-
The mechanism at the quark level for the
$\bar{B}_s^0 \to J/\psi \pi^+ \pi^- (\pi^0 \eta)$ reaction is depicted in Fig. 1, having an$ s\bar{s} $ pair with isospin$ I = 0 $ at the end. Note that the light scalars$ f_0(980) $ and$ a_0(980) $ have$ I = 0, 1 $ , respectively. The production of$ f_0(980) $ is isospin conserved, while the production of$ a_0(980) $ is isospin forbidden and involves isospin violation.To obtain
$ \pi^+\pi^- $ or$ \pi^0\eta $ in the final state in Fig. 1, we need to hadronize the$ s\bar s $ pair by introducing an extra$ \bar q q $ pair with vacuum quantum numbers. We start with the$ q \bar q $ matrix M in SU(3),$ M = \left( {\begin{array}{*{20}{c}} {u\bar u}&{u\bar d}&{u\bar s}\\ {d\bar u}&{d\bar d}&{d\bar s}\\ {s\bar u}&{s\bar d}&{s\bar s} \end{array}} \right).$
(1) Next, we write the matrix M in terms of pseudoscalar mesons, assuming that the
$ \eta $ is$ \eta_8 $ of SU(3),$ M \to {\cal P} = \left( {\begin{array}{*{20}{c}}\\[-5pt] {\dfrac{1}{{\sqrt 2 }}{\pi ^0} + \dfrac{1}{{\sqrt 6 }}\eta + \dfrac{1}{{\sqrt 3 }}\eta '}&{{\pi ^ + }}&{{K^ + }}\\ {{\pi ^ - }}&{ - \dfrac{1}{{\sqrt 2 }}{\pi ^0} + \dfrac{1}{{\sqrt 6 }}\eta + \dfrac{1}{{\sqrt 3 }}\eta '}&{{K^0}}\\ {{K^ - }}&{{{\bar K}^0}}&{ - \sqrt {\dfrac{2}{3}} \eta + \sqrt {\dfrac{1}{3}} \eta '}\\ \end{array}} \right),$ (2) which is often used in chiral perturbation theory [4]. On the other hand, when we consider the Bramon
$ \eta-\eta' $ mixing [41], the matrix M can be written as$ M \to {{\cal P}^{({m})}} = \left( {\begin{array}{*{20}{c}} {\dfrac{1}{{\sqrt 2 }}{\pi ^0} + \dfrac{1}{{\sqrt 3 }}\eta + \dfrac{1}{{\sqrt 6 }}\eta '}&{{\pi ^ + }}&{{K^ + }}\\ {{\pi ^ - }}&{ - \dfrac{1}{{\sqrt 2 }}{\pi ^0} + \dfrac{1}{{\sqrt 3 }}\eta + \dfrac{1}{{\sqrt 6 }}\eta '}&{{K^0}}\\ {{K^ - }}&{{{\bar K}^0}}&{ - \dfrac{1}{{\sqrt 3 }}\eta + \sqrt {\dfrac{2}{3}} \eta '} \end{array}} \right).$
(3) Since the
$ \eta' $ is inessential in the dynamical generation of the$ f_0(980) $ and$ a_0(980) $ resonances [4], we will ignore the$ \eta' $ in the present work.After hadronization of the
$ s \bar s $ component, we obtain$ s\bar s \to H = \sum\limits_i s\, \bar q_i q_i \, \bar s = \sum\limits_i {\mathcal{P}}_{3i} \; {\mathcal{P}}_{i3}\, = ({\mathcal{P}}^2)_{33}. $
(4) In the case without
$ \eta-\eta' $ mixing, the matrix$ \mathcal{P} $ of Eq. (2) is used, and then the hadron component H in Eq. (4) is given by$ H = K^-K^+ + \bar K^0 K^0 + \frac{2}{3}\, \eta \eta. $
(5) In the case with
$ \eta-\eta' $ mixing, one uses matrix$ \mathcal{P}^{(m)} $ of Eq. (3), and obtains$ H = K^-K^+ + \bar K^0 K^0 + \frac{1}{3}\, \eta \eta, $
(6) differing only in the
$ \eta\eta $ component, which affects the production of$ f_0 $ but not the production of$ a_0 $ . We define the weight of the$ PP $ components in H as$ h_{K^+K^-} = 1,\; \; \; \; h_{K^0 \bar K^0} = 1,\; \; \; \; h_{\eta\eta} = \frac{2}{3},\; \; \; \; h_{\eta\eta}^{(\mathrm{m})} = \frac{1}{3}. $
