Strong decays of $T^a_{c{\bar{s}0}}(2900)^{++/0}$ as a fully open-flavor tetraquark state

Abstract

We have studied the strong decay properties of the recently observed $T^a_{c{\bar{s}0}}(2900)^{++/0}$ by considering it as a $cu{\bar{d}}{\bar{s}}/cd{\bar{u}}{\bar{s}}$ fully open-flavor tetraquark state with $I(J^P)=1(0^+)$. In the framework of QCD sum rules, we have calculated the three-point correlation functions of the two-body strong decay processes $T^a_{c\bar{s}0}(2900)^{++}\rightarrow D_s^+\pi ^+$, $D^+K^+, D_s^{*+}\rho ^+$ and $D_{s1}^+\pi ^+$. The full width of $T^a_{c{\bar{s}0}}(2900)^{++/0}$ is obtained as $161.7\pm 94.8$ MeV, which is consistent with the experimental observation. We predict the relative branching ratios as $\Gamma (T\rightarrow D_s\pi ): \Gamma (T\rightarrow DK):\Gamma (T\rightarrow D_s^{*} \rho ):\Gamma (T\rightarrow D_{s1}\pi )\approx 1.00:1.10:0.04:0.43$, implying that the main decay modes of $T^a_{c{\bar{s}0}}(2900)^{++/0}$ state are $D_s\pi$ and DK channels in our calculations. However, the P-wave decay mode $D_{s1}\pi$ is also comparable and important by including the uncertainties. To further identify the nature of $T^a_{c\bar{s}0}(2900)^{++/0}$, we suggest confirming them in the DK and $D_{s}\pi$ final states, and measuring the above ratios in future experiments.

1 Introduction

With significant experimental and theoretical progress in the past one and a half decades, more and more charmoniumlike and bottomoniumlike states (the XYZ states) have been observed, including the hidden-charm pentaquark $P_c$ and $P_{cs}$ states, the doubly charmed $T_{cc}$ state, and the fully-charm tetraquark states [1,2,3,4,5,6]. These states provide a very good platform for identifying exotic states and for understanding the nonperturbative behavior of QCD. The story is ongoing with recent observation of $T^a_{c\bar{s}0}(2900)^{++/0}$ states [7, 8].

Very recently, the LHCb Collaboration reported two new tetraquark candidates $T^a_{c{\bar{s}0}}(2900)^{++}$ and $T^a_{c\bar{s}0}(2900)^{0}$ with significance more than $9\sigma$ in the $D_s^+\pi ^+$ and $D_s^+\pi ^-$ invariant mass spectra of B decay processes, respectively [7, 8]. Their mass and decay width are measured respectively as

$$\begin{aligned} \begin{aligned} T^a_{c{\bar{s}0}}(2900)^{0}:\,&M=2.892\pm 0.014\pm 0.015~\text {GeV}\, ,\\&\Gamma =0.119\pm 0.026\pm 0.013 ~\text {GeV}\, ,\\ T^a_{c{\bar{s}0}}(2900)^{++}:\,&M=2.921\pm 0.017\pm 0.020~\text {GeV}\, ,\\&\Gamma =0.137\pm 0.032\pm 0.017 ~\text {GeV}\, . \end{aligned} \end{aligned}$$

(1)

Assuming $T^a_{c{\bar{s}0}}(2900)^{++}$ and $T^a_{c{\bar{s}0}}(2900)^{0}$ belong to the same isospin triplet, the common mass and width are determined to be

$$\begin{aligned} \begin{aligned} M^{com}&=2.908\pm 0.011\pm 0.020~\text {GeV}\, , \\ \Gamma ^{com}&=0.136\pm 0.023\pm 0.011~\text {GeV}\, , \end{aligned} \end{aligned}$$

(2)

while their spin-parity was determined as $J^P=0^+$ over other possibilities by at least $7.5\sigma$. Observed in the $D_s\pi$ final states, the $T^a_{c{\bar{s}0}}(2900)^{++}$ and $T^a_{c\bar{s}0}(2900)^{0}$ states are composed of four different flavor quarks. They are fully open-flavor isovector $cu{\bar{d}}{\bar{s}}$ and $cd\bar{u}{\bar{s}}$ tetraquark states.

The observations of $T^a_{c{\bar{s}0}}(2900)^{++}$ and $T^a_{c\bar{s}0}(2900)^{0}$ are in excellent agreement with our earlier theoretical calculations for the masses of fully open-flavor charmed $cu{\bar{d}}{\bar{s}}$ and $cd{\bar{u}}{\bar{s}}$ tetraquark states in QCD sum rules [9]. In Ref. [9], we used the following interpolating currents with $I(J^P)=1(0^+)$ to study the fully open-flavor $sd{\bar{u}}{\bar{c}}$ and $su{\bar{d}}{\bar{c}}$ tetraquarks

$$\begin{aligned} J_2&=s^T_aC\gamma _\mu q_{1b}({\bar{q}}_{2a}\gamma ^\mu C{\bar{c}}^T_b-{\bar{q}}_{2b}\gamma ^\mu C{\bar{c}}^T_a)\, , \end{aligned}$$

(3)

$$\begin{aligned} J_{5\mu \nu }&=s^T_aC\gamma _\mu q_{1b}({\bar{q}}_{2a}\gamma _\nu C{\bar{c}}^T_b-{\bar{q}}_{2b}\gamma _\nu C{\bar{c}}^T_a)\, , \end{aligned}$$

(4)

in which $q_{1(2)}$ denotes up(down) quark field, and a, b are color indices. These two interpolating currents are composed of a pair of axial-vector diquark and antidiquark fields ($C\gamma _\mu \times \gamma _\nu C$) in the antisymmetric $[\varvec{{\bar{3}}_c}]_{sq_1} \otimes [\mathbf {3_c}]_{\bar{q_2}\bar{c}}$ color structure. The tensor current $J_{5\mu \nu }$ can couple to the tetraquarks with quantum numbers $I(J^P)=1(0^+)$ via its traceless symmetric part $J_{5\mu \nu }(S)$ and trace part $J_{5\mu \nu }(T)$. Using these two currents, the mass sum rules in Ref. [9] gave hadron masses as $2.91\pm 0.14$ GeV ($J_2$) and $2.88\pm 0.15$ GeV ($J_{5\mu \nu }(T)$) for the $sd\bar{u}{\bar{c}}$/$su{\bar{d}}{\bar{c}}$ tetraquark state, which are in good agreement with the measured mass of $T^a_{c{\bar{s}0}}(2900)^{++/0}$ in Eq. (2). Furthermore, in Ref. [9] we systematically analyzed the possible decay behaviors of these fully open-flavor charmed tetraquarks. In particular, we suggested searching for the doubly-charged charmed tetraquarks in the $D_s\pi$ final states in the abstract of Ref. [9], considering they are absolute exotic states that cannot be composed in the quark-antiquark formalism. The compact tetraquark interpretations of $T^a_{c{\bar{s}0}}(2900)^{++/0}$ are also supported by the investigations of their masses and decays in the methods of the nonrelativistic potential quark model [10] and the color flux-tube model [11]. These fully open-flavor charmed tetraquark states were also studied in the framework of QCD sum rules in Refs. [12,13,14].

Shortly after their observations, the mass of $T^a_{c\bar{s}0}(2900)^{++/0}$ has been reproduced by considering it as the $D_s^{*}\rho$ hadronic molecule in Ref. [15]. Since the mass of the $T^a_{c{\bar{s}0}}(2900)^{++/0}$ is also close to the threshold of $D^{*}K^{*}$, the $T^a_{c\bar{s}0}(2900)^{++/0}$ is interpreted as the $D^{*}K^{*}$ hadronic molecule in Refs. [16,17,18], in which both the mass and decay behaviors are studied. However, the calculations within the Bethe–Salpeter framework indicate that the $I=1$ bound state can not exist in the $D^{*}K^{*}$ system [19], which is in conflict with the conclusions of Refs. [16,17,18]. In addition, the $T^a_{c{\bar{s}0}}(2900)^{++/0}$ may also be explained as a resonance-like structure around the $D^{*}K^{*}$ threshold [20], or as a threshold effect from the interaction of the $D^{*}K^{*}$ and $D_{s}^{*}\rho$ channels [21].

