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The effective weak Hamiltonian for CKM enhanced B-meson decays is given by
$ {H_w} = \frac{G_{\rm F}}{{\sqrt 2 }}{V_{cb}}V_{ud}^ * \left[ {{c_1}\left( {\overline d u} \right)\left( {\overline c b} \right) + {c_2}\left( {\overline c u} \right)\left( {\overline d b} \right)} \right], $
(1) where
${\bar q_1}{q_2} = {\bar q_1}{\gamma _\mu }\left( {1 - {\gamma _5}} \right){q_2}$ denotes the color singlet V−A Dirac current, and the QCD coefficients [23, 24] on the bottom mass scale are$ {c_1} = 1.132,\quad \quad {c_2} = - 0.287. $
(2) Because the current operators in the weak Hamiltonian are expressed in terms of fundamental quark fields, it is appropriate to have the Hamiltonian in a form such that one of these currents carries the same quantum numbers as one of the mesons emitted in the final state of bottom meson decays. Consequently, the hadronic matrix elements of an operator O receives contributions from the operator itself and the Fierz transformation of O, which generates the factorizable and nonfactorizable parts through the Fierz identity,
$\begin{aligned}[b](\bar du)(\bar cb) =& \frac{1}{{{N_c}}} (\bar cu)(\bar db) \\&+ \frac{1}{2}\left( {\overline c {\lambda ^a}\,u} \right)\left( {\overline d {\kern 1pt} {\lambda ^a}b} \right), \end{aligned}$
(3) where
${\bar q_1}{\lambda ^a}\,{q_2} \equiv \,{\bar q_1}{\kern 1pt} {\gamma _\mu }\,\left( {1 - {\gamma _5}\,} \right){\lambda ^a}{q_2}$ represents the color octet current. Performing a similar treatment on the other operator$(\overline c u)(\overline d b)$ , the weak Hamiltonian becomes$ H_w^{\rm CF} = \frac{G_{\rm F}}{{\sqrt 2 }}{V_{cb}}V_{ud}^ * \left[ {{a_1}{{\left( {\overline d u} \right)}_H}{{\left( {\overline c b} \right)}_H} + {c_2}H_w^8} \right], $
(4) $ H_w^{\rm CS} = \frac{G_{\rm F}}{{\sqrt 2 }}{V_{cb}}V_{ud}^ * \left[ {{a_2}{{\left( {\overline c u} \right)}_H}{{\left( {\overline d b} \right)}_H} + {c_1}\tilde H_w^8} \right], $
(5) $ {a_{1,2}} = {c_{1,2}} + \frac{{{c_{2,1}}}}{{{N_c}}}, $
(6) $\begin{aligned}[b]& H_w^8 = \frac{1}{2}\sum\limits_{a = 1}^8 {\left( {\overline c {\lambda ^a}\,u} \right)\left( {\overline d {\kern 1pt} {\lambda ^a}b} \right)} ,\\& \tilde H_w^8 = \frac{1}{2}\sum\limits_{a = 1}^8 {\left( {\overline d {\kern 1pt} {\lambda ^a}u} \right)\left( {\overline c {\lambda ^a}\,b} \right)} , \end{aligned}$
(7) which describe the color-favored (CF) and color-suppressed (CS) processes, respectively. Here, the index H in (4) and (5) indicates the change from the quark current to hadron field operator [4]. The matrix elements of the first terms in (4) and (5) lead to the factorizable contributions [4], and the second terms, involving the color octet currents, generate nonfactorized contributions.
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The branching fraction for B-meson decay into two pseudoscalar mesons is related to its decay amplitude as follows:
$\begin{aligned}[b] B\left( {\bar B \to {P_1}{P_2}} \right) =& {\tau _B}{\left| {\frac{G_{\rm F}}{{\sqrt 2 }}{V_{cb}}V_{ud}^ * } \right|^2}\frac{p}{{8\pi m_B^2}}\\&\times{\left| {A\left( {\bar B \to {P_1}{P_2}} \right)} \right|^2} , \end{aligned}$
(8) where
$ {\tau _B} $ denotes the lifetime of B-mesons taken from [1],$\begin{aligned}[b] {\tau _{{{\bar B}^0}}} = (1.519 \pm 0.004) \times {10^{ - 12}}\;{\rm s}, \\ {\tau _{{B^ - }}} = (1.638 \pm 0.004) \times {10^{ - 12}}\;{\rm s}, \end{aligned}$
$ {V_{ud}}{V_{cb}}$ is the product of the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements [1],$ {V_{ud}} = 0.975,{\kern 1pt} \;\;\;\;\;\;{V_{cb}} = 0.041 , $
and p is the magnitude of the three-momentum of the final state particles in the rest frame of the parent B- meson,
$\begin{aligned}[b] p =& \left| {{p_1}} \right| = \left| {{p_2}} \right| = \frac{1}{{2{m_B}}}\left[ \left\{ {m_B^2 - {{\left( {{m_1} + {m_2}} \right)}^2}} \right\}\right.\\&\times\left.\left\{ {m_B^2 - {{\left( {{m_1} - {m_2}} \right)}^2}} \right\} \right]^{1/2}. \end{aligned}$
(9) In heavy flavor meson decays, it has been observed that long distance strong FSI rescattering [20–21] of out-going mesons significantly affects their branching fractions. In general, such FSI phenomena can affect a decay amplitude in two ways: The decay amplitude may itself be modulated or it may acquire a phase. It has been shown by Kamal [25] that, in the weak scattering limit, the elastic FSI effect is mainly used to obtain a phase factor, i.e.,
$ {A^{\rm FSI}} = A\;{{\rm e}^{{\rm i}\delta }}. $
(10) Consequently, mixing of final states with the same quantum numbers can take place. Initially, it was expected that bottom meson decays may not be affected by FSI because the produced particles may not have sufficient time to interact, and there are no meson resonances near the B- meson mass corresponding to the quantum numbers of the final state. However, experimental data do not fulfill this naïve expectation [26].
