Exact Solutions v.s. Perturbative Calculations of Finite Φ3-Φ4 Hybrid-Matrix-Model

Abstract

There is a matrix model corresponding to a scalar field theory on noncommutative spaces called Grosse-Wulkenhaar model (Φ4 matrix model), which is renormalizable by adding a harmonic oscillator potential to scalar Φ4 theory on Moyal spaces. There are more unknowns in Φ4 matrix model than in Φ3 matrix model, for example, in terms of integrability. We then construct a one-matrix model (Φ3-Φ4 Hybrid-Matrix-Model) with multiple potentials, which is a combination of a 3-point interaction and a 4-point interaction, where the 3-point interaction of Φ3 is multiplied by some positive definite diagonal matrix M. This model is solvable due to the effect of this M. In particular, the connected ∑i=1BNi-point function G|aN11⋯aN11|⋯|a1B⋯aNBB| of Φ3-Φ4 Hybrid-Matrix-Model is studied in detail. This ∑i=1BNi-point function can be interpreted geometrically and corresponds to the sum over all Feynman diagrams (ribbon graphs) drawn on Riemann surfaces with B boundaries (punctures). Each |a1i⋯aNii| represents Ni external lines coming from the i-th boundary (puncture) in each Feynman diagram. First, we construct Feynman rules for Φ3-Φ4 Hybrid-Matrix-Model and calculate perturbative expansions of some multipoint functions in ordinary methods. Second, we calculate the path integral of the partition function Z[J] and use the result to compute exact solutions for 1-point function G|a| with 1-boundary, 2-point function G|ab| with 1-boundary, 2-point function G|a|b| with 2-boundaries, and n-point function G|a1|a2|⋯|an| with n-boundaries. They include contributions from Feynman diagrams corresponding to nonplanar Feynman diagrams or higher genus surfaces.