GKZ-hypergeometric systems for Feynman integrals

Feng, Tao; Chang, Chao-Hsi; Chen, Jianbin; Zhang, Hai-Bin

doi:10.1016/j.nuclphysb.2020.114952

Cited by 33 publications

(20 citation statements)

References 55 publications

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“…We warn the reader that, apart from restricting to the kinematic space or even smaller subspaces E of P |A|−1 , an important difference with A-discriminant analysis is that we discard singularities of {F G = 0} inside the locus {U G = 0}. Related recent applications of A-discriminants to holonomic systems for Feynman integrals include [53][54][55][56].…”

Section: Motivationmentioning

confidence: 99%

Landau Discriminants

Mizera¹,

Telen²

2021

Preprint

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Scattering amplitudes in quantum field theories have intricate analytic properties as functions of the energies and momenta of the scattered particles. In perturbation theory, their singularities are governed by a set of nonlinear polynomial equations, known as Landau equations, for each individual Feynman diagram. The singularity locus of the associated Feynman integral is made precise with the notion of the Landau discriminant, which characterizes when the Landau equations admit a solution. In order to compute this discriminant, we present approaches from classical elimination theory, as well as a numerical algorithm based on homotopy continuation. These methods allow us to compute Landau discriminants of various Feynman diagrams up to 3 loops, which were previously out of reach. For instance, the Landau discriminant of the envelope diagram is a reducible surface of degree 45 in the three-dimensional space of kinematic invariants. We investigate geometric properties of the Landau discriminant, such as irreducibility, dimension and degree. In particular, we find simple examples in which the Landau discriminant has codimension greater than one. Furthermore, we describe a numerical procedure for determining which parts of the Landau discriminant lie in the physical regions. In order to study degenerate limits of Landau equations and bounds on the degree of the Landau discriminant, we introduce Landau polytopes and study their facet structure. Finally, we provide an efficient numerical algorithm for the computation of the number of master integrals based on the connection to algebraic statistics. The algorithms used in this work are implemented in the open-source Julia package Landau.jl available at https://mathrepo.mis.mpg.de/Landau/.

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Section: Motivationmentioning

confidence: 99%

Landau Discriminants

Mizera¹,

Telen²

2021

Preprint

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“…It is well known to all that, there are a lot of two loop diagrams [25,26] contributing to muon MDM in SUSY models. So, to identify the important two loop diagrams is a meaning- Fig.…”

Section: Introductionmentioning

confidence: 99%

The order analysis for the two loop corrections to lepton MDM

Zhao

Dong

et al. 2020

Eur. Phys. J. C

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The experimental data of the magnetic dipole moment(MDM) of lepton(e, μ) is very exact. The deviation between the experimental data and the standard model prediction maybe come from new physics contribution. In the supersymmetric models, there are very many two loop diagrams contributing to the lepton MDM. In supersymmetric models, we suppose two mass scales M S H and M with M S H M for supersymmetric particles. Squarks belong to M S H and the other supersymmetric particles belong to M. We analyze the order of the contributions from the two loop diagrams. The two loop triangle diagrams corresponding to the two loop self-energy diagram satisfy Ward-identity, and their contributions possess particular factors. This work can help to distinguish the important two loop diagrams giving corrections to lepton MDM.

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“…(4.7) 30 The terminology of -vectors employed here is of course not be confused with vectors that have l components or the like, which is why have chosen a different symbol .…”

Section: Relating the Frobenius Basis Of M L−1 To The Basis Of Kähler Classes On W L−1mentioning

confidence: 99%

Analytic structure of all loop banana integrals

Bönisch¹,

Fischbach²,

Klemm

et al. 2021

J. High Energ. Phys.

102

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Using the Gelfand-Kapranov-Zelevinskĭ system for the primitive cohomology of an infinite series of complete intersection Calabi-Yau manifolds, whose dimension is the loop order minus one, we completely clarify the analytic structure of all banana integrals with arbitrary masses. In particular, we find that the leading logarithmic structure in the high energy regime, which corresponds to the point of maximal unipotent monodromy, is determined by a novel $$ \hat{\Gamma}\hbox{-} \mathrm{class} $$ Γ ̂ ‐ class evaluation in the ambient spaces of the mirror, while the imaginary part of the integral in this regime is determined by the $$ \hat{\Gamma}\hbox{-} \mathrm{class} $$ Γ ̂ ‐ class of the mirror Calabi-Yau manifold itself. We provide simple closed all loop formulas for the former as well as for the Frobenius κ-constants, which determine the behaviour of the integrals when the momentum square equals the sum of the masses squared, in terms of zeta values. We extend our previous work from three to four loops by providing for the latter case a complete set of (inhomogeneous) Picard-Fuchs differential equations for arbitrary masses. This allows to evaluate the banana integral in very short time to very high numerical precision for all values of the physical parameters. Using modular properties of the periods we determine the value of the maximal cut equal mass four-loop integral at the attractor points in terms of periods of modular weight two and four Hecke eigenforms and the quasiperiods of their meromorphic cousins.

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GKZ-hypergeometric systems for Feynman integrals

Cited by 33 publications

References 55 publications

Landau Discriminants

Landau Discriminants

The order analysis for the two loop corrections to lepton MDM

Analytic structure of all loop banana integrals

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