Decay properties of $D_{s0}^{*}(2317)^{+}$ as a conventional $c\bar{s}$ meson

When both quarks in the initial state appear in the final state meson, the strong decay of $D_{s0}^{*}(2317)^{+}$ can be expressed by a Feynman diagram, as shown in Fig.1. The transition matrix element of the two-body strong decay $D_{s0}^{*}(2317)^{+}\to D_{s}^{+}\eta$ can be expressed as (we compute the $D_{s0}^{*}(2317)^{+}\to D_{s}^{+}\eta$ process first, and then we import $\eta-\pi^{0}$ mixing for $D_{s}^{+}\eta\to D_{s}^{+}\pi^{0}$)[41]

$$\left<D_{s}^{+}(P_{{}_{f}})\eta(P^{{}^{\prime}})|D_{s0}^{*+}(P)\right>=\int d^{4}xe^{iP^{{}^{\prime}}x}(M_{\eta}^{2}-P^{{}^{\prime}2})\left<D_{s}^{+}(P_{{}_{f}})|\Phi_{\eta}(x)|D_{s0}^{*+}(P)\right>,$$

(1)

By combining Eqs.(1) and (2), we obtain

$$\left<D_{s}^{+}(P_{{}_{f}})\eta(P^{{}^{\prime}})|D_{s0}^{*+}(P)\right>=\frac{M_{\eta}^{2}-P^{{}^{\prime}2}}{M_{\eta}^{2}f_{{}_{P}}}\int d^{4}xe^{iP^{{}^{\prime}}x}\left<D_{s}^{+}(P_{{}_{f}})|\partial^{\mu}\left(\bar{s}\gamma_{\mu}\gamma_{5}s\right)|D_{s0}^{*+}(P)\right>.$$

(3)

By using partial integration and applying the low-energy theorem, we can obtain the form of the transition matrix element in the momentum space. It can be written as

	$\displaystyle\left<D_{s}^{+}(P_{{}_{f}})\eta(P^{{}^{\prime}})|D_{s0}^{*}(P)\right>$	$\displaystyle=\frac{M_{\eta}^{2}-P^{{}^{\prime}2}}{M_{\eta}^{2}f_{{}_{P}}}\int d^{4}xe^{iP^{{}^{\prime}}x}\left<D_{s}^{+}(P_{{}_{f}})|\partial^{\mu}\left(\bar{s}\gamma_{\mu}\gamma_{5}s\right)|D_{s0}^{*+}(P)\right>$		(4)
		$\displaystyle=\frac{-iP^{{}^{\prime}\mu}\left(M_{\eta}^{2}-P^{{}^{\prime}2}\right)}{M_{\eta}^{2}f_{{}_{P}}}\int d^{4}xe^{iP^{{}^{\prime}}x}\left<D_{s}^{+}(P_{{}_{f}})|\bar{s}\gamma_{\mu}\gamma_{5}s|D_{s0}^{*+}(P)\right>$
		$\displaystyle\approx(2\pi)^{4}\delta^{4}(P-P_{{}_{f}}-P^{{}^{\prime}})\frac{-iP^{{}^{\prime}\mu}}{f_{{}_{P}}}\left<D_{s}^{+}(P_{{}_{f}})|\bar{s}\gamma_{\mu}\gamma_{5}s|D_{s0}^{*+}(P)\right>.$

Then, the transition amplitude for the process $D_{s0}^{*}(2317)^{+}\to D_{s}^{+}+\eta$ is

$$T=\frac{-iP^{{}^{\prime}\mu}}{f_{{}_{P}}}\left<D_{s}^{+}(P_{{}_{f}})|\bar{s}\gamma_{\mu}\gamma_{5}s|D_{s0}^{*+}(P)\right>.$$

(5)

In the Mandelstam form [42], the transition amplitude can be written as an overlapping integral of the initial and final meson wave functions [43]

