# Comments on the ”Optimal strategy of deteriorating items with capacity
constraints under two-levels of trade credit policy”

Sunil Tiwari<EMAIL_ADDRESS>Masih Fadaki<EMAIL_ADDRESS>Anuj Kumar Sharma<EMAIL_ADDRESS>”Department of Industrial Systems
Engineering and Management, National University of Singapore, 1 Engineering
Drive 2, 117576, Singapore” ”School of Business IT & Logistics, RMIT
University, Melbourne, VIC 3000, Australia” Department of Mathematics, Shyam
Lal College, University of Delhi, India

###### Abstract

This technical note rectified the mathematical and conceptual errors present
in Liao et al. [2014]. Liao et al. [2014] proposed an EOQ model under two-
levels trade credit policy considering limited storage capacity whereby the
supplier provides a permissible delay period ($M$) to the retailer, and the
retailer also offers a permissible delay period ($N$) (where $M>N$) to its
customers. In the current technical note, we point out some defects of their
model from the logical viewpoints of mathematics regarding both interest
charged and interest earned. Furthermore, as an example, one of the affected
numerical results is re-evaluated.

###### keywords:

Trade credit financing , Permissible delay in payments , Deterioration ,
Limited storage capacity , Inventory.

##  1 Introduction

To reflect the real business environment and addressing the inter-dependencies
among the operations and financing processes, there is recently a trend of
optimising inventory while the trade credit is also taken into consideration.
Liao et al. [2014] presented an inventory model for the perishable items with
two-levels of trade credit and limited storage capacity. In this two-echelon
supply chain inventory model, a supplier provides the products to a retailer
and the retailer sells them to its customers. Under two levels of trade
credit, supplier offers a fixed credit period to the retailer to settle the
account and the retailer provides some trade credit to his customers. Beyond
the allowable credit period, the retailer is charged as per agreed interest
rate while she receives the interest regarding the customer’s ”outstanding
amount if it is not paid within the allowable delay period.”

While the study contributed an interesting idea of considering the capacity
constrained in the context of inventory optimisation under trade credit
policy, we have identified a major issue in developing the mathematical
expression regarding the interest earned of the second case ($N<T\leq M$).
Furthermore, a number of minor issues including an error in the Notations
section and an incorrect mathematical expression for the annual deteriorating
cost is spotted. Since the outlined major issue has a significant impact on
the numerical results and the corresponding recommendations, we also amend the
impacted computational results.

##  2 Notations

We have reviewed the notations of the original paper and compared the provided
notations with the terms used in the paper. A minor issue exists in the
notations with respect to the formula of $W^{*}$. The unit selling price of
the item is denoted by ’$s$’ in the notation list while the $W^{*}$ is
formulated as $pDM+\frac{pI_{e}DM^{2}}{2}$ in which ’$p$’ is used for the unit
selling price. This expression should be updated to
$sDM+\frac{sI_{e}DM^{2}}{2}$.

##  3 Model

Suppose there is a delay in payment time, denoted as M. The inventory level
$I(t)$ at a time $t\in\left[0,{\rm\;}T\right]$ can be formulated as:

$\displaystyle{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}"}\frac{dI\left(t\right)}{dt}+\theta
I\left(t\right)=-D;{\rm\;\;\;\;}0\leq t\leq
T{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}"}$
(1)

” Given the boundary condition as $I(T)=0$, we get ”

$\displaystyle
I\left(t\right)=\frac{D}{\theta}\left(e^{\theta\left(T-t\right)}-1\right);{\rm\;\;\;\;}0\leq
t\leq T$ (2)

At time $t=0$, I(0) = Q and thus:

$\displaystyle Q$ $\displaystyle=\frac{D}{\theta}\left(e^{\theta T}-1\right)$
(3) $\displaystyle\Rightarrow\;\frac{dQ}{dT}$ $\displaystyle=De^{\theta T}$

Considering Equation 3, the time at which W inventory is exhausted (Ta) can be
computed as:

$\displaystyle W$ $\displaystyle=\frac{D}{\theta}\left(e^{\theta
T_{a}}-1\right)$ (4) $\displaystyle\Rightarrow T_{a}$
$\displaystyle=\frac{1}{\theta}\ln\left(\frac{\theta W}{D}+1\right)$

###  3.1 Estimation of total annual cost components

There is another minor issue regarding the mathematical expression of annual
deteriorating cost as it has been developed as $\frac{Dh(e^{\theta T}-\theta
T-1)}{\theta T}$ in Liao et al.’s [2014] model; however, the correct term is:

$\displaystyle D_{T}$ $\displaystyle=\frac{c\left(Q-DT\right)}{T}$
$\displaystyle=\frac{cD}{\theta T}\left\\{e^{\theta T}-\theta T-1\right\\}$

The interest earned per year, Case 2: $N<T\leq M$

The incomplete term in Liao et al.’s [2014] model pertains to computing the
annual interest earned for the second case where ”$N<T\leq M$”. This is the
case where Liao et al. [2014] did not consider the interest earned during N to
T. As depicted in Figure 1, annual interest earned should be calculated for
both periods of $N$ to $T$ and $T$ to $M$. They formulated the interest earned
for this period as $\frac{sI_{e}D(2MT-N^{2}-T^{2})}{2T}$; however, the correct
interest earned for this case should be formulated as:

”

$\displaystyle=\frac{sI_{e}\left(DN+DT\right)\left(T-N\right)}{2T}+\frac{1}{T}\left\\{sDT+\frac{sDI_{e}\left(T^{2}-N^{2}\right)}{2}\right\\}\left(M-T\right)I_{e}$
$\displaystyle=\frac{sDI_{e}\left(T^{2}-N^{2}\right)+\left(2sDT+sD\left(T^{2}-N^{2}\right)I_{e}\right)I_{e}\left(M-T\right)}{2T}$

”

time$0$$N$$T$$M$$DN$$DT$Interest earnedInterest charged Figure 1:
Demonstration of annual interest earned for case 2.

###  3.2 Implications of corrected term on the total annual cost

Putting together all components of retailer’s total annual cost, is can be
expressed as:

> TC(T) = | ”ordering cost + deterioration cost + stock-holding cost in RW
> ---|---
> | \+ stock-holding cost in OW + interest paid – interest earned”

Given the possible values of N, M, and T, the retailer’s total annual cost is
estimated for the following configurations:

Configuration 1: ”$T_{a}<N<M<W^{*}$”

Regarding this configuration, the total annual cost can be estimated as
follows:

$TC\left(T\right)=\left\\{\begin{array}[]{l}{TC_{1}\left(T\right);{\rm\;\;\;\;}if{\rm\;}0<T\leq
T_{a}}\\\ {TC_{2}\left(T\right);{\rm\;\;\;\;}if{\rm\;}T_{a}<T\leq N}\\\
{TC_{3}\left(T\right);{\rm\;\;\;\;}if{\rm\;}N<T\leq M}\\\
{TC_{4}\left(T\right);{\rm\;\;\;\;}if{\rm\;}M<T\leq W^{*}}\\\
{TC_{5}\left(T\right);{\rm\;\;\;\;}if{\rm\;}W^{*}<T}\end{array}\right.$

In this configuration, the total cost regarding the third time window
($N<T\leq M$) should be amended to:

$\displaystyle{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}"}TC_{3}\left(T\right)$
$\displaystyle=\frac{A}{T}+\frac{D\left(k+c\theta\right)}{\theta^{2}T}\left(e^{\theta
T}-\theta
T-1\right)-\frac{\left(k-h\right)}{\theta^{2}T}\left\\{D\left(e^{\theta
T_{a}}-\theta T_{a}-1\right)+\theta^{2}W\left(T-T_{a}\right)\right\\}$ (5)
$\displaystyle\hphantom{{}=}-\frac{1}{2T}\left\\{sDI_{e}\left(T^{2}-N^{2}\right)+\left(2sDT+sD\left(T^{2}-N^{2}\right)I_{e}\right)I_{e}\left(M-T\right)\right\\}$
$\displaystyle{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}"}$

Configuration 2: ”$N<T_{a}<M<W^{*}$”

With respect to this configuration, the total annual cost can be estimated as:

”

$TC\left(T\right)=\left\\{\begin{array}[]{l}{TC_{1}\left(T\right);{\rm\;\;\;\;}if{\rm\;}0<T\leq
N}\\\ {TC_{6}\left(T\right);{\rm\;\;\;\;}if{\rm\;}N<T\leq T_{a}}\\\
{TC_{3}\left(T\right);{\rm\;\;\;\;}if{\rm\;}T_{a}<T\leq M}\\\
{TC_{4}\left(T\right);{\rm\;\;\;\;}if{\rm\;}M<T{\rm\;}\leq W^{*}}\\\
{TC_{5}\left(T\right);{\rm\;\;\;\;}if{\rm\;}W^{*}<T}}\rm{\end{array}\right.$

” where

$\displaystyle{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}"}TC_{3}\left(T\right)$
$\displaystyle=\frac{A}{T}+\frac{D\left(k+c\theta\right)}{\theta^{2}T}\left(e^{\theta
T}-\theta
T-1\right)-\frac{\left(k-h\right)}{\theta^{2}T}\left\\{D\left(e^{\theta
T_{a}}-\theta T_{a}-1\right)+\theta^{2}W\left(T-T_{a}\right)\right\\}$ (6)
$\displaystyle\hphantom{{}=}-\frac{1}{2T}\left\\{sDI_{e}\left(T^{2}-N^{2}\right)+\left(2sDT+sD\left(T^{2}-N^{2}\right)I_{e}\right)I_{e}\left(M-T\right)\right\\}$
$\displaystyle{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}"}$

###  3.3 Probing the convexity for total annual cost function

In this section, we investigate the convexity of the total annual cost
functions ($TC_{3}(T)$).

######  Theorem 3.1.

$TC_{3}\left(T\right)$ is convex on $T>0$.

