DESY 24-061

Signature decay modes of the compact doubly-heavy tetraquarks
$T_{bb\bar{u}\bar{d}}$ and $T_{bc\bar{u}\bar{d}}$

Ahmed Ali ,

Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg,Germany

Ishtiaq Ahmed ,

National Center for Physics, Quaid-i-Azam University Campus, Islamabad, 44000,
Pakistan

Muhammad Jamil Aslam ,

Department of Physics, Quaid-i-Azam University, Islamabad, 45320, Pakistan

Abstract
Based on the expectations that the lowest-lying double-bottom tetraquark
$T_{bb\bar{u}\bar{d}}$ ($J^{P}=1^{+}$) and the bottom-charm tetraquark
$T_{bc\bar{u}\bar{d}}$ ($J^{P}=0^{+}$) are stable against strong and
electromagnetic decays, we work out a number of semileptonic and non-leptonic
weak decays of these hadrons, making use of the heavy quark symmetry. In doing
this, we concentrate on the exclusive decays involving also tetraquarks in the
final states, i.e., transitions such as $T_{bb\bar{u}\bar{d}}\to
T_{bc\bar{u}\bar{d}}(\ell^{-}\nu_{\ell},h^{-})$ and $T_{bc\bar{u}\bar{d}}\to
T_{cc\bar{u}\bar{d}}(\ell^{-}\nu_{\ell},h^{-})$, where
$h^{-}=\pi^{-},\rho^{-},a_{1}^{-}$. So far, only the $J^{P}=1^{+}$ tetraquark
$T_{cc\bar{u}\bar{d}}$ has been discovered, which we identify with the $I=0$
$T_{cc}^{+}$ object, compatible with $J^{P}=1^{+}$ and having the mass and
decay widths $\delta m=M(T_{cc}^{+})-(M(D^{*+})-M(D^{0}))=-360\pm
40^{+4}_{-0}$ keV and $\Gamma(T_{cc}^{+})=48^{+2}_{-14}$ keV. Experimental
discoveries of the transitions worked out here will go a long way in
establishing the nature of these tetraquarks as (mainly) compact four-quark
objects.

##  I Introduction

The physics of the multiquark hadrons has emerged as a recurrent theme in
high-energy physics, in particular, at the LHC. Prominent among these hadrons
is the double-charm tetraquark $T^{+}_{cc\bar{u}\bar{d}}$, called
$T_{cc}^{+}(3875)$, discovered by LHCb LHCb:2021vvq  in prompt proton-proton
collisions as a very narrow state (${\rm FWHM}=48$ keV) in the final state
$D^{0}D^{0}\pi^{+}$. It is found just below the $D^{*+}D^{0}$ threshold, has a
mass $m(T_{cc\bar{u}\bar{d}}^{+})\simeq 3,875$ GeV, and is compatible with
being an isoscalar ($I=0$) with the spin-parity $J^{P}=1^{+}$. Its
characteristic size, calculated from the binding energy $\Delta E=-360\pm
40^{+4}_{-0}$ keV, yields $R_{\Delta E}=7.49\pm 0.42$ fm, too large for a
compact hadron, nevertheless having a sizeable cross-section, estimated as
$\sigma(pp\to T_{cc\bar{u}\bar{d}}+X)=(45\pm 20)$ nb at $\sqrt{s}=13$ TeV for
the typical LHCb acceptance ($2<p_{T}<20~{}{\rm GeV};2<y<4.5)$ LHCb:2021auc .
Its nature (hadron molecule or a compact tetraquark) is still being debated.

With this discovery, as well as that of the double-charm $(C=2)$ baryon
$\Xi_{cc}^{++}$, having the quark content $(ucc)$ LHCb:2017iph , the focus of
the experimental and theoretical research is now on their heavier
counterparts, the bottom-charm hadrons ($C=1,B=1$), $\Xi_{bcq}$ and
$T_{bc\bar{u}\bar{d}}$, and eventually on the double-bottom ($B=2$) baryons
$\Xi_{bbq}$, $(q=u,d,s)$, and the double-bottom tetraquarks
$T_{bb\bar{u}\bar{d}}$ and $T_{bb\bar{u}\bar{s}}$. Theoretically, stable
(w.r.t. strong decays) heavy multiquark states were predicted a long time ago
Ader:1981db ; Manohar:1992nd ; Ebert:2007rn . Their phenomenology has been
studied in several competing theoretical approaches. Among other frameworks,
the ones based on diquarks Jaffe:1976ig ; Jaffe:2003sg  have been extensively
used to model compact multiquark states Ali:2019roi . More recently, heavy
quark-heavy diquark symmetry has been invoked to relate singly heavy mesons,
anti-baryons, and double-heavy baryons and tetraquarks Mehen:2017nrh ;
Eichten:2017ffp . Lately, the role of the local diquark-antidiquark operators
in the spectroscopy of tetraquark hadrons has also been investigated using the
Lattice-QCD framework Bicudo:2021qxj ; Alexandrou:2023cqg ;
Radhakrishnan:2024ihu ; Alexandrou:2024iwi . In particular, the masses of the
ground-state doubly-heavy tetraquarks have been calculated using both the
meson-meson and the diquark-antidiquark operators. In one such study, carried
out for the $T_{bb\bar{u}\bar{d}}$ and $T_{bb\bar{u}\bar{s}}$ tetraquarks, an
approximately even mix of meson-meson and diquark-antidiquark component are
found Bicudo:2021qxj . Further studies are needed to quantify this prediction,
and experimental proofs of the compact nature of the multiquark states
unambiguously are required. We argue here that the weak transitions of doubly-
heavy tetraquarks may also reveal the existence of heavy diquarks (and
antidiquarks), and we work out some characteristic decays reflecting the
underlying compact structures.

We note that the lowest-mass double-bottom tetraquarks $T_{bb\bar{u}\bar{d}}$
and $T_{bb\bar{u}\bar{s}}$ are estimated lie below their respective strong-
decay thresholds. A recent Lattice-QCD simulation Alexandrou:2024iwi , puts
their masses (measured w.r.t. their respective thresholds) as $\delta
m=-100\pm 10^{+36}_{-43}$ MeV for the $T_{bb\bar{u}\bar{d}}$ and $\delta
m=-30\pm 3^{+11}_{-31}$ MeV for the $T_{bb\bar{u}\bar{s}}$, similar to earlier
estimates Alexandrou:2024iwi-Fig-13 . Some weak decays of these tetraquarks
have been worked out in a number of papers Ali:2018ifm ; Agaev:2020dba ;
Agaev:2020mqq ; Hernandez:2019eox , providing estimates for the lifetimes and
branching ratios. Of particular interest are the inclusive decays of doubly-
bottom tetraquarks $T_{bb\bar{u}\bar{d}}\to B_{c}^{(*)}+X$, as well as of the
corresponding baryons Kiselev:2001fw ; Ridgway:2019zks  $\Xi_{bbq}\to
B_{c}^{(*)}+X$, with detached $B_{c}$-vertex Gershon:2018gda  due to the
predicted long lifetimes of these hadrons, estimated to lie in the range
$0.4-0.8$ ps Ali:2018ifm ; Kiselev:2001fw ; Karliner:2014gca ;
Berezhnoy:2018bde .

Until recently, there was no consensus on the issue of whether the lowest-mass
($C=1,B=1$) tetraquark state is stable against strong and electromagnetic
decays, as well as on the assignment of the quantum numbers
Radhakrishnan:2024ihu-Fig-8 . However, two recent Lattice-based estimates
Radhakrishnan:2024ihu ; Alexandrou:2023cqg  have posted the mass of the
lowest-lying state, with spin-parity $J^{P}=0^{+}$, below the $DB$-threshold.
If confirmed by further theoretical developments, this would make a strong
case for the $T_{bc\bar{u}\bar{d}}$ tetraquark as the first multiquark state
to decay weakly. The corresponding $J^{P}=1^{+}$ tetraquark is also found in
these studies Radhakrishnan:2024ihu ; Alexandrou:2023cqg  to be below the
$DB^{*}$-threshold, though it would decay via electromagnetic transition to
$DB\gamma$ or radiatively to the lower-mass $J^{P}=0^{+}$ state. This
information is helpful for the experimental searches of $T_{bc\bar{u}\bar{d}}$
tetraquark in focusing on the final states, which can only be reached by
charged current weak interactions.

With this hindsight, we shall concentrate here on the weak decays of the
double-bottom $T_{bb\bar{u}\bar{d}}$ and the bottom-charm
$T_{bc\bar{u}\bar{d}}$ tetraquarks in which the meson-meson and the diquark-
antidiquark components are established on the Lattice. Weak decays from the
meson-meson components (such as $BB^{(*)}$, $DB^{(*)}$) follow the known
patterns, well documented in the Particle data Group tables
ParticleDataGroup:2022pth  and are included in the current experimental search
strategies Blusk:21 ; Polyakov:23 . The ones, following from the diquark-
antidiquark component discussed in this Letter are new, or at least have not
been studied so far quantitatively. In particular, they lead to weak decays
involving tetraquarks both in the initial and final states, i.e., they induce
transitions, such as $T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}+X$ and
$T_{bc\bar{u}\bar{d}}\to T_{cc\bar{u}\bar{d}}+X$. We work out some exclusive
decays, $T_{bb\bar{u}\bar{d}}\to
T_{bc\bar{u}\bar{d}}(\ell^{-}\nu_{\ell},h^{-})$ and $T_{bc\bar{u}\bar{d}}\to
T_{cc\bar{u}\bar{d}}(\ell^{-}\nu_{\ell},h^{-})$, where
$\ell^{-}\nu_{\ell}=e^{-}\nu_{e},\mu^{-}\nu_{\mu},\tau^{-}\nu_{\tau}$ and
$h^{-}=\pi^{-},\rho^{-},a_{1}^{-}$. They represent signature decay modes of
compact tetraquarks, reflecting the heavy diquark configurations in their
wave-functions, which dominate over the mesonic configurations at short inter-
heavy quark distances Bicudo:2021qxj . We hope that these weak decay modes of
the doubly-heavy hadrons reflecting their diquark components will be included
in the experimental search strategies.

Since there is no annihilation or $W$-exchange diagrams allowed in these
transitions, these decays take place via the so-called color-allowed tree
diagrams, shown in Figs. 1, 2. Here, $b$-quark acts as the active (or valence)
quark. There are two of them in the double-bottom tetraquarks
$T_{bb\bar{u}\bar{d}}$ and one in $T_{bc\bar{u}\bar{d}}$. Weak interaction
induces the (dominant) $b\to c$ transition, but the crucial difference is that
the $bb$-diquark emerges in the weak decays as an intact $bc$ diquark.
Invoking the heavy diquark - heavy quark symmetry, we relate the weak decays
of these tetraquarks to the corresponding $B\to(D,D^{*})$ weak decays of the
$B$-mesons. The decay rates are worked out in the heavy quark symmetry limit,
using the HQET (heavy quark effective theory) framework. The reported
branching ratios are encouraging to be measured. We note that the High-
luminosity LHC (HL-LHC) at CERN Azzi:2019yne ; Cerri:2018ypt , and the planned
Tera-$Z$ factories FCC:2018evy ; CEPCStudyGroup:2018ghi  are estimated to have
large enough data samples Ali:2018ifm ; Ali:2018xfq  to carry out the required
measurements.

