
can be built out of the three operators utilized in this work.

In Figure 3, we show the operator basis dependence as determined for
$M_{ps}\sim$ 700 MeV in the $L_{1}$ ensemble, for various operator basis build
out of $\mathcal{O}_{1}$, $\mathcal{O}_{2}$, and $\mathcal{O}_{3}$ operators
as defined in Eq. (2) of the main draft. The digital indexing on the $x$-axis
tick labels refers to various operator basis in the order
$\\{\mathcal{O}_{1},\mathcal{O}_{2},\mathcal{O}_{3}\\}$, with an overline on
the third index as a visual aid within the plot to highlight the diquark-
antidiqaurk interpolator. 1 (0) indicates an operator is included in (excluded
from) the basis. The horizontal lines refer to the $D\bar{B}^{*}$,
$\bar{B}D^{*}$ and $\bar{B}^{*}D^{*}$ thresholds. The gray horizontal bands
refer to the two lowest levels in the full basis indicated by
$11\overline{1}$. A level below the threshold appears only when
$\mathcal{O}_{1}$ is present in the basis. The first excited eigenenergy in
the full basis $11\overline{1}$ is faithfully reproduced in those bases where
$\mathcal{O}_{2}$ is included. $\mathcal{O}_{3}$ alone does not precisely
determine any level among the GEVP eigenenergies using full basis. Similar
observations are also made on other ensembles. In summary, the ground state in
the full basis $11\overline{1}$ is reliably determined with $\mathcal{O}_{1}$
and is unaffected by the inclusion of $\mathcal{O}_{2}$ and $\mathcal{O}_{3}$
operators.

$M_{ps}$ [GeV] |  $\chi^{2}/d.o.f$ | Linear |  $\chi^{2}/d.o.f$ | Quadratic
---|---|---|---|---
|  |  $A^{[0]}/E_{D\bar{B}^{*}}$ |  $A^{[1]}/E_{D\bar{B}^{*}}$ |  |  $A^{[0]}/E_{D\bar{B}^{*}}$ |  $A^{[2]}/E_{D\bar{B}^{*}}$
0.5 |  2.1/2 |  $-0.05(1)$ |  $~{}0.17(_{-11}^{+13})$ |  2.1/2 |  $-0.045(6)$ |  $0.9(_{-6}^{+7})$
0.6 |  0.5/2 |  $-0.044(_{-8}^{+9})$ |  $~{}0.10(_{-9}^{+9})$ |  0.5/2 |  $-0.040(_{-5}^{+6})$ |  $0.6(5)$
0.7 |  3.0/2 |  $-0.042(_{-6}^{+8})$ |  $~{}0.09(_{-7}^{+6})$ |  3.7/2 |  $-0.037(_{-4}^{+5})$ |  $0.5(_{-4}^{+3})$
1.0 |  2.9/2 |  $-0.043(4)$ |  $~{}0.11(_{-5}^{+5})$ |  2.9/2 |  $-0.038(_{-3}^{+3})$ |  $0.8(3)$
3.0 |  3.6/2 |  $~{}0.006(_{-5}^{+6})$ |  $-0.20(_{-5}^{+4})$ |  3.6/2 |  $-0.002(_{-3}^{+4})$ |  $-1.2(_{-3}^{+2})$
Table 1: Results from amplitude fits at different light quark mass cases
indicated in terms of $M_{ps}$ in the first column. The amplitude is
approximated to be determined by the scattering length, with a linear or
quadratic lattice spacing dependence as discussed in the main draft. The
optimized parameter values in the table are presented in units of energy of
the $D\bar{B}^{*}$ threshold, $E_{D\bar{B}^{*}}$. The $A^{[0]}$ parameter in
either parametrization is negative of the inverse scattering length in the
continuum limit.

Figure 4: Left: $k{\mathrm{cot}}\delta_{0}$, in units of the elastic threshold
$E_{D\bar{B}^{*}}$, versus $a$ (lattice spacing) for all $M_{ps}$ values. We
follow the marker/color coding in Table I of the main draft for the data
points referring to the simulated data. The colored/gray bands indicate the
fit results with linear and quadratic lattice spacing dependence,
respectively. Right: $k{\mathrm{cot}}\delta_{0}$ versus $k^{2}$ for all
$M_{ps}$ values studied in units of the elastic threshold $E_{D\bar{B}^{*}}$.
The dashed orange (cyan) curve indicates the constraint for the existence of a
sub-threshold pole in the scattering amplitude. The horizontal bands are the
continuum extrapolated estimates of $k{\mathrm{cot}}\delta_{0}$ for the
respective $M_{ps}$. The black dashed vertical lines in the plots on the right
indicate the location of the branch point associated with the left hand cut
arising from the $D\bar{B}\pi$ channel.

##  IV Results on scattering amplitude

In Table 1, we tabulate the results from different amplitude fits that were
performed. In Figure 4, we present the quality of these fits by comparing the
fit results with the data points (see the figure caption for details). On the
left of Figure 5, we present the light quark mass dependence in the chiral
regime determined from continuum extrapolated elastic $D\bar{B}^{*}$
scattering amplitudes following a linear and quadratic lattice spacing
dependence. We present the final estimate (black star) from the linear fit
form considering the presence of heavy quarks in our system, whereas the
difference in the fit results are accounted in the systematics quoted in the
main draft. On the right of Figure 5, we present a comparison of the light
quark mass dependence in the chiral regime between the fit involving all
$M_{ps}$ datasets and the fit involving the lightest four $M_{ps}$ datasets.
Either fitting procedures can be seen to be consistent with our final estimate
in the chiral limit is shown by black star.

Figure 5: Left: Comparison of light quark mass dependence of scattering
amplitudes in the chiral regime determined from a linear (red band) and
quadratic (green band) dependence of $k{\mathrm{cot}}\delta_{0}$ on the
lattice spacing. Right: Comparison of light quark mass dependence of
scattering amplitude $k{\mathrm{cot}}\delta_{0}$ in the chiral regime
determined using the results from all five light quark masses (red band) and
the results from four light light quark masses (black). The results in the
physical limit in either cases can be seen to be consistent with the main
result indicated by the star symbol.