1. Introduction
Since the discovery of the X(3872) in 2003 by Belle [1], many charmonium-like XYZ states have been discovered in the past twenty years [2]. They are good candidates of four-quark states consisting of two quarks and two antiquarks, and their experimental and theoretical studies have significantly improved our understanding of the strong interaction at the low energy region. Although there is still a long way to go to fully understand how the strong interaction binds these quarks and antiquarks together with gluons, this subject has become and will continuously be one of the most intriguing research topics in hadron physics [3–10].
The X(3872) is the most puzzling state among all the charmonium-like XYZ states. Although it is denoted as the χc1(3872) in PDG2018 [2], the mass of the charmonium state χc1(2P) was estimated to be 3.95 GeV [11], significantly higher than the X(3872). This challenges the interpretation of the X(3872) as a conventional charmonium state, and various interpretations were proposed to explain it, such as a compact tetraquark state composed of a diquark and an antidiquark [12–17], a loosely-bound hadronic molecular state composed of two charmed mesons [18–24], and a hybrid charmonium state with constituents $c\bar{c}g$ [25, 26], etc. There were also some studies of the X(3872) as a conventional $c\bar{c}$ state [27–30], and it was considered as the mixture of a $c\bar{c}$ state with a $D{\bar{D}}^{* }$ component in [31–33]. We refer to reviews [3–10] for detailed discussions.
The charged charmonium-like state X(3872) of JPC = 1++ [34] has been observed in the ${D}^{0}{\bar{D}}^{* 0}$, J/ψππ, J/ψω, and γJ/ψ decay channels [35–44], and there have been some evidences for the X(3872) → γψ(2S) decay [42, 44]. Especially, its decay channels J/ψππ and J/ψω have comparable branching ratios [38–41], implying a large isospin violation. In a recent BESIII experiment [45], evidence for the X(3872) → χc1π decay was reported with a statistical significance of 5.2σ using data at center-of-mass energies between 4.15 and 4.30 GeV, while this was not confirmed in the later Belle experiment [46]. We refer to a recent paper [47], where the authors presented a complete analysis of all the existing experimental data and determine the absolute branching fractions of the X(3872) decays. We also refer to another recent paper [48], which studies branching fractions of the X(3872) from a theoretical point of view. There have been many experimental and theoretical studies on this subject, and we refer to reviews [3–10] for more discussions.
In [49] we studied decay properties of the Zc(3900) through the Fierz rearrangement of the Dirac and color indices, and in this paper we shall apply the same method to study decay properties of the X(3872). Both of these two studies are based on our previous finding that the diquark-antidiquark currents ($[{qq}][\bar{q}\bar{q}]$) and the meson-meson currents ($[\bar{q}q][\bar{q}q]$) are related to each other through the Fierz rearrangement of the Dirac and color indices [50–53]. More studies on light baryon operators can be found in [54, 55]. The present study follows the idea of the QCD factorization method [56–58], which has been widely and successfully applied to study weak decay properties of (heavy) hadrons.
The X(3872), as either a compact tetraquark state or a hadronic molecular state, contains four quarks. There can be three configurations (q = u/d):
$\begin{eqnarray*}[{cq}][\bar{c}\bar{q}],\,\,[\bar{c}q][\bar{q}c],\,\,\mathrm{and}\,\,[\bar{c}c][\bar{q}q].\end{eqnarray*}$
In this paper we shall apply the Fierz rearrangement to relate them, and extract some strong decay properties of the X(3872) under both the compact tetraquark and hadronic molecule interpretations. We shall not calculate the absolute values of these decay widths, but extract their relative branching ratios, which are also useful to understand the nature of the X(3872) [59]. A similar arrangement in the nonrelativistic case was used to study strong decay properties of the X(3872) and Zc(3900) in [60–62].This paper is organized as follows. In section 2 we systematically construct all the tetraquark currents of JPC = 1++ with the quark content $c\bar{c}q\bar{q}$. We consider three configurations, $[{cq}][\bar{c}\bar{q}]$, $[\bar{c}q][\bar{q}c]$, and $[\bar{c}c][\bar{q}q]$, and we derive their relations using the Fierz rearrangement of the Dirac and color indices. In section 3 and section 4 we extract some isoscalar decay channels of the X(3872), separately for the compact tetraquark and hadronic molecule interpretations, and in section 5 we investigate its isovector decay channels. The obtained results are discussed in section 6 , and formulae of decay amplitudes and decay widths are given in appendix A .
2. Tetraquark currents of JPC = 1++ and their relations
Similar to [49], we can use the c, $\bar{c}$, q, $\bar{q}$ quarks (q = u/d) to construct three types of tetraquark currents of JPC = 1++, as illustrated in figure 1:
$\begin{eqnarray}\begin{array}{rcl}\eta (x,y) & = & [{q}_{a}^{{\rm{T}}}(x)\,{\mathbb{C}}{{\rm{\Gamma }}}_{1}\,{c}_{b}(x)]\times [{\bar{q}}_{c}(y)\,{{\rm{\Gamma }}}_{2}{\mathbb{C}}\,{\bar{c}}_{d}^{{\rm{T}}}(y)],\\ \xi (x,y) & = & [{\bar{c}}_{a}(x)\,{{\rm{\Gamma }}}_{3}\,{q}_{b}(x)]\times [{\bar{q}}_{c}(y)\,{{\rm{\Gamma }}}_{4}\,{c}_{d}(y)],\\ \theta (x,y) & = & [{\bar{c}}_{a}(x)\,{{\rm{\Gamma }}}_{5}\,{c}_{b}(x)]\times [{\bar{q}}_{c}(y)\,{{\rm{\Gamma }}}_{6}\,{q}_{d}(y)],\end{array}\end{eqnarray}$
where Γi are Dirac matrices, and the subscripts a, b, c, d are color indices. We separately construct them in the following subsections.
Figure 1. Three types of tetraquark currents. Quarks and antiquarks are shown in red, green, and blue color. |
Generally speaking, one can apply the Fierz rearrangement to relate the local diquark-antidiquark currents η(x, x) and the local meson-meson currents ξ(x, x) and θ(x, x), but this equivalence is just between diquark-antidiquark and mesonic-mesonic currents, while compact diquark-antidiquark tetraquark states and weakly-bound meson-meson molecular states are totally different. To exactly describe them, one needs to explicitly use non-local currents to perform QCD sum rule analyses, but we are still not able to do this.
2.1. $[{qc}][\bar{q}\bar{c}]$ currents ${\eta }_{\mu }^{i}(x,y)$
There are eight independent $[{qc}][\bar{q}\bar{c}]$ currents of JPC = 1++ [63]:
$\begin{eqnarray}\begin{array}{rcl}{\eta }_{\mu }^{1} & = & {q}_{a}^{{\rm{T}}}{\mathbb{C}}{\gamma }_{\mu }{c}_{b}\,{\bar{q}}_{a}{\gamma }_{5}{\mathbb{C}}{\bar{c}}_{b}^{{\rm{T}}}+{q}_{a}^{{\rm{T}}}{\mathbb{C}}{\gamma }_{5}{c}_{b}\,{\bar{q}}_{a}{\gamma }_{\mu }{\mathbb{C}}{\bar{c}}_{b}^{{\rm{T}}},\\ {\eta }_{\mu }^{2} & = & {q}_{a}^{{\rm{T}}}{\mathbb{C}}{\gamma }_{\mu }{c}_{b}\,{\bar{q}}_{b}{\gamma }_{5}{\mathbb{C}}{\bar{c}}_{a}^{{\rm{T}}}+{q}_{a}^{{\rm{T}}}{\mathbb{C}}{\gamma }_{5}{c}_{b}\,{\bar{q}}_{b}{\gamma }_{\mu }{\mathbb{C}}{\bar{c}}_{a}^{{\rm{T}}},\\ {\eta }_{\mu }^{3} & = & {q}_{a}^{{\rm{T}}}{\mathbb{C}}{\gamma }^{\nu }{c}_{b}\,{\bar{q}}_{a}{\sigma }_{\mu \nu }{\gamma }_{5}{\mathbb{C}}{\bar{c}}_{b}^{{\rm{T}}}+{q}_{a}^{{\rm{T}}}{\mathbb{C}}{\sigma }_{\mu \nu }{\gamma }_{5}{c}_{b}\,{\bar{q}}_{a}{\gamma }^{\nu }{\mathbb{C}}{\bar{c}}_{b}^{{\rm{T}}},\\ {\eta }_{\mu }^{4} & = & {q}_{a}^{{\rm{T}}}{\mathbb{C}}{\gamma }^{\nu }{c}_{b}\,{\bar{q}}_{b}{\sigma }_{\mu \nu }{\gamma }_{5}{\mathbb{C}}{\bar{c}}_{a}^{{\rm{T}}}+{q}_{a}^{{\rm{T}}}{\mathbb{C}}{\sigma }_{\mu \nu }{\gamma }_{5}{c}_{b}\,{\bar{q}}_{b}{\gamma }^{\nu }{\mathbb{C}}{\bar{c}}_{a}^{{\rm{T}}},\\ {\eta }_{\mu }^{5} & = & {q}_{a}^{{\rm{T}}}{\mathbb{C}}{\gamma }_{\mu }{\gamma }_{5}{c}_{b}\,{\bar{q}}_{a}{\mathbb{C}}{\bar{c}}_{b}^{{\rm{T}}}+{q}_{a}^{{\rm{T}}}{\mathbb{C}}{c}_{b}\,{\bar{q}}_{a}{\gamma }_{\mu }{\gamma }_{5}{\mathbb{C}}{\bar{c}}_{b}^{{\rm{T}}},\\ {\eta }_{\mu }^{6} & = & {q}_{a}^{{\rm{T}}}{\mathbb{C}}{\gamma }_{\mu }{\gamma }_{5}{c}_{b}\,{\bar{q}}_{b}{\mathbb{C}}{\bar{c}}_{a}^{{\rm{T}}}+{q}_{a}^{{\rm{T}}}{\mathbb{C}}{c}_{b}\,{\bar{q}}_{b}{\gamma }_{\mu }{\gamma }_{5}{\mathbb{C}}{\bar{c}}_{a}^{{\rm{T}}},\\ {\eta }_{\mu }^{7} & = & {q}_{a}^{{\rm{T}}}{\mathbb{C}}{\gamma }^{\nu }{\gamma }_{5}{c}_{b}\,{\bar{q}}_{a}{\sigma }_{\mu \nu }{\mathbb{C}}{\bar{c}}_{b}^{{\rm{T}}}+{q}_{a}^{{\rm{T}}}{\mathbb{C}}{\sigma }_{\mu \nu }{c}_{b}\,{\bar{q}}_{a}{\gamma }^{\nu }{\gamma }_{5}{\mathbb{C}}{\bar{c}}_{b}^{{\rm{T}}},\\ {\eta }_{\mu }^{8} & = & {q}_{a}^{{\rm{T}}}{\mathbb{C}}{\gamma }^{\nu }{\gamma }_{5}{c}_{b}\,{\bar{q}}_{b}{\sigma }_{\mu \nu }{\mathbb{C}}{\bar{c}}_{a}^{{\rm{T}}}+{q}_{a}^{{\rm{T}}}{\mathbb{C}}{\sigma }_{\mu \nu }{c}_{b}\,{\bar{q}}_{b}{\gamma }^{\nu }{\gamma }_{5}{\mathbb{C}}{\bar{c}}_{a}^{{\rm{T}}}.\end{array}\end{eqnarray}$
In the above expressions we have omitted the coordinates x and y. The color structures of ${\eta }_{\mu }^{1}-{\eta }_{\mu }^{2}$, ${\eta }_{\mu }^{3}-{\eta }_{\mu }^{4}$, ${\eta }_{\mu }^{5}-{\eta }_{\mu }^{6}$, and ${\eta }_{\mu }^{7}-{\eta }_{\mu }^{8}$ are all antisymmetric ${[{qc}]}_{{\bar{{\bf{3}}}}_{c}}{[\bar{q}\bar{c}]}_{{{\bf{3}}}_{c}}\to {[{qc}\bar{q}\bar{c}]}_{{{\bf{1}}}_{c}}$, and those of ${\eta }_{\mu }^{1}+{\eta }_{\mu }^{2}$, ${\eta }_{\mu }^{3}+{\eta }_{\mu }^{4}$, ${\eta }_{\mu }^{5}+{\eta }_{\mu }^{6}$, and ${\eta }_{\mu }^{7}+{\eta }_{\mu }^{8}$ are all symmetric ${[{qc}]}_{{{\bf{6}}}_{c}}{[\bar{q}\bar{c}]}_{{\bar{{\bf{6}}}}_{c}}\to {[{qc}\bar{q}\bar{c}]}_{{{\bf{1}}}_{c}}$.In the diquark-antidiquark model proposed in [12, 13, 64], the authors use $| {s}_{{qc}},{s}_{\bar{q}\bar{c}}{\rangle }_{J}$ to denote ground-state tetraquarks, where sqc and ${s}_{\bar{q}\bar{c}}$ are the charmed diquark and antidiquark spins, respectively. They interpret the X(3872) as a compact tetraquark state of JPC = 1++:
$\begin{eqnarray}| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle =\displaystyle \frac{1}{\sqrt{2}}\left(| {0}_{{qc}},{1}_{\bar{q}\bar{c}}{\rangle }_{J=1}+| {1}_{{qc}},{0}_{\bar{q}\bar{c}}{\rangle }_{J=1}\right).\end{eqnarray}$
The interpolating current having the identical internal structure is the current ${\eta }_{\mu }^{1}-{\eta }_{\mu }^{2}$, which has been studied in [63, 65–69]: $\begin{eqnarray}\begin{array}{rcl}{\eta }_{\mu }^{{ \mathcal X }}(x,y) & = & {\eta }_{\mu }^{1}(x,y)-{\eta }_{\mu }^{2}(x,y)\\ & = & {q}_{a}^{{\rm{T}}}(x){\mathbb{C}}{\gamma }_{\mu }{c}_{b}(x)\left({\bar{q}}_{a}(y){\gamma }_{5}{\mathbb{C}}{\bar{c}}_{b}^{{\rm{T}}}(y)-\{a\leftrightarrow b\}\right)\\ & & +\{{\gamma }_{\mu }\leftrightarrow {\gamma }_{5}\}.\end{array}\end{eqnarray}$
2.2. $[\bar{c}q][\bar{q}c]$ currents ${\xi }_{\mu }^{i}(x,y)$
There are eight independent $[\bar{c}q][\bar{q}c]$ currents of JPC = 1++:
