The role of triangle singularity in the $ {B^0 \to D^- \pi^+ a_0(980)(\pi^0 \eta)}$ decay

Over the past few decades, significant progress has been made in studies of the hadron spectrum, driven by the observation of numerous exotic states [1]. Although many of these exotic states can be interpreted as multiquark states or hadronic molecules, some observed structures may not correspond to poles of the S-matrix but may instead arise from kinematic singularities [2−4]. Among such kinematic effects, the triangle singularity (TS), first proposed by Landau in 1959 [5, 6], has attracted increasing interest. A triangle mechanism originates from the following sequential process: an initial particle A decays into two internal particles, labeled 1 and 2, which move back-to-back in the rest frame of A. Particle 2 subsequently decays into an internal particle 3 and an external particle B, with particle 3 moving in the same direction as particle 1. The two internal particles 1 and 3 then undergo rescattering and form an external particle C. According to the Coleman-Norton theorem [7], the emergence of a TS depends on whether the above processes can be interpreted as a classical scattering process and whether all three internal particles can simultaneously go on shell and become collinear in the rest frame of the decaying particle [5]. In reality, the internal particles have finite widths, which smear the singular behavior and transform the TS into a finite peak that can be observed experimentally.

Phenomenological studies have successfully applied the TS mechanism to explain a variety of long-standing puzzles in hadron physics. The anomalously large isospin violation observed in the decay $ J/\psi \to \gamma \eta(1405/ 1475) \to \gamma \pi^0 f_0(980) \to \gamma 3\pi $, reported by the BESIII Collaboration [8], was interpreted in terms of the TS in Ref. [9]. This mechanism has contributed to a better understanding of the nature of the two nearby states $ \eta(1405) $ and $ \eta(1475) $, as well as the mixing between the $ a_0(980) $ and $ f_0(980) $ resonances, which has been further investigated in a series of subsequent works [10−14]. Meanwhile, the triangle mechanism has also been employed to interpret the properties of heavy exotic hadronic states, such as the $ Y(4260) $, $ Z_c(3900) $, $ Z_b(10610) $, and $ Z_b(10650) $, in various processes [15−23]. A milestone in the development of the triangle mechanism was achieved in Ref. [24], where a detailed analysis of the singularities in a triangle loop integral was presented and a compact formula was derived for evaluating the TS on the physical boundary. This formalism was successfully applied to the decay $ \Lambda_b \to J/\psi K^- p $ through a $ \Lambda^* $-charmonium-proton triangle loop. Thereafter, the triangle mechanism involving an $ f_0(980) $ or $ a_0(980) $ final state was further explored in a variety of processes [25−31]. In Ref. [26], the authors identified two nonresonant peaks associated with the TS at approximately $ 2850 $ and $ 3000\; \mathrm{MeV} $ in the invariant mass distributions of the $ \pi D_{s0} $ and $ \pi D_{s1} $ systems, respectively. That work also demonstrated the relation between the structure of the triangle amplitude and the finite widths of the internal particles in the loop. A comprehensive review of threshold cusps and various TS structures in hadronic reactions can be found in Ref. [32], where their roles in phenomena related to exotic hadron candidates are systematically summarized.

The triangle mechanism has been extensively discussed in a wide range of hadronic decays involving charmonium [33−39], the Λ baryon family [40−46], and the D-meson sector [47−50]. In addition, decays of the B-meson family provide a promising platform for probing this mechanism, such as $ B^- \to D^{*0} \pi^- \pi^+ \pi^-(\pi^0 \eta) $ [26], $ B^- \to K^- \pi^0 X(3872) $ [51], $ B^- \to K^- X(3872) $ with $ X(3872) \to \pi^0 \pi^+ \pi^- $ [52], $ B^- \to K^- \pi^- D_{s0/s1}^+ $ [53], $ B^+ \to J/\psi \phi K^+ $ [54], $ B^+ \to J/\psi \pi^{+} \pi^0 K^0 $ and $ B^+ \to J/\psi \pi^{+} \pi^- K^+ $ [55], $ B^+ \to D^- D_s^+ \pi^+ $ and $ B^0 \to \bar{D}^0 D_s^+ \pi^- $ [56], $ B^0 \to J/\psi K^0 f_0(980)(a_0(980)) $ [31], $ \bar{B}^0 \to \chi_{c1} K^- \pi^+ $ [57], $ B \to (J/\psi \pi^+ \pi^-) K \pi $ [58], $ \bar{B}_s^0 \to J/\psi \pi^0 f_0(980) $ [29], and $ B_c \to B_s \pi \pi $ [59]. Beyond hadronic decays, the triangle mechanism has also been explored in a variety of other processes, including semileptonic τ-lepton decays [60−62], photon-proton collisions [63, 64], electron-positron annihilation [65, 66], and proton-proton collisions [67, 68]. Furthermore, an interesting study [69] discussed the structure of TS involving new-physics particles at high-energy colliders. In addition, numerous other investigations have been devoted to the triangle mechanism and its phenomenological implications [70−81].

In particular, the TS explanation for the observation of the $ a_1(1420) $ reported by the COMPASS Collaboration [82, 83] has been widely accepted. In this mechanism, the $ a_1(1260) $ first decays into $ K^{*}\bar{K} $, with $ K^{*}\to \pi K $, after which the $ K\bar{K} $ pair fuses to form the $ f_0(980) $, giving the observed $ \pi f_0(980) $ decay mode. In addition, it has also been studied in various processes [28, 29, 37, 40, 60, 84]. Based on previous studies, in this work we explore the physical effects of the TS and identify its contribution to the $ B^0 \to D^- \pi^+ a_0(980) $ decay. The observation of the weak decay $ B^0 \to D^- K^+ \bar{K}^{*0} $ by the Belle Collaboration [85], together with the cascade decay $ \bar{K}^{*0} \to K^- \pi^+ $, allows the construction of a $ \bar{K}^{*0} $-K-K triangle loop. As we will show, the invariant mass of the $ \bar{K}^{*0} K^+ $ system in the $ B^0 $ decay allows the production of the $ a_0(980) $ in the region where the TS condition is satisfied. This leads to a pronounced peak structure in the $ \pi^+ a_0(980) $ invariant mass spectrum around $ 1420\; \mathrm{MeV} $. In the above calculation, the triangle amplitude is evaluated following the analytical formulations developed in Refs. [24, 32]. Furthermore, within the chiral unitary approach, the $ a_0(980) $ can be regarded as a dynamically generated state arising from meson-meson interactions [86−88]. An enhancement around $ 980\; \mathrm{MeV} $ is expected to be reproduced in the $ \pi^0 \eta $ invariant mass distribution. Consequently, the $ B^0 \to D^- \pi^+ a_0(980)(\pi^0\eta) $ decay provides a potential platform to investigate both the TS structure associated with the $ \bar{K}^{*0} $-K-K triangle loop and the nature of the $ a_0(980) $ resonance.

