Abstract
We study \(D_{s}^{+}\rightarrow \rho ^{0(+)}a^{+(0)}_{0}\), \(D_{s}^{+}\rightarrow \omega a^{+}_{0}\), and the resonant \(D_{s}^{+}\rightarrow \rho a_0\), \(a_{0}\rightarrow \eta \pi (KK)\) decays. In the final state interaction, where \(D_s^+\rightarrow (\eta ^{(\prime )}\pi ^+,K^+{{\bar{K}}}^0)\) are followed by the \((\eta ^{(\prime )}\pi ^+,K^+{{\bar{K}}}^0)\) to \(\rho ^{0(+)}a^{+(0)}_{0}\) rescatterings, we predict \(\mathcal{B}(D_{s}^{+}\rightarrow \rho ^{0(+)}a^{+(0)}_{0})=(3.0\pm 0.3\pm 1.0)\times 10^{-3}\). Due to the cancellation of the rescattering effects and the suppressed short-distance W annihilation contribution, we expect that \({{\mathcal {B}}}(D_{s}^{+}\rightarrow \omega a^{+}_{0}) \simeq \mathcal{B}(D_s^+\rightarrow \pi ^+\pi ^0)<3.4\times 10^{-4}\). In our calculation, \({{\mathcal {B}}}(D_{s}^{+}\rightarrow \rho ^{0}(a^{+}_{0}\rightarrow )\eta \pi ^{+}) =(1.6^{+0.2}_{-0.3}\pm 0.6)\times 10^{-3}\) agrees with the data, whereas \({{\mathcal {B}}}(D_{s}^{+}\rightarrow \rho ^{+}(a^{0}_{0}\rightarrow )K^+K^-)\) is 10 times smaller than the observation, which requires a careful examination.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The two-body \(D_s^+\rightarrow PP,PV\) decays with P(V) denoting the strangeless pesudoscalar (vector) meson have no configurations from the W-boson emission processes because \({{\bar{s}}}\) in \(D_s^+\) cannot be eliminated, as drawn in Fig. 1 for the topological diagrams T and C. Interestingly, it leads to a specific exploration for the annihilation mechanism applied to \(D_s^+\rightarrow \pi ^+\pi ^0,\pi ^{+}\rho ^{0},\pi ^+\omega \) [1,2,3,4,5,6].
In the short-distance W-boson annihilation (WA) \(D_s^+(c{{\bar{s}}})\rightarrow W^+\rightarrow u{{\bar{d}}}\) process, \(u{{\bar{d}}}\) can be seen to move in the opposite directions in the \(D_s^+\) rest frame, such that there exists no orbital angular momentum between them. It indicates that \(G(u{{\bar{d}}})=G(\pi ^+)=+1\) with G denoting the G-parity symmetry [3]. Since G-parity is a multiplicative quantum number, one obtains \(G(\pi ^+\rho ^0,\pi ^+\pi ^0)=(+1,-1)\). Consequently, the WA \(D_s^+\rightarrow \pi ^+\rho ^0(\pi ^+\pi ^0)\) decay due to \(G(u{{\bar{d}}})=G(\pi ^+\rho ^0)[=-G(\pi ^+\pi ^0)]\) is a G-parity conserved (violated) process, which corresponds to the experimental result \({{\mathcal {B}}}(D_s^+\rightarrow \pi ^+\rho ^0)=(1.9\pm 1.2)\times 10^{-4}\) \([{{\mathcal {B}}}(D_s^+\rightarrow \pi ^+\pi ^0)<3.4\times 10^{-4}]\) [7]. By contrast, although the WA \(D_s^+\rightarrow \pi ^+\omega \) decay violates the G-parity symmetry, \({{\mathcal {B}}}(D_s^+\rightarrow \pi ^+\omega )=(1.9\pm 0.3)\times 10^{-3}\) shows no suppression [7]. It is hence considered to receive the long-distance annihilation contribution [1, 3].
