# Study of isoscalar scalar $bc\bar{u}\bar{d}$ tetraquark $T_{bc}$ from
lattice QCD

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

###### Abstract

We present a lattice QCD study of the elastic $S$-wave $D\bar{B}$ scattering
in search of tetraquark candidates with explicitly exotic flavor content
$bc\bar{u}\bar{d}$ in the isospin $I\\!=\\!0$ and $J^{P}=0^{+}$ channel. We
use four lattice QCD ensembles with dynamical $u/d$, $s$, and $c$ quark fields
generated by the MILC Collaboration. A non-relativistic QCD Hamiltonian,
including improvement coefficients up to $\mathcal{O}(\alpha_{s}v^{4})$, is
utilized for the bottom quarks. For the rest of the valence quarks we employ a
relativistic overlap action. Five different valence quark masses are utilized
to study the light quark mass dependence of the $D\bar{B}$ scattering
amplitude. The finite volume energy spectra are extracted following a
variational approach. The elastic $D\bar{B}$ scattering amplitudes are
extracted employing Lüscher’s prescription. The light quark mass dependence of
the continuum extrapolated amplitudes suggests an attractive interaction
between the $\bar{B}$ and $D$ mesons. At the physical pseudoscalar meson mass
($M_{ps}=M_{\pi}$) the $D\bar{B}$ scattering amplitude has a sub-threshold
pole corresponding to a binding energy of
$-39(^{+4}_{-6})(^{~{}+8}_{-18})\mbox{~{}MeV}$ with respect to the $D\bar{B}$
threshold. The critical $M_{ps}$ at which the $D\bar{B}$ scattering length
diverges and the system becomes unbound corresponds to
$M^{*}_{ps}=2.94(15)(5)\mbox{~{}GeV}$. This result can hold significant
experimental relevance in the search for a bound scalar $T_{bc}$ tetraquark,
which could well be the next “doubly heavy” bound tetraquark to be discovered
with only weak decay modes.

††preprint: IMSc/2024/2 TIFR/TH/24-2

##  I Introduction

The study of exotic hadrons is one of the most prominent areas of research in
contemporary particle physics. The proliferating list of discovered exotic
hadrons, exhibiting properties that demand interpretations beyond conventional
meson and/or baryon models, continues to captivate the scientific interest. A
compilation of various exotic hadrons discovered till now and their properties
can be found in Ref. [1]. Among these exotic hadrons, those with manifestly
exotic flavor content having four quarks in their valence structures are
particularly popular. Several of them are proximal to open flavor thresholds
pointing to a connection with the corresponding scattering channel for their
existence, and possibly their nature. These four quark states could be compact
tetraquarks, or meson-meson molecular excitations, or a mixture of both or
something more intriguing: a much enviable research topic at this time. An in-
depth understanding of the binding mechanism governing these hadrons can play
a crucial role in elucidating the non-perturbative QCD dynamics.

A particularly notable common feature among the discovered four quark hadrons
is the presence of at least one heavy quark constituent in their valence
structure. Phenomenologically it has been hypothesized and discussed that a
color-singlet combination of two very heavy quarks (anti-quarks) and two light
anti-quarks (quarks) can form a $QQ\bar{q}_{1}\bar{q}_{2}$ bound state [2, 3].
Recently a handful of calculations using first principles method of lattice
QCD also strongly indicate the presence of deeply bound states with the quark
contents $bb\bar{q}_{1}\bar{q}_{2}$; $q_{1}\in u,d$; $q_{2}\in d(s),u(s)$ [4,
5, 6, 7, 8, 9]. Very interestingly a doubly-charmed four quark hadron, coined
as $T_{cc}^{++}$, with the quark content $cc\bar{u}_{1}\bar{d}_{2}$ and
unusually long lifetime, has recently been discovered by LHCb [10]. Lattice
QCD calculations have also investigated $T_{cc}^{++}$ and suggested that the
existence of this hadron could be the result of a delicate fine tuning between
the light and heavy quark masses [11, 12, 13, 14]. In summary, lattice QCD
calculations and phenomenological investigations consistently suggest the
existence of deeply bound states in doubly bottom four quark system, referred
to as $T_{bb}\in bb\bar{q}_{1}\bar{q}_{2}$, while experimental evidence has
been reported for a four quark hadron ($T_{cc}$) with the quark content
$cc\bar{u}\bar{d}$. Notably, the charm quark mass is comparatively lighter
than the bottom quark mass, suggesting potentially differing binding strengths
for doubly-bottomed and doubly-charmed four quark states due to QCD dynamics
operating at multiple scales.

In this respect, four quark systems ($T_{bc}\in bc\bar{q}_{1}\bar{q}_{2}$)
which are in between $T_{bb}$ and $T_{cc}$, i.e., with a bottom and a charm
valence quarks, are of particularly interesting. Phenomenologically,
predictions on the existence of such states are ambiguous with their energies
exhibiting considerable spread over several hundreds of MeV around the
relevant two-meson threshold. Several model studies based on heavy quark
symmetry [15, 16, 17] suggest no binding. Numerous non-chiral models [18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29] also suggest either a weak binding or
an unbound system. However, certain chiral models [30, 31, 32, 33] and QCD sum
rule investigations [34, 35, 36, 37, 38] propose a more pronounced binding.

