# Masses of doubly heavy tetraquark states with isospin = $\frac{1}{2}$ and 1
and spin-parity $1^{+\pm}$

Di Gao1 Duojie Jia1<EMAIL_ADDRESS>Yan-Jun Sun1<EMAIL_ADDRESS>Zhan
Zhang1 Wen-Nian Liu1 Qing Mei1 1Institute of Theoretical Physics, College of
Physics and Electronic Engineering, Northwest Normal University, Lanzhou
730070, China

###### Abstract

We apply the method of QCD sum rules to study the doubly heavy tetraquark
states($QQ\bar{q}\bar{q}$) with the isospin $I=\frac{1}{2}$ and 1 and spin-
parity $J^{PC}=1^{++}$ and $J^{PC}=1^{+-}$ by constructing all the
$QQ\bar{q}\,\bar{q}$ tetraquark currents. The masses of the doubly bottom and
charm tetraquark states are computed in the context of the two-point sum rule
method incorporating the quark, gluon and mixed condensates up to dimension
$10$. By the way, weak decay widths of the doubly bottom tetraquark
$bb\bar{u}\bar{d}$ are also given.

##  I Introduction

In recent years, a large number of unknown strongly interacting hadrons such
as X, Y and Z have been discovered experimentally. Compared with the
conventional quark-antiquark mesons and three-quark baryons, these XYZ
particles are more difficult to identify due to their potential possibility of
mixing exotic multiquark components in them, so understanding these particles
via exotic multiquarks has attracted much attention. In 2015, two hidden-charm
pentaquarks, $P_{c}(4380)$ and $P_{c}(4450)$, were reported in the $J/\psi p$
invariant mass spectrum by LHCb collaboration[1], both of which provide a good
chance to research multiquarks. In addition, some multiquarks have a specific
features that their masses are near threshold of two charm or bottom mesons,
which implies frequently that they could be the molecule states of these heavy
mesons.

The mass of the doubly-charmed baryon $\Xi_{cc}^{++}(3620)$ has been predicted
in 2002[2]. Its observation in 2017, by LHCb with a mass of $3621.40\pm
0.78MeV$[3], inspires a set of works on double charm, doubly bottom tetraquark
and so on[4-12]. For instance, Eichtan and Quigg argue via heavy-quark
symmetry[6] that the configurations $bb\bar{u}\bar{d}$ and
$bb\bar{q}_{k}\bar{s}$ are stable heavy tetraquark states, and Karliner and
Rosner[4], based on the research of the $\Xi_{cc}$, predict the mass of the
tetraquark $bb\bar{u}\bar{d}$ and its weak decay width. This enhances our
belief about the existence of heavy, stable tetraquark states.

On the other hand, observation and mass spectra of the $Z_{b}(10610)$ and
$Z_{b}(10650)$ [13] implies that there are new particles exist between
$10300MeV$$<$ $\sqrt{s}$ $<$$10400MeV$ and $10500MeV$$<$ $\sqrt{s}$
$<$$10600MeV$. By analogy with the doubly charm tetraquark, one can speculate
that the former is $bb\bar{u}\bar{d}$ tetraquark states with $J^{P}=1^{+}$
while the latter is $bb\bar{q}_{k}\bar{s}$ with $J^{P}=1^{+}$.

In this work, we compute the mass of $bb\bar{q}_{k}\bar{q}$ and give the mass
analysis of their charm partners using the QCD sum rule approach, where the
light quarks $q_{k/l}$ and $q$ can be $u/d$ or $s$. In our estimation of the
continuum threshold parameter $s_{0}$, we have checked that our chosen $s_{0}$
is acceptable within the standard limits. Further, the Borel parameter
$M_{b}^{2}$ in our computation is limited to the range $[15,20]GeV^{2}$ by
tuning the pole contribution to ensure pole dominance at the phenomenological
side, and tuning the convergence ratio to make sure the operator product
expansion(OPE) convergent.