The \(D_s^+\rightarrow SP,SV\) decays can help to investigate the short and long-distance annihilation mechanisms [8,9,10], where S stands for a non-strange scalar meson. For example, the WA process for \(D_s^+\rightarrow a_0^{0(+)}\pi ^{+(0)}\) violates G-parity [9, 11], such that its branching fraction is expected as small as \({{\mathcal {B}}}(D_s^+\rightarrow \pi ^+\pi ^0)\). Nonetheless, one measures that \({{\mathcal {B}}}(D_s^+\rightarrow a_0^{0(+)}\pi ^{+(0)})\simeq 100 {{\mathcal {B}}}(D_s^+\rightarrow \pi ^+\rho ^0)\) [7, 12]. Clearly, it indicates the main contribution from the long-distance annihilation process [9]. Explicitly, the long-distance annihilation process for \(D_s^+\rightarrow a_0^{0(+)}\pi ^{+(0)}\) starts with the \(D_s^+\rightarrow \eta ^{(\prime )}\rho ^+\) weak decay, followed by the \(\eta ^{(\prime )}\) and \(\rho ^+\) rescattering. With the \(\pi ^{+(0)}\) exchange, \(\eta ^{(\prime )}\) and \(\pi ^+\) are turned into \(a_0^{+(0)}\) and \(\pi ^{0(+)}\), respectively. Since BESIII has recently reported the first observation of the branching fractions of \(D_s^+\rightarrow SV,S\rightarrow PP\) as [13, 14]
we are wondering which of the short and long-distance annihilation processes can be the dominant contribution. Hence, we propose to study \(D_s^+\rightarrow a_0^{+(0)} \rho ^{0(+)}\), \(D_s^+\rightarrow a_0^+\omega \), and the resonant three-body \(D_s^+\rightarrow a_0^{+(0)} \rho ^{0(+)}, a_0^{+(0)}\rightarrow [\eta \pi ^{+(0)},K^+{{\bar{K}}}^0(K^+ K^-)]\) decays, in order to analyze the data in Eq. (1). We will also test if \({{\mathcal {B}}}(D_s^+\rightarrow a_0^+ \rho ^0, a_0^+ \rho ^0)\) have nearly equal sizes as \({{\mathcal {B}}}(D_s^+\rightarrow a_0^+ \pi ^0)\simeq {{\mathcal {B}}}(D_s^+\rightarrow a_0^0 \pi ^+)\) that respects the isospin symmetry.
2 Formalism
Considering the short-distance WA processes, \(D_s^+\rightarrow PP(PV)\) and \(D_s^+\rightarrow SP(SV)\) both get an A term as the annihilation amplitude in Fig. 1. According to \({{\mathcal {B}}}(D_s^+\rightarrow \pi ^+\rho ^0)\simeq 10^{-4}\) that receives the short-distance WA contribution [7], one regards A to give \({{\mathcal {B}}}(D_s^+\rightarrow a_0^{+(0)} \rho ^{0(+)})\) not larger than \(10^{-4}\). However, \(\mathcal{B}_+\) in Eq. (1) suggests \({{\mathcal {B}}}(D_s^+\rightarrow a_0^+ \rho ^0)\simeq 10^{-3}\). This strongly suggests that the main contribution to \(D_s^+\rightarrow a_0^{+(0)} \rho ^{0(+)}\) is from the long-distance annihilation process. For \(D_s^+\rightarrow a_0^+\omega \), the WA contribution is suppressed with the G-parity violation, such that \(A\simeq 0\). Therefore, we start with the triangle rescattering processes for \(D_s^+\rightarrow a_0^{+(0)} \rho ^{0(+)}\) and \(D_s^+\rightarrow a_0^+\omega \) as the most possible main contributions.
