Assignment of charmed-strange $D_{s0}(2590)^{+}$ and $D_{sJ}(3040)^{+}$

A charmed-strange meson consists of a heavy charm quark/antiquark and a light strange antiquark/quark. The investigation of the mass spectrum, decays and productions of charmed-strange mesons will help people to understand the properties of quark dynamics and confinement inside mesons.

There are $11$ charmed-strange meson candidates listed in the Review of Particles Physics [1]. The orbital $S-$wave and $P-$wave charmed-strange excitations except for $D^{*}_{s0}(2317)^{\pm}$ and $D_{s1}(2460)^{\pm}$ have been established. Regarding $D^{*}_{s0}(2317)^{\pm}$ and $D_{s1}(2460)^{\pm}$, their masses are much lower than relevant theoretical predictions of $c\bar{s}$ mesons, and there exist many explanations of these two states outside the normal $c\bar{s}$ meson configuration. In recent year, more higher charmed-strange states were observed [1]. These observations arise people’s great interest and provide more information on the quark structure and dynamics in mesons.

In $2021$, a new excited $D_{s}$ denoted as $D_{s0}(2590)^{+}$ in the $D^{+}K^{+}\pi^{-}$ final state was observed by the LHCb collaboration in $pp$ collision [2]. Its mass and decay width have been measured with $M=2591\pm 6\pm 7$ MeV and $\Gamma=89\pm 16\pm 12$ MeV. Its spin parity was also determined to be $J^{P}=0^{-}$. The LHCb collaboration suggested it as a strong candidate for $D_{s}(2~{}^{1}S_{0})$ state.

In the relativized Godfrey-Isgur quark model [3], the predicted mass for $D_{s}(2~{}^{1}S_{0})$ state is about $80$ MeV higher than the measured one of $D_{s0}(2590)^{+}$. To make the theoretical mass predictions consistent with experimental data, coupled channels calculations [4, 5, 6, 7] were proposed. In these coupled channels calculations, the interaction between meson pairs and mixing between $q\bar{q}$ meson and meson pair system were taken into account. In the coupled channels model [6], $D_{s0}(2590)^{+}$ is not thought a bare $c\bar{s}$ meson but a mixture with a large probability of meson pair system (coupled-channels). The mass of the mixed $D_{s}(2~{}^{1}S_{0})$ was predicted with $2616$ MeV and its decay width was predicted with $112$ MeV in the ${}^{3}P_{0}$ model. $DK^{*}~{}(20.4\%)$ and $D^{*}K^{*}~{}(26.2\%)$ are its dominant coupling channels. In the same coupled channels model with a different potential [7], the predicted $D_{s}(2~{}^{1}S_{0})$ mass is about $50$ MeV higher than the mass of $D_{s0}(2590)^{+}$, while the predicted width $87$ MeV agrees well with the experimental data.

Instead of the coupled channels, an alternate modified potential model with screening effects [8] was employed to calculate the mass spectrum of the charmed-strange mesons. The mass of pure $D_{s}(2~{}^{1}S_{0})$ was predicted with $2620$ MeV and the strong decay width of $D_{s0}(2590)^{+}$ was predicted with $74.9$ MeV with the wave function obtained from a modified relativized quark model including the screening effects at $\gamma=9.32$ in the ${}^{3}P_{0}$ model. The partial decay widths to $D^{*+}K^{0}$ and $D^{*0}K^{+}$ are $35.5$ MeV and $39.4$ MeV, respectively.

Since the color hyperfine interaction, there is a mixing between ${}^{3}L_{J}$ meson and ${}^{3}L^{\prime}_{J}$ meson. There is hence a mixing between $2~{}^{3}S_{1}$ and $1~{}^{3}D_{1}$ charmed mesons and charmed-strange mesons. $D^{*}_{s1}(2700)^{\pm}$ and $D^{*}_{s1}(2860)^{\pm}$ are popularly regarded as the mixtures of $2~{}^{3}S_{1}$ and $1~{}^{3}D_{1}$ $D_{s}$ states. As indicated in Ref. [9], the mass of the mixed radially excited state $D_{s1}^{*}$ is approximately the mass of the pure radially excited $2~{}^{3}S_{1}$ $D_{s}$ meson in a relativized quark model no matter how large the mixing is. A similar behavior exists in the charmonium system where the mixing of states with different $L$ can change the decay modes of the charmonium mesons, but cannot change dramatically the values of their masses [10]. Though there are many explanations of $D^{*}_{s1}(2700)^{\pm}$, it is often identified with the radial excitation of $D_{s}^{*\pm}$ with $J^{P}=1^{-}$ apart from the mixing detail and quark dynamics.

