Studying the Fourth Generation Quark Contributions to the Double Charm Decays $B_{(s)} \to D_{(s)}^{(*)} D_{s}^{(*)}$

Abstract

Almost all branching ratios and longitudinal polarization fractions of the double charm decays $B_{(s)} \to D_{(s)}^{(*)} D_{s}^{(*)}$ have been measured, and the experimental central value of $f_{L}({B^{0}_{s}}\to D^{*+}_{s}D^{*-}_{s})$ is quite small comparing to its Standard Model prediction. We study the fourth generation quark contributions to the double charm decays $B_{(s)} \to D_{(s)}^{(*)} D_{s}^{(*)}$. We find that the loop diagrams involving the fourth generation quark t′ have great effects on all branching ratios and CP asymmetries, which are very sensitive to the fourth generation parameter $\lambda ^{s}_{t^{\prime }}$ and $\phi _{t^{\prime }}$. Nevertheless, the experimental measurements of all branching ratios can not give effective constraints on relevant new physics parameters. In addition, they have no obvious effect on the relevant polarization fractions. These results could be used to search for the fourth heavy quark t′ via its indirect manifestations in loop diagrams.

Appendices

Appendix A: FF1

The form factors calculated in the framework of Heavy Quark Effective Theory [39, 40].

The form factors of B → D transition are [40]

$$\begin{array}{@{}rcl@{}} F_{1}(q^{2})&=&\frac{G_{1}(\omega)}{R_{D}},\\ F_{0}(q^{2})&=&R_{D}\frac{1+\omega}{2}G_{1}(\omega)\frac{1+r}{1-r}{\Delta} (\omega), \end{array} $$

(14)

with $R_{D^{(*)}}=2\sqrt {m_{B}m_{D^{(*)}}}/(m_{B}+m_{D^{(*)}})$, $r=m_{D^{(*)}}/m_{B}$ and $\omega =({m_{B}^{2}}+m_{D^{(*)}}^{2}-q^{2})/(2m_{B}m_{D^{(*)}})$, therefore, we take Δ(ω) = 0.46 ± 0.02 [44], and

$$\begin{array}{@{}rcl@{}} G_{1}(\omega)&=&G_{1}(1)\left[1-8{\rho_{1}^{2}}z(\omega)+(51{\rho^{2}_{1}}-10)z(\omega)^{2}-(252{\rho^{2}_{1}}-84)z(\omega)^{3}\right], \end{array} $$

(15)

with $z(\omega ) = (\sqrt {\omega +1}-\sqrt {2})/(\sqrt {\omega +1}+\sqrt {2})$ [39] and ${\rho _{1}^{2}}=1.186\pm 0.055$ [45].

The form factors of B → D ^∗ transition are [39]

$$\begin{array}{@{}rcl@{}} V(q^{2})&=&\frac{R_{1}(\omega)}{R_{D^{*}}}h_{A_{1}}(\omega),\\ A_{1}(q^{2})&=&R_{D^{*}}\frac{\omega+1}{2}h_{A_{1}}(\omega),\\ A_{2}(q^{2})&=&\frac{R_{2}(\omega)}{R_{D^{*}}}h_{A_{1}}(\omega),\\ A_{0}(q^{2})&=&\frac{R_{0}(\omega)}{R_{D^{*}}}h_{A_{1}}(\omega),\\ A_{3}(q^{2})&=&\frac{m_{B}+m_{D^{*}}}{2m_{D^{*}}}A_{1}(q^{2})-\frac{m_{B}-m_{D^{*}}}{2m_{D^{*}}}A_{2}(q^{2}), \end{array} $$

(16)

with

$$\begin{array}{@{}rcl@{}} h_{A_{1}}(\omega)&=&h_{A_{1}}(1)\left[1-8\rho^{2}z(\omega)+(53\rho^{2}-15)z(\omega)^{2}-(231\rho^{2}-91)z(\omega)^{3}\right],\\ R_{0}(\omega)&=&R_{0}(1)-0.11(\omega-1)+0.01(\omega-1)^{2},\\ R_{1}(\omega)&=&R_{1}(1)-0.12(\omega-1)+0.05(\omega-1)^{2},\\ R_{2}(\omega)&=&R_{2}(1)-0.11(\omega-1)-0.06(\omega-1)^{2}. \end{array} $$

(17)

For the free parameters, we take $h_{A_{1}}(1)|V_{cb}|=(35.90\pm 0.45)\times 10^{-3}$, ρ ²=1.207±0.026, R ₁(1) = 1.403±0.033 and R ₂(1) = 0.854±0.020 from [45], in addition, R ₃(1) = 0.97±0.10 taken from [46] and $R_{0}(1) = \frac {[R_{3}(1)(1-r)^{2}-R_{2}(1)(1-r)]/r+2}{(1+r)}$ taken from the HQET prediction for the linear combination [40, 42, 47].

