& \simeq & \int_0^{\sqrt{s}} dm ~\rho(m) ~\sigma(m)\end{aligned}$$ approximating the discrete (but closely-spaced) tower of states by a continuum. The density of states is given by[@HLZ] $$\rho(m) = \frac{2~R_c^d ~m^{d-1}}{(4\pi)^{d/2} ~\Gamma(d/2)} \ ,$$ and the integration is cut off at the kinematic limit $\sqrt{s}$.

As the calculation, even using HELAS, is long and cumbersome, some checks on the numerical results are called for. The most useful check is provided by the Ward identities arising from general coordinate invariance, which constitute an essential feature of any theory involving gravity. We can write the amplitude for the emission of any graviton in the form $$M_n(\lambda_1, \lambda_2) 
= T^{\mu\nu}(\lambda_1, \lambda_2) ~\epsilon^{(n)*}_{\mu\nu}(p_1)$$ where $\epsilon^{(n)}_{\mu\nu}(p_1)$ is the polarization tensor for the $n$th (massive) graviton mode. The tensor $T^{\mu\nu}(\lambda_1, \lambda_2)$ is [*the same for every mode*]{}, including the massless mode $\epsilon^{(0)}_{\mu\nu}(p_1)$, which is the usual graviton of four-dimensional Einstein gravity. This must now satisfy the Ward identities $$p_1^\mu T_{\mu\nu}(\lambda_1,\lambda_2)
= p_1^\nu T_{\mu\nu}(\lambda_1,\lambda_2) = 0 \ ,$$ where we note that $$\begin{aligned}
T^{\mu\nu}(\lambda_1, \lambda_2) = \sum_{i = 1}^{28}
T_i^{\mu\nu}(\lambda_1, \lambda_2)\end{aligned}$$ with $i$ indicating the $i$th diagram, as above. The consistency check therefore requires a perfect cancellation, for each choice of $\lambda_1$ and $\lambda_2$, between all such terms[^1], which 