$D_{sJ}(3040)^{\pm}$ was observed in inclusive production of $D^{*}K$ (not observed in the $DK$ channel) in $e^{+}e^{-}$ annihilation by BaBar collaboration [11], this state was not observed in $D^{*0}K^{+}X$ final states in $pp$ collision by LHCb collaboration [12]. As well known, there exists a mixing between the $~{}^{3}P_{1}$ and the $~{}^{1}P_{1}$ states via the spin orbit interaction or some other mechanism. In addition to some exotic explanations of $D_{s1}(2460)$, the two $J^{P}=1^{+}$ $D_{s}$ mesons ($D_{s1}(2460)$ and $D_{s1}(2536)$) are often regarded as the axial-vector mixing states between the $1~{}^{3}P_{1}$ and the $1~{}^{1}P_{1}$ $D_{s}$. Based on the analyses of mass and decay features, $D_{sJ}(3040)^{+}$ was popularly suggested as the mixing state between the $2~{}^{3}P_{1}$ and the $2~{}^{1}P_{1}$ $D_{s}$ with $J^{P}=1^{+}$.

However, there are different opinions on whose radial excitation $D_{sJ}(3040)^{+}$ is. $D_{sJ}(3040)^{+}$ was assigned as the first radial excitation of $D_{s}(2460)$ in Refs. [6, 13, 14], but was suggested as the first radial excitation of $D_{s1}(2536)$ in Ref. [15]. Apart from the assignment of the basic $1P$ mixing $D_{s}$, $D_{sJ}(3040)^{+}$ was identified as the radially excited low-mass mixing state in Ref. [16], while was identified as the radially excited high-mass mixing state in Ref. [17, 18]. Due to large theoretical and experimental uncertainties on the width, some analyses [19, 20, 21] indicated that $D_{sJ}(3040)^{+}$ may be identified with the radial excitation of $D_{s1}(2460)$ or the radial excitation of $D_{s1}(2536)$. That is to say, both identifications are possible. Outside the normal meson interpretation, $D_{sJ}(3040)^{+}$ was described as a $D^{*}(2600)K$ bound state [22].

It is observed that light $q\bar{q}$ mesons ($M<2400$ MeV) with the same $IJ^{PC}$ but different radial excitations form a Regge trajectory on $(n,~{}M^{2})$ plots[23]

$$M^{2}=M^{2}_{0}+(n-1)\mu^{2}$$

(1)

where $n$ is the radial quantum numbers, $M_{0}$ is the mass of the basic meson and $\mu^{2}$ is the trajectory slope parameter which is approximately the same for all trajectories. This linear Regge trajectories feature was also observed in heavy quarkonia system [10] and mentioned in $D_{s}$ mesons [13].

No matter whether $D_{s0}(2590)^{+}$, $D^{*}_{s1}(2700)^{\pm}$ and $D_{sJ}(3040)^{+}$ are pure ${}^{2S+1}L_{J}$ states or mixing states, they have interesting mass relations with $D_{s}$, $D^{*\pm}_{s}$ and $D_{s1}(2536)^{\pm}$, respectively

	$\displaystyle M^{2}(D_{s0}(2590)^{+})-M^{2}(D^{\pm}_{s})=2.840~{}\mathrm{GeV^{2}},$
	$\displaystyle M^{2}(D^{*}_{s1}(2700)^{\pm})-M^{2}(D^{*\pm}_{s})=2.905~{}\mathrm{GeV^{2}},$
	$\displaystyle M^{2}(D_{sJ}(3040)^{+})-M^{2}(D_{s1}(2536)^{\pm})=2.840~{}\mathrm{GeV^{2}}.$		(2)

Obviously, once $D_{s0}(2590)^{+}$, $D^{*}_{s1}(2700)^{\pm}$ and $D_{sJ}(3040)^{+}$ are identified with the radial excitations of $D_{s}$, $D^{*\pm}_{s}$ and $D_{s1}(2536)^{\pm}$, respectively, these relations meet Eq. [1]. In other words, the masses of these radial $D_{s}$ mesons meet the linear Regge trajectories on the $(n,~{}M^{2})$ plots very mell with a similar slope. This slope in $D_{s}$ system lies between the one in the light-meson system and the one in heavy quarkonia system.