The paper is organized as follows. In Sec. II, we present the theoretical formalism and detail the kinematic conditions for the $ \bar{K}^{*0} $-K-K triangle loop considered here. In Sec. III, the weak decay mechanism for $ B^0 \to D^- K^+ \bar{K}^{*0} $ is described, and the effective coupling strength is extracted from current experimental measurements. In Sec. IV, we formulate the $ B^0 \to D^- \pi^+ a_0(980) $ decay by incorporating the triangle mechanism and provide the details of the derivation. The $ a_0(980) $ is further treated as a dynamically generated state, and the sequential process $ B^0 \to D^- \pi^+ a_0(980) $ with $ a_0(980) \to \pi^0 \eta $ is analyzed. In Sec. V, we present the numerical results for these processes. The peaks around $ 1420 $ and $ 980\; \mathrm{MeV} $ are reproduced in the invariant mass distributions of the $ \pi a_0 $ and $ \pi^0 \eta $ systems, respectively. Finally, a brief summary is provided in Sec. VI.

In this section, we show that a peak structure around $ 1420\; \mathrm{MeV} $, induced by the TS, can be generated in the $ B^0 \to D^- \pi^+ a_0(980) $ decay. The Feynman diagram is shown in the left panel of Fig. 1. In this process, the $ B^0 $ first decays into $ D^- \bar{K}^{*0} K^+ $, after which the $ \bar{K}^{*0} $ decays into $ K^- $ and $ \pi^+ $. The $ K^+ $ and $ K^- $ move in the same direction, with the $ K^- $ moving faster than the $ K^+ $, allowing the $ K^+K^- $ pair to fuse into the $ a_0(980) $. The channel $ B^0 \to D^{-} \bar{K}^{0} K^{*+} $ has not yet been observed experimentally. Therefore, the contribution of the $ K^{*+} $-$ \bar{K}^{0} $-$ K^{0} $ triangle loop from the $ B^0 \to D^{-} \bar{K}^{0} K^{*+} $ decay is not considered in the present work. Furthermore, the $ a_0(980) $ can be interpreted as a dynamically generated state with isospin $ I=1 $, arising from the coupled channels $ \pi^0\eta $, $ K^+K^- $, and $ K^0\bar{K}^0 $ within the chiral unitary approach [87, 88]. The decay of $ a_0(980) $ is shown in the right panel of Fig. 1.

Figure 1.  The Feynman diagrams for the $ B^0 \to D^- \pi^+ a_0(980) $ and $ B^0 \to D^- \pi^+ a_0(980), a_0(980)\to\pi^0 \eta $ processes involving a $ \bar{K}^{*} $-K-$ \bar{K} $ triangle loop are shown in the left and right panels, respectively.

The TS occurs when the kinematic variables of the internal particles satisfy the following condition

where $ q_{+}^{\text{on}} $ denotes the on-shell three-momentum of $ K^{+} $ in the center-of-mass frame (COM) of the $ \bar{K}^{*0} K^+ $ system. The quantity $ m_{\text{inv}}(\pi a_0) $ is the invariant mass of the $ \bar{K}^{*0} K^+ $ system, and $ \lambda(x, y, z)=x^2 +y^2 +z^2 -2xy-2yz -2xz $ is the Källén function. Meanwhile, $ q_-^{\text{a}} $ is given by

with the definitions

where $ E_{K^{-}}^* $ and $ p_{K^{-}}^* $ are the energy and momentum of the $ K^{-} $ meson in the rest frame of $ a_0 $. ν and γ denote the velocity of the $ K^+ K^- $ system and the Lorentz boost factor, respectively. Eq. (1) implies that all three particles in the loop are on shell and that $ K^+ $ in the rest frame of $ a_0(980) $ and $ a_0(980) $ in the COM frame of $ \pi a_0(980) $ move in the same direction. The formation of $ a_0(980) $ through $ K^+ K^- $ fusion further requires that the momentum of $ K^+ $ in the COM frame of $ \pi a_0 $ be smaller than that of $ K^- $ in the rest frame of $ \bar{K}^{*0} $. Equivalently, Eq. (1) can be understood as the condition that $ q^a_- $ and $ q^{\text{on}}_+ $ represent the singularities of the triangle loop function in the upper and lower halves of the complex-q plane, respectively. The integration contour of the loop function in Eq. (21) is then pinched between $ q^a_- $ and $ q^{\text{on}}_+ $ at the same point on the real axis. Consequently, solving Eq. (1) yields a TS around $ 1420 $ MeV in the $ m_{\text{inv}}(\pi a_0) $ invariant mass distribution. When the finite width of the $ \bar{K}^{*0} $ is taken into account by replacing $m_{\bar{K}^{*0}} \to m_{\bar{K}^{*0}} - {\rm i}\Gamma_{\bar{K}^{*0}}/2$ with $ \Gamma_{\bar{K}^{*0}} = 47\; \mathrm{MeV} $, the TS moves into the complex plane, yielding $1420.7 - {\rm i}\,28.0\; \mathrm{MeV}$.

In this section, we show that a peak structure around $ 1420\; \mathrm{MeV} $, induced by the TS, can be generated in the $ B^0 \to D^- \pi^+ a_0(980) $ decay. The Feynman diagram is shown in the left panel of Fig. 1. In this process, the $ B^0 $ first decays into $ D^- \bar{K}^{*0} K^+ $, after which the $ \bar{K}^{*0} $ decays into $ K^- $ and $ \pi^+ $. The $ K^+ $ and $ K^- $ move in the same direction, with the $ K^- $ moving faster than the $ K^+ $, allowing the $ K^+K^- $ pair to fuse into the $ a_0(980) $. The channel $ B^0 \to D^{-} \bar{K}^{0} K^{*+} $ has not yet been observed experimentally. Therefore, the contribution of the $ K^{*+} $-$ \bar{K}^{0} $-$ K^{0} $ triangle loop from the $ B^0 \to D^{-} \bar{K}^{0} K^{*+} $ decay is not considered in the present work. Furthermore, the $ a_0(980) $ can be interpreted as a dynamically generated state with isospin $ I=1 $, arising from the coupled channels $ \pi^0\eta $, $ K^+K^- $, and $ K^0\bar{K}^0 $ within the chiral unitary approach [87, 88]. The decay of $ a_0(980) $ is shown in the right panel of Fig. 1.

Figure 1.  The Feynman diagrams for the $ B^0 \to D^- \pi^+ a_0(980) $ and $ B^0 \to D^- \pi^+ a_0(980), a_0(980)\to\pi^0 \eta $ processes involving a $ \bar{K}^{*} $-K-$ \bar{K} $ triangle loop are shown in the left and right panels, respectively.