(7) One can see that neither Eq. (5) nor Eq. (6) contains
$ \pi^+ \pi^- $ or$ \pi^0 \eta $ , but they can be produced by the final state interaction of the$ K \bar K $ and$ \eta \eta $ components, as depicted in Fig. 2. The transition matrix from the$ PP $ state to$ \pi^+ \pi^- $ or$ \pi^0 \eta $ is represented by the circle behind the meson–meson loop in Fig. 2, which contains the information of$ f_0(980) $ and$ a_0(980) $ respectively. According to the method in Ref. [4] (the chiral unitary approach), these resonances are the result of the$ PP $ interaction in the coupled channels$ K\bar K, \pi\pi, \pi\eta, \eta\eta $ .Figure 2. Final state interaction of the hadron components leading to
$\pi^+\pi^-$ or$\pi^0 \eta$ in the final state.By using the unitary normalization [4, 38], the amplitude for the
$ \bar{B}_s^0 \to J/\psi \pi^+ \pi^- $ decay, as a function of the$ \pi^+\pi^- $ invariant mass$ M_{\rm{inv}}(\pi^+\pi^-) $ , is given by [38]$ \begin{aligned}[b] t_{\pi^+\pi^-} =& {\mathcal{C}}\left[ h_{K^+K^-} \cdot G_{K^+K^-} (M_{\rm{inv}}(\pi^+\pi^-)) \cdot T_{K^+K^-,\pi^+\pi^-}(M_{\rm{inv}}(\pi^+\pi^-)) \right. \\ & + h_{K^0 \bar K^0} \cdot G_{K^0 \bar K^0} (M_{\rm{inv}}(\pi^+\pi^-)) \cdot T_{K^0 \bar K^0,\pi^+\pi^-}(M_{\rm{inv}}(\pi^+\pi^-)) \\ & + h_{\eta\eta} \times 2\times \frac{1}{2} \cdot \left. G_{\eta\eta} (M_{\rm{inv}}(\pi^+\pi^-)) \cdot T_{\eta\eta,\pi^+\pi^-}(M_{\rm{inv}}(\pi^+\pi^-))\right], \end{aligned} $
(8) and the amplitude for the
$ \bar{B}_s^0 \to J/\psi \pi^0 \eta $ decay, as a function of the$ \pi^0\eta $ invariant mass$ M_{\rm{inv}}(\pi^0\eta) $ , is given by [38]$ \begin{aligned}[b] t_{\pi^0\eta} = & {\mathcal{C}}\left[ h_{K^+K^-} \cdot G_{K^+K^-} (M_{\rm{inv}}(\pi^0\eta)) \cdot T_{K^+K^-,\pi^0\eta}(M_{\rm{inv}}(\pi^0\eta)) \right. \\ & + h_{K^0 \bar K^0} \cdot G_{K^0 \bar K^0} (M_{\rm{inv}}(\pi^0\eta)) \cdot T_{K^0 \bar K^0,\pi^0\eta}(M_{\rm{inv}}(\pi^0\eta)) \\ & + h_{\eta\eta} \times 2\times \frac{1}{2} \cdot \left. G_{\eta\eta} (M_{\rm{inv}}(\pi^0\eta)) \cdot T_{\eta\eta,\pi^0\eta}(M_{\rm{inv}}(\pi^0\eta))\right], \end{aligned} $
(9) with
$ \mathcal{C} $ an arbitrary normalization constant which is canceled in the ratio of the$ f_0 $ and$ a_0 $ production rates. For the case with$ \eta-\eta' $ mixing, the corresponding amplitudes can be obtained by replacing$ h_{\eta\eta} $ with$ h_{\eta\eta}^{(m)} $ in Eqs. (8) and (9).In Eqs. (8) and (9),
$ G_i $ is the loop function of the two intermediate pseudoscalar mesons, which is regularized with a three momentum cut-off$ q_{\rm{max}} $ [4],$ G_i(\sqrt{s}) = \int_0^{q_{\rm{max}}} \frac{q^2\; {\rm{d}}q}{(2\pi)^2}\; \frac{w_1+w_2}{w_1\, w_2 \,[s-(w_1 +w_2)^2+{\rm i} \epsilon]}, $
(10) with
