-
In the framework of the QCD factorization approach [38, 39], we can obtain the matrix element B decaying to two mesons
$ M_1 $ and$ M_2 $ by matching the effective weak Hamiltonian onto a transition operator, which is summarized as follows ($ \lambda_p^{(D)}=V_{pb}V_{pD}^* $ with$ D=d $ or s)$ \langle{M_1M_2}|{\cal{H}}_{\rm eff}|B\rangle=\sum\limits_{p=u,c}\lambda_{p}^{(D)}\langle{M_1M_2}|{\cal{T}}_A^p+{\cal{T}}_B^p|B\rangle, $
(1) where
$ {\cal{T}}_A^p $ and$ {\cal{T}}_B^p $ describe the contributions from non-annihilation and annihilation topology amplitudes, respectively, which can be expressed in terms of the parameters$ a_i^p $ and$ b_i^p $ , respectively, both of which are defined in detail in Ref. [38].With the operator product expansion, the effective weak Hamiltonian can be expressed as [38]
$\begin{aligned}[b] {\cal{H}}_{\rm eff}=&\frac{G_{\rm F}}{\sqrt{2}}\sum\limits_{p=u,c}\lambda_{p}^{(D)}{\bigg(c_1Q_1^p+c_2Q_2^p+\sum\limits_{i=3}^{10}c_iQ_i}\\&+c_{7\gamma}Q_{7\gamma}+c_{8g}Q_{8g}\bigg)+{\rm h.c.}, \end{aligned}$
(2) where
$G_{\rm F}}$ represents the Fermi constant,$ \lambda_p^{(D)}=V_{pb}V_{pD}^* $ ($ V_{pb} $ and$ V_{pD} $ are the CKM matrix elements),$ D=d,s $ can be a down or strange quark, and$ c_i(i=1,2,\cdots,10) $ are the Wilson coefficients. The operators$ Q_i $ are given by [39]$ \begin{aligned}[b] Q_1^p=&(\bar{p}b)_{V-A}(\bar{s}p)_{V-A},\qquad Q_2^p=(\bar{p}_\alpha b_\beta)_{V-A}(\bar{s}_\beta p_\alpha)_{V-A}, \\ Q_3 =&(\bar{s}b)_{V-A}\sum\limits_q(\bar{q}q)_{V-A}, \quad Q_4 =(\bar{s}_\alpha b_\beta)_{V-A}\sum\limits_q(\bar{q}_\beta q_\alpha)_{V-A},\\ Q_5=&(\bar{s}b)_{V-A}\sum\limits_q(\bar{q}q)_{V+A},\quad Q_6 =(\bar{s}_\alpha b_\beta)_{V-A}\sum\limits_q(\bar{q}_\beta q_\alpha)_{V+A},\end{aligned} $
$ \begin{aligned}[b] Q_7=&(\bar{s}b)_{V-A}\sum\limits_q\frac{3}{2}e_q(\bar{q}q)_{V+A},\\ Q_8 =&(\bar{s}_\alpha b_\beta)_{V-A}\sum\limits_q\frac{3}{2}e_q(\bar{q}_\beta q_\alpha)_{V+A}, \\ Q_9=&(\bar{s}b)_{V-A}\sum\limits_q\frac{3}{2}e_q(\bar{q}q)_{V-A},\\ Q_{10} =&(\bar{s}_\alpha b_\beta)_{V-A}\sum\limits_q\frac{3}{2}e_q(\bar{q}_\beta q_\alpha)_{V-A}, \\ Q_{7\gamma}=&\frac{-e}{8\pi^2}m_b\bar{s}\sigma_{\mu\nu}(1+\gamma_5)F^{\mu\nu}b, \\ Q_{8g}=&\frac{-g_s}{8\pi^2}m_b\bar{s}\sigma_{\mu\nu}(1+\gamma_5)G^{\mu\nu}b,\end{aligned} $
(3) where
$ Q_{1,2}^p $ and$ Q_{3-10} $ are the tree and penguin operators, respectively,$ Q_{7\gamma} $ is the electromagnetic dipole operator,$ Q_{8g} $ is the chromomagnetic dipole operators, α and β are color indices, and$ q=u,d,s,c,b $ quarks.Generally,