###### Proof.

Considering the first and the second derivatives of the revised $TC_{3}(T)$:

$\displaystyle{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}"}TC_{3}\left(T\right)$
$\displaystyle=\frac{A}{T}+\frac{D\left(k+c\theta\right)}{\theta^{2}T}\left(e^{\theta
T}-\theta
T-1\right)-\frac{\left(k-h\right)}{\theta^{2}T}\left\\{D\left(e^{\theta
T_{a}}-\theta T_{a}-1\right)+\theta^{2}W\left(T-T_{a}\right)\right\\}$
$\displaystyle\hphantom{{}=}-\frac{1}{2T}\left\\{DsI_{e}\left(T^{2}-N^{2}\right)+\left(2DTs+Ds\left(T^{2}-N^{2}\right)I_{e}\right)I_{e}\left(M-T\right)\right\\}$

”

”

$\displaystyle TC_{3}^{{}^{\prime}}\left(T\right)$
$\displaystyle=\frac{1}{\left(\theta
T\right)^{2}}\left[-\theta^{2}A+D\left(k+c\theta\right)\left(\theta Te^{\theta
T}-e^{\theta T}+1\right)-D\left(k-h\right)\left(\theta T_{a}e^{\theta
T_{a}}-e^{\theta T_{a}}+1\right)\right]$
$\displaystyle\hphantom{{}=}-\frac{DsI_{e}}{2T^{2}}\left\\{\left(N^{2}-T^{2}\right)+\left(1+TI_{e}\right)+\left(N^{2}+T^{2}\right)I_{e}\right\\}$

”

”

$\displaystyle TC_{3}^{{}^{\prime\prime}}\left(T\right)$
$\displaystyle=\frac{2A}{T^{3}}+\frac{D\left(k+c\theta\right)}{\theta^{2}T^{3}}\left\\{\theta^{2}T^{2}e^{\theta
T}-2\theta Te^{\theta T}+2e^{\theta
T}-2\right\\}+\frac{2D\left(k-h\right)}{\theta^{2}T^{3}}\left(\theta
T_{a}e^{\theta T_{a}}-e^{\theta T_{a}}+1\right)$
$\displaystyle\hphantom{{}=}+\frac{DsI_{e}}{2T^{3}}\left\\{T^{2}+N^{2}+I_{e}\left(N^{2}-T^{2}+2T^{3}\right)\right\\}$

”

”

$\displaystyle TC_{3}^{{}^{\prime}}\left(T\right)$
$\displaystyle=TC_{2}^{{}^{\prime}}\left(T\right)-\frac{DsI_{e}}{2T^{2}}\left\\{\left(N^{2}-T^{2}\right)+\left(1+TI_{e}\right)+\left(N^{2}+T^{2}\right)I_{e}\right\\}$

”

”

$\displaystyle TC_{3}^{{}^{\prime\prime}}\left(T\right)$
$\displaystyle=TC_{2}^{{}^{\prime\prime}}\left(T\right)+\frac{DsI_{e}}{2T^{3}}\left\\{T^{2}+N^{2}+I_{e}\left(N^{2}-T^{2}+2T^{3}\right)\right\\}$
(7)

”

######  Lemma 3.2.

$T^{2}+N^{2}+I_{e}\left(N^{2}-T^{2}+2T^{3}\right)$$\mathrm{>}$0

###### Proof.

Let

$f(T)=T^{2}+N^{2}+I_{e}\left(N^{2}-T^{2}+2T^{3}\right)$ (8)

$f^{{}^{\prime}}(T)=2T-2I_{e}T+6T^{2}$ $f^{{}^{\prime}}(T)=2T(1-I_{e})+6T^{2}$
$f^{{}^{\prime}}(T)>0\,\,\therefore\,\,0<I_{e}<1$

This implies that $f(T)$ is increasing function of $T$ and
$f(0)=N^{2}(1+I_{e})>0$.

Hence, $f(T)>0$, when $T>0$

This completes the proof of Lemma 3.2. ∎

Continuing the proof of Theorem 3.1.

From Equations 7 and 8:

$\displaystyle TC_{3}^{{}^{\prime\prime}}\left(T\right)$
$\displaystyle=TC_{2}^{{}^{\prime\prime}}\left(T\right)+\frac{DsI_{e}}{2T^{3}}f(T)$

From Lemma 3.2 and as $TC_{2}\left(T\right)$ is convex (See Liao et al.
[2014]’s paper), therefore $TC_{3}^{{}^{\prime\prime}}\left(T\right)>0$

Hence, $TC_{3}\left(T\right)$ is convex on $T>0$.

This completes the proof.

∎

##  4 Updated Decision rule

From Theorem 3.1, we get,

”

$\displaystyle TC_{3}^{{}^{\prime}}\left(T\right)$
$\displaystyle=\frac{1}{\left(\theta
T\right)^{2}}\left[-\theta^{2}A+D\left(k+c\theta\right)\left(\theta Te^{\theta
T}-e^{\theta T}+1\right)-D\left(k-h\right)\left(\theta T_{a}e^{\theta
T_{a}}-e^{\theta T_{a}}+1\right)\right]$
$\displaystyle\hphantom{{}=}-\frac{DsI_{e}}{2T^{2}}\left\\{\left(N^{2}-T^{2}\right)+\left(1+TI_{e}\right)+\left(N^{2}+T^{2}\right)I_{e}\right\\}$

”

After simplifying above equation, we get

”

$\displaystyle TC_{3}^{{}^{\prime}}(T)$ $\displaystyle=\frac{1}{(\theta
T)^{2}}[-\theta^{2}A+D(k+c\theta)(\theta Te^{\theta T}-e^{\theta
T}+1)-D(k-h)(\theta T_{a}e^{\theta T_{a}}-e^{\theta T_{a}}+1)$
$\displaystyle\hphantom{{}=}-\frac{DsI_{e}\theta^{2}}{2}\\{(N^{2}-T^{2})+(1+TI_{e})+(N^{2}+T^{2})I_{e}\\}]$

”

Now, replacing $T=M$ in above equation, we get

”

$\displaystyle TC_{3}^{{}^{\prime}}(M)$ $\displaystyle=\frac{1}{(\theta
M)^{2}}[-\theta^{2}A+D(k+c\theta)(\theta Me^{\theta M}-e^{\theta
M}+1)-D(k-h)(\theta T_{a}e^{\theta T_{a}}-e^{\theta T_{a}}+1)$
$\displaystyle\hphantom{{}=}-\frac{DsI_{e}\theta^{2}}{2}\\{(N^{2}-M^{2})+(1+MI_{e})+(N^{2}+M^{2})I_{e}\\}]$

”

$\displaystyle TC_{3}^{{}^{\prime}}\left(M\right)$
$\displaystyle=\frac{\Delta_{3}}{\left(\theta M\right)^{2}}$

where

”

$\displaystyle\Delta_{3}$ $\displaystyle=[-\theta^{2}A+D(k+c\theta)(\theta
Me^{\theta M}-e^{\theta M}+1)-D(k-h)(\theta T_{a}e^{\theta T_{a}}-e^{\theta
T_{a}}+1)$
$\displaystyle\hphantom{{}=}-\frac{DsI_{e}\theta^{2}}{2}\\{(N^{2}-M^{2})+(1+MI_{e})+(N^{2}+M^{2})I_{e}\\}]$

”

##  5 Implications on the Final Result

In Tables 1 and 2 of Liao’s study, where the optimal value of $T$ obtains from
$\mathrm{TC_{3}}$, the value of optimal $T$, the corresponding decision rule,
and value of total cost need to be updated. To illustrate this matter, we have
re-calculated the values of optimal $T$ and the corresponding total cost for
row 3 of Table 1 in Liao’s study (Table 1).

Parameters of experiment regarding row 3 of Table 1

$\theta$ | = | 0.00001 |  | $h$ | = | 1 |
---|---|---|---|---|---|---|---
$D$ | = | 9000000 |  | $\mathrm{I_{e}}$ | = | 0.000005 |
$A$ | = | 1224.04585 |  | $\mathrm{I_{p}}$ | = | 0.15 |
$c$ | = | 1.9999 |  | $N$ | = | 0.0161 |
$s$ | = | 2 |  | $M$ | = | 0.0165 |
$k$ | = | 1.1 |  | $W$ | = | 144900 |

Table 1: Implications of updated $\mathrm{TC_{3}}$ on the final result.

| Table | Row No | Dec. Rule | $T_{a}$ | W* | $\Delta_{1}$ | $\Delta_{2}$ | $\Delta_{3}$ | $\Delta_{4}$ | $\Delta_{5}$ | T* | TVC(T*)
---|---|---|---|---|---|---|---|---|---|---|---|---
Liao | 1 | 3 | (B1) | 0.0161 | 0.0165 | <0 | <0 | >0 | <0 | <0 |  $\mathrm{T^{*}_{3}}$=0.0161 | 148020
Updated |  |  | (B2) |  |  |  |  |  $\sim$0 |  | >0 |  $\mathrm{T^{*}_{3}}$=1.000129 | 4936419.16

##  6 Conclusion

In this technical note, we reviewed the Liao et al.’s [2014] model in which an
inventory model was developed by considering that the storage capacity is
limited and there are two levels of trade credit. The unavoidable complexity
emanating from the difference between permissible delay regarding the periods
of interest earned and interest charged requires the decision maker to seek
for an optimal inventory model. We identified some minor and one major issue
regarding the formulation of $\mathrm{TC_{3}}$ as the original study did not
consider the interest earned during N to T. The errors have been corrected and
the impact of updated terms on the computational result has been presented.

## References

## References

  * Liao et al. [2014] Liao, J.-J., Huang, K.-N., Ting, P.-S., 2014. Optimal strategy of deteriorating items with capacity constraints under two-levels of trade credit policy. Applied Mathematics and Computation 233, 647–658.


# Semileptonic and nonleptonic decays of the axial-vector tetraquark
$T_{bb;\overline{u}\overline{d}}^{-}$

S. S. Agaev  Institute for Physical Problems, Baku State University, Az–1148
Baku, Azerbaijan  K. Azizi  Department of Physics, University of Tehran, North
Karegar Avenue, Tehran 14395-547, Iran  Department of Physics, Doǧuş
University, Acibadem-Kadiköy, 34722 Istanbul, Turkey  B. Barsbay  Department
of Physics, Doǧuş University, Acibadem-Kadiköy, 34722 Istanbul, Turkey
Department of Physics, Kocaeli University, 41380 Izmit, Turkey  H. Sundu
Department of Physics, Kocaeli University, 41380 Izmit, Turkey

###### Abstract

The semileptonic and nonleptonic decays of the double-beauty axial-vector
tetraquark $T_{bb;\overline{u}\overline{d}}^{-}$ to a state
$\widetilde{T}_{bc;\overline{u}\overline{d}}^{0}$ (hereafter $T_{bb}^{-}$ and
$\widetilde{T}_{bc}^{0}$, respectively) are investigated in the context of the
QCD sum rule method. The final-state tetraquark $\widetilde{T}_{bc}^{0}$ is
treated as an axial-vector particle built of a heavy axial-vector diquark
$b^{T}\gamma_{\mu}Cc$ and light scalar antidiquark
$\overline{u}C\gamma_{5}\overline{d}^{T}$. Its spectroscopic parameters are
calculated using the two-points sum rules by taking into account contributions
of quark, gluon and mixed condensates up to dimension $10$. We study the
dominant semileptonic $T_{bb}^{-}\to\widetilde{T}_{bc}^{0}l\overline{\nu}_{l}$
and nonleptonic decays $T_{bb}^{-}\to\widetilde{T}_{bc}^{0}M$, where $M$ is
one of the pseudoscalar mesons $\pi^{-},K^{-},D^{-}$ and $D_{s}^{-}$. The
partial widths of these processes are computed in terms of weak form factors
$G_{i}(q^{2}),\ i=1,2,3,4$, extracted by employing the QCD three-point sum
rule approach. Predictions obtained for partial widths of considered decays
are used to improve accuracy of theoretical predictions for full width and
lifetime of the tetraquark $T_{bb}^{-}$, which are important for experimental
exploration of this exotic meson.

##  I Introduction

The four-quark exotic mesons containing a few heavy $Q=b$, $c$ quarks are
particles, investigation of which attracted interest of scientists more than
thirty years ago Iwasaki:1977qw ; Chao:1980dv ; Ader:1981db ; Lipkin:1986dw ;
Zouzou:1986qh ; Heller:1985cb ; Carlson:1987hh . Among these particles most
interesting are states composed of heavy diquarks $QQ^{\prime}$ and light
$\overline{q}\overline{q}^{\prime}$ antidiquarks, because they are real
candidates to stable exotic mesons. First qualitative results concerning a
stability of the compounds $QQ^{\prime}\overline{q}\overline{q}^{\prime}$
against strong decays were obtained already in Refs. Ader:1981db ;
Lipkin:1986dw ; Zouzou:1986qh ; Carlson:1987hh . Exotic mesons
$QQ^{\prime}\overline{q}\overline{q}^{\prime}$ may decay through strong
interaction to mesons $Q\overline{q}$ and $Q^{\prime}\overline{q}^{\prime}$ or
to $Q\overline{q}^{\prime}$ and $Q^{\prime}\overline{q}$ if the mass of the
master particle exceeds masses of final-state conventional mesons. It was
demonstrated that such four-quark states would be stable provided that the
ratio $m_{Q}/m_{q}$ is sufficiently large. A well known particle from this
range is the axial-vector tetraquark $T_{bb}^{-}$: Calculations carried out in
the context of different models proved that its mass is below the
$B\overline{B}^{\ast}$ threshold, and $T_{bb}^{-}$ is the particle stable
against strong decays Carlson:1987hh ; Navarra:2007yw .