$\begin{eqnarray}\begin{array}{rcl}{\xi }_{\mu }^{1} & = & {\bar{c}}_{a}{\gamma }_{\mu }{q}_{a}\,{\bar{q}}_{b}{\gamma }_{5}{c}_{b}-{\bar{c}}_{a}{\gamma }_{5}{q}_{a}\,{\bar{q}}_{b}{\gamma }_{\mu }{c}_{b},\\ {\xi }_{\mu }^{2} & = & {\bar{c}}_{a}{\gamma }^{\nu }{q}_{a}\,{\bar{q}}_{b}{\sigma }_{\mu \nu }{\gamma }_{5}{c}_{b}+{\bar{c}}_{a}{\sigma }_{\mu \nu }{\gamma }_{5}{q}_{a}\,{\bar{q}}_{b}{\gamma }^{\nu }{c}_{b},\\ {\xi }_{\mu }^{3} & = & {\bar{c}}_{a}{\gamma }_{\mu }{\gamma }_{5}{q}_{a}\,{\bar{q}}_{b}{c}_{b}+{\bar{c}}_{a}{q}_{a}\,{\bar{q}}_{b}{\gamma }_{\mu }{\gamma }_{5}{c}_{b},\\ {\xi }_{\mu }^{4} & = & {\bar{c}}_{a}{\gamma }^{\nu }{\gamma }_{5}{q}_{a}\,{\bar{q}}_{b}{\sigma }_{\mu \nu }{c}_{b}-{\bar{c}}_{a}{\sigma }_{\mu \nu }{q}_{a}\,{\bar{q}}_{b}{\gamma }^{\nu }{\gamma }_{5}{c}_{b},\\ {\xi }_{\mu }^{5} & = & {\lambda }_{{ab}}^{n}{\lambda }_{{cd}}^{n}\times \left({\bar{c}}_{a}{\gamma }_{\mu }{q}_{b}\,{\bar{q}}_{c}{\gamma }_{5}{c}_{d}-{\bar{c}}_{a}{\gamma }_{5}{q}_{b}\,{\bar{q}}_{c}{\gamma }_{\mu }{c}_{d}\right),\\ {\xi }_{\mu }^{6} & = & {\lambda }_{{ab}}^{n}{\lambda }_{{cd}}^{n}\times \left({\bar{c}}_{a}{\gamma }^{\nu }{q}_{b}\,{\bar{q}}_{c}{\sigma }_{\mu \nu }{\gamma }_{5}{c}_{d}+{\bar{c}}_{a}{\sigma }_{\mu \nu }{\gamma }_{5}{q}_{b}\,{\bar{q}}_{c}{\gamma }^{\nu }{c}_{d}\right),\\ {\xi }_{\mu }^{7} & = & {\lambda }_{{ab}}^{n}{\lambda }_{{cd}}^{n}\times \left({\bar{c}}_{a}{\gamma }_{\mu }{\gamma }_{5}{q}_{b}\,{\bar{q}}_{c}{c}_{d}+{\bar{c}}_{a}{q}_{b}\,{\bar{q}}_{c}{\gamma }_{\mu }{\gamma }_{5}{c}_{d}\right),\\ {\xi }_{\mu }^{8} & = & {\lambda }_{{ab}}^{n}{\lambda }_{{cd}}^{n}\times \left({\bar{c}}_{a}{\gamma }^{\nu }{\gamma }_{5}{q}_{b}\,{\bar{q}}_{c}{\sigma }_{\mu \nu }{c}_{d}-{\bar{c}}_{a}{\sigma }_{\mu \nu }{q}_{b}\,{\bar{q}}_{c}{\gamma }^{\nu }{\gamma }_{5}{c}_{d}\right).\end{array}\end{eqnarray}$
The former four ${\xi }_{\mu }^{1,2,3,4}$ have the color structure ${[\bar{c}q]}_{{{\bf{1}}}_{c}}{[\bar{q}c]}_{{{\bf{1}}}_{c}}\to {[c\bar{c}q\bar{q}]}_{{{\bf{1}}}_{c}}$, and the latter four ${\xi }_{\mu }^{5,6,7,8}$ have thecolor structure ${[\bar{c}q]}_{{{\bf{8}}}_{c}}{[\bar{q}c]}_{{{\bf{8}}}_{c}}\to {[c\bar{c}q\bar{q}]}_{{{\bf{1}}}_{c}}$.In the molecular picture the X(3872) is interpreted as the $D{\bar{D}}^{* }$ hadronic molecular state of JPC = 1++ [18–24]:
$\begin{eqnarray}| D{\bar{D}}^{* };{1}^{++}\rangle =\displaystyle \frac{1}{\sqrt{2}}\left(| D{\bar{D}}^{* }{\rangle }_{J=1}+| \bar{D}{D}^{* }{\rangle }_{J=1}\right),\end{eqnarray}$
and the relevant interpolating current is just ${\xi }_{\mu }^{1}(x,y)$, which has been studied in [70–73]: $\begin{eqnarray}\begin{array}{l}{\xi }_{\mu }^{{ \mathcal X }}(x,y)={\xi }_{\mu }^{1}(x,y)\\ \quad ={\bar{c}}_{a}(x){\gamma }_{\mu }{q}_{a}(x)\,{\bar{q}}_{b}(y){\gamma }_{5}{c}_{b}(y)-\{{\gamma }_{\mu }\leftrightarrow {\gamma }_{5}\}.\end{array}\end{eqnarray}$
2.3. $[\bar{c}c][\bar{q}q]$ currents ${\theta }_{\mu }^{i}(x,y)$
There are eight independent $[\bar{c}c][\bar{q}q]$ currents of JPC = 1++:
$\begin{eqnarray}\begin{array}{rcl}{\theta }_{\mu }^{1}(x,y) & = & {\bar{c}}_{a}(x){c}_{a}(x)\,{\bar{q}}_{b}(y){\gamma }_{\mu }{\gamma }_{5}{q}_{b}(y),\\ {\theta }_{\mu }^{2}(x,y) & = & {\bar{c}}_{a}(x){\gamma }_{\mu }{\gamma }_{5}{c}_{a}(x)\,{\bar{q}}_{b}(y){q}_{b}(y),\\ {\theta }_{\mu }^{3}(x,y) & = & {\bar{c}}_{a}(x){\gamma }^{\nu }{c}_{a}(x)\,{\bar{q}}_{b}(y){\sigma }_{\mu \nu }{\gamma }_{5}{q}_{b}(y),\\ {\theta }_{\mu }^{4}(x,y) & = & {\bar{c}}_{a}(x){\sigma }_{\mu \nu }{\gamma }_{5}{c}_{a}(x)\,{\bar{q}}_{b}(y){\gamma }^{\nu }{q}_{b}(y),\\ {\theta }_{\mu }^{5}(x,y) & = & {\lambda }_{{ab}}^{n}{\lambda }_{{cd}}^{n}\times \left({\bar{c}}_{a}(x){c}_{b}(x)\,{\bar{q}}_{c}(y){\gamma }_{\mu }{\gamma }_{5}{q}_{d}(y)\right),\\ {\theta }_{\mu }^{6}(x,y) & = & {\lambda }_{{ab}}^{n}{\lambda }_{{cd}}^{n}\times \left({\bar{c}}_{a}(x){\gamma }_{\mu }{\gamma }_{5}{c}_{b}(x)\,{\bar{q}}_{c}(y){q}_{d}(y)\right),\\ {\theta }_{\mu }^{7}(x,y) & = & {\lambda }_{{ab}}^{n}{\lambda }_{{cd}}^{n}\times \left({\bar{c}}_{a}(x){\gamma }^{\nu }{c}_{b}(x)\,{\bar{q}}_{c}(y){\sigma }_{\mu \nu }{\gamma }_{5}{q}_{d}(y)\right),\\ {\theta }_{\mu }^{8}(x,y) & = & {\lambda }_{{ab}}^{n}{\lambda }_{{cd}}^{n}\times \left({\bar{c}}_{a}(x){\sigma }_{\mu \nu }{\gamma }_{5}{c}_{b}(x)\,{\bar{q}}_{c}(y){\gamma }^{\nu }{q}_{d}(y)\right).\end{array}\end{eqnarray}$
The former four ${\theta }_{\mu }^{1,2,3,4}$ have the color structure ${[\bar{c}c]}_{{{\bf{1}}}_{c}}{[\bar{q}q]}_{{{\bf{1}}}_{c}}\to {[c\bar{c}q\bar{q}]}_{{{\bf{1}}}_{c}}$, and the latter four ${\theta }_{\mu }^{5,6,7,8}$ havethe color structure ${[\bar{c}c]}_{{{\bf{8}}}_{c}}{[\bar{q}q]}_{{{\bf{8}}}_{c}}\to {[c\bar{c}q\bar{q}]}_{{{\bf{1}}}_{c}}$.2.4. Fierz rearrangement
The Fierz rearrangement [74] of the Dirac and color indices has been systematically applied to study light baryon and tetraquark operators/currents in [49–55]. All the necessary equations can be found in section 3.3.2 of [75]. More studies can be found in [69, 76]. In the present study we apply it to relate the above three types of tetraquark currents.
The Fierz rearrangement is usually applied to local operators/currents. However, it is actually a matrix identity, and is valid if the same quark field in the initial and final operators is at the same location. As an example, we can apply the Fierz rearrangement to transform the non-local current with the quark fields $\eta ({x}^{{\prime} },x;{y}^{{\prime} },y)=[q({x}^{{\prime} })c(x)][\bar{q}({y}^{{\prime} })\bar{c}(y)]$ into a combination of several non-local currents with the quark fields at same locations $\theta (y,x;{y}^{{\prime} },{x}^{{\prime} })=[\bar{c}(y)c(x)][\bar{q}({y}^{{\prime} })q({x}^{{\prime} })]$.
To apply it to study the decay process, we need to add two overall dynamical processes in the first and third steps:
$\begin{eqnarray}\begin{array}{rcl}\eta (x,y) & = & [q(x)\,c(x)]\,\times \,[\bar{q}(y)\,\bar{c}(y)]\\ \to \eta ({x}^{{\prime} },x;{y}^{{\prime} },y) & =\, & [q({x}^{{\prime} })\,c(x)]\,\times \,[\bar{q}({y}^{{\prime} })\,\bar{c}(y)]\\ \to \theta (y,x;{y}^{{\prime} },{x}^{{\prime} }) & =\, & [\bar{c}(y)\,c(x)]\,\times \,[\bar{q}({y}^{{\prime} })\,q({x}^{{\prime} })]\\ \to \theta ({x}^{{\prime\prime} };{y}^{{\prime\prime} }) & = & [\bar{c}({x}^{{\prime\prime} })\,c({x}^{{\prime\prime} })]\,+\,[\bar{q}({y}^{{\prime\prime} })\,q({y}^{{\prime\prime} })],\end{array}\end{eqnarray}$
which will be discussed in detail in the next section. The second step is the Fierz rearrangement whose explicit expressions are given as follows.Altogether, we obtain the following relation between the currents ${\eta }_{\mu }^{i}({x}^{{\prime} },x;{y}^{{\prime} },y)$ and ${\theta }_{\mu }^{i}(y,x;{y}^{{\prime} },{x}^{{\prime} })$:
$\begin{eqnarray}\left(\begin{array}{c}{\eta }_{\mu }^{1}\\ {\eta }_{\mu }^{2}\\ {\eta }_{\mu }^{3}\\ {\eta }_{\mu }^{4}\\ {\eta }_{\mu }^{5}\\ {\eta }_{\mu }^{6}\\ {\eta }_{\mu }^{7}\\ {\eta }_{\mu }^{8}\end{array}\right)=\left(\begin{array}{cccccccc}-1/2 & 1/2 & -{\rm{i}}/2 & {\rm{i}}/2 & 0 & 0 & 0 & 0\\ -1/6 & 1/6 & -{\rm{i}}/6 & {\rm{i}}/6 & -1/4 & 1/4 & -{\rm{i}}/4 & {\rm{i}}/4\\ -3{\rm{i}}/2 & -3{\rm{i}}/2 & 1/2 & 1/2 & 0 & 0 & 0 & 0\\ -{\rm{i}}/2 & -{\rm{i}}/2 & 1/6 & 1/6 & -3{\rm{i}}/4 & -3{\rm{i}}/4 & 1/4 & 1/4\\ -1/2 & -1/2 & {\rm{i}}/2 & {\rm{i}}/2 & 0 & 0 & 0 & 0\\ -1/6 & -1/6 & {\rm{i}}/6 & {\rm{i}}/6 & -1/4 & -1/4 & {\rm{i}}/4 & {\rm{i}}/4\\ -3{\rm{i}}/2 & 3{\rm{i}}/2 & -1/2 & 1/2 & 0 & 0 & 0 & 0\\ -{\rm{i}}/2 & {\rm{i}}/2 & -1/6 & 1/6 & -3{\rm{i}}/4 & 3{\rm{i}}/4 & -1/4 & 1/4\end{array}\right)\times \left(\begin{array}{c}{\theta }_{\mu }^{1}\\ {\theta }_{\mu }^{2}\\ {\theta }_{\mu }^{3}\\ {\theta }_{\mu }^{4}\\ {\theta }_{\mu }^{5}\\ {\theta }_{\mu }^{6}\\ {\theta }_{\mu }^{7}\\ {\theta }_{\mu }^{8}\end{array}\right),\end{eqnarray}$
the following relation between ${\eta }_{\mu }^{i}({x}^{{\prime} },x;{y}^{{\prime} },y)$ and ${\xi }_{\mu }^{i}(y,{x}^{{\prime} };{y}^{{\prime} },x)$: $\begin{eqnarray}\left(\begin{array}{c}{\eta }_{\mu }^{1}\\ {\eta }_{\mu }^{2}\\ {\eta }_{\mu }^{3}\\ {\eta }_{\mu }^{4}\\ {\eta }_{\mu }^{5}\\ {\eta }_{\mu }^{6}\\ {\eta }_{\mu }^{7}\\ {\eta }_{\mu }^{8}\end{array}\right)=\left(\begin{array}{cccccccc}1/6 & 0 & 0 & -{\rm{i}}/6 & 1/4 & 0 & 0 & -{\rm{i}}/4\\ 1/2 & 0 & 0 & -{\rm{i}}/2 & 0 & 0 & 0 & 0\\ 0 & -1/6 & {\rm{i}}/2 & 0 & 0 & -1/4 & 3{\rm{i}}/4 & 0\\ 0 & -1/2 & 3{\rm{i}}/2 & 0 & 0 & 0 & 0 & 0\\ 0 & {\rm{i}}/6 & -1/6 & 0 & 0 & {\rm{i}}/4 & -1/4 & 0\\ 0 & {\rm{i}}/2 & -1/2 & 0 & 0 & 0 & 0 & 0\\ -{\rm{i}}/2 & 0 & 0 & 1/6 & -3{\rm{i}}/4 & 0 & 0 & 1/4\\ -3{\rm{i}}/2 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0\end{array}\right)\times \left(\begin{array}{c}{\xi }_{\mu }^{1}\\ {\xi }_{\mu }^{2}\\ {\xi }_{\mu }^{3}\\ {\xi }_{\mu }^{4}\\ {\xi }_{\mu }^{5}\\ {\xi }_{\mu }^{6}\\ {\xi }_{\mu }^{7}\\ {\xi }_{\mu }^{8}\end{array}\right),\end{eqnarray}$
the following relation among ${\eta }_{\mu }^{i}({x}^{{\prime} },x;{y}^{{\prime} },y)$, ${\xi }_{\mu }^{1,2,3,4}(y,{x}^{{\prime} };{y}^{{\prime} },x)$, and ${\theta }_{\mu }^{1,2,3,4}(y,x;{y}^{{\prime} },{x}^{{\prime} })$: $\begin{eqnarray}\left(\begin{array}{c}{\eta }_{\mu }^{1}\\ {\eta }_{\mu }^{2}\\ {\eta }_{\mu }^{3}\\ {\eta }_{\mu }^{4}\\ {\eta }_{\mu }^{5}\\ {\eta }_{\mu }^{6}\\ {\eta }_{\mu }^{7}\\ {\eta }_{\mu }^{8}\end{array}\right)=\left(\begin{array}{cccccccc}0 & 0 & 0 & 0 & -1/2 & 1/2 & -{\rm{i}}/2 & {\rm{i}}/2\\ 1/2 & 0 & 0 & -{\rm{i}}/2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -3{\rm{i}}/2 & -3{\rm{i}}/2 & 1/2 & 1/2\\ 0 & -1/2 & 3{\rm{i}}/2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/2 & -1/2 & {\rm{i}}/2 & {\rm{i}}/2\\ 0 & {\rm{i}}/2 & -1/2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -3{\rm{i}}/2 & 3{\rm{i}}/2 & -1/2 & 1/2\\ -3{\rm{i}}/2 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0\end{array}\right)\times \left(\begin{array}{c}{\xi }_{\mu }^{1}\\ {\xi }_{\mu }^{2}\\ {\xi }_{\mu }^{3}\\ {\xi }_{\mu }^{4}\\ {\theta }_{\mu }^{1}\\ {\theta }_{\mu }^{2}\\ {\theta }_{\mu }^{3}\\ {\theta }_{\mu }^{4}\end{array}\right),\end{eqnarray}$
and the following relation between ${\xi }_{\mu }^{i}(y,{x}^{{\prime} };{y}^{{\prime} },x)$ and ${\theta }_{\mu }^{i}(y,x;{y}^{{\prime} },{x}^{{\prime} })$: $\begin{eqnarray}\left(\begin{array}{c}{\xi }_{\mu }^{1}\\ {\xi }_{\mu }^{2}\\ {\xi }_{\mu }^{3}\\ {\xi }_{\mu }^{4}\\ {\xi }_{\mu }^{5}\\ {\xi }_{\mu }^{6}\\ {\xi }_{\mu }^{7}\\ {\xi }_{\mu }^{8}\end{array}\right)=\left(\begin{array}{cccccccc}1/6 & -1/6 & -{\rm{i}}/6 & {\rm{i}}/6 & 1/4 & -1/4 & -{\rm{i}}/4 & {\rm{i}}/4\\ {\rm{i}}/2 & {\rm{i}}/2 & 1/6 & 1/6 & 3{\rm{i}}/4 & 3{\rm{i}}/4 & 1/4 & 1/4\\ -1/6 & -1/6 & -{\rm{i}}/6 & -{\rm{i}}/6 & -1/4 & -1/4 & -{\rm{i}}/4 & -{\rm{i}}/4\\ -{\rm{i}}/2 & {\rm{i}}/2 & 1/6 & -1/6 & -3{\rm{i}}/4 & 3{\rm{i}}/4 & 1/4 & -1/4\\ 8/9 & -8/9 & -8{\rm{i}}/9 & 8{\rm{i}}/9 & -1/6 & 1/6 & {\rm{i}}/6 & -{\rm{i}}/6\\ 8{\rm{i}}/3 & 8{\rm{i}}/3 & 8/9 & 8/9 & -{\rm{i}}/2 & -{\rm{i}}/2 & -1/6 & -1/6\\ -8/9 & -8/9 & -8{\rm{i}}/9 & -8{\rm{i}}/9 & 1/6 & 1/6 & {\rm{i}}/6 & {\rm{i}}/6\\ -8{\rm{i}}/3 & 8{\rm{i}}/3 & 8/9 & -8/9 & {\rm{i}}/2 & -{\rm{i}}/2 & -1/6 & 1/6\end{array}\right)\times \left(\begin{array}{c}{\theta }_{\mu }^{1}\\ {\theta }_{\mu }^{2}\\ {\theta }_{\mu }^{3}\\ {\theta }_{\mu }^{4}\\ {\theta }_{\mu }^{5}\\ {\theta }_{\mu }^{6}\\ {\theta }_{\mu }^{7}\\ {\theta }_{\mu }^{8}\end{array}\right).\end{eqnarray}$
2.5. Isospin of the X(3872) and decay constants
In the present study we shall first use isoscalar tetraquark currents to study decay properties of the X(3872), for example,