In such a scenario, first principles lattice QCD calculations can provide much
needed information on the bindings of $T_{bc}$ states. However, previous
lattice QCD calculations [39, 40, 41] claim either no evidence for a bound
$T_{bc}$ state or insufficient statistics to conclude its existence. In Ref.
[41], the authors do not come up to a conclusion due to large uncertainties.
In a recent work, we investigated the $J^{P}=1^{+}$ channel considering chiral
as well as continuum extrapolations and found an attractive interaction
between the $\bar{B}^{*}$ and $D$ mesons indicating the possible existence of
a bound $T_{bc}$ state with a binding energy of
$-43(^{+6}_{-7})(^{+14}_{-24})$ MeV with respect to the $D\bar{B}^{*}$
threshold [42]. Afterwards, in another recent calculation [43], some of the
authors of Ref. [41] studied $J^{P}=0^{+}$ as well as $1^{+}$ channels,
involving bilocal two-meson interpolators corresponding to the elastic
excitations of $D\bar{B}^{(*)}$ scattering. They found attractive interactions
in both channels, and subsequently pointed to the existence of shallow bound
states. Given the coarse lattices they utilize for these hadrons with two
heavy quarks, it will be important to check whether the binding of these
states observed in Ref. [43] will survive or enhance with continuum
extrapolation.

Motivated by the recent progress, and building upon our previous work for
$J^{P}=1^{+}$ channel [42], in this work we perform a lattice QCD calculation
of elastic $D\bar{B}$ mesons 111We assume $m_{u}=m_{d}$, ignore QED effects,
and refer to the degenerate ($D^{+}B^{-},~{}D^{0}\bar{B}^{0}$) threshold as
$D\bar{B}$. scattering in the isoscalar scalar channel $I(J^{P})=0(0^{+})$.
Following a partially quenched approach, we investigate the light quark mass
dependence of the $D\bar{B}$ mesons scattering. The lattice-extracted
scattering amplitudes, employing Lüscher’s finite-volume prescription, are
extrapolated to the continuum limit. The amplitude at the physical pion mass
is deduced following a study of the light quark mass dependence of these
continuum-extrapolated results. Finally, the hadronic pole information in this
physical amplitude is studied towards identification of bound state poles.

Experimentally, $J^{P}=0^{+}$ channel is also more interesting as it could be
the next “doubly heavy” tetraquark to discover since it has a reduced heavy
diquark mass that is lower than that for the $bb\bar{q}_{1}\bar{q}_{2}$
system. It would also likely be the first tetraquark that would unambiguously
decay only weakly.

The remainder of the manuscript is structured as follows. A brief overview of
our lattice setup is provided in Section II. In Section III, we discuss
various relevant technical details involved in our calculation such as the
observable measured, the interpolating operators utilized and the extraction
of finite volume energy spectra, which are presented in Section IV. The
extraction of scattering amplitudes, continuum extrapolations, and chiral
extrapolations made are presented in Section V. In section VI we present a
discussion on the bindings of $T_{bc}$ four-quark states, in perspectives of
available lattice and non-lattice results, along with a comparison of
scattering lengths for $DD^{*}$, $D\bar{B^{*}}$, $BB^{*}$ and $D\bar{B}$
scatterings. Finally we summarized our results in section VII.

##  II Lattice Details

The computational setup used in this calculation is similar to the one in
several of our previous publications [44, 7, 45, 46, 47, 48, 49, 50, 51, 52,
53] and most recently in Ref. [42]. We use four $N_{f}=2+1+1$ ensembles with
dynamical quark fields respecting a Highly Improved Staggered Quark (HISQ)
action generated by the MILC collaboration [54]. Other relevant details of
various lattice QCD ensembles used are listed in Table 1. The lattices have
different volumes and lattice spacings $a$, which are estimated using the
$r_{1}$ parameter [54]. The gauge fields respect one-loop and follow the
tadpole-improved Symanzik gauge action with tuned coefficients through
$\mathcal{O}(\alpha_{s}a^{2},n_{f}\alpha_{s}a^{2})$ [55].

The valence quark masses upto the charm quark are realized using an overlap
fermion action that is $\mathcal{O}(am)$ improved [56, 57]. The bare charm
quark mass on each ensemble was tuned using the kinetic mass of spin averaged
$1S$ charmonia
$\\{a\overline{M}_{kin}^{\bar{c}c}=0.75aM_{kin}(J/\psi)+0.25aM_{kin}(\eta_{c})\\}$
determined for the respective ensembles following the Fermilab prescription
[58] (for more details see Refs. [47, 48]). The bare strange quark mass is
tuned to the physical point such that the lattice estimate for the fictitious
pseudoscalar $\bar{s}s$ equals 688.5 MeV [59].

Our setup assumes an exact isospin symmetry $m_{u}=m_{d}$ over a range of
light quark masses corresponding to $M_{ps}\sim$0.5, 0.6, 0.7 (equivalent to
the strange quark mass), 1.0, and 3.0 (equivalent to the charm quark mass)
GeV, to map the light quark mass dependence over a wide range of $M_{ps}$
values. In Figure 1, we present the landscape of different light quark masses
(in terms of $M_{ps}$) studied in the lattice ensembles employed. We utilize a
wall-smearing procedure at the quark source for our propagator measurements
which is described in Refs. [50, 7, 53].

We use a nonrelativistic QCD (NRQCD) Hamiltonian approach for the bottom quark
[60]. The bottom quark mass was tuned following the Fermilab prescription
[58], matching the lattice-determined kinetic mass of the spin-averaged $1S$
bottomonium state to its experimental value. For details regarding the NRQCD
Hamiltonian, improvement coefficients, and bottom quark mass tuning specific
to our setup, see Ref. [49].

Ensemble |  Symbol |  Lattice Spacing |  Dimensions |  $M_{ps}^{sea}$
---|---|---|---|---
|  ($a$) [fm] |  ($N_{s}^{3}\times N_{t}$) |  |
$S_{1}$ |  |  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: Details of lattice QCD ensembles employed. $M_{ps}^{sea}$ refers to
the sea pion mass. $S_{1}$, $S_{1}$, and $S_{1}$ refer to small spatial volume
ensembles and $L_{1}$ refers to the large volume ensemble. Figure 1: A
landscape plot of the pseudoscalar masses used across the different lattice
ensembles. The light quark masses are varied across these five values, while
charm and bottom quarks are tuned to their physical values. The horizontal
gray bands represent estimates of $M_{ps}$, to facilitate a comparison of
pseudoscalar meson masses across all four ensembles.