$ a_i^p $ includes the contributions from the naive factorization, vertex correction, penguin amplitude, and spectator scattering terms, which have the following expressions [38]:$\begin{aligned}[b] a_i^p{(M_1M_2)}=&{\left(c_i+\frac{c_{i\pm1}}{N_c}\right)}N_i{(M_2)}+\frac{c_{i\pm1}}{N_c}\frac{C_{\rm F}\alpha_s}{4\pi}\\&\times{\bigg[V_i{(M_2)}+\frac{4\pi^2}{N_c}H_i{(M_1M_2)}\bigg]+P_i^p{(M_2)}}, \end{aligned}$
(4) where
$ N_i{(M_2)} $ are the leading-order coefficients,$ V_i{(M_2)} $ ,$ H_i{(M_1M_2)} $ , and$ P_i^p{(M_1M_2)} $ are from one-loop vertex corrections, hard spectator interactions with a hard gluon exchange between the emitted meson, and the spectator quark of the B meson and penguin contractions, respectively, and their specific forms and derivations are presented in Refs. [5, 38, 40],$C_{\rm F}={(N_c^2-1)}/{2N_c}$ with$ N_c=3 $ [38].The weak annihilation contributions can be expressed in terms of
$ b_i $ and$b_{i,\rm EW}$ , which are$ \begin{aligned}[b] b_1=&\frac{C_{\rm F}}{N_c^2}c_1A_1^i, \quad b_2=\frac{C_{\rm F}}{N_c^2}c_2A_1^i, \\ b_3^p=&\frac{C_{\rm F}}{N_c^2}\bigg[c_3A_1^i+c_5(A_3^i+A_3^f)+N_cc_6A_3^f \bigg],\\ b_4^p=&\frac{C_{\rm F}}{N_c^2}\bigg[c_4A_1^i+c_6A_2^i \bigg], \\ b_{3,\rm EW}^p=&\frac{C_{\rm F}}{N_c^2}\bigg[c_9A_1^i+C_7(A_3^i+A_3^f)+N_cc_8A_3^f \bigg],\\ b_{4,\rm EW}^p=&\frac{C_{\rm F}}{N_c^2}\bigg[c_{10}A_1^i+c_8A_2^i \bigg], \end{aligned}$
(5) where the subscripts 1, 2, 3 of
$ A_n^{i,f}(n=1,2,3) $ denote the annihilation amplitudes induced from$ (V-A)(V-A) $ ,$ (V-A)(V+A) $ , and$ (S-P)(S+P) $ operators, respectively, the superscripts i and f refer to gluon emission from the initial- and final-state quarks, respectively. The explicit expressions for$ A_n^{i,f} $ are provided in Ref. [41].Concretely,
$ {\cal{T}}_A^p $ contains the contributions from naive factorization, vertex correction, penguin amplitude, and spectator scattering and can be expressed as$ \begin{aligned}[b] {\cal{T}}_A^p=&\delta_{pu}\alpha_1(M_1M_2)A([\bar{q}_su][\bar{u}D])\\&+\delta_{pu}\alpha_2(M_1M_2)A([\bar{q}_sD][\bar{u}u])\\ & +\alpha_3^p(M_1M_2)\sum\limits_qA([\bar{q}_sD][\bar{q}q])\\&+\alpha_4^p(M_1M_2)\sum\limits_qA([\bar{q}_sq][\bar{q}D])\\ & +\alpha_{3,\rm EW}^p(M_1M_2)\sum\limits_q\frac{3}{2}e_qA([\bar{q}_sD][\bar{q}q])\\&+\alpha_{4,\rm EW}^p(M_1M_2)\sum\limits_q\frac{3}{2}e_qA([\bar{q}_sq][\bar{q}D]),\end{aligned} $
(6) where the sums extend over
$ q=u,\,d,\,s $ , and$ \bar{q}_s(=\bar{u},\, \bar{d}\, {\rm{or}}\,\bar{s}) $ denotes the spectator antiquark. The coefficients$ \alpha_i^p(M_1M_2) $ and$\alpha_{i,\rm EW}^p(M_1M_2)$ contain all dynamical information and can be expressed in terms of the coefficients$ a_i^p $ .For the power-suppressed annihilation part, we can parameterize it into the following form:
$ \begin{aligned}[b] {\cal{T}}_B^p=\delta_{pu}b_1(M_1M_2)\sum\limits_{q'}B([\bar{u}q'][\bar{q}'u][\bar{D}b])\end{aligned} $
$ \begin{aligned}[b] \quad&+\delta_{pu}b_2(M_1M_2)\sum\limits_{q'}B([\bar{u}q'][\bar{q}'D][\bar{u}b]) \\& +b_3^p(M_1M_2)\sum\limits_{q,q'}B([\bar{q}q'][\bar{q}'D][\bar{q}b])\\&+b_4^p(M_1M_2)\sum\limits_{q,q'}B([\bar{q}q'][\bar{q}'q][\bar{D}b]\\ & +b_{3,\rm EW}^p(M_1M_2)\sum\limits_{q,q'}\frac{3}{2}e_qB([\bar{q}q'][\bar{q}'D][\bar{q}b])\\&+b_{4,\rm EW}^p(M_1M_2)\sum\limits_{q,q'}\frac{3}{2}e_qB([\bar{q}q'][\bar{q}'q][\bar{D}b]), \end{aligned} $
(7) where
$ q, q'=u, d, s $ and the sums extend over$ q, q' $ . The sum over$ q' $ occurs because a quark-antiquark pair must be created via$ g\rightarrow \bar{q}'q' $ after the spectator quark is annihilated. -
In the condition of turning on the
$ a_0^0(980)-f_0(980) $ mixing mechanism, we can obtain the propagator matrix of$ a_0^0(980) $ and$ f_0(980) $ by summing up all the contributions of$ a_0^0(980)\rightarrow f_0(980)\rightarrow \cdot\cdot\cdot\rightarrow a_0^0(980) $ and$ f_0(980)\rightarrow a_0^0(980) \rightarrow \cdot\cdot\cdot\rightarrow f_0(980) $ , respectively, which are expressed as [33]$ \left( {\begin{array}{*{20}{c}} { P_{a_0}(s)}&{ P_{a_0f_0}(s)}\\ {P_{f_0 a_0}(s)}&{P_{f_0}(s)} \end{array}} \right) = \frac{1}{D_{f_0}(s)D_{a_0}(s)-|\Lambda(s)|^2} \left( {\begin{array}{*{20}{c}} {D_{a_0}(s)}&{\Lambda(s)}\\ {\Lambda(s)}&{D_{f_0}(s)} \end{array}} \right), $ (8) where
$ P_{a_0}(s) $ and$ P_{f_0}(s) $ are the propagators of$ a_0 $ and$ f_0 $ , respectively,$ P_{a_0f_0}(s) $ ,$ P_{f_0 a_0}(s) $ , and$ \Lambda(s) $ occur due to the$ a_0^0(980)-f_0(980) $ mixing effect, and$ D_{a_0}(s) $ and$ D_{f_0}(s) $ are the denominators for the propagators of$ a_0 $ and$ f_0 $ when the$ a_0^0(980)-f_0(980) $ mixing effect is absent, respectively, which can be expressed as follows in the Flatté parametrization:$\begin{aligned}[b] D_{a_0}(s)=&m_{a_0}^2-s-{\rm i}\sqrt{s}[\Gamma_{\eta\pi}^{a_0}(s)+\Gamma_{K\bar{K}}^{a_0}(s)],\\ D_{f_0}(s)=&m_{f_0}^2-s-{\rm i}\sqrt{s}[\Gamma_{\pi\pi}^{f_0}(s)+\Gamma_{K\bar{K}}^{f_0}(s)],\end{aligned} $
(9) where
$ m_{a_0} $ and$ m_{f_0} $ are the masses of the$ a_0 $ and$ f_0 $ mesons, respectively, with the decay width$ \Gamma^a_{bc} $ expressed as$\begin{aligned}[b] \Gamma_{bc}^a(s)=\frac{g_{abc}^2}{16\pi\sqrt{s}}\rho_{bc}(s)\quad {\rm{with}}\end{aligned} $
$\begin{aligned}[b] \\[-8pt] \rho_{bc}(s)=\sqrt{\left[1-\frac{(m_b-m_c)^2}{s}\right]\left[1+\frac{(m_b-m_c)^2}{s}\right]}.\end{aligned} $
(10) Scholars have indicated that the contribution from the amplitude of
