# Decay behaviors of the fully-bottom and fully-charm tetraquark states

Hua-Xing Chen<EMAIL_ADDRESS>School of Physics, Southeast University,
Nanjing 210094, China  Yi-Xin Yan<EMAIL_ADDRESS>School of Physics,
Southeast University, Nanjing 210094, China  Wei Chen
<EMAIL_ADDRESS>School of Physics, Sun Yat-Sen University,
Guangzhou 510275, China

###### Abstract

We study the decay behaviors of the fully-bottom tetraquark states within the
diquark-antidiquark picture, and calculate their relative branching ratios
through the Fierz rearrangement. Our results suggest that the $C=+$ states can
be searched for in the $\mu^{+}\mu^{-}\Upsilon(1S)$ and
$\mu^{+}\mu^{-}\Upsilon(2S)$ channels with the relative branching ratio
$\mathcal{B}(X\to\mu\mu\Upsilon(2S))/\mathcal{B}(X\to\mu\mu\Upsilon(1S))\approx
0.4$. Our results also suggest that the $C=-$ states can be searched for in
the $\mu^{+}\mu^{-}\eta_{b}(1S)$ and $\mu^{+}\mu^{-}\eta_{b}(2S)$ channels
with the similar relative branching ratio
$\mathcal{B}(X\to\mu\mu\eta_{b}(2S))/\mathcal{B}(X\to\mu\mu\eta_{b}(1S))\approx
0.4$. We also reanalysis the fully-charm tetraquark states, and study the
$X(6900)$ decay into the $J/\psi\psi(2S)$ channel to obtain the relative
branching ratio $\mathcal{B}(X\to J/\psi\psi(2S))/\mathcal{B}(X\to J/\psi
J/\psi)\approx 0.1$.

exotic hadron, fully-bottom tetraquark, fully-charm tetraquark, Fierz
rearrangement

###### pacs:

12.39.Mk, 12.38.Lg, 12.40.Yx

##  I Introduction

In 2020 the LHCb collaboration reported their observation of two exotic
structures in the di-$J/\psi$ invariant mass spectrum LHCb:2020bwg , i.e., a
broad structure ranging from 6.2 to 6.8 GeV and a narrow structure at around
6.9 GeV. They described the latter as a resonance with the Breit-Wigner
lineshape, whose mass and width were measured to be

$\displaystyle X(6900)$ $\displaystyle:$ $\displaystyle M=6905\pm 11\pm
7{\rm~{}MeV}\,,$ $\displaystyle\Gamma=80\pm 19\pm 33{\rm~{}MeV}\,.$

These values were obtained under the assumption that no interference with the
non-resonant single-parton scattering continuum is present. Assuming that the
continuum interferes with the broad structure, the above values were shifted
to be

$\displaystyle X(6900)$ $\displaystyle:$ $\displaystyle M=6886\pm 11\pm
11{\rm~{}MeV}\,,$ $\displaystyle\Gamma=168\pm 33\pm 69{\rm~{}MeV}\,.$

The above two structures are good candidates for the fully-charm tetraquark
states, and their observation immediately attracted much attention from the
particle physics community liu:2020eha ; Tiwari:2021tmz ; Lu:2020cns ;
Faustov:2020qfm ; Zhang:2020xtb ; Li:2021ygk ; Bedolla:2019zwg ; Weng:2020jao
; Liu:2021rtn ; Giron:2020wpx ; Karliner:2020dta ; Zhao:2020zjh ;
Mutuk:2021hmi ; Wang:2021kfv ; Wang:2020ols ; Ke:2021iyh ; Zhu:2020xni ;
Jin:2020jfc ; Yang:2021hrb ; Albuquerque:2020hio ; Albuquerque:2021erv ;
Wu:2022qwd ; Asadi:2021ids ; Yang:2020wkh ; Huang:2020dci ; Feng:2020riv ;
Huang:2021vtb ; Ma:2020kwb ; Maciula:2020wri ; Goncalves:2021ytq ;
Wang:2020gmd ; Esposito:2021ptx ; Zhuang:2021pci ; Zhao:2020nwy ;
Becchi:2020uvq ; Andrade:2022rbn ; Lansberg:2020ejc ; Sonnenschein:2020nwn ;
Zhu:2020snb ; Wan:2020fsk ; Gordillo:2020sgc ; Liu:2020tqy ; Majarshin:2021hex
; Sombillo:2021rxv ; Kuang:2022vdy ; Yang:2020rih ; Yang:2021zrc ;
Wang:2021taf ; Wang:2021mma . We refer to our recent review Chen:2022asf  as
well as the reviews Chen:2016qju ; Liu:2019zoy ; Lebed:2016hpi ;
Esposito:2016noz ; Hosaka:2016pey ; Guo:2017jvc ; Ali:2017jda ; Olsen:2017bmm
; Karliner:2017qhf ; Bass:2018xmz ; Brambilla:2019esw ; Guo:2019twa ;
Ketzer:2019wmd ; Yang:2020atz ; Roberts:2021nhw ; Fang:2021wes ; Jin:2021vct ;
JPAC:2021rxu ; Meng:2022ozq ; Mai:2022eur ; Maiani:2022psl  and the reports
Maiani:2020pur ; Chao:2020dml ; Richard:2020hdw  for their detailed
discussions. Especially, some theorists reanalysed the LHCb data on the
di-$J/\psi$ spectrum LHCb:2020bwg  and proposed the existence of more
structures, e.g., the authors of Ref. Wang:2020wrp  reproduced three peak
structures at near 6.5, 6.9, and 7.3 GeV, while the authors of Ref.
Dong:2020nwy  proposed the existence of a near-threshold state in the
di-$J/\psi$ system at near 6.2 GeV. We refer to Refs. Wang:2020tpt ;
Dong:2020hxe ; Liang:2021fzr ; Nefediev:2021pww ; Gong:2020bmg ; Dong:2021lkh
; Guo:2020pvt ; Cao:2020gul ; Wang:2022jmb ; Zhou:2022xpd  and the reviews
Chen:2022asf ; Mai:2022eur  for more discussions.

Very recently, the CMS and ATLAS collaborations also investigated the
di-$J/\psi$ invariant mass spectrum, and both of them confirmed the existence
of the $X(6900)$ CMS ; ATLAS . Besides, the CMS collaboration observed two new
structures, the $X(6600)$ and $X(7200)$, in the di-$J/\psi$ invariant mass
spectrum. Their masses and widths were measured to be CMS :

$\displaystyle X(6600)$ $\displaystyle:$ $\displaystyle M=6552\pm 10\pm
12{\rm~{}MeV}\,,$ $\displaystyle\Gamma=124\pm 29\pm 34{\rm~{}MeV}\,;$
$\displaystyle X(6900)$ $\displaystyle:$ $\displaystyle M=6927\pm 9\pm
5{\rm~{}MeV}\,,$ $\displaystyle\Gamma=122\pm 22\pm 19{\rm~{}MeV}\,;$
$\displaystyle X(7200)$ $\displaystyle:$ $\displaystyle M=7287\pm 19\pm
5{\rm~{}MeV}\,,$ $\displaystyle\Gamma=95\pm 46\pm 20{\rm~{}MeV}\,.$

The ATLAS collaboration investigated the di-$J/\psi$ invariant mass spectrum,
and their best fit was performed with three interfering resonances, whose
masses and widths were measured to be ATLAS :

$\displaystyle X(6200)$ $\displaystyle:$ $\displaystyle M=6.22\pm
0.05^{+0.04}_{-0.05}{\rm~{}GeV}\,,$ $\displaystyle\Gamma=0.31\pm
0.12^{+0.07}_{-0.08}{\rm~{}GeV}\,;$ $\displaystyle X(6600)$ $\displaystyle:$
$\displaystyle M=6.62\pm 0.03^{+0.02}_{-0.01}{\rm~{}GeV}\,,$
$\displaystyle\Gamma=0.31\pm 0.09^{+0.06}_{-0.11}{\rm~{}GeV}\,;$
$\displaystyle X(6900)$ $\displaystyle:$ $\displaystyle M=6.87\pm
0.03^{+0.06}_{-0.01}{\rm~{}GeV}\,,$ $\displaystyle\Gamma=0.12\pm
0.04^{+0.03}_{-0.01}{\rm~{}GeV}\,.$