##  II QCD sum rule analyses

As a powerful and successful non-perturbative method, QCD sum rules [14-16]
have been applied to study multifarious hadrons [17-24]. In this method one
uses the quark-hadron duality to balance the (integrated) correlation
function.

$\displaystyle\Pi_{\mu\nu}(q^{2})$
$\displaystyle\equiv\int{ie^{iqx}d^{4}x\langle
0|T[J_{\mu}(x)J^{\dagger}_{\nu}(0)]|0\rangle}$ (1)
$\displaystyle=(\frac{q_{\mu}q_{\nu}}{q^{2}}-g_{\mu\nu})\Pi(q^{2}).$

In order to study the $bb\overline{u}\bar{d}$ and $bb\bar{q}_{k}\bar{s}$
tetraquarks, one constructs the four-quark($QQ\bar{q}\bar{q}$) interpolating
currents in the “diquark-antidiquark” configuration and considers the Pauli
principle to let all diquark fields to have certain color and spin-flavor
structure, composing the tetraquark operator with quantum number $J^{P}$. The
$bb\bar{q}\,\bar{q}$ tetraquark interpolating currents with $J^{PC}=1^{++}$
and $1^{+-}$[25] are:

$J_{1\mu}^{\pm}=Q_{a}^{T}C\gamma_{\mu}\gamma_{5}Q_{b}\bar{q}_{a}C\bar{q}_{b}^{T}\pm
Q_{a}^{T}CQ_{b}\bar{q}_{a}\gamma_{\mu}\gamma_{5}C\bar{q}_{b}^{T},$ (2)
$J_{2\mu}^{\pm}=Q_{a}^{T}C\sigma_{\mu\nu}\gamma_{5}Q_{b}\bar{q}_{a}\gamma^{\nu}C\bar{q}_{b}^{T}\pm
Q_{a}^{T}C\gamma^{\nu}Q_{b}\bar{q}_{a}\sigma_{\mu\nu}\gamma_{5}C\bar{q}_{b}^{T}.$
(3)

The current in Eq.(2) belongs to symmetric color structure while that in
Eq.(3), respectively, belongs to antisymmetric color structure. At the hadron
level, we can express $\Pi(q^{2})$ in the form of the dispersion relation with
a spectral function $\rho_{phen}(s)$ :

$\Pi(q^{2})=\int_{(2m_{b}+2m_{q})^{2}}^{\infty}ds\frac{\rho_{phen}(s)}{s-q^{2}-i\varepsilon}$
(4)

with the integration starting from the physical threshold,
$(2m_{b}+2m_{q})^{2}$. Here, the spectral density $\rho_{phen}(s)$ is the
imaginary part of the correlating function,
$\rho_{phen}(s)=\frac{1}{\pi}Im\Pi(s)$ .

A parameterization of one pole dominance and a continuum contribution gives

$\displaystyle\rho_{phen}(s)$ $\displaystyle\equiv\frac{1}{\pi}Im\Pi(s)$ (5)
$\displaystyle=\Sigma\delta(s-M_{n}^{2})\langle 0|J|n\rangle\langle
n|J^{\dagger}|0\rangle$ $\displaystyle=f_{x}^{2}\delta(s-M_{x}^{2})+higher\
states,$

where $f_{x}$ is the coupling strength of the hadron with $J(x)$ in the hadron
spectrum expansion and $M_{x}$ is the mass of hadron of ground state.