##  III Determining the Finite Volume Spectra using Lattice QCD

The time dependence of Euclidean two-point correlation functions

$\mathcal{C}_{ij}(t)=\sum_{\mathbf{x}}\left<\mathcal{O}_{i}(\mathbf{x},t)\mathcal{\tilde{O}}_{j}^{\dagger}(0)\right>=\sum_{n}\frac{Z_{i}^{n}\mathcal{Z}_{j}^{n\dagger}}{2E^{n}}e^{-E^{n}t}.$
(1)

featuring operators $\mathcal{O}_{i}(\mathbf{x},t)$ with the desired quantum
numbers dictate the time evolution of finite volume spectral levels. Here the
operator-state overlap $Z_{i}^{n}=\bra{0}\mathcal{O}_{i}\ket{n}$ determines
the coupling of the operator $\mathcal{O}_{i}$ with the state $n$. The wall-
smearing at the quark source in our setup filters out all the high-momentum
modes at the source, whereas at the sink time slice we utilize a point sink
for the quark fields and project the correlation function to its rest frame as
shown in Eq. (1). This asymmetric nature of the wall-source point-sink setup
is emphasized in Eq. (1) with different operators and the operator-state
overlaps at the source and at the sink.

For the $\bar{B}$ and $D$ mesons, we compute two-point correlation functions
using the standard local quark bilinear interpolators
($\overline{Q}~{}\Gamma~{}q$) with spin structure $\Gamma\sim\gamma_{5}$.
Since we are only interested in the rest frame ground state, single meson
correlation functions are evaluated only for the $A_{1}^{-}$ irrep in the
finite volume.

Our study focuses on the $S-$wave $D\bar{B}$ scattering in the rest frame
leading to infinite volume quantum numbers $J^{P}=0^{+}$, which reduces to the
$A_{1}^{+}$ finite-volume irrep. The elastic two-meson threshold is at
$E_{D\bar{B}}=m_{D}+m_{\bar{B}}$, whereas the lowest inelastic threshold
corresponds to the $D^{*}\bar{B}^{*}$ scattering channel, which is
sufficiently high to assume a purely elastic $D\bar{B}$ scattering in
$S-$wave. There are no relevant low lying three particle thresholds in this
channel and the lowest multi-particle inelastic threshold corresponds to
$D\bar{B}\pi\pi$.

In the present analysis, we use both a meson-meson type of operator and a
local diquark-antidiquark kind of operator as in Ref. [40].

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

Here $\mathcal{O}_{1}(x)$ is a meson-meson operator associated with the
$D\bar{B}$ threshold with the individual $D$ and $\bar{B}$ forming a color
singlet. We do not include any other scattering operators since the next one,
corresponding to the $D^{*}\bar{B}^{*}$ is sufficiently higher up in energy
and is assumed to have negligible effects on the low-lying spectrum. Excited
elastic two-meson operators of $D\bar{B}$ system with nonzero relative meson
momenta, such as those used in Ref. [43], are also not utilized in this study.
The wall-smearing setup we utilize disallows construction of such elastic
scattering operators.

$\mathcal{O}_{2}(x)$ is a local diquark-antidiquark type operator where all
the (anti)quark fields are jointly projected to zero momentum. In the color
space, diquarks/antidiquarks are built in the antitriplet/triplet
representations of $SU(3)_{c}$. In Eq. (2), $\Gamma_{k}=C\gamma_{k}$ with
$C=i\gamma_{y}\gamma_{t}$ being the charge conjugation matrix.
Phenomenologically, doubly heavy tetraquarks are expected to be deeply bound
and compact in the heavy quark limit, which motivates the use of this operator
[5, 61]. Such compact local operators were also considered in our previous
study [42] of axial-vector bottom-charm tetraquarks. It is also empirically
known from several other studies of doubly-bottom tetraquarks that such
operators have a rich overlap with the ground state [40, 41, 62, 63, 64, 65,
66, 67].

Figure 2: Effective energy plot for the eigenvalue of the ground state
$\lambda^{0}(t)$ in circular black markers and for the lowest threshold which
is the product of single-meson correlators for the $D$ and $\bar{B}$ meson,
$\mathcal{C}_{D}(t)\mathcal{C}_{\bar{B}}(t)$ in square red markers. The
corresponding blue and orange bands are the energy fit estimates using single
exponential fit forms on $\lambda^{0}(t)$ and the single-meson correlation
functions respectively.

With these two operators, we find a suitable linear combination that overlaps
maximally to the ground state by solving for the Generalized Eigenvalue
Problem (GEVP) [68],

$\mathcal{C}(t)v^{n}(t)=\lambda^{n}(t)\mathcal{C}(t_{0})v^{n}(t),$ (3)

The eigenvalues, $\lambda^{n}(t)$ correspond to the $n^{th}$ lowest
eigenstates with energy $E^{n}$, where $n\leq 1$ in our case. We are only
interested in the ground state $E^{0}$. The time evolution of the lowest
eigenvalue, $\lim_{t\to\infty}\lambda^{0}(t)\sim A_{0}e^{-E^{0}t}$, gives the
value of the ground state energy in the large time limit, whereas the
magnitude of the operator-state-overlaps

$Z_{i}^{0}=\bra{0}\mathcal{O}_{i}\ket{0}=\sqrt{2E^{0}}(V^{-1})_{i}^{0}e^{E^{0}(t_{0})/2},$
(4)

indicates the coupling of the operators to the ground state. Here $V$ is the
matrix of eigensolutions $v^{n}(t)$, which are expected to be time independent
in the large time limit, where $\mathcal{C}(t)$ is saturated by the lowest $N$
eigenstates.