The ATLAS collaboration also investigated the $J/\psi\psi(2S)$ invariant mass
spectrum. They reported the evidence for an enhancement at 6.9 GeV and a
resonance at 7.2 GeV, whose masses and widths were measured to be ATLAS :

$\displaystyle X(6900)$ $\displaystyle:$ $\displaystyle M=6.78\pm
0.36^{+0.35}_{-0.54}{\rm~{}GeV}\,,$ $\displaystyle\Gamma=0.39\pm
11^{+0.11}_{-0.07}{\rm~{}GeV}\,;$ $\displaystyle X(7200)$ $\displaystyle:$
$\displaystyle M=7.22\pm 0.03^{+0.02}_{-0.03}{\rm~{}GeV}\,,$
$\displaystyle\Gamma=0.10{{}^{+0.13}_{-0.07}}{{}^{+0.06}_{-0.05}}{\rm~{}GeV}\,.$

Actually, the fully-heavy tetraquark states were already studied by some
theorists in the 1980s Chao:1980dv ; Iwasaki:1975pv ; Ader:1981db ;
Heller:1985cb ; Badalian:1985es ; Zouzou:1986qh ; Lloyd:2003yc ; Barnea:2006sd
; Berezhnoy:2011xn , but there have not been many relevant experiments from
that time till now. Besides the above experiments LHCb:2020bwg ; CMS ; ATLAS ,
in 2017 the CMS collaboration found an excess in the
$\Upsilon(1S)\mu^{+}\mu^{-}$ invariant mass spectrum near 18.5 GeV with a
global significance of 3.6$\sigma$ Durgut:2018lmn ; Yi:2018fxo , and in 2019
the ANDY collaboration at RHIC reported an evidence of a significance peak at
around 18.12 GeV ANDY:2019bfn . These structures are good candidates for the
fully-bottom tetraquark states, and they also attracted some attention from
the particle physics community Anwar:2017toa ; Esposito:2018cwh ;
Hughes:2017xie ; Karliner:2016zzc ; Wu:2016vtq ; Richard:2017vry ; Bai:2016int
; Chen:2019dvd ; Debastiani:2017msn ; Wang:2019rdo ; Liu:2019zuc ;
Deng:2020iqw ; Richard:2018yrm ; Wang:2017jtz ; Wang:2018poa ; Chiu:2007km .
Since these structures were not confirmed by some other experiments
LHCb:2018uwm ; CMS:2020qwa , they require more investigations crucially.

Table 1: Mass spectra of the fully-charm and fully-bottom tetraquark states, calculated in Ref. Chen:2016jxd  through the QCD sum rule method.  $J^{PC}$ |  Currents |  $T_{cc\bar{c}\bar{c}}$ (GeV) |  $T_{bb\bar{b}\bar{b}}$ (GeV)
---|---|---|---
$0^{++}$ | $J^{0^{++}}_{1}$ | $6.44\pm 0.15$ | $18.45\pm 0.15$
$J^{0^{++}}_{2}$ | $6.46\pm 0.16$ | $18.46\pm 0.14$
$1^{+-}$ | $J^{1^{+-}}_{3\alpha}$ | $6.51\pm 0.15$ | $18.54\pm 0.15$
$2^{++}$ | $J^{2^{++}}_{4\alpha\beta}$ | $6.51\pm 0.15$ | $18.53\pm 0.15$
$0^{-+}$ | $J^{0^{-+}}_{5}$ | $6.84\pm 0.18$ | $18.77\pm 0.18$
$J^{0^{-+}}_{6}$ | $6.85\pm 0.18$ | $18.79\pm 0.18$
$0^{--}$ | $J^{0^{--}}_{7}$ | $6.84\pm 0.18$ | $18.77\pm 0.18$
$1^{-+}$ | $J^{1^{-+}}_{8\alpha}$ | $6.84\pm 0.18$ | $18.80\pm 0.18$
$J^{1^{-+}}_{9\alpha}$ | $6.88\pm 0.18$ | $18.83\pm 0.18$
$1^{--}$ | $J^{1^{--}}_{10\alpha}$ | $6.84\pm 0.18$ | $18.77\pm 0.18$
$J^{1^{--}}_{11\alpha}$ | $6.83\pm 0.18$ | $18.77\pm 0.16$

In Ref. Chen:2020xwe  we have studied the two-body decay behaviors of the
fully-charm tetraquark states within the diquark-antidiquark picture, and
calculated their relative branching ratios through the Fierz rearrangement of
the Dirac and color indices. In this paper we shall further study the decay
behaviors of the fully-bottom tetraquark states. As summarized in Table 1, our
previous QCD sum rule results suggest that the fully-charm tetraquark states
lie above the di-charmonium thresholds, while some of the fully-bottom
tetraquark states lie below the di-bottomonium thresholds Chen:2016jxd .
Accordingly, in this paper we shall investigate not the fall-apart two-body
decays, but the three-body decays of the fully-bottom tetraquark states. We
shall study their decays into one bottomonium meson and one muon-antimuon
pair, with the muon-antimuon pair produced by another intermediate vector
bottomonium meson. This method has been applied in Refs. Chen:2019wjd ;
Chen:2019eeq ; Chen:2020pac ; Chen:2020opr ; Chen:2021erj  to investigate some
other exotic hadrons, such as the $Z_{c}(3900)$ and the $P_{c}$ states, etc.
Especially, our results obtained in Ref. Chen:2019wjd  for the $Z_{c}(3900)$
are consistent with those obtained in Refs. Dias:2013xfa ; Agaev:2016dev ;
Esposito:2014hsa  using the QCD sum rule method and the non-relativistic
effective field theory. Besides, a similar arrangement of the spin and color
indices in the nonrelativistic case was applied in Refs. Voloshin:2019aut ;
Voloshin:2013dpa ; Maiani:2017kyi ; Wang:2019spc ; Xiao:2019spy ;
Cheng:2020nho  to study the decay properties of exotic hadrons, and our
results obtained in Ref. Chen:2020pac  for the $P_{c}$ states are consistent
with those obtained in Ref. Voloshin:2019aut  through the heavy quark spin
symmetry.

This paper is organized as follows. In Sec. II we construct the fully-bottom
tetraquark currents within the diquark-antidiquark picture, and apply the
Fierz rearrangement to transform them into the meson-meson currents. Based on
the obtained Fierz identities, we study the decay behaviors of the fully-
bottom tetraquark states in Sec. III, and calculate their relative branching
ratios. The results are summarized and discussed in Sec. IV.

##  II Currents and Fierz Identities

In Refs. Chen:2016jxd ; Chen:2020xwe  we have systematically constructed all
the fully-heavy tetraquark currents without derivatives. We briefly summarize
them in this section, which will be used to study the decay behaviors of the
fully-bottom tetraquark states in the next section.

Besides, in Ref. Su:2022eun  we have systematically constructed all the
$P$-wave fully-strange tetraquark currents by explicitly adding the covariant
derivative operator. Their corresponding fully-heavy tetraquark currents with
derivatives can be similarly constructed. See Refs. Chen:2020aos ;
Chen:2020uif  for more discussions. However, we shall not investigate them in
the present study, since many relevant decay constants are not known yet. We
also refer to Refs. liu:2020eha ; Liu:2020lpw  where the authors
systematically classified the $S$-/$P$-wave fully-heavy/strange tetraquark
states using the nonrelativistic quark model within the diquark-antidiquark
picture.