The TS occurs when the kinematic variables of the internal particles satisfy the following condition

where $ q_{+}^{\text{on}} $ denotes the on-shell three-momentum of $ K^{+} $ in the center-of-mass frame (COM) of the $ \bar{K}^{*0} K^+ $ system. The quantity $ m_{\text{inv}}(\pi a_0) $ is the invariant mass of the $ \bar{K}^{*0} K^+ $ system, and $ \lambda(x, y, z)=x^2 +y^2 +z^2 -2xy-2yz -2xz $ is the Källén function. Meanwhile, $ q_-^{\text{a}} $ is given by

with the definitions

where $ E_{K^{-}}^* $ and $ p_{K^{-}}^* $ are the energy and momentum of the $ K^{-} $ meson in the rest frame of $ a_0 $. ν and γ denote the velocity of the $ K^+ K^- $ system and the Lorentz boost factor, respectively. Eq. (1) implies that all three particles in the loop are on shell and that $ K^+ $ in the rest frame of $ a_0(980) $ and $ a_0(980) $ in the COM frame of $ \pi a_0(980) $ move in the same direction. The formation of $ a_0(980) $ through $ K^+ K^- $ fusion further requires that the momentum of $ K^+ $ in the COM frame of $ \pi a_0 $ be smaller than that of $ K^- $ in the rest frame of $ \bar{K}^{*0} $. Equivalently, Eq. (1) can be understood as the condition that $ q^a_- $ and $ q^{\text{on}}_+ $ represent the singularities of the triangle loop function in the upper and lower halves of the complex-q plane, respectively. The integration contour of the loop function in Eq. (21) is then pinched between $ q^a_- $ and $ q^{\text{on}}_+ $ at the same point on the real axis. Consequently, solving Eq. (1) yields a TS around $ 1420 $ MeV in the $ m_{\text{inv}}(\pi a_0) $ invariant mass distribution. When the finite width of the $ \bar{K}^{*0} $ is taken into account by replacing $m_{\bar{K}^{*0}} \to m_{\bar{K}^{*0}} - {\rm i}\Gamma_{\bar{K}^{*0}}/2$ with $ \Gamma_{\bar{K}^{*0}} = 47\; \mathrm{MeV} $, the TS moves into the complex plane, yielding $1420.7 - {\rm i}\,28.0\; \mathrm{MeV}$.

To evaluate the amplitude shown in Fig. 1, the effective coupling strength of the $ B^0 \to D^- K^+ \bar{K}^{*0} $ vertex is required. The $ B^0 $ meson can decay into the $ D^- K^+ \bar{K}^{*0} $ final state through the weak decay of the b quark, as illustrated in Fig. 2. At the quark level, the b quark undergoes a weak transition into a c quark through the emission of a W boson and hadronizes into the $ D^- $ meson. The emitted W boson subsequently decays into a $ u\bar{d} $ pair, which combines with an additional $ q\bar{q} $ pair from the vacuum to form the $ \bar{K}^{*0}K^+ $ final state.

Figure 2.  (color online) The quark-level and hadron-level Feynman diagrams for the decay $ B^0 \to D^- K^+ \bar{K}^{*0} $ are shown in the left and right panels, respectively.

The interaction of the effective vertex in Fig. 2 can be constructed using the P-wave interaction. Following the convention adopted in Refs. [28, 53], we write

where $ \vec{\epsilon}_{\bar{K}^{*0}} $ and $ \vec{p}_{D} $ are the polarization vector of the $ \bar{K}^{*0} $ and the momentum of the $ D^- $, respectively. Below, we present the details of the calculation of the vertex coupling strength C. By solving Eq. (1), a TS around $1420\; \mathrm{MeV}$ is obtained. The three-momentum of the $ \bar{K}^{*0} $ in the $ \pi a_0 $ rest frame is approximately $ 135.66\; \mathrm{MeV} $, which is smaller than the $ \bar{K}^{*0} $ mass of $ 895.81 $ MeV. Therefore, the time component $ \epsilon^0 $ of the $ \bar{K}^{*0} $ polarization vector can be safely neglected. We take the polarization sum for the $ \bar{K}^{*0} $ as

The decay width for the process $ B \to D K \bar{K}^{*0} $ is given by

where $ m_{\mathrm{inv}}(\bar{K}^{*0}K^+) $ is the invariant mass of the $ \bar{K}^{*0}K^+ $ system. Here, $ \vec{\tilde{p}}^{\,\prime}_{\bar{K}^{*0}} $ and $ \vec{\tilde{p}}_{D} $ denote the three-momenta of the $ \bar{K}^{*0} $ in the $ \bar{K}^{*0}K^+ $ COM frame and of the $ D^- $ in the $ B^0 $ rest frame, respectively. They are given by

After squaring the amplitude $ t_{B \to D K \bar{K}^{*0} } $ and applying the polarization sum in Eq. (5), we obtain

where $ \vec{\tilde{p}}_{D}^\prime $ is the three-momentum of the $ D^- $ in the COM frame of the $ \bar{K}^{*0}K^+ $ system.

From recent measurements [1, 85], the partial decay branching fraction is given by Br($ B^0 \to D^- K^+ \bar{K}^{*0} $) =$ (7.7\pm 0.6)\times 10^{-4} $. Combining Eq. (6) with Eq. (8), we obtain

To evaluate the amplitude shown in Fig. 1, the effective coupling strength of the $ B^0 \to D^- K^+ \bar{K}^{*0} $ vertex is required. The $ B^0 $ meson can decay into the $ D^- K^+ \bar{K}^{*0} $ final state through the weak decay of the b quark, as illustrated in Fig. 2. At the quark level, the b quark undergoes a weak transition into a c quark through the emission of a W boson and hadronizes into the $ D^- $ meson. The emitted W boson subsequently decays into a $ u\bar{d} $ pair, which combines with an additional $ q\bar{q} $ pair from the vacuum to form the $ \bar{K}^{*0}K^+ $ final state.

Figure 2.  (color online) The quark-level and hadron-level Feynman diagrams for the decay $ B^0 \to D^- K^+ \bar{K}^{*0} $ are shown in the left and right panels, respectively.

The interaction of the effective vertex in Fig. 2 can be constructed using the P-wave interaction. Following the convention adopted in Refs. [28, 53], we write

where $ \vec{\epsilon}_{\bar{K}^{*0}} $ and $ \vec{p}_{D} $ are the polarization vector of the $ \bar{K}^{*0} $ and the momentum of the $ D^- $, respectively. Below, we present the details of the calculation of the vertex coupling strength C. By solving Eq. (1), a TS around $1420\; \mathrm{MeV}$ is obtained. The three-momentum of the $ \bar{K}^{*0} $ in the $ \pi a_0 $ rest frame is approximately $ 135.66\; \mathrm{MeV} $, which is smaller than the $ \bar{K}^{*0} $ mass of $ 895.81 $ MeV. Therefore, the time component $ \epsilon^0 $ of the $ \bar{K}^{*0} $ polarization vector can be safely neglected. We take the polarization sum for the $ \bar{K}^{*0} $ as

The decay width for the process $ B \to D K \bar{K}^{*0} $ is given by

where $ m_{\mathrm{inv}}(\bar{K}^{*0}K^+) $ is the invariant mass of the $ \bar{K}^{*0}K^+ $ system. Here, $ \vec{\tilde{p}}^{\,\prime}_{\bar{K}^{*0}} $ and $ \vec{\tilde{p}}_{D} $ denote the three-momenta of the $ \bar{K}^{*0} $ in the $ \bar{K}^{*0}K^+ $ COM frame and of the $ D^- $ in the $ B^0 $ rest frame, respectively. They are given by

After squaring the amplitude $ t_{B \to D K \bar{K}^{*0} } $ and applying the polarization sum in Eq. (5), we obtain

where $ \vec{\tilde{p}}_{D}^\prime $ is the three-momentum of the $ D^- $ in the COM frame of the $ \bar{K}^{*0}K^+ $ system.

