{"arxiv_id":"1704.02187","incoming":{"papers":7,"sentences":21,"groups":[{"kind":"extends","label":"Builds on it","papers":1,"sentences":2,"quotes":[{"arxiv_id":"2009.04947","title":"Numerator Seagull and Extended Symmetries of Feynman Integrals","url":"/paper/2009.04947","section":"Summary and discussion","sentence":"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.","intent_p":1.0,"contests_p":null}]},{"kind":"method_or_tool","label":"Uses its method or tool","papers":1,"sentences":1,"quotes":[{"arxiv_id":"1809.05040","title":"The propagator seagull: general evaluation of a two loop diagram","url":"/paper/1809.05040","section":"Introduction","sentence":"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).","intent_p":0.87,"contests_p":null}]},{"kind":"data_used","label":"Uses its data or values","papers":1,"sentences":1,"quotes":[{"arxiv_id":"2009.04947","title":"Numerator Seagull and Extended Symmetries of Feynman Integrals","url":"/paper/2009.04947","section":"Epsilon expansion of the 3 scale sector of the vacuum seagull","sentence":"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) .","intent_p":0.85,"contests_p":null}]},{"kind":"background","label":"Cites it as background","papers":3,"sentences":4,"quotes":[{"arxiv_id":"2009.04947","title":"Numerator Seagull and Extended Symmetries of Feynman Integrals","url":"/paper/2009.04947","section":"Introduction","sentence":"By now, SFI was further developed and numerous diagrams have been analyzed within it [ref] .","intent_p":0.97,"contests_p":null},{"arxiv_id":"1808.02494","title":"Kite diagram through Symmetries of Feynman Integrals","url":"/paper/1808.02494","section":"Introduction","sentence":"Since its introduction the SFI method was developed and applied to several diagrams in [ref] .","intent_p":0.92,"contests_p":null},{"arxiv_id":"1807.07471","title":"Two-loop vacuum diagram through the Symmetries of Feynman Integrals method","url":"/paper/1807.07471","section":"Introduction","sentence":"The above-mentioned 3-loop diagram, the vacuum seagull [ref] , is on the third column from the left at the bottom.","intent_p":0.89,"contests_p":null}]}],"unlabelled_sentences":13},"outgoing":{"papers":9,"sentences":23,"groups":[{"kind":"extends","label":"Builds on","papers":1,"sentences":1,"quotes":[{"arxiv_id":"1610.07720","title":"Evaluation of the general 3-loop vacuum Feynman integral","url":"/paper/1610.07720","section":"Summary","sentence":"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.","intent_p":0.91,"contests_p":null}]},{"kind":"comparison","label":"Compares with","papers":3,"sentences":7,"quotes":[{"arxiv_id":"1610.07720","title":"Evaluation of the general 3-loop vacuum Feynman integral","url":"/paper/1610.07720","section":"Epsilon expansion","sentence":"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 $m_3 = m_5 = 0$.","intent_p":0.99,"contests_p":null},{"arxiv_id":"1609.09159","title":"Three-loop vacuum integrals with arbitrary masses","url":"/paper/1609.09159","section":"Epsilon expansion","sentence":"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 $m_3 = m_5 = 0$.","intent_p":0.99,"contests_p":null},{"arxiv_id":"1408.2372","title":"FIRE5: a C++ implementation of Feynman Integral REduction","url":"/paper/1408.2372","section":"Algebraic locus","sentence":"For example, when additionally $x_3=x_5=0$ we get [equation] We confirm that this result coincides with the answer obtained via [ref] .","intent_p":0.95,"contests_p":null}]},{"kind":"background","label":"Cites as background","papers":4,"sentences":7,"quotes":[{"arxiv_id":"1507.01359","title":"Symmetries of Feynman integrals and the Integration By Parts method","url":"/paper/1507.01359","section":"Introduction","sentence":"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.","intent_p":0.81,"contests_p":null},{"arxiv_id":"1610.07720","title":"Evaluation of the general 3-loop vacuum Feynman integral","url":"/paper/1610.07720","section":"Introduction","sentence":"[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.","intent_p":0.84,"contests_p":null},{"arxiv_id":"1609.09159","title":"Three-loop vacuum integrals with arbitrary masses","url":"/paper/1609.09159","section":"Introduction","sentence":"[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.","intent_p":0.87,"contests_p":null},{"arxiv_id":"1702.02996","title":"TVID: Three-loop Vacuum Integrals from Dispersion relations","url":"/paper/1702.02996","section":"Introduction","sentence":"[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.","intent_p":0.87,"contests_p":null}]}],"unlabelled_sentences":8},"claims":[],"thresholds":{"intent_p":0.8,"supports_p":0.95,"contests_p":0.7},"method":"Citing sentences come from each paper's LaTeX source; every \\cite marks the sentence and the works it cites. Pith's classifier typed each sentence. A label is shown at intent probability 0.8 or more (0.95 for support), a contest at 0.7 or more. Measured accuracy against two blind model families is at https://pith.science/accuracy. The quoted sentences are the citing authors' own.","held_up":null,"_links":{"html":"https://pith.science/paper/1704.02187"}}