###  II.1 Currents of the positive parity

There are altogether twelve fully-bottom tetraquark currents of the positive
parity, four of which correspond to the $S$-wave fully-bottom tetraquark
states within the diquark-antidiquark picture:

$\displaystyle J^{0^{++}}_{1}$ $\displaystyle=$ $\displaystyle
b_{a}^{T}C\gamma_{5}b_{b}~{}\bar{b}_{a}\gamma_{5}C\bar{b}_{b}^{T}\,,$ (11)
$\displaystyle J^{0^{++}}_{2}$ $\displaystyle=$ $\displaystyle
b_{a}^{T}C\gamma_{\mu}b_{b}~{}\bar{b}_{a}\gamma^{\mu}C\bar{b}_{b}^{T}\,,$ (12)
$\displaystyle J^{1^{+-}}_{3\alpha}$ $\displaystyle=$ $\displaystyle
b_{a}^{T}C\gamma^{\mu}b_{b}~{}\bar{b}_{a}\sigma_{\alpha\mu}\gamma_{5}C\bar{b}_{b}^{T}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-b_{a}^{T}C\sigma_{\alpha\mu}\gamma_{5}b_{b}~{}\bar{b}_{a}\gamma^{\mu}C\bar{b}_{b}^{T}\,,$
$\displaystyle J^{2^{++}}_{4\alpha\beta}$ $\displaystyle=$
$\displaystyle\mathcal{P}_{\alpha\beta}^{\mu\nu}~{}b_{a}^{T}C\gamma_{\mu}b_{b}~{}\bar{b}_{a}\gamma_{\nu}C\bar{b}_{b}^{T}\,.$
(14)

In the above expressions $b_{a}$ is the bottom quark field with the color
index $a$, and $\mathcal{P}_{\alpha\beta}^{\mu\nu}$ is the projection
operator,

$\displaystyle\mathcal{P}^{\alpha\beta;\mu\nu}=g^{\alpha\mu}g^{\beta\nu}+g^{\alpha\nu}g^{\beta\mu}-{1\over
2}g^{\alpha\beta}g^{\mu\nu}\,.$ (15)

After applying the Fierz transformation, we obtain:

$\displaystyle J^{0^{++}}_{1}$ $\displaystyle=$ $\displaystyle-{1\over
4}\xi_{1}^{0^{++}}-{1\over 4}\xi_{2}^{0^{++}}-{1\over
4}\xi_{3}^{0^{++}}-{1\over 4}\xi_{4}^{0^{++}}+{1\over 8}\xi_{5}^{0^{++}}\,,$
$\displaystyle J^{0^{++}}_{2}$ $\displaystyle=$
$\displaystyle\xi_{1}^{0^{++}}-\xi_{2}^{0^{++}}+{1\over
2}\xi_{3}^{0^{++}}-{1\over 2}\xi_{4}^{0^{++}}\,,$ (17) $\displaystyle
J^{1^{+-}}_{3\alpha}$ $\displaystyle=$
$\displaystyle{3i}\xi_{6\alpha}^{1^{+-}}-\xi_{7\alpha}^{1^{+-}}\,,$ (18)
$\displaystyle J^{2^{++}}_{4\alpha\beta}$ $\displaystyle=$
$\displaystyle{1\over 2}\xi_{8\alpha\beta}^{2^{++}}-{1\over
2}\xi_{9\alpha\beta}^{2^{++}}+{1\over 2}\xi_{10\alpha\beta}^{2^{++}}\,,$ (19)

where

$\displaystyle\xi_{1}^{0^{++}}$ $\displaystyle=$
$\displaystyle\bar{b}_{a}b_{a}~{}\bar{b}_{b}b_{b}\,,$
$\displaystyle\xi_{2}^{0^{++}}$ $\displaystyle=$
$\displaystyle\bar{b}_{a}\gamma_{5}b_{a}~{}\bar{b}_{b}\gamma_{5}b_{b}\,,$
$\displaystyle\xi_{3}^{0^{++}}$ $\displaystyle=$
$\displaystyle\bar{b}_{a}\gamma_{\mu}b_{a}~{}\bar{b}_{b}\gamma^{\mu}b_{b}\,,$
$\displaystyle\xi_{4}^{0^{++}}$ $\displaystyle=$
$\displaystyle\bar{b}_{a}\gamma_{\mu}\gamma_{5}b_{a}~{}\bar{b}_{b}\gamma^{\mu}\gamma_{5}b_{b}\,,$
$\displaystyle\xi_{5}^{0^{++}}$ $\displaystyle=$
$\displaystyle\bar{b}_{a}\sigma_{\mu\nu}b_{a}~{}\bar{b}_{b}\sigma^{\mu\nu}b_{b}\,,$
(20) $\displaystyle\xi_{6\alpha}^{1^{+-}}$ $\displaystyle=$
$\displaystyle\bar{b}_{a}\gamma_{5}b_{a}~{}\bar{b}_{b}\gamma_{\alpha}b_{b}\,,$
$\displaystyle\xi_{7\alpha}^{1^{+-}}$ $\displaystyle=$
$\displaystyle\bar{b}_{a}\gamma^{\mu}\gamma_{5}b_{a}~{}\bar{b}_{b}\sigma_{\alpha\mu}b_{b}\,,$
$\displaystyle\xi_{8\alpha\beta}^{2^{++}}$ $\displaystyle=$
$\displaystyle\mathcal{P}_{\alpha\beta}^{\mu\nu}~{}\bar{b}_{a}\gamma_{\mu}\gamma_{5}b_{a}~{}\bar{b}_{b}\gamma_{\nu}\gamma_{5}b_{b}\,,$
$\displaystyle\xi_{9\alpha\beta}^{2^{++}}$ $\displaystyle=$
$\displaystyle\mathcal{P}_{\alpha\beta}^{\mu\nu}~{}\bar{b}_{a}\gamma_{\mu}b_{a}~{}\bar{b}_{b}\gamma_{\nu}b_{b}\,,$
$\displaystyle\xi_{10\alpha\beta}^{2^{++}}$ $\displaystyle=$
$\displaystyle\mathcal{P}_{\alpha\beta}^{\mu\nu}~{}\bar{b}_{a}\sigma_{\mu\rho}b_{a}~{}\bar{b}_{b}\sigma_{\nu\rho}b_{b}\,.$

###  II.2 Currents of the negative parity

There are altogether seven fully-bottom tetraquark currents of the negative
parity:

$\displaystyle J^{0^{-+}}_{5}$ $\displaystyle=$ $\displaystyle
b^{T}_{a}Cb_{b}~{}\bar{b}_{a}\gamma_{5}C\bar{b}_{b}^{T}+b^{T}_{a}C\gamma_{5}b_{b}~{}\bar{b}_{a}C\bar{b}_{b}^{T}\,,$
(21) $\displaystyle J^{0^{-+}}_{6}$ $\displaystyle=$ $\displaystyle
b^{T}_{a}C\sigma_{\mu\nu}b_{b}~{}\bar{b}_{a}\sigma^{\mu\nu}\gamma_{5}C\bar{b}^{T}_{b}\,,$
(22) $\displaystyle J^{0^{--}}_{7}$ $\displaystyle=$ $\displaystyle
b^{T}_{a}Cb_{b}~{}\bar{b}_{a}\gamma_{5}C\bar{b}_{b}^{T}-b^{T}_{a}C\gamma_{5}b_{b}~{}\bar{b}_{a}C\bar{b}_{b}^{T}\,,$
(23) $\displaystyle J^{1^{-+}}_{8\alpha}$ $\displaystyle=$ $\displaystyle
b^{T}_{a}C\gamma_{\alpha}\gamma_{5}b_{b}~{}\bar{b}_{a}\gamma_{5}C\bar{b}_{b}^{T}+b^{T}_{a}C\gamma_{5}b_{b}~{}\bar{b}_{a}\gamma_{\alpha}\gamma_{5}C\bar{b}_{b}^{T}\,,$
$\displaystyle J^{1^{-+}}_{9\alpha}$ $\displaystyle=$ $\displaystyle
b^{T}_{a}C\sigma_{\alpha\mu}b_{b}~{}\bar{b}_{a}\gamma^{\mu}C\bar{b}^{T}_{b}+b^{T}_{a}C\gamma^{\mu}b_{b}~{}\bar{b}_{a}\sigma_{\alpha\mu}C\bar{b}^{T}_{b}\,,$
$\displaystyle J^{1^{--}}_{10\alpha}$ $\displaystyle=$ $\displaystyle
b^{T}_{a}C\gamma_{\alpha}\gamma_{5}b_{b}~{}\bar{b}_{a}\gamma_{5}C\bar{b}_{b}^{T}-b^{T}_{a}C\gamma_{5}b_{b}~{}\bar{b}_{a}\gamma_{\alpha}\gamma_{5}C\bar{b}_{b}^{T}\,,$
$\displaystyle J^{1^{--}}_{11\alpha}$ $\displaystyle=$ $\displaystyle
b^{T}_{a}C\sigma_{\alpha\mu}b_{b}~{}\bar{b}_{a}\gamma^{\mu}C\bar{b}^{T}_{b}-b^{T}_{a}C\gamma^{\mu}b_{b}~{}\bar{b}_{a}\sigma_{\alpha\mu}C\bar{b}^{T}_{b}\,.$

After applying the Fierz transformation, we obtain:

$\displaystyle J^{0^{-+}}_{5}$ $\displaystyle=$
$\displaystyle-\xi_{11}^{0^{-+}}+{1\over 4}\xi_{12}^{0^{-+}}\,,$ (28)
$\displaystyle J^{0^{-+}}_{6}$ $\displaystyle=$ $\displaystyle
6\xi_{11}^{0^{-+}}-{1\over 2}\xi_{12}^{0^{-+}}\,,$ (29) $\displaystyle
J^{0^{--}}_{7}$ $\displaystyle=$ $\displaystyle-\xi_{13}^{0^{--}}\,,$ (30)
$\displaystyle J^{1^{-+}}_{8\alpha}$ $\displaystyle=$
$\displaystyle-\xi_{14\alpha}^{1^{-+}}+i\xi_{15\alpha}^{1^{-+}}\,,$ (31)
$\displaystyle J^{1^{-+}}_{9\alpha}$ $\displaystyle=$
$\displaystyle-3i\xi_{14\alpha}^{1^{-+}}+\xi_{15\alpha}^{1^{-+}}\,,$ (32)
$\displaystyle J^{1^{--}}_{10\alpha}$ $\displaystyle=$
$\displaystyle\xi_{16\alpha}^{1^{--}}-i\xi_{17\alpha}^{1^{--}}\,,$ (33)
$\displaystyle J^{1^{--}}_{11\alpha}$ $\displaystyle=$
$\displaystyle-3i\xi_{16\alpha}^{1^{--}}+\xi_{17\alpha}^{1^{--}}\,,$ (34)

where

$\displaystyle\xi_{11}^{0^{-+}}$ $\displaystyle=$
$\displaystyle\bar{b}_{a}b_{a}~{}\bar{b}_{b}\gamma_{5}b_{b}\,,$
$\displaystyle\xi_{12}^{0^{-+}}$ $\displaystyle=$
$\displaystyle\bar{b}_{a}\sigma_{\mu\nu}b_{a}~{}\bar{b}_{b}\sigma^{\mu\nu}\gamma_{5}b_{b}\,,$
$\displaystyle\xi_{13}^{0^{--}}$ $\displaystyle=$
$\displaystyle\bar{b}_{a}\gamma_{\mu}b_{a}~{}\bar{b}_{b}\gamma^{\mu}\gamma_{5}b_{b}\,,$
$\displaystyle\xi_{14\alpha}^{1^{-+}}$ $\displaystyle=$
$\displaystyle\bar{b}_{a}\gamma_{5}b_{a}~{}\bar{b}_{b}\gamma_{\alpha}\gamma_{5}b_{b}\,,$
(35) $\displaystyle\xi_{15\alpha}^{1^{-+}}$ $\displaystyle=$
$\displaystyle\bar{b}_{a}\gamma^{\mu}b_{a}~{}\bar{b}_{b}\sigma_{\alpha\mu}b_{b}\,,$
$\displaystyle\xi_{16\alpha}^{1^{--}}$ $\displaystyle=$
$\displaystyle\bar{b}_{a}b_{a}~{}\bar{b}_{b}\gamma_{\alpha}b_{b}\,,$
$\displaystyle\xi_{17\alpha}^{1^{--}}$ $\displaystyle=$
$\displaystyle\bar{b}_{a}\gamma^{\mu}\gamma_{5}b_{a}~{}\bar{b}_{b}\sigma_{\alpha\mu}\gamma_{5}b_{b}\,.$

##  III Relative Branching Ratios

In this section we study possible decay channels of the fully-bottom
tetraquark states, and calculate their relative branching ratios. The same
method has been applied in Ref. Chen:2020xwe  to study the fully-charm
tetraquark states. We separately investigate the $S$\- and $P$-wave fully-
bottom tetraquark states as follows:

  * •

Accordingly to our previous QCD sum rule calculations summarized in Table 1,
we assume the masses of the $S$-wave fully-bottom tetraquark states to be
about 18.5 GeV. This value is below the
$\eta_{b}(1S)\eta_{b}(1S)/\eta_{b}(1S)\Upsilon(1S)/\Upsilon(1S)\Upsilon(1S)$
thresholds, so the $S$-wave fully-bottom tetraquark states can not fall-apart
decay into these two-body channels. Instead, they can decay into one
bottomonium meson and one intermediate $\Upsilon(1S)/\Upsilon(2S)/\cdots$
meson, with the intermediate meson annihilating to be a photon and then
transferring into a muon-antimuon pair. We depict this decay process in Fig.
1.

Figure 1: Decay mechanism of a fully-bottom tetraquark state into one
bottomonium meson and one intermediate $\Upsilon(1S)/\Upsilon(2S)/\cdots$
meson, with the intermediate $\Upsilon(1S)/\Upsilon(2S)/\cdots$ meson
annihilating to be a photon and then transferring into a muon-antimuon pair.

  * •

Accordingly to our previous QCD sum rule calculations summarized in Table 1,
we assume the masses of the $P$-wave fully-bottom tetraquark states to be
about 18.8 GeV. This value is below the
$\eta_{b}(1S)\Upsilon(1S)/\Upsilon(1S)\Upsilon(1S)$ thresholds, so the
$P$-wave fully-bottom tetraquark states can not fall-apart decay into these
two-body channels. This value is above the $\eta_{b}(1S)\eta_{b}(1S)$
threshold, but the $P$-wave fully-bottom tetraquark states of
$J^{PC}=0^{-\pm}/1^{-\pm}$ can not decay into this two-body channel neither,
due to either the $C$-parity conservation or the Bose-Einstein statistics.
Accordingly, we shall also investigate the three-body decay process depicted
in Fig. 1.

