Solution of the self-dual Φ4 QFT-model on four-dimensional Moyal space

Abstract

Previously the exact solution of the planar sector of the self-dual Φ 4 -model on 4dimensional Moyal space was established up to the solution of a Fredholm integral equation. This paper solves, for any coupling constant λ > − 1 π , the Fredholm equation in terms of a hypergeometric function and thus completes the construction of the planar sector of the model. We prove that the interacting model has spectral dimension 4 − 2 arcsin(λπ) π for |λ| < 1 π . It is this dimension drop which for λ > 0 avoids the triviality problem of the matricial Φ 4 4 -model. We also establish the power series approximation of the Fredholm solution to all orders in λ. The appearing functions are hyperlogarithms defined by iterated integrals, here of alternating letters 0 and −1. We identify the renormalisation parameter which gives the same normalisation as the ribbon graph expansion.MSC 2010: 33C05, 45B05, 81Q80, 81Q30

Asymptotic freedom for $$\lambda \phi ^4_{\star }$$ QFT in Snyder–de Sitter space

“…The main peculiarity of this model is the presence in the action of an harmonic oscillator term that smooths the infrared behavior and in the end gives rise to an all-order perturbatively renormalizable theory. According to the latest results, in a certain limit it could provide the first example of a solvable model in four dimensions [11].…”

Perturbative and Geometric Analysis of the Quartic Kontsevich Model

“…The initial non-linear Dyson-Schwinger equation has been solved implicitly [21], but in full generality. An explicit solution in terms of special functions succeeded for the 2D Moyal space (where it gives the Lambert function [31]) and for the 4D Moyal space (where it gives the inverse of a Gauß hypergeometric function [22]). In these two cases all the renormalised correlation functions of disk topology can be written down (thanks to [16]) as integral representations.…”

“…An integral representation for the planar 2-point function is only consistent for λ > − 1 log(4) . Let d → ∞ with spectral measure ϱ(t) = t, the four-dimensional Moyal plane: One finds R(z) = z 2 F 1 (α λ , 1 − α λ , 2; −z), where α λ = arcsin(λπ)/π for |λ| ≤ 1 π and α λ = 1 2 + i arcosh(λπ)/π for λ ≥ 1 π [22]. The singular value is λ s = −1/π.…”

“…We are working on a programme which achieves and investigates the exact solution of a quantum field theory toy model, namely of a matrix model with quartic interaction and non-trivial covariance [24]. It is already known that for particular choices of parameters the exact solution of the planar sector expands into number-theoretic objects such as Nielsen polylogarithms [31] and hyperlogarithms [22], respectively.…”

Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme

“…So far, explicit calculations of renormalized amplitudes in cNLFT depending on external kinematic variables are only known for noncommutative field theory and its matrix-field representation using the BPHZ momentum scheme [6,22,28]. In fact, exact solutions for matrix field theory have been found recently [11,21,41] and the perturbative results are mainly used as a check of consistency [22,28]. For other examples of cNLFT, renormalizability has been proven for various field theories with tensorial interactions [2,3,4,13,14,15] but no explicit results of amplitudes, let alone of full renormalized correlation functions, are known.…”