Fourth generation bound states

We investigate the spectrum and wave functions of ${\overline{q}}^{\prime }{q}^{\prime }$ bound states for heavy fourth generation quarks ($q^{\prime }$) that have a very small mixing with the three observed generations of standard model quarks. Such bound states come with different color, spin and flavor quantum numbers. Since the fourth generation Yukawa coupling, ${\lambda }_{q^{\prime }}$, is large we include all perturbative corrections to the potential between the heavy quark and antiquark of order ${\lambda }_{q^{\prime }}^2{N}_c/16{\pi }^2$ where $N_c$ is the number of colors, as well as relativistic corrections suppressed by $(v/c{)}^2$. We find that the lightest fourth generation quark masses for which a bound state exists for color octet states. For the color singlet states, which always have a bound state, we analyze the influence that the Higgs couplings have on the size and binding energy of the bound states.

Figure 1

Diagrams of Higgs self-energy (left) and $W$ boson vacuum polarization (right).

Figure 2

$t$-channel Higgs exchange.

Figure 3

$s$- and $t$-channel neutral fictitious scalar exchange.

Figure 4

$t$-channel charged fictitious scalar exchange.

Figure 5

Gluon exchange.

Figure 6

(i) $a_0/a_{\mathrm{QCD}}$ as the function of quark mass in color singlet channels. Here we take $m_h=130\mathrm{GeV}$ and $a_0$ is the value for which $E[a]$ is minimized (and negative) for fixed quark mass. (ii) Variational binding energy of color singlet channels as the function of the heavy quark mass. Here we set $a=a_0$.

Figure 7

Variational binding energy of color octet channels plotted as the function of the heavy quark mass. In the plot, we use $m_h=130\mathrm{GeV}$ and take $a$ as the value which gives the lowest binding energy for fixed $m_{q^{\prime }}$. Note that $(\mathbf{8},0,1)$ and $(\mathbf{8},1,0)$ channels give almost the same results.

Figure 8

The same as Fig. 6, except for taking $m_h=m_{q^{\prime }}$.

Figure 9

The same as Fig. 7, except for taking $m_h=m_{q^{\prime }}$.