The quality of signals in the energy estimates are assessed using the
effective energies,

$aE_{eff}=[ln(\mathcal{C}(t)/\mathcal{C}(t+\delta t))]/\delta t,$ (5)

c.f. Figure 2, where we plot present $aE_{eff}$ for the case $M_{ps}\sim 3$
GeV on the finest lattice. The black points represent the effective energy of
the interacting system $\mathcal{C}(t)=\lambda^{0}(t)$, whereas the red points
indicate the effective energy of the correlator
$\mathcal{C}(t)=\mathcal{C}_{D}(t)\mathcal{C}_{\bar{B}}(t)$ of the
noninteracting system of $D$ and $\bar{B}$ mesons and serve as a reference. A
negative shift of the interacting energy level with respect to the
noninteracting ones is evident in Figure 2.

The energy estimates are extracted from the correlator data from fitting them
with their expected asymptotic forms. This can be performed in two ways: the
obvious way of fitting the interacting correlator $\lambda^{0}(t)$ directly,
or to fit the ratio of correlators

$R^{0}(t)=\frac{\lambda^{0}(t)}{\mathcal{C}_{D}(t)\mathcal{C}_{\bar{B}}(t)},$
(6)

with a single exponential form in the large time limit. We are primarily
interested in determining the energy splittings between the interacting data
and the noninteracting one, $\Delta E^{0}=E^{0}-M_{D}-M_{\bar{B}}$. Fits to
$R^{0}(t)$ directly leads to the estimates for $\Delta E^{0}$. Alternatively,
these splittings can be evaluated as differences between the estimates for
energy $E^{0}$ from fits to $\lambda^{0}(t)$ and for ($M_{D}$, $M_{\bar{B}}$)
from separate fits to $\mathcal{C}_{D}(t)$ and $\mathcal{C}_{\bar{B}}(t)$,
respectively. A comparison of estimates from these two procedures assures that
the ground state energy splittings we extract are not influenced by any
conspired cancellation of noises leading to any fake energy plateaus. We
present a demonstration of such a comparison in Figure 3, where it is evident
that the value of $\Delta E^{0}$ estimated from the two different procedures
agree with each other within error-bars. This trend is observed throughout all
the correlators examined. The final results quoted in this paper are based on
fitting the ratio correlators defined in Eq. (6).

Figure 3: $a\Delta E^{0}$ vs. $t/a$ plot, for $M_{ps}\sim 3$ GeV on the finest
lattice. Here $a\Delta E^{0}(t)$, shown in the circular black data data-
points, is the effective energy splitting determined using Eq. (5) with
$\mathcal{C}(t)=R^{0}(t)$. The fit estimates determined from the single
exponential fits to $\lambda^{0}(t)$ and $R^{0}(t)$ is shown in grey and blue
bands respectively.

##  IV Finite Volume Spectra

Figure 4: The ground state finite volume energies in the $0(0^{+})$
$bc\bar{u}\bar{d}$ channel. Different panels stand for different $M_{ps}$
values indicated on the top of the respective channel. The $y$-axis indicates
the energy in the center-of-mass frame, in units of energy of the $D\bar{B}$
threshold. The $x$-axis in each panel indicates the spatial extent of the
lattice ensembles used.

In Figure 4, we present the extracted finite-volume energy spectra of the
$0(0^{+})$ $bc{\bar{u}}{\bar{d}}$ channel on the four ensembles listed in
Table 1, at the five different $m_{u/d}$ values corresponding to roughly,
$M_{ps}\sim$ 0.5, 0.6, 0.7, 1.0, and 3.0 GeV. The energy spectrum shown is
normalized by the threshold $M_{D}+M_{\bar{B}}$, such that center-of-mass
energy at threshold is unity in these units. In each panel, the $x$-axis
represents the spatial extension of the lattice.

The finite-volume energies are determined from energy splittings extracted
from the ratio correlators given in Eq. (6). These energy splittings are free
of the additive offsets, inherent to the NRQCD formulation, as the numerator
and denominator in Eq. (6) carries same number of valence NRQCD-based bottom
quarks. The reconstruction of the finite-volume energies from the energy
splittings follow the same lines as in Ref. [42].

A clear trend of negative shifts for the ground state energies with respect to
the $D\bar{B}$ threshold can be observed for all the lattices and for all the
quark masses studied. It is also evident that this negative shifts decreases
in magnitude with increasing $M_{ps}$, as expected for a doubly heavy
tetraquark system [5, 61]. The variation in this splitting across different
lattice spacings for any given $M_{ps}$ is not transparent due to large
uncertainties, whereas unlike in our study of axial-vector $T_{bc}$
tetraquark, a moderate trend of decreasing splitting with increasing volume
can be observed as expected. However, it is too early to substantiate this
behavior considering the large uncertainties. Despite the large uncertainties,
the consistent negative shifts clearly point to an attractive interaction
between the $D$ and $\bar{B}$-meson in the scalar channel.

In the wall-smearing setup we use, the elastic $D\bar{B}$ excitations
involving nonzero relative meson momenta are suppressed. This should not
affect the ground state determination because it is unlikely that operators
with relative momenta contribute to the ground state. Additionally, we employ
various cross checks that helps us estimate the excited state contaminations
in the ground state energy, that are then included in the systematic
uncertainties. We refrain from using or plotting the excited states determined
from the solutions of Eq. (3) in Figure 4, as they do not represent the the
elastic $D\bar{B}$ excitations in the wall-smearing setup. Another significant
limitation of the wall-smearing setup is its asymmetry, leading to the
possibility of the ground-state energy plateau being approached from below.
The agreement observed between the energy splittings calculated from ratios of
correlators and those determined from the difference in energy fit estimates
for individual single meson and interacting two-meson correlators indicates
that our correlator-based fitting estimates effectively manage contaminations
from excited states which are then incorporated into the systematic errors.