During the last decade properties of tetraquarks composed of heavy $bb$ and
$bc$ diquarks were investigated in numerous articles using various methods
(see, for example Refs. Du:2012wp ; Chen:2013aba ; Francis:2018jyb ;
Caramees:2018oue  and references therein). Recent analysis of the heavy-light
particles $QQ^{\prime}\overline{q}\overline{q}^{\prime}$ confirmed stable
nature of the tetraquark $T_{bb}^{-}$ Karliner:2017qjm ; Eichten:2017ffp ;
Agaev:2018khe . In our work Agaev:2018khe  the axial-vector particle
$T_{bb}^{-}$ was studied by means of the QCD sum rule method. In accordance
with our result, the mass of $T_{bb}^{-}$ is equal to $m=(10035\pm
260)~{}\mathrm{MeV}$ which is below the $B^{-}\overline{B}^{\ast 0}$ and
$B^{-}\overline{B}^{0}\gamma$ thresholds. In other words, this particle is
stable against the strong and radiative decays. Hence, it dissociates to
conventional mesons via weak processes considered also in Ref. Agaev:2018khe .
We explored the semileptonic decays $T_{bb}^{-}$ $\rightarrow
Z_{bc}^{0}l\overline{\nu}_{l}$, where the final-state tetraquark
$Z_{bc}^{0}=[bc][\overline{u}\overline{d}]$ was treated as a scalar particle.
By computing partial widths of these decays, we estimated the width $\Gamma$
and mean lifetime $\tau$ of the axial-vector tetraquark $T_{bb}^{-}$.
Predictions obtained for these parameters $\Gamma=(7.17\pm 1.23)\times
10^{-8}~{}\mathrm{MeV}$ and $\tau=9.18_{-1.34}^{+1.90}~{}\mathrm{fs}$ may be
useful for experimental investigation of double-heavy exotic mesons. Problems
connected with calculation of parameters and weak decay channels of $\
T_{bb}^{-}$ were addressed in Ref. Hernandez:2019eox , as well.

Investigations showed that not only the tetraquark $T_{bb}^{-}$, but also
other double-beauty states may be strong and electromagnetic interactions
stable particles. Thus, the scalar counterparts of $T_{bb}^{-}$, i.e., the
tetraquark $T_{b:\overline{d}}^{-}$, the scalar and axial-vector four-quark
mesons $T_{bb;\overline{u}\overline{s}}^{-}$ are stable against strong and
radiative decays. The spectroscopic parameters, widths and lifetimes of these
exotic mesons were computed in Refs. Agaev:2020dba ; Agaev:2019lwh ;
Agaev:2020zag .

As we have mentioned above, tetraquarks containing a diquark $bc$ are also
interesting object for studies, because some of them may be stable particles.
Thus, the mass of the scalar exotic meson $Z_{bc}^{0}$ equals to
$m_{Z}=(6660\pm 150)~{}\mathrm{MeV}$, which is below thresholds for strong and
radiative decays of $Z_{bc}^{0}$ to conventional mesons Agaev:2018khe . As a
result, $Z_{bc}^{0}$ is the strong- and electromagnetic-interaction stable
compound weak decays of which were studied in Ref. Sundu:2019feu . Predictions
for width and lifetime of $Z_{bc}^{0}$ obtained there provide valuable
information on features of this particle. The spectroscopic parameters and
possible strong and weak decay channels of the axial-vector tetraquark
$T_{bc}^{0}=bc\overline{u}\overline{d}$ was studied, as well Agaev:2019kkz .

In the present article, we extend our analysis of the tetraquark $T_{bb}^{-}$
by considering its new weak decay channels
$T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}l\overline{\nu}_{l}$ and
$T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}M$, where $\widetilde{T}_{bc}^{0}$
is an axial-vector state. This investigation will allow us to improve
estimates for the full width and lifetime of $T_{bb}^{-}$.

We treat $\widetilde{T}_{bc}^{0}$ as a tetraquark composed of color-
antitriplet heavy axial-vector diquark $bc$ and light color-triplet scalar
antidiquark $\overline{u}\overline{d}$. It is worth noting that quark contents
and quantum numbers of the tetraquarks $T_{bc}^{0}$ and
$\widetilde{T}_{bc}^{0}$ are the same, and both of them have the antisymmetric
color structure
$[\overline{\mathbf{3}}_{c}]_{bc}\otimes[\mathbf{3}_{c}]_{\overline{u}\overline{d}}$.
But $T_{bc}^{0}$ and $\widetilde{T}_{bc}^{0}$ differ from each another due to
their internal organizations. In fact, the heavy diquark in $T_{bc}^{0}$ is a
scalar state, whereas the tetraquark $\widetilde{T}_{bc}^{0}$ is made of an
axial-vector heavy diquark $bc$. The reason for such choice of the final-state
tetraquark $\widetilde{T}_{bc}^{0}$ will be explained later. The axial-vector
tetraquarks $[bc][\overline{q}\overline{q}^{\prime}]$ with definite isospins
can be modeled using a structure $\widetilde{T}_{bc}^{0}$. But in the limit
$m_{u}=m_{d}=0$, adopted in the present study, these states are degenerate and
their parameters do not differ from the ones of $\widetilde{T}_{bc}^{0}$. This
conclusion is valid for the tetraquarks $T_{bb}^{-}$ and $T_{bc}^{0}$ as well.

To compute partial widths of the initial $T_{bb}^{-}$ particle’s weak decays,
apart from its mass and current coupling, one needs also spectral parameters
of the tetraquark $\widetilde{T}_{bc}^{0}$. The mass $\widetilde{m}$ and
coupling $\widetilde{f}$ of the state $\widetilde{T}_{bc}^{0}$ are calculated
using the QCD sum rule method Shifman:1978bx ; Shifman:1978by , which is a
powerful tool to calculate parameters of conventional mesons and baryons. It
can be successfully applied to analyze multiquark hadrons, as well
Agaev:2016dev ; Albuquerque:2018jkn . We calculate the mass and current
coupling of the tetraquark $\widetilde{T}_{bc}^{0}$ using relevant
interpolating current by taking into account various quark, gluon, and mixed
condensates up to dimension $10$. Spectral parameters of
$\widetilde{T}_{bc}^{0}$ extracted from such analysis are also of particular
interest to explore the family of tetraquarks
$bc\overline{q}\overline{q}^{\prime}$.

There are different weak decays of $T_{bb}^{-}$, but dominant ones are
processes triggered by a subprocess $b\rightarrow W^{-}c$ responsible for
transformation of $T_{bb}^{-}$ to the final axial-vector state
$\widetilde{T}_{bc}^{0}$. In semileptonic decays
$T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}l\overline{\nu}_{l}$ the
tetraquark $\widetilde{T}_{bc}^{0}$ is accompanied by a lepton pair
$l\overline{\nu}_{l}$, whereas in nonleptonic processes
$T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}M$ there is an additional ordinary
meson $M$ in the final phase of the process. We consider decays in which $M$
is one of the conventional pseudoscalar mesons $\pi^{-}$, $K^{-}$, $D^{-}$ and
$D_{s}^{-}$. To evaluate partial widths of weak decays one has to determine
form factors $G_{i}(q^{2}),\ i=1,2,3,4$ which govern weak transitions: They
enter to differential rate $d\Gamma/dq^{2}$ of semileptonic and partial width
of nonleptonic processes. To this end, we employ the QCD three-point sum rule
approach, and extract $G_{i}(q^{2})$ at $q^{2\text{ }}$ accessible for sum
rule calculations. As usual, these $q^{2}$ do not cover a full region
$m_{l}^{2}\leq q^{2}\leq(m-\widetilde{m})^{2}$ necessary to integrate the
differential rates $d\Gamma/dq^{2}$ of semileptonic decays. Therefore, one has
to introduce model functions $\mathcal{G}_{i}(q^{2})$ that coincide with the
sum rule predictions when they are accessible, and can be easily extrapolated
to all $q^{2}$: Usage of $\mathcal{G}_{i}(q^{2})$ in calculations solves these
technical problems.

This article is structured in the following way: In Sec. II, we evaluate the
mass and current coupling of the tetraquark $\widetilde{T}_{bc}^{0}$ by
employing the QCD two-point sum rule method. Calculations of the weak form
factors $G_{i}(q^{2})$ in the framework of the three-point sum rule approach
are performed in section III. Here, we determine the model functions
$\mathcal{G}_{i}(q^{2})$ and also find partial widths of the semileptonic
decays $T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}l\overline{\nu}_{l}$.
Section IV is devoted to analysis of the nonleptonic weak transformations of
the tetraquark $T_{bb}^{-}$. In section V we calculate the full width and
lifetime of $T_{bb}^{-}$, and discuss obtained results. This section contains
also our concluding notes.

##  II Mass and current coupling of the axial-vector tetraquark
$\widetilde{T}_{bc}^{0}$

The mass $\widetilde{m}$, and coupling $\widetilde{f}$ of the tetraquark
$\widetilde{T}_{bc}^{0}$ are important parameters of the problem under
consideration: they are required to find partial widths of the weak processes
$T_{bb}^{-}$ $\rightarrow\widetilde{T}_{bc}^{0}l\overline{\nu}_{l}$ and
$T_{bb}^{-}$ $\rightarrow\widetilde{T}_{bc}^{0}M$. Besides, the axial-vector
tetraquark $\widetilde{T}_{bc}^{0}$, as its partner state $T_{bc}^{0}$, may be
strong- and/or electromagnetic-interaction stable particle, which is
interesting in itself.

The sum rules to extract spectroscopic parameters of $\widetilde{T}_{bc}^{0}$
can be derived from analysis of the two-point correlation function
$\Pi_{\mu\nu}(p)$ given by the expression

$\Pi_{\mu\nu}(p)=i\int d^{4}xe^{ipx}\langle
0|\mathcal{T}\\{\widetilde{J}_{\mu}(x)\widetilde{J}_{\nu}^{{\dagger}}(0)\\}|0\rangle,$
(1)

where $\widetilde{J}_{\mu}(x)$ is the interpolating current to the axial-
vector tetraquark $\widetilde{T}_{bc}^{0}$. The structure of this current is
determined, in some respects, by the organization of the initial particle
$T_{bb}^{-}$. It is instructive to consider the structure and interpolating
current $J_{\mu}(x)$ of the teraquark $T_{bb}^{-}$

$J_{\mu}(x)=b_{a}^{T}(x)\gamma_{\mu}Cb_{b}(x)\overline{u}_{a}(x)C\gamma_{5}\overline{d}_{b}^{T}(x),$
(2)

where $a$ and $b$ are color indices and $C$ is charge-conjugation operator.
The current $J_{\mu}(x)$ was used in Ref. Agaev:2018khe  to study the exotic
meson $T_{bb}^{-}$. As is seen, $T_{bb}^{-}$ is built of the axial-vector
diquark $b^{T}\gamma_{\mu}Cb$ and light scalar antidiquark
$\overline{u}C\gamma_{5}\overline{d}^{T}$, and the current $J_{\mu}$ belongs
to the
$[\overline{\mathbf{3}}_{c}]_{bb}\otimes[\mathbf{3}_{c}]_{\overline{u}\overline{d}}$
representation of the color group $SU_{c}(3)$.