One and a half decades ago, the exotic $cq{\bar{s}}{\bar{q}}$ tetraquark state with $I(J^P)=0(0^+)$ was extensively studied to understand the structure of the charm-strange state $D_{s0}^*(2317)$ [22,23,24,25], in which the mass of such isoscalar scalar $cq{\bar{s}}{\bar{q}}$ tetraquark was given around 2.3$-$2.4 GeV. In Ref. [26], the masses of the scalar $cq{\bar{s}}\bar{q}$ tetraquarks with $I=0$ and $I=1$ were calculated as 2731 MeV and 2699 MeV respectively. All these results are much lower than the mass of $T^a_{c{\bar{s}0}}(2900)^{++/0}$. In Ref. [27], the authors predicted four isoscalar scalar $cu{\bar{s}}{\bar{d}}$ tetraquarks with masses (2213, 2537, 2919, 3150) MeV and decay widths (70, >110, 40, >110) MeV by using a color-magnetic interaction model, in which only the second-highest state lies close to the $T^a_{c{\bar{s}0}}(2900)^{++}$ with much narrower decay width.

To identify the nature of $T^a_{c{\bar{s}0}}(2900)^{++}$, we investigate its hadronic decay processes by calculating the three-point correlation functions in the framework of the QCD sum rules: $T^a_{c{\bar{s}0}}(2900)^{++}\rightarrow D_s^+\pi ^+$, $D^+K^+, D_s^{*+}\rho ^+$, $D_{s1}^+\pi ^+$. The first three decay processes are in S-wave decay modes while the last one is in P-wave. Inspired by the accurate prediction on the mass of $T^a_{c\bar{s}0}(2900)^{++}$ in Ref. [9], we study these decays by using the interpolating current $J_{5\mu \nu }$ (actually its charge conjugate current with two positive charges) in Eq. (4) and considering $T^a_{c\bar{s}0}(2900)^{++}$ as a $cu{\bar{d}}{\bar{s}}$ compact tetraquark with $I(J^P)=1(0^+)$. As an isospin partner, the neutral state $T^a_{c{\bar{s}0}}(2900)^{0}$ shall have the same mass and decay width with $T^a_{c{\bar{s}0}}(2900)^{++}$ in our calculations. The light-cone sum rule is another useful tool to study the strong couplings and hadron properties [18, 28,29,30,31]. We shall also consider to investigate the strong decay properties of $T^a_{c{\bar{s}0}}(2900)^{++/0}$ by using the light-cone sum rules in the near future.

This paper is organized as follows. In Sect. 2, we use QCD sum rules to study the three-point correlation functions for $T^{a++}_{c{\bar{s}0}}D_s^+\pi ^+$, $T^{a++}_{c{\bar{s}0}}D^+K^+$, $T^{a++}_{c{\bar{s}0}}D_s^{*+}\rho ^+$, $T^{a++}_{c\bar{s}0}D_{s1}^+\pi ^+$ vertices. We will study the operator product expansion (OPE) series up to dimension 8 condensates with various tensor structures, and accordingly calculate the coupling constants and partial decay widths of these channels. Finally, we summarize our results on the decay behavior of $T^a_{c{\bar{s}0}}(2900)^{++/0}$.

2 QCD sum rules and three-point correlation function

Over past several decades, the method of QCD sum rules has been proven to be very powerful to study hadron properties [32,33,34,35,36]. In this work, we shall study the three-point correlation function of two-body strong decay process $T^a_{c{\bar{s}0}}(2900)^{++}\rightarrow A+B$ as

$$\begin{aligned} \Pi (p, p^{\prime }, q)=&\int d^4xd^4y\,e^{ip^{\prime } \cdot x}e^{iq \cdot y} \nonumber \\&\times \langle 0|{{\mathbb {T}}}[J_A(x)J_B(y)J_{T^a_{c\bar{s}0}}^\dag (0)]|0\rangle , \end{aligned}$$

(5)

where $J_{T^a_{c{\bar{s}0}}}$ is the interpolating current for the $T^a_{c{\bar{s}0}}(2900)^{++}$ state while $J_A$ and $J_B$ are the currents for the final states A and B, respectively. The three point function in Eq. (5) associates the momentum p with $T^a_{c{\bar{s}0}}$, q with B, and $p'=p-q$ with A. We shall consider the $T^a_{c{\bar{s}0}}(2900)^{++}\rightarrow D_s^+\pi ^+$, $D^+K^+, D_s^{*+}\rho ^+$, $D_{s1}^+\pi ^+$ decay processes in the following.

We use the following interpolating current for $T^a_{c\bar{s}0}(2900)^{++}$ by considering it as a $cu{\bar{d}}{\bar{s}}$ compact tetraquark with $I(J^P)=1(0^+)$

$$\begin{aligned} J_{T^{a++}_{c{\bar{s}0}}}^{\mu \nu }=c^T_aC\gamma ^\mu u_b({\bar{d}}_{a}\gamma ^\nu C{\bar{s}}^T_b-{\bar{d}}_{b}\gamma ^\nu C{\bar{s}}^T_a)\, . \end{aligned}$$

(6)

As mentioned above, the hadron mass extracted from this current is $2.88\pm 0.15$ GeV in Ref. [9], which is in excellent agreement with the observed mass of $T^a_{c\bar{s}0}(2900)^{++}$ [7, 8]. The current in Eq. (6) can couple to the $T^a_{c{\bar{s}0}}(2900)^{++}$ state via

$$\begin{aligned} \langle 0|J_{T^{a++}_{c{\bar{s}0}}}^{\mu \nu }|T^{a++}_{c\bar{s}0}(p)\rangle =f_{0T}g^{\mu \nu }\,, \end{aligned}$$

(7)

in which the coupling constant $f_{0T}$ can be determined from the two-point mass sum rules established in Ref. [9]. Using the same QCD parameters and the same working regions of the Borel mass and continuum threshold as Ref. [9], we can obtain the value of this coupling constant as

$$\begin{aligned} f_{0T}=(3.0\pm 0.4)\times 10^{-3}~\text {GeV}^5 . \end{aligned}$$

(8)

In the following analyses, we shall adopt this value to calculate the decay width of the $T^a_{c{\bar{s}0}}(2900)^{++}$ state.

2.1 Decay mode of $T^a_{c{\bar{s}0}}(2900)^{++}\rightarrow D_s^+\pi ^+$

In this subsection, we study the decay process $T^a_{c\bar{s}0}(2900)^{++}\rightarrow D_s^+\pi ^+$, the mode in which the $T^a_{c{\bar{s}0}}(2900)^{++}$ state was observed. To calculate the three-point correlation function in Eq. (5) for the vertex $T^a_{c{\bar{s}0}}(2900)^{++}D_s^+\pi ^+$, we need the interpolating currents for $D_s^+$ and $\pi ^+$ mesons

$$\begin{aligned} J_{D_s^+}= & {} {\bar{s}}_ai\gamma _5c_a, \end{aligned}$$

(9)

$$\begin{aligned} J_{\pi ^+}= & {} {\bar{d}}_ai\gamma _5 u_a. \end{aligned}$$

(10)

They can couple to the pseudoscalar $D_s^+$ and $\pi ^+$ mesons via the following relations [37]

$$\begin{aligned} \langle 0|J_{D_s^+}|D_s^+(p^{\prime })\rangle= & {} \frac{f_{D_s}m^2_{D_s}}{m_c+m_s}\equiv \lambda _{D_s}\,, \end{aligned}$$

(11)

$$\begin{aligned} \langle 0|J_{\pi ^+}|\pi ^+(q)\rangle= & {} \frac{f_{\pi }m^2_{\pi }}{m_u+m_d}\equiv \lambda _{\pi }\,, \end{aligned}$$