$ a_0^0(980)-f_0(980) $ mixing is convergent and can be expressed as an expansion in the$ K\bar{K} $ phase space when only$ K\bar{K} $ loop contributions are considered [12, 42],$\begin{aligned}[b] \Lambda(s)_{K\bar{K}}=&\frac{g_{a_0K^+K^-}g_{f_0K^+K^-}}{16\pi}\bigg\{{\rm i}\bigg[\rho_{K^+K^-}(s)-\rho_{K^0\bar{K}^0}(s)\bigg] \\&-{\cal{O}}(\rho_{K^+K^-}^2(s)-\rho_{K^0\bar{K}^0}^2(s))\bigg\}, \end{aligned}$
(11) where
$ g_{a_0K^+K^-} $ and$ g_{f_0K^+K^-} $ are the effective coupling constants. Since the mixing mainly results from the$ K\bar{K} $ loops, we can adopt$ \Lambda (s)\approx \Lambda_{K\bar{K}}(s) $ . -
With the
$ a_0^0(980)-f_0(980) $ mixing being considered, the process of the$ B^-\rightarrow K^-\pi^+\pi^- $ decay is shown in Fig. 1 and the amplitude can be expressed asFigure 1. Feynman diagram for the
$ B^-\rightarrow K^-\pi^+\pi^- $ decay with the$ a_0^0(980)-f_0(980) $ mixing mechanism.$ {\cal{M}}=\langle K^-\pi^+\pi^-|{\cal{H}}^T|B^-\rangle+\langle K^-\pi^+\pi^-|{\cal{H}}^P|B^-\rangle, $
(12) in which
$ {\cal{H}}^T $ and$ {\cal{H}}^P $ are the tree and penguin operators, respectively, and we obtain$\begin{aligned}[b] \langle K^-\pi^+\pi^-|{\cal{H}}^T|B^-\rangle =&\frac{g_{f_0\pi\pi}T_{f_0}}{D_{f_0}}+\frac{g_{f_0\pi\pi}T_{a_0}\Lambda}{D_{a_0}D_{f_0}-\Lambda^2},\\ \langle K^-\pi^+\pi^-|{\cal{H}}^P|B^-\rangle=&\frac{g_{f_0\pi\pi}P_{f_0}}{D_{f_0}}+\frac{g_{f_0\pi\pi}P_{a_0}\Lambda}{D_{a_0}D_{f_0}-\Lambda^2},\end{aligned} $
(13) where
$ T_{a_0(f_0)} $ and$ P_{a_0(f_0)} $ represent the tree and penguin diagram amplitudes for$ B\rightarrow K a_0(f_0) $ decay, respectively. Substituting Eq. (13) into Eq. (12), the total amplitude of the decay$ B^-\rightarrow K^- f_0(a_0) \rightarrow K^-\pi^+\pi^- $ can be expressed as$\begin{aligned}[b] {\cal{M}}(B^-\rightarrow K^-\pi^+\pi^-)=&\frac{g_{f_0\pi\pi}}{D_{f_0}}{\cal{M}}(B^-\rightarrow K^-f_0)\\& +\frac{g_{f_0\pi\pi}\Lambda}{D_{a_0}D_{f_0}-\Lambda^2}{\cal{M}}(B^-\rightarrow K^-a_0).\end{aligned} $
(14) In the QCD factorization approach, we derive the amplitudes of the
$ B^-\rightarrow K^-f_0 $ and$ B^-\rightarrow K^-a_0 $ decays, which are$ \begin{aligned}[b] {\cal{M}}(B^-\rightarrow K^-f_0)=&-\frac{G_{\rm F}}{\sqrt{2}}\sum\limits_{p=u,c}\lambda_p^{(s)}\bigg\{(\delta_{pu}a_1+a_4^p-r_\chi^K a_6^p+a_{10}^p-r_\chi^K a_8^p)_{f_0^uK}(m_B^2-m_{f_0}^2)f_KF_0^{Bf_0^u}(m_K^2)\\& -\left(\delta_{pu}a_2+2a_3^p+2a_5^p+\frac{1}{2}a_9^p+\frac{1}{2}a_7^p\right)_{Kf_0^u}(m_B^2-m_{K}^2)\bar{f}_{f_0^u}F_0^{BK}(m_{f_0}^2)\\ & -\left(a_3^p+a_5^p+a_4^p-r_\chi^fa_6^p-\frac{1}{2}a_9^p-\frac{1}{2}a_7^p-\frac{1}{2}a_{10}^p+\frac{1}{2}r_\chi^fa_8^p\right)_{Kf_0^s}(m_B^2-m_K^2)\bar{f}_{f_0^s}F_0^{BK}(m_{f_0}^2)\\&+(\delta_{pu}b_2+b_3^p+b_{3,\rm EW})_{Kf_0^u}f_B\bar{f}_{f_0^u}f_K+\left(\delta_{p,u}b_2+b_3^p-\frac{1}{2}b_{3,\rm EW}\right)_{Kf_0^s}f_B\bar{f}_{f_0^s}f_K\bigg\}, \end{aligned} $ (15) and
