and u are all large. Sect. 6 deals with the expansion of the invariant amplitudes into the perturbative series in order to extract the first-loop and the second-loop contributions. Then we compare these contributions to the analogous terms obtained when one universal cut-off is used and study their difference. The effect of high-order contributions in the two approaches is further studied in Sect. 7 where the asymptotic expressions of the amplitudes are compared. Finally, Sect. 8 contains our concluding remarks.

Invariant amplitudes for the annihilation processes {#ELAST}
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Let us consider a general process where the lepton $l^k(p_1)$ and its anti-particle $\bar{l}_i(p_2)$ annihilate into a quark or a lepton $q ^{k'}(p'_1)$ and its anti-particle $\bar{q}_{i'}({p'}_2)$ (see Fig. 1): $$\label{annih}
l^k(p_1)\bar{l}_i(p_2) \to q^{k'}({p'}_1) \bar{q}_{i'}({p'}_2) ~.$$

![Scattering amplitude of the annihilation of Eq. 3.[]{data-label="fig.T1"}](Ampfig1.epsi)

For this process, the most complicate case occurs when both the initial and the final particles (anti-particles) are left-handed (right-handed). The scattering amplitudes for other helicities can be obtained easily from the formulae derived for this case. As there is no technical difference when considering the annihilation into quarks or leptons, we present parallel results for the annihilation into a quark-antiquark or a lepton-antilepton pair. According to our 