Weak decays of $T_{bb}^{-}$ run through transition of $b$-quark $b\rightarrow
c$, therefore expected organization of the final diquark field is
$b_{a}^{T}\gamma_{\mu}Cc_{b}$, whereas the antidiquark field preserves its
quark content and scalar nature. Then, at the final state we get the axial-
vector tetraquark, which is described by the current

$\displaystyle\widetilde{J}_{\mu}(x)$ $\displaystyle=$ $\displaystyle
b_{a}^{T}(x)\gamma_{\mu}Cc_{b}(x)\left[\overline{u}_{a}(x)C\gamma_{5}\overline{d}_{b}^{T}(x)\right.$
(3)
$\displaystyle\left.-\overline{u}_{b}(x)\gamma_{\mu}C\gamma_{5}\overline{d}_{a}^{T}(x)\right].$

The current $\widetilde{J}_{\mu}$ is antisymmetric in color indices and has
color-triplet structure
$[\overline{\mathbf{3}}_{c}]_{bc}\otimes[\mathbf{3}_{c}]_{\overline{u}\overline{d}}$.
It is known that scalar and axial vector triplet diquarks are most favorable
two-quark states to construct tetraquarks with $J^{\mathrm{P}}=1^{+}$
Jaffe:2004ph . The current $\widetilde{J}_{\mu}$ corresponds to lower lying
tetraquark with structure $\gamma_{\mu}C\otimes C\gamma_{5}$ and spin-parity
$J^{\mathrm{P}}=1^{+}$. Of course, there is an alternative choice for
$\widetilde{J}_{\mu}$ composed of a heavy scalar diquark and an axial-vector
light antidiquark. Properties of such state $T_{bc}^{0}$ with a composition
$C\gamma_{5}\otimes\gamma_{\mu}C$, its weak and strong decays were
investigated in Ref. Agaev:2019kkz . To model $\widetilde{T}_{bc}^{0}$ we
choose $\widetilde{J}_{\mu}$ given by Eq. (3) as a current stemming naturally
from organization of the master particle $T_{bb}^{-}$.

To find the sum rules for the mass $\widetilde{m}$ and coupling
$\widetilde{f}$ of the tetraquark $\widetilde{T}_{bc}^{0}$, we write down the
correlation function $\Pi_{\mu\nu}^{\mathrm{Phys}}(p)$ using physical
parameters of $\widetilde{T}_{bc}^{0}$. We treat $\widetilde{T}_{bc}^{0}$ as a
ground-state particle, and separate its contribution to
$\Pi_{\mu\nu}^{\mathrm{Phys}}(p)$ from other terms

$\Pi_{\mu\nu}^{\mathrm{Phys}}(p)=\frac{\langle
0|\widetilde{J}_{\mu}|\widetilde{T}_{bc}^{0}(p)\rangle\langle\widetilde{T}_{bc}^{0}(p)|\widetilde{J}_{\nu}^{\dagger}|0\rangle}{\widetilde{m}^{2}-p^{2}}+\cdots.$
(4)

Effects of higher resonances and continuum states are denoted in
$\Pi_{\mu\nu}^{\mathrm{Phys}}(p)$ by dots. The expression (4) is derived by
saturating the correlation function with a complete set of $J^{P}=1^{+}$
states with required quark content and performing integration in
$\Pi_{\mu\nu}(p)$ over $x$.

The correlator $\Pi_{\mu\nu}^{\mathrm{Phys}}(p)$ can be simplified by
introducing the matrix element $\langle
0|\widetilde{J}_{\mu}|\widetilde{T}_{bc}^{0}(p)\rangle$

$\langle
0|\widetilde{J}_{\mu}|\widetilde{T}_{bc}^{0}(p)\rangle=\widetilde{m}\widetilde{f}\widetilde{\epsilon}_{\mu},$
(5)

where $\widetilde{\epsilon}_{\mu}$ is the polarization vector of the
tetraquark $\widetilde{T}_{bc}^{0}$. In terms of the mass $\widetilde{m}$ and
coupling $\widetilde{f}$ the function $\Pi_{\mu\nu}^{\mathrm{Phys}}(p)$ takes
the form

$\Pi_{\mu\nu}^{\mathrm{Phys}}(p)=\frac{\widetilde{m}^{2}\widetilde{f}^{2}}{\widetilde{m}^{2}-p^{2}}\left(-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{\widetilde{m}^{2}}\right)+\cdots.$
(6)

The sum rules require computation of $\Pi_{\mu\nu}(p)$ in terms of quark
propagators, as well. To this end, one needs to substitute
$\widetilde{J}_{\mu}(x)$ into the correlation function (1) and contract
relevant light and heavy quark fields. These manipulations yields

$\displaystyle\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$ $\displaystyle=$ $\displaystyle
i\int
d^{4}xe^{ipx}\mathrm{Tr}\left[\gamma_{\mu}\widetilde{S}_{b}^{aa^{\prime}}(x)\gamma_{\nu}S_{c}^{bb^{\prime}}(x)\right]$
(7)
$\displaystyle\times\left\\{\mathrm{Tr}\left[\gamma_{5}\widetilde{S}_{d}^{a^{\prime}b}(-x)\gamma_{5}S_{u}^{b^{\prime}a}(-x)\right]\right.$
$\displaystyle-\mathrm{Tr}\left[\gamma_{5}\widetilde{S}_{d}^{b^{\prime}b}(-x)\gamma_{5}S_{u}^{a^{\prime}a}(-x)\right]$
$\displaystyle+\mathrm{Tr}\left[\gamma_{5}\widetilde{S}_{d}^{b^{\prime}a}(-x)\gamma_{5}S_{u}^{a^{\prime}b}(-x)\right]$
$\displaystyle\left.-\mathrm{Tr}\left[\gamma_{5}\widetilde{S}_{d}^{a^{\prime}a}(-x)\gamma_{5}S_{u}^{b^{\prime}b}(-x)\right]\right\\},$

where $S_{Q}^{ab}(x)$ and $S_{q}^{ab}(x)$ are the heavy $Q=b(c)$ and light
$q=d(u)$ quark propagators, respectively. Here, we have introduced also the
notation

$\widetilde{S}_{Q(q)}(x)=CS_{Q(q)}^{T}(x)C.$ (8)

In the present work, we use the light quark propagator given by the expression
Agaev:2020zad

$\displaystyle
S_{q}^{ab}(x)=i\frac{\not{x}\delta_{ab}}{2\pi^{2}x^{4}}-\frac{m_{q}\delta_{ab}}{4\pi^{2}x^{2}}-\frac{\langle\overline{q}q\rangle}{12}\left(1-i\frac{m_{q}}{4}\not{x}\right)\delta_{ab}$
$\displaystyle-\frac{x^{2}}{192}\langle\overline{q}g_{s}\sigma
Gq\rangle\left(1-i\frac{m_{q}}{6}\not{x}\right)\delta_{ab}-\frac{\not{x}x^{2}g_{s}^{2}}{7776}\langle\overline{q}q\rangle^{2}\delta_{ab}$
$\displaystyle-\frac{ig_{s}G_{ab}^{\mu\nu}}{32\pi^{2}x^{2}}\left[\not{x}\sigma_{\mu\nu}+\sigma_{\mu\nu}\not{x}\right]-\frac{x^{4}\langle\overline{q}q\rangle\langle
g_{s}^{2}G^{2}\rangle}{27648}\delta_{ab}+\cdots.$ (9)

The propagator of the heavy quarks $Q$ is determined by the formula

$\displaystyle
S_{Q}^{ab}(x)=i\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ikx}\Bigg{\\{}\frac{\delta_{ab}\left({\not{k}}+m_{Q}\right)}{k^{2}-m_{Q}^{2}}$
$\displaystyle-\frac{g_{s}G_{ab}^{\alpha\beta}}{4}\frac{\sigma_{\alpha\beta}\left({\not{k}}+m_{Q}\right)+\left({\not{k}}+m_{Q}\right)\sigma_{\alpha\beta}}{(k^{2}-m_{Q}^{2})^{2}}$
$\displaystyle+\frac{g_{s}^{2}G^{2}}{12}\delta_{ab}m_{Q}\frac{k^{2}+m_{Q}{\not{k}}}{(k^{2}-m_{Q}^{2})^{4}}+\frac{g_{s}^{3}G^{3}}{48}\delta_{ab}\frac{\left({\not{k}}+m_{Q}\right)}{(k^{2}-m_{Q}^{2})^{6}}$
$\displaystyle\times\left[{\not{k}}\left(k^{2}-3m_{Q}^{2}\right)+2m_{Q}\left(2k^{2}-m_{Q}^{2}\right)\right]\left({\not{k}}+m_{Q}\right)+\cdots\Bigg{\\}}.$
(10)

In expressions (9) and (10)

$\displaystyle G_{ab}^{\alpha\beta}$ $\displaystyle=$ $\displaystyle
G_{A}^{\alpha\beta}t_{ab}^{A},\,\,~{}~{}G^{2}=G_{\alpha\beta}^{A}G_{\alpha\beta}^{A},$
$\displaystyle G^{3}$ $\displaystyle=$
$\displaystyle\,\,f^{ABC}G_{\mu\nu}^{A}G_{\nu\delta}^{B}G_{\delta\mu}^{C},$
(11)

where $a,\,b=1,2,3$ are color indices and $A,B,C=1,\,2\,\cdots 8$. Here
$t^{A}=\lambda^{A}/2$, where $\lambda^{A}$ are the Gell-Mann matrices. The
gluon field strength tensor is fixed at $x=0$, i. e.,
$G_{\alpha\beta}^{A}\equiv G_{\alpha\beta}^{A}(0)$.

To proceed one should choose invariant amplitudes corresponding to the same
Lorentz structures from both $\Pi_{\mu\nu}^{\mathrm{Phys}}(p)$ and
$\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$. There are two Lorentz structures
proportional to $g_{\mu\nu}$ and $p_{\mu}p_{\nu}$ in these correlation
functions. Because invariant amplitudes $\Pi^{\mathrm{Phys}}(p^{2})$ and
$\Pi^{\mathrm{OPE}}(p^{2})$ corresponding to terms $\sim g_{\mu\nu}$ do not
contain contributions of scalar particles, we work with these functions. The
sum rules for $\widetilde{m}$ and $\widetilde{f}$ can be obtained by equating
these invariant amplitudes and performing standard prescriptions of the sum
rule method. As the first step, one applies the Borel transformation to both
sides of obtained equality, which is necessary to suppress contributions due
to higher resonances and continuum states. Afterwards, these contributions
should be subtracted from the physical side of this equality by employing the
hypothesis on quark-hadron duality. After these manipulations, a final
expression becomes a function of the Borel $M^{2}$ and continuum threshold
$s_{0}$ parameters. The second expression required to derive sum rules for
$\widetilde{m}$ and $\widetilde{f}$ can be obtained from the first equality by
acting on it by the operator $d/d(-1/M^{2})$. As a result, for $\widetilde{m}$
and $\widetilde{f}$, we get the sum rules

$\widetilde{m}^{2}=\frac{\Pi^{\prime}(M^{2},s_{0})}{\Pi(M^{2},s_{0})},$ (12)

and

$\widetilde{f}^{2}=\frac{e^{\widetilde{m}^{2}/M^{2}}}{\widetilde{m}^{2}}\Pi(M^{2},s_{0}).$
(13)

In expressions above $\Pi(M^{2},s_{0})$ is the Borel transformed and continuum
subtracted invariant amplitude $\Pi^{\mathrm{OPE}}(p^{2})$, and
$\Pi^{\prime}(M^{2},s_{0})=d/d(-1/M^{2})\Pi(M^{2},s_{0})$.