$\begin{eqnarray}\begin{array}{l}{\eta }_{\mu }^{1}(I=0)=\displaystyle \frac{1}{\sqrt{2}}\times \left({\eta }_{\mu }^{1}([{uc}][\bar{u}\bar{c}])+{\eta }_{\mu }^{1}([{dc}][\bar{d}\bar{c}])\right)\\ \quad =\displaystyle \frac{1}{\sqrt{2}}\times \left({u}_{a}^{{\rm{T}}}{\mathbb{C}}{\gamma }_{\mu }{c}_{b}\,{\bar{u}}_{a}{\gamma }_{5}{\mathbb{C}}{\bar{c}}_{b}^{{\rm{T}}}+\{{\gamma }_{\mu }\leftrightarrow {\gamma }_{5}\}\right.\\ \quad \left.+\{u/\bar{u}\to d/\bar{d}\}\right).\end{array}\end{eqnarray}$
Hence, we need couplings of light isoscalar meson operators to light isoscalar meson states, which are summarized in table 1. We also need couplings of charmonium operators to charmonium states as well as those of charmed meson operators to charmed meson states, which are also summarized in table 1. We refer to [49] for detailed discussions.Table 1. Couplings of meson operators to meson states. All the light isovector meson operators ${J}_{(\mu \nu )}^{S/P/V/A/T}$ have the quark content $\bar{q}{\rm{\Gamma }}q=\left(\bar{u}{\rm{\Gamma }}u-\bar{d}{\rm{\Gamma }}d\right)/\sqrt{2}$, and all the light isoscalar meson operators ${P}_{(\mu \nu )}^{S/P/V/A/T}$ have the quark content $\bar{q}{\rm{\Gamma }}q=\left(\bar{u}{\rm{\Gamma }}u+\bar{d}{\rm{\Gamma }}d\right)/\sqrt{2}$. Color indices are omitted for simplicity. |
| Operators | IGJPC | Mesons | IGJPC | Couplings | Decay Constants |
|---|---|---|---|---|---|
| ${J}^{S}=\bar{q}q$ | 1−0++ | — | 1−0++ | — | — |
| ${J}^{P}=\bar{q}{\rm{i}}{\gamma }_{5}q$ | 1−0−+ | π0 | 1−0−+ | ⟨0∣JP∣π0⟩ = λπ | ${\lambda }_{\pi }=\displaystyle \frac{{f}_{\pi }{m}_{\pi }^{2}}{{m}_{u}+{m}_{d}}$ |
| ${J}_{\mu }^{V}=\bar{q}{\gamma }_{\mu }q$ | 1+1− | ρ0 | 1+1− | $\langle 0| {J}_{\mu }^{V}| {\rho }^{0}\rangle ={m}_{\rho }{f}_{\rho }{\epsilon }_{\mu }$ | fρ = 216 MeV [94] |
| ${J}_{\mu }^{A}=\bar{q}{\gamma }_{\mu }{\gamma }_{5}q$ | 1−1++ | π0 | 1−0−+ | $\langle 0| {J}_{\mu }^{A}| {\pi }^{0}\rangle ={\rm{i}}{p}_{\mu }{f}_{\pi }$ | fπ = 130.2 MeV [2] |
| a1(1260) | 1−1++ | $\langle 0| {J}_{\mu }^{A}| {a}_{1}\rangle ={m}_{{a}_{1}}{f}_{{a}_{1}}{\epsilon }_{\mu }$ | ${f}_{{a}_{1}}=254$ MeV [95] | ||
| ${J}_{\mu \nu }^{T}=\bar{q}{\sigma }_{\mu \nu }q$ | 1+1±−−− | ρ0 | 1+1− | $\langle 0| {J}_{\mu \nu }^{T}| {\rho }^{0}\rangle ={\rm{i}}{f}_{\rho }^{T}({p}_{\mu }{\epsilon }_{\nu }-{p}_{\nu }{\epsilon }_{\mu })$ | ${f}_{\rho }^{T}=159$ MeV [94] |
| b1(1235) | 1+1+− | $\langle 0| {J}_{\mu \nu }^{T}| {b}_{1}\rangle ={\rm{i}}{f}_{{b}_{1}}^{T}{\epsilon }_{\mu \nu \alpha \beta }{\epsilon }^{\alpha }{p}^{\beta }$ | ${f}_{{b}_{1}}^{T}=180$ MeV [96] | ||
| ${P}^{S}=\bar{q}q$ | 0+0++ | f0(500) (?) | 0+0++ | $\langle 0| {P}^{S}| {f}_{0}\rangle ={m}_{{f}_{0}}{f}_{{f}_{0}}$ | ${f}_{{f}_{0}}\sim 380$ MeV (?) |
| ${P}^{P}=\bar{q}{\rm{i}}{\gamma }_{5}q$ | 0+0−+ | η | 0+0−+ | — | — |
| ${P}_{\mu }^{V}=\bar{q}{\gamma }_{\mu }q$ | 0−1− | ω | 0−1− | $\langle 0| {P}_{\mu }^{V}| \omega \rangle ={m}_{\omega }{f}_{\omega }{\epsilon }_{\mu }$ | fω ≈ fρ = 216 MeV [94] |
| ${P}_{\mu }^{A}=\bar{q}{\gamma }_{\mu }{\gamma }_{5}q$ | 0+1++ | η | 0+0−+ | $\langle 0| {P}_{\mu }^{A}| \eta \rangle ={\rm{i}}{p}_{\mu }{f}_{\eta }$ | fη = 97 MeV [97, 98] |
| f1(1285) | 0+1++ | — | — | ||
| ${P}_{\mu \nu }^{T}=\bar{q}{\sigma }_{\mu \nu }q$ | 0−1±−−− | ω | 0−1− | $\langle 0| {P}_{\mu \nu }^{T}| \omega \rangle ={\rm{i}}{f}_{\omega }^{T}({p}_{\mu }{\epsilon }_{\nu }-{p}_{\nu }{\epsilon }_{\mu })$ | ${f}_{\omega }^{T}\approx {f}_{\rho }^{T}=159$ MeV [94] |
| h1(1170) | 0−1+− | $\langle 0| {P}_{\mu \nu }^{T}| {h}_{1}\rangle ={\rm{i}}{f}_{{h}_{1}}^{T}{\epsilon }_{\mu \nu \alpha \beta }{\epsilon }^{\alpha }{p}^{\beta }$ | ${f}_{{h}_{1}}^{T}\approx {f}_{{b}_{1}}^{T}=180$ MeV [96] | ||
| ${I}^{S}=\bar{c}c$ | 0+0++ | χc0(1P) | 0+0++ | $\langle 0| {I}^{S}| {\chi }_{c0}\rangle ={m}_{{\chi }_{c0}}{f}_{{\chi }_{c0}}$ | ${f}_{{\chi }_{c0}}=343$ MeV [99] |
| ${I}^{P}=\bar{c}{\rm{i}}{\gamma }_{5}c$ | 0+0−+ | ηc | 0+0−+ | $\langle 0| {I}^{P}| {\eta }_{c}\rangle ={\lambda }_{{\eta }_{c}}$ | ${\lambda }_{{\eta }_{c}}=\displaystyle \frac{{f}_{{\eta }_{c}}{m}_{{\eta }_{c}}^{2}}{2{m}_{c}}$ |
| ${I}_{\mu }^{V}=\bar{c}{\gamma }_{\mu }c$ | 0−1− | J/ψ | 0−1− | $\langle 0| {I}_{\mu }^{V}| J/\psi \rangle ={m}_{J/\psi }{f}_{J/\psi }{\epsilon }_{\mu }$ | fJ/ψ = 418 MeV [100] |
| ${I}_{\mu }^{A}=\bar{c}{\gamma }_{\mu }{\gamma }_{5}c$ | 0+1++ | ηc | 0+0−+ | $\langle 0| {I}_{\mu }^{A}| {\eta }_{c}\rangle ={\rm{i}}{p}_{\mu }{f}_{{\eta }_{c}}$ | ${f}_{{\eta }_{c}}=387$ MeV [100] |
| χc1(1P) | 0+1++ | $\langle 0| {I}_{\mu }^{A}| {\chi }_{c1}\rangle ={m}_{{\chi }_{c1}}{f}_{{\chi }_{c1}}{\epsilon }_{\mu }$ | ${f}_{{\chi }_{c1}}=335$ MeV [101] | ||
| ${I}_{\mu \nu }^{T}=\bar{c}{\sigma }_{\mu \nu }c$ | 0−1±−−− | J/ψ | 0−1− | $\langle 0| {I}_{\mu \nu }^{T}| J/\psi \rangle ={\rm{i}}{f}_{J/\psi }^{T}({p}_{\mu }{\epsilon }_{\nu }-{p}_{\nu }{\epsilon }_{\mu })$ | ${f}_{J/\psi }^{T}=410$ MeV [100] |
| hc(1P) | 0−1+− | $\langle 0| {I}_{\mu \nu }^{T}| {h}_{c}\rangle ={\rm{i}}{f}_{{h}_{c}}^{T}{\epsilon }_{\mu \nu \alpha \beta }{\epsilon }^{\alpha }{p}^{\beta }$ | ${f}_{{h}_{c}}^{T}=235$ MeV [100] | ||
| ${O}^{S}=\bar{q}c$ | 0+ | ${D}_{0}^{* }$ | 0+ | $\langle 0| {O}^{S}| {D}_{0}^{* }\rangle ={m}_{{D}_{0}^{* }}{f}_{{D}_{0}^{* }}$ | ${f}_{{D}_{0}^{* }}=410$ MeV [102] |
| ${O}^{P}=\bar{q}{\rm{i}}{\gamma }_{5}c$ | 0− | D | 0− | ⟨0∣OP∣D⟩ = λD | ${\lambda }_{D}=\displaystyle \frac{{f}_{D}{m}_{D}^{2}}{{m}_{c}+{m}_{d}}$ |
| ${O}_{\mu }^{V}=\bar{c}{\gamma }_{\mu }q$ | 1− | ${\bar{D}}^{* }$ | 1− | $\langle 0| {O}_{\mu }^{V}| {\bar{D}}^{* }\rangle ={m}_{{D}^{* }}{f}_{{D}^{* }}{\epsilon }_{\mu }$ | ${f}_{{D}^{* }}=253$ MeV [103] |
| ${O}_{\mu }^{A}=\bar{c}{\gamma }_{\mu }{\gamma }_{5}q$ | 1+ | $\bar{D}$ | 0− | $\langle 0| {O}_{\mu }^{A}| \bar{D}\rangle ={\rm{i}}{p}_{\mu }{f}_{D}$ | fD = 211.9 MeV [2] |
| D1 | 1+ | $\langle 0| {O}_{\mu }^{A}| {D}_{1}\rangle ={m}_{{D}_{1}}{f}_{{D}_{1}}{\epsilon }_{\mu }$ | ${f}_{{D}_{1}}=356$ MeV [102] | ||
| ${O}_{\mu \nu }^{T}=\bar{q}{\sigma }_{\mu \nu }c$ | 1± | ${\bar{D}}^{* }$ | 1− | $\langle 0| {O}_{\mu \nu }^{T}| {D}^{* }\rangle ={\rm{i}}{f}_{{D}^{* }}^{T}({p}_{\mu }{\epsilon }_{\nu }-{p}_{\nu }{\epsilon }_{\mu })$ | ${f}_{{D}^{* }}^{T}\approx 220$ MeV [49] |
| — | 1+ | — | — |
Since light scalar mesons have a complicated nature [77], couplings of the light scalar-isoscalar meson operator ${P}^{S}=\left(\bar{u}u+\bar{d}d\right)/\sqrt{2}$ to f0 mesons are quite ambiguous, where f0 can be either the f0(500) or f0(1370), etc. In this paper we shall simply use the f0(500) meson to estimate relevant partial decay widths, whose coupling to PS is assumed to be
$\begin{eqnarray}\langle 0| \displaystyle \frac{\bar{u}u+\bar{d}d}{\sqrt{2}}| {f}_{0}(p)\rangle ={m}_{{f}_{0}}{f}_{{f}_{0}}.\end{eqnarray}$
In the present study we simply average among the decay constants ${f}_{{\chi }_{c0}}$ and ${f}_{{D}_{0}^{* }}$ to obtain $\begin{eqnarray}{f}_{{f}_{0}}\sim 380\,\mathrm{MeV}.\end{eqnarray}$
The isospin breaking effect of the X(3872) is significant and important to understand its nature. There have been many studies on this, and we refer to reviews [3–10] for detailed discussions. In the present study we shall study this effect by freely choosing the quark content of the X(3872) [12, 65, 66], for example,5 , so that the X(3872) is assumed not to be a purely isoscalar state. To study this, we need couplings of light isovector meson operators to light isovector meson states, which are also summarized in table 1.
$\begin{eqnarray}\begin{array}{l}{\eta }_{\mu }^{1}(\theta /{\theta }^{{\prime} })=\cos \theta \,{\eta }_{\mu }^{1}([{uc}][\bar{u}\bar{c}])+\sin \theta \,{\eta }_{\mu }^{1}([{dc}][\bar{d}\bar{c}])\\ \quad =\,\cos \theta \times \left({u}_{a}^{{\rm{T}}}{\mathbb{C}}{\gamma }_{\mu }{c}_{b}\,{\bar{u}}_{a}{\gamma }_{5}{\mathbb{C}}{\bar{c}}_{b}^{{\rm{T}}}+\{{\gamma }_{\mu }\leftrightarrow {\gamma }_{5}\}\right)\\ \quad +\,\sin \theta \times \{u/\bar{u}\to d/\bar{d}\}\\ \quad \Rightarrow \,\cos {\theta }^{{\prime} }\,{\eta }_{\mu }^{1}(I=0)+\sin {\theta }^{{\prime} }\,{\eta }_{\mu }^{1}(I=1),\end{array}\end{eqnarray}$
where θ and ${\theta }^{{\prime} }$ are the two related mixing angles. We shall fine-tune them to be different from $\theta =45^\circ /{\theta }^{{\prime} }=0^\circ $ in section 3. Decay properties of the X(3872) as a diquark-antidiquark state.
In this section and the next we shall use equations (10 )–(13 ) derived in the previous section to study decay properties of the X(3872) as a purely isoscalar state. Its two possible interpretations are: a) the compact tetraquark state of JPC = 1++ composed of a JP = 0+ diquark/antidiquark and a JP = 1+ antidiquark/diquark [12–17], i.e., $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle $ defined in equation (3 ); and (b) the $D{\bar{D}}^{* }$ hadronic molecular state of JPC = 1++ [18–24], i.e., $| D{\bar{D}}^{* };{1}^{++}\rangle $ defined in equation (6 ).
In this section we investigate the former compact tetraquark interpretation using the isoscalar current ${\eta }_{\mu }^{{ \mathcal X }}(x,y;I=0)$ defined in equation (4 ). We can transform it to ${\theta }_{\mu }^{i}(x,y;I=0)$ and ${\xi }_{\mu }^{i}(x,y;I=0)$ according to equations (10 )–(12 ), through which we study decay properties of the X(3872) as an isoscalar compact tetraquark state in the following subsections.
3.1. ${\eta }_{\mu }^{{ \mathcal X }}([{qc}][\bar{q}\bar{c}]) \rightarrow {\theta }_{\mu }^{i}([\bar{c}c]\,+\,[\bar{q}q])$
As depicted in figure 2, when the q and $\bar{q}$ quarks meet each other and the c and $\bar{c}$ quarks meet each other at the same time, a compact tetraquark state decays into one charmonium meson and one light meson:10 ):
$\begin{eqnarray}\begin{array}{l}[q(x)c(x)]\,[\bar{q}(y)\bar{c}(y)]\\ \Longrightarrow [q(x\to {y}^{{\prime} })\,c(x\to {x}^{{\prime} })]\,[\bar{q}(y\to {y}^{{\prime} })\,\bar{c}(y\to {x}^{{\prime} })]\\ \Longrightarrow [\bar{c}({x}^{{\prime} })c({x}^{{\prime} })]\,+\,[\bar{q}({y}^{{\prime} })q({y}^{{\prime} })].\end{array}\end{eqnarray}$
The first process is a dynamical process, and the second process can be described through the transformation ( $\begin{eqnarray}\begin{array}{l}{\eta }_{\mu }^{{ \mathcal X }}(x,y;I=0)\\ \Longrightarrow -\displaystyle \frac{1}{3}\,{\theta }_{\mu }^{1}({x}^{{\prime} },{y}^{{\prime} };I=0)+\displaystyle \frac{1}{3}\,{\theta }_{\mu }^{2}({x}^{{\prime} },{y}^{{\prime} };I=0)\\ -\displaystyle \frac{{\rm{i}}}{3}\,{\theta }_{\mu }^{3}({x}^{{\prime} },{y}^{{\prime} };I=0)+\displaystyle \frac{{\rm{i}}}{3}\,{\theta }_{\mu }^{4}({x}^{{\prime} },{y}^{{\prime} };I=0)\,+\,\cdots \\ =-\displaystyle \frac{1}{3}\,{I}^{S}({x}^{{\prime} })\,{P}_{\mu }^{A}({y}^{{\prime} })+\displaystyle \frac{1}{3}\,{I}_{\mu }^{A}({x}^{{\prime} })\,{P}^{S}({y}^{{\prime} })\\ +\displaystyle \frac{1}{6}\,{\epsilon }_{\mu \nu \rho \sigma }\,{I}^{V,\nu }({x}^{{\prime} })\,{P}^{T,\rho \sigma }({y}^{{\prime} })\\ -\displaystyle \frac{1}{6}\,{\epsilon }_{\mu \nu \rho \sigma }\,{I}^{T,\rho \sigma }({x}^{{\prime} })\,{P}^{V,\nu }({y}^{{\prime} })\,+\,\cdots ,\end{array}\end{eqnarray}$
where we have used $\begin{eqnarray}{\sigma }_{\mu \nu }{\gamma }_{5}=\displaystyle \frac{{\rm{i}}}{2}{\epsilon }_{\mu \nu \rho \sigma }{\sigma }^{\rho \sigma }.\end{eqnarray}$
In the above expression we keep only the direct fall-apart process described by ${\theta }_{\mu }^{1,2,3,4}$, but neglect the ${ \mathcal O }({\alpha }_{s})$ corrections described by ${\theta }_{\mu }^{5,6,7,8}$.