From recent measurements [1, 85], the partial decay branching fraction is given by Br($ B^0 \to D^- K^+ \bar{K}^{*0} $) =$ (7.7\pm 0.6)\times 10^{-4} $. Combining Eq. (6) with Eq. (8), we obtain

The amplitude for the $ B^0 \to D^- \pi^+ a_0(980) $ decay (left panel of Fig. 1) can be written as

in the COM frame of the $ \pi a_0 $ system, where the $ \pi^+ $ originates from the $ \bar{K}^{*0} $ decay. In Eq. (11), the amplitude $ t_{B \to D K \bar{K}^{*0}} $ was calculated in the previous section (Sec. III). The amplitude $ t_{\bar{K}^{*0} K^- \pi^+} $ can be calculated using the chiral-invariant Lagrangian with local hidden symmetry [89, 90] and is given by

where bracket $ \langle \cdots \rangle $ denotes the SU(3) trace, and the coupling constant is given by $ g=m_V/2f_\pi $, with $ m_V=800 $ MeV and $ f_\pi=93 $ MeV, within the local hidden gauge formalism. P and $ V^\mu $ represent the pseudoscalar and vector meson octets, respectively, which are given by

Then, the amplitude for the decay $ \bar{K}^{*0} \to K^- \pi^+ $ can be written as

As in Eq. (8), we neglect the time component of the polarization vector in Eq. (14) and employ the polarization sum given in Eq. (5). Here, $ \vec{p}_{\pi^+}^{\,\prime} $ and $ \vec{p}_{K^-}^{\,\prime} $ denote the three-momenta of the $ \pi^+ $ and $ K^- $ in the COM frame of the $ \pi a_0 $ system, respectively. The momentum $ \vec{p}_{\pi^+}^{\,\prime} $ is given by

As discussed in Sec. II, the $ a_0(980) $ is considered to be a dynamically generated state. The amplitude $ t_{K^+ K^- a_0 } $ can be written directly as

Above, we set $ g_{K^+ K^- a_0 } = 3875 $ MeV [87]. Thus, the amplitude expression in Eq. (11) can be simplified to

where $ \vec{p}_{\pi^+}^{\,\prime} $ is given in Eq. (15). The quantity $ \vec{p}_{D}^{\,\prime} $ is the momentum of the $ D^- $ in the COM frame of $ \pi a_0 $, which originates from the $ B^0 $ decay.

Equation (5) is used to perform the polarization sum in Eq. (17), yielding

where $ \vec{p}_{\pi^+}^{\,\prime}-\vec{p}_{K^-}^{\,\prime}=\vec{k}-(-\vec{k}-\vec{q})=2\vec{k}+\vec{q} $. P is the momentum of the $ \pi a_0 $ system in its rest frame.

The quantity $ t_T $ is defined to describe the loop integral in Eq. (19) as

Following Refs. [24, 32], the negative-energy part of the $ \bar{K}^{*0} $ propagator in $ t_T $ can be neglected. The integration over $ dq^0 $ in $ t_T $ is performed by applying the residue theorem and using the following formula

Now, the triangle loop amplitude $ t_T $ reduces to

where $ \omega_{K^+}(\vec{q})=\sqrt{m_{K}^2 + \vec{q}^2} $, $ \omega_{K^{*0}}(\vec{q})= \sqrt{m_{\bar{K}^{*0}}^2 + \vec{q}^2} $, and $ \omega_{K^-}(\vec{q}+\vec{k}) = \sqrt{m_{K}^2 + (\vec{k}+\vec{q})^2} $. From Eq. (20), $ P^0 = m_{\mathrm{inv}}(\pi a_0) $ is the invariant mass of the $ \pi a_0 $ system. Although the integral in Eq. (23) is convergent, the chiral unitary approach naturally provides an upper limit $ q_{\mathrm{max}} $ [10, 13]. In this work, we use $ q_{\mathrm{max}} = 600\; \mathrm{MeV} $ for $ a_0(980) $ production [86, 91]. To reproduce the peak observed in experiments, the finite width of the $ \bar{K}^{*0} $ in the triangle loop must be taken into account. This is implemented by replacing $ \omega_{K^{*}} $ with $\omega_{K^{*}} - {\rm i} \Gamma_{\bar{K}^{*0}}/2$ in the denominator of Eq. (23).

Finally, the differential decay width for the three-body final state $ B^0 \to D^- \pi^+ a_0(980) $ is given by

The factor $ 2/3 $ in Eq. (24) originates from the angular phase-space integration of the integrand in $ \vec{p}_{D}^{\,\prime} \cdot \vec{k}= |\vec{p}_{D}^{\,\prime}| | \vec{k}|\cos \theta $. Here, $ \vec{p}_{D} $ denotes the momentum of the $ D^- $ in the $ B^0 $ rest frame.

In addition, we further explore the possibility of observing the $ a_0(980) $ through the $ \pi^0 \eta $ final state. Following Ref. [53], the $ a_0(980) $ appears as an unstable resonance in the $ B^0 $ decay and subsequently decays into $ \pi^0 \eta $. Therefore, we consider the corresponding four-body decay, as illustrated in the right panel of Fig. 1. In this process, the $ a_0(980) $ resonance can be treated as an intermediate particle, allowing us to include an additional differential distribution with respect to the $ \pi\eta $ invariant mass in Eq. 24. The differential decay width for $ B^0 \to D^- \pi^+ \pi^0 \eta $ is then given by

with

where $ m_{\text{inv}}(\pi \eta) $ is the invariant mass of the $ \pi \eta $ system produced in the $ a_0(980) $ decay, and $ m_{a_0} $ is the mass of $ a_0(980) $. Eq. (26) reduces to Eq. (24) in the limit $ \Gamma_{a_0} \to 0 $, where ${\rm i}\,\mathrm{Im}\,D \to -{\rm i}\pi \, \delta\big(m_{\mathrm{inv}}^2(\pi \eta) - m_{a_0}^2 \big)$. Thus, Eq. (26) simplifies to

The decay width $ \Gamma_{a_0} $ is calculated by considering the $ a_0 \to \pi^0 \eta $ decay

where $ |\vec{\tilde{q}}_{\eta}| $ denotes the magnitude of the η momentum in the rest frame of $ a_0(980) $ and is given by

By substituting Eq. (29) into Eq. (28), we obtain

Meanwhile, the amplitude for the $ K^+ K^- \to \pi \eta $ process mediated by exchange of the $ a_0(980) $ resonance can be written as

In contrast, when the $ a_0(980) $ is treated as a dynamically generated state, the amplitude $ t_{K^+ K^-\to \pi \eta} $ can be calculated by solving the Bethe–Salpeter (BS) equation within the chiral unitary approach. The BS equation is given by

Here, G denotes the loop function for two meson propagators [86].

and is regularized by a cutoff $ q_{\text{max}} $ as

with

A value of $ q_{\text{max}}=600 $ MeV is used to reproduce the $ a_0(980) $ around 980 MeV. The meson-meson interaction potentials for the case $ I=1 $ are given as follows [86]:

Finally, the double-differential distribution of the $ B^0 $ decay can be written as

