## SciPost Submission Page

# The High-Temperature Expansion of the Thermal Sunset

### by Andreas Ekstedt, Johan Löfgren

### Submission summary

As Contributors: | Andreas Ekstedt · Johan Löfgren |

Arxiv Link: | https://arxiv.org/abs/2006.02179v3 (pdf) |

Date submitted: | 2020-09-15 16:13 |

Submitted by: | Löfgren, Johan |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | High-Energy Physics - Theory |

Approaches: | Theoretical, Computational |

### Abstract

We give a prescription for calculating the high-temperature expansion of the thermal sunset integral to arbitrary order. We derive all terms odd in $T$, and rederive previous results up to $\mathcal{O}(T^0) $ for both bosonic and fermionic thermal sunsets in dimensional regularisation. We perform analytical and numerical cross-checks. Intermediate steps involve integrals over three Bessel functions.

###### Current status:

Editor-in-charge assigned

### Author comments upon resubmission

We thank the referee for their comments. And the updated version of the paper takes into account their suggestions. As such we have included references to previous work dealing with IBP methods.

Though we were aware of IBP methods in general, we had set them aside because we did not need them. But in retrospect we agree with the referee's assessment that showing a holistic picture is desirable. We now discuss IBP methods, and in this updated version we have made it clear which parts have been calculated before, with appropriate references to the IBP literature.

Regarding the referee's further comments, we agree that the expansion is asymptotic. Some indications of this is that the integral depends on several scales, and that the expansion contains an infinite number of non-analytic terms. The expansion's usefulness stems from that, as with Feynman diagrams, it provides a good approximation for the first couple of terms. We now briefly comment on this in the paper.

Though we have disparate views from the referee as to what constitutes scientific writing, we have taken the referee's suggestions to heart and altered the text accordingly.

We again thank the referee for reading through the manuscript, and for giving insightful comments.

Though we were aware of IBP methods in general, we had set them aside because we did not need them. But in retrospect we agree with the referee's assessment that showing a holistic picture is desirable. We now discuss IBP methods, and in this updated version we have made it clear which parts have been calculated before, with appropriate references to the IBP literature.

Regarding the referee's further comments, we agree that the expansion is asymptotic. Some indications of this is that the integral depends on several scales, and that the expansion contains an infinite number of non-analytic terms. The expansion's usefulness stems from that, as with Feynman diagrams, it provides a good approximation for the first couple of terms. We now briefly comment on this in the paper.

Though we have disparate views from the referee as to what constitutes scientific writing, we have taken the referee's suggestions to heart and altered the text accordingly.

We again thank the referee for reading through the manuscript, and for giving insightful comments.

### List of changes

1. Included discussion of IBP methods, with references.

2. Altered the text to be more formal.

3. Provided brief discussion regarding asymptotic expansion.

### Submission & Refereeing History

Submission 2006.02179v2 on 25 June 2020