Multi-valued Feynman Graphs and Scattering Theory

Kreimer, Dirk

doi:10.1007/978-3-030-04480-0_13

Cited by 5 publications

(6 citation statements)

References 23 publications

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“…Such non-principal sheet monodromies need to be studied to understand the mixed Hodge theory of D R (b n ) as a multi-valued function in future work. See [21] for some preliminary considerations.…”

Section: Normal and Pseudo-thresholds For B Nmentioning

confidence: 99%

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Bananas: multi-edge graphs and their Feynman integrals

Kreimer

2023

Lett Math Phys

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We consider multi-edge or banana graphs $$b_n$$ b n on n internal edges $$e_i$$ e i with different masses $$m_i$$ m i . We focus on the cut banana graphs $$\Im (\Phi _R(b_n))$$ ℑ ( Φ R ( b n ) ) from which the full result $$\Phi _R(b_n)$$ Φ R ( b n ) can be derived through dispersion. We give a recursive definition of $$\Im (\Phi _R(b_n))$$ ℑ ( Φ R ( b n ) ) through iterated integrals. We discuss the structure of this iterated integral in detail. A discussion of accompanying differential equations, of monodromy and of a basis of master integrals is included.

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Section: Normal and Pseudo-thresholds For B Nmentioning

confidence: 99%

“…See, for example, [5,21] for a discussion of its analytic structure and behaviour off the principal sheet. The threshold divisor defined by the intersection L 1 ∩ L 2 where the zero loci…”

Section: Normal and Pseudo-thresholds For B Nmentioning

confidence: 99%

Bananas: multi-edge graphs and their Feynman integrals

Kreimer

2023

Lett Math Phys

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“…provide an integrand which is to be integrated over a compact domain only [21,4]. For the definition of the renormalization scheme R we refer the reader to [4,22] and Appendix B.…”

Section: Remark 43mentioning

confidence: 99%

“…This coaction concerns variations on non-principal sheets [21]. It relates to the jewels of Vogtmann and collaborators [38].…”

Section: Note Thatmentioning

confidence: 99%

Algebraic Interplay between Renormalization and Monodromy

Kreimer¹,

Yeats²

2021

Preprint

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We investigate combinatorial and algebraic aspects of the interplay between renormalization and monodromies for Feynman amplitudes. We clarify how extraction of subgraphs from a Feynman graph interacts with putting edges onshell or with contracting them to obtain reduced graphs. Graph by graph this leads to a study of cointeracting bialgebras. One bialgebra comes from extraction of subgraphs and hence is needed for renormalization. The other bialgebra is an incidence bialgebra for edges put either onor offshell. It is hence related to the monodromies of the multivalued function to which a renormalized graph evaluates. Summing over infinite series of graphs, consequences for Green functions are derived using combinatorial Dyson-Schwinger equations. Contents 2.1. Graphs 2.2. Cuts 2.3. Sub-and co-graphs 3. Hopf algebras 3.1. The core Hopf algebra H core 3.2. Quotient Hopf algebras 3.3. The Hopf algebra H pC 3.4. Pre-Cutkosky Necklaces 3.5. Extension to a core coproduct for pairs (Γ, F ) 3.6. Counting spanning trees 4. Coactions 4.1. H pC → H core ⊗ H pC 4.2. Cointeracting bialgebras 4.3. Sector Decomposition 4.4. Two more coactions 5. DSEs in core and quotient Hopf algebras 5.1. Dyson-Schwinger and cut Dyson-Schwinger setup 5.2. Assembly maps vs Hochschild 1-cocycles 5.3. Core, no cuts 5.4. Core, with cuts 5.5. The coaction on Green functions 5.6. Pairs (Γ, F ) or H C 1

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“…An especially suggestive candidate for such a superior object in the case of scalar quantum field theories is Outer space [39] and its quotient formed under the action of Out(F n ) which is the moduli space of graphs. For instance, unitarity and branch cut properties of Feynman integrals can be understood using Outer space [13,68,8]. This space can be seen as a tropical analogue of Teichmüller space [35] and the moduli space of curves, which holds a similar superior role in string theory.…”

Section: (Tropical Sampling Applied To Sums Of Feynman Diagrams)mentioning

confidence: 99%

Tropical Monte Carlo quadrature for Feynman integrals

Borinsky¹

2020

Preprint

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We introduce a new method to evaluate algebraic integrals over the simplex numerically. It improves upon geometric sector decomposition by employing tools from tropical geometry. The method can be improved further by exploiting the geometric structure of the underlying integrand. As an illustration of this, we give a specialized algorithm for a class of integrands that exhibit the form of a generalized permutahedron. This class includes integrands for scattering amplitudes and parametric Feynman integrals with tame kinematics. A proof-of-concept implementation is provided with which Feynman integrals up to loop order 17 can be evaluated.

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Multi-valued Feynman Graphs and Scattering Theory

Cited by 5 publications

References 23 publications

Bananas: multi-edge graphs and their Feynman integrals

Bananas: multi-edge graphs and their Feynman integrals

Algebraic Interplay between Renormalization and Monodromy

Tropical Monte Carlo quadrature for Feynman integrals

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