-
First, we define the kinematic variables and discuss their relations. The four-body semi-leptonic decay
$ D\to M_{1}M_{2}\ell^+\nu_\ell $ is considered, where D is the parent meson,$ M_{1(2)} $ is the product meson, and$ \ell=e,\,\mu $ . The momentum four-vectors and invariant masses are denoted by p and m, respectively. For convenience, the independent four-vector combinations are defined as$ \begin{aligned}[b]& P^\mu=p^\mu_{M_1}+p^\mu_{M_2},\quad Q^\mu=p^\mu_{M_1}-p^\mu_{M_2},\\ & L^\mu=p^\mu_{\ell}+p^\mu_{\nu},\quad N^\mu=p^\mu_{\ell}-p^\mu_{\nu}. \end{aligned} $
(1) A four-body decay can be uniquely described via kinematic parameterization with five variables (besides spin). The squared masses of the hadronic system
$ M_1M_2 $ and leptonic system$ \ell^+\nu_\ell $ are chosen as two of the five variables,$ \begin{eqnarray} s_M&=&P^2, \,\,\,\ s_L=L^2, \end{eqnarray} $
(2) and the following relations can be easily derived:
$ \begin{eqnarray} Q^2 =2 m_{M_1}^2+2m_{M_2}^2-s_M,~~~ N^2 =2m_{\ell}^2+2m_{\nu}^2-s_L, \end{eqnarray} $
(3) $ \begin{eqnarray} L\cdot P =\frac{ m_D^2-s_M-s_L}{2},\,\,\,\, L\cdot N = m_{\ell}^2-m_{\nu}^2,\,\,\,\, P\cdot Q = m_{M_1}^2-m_{M_2}^2. \end{eqnarray} $
(4) The other three variables are chosen as the angle between the
$ M_2 $ three-momentum and the D direction in the$ M_1M_2 $ rest frame ($ \theta_M $ ), the angle between$ \nu_\ell $ and the D direction in the$ \ell^+\nu_\ell $ rest frame ($ \theta_L $ ), and the angle between the two decay planes (ϕ)2 . The angles$ \theta_M $ ,$ \theta_L $ , and ϕ are illustrated in Fig. 1. The various relationships between scalar-product invariants can be written asFigure 1. (color online) Definition of angles
$ \theta_L $ ,$ \theta_M $ , and ϕ in the cascade decay$ D\to M_1M_2 \ell^+\nu_\ell $ .$ \begin{aligned}[b] L\cdot Q =& L\cdot P\frac{m_{M_1}^2-m_{M_2}^2}{s_M} +X\beta_M \cos\theta_M\,,\\ N\cdot P =& L\cdot P\frac{m_{\ell}^2-m_{\nu}^2}{s_L} +X\beta_L \cos\theta_L\,,\\ N\cdot Q =&L\cdot P\beta_M \beta_L \cos\theta_M \cos\theta_L +\frac{m_{\ell}^2-m_{\nu}^2}{s_L}X\beta_M \cos\theta_M\\&+\frac{ m_{M_1}^2-m_{M_2}^2}{s_M}X\beta_L \cos\theta_L+L\cdot P\frac{ m_{M_1}^2-m_{M_2}^2}{s_M}\frac{m_{\ell}^2-m_{\nu}^2}{s_L}\\ & -\sqrt{s_M}\sqrt{s_L}\beta_M \beta_L \sin\theta_M \sin\theta_L \cos\phi\,,\\ \epsilon_{\mu\nu\rho\sigma}L^{\mu}&N^{\nu}P^{\rho}Q^{\sigma}= X\sqrt{s_M}\sqrt{s_L}\beta_M \beta_L \sin\theta_M \sin\theta_L \sin\phi\,, \end{aligned} $
(5) where
$ \beta_M $ is the three-momentum modulus of the meson in the center-of-mass frame of the meson-meson system,$ \beta_L $ the three-momentum modulus of the lepton in the center-of-mass frame of the lepton-neutrino system, and X an element of phase space,$ \begin{aligned}[b] \beta_M=&\sqrt{(s_M-m^2_+)(s_M-m^2_-)}/s_M \,,\\ \beta_L=&\sqrt{(s_L-m^2_{L^+})(s_L-m^2_{L^-})}/s_L \,,\\ X=&\sqrt{m_D^4+s_L^2+s_M^2-2s_Dm_L^2-2s_Mm_D^2-2s_Ms_L}/2 \,, \end{aligned} $
(6) with
$ \begin{aligned}[b] m_{L^+}= & m_{\ell}+m_{\nu};~~ m_{L^-}=m_{\ell}-m_{\nu}; \\m_+ =&m_{M_1}+m_{M_2};~~ m_-=m_{M_1}-m_{M_2}. \end{aligned} $
(7) Compared to Refs. [6, 7], we do not neglect the lepton mass.