	$\displaystyle\mathcal{M}$	$\displaystyle=\left<D_{s}^{+}(P_{{}_{f}})|\bar{s}\gamma_{\mu}\gamma_{5}s|D_{s0}^{*+}(P)\right>$		(6)
		$\displaystyle=\int\frac{d^{4}q}{\left(2\pi\right)^{4}}\frac{d^{4}q_{{}_{f}}}{\left(2\pi\right)^{4}}Tr\left[\bar{\chi}_{P_{{}_{f}}}\left(q_{{}_{f}}\right)S_{1}^{-1}\chi_{P}\left(q\right)\gamma_{\mu}\gamma_{5}\right]\left(2\pi\right)^{4}\delta^{4}\left(p_{{}_{1}}-p_{{}_{1f}}\right)$
		$\displaystyle\approx\int\frac{d^{3}q_{\perp}}{2\pi^{3}}Tr\left[\bar{\varphi}_{P_{{}_{f}}}^{++}(q_{\perp}-\alpha_{1}P_{{}_{f\perp}})\frac{\not{P}}{M}\varphi_{P}^{++}\left(q_{\perp}\right)\gamma_{\mu}\gamma_{5}\right],$

where $\chi_{{}_{P}}\left(q\right)$ and ${\chi}_{{}_{P_{{}_{f}}}}\left(q_{{}_{f}}\right)$ represent the relativistic BS wave functions of $D_{s0}^{*}(2317)^{+}$ and $D_{s}^{+}$, respectively. $q$ and $q_{{}_{f}}$ represent the internal relative momenta of the initial and final mesons, respectively. $p_{{}_{1}}$, $p_{{}_{2}}$, $p_{{}_{1f}}$ and $p_{{}_{2f}}$ represent the momenta of the quarks and anti-quarks of the initial and final mesons, respectively. $S_{1}$ represents the propagator of the quarks. $M$ represents the mass of $D_{s0}^{*}(2317)^{+}$, and $\alpha_{1}=\frac{m_{c}}{m_{c}+m_{s}}$ with the quark mass $m_{c}$ and the anti-quark mass $m_{\bar{s}}$; $\varphi_{P_{{}_{f}}}^{++}$ and $\varphi_{P}^{++}$ represent the positive-energy wave functions of $D_{s}^{+}$ and $D_{s0}^{*}(2317)^{+}$, respectively. $P_{{}_{f\perp}}$ is defined as $P_{{}_{f\perp}}^{\mu}=P_{{}_{f}}^{\mu}-(\frac{P\cdot P_{{}_{f}}}{M^{2}})P^{\mu}$ (similar definition to $q_{\perp}$), and $\bar{\varphi}_{P_{{}_{f}}}^{++}$ is defined as $\gamma_{0}(\varphi_{P_{{}_{f}}}^{++})^{\dagger}\gamma_{0}$.

In our model, we use the modified BS method based on the constituent quark model to give the wave functions for $J^{P}$ or $J^{PC}$ states. Here, we will directly give the corresponding positive energy wave functions of initial and final states.

The two-body decay width formula can be expressed (import $\eta-\pi^{0}$ mixing) as

$$\Gamma=\frac{1}{8\pi}\frac{\left|\vec{P_{{}_{f}}}\right|}{M^{2}}\frac{1}{2J+1}\sum_{\lambda}\left|\frac{Tt_{\pi\eta}}{m^{2}_{\pi}-m^{2}_{\eta}}\right|^{2},$$

(9)

where $|\vec{P_{{}_{f}}}|=\sqrt{[M^{2}-(M_{f}-m_{\pi})^{2}][M^{2}-(M_{f}+m_{\pi})^{2}]}/(2M)$, which is the three-momenta of the final meson $D_{s}^{+}$. $M$, $M_{f}$, and $m_{\pi}$ represent the masses of the initial state and two final states, respectively. $J$ represents the spin quantum number of the initial meson, which is 0 in this case. $t_{\pi\eta}$ is the mixing matrix entry of $\eta-\pi^{0}$, and $t_{\pi\eta}=\left<\pi^{0}\left|\mathcal{H}\right|\eta\right>=-0.003$ GeV² [46] is chosen.

The transition matrix element of the EM decay $D_{s0}^{*}(2317)^{+}\rightarrow D_{s}^{*}\gamma$ can be written as

$$\left<D_{s}^{*}\left(P_{{}_{f}},\epsilon\right)\gamma\left(k,\epsilon_{0}\right)|D_{s0}^{*+}\right>=(2\pi)^{4}\delta^{4}\left(P-P_{{}_{f}}-k\right)\epsilon_{0\mu}\mathcal{M}^{\mu},$$

(10)

where $\epsilon_{0}$ and $\epsilon$ represent the polarization vectors of the photon and final meson, respectively. $P$, $P_{{}_{f}}$, and $k$ represent the momenta of the initial meson, final meson, and photon, respectively.