(12)

where $f_{D_s}$ and $f_{\pi }$ are coupling constants for $D_s^+$ and $\pi ^+$, respectively. From the definition of the three-point correlation function, momentum conservation implies $p=p^{\prime }+q$ in the two-body decay process. Using the above two relations and Eq. (7), we can write down the three-point correlation function Eq. (5) on the phenomenological side

$$\begin{aligned}&\Pi ^{\mu \nu (\textrm{PH})}_{D_s \pi }(p, p^\prime , q) \nonumber \\&\quad =\int d^4x d^4y e^{ip^\prime x} e ^{i q y} \langle 0|{{\mathbb {T}}} [J_{D_s^+}(x) J_{\pi ^+}(y) J^{\mu \nu \dagger }_{T^{a++}_{c\bar{s}0}}(0)] |0\rangle \nonumber \\&\quad = \frac{ g^{\mu \nu }f_{0T} \lambda _{D_s}\lambda _{\pi }\langle D_s^+(p^\prime )\pi ^+(q)|T^{a++}_{c\bar{s}0}(p)\rangle }{(p^2-m_{T^a_{c{\bar{s}0}}}^2 +i\epsilon ) (p^{\prime 2}-m_{D_s}^2 +i\epsilon )(q^2-m_\pi ^2 +i\epsilon )} +\cdots \, , \end{aligned}$$

(13)

in which the “$\cdots$” represents the contributions from higher excited states. We insert the intermediate hadronic states $\sum \limits _{X} |X\rangle \langle X|$ into the correlation function in the last step of Eq. (13) to obtain a representation of $\Pi ^{\mu \nu (\textrm{PH})}_{D_s \pi }(p, p^\prime , q)$ in terms of hadron parameters that correspond to the currents $J_{D_s^+}$, $J_{\pi ^+}$ and $J^{\mu \nu }_{T^{a++}_{c{\bar{s}0}}}$. The coupling constant $g_{T^a_{c{\bar{s}0}}D_s \pi }$ is defined via the effective Lagrangian [38, 39]

$$\begin{aligned} \mathcal{L}_{T^a_{c{\bar{s}0}}D_s\pi }=-g_{T^a_{c{\bar{s}0}}D_s \pi } T_{c\bar{s}0}^{a++} \partial ^\mu \pi ^+ \partial _\mu D_s^+ + h.c. \,. \end{aligned}$$

(14)

The transition matrix element $\langle D_s^+(p^\prime )\pi ^+(q)|T^{a++}_{c{\bar{s}0}}(p)\rangle$ in Eq. (13) can be thus obtained as

$$\begin{aligned} \langle D_s^+(p^\prime )\pi ^+(q)|T^{a++}_{c\bar{s}0}(p)\rangle =g_{T^a_{c{\bar{s}0}}D_s\pi }p^{\prime }\cdot q,\, \end{aligned}$$

(15)

Then the three-point correlation function is written as

$$\begin{aligned}&\Pi ^{\mu \nu (\textrm{PH})}_{D_s \pi }(p, p^\prime , q)\nonumber \\&\quad =\frac{ g^{\mu \nu }f_{0T} \lambda _{D_s}\lambda _{\pi } g_{T^a_{c\bar{s}0}D_s\pi }p^{\prime }\cdot q}{(p^2-m_{T^a_{c{\bar{s}0}}}^2 +i\epsilon ) (p^{\prime 2}-m_{D_s}^2 +i\epsilon )(q^2-m_\pi ^2 +i\epsilon )}+\cdots \, . \end{aligned}$$

(16)

To calculate the three-point correlation function at the quark-gluon level, we adopt the coordinate-space expression for the light quark propagators and momentum-space expression for the charm quark propagator

$$\begin{aligned} iS_{ab}^q(x)= & {} \frac{i\delta _{ab}}{2\pi ^2x^4}{\hat{x}} -\frac{i}{32\pi ^2}\frac{\lambda ^n_{ab}}{2}g_sG_{\mu \nu }^n\frac{1}{x^2}(\sigma ^{\mu \nu }{\hat{x}} +{\hat{x}}\sigma ^{\mu \nu }) \nonumber \\{} & {} -\frac{\delta _{ab}}{12}\langle {\bar{q}}q\rangle - \frac{\delta _{ab}x^2}{192}\langle g_s {\bar{q}}\sigma Gq\rangle , \end{aligned}$$

(17)

$$\begin{aligned} iS_{ab}^{c}(p)= & {} \frac{i\delta _{ab}(p\!\!\!/+m_c)}{p^2-m_c^2}-\frac{i}{4}g_s\frac{\lambda ^n_{ab}}{2}G^ n_{\mu \nu }\frac{1}{(p^2-m_c^2)^2} \nonumber \\{} & {} \times \{\sigma _{\mu \nu }(p\!\!\!/+m_c)+(p\!\!\!/+m_c) \sigma _{\mu \nu }\}, \end{aligned}$$

(18)

where $m_c$ is the mass of the charm quark. We have neglected the chirally-suppressed contributions from the current light quark masses ($m_q=0$ in the chiral limit) in the OPE calculations because they are numerically insignificant. To establish a sum rule for the coupling constant $g_{T^a_{c{\bar{s}0}}D_s\pi }$, we will pick out the $1/q^2$ terms with the structure $g^{\mu \nu }p^{\prime }\cdot q$ in the OPE series and then equate it with the three-point correlation function in Eq. (16) at the hadron level. For this tensor structure, it is found that the quark condensate $\langle {\bar{qq}}\rangle$, the quark-gluon mixed condensate $\langle {\bar{qg}}_s\sigma \cdot Gq\rangle$ and the $\langle g_s^2GG\rangle \langle {\bar{qq}}\rangle$ condensate give contributions to the OPE side at order $1/q^2$ ($q^2$ pole)

$$\begin{aligned} \rho _{_{D_s\pi }}(s)= & {} \left( (m_c-m_s)^2-s\right) (m_c+m_s)\nonumber \\{} & {} \times \left( \frac{\langle {\bar{qq}}\rangle f(s)}{8\pi ^2 s}-\frac{\langle {\bar{qg}}_s\sigma \cdot Gq\rangle }{96\pi ^2 s^2 f(s)}\right) , \end{aligned}$$

(19)

$$\begin{aligned} \Pi _{_{D_s\pi }}^{\langle g_s^2GG\rangle \langle {\bar{qq}}\rangle }(p'^2)&=\frac{\langle g_s^2GG\rangle \langle {\bar{qq}}\rangle }{288 \pi ^2}\int _{0}^{1}dx\frac{1}{ \Delta (p'^2)^3 (x-1)^2 x^2}\\&\quad \times \left( 2 m_c^3 x^3+2 m_c^2 m_s \left( x^2-x+1\right) x\right. \\&\quad -\Delta (p'^2) m_s (x-1)^2 \left( x^2+2\right) \\&\quad +2 m_c m_s^2 (x-1)^2\\&\quad \left. -\Delta (p'^2) m_c \left( x^2-2 x+3\right) x^2\right) , \end{aligned}$$

where

$$\begin{aligned} f(s)\equiv&\sqrt{\left( 1-\frac{(m_c-m_s)^2}{s}\right) \left( 1-\frac{(m_c+m_s)^2}{s}\right) }\, ,\end{aligned}$$

(20)

$$\begin{aligned} \Delta (p'^2)\equiv&\frac{m_c^2}{1-x}-p'^2\, , \end{aligned}$$

(21)

in which $m_s$ is the strange quark mass.

The three-point correlation function in the OPE side can be written as

$$\begin{aligned} \Pi ^{\mu \nu (\textrm{OPE})}_{D_s \pi }=\frac{g^{\mu \nu }p^{\prime }\cdot q}{q^2}\left[ \int _{<}^{s_0}ds \frac{\rho ^{OPE}(s)}{s-p'^2}+\int _{s_0}^{\infty }ds \frac{\rho ^{OPE}(s)}{s-p'^2}\right] \,. \end{aligned}$$

(22)

Comparing to Eq. (16), it is assumed that the contributions from the higher excited states of “$\cdots$” can be compensated by the second term of Eq. (22) above the continuum threshold $s_0$, as usual in the QCD sum rule method. Using the soft-pion approximation [33] in Eq. (16), one can set up $p^2=p^{\prime 2}=P^2$ ($q\sim 0$) and derive a sum rule for the coupling constant $g_{T^a_{c{\bar{s}0}}D_s \pi }$ at the $q^2$ pole after performing the Borel transform ($P^2\rightarrow M_B^2$) on both the phenomenological and OPE sides

$$\begin{aligned} g_{T^a_{c{\bar{s}0}}D_s \pi }(s_0, M_B^2)&= \frac{1}{f_{0T} \lambda _{D_s}\lambda _{\pi }}\frac{m^2_{T^a_{c\bar{s}0}}-m^2_{D_s}}{e^{-m^2_{D_s}/M_B^2}-e^{-m^2_{T^a_{c\bar{s}0}}/M_B^2}}\nonumber \\&\quad \times \Bigg (\int _{(m_c+m_s)^2}^{s_0}\rho _{_{D_s\pi }}(s)e^{-s/M_B^2}ds\nonumber \\&\quad +\Pi _{_{D_s\pi }}^{\langle g_s^2GG\rangle \langle {\bar{qq}}\rangle }(M_B^2)\Bigg )\, , \end{aligned}$$

(23)

in which $M^2_B$ and $s_0$ are the Borel parameter and continuum threshold, respectively. $\Pi _{_{D_s\pi }}^{\langle g_s^2GG\rangle \langle {\bar{qq}}\rangle }(M_B^2)$ is the Borel transformation of $\Pi _{_{D_s\pi }}^{\langle g_s^2GG\rangle \langle {\bar{qq}}\rangle }(p'^2)$.