The amplitude for the $ B^0 \to D^- \pi^+ a_0(980) $ decay (left panel of Fig. 1) can be written as

in the COM frame of the $ \pi a_0 $ system, where the $ \pi^+ $ originates from the $ \bar{K}^{*0} $ decay. In Eq. (11), the amplitude $ t_{B \to D K \bar{K}^{*0}} $ was calculated in the previous section (Sec. III). The amplitude $ t_{\bar{K}^{*0} K^- \pi^+} $ can be calculated using the chiral-invariant Lagrangian with local hidden symmetry [89, 90] and is given by

where bracket $ \langle \cdots \rangle $ denotes the SU(3) trace, and the coupling constant is given by $ g=m_V/2f_\pi $, with $ m_V=800 $ MeV and $ f_\pi=93 $ MeV, within the local hidden gauge formalism. P and $ V^\mu $ represent the pseudoscalar and vector meson octets, respectively, which are given by

Then, the amplitude for the decay $ \bar{K}^{*0} \to K^- \pi^+ $ can be written as

As in Eq. (8), we neglect the time component of the polarization vector in Eq. (14) and employ the polarization sum given in Eq. (5). Here, $ \vec{p}_{\pi^+}^{\,\prime} $ and $ \vec{p}_{K^-}^{\,\prime} $ denote the three-momenta of the $ \pi^+ $ and $ K^- $ in the COM frame of the $ \pi a_0 $ system, respectively. The momentum $ \vec{p}_{\pi^+}^{\,\prime} $ is given by

As discussed in Sec. II, the $ a_0(980) $ is considered to be a dynamically generated state. The amplitude $ t_{K^+ K^- a_0 } $ can be written directly as

Above, we set $ g_{K^+ K^- a_0 } = 3875 $ MeV [87]. Thus, the amplitude expression in Eq. (11) can be simplified to

where $ \vec{p}_{\pi^+}^{\,\prime} $ is given in Eq. (15). The quantity $ \vec{p}_{D}^{\,\prime} $ is the momentum of the $ D^- $ in the COM frame of $ \pi a_0 $, which originates from the $ B^0 $ decay.

Equation (5) is used to perform the polarization sum in Eq. (17), yielding

where $ \vec{p}_{\pi^+}^{\,\prime}-\vec{p}_{K^-}^{\,\prime}=\vec{k}-(-\vec{k}-\vec{q})=2\vec{k}+\vec{q} $. P is the momentum of the $ \pi a_0 $ system in its rest frame.

The quantity $ t_T $ is defined to describe the loop integral in Eq. (19) as

Following Refs. [24, 32], the negative-energy part of the $ \bar{K}^{*0} $ propagator in $ t_T $ can be neglected. The integration over $ dq^0 $ in $ t_T $ is performed by applying the residue theorem and using the following formula

Now, the triangle loop amplitude $ t_T $ reduces to

where $ \omega_{K^+}(\vec{q})=\sqrt{m_{K}^2 + \vec{q}^2} $, $ \omega_{K^{*0}}(\vec{q})= \sqrt{m_{\bar{K}^{*0}}^2 + \vec{q}^2} $, and $ \omega_{K^-}(\vec{q}+\vec{k}) = \sqrt{m_{K}^2 + (\vec{k}+\vec{q})^2} $. From Eq. (20), $ P^0 = m_{\mathrm{inv}}(\pi a_0) $ is the invariant mass of the $ \pi a_0 $ system. Although the integral in Eq. (23) is convergent, the chiral unitary approach naturally provides an upper limit $ q_{\mathrm{max}} $ [10, 13]. In this work, we use $ q_{\mathrm{max}} = 600\; \mathrm{MeV} $ for $ a_0(980) $ production [86, 91]. To reproduce the peak observed in experiments, the finite width of the $ \bar{K}^{*0} $ in the triangle loop must be taken into account. This is implemented by replacing $ \omega_{K^{*}} $ with $\omega_{K^{*}} - {\rm i} \Gamma_{\bar{K}^{*0}}/2$ in the denominator of Eq. (23).

Finally, the differential decay width for the three-body final state $ B^0 \to D^- \pi^+ a_0(980) $ is given by

The factor $ 2/3 $ in Eq. (24) originates from the angular phase-space integration of the integrand in $ \vec{p}_{D}^{\,\prime} \cdot \vec{k}= |\vec{p}_{D}^{\,\prime}| | \vec{k}|\cos \theta $. Here, $ \vec{p}_{D} $ denotes the momentum of the $ D^- $ in the $ B^0 $ rest frame.

In addition, we further explore the possibility of observing the $ a_0(980) $ through the $ \pi^0 \eta $ final state. Following Ref. [53], the $ a_0(980) $ appears as an unstable resonance in the $ B^0 $ decay and subsequently decays into $ \pi^0 \eta $. Therefore, we consider the corresponding four-body decay, as illustrated in the right panel of Fig. 1. In this process, the $ a_0(980) $ resonance can be treated as an intermediate particle, allowing us to include an additional differential distribution with respect to the $ \pi\eta $ invariant mass in Eq. 24. The differential decay width for $ B^0 \to D^- \pi^+ \pi^0 \eta $ is then given by

with

where $ m_{\text{inv}}(\pi \eta) $ is the invariant mass of the $ \pi \eta $ system produced in the $ a_0(980) $ decay, and $ m_{a_0} $ is the mass of $ a_0(980) $. Eq. (26) reduces to Eq. (24) in the limit $ \Gamma_{a_0} \to 0 $, where ${\rm i}\,\mathrm{Im}\,D \to -{\rm i}\pi \, \delta\big(m_{\mathrm{inv}}^2(\pi \eta) - m_{a_0}^2 \big)$. Thus, Eq. (26) simplifies to

The decay width $ \Gamma_{a_0} $ is calculated by considering the $ a_0 \to \pi^0 \eta $ decay

where $ |\vec{\tilde{q}}_{\eta}| $ denotes the magnitude of the η momentum in the rest frame of $ a_0(980) $ and is given by

By substituting Eq. (29) into Eq. (28), we obtain

Meanwhile, the amplitude for the $ K^+ K^- \to \pi \eta $ process mediated by exchange of the $ a_0(980) $ resonance can be written as

In contrast, when the $ a_0(980) $ is treated as a dynamically generated state, the amplitude $ t_{K^+ K^-\to \pi \eta} $ can be calculated by solving the Bethe–Salpeter (BS) equation within the chiral unitary approach. The BS equation is given by

Here, G denotes the loop function for two meson propagators [86].

and is regularized by a cutoff $ q_{\text{max}} $ as

with

A value of $ q_{\text{max}}=600 $ MeV is used to reproduce the $ a_0(980) $ around 980 MeV. The meson-meson interaction potentials for the case $ I=1 $ are given as follows [86]:

Finally, the double-differential distribution of the $ B^0 $ decay can be written as

In Fig. 3, we first present the triangle amplitudes $ \tilde{t}_T $, $ \mathrm{Im}(\tilde{t}_T) $, $ \mathrm{Re}(\tilde{t}_T) $, and $ |\tilde{t}_T|^2 \times 10^{7} $ as functions of the invariant mass $ m_{\rm{inv}}(\pi a_0) $, with the $ a_0(980) $ mass fixed at $ m_{a_0} = 980\; \mathrm{MeV} $. The results for $ |\tilde{t}_T|^2 $ exhibit a clear peak around $ 1420\; \mathrm{MeV} $, consistent with the condition in Eq. (1). The peaks of $ \mathrm{Im}(\tilde{t}_T) $ and $ \mathrm{Re}(\tilde{t}_T) $ appear at $ 1440\; \mathrm{MeV} $ and $ 1390\; \mathrm{MeV} $, respectively, arising from the TS and the $ \bar{K}^{*0} K^+ $ threshold. From top to bottom, the results correspond to the cutoff parameters $ q_{\rm{max}} = 600, \; 800 $ and $ 1000\; \mathrm{MeV} $, respectively. As $ q_{\rm{max}} $ increases, the peak positions in $ \mathrm{Im}(\tilde{t}_T) $, $ \mathrm{Re}(\tilde{t}_T) $, and $ |\tilde{t}_T|^2 $ remain unchanged, while their magnitudes generally increase. The triangle amplitudes for different masses of $ a_0(980) $ are shown in Fig. 4. The results indicate that the peaks remain near $ 1420\; \mathrm{MeV} $, while the strength of the triangle amplitude increases as $ m_{a_0} $ increases. Next, we examine the behavior of the triangle amplitudes $ \mathrm{Im}(\tilde{t}_T) $ and $ \mathrm{Re}(\tilde{t}_T) $ as functions of the internal particle width.