Figure 2. The decay of a compact tetraquark state into one charmonium meson and one light meson, which can happen through either (b) a direct fall-apart process, or (c) a process with gluons exchanged. |
Together with table 1, we extract the following decay channels:
| 1. The decay of $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle $ into χc0η is contributed by ${I}^{S}\times {P}_{\mu }^{A}$: $\begin{eqnarray}\begin{array}{l}\langle X(p,\epsilon )| {\chi }_{c0}({p}_{1})\,\eta ({p}_{2})\rangle \\ \approx -\displaystyle \frac{{\rm{i}}{c}_{1}}{3}\,{m}_{{\chi }_{c0}}{f}_{{\chi }_{c0}}{f}_{\eta }\,\epsilon \cdot {p}_{2}\equiv {g}_{{\chi }_{c0}\eta }\,\epsilon \cdot {p}_{2},\end{array}\end{eqnarray}$ where c1 is an overall factor, related to the coupling of ${\eta }_{\mu }^{{ \mathcal X }}(x,y)$ to the X(3872) as well as the dynamical process $(x,y)\Longrightarrow ({x}^{{\prime} },{y}^{{\prime} })$ shown in figure 2. This decay is kinematically forbidden. | |
| 2. According to ${I}^{S}\times {P}_{\mu }^{A}$, $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle $ can also decay into χc0f1(1285): $\begin{eqnarray}| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to {\chi }_{c0}{f}_{1}.\end{eqnarray}$ This decay is kinematically forbidden. | |
| 3. Decays of $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle $ into ηcf0(500) and χc1f0(500) are both contributed by ${I}_{\mu }^{A}\times {P}^{S}$: $\begin{eqnarray}\begin{array}{l}\langle X(p,\epsilon )| {\eta }_{c}({p}_{1})\,{f}_{0}({p}_{2})\rangle \\ \approx \,+\,\displaystyle \frac{{\rm{i}}{c}_{1}}{3}\,{m}_{{f}_{0}}{f}_{{f}_{0}}{f}_{{\eta }_{c}}\,\epsilon \cdot {p}_{1}\equiv {g}_{{\eta }_{c}{f}_{0}}\,\epsilon \cdot {p}_{1},\end{array}\end{eqnarray}$ $\begin{eqnarray}\begin{array}{l}\langle X(p,\epsilon )| {\chi }_{c1}({p}_{1},{\epsilon }_{1})\,{f}_{0}({p}_{2})\rangle \\ \approx \,+\,\displaystyle \frac{{c}_{1}}{3}\,{m}_{{f}_{0}}{f}_{{f}_{0}}{m}_{{\chi }_{c1}}{f}_{{\chi }_{c1}}\,\epsilon \cdot {\epsilon }_{1}\equiv {g}_{{\chi }_{c1}{f}_{0}}\,\epsilon \cdot {\epsilon }_{1}.\end{array}\end{eqnarray}$ Because it is difficult to observe the f0(500) in experiments, in the present study we shall calculate widths of the $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to {\eta }_{c}{f}_{0}\to {\eta }_{c}\pi \pi $ and $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to {\chi }_{c1}{f}_{0}\to {\chi }_{c1}\pi \pi $ processes. | |
| 4. The decay of $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle $ into J/ψω is contributed by both IV,ν × PT,ρσ and IT,ρσ × PV,ν: $\begin{eqnarray}\begin{array}{l}\langle X(p,\epsilon )| J/\psi ({p}_{1},{\epsilon }_{1})\,\omega ({p}_{2},{\epsilon }_{2})\rangle \\ \approx -\displaystyle \frac{{\rm{i}}{c}_{1}}{3}\,{\epsilon }_{\mu \nu \rho \sigma }{\epsilon }^{\mu }{\epsilon }_{1}^{\nu }{\epsilon }_{2}^{\rho }{p}_{2}^{\sigma }\,{m}_{J/\psi }{f}_{J/\psi }{f}_{\omega }^{T}\\ -\displaystyle \frac{{\rm{i}}{c}_{1}}{3}\,{\epsilon }_{\mu \nu \rho \sigma }{\epsilon }^{\mu }{\epsilon }_{1}^{\nu }{\epsilon }_{2}^{\rho }{p}_{1}^{\sigma }\,{m}_{\omega }{f}_{\omega }{f}_{J/\psi }^{T}\\ \equiv \,{g}_{\psi \omega }^{A}\,{\epsilon }_{\mu \nu \rho \sigma }{\epsilon }^{\mu }{\epsilon }_{1}^{\nu }{\epsilon }_{2}^{\rho }{p}_{2}^{\sigma }+{g}_{\psi \omega }^{B}\,{\epsilon }_{\mu \nu \rho \sigma }{\epsilon }^{\mu }{\epsilon }_{1}^{\nu }{\epsilon }_{2}^{\rho }{p}_{1}^{\sigma }.\end{array}\end{eqnarray}$ This decay is kinematically forbidden, but the $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \omega \to J/\psi \pi \pi \pi $ process is kinematically allowed. | |
| 5. According to IV,ν × PT,ρσ, $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle $ can also decay into J/ψh1(1170): $\begin{eqnarray}\begin{array}{l}\langle X(p,\epsilon )| J/\psi ({p}_{1},{\epsilon }_{1})\,{h}_{1}({p}_{2},{\epsilon }_{2})\rangle \\ \approx \,+\,\displaystyle \frac{{\rm{i}}{c}_{1}}{6}\,{\epsilon }_{\mu \nu \rho \sigma }{\epsilon }_{\rho \sigma \alpha \beta }{\epsilon }^{\mu }{\epsilon }_{1}^{\nu }{\epsilon }_{2}^{\alpha }{p}_{2}^{\beta }\,{m}_{J/\psi }{f}_{J/\psi }{f}_{{h}_{1}}^{T}\\ \equiv \,{g}_{\psi {h}_{1}}\,{\epsilon }_{\mu \nu \rho \sigma }{\epsilon }_{\rho \sigma \alpha \beta }{\epsilon }^{\mu }{\epsilon }_{1}^{\nu }{\epsilon }_{2}^{\alpha }{p}_{2}^{\beta }.\end{array}\end{eqnarray}$ This decay is kinematically forbidden, but the $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi {h}_{1}\to J/\psi \rho \pi \to J/\psi \pi \pi \pi $ process is kinematically allowed. | |
| 6. According to IT,ρσ × PV,ν, $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle $ can also decay into hcω: $\begin{eqnarray}| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to {h}_{c}\omega .\end{eqnarray}$ This decay is kinematically forbidden. |
Summarizing the above results, we obtain numericallyA .
$\begin{eqnarray}\begin{array}{l}{g}_{{\eta }_{c}{f}_{0}}\sim +{\rm{i}}{c}_{1}\,2.51\times {10}^{7}\,{\mathrm{MeV}}^{3},\\ {g}_{{\chi }_{c1}{f}_{0}}\sim +{c}_{1}\,0.76\times {10}^{11}\,{\mathrm{MeV}}^{4},\\ {g}_{\psi \omega }^{A}=-{\rm{i}}{c}_{1}\,6.86\times {10}^{7}\,{\mathrm{MeV}}^{3},\\ {g}_{\psi \omega }^{B}=-{\rm{i}}{c}_{1}\,2.31\times {10}^{7}\,{\mathrm{MeV}}^{3},\\ {g}_{\psi {h}_{1}}=+{\rm{i}}{c}_{1}\,3.88\times {10}^{7}\,{\mathrm{MeV}}^{3},\end{array}\end{eqnarray}$
from which we further obtain $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{ \mathcal B }(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to {\eta }_{c}{f}_{0}\to {\eta }_{c}\pi \pi )}{{ \mathcal B }(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \omega \to J/\psi \pi \pi \pi )}\sim 0.091,\\ \displaystyle \frac{{ \mathcal B }(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to {\chi }_{c1}{f}_{0}\to {\chi }_{c1}\pi \pi )}{{ \mathcal B }(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \omega \to J/\psi \pi \pi \pi )}\sim 0.086,\\ \displaystyle \frac{{ \mathcal B }(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi {h}_{1}\to J/\psi \pi \pi \pi )}{{ \mathcal B }(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \omega \to J/\psi \pi \pi \pi )}=1.4\times {10}^{-3}.\end{array}\end{eqnarray}$
Detailed calculations can be found in appendix 3.2. ${\eta }_{\mu }^{{ \mathcal X }}([{qc}][\bar{q}\bar{c}]) \rightarrow {\xi }_{\mu }^{i}([\bar{c}q]\,+\,[\bar{q}c])$
As depicted in figure 3, when the c and $\bar{q}$ quarks meet each other and the q and $\bar{c}$ quarks meet each other at the same time, a compact tetraquark state decays into two charmed mesons. This process can be described by the transformation (11 ):
$\begin{eqnarray}\begin{array}{l}{\eta }_{\mu }^{{ \mathcal X }}(x,y;I=0)\\ \Longrightarrow -\displaystyle \frac{1}{3}\,{\xi }_{\mu }^{1}({x}^{{\prime} },{y}^{{\prime} };I=0)+\displaystyle \frac{{\rm{i}}}{3}\,{\xi }_{\mu }^{4}({x}^{{\prime} },{y}^{{\prime} };I=0)\,+\,\cdots \\ =\,+\,\displaystyle \frac{{\rm{i}}}{3}\,{O}_{\mu }^{V}({x}^{{\prime} })\,{O}^{P}({y}^{{\prime} })+\displaystyle \frac{{\rm{i}}}{3}\,{O}^{A,\nu }({x}^{{\prime} })\,{O}_{\mu \nu }^{T}({y}^{{\prime} })\\ +\,{\rm{c}}.{\rm{c}}.\,+\,\cdots .\end{array}\end{eqnarray}$
Again, we keep only the direct fall-apart process described by ${\xi }_{\mu }^{1,4}$, but neglect the ${ \mathcal O }({\alpha }_{s})$ corrections described by ${\xi }_{\mu }^{5,8}$.
Figure 3. The decay of a compact tetraquark state into two charmed mesons, which can happen through either (b) a direct fall-apart process, or (c) a process with gluons exchanged. |
The decay of $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle $ into the $D{\bar{D}}^{* }$ final state is contributed by both ${O}_{\mu }^{V}\times {O}^{P}$ and ${O}^{A,\nu }\times {O}_{\mu \nu }^{T}$:
$\begin{eqnarray}\begin{array}{l}\langle X(p,\epsilon )| {D}^{0}({p}_{1}){\bar{D}}^{* 0}({p}_{2},{\epsilon }_{2})\rangle \\ \approx \,+\,\displaystyle \frac{{\rm{i}}{c}_{2}}{3\sqrt{2}}\,{\lambda }_{D}{m}_{{D}^{* }}{f}_{{D}^{* }}\,\epsilon \cdot {\epsilon }_{2}\\ -\displaystyle \frac{{\rm{i}}{c}_{2}}{3\sqrt{2}}\,{f}_{D}{f}_{{D}^{* }}^{T}\,(\epsilon \cdot {p}_{2}\,{\epsilon }_{2}\cdot {p}_{1}-{p}_{1}\cdot {p}_{2}\,\epsilon \cdot {\epsilon }_{2})\\ \equiv \,{g}_{D{\bar{D}}^{* }}^{S}\,\epsilon \cdot {\epsilon }_{2}+{g}_{D{\bar{D}}^{* }}^{D}\,(\epsilon \cdot {p}_{2}\,{\epsilon }_{2}\cdot {p}_{1}-{p}_{1}\cdot {p}_{2}\,\epsilon \cdot {\epsilon }_{2}),\end{array}\end{eqnarray}$
where c2 is an overall factor, related to the coupling of ${\eta }_{\mu }^{{ \mathcal X }}(x,y)$ to the X(3872) as well as the dynamical process $(x,y)\Longrightarrow ({x}^{{\prime} },{y}^{{\prime} })$ shown in figure 3. This decay might be kinematically forbidden, but the $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to {D}^{0}{\bar{D}}^{* 0}+{D}^{* 0}{\bar{D}}^{0}\to {D}^{0}{\bar{D}}^{0}{\pi }^{0}$ process is surely kinematically allowed. The two coupling constants ${g}_{D{\bar{D}}^{* }}^{S}$ and ${g}_{D{\bar{D}}^{* }}^{D}$ are defined for the S- and D-wave $X(3872)\to D{\bar{D}}^{* }$ decays: $\begin{eqnarray}{{ \mathcal L }}_{D{\bar{D}}^{* }}^{S}={g}_{D{\bar{D}}^{* }}^{S}\,{X}^{\mu }\,{D}^{0}\,{\bar{D}}_{\mu }^{* 0}\,+\,\cdots ,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{{ \mathcal L }}_{D{\bar{D}}^{* }}^{D}={g}_{D{\bar{D}}^{* }}^{D}\times \left({g}^{\mu \sigma }{g}^{\nu \rho }-{g}^{\mu \nu }{g}^{\rho \sigma }\right)\\ \times {X}^{\mu }\,{\partial }_{\rho }{D}^{0}\,{\partial }_{\sigma }{\bar{D}}_{\nu }^{* 0}+\cdots .\end{array}\end{eqnarray}$
Numerically, we obtainA . As proposed in [61], when the X(3872) decays, a constituent of the diquark must tunnel through the barrier of the diquark-antidiquark potential. However, this tunnelling for heavy quarks is exponentially suppressed compared to that for light quarks, so the compact tetraquark couplings are expected to favour the open charm modes with respect to charmonium ones. Accordingly, c2 may be significantly larger than c1, so that $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle $ may mainly decay into the ${D}^{0}{\bar{D}}^{0}{\pi }^{0}$ final state.
$\begin{eqnarray}\begin{array}{rcl}{g}_{D{\bar{D}}^{* }}^{S} & = & +{\rm{i}}{c}_{2}\,0.69\times {10}^{11}\,{\mathrm{MeV}}^{4},\\ {g}_{D{\bar{D}}^{* }}^{D} & = & -{\rm{i}}{c}_{2}\,1.10\times {10}^{4}\,{\mathrm{MeV}}^{2}.\end{array}\end{eqnarray}$
Comparing this decay with the $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \omega \to J/\psi \pi \pi \pi $ decay studied in the previous subsection, we further obtain $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{ \mathcal B }(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to {D}^{0}{\bar{D}}^{* 0}+{D}^{* 0}{\bar{D}}^{0}\to {D}^{0}{\bar{D}}^{0}{\pi }^{0})}{{ \mathcal B }(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \omega \to J/\psi \pi \pi \pi )}\\ =\,0.32\times \displaystyle \frac{{c}_{2}^{2}}{{c}_{1}^{2}}.\end{array}\end{eqnarray}$
Detailed calculations can also be found in appendix 3.3. ${\eta }_{\mu }^{{ \mathcal X }}([{qc}][\bar{q}\bar{c}]) \rightarrow {\theta }_{\mu }^{1,2,3,4}([\bar{c}c]\,+\,[\bar{q}q])+{\xi }_{\mu }^{1,2,3,4}([\bar{c}q]\,+\,[\bar{q}c])$
If the above two processes investigated in section 3.1 and section 3.2 happen at the same time, i.e., $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle $ decays into one charmonium meson and one light meson as well as two charmed mesons simultaneously, we can use the transformation (12 ), which contains the color-singlet-color-singlet currents ${\theta }_{\mu }^{1,2,3,4}$ and ${\xi }_{\mu }^{1,2,3,4}$ together:19 ) and (30 ), we obtain the same relative branching ratios as section 3.1 and section 3.2 , just with the overall factors c1 and c2 replaced by others.
$\begin{eqnarray}\begin{array}{l}{\eta }_{\mu }^{{ \mathcal X }}(x,y;I=0)\\ \Longrightarrow -\displaystyle \frac{1}{2}\,{\theta }_{\mu }^{1}({x}^{{\prime} },{y}^{{\prime} };I=0)+\displaystyle \frac{1}{2}\,{\theta }_{\mu }^{2}({x}^{{\prime} },{y}^{{\prime} };I=0)\\ -\displaystyle \frac{{\rm{i}}}{2}\,{\theta }_{\mu }^{3}({x}^{{\prime} },{y}^{{\prime} };I=0)+\displaystyle \frac{{\rm{i}}}{2}\,{\theta }_{\mu }^{4}({x}^{{\prime} },{y}^{{\prime} };I=0)\\ -\displaystyle \frac{1}{2}\,{\xi }_{\mu }^{1}({x}^{{\prime\prime} },{y}^{{\prime\prime} };I=0)+\displaystyle \frac{{\rm{i}}}{2}\,{\xi }_{\mu }^{4}({x}^{{\prime\prime} },{y}^{{\prime\prime} };I=0).\end{array}\end{eqnarray}$
In the above expression we keep all terms, and there is no ⋯any more. Comparing this equation with equations (4. Decay properties of the X(3872) as a hadronic molecular state
Another possible interpretation of the X(3872) is the $D{\bar{D}}^{* }$ hadronic molecular state of JPC = 1++ [18–24], i.e., $| D{\bar{D}}^{* };{1}^{++}\rangle $ defined in equation (6 ). Its relevant isoscalar current ${\xi }_{\mu }^{{ \mathcal X }}(x,y;I=0)$ is given in equation (7 ). We can transform it to ${\theta }_{\mu }^{i}(x,y;I=0)$ according to equation (13 ), through which we study decay properties of the X(3872) as an isoscalar hadronic molecular state in the following subsections.
4.1. ${\xi }_{\mu }^{{ \mathcal X }}([\bar{c}q][\bar{q}c])\to {\theta }_{\mu }^{i}([\bar{c}c]\,+\,[\bar{q}q])$
As depicted in figure 4, when the q and $\bar{q}$ quarks meet each other and the c and $\bar{c}$ quarks meet each other at the same time, a hadronic molecular state decays into one charmonium meson and one light meson. This process can be described by the transformation (13 ):
$\begin{eqnarray}\begin{array}{l}{\xi }_{\mu }^{{ \mathcal X }}(x,y;I=0)\\ \Longrightarrow +\displaystyle \frac{1}{6}\,{\theta }_{\mu }^{1}({x}^{{\prime} },{y}^{{\prime} };I=0)-\displaystyle \frac{1}{6}\,{\theta }_{\mu }^{2}({x}^{{\prime} },{y}^{{\prime} };I=0)\\ -\displaystyle \frac{{\rm{i}}}{6}\,{\theta }_{\mu }^{3}({x}^{{\prime} },{y}^{{\prime} };I=0)+\displaystyle \frac{{\rm{i}}}{6}\,{\theta }_{\mu }^{4}({x}^{{\prime} },{y}^{{\prime} };I=0)\,+\,\cdots \\ =+\displaystyle \frac{1}{6}\,{I}^{S}({x}^{{\prime} })\,{P}_{\mu }^{A}({y}^{{\prime} })-\displaystyle \frac{1}{6}\,{I}_{\mu }^{A}({x}^{{\prime} })\,{P}^{S}({y}^{{\prime} })\\ +\displaystyle \frac{1}{12}\,{\epsilon }_{\mu \nu \rho \sigma }\,{I}^{V,\nu }({x}^{{\prime} })\,{P}^{T,\rho \sigma }({y}^{{\prime} })\\ -\displaystyle \frac{1}{12}\,{\epsilon }_{\mu \nu \rho \sigma }\,{I}^{T,\rho \sigma }({x}^{{\prime} })\,{P}^{V,\nu }({y}^{{\prime} })\,+\,\cdots .\end{array}\end{eqnarray}$
Here we keep only the direct fall-apart process described by ${\theta }_{\mu }^{1,2,3,4}$, but neglect the ${ \mathcal O }({\alpha }_{s})$ corrections described by ${\theta }_{\mu }^{5,6,7,8}$.