The linear behavior of the Regge trajectories on $(n,~{}M^{2})$ plane is a phenomenological observation of the experimental data in light-meson and heavy quarkonia systems, which indicates the ’string’ nature of the mesons. The linear behavior of the Regge trajectories has also been explored and verified in some theoretical models [24, 25, 26, 27]. Though a $0^{-}$ radially excited $D_{s}$ denoted as $D^{\prime}_{s}(2599)$ was already predicted ten years before the observation of $D_{s0}(2590)^{+}$ based only on a Regge trajectory analysis  [28], whether there exists the linear behavior in heavy-light mesons has not been confirmed. Obviously, a test of the assignment of the radial excitations in other ways such as their strong decay features would provide an evidence.

${}^{3}P_{0}$ strong decay model as an effective phenomenological method has been employed extensively and successfully to calculate the Okubo-Zweig-Iizuka(OZI)-allowed hadronic decay widths of mesons, baryons and even exotic multiquark states. In addition to the mass analysis, ${}^{3}P_{0}$ model was also employed to compute the strong decay widths of $D_{s0}(2590)^{+}$ [4, 5, 6, 7, 8] and $D_{sJ}(3040)^{+}$ [6, 14, 16, 19, 21]. However, the predictions of the strong decay widths are different for different assumptions of $D_{s0}(2590)^{+}$ and $D_{sJ}(3040)^{+}$. Further investigation of $D_{s0}(2590)^{+}$ and $D_{sJ}(3040)^{+}$ in the ${}^{3}P_{0}$ model is required.

The rest of this paper is organized as follows. The ${}^{3}P_{0}$ model is briefly introduced in Sec. 2. We will present the numerical results and analysis in Sec. 3. A summary is presented in the last section.

The phenomenological ${}^{3}P_{0}$ model is often known as the quark pair creation model. It was first proposed by Micu [29] and subsequently developed by Yaouanc et al. [30, 31]. The main idea of the ${}^{3}P_{0}$ model for a meson decay may be described in a simple picture as shown in Fig. 1. A pair of quark and antiquark in a flavor singlet with $J^{PC}=0^{++}$ (in a ${}^{3}P_{0}$ configuration) are assumed to be created from vacuum, and then regroup with the antiquark and the quark from an initial meson $A$ into final mesons $B$ and $C$.

In the ${}^{3}P_{0}$ model, the decay width of $A\rightarrow B+C$ is

$$\Gamma=\pi^{2}\frac{\mid\vec{p}\mid}{m^{2}_{A}}\sum_{JL}|\mathcal{M}^{JL}|^{2}$$

(3)

where $\vec{p}$ is the momentum of mesons $B$ and $C$ in the initial meson $A$’s center-of-mass frame

$$|\vec{p}|=\frac{\sqrt{[m_{A}^{2}-(m_{B}-m_{C})^{2}][m_{A}^{2}-(m_{B}+m_{C})^{2}]}}{2m_{A}}.$$

(4)

$\mathcal{M}^{JL}$ is the partial wave amplitude of $A\rightarrow B+C$ with $\vec{J}=\vec{J}_{A}+\vec{J}_{B}$, $\vec{J}_{A}=\vec{J}_{B}+\vec{J}_{C}+\vec{L}$ and $M_{J_{A}}=M_{J_{B}}+M_{J_{C}}$. In terms of the Jacob-Wick formula, one can derive the partial wave amplitude $\mathcal{M}^{JL}$ from the helicity amplitude $\mathcal{M}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}$

	$\displaystyle\mathcal{M}^{JL}(A\rightarrow B+C)=$	$\displaystyle\frac{\sqrt{2L+1}}{2J_{A}+1}\sum_{M_{J_{B}}M{J_{C}}}\langle L0JM_{J_{A}}|J_{A}M_{J_{A}}\rangle$
		$\displaystyle\times\langle J_{B}M_{J_{B}}J_{C}M_{J_{C}}|JM_{J_{A}}\rangle$
		$\displaystyle\times\mathcal{M}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}.$		(5)