Figure 3.  (color online) The $ \bar{K}^{*0} K^+ K^- $ triangle amplitudes $ \tilde{t}_T $ for $ q_{\text{max}}=600,800, 1000 $ MeV are shown from top to bottom, respectively. We set $ m_{a_0}=980 $ MeV, and $ t_T^2 $ is multiplied by $ 10^{7} $.

Figure 4.  (color online) As in Fig. 3, the triangle amplitudes for $ m_{a_0}=983 $, $ 985 $, and $ 987 $ MeV with $ q_{\text{max}}=600 $ MeV are shown.

Following Ref. [53], Fig. 5 illustrates the development of the TS by fixing $ \Gamma_{K^*}/2 = \epsilon $ at different finite values close to zero. The formation of the TS in the $ \bar{K}^{*0} $-K-K triangle loop requires the $ a_0(980) $ mass to be slightly above the $ K^- K^+ $ threshold; therefore, we take $ m_{a_0} = 990\; \mathrm{MeV} $. The distinct origins of the peaks in $ \mathrm{Re}(\tilde{t}_T) $ and $ \mathrm{Im}(\tilde{t}_T) $ indicate that these two components exhibit different behaviors, as shown in the upper and lower panels of Fig. 5. Specifically, $ \mathrm{Re}(\tilde{t}_T) $ exhibits a cusp at the $ \bar{K}^{*0} K $ threshold around $ 1390\; \mathrm{MeV} $, followed by a sharp decrease near the TS at $ 1420\; \mathrm{MeV} $. In contrast, $ \mathrm{Im}(\tilde{t}_T) $ develops a narrow peak at $ 1420\; \mathrm{MeV} $, directly reflecting the emergence of the triangle singularity. Fig. 5 also shows that, as ϵ decreases, the decrease in $ \mathrm{Re}(\tilde{t}_T) $ near $ 1420\; \mathrm{MeV} $ becomes steeper, the cusp near $ 1390\; \mathrm{MeV} $ becomes more pronounced, and the peak in $ \mathrm{Im}(\tilde{t}_T) $ becomes increasingly sharp. In the limit $ \Gamma_{K^*}/2 = \epsilon = 0 $, the peak of $ \mathrm{Im}(\tilde{t}_T) $ develops into a genuine singularity.

Figure 5.  (color online) The dependence of the triangle amplitude Re$ (\tilde{t}_T) $ and Im$ (\tilde{t}_T) $ on the widths of the internal particles is shown in the upper and lower panels, respectively. We set $ m_{a_0}=990 $ MeV, which is slightly larger than the $ K^- K^+ $ threshold, and use $ \Gamma_{\bar{K}^{*0}}/2=\epsilon=0.1,\ 0.3,\ 0.5 $, and 0.7 MeV.

In Fig. 6, the differential distributions $\dfrac{{\rm d} \Gamma^\prime}{{\rm d} m_{\text{inv}(\pi a_0)}}$, Eq. (24), and $\dfrac{1}{\Gamma_{B^0}}\dfrac{{\rm d}^2 \Gamma^\prime}{{\rm d} m_{\text{inv}}(\pi a_0) {\rm d} m_{\text{inv}}(\pi \eta)}$, Eq. (36), are shown as functions of the invariant masses $ m_{\text{inv}}(\pi a_0) $ and $ m_{\rm{inv}}(\pi \eta) $ in the upper and lower panels, respectively. In the upper panel of Fig. 6, the differential mass distribution for the $ B^0 \to D^- \pi^+ a_0(980) $ decay with respect to $ m_{\rm{inv}}(\pi a_0) $ is shown for fixed values $ m_{\rm{inv}}(\pi \eta) = 980,985,987 $ and $ 988\; \mathrm{MeV} $ with $ q_{\rm{max}} = 600\; \mathrm{MeV} $. It is evident that increasing $ m_{a_0} $ enhances the decay width of $ B^0 $, while the peak position remains at the TS, $ m_{\rm{inv}}(\pi a_0) \simeq 1420\; \mathrm{MeV} $. In the lower panel of Fig. 6, the dependence of $\dfrac{1}{\Gamma_{B^0}}\dfrac{{\rm d}^2 \Gamma^\prime}{{\rm d} m_{\text{inv}}(\pi a_0){\rm d} m_{\text{inv}}(\pi \eta)}$ on $ m_{\rm{inv}}(\pi \eta) $ is shown for fixed values $ m_{\rm{inv}}(\pi a_0) = 1400, 1410, 1420 $ and $ 1440\; \mathrm{MeV} $ around the TS. A pronounced peak appears around $ 980\; \mathrm{MeV} $, indicating that the main contribution to the decay width comes from the region $ m_{\rm{inv}}(\pi \eta) \simeq 980\; \mathrm{MeV} $. Moreover, the peak for $ m_{\rm{inv}}(\pi a_0) = 1420\; \mathrm{MeV} $, close to the TS, is significantly sharper than those at the other values. Therefore, when focusing on the $ a_0(980) $ region, we can integrate the double differential distribution $\dfrac{1}{\Gamma_B} \dfrac{{\rm d}^2 \Gamma^\prime}{{\rm d} m_{\rm{inv}}(\pi a_0)\, {\rm d} m_{\rm{inv}}(\pi \eta)}$ over $ m_{\rm{inv}}(\pi \eta) $ around $ m_{\rm{inv}}(\pi \eta) = 980\; \mathrm{MeV} $. In Fig. 7, we present the results obtained by integrating over $ m_{\mathrm{inv}}(\pi\eta) \in [980\; \mathrm{MeV} - \Delta_m,\; 980\; \mathrm{MeV} + \Delta_m] $ with $ \Delta_m = 30,\, 50,\,100 $ and $ 150\; \mathrm{MeV} $. The results show a clear peak at $ 1420\; \mathrm{MeV} $ when the $ a_0(980) $ is considered a dynamically generated state, consistent with the results in Fig. 6. A comparison of the subfigures for $ q_{\rm{max}} = 600,800 $ and $ 1000\; \mathrm{MeV} $ shows that the decay width exhibits a slight enhancement as the cutoff parameter increases.