$ w_j = \sqrt{m_j^2+\vec q^{\,2}} $ and$ \sqrt{s} $ the centre-of-mass energy of the two mesons in the loop.$ T_{i,j} $ is the total amplitude for the$ i\to j $ transition and can be obtained by solving the Bethe–Salpeter (BS) equation with six$ PP $ coupled channels$ \pi^+\pi^- $ ,$ \pi^0\pi^0 $ ,$ K^+K^- $ ,$ K^0\bar{K}^0 $ ,$ \eta\eta $ and$ \pi^0\eta $ , in a matrix form,$ T = [1-V\,G]^{-1}\, V, $
(11) where the matrix V is the kernel of the BS equation. Its elements
$ V_{ij} $ are the s-wave transition potentials which can be taken from Eq. (A3) and Eq. (A4) of Ref. [38], corresponding to the cases without and with$ \eta-\eta' $ mixing, respectively.The differential decay width for
$ \bar B_s^0\to J/\psi\pi^0\eta $ or$ \bar B_s^0\to J/\psi\pi^+\pi^- $ decay is given by$ \frac{{\rm{d}} \Gamma}{{\rm{d}} M_{\rm{inv}}(ij)} = \frac{1}{(2\pi)^3} \; \frac{1}{4M_{\bar B_s^0}^2}\; \frac{1}{3}\; p_{J/\psi}^2 \; p_{J/\psi}\; \tilde{p}_{\pi}\; |t_{ij}|^2, $
(12) where
$ ij = \pi^+ \pi^- $ or$ \pi^0 \eta $ ,$ M_{\rm{inv}}(ij) $ is the invariant mass of the final$ \pi^+ \pi^- $ or$ \pi^0 \eta $ ,$ t_{\pi^+ \pi^-} $ and$ t_{\pi^0 \eta} $ are the amplitudes from Eq. (8) and Eq. (9) respectively,$ p_{J/\psi} $ is the$ J/\psi $ momentum in the$ \bar B_s^0 $ rest frame, and$ \tilde{p}_{\pi} $ is the pion momentum in the rest frame of the$ \pi^+ \pi^- $ or$ \pi^0 \eta $ system,$ p_{J/\psi} = \frac{\lambda^{1/2}(M^2_{\bar B^0_s},M^2_{J/\psi},M_{\rm{inv}}^2)}{2M_{\bar B^0_s}}, $
(13) $ {\tilde p_\pi } = \left\{ {\begin{array}{*{20}{l}} {\dfrac{{{\lambda ^{1/2}}(M_{{\rm{inv}}}^2,m_\pi ^2,m_\pi ^2)}}{{2{M_{{\rm{inv}}}}}},}&{{\rm{for}}\;{\pi ^ + }{\pi ^ - }\;{\rm{production}},}\\ {\dfrac{{{\lambda ^{1/2}}(M_{{\rm{inv}}}^2,m_\pi ^2,m_\eta ^2)}}{{2{M_{{\rm{inv}}}}}},}&{{\rm{for}}\;{\pi ^0}\eta \;{\rm{production}},} \end{array}} \right.$
(14) with
$ \lambda(x^2, y^2, z^2) = x^2+y^2+z^2-2xy-2yz-2zx $ the Källen function. In Eq. (12), the factor$ \dfrac{1}{3}\, p_{J/\psi}^2 $ stems from the fact that we need a p-wave to match angular momentum in the$ 0^- \to 1^-\, 0^+ $ transition and we take a vertex of type$ p_{J/\psi}\, \cos \theta $ . -
We follow Ref. [38] and take the cut-off
$ q_{\rm{max}} = 600 $ MeV and$ 650 $ MeV for the cases without$ \eta-\eta' $ mixing and with$ \eta-\eta' $ mixing respectively, with which the$ f_0(980) $ and$ a_0(980) $ resonances can be dynamically produced well from the PP interaction. The$ \pi^+ \pi^- $ and$ \pi^0 \eta $ mass distributions$ \frac{{\rm{d}} \Gamma}{{\rm{d}} M_{\rm{inv}}(ij)} $ are shown in Fig. 3 for the case without$ \eta-\eta' $ mixing and in Fig. 4 for the case with$ \eta-\eta' $ mixing, respectively. By comparing Fig. 3 and Fig. 4, one finds that the results of the two figures are very similar, and the difference between them can serve as an estimate of the uncertainties of our formalism.Figure 3. (color online)