In the equation, the helicity amplitude is written as

	$\displaystyle\mathcal{M}$	${}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}$
	$\displaystyle=$	$\displaystyle\sqrt{8E_{A}E_{B}E_{C}}\gamma\sum_{\begin{subarray}{c}M_{L_{A}},M_{S_{A}}\\ M_{L_{B}},M_{S_{B}}\\ M_{L_{C}},M_{S_{C}},m\end{subarray}}\langle L_{A}M_{L_{A}}S_{A}M_{S_{A}}|J_{A}M_{J_{A}}\rangle$
		$\displaystyle\times\langle L_{B}M_{L_{B}}S_{B}M_{S_{B}}|J_{B}M_{J_{B}}\rangle\langle L_{C}M_{L_{C}}S_{C}M_{S_{C}}|J_{C}M_{J_{C}}\rangle$
		$\displaystyle\times\langle 1m;1-m|00\rangle\langle\chi^{13}_{S_{B}M_{S_{B}}}\chi^{24}_{S_{C}M_{S_{C}}}|\chi^{12}_{S_{A}M_{S_{A}}}\chi^{34}_{1-m}\rangle$
		$\displaystyle\times\langle\varphi^{13}_{B}\varphi^{24}_{C}|\varphi^{12}_{A}\varphi^{34}_{0}\rangle I^{M_{L_{A}},m}_{M_{L_{B}},M_{L_{C}}}(\vec{p})$		(6)

where $\gamma$ is a dimensionless parameter reflecting the strength of the quark/antiquark creation from vacuum and $I^{M_{L_{A}},m}_{M_{L_{B}},M_{L_{C}}}(\vec{p})$ is the momentum integral

	$\displaystyle I^{M_{L_{A}},m}_{M_{L_{B}},M_{L_{C}}}(\vec{p})=$	$\displaystyle\int d\vec{k}_{1}d\vec{k}_{2}d\vec{k}_{3}d\vec{k}_{4}$
		$\displaystyle\times\delta^{3}(\vec{k}_{1}+\vec{k}_{2}-\vec{p}_{A})\delta^{3}(\vec{k}_{3}+\vec{k}_{4})$
		$\displaystyle\times\delta^{3}(\vec{p}_{B}-\vec{k}_{1}-\vec{k}_{3})\delta^{3}(\vec{p}_{C}-\vec{k}_{2}-\vec{k}_{4})$
		$\displaystyle\times\Psi^{*}_{n_{B}L_{B}M_{L_{B}}}(\vec{k}_{13})\Psi^{*}_{n_{C}L_{C}M_{L_{C}}}(\vec{k}_{24})$
		$\displaystyle\times\Psi_{n_{A}L_{A}M_{L_{A}}}(\vec{k}_{12})y_{1m}(\vec{k}_{34}).$		(7)

In Eq. [II], $\vec{k}_{ij}=\frac{m_{j}\vec{k}_{i}-m_{i}\vec{k}_{j}}{\vec{k}_{i}+\vec{k}_{j}}$ is the relative momentum of quark $i$ and quark $j$. $y_{1m}(\vec{k})$ denotes the solid harmonic polynomial corresponding to the quark pair.

Taking into account the symmetry of the total wave function (spacial, spin, flavor (isospin) and colour etc) of each meson, the flavor matrix element could be written as a product of the isospin matrix element with the Wigner $9j$ symbol after recoupling calculation [32, 33]

	$\displaystyle\langle\varphi^{13}_{B}\varphi^{24}_{C}|\varphi^{12}_{A}\varphi^{34}_{0}\rangle=$	$\displaystyle\sum_{I,I^{3}}\langle I_{C}I_{C}^{3};I_{B}I_{B}^{3}|I_{A},I_{A}^{3}\rangle$
		$\displaystyle\times[(2I_{B}+1)(2I_{C}+1)(2I_{A}+1)]^{\frac{1}{2}}$
		$\displaystyle\times\begin{Bmatrix}I_{1}&I_{3}&I_{B}\\ I_{2}&I_{4}&I_{C}\\ I_{A}&0&I_{A}\end{Bmatrix}$		(8)

where $I_{i}$ is the isospin of $u,~{}d,~{}s,~{}c$ quark, $I_{A},~{}I_{B},~{}I_{C}$ are the isospin of the mesons $A,~{}B$ and $C$. $I_{A}^{3},~{}I_{B}^{3},~{}I_{C}^{3}$ are the third isospin components of these three mesons. Isospin $I_{P}=0$ is assumed for the created quark pair.

Similarly, the spin matrix element is written as

	$\displaystyle\langle\chi^{13}_{S_{B}M_{S_{B}}}$	$\displaystyle\chi^{24}_{S_{C}M_{S_{C}}}|\chi^{12}_{S_{A}M_{S_{A}}}\chi^{34}_{1-m}\rangle$
	$\displaystyle=$	$\displaystyle(-1)^{S_{C}+1}[3(2S_{B}+1)(2S_{C}+1)(2S_{A}+1)]^{\frac{1}{2}}$
		$\displaystyle\times\sum_{S,M_{S}}\langle S_{B}M_{S_{B}}S_{C}M_{S_{C}}|SM_{S}\rangle\langle SM_{S}|S_{A}M_{S_{A}};1,-m\rangle$
		$\displaystyle\times\begin{Bmatrix}1/2&1/2&S_{B}\\ 1/2&1/2&S_{C}\\ S_{A}&1&S\end{Bmatrix}$		(9)