Figure 6.  (color online) The upper and lower panels show, respectively, the differential distribution $\dfrac{{\rm d} \Gamma^\prime}{{\rm d} m_{\text{inv}(\pi a_0)}}$ (Eq. 24) as a function of the invariant mass $ m_{\text{inv}}(\pi a_0) $ and the double differential distribution $\dfrac{1}{\Gamma_{B^0}}\dfrac{{\rm d}^2 \Gamma^\prime}{{\rm d} m_{\text{inv}}(\pi a_0) {\rm d} m_{\text{inv}}(\pi \eta)}$ (Eq. 36) as a function of the invariant masses $ m_{\text{inv}}(\pi a_0) $ and $ m_{\rm{inv}}(\pi \eta) $.

Figure 7.  (color online) The differential decay width $\dfrac{1}{\Gamma_{B^0}}\dfrac{{\rm d}^2 \Gamma^\prime}{{\rm d} m_{\text{inv}}(\pi a_0) {\rm d} m_{\text{inv}}(\pi \eta)}$ is shown as a function of the invariant mass $ m_{\rm{inv}}(\pi a_0) $, integrated over $ m_{\mathrm{inv}}(\pi\eta) \in [980\; \mathrm{MeV} - $$ \Delta_m,\; 980\; \mathrm{MeV} + \Delta_m] $. From top to bottom, the curves correspond to $ q_{\mathrm{max}} = 600,800 $ and $ 1000\; \mathrm{MeV} $, respectively.

In Fig. 3, we first present the triangle amplitudes $ \tilde{t}_T $, $ \mathrm{Im}(\tilde{t}_T) $, $ \mathrm{Re}(\tilde{t}_T) $, and $ |\tilde{t}_T|^2 \times 10^{7} $ as functions of the invariant mass $ m_{\rm{inv}}(\pi a_0) $, with the $ a_0(980) $ mass fixed at $ m_{a_0} = 980\; \mathrm{MeV} $. The results for $ |\tilde{t}_T|^2 $ exhibit a clear peak around $ 1420\; \mathrm{MeV} $, consistent with the condition in Eq. (1). The peaks of $ \mathrm{Im}(\tilde{t}_T) $ and $ \mathrm{Re}(\tilde{t}_T) $ appear at $ 1440\; \mathrm{MeV} $ and $ 1390\; \mathrm{MeV} $, respectively, arising from the TS and the $ \bar{K}^{*0} K^+ $ threshold. From top to bottom, the results correspond to the cutoff parameters $ q_{\rm{max}} = 600, \; 800 $ and $ 1000\; \mathrm{MeV} $, respectively. As $ q_{\rm{max}} $ increases, the peak positions in $ \mathrm{Im}(\tilde{t}_T) $, $ \mathrm{Re}(\tilde{t}_T) $, and $ |\tilde{t}_T|^2 $ remain unchanged, while their magnitudes generally increase. The triangle amplitudes for different masses of $ a_0(980) $ are shown in Fig. 4. The results indicate that the peaks remain near $ 1420\; \mathrm{MeV} $, while the strength of the triangle amplitude increases as $ m_{a_0} $ increases. Next, we examine the behavior of the triangle amplitudes $ \mathrm{Im}(\tilde{t}_T) $ and $ \mathrm{Re}(\tilde{t}_T) $ as functions of the internal particle width.

Figure 3.  (color online) The $ \bar{K}^{*0} K^+ K^- $ triangle amplitudes $ \tilde{t}_T $ for $ q_{\text{max}}=600,800, 1000 $ MeV are shown from top to bottom, respectively. We set $ m_{a_0}=980 $ MeV, and $ t_T^2 $ is multiplied by $ 10^{7} $.

Figure 4.  (color online) As in Fig. 3, the triangle amplitudes for $ m_{a_0}=983 $, $ 985 $, and $ 987 $ MeV with $ q_{\text{max}}=600 $ MeV are shown.

Following Ref. [53], Fig. 5 illustrates the development of the TS by fixing $ \Gamma_{K^*}/2 = \epsilon $ at different finite values close to zero. The formation of the TS in the $ \bar{K}^{*0} $-K-K triangle loop requires the $ a_0(980) $ mass to be slightly above the $ K^- K^+ $ threshold; therefore, we take $ m_{a_0} = 990\; \mathrm{MeV} $. The distinct origins of the peaks in $ \mathrm{Re}(\tilde{t}_T) $ and $ \mathrm{Im}(\tilde{t}_T) $ indicate that these two components exhibit different behaviors, as shown in the upper and lower panels of Fig. 5. Specifically, $ \mathrm{Re}(\tilde{t}_T) $ exhibits a cusp at the $ \bar{K}^{*0} K $ threshold around $ 1390\; \mathrm{MeV} $, followed by a sharp decrease near the TS at $ 1420\; \mathrm{MeV} $. In contrast, $ \mathrm{Im}(\tilde{t}_T) $ develops a narrow peak at $ 1420\; \mathrm{MeV} $, directly reflecting the emergence of the triangle singularity. Fig. 5 also shows that, as ϵ decreases, the decrease in $ \mathrm{Re}(\tilde{t}_T) $ near $ 1420\; \mathrm{MeV} $ becomes steeper, the cusp near $ 1390\; \mathrm{MeV} $ becomes more pronounced, and the peak in $ \mathrm{Im}(\tilde{t}_T) $ becomes increasingly sharp. In the limit $ \Gamma_{K^*}/2 = \epsilon = 0 $, the peak of $ \mathrm{Im}(\tilde{t}_T) $ develops into a genuine singularity.

Figure 5.  (color online) The dependence of the triangle amplitude Re$ (\tilde{t}_T) $ and Im$ (\tilde{t}_T) $ on the widths of the internal particles is shown in the upper and lower panels, respectively. We set $ m_{a_0}=990 $ MeV, which is slightly larger than the $ K^- K^+ $ threshold, and use $ \Gamma_{\bar{K}^{*0}}/2=\epsilon=0.1,\ 0.3,\ 0.5 $, and 0.7 MeV.