The function $\Pi(M^{2},s_{0})$ has the following form

$\Pi(M^{2},s_{0})=\int_{\mathcal{M}^{2}}^{s_{0}}ds\rho^{\mathrm{OPE}}(s)e^{-s/M^{2}}+\Pi(M^{2}),$
(14)

where $\mathcal{M}=m_{b}+m_{c}$. Here, $\rho^{\mathrm{OPE}}(s)$ is the two-
point spectral density computed as an imaginary part of the correlation
function. The second term in Eq. (14) contains nonperturbative contributions
computed directly from $\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$. In the present work,
we calculate $\Pi(M^{2},s_{0})$ by taking into account nonperturbative terms
up to dimension $10$. The explicit expression of the function
$\Pi(M^{2},s_{0})$ is rather lengthy, therefore we do not provide it here.

The obtained sum rules contain numerous input parameters, which have to be
specified in order to carry out computations. The vacuum condensates and
masses of $b$, and $c$ quarks are universal parameters and do not depend on
the problem under analysis: Their values are listed below

$\displaystyle\langle\overline{q}q\rangle=-(0.24\pm
0.01)^{3}~{}\mathrm{GeV}^{3},\ $ $\displaystyle\langle\overline{q}g_{s}\sigma
Gq\rangle=m_{0}^{2}\langle\overline{q}q\rangle,\ m_{0}^{2}=(0.8\pm
0.1)~{}\mathrm{GeV}^{2},$
$\displaystyle\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle=(0.012\pm
0.004)~{}\mathrm{GeV}^{4},$ $\displaystyle\langle
g_{s}^{3}G^{3}\rangle=(0.57\pm 0.29)~{}\mathrm{GeV}^{6},\ $ $\displaystyle
m_{c}=1.27\pm
0.2~{}\mathrm{GeV},~{}m_{b}=4.18_{-0.02}^{+0.03}~{}\mathrm{GeV}.$ (15)

The mass and coupling of the tetraquark $\widetilde{T}_{bc}^{0}$ depend on the
auxiliary parameters $M^{2}$ and $s_{0}$, and their correct choice is one of
important problems of our studies. We fix the upper allowed value of the Borel
parameter from a restriction $\mathrm{PC}>0.2$, where $\mathrm{PC}$ is a pole
contribution to the sum rules. The lower bound is found from convergence of
the sum rules. Additionally, quantities extracted from Eqs. (12) and (13)
should be as stable as possible against variations of $M^{2}$. The continuum
threshold parameter $s_{0}$ divides the ground-state contribution and effects
of higher resonances and continuum states. Therefore, $s_{0}$ should be below
the first excited state of the tetraquark $\widetilde{T}_{bc}^{0}$ and obey
$\sqrt{s_{0}}-\widetilde{m}$ $\approx(400-600)~{}\mathrm{MeV}$, which can be
considered as a reasonable restriction for heavy tetraquarks.

Performed numerical analyses demonstrate that regions

$M^{2}\in[5.5,7]\ \mathrm{GeV}^{2},\ s_{0}\in[58,60]\ \mathrm{GeV}^{2},$ (16)

satisfy all aforementioned constraints on $M^{2}$ and $s_{0}$. Indeed, at
$M^{2}=7~{}\mathrm{GeV}^{2}$ the pole contribution is $37\%$, whereas at
$M^{2}=5.5~{}\mathrm{GeV}^{2}$ amounts to $79\%$ of the whole result. These
values of $M^{2}$ fix the boundaries of the region in which the Borel
parameter can be varied. At the minimum of $M^{2}=5.5~{}\mathrm{GeV}^{2}$
contributions of last three terms to $\Pi(M^{2},s_{0})$ do not exceed $1\%$ of
its value.

For $\widetilde{m}$ and $\widetilde{f}$ we find

$\displaystyle\widetilde{m}$ $\displaystyle=$ $\displaystyle(7050\pm
125)~{}\mathrm{MeV},$ $\displaystyle\widetilde{f}$ $\displaystyle=$
$\displaystyle(8.3\pm 1.3)\times 10^{-3}~{}\mathrm{GeV}^{4}.$ (17)

In Fig. 1 we plot the sum rule’s prediction for $\widetilde{m}$, where its
dependence on the Borel $M^{2}$ and continuum threshold parameter $s_{0}$ is
seen explicitly. Theoretical errors in the case of $\widetilde{m}$ amount to
$\pm 1.8\%$, which confirms a nice accuracy of performed computations. The
ambiguities in deriving of the coupling $\widetilde{f}$ are equal to $\pm
16\%$ of the central value: they are larger than that for $\widetilde{m}$, but
still within limits accepted in sum rule computations. Reasons behind of these
effects are quite clear. Indeed, the sum rule for the mass $\widetilde{m}$ is
given by the ratio (12) which smooths the dependence of $\widetilde{m}$ on the
parameter $M^{2}$, whereas the sum rule for $\widetilde{f}$ (13) contains only
the correlator $\Pi(M^{2},s_{0})$.

Figure 1:  The mass of the tetraquark $\widetilde{T}_{bc}^{0}$ as a function
of the Borel parameter $M^{2}$ at fixed $s_{0}$ (left panel) and as a function
of the continuum threshold $s_{0}$ at fixed $M^{2}$ (right panel).

It is interesting to compare the result obtained for the mass of the axial-
vector tetraquark $\widetilde{T}_{bc}^{0}$ with the mass of $T_{bc}^{0}$. Let
us remind that the latter has the same quark content and quantum numbers, but
is composed of the heavy scalar diquark $b^{T}C\gamma_{5}c$ and light axial-
vector antidiquark $\overline{u}C\gamma_{\mu}\overline{d}^{T}$. This particle
has the mass $(7105\pm 155)~{}\mathrm{MeV}$ and is $\Delta m\approx
50~{}\mathrm{MeV}$ ”heavier” than $\widetilde{T}_{bc}^{0}$. Our estimate for
the mass splitting between $\widetilde{T}_{bc}^{0}$ and $T_{bc}^{0}$ is
obtained using central values of their masses. Spectroscopic parameters of
these tetraquarks have been computed in the context of the QCD sum rule
method, predictions of which contain theoretical errors. It is evident that
the mass difference $\Delta m$ is smaller than uncertainties of this analysis
and does not allow one to distinguish these particles from each other
reliably. Real exotic mesons may be superpositions of these tetraquarks.

An important feature of $\widetilde{T}_{bc}^{0}$ is that it is stable against
strong decays. In fact, the sum rule method predicts not only the mass of
tetraquarks with some accuracy, but also allows one to estimate theoretical
errors of performed computations. The central values of the masses
$7105~{}\mathrm{MeV}$ and $7050~{}\mathrm{MeV}$ for $T_{bc}^{0}$ and
$\widetilde{T}_{bc}^{0}$ are below the threshold $\approx 7190~{}\mathrm{MeV}$
for strong decays to a pair of conventional mesons $B^{\ast-}D^{+}$ and
$\overline{B}^{\ast 0}D^{0}$. But the maximal prediction for the mass of
$T_{bc}^{0}$ equals to $7260~{}\mathrm{MeV}$ and exceeds this threshold.
Stated differently, the sum rule results do not exclude strong decays of
$T_{bc}^{0}$ to conventional mesons Agaev:2019kkz . In the case of
$\widetilde{T}_{bc}^{0}$, we get for the upper bound of the mass
$7175~{}\mathrm{MeV}$ which is still below $7190~{}\mathrm{MeV}$, and hence
$\widetilde{T}_{bc}^{0}$ is strong-interaction stable particle.

The first excited state of the tetraquark $\widetilde{T}_{bc}^{0}$ has
presumably a mass $\sqrt{s_{0}}$
$\approx\widetilde{m}+(400-600)~{}\mathrm{MeV}$ which makes it unstable
against decays not only to aforementioned mesons $B^{\ast-}D^{+}$ and
$\overline{B}^{\ast 0}D^{0}$, but also to pairs $B^{-}D^{\ast+}$ and
$\overline{B}^{0}D^{\ast 0}$. These two-hadron reducible terms are common
problem when studying multiquark systems in the framework of the QCD sum rule
approach Kondo:2004cr ; Lee:2004xk . In the case of tetraquarks two-meson
contributions generate the finite width $\Gamma(p^{2})$ of these particles and
lead to rescaling of their coupling constants. Nevertheless these effects are
small and, as usual, can be neglected Wang:2015nwa ; Agaev:2018vag ;
Sundu:2018nxt .

##  III Semileptonic decay
$T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}l\overline{\nu}_{l}$

Weak decays of $T_{bb}^{-}$ can be triggered by subprocesses $b\rightarrow
W^{-}c$ and $b\rightarrow W^{-}u$. The processes generated by the transition
$b\rightarrow W^{-}c$ are dominant decay modes of $T_{bb}^{-}$. The reason is
that the subprocess $b\rightarrow W^{-}u$ leads to decays suppressed relative
to dominant ones by a factor $|V_{bu}|^{2}/|V_{bc}|^{2}$ $\simeq 0.01$, where
$V_{q_{1}q_{2}}$ are the Cabibbo-Khobayasi-Maskawa (CKM) matrix elements. In
the present work, we consider only dominant weak decays of $T_{bb}^{-}$ to
axial-vector tetraquark $[bc][\overline{u}\overline{d}]$, which, in general,
may contain components $T_{bc}^{0}$ and $\widetilde{T}_{bc}^{0}$. Our analysis
proves that the matrix element for the weak transition $T_{bb}^{-}\rightarrow
T_{bc}^{0}$ vanishes and does not contribute to relevant decay processes. This
fact maybe is connected with spin flips in initial diquark and antidiquark
necessary to form the final state $T_{bc}^{0}.$ Therefore we restrict
ourselves by exploring the weak transformations
$T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}$, which consist of the
semileptonic $T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}l\overline{\nu}_{l}$
and nonleptonic $T_{bb}^{-}$ $\rightarrow\widetilde{T}_{bc}^{0}M$ processes.

Here, we concentrate on semileptonic decays
$T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}l\overline{\nu}_{l}$ of the
tetraquark $T_{bb}^{-}$. The analysis carried out in the section II has
allowed us to calculate the spectroscopic parameters of the tetraquark
$\widetilde{T}_{bc}^{0}$, which are input information to investigate decays of
the initial particle $T_{bb}^{-}$. A big mass gap between the initial and
final-state tetraquarks makes kinematically possible semileptonoc decays with
all lepton spices $l=e,\mu$, and $\tau$.