In Section II, we calculate the semileptonic decays
$T_{bb\bar{u}\bar{d}}\left(J^{P}=1^{+}\right)\to
T_{bc\bar{u}\bar{d}}\left(J^{P}=0^{+}\right)\ell^{-}\nu_{\ell}$. The decay
$T_{bb\bar{u}\bar{d}}\left(J^{P}=1^{+}\right)\to
T_{bc\bar{u}\bar{d}}\left(J^{P}=1^{+}\right)\ell^{-}\nu_{\ell}$ are presented
in Section III. Non-Leptonic decays $T_{bb\bar{u}\bar{d}}\to
T_{bc\bar{u}\bar{d}}h^{-}$ ($h^{-}=\pi^{-},\rho^{-},a_{1}^{-}$) are discussed
in Section IV. The semileptonic and non-leptonic decays
$T_{bc\bar{u}\bar{d}}\left(J^{P}=0^{+}\right)\to
T_{cc\bar{u}\bar{d}}\left(J^{P}=1^{+}\right)(\ell^{-}\nu_{\ell},h^{-})$ are
discussed in Section V. We conclude with a summary in Section VI.

##  II Semileptonic decays $T_{bb\bar{u}\bar{d}}\left(J^{P}=1^{+}\right)\to
T_{bc\bar{u}\bar{d}}\left(J^{P}=0^{+}\right)\ell^{-}\nu_{\ell}$

The decay $T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}\ell^{-}\nu_{\ell}$ is
governed by the following effective Hamiltonian

$\mathcal{H}_{\text{eff}}=4\frac{G_{F}}{\sqrt{2}}V_{cb}\left(\bar{c}\gamma^{\mu}P_{L}b\right)\left(\bar{\ell}\gamma_{\mu}P_{L}\nu_{\ell}\right),$
(1)

where $P_{L}=\frac{1-\gamma_{5}}{2}$, and the corresponding Feynman diagram is
shown in Fig. 1 (a).

|
---|---
(a) |  (b)

Figure 1: Feynman diagram for (a) $T_{bb\bar{u}\bar{d}}\to
T_{bc\bar{u}\bar{d}}(\ell^{-}\nu_{\ell})$,and (b) $T_{bc\bar{u}\bar{d}}\to
T_{cc\bar{u}\bar{d}}(\ell^{-}\nu_{\ell})$ where $\ell,e,\mu,\;\tau$.

As discussed in the introduction, we assume that the initial state tetraquark
$T_{bb\bar{u}\bar{d}}$ has the spin-parity $J^{P}=1^{+}$ (axial-vector) and
the final state $T_{bcud}$ has $J^{P}=0^{+}$ (scalar).

We shall use the heavy quark effective theory (HQET) Manohar:2000dt ;
Neubert:1993mb  to calculate the matrix elements $\left\langle
T_{bc\bar{u}\bar{d}}\left(p^{\prime}\right)\left|\bar{c}\gamma^{\mu}b\right|T_{bb\bar{u}\bar{d}}\left(p,\varepsilon\right)\right\rangle$
and $\left\langle
T_{bc\bar{u}\bar{d}}\left(p^{\prime}\right)\left|\bar{c}\gamma^{\mu}\gamma_{5}b\right|T_{bb\bar{u}\bar{d}}\left(p,\varepsilon\right)\right\rangle$.
To that end, we recall that the superfield combining the pseudo-scalar and
vector mesons in HQET can be written as

$\mathcal{H}_{v}^{\left(Q\right)}=\frac{1+\not{v}}{2}\left[P_{v}^{*\mu\left(Q\right)}\gamma_{\mu}-P_{v}^{\left(Q\right)}\gamma_{5}\right],$
(2)

where $Q=b,c$ and for the vector field $v\cdot P_{v}^{\left(Q\right)}=0.$
Similarly, for the axial-vector and the scalar, we just have to multiply it
with $\gamma_{5}$ from the right, i.e.,

$\mathcal{H}_{v}^{\left(Q\right)}=\frac{1+\not{v}}{2}\left[P_{v}^{*\mu\left(Q\right)}\gamma_{\mu}\gamma_{5}-P_{v}^{\left(Q\right)}\right],$
(3)

where the axial vector part is

$\mathcal{H}_{A}^{\left(Q\right)}=\frac{1+\not{v}}{2}\left[P_{v}^{*\mu\left(Q\right)}\gamma_{\mu}\gamma_{5}\right].$
(4)

The corresponding conjugate field can be obtained as

$\displaystyle\bar{\mathcal{H}}_{A}^{\left(Q\right)}$
$\displaystyle=\gamma^{0}\mathcal{H}_{A}^{\dagger\left(Q\right)}\gamma^{0}=\left[P_{v}^{*\mu\dagger\left(Q\right)}\gamma^{0}\gamma_{5}\gamma_{\mu}^{\dagger}\right]\frac{1+\not{v}^{\dagger}}{2}\gamma^{0}=-\left[P_{v}^{*\mu\dagger\left(Q\right)}\gamma_{5}\gamma^{0}\gamma_{\mu}\right]\gamma^{0}\gamma^{0}\frac{1+\not{v}^{\dagger}}{2}\gamma^{0},$
$\displaystyle=\left[P_{v}^{*\mu\dagger\left(Q\right)}\gamma_{\mu}\gamma_{5}\right]\frac{1+\not{v}}{2}$
(5)

since $\gamma^{0}\gamma_{\mu}^{\dagger}\gamma^{0}=\gamma_{\mu}$ and
$\left\\{\gamma_{0,\mu},\gamma_{5}\right\\}=0.$ The important properties are

$\not{v}\mathcal{H}_{A}^{\left(Q\right)}=\mathcal{H}_{A}^{\left(Q\right)},\quad\mathcal{H}_{A}^{\left(Q\right)}\not{v}=\mathcal{H}_{A}^{\left(Q\right)}.$
(6)

The conjugate of the scalar part can be written as

$\bar{\mathcal{H}}_{S}^{\left(Q\right)}=\gamma^{0}\frac{1+\not{v}^{\dagger}}{2}P_{v}^{{}^{\dagger}\left(Q\right)}\gamma^{0}=\frac{1+\not{v}}{2}P_{v}^{{}^{\dagger}\left(Q\right)}.$
(7)

In the HQET, the matrix elements can be parameterized as

$\displaystyle\frac{\left\langle
T_{bc\bar{u}\bar{d}}\left(v^{\prime}\right)\left|\bar{c}\gamma^{\mu}b\right|T_{bb\bar{u}\bar{d}}\left(v,\varepsilon\right)\right\rangle}{\sqrt{m_{T}m_{T^{\prime}}}}$
$\displaystyle=h_{1}\left(w\right)i\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{\nu}v_{\alpha}v_{\beta}^{\prime},$
$\displaystyle\frac{\left\langle
T_{bc\bar{u}\bar{d}}\left(v^{\prime}\right)\left|\bar{c}\gamma^{\mu}\gamma_{5}b\right|T_{bb\bar{u}\bar{d}}\left(v,\varepsilon\right)\right\rangle}{\sqrt{m_{T}m_{T^{\prime}}}}$
$\displaystyle=\left(h_{2}\left(w\right)\left(w+1\right)\varepsilon^{\mu}-h_{3}\left(w\right)\left(\varepsilon\cdot
v^{\prime}\right)v^{\mu}-h_{4}\left(w\right)\left(\varepsilon\cdot
v\right)v^{\prime\mu}\right).$ (8)

where $m_{T}$ and $m_{T^{\prime}}$ denote the masses of initial
$\left(T_{bb\bar{u}\bar{d}}\right)$ and final state
$\left(T_{bc\bar{u}\bar{d}}\right)$ tetraquarks, respectively. The
corresponding form factors for the transitions
$B\to(D,D^{*})\ell^{-}\nu_{\ell}$ have been investigated at length, but the
ones for the tetraquarks are yet to be studied.

Using the spin symmetry to relate the various form factors, at the leading
order (LO) in heavy quark mass, we can write

$\left\langle\mathcal{H}_{S}^{\left(Q^{\prime}\right)}\left|\bar{Q}^{\prime}\Gamma
Q\right|\mathcal{H}_{A}^{\left(Q\right)}\right\rangle=-\xi\left(w\right)\text{Tr}\left\\{\bar{\mathcal{H}}_{S}^{\left(Q^{\prime}\right)}\Gamma\mathcal{H}_{A}^{\left(Q\right)}\right\\},$
(9)

where $\Gamma$ is a particular combination of $\gamma-$matrices and
$Q^{\prime},Q$ are the $c,b$ quarks with velocities $v^{\prime},v$,
respectively. It involves a single non-perturbative function $\xi\left(w\equiv
v\cdot v^{\prime}\right)$, i.e., the Isgur-Wise function. We note that there
are two $b$ quarks present in $T_{bb\bar{u}\bar{d}}$, and in the weak
transition only one active $b$-quark is involved at a time, with the other
treated as a spectator. There are no annihilation or $W^{\pm}$-exchange
diagrams involving the two $b$ quarks. However, due to the multiplicity of the
$b$-quarks in the initial hadron, the decay rate is to be multiplied by 2.