In Fig. 6, the differential distributions $\dfrac{{\rm d} \Gamma^\prime}{{\rm d} m_{\text{inv}(\pi a_0)}}$, Eq. (24), and $\dfrac{1}{\Gamma_{B^0}}\dfrac{{\rm d}^2 \Gamma^\prime}{{\rm d} m_{\text{inv}}(\pi a_0) {\rm d} m_{\text{inv}}(\pi \eta)}$, Eq. (36), are shown as functions of the invariant masses $ m_{\text{inv}}(\pi a_0) $ and $ m_{\rm{inv}}(\pi \eta) $ in the upper and lower panels, respectively. In the upper panel of Fig. 6, the differential mass distribution for the $ B^0 \to D^- \pi^+ a_0(980) $ decay with respect to $ m_{\rm{inv}}(\pi a_0) $ is shown for fixed values $ m_{\rm{inv}}(\pi \eta) = 980,985,987 $ and $ 988\; \mathrm{MeV} $ with $ q_{\rm{max}} = 600\; \mathrm{MeV} $. It is evident that increasing $ m_{a_0} $ enhances the decay width of $ B^0 $, while the peak position remains at the TS, $ m_{\rm{inv}}(\pi a_0) \simeq 1420\; \mathrm{MeV} $. In the lower panel of Fig. 6, the dependence of $\dfrac{1}{\Gamma_{B^0}}\dfrac{{\rm d}^2 \Gamma^\prime}{{\rm d} m_{\text{inv}}(\pi a_0){\rm d} m_{\text{inv}}(\pi \eta)}$ on $ m_{\rm{inv}}(\pi \eta) $ is shown for fixed values $ m_{\rm{inv}}(\pi a_0) = 1400, 1410, 1420 $ and $ 1440\; \mathrm{MeV} $ around the TS. A pronounced peak appears around $ 980\; \mathrm{MeV} $, indicating that the main contribution to the decay width comes from the region $ m_{\rm{inv}}(\pi \eta) \simeq 980\; \mathrm{MeV} $. Moreover, the peak for $ m_{\rm{inv}}(\pi a_0) = 1420\; \mathrm{MeV} $, close to the TS, is significantly sharper than those at the other values. Therefore, when focusing on the $ a_0(980) $ region, we can integrate the double differential distribution $\dfrac{1}{\Gamma_B} \dfrac{{\rm d}^2 \Gamma^\prime}{{\rm d} m_{\rm{inv}}(\pi a_0)\, {\rm d} m_{\rm{inv}}(\pi \eta)}$ over $ m_{\rm{inv}}(\pi \eta) $ around $ m_{\rm{inv}}(\pi \eta) = 980\; \mathrm{MeV} $. In Fig. 7, we present the results obtained by integrating over $ m_{\mathrm{inv}}(\pi\eta) \in [980\; \mathrm{MeV} - \Delta_m,\; 980\; \mathrm{MeV} + \Delta_m] $ with $ \Delta_m = 30,\, 50,\,100 $ and $ 150\; \mathrm{MeV} $. The results show a clear peak at $ 1420\; \mathrm{MeV} $ when the $ a_0(980) $ is considered a dynamically generated state, consistent with the results in Fig. 6. A comparison of the subfigures for $ q_{\rm{max}} = 600,800 $ and $ 1000\; \mathrm{MeV} $ shows that the decay width exhibits a slight enhancement as the cutoff parameter increases.

Figure 6.  (color online) The upper and lower panels show, respectively, the differential distribution $\dfrac{{\rm d} \Gamma^\prime}{{\rm d} m_{\text{inv}(\pi a_0)}}$ (Eq. 24) as a function of the invariant mass $ m_{\text{inv}}(\pi a_0) $ and the double differential distribution $\dfrac{1}{\Gamma_{B^0}}\dfrac{{\rm d}^2 \Gamma^\prime}{{\rm d} m_{\text{inv}}(\pi a_0) {\rm d} m_{\text{inv}}(\pi \eta)}$ (Eq. 36) as a function of the invariant masses $ m_{\text{inv}}(\pi a_0) $ and $ m_{\rm{inv}}(\pi \eta) $.

Figure 7.  (color online) The differential decay width $\dfrac{1}{\Gamma_{B^0}}\dfrac{{\rm d}^2 \Gamma^\prime}{{\rm d} m_{\text{inv}}(\pi a_0) {\rm d} m_{\text{inv}}(\pi \eta)}$ is shown as a function of the invariant mass $ m_{\rm{inv}}(\pi a_0) $, integrated over $ m_{\mathrm{inv}}(\pi\eta) \in [980\; \mathrm{MeV} - $$ \Delta_m,\; 980\; \mathrm{MeV} + \Delta_m] $. From top to bottom, the curves correspond to $ q_{\mathrm{max}} = 600,800 $ and $ 1000\; \mathrm{MeV} $, respectively.

In this work, we investigate the triangle mechanism in the $ B^0 \to D^- \pi^+ a_0(980){(\pi^0 \eta)} $ decay and explore the nature of $ a_0(980) $ as a dynamically generated state arising from meson-meson interactions. The $ \bar{K}^{*0} $-$ K^+ $-$ K^- $ triangle loop originates from the weak decay $ B^0 \to D^- K^+ \bar{K}^{*0} $, followed by $ \bar{K}^{*0} \to \pi^+ K^- $ and the subsequent rescattering of $ K^+ K^- $ to form $ a_0(980) $. We provide analytical expressions for the differential decay widths $\dfrac{{\rm d}\Gamma}{{\rm d} m_{\rm{inv}}(\pi a_0)}$ and $\dfrac{{\rm d}^2 \Gamma}{{\rm d} m_{\rm{inv}}(\pi a_0)\, {\rm d} m_{\rm{inv}}(\pi \eta)}$ corresponding to the three-body and four-body final states, respectively, including the $ \bar{K}^{*0} $-$ K^+ $-$ K^- $ triangle loop amplitude. The triangle amplitude produces a pronounced peak near $ 1420\; \mathrm{MeV} $ in the $ \pi a_0(980) $ invariant mass distribution, which can be interpreted as a TS signal. The dependence of the TS on the internal $ \bar{K}^{*0} $ width is also investigated. The $ K^+K^- $ rescattering generates a pronounced peak near $ 980\; \mathrm{MeV} $ in the $ \pi^0\eta $ invariant mass distribution, reflecting the formation of the $ a_0(980) $. Meanwhile, the invariant mass spectrum of the $ \pi^+\pi^0\eta $ system exhibits a peak around $ 1420\; \mathrm{MeV} $, consistent with the results obtained from the three-body final state. Based on the current observations and analyses by the COMPASS Collaboration, we expect that $ B^0 $ decays can serve as a potential platform for further confirming the TS nature of the $ a_1(1420) $ in future experiments.

In this work, we investigate the triangle mechanism in the $ B^0 \to D^- \pi^+ a_0(980){(\pi^0 \eta)} $ decay and explore the nature of $ a_0(980) $ as a dynamically generated state arising from meson-meson interactions. The $ \bar{K}^{*0} $-$ K^+ $-$ K^- $ triangle loop originates from the weak decay $ B^0 \to D^- K^+ \bar{K}^{*0} $, followed by $ \bar{K}^{*0} \to \pi^+ K^- $ and the subsequent rescattering of $ K^+ K^- $ to form $ a_0(980) $. We provide analytical expressions for the differential decay widths $\dfrac{{\rm d}\Gamma}{{\rm d} m_{\rm{inv}}(\pi a_0)}$ and $\dfrac{{\rm d}^2 \Gamma}{{\rm d} m_{\rm{inv}}(\pi a_0)\, {\rm d} m_{\rm{inv}}(\pi \eta)}$ corresponding to the three-body and four-body final states, respectively, including the $ \bar{K}^{*0} $-$ K^+ $-$ K^- $ triangle loop amplitude. The triangle amplitude produces a pronounced peak near $ 1420\; \mathrm{MeV} $ in the $ \pi a_0(980) $ invariant mass distribution, which can be interpreted as a TS signal. The dependence of the TS on the internal $ \bar{K}^{*0} $ width is also investigated. The $ K^+K^- $ rescattering generates a pronounced peak near $ 980\; \mathrm{MeV} $ in the $ \pi^0\eta $ invariant mass distribution, reflecting the formation of the $ a_0(980) $. Meanwhile, the invariant mass spectrum of the $ \pi^+\pi^0\eta $ system exhibits a peak around $ 1420\; \mathrm{MeV} $, consistent with the results obtained from the three-body final state. Based on the current observations and analyses by the COMPASS Collaboration, we expect that $ B^0 $ decays can serve as a potential platform for further confirming the TS nature of the $ a_1(1420) $ in future experiments.