$ \begin{aligned}[b] {\cal{M}}(B^-\rightarrow K^-a_0)=&-\frac{G_{\rm F}}{\sqrt{2}}\sum_{p=u,c}\lambda_p^{(s)}\bigg\{(\delta_{pu}a_1+a_4^p-r_\chi^K a_6^p+a_{10}^p-r_\chi^K a_8^p)_{a_0K}(m_B^2-m_{a_0}^2)F_0^{Ba_0}(m_K^2)f_K\\ & -\left(\delta_{pu}a_2+\frac{3}{2}a_9^p+\frac{3}{2}a_7^p\right)_{Ka_0}(m_B^2-m_K^2)F_0^{B\rightarrow K}(m_{a_0}^2)\bar{f}_{a_0} +(\delta_{pu}b_2+b_3^p+b_{3,\rm EW}^p)_{a_0K}f_B\bar{f}_{a_0}f_K\bigg\}, \end{aligned} $
(16) respectively, where
$G_{\rm F}$ represents the Fermi constant;$ f_B $ ,$ f_K $ ,$ \bar{f}_{f_0} $ , and$ \bar{f}_{a_0} $ are the decay constants of B, K,$ f_0 $ , and$ a_0 $ , respectively;$ F_0^{Bf_0^u}(m_K^2) $ ,$ F_0^{BK}(m_{f_0}^2) $ , and$ F_0^{Ba_0}(m_K^2) $ are the form factors for the B to$ f_0 $ , K and$ a_0 $ transitions, respectively.By integrating the numerator and denominator of the differential
$ CP $ asymmetry parameter, we can obtain the localized integrated$ CP $ asymmetry, which can be measured using experiments and takes the following form in the region R:$ A_{CP}^R=\frac{\int_R{\rm d}s{\rm d}s'(\mid{\cal{M}}\mid^2-\mid{\cal{\bar{M}}}\mid^2)}{\int_R{\rm d}s{\rm d}s'(\mid{\cal{M}}\mid^2+\mid{\cal{\bar{M}}}\mid^2)}, $
(17) where s and
$ s' $ are the invariant masses squared of$ \pi\pi $ or$ K\pi $ pair in our case, and$ {\cal{\bar{M}}} $ is the decay amplitude of the$ CP $ -conjugate process.Since the decay process
$ B^-\rightarrow K^-\pi^+\pi^- $ has a three-body final state, the branching fraction of this decay can be expressed as [43]$ {\cal{B}}=\frac{\tau_B}{(2\pi)^5 16m_B^2}\int {\rm d}s |{\bf{p}}_1^*||{\bf{p}}_3|\int {\rm d}\Omega_1^* \int {\rm d}\Omega_3 |{\cal{M}}|^2, $
(18) in which
$ \Omega_1^* $ and$ \Omega_3 $ are the solid angles for the final π in the$ \pi\pi $ rest frame and for the final K in the B meson rest frame, respectively,$ |{\bf{p}}_1^*| $ and$ |{\bf{p}}_3| $ are the norms of the three-momenta of final-state π in the$ \pi\pi $ rest frame and K in the B rest frame, respectively, which take the following forms:$ |{\bf{p}}_1^*|=\frac{\sqrt{\lambda(s,m_\pi^2,m_\pi^2)}}{2\sqrt{s}},\\ |{\bf{p}}_3|=\frac{\sqrt{\lambda(m_B^2,m_K^2,s)}}{2m_B}, $
(19) where
$ \lambda(a,b,c) $ is the Källén function and with the form$ \lambda(a,b,c)=a^2+b^2+c^2-2(ab+ac+bc) $ . -
In the numerical calculations, we should input distribution amplitudes and the CKM matrix elements in the Wolfenstein parametrization. For the CKM matrix elements, which are determined from experiments, we use the results in Ref. [43]:
