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Numerator Seagull and Extended Symmetries of Feynman Integrals
High Energy Physics - Theory · Submitted 10 Sep 2020
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We solve the equation system in two cases: over the singular locus and in a certain 3 scale sector where we obtain novel closed-form evaluations and epsilon expansions, thereby extending previous results for the numerator-free case.
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The abstract, sentence by sentence
The Symmetries of Feynman Integrals (SFI) method is extended for the first time to incorporate an irreducible numerator. This is done in the context of the so-called vacuum and propagator seagull diagrams, which have 3 and 2 loops, respectively, and both have a single irreducible numerator. For this purpose, an extended version of SFI (xSFI) is developed. For the seagull diagrams with general masses, the SFI equation system is found to extend by two additional equations. The first is a recursion equation in the numerator power, which has an alternative form as a differential equation for the generating function. The second equation applies only to the propagator seagull and does not involve the numerator. We solve the equation system in two cases: over the singular locus and in a certain 3 scale sector where we obtain novel closed-form evaluations and epsilon expansions, thereby extending previous results for the numerator-free case.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The Symmetries of Feynman Integrals (SFI) method is extended for the first time to incorporate an irreducible numerator. | contribution 1.00 | 0.87 | 0.49 | asserted 1.00 |
| s1 | This is done in the context of the so-called vacuum and propagator seagull diagrams, which have 3 and 2 loops, respectively, and both have a single irreducible numerator. | background 0.53 | 0.43 | 0.08 | asserted 1.00 |
| s2 | For this purpose, an extended version of SFI (xSFI) is developed. | contribution 0.52 | 0.92 | 0.11 | asserted 1.00 |
| s3 | For the seagull diagrams with general masses, the SFI equation system is found to extend by two additional equations. | result 0.83 | 0.92 | 0.14 | asserted 1.00 |
| s4 | The first is a recursion equation in the numerator power, which has an alternative form as a differential equation for the generating function. | result 0.53 | 0.87 | 0.09 | asserted 1.00 |
| s5 | The second equation applies only to the propagator seagull and does not involve the numerator. | result 0.78 | 0.87 | 0.05 | asserted 1.00 |
| s6 | We solve the equation system in two cases: over the singular locus and in a certain 3 scale sector where we obtain novel closed-form evaluations and epsilon expansions, thereby extending previous results for the numerator-free case. | result 0.92 | 0.96 | 0.59 | asserted 1.00 |
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The paper
Original abstract
The Symmetries of Feynman Integrals (SFI) method is extended for the first time to incorporate an irreducible numerator. This is done in the context of the so-called vacuum and propagator seagull diagrams, which have 3 and 2 loops, respectively, and both have a single irreducible numerator. For this purpose, an extended version of SFI (xSFI) is developed. For the seagull diagrams with general masses, the SFI equation system is found to extend by two additional equations. The first is a recursion equation in the numerator power, which has an alternative form as a differential equation for the generating function. The second equation applies only to the propagator seagull and does not involve the numerator. We solve the equation system in two cases: over the singular locus and in a certain 3 scale sector where we obtain novel closed-form evaluations and epsilon expansions, thereby extending previous results for the numerator-free case.
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How the literature uses it
1 arXiv paper cite this in 1 sentence
What it cites · 26 sentences without a confident label
Builds on 1 paper
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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.
The vacuum seagull: evaluating a 3-loop Feynman diagram with 3 mass scales · Summary and discussion · 1.00
Uses the method or tool of 1 paper
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We will examine each of these hypersurfaces separately, using the method of maximal minors [ref] .
Algebraic aspects of when and how a Feynman diagram reduces to simpler ones · Singular loci · 1.00
Uses data or values from 3 papers
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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) .