Next, from the effective Hamiltonian at the quark level for
$ D\to M_{1}M_{2}\ell^+\nu_{\ell} $ , the decay amplitude is given by$ \begin{aligned}[b] & {\cal A}(D\to M_{1}M_{2}\ell^+\nu_{\ell}) \\=& \frac{G_{\rm F}}{\sqrt{2}}V_{q_1q_2}\langle M_{2}M_{1}|\bar{q}_1\gamma_{\mu}(1-\gamma_{5})q_2|D\rangle \bar{u}(p_\ell)\gamma^{\mu}(1-\gamma_{5})v(p_\nu)\,, \end{aligned} $
(8) where
$G_{\rm F}$ is the Fermi constant, and$ V_{q_1q_2} $ is the element of the Cabibbo-Kobayashi-Maskawa matrix. The hadronic matrix element can be written in terms of four form factors,$ w_\pm $ , r, and h, which are defined by$ \begin{aligned}[b]& \langle M_{2}M_{1}|\bar{q}_1\gamma_{\mu}(1-\gamma_{5})q_2|D\rangle\\=& h\epsilon^{\mu\nu\alpha\beta}p_D^\nu P^\alpha Q^\beta+ {\rm i} rL^\mu+{\rm i}w_+ P^\mu+{\rm i} w_-Q^\mu\,, \end{aligned} $
(9) where the form factors
$ w_\pm $ , r, and h are functions of$ s_M $ ,$ s_L $ , and$ \cos\theta_M $ , and$ \epsilon^{\mu\nu\alpha\beta} $ is the Levi-Civita symbol.The differential decay rate takes the form
$\begin{aligned}[b] {\rm d}\Gamma =& \frac{G_{\rm F}^2|V_{q_1q_2}|^2}{(4\pi)^6 m_{D}^3}X\beta_{M}{I}(s_M, s_L, \theta_M, \theta_L, \phi)\\ & \times {\rm d}s_{M} {\rm d} s_{L} {\rm d} {\rm cos} \theta_{M} {\rm d}{\rm cos}\theta_{L} {\rm d}\phi\,.\end{aligned} $
(10) To study the structure of the hadron system, that is, the form factors of the
$ M_1M_2 $ system, the decay intensity I is decomposed with respect to$ \theta_L $ and ϕ, which is written as$ \begin{aligned}[b] I =& I_1+I_2\cos2\theta_L+I_3\sin^2\theta_L\cos2\phi+I_4\sin2\theta_L\cos\phi\\&+I_5\sin\theta_L\cos\phi+I_6\cos\theta_L +I_7\sin\theta_L\sin\phi \\& +I_8\sin2\theta_L\sin\phi+I_9\sin^2\theta_L\sin2\phi, \end{aligned} $
(11) where
$I_{1,2,...,9}$ depend only on$ s_M $ ,$ s_L $ , and ϕ. We can further express$I_{1,2,...,9}$ in terms of form factors,$ \begin{aligned}[b] I_1 =& \frac{1}{4}(2-\beta_L)\beta_L|F_1|^2 +\left(\frac{\beta_L}{2}-\frac{\beta_L^2}{8}\right)\sin^2\theta_M(|F_2|^2+|F_3|^2)\\&+\frac{1}{2}(1-\beta_L)\beta_L|F_4|^2\,,\\ I_2 =& -\frac{\beta_L^2}{4}\left[|F_1|^2-\frac{1}{2}\sin^2\theta_M(|F_2|^2+|F_3|^2)\right]\,,\\ I_3 =& -\frac{\beta_L^2}{4}\left[\sin^2\theta_M(|F_2|^2-|F_3|^2)\right]\,,\\ I_4 =& \frac{\beta_L^2}{2}\sin\theta_M {\rm Re}(F_1F_2^\star)\,,\\ I_5 =& -\beta_L\sin\theta_M\left\{{\rm Re}(F_1F_3^\star)-(1-\beta_L){\rm Re}(F_2F_4^\star)\right\}\,,\\ I_6 =& -\beta_L\sin^2\theta_M {\rm Re}(F_2F_3^\star)-\beta_L(1-\beta_L) {\rm Re}(F_1F_4^\star)\,,\\ I_7=& \beta_L\sin\theta_M \left\{{\rm Im}(F_1F_2^\star)+(1-\beta_L) {\rm Im}(F_3F_4^\star)\right\}\,,\\ I_8 =&-\frac{\beta_L^2}{2}\sin\theta_M {\rm Im}(F_1F_3^\star)\,,\\ I_9 =&\frac{\beta_L^2}{2}\sin^2\theta_M {\rm Im}(F_2F_3^\star)\,,\\[-10pt] \end{aligned} $