At the quark-gluonic level, Eq. (1) are calculated with the OPE. Performing
the Borel transformation at hadron and quark-gluon levels, one finds

$\Pi(M_{b}^{2},\infty)\equiv\widehat{B}_{M_{b}^{2}}\Pi(q^{2})=\int_{(2m_{b}+2m_{q})^{2}}^{\infty}dse^{-s/M_{x}^{2}}\rho(s).$
(6)

Approximating the contribution from the continuum states by the spectral
density above a threshold value $s_{0}$, one obtains the sum rule relation

$\Pi(M_{b}^{2},s_{0})\equiv
f_{x}^{2}e^{-M_{x}^{2}/M_{b}^{2}}=\int_{(2m_{b}+2m_{q})^{2}}^{s_{0}}dse^{-s/M_{b}^{2}}\rho(s),$
(7)

from which one can extract the hadron mass of the lowest lying resonance
$M_{x}$ to be

$M_{x}^{2}(M_{b}^{2},s_{0})=\frac{\frac{\partial}{\partial(-\frac{1}{M_{b}^{2}})}{\Pi(M_{b}^{2},s_{0})}}{\Pi(M_{b}^{2},s_{0})}=\frac{\int_{(2m_{b}+2m_{q})^{2}}^{s_{0}}dse^{-s/M_{b}^{2}}\rho(s)s}{\int_{(2m_{b}+2m_{q})^{2}}^{s_{0}}dse^{-s/M_{b}^{2}}\rho(s)}.$
(8)

We compute the heavy tetraquark mass via computing the RHS of Eq. (8). For
integration in Eq. (8), we consider all Feynman diagrams of the quark, gluon
and mixed condensates up to dimension 10, as explicitly shown in FIG.1.

(a) $pert$

(b) $\langle qq\rangle$

(c) $\langle GG\rangle(a)$

(d) $\langle GG\rangle>(b)$

(e) $\langle GG\rangle>(c)$

(f) $\langle GG\rangle(d)$

(g) $\langle qGq\rangle(a)$

(h) $\langle qGq\rangle(b)$

(i) $\langle qGq\rangle(c)$

(j) $\langle qGq\rangle(d)$

(k) $\langle qq\rangle^{2}$

(l) $\langle qq\rangle\langle GG\rangle(a)$

(m) $\langle qq\rangle\langle GG\rangle(b)$

(n) $\langle qq\rangle\langle GG\rangle(c)$

(o) $\langle qq\rangle\langle GG\rangle(d)$

(p) $\langle qq\rangle\langle qGq\rangle(a)$

(q) $\langle qq\rangle\langle qGq\rangle(b)$

(r) $\langle qq\rangle\langle qGq\rangle(c)$

(s) $\langle qq\rangle\langle qGq\rangle(d)$

(t) $\langle qGq\rangle\langle qGq\rangle(a)$

(u) $\langle qGq\rangle\langle qGq\rangle(b)$

(v) $\langle qGq\rangle\langle qGq\rangle(c)$

(w) $\langle qGq\rangle\langle qGq\rangle(d)$

(x) $\langle qGq\rangle\langle qGq\rangle(e)$

Figure 1: Feynman diagrams for the two-point function of the tetraquark
current $\Pi_{\mu\nu}$. Squares represent events, straight line represent
heavy quarks, curves represent light quarks, and helical lines represent
gluons.

##  III Numerical analyses and discussion

Before numerical analyses we use the following values of parameters for quark
masses and various QCD condensates[26-32]:

$m_{b}=4.18_{-0.03}^{+0.04}GeV$,

$m_{s}=96_{-4}^{+8}MeV$,

$\langle\bar{q}q\rangle=-(240\pm 10MeV)^{3}$,

$\langle\bar{q}g_{s}\sigma\cdot Gq\rangle=-M_{0}^{2}\langle\bar{q}q\rangle$,

$\langle g_{s}^{2}GG\rangle=(0.48\pm 0.14)GeV^{4}$,

$M_{0}^{2}=(0.8\pm 0.2)GeV^{2}$.