To demonstrate this, we employ the isospin framework, in which
$ \bar B \to \pi D $ decay amplitudes are represented in terms of isospin reduced amplitudes, including the strong interaction phases$ \delta _{1/2}^{\pi D},\,\;\delta _{3/2}^{\pi D} $ in the Isospin -1/2 and 3/2 final states, respectively, as$ \begin{aligned}[b] A({{\bar B}^0} \to {\pi ^ - }{D^ + }) =& \frac{1}{{\sqrt 3 }}\left[ {A_{3/2}^{\pi D}{{\rm e}^{{\rm i}\delta _{3/2}^{\pi D}}} + \sqrt 2 A_{1/2}^{\pi D}{{\rm e}^{{\rm i}\delta _{1/2}^{\pi D}}}} \right], \\ A({{\bar B}^0} \to {\pi ^0}{D^0}) =& \frac{1}{{\sqrt 3 }}\left[ {\sqrt 2 A_{3/2}^{\pi D}{{\rm e}^{{\rm i}\delta _{3/2}^{\pi D}}} - A_{1/2}^{\pi D}{{\rm e}^{{\rm i}\delta _{1/2}^{\pi D}}}} \right], \\ A({B^ - } \to {\pi ^ - }{D^0}) =& \sqrt 3 A_{3/2}^{\pi D}{{\rm e}^{{\rm i}\delta _{3/2}^{\pi D}}}. \end{aligned} $
(11) These lead to the following relations:
$ \begin{aligned}[b] A_{1/2}^{\pi D} =& \left[{ \left| A({{\bar B}^0} \to {\pi ^ - }{D^ + }) \right|}^2 + \left| {A({{\bar B}^0} \to {\pi ^0}{D^0})} \right|^2\right. \\&\left. - \frac{1}{3}\left| {A({B^ - } \to {\pi ^ - }{D^0})} \right|^2 \right]^{1/2}, \\ A_{3/2}^{\pi D} = &\sqrt {\frac{1}{3}} \left| {A({B^ - } \to {\pi ^ - }{D^0})} \right|, \end{aligned} $
(12) and the relative phase difference,
$ \delta _{}^{\pi D} = \delta _{1/2}^{\pi D} - \delta _{3/2}^{\pi D}, $ is given by$\begin{aligned}[b]\\[-5pt] {Cos} \delta _{}^{\pi D} = \frac{{(3{{\left| {A({{\bar B}^0} \to {\pi ^ - }{D^ + })} \right|}^2} - 6{{\left| {A({{\bar B}^0} \to {\pi ^0}{D^0})} \right|}^2} + {{\left| {A({B^ - } \to {\pi ^ - }{D^0})} \right|}^2})}}{{6\sqrt 2 \left| {A_{1/2}^{\pi D}} \right|\left| {A_{3/2}^{\pi D}} \right|}}.\\ \end{aligned}$ (13) Thus,
$ A_{1/2}^{\pi D} $ and$ A_{3/2}^{\pi D} $ can be treated as real quantities in the following analysis:Using the experimental values [1]
$ B\left( {{{\bar B}^0} \to {\pi ^ - }{D^ + }} \right) = \left( {2.52 \pm 0.13} \right) \times {10^{ - 3}}, $
$ B\left( {{{\bar B}^0} \to {\pi ^0}{D^0}} \right) = \left( {2.63 \pm 0.14} \right) \times {10^{ - 4}}, $
$ B\left( {{B^ - } \to {\pi ^ - }{D^0}} \right) = \left( {4.68 \pm 0.13} \right) \times {10^{ - 3}}, $
we obtain
$\begin{aligned}[b] A_{1/2}^{\pi D}{\,^{\exp }} = &\pm \left( {1.273 \pm 0.065} \right){\;\rm Ge{V^3}},\\A{_{3/2}^{\pi D\,}\,^{\exp }} =& \pm \left( {1.323 \pm 0.018} \right){\;\rm Ge{V^3}}, \end{aligned}$
(14) and the phase difference
$ {\delta ^{\pi D}} = \left( {28 \pm 7} \right){^\circ } , $
(15) which agrees with the final state rescattering analysis [22]. Although this phase difference is relatively smaller than that of the
$ D \to \bar K\pi $ mode$ {\delta ^{}} = \left( {86 \pm 7} \right){^\circ } , $ it certainly indicates the presence of non-vanishing strong phases in the B- meson sector.We express the decay amplitude as a sum of the factorizable and nonfactorizable parts,
$ A(\bar B \to \pi D) = {A^f}(\bar B \to \pi D) + {A^{n{\kern 1pt} f}}(\bar B \to \pi D), $
(16) arising from the respective terms of the weak Hamiltonian given in (4) and (5).