Table 2: Couplings of the charmonium/bottomonium operators to the charmonium/bottomonium states. Operators |  $J^{PC}$ |  Mesons |  $J^{PC}$ |  Couplings |  Decay Constants
---|---|---|---|---|---
$I^{S}=\bar{c}c$ | $0^{++}$ | $\chi_{c0}(1P)$ | $0^{++}$ | $\langle 0|I^{S}|\chi_{c0}(1P)\rangle=m_{\chi_{c0}(1P)}f_{\chi_{c0}(1P)}$ |  $f_{\chi_{c0}(1P)}=343$ MeV Veliev:2010gb
$I^{P}=\bar{c}i\gamma_{5}c$ | $0^{-+}$ | $\eta_{c}(1S)$ | $0^{-+}$ | $\langle 0|I^{P}|\eta_{c}(1S)\rangle=\lambda_{\eta_{c}(1S)}$ | $\lambda_{\eta_{c}(1S)}={f_{\eta_{c}(1S)}m_{\eta_{c}(1S)}^{2}\over 2m_{c}}$
$\eta_{c}(2S)$ | $0^{-+}$ | $\langle 0|I^{P}|\eta_{c}(2S)\rangle=\lambda_{\eta_{c}(2S)}$ | $\lambda_{\eta_{c}(2S)}=\lambda_{\eta_{c}(1S)}\times{f_{\psi(2S)}\over f_{J/\psi}}$
$I^{V}_{\mu}=\bar{c}\gamma_{\mu}c$ | $1^{--}$ | $J/\psi$ | $1^{--}$ | $\langle 0|I^{V}_{\mu}|J/\psi\rangle=m_{J/\psi}f_{J/\psi}\epsilon_{\mu}$ |  $f_{J/\psi}=418$ MeV Becirevic:2013bsa
$\psi(2S)$ | $1^{--}$ | $\langle 0|I^{V}_{\mu}|\psi(2S)\rangle=m_{\psi(2S)}f_{\psi(2S)}\epsilon_{\mu}$ |  $f_{\psi(2S)}=294.5$ MeV MaiordeSousa:2012vv
$I^{A}_{\mu}=\bar{c}\gamma_{\mu}\gamma_{5}c$ | $1^{++}$ | $\eta_{c}(1S)$ | $0^{-+}$ | $\langle 0|I^{A}_{\mu}|\eta_{c}(1S)\rangle=ip_{\mu}f_{\eta_{c}(1S)}$ |  $f_{\eta_{c}(1S)}=387$ MeV Becirevic:2013bsa
$\eta_{c}(2S)$ | $0^{-+}$ | $\langle 0|I^{A}_{\mu}|\eta_{c}(2S)\rangle=ip_{\mu}f_{\eta_{c}(2S)}$ | $f_{\eta_{c}(2S)}=f_{\eta_{c}(1S)}\times{f_{\psi(2S)}\over f_{J/\psi}}$
$\chi_{c1}(1P)$ | $1^{++}$ | $\langle 0|I^{A}_{\mu}|\chi_{c1}(1P)\rangle=m_{\chi_{c1}(1P)}f_{\chi_{c1}(1P)}\epsilon_{\mu}$ |  $f_{\chi_{c1}(1P)}=335$ MeV Novikov:1977dq
$I^{T}_{\mu\nu}=\bar{c}\sigma_{\mu\nu}c$ | $1^{\pm-}$ | $J/\psi$ | $1^{--}$ | $\langle 0|I^{T}_{\mu\nu}|J/\psi\rangle=if^{T}_{J/\psi}(p_{\mu}\epsilon_{\nu}-p_{\nu}\epsilon_{\mu})$ |  $f_{J/\psi}^{T}=410$ MeV Becirevic:2013bsa
$\psi(2S)$ | $1^{--}$ | $\langle 0|I^{T}_{\mu\nu}|\psi(2S)\rangle=if^{T}_{\psi(2S)}(p_{\mu}\epsilon_{\nu}-p_{\nu}\epsilon_{\mu})$ | $f_{\psi(2S)}^{T}=f_{J/\psi}^{T}\times{f_{\psi(2S)}\over f_{J/\psi}}$
$h_{c}(1P)$ | $1^{+-}$ | $\langle 0|I^{T}_{\mu\nu}|h_{c}(1P)\rangle=if^{T}_{h_{c}(1P)}\epsilon_{\mu\nu\alpha\beta}\epsilon^{\alpha}p^{\beta}$ |  $f_{h_{c}(1P)}^{T}=235$ MeV Becirevic:2013bsa
$J^{S}=\bar{b}b$ | $0^{++}$ | $\chi_{b0}(1P)$ | $0^{++}$ | $\langle 0|J^{S}|\chi_{b0}(1P)\rangle=m_{\chi_{b0}(1P)}f_{\chi_{b0}(1P)}$ |  $f_{\chi_{b0}(1P)}=175$ MeV Veliev:2010gb
$J^{P}=\bar{b}i\gamma_{5}b$ | $0^{-+}$ | $\eta_{b}(1S)$ | $0^{-+}$ | $\langle 0|J^{P}|\eta_{b}(1S)\rangle=\lambda_{\eta_{b}(1S)}$ | $\lambda_{\eta_{b}(1S)}={f_{\eta_{b}(1S)}m_{\eta_{b}(1S)}^{2}\over 2m_{b}}$
$\eta_{b}(2S)$ | $0^{-+}$ | $\langle 0|J^{P}|\eta_{b}(2S)\rangle=\lambda_{\eta_{b}(2S)}$ | $\lambda_{\eta_{b}(2S)}=\lambda_{\eta_{b}(1S)}\times{f_{\Upsilon(2S)}\over f_{\Upsilon(1S)}}$
$J^{V}_{\mu}=\bar{b}\gamma_{\mu}b$ | $1^{--}$ | $\Upsilon(1S)$ | $1^{--}$ | $\langle 0|J^{V}_{\mu}|\Upsilon(1S)\rangle=m_{\Upsilon(1S)}f_{\Upsilon(1S)}\epsilon_{\mu}$ |  $f_{\Upsilon(1S)}=715$ MeV MaiordeSousa:2012vv
$\Upsilon(2S)$ | $1^{--}$ | $\langle 0|J^{V}_{\mu}|\Upsilon(2S)\rangle=m_{\Upsilon(2S)}f_{\Upsilon(2S)}\epsilon_{\mu}$ |  $f_{\Upsilon(2S)}=497.5$ MeV MaiordeSousa:2012vv
$J^{A}_{\mu}=\bar{b}\gamma_{\mu}\gamma_{5}b$ | $1^{++}$ | $\eta_{b}(1S)$ | $0^{-+}$ | $\langle 0|J^{A}_{\mu}|\eta_{b}(1S)\rangle=ip_{\mu}f_{\eta_{b}(1S)}$ |  $f_{\eta_{b}(1S)}=801$ MeV Chiu:2007km
$\eta_{b}(2S)$ | $0^{-+}$ | $\langle 0|J^{A}_{\mu}|\eta_{b}(2S)\rangle=ip_{\mu}f_{\eta_{b}(2S)}$ | $f_{\eta_{b}(2S)}=f_{\eta_{b}(1S)}\times{f_{\Upsilon(2S)}\over f_{\Upsilon(1S)}}$
$\chi_{b1}(1P)$ | $1^{++}$ | $\langle 0|J^{A}_{\mu}|\chi_{b1}(1P)\rangle=m_{\chi_{b1}(1P)}f_{\chi_{b1}(1P)}\epsilon_{\mu}$ | $f_{\chi_{b1}(1P)}=f_{\chi_{b0}(1P)}\times{f_{\chi_{c1}(1P)}\over f_{\chi_{c0}(1P)}}$
$J^{T}_{\mu\nu}=\bar{b}\sigma_{\mu\nu}b$ | $1^{\pm-}$ | $\Upsilon(1S)$ | $1^{--}$ | $\langle 0|J^{T}_{\mu\nu}|\Upsilon(1S)\rangle=if^{T}_{\Upsilon(1S)}(p_{\mu}\epsilon_{\nu}-p_{\nu}\epsilon_{\mu})$ | $f_{\Upsilon(1S)}^{T}=f^{T}_{J/\psi}\times{f_{\Upsilon(1S)}\over f_{J/\psi}}$
$\Upsilon(2S)$ | $1^{--}$ | $\langle 0|J^{T}_{\mu\nu}|\Upsilon(2S)\rangle=if^{T}_{\Upsilon(2S)}(p_{\mu}\epsilon_{\nu}-p_{\nu}\epsilon_{\mu})$ | $f_{\Upsilon(2S)}^{T}=f^{T}_{\Upsilon(1S)}\times{f_{\Upsilon(2S)}\over f_{\Upsilon(1S)}}$
$h_{b}(1P)$ | $1^{+-}$ | $\langle 0|J^{T}_{\mu\nu}|h_{b}(1P)\rangle=if^{T}_{h_{b}(1P)}\epsilon_{\mu\nu\alpha\beta}\epsilon^{\alpha}p^{\beta}$ | $f_{h_{b}(1P)}^{T}=f_{h_{c}(1P)}^{T}\times{f_{\Upsilon(1S)}\over f_{J/\psi}}$

As an example, we apply the Fierz rearrangement given in Eq. (LABEL:eq:tran1)
to investigate the decay properties of the fully-bottom tetraquark state of
$J^{PC}=0^{++}$ corresponding to the current $J^{0^{++}}_{1}$ defined in Eq.
(11). We denote this state as $|X_{1};0^{++}\rangle$, and assume the coupling
to be

$\langle 0|J^{0^{++}}_{1}|X_{1};0^{++}\rangle=f_{X_{1}}\,,$ (36)

with $f_{X_{1}}$ the decay constant. As summarized in Table 1, its mass has
been calculated in Ref. Chen:2020xwe  through the QCD sum rule method to be
$18.45\pm 0.15$ GeV.

Figure 2: The fall-apart decay of a compact diquark-antidiquark
$[bb][\bar{b}\bar{b}]$ state into two bottomonium states. Quarks are shown in
red/green/blue color and antiquarks are shown in cyan/magenta/yellow color.

