Fermionic QFT, Grassmann calculus, and combinatorics

2021 ◽  
pp. 56-71
Author(s):  
Adrian Tanasa
Keyword(s):  
Tutte Polynomial ◽  
Grassmann Algebra ◽  
Parametric Representation ◽  
Feynman Integrals ◽  
Intersecting Cycles

In the first section of this chapter, we use Grassmann calculus, used in fermionic QFT, to give, first a reformulation of the Lingström–Gesse–Viennot lemma proof. We further show that this proof generalizes to graphs with cycles. We then use the same Grassmann calculus techniques to give new proofs of Stembridge's identities relating appropriate graph Pfaffians to sum over non-intersecting paths. The results presented here go further than the ones of Stembridge, because Grassmann algebra techniques naturally extend (without any cost!) to graphs with cycles. We thus obtain, instead of sums over non-intersecting paths, sums over non-intersecting paths and non-intersecting cycles. In the fifth section of the chapter, we give a generalization of these results. In the sixth section of this chapter we use Grassmann calculus to exhibit the relation between a multivariate version of Tutte polynomial and the Kirchhoff-Symanzik polynomials of the parametric representation of Feynman integrals, polynomials already introduced in Chapters 1 and Chapter 3.

scholarly journals Reduction of Feynman integrals in the parametric representation II: reduction of tensor integrals

2021 ◽  
Vol 81 (3) ◽  
Author(s):  
Wen Chen
Keyword(s):  
Gröbner Basis ◽  
Parametric Representation ◽  
Polynomial Equations ◽  
Grobner Basis ◽  
Spacetime Dimension ◽  
Integration By Parts ◽  
Feynman Integrals ◽  
Parametric Integrals ◽  
Parametric Integration

AbstractIn a recent paper by the author (Chen in JHEP 02:115, 2020), the reduction of Feynman integrals in the parametric representation was considered. Tensor integrals were directly parametrized by using a generator method. The resulting parametric integrals were reduced by constructing and solving parametric integration-by-parts (IBP) identities. In this paper, we furthermore show that polynomial equations for the operators that generate tensor integrals can be derived. Based on these equations, two methods to reduce tensor integrals are developed. In the first method, by introducing some auxiliary parameters, tensor integrals are parametrized without shifting the spacetime dimension. The resulting parametric integrals can be reduced by using the standard IBP method. In the second method, tensor integrals are (partially) reduced by using the technique of Gröbner basis combined with the application of symbolic rules. The unreduced integrals can further be reduced by solving parametric IBP identities.

scholarly journals Reduction of Feynman integrals in the parametric representation III: integrals with cuts

2020 ◽  
Vol 80 (12) ◽  
Author(s):  
Wen Chen
Keyword(s):  
Phase Space ◽  
Differential Equations ◽  
Momentum Space ◽  
Theta Functions ◽  
Parametric Representation ◽  
Systematic Method ◽  
Integration By Parts ◽  
Feynman Integrals

AbstractPhase space cuts are implemented by inserting Heaviside theta functions in the integrands of momentum-space Feynman integrals. By directly parametrizing theta functions and constructing integration-by-parts (IBP) identities in the parametric representation, we provide a systematic method to reduce integrals with cuts. Since the IBP method is available, it becomes possible to evaluate integrals with cuts by constructing and solving differential equations.

QFT on the non-commutative Moyal space and combinatorics

2021 ◽  
pp. 95-120
Author(s):  
Adrian Tanasa
Keyword(s):  
Heat Kernel ◽  
High Energy Physics ◽  
Parametric Representation ◽  
High Energy ◽  
Translation Invariance ◽  
Translation Invariant ◽  
Feynman Integrals ◽  
Usual Property ◽  
Mehler Kernel ◽  
Energy Physics

In this chapter we present the Phi? QFT model on the non-commutative Moyal space and the UV/IR mixing issue, which prevents it from being renormalizable. We then present the Grosse–Wulkenhaar Phi? QFT model on the non-commutative Moyal space, which changes the usual propagator of the Phi? model (based on the heat kernel formula) to a Mehler kernel based propagator. This Grosse–Wulkenhaar model is perturbatively renormalizable but it is not translation-invariant (translation-invariance being a usual property of high-energy physics models). We then show how the Mellin transform technique can be used to express the Feynman integrals of the Grosse-Wulkenhaar model. In the last part of the chapter, we present another Phi? QFT model on the non-commutative Moyal space, which is however both renormalizable and translation-invariant. We show the relation between the parametric representation of this model and the Bollobás–Riordan polynomial.

scholarly journals Two-loop helicity amplitudes for gg → ZZ with full top-quark mass effects

