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The vacuum seagull: evaluating a 3-loop Feynman diagram with 3 mass scales

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Philipp Burda, Barak Kol, Ruth Shir

High Energy Physics - Theory · Submitted 7 Apr 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We find the generic $G$-orbit to be co-dimension 0 and hence the method is maximally effective, and the diagram reduces to a line integral over simpler diagrams.

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arXiv ID1704.02187 (version 2)
Pith IDOGTXE4TV, Pith's permanent record of this paper and its review
CitationBarak Kol, Philipp Burda, Ruth Shir. "The vacuum seagull: evaluating a 3-loop Feynman diagram with 3 mass scales." arXiv:1704.02187 (2017). https://pith.science/paper/OGTXE4TV
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,474 input tokens.

What the abstract itself states, as probabilities: claims first 0.73 · states limitation 0.35 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.76 · compares with prior results 0.16 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We study a 3-loop 5-propagator Feynman Integral, which we call the vacuum seagull, with arbitrary masses and spacetime dimension using the Symmetries of Feynman Integrals method. It is our first example with potential numerators. We determine the associated group $G \subset GL(3)$ which happens to be 5 dimensional and the associated set of 5 differential equations. $G$ is determined by a geometric approach which we term "current freedom". We find the generic $G$-orbit to be co-dimension 0 and hence the method is maximally effective, and the diagram reduces to a line integral over simpler diagrams. For a reduced parameter space with 3 mass scales we are able to present explicit results in terms of special functions. This might be the first such example.

background contribution method result interpretation main claim

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study a 3-loop 5-propagator Feynman Integral, which we call the vacuum seagull, with arbitrary masses and spacetime dimension using the Symmetries of Feynman Integrals method. contribution 1.00 0.94 0.11 asserted 0.99
s1 It is our first example with potential numerators. contribution 0.57 0.51 0.03 asserted 1.00
s2 We determine the associated group $G \subset GL(3)$ which happens to be 5 dimensional and the associated set of 5 differential equations. result 0.69 0.96 0.11 asserted 1.00
s3 $G$ is determined by a geometric approach which we term "current freedom". method 0.97 0.87 0.04 asserted 1.00
s4 We find the generic $G$-orbit to be co-dimension 0 and hence the method is maximally effective, and the diagram reduces to a line integral over simpler diagrams. result 0.86 0.96 0.45 asserted 1.00
s5 For a reduced parameter space with 3 mass scales we are able to present explicit results in terms of special functions. result 0.94 0.96 0.26 asserted 0.99
s6 This might be the first such example. interpretation 0.67 0.44 0.08 speculative 0.95

What kind of paper the classifier reports this is: theoretical result 0.78 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We study a 3-loop 5-propagator Feynman Integral, which we call the vacuum seagull, with arbitrary masses and spacetime dimension using the Symmetries of Feynman Integrals method. It is our first example with potential numerators. We determine the associated group $G \subset GL(3)$ which happens to be 5 dimensional and the associated set of 5 differential equations. $G$ is determined by a geometric approach which we term "current freedom". We find the generic $G$-orbit to be co-dimension 0 and hence the method is maximally effective, and the diagram reduces to a line integral over simpler diagrams. For a reduced parameter space with 3 mass scales we are able to present explicit results in terms of special functions. This might be the first such example.

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How the literature uses it

7 arXiv papers cite this in 21 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 13 sentences without a confident label

Builds on it 1 paper
  • The main novel evaluation extends the evaluation of the numerator-free vacuum seagull within a 3 mass scale sector [ref] to $I_1^VS$, which includes the numerator to the first power: * Eq. [ref] presents $I_1^VS$ in terms of $I^VS, J_1^VS$, which are detailed below it.

    Numerator Seagull and Extended Symmetries of Feynman Integrals · Summary and discussion · 1.00

Uses its method or tool 1 paper
  • The Symmetries of Feynman Integrals method (SFI) [ref] , see also developments in [ref] , reduces the diagram to its value at some allowed and more convenient base point in parameter space, namely the space $X$ of masses and kinematical invariants, plus a line integral in $X$ over simpler diagrams (with one edge contracted).