Using the factorization scheme, the spectator-quark parts of the decay amplitudes arising from W- emission① diagrams are derived for the following classes of
$ \bar B \to \pi D $ decays:(a) Class I: Color favored (CF)
$ {A^f}({\bar B^0} \to {\pi ^ - }{D^ + }) = {a_1}{f_\pi }\left( {m_B^2 - m_D^2} \right){\kern 1pt} F_0^{\bar BD}\left( {m_\pi ^2\,} \right), $
(17) (b) Class II: Color Suppressed (CS)
$ {A^f}({\bar B^0} \to {\pi ^0}{D^0}) = - \frac{1}{{\sqrt 2 }}{a_2}{f_D}\left( {m_B^2 - m_\pi ^2} \right){\kern 1pt} F_0^{\bar B\pi }\left( {m_D^2\,} \right), $
(18) (c) Class III : Interference of CF and CS
$\begin{aligned}[b] {A^f}({B^ - } \to {\pi ^ - }{D^0}) =& {a_1}{f_\pi }\left( {m_B^2 - m_D^2} \right){\kern 1pt} F_0^{\bar BD}\left( {m_\pi ^2\,} \right) \\&+ {a_2}{f_D}\left( {m_B^2 - m_\pi ^2} \right){\kern 1pt} F_0^{\bar B\pi }\left( {m_D^2\,} \right)\,. \end{aligned}$
(19) We calculate the values of the factorization contributions for
$ {N_c} = 3 $ (real value) using numerical inputs for decay constants taken as$\begin{aligned}[b] {f_D} =& \left( {0.207 \pm 0.009} \right){\;\rm GeV},\\{f_\pi } =& \left( {0.131 \pm 0.002} \right)\;{\rm GeV}, \end{aligned}$
(20) from the leptonic decays of D and π mesons, respectively [27].
Assuming nearest pole dominance, momentum dependence of the form-factors, appearing in the decay amplitudes given in (17–19), is taken as
$ {F_0}\left( {{q^2}} \right) = \frac{{{F_0}\left( 0 \right)}}{{{{\left( {1 - {\raise0.7ex\hbox{${{q^2}}$} \mathord{\left/ {\vphantom {{{q^2}} {m_s^2}}}\right.} \lower0.7ex\hbox{${m_s^2}$}}} \right)}^n}}}, $
(21) where the pole masses are given by the scalar meson carrying the quantum number of the corresponding weak current, which are ms = 5.78 GeV and ms = 6.80 GeV, and n = 1 for the monopole formula. The form-factors
$ F _0^{}\left( {0\,} \right) $ at q2 =0 are taken from [28], as given below.$ \begin{aligned}[b] {\kern 1pt} F_0^{\bar B\pi }\left( {0\,} \right) =& \left( {0.27 \pm 0.05} \right), \\ F_0^{\bar BD}\left( {0\,} \right) =& \left( {0.66 \pm 0.03} \right). \end{aligned} $
(22) We finally obtain
$ \begin{aligned}[b] {A^f}({{\bar B}^0} \to {\pi ^ - }{D^ + }) = 2.180 \pm \;0.099\;{{\rm{GeV}}^3}, \hfill \\ {A^f}({{\bar B}^0} \to {\pi ^0}{D^0}) = - 0.111 \pm \;0.021\; {{\rm{GeV}}^3}, \hfill \\ {A^f}({B^ - } \to {\pi ^ - }{D^0}) = 2.339 \pm \;0.103 \;{{\rm{GeV}}^3}. \hfill \\ \end{aligned} $
(23) Exploiting the following isospin relations:
$ \begin{gathered} A_{1/2}^f(\bar B \to \pi D) = \frac{1}{{\sqrt 3 }}\left\{ {\sqrt 2 {A^f}({{\bar B}^0} \to {\pi ^ - }{D^ + }) - {A^f}({{\bar B}^0} \to {\pi ^0}{D^0})} \right\}, \hfill \\ A_{3/2}^f(\bar B \to \pi D) = \frac{1}{{\sqrt 3 }}\left\{ {{A^f}({{\bar B}^0} \to {\pi ^ - }{D^ + }) + \sqrt 2 {A^f}({{\bar B}^0} \to {\pi ^0}{D^0})} \right\}, \hfill \\ \end{gathered} $
(24) we obtain
$\begin{aligned}[b] A_{1/2}^f =& \left( {1.845 \pm 0.082} \right){\kern 1pt} {\kern 1pt} {{\rm{GeV}}^3},\\ A_{3/2}^f =& \left( {1.168 \pm 0.060} \right){\kern 1pt} {\rm{GeV}}{^3}. \end{aligned}$
(25) Using isospin C. G. coefficients with the convention used in [17, 18], the nonfactorizable part of the decay amplitudes can be expressed in terms of the scattering amplitudes for the spurion +