$M_{\rm inv}(\pi^+ \pi^-)$ mass distribution for$\bar B_s^0 \to $ $ J/\psi f_0(980), f_0(980) \to \pi^+ \pi^-$ decay, and$M_{\rm inv}(\pi^0 \eta)$ mass distribution for$\bar B_s^0 \to J/\psi a_0(980), a_0(980) \to \pi^0 \eta$ decay. Inset: Magnified$\pi^0 \eta$ . (Without$\eta-\eta'$ mixing).Figure 4. (color online)
$M_{\rm inv}(\pi^+ \pi^-)$ mass distribution for$\bar B_s^0 \to $ $ J/\psi f_0(980), f_0(980) \to \pi^+ \pi^-$ decay, and$M_{\rm inv}(\pi^0 \eta)$ mass distribution for$\bar B_s^0 \to J/\psi a_0(980), a_0(980) \to \pi^0 \eta$ decay. Inset: Magnified$\pi^0 \eta$ . (With$\eta-\eta'$ mixing).Now, let us look at the
$ \pi^+ \pi^- $ and$ \pi^0 \eta $ mass distributions in Fig. 4 with$ \eta-\eta' $ mixing. One can see a strong peak for$ f_0(980) $ production in the$ \pi^+ \pi^- $ mass distribution and a small peak for$ a_0(980) $ production in the$ \pi^0 \eta $ mass distribution. Here the shape of$ a_0(980) $ resonance is quite narrow, considerably different to the standard cusp-like shape (with a width of about 120 MeV) of the ordinary production of$ a_0(980) $ in an isospin allowed reaction [42]. If isospin were conserved, one would find the$ a_0(980) $ production with zero strength. The small peak of$ a_0(980) $ in Fig. 4 indicates that isospin violation takes places in the$ \bar{B}_s^0 \to J/\psi \pi^0 \eta $ reaction. According to Eq. (A4) of Ref. [38], we have$ V_{K^+K^-,\pi^0\eta} = -V_{K^0 \bar K^0,\pi^0\eta} $ for the transition potentials. Hence, if we use average masses for kaons, there will be a precise cancellation of the first two terms of the amplitude$ t_{\pi^0\eta} $ in Eq. (9), resulting on zero strength for$ a_0(980) $ production. On the contrary, using the physical masses for the neutral$ K^0 $ and the charged$ K^+ $ in the formalism results in the production of the$ a_0(980) $ resonance with a narrow shape related to the difference of mass between the charged and neutral kaons. In our picture, there are two sources of isospin violation: one is the$ K^+ $ ,$ K^0 $ mass difference for the explicit$ K^+ K^- $ and$ K^0 \bar K^0 $ loops in Fig. 2, and the other is from the T matrix involving rescattering in Fig. 2.It is interesting to investigate the effects of these two sources of isospin violation. For that, we follow Ref. [38] and define the ratio R, which reflects the amount of the isospin violation, as
$ R = \frac{\Gamma(\bar B_s^0 \to J/\psi a_0(980), a_0(980) \to \pi^0 \eta )} {\Gamma(\bar B_s^0 \to J/\psi f_0(980), f_0(980) \to \pi^+ \pi^-)}, $
(15) with decay widths
$ \Gamma[\bar B_s^0 \to J/\psi a_0(980) $ ,$a_0(980) \to \pi^0 \eta ] $ and$ \Gamma[\bar B_s^0 \to J/\psi f_0(980), f_0(980) \to \pi^+ \pi^-] $ obtained by integrating Eq. (12) over the invariant mass$ M_{\rm{inv}}(ij) $ .Under several different assumptions related to the two sources of isospin violation, we evaluate the ratio R. The results are shown in Table 1.