##  V $\mathbf{D\bar{B}}$ scattering amplitude from the finite-volume spectra

In this section, we present $S$-wave elastic $D\bar{B}$ scattering amplitudes
determined following Lüscher’s finite-volume prescription [69]. We use only
the ground state energies to constrain the amplitudes, since the wall-smearing
procedure that we utilize for quark sources is not suited to extract the
elastic excitations, but only the ground states [42]. For the scalar channel
considered in this work, the lowest inelastic threshold is $D^{*}\bar{B}^{*}$,
which is significantly high in energy and there are no higher partial wave
that can mix with the $S$-wave, justifying an elastic $S$-wave analysis.

A topical aspect in the study of doubly heavy hadrons is the influence of
left-hand cuts (lhc) due to off-shell pion exchanges[70]. Recently, there has
been efforts to accommodate the lhc effects arising from single pion exchanges
[71, 72, 73]. In $D\bar{B}$ scattering, the closest non-analyticity below the
threshold can happen from an off-shell two-pion exchange, which has its branch
point well below the elastic threshold. Hence we ignore any effects of such
left-hand nonanalyticities in our analysis.

###  V.1 Amplitude fits using Lüscher’s finite-volume formalism

The Lüscher’s finite-volume formalism relates the amplitude of two-particle
scattering to the finite volume-spectrum in a cubic box. Particularly for the
elastic $S$-wave scattering of $B$ and $D$ mesons,

$k\cot[\delta_{0}(k^{2})]=\frac{2Z_{00}[1;(\frac{kL}{2\pi})^{2})]}{L\sqrt{\pi}},$
(7)

where $Z_{00}$ is the generalized zeta function described in Ref. [69], $L$ is
the spatial extent of the cubic box and $\delta_{0}(k)$ is the $S$-wave phase
shift as a function of $k$, which is the momentum of either mesons in the
center of momentum frame related to the center of momentum energy
$E_{cm}=\sqrt{s}$ through
$4sk^{2}=(s-(M_{D}+M_{\bar{B}})^{2})(s-(M_{D}-M_{\bar{B}})^{2})$. From Eq.
(7), it is clear that there is a one-to-one correspondence between the energy
level and the $\delta_{0}(k)$, _i.e._ each finite-volume energy level provides
a specific value of the $S$-wave elastic phase shift with which one can
constrain the energy or $k$ dependence of the phase shift.

Figure 5: $k\cot[\delta_{0}]$ normalized by the elastic threshold
$E_{D\bar{B}}$, versus lattice spacing, $a$, for the $M_{ps}$ values studied
in this analysis as indicated in the different panels. The colored bands
indicate the fit results to the amplitude parameterization given in Eq. (9).
The marker conventions are as listed in Table 1. For all the $M_{ps}$ values
except when $M_{ps}\sim$ 3.0 GeV, the fits show a positive scattering length
$a_{0}$, indicating an attractive interaction.

We perform the amplitude fits with the ground states from all four ensembles
listed in Table 1, and repeat this for all five values of $M_{ps}$ indicated
in Figure 1. The fits follow minimization of a cost function defined as

$\displaystyle\chi^{2}$ $\displaystyle=$
$\displaystyle\sum_{i,j}(f(k_{i}^{2})-f(\\{A\\},k_{i}^{2}))$ (8)
$\displaystyle~{}~{}~{}~{}~{}(\mathcal{C}^{-1})_{ij}(f(k_{j}^{2})-f(\\{A\\},k_{j}^{2}))$

where $f(k_{i}^{2})$ is the amplitude (lhs of Eq. (7)) extracted from the
simulations at $k_{i}^{2}$, and $f(\\{A\\},k_{i}^{2})$ is the parameterization
of the energy dependence of the amplitude. $\mathcal{C}$ is the covariance
matrix defined as in Ref. [74]. We verify that the results determined from the
$\chi^{2}$ defined in Eq. (8) are consistent with that one gets from the
procedure outlined in Appendix B of Ref. [14]. Considering the smallness of
$(k/E_{D\bar{B}})^{2}$ and the limited energy range over which the ground
states are placed, we assume a zero-range approach for the amplitude
parameterization. Additionally, we include a linear lattice spacing dependence
to account for the cutoff effects in the extracted amplitude, which takes the
form

$k\cot[\delta_{0}]=A^{[0]}+aA^{[1]}$ (9)

where $A^{[0]}=-1/a_{0}$, with $a_{0}$ being the scattering length in the
continuum limit.

$M_{ps}$ [GeV] | $\chi^{2}/d.o.f$ | $A^{[0]}/E_{D\bar{B}}$ | $A^{[1]}/E_{D\bar{B}}$
---|---|---|---
0.5 | 0.04/2 | $-0.038(_{-11}^{+15})$ | $~{}0.004(_{-134}^{+122})$
0.6 | 0.43/2 | $-0.044(_{-7}^{+8})$ | $~{}0.06(7)$
0.7 | 3.76/2 | $-0.042(_{-4}^{+5})$ | $~{}0.05(_{-4}^{+5})$
1.0 | 0.67/2 | $-0.043(4)$ | $~{}0.12(4)$
3.0 | 5.32/2 | $~{}0.002(3)$ | $-0.17(_{-3}^{+2})$
Table 2: Fit results for amplitude with parameterization given in Eq. (9) at
various light quark masses, corresponding to $M_{ps}$ in the first column. The
optimized parameter values in the table are expressed in units of the energy
of the threshold, $E_{D\bar{B}}$. The numbers within the parenthesis indicate
the statistical errors.

