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 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=1∑B​Ni​-point function
G∣aN1​1​⋯aN1​1​∣⋯∣a1B​⋯aNB​B​∣​
of Φ3-Φ4 Hybrid-Matrix-Model is studied in detail. This
i=1∑B​Ni​-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​⋯aNi​i​∣ 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.Comment: 28 pages. Typos were corrected, three references were added, and an
annotation was added in Section