As depicted in Fig. 2, when the magenta $\bar{b}$ antiquark and the green $b$
quark meet each other, and the yellow $\bar{b}$ antiquark and the blue $b$
quark meet each other at the same time, $|X_{1};0^{++}\rangle$ can decay into
two bottomonium states:

$\displaystyle\left[b_{a}(x)b_{b}(x)\right]~{}\left[\bar{b}_{a}(x)\bar{b}_{b}(x)\right]$
$\displaystyle\xrightarrow{~{}\rm Fierz~{}}$
$\displaystyle\left[\bar{b}_{a}(x)b_{a}(x)\right]~{}\left[\bar{b}_{b}(x)b_{b}(x)\right]$
$\displaystyle\xrightarrow{~{}\rm decay~{}}$
$\displaystyle\left[\bar{b}_{a}(y)b_{a}(y)\right]~{}\left[\bar{b}_{b}(z)b_{b}(z)\right]\,.$

This decay process can be described by the Fierz rearrangement given in Eq.
(LABEL:eq:tran1), i.e.,

$\displaystyle J^{0^{++}}_{1}$ $\displaystyle=$ $\displaystyle
b_{a}^{T}C\gamma_{5}b_{b}~{}\bar{b}_{a}\gamma_{5}C\bar{b}_{b}^{T}$
$\displaystyle\rightarrow$ $\displaystyle-{1\over
4}\bar{b}_{a}b_{a}~{}\bar{b}_{b}b_{b}-{1\over
4}\bar{b}_{a}\gamma_{5}b_{a}~{}\bar{b}_{b}\gamma_{5}b_{b}$
$\displaystyle-{1\over
4}\bar{b}_{a}\gamma_{\mu}\gamma_{5}b_{a}~{}\bar{b}_{b}\gamma^{\mu}\gamma_{5}b_{b}-{1\over
4}\bar{b}_{a}\gamma_{\mu}b_{a}~{}\bar{b}_{b}\gamma^{\mu}b_{b}$
$\displaystyle+{1\over
8}\bar{b}_{a}\sigma_{\mu\nu}b_{a}~{}\bar{b}_{b}\sigma^{\mu\nu}b_{b}\,.$

In principle, we need the decay constant $f_{X_{1}}$ as an input to calculate
the partial decay widths, but it is not necessary any more if we only
calculate the relative branching ratios. Moreover, because the couplings of
meson operators to meson states have been well studied in the literature, but
the couplings of tetraquark currents to tetraquark states have not, the decay
constant $f_{X_{1}}$ is not so well determined compared to the bottomonium
decay constants listed in Table 2. Therefore, we can calculate the relative
branching ratios more reliably than the partial decay widths.

To do this we apply Eq. (III) to derive the couplings of the current
$J^{0^{++}}_{1}$ to both the $\Upsilon(1S)\Upsilon(1S)$ and
$\Upsilon(1S)h_{b}(1P)$ channels to be:

$\displaystyle\langle
0|J^{0^{++}}_{1}|\Upsilon(p_{1},\epsilon_{1})~{}\Upsilon(p_{2},\epsilon_{2})\rangle$
$\displaystyle=$
$\displaystyle\epsilon_{1}^{\mu}\epsilon_{2}^{\nu}~{}\Big{(}-{1\over
2}m_{\Upsilon}^{2}f_{\Upsilon}^{2}g_{\mu\nu}$
$\displaystyle~{}~{}~{}~{}~{}-{1\over 2}(f_{\Upsilon}^{T})^{2}p_{1}\cdot
p_{2}g_{\mu\nu}+{1\over 2}(f_{\Upsilon}^{T})^{2}p_{1\nu}p_{2\mu}\Big{)}\,,$
$\displaystyle\langle
0|J^{0^{++}}_{1}|\Upsilon(p_{1},\epsilon_{1})~{}h_{b}(p_{2},\epsilon_{2})\rangle$
$\displaystyle=$ $\displaystyle-{1\over
2}\times\epsilon_{1}^{\mu}\epsilon_{2}^{\nu}~{}f_{\Upsilon}^{T}f_{h_{b}}^{T}\epsilon_{\mu\nu\rho\sigma}p_{1}^{\rho}p_{2}^{\sigma}\,,$

from which we further extract the couplings of $|X_{1};0^{++}\rangle$ to both
the $\Upsilon(1S)\Upsilon(1S)$ and $\Upsilon(1S)h_{b}(1P)$ channels to be:

$\displaystyle\langle
X_{1}(p);0^{++}|\Upsilon(p_{1},\epsilon_{1})~{}\Upsilon(p_{2},\epsilon_{2})\rangle$
$\displaystyle=$ $\displaystyle
c\times\epsilon_{1}^{\mu}\epsilon_{2}^{\nu}~{}\Big{(}-{1\over
2}m_{\Upsilon}^{2}f_{\Upsilon}^{2}g_{\mu\nu}$
$\displaystyle~{}~{}~{}~{}~{}-{1\over 2}(f_{\Upsilon}^{T})^{2}p_{1}\cdot
p_{2}g_{\mu\nu}+{1\over 2}(f_{\Upsilon}^{T})^{2}p_{1\nu}p_{2\mu}\Big{)}\,,$
$\displaystyle\langle
X_{1}(p);0^{++}|\Upsilon(p_{1},\epsilon_{1})~{}h_{b}(p_{2},\epsilon_{2})\rangle$
$\displaystyle=$ $\displaystyle-{c\over
2}\times\epsilon_{1}^{\mu}\epsilon_{2}^{\nu}~{}f_{\Upsilon}^{T}f_{h_{b}}^{T}\epsilon_{\mu\nu\rho\sigma}p_{1}^{\rho}p_{2}^{\sigma}\,.$

The decay constants $f_{\Upsilon}\equiv f_{\Upsilon(1S)}$,
$f_{\Upsilon}^{T}\equiv f_{\Upsilon(1S)}^{T}$, and $f_{h_{b}}^{T}\equiv
f_{h_{b}(1P)}^{T}$ of the $\Upsilon(1S)$ and $h_{b}(1P)$ mesons are given in
Table 2. The overall factor $c$ is related to the decay constant $f_{X_{1}}$,
which will be eliminated when calculating the relative branching ratios.

Based on Eq. (III), we can write the decay amplitude of the three-body decay
process
$|X_{1};0^{++}\rangle\rightarrow\Upsilon(1S)\Upsilon(1S)\rightarrow\Upsilon(1S)\mu^{+}\mu^{-}$
as

$\displaystyle\mathcal{M}\Big{(}X_{1}(p)\to\Upsilon(p_{1},\epsilon_{1})\Upsilon(q,\epsilon_{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\rightarrow\Upsilon(p_{1},\epsilon_{1})\mu^{-}(p_{2})\mu^{+}(p_{3})\Big{)}$
$\displaystyle=$ $\displaystyle
cc^{\prime}e\times\epsilon_{1}^{\mu}~{}\Big{(}-{1\over
2}m_{\Upsilon}^{2}f_{\Upsilon}^{2}g_{\mu\nu}$
$\displaystyle~{}~{}~{}~{}~{}-{1\over 2}(f_{\Upsilon}^{T})^{2}p_{1}\cdot
qg_{\mu\nu}+{1\over 2}(f_{\Upsilon}^{T})^{2}p_{1\nu}q_{\mu}\Big{)}$
$\displaystyle\times~{}{\bar{u}(p_{2})\gamma_{\alpha}v(p_{3})\over
q^{2}\left(q^{2}-m_{\Upsilon}^{2}+{\rm
i}m_{\Upsilon}\Gamma_{\Upsilon}\right)}~{}\left(g^{\alpha\nu}-{q^{\alpha}q^{\nu}\over
m_{\Upsilon}^{2}}\right)\,,$

where $u(p_{2})$ and $v(p_{3})$ are the Dirac spinors of the $\mu^{-}$ and
$\mu^{+}$, respectively. The overall factor $c^{\prime}$ is related to the
coupling of $\Upsilon(1S)$ to the photon, which will also be eliminated when
calculating the relative branching ratios.

