On $\varepsilon$-factorised bases and pure Feynman integrals

Adds interesting results to the understanding of mathematical structures in Feynman integrals

The concept of epsilon-factorised differential equations for master integrals, originally proposed by Henn in 2013, has catalyzed the analysis of the mathematical structure of Feynman integrals tremendously. While for rational weight functions which iteratively integrate to multiple (or Goncharov) polylogarithms, the situation is well understood, less is known in the case when ellptic polylogarithms are involved. The example for the latter case that is easiest to study is the two-loop equal-mass sunrise integral.

In the present paper, the authors investigate a couple of questions in the context of the connection between epsilon-factorised differential equations in the elliptic case and uniform-weight / purity properties of the Feynman integrals involved. The two-loop equal-mass sunrise integral, which has been investigated in great detail in the past (in particular by the authors), serves as an example. To this end, the authors consider three bases of master integrals, two of which (called J and K) result in epsilon-factorised differential equations. The first question the authors address is whether this implies that the masters in these two bases are of uniform weight. The second question is related to a definition of purity and logarithmic singularities. I consider the research in this paper timely and relevant. The paper addresses questions whose answers represent a clear step towards a better understanding of the mathematical structures of Feynman integrals.

The paper is organised as follows: After an introduction, the authors exemplify in section 2, using a toy integral, that an epsilon-factorised differential equation alone does not necessarily render a solution of uniform weight, but the boundary conditions have to satisfy this condition as well. Although in my opnion this property somewhere in between "well-known" and "trivial", it adds to the self-consistency of the paper. In section 3, the two-loop equal-mass sunrise integral is reviewed, including three bases of master integrals, the "Laporta basis" I, and the aforementioned bases J and K. This section builds on earlier findings in Refs. [9-11,21-23], but is required to introduce notation and to make the paper self-consistent. Section 4 discusses the differential equation in the three bases, their solutions, and in particular the relation between bases J and K. Part of the results in this section have also appeared earlier (e.g. the solution to $J_2$ was given in [10]), but it contains also new material. In section 5 the authors give a definition of "purity", and investigate whether, according to this definition, the solutions are pure. They find that for J, the given definition of purity works locally, but not globally. Moreover, it is investigated whether a modification of the definition of purity can remedy the situation, with negative outcome. In the context of revealing the mathematical structure of Feynman integrals, these are the most important and most interesting results of the paper. The authors conclude in section 6, and relegate supplemental material to two appendices.

To summarize, the paper contains enough new and interesting material to justify publication in Scipost. Besides the new material, the paper repeats also quite a number of results from earlier publications. However, I find this appropriate since it makes the paper self-contained. I therefore recommend to publish the paper after the following minor improvements have been implemented in the manuscript.

p. 2, 1st paragraph: ... modern techniques for computing Feynman integrals --> ... modern techniques for analytically computing Feynman integrals

p. 2, 4th paragraph: the the --> the

p. 4, between (3) and (4): Please specify that the boundary condition is taken in $x=1$.

p. 5, bottom: Please define $E(k)$ and $K(k)$, similar to eq. (105) in [10]

p. 2, 1st paragraph: ... modern techniques for computing Feynman integrals --> ... modern techniques for analytically computing Feynman integrals

p. 2, 4th paragraph: the the --> the

p. 4, between (3) and (4): Please specify that the boundary condition is taken in $x=1$.

p. 5, bottom: Please define $E(k)$ and $K(k)$, similar to eq. (105) in [10]

1. Interesting insights into an important and urgent problem in the computation of Feynman integrals.
2. Explicit examples.

1. Clarity.

In this work, the authors provide interesting insights into an important and urgent problem in the computation of Feynman integrals through the method of differential equations (DEs). I have no doubt this paper should be accepted for publication in SciPost, but in its current form I believe it would be understandable only to a very small part of the potentially interested readers. I recommend that the authors take into account the following comments to improve the clarity of the presentation.

1. In the introduction the authors say

"Functions of uniform weight are also called pure functions".

