Abstract
In this paper, we suggest a method of a complete calculation of the Sommerfeld effect in both the s-wave and p-wave annihilation situations. The basic idea is to uniformly consider the equal-time Beth-Salpeter wave functions of both the dark matter and the final state particles, then short-distance interactions can be parametrized as a cross-term considering the equations. Solving these equations will result in the complete formulas of the cross sections with the Sommerfeld effects.
Highlights
If the dark matter particles interact with themselves through exchanging some bosonic light mediator, the annihilation behavior of a dark matter particle pair as a function of their relative velocity can be significantly altered
We try to regard both these interactions in a unified framework of wave functions in which both the Sommerfeld effect and the annihilation processes are all calculated by solving the equations of wave function
If we uniformly describe the annihilating dark matter and the final state particles in the form of Bethe-Salpeter wave functions, we can parametrize and write down the short-distance interaction terms between them in the wave function equations
Summary
If the dark matter particles interact with themselves through exchanging some bosonic light mediator, the annihilation behavior of a dark matter particle pair as a function of their relative velocity can be significantly altered. We try to regard both these interactions in a unified framework of wave functions in which both the Sommerfeld effect and the annihilation processes are all calculated by solving the equations of wave function. If we uniformly describe the annihilating dark matter and the final state particles in the form of Bethe-Salpeter wave functions, we can parametrize and write down the short-distance interaction terms between them in the wave function equations. We will prove in our paper that the Bethe-Salpeter wave function of the relative motion of a pair of particles in many cases satisfies the d’Alembert equation. We will solve these equations and extract the cross sections from the results. For L 1⁄4 1, S 1⁄4 0, J 1⁄4 1 dark matter annihilating to the L 1⁄4 1, S 1⁄4 0, J 1⁄4 1 final states, we have
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