# The Sun Also Rises: the High-Temperature Expansion of the Thermal Sunset

### Submission summary

 As Contributors: Andreas Ekstedt · Johan Löfgren Arxiv Link: https://arxiv.org/abs/2006.02179v2 (pdf) Date submitted: 2020-06-25 02:00 Submitted by: Löfgren, Johan Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory High-Energy Physics - Phenomenology Approaches: Theoretical, Computational

### Abstract

We give a prescription for calculating the high-temperature expansion of the thermal sunset integral to arbitrary order. Results up to $\mathcal{O}(T^0)$ are given for both bosonic and fermionic thermal sunsets in dimensional regularisation, and for all odd powers of $T$ up to order $\epsilon^0$. The methods used generalize to non-zero external momentum. We verify the results with sundry analytical and numerical cross-checks. Intermediate steps involve integrals over three Bessel functions.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission 2006.02179v3 on 15 September 2020

Submission 2006.02179v2 on 25 June 2020

## Reports on this Submission

### Report

In this paper thermal 2-loop sum-integrals are considered. Going to
the high-temperature limit, the authors derive subleading terms in an
expansion in masses over the temperature.

As far as I can see, the technical computations look sound, and I have
no immediate reason to doubt their correctness.

A much bigger worry is whether the results are novel and represent the
state of the art. Usually, this kind of expressions are reported in an
appendix of a paper whose main focus is on a physics application, so
it is not easy to carry out a comprehensive literature scan. However,
by rapidly checking who has cited the classic hep-ph/9408276,
hep-ph/9410360 by Arnold and Zhai in recent years, I located eq.(A.22)
of 1911.09123, which precedes what the authors claim as a new result
on the last line of their (2.14). Even more importantly, from (A.6) of
1911.09123, I infer that such terms can be given in $d$ dimensions in
closed form, after making use of integration-by-parts (IBP) identities,
so it looks that the authors have missed this modern tool of choice.

Related to the above, the state of the art of massless
high-temperature sum-integrals has been on the 3-loop level since more
than 25 years. Given that the current paper has its sole focus on
technical aspects at the lower 2-loop level, I think that pioneering
references, like Arnold and Zhai, or more recent works by Schroder
[e.g. 1207.5666 and references therein], who introduced IBP
tools for this problem, should be mentioned for context.

On a conceptual note, figs. 1 and 2 suggest that the mass expansion
considered is an 'asymptotic' one (non-convergent in a mathematical
sense). Perhaps the authors could explain why the expansion might

Finally, the presentation seems rather careless, with a silly title,
many incomplete sentences (without a verb), overlong lines like the
one above (3.22), colloquial wordings, etc. The authors would be well
advised to try and render their presentation somewhat more 'scientific'.

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