$ \bar B \to \pi D $ process.$ \begin{aligned}[b] &{A^{n{\kern 1pt} f}}({{\bar B}^0} \to {\pi ^ - }{D^ + }) = \frac{1}{3}{c_2}\left( {\left\langle {\pi D\left\| {\left. {H_w^8} \right\|} \right.} \right.{{\left. {\bar B} \right\rangle }_{3/2}} + 2\left\langle {\pi D\left\| {\left. {H_w^8} \right\|} \right.} \right.{{\left. {\bar B} \right\rangle }_{1/2}}} \right), \\ &{A^{n{\kern 1pt} f}}({{\bar B}^0} \to {\pi ^0}{D^0}) = \frac{{\sqrt 2 }}{3}{c_1}\left( {\left\langle {\pi D\left\| {\left. {\tilde H_w^8} \right\|} \right.} \right.{{\left. {\bar B} \right\rangle }_{3/2}} - \left\langle {\pi D\left\| {\left. {\tilde H_w^8} \right\|} \right.} \right.{{\left. {\bar B} \right\rangle }_{1/2}}} \right), \\ & {A^{nf}}({B^ - } \to {\pi ^ - }{D^0}) = {c_2}\left\langle {\pi D\left\| {\left. {H_w^8} \right\|} \right.} \right.{\left. {\bar B} \right\rangle _{3/2}} + {c_1}\left\langle {\pi D\left\| {\left. {\tilde H_w^8} \right\|} \right.} \right.{\left. {\bar B} \right\rangle _{3/2}}. \end{aligned} $
(26) At present, there is no available technique to exactly calculate these quantities from the theory of strong interactions. Therefore, by subtracting the factorizable part (25) from the experimental decay amplitude (14), we determine the nonfactorizable isospin reduced amplitudes,
$\begin{aligned}[b] A_{1/2}^{nf} =& - \left( {0.572 \pm 0.105} \right)\;{\rm Ge{V^3}},\\ A_{3/2}^{nf} =& - \left( {2.491 \pm 0.062} \right)\;{\rm Ge{V^3}},\end{aligned} $
(27) by choosing positive and negative values for
$ A_{1/2}^{nf} $ and$ A_{3/2}^{nf} $ , respectively. Their ratio is$ \alpha = {{A_{1/2}^{nf}}}/{{A_{3/2}^{nf}}} = 0.229 \pm 0.042. $
(28) There are several calculations for form factors, obtained from different approaches in literature, which are given in Table 1.
Table 1. Form-factor of
$ \bar B \to D $ and$ \bar B \to \pi $ transitions at maximum recoil (q2 =0).To observe the effect of form-factor variation on our analysis, we give the ratio
$ \alpha $ in Table 2 for the maximum and minimum values of the form-factors, which are consistent with (28) within errors.$ F_0^{\bar BD}\left( {0\,} \right) $ 0.69 0.69 0.63 0.63 $ F_0^{\bar B\pi }\left( {0\,} \right) $ 0.32 0.22 0.32 0.22 $ \alpha $ 0.262 0.249 0.207 0.195 Table 2. Ratio
$ \alpha = {{A_{1/2}^{nf}}}/{{A_{3/2}^{nf}}} $ for maximum and minimum values of form-factors.We also plot the dependence of
$ \alpha $ on form-factors$ F_0^{\bar BD}(0) $ and$ F_0^{\bar B\pi }(0) $ in Fig. 1, which shows that$ \alpha $ is not quite sensitive to them. -
Using the branching fraction,
$ B\left( {\bar B \to PV} \right) = {\tau _B}{\left| {\frac{G_{\rm F}}{{\sqrt 2 }}{V_{cb}}V_{ud}^ * } \right|^2}\frac{{{p^3}}}{{8\pi m_V^2}}{\left| {A\left( {\bar B \to PV} \right)} \right|^2}. $
(29) Because the isospin structure of
$ \bar B \to \rho D $ decays is exactly the same as that of$ \bar B \to \pi D $ decays,$ \begin{aligned}[b] &A({{\bar B}^0} \to {\rho ^ - }{D^ + }) = \frac{1}{{\sqrt 3 }}\left[ {A_{3/2}^{\rho D}{{\rm e}^{{\rm i}\delta _{3/2}^{\rho D}}} + \sqrt 2 A_{1/2}^{\rho D}{{\rm e}^{{\rm i}\delta _{1/2}^{\rho D}}}} \right], \hfill \\ & A({{\bar B}^0} \to {\rho ^0}{D^0}) = \frac{1}{{\sqrt 3 }}\left[ {\sqrt 2 A_{3/2}^{\rho D}{{\rm e}^{{\rm i}\delta _{3/2}^{\rho D}}} - A_{1/2}^{\rho D}{{\rm e}^{{\rm i}\delta _{1/2}^{\rho D}}}} \right], \hfill \\ & A({B^ - } \to {\rho ^ - }{D^0}) = \sqrt 3 A_{3/2}^{\rho D}{{\rm e}^{{\rm i}\delta _{1/2}^{\rho D}}}. \hfill \\ \end{aligned} $
(30) We repeat the same procedure as before. Using the experimental branching fractions
$ \begin{gathered} B\left( {{{\bar B}^0} \to {\rho ^ - }{D^ + }} \right) = \left( {7.6 \pm 1.2} \right) \times {10^{ - 3}}, \hfill \\ B\left( {{{\bar B}^0} \to {\rho ^0}{D^0}} \right) = \left( {3.21 \pm 0.21} \right) \times {10^{ - 4}}, \hfill \\ B\left( {{B^ - } \to {\rho ^ - }{D^0}} \right) = \left( {1.34 \pm 0.18} \right) \times {10^{ - 2}}, \hfill \\ \end{gathered} $