Using Eqs. (4) and (7) in Eq. (9), for vector and axial-vector currents, we
have

$\displaystyle\left\langle\mathcal{H}_{S}^{\left(Q^{\prime}\right)}\left|\bar{c}_{v^{\prime}}\gamma^{\mu}b_{v}\right|\mathcal{H}_{A}^{\left(Q\right)}\right\rangle$
$\displaystyle=-\xi\left(w\right)\text{Tr}\left\\{P_{v^{\prime}}^{{}^{\dagger}\left(Q\right)}\frac{1+\not{v}^{\prime}}{2}\gamma^{\mu}\frac{1+\not{v}}{2}\left[P_{v}^{*\alpha\left(Q\right)}\gamma_{\alpha}\gamma_{5}\right]\right\\}$
$\displaystyle=i\xi\left(w\right)\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{\nu}v_{\alpha}v_{\beta}^{\prime},$
$\displaystyle\left\langle\mathcal{H}_{S}^{\left(Q^{\prime}\right)}\left|\bar{c}_{v^{\prime}}\gamma^{\mu}\gamma_{5}b_{v}\right|\mathcal{H}_{A}^{\left(Q\right)}\right\rangle$
$\displaystyle=-\xi\left(w\right)\text{Tr}\left\\{P_{v^{\prime}}^{{}^{\dagger}\left(Q\right)}\frac{1+\not{v}^{\prime}}{2}\gamma^{\mu}\gamma_{5}\frac{1+\not{v}}{2}\left[P_{v}^{*\alpha\left(Q\right)}\gamma_{\alpha}\gamma_{5}\right]\right\\}$
$\displaystyle=\xi\left(w\right)\left[\left(w+1\right)\varepsilon^{\mu}-\left(\varepsilon\cdot
v^{\prime}\right)v^{\mu}\right].$ (10)

Comparing Eq. (8) and Eq. (10), one gets

$h_{1}=h_{2}=h_{3}=\xi\left(w\right)$ (11)

and $h_{4}=0.$

Therefore, the hadronic part of the transition amplitude becomes

$\displaystyle\mathcal{M}^{\mu}$
$\displaystyle=-\frac{G_{F}}{\sqrt{2}}V_{cb}\xi\left(w\right)\sqrt{m_{T}m_{T^{\prime}}}\left[\left(w+1\right)\varepsilon^{\mu}-\left(\varepsilon\cdot
v^{\prime}\right)v^{\mu}-i\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{\nu}v_{\alpha}v_{\beta}^{\prime}\right].$
(12)

In $B\to\left(D,D^{*}\right)\ell^{-}\nu_{\ell}$ decays, one has

$\displaystyle\frac{\left\langle
D\left(v^{\prime}\right)\left|\bar{c}\gamma^{\mu}b\right|B\left(v\right)\right\rangle}{\sqrt{m_{B}m_{D}}}$
$\displaystyle=h_{+}\left(w\right)\left(v+v^{\prime}\right)^{\mu}+h_{-}\left(w\right)\left(v-v^{\prime}\right)^{\mu},$
$\displaystyle\frac{\left\langle
D^{*}\left(v^{\prime},\varepsilon^{\prime}\right)\left|\bar{c}\gamma^{\mu}b\right|B\left(v\right)\right\rangle}{\sqrt{m_{B}m_{D^{*}}}}$
$\displaystyle=ih_{V}\left(w\right)\varepsilon^{\mu\nu\alpha\beta}\varepsilon^{*\prime\nu}v^{\prime\alpha}v^{\beta},$
$\displaystyle\frac{\left\langle
D^{*}\left(v^{\prime},\varepsilon^{\prime}\right)\left|\bar{c}\gamma^{\mu}\gamma_{5}b\right|B\left(v\right)\right\rangle}{\sqrt{m_{B}m_{D^{*}}}}$
$\displaystyle=h_{A_{1}}\left(w\right)\left(w+1\right)\varepsilon^{*\prime\mu}-h_{A_{2}}\left(w\right)\varepsilon^{*\prime}\cdot
vv^{\mu}-h_{A_{3}}\left(w\right)\varepsilon^{*\prime}\cdot vv^{\prime\mu}.$
(13)

Using the procedure adopted in Eq. (10), with the field defined in Eq. (2),
one gets the well-known Isgur-Wise function for the weak transitions in
$B$-meson decays Isgur:1989vq ; Isgur:1990yhj , i.e.,

$h_{+}\left(w\right)=h_{1}\left(w\right)=h_{2}\left(w\right)=h_{3}\left(w\right)=\xi^{B\to
D^{\left(*\right)}}\left(w\right).$ (14)

In the symmetry limit, there is a single Isgur-Wise function, for both the
$B\to D$ and $B\to D^{*}$ transitions, which is normalised at the symmetry-
point: $\xi^{B\to D^{\left(*\right)}}\left(w=1\right)=1$. Symmetry-breaking
corrections of ${\cal O}(\alpha_{s}(m_{b})$ and power corrections of ${\cal
O}(\Lambda_{\rm QCD}/m_{b})$ have been calculated Falk:1990yz . They yield
$\xi^{B\to D}(w=1)=0.98\pm 0.07$ Ligeti:1993hw . For the $B\to
D^{*}\ell^{-}\nu_{\ell}$, Luke’s theorem protects the leading order
corrections Luke:1990eg , with the second-order yielding $\xi^{B\to
D^{*}}(w=1)=0.91\pm 0.03$ Neubert:1991td ; Shifman:1994jh ; Czarnecki:1996gu .
They have been used in the precise determination of the CKM matrix element
$|V_{cb}|$ from the exclusive $B$-meson decays Mannel:2023pdg .

Thus, in the symmetry limit, HQET relates the two form factors appearing in
the semileptonic decays of the double-bottom tetraquark and $B$ mesons

$\xi_{T\to
T^{\prime}}\left(w\right)=\frac{\sqrt{m_{B}m_{D^{\left(*\right)}}}}{\sqrt{m_{T}m_{T^{\prime}}}}\xi^{B\to
D^{\left(*\right)}}\left(w\right).$ (15)

This relation will be modified including the process-dependent symmetry-
breaking effects. They pertain to the interactions involving the spectators,
which differ for the $B$-mesons and the double-heavy tetraquarks. We expect
them to be subdominant and the relation in Eq. (15) a good approximation.
Given a model for the double-heavy tetraquark wave-functions, they can be
calculated, in principle, but are beyond the scope of this Letter.

We now discuss the overall normalization related to the Fock-space composition
of the tetraquarks in question. This is also related to the tetraquark wave-
function. In this, we follow the formulation used in the study of the creation
operators for the $T_{bb\bar{u}\bar{d}}$ tetraquark, whose ground state has
the quantum numbers $J^{P}=1^{+}$ Bicudo:2021qxj , as also assumed here. The
wave-function $|\Phi_{b,d}\rangle$ of this tetraquark is spanned by the two
components in question, meson-meson
$|\Phi_{BB,(1+\gamma_{0})\gamma_{5}}\rangle$, and diquark-antidiquark
$|\Phi_{Dd,(1+\gamma_{0})\gamma_{5}}\rangle$. Writing the wave-function

$|\Phi_{b,d}\rangle=b|\Phi_{b,d}\rangle+d|\Phi_{Dd,(1+\gamma_{0})\gamma_{5}}\rangle,$
(16)

allows to define the ratios

$\omega_{BB}(r)=\frac{|b|^{2}}{|b|^{2}+|d|^{2}};~{}~{}~{}\omega_{Dd}(r)=\frac{|d|^{2}}{|b|^{2}+|d|^{2}},$
(17)

with $\omega_{Dd}(r)=1-\omega_{BB}(r)$, which can be interpreted as the
relative weights of a meson-meson and diquark-antidiquark at the
$bb$-separation $r$ in the ground state with $J^{P}=1^{+}$. Lattice studies
yield that the diquark-antidiquark component $\omega_{Dd}(r)$ dominates over
the meson-meson component for $r<0.20$ fm.

For the general case in which both the diquarkonic and mesonic components are
present in the Fock-space of the tetraquarks, one has to project out the
diquark component to determine the normalization of the decay rates discussed
here. It is related to the fraction $\omega_{Dd}(r)$, being its integral
obtained by integrating $\omega_{Dd}(r)$ over the size of the tetraquarks.
This is a priori not known, but the general expectations are that the compact
double-bottom tetrahadrons should have a similar hadronic size as a $B$ meson,
and the diquarkonic fraction could even be dominant. We call this quantity
$f_{Dd}(bb)^{2}$, where we admit the possibility that this fraction may depend
on the heavy quark flavor, and take that into account as the compact-hadronic
fraction of the tetraquark in estimating the transition rates. Thus, for the
$T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}$ transition, Eq. (12) takes the
form

$\displaystyle\mathcal{M}_{T_{bb}\to T_{bc}}^{\mu}$
$\displaystyle=-\frac{G_{F}}{\sqrt{2}}V_{cb}\sqrt{m_{T}m_{T^{\prime}}}\xi_{T\to
T^{\prime}}\left(w\right)\left(f_{Dd}(bb)\right)\left[\left(w+1\right)\varepsilon^{\mu}-\left(\varepsilon\cdot
v^{\prime}\right)v^{\mu}-i\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{\nu}v_{\alpha}v_{\beta}^{\prime}\right].$
(18)

The corresponding leptonic part is

$\mathcal{L}_{\mu}=\bar{u}\left(p_{1},m_{\ell}\right)\gamma_{\mu}\left(1-\gamma_{5}\right)v\left(p_{2},m_{\nu_{\ell}}\right).$
(19)

The decay rate is determined by the expression

$\frac{d\Gamma}{dw}=\frac{1}{(2\pi)^{3}}\frac{1}{32m_{T}^{3}}\left|\bar{\mathcal{A}}_{T\to
T^{\prime}}\right|^{2},$ (20)

where $\left|\bar{\mathcal{A}}_{T\to T^{\prime}}\right|^{2}$ can be calculated
from the square of the amplitude (18) after summing over the polarizations of
initial state tetraquark, i.e.,
$\sum_{\lambda}\varepsilon_{\mu}\left(v,\lambda\right)\varepsilon_{\nu}^{*}\left(v,\lambda\right)=-g_{\mu\nu}+v_{\mu}v_{\nu}/v^{2}$,
and contracting it with the square of the spin-summed leptonic amplitude.

Using this, along with the decay kinematics in Eq. (20), the differential
decay rate is expressed as

$\displaystyle\frac{d\Gamma}{dw}$ $\displaystyle=$ $\displaystyle
N_{b}\frac{G{{}_{F}}^{2}\left|V_{bc}\right|^{2}m_{T^{\prime}}}{384\pi^{3}m_{T}^{2}}\left|\xi_{T\to
T^{\prime}}\left(w\right)\right|^{2}\left(f_{Dd}(bb)\right)^{2}\sqrt{w^{2}-1}\sqrt{1-\frac{m_{\ell}^{2}}{s}}\bigg{[}3m_{\ell}^{4}\left(\left(m_{T}+m_{T^{\prime}}\right)^{2}+4m_{T}m_{T^{\prime}}w\right)$
(21)
$\displaystyle-\frac{4m_{\ell}^{2}}{s}m_{T}m_{T^{\prime}}\left(w+1\right)\left(6m_{T}^{4}+m_{T}^{3}m_{T^{\prime}}\left(5-11w\right)-2m_{T}^{2}m_{T^{\prime}}^{2}\left(2w^{2}+5w-1\right)+m_{T}m_{T^{\prime}}^{3}\left(13w+5\right)-6m_{T^{\prime}}^{4}\right)$
$\displaystyle+8m_{T}^{2}m_{T^{\prime}}^{2}\left(w+1\right)\left(\left(m_{T}^{2}+m_{T^{\prime}}^{2}\right)\left(5w+1\right)-2m_{T}m_{T^{\prime}}\left(4w^{2}+w+1\right)\right)\bigg{]},$

where $s=q^{2}=m_{T}^{2}+m_{T^{\prime}}^{2}-2m_{T}m_{T^{\prime}}w$ is the
square of momentum transfer, and $N_{b}=2$ for $T_{bb\bar{u}\bar{d}}$ state.