no $\eta-\eta'$ mixingI.V. both in T matrix and in explicit $K\bar K$ loops (Case 1)$3.1\times 10^{-2}$ I.V. only in T matrix (Case 2) $3.5\times 10^{-2}$ I.V. only in explicit $K\bar K$ loops (Case 3)$7.0\times 10^{-4}$ with $\eta-\eta'$ mixingI.V. both in T matrix and in explicit $K\bar K$ loops (Case 4)$3.7\times 10^{-2}$ I.V. only in T matrix (Case 5) $4.1\times 10^{-2}$ I.V. only in explicit $K\bar K$ loops (Case 6)$9.7\times 10^{-4}$ Table 1. Values of R with different assumptions. (In the table, I.V. denotes isospin violation.)
From Table 1, we observe that the ratio R with
$ \eta-\eta' $ mixing (Case 4) is about 20% bigger than that without$ \eta-\eta' $ mixing (Case 1). By comparing the values of R for Case 2 and Case 3 (or, for Case 5 and Case 6), we find that the isospin violation in the T matrix has a more important effect than that in the explicit$ K\bar K $ loops, being at least one order of magnitude larger. This fact is interesting, since in our picture the$ f_0(980) $ and$ a_0(980) $ resonances are dynamically generated from the$ PP $ interaction with the information on their nature contained in the T matrix. For the$ \bar B_s^0 \to J/\psi \pi^+ \pi^- (\pi^0 \eta) $ decay, neither the$ \pi^+ \pi^- $ nor the$ \pi^0 \eta $ can be directly produced from$ s\bar s $ hadronization [see Eqs. (5) and (6)], hence there is no contribution from the tree level. Instead, they are produced through the rescattering mechanism of Fig. 2, with$ f_0(980) $ and$ a_0(980) $ resonances as dynamically generated states from the$ PP $ interaction. The production rate of the$ f_0(980) $ ($ a_0(980) $ ) resonance in the$ \bar B_s^0 \to J/\psi \pi^+ \pi^- (\pi^0 \eta) $ decay is sensitive to the resonance information contained in the T matrix. Therefore, this mode is particularly suitable to test the nature of$ f_0(980) $ and$ a_0(980) $ resonances and to investigate the isospin violation.From the PDG [40], the experimental branching ratio of the
$ \bar B_s^0 \to J/\psi f_0(980), f_0(980) \to \pi^+ \pi^- $ decay reads$ {\mathrm{Br}}[\bar B_s^0 \to J/\psi f_0(980), f_0(980) \to \pi^+ \pi^-] = (1.28 \pm 0.18)\times 10^{-4}. $
(16) By using the ratio R in Table 1 and the branching ratio of Eq. (16), the branching ratio for
$ a_0(980) $ production can be obtained,$\begin{aligned}[b]& {\rm{Br}}[\bar B_s^0 \to J/\psi {a_0}(980),{a_0}(980) \to {\pi ^0}\eta ] \\=& \left\{ {\begin{array}{*{20}{l}} {(3.95 \pm 0.56) \times {{10}^{ - 6}},}&{{\rm{for}}\;{\rm{Case}}\;1;}\\ {(4.74 \pm 0.67) \times {{10}^{ - 6}},}&{{\rm{for}}\;{\rm{Case}}\;4.} \end{array}} \right. \end{aligned}$
(17) This branching ratio is of the order of
$ 5\times 10^{-6} $ , not too small considering that several rates of the order of$ 10^{-7} $ are tabulated in the PDG [40]. The branching ratio and the shape of the$ \pi^0 \eta $ mass distribution of the$ \bar B_s^0 \to J/\psi \pi^0 \eta $ decay provide relevant information on the nature of the$ a_0(980) $ resonance. Experimental measurements will be very valuable.
The ${\bar{\boldsymbol B}_{\boldsymbol s}^{\bf 0} \to {\boldsymbol J}/\boldsymbol\psi \boldsymbol\pi^{\bf 0}\boldsymbol \eta}$ decay and the $ {{\boldsymbol a}_{\bf 0}{\bf (980)}-{\boldsymbol f}_{\bf 0}{\bf (980)}} $ mixing
- Received Date: 2022-04-13
- Available Online: 2022-08-15
Abstract: We study the