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Bakul Agarwal ◽  
Stephen P. Jones ◽  
Andreas von Manteuffel
Keyword(s):  
Top Quark ◽  
Quark Mass ◽  
Parametric Representation ◽  
Top Quark Mass ◽  
Quark Loop ◽  
Linear Combinations ◽  
Feynman Integrals ◽  
Field Techniques ◽  
Helicity Amplitudes ◽  
Finite Integrals

Abstract We calculate the two-loop QCD corrections to gg → ZZ involving a closed top-quark loop. We present a new method to systematically construct linear combinations of Feynman integrals with a convergent parametric representation, where we also allow for irreducible numerators, higher powers of propagators, dimensionally shifted integrals, and subsector integrals. The amplitude is expressed in terms of such finite integrals by employing syzygies derived with linear algebra and finite field techniques. Evaluating the amplitude using numerical integration, we find agreement with previous expansions in asymptotic limits and provide ab initio results also for intermediate partonic energies and non-central scattering at higher energies.

A Parametric Representation of Wind-Driven Rain in Experimental Setups

2012 ◽  
Author(s):  
Thomas Baheru ◽  
Arindam Gan Chowdhury ◽  
Girma Bitsuamlak ◽  
Ali Tokay
Keyword(s):  
Parametric Representation

Pattern Theory

2006 ◽  
Author(s):  
Ulf Grenander ◽  
Michael I. Miller
Keyword(s):  
Computational Linguistics ◽  
Parametric Representation ◽  
Bayes Estimation ◽  
Geometric Transformation ◽  
Central Component ◽  
Pattern Theory ◽  
Infinite Dimensional ◽  
Engineering Mathematics ◽  
Dimensional Shape ◽  
The Continuum

Pattern Theory provides a comprehensive and accessible overview of the modern challenges in signal, data, and pattern analysis in speech recognition, computational linguistics, image analysis and computer vision. Aimed at graduate students in biomedical engineering, mathematics, computer science, and electrical engineering with a good background in mathematics and probability, the text includes numerous exercises and an extensive bibliography. Additional resources including extended proofs, selected solutions and examples are available on a companion website. The book commences with a short overview of pattern theory and the basics of statistics and estimation theory. Chapters 3-6 discuss the role of representation of patterns via condition structure. Chapters 7 and 8 examine the second central component of pattern theory: groups of geometric transformation applied to the representation of geometric objects. Chapter 9 moves into probabilistic structures in the continuum, studying random processes and random fields indexed over subsets of Rn. Chapters 10 and 11 continue with transformations and patterns indexed over the continuum. Chapters 12-14 extend from the pure representations of shapes to the Bayes estimation of shapes and their parametric representation. Chapters 15 and 16 study the estimation of infinite dimensional shape in the newly emergent field of Computational Anatomy. Finally, Chapters 17 and 18 look at inference, exploring random sampling approaches for estimation of model order and parametric representing of shapes.

scholarly journals Minkowski box from Yangian bootstrap

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Luke Corcoran ◽  
Florian Loebbert ◽  
Julian Miczajka ◽  
Matthias Staudacher
Keyword(s):  
Minkowski Space ◽  
Wigner Function ◽  
Functional Form ◽  
A Priori ◽  
Alternative Methods ◽  
Feynman Integrals ◽  
The One

Abstract We extend the recently developed Yangian bootstrap for Feynman integrals to Minkowski space, focusing on the case of the one-loop box integral. The space of Yangian invariants is spanned by the Bloch-Wigner function and its discontinuities. Using only input from symmetries, we constrain the functional form of the box integral in all 64 kinematic regions up to twelve (out of a priori 256) undetermined constants. These need to be fixed by other means. We do this explicitly, employing two alternative methods. This results in a novel compact formula for the box integral valid in all kinematic regions of Minkowski space.

Error analysis of higher order trace finite element methods for the surface Stokes equation

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Thomas Jankuhn ◽  
Maxim A. Olshanskii ◽  
Arnold Reusken ◽  
Alexander Zhiliakov
Keyword(s):  
Finite Element ◽  
Stokes System ◽  
Parametric Representation ◽  
Higher Order ◽  
Optimal Order ◽  
Helmholtz Instability ◽  
Surface Stability ◽  
Instability Problem ◽  
Convergence Results ◽  
Optimal Order Convergence

AbstractThe paper studies a higher order unfitted finite element method for the Stokes system posed on a surface in ℝ3. The method employs parametric Pk-Pk−1 finite element pairs on tetrahedral bulk mesh to discretize the Stokes system on embedded surface. Stability and optimal order convergence results are proved. The proofs include a complete quantification of geometric errors stemming from approximate parametric representation of the surface. Numerical experiments include formal convergence studies and an example of the Kelvin--Helmholtz instability problem on the unit sphere.

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