In Figures 5 and 6, we present the fit results to $k\cot[\delta_{0}]$ (the
bands) as a function of the lattice spacing and $(k/E_{D\bar{B}})^{2}$,
respectively, along with the lattice data. The bands in Figure 6 are the
continuum extrapolated results given by the parameter $A^{[0]}$. Different
horizontal panels represent different $M_{ps}$ values. The best fit parameters
and corresponding quality of fits are tabulated in Table 2.

Given the negative energy shifts and the sign of $A^{[0]}=-1/a_{0}$,
determines the nature of the near-threshold poles, if any. Note that for the
non-charm light quark masses, $a_{0}$ is consistently positive suggesting that
the strength of interaction to be sufficient enough to house a bound state.
Whereas at the charm point $a_{0}$ is negative, despite negative energy
shifts, suggesting only a feeble interaction that cannot hold a subthreshold
pole with square-integrable wave-function. This is similar to our observation
in the axial-vector channel using the same setup and formalism in Ref. [42],
as well as to the phenomenological expectation for doubly heavy four quark
systems, here the binding energy is expected to decrease with increasing light
quark masses for fixed heavy quark masses.

Another interesting observation is on the variation in the cut off dependence
of the amplitudes as the light quark masses are varied. The cut off dependence
is accounted by the parameter $A^{[1]}$, which shows a signature change as the
light quark mass increases towards the charm point. This suggests that for a
doubly heavy four quark ($QQ^{\prime}l_{1}l_{2}$) system with
$(m_{l_{1}}=m_{l_{2}},~{}m_{Q},m_{Q^{\prime}}>>m_{l})$, the cut off effects
weaken the finite-volume energy splitting of the ground state with the elastic
threshold. On the other hand, close to the charm point (where
$m_{Q},m_{Q^{\prime}}\sim m_{l}$) such effects enhance this energy splitting
in the $QQ^{\prime}l_{1}l_{2}$ system determined in a finite-volume.
Relatively large errors at the non-charm $M_{ps}$ values partially obscure
these effects, if any exist, while at the charm point such effects are clearly
reflected. Any further quantified comments on this lattice spacing dependence
is currently beyond the scope of the current work, particularly considering
the large uncertainties.

###  V.2 Extrapolation to physical light quark mass

Following the extraction of the continuum extrapolated amplitude at different
$M_{ps}$ values, we delve into the light quark mass dependence of the fitted
parameters. The leading order $M_{ps}$ term in the chiral expansion suggests
the $M_{ps}$ dependence of hadron masses for light $m_{u/d}$ values
($m_{q}\lesssim\Lambda_{QCD}$) to be $M_{ps}^{2}$. Whereas in the heavy light
quark mass regime ($m_{q}>>\Lambda_{QCD}$) heavy hadron masses are expected to
be linear in $M_{ps}$ [75]. With these phenomenological expectations, we use
three fit forms like [42],

$\displaystyle f_{l}(M_{ps})$ $\displaystyle=$
$\displaystyle\alpha_{c}+\alpha_{l}M_{ps},$ $\displaystyle f_{s}(M_{ps})$
$\displaystyle=$
$\displaystyle\beta_{c}+\beta_{s}M_{ps}^{2},\mbox{~{}~{}~{}and}$
$\displaystyle f_{q}(M_{ps})$ $\displaystyle=$
$\displaystyle\theta_{c}+\theta_{l}M_{ps}+\theta_{s}M_{ps}^{2}.$ (10)

Figure 6: $k\cot[\delta_{0}]$ versus $k^{2}$ for the different $M_{ps}$ values
shown in the legend. The scales in either axis are plotted in units of elastic
$D\bar{B}$ threshold $E_{D\bar{B}}$. The dashed orange curve is the unitarity
parabola related to the existence of a real bound state pole in the scattering
amplitude. The horizontal bands are the continuum extrapolated amplitudes in
Eq. (7) for each $M_{ps}$, also listed in Table 2. Figure 7: The elastic
$D\bar{B}$ scattering amplitude in the $S$-wave as a function of the light
quark mass, in terms of the pseudoscalar mass squared $M_{ps}^{2}$. The
amplitude and $M_{ps}^{2}$ are presented in units of the energy of $D\bar{B}$
threshold ($E_{D\bar{B}}$). The bands indicate fits with different functional
dependencies listed in Eq. (10). The vertical dotted line near the y-axis
represents the physical pion mass and the black star on it indicates the
scattering amplitude in the physical limit. Another star symbol on the
$x$-axis indicates the critical $M_{ps}$ where the $D\bar{B}$ system becomes
unbound.

The light quark mass dependence is determined by minimizing another cost
function. The function is defined in terms of the differences in the data with
the phenomenologically motivated parameterizations (c.f Eq. (10)) for its
$M_{ps}$ dependence and the data covariance. We present the results for this
quark mass dependence in Figure 7 together with the lattice extracted
amplitudes as a function of $M_{ps}^{2}/E_{D\bar{B}}^{2}$. The two black
symbols represents amplitude at the physical pion mass limit ($y$-axis
intercept; $M_{ps}=M_{\pi}$) and the critical mass ($x$-axis intercept;
$M_{ps}^{*}$) at which the system is close to unitarity branch point. The
inner errors associated with these black symbols represent the statistical
errors, whereas the outer errors also include systematic uncertainties added
in quadrature.