(12) where
$F_{1,2,3,4}$ are the form factors$ \begin{aligned}[b] F_1 =&X w_++(\beta_M P\cdot L\cos\theta_M+\frac{m_+m_-}{s_M}X)w_-\,,\\ F_2 =&\beta_M\sqrt{s_M}\sqrt{s_L}w_-\,,\\ F_3 =&X\beta_M\sqrt{s_M}\sqrt{s_L} h\,,\\ F_4 =&s_L r+P\cdot Lw_++(X\beta_M\cos\theta_M+\frac{m_+m_-}{s_M}P\cdot L)w_-\,. \end{aligned} $
(13) For the purpose of discussing the angular momentum of
$ M_1M_2 $ , for example, S- and P-waves, the partial wave expansions in spherical harmonics for the form factors$F_{1,2,3,4}$ are written as$ \begin{aligned}[b] F_1(s_M,s_L,\cos\theta_M) =& \sum_{l=0}^{\infty} F_{1l}(s_M,s_L)P_l(\cos\theta_M)\,,\\ F_2(s_M,s_L,\cos\theta_M) =& \sum_{l=1}^{\infty}\frac{1}{\sqrt{l(l+1)}} F_{2l}(s_M,s_L)\frac{{\rm d} P_l(\cos\theta_M)}{{\rm d}\cos\theta_M}\,,\\ F_3(s_M,s_L,\cos\theta_M) =& \sum_{l=1}^{\infty}\frac{1}{\sqrt{l(l+1)}} F_{3l}(s_M,s_L)\frac{{\rm d}P_l(\cos\theta_M)}{{\rm d}\cos\theta_M}\,,\\ F_4(s_M,s_L,\cos\theta_M) =&\sum_{l=0}^{\infty} F_{4l}(s_M,s_L)P_l(\cos\theta_M)\,. \end{aligned} $
(14) Moreover, the decay
$ D\to M_{1}M_{2}\ell^+\nu_\ell $ may occur via intermediate states, such as vector or scalar mesons, which provides information about the intermediate resonances [8, 9]. The amplitudes of$ D\to(S\to M_{1}M_{2})\ell^+\nu_{\ell} $ and$ D\to(V\to M_{1}M_{2})\ell^+\nu_{\ell} $ can be given by$ \begin{aligned}[b]& {\cal A}(D\to(V\to M_{1}M_{2})\ell^+\nu_{\ell}) \\=& \langle V|\bar{q}_1\gamma_{\mu}(1-\gamma_{5})q_2|D\rangle\epsilon\cdot Q g_{VM_{1}M_{2}} D_{FV} \bar{u}(p_\ell)\gamma^{\mu}(1-\gamma_{5})v(p_\nu)\,,\\ &{\cal A}(D\to(S\to M_{1}M_{2})\ell^+\nu_{\ell}) \\=& \langle S|\bar{q}_1\gamma_{\mu}(1-\gamma_{5})q_2|D\rangle g_{SM_{1}M_{2}}D_{FS} \bar{u}(p_\ell)\gamma^{\mu}(1-\gamma_{5})v(p_\nu)\,, \end{aligned} $
(15) where
$ \begin{aligned}[b] &\langle V|\bar{q}_1\gamma_{\mu}(1-\gamma_{5})q_2|D\rangle= \\& -\epsilon_{\mu\nu\alpha\beta}\epsilon^{\nu\ast}P_{D}^{\alpha}P^\beta\frac{2V_0(s_L)}{m_V+m_D}-{\rm i}\left(\epsilon^{\ast}_\mu-\frac{\epsilon^{\ast}\cdot L}{L^2}L_\mu\right)(m_V+m_D)A_1(s_L)\,,\\ &+{\rm i} \left({P_{D}}_\mu+P_{\mu}-\frac{m^2_D-m^2_V}{L^2}L_\mu\right)\epsilon^{\ast}\cdot L \frac{A_2(s_L)}{m_V+m_D}\\&-{\rm i}\frac{2m_V\epsilon^{\ast}\cdot L}{L^2}L_\mu A_0(s_L)\,,\\ & \langle S|\bar{q}_1\gamma_{\mu}(1-\gamma_{5})q_2|D\rangle = {\rm i}(f^+(s_L)P_\mu+f^-(s_L)L_\mu)\,, \end{aligned} $
(16) with
