A Convergent Kinetic-Term Perturbation Expansion for $φ^4$ Theory

Abstract

We revisit scalar $φ^4$ theory and construct a reorganized perturbative expansion in which the kinetic operator, rather than the quartic interaction, is treated as the perturbation. Starting from the exactly solvable $0$-dimensional model, we show that the resulting series is convergent for positive coupling and can be written as an expansion in negative powers of the quartic coupling $λ$. We extend the construction to higher-dimensional field theory using an auxiliary field, and we formulate a discrete lattice version in which multi-site contributions are systematically organized. We explicitly compute the leading terms in the expansion, study their continuum limit, and compare against brute-force numerical evaluations of the partition function. We discuss the relation of this expansion to standard weak-coupling perturbation theory, strong-coupling expansions, and resummation techniques, and we outline possible applications to nonperturbative studies of scalar field theories.

Figures (7)

• Figure 1: Partition function $Z(J)$ of the $0$-dimensional $\phi^4$ theory as a function of the source $J$. The solid black curve shows the exact result obtained by direct numerical quadrature, while the dashed colored curves show partial sums of the reorganized expansion for increasing truncation orders $(n_{\max},n'_{\max})$. The uniform convergence of the truncated series to the exact result provides a direct numerical confirmation of the convergent nature of the reorganized expansion.
• Figure 2: Diagrammatic representation of the local factor $\widehat{\mathcal{I}}(\mathrm{i} X(x)+J(x))$ obtained after integrating out the field $\phi(x)$ exactly. The vertex corresponds to a single spacetime point $x$ and carries arbitrary numbers of insertions of the external source $J(x)$ (open circles) and the auxiliary field $X(x)$ (filled circles). The associated weight is determined by the exact local integral $\widehat{\mathcal{I}}$ and is symmetric under permutations of the legs.
• Figure 3: Contraction (Gaussian propagator) of the auxiliary field $X$. Since $X$ appears quadratically, its two-point function is given by the inverse kinetic operator $A^{-1}(x-y) = (-\partial^2 + m^2)^{-1}(x-y)$. Dashed lines denote contractions of auxiliary-field insertions generated by the Gaussian measure.
• Figure 4: Partition function $Z[0]$ of the lattice $\phi^4$ theory as a function of the quartic coupling $\lambda$ for $N=1$, $\Delta x=0.7$, and $m^2=1$. The blue curve shows the exact result obtained from brute-force numerical integration over the lattice fields. The remaining curves show successive truncations of the reorganized kinetic-term expansion: the strong-coupling prefactor only, the prefactor plus one- and two-site contributions, and the prefactor plus one-, two-, and three-site contributions. The progressive improvement demonstrates the systematic convergence of the expansion as higher multi-site terms are included.
• Figure 5: Sunset diagram contributing to the two-point function in the auxiliary-field formulation. Internal lines represent resummed auxiliary-field propagators, and the two vertices correspond to local $JX^3$ insertions generated by the expansion of $\widehat{\mathcal{I}}$.