The effective Hamiltonian to study processes $b\rightarrow W^{-}c$ at the
tree-level is determined by the expression

$\mathcal{H}^{\mathrm{eff}}=\frac{G_{F}}{\sqrt{2}}V_{bc}\overline{c}\gamma_{\alpha}(1-\gamma_{5})b\overline{l}\gamma^{\alpha}(1-\gamma_{5})\nu_{l},$
(18)

where $G_{F}$ and $V_{bc}$ are the Fermi coupling constant and CKM matrix
element, respectively. The matrix element of $\mathcal{H}^{\mathrm{eff}}$
placed between the initial and final tetraquarks contains the leptonic and
hadronic factors

$\langle\widetilde{T}_{bc}^{0}(p^{\prime})|\mathcal{H}^{\mathrm{eff}}|T_{bb}^{-}(p)\rangle=L^{\alpha}H_{\alpha}.$
(19)

We are interested in calculation of $H_{\alpha}$, because the leptonic part of
the matrix element $L^{\alpha}$ is universal for all semileptonic decays and
does not contain information on tetraquarks. Then, $H_{\alpha}$ is nothing
more than the matrix element of the current

$J_{\alpha}^{\mathrm{tr}}=\overline{c}\gamma_{\alpha}(1-\gamma_{5})b,$ (20)

sandwiched between the initial and final particles. It can be modeled in terms
of the form factors $G_{i}(q^{2})$ which parametrize the long-distance
dynamics of the weak transition

$\displaystyle\langle\widetilde{T}_{bc}^{0}(p^{\prime},\epsilon^{\prime})|J_{\alpha}^{\mathrm{tr}}|T_{bb}^{-}(p,\epsilon)\rangle=\epsilon^{\mu}\epsilon^{\prime\nu}\left[G_{1}(q^{2})g_{\mu\nu}P_{\alpha}\right.$
$\displaystyle\left.+G_{2}(q^{2})\left(q_{\mu}g_{\alpha\nu}-q_{\nu}g_{\alpha\mu}\right)-\frac{G_{3}(q^{2})}{2m^{2}}q_{\mu}q_{\nu}P_{\alpha}\right]$
$\displaystyle+G_{4}(q^{2})\varepsilon_{\alpha\mu\rho\nu}\epsilon^{\mu}\epsilon^{\prime\rho}P^{\nu},$
(21)

where $(p,\epsilon)$ and $(p^{\prime},\epsilon^{\prime})$ are momenta and
polarization vectors of $T_{bb}^{-}$ and $\widetilde{T}_{bc}^{0}$,
respectively. Here, we also use $P=p+p^{\prime}$ and $q=p-p^{\prime}$. The
momentum transfer in the weak process $q^{2}$ changes within the limits
$m_{l}^{2}\leq q^{2}\leq(m-\widetilde{m})^{2},$ where $m_{l}$ is the mass of
the lepton $l$.

The weak form factors $G_{i}(q^{2})$ are key ingredients of our
investigations. They should be determined from the QCD three-point sum rules,
which can be derived using the correlation function

$\displaystyle\Pi_{\mu\alpha\nu}(p,p^{\prime})=i^{2}\int
d^{4}xd^{4}ye^{i(p^{\prime}y-px)}$ $\displaystyle\times\langle
0|\mathcal{T}\\{\widetilde{J}_{\nu}(y)J_{\alpha}^{\mathrm{tr}}(0)J_{\mu}^{{}^{\dagger}}(x)\\}|0\rangle.$
(22)

The standard methods of the sum rule analysis require calculation of the
correlation function $\Pi_{\mu\alpha\nu}(p,p^{\prime})$ using the physical
parameters of the tetraquarks and, by this way, to find the physical side of
the sum rules. At the next phase of studies, one has to determine
$\Pi_{\mu\alpha\nu}(p,p^{\prime})$ by employing the quark propagators, and
express $\Pi_{\mu\alpha\nu}^{\mathrm{OPE}}(p,p^{\prime})$ in terms of quark,
gluon and mixed vacuum condensates. By equating obtained results and using the
assumption about the quark-hadron duality, it is possible to derive sum rules
and compute the form factors of interest.

The physical side of the sum rules
$\Pi_{\mu\alpha\nu}^{\mathrm{Phys}}(p,p^{\prime})$ can be written down in the
following form

$\displaystyle\Pi_{\mu\alpha\nu}^{\mathrm{Phys}}(p,p^{\prime})=\frac{\langle
0|\widetilde{J}_{\nu}|\widetilde{T}_{bc}^{0}(p^{\prime},\epsilon^{\prime})\rangle\langle\widetilde{T}_{bc}^{0}(p^{\prime},\epsilon^{\prime})|J_{\alpha}^{\mathrm{tr}}|T_{bb}^{-}(p,\epsilon)\rangle}{(p^{2}-m^{2})(p^{\prime
2}-\widetilde{m}^{2})}$ $\displaystyle\times\langle
T_{bb}^{-}(p,\epsilon)|J_{\mu}^{{}^{\dagger}}|0\rangle+\cdots,$ (23)

where the contribution of the ground-state particles is shown explicitly,
whereas other terms are denoted by dots.

Calculation of $\Pi_{\mu\alpha\nu}^{\mathrm{Phys}}(p,p^{\prime})$ can be
finished by taking into account Eq. (5), the explicit expression of the matrix
element
$\langle\widetilde{T}_{bc}^{0}(p^{\prime},\epsilon^{\prime})|J_{\alpha}^{\mathrm{tr}}|T_{bb}^{-}(p,\epsilon)\rangle$,
and the formula

$\langle
T_{bb}^{-}(p,\epsilon)|J_{\mu}^{{}^{\dagger}}|0\rangle=fm\epsilon_{\mu}^{\ast},$
(24)

where $f$ is the coupling of the state $T_{bb}^{-}$. Having substituted the
relevant matrix elements into Eq. (23), we find the final expression for
$\Pi_{\mu\alpha\nu}^{\mathrm{Phys}}(p,p^{\prime},q^{2})$

$\displaystyle\Pi_{\mu\alpha\nu}^{\mathrm{Phys}}(p,p^{\prime})=\frac{fm\widetilde{f}\widetilde{m}}{(p^{2}-m^{2})(p^{\prime
2}-\widetilde{m}^{2})}\left\\{G_{1}(q^{2})p_{\alpha}g_{\mu\nu}\right.$
$\displaystyle+G_{2}(q^{2})\left[1-\frac{m^{2}-\widetilde{m}^{2}+q^{2}}{2m^{2}}\right]p_{\mu}g_{\alpha\nu}$
$\displaystyle\left.-\frac{G_{3}(q^{2})}{2m^{2}}p_{\alpha}p_{\nu}p_{\mu}^{\prime}+G_{4}(q^{2})\varepsilon_{\theta\alpha\mu\nu}p_{\theta}\right\\}+\cdots.$
(25)

The dots in $\Pi_{\mu\alpha\nu}^{\mathrm{Phys}}(p,p^{\prime})$ stand for not
only effects due to excited and continuum states, but also for contributions
of structures which will not be used in following analysis.

The QCD side of the sum rules can be derived from Eq. (22) by using the
interpolating currents and contracting relevant quark fields. The result of
these computations is given by the following formula

$\displaystyle\Pi_{\mu\alpha\nu}^{\mathrm{OPE}}(p,p^{\prime})=i^{2}\int
d^{4}xd^{4}ye^{i(p^{\prime}y-px)}\left\\{\left[\mathrm{Tr}\left[\gamma_{5}\widetilde{S}_{d}^{b^{\prime}b}(x-y)\right.\right.\right.$
$\displaystyle\left.\left.\times\gamma_{5}S_{u}^{a^{\prime}a}(x-y)\right]-\mathrm{Tr}\left[\gamma_{5}\widetilde{S}_{d}^{b^{\prime}a}(x-y)\gamma_{5}S_{u}^{a^{\prime}b}(x-y)\right]\right]$
$\displaystyle\times\left[\mathrm{Tr}\left[\gamma_{\nu}\widetilde{S}_{b}^{ia^{\prime}}(-x)(1-\gamma_{5})\gamma_{\alpha}\widetilde{S}_{c}^{bi}(y)\gamma_{\mu}S_{b}^{ab^{\prime}}(y-x)\right]\right.$
$\displaystyle\left.\left.-\mathrm{Tr}\left[\gamma_{\nu}\widetilde{S}_{b}^{aa^{\prime}}(y-x)\gamma_{\mu}S_{c}^{bi}(y)\gamma_{\alpha}(1-\gamma_{5})S_{b}^{ib^{\prime}}(-x)\right]\right]\right\\}.$

The correlation function $\Pi_{\mu\alpha\nu}^{\mathrm{OPE}}(p,p^{\prime})$
contains the same Lorentz structures as its counterpart
$\Pi_{\mu\alpha\nu}^{\mathrm{Phys}}(p,p^{\prime})$. We use corresponding
invariant amplitudes to obtain the required sum rules for the form factors
$G_{i}(q^{2})$. But before this final operation, we make double Borel
transformation over variables $p^{2}$ and $p^{\prime 2}$ to suppress
contributions of the higher excited and continuum states, and perform
continuum subtraction. These rather routine manipulations give the sum rules
for the form factors $G_{i}(q^{2})$. For $G_{i}(q^{2}),$ $i=1$ and $4$ we get
the similar sum rules

$\displaystyle
G_{i}(M^{2},s_{0},q^{2})=\frac{1}{fm\widetilde{f}\widetilde{m}}\int_{4m_{b}^{2}}^{s_{0}}ds\int_{\mathcal{M}^{2}}^{s_{0}^{\prime}}ds^{\prime}$
$\displaystyle\times\rho_{i}(s,s^{\prime},q^{2})e^{(m^{2}-s)/M_{1}^{2}}e^{(\widetilde{m}^{2}-s^{\prime})/M_{2}^{2}},$
(27)

where $M_{1}^{2},\ M_{2}^{2}$ and $s_{0},\ s_{0}^{\prime}$ are the Borel and
continuum threshold parameters, respectively. The pair of parameters
$(M_{1}^{2},s_{0})$ corresponds to a channel of the initial tetraquark
$T_{bb}^{-}$, whereas $(M_{2}^{2},s_{0}^{\prime})$ describe the final-state
particle $\widetilde{T}_{bc}^{0}$. The remaining two sum rules read:

$\displaystyle
G_{2}(M^{2},s_{0},q^{2})=\frac{2m}{\widetilde{f}\widetilde{m}f(m^{2}+\widetilde{m}^{2}-q^{2})}$
$\displaystyle\times\int_{4m_{b}^{2}}^{s_{0}}ds\int_{\mathcal{M}^{2}}^{s_{0}^{\prime}}ds^{\prime}\rho_{2}(s,s^{\prime},q^{2})e^{(m^{2}-s)/M_{1}^{2}}e^{(\widetilde{m}^{2}-s^{\prime})/M_{2}^{2}},$

and

$\displaystyle
G_{3}(M^{2},s_{0},q^{2})=-\frac{2m}{\widetilde{f}\widetilde{m}f}\int_{4m_{b}^{2}}^{s_{0}}ds\int_{\mathcal{M}^{2}}^{s_{0}^{\prime}}ds^{\prime}$
$\displaystyle\times\rho_{3}(s,s^{\prime},q^{2})e^{(m^{2}-s)/M_{1}^{2}}e^{(\widetilde{m}^{2}-s^{\prime})/M_{2}^{2}}.$
(29)

As is seen the sum rules are written down using the spectral densities
$\rho_{i}(s,s^{\prime},q^{2})$ which are proportional to the imaginary part of
the corresponding invariant amplitudes in
$\Pi_{\mu\alpha\nu}^{\mathrm{OPE}}(p,p^{\prime})$. All of them contain both
the perturbative and nonperturbative contributions and are calculated with
dimension-5 accuracy. Explicit expressions of $\rho_{i}(s,s^{\prime},q^{2})$
are cumbersome, therefore we refrain from providing them here.