I find this confusing, because these two notions are - to my understanding - typically distinct in the literature (see e.g. the pioneering paper by Henn on this topic, https://arxiv.org/pdf/1304.1806.pdf). Purity is a stronger property than uniform weight, as it implies the absence of algebraic coefficients from uniform-weight functions. With this definition, for example, $x^{\epsilon}$ is a pure function, whereas $x^{\epsilon+1}$ has uniform weight but is not pure. My guess is that the authors are identifying both notions with that of purity, i.e. for them $x^{\epsilon+1}$ would not even have uniform weight. I suggest that the authors either adopt this distinction or at least say explicitly that they are not following the conventional definitions in this regard, to avoid the confusion in those readers who are already familiar with this topic.

2. In section 2, the authors present a toy example which illustrates that the canonical form of the DEs is not enough for the solution to be pure; the boundary values need to be pure as well. This has nothing to do with the presence of elliptic or more complicated functions, and may happen even for polylogarithmic solutions (as in their example). However, in this example and in the polylogarithmic cases in general, this situation arises only when the master integrals are pure up to an overall $\epsilon$-dependent normalisation. The two bases of elliptic master integrals considered here instead differ by a non-trivial rotation which depends also on the kinematics. The situation is therefore completely different from the toy example. A natural question therefore is whether also in the polylogarithmic case it is possible to have two sets of master integrals which both satisfy canonical DEs and which are related by a kinematic-dependent rotation. I recommend that the authors emphasise and comment on this important aspect, because - as is - the toy example is somewhat misleading, in the sense that it does not clarify why this peculiar situation may arise for elliptic integrals.

3. In section 3, eq. (7) defines the elliptic curve, which the authors say is "given by a quartic polynomial $P(u,v)$". It therefore appears that the quartic polynomial $P(u,v)$ is what's on the LHS of eq. (7). However, below they give the "roots of the quartic polynomial" in eq. (8), but these are the roots only of the second term of the $P(u,v)$ given in eq. (7), i.e. the part which does not depend on v. I believe they are talking of two quartic polynomials, and they should clarify this.

4. In eq. (12) they introduce the functions $E$ and $K$ without definition. They should spell out that they are the complete elliptic integrals.

5. In section 4.2, below eq. (32), the authors introduce for the first time the congruence subgroup $\Gamma_1(6)$, without defining it. This makes it impossible for anyone who's not already familiar with the technical details of this topic to follow the discussion hereon. The notion of congruence subgroup and in general the properties of modular forms are in fact crucial for the rest of the paper, but are taken entirely for granted, to the point of not defining the symbols in the equations (e.g. $\Gamma_1(6)$). I believe that the authors should define what a congruence subgroup is, and how modular forms are defined/transform, as these properties are crucial to understand the paper. A reference to a more pedagogical introduction would also be helpful.

6. In section 4.3, below eq. (41), the authors say:

"As advertised, we see that $P^{K, leading}$ is proportional to the unit matrix."

Isn't this true by construction for the entire period matrix $P^{K}$? From this phrasing it seems that it holds only for the leading terms.

7. In section 5.2, below eq. (73), the authors "assume that modular forms are normalised such that the coefficients of their $q$-expansion are algebraic numbers". They should clarify if this implies any loss of generality. A reference to where this assumption is discussed would be helpful as well.

8. The last paragraph of section 5.2 is rather obscure to those who do not fully master the topic. In addition to what I already pointed out in point 5, here the argument also relies on the notions of Eisenstein series and cusp forms, which are taken completely for granted. They make a number of non-trivial statements, in particular that "The space of modular forms ... of modular weight 3 for $\Gamma_1(6)$ is four-dimensional and consists solely of the Eisenstein subspace", without motivating them nor giving any reference for them. The minimal fix would be to provide such references, though an expanded explanation in this paragraph would improve considerably the clarity of the proof.

I reiterate that this is an interesting paper on an important matter. This is why I insist so much on improving its clarity such that more than a few readers can appreciate it. I will be happy to recommend it for publication once the above comments are taken into account.

See report.