In the massless lepton case, Eq. (21) simplifies to

$\frac{d\Gamma}{dw}=N_{b}\frac{G_{F}^{2}}{48\pi^{3}}V_{cb}^{2}m_{T^{\prime}}^{3}\left|\xi_{T\to
T^{\prime}}\left(w\right)\right|^{2}\left(w+1\right)^{\frac{3}{2}}\sqrt{w-1}\left(f_{Dd}(bb)\right)^{2}\left[\left(m_{T}^{2}+m_{T^{\prime}}^{2}\right)\left(5w+1\right)-2m_{T}m_{T^{\prime}}\left(4w^{2}+w+1\right)\right].$
(22)

Using the form factors relation defined in Eq. (15), Eq. (22) yields

$\displaystyle\frac{d\Gamma}{dw}$
$\displaystyle=N_{b}\frac{G_{F}^{2}}{48\pi^{3}}V_{cb}^{2}\frac{m_{B}m_{D}}{m_{T}}m_{T^{\prime}}^{2}\left|\xi^{B\to
D}\left(w\right)\right|^{2}\left(w+1\right)^{\frac{3}{2}}\sqrt{w-1}\left(f_{Dd}(bb)\right)^{2}$
$\displaystyle\times\left[\left(m_{T}^{2}+m_{T^{\prime}}^{2}\right)\left(5w+1\right)-2m_{T}m_{T^{\prime}}\left(4w^{2}+w+1\right)\right].$
(23)

To calculate the lepton energy spectrum, and the semileptonic decay rate, we
need to parametrize the Isgur-Wise function for the weak decays of the heavy-
to-heavy tetraquarks. We use the corresponding form for the $B\to(D,D^{*})$
form factors in the zero recoil expansion (c.f. Eq. (41) Caprini:1997mu ):

$\xi^{B\to D}\left(w\right)=\xi^{B\to
D}\left(1\right)\left[1-8\rho^{2}_{1}z+\left(51\rho^{2}_{1}-10\right)z^{2}-\left(252\rho^{2}_{1}-84\right)z^{3}\right],$
(24)

where $z=\left(\sqrt{w+1}-\sqrt{2}\right)/\left(\sqrt{w+1}+\sqrt{2}\right)$,
and $\rho^{2}_{1}$ is a slope parameter at zero-recoil, bounded between
$-0.14<\rho^{2}_{1}<1.54$. We anticipate that the slope of the Isgur-Wise
functions in the two cases, namely $B\to(D,D^{*})$ and
$T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}$, are expected to differ from
each other, but for the sake of definiteness, we shall use the same range for
the two cases. The rationale behind this assumption is that in both cases, the
recoil momenta of the spectators are similar and small.

Using the numerical values for the various input parameters given in Table 1,
and with $\left(f_{Dd}(bb)\right)^{2}=1$, yields the following decay rates:

$\displaystyle\Gamma\left(T_{bb\bar{u}\bar{d}}(J^{P}=1^{+})\to
T_{bc\bar{u}\bar{d}}(J^{P}=0^{+})\ell^{-}\nu_{\ell}\right)$
$\displaystyle=\left(1.94,\;0.68\right)\times 10^{-11}\text{MeV}.$ (25)

The first number corresponds to the $e^{-}\nu_{e}$ and $\mu^{-}\nu_{\mu}$
cases, and the second for the $\tau^{-}\nu_{\tau}$ case. In a more realistic
case, $\left(f_{Dd}(bb)\right)^{2}<1$, and the decay rates will be decreased
by this amount. Further studies are needed to quantify this quantity. However,
this can also be determined by the branching ratios discussed here, once they
are measured.

Using the total width $\Gamma_{\rm total}(T_{bb\bar{u}\bar{d}})=8.2\times
10^{-10}$ MeV, which is derived from the lifetime
$\tau(T_{bb\bar{u}\bar{d}})=0.8$ps Ali:2018ifm , we get the following
branching ratios:

$\displaystyle{\cal B}\left(T_{bb\bar{u}\bar{d}}(J^{P}=1^{+})\to
T_{bc\bar{u}\bar{d}}(J^{P}=0^{+})\ell^{-}\nu_{\ell}\right)$
$\displaystyle=2.4\%~{}({\text{for}}~{}\ell^{-}\nu_{\ell}=e^{-}\nu_{e},\;\mu^{-}\nu_{\mu}),$
$\displaystyle{\cal B}\left(T_{bb\bar{u}\bar{d}}(J^{P}=1^{+})\to
T_{bc\bar{u}\bar{d}}(J^{P}=0^{+})\tau^{-}\nu_{\tau}\right)$
$\displaystyle=8.2\times 10^{-3}.$ (26)

##  III Semileptonic decays $T_{bb\bar{u}\bar{d}}\left(J^{P}=1^{+}\right)\to
T_{bc\bar{u}\bar{d}}\left(J^{P}=1^{+}\right)\ell^{-}\nu_{\ell}$

In this section, we will consider the final state tetraquark
$T_{bc\bar{u}\bar{d}}$ with $J^{P}=1^{+}$, therefore, $T_{bb\bar{u}\bar{d}}\to
T_{bc\bar{u}\bar{d}}$ correspond to an axial-vector to axial-vector
transition. With the effective Hamiltonian describing the $b\to
c\ell^{-}\nu_{\ell}$ transition given in Eq. (1), the matrix elements in HQET
can be parameterized as

$\displaystyle\frac{\left\langle
T_{bc\bar{u}\bar{d}}\left(v^{\prime},\varepsilon^{\prime}\right)\left|\bar{c}\gamma^{\mu}b\right|T_{bb\bar{u}\bar{d}}\left(v,\varepsilon\right)\right\rangle}{\sqrt{m_{T}m_{T^{\prime}}}}$
$\displaystyle=-\varepsilon^{*\prime}\cdot\varepsilon\left[h_{1}\left(w\right)\left(v+v^{\prime}\right)^{\mu}+h_{2}\left(w\right)\left(v-v^{\prime}\right)^{\mu}\right]$
$\displaystyle+h_{3}\left(w\right)\varepsilon^{*\prime}\cdot
v\varepsilon^{\mu}+h_{4}\left(w\right)\varepsilon\cdot
v^{\prime}\varepsilon^{*\prime\mu}-\varepsilon\cdot
v^{\prime}\varepsilon^{*\prime}\cdot
v\left[h_{5}\left(w\right)v^{\mu}+h_{6}\left(w\right)v^{\prime\mu}\right],$
$\displaystyle\frac{\left\langle
T_{bc\bar{u}\bar{d}}\left(v^{\prime},\varepsilon^{\prime}\right)\left|\bar{c}\gamma^{\mu}\gamma_{5}b\right|T_{bb\bar{u}\bar{d}}\left(v,\varepsilon\right)\right\rangle}{\sqrt{m_{T}m_{T^{\prime}}}}$
$\displaystyle=i\varepsilon^{\mu\nu\alpha\beta}\left\\{\varepsilon_{\alpha}\varepsilon_{\beta}^{*\prime}\left[h_{7}\left(w\right)\left(v+v^{\prime}\right)_{\nu}+h_{8}\left(w\right)\left(v-v^{\prime}\right)_{\nu}\right]\right.$
$\displaystyle+v_{\alpha}^{\prime}v_{\beta}\left[h_{9}\left(w\right)\varepsilon^{*\prime}\cdot
v\varepsilon_{\nu}+h_{10}\left(w\right)\varepsilon\cdot
v^{\prime}\varepsilon_{\nu}^{*}\right],$ (27)

where $m_{T}$ and $m_{T^{\prime}}$ denote the masses of the initial and final
state tetraquarks, $T_{bbud}$ and $T_{bcud}$, respectively.

For $1^{+}\to 1^{+}$ transitions, the heavy quark spin symmetry implies

$\left\langle\mathcal{H}_{A}^{\left(Q^{\prime}\right)}\left|\bar{Q}^{\prime}\Gamma
Q\right|\mathcal{H}_{A}^{\left(Q\right)}\right\rangle=-\xi\left(w\right)\text{Tr}\left\\{\bar{\mathcal{H}}_{A}^{\left(Q^{\prime}\right)}\Gamma\mathcal{H}_{A}^{\left(Q\right)}\right\\},$
(28)

which on using Eqs. (4) and (5) gives

$\displaystyle\left\langle\mathcal{H}_{A}^{\left(Q^{\prime}\right)}\left|\bar{c}_{v^{\prime}}\gamma^{\mu}b_{v}\right|\mathcal{H}_{A}^{\left(Q\right)}\right\rangle$
$\displaystyle=-\xi\left(w\right)\text{Tr}\left\\{\left[P_{v^{\prime}}^{*\nu\dagger\left(Q^{\prime}\right)}\gamma_{\nu}\gamma_{5}\right]\frac{1+\not{v}}{2}\gamma^{\mu}\frac{1+\not{v}}{2}\left[P_{v}^{*\alpha\left(Q\right)}\gamma_{\alpha}\gamma_{5}\right]\right\\}$
$\displaystyle=\xi\left(w\right)\left\\{\varepsilon^{*\prime}\cdot\varepsilon\left(v+v^{\prime}\right)^{\mu}-\varepsilon^{*\prime}\cdot
v\varepsilon^{\mu}-\varepsilon\cdot
v^{\prime}\varepsilon^{*\prime\mu}\right\\}$
$\displaystyle\left\langle\mathcal{H}_{A}^{\left(Q^{\prime}\right)}\left|\bar{c}_{v^{\prime}}\gamma^{\mu}\gamma_{5}b_{v}\right|\mathcal{H}_{A}^{\left(Q\right)}\right\rangle$
$\displaystyle=-\xi\left(w\right)\text{Tr}\left\\{\left[P_{v^{\prime}}^{*\nu\dagger\left(Q^{\prime}\right)}\gamma_{\nu}\gamma_{5}\right]\frac{1+\not{v}}{2}\gamma^{\mu}\gamma_{5}\frac{1+\not{v}}{2}\left[P_{v}^{*\alpha\left(Q\right)}\gamma_{\alpha}\gamma_{5}\right]\right\\},$
$\displaystyle=i\xi\left(w\right)\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{\alpha}\varepsilon_{\beta}^{*\prime}\left(v+v^{\prime}\right)_{\nu}.$
(29)

Here, we have written $P_{v^{\prime}}^{*\nu\dagger\left(Q^{\prime}\right)}$
and $P_{v}^{*\alpha\left(Q\right)}$ as the polarization vectors
$\varepsilon^{*\prime}$ and $\varepsilon$ and used $\varepsilon^{*\prime}\cdot
v^{\prime}=\varepsilon\cdot v=0.$

Comparing Eq. (27) and Eq. (29), we get

$h_{1}=h_{3}=h_{4}=h_{7}=\xi\left(w\right),$

and the others vanish in this limit. Hence, the relation given in Eq. (15)
holds in this case too.