To perform the QCD sum-rule numerical analysis, the following QCD parameter values of quark masses and various condensates are adopted [33, 40,41,42]

$$\begin{aligned}&m_u=2.16^{+0.49}_{-0.26} ~\text {MeV}\, ,\nonumber \\&m_d=4.67^{+0.48}_{-0.17} ~\text {MeV}\, ,\nonumber \\&m_c(\mu =m_c)={\overline{m}}_c=1.27\pm 0.02~\text {GeV}\, , \nonumber \\&m_c/m_s=11.76^{+0.05}_{-0.10} \, , \nonumber \\&\langle {\bar{qq}}\rangle =-(0.24\pm 0.01)^3~\text {GeV}^3\, ,\nonumber \\&\langle {\bar{ss}}\rangle =(0.8\pm 0.1)\langle {\bar{qq}}\rangle \, ,\nonumber \\&\langle {\bar{qg}}_s\sigma \cdot Gq\rangle =M_0^2\langle {\bar{qq}}\rangle \, ,\nonumber \\&\langle {\bar{sg}}_s\sigma \cdot Gs\rangle =M_0^2\langle {\bar{ss}}\rangle \, ,\nonumber \\&M_0^2=0.8\pm 0.2~\text {GeV}^2\, ,\nonumber \\&\langle g_s^2GG\rangle =0.48\pm 0.14~\text {GeV}^4\, , \end{aligned}$$

(24)

where the mass of the charm quark is the “running” mass in the $\overline{\text {MS}}$ scheme. We use the renormalization scheme and scale independent $m_c/m_s$ mass ratio from PDG [40] to make sure that the strange quark mass $m_s$ is at the same renormalization scale with $m_c$.

In Fig. 1, we show the contributions of each term in OPE series to the correlation function after doing the Borel transform. It is shown that the dominant contribution is from the quark condensate $\langle {\bar{qq}}\rangle$, while the $\langle g_s^2GG\rangle \langle {\bar{qq}}\rangle$ term is much smaller.

Fig. 1

The contributions of each term in OPE series to the correlation function

There are also some hadronic parameters [40]

$$\begin{aligned} m_\pi&=139.57094\pm 0.00018~\text {MeV}, \nonumber \\ f_\pi&=130.56~\text {MeV} , \nonumber \\ m_{D_s}&=1968.35\pm 0.07~\text {MeV},\nonumber \\ f_{D_s}&=246.7~\text {MeV} ,\nonumber \\ m_{T^a_{c{\bar{s}0}}}&=2908\pm 11\pm 20~\text {MeV}, \end{aligned}$$

(25)

in which we use the experimental value of the mass of $T^a_{c\bar{s}0}$ state [7, 8].

In Eq. (23), the coupling constant $g_{T^a_{c{\bar{s}0}}D_s \pi }$ depends on the continuum threshold $s_0$ and the non-physical parameter $M_B^2$ in the limit $Q^2\rightarrow 0$. The continuum threshold $s_0$ as a physical parameter should have the same value as in the mass sum rule in Ref. [9], i.e., $s_0=10~\text {GeV}^2$. With this value of $s_0$ and the parameters given in Eqs. (24) and (25), we show the variation of the coupling constant $g_{T^a_{c{\bar{s}0}}D_s \pi }$ with the Borel parameter $M_B^2$ in Fig. 2. We find that the sum rule gives a minimum value of the coupling constant $g_{T^a_{c{\bar{s}0}}D_s \pi }$ at $M_B^2\sim 1.12~\text {GeV}^2$, around which it has minimal dependence on the non-physical parameter $M_B^2$. Considering the errors from all parameters, we obtain the coupling constant as

$$\begin{aligned} g_{T^a_{c{\bar{s}0}}D_s \pi }=1.82\pm 0.52~\text {GeV}^{-1}\, . \end{aligned}$$

(26)

Fig. 2

The dependence of the coupling constant $g_{T^a_{c{\bar{s}0}}D_s \pi }$ on the Borel parameter $M_B^2$

The decay width of the two-body strong decay process $T^{a++}_{c\bar{s}0}\rightarrow D_s^+\pi ^+$ ($S\rightarrow PP$) can be derived from Eq. (15) as

$$\begin{aligned} \Gamma (T^{a++}_{c{\bar{s}0}}\rightarrow D_s^+\pi ^+)&=\frac{g^2_{T^a_{c{\bar{s}0}}D_s \pi }p^*(m_{T^a_{c\bar{s}0}},m_{D_s},m_\pi )}{32 \pi m^2_{T_{c{\bar{s}}}}}\nonumber \\&\quad \times \left( m^2_{T^a_{c{\bar{s}0}}}-m^2_{D_s}-m^2_\pi \right) ^2\,, \end{aligned}$$

(27)

where the magnitude of outgoing momentum $p^*(a,b,c)$ is defined as

$$\begin{aligned} p^*(a,b,c)\equiv \frac{\sqrt{a^4+b^4+c^4-2a^2b^2-2b^2c^2-2c^2a^2}}{2a}. \end{aligned}$$

(28)

Then we can calculate the partial decay width of the process $T^{a++}_{c{\bar{s}0}}\rightarrow D_s^+\pi ^+$ as

$$\begin{aligned} \Gamma (T^{a++}_{c{\bar{s}0}}\rightarrow D_s^+\pi ^+)=62.9\pm 36.7~\text {MeV}\, , \end{aligned}$$

(29)

where the error is mainly from the uncertainties of the quark condensate and the quark-gluon mixed condensate.

2.2 Decay mode of $T^a_{c{\bar{s}0}}(2900)^{++}\rightarrow D^+K^+$

In this subsection, we study the decay process $T^a_{c\bar{s}0}(2900)^{++}\rightarrow D^+K^+$, which is also the two pseudoscalar mesons channel. We use the following pseudoscalar interpolating currents for D and K mesons:

$$\begin{aligned} J_{D^+}= & {} {\bar{d}}_ai\gamma _5c_a\,, \end{aligned}$$

(30)

$$\begin{aligned} J_{K^+}= & {} {\bar{s}}_ai\gamma _5u_a, \end{aligned}$$

(31)

with the current-meson coupling

$$\begin{aligned} \langle 0|J_{D^+}|D^+(p^{\prime })\rangle= & {} \frac{f_{D}m^2_{D}}{m_c+m_d}\equiv \lambda _{D}\,, \end{aligned}$$

(32)

$$\begin{aligned} \langle 0|J_{K^+}|K^+(q)\rangle= & {} \frac{f_{K}m^2_{K}}{m_u+m_s}\equiv \lambda _{K}\,, \end{aligned}$$