Here, the listed vacuum condensates are fixed, whereas the Borel parameter and
threshold can be varied within some regions, which have to satisfy the
standard restrictions of the sum rules computations. The window for
$M_{b}^{2}\in[{M_{b}^{2}}_{min},{M_{b}^{2}}_{max}]$ is fixed from the
constraints imposed on the pole contribution (PC) which determines
${M_{b}^{2}}_{max}$ and the (convergence) ratio $R({M_{b}^{2}}_{min})$
necessary to find ${M_{b}^{2}}_{min}$. The PC is defined as

$PC\equiv\frac{\Pi({M_{b_{max}}^{2}},s_{0})}{\Pi({M_{b_{max}}^{2}},\infty)}=\frac{\int_{(2m_{b}+2m_{q})^{2}}^{s_{0}}dse^{-s/M_{b}^{2}}\rho(s)}{\int_{(2m_{b}+2m_{q})^{2}}^{\infty}dse^{-s/M_{b}^{2}}\rho(s)},$
(9)

and $R({M_{b}^{2}}_{min})$ as

$R({M_{b}^{2}}_{min})\equiv\frac{\Pi^{DimN}({M_{b}^{2}}_{min},s_{0})}{\Pi({M_{b}^{2}}_{min},s_{0})},$
(10)

where $\Pi^{DimN}({M_{b}^{2}}_{min},s_{0})$ is the contribution of the higher
orders.

We carry out the numerical analyses by taking into account all of the
aforementioned constraints. This allows us to determine the optimal regions
for $M_{b}^{2}$ and $s_{0}$. During the search for the Borel parameter
$M_{b}^{2}$ and the continuum threshold parameter $s_{0}$ the following
criteria are used:

(1) Pole dominantes at the phenomenological side.

(2) The OPE is convergent.

(2) Borel platforms appear.

Figure 2: The PC($J_{1\mu}^{+}$), defined in Eq.(9), as a function of the
Borel parameter $M_{b}$. The curve is obtained by taking
$s_{0}(bb\bar{u}\bar{d})$=$133.8GeV^{2}$ and
$s_{0}(bb\bar{q}_{k}\bar{s})$=$138.5GeV^{2}$.

We can estimate, by Eq. (9), the lower limits of the PC for every $M_{b}^{2}$
in FIG. 2 with the help of a set of mass inequalities,
$max\\{m_{bb\bar{q}_{k}{\bar{q}}}\\}$ $<$
$min_{Q\bar{Q}}\\{min\\{m_{bb\bar{q}_{k}{\bar{q}}Q\bar{Q}}\\}\\}$ $<$
$max_{Q\bar{Q}}\\{min\\{m_{bb\bar{q}_{k}{\bar{q}}Q\bar{Q}}\\}\\}$ $\leq$
$min\\{m_{bb\bar{q}_{k}{\bar{q}}b\bar{b}}\\}$ $<$
$2m_{\bar{B}/\bar{B}_{s}}+m_{\eta_{b}/h_{b}/\Upsilon/\chi_{b}}$ $\approx$
$20GeV$ (Appendix A) due to the quantum vacuum (containing sea-quarks) and
confining features of QCD. In obtaining inequalities, we assumed a hexaquark
configuration $bb\bar{q}_{k}{q}b\bar{b}$ to exist at least and have used the
notations $max\\{m_{bb\bar{q}_{k}{\bar{q}}}\\}$ to stand for the heaviest mass
of all states with given flavor configurations and
$min\\{m_{bb\bar{q}_{k}{\bar{q}}Q\bar{Q}}\\}$ for that of the lowest(ground)
states for the given configuration. $Q$ stand for the $u,d,s$ or $c,b$.

Figure 3: The convergence ratio R($J_{1\mu}^{+}$), defined in Eq.(10) (here,
$DimN=Dim8+Dim9+Dim10$), as a function of the Borel parameter $M^{2}_{b}$. The
curve is obtained by taking $s_{0}(bb\bar{u}\bar{d})$=$133.8GeV^{2}$ and
$s_{0}(bb\bar{q}_{k}\bar{s})$=$138.5GeV^{2}$.