The corresponding hadronic part of transition amplitude can be written as

$\displaystyle\mathcal{M}^{\mu}_{T_{bb}\to T_{bc}^{\prime}}$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{cb}\left\langle
T_{bc\bar{u}\bar{d}}\left(v^{\prime},\varepsilon^{\prime}\right)\left|\bar{c}_{v^{\prime}}\gamma^{\mu}\left(1-\gamma_{5}\right)b_{v}\right|T_{bb\bar{u}\bar{d}}\left(v,\varepsilon\right)\right\rangle$
$\displaystyle=-\frac{G_{F}}{\sqrt{2}}V_{cb}\xi_{T\to
T^{\prime}}\left(w\right)\left[\varepsilon^{*\prime}\cdot\varepsilon\left(v+v^{\prime}\right)^{\mu}-\varepsilon^{*\prime}\cdot
v\varepsilon^{\mu}-\varepsilon\cdot
v^{\prime}\varepsilon^{*\prime\mu}+i\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{\alpha}\varepsilon_{\beta}^{*\prime}\left(v+v^{\prime}\right)_{\nu}\right]f_{Dd}(bb).$
(30)

Contracting the above hadronic amplitude with the leptonic part (19) and
summing over the polarization, i.e.,
$\sum_{\lambda}\varepsilon_{\mu}\left(v^{(\prime)},\lambda\right)\varepsilon_{\nu}^{*}\left(v^{(\prime)},\lambda\right)=-g_{\mu\nu}+v^{(\prime)}_{\mu}v^{(\prime)}_{\nu}/v^{(\prime)2}$
and spins of the leptons, the differential decay rate becomes

$\displaystyle\frac{d\Gamma^{1^{+}\to 1^{+}}}{dw}$ $\displaystyle=$
$\displaystyle
N_{b}\frac{G{{}_{F}}^{2}\left|V_{bc}\right|^{2}m_{T^{\prime}}}{384\pi^{3}m_{T}^{2}}\left|\xi_{T\to
T^{\prime}}\left(w\right)\right|^{2}\sqrt{w^{2}-1}\sqrt{1-\frac{m_{\ell}^{2}}{s}}\bigg{[}-3m_{\ell}^{4}\left(m_{T}^{2}+m_{T^{\prime}}^{2}+2m_{T}m_{T^{\prime}}\left(4w+5\right)\right)$
(31)
$\displaystyle-\frac{4m_{\ell}^{2}}{s}m_{T}m_{T^{\prime}}\left(w+1\right)\left(12m_{T}^{4}+m_{T}m_{T^{\prime}}\left(m_{T}^{2}+m_{T^{\prime}}^{2}\right)\left(1-49w\right)+2m_{T}^{2}m_{T^{\prime}}^{2}\left(20w^{2}-w+17\right)+12m_{T^{\prime}}^{4}\right)$
$\displaystyle+8m_{T}^{2}m_{T^{\prime}}^{2}\left(w+1\right)\left(\left(m_{T}^{2}+m_{T^{\prime}}^{2}\right)\left(13w-1\right)-2m_{T}m_{T^{\prime}}\left(8w^{2}-w+5\right)\right)\bigg{]}\left(f_{Dd}(bb)\right)^{2},$

which in the massless lepton case reduces to

$\displaystyle\frac{d\Gamma^{1^{+}\to 1^{+}}}{dw}$ $\displaystyle=$
$\displaystyle
N_{b}\frac{G{{}_{F}}^{2}\left|V_{bc}\right|^{2}}{48\pi^{3}}\left|\xi_{T\to
T^{\prime}}\left(w\right)\right|^{2}m^{3}_{T^{\prime}}(1+w)^{3/2}\sqrt{w-1}\times$
(32)
$\displaystyle\left[\left(m^{2}_{T^{\prime}}+m^{2}_{T}\right)\left(13w-1\right)-2m_{T}m_{T^{\prime}}\left(8w^{2}-w+5\right)\right]\left(f_{Dd}(bb)\right)^{2}.$

Using the numerical values of the input parameters from Table 1, and
integrating over $w$ in the allowed kinematic range gives

$\displaystyle\Gamma\left(T_{bb\bar{u}\bar{d}}\left(J^{P}=1^{+}\right)\to
T_{bc\bar{u}\bar{d}}\left(J^{P}=1^{+}\right)\ell^{-}\nu_{\ell}\right)$
$\displaystyle=$ $\displaystyle 5.77\times
10^{-11}\;\text{MeV};\;({\text{for}}~{}\ell^{-}\nu_{\ell}=e^{-}\nu_{e},\;\mu^{-}\nu_{\mu})$
$\displaystyle\Gamma\left(T_{bb\bar{u}\bar{d}}\left(J^{P}=1^{+}\right)\to
T_{bc\bar{u}\bar{d}}\left(J^{P}=1^{+}\right)\tau^{-}\nu_{\tau}\right)$
$\displaystyle=$ $\displaystyle 2.64\times 10^{-11}\;\text{MeV}.$ (33)

They yield the following branching ratios:

$\displaystyle{\cal B}(T_{bb\bar{u}\bar{d}}(J^{P}=1^{+})\to
T_{bc\bar{u}\bar{d}}(J^{P}=1^{+})\ell^{-}\nu_{\ell})$
$\displaystyle=7.0\%~{}({\text{for}}~{}\ell^{-}\nu_{\ell}=e^{-}\nu_{e},\;\mu^{-}\nu_{\mu}),$
$\displaystyle{\cal B}(T_{bb\bar{u}\bar{d}}(J^{P}=1^{+})\to
T_{bc\bar{u}\bar{d}}(J^{P}=1^{+})\tau^{-}\nu_{\tau})$ $\displaystyle=3.2\%.$
(34)

Thus, ${\mathcal{B}}\left(T_{bb\bar{u}\bar{d}}\left(J^{P}=0^{+}\right)\to
T_{bc\bar{u}\bar{d}}\left(J^{P}=1^{+}\right)\ell^{-}\nu_{\ell}\right)$ and
$\mathcal{B}\left(T_{bb\bar{u}\bar{d}}\left(J^{P}=1^{+}\right)\to
T_{bc\bar{u}\bar{d}}\left(J^{P}=1^{+}\right)\ell^{-}\nu_{\ell}\right)$ almost
saturate the inclusive semileptonic branching ratios for the double-bottom
tetraquarks. As anticipated, this is not too dissimilar to the corresponding
$B$-meson decays $B\to D^{(*)}\ell\nu_{\ell}$ ParticleDataGroup:2022pth .
However, these transitions are not foreseen for the hadron molecular picture,
as in this case, the $BB^{*}$ mesons will decay essentially independently.

$m_{T_{bbud}}\left(1^{+}\right)=10,504.4^{+37.36}_{-44.15}~{}\mathrm{MeV}$ ,
$m_{T_{bcud}}\left(0^{+}\right)=7155^{+9}_{-19}~{}\mathrm{MeV}$ ,
$m_{T_{bcud}}\left(1^{+}\right)=7152^{+9}_{-19}~{}\mathrm{MeV}$ ,
$m_{T_{ccud}}\left(1^{+}\right)=3875~{}\mathrm{MeV}$
---
$m_{\pi}=\left(139.57\pm 0.00018\right)~{}\mathrm{MeV}$,
$m_{\rho}=\left(775.26\pm 0.23\right)~{}\mathrm{MeV}$ ,
$m_{a_{1}}=\left(1230\pm 40\right)~{}\mathrm{MeV}$ , $m_{D}=(1869.61\pm
0.10)~{}\mathrm{MeV}$
$m_{B}=(5279.66\pm 0.12)~{}\mathrm{MeV}$ , $m_{B^{*}}=(5324.71\pm
0.21)~{}\mathrm{MeV}$ , $f_{\pi}=\left(130.41\pm 0.03\pm
0.20\right)~{}\mathrm{MeV}$ , $f_{\rho}=\left(210\pm 4\right)~{}\mathrm{MeV}$
$f_{a_{1}}=234~{}\mathrm{MeV}$ , $\rho_{1}^{2}=1.3$ ,
$C_{1}\left(m_{b}\right)\left[C_{2}\left(m_{b}\right)\right]=1.117\left[-0.257\right]$
, $G_{F}=1.16637\times 10^{-11}~{}\mathrm{MeV}^{-2}$,
$m_{e}=0.511~{}\mathrm{MeV}$
$m_{\mu}=105.7~{}\mathrm{MeV}$, $m_{\tau}=1777~{}\mathrm{MeV}$,
$V_{ud}=0.97420\pm 0.3021$, $V_{cb}=\left(39.5\pm 0.08\right)\times 10^{-3}$
Table 1: Numerical values of the input parameters. The decay rates are
calculated for the central values in this table.

##  IV Non-Leptonic decays $T_{bb\bar{u}\bar{d}}\to
T_{bc\bar{u}\bar{d}}\;h^{-}$

The simplest non-leptonic decay of the double-bottom tetraquark is
$T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}h^{-}$, where $h^{-}$ is a light
meson, such as $h^{-}=\pi^{-},\rho^{-},a_{1}^{-}$. At the quark level, these
decays take place via the weak transition $b\to cW^{-}(\to\bar{u}d)$. This is
similar to the $B$-meson decays, such as $B\to D^{(*)}\pi$, which were
calculated in the so-called ”naive factorization approach Bauer:1986bm .
Subsequent improvements have provided a QCD basis for this class of factorized
amplitudes, and the resulting corrections are found to be small Dugan:1990de .
We assume that this factorization approach can also be used to study the non-
leptonic decays of the tetraquarks, though a formal proof is lacking.

The effective Hamiltonian for this decay at the tree level is given by

$\mathcal{H}_{\text{eff}}=\frac{G_{F}}{\sqrt{2}}V_{cb}V_{ud}^{*}\left[C_{1}\left(\mu\right)\mathcal{O}_{1}+C_{2}\left(\mu\right)\mathcal{O}_{2}\right],$
(35)

where $C_{1,2}\left(\mu\right)$ are the Wilson coefficients, calculated at the
factorization scale $\mu$. The operators are:

$\displaystyle\mathcal{O}_{1}$
$\displaystyle=\left(\bar{d}_{i}\gamma_{\mu}P_{L}u_{i}\right)\left(\bar{c}_{j}\gamma^{\mu}P_{L}b_{j}\right)$
$\displaystyle\mathcal{O}_{2},$
$\displaystyle=\left(\bar{d}_{i}\gamma_{\mu}P_{L}u_{j}\right)\left(\bar{c}_{j}\gamma^{\mu}P_{L}b_{i}\right),$
(36)

with $i,j$ representing the color indices. The current in each bracket for
$\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ is a color singlet and octet,
respectively. The corresponding Feynman diagrams are shown in Figure 2(a).

|
---|---
(a) |  (b)

Figure 2: Feynman diagrams for (a) $T_{bb\bar{u}\bar{d}}\to
T_{bc\bar{u}\bar{d}}h^{-}$, and (b) $T_{bc\bar{u}\bar{d}}\to
T_{cc\bar{u}\bar{d}}h^{-}$, where $h^{-}=\pi^{-},\rho^{-},a_{1}^{-}$.