$\mathcal{M}$ represents the invariant amplitude, which is expresed as

	$\displaystyle\mathcal{M}^{\mu}=\int\frac{d^{4}q}{\left(2\pi\right)^{4}}\frac{d^{4}q_{{}_{f}}}{\left(2\pi\right)^{4}}Tr$	$\displaystyle\left[\bar{\chi}_{P_{{}_{f}}}\left(q_{{}_{f}}\right)Q_{1}\gamma^{\mu}\chi_{P}\left(q\right)\left(2\pi\right)^{4}\delta^{4}(p_{2}-p_{2}^{{}^{\prime}})S_{2}^{-1}\left(p_{2}\right)\right.$		(11)
		$\displaystyle\left.+\bar{\chi}_{P_{{}_{f}}}\left(q_{{}_{f}}\right)\left(2\pi\right)^{4}\delta^{4}(p_{1}-p_{1}^{{}^{\prime}})S_{1}^{-1}\left(p_{1}\right)\chi_{P}\left(q\right)Q_{2}\gamma^{\mu}\right].$

Here, $Q_{1}$ and $Q_{2}$ represent the electric charges (in unit of $e$) of quark and anti-quark, respectively. We use the instantaneous approximation to obtain the following form of the amplitude [43]:

	$\displaystyle\mathcal{M}^{\mu}=\int\frac{d^{3}q}{\left(2\pi\right)^{3}}Tr$	$\displaystyle\left[Q_{1}\bar{\varphi}_{P_{{}_{f}}}^{++}\left(q_{\perp}+\alpha_{2}P_{f\perp}\right)\gamma^{\mu}\varphi_{P}^{++}\left(q_{\perp}\right)\frac{\not{P}}{M}\right.$		(12)
		$\displaystyle\left.+Q_{2}\bar{\varphi}_{P_{{}_{f}}}^{++}\left(q_{\perp}-\alpha_{1}P_{f\perp}\right)\frac{\not{P}}{M}\varphi_{P}^{++}\gamma^{\mu}\right],$

where $M$ represents the mass of $D_{s0}^{*}(2317)^{+}$, and $q_{f\perp}=q_{\perp}+\alpha_{2}P_{f\perp}$ and $q_{f\perp}=q_{\perp}-\alpha_{1}P_{f\perp}$ correspond to two different processes. $\alpha_{1}=\frac{m_{c}}{m_{c}+m_{s}}$ and $\alpha_{2}=\frac{m_{s}}{m_{c}+m_{s}}$ are substituted into the above formula with the quark mass $m_{c}$ and the anti-quark mass $m_{\bar{s}}$, respectively.

The relativistic positive energy Salpeter wave function for $D_{s}^{*}$ (${}^{3}S_{1}$ state) can be written as

	$\displaystyle\varphi_{1^{-}}^{++}\left(P,q_{\perp}\right)=$	$\displaystyle C_{1}\not{\varepsilon}+C_{2}\not{\varepsilon}\not{P}+C_{3}\left(\not{q}_{\perp}\not{\varepsilon}-q_{\perp}\cdot\varepsilon\right)+C_{4}\left(\not{P}\not{\varepsilon}\not{q}_{\perp}-\not{P}q_{\perp}\cdot\varepsilon\right)$		(13)
		$\displaystyle+q_{\perp}\cdot\varepsilon\left(C_{5}+C_{6}\not{P}+C_{7}\not{q}_{\perp}+C_{8}\not{q}_{\perp}\not{P}\right),$

where we define the parameters $C_{i}$ using the radial wave functions $f_{i}$ (i=3,4,5,6) for a $1^{-}$ state [44]:

	$\displaystyle C_{1}=\frac{M}{2}\left(f_{5}-f_{6}\frac{\omega_{c}+\omega_{s}}{m_{c}+m_{s}}\right),$
	$\displaystyle C_{2}=-\frac{1}{2}\frac{m_{c}+m_{s}}{\omega_{c}+\omega_{s}}\left(f_{5}-f_{6}\frac{\omega_{c}+\omega_{s}}{m_{c}+m_{s}}\right),$
	$\displaystyle C_{3}=\frac{M}{2}\frac{\omega_{s}-\omega_{c}}{m_{c}\omega_{s}+m_{s}\omega_{c}}\left(f_{5}-f_{6}\frac{\omega_{c}+\omega_{s}}{m_{c}+m_{s}}\right),$
	$\displaystyle C_{4}=\frac{1}{2}\frac{\omega_{c}+\omega_{s}}{\omega_{c}\omega_{s}+m_{c}m_{s}-q_{\perp}^{2}}\left(f_{5}-f_{6}\frac{\omega_{c}+\omega_{s}}{m_{c}+m_{s}}\right),$
	$\displaystyle C_{5}=\frac{1}{2M}\frac{m_{c}+m_{s}}{\omega_{c}\omega_{s}+m_{c}m_{s}+q_{\perp}^{2}}\left[M^{2}\left(f_{5}-f_{6}\frac{m_{c}+m_{s}}{\omega_{c}+\omega_{s}}\right)+q_{\perp}^{2}\left(f_{3}+f_{4}\frac{m_{c}+m_{s}}{\omega_{c}+\omega_{s}}\right)\right],$
	$\displaystyle C_{6}=\frac{1}{2M^{2}}\frac{\omega_{c}+\omega_{s}}{\omega_{c}\omega_{s}+m_{c}m_{s}+q_{\perp}^{2}}\left[M^{2}\left(f_{5}-f_{6}\frac{m_{c}+m_{s}}{\omega_{c}+\omega_{s}}\right)+q_{\perp}^{2}\left(f_{3}+f_{4}\frac{m_{c}+m_{s}}{\omega_{c}+\omega_{s}}\right)\right],$
	$\displaystyle C_{7}=\frac{1}{2M}\left(f_{3}+f_{4}\frac{m_{c}+m_{s}}{\omega_{c}+\omega_{s}}\right)-\frac{f_{6}M}{m_{c}\omega_{s}+m_{s}\omega_{c}},$
	$\displaystyle C_{8}=\frac{1}{2M^{2}}\frac{\omega_{c}+\omega_{s}}{m_{c}+m_{s}}\left(f_{3}+f_{4}\frac{m_{c}+m_{s}}{\omega_{c}+\omega_{s}}\right)-f_{5}\frac{\omega_{c}+\omega_{s}}{\left(m_{c}+m_{s}\right)\left(\omega_{c}\omega_{s}+m_{c}m_{s}-q_{\perp}^{2}\right)}.$

The positive energy wave function $\varphi_{1^{-}}^{++}\left(P,q_{\perp}\right)$ for $D_{s}^{*}$ includes three partial waves: the $C_{1}$ and $C_{2}$ terms are $S$ $waves$; the $C_{3}$, $C_{4}$, $C_{5}$, and $C_{6}$ terms are $P$ $waves$; and the $C_{7}$ and $C_{8}$ terms are $D$ $waves$ [45].

As mentioned previously, the complete relativistic wave function for a $J^{P}$ state is not a pure wave. It includes different partial waves [45]: both the non-relativistic main part and the relativistic correction terms are included. We calculate the contributions (to the decay width) of different partial waves in the initial and final states. The results for strong decay are presented in Table 3, and those for EM decay are presented in Table 4. In the tables, “$complete$” means that the complete wave function is used, “$S$ $wave$” means that only the $S$ partial wave makes a contribution and other partial waves are ignored.

In this paper, taking the particle $D_{s0}^{*}(2317)^{+}$ as the conventional $c\bar{s}$ meson, we calculate its main decay processes in the framework of the Bethe-Salpeter method, $i.e.$, the strong decay and electromagnetic decay. Our results are $\Gamma(D_{s0}^{*+}\to D_{s}^{+}\pi^{0})=7.83^{+1.97}_{-1.55}$ keV and $\Gamma(D_{s0}^{*+}\to D_{s}^{*+}\gamma)=2.55^{+0.37}_{-0.45}$ keV. Because of the low mass, $D_{s0}^{*}(2317)^{+}$ has no direct OZI-allowed strong decay channel. Its strong decay channel $D_{s0}^{*}(2317)^{+}\to D_{s}^{+}\pi^{0}$ violates the isospin conservation, and the $\eta-\pi^{0}$ mixing is needed. Owing to this mixing effect, the strong decay width is relatively small.

In addition, we calculated the contributions of different partial waves. For the decay of $D_{s0}^{*+}\to D_{s}^{+}\pi^{0}$, the main contribution comes from the dominant partial waves of initial and final states, while for the $D_{s0}^{*+}\to D_{s}^{*}\gamma$ decay, the main contribution comes from the dominant partial wave in the initial state and the small $P$ partial wave in the final state. In both cases, the relativistic corrections are very large.