The scattering length at the physical pion mass $M_{ps}=M_{\pi}$

$a_{0}^{phys}=0.61(^{+3}_{-4})(18)\mbox{~{}fm}$ (11)

together with the observed negative energy shifts in the interacting lattice
levels indicate an attractive interaction between the $B$ and $D$ mesons,
similar to the observation in Ref. [42]. This attraction is sufficiently
strong enough to hold a real bound state with a binding energy

$\delta
E_{T_{bc}}=E_{T_{bc}}-E_{D\bar{B}}=-39(^{+4}_{-6})(^{~{}+8}_{-18})\mbox{~{}MeV}.$
(12)

When $m_{u/d}>>\Lambda_{QCD}$, the leading linear behavior in $M_{ps}$ is
expected to be a good description. The black star at the $x$-axis intercept
based on the linear $M_{ps}$ dependence in Figure 7 indicates the critical
point

$M^{*}_{ps}=2.94(15)(5)\mbox{~{}GeV}.$ (13)

at which $a_{0}$ changes its sign from negative to positive. $M^{*}_{ps}$ and
the associated errors are evaluated from the fit form $f_{l}(M_{ps})$ inspired
by the leading linear behavior based on heavy quark effective field theory
[75]. Note that the inverse scattering length at the charm point is consistent
with zero and any fit form is constrained by the data at the charm point.
Hence systematics associated with the critical mass estimates are
significantly small compare to the statistical errors.

Source | Error [fm]$\times 10^{2}$
---|---
Statistical Errors | $\left({}^{+3}_{-4}\right)$
scale setting | 3
$m_{b/c}$ tuning | 3
excited states | 4
continuum extrapolation | 8
chiral extrapolation | 15
Total systematics | 18
Table 3: The error budget in the calculation of the scattering length,
$a_{0}^{phys}$. This includes the systematics involved as a result of scale
setting, excited state effects, heavy quark mass tuning, and uncertainties
related to chiral and continuum extrapolations. The total systematics is
determined by adding differential estimates in quadrature.

###  V.3 Systematic Uncertainties

In this section we discuss various sources of uncertainties in this
calculation that are summarized in Table 3. We follow the bootstrap procedure
to carefully carry the statistical errors. The most dominant systematics are
observed to be associated with the light quark mass dependence in the chiral
regime. Different chiral extrapolation fit forms lead to different estimates
for the physical scattering length more significant than the statistical
precision. The combination of $N_{f}=2+1+1$ MILC lattice QCD ensembles we
employ, together with the partially quenched setup using an overlap fermion
action for light and charm quarks, and an NRQCD formulation for bottom quarks,
and a rigorous heavy quark mass tuning procedure has been demonstrated to be
quite efficient in extracting the ground states from finite volume. This setup
also reproduces the $1S$ hyperfine splittings in quarkonia very precisely with
uncertainties less than 6 MeV [53, 49]. The energy splittings and mass ratios
we have adopted to work with, efficiently mitigate the systematics associated
with the lattice realization of heavy quark dynamics [50, 53]. We have also
included the errors due to fit-window which includes the excited-state
contamination. The values within the second parenthesis in Equations 11, 12,
and 13 represent the cumulative systematic uncertainties added in quadrature
where the uncertainties arising from chiral extrapolation fit forms can be
observed to be dominant from Table 3 [50, 53].

##  VI Discussion on the bindings of $T_{bc}$

Figure 8: The binding energy calculated in this work in comparison to the
recent lattice QCD calculation [43] and other non-lattice determinations.
Estimates from non-lattice approaches seem to have a mixed conclusion where
several of them show shallow/deep binding and many others predicting an
unbound state.

At this stage, it is natural to assess, where our results stand among other
existing lattice QCD-based and phenomenological calculations of
$D\bar{B}$($D\bar{B}^{*}$) scattering in the isoscalar channels. Our
investigations presented in this work (Ref. [42]) indicate negative finite-
volume energy shifts in S-wave elastic scattering in
$D\bar{B}$($D\bar{B}^{*}$) meson systems. Further analysis of scattering
amplitude using finite volume Lüscher method points to the existence of a real
square integrable bound state with binding energy of approximately 40 MeV in
scalar (see Eq. (12)) and axialvector channels in Ref. [42]. While the
erorrbar is large in the estimation in the binding energy, the conclusion on
the attractiveness is robust. Recently another lattice QCD calculation with a
different lattice setup has also confirms the attractive nature of
interactions in both the channels, however, with a much lower binding, just
below the respective threshold energies [42].

In Figure 8, we present the results from various calculations on the binding
of $T_{bc}$ that have been predicted over the years. The results presented
include those determined using lattice QCD and the non-lattice methods,
separated by a horizontal line. In each plot, the vertical dashed lines are
the respective elastic thresholds ($D\bar{B}$ for $0^{+}$ and $D\bar{B}^{*}$
for $1^{+}$). Results to the left of this vertical line suggests a bound
state, whereas those lying to the right points to an unbound system. The
vertical green bands are the results from our calculations (left: this work,
right is from Ref. [42]) in perspective to those of other calculations. The
left plot shows the results for $0(0^{+})$ channel while the right one is for
those of $0(1^{+})$ channel. Estimates from non-lattice approaches seem to
have a large spread of the order of several hundred MeV across the threshold.
Both the lattice QCD results point towards the existence of bound states of
$T_{bc}$. However, more detailed lattice calculations are necessary to find
the exact locations and the nature of the bound state poles. Given these
predictions from lattice QCD calculations, and considering the importance of
the $T_{bc}$ states as discussed in the introduction, experimental searches
for these states would indeed be highly worthwhile in the near future.