We use Eq. (III) to further evaluate the partial decay width to be:

$\displaystyle\Gamma\left(X_{1}(p)\to\Upsilon(p_{1},\epsilon_{1})\mu^{+}(p_{2})\mu^{-}(p_{3})\right)$
$\displaystyle=$ $\displaystyle{1\over(2\pi)^{3}}{c^{2}c^{\prime 2}e^{2}\over
32m_{X_{1}}^{3}}\int{\rm d}m_{12}^{2}{\rm d}m_{23}^{2}$
$\displaystyle\times\left|{1\over q^{2}-m_{\Upsilon}^{2}+{\rm
i}m_{\Upsilon}\Gamma_{\Upsilon}}\right|^{2}\left|{1\over q^{2}}\right|^{2}$
$\displaystyle\times{\rm
Tr}[(p\\!\\!\\!/_{2}+m_{\mu^{-}})\gamma_{\alpha}(p\\!\\!\\!/_{3}-m_{\mu^{+}})\gamma_{\alpha^{\prime}}]$
$\displaystyle\times\Big{(}-{1\over
2}m_{\Upsilon}^{2}f_{\Upsilon}^{2}g_{\mu\nu}$
$\displaystyle~{}~{}~{}~{}~{}-{1\over 2}(f_{\Upsilon}^{T})^{2}p_{1}\cdot
qg_{\mu\nu}+{1\over 2}(f_{\Upsilon}^{T})^{2}p_{1\nu}q_{\mu}\Big{)}$
$\displaystyle\times\Big{(}-{1\over
2}m_{\Upsilon}^{2}f_{\Upsilon}^{2}g_{\mu^{\prime}\nu^{\prime}}$
$\displaystyle~{}~{}~{}~{}~{}-{1\over 2}(f_{\Upsilon}^{T})^{2}p_{1}\cdot
qg_{\mu^{\prime}\nu^{\prime}}+{1\over
2}(f_{\Upsilon}^{T})^{2}p_{1\nu^{\prime}}q_{\mu^{\prime}}\Big{)}$
$\displaystyle\times\left(g^{\alpha\nu}-{q^{\alpha}q^{\nu}\over
m_{\Upsilon}^{2}}\right)~{}\left(g^{\alpha^{\prime}\nu^{\prime}}-{q^{\alpha^{\prime}}q^{\nu^{\prime}}\over
m_{\Upsilon}^{2}}\right)$
$\displaystyle\times\left(g^{\mu\mu^{\prime}}-{p_{1}^{\mu}p_{1}^{\mu^{\prime}}\over
m_{\Upsilon}^{2}}\right)\,.$

Similarly, we study the three-body decay process
$|X_{1};0^{++}\rangle\rightarrow h_{b}(1P)\Upsilon(1S)\rightarrow
h_{b}(1P)\mu^{+}\mu^{-}$ and calculate its partial decay width. After
eliminating the overall factors $c$ and $c^{\prime}$, we obtain:

${\mathcal{B}(|X_{1};0^{++}\rangle\rightarrow h_{b}(1P)\Upsilon(1S)\rightarrow
h_{b}(1P)\mu^{+}\mu^{-})\over\mathcal{B}(|X_{1};0^{++}\rangle\rightarrow\Upsilon(1S)\Upsilon(1S)\rightarrow\Upsilon(1S)\mu^{+}\mu^{-})}=0.002\,.$
(45)

The above procedures are applied to investigate the process with the
intermediate $\Upsilon(1S)$ meson. We can apply the same procedures to
investigate the process with the intermediate
$\Upsilon(2S)/\Upsilon(3S)/\cdots$ mesons, and the obtained results are
approximately the same, while we do not consider other intermediate
bottomonium mesons in the present study, such as the $\Upsilon(1D)$ meson,
etc. Assuming that the
$|X_{1};0^{++}\rangle\rightarrow\Upsilon(1S)\mu^{+}\mu^{-}$ and
$|X_{1};0^{++}\rangle\rightarrow h_{b}(1P)\mu^{+}\mu^{-}$ decays are dominated
by these processes, we finally obtain

${\mathcal{B}(|X_{1};0^{++}\rangle\rightarrow
h_{b}(1P)\mu^{+}\mu^{-})\over\mathcal{B}(|X_{1};0^{++}\rangle\rightarrow\Upsilon(1S)\mu^{+}\mu^{-})}\approx
0.002\,.$ (46)

After considering several relevant channels, we obtain:

$\displaystyle\mathcal{B}(|X_{1};0^{++}\rangle$ $\displaystyle\rightarrow$
$\displaystyle\Upsilon(1S)\mu\mu:\Upsilon(2S)\mu\mu:h_{b}(1P)\mu\mu\,)$ (47)
$\displaystyle\approx$
$\displaystyle~{}~{}~{}~{}~{}1~{}~{}~{}~{}~{}\,:~{}~{}~{}0.42~{}~{}~{}:~{}~{}~{}0.002\,.$

Similarly, we apply the above procedures to investigate the $S$\- and $P$-wave
fully-bottom tetraquark states $|X_{2\cdots 11};J^{PC}\rangle$ through the
currents $J^{\cdots}_{2\cdots 11}$. The obtained results are summarized in
Table 3, which we shall use to draw conclusions in the next section. It is
interesting to notice that some relative branching ratios are significantly
larger/smaller than the others, which is partly due to that these ratios are
proportional to the square of the Fierz coefficients given in Eqs.
(LABEL:eq:tran1-19) and Eqs. (28-34). For example, the relative branching
ratio of $|X_{11};1^{--}\rangle$ decaying into the
$\mu^{+}\mu^{-}\chi_{b0}(1P)$ channel is significantly larger than that of the
$\mu^{+}\mu^{-}\eta_{b}(1S)$ channel:

${\mathcal{B}(|X_{11};1^{--}\rangle\rightarrow\chi_{b0}(1P)\mu^{+}\mu^{-})\over\mathcal{B}(|X_{11};1^{--}\rangle\rightarrow\eta_{b}(1S)\mu^{+}\mu^{-})}\approx
12\,,$ (48)

where the factor contributed by the Fierz coefficients is 9. Besides, the
relative branching ratios are also contributed by the decay constants as well
as the kinematics. For example, the relative branching ratio of
$|X_{8};1^{-+}\rangle$ decaying into the $\mu^{+}\mu^{-}h_{b}(1P)$ channel is
not far from that of the $\mu^{+}\mu^{-}\Upsilon(1S)$ channel:

${\mathcal{B}(|X_{8};1^{-+}\rangle\rightarrow
h_{b}(1P)\mu^{+}\mu^{-})\over\mathcal{B}(|X_{8};1^{-+}\rangle\rightarrow\Upsilon(1S)\mu^{+}\mu^{-})}\approx
0.27\,.$ (49)

The Fierz coefficients do not contribute to this ratio, while the factors
contributed by the decay constants and the kinematics are about 0.33 and 0.81,
respectively.

In our previous study Chen:2020xwe  we have studied the decays of the fully-
charm tetraquark states into the $1S$ and $1P$ double-charmonium channels
$J/\psi J/\psi$, $J/\psi\eta_{c}(1S)$, and $\eta_{c}(1S)\eta_{c}(1S)$, etc. In
the present study we further take into account the $2S$ double-charmonium
channels $J/\psi\psi(2S)$, $\eta_{c}(1S)\eta_{c}(2S)$, $J/\psi\eta_{c}(2S)$,
and $\eta_{c}(1S)\psi(2S)$. The obtained results are summarized in Table 4,
which we shall also use to draw conclusions in the next section.