For the color-allowed tree amplitudes, factorization of the decay amplitude is
expected to be a good approximation, and the decay amplitude for
$T_{bb}\left(1^{+}\right)\to T_{bc}\left(0^{+};\;1^{+}\right)\pi^{-}$ can be
written in the form:

$\displaystyle\mathcal{M}$
$\displaystyle=4\frac{G_{F}}{\sqrt{2}}V_{cb}V_{ud}^{*}a_{1}\left(\mu\right)\left\langle\pi^{-}\left(q\right)\left|\left(\bar{d}_{i}\gamma_{\mu}P_{L}u_{i}\right)\right|0\right\rangle\left\langle
T_{bc\bar{u}\bar{d}}\left(p^{\prime}\right)\left|\left(\bar{c}_{j}\gamma^{\mu}P_{L}b_{j}\right)\right|T_{bb\bar{u}\bar{d}}\left(p\right)\right\rangle.$
$\displaystyle=-2\frac{G_{F}}{\sqrt{2}}V_{cb}V_{ud}^{*}a_{1}\left(\mu\right)\left\langle\pi^{-}\left(q\right)\left|\left(\bar{d}_{i}\gamma_{\mu}\gamma_{5}u_{i}\right)\right|0\right\rangle\left\langle
T_{bc\bar{u}\bar{d}}\left(p^{\prime}\right)\left|\left(\bar{c}_{j}\gamma^{\mu}P_{L}b_{j}\right)\right|T_{bb\bar{u}\bar{d}}\left(p\right)\right\rangle$
$\displaystyle=-2i\frac{G_{F}}{\sqrt{2}}V_{cb}V_{ud}^{*}a_{1}\left(\mu\right)f_{\pi}q_{\mu}\left\langle
T_{bc\bar{u}\bar{d}}\left(p^{\prime}\right)\left|\left(\bar{c}_{j}\gamma^{\mu}P_{L}b_{j}\right)\right|T_{bb\bar{u}\bar{d}}\left(p\right)\right\rangle,$
(37)

where $f_{\pi}$ is the decay constant of $\pi^{-}$, $q=p-p^{\prime}$, and
$a_{1}\left(\mu\right)=C_{1}\left(\mu\right)+C_{2}\left(\mu\right)/3.$ Using
the result of the matrix element from Eq. (18), which holds in the heavy quark
symmetry limit, and noting that the tetraquark in the final state has
$J^{P}=0^{+}$, the HQET version of the amplitude given in Eq. (37) takes the
form

$\displaystyle\mathcal{M}^{1^{+}\to 0^{+}}$
$\displaystyle=-i\frac{G_{F}}{\sqrt{2}}V_{cb}V_{ud}^{*}a_{1}\left(\mu\right)f_{\pi}\xi_{T\to
T^{\prime}}\left(w\right)q_{\mu}\left(f_{Dd}(bb)\right)\left[\left(w+1\right)\varepsilon^{\mu}-\left(\varepsilon\cdot
v^{\prime}\right)v^{\mu}-i\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{\nu}v_{\alpha}v_{\beta}^{\prime}\right].$
(38)

Along the same lines, for the final state $T_{bcud}$ having $J^{P}=1^{+}$, we
use Eq. (30) in Eq. (37), which gives

$\mathcal{M}^{1^{+}\to
1^{+}}=i\frac{G_{F}}{\sqrt{2}}V_{cb}V_{ud}^{*}a_{1}\left(\mu\right)f_{\pi}\xi_{T\to
T^{\prime}}\left(w\right)q_{\mu}\left(f_{Dd}\left(bb\right)\right)\left[\varepsilon^{*\prime}\cdot\varepsilon\left(v+v^{\prime}\right)^{\mu}-\varepsilon^{*\prime}\cdot
v\varepsilon^{\mu}-\varepsilon\cdot
v^{\prime}\varepsilon^{*\prime\mu}+i\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{\alpha}\varepsilon_{\beta}^{*\prime}\left(v+v^{\prime}\right)_{\nu}\right].$
(39)

For $h^{-}=\rho^{-}$ and $h^{-}=a_{1}$, which are light vector and axial-
vector mesons, respectively, the relevant matrix elements are

$\displaystyle\left\langle\rho\left(q\right)\left|\left(\bar{d}_{i}\gamma_{\mu}u_{i}\right)\right|0\right\rangle$
$\displaystyle=$ $\displaystyle
f_{\rho}m_{\rho}\varepsilon^{*}_{1\;\mu}\left(q\right),$
$\displaystyle\left\langle
a_{1}\left(q\right)\left|\left(\bar{d}_{i}\gamma_{\mu}\gamma_{5}u_{i}\right)\right|0\right\rangle$
$\displaystyle=$ $\displaystyle
f_{a_{1}}m_{a_{1}}\varepsilon^{*}_{1\;\mu}\left(q\right),$ (40)

where $f_{\rho}$ and $f_{a_{1}}$ are their respective decay constants and
$\varepsilon^{*}_{1\;\mu}\left(q\right)$ denote their polarization vector. The
corresponding amplitudes involving spin-1 particles in the final state can be
obtained by replacing $f_{\pi}$ with $f_{\rho}$ and $f_{a_{1}}$ for $\rho$ and
$a_{1}$ mesons in Eqs. (38) and (39), respectively. Also, the momentum vector
$q_{\mu}$ is to be replaced with their mass times the polarization vector
$\varepsilon^{*}_{1\;\mu}\left(q\right)$.

The decay rate of a two-body process has the standard form:

$d\Gamma=\frac{1}{8\pi}\left|\mathcal{\bar{M}}\right|^{2}\frac{\left|\bf{p}_{1}\right|}{M^{2}},$
(41)

with the three-momenta of the final state particle in the rest frame of the
parent hadron given by

$\left|\bf{p}_{1}\right|=\left|\bf{p}_{2}\right|=\frac{\left[\left(M^{2}-\left(m_{1}+m_{2}\right)^{2}\right)\left(M^{2}-\left(m_{1}-m_{2}\right)^{2}\right)\right]^{\frac{1}{2}}}{2M},$
(42)

where $m_{1}$ and $m_{2}$ are the masses of final state particles and $M$ is
the mass of the decaying particle. Using Eq. (38), we get the decay rate for
$T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}\pi^{-}$

$\Gamma^{1^{+}\to
0^{+}}=N_{b}\frac{G_{F}^{2}\left|V_{cb}V_{ud}^{*}\right|^{2}a_{1}(\mu)^{2}f_{\pi}^{2}\left(f_{Dd}(bb)\right)^{2}}{512\pi
m_{T}^{6}m_{T^{\prime}}}\left|\xi_{T\to
T^{\prime}}\left(w\right)\right|^{2}\lambda\left(m_{T}^{2},m_{T^{\prime}}^{2},m_{\pi^{-}}^{2}\right)^{\frac{3}{2}}\left(m_{T}^{2}+2m_{T}m_{T^{\prime}}\left(w+1\right)-m_{T^{\prime}}^{2}+m_{\pi^{-}}^{2}\right)^{2},$
(43)

where
$\lambda\left(m_{T}^{2},m_{T^{\prime}}^{2},m_{\pi^{-}}^{2}\right)=\left(m_{T}^{2}-\left(m_{T^{\prime}}+m_{\pi^{-}}\right)^{2}\right)\left(m_{T}^{2}-\left(m_{T^{\prime}}-m_{\pi^{-}}\right)^{2}\right)$,
and $w$ is evaluated at the final state hadron $\left(\pi^{-}\right)$ mass,
i.e.,

$w=\frac{m_{T}^{2}+m_{T^{\prime}}^{2}-m_{\pi^{-}}^{2}}{2m_{T}m_{T^{\prime}}}.$
(44)

For the $1^{+}\to 1^{+}$ decay, we use Eq. (39) to get

$\displaystyle\Gamma^{1^{+}\to 1^{+}}$ $\displaystyle=$ $\displaystyle
N_{b}\frac{G_{F}^{2}\left|V_{cb}V_{ud}^{*}\right|^{2}a_{1}(\mu)^{2}f_{\pi}^{2}\left(f_{Dd}(bb)\right)^{2}}{128\pi
m_{T}^{4}m_{T^{\prime}}}\left|\xi_{T\to
T^{\prime}}\left(w\right)\right|^{2}\times$ (45)
$\displaystyle\lambda\left(m_{T}^{2},m_{T^{\prime}}^{2},m_{\pi^{-}}^{2}\right)^{\frac{1}{2}}\left(\left(m_{T}+m_{T^{\prime}}\right)^{2}-m_{\pi^{-}}^{2}\right)\left(m_{\pi^{-}}^{2}\left(2m_{T}m_{T^{\prime}}-5\left(m_{T}^{2}+m_{T^{\prime}}^{2}\right)\right)+5\left(m_{T}^{2}-m_{T^{\prime}}^{2}\right)^{2}\right).$

Using the values of decay constant $f_{\pi}$ and the other input parameters
from Table 1, setting $N_{b}=2$ and $\left(f_{Dd}(bb)\right)^{2}=1$, we get

$\displaystyle\Gamma\left(T_{bb\bar{u}\bar{d}}\left(1^{+}\right)\to
T_{bc\bar{u}\bar{d}}\left(0^{+}\right)\pi^{-}\right)$ $\displaystyle=$
$\displaystyle 1.40\times 10^{-12}\;\text{MeV}.$ (46)

Similarly for the $1^{+}\to 1^{+}$ case, the result from Eq. (45) is:

$\Gamma\left(T_{bb\bar{u}\bar{d}}\left(1^{+}\right)\to
T_{bc\bar{u}\bar{d}}\left(1^{+}\right)\pi^{-}\right)=6.94\times
10^{-12}\;\text{MeV}.$ (47)

In the case of a vector or an axial-vector meson in the final state, the
expressions for the decay rates are:

$\displaystyle\Gamma^{1^{+}\to 0^{+}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}\left|V_{cb}V_{ud}^{*}\right|^{2}a_{1}(\mu)^{2}f_{V}^{2}\left(f_{Dd}(bb)\right)^{2}}{64\pi
m_{T}^{4}m_{T^{\prime}}}\left|\xi_{T\to
T^{\prime}}\left(w\right)\right|^{2}\lambda\left(m_{T}^{2},m_{T^{\prime}}^{2},m_{V}^{2}\right)^{\frac{1}{2}}\left(\left(m_{T}+m_{T^{\prime}}\right)^{2}-m_{V}^{2}\right)$
(48)
$\displaystyle\left(\left(m_{T}^{2}+m_{T^{\prime}}^{2}\right)^{2}-4m_{V}^{4}+m_{V}^{2}\left(3m_{T}^{2}+2m_{T}m_{T^{\prime}}+3m_{T^{\prime}}^{2}\right)\right),$
$\displaystyle\Gamma^{1^{+}\to 1^{+}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}\left|V_{cb}V_{ud}^{*}\right|^{2}a_{1}(\mu)^{2}f_{V}^{2}\left(f_{Dd}(bb)\right)^{2}}{64\pi
m_{T}^{4}m_{T^{\prime}}}\left|\xi_{T\to
T^{\prime}}\left(w\right)\right|^{2}\lambda\left(m_{T}^{2},m_{T^{\prime}}^{2},m_{V}^{2}\right)^{\frac{1}{2}}\left(\left(m_{T}+m_{T^{\prime}}\right)^{2}-m_{V}^{2}\right)$
(49)
$\displaystyle\left(5\left(m_{T}^{2}+m_{T^{\prime}}^{2}\right)^{2}-8m_{V}^{4}+m_{V}^{2}\left(3m_{T}^{2}-2m_{T}m_{T^{\prime}}+3m_{T^{\prime}}^{2}\right)\right).$

where the subscript $V$ defines $\rho^{-},\;a^{-}_{1}$. Using the numerical
values from Table 1, we get the partial decay rates for the $1^{+}\to 0^{+}$
transitions