We compute the PC and find it to be
$38.97$%$<PC(bb\bar{q}_{k}\bar{s}(J^{+}_{1\mu}))<40.65$%($37.93$%$<PC(bb\bar{u}\bar{d}(J^{+}_{1\mu}))<39.60$%)
in the regions $15GeV^{2}$ $<$ $M_{b}^{2}$ $<$ $20GeV^{2}$ with
$J^{PC}=1^{++}$, as shown in FIG. $2$. We also calculate the ratio
R($J_{1\mu}^{+}$) and find it to be
$1.98$%$<R(bb\bar{q}_{k}\bar{s}(J^{+}_{1\mu}))<4.92$%($2.02$%$<R(bb\bar{u}\bar{d}(J^{+}_{1\mu}))<4.99$%)
in the regions $15GeV^{2}$ $<$ $M_{b}^{2}$ $<$ $20GeV^{2}$(shown in FIG. $3$).
Similar calculations yield the PC and R for other configurations,

$\displaystyle bb\bar{u}\bar{d}(J^{-}_{1\mu})\quad:\quad
37.11\%<PC<38.49\%,2.05\%<R<5.00\%;$ (11) $\displaystyle
bb\bar{q}_{k}\bar{s}(J^{-}_{1\mu})\quad:\quad
38.09\%<PC<39.57\%,1.99\%<R<4.93\%;$ $\displaystyle
bb\bar{u}\bar{d}(J^{+}_{2\mu})\quad:\quad 38.09\%<PC<39.57\%,2.07\%<R<5.02\%;$
$\displaystyle bb\bar{q}_{k}\bar{s}(J^{+}_{2\mu})\quad:\quad
38.11\%<PC<39.59\%,2.00\%<R<4.95\%;$ $\displaystyle
bb\bar{u}\bar{d}(J^{-}_{2\mu})\quad:\quad 37.08\%<PC<38.49\%,2.06\%<R<4.96\%;$
$\displaystyle bb\bar{q}_{k}\bar{s}(J^{-}_{2\mu})\quad:\quad
38.12\%<PC<39.55\%,1.99\%<R<4.93\%.$

Considering all together, the ranges we choose turn out to be appropriate in
view of appearance of Borel platforms. The ranges obtained are:

$\displaystyle bb\bar{u}\bar{d}\quad:\quad
M_{b}^{2}=(15-20)GeV^{2},s_{0}=(131.3-136.3)GeV^{2},$ (12) $\displaystyle
bb\bar{q}_{k}\bar{s}\quad:\quad
M_{b}^{2}=(15-20)GeV^{2},s_{0}=(136.0-141.0)GeV^{2}.$

To reduce the uncertainty from the PC and R, we plot dependence of the mass of
$bb\bar{u}\bar{d}$ and $bb\bar{q}_{k}\bar{s}$ tetraquarks upon $M_{b}^{2}$ and
$s_{0}$ in FIG.$4$ and FIG.$5$.

(a) $bb\bar{u}\bar{d}$

(b) $bb\bar{q}_{k}\bar{s}$

Figure 4: Dependence of the mass of the tetraquarks $bb\bar{u}\bar{d}$ (a) and
$bb\bar{q}_{k}\bar{s}$(b) upon the Borel parameters $M_{b}^{2}$. The curves
are obtained with $s_{0}(bb\bar{u}\bar{d})$=$133.8.0GeV^{2}$ and
$s_{0}(bb\bar{q}_{k}\bar{s})$=$138.5GeV^{2}$.

(a) $bb\bar{u}\bar{d}$

(b) $bb\bar{q}_{k}\bar{s}$

Figure 5: Dependence of the mass of the tetraquarks $bb\bar{u}\bar{d}$ (a) and
$bb\bar{q}_{k}\bar{s}$(b) upon the threshold parameters $s_{0}$. The curves
are obtained with $M_{b}^{2}=17GeV^{2}$.