See Fig. 2, the rescattering processes for \(D_s^+\rightarrow a_0^{+(0)} \rho ^{0(+)}\) include both the weak and strong decays. In our case, the weak decays come from \(D^{+}_{s}\rightarrow \pi ^{+}\eta ^{(\prime )},K^{+}{\bar{K}}^0\), and the amplitudes are given by [2,3,4, 6]
where \(G_F\) is the Fermi constant, \(V^*_{cs}V_{ud}\) the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements, and \(V^*_{cs}V_{ud}\simeq 1\) presents the Cabibbo-allowed decay modes. In addition, (T, C, A) are the topological amplitudes, along with the mixing angle \(\phi =43.5^\circ \) from the \(\eta -\eta ^\prime \) mixing matrix [15, 16]:
For the strong decays \(V,S\rightarrow PP\), the amplitudes are given by [9]
where \(\epsilon _{\mu }\) is the polarization four vector of the \(\rho \) meson, and \(q_{1,2}^\mu \) are the four momenta of \(\pi ^+\pi ^-(\pi ^0\pi ^+)\), respectively. The SU(3) flavor symmetry is able to relate different \(V(S)\rightarrow PP\) decay channels [17, 18], such that we obtain \(g_\rho /2\) for \({{\mathcal {M}}}_{\rho ^0\rightarrow K^+ K^-,{{\bar{K}}}^0 K^0}\), \( g_\rho /\sqrt{2}\) for \({{\mathcal {M}}}_{\rho ^+\rightarrow K^+{{\bar{K}}}^0}\), and \((-)g_K/\sqrt{2} \) for \({{\mathcal {M}}}_{a^0_0\rightarrow K^+ K^-({{\bar{K}}}^0 K^0)}\), together with \((-)g_\rho /2\) for \({{\mathcal {M}}}_{\omega \rightarrow K^+ K^-(\bar{K}^0 K^0)}\).
By assembling the weak and strong couplings in the rescattering processes, we derive that
with \(M_a^{(\prime )}\) and \(M_{b(c)}[{\hat{M}}_{b(c)}]\) for Fig. 2a, b(c), respectively. More explicitly, \({{\mathcal {M}}}_{a,b,c}\) are given by [9, 19, 20]
with \(q_2=q_1-p_2\) and \(q_3=q_1-p_1\) following the momentum flows in Fig. 2. The form factor \(F_M(q_{2}^2)\equiv (\Lambda _M^{2}-m^{2}_M)/(\Lambda _M^{2}-q^{2}_{2})\) with the cutoff parameter \(\Lambda _M\) [\(M=(\pi ,K)\)] is to avoid the overestimation with \(q_2\) to \(\pm \infty \) [21]. Substituting \(\eta ^\prime \) and \(\omega \) for \(\eta \) in \({{\mathcal {M}}}_a\) and \(\rho ^0\) in \({{\mathcal {M}}}_{b(c)}\) leads to \({{\mathcal {M}}}_a^\prime \) and \(\hat{{\mathcal {M}}}_{b(c)}\), respectively.
As a consequence, we find that
where the first relation respects the isospin symmetry, whereas \(\hat{{\mathcal {M}}}_{b}=- \hat{{\mathcal {M}}}_{c}\) due to \({{\mathcal {M}}}_{\omega \rightarrow K^+ K^-}=-{{\mathcal {M}}}_{\omega \rightarrow {{\bar{K}}}^0 K^0}\) cancels the rescattering contributions to \(D_{s}^{+}\rightarrow \omega a^{+}_{0}\), which causes \({{\mathcal {M}}}(D_{s}^{+}\rightarrow \omega a^{+}_{0})=0\).
To deal with the triangle loops in Eq. (6), the equation in Refs. [22,23,24,25] can be useful, given by
By calculating the triangle rescattering processes, we obtain
with \(C_{\eta ^{(\prime )}}=C_{2,\eta ^{(\prime )}}-C_{2,\eta ^{(\prime )}}^*\) and \(C_K=C_{2,K}-C_{2,K}^*\) as the integrated results of the \(C_2\) terms in Eq. (8), where \(C_{2,M}^{(*)}\) are defined by
However, the \(C_1\) terms have been disappearing due to \(p_\rho ^\mu \cdot \epsilon =0\) in the amplitudes.
For the three-body decays \(D_{s}^{+}\rightarrow \rho ^{0(+)}(a^{+(0)}_{0}\rightarrow ) \eta \pi ^{+(0)}, K^+{{\bar{K}}}^0(K^+ K^-)\), we present [7]
with \(s=(p_{\eta (K)}+p_{\pi (K)})^{2}\) and \(t=(p_{\rho }+p_{\pi (K)})^{2}\), such that the theoretical results can be compared to the data in Eq. (1). In the above equation, \(1/D_{a_{0}}(s)\) presents the propagator for \(a_0\), and we define [26]
where
with \(G_{\alpha \beta }=(g_{\eta ^{(\prime )}},g_K)\) for \(\alpha \beta ={(\eta ^{(\prime )}\pi ,K {{\bar{K}}})}\) and \(m_{\alpha \beta }^\pm =m_\alpha \pm m_\beta \).
