$CP$ violation and the fourth generation

Within the standard model with the 4th generation quarks $b^{\prime }$ and $t^{\prime }$ we have analyzed $CP$-violating flavor changing neutral current processes $t\to cX$, $b^{\prime }\to sX$, $b^{\prime }\to bX$, $t^{\prime }\to cX$, and $t^{\prime }\to tX$, with $X=H,Z,\gamma ,g$, by constructing and employing a global, unique fit for the 4th generation mass mixing matrix (CKM4) at $300\le m_{t^{\prime }}\le 700\mathrm{GeV}$. All quantities appearing in the CKM4 were subject to our fitting procedure. We have found that our fit produces the following $CP$ partial rate asymmetry dominance: $a_{CP}(b^{\prime }\to s(H,Z;\gamma ,g))\simeq (94,62;47,41)\%$, at $m_{t^{\prime }}\simeq 300,350\mathrm{GeV}$, respectively. From the experimental point of view the best decay mode, out of the above four, is certainly $b^{\prime }\to s\gamma$, due to the presence of the high energy single photon in the final state. We have also obtained relatively large asymmetry $a_{CP}(t\to cg)\simeq (8-18)\%$ for $t^{\prime }$ running in the loops. There are fair chances that the 4th generation quarks will be discovered at LHC and that some of their decay rates will be measured. If $b^{\prime }$ and $t^{\prime }$ exist at energies we assumed, with well executed tagging, large $a_{CP}$ could be found too.

Figure 1

Generic diagrams for FCNC decays of the 4th generation quarks. $X$ denotes possible decays to $X=H,Z,\gamma ,g$ and quarks running in the loops are $q_U=\{u,c,t,t^{\prime }\}$ and $q_D=\{d,s,b,b^{\prime }\}$. (a) $b^{\prime }$ decays and (b) $t^{\prime }$ decays.

Figure 2

Branching ratios for rare top decays in the model with 4th generation as a function of $m_{t^{\prime }}$, for $b^{\prime }$ of the mass $m_{b^{\prime }}=m_{t^{\prime }}-55\mathrm{GeV}$ running in the loops. $X$ denotes possible decays to $X=H$ (dashed line), $Z$ (solid line), $\gamma$ (dotted line), $g$ (dash-dotted line).

Figure 3

Branching ratios of $t^{\prime }\to (c,t)X$ as a function of $m_{t^{\prime }}=m_{b^{\prime }}+55\mathrm{GeV}$. $X$ denotes possible decays to $X=H$ (dashed line), $Z$ (solid line), $\gamma$ (dotted line), $g$ (dash-dotted line). (a) $\mathrm{BR}(t^{\prime }\to cX)$ and (b) $\mathrm{BR}(t^{\prime }\to tX)$.

Figure 4

Branching ratios of $b^{\prime }\to (s,b)X$ as a function of $m_{t^{\prime }}=m_{b^{\prime }}+55\mathrm{GeV}$. $X$ denotes possible decays to $X=H$ (dashed line), $Z$ (solid line), $\gamma$ (dotted line), $g$ (dash-dotted line). (a) $\mathrm{BR}(b^{\prime }\to sX)$ and (b) $\mathrm{BR}(b^{\prime }\to bX)$.

Figure 5

Fourth generation effect on the $a_{CP}$ of the rare top decays as a function of $m_{t^{\prime }}$, for $b^{\prime }$ of the mass $m_{b^{\prime }}=m_{t^{\prime }}-55\mathrm{GeV}$ running in the loops. $X$ denotes possible decays to $X=H$ (●), $Z$ (▴), $\gamma$ (■), $g$ (◆).

Figure 6

$CP$ asymmetries in $t^{\prime }\to (c,t)X$ decays as a function of $m_{t^{\prime }}=m_{b^{\prime }}+55\mathrm{GeV}$. $X$ denotes possible decays to $X=H$ (●), $Z$ (▴), $\gamma$ (■), $g$ (◆). (a) $a_{CP}(t^{\prime }\to cX)$ and (b) $a_{CP}(t^{\prime }\to tX)$.

Figure 7

$CP$ asymmetries in $b^{\prime }\to (s,b)X$ decays as a function of $m_{t^{\prime }}=m_{b^{\prime }}+55\mathrm{GeV}$. $X$ denotes possible decays to $X=H$ (●), $Z$ (▴), $\gamma$ (■), $g$ (◆). (a) $a_{CP}(b^{\prime }\to sX)$ and (b) $a_{CP}(b^{\prime }\to bX)$.