Figure 4. The decay of a hadronic molecular state into one charmonium meson and one light meson, which can happen through either (b) a direct fall-apart process, or (c) a process with gluons exchanged. |
We repeat the same procedures as those done in section 3.1 , and extract the following coupling constants from this transformation:
$\begin{eqnarray}\begin{array}{rcl}{h}_{{\eta }_{c}{f}_{0}} & \sim & -{\rm{i}}{c}_{1}\,1.26\times {10}^{7}\,{\mathrm{MeV}}^{3},\\ {h}_{{\chi }_{c1}{f}_{0}} & \sim & -{c}_{1}\,0.38\times {10}^{11}\,{\mathrm{MeV}}^{4},\\ {h}_{\psi \omega }^{A} & = & -{\rm{i}}{c}_{4}\,3.43\times {10}^{7}\,{\mathrm{MeV}}^{3},\\ {h}_{\psi \omega }^{B} & = & -{\rm{i}}{c}_{4}\,1.16\times {10}^{7}\,{\mathrm{MeV}}^{3},\\ {h}_{\psi {h}_{1}} & = & +{\rm{i}}{c}_{4}\,1.94\times {10}^{7}\,{\mathrm{MeV}}^{3}.\end{array}\end{eqnarray}$
They are defined for the $| D{\bar{D}}^{* };{1}^{++}\rangle \to {\eta }_{c}{f}_{0}$, χc1f0, J/ψω, and J/ψh1 decays. They all contain an overall factor c4, which is related to the coupling of ${\xi }_{\mu }^{{ \mathcal X }}(x,y)$ to the X(3872) as well as the dynamical process $(x,y)\Longrightarrow ({x}^{{\prime} },{y}^{{\prime} })$ shown in figure 4.Using these coupling constants, we further obtain29 ), obtained in section 3.1 for the compact tetraquark state $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle $.
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{ \mathcal B }(| D{\bar{D}}^{* };{1}^{++}\rangle \to {\eta }_{c}{f}_{0}\to {\eta }_{c}\pi \pi )}{{ \mathcal B }(| D{\bar{D}}^{* };{1}^{++}\rangle \to J/\psi \omega \to J/\psi \pi \pi \pi )}\sim 0.091,\\ \displaystyle \frac{{ \mathcal B }(| D{\bar{D}}^{* };{1}^{++}\rangle \to {\chi }_{c1}{f}_{0}\to {\chi }_{c1}\pi \pi )}{{ \mathcal B }(| D{\bar{D}}^{* };{1}^{++}\rangle \to J/\psi \omega \to J/\psi \pi \pi \pi )}\sim 0.086,\\ \displaystyle \frac{{ \mathcal B }(| D{\bar{D}}^{* };{1}^{++}\rangle \to J/\psi {h}_{1}\to J/\psi \pi \pi \pi )}{{ \mathcal B }(| D{\bar{D}}^{* };{1}^{++}\rangle \to J/\psi \omega \to J/\psi \pi \pi \pi )}=1.4\times {10}^{-3},\end{array}\end{eqnarray}$
which ratios are the same as equations (4.2. ${\xi }_{\mu }^{{ \mathcal X }}([\bar{c}q][\bar{q}c])\to {\xi }_{\mu }^{i}([\bar{c}q]\,+\,[\bar{q}c])$
Assuming the X(3872) to be the $D{\bar{D}}^{* }$ hadronic molecular state of JPC = 1++, it can naturally decay into the $D{\bar{D}}^{* }$ final state, which fall-apart process can be described by itself:
$\begin{eqnarray}\begin{array}{l}{\xi }_{\mu }^{{ \mathcal Z }}(x,y;I=0)\Longrightarrow {\xi }_{\mu }^{1}({x}^{{\prime} },{y}^{{\prime} };I=0)\\ =-{\rm{i}}\,{O}_{\mu }^{V}({x}^{{\prime} })\,{O}^{P}({y}^{{\prime} })+{\rm{c}}.{\rm{c}}.,\end{array}\end{eqnarray}$
and so $\begin{eqnarray}\begin{array}{l}\langle X(p,\epsilon )| {D}^{0}({p}_{1}){\bar{D}}^{* 0}({p}_{2},{\epsilon }_{2})\rangle \approx -\displaystyle \frac{{\rm{i}}{c}_{5}}{\sqrt{2}}\,{\lambda }_{D}{m}_{{D}^{* }}{f}_{{D}^{* }}\,\epsilon \cdot {\epsilon }_{2}\\ \equiv \,{h}_{D{\bar{D}}^{* }}\,\epsilon \cdot {\epsilon }_{2},\end{array}\end{eqnarray}$
where c5 is an overall factor. Again, this decay might be kinematically forbidden, but the $| D{\bar{D}}^{* };{1}^{++}\rangle \to {D}^{0}{\bar{D}}^{* 0}\,+{D}^{* 0}{\bar{D}}^{0}\to {D}^{0}{\bar{D}}^{0}{\pi }^{0}$ process is surely kinematically allowed.Numerically, we obtain
$\begin{eqnarray}\begin{array}{rcl}{h}_{D{\bar{D}}^{* }}^{S} & = & -{\rm{i}}{c}_{5}\,2.1\times {10}^{11}\,{\mathrm{MeV}}^{4},\\ {h}_{D{\bar{D}}^{* }}^{D} & = & 0.\end{array}\end{eqnarray}$
Comparing this decay with the $| D{\bar{D}}^{* };{1}^{++}\rangle \to J/\psi \omega \to J/\psi \pi \pi \pi $ decay studied in the previous subsection, we further obtain $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{ \mathcal B }(| D{\bar{D}}^{* };{1}^{++}\rangle \to {D}^{0}{\bar{D}}^{* 0}+{D}^{* 0}{\bar{D}}^{0}\to {D}^{0}{\bar{D}}^{0}{\pi }^{0})}{{ \mathcal B }(| D{\bar{D}}^{* };{1}^{++}\rangle \to J/\psi \omega \to J/\psi \pi \pi \pi )}\\ \quad =\,4.5\times \displaystyle \frac{{c}_{5}^{2}}{{c}_{4}^{2}}.\end{array}\end{eqnarray}$
Therefore, $| D{\bar{D}}^{* };{1}^{++}\rangle $ mainly decays into two charmed mesons, because c5 is probably larger than c4.5. Isospin of the X(3872)
The isospin breaking effect of the X(3872) is significant and important to understand its nature [3–10]. As proposed in [78], this can be simply because the close proximity of the mass of the isoscalar X(3872) to the neutral ${D}^{0}{\bar{D}}^{* 0}$ threshold. We argue that in this case the X(3872) and the ${D}^{0}{\bar{D}}^{* 0}$ threshold together can be considered as a ‘mixed’ state, not purely isoscalar any more, through which we can investigate the isovector decay channels of the X(3872).
In this section we shall investigate the isospin breaking effect of the X(3872) by freely choosing its quark content [12, 65, 66], for example,
$\begin{eqnarray}\begin{array}{l}{\eta }_{\mu }^{1}(\theta /{\theta }^{{\prime} })=\cos \theta \,{\eta }_{\mu }^{1}([{uc}][\bar{u}\bar{c}])+\sin \theta \,{\eta }_{\mu }^{1}([{dc}][\bar{d}\bar{c}])\\ =\,\cos \theta \times \left({u}_{a}^{{\rm{T}}}{\mathbb{C}}{\gamma }_{\mu }{c}_{b}\,{\bar{u}}_{a}{\gamma }_{5}{\mathbb{C}}{\bar{c}}_{b}^{{\rm{T}}}+\{{\gamma }_{\mu }\leftrightarrow {\gamma }_{5}\}\right)\\ +\,\sin \theta \times \{u/\bar{u}\to d/\bar{d}\}\\ \Rightarrow \,\cos {\theta }^{{\prime} }\,{\eta }_{\mu }^{1}(I=0)+\sin {\theta }^{{\prime} }\,{\eta }_{\mu }^{1}(I=1),\end{array}\end{eqnarray}$
where θ and ${\theta }^{{\prime} }$ are the two related mixing angles. We shall fine-tune them to be different from $\theta =45^\circ /{\theta }^{{\prime} }=0^\circ $, so that the X(3872) is assumed not to be a purely isoscalar state. We shall study this effect separately for the compact tetraquark and hadronic molecule scenarios in the following subsections.5.1. Isospin breaking effect of $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle $
We repeat the same procedures as those done in section 3.1 , and transform ${\eta }_{\mu }^{{ \mathcal X }}(x,y;{\theta }_{1}^{{\prime} })$ to ${\theta }_{\mu }^{i}(x,y;{\theta }_{1}^{{\prime} })$ according to equation (10 ), from which we extract the following isovector decay channels:
Numerically, we obtain3.1 (a factor ${\cos }^{2}{\theta }_{1}^{{\prime} }$ needs to be multiplied there), we can use ${\theta }_{1}^{{\prime} }=\pm 15^\circ $ to obtain
| 1. The decay of $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle $ into χc0π is contributed by ${I}^{S}\times {J}_{\mu }^{A}$: $\begin{eqnarray}\begin{array}{l}\langle X(p,\epsilon )| {\chi }_{c0}({p}_{1})\,\pi ({p}_{2})\rangle \\ \approx -\displaystyle \frac{{\rm{i}}{c}_{1}\sin {\theta }_{1}^{{\prime} }}{3}\,{m}_{{\chi }_{c0}}{f}_{{\chi }_{c0}}{f}_{\pi }\,\epsilon \cdot {p}_{2}\equiv {g}_{{\chi }_{c0}\pi }\,\epsilon \cdot {p}_{2}.\end{array}\end{eqnarray}$ This process is kinematically allowed. | |
| 2. The decay of $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle $ into J/ψρ is contributed by both IV,ν × JT,ρσ and IT,ρσ × JV,ν: $\begin{array}{l}\langle X(p,\epsilon )| J/\psi ({p}_{1},{\epsilon }_{1})\,\rho ({p}_{2},{\epsilon }_{2})\rangle \\ \approx -\displaystyle \frac{{\rm{i}}{c}_{1}\sin {\theta }_{1}^{{\prime} }}{3}\,{\epsilon }_{\mu \nu \rho \sigma }{\epsilon }^{\mu }{\epsilon }_{1}^{\nu }{\epsilon }_{2}^{\rho }{p}_{2}^{\sigma }\,{m}_{J/\psi }{f}_{J/\psi }{f}_{\rho }^{T}\\ -\displaystyle \frac{{\rm{i}}{c}_{1}\sin {\theta }_{1}^{{\prime} }}{3}\,{\epsilon }_{\mu \nu \rho \sigma }{\epsilon }^{\mu }{\epsilon }_{1}^{\nu }{\epsilon }_{2}^{\rho }{p}_{1}^{\sigma }\,{m}_{\rho }{f}_{\rho }{f}_{J/\psi }^{T}\\ \equiv \,{g}_{\psi \rho }^{A}\,{\epsilon }_{\mu \nu \rho \sigma }{\epsilon }^{\mu }{\epsilon }_{1}^{\nu }{\epsilon }_{2}^{\rho }{p}_{2}^{\sigma }+{g}_{\psi \rho }^{B}\,{\epsilon }_{\mu \nu \rho \sigma }{\epsilon }^{\mu }{\epsilon }_{1}^{\nu }{\epsilon }_{2}^{\rho }{p}_{1}^{\sigma }.\end{array}$ If we use ${m}_{{\rho }^{0}}=775.26\,\mathrm{MeV}$ [2], this decay would be kinematically forbidden, but the $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \,\to J/\psi \rho \to J/\psi \pi \pi $ process is surely kinematically allowed. |
$\begin{eqnarray}\begin{array}{rcl}{g}_{{\chi }_{c0}\pi } & = & -{\rm{i}}{c}_{1}\sin {\theta }_{1}^{{\prime} }\,5.08\times {10}^{7}\,{\mathrm{MeV}}^{3},\\ {g}_{\psi \rho }^{A} & = & -{\rm{i}}{c}_{1}\sin {\theta }_{1}^{{\prime} }\,6.86\times {10}^{7}\,{\mathrm{MeV}}^{3},\\ {g}_{\psi \rho }^{B} & = & -{\rm{i}}{c}_{1}\sin {\theta }_{1}^{{\prime} }\,2.29\times {10}^{7}\,{\mathrm{MeV}}^{3}.\end{array}\end{eqnarray}$
Comparing these decays with the $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \omega \to J/\psi \pi \pi \pi $ decay studied in section $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{ \mathcal B }(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to {\chi }_{c0}\pi )}{{ \mathcal B }(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \rho \to J/\psi \pi \pi )}=0.024,\\ \displaystyle \frac{{ \mathcal B }(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \omega \to J/\psi \pi \pi \pi )}{{ \mathcal B }(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \rho \to J/\psi \pi \pi )}=1.6,\end{array}\end{eqnarray}$
where the latter ratio has been fine-tuned to be the same as the recent BESIII experiment [41].The isospin breaking effect can also affect the branching ratio of the $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to {D}^{0}{\bar{D}}^{* 0}+{D}^{* 0}{\bar{D}}^{0}\to {D}^{0}{\bar{D}}^{0}{\pi }^{0}$ decay. Using ${\theta }_{1}^{{\prime} }=+15^\circ $, we obtain
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{ \mathcal B }(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to {D}^{0}{\bar{D}}^{* 0}+{D}^{* 0}{\bar{D}}^{0}\to {D}^{0}{\bar{D}}^{0}{\pi }^{0})}{{ \mathcal B }(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \omega \to J/\psi \pi \pi \pi )}\,=\,0.52\times \displaystyle \frac{{c}_{2}^{2}}{{c}_{1}^{2}},\end{array}\end{eqnarray}$
while using ${\theta }_{1}^{{\prime} }=-15^\circ $, this ratio is calculated to be around $0.17\times {c}_{2}^{2}/{c}_{1}^{2}$.5.2. Isospin breaking effect of $| D{\bar{D}}^{* };{1}^{++}\rangle $
We repeat the same procedures as those done in section 4.1 , and transform ${\xi }_{\mu }^{{ \mathcal X }}(x,y;{\theta }_{2}^{{\prime} })$ to ${\theta }_{\mu }^{i}(x,y;{\theta }_{2}^{{\prime} })$ according to equation (13 ), from which we extract the following isovector coupling constants:
$\begin{eqnarray}\begin{array}{rcl}{h}_{{\chi }_{c0}\pi } & = & +{\rm{i}}{c}_{4}\sin {\theta }_{2}^{{\prime} }\,2.54\times {10}^{7}\,{\mathrm{MeV}}^{3},\\ {h}_{\psi \rho }^{A} & = & -{\rm{i}}{c}_{4}\sin {\theta }_{2}^{{\prime} }\,3.43\times {10}^{7}\,{\mathrm{MeV}}^{3},\\ {h}_{\psi \rho }^{B} & = & -{\rm{i}}{c}_{4}\sin {\theta }_{2}^{{\prime} }\,1.14\times {10}^{7}\,{\mathrm{MeV}}^{3}.\end{array}\end{eqnarray}$
They are defined for the $| D{\bar{D}}^{* };{1}^{++}\rangle \to {\chi }_{c0}\pi $ and J/ψρ decays.We can use the same angle ${\theta }_{2}^{{\prime} }=\pm 15^\circ $ to obtain48 ), obtained in the previous subsection for the compact tetraquark state $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle $.