3 Numerical results
In the numerical analysis, we adopt \(V_{cs}=V_{ud}=1-\lambda ^2/2\) with \(\lambda =0.22453\pm 0.00044\) in the Wolfenstein parameterization [7], along with \(m_{a_0^0}=0.987\) GeV [27, 28]. The topological parameters (T, C, A) have been extracted as [6]
where \(\delta _{C,A}\) are the relative strong phases. According to the extraction, we obtain \({{\mathcal {B}}}(D_s^+\rightarrow \pi ^{+}\eta ,\pi ^+\eta ^{\prime },K^{+}{\bar{K}}^0)= (1.8\pm 0.2,4.2\pm 0.5,3.1\pm 0.5)\times 10^{-2}\), consistent with the experimental values of \((1.68\pm 0.10,3.94\pm 0.25,2.95\pm 0.14)\times 10^{-2}\), respectively [7]. For \(V,S\rightarrow PP\), the strong coupling constants read [7, 27, 28]
Empirically, \(\Lambda _M\) of \({{\mathcal {O}}}(1.0~\text {GeV})\) is commonly used to explain the data [29,30,31]; besides, it is obtained that \(\Lambda _K-\Lambda _\pi =m_K-m_\pi \) [32]. Therefore, we are allowed to use \((\Lambda _\pi ,\Lambda _K)=(1.25\pm 0.25,1.60\pm 0.25)\) GeV, which result in
with \(p_{a_0}^2=m_{a_0}^2\). Subsequently, we predict
where the first error in \(\mathcal{B}(D_{s}^{+}\rightarrow \rho ^{0(+)}a^{+(0)}_{0})\) takes into account the uncertainties from \(\Lambda _{\pi }\) and \(\Lambda _{K}\), and the second one combines those from \(V_{cs}^*\), \(V_{ud}\), (T, C, A), and the strong coupling constants. For the resonant three-body decays, we obtain
where the sources of the two errors are the same as those in Eq. (17).
4 Discussions and conclusions
In the triangle loop, when the momentum flow approaches the mass shell for one of the three propagators, the integration with \(i\epsilon \simeq im\Gamma \) gives rise to the imaginary parts in Eq. (16), where \(\Gamma \) is a very tiny decay width for \(\pi \), \(\eta ^{(\prime )}\) or K. The off-shell integrations are responsible for the real parts in Eq. (16), for which we take \({{\mathcal {M}}}_a\) as our description. In principle, the integration allows a momentum flow from \(-\infty \) to \(+\infty \). However, when the exchange particle proceeds with \(q_2^2\) around several GeV\(^2\), instead of the infinity, the integration with \(q_2^2\sim \pm \infty \) causes an overestimation [21]. We have accordingly introduced the form factor \(F_\pi \) in Eq. (6) to cut off the contribution from \(q_2^2\sim \pm \infty \). While \(q_1^2\), \(q_2^2\) and \(q_3^2\) have been associated in the loop, \(F_\pi \) also works to cut off the contributions from the propagators of the rescattering particles \(\pi ^+\) and \(\eta ^{(\prime )}\). Note that the single cutoff form factor has also been commonly used elsewhere [33,34,35].Footnote 1
The smallness of the WA \(D_s^+\rightarrow MM\) decay can be traced back to its amplitude, given by [37,38,39]
where \(q^\mu \langle MM|u\gamma _\mu d|0\rangle =0\) corresponds to the conservation of the vector current (CVC); most importantly, \(m_{u(d)}\simeq 0\) causes the chiral suppression of the WA \(D_s^+\rightarrow MM\) decay. According to the data, \({{\mathcal {B}}}(D_s^+\rightarrow \pi ^+\rho ^0)=(1.9\pm 1.2)\times 10^{-4}\) indicates that \(\mathcal{B}_{WA}(D_s^+\rightarrow MM)\) should be around \(10^{-4}\). In addition, \({{\mathcal {B}}}(D_s^+\rightarrow \pi ^+\pi ^0)<3.4\times 10^{-4}\) suggests that the G-parity violation suppresses the WA process even more. Therefore, since \(D_s^+\rightarrow \rho a_0,\rho \pi \) are both the G-parity conserved processes, it is reasonable to present that \({{\mathcal {B}}}_{WA}(D_s^+\rightarrow \rho a_0) \simeq {{\mathcal {B}}}_{WA}(D_s^+\rightarrow \rho \pi )\sim 10^{-4}\). As a theoretical support, we present
in agreement with our estimation, where \(R_f=3.1\times 10^{-2}\) is mostly from the phase space factors, and \({{\mathcal {B}}}_{WA}(B_c^+\rightarrow \rho a_0)\simeq 1.0\times 10^{-5}\) is adopted from Ref. [40].
