Tensor structure from scalar Feynman matroids
We show how to interpret the scalar Feynman integrals which appear when reducing tensor integrals as scalar Feynman integrals coming from certain nice matroids.
math-ph↗
arXiv subjects
Karen Yeats
Publications and source records attributed to Karen Yeats.
Monadic Second-Order Classes of Forests with a Monadic Second-Order 0-1 Law
Let $\cT$ be a monadic-second order class of finite trees, and let $\bT(x)$ be its (ordinary) generating function, with radius of convergence $ρ$. If $ρ\ge 1$ then $\cT$ has an explicit specification (without using recursion) in terms of the operations of union, sum, stack, and the multiset operators $(n)$ and $(\ge n)$. Using this, one has an explicit expression for $\bT(x)$ in terms of the initial functions $x$ and $x\cdot \big(1-x^n\big)^{-1}$, the operations of addition and multiplication, and the Pólya exponentiation operators $\sE_n, \sE_{\ge n}$. Let $\cF$ be a monadic-second order class of finite forests, and let $\bF(x)=\sum_n f(n) x^n$ be its (ordinary) generating function. Suppose $\cF$ is closed under extraction of component trees and sums of forests. Using the above-mentioned structure theory for the class $\cT$ of trees in $\cF$, Compton's theory of 0--1 laws, and a significantly strengthened version of 2003 results of Bell and Burris on generating functions, we show that $\cF$ has a monadic second-order 0--1 law iff the radius of convergence of $\bF(x)$ is 1 iff the radius of convergence of $\bT(x)$ is $\ge 1$.
math.LO↗
Spectra and Systems of Equations
In a previous work we introduced an elementary method to analyze the periodicity of a generating function defined by a single equation y=G(x,y). This was based on deriving a single set-equation Y = Gammma(Y) defining the spectrum of the generating function. This paper focuses on extending the analysis of periodicity to generating functions defined by a system of equations y = G(x,y). The final section looks at periodicity results for the spectra of monadic second-order classes whose spectrum is determined by an equational specification - an observation of Compton shows that monadic-second order classes of trees have this property. This section concludes with a substantial simplification of the proofs in the 2003 foundational paper on spectra by Gurevich and Shelah, namely new proofs are given of: (1) every monadic second-order class of $m$-colored functional digraphs is eventually periodic, and (2) the monadic second-order theory of finite trees is decidable.
math.LO↗
Spanning forest polynomials and the transcendental weight of Feynman graphs
We give combinatorial criteria for predicting the transcendental weight of Feynman integrals of certain graphs in $ϕ^4$ theory. By studying spanning forest polynomials, we obtain operations on graphs which are weight-preserving, and a list of subgraphs which induce a drop in the transcendental weight.
math-ph↗
The QCD beta-function from global solutions to Dyson-Schwinger equations
We study quantum chromodynamics from the viewpoint of untruncated Dyson-Schwinger equations turned to an ordinary differential equation for the gluon anomalous dimension. This nonlinear equation is parameterized by a function P(x) which is unknown beyond perturbation theory. Still, very mild assumptions on P(x) lead to stringent restrictions for possible solutions to Dyson-Schwinger equations. We establish that the theory must have asymptotic freedom beyond perturbation theory and also investigate the low energy regime and the possibility for a mass gap in the asymptotically free theory.