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{ \mathcal B }(| D{\bar{D}}^{* };{1}^{++}\rangle \to {\chi }_{c0}\pi )}{{ \mathcal B }(| D{\bar{D}}^{* };{1}^{++}\rangle \to J/\psi \rho \to J/\psi \pi \pi )}=0.024,\\ \displaystyle \frac{{ \mathcal B }(| D{\bar{D}}^{* };{1}^{++}\rangle \to J/\psi \omega \to J/\psi \pi \pi \pi )}{{ \mathcal B }(| D{\bar{D}}^{* };{1}^{++}\rangle \to J/\psi \rho \to J/\psi \pi \pi )}=1.6,\end{array}\end{eqnarray}$
which ratios are the same as equations (Again, the isospin breaking effect can affect the branching ratio of the $| D{\bar{D}}^{* };{1}^{++}\rangle \to {D}^{0}{\bar{D}}^{* 0}+{D}^{* 0}{\bar{D}}^{0}\to {D}^{0}{\bar{D}}^{0}{\pi }^{0}$ decay. Using ${\theta }_{2}^{{\prime} }=+15^\circ $, we obtain
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{ \mathcal B }(| D{\bar{D}}^{* };{1}^{++}\rangle \to {D}^{0}{\bar{D}}^{* 0}+{D}^{* 0}{\bar{D}}^{0}\to {D}^{0}{\bar{D}}^{0}{\pi }^{0})}{{ \mathcal B }(| D{\bar{D}}^{* };{1}^{++}\rangle \to J/\psi \omega \to J/\psi \pi \pi \pi )}\,=\,7.4\times \displaystyle \frac{{c}_{5}^{2}}{{c}_{4}^{2}},\end{array}\end{eqnarray}$
while using ${\theta }_{2}^{{\prime} }=-15^\circ $, this ratio is calculated to be around $2.4\times {c}_{5}^{2}/{c}_{4}^{2}$.6. Summary and discussions
In this paper we systematically construct the tetraquark currents of JPC = 1++ with the quark content $c\bar{c}q\bar{q}$ (q=u/d). We consider three configurations, $[{cq}][\bar{c}\bar{q}]$, $[\bar{c}q][\bar{q}c]$, and $[\bar{c}c][\bar{q}q]$, and we construct eight independent currents for each of them. Their relations are derived using the Fierz rearrangement of the Dirac and color indices, through which we study decay properties of the X(3872):
| • | Based on the transformation of $[{qc}][\bar{q}\bar{c}]\to [\bar{c}c][\bar{q}q]$, we study decay properties of the X(3872) as a compact tetraquark state into one charmonium meson and one light meson. |
| • | Based on the transformation of $[{qc}][\bar{q}\bar{c}]\to [\bar{c}q][\bar{q}c]$, we study decay properties of the X(3872) as a compact tetraquark state into two charmed mesons. |
| • | Based on the transformation of the $[{qc}][\bar{q}\bar{c}]$ currents to the color-singlet-color-singlet $[\bar{c}c][\bar{q}q]$ and $[\bar{c}q][\bar{q}c]$ currents, we obtain the same relative branching ratios as those obtained using the above two transformations. |
| • | Based on the transformation of $[\bar{c}q][\bar{q}c]\to [\bar{c}c][\bar{q}q]$, we study decay properties of the X(3872) as a hadronic molecular state into one charmonium meson and one light meson. |
| • | Based on the $[\bar{c}q][\bar{q}c]$ currents themselves, we study decay properties of the X(3872) as a hadronic molecular state into two charmed mesons. |
We first use isoscalar tetraquark currents to study decay properties of the X(3872) as a purely isoscalar state, and then use isovector tetraquark currents to investigate its isospin breaking effect. The extracted relative branching ratios are summarized in table 2, where we have investigated the following interpretations of the X(3872):
| • | In the second, third, and fourth columns of table 2, $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle $ denotes the compact tetraquark state of JPC = 1++, defined in equation ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{ \mathcal B }\left(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \omega (\to \pi \pi \pi ):\,\,J/\psi \rho (\to \pi \pi )\,:\,\,{\chi }_{c0}\pi \,:{\eta }_{c}{f}_{0}(\to \pi \pi ):{\chi }_{c1}{f}_{0}(\to \pi \pi ):{D}^{0}{\bar{D}}^{* 0}(\to {D}^{0}{\bar{D}}^{0}{\pi }^{0})\,\right)}{{ \mathcal B }\left(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \omega (\to \pi \pi \pi )\right)}\\ \qquad \sim 1:0.63\,(\mathrm{input})\,\,:\,0.015\,:\,\,0.091\,(?):0.086(?):0.52\,{t}_{1}\,,\end{array}\end{eqnarray}$ while using the mixing angle ${\theta }_{1}^{{\prime} }=-{15}^{{\rm{o}}}$, we obtain $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{ \mathcal B }\left(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \omega (\to \pi \pi \pi ):\,\,J/\psi \rho (\to \pi \pi )\,:\,\,{\chi }_{c0}\pi \,:{\eta }_{c}{f}_{0}(\to \pi \pi ):{\chi }_{c1}{f}_{0}(\to \pi \pi ):{D}^{0}{\bar{D}}^{* 0}(\to {D}^{0}{\bar{D}}^{0}{\pi }^{0})\,\right)}{{ \mathcal B }\left(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \omega (\to \pi \pi \pi )\right)}\\ \sim 1:\,\,0.63\,(\mathrm{input})\,\,:\,0.015\,:\,\,0.091\,(?)\,:\,\,\,0.086\,(?)\,\,:\,\,\,\,\,\,\,0.17\,{t}_{1}\,.\end{array}\end{eqnarray}$ |
| • | In the fifth, sixth, and seventh columns of table 2, $| D{\bar{D}}^{* };{1}^{++}\rangle $ denotes the hadronic molecular state of JPC = 1++, defined in equation ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{ \mathcal B }\left(| D{\bar{D}}^{* };{1}^{++}\rangle \to J/\psi \omega (\to \pi \pi \pi ):\,\,J/\psi \rho (\to \pi \pi )\,:\,{\chi }_{c0}\pi \,:{\eta }_{c}{f}_{0}(\to \pi \pi ):{\chi }_{c1}{f}_{0}(\to \pi \pi ):{D}^{0}{\bar{D}}^{* 0}(\to {D}^{0}{\bar{D}}^{0}{\pi }^{0})\,\right)}{{ \mathcal B }\left(| D{\bar{D}}^{* };{1}^{++}\rangle \to J/\psi \omega (\to \pi \pi \pi )\right)}\\ \sim 1:\,\,0.63\,(\mathrm{input})\,\,:\,0.015\,:\,\,0.091\,(?)\,:\,\,\,0.086\,(?)\,\,:\,\,\,\,\,\,\,7.4\,{t}_{2}\,,\end{array}\end{eqnarray}$ while using the mixing angle ${\theta }_{2}^{{\prime} }=-{15}^{{\rm{o}}}$, we obtain $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{ \mathcal B }\left(| D{\bar{D}}^{* };{1}^{++}\rangle \to J/\psi \omega (\to \pi \pi \pi ):\,\,J/\psi \rho (\to \pi \pi )\,:\,{\chi }_{c0}\pi \,:{\eta }_{c}{f}_{0}(\to \pi \pi ):{\chi }_{c1}{f}_{0}(\to \pi \pi ):{D}^{0}{\bar{D}}^{* 0}(\to {D}^{0}{\bar{D}}^{0}{\pi }^{0})\,\right)}{{ \mathcal B }\left(| D{\bar{D}}^{* };{1}^{++}\rangle \to J/\psi \omega (\to \pi \pi \pi )\right)}\\ \sim 1:\,\,0.63\,(\mathrm{input})\,\,:\,0.015\,:\,\,0.091\,(?)\,:\,\,\,0.086\,(?)\,\,:\,\,\,\,\,\,\,2.4\,{t}_{2}\,.\end{array}\end{eqnarray}$ |
Table 2. Relative branching ratios of the X(3872) evaluated through the Fierz rearrangement. ${\theta }_{1,2}^{{\prime} }$ are the two angles related to the isospin breaking effect, which are fine-tuned to be ${\theta }_{1}^{{\prime} }={\theta }_{2}^{{\prime} }=\pm {15}^{{\rm{o}}}$, so that $\displaystyle \frac{{ \mathcal B }(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \omega \to J/\psi \pi \pi \pi )}{{ \mathcal B }(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \rho \to J/\psi \pi \pi )}$ $=\displaystyle \frac{{ \mathcal B }(| D{\bar{D}}^{* };{1}^{++}\rangle \to J/\psi \omega \to J/\psi \pi \pi \pi )}{{ \mathcal B }(| D{\bar{D}}^{* };{1}^{++}\rangle \to J/\psi \rho \to J/\psi \pi \pi )}=1.6$ [41]. |
| Channels | $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle $ | $| D{\bar{D}}^{* };{1}^{++}\rangle $ | ||||
|---|---|---|---|---|---|---|
| $I=0/{\theta }_{1}^{{\prime} }={0}^{{\rm{o}}}$ | ${\theta }_{1}^{{\prime} }=+{15}^{{\rm{o}}}$ | ${\theta }_{1}^{{\prime} }=-{15}^{{\rm{o}}}$ | $I=0/{\theta }_{2}^{{\prime} }={0}^{{\rm{o}}}$ | ${\theta }_{2}^{{\prime} }=+{15}^{{\rm{o}}}$ | ${\theta }_{2}^{{\prime} }=-{15}^{{\rm{o}}}$ | |
| $\displaystyle \frac{{ \mathcal B }(X\to {\eta }_{c}{f}_{0}\to {\eta }_{c}\pi \pi )}{{ \mathcal B }(X\to J/\psi \omega \to J/\psi \pi \pi \pi )}$ | ∼0.091 | ∼0.091 | ∼0.091 | ∼0.091 | ∼0.091 | ∼0.091 |
| $\displaystyle \frac{{ \mathcal B }(X\to {\chi }_{c1}{f}_{0}\to {\chi }_{c1}\pi \pi )}{{ \mathcal B }(X\to J/\psi \omega \to J/\psi \pi \pi \pi )}$ | ∼0.086 | ∼0.086 | ∼0.086 | ∼0.086 | ∼0.086 | ∼0.086 |
| $\displaystyle \frac{{ \mathcal B }(X\to J/\psi {h}_{1}\to J/\psi \pi \pi \pi )}{{ \mathcal B }(X\to J/\psi \omega \to J/\psi \pi \pi \pi )}$ | 1.4 × 10−3 | 1.4 × 10−3 | 1.4 × 10−3 | 1.4 × 10−3 | 1.4 × 10−3 | 1.4 × 10−3 |
| $\displaystyle \frac{{ \mathcal B }(X\to {\chi }_{c0}\pi )}{{ \mathcal B }(X\to J/\psi \rho \to J/\psi \pi \pi )}$ | — | 0.024 | 0.024 | — | 0.024 | 0.024 |
| $\displaystyle \frac{{ \mathcal B }(X\to J/\psi \omega \to J/\psi \pi \pi \pi )}{{ \mathcal B }(X\to J/\psi \rho \to J/\psi \pi \pi )}$ | — | 1.6 (input) | 1.6 (input) | — | 1.6 (input) | 1.6 (input) |
| $\displaystyle \frac{{ \mathcal B }(X\to {D}^{0}{\bar{D}}^{* 0}+{D}^{* 0}{\bar{D}}^{0}\to {D}^{0}{\bar{D}}^{0}{\pi }^{0})}{{ \mathcal B }(X\to J/\psi \omega \to J/\psi \pi \pi \pi )}$ | 0.32 t1 | 0.52 t1 | 0.17 t1 | 4.5 t2 | 7.4 t2 | 2.4 t2 |
In the above expressions, we define the ratio ${t}_{1}\equiv {c}_{2}^{2}/{c}_{1}^{2}$ to be the parameter measuring which process happens more easily, the process depicted in figure 2(b) or the process depicted in figure 3(b). Because the exchange of one light quark with another light quark seems to be easier than the exchange of one light quark with another heavy quark [61, 79], it can be the case that t1 ≥ 1. As discussed in section 4.2 , c5 is probably larger than c4, so that the other ratio ${t}_{2}\equiv {c}_{5}^{2}/{c}_{4}^{2}\geqslant 1$.
The above relative branching ratios extracted in the present study turn out to be very much different. This might be one of the reasons why many multiquark states were observed only in a few decay channels [49]. We note that in this paper we only consider the leading-order fall-apart decays described by color-singlet-color-singlet meson-meson currents, but neglect the ${ \mathcal O }({\alpha }_{s})$ corrections described by color-octet-color-octet meson-meson currents, so there can be other possible decay channels, such as X(3872) → χc1π [45]. Besides, there is still one parameter not considered in above analyses, that is the phase angle between S- and D-wave coupling constants. We shall investigate its relevant uncertainty in B .
Based on table 2, we conclude this paper. Generally speaking, compared to the Zc(3900) studied in [49], the results of this paper suggest that decay channels of the X(3872) are quite limited:
| • | The X(3872) can couple to the χc0η, χc0f1(1285), and hcω channels, but all of them are kinematically forbidden. |
| • | The X(3872) can couple to the isovector channels J/ψρ and χc0π, but both of them are due to the isospin breaking effect. |
| • | The X(3872) can couple to the ${D}^{0}{\bar{D}}^{* 0}$ and J/ψρ channels, but its mass is very close to the relevant thresholds. Hence, in the present study we calculate widths of the three-body decays $X\to {D}^{0}{\bar{D}}^{* 0}+{D}^{* 0}{\bar{D}}^{0}\to {D}^{0}{\bar{D}}^{0}{\pi }^{0}$ and X → J/ψρ → J/ψππ. |
| • | The X(3872) can couple to the J/ψω and J/ψh1(1170) channels, but both of them are kinematically forbidden. Hence, in the present study we calculate widths of the four-body decays X → J/ψω → J/ψπππ and X →J/ψh1 → J/ψπππ. |
| • | The decay processes X → ηcf0 → ηcππ and X → χc1f0 → χc1ππ might be possible. In this paper we simply use the f0(500) to estimate widths of these two processes, but note that the obtained results do significantly depend on the nature of light scalar mesons, which are still quite ambiguous [77]. |
To end this paper, we give several comments and proposals:
| • | The hadronic molecular state $| D{\bar{D}}^{* };{1}^{++}\rangle $ mainly decays into two charmed mesons, because c5 is probably larger than c4. The compact tetraquark state $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle $ may also mainly decay into two charmed mesons after taking into account the barrier of the diquark-antidiquark potential (see detailed discussions in [61] proposing c2 ≫ c1). |
| • | The isospin breaking effect of the X(3872) is significant and important to understand its nature [3–10]. The isovector decay channel X(3872) → J/ψρ → J/ψππ has been well observed in experiments, and recently measured by the BESIII experiment [41] to be: $\begin{eqnarray}\displaystyle \frac{{ \mathcal B }(X\to J/\psi \omega \to J/\psi \pi \pi \pi )}{{ \mathcal B }(X\to J/\psi \rho \to J/\psi \pi \pi )}={1.6}_{-0.3}^{+0.4}\pm 0.2.\end{eqnarray}$ In the present study we can well reproduce this value under both the compact tetraquark and hadronic molecule interpretations. Besides this, our result suggests that there can be another isovector decay channel X(3872) → χc0π. Under both the compact tetraquark and hadronic molecule interpretations, we obtain $\begin{eqnarray}\displaystyle \frac{{ \mathcal B }(X\to {\chi }_{c0}\pi )}{{ \mathcal B }(X\to J/\psi \rho \to J/\psi \pi \pi )}=0.024.\end{eqnarray}$ We refer to [80–87] for more theoretical studies, and propose to study the X(3872) → χc0π decay in the BESIII, Belle-II, and LHCb experiments to better understand the isospin breaking effect of the X(3872). |
| • | Our result suggests that the decay processes X(3872) → ηcf0 → ηcππ and X(3872) → χc1f0 → χc1ππ might be possible. We note that light scalar mesons have a complicated nature, so our results on these processes are just roughly estimations. We notice that the BaBar experiment [88] did not observe the γγ → X(3872) → ηcππ process, but that experiment was performed after assuming X(3872) to be a spin-2 state. Moreover, there seems to be a dip structure just at the mass of the X(3872) in the ηcππ invariant mass spectrum, as shown in figure 6(f) of [88]. We also notice that the Belle experiment [89] did not observe the X(3872) → χc1ππ decay. They extracted the following upper limit $\begin{eqnarray*}{ \mathcal B }({B}^{+}\to {K}^{+}X){ \mathcal B }(X\to {\chi }_{c1}\pi \pi )\lt 1.5\times {10}^{-6},\end{eqnarray*}$ at 90% C.L. Together with another Belle experiment [90] measuring $\begin{eqnarray*}{ \mathcal B }({B}^{+}\to {K}^{+}X)=(1.2\pm 1.1\pm 0.1)\times {10}^{-4},\end{eqnarray*}$ one may roughly estimate $\begin{eqnarray}{ \mathcal B }(X\to {\chi }_{c1}\pi \pi )\lt 1.3\times {10}^{-2},\end{eqnarray}$ which value seems not small enough to rule out the X(3872) → χc1ππ decay channel. Again, we refer to [80–87, 91] for more discussions, and propose to reanalysis the X(3872) → ηcf0 → ηcππ and X(3872) → χc1f0 → χc1ππ processes in the BESIII, Belle-II, and LHCb experiments to search for more decay channels of the X(3872). |
Acknowledgments
This project is supported by the National Natural Science Foundation of China under Grant No. 11722540 and No. 12075019, the Jiangsu Provincial Double-Innovation Program under Grant No. JSSCRC2021488, and the Fundamental Research Funds for the Central Universities.