The vacuum seagull: evaluating a 3-loop Feynman diagram with 3 mass scales · Epsilon expansion of the 3 scale sector of the vacuum seagull · 0.85
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The $ϵ$-expansions of the diagrams which appear in the source terms ([ref]), are [ref] Tad_1(x) = 1/ϵx - A(x) - ϵA_ϵ(x) Dia_1,1,1(x,y,z) = 1/2 ϵ^2 (x+y+z) - 1/ϵ (A(x) + A(y) + A(z) -1/2(x+y+z) ) -I_0(x,y,z) Melon_1,1,1,1(x,y,z,u) = 1/3 ϵ^3(xy + xz + xu + yz + yu + zu) +1/ϵ^2 [-1/2((y+z+u)A(x)+(x+z+u)A(y) +(x+y+u)A(z)+(x+y+z)A(u)) +1/3(xy + xz + xu + yz + yu + zu) -1/12(x^2+y^2+z^2+u^2) ] + 1/ϵ [A(x)A(y)+A(x)A(z)+A(x)A(u)+A(y)A(z)+A(y)A(u)+A(z)A(u) -(y+z+u)(A_ϵ(x)+A(x))/2 - cyclic +(x A(x)+ y A(y)+ z A(z) + u A(u))/4 +1/3(xy + xz + xu + yz + yu + zu) -3 (x^2+y^2+z^2+u^2)/8].
Evaluation of the general 3-loop vacuum Feynman integral · Epsilon expansion of the 3 scale sector of the vacuum seagull · 1.00
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The $ϵ$-expansions of the diagrams which appear in the source terms ([ref]), are [ref] Tad_1(x) = 1/ϵx - A(x) - ϵA_ϵ(x) Dia_1,1,1(x,y,z) = 1/2 ϵ^2 (x+y+z) - 1/ϵ (A(x) + A(y) + A(z) -1/2(x+y+z) ) -I_0(x,y,z) Melon_1,1,1,1(x,y,z,u) = 1/3 ϵ^3(xy + xz + xu + yz + yu + zu) +1/ϵ^2 [-1/2((y+z+u)A(x)+(x+z+u)A(y) +(x+y+u)A(z)+(x+y+z)A(u)) +1/3(xy + xz + xu + yz + yu + zu) -1/12(x^2+y^2+z^2+u^2) ] + 1/ϵ [A(x)A(y)+A(x)A(z)+A(x)A(u)+A(y)A(z)+A(y)A(u)+A(z)A(u) -(y+z+u)(A_ϵ(x)+A(x))/2 - cyclic +(x A(x)+ y A(y)+ z A(z) + u A(u))/4 +1/3(xy + xz + xu + yz + yu + zu) -3 (x^2+y^2+z^2+u^2)/8].
Effective potential at three loops · Epsilon expansion of the 3 scale sector of the vacuum seagull · 1.00
Compares with 1 paper
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In this case we get the solution I^PS|_x_5=0 = (O_2-O_5)I^PS/x_12 +1/d-2[2x_3∂_3 -2x_4∂_4/x_34(O_3+O_4-O_5)I^PS|_x_5=0 -2x_1∂_1-2x_2∂_2/x_12O_5I^PS]_x_5=0 .Using equations for the sources in appendix [ref] we reproduce the result in equation (6) of [ref] , namely I^PS|_x_5=0 = [O_2I^PS/x_12-(O_3-O_4)I^PS/x_34 -x_12(2x_3∂_3 -2x_4∂_4 )-x_34(4x_2 ∂_2 + 2x_3 ∂_3 + 2x_4 ∂_4 + 8-3d)/(d-2)x_12x_34O_5 I^PS ]_x_5=0 .
Counting master integrals: Integration by parts vs. functional equations · $x_5=0$ · 0.88
Cites as background 22 papers
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Symmetries of Feynman Integrals (SFI) is a general method introduced in [ref] which associates with any given Feynman diagram a system of partial differential equations.
Symmetries of Feynman integrals and the Integration By Parts method · Introduction · 0.90
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By now, SFI was further developed and numerous diagrams have been analyzed within it [ref] .
Algebraic aspects of when and how a Feynman diagram reduces to simpler ones · Introduction · 0.97
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By now, SFI was further developed and numerous diagrams have been analyzed within it [ref] .
Bubble diagram through the Symmetries of Feynman Integrals method · Introduction · 0.97
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