$\displaystyle\Gamma\left(T_{bb\bar{u}\bar{d}}\left(1^{+}\right)\to
T_{bc\bar{u}\bar{d}}\left(0^{+}\right)\rho^{-}\right)$ $\displaystyle=$
$\displaystyle 3.94\times 10^{-12}\;\text{MeV}$
$\displaystyle\Gamma\left(T_{bb\bar{u}\bar{d}}\left(1^{+}\right)\to
T_{bc\bar{u}\bar{d}}\left(0^{+}\right)a_{1}^{-}\right)$ $\displaystyle=$
$\displaystyle 5.43\times 10^{-12}\;\text{MeV}$ (50)

The numerical values of the decay rates for the $1^{+}\to 1^{+}$ transitions
are:

$\displaystyle\Gamma\left(T_{bb\bar{u}\bar{d}}\left(1^{+}\right)\to
T_{bc\bar{u}\bar{d}}\left(1^{+}\right)\rho^{-}\right)$ $\displaystyle=$
$\displaystyle 1.80\times 10^{-11}\;\text{MeV},$
$\displaystyle\Gamma\left(T_{bb\bar{u}\bar{d}}\left(1^{+}\right)\to
T_{bc\bar{u}\bar{d}}\left(1^{+}\right)a_{1}^{-}\right)$ $\displaystyle=$
$\displaystyle 2.20\times 10^{-11}\;\text{MeV}.$ (51)

The resulting branching ratios for the decays
$T_{bb\bar{u}\bar{d}}\left(1^{+}\right)\to
T_{bc\bar{u}\bar{d}}\left(0^{+}\right)(\pi^{-},\rho^{-},a_{1}^{-})$ and
$T_{bb\bar{u}\bar{d}}\left(1^{+}\right)\to
T_{bc\bar{u}\bar{d}}\left(1^{+}\right)(\pi^{-},\rho^{-},a_{1}^{-})$ are given
in Table II. As for the semileptonic branching ratios, we use the total width
$\Gamma_{\rm total}(T_{bb\bar{u}\bar{d}})=8.2\times 10^{-10}$ MeV, derived
from the lifetime $\tau(T_{bb\bar{u}\bar{d}})=0.8$ps Ali:2018ifm , Since the
$J^{P}=1^{+}$ tetraquark $T_{bc\bar{u}\bar{d}}\left(0^{+}\right)$ is expected
to decay radiatively to the $J^{P}=0^{+}$ state, the branching ratios for
$T_{bb\bar{u}\bar{d}}\left(1^{+}\right)\to
T_{bc\bar{u}\bar{d}}\left(0^{+}\right)(\pi^{-},\rho^{-},a_{1}^{-})+(\gamma)$
are in the range of $(1-3)$%.

Decay Process | $T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}\;\ell^{-}\nu_{\ell}$ | $T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}\;\tau^{-}\nu_{\tau}$ | $T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}\;\pi^{-}$ | $T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}\;\rho^{-}$ | $T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}\;a_{1}^{-}$
---|---|---|---|---|---
$\mathcal{B}\left(1^{+}\to 0^{+}\right)$ | $2.4\times 10^{-2}$ | $8.2\times 10^{-3}$ | $1.7\times 10^{-3}$ | $4.8\times 10^{-3}$ | $6.6\times 10^{-3}$
Decay Process | $T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}\;\ell^{-}\nu_{\ell}$ | $T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}\;\tau^{-}\nu_{\tau}$ | $T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}\;\pi^{-}$ | $T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}\;\rho^{-}$ | $T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}\;a_{1}^{-}$
$\mathcal{B}\left(1^{+}\to 1^{+}\right)$ | $7.0\times 10^{-2}$ | $3.2\times 10^{-2}$ | $8.5\times 10^{-3}$ | $2.2\times 10^{-2}$ | $2.7\times 10^{-2}$
Decay Process | $T_{bc\bar{u}\bar{d}}\to T_{cc\bar{u}\bar{d}}\;\ell^{-}\nu_{\ell}$ | $T_{bc\bar{u}\bar{d}}\to T_{cc\bar{u}\bar{d}}\;\tau^{-}\nu_{\tau}$ | $T_{bc\bar{u}\bar{d}}\to T_{cc\bar{u}\bar{d}}\;\pi^{-}$ | $T_{bc\bar{u}\bar{d}}\to T_{cc\bar{u}\bar{d}}\;\rho^{-}$ | $T_{bc\bar{u}\bar{d}}\to T_{cc\bar{u}\bar{d}}\;a_{1}^{-}$
$\mathcal{B}\left(0^{+}\to 1^{+}\right)$ | $7.1\times 10^{-3}$ | $2.2\times 10^{-3}$ | $5.0\times 10^{-4}$ | $1.4\times 10^{-3}$ | $2.0\times 10^{-3}$

Table 2: Semileptonic and Non-leptonic branching ratios of the double-bottom
$T_{bb\bar{u}\bar{d}}$ and bottom-charm $T_{bc\bar{u}\bar{d}}$ tetraquarks
with the indicated spin-parity.

##  V Semileptonic and Non-leptonic decays
$T_{bc\bar{u}\bar{d}}\left(J^{P}=0^{+}\right)\to
T_{cc\bar{u}\bar{d}}\left(J^{P}=1^{+}\right)(\ell^{-}\nu_{\ell},h^{-})$

The decay $T_{bc\bar{u}\bar{d}}\to T_{cc\bar{u}\bar{d}}\ell^{-}\nu_{\ell}$ is
governed by the effective Hamiltonian given in Eq. (1), and the corresponding
Feynman diagram is drawn in Fig. 1(b). We identify the tetraquark
$T_{cc\bar{u}\bar{d}}$ with the double-charm hadron $T_{cc}^{+}(3875)$,
discovered by the LHCb, having $J^{P}=1^{+}$ LHCb:2021vvq , which implies that
we have a scalar to axial-vector transition. The HQET matrix element in this
case is parameterized as

$\displaystyle\frac{\left\langle
T_{cc\bar{u}\bar{d}}\left(v^{\prime},\varepsilon^{\prime}\right)\left|\bar{c}\gamma^{\mu}b\right|T_{bc\bar{u}\bar{d}}\left(v\right)\right\rangle}{\sqrt{m_{T}m_{T^{\prime}}}}$
$\displaystyle=h_{1}\left(w\right)i\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{\nu}^{\prime}v_{\alpha}v_{\beta}^{\prime}$
$\displaystyle\frac{\left\langle
T_{cc\bar{u}\bar{d}}\left(v^{\prime},\varepsilon^{\prime}\right)\left|\bar{c}\gamma^{\mu}\gamma_{5}b\right|T_{bb\bar{u}\bar{d}}\left(v\right)\right\rangle}{\sqrt{m_{T}m_{T^{\prime}}}}$
$\displaystyle=\left(h_{2}\left(w\right)\left(w+1\right)\varepsilon^{\prime\mu}-h_{3}\left(w\right)\left(\varepsilon^{\prime}\cdot
v\right)v^{\prime\mu}-h_{4}\left(w\right)\left(\varepsilon^{\prime}\cdot
v^{\prime}\right)v^{\mu}\right).$ (52)

The spin symmetry relating the form factors in this case becomes

$\left\langle\mathcal{H}_{A}^{\left(Q^{\prime}\right)}\left|\bar{Q}^{\prime}\Gamma
Q\right|\mathcal{H}_{S}^{\left(Q\right)}\right\rangle=-\xi\left(w\right)\text{Tr}\left\\{\bar{\mathcal{H}}_{A}^{\left(Q^{\prime}\right)}\Gamma\mathcal{H}_{S}^{\left(Q\right)}\right\\},$
(53)

and by using Eqs. (5) and (8) in Eq. (53), the matrix elements for vector and
axial-vector currents become

$\displaystyle\left\langle\mathcal{H}_{A}^{\left(Q^{\prime}\right)}\left|\bar{c}_{v^{\prime}}\gamma^{\mu}b_{v}\right|\mathcal{H}_{S}^{\left(Q\right)}\right\rangle$
$\displaystyle=-\xi\left(w\right)\text{Tr}\left\\{P_{v^{\prime}}^{*\alpha\dagger\left(Q^{\prime}\right)}\gamma_{\alpha}\gamma_{5}\frac{1+\not{v}^{\prime}}{2}\gamma^{\mu}\frac{1+\not{v}}{2}P_{v}^{*\left(Q\right)}\right\\}$
$\displaystyle=i\xi\left(w\right)\varepsilon^{\mu\nu\alpha\beta}\varepsilon^{\prime}_{\nu}v_{\alpha}v_{\beta}^{\prime}$
$\displaystyle\left\langle\mathcal{H}_{A}^{\left(Q^{\prime}\right)}\left|\bar{c}_{v^{\prime}}\gamma^{\mu}b_{v}\right|\mathcal{H}_{S}^{\left(Q\right)}\right\rangle$
$\displaystyle=-\xi\left(w\right)\text{Tr}\left\\{P_{v^{\prime}}^{*\alpha\dagger\left(Q^{\prime}\right)}\gamma_{\alpha}\gamma_{5}\frac{1+\not{v}^{\prime}}{2}\gamma^{\mu}\gamma_{5}\frac{1+\not{v}}{2}P_{v}^{*\left(Q\right)}\right\\}$
$\displaystyle=\xi\left(w\right)\left[\left(w+1\right)\varepsilon^{\prime\mu}-\left(\varepsilon^{\prime}\cdot
v\right)v^{\prime\mu}\right],$ (54)

Comparing Eq. (53) and Eq. (54), one gets

$h_{1}=h_{2}=h_{3}=\xi\left(w\right)$ (55)

and $h_{4}=0.$

Therefore, the hadronic transition amplitude in this case becomes

$\displaystyle\mathcal{M}^{\mu}$
$\displaystyle=-\frac{G_{F}}{\sqrt{2}}V_{cb}\xi\left(w\right)\sqrt{m_{T}m_{T^{\prime}}}\left[\left(w+1\right)\varepsilon^{\prime\mu}-\left(\varepsilon^{\prime}\cdot
v\right)v^{\prime\mu}-i\varepsilon^{\mu\nu\alpha\beta}\varepsilon^{\prime}_{\nu}v_{\alpha}v_{\beta}^{\prime}\right].$
(56)

As discussed earlier, the wave-function of the tetraquark
$T_{bc\bar{u}\bar{d}}$ may involve both the compact and hadron molecular
component. Including the compact fraction $f_{Dd}(bc)$, the above equation
takes the form

$\displaystyle\mathcal{M}_{T_{bc}\to T_{cc}}^{\mu}$
$\displaystyle=-\frac{G_{F}}{\sqrt{2}}V_{cb}\xi_{T\to
T^{\prime}}\left(w\right)\left(f_{Dd}(bc)\right)\sqrt{m_{T}m_{T^{\prime}}}\left[\left(w+1\right)\varepsilon^{\prime\mu}-\left(\varepsilon^{\prime}\cdot
v\right)v^{\prime\mu}-i\varepsilon^{\mu\nu\alpha\beta}\varepsilon^{\prime}_{\nu}v_{\alpha}v_{\beta}^{\prime}\right].$
(57)

The relation between the form factors given in Eq. (15) holds in this case as
well. Contracting the hadronic part with the leptonic part given in Eq. (19),
and summing over the polarization, i.e.,
$\sum_{\lambda}\varepsilon^{\prime}_{\mu}\left(v^{\prime},\lambda\right)\varepsilon_{\nu}^{\prime*}\left(v^{\prime},\lambda\right)=-g_{\mu\nu}+v^{\prime}_{\mu}v^{\prime}_{\nu}/v^{\prime
2}$, it is easy to see that the expression for the decay rate is again given
by Eq. (21), but one has to replace $\left(f_{Dd}(bb)\right)$ with
$\left(f_{Dd}(bc)\right)$.