we obtain the total isospin reduced amplitudes
$ \begin{aligned}[b]&A{_{1/2}^{\rho D}\,^{\exp }} = \pm \left( {0.143 \pm 0.025} \right)\;{\rm Ge{V^2}},\\&A{_{3/2}^{\rho D}\,^{\exp }} = \pm \left( {0.149 \pm 0.010} \right)\;\rm Ge{V^2},\end{aligned} $
(31) and the phase difference
$ {\delta ^{\rho {\kern 1pt} {\kern 1pt} D}} \equiv \delta _{1/2}^{\rho D} - \delta _{3/2}^{\rho D} = {\left( {8\;_{ - 8}^{ + 30}} \right)^ \circ }. $
(32) The factorizable decay amplitudes of the spectator-quark diagrams can be expressed as
$ \begin{gathered} {A^f}({{\bar B}^0} \to {\rho ^ - }{D^ + }) = 2{a_1}{m_\rho }{f_\rho }F_1^{\bar BD}\left( {m_\rho ^2} \right), \hfill \\ {A^f}({{\bar B}^0} \to {\rho ^0}{D^0}) = - \sqrt 2 {a_2}{f_D}{m_\rho }A_0^{\bar B\rho }\left( {m_D^2} \right), \hfill \\ {A^f}({B^ - } \to {\rho ^ - }{D^0}) = {a_1}2{m_\rho }{f_\rho }F_1^{\bar BD}\left( {m_\rho ^2} \right) + {a_2}{f_D}2{m_\rho }A_0^{\bar B\rho }\left( {m_D^2} \right). \hfill \\ \end{gathered} $
(33) It has been noted in the BSW II model [3] that consistency with heavy quark symmetry requires certain form- factors, such as
$ {F_1}(0) $ and$ {A_0}(0), $ to have dipole q2 dependence (n=2) in$\begin{aligned}[b] {F_1}\left( {{q^2}} \right) = \frac{{{F_1}\left( 0 \right)}}{{{{\left( {1 - {\raise0.7ex\hbox{${{q^2}}$} \mathord{\left/ {\vphantom {{{q^2}} {m_V^2}}}\right.} \lower0.7ex\hbox{${m_V^2}$}}} \right)}^n}}},\;\;\; {A_0}\left( {{q^2}} \right) = \frac{{{A_0}\left( 0 \right)}}{{{{\left( {1 - {\raise0.7ex\hbox{${{q^2}}$} \mathord{\left/ {\vphantom {{{q^2}} {m_P^2}}}\right.} \lower0.7ex\hbox{${m_P^2}$}}} \right)}^n}}}, \end{aligned}$
(34) where the vector V(1–) meson and pseudoscalar P(0–) meson pole masses are 6.34 and 5.27 GeV, respectively.
Decay constant values are taken from [27] as
$\begin{aligned}[b]& {f_D} = \left( {0.207 \pm 0.009} \right)\;{\rm GeV},\\&{f_\rho } = \left( {0.215 \pm 0.005} \right)\;{\rm GeV}, \end{aligned}$
(35) and form-factors for
$ \bar B \to V $ transitions are chosen from [33],$ {\kern 1pt} A_0^{\bar B\rho }\left( {0\,} \right) = 0.356 \pm 0.042, $
(36) where the
$ F_0^{\bar BD}\left( {0\,} \right) $ value is taken from Eq. (22).$ {\kern 1pt} F_1^{\bar BD}\left( {0\,} \right) = F_0^{\bar BD}\left( {0\,} \right) = 0.66 \pm 0.03. $
(37) Thus, we calculate the factorizable contributions to the decay amplitudes,
$ \begin{aligned}[b] & {A^f}({{\bar B}^0} \to {\rho ^ - }{D^ + }) = \left( {0.235 \pm 0.011} \right)\;{\rm Ge{V^2}}, \hfill \\ & {A^f}({{\bar B}^0} \to {\rho ^0}{D^0}) = - \left( {0.010 \pm 0.001} \right)\;{\rm Ge{V^2}}, \hfill \\ & {A^f}({B^ - } \to {\rho ^ - }{D^0}) = \left( {0.248 \pm 0.011} \right)\;{\rm Ge{V^2}}, \hfill \\ \end{aligned} $
(38) thereby the isospin reduced amplitudes of the factorized amplitudes are calculated as
$\begin{aligned}[b]& A_{1/2}^f = \left( {0.197 \pm 0.009} \right) \;{\rm Ge{V^2}},\\& A_{3/2}^f = \left( {0.127 \pm 0.006} \right) \;{\rm Ge{V^2}}. \end{aligned}$
(39) Following the procedure discussed for
$ \bar B \to \pi D $ , we determine the nonfactorizable reduced isospin amplitudes$\begin{aligned}& A_{1/2}^{nf} = - \left( {0.054 \pm 0.026} \right){\kern 1pt} {\kern 1pt} \;{\rm Ge{V^2}},\\& A_{3/2}^{nf} = - \left( {0.277 \pm 0.012} \right){\kern 1pt} \;{\rm Ge{V^2}}, \end{aligned}$
(40) which bear the following ratio:
$ \alpha = \frac{{A_{1/2}^{nf}}}{{A_{3/2}^{nf}}} = 0.200 \pm 0.096. $
(41) There are also existing calculations for
