the virtual particle, and the gravitational self-energy $\Delta\cal E$ of the latter one yields $$\Delta E_1\approx-2\,\frac{m\,{\cal E}^2}{m_P^2}\approx
-8\,\frac{mp^2}{m_P^2}\left[1-2\frac{m}{p}\left(1+\frac{1}{2}\frac{p^2}{m^2}
\cos^2\frac{\vartheta_0}{2}\right)\right]$$ and $$2\Delta E_2=\Delta{\cal E}\approx-\frac{6}{5}\frac{{\cal
E}^3}{m_P^2}\approx-\frac{48}{5}\frac{p^3}{m_P^2}\left[1-3\frac{m}{p}\left(1+\frac{1}{2}
\frac{p^2}{m^2}\cos^2\frac{\vartheta_0}{2}\right)\right].$$

With the leading contributions of these corrections the effective energy of the virtual particle, ${\cal E}_{\rm eff}={\cal E}-2\Delta
E_1-\Delta\cal E$, becomes $$\label{aff}
{\cal E}_{\rm eff}\approx
2p\left[1+\frac{24}{5}\frac{p^2}{m_P^2}-\frac{4}{5}\frac{mp}{m_P^2}\left(
4+9\frac{p^2}{m^2}\cos^2\frac{\vartheta_0}{2}\right)\right].$$ The last two terms in parenthesis are of the same order, because $\cos^2\frac{\vartheta_0}{2}$ is of order $m^2/p^2$. Importantly, there is a leading order correction, quadratic in $p/m_P$, independent from the scattering angle, and a smaller one, of order $mp/m_P^2$, depending on $\vartheta_0$.

When gravity is again replaced by introducing the effective energy of the exchange particle, the wavelength of the latter one becomes in leading order (coming from $\Delta E_2$) $$\lambda\approx\frac{1}{{\cal E}_{\rm
eff}}\approx\frac{1}{2p\left(1+\beta\frac{p^2}{m_P^2}\right)},$$ where, for convenience, the fancy numerical factor $24/5$ is again replaced by $\beta$.

In the figure this means that the particle comes closer to the centre C, the vertex V is shifted a small distance to the left, so that the distance $\overline{\rm CV}$ becomes $\lambda/2$ instead of $\lambda_0/2$ and the scattering angle becomes modified from $\vartheta_0$ to $\vartheta$. From the relation $$\label{imp}
2b=\lambda_0\cos\frac{\vartheta_0}{2}=\lambda\cos\frac{\vartheta}{2}$$ we obtain the modification of the scattering angle $$\label{teta}
\cos\frac{\vartheta}{2}\approx\left(1+\beta\,\frac{p^2}{m_P^2}\right)
\cos\frac{\vartheta_0}{2}.$$ 