The differential rate of the semileptonic decay
$T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}l\overline{\nu}_{l}$ is determined
by the weak form factors $G_{i}(q^{2})$ and is given by the expression
Sundu:2018uyi

$\displaystyle\frac{d\Gamma}{dq^{2}}=\frac{G_{F}^{2}|V_{cb}|^{2}}{3\cdot
2^{9}\pi^{3}m^{3}}\left(\frac{q^{2}-m_{l}^{2}}{q^{2}}\right)\lambda\left(m^{2},\widetilde{m}^{2},q^{2}\right)$
$\displaystyle\times\left[\sum_{i=1}^{i=4}G_{i}^{2}(q^{2})\mathcal{A}_{i}(q^{2})+G_{1}(q^{2})G_{2}(q^{2})\mathcal{A}_{12}(q^{2})\right.$
$\displaystyle\left.+G_{1}(q^{2})G_{3}(q^{2})\mathcal{A}_{13}(q^{2})+G_{2}(q^{2})G_{3}(q^{2})\mathcal{A}_{23}(q^{2})\right.\bigg{]},$
(30)

where

$\displaystyle\lambda\left(m^{2},\widetilde{m}^{2},q^{2}\right)=\left[m^{4}+\widetilde{m}^{4}+q^{4}\right.$
$\displaystyle\left.-2(m^{2}\widetilde{m}^{2}+m^{2}q^{2}+\widetilde{m}^{2}q^{2})\right]^{1/2}.$
(31)

The decay rate $d\Gamma/dq^{2}$ depends also on functions
$\mathcal{A}_{i}(q^{2})$ and $\mathcal{A}_{ij}(q^{2})$ which can be found in
Ref. Sundu:2018uyi .

Sum rules for $G_{i}(q^{2})$ are necessary to find corresponding fit functions
$\mathcal{G}_{i}(q^{2})$ and calculate the width of the semileptonic decays.
Technical details of numerical computations to extract weak form factors are
well known. In fact, Eqs. (27), (LABEL:eq:FF2) and (29) through the spectral
densities $\rho_{i}(s,s^{\prime},q^{2})$ depend on the quark, gluon and mixing
condensates, as well as masses of the $c$ and $b$-quarks: these parameters
have been specified in the previous section. Besides, the sum rules contain
also masses and couplings of the tetraquarks $T_{bb}^{-}$ and
$\widetilde{T}_{bc}^{0}$. The mass and coupling of $T_{bb}^{-}$ were evaluated
in Ref. Agaev:2018khe

$\displaystyle m$ $\displaystyle=$ $\displaystyle(10035\pm
260)~{}\mathrm{MeV},$ $\displaystyle f$ $\displaystyle=$
$\displaystyle(1.38\pm 0.27)\times 10^{-2}~{}\mathrm{GeV}^{4}.$ (32)

The spectroscopic parameters of the tetraquark $\widetilde{T}_{bc}^{0}$ have
been found in the present work. We need also to fix Borel and continuum
threshold parameters to carry out numerical analysis. In the intial particle
channel $(M_{1}^{2},s_{0})$ are chosen as in Ref. Agaev:2018khe , in which the
mass and coupling of $T_{bb}^{-}$ were calculated

$M^{2}\in[9,13]\ \mathrm{GeV}^{2},\ s_{0}\in[115,120]\ \mathrm{GeV}^{2}.$ (33)

For the next pair $(M_{2}^{2},s_{0}^{\prime})$ we use parameters from Eq.
(16).

The sum rules give reliable results for $G_{i}(q^{2})$ in the region
$m_{l}^{2}\leq q^{2}\leq 7~{}\mathrm{GeV}^{2}$, which is not enough to find
the partial width of the process
$T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}l\overline{\nu}_{l}$ under
consideration. To calculate the width of the semileptonic decay
$d\Gamma/dq^{2}$ must be integrated over $q^{2}$ in the boundaries
$m_{l}^{2}\leq q^{2}\leq(m-\widetilde{m})^{2}$, i.e., in the limits
$m_{l}^{2}\leq q^{2}\leq 8.9~{}\mathrm{GeV}^{2}$. But this region is wider
than the one where the sum rules lead to strong results. This problem can be
solved by introducing extrapolating (fit) functions $\mathcal{G}_{i}(q^{2})$.
At the momentum transfers $q^{2}$ accessible for the sum rule computations
they must coincide with $G_{i}(q^{2})$, but have analytic forms suitable to
carry out integrations over $q^{2}$.

Figure 2: Sum rule predictions for the weak form factors $G_{1}(q^{2})$ (the
upper blue circles) and $|G_{2}(q^{2})|$ (the lower red squares). The lines
denote the fit functions $\mathcal{G}_{1}(q^{2})$ and
$|\mathcal{G}_{2}(q^{2})|$, respectively.

To this end, we use the functions

$\mathcal{G}_{i}(q^{2})=\mathcal{G}_{0}^{i}\exp\left[g_{1}^{i}\frac{q^{2}}{m^{2}}+g_{2}^{i}\left(\frac{q^{2}}{m^{2}}\right)^{2}\right],$
(34)

where parameters $\mathcal{G}_{0}^{i},~{}g_{1}^{i}$, and $g_{2}^{i}$ should be
fitted to satisfy sum rules’ predictions. The parameters of the functions
$\mathcal{G}_{i}(q^{2})$ extracted from numerical analysis are collected in
Table 1. As an example, the functions $\mathcal{G}_{1}(q^{2})$ and
$|\mathcal{G}_{2}(q^{2})|$ are depicted in Fig. 2, in which we see a quite
nice agreement between the sum rule predictions and fit functions.

To calculate the partial widths of the semileptonic decays, apart from the
form factors, we also use the parameters

$\displaystyle G_{F}$ $\displaystyle=$ $\displaystyle 1.16637\times
10^{-5}~{}\mathrm{GeV}^{-2},$ $\displaystyle|V_{bc}|$ $\displaystyle=$
$\displaystyle(42.2\pm 0.08)\times 10^{-3}.$ (35)

$\mathcal{G}_{i}(q^{2})$ | $\mathcal{G}_{0}^{i}$ | $g_{1}^{i}$ | $g_{2}^{i}$
---|---|---|---
$\mathcal{G}_{1}(q^{2})$ | $2.13$ | $3.28$ | $-4.33$
$\mathcal{G}_{2}(q^{2})$ | $-0.72$ | $6.26$ | $-17.13$
$\mathcal{G}_{3}(q^{2})$ | $334.57$ | $3.31$ | $-5.71$
$\mathcal{G}_{4}(q^{2})$ | $-1.33$ | $3.36$ | $0.35$
Table 1: Parameters of the extrapolating functions $\mathcal{G}_{i}(q^{2})$.

Our results for the partial widths of the semileptonic decay channels are
presented below

$\displaystyle\Gamma(T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}e^{-}\overline{\nu}_{e})=(2.02\pm
0.39)\times 10^{-9}~{}\mathrm{MeV},$
$\displaystyle\Gamma(T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}\mu^{-}\overline{\nu}_{\mu})=(1.96\pm
0.37)\times 10^{-9}~{}\mathrm{MeV},$
$\displaystyle\Gamma(T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}\tau^{-}\overline{\nu}_{\tau})=(1.03\pm
0.19)\times 10^{-10}~{}\mathrm{MeV}.$ (36)

Obtained information on partial widths of semileptonic channels can be used to
improve predictions for the full width and lifetime of the exotic meson
$T_{bb}^{-}$.

##  IV  Nonleptonic processes $T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}M$

Dominant nonleptonic decays of $T_{bb}^{-}$ are triggered by the subprocess
$b\rightarrow W^{-}c$, whereas the transition $b\rightarrow W^{-}u$ leads to
decays suppressed relative to main ones, as it has been explained in the
previous section. Therefore, in this section we consider nonleptonic decays
$T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}M$ of the tetraquark $T_{bb}^{-}$.
In these processes $M$ is one of the pseudoscalar mesons $\pi^{-}$, $K^{-}$,
$\ D^{-}$, and $\ D_{s}^{-}$. At the final state of the process they appear
due to decays of $W^{-}$ to pairs of quark-antiquark $d\overline{u}$,
$s\overline{u}$, $d\overline{c}$, and $s\overline{c}$, respectively. The
masses and decay constants of the mesons $\pi^{-}$, $K^{-}$, $D^{-}$, and
$D_{s}^{-}$ are presented in Table 2. It is not difficult to see, that the
mass of $T_{bb}^{-}$ obeys a requirement $m>\widetilde{m}+m_{M}$ for all of
these mesons, and these decays are kinematically allowed processes.

We describe production of mesons $M$ by employing the effective Hamiltonian,
and introducing relevant effective weak vertices. To study the nonleptonic
decays $T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}M$, we use also the QCD
factorization approach. This method is fruitful for studying of ordinary
mesons’ nonleptonic decays Beneke:1999br ; Beneke:2000ry , but can be applied
to investigate weak decays of tetraquarks as well Agaev:2019lwh ;
Sundu:2019feu .

We present in a detailed form the decay
$T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}\pi^{-}$, and write down final
results for other channels. The effective Hamiltonian
$\widetilde{\mathcal{H}}^{\mathrm{eff}}$ for this decay at the tree-level is
given by the expression

$\widetilde{\mathcal{H}}^{\mathrm{eff}}=\frac{G_{F}}{\sqrt{2}}V_{bc}V_{ud}^{\ast}\left[c_{1}(\mu)Q_{1}+c_{2}(\mu)Q_{2}\right],$
(37)

where

$\displaystyle Q_{1}$ $\displaystyle=$
$\displaystyle\left(\overline{d}_{i}u_{i}\right)_{\mathrm{V-A}}\left(\overline{c}_{j}b_{j}\right)_{\mathrm{V-A}},$
$\displaystyle Q_{2}$ $\displaystyle=$
$\displaystyle\left(\overline{d}_{i}u_{j}\right)_{\mathrm{V-A}}\left(\overline{c}_{j}b_{i}\right)_{\mathrm{V-A}},$
(38)

and $i$ , $j$ are the color indices. In Eq. (37) the abbreviation
$\left(\overline{q}_{1}q_{2}\right)_{\mathrm{V-A}}$ means

$\left(\overline{q}_{1}q_{2}\right)_{\mathrm{V-A}}=\overline{q}_{1}\gamma_{\mu}(1-\gamma_{5})q_{2}.$
(39)

Let us note that, we do not include into Eq. (37) current-current operators
appearing due the QCD penguin and electroweak-penguin diagrams. The short-
distance Wilson coefficients $c_{1}(\mu)$ and $c_{2}(\mu)$ are given at the
factorization scale $\mu$.

The amplitude of the decay
$T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}\pi^{-}$ is determined by the
following expression

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{bc}V_{ud}^{\ast}a_{1}(\mu)\langle\pi^{-}(q)|\left(\overline{d}_{i}u_{i}\right)_{\mathrm{V-A}}|0\rangle$
(40)
$\displaystyle\times\langle\widetilde{T}_{bc}^{0}(p^{\prime})|\left(\overline{c}_{j}b_{j}\right)_{\mathrm{V-A}}|T_{bb}^{-}(p)\rangle,$

where

$a_{1}(\mu)=c_{1}(\mu)+\frac{1}{N_{c}}c_{2}(\mu),$ (41)

with $N_{c}=3$ being the number of quark colors.