Disregarding the WA contributions, we predict \(\mathcal{B}(D_{s}^{+}\rightarrow \rho ^{0(+)}a^{+(0)}_{0})=3.0\times 10^{-3}\) in Eq. (17). It is found that the \(\eta \pi ,\eta '\pi ,K^+\bar{K}^0\) rescatterings and their interferences give 6%, 7%, 30% and 59% of \({{\mathcal {B}}}(D_s^+\rightarrow a_0^{+(0)}\rho ^{0(+)})\), respectively. By contrast, the \(\rho \eta ^{(\prime )}\) rescatterings from \(D_s^+\rightarrow \rho \eta ^{(\prime )}\) dominantly contribute to \(D_s^+\rightarrow a_0^{+(0)}\pi ^{0(+)}\), instead of \(D_s^+\rightarrow K^* K\) [9]. Since \(D_{s}^{+}\rightarrow \omega a^{+}_{0}\) has no rescattering effects; besides, the WA contribution is suppressed by the G-parity violation, we anticipate that \(\mathcal{B}(D_{s}^{+}\rightarrow \omega a^{+}_{0}) \simeq \mathcal{B}(D_s^+\rightarrow \pi ^+\pi ^0)<3.4\times 10^{-4}\) [7].
In Eq. (18), \(\mathcal{B}(D_{s}^{+}\rightarrow \rho ^{0}(a^{+}_{0}\rightarrow )\eta \pi ^{+}) =(1.6^{+0.2}_{-0.3}\pm 0.6)\times 10^{-3}\) is able to explain the data [see Eq. (1)], demonstrating the sufficient long-distance annihilation contribution. It is confusing that \(\mathcal{B}(D_s^+\rightarrow a_0^0 \rho ^+,a_0^0\rightarrow K^+ K^-)\) is 10 times smaller than \({{\mathcal {B}}}_0\) in Eq. (1). For clarification, we take the approximate form of the resonant branching fraction: \(\mathcal{B}(D_s^+\rightarrow SV,S\rightarrow PP)\simeq {{\mathcal {B}}}(D_s^+\rightarrow SV){{\mathcal {B}}}(S\rightarrow PP)\), together with the isospin relation: \(\mathcal{B}(D_{s}^{+}\rightarrow \rho ^0 a^+_0)\simeq {{\mathcal {B}}}(D_{s}^{+}\rightarrow \rho ^+ a^0_{0})\), such that \({{\mathcal {B}}}_0/{{\mathcal {B}}}_+\) is reduced as \(\mathcal{B}(a_0^0\rightarrow K^+ K^-)/{{\mathcal {B}}}(a_0^+\rightarrow \eta \pi ^+)\simeq 1/3\), disagreeing with \({{\mathcal {B}}}(a_0^0\rightarrow K^+ K^-)/{{\mathcal {B}}}(a_0^+\rightarrow \eta \pi ^+)=1/10\) from the experimental extraction [8]. Therefore, we conclude that there exists a possible contradiction between the observations in Eq. (1).
