This paper is available on arxiv under CC 4.0 license.

**Authors:**

(1) Nicolas Bernal, New York University Abu Dhabi;

(2) Partha Konar, Physical Research Laboratory;

(3) Sudipta Show, Physical Research Laboratory.

## Table of Links

- Abstract and Intro
- S-matrix: Unitarity and its Consequences
- Dark Matter Annihilation and Unitarity Bound
- Low-temperature Reheating
- Freeze-out with a Low-temperature Reheating
- Summary and Conclusion
- Acknowledgments and References

## 2. S-matrix: Unitarity and its Consequences

For the sake of simplicity, here we will focus on the collisions of nonidentical spin-0 particles. However, the case of other spins can be generalized simply by using Refs. [13, 114]. We carry out the basis transformation from the momentum eigenstates to a general basis state by using rotational invariance.

Now, we derive the total cross section, the sum of the elastic and inelastic cross sections, in the channel n with the help of the optical theorem in Eq. (2.1) and the general expression of the matrix element in Eq. (2.2)

Then, one can easily obtain the inelastic cross section by subtracting the elastic in Eq. (2.3) from the total cross sections in Eq. (2.4)

Finally, we note that for a collision among identical particles, an additional multiplicative factor of 2 in the cross section is required to avoid double counting [114].