Another interesting quantity to compare is the scattering length determining
the small momentum meson-meson interactions in different doubly heavy quark
systems ($T_{bb}$, $T_{bc}$ and $T_{cc}$) across various LQCD calculations. On
the left hand side of Figure 9, we present the inverse scattering length
($1/a_{0}$) at the physical $M_{ps}$ determined for these three exotic systems
from different lattice calculations [8, 12, 76, 42], where the [42] is our
previous study using the same setup as the present study, together with the
scattering length for the discovered $T_{cc}$ [10]. The only other LQCD study
of $D\bar{B}$ scattering [43] has also been included (faded point) for
completeness albeit the analysis not being extrapolated to physical pion mass
and to the continuum limit. The subscripts $(H)$ and $(L)$ in the $x$-axis
tick labels refer to two distinct procedures, the HALQCD and Lüscher-type
finite-volume prescription followed respectively, in extracting the scattering
length. The HALQCD procedure followed in Refs. [12, 76] provides quite precise
estimates, whereas the large uncertainty in the $BB^{*}$ scattering using
Lüscher-type procedure obscures extracting a possible trend, if any exist.
Subduing these uncertainties require more finite-volume energy levels to
constrain the amplitudes, which can be achieved either by extracting higher
excited states, or by studying more ensembles at different volumes or at
nonzero lab frame momenta [77]. In short, more followup studies involving
rigorous Lüscher-type finite-volume treatments with precise estimates are
highly desirable to make concrete procedure-independent statements on the
bindings in different doubly heavy systems. A similar comparison of the
scattering length in the S-wave scalar $D\bar{B}$ channel is shown on the
right panel of Figure 9.

Considering the differences in systematics between the two evaluations (this
work and Ref. [43]) for the $D\bar{B}^{(*)}$ systems, it is too early to argue
on the reasons for the observed discrepancies in the magnitude of scattering
length and binding energy. It could possibly be related to the fact that the
results from Ref. [43] lacks any chiral or continuum extrapolations or related
to the lack of access to the excited elastic excitations in our work, which
needs to be investigated further. Despite this discrepancy in the magnitude of
scattering length, either calculations support attractive interactions in
these systems. The large errors from our current study naturally indicate
equally large uncertainty in the binding energy, however the fact that a bound
state is seen is expected to be robust and consistent given the statistical
relevance. Here again, more followup studies with a large number of
interpolating operators and large statistics with rigorous Lüscher finite-
volume analysis is highly desirable to obtain precise results.

Figure 9: Left plot: The inverse scattering length for $DD^{*}$,
$D\bar{B}^{*}$ and $BB^{*}$ scatterings at the physical pion mass as
determined in Refs. [12, 42, 8, 76]. The faded point corresponds to a recent
lattice evaluation at an unphysically heavy pion mass [43]. Right plot: The
inverse scattering length ($1/a_{0}$) in $D\bar{B}$ scattering compared
between this work and Ref. [43].

##  VII Summary

In this work, we present a lattice QCD simulation of elastic S-wave $D\bar{B}$
scattering with explicitly exotic flavor $bc\bar{u}\bar{d}$ in the isoscalar
scalar quantum numbers [$I(J^{P})=0(0^{+})$]. We use four $N_{f}=2+1+1$
ensembles with dynamical Highly Improved Staggered Quark (HISQ) action
generated by the MILC collaboration, with the valence quarks, up to the charm
quark mass, realized using an overlap fermion action. The valence bottom
quarks are described using an improved NRQCD formulation.

Using the ground state energy levels, presented in Figure 4, we perform a
rigorous finite-volume amplitude analysis using Lüscher’s prescription. The
analysis accounts for the lattice spacing effects by parameterizing the
amplitude with a lattice spacing dependence, and taking the continuum limit
separately for the five light quark masses studied. The quark mass dependence
is then investigated to determine the elastic $D\bar{B}$ scattering length
$a_{0}^{phys}$ at the physical pion mass and the critical pseudoscalar mass
$M^{*}_{ps}$ at which $a_{0}$ diverges. The negative energy shifts in the
ground state finite-volume energies taken together with the positive estimates
for $a_{0}^{phys}$ (presented in equation 11) suggests an attractive
interaction between the $D$ and $\bar{B}$ mesons, that is strong enough to
form a real square integrable bound state with binding energy of
$-39(^{+4}_{-6})(^{~{}+8}_{-18})$ MeV.

Recently another lattice QCD calculation on the $D\bar{B}$ systems also
supports an attractive interaction between the mesons, however, with a smaller
binding and closer to the threshold [43]. Note that this calculation employed
bilocal two-meson-type operators at the source and sink and in extracting the
relevant elastic excitations in the $D\bar{B}$ channel. However, the
investigation is limited to two lattice ensembles with approximately similar
lattice spacings ($\sim 0.12$ fm), that is comparable to our coarsest lattice.
The apparent discrepancy in the binding energy, whether it is a result of
uncontrolled excited state contamination due to an asymmetric setup or if it
is a result of uncontrolled discretization effects, remains to be understood.
We leave this issue for the future lattice investigations.

In this study we are limited to rest frame ground states. While we are able to
extract the amplitude with a zero-range approximation, future investigations
with more rigor in extracting elastic excitations are necessary to constrain
the energy dependence of the amplitude over a wider energy range. This would
require meson-meson operators with zero overall momentum but individual
momentum projected mesons like in Ref. [43] to extract the elastic excitations
as well as meson-meson operators with non-zero overall momentum. Inclusion of
such operators is beyond the scope of our current setup. Additionally, future
studies involving fully dynamical simulations on a wider range of ensembles
with different fermion actions, high-statistics studies with lighter up and
down quark masses, and other improvements. These additional efforts would help
constrain the relevant scattering amplitude in a framework-independent manner.
In that journey, our calculation is an important step ahead where we have
clearly shown the presence of an attractive interaction within the $D\bar{B}$
system, with controlled cut-off uncertainties and finite volume effects. Our
findings offer a stride towards understanding the existence of $0(0^{+})$
$T_{bc}$, which could well be the next doubly-heavy bound tetraquark to be
discovered in the near future.

###### 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 the authors of Ref. [78] for making the
TwoHadronsInBox package utilized in this work. Computations were carried out
on the Cray-XC30 of ILGTI, TIFR. N. M. would also like to thank A. Salve and
K. Ghadiali for computational support.

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