$ A_0^{\bar B\rho }\left( {0\,} \right) $ , which are given in Table 3. To show the effect of form-factors on our analysis, we obtain the ratio$ \alpha = {{A_{1/2}^{nf}}}/{{A_{3/2}^{nf}}} $ for the maximum and minimum value of the form factors given in Table 4, which are consistent with (41) within errors. This is also shown in Fig. 2.Figure 2. (color online) Variation in
$ \alpha $ with form factors$ F_0^{\bar BD}(0) $ and$ F_0^{\bar B\rho }(0) $ .Table 3. Form-factor of
$ \bar B \to \rho $ transitions at maximum recoil (q2 =0).$ F_0^{\bar BD}\left( {0\,} \right) $ 0.69 0.69 0.63 0.63 $ A_0^{\bar B\rho }\left( {0\,} \right) $ 0.40 0.31 0.40 0.31 $ \alpha $ 0.226 0.219 0.171 0.158 Table 4. Ratio
$ \alpha = {{A_{1/2}^{nf}}}/{{A_{3/2}^{nf}}} $ for maximum and minimum values of form-factors. -
Including the strong phases between the isospin I=1/2 and 3/2 states, the decay amplitudes are given by
$ \begin{aligned}[b] &A({{\bar B}^0} \to {\pi ^ - }{D^*}^ + ) = \frac{1}{{\sqrt 3 }}\left[ {A_{3/2}^{\pi {D^*}}{{\rm e}^{{\rm i}\delta _{3/2}^{\pi {D^*}}}} + \sqrt 2 A_{1/2}^{\pi {D^*}}{{\rm e}^{{\rm i}\delta _{1/2}^{\pi {D^*}}}}} \right], \hfill \\ &A({{\bar B}^0} \to {\pi ^0}{D^*}^0) = \frac{1}{{\sqrt 3 }}\left[ {\sqrt 2 A_{3/2}^{\pi {D^*}}{{\rm e}^{{\rm i}\delta _{3/2}^{\pi {D^*}}}} - A_{1/2}^{\pi {D^*}}{{\rm e}^{{\rm i}\delta _{1/2}^{\pi {D^*}}}}} \right], \hfill \\ &A({B^ - } \to {\pi ^ - }{D^*}^0) = \sqrt 3 A_{3/2}^{\pi {D^*}}{{\rm e}^{{\rm i}\delta _{3/2}^{\pi {D^*}}}}. \hfill \\ \end{aligned} $
(42) Using the experimental values of branching fractions [1],
$ \begin{aligned}[b] & B\left( {{{\bar B}^0} \to {\pi ^ - }{D^{* + }}} \right) = \left( {2.74 \pm 0.13} \right) \times {10^{ - 3}}, \hfill \\ & B\left( {{{\bar B}^0} \to {\pi ^0}{D^{*0}}} \right) = \left( {2.20 \pm 0.60} \right) \times {10^{ - 4}}, \hfill \\ & B\left( {{B^ - } \to {\pi ^ - }{D^{*0}}} \right) = \left( {4.90 \pm 0.17} \right) \times {10^{ - 3}}, \hfill \\ \end{aligned} $
we calculate the total isospin reduced amplitudes
$\begin{aligned}[b]& A{_{1/2}^{\pi {D^*}}\,^{\exp }} = \pm \left( {0.226 \pm 0.042} \right)\;{\rm Ge{V^2}},\\&A{_{3/2}^{\pi {D^*}}\,^{\exp }} = \pm \left( {0.231 \pm 0.040} \right)\;{\rm Ge{V^2}},\end{aligned} $
(43) and the phase difference
$ {\delta ^{\,\pi {D^*}}} = {\left( {24 \pm 24} \right)^ \circ }. $
(44) Therefore, the
$ \bar B \to PV $ decays also indicate the presence of strong FSI phases, as also observed in [17, 18].The factorizable amplitudes for this mode are
$ \begin{aligned}[b] {A^f}({{\bar B}^0} \to {\pi ^ - }{D^{ * + }}) =& 2{a_1}{m_{{D^ * }}}{f_\pi }A_0^{\bar B{D^ * }}\left( {m_\pi ^2} \right), \hfill \\ {A^f}({{\bar B}^0} \to {\pi ^0}{D^{ * 0}}) =& - \sqrt 2 {a_2}{f_{{D^ * }}}{m_{{D^ * }}}F_1^{\bar B\pi }\left( {m_{{D^ * }}^2} \right), \hfill \\ {A^f}({B^ - } \to {\pi ^ - }{D^{ * 0}}) =& {a_1}2{m_{{D^ * }}}{f_\pi }A_0^{\bar B{D^ * }}\left( {m_\pi ^2} \right) \\&+ {a_2}{f_{{D^ * }}}2{m_{{D^ * }}}F_1^{\bar B\pi }\left( {m_{{D^ * }}^2} \right). \hfill \\ \end{aligned} $
(45) Using the decay constant values [27]
$\begin{aligned}[b]& {f_{{D^*}}} = \left( {0.245 \pm 0.034} \right)\;{\rm GeV},\\& {f_\pi } = \left( {0.131 \pm 0.002} \right)\;{\rm GeV}, \end{aligned}$
(46) and the form-factor
$ A_0^{\bar B{D^*}}\left( {0\,} \right) = \left( {0.68 \pm 0.04} \right), $
(47) taken from [27], with pole masses
$ V({1^ - }) = 5.32 $ GeV and$ P({0^ - }) = 6.28 $ GeV for$ {q^2} $ -dependence (34),$ F_1^{\bar B\pi }\left( {0\,} \right) = F_0^{\bar B\pi }\left( {0\,} \right) = (0.27 \pm 0.05), $
we calculate the factorized amplitudes as
