(0,15)\[l\][$e^-$]{} (0,70)\[l\][$e^+$]{} (45,25)\[c\][$\gamma ,Z$]{} (100,15)\[r\][$e^+$]{} (100,70)\[r\][$e^-$]{} (105,40)\[r\][$G$]{} (45,8)\[c\][$(c)$]{}

Figure 1. [*Representative Feynman diagrams for the process $e^+ e^- \to  e^+ e^- G_n$.*]{}

Considering the process $$e^-(k_1,\lambda_1) ~e^+(k_2,\lambda_2) \to  G_n(p_1) ~e^+(p_2) ~e^-(p_3)$$ with a single Kaluza-Klein mode $G_n(p_1)$ in the final state, we obtain the unpolarized cross-section $$\sigma(m_n) = \frac{1}{2s} \int 
\frac{d^3\vec{p_1}}{(2\pi)^3 ~2E_1}  
~\frac{d^3\vec{p_2}}{(2\pi)^3 ~2E_2}
~\frac{d^3\vec{p_3}}{(2\pi)^3 ~2E_3} 
~\delta^4(k_1 + k_2 - p_1 - p_2 - p_3)
~~\frac{1}{4}\sum_{\lambda_1, \lambda_2} |M_n(\lambda_1, \lambda_2)|^2
\label{cross}$$ where the final state helicities are summed over. This cross-section is a function of the mass $m_n$ of the graviton excitation.

The squared matrix element $|M_n(\lambda_1, \lambda_2)|^2$ for the process can be written as the sum $$|M_n(\lambda_1, \lambda_2)|^2 = 
\left|\sum_{i = 1}^{28} M_n^{(i)}(\lambda_1, \lambda_2)\right|^2$$ where $|M_n^{(i)}(\lambda_1, \lambda_2)|^2$ represents the squared helicity amplitude arising from the $i$th diagram. In our calculation, we make use of the helicity amplitude technique to write down amplitudes for all these 28 diagrams and evaluate the individual terms in the above sum using the subroutine HELAS[@Hagiwara].

Recalling that the observed final state, namely, $e^+ e^-$ plus missing energy, consists of an incoherent sum over the tower of graviton modes, we get $$\begin{aligned}
\sigma(e^+ e^- \to e^+ e^- + \not{\!\!E}) 
& = & \sum_n \sigma(m_n) \nonumber \\
