Exact Solutions v.s. Perturbative Calculations of Finite ${\Phi }^3$-${\Phi }^4$ Hybrid-Matrix-Model

Abstract

There is a matrix model corresponding to a scalar field theory called Grosse-Wulkenhaar model, which is renormalizable by adding a harmonic oscillator potential to scalar ${\Phi }^4$ theory on Moyal spaces. There are more unknowns in ${\Phi }^4$ matrix model than in ${\Phi }^3$ matrix model, for example, in terms of integrability. We then construct a one-matrix model (${\Phi }^3$-${\Phi }^4$ 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 ${\Phi }^3$ is multiplied by some positive definite diagonal matrix $M$. This model is solvable due to the effect of this $M$. In particular, the connected $\sum _{i=1}^B{N}_i$-point function $G_{|a_{N_1}^1\cdots a_{N_1}^1|\cdots |a_1^B\cdots a_{N_B}^B|}$ of ${\Phi }^3$-${\Phi }^4$ Hybrid-Matrix-Model is studied in detail. This $\sum _{i=1}^B{N}_i$-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 $|a_1^i\cdots a_{N_i}^i|$ represents $N_i$ external lines coming from the $i$-th boundary (puncture) in each Feynman diagram. First, we construct Feynman rules for ${\Phi }^3$-${\Phi }^4$ Hybrid-Matrix-Model and calculate perturbative expansions of some multipoint functions in ordinary methods. Second, we calculate the path integral of the partition function $\mathcal{Z}[J]$ 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 $G_{|a^1|a^2|\cdots |a^n|}$ with $n$-boundaries. They include contributions from Feynman diagrams corresponding to nonplanar Feynman diagrams or higher genus surfaces.

arXiv comment28 pages. Typos were corrected, three references were added, and an annotation was added in Section 3

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