$ \begin{aligned}[b] \sum\epsilon^{\mu\ast}\epsilon^{\nu} =& -g^{\mu\nu}+\frac{P^\mu P^\nu}{P^2}\,,\\ {\cal A}(D\to M_{1}M_{2}\ell^+\nu_{\ell}) =& {\cal A}(D\to(V\to M_{1}M_{2})\ell^+\nu_{\ell})\\&+{\cal A}(D\to(S\to M_{1}M_{2})\ell^+\nu_{\ell})\,. \end{aligned} $
(17) Here,
$ f^\pm(s_L) $ is the$ D\to S $ form factor,$ A_{0,1,2} $ are the$ D\to V $ axial-vector form factors, and$ V_0 $ is the$ D\to V $ vector form factor [10, 11]. Finally, we can obtain$ F_{1-4} $ in the helicity basis (for S- and P-waves only),$ \begin{aligned}[b] F_1(s_M,s_L,\cos\theta_M) = & Xf^+(s_L)g_{SM_{1}M_{2}}D_{FS}\\ &+\cos\theta_M\beta_Mg_{VM_{1}M_{2}} D_{FV}\sqrt{s_Ls_M} H_0(s_L)\,,\\ F_2(s_M,s_L,\cos\theta_M) =& \frac{1}{2}\beta_Mg_{VM_{1}M_{2}} D_{FV}\sqrt{s_Ls_M} \\ &\times (H_+(s_L)+H_-(s_L))\,,\\ F_3(s_M,s_L,\cos\theta_M) =& \frac{1}{2}\beta_Mg_{VM_{1}M_{2}} D_{FV}\sqrt{s_Ls_M}\\ &\times (H_+(s_L)-H_-(s_L))\,,\\ F_4(s_M,s_L,\cos\theta_M) =&s_Lf^-(s_L)g_{SM_{1}M_{2}}D_{FS}\\ &+2\cos\theta_M\beta_Mg_{VM_{1}M_{2}} D_{FV} \sqrt{s_Ls_M} H_t(s_L)\,, \end{aligned} $
(18) with
$ \begin{aligned}[b] H_0(s_L) =& \frac{1}{\sqrt{s_Ls_M}}\left[P\cdot L(m_V+m_D)A_1(s_L)-2\frac{X^2}{m_V+m_D}A_2(s_L)\right]\,,\\ H_\pm(s_L) =& (m_V+m_D)A_1(s_L)\mp\frac{2X}{m_V+m_D}V_0(s_L)\,,\\ H_t(s_L) =& \frac{X}{\sqrt{s_L}}[A_0(s_L)+A_1(s_L)+A_2(s_L)]\,, \end{aligned} $
(19) where
$ g_{SM_{1}M_{2}}(g_{VM_{1}M_{2}}) $ is the coupling constant,$ D_{FS}(D_{FV}) $ is derived from the propagator for$ S(V) $ . In the case of Breit-Wigner lineshapes,$D_{FS}= 1/(s_M-m^2_S+{\rm i}m_S\Gamma_S)$ and$D_{FV}= 1/(s_M-m^2_V+{\rm i}m_V\Gamma_V)$ , or$D_{FS}=1/(s_M-m^2_S+ {\rm i}\dfrac{s_M}{m_S}\Gamma_S)$ and$D_{FV}=1/(s_M-m^2_V+{\rm i}\dfrac{s_M}{m_V}\Gamma_V)$ if decay width$ \Gamma_{S}(\Gamma_{V}) $ is not negligible [12]. For the scalar meson f0(980), it couples to the channels$K\bar{K} $ and$ \pi\pi $ strongly. Likewise,$ a_0(980) $ couples to$K\bar{K} $ and$ \pi\eta $ strongly. One needs to replace the usual Breit-Wigner lineshape by the Flatté formula [13] to take into account the coupled channel effect. For the broad$ \rho(770) $ resonance, it is customary to adapt the Gounaris-Sakurai model [14] to account for the$\pi\pi$ rescattering.
Lepton mass correction in partial wave analyses of charmed meson semi-leptonic decays
- Received Date: 2023-02-22
- Available Online: 2023-06-15
Abstract: We derive a parameterization formula for the partial wave analyses of charmed meson semi-leptonic decays while considering the effects of lepton mass. Because the proposed super-tau-charm factory will reach a significantly enhanced luminosity and BESIII is collecting new