In our reasoning, the contradiction might be caused by \(D_s^+\rightarrow \rho ^+ f_0,f_0\rightarrow K^+ K^-\) with \(f_0\equiv f_0(980)\), which can be mistaken as \(D_s^+\rightarrow \rho ^+ a_0^0,a_0^0\rightarrow K^+ K^-\) [41]. First, since \(D_s^+\rightarrow \rho ^+ f_0\) is an external W-boson emission process, its branching fraction can be of order \(10^{-3}\). Second, \(a_0\) and \(f_0\) are both scalar mesons, and have nearly the same masses and overlapped decay widths. As a result, it is possible that one cannot distinguish between the resonant signals of \(D_s^+\rightarrow \rho ^+(f_0,a_0), (f_0,a_0)\rightarrow K^+ K^-\) in the \(K^+ K^-\) invariant mass spectrum. For a careful examination, we suggest a measurement of \({{\mathcal {B}}}(D_{s}^{+}\rightarrow \rho ^{0}(a^{+}_{0}\rightarrow )K^+{\bar{K}}^0)/ \mathcal{B}(D_{s}^{+}\rightarrow \rho ^{+}(a^{0}_{0}\rightarrow )K^+ K^-)\), which will be observed around 2 if there exists no resonant \(f_0\rightarrow K^+ K^-\) decay to be involved in \({{\mathcal {B}}}(D_{s}^{+}\rightarrow \rho ^+ K^+ K^-)\).
In summary, we have studied \(D_{s}^{+}\rightarrow \rho ^{0(+)}a^{+(0)}_{0}\), \(D_{s}^{+}\rightarrow \omega a_0^0\), and the resonant \(D_{s}^{+}\rightarrow \rho a_0\), \(a_{0}\rightarrow \eta \pi (KK)\) decays. In the final state interaction, where \(D_s^+\rightarrow \eta ^{(\prime )}\pi ^+ (K^+{{\bar{K}}}^0)\) is followed by the \(\eta ^{(\prime )}\pi ^+\) (\(K^+{{\bar{K}}}^0\)) to \(\rho a_0\) rescattering, we have predicted \(\mathcal{B}(D_{s}^{+}\rightarrow \rho ^{0(+)}a^{+(0)}_{0})=(3.0\pm 0.3\pm 1.0)\times 10^{-3}\). Because of the non-contribution of the rescattering effects and the suppressed short-distance W annihilation, it has been expected that \({{\mathcal {B}}}(D_{s}^{+}\rightarrow \omega a^{+}_{0}) \simeq {{\mathcal {B}}}(D_s^+\rightarrow \pi ^+\pi ^0)<3.4\times 10^{-4}\). For the resonant three-body decay, \(\mathcal{B}(D_{s}^{+}\rightarrow \rho ^{0}(a^{+}_{0}\rightarrow )\eta \pi ^{+}) =(1.6^{+0.2}_{-0.3}\pm 0.6)\times 10^{-3}\) has been shown to agree with the data. We have presented \(\mathcal{B}(D_{s}^{+}\rightarrow \rho ^{+}(a^{0}_{0}\rightarrow )K^+K^-)\) 10 times smaller than the observation, indicating that a more careful examination is needed.
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The data used in this paper are publicly available and they can be found in the corresponding references.]
Notes
Please also consult Refs. [36], where one considers three cutoff form factors.