hep-th↗
Growth estimates for Dyson-Schwinger equations
Dyson-Schwinger equations are integral equations in quantum field theory that describe the Green functions of a theory and mirror the recursive decomposition of Feynman diagrams into subdiagrams. Taken as recursive equations, the Dyson-Schwinger equations describe perturbative quantum field theory. However, they also contain non-perturbative information. Using the Hopf algebra of Feynman graphs we will follow a sequence of reductions to convert the Dyson-Schwinger equations to the following system of differential equations, \[ γ_1^r(x) = P_r(x) - \sgn(s_r)γ_1^r(x)^2 + (\sum_{j \in \mathcal{R}}|s_j|γ_1^j(x)) x \partial_x γ_1^r(x) \] where $r \in \mathcal{R}$, $\mathcal{R}$ is the set of amplitudes of the theory which need renormalization, $γ_1^r$ is the anomalous dimension associated to $r$, $P_r(x)$ is a modified version of the function for the primitive skeletons contributing to $r$, and $x$ is the coupling constant. Next, we approach the new system of differential equations as a system of recursive equations by expanding $γ_1^r(x) = \sum_{n \geq 1}γ^r_{1,n} x^n$. We obtain the radius of convergence of $\sum γ^r_{1,n}x^n/n!$ in terms of that of $\sum P_r(n)x^n/n!$. In particular we show that a Lipatov bound for the growth of the primitives leads to a Lipatov bound for the whole theory. Finally, we make a few observations on the new system considered as differential equations.
math-ph↗
The QED beta-function from global solutions to Dyson-Schwinger equations
We discuss the structure of beta functions as determined by the recursive nature of Dyson--Schwinger equations turned into an analysis of ordinary differential equations, with particular emphasis given to quantum electrodynamics. In particular we determine when a separatrix for solutions to such ODEs exists and clarify the existence of Landau poles beyond perturbation theory. Both are determined in terms of explicit conditions on the asymptotics for the growth of skeleton graphs.
hep-th↗
Unbounded regions of Infinitely Logconcave Sequences
We study the properties of a logconcavity operator on a symmetric, unimodal subset of finite sequences. In doing so we are able to prove that there is a large unbounded region in this subset that is $\infty$-logconcave. This problem was motivated by the conjecture of Moll and Boros in that the binomial coefficients are $\infty$-logconcave.
math.CO↗
Recursion and growth estimates in renormalizable quantum field theory
In this paper we show that there is a Lipatov bound for the radius of convergence for superficially divergent one-particle irreducible Green functions in a renormalizable quantum field theory if there is such a bound for the superficially convergent ones. The radius of convergence turns out to be ${\rm min}\{ρ,1/b_1\}$, where $ρ$ is the bound on the convergent ones, the instanton radius, and $b_1$ the first coefficient of the $β$-function.
hep-th↗
Sufficient Conditions for Labelled 0-1 Laws
If F(x) = e^G(x), where F(x) = \sum f(n)x^n and $G(x) = \sum g(n)x^n, with 0 \le g(n) = O(n^{theta n}/n!),theta in (0,1), and gcd(n : g(n) > 0)=1, then f(n) = o(f(n-1)). This gives an answer to Compton's request in Question 8.3 for an ``easily verifiable sufficient condition'' to show that an adequate class of structures has a labelled first-order 0-1 law, namely it suffices to show that the labelled component count function is O(n^{theta n}) for some theta in (0,1). It also provides the means to recursively construct an adequate class of structures with a labelled 0-1 law but not an unlabelled 0-1 law, answering Compton's Question 8.4.
math.CO↗
An Étude in non-linear Dyson--Schwinger Equations
We show how to use the Hopf algebra structure of quantum field theory to derive nonperturbative results for the short-distance singular sector of a renormalizable quantum field theory in a simple but generic example. We discuss renormalized Green functions $G_R(α,L)$ in such circumstances which depend on a single scale $L=\ln q^2/μ^2$ and start from an expansion in the scale $G_R(α,L)=1+\sum_k γ_k(α)L^k$. We derive recursion relations between the $γ_k$ which make full use of the renormalization group. We then show how to determine the Green function by the use of a Mellin transform on suitable integral kernels. We exhibit our approach in an example for which we find a functional equation relating weak and strong coupling expansions.
hep-th↗
Admissible Dirichlet Series
We propose a definition of "Admissible Dirichlet Series" as the analog of Hayman's 1956 definition of "Admissible Power Series".
math.CA↗