Planar matrices and arrays of Feynman diagrams:poles for higher k

Planar arrays of tree diagrams were introduced as a generalization of Feynman diagrams that enable the computation of biadjoint amplitudes m_(n)(^(k))for k>2.In this follow-up work,we investigate the poles of m_(n)(^(k))from the perspective of such arrays.For general k,we characterize the underlying polytope as a Flag Complex and propose a computation of the amplitude-based solely on the knowledge of the poles,whose number is drastically less than the number of the full arrays.As an example,we first provide all the poles for the cases(k,n)=(3,7),(3,8),(3,9),(3,10),(4,8)and(4,9)in terms of their planar arrays of degenerate Feynman diagrams.We then implement simple compatibility criteria together with an addition operation between arrays and recover the full collections/arrays for such cases.Along the way,we implement hard and soft kinematical limits,which provide a map between the poles in kinematic space and their combinatoric arrays.We use the operation to give a proof of a previously conjectured combinatorial duality for arrays in(k,n)and(n-k,n).We also outline the relation to boundary maps of the hypersimplex Δ_(k,n) and rays in the tropical Grassmannian Tr(k,n).

Planar matrices and arrays of Feynman diagrams

Recently,planar collections of Feynman diagrams were proposed by Borges and one of the authors as the natural generalization of Feynman diagrams for the computation of k=3 biadjoint amplitudes.Planar collections are one-dimensional arrays of metric trees satisfying an induced planarity and compatibility condition.In this work,we introduce planar matrices of Feynman diagrams as the objects that compute k=4 biadjoint amplitudes.These are symmetric matrices of metric trees satisfying compatibility conditions.We introduce two notions of combinatorial bootstrap techniques for finding collections from Feynman diagrams and matrices from collections.As applications of the first,we find all 693,13612 and 346710 collections for(k,n)=(3,7),(3,8)and(3,9),respectively.As applications of the second kind,we find all90608 and 30659424 planar matrices that compute(k,n)=(4,8)and(4,9)biadjoint amplitudes,respectively.As an example of the evaluation of matrices of Feynman diagrams,we present the complete form of the(4,8)and(4,9)biadjoint amplitudes.We also start a study of higher-dimensional arrays of Feynman diagrams,including the combinatorial version of the duality between(k,n)and(n-k,n)objects.

The symmetry group of Feynman diagrams and consistency of the BPHZ renormalization scheme

We study the relation between the symmetry group of a Feynman diagram and its reduced diagrams.We then prove that the counterterms in the BPHZ renormalization scheme are consistent with adding counterterms to the interaction Hamiltonian in all cases,including that of Feynman diagrams with symmetry factors.

Exact Calculations of Vertex ^-sγb and ,^-sZb in the Unitary Gauge

In this paper, we present the exact calculations for the vertex ^-sγb and ^sZb in the unitary gauge. We find that （a） the divergent- and μ-dependent terms are left in the effective vertex function Г^γμ（p, k） for b → sγ transition even after we sum up the contributions from four related Feynman diagrams; （b） for an on-shell photon, such terms do not contribute et al.; （c） for off-shell photon, these terms will be canceled when the contributions from both vertex ^sγb and ^sZb are taken into account simultaneously, and therefore the finite and gauge-independent function Zo（xt） = Co（xt） ＋ Do（xt）/4, which governs the semi-leptonic decay b → sl^- l^＋, is derived in the unitary gauge.

State space formalism of Feynman diagram theory for acousto-optic interactions:I.single-frequency case

A Feynman diagram theory for acousto-optic (AO) interactions is established, which provides a general method to calculate the scattering amplitudes and intensities for both single-frequency and multifrequency AO interactions. The method is based on counting the number of allowable Feynman diagrams. Some important assertions have been proved rigorously in this paper.

State space formalism of Feynman diagram theory for acousto-optic interactions:Ⅱ. multifrequency case

A Feynman diagram theory for acousto-optic (AO) interactions is established, which provides a general method to calculate the scattering amplitudes and intensities for both single-frequency and multifrequency AO interactions. The method is based on counting the number of allowable Feynman diagrams. The following important assertion has been proved rigorously in this paper. The ratios of the numbers of Feynman diagrams allowable in various Bragg diffractions (isotropic, nondegenerate birefringent, and degenerate birefringent) to that in Raman-Nath diffraction are independent of the number of different acoustic frequencies, being a function only of the order of the Feynman diagram and the diffraction order of the final state. General expressions for these ratios are obtained. Based on this, complete perturbation solutions for the scattering amplitudes and intensities are obtained for any kind of AO interactions, any number of acoustic frequencies, and any final state. This theory gives all results obtained previously by the theory of coupled-wave equation. The theory is also verified by comparing with experiments.

Feynman diagram theory for multifrequency nonlinear acoustics

The Feynman diagram theory with the state-space formalism is adopted to study the multifrequency nonlinear acoustics effects. By establishing the relation between the strain magnitude corresponding to any final state S(m1,…,mn; x) and the number of paths from the initial state of the interactingphonons to the final state, not only the complete perturbation solutions but also the corresponding analytical expressions of the acoustic harmonics and intermodulation products have been obtained. For a few special cases, results of our theory is consistent with those obtained by conventional methods. While the general solution for any number of frequencies can easily be obtained by our theory, this is impossible by using conventional methods.

Are operators describing b→sγ?

The operators of b→ sγ, b →sl＋l- are usually regarded as being sufficient to describe b →sγγ, b → sl＋l-T with the statement that contributions from diagrams without an effective vertex b --＊ ST to processes b →sγγ and b →sl＋l-T are negligible. In this work we present a comprehensive analysis of the transition b→sγγ and find that 1） Effects due to off-shell quarks in vertex b → sγ on b →sγγ are large; 2） Contributions from diagrams without an effective vertex b →sγ to b →sγγ are not negligible compared with others; 3） These effects cancel each other out exactly, so the operators of b→sγ can safely be used to describe b →sγγ.