##  IV Summary and discussions

Table 3: Relative branching ratios of the $S$\- and $P$-wave fully-bottom tetraquark states $|X_{1\cdots 11};J^{PC}\rangle$ corresponding to the currents $J^{\cdots}_{1\cdots 11}$. In the 3rd-5th columns we show the branching ratios relative to the $\mu^{+}\mu^{-}\Upsilon(1S)$ channel, and in the 6th-9th columns we show the branching ratios relative to the $\mu^{+}\mu^{-}\eta_{b}(1S)$ channel. $J^{PC}$ |  Current | Decay Channels
---|---|---
$\mu^{+}\mu^{-}\Upsilon(1S)$ |  $\mu^{+}\mu^{-}\Upsilon(2S)$ |  $\mu^{+}\mu^{-}h_{b}(1P)$ |  $\mu^{+}\mu^{-}\eta_{b}(1S)$ |  $\mu^{+}\mu^{-}\eta_{b}(2S)$ |  $\mu^{+}\mu^{-}\chi_{b0}(1P)$ |  $\mu^{+}\mu^{-}\chi_{b1}(1P)$
$0^{++}$ | $J^{0^{++}}_{1}$ | $1$ | $0.42$ | $0.002$ | – | – | – | –
$J^{0^{++}}_{2}$ | $1$ | $0.42$ | – | – | – | – | –
$1^{+-}$ | $J^{1^{+-}}_{3\alpha}$ | – | – | – | $1$ | $0.42$ | – | $1\times 10^{-4}$
$2^{++}$ | $J^{2^{++}}_{4\alpha\beta}$ | $1$ | $0.42$ | $0.002$ | – | – | – | –
$0^{-+}$ | $J^{0^{-+}}_{5}$ | $1$ | $0.39$ | $0.090$ | – | – | – | –
$J^{0^{-+}}_{6}$ | $1$ | $0.39$ | $0.090$ | – | – | – | –
$0^{--}$ | $J^{0^{--}}_{7}$ | – | – | – | $1$ | $0.42$ | – | $0.041$
$1^{-+}$ | $J^{1^{-+}}_{8\alpha}$ | $1$ | $0.43$ | $0.27$ | – | – | – | –
$J^{1^{-+}}_{9\alpha}$ | $1$ | $0.43$ | $0.27$ | – | – | – | –
$1^{--}$ | $J^{1^{--}}_{10\alpha}$ | – | – | – | $1$ | $0.38$ | $1.3$ | $0.070$
$J^{1^{--}}_{11\alpha}$ | – | – | – | $1$ | $0.38$ | $12$ | $0.070$
Table 4: Relative branching ratios of the $S$\- and $P$-wave fully-charm tetraquark states, calculated through the fully-charm tetraquark currents $J^{\cdots}_{1\cdots 11}|_{b/\bar{b}\to c/\bar{c}}$. In the 3rd-9th columns we show the branching ratios relative to the $J/\psi J/\psi$ channel, and in the 10th-15th columns we show the branching ratios relative to the $J/\psi\eta_{c}$ channel. The notations $\psi^{\prime}\equiv\psi(2S)$ and $\eta_{c}^{\prime}\equiv\eta_{c}(2S)$ are used here. $J^{PC}$ |  Current | Decay Channels
---|---|---
$J/\psi J/\psi$ |  $J/\psi\psi^{\prime}$ |  $\eta_{c}\eta_{c}$ |  $\eta_{c}\eta_{c}^{\prime}$ |  $J/\psi h_{c}$ |  $\eta_{c}\chi_{c0}$ |  $\eta_{c}\chi_{c1}$ |  $J/\psi\eta_{c}$ |  $J/\psi\eta_{c}^{\prime}$ |  $\psi^{\prime}\eta_{c}$ |  $J/\psi\chi_{c0}$ |  $J/\psi\chi_{c1}$ |  $\eta_{c}h_{c}$
$0^{++}$ | $J^{0^{++}}_{1}$ | $1$ | – | $0.45$ | – | – | – | $2\times 10^{-5}$ | – | – | – | – | – | –
$J^{0^{++}}_{2}$ | $1$ | – | $4.1$ | – | – | – | $9\times 10^{-5}$ | – | – | – | – | – | –
$1^{+-}$ | $J^{1^{+-}}_{3\alpha}$ | – | – | – | – | – | – | – | $1$ | – | – | – | – | –
$2^{++}$ | $J^{2^{++}}_{4\alpha\beta}$ | $1$ | – | $0.036$ | – | – | – | $0.003$ | – | – | – | – | – | –
$0^{-+}$ | $J^{0^{-+}}_{5}$ | $1$ | $0.071$ | – | – | $0.21$ | $0.69$ | – | – | – | – | – | – | –
$J^{0^{-+}}_{6}$ | $1$ | $0.071$ | – | – | $0.21$ | $6.2$ | – | – | – | – | – | – | –
$0^{--}$ | $J^{0^{--}}_{7}$ | – | – | – | – | – | – | – | $1$ | $0.048$ | $0.078$ | – | $1.4$ | –
$1^{-+}$ | $J^{1^{-+}}_{8\alpha}$ | $1$ | $0.071$ | – | – | $0.78$ | – | $0.94$ | – | – | – | – | – | –
$J^{1^{-+}}_{9\alpha}$ | $1$ | $0.071$ | – | – | $0.78$ | – | $8.4$ | – | – | – | – | – | –
$1^{--}$ | $J^{1^{--}}_{10\alpha}$ | – | – | – | – | – | – | – | $1$ | $0.048$ | $0.078$ | $0.79$ | $1.5$ | $0.43$
$J^{1^{--}}_{11\alpha}$ | – | – | – | – | – | – | – | $1$ | $0.048$ | $0.078$ | $7.1$ | $1.5$ | $0.43$

In this paper we systematically study the decay behaviors of the fully-bottom
and fully-charm tetraquark states through their corresponding interpolating
currents without derivatives. We work within the diquark-antidiquark picture,
and apply the Fierz rearrangement of the Dirac and color indices to transform
the diquark-antidiquark currents into the meson-meson currents. The obtained
Fierz identities are given in Eqs. (LABEL:eq:tran1-19) and Eqs. (28-34).

Based on these Fierz identities, we study the decay mechanism depicted in Fig.
1, where a fully-bottom tetraquark state decays into one bottomonium meson and
one intermediate $\Upsilon(1S)/\Upsilon(2S)/\cdots$ meson, with the
intermediate $\Upsilon(1S)/\Upsilon(2S)/\cdots$ meson annihilating to be a
photon and then transferring into a muon-antimuon pair. We consider several
possible decay channels and calculate their relative branching ratios. The
obtained results are summarized in Table 3, where the masses of the $S$\- and
$P$-wave fully-bottom tetraquark states are assumed to be 18.5 GeV and 18.8
GeV, respectively Chen:2016jxd . In the calculations we work within the naive
factorization scheme, so the uncertainty of our results is significantly
larger than the well-developed QCD factorization scheme (about 5% when
studying the weak and radiative decays of the conventional hadrons)
Beneke:1999br ; Beneke:2000ry ; Beneke:2001ev ; Li:2020rcg . However, we
calculate the relative branching ratios after eliminating several ambiguous
overall factors, such as the decay constant $f_{X_{1}}$ and the coupling of
the $\Upsilon(1S)$ to the photon. This largely reduces our uncertainty, e.g.,
we roughly estimate the uncertainty of Eq. (48) to be

${\mathcal{B}(|X_{11};1^{--}\rangle\rightarrow\chi_{b0}(1P)\mu^{+}\mu^{-})\over\mathcal{B}(|X_{11};1^{--}\rangle\rightarrow\eta_{b}(1S)\mu^{+}\mu^{-})}\approx
12^{+24}_{-~{}8}\,,$ (50)

based on our previous systematical QCD sum rule studies on the decay
properties of the excited heavy baryons Yang:2020zrh ; Yang:2021lce .

Our results suggest that the fully-bottom tetraquark states of
$J^{PC}=0^{++}/2^{++}/0^{-+}/1^{-+}$ can be searched for in the
$\mu^{+}\mu^{-}\Upsilon(1S)$ channel, and they can also be searched for in the
$\mu^{+}\mu^{-}\Upsilon(2S)$ channel, with the relative branching ratio
$\mathcal{B}(X\to\mu\mu\Upsilon(2S))/\mathcal{B}(X\to\mu\mu\Upsilon(1S))\approx
0.4$. Our results also suggest that the fully-bottom tetraquark states of
$J^{PC}=1^{+-}/0^{--}/1^{--}$ can be searched for in the
$\mu^{+}\mu^{-}\eta_{b}(1S)$ channel, and they can also be searched for in the
$\mu^{+}\mu^{-}\eta_{b}(2S)$ channel, with the similar relative branching
ratio
$\mathcal{B}(X\to\mu\mu\eta_{b}(2S))/\mathcal{B}(X\to\mu\mu\eta_{b}(1S))\approx
0.4$. We propose to examine these decay channels to search for the fully-
bottom tetraquark states in future CMS experiments.

In this paper we also update our previous study of Ref. Chen:2020xwe  and
reanalysis the fall-apart two-body decays of the fully-charm tetraquark
states. The obtained results are summarized in Table 4, where the masses of
the $S$\- and $P$-wave fully-charm tetraquark states are assumed to be 6.5 GeV
and 6.9 GeV, respectively Chen:2016jxd . These states were used in Ref.
Chen:2020xwe  to explain the broad structure at around 6.2-6.8 GeV and the
narrow structure at around 6.9 GeV observed by LHCb in the di-$J/\psi$
invariant mass spectrum LHCb:2020bwg . Based on the results of the present
study, we calculate the $X(6900)$ decay into the $J/\psi\psi(2S)$ channel and
obtain the relative branching ratio $\mathcal{B}(X\to
J/\psi\psi(2S))/\mathcal{B}(X\to J/\psi J/\psi)\approx 0.1$.

## Acknowledgments

We thank Xiang Liu and Shi-Lin Zhu for helpful discussion. This project is
supported by the National Natural Science Foundation of China under Grants No.
12075019 and No. 12175318, the Jiangsu Provincial Double-Innovation Program
under Grant No. JSSCRC2021488, the Natural Science Foundation of Guangdong
Province of China under Grant No. 2022A1515011922, and the Fundamental
Research Funds for the Central Universities.

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