Appendix B: FF2

The form factors which include perturbative QCD corrections induced by hard gluon vertex corrections of b → c transitions and power corrections in orders of 1/m _{b, c} [41, 42].

$$\begin{array}{@{}rcl@{}} F_{1}(q^{2})&=&\frac{m_{B}+m_{D}}{2\sqrt{m_{B}m_{D}}}\left[\xi_{+}(\omega)-\frac{m_{B}-m_{D}}{m_{B}+m_{D}}\xi_{-}(\omega)\right],\\ F_{0}(q^{2})&=&\frac{m_{B}+m_{D}}{2\sqrt{m_{B}m_{D}}}\left[1-\frac{q^{2}}{(m_{B}+m_{D})^{2}}\right]\left\{\xi_{+}(\omega)-\frac{m_{B}+m_{D}}{m_{B}-m_{D}}\left( \frac{\omega-1}{\omega+1}\right)\xi_{-}(\omega)\right\},\\ V(q^{2})&=&\frac{m_{B}+m_{D^{*}}}{2\sqrt{m_{B}m_{D^{*}}}} \xi_{V}(\omega),\\ A_{1}(q^{2})&=&\frac{m_{B}+m_{D^{*}}}{2\sqrt{m_{B}m_{D^{*}}}}\left[1-\frac{q^{2}}{(m_{B}+m_{D^{*}})^{2}}\right]\xi_{A_{1}}(\omega),\\ A_{2}(q^{2})&=&\frac{m_{B}+m_{D^{*}}}{2\sqrt{m_{B}m_{D^{*}}}}\left[\xi_{A_{3}}(\omega)+\frac{m_{D^{*}}}{m_{B}}\xi_{A_{2}}(\omega)\right],\\ A_{3}(q^{2})&=&\frac{m_{B}+m_{D^{*}}}{2\sqrt{m_{B}m_{D^{*}}}}\left\{\frac{m_{B}(\omega+1)}{m_{B}+m_{D^{*}}}\xi_{A_{1}}(\omega)-\frac{m_{B}-m_{D^{*}}}{2m_{D^{*}}}\left[\xi_{A_{3}}(\omega)+\frac{m_{D^{*}}}{m_{B}}\xi_{A_{2}}(\omega)\right]\right\},\\ A_{0}(q^{2})&=&A_{3}(q^{2})+\frac{q^{2}}{4m_{B}m_{D^{*}}}\sqrt{\frac{m_{B}}{m_{D^{*}}}}\left[\xi_{A_{3}}(\omega)-\frac{m_{D^{*}}}{m_{B}}\xi_{A_{2}}(\omega)\right], \end{array} $$

(18)

with

$$\begin{array}{@{}rcl@{}} &&\xi_{i}(x) = (\alpha_{i}+\beta_{i}(m_{b},m_{c},x)+\gamma_{i}(x))\xi(x), \end{array} $$

(19)

$$\begin{array}{@{}rcl@{}} &&\alpha_{+,V,A_{1},A_{3}}=1, \alpha_{-,A_{2}}=0, \end{array} $$

(20)

$$\begin{array}{@{}rcl@{}} &&\beta_{+}=\hat{C}_{1}-\frac{x+1}{2}(\hat{C}_{2}+\hat{C}_{3})-1,\\ &&\beta_{-}=-\frac{x+1}{2}(\hat{C}_{2}-\hat{C}_{3}),\\ &&\beta_{V}=\hat{C}_{1}-1,\\ &&\beta_{A_{1}}=\hat{C}_{1}^{5}-1,\\ &&\beta_{A_{2}}=-\hat{C}_{2}^{5},\\ &&\beta_{A_{3}}=\hat{C}_{1}^{5}-\hat{C}_{3}^{5}-1, \end{array} $$

(21)

$$\begin{array}{@{}rcl@{}} &&\gamma_{+}(x) = \left( \frac{1}{m_{c}}+\frac{1}{m_{b}}\right)\rho_{1}(x),\\ &&\gamma_{-}(x) = \left( \frac{1}{m_{c}}-\frac{1}{m_{b}}\right)\left[-\frac{\bar{\Lambda}}{2}+\rho_{4}(x)\right],\\ &&\gamma_{V}(x) = \frac{\bar{\Lambda}}{2}\left( \frac{1}{m_{c}}+\frac{1}{m_{b}}\right)+\frac{1}{m_{c}}\rho_{2}(x)+\frac{1}{m_{b}}\left[\rho_{1}(x)-\rho_{4}(x)\right],\\ &&\gamma_{A_{1}}(x) = \frac{\bar{\Lambda}}{2}\frac{y-1}{y+1}\left( \frac{1}{m_{c}}+\frac{1}{m_{b}}\right)+\frac{1}{m_{c}}\rho_{2}(x)+\frac{1}{m_{b}}\left[\rho_{1}(x)-\frac{y-1}{y+1}\rho_{4}(x)\right],\\ \end{array} $$

$$\begin{array}{@{}rcl@{}} &&\gamma_{A_{2}}(x) = \frac{1}{y+1}\frac{1}{m_{c}}\left[-\bar{\Lambda}+(y+1)\rho_{3}(x)-\rho_{4}(x)\right],\\ &&\gamma_{A_{3}}(x) = \frac{\bar{\Lambda}}{2}\left( \frac{y-1}{y+1}\frac{1}{m_{c}}+\frac{1}{m_{b}}\right)+\frac{1}{m_{c}}\left[\rho_{2}(x)-\rho_{3}(x)-\frac{1}{y+1}\rho_{4}(x)\right]+\frac{1}{m_{b}}\left[\rho_{1}(x)-\rho_{4}(x)\right],\\ \end{array} $$

(22)

where Isgur-Wise function ξ(x) = 1−1.22(x−1)+0.85(x−1)² taken from Ref. [43], in addition, $\hat {C}_{i}$, ρ _i (x) and $\bar {\Lambda }$ can be found in Ref. [41].