$ \begin{aligned}[b] & {A^f}({{\bar B}^0} \to {\pi ^ - }{D^{ * + }}) = \left( {0.371 \pm 0.022} \right)\;{\rm Ge{V^2}}, \hfill \\ & {A^f}({{\bar B}^0} \to {\pi ^0}{D^{ * 0}}) = - \left( {0.023 \pm 0.004} \right)\;{\rm Ge{V^2}}, \hfill \\ & {A^f}({B^ - } \to {\pi ^ - }{D^{ * 0}}) = \left( {0.403 \pm 0.023} \right)\;{\rm Ge{V^2}}, \hfill \\ \end{aligned} $
(48) which in turn yield the isospin reduced amplitudes
$\begin{aligned}[b]& A_{1/2}^f = \left( {0.317 \pm 0.018} \right){\kern 1pt} \;{\rm Ge{V^2}},\\& A_{3/2}^f = \left( {0.196 \pm 0.013} \right){\kern 1pt} \;{\rm Ge{V^2}}. \end{aligned}$
(49) Subtracting the factorizable parts from the total experimental amplitudes, we calculate
$\begin{aligned}[b] &A_{1/2}^{nf} = - \left( {0.090 \pm 0.046} \right){\kern 1pt} \;{\rm Ge{V^2}},\\& A_{3/2}^{nf} = - \left( {0.426 \pm 0.042} \right){\kern 1pt} \;{\rm Ge{V^2}}, \end{aligned}$
(50) with the following ratio:
$ \alpha = \frac{{A_{1/2}^{nf}}}{{A_{3/2}^{nf}}} = 0.211 \pm 0.109. $
(51) In literature, we find different values of the
$ A_0^{\bar B{D^ * }}\left( {0\,} \right) $ form-factor, as shown in Table 5. We calculate the ratio$ \alpha $ for the maximum and minimum values of the form-factors, as shown in Table 6, and plot the variation in$ \alpha $ in Fig. 3. Although$ \alpha $ remains insensitive to the$ F_0^{\bar B\pi }(0) $ form-factor, it increases slowly for large values of$ A_0^{\bar B{D^*}}(0) $ . However, considering the near equality of the ratio$ \alpha $ for$ \bar B \to \pi D/\rho D/\pi {D^*}, $ we expect a higher value of$ A_0^{\bar B{D^*}}(0) $ is less likely.Figure 3. (color online) Variation in
$ \alpha $ with form factors$ A_0^{\bar B{D^*}}(0) $ and$ F_0^{\bar B\pi }(0) $ Table 5. Form-factor of the
$ \bar B \to {D^*} $ transitions at maximum recoil (q2 =0).$ A_0^{\bar B{D^*}}\left( {0\,} \right) $ 0.72 0.72 0.64 0.64 $ F_0^{\bar B\pi }\left( {0\,} \right) $ 0.32 0.22 0.32 0.22 $ \alpha $ 0.264 0.245 0.192 0.173 Table 6. Ratio
$\alpha = {{A_{1/2}^{nf}}}/{{A_{3/2}^{nf}}}$ for maximum and minimum values of form-factors. -
The purpose of performing an isospin analysis on the
$ \bar B \to \pi D $ and$ \bar B \to \rho D/\pi {D^*} $ decays is to search for systematics, which have previously been identified in the charm sector [17, 18]. By choosing a positive sign for$ A_{1/2}^{\exp } $ and a negative sign for$ A_{3/2}^{\exp } $ in each case, we obtain the same value of the ratio of the corresponding nonfactorizable reduced matrix elements$ A_{1/2}^{nf} $ and$ A_{3/2}^{nf} $ , i.e.,$ \begin{gathered} \frac{{A_{1/2}^{nf}(\bar B \to \pi D)}}{{A_{3/2}^{nf}(\bar B \to \pi D)}} = \frac{{A_{1/2}^{nf}(\bar B \to \rho D)}}{{A_{3/2}^{nf}(\bar B \to \rho D)}}\, = \frac{{A_{1/2}^{nf}(\bar B \to \pi {D^*})}}{{A_{3/2}^{nf}(\bar B \to \pi {D^*})}}\quad , \hfill \\ 0.229 \pm 0.042\;\;\;\;\;\;\;0.200 \pm 0.096\;\;\;\quad \;0.211 \pm 0.109 \\[-10pt] \end{gathered} $
(52) and note that
$ A_{1/2}^{nf} $ has a negative sign for the cases$ A_{1/2}^{nf}(\bar B \to \pi D) = - \left( {0.572 \pm 0.105} \right)\;{\rm Ge{V^3}}, $
(53) $ A_{1/2}^{nf}(\bar B \to \rho D) = - \left( {0.054 \pm 0.026} \right)\;{\rm Ge{V^2}}, $
(54) $ A_{1/2}^{nf}(\bar B \to \pi {D^ * }) = - \left( {0.090 \pm 0.046} \right)\;{\rm Ge{V^2}}. $
(55) We can generically predict the sum of the branching fractions of the
$ {\bar B^0} - $ meson decays in the respective modes considered here as$ \begin{aligned}[b]&{B_{ - \, + }} + {B_{0\,0}}\\=& \frac{{{\tau _{{{\bar B}^0}}}}}{{3{\tau _{{B^ - }}}}}{B_{0 - }}\left[ {1 + {{\left\{ {\alpha + \frac{{\left( {\sqrt 2 - \alpha } \right)A_{ - + }^f - \left( {1 + \sqrt 2 \alpha } \right)A_{00}^f}}{{{A_{0 - }}}}} \right\}}^2}} \right], \end{aligned}$