(33)

where $f_{D}$ and $f_{K}$ are the decay constants for D and K mesons, respectively. Then the three-point function at the hadron level can be written as

$$\begin{aligned}&\Pi ^{\mu \nu (\textrm{PH})}_{D K}(p, p^\prime , q)\nonumber \\&\quad =\int d^4x d^4y e^{ip^\prime x} e ^{i q y} \langle 0|{{\mathbb {T}}} [J_{D^+}(x) J_{K^+}(y) J^{\mu \nu \dagger }_{T^{a++}_{c{\bar{s}0}}}(0) ]|0\rangle \nonumber \\&\quad = \frac{g^{\mu \nu }f_{0T} \lambda _{D}\lambda _{K}\langle D^+(p^\prime )K^+(q)|T^{a++}_{c{\bar{s}0}}(p)\rangle }{(p^2-m_{T^a_{c\bar{s}0}}^2 +i\epsilon ) (p^{\prime 2}-m_{D}^2 +i\epsilon )(q^2-m_K^2 +i\epsilon )} +\cdots \nonumber \\&\quad =\frac{ g^{\mu \nu }f_{0T} \lambda _{D}\lambda _{K} g_{T^a_{c\bar{s}0}DK}p^{\prime }\cdot q}{(p^2-m_{T^a_{c{\bar{s}0}}}^2 +i\epsilon ) (p^{\prime 2}-m_{D}^2 +i\epsilon )(q^2-m_K^2 +i\epsilon )}+\cdots \, , \end{aligned}$$

(34)

where the coupling constant $g_{T^a_{c{\bar{s}0}}DK}$ is defined in the same way as $g_{T^a_{c\bar{s}0}D_s\pi }$ [38, 39]

$$\begin{aligned} \mathcal{L}_{T^a_{c{\bar{s}0}}DK}=-g_{T^a_{c{\bar{s}0}}DK} T_{c{\bar{s}0}}^{a++} \partial ^\mu K^+ \partial _\mu D^+ + h.c. \, . \end{aligned}$$

(35)

The calculation of the three-point correlation function at the quark-gluon level for this decay mode is similar to the $D_s\pi$ mode. However, we cannot simply ignore the mass of K meson in this situation as we did for $\pi$ meson. To apply sum rules appropriately, we still extract the terms proportional to $1/q^2$ with the structure $g^{\mu \nu }p^{\prime }\cdot q$ in the OPE series, implying that the coupling constant $g_{T^a_{c{\bar{s}0}}DK}$ as a function of $Q^2(=-q^2)$ is calculated at $Q^2$ far away from the on-shell mass $-m_K^2$. Then we will extrapolate the coupling constant $g_{T^a_{c{\bar{s}0}}DK}$ at $Q^2=-m_K^2$ from the QCD sum rule working region [43, 44].

We obtain the following OPE for this channel

$$\begin{aligned}{} & {} \rho _{_{DK}}(s)=-\frac{m_c(m_c^2-s)^2(\langle {\bar{qq}}\rangle +\langle {\bar{ss}}\rangle )}{16\pi ^2s^2}\nonumber \\{} & {} \quad +\frac{m_c^3\left( \langle {\bar{qg}}_s\sigma \cdot Gq\rangle -\langle {\bar{sg}}_s\sigma \cdot Gs\rangle \right) }{192\pi ^2s^2}\nonumber \\{} & {} \quad +\frac{m_s\langle {\bar{qg}}_s\sigma \cdot Gq\rangle \delta (s-m_c^2)}{192\pi ^2}+\frac{m_c\langle {\bar{sg}}_s\sigma \cdot Gs\rangle }{96\pi ^2s}\nonumber \\{} & {} \quad +\frac{\langle g_s^2GG\rangle m_cm_s(2m_c^2-7s)}{4608\pi ^2s^2}\,, \end{aligned}$$

(36)

$$\begin{aligned}{} & {} \Pi _{_{DK}}^{\langle g_s^2GG\rangle \langle {\bar{qq}}\rangle }(p'^2)=\frac{\langle g_s^2GG\rangle \langle {\bar{qq}}\rangle m_s}{1152\pi ^2(p'^2-m_c^2)^2}\nonumber \\{} & {} \quad + \frac{\langle g_s^2GG\rangle (\langle {\bar{qq}}\rangle +\langle {\bar{ss}}\rangle ) m_c}{576 \pi ^2}\int _{0}^{1}\frac{dx}{ \Delta (p'^2) ^3 (x-1)^2}\nonumber \\{} & {} \quad \times \left( 2 m_c^2 x-\Delta (p'^2) \left( x^2-2 x+3\right) \right) \,, \end{aligned}$$

(37)

which contain the quark condensates, quark-gluon mixed condensates and gluon condensates. Then we can establish the sum rule for $g_{T^a_{c{\bar{s}0}}DK}$ by assuming $p^2=p^{\prime 2}=P^2$ and performing the Borel transform on both the phenomenological and OPE sides

$$\begin{aligned} g_{T^a_{c\bar{s}0}DK}(s_0,M_B^2,Q^2)=&\frac{1}{f_{0T}\lambda _D\lambda _K}\frac{m^2_ {T^a_{c{\bar{s}}0}}-m_D^2}{e^{-m^2_{D}/M_B^2}-e^{-m^2_{T^a_{c\bar{s}0}}/M_B^2}}\nonumber \\&\times \left( \frac{Q^2+m_K^2}{Q^2}\right) \Bigg (\int _{m_c^2}^{s_0} \rho _{_{DK}}(s)e^{-s/M_B^2}ds\nonumber \\&+\Pi _{_{DK}}^{\langle g_s^2GG\rangle \langle {\bar{qq}}\rangle }(M_B^2)\Bigg )\, . \end{aligned}$$

(38)

The following parameters for the D and K mesons are adopted to perform numerical analysis [40]

$$\begin{aligned} \begin{aligned} m_K=493.677\pm 0.016~\text {MeV},\quad&f_K=155.7~\text {MeV},\\ m_{D}=1869.66\pm 0.05~\text {MeV},\quad&f_{D}=208.9~\text {MeV}. \end{aligned} \end{aligned}$$

(39)

Fig. 3

Variation of the coupling constant $g_{T^a_{c\bar{s}0}DK}(Q^2)$ at $Q^2=3.5~\text {GeV}^2$ with the Borel parameter $M_B^2$

In Fig. 3, we show the variation of the coupling constant $g_{T^a_{c{\bar{s}0}}DK}(Q^2)$ with the Borel parameter $M_B^2$ at the Euclidean momentum point $Q^2=m_D^2\approx 3.5~\text {GeV}^2$ and the continuum threshold $s_0=10~\text {GeV}^2$. Such a momentum point is chosen far away from $m_K^2$ so that it can be safely ignored and the OPE series is valid in this region. We can see that the minimum value of $g_{T^a_{c{\bar{s}0}}DK}$ appears around $M_B^2=1.01~\text {GeV}^2$. To obtain the value of $g_{T^a_{c\bar{s}0}DK}$ at $Q^2=-m_K^2$, we extrapolate the coupling constant $g_{T^a_{c{\bar{s}0}}DK}$ from the valid QCD sum rule working region to the physical pole $Q^2=-m_K^2$ by using the exponential model [43, 44]

$$\begin{aligned} g_{T^a_{c{\bar{s}0}}DK}=g_1e^{-g_2Q^2}\,, \end{aligned}$$

(40)

where the parameters $g_1$ and $g_2$ can be determined by fitting the QCD sum rule result for $g_{T^a_{c{\bar{s}0}}DK}$. In Fig. 4, we fit the QCD sum rule result for $s_0=10~\text {GeV}^2$ and $M_B^2=1.01~\text {GeV}^2$ well described by the model Eq. (40) when $g_1=1.87~\text {GeV}^{-1}$ and $g_2=0.02~\text {GeV}^{-2}$. Then we can obtain the value of the coupling constant $g_{T^a_{c{\bar{s}0}}DK}$ at the physical pole $Q^2=-m_K^2$

$$\begin{aligned} g_{T^a_{c{\bar{s}0}}DK}(Q^2=-m_K^2)=1.88\pm 0.55~\text {GeV}^{-1}\, . \end{aligned}$$

(41)

As the $S\rightarrow PP$ decay process, the decay width for $T^a_{c\bar{s}0}(2900)^{++}\rightarrow D^+K^+$ can be evaluated by Eq. (27). Using the parameter values mentioned above, we obtain the partial decay width of this process as

$$\begin{aligned} \Gamma (T_{c{\bar{s}0}}^{a++}\rightarrow D^+K^+)=69.2\pm 40.8~\text {MeV}\, , \end{aligned}$$

(42)

where the error is mainly from the uncertainties of the quark condensates $\langle {\bar{qq}}\rangle$ and $\langle {\bar{ss}}\rangle$.