In FIG.$4$, we plot the prediction for $bb\bar{q}_{k}\bar{q}$, which
demonstrate on used values of $M_{b}^{2}$ . It is seen that the dependence of
the mass on Borel parameter is very weak: the predictions for
$bb\bar{q}_{k}\bar{q}$ demonstrate a high stability against changes of
$M_{b}^{2}$ in the optimized working interval. In FIG.$5$, we plot the
prediction for $bb\bar{q}_{k}\bar{q}$, which demonstrate on used values of
$s_{0}$. While the computed masses are sensitive to the choice of the
continuum threshold parameter $s_{0}$, which implies that variation of $s_{0}$
generates a main part of uncertainties, it remains a constant approximately,
within standard limits acceptable for our computations.

Given all above considerations, the masses of the doubly bottom tetraquarks
are obtained and compared with Ref. [6] and Ref. [4]. The results are
collected in Table I.

Table 1: The masses of $bb\bar{q}_{k}\bar{q}$ tetraquark states with $J^{PC}=1^{++}$ and $J^{PC}=1^{+-}$. State |  Current |  Our work($GeV$) |  Ref($MeV$)[6] |  Ref($MeV$)[4]
---|---|---|---|---
$bb\bar{u}\bar{d}$ |  $J^{+}_{1\mu}$ |  $10.325^{+0.015+0.02}_{-0.014-0.02}$ |  10482 |  10389.4$\pm 12$
|  $J^{-}_{1\mu}$ |  $10.380^{+0.015+0.03}_{-0.014-0.02}$
|  $J^{+}_{2\mu}$ |  $10.381^{+0.014+0.03}_{-0.015-0.02}$
|  $J^{-}_{2\mu}$ |  $10.380^{+0.013+0.03}_{-0.015-0.02}$
$bb\bar{q}_{k}\bar{s}$ |  $J^{+}_{1\mu}$ |  $10.519^{+0.015+0.01}_{-0.016-0.02}$ |  10643 |  -
|  $J^{-}_{1\mu}$ |  $10.572^{+0.014+0.01}_{-0.016-0.02}$
|  $J^{+}_{2\mu}$ |  $10.573^{+0.014+0.01}_{-0.016-0.02}$
|  $J^{-}_{2\mu}$ |  $10.571^{+0.014+0.01}_{-0.015-0.02}$

In Ref. [4], it is suggested that the tetraquark $bb\bar{u}\bar{d}$ decays
weakly since it is deeply bounded. Assuming a final state $\bar{B}D$ for weak
decay of a given tetraquark $bb\bar{u}\bar{d}$ with a charged weak current
giving rise to $e\bar{\nu}_{e}$, $\mu\bar{\nu}_{\mu}$, $\tau\bar{\nu}_{\tau}$,
one can use color factor(=3) of $\bar{u}d$ and $\bar{c}s$, a CKM matrix
element $|V_{cb}|=0.04$[33] and a factor(=2) counting each decaying of $b$
quark to compute its decay rate. The widths for all tetraquark
$QQ\bar{q}\bar{q}$ states with $J^{PC}=1^{++}$ and $J^{PC}=1^{+-}$ are[4]

$\Gamma(bb\bar{u}\bar{d})=\frac{18G_{F}^{2}M(bb\bar{u}\bar{d})^{5}}{192\pi^{3}}F(x)|V_{cb}|^{2},$
(13)

in which the kinematic suppression factor $F(x)$ is given by

$F(x)=1-8x+8x^{3}-x^{4}+12x^{2}\rm{ln}\frac{1}{x},x\equiv(\frac{M(\bar{B})+M(D)}{M(bb\bar{u}\bar{d})})^{2},$
(14)

with $M(\bar{B})$,$M(D)$ and $M(bb\bar{u}\bar{d})$ the masses of the heavy
meson $B$,$D$ and the $bb\bar{u}\bar{d}$, respectively. The results obtained
thereby are collected in Table II.

