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A Symbol and Coaction for Higher-Loop Sunrise Integrals

by Andreas Forum, Matt von Hippel

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Submission summary

Authors (as registered SciPost users): Matt von Hippel
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-06-15
Date submitted: 2023-05-15 08:46
Submitted by: von Hippel, Matt
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • High-Energy Physics - Phenomenology
Approach: Theoretical


We construct a symbol and coaction for $l$-loop sunrise integrals, both for the equal-mass and generic-mass cases. These constitute the first concrete examples of symbols and coactions for integrals involving Calabi-Yau threefolds and higher. In order to achieve a symbol of finite length, we recast the differential equations satisfied by the master integrals of this topology in the form of a unipotent differential equation. We augment the basis of master integrals in a natural way by including ratios of maximal cuts $\tau_i$. We discuss the relationship of this construction to constructions of symbols and coactions for multiple polylogarithms and elliptic multiple polylogarithms, in particular its connection to notions of transcendental weight.

Author comments upon resubmission

As recommended by the editor and referees, we resubmit with the minor changes requested.

List of changes

We have added additional discussion regarding the relationship of this notion of the symbol to others, in particular regarding its length, clarifying in particular which aspects relate to a choice of basis of functions and which to the division between semi-simple and unipotent functions. This discussion appears in the Conclusions and at the end of section 3, and references the subsequent work mentioned by the second referee. We believe that this expanded discussion should address both referees' concerns regarding motivation of the formalism and comparison to the existing literature and related formalisms, including point (5) of the second referee.

We have addressed the typos mentioned in the second referee report as (1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17), two of which were also mentioned in the first referee report as (1, 4). In addition we have corrected several cases of missing underlines on vector quantities.

We have addressed comment (2) of the first referee and comment (2) of the second referee by adding a definition of the intersection matrix after eq. (33). As recommended by the second referee, we have added a footnote on page 9 to mitigate confusion of notation.

We have addressed comment (3) of the first referee by clarifying the origin of the points made at the end of section 3.1.

We have included the references suggested in comment (6) of the first referee in a footnote on page 7.

Published as SciPost Phys. Core 6, 050 (2023)

Reports on this Submission

Anonymous Report 2 on 2023-6-1 (Invited Report)


All points in my report have been addressed, my opinion is that the paper can be published in present form on Scipost.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2023-5-30 (Invited Report)


The comments and requirement for changes have been implemented.

  • validity: good
  • significance: good
  • originality: high
  • clarity: top
  • formatting: excellent
  • grammar: perfect

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