Appendix A. Formulae of decay amplitudes and decay widths
In this appendix we give formulae of decay amplitudes and decay widths used in the present study. Specifically, the mass of the X(3872) is taken from PDG [2] to be
$\begin{eqnarray}{m}_{X}=3871.69\,\mathrm{MeV}.\end{eqnarray}$
A.1. Two-body decay X → χc0π0
The decay amplitude of the two-body decay X(3872) → χc0π0 is
$\begin{eqnarray}{ \mathcal M }\left(X(\epsilon ,p)\to {\chi }_{c0}({p}_{1}){\pi }^{0}({p}_{2})\right)={g}_{{\chi }_{c0}\pi }\,\epsilon \cdot {p}_{2}.\end{eqnarray}$
This amplitude can be used to evaluate its decay width: $\begin{eqnarray}{\rm{\Gamma }}\left(X\to {\chi }_{c0}{\pi }^{0}\right)=\displaystyle \frac{\left|{\vec{p}}_{2}\right|}{8\pi {m}_{X}^{2}}\left|{g}_{{\chi }_{c0}\pi }^{2}\right|\displaystyle \frac{{p}_{2}^{\mu }{p}_{2}^{\nu }}{3}\left({g}_{\mu \nu }-\displaystyle \frac{{p}_{\mu }{p}_{\nu }}{{m}_{X}^{2}}\right),\end{eqnarray}$
where we have used the following formula for the vector meson $\begin{eqnarray}\sum {\epsilon }_{\mu }{\epsilon }_{\nu }^{* }={g}_{\mu \nu }-\displaystyle \frac{{p}_{\mu }{p}_{\nu }}{{m}_{X}^{2}}.\end{eqnarray}$
A.2. Three-body decay X → J/ψρ0 → J/ψπ+π−
First we need to investigate the two-body decay ρ0 → π+π−, whose amplitude is
$\begin{eqnarray}{ \mathcal M }\left({\rho }^{0}(\epsilon ,p)\to {\pi }^{+}({p}_{1}){\pi }^{-}({p}_{2})\right)={g}_{\rho \pi \pi }\,\epsilon \cdot \left({p}_{1}-{p}_{2}\right),\end{eqnarray}$
so that $\begin{eqnarray}\begin{array}{l}{\rm{\Gamma }}\left({\rho }^{0}\to {\pi }^{+}{\pi }^{-}\right)=\displaystyle \frac{1}{3}\displaystyle \frac{\left|{\vec{p}}_{2}\right|}{8\pi {m}_{\rho }^{2}}\left|{g}_{\rho \pi \pi }^{2}\right|\left({g}_{\mu \nu }-\displaystyle \frac{{p}_{\mu }{p}_{\nu }}{{m}_{\rho }^{2}}\right)\\ \times \,\left({p}_{1}^{\mu }-{p}_{2}^{\mu }\right)\left({p}_{1}^{\nu }-{p}_{2}^{\nu }\right).\end{array}\end{eqnarray}$
We can use the experimental parameters ${{\rm{\Gamma }}}_{{\rho }^{0}}=147.8\,\mathrm{MeV}$ and ${ \mathcal B }({\rho }^{0}\to {\pi }^{+}{\pi }^{-})\approx 100 \% $ [2] to extract $\begin{eqnarray}{g}_{\rho \pi \pi }=5.94.\end{eqnarray}$
The decay amplitude of the three-body decay X(3872) →J/ψρ0 → J/ψπ+π− is
$\begin{eqnarray}\begin{array}{l}{ \mathcal M }\left(X(\epsilon ,p)\to J/\psi ({\epsilon }_{1},{p}_{1}){\rho }^{0}({\epsilon }^{{\prime} },q)\right.\\ \left.\to J/\psi ({\epsilon }_{1},{p}_{1}){\pi }^{+}({p}_{2}){\pi }^{-}({p}_{3}\right)\\ ={g}_{\rho \pi \pi }\,\left({g}_{\psi \rho }^{A}\,{\epsilon }_{\mu \nu \rho \sigma }{\epsilon }^{\mu }{\epsilon }_{1}^{\nu }{q}^{\sigma }+{g}_{\psi \rho }^{B}\,{\epsilon }_{\mu \nu \rho \sigma }{\epsilon }^{\mu }{\epsilon }_{1}^{\nu }{p}_{1}^{\sigma }\right)\\ \times \,\displaystyle \frac{{p}_{2,\alpha }-{p}_{3,\alpha }}{{q}^{2}-{m}_{\rho }^{2}+{\rm{i}}{m}_{\rho }{{\rm{\Gamma }}}_{\rho }}\,\left({g}^{\rho \alpha }-\displaystyle \frac{{q}^{\rho }{q}^{\alpha }}{{m}_{\rho }^{2}}\right).\end{array}\end{eqnarray}$
This amplitude can be used to evaluate its decay width: $\begin{eqnarray}\begin{array}{l}{\rm{\Gamma }}\left(X\to J/\psi {\rho }^{0}\to J/\psi {\pi }^{+}{\pi }^{-}\right)\\ =\displaystyle \frac{1}{3}\displaystyle \frac{1}{{\left(2\pi \right)}^{3}}\displaystyle \frac{{g}_{\rho \pi \pi }^{2}}{32{m}_{X}^{3}}\displaystyle \int {\rm{d}}{m}_{12}^{2}{\rm{d}}{m}_{23}^{2}\,{\left|\displaystyle \frac{1}{{q}^{2}-{m}_{\rho }^{2}+{\rm{i}}{m}_{\rho }{{\rm{\Gamma }}}_{\rho }}\right|}^{2}\\ \times \left({g}_{\psi \rho }^{A}\,{\epsilon }_{\mu \nu \rho \sigma }{q}^{\sigma }+{g}_{\psi \rho }^{B}\,{\epsilon }_{\mu \nu \rho \sigma }{p}_{1}^{\sigma }\right)\\ \times \left({g}_{\psi \rho }^{A* }\,{\epsilon }_{{\mu }^{{\prime} }{\nu }^{{\prime} }{\rho }^{{\prime} }{\sigma }^{{\prime} }}{q}^{{\sigma }^{{\prime} }}+{g}_{\psi \rho }^{B* }\,{\epsilon }_{{\mu }^{{\prime} }{\nu }^{{\prime} }{\rho }^{{\prime} }{\sigma }^{{\prime} }}{p}_{1}^{{\sigma }^{{\prime} }}\right)\\ \times \left({g}^{\mu {\mu }^{{\prime} }}-\displaystyle \frac{{p}^{\mu }{p}^{{\mu }^{{\prime} }}}{{m}_{X}^{2}}\right)\,\left({g}^{\nu {\nu }^{{\prime} }}-\displaystyle \frac{{p}_{1}^{\nu }{p}_{1}^{{\nu }^{{\prime} }}}{{m}_{J/\psi }^{2}}\right)\\ \times \left({g}^{\rho \alpha }-\displaystyle \frac{{q}^{\rho }{q}^{\alpha }}{{m}_{\rho }^{2}}\right)\,\left({g}^{{\rho }^{{\prime} }{\alpha }^{{\prime} }}-\displaystyle \frac{{q}^{{\rho }^{{\prime} }}{q}^{{\alpha }^{{\prime} }}}{{m}_{\rho }^{2}}\right)\\ \times \left({p}_{2,\alpha }-{p}_{3,\alpha }\right)\,\left({p}_{2,{\alpha }^{{\prime} }}-{p}_{3,{\alpha }^{{\prime} }}\right).\end{array}\end{eqnarray}$
A.3. Three-body decay X → ηcf0 → ηcππ
First we need to investigate the two-body decay f0(500) →ππ, whose amplitude is
$\begin{eqnarray}{ \mathcal M }\left({f}_{0}(p)\to \pi ({p}_{1})\pi ({p}_{2})\right)={g}_{{f}_{0}\pi \pi }.\end{eqnarray}$
In this case we do not differentiate π±,0. The above amplitude can be used to evaluate its decay width: $\begin{eqnarray*}{\rm{\Gamma }}\left({f}_{0}\to \pi \pi \right)=\displaystyle \frac{\left|{\vec{p}}_{2}\right|}{8\pi {m}_{{f}_{0}}^{2}}\,\left|{g}_{{f}_{0}\pi \pi }^{2}\right|.\end{eqnarray*}$
We can use the experimental parameters ${m}_{{f}_{0}}=512\,\mathrm{MeV}$ and ${{\rm{\Gamma }}}_{{f}_{0}}=376\,\mathrm{MeV}$ [92] to extract $\begin{eqnarray}{g}_{{f}_{0}\pi \pi }=3380\,\mathrm{MeV}.\end{eqnarray}$
The decay amplitude of the three-body decay X(3872) →ηcf0 → ηcππ is
$\begin{eqnarray}\begin{array}{l}{ \mathcal M }\left(X(\epsilon ,p)\to {\eta }_{c}({p}_{1}){f}_{0}(q)\to {\eta }_{c}({p}_{1})\pi ({p}_{2})\pi ({p}_{3})\right)\\ =\,{g}_{{f}_{0}\pi \pi }\,{g}_{{\eta }_{c}{f}_{0}}\,\displaystyle \frac{\epsilon \cdot {p}_{1}}{{q}^{2}-{m}_{{f}_{0}}^{2}+{\rm{i}}{m}_{{f}_{0}}{{\rm{\Gamma }}}_{{f}_{0}}}.\end{array}\end{eqnarray}$
This amplitude can be used to evaluate its decay width: $\begin{eqnarray}\begin{array}{l}{\rm{\Gamma }}\left(X\to {\eta }_{c}{f}_{0}\to {\eta }_{c}\pi \pi \right)\\ =\displaystyle \frac{1}{3}\displaystyle \frac{1}{{\left(2\pi \right)}^{3}}\displaystyle \frac{{g}_{{f}_{0}\pi \pi }^{2}}{32{m}_{X}^{3}}\displaystyle \int {\rm{d}}{m}_{12}^{2}{\rm{d}}{m}_{23}^{2}\,{\left|\displaystyle \frac{1}{{q}^{2}-{m}_{{f}_{0}}^{2}+{\rm{i}}{m}_{{f}_{0}}{{\rm{\Gamma }}}_{{f}_{0}}}\right|}^{2}\\ \times {g}_{{\eta }_{c}{f}_{0}}{g}_{{\eta }_{c}{f}_{0}}^{* }\,\left({g}^{\mu \nu }-\displaystyle \frac{{p}^{\mu }{p}^{\nu }}{{m}_{X}^{2}}\right)\,{p}_{1,\mu }{p}_{1,\nu }.\end{array}\end{eqnarray}$
The X(3872) → χc1f0 → χc1ππ decay can be similarly studied.A.4. Three-body decay $X \rightarrow {D}^{0}{\bar{D}}^{* 0} \rightarrow {D}^{0}{\bar{D}}^{0}{\pi }^{0}$
First we need to investigate the two-body decay D*0 → D0π0, whose amplitude is
$\begin{eqnarray}{ \mathcal M }\left({D}^{* 0}(\epsilon ,p)\to {D}^{0}({p}_{1}){\pi }^{0}({p}_{2})\right)={g}_{{D}^{* }D\pi }\,\epsilon \cdot {p}_{2},\end{eqnarray}$
so that $\begin{eqnarray}{\rm{\Gamma }}\left({D}^{* 0}\to {D}^{0}{\pi }^{0}\right)=\displaystyle \frac{\left|{\vec{p}}_{2}\right|\left|{g}_{{D}^{* }D\pi }^{2}\right|}{8\pi {m}_{{D}^{* }}^{2}}\displaystyle \frac{{p}_{2}^{\mu }{p}_{2}^{\nu }}{3}\left({g}_{\mu \nu }-\displaystyle \frac{{p}_{\mu }{p}_{\nu }}{{m}_{{D}^{* }}^{2}}\right).\end{eqnarray}$
We can use the parameters ${{\rm{\Gamma }}}_{{D}^{* 0}}=83.3\,\mathrm{keV}$ [93] and ${ \mathcal B }({D}^{* 0}\to {D}^{0}{\pi }^{0})=64.7 \% $ [2] to extract $\begin{eqnarray}{g}_{{D}^{* }D\pi }=14.6.\end{eqnarray}$
The decay amplitude of the three-body decay $X(3872)\to {D}^{0}{\bar{D}}^{* 0}\to {D}^{0}{\bar{D}}^{0}{\pi }^{0}$ is
$\begin{eqnarray}\begin{array}{l}{ \mathcal M }\left(X(\epsilon ,p)\to {D}^{0}({p}_{1}){\bar{D}}^{* 0}({\epsilon }^{{\prime} },q)\right.\\ \left.\to {D}^{0}({p}_{1}){\bar{D}}^{0}({p}_{2}){\pi }^{0}({p}_{3}\right)\\ ={g}_{{D}^{* }D\pi }\,\left({g}_{D{\bar{D}}^{* }}^{S}\,{\epsilon }_{\mu }+{g}_{D{\bar{D}}^{* }}^{D}\,(\epsilon \cdot q\,{p}_{1,\mu }-{p}_{1}\cdot q\,{\epsilon }_{\mu })\right)\\ \times \,\displaystyle \frac{{p}_{3,\nu }}{{q}^{2}-{m}_{{D}^{* }}^{2}+{\rm{i}}{m}_{{D}^{* }}{{\rm{\Gamma }}}_{{D}^{* }}}\,\left({g}^{\mu \nu }-\displaystyle \frac{{q}^{\mu }{q}^{\nu }}{{m}_{{D}^{* }}^{2}}\right).\end{array}\end{eqnarray}$
This amplitude can be used to evaluate its decay width: $\begin{eqnarray}\begin{array}{l}{\rm{\Gamma }}\left(X\to {D}^{0}{\bar{D}}^{* 0}\to {D}^{0}{\bar{D}}^{0}{\pi }^{0}\right)\\ =\displaystyle \frac{1}{3}\displaystyle \frac{1}{{\left(2\pi \right)}^{3}}\displaystyle \frac{{g}_{{D}^{* }D\pi }^{2}}{32{m}_{X}^{3}}\displaystyle \int {\rm{d}}{m}_{12}^{2}{\rm{d}}{m}_{23}^{2}\\ \times {\left|\displaystyle \frac{1}{{q}^{2}-{m}_{{D}^{* }}^{2}+{\rm{i}}{m}_{{D}^{* }}{{\rm{\Gamma }}}_{{D}^{* }}}\right|}^{2}\,\left({g}^{\mu {\mu }^{{\prime} }}-\displaystyle \frac{{p}^{\mu }{p}^{{\mu }^{{\prime} }}}{{m}_{X}^{2}}\right)\\ \times \left({g}_{D{\bar{D}}^{* }}^{S}\,{g}_{\mu \nu }+{g}_{D{\bar{D}}^{* }}^{D}\,({p}_{1,\nu }{q}_{\mu }-{p}_{1}\cdot q\,{g}_{\mu \nu })\right)\\ \times \left({g}_{D{\bar{D}}^{* }}^{S* }\,{g}_{{\mu }^{{\prime} }{\nu }^{{\prime} }}+{g}_{D{\bar{D}}^{* }}^{D* }\,({p}_{1,{\nu }^{{\prime} }}{q}_{{\mu }^{{\prime} }}-{p}_{1}\cdot q\,{g}_{{\mu }^{{\prime} }{\nu }^{{\prime} }})\right)\\ \times \left({g}^{\nu \rho }-\displaystyle \frac{{q}^{\nu }{q}^{\rho }}{{m}_{{D}^{* }}^{2}}\right)\,\left({g}^{{\nu }^{{\prime} }{\rho }^{{\prime} }}-\displaystyle \frac{{q}^{{\nu }^{{\prime} }}{q}^{{\rho }^{{\prime} }}}{{m}_{{D}^{* }}^{2}}\right)\,{p}_{3,\rho }{p}_{3,{\rho }^{{\prime} }}.\end{array}\end{eqnarray}$
The width of the $X(3872)\to {D}^{* 0}{\bar{D}}^{0}\to {D}^{0}{\bar{D}}^{0}{\pi }^{0}$ decay is the same as the $X(3872)\to {D}^{0}{\bar{D}}^{* 0}\to {D}^{0}{\bar{D}}^{0}{\pi }^{0}$ decay, and we assume them to be non-coherent in the present study.A.5. Four-body decay X → J/ψω → J/ψπ+π−π0
First we need to investigate the three-body decay ω → π+π−π0, whose amplitude is
$\begin{eqnarray}\begin{array}{l}{ \mathcal M }\left(\omega (\epsilon ,p)\to {\pi }^{+}({p}_{1}){\pi }^{-}({p}_{2}){\pi }^{0}({p}_{3})\right)\\ =\,{g}_{\omega 3\pi }\,{\epsilon }_{\mu \nu \rho \sigma }{\epsilon }^{\mu }{p}_{1}^{\nu }{p}_{2}^{\rho }{p}_{3}^{\sigma },\end{array}\end{eqnarray}$
so that $\begin{eqnarray}\begin{array}{l}{\rm{\Gamma }}\left(\omega \to {\pi }^{+}{\pi }^{-}{\pi }^{0}\right)\\ =\displaystyle \frac{1}{3}\displaystyle \frac{1}{{\left(2\pi \right)}^{3}}\displaystyle \frac{{g}_{\omega 3\pi }^{2}}{32{m}_{\omega }^{3}}\displaystyle \int {\rm{d}}{m}_{12}^{2}{\rm{d}}{m}_{23}^{2}\,\left({g}^{\mu {\mu }^{{\prime} }}-\displaystyle \frac{{p}^{\mu }{p}^{{\mu }^{{\prime} }}}{{m}_{\omega }^{2}}\right)\\ \times \,{\epsilon }_{\mu \nu \rho \sigma }{p}_{1}^{\nu }{p}_{2}^{\rho }{p}_{3}^{\sigma }\,{\epsilon }_{{\mu }^{{\prime} }{\nu }^{{\prime} }{\rho }^{{\prime} }{\sigma }^{{\prime} }}{p}_{1}^{{\nu }^{{\prime} }}{p}_{2}^{{\rho }^{{\prime} }}{p}_{3}^{{\sigma }^{{\prime} }}.\end{array}\end{eqnarray}$
We can use the experimental parameters Γω = 8.49 MeV and ${ \mathcal B }(\omega \to {\pi }^{+}{\pi }^{-}{\pi }^{0})=89.3 \% $ [2] to extract $\begin{eqnarray}{g}_{\omega 3\pi }=1.4\times {10}^{-6}\,{\mathrm{MeV}}^{-3}.\end{eqnarray}$
The decay amplitude of the four-body decay X(3872) → J/ψω → J/ψπ+π−π0 is