    The propagator seagull: general evaluation of a two loop diagram · Introduction · 0.87

Uses its data or values 1 paper
  • The other ingredient for computing the $ϵ$-expansion of the vacuum seagull with numerator in the 3-mass sector through Equation ([ref]) is the $ϵ$-expansion for the vacuum seagull in this sector, which can be found in [ref] , and is repeated here for completeness: I^VS = 1/ϵ^3 I^(-3) + 1/ϵ^2 I^(-2) + 1/ϵ I^(-1) + O(ϵ^0), where I^(-3) = 1/6 [x_1+x_3 +2x_5] I^(-2) = x_1(1-1/2 ln(x_1/Q^2)) + x_3(1-1/2ln(x_3/Q^2)) +x_5(5/3-ln(x_5/Q^2))] I^(-2) = -2/3x_5 -2/3(x_1+x_3 +2x_5)ln(1-x_3/x_1)ln(1-x_5/x_1) + 3(x_1(1-1/2 lnx_1) + x_3(1-1/2lnx_3) +x_5(5/3-ln(x_5/Q^2)))(2-ln(x_5/Q^2)) + x_5/4( x_1/x_5 ln^2(x_1/x_5) + x_3/x_5 ln^2(x_3/x_5)) + 1/6(x_1+x_3 +2x_5)(7 +π^2/4 -9/2(2-ln(x_5/Q^2))^2) + x_5(1-x_1/x_5)Li_2(1-x_1/x_5) + x_5 (1-x_3/x_5)Li_2(1-x_3/x_5) + 2/3 (x_1+x_3 +2x_5) ( Li_1,1(x_1/x_3, x_3/x_5) + Li_1,2(x_3/x_1, x_1/x_5) .

    Numerator Seagull and Extended Symmetries of Feynman Integrals · Epsilon expansion of the 3 scale sector of the vacuum seagull · 0.85

Cites it as background 3 papers

What it cites · 8 sentences without a confident label

Builds on 1 paper
  • This work adds to that of the closely related [ref] not only by the last item above, but also by presenting the equation set explicitly, as well as by determining the group structure and the $G$-foliation of parameter space, thereby exposing the underlying geometry.

    Evaluation of the general 3-loop vacuum Feynman integral · Summary · 0.91

Compares with 3 papers
  • Using explicit expressions for the sources ([ref]), ([ref]), one finds [equation] performing the integral over $t$ from $0$ to $1$ and combining all terms together we arrive to the following very simple expression [equation] which coincide with the expressions from [ref] and [ref] , after proper redefinitions and in the limit of $m_3 = m_5 = 0$.

    Evaluation of the general 3-loop vacuum Feynman integral · Epsilon expansion · 0.99

  • Using explicit expressions for the sources ([ref]), ([ref]), one finds [equation] performing the integral over $t$ from $0$ to $1$ and combining all terms together we arrive to the following very simple expression [equation] which coincide with the expressions from [ref] and [ref] , after proper redefinitions and in the limit of $m_3 = m_5 = 0$.

    Three-loop vacuum integrals with arbitrary masses · Epsilon expansion · 0.99

  • For example, when additionally $x_3=x_5=0$ we get [equation] We confirm that this result coincides with the answer obtained via [ref] .

    FIRE5: a C++ implementation of Feynman Integral REduction · Algebraic locus · 0.95

Cites as background 4 papers
  • Feynman diagrams and their evaluation have been known for long and much is known about them, yet the Symmetries of Feynman Integrals (SFI) method [ref] is a recent contribution which is rather general and natural.

    Symmetries of Feynman integrals and the Integration By Parts method · Introduction · 0.81

  • [ref] studied the vacuum seagull around $d=4$ by using the method of differential equations (essentially the same as SFI) and introduced a computer package which solves an associated set of ordinary differential equations.

    Evaluation of the general 3-loop vacuum Feynman integral · Introduction · 0.84

  • [ref] studied this diagram with general masses in an $ϵ$ expansion and used dispersion relations to reduce the computation to a one dimensional integral suitable for numerical integration.

    Three-loop vacuum integrals with arbitrary masses · Introduction · 0.87

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