the t -channel, shown in Fig. (A). The cross-section for these scatterings is ∼ α | t | − $^{1}$. Here ′′ t ′′ is related to the exchanged energy, Δ E , and the momentum, −→ Δ p , through t = Δ E 2 − | −→ Δ p |$^{2}$ . The fine structure constant is denoted by α (note that α ≥ 10 − 2 in the SM/MSSM). This cross section can be understood as follows: the gauge boson propagator introduces a factor of | t | − $^{2}$, while phase space integration results in an extra factor of | t | . Scalar exchange in t -channel diagrams are usually suppressed, similarly a fermion-fermion-scalar vertex, which arises from a Yukawa coupling, flips the chirality of the scattered fermion, are also suppressed. Due to an infrared singularity, these scatterings are very efficient even in a dilute plasma [120, 122]. • Chemical equilibrium: In addition one also needs to achieve chemical equilibrium by changing the number of particles in the reheat plasma. The relative chemical equilibrium among different degrees of freedom is built through 2 → 2 annihilation processes, occurring through s − channel diagrams. Hence they have a much smaller cross-section ∼ αs − $^{1}$. More importantly the total number of particles in the plasma must also change. It turns out from Eq. (375) that in order to reach full equilibrium, the total number of particles must increase by a factor of: n$_{eq}$/n , where n ≈ ρ/m$_{φ}$ and the equilibrium value is: n$_{eq}$ ∼ ρ 3 / $^{4}$. This can be a very large number, i.e. n$_{eq}$/n ∼ O (10 $^{3}$). It was recognized in [120, 121], see also [748–750], that the most relevant processes are 2 → 3 scatterings with gauge-boson exchange in the t − channel. Again the key issue is the infrared singularity of such diagrams shown in Fig. (A). The cross-section for emitting a gauge boson, whose energy is | t | 1 / 2 ≪ E , from the scattering of two fermions is ∼ α $^{3}$| t | − $^{1}$. When these inelastic scatterings become efficient, i.e., their rate exceeds the Hubble expansion rate, the number of particles increases very rapidly [751], because the produced gauge bosons subsequently participate in similar 2 → 3 scatterings. Decays (which have been considered in [750]) are helpful, but in general they cannot increase the number of particles to the required level. The full thermal equilibrium will be established shortly after the 2 → 3 scatterings become efficient. For this reason, to a very good approximation, one can use the rate for 174