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The Sun Also Rises: the HighTemperature Expansion of the Thermal Sunset
by Andreas Ekstedt, Johan Löfgren
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Submission summary
As Contributors:  Andreas Ekstedt · Johan Löfgren 
Arxiv Link:  https://arxiv.org/abs/2006.02179v2 (pdf) 
Date submitted:  20200625 02:00 
Submitted by:  Löfgren, Johan 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We give a prescription for calculating the hightemperature 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 nonzero external momentum. We verify the results with sundry analytical and numerical crosschecks. Intermediate steps involve integrals over three Bessel functions.
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Anonymous Report 1 on 2020718 Invited Report
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In this paper thermal 2loop sumintegrals are considered. Going to
the hightemperature 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 hepph/9408276,
hepph/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 integrationbyparts (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
hightemperature sumintegrals has been on the 3loop level since more
than 25 years. Given that the current paper has its sole focus on
technical aspects at the lower 2loop 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 (nonconvergent in a mathematical
sense). Perhaps the authors could explain why the expansion might
nevertheless be helpful?
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'.