We study interactions amongst topologically conserved excitations of quantum theories of gravity, in particular the braid excitations of four-valent spin networks. These have been shown previously to propagate and interact under evolution rules of spin foam models. We show that the dynamics of these braid excitations can be described by an effective theory based on Feynman diagrams. In this language, braids which are actively interacting are analogous to bosons, in that the topological conservation laws permit them to be singly created and destroyed. Exchanges of these excitations give rise to interactions between braids which are charged under the topological conservation rules.
The method of Symmetries of Feynman Integrals defines for any Feynman diagram a set of partial differential equations. On some locus in parameter space the equations imply that the diagram can be reduced to a linear combination of simpler diagrams. This paper provides a systematic method to determine this locus and the associated reduction through an algebraic method involving factorization of maximal minors.
The Symmetries of Feynman Integrals (SFI) is a method for evaluating Feynman Integrals which exposes a novel continuous group associated with the diagram which depends only on its topology and acts on its parameters. Using this method we study the kite diagram, a two-loop diagram with two external legs, with arbitrary masses and spacetime dimension. Generically, this method reduces a Feynman integral into a line integral over simpler diagrams. We identify a locus in parameter space where the integral further reduces to a mere linear combination of simpler diagrams, thereby maximally generalizi…
We derive the simplest commutation relations of operator algebras associated to M2 branes and an M5 brane in the $Ω$-deformed M-theory, which is a natural set-up for Twisted holography. Feynman diagram 1-loop computations in the twisted-holographic dual side reproduce the same algebraic relations.
A short review of the Operator/Feynman diagram/dessin d'enfants correspondence in the rank 3 tensor model is presented, and the cut & join operation is given in the language of dessin d'enfants as a straightforward development. We classify operators of the rank 3 tensor model up to level 5 with dessin d'enfants.
We review the construction of the adiabatic expansion for Bose and Fermi systems and show how it may be used to explore the chiral and parity anomalies for Dirac fermions without the need to compute Feynman diagrams.
The Symmetries of Feynman Integrals method (SFI) associates a natural Lie group with any diagram, depending only on its topology. The group acts on parameter space and the method determines the integral's dependence within group orbits. This paper analyzes the two-loop vacuum diagram. It is shown how the solution of the SFI equations practically reproduces the most general value of the integral. On the way certain novel derivations are found, a geometrical interpretation is described, and divergences in general dimension are analyzed. These would hopefully be useful for engaging with more invo…
Tree-level Feynman diagrams in a cubic scalar theory can be given a metric such that each edge has a length. The space of metric trees is made out of orthants joined where a tree degenerates. Here we restrict to planar trees since each degeneration of a tree leads to a single planar neighbor. Amplitudes are computed as an integral over the space of metrics where edge lengths are Schwinger parameters. In this work we propose that a natural generalization of Feynman diagrams is provided by what are known as metric tree arrangements. These are collections of metric trees subject to a compatibilit…
Recursive Graphical Solution of Closed Schwinger-Dyson Equations in phi^4-Theory -- Part1: Generation of Connected and One-Particle Irreducible Feynman Diagrams
We investigate a system of differential equations for the beta function of massless scalar $ϕ^4$ theory and continue the combinatorial investigation of the cut structure of Feynman diagrams.
We discuss the symmetry factors of Feynman diagrams of scalar field theories with polynomial potential. After giving a concise general formula for them, we present an elementary and direct proof that when computing scattering amplitudes using the homological perturbation lemma, each contributing Feynman diagram is indeed included with the correct symmetry factor.
We describe a systematic approach for the evaluation of Witten diagrams for multi-loop scattering amplitudes of a conformally coupled scalar $ϕ^4$-theory in Euclidean AdS$_4$, by recasting the Witten diagrams as flat space Feynman integrals. We derive closed form expressions for the anomalous dimensions for all double-trace operators up to the second order in the coupling constant. We explain the relation between the flat space unitarity methods and the discontinuities of the short distance expansion on the boundary of Witten diagrams.
We study the most general triangle diagram through the Symmetries of Feynman Integrals (SFI) approach. The SFI equation system is obtained and presented in a simple basis. The system is solved providing a novel derivation of an essentially known expression. We stress a description of the underlying geometry in terms of the Distance Geometry of a tetrahedron discussed by Davydychev-Delbourgo [1], a tetrahedron which is the dual on-shell diagram. In addition, the singular locus is identified and the diagram's value on the locus's two components is expressed as a linear combination of descendant …
It is shown that the usual expression for a Feynman diagram in terms of the Feynman propagator $Δ_F(x-y)$ can be replaced by an equivalent expression involving the positive-energy on-shell propagator $Δ^+(x-y)$, supplemented by appropriate functions associated with time-ordering. When this alternate way of expressing a Feynman diagram is Fourier transformed into momentum space, the momentum associated with each function $Δ^+(x-y)$ is on-shell, and is only conserved at each vertex if an energy is attributed to the contributions of the time-ordering functions. The resulting expression is analogo…
Reduction technique for expanding the Feynman diagrams in AdS$_2$
I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute（P.N. 列别捷夫物理研究所 I.E. 塔姆理论物理系）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）
Feynman diagrams for gluon tree amplitudes are studied in the Feynman gauge and in any number of spacetime dimensions. The color-kinematics combinations $Δ=n_s-n_t-n_u$ of numerators are explicitly calculated for $N=4,5,6$ gluons to see whether the color-kinematics relation $Δ=0$ is satisfied. This is a tedious task because of the presence of four-gluon vertices, and the large number of Feynman diagrams, numerators, and $Δ$ combinations involved, especially when $N=6$. For on-shell amplitudes, it is found that $Δ=0$ for $N=4$, but $Δ\not=0$ for $N=5$ and $N=6$ owing to the presence of the four…
We propose a general coaction for families of integrals appearing in the evaluation of Feynman diagrams, such as multiple polylogarithms and generalized hypergeometric functions. We further conjecture a link between this coaction and graphical operations on Feynman diagrams. At one-loop order, there is a basis of integrals for which this correspondence is fully explicit. We discuss features and present examples of the diagrammatic coaction on two-loop integrals. We also present the coaction for the functions ${}_{p+1}F_p$ and Appell $F_1$.
The ASPIRE program, which is based on the Landau singularities and the method of power geometry to unveil the regions required for the evaluation of a given Feynman diagram asymptotically in a given limit, also allows for the evaluation of scaling coming from the top facets. In this work, we relate the scaling having equal components of the top facets of the Newton polytope to the maximal cut of given Feynman integrals. We have therefore connected two independent approaches to the analysis of Feynman diagrams.