Fig. 4

Variation of the coupling constant $g_{T^a_{c\bar{s}0}DK}$ with $Q^2$. The dotted points represent the QCD sum rule results for $g_{T^a_{c{\bar{s}0}}DK}$ with $s_0=10~\text {GeV}^2$ and $M_B^2=1.01~\text {GeV}^2$. The solid line is the fit of the QCD sum rule result through model Eq. (40) and the extrapolation to the physical pole $Q^2=-m_K^2$

2.3 Decay mode of $T^a_{c{\bar{s}0}}(2900)^{++}\rightarrow D_s^{*+} \rho ^+$

In this subsection, we study the decay mode $T^a_{c\bar{s}0}(2900)^{++}\rightarrow D_s^{*+} \rho ^+$ as an $S\rightarrow VV$ process. We use the vector interpolating currents for the $D_s^{*+}$ and $\rho ^+$ mesons

$$\begin{aligned} J_{D_s^{*+}}^\alpha= & {} {\bar{s}}_a\gamma ^\alpha c_a, \end{aligned}$$

(43)

$$\begin{aligned} J_{\rho ^+}^\beta= & {} {\bar{d}}_a\gamma ^\beta u_a\,. \end{aligned}$$

(44)

The corresponding current-meson couplings are

$$\begin{aligned} \langle 0|J_{D_s^{*+}}^\alpha |D_s^{*+}(p^{\prime })\rangle= & {} f_{D^{*}_s}m_{D^{*}_s}\epsilon ^\alpha (p^\prime )=\lambda _{D_s^{*}}\epsilon ^\alpha (p^\prime )\,, \end{aligned}$$

(45)

$$\begin{aligned} \langle 0|J_{\rho ^+}^\beta |\rho ^+(q)\rangle= & {} f_{\rho }m_{\rho }\epsilon ^\beta (q)=\lambda _{\rho }\epsilon ^\beta (q)\,, \end{aligned}$$

(46)

where $f_{D^{*}_s}$ and $\epsilon ^\alpha (p^\prime )$ are the decay constant and polarization vector for $D^{*}_s$ respectively, while $f_{\rho }$ and $\epsilon ^\beta (q)$ are for the $\rho$ meson.

To match the structure of the SVV three-point correlation function, one can use the following $T^{a++}_{c{\bar{s}0}}D^{*+}_s\rho ^+$ interaction vertex [45, 46]

$$\begin{aligned} \langle D_s^{*+}(p')\rho ^+(q)\vert T^{a++}_{c{\bar{s}} 0}(p)\rangle =-g_{T^a_{c{\bar{s}} 0}D_s^*\rho } \epsilon ^{*}(p')\cdot \epsilon ^{*}(q)\,, \end{aligned}$$

(47)

in which and the coupling constant $g_{T^a_{c{\bar{s}} 0}D_s^*\rho }$ is defined through

$$\begin{aligned} \mathcal{L}_{T^a_{c{\bar{s}0}}D_s^*\rho }=-g_{T^a_{c{\bar{s}} 0}D_s^*\rho } T_{c{\bar{s}0}}^{a++} D_{s}^{*+\mu }\rho ^{+}_{\mu }+ h.c. \, . \end{aligned}$$

(48)

Then the three-point correlation function at the hadron level can be written as

$$\begin{aligned}&\Pi ^{\mu \nu , \alpha \beta (\textrm{PH})}_{D_{s}^{*} \rho }(p, p^\prime , q)\nonumber \\&\quad =\int d^4x d^4y e^{ip^\prime x} e ^{i q y} \langle 0|{{\mathbb {T}}} [J_{D_{s}^{*+}}^\alpha (x) J_{\rho ^+}^\beta (y) J^{\mu \nu \dagger }_{T^{a++}_{c{\bar{s}0}}}(0) ]|0\rangle \nonumber \\&\quad = \frac{ g^{\mu \nu }f_{0T} \lambda _{D_{s}^{*}}\lambda _{\rho }\epsilon ^\alpha (p^\prime )\epsilon ^\beta (q)\langle D_s^{*^+}(p')\rho ^+(q)\vert T^{a++}_{c{\bar{s}} 0}(p)\rangle }{(p^2-m_{T^a_{c{\bar{s}0}}}^2 +i\epsilon ) (p^{\prime 2}-m_{D_{s}^{*}}^2 +i\epsilon )(q^2-m_\rho ^2 +i\epsilon )} +\cdots \nonumber \\&\quad =\frac{ g^{\mu \nu }f_{0T} \lambda _{D_{s}^{*}}\lambda _{\rho } g_{T^a_{c{\bar{s}} 0}D_s^*\rho }\left( \frac{p^{\prime \alpha }p^{\prime \beta }}{p^{\prime 2}} +\frac{q^{\alpha }q^{\beta }}{q^{2}}-\frac{p^{\prime \alpha }q^\beta p^\prime \cdot q}{p^{\prime 2}q^2}-g^{\alpha \beta }\right) }{(p^2-m_{T^a_{c{\bar{s}0}}}^2 +i\epsilon ) (p^{\prime 2}-m_{D_{s}^{*}}^2 +i\epsilon )(q^2-m_\rho ^2 +i\epsilon )}+\cdots \, . \end{aligned}$$

(49)

There are four different tensor structures $p^{\prime \alpha }p^{\prime \beta }$, $q^{\alpha }q^{\beta }$, $p^{\prime \alpha }q^\beta$ and $g^{\alpha \beta }$ in Eq. (49). In our calculations on the OPE side, we shall use the terms proportional to $1/q^2$ with tensor structure $g^{\alpha \beta }$ to match those on the phenomenological side. Then we find that only the gluon condensate $\langle g_s^2GG\rangle$ contributes to the spectral density

$$\begin{aligned} \rho _{_{D_{s}^{*} \rho }}(s)=-\frac{\langle g_s^2GG\rangle \left( (m_c-m_s)^2-s\right) \left( (m_c+m_s)^2+2s\right) f(s)}{2304\pi ^2s}\, . \end{aligned}$$

(50)

The QCD sum rules for the coupling $g_{T^a_{c{\bar{s}} 0}D_s^*\rho }$ can be established after performing Borel transform

$$\begin{aligned} g_{T^a_{c{\bar{s}} 0}D_s^*\rho }(s_0,M_B^2,Q^2)&=\frac{1}{f_{0T}\lambda _{D_s^*}\lambda _\rho } \frac{m^2_{T^a_{c{\bar{s}}0}}-m_{D_s^*}^2}{e^{-m^2_{D_s^*}/M_B^2}-e^{-m^2_{T^a_{c\bar{s}0}}/M_B^2}}\nonumber \\&\quad \times \left( \frac{Q^2+m_\rho ^2}{Q^2}\right) \int _{(m_c+m_s)^2}^{s_0}\rho _{_{D_{s}^{*} \rho }}(s)e^{-s/M_B^2}ds. \end{aligned}$$

(51)

The hadron parameters for $D_s^*$ and $\rho$ are [40, 44, 47]

$$\begin{aligned} \begin{aligned} m_\rho =775.26\pm 0.23~\text {MeV},\quad&f_\rho =157~\text {MeV},\\ m_{D_s^*}=2112.2\pm 0.4~\text {MeV},\quad&f_{D_s^*}=305.5~\text {MeV}. \end{aligned} \end{aligned}$$

(52)

In Fig. 5, we show the variation of the coupling constant $g_{T^a_{c{\bar{s}} 0}D_s^*\rho }(Q^2)$ with the Borel parameter $M_B^2$ at $Q^2=5~\text {GeV}^2$, which is far away from $m_\rho ^2$. The minimum value of $g_{T^a_{c{\bar{s}} 0}D_s^*\rho }(Q^2)$ appears around $M_B^2=1.22~\text {GeV}^2$. To extrapolate $g_{T^a_{c{\bar{s}} 0}D_s^*\rho }$ to the physical pole $Q^2=-m_\rho ^2$, we fit the QCD sum rule result with $s_0=10~\text {GeV}^2$ and $M_B^2=1.22~\text {GeV}^2$ by using the exponential model Eq. (40) with $g_1=1.02~\text {GeV}$ and $g_2=0.02~\text {GeV}^{-2}$, as shown in Fig. 6. The coupling constant at the physical pole is obtained as

$$\begin{aligned} g_{T^a_{c{\bar{s}} 0}D_s^*\rho }(Q^2=-m_\rho ^2)=1.04\pm 0.48~\text {GeV}\, . \end{aligned}$$