$ \begin{aligned}[b]&\bar{\rho}=0.117\pm0.021, \quad \bar{\eta}=0.353\pm0.013, \\& \lambda=0.225\pm0.00061,\quad A=0.811^{+0.023}_{-0.024},\end{aligned}\tag{A1}$
where
$ \bar{\rho}=\rho\left(1-\frac{\lambda^2}{2}\right), \quad \bar{\eta}=\eta\left(1-\frac{\lambda^2}{2}\right).\tag{A2} $
The Wilson coefficients used in our calculations are obtained from Refs. [46–49]. It should be noted that the convention in this work is different from that in Ref. [46] for the effective Hamiltonian (from the expressions of the
$ Q_{1, 2}^p $ ), so we adopt$ c_1=1.1502 $ and$ c_2=-0.3125 $ :$\begin{aligned}[b] &c_1=1.1502, \quad c_2=-0.3125,\quad c_3=0.0174,\\& c_4=-0.0373,\quad c_5=0.0104,\quad c_6=-0.0459,\\&c_7=-1.050\times10^{-5},\quad c_8=3.839\times10^{-4}, \\& c_9=-0.0101,\quad c_{10}=1.959\times10^{-3}. \end{aligned}\tag{A3}$
For the masses of mesons appeared in B decays, we use the following values [43] (in units of GeV):
$ \begin{aligned}[b]&m_{B^-}=5.279,\quad m_{K^-}=0.494,\quad m_{f_0(980)}=0.990,\\& m_{a_0^0(980)}=0.980,\quad m_{\pi^\pm}=0.14, \end{aligned}\tag{A4}$
whereas, for the widths we use (in
$ {\rm{GeV}} $ ) [43]$ \Gamma_{f_0(980)}=0.074,\quad\Gamma_{a_0^0(980)}=0.092. \tag{A5}$
The pole masses of quarks are [43] (in
$ {\rm{GeV}} $ ):$ \begin{aligned}[b]&m_u=m_d=0.0035, \quad m_b=4.78,\\& m_q=\frac{m_u+m_d}{2},\quad m_c=1.67. \end{aligned}\tag{A6}$
The running masses of quarks are [5, 43] (in
$ {\rm{GeV}} $ ):$ \begin{aligned}[b]&m_s(1{\rm{GeV}})=0.119,\quad m_c(m_c)=1.30, \\& m_b(m_b)=4.20, \quad \frac{m_s(\mu)}{m_{u,d}(\mu)}=27.5. \end{aligned}\tag{A7}$
The following numerical values for the decay constants are used [5, 50–52](in
$ {\rm{GeV}} $ ):$ \begin{aligned}[b]& f_{\pi^\pm}=0.131,\quad f_{B^-}=0.21\pm0.02, \\& f_{K^-}=0.156\pm0.007,\\& \bar{f}_{f_0(980)}=0.370\pm0.02, \\&\bar{f}_{a_0^0(980)}=0.365\pm0.02. \end{aligned}\tag{A8} $
For the form factors, we use [5]
$ \begin{aligned}[b]&F_0^{B\rightarrow K}(0)=0.35\pm0.04,\\& F_0^{B\rightarrow f_0(980)}(0)=0.25, \\& F_0^{B\rightarrow a_0^0(980)}(0)=0.25.\end{aligned}\tag{A9} $
The values of Gegenbauer moments at
$ \mu=1 {\rm{GeV}} $ are obtained from [5]:$ \begin{aligned}[b]& B_{1,f_0(980)}=-0.78\pm0.08,\quad B_{3,f_0(980)}=0.02\pm0.07,\\& B_{1,a_0^0(980)}=-0.93\pm0.10,\quad B_{3,a_0^0(980)}=0.14\pm0.08. \end{aligned} \tag{A10}$
Impact of ${ {\boldsymbol a}_{\bf 0}^{\bf {0}}(980)-{\boldsymbol f}_{\bf {0}}(980) }$ mixing on the localized CP violations of the ${{\boldsymbol B}^-{\bf\rightarrow} {\boldsymbol K}^- \boldsymbol\pi^+\boldsymbol\pi^- }$ decay
- Received Date: 2022-02-28
- Available Online: 2022-08-15
Abstract: In the framework of the QCD factorization approach, we study the localized