Table 2: The decay widths of $bb\bar{q}_{k}\bar{q}$$\rightarrow$$\bar{B}D$$/$$\bar{B}D_{s}$. decay channel |  current |  our work($GeV$) |  Ref($GeV$)[4]
---|---|---|---
$bb\bar{u}\bar{d}$$\rightarrow$$\bar{B}$$D$ |  $J^{+}_{1\mu}$ |  $17.42\times 10^{-13}$ |  $17.9\times 10^{-13}$
|  $J^{-}_{1\mu}$ |  $17.89\times 10^{-13}$
|  $J^{+}_{2\mu}$ |  $17.74\times 10^{-13}$
|  $J^{-}_{2\mu}$ |  $17.89\times 10^{-13}$
$bb\bar{q}_{k}\bar{s}$$\rightarrow$$\bar{B}$$D_{s}$ |  $J^{+}_{1\mu}$ |  $18.00\times 10^{-13}$ |  -
|  $J^{-}_{1\mu}$ |  $19.45\times 10^{-13}$
|  $J^{+}_{2\mu}$ |  $19.26\times 10^{-13}$
|  $J^{-}_{2\mu}$ |  $20.20\times 10^{-13}$

In obtaining Table II, we have used the following masses $M(bb\bar{u}\bar{d})$
of the initial decaying tetraquark states: with
${m_{bb\bar{u}\bar{d}}}_{J^{+}_{1\mu}}=10.325GeV$,
${m_{bb\bar{u}\bar{d}}}_{J^{-}_{1\mu}}=10.380GeV$,
${m_{bb\bar{u}\bar{d}}}_{J^{+}_{2\mu}}=10.381GeV$,
${m_{bb\bar{u}\bar{d}}}_{J^{-}_{2\mu}}=10.380GeV$,
${m_{bb\bar{q}_{k}\bar{s}}}_{J^{+}_{1\mu}}=10.519GeV$,
${m_{bb\bar{q}_{k}\bar{s}}}_{J^{-}_{1\mu}}=10.572GeV$,
${m_{bb\bar{q}_{k}\bar{s}}}_{J^{+}_{2\mu}}=10.573GeV$, and
${m_{bb\bar{q}_{k}\bar{s}}}_{J^{-}_{2\mu}}=10.571GeV$.

Similar analysis can apply to the strange partners $bb\bar{q}_{k}\bar{s}$ of
the above tetraquarks, which gives rise to the weak decay widths listed in
Table II. In both of Table I and II, a reasonable agreements of our work with
that in Ref. [4] are achieved.

##  IV Summary

In this work, we compute the masses of the doubly heavy tetraquark states of
$bb\bar{q}_{k}\bar{q}$ with $J^{PC}=1^{++}$ and $J^{PC}=1^{+-}$ by employing
the QCD sum rules, with multifarious condensates taken into account up to
dimension $10$. The care is taken to estimate $M_{b}^{2}$ and $s_{0}$, which
has allowed us to improve the accuracy.

In Table I, the central values correspond to $M_{b}^{2}=17GeV^{2}$,
$s_{0}(bb\bar{u}\bar{d})=133.8GeV^{2}$ and
$s_{0}(bb\bar{q}_{k}\bar{s})=138.5GeV^{2}$, and the uncertainties are due to
the Borel parameter $M_{b}^{2}$ and the threshold parameter $s_{0}$. In our
computations, we have not considered the uncertainty due to other parameters
such as $m_{b}$, $m_{q}$, multifarious condensates and so on. Note also that
the mass calculation of the tetraquark state alone can not provide full
information to distinguish it unambiguously as a bound state of a diquark and
an antidiquark from a pair mesons with the same quantum numbers. Additional
important information can be extracted from analysis of its decay channels.
Angular momentum and parity conservation in strong and electroweak
interactions forbid a state with $J^{P}=1^{+}$ to decay strongly or
electromagnetically into two pseudoscalar mesons in any partial wave. We
therefore calculate the decay width of $bb\bar{q}_{k}\bar{q}$ through its weak
decays and compare it with Ref. [4], as shown in Table II:

Table 3: Masses of $cc\bar{u}\bar{d}$ with $J^{PC}=1^{++}$ and $J^{PC}=1^{+-}$.  State |  Current |  Our work($GeV$) |  Lattice($MeV$)[34] |  Ref. [4]($MeV$)
---|---|---|---|---
$cc\bar{u}\bar{d}$ |  $J^{+}_{1\mu}$ |  $3.624^{+0.002+0.03}_{-0.001-0.03}$ |  3805($m_{\pi}=411)$ |  $3882.2\pm 12$
|  $J^{-}_{1\mu}$ |  $3.860^{+0.001+0.03}_{-0.001-0.02}$ |  3893($m_{\pi}=572$)
|  $J^{+}_{2\mu}$ |  $3.877^{+0.001+0.03}_{-0.001-0.02}$ |  3999($m_{\pi}=699$)
|  $J^{-}_{2\mu}$ |  $3.849^{+0.001+0.03}_{-0.001-0.02}$

For completeness, we list also the masses calculated with Lattice[34] and that
in Ref. [4] for the doubly charm tetraquark $cc\bar{u}\bar{d}$, as shown in
Table III. In Table III, the central values correspond to
$M_{b}^{2}=17GeV^{2}$ and ${s_{0}}(cc\bar{u}\bar{d})=19.36GeV^{2}$, and the
uncertainties are due to the Borel parameter $M_{b}^{2}$ and the threshold
parameter $s_{0}$, where $M_{b}^{2}$ ranges in $[15,20]GeV^{2}$ and $s_{0}$ in
$[18.9,19.8]GeV^{2}$. The uncertainty treatment due to the parameters is same
with for the doubly bottom tetraquark states.

## ACKNOWLEDGEMENTS

D.J. is supported by the National Natural Science Foundation of China under
the No. 11565023 and the Feitian Distinguished Professor Program of Gansu
(2014-2016). Y-J.S. is supported in part by the National Natural Science
Foundation of China under the Grant No. 11365018 and No. 11375240.

## APPENDIX A

For the infinity ($``\infty"$) of the denominator in PC, one aim to integrate
over all excited states of $bb\bar{q}\bar{q}$. In reality, it is enough to
somehow use a upper limit of the integral to replace the
infinity($``\infty"$). To find a rough estimate of this upper limit, we assume
a hypothetical process: A gradually increasing energy is provided to the
tetraquark states $bb\bar{q}\bar{q}$ to obtain all its excited states and this
makes some pair of quark $Q\bar{Q}$ to be created from the QCD vacuum to form
a $bb\bar{q}\bar{q}Q\bar{Q}$ hexaquark state. This process stops when there is
no further higher excited states of $bb\bar{q}\bar{q}$ created due to the pair
creation in QCD vacuum. It is unknown which states of
$bb\bar{q}\bar{q}Q\bar{Q}$ hexaquark is stable against strong decays. We
assume the heaviest configuration of $bb\bar{q}\bar{q}Q\bar{Q}$, that is,
$bb\bar{q}\bar{q}b\bar{b}$, are stable against strong decays. Instead of
evaluating masses of the $bb\bar{q}\bar{q}b\bar{b}$ hexaquark states, we rest
content with finding it upper limit. Note that there should be at least one of
the $bb\bar{q}\bar{q}b\bar{b}$ hexaquark states with mass less than the mass
sum of the final products from its strong decay. One can infer then that some
of $bb\bar{q}\bar{q}b\bar{b}$ hexaquark states has mass smaller than
$2m_{\bar{B}/\bar{B}_{s}}+m_{\eta_{b}/h_{b}/\Upsilon/\chi_{b}}$ $=2\cdot
5367\text{MeV}+9460\text{MeV}\approx 20\text{GeV}$.

Accepting that the above process are valid generally, we conclude that the
highest excited state of any hadron is lighter than the least massive
multiquark generated via the process described above.

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