(56) where
$ \alpha $ has been defined previously (27), and the experimental decay amplitude of the$ {B^ - } $ decays is$ {A_{0 - }} = \sqrt {\frac{{{B_{0 - }}}}{{{\tau _{{B^ - }}} \times \left( {\rm kinematic\;factor} \right)}}} \;, $
where the subscripts -+, 00, and 0- denote the charge states of the non-charm and charm mesons emitted in each case. Taking the average value of α = 0.22, we predict
$ \begin{aligned}[b] B\left( {{{\bar B}^0} \to {\pi ^ - }{D^ + }} \right) + B\left( {{{\bar B}^0} \to {\pi ^0}{D^0}} \right) =& (0.28 \pm 0.02){\text{%}}\quad {\rm Theo}, \\ = &(0.28 \pm 0.01){\text{%}}\quad {\rm Expt}; \end{aligned} $
(57) $ \begin{aligned}[b] B\left( {{{\bar B}^0} \to {\rho ^ - }{D^ + }} \right) + B\left( {{{\bar B}^0} \to {\rho ^0}{D^0}} \right) =& (0.76 \pm 0.13){\text{%}}\quad {\rm Theo}, \\ =& (0.79 \pm 0.12){\text{%}}\quad {\rm Expt}; \end{aligned} $
(58) $ \begin{aligned}[b] B\left( {{{\bar B}^0} \to {\pi ^ - }{D^{* + }}} \right) + B\left( {{{\bar B}^0} \to {\pi ^0}{D^{*0}}} \right) =& (0.29 \pm 0.04){\text{%}}\quad {\rm Theo}, \\ =& (0.30 \pm 0.01){\text{%}}\quad {\rm Expt}; \\ \end{aligned} $
(59) which are in good agreement with the experiment. To show that this agreement is not coincidental and to study the sensitivity of the sum of the
$ {\bar B^0} $ branching fractions with the ratio α, we plot$\sum {B\left( {{{\bar B}^0} \to\rm decays} \right)}$ against α by treating it as a free parameter for all three cases, which are shown in Figs. 4, 5, and 6. Clearly, the experiment data indicate α = 0.22 consistently. The broken curves represent the errors due to the decay constant, form factors, and branching fractions. The horizontal lines correspond to the experimental value of the sum, and its errors are indicated by broken lines.Figure 4. (color online) Variation in the sum of
$ B\left( {{{\bar B}^0} \to {\pi ^ - }{D^ + }} \right) $ and$ B\left( {{{\bar B}^0} \to {\pi ^0}{D^0}} \right) $ with the ratio$ \alpha = {{A_{1/2}^{nf}} \mathord{\left/ {\vphantom {{A_{1/2}^{nf}} {A_{3/2}^{nf}}}} \right. } {A_{3/2}^{nf}}}. $ Figure 5. (color online) Variation in the sum of
$ B\left( {{{\bar B}^0} \to {\rho ^ - }{D^ + }} \right) $ and$ B\left( {{{\bar B}^0} \to {\rho ^0}{D^0}} \right) $ with the ratio$ \alpha = {{A_{1/2}^{nf}} \mathord{\left/ {\vphantom {{A_{1/2}^{nf}} {A_{3/2}^{nf}}}} \right. } {A_{3/2}^{nf}}}. $ Figure 6. (color online) Variation in the sum of
$ B\left( {{{\bar B}^0} \to {\pi ^ - }{D^{* + }}} \right) $ and$ B\left( {{{\bar B}^0} \to {\pi ^0}{D^{*{\kern 1pt} 0}}} \right) $ with the ratio$ \alpha = {{A_{1/2}^{nf}} \mathord{\left/ {\vphantom {{A_{1/2}^{nf}} {A_{3/2}^{nf}}}} \right. } {A_{3/2}^{nf}}}. $ We wish to remark that similar observations have also been made in the FAT approach [23] analysis used for B- meson decays, which separates the factorizable and nonfactorizable contributions in each topological quark level diagram. The most important result in this approach is that the non-perturbative parameters
$ {\chi ^{C,{\kern 1pt} E}} $ and$ {\varphi ^{C,{\kern 1pt} E}} $ , representing the nonfactorizable contributions, are found to be universal for all the$ \bar B \to \pi D/\rho D/\pi {D^*} $ decay modes, which is consistent with the systematics recognized in our analysis.
Searching systematics for nonfactorizable contributions to $ {{\boldsymbol B^ - }} $ and $ {\bar {\boldsymbol B}^{\bf 0}}$ hadronic decays
- Received Date: 2021-11-23
- Available Online: 2022-07-15
Abstract: The two-body weak decays