References
S. Fajfer, A. Prapotnik, P. Singer, J. Zupan, Phys. Rev. D 68, 094012 (2003)
B. Bhattacharya, J.L. Rosner, Phys. Rev. D 81, 014026 (2010)
H.Y. Cheng, C.W. Chiang, Phys. Rev. D 81, 074021 (2010)
F.S. Yu, X.X. Wang, C.D. Lu, Phys. Rev. D 84, 074019 (2011)
H.N. Li, C.D. Lu, Q. Qin, F.S. Yu, Phys. Rev. D 89, 054006 (2014)
H.Y. Cheng, C.W. Chiang, Phys. Rev. D 100, 093002 (2019)
P.A. Zyla et al., [Particle Data Group], PTEP. 2020, 083C01 (2020)
H.Y. Cheng, C.W. Chiang, Phys. Rev. D 81, 074031 (2010)
Y.K. Hsiao, Y. Yu, B.C. Ke, Eur. Phys. J. C 80, 895 (2020)
X.Z. Ling, M.Z. Liu, J.X. Lu, L.S. Geng, J.J. Xie, Phys. Rev. D 103, 116016 (2021)
N.N. Achasov, G.N. Shestakov, Phys. Rev. D 96, 036013 (2017)
M. Ablikim et al., BESIII, Phys. Rev. Lett. 123, 112001 (2019)
M. Ablikim et al., BESIII, Phys. Rev. D 104, 071101 (2021)
M. Ablikim et al., BESIII, Phys. Rev. D 104, 032011 (2021)
T. Feldmann, P. Kroll, B. Stech, Phys. Rev. D 58, 114006 (1998)
T. Feldmann, P. Kroll, B. Stech, Phys. Lett. B 449, 339 (1999)
N.A. Tornqvist, Ann. Phys. 123, 1 (1979)
Fayyazuddin and Riazuddin, A Modern Introduction To Particle Physics, 3rd edn. (World Scientific, Singapore, 2011)
Y. Yu, Y.K. Hsiao, Phys. Lett. B 820, 136586 (2021)
Y.K. Hsiao, Y. Yu, Phys. Rev. D 104, 034008 (2021)
M.C. Du, Q. Zhao, Phys. Rev. D 104, 036008 (2021)
G.T. Hooft, M.J.G. Veltman, Nucl. Phys. B 153, 365 (1979)
G. Passarino, M.J.G. Veltman, Nucl. Phys. B 160, 151 (1979)
T. Hahn, M. Perez-Victoria, Comput. Phys. Commun. 118, 153 (1999)
A. Denner, S. Dittmaier, Nucl. Phys. B 734, 62 (2006)
N.N. Achasov, A.V. Kiselev, Phys. Rev. D 70, 111901 (2004)
M. Ablikim et al., BESIII, Phys. Rev. D 95, 032002 (2017)
D.V. Bugg, Phys. Rev. D 78, 074023 (2008)
N.A. Tornqvist, Z. Phys. C 61, 525 (1994)
X.Q. Li, D.V. Bugg, B.S. Zou, Phys. Rev. D 55, 1421 (1997)
Q. Wu, D.Y. Chen, X.J. Fan, G. Li, Eur. Phys. J. C 79, 265 (2019)
H.Y. Cheng, C.K. Chua, A. Soni, Phys. Rev. D 71, 014030 (2005)
Q. Wu, D.Y. Chen, T. Matsuki, Eur. Phys. J. C 81, 193 (2021)
Q. Wu, D.Y. Chen, Phys. Rev. D 100, 114002 (2019)
G. Li, X.H. Liu, Z. Zhou, Phys. Rev. D 90, 054006 (2014)
Y. Cheng, Q. Zhao, arXiv:2106.12483 [hep-ph]
Y.K. Hsiao, C.Q. Geng, Phys. Rev. D 91, 077501 (2015)
Y.K. Hsiao, S.Y. Tsai, C.C. Lih, E. Rodrigues, JHEP 04, 035 (2020)
X. Huang, Y.K. Hsiao, J. Wang, L. Sun, arXiv:2109.02897 [hep-ph]
X. Liu, Z.J. Xiao, Phys. Rev. D 82, 054029 (2010)
Private communications with the group members at BESIII
Acknowledgements
We would like to thank Prof. Liang Sun for useful discussions. YKH was supported in part by NSFC (Grant nos. 11675030 and 12175128). YY was supported in part by NSFC (Grant nos. 11905023 and 12047564), the Fundamental Research Funds for the Central Universities (Grant no. 2020CDJQY-Z003) and CQCSTC (Grant nos. cstc2020jcyj-msxmX0555 and cstc2020jcyj-msxmX0810). BCK was supported in part by NSFC (Grant no. 11875054) and the Chinese Academy of Sciences (CAS) Large-scale Scientific Facility Program; Joint Large-Scale Scientific Facility Fund of the NSFC and CAS (Contract no. U2032104).
Author information
Authors and Affiliations
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Funded by SCOAP3
About this article
Cite this article
Yu, Y., Hsiao, YK. & Ke, BC. Study of the \(D_s^+\rightarrow a_0(980) \rho \) and \(a_0(980) \omega \) decays. Eur. Phys. J. C 81, 1093 (2021). https://doi.org/10.1140/epjc/s10052-021-09895-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjc/s10052-021-09895-y