Abstract
Models explaining dark matter typically include interactions with charged scalar and fermion fields. The infra-red (IR) finiteness of thermal field theories of charged fermions (fermionic QED) has been proven to all orders in perturbation theory. Here we reexamine the IR behaviour of charged scalar theories at finite temperature. Using the method of Grammer and Yennie, we identify and factorise the infra-red divergences to all orders in perturbation theory. The inclusion of IR finite pieces arising from the 4-point interaction terms of scalars with photon fields is key to the exponentiation. We use this in a companion paper to prove the IR finiteness of the corresponding thermal theory which is of relevance in dark matter calculations.
Highlights
At zero temperature, Bloch and Nordsieck [1] were among the first to study the infra-red (IR) behaviour of fermionic QED
The terms that depend on the kk K photon momenta are combined with the factor for every K insertion
The IR finiteness of pure scalar QED at finite temperature was explicitly shown here to all orders in perturbation theory using the technique of Grammer and Yennie
Summary
Bloch and Nordsieck [1] were among the first to study the infra-red (IR) behaviour of fermionic QED. The infra-red finiteness of such thermal QED with purely charged fermions has been shown [12,13] to all orders in the theory Both absorption and emission of photons with respect to the heat bath are required [12,14] in order to cancel the linear divergences as well as the logarithmic subdivergences. We show that the IR divergent parts will cancel between the virtual and real diagrams, that is, between the K and K contributions, when they are added, order by order, in the theory This is achieved only when both real photon emission into, and absorption from, the heat bath is taken into account. Details of the result for the insertion of a virtual G photon in all possible ways into an nth order graph is found in Appendix D; it is shown that all such virtual G photon contributions are IR finite
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