The monodromy rings of one loop Feynman integrals

Ponzano, G.; Regge, T.; Speer, Eugène R.; Westwater, M. J.

doi:10.1007/bf01649638

Cited by 18 publications

(12 citation statements)

References 9 publications

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“…In particular, the projective nature of Feynman parameter integrands, and the importance of the monodromy properties of Feynman integrals under analytic continuation around their singularities, were soon uncovered, and attracted the attention of mathematicians [14,15] and physicists [16]. In this context, Tullio Regge and collaborators published a series of papers [17][18][19] studying the 'monodromy ring' of interesting classes of Feynman graphs: first the ones we would at present describe as 'multi-loop sunrise' graphs in ref. [17], then generic one-particle irreducible n-point one-loop graphs in ref.…”

Section: Jhep03(2024)096mentioning

confidence: 99%

“…[17], then generic one-particle irreducible n-point one-loop graphs in ref. [18], and finally the natural combination of these two classes, in which each propagator of the one-loop n-point diagram is replaced by a k-loop sunrise [19]. All of these papers employ the parameter representation as a starting point, and make heavy use of the projective nature of the integrand.…”

Section: Jhep03(2024)096mentioning

confidence: 99%

“…In the present paper, we start from the ideas of refs. [16][17][18][19] and the concrete results of Barucchi and Ponzano [30,31] to propose a projective framework to derive IBP identities and systems of linear differential equations for Feynman integrals. In order to do so, we JHEP03(2024)096 need to generalise the Barucchi-Ponzano results in several directions.…”

Section: Jhep03(2024)096mentioning

confidence: 99%

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Integration-by-parts identities and differential equations for parametrised Feynman integrals

Artico,

Magnea

2024

J. High Energ. Phys.

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Integration-by-parts (IBP) identities and differential equations are the primary modern tools for the evaluation of high-order Feynman integrals. They are commonly derived and implemented in the momentum-space representation. We provide a different viewpoint on these important tools by working in Feynman-parameter space, and using its projective geometry. Our work is based upon little-known results pre-dating the modern era of loop calculations [16–19, 30, 31]: we adapt and generalise these results, deriving a very general expression for sets of IBP identities in parameter space, associated with a generic Feynman diagram, and valid to any loop order, relying on the characterisation of Feynman-parameter integrands as projective forms. We validate our method by deriving and solving systems of differential equations for several simple diagrams at one and two loops, providing a unified perspective on a number of existing results.

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Section: Jhep03(2024)096mentioning

confidence: 99%

Section: Jhep03(2024)096mentioning

confidence: 99%

Section: Jhep03(2024)096mentioning

confidence: 99%

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Integration-by-parts identities and differential equations for parametrised Feynman integrals

Artico,

Magnea

2024

J. High Energ. Phys.

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“…They are often formulated in terms of a potential function on the cotangent bundle of Minkowski space. Landau equations turn out to hide rich geometric structure, whose study has been undertaken, e.g., from the perspective of the theory of hyperfunctions and holonomic systems [80][81][82]; monodromy groups [83,84]; motives and Morse theory [85,86]; and combinatorics of hypersphere arrangements [87,88]. For reviews see [41-43, 89, 90] and especially [91][92][93].…”

Section: What Is Known About Solutions Of Landau Equations?mentioning

confidence: 99%

Crossing Symmetry in the Planar Limit

Mizera

2021

Preprint

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Crossing symmetry asserts that particles are indistinguishable from anti-particles traveling back in time. In quantum field theory, this statement translates to the longstanding conjecture that probabilities for observing the two scenarios in a scattering experiment are described by one and the same function. Why could we expect it to be true? In this work we examine this question in a simplified setup and take steps towards illuminating a possible physical interpretation of crossing symmetry. To be more concrete, we consider planar scattering amplitudes involving any number of particles with arbitrary spins and masses to all loop orders in perturbation theory. We show that by deformations of the external momenta one can smoothly interpolate between pairs of crossing channels without encountering singularities or violating mass-shell conditions and momentum conservation. The analytic continuation can be realized using two types of moves. The first one makes use of an iε prescription for avoiding singularities near the physical kinematics and allows us to adjust the momenta of the external particles relative to one another within their lightcones. The second, more violent, step involves a rotation of subsets of particle momenta via their complexified lightcones from the future to the past and vice versa. We show that any singularity along such a deformation would have to correspond to two beams of particles scattering off each other. For planar Feynman diagrams, these kinds of singularities are absent because of the particular flow of energies through their propagators. We prescribe a five-step sequence of such moves that combined together proves crossing symmetry for planar scattering amplitudes in perturbation theory, paving a way towards settling this question for more general scattering processes in quantum field theories.

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“…where the R e are generic rational functions and the A e form a set of N independent (over the field of rational functions) solutions of the problem. dA e The derivative -with respect to any variable z has still the same dz monodromy group, therefore we must have a differential equation of the type: (4) dz *…”

Section: ) the Construction Of An ^-Matrix Is Enormously Complicated Bymentioning

confidence: 99%

Old Problems and New Hopes in $S$-Matrix Theory

Regge¹

1976

Publ. Res. Inst. Math. Sci.

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1) The problem of constructing a convenient ser of axioms for the /S-matrix directly in terms of the scattering variables and without recourse to conventional field theory is still receiving wide attention by theoretiticians in recent years. The ingredients which go into these attempts are well-known. There is relativistic invariance, unitarity, and finally some degree of analyticity. The hope is that by combining them according to some recipe one obtains the *S-matrix. The trouble with this program is that really no one seems to know how to combine the ingredients in the proper order; moreover there is wide disagreement as to what analyticity really means. It is on this last point that I wish to concentrate the discussion. Obviously Lorentz invariance and unitarity have quite an ambiguous meaning. Every attempt toward 5-matrix theory usually starts with the rather ambitious goal of doing away with conventional field theory. The basic philosophy is opposite to that of constructive field theory whose remarkable progress in recent years has overshadowed any alternative approach.My personal belief however, is that the possibility of different approaches originates from a basic inadequacy of the theory and that ultimately field theory and ^-matrix theory will be unified. Secondly the constructive approach is still facing tremendous difficulties and is still far removed from any physical application; but basically I feel that committing ourselves to any particular philosophy at this early stage is an essentially unsound strategy.

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The monodromy rings of one loop Feynman integrals

Abstract: The monodromy rings of Feynman integrals for one loop graphs with an arbitrary number of lines are determined.

Cited by 18 publications

References 9 publications

Integration-by-parts identities and differential equations for parametrised Feynman integrals

Integration-by-parts identities and differential equations for parametrised Feynman integrals

Crossing Symmetry in the Planar Limit

Old Problems and New Hopes in $S$-Matrix Theory

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