(53)

The decay width of $T^a_{c{\bar{s}0}}(2900)^{++}\rightarrow D_s^{*+} \rho ^+$ as an $S\rightarrow VV$ process can be derived from Eq. (47)

$$\begin{aligned} \Gamma (T_{c{\bar{s}0}}^{a++}\rightarrow D_s^{*+} \rho ^+)&= \frac{p^*(m_{T^a_{c{\bar{s}0}}},m_{D_s^{*}},m_\rho )}{8\pi }g^2_{T^a_{c{\bar{s}} 0}D_s^*\rho }\nonumber \\&\quad \times \left( \frac{3}{m_{T^a_{c{\bar{s}0}}}^2}+\frac{p^*(m_{T^a_{c{\bar{s}0}}},m_{D_s^{*}},m_\rho )^2}{m_{D_s^{*}}^2m_\rho ^2}\right) \,. \end{aligned}$$

(54)

Then the partial decay width of this decay mode can be obtained as

$$\begin{aligned} \Gamma (T_{c{\bar{s}0}}^{a++}\rightarrow D_s^{*+} \rho ^+)=2.4\pm 2.3~\text {MeV}\, , \end{aligned}$$

(55)

where the error is mainly from the uncertainty of the gluon condensate. This partial decay width is much smaller than those for the $D_s^+\pi ^+$ and $D^+K^+$ decay modes due to the suppressed phase space.

Fig. 5

Variation of the coupling constant $g_{T^a_{c{\bar{s}} 0}D_s^*\rho }(Q^2)$ at $Q^2=5~\text {GeV}^2$ with the Borel parameter $M_B^2$

Fig. 6

Variation of the coupling constant $g_{T^a_{c{\bar{s}} 0}D_s^*\rho }$ with $Q^2$. The dotted points represent the QCD sum rule results for $g_{T^a_{c{\bar{s}} 0}D_s^*\rho }$ with $s_0=10~\text {GeV}^2$ and $M_B^2=1.22~\text {GeV}^2$. The solid line is the fit of the QCD sum rule result through model Eq. (40) and the extrapolation to the physical pole $Q^2=-m_\rho ^2$

2.4 Decay mode of $T^a_{c{\bar{s}0}}(2900)^{++}\rightarrow D_{s1}^{+} \pi ^+$

In this subsection, we study the P-wave decay mode $T^a_{c\bar{s}0}(2900)^{++}\rightarrow D_{s1}^{+} \pi ^+$ ($S\rightarrow AP$). The axial-vector interpolating current for $D_{s1}^+$ meson is

$$\begin{aligned} J_{D_{s1}^+}^\alpha ={\bar{s}}\gamma ^\alpha \gamma _5c\,, \end{aligned}$$

(56)

with the current-meson coupling

$$\begin{aligned} \langle 0|J_{D_{s1}^{+}}^\alpha |D_{s1}^{+}(p^{\prime })\rangle =f_{D_{s1}}m_{D_{s1}} \epsilon ^\alpha (p^\prime )=\lambda _{D_{s1}}\epsilon ^\alpha (p^\prime )\, . \end{aligned}$$

(57)

The effective Lagrangian of $T_{c{\bar{s}0}}^{a++}D_{s1}^+\pi ^+$ vertex is constructed as

$$\begin{aligned} \mathcal{L}_{T^a_{c{\bar{s}0}}D_{s1}\pi }=g_{T^a_{c{\bar{s}0}}D_{s1} \pi } T_{c{\bar{s}0}}^{a++} D_{s1}^{+\mu }\partial _\mu \pi ^+ + h.c. \,. \end{aligned}$$

(58)

One can immediately read out the transition matrix element

$$\begin{aligned} \langle D_{s1}^+(p^\prime )\pi ^+(q)|T_{c{\bar{s}0}}^{a++}(p)\rangle =i g_{T^a_{c{\bar{s}0}}D_{s1} \pi } q\cdot \epsilon ^*(p^\prime )\, , \end{aligned}$$

(59)

where $g_{T^a_{c{\bar{s}0}}D_{s1} \pi }$ is the coupling constant of the $T_{c{\bar{s}0}}^{a++}D_{s1}^+\pi ^+$ vertex, and $\epsilon ^*(p^\prime )$ is the polarization vector of $D_{s1}^+$ meson. Using these definitions, the three-point correlation function for $T^a_{c{\bar{s}0}}(2900)^{++}\rightarrow D_{s1}^{+} \pi ^+$ process at the hadron level can be written as

$$\begin{aligned}&\Pi ^{\mu \nu , \alpha (\textrm{PH})}_{D_{s1} \pi }(p, p^\prime , q)\nonumber \\&\quad =\int d^4x d^4y e^{ip^\prime x} e ^{i q y} \langle 0|{{\mathbb {T}}} [J_{D_{s1}^+}^\alpha (x) J_{\pi ^+}(y) J^{\mu \nu \dagger }_{T^{a++}_{c{\bar{s}0}}}(0) ]|0\rangle \nonumber \\&\quad = \frac{ ig^{\mu \nu }f_{0T} \lambda _{D_{s1}}\lambda _{\pi }\epsilon ^\alpha (p^\prime )\langle D_{s1}^{+}(p')\pi ^+(q)\vert T^{a++}_{c{\bar{s}} 0}(p)\rangle }{(p^2-m_{T^a_{c{\bar{s}0}}}^2 +i\epsilon ) (p^{\prime 2}-m_{D_{s1}}^2 +i\epsilon )(q^2-m_\pi ^2 +i\epsilon )} +\cdots \nonumber \\&\quad = \frac{ig^{\mu \nu }f_{0T} \lambda _{D_{s1}}\lambda _{\pi } g_{T^a_{c{\bar{s}0}}D_{s1} \pi }\left( -q^\alpha +\frac{p^\prime \cdot qp^{\prime \alpha }}{p^{\prime 2}}\right) }{(p^2-m_{T^a_{c{\bar{s}0}}}^2 +i\epsilon ) (p^{\prime 2}-m_{D_{s1}}^2 +i\epsilon )(q^2-m_\pi ^2 +i\epsilon )}+\cdots \, . \end{aligned}$$

(60)

There are two different tensor structures $q^\alpha$ and $p^{\prime \alpha }$ in the three-point correlation function. To study the $S\rightarrow AP$ decay mode, we use the $q^\alpha$ structure in the three-point correlation function on both the phenomenological and OPE sides in the $m_\pi \rightarrow 0$ limit. At the quark-gluon level, we obtain the terms proportional to $1/q^2$ in the OPE series as

$$\begin{aligned}&\rho _{_{D_{s1}\pi }}(s)=\frac{\langle {\bar{qq}}\rangle \left( (m_c+m_s)^2-s\right) \left( (m_c-m_s)^2+2s\right) f(s)}{24\pi ^2s}\nonumber \\&\quad -\frac{3\langle {\bar{qg}}_s\sigma \cdot Gq\rangle (m_c-m_s)^2\left( (m_c+m_s)^2-s\right) }{288\pi ^2s^2f(s)}\, , \end{aligned}$$

(61)

$$\begin{aligned}&\Pi _{_{D_{s1}\pi }}^{\langle g_s^2GG\rangle \langle {\bar{qq}}\rangle }(p'^2)=\frac{\langle g_s^2GG\rangle \langle {\bar{qq}}\rangle }{288 \pi ^2}\int _{0}^{1}\frac{dx}{\Delta (p'^2) ^3 (x-1)^3 x^2}\nonumber \\&\quad \times \bigg (-2 m_c^4 x^3-2 m_c^3 m_s (x-1) x \nonumber \\&\quad + m_c^2 (x-1) \left( 2 m_s^2 (x-1)^2+\Delta (p'^2) x^2 (1-2 x)\right) \nonumber \\&\quad +2 \Delta (p'^2) m_c m_s \left( 2 x^3-4 x^2+3 x-1\right) \nonumber \\&\quad +\Delta (p'^2) (x-1)^3 \left( m_s^2 (x-1)+2 \Delta (p'^2) x^2\right) \bigg )\, , \end{aligned}$$

(62)

in which the quark condensate, the quark-gluon mixed condensate and the gluon condensate are included.