The matrix element
$\langle\widetilde{T}_{bc}^{0}(p^{\prime})|\left(\overline{c}_{j}b_{j}\right)_{\mathrm{V-A}}|T_{bb}^{-}(p)\rangle$
in terms of the weak form factors is given by Eq. (21). The matrix element
$\langle\pi^{-}(q)|\left(\overline{d}_{i}u_{i}\right)_{\mathrm{V-A}}|0\rangle$
in $\mathcal{A}$ can be written down in the following form

$\langle\pi^{-}(q)|\left(\overline{d}_{i}u_{i}\right)_{\mathrm{V-A}}|0\rangle=if_{\pi}q_{\mu},$
(42)

where $f_{\pi}$ is the decay constant of the pion. Then, the amplitude
$\mathcal{A}$ of the nonleptonic weak decay is determined by the expression

$\displaystyle\mathcal{A}=i\frac{G_{F}}{\sqrt{2}}f_{\pi}V_{bc}V_{ud}^{\ast}a_{1}(\mu)\left\\{Pq\left[G_{1}(q^{2})\epsilon\cdot\epsilon^{\prime}\right.\right.$
$\displaystyle\left.\left.-\frac{G_{3}(q^{2})}{2m^{2}}q\cdot\epsilon
q\cdot\epsilon^{\prime}\right]+G_{4}(q^{2})\varepsilon_{\alpha\mu\rho\nu}\epsilon^{\mu}\epsilon^{\prime\rho}P^{\nu}q^{\alpha}\right\\}.$
(43)

For completeness we provide below the partial width of this process

$\Gamma(T_{b:\overline{d}}^{-}\rightarrow\widetilde{Z}_{bc}^{0}\pi^{-})=\frac{|\mathcal{A}|^{2}}{48\pi
m^{3}}\lambda\left(m^{2},\widetilde{m}_{Z}^{2},q^{2}\right),$ (44)

where

$\displaystyle|\mathcal{A}|^{2}=\frac{G_{F}^{2}}{2}f_{\pi}^{2}|V_{bc}|^{2}|V_{ud}|^{2}a_{1}^{2}(\mu)\left\\{G_{1}^{2}(q^{2})\left(m^{2}-\widetilde{m}^{2}\right)^{2}\right.$
$\displaystyle\times\frac{\left[m^{4}+\left(\widetilde{m}^{2}-q^{2}\right)^{2}-2m^{2}\left(5\widetilde{m}^{2}-q^{2}\right)\right]}{4m^{2}\widetilde{m}^{2}}$
$\displaystyle+G_{3}^{2}(q^{2})\frac{\left(m^{2}-\widetilde{m}^{2}\right)^{2}}{64m^{6}\widetilde{m}^{2}}\left[m^{4}+\left(\widetilde{m}^{2}-q^{2}\right)^{2}\right.$
$\displaystyle\left.-2m^{2}\left(\widetilde{m}^{2}+q^{2}\right)\right]^{2}+2G_{4}^{2}(q^{2})\left[m^{4}+\left(\widetilde{m}^{2}-q^{2}\right)^{2}\right.$
$\displaystyle\left.-2m^{2}\left(\widetilde{m}^{2}+q^{2}\right)\right]+G_{1}(q^{2})G_{3}(q^{2})\frac{\left(m^{2}-\widetilde{m}^{2}\right)^{2}}{8m^{4}\widetilde{m}^{2}}$
$\displaystyle\times\left[m^{6}+\left(\widetilde{m}^{2}-q^{2}\right)^{3}-m^{4}\left(\widetilde{m}^{2}+3q^{2}\right)\right.$
$\displaystyle\left.\left.-m^{2}\left(\widetilde{m}^{4}+2\widetilde{m}^{2}q^{2}-3q^{4}\right)\right]\right\\}.$
(45)

In Eqs. (44) and (45) the weak form factors $G_{i}(q^{2})$ and
$\lambda\left(m^{2},\widetilde{m}_{Z}^{2},q^{2}\right)$ are computed at
$q^{2}=m_{\pi}^{2}$. The decay modes $T_{bb}^{-}$
$\rightarrow\widetilde{T}_{bc}^{0}K^{-}(D^{-},\ D_{s}^{-})$ can be studied in
a similar way. For these purposes, in expressions above one should replace
($m_{\pi},f_{\pi}$) by the masses and decay constants of the mesons $K^{-}$,
$D^{-}$, and $D_{s}^{-}$, make the substitutions $V_{ud}\rightarrow V_{us}$,
$V_{cd}$, and $V_{cs}$, and fix the form factors $G_{i}$ and $\lambda$ at
$q^{2}=m_{M}^{2}$.

Input parameters required for numerical computations are collected in Table 2.
This table contains the masses and decay constants of the final state mesons,
and relevant CKM matrix elements. The coefficients $c_{1}(m_{b})$, and
$c_{2}(m_{b})$ with next-to-leading order QCD corrections are borrowed from
Refs. Buras:1992zv ; Ciuchini:1993vr ; Buchalla:1995vs

$c_{1}(m_{b})=1.117,\ c_{2}(m_{b})=-0.257.$ (46)

Quantity | Value
---|---
$m_{\pi}$ | $139.570~{}\mathrm{MeV}$
$m_{K}$ | $(493.677\pm 0.016)~{}\mathrm{MeV}$
$m_{D}$ | $(1869.61\pm 0.10)~{}\mathrm{MeV}$
$m_{D_{s}}$ | $(1968.30\pm 0.11)~{}\mathrm{MeV}$
$f_{\pi}$ | $131~{}\mathrm{MeV}$
$f_{K}$ | $(155.72\pm 0.51)~{}\mathrm{MeV}$
$f_{D}$ | $(203.7\pm 4.7)~{}\mathrm{MeV}$
$f_{D_{s}}$ | $(257.8\pm 4.1)~{}\mathrm{MeV}$
$|V_{ud}|$ | $0.97420\pm 0.00021$
$|V_{us}|$ | $0.2243\pm 0.0005$
$|V_{cd}|$ | $0.218\pm 0.004$
$|V_{cs}|$ | $0.997\pm 0.017$
Table 2: Masses and decay constants of the final state pseudoscalar mesons.
The CKM matrix elements are also included.

For the decay $T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}\pi^{-}$
calculations yield

$\displaystyle\Gamma(T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}\pi^{-})=\left(5.84\pm
1.11\right)\times 10^{-10}~{}\mathrm{MeV}.$ (47)

Partial widths of other nonleptonic decays of the tetraquark $T_{bb}^{-}$ are

$\displaystyle\Gamma(T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}K^{-})=\left(6.43\pm
1.32\right)\times 10^{-11}~{}\mathrm{MeV},$
$\displaystyle\Gamma(T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}D^{-})=\left(3.01\pm
0.64\right)\times 10^{-11}~{}\mathrm{MeV},$
$\displaystyle\Gamma(T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}D_{s}^{-})=\left(7.80\pm
1.54\right)\times 10^{-10}~{}\mathrm{MeV}.$ (48)

Results of this section are second part of required information.

##  V Discussion and concluding notes

Results of the previous two sections allow us to improve predictions for the
full width and mean lifetime of the tetraquark $T_{bb}^{-}$. The semileptonic
decays of this particle $T_{bb}^{-}\rightarrow Z_{bc}^{0}l\overline{\nu}_{l}$
were analyzed in Ref. Agaev:2018khe . Using partial widths of these processes
and widths of the semileptonic and nonleptonic decays
$T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}l\overline{\nu}_{l}$ and
$T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}M$, it is not difficult to
evaluate relevant parameters. Thus, for the full width of $T_{bb}^{-}$, we get

$\widetilde{\Gamma}=(7.72\pm 1.23)\times 10^{-8}~{}\mathrm{MeV}.$ (49)

The lifetime of $T_{bb}^{-}$ is estimated in the range

$\widetilde{\tau}=8.53_{-1.18}^{+1.57}~{}\mathrm{fs}.$ (50)

Comparing improved estimates for $\widetilde{\Gamma}$ and $\widetilde{\tau}$
with previous results

$\displaystyle\Gamma$ $\displaystyle=$ $\displaystyle(7.17\pm 1.23)\times
10^{-8}~{}\mathrm{MeV},$ $\displaystyle\tau$ $\displaystyle=$ $\displaystyle
9.18_{-1.34}^{+1.90}~{}\mathrm{fs},$ (51)

one can see that semileptonic processes $T_{bb}^{-}\rightarrow
Z_{bc}^{0}l\overline{\nu}_{l}$ are dominant decay channels of the tetraquark
$T_{bb}^{-}$. In fact, a difference
$\Delta=\widetilde{\Gamma}-\Gamma=0.55\times 10^{-8}~{}\mathrm{MeV}$ is equal
to $8\%$ of $\Gamma$. In other words, $7$ new decay modes considered in the
present work constitute approximately $8\%$ part of the full width $\Gamma$.
The branching ratios of different weak decay modes of $T_{bb}^{-}$ are
presented in Table 3, excluding two nonleptonic decays $\mathcal{BR}$s of
which are negligible.

Channels | $\mathcal{BR}(\%)$
---|---
$T_{bb}^{-}\rightarrow Z_{bc}^{0}e^{-}\overline{\nu}_{e}$ | $34.3$
$T_{bb}^{-}\rightarrow Z_{bc}^{0}\mu^{-}\overline{\nu}_{\mu}$ | $34.2$
$T_{bb}^{-}\rightarrow Z_{bc}^{0}\tau^{-}\overline{\nu}_{\tau}$ | $24.4$
$T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}e^{-}\overline{\nu}_{e}$ | $2.6$
$T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}\mu^{-}\overline{\nu}_{\mu}$ | $2.5$
$T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}\tau^{-}\overline{\nu}_{\tau}$ | $0.13$
$T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}\pi^{-}$ | $0.76$
$T_{bb}^{-}\rightarrow\widetilde{T}_{bc}^{0}D_{s}^{-}$ | $1.01$
Table 3: Branching ratios of the weak decay channels of the tetraquark
$T_{bb}^{-}$.

We have explored the weak decays of $T_{bb}^{-}$, where the final-state
tetraquark $\widetilde{T}_{bc}^{0}=[bc][\overline{u}\overline{d}]$ has been
considered as an axial-vector particle. By computing partial widths of these
decays, we have calculated the full width $\widetilde{\Gamma}$ and mean
lifetime $\widetilde{\tau}$ of the axial-vector tetraquark $T_{bb}^{-}$ and
improved existing estimations for them. Up to now experimental collaborations
did not discover weak decays of tetraquarks. But some of active experiments,
such as LHC, have a certain potential to observe weak decay channels of
tetraquarks $T_{bb}$ Ali:2018xfq : In Ref. Ali:2018xfq  the authors
demonstrated that during $1-4$ runs of LHC one may expect
$\mathcal{O}(10^{8})$ events with $T_{bb}^{-}$. Such potential will have also
a Tera-$Z$ factory Ali:2018ifm . Predictions obtained for parameters of the
tetraquark $T_{bb}^{-}$ may be useful for analysis of these processes.

## ACKNOWLEDGEMENTS

The work of K. A, B. B., and H. S was supported in part by TUBITAK under the
grant No: 119F050.

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