scholarly journals Yangian bootstrap for massive Feynman integrals

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Florian Loebbert ◽  
Julian Miczajka ◽  
Dennis Müller ◽  
Hagen Münkler
Keyword(s):  
Momentum Space ◽  
Conformal Symmetry ◽  
Feynman Integrals ◽  
Series Representations ◽  
Yangian Symmetry

We extend the study of the recently discovered Yangian symmetry of massive Feynman integrals and its relation to massive momentum space conformal symmetry. After proving the symmetry statements in detail at one and two loop orders, we employ the conformal and Yangian constraints to bootstrap various one-loop examples of massive Feynman integrals. In particular, we explore the interplay between Yangian symmetry and hypergeometric expressions of the considered integrals. Based on these examples we conjecture single series representations for all dual conformal one-loop integrals in D spacetime dimensions with generic massive propagators.

scholarly journals Massive fishnets

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Florian Loebbert ◽  
Julian Miczajka
Keyword(s):  
Symmetry Breaking ◽  
Spontaneous Symmetry Breaking ◽  
Conformal Symmetry ◽  
Scaling Limit ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Coulomb Branch ◽  
Feynman Integrals ◽  
Four Dimensions ◽  
Yangian Symmetry

Abstract Recently, infinite families of massive Feynman integrals were found to feature an unexpected Yangian symmetry. In the massless case, similar integrability properties are understood via the interpretation of individual Feynman integrals as correlators in the massless fishnet theory introduced by Gürdoğan and Kazakov. Here we seek for an analogous interpretation of the integrability of massive Feynman integrals. We contrast two approaches to define simple massive quantum field theories in four dimensions. First, we discuss spontaneous symmetry breaking in the massless bi-scalar fishnet theory. We then propose an alternative route to a massive fishnet theory by taking a double-scaling limit of $$ \mathcal{N} $$ N = 4 SYM theory on the Coulomb branch. Both approaches lead to a massive extension of the massless fishnet theory, differing in how masses enter into the propagators. In the latter theory, planar off-shell amplitudes are in one-to-one correspondence with precisely those massive Feynman integrals that were shown to be invariant under the Yangian. This suggests a re-investigation of Coulomb branch $$ \mathcal{N} $$ N = 4 SYM theory with regard to integrability. Finally, we demonstrate that in the case of spontaneous symmetry breaking, the original conformal symmetry leads to soft theorems for scattering amplitudes in the broken phase.

scholarly journals Massive Conformal Symmetry and Integrability for Feynman Integrals

2020 ◽  
Vol 125 (9) ◽  
Author(s):  
Florian Loebbert ◽  
Julian Miczajka ◽  
Dennis Müller ◽  
Hagen Münkler
Keyword(s):  
Conformal Symmetry ◽  
Feynman Integrals

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.

scholarly journals STR: A Mathematica package for the method of uniqueness

2020 ◽  
Vol 31 (10) ◽  
pp. 2050146
Author(s):  
Michelangelo Preti
Keyword(s):  
Conformal Symmetry ◽  
Scaling Limit ◽  
Space Time ◽  
Time Dimension ◽  
Feynman Integrals ◽  
Master Integral ◽  
Graphic Interface ◽  
Graphical Environment ◽  
Feynman Graphs ◽  
Double Scaling Limit

We present Star–Triangle Relations (STRs), a Mathematica® package designed to solve Feynman diagrams by means of the method of uniqueness in any Euclidean space-time dimension. The method of uniqueness is a powerful technique to solve multi-loop Feynman integrals in theories with conformal symmetry imposing some relations between the powers of propagators and the space-time dimension. In our algorithm, we include both identities for scalar and Yukawa type integrals. The package provides a graphical environment in which it is possible to draw the desired diagram with the mouse input and a set of tools to modify and compute it. Throughout the use of a graphic interface, the package should be easily accessible to users with little or no previous experience on diagrams computation. This manual includes some pedagogical examples of computation of Feynman graphs as the scalar two-loop kite master integral and a fermionic diagram appearing in the computation of the spectrum of the [Formula: see text]-deformed [Formula: see text] SYM in the double scaling limit.

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