A simple harmonic oscillator(SHO) wave function is employed in our calculation as follows

	$\displaystyle\Psi_{nLM_{L}}(\vec{k})=$	$\displaystyle\frac{(-1)^{n}(-i)^{L}}{\beta^{3/2}}\sqrt{\frac{2n!}{\Gamma(n+L+3/2)}}(\frac{\vec{k}}{\beta})^{L}$
		$\displaystyle\times\text{exp}(-\frac{\vec{k}^{2}}{2\beta^{2}})L^{L+1/2}_{n}(\frac{\vec{k}^{2}}{\beta^{2}})Y_{LM_{L}}(\Omega),$		(10)

where $L^{L+1/2}_{n}(\frac{\vec{k}^{2}}{\beta^{2}})$ is the Lagueere polynomial function and $Y_{LM_{L}}(\Omega)$ is the spherical harmonic functions. $\beta$ is a harmonic oscillator dimensionless parameter. Further details of the equations, indices, matrix elements, and other indications for meson decay in the ${}^{3}P_{0}$ model can be found in Refs. [33, 34].

$D_{s0}(2590)^{+}$ was observed in the $D^{+}K^{+}\pi^{-}$ final state in $pp$ collision by LHCb collaboration. When its mass, decay width and spin-parity have been determined, $D_{s0}(2590)^{+}$ was suggested as a strong candidate for $D_{s}(2~{}^{1}S_{0})$ state by LHCb collaboration. In existed literatures, the spectroscopy of $D_{s0}(2590)^{+}$ is explained by coupling-channels methods or modified quark potential model with screening effects.

$D_{sJ}(3040)^{\pm}$ was observed in inclusive production of $D^{*}K$ in $e^{+}e^{-}$ annihilation by BaBar collaboration but not observed in $DK$ channel. It was explained as the mixing axial-vector state between the $2~{}^{3}P_{1}$ and the $2~{}^{1}P_{1}$ states. However, people is not sure whether $D_{sJ}(3040)^{\pm}$ is the radial excitation of $D_{s1}(2460)$ or $D_{s1}(2536)^{+}$.

With a simple Regge trajectory analysis, we found that the masses of $D_{s0}(2590)^{+}$ and $D_{sJ}(3040)^{\pm}$ meet the radial linear trajectories on the $(n,~{}M^{2})$ plane very well. In other words, $D_{s0}(2590)^{+}$ and $D_{sJ}(3040)^{\pm}$ are very possibly the radial excitations of the pseudoscalar $D_{s}$ and the axial-vector $D_{s1}(2536)^{+}$, respectively. Of course, the linear behavior of Regge trajectories in the heavy-light $D_{s}$ system requires more experimental data to be confirmed.

Under these assumptions, the strong decay features of $D_{s0}(2590)^{+}$ and $D_{sJ}(3040)^{\pm}$ are studied in the ${}^{3}P_{0}$ model. The numerical strong decay results support the assumption that $D_{s0}(2590)^{+}$ is the radial excitation of the pseudoscalar $D_{s}$. $D_{s0}(2590)^{+}$ is possibly the pure $D_{s}(2~{}^{0}S_{1})$ meson. Both $D_{sJ}(3040)^{\pm}$ and $D_{s1}(2536)^{+}$ are very possibly the mixing states $|nP_{1}\rangle$ between the spin singlet $D_{s}(^{1}P_{1})$ and the spin triplet $D_{s}(^{3}P_{1})$. $D_{sJ}(3040)^{\pm}$ is the radial excitation of $D_{s1}(2536)^{+}$. In this assignment, $D_{s0}(2590)^{+}$ has $D^{*0}K^{+}$ and $D^{*+}K^{0}$ two decay channels. The total decay width of $D_{s0}(2590)^{+}$ is about $76.12$ MeV. Once the three-body decay contribution and uncertainties from theory and experiment have been taken into account, the predicted decay width agree with the experimental data very well. $D_{sJ}(3040)^{+}$ has main decay channels: $D^{*}K$ and $D^{*}K^{*}$. The total decay width of $D_{sJ}(3040)^{+}$ is about $283.46$ MeV. A forthcoming measurement of the branching fraction ratio $\Gamma_{D^{*}K}/\Gamma_{D^{*}_{2}(2460)K}$ will be important for the assignment of $D_{sJ}(3040)^{+}$.