$\begin{eqnarray}\begin{array}{l}{ \mathcal M }\left(X(\epsilon ,p)\to J/\psi ({\epsilon }_{1},{p}_{1})\omega ({\epsilon }^{{\prime} },q)\right.\\ \left.\to \,J/\psi ({\epsilon }_{1},{p}_{1}){\pi }^{+}({p}_{2}){\pi }^{-}({p}_{3}){\pi }^{0}({p}_{4}\right)\\ =\,{g}_{\omega 3\pi }\,\left({g}_{\psi \omega }^{A}\,{\epsilon }_{\mu \nu \rho \sigma }{\epsilon }^{\mu }{\epsilon }_{1}^{\nu }{q}^{\sigma }+{g}_{\psi \omega }^{B}\,{\epsilon }_{\mu \nu \rho \sigma }{\epsilon }^{\mu }{\epsilon }_{1}^{\nu }{p}_{1}^{\sigma }\right)\\ \times \,\displaystyle \frac{1}{{q}^{2}-{m}_{\omega }^{2}+{\rm{i}}{m}_{\omega }{{\rm{\Gamma }}}_{\omega }}\,\left({g}^{\rho \alpha }-\displaystyle \frac{{q}^{\rho }{q}^{\alpha }}{{m}_{\omega }^{2}}\right)\,{\epsilon }_{\alpha \beta \gamma \zeta }{p}_{2}^{\beta }{p}_{3}^{\gamma }{p}_{4}^{\zeta }.\end{array}\end{eqnarray}$
This amplitude can be used to evaluate its decay width: $\begin{eqnarray}\begin{array}{l}{\rm{\Gamma }}\left(X\to J/\psi \omega \to J/\psi {\pi }^{+}{\pi }^{-}{\pi }^{0}\right)\\ =\displaystyle \frac{{g}_{\omega 3\pi }^{2}}{3}\displaystyle \frac{{\left(2\pi \right)}^{4}}{2{m}_{X}}\displaystyle \int \displaystyle \frac{{{\rm{d}}}^{3}{p}_{1}}{{\left(2\pi \right)}^{3}2{E}_{1}}\displaystyle \frac{{{\rm{d}}}^{3}{p}_{2}}{{\left(2\pi \right)}^{3}2{E}_{2}}\displaystyle \frac{4\pi {p}_{3x}^{2}}{{\left(2\pi \right)}^{6}2{E}_{3}2{E}_{4}}\\ \times \left|\displaystyle \frac{1}{\tfrac{{p}_{3x}}{{E}_{3}}+\tfrac{{p}_{1x}+{p}_{2x}+{p}_{3x}}{{E}_{4}}}\right|\,{\left|\displaystyle \frac{1}{{q}^{2}-{m}_{\omega }^{2}+{\rm{i}}{m}_{\omega }{{\rm{\Gamma }}}_{\omega }}\right|}^{2}\\ \times \left({g}_{\psi \omega }^{A}\,{\epsilon }_{\mu \nu \rho \sigma }{q}^{\sigma }+{g}_{\psi \omega }^{B}\,{\epsilon }_{\mu \nu \rho \sigma }{p}_{1}^{\sigma }\right)\\ \times \left({g}_{\psi \omega }^{A* }\,{\epsilon }_{{\mu }^{{\prime} }{\nu }^{{\prime} }{\rho }^{{\prime} }{\sigma }^{{\prime} }}{q}^{{\sigma }^{{\prime} }}+{g}_{\psi \omega }^{B* }\,{\epsilon }_{{\mu }^{{\prime} }{\nu }^{{\prime} }{\rho }^{{\prime} }{\sigma }^{{\prime} }}{p}_{1}^{{\sigma }^{{\prime} }}\right)\\ \times \left({g}^{\mu {\mu }^{{\prime} }}-\displaystyle \frac{{p}^{\mu }{p}^{{\mu }^{{\prime} }}}{{m}_{X}^{2}}\right)\,\left({g}^{\nu {\nu }^{{\prime} }}-\displaystyle \frac{{p}_{1}^{\nu }{p}_{1}^{{\nu }^{{\prime} }}}{{m}_{J/\psi }^{2}}\right)\\ \times \left({g}^{\rho \alpha }-\displaystyle \frac{{q}^{\rho }{q}^{\alpha }}{{m}_{\omega }^{2}}\right)\,\left({g}^{{\rho }^{{\prime} }{\alpha }^{{\prime} }}-\displaystyle \frac{{q}^{{\rho }^{{\prime} }}{q}^{{\alpha }^{{\prime} }}}{{m}_{\omega }^{2}}\right)\\ \times \,{\epsilon }_{\alpha \beta \gamma \zeta }{p}_{2}^{\beta }{p}_{3}^{\gamma }{p}_{4}^{\zeta }\,{\epsilon }_{{\alpha }^{{\prime} }{\beta }^{{\prime} }{\gamma }^{{\prime} }{\zeta }^{{\prime} }}{p}_{2}^{{\beta }^{{\prime} }}{p}_{3}^{{\gamma }^{{\prime} }}{p}_{4}^{{\zeta }^{{\prime} }}.\end{array}\end{eqnarray}$
The phase space integration is done in the reference frame where ${p}_{3}=\left({E}_{3},{p}_{3x},0,0\right)$, and p3x satisfies p3x > 0 as well as $\begin{eqnarray}{E}_{1}+{E}_{2}+{E}_{3}+{E}_{4}={m}_{X}.\end{eqnarray}$
A.6. Four-body decay X → J/ψh1 → J/ψπ+π−π0
First we need to investigate the three-body decay h1(1170) → ρπ → π+π−π0, whose amplitude is simply assumed to be
$\begin{eqnarray}{ \mathcal M }\left({h}_{1}(\epsilon ,p)\to {\pi }^{+}({p}_{1}){\pi }^{-}({p}_{2}){\pi }^{0}({p}_{3})\right)={g}_{{h}_{1}3\pi }\,\epsilon \cdot {p}_{3},\end{eqnarray}$
so that $\begin{eqnarray*}\begin{array}{l}{\rm{\Gamma }}\left({h}_{1}\to {\pi }^{+}{\pi }^{-}{\pi }^{0}\right)=\displaystyle \frac{1}{3}\displaystyle \frac{1}{{\left(2\pi \right)}^{3}}\displaystyle \frac{{g}_{{h}_{1}3\pi }^{2}}{32{m}_{{h}_{1}}^{3}}\displaystyle \int {\rm{d}}{m}_{12}^{2}{\rm{d}}{m}_{23}^{2}\\ \times \,{p}_{3,\mu }{p}_{3,{\mu }^{{\prime} }}\,\left({g}^{\mu {\mu }^{{\prime} }}-\displaystyle \frac{{p}^{\mu }{p}^{{\mu }^{{\prime} }}}{{m}_{{h}_{1}}^{2}}\right).\end{array}\end{eqnarray*}$
We can use the experimental parameters ${{\rm{\Gamma }}}_{{h}_{1}}=360\,\mathrm{MeV}$ [2] to estimate $\begin{eqnarray}{g}_{{h}_{1}3\pi }\approx 0.39\,{\mathrm{MeV}}^{-1}.\end{eqnarray}$
The decay amplitude of the four-body decay X(3872) → J/ψh1 → J/ψπ+π−π0 is
$\begin{eqnarray}\begin{array}{l}{ \mathcal M }\left(X(\epsilon ,p)\to J/\psi ({\epsilon }_{1},{p}_{1}){h}_{1}({\epsilon }^{{\prime} },q)\right.\\ \left.\to J/\psi ({\epsilon }_{1},{p}_{1}){\pi }^{+}({p}_{2}){\pi }^{-}({p}_{3}){\pi }^{0}({p}_{4}\right)\\ ={g}_{{h}_{1}3\pi }\,{g}_{\psi {h}_{1}}\,{\epsilon }_{\mu \nu \rho \sigma }{\epsilon }^{\rho \sigma \alpha \beta }{\epsilon }^{\mu }{\epsilon }_{1}^{\nu }{q}_{\beta }\\ \times \,\displaystyle \frac{1}{{q}^{2}-{m}_{{h}_{1}}^{2}+{\rm{i}}{m}_{{h}_{1}}{{\rm{\Gamma }}}_{{h}_{1}}}\,\left({g}_{\alpha \gamma }-\displaystyle \frac{{q}_{\alpha }{q}_{\gamma }}{{m}_{{h}_{1}}^{2}}\right)\,{p}_{4}^{\gamma }.\end{array}\end{eqnarray}$
This amplitude can be used to evaluate its decay width: $\begin{eqnarray}\begin{array}{l}{\rm{\Gamma }}\left(X\to J/\psi {h}_{1}\to J/\psi {\pi }^{+}{\pi }^{-}{\pi }^{0}\right)\\ =\displaystyle \frac{{g}_{{h}_{1}3\pi }^{2}}{3}\displaystyle \frac{{\left(2\pi \right)}^{4}}{2{m}_{X}}\displaystyle \int \displaystyle \frac{{{\rm{d}}}^{3}{p}_{1}}{{\left(2\pi \right)}^{3}2{E}_{1}}\displaystyle \frac{{{\rm{d}}}^{3}{p}_{2}}{{\left(2\pi \right)}^{3}2{E}_{2}}\displaystyle \frac{4\pi {p}_{3x}^{2}}{{\left(2\pi \right)}^{6}2{E}_{3}2{E}_{4}}\\ \times \left|\displaystyle \frac{1}{\tfrac{{p}_{3x}}{{E}_{3}}+\tfrac{{p}_{1x}+{p}_{2x}+{p}_{3x}}{{E}_{4}}}\right|\,{\left|\displaystyle \frac{1}{{q}^{2}-{m}_{{h}_{1}}^{2}+{\rm{i}}{m}_{{h}_{1}}{{\rm{\Gamma }}}_{{h}_{1}}}\right|}^{2}\\ \times \,{g}_{\psi {h}_{1}}\,{\epsilon }_{\mu \nu \rho \sigma }{\epsilon }^{\rho \sigma \alpha \beta }{q}_{\beta }\,{g}_{\psi {h}_{1}}^{* }\,{\epsilon }_{{\mu }^{{\prime} }{\nu }^{{\prime} }{\rho }^{{\prime} }{\sigma }^{{\prime} }}{\epsilon }^{{\rho }^{{\prime} }{\sigma }^{{\prime} }{\alpha }^{{\prime} }{\beta }^{{\prime} }}{q}_{{\beta }^{{\prime} }}\\ \times \left({g}^{\mu {\mu }^{{\prime} }}-\displaystyle \frac{{p}^{\mu }{p}^{{\mu }^{{\prime} }}}{{m}_{X}^{2}}\right)\,\left({g}^{\nu {\nu }^{{\prime} }}-\displaystyle \frac{{p}_{1}^{\nu }{p}_{1}^{{\nu }^{{\prime} }}}{{m}_{J/\psi }^{2}}\right)\\ \times \left({g}_{\alpha \gamma }-\displaystyle \frac{{q}_{\alpha }{q}_{\gamma }}{{m}_{{h}_{1}}^{2}}\right)\,\left({g}_{{\alpha }^{{\prime} }{\gamma }^{{\prime} }}-\displaystyle \frac{{q}_{{\alpha }^{{\prime} }}{q}_{{\gamma }^{{\prime} }}}{{m}_{{h}_{1}}^{2}}\right)\,{p}_{4}^{\gamma }{p}_{4}^{{\gamma }^{{\prime} }}.\end{array}\end{eqnarray}$
Again, the phase space integration is done in the reference frame where ${p}_{3}=\left({E}_{3},{p}_{3x},0,0\right)$, and p3x satisfies p3x > 0 as well as $\begin{eqnarray}{E}_{1}+{E}_{2}+{E}_{3}+{E}_{4}={m}_{X}.\end{eqnarray}$
Appendix B. Uncertainties due to the phase angle
There are two different effective Lagrangians for the X(3872) decay into the $D{\bar{D}}^{* }$ final state, as given in equations (32 ) and (33 ):
$\begin{eqnarray}{{ \mathcal L }}_{D{\bar{D}}^{* }}^{S}={g}_{D{\bar{D}}^{* }}^{S}\,{X}^{\mu }\,{D}^{0}\,{\bar{D}}_{\mu }^{* 0}\,+\,\cdots ,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{{ \mathcal L }}_{D{\bar{D}}^{* }}^{D}={g}_{D{\bar{D}}^{* }}^{D}\times \left({g}^{\mu \sigma }{g}^{\nu \rho }-{g}^{\mu \nu }{g}^{\rho \sigma }\right)\\ \times {X}^{\mu }\,{\partial }_{\rho }{D}^{0}\,{\partial }_{\sigma }{\bar{D}}_{\nu }^{* 0}+\cdots .\end{array}\end{eqnarray}$
There can be a phase angle θ between ${g}_{D{\bar{D}}^{* }}^{S}$ and ${g}_{D{\bar{D}}^{* }}^{D}$, which parameter is not fixed. We rotate it to be φ = π, and redo the previous calculations. The results are summarized in table 3, where only the relative branching ratio $\displaystyle \frac{{ \mathcal B }(| D{\bar{D}}^{* };{1}^{++}\rangle \to {D}^{0}{\bar{D}}^{* 0}(\to {D}^{0}{\bar{D}}^{0}{\pi }^{0}))}{{ \mathcal B }(| D{\bar{D}}^{* };{1}^{++}\rangle \to J/\psi \omega (\to \pi \pi \pi ))}$ is influenced by this parameter.Table 3. Relative branching ratios of the X(3872) evaluated through the Fierz rearrangement. In this table we fix the phase angle θ between the S- and D-wave coupling constants, ${g}_{D{\bar{D}}^{* }}^{S}$ and ${g}_{D{\bar{D}}^{* }}^{D}$, to be θ = π. |
| Channels | $| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle $ | $| D{\bar{D}}^{* };{1}^{++}\rangle $ | ||||
|---|---|---|---|---|---|---|
| $I=0/{\theta }_{1}^{{\prime} }={0}^{{\rm{o}}}$ | ${\theta }_{1}^{{\prime} }=+{15}^{{\rm{o}}}$ | ${\theta }_{1}^{{\prime} }=-{15}^{{\rm{o}}}$ | $I=0/{\theta }_{2}^{{\prime} }={0}^{{\rm{o}}}$ | ${\theta }_{2}^{{\prime} }=+{15}^{{\rm{o}}}$ | ${\theta }_{2}^{{\prime} }=-{15}^{{\rm{o}}}$ | |
| $\displaystyle \frac{{ \mathcal B }(X\to {\eta }_{c}{f}_{0}\to {\eta }_{c}\pi \pi )}{{ \mathcal B }(X\to J/\psi \omega \to J/\psi \pi \pi \pi )}$ | ∼0.091 | ∼0.091 | ∼0.091 | ∼0.091 | ∼0.091 | ∼0.091 |
| $\displaystyle \frac{{ \mathcal B }(X\to {\chi }_{c1}{f}_{0}\to {\chi }_{c1}\pi \pi )}{{ \mathcal B }(X\to J/\psi \omega \to J/\psi \pi \pi \pi )}$ | ∼0.086 | ∼0.086 | ∼0.086 | ∼0.086 | ∼0.086 | ∼0.086 |
| $\displaystyle \frac{{ \mathcal B }(X\to J/\psi {h}_{1}\to J/\psi \pi \pi \pi )}{{ \mathcal B }(X\to J/\psi \omega \to J/\psi \pi \pi \pi )}$ | 1.4 × 10−3 | 1.4 × 10−3 | 1.4 × 10−3 | 1.4 × 10−3 | 1.4 × 10−3 | 1.4 × 10−3 |
| $\displaystyle \frac{{ \mathcal B }(X\to {\chi }_{c0}\pi )}{{ \mathcal B }(X\to J/\psi \rho \to J/\psi \pi \pi )}$ | — | 0.024 | 0.024 | — | 0.024 | 0.024 |
| $\displaystyle \frac{{ \mathcal B }(X\to J/\psi \omega \to J/\psi \pi \pi \pi )}{{ \mathcal B }(X\to J/\psi \rho \to J/\psi \pi \pi )}$ | — | 1.6 (input) | 1.6 (input) | — | 1.6 (input) | 1.6 (input) |
| $\displaystyle \frac{{ \mathcal B }(X\to {D}^{0}{\bar{D}}^{* 0}+{D}^{* 0}{\bar{D}}^{0}\to {D}^{0}{\bar{D}}^{0}{\pi }^{0})}{{ \mathcal B }(X\to J/\psi \omega \to J/\psi \pi \pi \pi )}$ | 0.021 t1 | 0.034 t1 | 0.011 t1 | 4.5 t2 | 7.4 t2 | 2.4 t2 |
Using the mixing angle ${\theta }_{1}^{{\prime} }=+{15}^{{\rm{o}}}$, we obtain
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{ \mathcal B }\left(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \omega (\to \pi \pi \pi ):\,\,J/\psi \rho (\to \pi \pi )\,:\,\,{\chi }_{c0}\pi \,:{\eta }_{c}{f}_{0}(\to \pi \pi ):{\chi }_{c1}{f}_{0}(\to \pi \pi ):{D}^{0}{\bar{D}}^{* 0}(\to {D}^{0}{\bar{D}}^{0}{\pi }^{0})\,\right)}{{ \mathcal B }(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \omega (\to \pi \pi \pi )}\\ \sim 1:\,\,0.63\,(\mathrm{input})\,\,:\,0.015\,:\,\,0.091\,(?)\,:\,\,\,0.086\,(?)\,\,:\,\,\,\,\,\,\,0.034\,{t}_{1}\,,\end{array}\end{eqnarray}$
while using the mixing angle ${\theta }_{1}^{{\prime} }=-{15}^{{\rm{o}}}$, we obtain $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{ \mathcal B }\left(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \omega (\to \pi \pi \pi ):\,\,J/\psi \rho (\to \pi \pi )\,:\,\,{\chi }_{c0}\pi \,:{\eta }_{c}{f}_{0}(\to \pi \pi ):{\chi }_{c1}{f}_{0}(\to \pi \pi ):{D}^{0}{\bar{D}}^{* 0}(\to {D}^{0}{\bar{D}}^{0}{\pi }^{0})\,\right)}{{ \mathcal B }\left(| {0}_{{qc}}{1}_{\bar{q}\bar{c}};{1}^{++}\rangle \to J/\psi \omega (\to \pi \pi \pi )\right)}\\ \sim 1:\,\,0.63\,(\mathrm{input})\,\,:\,0.015\,:\,\,0.091\,(?)\,:\,\,\,0.086\,(?)\,\,:\,\,\,\,\,\,\,0.011\,{t}_{1}\,.\end{array}\end{eqnarray}$