We establish the sum rules for the coupling constant $g_{T^a_{c\bar{s}0}D_{s1} \pi }$ at the $q^2$ pole as

$$\begin{aligned}&g_{T^a_{c{\bar{s}0}}D_s \pi }(s_0, M_B^2) \nonumber \\ {}&\quad =\frac{1}{f_{0T} \lambda _{D_{s1}}\lambda _{\pi }} \frac{m^2_{T^a_{c\bar{s}0}}-m^2_{D_{s1}}}{e^{-m^2_{D_{s1}}/M_B^2}-e^{-m^2_{T^a_{c\bar{s}0}}/M_B^2}}\Bigg (\int _{(m_c+m_s)^2}^{s_0}\rho _{_{D_{s1}\pi }}(s)e^{-s/M_B^2}ds\nonumber \\&\qquad +\Pi _{_{D_{s1}\pi }}^{\langle g_s^2GG\rangle \langle {\bar{qq}}\rangle }(M_B^2) \Bigg )\, . \end{aligned}$$

(63)

Fig. 7

The dependence of the coupling constant $g_{T^a_{c{\bar{s}0}}D_{s1} \pi }$ on the Borel parameter $M_B^2$

To perform numerical analysis, the hadron parameters for the $D_{s1}$ meson are [40, 48]

$$\begin{aligned} \begin{aligned} m_{D_{s1}}=2459.5\pm 0.6~\text {MeV},\quad&f_{D_{s1}}=345~\text {MeV}\, , \end{aligned} \end{aligned}$$

(64)

in which the coupling constant $f_{D_{s1}}$ is the theoretical predicted value by QCD sum rule method [48].

In Fig. 7, we show the variation of the coupling constant $g_{T^a_{c{\bar{s}0}}D_{s1} \pi }$ with $M_B^2$ for $s_0=10~\text {GeV}^2$, in which the minimal dependence of $g_{T^a_{c{\bar{s}0}}D_{s1} \pi }$ on $M_B^2$ appears around $M_B^2\sim 1.62~\text {GeV}^2$. The coupling constant at this point is

$$\begin{aligned} g_{T^a_{c{\bar{s}0}}D_{s1} \pi }=8.24\pm 2.23\, . \end{aligned}$$

(65)

The decay width of $T^a_{c{\bar{s}0}}(2900)^{++}\rightarrow D_{s1}^{+} \pi ^+$ as an $S\rightarrow AP$ process can be derived from Eq. (59) as

$$\begin{aligned} \Gamma (T^{a++}_{c{\bar{s}0}}\rightarrow D_{s1}^{+} \pi ^+)=\frac{p^*(m_{T^a_{c{\bar{s}0}}},m_{D_{s1}},m_\pi )^3}{8\pi m^2_{D_{s1}}}g^2_{T^a_{c{\bar{s}0}}D_{s1} \pi }\, , \end{aligned}$$

(66)

from which the numerical value of this partial decay width is

$$\begin{aligned} \Gamma (T^{a++}_{c{\bar{s}0}}\rightarrow D_{s1}^{+} \pi ^+)=27.2\pm 15.0~\text {MeV}\, , \end{aligned}$$

(67)

where the error is mainly from the uncertainty of the quark condensate. This partial decay width is surprisingly large because the phase space in this channel is sizable, although it is in P-wave.

3 Conclusions and discussions

In this work, we have investigated the recent observed resonance $T^a_{c{\bar{s}0}}(2900)^{++}$ ($T^a_{c{\bar{s}0}}(2900)^{0}$) as a fully open-flavor $cu{\bar{d}}{\bar{s}}$ ($cd{\bar{u}}{\bar{s}}$) tetraquark state with $I(J^P)=1(0^+)$. In Ref. [9], we have successfully predicted the existence of such open-flavor tetraquark states by studying their mass spectra and suggested searching for them in the $D_s\pi$ final states, which are exactly the observed channels of $T^a_{c{\bar{s}0}}(2900)^{++}$ and $T^a_{c\bar{s}0}(2900)^{0}$. To further understand the nature of $T^a_{c\bar{s}0}(2900)^{++}$, we have studied its two-body strong decay processes $T^a_{c{\bar{s}0}}(2900)^{++}\rightarrow D_s^+\pi ^+$, $D^+K^+, D_s^{*+}\rho ^+$, $D_{s1}^+\pi ^+$ in the compact tetraquark picture using the method of QCD sum rules.

We calculated the three-point correlation functions up to dimension eight condensates for the $T^{a++}_{c{\bar{s}0}}D_s^+\pi ^+$, $T^{a++}_{c{\bar{s}0}}D^+K^+$, $T^{a++}_{c{\bar{s}0}}D_s^{*+}\rho ^+$, $T^{a++}_{c{\bar{s}0}}D_{s1}^+\pi ^+$ vertices. By extracting the $1/Q^2$ terms in the OPE series, we can establish QCD sum rules for the coupling constant and then extrapolate its value to the physical pole. After the numerical analyses, we obtain the partial decay widths of the four decay channels as

$$\begin{aligned}&\Gamma (T^{a++}_{c{\bar{s}0}}\rightarrow D_s^+\pi ^+)=62.9\pm 36.7~\text {MeV}\, , \nonumber \\&\Gamma (T_{c{\bar{s}0}}^{a++}\rightarrow D^+K^+)=69.2\pm 40.8~\text {MeV}\,,\nonumber \\&\Gamma (T_{c{\bar{s}0}}^{a++}\rightarrow D_s^{*+} \rho ^+)=2.4\pm 2.3~\text {MeV}\, ,\nonumber \\&\Gamma (T^{a++}_{c{\bar{s}0}}\rightarrow D_{s1}^{+} \pi ^+)=27.2\pm 15.0~\text {MeV}\, . \end{aligned}$$

(68)

The resulting full width of $T^a_{c{\bar{s}0}}(2900)^{++}$ is

$$\begin{aligned} \Gamma _{T^{a++}_{c{\bar{s}0}}}=161.7\pm 94.8~\text {MeV}\,, \end{aligned}$$

(69)

which is in good accord with the experimental value [7, 8]. This result supports the compact tetraquark explanation of $T^a_{c{\bar{s}0}}(2900)^{++}$ and $T^a_{c{\bar{s}0}}(2900)^{0}$, as predicted in Ref. [9].

It is useful to give the relative branching ratios from Eq. (68)

$$\begin{aligned}&\Gamma (T^{a++}_{c{\bar{s}0}}\rightarrow D_s^+\pi ^+): \Gamma (T_{c{\bar{s}0}}^{a++}\rightarrow D^+K^+): \Gamma (T_{c{\bar{s}0}}^{a++}\rightarrow D_s^{*+} \rho ^+): \nonumber \\&\Gamma (T^{a++}_{c{\bar{s}0}}\rightarrow D_{s1}^{+} \pi ^+)\approx 1.00:1.10:0.04:0.43\, , \end{aligned}$$

(70)

where only the central values of these partial decay widths are adopted. These relative branching ratios can vary in a broad range considering the uncertainties of the partial decay widths in Eq. (68). Our results show that the main decay modes of $T^a_{c{\bar{s}0}}(2900)^{++/0}$ state are $D_s\pi$ and DK channels. However, the P-wave decay mode $D_{s1}\pi$ is also comparable and important by including the uncertainties. The investigations of nonrelativistic potential quark model also show that the main decay modes are the same with our results with similar relative branching ratios $\Gamma (T^{a++}_{c{\bar{s}0}}\rightarrow D_s\pi ): \Gamma (T^{a++}_{c{\bar{s}0}}\rightarrow DK):\Gamma (T^{a++}_{c{\bar{s}0}}\rightarrow D_s^{*} \rho )=1.00:1.41:0.01$ [10], although their total decay width is much smaller. By considering $T^a_{c{\bar{s}0}}(2900)^{++/0}$ as a $D^{*}K^{*}$ molecule, the decay behavior is also very different because the partial width of the DK channel is much larger than those in other channels [17].

To conclude this paper, we suggest confirming these exotic $T^a_{c{\bar{s}0}}(2900)^{++}$ and $T^a_{c{\bar{s}0}}(2900)^{0}$ structures in the DK and $D_{s} \pi$ final states in future experiments. The relative branching ratios between these decay channels will be very useful for understanding the nature of these states.