Using the numerical values of the various input parameters from Table 1, the
total width $\Gamma_{\rm total}(T_{bc\bar{u}\bar{d}})=2.2\times 10^{-9}$ MeV,
which is derived from the lifetime $\tau(T_{bc\bar{u}\bar{d}})=0.3$ps, based
on the assumption that it is expected to be similar to the one calculated for
the bottom-charm baryon $\tau(\Xi(cbu)=(0.28\pm 0.33)$ps Kiselev:2001fw ,
together with $N_{b}=1$, and setting $\left(f_{Dd}(bc)\right)=1$, we get the
following branching ratios:

$\displaystyle{\cal B}\left(T_{bc\bar{u}\bar{d}}(J^{P}=0^{+})\to
T_{cc\bar{u}\bar{d}}(J^{P}=1^{+})\ell^{-}\nu_{\ell}\right)$
$\displaystyle=7.1\times
10^{-3}~{}({\text{for}}~{}\ell^{-}\nu_{\ell}=e^{-}\nu_{e},\;\mu^{-}\nu_{\mu})$
$\displaystyle{\cal B}\left(T_{bb\bar{u}\bar{d}}(J^{P}=0^{+})\to
T_{bc\bar{u}\bar{d}}(J^{P}=1^{+})\tau^{-}\nu_{\tau}\right)$
$\displaystyle=2.2\times 10^{-3}$ (58)

In the case of the non-leptonic decays, following the same procedure as
adopted for the $T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}$ case, the
amplitude for $h^{-}=\pi^{-}$ becomes

$\displaystyle\mathcal{M}^{0^{+}\to 1^{+}}$
$\displaystyle=-i\frac{G_{F}}{\sqrt{2}}V_{cb}V_{ud}^{*}a_{1}\left(\mu\right)f_{\pi}\xi_{T\to
T^{\prime}}\left(w\right)q_{\mu}\left(f_{Dd}(bc)\right)\left[\left(w+1\right)\varepsilon^{\prime\mu}-\left(\varepsilon^{\prime}\cdot
v\right)v^{\prime\mu}-i\varepsilon^{\mu\nu\alpha\beta}\varepsilon^{\prime}_{\nu}v_{\alpha}v_{\beta}^{\prime}\right].$
(59)

From Eq. (41), after performing the polarization sum, one gets the expression
for the decay rate given in Eq. (43) with the obvious replacements:
$\left(f_{Dd}(bb)\right)$ with $\left(f_{Dd}(bc)\right)$, and $N_{b}=1$. Using
the values of the input parameters from Table 1, we get

$\Gamma\left(T_{bc\bar{u}\bar{d}}\left(0^{+}\right)\to
T_{cc\bar{u}\bar{d}}\left(1^{+}\right)\pi^{-}\right)=1.10\times
10^{-12}\;\text{MeV},$ (60)

where we have set $\left(f_{Dd}(bc)\right)=1$. Similarly, for the cases
$h^{-}=\rho^{-},\;a_{1}^{-}$, from Eq. (48), the numerical values of the decay
rates are:

$\displaystyle\Gamma\left(T_{bc\bar{u}\bar{d}}\left(0^{+}\right)\to
T_{cc\bar{u}\bar{d}}\left(1^{+}\right)\rho^{-}\right)$ $\displaystyle=$
$\displaystyle 3.11\times 10^{-12}\;\text{MeV}$
$\displaystyle\Gamma\left(T_{bc\bar{u}\bar{d}}\left(0^{+}\right)\to
T_{cc\bar{u}\bar{d}}\left(1^{+}\right)a_{1}^{-}\right)$ $\displaystyle=$
$\displaystyle 4.37\times 10^{-12}\;\text{MeV}.$ (61)

This completes the numerical part of this Letter.

##  VI Summary and prospects of experimental measurements

Spectroscopic calculations based on Lattice QCD support a substantial diquark-
antidiquark component in the Fock space of the tetraquarks
$T_{bb\bar{u}\bar{d}}$ and $T_{bb\bar{u}\bar{s}}$. Motivated by this input and
the theoretical consensus that the double-bottom tetraquarks
$T_{bb\bar{u}\bar{d}}$ and $T_{bb\bar{u}\bar{s}}$ are stable against strong
and electromagnetic interactions, we have worked out some signature decays,
reflecting the diquark-antidiquark component of the $T_{bb\bar{u}\bar{d}}$
tetraquark wave-function. They are characterised by the semi-inclusive
transitions $T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}+X$, where
$X=\ell^{-}\nu_{\ell},{\rm hadrons}$. We work out the branching ratios for the
semileptonic decays for the assumed $J^{P}$ quantum numbers of the initial and
the final state tetraquarks. This requires the knowledge of the weak current
matrix elements (form factors) for the tetraquarks, which are not at hand. We
use heavy quark symmetry to relate the corresponding form factors at the
symmetry point and have argued that this should be a good approximation, as
the underlying weak Hamiltonian for the $B\to(D,D^{*})\ell\nu_{\ell}$ and
$T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}\ell\nu_{\ell}$ is the same,
resulting from the $b\to cW^{-}$ transition, involving heavy quarks in the
initial and final states. This implies that the momentum-transfer to the
spectators is small in both cases. A relation between the Isgur-Wise form
factors of the double-bottom tetraquarks and the $B$-mesons follows in the
symmetry limit.

For the non-leptonic decays, we have concentrated for the cases where $X$ is a
single light hadron $X=\pi^{-},\rho^{-},a_{1}^{-}$, which represent the so-
called color-allowed tree diagrams, and have used factorization to write the
hadronic matrix elements - a method well-known from the decays
$B\to(D,D^{*})\pi$. Of course, it remains to be seen if the QCD corrections
are small in this case as well. The largest uncertainty in the decay rates
presented here, however, lies in the composition of the tetraquark wave-
function. Following a Lattice-QCD study Bicudo:2021qxj , this is composed of
diquarkonic and mesonic components. In the present context, we parametrize it
by the diquarkonic fraction $\left(f_{Dd}(bb)\right)^{2}$, with the mesonic
fraction given by $1-\left(f_{Dd}(bb)\right)^{2}$. Further studies using the
Lattice-QCD techniques are needed to quantify this fraction. We argue that
experimental measurements of some of the branching ratios presented here may
also determine this fraction. Assuming the dominance of the diquarkonic
component, i.e., setting $\left(f_{Dd}(bb)\right)^{2}=1$, the decay rates are
shown in Table II. They are large enough to be measured at the HL-LHC, and in
the long run at the proposed Tera-$Z$ factories. Since these measurements
involve the bottom-charm tetraquarks in the final state, they have to be
measured first in sufficient numbers. Detailed studies at the LHCb are
encouraging Blusk:21 ; Polyakov:23 . There are good prospects at the two large
experiments (ATLAS and CMS) at the LHC due to their larger acceptance and much
higher integrated luminosity. The extension to the non-leptonic decays of the
tetraquarks resulting from the weak decay $b\to c\bar{c}s$ are obvious. Of
course, many more decay modes of the tetraquarks following from the $b$-quark
decays $b\to c(\bar{u}d,\bar{c}s)$ can be calculated.

We have also worked out the weak decays of the $J^{P}=0^{+}$ bottom-charm
tetraquark $T_{bc\bar{u}\bar{d}}\to T_{cc\bar{u}\bar{d}}+X$, where
$X=\ell^{-}\nu_{\ell},{\rm hadrons}$, with the final-state tetraquark
identified with the observed double-charm narrow state $T_{cc}^{+}(3875)$,
having $J^{P}=1^{+}$, decaying to $DD^{*}$ LHCb:2021vvq . Their branching
ratios are also shown in Table 2. Compared to the corresponding decays of the
double-bottom tetraquarks, they are smaller due to the anticipated larger
decay width of the $T_{bc\bar{u}\bar{d}}$ tetraquark. We recall that there are
three branches of the weak decays of $T_{bc\bar{u}\bar{d}}$. Their relative
fractions, based on the anticipated similarities with the bottom-charm baryon
$\Xi_{bcu}$ decays Kiselev:2001fw  are estimated as $b\to c$; $f_{b}=0.22\pm
0.04$, $c\to s$; $f_{c}=0.72\pm 0.04$, and $W^{\pm}$-exchange, $f_{W}=0.06\pm
0.04$ Polyakov:23 . Of these, only the $b\to c$ transition will yield the
observed tetraquark $T_{cc}^{+}(3875)$. Other branches will lead to different
final states, of which we expect that the $c\to s$ transition may also lead to
a tetraquark with the quark content $T_{bs\bar{u}\bar{d}}$ from the decays
$T_{bc\bar{u}\bar{d}}\to T_{bs\bar{u}\bar{d}}+(\ell^{+}\bar{\nu}_{\ell},X)$.
Due to the larger fraction of the $c\to s$ decay, $f_{c}$, this might be a
promising place to discover a tetraquark with four different quark flavors.

In summary, weak decays of the double-heavy tetraquarks, some of which are
discussed here, are anticipated to induce tetraquark $\to$ tetraquark
transitions. This is due to the presence of a doubly-heavy diquark component
in the Fock-space of these hadrons. These signature decays of the compact
tetraquarks may be observed at the high-luminosity LHC experiments (LHCb,
ATLAS and CMS) Azzi:2019yne ; Cerri:2018ypt ; LHCb:2018roe , and eventually at
the Tera-$Z$ factories, being considered at the future circular $e^{+}e^{-}$
machines at CERN FCC:2018evy  and China CEPCStudyGroup:2018ghi . Establishing
these decays will provide a proof of the existence of double-heavy diquarks,
long anticipated in the context of the heavy tetraquarks Manohar:1992nd ,
which have now received additional support from the Lattice-QCD studies.

## Acknowledgements

This project was initiated during the 5th. International Workshop on Heavy
Quark Physics at NCP, Islamabad. One of us (AA) would like to thank NCP for
the kind hospitality. We acknowledge helpful discussions with Timothy Gershon,
Luciano Maiani and Ivan Polyakov.

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